Finding analytic stationary solutions to the chemical master equation by gluing state spaces at one or two states recursively

Noise is often indispensable to key cellular activities, such as gene expression, necessitating the use of stochastic models to capture its dynamics. The chemical master equation (CME) is a commonly used stochastic model that describes how the probability distribution of a chemically reacting system varies with time. Knowing analytic solutions to the CME can have benefits, such as expediting simulations of multiscale biochemical reaction networks and aiding the design of distributional responses. However, analytic solutions are rarely known. A recent method of computing analytic stationary solutions relies on gluing simple state spaces together recursively at one or two states. We explore the capabilities of this method and introduce algorithms to derive analytic stationary solutions to the CME. We first formally characterise state spaces that can be constructed by performing single-state gluing of paths, cycles or both sequentially. We then study stochastic biochemical reaction networks that consist of reversible, elementary reactions with two-dimensional state spaces. We also discuss extending the method to infinite state spaces and designing stationary distributions that satisfy user-specified constraints. Finally, we illustrate the aforementioned ideas using examples that include two interconnected transcriptional components and chemical reactions with two-dimensional state spaces.Subject AreasSystems biology, synthetic biology, biomathematics, bioengineering