19. Genetic Drift. The biological context. There are four basic consequences of genetic drift:

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1 9. Genetic Drift Genetic drift is the alteration of gene frequencies due to sampling variation from one generation to the next. It operates to some degree in all finite populations, but can be significant as an evolutionary process in natural populations only when they are relatively small. The biological context There are four basic consequences of genetic drift: ) Random change: If the allele frequencies in a single small population are followed, they may be seen to change in an erratic manner from generation to generation, with no tendency to revert to their original values. 2) Differentiation among subpopulations: Genetic drift occurring independently in different subpopulations leads to genetic differentiation between them. Large populations in nature are seldom homogeneous because mating occurs more often between inhabitats of the same region; such subpopulations, if small, tend to differentiate by drift. Domesticated or laboratory populations are often subdivided (into herds or strains) and to become different over time. 3) Uniformity within subpopulations: Genetic variation within each subpopulation becomes progressively reduced, and individuals become more and more alike in genotype. This genetic uniformity is the reason for the widespread use of inbred strains of laboratory animals in many areas of biological research. 4) Decreased heterozygosity: Homozygotes increase in frequency at the expense of heterozygotes. This, coupled with the general tendency for deleterious alleles to be recessive, is the genetic basis for the loss of fertility and viability that almost always results from inbreeding. In theory, there are four situations in which genetic drift might play an effective role in the evolution of natural populations: ) Continuous drift: The population remains small in size, and sampling variation is effective in each generation. This is particularly likely to occur with endangered species.

2 9. Genetic drift 2 2) Intermittent drift: The population is occasionally reduced to a size sufficiently small to allow drift to operate. Reduction can be effective in one of two ways: a) Bottleneck effect: If mortality is random at the time of reduction in population size, the sample of survivors can have a different genetic composition, and reduced heterozygosity, due to chance alone. b) If the population remains small over two or more consecutive generations, the process of continuous drift is initiated. 3) Founder effect: New populations are often started by small numbers of migratory individuals, which carry only a fraction of the genetic variability of the parental population and thus differ randomly from it. Thus new populations tend to differ both from the parent population and from each other. 4) Wahlund effect: If a population is subdivided into small, effectively isolated demes, then genetic drift will be more effective than if the population were undergoing random mating. As a consequence, heterozygosity of the overall population is reduced relative to Hardy-Weinberg expectations. The quantitative model ) The probabilities of drawing various possible combinations of gametes (alleles) to form N progeny, based on the random selection of gametes from a parental gene pool having allele frequencies p and q (i.e., random mating), is given by the binomial distribution ( ) B(2 N, p) = p + q = = p q = C p q n n2 n n2 n n n = n= where n is the number of A alleles sampled, which varies from (the condition in which the progeny are fixed for A 2 ) to (the condition in which the progeny are fixed for A ), and n 2 is the n n2 number of A 2 allele sampled (n + n 2 = ). The term p q is the probability of combining n 2 N (2 N)! A alleles with n 2 A 2 alleles to form the progeny; the term = Cn = is the number of n n! n2! possible ways in which n A alleles and n 2 A 2 alleles can be combined to form the genotypes of the progeny. For example, the following sets of terms are the binomial expansions of the equation above. As above, N is the population size, and is the number of gametes sampled from the gene pool to form the offspring generation.

3 9. Genetic drift 3 N Binomial expansion 2 ( ) p+ q = pq + 2pq + pq = = p + 2pq+ q ( ) p+ q = pq + 4pq + 6pq + 4 pq + pq = = p + 4pq+ 6pq + 4pq + q ( ) p+ q = pq + 6pq + 5pq + 2pq = pq 6 pq pq = p + 6pq+ 5pq + 2pq + 5pq + 6pq + q ( ) p+ q = pq + 8pq + 28pq + 56pq + 7 pq = pq 28pq 8pq pq = p + 8pq+ 28pq + 56pq + 7 pq pq + 28pq + 8pq + q ( ) p+ q = p q + pq + 45pq + 2pq + 2pq pq = pq 2pq 45pq pq pq = p + pq+ 45pq + 2pq + 2pq pq pq + 2pq + 45pq + pq + q

4 9. Genetic drift 4 Context for the Model of Genetic Drift Model for analyzing the effects of random genetic drift on the differentiation of independent replicate subpopulations. Each of the subpopulations (vertical columns of boxes and arrows) founded from the initial large population is assumed to be genetically isolated from the others. Each subpopulation produces an infinite number of gametes, of which are chosen at random to form the next generation s breeding population. Random genetic drift results from sampling variation in this process. (From Hartl, 988).

5 9. Genetic drift 5 2) When a random sample of gametes is selected from the parental gene pool in accordance with the binomial probabilities, the allele frequencies of the sample are likely to be slightly different from those of the parents. This random change in allele frequencies from parents to offspring due solely to sampling variation is genetic drift. Although drift produces changes in allele frequencies, the subpopulation can be in Hardy-Weinberg-Castle equilibrium at each generation. 3) δq t = change in allele frequency, over t generations, due to genetic drift. This term is analogous to the q of previous deterministic models, but is a statistical rather than fixed value. It has an expected value (mean) and variance, but it s direction and magnitude can t be predicted in any particular case. 4) The expected net change of δq t (over numerous subpopulation replicates) is zero: E(δq t ) = 5) The expected variance in change (over numerous subpopulation replicates) in a single generation is 2 pq σ δ q =, where N is the subpopulation size (assumed to remain constant across generations). This value is the sampling variance (standard error of the mean) of the corresponding binomial distribution. 6) The expected variance in change (over numerous replicates) across t generations is 2 σ, ( ) t δ qt = pq. The form of this equation predicts that the variance due to drift: a) is large for small N, and small for large N (i.e., is inversely proportional to N ); b) increases with time, t ; c) is greatest for intermediate values of p and q (near.5), is smaller for very rare and very common alleles, and vanishes when one of the alleles becomes fixed. Note that, when t = [as in note (5), above], 2 σδ q, = pq ( ) = pq ( ) = pq + pq =

6 9. Genetic drift 6 7) The mean allele frequency among subpopulations, q, is the weighted mean allele frequency among the k subpopulations, weighted by subpopulation size, N i : q k = Nq. i i i= The variance in allele frequency among subpopulations is denoted heterozygotes in generation t is H = 2pq = 2pq σ. 2 t t t qt, 2 σ q. The expected number of 8) The probability of fixation or loss of an allele between one generation and the next is p + q (the two extreme terms of the binomial distribution). This implies that, on average, fixation and loss of alleles occur at a rate of roughly ¼ N per locus per generation. Thus the mean time to fixation or extinction of any given allele is approximately 4N generations (but this is a very rough approximation). N = [2, 5,, 5, 2, 25, ].9 Probability of fixation ( gen)

7 9. Genetic drift 7 9) The binomial distribution predicts the exact probabilities of forming specified combinations of genotypes, but becomes impractical to use for N > 2 or so. However, the central limit theorem of probability theory predicts that, as N becomes large, δq approaches a normal distribution having mean and standard deviation σ δ q. Thus, for larger population sizes, we can use normal distributions to provide approximate results. (However, the exact results are always provided by the binomial distribution, which are used in computer simulations for any value of N, no matter how large.)

8 9. Genetic drift 8 Computer simulations of genetic drift These examples portray the results of a stochastic computer model of genetic drift, as a function of population size and initial allele frequencies. In each case, replicates ( locus for populations, or loci for one population) were tracked independently for generations..9.8 N = p = N = p =. (q) (q) Generation Generation.9.8 N = 5 p = N = 5 p = Generation Generation.9.8 p = p = Generation Generation

9 9. Genetic drift 9 Binomial Distributions Centered on p = q =.5.5 N =.4.35 N = (p) (p).25 N = N = (p) (p) N = (p) (p)

10 9. Genetic drift Uncentered Binomial Distributions.3.25 N = 2 p =..2 N = 2 p = N = 2 p = N = 2 p = N = 2 p = N = 2 p =

11 9. Genetic drift Normal Approximations to the Binomial Distribution N = 5, p = N = 5, p = N = 5, p =

12 9. Genetic drift 2 Increase in Expected Variance due to Genetic Drift p =.3 t =.35 p =.3 t = 2 Frequency.4.3 Frequency Frequency p =.3 t = 3 Frequency p =.3 t = p =.3 t =.4 p =.3 t = 2 Frequency.5. Frequency

13 9. Genetic drift 3 The Standard Normal Distribution.4.35 Relative Frequency Standard Deviation Units % Relative Frequency %.5 95% 99% Standard Deviation Units

14 9. Genetic drift 4 One-tailed normal probabilities: proportion of area of the normal distribution that is equal to or more extreme than the specified z value. z <. <. <. <. <. <. <. <. <. <. 4. <. <. <. <. <. <. <. <. <. <.