Team:Technion-Israel/Modeling

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Safie by Technion-Israel

Why it Should 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:

figure AlphaNorm.jpg
Figure 1 On the left:a map of the values of the parameters of the system for which it is bi-stable (bi-stable in red, mono-stable in blue). On the right: a map of the normalized bi-stability parameter we have defined as a function of its parameters.

It is clear from this graph, that the alpha system is bi-stable for a large part of the range of possible inputs.

Why it Should 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)

figure 1,2,0.jpg figure 1-2-0.gif figure 1,2,1.jpg figure 1-2-1.gif
Figure 1 From Top To Bottom: Fokker Planck in the Alpha System when the system begins off, and then when it begins off: On the left is a heat map of the probability distribution function, as a function of time. On the right is a gif showing the probability distribution function over 100 timelapses
Clearly the “on” state (high AHL concentration - low on the graph) of our system is the more stable state of our system - so much so that it can spontaneously switch to the on state. This means that our system is bound to have a high likelihood of false positives.
Synthetic Biofilm Formation

A Simulated Model for the Azobenzene


We aimed to create a dynamic simulation of bacteria with Azobenzene molecules attached to their membranes. These molecules, once activated by an outside stimulus (usually a certain wavelength of photons) - will act as a sort of “Velcro” between the bacteria; they attach to other bacteria upon contact forming clusters.
The clusters of bacteria will thereafter act as one unit - a biofim.
With this model we opted for a "brute-force" simulation of particles in a fluid under the following terms:

• The simulation “Playground” will be a discreet matrix of the dimentions x × y × z.
• Each bacterium will occupy a 1 × 1 × 1 point in in space.
• For every t=t+1 passage of time, each bacterium “tumbles” a random amount of steps in a random direction, we called this a "Tumble Vector"
• Each bacterium can have either a “sticky” or “non-sticky” value corresponding to it. This is equivalent of assuming that all azobenzene molecules “switch on” at once in all directions.
• Each sticky bacterium (i.e. with a “sticky” value) will “attach” to any “neighbor” (i.e. a bacterium with a location of 0, ± 1 in either direction), after which they will “tumble” together as one cluster, with their direction being determined by summing up all the bacteria's "Tumble Vectors" together.
• Once a bacterium has a neighbor attached to it, they cannot separate and that neighbor's location is forever occupied by the same bacterium, it cannot be overridden.
• A sticky bacterium on the edge of a cluster can stick to any neighboring bacterium. If said neighbor is already a part of a cluster we now have two clusters joining to form a "super-cluster" – which does not vary in definition from a normal cluster programming-wise.

The simulation was written using C++, using tumble and playground sizes values to simulate the world of actual bacteria. The results were then rendered in MATLAB:

figure Nucleation.gif
Figure 3 A simulation of the clustering of cells in the presence of AB. The simulation contains 10,000 cells of which 2,000 are sticky simulated over 400 secs, with a time-lapse of 4 seconds per image. We can clearly see that over half of the cells are joined into 1 cluster at the end of the simulation, leading us to believe that the clustering would have a visible effect on the OD of the sample.