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}

Those numbers represent the amount of mollecules of each chemical species in the cell. Respectively, the amount of ethanol, methanol, free transcription factor, ethanol-binded transcription factor, methanol-binded transcription factor, free promoter, ethanol-binded promoter, methanol-binded promoter and RFP molecules inside the cell.

Continuous Time Markov Chain

The main idea behind the Stochastic Modeling is that given an chemical reaction, e.g.: $$A \longrightarrow B$$ A single mollecule will react after some time $t$, following a probability distribution of the form $$p(t) = \lambda e^{-\lambda t}$$

These are the only kind of distribution in continuous time which do not have a "memory" or in Mathamatical language:

\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 there are $[A]$ os molecules "trying" to react the first reaction will occur with a probability distribution (as we see on sections 5.4.1 and 5.7.3 in [1]) given by:

$$p(t) = [A] \lambda e^{- [A] \lambda t}$$

We can see that this $\lambda$ parameter really looks like a parameter of a deterministic equation: $$\frac{d[A]}{dt} = - k_A [A]$$

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

\begin{equation} X_f + et \rightleftharpoons X_et \end{equation}

The Probability of each reaction happens before a time t<\i> follows the distribution:

\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=====

If we have some reactions happening at the rate

$$p_i(t) = - a_i e^{-a_i t}$$

Then the probability of a reaction occurring at a time previous to $t_A$ is:

\begin{equation} P_i(t \leq t_A)= \int_0^{t_A} p_i(t) = 1 - e^{-a_i t} \end{equation}

And the probability that the reaction does not occour before this time is:

$$P_i(t \geq t_A) = 1 - P(t \leq t_A)= e^{-a_i t} $$

Being all the reactions independents, the probability of no reaction occurring before a given time point $t_* $is equal to the probability all reactions occurring before $t_A $. That is:

$$ P_0(t \geq t_A) = \prod_{i=1}^{11} P_i(t \geq t_A) = \prod_{i=1}^{11} e^{- a_i t_A} $$

Defining

$$a_0 = \sum_{i=1}^{11} a_i $$

So we realize that the probability of one reaction occurring at a time before $t_A$ is

$$ P_0(t \geq t_A) = e^{a_0 t_A}$$

And therefore the time until the first reaction will be given by the distribution:

$$p(t) = - a_0 e^{a_0t} $$

Through a parameter change, it is possible to calculate the distribution of the time interval between two reactions. Given a random variable $r$, uniformly distributed in $[0,1]$. The quantity

$$ \Delta t = \frac{1}{a_0}\ln (1/r)$$ has a probability distribution given by: $$ - a_0 e^{a_0t} $$ Once the the time $t + \Delta t$ when the next reaction occurs is calculated, all that remains is to choose which reaction should happen. Another random variable uniformly distributed in $[0,1]$ is used, and the probability correspondent to each reaction $i$ is $P_i = a_i/a_0 $.

Guilespe

This stochastic simulation was implemented by the Gillespie algorithm, described above and well explained in [2]. This algorithm consists in repeating the instructions below beginning at some initial condition for the metabolites, and updating the amount of each kind of mollecule after each time interval. \begin{itemize} \item[•] Generate two random numbers $r_1, r_2$ uniformly distributed in $[0,1]$ \item[•] Compute the time when the next chemical reaction takes place as $t+\Delta t$ where $$ \Delta t = \frac{1}{a_0}\ln (1/r)$$ \item[•] Choose with reaction occurs using $r_2$ and giving each reaction $i$ a probabilty $P_i = a_i/a_0 $ \item[•] Update the amount of molecules at time $t + \Delta t.$ according to the table above. \end{itemize} \begin{equation} \end{equation} \begin{equation} \end{equation} \begin{equation} \end{equation} \begin{equation} \end{equation}

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