Glasgow/Modeling
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Summary
Framework
We have used a framework which unifies the qualitative, stochastic and continuous worlds, as a basis for our overall approach to modelling and analysing the biochemical pathways. Each perspective adds its contribution to the understanding of the system, thus the three approaches do not compete, but complement each other. Qualitative descriptions are abstractions over stochastic or continuous descriptions, and the stochastic and continuous models approximate each other. Note: this framework is based on David Gilbert, Monika Heiner and Sebastian Lehrack (2007). [[Media:Edinburgh_proceedings.pdf "A Unifying Framework for Modelling and Analysing Biochemical Pathways Using Petri Nets"]]. In proceedings Computational Methods in Systems Biology CMSB 2007 (Computational Methods in Systems Biology), Springer-Verlag LNCS/LNBI Volume 4695, pp. 200-216.
Our overall framework is illustrated in Figure (frameworkfig) that relates the three major ways of modelling and analysing biochemical networks that we have used: qualitative, stochastic and continuous.
The most abstract representation of a biochemical network is qualitative and is minimally described by its topology. Initial descriptions can be obtained from biochemists, and are often in some semiformal representation. These can easily be transformed into a formal description at this stage which is usually a bipartite directed graph with nodes representing biochemical entities or reactions, or in Petri net terminology places and transitions (link to petrinetsection).
The qualitative description can be further enhanced by the abstract representation of discrete quantities of species, achieved in Petri nets by the use of tokens at places. These can represent the number of molecules, or the level of concentration, of a species. A particular arrangement of tokens over a network is called a marking. The standard semantics for these qualitative Petri nets (QPN) does not associate a time with transitions or the sojourn of tokens at places, and thus these descriptions are time-free. The qualitative analysis considers however all possible behaviour of the system under any timing. The behaviour of such a net forms a discrete state space.
Timed information can be added to the qualitative description in two ways -- stochastic and continuous. The continuous model replaces the discrete values of species with continuous values, and hence is not able to describe the behaviour of species at the level of individual molecules, but only the overall behaviour via concentrations. We can regard the discrete description of concentration levels as abstracting over the continuous description of concentrations. Timed information is introduced by the association of a particular deterministic rate information with each transition, permitting the continuous model to be represented as a set of ordinary differential equations (ODEs) (link to ODE section). The concentration of a particular species in such a model will have the same value at each point of time for repeated experiments. The state space of such models is continuous and linear. It is also possible to linearise ODE descriptions, by for example Laplace transforms, in an attempt to increase modularity in the system description and hence to facilitate model construction (link to carolis' section). This approach results in transformations from the time-domain, in which inputs and outputs are functions of time, to the frequency-domain.
The stochastic Petri net (SPN) description preserves the discrete state description, but in addition associates a probabilistically distributed firing rate or rate constant (waiting time) with each reaction and an associated rate law or hazard function which depends on the state of the system. Thus, the time-evolution of a reaction can be regarded as a stochastic process, and the time-evolution of the states of such a reaction system can be regarded as a Markov process with a discrete state space. In this context we can refer to the Chemical Master Equation (CME), which is equivalent to Kolmorogov's forward erquation for a stochastic kinetic process (Wilkinson 2006). It is quite straightforward to simulate such a system, and this is usually done with the standard discrete event simulation procedure known as "the Gillespie algorithm" (Gillespie, 1977). In our case we have directly simulated the system from CMEs using the Gillespie algorithm (link to stochastic section).
In summary, the qualitative time-free description is the most basic, with discrete values representing numbers of molecules or levels of concentrations. The qualitative description abstracts over two timed, quantitative models. In the stochastic description, discrete values for the amounts of species are retained, but a stochastic rate is associated with each reaction. The continuous model describes amounts of species using continuous values and associates a deterministic rate with each reaction. These two time-dependent models can be mutually approximated by hazard functions belonging to the stochastic world. The QPN is an abstraction of the SPN, sharing the same state space and transition relation with the stochastic model, with the probabilistic information removed. All qualitative properties valid in the QPN are also valid in the SPN, and vice versa.
Detailed reports
Minicap Sensitivity Analysis Program PackageThe Multi-Parametric and Initial Concentration Sensitivity analysis (Minicap) package is a Matlab function which executes a chosen Dynamic or Stochastic System Function for a defined number of different variable values across any desired range. The subject of analysis can either be constants in the user's system eg. parameters in a biological system (MPSA) or initial values of the variables in the user's system eg. initial substrate concentrations (ISCSA). The program will output a plot for each variable showing a comparison of acceptable and unacceptable samples across the subject's range with 3 calculated quantitative comparison figures: the Correlation Coefficient, the Area between acceptable and unacceptable curves, and the Standard Deviation of the gradient of the acceptable plot. The comparative and intrinsic sensitivity of each chosen subject is thus highlighted. A plot showing the trend of the Substrate of Interest over time is also displayed. As well as this report, the Minicap package contains a User Manual in html, a number of example codes and all the novel (i.e. not ode15s) function and text files required to run Minicap. To see the full PDF report on Minicap [click here]. |
BioBrick library [http://compbio.dcs.gla.ac.uk/iGEM2007/BioBricklibrary.zip BioBrickLibrary.zip] An add on Library to Simulink for modelling dynamical biological systems at Brick (Gene) level. Simulink is a program dedicated for dynamical system simulation, however in depth knowledge of dynamics is needed if one is to simulate system mentioned above. The BioBrick library has all the blocks as well as GUI (Graphical User Interface) needed to do the job without understanding how Simulink works. It uses drag and drop system and shares all constants in Matlab’s .m file, so it is easy to store and update them. BioBrick library’s main aim is to tell whether and how different topology will influence the output of the system. If actual rate constants are known it can be used instead or as complimentary to ODE modelling. However it must be noted that ODE rate constants ARE NOT TRANSFERABLE to BioBrick library. More information can be found in ElectrEcoBluSimulinkManual document. |
Microbial Fuel Cell Evaluation During the course of the project some introductory work has been done with microbial fuel cells in order to prepare us for the envisaged final stage of the project. We have had three fuel cells at our disposition supplied by the UK's NCBE. The experience we have gained and some of the results have been aggregated in this short work. | other |
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