Kinetics Module

Introduction

Based on a set of ordinary differential equations (ODEs) describing the kinetics of the cells diffenrenciation, we design a model to find the best diffenrenciation rate ie find the constant reaction k1 that optimize the vitamin production.
We find out that a stochastic algorithm is an other solution to solve our problem.
For system involving large molecular counts, ODE model give an accurate representation of the behavior. But with small molecular counts, the stochastic and discrete method have a significant influence on the observed behaviour.
For this reason we write a stochastic programm based on the Gillespie’s stochastic simulation algorithm (SSA) and obtain an accurate analysis.

Equations

\[MC \xrightarrow[]{k_{1}} DC\] \[MC \xrightarrow[]{k_{2}} 2.MC\] \[DC \xrightarrow[]{k_{3}} 2.DC\] \[DC \xrightarrow[]{k_{4}} DC + Vitamin\]

\[\frac{d[MC]}{dt}(t) = (k_{2} - k_{1}).[MC](t)\] \[\frac{d[DC]}{dt}(t) = k_{1}.[MC](t) + k_{3}.[DC](t)\] \[\frac{d[Vitamin]}{dt}(t) = k_{4}.[DC](t)\] \[[Vitamin](t) = \int_{0}^{t}{[DC](t').dt'}\]