Modularity and anti-modularity in networks with arbitrary degree distribution

Networks describing the interaction of the elements that constitute a complex system grow and develop via a number of different mechanisms, such as the addition and deletion of nodes, the addition and deletion of edges, as well as the duplication or fusion of nodes. While each of these mechanisms can have a different cause depending on whether the network is biological, technological, or social, their impact on the network's structure, as well as its local and global properties, is similar. This allows us to study how each of these mechanisms affects networks either alone or together with the other processes, and how they shape the characteristics that have been observed. We study how a network's growth parameters impact the distribution of edges in the network, how they affect a network's modularity, and point out that some parameters will give rise to networks that have the opposite tendency, namely to display anti-modularity. Within the model we are describing, we can search the space of possible networks for parameter sets that generate networks that are very similar to well-known and well-studied examples, such as the brain of a worm, and the network of interactions of the proteins in baker's yeast.Comment: 23 pages. 13 figures, 1 table. Includes Supplementary tex

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

Critical properties of complex fitness landscapes

Evolutionary adaptation is the process that increases the fit of a population to the fitness landscape it inhabits. As a consequence, evolutionary dynamics is shaped, constrained, and channeled, by that fitness landscape. Much work has been expended to understand the evolutionary dynamics of adapting populations, but much less is known about the structure of the landscapes. Here, we study the global and local structure of complex fitness landscapes of interacting loci that describe protein folds or sets of interacting genes forming pathways or modules. We find that in these landscapes, high peaks are more likely to be found near other high peaks, corroborating Kauffman's "Massif Central" hypothesis. We study the clusters of peaks as a function of the ruggedness of the landscape and find that this clustering allows peaks to form interconnected networks. These networks undergo a percolation phase transition as a function of minimum peak height, which indicates that evolutionary trajectories that take no more than two mutations to shift from peak to peak can span the entire genetic space. These networks have implications for evolution in rugged landscapes, allowing adaptation to proceed after a local fitness peak has been ascended.Comment: 7 pages, 6 figures, requires alifex11.sty. To appear in Proceedings of 12th International Conference on Artificial Lif

Does self-replication imply evolvability?

The most prominent property of life on Earth is its ability to evolve. It is often taken for granted that self-replication--the characteristic that makes life possible--implies evolvability, but many examples such as the lack of evolvability in computer viruses seem to challenge this view. Is evolvability itself a property that needs to evolve, or is it automatically present within any chemistry that supports sequences that can evolve in principle? Here, we study evolvability in the digital life system Avida, where self-replicating sequences written by hand are used to seed evolutionary experiments. We use 170 self-replicators that we found in a search through 3 billion randomly generated sequences (at three different sequence lengths) to study the evolvability of generic rather than hand-designed self-replicators. We find that most can evolve but some are evolutionarily sterile. From this limited data set we are led to conclude that evolvability is a likely--but not a guaranteed-- property of random replicators in a digital chemistry.Comment: 8 pages, 5 figures. To appear in "Advances in Artificial Life": Proceedings of the 13th European Conference on Artificial Life (ECAL 2015

Evolution of complex modular biological networks

Biological networks have evolved to be highly functional within uncertain environments while remaining extremely adaptable. One of the main contributors to the robustness and evolvability of biological networks is believed to be their modularity of function, with modules defined as sets of genes that are strongly interconnected but whose function is separable from those of other modules. Here, we investigate the in silico evolution of modularity and robustness in complex artificial metabolic networks that encode an increasing amount of information about their environment while acquiring ubiquitous features of biological, social, and engineering networks, such as scale-free edge distribution, small-world property, and fault-tolerance. These networks evolve in environments that differ in their predictability, and allow us to study modularity from topological, information-theoretic, and gene-epistatic points of view using new tools that do not depend on any preconceived notion of modularity. We find that for our evolved complex networks as well as for the yeast protein-protein interaction network, synthetic lethal pairs consist mostly of redundant genes that lie close to each other and therefore within modules, while knockdown suppressor pairs are farther apart and often straddle modules, suggesting that knockdown rescue is mediated by alternative pathways or modules. The combination of network modularity tools together with genetic interaction data constitutes a powerful approach to study and dissect the role of modularity in the evolution and function of biological networks.Comment: 28 pages, 10 figures, 8 supplemental figures, and one supplementary table. Final version to appear in PLoS Comp Bi

Origin of life in a digital microcosm

While all organisms on Earth descend from a common ancestor, there is no consensus on whether the origin of this ancestral self-replicator was a one-off event or whether it was only the final survivor of multiple origins. Here we use the digital evolution system Avida to study the origin of self-replicating computer programs. By using a computational system, we avoid many of the uncertainties inherent in any biochemical system of self-replicators (while running the risk of ignoring a fundamental aspect of biochemistry). We generated the exhaustive set of minimal-genome self-replicators and analyzed the network structure of this fitness landscape. We further examined the evolvability of these self-replicators and found that the evolvability of a self-replicator is dependent on its genomic architecture. We studied the differential ability of replicators to take over the population when competed against each other (akin to a primordial-soup model of biogenesis) and found that the probability of a self-replicator out-competing the others is not uniform. Instead, progenitor (most-recent common ancestor) genotypes are clustered in a small region of the replicator space. Our results demonstrate how computational systems can be used as test systems for hypotheses concerning the origin of life.Comment: 20 pages, 7 figures. To appear in special issue of Philosophical Transactions of the Royal Society A: Re-Conceptualizing the Origins of Life from a Physical Sciences Perspectiv