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Scott AD, King DM, Ordway SW, Bahar S. Phase transitions in evolutionary dynamics. CHAOS (WOODBURY, N.Y.) 2022; 32:122101. [PMID: 36587338 DOI: 10.1063/5.0124274] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/04/2022] [Accepted: 12/05/2022] [Indexed: 06/17/2023]
Abstract
Sharp changes in state, such as transitions from survival to extinction, are hallmarks of evolutionary dynamics in biological systems. These transitions can be explored using the techniques of statistical physics and the physics of nonlinear and complex systems. For example, a survival-to-extinction transition can be characterized as a non-equilibrium phase transition to an absorbing state. Here, we review the literature on phase transitions in evolutionary dynamics. We discuss directed percolation transitions in cellular automata and evolutionary models, and models that diverge from the directed percolation universality class. We explore in detail an example of an absorbing phase transition in an agent-based model of evolutionary dynamics, including previously unpublished data demonstrating similarity to, but also divergence from, directed percolation, as well as evidence for phase transition behavior at multiple levels of the model system's evolutionary structure. We discuss phase transition models of the error catastrophe in RNA virus dynamics and phase transition models for transition from chemistry to biochemistry, i.e., the origin of life. We conclude with a review of phase transition dynamics in models of natural selection, discuss the possible role of phase transitions in unraveling fundamental unresolved questions regarding multilevel selection and the major evolutionary transitions, and assess the future outlook for phase transitions in the investigation of evolutionary dynamics.
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Affiliation(s)
- Adam D Scott
- Department of Physics and Astronomy and Center for Neurodynamics, University of Missouri at St. Louis, One University Blvd., St. Louis, Missouri 63121, USA
| | - Dawn M King
- Department of Physics and Astronomy and Center for Neurodynamics, University of Missouri at St. Louis, One University Blvd., St. Louis, Missouri 63121, USA
| | - Stephen W Ordway
- Department of Physics and Astronomy and Center for Neurodynamics, University of Missouri at St. Louis, One University Blvd., St. Louis, Missouri 63121, USA
| | - Sonya Bahar
- Department of Physics and Astronomy and Center for Neurodynamics, University of Missouri at St. Louis, One University Blvd., St. Louis, Missouri 63121, USA
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2
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Zhou Q, Sun X, Xia X, Fan Z, Luo Z, Zhao S, Shakhnovich E, Liang H. Exploring the Mutational Robustness of Nucleic Acids by Searching Genotype Neighborhoods in Sequence Space. J Phys Chem Lett 2017; 8:407-414. [PMID: 28045264 DOI: 10.1021/acs.jpclett.6b02769] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
To assess the mutational robustness of nucleic acids, many genome- and protein-level studies have been performed, where nucleic acids are treated as genetic information carriers and transferrers. However, the molecular mechanisms through which mutations alter the structural, dynamic, and functional properties of nucleic acids are poorly understood. Here we performed a SELEX in silico study to investigate the fitness distribution of the l-Arm-binding aptamer genotype neighborhoods. Two novel functional genotype neighborhoods were isolated and experimentally verified to have comparable fitness as the wild-type. The experimental aptamer fitness landscape suggests the mutational robustness is strongly influenced by the local base environment and ligand-binding mode, whereas bases distant from the binding pocket provide potential evolutionary pathways to approach the global fitness maximum. Our work provides an example of successful application of SELEX in silico to optimize an aptamer and demonstrates the strong sensitivity of mutational robustness to the site of genetic variation.
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Affiliation(s)
- Qingtong Zhou
- iHuman Institute, ShanghaiTech University , Shanghai 201210, China
| | - Xianbao Sun
- CAS Key Laboratory of Soft Matter Chemistry, Collaborative Innovation Center of Chemistry for Energy Materials, Department of Polymer Science and Engineering, Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China , Hefei, Anhui 230026, China
| | - Xiaole Xia
- Key Laboratory of Industrial Biotechnology, Ministry of Education, School of Biotechnology, Jiangnan University , Wuxi, Jiangsu 214122, China
| | - Zhou Fan
- iHuman Institute, ShanghaiTech University , Shanghai 201210, China
- Key Laboratory of Computational Biology, Max Planck Independent Research Group on Population Genomics, CAS-MPG Partner Institute for Computational Biology, Shanghai Institutes for Biological Sciences, Chinese Academy of Sciences , Shanghai 200031, China
- University of Chinese Academy of Sciences , Beijing 100049, China
- School of Life Science and Technology, ShanghaiTech University , Shanghai 201210, China
| | - Zhaofeng Luo
- School of Life Science, University of Science and Technology of China , Hefei, Anhui 230026, China
| | - Suwen Zhao
- iHuman Institute, ShanghaiTech University , Shanghai 201210, China
- School of Life Science and Technology, ShanghaiTech University , Shanghai 201210, China
| | - Eugene Shakhnovich
- Department of Chemistry and Chemical Biology, Harvard University , Cambridge, Massachusetts 02138, United States
| | - Haojun Liang
- CAS Key Laboratory of Soft Matter Chemistry, Collaborative Innovation Center of Chemistry for Energy Materials, Department of Polymer Science and Engineering, Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China , Hefei, Anhui 230026, China
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Wagner N, Atsmon-Raz Y, Ashkenasy G. Theoretical Models of Generalized Quasispecies. Curr Top Microbiol Immunol 2016; 392:141-59. [PMID: 26373410 DOI: 10.1007/82_2015_456] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/11/2023]
Abstract
Theoretical modeling of quasispecies has progressed in several directions. In this chapter, we review the works of Emmanuel Tannenbaum, who, together with Eugene Shakhnovich at Harvard University and later with colleagues and students at Ben-Gurion University in Beersheva, implemented one of the more useful approaches, by progressively setting up various formulations for the quasispecies model and solving them analytically. Our review will focus on these papers that have explored new models, assumed the relevant mathematical approximations, and proceeded to analytically solve for the steady-state solutions and run stochastic simulations . When applicable, these models were related to real-life problems and situations, including changing environments, presence of chemical mutagens, evolution of cancer and tumor cells , mutations in Escherichia coli, stem cells , chromosomal instability (CIN), propagation of antibiotic drug resistance , dynamics of bacteria with plasmids , DNA proofreading mechanisms, and more.
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Affiliation(s)
- Nathaniel Wagner
- Department of Chemistry, Ben-Gurion University of the Negev, Beer Sheva, 84105, Israel
| | - Yoav Atsmon-Raz
- Department of Chemistry, Ben-Gurion University of the Negev, Beer Sheva, 84105, Israel
| | - Gonen Ashkenasy
- Department of Chemistry, Ben-Gurion University of the Negev, Beer Sheva, 84105, Israel.
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Kussell E, Vucelja M. Non-equilibrium physics and evolution--adaptation, extinction, and ecology: a key issues review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2014; 77:102602. [PMID: 25303141 DOI: 10.1088/0034-4885/77/10/102602] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Evolutionary dynamics in nature constitute an immensely complex non-equilibrium process. We review the application of physical models of evolution, by focusing on adaptation, extinction, and ecology. In each case, we examine key concepts by working through examples. Adaptation is discussed in the context of bacterial evolution, with a view toward the relationship between growth rates, mutation rates, selection strength, and environmental changes. Extinction dynamics for an isolated population are reviewed, with emphasis on the relation between timescales of extinction, population size, and temporally correlated noise. Ecological models are discussed by focusing on the effect of spatial interspecies interactions on diversity. Connections between physical processes--such as diffusion, turbulence, and localization--and evolutionary phenomena are highlighted.
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Affiliation(s)
- E Kussell
- Department of Biology and Center for Genomics and Systems Biology, New York University, 12 Waverly Place, New York, NY 10003, USA. Department of Physics, New York University, New York, NY 10003, USA
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Woo HJ, Vijaya Satya R, Reifman J. Thermodynamic basis for the emergence of genomes during prebiotic evolution. PLoS Comput Biol 2012; 8:e1002534. [PMID: 22693440 PMCID: PMC3364946 DOI: 10.1371/journal.pcbi.1002534] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2011] [Accepted: 04/08/2012] [Indexed: 01/02/2023] Open
Abstract
The RNA world hypothesis views modern organisms as descendants of RNA molecules. The earliest RNA molecules must have been random sequences, from which the first genomes that coded for polymerase ribozymes emerged. The quasispecies theory by Eigen predicts the existence of an error threshold limiting genomic stability during such transitions, but does not address the spontaneity of changes. Following a recent theoretical approach, we applied the quasispecies theory combined with kinetic/thermodynamic descriptions of RNA replication to analyze the collective behavior of RNA replicators based on known experimental kinetics data. We find that, with increasing fidelity (relative rate of base-extension for Watson-Crick versus mismatched base pairs), replications without enzymes, with ribozymes, and with protein-based polymerases are above, near, and below a critical point, respectively. The prebiotic evolution therefore must have crossed this critical region. Over large regions of the phase diagram, fitness increases with increasing fidelity, biasing random drifts in sequence space toward ‘crystallization.’ This region encloses the experimental nonenzymatic fidelity value, favoring evolutions toward polymerase sequences with ever higher fidelity, despite error rates above the error catastrophe threshold. Our work shows that experimentally characterized kinetics and thermodynamics of RNA replication allow us to determine the physicochemical conditions required for the spontaneous crystallization of biological information. Our findings also suggest that among many potential oligomers capable of templated replication, RNAs may have evolved to form prebiotic genomes due to the value of their nonenzymatic fidelity. A leading hypothesis for the origin of life describes a prebiotic world where RNA molecules started carrying genetic information for catalyzing their own replication. This origin of biological information is akin to the crystallization of ice from water, where ‘order’ emerges from ‘disorder.’ What does the science of such phase transformations tell us about the emergence of genomes? In this paper, we show that such thermodynamic considerations of RNA synthesis, when combined with kinetics and population dynamics, lead to the conclusion that the ‘crystallization’ of genomes from its basic elements would have been spontaneous for RNAs, but not necessarily for other potential building blocks of genomes in the prebiotic soup.
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Affiliation(s)
| | | | - Jaques Reifman
- DoD Biotechnology High Performance Computing Software Applications Institute, Telemedicine and Advanced Technology Research Center, United States Army Medical Research and Materiel Command, Fort Detrick, Maryland, United States of America
- * E-mail:
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Kama A, Tannenbaum E. Effect of the SOS response on the mean fitness of unicellular populations: a quasispecies approach. PLoS One 2010; 5:e14113. [PMID: 21152423 PMCID: PMC2994707 DOI: 10.1371/journal.pone.0014113] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2008] [Accepted: 10/26/2010] [Indexed: 11/23/2022] Open
Abstract
The goal of this paper is to develop a mathematical model that analyzes the selective advantage of the SOS response in unicellular organisms. To this end, this paper develops a quasispecies model that incorporates the SOS response. We consider a unicellular, asexually replicating population of organisms, whose genomes consist of a single, double-stranded DNA molecule, i.e. one chromosome. We assume that repair of post-replication mismatched base-pairs occurs with probability , and that the SOS response is triggered when the total number of mismatched base-pairs is at least . We further assume that the per-mismatch SOS elimination rate is characterized by a first-order rate constant . For a single fitness peak landscape where the master genome can sustain up to mismatches and remain viable, this model is analytically solvable in the limit of infinite sequence length. The results, which are confirmed by stochastic simulations, indicate that the SOS response does indeed confer a fitness advantage to a population, provided that it is only activated when DNA damage is so extensive that a cell will die if it does not attempt to repair its DNA.
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Affiliation(s)
- Amit Kama
- Department of Chemistry, Ben-Gurion University of the Negev, Be'er-Sheva, Israel
| | - Emmanuel Tannenbaum
- Department of Chemistry, Ben-Gurion University of the Negev, Be'er-Sheva, Israel
- * E-mail:
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7
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Itan E, Tannenbaum E. Semiconservative quasispecies equations for polysomic genomes: the general case. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:061915. [PMID: 20866448 DOI: 10.1103/physreve.81.061915] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2009] [Indexed: 05/29/2023]
Abstract
This paper develops a formulation of the quasispecies equations appropriate for polysomic, semiconservatively replicating genomes. This paper is an extension of previous work on the subject, which considered the case of haploid genomes. Here, we develop a more general formulation of the quasispecies equations that is applicable to diploid and even polyploid genomes. Interestingly, with an appropriate classification of population fractions, we obtain a system of equations that is formally identical to the haploid case. As with the work for haploid genomes, we consider both random and immortal DNA strand chromosome segregation mechanisms. However, in contrast to the haploid case, we have found that an analytical solution for the mean fitness is considerably more difficult to obtain for the polyploid case. Accordingly, whereas for the haploid case we obtained expressions for the mean fitness for the case of an analog of the single-fitness-peak landscape for arbitrary lesion repair probabilities (thereby allowing for noncomplementary genomes), here we solve for the mean fitness for the restricted case of perfect lesion repair.
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Affiliation(s)
- Eran Itan
- Department of Chemistry, Ben-Gurion University of the Negev, Be'er-Sheva, Israel
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8
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Gorodetsky P, Tannenbaum E. Effect of mutators on adaptability in time-varying fitness landscapes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:042901. [PMID: 18517675 DOI: 10.1103/physreve.77.042901] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/15/2008] [Indexed: 05/26/2023]
Abstract
This Brief Report studies the quasispecies dynamics of a population capable of genetic repair evolving on a time-dependent fitness landscape. We develop a model that considers an asexual population of single-stranded, conservatively replicating genomes, whose only source of genetic variation is due to copying errors during replication. We consider a time-dependent, single-fitness-peak landscape where the master sequence changes by a single point mutation at every time tau. We are able to analytically solve for the evolutionary dynamics of the population in the point-mutation limit. In particular, our model provides an analytical expression for the fraction of mutators in the dynamic fitness landscape that agrees well with results from stochastic simulations.
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Lee B, Tannenbaum E. Asexual and sexual replication in sporulating organisms. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:021909. [PMID: 17930067 DOI: 10.1103/physreve.76.021909] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2006] [Revised: 04/18/2007] [Indexed: 05/25/2023]
Abstract
Replication via sporulation is the replication strategy for all multicellular life, and may even be observed in unicellular life (such as with budding yeast). We consider diploid populations replicating via one of two possible sporulation mechanisms. (1) Asexual sporulation, whereby adult organisms produce single-celled diploid spores that grow into adults themselves. (2) Sexual sporulation, whereby adult organisms produce single-celled diploid spores that divide into haploid gametes. The haploid gametes enter a haploid "pool," where they may recombine with other haploids to form a diploid spore that then grows into an adult. We consider a haploid fusion rate given by second-order reaction kinetics. We work with a simplified model where the diploid genome consists of only two chromosomes, each of which may be rendered defective with a single point mutation of the wild-type. We find that the asexual strategy is favored when the rate of spore production is high compared to the characteristic growth rate from a spore to a reproducing adult. Conversely, the sexual strategy is favored when the rate of spore production is low compared to the characteristic growth rate from a spore to a reproducing adult. As the characteristic growth time increases, or as the population density increases, the critical ratio of spore production rate to organism growth rate at which the asexual strategy overtakes the sexual one is pushed to higher values. Therefore, the results of this model suggest that, for complex multicellular organisms, sexual replication is favored at high population densities and low growth and sporulation rates.
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Affiliation(s)
- Bohyun Lee
- School of Biology, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
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Tannenbaum E. Selective advantage for sexual reproduction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:061925. [PMID: 16906882 DOI: 10.1103/physreve.73.061925] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2005] [Revised: 12/08/2005] [Indexed: 05/11/2023]
Abstract
This paper develops a simplified model for sexual reproduction within the quasispecies formalism. The model assumes a diploid genome consisting of two chromosomes, where the fitness is determined by the number of chromosomes that are identical to a given master sequence. We also assume that there is a cost to sexual reproduction, given by a characteristic time tau(seek) during which haploid cells seek out a mate with which to recombine. If the mating strategy is such that only viable haploids can mate, then when tau(seek) = 0, it is possible to show that sexual reproduction will always out compete asexual reproduction. However, as tau(seek) increases, sexual reproduction only becomes advantageous at progressively higher mutation rates. Once the time cost for sex reaches a critical threshold, the selective advantage for sexual reproduction disappears entirely. The results of this paper suggest that sexual reproduction is not advantageous in small populations per se, but rather in populations with low replication rates. In this regime, the cost for sex is sufficiently low that the selective advantage obtained through recombination leads to the dominance of the strategy. In fact, at a given replication rate and for a fixed environment volume, sexual reproduction is selected for in high populations because of the reduced time spent finding a reproductive partner.
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Affiliation(s)
- Emmanuel Tannenbaum
- Department of Chemistry, Ben-Gurion University of the Negev, Be'er-Sheva 84105, Israel.
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Tannenbaum E, Sherley JL, Shakhnovich EI. Semiconservative quasispecies equations for polysomic genomes: the haploid case. J Theor Biol 2006; 241:791-805. [PMID: 16527313 DOI: 10.1016/j.jtbi.2006.01.016] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2005] [Revised: 12/12/2005] [Accepted: 01/13/2006] [Indexed: 11/23/2022]
Abstract
This paper develops the semiconservative quasispecies equations for genomes consisting of an arbitrary number of chromosomes. We assume that the chromosomes are distinguishable, so that we are effectively considering haploid genomes. We derive the quasispecies equations under the assumption of arbitrary lesion repair efficiency, and consider the cases of both random and immortal strand chromosome segregation. We solve the model in the limit of infinite sequence length for the case of the static single fitness peak landscape, where the master genome has a first-order growth rate constant of k>1, and all other genomes have a first-order growth rate constant of 1. If we assume that each chromosome can tolerate an arbitrary number of lesions, so that only one master copy of the strands is necessary for a functional chromosome, then for random chromosome segregation we obtain an equilibrium mean fitness of [equation in text] below the error catastrophe, while for immortal strand co-segregation we obtain kappa (t=infinity)=k[e(-mu(1-lambda/2))+e(-mulambda/2)-1] (N denotes the number of chromosomes, lambda denotes the lesion repair efficiency, and mu is identical with epsilonL, where epsilon is the per base-pair mismatch probability, and L is the total genome length). It follows that immortal strand co-segregation leads to significantly better preservation of the master genome than random segregation when lesion repair is imperfect. Based on this result, we conjecture that certain classes of tumor cells exhibit immortal strand co-segregation.
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Affiliation(s)
- Emmanuel Tannenbaum
- Department of Chemistry, Ben-Gurion University of the Negev, Be'er-Sheva 84105, Israel.
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André JB, Godelle B. The evolution of mutation rate in finite asexual populations. Genetics 2006; 172:611-26. [PMID: 16157667 PMCID: PMC1456187 DOI: 10.1534/genetics.105.046680] [Citation(s) in RCA: 90] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2005] [Accepted: 09/08/2005] [Indexed: 11/18/2022] Open
Abstract
In this article, we model analytically the evolution of mutation rate in asexual organisms. Three selective forces are present. First, everything else being equal, individuals with higher mutation rate have a larger fitness, thanks to the energy and time saved by not replicating DNA accurately. Second, as a flip side, the genome of these individuals is replicated with errors that may negatively affect fitness. Third, and conversely, replication errors have a potential benefit if beneficial mutations are to be generated. Our model describes the fate of modifiers of mutation rate under the three forces and allows us to predict the long-term evolutionary trajectory of mutation rate. We obtain three major results. First, in asexuals, the needs for both adaptation and genome preservation are not evolutionary forces that can stabilize mutation rate at an intermediate optimum. When adaptation has a significant role, it primarily destabilizes mutation rate and yields the emergence of strong-effect mutators. Second, in contrast to what is usually believed, the appearance of modifiers with large mutation rate is more likely when the fitness cost of each deleterious mutation is weak, because the cost of replication errors is then paid after a delay. Third, in small populations, and even if adaptations are needed, mutation rate is always blocked at the minimum attainable level, because the rate of adaptation is too slow to play a significant role. Only populations whose size is above a critical mass see their mutation rate affected by the need for adaptation.
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Affiliation(s)
- Jean-Baptiste André
- Laboratoire Génome, Populations, Interactions, Adaptation, USTL-IFREMER-CNRS UMR 5171, Université des Sciences et Techniques du Languedoc, 34095 Montpellier, France.
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Tannenbaum E, Shakhnovich EI. Semiconservative replication, genetic repair, and many-gened genomes: Extending the quasispecies paradigm to living systems. Phys Life Rev 2005. [DOI: 10.1016/j.plrev.2005.08.001] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Wilke CO. Quasispecies theory in the context of population genetics. BMC Evol Biol 2005; 5:44. [PMID: 16107214 PMCID: PMC1208876 DOI: 10.1186/1471-2148-5-44] [Citation(s) in RCA: 176] [Impact Index Per Article: 9.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2005] [Accepted: 08/17/2005] [Indexed: 12/17/2022] Open
Abstract
BACKGROUND A number of recent papers have cast doubt on the applicability of the quasispecies concept to virus evolution, and have argued that population genetics is a more appropriate framework to describe virus evolution than quasispecies theory. RESULTS I review the pertinent literature, and demonstrate for a number of cases that the quasispecies concept is equivalent to the concept of mutation-selection balance developed in population genetics, and that there is no disagreement between the population genetics of haploid, asexually-replicating organisms and quasispecies theory. CONCLUSION Since quasispecies theory and mutation-selection balance are two sides of the same medal, the discussion about which is more appropriate to describe virus evolution is moot. In future work on virus evolution, we would do good to focus on the important questions, such as whether we can develop accurate, quantitative models of virus evolution, and to leave aside discussions about the relative merits of perfectly equivalent concepts.
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Affiliation(s)
- Claus O Wilke
- Keck Graduate Institute of Applied Life Sciences, 535 WatsonDrive, Claremont, California 91711, USA
- Digital Life Laboratory, California Institute of Technology, Mail Code 136-93, Pasadena, California 91125, USA
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15
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Tannenbaum E, Sherley JL, Shakhnovich EI. Imperfect DNA lesion repair in the semiconservative quasispecies model: derivation of the Hamming class equations and solution of the single-fitness peak landscape. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:061915. [PMID: 15697410 DOI: 10.1103/physreve.70.061915] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2004] [Indexed: 05/24/2023]
Abstract
This paper develops a Hamming class formalism for the semiconservative quasispecies equations with imperfect lesion repair, first presented and analytically solved in Y. Brumer and E.I. Shakhnovich (q-bio.GN/0403018, 2004). Starting from the quasispecies dynamics over the space of genomes, we derive an equivalent dynamics over the space of ordered sequence pairs. From this set of equations, we are able to derive the infinite sequence length form of the dynamics for a class of fitness landscapes defined by a master genome. We use these equations to solve for a generalized single-fitness-peak landscape, where the master genome can sustain a maximum number of lesions and remain viable. We determine the mean equilibrium fitness and error threshold for this class of landscapes, and show that when lesion repair is imperfect, semiconservative replication displays characteristics from both conservative replication and semiconservative replication with perfect lesion repair. The work presented here provides a formulation of the model which greatly facilitates the analysis of a relatively broad class of fitness landscapes, and thus serves as a convenient springboard into biological applications of imperfect lesion repair.
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Affiliation(s)
- Emmanuel Tannenbaum
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA.
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16
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Tannenbaum E, Shakhnovich EI. Solution of the quasispecies model for an arbitrary gene network. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:021903. [PMID: 15447511 DOI: 10.1103/physreve.70.021903] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2004] [Indexed: 05/24/2023]
Abstract
In this paper, we study the equilibrium behavior of Eigen's quasispecies equations for an arbitrary gene network. We consider a genome consisting of N genes, so that the full genome sequence sigma may be written as sigma= sigma1sigma2...sigmaN, where sigma(i) are sequences of individual genes. We assume a single fitness peak model for each gene, so that gene i has some "master" sequence sigma(i,0) for which it is functioning. The fitness landscape is then determined by which genes in the genome are functioning and which are not. The equilibrium behavior of this model may be solved in the limit of infinite sequence length. The central result is that, instead of a single error catastrophe, the model exhibits a series of localization to delocalization transitions, which we term an "error cascade." As the mutation rate is increased, the selective advantage for maintaining functional copies of certain genes in the network disappears, and the population distribution delocalizes over the corresponding sequence spaces. The network goes through a series of such transitions, as more and more genes become inactivated, until eventually delocalization occurs over the entire genome space, resulting in a final error catastrophe. This model provides a criterion for determining the conditions under which certain genes in a genome will lose functionality due to genetic drift. It also provides insight into the response of gene networks to mutagens. In particular, it suggests an approach for determining the relative importance of various genes to the fitness of an organism, in a more accurate manner than the standard "deletion set" method. The results in this paper also have implications for mutational robustness and what C.O. Wilke termed "survival of the flattest."
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Affiliation(s)
- Emmanuel Tannenbaum
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA.
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17
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Tannenbaum E, Deeds EJ, Shakhnovich EI. Semiconservative replication in the quasispecies model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:061916. [PMID: 15244626 DOI: 10.1103/physreve.69.061916] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/2003] [Revised: 01/23/2004] [Indexed: 05/24/2023]
Abstract
This paper extends Eigen's quasispecies equations to account for the semiconservative nature of DNA replication. We solve the equations in the limit of infinite sequence length for the simplest case of a static, sharply peaked fitness landscape. We show that the error catastrophe occurs when micro, the product of sequence length and per base pair mismatch probability, exceeds 2 ln [2/ ( 1+1/k ) ], where k>1 is the first-order growth rate constant of the viable "master" sequence (with all other sequences having a first-order growth rate constant of 1 ). This is in contrast to the result of ln k for conservative replication. In particular, as k--> infinity, the error catastrophe is never reached for conservative replication, while for semiconservative replication the critical micro approaches 2 ln 2. Semiconservative replication is therefore considerably less robust than conservative replication to the effect of replication errors. We also show that the mean equilibrium fitness of a semiconservatively replicating system is given by k ( 2 e(-micro/2) -1 ) below the error catastrophe, in contrast to the standard result of k e(-micro) for conservative replication (derived by Kimura and Maruyama in 1966). From this result it is readily shown that semiconservative replication is necessary to account for the observation that, at sufficiently high mutagen concentrations, faster replicating cells will die more quickly than more slowly replicating cells. Thus, in contrast to Eigen's original model, the semiconservative quasispecies equations are able to provide a mathematical basis for explaining the efficacy of mutagens as chemotherapeutic agents.
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Tannenbaum E, Shakhnovich EI. Error and repair catastrophes: A two-dimensional phase diagram in the quasispecies model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:011902. [PMID: 14995642 DOI: 10.1103/physreve.69.011902] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2003] [Revised: 09/04/2003] [Indexed: 05/24/2023]
Abstract
This paper develops a two-gene, single fitness peak model for determining the equilibrium distribution of genotypes in a unicellular population which is capable of genetic damage repair. The first gene, denoted by sigma(via), yields a viable organism with first-order growth rate constant k>1 if it is equal to some target "master" sequence sigma(via,0). The second gene, denoted by sigma(rep), yields an organism capable of genetic repair if it is equal to some target "master" sequence sigma(rep,0). This model is analytically solvable in the limit of infinite sequence length, and gives an equilibrium distribution which depends on micro identical with Lepsilon, the product of sequence length and per base pair replication error probability, and epsilon(r), the probability of repair failure per base pair. The equilibrium distribution is shown to exist in one of the three possible "phases." In the first phase, the population is localized about the viability and repairing master sequences. As epsilon(r) exceeds the fraction of deleterious mutations, the population undergoes a "repair" catastrophe, in which the equilibrium distribution is still localized about the viability master sequence, but is spread ergodically over the sequence subspace defined by the repair gene. Below the repair catastrophe, the distribution undergoes the error catastrophe when micro exceeds ln k/epsilon(r), while above the repair catastrophe, the distribution undergoes the error catastrophe when micro exceeds ln k/f(del), where f(del) denotes the fraction of deleterious mutations.
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