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Ventre E, Espinasse T, Bréhier CE, Calvez V, Lepoutre T, Gandrillon O. Reduction of a stochastic model of gene expression: Lagrangian dynamics gives access to basins of attraction as cell types and metastabilty. J Math Biol 2021; 83:59. [PMID: 34739605 DOI: 10.1007/s00285-021-01684-1] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2020] [Revised: 09/02/2021] [Accepted: 10/13/2021] [Indexed: 12/16/2022]
Abstract
Differentiation is the process whereby a cell acquires a specific phenotype, by differential gene expression as a function of time. This is thought to result from the dynamical functioning of an underlying Gene Regulatory Network (GRN). The precise path from the stochastic GRN behavior to the resulting cell state is still an open question. In this work we propose to reduce a stochastic model of gene expression, where a cell is represented by a vector in a continuous space of gene expression, to a discrete coarse-grained model on a limited number of cell types. We develop analytical results and numerical tools to perform this reduction for a specific model characterizing the evolution of a cell by a system of piecewise deterministic Markov processes (PDMP). Solving a spectral problem, we find the explicit variational form of the rate function associated to a large deviations principle, for any number of genes. The resulting Lagrangian dynamics allows us to define a deterministic limit of which the basins of attraction can be identified to cellular types. In this context the quasipotential, describing the transitions between these basins in the weak noise limit, can be defined as the unique solution of an Hamilton-Jacobi equation under a particular constraint. We develop a numerical method for approximating the coarse-grained model parameters, and show its accuracy for a symmetric toggle-switch network. We deduce from the reduced model an approximation of the stationary distribution of the PDMP system, which appears as a Beta mixture. Altogether those results establish a rigorous frame for connecting GRN behavior to the resulting cellular behavior, including the calculation of the probability of jumps between cell types.
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Affiliation(s)
- Elias Ventre
- ENS de Lyon, CNRS UMR 5239, Laboratory of Biology and Modelling of the Cell, Lyon, France. .,Inria Center Grenoble Rhone-Alpes, Team Dracula, Villeurbanne, France. .,Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Villeurbanne, France.
| | - Thibault Espinasse
- Inria Center Grenoble Rhone-Alpes, Team Dracula, Villeurbanne, France.,Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Villeurbanne, France
| | - Charles-Edouard Bréhier
- Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Villeurbanne, France
| | - Vincent Calvez
- Inria Center Grenoble Rhone-Alpes, Team Dracula, Villeurbanne, France.,Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Villeurbanne, France
| | - Thomas Lepoutre
- Inria Center Grenoble Rhone-Alpes, Team Dracula, Villeurbanne, France.,Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, Villeurbanne, France
| | - Olivier Gandrillon
- ENS de Lyon, CNRS UMR 5239, Laboratory of Biology and Modelling of the Cell, Lyon, France.,Inria Center Grenoble Rhone-Alpes, Team Dracula, Villeurbanne, France
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Brackston RD, Lakatos E, Stumpf MPH. Transition state characteristics during cell differentiation. PLoS Comput Biol 2018; 14:e1006405. [PMID: 30235202 PMCID: PMC6168170 DOI: 10.1371/journal.pcbi.1006405] [Citation(s) in RCA: 27] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2018] [Revised: 10/02/2018] [Accepted: 07/27/2018] [Indexed: 12/11/2022] Open
Abstract
Models describing the process of stem-cell differentiation are plentiful, and may offer insights into the underlying mechanisms and experimentally observed behaviour. Waddington's epigenetic landscape has been providing a conceptual framework for differentiation processes since its inception. It also allows, however, for detailed mathematical and quantitative analyses, as the landscape can, at least in principle, be related to mathematical models of dynamical systems. Here we focus on a set of dynamical systems features that are intimately linked to cell differentiation, by considering exemplar dynamical models that capture important aspects of stem cell differentiation dynamics. These models allow us to map the paths that cells take through gene expression space as they move from one fate to another, e.g. from a stem-cell to a more specialized cell type. Our analysis highlights the role of the transition state (TS) that separates distinct cell fates, and how the nature of the TS changes as the underlying landscape changes-change that can be induced by e.g. cellular signaling. We demonstrate that models for stem cell differentiation may be interpreted in terms of either a static or transitory landscape. For the static case the TS represents a particular transcriptional profile that all cells approach during differentiation. Alternatively, the TS may refer to the commonly observed period of heterogeneity as cells undergo stochastic transitions.
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Affiliation(s)
- Rowan D. Brackston
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London, United Kingdom
| | - Eszter Lakatos
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London, United Kingdom
| | - Michael P. H. Stumpf
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London, United Kingdom
- School of BioScience and School of Mathematics and Statistics, University of Melbourne, Melbourne, Australia
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