Peter Schuster

Paper #: 12-06-006

Evolution through mutation and selection in populations of asexually replicating entities is modeled by ordinary differential equations (ODEs) that are derived from chemical kinetics of replication. The solutions of the mutation-selection equation are obtained in terms of the eigenvectors of the value matrix W = Q.F with Q being the matrix of mutation frequencies and F the diagonal matrix of fitness values. The stationary mutant distribution of the population is given by the largest eigenvector of W called quasispecies Y. In absence of neutrality a single variant, the master sequence Xm, is present at highest concentration. The stationary frequency of mutants is determined by their Hamming distance from the master, by their fitness values, and by the fitness of their neighbors in sequence space. The quasispecies as a function of the mutation rate, Y(p), may show a sharp transition from an ordered regime into the uniform distribution at p = pcr that is called error threshold. Three phenomena that are separable on model landscapes coincide at p=pcr: (i) steep decay in the concentration of the master sequence, (ii) phase transition like behavior, and (iii) a wide range of random replication where Y is the uniform distribution. "Realistic'' model landscapes based on current knowledge of nucleic acid structures and functions show error thresholds but also other sharp transitions, where one quasispecies distribution is replaced by another quasispecies with a different master sequence at critical mutation rates p = ptr. Groups of nearest neighbors of high fitness are strongly coupled by mutation, behave like a single entity and are unlikely to be replaced in phase transitions. Such ``strong quasispecies'' -- consisting of a master sequence and its most frequent mutants -- maintain their identity over the entire range of mutation frequencies from p = 0 to the error threshold at p = pcr. Neutrality in the sense of identical fitness values for two or more sequences with Hamming distances dH < 3 leads to strongly coupled clusters of variants, which remain stable in the limit lim p -> 0. Nearest neighbor or next nearest neighbor sequences appear at fixed ratios in the stationary distributions. Random selection of sequences by random drift occurs only at Hamming distances dH ≥ 3.