Compartmentalization of biochemical processes underlies all biological systems, from the organelle to the tissue scale. Theoretical models to study the interplay between noisy reaction dynamics and compartmentalization are sparse, and typically very challenging to analyze computationally. Recent studies have made progress toward addressing this problem in the context of specific biological systems, but a general and sufficiently effective approach remains lacking. In this work, we propose a mathematical framework based on counting processes that allows us to study dynamic compartment populations with arbitrary interactions and internal biochemistry. We derive an efficient description of the dynamics in terms of differential equations which capture the statistics of the population. We demonstrate the relevance of our approach by analyzing models inspired by different biological processes, including subcellular compartmentalization and tissue homeostasis.

Stochastic simulations of biochemical networks are of vital importance for understanding complex dynamics in cells and tissues. However, existing methods to perform such simulations are associated with computational difficulties and addressing those remains a daunting challenge to the present. Here we introduce the selected-node stochastic simulation algorithm (snSSA), which allows us to exclusively simulate an arbitrary, selected subset of molecular species of a possibly large and complex reaction network. The algorithm is based on an analytical elimination of chemical species, thereby avoiding explicit simulation of the associated chemical events. These species are instead described continuously in terms of statistical moments derived from a stochastic filtering equation, resulting in a substantial speedup when compared to Gillespie's stochastic simulation algorithm (SSA). Moreover, we show that statistics obtained via snSSA profit from a variance reduction, which can significantly lower the number of Monte Carlo samples needed to achieve a certain performance. We demonstrate the algorithm using several biological case studies for which the simulation time could be reduced by orders of magnitude.

Living cells encode and transmit information in the temporal dynamics of biochemical components. Gaining a detailed understanding of the input-output relationship in biological systems therefore requires quantitative measures that capture the interdependence between complete time trajectories of biochemical components. Mutual information provides such a measure but its calculation in the context of stochastic reaction networks is associated with mathematical challenges. Here we show how to estimate the mutual information between complete paths of two molecular species that interact with each other through biochemical reactions. We demonstrate our approach using three simple case studies.