Spectral analysis of neutral evolution

Abstract

It has been argued that much of evolution takes place in the absence of fitness gradients. Such periods of evolution can be analysed by examining the mutational network formed by sequences of equal fitness, that is, the neutral network. It has been demonstrated that, in large populations under a high mutation rate, the population distribution over the neutral network and average mutational robustness are given by the principal eigenvector and eigen- value, respectively, of the network's adjacency matrix. However, little progress has been made towards understanding the manner in which the topology of the neutral network influences the resulting population distribution and robustness. In this work, we build on recent results from spectral graph theory and utilize numerical methods to enhance our understanding of how populations distribute themselves over neutral networks. We demonstrate that, in the presence of certain topological features, the population will undergo an exploration catastrophe and become confined to a small portion of the network. We further derive approximations, in terms of mutational biases, for the population distribution and average robustness in networks with a homogeneous structure. The applicability of these results is explored, first, by a detailed review of the literature in both evolutionary computing and biology concerning the structure of neutral networks. This is extended by studying the actual and predicted population distribution over the neutral networks of H1N1 and H3N2 influenza haemagglutinin during seasons between 2005 and 2016. It is shown that, in some instances, these populations experience an exploration catastrophe. These results provide insight into the behaviour of populations on neutral networks, demonstrating that neutrality does not necessarily lead to an exploration of genotype/phenotype space or an associated increase in population diversity. Moreover, they provide a plausible explanation for conflicting results concerning the relationship between robustness and evolvability.