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Introduction to Genetic Theory
Pak Sham Twin Workshop, March 2003
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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
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Mendel’s Experiments AA aa Pure Lines F1 Aa Aa Intercross Aa Aa aa AA
3:1 Segregation Ratio
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Mendel’s Experiments F1 Pure line Aa aa Back cross Aa aa
1:1 Segregation ratio
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Mendel’s Law of Segregation
Parental genotype Meiosis/Segregation Gametes A1 A2
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Mendel’s Law of Segregation
Maternal A1 A2 Paternal A4 A3 A1 A2
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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
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IBD: Parent-Offspring
AB CD AC If the parents are unrelated, then parent-offspring pairs always share 1 allele IBD
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MZ twins always share 2 alleles IBD
IBD: MZ Twins AB CD AC AC MZ twins always share 2 alleles IBD
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IBD: Half Sibs AB CD EE AC CE/DE IBD Sharing Probability 0 ½ 1 ½
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IBD: Full Sibs IBD of paternal alleles 1 1 2 IBD of maternal alleles 1
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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
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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 Parent-offspring Full sibs Half sibs
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Proportion of Alleles IBD ()
Proportion of alleles IBD = Number of alleles IBD / 2 Relatiobship E() Var() MZ Parent-Offspring Full sibs Half sibs Most relationships demonstrate variation in across the chromosomes
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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
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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
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Biometrical Genetic Model
Genotype means AA m + a -a d +a Aa m + d aa m – a
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Continuous Variation 95% probability 2.5% 2.5% -1.96 1.96
1.96 Normal distribution Mean , variance 2
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Bivariate normal
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Familial Covariation Bivariate normal disttribution Relative 2
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Means, Variances and Covariances
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Covariance Algebra Forms Basis for Path Tracing Rules
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Covariance and Correlation
Correlation is covariance scaled to range [-1,1]. For two traits with the same variance: Cov(X1,X2) = r12 Var(X)
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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
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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
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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)2q = 2pq[a+(q-p)d]2 + (2pqd)2 = VA + VD
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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 = 2pq2
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Additive and Dominance Variance
aa Aa AA Total Variance = Regression Variance + Residual Variance = Additive Variance + Dominance Variance
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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.
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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
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Covariance for Parent-offspring (P-O)
AA Aa aa AA p3 Aa p2q pq aa 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
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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
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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 P-O U
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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
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Covariance: General Relative Pair
Genetic covariance = 2VA + VD
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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) VD ()
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Environmental components
Shared (C) Correlation = 1 Nonshared (E) Correlation = 0
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ACE Model for twin data 1 [0.5/1] E C A A C E e c a a c e PT1 PT2
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Implied covariance matrices
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Decomposing variance E Covariance A C Adoptive Siblings 0.5 1 DZ MZ
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QTL Mapping Heritability analysis:
Relates genome-wide average IBD sharing to phenotypic similarity QTL analysis: Relates locus-specific IBD sharing to phenotypic similarity
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No linkage
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Under linkage
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Path Diagram for QTL model
1 [0 / 0.5 / 1] N S Q Q S N n s q q s n PT1 PT2
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Exercise Write down to covariance matrices implied by the QTL path model, for sib pairs sharing 0, 1 and 2 alleles IBD.
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Components of variance
Phenotypic Variance Environmental Genetic GxE interaction and correlation
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Components of variance
Phenotypic Variance Environmental Genetic GxE interaction Additive Dominance Epistasis and correlation
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Components of variance
Phenotypic Variance Environmental Genetic GxE interaction Additive Dominance Epistasis Quantitative trait loci and correlation
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