On constrained Langevin equations and (bio)chemical reaction networks

Stochastic effects play an important role in modeling the time evolution of chemical reaction systems in fields such as systems biology, where the concentrations of some constituent molecules can be low. The most common stochastic models for these systems are continuous time Markov chains, which track the molecular abundance of each chemical species. Often, these stochastic models are studied by computer simulations, which can quickly become computationally expensive. A common approach to reduce computational effort is to approximate the discrete valued Markov chain by a continuous valued diffusion process. However, existing diffusion approximations either do not respect the constraint that chemical concentrations are never negative (linear noise approximation) or are typically only valid until the concentration of some chemical species first becomes zero (chemical Langevin equation). In this paper, we propose (obliquely) reflected diffusions, which respect the non-negativity of chemical concentrations, as approximations for Markov chain models of chemical reaction networks. These reflected diffusions satisfy “constrained Langevin equations,” in that they behave like solutions of chemical Langevin equations in the interior of the positive orthant and are constrained to the orthant by instantaneous oblique reflection at the boundary. To motivate their form, we first illustrate our constrained Langevin approximations for two simple examples. We then describe the general form of our proposed approximation. We illustrate the performance of our approximations through comparison of their stationary distributions for the two examples with those of the Markov chain model and through simulations of more complex examples