http://2007.igem.org/wiki/index.php?title=Special:Contributions/Uhrm&feed=atom&limit=50&target=Uhrm&year=&month=2007.igem.org - User contributions [en]2024-03-29T12:26:33ZFrom 2007.igem.orgMediaWiki 1.16.5http://2007.igem.org/wiki/index.php/ETHZ/ModelETHZ/Model2007-10-25T13:15:28Z<p>Uhrm: /* Introduction */</p>
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=Introduction=<br />
<br />
As previously discussed in the main page, we are interested in designing a system that is able to adapt to its environment. Our ideas are based on discussions about neural networks, and how we can create a biological system that exhibits the behavior of learning without having to resort to evolutionary processes. <br />
<br />
[[Image:ETHzFlowdiagram2.png|thumb|<b>Fig. 1</b>: Flow diagram. This figure shows the protocol with which the final system should be tested as well as the test results in form of the reported colors. The protocol is divided into three phases: (1) a training or learning phase in which the system learns an input and stores it in its memory, (2) a memory phase in which the system keeps the content of its memory, and finally (3) a recognition phase where the output of the system depends on the content of its memory as well as the current input.|450px]]<br />
<br />
Learning can be considered as a switching of behavior, based on some external stimuli. Thus, it comes naturally to work on existing ideas of toggle switches and [[ETHZ/FSM | finite state machines]]. <br />
<br />
The proposed system is able to distinguish between two chemicals. It represents a minimal test system that is intended as a proof of concept. By introducing the ability to distinguish more than two chemical and thereby introducing new [[ETHZ/FSM | system states]], the power of the system or its "intelligence" can be scaled. A protocol depicting how the system should react according to an input is shown in Fig. 1.<br />
<br />
The idea behind this protocol is that:<br />
* The system will be able to learn one of two input signals - ATC or IPTG - during a learning phase, while a "learning signal" (AHL) is not yet present. Depending on the input it will report by producing either cyan or yellow florescence. <br />
* Once the system has learned, the inputs - ATC or IPTG - can be removed and the system goes into a memory state in the presence of AHL. In this state, no output color is reported. Memorizing is guaranteed by removing the input chemicals.<br />
* During the recognition phase, the inputs ATC or IPTG are (re-)inserted. The system reports by changing its color depending on the input and its current memory state. This is why the system has different florescence properties even in the presence of the same input. The recognition phase takes place in the presence of AHL, to keep the memory enabled and avoid another learning phase. Since we would like to separate four different end states, we use four different fluorescent proteins to encode them.<br />
<br />
==Model Overview==<br />
<br />
The model for the proposed system is developed using a top-down approach. We start with a black box approach as shown in Fig. 2. <br />
<br />
[[Image:ETHZBlackbox.png|thumb|<b>Fig. 2</b>: Black box |280px]]<br />
<br />
The system is sketched in Fig. 3. It can be summarized as follows:<br />
* There are two inputs to be learned/detected/adapted to.<br />
* There is one separate input to switch on the memory.<br />
* The system has to alternate between at least three states. Hence, we decided to use two state variables - CI and P22CII (when interpreted as binary variables, in principle allowing for up to four different states).<br />
* There are four different output signals (synthesis of four fluorescent proteins). One could also decide to take six output signals into account to further distinguish the learning phase from the recognition phase. However, we restricted ourselves to four outputs to reduce the number of genes that are needed to implement the signals.<br />
<br />
[[Image:ETHZFullsystemmodel.png|left|thumb|<b>Fig. 3</b>: System overview. AHL, IPTG and ATC pass the cell membrane where they build complexes with the sensor proteins LuxR, LacI and TetR. These sensor proteins and/or complexes are used to control the internal system state: the memory represented by the proteins CI and P22CII (mutually repressing their synthesis) and the sensed input (IPTG, ATC). CFP, RFP, GFP and YFP stand for yellow, red, cyan and green florescent protein, respectively.|420px]]<br />
<br />
However, we had to keep in mind that the proposed system should be implemented in DNA, and that it would be sensitive to noise. As a result, we took several actions to achieve better experimental results and easier DNA construction:<br />
* To be more robust against perturbations, we coupled the state variables CI and P22CII like it is known from toggle switches [1]. Based on this approach, one state variable is depressing the other one, and the system's internal toggle has the possibility of reaching two stable states.<br />
* Since - due to their size - proteins can only hardly pass the cell membrane (if they are not actively transported through the cell membrane), we decided to use the much smaller inducer molecules AHL, IPTG and ATC as inputs. However, since these inducers cannot directly act on the transcription of the DNA nor on the production of proteins, we need to produce the sensor proteins LuxR, LacI and TetR that build complexes with AHL, IPTG and ATC, respectively.<br />
* The sensor proteins and complexes are used to control the memory formation and the production of the florescent reporter proteins CFP, RFP, GFP and YFP.<br><br><br />
<br />
==Detailed Model==<br />
<br />
In order to test our ideas, we came up with a detailed model of all the interactions in the system. <br />
After defining the desired behavior of our system (as shown in the introduction) and a [[ETHZ/FSM | formalized desciption of the system]] we identified necessary biological components and their interactions. As we can observe in Fig. 3, our system is composed from three basic subparts:<br />
* sensors,<br />
* memory, and<br />
* reporters.<br />
<br />
===Sensors===<br />
<br />
The first part contains the sensors. Our sensors are the proteins LacI, luxR and TetR, which are constitutively produced. The sensing subsystem is shown in Fig. 4.<br />
<br />
[[Image:Model01b.png|center|thumb|<b>Fig. 4</b>: The proteins that act as sensors are constitutively produced.|140px]]<br />
<br />
===Memory===<br />
<br />
The second subsystem implements the memory. The memory control is based on the following underlying mechanisms:<br />
* The sensor proteins form complexes together with the inducers. These complexes are used to activate the transcription of the genes for the proteins CI and P22CII. <br />
* P22CII and CI repress the DNA transcription of each other, so that the closed loop system behaves as a toggle; a dynamic system with only two possible steady states (see Fig. 6).<br />
<br />
[[Image:ETHZModelLearning.png|center|thumb|<b>Fig. 5</b>: Learning system: Depending on the inputs IPTG or ATC the proteins CI and P22CII are produced.|300px]]<br />
<br />
* Fig. 5 shows the protein production system that is used during the learning phase. During the learning phase, there is still no CI or P22CII produced. They are produced, only if either IPTG or ATC is added, respectively. Since no AHL is present, the inner toggle switch (see Figure 6) is turned off.<br />
<br />
[[Image:ETHZModelMemory.png|center|thumb|<b>Fig. 6</b>: Memory system. If AHL is present the production of either CI or P22CII is continued.|420px]]<br />
<br />
* During the memory phase, AHL is added and the IPTG and ATC are removed. That is why only the inner toggle switch (see Fig. 6) is turned on while the protein production systems shown in Fig. 5 are deactivated. Depending on what was produced during the learning phase, the production of either CI or P22CII is continued. That is why the system can act as memory, effectively storing the information it is exposed to.<br />
<br />
Based on all the above, we present the final assembly of the memory subsystem in Fig. 7.<br />
<br />
[[Image:Model02b.png|center|thumb|<b>Fig. 7</b>: Final interaction of the learning and memory system. The memory content is represented by the concentrations of the proteins CI and P22CII.|560px]]<br />
<br />
===Reporters===<br />
<br />
Fig. 8 gives an overview of the reporter subsystem. Florescent reporter proteins are expressed depending on the inducer concentrations, and the concentrations of CI and P22CII. For example, the presence of either TetR or CI will repress the production of YFP. However, if the inducer ATC is present, ATC will bind to TetR which can no longer block the production of YFP. We are using four fluorescent proteins, to encode the steady states of our system at the final recognition stage. In this way, we are able to distinguish between all the different transition paths in the system.<br />
<br />
[[Image:Model03b.png|center|thumb|<b>Fig. 8</b>: The production of the florescent reporter proteins depends on the memory content (CI or P22CII) and the current input (ATC or IPTG).|600px]]<br />
<br />
==Final Model==<br />
<br />
So far, we have presented all parts needed to model and simulate the behavior of the proposed system with. By following the details presented in the previous section, we have all the necessary information to fully understand the interior of the black boxes that were presented in Fig. 2 and Fig. 3. Our overall system model is presented in Fig. 9.<br />
<br />
[[Image:ETHZFullsystem.png|center|thumb|<b>Fig. 9</b>: Final model of the educatETH <i>E. coli</i> system.|900px]]<br />
<br />
In contrast to this biological implementation, an alternative implementation using an engineering approach can be found on our [[ETHZ/FlipFlop | 'Engineer's View' page]].<br />
<br />
==Mathematical Model==<br />
<br />
Based on the modeling done so far, we can derive the equations that govern the behavior of our system. The model is given by sets of coupled [[ETHZ/Modeling_Basics | ordinary differential equations]] which are presented below. We use a simple notation for the different elements of the equations. Namely: <br />
* All concentrations are given in brackets (for example [CI]). <br />
* All decay constants are described by a variable d followed by the name of the protein they refer to. <br />
* The production of the proteins is described by a basic constant production level named 'a' that models the leak of the production system, and a factor of l and c<sub>max</sub> that describe the maximum production of a protein, given in [M/min]. <br />
* Depending on whether the DNA for a protein is implemented on a low or a high copy plasmid, we distinguish between l<sub>lo</sub> and l<sub>hi</sub>, respectively.<br />
* Dissociation constants are given by 'K' followed by the name of the protein they refer to.<br />
* The Hill cooperabilities are described by the constants 'n' followed by the name of the protein they refer to.<br />
<br />
For a more detailed introduction into how we transferred our model into equations, see the section [[ETHZ/Modeling_Basics|Modeling Basics]].<br><br />
<br />
===Allosteric regulation===<br />
<br />
These equations describe the formation of complexes between the inducers and sensor proteins. We do not use differential equations, but describe directly the concentrations of the complexes. This is a valid assumption, provided that we always wait a sufficient time, and the system reaches a steady state.<br />
We describe the total amount of proteins with the index 't', while we use the index '*' for proteins that build a complex with their respective inducer. For example: <br />
* [TetR<sub>t</sub>] describes the total concentration of TetR that is available. It is the sum of the free TetR proteins and the TetR proteins that form a complex with ATC. <br />
* [TetR<sub>*</sub>] describes the proteins that are available as a complex with ATC, and<br />
* [TetR] gives the concentration of free TetR proteins.<br />
<br />
[[Image:Eq04.png|238px]]<br />
<br />
===Constitutively produced proteins===<br />
<br />
The differential equations for the constitutively produced proteins are very simple, since there is no dependence on other proteins. They are designed so that the protein concentration reaches the value l<sub>hi</sub>*c<sub>max</sub>/d at steady state.<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
===Learning and memory subsystem===<br />
<br />
The learning and memory subsystem is the core of the system that we are trying to model and implement. It is characterized by the feedback between its state variables/proteins CI and P22CII. Its behavior is further complicated by the variation of the production of the aforementioned proteins because of the inputs. The following equations describe the concentrations of the memory proteins as a system of coupled differential equations. The equations consist of two major production parts and a decay part. <br />
* The first production part models the production of either CI or P22CII during the learning phase, and corresponds to the model in Fig. 5.<br />
* The second production part describes the inner toggle switch that was shown in Fig. 6.<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
===Reporting subsystem===<br />
<br />
The equations for the reporting subsystem finally describe the production of the florescence proteins depending on the inputs and memory proteins as modeled in Figure 8. Note that both the free constitutively produced proteins and the memory proteins repress the production of the florescence proteins. So e.g. YFP is only produced when there is both no CI and all TetR is bound in a complex together with ATC.<br />
<br />
[[Image:Reporter_braced.png|778px]]<br />
<br />
The systems of equations presented above describe and predict the behavior of our system. We have simulated the behavior of our system at steady states, and the results can be seen in the section [[ETHZ/Simulation|Simulations]]. In order to increase the accuracy of our results, we conducted an extensive literature survey, in order to isolate and find the parameters of our system. Since this is a burden for every team undertaking a complicated project in synthetic biology, we are presenting our full table of parameters in the [[ETHZ/Parameters|Parameters]] page.<br />
<br />
== References ==<br />
<p><br />
[http://www.nature.com/nature/journal/v403/n6767/abs/403339a0.html &#91;1&#93; Gardner TS, Cantor CR and Collins JJ] <i>"Construction of a genetic toggle switch in Escherichia coli"</i>, Nature 403:339–342, 2000<br /></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/ModelETHZ/Model2007-10-25T13:13:09Z<p>Uhrm: /* Reporting subsystem */</p>
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=Introduction=<br />
<br />
As previously discussed in the main page, we are interested in designing a system that is able to adapt to its environment. Our ideas are based on discussions about neural networks, and how we can create a biological system that exhibits the behavior of learning without having to resort to evolutionary processes. <br />
<br />
[[Image:ETHzFlowdiagram2.png|thumb|<b>Fig. 1</b>: Flow diagram. This figure shows the protocol with which the final system should be tested as well as the test results in form of the reported colors. The protocol is divided into three phases: (1) a training or learning phase in which the system learns an input and stores it in its memory, (2) a memory phase in which the system keeps the content of its memory, and finally (3) a recognition phase where the output of the system depends on the content of its memory as well as the current input.|450px]]<br />
<br />
Learning can be considered as a switching of behavior, based on some external stimuli. Thus, it comes naturally to work on existing ideas of toggle switches and [[ETHZ/FSM | finite state machines]]. <br />
<br />
The proposed system is able to distinguish between two chemicals. It represents a minimal test system that is intended as a proof of concept. By introducing the ability to distinguish more than two chemical and thereby introducing new [[ETHZ/FSM | system states]], the power of the system or its "intelligence" can be scaled. A protocol depicting how the system should react according to an input is shown in Fig. 1.<br />
<br />
The idea behind this protocol is that:<br />
* The system will be able to learn one of two input signals - aTc or IPTG - during a learning phase, while a "learning signal" (AHL) is not yet present. Depending on the input it will report by producing either cyan or yellow florescence. <br />
* Once the system has learned, the inputs - aTc or IPTG - can be removed and the system goes into a memory state in the presence of AHL. In this state, no output color is reported. Memorizing is guaranteed by removing the input chemicals.<br />
* During the recognition phase, the inputs aTc or IPTG are (re-)inserted. The system reports by changing its color depending on the input and its current memory state. This is why the system has different florescence properties even in the presence of the same input. The recognition phase takes place in the presence of AHL, to keep the memory enabled and avoid another learning phase. Since we would like to separate four different end states, we use four different fluorescent proteins to encode them.<br />
<br />
==Model Overview==<br />
<br />
The model for the proposed system is developed using a top-down approach. We start with a black box approach as shown in Fig. 2. <br />
<br />
[[Image:ETHZBlackbox.png|thumb|<b>Fig. 2</b>: Black box |280px]]<br />
<br />
The system is sketched in Fig. 3. It can be summarized as follows:<br />
* There are two inputs to be learned/detected/adapted to.<br />
* There is one separate input to switch on the memory.<br />
* The system has to alternate between at least three states. Hence, we decided to use two state variables - cI and p22cII (when interpreted as binary variables, in principle allowing for up to four different states).<br />
* There are four different output signals (synthesis of four fluorescent proteins). One could also decide to take six output signals into account to further distinguish the learning phase from the recognition phase. However, we restricted ourselves to four outputs to reduce the number of genes that are needed to implement the signals.<br />
<br />
[[Image:ETHZFullsystemmodel.png|left|thumb|<b>Fig. 3</b>: System overview. AHL, IPTG and ATC pass the cell membrane where they build complexes with the sensor proteins LuxR, LacI and TetR. These sensor proteins and/or complexes are used to control the internal system state: the memory represented by the proteins cI and p22cII (mutually repressing their synthesis) and the sensed input (IPTG, ATC). CFP, RFP, GFP and YFP stand for yellow, red, cyan and green florescent protein, respectively.|420px]]<br />
<br />
However, we had to keep in mind that the proposed system should be implemented in DNA, and that it would be sensitive to noise. As a result, we took several actions to achieve better experimental results and easier DNA construction:<br />
* To be more robust against perturbations, we coupled the state variables cI and p22cII like it is known from toggle switches [1]. Based on this approach, one state variable is depressing the other one, and the system's internal toggle has the possibility of reaching two stable states.<br />
* Since - due to their size - proteins can only hardly pass the cell membrane (if they are not actively transported through the cell membrane), we decided to use the much smaller inducer molecules AHL, IPTG and ATC as inputs. However, since these inducers cannot directly act on the transcription of the DNA nor on the production of proteins, we need to produce the sensor proteins LuxR, LacI and TetR that build complexes with AHL, IPTG and ATC, respectively.<br />
* The sensor proteins and complexes are used to control the memory formation and the production of the florescent reporter proteins CFP, RFP, GFP and YFP.<br><br><br />
<br />
==Detailed Model==<br />
<br />
In order to test our ideas, we came up with a detailed model of all the interactions in the system. <br />
After defining the desired behavior of our system (as shown in the introduction) and a [[ETHZ/FSM | formalized desciption of the system]] we identified necessary biological components and their interactions. As we can observe in Fig. 3, our system is composed from three basic subparts:<br />
* sensors,<br />
* memory, and<br />
* reporters.<br />
<br />
===Sensors===<br />
<br />
The first part contains the sensors. Our sensors are the proteins LacI, luxR and TetR, which are constitutively produced. The sensing subsystem is shown in Fig. 4.<br />
<br />
[[Image:Model01b.png|center|thumb|<b>Fig. 4</b>: The proteins that act as sensors are constitutively produced.|140px]]<br />
<br />
===Memory===<br />
<br />
The second subsystem implements the memory. The memory control is based on the following underlying mechanisms:<br />
* The sensor proteins form complexes together with the inducers. These complexes are used to activate the transcription of the genes for the proteins CI and P22CII. <br />
* P22CII and CI repress the DNA transcription of each other, so that the closed loop system behaves as a toggle; a dynamic system with only two possible steady states (see Fig. 6).<br />
<br />
[[Image:ETHZModelLearning.png|center|thumb|<b>Fig. 5</b>: Learning system: Depending on the inputs IPTG or ATC the proteins CI and P22CII are produced.|300px]]<br />
<br />
* Fig. 5 shows the protein production system that is used during the learning phase. During the learning phase, there is still no CI or P22CII produced. They are produced, only if either IPTG or ATC is added, respectively. Since no AHL is present, the inner toggle switch (see Figure 6) is turned off.<br />
<br />
[[Image:ETHZModelMemory.png|center|thumb|<b>Fig. 6</b>: Memory system. If AHL is present the production of either CI or P22CII is continued.|420px]]<br />
<br />
* During the memory phase, AHL is added and the IPTG and ATC are removed. That is why only the inner toggle switch (see Fig. 6) is turned on while the protein production systems shown in Fig. 5 are deactivated. Depending on what was produced during the learning phase, the production of either CI or P22CII is continued. That is why the system can act as memory, effectively storing the information it is exposed to.<br />
<br />
Based on all the above, we present the final assembly of the memory subsystem in Fig. 7.<br />
<br />
[[Image:Model02b.png|center|thumb|<b>Fig. 7</b>: Final interaction of the learning and memory system. The memory content is represented by the concentrations of the proteins CI and P22CII.|560px]]<br />
<br />
===Reporters===<br />
<br />
Fig. 8 gives an overview of the reporter subsystem. Florescent reporter proteins are expressed depending on the inducer concentrations, and the concentrations of CI and P22CII. For example, the presence of either TetR or cI will repress the production of YFP. However, if the inducer ATC is present, ATC will bind to TetR which can no longer block the production of YFP. We are using four fluorescent proteins, to encode the steady states of our system at the final recognition stage. In this way, we are able to distinguish between all the different transition paths in the system.<br />
<br />
[[Image:Model03b.png|center|thumb|<b>Fig. 8</b>: The production of the florescent reporter proteins depends on the memory content (CI or P22CII) and the current input (ATC or IPTG).|600px]]<br />
<br />
==Final Model==<br />
<br />
So far, we have presented all parts needed to model and simulate the behavior of the proposed system with. By following the details presented in the previous section, we have all the necessary information to fully understand the interior of the black boxes that were presented in Fig. 2 and Fig. 3. Our overall system model is presented in Fig. 9.<br />
<br />
[[Image:ETHZFullsystem.png|center|thumb|<b>Fig. 9</b>: Final model of the educatETH <i>E. coli</i> system.|900px]]<br />
<br />
In contrast to this biological implementation, an alternative implementation using an engineering approach can be found on our [[ETHZ/FlipFlop | 'Engineer's View' page]].<br />
<br />
==Mathematical Model==<br />
<br />
Based on the modeling done so far, we can derive the equations that govern the behavior of our system. The model is given by sets of coupled [[ETHZ/Modeling_Basics | ordinary differential equations]] which are presented below. We use a simple notation for the different elements of the equations. Namely: <br />
* All concentrations are given in brackets (for example [CI]). <br />
* All decay constants are described by a variable d followed by the name of the protein they refer to. <br />
* The production of the proteins is described by a basic constant production level named 'a' that models the leak of the production system, and a factor of l and c<sub>max</sub> that describe the maximum production of a protein, given in [M/min]. <br />
* Depending on whether the DNA for a protein is implemented on a low or a high copy plasmid, we distinguish between l<sub>lo</sub> and l<sub>hi</sub>, respectively.<br />
* Dissociation constants are given by 'K' followed by the name of the protein they refer to.<br />
* The Hill cooperabilities are described by the constants 'n' followed by the name of the protein they refer to.<br />
<br />
For a more detailed introduction into how we transferred our model into equations, see the section [[ETHZ/Modeling_Basics|Modeling Basics]].<br><br />
<br />
===Allosteric regulation===<br />
<br />
These equations describe the formation of complexes between the inducers and sensor proteins. We do not use differential equations, but describe directly the concentrations of the complexes. This is a valid assumption, provided that we always wait a sufficient time, and the system reaches a steady state.<br />
We describe the total amount of proteins with the index 't', while we use the index '*' for proteins that build a complex with their respective inducer. For example: <br />
* [TetR<sub>t</sub>] describes the total concentration of TetR that is available. It is the sum of the free TetR proteins and the TetR proteins that form a complex with ATC. <br />
* [TetR<sub>*</sub>] describes the proteins that are available as a complex with ATC, and<br />
* [TetR] gives the concentration of free TetR proteins.<br />
<br />
[[Image:Eq04.png|238px]]<br />
<br />
===Constitutively produced proteins===<br />
<br />
The differential equations for the constitutively produced proteins are very simple, since there is no dependence on other proteins. They are designed so that the protein concentration reaches the value l<sub>hi</sub>*c<sub>max</sub>/d at steady state.<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
===Learning and memory subsystem===<br />
<br />
The learning and memory subsystem is the core of the system that we are trying to model and implement. It is characterized by the feedback between its state variables/proteins CI and P22CII. Its behavior is further complicated by the variation of the production of the aforementioned proteins because of the inputs. The following equations describe the concentrations of the memory proteins as a system of coupled differential equations. The equations consist of two major production parts and a decay part. <br />
* The first production part models the production of either CI or P22CII during the learning phase, and corresponds to the model in Fig. 5.<br />
* The second production part describes the inner toggle switch that was shown in Fig. 6.<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
===Reporting subsystem===<br />
<br />
The equations for the reporting subsystem finally describe the production of the florescence proteins depending on the inputs and memory proteins as modeled in Figure 8. Note that both the free constitutively produced proteins and the memory proteins repress the production of the florescence proteins. So e.g. YFP is only produced when there is both no CI and all TetR is bound in a complex together with ATC.<br />
<br />
[[Image:Reporter_braced.png|778px]]<br />
<br />
The systems of equations presented above describe and predict the behavior of our system. We have simulated the behavior of our system at steady states, and the results can be seen in the section [[ETHZ/Simulation|Simulations]]. In order to increase the accuracy of our results, we conducted an extensive literature survey, in order to isolate and find the parameters of our system. Since this is a burden for every team undertaking a complicated project in synthetic biology, we are presenting our full table of parameters in the [[ETHZ/Parameters|Parameters]] page.<br />
<br />
== References ==<br />
<p><br />
[http://www.nature.com/nature/journal/v403/n6767/abs/403339a0.html &#91;1&#93; Gardner TS, Cantor CR and Collins JJ] <i>"Construction of a genetic toggle switch in Escherichia coli"</i>, Nature 403:339–342, 2000<br /></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/ModelETHZ/Model2007-10-25T13:12:28Z<p>Uhrm: /* Mathematical Model */</p>
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__NOTOC__<br />
<br />
=Introduction=<br />
<br />
As previously discussed in the main page, we are interested in designing a system that is able to adapt to its environment. Our ideas are based on discussions about neural networks, and how we can create a biological system that exhibits the behavior of learning without having to resort to evolutionary processes. <br />
<br />
[[Image:ETHzFlowdiagram2.png|thumb|<b>Fig. 1</b>: Flow diagram. This figure shows the protocol with which the final system should be tested as well as the test results in form of the reported colors. The protocol is divided into three phases: (1) a training or learning phase in which the system learns an input and stores it in its memory, (2) a memory phase in which the system keeps the content of its memory, and finally (3) a recognition phase where the output of the system depends on the content of its memory as well as the current input.|450px]]<br />
<br />
Learning can be considered as a switching of behavior, based on some external stimuli. Thus, it comes naturally to work on existing ideas of toggle switches and [[ETHZ/FSM | finite state machines]]. <br />
<br />
The proposed system is able to distinguish between two chemicals. It represents a minimal test system that is intended as a proof of concept. By introducing the ability to distinguish more than two chemical and thereby introducing new [[ETHZ/FSM | system states]], the power of the system or its "intelligence" can be scaled. A protocol depicting how the system should react according to an input is shown in Fig. 1.<br />
<br />
The idea behind this protocol is that:<br />
* The system will be able to learn one of two input signals - aTc or IPTG - during a learning phase, while a "learning signal" (AHL) is not yet present. Depending on the input it will report by producing either cyan or yellow florescence. <br />
* Once the system has learned, the inputs - aTc or IPTG - can be removed and the system goes into a memory state in the presence of AHL. In this state, no output color is reported. Memorizing is guaranteed by removing the input chemicals.<br />
* During the recognition phase, the inputs aTc or IPTG are (re-)inserted. The system reports by changing its color depending on the input and its current memory state. This is why the system has different florescence properties even in the presence of the same input. The recognition phase takes place in the presence of AHL, to keep the memory enabled and avoid another learning phase. Since we would like to separate four different end states, we use four different fluorescent proteins to encode them.<br />
<br />
==Model Overview==<br />
<br />
The model for the proposed system is developed using a top-down approach. We start with a black box approach as shown in Fig. 2. <br />
<br />
[[Image:ETHZBlackbox.png|thumb|<b>Fig. 2</b>: Black box |280px]]<br />
<br />
The system is sketched in Fig. 3. It can be summarized as follows:<br />
* There are two inputs to be learned/detected/adapted to.<br />
* There is one separate input to switch on the memory.<br />
* The system has to alternate between at least three states. Hence, we decided to use two state variables - cI and p22cII (when interpreted as binary variables, in principle allowing for up to four different states).<br />
* There are four different output signals (synthesis of four fluorescent proteins). One could also decide to take six output signals into account to further distinguish the learning phase from the recognition phase. However, we restricted ourselves to four outputs to reduce the number of genes that are needed to implement the signals.<br />
<br />
[[Image:ETHZFullsystemmodel.png|left|thumb|<b>Fig. 3</b>: System overview. AHL, IPTG and ATC pass the cell membrane where they build complexes with the sensor proteins LuxR, LacI and TetR. These sensor proteins and/or complexes are used to control the internal system state: the memory represented by the proteins cI and p22cII (mutually repressing their synthesis) and the sensed input (IPTG, ATC). CFP, RFP, GFP and YFP stand for yellow, red, cyan and green florescent protein, respectively.|420px]]<br />
<br />
However, we had to keep in mind that the proposed system should be implemented in DNA, and that it would be sensitive to noise. As a result, we took several actions to achieve better experimental results and easier DNA construction:<br />
* To be more robust against perturbations, we coupled the state variables cI and p22cII like it is known from toggle switches [1]. Based on this approach, one state variable is depressing the other one, and the system's internal toggle has the possibility of reaching two stable states.<br />
* Since - due to their size - proteins can only hardly pass the cell membrane (if they are not actively transported through the cell membrane), we decided to use the much smaller inducer molecules AHL, IPTG and ATC as inputs. However, since these inducers cannot directly act on the transcription of the DNA nor on the production of proteins, we need to produce the sensor proteins LuxR, LacI and TetR that build complexes with AHL, IPTG and ATC, respectively.<br />
* The sensor proteins and complexes are used to control the memory formation and the production of the florescent reporter proteins CFP, RFP, GFP and YFP.<br><br><br />
<br />
==Detailed Model==<br />
<br />
In order to test our ideas, we came up with a detailed model of all the interactions in the system. <br />
After defining the desired behavior of our system (as shown in the introduction) and a [[ETHZ/FSM | formalized desciption of the system]] we identified necessary biological components and their interactions. As we can observe in Fig. 3, our system is composed from three basic subparts:<br />
* sensors,<br />
* memory, and<br />
* reporters.<br />
<br />
===Sensors===<br />
<br />
The first part contains the sensors. Our sensors are the proteins LacI, luxR and TetR, which are constitutively produced. The sensing subsystem is shown in Fig. 4.<br />
<br />
[[Image:Model01b.png|center|thumb|<b>Fig. 4</b>: The proteins that act as sensors are constitutively produced.|140px]]<br />
<br />
===Memory===<br />
<br />
The second subsystem implements the memory. The memory control is based on the following underlying mechanisms:<br />
* The sensor proteins form complexes together with the inducers. These complexes are used to activate the transcription of the genes for the proteins CI and P22CII. <br />
* P22CII and CI repress the DNA transcription of each other, so that the closed loop system behaves as a toggle; a dynamic system with only two possible steady states (see Fig. 6).<br />
<br />
[[Image:ETHZModelLearning.png|center|thumb|<b>Fig. 5</b>: Learning system: Depending on the inputs IPTG or ATC the proteins CI and P22CII are produced.|300px]]<br />
<br />
* Fig. 5 shows the protein production system that is used during the learning phase. During the learning phase, there is still no CI or P22CII produced. They are produced, only if either IPTG or ATC is added, respectively. Since no AHL is present, the inner toggle switch (see Figure 6) is turned off.<br />
<br />
[[Image:ETHZModelMemory.png|center|thumb|<b>Fig. 6</b>: Memory system. If AHL is present the production of either CI or P22CII is continued.|420px]]<br />
<br />
* During the memory phase, AHL is added and the IPTG and ATC are removed. That is why only the inner toggle switch (see Fig. 6) is turned on while the protein production systems shown in Fig. 5 are deactivated. Depending on what was produced during the learning phase, the production of either CI or P22CII is continued. That is why the system can act as memory, effectively storing the information it is exposed to.<br />
<br />
Based on all the above, we present the final assembly of the memory subsystem in Fig. 7.<br />
<br />
[[Image:Model02b.png|center|thumb|<b>Fig. 7</b>: Final interaction of the learning and memory system. The memory content is represented by the concentrations of the proteins CI and P22CII.|560px]]<br />
<br />
===Reporters===<br />
<br />
Fig. 8 gives an overview of the reporter subsystem. Florescent reporter proteins are expressed depending on the inducer concentrations, and the concentrations of CI and P22CII. For example, the presence of either TetR or cI will repress the production of YFP. However, if the inducer ATC is present, ATC will bind to TetR which can no longer block the production of YFP. We are using four fluorescent proteins, to encode the steady states of our system at the final recognition stage. In this way, we are able to distinguish between all the different transition paths in the system.<br />
<br />
[[Image:Model03b.png|center|thumb|<b>Fig. 8</b>: The production of the florescent reporter proteins depends on the memory content (CI or P22CII) and the current input (ATC or IPTG).|600px]]<br />
<br />
==Final Model==<br />
<br />
So far, we have presented all parts needed to model and simulate the behavior of the proposed system with. By following the details presented in the previous section, we have all the necessary information to fully understand the interior of the black boxes that were presented in Fig. 2 and Fig. 3. Our overall system model is presented in Fig. 9.<br />
<br />
[[Image:ETHZFullsystem.png|center|thumb|<b>Fig. 9</b>: Final model of the educatETH <i>E. coli</i> system.|900px]]<br />
<br />
In contrast to this biological implementation, an alternative implementation using an engineering approach can be found on our [[ETHZ/FlipFlop | 'Engineer's View' page]].<br />
<br />
==Mathematical Model==<br />
<br />
Based on the modeling done so far, we can derive the equations that govern the behavior of our system. The model is given by sets of coupled [[ETHZ/Modeling_Basics | ordinary differential equations]] which are presented below. We use a simple notation for the different elements of the equations. Namely: <br />
* All concentrations are given in brackets (for example [CI]). <br />
* All decay constants are described by a variable d followed by the name of the protein they refer to. <br />
* The production of the proteins is described by a basic constant production level named 'a' that models the leak of the production system, and a factor of l and c<sub>max</sub> that describe the maximum production of a protein, given in [M/min]. <br />
* Depending on whether the DNA for a protein is implemented on a low or a high copy plasmid, we distinguish between l<sub>lo</sub> and l<sub>hi</sub>, respectively.<br />
* Dissociation constants are given by 'K' followed by the name of the protein they refer to.<br />
* The Hill cooperabilities are described by the constants 'n' followed by the name of the protein they refer to.<br />
<br />
For a more detailed introduction into how we transferred our model into equations, see the section [[ETHZ/Modeling_Basics|Modeling Basics]].<br><br />
<br />
===Allosteric regulation===<br />
<br />
These equations describe the formation of complexes between the inducers and sensor proteins. We do not use differential equations, but describe directly the concentrations of the complexes. This is a valid assumption, provided that we always wait a sufficient time, and the system reaches a steady state.<br />
We describe the total amount of proteins with the index 't', while we use the index '*' for proteins that build a complex with their respective inducer. For example: <br />
* [TetR<sub>t</sub>] describes the total concentration of TetR that is available. It is the sum of the free TetR proteins and the TetR proteins that form a complex with ATC. <br />
* [TetR<sub>*</sub>] describes the proteins that are available as a complex with ATC, and<br />
* [TetR] gives the concentration of free TetR proteins.<br />
<br />
[[Image:Eq04.png|238px]]<br />
<br />
===Constitutively produced proteins===<br />
<br />
The differential equations for the constitutively produced proteins are very simple, since there is no dependence on other proteins. They are designed so that the protein concentration reaches the value l<sub>hi</sub>*c<sub>max</sub>/d at steady state.<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
===Learning and memory subsystem===<br />
<br />
The learning and memory subsystem is the core of the system that we are trying to model and implement. It is characterized by the feedback between its state variables/proteins CI and P22CII. Its behavior is further complicated by the variation of the production of the aforementioned proteins because of the inputs. The following equations describe the concentrations of the memory proteins as a system of coupled differential equations. The equations consist of two major production parts and a decay part. <br />
* The first production part models the production of either CI or P22CII during the learning phase, and corresponds to the model in Fig. 5.<br />
* The second production part describes the inner toggle switch that was shown in Fig. 6.<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
===Reporting subsystem===<br />
<br />
The equations for the reporting subsystem finally describe the production of the florescence proteins depending on the inputs and memory proteins as modeled in Figure 8. Note that both the free constitutively produced proteins and the memory proteins repress the production of the florescence proteins. So e.g. YFP is only produced when there is both no cI and all TetR is bound in a complex together with ATC.<br />
<br />
[[Image:Reporter_braced.png|778px]]<br />
<br />
The systems of equations presented above describe and predict the behavior of our system. We have simulated the behavior of our system at steady states, and the results can be seen in the section [[ETHZ/Simulation|Simulations]]. In order to increase the accuracy of our results, we conducted an extensive literature survey, in order to isolate and find the parameters of our system. Since this is a burden for every team undertaking a complicated project in synthetic biology, we are presenting our full table of parameters in the [[ETHZ/Parameters|Parameters]] page.<br />
<br />
== References ==<br />
<p><br />
[http://www.nature.com/nature/journal/v403/n6767/abs/403339a0.html &#91;1&#93; Gardner TS, Cantor CR and Collins JJ] <i>"Construction of a genetic toggle switch in Escherichia coli"</i>, Nature 403:339–342, 2000<br /></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/ModelETHZ/Model2007-10-25T13:10:54Z<p>Uhrm: /* Reporters */</p>
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<br />
=Introduction=<br />
<br />
As previously discussed in the main page, we are interested in designing a system that is able to adapt to its environment. Our ideas are based on discussions about neural networks, and how we can create a biological system that exhibits the behavior of learning without having to resort to evolutionary processes. <br />
<br />
[[Image:ETHzFlowdiagram2.png|thumb|<b>Fig. 1</b>: Flow diagram. This figure shows the protocol with which the final system should be tested as well as the test results in form of the reported colors. The protocol is divided into three phases: (1) a training or learning phase in which the system learns an input and stores it in its memory, (2) a memory phase in which the system keeps the content of its memory, and finally (3) a recognition phase where the output of the system depends on the content of its memory as well as the current input.|450px]]<br />
<br />
Learning can be considered as a switching of behavior, based on some external stimuli. Thus, it comes naturally to work on existing ideas of toggle switches and [[ETHZ/FSM | finite state machines]]. <br />
<br />
The proposed system is able to distinguish between two chemicals. It represents a minimal test system that is intended as a proof of concept. By introducing the ability to distinguish more than two chemical and thereby introducing new [[ETHZ/FSM | system states]], the power of the system or its "intelligence" can be scaled. A protocol depicting how the system should react according to an input is shown in Fig. 1.<br />
<br />
The idea behind this protocol is that:<br />
* The system will be able to learn one of two input signals - aTc or IPTG - during a learning phase, while a "learning signal" (AHL) is not yet present. Depending on the input it will report by producing either cyan or yellow florescence. <br />
* Once the system has learned, the inputs - aTc or IPTG - can be removed and the system goes into a memory state in the presence of AHL. In this state, no output color is reported. Memorizing is guaranteed by removing the input chemicals.<br />
* During the recognition phase, the inputs aTc or IPTG are (re-)inserted. The system reports by changing its color depending on the input and its current memory state. This is why the system has different florescence properties even in the presence of the same input. The recognition phase takes place in the presence of AHL, to keep the memory enabled and avoid another learning phase. Since we would like to separate four different end states, we use four different fluorescent proteins to encode them.<br />
<br />
==Model Overview==<br />
<br />
The model for the proposed system is developed using a top-down approach. We start with a black box approach as shown in Fig. 2. <br />
<br />
[[Image:ETHZBlackbox.png|thumb|<b>Fig. 2</b>: Black box |280px]]<br />
<br />
The system is sketched in Fig. 3. It can be summarized as follows:<br />
* There are two inputs to be learned/detected/adapted to.<br />
* There is one separate input to switch on the memory.<br />
* The system has to alternate between at least three states. Hence, we decided to use two state variables - cI and p22cII (when interpreted as binary variables, in principle allowing for up to four different states).<br />
* There are four different output signals (synthesis of four fluorescent proteins). One could also decide to take six output signals into account to further distinguish the learning phase from the recognition phase. However, we restricted ourselves to four outputs to reduce the number of genes that are needed to implement the signals.<br />
<br />
[[Image:ETHZFullsystemmodel.png|left|thumb|<b>Fig. 3</b>: System overview. AHL, IPTG and ATC pass the cell membrane where they build complexes with the sensor proteins LuxR, LacI and TetR. These sensor proteins and/or complexes are used to control the internal system state: the memory represented by the proteins cI and p22cII (mutually repressing their synthesis) and the sensed input (IPTG, ATC). CFP, RFP, GFP and YFP stand for yellow, red, cyan and green florescent protein, respectively.|420px]]<br />
<br />
However, we had to keep in mind that the proposed system should be implemented in DNA, and that it would be sensitive to noise. As a result, we took several actions to achieve better experimental results and easier DNA construction:<br />
* To be more robust against perturbations, we coupled the state variables cI and p22cII like it is known from toggle switches [1]. Based on this approach, one state variable is depressing the other one, and the system's internal toggle has the possibility of reaching two stable states.<br />
* Since - due to their size - proteins can only hardly pass the cell membrane (if they are not actively transported through the cell membrane), we decided to use the much smaller inducer molecules AHL, IPTG and ATC as inputs. However, since these inducers cannot directly act on the transcription of the DNA nor on the production of proteins, we need to produce the sensor proteins LuxR, LacI and TetR that build complexes with AHL, IPTG and ATC, respectively.<br />
* The sensor proteins and complexes are used to control the memory formation and the production of the florescent reporter proteins CFP, RFP, GFP and YFP.<br><br><br />
<br />
==Detailed Model==<br />
<br />
In order to test our ideas, we came up with a detailed model of all the interactions in the system. <br />
After defining the desired behavior of our system (as shown in the introduction) and a [[ETHZ/FSM | formalized desciption of the system]] we identified necessary biological components and their interactions. As we can observe in Fig. 3, our system is composed from three basic subparts:<br />
* sensors,<br />
* memory, and<br />
* reporters.<br />
<br />
===Sensors===<br />
<br />
The first part contains the sensors. Our sensors are the proteins LacI, luxR and TetR, which are constitutively produced. The sensing subsystem is shown in Fig. 4.<br />
<br />
[[Image:Model01b.png|center|thumb|<b>Fig. 4</b>: The proteins that act as sensors are constitutively produced.|140px]]<br />
<br />
===Memory===<br />
<br />
The second subsystem implements the memory. The memory control is based on the following underlying mechanisms:<br />
* The sensor proteins form complexes together with the inducers. These complexes are used to activate the transcription of the genes for the proteins CI and P22CII. <br />
* P22CII and CI repress the DNA transcription of each other, so that the closed loop system behaves as a toggle; a dynamic system with only two possible steady states (see Fig. 6).<br />
<br />
[[Image:ETHZModelLearning.png|center|thumb|<b>Fig. 5</b>: Learning system: Depending on the inputs IPTG or ATC the proteins CI and P22CII are produced.|300px]]<br />
<br />
* Fig. 5 shows the protein production system that is used during the learning phase. During the learning phase, there is still no CI or P22CII produced. They are produced, only if either IPTG or ATC is added, respectively. Since no AHL is present, the inner toggle switch (see Figure 6) is turned off.<br />
<br />
[[Image:ETHZModelMemory.png|center|thumb|<b>Fig. 6</b>: Memory system. If AHL is present the production of either CI or P22CII is continued.|420px]]<br />
<br />
* During the memory phase, AHL is added and the IPTG and ATC are removed. That is why only the inner toggle switch (see Fig. 6) is turned on while the protein production systems shown in Fig. 5 are deactivated. Depending on what was produced during the learning phase, the production of either CI or P22CII is continued. That is why the system can act as memory, effectively storing the information it is exposed to.<br />
<br />
Based on all the above, we present the final assembly of the memory subsystem in Fig. 7.<br />
<br />
[[Image:Model02b.png|center|thumb|<b>Fig. 7</b>: Final interaction of the learning and memory system. The memory content is represented by the concentrations of the proteins CI and P22CII.|560px]]<br />
<br />
===Reporters===<br />
<br />
Fig. 8 gives an overview of the reporter subsystem. Florescent reporter proteins are expressed depending on the inducer concentrations, and the concentrations of CI and P22CII. For example, the presence of either TetR or cI will repress the production of YFP. However, if the inducer ATC is present, ATC will bind to TetR which can no longer block the production of YFP. We are using four fluorescent proteins, to encode the steady states of our system at the final recognition stage. In this way, we are able to distinguish between all the different transition paths in the system.<br />
<br />
[[Image:Model03b.png|center|thumb|<b>Fig. 8</b>: The production of the florescent reporter proteins depends on the memory content (CI or P22CII) and the current input (ATC or IPTG).|600px]]<br />
<br />
==Final Model==<br />
<br />
So far, we have presented all parts needed to model and simulate the behavior of the proposed system with. By following the details presented in the previous section, we have all the necessary information to fully understand the interior of the black boxes that were presented in Fig. 2 and Fig. 3. Our overall system model is presented in Fig. 9.<br />
<br />
[[Image:ETHZFullsystem.png|center|thumb|<b>Fig. 9</b>: Final model of the educatETH <i>E. coli</i> system.|900px]]<br />
<br />
In contrast to this biological implementation, an alternative implementation using an engineering approach can be found on our [[ETHZ/FlipFlop | 'Engineer's View' page]].<br />
<br />
==Mathematical Model==<br />
<br />
Based on the modeling done so far, we can derive the equations that govern the behavior of our system. The model is given by sets of coupled [[ETHZ/Modeling_Basics | ordinary differential equations]] which are presented below. We use a simple notation for the different elements of the equations. Namely: <br />
* All concentrations are given in brackets (for example [cI]). <br />
* All decay constants are described by a variable d followed by the name of the protein they refer to. <br />
* The production of the proteins is described by a basic constant production level named 'a' that models the leak of the production system, and a factor of l and c<sub>max</sub> that describe the maximum production of a protein, given in [M/min]. <br />
* Depending on whether the DNA for a protein is implemented on a low or a high copy plasmid, we distinguish between l<sub>lo</sub> and l<sub>hi</sub>, respectively.<br />
* Dissociation constants are given by 'K' followed by the name of the protein they refer to.<br />
* The Hill cooperabilities are described by the constants 'n' followed by the name of the protein they refer to.<br />
<br />
For a more detailed introduction into how we transferred our model into equations, see the section [[ETHZ/Modeling_Basics|Modeling Basics]].<br><br />
<br />
===Allosteric regulation===<br />
<br />
These equations describe the formation of complexes between the inducers and sensor proteins. We do not use differential equations, but describe directly the concentrations of the complexes. This is a valid assumption, provided that we always wait a sufficient time, and the system reaches a steady state.<br />
We describe the total amount of proteins with the index 't', while we use the index '*' for proteins that build a complex with their respective inducer. For example: <br />
* [TetR<sub>t</sub>] describes the total concentration of TetR that is available. It is the sum of the free TetR proteins and the TetR proteins that form a complex with aTc. <br />
* [TetR<sub>*</sub>] describes the proteins that are available as a complex with aTc, and<br />
* [TetR] gives the concentration of free TetR proteins.<br />
<br />
[[Image:Eq04.png|238px]]<br />
<br />
===Constitutively produced proteins===<br />
<br />
The differential equations for the constitutively produced proteins are very simple, since there is no dependence on other proteins. They are designed so that the protein concentration reaches the value l<sub>hi</sub>*c<sub>max</sub>/d at steady state.<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
===Learning and memory subsystem===<br />
<br />
The learning and memory subsystem is the core of the system that we are trying to model and implement. It is characterized by the feedback between its state variables/proteins cI and p22cII. Its behavior is further complicated by the variation of the production of the aforementioned proteins because of the inputs. The following equations describe the concentrations of the memory proteins as a system of coupled differential equations. The equations consist of two major production parts and a decay part. <br />
* The first production part models the production of either cI or p22cII during the learning phase, and corresponds to the model in Fig. 5.<br />
* The second production part describes the inner toggle switch that was shown in Fig. 6.<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
===Reporting subsystem===<br />
<br />
The equations for the reporting subsystem finally describe the production of the florescence proteins depending on the inputs and memory proteins as modeled in Figure 8. Note that both the free constitutively produced proteins and the memory proteins repress the production of the florescence proteins. So e.g. YFP is only produced when there is both no cI and all TetR is bound in a complex together with aTc.<br />
<br />
[[Image:Reporter_braced.png|778px]]<br />
<br />
The systems of equations presented above describe and predict the behavior of our system. We have simulated the behavior of our system at steady states, and the results can be seen in the section [[ETHZ/Simulation|Simulations]]. In order to increase the accuracy of our results, we conducted an extensive literature survey, in order to isolate and find the parameters of our system. Since this is a burden for every team undertaking a complicated project in synthetic biology, we are presenting our full table of parameters in the [[ETHZ/Parameters|Parameters]] page.<br />
<br />
== References ==<br />
<p><br />
[http://www.nature.com/nature/journal/v403/n6767/abs/403339a0.html &#91;1&#93; Gardner TS, Cantor CR and Collins JJ] <i>"Construction of a genetic toggle switch in Escherichia coli"</i>, Nature 403:339–342, 2000<br /></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/ModelETHZ/Model2007-10-25T13:09:31Z<p>Uhrm: /* Memory */</p>
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<br />
=Introduction=<br />
<br />
As previously discussed in the main page, we are interested in designing a system that is able to adapt to its environment. Our ideas are based on discussions about neural networks, and how we can create a biological system that exhibits the behavior of learning without having to resort to evolutionary processes. <br />
<br />
[[Image:ETHzFlowdiagram2.png|thumb|<b>Fig. 1</b>: Flow diagram. This figure shows the protocol with which the final system should be tested as well as the test results in form of the reported colors. The protocol is divided into three phases: (1) a training or learning phase in which the system learns an input and stores it in its memory, (2) a memory phase in which the system keeps the content of its memory, and finally (3) a recognition phase where the output of the system depends on the content of its memory as well as the current input.|450px]]<br />
<br />
Learning can be considered as a switching of behavior, based on some external stimuli. Thus, it comes naturally to work on existing ideas of toggle switches and [[ETHZ/FSM | finite state machines]]. <br />
<br />
The proposed system is able to distinguish between two chemicals. It represents a minimal test system that is intended as a proof of concept. By introducing the ability to distinguish more than two chemical and thereby introducing new [[ETHZ/FSM | system states]], the power of the system or its "intelligence" can be scaled. A protocol depicting how the system should react according to an input is shown in Fig. 1.<br />
<br />
The idea behind this protocol is that:<br />
* The system will be able to learn one of two input signals - aTc or IPTG - during a learning phase, while a "learning signal" (AHL) is not yet present. Depending on the input it will report by producing either cyan or yellow florescence. <br />
* Once the system has learned, the inputs - aTc or IPTG - can be removed and the system goes into a memory state in the presence of AHL. In this state, no output color is reported. Memorizing is guaranteed by removing the input chemicals.<br />
* During the recognition phase, the inputs aTc or IPTG are (re-)inserted. The system reports by changing its color depending on the input and its current memory state. This is why the system has different florescence properties even in the presence of the same input. The recognition phase takes place in the presence of AHL, to keep the memory enabled and avoid another learning phase. Since we would like to separate four different end states, we use four different fluorescent proteins to encode them.<br />
<br />
==Model Overview==<br />
<br />
The model for the proposed system is developed using a top-down approach. We start with a black box approach as shown in Fig. 2. <br />
<br />
[[Image:ETHZBlackbox.png|thumb|<b>Fig. 2</b>: Black box |280px]]<br />
<br />
The system is sketched in Fig. 3. It can be summarized as follows:<br />
* There are two inputs to be learned/detected/adapted to.<br />
* There is one separate input to switch on the memory.<br />
* The system has to alternate between at least three states. Hence, we decided to use two state variables - cI and p22cII (when interpreted as binary variables, in principle allowing for up to four different states).<br />
* There are four different output signals (synthesis of four fluorescent proteins). One could also decide to take six output signals into account to further distinguish the learning phase from the recognition phase. However, we restricted ourselves to four outputs to reduce the number of genes that are needed to implement the signals.<br />
<br />
[[Image:ETHZFullsystemmodel.png|left|thumb|<b>Fig. 3</b>: System overview. AHL, IPTG and ATC pass the cell membrane where they build complexes with the sensor proteins LuxR, LacI and TetR. These sensor proteins and/or complexes are used to control the internal system state: the memory represented by the proteins cI and p22cII (mutually repressing their synthesis) and the sensed input (IPTG, ATC). CFP, RFP, GFP and YFP stand for yellow, red, cyan and green florescent protein, respectively.|420px]]<br />
<br />
However, we had to keep in mind that the proposed system should be implemented in DNA, and that it would be sensitive to noise. As a result, we took several actions to achieve better experimental results and easier DNA construction:<br />
* To be more robust against perturbations, we coupled the state variables cI and p22cII like it is known from toggle switches [1]. Based on this approach, one state variable is depressing the other one, and the system's internal toggle has the possibility of reaching two stable states.<br />
* Since - due to their size - proteins can only hardly pass the cell membrane (if they are not actively transported through the cell membrane), we decided to use the much smaller inducer molecules AHL, IPTG and ATC as inputs. However, since these inducers cannot directly act on the transcription of the DNA nor on the production of proteins, we need to produce the sensor proteins LuxR, LacI and TetR that build complexes with AHL, IPTG and ATC, respectively.<br />
* The sensor proteins and complexes are used to control the memory formation and the production of the florescent reporter proteins CFP, RFP, GFP and YFP.<br><br><br />
<br />
==Detailed Model==<br />
<br />
In order to test our ideas, we came up with a detailed model of all the interactions in the system. <br />
After defining the desired behavior of our system (as shown in the introduction) and a [[ETHZ/FSM | formalized desciption of the system]] we identified necessary biological components and their interactions. As we can observe in Fig. 3, our system is composed from three basic subparts:<br />
* sensors,<br />
* memory, and<br />
* reporters.<br />
<br />
===Sensors===<br />
<br />
The first part contains the sensors. Our sensors are the proteins LacI, luxR and TetR, which are constitutively produced. The sensing subsystem is shown in Fig. 4.<br />
<br />
[[Image:Model01b.png|center|thumb|<b>Fig. 4</b>: The proteins that act as sensors are constitutively produced.|140px]]<br />
<br />
===Memory===<br />
<br />
The second subsystem implements the memory. The memory control is based on the following underlying mechanisms:<br />
* The sensor proteins form complexes together with the inducers. These complexes are used to activate the transcription of the genes for the proteins CI and P22CII. <br />
* P22CII and CI repress the DNA transcription of each other, so that the closed loop system behaves as a toggle; a dynamic system with only two possible steady states (see Fig. 6).<br />
<br />
[[Image:ETHZModelLearning.png|center|thumb|<b>Fig. 5</b>: Learning system: Depending on the inputs IPTG or ATC the proteins CI and P22CII are produced.|300px]]<br />
<br />
* Fig. 5 shows the protein production system that is used during the learning phase. During the learning phase, there is still no CI or P22CII produced. They are produced, only if either IPTG or ATC is added, respectively. Since no AHL is present, the inner toggle switch (see Figure 6) is turned off.<br />
<br />
[[Image:ETHZModelMemory.png|center|thumb|<b>Fig. 6</b>: Memory system. If AHL is present the production of either CI or P22CII is continued.|420px]]<br />
<br />
* During the memory phase, AHL is added and the IPTG and ATC are removed. That is why only the inner toggle switch (see Fig. 6) is turned on while the protein production systems shown in Fig. 5 are deactivated. Depending on what was produced during the learning phase, the production of either CI or P22CII is continued. That is why the system can act as memory, effectively storing the information it is exposed to.<br />
<br />
Based on all the above, we present the final assembly of the memory subsystem in Fig. 7.<br />
<br />
[[Image:Model02b.png|center|thumb|<b>Fig. 7</b>: Final interaction of the learning and memory system. The memory content is represented by the concentrations of the proteins CI and P22CII.|560px]]<br />
<br />
===Reporters===<br />
<br />
Fig. 8 gives an overview of the reporter subsystem. Florescent reporter proteins are expressed depending on the inducer concentrations, and the concentrations of cI and p22cII. For example, the presence of either TetR or cI will repress the production of YFP. However, if the inducer aTc is present, aTc will bind to TetR which can no longer block the production of YFP. We are using four fluorescent proteins, to encode the steady states of our system at the final recognition stage. In this way, we are able to distinguish between all the different transition paths in the system.<br />
<br />
[[Image:Model03b.png|center|thumb|<b>Fig. 8</b>: The production of the florescent reporter proteins depends on the memory content (cI or p22cII) and the current input (aTc or IPTG).|600px]]<br />
<br />
==Final Model==<br />
<br />
So far, we have presented all parts needed to model and simulate the behavior of the proposed system with. By following the details presented in the previous section, we have all the necessary information to fully understand the interior of the black boxes that were presented in Fig. 2 and Fig. 3. Our overall system model is presented in Fig. 9.<br />
<br />
[[Image:ETHZFullsystem.png|center|thumb|<b>Fig. 9</b>: Final model of the educatETH <i>E. coli</i> system.|900px]]<br />
<br />
In contrast to this biological implementation, an alternative implementation using an engineering approach can be found on our [[ETHZ/FlipFlop | 'Engineer's View' page]].<br />
<br />
==Mathematical Model==<br />
<br />
Based on the modeling done so far, we can derive the equations that govern the behavior of our system. The model is given by sets of coupled [[ETHZ/Modeling_Basics | ordinary differential equations]] which are presented below. We use a simple notation for the different elements of the equations. Namely: <br />
* All concentrations are given in brackets (for example [cI]). <br />
* All decay constants are described by a variable d followed by the name of the protein they refer to. <br />
* The production of the proteins is described by a basic constant production level named 'a' that models the leak of the production system, and a factor of l and c<sub>max</sub> that describe the maximum production of a protein, given in [M/min]. <br />
* Depending on whether the DNA for a protein is implemented on a low or a high copy plasmid, we distinguish between l<sub>lo</sub> and l<sub>hi</sub>, respectively.<br />
* Dissociation constants are given by 'K' followed by the name of the protein they refer to.<br />
* The Hill cooperabilities are described by the constants 'n' followed by the name of the protein they refer to.<br />
<br />
For a more detailed introduction into how we transferred our model into equations, see the section [[ETHZ/Modeling_Basics|Modeling Basics]].<br><br />
<br />
===Allosteric regulation===<br />
<br />
These equations describe the formation of complexes between the inducers and sensor proteins. We do not use differential equations, but describe directly the concentrations of the complexes. This is a valid assumption, provided that we always wait a sufficient time, and the system reaches a steady state.<br />
We describe the total amount of proteins with the index 't', while we use the index '*' for proteins that build a complex with their respective inducer. For example: <br />
* [TetR<sub>t</sub>] describes the total concentration of TetR that is available. It is the sum of the free TetR proteins and the TetR proteins that form a complex with aTc. <br />
* [TetR<sub>*</sub>] describes the proteins that are available as a complex with aTc, and<br />
* [TetR] gives the concentration of free TetR proteins.<br />
<br />
[[Image:Eq04.png|238px]]<br />
<br />
===Constitutively produced proteins===<br />
<br />
The differential equations for the constitutively produced proteins are very simple, since there is no dependence on other proteins. They are designed so that the protein concentration reaches the value l<sub>hi</sub>*c<sub>max</sub>/d at steady state.<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
===Learning and memory subsystem===<br />
<br />
The learning and memory subsystem is the core of the system that we are trying to model and implement. It is characterized by the feedback between its state variables/proteins cI and p22cII. Its behavior is further complicated by the variation of the production of the aforementioned proteins because of the inputs. The following equations describe the concentrations of the memory proteins as a system of coupled differential equations. The equations consist of two major production parts and a decay part. <br />
* The first production part models the production of either cI or p22cII during the learning phase, and corresponds to the model in Fig. 5.<br />
* The second production part describes the inner toggle switch that was shown in Fig. 6.<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
===Reporting subsystem===<br />
<br />
The equations for the reporting subsystem finally describe the production of the florescence proteins depending on the inputs and memory proteins as modeled in Figure 8. Note that both the free constitutively produced proteins and the memory proteins repress the production of the florescence proteins. So e.g. YFP is only produced when there is both no cI and all TetR is bound in a complex together with aTc.<br />
<br />
[[Image:Reporter_braced.png|778px]]<br />
<br />
The systems of equations presented above describe and predict the behavior of our system. We have simulated the behavior of our system at steady states, and the results can be seen in the section [[ETHZ/Simulation|Simulations]]. In order to increase the accuracy of our results, we conducted an extensive literature survey, in order to isolate and find the parameters of our system. Since this is a burden for every team undertaking a complicated project in synthetic biology, we are presenting our full table of parameters in the [[ETHZ/Parameters|Parameters]] page.<br />
<br />
== References ==<br />
<p><br />
[http://www.nature.com/nature/journal/v403/n6767/abs/403339a0.html &#91;1&#93; Gardner TS, Cantor CR and Collins JJ] <i>"Construction of a genetic toggle switch in Escherichia coli"</i>, Nature 403:339–342, 2000<br /></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/ModelETHZ/Model2007-10-25T13:07:19Z<p>Uhrm: /* Model Overview */</p>
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<br />
=Introduction=<br />
<br />
As previously discussed in the main page, we are interested in designing a system that is able to adapt to its environment. Our ideas are based on discussions about neural networks, and how we can create a biological system that exhibits the behavior of learning without having to resort to evolutionary processes. <br />
<br />
[[Image:ETHzFlowdiagram2.png|thumb|<b>Fig. 1</b>: Flow diagram. This figure shows the protocol with which the final system should be tested as well as the test results in form of the reported colors. The protocol is divided into three phases: (1) a training or learning phase in which the system learns an input and stores it in its memory, (2) a memory phase in which the system keeps the content of its memory, and finally (3) a recognition phase where the output of the system depends on the content of its memory as well as the current input.|450px]]<br />
<br />
Learning can be considered as a switching of behavior, based on some external stimuli. Thus, it comes naturally to work on existing ideas of toggle switches and [[ETHZ/FSM | finite state machines]]. <br />
<br />
The proposed system is able to distinguish between two chemicals. It represents a minimal test system that is intended as a proof of concept. By introducing the ability to distinguish more than two chemical and thereby introducing new [[ETHZ/FSM | system states]], the power of the system or its "intelligence" can be scaled. A protocol depicting how the system should react according to an input is shown in Fig. 1.<br />
<br />
The idea behind this protocol is that:<br />
* The system will be able to learn one of two input signals - aTc or IPTG - during a learning phase, while a "learning signal" (AHL) is not yet present. Depending on the input it will report by producing either cyan or yellow florescence. <br />
* Once the system has learned, the inputs - aTc or IPTG - can be removed and the system goes into a memory state in the presence of AHL. In this state, no output color is reported. Memorizing is guaranteed by removing the input chemicals.<br />
* During the recognition phase, the inputs aTc or IPTG are (re-)inserted. The system reports by changing its color depending on the input and its current memory state. This is why the system has different florescence properties even in the presence of the same input. The recognition phase takes place in the presence of AHL, to keep the memory enabled and avoid another learning phase. Since we would like to separate four different end states, we use four different fluorescent proteins to encode them.<br />
<br />
==Model Overview==<br />
<br />
The model for the proposed system is developed using a top-down approach. We start with a black box approach as shown in Fig. 2. <br />
<br />
[[Image:ETHZBlackbox.png|thumb|<b>Fig. 2</b>: Black box |280px]]<br />
<br />
The system is sketched in Fig. 3. It can be summarized as follows:<br />
* There are two inputs to be learned/detected/adapted to.<br />
* There is one separate input to switch on the memory.<br />
* The system has to alternate between at least three states. Hence, we decided to use two state variables - cI and p22cII (when interpreted as binary variables, in principle allowing for up to four different states).<br />
* There are four different output signals (synthesis of four fluorescent proteins). One could also decide to take six output signals into account to further distinguish the learning phase from the recognition phase. However, we restricted ourselves to four outputs to reduce the number of genes that are needed to implement the signals.<br />
<br />
[[Image:ETHZFullsystemmodel.png|left|thumb|<b>Fig. 3</b>: System overview. AHL, IPTG and ATC pass the cell membrane where they build complexes with the sensor proteins LuxR, LacI and TetR. These sensor proteins and/or complexes are used to control the internal system state: the memory represented by the proteins cI and p22cII (mutually repressing their synthesis) and the sensed input (IPTG, ATC). CFP, RFP, GFP and YFP stand for yellow, red, cyan and green florescent protein, respectively.|420px]]<br />
<br />
However, we had to keep in mind that the proposed system should be implemented in DNA, and that it would be sensitive to noise. As a result, we took several actions to achieve better experimental results and easier DNA construction:<br />
* To be more robust against perturbations, we coupled the state variables cI and p22cII like it is known from toggle switches [1]. Based on this approach, one state variable is depressing the other one, and the system's internal toggle has the possibility of reaching two stable states.<br />
* Since - due to their size - proteins can only hardly pass the cell membrane (if they are not actively transported through the cell membrane), we decided to use the much smaller inducer molecules AHL, IPTG and ATC as inputs. However, since these inducers cannot directly act on the transcription of the DNA nor on the production of proteins, we need to produce the sensor proteins LuxR, LacI and TetR that build complexes with AHL, IPTG and ATC, respectively.<br />
* The sensor proteins and complexes are used to control the memory formation and the production of the florescent reporter proteins CFP, RFP, GFP and YFP.<br><br><br />
<br />
==Detailed Model==<br />
<br />
In order to test our ideas, we came up with a detailed model of all the interactions in the system. <br />
After defining the desired behavior of our system (as shown in the introduction) and a [[ETHZ/FSM | formalized desciption of the system]] we identified necessary biological components and their interactions. As we can observe in Fig. 3, our system is composed from three basic subparts:<br />
* sensors,<br />
* memory, and<br />
* reporters.<br />
<br />
===Sensors===<br />
<br />
The first part contains the sensors. Our sensors are the proteins LacI, luxR and TetR, which are constitutively produced. The sensing subsystem is shown in Fig. 4.<br />
<br />
[[Image:Model01b.png|center|thumb|<b>Fig. 4</b>: The proteins that act as sensors are constitutively produced.|140px]]<br />
<br />
===Memory===<br />
<br />
The second subsystem implements the memory. The memory control is based on the following underlying mechanisms:<br />
* The sensor proteins form complexes together with the inducers. These complexes are used to activate the transcription of the genes for the proteins cI and p22cII. <br />
* p22cII and cI repress the DNA transcription of each other, so that the closed loop system behaves as a toggle; a dynamic system with only two possible steady states (see Fig. 6).<br />
<br />
[[Image:ETHZModelLearning.png|center|thumb|<b>Fig. 5</b>: Learning system: Depending on the inputs IPTG or aTc the proteins cI and p22cII are produced.|300px]]<br />
<br />
* Fig. 5 shows the protein production system that is used during the learning phase. During the learning phase, there is still no cI or p22cII produced. They are produced, only if either IPTG or aTc is added, respectively. Since no AHL is present, the inner toggle switch (see Figure 6) is turned off.<br />
<br />
[[Image:ETHZModelMemory.png|center|thumb|<b>Fig. 6</b>: Memory system. If AHL is present the production of either cI or p22cII is continued.|420px]]<br />
<br />
* During the memory phase, AHL is added and the IPTG and aTc are removed. That is why only the inner toggle switch (see Fig. 6) is turned on while the protein production systems shown in Fig. 5 are deactivated. Depending on what was produced during the learning phase, the production of either cI or p22cII is continued. That is why the system can act as memory, effectively storing the information it is exposed to.<br />
<br />
Based on all the above, we present the final assembly of the memory subsystem in Fig. 7.<br />
<br />
[[Image:Model02b.png|center|thumb|<b>Fig. 7</b>: Final interaction of the learning and memory system. The memory content is represented by the concentrations of the proteins cI and p22cII.|560px]]<br />
<br />
===Reporters===<br />
<br />
Fig. 8 gives an overview of the reporter subsystem. Florescent reporter proteins are expressed depending on the inducer concentrations, and the concentrations of cI and p22cII. For example, the presence of either TetR or cI will repress the production of YFP. However, if the inducer aTc is present, aTc will bind to TetR which can no longer block the production of YFP. We are using four fluorescent proteins, to encode the steady states of our system at the final recognition stage. In this way, we are able to distinguish between all the different transition paths in the system.<br />
<br />
[[Image:Model03b.png|center|thumb|<b>Fig. 8</b>: The production of the florescent reporter proteins depends on the memory content (cI or p22cII) and the current input (aTc or IPTG).|600px]]<br />
<br />
==Final Model==<br />
<br />
So far, we have presented all parts needed to model and simulate the behavior of the proposed system with. By following the details presented in the previous section, we have all the necessary information to fully understand the interior of the black boxes that were presented in Fig. 2 and Fig. 3. Our overall system model is presented in Fig. 9.<br />
<br />
[[Image:ETHZFullsystem.png|center|thumb|<b>Fig. 9</b>: Final model of the educatETH <i>E. coli</i> system.|900px]]<br />
<br />
In contrast to this biological implementation, an alternative implementation using an engineering approach can be found on our [[ETHZ/FlipFlop | 'Engineer's View' page]].<br />
<br />
==Mathematical Model==<br />
<br />
Based on the modeling done so far, we can derive the equations that govern the behavior of our system. The model is given by sets of coupled [[ETHZ/Modeling_Basics | ordinary differential equations]] which are presented below. We use a simple notation for the different elements of the equations. Namely: <br />
* All concentrations are given in brackets (for example [cI]). <br />
* All decay constants are described by a variable d followed by the name of the protein they refer to. <br />
* The production of the proteins is described by a basic constant production level named 'a' that models the leak of the production system, and a factor of l and c<sub>max</sub> that describe the maximum production of a protein, given in [M/min]. <br />
* Depending on whether the DNA for a protein is implemented on a low or a high copy plasmid, we distinguish between l<sub>lo</sub> and l<sub>hi</sub>, respectively.<br />
* Dissociation constants are given by 'K' followed by the name of the protein they refer to.<br />
* The Hill cooperabilities are described by the constants 'n' followed by the name of the protein they refer to.<br />
<br />
For a more detailed introduction into how we transferred our model into equations, see the section [[ETHZ/Modeling_Basics|Modeling Basics]].<br><br />
<br />
===Allosteric regulation===<br />
<br />
These equations describe the formation of complexes between the inducers and sensor proteins. We do not use differential equations, but describe directly the concentrations of the complexes. This is a valid assumption, provided that we always wait a sufficient time, and the system reaches a steady state.<br />
We describe the total amount of proteins with the index 't', while we use the index '*' for proteins that build a complex with their respective inducer. For example: <br />
* [TetR<sub>t</sub>] describes the total concentration of TetR that is available. It is the sum of the free TetR proteins and the TetR proteins that form a complex with aTc. <br />
* [TetR<sub>*</sub>] describes the proteins that are available as a complex with aTc, and<br />
* [TetR] gives the concentration of free TetR proteins.<br />
<br />
[[Image:Eq04.png|238px]]<br />
<br />
===Constitutively produced proteins===<br />
<br />
The differential equations for the constitutively produced proteins are very simple, since there is no dependence on other proteins. They are designed so that the protein concentration reaches the value l<sub>hi</sub>*c<sub>max</sub>/d at steady state.<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
===Learning and memory subsystem===<br />
<br />
The learning and memory subsystem is the core of the system that we are trying to model and implement. It is characterized by the feedback between its state variables/proteins cI and p22cII. Its behavior is further complicated by the variation of the production of the aforementioned proteins because of the inputs. The following equations describe the concentrations of the memory proteins as a system of coupled differential equations. The equations consist of two major production parts and a decay part. <br />
* The first production part models the production of either cI or p22cII during the learning phase, and corresponds to the model in Fig. 5.<br />
* The second production part describes the inner toggle switch that was shown in Fig. 6.<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
===Reporting subsystem===<br />
<br />
The equations for the reporting subsystem finally describe the production of the florescence proteins depending on the inputs and memory proteins as modeled in Figure 8. Note that both the free constitutively produced proteins and the memory proteins repress the production of the florescence proteins. So e.g. YFP is only produced when there is both no cI and all TetR is bound in a complex together with aTc.<br />
<br />
[[Image:Reporter_braced.png|778px]]<br />
<br />
The systems of equations presented above describe and predict the behavior of our system. We have simulated the behavior of our system at steady states, and the results can be seen in the section [[ETHZ/Simulation|Simulations]]. In order to increase the accuracy of our results, we conducted an extensive literature survey, in order to isolate and find the parameters of our system. Since this is a burden for every team undertaking a complicated project in synthetic biology, we are presenting our full table of parameters in the [[ETHZ/Parameters|Parameters]] page.<br />
<br />
== References ==<br />
<p><br />
[http://www.nature.com/nature/journal/v403/n6767/abs/403339a0.html &#91;1&#93; Gardner TS, Cantor CR and Collins JJ] <i>"Construction of a genetic toggle switch in Escherichia coli"</i>, Nature 403:339–342, 2000<br /></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/ModelETHZ/Model2007-10-25T13:05:14Z<p>Uhrm: /* Allosteric regulation */</p>
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<br />
=Introduction=<br />
<br />
As previously discussed in the main page, we are interested in designing a system that is able to adapt to its environment. Our ideas are based on discussions about neural networks, and how we can create a biological system that exhibits the behavior of learning without having to resort to evolutionary processes. <br />
<br />
[[Image:ETHzFlowdiagram2.png|thumb|<b>Fig. 1</b>: Flow diagram. This figure shows the protocol with which the final system should be tested as well as the test results in form of the reported colors. The protocol is divided into three phases: (1) a training or learning phase in which the system learns an input and stores it in its memory, (2) a memory phase in which the system keeps the content of its memory, and finally (3) a recognition phase where the output of the system depends on the content of its memory as well as the current input.|450px]]<br />
<br />
Learning can be considered as a switching of behavior, based on some external stimuli. Thus, it comes naturally to work on existing ideas of toggle switches and [[ETHZ/FSM | finite state machines]]. <br />
<br />
The proposed system is able to distinguish between two chemicals. It represents a minimal test system that is intended as a proof of concept. By introducing the ability to distinguish more than two chemical and thereby introducing new [[ETHZ/FSM | system states]], the power of the system or its "intelligence" can be scaled. A protocol depicting how the system should react according to an input is shown in Fig. 1.<br />
<br />
The idea behind this protocol is that:<br />
* The system will be able to learn one of two input signals - aTc or IPTG - during a learning phase, while a "learning signal" (AHL) is not yet present. Depending on the input it will report by producing either cyan or yellow florescence. <br />
* Once the system has learned, the inputs - aTc or IPTG - can be removed and the system goes into a memory state in the presence of AHL. In this state, no output color is reported. Memorizing is guaranteed by removing the input chemicals.<br />
* During the recognition phase, the inputs aTc or IPTG are (re-)inserted. The system reports by changing its color depending on the input and its current memory state. This is why the system has different florescence properties even in the presence of the same input. The recognition phase takes place in the presence of AHL, to keep the memory enabled and avoid another learning phase. Since we would like to separate four different end states, we use four different fluorescent proteins to encode them.<br />
<br />
==Model Overview==<br />
<br />
The model for the proposed system is developed using a top-down approach. We start with a black box approach as shown in Fig. 2. <br />
<br />
[[Image:ETHZBlackbox.png|thumb|<b>Fig. 2</b>: Black box |280px]]<br />
<br />
The system is sketched in Fig. 3. It can be summarized as follows:<br />
* There are two inputs to be learned/detected/adapted to.<br />
* There is one separate input to switch on the memory.<br />
* The system has to alternate between at least three states. Hence, we decided to use two state variables - cI and p22cII (when interpreted as binary variables, in principle allowing for up to four different states).<br />
* There are four different output signals (synthesis of four fluorescent proteins). One could also decide to take six output signals into account to further distinguish the learning phase from the recognition phase. However, we restricted ourselves to four outputs to reduce the number of genes that are needed to implement the signals.<br />
<br />
[[Image:ETHZFullsystemmodel.png|left|thumb|<b>Fig. 3</b>: System overview. AHL, IPTG and aTc pass the cell membrane where they build complexes with the sensor proteins LuxR, LacI and TetR. These sensor proteins and/or complexes are used to control the internal system state: the memory represented by the proteins cI and p22cII (mutually repressing their synthesis) and the sensed input (IPTG, aTc). CFP, RFP, GFP and YFP stand for yellow, red, cyan and green florescent protein, respectively.|420px]]<br />
<br />
However, we had to keep in mind that the proposed system should be implemented in DNA, and that it would be sensitive to noise. As a result, we took several actions to achieve better experimental results and easier DNA construction:<br />
* To be more robust against perturbations, we coupled the state variables cI and p22cII like it is known from toggle switches [1]. Based on this approach, one state variable is depressing the other one, and the system's internal toggle has the possibility of reaching two stable states.<br />
* Since - due to their size - proteins can only hardly pass the cell membrane (if they are not actively transported through the cell membrane), we decided to use the much smaller inducer molecules AHL, IPTG and aTc as inputs. However, since these inducers cannot directly act on the transcription of the DNA nor on the production of proteins, we need to produce the sensor proteins LuxR, LacI and TetR that build complexes with AHL, IPTG and aTc, respectively.<br />
* The sensor proteins and complexes are used to control the memory formation and the production of the florescent reporter proteins CFP, RFP, GFP and YFP.<br><br><br />
<br />
==Detailed Model==<br />
<br />
In order to test our ideas, we came up with a detailed model of all the interactions in the system. <br />
After defining the desired behavior of our system (as shown in the introduction) and a [[ETHZ/FSM | formalized desciption of the system]] we identified necessary biological components and their interactions. As we can observe in Fig. 3, our system is composed from three basic subparts:<br />
* sensors,<br />
* memory, and<br />
* reporters.<br />
<br />
===Sensors===<br />
<br />
The first part contains the sensors. Our sensors are the proteins LacI, luxR and TetR, which are constitutively produced. The sensing subsystem is shown in Fig. 4.<br />
<br />
[[Image:Model01b.png|center|thumb|<b>Fig. 4</b>: The proteins that act as sensors are constitutively produced.|140px]]<br />
<br />
===Memory===<br />
<br />
The second subsystem implements the memory. The memory control is based on the following underlying mechanisms:<br />
* The sensor proteins form complexes together with the inducers. These complexes are used to activate the transcription of the genes for the proteins cI and p22cII. <br />
* p22cII and cI repress the DNA transcription of each other, so that the closed loop system behaves as a toggle; a dynamic system with only two possible steady states (see Fig. 6).<br />
<br />
[[Image:ETHZModelLearning.png|center|thumb|<b>Fig. 5</b>: Learning system: Depending on the inputs IPTG or aTc the proteins cI and p22cII are produced.|300px]]<br />
<br />
* Fig. 5 shows the protein production system that is used during the learning phase. During the learning phase, there is still no cI or p22cII produced. They are produced, only if either IPTG or aTc is added, respectively. Since no AHL is present, the inner toggle switch (see Figure 6) is turned off.<br />
<br />
[[Image:ETHZModelMemory.png|center|thumb|<b>Fig. 6</b>: Memory system. If AHL is present the production of either cI or p22cII is continued.|420px]]<br />
<br />
* During the memory phase, AHL is added and the IPTG and aTc are removed. That is why only the inner toggle switch (see Fig. 6) is turned on while the protein production systems shown in Fig. 5 are deactivated. Depending on what was produced during the learning phase, the production of either cI or p22cII is continued. That is why the system can act as memory, effectively storing the information it is exposed to.<br />
<br />
Based on all the above, we present the final assembly of the memory subsystem in Fig. 7.<br />
<br />
[[Image:Model02b.png|center|thumb|<b>Fig. 7</b>: Final interaction of the learning and memory system. The memory content is represented by the concentrations of the proteins cI and p22cII.|560px]]<br />
<br />
===Reporters===<br />
<br />
Fig. 8 gives an overview of the reporter subsystem. Florescent reporter proteins are expressed depending on the inducer concentrations, and the concentrations of cI and p22cII. For example, the presence of either TetR or cI will repress the production of YFP. However, if the inducer aTc is present, aTc will bind to TetR which can no longer block the production of YFP. We are using four fluorescent proteins, to encode the steady states of our system at the final recognition stage. In this way, we are able to distinguish between all the different transition paths in the system.<br />
<br />
[[Image:Model03b.png|center|thumb|<b>Fig. 8</b>: The production of the florescent reporter proteins depends on the memory content (cI or p22cII) and the current input (aTc or IPTG).|600px]]<br />
<br />
==Final Model==<br />
<br />
So far, we have presented all parts needed to model and simulate the behavior of the proposed system with. By following the details presented in the previous section, we have all the necessary information to fully understand the interior of the black boxes that were presented in Fig. 2 and Fig. 3. Our overall system model is presented in Fig. 9.<br />
<br />
[[Image:ETHZFullsystem.png|center|thumb|<b>Fig. 9</b>: Final model of the educatETH <i>E. coli</i> system.|900px]]<br />
<br />
In contrast to this biological implementation, an alternative implementation using an engineering approach can be found on our [[ETHZ/FlipFlop | 'Engineer's View' page]].<br />
<br />
==Mathematical Model==<br />
<br />
Based on the modeling done so far, we can derive the equations that govern the behavior of our system. The model is given by sets of coupled [[ETHZ/Modeling_Basics | ordinary differential equations]] which are presented below. We use a simple notation for the different elements of the equations. Namely: <br />
* All concentrations are given in brackets (for example [cI]). <br />
* All decay constants are described by a variable d followed by the name of the protein they refer to. <br />
* The production of the proteins is described by a basic constant production level named 'a' that models the leak of the production system, and a factor of l and c<sub>max</sub> that describe the maximum production of a protein, given in [M/min]. <br />
* Depending on whether the DNA for a protein is implemented on a low or a high copy plasmid, we distinguish between l<sub>lo</sub> and l<sub>hi</sub>, respectively.<br />
* Dissociation constants are given by 'K' followed by the name of the protein they refer to.<br />
* The Hill cooperabilities are described by the constants 'n' followed by the name of the protein they refer to.<br />
<br />
For a more detailed introduction into how we transferred our model into equations, see the section [[ETHZ/Modeling_Basics|Modeling Basics]].<br><br />
<br />
===Allosteric regulation===<br />
<br />
These equations describe the formation of complexes between the inducers and sensor proteins. We do not use differential equations, but describe directly the concentrations of the complexes. This is a valid assumption, provided that we always wait a sufficient time, and the system reaches a steady state.<br />
We describe the total amount of proteins with the index 't', while we use the index '*' for proteins that build a complex with their respective inducer. For example: <br />
* [TetR<sub>t</sub>] describes the total concentration of TetR that is available. It is the sum of the free TetR proteins and the TetR proteins that form a complex with aTc. <br />
* [TetR<sub>*</sub>] describes the proteins that are available as a complex with aTc, and<br />
* [TetR] gives the concentration of free TetR proteins.<br />
<br />
[[Image:Eq04.png|238px]]<br />
<br />
===Constitutively produced proteins===<br />
<br />
The differential equations for the constitutively produced proteins are very simple, since there is no dependence on other proteins. They are designed so that the protein concentration reaches the value l<sub>hi</sub>*c<sub>max</sub>/d at steady state.<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
===Learning and memory subsystem===<br />
<br />
The learning and memory subsystem is the core of the system that we are trying to model and implement. It is characterized by the feedback between its state variables/proteins cI and p22cII. Its behavior is further complicated by the variation of the production of the aforementioned proteins because of the inputs. The following equations describe the concentrations of the memory proteins as a system of coupled differential equations. The equations consist of two major production parts and a decay part. <br />
* The first production part models the production of either cI or p22cII during the learning phase, and corresponds to the model in Fig. 5.<br />
* The second production part describes the inner toggle switch that was shown in Fig. 6.<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
===Reporting subsystem===<br />
<br />
The equations for the reporting subsystem finally describe the production of the florescence proteins depending on the inputs and memory proteins as modeled in Figure 8. Note that both the free constitutively produced proteins and the memory proteins repress the production of the florescence proteins. So e.g. YFP is only produced when there is both no cI and all TetR is bound in a complex together with aTc.<br />
<br />
[[Image:Reporter_braced.png|778px]]<br />
<br />
The systems of equations presented above describe and predict the behavior of our system. We have simulated the behavior of our system at steady states, and the results can be seen in the section [[ETHZ/Simulation|Simulations]]. In order to increase the accuracy of our results, we conducted an extensive literature survey, in order to isolate and find the parameters of our system. Since this is a burden for every team undertaking a complicated project in synthetic biology, we are presenting our full table of parameters in the [[ETHZ/Parameters|Parameters]] page.<br />
<br />
== References ==<br />
<p><br />
[http://www.nature.com/nature/journal/v403/n6767/abs/403339a0.html &#91;1&#93; Gardner TS, Cantor CR and Collins JJ] <i>"Construction of a genetic toggle switch in Escherichia coli"</i>, Nature 403:339–342, 2000<br /></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-25T12:32:46Z<p>Uhrm: /* Inducer Molecules */</p>
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<br />
=Modeling Basics=<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E.coli'' cells, based on genes that we introduced. To understand the system, it is crucial to model the gene expression, gene regulation, and the resulting gene product concentrations accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple case, proteins are produced through continuous transcription of genes. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|<b>Fig. 1</b>: A system constitutively producing protein P. The production rate is c<sup>max</sup> and the degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equation] (ODE):<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer: The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|<b>Fig. 2</b>: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
Often, genes are not constitutively expressed but their expression depends on the presence of other proteins R (i.e., transcription factors). These transcription factors can activate or inhibit the promoter of the gene in question. To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. Example systems for inhibition and activation of a promoter by R are given in Fig. 3 and Fig.4, respectively.<br />
<br />
[[Image:basic_fig03.png|thumb|<b>Fig. 3</b>: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|<b>Fig. 4</b>: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model assumes that the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, forward and backward reaction rates are identical, so we can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (i.e., there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, all elements are in place to write down an ODE for the concentration of protein P whose expression is regulated by the regulatory protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start with the same assumption: a transcription factor reversibly binding to DNA.<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Here, we are interested in the DNA - transcription factor complex because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Again assuming equilibrium conditions, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
== Basic Production ==<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|<b>Fig. 5</b>: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place. For inhibition we have<br />
<br />
[[Image:basic_eq08.png|center|361px]]<br />
<br />
and for activation<br />
<br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'' which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation of transcription by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.<br />
<br />
== Inducer Molecules ==<br />
<br />
A problem in biology is that regulatory proteins (such as protein R in the previous sections) cannot be used as system inputs directly. It is not possible to add such proteins to an assay to steer the behavior of a biological system because these proteins are big and do not diffuse through cell walls. Therefore, they cannot enter the cells.<br />
<br />
To circumvent this limitation, one has to produce these proteins directly in the cell. But then, one needs a possibility to switch the functionality of these proteins on and off. This is where inducer molecules become useful. Inducer molecules are small molecules that can diffuse through cell walls freely. Furthermore, they are able to bind to the regulatory proteins and switch the functionality on or off.<br />
<br />
Thus, we need a model to describe the binding of the inducer to the protein. We again assume that the inducer I binds reversibly to the protein R<br />
<br />
[[Image:basic_eq10.png|center]]<br />
<br />
Further on, again we assume that the reaction is in equilibrium. We thus have<br />
<br />
[[Image:basic_eq11.png|center|157px]]<br />
<br />
Depending on the species at hand we are either interested in the protein-inducer complex concentration or in the concentration of 'free protein'. For some species (e.g., LuxR), the complex acts as an activator. For others, complex formation relieves repression (e.g., TetR).<br />
<br />
It is again possible to derive a formula for both, the concentration of free protein R and the complex concentration R-nI as a function of total inducer concentration (for notational convenience, we write R* for the complex R-nI in the following equations).<br />
<br />
[[Image:basic_eq12.png|center|157px]]<br />
<br />
<br><br />
<br />
[[Image:basic_eq13.png|center|157px]]<br />
<br />
== Caveats ==<br />
<br />
With this approach to modeling, the equations turn out to be both, easy to understand and to simulate. But to arrive at this point, a certain number of assumptions must be made. This section points out possible problems with these assumptions and ideas to further improve the modeling.<br />
<br />
* '''Equilibrium''': This is a very basic assumption that was the basis of all results in the above discussion. At the same time, it is also the least problematic assumption. This can be seen by remembering the massive machinery involved in transcription and translation of DNA/RNA compared to simple reversible binding of molecules. It is plausible to assume that the latter indeed happens on a much shorter time scale than the first.<br />
<br />
* '''Excess of substrate''': This assumption was implicitly made when we wrote down the equations of regulation and inducer binding. The basic idea is that the total inducer concentration is very close to the concentration of unbound inducer. Mathematically more precise, this condition is met when K<sub>I</sub> &gt;&gt; [R]<sub>t</sub>. As the uncertainty in the values of dissociation constants and steady state concentrations is sometimes very large, it is difficult to say whether this assumption is justified.<br />
<br />
* '''Mechanism dependence''': If the [http://en.wikipedia.org/wiki/Hill_coefficient Hill cooperativity coefficient ]''n'' is greater than one, the above formulas become in fact dependent on the mechanism of the binding process. In these cases, the formulas can be off up to a factor ''n''. As the true mechanism of binding is often unknown, it is practically impossible to take this fact into account.<br />
<br />
* '''Low protein concentration'''. Whenever the concentration is very low (say, below 100 nM in the case of ''E.coli''), the number of molecules per cell becomes small. In this situation, the assumptions behind the ODE modeling approach (e.g., well mixed compartment where each molecule can freely interact with other molecules) are not met any more and simulation results become inaccurate. Then, one would have to resort to stochastic simulations.</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-25T12:30:25Z<p>Uhrm: /* Inhibition */</p>
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=Modeling Basics=<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E.coli'' cells, based on genes that we introduced. To understand the system, it is crucial to model the gene expression, gene regulation, and the resulting gene product concentrations accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple case, proteins are produced through continuous transcription of genes. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|<b>Fig. 1</b>: A system constitutively producing protein P. The production rate is c<sup>max</sup> and the degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equation] (ODE):<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer: The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|<b>Fig. 2</b>: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
Often, genes are not constitutively expressed but their expression depends on the presence of other proteins R (i.e., transcription factors). These transcription factors can activate or inhibit the promoter of the gene in question. To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. Example systems for inhibition and activation of a promoter by R are given in Fig. 3 and Fig.4, respectively.<br />
<br />
[[Image:basic_fig03.png|thumb|<b>Fig. 3</b>: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|<b>Fig. 4</b>: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model assumes that the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, forward and backward reaction rates are identical, so we can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (i.e., there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, all elements are in place to write down an ODE for the concentration of protein P whose expression is regulated by the regulatory protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start with the same assumption: a transcription factor reversibly binding to DNA.<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Here, we are interested in the DNA - transcription factor complex because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Again assuming equilibrium conditions, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
== Basic Production ==<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|<b>Fig. 5</b>: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place. For inhibition we have<br />
<br />
[[Image:basic_eq08.png|center|361px]]<br />
<br />
and for activation<br />
<br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'' which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation of transcription by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.<br />
<br />
== Inducer Molecules ==<br />
<br />
A problem in biology is that regulatory proteins (such as protein R in the previous sections) cannot be used as system inputs directly. It is not possible to add such proteins to an assay to steer the behavior of a biological system because these proteins are big and do not diffuse through cell walls. Therefore, they cannot enter the cells.<br />
<br />
To circumvent this limitation, one has to produce these proteins directly in the cell. But then, one needs a possibility to switch the functionality of these proteins on and off. This is where inducer molecules become useful. Inducer molecules are small molecules that can diffuse through cell walls freely. Furthermore, they are able to bind to the regulatory proteins and switch the functionality on or off.<br />
<br />
Thus, we need a model to describe the binding of the inducer to the protein. We again assume that the inducer I binds reversibly to the protein R<br />
<br />
[[Image:basic_eq10.png|center]]<br />
<br />
Further on, again we assume that the reaction is in equilibrium. We thus have<br />
<br />
[[Image:basic_eq11.png|center|157px]]<br />
<br />
Depending on the species at hand we are either interested in the protein-inducer complex concentration or in the concentration of 'free protein'. For some species (e.g., LuxR), the complex form is functional and for others it is the free protein (e.g., TetR).<br />
<br />
It is again possible to derive a formula for both, the concentration of free protein R and the complex concentration R-nI as a function of total inducer concentration (for notational convenience, we write R* for the complex R-nI in the following equations).<br />
<br />
[[Image:basic_eq12.png|center|157px]]<br />
<br />
<br><br />
<br />
[[Image:basic_eq13.png|center|157px]]<br />
<br />
<br />
== Caveats ==<br />
<br />
With this approach to modeling, the equations turn out to be both, easy to understand and to simulate. But to arrive at this point, a certain number of assumptions must be made. This section points out possible problems with these assumptions and ideas to further improve the modeling.<br />
<br />
* '''Equilibrium''': This is a very basic assumption that was the basis of all results in the above discussion. At the same time, it is also the least problematic assumption. This can be seen by remembering the massive machinery involved in transcription and translation of DNA/RNA compared to simple reversible binding of molecules. It is plausible to assume that the latter indeed happens on a much shorter time scale than the first.<br />
<br />
* '''Excess of substrate''': This assumption was implicitly made when we wrote down the equations of regulation and inducer binding. The basic idea is that the total inducer concentration is very close to the concentration of unbound inducer. Mathematically more precise, this condition is met when K<sub>I</sub> &gt;&gt; [R]<sub>t</sub>. As the uncertainty in the values of dissociation constants and steady state concentrations is sometimes very large, it is difficult to say whether this assumption is justified.<br />
<br />
* '''Mechanism dependence''': If the [http://en.wikipedia.org/wiki/Hill_coefficient Hill cooperativity coefficient ]''n'' is greater than one, the above formulas become in fact dependent on the mechanism of the binding process. In these cases, the formulas can be off up to a factor ''n''. As the true mechanism of binding is often unknown, it is practically impossible to take this fact into account.<br />
<br />
* '''Low protein concentration'''. Whenever the concentration is very low (say, below 100 nM in the case of ''E.coli''), the number of molecules per cell becomes small. In this situation, the assumptions behind the ODE modeling approach (e.g., well mixed compartment where each molecule can freely interact with other molecules) are not met any more and simulation results become inaccurate. Then, one would have to resort to stochastic simulations.</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-25T12:29:36Z<p>Uhrm: /* Inhibition */</p>
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__NOTOC__<br />
<br />
=Modeling Basics=<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E.coli'' cells, based on genes that we introduced. To understand the system, it is crucial to model the gene expression, gene regulation, and the resulting gene product concentrations accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple case, proteins are produced through continuous transcription of genes. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|<b>Fig. 1</b>: A system constitutively producing protein P. The production rate is c<sup>max</sup> and the degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as [http://en.wikipedia.org/wiki/Ordinary_differential_equation ordinary differential equation] (ODE):<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer: The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|<b>Fig. 2</b>: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
Often, genes are not constitutively expressed but their expression depends on the presence of other proteins R (i.e., transcription factors). These transcription factors can activate or inhibit the promoter of the gene in question. To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. Example systems for inhibition and activation of a promoter by R are given in Fig. 3 and Fig.4, respectively.<br />
<br />
[[Image:basic_fig03.png|thumb|<b>Fig. 3</b>: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|<b>Fig. 4</b>: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model assumes that the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, forward and backward reaction rates are identical, so we can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (i.e., there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we basically have everything together to write down an ODE for the concentration of protein P whose expression is regulated by the regulatory protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start with the same assumption: a transcription factor reversibly binding to DNA.<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Here, we are interested in the DNA - transcription factor complex because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Again assuming equilibrium conditions, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
== Basic Production ==<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|<b>Fig. 5</b>: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place. For inhibition we have<br />
<br />
[[Image:basic_eq08.png|center|361px]]<br />
<br />
and for activation<br />
<br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'' which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation of transcription by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.<br />
<br />
== Inducer Molecules ==<br />
<br />
A problem in biology is that regulatory proteins (such as protein R in the previous sections) cannot be used as system inputs directly. It is not possible to add such proteins to an assay to steer the behavior of a biological system because these proteins are big and do not diffuse through cell walls. Therefore, they cannot enter the cells.<br />
<br />
To circumvent this limitation, one has to produce these proteins directly in the cell. But then, one needs a possibility to switch the functionality of these proteins on and off. This is where inducer molecules become useful. Inducer molecules are small molecules that can diffuse through cell walls freely. Furthermore, they are able to bind to the regulatory proteins and switch the functionality on or off.<br />
<br />
Thus, we need a model to describe the binding of the inducer to the protein. We again assume that the inducer I binds reversibly to the protein R<br />
<br />
[[Image:basic_eq10.png|center]]<br />
<br />
Further on, again we assume that the reaction is in equilibrium. We thus have<br />
<br />
[[Image:basic_eq11.png|center|157px]]<br />
<br />
Depending on the species at hand we are either interested in the protein-inducer complex concentration or in the concentration of 'free protein'. For some species (e.g., LuxR), the complex form is functional and for others it is the free protein (e.g., TetR).<br />
<br />
It is again possible to derive a formula for both, the concentration of free protein R and the complex concentration R-nI as a function of total inducer concentration (for notational convenience, we write R* for the complex R-nI in the following equations).<br />
<br />
[[Image:basic_eq12.png|center|157px]]<br />
<br />
<br><br />
<br />
[[Image:basic_eq13.png|center|157px]]<br />
<br />
<br />
== Caveats ==<br />
<br />
With this approach to modeling, the equations turn out to be both, easy to understand and to simulate. But to arrive at this point, a certain number of assumptions must be made. This section points out possible problems with these assumptions and ideas to further improve the modeling.<br />
<br />
* '''Equilibrium''': This is a very basic assumption that was the basis of all results in the above discussion. At the same time, it is also the least problematic assumption. This can be seen by remembering the massive machinery involved in transcription and translation of DNA/RNA compared to simple reversible binding of molecules. It is plausible to assume that the latter indeed happens on a much shorter time scale than the first.<br />
<br />
* '''Excess of substrate''': This assumption was implicitly made when we wrote down the equations of regulation and inducer binding. The basic idea is that the total inducer concentration is very close to the concentration of unbound inducer. Mathematically more precise, this condition is met when K<sub>I</sub> &gt;&gt; [R]<sub>t</sub>. As the uncertainty in the values of dissociation constants and steady state concentrations is sometimes very large, it is difficult to say whether this assumption is justified.<br />
<br />
* '''Mechanism dependence''': If the [http://en.wikipedia.org/wiki/Hill_coefficient Hill cooperativity coefficient ]''n'' is greater than one, the above formulas become in fact dependent on the mechanism of the binding process. In these cases, the formulas can be off up to a factor ''n''. As the true mechanism of binding is often unknown, it is practically impossible to take this fact into account.<br />
<br />
* '''Low protein concentration'''. Whenever the concentration is very low (say, below 100 nM in the case of ''E.coli''), the number of molecules per cell becomes small. In this situation, the assumptions behind the ODE modeling approach (e.g., well mixed compartment where each molecule can freely interact with other molecules) are not met any more and simulation results become inaccurate. Then, one would have to resort to stochastic simulations.</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/ParametersETHZ/Parameters2007-10-23T12:48:08Z<p>Uhrm: /* Model Parameters */</p>
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<br />
= Parameters for the EducatETH <i>E. coli</i> system =<br />
<br />
<p><br />
In order to provide as realistic simulation results as possible, and to find good estimates for the simulation parameters, we performed an intensive literature review. However, not all parameters could be found in the literature. Furthermore, one has to take into account that biological parameters cannot be estimated to a very high precision.<br />
</p><br><br />
<br />
== Model Parameters ==<br />
<br />
=== General parameters ===<br />
{| class="wikitable" width="100%" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
!width="10%"| Parameter <br />
!width="10%"| Value<br />
!width="20%"| Description<br />
!width="60%"| Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of LuxR-activated promoter (per gene)<br />
| Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Estimate<br />
|-<br />
| a<br />
| 1%<br />
| basic production levels<br />
| Estimate<br />
|-<br />
|}<br />
<br />
=== Degradation constants ===<br />
{| class="wikitable" width="100%" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
!width="10%"| Parameter <br />
!width="10%"| Value<br />
!width="20%"| Description<br />
!width="60%"| Comments<br />
|-<br />
| d<sub>LacI</sub><br />
| 2.31e-3 [1/s]<br />
| degradation of LacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>TetR</sub><br />
| <br />
*1e-5 [1/s]<br />
*2.31e-3 [1/s]<br />
| degradation of TetR<br />
| <br />
*Ref. [9]<br />
*Ref. [10]<br />
|-<br />
| d<sub>LuxR</sub><br />
| 1e-3 - 1e-4 [1/s]<br />
| degradation of LuxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>CI</sub><br />
| 7e-4 [1/s]<br />
| degradation of CI<br />
| Ref. [7]<br />
|-<br />
| d<sub>P22CII</sub><br />
| <br />
| degradation of P22CII<br />
| <br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
|}<br />
<br />
=== Dissociation constants ===<br />
<br />
{| class="wikitable" width="100%" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
!width="10%"| Parameter <br />
!width="10%"| Value<br />
!width="20%"| Description<br />
!width="60%"| Comments<br />
|-<br />
| K<sub>LacI</sub><br />
| <br />
* 0.1 - 1 [pM]<br />
* 800 [nM]<br />
| LacI repressor dissociation constant<br />
| <br />
* Ref. [2]<br />
* Ref. [12]<br />
|-<br />
| K<sub>IPTG</sub><br />
| 1.3 [&#181;M]<br />
| IPTG-LacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>TetR</sub><br />
| 179 [pM]<br />
| TetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>ATC</sub><br />
| 893 [pM]<br />
| ATC-TetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>LuxR</sub><br />
| 55 - 520 [nM]<br />
| LuxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>AHL</sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-LuxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>CI</sub><br />
|<br />
*8 [pM]<br />
*50 [nM]<br />
| CI repressor dissociation constant<br />
|<br />
*Ref. [12]<br />
*starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>P22CII</sub><br />
| 0.577 [&#181;M]<br />
| P22CII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein CII and we have P22CII. Does that match?<br />
|-<br />
|}<br />
<br />
=== Hill cooperativity ===<br />
{| class="wikitable" width="100%" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
!width="10%"| Parameter <br />
!width="10%"| Value<br />
!width="20%"| Description<br />
!width="60%"| Comments<br />
|-<br />
| n<sub>LacI</sub><br />
| <br />
* 1<br />
* 2<br />
| LacI repressor Hill cooperativity<br />
| <br />
* Ref. [5]<br />
* Ref. [12]<br />
|-<br />
| n<sub>IPTG</sub><br />
| 2<br />
| IPTG-LacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>TetR</sub><br />
| 3<br />
| TetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>ATC</sub><br />
| 2 (1.5-2.5)<br />
| ATC-TetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>LuxR</sub><br />
| 2<br />
| LuxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>AHL</sub><br />
| 1<br />
| AHL-LuxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>CI</sub><br />
| 2<br />
| CI repressor Hill cooperativity<br />
| Ref. [12]<br />
|-<br />
| n<sub>P22CII</sub><br />
| 4<br />
| P22CII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein CII and we have P22CII. Does that match?<br />
|-<br />
|}<br />
<br />
<br><br />
<br />
== References ==<br />
<p><br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/ParametersETHZ/Parameters2007-10-23T12:28:12Z<p>Uhrm: /* Model Parameters */</p>
<hr />
<div><center>[[Image:Eth_zh_logo_4.png|830px]]</center><br />
<br />
<center>[[ETHZ | Main Page]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Model | System Modeling]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Simulation | Simulations]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology | System Implementation]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology/Lab| Lab Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Meet_the_team | Meet the Team]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Internal | Team Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Pictures | Pictures!]]</center><br><br />
<br />
__NOTOC__<br />
<br />
= Parameters for the EducatETH <i>E. coli</i> system =<br />
<br />
<p><br />
In order to provide as realistic simulation results as possible, and to find good estimates for the simulation parameters, we performed an intensive literature review. However, not all parameters could be found in the literature. Furthermore, one has to take into account that biological parameters cannot be estimated to a very high precision.<br />
</p><br><br />
<br />
== Model Parameters ==<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
!<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Estimate<br />
|<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of LuxR-activated promoter (per gene)<br />
| Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Estimate<br />
|<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Estimate<br />
|-<br />
| a<br />
| 1%<br />
| basic production levels<br />
| Estimate<br />
|<br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Degradation constants <br />
|<br />
| <br />
| <br />
| <br />
|<br />
|<br />
| <br />
|-<br />
| d<sub>LacI</sub><br />
| 2.31e-3 [1/s]<br />
| degradation of LacI<br />
| Ref. [10]<br />
|<br />
| d<sub>TetR</sub><br />
| <br />
*1e-5 [1/s]<br />
*2.31e-3 [1/s]<br />
| degradation of TetR<br />
| <br />
*Ref. [9]<br />
*Ref. [10]<br />
|-<br />
| d<sub>LuxR</sub><br />
| 1e-3 - 1e-4 [1/s]<br />
| degradation of LuxR<br />
| Ref: [6]<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| d<sub>CI</sub><br />
| 7e-4 [1/s]<br />
| degradation of CI<br />
| Ref. [7]<br />
|<br />
| d<sub>P22CII</sub><br />
| <br />
| degradation of P22CII<br />
| <br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| Dissociation constants<br />
| <br />
| <br />
| <br />
|<br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| K<sub>LacI</sub><br />
| <br />
* 0.1 - 1 [pM]<br />
* 800 [nM]<br />
| LacI repressor dissociation constant<br />
| <br />
* Ref. [2]<br />
* Ref. [12]<br />
|<br />
| K<sub>IPTG</sub><br />
| 1.3 [&#181;M]<br />
| IPTG-LacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>TetR</sub><br />
| 179 [pM]<br />
| TetR repressor dissociation constant<br />
| Ref. [1]<br />
|<br />
| K<sub>ATC</sub><br />
| 893 [pM]<br />
| ATC-TetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>LuxR</sub><br />
| 55 - 520 [nM]<br />
| LuxR activator dissociation constant<br />
| Ref: [6]<br />
|<br />
| K<sub>AHL</sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-LuxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>CI</sub><br />
|<br />
*8 [pM]<br />
*50 [nM]<br />
| CI repressor dissociation constant<br />
|<br />
*Ref. [12]<br />
*starting with values of Ref. [6] and using Ref. [3]<br />
|<br />
| K<sub>P22CII</sub><br />
| 0.577 [&#181;M]<br />
| P22CII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein CII and we have P22CII. Does that match?<br />
|-<br />
|Hill cooperativity<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| n<sub>LacI</sub><br />
| <br />
* 1<br />
* 2<br />
| LacI repressor Hill cooperativity<br />
| <br />
* Ref. [5]<br />
* Ref. [12]<br />
|<br />
| n<sub>IPTG</sub><br />
| 2<br />
| IPTG-LacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>TetR</sub><br />
| 3<br />
| TetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|<br />
| n<sub>ATC</sub><br />
| 2 (1.5-2.5)<br />
| ATC-TetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>LuxR</sub><br />
| 2<br />
| LuxR activator Hill cooperativity<br />
| Ref: [6]<br />
|<br />
| n<sub>AHL</sub><br />
| 1<br />
| AHL-LuxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>CI</sub><br />
| 2<br />
| CI repressor Hill cooperativity<br />
| Ref. [12]<br />
|<br />
| n<sub>P22CII</sub><br />
| 4<br />
| P22CII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein CII and we have P22CII. Does that match?<br />
|-<br />
|}<br />
<br />
<br><br />
<br />
== References ==<br />
<p><br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/ParametersETHZ/Parameters2007-10-23T12:26:38Z<p>Uhrm: /* Model Parameters */</p>
<hr />
<div><center>[[Image:Eth_zh_logo_4.png|830px]]</center><br />
<br />
<center>[[ETHZ | Main Page]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Model | System Modeling]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Simulation | Simulations]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology | System Implementation]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology/Lab| Lab Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Meet_the_team | Meet the Team]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Internal | Team Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Pictures | Pictures!]]</center><br><br />
<br />
__NOTOC__<br />
<br />
= Parameters for the EducatETH <i>E. coli</i> system =<br />
<br />
<p><br />
In order to provide as realistic simulation results as possible, and to find good estimates for the simulation parameters, we performed an intensive literature review. However, not all parameters could be found in the literature. Furthermore, one has to take into account that biological parameters cannot be estimated to a very high precision.<br />
</p><br><br />
<br />
== Model Parameters ==<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
!<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Estimate<br />
|<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of LuxR-activated promoter (per gene)<br />
| Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Estimate<br />
|<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Estimate<br />
|-<br />
| a<br />
| 1%<br />
| basic production levels<br />
| Estimate<br />
|<br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| Degradation constants <br />
|<br />
| <br />
| <br />
| <br />
|<br />
|<br />
| <br />
|-<br />
| d<sub>LacI</sub><br />
| 2.31e-3 [1/s]<br />
| degradation of LacI<br />
| Ref. [10]<br />
|<br />
| d<sub>TetR</sub><br />
| <br />
*1e-5 [1/s]<br />
*2.31e-3 [1/s]<br />
| degradation of TetR<br />
| <br />
*Ref. [9]<br />
*Ref. [10]<br />
|-<br />
| d<sub>LuxR</sub><br />
| 1e-3 - 1e-4 [1/s]<br />
| degradation of LuxR<br />
| Ref: [6]<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| d<sub>CI</sub><br />
| 7e-4 [1/s]<br />
| degradation of CI<br />
| Ref. [7]<br />
|<br />
| d<sub>P22CII</sub><br />
| <br />
| degradation of P22CII<br />
| <br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [1/min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| Dissociation constants<br />
| <br />
| <br />
| <br />
|<br />
| <br />
| <br />
| <br />
| <br />
|-<br />
| K<sub>LacI</sub><br />
| 0.1 - 1 [pM]<br />
| LacI repressor dissociation constant<br />
| Ref. [2]<br />
|<br />
| K<sub>IPTG</sub><br />
| 1.3 [&#181;M]<br />
| IPTG-LacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>TetR</sub><br />
| 179 [pM]<br />
| TetR repressor dissociation constant<br />
| Ref. [1]<br />
|<br />
| K<sub>ATC</sub><br />
| 893 [pM]<br />
| ATC-TetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>LuxR</sub><br />
| 55 - 520 [nM]<br />
| LuxR activator dissociation constant<br />
| Ref: [6]<br />
|<br />
| K<sub>AHL</sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-LuxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>CI</sub><br />
|<br />
*8 [pM]<br />
*50 [nM]<br />
| CI repressor dissociation constant<br />
|<br />
*Ref. [12]<br />
*starting with values of Ref. [6] and using Ref. [3]<br />
|<br />
| K<sub>P22CII</sub><br />
| 0.577 [&#181;M]<br />
| P22CII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein CII and we have P22CII. Does that match?<br />
|-<br />
|Hill cooperativity<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
| n<sub>LacI</sub><br />
| <br />
* 1<br />
* 2<br />
| LacI repressor Hill cooperativity<br />
| <br />
* Ref. [5]<br />
* Ref. [12]<br />
|<br />
| n<sub>IPTG</sub><br />
| 2<br />
| IPTG-LacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>TetR</sub><br />
| 3<br />
| TetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|<br />
| n<sub>ATC</sub><br />
| 2 (1.5-2.5)<br />
| ATC-TetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>LuxR</sub><br />
| 2<br />
| LuxR activator Hill cooperativity<br />
| Ref: [6]<br />
|<br />
| n<sub>AHL</sub><br />
| 1<br />
| AHL-LuxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>CI</sub><br />
| 2<br />
| CI repressor Hill cooperativity<br />
| Ref. [12]<br />
|<br />
| n<sub>P22CII</sub><br />
| 4<br />
| P22CII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein CII and we have P22CII. Does that match?<br />
|-<br />
|}<br />
<br />
<br><br />
<br />
== References ==<br />
<p><br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-22T12:46:37Z<p>Uhrm: /* Caveats */</p>
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<br />
__NOTOC__<br />
<br />
=Modeling Basics=<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells, based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|<b>Fig. 1</b>: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as ordinary differential equation (ODE)<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|<b>Fig. 2</b>: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|<b>Fig. 3</b>: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|<b>Fig. 4</b>: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start from the same model of the transcription factor binding reversibly to DNA<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
But in case of activation, we are interested in the DNA - transcription factor complex, because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Assuming equilibrium conditions as in the inhibition case, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
== Basic Production ==<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|<b>Fig. 5</b>: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place. For inhibition we have<br />
<br />
[[Image:basic_eq08.png|center|361px]]<br />
<br />
and for activation<br />
<br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation of transcription by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.<br />
<br />
<br />
== Inducer Molecules ==<br />
<br />
A problem in biology is that regulatory proteins (such as protein R in the previous sections) cannot be used as system inputs directly. It is not possible to add such proteins to an assay to steer the behavior of a biological system, because these proteins are big and do not diffuse through cell walls. They cannot enter the cell.<br />
<br />
To circumvent this limitation, one has to produce these proteins directly in the cell. But then, one needs a possibility to switch the functionality of these proteins on and off. This is where inducer molecules become useful. Inducer molecules are small molecules that can diffuse through cell walls freely. Furthermore, they are able to bind to the regulatory proteins and switch the functionality on or off.<br />
<br />
Thus, we need a model to describe the binding of the inducer to the protein. We assume again that the inducer I binds reversibly to the protein R<br />
<br />
[[Image:basic_eq10.png|center]]<br />
<br />
We assume once more that this reaction is in equilibrium. We thus have<br />
<br />
[[Image:basic_eq11.png|center|157px]]<br />
<br />
Depending on the species at hand we are either interested in the protein-inducer complex concentration of in the concentration of 'free protein'. For some species (e.g. LuxR), the complex form is functional and for others it is the free protein (e.g. TetR).<br />
<br />
It is again possible to derive a formula for both the concentration of free protein R and the complex concentration R-nI as a function of total inducer concentration (for notational convenience, we write R* for the complex R-nI in the following equations).<br />
<br />
[[Image:basic_eq12.png|center|157px]]<br />
<br />
<br><br />
<br />
[[Image:basic_eq13.png|center|157px]]<br />
<br />
<br />
== Caveats ==<br />
<br />
With this approach to modeling, the equations turn out to be easy both to understand and to simulate. But to arrive at this point, I certain number of assumptions must be met. This section points out possible problems with these assumptions and ideas to further improve the modeling.<br />
<br />
* '''Equilibrium'''. This is a very basic assumption that was the basis of all results in the above discussion. At the same time, it is also the least problematic assumption. This can be seen by remembering the massive machinery involved in transcription and translation of DNA/RNA compared to simple reversible binding of molecules. It is not difficult to imagine that the latter happens indeed on a much shorter time scale than the first.<br />
<br />
* '''Excess of substrate'''. This assumption was implicitly made, when we wrote down the equations of regulation and inducer binding. The basic idea is that the total inducer concentration is very close to the concentration of unbound inducer. Mathematically more precise, this condition is met when K<sub>I</sub> &gt;&gt; [R]<sub>t</sub>. As the uncertainty in the values of dissociation constants and steady state concentrations is sometimes very large, it is difficult to say whether this assumption is justified.<br />
<br />
* '''Mechanism dependence'''. If the Hill cooperativity coefficient ''n'' is greater than one, the above formulas become in fact dependent on the mechanism of the binding process. In these cases, the formulas can be off up to a factor ''n''. As the true mechanism of binding is often unknown, it is practically impossible to take this fact into account.<br />
<br />
* '''Low protein concentration'''. Whenever the concentration is very low (say, below 100 nM in the case of ''E.&nbsp;coli''), the number of molecules per cell becomes small. In this situation, the assumptions behind the ODE modeling approach are not met any more and simulation results become inaccurate. Then, one would have to resort to stochastic simulations.</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-22T12:38:51Z<p>Uhrm: /* Inducer Molecules */</p>
<hr />
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<br />
__NOTOC__<br />
<br />
=Modeling Basics=<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells, based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|<b>Fig. 1</b>: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as ordinary differential equation (ODE)<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|<b>Fig. 2</b>: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|<b>Fig. 3</b>: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|<b>Fig. 4</b>: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start from the same model of the transcription factor binding reversibly to DNA<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
But in case of activation, we are interested in the DNA - transcription factor complex, because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Assuming equilibrium conditions as in the inhibition case, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
== Basic Production ==<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|<b>Fig. 5</b>: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place. For inhibition we have<br />
<br />
[[Image:basic_eq08.png|center|361px]]<br />
<br />
and for activation<br />
<br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation of transcription by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.<br />
<br />
<br />
== Inducer Molecules ==<br />
<br />
A problem in biology is that regulatory proteins (such as protein R in the previous sections) cannot be used as system inputs directly. It is not possible to add such proteins to an assay to steer the behavior of a biological system, because these proteins are big and do not diffuse through cell walls. They cannot enter the cell.<br />
<br />
To circumvent this limitation, one has to produce these proteins directly in the cell. But then, one needs a possibility to switch the functionality of these proteins on and off. This is where inducer molecules become useful. Inducer molecules are small molecules that can diffuse through cell walls freely. Furthermore, they are able to bind to the regulatory proteins and switch the functionality on or off.<br />
<br />
Thus, we need a model to describe the binding of the inducer to the protein. We assume again that the inducer I binds reversibly to the protein R<br />
<br />
[[Image:basic_eq10.png|center]]<br />
<br />
We assume once more that this reaction is in equilibrium. We thus have<br />
<br />
[[Image:basic_eq11.png|center|157px]]<br />
<br />
Depending on the species at hand we are either interested in the protein-inducer complex concentration of in the concentration of 'free protein'. For some species (e.g. LuxR), the complex form is functional and for others it is the free protein (e.g. TetR).<br />
<br />
It is again possible to derive a formula for both the concentration of free protein R and the complex concentration R-nI as a function of total inducer concentration (for notational convenience, we write R* for the complex R-nI in the following equations).<br />
<br />
[[Image:basic_eq12.png|center|157px]]<br />
<br />
<br><br />
<br />
[[Image:basic_eq13.png|center|157px]]<br />
<br />
<br />
== Caveats ==<br />
<br />
With this approach to modeling, the equations turn out to be easy both to understand and to simulate. But to arrive at this point, I certain number of assumptions must be met. This section points out possible problems with these assumptions and ideas to further improve the modeling.<br />
<br />
* '''Equilibrium'''. This is a very basic assumption that was the basis of all results in the above discussion. At the same time, it is also the least problematic assumption. This can be seen by remembering the massive machinery involved in transcription and translation of DNA/RNA compared to simple reversible binding of molecules. It is not difficult to imagine that the latter happens indeed on a much shorter time scale than the first.<br />
<br />
* '''Excess of substrate'''. This assumption was implicitly made, when we wrote down the equations of regulation and inducer binding. The basic idea is that the total inducer concentration is very close to the concentration of unbound inducer. Mathematically more precise, this condition is met when K<sub>I</sub> &gt;&gt; [R]<sub>t</sub>. As the uncertainty in the values of dissociation constants and steady state concentrations is sometimes very large, it is difficult to say whether this assumption is justified.<br />
<br />
* '''Mechanism dependence'''. If the Hill cooperativity coefficient ''n'' is greater than one, the above formulas become in fact dependent on the mechanism of the binding process. In these cases, the formulas can be off up to a factor ''n''. As the true mechanism of binding is often unknown, it is practically impossible to take this fact into account.</div>Uhrmhttp://2007.igem.org/wiki/index.php/File:Basic_eq12.pngFile:Basic eq12.png2007-10-22T12:16:50Z<p>Uhrm: </p>
<hr />
<div></div>Uhrmhttp://2007.igem.org/wiki/index.php/File:Basic_eq13.pngFile:Basic eq13.png2007-10-22T12:15:09Z<p>Uhrm: </p>
<hr />
<div></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-22T11:11:57Z<p>Uhrm: /* Inducer Molecules */</p>
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<br />
__NOTOC__<br />
<br />
=Modeling Basics=<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells, based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|<b>Fig. 1</b>: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as ordinary differential equation (ODE)<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|<b>Fig. 2</b>: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|<b>Fig. 3</b>: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|<b>Fig. 4</b>: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start from the same model of the transcription factor binding reversibly to DNA<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
But in case of activation, we are interested in the DNA - transcription factor complex, because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Assuming equilibrium conditions as in the inhibition case, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
== Basic Production ==<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|<b>Fig. 5</b>: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place. For inhibition we have<br />
<br />
[[Image:basic_eq08.png|center|361px]]<br />
<br />
and for activation<br />
<br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation of transcription by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.<br />
<br />
<br />
== Inducer Molecules ==<br />
<br />
A problem in biology is that regulatory proteins (such as protein R in the previous sections) cannot be used as system inputs directly. It is not possible to add such proteins to an assay to steer the behavior of a biological system, because these proteins are big and do not diffuse through cell walls. They cannot enter the cell.<br />
<br />
To circumvent this limitation, one has to produce these proteins directly in the cell. But then, one needs a possibility to switch the functionality of these proteins on and off. This is where inducer molecules become useful. Inducer molecules are small molecules that can diffuse through cell walls freely. Furthermore, they are able to bind to the regulatory proteins and switch the functionality on or off.<br />
<br />
Thus, we need a model to describe the binding of the inducer to the protein. We assume again that the inducer I binds reversibly to the protein R<br />
<br />
[[Image:basic_eq10.png|center]]<br />
<br />
We assume once more that this reaction is in equilibrium. We thus have<br />
<br />
[[Image:basic_eq11.png|center|157px]]<br />
<br />
Depending on the species at hand we are either interested in the protein-inducer complex concentration of in the concentration of 'free protein'. For some species (e.g. LuxR), the complex form is functional and for others it is the free protein (e.g. TetR).<br />
<br />
It is again possible to derive a formula for both the concentration of free protein R and the complex concentration R-nI as a function of total inducer concentration (for notational convenience, we write R* for the complex R-nI in the following equations).<br />
<br />
[[Image:basic_eq12.png|center|157px]]<br />
<br />
[[Image:basic_eq13.png|center|157px]]</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-22T11:07:50Z<p>Uhrm: /* Inducer Molecules */</p>
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<br />
<center>[[ETHZ | Main Page]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Model | System Modeling]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Simulation | Simulations]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology | System Implementation]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology/Lab| Lab Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Meet_the_team | Meet the Team]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Internal | Team Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Pictures | Pictures!]]</center><br><br />
<br />
__NOTOC__<br />
<br />
=Modeling Basics=<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells, based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|<b>Fig. 1</b>: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as ordinary differential equation (ODE)<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|<b>Fig. 2</b>: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|<b>Fig. 3</b>: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|<b>Fig. 4</b>: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start from the same model of the transcription factor binding reversibly to DNA<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
But in case of activation, we are interested in the DNA - transcription factor complex, because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Assuming equilibrium conditions as in the inhibition case, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
== Basic Production ==<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|<b>Fig. 5</b>: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place. For inhibition we have<br />
<br />
[[Image:basic_eq08.png|center|361px]]<br />
<br />
and for activation<br />
<br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation of transcription by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.<br />
<br />
<br />
== Inducer Molecules ==<br />
<br />
A problem in biology is that regulatory proteins (such as protein R in the previous sections) cannot be used as system inputs directly. It is not possible to add such proteins to an assay to steer the behavior of a biological system, because these proteins are big and do not diffuse through cell walls. They cannot enter the cell.<br />
<br />
To circumvent this limitation, one has to produce these proteins directly in the cell. But then, one needs a possibility to switch the functionality of these proteins on and off. This is where inducer molecules become useful. Inducer molecules are small molecules that can diffuse through cell walls freely. Furthermore, they are able to bind to the regulatory proteins and switch the functionality on or off.<br />
<br />
Thus, we need a model to describe the binding of the inducer to the protein. We assume again that the inducer I binds reversibly to the protein R<br />
<br />
[[Image:basic_eq10.png|center]]<br />
<br />
We assume once more that this reaction is in equilibrium. We thus have<br />
<br />
[[Image:basic_eq11.png|center|157px]]<br />
<br />
Depending on the species at hand we are either interested in the protein-inducer complex concentration of in the concentration of 'free protein'. For some species (e.g. luxR), the complex form is functional and for others it is the free protein (e.g. tetR).</div>Uhrmhttp://2007.igem.org/wiki/index.php/File:Basic_eq11.pngFile:Basic eq11.png2007-10-22T11:07:10Z<p>Uhrm: </p>
<hr />
<div></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-22T11:04:09Z<p>Uhrm: /* Inducer Molecules */</p>
<hr />
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<br />
<center>[[ETHZ | Main Page]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Model | System Modeling]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Simulation | Simulations]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology | System Implementation]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology/Lab| Lab Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Meet_the_team | Meet the Team]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Internal | Team Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Pictures | Pictures!]]</center><br><br />
<br />
__NOTOC__<br />
<br />
=Modeling Basics=<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells, based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|<b>Fig. 1</b>: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as ordinary differential equation (ODE)<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|<b>Fig. 2</b>: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|<b>Fig. 3</b>: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|<b>Fig. 4</b>: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start from the same model of the transcription factor binding reversibly to DNA<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
But in case of activation, we are interested in the DNA - transcription factor complex, because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Assuming equilibrium conditions as in the inhibition case, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
== Basic Production ==<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|<b>Fig. 5</b>: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place. For inhibition we have<br />
<br />
[[Image:basic_eq08.png|center|361px]]<br />
<br />
and for activation<br />
<br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation of transcription by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.<br />
<br />
<br />
== Inducer Molecules ==<br />
<br />
A problem in biology is that regulatory proteins (such as protein R in the previous sections) cannot be used as system inputs directly. It is not possible to add such proteins to an assay to steer the behavior of a biological system, because these proteins are big and do not diffuse through cell walls. They cannot enter the cell.<br />
<br />
To circumvent this limitation, one has to produce these proteins directly in the cell. But then, one needs a possibility to switch the functionality of these proteins on and off. This is where inducer molecules become useful. Inducer molecules are small molecules that can diffuse through cell walls freely. Furthermore, they are able to bind to the regulatory proteins and switch the functionality on or off.<br />
<br />
Thus, we need a model to describe the binding of the inducer to the protein. We assume again that the inducer I binds reversibly to the protein R<br />
<br />
[[Image:basic_eq10.png|center]]<br />
<br />
We assume once more that this reaction is in equilibrium. We thus have<br />
<br />
[[Image:basic_eq11.png|center|361px]]<br />
<br />
Depending on the species at hand we are either interested in the protein-inducer complex concentration of in the concentration of 'free protein'. For some species (e.g. luxR), the complex form is functional and for others it is the free protein (e.g. tetR).</div>Uhrmhttp://2007.igem.org/wiki/index.php/File:Basic_eq10.pngFile:Basic eq10.png2007-10-22T11:03:36Z<p>Uhrm: </p>
<hr />
<div></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-22T10:59:58Z<p>Uhrm: /* Inducer Molecules */</p>
<hr />
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<br />
<center>[[ETHZ | Main Page]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Model | System Modeling]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Simulation | Simulations]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology | System Implementation]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology/Lab| Lab Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Meet_the_team | Meet the Team]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Internal | Team Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Pictures | Pictures!]]</center><br><br />
<br />
__NOTOC__<br />
<br />
=Modeling Basics=<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells, based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|<b>Fig. 1</b>: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as ordinary differential equation (ODE)<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|<b>Fig. 2</b>: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|<b>Fig. 3</b>: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|<b>Fig. 4</b>: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start from the same model of the transcription factor binding reversibly to DNA<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
But in case of activation, we are interested in the DNA - transcription factor complex, because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Assuming equilibrium conditions as in the inhibition case, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
== Basic Production ==<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|<b>Fig. 5</b>: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place. For inhibition we have<br />
<br />
[[Image:basic_eq08.png|center|361px]]<br />
<br />
and for activation<br />
<br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation of transcription by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.<br />
<br />
<br />
== Inducer Molecules ==<br />
<br />
A problem in biology is that regulatory proteins (such as protein R in the previous sections) cannot be used as system inputs directly. It is not possible to add such proteins to an assay to steer the behavior of a biological system, because these proteins are big and do not diffuse through cell walls. They cannot enter the cell.<br />
<br />
To circumvent this limitation, one has to produce these proteins directly in the cell. But then, one needs a possibility to switch the functionality of these proteins on and off. This is where inducer molecules become useful. Inducer molecules are small molecules that can diffuse through cell walls freely. Furthermore, they are able to bind to the regulatory proteins and switch the functionality on or off.<br />
<br />
Thus, we need a model to describe the binding of the inducer to the protein. We assume again that the inducer I binds reversibly to the protein R<br />
<br />
[[Image:basic_eq10.png|center|361px]]<br />
<br />
We assume once more that this reaction is in equilibrium. We thus have<br />
<br />
[[Image:basic_eq11.png|center|361px]]<br />
<br />
Depending on the species at hand we are either interested in the protein-inducer complex concentration of in the concentration of 'free protein'. For some species (e.g. luxR), the complex form is functional and for others it is the free protein (e.g. tetR).</div>Uhrmhttp://2007.igem.org/wiki/index.php/File:Markus_Uhr.jpgFile:Markus Uhr.jpg2007-10-20T07:57:25Z<p>Uhrm: </p>
<hr />
<div></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Meet_the_teamETHZ/Meet the team2007-10-20T07:55:14Z<p>Uhrm: /* Graduate students */</p>
<hr />
<div><center>[[Image:Eth_zh_logo_4.png|830px]]</center><br />
<br />
<center>[[ETHZ | Main Page]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Model | System Modeling]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Simulation | Simulations]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology | System Implementation]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Biology/Lab| Lab Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Meet_the_team | Meet the Team]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Internal | Team Notes]] &nbsp;&nbsp;&nbsp;&nbsp; [[ETHZ/Pictures | Pictures!]]</center><br><br />
<br />
__NOTOC__<br />
<br />
=Meet the ETH Zurich 07 team=<br />
<br />
In this page, you can find more information on this year's team of ETH Zurich. If you also want to see photos taken during the preparation for this year's iGEM, don't forget to see [[ETHZ/Pictures | Pictures!]] as well.<br />
<br />
[[Image:ETHZ_Group_Photo2.jpg|left|thumb|The ETHZ iGEM2007 Team, posing!|420px]] [[Image:ETHZ_Group_Photo3.jpg|right|thumb|The ETHZ iGEM2007 Team, with its missing members!|420px]]<br />
<br />
<br />
==Team description==<br />
<br />
Our team is a combination of hard-working biologists and engineers! :)<br />
===Undergraduate students===<br />
*[[Image:Martin_Brutsche.jpg|left|40px]] Hi, my name is '''Martin''', and I'm a Master Student in Biomedical Engineering at ETH Zurich. Before that, I did my Diploma studies in Mechanical Engineering, at the University of Applied Sciences in Constance, Germany. I love sailing, snooker and holidays in Denmark. I joined iGEM to learn more about synthetic biology and labwork ([https://2007.igem.org/User:brutsche more info]).<br><br />
*[[Image:Katerina_Dikaiou.jpg|left|40px]]: Hi, I am '''Katerina'''! I'm one of the engineers in the group - I hold a Diploma in Electrical and Computer Engineering from the Aristotle University of Thessaloniki, Greece, and I am currently a Master student of Biomedical Engineering at ETH Zurich. I enjoy working at the interface between engineering and biology, and I also love music, literature and (very) long walks([https://2007.igem.org/User:kdikaiou more info]).<br><br />
*[[Image:Raphael_Gubeli.jpg|left|40px]] Hi, I'm '''Raphael''', Master Student of Biotechnology at ETH Zurich. So, that's why you can find me mainly in the lab (day and night!). Anyway, if there is time to see the daylight, you can find me on my bicycle in the city, or skiing down the mountains. I joined iGEM because I think that there is really a need to find international standards like the "biobricks". Of course, despite having a lot of work, it is a lot of pleasure and fun, working on this project ([https://2007.igem.org/User:Raphael more info]).<br><br />
*[[Image:Foto_iGem_hoehnels.png|left|38px]] Hello! I'm '''Sylke Hoehnel''', B.Sc.-Biotech Student at the ETH Zurich. I was born in Germany, and have lived in London for four years. So, as you can see, I like jetting around the world. I found iGEM as a great opportunity to be part of a team, working on a fun project that's also related to my studies. Apart from Bio-books, I like art, swimming and snowboarding ([https://2007.igem.org/User:hoehnels more info]).<br><br />
*[[Image:Nan_Li.jpg|left|40px]] Hello, I am '''Nan'''! I grew up in China and got my Bachelor degree in [http://en.wikipedia.org/wiki/Zhejiang_University Zhejiang University] located in the Heaven City - Hangzhou, Zhejiang, China. Right now I am doing my Master study in Biomedical Engineering at [http://www.ethz.ch ETH Zürich]. During my spare time, I enjoy the painting in silence. I like hiking with friends and traveling around. The only means to keep my life refreshing is to experience things new ([https://2007.igem.org/Nan_Li more info]).<br><br />
*Stefan Luzi: My name is '''Stefan Luzi''', M.Sc-Biotechnology student at ETH Zurich, and member of the ETH iGEM team 2007. During my spare time (yes, I have a life besides my studies) I like hiking, rowing or cycling, and can never reject a cold jug of beer ;-) ([https://2007.igem.org/User:Stefan more info]).<br />
<br />
===Graduate students===<br />
*[[Image:Christos_Bergeles.jpg|left|40px]] Hoi zaeme, I am '''Christos'''. I graduated with a Diploma in Electrical and Computer Engineering, from the National Technical University of Athens, Greece, and currently I am a PhD Student at ETH Zurich, working with nice, little, autonomous, magnetic, wireless, reconfigurable, fluorescent, swallowable, biocompatible super geeky microrobots. I sleep in my lab currently, but all is good! ([http://christos.bergeles.net more info]).<br />
*[[Image:Tim_Hohm.jpg|left|35px]] Hello everyone, my name is '''Tim Hohm'''. I received a diploma in computer science in 2003, from the University of Dortmund, Germany. After staying for two years in the research institute caesar (in Bonn, Germany), 2006 I joined the Systems Optimization Group at ETH Zurich headed by Prof. Zitzler. My research focuses on the application of bio-inspired optimization techniques on systems biology problems ([http://www.tik.ee.ethz.ch/~sop/people/thohm/ Tim Hohm]).<br />
*Christian Kemmer: ([http://fm-eth.ethz.ch/eth/peoplefinder/FMPro?-db=phonebook.fp5&-format=pf%5fdetail%5fde.html&-lay=html&-op=cn&Typ=Staff&Suche%5fText=kemmer&Suche%5fText%5fpre=kemmer&-recid=3772770936&-find=/ more info])<br />
*[[Image:Joe_Knight.jpg|left|40px]] '''Joe Knight''' here. I am a PhD candidate performing research in biomedical engineering at the ETH. Before coming to the ETH I was the 2005-06 Grube Fellow at Stanford University's Biodesign Innovation Program. I joined iGEM because I believe some of the greatest innovations in biotechnology of tomorrow will be made in synthetic biology, and I hope to be part of it all! ([https://2007.igem.org/User:JoeKnight more info])<br />
*[[Image:Markus_Uhr.jpg|left|40px]] I'm '''Markus Uhr''', a PhD candidate in Computational Systems Biology, at ETH Zurich. I did my MSc in Computational Science and Engineering, so I'm the guy that likes big equations and solving them on a computer ;-). Besides the geeky stuff I do to make a living I also enjoy running and orienteering, or listening to good music. ([https://2007.igem.org/User:uhrm more info])<br />
*[[Image:Rico_Mockel.jpg|left|40px]] Hi, I am '''Rico'''. I am a Ph.D student of the ETH Zurich, Switzerland. I am working at the Institute of Neuroinformatics. I received a diploma (equivalent to MSc) in Electrical Engineering from the University of Rostock, Germany. I did my diploma thesis at the Institute of Neuroinformatics in Switzerland where I developed a STDP floating-gate synapse in CMOS VLSI ([http://www.ricomoeckel.de more info]).<br />
<br />
===Advisors===<br />
*[[Image:Sven_Panke.jpg|left|thumb|Sven Panke|80px]]<br />
*Joerg Stelling: (homepage)</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-17T15:56:28Z<p>Uhrm: /* Reporter system */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Reporter_braced.png|778px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| <br />
* 1e-5 [per sec]<br />
* 2.31e-3 [per sec]<br />
| degradation of tetR<br />
| <br />
* Ref. [9]<br />
* Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
|<br />
* 0.1 - 1 [pM]<br />
* 800 [nM]<br />
| lacI repressor dissociation constant<br />
|<br />
* Ref. [2]<br />
* Ref. [12]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
|<br />
* 55 - 520 [nM]<br />
* 10 [nM]<br />
| luxR activator dissociation constant<br />
|<br />
* Ref: [6]<br />
* Ref: [12]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
|<br />
* 1<br />
* 2<br />
| lacI repressor Hill cooperativity<br />
|<br />
* Ref. [5]<br />
* Ref. [12]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 2<br />
| cI repressor Hill cooperativity<br />
| Ref. [12]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /><br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-17T15:56:18Z<p>Uhrm: /* Learning system */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Eq03.png|586px]]<br />
<br />
[[Image:Reporter_braced.png|778px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| <br />
* 1e-5 [per sec]<br />
* 2.31e-3 [per sec]<br />
| degradation of tetR<br />
| <br />
* Ref. [9]<br />
* Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
|<br />
* 0.1 - 1 [pM]<br />
* 800 [nM]<br />
| lacI repressor dissociation constant<br />
|<br />
* Ref. [2]<br />
* Ref. [12]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
|<br />
* 55 - 520 [nM]<br />
* 10 [nM]<br />
| luxR activator dissociation constant<br />
|<br />
* Ref: [6]<br />
* Ref: [12]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
|<br />
* 1<br />
* 2<br />
| lacI repressor Hill cooperativity<br />
|<br />
* Ref. [5]<br />
* Ref. [12]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 2<br />
| cI repressor Hill cooperativity<br />
| Ref. [12]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /><br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-17T15:56:07Z<p>Uhrm: /* Constitutively produced proteins */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Eq02.png|475px]]<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Eq03.png|586px]]<br />
<br />
[[Image:Reporter_braced.png|778px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| <br />
* 1e-5 [per sec]<br />
* 2.31e-3 [per sec]<br />
| degradation of tetR<br />
| <br />
* Ref. [9]<br />
* Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
|<br />
* 0.1 - 1 [pM]<br />
* 800 [nM]<br />
| lacI repressor dissociation constant<br />
|<br />
* Ref. [2]<br />
* Ref. [12]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
|<br />
* 55 - 520 [nM]<br />
* 10 [nM]<br />
| luxR activator dissociation constant<br />
|<br />
* Ref: [6]<br />
* Ref: [12]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
|<br />
* 1<br />
* 2<br />
| lacI repressor Hill cooperativity<br />
|<br />
* Ref. [5]<br />
* Ref. [12]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 2<br />
| cI repressor Hill cooperativity<br />
| Ref. [12]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /><br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrmhttp://2007.igem.org/wiki/index.php/File:Reporter_braced.pngFile:Reporter braced.png2007-10-17T15:55:36Z<p>Uhrm: </p>
<hr />
<div></div>Uhrmhttp://2007.igem.org/wiki/index.php/File:Toggle_braced.pngFile:Toggle braced.png2007-10-17T15:55:11Z<p>Uhrm: </p>
<hr />
<div></div>Uhrmhttp://2007.igem.org/wiki/index.php/File:Constitutive_braced.pngFile:Constitutive braced.png2007-10-17T15:54:43Z<p>Uhrm: </p>
<hr />
<div></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-17T15:54:20Z<p>Uhrm: /* Reporter system */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Eq01.png|171px]]<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Eq02.png|475px]]<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Eq03.png|586px]]<br />
<br />
[[Image:Reporter_braced.png|778px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| <br />
* 1e-5 [per sec]<br />
* 2.31e-3 [per sec]<br />
| degradation of tetR<br />
| <br />
* Ref. [9]<br />
* Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
|<br />
* 0.1 - 1 [pM]<br />
* 800 [nM]<br />
| lacI repressor dissociation constant<br />
|<br />
* Ref. [2]<br />
* Ref. [12]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
|<br />
* 55 - 520 [nM]<br />
* 10 [nM]<br />
| luxR activator dissociation constant<br />
|<br />
* Ref: [6]<br />
* Ref: [12]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
|<br />
* 1<br />
* 2<br />
| lacI repressor Hill cooperativity<br />
|<br />
* Ref. [5]<br />
* Ref. [12]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 2<br />
| cI repressor Hill cooperativity<br />
| Ref. [12]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /><br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-17T15:53:41Z<p>Uhrm: /* Learning system */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Eq01.png|171px]]<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Eq02.png|475px]]<br />
<br />
[[Image:Toggle_braced.png|770px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Eq03.png|586px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| <br />
* 1e-5 [per sec]<br />
* 2.31e-3 [per sec]<br />
| degradation of tetR<br />
| <br />
* Ref. [9]<br />
* Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
|<br />
* 0.1 - 1 [pM]<br />
* 800 [nM]<br />
| lacI repressor dissociation constant<br />
|<br />
* Ref. [2]<br />
* Ref. [12]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
|<br />
* 55 - 520 [nM]<br />
* 10 [nM]<br />
| luxR activator dissociation constant<br />
|<br />
* Ref: [6]<br />
* Ref: [12]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
|<br />
* 1<br />
* 2<br />
| lacI repressor Hill cooperativity<br />
|<br />
* Ref. [5]<br />
* Ref. [12]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 2<br />
| cI repressor Hill cooperativity<br />
| Ref. [12]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /><br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-17T15:52:51Z<p>Uhrm: /* Constitutively produced proteins */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Eq01.png|171px]]<br />
<br />
[[Image:Constitutive_braced.png|330px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Eq02.png|475px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Eq03.png|586px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| <br />
* 1e-5 [per sec]<br />
* 2.31e-3 [per sec]<br />
| degradation of tetR<br />
| <br />
* Ref. [9]<br />
* Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
|<br />
* 0.1 - 1 [pM]<br />
* 800 [nM]<br />
| lacI repressor dissociation constant<br />
|<br />
* Ref. [2]<br />
* Ref. [12]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
|<br />
* 55 - 520 [nM]<br />
* 10 [nM]<br />
| luxR activator dissociation constant<br />
|<br />
* Ref: [6]<br />
* Ref: [12]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
|<br />
* 1<br />
* 2<br />
| lacI repressor Hill cooperativity<br />
|<br />
* Ref. [5]<br />
* Ref. [12]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 2<br />
| cI repressor Hill cooperativity<br />
| Ref. [12]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /><br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-15T13:53:08Z<p>Uhrm: /* Model Parameters */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Eq01.png|171px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Eq02.png|475px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Eq03.png|586px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| <br />
* 1e-5 [per sec]<br />
* 2.31e-3 [per sec]<br />
| degradation of tetR<br />
| <br />
* Ref. [9]<br />
* Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
|<br />
* 0.1 - 1 [pM]<br />
* 800 [nM]<br />
| lacI repressor dissociation constant<br />
|<br />
* Ref. [2]<br />
* Ref. [12]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
|<br />
* 55 - 520 [nM]<br />
* 10 [nM]<br />
| luxR activator dissociation constant<br />
|<br />
* Ref: [6]<br />
* Ref: [12]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
|<br />
* 1<br />
* 2<br />
| lacI repressor Hill cooperativity<br />
|<br />
* Ref. [5]<br />
* Ref. [12]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 2<br />
| cI repressor Hill cooperativity<br />
| Ref. [12]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /><br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-15T13:26:06Z<p>Uhrm: /* Model Parameters */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Eq01.png|171px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Eq02.png|475px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Eq03.png|586px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| <br />
* 1e-5 [per sec]<br />
* 2.31e-3 [per sec]<br />
| degradation of tetR<br />
| <br />
* Ref. [9]<br />
* Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
| 0.1 - 1 [pM]<br />
| lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
|<br />
* 55 - 520 [nM]<br />
* 10 [nM]<br />
| luxR activator dissociation constant<br />
|<br />
* Ref: [6]<br />
* Ref: [12]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
|<br />
* 1<br />
* 2<br />
| lacI repressor Hill cooperativity<br />
|<br />
* Ref. [5]<br />
* Ref. [12]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 2<br />
| cI repressor Hill cooperativity<br />
| Ref. [12]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /><br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-15T13:20:19Z<p>Uhrm: /* Model Parameters */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Eq01.png|171px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Eq02.png|475px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Eq03.png|586px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| <br />
* 1e-5 [per sec]<br />
* 2.31e-3 [per sec]<br />
| degradation of tetR<br />
| <br />
* Ref. [9]<br />
* Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
| 0.1 - 1 [pM]<br />
| lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
| 55 - 520 [nM]<br />
| luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
|<br />
* 1<br />
* 2<br />
| lacI repressor Hill cooperativity<br />
|<br />
* Ref. [5]<br />
* Ref. [12]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 2<br />
| cI repressor Hill cooperativity<br />
| Ref. [12]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
[http://www.pnas.org/cgi/content/abstract/104/8/2643 &#91;1&#93; Weber W et al.] <i>"A synthetic time-delay circuit in mammalian cells and mice"</i>, P Natl Acad Sci USA 104(8):2643-2648, 2007<br /><br />
[http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck &#91;2&#93; Setty Y et al.] <i>"Detailed map of a cis-regulatory input function"</i>, P Natl Acad Sci USA 100(13):7702-7707, 2003<br /><br />
[http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf &#91;3&#93; Braun D et al.] <i>"Parameter Estimation for Two Synthetic Gene Networks: A Case Study"</i>, ICASSP 5:769-772, 2005<br /><br />
[http://www.nature.com/nature/journal/v435/n7038/suppinfo/nature03508.html &#91;4&#93; Fung E et al.] <i>"A synthetic gene--metabolic oscillator"</i>, Nature 435:118-122, 2005 (supplementary material)<br /><br />
[http://dx.doi.org/10.1016/j.jbiotec.2005.08.030 &#91;5&#93; Iadevaia S and Mantzais NV] <i>"Genetic network driven control of PHBV copolymer composition"</i>, J Biotechnol 122(1):99-121, 2006<br /><br />
[http://dx.doi.org/10.1016/j.biosystems.2005.04.006 &#91;6&#93; Goryachev AB et al.] <i>"Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants"</i>, Biosystems 83(2-3):178-187, 2004<br /><br />
[http://www.genetics.org/cgi/content/abstract/149/4/1633 &#91;7&#93; Arkin A et al.] <i>"Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-Infected Escherichia coli cells"</i>, Genetics 149: 1633-1648, 1998<br /><br />
[http://download.cell.com/supplementarydata/cell/107/6/739/DC1/index.htm &#91;8&#93; Colman-Lerner A et al.] <i>"Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates"</i>, Cell 107(6): 739-750, 2001 (supplementary material)<br /><br />
[http://www.nature.com/nature/journal/v405/n6786/abs/405590a0.html &#91;9&#93; Becskei A and Serrano L] <i>"Engineering stability in gene networks by autoregulation"</i>, Nature 405: 590-593, 2000<br /><br />
[http://www.biophysj.org/cgi/content/full/89/6/3873?maxtoshow=&HITS=10&hits=10&RESULTFORMAT=&searchid=1&FIRSTINDEX=0&volume=89&firstpage=3873&resourcetype=HWCIT &#91;10&#93; Tuttle et al.] <i>"Model-Driven Designs of an Oscillating Gene Network"</i>, Biophys J 89(6):3873-3883, 2005<br /><br />
[http://www.pnas.org/cgi/reprint/99/2/679 &#91;11&#93; McMillen LM et al.] <i>"Synchronizing genetic relaxation oscillators by intercell signaling"</i>, P Natl Acad Sci USA 99(2):679-684, 2002<br /><br />
[http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html &#91;12&#93; Basu S et al.] <i>"A synthetic multicellular system for programmed pattern formation"</i>, Nature 434:1130-1134, 2005<br /><br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-12T08:33:05Z<p>Uhrm: /* Basic Production */</p>
<hr />
<div>== Introduction ==<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|Fig. 1: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as an ODE<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|Fig. 2: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|Fig. 3: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|Fig. 4: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start from the same model of the transcription factor binding reversibly to DNA<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
But in case of activation, we are interested in the DNA - transcription factor complex, because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Assuming equilibrium conditions as in the inhibition case, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
=== Basic Production ===<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|Fig. 5: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place. For inhibition we have<br />
<br />
[[Image:basic_eq08.png|center|361px]]<br />
<br />
and for activation<br />
<br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation of transcription by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.<br />
<br />
<br />
== Inducer Molecules ==<br />
<br />
A problem in biology is that regulatory proteins (such as protein R in the previous sections) cannot be used as system inputs directly. It is not possible to add such proteins to an assay to steer the behavior of a biological system, because these proteins are big and do not diffuse through cell walls. They cannot enter the cell.<br />
<br />
To circumvent this limitation, one has to produce these proteins directly in the cell. But then, one needs a possibility to switch the functionality of these proteins on and off. This is where inducer molecules become useful. Inducer molecules are small molecules that can diffuse through cell walls freely. Furthermore, they are able to bind to the regulatory proteins and switch the functionality on or off.<br />
<br />
Thus, we need a model to describe the binding of the inducer to the protein. We assume again, that the inducer binds reversibly to the protein<br />
<br />
[[Image:basic_eq10.png|center|361px]]<br />
<br />
We assume once more that this reaction is in equilibrium. We thus have<br />
<br />
[[Image:basic_eq11.png|center|361px]]<br />
<br />
Depending on the species at hand we are either interested in the protein-inducer complex concentration of in the concentration of 'free protein'. For some species (e.g. luxR), the complex form is functional and for others it is the free protein (e.g. tetR).</div>Uhrmhttp://2007.igem.org/wiki/index.php/File:Basic_eq09.pngFile:Basic eq09.png2007-10-12T08:05:03Z<p>Uhrm: </p>
<hr />
<div></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-12T08:03:06Z<p>Uhrm: /* Basic Production */</p>
<hr />
<div>== Introduction ==<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|Fig. 1: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as an ODE<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|Fig. 2: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|Fig. 3: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|Fig. 4: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start from the same model of the transcription factor binding reversibly to DNA<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
But in case of activation, we are interested in the DNA - transcription factor complex, because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Assuming equilibrium conditions as in the inhibition case, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
=== Basic Production ===<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|Fig. 5: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place. For inhibition we have<br />
<br />
[[Image:basic_eq08.png|center|361px]]<br />
<br />
and for activation<br />
<br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.</div>Uhrmhttp://2007.igem.org/wiki/index.php/File:Basic_eq07.pngFile:Basic eq07.png2007-10-12T08:01:44Z<p>Uhrm: </p>
<hr />
<div></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-12T07:59:44Z<p>Uhrm: /* Activation */</p>
<hr />
<div>== Introduction ==<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|Fig. 1: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as an ODE<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|Fig. 2: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|Fig. 3: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|Fig. 4: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start from the same model of the transcription factor binding reversibly to DNA<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
But in case of activation, we are interested in the DNA - transcription factor complex, because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Assuming equilibrium conditions as in the inhibition case, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|175px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
=== Basic Production ===<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|Fig. 5: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place:<br />
<br />
[[Image:basic_eq08.png|center|361px]] <br><br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.</div>Uhrmhttp://2007.igem.org/wiki/index.php/File:Basic_eq08.pngFile:Basic eq08.png2007-10-12T07:52:42Z<p>Uhrm: </p>
<hr />
<div></div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-12T07:51:35Z<p>Uhrm: /* Basic Production */</p>
<hr />
<div>== Introduction ==<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|Fig. 1: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as an ODE<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|Fig. 2: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|Fig. 3: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|Fig. 4: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start from the same model of the transcription factor binding reversibly to DNA<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
But in case of activation, we are interested in the DNA - transcription factor complex, because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Assuming equilibrium conditions as in the inhibition case, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|152px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
=== Basic Production ===<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|Fig. 5: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place:<br />
<br />
[[Image:basic_eq08.png|center|361px]] <br><br />
[[Image:basic_eq09.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-12T07:50:21Z<p>Uhrm: /* Activation */</p>
<hr />
<div>== Introduction ==<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|Fig. 1: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as an ODE<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|Fig. 2: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|Fig. 3: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|Fig. 4: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
To understand activation, we start from the same model of the transcription factor binding reversibly to DNA<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
But in case of activation, we are interested in the DNA - transcription factor complex, because the transcription rate is proportional to the probability that a protein is bound to DNA.<br />
<br />
Assuming equilibrium conditions as in the inhibition case, we can derive an expression for the DNA-protein complex concentration as a function of protein concentration:<br />
<br />
[[Image:basic_eq06.png|center|152px]]<br />
<br />
Analogously, we can now write the whole differential equation for protein concentration if transcription is activated by protein R<br />
<br />
[[Image:basic_eq07.png|center|253px]]<br />
<br />
=== Basic Production ===<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|Fig. 5: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place:<br />
<br />
[[Image:basic_eq0x.png|center|361px]] <br><br />
[[Image:basic_eq0x.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-12T07:41:08Z<p>Uhrm: /* Basic Production */</p>
<hr />
<div>== Introduction ==<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|Fig. 1: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as an ODE<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|Fig. 2: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|Fig. 3: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|Fig. 4: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
<br />
<br />
=== Basic Production ===<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|Fig. 5: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place:<br />
<br />
[[Image:basic_eq0x.png|center|361px]] <br><br />
[[Image:basic_eq0x.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-12T07:40:45Z<p>Uhrm: /* Activation */</p>
<hr />
<div>== Introduction ==<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|Fig. 1: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as an ODE<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|Fig. 2: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|Fig. 3: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|Fig. 4: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===<br />
<br />
<br />
<br />
=== Basic Production ===<br />
<br />
The equations so far assume perfect inhibition/activation. This means, in case of inhibition, that if the inhibitor concentration is high enough, the transcription of protein P is practically zero. Or, in the case of activation, that in the absence of activator protein there is no transcription. In reality, one observes always some ''basic production'', despite high inhibitor concentrations or absence of activator protein. The basic transcription is usually around 10-20% of the maximum transcription rate. The case of inhibition is illustrated in Fig. 5. [[Image:basic_fig05.png|thumb|Fig. 5: Inhibition of transcription factor is only effective between basic transcription and maximum transcription. That is, protein production cannot be shut off completely by inhibition.]] <br />
<br />
Thus we have to introduce some 'basic transcription' that always takes place:<br />
<br />
[[Image:basic_eq0x.png|center|361px]]<br />
[[Image:basic_eq0x.png|center|361px]]<br />
<br />
Note that the basic transcription rate is introduced as a ''leakiness factor'' ''a'', which is a percentage of the maximum transcription rate ''c''<sup>max</sup>. Regulation by protein R now is only effective in the range between ''a''&middot;''c''<sup>max</sup> and ''c''<sup>max</sup>.</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/Modeling_BasicsETHZ/Modeling Basics2007-10-12T07:29:52Z<p>Uhrm: /* Inhibition */</p>
<hr />
<div>== Introduction ==<br />
<br />
The functioning of our model depends mostly on ''protein concentrations''. These proteins are produced within the ''E. coli'' cells based on genes that we introduced into the cell. To understand the system, it is crucial to model the gene expression and regulation accurately.<br />
<br />
This Wiki page is intended to present the basic mechanisms and assumptions that went into the mathematical description of our iGEM model.<br />
<br />
== Constitutive Protein Production ==<br />
<br />
The most simple parts in the system are genes that are continuously transcribed to produce a protein. At the same time, proteins have a certain half-life time, which means that they are degraded. This leads to the following simple model of protein production/degradation shown in Fig. 1.<br />
<br />
[[Image:basic_fig01.png|thumb|Fig. 1: System of a constitutively produced protein P. Production rate is c<sup>max</sup> and degradation rate is d<sub>P</sub>.]]<br />
<br />
To find the concentration of protein P (as a function of time), the system of Fig. 1 can be written as an ODE<br />
<br />
[[Image:basic_eq01.png|center|150px]]<br />
<br />
This equation states that the change of protein concentration is a function of protein production (c<sup>max</sup>) and protein degradation (d<sub>P</sub>[P]). <br />
<br />
It is worth looking at the protein production a bit closer. The production of protein P depends on the expression of a gene that codes for this protein. In the case of constant protein production, the gene can be modeled by a ''constitutive promoter'' and a coding region for the protein (Fig. 2).<br />
<br />
[[Image:basic_fig02.png|thumb|Fig. 2: Model for the gene coding for protein P. The promoter of the gene is continuously expressed.]]<br />
<br />
== Regulated Protein Production ==<br />
<br />
To build a system with logic functionality, it is necessary to produce a protein depending on the concentration of another protein (i.e. a transcription factor). To model such a system, the promoter of the constitutive production system in Fig. 2 must be extended to take into account the presence of the regulatory protein R. There are two possible cases: the regulatory protein either ''inhibits'' or ''activates'' the expression of the gene (see Figs. 3 and 4, respectively).<br />
<br />
[[Image:basic_fig03.png|thumb|Fig. 3: Regulated protein production where the regulatory protein has inhibitory effect. That is, the higher the concentration of R, the smaller the expression of protein P.]]<br />
<br />
[[Image:basic_fig04.png|thumb|Fig. 4: Regulated protein production where the regulatory protein activates protein production. That is, the higher the concentration of R, the larger the expression of protein P.]]<br />
<br />
=== Inhibition ===<br />
<br />
To derive the equations describing the regulated transcription, we first need a model how the transcription factor interacts with DNA. The most simple model is when the protein binds reversibly to the DNA:<br />
<br />
[[Image:basic_eq02.png|center]]<br />
<br />
Note that the equation above involves ''n'' transcription factors. For certain transcription factors the number of proteins involved is indeed greater than 1. These are interesting cases that enable applications such as toggle switches.<br />
<br />
When the transcription factor binds to DNA, it blocks the enzymes transcribing the gene. Thus, the higher the concentration of R, the smaller the transcription of the gene. By controlling the concentration of the regulatory protein R, the expression of protein P can effectively be regulated.<br />
<br />
To understand this process in more detail, we make the simplification that the binding of R to DNA is in ''equilibrium''. That is, one can write<br />
<br />
[[Image:basic_eq03.png|center|225px]]<br />
<br />
In case of inhibition, the expression of the gene is proportional to the probability that the DNA is 'free' (meaning that there is no transcription factor bound to it). After some algebraic manipulation of the above equation an expression for the 'free DNA' as a function of transcription factor concentration can be derived:<br />
<br />
[[Image:basic_eq04.png|center|152px]]<br />
<br />
Now, we have basically everything together to write down a differential equation for the concentration of protein P whose expression is regulated by protein R:<br />
<br />
[[Image:basic_eq05.png|center|253px]]<br />
<br />
The transcription does not always take place at the maximum rate ''c''<sup>max</sup> as was the case for the constitutively produced proteins, but is modulated by the concentration of protein R.<br />
<br />
=== Activation ===</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-11T15:48:13Z<p>Uhrm: /* Model Parameters */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Eq01.png|171px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Eq02.png|475px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Eq03.png|586px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| 1e-5 [pro sec]/2.31e-3 [per sec]<br />
| degradation of tetR<br />
| Ref. [9]/ Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
| 0.1 - 1 [pM]<br />
| lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
| 55 - 520 [nM]<br />
| luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
| 1<br />
| lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 2<br />
| cI repressor Hill cooperativity<br />
| Ref. [12]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
<br />
# A synthetic time-delay circuit in mammalian cells and mice (http://www.pnas.org/cgi/content/abstract/104/8/2643)<br />
# Detailed map of a cis-regulatory input function (http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck)<br />
# Parameter Estimation for two synthetic gene networks (http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf)<br />
# Supplementary on-line information for "A Synthetic gene-metabolic oscillator" (no link)<br />
# Genetic network driven control of PHBV copolymer composition (http://dx.doi.org/10.1016/j.jbiotec.2005.08.030)<br />
# Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants (http://dx.doi.org/10.1016/j.biosystems.2005.04.006)<br />
# Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Phage λ-Infected <i>E. coli</i> Cells<br />
# Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates<br />
# Engineering stability in gene networks by autoregulation<br />
# Model-Driven Designs of an Oscillating Gene Network<br />
# Synchronizing genetic relaxation oscillators by intercell signaling (http://www.pnas.org/cgi/reprint/99/2/679)<br />
# A synthetic multicellular system for programmed pattern formation (http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html)<br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-11T15:47:15Z<p>Uhrm: /* References */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Eq01.png|171px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Eq02.png|475px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Eq03.png|586px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| 1e-5 [pro sec]/2.31e-3 [per sec]<br />
| degradation of tetR<br />
| Ref. [9]/ Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
| 0.1 - 1 [pM]<br />
| lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
| 55 - 520 [nM]<br />
| luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
| 1<br />
| lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 1.9<br />
| cI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
<br />
# A synthetic time-delay circuit in mammalian cells and mice (http://www.pnas.org/cgi/content/abstract/104/8/2643)<br />
# Detailed map of a cis-regulatory input function (http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck)<br />
# Parameter Estimation for two synthetic gene networks (http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf)<br />
# Supplementary on-line information for "A Synthetic gene-metabolic oscillator" (no link)<br />
# Genetic network driven control of PHBV copolymer composition (http://dx.doi.org/10.1016/j.jbiotec.2005.08.030)<br />
# Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants (http://dx.doi.org/10.1016/j.biosystems.2005.04.006)<br />
# Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Phage λ-Infected <i>E. coli</i> Cells<br />
# Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates<br />
# Engineering stability in gene networks by autoregulation<br />
# Model-Driven Designs of an Oscillating Gene Network<br />
# Synchronizing genetic relaxation oscillators by intercell signaling (http://www.pnas.org/cgi/reprint/99/2/679)<br />
# A synthetic multicellular system for programmed pattern formation (http://www.nature.com/nature/journal/v434/n7037/full/nature03461.html)<br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrmhttp://2007.igem.org/wiki/index.php/ETHZ/SimulationsETHZ/Simulations2007-10-11T15:45:30Z<p>Uhrm: /* Model Parameters */</p>
<hr />
<div>== Basic Model ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Model01b.png]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Model02b.png]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Model03b.png]]<br />
<br />
== System Equations ==<br />
<br />
==== Constitutively produced proteins ====<br />
<br />
[[Image:Eq01.png|171px]]<br />
<br />
==== Learning system ====<br />
<br />
[[Image:Eq02.png|475px]]<br />
<br />
==== Reporter system ====<br />
<br />
[[Image:Eq03.png|586px]]<br />
<br />
==== Allosteric regulation ====<br />
<br />
[[Image:Eq04.png|208px]]<br />
<br />
<br />
==== Comments ====<br />
<br />
Note that the three constitutively produced proteins lacI, tetR and luxR exist in two different forms: as free proteins and in complexes they build with IPTG, aTc and AHL, respectively.<br />
<br />
In this new formulation of the model equations, the characterization is more amenable to human interpretation (although equivalent to the previous formuation). The promoters are now characterized by their ''maximum transcription rate'' (c<sub>i</sub><sup>max</sup>) and the ''basic production'' (a<sub>X</sub>), which gives the 'leakage' if the gene is fully inhibited. Note that in the given mathematical formulation the ''basic production'' is specified as a percentage of the ''max. transcription rate'' and is therefore unitless.<br />
<br />
The max. transcription rate is given ''per gene'' (as agreed with Sven during the meeting at Sep 20.). This means that to get the total transcription rate we need to multiply with the number of gene copies per cell which is represented as l<sup>lo</sup>/l<sup>hi</sup> in the model equations.<br />
<br />
== Model Parameters ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Parameter <br />
! Value<br />
! Description<br />
! Comments<br />
|-<br />
| c<sub>1</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of constitutive promoter (per gene)<br />
| promoter no. J23105; Reference: Estimate<br />
|-<br />
| c<sub>2</sub><sup>max</sup><br />
| 0.01 [mM/h]<br />
| max. transcription rate of luxR-activated promoter (per gene)<br />
| Reference: Estimate<br />
|-<br />
| l<sup>hi</sup><br />
| 25<br />
| high-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| l<sup>lo</sup><br />
| 5<br />
| low-copy plasmid number<br />
| Reference: Estimate<br />
|-<br />
| a<sub>Q<sub>2</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub></sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>2</sub>,S</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>2</sub>/S-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| a<sub>Q<sub>1</sub>,R</sub><br />
| 0.1 - 0.2<br />
| basic production of Q<sub>1</sub>/R-inhibited genes<br />
| Reference: Conclusions after discussion<br />
|-<br />
| d<sub>R</sub><br />
| 2.31e-3 [per sec]<br />
| degradation of lacI<br />
| Ref. [10]<br />
|-<br />
| d<sub>S</sub><br />
| 1e-5 [pro sec]/2.31e-3 [per sec]<br />
| degradation of tetR<br />
| Ref. [9]/ Ref. [10]<br />
|-<br />
| d<sub>L</sub><br />
| 1e-3 - 1e-4 [per sec]<br />
| degradation of luxR<br />
| Ref: [6]<br />
|-<br />
| d<sub>Q<sub>1</sub></sub><br />
| 7e-4 [per sec]<br />
| degradation of cI<br />
| Ref. [7]<br />
|-<br />
| d<sub>Q<sub>2</sub></sub><br />
| <br />
| degradation of p22cII<br />
|-<br />
| d<sub>YFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of YFP<br />
| suppl. mat. to Ref. [8] corresponding to a half life of 110min<br />
|-<br />
| d<sub>GFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of GFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>RFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of RFP<br />
| in analogy to YFP<br />
|-<br />
| d<sub>CFP</sub><br />
| 6.3e-3 [per min]<br />
| degradation of CFP<br />
| in analogy to YFP<br />
|-<br />
| K<sub>R</sub><br />
| 0.1 - 1 [pM]<br />
| lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>I<sub>R</sub></sub><br />
| 1.3 [&#181;M]<br />
| IPTG-lacI repressor dissociation constant<br />
| Ref. [2]<br />
|-<br />
| K<sub>S</sub><br />
| 179 [pM]<br />
| tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>I<sub>S</sub></sub><br />
| 893 [pM]<br />
| aTc-tetR repressor dissociation constant<br />
| Ref. [1]<br />
|-<br />
| K<sub>L</sub><br />
| 55 - 520 [nM]<br />
| luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>I<sub>L</sub></sub><br />
| 0.09 - 1 [&#181;M]<br />
| AHL-luxR activator dissociation constant<br />
| Ref: [6]<br />
|-<br />
| K<sub>Q<sub>1</sub></sub><br />
|<br />
* 8 [pM]<br />
* 50 [nM]<br />
| cI repressor dissociation constant<br />
|<br />
* Ref. [12]<br />
* starting with values of Ref. [6] and using Ref. [3]<br />
|-<br />
| K<sub>Q<sub>2</sub></sub><br />
| 0.577 [&#181;M]<br />
| p22cII repressor dissociation constant<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
| n<sub>R</sub><br />
| 1<br />
| lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>I<sub>R</sub></sub><br />
| 2<br />
| IPTG-lacI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>S</sub><br />
| 3<br />
| tetR repressor Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>I<sub>S</sub></sub><br />
| 2 (1.5-2.5)<br />
| aTc-tetR repressor Hill cooperativity<br />
|Ref. [3]<br />
|-<br />
| n<sub>L</sub><br />
| 2<br />
| luxR activator Hill cooperativity<br />
| Ref: [6]<br />
|-<br />
| n<sub>I<sub>L</sub></sub><br />
| 1<br />
| AHL-luxR activator Hill cooperativity<br />
| Ref. [3]<br />
|-<br />
| n<sub>Q<sub>1</sub></sub><br />
| 1.9<br />
| cI repressor Hill cooperativity<br />
| Ref. [5]<br />
|-<br />
| n<sub>Q<sub>2</sub></sub><br />
| 4<br />
| p22cII repressor Hill cooperativity<br />
| Ref. [11]. Note that they use a protein cII and we have p22cII. Does that match?<br />
|-<br />
|}<br />
<br />
== References ==<br />
<br />
# A synthetic time-delay circuit in mammalian cells and mice (http://www.pnas.org/cgi/content/abstract/104/8/2643)<br />
# Detailed map of a cis-regulatory input function (http://www.pnas.org/cgi/content/full/100/13/7702?ck=nck)<br />
# Parameter Estimation for two synthetic gene networks (http://ieeexplore.ieee.org/iel5/9711/30654/01416417.pdf)<br />
# Supplementary on-line information for "A Synthetic gene-metabolic oscillator" (no link)<br />
# Genetic network driven control of PHBV copolymer composition (http://dx.doi.org/10.1016/j.jbiotec.2005.08.030)<br />
# Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants (http://dx.doi.org/10.1016/j.biosystems.2005.04.006)<br />
# Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Phage λ-Infected <i>E. coli</i> Cells<br />
# Yeast Cbk1 and Mob2 Activate Daughter-Specific Genetic Programs to Induce Asymmetric Cell Fates<br />
# Engineering stability in gene networks by autoregulation<br />
# Model-Driven Designs of an Oscillating Gene Network<br />
# Synchronizing genetic relaxation oscillators by intercell signaling (http://www.pnas.org/cgi/reprint/99/2/679)<br />
<br />
== Variable Mapping ==<br />
<br />
{| class="wikitable" border="1" cellspacing="0" cellpadding="2" style="text-align:left; margin: 1em 1em 1em 0; background: #f9f9f9; border: 1px #aaa solid; border-collapse: collapse;"<br />
! Variable <br />
! Compound<br />
|-<br />
| R<br />
| lacI<br />
|-<br />
| I<sub>R</sub><br />
| IPTG<br />
|-<br />
| S<br />
| tetR<br />
|-<br />
| I<sub>S</sub><br />
| aTc<br />
|-<br />
| L<br />
| luxR<br />
|-<br />
| I<sub>L</sub><br />
| AHL<br />
|-<br />
| Q<sub>1</sub><br />
| cI<br />
|-<br />
| Q<sub>2</sub><br />
| p22cII<br />
|-<br />
|}</div>Uhrm