Modeling stochasticity in gene regulation: characterization in the terms of the underlying distribution function

Intrinsic stochasticity plays an essential role in gene regulation because of a small number of involved molecules of DNA, mRNA and protein of a given species. To better understand this phenomenon, small gene regulatory systems are mathematically modeled as systems of coupled chemical reactions, but the existing exact description utilizing a Chapman-Kolmogorov equation or simulation algorithms is limited and inefficient. The present work considers a much more efficient yet accurate modeling approach, which allows analyzing stochasticity in the system in the terms of the underlying distribution function. We depart from the analysis of a single gene regulatory module to find that the mRNA and protein variance is decomposable into additive terms resulting from respective sources of stochasticity. This variance decomposition is asserted by constructing two approximations to the exact stochastic description: First, the continuous approximation, which considers only the stochasticity due to the intermittent gene activity. Second, the mixed approximation, which in addition attributes stochasticity to the mRNA transcription/decay process. Considered approximations yield systems of first order partial differential equations for the underlying distribution function, which can be efficiently solved using developed numerical methods. Single cell simulations and numerical two-dimensional mRNA-protein stationary distribution functions are presented to confirm accuracy of approximating models.