  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.
