Arne Traulsen

Comparative lesion sequencing provides insights into tumor evolution

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 ≈17 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.

Via Freedom to Coercion: The Emergence of Costly Punishment

In human societies, cooperative behavior 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 behavior are stable—once established, they prevent dissident minorities from spreading. But how can such costly punishing behavior 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 endeavor, this paves the way for the emergence and establishment of cooperative behavior based on the punishment of defectors. Paradoxically, the freedom to withdraw from the common enterprise leads to enforcement of social norms. Joint enterprises that are compulsory rather than voluntary are less likely to lead to cooperation.

Coevolution of strategy and structure in complex networks with dynamical linking.

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.

Genetic Progression and the Waiting Time to Cancer

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.

Stochastic dynamics of invasion and fixation

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.

Social learning promotes institutions for governing the commons

Theoretical and empirical research highlights the role of punishment in promoting collaborative efforts. However, both the emergence and the stability of costly punishment are problematic issues. It is not clear how punishers can invade a society of defectors by social learning or natural selection, or how second-order free-riders (who contribute to the joint effort but not to the sanctions) can be prevented from drifting into a coercion-based regime and subverting cooperation. Here we compare the prevailing model of peer-punishment with pool-punishment, which consists in committing resources, before the collaborative effort, to prepare sanctions against free-riders. Pool-punishment facilitates the sanctioning of second-order free-riders, because 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.

Coevolutionary dynamics: from finite to infinite populations.

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.

Human strategy updating in evolutionary games

Evolutionary game dynamics describe not only frequency-dependent genetic evolution, but also cultural evolution in humans. In this context, successful strategies spread by imitation. It has been shown that the details of strategy update rules can have a crucial impact on evolutionary dynamics in theoretical models and, for example, can significantly alter the level of cooperation in social dilemmas. What kind of strategy update rules can describe imitation dynamics in humans? Here, we present a way to measure such strategy update rules in a behavioral experiment. We use a setting in which individuals are virtually arranged on a spatial lattice. This produces a large number of different strategic situations from which we can assess strategy updating. Most importantly, spontaneous strategy changes corresponding to mutations or exploration behavior are more frequent than assumed in many models. Our experimental approach to measure properties of the update mechanisms used in theoretical models will be useful for mathematical models of cultural evolution.

Exploration dynamics in evolutionary games

Evolutionary game theory describes systems where individual success is based on the interaction with others. We consider a system in which players unconditionally imitate more successful strategies but sometimes also explore the available strategies at random. Most research has focused on how strategies spread via genetic reproduction or cultural imitation, but random exploration of the available set of strategies has received less attention so far. In genetic settings, the latter corresponds to mutations in the DNA, whereas in cultural evolution, it describes individuals experimenting with new behaviors. Genetic mutations typically occur with very small probabilities, but random exploration of available strategies in behavioral experiments is common. We term this phenomenon “exploration dynamics” to contrast it with the traditional focus on imitation. As an illustrative example of the emerging evolutionary dynamics, we consider a public goods game with cooperators and defectors and add punishers and the option to abstain from the enterprise in further scenarios. For small mutation rates, cooperation (and punishment) is possible only if interactions are voluntary, whereas moderate mutation rates can lead to high levels of cooperation even in compulsory public goods games. This phenomenon is investigated through numerical simulations and analytical approximations.

Evolutionary games in the multiverse

Evolutionary game dynamics of two players with two strategies has been studied in great detail. These games have been used to model many biologically relevant scenarios, ranging from social dilemmas in mammals to microbial diversity. Some of these games may, in fact, take place between a number of individuals and not just between two. Here we address one-shot games with multiple players. As long as we have only two strategies, many results from two-player games can be generalized to multiple players. For games with multiple players and more than two strategies, we show that statements derived for pairwise interactions no longer hold. For two-player games with any number of strategies there can be at most one isolated internal equilibrium. For any number of players with any number of strategies , there can be at most isolated internal equilibria. Multiplayer games show a great dynamical complexity that cannot be captured based on pairwise interactions. Our results hold for any game and can easily be applied to specific cases, such as public goods games or multiplayer stag hunts.