Inferring extrinsic noise from single-cell gene expression data using approximate Bayesian computation

Abstract

Background: Gene expression is known to be an intrinsically stochastic process which can involve single-digit numbers of mRNA molecules in a cell at any given time. The modelling of such processes calls for the use of exact stochastic simulation methods, most notably the Gillespie algorithm. However, this stochasticity, also termed "intrinsic noise", does not account for all the variability between genetically identical cells growing in a homogeneous environment. Despite substantial experimental efforts, determining appropriate model parameters continues to be a challenge. Methods based on approximate Bayesian computation can be used to obtain posterior parameter distributions given the observed data. However, such inference procedures require large numbers of simulations of the model and exact stochastic simulation is computationally costly. In this work we focus on the specific case of trying to infer model parameters describing reaction rates and extrinsic noise on the basis of measurements of molecule numbers in individual cells at a given time point. Results: To make the problem computationally tractable we develop an exact, model-specific, stochastic simulation algorithm for the commonly used two-state model of gene expression. This algorithm relies on certain assumptions and favourable properties of the model to forgo the simulation of the whole temporal trajectory of protein numbers in the system, instead returning only the number of protein and mRNA molecules present in the system at a specified time point. The computational gain is proportional to the number of protein molecules created in the system and becomes significant for systems involving hundreds or thousands of protein molecules. Conclusions: We employ this simulation algorithm with approximate Bayesian computation to jointly infer the model's rate and noise parameters from published gene expression data. Our analysis indicates that for most genes the extrinsic contributions to noise will be small to moderate but certainly are non-negligible.

Coloured Noise from Stochastic Inflows in Reaction–Diffusion Systems

“…Therefore, the effect of red noise is a deterministic one and to get an accurate description of the system behaviour a full nonlinear, deterministic compartment model with generic (random) inflow parameters needs to be analysed. The red noise case is similar to models which assume a cell-to-cell variability where a parameter is perturbed by a constant, sampled from a given distribution [21,44]. In contrast to conventional Turing theory in which the inflow parameter is constant throughout the domain, our analysis indicates that randomly varying inflows across the domain might actually facilitate pattern formation.…”

Identification of cell‐to‐cell heterogeneity through systems engineering approaches

“…On the other hand, Stumpf and his coworkers 23,27 proposed an approximate Bayesian computation scheme for inferring the parameter distributions, which was subsequently used in different biological systems. 17,28,29 In many of these studies, PDFs of all the parameters were estimated simultaneously. Or, the PDFs of only a subset of model parameters from the entire set were selected based on prior knowledge, and only these were estimated.…”

Efficient Exact Inference for Dynamical Systems with Noisy Measurements using Sequential Approximate Bayesian Computation

“…gene expression, signal transduction, and multi-cellular systems (e.g. Lenive et al (2016); Picchini (2014); Imle et al (2019)).…”

“…Contrarily, in likelihood-free methods, particularly ABC, it is easy to disregard any noise due to the unnecessity of even formulating a likelihood and the various inherent approximation levels, so that error sources can be difficult to pinpoint from the result. In the past, it has repeatedly not been included in ABC analyses (Toni et al, 2009;Lenive et al, 2016;Jagiella et al, 2017;Imle et al, 2019;Eriksson et al, 2019). Asymptotic unbiasedness of ABC is however granted only if the data-generation process is perfectly reproduced.…”