Approximate probability distributions of the master equation

Philipp Thomas and Ramon Grima

Phys. Rev. E 92, 012120 – Published 13 July 2015

Abstract

Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation using orthogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support, both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master equation, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems.

Received 13 November 2014
Revised 3 April 2015

DOI:https://doi.org/10.1103/PhysRevE.92.012120

Authors & Affiliations

Philipp Thomas^*

School of Mathematics and School of Biological Sciences, University of Edinburgh, Edinburgh EH8 9YL, United Kingdom

Ramon Grima^†

School of Biological Sciences, University of Edinburgh, Edinburgh EH8 9YL, United Kingdom

^*philipp.thomas@ed.ac.uk
^†ramon.grima@ed.ac.uk

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Vol. 92, Iss. 1 — July 2015

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Figure 1
Linear birth-death process. We consider the reaction system (29) in stationary conditions. (a) We compare the exact Poisson distribution (gray) to the continuous SSE approximation [Eq. (18) together with Eqs. (25) and (30)] truncated after the $Ω^{0}$ (LNA, blue line), $Ω^{- 1}$ (green), and $Ω^{- 3}$ terms (red) for parameter values $k_{0} = 0.5, k_{1} = 1$ , and $Ω = 1$ giving half a molecule on average. We observe that the continuous approximation becomes increasingly negative and tends to oscillations with increasing truncation order. (b) In contrast, the discrete approximation shows no oscillations, and the overall agreement with the exact Poisson distribution (gray bars) improves with increasing truncation order.
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Figure 2
Nonlinear birth-death process. A metabolic reaction with Michaelis-Menten kinetics, scheme (37), is studied using the reduced model described in Sec. 4b. The exact stationary distribution is a negative binomial (shown in gray). (a) The discrete SSE approximation given by Eq. (36) with Eq. (25) and (30) is shown in the low-molecule-number regime ( $k_{0} / k_{1} = 0.25$ , 1 molecule on average) when truncated after the $Ω^{0}$ (blue), $Ω^{- 3 / 2}$ (green), and $Ω^{- 4}$ terms (red dots). We observe that the expansion tends to oscillations and negative values of probability as the truncation order is increased. (b) Similar oscillations are observed for moderate molecule numbers ( $k_{0} / k_{1} = 0.9$ , 27 molecules on average) for the discrete series truncated after the $Ω^{0}$ (blue), $Ω^{- 3 / 2}$ (green), and $Ω^{- 3}$ terms (red lines). In (c) and (d) we show the approximations corresponding to the same parameters used in (a) and (b), respectively, but obtained using the renormalization procedure given by Eq. (41) with Eq. (42) as described in the main text. The renormalized approximations avoid oscillations and are in excellent agreement with the true probability distributions (gray bars). We note that for the cases (b) and (d) the continuous and discrete approximations give essentially the same result. The remaining parameters are given by $Ω = 10$ and $K = 0.1$ .
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Figure 3
Predicting transient distributions of gene expression. The dynamics of protein synthesis with enzymatic degradation, scheme (54), is studied using the burst approximation (55). (a) We compare the time dependence of the renormalized discrete approximations to exact stochastic simulations at times 1, 2, and 14 min. The overall shape (mode, skewness, distribution tails) of the simulated distributions (bars) is in excellent agreement with the series approximation when truncated after the $Ω^{- 3}$ terms (solid lines) but not when only $Ω^{0}$ are taken into account (dashed lines). This agreement is also observed for the first two moments shown in the inset: While the $Ω^{- 3}$ approximation (blue solid line) agrees with the moment dynamics of the simulated distributions (dots) of the reduced model (55), the $Ω^{0}$ approximation underestimates the mean (gray solid line) and variance by $25 %$ . The area within one standard deviation of the mean obtained from simulations is shown in blue, and the boundary obtained from the approximations are shown as dashed lines ( $Ω^{0}$ gray, $Ω^{- 3}$ blue). (b) Despite the good agreement shown in (a) we found that at very short times (12 s, blue solid line) the series truncated after the $Ω^{- 3}$ terms tends to oscillations which quickly disappear for later times (24 s, green; 48 s, red solid line). See the main text for discussion. Parameters are $k_{0} Ω = 15 \min^{- 1}, k_{1} Ω = 100 \min^{- 1}, K Ω = 20, Ω = 100$ , and $b = 5$ . Histograms were obtained from $10 000$ stochastic simulations.
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