Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular population structure which suggests a natural continuous state-space endowed with a dynamics for the approximating process. We construct here a general method based on spectral analysis of the transition matrix of the CTMC, without the need for a population structure. Specifically, we use the popular manifold learning method of diffusion maps to analyse the transition matrix as the operator of a hidden continuous process. An embedding of states in a continuous space is recovered, and the space is endowed with a drift vector field inferred via Gaussian process regression. In this manner, we construct an ODE whose solution approximates the evolution of the CTMC mean, mapped onto the continuous space (known as the fluid limit).
In the case of pCTMCs, a more concise description in terms of the collective dynamics of population averages is however available. Starting with the seminal work of van Kampen @cite_0 , and motivated by the interpretation of pCTMCs as chemical reaction systems, several approximation schemes have been developed which relax the original pCTMC to a continuous stochastic process; see @cite_8 for a recent review.
@cite_0 @cite_8
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In order to solve the master equation by a systematic approximation method, an expansion in powers of some parameter is needed. The appropriate parameter is the reciprocal size of the system, defined as the ratio of intensive and extensive variables. The lowest approximation yields the phenomenological law for the approach to equilibrium. The next approximation determines the mean square of the fluctuations about the phenomenological behavior. In equilibrium this approximation has the form of a linear Fokker–Planck equation. The higher approximations describe the effect of the non-linearity on the fluctuations, in particular on their spectral density. The method is applied to three examples: density fluctuations, Alkemade's diode, and Rayleigh's piston. The relation to the expansion recently given by Siegel is also discussed. Stochastic fluctuations of molecule numbers are ubiquitous in biological systems. Important examples include gene expression and enzymatic processes in living cells. Such systems are typically modelled as chemical reaction networks whose dynamics are governed by the Chemical Master Equation. Despite its simple structure, no analytic solutions to the Chemical Master Equation are known for most systems. Moreover, stochastic simulations are computationally expensive, making systematic analysis and statistical inference a challenging task. Consequently, significant effort has been spent in recent decades on the development of efficient approximation and inference methods. This article gives an introduction to basic modelling concepts as well as an overview of state of the art methods. First, we motivate and introduce deterministic and stochastic methods for modelling chemical networks, and give an overview of simulation and exact solution methods. Next, we discuss several approximation methods, including the chemical Langevin equation, the system size expansion, moment closure approximations, time-scale separation approximations and hybrid methods. We discuss their various properties and review recent advances and remaining challenges for these methods. We present a comparison of several of these methods by means of a numerical case study and highlight some of their respective advantages and disadvantages. Finally, we discuss the problem of inference from experimental data in the Bayesian framework and review recent methods developed the literature. In summary, this review gives a self-contained introduction to modelling, approximations and inference methods for stochastic chemical kinetics.