Lecture 3: Allele Frequencies and Hardy-Weinberg Equilibrium August 24, 2015

Last Time uReview of genetic variation and Mendelian Genetics  Sample calculations for Mendelian expectations: see solutions in excel file on website uMethods for detecting variation  Morphology  Allozymes  DNA Markers (deferred to Friday: Guest lecture) äAnonymous äSequence-tagged

Today uIntroduction to statistical distributions uEstimating allele frequencies uIntroduction to Hardy-Weinberg Equilibrium uUsing Hardy-Weinberg: Estimating allele frequencies for dominant loci

Statistical Distributions: Normal Distribution uMany types of estimates follow normal distribution  Can be visualized as a frequency distribution (histogram)  Can interpret as a probability density function Variance (V x ): A measure of the dispersion around the mean: Expected Value (Mean): where n is the number of samples Standard Deviation (sd): A measure of dispersion around the mean that is on same scale as mean 1 sd 2 sd

Standard Error of Mean uStandard Deviation is a measure of how individual points differ from the mean estimates in a single sample uStandard Error is a measure of how much the estimate differs from the true parameter value (in the case of means, μ)  If you repeated the experiment, how close would you expect the mean estimate to be to your previous estimate? Standard Error of the Mean (se): 95% Confidence Interval:

Estimating Allele Frequencies, Codominant Loci uMeasured allele frequency is maximum likelihood estimator of the true frequency of the allele in the population (See Hedrick, pp for derivation) uExpected number of observations of allele A 1 : E(Y)=np  Where n is number of samples  For diploid organisms, n = 2N, where N is number of individuals sampled uExpected number of observations of allele A 1 is analogous to the mean of a sample from a normal distribution uAllele frequency can also be interpreted as an estimate of the mean

uAssume a population of Mountain Laurel (Kalmia latifolia) at Cooper’s Rock, WV Allele Frequency Example Red buds: 5000 Pink buds: 3000 White buds: 2000 uPhenotype is determined by a single, codominant locus: Anthocyanin uWhat is frequency of “red” alleles (A 1 ), and “white” alleles (A 2 )? A1A1A1A2A2A2A1A1A1A2A2A2 Frequency of A 1 = p Frequency of A 2 = q

Allele Frequencies are Distributed as Binomials uBinomials are variables that can be interpreted as the number of successes and failures in a series of trials uBased on samples from a population  For two-allele system, each sample is like a “trial”  Does the individual contain Allele A 1 ?  Remember, q=1-p, so only one parameter is estimated Number of ways of observing y positive results in n trials Probability of observing y positive results in n trials once where s is the probability of a success, and f is the probability of a failure

Given the allele frequencies that you calculated earlier for Cooper’s Rock Kalmia latifolia, what is the probability of observing two “white” alleles in a sample of two plants?

Variation in Allele Frequencies, Codominant Loci uBinomial variance is pq or p(1-p) uVariance in number of observations of A 1 : V(Y) = np(1-p) uVariance in allele frequency estimates (codominant, diploid): uStandard Error of allele frequency estimates: uNotice that estimates get better as sample size increases uNotice also that variance is maximum at intermediate allele frequencies

Maximum variance as a function of allele frequency for a codominant locus

Why is variance highest at intermediate allele frequencies? p = 0.5 If this were a target, how variable would your outcome be in each case (red versus white hits)? Variance is constrained when value approaches limits (0 or 1) p = 0.125

What if there are more than 2 alleles? uGeneral formula for calculating allele frequencies in multiallelic system with codominant alleles: uVariance and Standard Error of allele frequency estimates remain:

How do we estimate allele frequencies for dominant loci? A2A2A2A2 CodominantlocusDominant locus A1A1A1A1 A1A2A1A2 A2A2A2A2 - + A1A1A1A1 A1A2A1A2 CodominantlocusDominant locus - +

Hardy-Weinberg Law uAfter one generation of random mating, single-locus genotype frequencies can be represented by a binomial (with 2 alleles) or a multinomial function of allele frequencies Frequency of A 2 A 2 (Q)Frequency of A 1 A 1 (P)Frequency of A 1 A 2 (H)

Hardy-Weinberg Law uHardy and Weinberg came up with this simultaneously in 1908 uAfter one generation of random mating, single-locus genotype frequencies can be represented by a binomial (with 2 alleles) or a multinomial function of allele frequencies Frequency of A 2 A 2 (Q)Frequency of A 1 A 1 (P)Frequency of A 1 A 2 (H)

Hardy-Weinberg Equilibrium uAfter one generation of random mating, genotype frequencies remain constant, as long as allele frequencies remain constant uProvides a convenient Neutral Model to test for departures from assumptions uAllows genotype frequencies to be represented by allele frequencies: simplification of calculations

New Notation GenotypeFrequency AAP Aa H aa Q Allele Frequency Ap aq

Hardy-Weinberg Assumptions uDiploid uLarge population uRandom Mating: equal probability of mating among genotypes uNo mutation uNo gene flow uEqual allele frequencies between sexes uNonoverlapping generations

Graphical Representation of Hardy-Weinberg Law (p+q) 2 = p 2 + 2pq + q 2 = 1

Relationship Between Allele Frequencies and Genotype Frequencies under Hardy-Weinberg

Hardy-Weinberg Law and Probability p 2 + 2pq + q 2 = 1 AA (p 2 ) aa (q 2 ) Aa (pq) aA (qp) A (p) a (q) A(p)

How does Hardy-Weinberg Work? uReproduction is a sampling process uExample: Mountain Laurel at Cooper’s Rock Red Flowers: 5000 Pink Flowers: 3000 White Flowers: 2000 A1A1A1A2A2A2A1A1A1A2A2A2 Frequency of A 1 = p = 0.65 Frequency of A 2 = q = 0.35 : A 2 =14 : A 1 =26 Alleles: : 4: 10 Genotypes: : 6 Phenotypes: : 4: 10: 6 What are expected numbers of phenotypes and genotypes in a sample of 20 trees? What are expected frequencies of alleles in pollen and ovules?

What will be the genotype and phenotype frequencies in the next generation? What assumptions must we make?

Pollen Gametes Ovule Gametes From Neal, D Introduction to Population Biology. A 1 (p)A 2 (q)A 3 (r) A1(p)A1(p) A2(q)A2(q) A3(r)A3(r) What about a 3-Allele System? uAlleles occur in gamete pool at same frequency as in adults  Probability of two alleles coming together to form a zygote is A B U A 1 A 1 = p 2 A 1 A 2 = 2pq A 2 A 3 = 2qr A 1 A 3 = 2pr A 2 A 2 = q 2 A 3 A 3 = r 2 uEquilibrium established with ONE GENERATION of random mating uGenotype frequencies remain stable as long as allele frequencies remain stable uRemember assumptions!