Construction of the model

PDF version of the next two sections (Equations for Toggle Switch and Quorum sensing)

As a proof of concept, we will consider anhydrotetracycline (aTc) instead of Mercury in our model, and the transcriptional regulator TetR in place of MerR. The TetR system is well characterized in E. coli, which facilitates the modelling. In addition, if the system works well with aTc, it should work as well with Mercury, Copper or lead for example.

In addition, since aTc diffuse freely accross the membrane of the cell, we do not have to take complex uptake mechanisms into considerations. Mercury is ionized and its entry in the cells depends on the presence of transporter protein.
So in the following chapter, we are going to use TetR instead of MerR and aTc instead of Mercury.

The Two modules of Modelling

Large-scale biochemical networks are classically decomposed into modules of smaller size to facilitate the model building process and the model calibration. The approach consists in modelling each module separately and integrating the different models into a larger model of the whole system. Like divided our genetic network in severals part (see genetic network), we also divided our model in two independant level of modelling:

• The Toggle Switch module: modelling this network module allows to predict in which state the bacteria are. In particular, we can determine the switching conditions in bacteria and the localization of the switching area on the device (that depend on both IPTG and aTc concentrations). This gives indications on how to adjust the size of the device to improve the aTc detection.
• The Quorum Sensing and Coloration module:the corresponding model is used to predict the localization and the width of the red line on the device. The modelling results are used to improve the device precision, for instance, by choosing between a device with a continuous bacterial layer and a device with strips containing bacteria. Each strip has the same concentration of aTc, but a different IPTG concentration.

The first level of the model will allow us to define the switch conditions in bacteria, depending on the concentrations of pollutant and IPTG. From these results, we could see where the switch zone will appear on the device and therefore, the position of the red line. These parts of the results indicate the range of mercury that can be detected.

The second level will provide us with the red line width, an indicator of the system precision. From these results, we could improve the precision of our device, for instance, by choosing between a device with a continuous bacterial layer and a device with strips containing bacteria. Each strip has a different IPTG concentration, but the same concentration of Mercury.

Two independent levels of modelling which are coupled in order to get a global simulation of the device.

Establishment of the equation

Toggle switch

In a first part, we define a model to caracterize the Toggle Switch module of our genetic network.

1. Biological Models

A toggle switch consists of two genes, each coding for a protein that represses the expression of the other gene. This double repression system ensures the basic function of a toggle switch: a bistable system which can be switched from one state to the other by putting, in our example, some IPTG or aTc molecules in the medium.

The biological system we are trying to implement is more complex, on both the biological and physical side. However, the toggle switch model is basically the same. In the study of the toggle switch itself, the system can be reduced to a simple two-state subsystem that we will then use for the rest of our modelling. However, our toggle switch model is basically the same: we can reduce it to a simple two-state subsystem that we will then use for the rest of our modelling. The toggle switch itself is not influenced by the rest of the system, if we do not consider the RsmA regulatory system that will be modeled at the very end of our work. Thus we will be able to model the toggle switch independently and then build the rest of the model upon this basis.

2. Mathematical Models

In order to get the state of the bacteria, we need to compute the concentration of both repressors. To obtain these variations, we develop a system of ordinary differential equations (ODES) which governs the behavior of the Toggle Switch.

$ \frac{d[TetR]}{dt} = \frac{k_{pLac}.[pLac]_{tot}}{1 + (\frac{[lacI]}{K_{pLac} + \frac{K_{pLac}.[IPTG]}{K_{lacI-IPTG}}.})^\beta} - \delta_{TetR}.[TetR]$
$\frac{d[lacI]}{dt} = \frac{k_{pTet}.[pTet]_{tot}}{1 + (\frac{[tetR]}{K_{pTet} + \frac{K_{pTet}.[aTc]}{K_{TetR-aTc}}.})^\gamma} - \delta_{lacI}.[lacI]$

With this ODE system, we follow the evolution of the concentration of the two toggle switch repressors, TetR and LacI. This gives us the state of the bacteria. A demonstration of this ODE system is available in the previous pdf file. We briefly explain the equations below.

Both equations are developed following the same approach. There is a term of production and a term of degradation. The parameters $k_{pLac}.[pLac]_{tot}$ represent the protein synthesis rate from the pLac promoter. K_{pLac} and K_{lacI−IPTG} are the dissociation constants of LacI and pLac, and LacI and IPTG, respectively. The parameters β and γ denote the cooperativity of repression, that is, the number of repressors bound to the promoter.

According to these equations, the rate of variation of repressor TetR is inhibited by the repressor LacI. The degree of inhibition is modulated by the IPTG concentration. Reciprocally in the second equation, the rate of variation of LacI depends on TetR, whose negative effect is modulated by aTc.

This indicates that the two equations are coupled. And also that only one repressor could be predominant, as shown later.

These two equations can be easily computed with a differential solver. We can precisely estimate the effects of each parameter.

Establishment of the equation

Quorum sensing

1. Mathematical Models

In a second part, we define a model able to characterize the quorum sensing module of our genetic network, which involves the quorum sensing genes CinI and CinR, as well as the signaling molecule AHL.
In order to model the Quorum Sensing module, we detail below the different reactions taking place inside this module. For reasons that will become clear later, we focus on a particular area of our device, where neighboring bacteria will have a different behavior, although they carry the same genetic circuit.


Figure 1:Mechanism of Quorum Sensing diffusion at the boundary

According to the figure, several steps must be taken into account:

1. the production of the Quorum Sensing molecule
2. the secretion of the molecule
3. the diffusion of the molecule in the medium
4. the penetration of the molecule
5. the complexation of the molecule with its receptor
6. the activation of the coloration

To model these mecanisms, we need to follow the evolution of the following quantities

• $[QS_i]$ concentration of the intracellular Quorum Sensing molecule.
• $[QS_e]$ concentration of the extracellular Quorum Sensing molecule.
• $[cinR]$ concentration of the Quorum Sensing receptor.
• $[cinI]$ concentration of the Quorum Sensing producer enzyme.
• $[cinR_{comp}]$ concentration of the complexe cinR-QS.

Based on the Bangalore 2007 iGEM team, we get the following system of equations:

$\frac{d[QS_i]}{dt} = \eta([QS_e]-[QS_i])-\delta_{QSi} [QS_i] + k_{QS-production}'[cinI]$

$\frac{d[QS_e]}{dt} = \rho v_c\eta([QS_i]-[QS_e])-\delta_{QSe} [QS_e] + D_{diff}\frac{\partial^2 [QS_e]}{\partial x^2}$

$\frac{d[cinR_{free}]}{dt} = \frac{k_{pLac}.[pLac]_{tot}}{1 + (\frac{[lacI_{total}]}{K_{pLac} + \frac{K_{pLac}.[IPTG]}{K_{lacI-IPTG}}.})^\beta} - \delta_{cinR}[cinR_{free}] - V_{complexation}$

$\frac{[cinR_{comp}]}{} = K_{comp}([cinR_{free}].[QS_i])$



In the first equation, expressing the evolution of the concentration of the intracellular Quorum Sensing molecule, 3 terms are involved: $[QSe]−[QSi]$, which describes the diffusion through the membrane, $\delta_{QSi}[QSi]$, the degradation and $k_{QS−production}[cinI]$, the production of Quorum Sensing molecule by the enzyme CinI. In the second equation, expressing the evolution of extracellular Quorum Sensing, there is no production term, but a spatial diffusion term $D_{diff}\frac{\partial^2 [QS_e]}{\partial x^2}$.

The concentrations of CinI and CinR can be obtained by using the toggle switch modelling. Indeed, since CinI is on the pathway of LacI and CinR on the pathway of Tet, their evolution follow the same equations. In addition we consider the complexation of the CinR protein and the Quorum Sensing molecule, which is expressed by the term $V_{complexation}$ in the CinR equation: \[V_{complexation} = K_{comp}[QS_i][CinR_{free}]\] with $K_{comp}$ the affinity constant of AHL for cinR.

In order to simplify the model, we considered that the complexation of CinR to AHL is total and depends only on $[cinR]$ and $[QS_i] which is the limitant factor$.

To solve this set of equations, we couldn't use a classical ODE solver from Matlab (Mathworks), because the equations are derived both in time and space. We therefore needed to use the finite difference method. To that aim, we build a matrix based on Taylor's series discretization.

2. Definition of the matrix

To solve this set of equations we have to use a matrix that will describe our system in both space and time. for example for the QS molecule outside of the cell :

$M_{QSe}(m,n) = [QS_e](x,t)$
$M_{QSe}(m+1,n+1) = [QS_e](x + dx,t+dt)$
• On the spatial point of view, we only consider the x dimension, as the IPTG gradient will be only evolving along this dimension. Thus we consider the state of our cells is the same along the width of our plate.
• With this Matrix, and after computation of all the terms, we can get the entire behaviour of CinI, CinR, QS inside and outside the cells.
• The first line of the Matrix equals 0. These are the initial conditions we set to 0 at time t = 0s.
• On the borders of the plate (x = 0 and x = L) the model used has to be different, limit conditions will be set.
• Of course, QSi, CinI and CinR matrices will be similarly implemented.
3. Discretization of the equation

With our continuous equations set, we want to obtain discrete definition of each of the matrices. interdependencies of the equations imply that the computation of the matrices will be performed on the entire CinI matrix first, then each line of the Qi and Qe matrices will be computed alternatively. The computation of the CinR will be eventually performed.

Parallel computation of all the matrices without proper control is not possible indeed, as the terms of Qi matrix will depend on the Qe terms of the preceding line (and vice-versa). Discretization is obtained with first order taylor series :

$ M_{QSi}(m,n+1) =\Delta t (\eta (M_{QSe}(m,n) - M_{QSi}(m,n)) - \delta_{QSi}.M_{QSi}(m,n) + k_{QSp}M_{CinI}(m,n)) + M_{QSi}(m,n) $

$ M_{QSe}(m,n+1) =\Delta t ( D_{m} + D_{diff} \frac{M_{QSe}(m+1,n) - 2 M_{QSe}(m,n) + M_{QSe}(m-1,n)}{\Delta x^2}) + M_{QSe}(m,n) $

$ with D_{m} =\rho v_c \eta .M_{QSi}(m,n) - M_{QSe}(m,n)(\delta_{QSe} + \rho v_c \eta) $

$ M_{CinR_{free}}(m,n+1) = \Delta t (\frac{k_{pTet}[P_{Tet total}]}{1+ (\frac{[TetR]}{K_{pTet}(1+\frac{[aTc]}{K_{TetR - aTc}})})^{n_{pTet}}} - M_{CinR_{free}}(m,n)(\delta_{CinR} - k_{comp}.M_{Qi}(m,n))) + M_{CinR_{free}}(m,n)$

With these discrete equations the 4 matrices can be computed through simple calculation loops over each line. The CinI matrix does not depend on space dimension, it is then possible to compute it without discretization with a differential solver.

Moreover to get the coupling of toggle switch and quorum sensing modules, we used the results of the first one (lacI and TetR evolution) to get the evolution of quorum sensing proteins (respectively cinI and cinR). It means that the output of the first module are used as input for the second module.

Our algorithms

An archive containing our Matlab scripts for the deterministic modeling can be found here (file Deterministic_archive.tar.gz). You can launch and ODE-based simulation(see our ODEs in the two previous sections) with the file biosenseur1Dmain.m.

Several dialog boxes will pop up to enter the specificities of the simulation: (physical specificities of the device, chemical species concentrations and IPTG gradient).

At the end of the simulation you obtain 3 matrices named M_stock, M_QS and M_comp containing the concentrations in each protein species at each time point and on each physical point of the plate. We wrote three Matlab scripts (DynamicplottingTS, DynamicplottingQS and DynamicplottingCP) that dynamically display the protein concentrations. For a good understanding of the models and of our results, we also wrote a script (Imageshow.m) to illustrate the plate coloration.