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 the term 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 (v1,v2) 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] (Van Kampen "Stochastic Processes in Physics and Chemistry", Third Edition), we produced the Fokker Planck variant of the equation for the AHL concentration shown above

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.

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