Evolutionary dynamics on networks of selectively neutral genotypes: Effects of topology and sequence stability Jacobo Aguirre, 1 Javier M. Buldú, 2 and Susanna C. Manrubia 1 1 Centro de Astrobiología, CSIC-INTA, Ctra. de Ajalvir km 4, 28850 Torrejón de Ardoz, Madrid, Spain 2 Complex Systems Group, Signal Theory and Communications Department, U.R.J.C., Camino del Molino s/n, 28943, Fuenlabrada, Madrid, Spain Received 17 July 2009; published 14 December 2009 Networks of selectively neutral genotypes underlie the evolution of populations of replicators in constant environments. Previous theoretical analysis predicted that such populations will evolve toward highly con- nected regions of the genome space. We first study the evolution of populations of replicators on simple networks and quantify how the transient time to equilibrium depends on the initial distribution of sequences on the neutral network, on the topological properties of the latter, and on the mutation rate. Second, network neutrality is broken through the introduction of an energy for each sequence. This allows to study the compe- tition between two features neutrality and energetic stability relevant for survival and subjected to different selective pressures. In cases where the two features are negatively correlated, the population experiences sudden migrations in the genome space for values of the relevant parameters that we calculate. The numerical study of larger networks indicates that the qualitative behavior to be expected in more realistic cases is already seen in representative examples of small networks. DOI: 10.1103/PhysRevE.80.066112 PACS numbers: 89.75.Hc, 87.23.Kg, 87.10.-e I. INTRODUCTION One of the tenets of the Darwinian theory of evolution is that the fittest variants in a population increase in number and might eventually get fixed, thus eliminating less fit forms. Fitness refers to the phenotype of individuals, to the measurable features that determine their suitability in a given environment. The phenotype is the target of selection, but random mutations, responsible for the generation of new variants, can only act on the genotype. A better comprehen- sion of the complex map between genotype and phenotype is an essential issue in the effort to understand the mechanisms behind evolution and adaptation of populations, among oth- ers their robustness in the face of perturbations or the appear- ance of novelty. There is abundant evidence of the existence of an ex- tremely large degeneration between genotype and phenotype. In other words, the same phenotype can be obtained from a huge number of different genotypes. This ensemble forms the neutral network of genotypes corresponding to a given phenotype. The idea of neutral evolution was first introduced by Kimura 1 in order to account for the known fact that a large number of mutations observed in proteins, DNA, or RNA, did not have any effect on fitness. RNA sequences folding into their minimum free-energy secondary structures are likely the most used model of the genotype-phenotype relationship 2–4. Analytical studies of the number of sequences of length l compatible with a fixed secondary structure used as a proxy for the phenotype have revealed that the average size of the corresponding neutral network grows as l 3/2 b l , where b is a constant 5. Hence, there should be about 10 28 sequences compatible with the structure of a transfer RNA which has length l =76, while the currently known smallest functional RNAs, of length l  14 6, could in principle be obtained from more than 10 6 different sequences. Neutral networks are astronomically large even for moderate values of the sequence length. Neutrality becomes particularly important in the evolution of quasispecies 7, populations of fast mutating replicators which are formed by a large number of different phenotypes—and many more genotypes—, and where high diversity and the concomitant steady exploration of the ge- nome space happen to be an adaptive strategy. Relevant ex- amples of quasispecies of RNA molecules are RNA viruses 8 and error-prone replicators in the context of the RNA world 9. Evolutionary innovation in quasispecies is facili- tated by the fact that most neutral networks span the whole space of genomes. Actually, taking again the case of the RNA sequence-structure map as example, all common struc- tures of length l can be found within a relatively small radius measured as the number of nucleotides that have to be changed of a randomly chosen sequence in genotype space 10, thus showing that neutral networks are deeply interwo- ven. The mutual proximity of neutral networks in genome space has received empirical support from studies showing how two sequences differing in only two nucleotides can fold and function as fully different ribozymes 11 and how diffusion on neutral networks promotes innovation in the evolution of influenza A 12. Diffusion through neutral net- works is thus regarded as an essential component of the ad- aptation of quasispecies to changing environments, which demand new functional phenotypes to guarantee survival. In the absence of environmental changes, quasispecies stay on the same neutral network and evolve toward regions denser in neutral genotypes. In these regions, the probability that upon replication a random mutation produces a sequence with a different fold is minimized, such that mutational ro- bustness is maximized. Models of evolution on neutral net- works use to define genotypes as the nodes of the network; two nodes are linked when their sequences are at a Hamming distance of one, that is, when they differ in only one nucle- otide 13,14. In this scenario, and when the dynamics are PHYSICAL REVIEW E 80, 066112 2009 1539-3755/2009/806/06611215 ©2009 The American Physical Society 066112-1