There is a growing interest in the study of evolutionary dynamics on populations with some non-homogeneous structure. In this paper we follow the model of Lieberman et al. Nature 433, 312-316) of evolutionary dynamics on a graph. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. We further demonstrate that finding similar solutions on graphs outside these classes is far more complex. Finally, we investigate our chosen classes numerically and discuss a number of features of the graphs; for example, we find the fixation probabilities for different initial starting positions and observe that average fixation probabilities are always increased for advantageous mutants as compared with those of unstructured populations.
A data set of 89 protein-RNA complexes has been extracted from the Protein Data Bank, and the nucleic acid recognition sites characterized through direct contacts, accessible surface area, and secondary structure motifs. The differences between RNA recognition sites that bind to RNAs in functional classes has also been analyzed. Analysis of the complete data set revealed that van der Waals interactions are more numerous than hydrogen bonds and the contacts made to the nucleic acid backbone occur more frequently than specific contacts to nucleotide bases. Of the base-specific contacts that were observed, contacts to guanine and adenine occurred most frequently. The most favored amino acid-nucleotide pairings observed were lysine-phosphate, tyrosine-uracil, arginine-phosphate, phenylalanine-adenine and tryptophan-guanine. The amino acid propensities showed that positively charged and polar residues were favored as expected, but also so were tryptophan and glycine. The propensities calculated for the functional classes showed trends similar to those observed for the complete data set. However, the analysis of hydrogen bond and van der Waal contacts showed that in general proteins complexed with messenger RNA, transfer RNA and viral RNA have more base specific contacts and less backbone contacts than expected, while proteins complexed with ribosomal RNA have less base-specific contacts than the expected. Hence, whilst the types of amino acids involved in the interfaces are similar, the distribution of specific contacts is dependent upon the functional class of the RNA bound.
Citation: Pattni, K., Broom, M., Rychtar, J. & Silvers, L. J. (2015). Evolutionary graph theory revisited: when is an evolutionary process equivalent to the Moran process?. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471, e2182. doi: 10.1098/rspa.2015 This is the accepted version of the paper.This version of the publication may differ from the final published version. Evolution in finite populations is often modelled using the classical Moran process. Over the last ten years this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population, is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated matrix of weights have been considered, as have a number of important dynamics. In this paper we revisit the original paper on evolutionary graph theory in light of these extensions to consider these developments in an integrated way. In particular we find general criteria for when an evolutionary graph with general weights satisfies the Moran probability for the set of six common evolutionary dynamics.
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