1. Introduction to Genetic Theory
Pak Sham
Twin Workshop, March 2003

2. 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

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

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

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

6. Mendel’s Law of Segregation
Maternal A1 A2
Paternal A4 A3 A1 A2

7. 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

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

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

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

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

12. 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

13. 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

14. 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

15. 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

16. 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

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

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

19. Bivariate normal

20. Familial Covariation Bivariate normal disttribution Relative 2

21. Means, Variances and Covariances

22. Covariance Algebra
Forms Basis for Path Tracing Rules

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

24. 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

25. 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

26. 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

27. 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

28. Additive and Dominance Varianceaa Aa AA
Total Variance = Regression Variance + Residual Variance
= Additive Variance + Dominance Variance

29. 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.

30. 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

31. 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

32. 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

33. 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

34. 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

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

36. 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 ()

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

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

39. Implied covariance matrices

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

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

42. No linkage

43. Under linkage

44. Path Diagram for QTL model1
[0 / 0.5 / 1] N S Q Q S N n s q q s n
PT1
PT2

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

46. Components of variance
Phenotypic Variance
Environmental Genetic GxE interaction
and correlation

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

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