Parameter inference with analytical propagators for stochastic models of autoregulated gene expression

Frits Veerman,Carsten Marr,Nikola Popović

doi:10.1515/ijnsns-2019-0258

Abstract

Abstract Stochastic gene expression in regulatory networks is conventionally modelled via the chemical master equation (CME). As explicit solutions to the CME, in the form of so-called propagators, are oftentimes not readily available, various approximations have been proposed. A recently developed analytical method is based on a separation of time scales that assumes significant differences in the lifetimes of mRNA and protein in the network, allowing for the efficient approximation of propagators from asymptotic expansions for the corresponding generating functions. Here, we showcase the applicability of that method to simulated data from a ‘telegraph’ model for gene expression that is extended with an autoregulatory mechanism. We demonstrate that the resulting approximate propagators can be applied successfully for parameter inference in the non-regulated model; moreover, we show that, in the extended autoregulated model, autoactivation or autorepression may be refuted under certain assumptions on the model parameters. These results indicate that our approach may allow for successful parameter inference and model identification from longitudinal single cell data.

Highlights

Introduction and backgroundGene expression in regulatory networks is an inherently stochastic process [1]
We demonstrate that the resulting approximate propagators can be applied successfully for parameter inference in the non-regulated model; we show that, in the extended autoregulated model, autoactivation or autorepression may be refuted under certain assumptions on the model parameters
In recent work [14], we presented an analytical method for obtaining explicit, general, time-dependent expressions for the generating function associated to the chemical master equation (CME) that arises from the model in (3.1)

Summary

Introduction

Introduction and backgroundGene expression in regulatory networks is an inherently stochastic process [1]. Mathematical models typically take the form of a chemical master equation (CME), which describes the temporal evolution of the probabilities of observing specific states in the network [2]. A typical data set, denoted by Q, consists of protein abundances ni at N + 1 different points in time; see Figure 1A. We can group these abundances into transitions ni → ni+1; cf Figure 1B. A model-derived propagator Pni+1| ni (Δt, Θ) allows for the calculation of the probabilities of such transitions for some set of

Methods

Results

Conclusion