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González Casanova A, Kurt N, Pérez JL. The ancestral selection graph for a Λ-asymmetric Moran model. Theor Popul Biol 2024; 159:91-107. [PMID: 38490495 DOI: 10.1016/j.tpb.2024.02.010] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2023] [Revised: 02/26/2024] [Accepted: 02/27/2024] [Indexed: 03/17/2024]
Abstract
Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanisms, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Λ-reproduction here means that a whole fraction of the population is replaced at a reproductive event. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Provided the measure are ordered stochastically, we can couple them. This allows us to construct the central object of this paper, the Λ-asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinuous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.
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Affiliation(s)
- Adrián González Casanova
- Instituto de Matematicas, Universidad Nacional Autonoma de Mexico (UNAM), Cuernavaca, Mexico; Department of Statistics, University of California at Berkeley, United States of America.
| | - Noemi Kurt
- Institut für Mathematik, Johann Wolfgang Goethe-Universität, 60325 Frankfurt am Main, Germany.
| | - José Luis Pérez
- Department of Probability and Statistics, Centro de Investigación en Matemáticas A.C., Calle Jalisco s/n. C.P. 36240, Guanajuato, Mexico.
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2
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Root-Bernstein RS, Bernstein MI. 'Evolutionary poker': an agent-based model of interactome emergence and epistasis tested against Lenski's long-term E. coli experiments. J Physiol 2024; 602:2511-2535. [PMID: 37707489 DOI: 10.1113/jp284421] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2023] [Accepted: 08/23/2023] [Indexed: 09/15/2023] Open
Abstract
A simple agent-based model is presented that produces results matching the experimental data found by Lenski's group for ≤50,000 generations of Escherichia coli bacteria under continuous selective pressure. Although various mathematical models have been devised previously to model the Lenski data, the present model has advantages in terms of overall simplicity and conceptual accessibility. The model also clearly illustrates a number of features of the evolutionary process that are otherwise not obvious, such as the roles of epistasis and historical contingency in adaptation and why evolution is time irreversible ('Dollo's law'). The reason for this irreversibility is that genomes become increasingly integrated or organized, and this organization becomes a novel selective factor itself, against which future generations must compete. Selection for integrated or synergistic networks, systems or sets of mutations or traits, not for individual mutations, confers the main adaptive advantage. The result is a punctuated form of evolution that follows a logarithmic occurrence probability, in which evolution proceeds very quickly when interactomes begin to form but which slows as interactomes become more robust and the difficulty of integrating new mutations increases. Sufficient parameters exist in the game to suggest not only how equilibrium or stasis is reached but also the conditions in which it will be punctuated, the factors governing the rate at which genomic organization occurs and novel traits appear, and how population size, genome size and gene variability affect these.
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Affiliation(s)
| | - Morton I Bernstein
- Department of Physiology, Michigan State University, East Lansing, MI, USA
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Hernandez‐Beltran JCR, Miró Pina V, Siri‐Jégousse A, Palau S, Peña‐Miller R, González Casanova A. Segregational instability of multicopy plasmids: A population genetics approach. Ecol Evol 2022; 12:e9469. [PMID: 36479025 PMCID: PMC9720003 DOI: 10.1002/ece3.9469] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Accepted: 10/14/2022] [Indexed: 12/11/2022] Open
Abstract
Plasmids are extra-chromosomal genetic elements that encode a wide variety of phenotypes and can be maintained in bacterial populations through vertical and horizontal transmission, thus increasing bacterial adaptation to hostile environmental conditions like those imposed by antimicrobial substances. To circumvent the segregational instability resulting from randomly distributing plasmids between daughter cells upon division, nontransmissible plasmids tend to be carried in multiple copies per cell, with the added benefit of exhibiting increased gene dosage and resistance levels. But carrying multiple copies also results in a high metabolic burden to the bacterial host, therefore reducing the overall fitness of the population. This trade-off poses an existential question for plasmids: What is the optimal plasmid copy number? In this manuscript, we address this question by postulating and analyzing a population genetics model to evaluate the interaction between selective pressure, the number of plasmid copies carried by each cell, and the metabolic burden associated with plasmid bearing in the absence of selection for plasmid-encoded traits. Parameter values of the model were estimated experimentally using Escherichia coli K12 carrying a multicopy plasmid encoding for a fluorescent protein and bla TEM-1, a gene conferring resistance to β-lactam antibiotics. By numerically determining the optimal plasmid copy number for constant and fluctuating selection regimes, we show that plasmid copy number is a highly optimized evolutionary trait that depends on the rate of environmental fluctuation and balances the benefit between increased stability in the absence of selection with the burden associated with carrying multiple copies of the plasmid.
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Affiliation(s)
- J. Carlos R. Hernandez‐Beltran
- Systems Biology Program, Center for Genomic SciencesUniversidad Nacional Autónoma de MéxicoCuernavacaMexico
- Department of Microbial Population BiologyMax Planck Institute for Evolutionary BiologyPlönGermany
| | - Verónica Miró Pina
- Centre for Genomic Regulation (CRG)The Barcelona Institute of Science and TechnologyBarcelonaSpain
- Universitat Pompeu Fabra (UPF)BarcelonaSpain
- Departamento de Probabilidad y Estadística, Instituto de Investigación en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de MéxicoCuernavacaMexico
| | - Arno Siri‐Jégousse
- Departamento de Probabilidad y Estadística, Instituto de Investigación en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de MéxicoCuernavacaMexico
| | - Sandra Palau
- Departamento de Probabilidad y Estadística, Instituto de Investigación en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de MéxicoCuernavacaMexico
| | - Rafael Peña‐Miller
- Systems Biology Program, Center for Genomic SciencesUniversidad Nacional Autónoma de MéxicoCuernavacaMexico
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Kingman’s model with random mutation probabilities: convergence and condensation I. ADV APPL PROBAB 2022. [DOI: 10.1017/apr.2021.33] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
Abstract
For a one-locus haploid infinite population with discrete generations, the celebrated model of Kingman describes the evolution of fitness distributions under the competition of selection and mutation, with a constant mutation probability. This paper generalises Kingman’s model by using independent and identically distributed random mutation probabilities, to reflect the influence of a random environment. The weak convergence of fitness distributions to the globally stable equilibrium is proved. Condensation occurs when almost surely a positive proportion of the population travels to and condenses at the largest fitness value. Condensation may occur when selection is favoured over mutation. A criterion for the occurrence of condensation is given.
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Boenkost F, González Casanova A, Pokalyuk C, Wakolbinger A. Haldane's formula in Cannings models: the case of moderately strong selection. J Math Biol 2021; 83:70. [PMID: 34870765 PMCID: PMC8648686 DOI: 10.1007/s00285-021-01698-9] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2020] [Revised: 11/03/2021] [Accepted: 11/19/2021] [Indexed: 11/28/2022]
Abstract
For a class of Cannings models we prove Haldane’s formula, \documentclass[12pt]{minimal}
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\begin{document}$$\pi (s_N) \sim \frac{2s_N}{\rho ^2}$$\end{document}π(sN)∼2sNρ2, for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for \documentclass[12pt]{minimal}
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\begin{document}$$s_N \sim N^{-b}$$\end{document}sN∼N-b and \documentclass[12pt]{minimal}
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\begin{document}$$0< b<1/2$$\end{document}0<b<1/2. Here, \documentclass[12pt]{minimal}
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\begin{document}$$s_N$$\end{document}sN is the selective advantage of an individual carrying the beneficial type, and \documentclass[12pt]{minimal}
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\begin{document}$$\rho ^2$$\end{document}ρ2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.
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Affiliation(s)
- Florin Boenkost
- Goethe-Universität Frankfurt, FB 12, 60054, Frankfurt, Germany.
| | - Adrián González Casanova
- Instituto de Matemáticas, Universidad Nacional Autónoma de México, Área de la Investigación Científica, Circuito Exterior, Ciudad Universitaria, 04510, Coyoacán, CDMX, Mexico
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Boenkost F, González Casanova A, Pokalyuk C, Wakolbinger A. Haldane’s formula in Cannings models: the case of moderately weak selection. ELECTRON J PROBAB 2021. [DOI: 10.1214/20-ejp572] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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7
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Blath J, Tóbiás A. Invasion and fixation of microbial dormancy traits under competitive pressure. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2020.07.018] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
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8
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Markov branching processes with disasters: Extinction, survival and duality to p-jump processes. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2019.07.011] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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González Casanova A, Miró Pina V, Pardo JC. The Wright-Fisher model with efficiency. Theor Popul Biol 2020; 132:33-46. [PMID: 32151657 DOI: 10.1016/j.tpb.2020.02.003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2019] [Revised: 02/22/2020] [Accepted: 02/27/2020] [Indexed: 10/24/2022]
Abstract
In this article, we propose a Wright-Fisher model with two types of individuals: the inefficient individuals, those who need more resources to reproduce and can have a higher growth rate, and the efficient individuals. In this model, the total amount of resource N is fixed, and the population size varies randomly depending on the number of efficient individuals. We show that, as N increases, the frequency process of efficient individuals converges to a diffusion which is a generalization of the Wright-Fisher diffusion with selection. The genealogy of this model is given by a branching-coalescing process that we call the Ancestral Selection/Efficiency Graph, and that is an extension of the Ancestral Selection Graph (Krone and Neuhauser, 1997a,b). The main contribution of this paper is that, in evolving populations, inefficiency can arise as a promoter of selective advantage and not necessarily as a trade-off.
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Affiliation(s)
- Adrián González Casanova
- Instituto de Matemáticas de la Universidad Nacional Autónoma de México, Área de la Investigación Científica, Circuito Exterior, C.U., 04510 Coyoacán, CDMX, Mexico.
| | - Verónica Miró Pina
- Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas, Universidad Nacional Autónoma de México, Circuito Escolar 3000, C.U., 04510 Coyoacán, CDMX, Mexico
| | - Juan Carlos Pardo
- Centro de Investigación en Matemáticas A.C., Calle Jalisco s/n. 36240 Guanajuato, Mexico
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Maintenance of diversity in a hierarchical host–parasite model with balancing selection and reinfection. Stoch Process Their Appl 2020. [DOI: 10.1016/j.spa.2019.04.009] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
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Modelling and simulating Lenski’s long-term evolution experiment. Theor Popul Biol 2019; 127:58-74. [DOI: 10.1016/j.tpb.2019.03.006] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2018] [Revised: 03/28/2019] [Accepted: 03/29/2019] [Indexed: 01/15/2023]
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12
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Yuan L. A generalization of Kingman’s model of selection and mutation and the Lenski experiment. Math Biosci 2017; 285:61-67. [DOI: 10.1016/j.mbs.2016.12.007] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2016] [Revised: 12/18/2016] [Accepted: 12/20/2016] [Indexed: 11/16/2022]
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