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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:
(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:
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:
It is clear from this graph, that the alpha system is bi-stable for a large part of the range of possible inputs.
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*
The RNA Splint – Deterministic and Stochastic Models of this Noise Reduction Method
When considering an additional design for our system, we thought of the idea for the RNA splint:
Basically, instead of directly producing the AHL, the cell will use a system called an RNA Splint (*link to RNA Splint*) to build the mRNA encoding for the production of AHL, in two parts and also produce a third component which would combine the two.
Adjusting the equation for the production of AHL
for this change (see [6]), we obtain:
We thought these changes would improve the bi-stability of the system (thereby reducing the odds of a false positive), because they enhance the non-linearity inherent in the system which has been shown to play a vital role in the bi-stability of the system (see [9] – Gardner et. al. [10] – Cold Spring Harbor Vernalization, other sources we copied from [10]). When producing a similar analysis for the phase plane of this gate as we did for the phase plane of the original equation (see [7]), we found the values of (v_A,v_B) for which the system is bi-stable, and compared this analysis to the results of the analysis of the original analysis.
Figure 2 From top to bottom: The Alpha System, the first model for the RNA Splint, and the second model for the RNA Splint. 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.
We then proceeded to produce the Fokker Planck equation for the new system (see [8]), and we got the following results:
*Fokker Planck Results With and without RNA Splint*
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:
*nuclation*
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 Azobenzene 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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