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Biometrical Genetics

This workshop delves into biometrical and Mendelian genetics models, covering allele frequencies, genetic effects, segregation ratios, and variance components. Learn about additive and dominance effects in genetic inheritance and practical applications. Explore twin data and the nuances of genetic variance. Gain insights into genotype frequencies and Hardy-Weinberg equilibrium. Dive deep into the genetic theories that shape our understanding of heredity and inheritance patterns.

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Biometrical Genetics

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  1. Biometrical Genetics Shaun Purcell Twin Workshop, March 2004

  2. Single locus model • Genetic effects → variance components • Genetic effects → familial covariances • Variance components → familial covariances

  3. ADE Model for twin data [0.25/1] [0.5/1] 1 1 1 1 1 1 E D A A D E e d a a d e PT1 PT2

  4. Some Components of a Genetic Theory • POPULATION MODEL • Allele & genotype frequencies • TRANSMISSION MODEL • Mendelian segregation • Identity by descent & genetic relatedness • PHENOTYPE MODEL • Biometrical model of quantitative traits • Additive & dominance components

  5. MENDELIAN GENETICS

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

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

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

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

  10. Maternal A3 A4 ½ ½ A1 A1 A3 A4 ¼ ¼ A2 A2 A3 A4 ¼ ¼ Mendel’s Law of Segregation Gametes ½ A1 Paternal A2 ½ Meiosis/Segregation

  11. PHENOTYPE MODEL

  12. Maternal D d ½ ½ D D D d 1 1 d d D d 1 0 Dominant Mendelian inheritance ½ D Paternal d ½

  13. Maternal D d ½ ½ D D D d 1 0 d d D d 0 0 Recessive Mendelian inheritance ½ D Paternal d ½

  14. Maternal D d ½ ½ D D D d Incomplete penetrance 60% 60% d d D d Phenocopies 60% 1% Dominant Mendelian inheritance ½ D Paternal d ½

  15. AA Aa aa Quantitative traits

  16. Biometrical Genetic Model P(X) Aa Genotypic means aa AA AA m + a X Aa m + d m aa m – a -a +a d

  17. POPULATION MODEL

  18. Population Frequencies • A single locus, with two alleles • Biallelic / diallelic • Single nucleotide polymorphism, SNP • Alleles A and a • Frequency of A is p • Frequency of a is q= 1 – p • Every individual inherits two copies • A genotype is the combination of the two alleles • e.g. AA, aa(the homozygotes) or Aa (the heterozygote)

  19. Genotype Frequencies (random mating) Aa Ap2 pqp aqp q2q p q Hardy-Weinberg Equilibrium frequencies P(AA) = p2 P(Aa) = 2pq P(aa) = q2

  20. Before we proceed,some basic statistical tools…

  21. Means, Variances and Covariances

  22. Biometrical Model for Single Locus GenotypeAAAaaa Frequencyp2 2pq q2 Effect (x)a d -a Residual var2 2 2 Meanm = p2(a) + 2pq(d) + q2(-a) = a(p-q) + 2pqd

  23. Biometrical Model for Single Locus GenotypeAA Aa aa Frequencyp2 2pq q2 (x-m)2(a-m)2 (d-m)2 (-a-m)2 Variance = (a-m)2p2 + (d-m)22pq + (-a-m)2q2 = VG (Broad-sense) heritability at this loci = VG / VTOT (Broad-sense) heritability = ΣLVG / VTOT

  24. Additive and dominance effects • Additive effects are the main effects of individual alleles: ‘gene-dosage’ • Parents transmit alleles, not genotypes • Dominance effects represent an interaction between the two alleles • i.e. if the heterozygote is not midway between the two homozygotes

  25. Practical 1 • H:\pshaun\biometric\sgene.exe • What determines additive genetic variance? • Under what conditions does VD > VA

  26. Some conclusions • Additive genetic variance depends on allele frequency p & additive genetic value a as well as dominance deviation d • Additive genetic variance typically greater than dominance variance

  27. Average allelic effect • Average allelic effect is the deviation of the allelic mean from the population mean, a(p-q)+2pqd • Of all the A alleles in the population: • A proportion (p) will be paired with another A • A proportion (q) will be paired with another a

  28. Average allelic effect • Denote the average allelic effects as α αA = q(a+d(q-p)) αa = -p(a+d(q-p)) • If only two alleles exist, we can define the average effect of allele substitution α = αA – αa α = (q-(-p))(a+d(q-p)) = (a+d(q-p)) • Therefore, αA = qα and αa = -pα

  29. Additive genetic variance • The variance of the average allelic effects Freq.Additive effect AA p2 2αA = 2qα Aa 2pq αA +αa = (q-p)α aa q2 2αa = -2pα VA = p2(2qα)2 + 2pq((q-p)α)2 + q2(-2pα)2 = 2pqα2 = 2pq(a+d(q-p))2

  30. Additive genetic variance • If there is no dominance VA = 2pqa2 • If p = q VA = ½a2

  31. a d m -a Additive and Dominance Variance aa Aa AA Total Variance = Regression Variance + Residual Variance = Additive Variance + Dominance Variance

  32. Biometrical Model for Single Locus GenotypeAA Aa aa Frequencyp2 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 VG = VA + VD

  33. VA

  34. Additive genetic variance VA -1 d -1 a +1 +1 Dominance genetic variance VD Allele frequency 0.01 0.05 0.1 0.2 0.3 0.5

  35. AA Aa aa -1 0 +1 d -1 0 +1 a VA > VD VA < VD Allele frequency 0.01 0.05 0.1 0.2 0.3 0.5

  36. 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:

  37. Covariance of MZ Twins AA Aa aa AAp2 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

  38. Covariance for Parent-offspring (P-O) AA Aa aa AA ? Aa ? ? aa ? ? ? • Exercise 2 : to calculate frequencies of parent-offspring combinations, in terms of allele frequencies p and q.

  39. Exercise 2 • e.g. given an AAfather, an AAoffspring can come from either AAx AAor AAx Aaparental mating types AAx AA will occur p2× p2 = p4 and have AA offspring Prob()=1 AAx Aa will occur p2× 2pq = 2p3q and have AA offspring Prob()=0.5 and have Aa offspring Prob()=0.5 Therefore, P(AA father & AAoffspring) = p4 + p3q = p3(p+q) = p3

  40. Covariance for Parent-offspring (P-O) AA Aa aa AAp3 Aa ? ? aa ? ? ? • AA offspring from AAparents = p4+p3q = p3(p+q) = p3

  41. Parental mating types

  42. Covariance for Parent-offspring (P-O) AA Aa aa AAp3 Aap2q ? aa ? ? ? • AA offspring from AAparents = p4+p3q = p3(p+q) = p3 • Aa offspring from AAparents = p3q+p2q2 = p2q(p+q) = p2q

  43. Parental mating types

  44. Covariance for Parent-offspring (P-O) AA Aa aa AAp3 Aap2q 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

  45. Covariance for Unrelated Pairs (U) AA Aa aa AAp4 Aa2p3q 4p2q2 aap2q2 2pq3 q4 Covariance = (a-m)2p4 + (d-m)24p2q2 + (-a-m)2q4 + (a-m)(d-m)4p3q+ (-a-m)(d-m)4pq + (a-m)(-a-m)2p2q2 = 0 ?

  46. IDENTITY BY DESCENT

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

  48. IBS  IBD A1A2 A1A3 IBS = 1 IBD = 0 A1A2 A1A3 IBS=Identity by State

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

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

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