Compensatory neutral mutations and the evolution of RNA

There are many examples of RNA molecules in which the secondary structure has been strongly conserved during evolution, but the base sequence is much less conserved, e.g., transfer RNA, ribosomal RNA, and ribonuclease P. A model of compensatory neutral mutations is used here to describe the evolution of the base sequence in RNA helices. There are two loci (i.e., the two sides of the pair) with four alleles at each locus (corresponding to A, C, G, U). Watson-Crick base pairs (AU, CG, GC, and UA) are each assigned a fitness 1, whilst all other pairs are treated as mismatches and assigned fitness 1-s. A population of N diploid individuals is considered with a mutation rate of u per base. For biologically reasonable parameter values, the frequency of mismatches is always small but the frequency of the four matching pairs can vary over a wide range. Using a diffusion model, the stationary distribution for the frequency x of any of the four matching pairs is calculated. The shape depends on the combination of variables beta = 8Nu2/9s. For small beta, the distribution diverges at the two extremes, x = 0 and x = 1-z, where z is the mean frequency of mismatches. The population typically consists almost entirely of one of the four types of matching pairs, but occasionally makes shifts between the four possible states. The mean rate at which these shifts occur is calculated here. The effect of recombination between the two loci is to decrease the probability density at intermediate x, and to increase the weight at the extremes. The rate of transition between the four states is slowed by recombination (as originally shown by Kimura in a two-allele model with irreversible mutation). A very small recombination rate r approximately u2/s is sufficient to increase the mean time between transitions dramatically. In addition to its application to RNA, this model is also relevant to the 'shifting balance' theory describing the drift of populations between alternative equilibria separated by low fitness valleys. Equilibrium values for the frequencies of the different allele combinations in an infinite population are also calculated. It is shown that for low recombination rates the equilibrium is symmetric, but there is a critical recombination rate above which alternative asymmetric equilibria become stable.