Pak Sham & Shaun Purcell Twin Workshop, March 2002 Biometrical Genetics Pak Sham & Shaun Purcell Twin Workshop, March 2002

Aims To gain an appreciation of the scope and rationale of biometric genetics To understand how the covariance structure of twin data is determined by genetic and environmental factors

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

Classical Mendelian Traits Dominant trait D, absence R AA, Aa  D; aa  R Recessive trait R, absence D AA  D; Aa, aa  R Co-dominant trait, X, Y, Z AA  X; Aa  Y; aa  Z

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

Quantitative Traits Mendel’s laws of inheritance apply to complex traits influenced by many genes Assume 2 alleles per loci acting additively Genotype AA Aa aa Phenotype 1 0 -1 Multiple loci 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

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

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].

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 Deviation (x) a d -a Residual var 2 2 2 Mean m = p2(a) + 2pq(d) + q2(-a) = (p-q)a + 2pqd

Genetic 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[d+(q-p)d]2 + (2pqd)2 = VA + VD

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 for 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[d+(q-p)d]2 + (2pqd)2 = VA + VD

Genotype tables : Parent-offspring AA Aa aa AA p2 Aa 0 2pq aa 0 0 q2

Genotype tables : Unrelated AA Aa aa AA p4 Aa 2p3q 4p2q2 aa p2q2 2pq3 q2 Covariance = … = 0

Genotype tables : DZ twins AA Aa aa AA p4 Aa 2p3q 4p2q2 aa p2q2 2pq3 q2 Weighted average ¼ MZ twins ½ Parent-offspring ¼ Unrelated Covariance = … = ½VA+¼VD

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

Components of variance Phenotypic Variance Environmental Genetic GxE interaction

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

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

Parent-offspring