Approximation and inference methods for stochastic biochemical kinetics—a tutorial review

Abstract

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.

Improvement of the memory function of a mutual repression network in a stochastic environment by negative autoregulation

“…where ( , ) was the probability density of protein concentration y. Next, the chemica l master equation was transformed into the Fokker-Planck equation [12,21,49,[65][66][67]:…”

“…Recent examples of opposing behavior further highlight the importance of considering the contribution of stochasticity to cellular circuitry. On the one hand, it has been shown that noise can induce multimodality and even stochastic memory in a system that, according to a deterministic description, lacks bistability [48,49]. On the other hand, stochastic fluctuations in gene expression levels often reduce or disrupt the memory function of biological networks [21,50,51].…”

Mixture distributions in a stochastic gene expression model with delayed feedback

“…Generally, however, explicit solutions are unavailable or intractable and one resorts to stochastic simulation or seeks a numerical solution to a finite truncation of the master equation (Munsky and Khammash, 2006;Borri et al, 2016;Gupta et al, 2017). An alternative approach, which often provides useful qualitative insights into the model behaviour, is based on reduction techniques such as quasi-steadystate (Srivastava et al, 2011;Kim et al, 2014) and adiabatic reductions (Bruna et al, 2014;Popovic et al, 2016), piecewise-deterministic framework (Lin and Doering, 2016;Lin and Buchler, 2018), linear-noise approximation (Schnoerr et al, 2017;Modi et al, 2018), or moment closure (Singh and Hespanha, 2007;Andreychenko et al, 2017;Gast et al, 2019).…”

A systems-biology approach to molecular machines: Exploration of alternative transporter mechanisms

“…1). These networks can be modeled using the chemical master equation: a set of differential equations describing the state probabilities and connected transition rates for each state (15). Biochemical networks are generally Markovian (16), have a number of different control patterns (17), and typically adhere to specific design principles (18).…”