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Stochastic approaches in systems biology

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Abstract The discrete and random occurrence of chemical reactions far from thermodynamic equilibrium, and low copy numbers of chemical species, in systems biology necessitate stochastic approaches. This review is an effort to give the reader a flavor of the most important stochastic approaches relevant to systems biology. Notions of biochemical reaction systems and the relevant concepts of probability theory are introduced side by side. This leads to an intuitive and easy‐to‐follow presentation of a stochastic framework for modeling subcellular biochemical systems. In particular, we make an effort to show how the notion of propensity, the chemical master equation (CME), and the stochastic simulation algorithm arise as consequences of the Markov property. Most stochastic modeling reviews focus on stochastic simulation approaches—the exact stochastic simulation algorithm and its various improvements and approximations. We complement this with an outline of an analytical approximation. The most common formulation of stochastic models for biochemical networks is the CME. Although stochastic simulations are a practical way to realize the CME, analytical approximations offer more insight into the influence of randomness on system's behavior. Toward that end, we cover the chemical Langevin equation and the related Fokker–Planck equation and the two‐moment approximation (2MA). Throughout the text, two pedagogical examples are used to key illustrate ideas. With extensive references to the literature, our goal is to clarify key concepts and thereby prepare the reader for more advanced texts. Copyright © 2009 John Wiley & Sons, Inc. This article is categorized under: Analytical and Computational Methods > Computational Methods

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Discrete and random nature of chemical reactions: (a) large copy numbers and frequent reactions allow for a continuous approximation leading to the chemical Langevin equation (CLE), which, for an infinitely large system, approaches deterministic rate equations. (b) small copy numbers and infrequent reactions require discrete stochastic approaches leading to the chemical master equation (CME) and stochastic simulations.

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The time courses of mean and mean ± SD for the isomerization reaction when the parameters are: (a) k1 = k2 = 2 sec−1, (b) k1 = 1 sec−1, k2 = 3 sec−1, and (c) k1 = 3 sec−1, k2 = 1 sec−1. All the parameter values satisfy k1 + k2 = 4. The scales on the vertical axes have been chosen to highlight the importance of SD relative to the mean.

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The time courses, for the isomerization reaction, of (a) standard deviation (SD), (b) coefficient of variation (CV), for the parameter pairs k1 = k2 = 2 sec−1 as solid lines, k1 = 1 sec−1, k2 = 3 sec−1 in dash‐dotted lines, and k1 = 3 sec−1, k2 = 1 sec−1 in dashed lines. All the parameter values satisfy k1 + k2 = 4.

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The time development of the histogram for the Schlögl reaction.

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