January 11, 2005 Bio 107/207 Winter 2005 Lecture 3 Random mating and the Hardy-Weinberg principle

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1 January 11, 2005 Bio 107/207 Winter 2005 Lecture 3 Random mating and the Hardy-Weinberg principle How much DNA polymorphism exists in natural populations? - in humans, the amount of nucleotide diversity is extremely small. - on page in the textbook, Hedrick describes a large study of human SNP polymorphism (a SNP is a Single Nucleotide Polymorphism) - the survey, based on 1.42 million SNPs (!), yields a estimate of π per site is only ! - how does this compare to other species? - once again, most data has been collected for Drosophila. - here are some estimates of π and θ for a few nuclear genes: D. melanogaster D. simulans Locus N π θ N π θ Adh Pgi Est Rh boss ci MEAN compared to humans, both Drosophila taxa show dramatically higher levels of DNA polymorphism. - D. simulans is much more polymorphic than D. melanogaster too! - why would this be? The Hardy-Weinberg-Castle Equilibrium - consider a single locus with two alleles A 1 and A 2. - let: p = frequency of A 1 allele q = frequency of A 2 allele - three genotypes are thus possible: A 1 A 1, A 1 A 2, A 2 A 2. - let: P = freq. of A 1 A 1 homozygote H = freq. of A 1 A 2 heterozygote Q = freq. of A 2 A 2 homozygote

2 - from the genotype frequencies, we can estimate allele frequencies: p = P + 1/2 H q = Q + 1/2 H - since only two alleles present, p + q = 1 Question: If mating occurs at random in the population, what will be the frequencies of A 1 and A 2 in the next generation? - what does random mating mean? - very simply, that matings take place in the population independently of the genotype of an individual. - if mating is random the frequency of matings among different genotypes is determined simply by their frequencies. - therefore, if random mating occurs in the population the following frequencies of matings will occur: Female genotypes Male genotypes A 1 A 1 (P) A 1 A 2 (H) A 2 A 2 (Q) A 1 A 1 (P) P 2 PH PQ A 1 A 2 (H) PH H 2 HQ A 2 A 2 (Q) PQ HQ Q 2 - let us now examine the progeny produced by this set of matings: Progeny Mating Frequency A 1 A 1 A 1 A 2 A 2 A 2 A 1 A 1 x A 1 A 1 P 2 P 2 A 1 A 1 x A 1 A 2 2PH PH PH A 1 A 1 x A 2 A 2 2PQ 2PQ A 1 A 2 x A 1 A 2 H 2 H 2 /4 H 2 /2 H 2 /4 A 1 A 2 x A 2 A 2 2HQ HQ HQ A 2 A 2 x A 2 A 2 Q 2 Q 2 (P + H + Q) 2 (P + H/2) 2 2(P + H/2)* (Q + H/2) 2 (Q + H/2) = 1 = p 2 = 2pq =q 2

3 - the frequencies of alleles have not changed. This leads to two important conclusions: 1. Allele frequencies in the population will remain constant indefinitely. 2. Genotypic proportions occur at Hardy-Weinberg proportions in the population as determined by the square law - for two alleles (p, q) these proportions are given by the expanding the term (p + q) 2. - for three alleles (p, q, r) these proportions are given by the expanding the term (p + q + r) 2. - the reason why the Hardy-Weinberg equilibrium is so important is that for evolutionary change to occur in a population it is necessary for one, or more, specific assumptions need to be violated. - what are these assumptions? Assumptions of Hardy-Weinberg Equilibrium: 1. Generations are discrete (i.e., non-overlapping) 2. Allele frequencies are the same in males and females 3. Random mating 4. Infinite population size (i.e., no genetic drift) 5. No migration (gene flow) 6. No mutation 7. No selection - the Hardy-Weinberg principle thus states that allele frequencies in populations will not change unless some evolutionary process is acting to result in a change of allele frequency. - it thus predicts that no evolution will occur unless one of the above assumptions is violated. - therefore, the study of microevolution is largely concerned with understanding the conditions under which the above assumptions are violated! - in other words, it involves determining the relative importance of random drift, migration, mutation, and natural selection in affecting the frequency of genetic polymorphism in natural populations. - given the large number of assumptions that are required for Hardy-Weinberg equilibrium to occur, it is reasonable to ask whether it is ever observed in natural populations. - surprisingly, it is. - here is an example involving 840 scallops sampled from a natural population off the east coast of Canada. - one SNP locus was segregating for two alleles ( A 1 and A 2 ), three genotypes were found in the following proportions: A 1 A 1 = 652

4 A 1 A 2 = 177 A 2 A 2 = Question: Is this population in Hardy-Weinberg equilibrium? Step 1. Estimate genotype frequencies: P = N A1A1 /N = 652/840 = H = N A1A2 /N = 177/840 = Q = N A2A2 /N = 11/840 = P + H + R = = 1 Step 2. Estimate allele frequencies: p = frequency of A1 allele = P + ½ H = ½ (0.2107) = q = frequency of A2 allele = Q + ½ H = ½ (0.2107) = Step 3. Determine expected number of genotypes Expected number of A 1 A 1 homozygotes Expected number of A 1 A 2 heterozygotes Expected number of A 2 A 2 homozygotes = p 2 x N = (0.882) 2 x 840 = = 2pq x N = 2(0.882)(0.118) x 840 = = q 2 x N = (0.118) 2 x 840 = 11.7

5 Step 4. Compare observed and expected numbers Genotype Observed Expected A 1 A A 1 A A 2 A the observed and expected numbers are very similar! Allele frequency differences between the sexes 1. Autosomal loci - if allele frequencies are the same in the two sexes, then H-W equilibrium will be established after one generation of random mating. - however, if frequencies differ between males and females then it will take more than one generation for H-W equilibrium will be established. - consider first an autosomal locus with two alleles, A 1 and A 2. - let: pf and pm = frequencies of the A 1 allele in females and males, respectively qf and qm = frequencies of the A 2 allele in females and males, respectively - genotype frequencies are now: P = p f p m H = p f q m + p m q f Q = q f q m - if frequencies differ between the two sexes, there will be an excess of heterozygotes and a deficiency of homozygotes relative to H-W expectations. - this can be seen by considering the mean frequencies: p = 1/2 (p f + p m ) q = 1/2 (q f + q m ) - this assumes that the sex ratio is 1:1 such that half the genes are in females and half in males. - the extent of deviation from HW proportions for the A1A1 homozygote is given by: P p 2 = p f p m [1/2(p f + p m )]2 = p f p m 1/4(p f 2 + 2p f p m + p m 2 )

6 = 1/4(p f 2 2p f p m + p m 2 ) = 1/4(p f p m ) 2 - deviations for the other two genotypes can be shown to be: H 2pq = 1/2(p f p m ) 2 Q q 2 = 1/4(p f p m ) 2 - these departures from HW proportions are expected to be small if frequencies are not that different between males and females. - they will also persist for only one generation. 2. X-linked loci or loci in haplo-diploid organisms - there are some interesting dynamics for sex-linked genes. - unlike the case for an autosomal locus, allele frequency differences between the sexes will disappear only over several generations - this is shown in Figure 2.5 of the textbook for an X-linked gene. - the deviation from the mean allele frequency is halved each generation. - because of the manner in which X chromosomes are passed between generations, frequencies oscillate around the mean frequency and slowly disappear over time.