Efficient Modeling of Stochastic Reaction–Diffusion Processes Using the Linear Noise Approximation

Stochastic fluctuations can strongly influence reaction–diffusion systems when the number of molecules is small. This is particularly important in nanoscale materials and chemical processes, where stochastic variations in molecular populations can propagate through diffusion and chemical reactions and affect the final spatial distributions of chemical species.

Although the stochastic simulation algorithm (SSA) provides an exact description of discrete reaction–diffusion events, its computational cost becomes prohibitive for large systems containing many spatial cells and chemical species. Therefore, in this study, we develop a computational framework based on the linear noise approximation (LNA) to efficiently predict both the mean molecular populations and their fluctuations.

The framework describes the evolution of the mean molecular populations and their fluctuations, i.e., variances and covariances, while incorporating the effects of both chemical reactions and molecular diffusion. The predicted mean and fluctuation profiles are validated against SSA simulations.

A major computational difficulty of the spatial LNA is the rapid growth of the covariance matrix with the number of spatial cells and chemical species. To reduce this computational cost, we introduce a localized LNA, in which covariance terms beyond a prescribed spatial distance are truncated. This approximation preserves important short-range spatial correlations while substantially reducing memory requirements and computational cost.

The accuracy and computational efficiency of the localized LNA are evaluated under various reaction–diffusion conditions. This framework may provide an efficient tool for analyzing molecular-scale fluctuations in nanoscale reaction–diffusion systems, such as those involved in advanced lithography.