Lecture 3: Allele Frequencies and Hardy-Weinberg Equilibrium August 4, 015
Last Time Review of genetic variation and Mendelian Genetics Ø Sample calculations for Mendelian expectations: see solutions in excel file on website Methods for detecting variation Ø Morphology Ø Allozymes Ø DA Markers (deferred to Friday: Guest lecture) Anonymous Sequence-tagged
Today Introduction to statistical distributions Estimating allele frequencies Introduction to Hardy-Weinberg Equilibrium Using Hardy-Weinberg: Estimating allele frequencies for dominant loci
Statistical Distributions: ormal Distribution Many types of estimates follow normal distribution Ø Can be visualized as a frequency distribution (histogram) Ø Can interpret as a probability density function 1 sd sd Expected Value (Mean): Variance (V x ): A measure of the dispersion around the mean: V x x 1 n 1 n i 1 1 n ( x i 1 i n x i where n is the number of samples Standard Deviation (sd): A measure of dispersion around the mean that is on same scale as mean sd V x x)
Standard Error of Mean Standard Deviation is a measure of how individual points differ from the mean estimates in a single sample Standard Error is a measure of how much the estimate differs from the true parameter value (in the case of means, µ) Ø 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): se Vx n 95% Confidence Interval: x ±1.96( se)
Estimating Allele Frequencies, Codominant Loci Measured allele frequency is maximum likelihood estimator of the true frequency of the allele in the population (See Hedrick, pp 8-83 for derivation) p 11 1 + 1 Expected number of observations of allele A 1 : E(Y)np Ø Where n is number of samples Ø For diploid organisms, n, where is number of individuals sampled Expected number of observations of allele A 1 is analogous to the mean of a sample from a normal distribution Allele frequency can also be interpreted as an estimate of the mean
Allele Frequency Example Assume a population of Mountain Laurel (Kalmia latifolia) at Cooper s Rock, WV Red buds: 5000 Pink buds: 3000 White buds: 000 A 1 A 1 A 1 A A A Phenotype is determined by a single, codominant locus: Anthocyanin What is frequency of red alleles (A 1 ), and white alleles (A )? p Frequency of A 1 p 11 1 + 1 + 11 1, q Frequency of A q 1 + 1 + 1,
Allele Frequencies are Distributed as Binomials Based on samples from a population Ø For two-allele system, each sample is like a trial Ø Does the individual contain Allele A 1? Ø Remember, q1-p, so only one parameter is estimated Binomials are variables that can be interpreted as the number of successes and failures in a series of trials P( Y y) n s y f y n y, where s is the probability of a success, and f is the probability of a failure umber of ways of observing y positive results in n trials n y Probability of observing y positive results in n trials once C n y n! y!( n y)!
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 Binomial variance is pq or p(1-p) Variance in number of observations of A 1 : V(Y) np(1-p) Variance in allele frequency estimates (codominant, diploid): p( 1 p) V p Standard Error of allele frequency estimates: p( 1 p) SE p otice that estimates get better as sample size increases otice also that variance is maximum at intermediate allele frequencies
Maximum variance as a function of allele frequency for a codominant locus 0.3 0.5 0. p (1-p ) 0.15 0.1 0.05 0 0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p
Why is variance highest at intermediate allele frequencies? p 0.5 p 0.15 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)
What if there are more than alleles? General formula for calculating allele frequencies in multiallelic system with codominant alleles: p i ii + 1 n j 1 ij, j i Variance and Standard Error of allele frequency estimates remain: V p i pi ( 1 pi ) SE p i pi ( 1 pi )
How do we estimate allele frequencies for dominant loci? - Codominant locus A 1 A 1 A 1 A A A Dominant locus A 1 A 1 A 1 A A A +
Hardy-Weinberg Law After one generation of random mating, single-locus genotype frequencies can be represented by a binomial (with alleles) or a multinomial function of allele frequencies ( p + q) p + pq + q Frequency of A 1 A 1 (P) Frequency of A 1 A (H) Frequency of A A (Q)
Hardy-Weinberg Law Hardy and Weinberg came up with this simultaneously in 1908 After one generation of random mating, single-locus genotype frequencies can be represented by a binomial (with alleles) or a multinomial function of allele frequencies ( p + q) p + pq + q Frequency of A 1 A 1 (P) Frequency of A 1 A (H) Frequency of A A (Q)
Hardy-Weinberg Equilibrium After one generation of random mating, genotype frequencies remain constant, as long as allele frequencies remain constant Provides a convenient eutral Model to test for departures from assumptions Allows genotype frequencies to be represented by allele frequencies: simplification of calculations
ew otation Genotype Frequency AA P Aa H aa Q Allele Frequency A p a q
Hardy-Weinberg Assumptions Diploid Large population Random Mating: equal probability of mating among genotypes o mutation o gene flow Equal allele frequencies between sexes onoverlapping generations
Graphical Representation of Hardy-Weinberg Law (p+q) p + pq + q 1
Relationship Between Allele Frequencies and Genotype Frequencies under Hardy-Weinberg
Hardy-Weinberg Law and Probability A(p) a(q) A (p) AA (p ) Aa (pq) a (q) aa (qp) aa (q ) p + pq + q 1
How does Hardy-Weinberg Work? Reproduction is a sampling process Example: Mountain Laurel at Cooper s Rock Red Flowers: 5000 Pink Flowers: 3000 White Flowers: 000 A 1 A 1 A 1 A A A Frequency of A 1 p 0.65 Frequency of A q 0.35 What are expected numbers of phenotypes and genotypes in a sample of 0 trees? What are expected frequencies of alleles in pollen and ovules? Alleles: : A 14 : A 1 6 Genotypes: : 4 : 10 : 6 Phenotypes: : 4 : 10 : 6
What will be the genotype and phenotype frequencies in the next generation? What assumptions must we make?
From eal, D. 004. Introduction to Population Biology. What about a 3-Allele System? Alleles occur in gamete pool at same frequency as in adults Probability of two alleles coming together to form a zygote is A B A 1 A 1 p U A 1 A pq A A 3 qr A 1 A 3 pr A A q A 3 A 3 r Ovule Gametes A 1 (p) A (q) A 3 (r) Pollen Gametes A 1 (p) A (q) A 3 (r) Equilibrium established with OE GEERATIO of random mating Genotype frequencies remain stable as long as allele frequencies remain stable Remember assumptions!