Team:Technion-Israel/Modeling

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Safie by Technion-Israel

Why Should it Work

Why Alpha System Should Work – a deterministic model of alpha system

When modelling our system, we began with the simplest method known – deterministic rate equations. Moreover, from the design it was clear that the most important benchmark for the signal within the system would be the concentration of AHL as a function of time, so we began by modelling this part of our system. It took only a simple derivation (see [1]) to obtain these equations which characterize this part of the system:

(d[mRNALuxI])/(dt) = (vB + vAkA[AHL]2)/(1 + kA[AHL]2) − γmRNALuxI[mRNALuxI] + GateI
(d[LuxI])/(dt) = αLuxI[mRNALuxI] − γLuxI[LuxI]
(d[AHL])/(dt) = αAHL[AHL] − γAHL[AHL]

(For a glossary see [1]).

We began to analyze this system by attempting to simplify it, by assuming a steady state solution wherever possible. Using this method (see [2]) we managed to obtain this equation:

(d[AHL])/(dt) = (vB + vAkA[AHL]2)/(1 + kA[AHL]2) − γAHL[AHL] + GateI

It is clear from the goals of our system, that we want to have some sort of bi-stability in the result, when Gate I is small (see [2]). The answer to whether this condition is met, would obviously depend on the constants of the system for which we could not find a reliable source, but using a simple geometric analysis of the phase space (see [3]), we were able to produce a graph showing for which values of (v_A,v_B) we could configure the system (by changing the IPTG concentration and the OD) to show bi-stability:

*map of bi-stability in the alpha system*

Why Should it Fail

Why Alpha System Should Fail – a stochastic model of alpha system

The above model assumes a low-noise system (as do all rate equation models), but especially when constructing a bi-stable network, it is important to consider the noise. To do this we need to create a stochastic model, which in our case, we based upon the commonly used Fokker Planck equation. Using the derivation found in [4] (book on stochastic models from Roee), we produced the Fokker Planck variant of the equation for the AHL concentration derived in *(link to “Why Alpha System Should Work”)*

After analyzing this equation as explained in [5], we produced the following results (using a point on the (v_A,v_B) plane which the previous analysis showed would be bi-stable)

*Fokker Planck Results*

Synthetic Biofilm Formation

A Simulated Model for the Azo-Benzene

*****Yair's part******

We improved this simulation by adding options for these clusters to degrade, initially by giving the connections the ability to be torn apart, from simple stress, but later on, when the size of these clusters became clear from the model, we decided to add another degradation due to problematic structural integrity of the cell clusters.

*Graph of large Cell Cluster with structurally weak point/s highlighted*

We define a cluster as having a problematic structural integrity, if it is very large, and can be divided into two large enough parts which are held together by only a small number of cells. When formulating the condition for which we want to search over the graph of cells (defined as connected if they have an Azo-Benzene link between them), we obtain an algorithmic problem which (while we have not proven is NP-hard), we do not know how to solve (or how to polynomialy verify).

As a result, we had to try to create a probabilistic algorithm which would solve the problem of locating these weak points in the cluster (see [11]). When running this algorithm several times on the above cluster, we were capable of finding the weak points indicated in the following graph:

*Graph of weak points*

When combining these degradation effects with our original model, we got the following behavior:

*Results of new AB simulation*

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