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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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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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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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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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Guo X, Tang T, Duan M, Zhang L, Ge H. The nonequilibrium mechanism of noise-enhanced drug synergy in HIV latency reactivation. iScience 2022; 25:104358. [PMID: 35620426 PMCID: PMC9127169 DOI: 10.1016/j.isci.2022.104358] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2021] [Revised: 03/04/2022] [Accepted: 04/29/2022] [Indexed: 11/29/2022] Open
Abstract
Noise-modulating chemicals can synergize with transcriptional activators in reactivating latent HIV to eliminate latent HIV reservoirs. To understand the underlying biomolecular mechanism, we investigate a previous two-gene-state model and identify two necessary conditions for the synergy: an assumption of the inhibition effect of transcription activators on noise enhancers; and frequent transitions to the gene non-transcription-permissive state. We then develop a loop-four-gene-state model with Tat transcription/translation and find that drug synergy is mainly determined by the magnitude and direction of energy input into the genetic regulatory kinetics of the HIV promoter. The inhibition effect of transcription activators is actually a phenomenon of energy dissipation in the nonequilibrium gene transition system. Overall, the loop-four-state model demonstrates that energy dissipation plays a crucial role in HIV latency reactivation, which might be useful for improving drug effects and identifying other synergies on lentivirus latency reactivation. The inhibition of Activator on Noise enhancer is necessary for their synergy in reactivating HIV The drug synergy is a nonequilibrium phenomenon in the gene regulatory system The magnitude and direction of energy input determine the drug synergy This nonequilibrium mechanism is general without regarding molecular details
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Detailed balance, local detailed balance, and global potential for stochastic chemical reaction networks. ADV APPL PROBAB 2021. [DOI: 10.1017/apr.2021.3] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
AbstractDetailed balance of a chemical reaction network can be defined in several different ways. Here we investigate the relationship among four types of detailed balance conditions: deterministic, stochastic, local, and zero-order local detailed balance. We show that the four types of detailed balance are equivalent when different reactions lead to different species changes and are not equivalent when some different reactions lead to the same species change. Under the condition of local detailed balance, we further show that the system has a global potential defined over the whole space, which plays a central role in the large deviation theory and the Freidlin–Wentzell-type metastability theory of chemical reaction networks. Finally, we provide a new sufficient condition for stochastic detailed balance, which is applied to construct a class of high-dimensional chemical reaction networks that both satisfies stochastic detailed balance and displays multistability.
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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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Fluctuating-rate model with multiple gene states. J Math Biol 2020; 81:1099-1141. [PMID: 33000313 DOI: 10.1007/s00285-020-01538-2] [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: 02/20/2019] [Revised: 08/28/2020] [Indexed: 10/23/2022]
Abstract
Multiple phenotypic states of single cells often co-exist in the presence of positive feedbacks. Stochastic gene-state switchings and low copy numbers of proteins in single cells cause considerable fluctuations. The chemical master equation (CME) is a powerful tool that describes the dynamics of single cells, but it may be overly complicated. Among many simplified models, a fluctuating-rate (FR) model has been proposed recently to approximate the full CME model in the realistic intermediate region of gene-state switchings. However, only the scenario with two gene states has been carefully analysed. In this paper, we generalise the FR model to the case with multiple gene states, in which the mathematical derivation becomes more complicated. The leading order of fluctuations around each phenotypic state, as well as the transition rates between phenotypic states, in the intermediate gene-state switching region is characterized by the rate function of the stationary distribution of the FR model in the Freidlin-Wentzell-type large deviation principle (LDP). Under certain reasonable assumptions, we show that the derivative of the rate function is equal to the unique nontrivial solution of a dominant generalised eigenvalue problem, leading to a new numerical algorithm for obtaining the LDP rate function directly. Furthermore, we prove the Lyapunov property of the rate function for the corresponding deterministic mean-field dynamics. Finally, through a tristable example, we show that the local fluctuations (the asymptotic variance of the stationary distribution at each phenotypic state) in the intermediate and rapid regions of gene-state switchings are different. Finally, a tri-stable example is constructed to illustrate the validity of our theory.
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Jia C, Grima R. Dynamical phase diagram of an auto-regulating gene in fast switching conditions. J Chem Phys 2020; 152:174110. [DOI: 10.1063/5.0007221] [Citation(s) in RCA: 25] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
Affiliation(s)
- Chen Jia
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
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Holehouse J, Cao Z, Grima R. Stochastic Modeling of Autoregulatory Genetic Feedback Loops: A Review and Comparative Study. Biophys J 2020; 118:1517-1525. [PMID: 32155410 PMCID: PMC7136347 DOI: 10.1016/j.bpj.2020.02.016] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/17/2019] [Revised: 01/27/2020] [Accepted: 02/11/2020] [Indexed: 02/08/2023] Open
Abstract
Autoregulatory feedback loops are one of the most common network motifs. A wide variety of stochastic models have been constructed to understand how the fluctuations in protein numbers in these loops are influenced by the kinetic parameters of the main biochemical steps. These models differ according to 1) which subcellular processes are explicitly modeled, 2) the modeling methodology employed (discrete, continuous, or hybrid), and 3) whether they can be analytically solved for the steady-state distribution of protein numbers. We discuss the assumptions and properties of the main models in the literature, summarize our current understanding of the relationship between them, and highlight some of the insights gained through modeling.
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Affiliation(s)
- James Holehouse
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
| | - Zhixing Cao
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom; The Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai, People's Republic of China
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom.
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Jia C, Grima R. Small protein number effects in stochastic models of autoregulated bursty gene expression. J Chem Phys 2020; 152:084115. [DOI: 10.1063/1.5144578] [Citation(s) in RCA: 23] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
Affiliation(s)
- Chen Jia
- Division of Applied and Computational Mathematics, Beijing Computational Science Research Center, Beijing 100193, China
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
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12
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Chen X, Jia C. Mathematical foundation of nonequilibrium fluctuation–dissipation theorems for inhomogeneous diffusion processes with unbounded coefficients. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2019.02.005] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Jia C, Wang LY, Yin GG, Zhang MQ. Single-cell stochastic gene expression kinetics with coupled positive-plus-negative feedback. Phys Rev E 2019; 100:052406. [PMID: 31869986 DOI: 10.1103/physreve.100.052406] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2019] [Indexed: 06/10/2023]
Abstract
Here we investigate single-cell stochastic gene expression kinetics in a minimal coupled gene circuit with positive-plus-negative feedback. A triphasic stochastic bifurcation is observed upon increasing the ratio of the positive and negative feedback strengths, which reveals a strong synergistic interaction between positive and negative feedback loops. We discover that coupled positive-plus-negative feedback amplifies gene expression mean but reduces gene expression noise over a wide range of feedback strengths when promoter switching is relatively slow, stabilizing gene expression around a relatively high level. In addition, we study two types of macroscopic limits of the discrete chemical master equation model: the Kurtz limit applies to proteins with large burst frequencies and the Lévy limit applies to proteins with large burst sizes. We derive the analytic steady-state distributions of the protein abundance in a coupled gene circuit for both the discrete model and its two macroscopic limits, generalizing the results obtained by Liu et al. [Chaos 26, 043108 (2016)CHAOEH1054-150010.1063/1.4947202]. We also obtain the analytic time-dependent protein distribution for the classical Friedman-Cai-Xie random bursting model [Friedman, Cai, and Xie, Phys. Rev. Lett. 97, 168302 (2006)PRLTAO0031-900710.1103/PhysRevLett.97.168302]. Our analytic results are further applied to study the structure of gene expression noise in a coupled gene circuit, and a complete decomposition of noise in terms of five different biophysical origins is provided.
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Affiliation(s)
- Chen Jia
- Division of Applied and Computational Mathematics, Beijing Computational Science Research Center, Beijing 100193, China
- Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA
| | - Le Yi Wang
- Department of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan 48202, USA
| | - George G Yin
- Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA
| | - Michael Q Zhang
- Department of Biological Sciences, Center for Systems Biology, University of Texas at Dallas, Richardson, Texas 75080, USA
- MOE Key Laboratory of Bioinformatics, Center for Synthetic and Systems Biology, Tsinghua University, Beijing 100084, China
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Atitey K, Loskot P, Rees P. Inferring distributions from observed mRNA and protein copy counts in genetic circuits. Biomed Phys Eng Express 2018. [DOI: 10.1088/2057-1976/aaef5c] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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