Lecture 2: Basic Population and Quantitative Genetics

1. Lecture 2: Basic Population and Quantitative Genetics

2. Allele and Genotype Frequencies Given genotype frequencies, we can always compute allele frequencies, e.g., The converse is not true: given allele frequencies we cannot uniquely determine the genotype frequencies For n alleles, there are n(n+1)/2 genotypes If we are willing to assume random mating, Hardy-Weinberg proportions

3. Hardy-Weinberg • Prediction of genotype frequencies from allele freqs • Allele frequencies remain unchanged over generations, provided: • Infinite population size (no genetic drift) • No mutation • No selection • No migration • Under HW conditions, a single generation of random mating gives genotype frequencies in Hardy-Weinberg proportions, and they remain forever in these proportions

4. Gametes and Gamete Frequencies When we consider two (or more) loci, we follow gametes Under random mating, gametes combine at random, e.g. Major complication: Even under HW conditions, gamete frequencies can change over time

5. ab AB AB ab AB ab AB ab In the F1, 50% AB gametes 50 % ab gametes If A and B are unlinked, the F2 gamete frequencies are AB 25% ab 25% Ab 25% aB 25% Thus, even under HW conditions, gamete frequencies change

6. Linkage disequilibrium Random mating and recombination eventually changes gamete frequencies so that they are in linkage equilibrium (LE). once in LE, gamete frequencies do not change (unless acted on by other forces) At LE, alleles in gametes are independent of each other: When linkage disequilibrium (LD) present, alleles are no longer independent --- knowing that one allele is in the gamete provides information on alleles at other loci The disequilibrium between alleles A and B is given by

7. Departure from LE LE value Initial LD value The Decay of Linkage Disequilibrium The frequency of the AB gamete is given by If recombination frequency between the A and B loci is c, the disequilibrium in generation t is Note that D(t) -> zero, although the approach can be slow when c is very small

8. Genotypic value Phenotypic value -- we will occasionally also use z for this value Environmental value Contribution of a locus to a trait Basic model: P = G + E G = average phenotypic value for that genotype if we are able to replicate it over the universe of environmental values G - E covariance -- higher performing animals may be disproportionately rewarded G x E interaction --- G values are different across environments. Basic model now becomes P = G + E + GE

9. C C C + a + d C + a(1+k) C + 2a C + 2a C -a C + d C + a 2a = G(Q2Q2) - G(Q1Q1) Alternative parameterizations of Genotypic values Q1Q1 Q2Q1 Q2Q2 d measures dominance, with d = 0 if the heterozygote is exactly intermediate to the two homozygotes d = ak =G(Q1Q2 ) - [G(Q2Q2) + G(Q1Q1) ]/2 k = d/a is a scaled measure of the dominance

10. Example: Booroola (B) gene 2a = G(BB) - G(bb) = 2.66 -1.46 --> a = 0.59 ak =d = G(Bb) - [ G(BB)+G(bb)]/2 = 0.10 k = d/a = 0.17

11. Dominance deviations --- the difference (for genotype AiAj) between the genotypic value predicted from the two single alleles and the actual genotypic value, Average contribution to genotypic value for allele i Mean value, with The genotypic value predicted from the individual allelic effects is thus Fisher’s Decomposition of G One of Fisher’s key insights was that the genotypic value consists of a fraction that can be passed from parent to offspring and a fraction that cannot. Since parents pass along single alleles to their offspring, the ai (the average effect of allele i) represent these contributions

12. Residual error Predicted value Intercept Independent (predictor) variable N = # of Q2 alleles Regression slope Regression residual Fisher’s decomposition is a Regression A notational change clearly shows this is a regression,

13. Allele Q1 common, a2 > a1 Allele Q2 common, a1 > a2 Both Q1 and Q2 frequent, a1 = a2 = 0 Slope = a2 - a1 G21 G22 G G11 0 1 2 N

14. Mean Allelic effects Dominance deviations Consider a diallelic locus, where p1 = freq(Q1)

15. ) ( Average effects and Breeding Values The a values are the average effects of an allele Breeders focus on breeding value (BV) Why all the fuss over the BV? Consider the offspring of a QxQy sire mated to a random dam. What is the expected value of the offspring?

16. For random w and z alleles, this has an expected value of zero For a random dam, these have expected value 0 Hence, The expected value of an offspring is the expected value of

17. We can thus estimate the BV for a sire by twice the deviation of his offspring from the pop mean, More generally, the expected value of an offspring is the average breeding value of its parents,

18. As Cov(a,d) = 0 Dominance Genetic Variance (or simply dominance variance) Additive Genetic Variance (or simply Additive Variance) Genetic Variances

19. Q1Q1 Q1Q2 Q2Q2 0 a(1+k) 2a When dominance present, asymmetric function of allele Frequencies Dominance effects additive variance Equals zero if k = 0 This is a symmetric function of allele frequencies Since E[a] = 0, Var(a) = E[(a -ma)2] = E[a2] One locus, 2 alleles: One locus, 2 alleles:

20. VA Allele frequency, p Additive variance, VA, with no dominance (k = 0)

21. Complete dominance (k = 1) VA VD Allele frequency, p

22. Zero additive variance Overdominance (k = 2) VA VD Allele frequency, p Allele frequency, p

23. Additive x Additive interactions -- interactions between a single allele at one locus with a single allele at another Dominance x dominance interaction --- the interaction between the dominance deviation at one locus with the dominance deviation at another. Additive x Dominant interactions -- interactions between an allele at one locus with the genotype at another, e.g. allele Ai and genotype Bkj Dominance value -- interaction between the two alleles at a locus Breeding value Epistasis These components are defined to be uncorrelated, (or orthogonal), so that