On constrained Langevin equations and (bio)chemical reaction
D. F. Anderson, D. J. Higham, S. C. Leite and R. J. Williams
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 simulation of a more complex example.
Appeared in SIAM Multiscale Modeling and Simulation Journal, Vol. 17 (2019), 1--30.
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