Punishment in Public Goods games leads to meta-stable phase transitions and hysteresis

The evolution of cooperation has been a perennial problem in evolutionary biology because cooperation can be undermined by selfish cheaters who gain an advantage in the short run, while compromising the long-term viability of the population. Evolutionary game theory has shown that under certain conditions, cooperation nonetheless evolves stably, for example if players have the opportunity to punish cheaters that benefit from a public good yet refuse to pay into the common pool. However, punishment has remained enigmatic because it is costly, and difficult to maintain. On the other hand, cooperation emerges naturally in the Public Goods game if the synergy of the public good (the factor multiplying the public good investment) is sufficiently high. In terms of this synergy parameter, the transition from defection to cooperation can be viewed as a phase transition with the synergy as the critical parameter. We show here that punishment reduces the critical value at which cooperation occurs, but also creates the possibility of meta-stable phase transitions, where populations can "tunnel" into the cooperating phase below the critical value. At the same time, cooperating populations are unstable even above the critical value, because a group of defectors that are large enough can "nucleate" such a transition. We study the mean-field theoretical predictions via agent-based simulations of finite populations using an evolutionary approach where the decisions to cooperate or to punish are encoded genetically in terms of evolvable probabilities. We recover the theoretical predictions and demonstrate that the population shows hysteresis, as expected in systems that exhibit super-heating and super-cooling. We conclude that punishment can stabilize populations of cooperators below the critical point, but it is a two-edged sword: it can also stabilize defectors above the critical point.Comment: 22 pages, 9 figures. Slight title change, version that appears in Physical Biolog

Strong Selection Significantly Increases Epistatic Interactions in the Long-Term Evolution of a Protein

Epistatic interactions between residues determine a protein's adaptability and shape its evolutionary trajectory. When a protein experiences a changed environment, it is under strong selection to find a peak in the new fitness landscape. It has been shown that strong selection increases epistatic interactions as well as the ruggedness of the fitness landscape, but little is known about how the epistatic interactions change under selection in the long-term evolution of a protein. Here we analyze the evolution of epistasis in the protease of the human immunodeficiency virus type 1 (HIV-1) using protease sequences collected for almost a decade from both treated and untreated patients, to understand how epistasis changes and how those changes impact the long-term evolvability of a protein. We use an information-theoretic proxy for epistasis that quantifies the co-variation between sites, and show that positive information is a necessary (but not sufficient) condition that detects epistasis in most cases. We analyze the "fossils" of the evolutionary trajectories of the protein contained in the sequence data, and show that epistasis continues to enrich under strong selection, but not for proteins whose environment is unchanged. The increase in epistasis compensates for the information loss due to sequence variability brought about by treatment, and facilitates adaptation in the increasingly rugged fitness landscape of treatment. While epistasis is thought to enhance evolvability via valley-crossing early-on in adaptation, it can hinder adaptation later when the landscape has turned rugged. However, we find no evidence that the HIV-1 protease has reached its potential for evolution after 9 years of adapting to a drug environment that itself is constantly changing.Comment: 25 pages, 9 figures, plus Supplementary Material including Supplementary Text S1-S7, Supplementary Tables S1-S2, and Supplementary Figures S1-2. Version that appears in PLoS Genetic

Evolution of genetic organization in digital organisms

We examine the evolution of expression patterns and the organization of genetic information in populations of self-replicating digital organisms. Seeding the experiments with a linearly expressed ancestor, we witness the development of complex, parallel secondary expression patterns. Using principles from information theory, we demonstrate an evolutionary pressure towards overlapping expressions causing variation (and hence further evolution) to sharply drop. Finally, we compare the overlapping sections of dominant genomes to those portions which are singly expressed and observe a significant difference in the entropy of their encoding.Comment: 18 pages with 5 embedded figures. Proc. of DIMACS workshop on "Evolution as Computation", Jan. 11-12, Princeton, NJ. L. Landweber and E. Winfree, eds. (Springer, 1999