Lecture 2: Basic Population and Quantitative Genetics

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

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

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

AB ab AB ab In the F 1, 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

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

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

Contribution of a locus to a trait Basic model: P = G + E Phenotypic value -- we will occasionally also use z for this value Genotypic value Environmental value 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

Alternative parameterizations of Genotypic values Q1Q1Q1Q1 Q2Q1Q2Q1 Q2Q2Q2Q2 CC + a(1+k)C + 2a CC + a + dC + 2a C -aC + dC + a 2a = G(Q 2 Q 2 ) - G(Q 1 Q 1 ) d = ak =G(Q 1 Q 2 ) - [G(Q 2 Q 2 ) + G(Q 1 Q 1 ) ]/2 d measures dominance, with d = 0 if the heterozygote is exactly intermediate to the two homozygotes k = d/a is a scaled measure of the dominance

Example: Booroola (B) gene GenotypebbBbBB Average Litter size a = G(BB) - G(bb) = > a = 0.59 ak =d = G(Bb) - [ G(BB)+G(bb)]/2 = 0.10 k = d/a = 0.17

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. Mean value, with Average contribution to genotypic value for allele i Since parents pass along single alleles to their offspring, the  i (the average effect of allele i) represent these contributions The genotypic value predicted from the individual allelic effects is thus Dominance deviations --- the difference (for genotype A i A j ) between the genotypic value predicted from the two single alleles and the actual genotypic value,

Fisher’s decomposition is a Regression Predicted value Residual error A notational change clearly shows this is a regression, Independent (predictor) variable N = # of Q 2 alleles Regression slope Intercept Regression residual

0 12 N G G 22 G 11 G 21 Allele Q 1 common,  2 >  1 Slope =  2 -  1 Allele Q 2 common,  1 >  2 Both Q 1 and Q 2 frequent,  1 =  2 = 0

GenotypeQ1Q1Q1Q1 Q2Q1Q2Q1 Q2Q2Q2Q2 Genotypic value 0a(1+k)2a Consider a diallelic locus, where p 1 = freq( Q 1 ) Mean Allelic effects Dominance deviations

Average effects and Breeding Values ( ) The  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 Q x Q y sire mated to a random dam. What is the expected value of the offspring?

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

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,

Genetic Variances As Cov(  ) = 0 Additive Genetic Variance (or simply Additive Variance) Dominance Genetic Variance (or simply dominance variance)

One locus, 2 alleles: Q 1 Q 1 Q 1 Q 2 Q 2 Q 2 0 a(1+k) 2a Dominance effects additive variance When dominance present, asymmetric function of allele Frequencies Equals zero if k = 0 This is a symmetric function of allele frequencies Since E[  ] = 0, Var(  ) = E[(  -  a ) 2 ] = E[  2 ]

Additive variance, V A, with no dominance (k = 0) Allele frequency, p VAVA

Complete dominance (k = 1) Allele frequency, p VAVA VDVD

Overdominance (k = 2) Allele frequency, p VAVA VDVD Zero additive variance

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