Chemical Master Equation and Langevin regimes for a gene transcription model

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 Chemical Master Equation to 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. Here, 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 Gadgil, Lee and Othmer, 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.