Introduction to Genetic Theory

1
Introduction to Genetic Theory

• Pak Sham
• Twin Workshop, March 2003

2
Aims

• To introduce Mendels 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
Mendels Experiments
AA
aa
Pure Lines
F1
Aa
Aa
Intercross
Aa
Aa
aa
AA
31 Segregation Ratio
4
Mendels Experiments
F1
Pure line
Aa
aa
Back cross
Aa
aa
11 Segregation ratio
5
Mendels Law of Segregation
Parental genotype
Meiosis/Segregation
A1
A2
Gametes
6
Mendels Law of Segregation
7
Identity by Descent (IBD)

• Two alleles are IBD if they are descended from
  and replicates of the same ancestral allele

2
1
aa
Aa
3
4
5
6
AA
Aa
Aa
Aa
7
8
AA
Aa
8
IBD Parent-Offspring
AB
CD
AC
If the parents are unrelated, then
parent-offspring pairs always share 1 allele IBD
9
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
0
1
0
IBD of maternal alleles
1
12
IBD Full Sibs
IBD Sharing Probability 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 0.5 1 Parent-off
spring 0.25 0 Full sibs 0.25 0.25 Half
sibs 0.125 0

14
Proportion of Alleles IBD (?)
Proportion of alleles IBD Number of alleles IBD
/ 2
Relatiobship ? E(?) Var(?) MZ 0.5 1
0 Parent-Offspring 0.25 0.5 0 Full
sibs 0.25 0.5 0.125 Half sibs 0.125 0.25 0.0625
Most relationships demonstrate variation in ?
across the chromosomes
15
Quantitative Traits

• Mendels 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
Central Limit Theorem ? Normal Distribution
17
Biometrical Genetic Model
Genotype means
0
AA
m a
d
a
-a
Aa
m d
aa
m a
18
Continuous Variation
95 probability
2.5
2.5
-1.96
1.96
0
Normal distribution Mean ?, variance ?2
19
Bivariate normal
20
Familial Covariation
Bivariate normal disttribution
Relative 2
Relative 1
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,X
2) 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)2q2
  2pqa(q-p)d2 (2pqd)2
• VA VD

27
Average Allelic Effect
Effect of gene substitution a ? A
If background allele is a, then effect is
(da) If background allele is A, then effect is
(a-d)

• Average effect of gene substitution is therefore
• q(da) p(a-d) a (q-p)d

Additive genetic variance is therefore VA 2pq?2
28
Additive and Dominance Variance
aa
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
2pqa(q-p)d2 (2pqd)2 VA
VD
31
Covariance for Parent-offspring (P-O)

• AA Aa aa
• AA p3
• Aa p2q 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
pqa(q-p)d2 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 2 1 1
• P-O 1 0.5 0
• U 0 0 0

34
Covariance for DZ twins

• Genotype frequencies are weighted averages
• ¼ MZ twins
• ½ Parent-offspring
• ¼ Unrelated
• Covariance ¼(VAVD) 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?) 1 0.5 0.5 0
• VD (?) 1 0.25 0 0

37
Environmental components

• Shared (C)
• Correlation 1
• Nonshared (E)
• Correlation 0

38
ACE Model for twin data
1
0.5/1
E
A
C
A
C
E
e
a
c
e
c
a
PT1
PT2
39
Implied covariance matrices
40

Decomposing variance
E
Covariance
A
C
0 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 model

1
0 / 0.5 / 1
N
Q
S
Q
S
N
n
q
s
n
s
q
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