The chemical Langevin equation for biochemical systems in dynamic environments.

Modeling and simulation of complex biochemical reaction networks form cornerstones of modern biophysics. Many of the approaches developed so far capture temporal fluctuations due to the inherent stochasticity of the biophysical processes, referred to as intrinsic noise. Stochastic fluctuations, however, predominantly stem from the interplay of the network with many other-and mostly unknown-fluctuating processes, as well as with various random signals arising from the extracellular world; these sources contribute extrinsic noise. Here, we provide a computational simulation method to probe the stochastic dynamics of biochemical systems subject to both intrinsic and extrinsic noise. We develop an extrinsic chemical Langevin equation (CLE)-a physically motivated extension of the CLE-to model intrinsically noisy reaction networks embedded in a stochastically fluctuating environment. The extrinsic CLE is a continuous approximation to the chemical master equation (CME) with time-varying propensities. In our approach, noise is incorporated at the level of the CME, and it can account for the full dynamics of the exogenous noise process, irrespective of timescales and their mismatches. We show that our method accurately captures the first two moments of the stationary probability density when compared with exact stochastic simulation methods while reducing the computational runtime by several orders of magnitude. Our approach provides a method that is practical, computationally efficient, and physically accurate to study systems that are simultaneously subject to a variety of noise sources.