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

Abstract

Noise is indispensible to key cellular activities, including gene expression coordination and probabilistic differentiation. Stochastic models, such as the chemical master equation (CME), are essential to model noise in the levels of cellular components. In the CME framework, each state is associated with the molecular counts of all component species, and specifies the probability for the system to have that set of molecular counts. Analytic solutions to the CME are rarely known but can bring exciting benefits. For instance, simulations of biochemical reaction networks that are multiscale in time can be sped up tremendously by incorporating analytic solutions of the slow time-scale dynamics. Analytic solutions also enable the design of stationary distributions with properties such as the modality of the distribution, the mean expression level, and the level of noise. One way to derive the analytic steady state response of a biochemical reaction network was recently proposed by (Mélykúti et al. 2014). The paper recursively glues simple state spaces together, for which we have analytic solutions, at one or two states. In this work, we explore the benefits and limitations of the gluing technique proposed by Mélykúti et al., and introduce recursive algorithms that use the technique to solve for the analytic steady state response of stochastic biochemical reaction networks. We give formal characterizations of the set of reaction networks whose state spaces can be obtained by carrying out single-point gluing of paths, cycles or both sequentially. We find that the dimension of the state space of a reaction network equals the maximum number of linearly independent reactions in the system. We then characterize the complete set of stochastic biochemical reaction networks that have elementary reactions and two-dimensional state spaces. As an example, we propose a recursive algorithm that uses the gluing technique to solve for the steady state response of a mass-conserving system with two connected monomolecular reversible reactions. Even though the gluing technique can only construct finite state spaces, we find that, by taking the size of a finite state space to infinity, the steady state response can converge to the analytic solution on the resulting infinite state space. Finally, we illustrate the aforementioned ideas with the example of two interconnected transcriptional components, which was first studied by (Ghaemi and Del Vecchio 2012).

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