![]() Thus, at large numbers of particles, the number of events per time step can become very large, and SSSAs become prohibitively slow. However, a more significant drawback is the fact that SSSAs are event-driven algorithms. For example, there are spatial resolution limits under which artefacts in particle interactions might occur, and also some effects at boundaries might not be accurately captured. While widely used, SSSAs suffer from several drawbacks. It is possible to generate sample paths consistent with the RDME using a variety of spatial Stochastic Simulation Algorithms (SSSAs). Mean-field approaches provide some analytical tools to help understand systems with bimolecular reactions, but these do not provide exact solutions. However, there do exist some closed form solutions for systems involving monomolecular reactions. The RDME is generally analytically intractable. Diffusion can then occur between different voxels, and reactions can occur within voxels on the assumption that reactants are well-mixed. A widely used approach to study stochastic spatial dynamics is the Reaction Diffusion Master Equation (RDME), in which space is partitioned discretely into a number of voxels. Spatial-stochastic effects are increasingly found to play important roles throughout a range of biological scales, from intracellular and intercellular processes, to ecological and epidemiological scales. However, there exist parameter sets where both the qualitative and quantitative behaviour of the SCLE can differ when compared to the RDME, so care should be taken in its use for applications demanding greater accuracy. This becomes very useful in search of quantitative parameters yielding desired qualitative solutions. The SCLE provides a fast alternative to existing methods for simulation of spatial stochastic biochemical networks, capturing many aspects of dynamics represented by the RDME. ![]() However, areas of the parameter space in the Gray-Scott model exist where either the SCLE and RDME give qualitatively different predictions, or the RDME predicts patterns, while the SCLE does not. As expected, the SCLE captures many dynamics of the RDME where deterministic methods fail to represent them. All approaches match at the leading order, and the RDME and SCLE match at the second leading order. For non-linear reaction networks, differential equations governing moments do not form a closed system, but a general moment equation can be compared term wise. Resultsįor linear reaction networks, it is well known that the first order moments of all three approaches match, that the RDME and SCLE match to the second moment, and that all approaches diverge at third order moments. ![]() We consider the Gray-Scott model, a well-known pattern generating system, and a predator–prey system with spatially inhomogeneous parameters as sample applications. Sample paths are generated computationally by the Next Subvolume method (RDME) and the Euler-Maruyama method (SCLE), while a deterministic solution is obtained with an Euler method. We investigate moment equations and correlation functions analytically, then we compare sample paths and moments of the SCLE to the RDME and associated deterministic solutions. Here we investigate an uncommon, but much faster alternative: the Spatial Chemical Langevin Equation (SCLE). However, simulating sample paths from the RDME can be computationally expensive, particularly at large populations. A popular method of representing such stochastic systems is the Reaction Diffusion Master Equation (RDME). ![]() It has been established that stochastic effects play an important role in spatio-temporal biochemical networks.
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