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

2. 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

3. Mendel’s Law of Segregation A3A3 A4A4 Maternal ½½ A1A1 A2A2 A1A1 A2A2 Paternal ½ ½ ¼ ¼ ¼ ¼ A4A4 A4A4 A3A3 A3A3 A1A1 A2A2

4. 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

5. Biometrical Genetic Model 0 d +a-a Genotype means AA Aa aa m + a m + d m – a

6. 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

7. Quantitative Traits 1 Gene  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

8. Continuous Variation 0 1.96 -1.96 2.5% 95% probability Normal distribution Mean , variance  2

9. Familial Covariation Relative 1 Relative 2 Bivariate normal disttribution

10. Means, Variances and Covariances

11. Covariance Algebra Forms Basis for Path Tracing Rules

12. Covariance and Correlation Correlation is covariance scaled to range [-1,1]. For two traits with the same variance: Cov(X 1,X 2 ) = r 12 Var(X)

13. Genotype Frequencies (random mating) Aa Ap 2 pq p aqp q 2 q p q Hardy-Weinberg frequencies p(AA) = p 2 p(Aa) = 2pq p(aa) = q 2

14. Biometrical Model for Single Locus GenotypeAAAaaa Frequencyp 2 2pqq 2 Effect (x)ad-a Residual var  2  2  2 Mean m = p 2 (a) + 2pq(d) + q 2 (-a) = (p-q)a + 2pqd

15. Single-locus Variance under Random Mating GenotypeAAAaaa Frequencyp 2 2pqq 2 (x-m) 2 (a-m) 2 (d-m) 2 (-a-m) 2 Variance = (a-m) 2 p 2 + (d-m) 2 2pq + (-a-m) 2 q 2 = 2pq[a+(q-p)d] 2 + (2pqd) 2 = V A + V D

16. 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 V A = 2pq  2

17. Additive and Dominance Variance aaAaAA m -a a d Total Variance = Regression Variance + Residual Variance = Additive Variance + Dominance Variance

18. Cross-Products of Deviations for Pairs of Relatives AAAaaa 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.

19. Covariance of MZ Twins AAAaaa AAp 2 Aa02pq aa00q 2 Covariance = (a-m) 2 p 2 + (d-m) 2 2pq + (-a-m) 2 q 2 = 2pq[a+(q-p)d] 2 + (2pqd) 2 = V A + V D

20. Covariance for Parent-offspring (P-O) AAAaaa AAp 3 Aap 2 qpq aa0 pq 2 q 3 Covariance = (a-m) 2 p 3 + (d-m) 2 pq + (-a-m) 2 q 3 + (a-m)(d-m)2p 2 q + (-a-m)(d-m)2pq 2 = pq[a+(q-p)d] 2 = V A / 2

21. Covariance for Unrelated Pairs (U) AAAaaa AAp 4 Aa2p 3 q4p 2 q 2 aap 2 q 2 2pq 3 q 4 Covariance = (a-m) 2 p 4 + (d-m) 2 4p 2 q 2 + (-a-m) 2 q 4 + (a-m)(d-m)4p 3 q + (-a-m)(d-m)4pq 3 + (a-m)(-a-m)2p 2 q 2 = 0

22. Segregation Variance A O = A P /2 + A M /2 + S P + S M E(A O | A P, A M ) = A P /2 + A M /2 Var (A O ) = Var(A P )/4 + Var(A M )/4 + Var(S P ) + Var(S M ) Segregation variance represents random variation among the gametes of an individual. For homozygous locus, Var(S) = 0 For heterozygous locus, Var(S) =  2 /4 (Binomial: 0,  with equal probability) Under random mating: Average segregation variance = 2pq  2 /4 = V A /4

23. Identity by Descent (IBD) Two alleles are IBD if they are descended from and replicates of the same ancestral allele 7 AaAa AaAa AaAa A aa AA Aa 1 2 3456 8

24. IBD and Correlation IBD  perfect correlation of allelic effect Non IBD  zero correlation of allelic effect # alleles IBD Correlation at each locusAllelicDom. MZ211 P-O10.50 U000

25. Mendel’s Law of Segregation A3A3 A4A4 Maternal ½½ A1A1 A2A2 A1A1 A2A2 Paternal ½ ½ ¼ ¼ ¼ ¼ A4A4 A4A4 A3A3 A3A3 A1A1 A2A2

26. Two offspring A 1 A 3 A 1 A 4 A 2 A 3 A 2 A 4 A1A3A1A4 A2A3A2A4A1A3A1A4 A2A3A2A4 A 1 A 3 A 2 A 4 A 1 A 4 A 2 A 3 A 2 A 3 A 1 A 4 A 2 A 4 A 1 A 3 A 1 A 3 A 1 A 4 A 1 A 3 A 2 A 3 A 1 A 4 A 1 A 3 A 1 A 4 A 2 A 4 A 2 A 3 A 1 A 3 A 2 A 3 A 2 A 4 A 2 A 4 A 1 A 4 A 2 A 4 A 2 A 3 A 1 A 3 A 1 A 4 A 2 A 3 A 2 A 4 Sib 2 Sib1Sib1

27. IBD sharing for two sibs A 1 A 3 A 1 A 4 A 2 A 3 A 2 A 4 A1A3A1A4 A2A3A2A4A1A3A1A4 A2A3A2A4 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 Pr(IBD=0) = 4 / 16 = 0.25 Pr(IBD=1) = 8 / 16 = 0.50 Pr(IBD=2) = 4 / 16 = 0.25 Expected IBD sharing = (2*0.25) + (1*0.5) + (0*0.25) = 1

28. Covariance for DZ twins Genotype frequencies are weighted averages: –¼ MZ twins –½ Parent-offspring –¼ Unrelated Covariance = ¼(V A +V D ) + ½(V A /2) + ¼ (0) = ½V A + ¼V D

29. Covariance: General Relative Pair Kinship coefficient: Probability of IBD between two alleles drawn at random, one from each individual, at the same locus (  ) Probability of sharing two alleles at the same locus IBD (  ) For non-inbred individuals: Average proportion of alleles IBD = 2  Genetic covariance = 2  V A +  V D

30. Total Genetic Variance Heritability is the combined effect of all loci –total component = sum of individual loci components V A = V A1 + V A2 + … + V AN V D = V D1 + V D2 + … + V DN CorrelationsMZDZP-OU –V A (2  )10.50.50 –V D (  ) 10.2500

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

32. P T1 AC E P T2 ACE 1 [0.5/1] eaceca ACE Model for twin data

33. Implied covariance matrices

34. Epistasis AA Aa aa BBm+a A +a B +aa m+d A +a B +da m-a A +a B -aa Bbm+a A +d B +ad m+d A +d B +dd m-a A +d B -ad bbm+a A -a B -aa m+d A -a B -da m-a A -a B +aa Calculation of variance involves summing over 9 terms. Calculation of covariance involves summing over 27 terms

35. A×A Epistasis X = aA + bB + iAB Var(X) = a 2 + b 2 + i 2 Cov(X 1,X 2 ) = a 2 Cov(A 1,A 2 ) + b 2 Cov(B 1,B 2 ) + i 2 Cov(A 1 B 1,A 2 B 2 ) MZ = a 2 + b 2 + i 2 DZ = a 2 /2 + b 2 /2 + i 2 /4 Note : Cov(A 1 B 1,A 2 B 2 ) = Cov(A 1,A 2 )Cov(B 1,B 2 )

36. A×C Interaction X = aA + cC + iAC Var(X) = a 2 + c 2 + i 2 Cov(X 1,X 2 ) = a 2 Cov(A 1,A 2 ) + c 2 Cov(C 1,C 2 ) + i 2 Cov(A 1 C 1,A 2 C 2 ) MZ = a 2 + c 2 + i 2 DZ = a 2 /2 + c 2 + i 2 /2 Note : Cov(A 1 C 1,A 2 C 2 ) = Cov(A 1,A 2 )Cov(C 1,C 2 ) = Cov(A 1,A 2 ) × 1

37. A×E Interaction X = aA + cE + iAE Var(X) = a 2 + e 2 + i 2 Cov(X 1,X 2 ) = a 2 Cov(A 1,A 2 ) + e 2 Cov(E 1,E 2 ) + i 2 Cov(A 1 E 1,A 2 E 2 ) MZ = a 2 DZ = a 2 /2 Note : Cov(A 1 E 1,A 2 E 2 ) = Cov(A 1,A 2 )Cov(E 1,E 2 ) = Cov(A 1,A 2 ) × 0

38. A×C Correlation X = aA + cC Var(X) = a 2 + c 2 + 2ac r AC Cov(X 1,X 2 ) = a 2 Cov(A 1,A 2 ) + c 2 Cov(C 1,C 2 ) + ac Cov(A 1,C 2 ) + ac Cov(A 2, C 1 ) MZ = a 2 + c 2 + 2ac r AC DZ = a 2 /2 + c 2 + 2ac r AC Note : Cov(A 1,C 2 ) = Cov(A 2,C 1 ) = r AC

39. A×E Correlation X = aA + eE Var(X) = a 2 + e 2 + 2ae r AE Cov(X 1,X 2 ) = a 2 Cov(A 1,A 2 ) + e 2 Cov(E 1,E 2 ) + ae Cov(A 1,E 2 ) + ae Cov(A 2, E 1 ) MZ = a 2 + 2ae r AE DZ = a 2 /2 + ae r AE

40. Components of variance Phenotypic Variance EnvironmentalGeneticGxE interaction and correlation

41. Components of variance Phenotypic Variance EnvironmentalGeneticGxE interaction Additive DominanceEpistasis and correlation

42. Components of variance Phenotypic Variance EnvironmentalGeneticGxE interaction Additive DominanceEpistasis Quantitative trait loci and correlation