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Published May 2017 | Submitted
Journal Article Open

Recursively constructing analytic expressions for equilibrium distributions of stochastic biochemical reaction networks


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 of Kolmogorov forward equations that describe how the probability distribution of a chemically reacting system varies with time. Finding 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 characterize 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 the stationary behaviour of stochastic biochemical reaction networks. Finally, we illustrate the aforementioned ideas using examples that include two interconnected transcriptional components and biochemical reactions with two-dimensional state spaces.

Additional Information

© 2017 The Author(s). Received March 2, 2017; Accepted May 2, 2017. Data accessibility: The data used in the paper is available in the electronic supplementary material. Authors' contributions: The project was formulated by A.-A.B., V.S. and R.M.M. X.F.M., A.-A.B. and V.S. performed the mathematical analyses. X.F.M. constructed the algorithms. A.-A.B. illustrated the design of stationary distributions. The paper was written by X.F.M., A.-A.B. and V.S. All the authors edited the paper. We declare we have no competing interests. This research is funded by the Air Force Office of Scientific Research through grant no. FA9550-14-1-0060 to the project Theory-Based Engineering of Biomolecular Circuits in Living Cells. The authors gratefully thank Anandh Swaminathan, Justin Bois, Enoch Yeung, Albert R. Chern, Scott C. Livingston and Charlie Erwall for insights and comments. We thank Reza Ghaemi and Domitilla Del Vecchio for granting us permission to use the figure of two interconnected transcriptional components.

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