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Numerical estimation of the stationary solution of the chemical master equation

Banff International Research Station for Mathematical Innovation and Discovery

The chemical master equation (CME) is frequently used in systems biology to quantify the effects of stochastic fluctuations that arise due to biomolecular species with low copy numbers. The CME is a system of ordinary differential equations that describes the evolution of probability density for each population vector in the state-space of the Markov chain that represents the stochastic reaction dynamics. Often this state-space is infinite in size, making it difficult to obtain the exact solutions of the CME. Hence these solutions need to be estimated by stochastic simulations or by solving an approximate CME over a finite truncated state-space. In many applications, one is interested in finding the stationary solution of the CME which corresponds to the stationary probability distribution of the underlying Markov chain. When the state-space is infinite, this stationary distribution satisfies an infinite-dimensional linear-algebraic system which cannot be directly solved. In this talk we will discuss how the stationary distribution can be accurately estimated by solving a finite linear-algebraic system that provides the stationary distribution of a suitably constructed Markov chain over a truncated state-space. We will argue that under certain stability conditions, like the irreducibility of the infinite state-space and exponential ergodicity of the reaction dynamics, the obtained approximations are guaranteed to converge to the exact stationary solution as the truncated state-space expands to the full state-space. We will explain how these stability conditions can be computationally checked for reaction networks and also describe how the approximation errors can be quantified. The presented ideas will be illustrated through many examples.

Mathematics; Dynamical systems and ergodic theory; Probability theory and stochastic processes; Mathematical biology

video/mp4

Author affiliation: ETH Zurich

2017-12-05

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63814

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/