Stationary Distributions of Continuous-Time Markov Chains: A Review of Theory and Truncation-Based Approximations

Shahrezaei

2021

J. R. Soc. Interface.

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The chemical master equation and the Gillespie algorithm are widely used to model the reaction kinetics inside living cells. It is thereby assumed that cell growth and division can be modelled through effective dilution reactions and extrinsic noise sources. We here re-examine these paradigms through developing an analytical agent-based framework of growing and dividing cells accompanied by an exact simulation algorithm, which allows us to quantify the dynamics of virtually any intracellular reaction network affected by stochastic cell size control and division noise. We find that the solution of the chemical master equation—including static extrinsic noise—exactly agrees with the agent-based formulation when the network under study exhibits stochastic concentration homeostasis , a novel condition that generalizes concentration homeostasis in deterministic systems to higher order moments and distributions. We illustrate stochastic concentration homeostasis for a range of common gene expression networks. When this condition is not met, we demonstrate by extending the linear noise approximation to agent-based models that the dependence of gene expression noise on cell size can qualitatively deviate from the chemical master equation. Surprisingly, the total noise of the agent-based approach can still be well approximated by extrinsic noise models.

Section: Discussionmentioning

confidence: 99%

Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations

Shahrezaei

2021

J. R. Soc. Interface.

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“…Such alternative methods include, for example, the conditional system size expansion including terms beyond the LNA, maximum entropy reconstructions using the method of conditional moments, or the finite state projection algorithm (Andreychenko et al, 2017;Gupta et al, 2017a), all of which can be readily incorporated into our mixture model strategy. These methods rely on different assumptions and their approximation quality will vary depending on the specific model parameters; in some cases, estimates for their approximation errors can be obtained with suitable methods, as discussed in a recent review on this topic (Kuntz et al, 2020).…”

Section: Discussionmentioning

confidence: 99%

Computation of Single-Cell Metabolite Distributions Using Mixture Models

Tonn

Barahona

et al. 2020

Front. Cell Dev. Biol.

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Metabolic heterogeneity is widely recognized as the next challenge in our understanding of non-genetic variation. A growing body of evidence suggests that metabolic heterogeneity may result from the inherent stochasticity of intracellular events. However, metabolism has been traditionally viewed as a purely deterministic process, on the basis that highly abundant metabolites tend to filter out stochastic phenomena. Here we bridge this gap with a general method for prediction of metabolite distributions across single cells. By exploiting the separation of time scales between enzyme expression and enzyme kinetics, our method produces estimates for metabolite distributions without the lengthy stochastic simulations that would be typically required for large metabolic models. The metabolite distributions take the form of Gaussian mixture models that are directly computable from single-cell expression data and standard deterministic models for metabolic pathways. The proposed mixture models provide a systematic method to predict the impact of biochemical parameters on metabolite distributions. Our method lays the groundwork for identifying the molecular processes that shape metabolic heterogeneity and its functional implications in disease.

Section: Author Contributionsmentioning

confidence: 99%

Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations

Shahrezaei

2020

Preprint

Self Cite

The chemical master equation and the stochastic simulation algorithm are widely used to model the reaction kinetics inside living cells. It is thereby assumed that cell growth and division can be modelled for through effective dilution reactions and extrinsic noise sources. We here re-examine these paradigms through developing an analytical agent-based framework of growing and dividing cells accompanied by an exact simulation algorithm, which allows us to quantify the dynamics of virtually any intracellular reaction network affected by stochastic cell size control and division noise in a growing population. We find that the solution of the chemical master equation - including static extrinsic noise - exactly agrees with the one of the agent-based formulation when a simple condition on the network's topology is met. We illustrate this result for a range of common gene expression networks. When these conditions are not met, we demonstrate using analytical solutions of the agent-based models that the dependence of gene expression noise on cell size can qualitatively deviate from the effective master equation. Surprisingly, the latter distorts total noise in gene regulatory networks by at most 8% independently of network parameters. Our results highlight the accuracy of extrinsic noise modelling within the chemical master equation framework.