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Zhang Q, Cao W, Wang J, Yin Y, Sun R, Tian Z, Hu Y, Tan Y, Zhang BG. Transcriptional bursting dynamics in gene expression. Front Genet 2024; 15:1451461. [PMID: 39346775 PMCID: PMC11437526 DOI: 10.3389/fgene.2024.1451461] [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] [Received: 06/19/2024] [Accepted: 08/30/2024] [Indexed: 10/01/2024] Open
Abstract
Gene transcription is a stochastic process that occurs in all organisms. Transcriptional bursting, a critical molecular dynamics mechanism, creates significant heterogeneity in mRNA and protein levels. This heterogeneity drives cellular phenotypic diversity. Currently, the lack of a comprehensive quantitative model limits the research on transcriptional bursting. This review examines various gene expression models and compares their strengths and weaknesses to guide researchers in selecting the most suitable model for their research context. We also provide a detailed summary of the key metrics related to transcriptional bursting. We compared the temporal dynamics of transcriptional bursting across species and the molecular mechanisms influencing these bursts, and highlighted the spatiotemporal patterns of gene expression differences by utilizing metrics such as burst size and burst frequency. We summarized the strategies for modeling gene expression from both biostatistical and biochemical reaction network perspectives. Single-cell sequencing data and integrated multiomics approaches drive our exploration of cutting-edge trends in transcriptional bursting mechanisms. Moreover, we examined classical methods for parameter estimation that help capture dynamic parameters in gene expression data, assessing their merits and limitations to facilitate optimal parameter estimation. Our comprehensive summary and review of the current transcriptional burst dynamics theories provide deeper insights for promoting research on the nature of cell processes, cell fate determination, and cancer diagnosis.
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Affiliation(s)
- Qiuyu Zhang
- Research Center of Nonlinear Sciences, School of Mathematical & Physical Sciences, Wuhan Textile University, Wu Han, China
| | - Wenjie Cao
- School of Mathematics, Sun Yat-sen University, Guangzhou, China
| | - Jiaqi Wang
- Research Center of Nonlinear Sciences, School of Mathematical & Physical Sciences, Wuhan Textile University, Wu Han, China
| | - Yihao Yin
- Research Center of Nonlinear Sciences, School of Mathematical & Physical Sciences, Wuhan Textile University, Wu Han, China
| | - Rui Sun
- Research Center of Nonlinear Sciences, School of Mathematical & Physical Sciences, Wuhan Textile University, Wu Han, China
| | - Zunyi Tian
- Research Center of Nonlinear Sciences, School of Mathematical & Physical Sciences, Wuhan Textile University, Wu Han, China
| | - Yuhan Hu
- Research Center of Nonlinear Sciences, School of Mathematical & Physical Sciences, Wuhan Textile University, Wu Han, China
| | - Yalan Tan
- School of Bioengineering & Health, Wuhan Textile University, Wu Han, China
| | - Ben-Gong Zhang
- Research Center of Nonlinear Sciences, School of Mathematical & Physical Sciences, Wuhan Textile University, Wu Han, China
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2
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Lo TW, Choi HJ, Huang D, Wiggins PA. Noise robustness and metabolic load determine the principles of central dogma regulation. SCIENCE ADVANCES 2024; 10:eado3095. [PMID: 39178264 PMCID: PMC11343026 DOI: 10.1126/sciadv.ado3095] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2024] [Accepted: 07/17/2024] [Indexed: 08/25/2024]
Abstract
The processes of gene expression are inherently stochastic, even for essential genes required for growth. How does the cell maximize fitness in light of noise? To answer this question, we build a mathematical model to explore the trade-off between metabolic load and growth robustness. The model provides insights for principles of central dogma regulation: Optimal protein expression levels for many genes are in vast overabundance. Essential genes are transcribed above a lower limit of one message per cell cycle. Gene expression is achieved by load balancing between transcription and translation. We present evidence that each of these regulatory principles is observed. These results reveal that robustness and metabolic load determine the global regulatory principles that govern gene expression processes, and these principles have broad implications for cellular function.
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Affiliation(s)
- Teresa W. Lo
- Department of Physics, University of Washington, Seattle, WA 98195, USA
| | - H. James Choi
- Department of Physics, University of Washington, Seattle, WA 98195, USA
| | - Dean Huang
- Department of Physics, University of Washington, Seattle, WA 98195, USA
| | - Paul A. Wiggins
- Department of Physics, University of Washington, Seattle, WA 98195, USA
- Department of Bioengineering, University of Washington, Seattle, WA 98195, USA
- Department of Microbiology, University of Washington, Seattle, WA 98195, USA
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3
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Choi HJ, Lo TW, Cutler KJ, Huang D, Will WR, Wiggins PA. Protein overabundance is driven by growth robustness. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2024:2024.08.14.607847. [PMID: 39185236 PMCID: PMC11343162 DOI: 10.1101/2024.08.14.607847] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Indexed: 08/27/2024]
Abstract
Protein expression levels optimize cell fitness: Too low an expression level of essential proteins will slow growth by compromising essential processes; whereas overexpression slows growth by increasing the metabolic load. This trade-off naïvely predicts that cells maximize their fitness by sufficiency, expressing just enough of each essential protein for function. We test this prediction in the naturally-competent bacterium Acinetobacter baylyi by characterizing the proliferation dynamics of essential-gene knockouts at a single-cell scale (by imaging) as well as at a genome-wide scale (by TFNseq). In these experiments, cells proliferate for multiple generations as target protein levels are diluted from their endogenous levels. This approach facilitates a proteome-scale analysis of protein overabundance. As predicted by the Robustness-Load Trade-Off (RLTO) model, we find that roughly 70% of essential proteins are overabundant and that overabundance increases as the expression level decreases, the signature prediction of the model. These results reveal that robustness plays a fundamental role in determining the expression levels of essential genes and that overabundance is a key mechanism for ensuring robust growth.
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Affiliation(s)
- H. James Choi
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - Teresa W. Lo
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - Kevin J. Cutler
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - Dean Huang
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - W. Ryan Will
- Department of Laboratory Medicine and Pathology, University of Washington, Seattle, Washington 98195, USA
| | - Paul A. Wiggins
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
- Department of Microbiology, University of Washington, Seattle, Washington 98195, USA
- Department of Bioengineering, University of Washington, Seattle, Washington 98195, USA
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4
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Lo TW, James Choi H, Huang D, Wiggins PA. Noise robustness and metabolic load determine the principles of central dogma regulation. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2024:2023.10.20.563172. [PMID: 38826369 PMCID: PMC11142067 DOI: 10.1101/2023.10.20.563172] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2024]
Abstract
The processes of gene expression are inherently stochastic, even for essential genes required for growth. How does the cell maximize fitness in light of noise? To answer this question, we build a mathematical model to explore the trade-off between metabolic load and growth robustness. The model predicts novel principles of central dogma regulation: Optimal protein expression levels for many genes are in vast overabundance. Essential genes are transcribed above a lower limit of one message per cell cycle. Gene expression is achieved by load balancing between transcription and translation. We present evidence that each of these novel regulatory principles is observed. These results reveal that robustness and metabolic load determine the global regulatory principles that govern gene expression processes, and these principles have broad implications for cellular function.
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Affiliation(s)
- Teresa W. Lo
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - Han James Choi
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - Dean Huang
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - Paul A. Wiggins
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
- Department of Bioengineering, University of Washington, Seattle, Washington 98195, USA
- Department of Microbiology, University of Washington, Seattle, Washington 98195, USA
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5
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Hong L, Zhang Z, Wang Z, Yu X, Zhang J. Phase separation provides a mechanism to drive phenotype switching. Phys Rev E 2024; 109:064414. [PMID: 39021038 DOI: 10.1103/physreve.109.064414] [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: 03/18/2024] [Accepted: 06/05/2024] [Indexed: 07/20/2024]
Abstract
Phenotypic switching plays a crucial role in cell fate determination across various organisms. Recent experimental findings highlight the significance of protein compartmentalization via liquid-liquid phase separation in influencing such decisions. However, the precise mechanism through which phase separation regulates phenotypic switching remains elusive. To investigate this, we established a mathematical model that couples a phase separation process and a gene expression process with feedback. We used the chemical master equation theory and mean-field approximation to study the effects of phase separation on the gene expression products. We found that phase separation can cause bistability and bimodality. Furthermore, phase separation can control the bistable properties of the system, such as bifurcation points and bistable ranges. On the other hand, in stochastic dynamics, the droplet phase exhibits double peaks within a more extensive phase separation threshold range than the dilute phase, indicating the pivotal role of the droplet phase in cell fate decisions. These findings propose an alternative mechanism that influences cell fate decisions through the phase separation process. As phase separation is increasingly discovered in gene regulatory networks, related modeling research can help build biomolecular systems with desired properties and offer insights into explaining cell fate decisions.
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Banerjee B, Das D. Effects of bursty synthesis in organelle biogenesis. Math Biosci 2024; 370:109156. [PMID: 38346665 DOI: 10.1016/j.mbs.2024.109156] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/22/2023] [Revised: 01/31/2024] [Accepted: 02/03/2024] [Indexed: 02/16/2024]
Abstract
A fundamental question of cell biology is how cells control the number of organelles. The processes of organelle biogenesis, namely de novo synthesis, fission, fusion, and decay, are inherently stochastic, producing cell-to-cell variability in organelle abundance. In addition, experiments suggest that the synthesis of some organelles can be bursty. We thus ask how bursty synthesis impacts intracellular organelle number distribution. We develop an organelle biogenesis model with bursty de novo synthesis by considering geometrically distributed burst sizes. We analytically solve the model in biologically relevant limits and provide exact expressions for the steady-state organelle number distributions and their means and variances. We also present approximate solutions for the whole model, complementing with exact stochastic simulations. We show that bursts generally increase the noise in organelle numbers, producing distinct signatures in noise profiles depending on different mechanisms of organelle biogenesis. We also find different shapes of organelle number distributions, including bimodal distributions in some parameter regimes. Notably, bursty synthesis broadens the parameter regime of observing bimodality compared to the 'non-bursty' case. Together, our framework utilizes number fluctuations to elucidate the role of bursty synthesis in producing organelle number heterogeneity in cells.
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Affiliation(s)
- Binayak Banerjee
- Department of Biological Sciences, Indian Institute of Science Education and Research Kolkata, Nadia 741 246, West Bengal, India
| | - Dipjyoti Das
- Department of Biological Sciences, Indian Institute of Science Education and Research Kolkata, Nadia 741 246, West Bengal, India.
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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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8
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Lo TW, Choi HKJ, Huang D, Wiggins PA. The one-message-per-cell-cycle rule: A conserved minimum transcription level for essential genes. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2023:2023.07.06.548020. [PMID: 37461493 PMCID: PMC10350078 DOI: 10.1101/2023.07.06.548020] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 07/30/2023]
Abstract
The inherent stochasticity of cellular processes leads to significant cell-to-cell variation in protein abundance. Although this noise has already been characterized and modeled, its broader implications and significance remain unclear. In this paper, we revisit the noise model and identify the number of messages transcribed per cell cycle as the critical determinant of noise. In yeast, we demonstrate that this quantity predicts the non-canonical scaling of noise with protein abundance, as well as quantitatively predicting its magnitude. We then hypothesize that growth robustness requires an upper ceiling on noise for the expression of essential genes, corresponding to a lower floor on the transcription level. We show that just such a floor exists: a minimum transcription level of one message per cell cycle is conserved between three model organisms: Escherichia coli, yeast, and human. Furthermore, all three organisms transcribe the same number of messages per gene, per cell cycle. This common transcriptional program reveals that robustness to noise plays a central role in determining the expression level of a large fraction of essential genes, and that this fundamental optimal strategy is conserved from E. coli to human cells.
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Affiliation(s)
- Teresa W. Lo
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - Han Kyou James Choi
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - Dean Huang
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - Paul A. Wiggins
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
- Department of Bioengineering, University of Washington, Seattle, Washington 98195, USA
- Department of Microbiology, University of Washington, Seattle, Washington 98195, USA
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9
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Molecular Origins of Transcriptional Heterogeneity in Diazotrophic Klebsiella oxytoca. mSystems 2022; 7:e0059622. [PMID: 36073804 PMCID: PMC9600154 DOI: 10.1128/msystems.00596-22] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/04/2023] Open
Abstract
Phenotypic heterogeneity in clonal bacterial batch cultures has been shown for a range of bacterial systems; however, the molecular origins of such heterogeneity and its magnitude are not well understood. Under conditions of extreme low-nitrogen stress in the model diazotroph Klebsiella oxytoca, we found remarkably high heterogeneity of nifHDK gene expression, which codes for the structural genes of nitrogenase, one key enzyme of the global nitrogen cycle. This heterogeneity limited the bulk observed nitrogen-fixing capacity of the population. Using dual-probe, single-cell RNA fluorescent in situ hybridization, we correlated nifHDK expression with that of nifLA and glnK-amtB, which code for the main upstream regulatory components. Through stochastic transcription models and mutual information analysis, we revealed likely molecular origins for heterogeneity in nitrogenase expression. In the wild type and regulatory variants, we found that nifHDK transcription was inherently bursty, but we established that noise propagation through signaling was also significant. The regulatory gene glnK had the highest discernible effect on nifHDK variance, while noise from factors outside the regulatory pathway were negligible. Understanding the basis of inherent heterogeneity of nitrogenase expression and its origins can inform biotechnology strategies seeking to enhance biological nitrogen fixation. Finally, we speculate on potential benefits of diazotrophic heterogeneity in natural soil environments. IMPORTANCE Nitrogen is an essential micronutrient for both plant and animal life and naturally exists in both reactive and inert chemical forms. Modern agriculture is heavily reliant on nitrogen that has been "fixed" into a reactive form via the energetically expensive Haber-Bosch process, with significant environmental consequences. Nitrogen-fixing bacteria provide an alternative source of fixed nitrogen for use in both biotechnological and agricultural settings, but this relies on a firm understanding of how the fixation process is regulated within individual bacterial cells. We examined the cell-to-cell variability in the nitrogen-fixing behavior of Klebsiella oxytoca, a free-living bacterium. The significance of our research is in identifying not only the presence of marked variability but also the specific mechanisms that give rise to it. This understanding gives insight into both the evolutionary advantages of variable behavior as well as strategies for biotechnological applications.
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Concentration fluctuations in growing and dividing cells: Insights into the emergence of concentration homeostasis. PLoS Comput Biol 2022; 18:e1010574. [PMID: 36194626 PMCID: PMC9565450 DOI: 10.1371/journal.pcbi.1010574] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/07/2022] [Revised: 10/14/2022] [Accepted: 09/14/2022] [Indexed: 11/19/2022] Open
Abstract
Intracellular reaction rates depend on concentrations and hence their levels are often regulated. However classical models of stochastic gene expression lack a cell size description and cannot be used to predict noise in concentrations. Here, we construct a model of gene product dynamics that includes a description of cell growth, cell division, size-dependent gene expression, gene dosage compensation, and size control mechanisms that can vary with the cell cycle phase. We obtain expressions for the approximate distributions and power spectra of concentration fluctuations which lead to insight into the emergence of concentration homeostasis. We find that (i) the conditions necessary to suppress cell division-induced concentration oscillations are difficult to achieve; (ii) mRNA concentration and number distributions can have different number of modes; (iii) two-layer size control strategies such as sizer-timer or adder-timer are ideal because they maintain constant mean concentrations whilst minimising concentration noise; (iv) accurate concentration homeostasis requires a fine tuning of dosage compensation, replication timing, and size-dependent gene expression; (v) deviations from perfect concentration homeostasis show up as deviations of the concentration distribution from a gamma distribution. Some of these predictions are confirmed using data for E. coli, fission yeast, and budding yeast.
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Chen L, Zhu C, Jiao F. A generalized moment-based method for estimating parameters of stochastic gene transcription. Math Biosci 2022; 345:108780. [DOI: 10.1016/j.mbs.2022.108780] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/21/2021] [Revised: 12/27/2021] [Accepted: 01/13/2022] [Indexed: 12/22/2022]
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12
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Alves R, Salvadó B, Milo R, Vilaprinyo E, Sorribas A. Maximization of information transmission influences selection of native phosphorelay architectures. PeerJ 2021; 9:e11558. [PMID: 34178454 PMCID: PMC8199921 DOI: 10.7717/peerj.11558] [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: 12/04/2020] [Accepted: 05/12/2021] [Indexed: 01/28/2023] Open
Abstract
Phosphorelays are signal transduction circuits that sense environmental changes and adjust cellular metabolism. Five different circuit architectures account for 99% of all phosphorelay operons annotated in over 9,000 fully sequenced genomes. Here we asked what biological design principles, if any, could explain selection among those architectures in nature. We began by studying kinetically well characterized phosphorelays (Spo0 of Bacillus subtilis and Sln1 of Saccharomyces cerevisiae). We find that natural circuit architecture maximizes information transmission in both cases. We use mathematical models to compare information transmission among the architectures for a realistic range of concentration and parameter values. Mapping experimentally determined phosphorelay protein concentrations onto that range reveals that the native architecture maximizes information transmission in sixteen out of seventeen analyzed phosphorelays. These results suggest that maximization of information transmission is important in the selection of native phosphorelay architectures, parameter values and protein concentrations.
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Affiliation(s)
- Rui Alves
- Ciències Mèdiques Bàsiques, Universitat de Lleida, Lleida, Spain
| | - Baldiri Salvadó
- Ciències Mèdiques Bàsiques, Universitat de Lleida, Lleida, Spain
| | - Ron Milo
- Plant and Environmental Science, Weizmann Institute of Science, Rehovot, Israel
| | - Ester Vilaprinyo
- Ciències Mèdiques Bàsiques, Universitat de Lleida, Lleida, Spain
| | - Albert Sorribas
- Ciències Mèdiques Bàsiques, Universitat de Lleida, Lleida, Spain
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Assaf M, Be'er S, Roberts E. Reconstructing an epigenetic landscape using a genetic pulling approach. Phys Rev E 2021; 103:062404. [PMID: 34271627 DOI: 10.1103/physreve.103.062404] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2021] [Accepted: 05/21/2021] [Indexed: 11/07/2022]
Abstract
Cells use genetic switches to shift between alternate stable gene expression states, e.g., to adapt to new environments or to follow a developmental pathway. Conceptually, these stable phenotypes can be considered as attractive states on an epigenetic landscape with phenotypic changes being transitions between states. Measuring these transitions is challenging because they are both very rare in the absence of appropriate signals and very fast. As such, it has proved difficult to experimentally map the epigenetic landscapes that are widely believed to underly developmental networks. Here, we introduce a nonequilibrium perturbation method to help reconstruct a regulatory network's epigenetic landscape. We derive the mathematical theory needed and then use the method on simulated data to reconstruct the landscapes. Our results show that with a relatively small number of perturbation experiments it is possible to recover an accurate representation of the true epigenetic landscape. We propose that our theory provides a general method by which epigenetic landscapes can be studied. Finally, our theory suggests that the total perturbation impulse required to induce a switch between metastable states is a fundamental quantity in developmental dynamics.
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Affiliation(s)
- Michael Assaf
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Shay Be'er
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Elijah Roberts
- Department of Biophysics, Johns Hopkins University, Baltimore, Maryland 21218, USA
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14
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Rammohan J, Lund SP, Alperovich N, Paralanov V, Strychalski EA, Ross D. Comparison of bias and resolvability in single-cell and single-transcript methods. Commun Biol 2021; 4:659. [PMID: 34079048 PMCID: PMC8172639 DOI: 10.1038/s42003-021-02138-6] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2020] [Accepted: 04/16/2021] [Indexed: 11/17/2022] Open
Abstract
Single-cell and single-transcript measurement methods have elevated our ability to understand and engineer biological systems. However, defining and comparing performance between methods remains a challenge, in part due to the confounding effects of experimental variability. Here, we propose a generalizable framework for performing multiple methods in parallel using split samples, so that experimental variability is shared between methods. We demonstrate the utility of this framework by performing 12 different methods in parallel to measure the same underlying reference system for cellular response. We compare method performance using quantitative evaluations of bias and resolvability. We attribute differences in method performance to steps along the measurement process such as sample preparation, signal detection, and choice of measurand. Finally, we demonstrate how this framework can be used to benchmark different methods for single-transcript detection. The framework we present here provides a practical way to compare performance of any methods.
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Affiliation(s)
- Jayan Rammohan
- National Institute of Standards and Technology, Gaithersburg, MD, USA.
| | - Steven P Lund
- National Institute of Standards and Technology, Gaithersburg, MD, USA
| | - Nina Alperovich
- National Institute of Standards and Technology, Gaithersburg, MD, USA
| | - Vanya Paralanov
- National Institute of Standards and Technology, Gaithersburg, MD, USA
| | | | - David Ross
- National Institute of Standards and Technology, Gaithersburg, MD, USA.
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15
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Jiang Q, Fu X, Yan S, Li R, Du W, Cao Z, Qian F, Grima R. Neural network aided approximation and parameter inference of non-Markovian models of gene expression. Nat Commun 2021; 12:2618. [PMID: 33976195 PMCID: PMC8113478 DOI: 10.1038/s41467-021-22919-1] [Citation(s) in RCA: 38] [Impact Index Per Article: 12.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2020] [Accepted: 04/07/2021] [Indexed: 02/03/2023] Open
Abstract
Non-Markovian models of stochastic biochemical kinetics often incorporate explicit time delays to effectively model large numbers of intermediate biochemical processes. Analysis and simulation of these models, as well as the inference of their parameters from data, are fraught with difficulties because the dynamics depends on the system's history. Here we use an artificial neural network to approximate the time-dependent distributions of non-Markovian models by the solutions of much simpler time-inhomogeneous Markovian models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters. The training of the neural network uses a relatively small set of noisy measurements generated by experimental data or stochastic simulations of the non-Markovian model. We show using a variety of models, where the delays stem from transcriptional processes and feedback control, that the Markovian models learnt by the neural network accurately reflect the stochastic dynamics across parameter space.
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Affiliation(s)
- Qingchao Jiang
- grid.28056.390000 0001 2163 4895Key Laboratory of Smart Manufacturing in Energy Chemical Process, Ministry of Education, East China University of Science and Technology, Shanghai, China
| | - Xiaoming Fu
- grid.28056.390000 0001 2163 4895Key Laboratory of Smart Manufacturing in Energy Chemical Process, Ministry of Education, East China University of Science and Technology, Shanghai, China ,grid.4305.20000 0004 1936 7988School of Biological Sciences, The University of Edinburgh, Edinburgh, Scotland UK
| | - Shifu Yan
- grid.28056.390000 0001 2163 4895Key Laboratory of Smart Manufacturing in Energy Chemical Process, Ministry of Education, East China University of Science and Technology, Shanghai, China
| | - Runlai Li
- grid.4280.e0000 0001 2180 6431Department of Chemistry, National University of Singapore, Singapore, Singapore
| | - Wenli Du
- grid.28056.390000 0001 2163 4895Key Laboratory of Smart Manufacturing in Energy Chemical Process, Ministry of Education, East China University of Science and Technology, Shanghai, China
| | - Zhixing Cao
- grid.28056.390000 0001 2163 4895Key Laboratory of Smart Manufacturing in Energy Chemical Process, Ministry of Education, East China University of Science and Technology, Shanghai, China ,grid.28056.390000 0001 2163 4895State Key Laboratory of Bioreactor Engineering, East China University of Science and Technology, Shanghai, China
| | - Feng Qian
- grid.28056.390000 0001 2163 4895Key Laboratory of Smart Manufacturing in Energy Chemical Process, Ministry of Education, East China University of Science and Technology, Shanghai, China
| | - Ramon Grima
- grid.4305.20000 0004 1936 7988School of Biological Sciences, The University of Edinburgh, Edinburgh, Scotland UK
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16
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Sorek M, Oweis W, Nissim-Rafinia M, Maman M, Simon S, Hession CC, Adiconis X, Simmons SK, Sanjana NE, Shi X, Lu C, Pan JQ, Xu X, Pouladi MA, Ellerby LM, Zhang F, Levin JZ, Meshorer E. Pluripotent stem cell-derived models of neurological diseases reveal early transcriptional heterogeneity. Genome Biol 2021; 22:73. [PMID: 33663567 PMCID: PMC7934477 DOI: 10.1186/s13059-021-02301-6] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/16/2020] [Accepted: 02/18/2021] [Indexed: 12/24/2022] Open
Abstract
BACKGROUND Many neurodegenerative diseases develop only later in life, when cells in the nervous system lose their structure or function. In many forms of neurodegenerative diseases, this late-onset phenomenon remains largely unexplained. RESULTS Analyzing single-cell RNA sequencing from Alzheimer's disease (AD) and Huntington's disease (HD) patients, we find increased transcriptional heterogeneity in disease-state neurons. We hypothesize that transcriptional heterogeneity precedes neurodegenerative disease pathologies. To test this idea experimentally, we use juvenile forms (72Q; 180Q) of HD iPSCs, differentiate them into committed neuronal progenitors, and obtain single-cell expression profiles. We show a global increase in gene expression variability in HD. Autophagy genes become more stable, while energy and actin-related genes become more variable in the mutant cells. Knocking down several differentially variable genes results in increased aggregate formation, a pathology associated with HD. We further validate the increased transcriptional heterogeneity in CHD8+/- cells, a model for autism spectrum disorder. CONCLUSIONS Overall, our results suggest that although neurodegenerative diseases develop over time, transcriptional regulation imbalance is present already at very early developmental stages. Therefore, an intervention aimed at this early phenotype may be of high diagnostic value.
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Affiliation(s)
- Matan Sorek
- Department of Genetics, The Alexander Silberman Institute of Life Sciences, Edmond J. Safra Campus, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel
- The Edmond and Lily Center for Brain Sciences, Edmond J. Safra Campus, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel
| | - Walaa Oweis
- Department of Genetics, The Alexander Silberman Institute of Life Sciences, Edmond J. Safra Campus, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel
| | - Malka Nissim-Rafinia
- Department of Genetics, The Alexander Silberman Institute of Life Sciences, Edmond J. Safra Campus, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel
| | - Moria Maman
- Department of Genetics, The Alexander Silberman Institute of Life Sciences, Edmond J. Safra Campus, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel
| | - Shahar Simon
- Department of Genetics, The Alexander Silberman Institute of Life Sciences, Edmond J. Safra Campus, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel
| | - Cynthia C Hession
- Stanley Center for Psychiatric Research, Broad Institute of MIT and Harvard, Cambridge, MA, USA
- Klarman Cell Observatory, Broad Institute of MIT and Harvard, Cambridge, MA, USA
| | - Xian Adiconis
- Stanley Center for Psychiatric Research, Broad Institute of MIT and Harvard, Cambridge, MA, USA
- Klarman Cell Observatory, Broad Institute of MIT and Harvard, Cambridge, MA, USA
| | - Sean K Simmons
- Stanley Center for Psychiatric Research, Broad Institute of MIT and Harvard, Cambridge, MA, USA
- Klarman Cell Observatory, Broad Institute of MIT and Harvard, Cambridge, MA, USA
| | - Neville E Sanjana
- Broad Institute of MIT and Harvard, Cambridge, MA, USA
- McGovern Institute for Brain Research, Massachusetts Institute of Technology, Cambridge, MA, USA
- New York Genome Center and Department of Biology, New York University, New York, NY, USA
| | - Xi Shi
- Broad Institute of MIT and Harvard, Cambridge, MA, USA
| | - Congyi Lu
- New York Genome Center and Department of Biology, New York University, New York, NY, USA
| | - Jen Q Pan
- Stanley Center for Psychiatric Research, Broad Institute of MIT and Harvard, Cambridge, MA, USA
| | - Xiaohong Xu
- Department of Neurology and Stroke Center, The First Affiliated Hospital, Jinan University, 613 Huangpu Avenue West, Guangzhou, 510632, Guangdong, China
- Translational Laboratory in Genetic Medicine (TLGM), Agency for Science, Technology and Research (A*STAR), 8A Biomedical Grove, Immunos, Level 5, Singapore, 138648, Singapore
| | - Mahmoud A Pouladi
- Translational Laboratory in Genetic Medicine (TLGM), Agency for Science, Technology and Research (A*STAR), 8A Biomedical Grove, Immunos, Level 5, Singapore, 138648, Singapore
- Department of Physiology, National University of Singapore, Singapore, 117597, Singapore
- British Columbia Children's Hospital Research Institute, Department of Medical Genetics, University of British Columbia, Vancouver, V5Z 4H4, Canada
| | - Lisa M Ellerby
- Buck Institute for Research on Aging, 8001 Redwood Blvd, Novato, CA, 94945, USA
| | - Feng Zhang
- Broad Institute of MIT and Harvard, Cambridge, MA, USA
- McGovern Institute for Brain Research, Massachusetts Institute of Technology, Cambridge, MA, USA
- Department of Biological Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
- Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Joshua Z Levin
- Stanley Center for Psychiatric Research, Broad Institute of MIT and Harvard, Cambridge, MA, USA
- Klarman Cell Observatory, Broad Institute of MIT and Harvard, Cambridge, MA, USA
| | - Eran Meshorer
- Department of Genetics, The Alexander Silberman Institute of Life Sciences, Edmond J. Safra Campus, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel.
- The Edmond and Lily Center for Brain Sciences, Edmond J. Safra Campus, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel.
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17
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Hsu IS, Moses AM. Stochastic models for single-cell data: Current challenges and the way forward. FEBS J 2021; 289:647-658. [PMID: 33570798 DOI: 10.1111/febs.15760] [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: 07/30/2020] [Revised: 12/22/2020] [Accepted: 02/10/2021] [Indexed: 11/28/2022]
Abstract
Although the quantity and quality of single-cell data have progressed rapidly, making quantitative predictions with single-cell stochastic models remains challenging. The stochastic nature of cellular processes leads to at least three challenges in building models with single-cell data: (a) because variability in single-cell data can be attributed to multiple different sources, it is difficult to rule out conflicting mechanistic models that explain the same data equally well; (b) the distinction between interesting biological variability and experimental variability is sometimes ambiguous; (c) the nonstandard distributions of single-cell data can lead to violations of the assumption of symmetric errors in least-squares fitting. In this review, we first discuss recent studies that overcome some of the challenges or set up a promising direction and then introduce some powerful statistical approaches utilized in these studies. We conclude that applying and developing statistical approaches could lead to further progress in building stochastic models for single-cell data.
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Affiliation(s)
- Ian S Hsu
- Department of Cell & Systems Biology, University of Toronto, ON, Canada
| | - Alan M Moses
- Department of Cell & Systems Biology, University of Toronto, ON, Canada
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18
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Hou XF, Zhou BQ, Zhou YF, Apata CO, Jiang L, Pei QM. Noisy signal propagation and amplification in phenotypic transition cascade of colonic cells. Phys Rev E 2020; 102:062411. [PMID: 33466057 DOI: 10.1103/physreve.102.062411] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/21/2020] [Accepted: 11/10/2020] [Indexed: 11/07/2022]
Abstract
Like genes and proteins, cells can use biochemical networks to sense and process information. The differentiation of the cell state in colonic crypts forms a typical unidirectional phenotypic transitional cascade, in which stem cells differentiate into the transit-amplifying cells (TACs), and TACs continue to differentiate into fully differentiated cells. In order to quantitatively describe the relationship between the noise of each compartment and the amplification of signals, the gain factor is introduced, and the gain-fluctuation relation is obtained by using the linear noise approximation of the master equation. Through the simulation of these theoretical formulas, the characters of noise propagation and amplification are studied. It is found that the transmitted noise is an important part of the total noise in each downstream cell. Therefore, a small number of downstream cells can only cause its small inherent noise, but the total noise may be very large due to the transmitted noise. The influence of the transmitted noise may be the indirect cause of colon cancer. In addition, the total noise of the downstream cells always has a minimum value. As long as a reasonable value of the gain factor is selected, the number of cells in colonic crypts will be controlled within the normal range. This may be a good method to intervene the uncontrollable growth of tumor cells and effectively control the deterioration of colon cancer.
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Affiliation(s)
- Xue-Fen Hou
- School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China
| | - Bin-Qian Zhou
- School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China
| | - Yi-Fan Zhou
- School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China
| | - Charles Omotomide Apata
- School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China
| | - Long Jiang
- School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China
| | - Qi-Ming Pei
- School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China
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19
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Ham L, Schnoerr D, Brackston RD, Stumpf MPH. Exactly solvable models of stochastic gene expression. J Chem Phys 2020; 152:144106. [PMID: 32295361 DOI: 10.1063/1.5143540] [Citation(s) in RCA: 21] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/14/2023] Open
Abstract
Stochastic models are key to understanding the intricate dynamics of gene expression. However, the simplest models that only account for active and inactive states of a gene fail to capture common observations in both prokaryotic and eukaryotic organisms. Here, we consider multistate models of gene expression that generalize the canonical Telegraph process and are capable of capturing the joint effects of transcription factors, heterochromatin state, and DNA accessibility (or, in prokaryotes, sigma-factor activity) on transcript abundance. We propose two approaches for solving classes of these generalized systems. The first approach offers a fresh perspective on a general class of multistate models and allows us to "decompose" more complicated systems into simpler processes, each of which can be solved analytically. This enables us to obtain a solution of any model from this class. Next, we develop an approximation method based on a power series expansion of the stationary distribution for an even broader class of multistate models of gene transcription. We further show that models from both classes cannot have a heavy-tailed distribution in the absence of extrinsic noise. The combination of analytical and computational solutions for these realistic gene expression models also holds the potential to design synthetic systems and control the behavior of naturally evolved gene expression systems in guiding cell-fate decisions.
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Affiliation(s)
- Lucy Ham
- School of BioSciences and School of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia
| | - David Schnoerr
- Department of Life Sciences, Imperial College London, South Kensington, London SW7 2AZ, United Kingdom
| | - Rowan D Brackston
- Department of Life Sciences, Imperial College London, South Kensington, London SW7 2AZ, United Kingdom
| | - Michael P H Stumpf
- School of BioSciences and School of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia
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20
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Ham L, Brackston RD, Stumpf MPH. Extrinsic Noise and Heavy-Tailed Laws in Gene Expression. PHYSICAL REVIEW LETTERS 2020; 124:108101. [PMID: 32216388 DOI: 10.1103/physrevlett.124.108101] [Citation(s) in RCA: 24] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/29/2019] [Accepted: 02/12/2020] [Indexed: 06/10/2023]
Abstract
Noise in gene expression is one of the hallmarks of life at the molecular scale. Here we derive analytical solutions to a set of models describing the molecular mechanisms underlying transcription of DNA into RNA. Our ansatz allows us to incorporate the effects of extrinsic noise-encompassing factors external to the transcription of the individual gene-and discuss the ramifications for heterogeneity in gene product abundance that has been widely observed in single cell data. Crucially, we are able to show that heavy-tailed distributions of RNA copy numbers cannot result from the intrinsic stochasticity in gene expression alone, but must instead reflect extrinsic sources of variability.
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Affiliation(s)
- Lucy Ham
- School of BioSciences and School of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia
| | - Rowan D Brackston
- Department Life Sciences, Imperial College London, SW7 2AZ, United Kingdom
| | - Michael P H Stumpf
- School of BioSciences and School of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia
- Department Life Sciences, Imperial College London, SW7 2AZ, United Kingdom
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21
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Jia C, Grima R. Small protein number effects in stochastic models of autoregulated bursty gene expression. J Chem Phys 2020; 152:084115. [DOI: 10.1063/1.5144578] [Citation(s) in RCA: 23] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
Affiliation(s)
- Chen Jia
- Division of Applied and Computational Mathematics, Beijing Computational Science Research Center, Beijing 100193, China
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
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22
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Sun L, Ashcroft P, Ackermann M, Bonhoeffer S. Stochastic Gene Expression Influences the Selection of Antibiotic Resistance Mutations. Mol Biol Evol 2020; 37:58-70. [PMID: 31504754 PMCID: PMC6984361 DOI: 10.1093/molbev/msz199] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022] Open
Abstract
Bacteria can resist antibiotics by expressing enzymes that remove or deactivate drug molecules. Here, we study the effects of gene expression stochasticity on efflux and enzymatic resistance. We construct an agent-based model that stochastically simulates multiple biochemical processes in the cell and we observe the growth and survival dynamics of the cell population. Resistance-enhancing mutations are introduced by varying parameters that control the enzyme expression or efficacy. We find that stochastic gene expression can cause complex dynamics in terms of survival and extinction for these mutants. Regulatory mutations, which augment the frequency and duration of resistance gene transcription, can provide limited resistance by increasing mean expression. Structural mutations, which modify the enzyme or efflux efficacy, provide most resistance by improving the binding affinity of the resistance protein to the antibiotic; increasing the enzyme's catalytic rate alone may contribute to resistance if drug binding is not rate limiting. Overall, we identify conditions where regulatory mutations are selected over structural mutations, and vice versa. Our findings show that stochastic gene expression is a key factor underlying efflux and enzymatic resistances and should be taken into consideration in future antibiotic research.
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Affiliation(s)
- Lei Sun
- Institute of Integrative Biology, ETH Zürich, Zürich, Switzerland
| | - Peter Ashcroft
- Institute of Integrative Biology, ETH Zürich, Zürich, Switzerland
| | - Martin Ackermann
- Institute of Biogeochemistry and Pollutant Dynamics, ETH Zürich, Zürich, Switzerland.,Department of Environmental Microbiology, EAWAG, Swiss Federal Institute of Aquatic Science and Technology, Dübendorf, Switzerland
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23
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Limit theorems for generalized density-dependent Markov chains and bursty stochastic gene regulatory networks. J Math Biol 2019; 80:959-994. [DOI: 10.1007/s00285-019-01445-1] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2019] [Revised: 08/22/2019] [Indexed: 11/26/2022]
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24
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Jia C, Wang LY, Yin GG, Zhang MQ. Single-cell stochastic gene expression kinetics with coupled positive-plus-negative feedback. Phys Rev E 2019; 100:052406. [PMID: 31869986 DOI: 10.1103/physreve.100.052406] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2019] [Indexed: 06/10/2023]
Abstract
Here we investigate single-cell stochastic gene expression kinetics in a minimal coupled gene circuit with positive-plus-negative feedback. A triphasic stochastic bifurcation is observed upon increasing the ratio of the positive and negative feedback strengths, which reveals a strong synergistic interaction between positive and negative feedback loops. We discover that coupled positive-plus-negative feedback amplifies gene expression mean but reduces gene expression noise over a wide range of feedback strengths when promoter switching is relatively slow, stabilizing gene expression around a relatively high level. In addition, we study two types of macroscopic limits of the discrete chemical master equation model: the Kurtz limit applies to proteins with large burst frequencies and the Lévy limit applies to proteins with large burst sizes. We derive the analytic steady-state distributions of the protein abundance in a coupled gene circuit for both the discrete model and its two macroscopic limits, generalizing the results obtained by Liu et al. [Chaos 26, 043108 (2016)CHAOEH1054-150010.1063/1.4947202]. We also obtain the analytic time-dependent protein distribution for the classical Friedman-Cai-Xie random bursting model [Friedman, Cai, and Xie, Phys. Rev. Lett. 97, 168302 (2006)PRLTAO0031-900710.1103/PhysRevLett.97.168302]. Our analytic results are further applied to study the structure of gene expression noise in a coupled gene circuit, and a complete decomposition of noise in terms of five different biophysical origins is provided.
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Affiliation(s)
- Chen Jia
- Division of Applied and Computational Mathematics, Beijing Computational Science Research Center, Beijing 100193, China
- Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA
| | - Le Yi Wang
- Department of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan 48202, USA
| | - George G Yin
- Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA
| | - Michael Q Zhang
- Department of Biological Sciences, Center for Systems Biology, University of Texas at Dallas, Richardson, Texas 75080, USA
- MOE Key Laboratory of Bioinformatics, Center for Synthetic and Systems Biology, Tsinghua University, Beijing 100084, China
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25
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Abstract
Numerous studies based on new single-cell and single-gene techniques show that individual genes can be transcribed in short bursts or pulses accompanied by changes in pulsing frequencies. Since so many examples of such discontinuous or fluctuating transcription have been found from prokaryotes to mammals, it now seems to be a common mode of gene expression. In this review we discuss the occurrence of the transcriptional fluctuations, the techniques used for their detection, their putative causes, kinetic characteristics, and probable physiological significance.
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Affiliation(s)
- Evgeny Smirnov
- a Institute of Biology and Medical Genetics , First Faculty of Medicine , Charles University and General University Hospital in Prague , Prague , Czech Republic
| | - Matúš Hornáček
- a Institute of Biology and Medical Genetics , First Faculty of Medicine , Charles University and General University Hospital in Prague , Prague , Czech Republic
| | - Tomáš Vacík
- a Institute of Biology and Medical Genetics , First Faculty of Medicine , Charles University and General University Hospital in Prague , Prague , Czech Republic
| | - Dušan Cmarko
- a Institute of Biology and Medical Genetics , First Faculty of Medicine , Charles University and General University Hospital in Prague , Prague , Czech Republic
| | - Ivan Raška
- a Institute of Biology and Medical Genetics , First Faculty of Medicine , Charles University and General University Hospital in Prague , Prague , Czech Republic
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26
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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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27
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Atitey K, Loskot P, Rees P. Inferring distributions from observed mRNA and protein copy counts in genetic circuits. Biomed Phys Eng Express 2018. [DOI: 10.1088/2057-1976/aaef5c] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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28
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Prajapat MK, Ribeiro AS. Added value of autoregulation and multi-step kinetics of transcription initiation. ROYAL SOCIETY OPEN SCIENCE 2018; 5:181170. [PMID: 30564410 PMCID: PMC6281912 DOI: 10.1098/rsos.181170] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/27/2018] [Accepted: 11/01/2018] [Indexed: 06/09/2023]
Abstract
Bacterial gene expression regulation occurs mostly during transcription, which has two main rate-limiting steps: the close complex formation, when the RNA polymerase binds to an active promoter, and the subsequent open complex formation, after which it follows elongation. Tuning these steps' kinetics by the action of e.g. transcription factors, allows for a wide diversity of dynamics. For example, adding autoregulation generates single-gene circuits able to perform more complex tasks. Using stochastic models of transcription kinetics with empirically validated parameter values, we investigate how autoregulation and the multi-step transcription initiation kinetics of single-gene autoregulated circuits can be combined to fine-tune steady state mean and cell-to-cell variability in protein expression levels, as well as response times. Next, we investigate how they can be jointly tuned to control complex behaviours, namely, time counting, switching dynamics and memory storage. Overall, our finding suggests that, in bacteria, jointly regulating a single-gene circuit's topology and the transcription initiation multi-step dynamics allows enhancing complex task performance.
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Affiliation(s)
- Mahendra Kumar Prajapat
- Laboratory of Biosystem Dynamics, Faculty of Biomedical Sciences and Engineering, BioMediTech Institute, Tampere University of Technology, 33101 Tampere, Finland
| | - Andre S. Ribeiro
- Laboratory of Biosystem Dynamics, Faculty of Biomedical Sciences and Engineering, BioMediTech Institute, Tampere University of Technology, 33101 Tampere, Finland
- Multi-scaled Biodata Analysis and Modelling Research Community, Tampere University of Technology, 33101 Tampere, Finland
- CA3 CTS/UNINOVA, Faculdade de Ciencias e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal
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29
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Vilk O, Assaf M. Population extinction under bursty reproduction in a time-modulated environment. Phys Rev E 2018; 97:062114. [PMID: 30011566 DOI: 10.1103/physreve.97.062114] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2018] [Indexed: 06/08/2023]
Abstract
In recent years nondemographic variability has been shown to greatly affect dynamics of stochastic populations. For example, nondemographic noise in the form of a bursty reproduction process with an a priori unknown burst size, or environmental variability in the form of time-varying reaction rates, have been separately found to dramatically impact the extinction risk of isolated populations. In this work we investigate the extinction risk of an isolated population under the combined influence of these two types of nondemographic variation. Using the so-called momentum-space Wentzel-Kramers-Brillouin (WKB) approach and accounting for the explicit time dependence in the reaction rates, we arrive at a set of time-dependent Hamilton equations. To this end, we evaluate the population's extinction risk by finding the instanton of the time-perturbed Hamiltonian numerically, whereas analytical expressions are presented in particular limits using various perturbation techniques. We focus on two classes of time-varying environments: periodically varying rates corresponding to seasonal effects and a sudden decrease in the birth rate corresponding to a catastrophe. All our theoretical results are tested against numerical Monte Carlo simulations with time-dependent rates and also against a numerical solution of the corresponding time-dependent Hamilton equations.
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Affiliation(s)
- Ohad Vilk
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Michael Assaf
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
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30
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Loewke NO, Pai S, Cordeiro C, Black D, King BL, Contag CH, Chen B, Baer TM, Solgaard O. Automated Cell Segmentation for Quantitative Phase Microscopy. IEEE TRANSACTIONS ON MEDICAL IMAGING 2018; 37:929-940. [PMID: 29610072 PMCID: PMC5907807 DOI: 10.1109/tmi.2017.2775604] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/13/2023]
Abstract
Automated cell segmentation and tracking is essential for dynamic studies of cellular morphology, movement, and interactions as well as other cellular behaviors. However, accurate, automated, and easy-to-use cell segmentation remains a challenge, especially in cases of high cell densities, where discrete boundaries are not easily discernable. Here, we present a fully automated segmentation algorithm that iteratively segments cells based on the observed distribution of optical cell volumes measured by quantitative phase microscopy. By fitting these distributions to known probability density functions, we are able to converge on volumetric thresholds that enable valid segmentation cuts. Since each threshold is determined from the observed data itself, virtually no input is needed from the user. We demonstrate the effectiveness of this approach over time using six cell types that display a range of morphologies, and evaluate these cultures over a range of confluencies. Facile dynamic measures of cell mobility and function revealed unique cellular behaviors that relate to tissue origins, state of differentiation, and real-time signaling. These will improve our understanding of multicellular communication and organization.
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31
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Li H, Sun X, Xiao J. Stochastic multiresonance in coupled excitable FHN neurons. CHAOS (WOODBURY, N.Y.) 2018; 28:043113. [PMID: 31906634 DOI: 10.1063/1.4997679] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
In this paper, effects of noise on Watts-Strogatz small-world neuronal networks, which are stimulated by a subthreshold signal, have been investigated. With the numerical simulations, it is surprisingly found that there exist several optimal noise intensities at which the subthreshold signal can be detected efficiently. This indicates the occurrence of stochastic multiresonance in the studied neuronal networks. Moreover, it is revealed that the occurrence of stochastic multiresonance has close relationship with the period of subthreshold signal Te and the noise-induced mean period of the neuronal networks T0. In detail, we find that noise could induce the neuronal networks to generate stochastic resonance for M times if Te is not very large and falls into the interval (M×T0,(M+1)×T0) with M being a positive integer. In real neuronal system, subthreshold signal detection is very meaningful. Thus, the obtained results in this paper could give some important implications on detecting subthreshold signal and propagating neuronal information in neuronal systems.
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Affiliation(s)
- Huiyan Li
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Xiaojuan Sun
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Jinghua Xiao
- School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
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32
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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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33
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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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34
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Be'er S, Assaf M. Reducing the extinction risk of stochastic populations via nondemographic noise. Phys Rev E 2018; 97:020302. [PMID: 29548157 DOI: 10.1103/physreve.97.020302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2017] [Indexed: 06/08/2023]
Abstract
We consider nondemographic noise in the form of uncertainty in the reaction step size and reveal a dramatic effect this noise may have on the stability of self-regulating populations. Employing the reaction scheme mA→kA but allowing, e.g., the product number k to be a priori unknown and sampled from a given distribution, we show that such nondemographic noise can greatly reduce the population's extinction risk compared to the fixed k case. Our analysis is tested against numerical simulations, and by using empirical data of different species, we argue that certain distributions may be more evolutionary beneficial than others.
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Affiliation(s)
- Shay Be'er
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Michael Assaf
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
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35
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Jia C, Zhang MQ, Qian H. Emergent Lévy behavior in single-cell stochastic gene expression. Phys Rev E 2018; 96:040402. [PMID: 29347590 DOI: 10.1103/physreve.96.040402] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2017] [Indexed: 11/07/2022]
Abstract
Single-cell gene expression is inherently stochastic; its emergent behavior can be defined in terms of the chemical master equation describing the evolution of the mRNA and protein copy numbers as the latter tends to infinity. We establish two types of "macroscopic limits": the Kurtz limit is consistent with the classical chemical kinetics, while the Lévy limit provides a theoretical foundation for an empirical equation proposed in N. Friedman et al., Phys. Rev. Lett. 97, 168302 (2006)PRLTAO0031-900710.1103/PhysRevLett.97.168302. Furthermore, we clarify the biochemical implications and ranges of applicability for various macroscopic limits and calculate a comprehensive analytic expression for the protein concentration distribution in autoregulatory gene networks. The relationship between our work and modern population genetics is discussed.
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Affiliation(s)
- Chen Jia
- Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Michael Q Zhang
- Department of Biological Sciences, University of Texas at Dallas, Richardson, Texas 75080, USA.,MOE Key Lab and Division of Bioinformatics, CSSB, TNLIST, Tsinghua University, Beijing 100084, China
| | - Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
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36
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Yan CCS, Chepyala SR, Yen CM, Hsu CP. Efficient and flexible implementation of Langevin simulation for gene burst production. Sci Rep 2017; 7:16851. [PMID: 29203832 PMCID: PMC5715166 DOI: 10.1038/s41598-017-16835-y] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2017] [Accepted: 11/16/2017] [Indexed: 12/14/2022] Open
Abstract
Gene expression involves bursts of production of both mRNA and protein, and the fluctuations in their number are increased due to such bursts. The Langevin equation is an efficient and versatile means to simulate such number fluctuation. However, how to include these mRNA and protein bursts in the Langevin equation is not intuitively clear. In this work, we estimated the variance in burst production from a general gene expression model and introduced such variation in the Langevin equation. Our approach offers different Langevin expressions for either or both transcriptional and translational bursts considered and saves computer time by including many production events at once in a short burst time. The errors can be controlled to be rather precise (<2%) for the mean and <10% for the standard deviation of the steady-state distribution. Our scheme allows for high-quality stochastic simulations with the Langevin equation for gene expression, which is useful in analysis of biological networks.
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Affiliation(s)
| | - Surendhar Reddy Chepyala
- Institute of Chemistry, Academia Sinica, Taipei, 115, Taiwan.,Bioinformatics Program, Taiwan International Graduate Program, Institute of Information Science, Academia Sinica, Taipei, 115, Taiwan.,Institute of Biomedical Informatics, National Yang-Ming University, Taipei, 112, Taiwan
| | - Chao-Ming Yen
- Institute of Chemistry, Academia Sinica, Taipei, 115, Taiwan.,Institute of Biochemical Sciences, College of Life Science, National Taiwan University, Taipei, 106, Taiwan.,Institute of Biological Chemistry, Academia Sinica, Taipei, 115, Taiwan
| | - Chao-Ping Hsu
- Institute of Chemistry, Academia Sinica, Taipei, 115, Taiwan. .,Genome and Systems Biology Degree Program, National Taiwan University, Taipei, 106, Taiwan.
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37
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Jia C, Xie P, Chen M, Zhang MQ. Stochastic fluctuations can reveal the feedback signs of gene regulatory networks at the single-molecule level. Sci Rep 2017; 7:16037. [PMID: 29167445 PMCID: PMC5700158 DOI: 10.1038/s41598-017-15464-9] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [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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Affiliation(s)
- Chen Jia
- Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080, USA
| | - Peng Xie
- Department of Biological Sciences, Center for Systems Biology, University of Texas at Dallas, Richardson, TX 75080, USA
| | - Min Chen
- Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080, USA.
| | - Michael Q Zhang
- Department of Biological Sciences, Center for Systems Biology, University of Texas at Dallas, Richardson, TX 75080, USA.
- MOE Key Lab and Division of Bioinformatics, CSSB, TNLIST, Tsinghua University, Beijing, 100084, China.
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38
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Falk J, Mendler M, Drossel B. A minimal model of burst-noise induced bistability. PLoS One 2017; 12:e0176410. [PMID: 28448638 PMCID: PMC5407650 DOI: 10.1371/journal.pone.0176410] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2016] [Accepted: 04/09/2017] [Indexed: 11/18/2022] Open
Abstract
We investigate the influence of intrinsic noise on stable states of a one-dimensional dynamical system that shows in its deterministic version a saddle-node bifurcation between monostable and bistable behaviour. The system is a modified version of the Schlögl model, which is a chemical reaction system with only one type of molecule. The strength of the intrinsic noise is varied without changing the deterministic description by introducing bursts in the autocatalytic production step. We study the transitions between monostable and bistable behavior in this system by evaluating the number of maxima of the stationary probability distribution. We find that changing the size of bursts can destroy and even induce saddle-node bifurcations. This means that a bursty production of molecules can qualitatively change the dynamics of a chemical reaction system even when the deterministic description remains unchanged.
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Affiliation(s)
- Johannes Falk
- Institut für Festkörperphysik, Technische Universität Darmstadt, Hochschulstr. 6, 64289 Darmstadt, Germany
- * E-mail:
| | - Marc Mendler
- Institut für Festkörperphysik, Technische Universität Darmstadt, Hochschulstr. 6, 64289 Darmstadt, Germany
| | - Barbara Drossel
- Institut für Festkörperphysik, Technische Universität Darmstadt, Hochschulstr. 6, 64289 Darmstadt, Germany
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39
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Jędrak J, Ochab-Marcinek A. Time-dependent solutions for a stochastic model of gene expression with molecule production in the form of a compound Poisson process. Phys Rev E 2016; 94:032401. [PMID: 27739798 DOI: 10.1103/physreve.94.032401] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2016] [Indexed: 06/06/2023]
Abstract
We study a stochastic model of gene expression, in which protein production has a form of random bursts whose size distribution is arbitrary, whereas protein decay is a first-order reaction. We find exact analytical expressions for the time evolution of the cumulant-generating function for the most general case when both the burst size probability distribution and the model parameters depend on time in an arbitrary (e.g., oscillatory) manner, and for arbitrary initial conditions. We show that in the case of periodic external activation and constant protein degradation rate, the response of the gene is analogous to the resistor-capacitor low-pass filter, where slow oscillations of the external driving have a greater effect on gene expression than the fast ones. We also demonstrate that the nth cumulant of the protein number distribution depends on the nth moment of the burst size distribution. We use these results to show that different measures of noise (coefficient of variation, Fano factor, fractional change of variance) may vary in time in a different manner. Therefore, any biological hypothesis of evolutionary optimization based on the nonmonotonic dependence of a chosen measure of noise on time must justify why it assumes that biological evolution quantifies noise in that particular way. Finally, we show that not only for exponentially distributed burst sizes but also for a wider class of burst size distributions (e.g., Dirac delta and gamma) the control of gene expression level by burst frequency modulation gives rise to proportional scaling of variance of the protein number distribution to its mean, whereas the control by amplitude modulation implies proportionality of protein number variance to the mean squared.
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Affiliation(s)
- Jakub Jędrak
- Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warsaw, Poland
| | - Anna Ochab-Marcinek
- Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warsaw, Poland
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40
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Soltani M, Vargas-Garcia CA, Antunes D, Singh A. Intercellular Variability in Protein Levels from Stochastic Expression and Noisy Cell Cycle Processes. PLoS Comput Biol 2016; 12:e1004972. [PMID: 27536771 PMCID: PMC4990281 DOI: 10.1371/journal.pcbi.1004972] [Citation(s) in RCA: 64] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/08/2015] [Accepted: 07/29/2016] [Indexed: 12/22/2022] Open
Abstract
Inside individual cells, expression of genes is inherently stochastic and manifests as cell-to-cell variability or noise in protein copy numbers. Since proteins half-lives can be comparable to the cell-cycle length, randomness in cell-division times generates additional intercellular variability in protein levels. Moreover, as many mRNA/protein species are expressed at low-copy numbers, errors incurred in partitioning of molecules between two daughter cells are significant. We derive analytical formulas for the total noise in protein levels when the cell-cycle duration follows a general class of probability distributions. Using a novel hybrid approach the total noise is decomposed into components arising from i) stochastic expression; ii) partitioning errors at the time of cell division and iii) random cell-division events. These formulas reveal that random cell-division times not only generate additional extrinsic noise, but also critically affect the mean protein copy numbers and intrinsic noise components. Counter intuitively, in some parameter regimes, noise in protein levels can decrease as cell-division times become more stochastic. Computations are extended to consider genome duplication, where transcription rate is increased at a random point in the cell cycle. We systematically investigate how the timing of genome duplication influences different protein noise components. Intriguingly, results show that noise contribution from stochastic expression is minimized at an optimal genome-duplication time. Our theoretical results motivate new experimental methods for decomposing protein noise levels from synchronized and asynchronized single-cell expression data. Characterizing the contributions of individual noise mechanisms will lead to precise estimates of gene expression parameters and techniques for altering stochasticity to change phenotype of individual cells. Inside individual cells, gene products often occur at low molecular counts and are subject to considerable stochastic fluctuations (noise) in copy numbers over time. An important consequence of noisy expression is that the level of a protein can vary considerably even among genetically identical cells exposed to the same environment. Such non-genetic phenotypic heterogeneity is physiologically relevant and critically influences diverse cellular processes. In addition to noise sources inherent in gene product synthesis, recent experimental studies have uncovered additional noise mechanisms that critically effect expression. For example, the time within the cell cycle when a gene duplicates, and the time taken to complete cell cycle are governed by random processes. The key contribution of this work is development of novel mathematical results quantifying how cell cycle-related noise sources combine with stochastic expression to drive intercellular variability in protein molecular counts. Derived formulas lead to many counterintuitive results, such as increasing randomness in the timing of cell division can lower noise in the level of a protein. Finally, these results inform experimental strategies to systematically dissect the contributions of different noise sources in the expression of a gene of interest.
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Affiliation(s)
- Mohammad Soltani
- Electrical and Computer Engineering Department, University of Delaware, Newark, Delaware, United States of America
| | - Cesar A. Vargas-Garcia
- Electrical and Computer Engineering Department, University of Delaware, Newark, Delaware, United States of America
| | - Duarte Antunes
- Mechanical Engineering Department, Eindhoven University of Technology, Eindhoven, Netherlands
| | - Abhyudai Singh
- Electrical and Computer Engineering Department, University of Delaware, Newark, Delaware, United States of America
- Biomedical Engineering Department, University of Delaware, Newark, Delaware, United States of America
- Mathematical Sciences Department, University of Delaware, Newark, Delaware, United States of America
- Center for Bioinformatics and Computational Biology, University of Delaware, Newark, Delaware, United States of America
- * E-mail:
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41
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Sharma R, Roberts E. Gradient sensing by a bistable regulatory motif enhances signal amplification but decreases accuracy in individual cells. Phys Biol 2016; 13:036003. [DOI: 10.1088/1478-3975/13/3/036003] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
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42
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Be'er S, Heller-Algazi M, Assaf M. Effect of reaction-step-size noise on the switching dynamics of stochastic populations. Phys Rev E 2016; 93:052117. [PMID: 27300840 DOI: 10.1103/physreve.93.052117] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2015] [Indexed: 06/06/2023]
Abstract
In genetic circuits, when the messenger RNA lifetime is short compared to the cell cycle, proteins are produced in geometrically distributed bursts, which greatly affects the cellular switching dynamics between different metastable phenotypic states. Motivated by this scenario, we study a general problem of switching or escape in stochastic populations, where influx of particles occurs in groups or bursts, sampled from an arbitrary distribution. The fact that the step size of the influx reaction is a priori unknown and, in general, may fluctuate in time with a given correlation time and statistics, introduces an additional nondemographic reaction-step-size noise into the system. Employing the probability-generating function technique in conjunction with Hamiltonian formulation, we are able to map the problem in the leading order onto solving a stationary Hamilton-Jacobi equation. We show that compared to the "usual case" of single-step influx, bursty influx exponentially decreases the population's mean escape time from its long-lived metastable state. In particular, close to bifurcation we find a simple analytical expression for the mean escape time which solely depends on the mean and variance of the burst-size distribution. Our results are demonstrated on several realistic distributions and compare well with numerical Monte Carlo simulations.
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Affiliation(s)
- Shay Be'er
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Metar Heller-Algazi
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Michael Assaf
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
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43
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Hilfinger A, Norman TM, Vinnicombe G, Paulsson J. Constraints on Fluctuations in Sparsely Characterized Biological Systems. PHYSICAL REVIEW LETTERS 2016; 116:058101. [PMID: 26894735 PMCID: PMC4834202 DOI: 10.1103/physrevlett.116.058101] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2015] [Indexed: 06/05/2023]
Abstract
Biochemical processes are inherently stochastic, creating molecular fluctuations in otherwise identical cells. Such "noise" is widespread but has proven difficult to analyze because most systems are sparsely characterized at the single cell level and because nonlinear stochastic models are analytically intractable. Here, we exactly relate average abundances, lifetimes, step sizes, and covariances for any pair of components in complex stochastic reaction systems even when the dynamics of other components are left unspecified. Using basic mathematical inequalities, we then establish bounds for whole classes of systems. These bounds highlight fundamental trade-offs that show how efficient assembly processes must invariably exhibit large fluctuations in subunit levels and how eliminating fluctuations in one cellular component requires creating heterogeneity in another.
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Affiliation(s)
- Andreas Hilfinger
- Department of Systems Biology, Harvard University, 200 Longwood Avenue, Boston, Massachusetts 02115, USA
| | - Thomas M. Norman
- Department of Systems Biology, Harvard University, 200 Longwood Avenue, Boston, Massachusetts 02115, USA
| | - Glenn Vinnicombe
- Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
| | - Johan Paulsson
- Department of Systems Biology, Harvard University, 200 Longwood Avenue, Boston, Massachusetts 02115, USA
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44
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Roberts E, Be'er S, Bohrer C, Sharma R, Assaf M. Dynamics of simple gene-network motifs subject to extrinsic fluctuations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062717. [PMID: 26764737 DOI: 10.1103/physreve.92.062717] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2015] [Indexed: 06/05/2023]
Abstract
Cellular processes do not follow deterministic rules; even in identical environments genetically identical cells can make random choices leading to different phenotypes. This randomness originates from fluctuations present in the biomolecular interaction networks. Most previous work has been focused on the intrinsic noise (IN) of these networks. Yet, especially for high-copy-number biomolecules, extrinsic or environmental noise (EN) has been experimentally shown to dominate the variation. Here, we develop an analytical formalism that allows for calculation of the effect of EN on gene-expression motifs. We introduce a method for modeling bounded EN as an auxiliary species in the master equation. The method is fully generic and is not limited to systems with small EN magnitudes. We focus our study on motifs that can be viewed as the building blocks of genetic switches: a nonregulated gene, a self-inhibiting gene, and a self-promoting gene. The role of the EN properties (magnitude, correlation time, and distribution) on the statistics of interest are systematically investigated, and the effect of fluctuations in different reaction rates is compared. Due to its analytical nature, our formalism can be used to quantify the effect of EN on the dynamics of biochemical networks and can also be used to improve the interpretation of data from single-cell gene-expression experiments.
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Affiliation(s)
- Elijah Roberts
- Department of Biophysics, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Shay Be'er
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
| | - Chris Bohrer
- Department of Biophysics, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Rati Sharma
- Department of Biophysics, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Michael Assaf
- Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
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45
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Pájaro M, Alonso AA, Vázquez C. Shaping protein distributions in stochastic self-regulated gene expression networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:032712. [PMID: 26465503 DOI: 10.1103/physreve.92.032712] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2015] [Indexed: 06/05/2023]
Abstract
In this work, we study connections between dynamic behavior and network parameters, for self-regulatory networks. To that aim, a method to compute the regions in the space of parameters that sustain bimodal or binary protein distributions has been developed. Such regions are indicative of stochastic dynamics manifested either as transitions between absence and presence of protein or between two positive protein levels. The method is based on the continuous approximation of the chemical master equation, unlike other approaches that make use of a deterministic description, which as will be shown can be misleading. We find that bimodal behavior is a ubiquitous phenomenon in cooperative gene expression networks under positive feedback. It appears for any range of transcription and translation rate constants whenever leakage remains below a critical threshold. Above such a threshold, the region in the parameters space which sustains bimodality persists, although restricted to low transcription and high translation rate constants. Remarkably, such a threshold is independent of the transcription or translation rates or the proportion of an active or inactive promoter and depends only on the level of cooperativity. The proposed method can be employed to identify bimodal or binary distributions leading to stochastic dynamics with specific switching properties, by searching inside the parameter regions that sustain such behavior.
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Affiliation(s)
- Manuel Pájaro
- Process Engineering Group, IIM-CSIC, Spanish Council for Scientific Research, Eduardo Cabello 6, 36208 Vigo, Spain
| | - Antonio A Alonso
- Process Engineering Group, IIM-CSIC, Spanish Council for Scientific Research, Eduardo Cabello 6, 36208 Vigo, Spain
| | - Carlos Vázquez
- Department of Mathematics, Faculty of Informatics, Campus de Elviña, s/n15071 A Coruña, Spain
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46
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Giampieri E, De Cecco M, Remondini D, Sedivy J, Castellani G. Active Degradation Explains the Distribution of Nuclear Proteins during Cellular Senescence. PLoS One 2015; 10:e0118442. [PMID: 26115222 PMCID: PMC4483236 DOI: 10.1371/journal.pone.0118442] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2014] [Accepted: 01/16/2015] [Indexed: 11/19/2022] Open
Abstract
The amount of cellular proteins is a crucial parameter that is known to vary between cells as a function of the replicative passages, and can be important during physiological aging. The process of protein degradation is known to be performed by a series of enzymatic reactions, ranging from an initial step of protein ubiquitination to their final fragmentation by the proteasome. In this paper we propose a stochastic dynamical model of nuclear proteins concentration resulting from a balance between a constant production of proteins and their degradation by a cooperative enzymatic reaction. The predictions of this model are compared with experimental data obtained by fluorescence measurements of the amount of nuclear proteins in murine tail fibroblast (MTF) undergoing cellular senescence. Our model provides a three-parameter stationary distribution that is in good agreement with the experimental data even during the transition to the senescent state, where the nuclear protein concentration changes abruptly. The estimation of three parameters (cooperativity, saturation threshold, and maximal velocity of the reaction), and their evolution during replicative passages shows that only the maximal velocity varies significantly. Based on our modeling we speculate the reduction of functionality of the protein degradation mechanism as a possible competitive inhibition of the proteasome.
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Affiliation(s)
- Enrico Giampieri
- Department of Physics and Astronomy, Bologna University, Bologna, Italy and INFN Bologna
- * E-mail:
| | - Marco De Cecco
- Department of Molecular Biology, Cell Biology and Biochemistry, Center for Genomics and Proteomics, Brown University, Providence, RI, USA
| | - Daniel Remondini
- Department of Physics and Astronomy, Bologna University, Bologna, Italy and INFN Bologna
| | - John Sedivy
- Department of Molecular Biology, Cell Biology and Biochemistry, Center for Genomics and Proteomics, Brown University, Providence, RI, USA
| | - Gastone Castellani
- Department of Physics and Astronomy, Bologna University, Bologna, Italy and INFN Bologna
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47
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Jones DL, Brewster RC, Phillips R. Promoter architecture dictates cell-to-cell variability in gene expression. Science 2014; 346:1533-6. [PMID: 25525251 DOI: 10.1126/science.1255301] [Citation(s) in RCA: 134] [Impact Index Per Article: 13.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023]
Abstract
Variability in gene expression among genetically identical cells has emerged as a central preoccupation in the study of gene regulation; however, a divide exists between the predictions of molecular models of prokaryotic transcriptional regulation and genome-wide experimental studies suggesting that this variability is indifferent to the underlying regulatory architecture. We constructed a set of promoters in Escherichia coli in which promoter strength, transcription factor binding strength, and transcription factor copy numbers are systematically varied, and used messenger RNA (mRNA) fluorescence in situ hybridization to observe how these changes affected variability in gene expression. Our parameter-free models predicted the observed variability; hence, the molecular details of transcription dictate variability in mRNA expression, and transcriptional noise is specifically tunable and thus represents an evolutionarily accessible phenotypic parameter.
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Affiliation(s)
- Daniel L Jones
- Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA
| | - Robert C Brewster
- Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA. Division of Biology, California Institute of Technology, Pasadena, CA 91125, USA
| | - Rob Phillips
- Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA. Division of Biology, California Institute of Technology, Pasadena, CA 91125, USA.
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48
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Singer ZS, Yong J, Tischler J, Hackett JA, Altinok A, Surani MA, Cai L, Elowitz MB. Dynamic heterogeneity and DNA methylation in embryonic stem cells. Mol Cell 2014; 55:319-31. [PMID: 25038413 PMCID: PMC4104113 DOI: 10.1016/j.molcel.2014.06.029] [Citation(s) in RCA: 212] [Impact Index Per Article: 21.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2013] [Revised: 04/04/2014] [Accepted: 06/18/2014] [Indexed: 01/16/2023]
Abstract
Cell populations can be strikingly heterogeneous, composed of multiple cellular states, each exhibiting stochastic noise in its gene expression. A major challenge is to disentangle these two types of variability and to understand the dynamic processes and mechanisms that control them. Embryonic stem cells (ESCs) provide an ideal model system to address this issue because they exhibit heterogeneous and dynamic expression of functionally important regulatory factors. We analyzed gene expression in individual ESCs using single-molecule RNA-FISH and quantitative time-lapse movies. These data discriminated stochastic switching between two coherent (correlated) gene expression states and burst-like transcriptional noise. We further showed that the “2i” signaling pathway inhibitors modulate both types of variation. Finally, we found that DNA methylation plays a key role in maintaining these metastable states. Together, these results show how ESC gene expression states and dynamics arise from a combination of intrinsic noise, coherent cellular states, and epigenetic regulation. smFISH in ESCs reveals two transcriptional states and highly stochastic expression Live-cell expression dynamics reveal the in situ transition rates between states DNA methylation regulates state-switching dynamics “2i” signaling inhibitors impact both gene expression noise and state transitions
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Affiliation(s)
- Zakary S Singer
- Computation and Neural Systems, California Institute of Technology, Pasadena, CA 91125, USA
| | - John Yong
- Division of Biology, California Institute of Technology, Pasadena, CA 91125, USA
| | - Julia Tischler
- The Wellcome Trust/Cancer Research UK Gurdon Institute, The Henry Wellcome Building of Cancer and Developmental Biology, University of Cambridge, Tennis Court Road, Cambridge CB2 1QN, UK
| | - Jamie A Hackett
- The Wellcome Trust/Cancer Research UK Gurdon Institute, The Henry Wellcome Building of Cancer and Developmental Biology, University of Cambridge, Tennis Court Road, Cambridge CB2 1QN, UK
| | - Alphan Altinok
- Division of Biology, California Institute of Technology, Pasadena, CA 91125, USA; Biological Network Modeling Center, California Institute of Technology, Pasadena, CA 91125, USA
| | - M Azim Surani
- The Wellcome Trust/Cancer Research UK Gurdon Institute, The Henry Wellcome Building of Cancer and Developmental Biology, University of Cambridge, Tennis Court Road, Cambridge CB2 1QN, UK
| | - Long Cai
- Program in Biochemistry and Molecular Biophysics and Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
| | - Michael B Elowitz
- Howard Hughes Medical Institute and Division of Biology and Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA.
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Ohno M, Karagiannis P, Taniguchi Y. Protein expression analyses at the single cell level. Molecules 2014; 19:13932-47. [PMID: 25197931 PMCID: PMC6270791 DOI: 10.3390/molecules190913932] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2014] [Revised: 08/13/2014] [Accepted: 08/29/2014] [Indexed: 01/07/2023] Open
Abstract
The central dogma of molecular biology explains how genetic information is converted into its end product, proteins, which are responsible for the phenotypic state of the cell. Along with the protein type, the phenotypic state depends on the protein copy number. Therefore, quantification of the protein expression in a single cell is critical for quantitative characterization of the phenotypic states. Protein expression is typically a dynamic and stochastic phenomenon that cannot be well described by standard experimental methods. As an alternative, fluorescence imaging is being explored for the study of protein expression, because of its high sensitivity and high throughput. Here we review key recent progresses in fluorescence imaging-based methods and discuss their application to proteome analysis at the single cell level.
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Affiliation(s)
- Masae Ohno
- Laboratory for Single Cell Gene Dynamics, Quantitative Biology Center, RIKEN Address, 6-2-3 Furuedai, Suita, Osaka 565-0874, Japan
| | - Peter Karagiannis
- Laboratory for Single Cell Gene Dynamics, Quantitative Biology Center, RIKEN Address, 6-2-3 Furuedai, Suita, Osaka 565-0874, Japan
| | - Yuichi Taniguchi
- Laboratory for Single Cell Gene Dynamics, Quantitative Biology Center, RIKEN Address, 6-2-3 Furuedai, Suita, Osaka 565-0874, Japan.
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Yu H, Guo X, Wang J, Deng B, Wei X. Effects of spike-time-dependent plasticity on the stochastic resonance of small-world neuronal networks. CHAOS (WOODBURY, N.Y.) 2014; 24:033125. [PMID: 25273205 DOI: 10.1063/1.4893773] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
The phenomenon of stochastic resonance in Newman-Watts small-world neuronal networks is investigated when the strength of synaptic connections between neurons is adaptively adjusted by spike-time-dependent plasticity (STDP). It is shown that irrespective of the synaptic connectivity is fixed or adaptive, the phenomenon of stochastic resonance occurs. The efficiency of network stochastic resonance can be largely enhanced by STDP in the coupling process. Particularly, the resonance for adaptive coupling can reach a much larger value than that for fixed one when the noise intensity is small or intermediate. STDP with dominant depression and small temporal window ratio is more efficient for the transmission of weak external signal in small-world neuronal networks. In addition, we demonstrate that the effect of stochastic resonance can be further improved via fine-tuning of the average coupling strength of the adaptive network. Furthermore, the small-world topology can significantly affect stochastic resonance of excitable neuronal networks. It is found that there exists an optimal probability of adding links by which the noise-induced transmission of weak periodic signal peaks.
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Affiliation(s)
- Haitao Yu
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People's Republic of China
| | - Xinmeng Guo
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People's Republic of China
| | - Jiang Wang
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People's Republic of China
| | - Bin Deng
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People's Republic of China
| | - Xile Wei
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, People's Republic of China
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