8. Genetic Diversity

Transcription

1 8. Genetic Diversity Many ways to measure the diversity of a population: For any measure of diversity, we expect an estimate to be: when only one kind of object is present; low when >1 kind of objects are present, one of which is dominant in abundance; high when >1 kind of objects are present, in roughly equal abundance; maximum value when >1 kind of objects are present in equal numbers. For genetic diversity, the objects are alleles; expect that the greater the frequency of heterozygotes, the more genetically diverse the population; should get maximum diversity when all alleles are present in equal frequencies. So let s use the frequency of heterozygotes as a measure of genetic diversity; can use either the observed relative frequencies of heterozygotes, or that frequency expected under the HWC model. Q: At what frequency of p, q will the frequency of heterozygotes be the highest, under HWC conditions (random mating, genetic equilibrium)? can assign various values to p ( range 1) and find expected frequency of heterozygotes: - plot pq vs. p to estimate maximum; p q pq

2 8. Genetic diversity Exp freq of heterozygotes (pq) Allele frequency (p) some simple calculus: let V = pq (frequency of heterozygotes); - find derivative of function V with respect to p; - set to zero and solve for p to find arg(maximum); V = pq = p( 1 p) = p p V = 4 p = p 4p = p = 5. q = 5. Another derivation of genetic diversity, based on frequencies of alleles rather than genotypes: Distributions are described by descriptive statistics, including mean and variance: mean is center of density, = expected value; discrete distribution: calculate mean as weighted average of X, where weights are frequencies; For genotype distributions: X = number of alleles (,1 ); weights = relative frequencies, sum to 1:

3 8. Genetic diversity 3 X = w X i w i

4 8. Genetic diversity 4 Assuming HW conditions: Genotype Frequency (weight) X = number of A 1 s A 1 A 1 p A 1 A pq 1 A A q µ = E( x) = ( p ) + 1( pq) + ( q ) = p + pq = p( p + q) = p That is, the mean number of alleles = allele frequency. In general, the mean of a binomial distribution is np, where n = for diploid genotypes. [ ] σ = E ( x µ ) = E( x ) µ E( x ) = ( p ) + 1 ( pq) + ( q ) = 4p + pq = p( p + q) = p( 1+ p) σ = E( x ) µ = p( 1+ p) ( p) = p + p 4p = p( 1 p) = pq

5 8. Genetic diversity 5 Thus we can define geneotypic variance as the frequency of heterozygotes in population. but note that: the frequencies of all homozygote and heterozygote frequencies must sum to one, since we are expressing them as relative frequencies; for homozygote frequency of allele i is P i ; therefore, for more than alleles, we can more conviently calculate the total heterozygote frequeny as one minus the total frequency of homogygotes: H = 1 k P i i= 1 for k alleles. this measure is known as the heterozygosity of a sample, which is used directly as a measure of genetic diversity; in general, the variance of a binomial distribution is npq, where n = ; but the expression holds whether or not the population is in HWC equilibrium. One final note about allelic variance: it is common to expresses allelic variance in terms of the relative frequency of alleles per genotype, rather than the number of alleles per genotype (as derived above); in this case the actual numeric value of the allelic variance is H/4, where H is the heterozygosity measure above.

6 8. Genetic diversity 6 Summary: Measures of Genetic Variability Genotype HW Frequency Number of A alleles Frequency of A alleles A 1 A 1 p A 1 A pq 1 ½ A A q 1 1. Genotypic variance, = heterozygosity (H =σ A ) = variance in the number (absolute frequency) of alleles of a particular kind (e.g., A ) among genotypes, weighted by the genotype frequencies. = variance of the binomial distribution, =pq. useful because, under Hardy-Weinberg conditions, it is equal to the relative frequency of heterozygotes in the population.. Allelic variance (σ q ) = variance in the relative frequency of alleles of a particular kind (e.g., A ) among genotypes, weighted by the genotype frequencies. useful because it is based directly on allele frequencies and can be decomposed into withinpopulation and among-population components. Heterozygosity and allelic variance are linearly proportional to one another, indicating that they are equivalent measures of genetic variability. Both attain their maximum values when p = q =.5, in which case maximum heterozygosity =.5 and maximum allelic variance = Heterozygosity Allele frequency Variance in allele frequency Variance in allele frequency Allele frequency Heterozygosity

7 8. Genetic diversity 7 Importance of Mendel s laws and the Hardy-Weinberg-Castle principle as null models: 1) Mendel s model predicts: that genotypes will be randomly assorted across generations; i.e., that variation within populations will be re-expressed across time. ) the Hardy-Weinberg-Castle model predicts: that allele frequencies will remain constant across generations (time) unless some biological process ( force ) acts to change them; i.e., that populations at rest are in genetic equilibrium. When populations are in genetic equilibrium: 1) allele frequencies remain constant across time; ) genetic diversity remains constant across time. Thus evolution is the result of genetic disequilibrium. Biological processes that can change allele frequencies: mutation assortative mating gene flow genetic drift selection (differentiat reproduction by different genotypes)