1
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Jia C, Grima R. Holimap: an accurate and efficient method for solving stochastic gene network dynamics. Nat Commun 2024; 15:6557. [PMID: 39095346 PMCID: PMC11297302 DOI: 10.1038/s41467-024-50716-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2024] [Accepted: 07/13/2024] [Indexed: 08/04/2024] Open
Abstract
Gene-gene interactions are crucial to the control of sub-cellular processes but our understanding of their stochastic dynamics is hindered by the lack of simulation methods that can accurately and efficiently predict how the distributions of gene product numbers vary across parameter space. To overcome these difficulties, here we present Holimap (high-order linear-mapping approximation), an approach that approximates the protein or mRNA number distributions of a complex gene regulatory network by the distributions of a much simpler reaction system. We demonstrate Holimap's computational advantages over conventional methods by applying it to predict the stochastic time-dependent dynamics of various gene networks, including transcriptional networks ranging from simple autoregulatory loops to complex randomly connected networks, post-transcriptional networks, and post-translational networks. Holimap is ideally suited to study how the intricate network of gene-gene interactions results in precise coordination and control of gene expression.
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Affiliation(s)
- Chen Jia
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing, China
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, UK.
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2
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Jiao F, Li J, Liu T, Zhu Y, Che W, Bleris L, Jia C. What can we learn when fitting a simple telegraph model to a complex gene expression model? PLoS Comput Biol 2024; 20:e1012118. [PMID: 38743803 PMCID: PMC11125521 DOI: 10.1371/journal.pcbi.1012118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/06/2024] [Revised: 05/24/2024] [Accepted: 04/27/2024] [Indexed: 05/16/2024] Open
Abstract
In experiments, the distributions of mRNA or protein numbers in single cells are often fitted to the random telegraph model which includes synthesis and decay of mRNA or protein, and switching of the gene between active and inactive states. While commonly used, this model does not describe how fluctuations are influenced by crucial biological mechanisms such as feedback regulation, non-exponential gene inactivation durations, and multiple gene activation pathways. Here we investigate the dynamical properties of four relatively complex gene expression models by fitting their steady-state mRNA or protein number distributions to the simple telegraph model. We show that despite the underlying complex biological mechanisms, the telegraph model with three effective parameters can accurately capture the steady-state gene product distributions, as well as the conditional distributions in the active gene state, of the complex models. Some effective parameters are reliable and can reflect realistic dynamic behaviors of the complex models, while others may deviate significantly from their real values in the complex models. The effective parameters can also be applied to characterize the capability for a complex model to exhibit multimodality. Using additional information such as single-cell data at multiple time points, we provide an effective method of distinguishing the complex models from the telegraph model. Furthermore, using measurements under varying experimental conditions, we show that fitting the mRNA or protein number distributions to the telegraph model may even reveal the underlying gene regulation mechanisms of the complex models. The effectiveness of these methods is confirmed by analysis of single-cell data for E. coli and mammalian cells. All these results are robust with respect to cooperative transcriptional regulation and extrinsic noise. In particular, we find that faster relaxation speed to the steady state results in more precise parameter inference under large extrinsic noise.
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Affiliation(s)
- Feng Jiao
- Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, China
| | - Jing Li
- Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, China
| | - Ting Liu
- Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, China
| | - Yifeng Zhu
- Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, China
| | - Wenhao Che
- Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, China
| | - Leonidas Bleris
- Bioengineering Department, The University of Texas at Dallas, Richardson, Texas, United States of America
- Center for Systems Biology, The University of Texas at Dallas, Richardson, Texas, United States of America
- Department of Biological Sciences, The University of Texas at Dallas, Richardson, Texas, United States of America
| | - Chen Jia
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing, China
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3
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Banerjee B, Das D. Effects of bursty synthesis in organelle biogenesis. Math Biosci 2024; 370:109156. [PMID: 38346665 DOI: 10.1016/j.mbs.2024.109156] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2023] [Revised: 01/31/2024] [Accepted: 02/03/2024] [Indexed: 02/16/2024]
Abstract
A fundamental question of cell biology is how cells control the number of organelles. The processes of organelle biogenesis, namely de novo synthesis, fission, fusion, and decay, are inherently stochastic, producing cell-to-cell variability in organelle abundance. In addition, experiments suggest that the synthesis of some organelles can be bursty. We thus ask how bursty synthesis impacts intracellular organelle number distribution. We develop an organelle biogenesis model with bursty de novo synthesis by considering geometrically distributed burst sizes. We analytically solve the model in biologically relevant limits and provide exact expressions for the steady-state organelle number distributions and their means and variances. We also present approximate solutions for the whole model, complementing with exact stochastic simulations. We show that bursts generally increase the noise in organelle numbers, producing distinct signatures in noise profiles depending on different mechanisms of organelle biogenesis. We also find different shapes of organelle number distributions, including bimodal distributions in some parameter regimes. Notably, bursty synthesis broadens the parameter regime of observing bimodality compared to the 'non-bursty' case. Together, our framework utilizes number fluctuations to elucidate the role of bursty synthesis in producing organelle number heterogeneity in cells.
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Affiliation(s)
- Binayak Banerjee
- Department of Biological Sciences, Indian Institute of Science Education and Research Kolkata, Nadia 741 246, West Bengal, India
| | - Dipjyoti Das
- Department of Biological Sciences, Indian Institute of Science Education and Research Kolkata, Nadia 741 246, West Bengal, India.
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4
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Wu B, Holehouse J, Grima R, Jia C. Solving the time-dependent protein distributions for autoregulated bursty gene expression using spectral decomposition. J Chem Phys 2024; 160:074105. [PMID: 38364008 DOI: 10.1063/5.0188455] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/21/2023] [Accepted: 01/19/2024] [Indexed: 02/18/2024] Open
Abstract
In this study, we obtain an exact time-dependent solution of the chemical master equation (CME) of an extension of the two-state telegraph model describing bursty or non-bursty protein expression in the presence of positive or negative autoregulation. Using the method of spectral decomposition, we show that the eigenfunctions of the generating function solution of the CME are Heun functions, while the eigenvalues can be determined by solving a continued fraction equation. Our solution generalizes and corrects a previous time-dependent solution for the CME of a gene circuit describing non-bursty protein expression in the presence of negative autoregulation [Ramos et al., Phys. Rev. E 83, 062902 (2011)]. In particular, we clarify that the eigenvalues are generally not real as previously claimed. We also investigate the relationship between different types of dynamic behavior and the type of feedback, the protein burst size, and the gene switching rate.
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Affiliation(s)
- Bingjie Wu
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China
| | - James Holehouse
- The Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, New Mexico 87501, USA
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3BF, United Kingdom
| | - Chen Jia
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China
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Adhikary R, Roy A, Jolly MK, Das D. Effects of microRNA-mediated negative feedback on gene expression noise. Biophys J 2023; 122:4220-4240. [PMID: 37803829 PMCID: PMC10645566 DOI: 10.1016/j.bpj.2023.09.019] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/24/2023] [Revised: 07/19/2023] [Accepted: 09/28/2023] [Indexed: 10/08/2023] Open
Abstract
MicroRNAs (miRNAs) are small noncoding RNAs that regulate gene expression post-transcriptionally in eukaryotes by binding with target mRNAs and preventing translation. miRNA-mediated feedback motifs are ubiquitous in various genetic networks that control cellular decision making. A key question is how such a feedback mechanism may affect gene expression noise. To answer this, we have developed a mathematical model to study the effects of a miRNA-dependent negative-feedback loop on mean expression and noise in target mRNAs. Combining analytics and simulations, we show the existence of an expression threshold demarcating repressed and expressed regimes in agreement with earlier studies. The steady-state mRNA distributions are bimodal near the threshold, where copy numbers of mRNAs and miRNAs exhibit enhanced anticorrelated fluctuations. Moreover, variation of negative-feedback strength shifts the threshold locations and modulates the noise profiles. Notably, the miRNA-mRNA binding affinity and feedback strength collectively shape the bimodality. We also compare our model with a direct auto-repression motif, where a gene produces its own repressor. Auto-repression fails to produce bimodal mRNA distributions as found in miRNA-based indirect repression, suggesting the crucial role of miRNAs in creating phenotypic diversity. Together, we demonstrate how miRNA-dependent negative feedback modifies the expression threshold and leads to a broader parameter regime of bimodality compared to the no-feedback case.
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Affiliation(s)
- Raunak Adhikary
- Department of Biological Sciences, Indian Institute of Science Education And Research Kolkata Mohanpur, Nadia, West Bengal, India
| | - Arnab Roy
- Department of Biological Sciences, Indian Institute of Science Education And Research Kolkata Mohanpur, Nadia, West Bengal, India
| | - Mohit Kumar Jolly
- Centre for BioSystems Science and Engineering, Indian Institute of Science, Bengaluru, India
| | - Dipjyoti Das
- Department of Biological Sciences, Indian Institute of Science Education And Research Kolkata Mohanpur, Nadia, West Bengal, India.
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6
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Öcal K, Sanguinetti G, Grima R. Model reduction for the Chemical Master Equation: An information-theoretic approach. J Chem Phys 2023; 158:114113. [PMID: 36948813 DOI: 10.1063/5.0131445] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/24/2023] Open
Abstract
The complexity of mathematical models in biology has rendered model reduction an essential tool in the quantitative biologist's toolkit. For stochastic reaction networks described using the Chemical Master Equation, commonly used methods include time-scale separation, Linear Mapping Approximation, and state-space lumping. Despite the success of these techniques, they appear to be rather disparate, and at present, no general-purpose approach to model reduction for stochastic reaction networks is known. In this paper, we show that most common model reduction approaches for the Chemical Master Equation can be seen as minimizing a well-known information-theoretic quantity between the full model and its reduction, the Kullback-Leibler divergence defined on the space of trajectories. This allows us to recast the task of model reduction as a variational problem that can be tackled using standard numerical optimization approaches. In addition, we derive general expressions for propensities of a reduced system that generalize those found using classical methods. We show that the Kullback-Leibler divergence is a useful metric to assess model discrepancy and to compare different model reduction techniques using three examples from the literature: an autoregulatory feedback loop, the Michaelis-Menten enzyme system, and a genetic oscillator.
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Affiliation(s)
- Kaan Öcal
- School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, United Kingdom
| | - Guido Sanguinetti
- Scuola Internazionale Superiore di Studi Avanzati, 34136 Trieste, Italy
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JH, United Kingdom
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7
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Ling MY, Chiu LJ, Hsieh CC, Shu CC. Dimerization induces bimodality in protein number distributions. Biosystems 2023; 223:104812. [PMID: 36427705 DOI: 10.1016/j.biosystems.2022.104812] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/27/2022] [Revised: 11/10/2022] [Accepted: 11/10/2022] [Indexed: 11/26/2022]
Abstract
We examined gene expression with DNA switching between two states, active and inactive. Subpopulations emerge from mechanisms that do not arise from trivial transcriptional heterogeneity. Although the RNA demonstrates a unimodal distribution, dimerization intriguingly causes protein bimodality. No control loop or deterministic bistability are present. In such a situation, increasing the degradation rate of the protein does not lead to bimodality. The bimodality is achieved through the interplay between the protein monomer and the formation of protein dimer. We applied Stochastic Simulation Algorithm (SSA) and found that cells spontaneously change states at the protein level. While sweeping parameters, decreasing the rate constant of dimerization severely impairs the bimodality. We also examined the influence of DNA switching. To have bimodality, the system requires a proper ratio of DNA in the active state to the inactive state. In addition to bimodality of the monomer, tetramerization also causes the bimodality of the dimer.
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Affiliation(s)
- Ming-Yang Ling
- Department of Chemical Engineering and Biotechnology, National Taipei University of Technology, Taipei City, Taiwan
| | - Lin-Jie Chiu
- Department of Chemical Engineering and Biotechnology, National Taipei University of Technology, Taipei City, Taiwan
| | - Ching-Chu Hsieh
- Department of Chemical Engineering and Biotechnology, National Taipei University of Technology, Taipei City, Taiwan
| | - Che-Chi Shu
- Department of Chemical Engineering and Biotechnology, National Taipei University of Technology, Taipei City, Taiwan.
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8
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Concentration fluctuations in growing and dividing cells: Insights into the emergence of concentration homeostasis. PLoS Comput Biol 2022; 18:e1010574. [PMID: 36194626 PMCID: PMC9565450 DOI: 10.1371/journal.pcbi.1010574] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2022] [Revised: 10/14/2022] [Accepted: 09/14/2022] [Indexed: 11/19/2022] Open
Abstract
Intracellular reaction rates depend on concentrations and hence their levels are often regulated. However classical models of stochastic gene expression lack a cell size description and cannot be used to predict noise in concentrations. Here, we construct a model of gene product dynamics that includes a description of cell growth, cell division, size-dependent gene expression, gene dosage compensation, and size control mechanisms that can vary with the cell cycle phase. We obtain expressions for the approximate distributions and power spectra of concentration fluctuations which lead to insight into the emergence of concentration homeostasis. We find that (i) the conditions necessary to suppress cell division-induced concentration oscillations are difficult to achieve; (ii) mRNA concentration and number distributions can have different number of modes; (iii) two-layer size control strategies such as sizer-timer or adder-timer are ideal because they maintain constant mean concentrations whilst minimising concentration noise; (iv) accurate concentration homeostasis requires a fine tuning of dosage compensation, replication timing, and size-dependent gene expression; (v) deviations from perfect concentration homeostasis show up as deviations of the concentration distribution from a gamma distribution. Some of these predictions are confirmed using data for E. coli, fission yeast, and budding yeast.
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9
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García-Tejera R, Schumacher L, Grima R. Regulation of stem cell dynamics through volume exclusion. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2022.0376] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022] Open
Abstract
The maintenance and regeneration of adult tissues rely on the self-renewal of stem cells. Regeneration without over-proliferation requires precise regulation of the stem cell proliferation and differentiation rates. The nature of such regulatory mechanisms in different tissues, and how to incorporate them in models of stem cell population dynamics, is incompletely understood. The critical birth-death (CBD) process is widely used to model stem cell populations, capturing key phenomena, such as scaling laws in clone size distributions. However, the CBD process neglects regulatory mechanisms. Here, we propose the birth-death process with volume exclusion (vBD), a variation of the birth-death process that considers crowding effects, such as may arise due to limited space in a stem cell niche. While the deterministic rate equations predict a single non-trivial attracting steady state, the master equation predicts extinction and transient distributions of stem cell numbers with three possible behaviours: long-lived quasi-steady state (QSS), and short-lived bimodal or unimodal distributions. In all cases, we approximate solutions to the vBD master equation using a renormalized system-size expansion, QSS approximation and the Wentzel–Kramers–Brillouin method. Our study suggests that the size distribution of a stem cell population bears signatures that are useful to detect negative feedback mediated via volume exclusion.
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Affiliation(s)
- Rodrigo García-Tejera
- Centre for Regenerative Medicine, University of Edinburgh, 5 Little France Dr, Edinburgh EH16 4UU, UK
- School of Biological Sciences, University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JF, UK
| | - Linus Schumacher
- Centre for Regenerative Medicine, University of Edinburgh, 5 Little France Dr, Edinburgh EH16 4UU, UK
- School of Biological Sciences, University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JF, UK
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JF, UK
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10
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Öcal K, Gutmann MU, Sanguinetti G, Grima R. Inference and uncertainty quantification of stochastic gene expression via synthetic models. JOURNAL OF THE ROYAL SOCIETY, INTERFACE 2022; 19:20220153. [PMID: 35858045 PMCID: PMC9277240 DOI: 10.1098/rsif.2022.0153] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 12/26/2022]
Abstract
Estimating uncertainty in model predictions is a central task in quantitative biology. Biological models at the single-cell level are intrinsically stochastic and nonlinear, creating formidable challenges for their statistical estimation which inevitably has to rely on approximations that trade accuracy for tractability. Despite intensive interest, a sweet spot in this trade-off has not been found yet. We propose a flexible procedure for uncertainty quantification in a wide class of reaction networks describing stochastic gene expression including those with feedback. The method is based on creating a tractable coarse-graining of the model that is learned from simulations, a synthetic model, to approximate the likelihood function. We demonstrate that synthetic models can substantially outperform state-of-the-art approaches on a number of non-trivial systems and datasets, yielding an accurate and computationally viable solution to uncertainty quantification in stochastic models of gene expression.
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Affiliation(s)
- Kaan Öcal
- School of Informatics, University of Edinburgh, Edinburgh EH9 3JH, UK.,School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JH, UK
| | - Michael U Gutmann
- School of Informatics, University of Edinburgh, Edinburgh EH9 3JH, UK
| | - Guido Sanguinetti
- Scuola Internazionale Superiore di Studi Avanzati, 34136 Trieste, Italy
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JH, UK
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11
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A Novel Dynamical Regulation of mRNA Distribution by Cross-Talking Pathways. MATHEMATICS 2022. [DOI: 10.3390/math10091515] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/10/2022]
Abstract
In this paper, we use a similar approach to the one proposed by Chen and Jiao to calculate the mathematical formulas of the generating function V(z,t) and the mass function Pm(t) of a cross-talking pathways model in large parameter regions. Together with kinetic rates from yeast and mouse genes, our numerical examples reveal novel bimodal mRNA distributions for intermediate times, whereby the mode of distribution Pm(t) displays unimodality with the peak at m=0 for initial and long times, which has not been obtained in previous works. Such regulation of mRNA distribution exactly matches the transcriptional dynamics for the osmosensitive genes in Saccharomyces cerevisiae, which has not been generated by those models with one single pathway or feedback loops. This paper may provide us with a novel observation on transcriptional distribution dynamics regulated by multiple signaling pathways in response to environmental changes and genetic perturbations.
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12
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Filatova T, Popović N, Grima R. Modulation of nuclear and cytoplasmic mRNA fluctuations by time-dependent stimuli: Analytical distributions. Math Biosci 2022; 347:108828. [DOI: 10.1016/j.mbs.2022.108828] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2022] [Revised: 04/15/2022] [Accepted: 04/15/2022] [Indexed: 10/18/2022]
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13
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Gorin G, Pachter L. Modeling bursty transcription and splicing with the chemical master equation. Biophys J 2022; 121:1056-1069. [PMID: 35143775 PMCID: PMC8943761 DOI: 10.1016/j.bpj.2022.02.004] [Citation(s) in RCA: 13] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2021] [Revised: 11/29/2021] [Accepted: 02/03/2022] [Indexed: 11/16/2022] Open
Abstract
Splicing cascades that alter gene products posttranscriptionally also affect expression dynamics. We study a class of processes and associated distributions that emerge from models of bursty promoters coupled to directed acyclic graphs of splicing. These solutions provide full time-dependent joint distributions for an arbitrary number of species with general noise behaviors and transient phenomena, offering qualitative and quantitative insights about how splicing can regulate expression dynamics. Finally, we derive a set of quantitative constraints on the minimum complexity necessary to reproduce gene coexpression patterns using synchronized burst models. We validate these findings by analyzing long-read sequencing data, where we find evidence of expression patterns largely consistent with these constraints.
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Affiliation(s)
- Gennady Gorin
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California
| | - Lior Pachter
- Division of Biology and Biological Engineering & Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California.
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14
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Chen L, Lin G, Jiao F. Using average transcription level to understand the regulation of stochastic gene activation. ROYAL SOCIETY OPEN SCIENCE 2022; 9:211757. [PMID: 35223065 PMCID: PMC8847896 DOI: 10.1098/rsos.211757] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/08/2021] [Accepted: 01/24/2022] [Indexed: 05/03/2023]
Abstract
Gene activation is a random process, modelled as a framework of multiple rate-limiting steps listed sequentially, in parallel or in combination. Together with suitably assumed processes of gene inactivation, transcript birth and death, the step numbers and parameters in activation frameworks can be estimated by fitting single-cell transcription data. However, current algorithms require computing master equations that are tightly correlated with prior hypothetical frameworks of gene activation. We found that prior estimation of the framework can be facilitated by the traditional dynamical data of mRNA average level M(t), presenting discriminated dynamical features. Rigorous theory regarding M(t) profiles allows to confidently rule out the frameworks that fail to capture M(t) features and to test potential competent frameworks by fitting M(t) data. We implemented this procedure for a large number of mouse fibroblast genes under tumour necrosis factor induction and determined exactly the 'cross-talking n-state' framework; the cross-talk between the signalling and basal pathways is crucial to trigger the first peak of M(t), while the following damped gentle M(t) oscillation is regulated by the multi-step basal pathway. This framework can be used to fit sophisticated single-cell data and may facilitate a more accurate understanding of stochastic activation of mouse fibroblast genes.
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Affiliation(s)
- Liang Chen
- Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, People’s Republic of China
- School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, People’s Republic of China
| | - Genghong Lin
- Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, People’s Republic of China
- School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, People’s Republic of China
| | - Feng Jiao
- Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, People’s Republic of China
- School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, People’s Republic of China
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15
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A Novel Approach for Calculating Exact Forms of mRNA Distribution in Single-Cell Measurements. MATHEMATICS 2021. [DOI: 10.3390/math10010027] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/21/2022]
Abstract
Gene transcription is a stochastic process manifested by fluctuations in mRNA copy numbers in individual isogenic cells. Together with mathematical models of stochastic transcription, the massive mRNA distribution data that can be used to quantify fluctuations in mRNA levels can be fitted by Pm(t), which is the probability of producing m mRNA molecules at time t in a single cell. Tremendous efforts have been made to derive analytical forms of Pm(t), which rely on solving infinite arrays of the master equations of models. However, current approaches focus on the steady-state (t→∞) or require several parameters to be zero or infinity. Here, we present an approach for calculating Pm(t) with time, where all parameters are positive and finite. Our approach was successfully implemented for the classical two-state model and the widely used three-state model and may be further developed for different models with constant kinetic rates of transcription. Furthermore, the direct computations of Pm(t) for the two-state model and three-state model showed that the different regulations of gene activation can generate discriminated dynamical bimodal features of mRNA distribution under the same kinetic rates and similar steady-state mRNA distribution.
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16
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Li Y, Jiang DQ, Jia C. Steady-state joint distribution for first-order stochastic reaction kinetics. Phys Rev E 2021; 104:024408. [PMID: 34525607 DOI: 10.1103/physreve.104.024408] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2021] [Accepted: 07/19/2021] [Indexed: 11/07/2022]
Abstract
While the analytical solution for the marginal distribution of a stochastic chemical reaction network has been extensively studied, its joint distribution, i.e., the solution of a high-dimensional chemical master equation, has received much less attention. Here we develop an alternative method of computing the exact joint distributions of a wide class of first-order stochastic reaction systems in steady-state conditions. The effectiveness of our method is validated by applying it to four gene expression models of biological significance, including models with 2A peptides, nascent mRNA, gene regulation, translational bursting, and alternative splicing.
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Affiliation(s)
- Youming Li
- LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China.,Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China
| | - Da-Quan Jiang
- LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China.,Center for Statistical Science, Peking University, Beijing 100871, China
| | - Chen Jia
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China
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17
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Wu Z, Wang Y, Wang K, Zhou D. Stochastic stem cell models with mutation: A comparison of asymmetric and symmetric divisions. Math Biosci 2021; 332:108541. [PMID: 33453222 DOI: 10.1016/j.mbs.2021.108541] [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: 10/08/2020] [Revised: 01/05/2021] [Accepted: 01/08/2021] [Indexed: 12/12/2022]
Abstract
In order to fulfill cell proliferation and differentiation through cellular hierarchy, stem cells can undergo either asymmetric or symmetric divisions. Recent studies pay special attention to the effect of different modes of stem cell division on the lifetime risk of cancer, and report that symmetric division is more beneficial to delay the onset of cancer. The fate uncertainty of symmetric division is considered to be the reason for the cancer-delaying effect. In this paper we compare asymmetric and symmetric divisions of stem cells via studying stochastic stem cell models with mutation. Specially, by using rigorous mathematical analysis we find that both the asymmetric and symmetric models show the same statistical average, but the symmetric model shows higher fluctuation than the asymmetric model. We further show that the difference between the two models would be more remarkable for lower mutation rates. Our work quantifies the uncertainty of cell division and highlights the significance of stochasticity for distinguishing between different modes of stem cell division.
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Affiliation(s)
- Zhijie Wu
- School of Mathematical Sciences, Xiamen University, Xiamen 361005, PR China
| | - Yuman Wang
- School of Mathematical Sciences, Xiamen University, Xiamen 361005, PR China
| | - Kun Wang
- School of Mathematical Sciences, Xiamen University, Xiamen 361005, PR China
| | - Da Zhou
- School of Mathematical Sciences, Xiamen University, Xiamen 361005, PR China.
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Holehouse J, Sukys A, Grima R. Stochastic time-dependent enzyme kinetics: Closed-form solution and transient bimodality. J Chem Phys 2020; 153:164113. [PMID: 33138415 DOI: 10.1063/5.0017573] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Abstract
We derive an approximate closed-form solution to the chemical master equation describing the Michaelis-Menten reaction mechanism of enzyme action. In particular, assuming that the probability of a complex dissociating into an enzyme and substrate is significantly larger than the probability of a product formation event, we obtain expressions for the time-dependent marginal probability distributions of the number of substrate and enzyme molecules. For delta function initial conditions, we show that the substrate distribution is either unimodal at all times or else becomes bimodal at intermediate times. This transient bimodality, which has no deterministic counterpart, manifests when the initial number of substrate molecules is much larger than the total number of enzyme molecules and if the frequency of enzyme-substrate binding events is large enough. Furthermore, we show that our closed-form solution is different from the solution of the chemical master equation reduced by means of the widely used discrete stochastic Michaelis-Menten approximation, where the propensity for substrate decay has a hyperbolic dependence on the number of substrate molecules. The differences arise because the latter does not take into account enzyme number fluctuations, while our approach includes them. We confirm by means of a stochastic simulation of all the elementary reaction steps in the Michaelis-Menten mechanism that our closed-form solution is accurate over a larger region of parameter space than that obtained using the discrete stochastic Michaelis-Menten approximation.
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Affiliation(s)
- James Holehouse
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
| | - Augustinas Sukys
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
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