PBG 650 Advanced Plant Breeding Module 6: Quantitative Genetics Environmental variance Heritability Covariance among relatives

More interactions For an individual G = A + D + I P = A + D + I + E For a population Two-locus interactions More than two loci…. Interlocus interactions are important, but difficult to quantify Many designs for genetic experiments lump dominance and epistatic interactions into one component called “non-additive” genetic variance

Genetic variances from a factorial model Bernardo, Chapt. 5

Environmental variance covariance would occur if better genotypes are given better environments randomization should generally remove this effect from genetic experiments in plants P = G + E P = G + E + GE genotype by environment interactions differences in relative performance of genotypes across environments experimentally, GE is part of E DeLacey et al., 1990 – summary of results from many crops and locations For a particular crop, only 10% of variation in phenotype is due to genotype! 70-20-10 rule E: GE: G

Repeatability Multiple observations on the same individuals May be repetitions in time or space (e.g. multiple fruit on a plant) variation among observations on the same individual due to temporary environmental effects ( = special environmental variance) variation among individuals due to genetic differences and permanent environmental effects ( = general environmental variance) Falconer & Mackay, pg 136

Repeatability Repeatability Sets an upper limit on heritabilities is easy to measure To separate and , you must evaluate repeatability of genetically uniform individuals

Gain from multiple measurements fyi Multiple measurements can increase precision and increase heritability (by reducing environmental and phenotypic variation) Greatest benefits are obtained for measurements that have low repeatability (large )

Heritability For an individual: P = A + D + I + E For a population: Broad sense heritability degree of genetic determination Narrow sense heritability extent to which phenotype is determined by genes transmitted from the parents “heritability” Falconer & Mackay, Chapter 8

Narrow sense heritability – another view h2 = the regression of breeding value on phenotypic value h2=0.5 +1 +2 h2=0.3 h2 is trait specific, population specific, and greatly influenced by the choice of testing environments

Narrow sense heritability Can be applied to individuals in a single environment (generally the case in animal breeding) In plants, it is commonly expressed on a family (plot) basis, which are often replicated within and across environments

Heritability in plants - complications Different mating systems, including varying degrees of selfing Different ploidy levels Annuals, perennials For many crops, measurement of some traits is only meaningful with competition, in a full stand variables such as yield are measured on a plot basis other traits are averages of multiple plants/plot plot size varies from one experiment to the next Replicates are evaluated in different microenvironments Genotype x environment interaction is prevalent for many important crop traits Nyquist, 1991; Holland et al., 2003

Heritability in plants - definition Fraction of the selection differential that is gained when selection is practiced on a defined reference unit (Hanson, 1963) Selection Differential S=s-0 Selection Response R=1-0 Y=bX R=Sbyx R/S=h2=byx Main purpose for estimating heritability is to make predictions about selection response under varying scenarios, in order to design the optimum selection strategy R=h2S High heritability – use mass selection, single environment Low heritabiltiy – progeny testing, family selection

Applications in plant breeding Selection in a cross-breeding population Selection among purelines (with or without subsequent recombination) Selection among clones Selection among testcross progeny in a hybrid breeding program Must specify the unit of selection, the selection method, and unit on which the response is measured

Heritability of a genotype mean   GXE Error variance   High heritability – use mass selection, single environment Low heritabiltiy – progeny testing, family selection broad sense heritability narrow sense heritability or “heritability”

Resemblance between Relatives Covariance between relatives measures degree of genetic resemblance Variance among groups = covariance within groups Intraclass correlation of phenotypic values Strategy: Determine expected covariance among relatives from theory, and compare to experimental observations Estimate genetic variances and heritabilities Falconer & Mackay, Chapt. 9

Covariance between offspring and one parent Genotype Frequency Genotypic Value Breeding Value Mean Genotypic Value of Offspring A1A1 p2 2q(-qd) 2q q A1A2 2pq (q-p)+2pqd (q - p) (1/2)(q - p) A2A2 q2 -2p(+pd) -2p -p CovOP=p2*2q(-qd)q+2pq[(q-p)+2pqd](1/2)(q - p) +q2[-2p(+pd)](-p) CovOP = pq2 = (1/2)σA2 This result is true for a single offspring and for the mean of any number of offspring

Resemblance between offspring and one parent For parents and offspring, observations occur in pairs Regression is more useful than the intraclass correlation as a measure of resemblance does not depend on the number of offspring does not require parents and offspring to have the same variance Get SE b from any standard stats book phenotypic variance of the parental population Estimate    

Resemblance between offspring and mid-parent CovO,MP = pq2 = (1/2)σA2 Regression on mid-parent is twice the regression of offspring on a single parent Number of offspring does not affect the covariance or the regression

Resemblance among half-sibs Genotype Frequency Breeding Value Mean Genotypic Value of Offspring Freq. x Value2 A1A1 p2 2q q p2q22 A1A2 2pq (q - p) (1/2)(q - p) (1/2)pq(q - p)22 A2A2 q2 -2p -p Covariance of half-sibs = variance among half-sib progeny CovHS = pq2[(1/2)(q - p)2+2pq] = pq2[(1/2)(p+q)2] = (1/2)pq2=(1/4)σA2

Resemblance among full-sibs Progeny Genotype of parents Frequency of mating A1A1 a A1A2 d A2A2 -a Mean Value of Progeny p4 1 4p3q 1/2 (1/2)(a+d) 2p2q2 4p2q2 1/4 (1/2)d 4pq3 (1/2)(d-a) q4 have to subtract the population mean because we’re working with the coded values CovFS= σFS2 = p4a2+4p3q[(1/2)(a+d)]2….+q4(-a)2 - 2 = pq[a+d(q-p)]2 + p2q2d2

Resemblance among full-sibs CovFS= σFS2 = p4a2+4p3q[(1/2)(a+d)]2….+q4(-a)2 - 2 = pq[a+d(q-p)]2 + p2q2d2

General formula for covariance of relatives Unilineal relatives Resemblance involves only Bilineal relatives Potential exist for relatives to have two common alleles that are identical by descent etc. (X1X3, X1X4, X2X3, or X2X4) A B Resemblance will also involve: X1X2 X3X4 etc. C D X1X3 X1X3

Covariance due to breeding values A B C D X Y (Ai Aj) (Ak Al)

Covariance due to dominance deviations A B C D X Y (Ai Aj) (Ak Al)

General formula for covariance of relatives A B C D X Y r = 2XY  = ACBD + ADBC Extended to include epistasis:

Adjusting coefficients for inbreeding Relatives r = 2XY  Parent-offspring 1/2 Half-sibs Common parent not inbred 1/4 Common parent inbred (1+F)/4 Full-sibs Parents not inbred Parents inbred (2+FA+FB)/4 (1+FA)(1+FB)/4