Introduction to Genetic Theory Pak Sham Twin Workshop, March 2003

Aims To introduce Mendel’s law and describe its consequences for genetic relationships To describe how the covariance structure of family data is influenced by genetic factors To describe how allele-sharing at QTL influences the covariance between relatives

Mendel’s Experiments AA aa Pure Lines F1 Aa Aa Intercross Aa Aa aa AA 3:1 Segregation Ratio

Mendel’s Experiments F1 Pure line Aa aa Back cross Aa aa 1:1 Segregation ratio

Mendel’s Law of Segregation ½ Parental genotype Meiosis/Segregation Gametes A1 A2

Mendel’s Law of Segregation Maternal ½ A1 A2 Paternal ½ A4 A3 A1 A2 ¼

Identity by Descent (IBD) Two alleles are IBD if they are descended from and replicates of the same ancestral allele 1 2 Aa aa 3 4 5 6 AA Aa Aa Aa 7 8 AA Aa

IBD: Parent-Offspring AB CD AC If the parents are unrelated, then parent-offspring pairs always share 1 allele IBD

MZ twins always share 2 alleles IBD IBD: MZ Twins AB CD AC AC MZ twins always share 2 alleles IBD

IBD: Half Sibs AB CD EE AC CE/DE IBD Sharing Probability 0 ½ 1 ½

IBD: Full Sibs IBD of paternal alleles 1 1 2 IBD of maternal alleles 1

IBD: Full Sibs IBD Sharing Probability 0 1/4 1 1/2 2 1/4 0 1/4 1 1/2 2 1/4 Average IBD sharing = 1

Genetic Relationships  (kinship coefficient): Probability of IBD between two alleles drawn at random, one from each individual, at the same locus. : Probability that both alleles at the same locus are IBD Relationship   MZ twins 0.5 1 Parent-offspring 0.25 0 Full sibs 0.25 0.25 Half sibs 0.125 0

Proportion of Alleles IBD () Proportion of alleles IBD = Number of alleles IBD / 2 Relatiobship  E() Var() MZ 0.5 1 0 Parent-Offspring 0.25 0.5 0 Full sibs 0.25 0.5 0.125 Half sibs 0.125 0.25 0.0625 Most relationships demonstrate variation in  across the chromosomes

Quantitative Traits Mendel’s laws of inheritance apply to complex traits influenced by many genes Polygenic Model: Multiple loci each of small and additive effects Normal distribution of continuous variation

Quantitative Traits 1 Gene 2 Genes 3 Genes 4 Genes  3 Genotypes  3 Phenotypes 2 Genes  9 Genotypes  5 Phenotypes 3 Genes  27 Genotypes  7 Phenotypes 4 Genes  81 Genotypes  9 Phenotypes Central Limit Theorem  Normal Distribution

Biometrical Genetic Model Genotype means AA m + a -a d +a Aa m + d aa m – a

Continuous Variation 95% probability 2.5% 2.5% -1.96 1.96 1.96 Normal distribution Mean , variance 2

Bivariate normal

Familial Covariation Bivariate normal disttribution Relative 2

Means, Variances and Covariances

Covariance Algebra Forms Basis for Path Tracing Rules

Covariance and Correlation Correlation is covariance scaled to range [-1,1]. For two traits with the same variance: Cov(X1,X2) = r12 Var(X)

Genotype Frequencies (random mating) A a A p2 pq p a qp q2 q p q Hardy-Weinberg frequencies p(AA) = p2 p(Aa) = 2pq p(aa) = q2

Biometrical Model for Single Locus Genotype AA Aa aa Frequency p2 2pq q2 Effect (x) a d -a Residual var 2 2 2 Mean m = p2(a) + 2pq(d) + q2(-a) = (p-q)a + 2pqd

Single-locus Variance under Random Mating Genotype AA Aa aa Frequency p2 2pq q2 (x-m)2 (a-m)2 (d-m)2 (-a-m)2 Variance = (a-m)2p2 + (d-m)22pq + (-a-m)2q2 = 2pq[a+(q-p)d]2 + (2pqd)2 = VA + VD

Average Allelic Effect Effect of gene substitution: a  A If background allele is a, then effect is (d+a) If background allele is A, then effect is (a-d) Average effect of gene substitution is therefore = q(d+a) + p(a-d) = a + (q-p)d Additive genetic variance is therefore VA = 2pq2

Additive and Dominance Variance aa Aa AA Total Variance = Regression Variance + Residual Variance = Additive Variance + Dominance Variance

Cross-Products of Deviations for Pairs of Relatives AA Aa aa AA (a-m)2 Aa (a-m)(d-m) (d-m)2 aa (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.

Covariance of MZ Twins AA Aa aa AA p2 Aa 0 2pq aa 0 0 q2 Covariance = (a-m)2p2 + (d-m)22pq + (-a-m)2q2 = 2pq[a+(q-p)d]2 + (2pqd)2 = VA + VD

Covariance for Parent-offspring (P-O) AA Aa aa AA p3 Aa p2q pq aa 0 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

Covariance for Unrelated Pairs (U) AA Aa aa AA p4 Aa 2p3q 4p2q2 aa p2q2 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

IBD and Correlation IBD  perfect correlation of allelic effect Non IBD  zero correlation of allelic effect # alleles IBD Correlation at each locus Allelic Dom. MZ 2 1 1 P-O 1 0.5 0 U 0 0 0

Covariance for DZ twins Genotype frequencies are weighted averages: ¼ MZ twins ½ Parent-offspring ¼ Unrelated Covariance = ¼(VA+VD) + ½(VA/2) + ¼ (0) = ½VA + ¼VD

Covariance: General Relative Pair Genetic covariance = 2VA + VD

Total Genetic Variance Heritability is the combined effect of all loci total component = sum of individual loci components VA = VA1 + VA2 + … + VAN VD = VD1 + VD2 + … + VDN Correlations MZ DZ P-O U VA (2) 1 0.5 0.5 0 VD () 1 0.25 0 0

Environmental components Shared (C) Correlation = 1 Nonshared (E) Correlation = 0

ACE Model for twin data 1 [0.5/1] E C A A C E e c a a c e PT1 PT2

Implied covariance matrices

Decomposing variance E Covariance A C Adoptive Siblings 0.5 1 DZ MZ

QTL Mapping Heritability analysis: Relates genome-wide average IBD sharing to phenotypic similarity QTL analysis: Relates locus-specific IBD sharing to phenotypic similarity

No linkage

Under linkage

Path Diagram for QTL model 1 [0 / 0.5 / 1] N S Q Q S N n s q q s n PT1 PT2

Exercise Write down to covariance matrices implied by the QTL path model, for sib pairs sharing 0, 1 and 2 alleles IBD.

Components of variance Phenotypic Variance Environmental Genetic GxE interaction and correlation

Components of variance Phenotypic Variance Environmental Genetic GxE interaction Additive Dominance Epistasis and correlation

Components of variance Phenotypic Variance Environmental Genetic GxE interaction Additive Dominance Epistasis Quantitative trait loci and correlation