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Overall ADJ, Waxman D. Influence of selection on the probability of fixation at a locus with multiple alleles. BMC Genomics 2024; 25:819. [PMID: 39215209 PMCID: PMC11365252 DOI: 10.1186/s12864-024-10733-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/09/2024] [Accepted: 08/22/2024] [Indexed: 09/04/2024] Open
Abstract
BACKGROUND Genes exist in a population in a variety of forms (alleles), as a consequence of multiple mutation events that have arisen over the course of time. In this work we consider a locus that is subject to either multiplicative or additive selection, and has n alleles, where n can take the values 2, 3, 4, … . We focus on determining the probability of fixation of each of the n alleles. For n = 2 alleles, analytical results, that are 'exact', under the diffusion approximation, can be found for the fixation probability. However generally there are no equally exact results for n ≥ 3 alleles. In the absence of such exact results, we proceed by finding results for the fixation probability, under the diffusion approximation, as a power series in scaled strengths of selection such asR i , j = 2 N e ( s i - s j ) , where N e is the effective population size, while s i and s j are the selection coefficients associated with alleles i and j, respectively. RESULTS We determined the fixation probability when all terms up to second order in the R i , j are kept. The truncation of the power series requires that the R i , j cannot be indefinitely large. For magnitudes of the R i , j up to a value of approximately 1, numerical evidence suggests that the results work well. Additionally, results given for the particular case of n = 3 alleles illustrate a general feature that holds for n ≥ 3 alleles, that the fixation probability of a particular allele depends on that allele's initial frequency, but generally, this fixation probability also depends on the initial frequencies of other alleles at the locus, as well as their selective effects. CONCLUSIONS We have analytically exposed the leading way the probability of fixation, at a locus with multiple alleles, is affected by selection. This result may offer important insights into CDCV traits that have extreme phenotypic variance due to numerous, low-penetrance susceptibility alleles.
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Affiliation(s)
- A D J Overall
- School of Applied Sciences, Huxley Building, University of Brighton, BN2 4GJ, Brighton, East Sussex, UK
| | - D Waxman
- Centre for Computational Systems Biology, ISTBI, Fudan University, 220 Handan Road, 200433, Shanghai, People's Republic of China.
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2
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Noland JK, Thorvaldsen S. The Exact Stochastic Process of the Haploid Multi-Allelic Wright-Fisher Mutation Model. IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 2024; 21:69-83. [PMID: 38010931 DOI: 10.1109/tcbb.2023.3336850] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/29/2023]
Abstract
Diffusion models are widely applied in population genetics, but their approximate solutions may not accurately capture the exact stochastic process. Nevertheless, this practice was necessary due to computing limitations, particularly for large populations. In this article, we develop the exact Markov chain algebra (MCA) for a discrete haploid multi-allelic Wright-Fisher model (MA-WFM) with a full mutation matrix to address this challenge. A special case of nonzero mutations between multiple alleles have not been captured by previous bi-allelic models. We formulated the mean allele frequencies for asymptotic equilibrium analytically for the tri- and quad-allelic case. We also evaluated the exact time-dependent Markov model numerically, presenting it concisely in terms of diffusion variables. The convergence with increasing population size to a diffusion limit is demonstrated for the population composition distribution. Our model shows that there will never be exact irreversible extinction when there are nonzero mutation rates into each allele and never be an exact irreversible fixation when there are nonzero mutation rates out of each allele. We only present results where there is no complete extinction and no complete fixation. Finally, we present detailed computations for the full Markov process, exposing the behavior near the boundaries for the compositional domains, which are non-singular boundaries according to diffusion theory.
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Ramirez L, San Miguel M, Galla T. Local and global ordering dynamics in multistate voter models. Phys Rev E 2022; 106:054307. [PMID: 36559520 DOI: 10.1103/physreve.106.054307] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/12/2022] [Accepted: 10/14/2022] [Indexed: 06/17/2023]
Abstract
We investigate the time evolution of the density of active links and of the entropy of the distribution of agents among opinions in multistate voter models with all-to-all interaction and on uncorrelated networks. Individual realizations undergo a sequence of eliminations of opinions until consensus is reached. After each elimination the population remains in a metastable state. The density of active links and the entropy in these states varies from realization to realization. Making some simple assumptions we are able to analytically calculate the average density of active links and the average entropy in each of these states. We also show that, averaged over realizations, the density of active links decays exponentially, with a timescale set by the size and geometry of the graph, but independent of the initial number of opinion states. The decay of the average entropy is exponential only at long times when there are at most two opinions left in the population. Finally, we show how metastable states comprising only a subset of opinions can be artificially engineered by introducing precisely one zealot in each of the prevailing opinions.
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Affiliation(s)
- Lucía Ramirez
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
- Departamento de Física, Universidad Nacional de San Luis, Ejército de Los Andes 950, D5700HHW, San Luis, Argentina
| | - Maxi San Miguel
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Tobias Galla
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
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4
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Phenomenology and dynamics of competitive ecosystems beyond the niche-neutral regimes. Proc Natl Acad Sci U S A 2022; 119:e2204394119. [PMID: 36251996 PMCID: PMC9618050 DOI: 10.1073/pnas.2204394119] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Structure, composition, and stability of ecological populations are shaped by the inter- and intraspecies interactions within their communities. It remains to be fully understood how the interplay of these interactions with other factors, such as immigration, controls the structure, the diversity, and the long-term stability of ecological systems in the presence of noise and fluctuations. We address this problem using a minimal model of interacting multispecies ecological communities that incorporates competition, immigration, and demographic noise. We find that a complete phase diagram exhibits rich behavior with multiple regimes that go beyond the classical "niche" and "neutral" regimes, extending and modifying the "rare biosphere" or "niche-like" dichotomy. In particular, we observe regimes that cannot be characterized as either niche or neutral where a multimodal species abundance distribution is observed. We characterize the transitions between the different regimes and show how these arise from the underlying kinetics of the species turnover, extinction, and invasion. Our model serves as a minimal null model of noisy competitive ecological systems, against which more complex models that include factors such as mutations and environmental noise can be compared.
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Monk T, van Schaik A. Martingales and the fixation time of evolutionary graphs with arbitrary dimensionality. ROYAL SOCIETY OPEN SCIENCE 2022; 9:220011. [PMID: 35573040 PMCID: PMC9091843 DOI: 10.1098/rsos.220011] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/06/2022] [Accepted: 04/01/2022] [Indexed: 05/03/2023]
Abstract
Evolutionary graph theory (EGT) investigates the Moran birth-death process constrained by graphs. Its two principal goals are to find the fixation probability and time for some initial population of mutants on the graph. The fixation probability of graphs has received considerable attention. Less is known about the distribution of fixation time. We derive clean, exact expressions for the full conditional characteristic functions (CCFs) of a close proxy to fixation and extinction times. That proxy is the number of times that the mutant population size changes before fixation or extinction. We derive these CCFs from a product martingale that we identify for an evolutionary graph with any number of partitions. The existence of that martingale only requires that the connections between those partitions are of a certain type. Our results are the first expressions for the CCFs of any proxy to fixation time on a graph with any number of partitions. The parameter dependence of our CCFs is explicit, so we can explore how they depend on graph structure. Martingales are a powerful approach to study principal problems of EGT. Their applicability is invariant to the number of partitions in a graph, so we can study entire families of graphs simultaneously.
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Affiliation(s)
- Travis Monk
- International Centre for Neuromorphic Systems, The MARCS Institute, Western Sydney University, Sydney, Australia
| | - André van Schaik
- International Centre for Neuromorphic Systems, The MARCS Institute, Western Sydney University, Sydney, Australia
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6
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Monk T, van Schaik A. Martingales and the characteristic functions of absorption time on bipartite graphs. ROYAL SOCIETY OPEN SCIENCE 2021; 8:210657. [PMID: 34703620 PMCID: PMC8527206 DOI: 10.1098/rsos.210657] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2021] [Accepted: 09/15/2021] [Indexed: 05/14/2023]
Abstract
Evolutionary graph theory investigates how spatial constraints affect processes that model evolutionary selection, e.g. the Moran process. Its principal goals are to find the fixation probability and the conditional distributions of fixation time, and show how they are affected by different graphs that impose spatial constraints. Fixation probabilities have generated significant attention, but much less is known about the conditional time distributions, even for simple graphs. Those conditional time distributions are difficult to calculate, so we consider a close proxy to it: the number of times the mutant population size changes before absorption. We employ martingales to obtain the conditional characteristic functions (CCFs) of that proxy for the Moran process on the complete bipartite graph. We consider the Moran process on the complete bipartite graph as an absorbing random walk in two dimensions. We then extend Wald's martingale approach to sequential analysis from one dimension to two. Our expressions for the CCFs are novel, compact, exact, and their parameter dependence is explicit. We show that our CCFs closely approximate those of absorption time. Martingales provide an elegant framework to solve principal problems of evolutionary graph theory. It should be possible to extend our analysis to more complex graphs than we show here.
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Affiliation(s)
- Travis Monk
- International Centre for Neuromorphic Systems, The MARCS Institute, Western Sydney University, Sydney, Australia
| | - André van Schaik
- International Centre for Neuromorphic Systems, The MARCS Institute, Western Sydney University, Sydney, Australia
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7
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Berríos-Caro E, Galla T, Constable GWA. Switching environments, synchronous sex, and the evolution of mating types. Theor Popul Biol 2021; 138:28-42. [PMID: 33639174 DOI: 10.1016/j.tpb.2021.02.001] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2020] [Revised: 02/11/2021] [Accepted: 02/12/2021] [Indexed: 01/31/2023]
Abstract
While facultative sex is common in sexually reproducing species, for reasons of tractability most mathematical models assume that such sex is asynchronous in the population. In this paper, we develop a model of switching environments to instead capture the effect of an entire population transitioning synchronously between sexual and asexual modes of reproduction. We use this model to investigate the evolution of the number of self-incompatible mating types in finite populations, which empirically can range from two to thousands. When environmental switching is fast, we recover the results of earlier studies that implicitly assumed populations were engaged in asynchronous sexual reproduction. However when the environment switches slowly, we see deviations from previous asynchronous theory, including a lower number of mating types at equilibrium and bimodality in the stationary distribution of mating types. We provide analytic approximations for both the fast and slow switching regimes, as well as a numerical scheme based on the Kolmogorov equations for the system to quickly evaluate the model dynamics at intermediate parameters. Our approach exploits properties of integer partitions in number theory. We also demonstrate how additional biological processes such as selective sweeps can be accounted for in this switching environment framework, showing that beneficial mutations can further erode mating type diversity in synchronous facultatively sexual populations.
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Affiliation(s)
- Ernesto Berríos-Caro
- Theoretical Physics, Department of Physics and Astronomy, School of Natural Sciences, Faculty of Science and Engineering, The University of Manchester, Manchester M13 9PL, United Kingdom.
| | - Tobias Galla
- Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), 07122 Palma de Mallorca, Spain; Theoretical Physics, Department of Physics and Astronomy, School of Natural Sciences, Faculty of Science and Engineering, The University of Manchester, Manchester M13 9PL, United Kingdom
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8
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Khalil N, Galla T. Zealots in multistate noisy voter models. Phys Rev E 2021; 103:012311. [PMID: 33601527 DOI: 10.1103/physreve.103.012311] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2020] [Accepted: 12/01/2020] [Indexed: 06/12/2023]
Abstract
The noisy voter model is a stylized representation of opinion dynamics. Individuals copy opinions from other individuals, and are subject to spontaneous state changes. In the case of two opinion states this model is known to have a noise-driven transition between a unimodal phase, in which both opinions are present, and a bimodal phase, in which one of the opinions dominates. The presence of zealots can remove the unimodal and bimodal phases in the model with two opinion states. Here we study the effects of zealots in noisy voter models with M>2 opinion states on complete interaction graphs. We find that the phase behavior diversifies, with up to six possible qualitatively different types of stationary states. The presence of zealots removes some of these phases, but not all. We analyze situations in which zealots affect the entire population, or only a fraction of agents, and show that this situation corresponds to a single-community model with a fractional number of zealots, further enriching the phase diagram. Our study is conducted analytically based on effective birth-death dynamics for the number of individuals holding a given opinion. Results are confirmed in numerical simulations.
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Affiliation(s)
- Nagi Khalil
- Escuela Superior de Ciencias Experimentales y Tecnología (ESCET) & GISC, Universidad Rey Juan Carlos, Móstoles 28933, Madrid, Spain
| | - Tobias Galla
- Instituto de Física Interdisciplinar y Sistemas Complejos, IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain and Department of Physics and Astronomy, School of Natural Sciences, The University of Manchester, Manchester M13 9PL, United Kingdom
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9
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Monk T, van Schaik A. Wald’s martingale and the conditional distributions of absorption time in the Moran process. Proc Math Phys Eng Sci 2020. [DOI: 10.1098/rspa.2020.0135] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moranbirth–death process, where that quantity is the number of mutants in a population. In such processes, we are often interested in their absorption (i.e. fixation) probabilities and the conditional distributions of absorption time. Those conditional time distributions can be very difficult to calculate, even for relatively simple processes like the Moran birth–death model. Instead of considering the time to absorption, we consider a closely related quantity: the number of mutant population size changes before absorption. We use Wald’s martingale to obtain the conditional characteristic functions of that quantity in the Moran process. Our expressions are novel, analytical and exact, and their parameter dependence is explicit. We use our results to approximate the conditional characteristic functions of absorption time. We state the conditions under which that approximation is particularly accurate. Martingales are an elegant framework to solve principal problems of evolutionary stochastic processes. They do not require us to evaluate recursion relations, so when they are applicable, we can quickly and tractably obtain absorption probabilities and times of evolutionary models.
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Affiliation(s)
- Travis Monk
- International Centre for Neuromorphic Engineering, MARCS Institute, Western Sydney University, Werrington, NSW 2747, Australia
| | - André van Schaik
- International Centre for Neuromorphic Engineering, MARCS Institute, Western Sydney University, Werrington, NSW 2747, Australia
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10
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Czuppon P, Constable GWA. Invasion and Extinction Dynamics of Mating Types Under Facultative Sexual Reproduction. Genetics 2019; 213:567-580. [PMID: 31391266 PMCID: PMC6781889 DOI: 10.1534/genetics.119.302306] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/09/2019] [Accepted: 08/04/2019] [Indexed: 01/08/2023] Open
Abstract
In sexually reproducing isogamous species, syngamy between gametes is generally not indiscriminate, but rather restricted to occurring between complementary self-incompatible mating types. A longstanding question regards the evolutionary pressures that control the number of mating types observed in natural populations, which ranges from two to many thousands. Here, we describe a population genetic null model of this reproductive system, and derive expressions for the stationary probability distribution of the number of mating types, the establishment probability of a newly arising mating type, and the mean time to extinction of a resident type. Our results yield that the average rate of sexual reproduction in a population correlates positively with the expected number of mating types observed. We further show that the low number of mating types predicted in the rare-sex regime is primarily driven by low invasion probabilities of new mating type alleles, with established resident alleles being very stable over long evolutionary periods. Moreover, our model naturally exhibits varying selection strength dependent on the number of resident mating types. This results in higher extinction and lower invasion rates for an increasing number of residents.
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Affiliation(s)
- Peter Czuppon
- Center for Interdisciplinary Research in Biology, CNRS, Collège de France, PSL Research University, 75231 Paris, France
- Institute of Ecology and Environmental Sciences of Paris, Sorbonne Université, UPEC, CNRS, IRD, INRA, 75252 Paris, France
| | - George W A Constable
- Department of Mathematical Sciences, The University of Bath, BA2 7AY, United Kingdom
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11
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Constable GWA, McKane AJ. Mapping of the stochastic Lotka-Volterra model to models of population genetics and game theory. Phys Rev E 2017; 96:022416. [PMID: 28950630 DOI: 10.1103/physreve.96.022416] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/21/2017] [Indexed: 06/07/2023]
Abstract
The relationship between the M-species stochastic Lotka-Volterra competition (SLVC) model and the M-allele Moran model of population genetics is explored via timescale separation arguments. When selection for species is weak and the population size is large but finite, precise conditions are determined for the stochastic dynamics of the SLVC model to be mappable to the neutral Moran model, the Moran model with frequency-independent selection, and the Moran model with frequency-dependent selection (equivalently a game-theoretic formulation of the Moran model). We demonstrate how these mappings can be used to calculate extinction probabilities and the times until a species' extinction in the SLVC model.
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Affiliation(s)
- George W A Constable
- Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom
- Department of Evolutionary Biology and Environmental Studies, University of Zurich, 8006 Zurich, Switzerland
| | - Alan J McKane
- Theoretical Physics Division, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
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12
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Michaud J. Continuous time limits of the utterance selection model. Phys Rev E 2017; 95:022308. [PMID: 28297849 DOI: 10.1103/physreve.95.022308] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2016] [Indexed: 11/07/2022]
Abstract
In this paper we derive alternative continuous time limits of the utterance selection model (USM) for language change [G. J. Baxter et al., Phys. Rev. E 73, 046118 (2006)PLEEE81539-375510.1103/PhysRevE.73.046118]. This is motivated by the fact that the Fokker-Planck continuous time limit derived in the original version of the USM is only valid for a small range of parameters. We investigate the consequences of relaxing these constraints on parameters. Using the normal approximation of the multinomial approximation, we derive a continuous time limit of the USM in the form of a weak-noise stochastic differential equation. We argue that this weak noise, not captured by the Kramers-Moyal expansion, cannot be neglected. We then propose a coarse-graining procedure, which takes the form of a stochastic version of the heterogeneous mean field approximation. This approximation groups the behavior of nodes of the same degree, reducing the complexity of the problem. With the help of this approximation, we study in detail two simple families of networks: the regular networks and the star-shaped networks. The analysis reveals and quantifies a finite-size effect of the dynamics. If we increase the size of the network by keeping all the other parameters constant, we transition from a state where conventions emerge to a state where no convention emerges. Furthermore, we show that the degree of a node acts as a time scale. For heterogeneous networks such as star-shaped networks, the time scale difference can become very large, leading to a noisier behavior of highly connected nodes.
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Affiliation(s)
- Jérôme Michaud
- School of Physics, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
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13
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Khatri BS. Quantifying evolutionary dynamics from variant-frequency time series. Sci Rep 2016; 6:32497. [PMID: 27616332 PMCID: PMC5018853 DOI: 10.1038/srep32497] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/19/2016] [Accepted: 08/08/2016] [Indexed: 12/05/2022] Open
Abstract
From Kimura's neutral theory of protein evolution to Hubbell's neutral theory of biodiversity, quantifying the relative importance of neutrality versus selection has long been a basic question in evolutionary biology and ecology. With deep sequencing technologies, this question is taking on a new form: given a time-series of the frequency of different variants in a population, what is the likelihood that the observation has arisen due to selection or neutrality? To tackle the 2-variant case, we exploit Fisher's angular transformation, which despite being discovered by Ronald Fisher a century ago, has remained an intellectual curiosity. We show together with a heuristic approach it provides a simple solution for the transition probability density at short times, including drift, selection and mutation. Our results show under that under strong selection and sufficiently frequent sampling these evolutionary parameters can be accurately determined from simulation data and so they provide a theoretical basis for techniques to detect selection from variant or polymorphism frequency time-series.
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Affiliation(s)
- Bhavin S. Khatri
- The Francis Crick Institute, Mill Hill Laboratory, The Ridgeway, London, NW7 1AA, UK
- Division of Infection and Immunity, University College London, London, WC1E 6BT, UK
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Hössjer O, Tyvand PA, Miloh T. Exact Markov chain and approximate diffusion solution for haploid genetic drift with one-way mutation. Math Biosci 2016; 272:100-12. [PMID: 26724565 DOI: 10.1016/j.mbs.2015.12.006] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2015] [Revised: 11/13/2015] [Accepted: 12/04/2015] [Indexed: 11/30/2022]
Abstract
The classical Kimura solution of the diffusion equation is investigated for a haploid random mating (Wright-Fisher) model, with one-way mutations and initial-value specified by the founder population. The validity of the transient diffusion solution is checked by exact Markov chain computations, using a Jordan decomposition of the transition matrix. The conclusion is that the one-way diffusion model mostly works well, although the rate of convergence depends on the initial allele frequency and the mutation rate. The diffusion approximation is poor for mutation rates so low that the non-fixation boundary is regular. When this happens we perturb the diffusion solution around the non-fixation boundary and obtain a more accurate approximation that takes quasi-fixation of the mutant allele into account. The main application is to quantify how fast a specific genetic variant of the infinite alleles model is lost. We also discuss extensions of the quasi-fixation approach to other models with small mutation rates.
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Affiliation(s)
- Ola Hössjer
- Department of Mathematics, Div. of Mathematical Statistics, Stockholm University, Stockholm SE 106 91, Sweden.
| | - Peder A Tyvand
- Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, Ås 1432, Norway.
| | - Touvia Miloh
- Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel.
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15
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Fontanari JF, Serva M. Nonlinear group survival in Kimura’s model for the evolution of altruism. Math Biosci 2014; 249:18-26. [DOI: 10.1016/j.mbs.2014.01.003] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/21/2013] [Revised: 01/03/2014] [Accepted: 01/09/2014] [Indexed: 11/15/2022]
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16
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Abstract
Models relating phenotype space to fitness (phenotype-fitness landscapes) have seen important developments recently. They can roughly be divided into mechanistic models (e.g., metabolic networks) and more heuristic models like Fisher's geometrical model. Each has its own drawbacks, but both yield testable predictions on how the context (genomic background or environment) affects the distribution of mutation effects on fitness and thus adaptation. Both have received some empirical validation. This article aims at bridging the gap between these approaches. A derivation of the Fisher model "from first principles" is proposed, where the basic assumptions emerge from a more general model, inspired by mechanistic networks. I start from a general phenotypic network relating unspecified phenotypic traits and fitness. A limited set of qualitative assumptions is then imposed, mostly corresponding to known features of phenotypic networks: a large set of traits is pleiotropically affected by mutations and determines a much smaller set of traits under optimizing selection. Otherwise, the model remains fairly general regarding the phenotypic processes involved or the distribution of mutation effects affecting the network. A statistical treatment and a local approximation close to a fitness optimum yield a landscape that is effectively the isotropic Fisher model or its extension with a single dominant phenotypic direction. The fit of the resulting alternative distributions is illustrated in an empirical data set. These results bear implications on the validity of Fisher's model's assumptions and on which features of mutation fitness effects may vary (or not) across genomic or environmental contexts.
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Steinrücken M, Wang YXR, Song YS. An explicit transition density expansion for a multi-allelic Wright-Fisher diffusion with general diploid selection. Theor Popul Biol 2012; 83:1-14. [PMID: 23127866 DOI: 10.1016/j.tpb.2012.10.006] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2012] [Revised: 10/14/2012] [Accepted: 10/15/2012] [Indexed: 10/27/2022]
Abstract
Characterizing time-evolution of allele frequencies in a population is a fundamental problem in population genetics. In the Wright-Fisher diffusion, such dynamics is captured by the transition density function, which satisfies well-known partial differential equations. For a multi-allelic model with general diploid selection, various theoretical results exist on representations of the transition density, but finding an explicit formula has remained a difficult problem. In this paper, a technique recently developed for a diallelic model is extended to find an explicit transition density for an arbitrary number of alleles, under a general diploid selection model with recurrent parent-independent mutation. Specifically, the method finds the eigenvalues and eigenfunctions of the generator associated with the multi-allelic diffusion, thus yielding an accurate spectral representation of the transition density. Furthermore, this approach allows for efficient, accurate computation of various other quantities of interest, including the normalizing constant of the stationary distribution and the rate of convergence to this distribution.
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Martínez-García R, Vazquez F, López C, Muñoz MA. Temporal disorder in up-down symmetric systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:051125. [PMID: 23004721 DOI: 10.1103/physreve.85.051125] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/24/2012] [Indexed: 06/01/2023]
Abstract
The effect of temporal disorder on systems with up-down Z_{2} symmetry is studied. In particular, we analyze two well-known families of phase transitions-the Ising and the generalized voter universality classes-and scrutinize the consequences of placing them under fluctuating global conditions. We observe that variability of the control parameter induces in both classes "temporal Griffiths phases" (TGPs). These recently uncovered phases are analogous to standard Griffiths phases appearing in systems with quenched spatial disorder, but where the roles of space and time are exchanged. TGPs are characterized by broad regions in parameter space in which (i) mean first-passage times scale algebraically with system size, and (ii) the system response (e.g., susceptibility) diverges. Our results confirm that TGPs are quite robust and ubiquitous in the presence of temporal disorder. Possible applications of our results to examples in ecology are discussed.
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Affiliation(s)
- Ricardo Martínez-García
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), E-07122 Palma de Mallorca, Spain
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Franzke B, Kosko B. Noise can speed convergence in Markov chains. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:041112. [PMID: 22181092 DOI: 10.1103/physreve.84.041112] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2011] [Revised: 06/29/2011] [Indexed: 05/31/2023]
Abstract
A new theorem shows that noise can speed convergence to equilibrium in discrete finite-state Markov chains. The noise applies to the state density and helps the Markov chain explore improbable regions of the state space. The theorem ensures that a stochastic-resonance noise benefit exists for states that obey a vector-norm inequality. Such noise leads to faster convergence because the noise reduces the norm components. A corollary shows that a noise benefit still occurs if the system states obey an alternate norm inequality. This leads to a noise-benefit algorithm that requires knowledge of the steady state. An alternative blind algorithm uses only past state information to achieve a weaker noise benefit. Simulations illustrate the predicted noise benefits in three well-known Markov models. The first model is a two-parameter Ehrenfest diffusion model that shows how noise benefits can occur in the class of birth-death processes. The second model is a Wright-Fisher model of genotype drift in population genetics. The third model is a chemical reaction network of zeolite crystallization. A fourth simulation shows a convergence rate increase of 64% for states that satisfy the theorem and an increase of 53% for states that satisfy the corollary. A final simulation shows that even suboptimal noise can speed convergence if the noise applies over successive time cycles. Noise benefits tend to be sharpest in Markov models that do not converge quickly and that do not have strong absorbing states.
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Affiliation(s)
- Brandon Franzke
- Center for Quantum Information Science and Technology, Signal and Image Processing Institute, Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA
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Russell DI, Blythe RA. Noise-induced dynamical transition in systems with symmetric absorbing states. PHYSICAL REVIEW LETTERS 2011; 106:165702. [PMID: 21599387 DOI: 10.1103/physrevlett.106.165702] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/09/2010] [Revised: 02/10/2011] [Indexed: 05/30/2023]
Abstract
We investigate the effect of noise strength on the macroscopic ordering dynamics of systems with symmetric absorbing states. Using an explicit stochastic microscopic model, we present evidence for a phase transition in the coarsening dynamics, from an Ising-like to a voter-like behavior, as the noise strength is increased past a nontrivial critical value. By mapping to a thermal diffusion process, we argue that the transition arises due to locally-absorbing states being entered more readily in the high-noise regime, which in turn prevents surface tension from driving the ordering process.
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Affiliation(s)
- D I Russell
- SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom
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Xie X. The Site-Frequency Spectrum of Linked Sites. Bull Math Biol 2010; 73:459-94. [DOI: 10.1007/s11538-010-9534-3] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2009] [Accepted: 03/08/2010] [Indexed: 11/27/2022]
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Vazquez F, López C. Systems with two symmetric absorbing states: relating the microscopic dynamics with the macroscopic behavior. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:061127. [PMID: 19256822 DOI: 10.1103/physreve.78.061127] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2008] [Indexed: 05/27/2023]
Abstract
We propose a general approach to study spin models with two symmetric absorbing states. Starting from the microscopic dynamics on a square lattice, we derive a Langevin equation for the time evolution of the magnetization field, that successfully explains coarsening properties of a wide range of nonlinear voter models and systems with intermediate states. We find that the macroscopic behavior only depends on the first derivatives of the spin-flip probabilities. Moreover, an analysis of the mean-field term reveals the three types of transitions commonly observed in these systems-generalized voter, Ising and directed percolation. Monte Carlo simulations of the spin dynamics qualitatively agree with theoretical predictions.
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Affiliation(s)
- Federico Vazquez
- IFISC, Instituto de Física Interdisciplinar y Sistemas Complejos (CSIC-UIB), E-07122 Palma de Mallorca, Spain.
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