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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 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:
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] (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 (v1,v2) plane which the previous analysis showed would be bi-stable)
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 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 two new potential models:
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 ([9],[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 (v1,v2) 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.
It is clear from this graph, that the RNA Splint is more bi-stable than the original alpha system. Moreover, it is clear that in both models of the RNA Splint, it is bi-stable in the same points, but not to the same degree.
We then proceeded to produce the Fokker Planck equation for the new system (see [8]), and we got the following results:
Figure 2 Left to Right: The Alpha System, The 1st Model for the RNA Splint, The 2nd Model. Top to Bottom: the Heat Map and then the Time-Lapse of the system starting at each of the modes of actiavtion.
It is clear that the “on” (high AHL concentration - low on the graph) state of all of these systems is more stable than their off state, and that there is a high likelihood of the systems spontaneously mocing to the “on” state (i.e. false-positive). However, we may notice that for both models of RNA Splint, the time for a shift from the “off” state to the “on” state is longer than for the Alpha System which is to say, that there is a far lower chance of recieving a false positive.
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:
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