Abstract
BackgroundThe chemical master equation is the fundamental equation of stochastic chemical kinetics. This differential-difference equation describes temporal evolution of the probability density function for states of a chemical system. A state of the system, usually encoded as a vector, represents the number of entities or copy numbers of interacting species, which are changing according to a list of possible reactions. It is often the case, especially when the state vector is high-dimensional, that the number of possible states the system may occupy is too large to be handled computationally. One way to get around this problem is to consider only those states that are associated with probabilities that are greater than a certain threshold level.ResultsWe introduce an algorithm that significantly reduces computational resources and is especially powerful when dealing with multi-modal distributions. The algorithm is built according to two key principles. Firstly, when performing time integration, the algorithm keeps track of the subset of states with significant probabilities (essential support). Secondly, the probability distribution that solves the equation is parametrised with a small number of coefficients using collocation on Gaussian radial basis functions. The system of basis functions is chosen in such a way that the solution is approximated only on the essential support instead of the whole state space.DiscussionIn order to demonstrate the effectiveness of the method, we consider four application examples: a) the self-regulating gene model, b) the 2-dimensional bistable toggle switch, c) a generalisation of the bistable switch to a 3-dimensional tristable problem, and d) a 3-dimensional cell differentiation model that, depending on parameter values, may operate in bistable or tristable modes. In all multidimensional examples the manifold containing the system states with significant probabilities undergoes drastic transformations over time. This fact makes the examples especially challenging for numerical methods.ConclusionsThe proposed method is a new numerical approach permitting to approximately solve a wide range of problems that have been hard to tackle until now. A full representation of multi-dimensional distributions is recovered. The method is especially attractive when dealing with models that yield solutions of a complex structure, for instance, featuring multi-stability.Electronic supplementary materialThe online version of this article (doi:10.1186/s12918-015-0210-y) contains supplementary material, which is available to authorized users.
Highlights
The chemical master equation is the fundamental equation of stochastic chemical kinetics
The method is especially attractive when dealing with models that yield solutions of a complex structure, for instance, featuring multi-stability
The matrix A is composed in such a way that the value of approximation sum (9) at points xi can be written as a column vector: u (xi, t) = Aα
Summary
The chemical master equation is the fundamental equation of stochastic chemical kinetics. In nearly every process, particle numbers are subject to random fluctuations caused by inherent stochastic noise Simulations of such systems are usually based on Monte Carlo (MC) simulations of the underlying Markov jump processes, such as Gillespie’s famous stochastic simulation algorithm (SSA) [1]. Even computing single realisations can be quite costly if many fast reactions are present; approximate MC methods like τ -leaping [2], averaging approaches [3, 4], and deterministic-stochastic hybrid formulations [5,6,7] have been introduced The applicability of these approaches depends on the existence of a permanent timescale gap that allows to clearly distinguish between fast and slow reactions
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