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Pan X, Chen H, Shen A, Zhao D, Su X. A Reliability Assessment Method for Complex Systems Based on Non-Homogeneous Markov Processes. SENSORS (BASEL, SWITZERLAND) 2024; 24:3446. [PMID: 38894237 PMCID: PMC11174667 DOI: 10.3390/s24113446] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2024] [Revised: 05/19/2024] [Accepted: 05/24/2024] [Indexed: 06/21/2024]
Abstract
The Markov method is a common reliability assessment method. It is often used to describe the dynamic characteristics of a system, such as its repairability, fault sequence and multiple degradation states. However, the "curse of dimensionality", which refers to the exponential growth of the system state space with the increase in system complexity, presents a challenge to reliability assessments for complex systems based on the Markov method. In response to this challenge, a novel reliability assessment method for complex systems based on non-homogeneous Markov processes is proposed. This method entails the decomposition of a complex system into multilevel subsystems, each with a relatively small state space, in accordance with the system function. The homogeneous Markov model or the non-homogeneous Markov model is established for each subsystem/system from bottom to top. In order to utilize the outcomes of the lower-level subsystem models as inputs to the upper-level subsystem model, an algorithm is proposed for converting the unavailability curve of a subsystem into its corresponding 2×2 dynamic state transition probability matrix (STPM). The STPM is then employed as an input to the upper-level system's non-homogeneous Markov model. A case study is presented using the reliability assessment of the Reactor Protection System (RPS) based on the proposed method, which is then compared with the models based on the other two contrast methods. This comparison verifies the effectiveness and accuracy of the proposed method.
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Affiliation(s)
- Xiaolei Pan
- College of Automation Engineering, Shanghai University of Electric Power, Shanghai 200090, China; (H.C.); (A.S.); (D.Z.); (X.S.)
- Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200090, China
| | - Hongxiao Chen
- College of Automation Engineering, Shanghai University of Electric Power, Shanghai 200090, China; (H.C.); (A.S.); (D.Z.); (X.S.)
- Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200090, China
| | - Ao Shen
- College of Automation Engineering, Shanghai University of Electric Power, Shanghai 200090, China; (H.C.); (A.S.); (D.Z.); (X.S.)
- Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200090, China
| | - Dongdong Zhao
- College of Automation Engineering, Shanghai University of Electric Power, Shanghai 200090, China; (H.C.); (A.S.); (D.Z.); (X.S.)
- Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200090, China
| | - Xiaoyan Su
- College of Automation Engineering, Shanghai University of Electric Power, Shanghai 200090, China; (H.C.); (A.S.); (D.Z.); (X.S.)
- Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200090, China
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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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5
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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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Wang X, Li Y, Jia C. Poisson representation: a bridge between discrete and continuous models of stochastic gene regulatory networks. J R Soc Interface 2023; 20:20230467. [PMID: 38016635 PMCID: PMC10684348 DOI: 10.1098/rsif.2023.0467] [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: 08/11/2023] [Accepted: 11/08/2023] [Indexed: 11/30/2023] Open
Abstract
Stochastic gene expression dynamics can be modelled either discretely or continuously. Previous studies have shown that the mRNA or protein number distributions of some simple discrete and continuous gene expression models are related by Gardiner's Poisson representation. Here, we systematically investigate the Poisson representation in complex stochastic gene regulatory networks. We show that when the gene of interest is unregulated, the discrete and continuous descriptions of stochastic gene expression are always related by the Poisson representation, no matter how complex the model is. This generalizes the results obtained in Dattani & Barahona (Dattani & Barahona 2017 J. R. Soc. Interface 14, 20160833 (doi:10.1098/rsif.2016.0833)). In addition, using a simple counter-example, we find that the Poisson representation in general fails to link the two descriptions when the gene is regulated. However, for a general stochastic gene regulatory network, we demonstrate that the discrete and continuous models are approximately related by the Poisson representation in the limit of large protein numbers. These theoretical results are further applied to analytically solve many complex gene expression models whose exact distributions are previously unknown.
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Affiliation(s)
- Xinyu Wang
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China
| | - Youming Li
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, People’s Republic of China
| | - Chen Jia
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China
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7
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Jia C, Grima R. Coupling gene expression dynamics to cell size dynamics and cell cycle events: Exact and approximate solutions of the extended telegraph model. iScience 2022; 26:105746. [PMID: 36619980 PMCID: PMC9813732 DOI: 10.1016/j.isci.2022.105746] [Citation(s) in RCA: 9] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2022] [Revised: 11/02/2022] [Accepted: 12/02/2022] [Indexed: 12/12/2022] Open
Abstract
The standard model describing the fluctuations of mRNA numbers in single cells is the telegraph model which includes synthesis and degradation of mRNA, and switching of the gene between active and inactive states. While commonly used, this model does not describe how fluctuations are influenced by the cell cycle phase, cellular growth and division, and other crucial aspects of cellular biology. Here, we derive the analytical time-dependent solution of an extended telegraph model that explicitly considers the doubling of gene copy numbers upon DNA replication, dependence of the mRNA synthesis rate on cellular volume, gene dosage compensation, partitioning of molecules during cell division, cell-cycle duration variability, and cell-size control strategies. Based on the time-dependent solution, we obtain the analytical distributions of transcript numbers for lineage and population measurements in steady-state growth and also find a linear relation between the Fano factor of mRNA fluctuations and cell volume fluctuations. We show that generally the lineage and population distributions in steady-state growth cannot be accurately approximated by the steady-state solution of extrinsic noise models, i.e. a telegraph model with parameters drawn from probability distributions. This is because the mRNA lifetime is often not small enough compared to the cell cycle duration to erase the memory of division and replication. Accurate approximations are possible when this memory is weak, e.g. for genes with bursty expression and for which there is sufficient gene dosage compensation when replication occurs.
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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 EH9 3JH, UK,Corresponding author
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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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Stochastic Transcription with Alterable Synthesis Rates. MATHEMATICS 2022. [DOI: 10.3390/math10132189] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Background: Gene transcription is a random bursting process that leads to large variability in mRNA numbers in single cells. The main cause is largely attributed to random switching between periods of active and inactive gene transcription. In some experiments, it has been observed that variation in the number of active transcription sites causes the initiation rate to vary during elongation. Results: We established a mathematical model based on the molecular reaction mechanism in single cells and studied a stochastic transcription system consisting of two active states and one inactive state, in which mRNA molecules are produced with two different synthesis rates. Conclusions: By calculation, we obtained the average mRNA expression level, the noise strength, and the skewness of transcripts. We gave a necessary and sufficient condition that causes the average mRNA level to peak at a limited time. The model could help us to distinguish an appropriate mechanism that may be employed by cells to transcribe mRNA molecules. Our simulations were in agreement with some experimental data and showed that the skewness can measure the deviation of the distribution of transcripts from the mean value. Especially for mature mRNAs, their distributions were almost able to be determined by the mean, the noise (or the noise strength), and the skewness.
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10
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Gao JX, Wang ZY, Zhang MQ, Qian MP, Jiang DQ. A data-driven method to learn a jump diffusion process from aggregate biological gene expression data. J Theor Biol 2022; 532:110923. [PMID: 34606876 DOI: 10.1016/j.jtbi.2021.110923] [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: 02/03/2021] [Revised: 08/15/2021] [Accepted: 08/29/2021] [Indexed: 10/20/2022]
Abstract
Dynamic models of gene expression are urgently required. In this paper, we describe the time evolution of gene expression by learning a jump diffusion process to model the biological process directly. Our algorithm needs aggregate gene expression data as input and outputs the parameters of the jump diffusion process. The learned jump diffusion process can predict population distributions of gene expression at any developmental stage, obtain long-time trajectories for individual cells, and offer a novel approach to computing RNA velocity. Moreover, it studies biological systems from a stochastic dynamic perspective. Gene expression data at a time point, which is a snapshot of a cellular process, is treated as an empirical marginal distribution of a stochastic process. The Wasserstein distance between the empirical distribution and predicted distribution by the jump diffusion process is minimized to learn the dynamics. For the learned jump diffusion process, its trajectories correspond to the development process of cells, the stochasticity determines the heterogeneity of cells, its instantaneous rate of state change can be taken as "RNA velocity", and the changes in scales and orientations of clusters can be noticed too. We demonstrate that our method can recover the underlying nonlinear dynamics better compared to previous parametric models and the diffusion processes driven by Brownian motion for both synthetic and real world datasets. Our method is also robust to perturbations of data because the computation involves only population expectations.
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Affiliation(s)
- Jia-Xing Gao
- LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
| | - Zhen-Yi Wang
- MOE Key Laboratory of Bioinformatics; Bioinformatics Division and Center for Synthetic and Systems Biology, BNRist; Department of Automation, Tsinghua University, Beijing 100084, China
| | - Michael Q Zhang
- MOE Key Laboratory of Bioinformatics; Bioinformatics Division and Center for Synthetic and Systems Biology, BNRist; School of Medicine, Tsinghua University, Beijing 100084, China; Department of Biological Sciences, Center for Systems Biology, The University of Texas, Richardson, TX 75080-3021, USA
| | - Min-Ping Qian
- LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China
| | - Da-Quan Jiang
- LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China; Center for Statistical Science, Peking University, Beijing 100871, China.
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11
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Chen L, Zhu C, Jiao F. A generalized moment-based method for estimating parameters of stochastic gene transcription. Math Biosci 2022; 345:108780. [DOI: 10.1016/j.mbs.2022.108780] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2021] [Revised: 12/27/2021] [Accepted: 01/13/2022] [Indexed: 12/22/2022]
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12
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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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13
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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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14
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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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15
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Limit theorems for generalized density-dependent Markov chains and bursty stochastic gene regulatory networks. J Math Biol 2019; 80:959-994. [DOI: 10.1007/s00285-019-01445-1] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2019] [Revised: 08/22/2019] [Indexed: 11/26/2022]
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16
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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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17
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Lin YT, Buchler NE. Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes. J R Soc Interface 2019; 15:rsif.2017.0804. [PMID: 29386401 PMCID: PMC5805981 DOI: 10.1098/rsif.2017.0804] [Citation(s) in RCA: 22] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2017] [Accepted: 01/08/2018] [Indexed: 11/12/2022] Open
Abstract
Single-cell experiments show that gene expression is stochastic and bursty, a feature that can emerge from slow switching between promoter states with different activities. In addition to slow chromatin and/or DNA looping dynamics, one source of long-lived promoter states is the slow binding and unbinding kinetics of transcription factors to promoters, i.e. the non-adiabatic binding regime. Here, we introduce a simple analytical framework, known as a piecewise deterministic Markov process (PDMP), that accurately describes the stochastic dynamics of gene expression in the non-adiabatic regime. We illustrate the utility of the PDMP on a non-trivial dynamical system by analysing the properties of a titration-based oscillator in the non-adiabatic limit. We first show how to transform the underlying chemical master equation into a PDMP where the slow transitions between promoter states are stochastic, but whose rates depend upon the faster deterministic dynamics of the transcription factors regulated by these promoters. We show that the PDMP accurately describes the observed periods of stochastic cycles in activator and repressor-based titration oscillators. We then generalize our PDMP analysis to more complicated versions of titration-based oscillators to explain how multiple binding sites lengthen the period and improve coherence. Last, we show how noise-induced oscillation previously observed in a titration-based oscillator arises from non-adiabatic and discrete binding events at the promoter site.
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Affiliation(s)
- Yen Ting Lin
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA .,School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK
| | - Nicolas E Buchler
- Department of Physics, Duke University, Durham, NC 27708, USA.,Department of Biology, Duke University, Durham, NC 27708, USA.,Center for Genomic and Computational Biology, Durham, NC 27710, USA
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Hufton PG, Lin YT, Galla T. Model reduction methods for population dynamics with fast-switching environments: Reduced master equations, stochastic differential equations, and applications. Phys Rev E 2019; 99:032122. [PMID: 30999395 DOI: 10.1103/physreve.99.032122] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2019] [Indexed: 11/07/2022]
Abstract
We study stochastic population dynamics coupled to fast external environments and combine expansions in the inverse switching rate of the environment and a Kramers-Moyal expansion in the inverse size of the population. This leads to a series of approximation schemes, capturing both intrinsic and environmental noise. These methods provide a means of efficient simulation and we show how they can be used to obtain analytical results for the fluctuations of population dynamics in switching environments. We place the approximations in relation to existing work on piecewise-deterministic and piecewise-diffusive Markov processes. Finally, we demonstrate the accuracy and efficiency of these model-reduction methods in different research fields, including systems in biology and a model of crack propagation.
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Affiliation(s)
- Peter G Hufton
- Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
| | - Yen Ting Lin
- Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom.,Center for Nonlinear Studies and Theoretical and Biophysics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - Tobias Galla
- Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
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Jia C, Qian H, Chen M, Zhang MQ. Relaxation rates of gene expression kinetics reveal the feedback signs of autoregulatory gene networks. J Chem Phys 2018. [DOI: 10.1063/1.5009749] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Affiliation(s)
- Chen Jia
- Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
| | - Min Chen
- Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Michael Q. Zhang
- Department of Biological Sciences, Center for Systems Biology, University of Texas at Dallas, Richardson, Texas 75080, USA
- MOE Key Lab and Division of Bioinformatics, CSSB, TNLIST, Tsinghua University, Beijing 100084, China
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Jia C, Zhang MQ, Qian H. Emergent Lévy behavior in single-cell stochastic gene expression. Phys Rev E 2018; 96:040402. [PMID: 29347590 DOI: 10.1103/physreve.96.040402] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2017] [Indexed: 11/07/2022]
Abstract
Single-cell gene expression is inherently stochastic; its emergent behavior can be defined in terms of the chemical master equation describing the evolution of the mRNA and protein copy numbers as the latter tends to infinity. We establish two types of "macroscopic limits": the Kurtz limit is consistent with the classical chemical kinetics, while the Lévy limit provides a theoretical foundation for an empirical equation proposed in N. Friedman et al., Phys. Rev. Lett. 97, 168302 (2006)PRLTAO0031-900710.1103/PhysRevLett.97.168302. Furthermore, we clarify the biochemical implications and ranges of applicability for various macroscopic limits and calculate a comprehensive analytic expression for the protein concentration distribution in autoregulatory gene networks. The relationship between our work and modern population genetics is discussed.
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Affiliation(s)
- Chen Jia
- Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Michael Q Zhang
- Department of Biological Sciences, University of Texas at Dallas, Richardson, Texas 75080, USA.,MOE Key Lab and Division of Bioinformatics, CSSB, TNLIST, Tsinghua University, Beijing 100084, China
| | - Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
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Stochastic fluctuations can reveal the feedback signs of gene regulatory networks at the single-molecule level. Sci Rep 2017; 7:16037. [PMID: 29167445 PMCID: PMC5700158 DOI: 10.1038/s41598-017-15464-9] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2017] [Accepted: 10/23/2017] [Indexed: 01/17/2023] Open
Abstract
Understanding the relationship between spontaneous stochastic fluctuations and the topology of the underlying gene regulatory network is of fundamental importance for the study of single-cell stochastic gene expression. Here by solving the analytical steady-state distribution of the protein copy number in a general kinetic model of stochastic gene expression with nonlinear feedback regulation, we reveal the relationship between stochastic fluctuations and feedback topology at the single-molecule level, which provides novel insights into how and to what extent a feedback loop can enhance or suppress molecular fluctuations. Based on such relationship, we also develop an effective method to extract the topological information of a gene regulatory network from single-cell gene expression data. The theory is demonstrated by numerical simulations and, more importantly, validated quantitatively by single-cell data analysis of a synthetic gene circuit integrated in human kidney cells.
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