Team:USP-Brazil/Model:Stochastic

Stochastic model

Introduction

We want to simulate a cell where is happening all the chemical reactions discribed in the Deterministic Model as a Stochastical Process whose states are determinated by a collection of nine numbers:

\begin{equation} (et, met, X_f,X_{et}, X_{met}, P_f,P_{et}, P_{met}, R) \end{equation} \begin{equation} A \longrightarrow B \end{equation} \begin{equation} p(t) = \lambda e^{-\lambda t} \end{equation}

These are the only kind of distribution in continuous time which do not have a "memory" that means:

\begin{equation} P(X (t+\delta t) = i | X(t) = j) =P(X (\delta t) = i | X(0) = j) \end{equation}

So the probability does not depend of which states the system was in the time interval [0,t], that is, the future state of the system only depends on its present state.

A larger explanation of that is in Chapter 5 of [1].

When we have a number $[A]$ os molecules "trying" to react the first reaction will ocour with probability distributtion, (as we see on sections 5.4.1 and 5.7.3 in [1]): \begin{equation} p(t) = [A] \lambda e^{- [A] \lambda t} \end{equation} \begin{equation} \frac{d[A]}{dt} = - k_A [A] \end{equation} Analising that the mean life time must be the same in both, to fit the same kind of experimental results, we realized that: \begin{equation} \lambda = k_A \end{equation} \begin{equation} \end{equation} \begin{equation} \end{equation} \begin{equation} \end{equation} \begin{equation} \end{equation} \begin{equation} \end{equation} \begin{equation} \end{equation} \begin{equation} \end{equation}

References

[1] Sheldon M. Ross, Stochastic Process, Wiley, New York 1996

[2] Radek Erban, S. Jonathan Chapman, Philip K. Maini: A prac tical guide to stochastic simulations of reaction-diffusion processes , http://arxiv.org/abs/0704.1908

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