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Chemical master equation and Langevin regimes for a gene transcription model. (English) Zbl 1151.92012

Summary: Gene transcription models must take account of intrinsic stochasticity. The Chemical Master Equation framework is based on modelling assumptions that are highly appropriate for this context, and the Stochastic Simulation Algorithm (also known as Gillespie’s algorithm) allows for practical simulations to be performed. However, for large networks and/or fast reactions, such computations can be prohibitatively expensive. The Chemical Langevin regime replaces the massive ordinary differential equation system with a small stochastic differential equation system that is more amenable to computation. Although the transition from the Chemical Master Equation to the Chemical Langevin Equation can be justified rigorously in the large system size limit, there is very little guidance available about how closely the two models match for a fixed system.
We consider a transcription model from the recent literature and show that it is possible to compare first and second moments in the two stochastic settings. To analyse the Chemical Master Equation we use some recent work of C. Gadgil, C. H. Lee and H. G. Othmer [A stochastic analysis of first-order reaction networks. Bull. Math. Biol. 67, 901–946 (2005)], and to analyse the Chemical Langevin Equation we use Ito’s lemma. We find that there is a perfect match, both modelling regimes give the same means, variances and correlations for all components in the system. The model that we analyse involves ‘unimolecular reactions’, and we finish with some numerical simulations involving dimerization to show that the means and variances in the two regimes can also be close when more general ‘bimolecular reactions’ are involved.

MSC:

92C40 Biochemistry, molecular biology
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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References:

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