Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Abstract

<jats:p> Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterizing stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealizations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time-course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with Matlab <jats:sup>®</jats:sup> implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community. </jats:p>

Identifying density-dependent interactions in collective cell behaviour

“…In the supporting material, we also investigate σ = ϕ = 24 µm, since this is a natural choice in a lattice-based framework where the migration distance and dispersal distance are also the same as the average agent diameter. We apply approximate Bayesian computation (ABC) [14,16,21,23] to update our knowledge of the parameters using experimental observations, X obs , from all nine experiments, to produce posterior distributions, π(θ|X obs ). Since this model is known to be computationally expensive [16] and we have a high-dimensional parameter space, we apply an ABC method based on sequential Monte-Carlo (SMC) [21,23,32].…”

“…The principle behind ABC SMC is to propagate a series of prior samples, called particles, through a sequence of distributions π(θ|ρ(X obs , X sim ) < ε u ), u = {1, ..., U} [21,23,32]. The thresholds ε u satisfy ε u > ε u+1 , so that the distribution gradually evolves to the target distribution π(θ|ρ(X obs , X sim ) < ε U ) ≈ π(θ|X obs ).…”

“…The thresholds ε u satisfy ε u > ε u+1 , so that the distribution gradually evolves to the target distribution π(θ|ρ(X obs , X sim ) < ε U ) ≈ π(θ|X obs ). To obtain a sequence of thresholds, and an estimate of the smallest discrepancy possible in all models, we first perform a pilot run using ABC rejection [16,23] with Model 1 (supporting material, section 1.1). From 100,000 prior samples, this provides an estimate of the probabilities Pr(ρ(X obs , X sim ) < ε u ), given θ is simulated from the prior.…”

“…The primary goal of this study is to identify interactions which enable the model to simultaneously describe experimental data across a range of initial densities. We take a Bayesian approach to parameter estimation [16,[21][22][23], and identify interactions using model selection [21]. We always force the model to simultaneously match data from all Experiment N(0) N(18) N(36 ) Fold change 1 183 322 652 3.56 2 299 528 927 3.10 3 354 549 893 2.52 4 404 636 1121 2.77 5 427 657 1077 2.52 6 522 849 1414 2.71 7 677 1200 2013 2.97 8 692 1285 1853 2.67 9 731 1155 1974 2.70…”

Practical parameter identifiability for spatiotemporal models of cell invasion

“…Such methods of model identifiability analysis are well-established in the field of systems biology [12][13][14][15][16][17][18][19][20]. In this context experimental data often take the form of time series describing temporal variations of different biochemical molecules in some kind of chemical reaction network or gene regulatory network and these data are modelled using ordinary or stochastic differential equation models [32]. In the present work we focus on apply methods of identifiability analysis to spatiotemporal partial differential equation models, reaction-diffusion equations in particular, which model both spatial and temporal variations in different quantities of interest.…”