markovian

Sella G & Hirsh AE 2005 The application of statistical physics to evolutionary biology. PNAS 102:9541-9546.

• we concentrate on the family of models that depict the evolutionary process as a succession of mutant fixations, each of which occurs on the genetic background of the population's previous common ancestor
• these models neglect linked polymorphism and the possibility of temporally overlapping fixations
• this condition holds when the product of the population size and the mutation rate is small, i.e., Nμ << 1

• Ohta's (16) suggestion that most substitutions are slightly deleterious does not lead necessarily to the (rather absurd) notion that all organisms are experiencing an ineluctable decline from an original state of perfect adaptation
• the alternative to such steady decay is simply that each adaptive fixation compensates for many slightly deleterious fixations
• resulting in a long-term steady state in which most substitutions are slightly deleterious
• in effect, many baby steps downhill in the fitness landscape are offset by a larger leap uphill

• when a single mutant with genotype j appears in a population with a wild-type genome i, the probability that it fixes depends on its fitness f_j, the fitness of the wild-type f_i, and the effective population size N
• the probability of fixation also depends on the sampling process that describes the replacement of parents by offspring
• for the Moran birth-death process, the exact probability of fixation is
• π(i → j) = (1 − f_i / f_j) / (1 − (f_i / f_j)^N) ... [1]
• we show that for the Wright-Fisher process, the equation
• π(i → j) = (1 − (f_i / f_j)^a) / (1 − (f_i / f_j)^2N), ... [2]
• where a = 2 in a haploid population, and a = 1 for a diploid population with multiplicative fitness within loci, provides a closer approximation to the probability of fixation than the canonical formula

• the results presented here suggest that a priori assumption of a particular distribution of mutant selection coefficients is inappropriate for steady-state models of nearly neutral evolution
• the distribution of mutant selection coefficients is determined by the operation of the evolutionary dynamic on the landscape of all possible genotypes and therefore cannot be assumed a priori
• the distribution determined by the evolutionary dynamic will differ in important ways from distributions assumed a priori
• although we find that the premise that most substitutions are slightly deleterious may be incorrect, many of the deductions that rely on this premise remain in tact