Introduction

Based on a set of ordinary differential equations (ODE) describing the kinetics of the cells differentiation, we designed a model to find the best differentiation rate.
First we developed a deterministic algorithm based on the ordinary differential equations solutions.
Then we find out that a stochastic algorithm could be an other solution to solve our problem.

For system involving large cell counts, the ordinary differential equations model give an accurate representation of the behavior. But with small cell counts, the stochastic and discrete method has a significant influence on the observed behaviour.
These reasons led us to write both a deterministic programm based on the mass action law and a stochastic programm based on the Gillespie’s stochastic simulation algorithm (SSA). With these two programms we obtain an accurate analysis of the vitamin production.

Mass action law model

Parameters

We conceived a simple model with the minimum number of parameters.
We find seven significant parameters for our model.

• \(t\) : fermentation period.
• \(k_{1}\) : rate constant of the mother cell differentiation.
• \(k_{2}\) : rate constant of the mother cell doubling time.
• \(k_{3}\) : rate constant of the differentiate cell doubling time.
• \(k_{4}\) : rate constant of the differentiate cell vitamin production.
• \([MC]_0\) : initial concentration of mother cells in the medium.
• \([DC]_0\) : initial concentration of differentiate cells in the medium.

Kinetic equations

We wrote four simple kinetic equations.
\[ \begin{align} MC \xrightarrow[]{k_{1}} DC \\ MC \xrightarrow[]{k_{2}} 2.MC \\ DC \xrightarrow[]{k_{3}} 2.DC \\ DC \xrightarrow[]{k_{4}} DC + Vitamin \end{align} \]

Formal mathematical solution

Translation in ordinary differential equations

We used the law of mass action to write the ordinary differential equations based on kinetic equations \((1)\), \((2)\), \((3)\) and \((4)\).
We found the following equations.
\[ \begin{align} \frac{d[MC]}{dt}(t) = (k_{2} - k_{1}).[MC](t) \end{align} \] \[ \begin{align} \frac{d[DC]}{dt}(t) = k_{1}.[MC](t) + k_{3}.[DC](t) \end{align} \] \[ \begin{align} \frac{d[Vitamin]}{dt}(t) = k_{4}.[DC](t) \Rightarrow [Vitamin](t) = k_{4}.\int_{0}^{t}{[DC](t').dt'} \end{align} \]

Mathematical resolution

Simple ordinary differential equations resolution methods are used to find the solutions of the previous equations \((5)\), \((6)\), and \((7)\).
We expressed the time evolution of the mother cell, differentiate cell and vitamin concentrations.
\[ \begin{align} [MC](t) = [MC]_{0}.e^{(k_{2} - k_{1}).t} \end{align} \]
\[ \begin{align} [DC](t) = \begin{cases} [DC]_{0}.e^{k_{3}.t} + [MC]_{0}.\frac{k_{1}}{k_{2}-k_{1}-k_{3}}.(e^{(k_{2} - k_{1}).t} - e^{k_{3}.t}) & \mbox{if } k_{1} \ne k_{3} - k_{2} \\ \\ ([MC]_{0}.(k_{2} - k_{3}).t + [DC]_{0}).e^{k_{3}.t} & \mbox{if } k_{1} = k_{3} - k_{2} \end{cases} \end{align} \]
\[ \begin{align} [Vitamin](t) = \begin{cases} [DC]_{0}.\frac{k_{4}}{k_{3}}.(e^{k_{3}.t} - 1) + [MC]_{0}.\frac{k_{4}.k_{1}}{(k_{2}-k_{1}-k_{3}).(k_{2} - k_{1})}.(e^{(k_{2} - k_{1}).t} - 1) + [MC]_{0}.\frac{k_{4}.k_{1}}{(k_{2}-k_{1}-k_{3}).k_{3}}.(1 - e^{k_{3}.t}) & \mbox{if } k_{1} \ne k_{3} - k_{2} \\ \\ [DC]_{0}.\frac{k_{4}}{k_{3}}.(e^{k_{3}.t} - 1) + [MC]_{0}.\frac{k_{4}}{k_{3}}.(k_{3}.t - 1).(\frac{k_{2}}{k_{3}} - 1).(e^{k_{3}.t} - 1) + [MC]_{0}.\frac{k_{4}}{k_{3}}.(\frac{k_{2}}{k_{3}} - 1) & \mbox{if } k_{1} = k_{3} - k_{2} \end{cases} \end{align} \]

Vitamin optimization

Our goal is to optimize the vitamin production. We can only change three parameters : \(k_{1}\), \([MC]_0\) and \([DC]_0\).
\(t\), \(k_{2}\), \(k_{3}\) and \(k_{4}\) are constants. \([MC]_0\) and \([DC]_0\) are not relevant parameters. It seems logical that the more cells are in the medium, the more vitamin are producted.
However, it could be interesting to compare different \(\frac{[MC]_0}{[DC]_0}\) ratios but we prefer to focus on the rate constant \(k_{1}\).
We try to find the best \(k_{1}\). To this end we maximize the vitamin function numerically with MATLAB.
As an example, lets consider the following parameters.

• \(t = 10\)
• \(k_{2} = 0.66\)
• \(k_{3} = 0.33\)
• \(k_{4} = 1\)
• \([MC]_0 = 5\)
• \([DC]_0 = 0\)

We obtain the following graph with a simple MATLAB programm available here.

As you can see, we are able to find the best \(k_{1}\) to optimize the vitamin production.
We noticed that two differents \(k_{1}\) exists to optimize the maximum concentration of differentiate cell \([DC]\) or the maximum concentration of vitamin \([Vitamin]\).
In our case, we chose of course the \(k_{1}\) that optimize the vitamin production ie \(k_{1} = 0.207\).

Deterministic evolution of mother cell, differentiate cell and vitamin concentrations

We wrote a deterministic algorithm with MATLAB using the previous solutions \((8)\), \((9)\) and \((10)\). For those interested, the source code is available here.
In order to optimize the vitamin production, we use the same parameters as previously and set \(k_{1}\) with the previous resulting value ie \(k_{1} = 0.207\).
We obtained the following graph.

As you can see, the mother cell, differentiate cell and vitamin concentrations follow an exponential law of time.
This result seems relevant. The model does not take into account the cells death and the nutrients present in the medium.

Stochastic model

Conclusion