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Liu Y, Zhang SY, Kleijn IT, Stumpf MPH. Approximate Bayesian computation for inferring Waddington landscapes from single-cell data. ROYAL SOCIETY OPEN SCIENCE 2024; 11:231697. [PMID: 39076359 PMCID: PMC11285904 DOI: 10.1098/rsos.231697] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/26/2023] [Accepted: 05/01/2024] [Indexed: 07/31/2024]
Abstract
Single-cell technologies allow us to gain insights into cellular processes at unprecedented resolution. In stem cell and developmental biology snapshot data allow us to characterize how the transcriptional states of cells change between successive cell types. Here, we show how approximate Bayesian computation (ABC) can be employed to calibrate mathematical models against single-cell data. In our simulation study, we demonstrate the pivotal role of the adequate choice of distance measures appropriate for single-cell data. We show that for good distance measures, notably optimal transport with the Sinkhorn divergence, we can infer parameters for mathematical models from simulated single-cell data. We show that the ABC posteriors can be used (i) to characterize parameter sensitivity and identify dependencies between different parameters and (ii) to construct representations of the Waddington or epigenetic landscape, which forms a popular and interpretable representation of the developmental dynamics. In summary, these results pave the way for fitting mechanistic models of stem cell differentiation to single-cell data.
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Affiliation(s)
- Yujing Liu
- School of Mathematics and Statistics, University of Melbourne, Melbourne, Australia
| | - Stephen Y. Zhang
- School of Mathematics and Statistics, University of Melbourne, Melbourne, Australia
| | | | - Michael P. H. Stumpf
- School of Mathematics and Statistics, University of Melbourne, Melbourne, Australia
- School of BioScience, University of Melbourne, Melbourne, Australia
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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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Noise distorts the epigenetic landscape and shapes cell-fate decisions. Cell Syst 2021; 13:83-102.e6. [PMID: 34626539 DOI: 10.1016/j.cels.2021.09.002] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/20/2021] [Revised: 06/21/2021] [Accepted: 09/02/2021] [Indexed: 12/24/2022]
Abstract
The Waddington epigenetic landscape has become an iconic representation of the cellular differentiation process. Recent single-cell transcriptomic data provide new opportunities for quantifying this originally conceptual tool, offering insight into the gene regulatory networks underlying cellular development. While many methods for constructing the landscape have been proposed, by far the most commonly employed approach is based on computing the landscape as the negative logarithm of the steady-state probability distribution. Here, we use simple models to highlight the complexities and limitations that arise when reconstructing the potential landscape in the presence of stochastic fluctuations. We consider how the landscape changes in accordance with different stochastic systems and show that it is the subtle interplay between the deterministic and stochastic components of the system that ultimately shapes the landscape. We further discuss how the presence of noise has important implications for the identifiability of the regulatory dynamics from experimental data. A record of this paper's transparent peer review process is included in the supplemental information.
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Kirunda JB, Yang L, Lu L, Jia Y. Effects of noise and time delay on E2F's expression level in a bistable Rb-E2F gene's regulatory network. IET Syst Biol 2021; 15:111-125. [PMID: 33881232 PMCID: PMC8675803 DOI: 10.1049/syb2.12017] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2021] [Revised: 03/18/2021] [Accepted: 03/24/2021] [Indexed: 12/15/2022] Open
Abstract
The bistable Rb-E2F gene regulatory network plays a central role in regulating cellular proliferation-quiescence transition. Based on Gillespie's chemical Langevin method, the stochastic bistable Rb-E2F gene's regulatory network with time delays is proposed. It is found that under the moderate intensity of internal noise, delay in the Cyclin E synthesis rate can greatly increase the average concentration value of E2F. When the delay is considered in both E2F-related positive feedback loops, within a specific range of delay (3-13) hr , the average expression of E2F is significantly increased. Also, this range is in the scope with that experimentally given by Dong et al. [65]. By analysing the quasi-potential curves at different delay times, simulation results show that delay regulates the dynamic behaviour of the system in the following way: small delay stabilises the bistable system; the medium delay is conducive to a high steady-state, making the system fluctuate near the high steady-state; large delay induces approximately periodic transitions between high and low steady-state. Therefore, by regulating noise and time delay, the cell itself can control the expression level of E2F to respond to different situations. These findings may provide an explanation of some experimental result intricacies related to the cell cycle.
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Affiliation(s)
- John Billy Kirunda
- Department of Physics and Institute of Biophysics, Central China Normal University, Wuhan, China
| | - Lijian Yang
- Department of Physics and Institute of Biophysics, Central China Normal University, Wuhan, China
| | - Lulu Lu
- Department of Physics and Institute of Biophysics, Central China Normal University, Wuhan, China
| | - Ya Jia
- Department of Physics and Institute of Biophysics, Central China Normal University, Wuhan, China
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Plant AL, Halter M, Stinson J. Probing pluripotency gene regulatory networks with quantitative live cell imaging. Comput Struct Biotechnol J 2020; 18:2733-2743. [PMID: 33101611 PMCID: PMC7560648 DOI: 10.1016/j.csbj.2020.09.025] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/01/2020] [Revised: 09/14/2020] [Accepted: 09/15/2020] [Indexed: 11/12/2022] Open
Abstract
Live cell imaging uniquely enables the measurement of dynamic events in single cells, but it has not been used often in the study of gene regulatory networks. Network components can be examined in relation to one another by quantitative live cell imaging of fluorescent protein reporter cell lines that simultaneously report on more than one network component. A series of dual-reporter cell lines would allow different combinations of network components to be examined in individual cells. Dynamical information about interacting network components in individual cells is critical to predictive modeling of gene regulatory networks, and such information is not accessible through omics and other end point techniques. Achieving this requires that gene-edited cell lines are appropriately designed and adequately characterized to assure the validity of the biological conclusions derived from the expression of the reporters. In this brief review we discuss what is known about the importance of dynamics to network modeling and review some recent advances in optical microscopy methods and image analysis approaches that are making the use of quantitative live cell imaging for network analysis possible. We also discuss how strategies for genetic engineering of reporter cell lines can influence the biological relevance of the data.
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Affiliation(s)
- Anne L Plant
- Biosystems and Biomaterials Division, National Institute of Standards and Technology, United States
| | - Michael Halter
- Biosystems and Biomaterials Division, National Institute of Standards and Technology, United States
| | - Jeffrey Stinson
- Biosystems and Biomaterials Division, National Institute of Standards and Technology, United States
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Abstract
Stochastic resonance (SR) is a prominent phenomenon in many natural and engineered noisy systems, whereby the response to a periodic forcing is greatly amplified when the intensity of the noise is tuned to within a specific range of values. We propose here a general mathematical framework based on large deviation theory and, specifically, on the theory of quasipotentials, for describing SR in noisy N-dimensional nonequilibrium systems possessing two metastable states and undergoing a periodically modulated forcing. The drift and the volatility fields of the equations of motion can be fairly general, and the competing attractors of the deterministic dynamics and the edge state living on the basin boundary can, in principle, feature chaotic dynamics. Similarly, the perturbation field of the forcing can be fairly general. Our approach is able to recover as special cases the classical results previously presented in the literature for systems obeying detailed balance and allows for expressing the parameters describing SR and the statistics of residence times in the two-state approximation in terms of the unperturbed drift field, the volatility field, and the perturbation field. We clarify which specific properties of the forcing are relevant for amplifying or suppressing SR in a system and classify forcings according to classes of equivalence. Our results indicate a route for a detailed understanding of SR in rather general systems.
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Affiliation(s)
- Valerio Lucarini
- Centre for the Mathematics of Planet Earth, University of Reading, Reading RG66AX, United Kingdom; Department of Mathematics and Statistics, University of Reading, Reading RG66AX, United Kingdom; and CEN, University of Hamburg, Hamburg 20144, Germany
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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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