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Szavits-Nossan J, Grima R. Solving stochastic gene-expression models using queueing theory: A tutorial review. Biophys J 2024; 123:1034-1057. [PMID: 38594901 PMCID: PMC11079947 DOI: 10.1016/j.bpj.2024.04.004] [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: 07/07/2023] [Revised: 02/12/2024] [Accepted: 04/02/2024] [Indexed: 04/11/2024] Open
Abstract
Stochastic models of gene expression are typically formulated using the chemical master equation, which can be solved exactly or approximately using a repertoire of analytical methods. Here, we provide a tutorial review of an alternative approach based on queueing theory that has rarely been used in the literature of gene expression. We discuss the interpretation of six types of infinite-server queues from the angle of stochastic single-cell biology and provide analytical expressions for the stationary and nonstationary distributions and/or moments of mRNA/protein numbers and bounds on the Fano factor. This approach may enable the solution of complex models that have hitherto evaded analytical solution.
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Affiliation(s)
- Juraj Szavits-Nossan
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom.
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2
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Berrocal A, Lammers NC, Garcia HG, Eisen MB. Kinetic sculpting of the seven stripes of the Drosophila even-skipped gene. eLife 2020; 9:61635. [PMID: 33300492 PMCID: PMC7864633 DOI: 10.7554/elife.61635] [Citation(s) in RCA: 17] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2020] [Accepted: 10/09/2020] [Indexed: 12/14/2022] Open
Abstract
We used live imaging to visualize the transcriptional dynamics of the Drosophila melanogaster even-skipped gene at single-cell and high-temporal resolution as its seven stripe expression pattern forms, and developed tools to characterize and visualize how transcriptional bursting varies over time and space. We find that despite being created by the independent activity of five enhancers, even-skipped stripes are sculpted by the same kinetic phenomena: a coupled increase of burst frequency and amplitude. By tracking the position and activity of individual nuclei, we show that stripe movement is driven by the exchange of bursting nuclei from the posterior to anterior stripe flanks. Our work provides a conceptual, theoretical and computational framework for dissecting pattern formation in space and time, and reveals how the coordinated transcriptional activity of individual nuclei shapes complex developmental patterns.
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Affiliation(s)
- Augusto Berrocal
- Department of Molecular & Cell Biology, University of California at Berkeley, Berkeley, United States
| | - Nicholas C Lammers
- Biophysics Graduate Group, University of California at Berkeley, Berkeley, United States
| | - Hernan G Garcia
- Department of Molecular & Cell Biology, University of California at Berkeley, Berkeley, United States.,Biophysics Graduate Group, University of California at Berkeley, Berkeley, United States.,Department of Physics, University of California at Berkeley, Berkeley, United States.,Institute for Quantitative Biosciences-QB3, University of California at Berkeley, Berkeley, United States
| | - Michael B Eisen
- Department of Molecular & Cell Biology, University of California at Berkeley, Berkeley, United States.,Biophysics Graduate Group, University of California at Berkeley, Berkeley, United States.,Institute for Quantitative Biosciences-QB3, University of California at Berkeley, Berkeley, United States.,Department of Integrative Biology, University of California at Berkeley, Berkeley, United States.,Howard Hughes Medical Institute, University of California at Berkeley, Berkeley, United States
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3
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Bakker R, Mani M, Carthew RW. The Wg and Dpp morphogens regulate gene expression by modulating the frequency of transcriptional bursts. eLife 2020; 9:e56076. [PMID: 32568073 PMCID: PMC7340504 DOI: 10.7554/elife.56076] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2020] [Accepted: 06/18/2020] [Indexed: 11/13/2022] Open
Abstract
Morphogen signaling contributes to the patterned spatiotemporal expression of genes during development. One mode of regulation of signaling-responsive genes is at the level of transcription. Single-cell quantitative studies of transcription have revealed that transcription occurs intermittently, in bursts. Although the effects of many gene regulatory mechanisms on transcriptional bursting have been studied, it remains unclear how morphogen gradients affect this dynamic property of downstream genes. Here we have adapted single molecule fluorescence in situ hybridization (smFISH) for use in the Drosophila wing imaginal disc in order to measure nascent and mature mRNA of genes downstream of the Wg and Dpp morphogen gradients. We compared our experimental results with predictions from stochastic models of transcription, which indicated that the transcription levels of these genes appear to share a common method of control via burst frequency modulation. Our data help further elucidate the link between developmental gene regulatory mechanisms and transcriptional bursting.
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Affiliation(s)
- Rachael Bakker
- Department of Molecular Biosciences, Northwestern UniversityEvanstonUnited States
- NSF-Simons Center for Quantitative Biology, Northwestern UniversityEvanstonUnited States
| | - Madhav Mani
- Department of Molecular Biosciences, Northwestern UniversityEvanstonUnited States
- NSF-Simons Center for Quantitative Biology, Northwestern UniversityEvanstonUnited States
- Department of Engineering Sciences and Applied Mathematics, Northwestern UniversityEvanstonUnited States
| | - Richard W Carthew
- Department of Molecular Biosciences, Northwestern UniversityEvanstonUnited States
- NSF-Simons Center for Quantitative Biology, Northwestern UniversityEvanstonUnited States
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4
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Binary Expression Enhances Reliability of Messaging in Gene Networks. ENTROPY 2020; 22:e22040479. [PMID: 33286254 PMCID: PMC7516962 DOI: 10.3390/e22040479] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/04/2020] [Revised: 04/20/2020] [Accepted: 04/20/2020] [Indexed: 01/31/2023]
Abstract
The promoter state of a gene and its expression levels are modulated by the amounts of transcription factors interacting with its regulatory regions. Hence, one may interpret a gene network as a communicating system in which the state of the promoter of a gene (the source) is communicated by the amounts of transcription factors that it expresses (the message) to modulate the state of the promoter and expression levels of another gene (the receptor). The reliability of the gene network dynamics can be quantified by Shannon's entropy of the message and the mutual information between the message and the promoter state. Here we consider a stochastic model for a binary gene and use its exact steady state solutions to calculate the entropy and mutual information. We show that a slow switching promoter with long and equally standing ON and OFF states maximizes the mutual information and reduces entropy. That is a binary gene expression regime generating a high variance message governed by a bimodal probability distribution with peaks of the same height. Our results indicate that Shannon's theory can be a powerful framework for understanding how bursty gene expression conciliates with the striking spatio-temporal precision exhibited in pattern formation of developing organisms.
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Lammers NC, Galstyan V, Reimer A, Medin SA, Wiggins CH, Garcia HG. Multimodal transcriptional control of pattern formation in embryonic development. Proc Natl Acad Sci U S A 2020; 117:836-847. [PMID: 31882445 PMCID: PMC6969519 DOI: 10.1073/pnas.1912500117] [Citation(s) in RCA: 47] [Impact Index Per Article: 11.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023] Open
Abstract
Predicting how interactions between transcription factors and regulatory DNA sequence dictate rates of transcription and, ultimately, drive developmental outcomes remains an open challenge in physical biology. Using stripe 2 of the even-skipped gene in Drosophila embryos as a case study, we dissect the regulatory forces underpinning a key step along the developmental decision-making cascade: the generation of cytoplasmic mRNA patterns via the control of transcription in individual cells. Using live imaging and computational approaches, we found that the transcriptional burst frequency is modulated across the stripe to control the mRNA production rate. However, we discovered that bursting alone cannot quantitatively recapitulate the formation of the stripe and that control of the window of time over which each nucleus transcribes even-skipped plays a critical role in stripe formation. Theoretical modeling revealed that these regulatory strategies (bursting and the time window) respond in different ways to input transcription factor concentrations, suggesting that the stripe is shaped by the interplay of 2 distinct underlying molecular processes.
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Affiliation(s)
| | - Vahe Galstyan
- Biochemistry and Molecular Biophysics Option, California Institute of Technology, Pasadena, CA 91126
- Department of Physics, Columbia University, New York, NY 10027
| | - Armando Reimer
- Biophysics Graduate Group, University of California, Berkeley, CA 94720
| | - Sean A Medin
- Department of Physics, University of California, Berkeley, CA 94720
| | - Chris H Wiggins
- Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027;
- Data Science Institute, Columbia University, New York, NY 10027
- Department of Systems Biology, Columbia University, New York, NY 10027
- Department of Statistics, Columbia University, New York, NY 10027
| | - Hernan G Garcia
- Biophysics Graduate Group, University of California, Berkeley, CA 94720;
- Department of Physics, University of California, Berkeley, CA 94720
- Department of Molecular and Cell Biology, University of California, Berkeley, CA 94720
- Institute for Quantitative Biosciences-QB3, University of California, Berkeley, CA 94720
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Lim B. Imaging transcriptional dynamics. Curr Opin Biotechnol 2018; 52:49-55. [PMID: 29501816 DOI: 10.1016/j.copbio.2018.02.008] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/06/2017] [Accepted: 02/11/2018] [Indexed: 01/02/2023]
Abstract
Recent advances in imaging techniques have enabled visualizations of nascent transcripts or individual protein molecules at high spatiotemporal resolution, revealing the complex nature of transcriptional regulation. Here, we highlight recent studies that have provided comprehensive insights to transcriptional dynamics using such quantitative imaging techniques. Specifically, they demonstrated that transcriptional activity is stochastic, and such transcriptional bursting is modulated by multiple components like chromatin environments, concentration of transcription factors, and enhancer-promoter interactions. Moreover, recent studies suggested that regulation of transcriptional activity is more complex than previously thought, by showing that transcription factors and RNA polymerases also move within the cell with distinct kinetics and sometimes form dynamic clusters to mediate transcriptional initiation.
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Affiliation(s)
- Bomyi Lim
- Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA.
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Wang H, Cheng X, Duan J, Kurths J, Li X. Likelihood for transcriptions in a genetic regulatory system under asymmetric stable Lévy noise. CHAOS (WOODBURY, N.Y.) 2018; 28:013121. [PMID: 29390613 DOI: 10.1063/1.5010026] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
This work is devoted to investigating the evolution of concentration in a genetic regulation system, when the synthesis reaction rate is under additive and multiplicative asymmetric stable Lévy fluctuations. By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that via examining the mean first exit time (MFET) and the first escape probability (FEP), the asymmetric fluctuations, interacting with nonlinearity in the system, lead to peculiar likelihood for transcription. This includes, in the additive noise case, realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index β, causing a stochastic bifurcation at the non-Gaussianity index value α = 1 (i.e., it is a separating point or line for the likelihood for transcription), and achieving a turning point at the threshold value β≈-0.5 (i.e., beyond which the likelihood for transcription suddenly reversed for α values). The stochastic bifurcation and turning point phenomena do not occur in the symmetric noise case (β = 0). While in the multiplicative noise case, non-Gaussianity index value α = 1 is a separating point or line for both the MFET and the FEP. We also investigate the noise enhanced stability phenomenon. Additionally, we are able to specify the regions in the whole parameter space for the asymmetric noise, in which we attain desired likelihood for transcription. We have conducted a series of numerical experiments in "regulating" the likelihood of gene transcription by tuning asymmetric stable Lévy noise indexes. This work offers insights for possible ways of achieving gene regulation in experimental research.
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Affiliation(s)
- Hui Wang
- Center for Mathematical Sciences and School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Xiujun Cheng
- Center for Mathematical Sciences and School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Jinqiao Duan
- Center for Mathematical Sciences and School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Jürgen Kurths
- Department of Physics, Humboldt University of Berlin, Newtonstrate 15, 12489 Berlin, Germany
| | - Xiaofan Li
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA
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