A fundamental aspect of all biological systems is cooperation. Cooperative interactions are required for many levels of biological organization ranging from single cells to groups of animals 1-4 . Human society is based to a large extent on mechanisms that promote cooperation 5-7 . It is well known that in unstructured populations, natural selection favors defectors over cooperators. There is much current interest, however, for studying evolutionary games in structured populations and on graphs 8-17 . These efforts recognize the fact that who-meets-whom is not random, but determined by spatial relationships or social networks 18-24 . Here we describe a surprisingly simple rule, which is a good approximation for all graphs that we have analyzed, including cycles, spatial lattices, random regular graphs, random graphs and scale-free networks 25,26 : natural selection favors cooperation, if the benefit of the altruistic act, b, divided by the cost, c, exceeds the average number of neighbors, k. Therefore, cooperation can evolve as a consequence of 'social viscosity' even in the absence of reputation effects or strategic complexity.A cooperator is someone who pays a cost, c, for another individual to receive a benefit, b. A defector pays no cost and does not distribute any benefits. In evolutionary biology, cost and benefit are measured in terms of fitness. Reproduction can be genetic or cultural. In the latter case, the strategy of someone who does well is imitated by others. In an unstructured population, where all individuals interact equally likely with each other, defectors have a higher average payoff than unconditional cooperators. Therefore, natural selection increases the relative abundance of defectors and drives cooperators to extinction. These evolutionary dynamics hold for the deterministic setting of the replicator equation 27,28 and for stochastic game dynamics of finite populations 29 .In our model, the players of an evolutionary game occupy the vertices of a graph. The edges denote links between individuals in terms of game dynamical interaction and biological reproduction. We assume that the graph is fixed for the duration of the evolutionary dynamics. Consider a population of N individuals consisting of cooperators and defectors. A cooperator helps all individuals to whom it is connected. If a cooperator is connected to k other individuals and i of those are cooperators, then its payoff is bi -ck. A defector does not provide any help, and therefore has no costs, but it can receive the benefit from neighboring cooperators. If a defector is connected to j cooperators, then its payoff is bj.The fitness of an individual is given by a constant term, denoting the baseline fitness, plus the payoff that arises from the game. Strong selection means that the payoff is large compared to the baseline fitness; weak selection means the payoff is small compared to the baseline fitness. The idea behind weak selection is that many different factors contribute to the overall fitness of an individual, and th...

Major efforts to sequence cancer genomes are now occurring throughout the world. Though the emerging data from these studies are illuminating, their reconciliation with epidemiologic and clinical observations poses a major challenge. In the current study, we provide a mathematical model that begins to address this challenge. We model tumors as a discrete time branching process that starts with a single driver mutation and proceeds as each new driver mutation leads to a slightly increased rate of clonal expansion. Using the model, we observe tremendous variation in the rate of tumor development-providing an understanding of the heterogeneity in tumor sizes and development times that have been observed by epidemiologists and clinicians. Furthermore, the model provides a simple formula for the number of driver mutations as a function of the total number of mutations in the tumor. Finally, when applied to recent experimental data, the model allows us to calculate the actual selective advantage provided by typical somatic mutations in human tumors in situ. This selective advantage is surprisingly small-0.004 AE 0.0004-and has major implications for experimental cancer research.genetics | mathematical biology

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Evolutionary game theory studies frequency dependent selection. The fitness of a strategy is not constant, but depends on the relative frequencies of strategies in the population. This type of evolutionary dynamics occurs in many settings of ecology, infectious disease dynamics, animal behavior and social interactions of humans. Traditionally evolutionary game dynamics are studied in well-mixed populations, where the interaction between any two individuals is equally likely. There have also been several approaches to study evolutionary games in structured populations. In this paper we present a simple result that holds for a large variety of population structures. We consider the game between two strategies, A and B, described by the payoff matrix (centeracenterbcenterccenterd). We study a mutation and selection process. If the payoffs are linear in a, b, c, d, then for weak selection strategy A is favored over B if and only if σa + b > c + σd. This means the effect of population structure on strategy selection can be described by a single parameter, σ. We present the values of σ for various examples including the well-mixed population, games on graphs and games in phenotype space. We give a proof for the existence of such a σ, which holds for all population structures and update rules that have certain (natural) properties. We assume weak selection, but allow any mutation rate. We discuss the relationship between σ and the critical benefit to cost ratio for the evolution of cooperation. The single parameter, σ, allows us to quantify the ability of a population structure to promote the evolution of cooperation or to choose efficient equilibria in coordination games.