We show that the times separating the birth of benign, invasive, and metastatic tumor cells can be determined by analysis of the mutations they have in common. When combined with prior clinical observations, these analyses suggest the following general conclusions about colorectal tumorigenesis: (i) It takes Ϸ17 years for a large benign tumor to evolve into an advanced cancer but <2 years for cells within that cancer to acquire the ability to metastasize; (ii) it requires few, if any, selective events to transform a highly invasive cancer cell into one with the capacity to metastasize; (iii) the process of cell culture ex vivo does not introduce new clonal mutations into colorectal tumor cell populations; and (iv) the rates at which point mutations develop in advanced cancers are similar to those of normal cells. These results have important implications for understanding human tumor pathogenesis, particularly those associated with metastasis.cancer genetics ͉ colorectal cancer ͉ metastasis ͉ stem cells

In human societies, cooperative behaviour in joint enterprises is often enforced through institutions that impose sanctions on defectors. Many experiments on so-called public goods games have shown that in the absence of such institutions, individuals are willing to punish defectors, even at a cost to themselves. Theoretical models confirm that social norms prescribing the punishment of uncooperative behaviour are stable: once established, they prevent dissident minorities from spreading. But how can such costly punishing behaviour gain a foothold in the population? A surprisingly simple model shows that if individuals have the option to stand aside and abstain from the joint endeavour, this paves the way for the emergence and establishment of cooperative behaviour based on the punishment of defectors. Paradoxically, the freedom to withdraw from the common enterprise leads to enforcement of social norms. Joint enterprises which are compulsory rather than voluntary are less likely to lead to cooperation. Keywordsevolutionary game theory; public goods games; cooperation; altruistic punishment; voluntary interactions An impressive body of evidence shows that many humans are willing to pay a personal cost in order to punish wrong-doers (1-8). In particular, punishment is an effective mechanism to ensure cooperation in public goods interactions (9-11). All human populations seem willing to use costly punishment to varying degrees, and their willingness to punish correlates with the propensity for altruistic contributions (12). This raises an evolutionary problem: in joint enterprises, free-riding individuals who do not contribute, but exploit the efforts of others, fare better than those who pay the cost of contributing. If successful behaviour spreads, for instance through imitation, these defectors will eventually take over. Punishment reduces the defectors' payoff, and thus may solve the social dilemma. But since punishment is costly, it also reduces the punishers' payoff. This raises a 'second order social dilemma': costly punishment seems to be an altruistic act, since individuals who contribute, but do not punish, are better off than the punishers.

We propose a minimalist stochastic model of multilevel (or group) selection. A population is subdivided into groups. Individuals interact with other members of the group in an evolutionary game that determines their fitness. Individuals reproduce, and offspring are added to the same group. If a group reaches a certain size, it can split into two. Faster reproducing individuals lead to larger groups that split more often. In our model, higher-level selection emerges as a byproduct of individual reproduction and population structure. We derive a fundamental condition for the evolution of cooperation by group selection: if b͞c > 1 ؉ n͞m, then group selection favors cooperation. The parameters b and c denote the benefit and cost of the altruistic act, whereas n and m denote the maximum group size and the number of groups. The model can be extended to more than two levels of selection and to include migration.finite populations ͉ prisoner's dilemma ͉ group selection ͉ fixation probability ͉ stochastic process C ompetition between groups can lead to selection of cooperative behavior. This idea can be traced back to Charles Darwin, who wrote in 1871: ''There can be no doubt that a tribe including many members who..were always ready to give aid to each other and to sacrifice themselves for the common good, would be victorious over other tribes; and this would be natural selection'' (1). The first mathematical model of group selection was proposed in 1945 by Sewall Wright (2). The enthusiastic attempt of early group selectionists to understand all of the evolution of altruism from this one perspective (3, 4) has led to vigorous criticism and a general denial of such ideas for decades (5-8). Only a small number of biologists continued to work in this area (9-19). Over many years, D. S. Wilson was the main proponent of the idea of group selection (20-22). Nowadays, there seems to be a renewed interest in the subject, as demonstrated by many empirical and theoretical studies (23-28). The current analysis of group selection is also closely related to the attempt at understanding the simultaneous effect of natural selection on multiple-levels (29-31). In our opinion, group selection is an important organizing principle that permeates evolutionary processes from the emergence of the first cells to eusociality and the economics of nations.Consider a population that is subdivided into groups. The fitness of individuals is determined by the payoff from an evolutionary game. Interactions occur between members of the same group. We model stochastic evolutionary dynamics. In any one time step, a single individual from the entire population is chosen for reproduction with a probability proportional to its fitness. The offspring is added to the same group. If the group reaches a critical size, n, it will divide into two groups with probability q. The members of the group are randomly distributed over the two daughter groups, see Fig. 1. With probability 1Ϫ q, the group does not divide, but a random individual of the group is eliminate...

Theoretical and empirical research highlights the role of punishment in promoting collaborative efforts 1,2,3,4,5 .However, both the emergence and the stability of costly punishment are problematic issues. How can punishers invade a society of defectors by social learning or natural selection, and how can second-order exploiters (who contribute to the joint effort but not to the sanctions) be prevented from drifting into a coercion-based regime and subverting cooperation? Here, we compare the prevailing model of peer-punishment 6,7,8 with pool-punishment, which consists in committing resources, prior to the collaborative effort, to prepare sanctions against free-riders. Pool punishment facilitates the sanctioning of second-order exploiters, since these are exposed even if everyone contributes to the common good. In the absence of such second-order punishment, peer-punishers do better than pool-punishers, but with second-order punishment, the situation is reversed. Efficiency is traded for stability. Neither other-regarding tendencies or preferences for reciprocity and equity, nor group selection or prescriptions from higher authorities are necessary for the emergence and stability of rudimentary forms of sanctioning institutions regulating common pool resources and enforcing collaborative efforts.Many economic experiments on 'public goods games' (PG games) have shown that a substantial fraction of players are willing to incur costs in order to impose fines on exploiters, i.e., those who do not contribute to the joint effort 1,2,3,4,5,6,7,8 . As a consequence, the threat of punishment looms credibly enough to increase the average level of pro-social contributions. However, the sanctioning system is itself a public good. Thus punishers are often seen as altruistic, since others benefit from their costly efforts 9,10,11,12,13 . Conversely, those who refrain from punishing exploiters are 'secondorder free-riders'. Among self-interested agents, second-order free-riding should spread and ultimately cause the collapse of cooperation.A solution is to also punish second-order free-riders 14 . But such 'second-order punishment' risks being subverted by third-order free-riders in turn, leading to infinite regress. Moreover, if everyone contributes to the public good, second-order free riders will not be spotted. Their number can grow through neutral drift, ultimately allowing defectors to invade with impunity.We show how a simple mechanism can overcome this objection. There exists a variety of sanctioning systems. Most experiments on PG with punishment have considered peer punishment: after the PG game, individuals can impose fines on exploiters, at a cost to themselves. Interestingly, the first experiment on PG with punishment 15 considered a different mechanism. Here, players decide whether to contribute to a 'punishment pool' before contributing to the PG. This can be viewed as a first step towards an institutionalized mechanism for punishing exploiters, and compared with the self-financed contract enforcement games in ...

We study evolutionary game dynamics in finite populations. We analyze an evolutionary process, which we call pairwise comparison, for which we adopt the ubiquitous Fermi distribution function from statistical mechanics. The inverse temperature in this process controls the intensity of selection, leading to a unified framework for evolutionary dynamics at all intensities of selection, from random drift to imitation dynamics. We derive a simple closed formula that determines the feasibility of cooperation in finite populations, whenever cooperation is modeled in terms of any symmetric two-person game. In contrast with previous results, the present formula is valid at all intensities of selection and for any initial condition. We investigate the evolutionary dynamics of cooperators in finite populations, and study the interplay between intensity of selection and the remnants of interior fixed points in infinite populations, as a function of a given initial number of cooperators, showing how this interplay strongly affects the approach to fixation of a given trait in finite populations, leading to counterintuitive results at different intensities of selection.

Traditionally, frequency dependent evolutionary dynamics is described by deterministic replicator dynamics assuming implicitly infinite population sizes. Only recently have stochastic processes been introduced to study evolutionary dynamics in finite populations. However, the relationship between deterministic and stochastic approaches remained unclear. Here we solve this problem by explicitly considering large populations. In particular, we identify different microscopic stochastic processes that lead to the standard or the adjusted replicator dynamics. Moreover, differences on the individual level can lead to qualitatively different dynamics in asymmetric conflicts and, depending on the population size, can even invert the direction of the evolutionary process.

Cancer results from genetic alterations that disturb the normal cooperative behavior of cells. Recent high-throughput genomic studies of cancer cells have shown that the mutational landscape of cancer is complex and that individual cancers may evolve through mutations in as many as 20 different cancer-associated genes. We use data published by Sjöblom et al. (2006) to develop a new mathematical model for the somatic evolution of colorectal cancers. We employ the Wright-Fisher process for exploring the basic parameters of this evolutionary process and derive an analytical approximation for the expected waiting time to the cancer phenotype. Our results highlight the relative importance of selection over both the size of the cell population at risk and the mutation rate. The model predicts that the observed genetic diversity of cancer genomes can arise under a normal mutation rate if the average selective advantage per mutation is on the order of 1%. Increased mutation rates due to genetic instability would allow even smaller selective advantages during tumorigenesis. The complexity of cancer progression can be understood as the result of multiple sequential mutations, each of which has a relatively small but positive effect on net cell growth.

We introduce a model in which individuals differ in the rate at which they seek new interactions with others, making rational decisions modeled as general symmetric two-player games. Once a link between two individuals has formed, the productivity of this link is evaluated. Links can be broken off at different rates. We provide analytic results for the limiting cases where linking dynamics is much faster than evolutionary dynamics and vice versa, and show how the individual capacity of forming new links or severing inconvenient ones maps into the problem of strategy evolution in a well-mixed population under a different game. For intermediate ranges, we investigate numerically the detailed interplay determined by these two time scales and show that the scope of validity of the analytical results extends to a much wider ratio of time scales than expected.