doi:10.1016/j.compbiolchem.2004.05.001 
Copyright © 2004 Elsevier Ltd. All rights reserved.
Review
Stochastic approaches for modelling in vivo reactions
a Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, 24–29 St. Giles’, Oxford OX1 3LB, UK
b Centre for Mathematical Biology, Mathematical Institute, 24–29 St Giles’, Oxford OX1 3LB, UK
c Christ Church, Oxford OX1 1DP, UK
d Department of Mathematics, Advanced Computational Modelling Centre, University of Queensland, Brisbane Qld 4072, Australia
Received 30 April 2004;
accepted 2 May 2004.
Available online 2 July 2004.
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Abstract
In recent years, stochastic modelling has emerged as a physically more realistic alternative for modelling in vivo reactions. There are numerous stochastic approaches available in the literature; most of these assume that observed random fluctuations are a consequence of the small number of reacting molecules. We review some important developments of the stochastic approach and consider its suitability for modelling intracellular reactions. We then describe recent efforts to include the fluctuation effects caused by the structural organisation of the cytoplasm and the limited diffusion of molecules due to macromolecular crowding.
Author Keywords: Intracellular reactions; Stochastic simulation algorithm; τ-leap method; Quasi-steady-state approximation; Fractal-like kinetics
Fig. 1. Stochastic algorithm simulation for the substrate density S in the Michaelis–Menten reaction (1). The blue/broken solid curves are individual simulations. The green/smooth solid curves are the results of numerically integrating the deterministic differential equation. The red/plus sign symbols are the mean for the 2000 runs of the stochastic simulation and the red/dash line corresponds to the mean plus (or minus) one standard deviation. Initial conditions are: C=P=0. (a) Results for initial molecular populations S0=100 and E0=10. Three individual simulations are shown explicitly. (b) Results for initial molecular populations S0=10 and E0=1. One individual simulation is shown explicitly.
Fig. 2. Mean results from 2000 runs of the stochastic algorithm simulating systems with varying molecular populations for the enzyme substrate complex C population in the Michaelis–Menten reaction (1). The green/smooth solid line is the numerical solution for the deterministic differential equations. The total number of molecules in the simulation increases from 10 molecules (cyan/broken solid curve), 110 molecules (blue/dash curve) to 1100 molecules (red/plus sign symbols). The initial molecular concentrations for the simulations are: S0=0.10, E0=0.01, C0=P0=0 molecules per unit volume.
Fig. 3. “Global S.D.’s” (i.e. the mean over the whole simulation time of the S.D. at each time-point) of molecular concentration data from 2000 runs of the stochastic algorithm for four different simulation volumes with equal initial molecular concentrations. Data for S (circle symbol) and P (cross symbol) are indistinguishable (blue/dash curve), as are data (red/solid curves) for C (circle symbol) and E (cross symbol).
Fig. 4. Benchmarking results for the stochastic simulations. We compared the simulation time (s) for the Gillespie algorithm simulations (blue/circle symbols), two-dimensional (green/plus sign symbols) and a three-dimensional (red/triangle symbols) lattice gas algorithms. (a) Average simulation time in seconds for various sizes of the simulation environment. (b) Average simulation time in seconds as a function of obstacle density, θ, for the lattice gas simulations.
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