From population genetics to variation among species: Computing the rate of fixations

Mutations and substitutions Substitutions are mutations that became fixed in the population. Polymorphism: mutations that are not fixed. When we look at two species, and all members of species A have nucleotide “T” in position 13 of a specific gene, and all members of species B have “A” in that position, we assume that at least one fixation (substitution) has occurred.

Evolutionary distance between two species d = the number of mutations that became fixed in the two populations from the time of divergence for a specific site. d = number of substitution per site. Sequence 2 Ancestral sequence Sequence 1 T T d

Substitution rate = average rate of fixation T Sequence 2 T Ancestral sequence Sequence 1 d = the number of mutations that became fixed in the two populations from the time of divergence for a specific site. r = the average number of mutations that became fixed in the two populations from the time of divergence for a specific site, per year (or per generation). d

Assuming that the differences reflect fixations, and that the substitution rate is the same for all sites we can conclude that there were at least two fixations per 8 sites since the time of divergence. The estimate for the number of fixation per site is 2/8 This method to estimate evolutionary distances is called the p distance. d can be estimated from sequence data SEQ1AACAAGCG SEQ2AACGAGCA There are 2 differences over 8 positions.

The average rate of fixation (substitution rate) How does the substitution rate depend on the population size and on the mutation rate?

Observation 1: allele with an initial frequency p of allele “A” has probability p to become fixed. This is because there is no selection and all current alleles have the same probability to finally be fixed. So the probability that any current allele is fixed is 1/2N and the probability that any one of the 2Np alleles with state “A” is fixed is thus Substitution rate

r = number of new mutations that enter the population each generation × probability of each such mutation to become fixed. This rate can be computed using the following formula:

Number of new mutations that enter the population each generation. Substitution rate For each such mutation there is a chance of 1/(2N) to be fixed.

The average rate of fixation (substitution rate) The rate of substitution of neutral alleles is equal to the mutation rate to neutral alleles, u. r = number of new mutations that enter the population each generation × probability of each such mutation to become fixed.

Gillespie wrote: “This is one of the most remarkable results in all of population genetics. At first it offends our intuition. The fixation of alleles is caused by genetic drift. The strength of the genetic drift depends on the population size. Our intuition wants the rate of fixation to depend on population size as well. However, the number of mutations entering the population each generation also depends on population size and does so in such a way as to cancel out drift’s dependency on population size when deriving the rate of substitution”.