Team:USP-Brazil/Model:Stochastic

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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}

Our Circuit by Markov Chains

\begin{equation} X_f + et \rightleftharpoons X_et \end{equation} The Probability of each reaction happens before a time $t_0$ is: \begin{equation} P((et,Xf,Xet) \rightarrow (et-1,Xf-1,Xet+1),t)= { \beta_{et}^{+}[Xf][et]} e^{- \beta_{et}^{+}[Xf][et] t} \end{equation} $$ P((et,Xf,Xet) \rightarrow (et+1,Xf+1,Xet-1) ,t)= { \beta_{et}^{-}[Xet]}e^{- \beta_{et}^{-}[Xf][et] t} $$ As we explained before, the Rates $beta_ET$. is the same one from the Deterministic model. ====AQUELA TABELA VAI POR AQUI===== $$p_i(t) = - a_i e^{-a_i t}$$ \begin{equation} P-i(t \leq t_A)= \int_0^{t_A} p_i(t) = 1 - e^{-a_i t} \end{equation} $$P_i(t \geq t_A) = 1 - P(t

See the deterministic model

See the experimental Results

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

RFP Visibility | Deterministic Model | Stochastic Model

@@ Line 80: / Line 80: @@
 \begin{equation}
-P-i(t <t_*)= \int_0^{t_*} p_i(t) = 1 - e^{-a_i t}
+P-i(t \leq t_A)= \int_0^{t_A} p_i(t) = 1 - e^{-a_i t}
 \end{equation}
+$$P_i(t \geq t_A) = 1 - P(t <t_A)=  e^{-a_i t}   $$
+$$ P_0(t \ geq t_A) = \prod_{i=1}^{11}  P_i(t \geq t_A) = \prod_{i=1}^{11} e^{- a_i t_A} $$
+$$a_0 = \sum_{i=1}^{11} a_i $$
+$$ P_0(t \geq t_A) =  e^{a_0 t_A}$$
+$$p(t) = - a_0 e^{a_0t} $$
 \begin{equation}