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 a stochastic programm based on the Gillespie’s stochastic simulation algorithm (SSA). With this programm we obtain an accurate analysis of the vitamin production.

Ordinary differential equations model

Parameters

We try to design a simple model with the minimum amount of parameters.

• \(t\) : 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 chosen.
\[ \begin{align} MC \xrightarrow[]{k_{1}} DC \\ MC \xrightarrow[]{k_{2}} 2.MC \\ DC \xrightarrow[]{k_{3}} 2.DC \\ DC \xrightarrow[]{k_{4}} DC + Vitamin \end{align} \]

Formal mathematical solution

Tranformation in ordinary differential equations

We use the law of mass action to write the ordinary differential 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.
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, in fact, not relevent parameters because it seems logical that the more cell you have, the more vitamin you product.
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.
This is a graph example obtained with a simple MATLAB programm disponible 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 \([Vitamin]\).

Results

Stochastic model

Conclusion