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.:: EducatETH E. coli - Engineering Perspective ::.

.:: Introduction ::.

In order to understand if it is possible to create the learning system that we wanted, we had to run some initial simulations, to see if we could reach the desirable steady states. After creating a basic framework on which to work on, we refined the parameters by searching the available literatures. In the next, we are presented the coupled differential equations that model our system, their parameters and the values that we picked, the results of our simulations, and lastly, we provide our references. For an introduction to system modeling in synthetic biology, please read our modeling tutorial here.

.:: System Model ::.

Based on [1], we modeled the biological system with differential equations. According to what presented in the Biology Perspective, our system is composed of three subsystems:

  1. A subsystem of constitutively produced proteins (see Fig. 1),
  2. The learning part (see Fig. 2), and
  3. The reporting subsystem (see Fig. 3).
The first two subsystems interact, and thus, they should be considered together. The third subsystem has no feedback with the other two, as it is only used for producing the appropriate fluorescent proteins. The subsystem with the constitutively produced proteins serves as a regulatory system, and can be modeled with three decoupled partial differential equations (see Fig. 1):
Subsystem 1: Constitutively produced proteins (Fig. 1)

The second subsystem is the main part of the biological model. This subsystem stores the information concerned the learned chemical, and drives the production of the appropriate reporter, during the recognition phase. It is actually a toggle switch, that reaches a steady state depending on the chemical that it is exposed to (see Fig. 2):

Subsystem 2: Basic learning subsystem (toggle) (Fig. 2)

The third subsystem reports the state that our system is, during the different phases of learning and recognition. During the learning phase, this subsystem reports which chemical the cells are exposed to, and during the recognition phase, it reports if the cells recognize the chemicals that they are currently exposed to (see Fig. 3):

Subsystem 3: Reporting subsystem (Fig. 3)

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. We need to model this complex-forming procedure, with another set of differential equations (Fig. 4):

Allosteric regulation (see Fig. 4)

In order to have meaningful results from our simulations, we browsed through the literature in order to find appropriate values for our parameters. We reduced our parameter space by joining parameters together, and we gave reasonable estimates, for the values that could not be extracted from available publications. Since this is a hard part that every team has to face, we present the table with the chosen parameters below:

Parameter Value Description Comments
c1max 0.01 [mM/h] max. transcription rate of constitutive promoter (per gene) promoter no. J23105; Reference: Estimate
c2max 0.01 [mM/h] max. transcription rate of luxR-activated promoter (per gene) Reference: Estimate
lhi 25 high-copy plasmid number Reference: Estimate
llo 5 low-copy plasmid number Reference: Estimate
ap22cII,LacI 0.1 - 0.2 basic production of p22cII/LacI-inhibited genes Reference: Conclusions after discussion
ap22cII 0.1 - 0.2 basic production of p22cII-inhibited genes Reference: Conclusions after discussion
acI,TetR 0.1 - 0.2 basic production of cI/TetR-inhibited genes Reference: Conclusions after discussion
acI 0.1 - 0.2 basic production of cI-inhibited genes Reference: Conclusions after discussion
ap22cII,TetR 0.1 - 0.2 basic production of p22cII/TetR-inhibited genes Reference: Conclusions after discussion
acI,LacI 0.1 - 0.2 basic production of cI/TetR-inhibited genes Reference: Conclusions after discussion
dLacI 2.31e-3 [pro sec] degradation of lacI Ref. [10]
dTetR 1e-5 [pro sec]/2.31e-3 [pro sec] degradation of TetR Ref. [9]/ Ref. [10]
dLuxR 1e-3 - 1e-4 [per sec] degradation of LuxR Ref: [6]
dcI 7e-4 [per sec] degradation of cI Ref. [7]
dp22cII degradation of p22cII
dYFP 6.3e-3 [per min] degradation of YFP suppl. mat. to Ref. [8] corresponding to a half life of 110min
dGFP 6.3e-3 [per min] degradation of GFP in analogy to YFP
dRFP 6.3e-3 [per min] degradation of RFP in analogy to YFP
dCFP 6.3e-3 [per min] degradation of CFP in analogy to YFP
KLacI 1.3e-3 - 2e-3 [mM/h] lacI repressor dissociation constant lower value is from Ref. [2], higher value is from Ref. [5]
KIPTG 1.5e-10 [mM/h] IPTG-lacI repressor dissociation constant Ref. [5]
KTetR 5.6 (+-2) [nM-1] tetR repressor dissociation constant Ref. [1] Note: This is for heptameric operator.
KaTc 1120 (+-400) [nM-1] aTc-tetR repressor dissociation constant Ref. [1] and Ref. [3]. According to Ref. [3], ratio between tetR repressor dissociation constant and aTc-tetR repressor dissociation constant is 0.4/80.
  • 0.003 [mM/s]
  • 55 - 520 [nM]
luxR activator dissociation constant Ref: [6] (Ratio of association/dissociation)
The values of the second bullet are what I've got using the data in the same reference, only considering the systems with dimerization (Uhrm 09:13, 5 October 2007 (EDT))
  • 0.009 [mM/s] - 0.1 [mM/s]
  • 0.09 - 1 [µM]
AHL-luxR activator dissociation constant Ref: [6] (Ratio of association/dissociation)
The values of the second bullet are what I've got using the data in the same reference (Uhrm 07:25, 5 October 2007 (EDT))
KcI 2e-3 [mM/h] cI repressor dissociation constant Ref. [5]
Kp22cII p22cII repressor dissociation constant
nLacI 1 lacI repressor Hill cooperativity Ref. [5]
nIPTG</sub> 2 IPTG-lacI repressor Hill cooperativity Ref. [5]
nTetR 3 tetR repressor Hill cooperativity Ref. [3]
naTc 2 (1.5-2.5) aTc-tetR repressor Hill cooperativity Ref. [3]
nLuxR 2 luxR activator Hill cooperativity Ref: [6]
nAHL 1 AHL-luxR activator Hill cooperativity Ref. [3]
ncI 1.9 cI repressor Hill cooperativity Ref. [5]
np22cII p22cII repressor Hill cooperativity

.:: Simulations ::.
.:: References ::.

[1] This book
[2] A synthetic time-delay circuit in mammalian cells and mice (
[3] Detailed map of a cis-regulatory input function (
[4] Parameter Estimation for two synthetic gene networks (
[5] Supplementary on-line information for "A Synthetic gene-metabolic oscillator" (no link)
[6] Genetic network driven control of PHBV copolymer composition (http://doi:10.1016/j.jbiotec.2005.08.030)

.:: To Do ::.