where U is a uniformly distributed random variable between zero and one, N0 is the initial population size, r is the rate of population growth, and T; is the sum of all the past coalescence waiting times up to the current number of lineages k that have not yet coalesced according to i+1

k=n as shown by Slatkin and Hudson (1991). In equation 3.91 time is scaled in units of r since T = rt. As T gets larger then more time has elapsed, meaning that the size of the population has changed more.

Note that equation 3.91 only applies to populations growing though time. In populations shrinking in size toward the present, there is a chance that there will never be coalescence to a MRCA since the probability of coalescence approaches zero as the population size approaches infinity going back in time from the present to the past.

• In finite populations, allele frequencies can change from generation to generation since the sample of gametes that found the next generation may not contain exactly the same number of each allele as the previous generation. The chances of a large change in the number of alleles decreases as the number of gametes sampled increases. Sampling error in allele frequency causes genetic drift, the random process whereby all alleles eventually reach fixation or loss.

• The Wright-Fisher model is a simplification of the biological life cycle used to model genetic drift. It makes assumptions identical to Hardy-Weinberg in addition to assuming that each generation is founded by sampling 2N gametes from an infinite pool of gametes. The binomial distribution can be used to predict the probability that a Wright-Fisher population goes from some initial number of alleles to any number of alleles in the next generation.

• The action of genetic drift in a very large number of identical replicate populations can be modeled with a Markov chain model (based on the binomial distribution). The model tracks the probabilities x of a population with any number of copies of a given allele transitioning to all possible numbers of copies of the same allele in the next generation. Markov models predict that the chances of fixation or loss are equal when p = q = lh and that genetic drift reduces genetic variation by 1 -

every generation.

The process of genetic drift in many replicate populations can be thought of as analogous to the diffusion of particles in space. This leads to the diffusion approximation of genetic drift, where the rate at which individual populations reach fixation or loss depends on the diffusion coefficient: a function of population size and allele frequency. The diffusion equation predicts that an allele at a initial frequency of 1/2 will remain segregating for an average of about 2.8N generations. The size of a population is defined by the behavior of allele frequencies over time. The effective population size (Ne) is the size of an ideal Wright-Fisher population that shows the same allele frequency behavior over time as an observed biological population regardless of its census population size (N).

Finite population size and consanguineous mating are analogous processes since both lead to increasing homozygosity and decreasing heterozygosity over time. The distinction is that genetic drift in finite populations causes changes in both genotype and allele frequencies (alleles are lost and fixed) while consanguineous mating changes only genotype frequencies.

Numerous models predict dynamics of genetic variation based on the effective population size (Ne). As a consequence, there are several definitions of Ne, including the variance effective population size, the inbreeding effective population size, and the breeding effective population size. The effective population size can be estimated from direct observation of genetic variation over time and Ne is often less than N in actual populations. The average time for a pair of lineages to coalesce is the same as the population size or 2 Ne with a large expected range around this average (the variance is 4N2). In a sample of k lineages the average time to the first coalescent event is 2Ne divided by the

M -1), number of unique pairs of lineages (-

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