Introduction

Based on a set of ordinary differential equations (ODE) describing the kinetics of the cells differentiation, we design a model to find the best differentiation rate ie find the constant reaction \(k_{1}\) to optimize the vitamin production.
We find out that a stochastic algorithm is 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 try to design a simple model with the minimum amount of parameters.
We find seven important parameters used in 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 media.
• \([DC]_0\) : initial concentration of differentiate cells in the media.

Kinetic equations

Four simple kinetic equations are writen.
\[ \begin{align} MC \xrightarrow[]{k_{1}} DC \\ MC \xrightarrow[]{k_{2}} 2.MC \\ DC \xrightarrow[]{k_{3}} 2.DC \\ DC \xrightarrow[]{k_{4}} DC + Vitamin \end{align} \]

Formal mathematical solution

Translation in ordinary differential equations

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

Mathematical resolution

Simple ordinary differential equations resolution methods are used to find the solution of the previous equations \((5)\), \((6)\), and \((7)\).
We express 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 in the media, the more vitamin are produced.
However, it could be interesting to compare different \(\frac{[MC]_0}{[DC]_0}\) ratio but we prefer to focus on the rate constant \(k_{1}\).
We try to find the best \(k_{1}\). 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 notice 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 the \(k_{1}\) that optimize the vitamin production ie \(k_{1} = 0.207\).

Deterministic evolution of mother cell, differentiate cell and vitamin concentrations

We write 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 obtain the following graph.

Stochastic model

Conclusion