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Reminder - Means, Variances and Covariances E ( X ) xi f xi i Var ( X ) E ( X ) 2 xi f xi 2 i Cov( X , Y ) E X X Y Y x y f x , y i i X i Y i i Covariance Algebra Cov(aX , Y ) aCov( X , Y ) Cov( X Y , Z ) Cov( X , Z ) Cov(Y , Z ) Var ( X Y ) Var ( X )Var (Y ) 2Cov( X , Y ) Covariance and Correlation Correlation is covariance scaled to range [-1,1]. rX ,Y Cov X , Y Var X Var Y For two traits with the same variance: Cov(X1,X2) = r12 Var(X) Phenotypes, Genotypes and environment A phenotype (P) is composed of genotypic values (G) and environmental deviations (E): P=G+E Whether we focus on mean, variance, or covariance, inference always comes from the measurement of the phenotype A distinction will be made: V will be used to indicate inferred components of variance s2 will be used to indicate observational components of variance Mean genotypic value is equal to the mean phenotypic value Genotypic values are expressed as deviations from the mid- homozygote point E(Gj) = EPj )= Pj Genotypic values Consider two alleles, A1 and A2, at a single locus. The two homozygous classes, A1A1 and A2A2, are assigned genotypic values +a and -a, respectively. Assume that the A1 allele increases the value of a phenotype while the A2 allele decreases the value. The heterozygous class, A1A2, is assigned a genotypic value of d Zero is midpoint between the two genotypic values of A1A1 and A2A2; d is measured as a deviation from this midpoint Genotype A2A2 -a Genotypic value A1A2 0 d A1A1 +a Properties of the Genotypic Values and Environmental Deviations The mean of environmental deviations is zero Ej = Pj – Gj , ,(E = 0) The correlation between genotypic values and environmental deviations for a population of subjects is zero (rGE = .00) Elements of a population mean Genotype Frequency Value Freq x Value A1A1 p2 +a p2a A1A2 2pq d 2pqd A2A2 q2 -a -q2a Sum = a(p - q) + 2dpq Population mean P = G = SGkpk Multiply frequency by genotypic value and sum Recall that p2 - q2 = (p + q)(p - q) = p - q P = a(p - q) + 2dpq Additive model Assume aA and aB correspond to A1A1 and B1B1 A1A1B1B1 = aA + aB So that P = Sa(p - q) + 2Sdpq Average effect for an allele (a) Population properties vis a vis family structure Transmission from parent to offspring; parents pass on genes and not genotypes Average effect of a particular gene (allele) is the mean deviation from the population mean of individuals which received that gene from one parent (assuming the gene transmitted from the other parent having come at random from the population) Average effect for an allele (a) Gamete A1 A2 Frequ. & Values A1A1 & +a A1A2 &d p q p G Minus pop. mean a pa + qd -[a(p-q) + 2dpq] q[(a + d(q-p)] -qa + pd -[a(p-q) + 2dpq] -p[a+d(q-p)] A2A2 & -a q Average effect for an allele (a) Thus, the average effect for each allele also can be calculated for A1 and A2 in the following manner (Falconer, 1989): a1 = pa + qd - [a(p - q) + 2dpq] and a2 = -p[a + d(q - p)] Average effect of a gene substitution Assume two alleles at a locus Select A2 genes at random from population; p in A1A2 and q in A2A2 A1A2 to A1A1 corresponds to a change of d to +a, i.e., (a - d); A2A2 to A1A2 corresponds to a change of -a to d, (d + a) On average, p(a - d) plus q(d + a) or a = a + d(q - p) When gene frequency is greater a is greater q = 0.10 q = 0.40 a1 = +0.24 +1.44 a2 = -2.16 -2.16 a = a1 - a2 2.40 3.60 Breeding Value (A) The average effects of the parents’ genes determine the mean genotypic value of its progeny Average effect can not be measured (gene substitution), while breeding value can Breeding value: Value of individual compared to mean value of its progeny Mate with a number of random partners; breeding value equals twice the mean deviation of the progeny from the population mean (provides only half the genes) Breeding value is interpretable only when we know in which population the individual is to be mated Breeding Value Genotype: Breeding value A1A1: A1A2: A2A2: 2a1 = 2qa a1 + a2 = (q - p)a 2a2 = -2pa Mean breeding value under HWC equilibrium is zero 2p2qa + 2pq(q - p)a - 2q2pa 2pqa(p + q - p - q) = 0 which equals... Dominance deviation Breeding values are referred to as “additive genotype”; variation due to additive effects of genes A symbolizes the breeding value of an individual Proportion of s2P attributable to s2A is called heritability (h2) G=A+D Statistically speaking, within-locus interaction Non-additive, within-locus effect A parent can not individually transmit dominance effects; it requires the gametic contribution of both parents Genotypic values, breeding values, and dominance deviation 2qa +a d 0 (q - p)a }a 0 -2pa -a Genotypic values Breeding values A2A2 q2 A1A2 2pq A1A1 p2 Genotypic values, breeding values, and dominance deviation Regression of genotypic value on gene dosage yields the genotypic values predicted by gene dosage average effect of an allele that which “breeds true” If there is dominance, this prediction of genotypic values from gene dosage will be slightly off dominance is deviation from the regression line Epitasis - Separate analysis locus A shows an association with the trait locus B appears unrelated AA Aa Locus A aa BB Bb Locus B bb Epitasis - Joint analysis locus B modifies the effects of locus A AA Aa aa BB Bb bb Genotypic Means Locus B BB Bb bb AA AABB AABb Aabb AA Locus A Aa AaBB AaBb Aabb Aa aa aaBB aaBb aabb aa BB Bb bb Partitioning of effects Locus A M P Locus B M P 4 main effects M P M P Additive effects 6 twoway interactions M M P M P M P P Additive-additive epistasis 4 threeway interactions M P M M P P M M P M P P Additivedominance epistasis 1 fourway interaction M P M P Dominancedominance epistasis Two loci AA BB m + aA + aB + aa Bb m + aA + dB + ad bb m + aA – aB – aa Aa aa m + dA + aB + da m – aA + aB – aa m + dA + dB + dd m – aA + dB – ad m + dA – aB – da m – aA – aB + aa Covariance matrix Sib 1 Sib 1 s2A + s2D + s2S + s2N Sib 2 s2A + zs2D + s2S Sib 2 s2A + zs2D + s2S s2A + s2D + s2S + s2N Sib 1 Sib 1 s2A + s2D + s2S + s2N Sib 2 ½s2A + ¼s2D + s2S Sib 2 ½s2A + ¼s2D + s2S s2A + s2D + s2S + s2N Detecting epistasis The test for epistasis is based on the difference in fit between - a model with single locus effects and epistatic effects and - a model with only single locus effects, Enables us to investigate the power of the variance components method to detect epistasis True Model A a Y Assumed Model B A b a* Y a* is the apparent co-efficient a* will deviate from a to the extent that A and B are correlated Phenotypic variance Again, assume P=G+E Thus differences in phenotypes, measured as variance and symbolized as VP, can be decomposed into both genetic and environmental variation, VG and VE, respectively. VP = VG + VE VG is comprised of three kinds of distinct variance: additive (VA), dominant (VD), and epistatic (VI). VP = (VA + VD + VI ) + VE Analysis of variance Variance Symbol Value Phenotypic VP Phenotypic value Genotypic VG Genotypic value Additive VA Breeding value Dominance VD Dominance deviation Epistasis VI Epistatic deviation Environment VE Environmental deviation Additive (VA) and dominance variance (VD) The covariance between breeding values and dominance deviations equals zero so that VG = VA + VD + VI VA = 2pq[a + d(q - p)]2 VD = d2(4q4p2 + 8p3q3 + 4p4q2) = (2pqd)2 Additive and dominance variance If d = 0, then VA = 2pqa2, where q is the recessive allele If d = a, then VA = 8pq3a2 If p = q = .50 (e.g., cross of two inbred strains) VA = 1/2a2 VD = 1/4d2 In general, genes at intermediate frequency contribute more variance than high or low frequencies Epistatic variance (VI) Epistatic variance beyond three or more loci do not contribute substantially to total variance Three types of two-factor interactions (breeding values by dominance deviations) additive x additive (VAA) additive x dominance (VAD) dominance x dominance (VDD) Environmental variance Special environmental variance (VEs) within-individual component temporary or localized circumstance General environmental variance (VEg) between-individual component permanent or non-localized circumstances Ratio of between-individual to total phenotypic is an intraclass correlation (r) Summary of variance partitioning Data needed Partition made Ratio Estimated Resemblance between relatives (VA):(VNA+VEg+VEs) Heritability, VA / VP Genetically uniform group (VA+VNA): (VEg+VEs) Degree of genetic = (VG):(VE) determination, VG/VP Multiple measurements (VG+VEg):VEs All three VA:VNA:VEg:VEs Repeatability, (VG+VEg)/VP Components of variance - Summary Phenotypic Variance Environmental Genetic GxE interaction and correlation Components of variance - Summary Phenotypic Variance Environmental Genetic GxE interaction and correlation Additive Dominance Epistasis Components of variance - Summary Phenotypic Variance Environmental Additive Genetic GxE interaction and correlation Dominance Quantitative trait loci Epistasis Resemblance of relatives Degree of relative resemblance is a function of additive variance, i.e, breeding values The proportionate amount of additive variance is an estimate of heritability (VA / VP) Intraclass correlation coefficient t = s2B / s2B + s2W Between and within full-sibships, for example Resemblance of relatives bOP = CovOP / s2P New property of the population is covariance of related individuals Cross-Products of Deviations for Pairs of Relatives AA Aa aa AA Aa aa (a-m)2 (a-m)(d-m) (d-m)2 (a-m)(-a-m) (-a-m)(d-m) (-a-m)2 The covariance between relatives of a certain class is the weighted average of these cross-products, where each cross-product is weighted by its frequency in that class. Offspring and one parent Individual genotypic values and those of their offspring produced by random mating When expressed as normal deviations, the mean value of the offspring is 1/2 the breeding value of the parent Covariance between individual’s genotypic value (G) with 1/2 its breeding value (A) Covariance for Parent-offspring (P-O) AA Aa aa AA p3 p2q 0 Aa aa pq pq2 q3 Covariance = (a-m)2p3 + (d-m)2pq + (-a-m)2q3 + (a-m)(d-m)2p2q + (-a-m)(d-m)2pq2 = pq[a+(q-p)d]2 = VA / 2 Offspring and one parent G = A + D so that covariance is between (A + D) and 1/2A; sum of cross products equal S1/2A(A + D) = 1/2SA2 + 1/2SAD CovOP = (1/2SA2 + 1/2SAD) / # of parents Recall that CovAD = 0 CovOP = 1/2VA (i.e., 1/2 the variance of breeding values) Offspring and one parent: Effects of a single locus Parents Offspring Genotype Frequency Genotypic value Mean genotypic value A1A1 p2 2q(a - qd) qa A1A2 2pq 1/2(q - p)a A2A2 q2 (q - p)a + 2pqd -2p(a + pd) -pa Offspring and one parent Mean genotypic values of the offspring are 1/2A of the parents Mean cross product equals Frequency X Genotypic value of the parent X Mean genotypic value of the offspring CovOP = pqa2(p2+2pq+q2)+2p2q2ad(-q+q-p+p)=pqa2=1/2VA (Note: VA = 2pqa2) Covariance of MZ Twins AA Aa aa AA p2 0 0 Aa aa 2pq 0 q2 Covariance = (a-m)2p2 + (d-m)22pq + (-a-m)2q2 = 2pq[a+(q-p)d]2 + (2pqd)2 = VA + VD Twins Dizygotic twins are fulls sibs and their genetic covariance is that of full sibs Monozygotic twins have identical genotypes, i.e., no genetic variance within pairs so that Cov(MZ) = VG Covariance for Unrelated Pairs (U) AA p4 2p3q p2q2 AA Aa aa Aa aa 4p2q2 2pq3 q4 Covariance = (a-m)2p4 + (d-m)24p2q2 + (-a-m)2q4 + (a-m)(d-m)4p3q + (-a-m)(d-m)4pq3 + (a-m)(-a-m)2p2q2 =0 Resemblance in general Let r be the fraction of VA and u the fraction of VD so that Cov = rVA + uVD P, Q two individual in relationship with parents A,B and C,D and f coancestry r = 2fPQ and u = fACfBD + fADfBC For inbred relatives, r = 2fPQ / [(1 + FP)(1 + FQ)]1/2 Resemblance in general Coefficient r of the additive variance is sometimes called the coefficient of relationship (the correlation between the breeding values A) Coefficient u represents the probability of the relatives having the same genotype through identity by descent It is zero unless the related individuals have paths of coancestry through both of their respective parents, (e.g., full sibs and double first cousins) Environmental covariance VE = VEc + VEw VEc; common, i.e., contributes to variance between means of families but not the variance within (covariance among related individuals) VEw; within, i.e., arises from independent of coefficient of relationship Maternal effects and competition Phenotypic resemblance between relatives Relatives Covariance Offspring and one parent Offspring and mid-parent Half sibs 1/2VA Regression (b) or intraclass correlation (t) b = 1/2(VA/VP) 1/2VA b = VA/VP 1/4VA t = 1/4(VA/VP) Full sibs 1/2VA+1/4VD+VEc t=(1/2VA+1/4VD+V Ec)/VP Heritability Regression of breeding value on phenotypic value Index of response to genetic selection Estimated with offspring-parent regression, sib analysis, intra-sire regression of offspring on dam, or combined estimates plus other methods (Markel et al., 1995, 1999) Heritability: Ratio of additive genetic variance to phenotypic variance h2 = VA / VP Regression of breeding value on phenotypic value h2 = bAP rAP = bAP sP / sA = h2(1/h) = h Heritability: Twins and human data Between pairs, s2b Within pairs, s2w Identical (MZ) VA + VD + VEc VEw Dizygotic (DZ) 1/2 VA + 1/4 VD + VEc 1/2 VA + 3/4 VD + VEw Difference 1/2 VA + 3/4 VD 1/2 VA + 3/4 VD Heritability: Twins and human data Correlations Trait Monozygotic Dizygotic Height 0.93 0.48 Intelligence 0.86 0.62 Personality 0.50 0.30 Alcohol consumption 0.64 0.27 C = B1P + B2 + B3P P B1 B2 C P B3 1.0 (MZT,DZT) or 0.0 (MZA, DZA) 1.0 (MZ) or .25 (DZ) 1.0 (MZ) or .5 (DZ) A1 C1 a E1 c e D1 d A2 a C2 c E2 e P1 P2 Twin 1 Twin 2 D2 d