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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)
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.
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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