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Section: New Results

Convergence of stochastic gene networks

Participants: A. Crudu, A. Debussche, A. Muller, Aurélie, O. Radulescu.

We propose simplified models for the stochastic dynamics of gene network models arising in molecular biology. Those gene networks are classically modeled by Markov jump processes, which are extremely time consuming. To overcome this drawback, we study the asymptotic behavior of multiscale stochastic gene networks using weak limits of Markov jump processes.

We consider a set of chemical reactions R r , r; is supposed to be finite. These reactions involve species indexed by a set S=1,,M, the number of molecules of the species i is denoted by n i and X M is the vector consisting of the n i 's. Each reaction R r has a rate λ r (X) which depends on the state of the system, described by X and corresponds to a change XX+γ r , γ r M .

Mathematically, this evolution can be described by the following Markov jump process. It is based on a sequence (τ k ) k1 of random waiting times with exponential distribution. Setting T 0 =0, T i =τ 1 ++τ i , X is constant on [T i-1 ,T i ) and has a jump at T i . The parameter of τ i is given by r λ r (X(T i-1 )):

𝐏(τ i >t)=exp- r λ r (X(T i-1 )) t.

At time T i , a reaction r is chosen with probability λ r (X(T i-1 ))/ r λ r (X(T i-1 )) and the state changes according to XX+γ r : X(T i )=X(T i-1 )+γ r . This Markov process has the following generator:

Af(X)= r f(X+γ r )-f(X)λ r (X).

In the applications we have in mind, the numbers of molecules have different scales. Some of the molecules are in small numbers and some are in large numbers. Accordingly, we split the set of species into two sets C and D with cardinals M C and M D . This induces the decomposition X=(X C ,X D ), γ r =(γ r C ,γ r D ). For iD, n i is of order 1 while for iC, n i is proportional to N where N is a large number. For iC, setting n ˜ i =n i /N, n ˜ i is of order 1. We define x C =X C /N and x=(x C ,X D ).

For this kind of system, we are able to give in [18] some relevant information on the asymptotic regime N when different type of reactions are involved. Depending on the time and concentration scales of the system we distinguish four types of limits:

  • Continuous piecewise deterministic processes (PDP) with switching.

  • PDP with jumps in the continuous variables.

  • Averaged PDP.

  • PDP with singular switching.

We justify rigorously the convergence for the four types of limits.