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The system-size expansion of the chemical master equation: developments in the past 5 years

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SDBW03 - Advances in numerical and analytic approaches for the study of non-spatial stochastic dynamical systems in molecular biology

Co-author: Philipp Thomas (Imperial College London)

The system-size expansion of the master equation, first developed by van Kampen, is a well known approximation technique for deterministically monostable systems. Its use has been mostly restricted to the lowest order terms of this expansion which lead to the deterministic rate equations and the linear-noise approximation (LNA). I will here describe recent developments concerning the system-size expansion, including (i) its use to obtain a general non-Gaussian expression for the probability distribution solution of the chemical master equation; (ii) clarification of the meaning of higher-order terms beyond the LNA and their use in stochastic models of intracellular biochemistry; (iii) the convergence of the expansion, at a fixed system-size, as one considers an increasing number of terms; (iv) extension of the expansion to describe gene-regulatory systems which exhibit noise-induced multimodality; (v) the conditions under which the LNA is exact up to second-order moments; (v i) the relationship between the system-size expansion, the chemical Fokker-Planck equation and moment-closure approximations.

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