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Kuo YP, Carja O. Evolutionary graph theory beyond single mutation dynamics: on how network structured populations cross fitness landscapes. Genetics 2024:iyae055. [PMID: 38639307 DOI: 10.1093/genetics/iyae055] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2024] [Revised: 03/28/2024] [Accepted: 04/01/2024] [Indexed: 04/20/2024] Open
Abstract
Spatially-resolved datasets are revolutionizing knowledge in molecular biology, yet are under-utilized for questions in evolutionary biology. To gain insight from these large-scale datasets of spatial organization, we need mathematical representations and modeling techniques that can both capture their complexity, but also allow for mathematical tractability. Evolutionary graph theory utilizes the mathematical representation of networks as a proxy for heterogeneous population structure and has started to reshape our understanding of how spatial structure can direct evolutionary dynamics. However, previous results are derived for the case of a single new mutation appearing in the population and the role of network structure in shaping fitness landscape crossing is still poorly understood. Here we study how network structured populations cross fitness landscapes and show that even a simple extension to a two-mutational landscape can exhibit complex evolutionary dynamics that cannot be predicted using previous single-mutation results. We show how our results can be intuitively understood through the lens of how the two main evolutionary properties of a network, the amplification and acceleration factors, change the expected fate of the intermediate mutant in the population and further discuss how to link these models to spatially-resolved datasets of cellular organization.
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Affiliation(s)
- Yang Ping Kuo
- Computational Biology Department, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 15232, USA
| | - Oana Carja
- Computational Biology Department, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 15232, USA
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2
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Chakraborty PP, Nemzer LR, Kassen R. Experimental evidence that network topology can accelerate the spread of beneficial mutations. Evol Lett 2023; 7:447-456. [PMID: 38045727 PMCID: PMC10693003 DOI: 10.1093/evlett/qrad047] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2023] [Revised: 07/26/2023] [Accepted: 09/28/2023] [Indexed: 12/05/2023] Open
Abstract
Whether and how the spatial arrangement of a population influences adaptive evolution has puzzled evolutionary biologists. Theoretical models make conflicting predictions about the probability that a beneficial mutation will become fixed in a population for certain topologies like stars, in which "leaf" populations are connected through a central "hub." To date, these predictions have not been evaluated under realistic experimental conditions. Here, we test the prediction that topology can change the dynamics of fixation both in vitro and in silico by tracking the frequency of a beneficial mutant under positive selection as it spreads through networks of different topologies. Our results provide empirical support that meta-population topology can increase the likelihood that a beneficial mutation spreads, broaden the conditions under which this phenomenon is thought to occur, and points the way toward using network topology to amplify the effects of weakly favored mutations under directed evolution in industrial applications.
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Affiliation(s)
| | - Louis R Nemzer
- Department of Chemistry and Physics, Halmos College of Arts and Sciences, Nova Southeastern University, Ft. Lauderdale, FL, United States
| | - Rees Kassen
- Department of Biology, University of Ottawa, Ottawa, ON, Canada
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Tkadlec J, Kaveh K, Chatterjee K, Nowak MA. Evolutionary dynamics of mutants that modify population structure. J R Soc Interface 2023; 20:20230355. [PMID: 38016637 PMCID: PMC10684346 DOI: 10.1098/rsif.2023.0355] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2023] [Accepted: 11/01/2023] [Indexed: 11/30/2023] Open
Abstract
Natural selection is usually studied between mutants that differ in reproductive rate, but are subject to the same population structure. Here we explore how natural selection acts on mutants that have the same reproductive rate, but different population structures. In our framework, population structure is given by a graph that specifies where offspring can disperse. The invading mutant disperses offspring on a different graph than the resident wild-type. We find that more densely connected dispersal graphs tend to increase the invader's fixation probability, but the exact relationship between structure and fixation probability is subtle. We present three main results. First, we prove that if both invader and resident are on complete dispersal graphs, then removing a single edge in the invader's dispersal graph reduces its fixation probability. Second, we show that for certain island models higher invader's connectivity increases its fixation probability, but the magnitude of the effect depends on the exact layout of the connections. Third, we show that for lattices the effect of different connectivity is comparable to that of different fitness: for large population size, the invader's fixation probability is either constant or exponentially small, depending on whether it is more or less connected than the resident.
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Affiliation(s)
- Josef Tkadlec
- Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
- Computer Science Institute, Charles University, Prague, Czech Republic
| | - Kamran Kaveh
- Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
| | - Krishnendu Chatterjee
- Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
| | - Martin A. Nowak
- Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
- Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA
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4
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Bhaumik J, Masuda N. Fixation probability in evolutionary dynamics on switching temporal networks. J Math Biol 2023; 87:64. [PMID: 37768362 PMCID: PMC10539469 DOI: 10.1007/s00285-023-01987-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/03/2023] [Revised: 08/03/2023] [Accepted: 08/13/2023] [Indexed: 09/29/2023]
Abstract
Population structure has been known to substantially affect evolutionary dynamics. Networks that promote the spreading of fitter mutants are called amplifiers of selection, and those that suppress the spreading of fitter mutants are called suppressors of selection. Research in the past two decades has found various families of amplifiers while suppressors still remain somewhat elusive. It has also been discovered that most networks are amplifiers of selection under the birth-death updating combined with uniform initialization, which is a standard condition assumed widely in the literature. In the present study, we extend the birth-death processes to temporal (i.e., time-varying) networks. For the sake of tractability, we restrict ourselves to switching temporal networks, in which the network structure deterministically alternates between two static networks at constant time intervals or stochastically in a Markovian manner. We show that, in a majority of cases, switching networks are less amplifying than both of the two static networks constituting the switching networks. Furthermore, most small switching networks, i.e., networks on six nodes or less, are suppressors, which contrasts to the case of static networks.
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Affiliation(s)
- Jnanajyoti Bhaumik
- Department of Mathematics, State University of New York at Buffalo, Buffalo, NY, 14260-2900, USA
| | - Naoki Masuda
- Department of Mathematics, State University of New York at Buffalo, Buffalo, NY, 14260-2900, USA.
- Computational and Data-Enabled Science and Engineering Program, State University of New York at Buffalo, Buffalo, NY, 14260-5030, USA.
- Center for Computational Social Science, Kobe University, Kobe, 657-8501, Japan.
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5
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Yagoobi S, Sharma N, Traulsen A. Categorizing update mechanisms for graph-structured metapopulations. J R Soc Interface 2023; 20:20220769. [PMID: 36919418 PMCID: PMC10015335 DOI: 10.1098/rsif.2022.0769] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/16/2023] Open
Abstract
The structure of a population strongly influences its evolutionary dynamics. In various settings ranging from biology to social systems, individuals tend to interact more often with those present in their proximity and rarely with those far away. A common approach to model the structure of a population is evolutionary graph theory. In this framework, each graph node is occupied by a reproducing individual. The links connect these individuals to their neighbours. The offspring can be placed on neighbouring nodes, replacing the neighbours-or the progeny of its neighbours can replace a node during the course of ongoing evolutionary dynamics. Extending this theory by replacing single individuals with subpopulations at nodes yields a graph-structured metapopulation. The dynamics between the different local subpopulations is set by an update mechanism. There are many such update mechanisms. Here, we classify update mechanisms for structured metapopulations, which allows to find commonalities between past work and illustrate directions for further research and current gaps of investigation.
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Affiliation(s)
- Sedigheh Yagoobi
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August-Thienemann Strasse 2, Plön 24306, Germany
| | - Nikhil Sharma
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August-Thienemann Strasse 2, Plön 24306, Germany
| | - Arne Traulsen
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, August-Thienemann Strasse 2, Plön 24306, Germany
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6
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Abstract
Evolutionary dynamics on graphs has remarkable features: For example, it has been shown that amplifiers of selection exist that-compared to an unstructured population-increase the fixation probability of advantageous mutations, while they decrease the fixation probability of disadvantageous mutations. So far, the theoretical literature has focused on the case of a single mutant entering a graph-structured population, asking how the graph affects the probability that a mutant takes over a population and the time until this typically happens. For continuously evolving systems, the more relevant case is that mutants constantly arise in an evolving population. Typically, such mutations occur with a small probability during reproduction events. We thus focus on the low mutation rate limit. The probability distribution for the fitness in this process converges to a steady state at long times. Intuitively, amplifiers of selection are expected to increase the population's mean fitness in the steady state. Similarly, suppressors of selection are expected to decrease the population's mean fitness in the steady state. However, we show that another set of graphs, called suppressors of fixation, can attain the highest population mean fitness. The key reason behind this is their ability to efficiently reject deleterious mutants. This illustrates the importance of the deleterious mutant regime for the long-term evolutionary dynamics, something that seems to have been overlooked in the literature so far.
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Coggan H, Page KM. The role of evolutionary game theory in spatial and non-spatial models of the survival of cooperation in cancer: a review. J R Soc Interface 2022; 19:20220346. [PMID: 35975562 PMCID: PMC9382458 DOI: 10.1098/rsif.2022.0346] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
Abstract
Evolutionary game theory (EGT) is a branch of mathematics which considers populations of individuals interacting with each other to receive pay-offs. An individual’s pay-off is dependent on the strategy of its opponent(s) as well as on its own, and the higher its pay-off, the higher its reproductive fitness. Its offspring generally inherit its interaction strategy, subject to random mutation. Over time, the composition of the population shifts as different strategies spread or are driven extinct. In the last 25 years there has been a flood of interest in applying EGT to cancer modelling, with the aim of explaining how cancerous mutations spread through healthy tissue and how intercellular cooperation persists in tumour-cell populations. This review traces this body of work from theoretical analyses of well-mixed infinite populations through to more realistic spatial models of the development of cooperation between epithelial cells. We also consider work in which EGT has been used to make experimental predictions about the evolution of cancer, and discuss work that remains to be done before EGT can make large-scale contributions to clinical treatment and patient outcomes.
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Affiliation(s)
- Helena Coggan
- Department of Mathematics, University College London, London, UK
| | - Karen M Page
- Department of Mathematics, University College London, London, UK
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8
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Abstract
Cooperation is prevalent in nature, not only in the context of social interactions within the animal kingdom but also on the cellular level. In cancer, for example, tumour cells can cooperate by producing growth factors. The evolution of cooperation has traditionally been studied for well-mixed populations under the framework of evolutionary game theory, and more recently for structured populations using evolutionary graph theory (EGT). The population structures arising due to cellular arrangement in tissues, however, are dynamic and thus cannot be accurately represented by either of these frameworks. In this work, we compare the conditions for cooperative success in an epithelium modelled using EGT, to those in a mechanical model of an epithelium—the Voronoi tessellation (VT) model. Crucially, in this latter model, cells are able to move, and birth and death are not spatially coupled. We calculate fixation probabilities in the VT model through simulation and an approximate analytic technique and show that this leads to stronger promotion of cooperation in comparison with the EGT model.
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Affiliation(s)
- Jessie Renton
- Department of Mathematics, University College London , Gower Street, London WC1E 6BT , UK
| | - Karen M Page
- Department of Mathematics, University College London , Gower Street, London WC1E 6BT , UK
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9
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Abstract
The incubation period for typhoid, polio, measles, leukemia and many other diseases follows a right-skewed, approximately lognormal distribution. Although this pattern was discovered more than sixty years ago, it remains an open question to explain its ubiquity. Here, we propose an explanation based on evolutionary dynamics on graphs. For simple models of a mutant or pathogen invading a network-structured population of healthy cells, we show that skewed distributions of incubation periods emerge for a wide range of assumptions about invader fitness, competition dynamics, and network structure. The skewness stems from stochastic mechanisms associated with two classic problems in probability theory: the coupon collector and the random walk. Unlike previous explanations that rely crucially on heterogeneity, our results hold even for homogeneous populations. Thus, we predict that two equally healthy individuals subjected to equal doses of equally pathogenic agents may, by chance alone, show remarkably different time courses of disease. When one child goes to school with a throat infection, many of his or her classmates will often start to come down with a sore throat after two or three days. A few of the children will get sick sooner, the very next day, while others may take about a week. As such, there is a distribution of incubation periods – the time from exposure to illness – across the children in the class. When plotted on a graph, the distribution of incubation periods is not the normal bell curve. Rather the curve looks lopsided, with a long tail on the right. Plotting the logarithms of the incubation periods, however, rather than the incubation periods themselves, does give a normal distribution. As such, statisticians refer to this kind of curve as a “lognormal distribution". Remarkably, many other, completely unrelated, diseases – like typhoid fever or bladder cancer – also have approximately lognormal distributions of incubation periods. This raised the question: why do such different diseases show such a similar curve? Working with a simple mathematical model in which chance plays a key role, Ottino-Löffler et al. calculate how long it takes for a bacterial infection or cancer cell to take over a network of healthy cells. The model explains why a lognormal-like distribution of incubation periods, modeled as takeover times, is so ubiquitous. It emerges from the random dynamics of the incubation process itself, as the disease-causing microbe or mutant cancer cell competes with the cells of the host. Intuitively, this new analysis builds on insights from the “coupon collector’s problem”: a classical problem in mathematics that describes the situation where a person collects items like baseball cards, stamps, or cartoon monsters in a videogame. If a random item arrives every day, and the collector’s luck is bad, they may have to wait a long time to collect those last few items. Similarly, in the model of Ottino-Löffler et al., the takeover time is dominated by dramatic slowdowns near the start or end of the infection process. These effects lead to an approximately lognormal distribution, with long waits, as seen in so many diseases. Ottino-Löffler et al. do not anticipate that their findings will have direct benefits for medicine or public health. Instead, they believe their results could help to advance basic research in the fields of epidemiology, evolutionary biology and cancer research. The findings might also make an impact outside biology. The term “contagion” has now become a familiar metaphor for the spread of everything from computer viruses to bank failures. This model sheds light on how long it takes for a contagion to take over a network, for a variety of idealized networks and spreading processes.
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Affiliation(s)
| | - Jacob G Scott
- Department of Translational Hematology and Oncology Research, Cleveland Clinic, Cleveland, United States.,Department of Radiation Oncology, Cleveland Clinic, Cleveland, United States
| | - Steven H Strogatz
- Center for Applied Mathematics, Cornell University, Ithaca, United States
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10
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Abstract
Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth-death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations.
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Affiliation(s)
- Laura Hindersin
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Plön, Germany
| | - Arne Traulsen
- Department of Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Plön, Germany
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11
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Abstract
The emergence of cooperation is a central question in evolutionary biology. Microorganisms often cooperate by producing a chemical resource (a public good) that benefits other cells. The sharing of public goods depends on their diffusion through space. Previous theory suggests that spatial structure can promote evolution of cooperation, but the diffusion of public goods introduces new phenomena that must be modeled explicitly. We develop an approach where colony geometry and public good diffusion are described by graphs. We find that the success of cooperation depends on a simple relation between the benefits and costs of the public good, the amount retained by a producer, and the average amount retained by each of the producer's neighbors. These quantities are derived as analytic functions of the graph topology and diffusion rate. In general, cooperation is favored for small diffusion rates, low colony dimensionality, and small rates of decay of the public good. DOI: http://dx.doi.org/10.7554/eLife.01169.001.
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Affiliation(s)
- Benjamin Allen
- Department of Mathematics, Emmanuel College, Boston, United States
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12
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Wyatt GAK, West SA, Gardner A. Can natural selection favour altruism between species? J Evol Biol 2013; 26:1854-65. [PMID: 23848844 DOI: 10.1111/jeb.12195] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/04/2013] [Revised: 04/09/2013] [Accepted: 05/06/2013] [Indexed: 11/27/2022]
Abstract
Darwin suggested that the discovery of altruism between species would annihilate his theory of natural selection. However, it has not been formally shown whether between-species altruism can evolve by natural selection, or why this could never happen. Here, we develop a spatial population genetic model of two interacting species, showing that indiscriminate between species helping can be favoured by natural selection. We then ask if this helping behaviour constitutes altruism between species, using a linear-regression analysis to separate the total action of natural selection into its direct and indirect (kin selected) components. We show that our model can be interpreted in two ways, as either altruism within species, or altruism between species. This ambiguity arises depending on whether or not we treat genes in the other species as predictors of an individual's fitness, which is equivalent to treating these individuals as agents (actors or recipients). Our formal analysis, which focuses upon evolutionary dynamics rather than agents and their agendas, cannot resolve which is the better approach. Nonetheless, because a within-species altruism interpretation is always possible, our analysis supports Darwin's suggestion that natural selection does not favour traits that provide benefits exclusively to individuals of other species.
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Affiliation(s)
- G A K Wyatt
- Department of Zoology, University of Oxford, Oxford OX1 3PS, UK.
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Allen B, Traulsen A, Tarnita CE, Nowak MA. How mutation affects evolutionary games on graphs. J Theor Biol 2012; 299:97-105. [PMID: 21473871 PMCID: PMC3150603 DOI: 10.1016/j.jtbi.2011.03.034] [Citation(s) in RCA: 67] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/20/2011] [Revised: 03/27/2011] [Accepted: 03/29/2011] [Indexed: 11/16/2022]
Abstract
Evolutionary dynamics are affected by population structure, mutation rates and update rules. Spatial or network structure facilitates the clustering of strategies, which represents a mechanism for the evolution of cooperation. Mutation dilutes this effect. Here we analyze how mutation influences evolutionary clustering on graphs. We introduce new mathematical methods to evolutionary game theory, specifically the analysis of coalescing random walks via generating functions. These techniques allow us to derive exact identity-by-descent (IBD) probabilities, which characterize spatial assortment on lattices and Cayley trees. From these IBD probabilities we obtain exact conditions for the evolution of cooperation and other game strategies, showing the dual effects of graph topology and mutation rate. High mutation rates diminish the clustering of cooperators, hindering their evolutionary success. Our model can represent either genetic evolution with mutation, or social imitation processes with random strategy exploration.
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Affiliation(s)
- Benjamin Allen
- Program for Evolutionary Dynamics, Department of Mathematics, Harvard University, One Brattle Square, Cambridge, MA 02138, United States.
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Fu F, Wang L, Nowak MA, Hauert C. Evolutionary dynamics on graphs: Efficient method for weak selection. Phys Rev E Stat Nonlin Soft Matter Phys 2009; 79:046707. [PMID: 19518380 PMCID: PMC2735202 DOI: 10.1103/physreve.79.046707] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2008] [Revised: 03/02/2009] [Indexed: 05/27/2023]
Abstract
Investigating the evolutionary dynamics of game theoretical interactions in populations where individuals are arranged on a graph can be challenging in terms of computation time. Here, we propose an efficient method to study any type of game on arbitrary graph structures for weak selection. In this limit, evolutionary game dynamics represents a first-order correction to neutral evolution. Spatial correlations can be empirically determined under neutral evolution and provide the basis for formulating the game dynamics as a discrete Markov process by incorporating a detailed description of the microscopic dynamics based on the neutral correlations. This framework is then applied to one of the most intriguing questions in evolutionary biology: the evolution of cooperation. We demonstrate that the degree heterogeneity of a graph impedes cooperation and that the success of tit for tat depends not only on the number of rounds but also on the degree of the graph. Moreover, considering the mutation-selection equilibrium shows that the symmetry of the stationary distribution of states under weak selection is skewed in favor of defectors for larger selection strengths. In particular, degree heterogeneity--a prominent feature of scale-free networks--generally results in a more pronounced increase in the critical benefit-to-cost ratio required for evolution to favor cooperation as compared to regular graphs. This conclusion is corroborated by an analysis of the effects of population structures on the fixation probabilities of strategies in general 2 x 2 games for different types of graphs. Computer simulations confirm the predictive power of our method and illustrate the improved accuracy as compared to previous studies.
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Affiliation(s)
- Feng Fu
- Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts 02138, USA
- Center for Systems and Control, State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China
| | - Long Wang
- Center for Systems and Control, State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China
| | - Martin A. Nowak
- Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts 02138, USA
- Department of Organismic and Evolutionary Biology, Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Christoph Hauert
- Program for Evolutionary Dynamics, Harvard University, Cambridge, Massachusetts 02138, USA
- Department of Mathematics, University of British Columbia, Vancouver B.C. Canada V6T 1Z2
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15
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Ohtsuki H, Pacheco JM, Nowak MA. Evolutionary graph theory: breaking the symmetry between interaction and replacement. J Theor Biol 2007; 246:681-94. [PMID: 17350049 PMCID: PMC2396517 DOI: 10.1016/j.jtbi.2007.01.024] [Citation(s) in RCA: 139] [Impact Index Per Article: 8.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2006] [Revised: 01/25/2007] [Accepted: 01/29/2007] [Indexed: 11/28/2022]
Abstract
We study evolutionary dynamics in a population whose structure is given by two graphs: the interaction graph determines who plays with whom in an evolutionary game; the replacement graph specifies the geometry of evolutionary competition and updating. First, we calculate the fixation probabilities of frequency dependent selection between two strategies or phenotypes. We consider three different update mechanisms: birth-death, death-birth and imitation. Then, as a particular example, we explore the evolution of cooperation. Suppose the interaction graph is a regular graph of degree h, the replacement graph is a regular graph of degree g and the overlap between the two graphs is a regular graph of degree l. We show that cooperation is favored by natural selection if b/c>hg/l. Here, b and c denote the benefit and cost of the altruistic act. This result holds for death-birth updating, weak-selection and large population size. Note that the optimum population structure for cooperators is given by maximum overlap between the interaction and the replacement graph (g=h=l), which means that the two graphs are identical. We also prove that a modified replicator equation can describe how the expected values of the frequencies of an arbitrary number of strategies change on replacement and interaction graphs: the two graphs induce a transformation of the payoff matrix.
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Affiliation(s)
- Hisashi Ohtsuki
- Program for Evolutionary Dynamics, Harvard University, Cambridge MA 02138, USA.
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16
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Abstract
We study evolutionary games on graphs. Each player is represented by a vertex of the graph. The edges denote who meets whom. A player can use any one of n strategies. Players obtain a payoff from interaction with all their immediate neighbors. We consider three different update rules, called 'birth-death', 'death-birth' and 'imitation'. A fourth update rule, 'pairwise comparison', is shown to be equivalent to birth-death updating in our model. We use pair approximation to describe the evolutionary game dynamics on regular graphs of degree k. In the limit of weak selection, we can derive a differential equation which describes how the average frequency of each strategy on the graph changes over time. Remarkably, this equation is a replicator equation with a transformed payoff matrix. Therefore, moving a game from a well-mixed population (the complete graph) onto a regular graph simply results in a transformation of the payoff matrix. The new payoff matrix is the sum of the original payoff matrix plus another matrix, which describes the local competition of strategies. We discuss the application of our theory to four particular examples, the Prisoner's Dilemma, the Snow-Drift game, a coordination game and the Rock-Scissors-Paper game.
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Affiliation(s)
- Hisashi Ohtsuki
- Department of Biology, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan.
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