1
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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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2
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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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3
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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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4
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Wang Y, He S. Inference on autoregulation in gene expression with variance-to-mean ratio. J Math Biol 2023; 86:87. [PMID: 37131095 PMCID: PMC10154285 DOI: 10.1007/s00285-023-01924-6] [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: 12/16/2022] [Revised: 04/14/2023] [Accepted: 04/18/2023] [Indexed: 05/04/2023]
Abstract
Some genes can promote or repress their own expressions, which is called autoregulation. Although gene regulation is a central topic in biology, autoregulation is much less studied. In general, it is extremely difficult to determine the existence of autoregulation with direct biochemical approaches. Nevertheless, some papers have observed that certain types of autoregulations are linked to noise levels in gene expression. We generalize these results by two propositions on discrete-state continuous-time Markov chains. These two propositions form a simple but robust method to infer the existence of autoregulation from gene expression data. This method only needs to compare the mean and variance of the gene expression level. Compared to other methods for inferring autoregulation, our method only requires non-interventional one-time data, and does not need to estimate parameters. Besides, our method has few restrictions on the model. We apply this method to four groups of experimental data and find some genes that might have autoregulation. Some inferred autoregulations have been verified by experiments or other theoretical works.
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Affiliation(s)
- Yue Wang
- Department of Computational Medicine, University of California, Los Angeles, CA, 90095, USA.
- Institut des Hautes Études Scientifiques (IHÉS), Bures-sur-Yvette, 91440, Essonne, France.
| | - Siqi He
- Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY, 11794, USA
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5
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Fralix B, Holmes M, Löpker A. A Markovian arrival stream approach to stochastic gene expression in cells. J Math Biol 2023; 86:79. [PMID: 37086292 DOI: 10.1007/s00285-023-01913-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2022] [Revised: 12/22/2022] [Accepted: 03/31/2023] [Indexed: 04/23/2023]
Abstract
We analyse a generalisation of the stochastic gene expression model studied recently in Fromion et al. (SIAM J Appl Math 73:195-211, 2013) and Robert (Probab Surv 16:277-332, 2019) that keeps track of the production of both mRNA and protein molecules, using techniques from the theory of point processes, as well as ideas from the theory of matrix-analytic methods. Here, both the activity of a gene and the creation of mRNA are modelled with an arbitrary Markovian Arrival Process governed by finitely many phases, and each mRNA molecule during its lifetime gives rise to protein molecules in accordance with a Poisson process. This modification is important, as Markovian Arrival Processes can be used to approximate many types of point processes on the nonnegative real line, meaning this framework allows us to further relax our assumptions on the overall process of transcription.
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Affiliation(s)
- Brian Fralix
- School of Mathematical and Statistical Sciences, Clemson University, Clemson, USA.
| | - Mark Holmes
- School of Mathematics and Statistics, The University of Melbourne, Melbourne, Australia
| | - Andreas Löpker
- Department of Computer Science and Mathematics, HTW Dresden, University of Applied Sciences, Dresden, Germany
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6
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mSphere of Influence: Learning Biology from the Radio. mSphere 2023; 8:e0002623. [PMID: 36728432 PMCID: PMC9942574 DOI: 10.1128/msphere.00026-23] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/03/2023] Open
Abstract
Daniel Schultz works at the intersection of microbiology and biological physics. In this mSphere of Influence article, he reflects on how two papers concerning the radio, "Can a biologist fix a radio? Or, what I learned while studying apoptosis" by Yuri Lazebnik and "The evolved radio and its implications for modeling the evolution of novel sensors" by Jon Bird and Paul Layzell, offered him complementary perspectives on how to bridge the gap between engineering and biology in the study of cellular circuits.
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7
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Koraichi MB, Touzel MP, Mazzolini A, Mora T, Walczak AM. NoisET: Noise Learning and Expansion Detection of T-Cell Receptors. J Phys Chem A 2022; 126:7407-7414. [PMID: 36178325 DOI: 10.1021/acs.jpca.2c05002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
High-throughput sequencing of T- and B-cell receptors makes it possible to track immune repertoires across time, in different tissues, in acute and chronic diseases and in healthy individuals. However, quantitative comparison between repertoires is confounded by variability in the read count of each receptor clonotype due to sampling, library preparation, and expression noise. We review methods for accounting for both biological and experimental noise and present an easy-to-use python package NoisET that implements and generalizes a previously developed Bayesian method. It can be used to learn experimental noise models for repertoire sequencing from replicates, and to detect responding clones following a stimulus. We test the package on different repertoire sequencing technologies and data sets. We review how such approaches have been used to identify responding clonotypes in vaccination and disease data. Availability: NoisET is freely available to use with source code at github.com/statbiophys/NoisET.
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Affiliation(s)
- Meriem Bensouda Koraichi
- Laboratoire de physique de l' École normale supérieure, CNRS, PSL University, Sorbonne Université, and Université de Paris, Paris75005, France
| | | | - Andrea Mazzolini
- Laboratoire de physique de l' École normale supérieure, CNRS, PSL University, Sorbonne Université, and Université de Paris, Paris75005, France
| | - Thierry Mora
- Laboratoire de physique de l' École normale supérieure, CNRS, PSL University, Sorbonne Université, and Université de Paris, Paris75005, France
| | - Aleksandra M Walczak
- Laboratoire de physique de l' École normale supérieure, CNRS, PSL University, Sorbonne Université, and Université de Paris, Paris75005, France
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8
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How does an organism extract relevant information from transcription factor concentrations? Biochem Soc Trans 2022; 50:1365-1376. [DOI: 10.1042/bst20220333] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/10/2022] [Revised: 08/16/2022] [Accepted: 08/17/2022] [Indexed: 11/17/2022]
Abstract
How does an organism regulate its genes? The involved regulation typically occurs in terms of a signal processing chain: an externally applied stimulus or a maternally supplied transcription factor leads to the expression of some downstream genes, which, in turn, are transcription factors for further genes. Especially during development, these transcription factors are frequently expressed in amounts where noise is still important; yet, the signals that they provide must not be lost in the noise. Thus, the organism needs to extract exactly relevant information in the signal. New experimental approaches involving single-molecule measurements at high temporal precision as well as increased precision in manipulations directly on the genome are allowing us to tackle this question anew. These new experimental advances mean that also from the theoretical side, theoretical advances should be possible. In this review, I will describe, specifically on the example of fly embryo gene regulation, how theoretical approaches, especially from inference and information theory, can help in understanding gene regulation. To do so, I will first review some more traditional theoretical models for gene regulation, followed by a brief discussion of information-theoretical approaches and when they can be applied. I will then introduce early fly development as an exemplary system where such information-theoretical approaches have traditionally been applied and can be applied; I will specifically focus on how one such method, namely the information bottleneck approach, has recently been used to infer structural features of enhancer architecture.
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9
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Chu X, Wang J. Dynamics and Pathways of Chromosome Structural Organizations during Cell Transdifferentiation. JACS AU 2022; 2:116-127. [PMID: 35098228 PMCID: PMC8791059 DOI: 10.1021/jacsau.1c00416] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/21/2021] [Indexed: 06/14/2023]
Abstract
Direct conversion of one differentiated cell type into another is defined as cell transdifferentiation. In avoidance of forming pluripotency, cell transdifferentiation can reduce the potential risk of tumorigenicity, thus offering significant advantages over cell reprogramming in clinical applications. Until now, the mechanism of cell transdifferentiation is still largely unknown. It has been well recognized that cell transdifferentiation is determined by the underlying gene expression regulation, which relies on the accurate adaptation of the chromosome structure. To dissect the transdifferentiation at the molecular level, we develop a nonequilibrium landscape-switching model to investigate the chromosome structural dynamics during the state transitions between the human fibroblast and neuron cells. We uncover the high irreversibility of the transdifferentiation at the local chromosome structural ranges, where the topologically associating domains form. In contrast, the pathways in the two opposite directions of the transdifferentiation projected onto the chromosome compartment profiles are highly overlapped, indicating that the reversibility vanishes at the long-range chromosome structures. By calculating the contact strengths in the chromosome at the states along the paths, we observe strengthening contacts in compartment A concomitant with weakening contacts in compartment B at the early stages of the transdifferentiation. This further leads to adapting contacts toward the ones at the embryonic stem cell. In light of the intimate structure-function relationship at the chromosomal level, we suggest an increase of "stemness" during the transdifferentiation. In addition, we find that the neuron progenitor cell (NPC), a cell developmental state, is located on the transdifferentiation pathways projected onto the long-range chromosome contacts. The findings are consistent with the previous single-cell RNA sequencing experiment, where the NPC-like cell states were observed during the direct conversion of the fibroblast to neuron cells. Thus, we offer a promising microscopic and physical approach to study the cell transdifferentiation mechanism from the chromosome structural perspective.
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Affiliation(s)
- Xiakun Chu
- Department
of Chemistry, State University of New York
at Stony Brook, Stony
Brook, New York 11794, United States
| | - Jin Wang
- Department
of Chemistry, State University of New York
at Stony Brook, Stony
Brook, New York 11794, United States
- Department
of Physics and Astronomy, State University
of New York at Stony Brook, Stony
Brook, New York 11794, United States
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10
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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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11
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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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12
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Bhattacharyya B, Wang J, Sasai M. Stochastic epigenetic dynamics of gene switching. Phys Rev E 2021; 102:042408. [PMID: 33212709 DOI: 10.1103/physreve.102.042408] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2020] [Accepted: 09/25/2020] [Indexed: 01/01/2023]
Abstract
Epigenetic modifications of histones crucially affect eukaryotic gene activity, while the epigenetic histone state is largely determined by the binding of specific factors such as the transcription factors (TFs) to DNA. Here, the way in which the TFs and the histone state are dynamically correlated is not obvious when the TF synthesis is regulated by the histone state. This type of feedback regulatory relation is ubiquitous in gene networks to determine cell fate in differentiation and other cell transformations. To gain insights into such dynamical feedback regulations, we theoretically analyze a model of epigenetic gene switching by extending the Doi-Peliti operator formalism of reaction kinetics to the problem of coupled molecular processes. Spin-1 and spin-1/2 coherent-state representations are introduced to describe stochastic reactions of histones and binding or unbinding of TFs in a unified way, which provides a concise view of the effects of timescale difference among these molecular processes; even in the case that binding or unbinding of TFs to or from DNA is adiabatically fast, the slow nonadiabatic histone dynamics gives rise to a distinct circular flow of the probability flux around basins in the landscape of the gene state distribution, which leads to hysteresis in gene switching. In contrast to the general belief that the change in the amount of TF precedes the histone state change, flux drives histones to be modified prior to the change in the amount of TF in self-regulating circuits. Flux-landscape analyses shed light on the nonlinear nonadiabatic mechanism of epigenetic cell fate decision making.
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Affiliation(s)
| | - Jin Wang
- Department of Chemistry, Physics and Applied Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794, USA
| | - Masaki Sasai
- Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
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13
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Waizmann T, Bortolussi L, Vandin A, Tribastone M. Improved estimations of stochastic chemical kinetics by finite-state expansion. Proc Math Phys Eng Sci 2021. [DOI: 10.1098/rspa.2020.0964] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite-state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations and multi-stability.
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Affiliation(s)
| | - Luca Bortolussi
- Department of Mathematics and Geosciences, University of Trieste, Trieste 34127, Italy
| | - Andrea Vandin
- Sant’Anna School of Advanced Studies, Pisa 56127, Italy
- Department of Applied Mathematics and Computer Science, DTU Technical University of Denmark, Kgs. Lyngby 2800, Denmark
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14
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Cappelletti D, Pal Majumder A, Wiuf C. The dynamics of stochastic mono-molecular reaction systems in stochastic environments. Stoch Process Their Appl 2021. [DOI: 10.1016/j.spa.2021.03.010] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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15
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Analytic solutions for stochastic hybrid models of gene regulatory networks. J Math Biol 2021; 82:9. [PMID: 33496854 DOI: 10.1007/s00285-021-01549-7] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2019] [Revised: 09/16/2020] [Accepted: 10/16/2020] [Indexed: 10/22/2022]
Abstract
Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is hindered by the fact that the underlying discrete state space is extremely large. Therefore hybrid models, in which protein counts are replaced by average protein concentrations, have become a popular alternative. The evolution of the corresponding probability density functions is given by a coupled system of hyperbolic PDEs. This system has Markovian nature but its hyperbolic structure makes it difficult to apply standard functional analytical methods. We are able to prove convergence towards the stationary solution and determine such equilibrium explicitly by combining abstract methods from the theory of positive operators and elementary ideas from potential analysis.
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16
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Sood A, Zhang B. Quantifying the Stability of Coupled Genetic and Epigenetic Switches With Variational Methods. Front Genet 2021; 11:636724. [PMID: 33552146 PMCID: PMC7862759 DOI: 10.3389/fgene.2020.636724] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2020] [Accepted: 12/29/2020] [Indexed: 01/23/2023] Open
Abstract
The Waddington landscape provides an intuitive metaphor to view development as a ball rolling down the hill, with distinct phenotypes as basins and differentiation pathways as valleys. Since, at a molecular level, cell differentiation arises from interactions among the genes, a mathematical definition for the Waddington landscape can, in principle, be obtained by studying the gene regulatory networks. For eukaryotes, gene regulation is inextricably and intimately linked to histone modifications. However, the impact of such modifications on both landscape topography and stability of attractor states is not fully understood. In this work, we introduced a minimal kinetic model for gene regulation that combines the impact of both histone modifications and transcription factors. We further developed an approximation scheme based on variational principles to solve the corresponding master equation in a second quantized framework. By analyzing the steady-state solutions at various parameter regimes, we found that histone modification kinetics can significantly alter the behavior of a genetic network, resulting in qualitative changes in gene expression profiles. The emerging epigenetic landscape captures the delicate interplay between transcription factors and histone modifications in driving cell-fate decisions.
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Affiliation(s)
- Amogh Sood
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA, United States
| | - Bin Zhang
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA, United States
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17
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Chu X, Wang J. Microscopic Chromosomal Structural and Dynamical Origin of Cell Differentiation and Reprogramming. ADVANCED SCIENCE (WEINHEIM, BADEN-WURTTEMBERG, GERMANY) 2020; 7:2001572. [PMID: 33101859 PMCID: PMC7578896 DOI: 10.1002/advs.202001572] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/28/2020] [Revised: 07/12/2020] [Accepted: 07/23/2020] [Indexed: 06/11/2023]
Abstract
As an essential and fundamental process of life, cell development involves large-scale reorganization of the 3D genome architecture, which forms the basis of gene regulation. Here, a landscape-switching model is developed to explore the microscopic chromosomal structural origin of embryonic stem cell (ESC) differentiation and somatic cell reprogramming. It is shown that chromosome structure exhibits significant compartment-switching in the unit of topologically associating domain. It is found that the chromosome during differentiation undergoes monotonic compaction with spatial repositioning of active and inactive chromosomal loci toward the chromosome surface and interior, respectively. In contrast, an overexpanded chromosome, which exhibits universal localization of loci at the chromosomal surface with erasing the structural characteristics formed in the somatic cells, is observed during reprogramming. An early distinct differentiation pathway from the ESC to the terminally differentiated cell, giving rise to early bifurcation on the Waddington landscape for the ESC differentiation is suggested. The theoretical model herein including the non-equilibrium effects, draws a picture of the highly irreversible cell differentiation and reprogramming processes, in line with the experiments. The predictions provide a physical understanding of cell differentiation and reprogramming from the chromosomal structural and dynamical perspective and can be tested by future experiments.
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Affiliation(s)
- Xiakun Chu
- Department of ChemistryState University of New York at Stony BrookStony BrookNY11794USA
| | - Jin Wang
- Department of ChemistryState University of New York at Stony BrookStony BrookNY11794USA
- Department of Physics and AstronomyState University of New York at Stony BrookStony BrookNY11794USA
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18
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Choudhary K, Narang A. Urn models for stochastic gene expression yield intuitive insights into the probability distributions of single-cell mRNA and protein counts. Phys Biol 2020; 17:066001. [PMID: 32650327 DOI: 10.1088/1478-3975/aba50f] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023]
Abstract
Fitting the probability mass functions from analytical solutions of stochastic models of gene expression to the single-cell count distributions of mRNA and protein molecules can yield valuable insights into mechanisms underlying gene expression. Solutions of chemical master equations are available for various kinetic schemes but, even for the basic ON-OFF genetic switch, they take complex forms with generating functions given as hypergeometric functions. Interpretation of gene expression dynamics in terms of bursts is not consistent with the complete range of parameters for these functions. Physical insights into the probability mass functions are essential to ensure proper interpretations but are lacking for models considering genetic switches. To fill this gap, we develop urn models for stochastic gene expression. We sample RNA polymerases or ribosomes from a master urn, which represents the cytosol, and assign them to recipient urns of two or more colors, which represent time intervals in which no switching occurs. Colors of the recipient urns represent sub-systems of the promoter states, and the assignments to urns of a specific color represent gene expression. We use elementary principles of discrete probability theory to solve a range of kinetic models without feedback, including the Peccoud-Ycart model, the Shahrezaei-Swain model, and models with an arbitrary number of promoter states. In the last case, we obtain a novel result for the protein distribution. For activated genes, we show that transcriptional lapses, which are events of gene inactivation for short time intervals separated by long active intervals, quantify the transcriptional dynamics better than bursts. We show that the intuition gained from our urn models may also be useful in understanding existing solutions for models with feedback. We contrast our models with urn models for related distributions, discuss a generalization of the Delaporte distribution for single-cell data analysis, and highlight the limitations of our models.
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Affiliation(s)
- Krishna Choudhary
- Gladstone Institute of Data Science and Biotechnology, Gladstone Institutes, San Francisco, CA, United States of America
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19
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Ali MZ, Parisutham V, Choubey S, Brewster RC. Inherent regulatory asymmetry emanating from network architecture in a prevalent autoregulatory motif. eLife 2020; 9:56517. [PMID: 32808926 PMCID: PMC7505660 DOI: 10.7554/elife.56517] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2020] [Accepted: 08/07/2020] [Indexed: 12/20/2022] Open
Abstract
Predicting gene expression from DNA sequence remains a major goal in the field of gene regulation. A challenge to this goal is the connectivity of the network, whose role in altering gene expression remains unclear. Here, we study a common autoregulatory network motif, the negative single-input module, to explore the regulatory properties inherited from the motif. Using stochastic simulations and a synthetic biology approach in E. coli, we find that the TF gene and its target genes have inherent asymmetry in regulation, even when their promoters are identical; the TF gene being more repressed than its targets. The magnitude of asymmetry depends on network features such as network size and TF-binding affinities. Intriguingly, asymmetry disappears when the growth rate is too fast or too slow and is most significant for typical growth conditions. These results highlight the importance of accounting for network architecture in quantitative models of gene expression.
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Affiliation(s)
- Md Zulfikar Ali
- Program in Systems Biology, University of Massachusetts Medical School, Worcester, United States.,Department of Microbiology and Physiological Systems, University of Massachusetts Medical School, Worcester, United States
| | - Vinuselvi Parisutham
- Program in Systems Biology, University of Massachusetts Medical School, Worcester, United States.,Department of Microbiology and Physiological Systems, University of Massachusetts Medical School, Worcester, United States
| | - Sandeep Choubey
- Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
| | - Robert C Brewster
- Program in Systems Biology, University of Massachusetts Medical School, Worcester, United States.,Department of Microbiology and Physiological Systems, University of Massachusetts Medical School, Worcester, United States
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20
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Wang Z, Zhang Z, Zhou T. Analytical results for non-Markovian models of bursty gene expression. Phys Rev E 2020; 101:052406. [PMID: 32575237 DOI: 10.1103/physreve.101.052406] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/11/2019] [Accepted: 03/24/2020] [Indexed: 11/07/2022]
Abstract
Modeling stochastic gene expression has long relied on Markovian hypothesis. In recent years, however, this hypothesis is challenged by the increasing availability of time-resolved data. Correspondingly, there is considerable interest in understanding how non-Markovian reaction kinetics of gene expression impact protein variations across a population of genetically identical cells. Here, we analyze a stochastic model of gene expression with arbitrary waiting-time distributions, which includes existing gene models as its special cases. We find that stationary probabilistic behavior of this non-Markovian system is exactly the same as that of an equivalent Markovian system with the same substrates. Based on this fact, we derive analytical results, which provide insight into the roles of feedback regulation and molecular memory in controlling the protein noise and properties of the steady states, which are inaccessible via existing methodology. Our results also provide quantitative insight into diverse cellular processes involving stochastic sources of gene expression and molecular memory.
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Affiliation(s)
- Zihao Wang
- Guangdong Province Key Laboratory of Computational Science School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
| | - Zhenquan Zhang
- Guangdong Province Key Laboratory of Computational Science School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
| | - Tianshou Zhou
- Guangdong Province Key Laboratory of Computational Science School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
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21
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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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22
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Fang X, Wang J. Nonequilibrium Thermodynamics in Cell Biology: Extending Equilibrium Formalism to Cover Living Systems. Annu Rev Biophys 2020; 49:227-246. [DOI: 10.1146/annurev-biophys-121219-081656] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
We discuss new developments in the nonequilibrium dynamics and thermodynamics of living systems, giving a few examples to demonstrate the importance of nonequilibrium thermodynamics for understanding biological dynamics and functions. We study single-molecule enzyme dynamics, in which the nonequilibrium thermodynamic and dynamic driving forces of chemical potential and flux are crucial for the emergence of non-Michaelis-Menten kinetics. We explore single-gene expression dynamics, in which nonequilibrium dissipation can suppress fluctuations. We investigate the cell cycle and identify the nutrition supply as the energy input that sustains the stability, speed, and coherence of cell cycle oscillation, from which the different vital phases of the cell cycle emerge. We examine neural decision-making processes and find the trade-offs among speed, accuracy, and thermodynamic costs that are important for neural function. Lastly, we consider the thermodynamic cost for specificity in cellular signaling and adaptation.
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Affiliation(s)
- Xiaona Fang
- Department of Chemistry, Stony Brook University, Stony Brook, New York 11794, USA
| | - Jin Wang
- Department of Chemistry, Stony Brook University, Stony Brook, New York 11794, USA
- Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA
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23
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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: 21] [Impact Index Per Article: 5.3] [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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24
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Hasan ABMSU, Kurata H, Pechmann S. Improvement of the memory function of a mutual repression network in a stochastic environment by negative autoregulation. BMC Bioinformatics 2019; 20:734. [PMID: 31881978 PMCID: PMC6935196 DOI: 10.1186/s12859-019-3315-2] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2019] [Accepted: 12/12/2019] [Indexed: 11/10/2022] Open
Abstract
BACKGROUND Cellular memory is a ubiquitous function of biological systems. By generating a sustained response to a transient inductive stimulus, often due to bistability, memory is central to the robust control of many important biological processes. However, our understanding of the origins of cellular memory remains incomplete. Stochastic fluctuations that are inherent to most biological systems have been shown to hamper memory function. Yet, how stochasticity changes the behavior of genetic circuits is generally not clear from a deterministic analysis of the network alone. Here, we apply deterministic rate equations, stochastic simulations, and theoretical analyses of Fokker-Planck equations to investigate how intrinsic noise affects the memory function in a mutual repression network. RESULTS We find that the addition of negative autoregulation improves the persistence of memory in a small gene regulatory network by reducing stochastic fluctuations. Our theoretical analyses reveal that this improved memory function stems from an increased stability of the steady states of the system. Moreover, we show how the tuning of critical network parameters can further enhance memory. CONCLUSIONS Our work illuminates the power of stochastic and theoretical approaches to understanding biological circuits, and the importance of considering stochasticity when designing synthetic circuits with memory function.
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Affiliation(s)
- A B M Shamim Ul Hasan
- Department of Biochemistry, Université de Montréal, 2900 Edouard Montpetit Blvd, Montreal, QC, H3T 1J4, Canada.,The Biomedical Informatics R&D Center, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka, 820-8502, Japan
| | - Hiroyuki Kurata
- The Biomedical Informatics R&D Center, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka, 820-8502, Japan.
| | - Sebastian Pechmann
- Department of Biochemistry, Université de Montréal, 2900 Edouard Montpetit Blvd, Montreal, QC, H3T 1J4, Canada.
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25
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Abstract
In order to respond to environmental signals, cells often use small molecular circuits to transmit information about their surroundings. Recently, motivated by specific examples in signaling and gene regulation, a body of work has focused on the properties of circuits that function out of equilibrium and dissipate energy. We briefly review the probabilistic measures of information and dissipation and use simple models to discuss and illustrate trade-offs between information and dissipation in biological circuits. We find that circuits with non-steady state initial conditions can transmit more information at small readout delays than steady state circuits. The dissipative cost of this additional information proves marginal compared to the steady state dissipation. Feedback does not significantly increase the transmitted information for out of steady state circuits but does decrease dissipative costs. Lastly, we discuss the case of bursty gene regulatory circuits that, even in the fast switching limit, function out of equilibrium.
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26
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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: 17] [Impact Index Per Article: 3.4] [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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27
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Ramos AF, Reinitz J. Physical implications of so(2, 1) symmetry in exact solutions for a self-repressing gene. J Chem Phys 2019; 151:041101. [PMID: 31370538 DOI: 10.1063/1.5105361] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We chemically characterize the symmetries underlying the exact solutions of a stochastic negatively self-regulating gene. The breaking of symmetry at a low molecular number causes three effects. Two branches of the solution exist, having high and low switching rates, such that the low switching rate branch approaches deterministic behavior and the high switching rate branch exhibits sub-Fano behavior. The average protein number differs from the deterministically expected value. Bimodal probability distributions appear as the protein number becomes a readout of the ON/OFF state of the gene.
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Affiliation(s)
- Alexandre F Ramos
- Escola de Artes, Ciências e Humanidades, Núcleo de Estudos Interdisciplinares em Sistemas Complexos, Departamento de Radiologia - Faculdade de Medicina, Universidade de São Paulo - Instituto do Câncer do Estado de São Paulo - Av. Arlindo Béttio, 1000 CEP 03828-000, São Paulo, SP, Brazil
| | - John Reinitz
- Departments of Statistics, Ecology and Evolution, Molecular Genetics and Cell Biology, Jones Laboratory, University of Chicago, 5747 South Ellis Ave., Chicago, Illinois 60637, USA
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28
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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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29
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Jiang Z, Tian L, Fang X, Zhang K, Liu Q, Dong Q, Wang E, Wang J. The emergence of the two cell fates and their associated switching for a negative auto-regulating gene. BMC Biol 2019; 17:49. [PMID: 31202264 PMCID: PMC6570905 DOI: 10.1186/s12915-019-0666-0] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/28/2018] [Accepted: 05/20/2019] [Indexed: 01/24/2023] Open
Abstract
Background Decisions in the cell that lead to its ultimate fate are important for fundamental cellular functions such as proliferation, growth, differentiation, development, and death. These cell fate decisions can be influenced by both the gene regulatory network and also environmental factors and can be modeled using simple gene feedback circuits. Negative auto-regulation is a common feedback motif in the gene circuits. It can act to reduce gene expression noise or induce oscillatory expression and is thought to lead to only one cell fate. Here, we present experimental and modeling data to suggest that a self-repressor circuit can lead to two cell fates under specific conditions. Results We show that the introduction of inducers capable of binding and unbinding to a self-repressing gene product (protein), thus regulating the associated gene, can lead to the emergence of two cell states. We suggest that the inducers can alter the effective regulatory binding and unbinding speed of the self-repressor regulatory protein to its destination DNA without changing the gene itself. The corresponding simulation results are consistent with the experimental findings. We propose physical and quantitative explanations for the origin of the two phenotypic cell fates. Conclusions Our results suggest a mechanism for the emergence of multiple cell fates. This may explain the heterogeneity often observed among cell states, while illustrating that altering gene regulation strength can influence cell fates and their decision-making processes without genetic changes. Electronic supplementary material The online version of this article (10.1186/s12915-019-0666-0) contains supplementary material, which is available to authorized users.
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Affiliation(s)
- Zhenlong Jiang
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin, 130022, China.,College of Physics, Jilin University, Changchun, Jilin, 130012, China
| | - Li Tian
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin, 130022, China
| | - Xiaona Fang
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin, 130022, China
| | - Kun Zhang
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin, 130022, China
| | - Qiong Liu
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin, 130022, China
| | - Qingzhe Dong
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin, 130022, China
| | - Erkang Wang
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin, 130022, China
| | - Jin Wang
- Department of Chemistry, Physics and Applied Mathematics, State University of New York at Stony Brook, Stony Brook, New York, 11794-3400, USA.
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30
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Choubey S, Kondev J, Sanchez A. Distribution of Initiation Times Reveals Mechanisms of Transcriptional Regulation in Single Cells. Biophys J 2019; 114:2072-2082. [PMID: 29742401 DOI: 10.1016/j.bpj.2018.03.031] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2017] [Revised: 03/18/2018] [Accepted: 03/29/2018] [Indexed: 11/25/2022] Open
Abstract
Transcription is the dominant point of control of gene expression. Biochemical studies have revealed key molecular components of transcription and their interactions, but the dynamics of transcription initiation in cells is still poorly understood. This state of affairs is being remedied with experiments that observe transcriptional dynamics in single cells using fluorescent reporters. Quantitative information about transcription initiation dynamics can also be extracted from experiments that use electron micrographs of RNA polymerases caught in the act of transcribing a gene (Miller spreads). Inspired by these data, we analyze a general stochastic model of transcription initiation and elongation and compute the distribution of transcription initiation times. We show that different mechanisms of initiation leave distinct signatures in the distribution of initiation times that can be compared to experiments. We analyze published data from micrographs of RNA polymerases transcribing ribosomal RNA genes in Escherichia coli and compare the observed distributions of interpolymerase distances with the predictions from previously hypothesized mechanisms for the regulation of these genes. Our analysis demonstrates the potential of measuring the distribution of time intervals between initiation events as a probe for dissecting mechanisms of transcription initiation in live cells.
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Affiliation(s)
- Sandeep Choubey
- Department of Physics, Brandeis University, Waltham, Massachusetts
| | - Jane Kondev
- Department of Physics, Brandeis University, Waltham, Massachusetts
| | - Alvaro Sanchez
- Rowland Institute at Harvard, Harvard University, Cambridge, Massachusetts; Department of Ecology and Evolutionary Biology, Microbial Sciences Institute, Yale University, New Haven, Connecticut.
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31
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Iida K, Obata N, Kimura Y. Quantifying heterogeneity of stochastic gene expression. J Theor Biol 2019; 465:56-62. [PMID: 30611711 DOI: 10.1016/j.jtbi.2019.01.003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2018] [Revised: 12/27/2018] [Accepted: 01/03/2019] [Indexed: 11/20/2022]
Abstract
The heterogeneity of stochastic gene expression, which refers to the temporal fluctuation in a gene product and its cell-to-cell variation, has attracted considerable interest from biologists, physicists, and mathematicians. The dynamics of protein production and degradation have been modeled as random processes with transition probabilities. However, there is a gap between theory and phenomena, particularly in terms of analytical formulation and parameter estimation. In this study, we propose a theoretical framework in which we present a basic model of a gene regulatory system, derive a steady-state solution, and provide a Bayesian approach for estimating the model parameters from single-cell experimental data. The proposed framework is demonstrated to be applicable for various scales of single-cell experiments at both the mRNA and protein levels and is useful for comparing kinetic parameters across species, genomes, and cell strains.
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Affiliation(s)
- Keita Iida
- Graduate School of Medicine, Tohoku University, Sendai 980-8575, Japan.
| | - Nobuaki Obata
- Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan.
| | - Yoshitaka Kimura
- Graduate School of Medicine, Tohoku University, Sendai 980-8575, Japan.
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32
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Ali Al-Radhawi M, Del Vecchio D, Sontag ED. Multi-modality in gene regulatory networks with slow promoter kinetics. PLoS Comput Biol 2019; 15:e1006784. [PMID: 30779734 PMCID: PMC6396950 DOI: 10.1371/journal.pcbi.1006784] [Citation(s) in RCA: 20] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2018] [Revised: 03/01/2019] [Accepted: 01/14/2019] [Indexed: 12/27/2022] Open
Abstract
Phenotypical variability in the absence of genetic variation often reflects complex energetic landscapes associated with underlying gene regulatory networks (GRNs). In this view, different phenotypes are associated with alternative states of complex nonlinear systems: stable attractors in deterministic models or modes of stationary distributions in stochastic descriptions. We provide theoretical and practical characterizations of these landscapes, specifically focusing on stochastic Slow Promoter Kinetics (SPK), a time scale relevant when transcription factor binding and unbinding are affected by epigenetic processes like DNA methylation and chromatin remodeling. In this case, largely unexplored except for numerical simulations, adiabatic approximations of promoter kinetics are not appropriate. In contrast to the existing literature, we provide rigorous analytic characterizations of multiple modes. A general formal approach gives insight into the influence of parameters and the prediction of how changes in GRN wiring, for example through mutations or artificial interventions, impact the possible number, location, and likelihood of alternative states. We adapt tools from the mathematical field of singular perturbation theory to represent stationary distributions of Chemical Master Equations for GRNs as mixtures of Poisson distributions and obtain explicit formulas for the locations and probabilities of metastable states as a function of the parameters describing the system. As illustrations, the theory is used to tease out the role of cooperative binding in stochastic models in comparison to deterministic models, and applications are given to various model systems, such as toggle switches in isolation or in communicating populations, a synthetic oscillator, and a trans-differentiation network.
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Affiliation(s)
- M. Ali Al-Radhawi
- Departments of Electrical and Computer Engineering and of Bioengineering, Northeastern University, Boston, Massachusetts, United States of America
| | - Domitilla Del Vecchio
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, United States of America
| | - Eduardo D. Sontag
- Departments of Electrical and Computer Engineering and of Bioengineering, Northeastern University, Boston, Massachusetts, United States of America
- Laboratory of Systems Pharmacology, Harvard Medical School, Boston, Massachusetts, United States of America
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33
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Probabilistic control of HIV latency and transactivation by the Tat gene circuit. Proc Natl Acad Sci U S A 2018; 115:12453-12458. [PMID: 30455316 DOI: 10.1073/pnas.1811195115] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022] Open
Abstract
The reservoir of HIV latently infected cells is the major obstacle for eradication of HIV infection. The "shock-and-kill" strategy proposed earlier aims to reduce the reservoir by activating cells out of latency. While the intracellular HIV Tat gene circuit is known to play important roles in controlling latency and its transactivation in HIV-infected cells, the detailed control mechanisms are not well understood. Here we study the mechanism of probabilistic control of the latent and the transactivated cell phenotypes of HIV-infected cells. We reconstructed the probability landscape, which is the probability distribution of the Tat gene circuit states, by directly computing the exact solution of the underlying chemical master equation. Results show that the Tat circuit exhibits a clear bimodal probability landscape (i.e., there are two distinct probability peaks, one associated with the latent cell phenotype and the other with the transactivated cell phenotype). We explore potential modifications to reactions in the Tat gene circuit for more effective transactivation of latent cells (i.e., the shock-and-kill strategy). Our results suggest that enhancing Tat acetylation can dramatically increase Tat and viral production, while increasing the Tat-transactivation response binding affinity can transactivate latent cells more rapidly than other manipulations. Our results further explored the "block and lock" strategy toward a functional cure for HIV. Overall, our study demonstrates a general approach toward discovery of effective therapeutic strategies and druggable targets by examining control mechanisms of cell phenotype switching via exactly computed probability landscapes of reaction networks.
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34
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Stochastic hybrid models of gene regulatory networks – A PDE approach. Math Biosci 2018; 305:170-177. [DOI: 10.1016/j.mbs.2018.09.009] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2018] [Revised: 09/14/2018] [Accepted: 09/19/2018] [Indexed: 12/14/2022]
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35
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Huang GR, Saakian DB, Hu CK. Accurate analytic solution of chemical master equations for gene regulation networks in a single cell. Phys Rev E 2018; 97:012412. [PMID: 29448337 DOI: 10.1103/physreve.97.012412] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/30/2017] [Indexed: 12/21/2022]
Abstract
Studying gene regulation networks in a single cell is an important, interesting, and hot research topic of molecular biology. Such process can be described by chemical master equations (CMEs). We propose a Hamilton-Jacobi equation method with finite-size corrections to solve such CMEs accurately at the intermediate region of switching, where switching rate is comparable to fast protein production rate. We applied this approach to a model of self-regulating proteins [H. Ge et al., Phys. Rev. Lett. 114, 078101 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.078101] and found that as a parameter related to inducer concentration increases the probability of protein production changes from unimodal to bimodal, then to unimodal, consistent with phenotype switching observed in a single cell.
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Affiliation(s)
- Guan-Rong Huang
- Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan
| | - David B Saakian
- Theoretical Physics Research Group, Advanced Institute of Materials Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam.,Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam.,Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan
| | - Chin-Kun Hu
- Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan.,Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan.,Department of Systems Science, University of Shanghai for Science and Technology, Shanghai 200093, China.,Department of Physics, National Dong Hwa University, Hualien 97401, Taiwan
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36
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Sabino AU, Vasconcelos MFS, Sittoni MY, Lautenschlager WW, Queiroga AS, Morais MCC, Ramos AF. Lessons and perspectives for applications of stochastic models in biological and cancer research. Clinics (Sao Paulo) 2018; 73:e536s. [PMID: 30281699 PMCID: PMC6131223 DOI: 10.6061/clinics/2018/e536s] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 04/11/2018] [Accepted: 06/14/2018] [Indexed: 11/18/2022] Open
Abstract
The effects of randomness, an unavoidable feature of intracellular environments, are observed at higher hierarchical levels of living matter organization, such as cells, tissues, and organisms. Additionally, the many compounds interacting as a well-orchestrated network of reactions increase the difficulties of assessing these systems using only experiments. This limitation indicates that elucidation of the dynamics of biological systems is a complex task that will benefit from the establishment of principles to help describe, categorize, and predict the behavior of these systems. The theoretical machinery already available, or ones to be discovered to help solve biological problems, might play an important role in these processes. Here, we demonstrate the application of theoretical tools by discussing some biological problems that we have approached mathematically: fluctuations in gene expression and cell proliferation in the context of loss of contact inhibition. We discuss the methods that have been employed to provide the reader with a biologically motivated phenomenological perspective of the use of theoretical methods. Finally, we end this review with a discussion of new research perspectives motivated by our results.
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Affiliation(s)
- Alan U Sabino
- Escola de Artes Ciências e Humanidades (EACH), Universidade de Sao Paulo, Sao Paulo, SP, BR
| | - Miguel FS Vasconcelos
- Escola de Artes Ciências e Humanidades (EACH), Universidade de Sao Paulo, Sao Paulo, SP, BR
| | - Misaki Yamada Sittoni
- Escola de Artes Ciências e Humanidades (EACH), Universidade de Sao Paulo, Sao Paulo, SP, BR
- Departamento de Radiologia e Oncologia, Instituto do Cancer do Estado de Sao Paulo (ICESP), Faculdade de Medicina FMUSP, Universidade de Sao Paulo, Sao Paulo, SP, BR
| | | | - Alexandre S Queiroga
- Escola de Artes Ciências e Humanidades (EACH), Universidade de Sao Paulo, Sao Paulo, SP, BR
- Departamento de Radiologia e Oncologia, Instituto do Cancer do Estado de Sao Paulo (ICESP), Faculdade de Medicina FMUSP, Universidade de Sao Paulo, Sao Paulo, SP, BR
| | - Mauro CC Morais
- Escola de Artes Ciências e Humanidades (EACH), Universidade de Sao Paulo, Sao Paulo, SP, BR
- Departamento de Radiologia e Oncologia, Instituto do Cancer do Estado de Sao Paulo (ICESP), Faculdade de Medicina FMUSP, Universidade de Sao Paulo, Sao Paulo, SP, BR
| | - Alexandre F Ramos
- Escola de Artes Ciências e Humanidades (EACH), Universidade de Sao Paulo, Sao Paulo, SP, BR
- Departamento de Radiologia e Oncologia, Instituto do Cancer do Estado de Sao Paulo (ICESP), Faculdade de Medicina FMUSP, Universidade de Sao Paulo, Sao Paulo, SP, BR
- *Corresponding author. E-mail:
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Arani BMS, Mahmoudi M, Lahti L, González J, Wit EC. Stability estimation of autoregulated genes under Michaelis-Menten-type kinetics. Phys Rev E 2018; 97:062407. [PMID: 30011543 DOI: 10.1103/physreve.97.062407] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/03/2017] [Indexed: 11/07/2022]
Abstract
Feedback loops are typical motifs appearing in gene regulatory networks. In some well-studied model organisms, including Escherichia coli, autoregulated genes, i.e., genes that activate or repress themselves through their protein products, are the only feedback interactions. For these types of interactions, the Michaelis-Menten (MM) formulation is a suitable and widely used approach, which always leads to stable steady-state solutions representative of homeostatic regulation. However, in many other biological phenomena, such as cell differentiation, cancer progression, and catastrophes in ecosystems, one might expect to observe bistable switchlike dynamics in the case of strong positive autoregulation. To capture this complex behavior we use the generalized family of MM kinetic models. We give a full analysis regarding the stability of autoregulated genes. We show that the autoregulation mechanism has the capability to exhibit diverse cellular dynamics including hysteresis, a typical characteristic of bistable systems, as well as irreversible transitions between bistable states. We also introduce a statistical framework to estimate the kinetics parameters and probability of different stability regimes given observational data. Empirical data for the autoregulated gene SCO3217 in the SOS system in Streptomyces coelicolor are analyzed. The coupling of a statistical framework and the mathematical model can give further insight into understanding the evolutionary mechanisms toward different cell fates in various systems.
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Affiliation(s)
- Babak M S Arani
- Department of Aquatic Ecology and Water Quality Management, Wageningen University, P.O. Box 47, NL-6700 AA Wageningen, The Netherlands
| | - Mahdi Mahmoudi
- Faculty of Mathematics, Statistics and Computer Science, Semnan University, P.O. Box 35195-363, Semnan, Iran
| | - Leo Lahti
- Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland
| | - Javier González
- Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, United Kingdom and Amazon Research Cambridge, Cambridge, United Kingdom
| | - Ernst C Wit
- Institute of Computational Science, USI, Via G. Buffi 13, Lugano 6900, Switzerland
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Fang X, Liu Q, Bohrer C, Hensel Z, Han W, Wang J, Xiao J. Cell fate potentials and switching kinetics uncovered in a classic bistable genetic switch. Nat Commun 2018; 9:2787. [PMID: 30018349 PMCID: PMC6050291 DOI: 10.1038/s41467-018-05071-1] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2018] [Accepted: 04/17/2018] [Indexed: 11/13/2022] Open
Abstract
Bistable switches are common gene regulatory motifs directing two mutually exclusive cell fates. Theoretical studies suggest that bistable switches are sufficient to encode more than two cell fates without rewiring the circuitry due to the non-equilibrium, heterogeneous cellular environment. However, such a scenario has not been experimentally observed. Here by developing a new, dual single-molecule gene-expression reporting system, we find that for the two mutually repressing transcription factors CI and Cro in the classic bistable bacteriophage λ switch, there exist two new production states, in which neither CI nor Cro is produced, or both CI and Cro are produced. We construct the corresponding potential landscape and map the transition kinetics among the four production states. These findings uncover cell fate potentials beyond the classical picture of bistable switches, and open a new window to explore the genetic and environmental origins of the cell fate decision-making process in gene regulatory networks.
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Affiliation(s)
- Xiaona Fang
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Changchun, 130022, China
- Department of Biophysics and Biophysical Chemistry, Johns Hopkins School of Medicine, Baltimore, MD, 21205, USA
- College of Physics, Jilin University, Changchun, 130012, China
- Department of Chemistry and Physics, Stony Brook University, Stony Brook, NY, 11790, USA
| | - Qiong Liu
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Changchun, 130022, China
| | - Christopher Bohrer
- Department of Biophysics and Biophysical Chemistry, Johns Hopkins School of Medicine, Baltimore, MD, 21205, USA
| | - Zach Hensel
- Department of Biophysics and Biophysical Chemistry, Johns Hopkins School of Medicine, Baltimore, MD, 21205, USA
- Instituto de Tecnologia Química e Biológica António Xavier, Universidade Nova de Lisboa, Av. da República, 2780-157, Oeiras, Portugal
| | - Wei Han
- College of Physics, Jilin University, Changchun, 130012, China
| | - Jin Wang
- State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry, Changchun, 130022, China.
- College of Physics, Jilin University, Changchun, 130012, China.
- Department of Chemistry and Physics, Stony Brook University, Stony Brook, NY, 11790, USA.
| | - Jie Xiao
- Department of Biophysics and Biophysical Chemistry, Johns Hopkins School of Medicine, Baltimore, MD, 21205, USA.
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Abstract
Gene expression is influenced by extrinsic noise (involving a fluctuating environment of cellular processes) and intrinsic noise (referring to fluctuations within a cell under constant environment). We study the standard model of gene expression including an (in-)active gene, mRNA and protein. Gene expression is regulated in the sense that the protein feeds back and either represses (negative feedback) or enhances (positive feedback) its production at the stage of transcription. While it is well-known that negative (positive) feedback reduces (increases) intrinsic noise, we give a precise result on the resulting fluctuations in protein numbers. The technique we use is an extension of the Langevin approximation and is an application of a central limit theorem under stochastic averaging for Markov jump processes (Kang et al. in Ann Appl Probab 24:721-759, 2014). We find that (under our scaling and in equilibrium), negative feedback leads to a reduction in the Fano factor of at most 2, while the noise under positive feedback is potentially unbounded. The fit with simulations is very good and improves on known approximations.
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Earnest TM, Cole JA, Luthey-Schulten Z. Simulating biological processes: stochastic physics from whole cells to colonies. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2018; 81:052601. [PMID: 29424367 DOI: 10.1088/1361-6633/aaae2c] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
The last few decades have revealed the living cell to be a crowded spatially heterogeneous space teeming with biomolecules whose concentrations and activities are governed by intrinsically random forces. It is from this randomness, however, that a vast array of precisely timed and intricately coordinated biological functions emerge that give rise to the complex forms and behaviors we see in the biosphere around us. This seemingly paradoxical nature of life has drawn the interest of an increasing number of physicists, and recent years have seen stochastic modeling grow into a major subdiscipline within biological physics. Here we review some of the major advances that have shaped our understanding of stochasticity in biology. We begin with some historical context, outlining a string of important experimental results that motivated the development of stochastic modeling. We then embark upon a fairly rigorous treatment of the simulation methods that are currently available for the treatment of stochastic biological models, with an eye toward comparing and contrasting their realms of applicability, and the care that must be taken when parameterizing them. Following that, we describe how stochasticity impacts several key biological functions, including transcription, translation, ribosome biogenesis, chromosome replication, and metabolism, before considering how the functions may be coupled into a comprehensive model of a 'minimal cell'. Finally, we close with our expectation for the future of the field, focusing on how mesoscopic stochastic methods may be augmented with atomic-scale molecular modeling approaches in order to understand life across a range of length and time scales.
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Affiliation(s)
- Tyler M Earnest
- Department of Chemistry, University of Illinois, Urbana, IL, 61801, United States of America. National Center for Supercomputing Applications, University of Illinois, Urbana, IL, 61801, United States of America
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Ge H, Wu P, Qian H, Xie XS. Relatively slow stochastic gene-state switching in the presence of positive feedback significantly broadens the region of bimodality through stabilizing the uninduced phenotypic state. PLoS Comput Biol 2018. [PMID: 29529037 PMCID: PMC5864076 DOI: 10.1371/journal.pcbi.1006051] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022] Open
Abstract
Within an isogenic population, even in the same extracellular environment, individual cells can exhibit various phenotypic states. The exact role of stochastic gene-state switching regulating the transition among these phenotypic states in a single cell is not fully understood, especially in the presence of positive feedback. Recent high-precision single-cell measurements showed that, at least in bacteria, switching in gene states is slow relative to the typical rates of active transcription and translation. Hence using the lac operon as an archetype, in such a region of operon-state switching, we present a fluctuating-rate model for this classical gene regulation module, incorporating the more realistic operon-state switching mechanism that was recently elucidated. We found that the positive feedback mechanism induces bistability (referred to as deterministic bistability), and that the parameter range for its occurrence is significantly broadened by stochastic operon-state switching. We further show that in the absence of positive feedback, operon-state switching must be extremely slow to trigger bistability by itself. However, in the presence of positive feedback, which stabilizes the induced state, the relatively slow operon-state switching kinetics within the physiological region are sufficient to stabilize the uninduced state, together generating a broadened parameter region of bistability (referred to as stochastic bistability). We illustrate the opposite phenotype-transition rate dependence upon the operon-state switching rates in the two types of bistability, with the aid of a recently proposed rate formula for fluctuating-rate models. The rate formula also predicts a maximal transition rate in the intermediate region of operon-state switching, which is validated by numerical simulations in our model. Overall, our findings suggest a biological function of transcriptional “variations” among genetically identical cells, for the emergence of bistability and transition between phenotypic states. Identifying the mechanism underlying the coexistence of multiple stable phenotypic states has been a challenging scientific problem for more than half a century, and an appropriate mathematical model at the single-cell level is also in high demand. Single-cell measurements conducted in the past ten years have shown that gene-state switching is slow relative to the typical rates of active transcription and translation; hence the recently proposed fluctuating-rate model is a good candidate for describing the single-cell dynamics. We use the classic gene regulation module of the lac operon as an archetype and build a specific fluctuating-rate model based on the recently identified operon-state switching mechanism. This model is analyzed to dissect the interplay between positive feedback and the stochastic switching of gene states in the emergence of bistability/multistablity and the transition between phenotypic states. We show that relatively slow operon-state switching stabilizes the uninduced state and that the positive feedback stabilizes the induced state. Thus, the parameter range for bistability is significantly broadened. In addition, recently proposed landscape theory and rate formula predict opposite phenotype-transition rate dependence on operon-state switching rates for the two types of bistability.
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Affiliation(s)
- Hao Ge
- Biodynamic Optical Imaging Center (BIOPIC), Peking University, Beijing, P.R.China
- Beijing International Center for Mathematical Research (BICMR), Peking University, Beijing, P.R.China
- * E-mail: (HG); (XSX)
| | - Pingping Wu
- School of Mathematical Sciences and Centre for Computational Systems Biology, Fudan University, Shanghai, P.R.China
| | - Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington, United States of America
| | - Xiaoliang Sunney Xie
- Biodynamic Optical Imaging Center (BIOPIC), Peking University, Beijing, P.R.China
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts, United States of America
- * E-mail: (HG); (XSX)
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42
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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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43
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Lin YT, Hufton PG, Lee EJ, Potoyan DA. A stochastic and dynamical view of pluripotency in mouse embryonic stem cells. PLoS Comput Biol 2018; 14:e1006000. [PMID: 29451874 PMCID: PMC5833290 DOI: 10.1371/journal.pcbi.1006000] [Citation(s) in RCA: 32] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2017] [Revised: 03/01/2018] [Accepted: 01/19/2018] [Indexed: 12/26/2022] Open
Abstract
Pluripotent embryonic stem cells are of paramount importance for biomedical sciences because of their innate ability for self-renewal and differentiation into all major cell lines. The fateful decision to exit or remain in the pluripotent state is regulated by complex genetic regulatory networks. The rapid growth of single-cell sequencing data has greatly stimulated applications of statistical and machine learning methods for inferring topologies of pluripotency regulating genetic networks. The inferred network topologies, however, often only encode Boolean information while remaining silent about the roles of dynamics and molecular stochasticity inherent in gene expression. Herein we develop a framework for systematically extending Boolean-level network topologies into higher resolution models of networks which explicitly account for the promoter architectures and gene state switching dynamics. We show the framework to be useful for disentangling the various contributions that gene switching, external signaling, and network topology make to the global heterogeneity and dynamics of transcription factor populations. We find the pluripotent state of the network to be a steady state which is robust to global variations of gene switching rates which we argue are a good proxy for epigenetic states of individual promoters. The temporal dynamics of exiting the pluripotent state, on the other hand, is significantly influenced by the rates of genetic switching which makes cells more responsive to changes in extracellular signals.
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Affiliation(s)
- Yen Ting Lin
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
- School of Physics and Astronomy, The University of Manchester, Manchester, United Kingdom
| | - Peter G. Hufton
- School of Physics and Astronomy, The University of Manchester, Manchester, United Kingdom
| | - Esther J. Lee
- Department of Bioengineering, Rice University, Houston, Texas, United States of America
| | - Davit A. Potoyan
- Department of Chemistry, Iowa State University, Ames, Iowa, United States of America
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44
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Jia C. Simplification of Markov chains with infinite state space and the mathematical theory of random gene expression bursts. Phys Rev E 2018; 96:032402. [PMID: 29346865 DOI: 10.1103/physreve.96.032402] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2017] [Indexed: 11/07/2022]
Abstract
Here we develop an effective approach to simplify two-time-scale Markov chains with infinite state spaces by removal of states with fast leaving rates, which improves the simplification method of finite Markov chains. We introduce the concept of fast transition paths and show that the effective transitions of the reduced chain can be represented as the superposition of the direct transitions and the indirect transitions via all the fast transition paths. Furthermore, we apply our simplification approach to the standard Markov model of single-cell stochastic gene expression and provide a mathematical theory of random gene expression bursts. We give the precise mathematical conditions for the bursting kinetics of both mRNAs and proteins. It turns out that random bursts exactly correspond to the fast transition paths of the Markov model. This helps us gain a better understanding of the physics behind the bursting kinetics as an emergent behavior from the fundamental multiscale biochemical reaction kinetics of stochastic gene expression.
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Affiliation(s)
- Chen Jia
- Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75080, USA
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45
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Time-dependent propagators for stochastic models of gene expression: an analytical method. J Math Biol 2017; 77:261-312. [PMID: 29247320 PMCID: PMC6061071 DOI: 10.1007/s00285-017-1196-4] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2017] [Revised: 11/27/2017] [Indexed: 12/01/2022]
Abstract
The inherent stochasticity of gene expression in the context of regulatory networks profoundly influences the dynamics of the involved species. Mathematically speaking, the propagators which describe the evolution of such networks in time are typically defined as solutions of the corresponding chemical master equation (CME). However, it is not possible in general to obtain exact solutions to the CME in closed form, which is due largely to its high dimensionality. In the present article, we propose an analytical method for the efficient approximation of these propagators. We illustrate our method on the basis of two categories of stochastic models for gene expression that have been discussed in the literature. The requisite procedure consists of three steps: a probability-generating function is introduced which transforms the CME into (a system of) partial differential equations (PDEs); application of the method of characteristics then yields (a system of) ordinary differential equations (ODEs) which can be solved using dynamical systems techniques, giving closed-form expressions for the generating function; finally, propagator probabilities can be reconstructed numerically from these expressions via the Cauchy integral formula. The resulting ‘library’ of propagators lends itself naturally to implementation in a Bayesian parameter inference scheme, and can be generalised systematically to related categories of stochastic models beyond the ones considered here.
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46
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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: 17] [Impact Index Per Article: 2.4] [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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47
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Dattani J, Barahona M. Stochastic models of gene transcription with upstream drives: exact solution and sample path characterization. J R Soc Interface 2017; 14:rsif.2016.0833. [PMID: 28053113 PMCID: PMC5310734 DOI: 10.1098/rsif.2016.0833] [Citation(s) in RCA: 39] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2016] [Accepted: 11/29/2016] [Indexed: 01/14/2023] Open
Abstract
Gene transcription is a highly stochastic and dynamic process. As a result, the mRNA copy number of a given gene is heterogeneous both between cells and across time. We present a framework to model gene transcription in populations of cells with time-varying (stochastic or deterministic) transcription and degradation rates. Such rates can be understood as upstream cellular drives representing the effect of different aspects of the cellular environment. We show that the full solution of the master equation contains two components: a model-specific, upstream effective drive, which encapsulates the effect of cellular drives (e.g. entrainment, periodicity or promoter randomness) and a downstream transcriptional Poissonian part, which is common to all models. Our analytical framework treats cell-to-cell and dynamic variability consistently, unifying several approaches in the literature. We apply the obtained solution to characterize different models of experimental relevance, and to explain the influence on gene transcription of synchrony, stationarity, ergodicity, as well as the effect of time scales and other dynamic characteristics of drives. We also show how the solution can be applied to the analysis of noise sources in single-cell data, and to reduce the computational cost of stochastic simulations.
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Affiliation(s)
- Justine Dattani
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
| | - Mauricio Barahona
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
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48
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Jansen M. Stochastic equations for a self-regulating gene. INT J BIOMATH 2017. [DOI: 10.1142/s1793524517500462] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Expression of cellular genes is regulated by binding of transcription factors to their promoter, either activating or inhibiting transcription of a gene. Particularly interesting is the case when the expressed protein regulates its own transcription. In this paper, the features of this self-regulating process are investigated. In the presented model here, the gene can be in two states. Either a protein is bound to its promoter or not. The steady state distributions of protein during and just before switching from one state to the next state are analyzed. Moreover, a powerful numerical method based on the corresponding master equation to compute the protein distribution in the steady state is presented and compared to an already-existing method. Additionally the special case of self-regulation, in which protein can only be produced, if one of these proteins is bound to the promoter region, is analyzed. Furthermore, a self-regulating gene is compared to a similar gene, which also has two states and produces the same amount of proteins but is not regulated by its protein-product.
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Affiliation(s)
- Martin Jansen
- University Medical Center Freiburg, Institute of Clinical Chemistry and Laboratory Medicine, Hugstetter Str. 55, D-79106 Freiburg, Germany
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49
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A stochastic analysis of autoregulation of gene expression. J Math Biol 2017; 75:1253-1283. [PMID: 28289838 DOI: 10.1007/s00285-017-1116-7] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/08/2015] [Revised: 12/14/2016] [Indexed: 01/15/2023]
Abstract
This paper analyzes, in the context of a prokaryotic cell, the stochastic variability of the number of proteins when there is a control of gene expression by an autoregulation scheme. The goal of this work is to estimate the efficiency of the regulation to limit the fluctuations of the number of copies of a given protein. The autoregulation considered in this paper relies mainly on a negative feedback: the proteins are repressors of their own gene expression. The efficiency of a production process without feedback control is compared to a production process with an autoregulation of the gene expression assuming that both of them produce the same average number of proteins. The main characteristic used for the comparison is the standard deviation of the number of proteins at equilibrium. With a Markovian representation and a simple model of repression, we prove that, under a scaling regime, the repression mechanism follows a Hill repression scheme with an hyperbolic control. An explicit asymptotic expression of the variance of the number of proteins under this regulation mechanism is obtained. Simulations are used to study other aspects of autoregulation such as the rate of convergence to equilibrium of the production process and the case where the control of the production process of proteins is achieved via the inhibition of mRNAs.
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50
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Chu BK, Tse MJ, Sato RR, Read EL. Markov State Models of gene regulatory networks. BMC SYSTEMS BIOLOGY 2017; 11:14. [PMID: 28166778 PMCID: PMC5294885 DOI: 10.1186/s12918-017-0394-4] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/27/2016] [Accepted: 01/13/2017] [Indexed: 11/22/2022]
Abstract
Background Gene regulatory networks with dynamics characterized by multiple stable states underlie cell fate-decisions. Quantitative models that can link molecular-level knowledge of gene regulation to a global understanding of network dynamics have the potential to guide cell-reprogramming strategies. Networks are often modeled by the stochastic Chemical Master Equation, but methods for systematic identification of key properties of the global dynamics are currently lacking. Results The method identifies the number, phenotypes, and lifetimes of long-lived states for a set of common gene regulatory network models. Application of transition path theory to the constructed Markov State Model decomposes global dynamics into a set of dominant transition paths and associated relative probabilities for stochastic state-switching. Conclusions In this proof-of-concept study, we found that the Markov State Model provides a general framework for analyzing and visualizing stochastic multistability and state-transitions in gene networks. Our results suggest that this framework—adopted from the field of atomistic Molecular Dynamics—can be a useful tool for quantitative Systems Biology at the network scale. Electronic supplementary material The online version of this article (doi:10.1186/s12918-017-0394-4) contains supplementary material, which is available to authorized users.
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Affiliation(s)
- Brian K Chu
- Department of Chemical Engineering and Materials Science, University of California Irvine, Irvine, CA, USA
| | - Margaret J Tse
- Department of Chemical Engineering and Materials Science, University of California Irvine, Irvine, CA, USA
| | - Royce R Sato
- Department of Chemical Engineering and Materials Science, University of California Irvine, Irvine, CA, USA
| | - Elizabeth L Read
- Department of Chemical Engineering and Materials Science, University of California Irvine, Irvine, CA, USA. .,Department of Molecular Biology and Biochemistry, University of California Irvine, Irvine, CA, USA.
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