1
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Nieto C, Vargas-García CA, Pedraza JM, Singh A. Mechanisms of cell size regulation in slow-growing Escherichia coli cells: discriminating models beyond the adder. NPJ Syst Biol Appl 2024; 10:61. [PMID: 38811603 PMCID: PMC11137094 DOI: 10.1038/s41540-024-00383-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/11/2023] [Accepted: 05/09/2024] [Indexed: 05/31/2024] Open
Abstract
Under ideal conditions, Escherichia coli cells divide after adding a fixed cell size, a strategy known as the adder. This concept applies to various microbes and is often explained as the division that occurs after a certain number of stages, associated with the accumulation of precursor proteins at a rate proportional to cell size. However, under poor media conditions, E. coli cells exhibit a different size regulation. They are smaller and follow a sizer-like division strategy where the added size is inversely proportional to the size at birth. We explore three potential causes for this deviation: degradation of the precursor protein and two models where the propensity for accumulation depends on the cell size: a nonlinear accumulation rate, and accumulation starting at a threshold size termed the commitment size. These models fit the mean trends but predict different distributions given the birth size. To quantify the precision of the models to explain the data, we used the Akaike information criterion and compared them to open datasets of slow-growing E. coli cells in different media. We found that none of the models alone can consistently explain the data. However, the degradation model better explains the division strategy when cells are larger, whereas size-related models (power-law and commitment size) account for smaller cells. Our methodology proposes a data-based method in which different mechanisms can be tested systematically.
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Affiliation(s)
- César Nieto
- Department of Physics, Universidad de los Andes, Bogotá, Colombia
- Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, 19716, USA
| | | | | | - Abhyudai Singh
- Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, 19716, USA.
- Department of Electrical and Computer Engineering, Biomedical Engineering, Mathematical Sciences, Center of Bioinformatic and Computational Biology, University of Delaware, Newark, DE, 19716, USA.
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2
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ElGamel M, Mugler A. Effects of Molecular Noise on Cell Size Control. PHYSICAL REVIEW LETTERS 2024; 132:098403. [PMID: 38489620 DOI: 10.1103/physrevlett.132.098403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2023] [Accepted: 02/12/2024] [Indexed: 03/17/2024]
Abstract
Cells employ control strategies to maintain a stable size. Dividing at a target size (the "sizer" strategy) is thought to produce the tightest size distribution. However, this result follows from phenomenological models that ignore the molecular mechanisms required to implement the strategy. Here we investigate a simple mechanistic model for exponentially growing cells whose division is triggered at a molecular abundance threshold. We find that size noise inherits the molecular noise and is consequently minimized not by the sizer but by the "adder" strategy, where a cell divides after adding a target amount to its birth size. We derive a lower bound on size noise that agrees with publicly available data from six microfluidic studies on Escherichia coli bacteria.
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Affiliation(s)
- Motasem ElGamel
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
| | - Andrew Mugler
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
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3
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Nieto C, Täuber S, Blöbaum L, Vahdat Z, Grünberger A, Singh A. Coupling Cell Size Regulation and Proliferation Dynamics of C. glutamicum Reveals Cell Division Based on Surface Area. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2023:2023.12.26.573217. [PMID: 38234762 PMCID: PMC10793411 DOI: 10.1101/2023.12.26.573217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/19/2024]
Abstract
Single cells actively coordinate growth and division to regulate their size, yet how this size homeostasis at the single-cell level propagates over multiple generations to impact clonal expansion remains fundamentally unexplored. Classical timer models for cell proliferation (where the duration of the cell cycle is an independent variable) predict that the stochastic variation in colony size will increase monotonically over time. In stark contrast, implementing size control according to adder strategy (where on average a fixed size added from cell birth to division) leads to colony size variations that eventually decay to zero. While these results assume a fixed size of the colony-initiating progenitor cell, further analysis reveals that the magnitude of the intercolony variation in population number is sensitive to heterogeneity in the initial cell size. We validate these predictions by tracking the growth of isogenic microcolonies of Corynebacterium glutamicum in microfluidic chambers. Approximating their cell shape to a capsule, we observe that the degree of random variability in cell size is different depending on whether the cell size is quantified as per length, surface area, or volume, but size control remains an adder regardless of these size metrics. A comparison of the observed variability in the colony population with the predictions suggests that proliferation matches better with a cell division based on the cell surface. In summary, our integrated mathematical-experimental approach bridges the paradigms of single-cell size regulation and clonal expansion at the population levels. This innovative approach provides elucidation of the mechanisms of size homeostasis from the stochastic dynamics of colony size for rod-shaped microbes.
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Affiliation(s)
- César Nieto
- Department of Electrical and Computing Engineering, University of Delaware. Newark, DE 19716, USA
| | - Sarah Täuber
- CeBiTec, Bielefeld University. Bielefeld, Germany
- Multiscale Bioengineering, Technical Faculty, Bielefeld University. Bielefeld, Germany
| | - Luisa Blöbaum
- CeBiTec, Bielefeld University. Bielefeld, Germany
- Multiscale Bioengineering, Technical Faculty, Bielefeld University. Bielefeld, Germany
| | - Zahra Vahdat
- Department of Electrical and Computing Engineering, University of Delaware. Newark, DE 19716, USA
| | - Alexander Grünberger
- CeBiTec, Bielefeld University. Bielefeld, Germany
- Multiscale Bioengineering, Technical Faculty, Bielefeld University. Bielefeld, Germany
- Institute of Process Engineering in Life Sciences: Microsystems in Bioprocess Engineering, Karlsruhe Institute of Technology. Karlsruhe, Germany
| | - Abhyudai Singh
- Department of Electrical and Computing Engineering, University of Delaware. Newark, DE 19716, USA
- Center for Bioinformatics and Computational Biology, University of Delaware, Newark, DE 19716 USA
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4
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Nieto C, Blanco SC, Vargas-García C, Singh A, Manuel PJ. PyEcoLib: a python library for simulating stochastic cell size dynamics. Phys Biol 2023; 20:10.1088/1478-3975/acd897. [PMID: 37224818 PMCID: PMC10665115 DOI: 10.1088/1478-3975/acd897] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2022] [Accepted: 05/24/2023] [Indexed: 05/26/2023]
Abstract
Recently, there has been an increasing need for tools to simulate cell size regulation due to important applications in cell proliferation and gene expression. However, implementing the simulation usually presents some difficulties, as the division has a cycle-dependent occurrence rate. In this article, we gather a recent theoretical framework inPyEcoLib, a python-based library to simulate the stochastic dynamics of the size of bacterial cells. This library can simulate cell size trajectories with an arbitrarily small sampling period. In addition, this simulator can include stochastic variables, such as the cell size at the beginning of the experiment, the cycle duration timing, the growth rate, and the splitting position. Furthermore, from a population perspective, the user can choose between tracking a single lineage or all cells in a colony. They can also simulate the most common division strategies (adder, timer, and sizer) using the division rate formalism and numerical methods. As an example of PyecoLib applications, we explain how to couple size dynamics with gene expression predicting, from simulations, how the noise in protein levels increases by increasing the noise in division timing, the noise in growth rate and the noise in cell splitting position. The simplicity of this library and its transparency about the underlying theoretical framework yield the inclusion of cell size stochasticity in complex models of gene expression.
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Affiliation(s)
- César Nieto
- Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, United States of America
- Department of Physics. Universidad de los Andes, Bogotá, Colombia
| | - Sergio Camilo Blanco
- Department of Mathematics and Engineering. Fundacion Universitaria Konrad Lorenz, Bogota, Colombia
| | | | - Abhyudai Singh
- Department of Electrical and Computer Engineering, Department of Biomedical Engineering and Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, United States of America
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5
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Nieto C, Vargas-García C, Pedraza JM. Continuous rate modeling of bacterial stochastic size dynamics. Phys Rev E 2021; 104:044415. [PMID: 34781449 DOI: 10.1103/physreve.104.044415] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/24/2021] [Accepted: 10/06/2021] [Indexed: 12/26/2022]
Abstract
Bacterial division is an inherently stochastic process with effects on fluctuations of protein concentration and phenotype variability. Current modeling tools for the stochastic short-term cell-size dynamics are scarce and mainly phenomenological. Here we present a general theoretical approach based on the Chapman-Kolmogorov equation incorporating continuous growth and division events as jump processes. This approach allows us to include different division strategies, noisy growth, and noisy cell splitting. Considering bacteria synchronized from their last division, we predict oscillations in both the central moments of the size distribution and its autocorrelation function. These oscillations, barely discussed in past studies, can arise as a consequence of the discrete time displacement invariance of the system with a period of one doubling time, and they do not disappear when including stochasticity on either division times or size heterogeneity on the starting population but only after inclusion of noise in either growth rate or septum position. This result illustrates the usefulness of having a solid mathematical description that explicitly incorporates the inherent stochasticity in various biological processes, both to understand the process in detail and to evaluate the effect of various sources of variability when creating simplified descriptions.
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Affiliation(s)
- César Nieto
- Department of Physics, Universidad de los Andes, Bogotá 111711, Colombia.,Department of Electrical and Computer Engineering, University of Delaware, Newark, Delaware 19716, USA
| | - César Vargas-García
- Corporacion Colombiana de Investigación Agropecuaria AGROSAVIA, Mosquera 250047, Colombia
| | - Juan M Pedraza
- Department of Physics, Universidad de los Andes, Bogotá 111711, Colombia
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6
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The effect of natural selection on the propagation of protein expression noise to bacterial growth. PLoS Comput Biol 2021; 17:e1009208. [PMID: 34280182 PMCID: PMC8321134 DOI: 10.1371/journal.pcbi.1009208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/25/2020] [Revised: 07/29/2021] [Accepted: 06/22/2021] [Indexed: 11/19/2022] Open
Abstract
In bacterial cells, protein expression is a highly stochastic process. Gene expression noise moreover propagates through the cell and adds to fluctuations in the cellular growth rate. A common intuition is that, due to their relatively high noise amplitudes, proteins with a low mean expression level are the most important drivers of fluctuations in physiological variables. In this work, we challenge this intuition by considering the effect of natural selection on noise propagation. Mathematically, the contribution of each protein species to the noise in the growth rate depends on two factors: the noise amplitude of the protein’s expression level, and the sensitivity of the growth rate to fluctuations in that protein’s concentration. We argue that natural selection, while shaping mean abundances to increase the mean growth rate, also affects cellular sensitivities. In the limit in which cells grow optimally fast, the growth rate becomes most sensitive to fluctuations in highly abundant proteins. This causes abundant proteins to overall contribute strongly to the noise in the growth rate, despite their low noise levels. We further explore this result in an experimental data set of protein abundances, and test key assumptions in an evolving, stochastic toy model of cellular growth. Gene expression in bacterial cells is intrinsically stochastic: copy numbers of all proteins vary between genetically identical cells, even in a homogeneous environment. Such noise in gene expression affects metabolic fluxes and even propagates to the cellular level. Indeed, also the growth rate of individual cells, an important proxy for fitness, fluctuates strongly. This beckons the question as to which protein species contribute most to the noise in the cell’s growth rate. We here derive general mathematical predictions stating that if cells have been optimised by evolution to grow fast, abundant protein species contribute most to the noise in the growth rate. This result is counter-intuitive, because the noise levels in the expression of those highly expressed proteins are small.
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7
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Jia C, Singh A, Grima R. Cell size distribution of lineage data: analytic results and parameter inference. iScience 2021; 24:102220. [PMID: 33748708 PMCID: PMC7961097 DOI: 10.1016/j.isci.2021.102220] [Citation(s) in RCA: 14] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/28/2020] [Revised: 01/29/2021] [Accepted: 02/17/2021] [Indexed: 01/06/2023] Open
Abstract
Recent advances in single-cell technologies have enabled time-resolved measurements of the cell size over several cell cycles. These data encode information on how cells correct size aberrations so that they do not grow abnormally large or small. Here, we formulate a piecewise deterministic Markov model describing the evolution of the cell size over many generations, for all three cell size homeostasis strategies (timer, sizer, and adder). The model is solved to obtain an analytical expression for the non-Gaussian cell size distribution in a cell lineage; the theory is used to understand how the shape of the distribution is influenced by the parameters controlling the dynamics of the cell cycle and by the choice of cell tracking protocol. The theoretical cell size distribution is found to provide an excellent match to the experimental cell size distribution of E. coli lineage data collected under various growth conditions.
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Affiliation(s)
- Chen Jia
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China
| | - Abhyudai Singh
- Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, EH9 3JH, UK
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8
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Vasdekis AE, Singh A. Microbial metabolic noise. WIREs Mech Dis 2020; 13:e1512. [PMID: 33225608 DOI: 10.1002/wsbm.1512] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/31/2019] [Revised: 09/23/2020] [Accepted: 10/26/2020] [Indexed: 11/06/2022]
Abstract
From the time a cell was first placed under the microscope, it became apparent that identifying two clonal cells that "look" identical is extremely challenging. Since then, cell-to-cell differences in shape, size, and protein content have been carefully examined, informing us of the ultimate limits that hinder two cells from occupying an identical phenotypic state. Here, we present recent experimental and computational evidence that similar limits emerge also in cellular metabolism. These limits pertain to stochastic metabolic dynamics and, thus, cell-to-cell metabolic variability, including the resulting adapting benefits. We review these phenomena with a focus on microbial metabolism and conclude with a brief outlook on the potential relationship between metabolic noise and adaptive evolution. This article is categorized under: Metabolic Diseases > Computational Models Metabolic Diseases > Biomedical Engineering.
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Affiliation(s)
| | - Abhyudai Singh
- Department of Electrical and Computer Engineering, University of Delaware, Newark, Delaware, USA
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9
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Vargas-Garcia CA, Björklund M, Singh A. Modeling homeostasis mechanisms that set the target cell size. Sci Rep 2020; 10:13963. [PMID: 32811891 PMCID: PMC7434900 DOI: 10.1038/s41598-020-70923-0] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/26/2020] [Accepted: 08/03/2020] [Indexed: 11/09/2022] Open
Abstract
How organisms maintain cell size homeostasis is a long-standing problem that remains unresolved, especially in multicellular organisms. Recent experiments in diverse animal cell types demonstrate that within a cell population, cellular proliferation is low for small and large cells, but high at intermediate sizes. Here we use mathematical models to explore size-control strategies that drive such a non-monotonic profile resulting in the proliferation capacity being maximized at a target cell size. Our analysis reveals that most models of size control yield proliferation capacities that vary monotonically with cell size, and non-monotonicity requires two key mechanisms: (1) the growth rate decreases with increasing size for excessively large cells; and (2) cell division occurs as per the Adder model (division is triggered upon adding a fixed size from birth), or a Sizer-Adder combination. Consistent with theory, Jurkat T cell growth rates increase with size for small cells, but decrease with size for large cells. In summary, our models show that regulation of both growth and cell-division timing is necessary for size control in animal cells, and this joint mechanism leads to a target cell size where cellular proliferation capacity is maximized.
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Affiliation(s)
- Cesar A Vargas-Garcia
- Corporación Colombiana de Investigación Agropecuaria-Agrosavia, Mosquera, Colombia.
- Fundación Universitaria Konrad Lorenz, Bogotá, Colombia.
| | - Mikael Björklund
- Zhejiang University-University of Edinburgh (ZJU-UoE) Institute, 718 East Haizhou Rd., Haining, 314400, Zhejiang, People's Republic of China
- Second Affiliated Hospital, Zhejiang University School of Medicine, Hangzhou, Zhejiang, People's Republic of China
| | - Abhyudai Singh
- Department of Biomedical Engineering, University of Delaware, Newark, Delaware, USA
- Department of Electrical and Computer Engineering, University of Delaware, Newark, Delaware, USA
- Department of Mathematical Sciences, University of Delaware, Newark, Delaware, USA
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10
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Nieto-Acuña C, Arias-Castro JC, Vargas-García C, Sánchez C, Pedraza JM. Correlation between protein concentration and bacterial cell size can reveal mechanisms of gene expression. Phys Biol 2020; 17:045002. [PMID: 32289764 DOI: 10.1088/1478-3975/ab891c] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2022]
Abstract
Classically, gene expression is modeled as a chemical process with reaction rates dependent on the concentration of the reactants (typically, DNA loci, plasmids, RNA, enzymes, etc). Other variables like cell size are in general ignored. Size dynamics can become an important variable due to the low number of many of these reactants, imperfectly symmetric cell partitioning and molecule segregation. In this work we measure the correlation between size and protein concentration by observing the gene expression of the RpOD gene from a low-copy plasmid in Escherichia coli during balanced growth in different media. A positive correlation was found, and we used it to examine possible models of cell size dynamics and plasmid replication. We implemented a previously developed model describing the full gene expression process including transcription, translation, loci replication, cell division and molecule segregation. By comparing with the observed correlation, we determine that the transcription rate must be proportional to the size times the number of plasmids. We discuss how fluctuations in plasmid segregation, due to the low copy number, can impose limits in this correlation.
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Affiliation(s)
| | - Juan Carlos Arias-Castro
- Department of Physics, Universidad de los Andes, Bogotá, Colombia.,Department of Systems Biology, Harvard Medical School, Boston, Massachusetts 02115, United States of America
| | - César Vargas-García
- Department of Mathematics and Engineering, Fundación universitaria Konrad Lorenz, Bogota, Colombia.,AGROSAVIA, Corporación Colombiana de Investigación Agropecuaria, Mosquera, Bogotá, Colombia
| | - Carlos Sánchez
- Department of Physics, Universidad de los Andes, Bogotá, Colombia.,Department of Systems Biology, Harvard Medical School, Boston, Massachusetts 02115, United States of America
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11
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Zatulovskiy E, Skotheim JM. On the Molecular Mechanisms Regulating Animal Cell Size Homeostasis. Trends Genet 2020; 36:360-372. [PMID: 32294416 PMCID: PMC7162994 DOI: 10.1016/j.tig.2020.01.011] [Citation(s) in RCA: 35] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/04/2019] [Revised: 01/28/2020] [Accepted: 01/28/2020] [Indexed: 12/19/2022]
Abstract
Cell size is fundamental to cell physiology because it sets the scale of intracellular geometry, organelles, and biosynthetic processes. In animal cells, size homeostasis is controlled through two phenomenologically distinct mechanisms. First, size-dependent cell cycle progression ensures that smaller cells delay cell cycle progression to accumulate more biomass than larger cells prior to cell division. Second, size-dependent cell growth ensures that larger and smaller cells grow slower per unit mass than more optimally sized cells. This decade has seen dramatic progress in single-cell technologies establishing the diverse phenomena of cell size control in animal cells. Here, we review this recent progress and suggest pathways forward to determine the underlying molecular mechanisms.
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Affiliation(s)
| | - Jan M Skotheim
- Department of Biology, Stanford University, Stanford, CA 94305, USA.
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12
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Xia M, Greenman CD, Chou T. PDE MODELS OF ADDER MECHANISMS IN CELLULAR PROLIFERATION. SIAM JOURNAL ON APPLIED MATHEMATICS 2020; 80:1307-1335. [PMID: 35221385 PMCID: PMC8871769 DOI: 10.1137/19m1246754] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/08/2023]
Abstract
Cell division is a process that involves many biochemical steps and complex biophysical mechanisms. To simplify the understanding of what triggers cell division, three basic models that subsume more microscopic cellular processes associated with cell division have been proposed. Cells can divide based on the time elapsed since their birth, their size, and/or the volume added since their birth-the timer, sizer, and adder models, respectively. Here, we propose unified adder-sizer models and investigate some of the properties of different adder processes arising in cellular proliferation. Although the adder-sizer model provides a direct way to model cell population structure, we illustrate how it is mathematically related to the well-known model in which cell division depends on age and size. Existence and uniqueness of weak solutions to our 2+1-dimensional PDE model are proved, leading to the convergence of the discretized numerical solutions and allowing us to numerically compute the dynamics of cell population densities. We then generalize our PDE model to incorporate recent experimental findings of a system exhibiting mother-daughter correlations in cellular growth rates. Numerical experiments illustrating possible average cell volume blowup and the dynamical behavior of cell populations with mother-daughter correlated growth rates are carried out. Finally, motivated by new experimental findings, we extend our adder model cases where the controlling variable is the added size between DNA replication initiation points in the cell cycle.
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Affiliation(s)
- Mingtao Xia
- Department of Mathematics, UCLA, Los Angeles, CA 90095-1555
| | - Chris D Greenman
- School of Computing Sciences, University of East Anglia, Norwich, UK NR4 7TJ
| | - Tom Chou
- Departments of Computational Medicine and Mathematics, UCLA, Los Angeles, CA 90095-1766
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13
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Nieto-Acuna CA, Vargas-Garcia CA, Singh A, Pedraza JM. Efficient computation of stochastic cell-size transient dynamics. BMC Bioinformatics 2019; 20:647. [PMID: 31881826 PMCID: PMC6933677 DOI: 10.1186/s12859-019-3213-7] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/09/2019] [Accepted: 11/12/2019] [Indexed: 12/16/2022] Open
Abstract
BACKGROUND How small, fast-growing bacteria ensure tight cell-size distributions remains elusive. High-throughput measurement techniques have propelled efforts to build modeling tools that help to shed light on the relationships between cell size, growth and cycle progression. Most proposed models describe cell division as a discrete map between size at birth and size at division with stochastic fluctuations assumed. However, such models underestimate the role of cell size transient dynamics by excluding them. RESULTS We propose an efficient approach for estimation of cell size transient dynamics. Our technique approximates the transient size distribution and statistical moment dynamics of exponential growing cells following an adder strategy with arbitrary precision. CONCLUSIONS We approximate, up to arbitrary precision, the distribution of division times and size across time for the adder strategy in rod-shaped bacteria cells. Our approach is able to compute statistical moments like mean size and its variance from such distributions efficiently, showing close match with numerical simulations. Additionally, we observed that these distributions have periodic properties. Our approach further might shed light on the mechanisms behind gene product homeostasis.
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Affiliation(s)
| | - Cesar Augusto Vargas-Garcia
- Mathematics and Engineering department, Fundación universitaria Konrad Lorenz, Bogotá, South America Colombia
| | - Abhyudai Singh
- Electrical and Compute Enginering Department, University of Delaware, Newark, Delaware USA
| | - Juan Manuel Pedraza
- Physics department, Universidad de los Andes, Bogotá, South America Colombia
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14
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Lee S, Wu LJ, Errington J. Microfluidic time-lapse analysis and reevaluation of the Bacillus subtilis cell cycle. Microbiologyopen 2019; 8:e876. [PMID: 31197963 PMCID: PMC6813450 DOI: 10.1002/mbo3.876] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2019] [Revised: 05/03/2019] [Accepted: 05/13/2019] [Indexed: 12/24/2022] Open
Abstract
Recent studies taking advantage of automated single-cell time-lapse analysis have reignited interest in the bacterial cell cycle. Several studies have highlighted alternative models, such as Sizer and Adder, which differ essentially in relation to whether cells can measure their present size or their amount of growth since birth. Most of the recent work has been done with Escherichia coli. We set out to study the well-characterized Gram-positive bacterium, Bacillus subtilis, at the single-cell level, using an accurate fluorescent marker for division as well as a marker for completion of chromosome replication. Our results are consistent with the Adder model in both fast and slow growth conditions tested, and with Sizer but only at the slower growth rate. We also find that cell size variation arises not only from the expected variation in size at division but also that division site offset from mid-cell contributes to a significant degree. Finally, although traditional cell cycle models imply a strong connection between the termination of a round of replication and subsequent division, we find that at the single-cell level these events are largely disconnected.
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Affiliation(s)
- Seoungjun Lee
- Centre for Bacterial Cell Biology, Institute for Cell and Molecular Biosciences, Medical SchoolNewcastle UniversityNewcastle‐upon‐TyneUK
- Present address:
Department of Basic and Clinical Neuroscience, Maurice Wohl Clinical Neuroscience Institute, Institute of Psychiatry, Psychology and NeuroscienceKing’s College LondonLondonUK
| | - Ling Juan Wu
- Centre for Bacterial Cell Biology, Institute for Cell and Molecular Biosciences, Medical SchoolNewcastle UniversityNewcastle‐upon‐TyneUK
| | - Jeff Errington
- Centre for Bacterial Cell Biology, Institute for Cell and Molecular Biosciences, Medical SchoolNewcastle UniversityNewcastle‐upon‐TyneUK
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15
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Modi S, Singh A. Controlling organism size by regulating constituent cell numbers. PROCEEDINGS OF THE ... IEEE CONFERENCE ON DECISION & CONTROL. IEEE CONFERENCE ON DECISION & CONTROL 2019; 2018:2685-2690. [PMID: 30886453 DOI: 10.1109/cdc.2018.8619546] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
How living cells employ counting mechanisms to regulate their numbers or density is a long-standing problem in developmental biology that ties directly with organism or tissue size. Diverse cells types have been shown to regulate their numbers via secretion of factors in the extracellular space. These factors act as a proxy for the number of cells and function to reduce cellular proliferation rates creating a negative feedback. It is desirable that the production rate of such factors be kept as low as possible to minimize energy costs and detection by predators. Here we formulate a stochastic model of cell proliferation with feedback control via a secreted extracellular factor. Our results show that while low levels of feedback minimizes random fluctuations in cell numbers around a given set point, high levels of feedback amplify Poisson fluctuations in secreted-factor copy numbers. This trade-off results in an optimal feedback strength, and sets a fundamental limit to noise suppression in cell numbers with short-lived factors providing more efficient noise buffering. We further expand the model to consider external disturbances in key physiological parameters, such as, proliferation and factor synthesis rates. Intriguingly, while negative feedback effectively mitigates disturbances in the proliferation rate, it amplifies disturbances in the synthesis rate. In summary, these results provide unique insights into the functioning of feedback-based counting mechanisms, and apply to organisms ranging from unicellular prokaryotes and eukaryotes to human cells.
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Affiliation(s)
- Saurabh Modi
- Department of Biomedical Engineering, University of Delaware, Newark, DE USA 19716.
| | - Abhyudai Singh
- Department of Electrical and Computer Engineering, Biomedical Engineering, Mathematical Sciences, Center for Bioinformatics and Computational Biology, University of Delaware, Newark, DE USA 19716.
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16
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Martins BMC, Tooke AK, Thomas P, Locke JCW. Cell size control driven by the circadian clock and environment in cyanobacteria. Proc Natl Acad Sci U S A 2018; 115:E11415-E11424. [PMID: 30409801 PMCID: PMC6275512 DOI: 10.1073/pnas.1811309115] [Citation(s) in RCA: 31] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022] Open
Abstract
How cells maintain their size has been extensively studied under constant conditions. In the wild, however, cells rarely experience constant environments. Here, we examine how the 24-h circadian clock and environmental cycles modulate cell size control and division timings in the cyanobacterium Synechococcus elongatus using single-cell time-lapse microscopy. Under constant light, wild-type cells follow an apparent sizer-like principle. Closer inspection reveals that the clock generates two subpopulations, with cells born in the subjective day following different division rules from cells born in subjective night. A stochastic model explains how this behavior emerges from the interaction of cell size control with the clock. We demonstrate that the clock continuously modulates the probability of cell division throughout day and night, rather than solely applying an on-off gate to division, as previously proposed. Iterating between modeling and experiments, we go on to identify an effective coupling of the division rate to time of day through the combined effects of the environment and the clock on cell division. Under naturally graded light-dark cycles, this coupling narrows the time window of cell divisions and shifts divisions away from when light levels are low and cell growth is reduced. Our analysis allows us to disentangle, and predict the effects of, the complex interactions between the environment, clock, and cell size control.
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Affiliation(s)
- Bruno M C Martins
- Sainsbury Laboratory, University of Cambridge, CB2 1LR Cambridge, United Kingdom
| | - Amy K Tooke
- Sainsbury Laboratory, University of Cambridge, CB2 1LR Cambridge, United Kingdom
| | - Philipp Thomas
- Department of Mathematics, Imperial College London, SW7 2AZ London, United Kingdom
| | - James C W Locke
- Sainsbury Laboratory, University of Cambridge, CB2 1LR Cambridge, United Kingdom;
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17
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Martins BMC, Tooke AK, Thomas P, Locke JCW. Cell size control driven by the circadian clock and environment in cyanobacteria. Proc Natl Acad Sci U S A 2018. [PMID: 30409801 DOI: 10.1002/(sici)1521-1878(200001)22:1¡10::aid-bies4¿3.0.co;2-a] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 04/29/2023] Open
Abstract
How cells maintain their size has been extensively studied under constant conditions. In the wild, however, cells rarely experience constant environments. Here, we examine how the 24-h circadian clock and environmental cycles modulate cell size control and division timings in the cyanobacterium Synechococcus elongatus using single-cell time-lapse microscopy. Under constant light, wild-type cells follow an apparent sizer-like principle. Closer inspection reveals that the clock generates two subpopulations, with cells born in the subjective day following different division rules from cells born in subjective night. A stochastic model explains how this behavior emerges from the interaction of cell size control with the clock. We demonstrate that the clock continuously modulates the probability of cell division throughout day and night, rather than solely applying an on-off gate to division, as previously proposed. Iterating between modeling and experiments, we go on to identify an effective coupling of the division rate to time of day through the combined effects of the environment and the clock on cell division. Under naturally graded light-dark cycles, this coupling narrows the time window of cell divisions and shifts divisions away from when light levels are low and cell growth is reduced. Our analysis allows us to disentangle, and predict the effects of, the complex interactions between the environment, clock, and cell size control.
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Affiliation(s)
- Bruno M C Martins
- Sainsbury Laboratory, University of Cambridge, CB2 1LR Cambridge, United Kingdom
| | - Amy K Tooke
- Sainsbury Laboratory, University of Cambridge, CB2 1LR Cambridge, United Kingdom
| | - Philipp Thomas
- Department of Mathematics, Imperial College London, SW7 2AZ London, United Kingdom
| | - James C W Locke
- Sainsbury Laboratory, University of Cambridge, CB2 1LR Cambridge, United Kingdom;
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18
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Collier J. Cell division control in Caulobacter crescentus. BIOCHIMICA ET BIOPHYSICA ACTA-GENE REGULATORY MECHANISMS 2018; 1862:685-690. [PMID: 29715525 DOI: 10.1016/j.bbagrm.2018.04.005] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/14/2018] [Revised: 04/26/2018] [Accepted: 04/26/2018] [Indexed: 12/31/2022]
Abstract
Caulobacter crescentus is a free-living Alphaproteobacterium that thrives in oligotrophic environments. This review focuses on the regulatory network used by this bacterium to control the levels of cell division proteins, their organization inside the cell and their activity as a function of the cell cycle. Strikingly, C. crescentus makes frequent use of master transcriptional regulators and epigenetic signals, most likely to synchronize cell division with other events of the cell cycle. In addition, cellular metabolism and DNA damage sensors emerge as central players regulating cell division in response to changing environmental conditions.
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Affiliation(s)
- Justine Collier
- Department of Fundamental Microbiology, Faculty of Biology and Medicine, University of Lausanne, Quartier UNIL/Sorge, Lausanne, CH 1015, Switzerland.
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19
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Vargas–Garcia CA, Ghusinga KR, Singh A. Cell size control and gene expression homeostasis in single-cells. CURRENT OPINION IN SYSTEMS BIOLOGY 2018; 8:109-116. [PMID: 29862376 PMCID: PMC5978733 DOI: 10.1016/j.coisb.2018.01.002] [Citation(s) in RCA: 46] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/06/2023]
Abstract
Growth of a cell and its subsequent division into daughters is a fundamental aspect of all cellular living systems. During these processes, how do individual cells correct size aberrations so that they do not grow abnormally large or small? How do cells ensure that the concentration of essential gene products are maintained at desired levels, in spite of dynamic/stochastic changes in cell size during growth and division? Both these questions have fascinated researchers for over a century. We review how advances in singe-cell technologies and measurements are providing unique insights into these questions across organisms from prokaryotes to human cells. More specifically, diverse strategies based on timing of cell-cycle events, regulating growth, and number of daughters are employed to maintain cell size homeostasis. Interestingly, size homeostasis often results in size optimality - proliferation of individual cells in a population is maximized at an optimal cell size. We further discuss how size-dependent expression or gene-replication timing can buffer concentration of a gene product from cell-to-cell size variations within a population. Finally, we speculate on an intriguing hypothesis that specific size control strategies may have evolved as a consequence of gene-product concentration homeostasis.
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Affiliation(s)
- Cesar A. Vargas–Garcia
- Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, USA
| | - Khem Raj Ghusinga
- Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, USA
| | - Abhyudai Singh
- Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, USA
- Department of Biomedical Engineering, University of Delaware, Newark, DE, USA
- Department of Mathematical Sciences, University of Delaware, Newark, DE, USA
- Center for Applications of Mathematics in Medicine, University of Delaware, Newark, DE, USA
- Center for Bioinformatics and Computational Biology, University of Delaware, Newark, DE, USA
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20
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Abstract
For many decades, the wedding of quantitative data with mathematical modeling has been fruitful, leading to important biological insights. Here, we review some of the ongoing efforts to gain insights into problems in microbiology - and, in particular, cell-cycle progression and its regulation - through observation and quantitative analysis of the natural fluctuations in the system. We first illustrate this idea by reviewing a classic example in microbiology - the Luria-Delbrück experiment - and discussing how, in that case, useful information was obtained by looking beyond the mean outcome of the experiment, but instead paying attention to the variability between replicates of the experiment. We then switch gears to the contemporary problem of cell cycle progression and discuss in more detail how insights into cell size regulation and, when relevant, coupling between the cell cycle and the circadian clock, can be gained by studying the natural fluctuations in the system and their statistical properties. We end with a more general discussion of how (in this context) the correct level of phenomenological model should be chosen, as well as some of the pitfalls associated with this type of analysis. Throughout this review the emphasis is not on providing details of the experimental setups or technical details of the models used, but rather, in fleshing out the conceptual structure of this particular approach to the problem. For this reason, we choose to illustrate the framework on a rather broad range of problems, and on organisms from all domains of life, to emphasize the commonality of the ideas and analysis used (as well as their differences).
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Affiliation(s)
- Ariel Amir
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA.
| | - Nathalie Q Balaban
- The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel
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21
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Bertaux F, Marguerat S, Shahrezaei V. Division rate, cell size and proteome allocation: impact on gene expression noise and implications for the dynamics of genetic circuits. ROYAL SOCIETY OPEN SCIENCE 2018; 5:172234. [PMID: 29657814 PMCID: PMC5882738 DOI: 10.1098/rsos.172234] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/05/2018] [Accepted: 02/15/2018] [Indexed: 05/12/2023]
Abstract
The cell division rate, size and gene expression programmes change in response to external conditions. These global changes impact on average concentrations of biomolecule and their variability or noise. Gene expression is inherently stochastic, and noise levels of individual proteins depend on synthesis and degradation rates as well as on cell-cycle dynamics. We have modelled stochastic gene expression inside growing and dividing cells to study the effect of division rates on noise in mRNA and protein expression. We use assumptions and parameters relevant to Escherichia coli, for which abundant quantitative data are available. We find that coupling of transcription, but not translation rates to the rate of cell division can result in protein concentration and noise homeostasis across conditions. Interestingly, we find that the increased cell size at fast division rates, observed in E. coli and other unicellular organisms, buffers noise levels even for proteins with decreased expression at faster growth. We then investigate the functional importance of these regulations using gene regulatory networks that exhibit bi-stability and oscillations. We find that network topology affects robustness to changes in division rate in complex and unexpected ways. In particular, a simple model of persistence, based on global physiological feedback, predicts increased proportion of persister cells at slow division rates. Altogether, our study reveals how cell size regulation in response to cell division rate could help controlling gene expression noise. It also highlights that understanding circuits' robustness across growth conditions is key for the effective design of synthetic biological systems.
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Affiliation(s)
- François Bertaux
- Department of Mathematics, Imperial College London, London SW7 2AZ,UK
- MRC London Institute of Medical Sciences (LMS), London W12 0NN, UK
- Institute of Clinical Sciences (ICS), Faculty of Medicine, Imperial College London, London W12 0NN, UK
| | - Samuel Marguerat
- MRC London Institute of Medical Sciences (LMS), London W12 0NN, UK
- Institute of Clinical Sciences (ICS), Faculty of Medicine, Imperial College London, London W12 0NN, UK
- Authors for correspondence: Samuel Marguerat e-mail:
| | - Vahid Shahrezaei
- Department of Mathematics, Imperial College London, London SW7 2AZ,UK
- Authors for correspondence: Vahid Shahrezaei e-mail:
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22
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Bertaux F, Marguerat S, Shahrezaei V. Division rate, cell size and proteome allocation: impact on gene expression noise and implications for the dynamics of genetic circuits. ROYAL SOCIETY OPEN SCIENCE 2018; 5:172234. [PMID: 29657814 DOI: 10.1101/209593] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/05/2018] [Accepted: 02/15/2018] [Indexed: 05/25/2023]
Abstract
The cell division rate, size and gene expression programmes change in response to external conditions. These global changes impact on average concentrations of biomolecule and their variability or noise. Gene expression is inherently stochastic, and noise levels of individual proteins depend on synthesis and degradation rates as well as on cell-cycle dynamics. We have modelled stochastic gene expression inside growing and dividing cells to study the effect of division rates on noise in mRNA and protein expression. We use assumptions and parameters relevant to Escherichia coli, for which abundant quantitative data are available. We find that coupling of transcription, but not translation rates to the rate of cell division can result in protein concentration and noise homeostasis across conditions. Interestingly, we find that the increased cell size at fast division rates, observed in E. coli and other unicellular organisms, buffers noise levels even for proteins with decreased expression at faster growth. We then investigate the functional importance of these regulations using gene regulatory networks that exhibit bi-stability and oscillations. We find that network topology affects robustness to changes in division rate in complex and unexpected ways. In particular, a simple model of persistence, based on global physiological feedback, predicts increased proportion of persister cells at slow division rates. Altogether, our study reveals how cell size regulation in response to cell division rate could help controlling gene expression noise. It also highlights that understanding circuits' robustness across growth conditions is key for the effective design of synthetic biological systems.
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Affiliation(s)
- François Bertaux
- Department of Mathematics, Imperial College London, London SW7 2AZ,UK
- MRC London Institute of Medical Sciences (LMS), London W12 0NN, UK
- Institute of Clinical Sciences (ICS), Faculty of Medicine, Imperial College London, London W12 0NN, UK
| | - Samuel Marguerat
- MRC London Institute of Medical Sciences (LMS), London W12 0NN, UK
- Institute of Clinical Sciences (ICS), Faculty of Medicine, Imperial College London, London W12 0NN, UK
| | - Vahid Shahrezaei
- Department of Mathematics, Imperial College London, London SW7 2AZ,UK
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23
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Kawade K, Tsukaya H. Probing the stochastic property of endoreduplication in cell size determination of Arabidopsis thaliana leaf epidermal tissue. PLoS One 2017; 12:e0185050. [PMID: 28926847 PMCID: PMC5605191 DOI: 10.1371/journal.pone.0185050] [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: 06/16/2017] [Accepted: 09/04/2017] [Indexed: 12/21/2022] Open
Abstract
Cell size distribution is highly reproducible, whereas the size of individual cells often varies greatly within a tissue. This is obvious in a population of Arabidopsis thaliana leaf epidermal cells, which ranged from 1,000 to 10,000 μm2 in size. Endoreduplication is a specialized cell cycle in which nuclear genome size (ploidy) is doubled in the absence of cell division. Although epidermal cells require endoreduplication to enhance cellular expansion, the issue of whether this mechanism is sufficient for explaining cell size distribution remains unclear due to a lack of quantitative understanding linking the occurrence of endoreduplication with cell size diversity. Here, we addressed this question by quantitatively summarizing ploidy profile and cell size distribution using a simple theoretical framework. We first found that endoreduplication dynamics is a Poisson process through cellular maturation. This finding allowed us to construct a mathematical model to predict the time evolution of a ploidy profile with a single rate constant for endoreduplication occurrence in a given time. We reproduced experimentally measured ploidy profile in both wild-type leaf tissue and endoreduplication-related mutants with this analytical solution, further demonstrating the probabilistic property of endoreduplication. We next extended the mathematical model by incorporating the element that cell size is determined according to ploidy level to examine cell size distribution. This analysis revealed that cell size is exponentially enlarged 1.5 times every endoreduplication round. Because this theoretical simulation successfully recapitulated experimentally observed cell size distributions, we concluded that Poissonian endoreduplication dynamics and exponential size-boosting are the sources of the broad cell size distribution in epidermal tissue. More generally, this study contributes to a quantitative understanding whereby stochastic dynamics generate steady-state biological heterogeneity.
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Affiliation(s)
- Kensuke Kawade
- Okazaki Institute for Integrative Bioscience, Okazaki, Aichi, Japan
- National Institute for Basic Biology, Okazaki, Aichi, Japan
- Department of Basic Biology, School of Life Science, Graduate University for Advanced Studies (SOKENDAI), Okazaki, Aichi, Japan
- * E-mail:
| | - Hirokazu Tsukaya
- Okazaki Institute for Integrative Bioscience, Okazaki, Aichi, Japan
- Department of Biological Sciences, Graduate School of Science, University of Tokyo, Bunkyo-ku, Tokyo, Japan
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24
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Priestman M, Thomas P, Robertson BD, Shahrezaei V. Mycobacteria Modify Their Cell Size Control under Sub-Optimal Carbon Sources. Front Cell Dev Biol 2017; 5:64. [PMID: 28748182 PMCID: PMC5506092 DOI: 10.3389/fcell.2017.00064] [Citation(s) in RCA: 31] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2017] [Accepted: 06/22/2017] [Indexed: 01/21/2023] Open
Abstract
The decision to divide is the most important one that any cell must make. Recent single cell studies suggest that most bacteria follow an “adder” model of cell size control, incorporating a fixed amount of cell wall material before dividing. Mycobacteria, including the causative agent of tuberculosis Mycobacterium tuberculosis, are known to divide asymmetrically resulting in heterogeneity in growth rate, doubling time, and other growth characteristics in daughter cells. The interplay between asymmetric cell division and adder size control has not been extensively investigated. Moreover, the impact of changes in the environment on growth rate and cell size control have not been addressed for mycobacteria. Here, we utilize time-lapse microscopy coupled with microfluidics to track live Mycobacterium smegmatis cells as they grow and divide over multiple generations, under a variety of growth conditions. We demonstrate that, under optimal conditions, M. smegmatis cells robustly follow the adder principle, with constant added length per generation independent of birth size, growth rate, and inherited pole age. However, the nature of the carbon source induces deviations from the adder model in a manner that is dependent on pole age. Understanding how mycobacteria maintain cell size homoeostasis may provide crucial targets for the development of drugs for the treatment of tuberculosis, which remains a leading cause of global mortality.
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Affiliation(s)
- Miles Priestman
- Department of Medicine, MRC Centre for Molecular Bacteriology and Infection, Imperial College LondonLondon, United Kingdom
| | - Philipp Thomas
- Department of Mathematics, Imperial College LondonLondon, United Kingdom
| | - Brian D Robertson
- Department of Medicine, MRC Centre for Molecular Bacteriology and Infection, Imperial College LondonLondon, United Kingdom
| | - Vahid Shahrezaei
- Department of Mathematics, Imperial College LondonLondon, United Kingdom
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