The diagram shows a cell that uses a food source X to drive an autocatalytic metabolism which produces in an AND-side reaction a molecule R. R is converted into M which is a membrane molecule that allows growth of the surface area of the cell. The mathematica code for modeling the above system is very simple and shown below. So far, I have not included a model of the concentration of expressed protein enzymes. This complicates matters. In a very abstract model one might wish to ignore the process of template replication, transcription and translation, and define [E] as a function (perhaps a delayed function) of A1, which could be considered to be some energetic co-factor essential to cellular processes.
No Chaos!?!?
The above dynamics are produced, which you can see are certainly not chaotic. The replication period reaches one stable value. What is the minimal change that must be made to obtain chaos in this system? Let us attempt to introduce a delay or a threshold in the conversion of R into M. This could occur due to the enzyme E that converts R into M being insensitive and cooperative, such that a steep sigmoidal function exists in its enzyme velocity curve. We model this using a Hill function with high n. This does not seem to produce chaos either.
Another Go Using Runge-Kutta in C++
Mathematica does not allow me to model discontinuous functions using its ODE solver, therefore I've written my own Runge-Kutta program, which is amazingly easy it turns out. Download the Runge-Kutter Outline C++ code here. Download the version used to generate the bifurcation plots here. For this version an even simpler cell model was considered having just three variables R, M and S(surface area), see below. CHAOS ARISES IN THE SYSTEM BELOW WHEN THE THRESHOLD FOR R CONVERSION TO M IS INCREASED, AND WHEN THE RATE OF R CONVERSION TO M IS WITHIN A NARROW RANGE.
The above cell feeds no X, has an autocatalytic core, R, which is also used in a synthetic reaction to produce membrane particls M, which get incorperated into the membrane (irreversibly). The reaction between R and X is reversible. There is a strict threshold Tr below which R is NOT converted into M. We examined the bifurcation diagrams of replication period vs. various parameters to look for chaos.
R threshold was increased from 10 (0) to 60 (500). The most interesting behavior occurs at 30 (200), so the threshold is set to this value in all further experiments.
The above graph shows the influence of altering the rate at which R is converted into M, from 0.230 (0) to 0.240 (100). There appears to be chaotic behaviour at 0.235 (50). We set kr to 0.235 in all further experiments. Let us plot this in finer resolution.
Is this really chaos? There is no clear bifurcation observed. Next we can observe the time evolution of R, M and S at various regions of parameter space.
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1. kr = 0.2348 (480), 2. kr = 0.235 (500), 3. kr = 0.2352 (550), 4. kr = 0.237 (700), 5. kr = 0.2388 (880).
The above graphs (2,5) do appear to show irregular replication periods. Let us examine the trajectory of the system in the space of M and R.
The x-axis shows the concentration of R, and the y-axis shows the concentration of M for kr = 0.2388 (880), (graph 5 above). The phase plot in the non-chaotic case is shown below for kr = 0.231.
