Biometrical Genetics

1
Biometrical Genetics

• Shaun Purcell
• Twin Workshop, March 2004

2
Single locus model

1. Genetic effects ? variance components
2. Genetic effects ? familial covariances
3. Variance components ? familial covariances

3
ADE Model for twin data
0.25/1
0.5/1
1
1
1
1
1
1
E
A
D
A
D
E
e
a
d
e
d
a
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
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

6
MENDELIAN GENETICS
7
Mendels Experiments
AA
aa
Pure Lines
F1
Aa
Aa
Intercross
AA
Aa
Aa
aa
31 Segregation Ratio
8
Mendels Experiments
F1
Pure line
Aa
aa
Back cross
Aa
aa
11 Segregation ratio
9
Mendels Experiments
AA
aa
Pure Lines
F1
Aa
Aa
Intercross
Aa
Aa
aa
AA
31 Segregation Ratio
10
Mendels Experiments
F1
Pure line
Aa
aa
Back cross
Aa
aa
11 Segregation ratio
11
Mendels Law of Segregation
Gametes
½
A1
Paternal
A2
½
Meiosis/Segregation
12
PHENOTYPE MODEL
13
Classical Mendelian Traits

• Dominant trait D, absence R
• AA, Aa ? D aa ? R
• Recessive trait R, absence D
• AA, Aa ? D aa ? R
• Co-dominant trait, X, Y, Z
• AA ? X Aa ? Y aa ? Z

14
Dominant Mendelian inheritance
½
D
Paternal
d
½
15
Recessive Mendelian inheritance
½
D
Paternal
d
½
16
Dominant Mendelian inheritance
½
D
Paternal
d
½
17
Quantitative traits
18
Biometrical Genetic Model
P(X)
Aa
Genotypic means
AA
aa
AA
m a
X
Aa
m d
m
aa
m a
a
-a
d
19
POPULATION MODEL
20
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)

21
Genotype Frequencies (random mating)

• A a
• A p2 pq p
• a qp q2 q
• p q
• Hardy-Weinberg Equilibrium frequencies
• P(AA) p2
• P(Aa) 2pq
• P(aa) q2

22
Before we proceed,some basic statistical tools
23
Means, Variances and Covariances
24
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)
• a(p-q) 2pqd

25
Biometrical Model for Single Locus

• 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
• VG
• (Broad-sense) heritability at this loci VG /
  VTOT
• (Broad-sense) heritability SLVG / VTOT

26
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

27
Practical 1

• H\pshaun\biometric\sgene.exe
• What determines additive genetic variance?
• Under what conditions does VD gt VA

28
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

29
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

AA Aa aa Allelic mean Average effect
a d -a
A p q paqd q(ad(q-p))
a p q qa-pd -p(ad(q-p))
30
Average allelic effect

• Denote the average allelic effects as a
• aA q(ad(q-p))
• aa -p(ad(q-p))
• If only two alleles exist, we can define the
  average effect of allele substitution
• a aA aa
• a (q-(-p))(ad(q-p)) (ad(q-p))
• Therefore, aA qa and aa -pa

31
Additive genetic variance

• The variance of the average allelic effects
• Freq. Additive effect
• AA p2 2aA 2qa
• Aa 2pq aA aa (q-p)a
• aa q2 2aa -2pa
• VA p2(2qa)2 2pq((q-p)a)2 q2(-2pa)2
• 2pqa2
• 2pq(ad(q-p))2

32
Additive genetic variance

• If there is no dominance VA 2pqa2
• If p q VA ½a2

33
Additive and Dominance Variance
aa
Aa
AA
Total Variance Regression Variance Residual
Variance Additive Variance Dominance
Variance
34
Biometrical Model for Single Locus

• 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
• VG VA VD

35
VA
36
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
37
-1 0 1
d
-1 0 1
a
VA gt VD
VA lt VD
Allele frequency
0.01
0.05
0.1
0.2
0.3
0.5
38
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
39
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
40
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.

41
Exercise 2

• e.g. given an AA father, an AA offspring can come
  from either AA x AA or AA x Aa parental
  mating types
• AA x AA will occur p2 p2 p4
• and have AA offspring Prob()1
• AA x Aa will occur p2 2pq 2p3q
• and have AA offspring Prob()0.5
• and have Aa offspring Prob()0.5
• Therefore, P(AA father AA offspring) p4
  p3q
• p3(pq)
• p3

42
Covariance for Parent-offspring (P-O)

• AA Aa aa
• AA p3
• Aa ? ?
• aa ? ? ?
• AA offspring from AA parents p4p3q p3(pq)
  p3

43
Parental mating types
Pat Mat P(PxM) AA Aa aa
AA AA p4 p4 0 ?
AA Aa 2p3q p3q p3q ?
AA aa p2q2 0 p2q2 ?

Aa AA 2p3q ? ? ?
Aa Aa 4p2q2 ? ? ?
Aa aa 2pq3 ? ? ?

aa AA p2q2 ? ? ?
aa Aa 2pq3 ? ? ?
aa aa q4 ? ? ?
44
Covariance for Parent-offspring (P-O)

• AA Aa aa
• AA p3
• Aa p2q ?
• aa ? ? ?
• AA offspring from AA parents p4p3q p3(pq)
  p3
• Aa offspring from AA parents p3qp2q2
  p2q(pq) p2q

45
Parental mating types
Pat Mat P(PxM) AA Aa aa
AA AA p4 p4
AA Aa 2p3q p3q p3q
AA aa p2q2 p2q2

Aa AA 2p3q p3q p3q
Aa Aa 4p2q2 p2q2 2p2q2 p2q2
Aa aa 2pq3 pq3 pq3

aa AA p2q2 p2q2
aa Aa 2pq3 pq3 pq3
aa aa q4 q4
46
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
47
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)4pq (a-m)(-a-m)2p2q2
0
?
48
IDENTITY BY DESCENT
49
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
50
IBS ? IBD
A1A2
A1A3
IBS 1 IBD 0
A1A2
A1A3
IBSIdentity by State
51
IBD MZ Twins
AB
CD
AC
AC
MZ twins always share 2 alleles IBD
52
IBD Parent-Offspring
AB
CD
AC
If the parents are unrelated, then
parent-offspring pairs always share 1 allele IBD
53
IBD Unrelated individuals
AB
CD
If two individuals are unrelated, they always
share 0 allele IBD
54
IBD Half Sibs
AB
CD
EE
AC
CE/DE
IBD Sharing Probability 0 ½ 1 ½
55
IBD Full Sibs
IBD of paternal alleles
0
1
0 1
1 2
0
IBD of maternal alleles
1
56
IBD and Correlation

• IBD ? perfect correlation of allelic effect
• Non IBD ? zero correlation of allelic effect
• alleles IBD Correlation
• at a locus Allelic Dom.
• MZ 2 1 1
• P-O 1 0.5 0
• U 0 0 0

57
Covariance between relatives

• Partition of variance ? Partition of covariance
• Overall covariance
• sum of covariances of all components
• Covariance of component between relatives
• correlation of component
• ? variance due to component

58
Average correlation in QTL effects

• MZ twins P(IBD 0) 0
• P(IBD 1) 0
• P(IBD 2) 1
• Average correlation
• Additive component 00 0½ 11
• 1
• Dominance component 00 00 11
• 1

59
Average correlation in QTL effects

• P-O P(IBD 0) 0
• P(IBD 1) 1
• P(IBD 2) 0
• Average correlation
• Additive component 00 1½ 01
• ½
• Dominance component 00 10 01
• 0

60
Average correlation in QTL effects

• Unrelated P(IBD 0) 1
• P(IBD 1) 0
• P(IBD 2) 0
• Average correlation
• Additive component 10 0½ 01
• 0
• Dominance component 10 00 01
• 0

61
Mendels Law of Segregation
62
IBD sharing for two sibs
Sib 2
A1A3 A1A4 A2A3 A2A4
A1A3 A1A4 A2A3 A2A4
S i b 1
63
IBD sharing for two sibs
A1A3 A1A4 A2A3 A2A4
0
1
1
2
A1A3 A1A4 A2A3 A2A4
1
0
1
2
1
0
1
2
2
0
1
1
Expected IBD sharing (20.25) (10.5)
(00.25) 1
Pr(IBD0) 4 / 16 0.25
Pr(IBD1) 8 / 16 0.50
Pr(IBD2) 4 / 16 0.25
64
Average correlation in QTL effects

• Sib pairs P(IBD 0) ¼
• P(IBD 1) ½
• P(IBD 2) ¼
• Average correlation
• Additive component ¼0 ½½ ¼1
• ½
• Dominance component ¼0 ½0 ¼1
• ¼

65
Summary of shared genetic variance

• MZ VA VD
• P-O ½ VA
• U 0
• DZ, FS ¼(VAVD) ½(VA/2) ¼ (0)
• ½VA ¼VD
• These single locus results can be summed over all
  loci to give total genetic variance, heritability.

66
(No Transcript)
67
Figure 3. BIOLOGICAL PARENT - OFFSPRING
1
1
1
1
1
1
1
1
E
C
A
D
E
C
A
D
d
a
c
e
d
a
c
e
0.5
0.5
0.5
0.5
P
P
MOTHER
FATHER
1
0.5
R
R
0.5
A
0.25
A
1
1
E
C
A
D
D
A
C
E
1
1
1
1
1
1
d
h
c
e
e
c
h
d
P
P
CHILD 2
CHILD 1
68
Segregation variance

• If both parents transmit 1 allele
• AO AP/2 AM/2
• which means
• Var(AO) ¼ Var(AP) ¼ Var(AM)
• But this violates Var(AO)Var(AP)Var(AM)
• So this explains the extra path in the P-O model
  as Var(A)1, arbitrarily, then
• Var(AO) ¼ Var(AP) ¼ Var(AM) ½

69
Segregation variance

• Segregation variance (SV) represents the random
  variation among the gametes of an individual
• i.e. an Aa parent could transmit either A or a
• AO AP/2 AM/2 SP SM
• Var (AO) Var(AP)/4 Var(AM)/4 Var(SP)
  Var(SM)
• For homozygous locus, Var(S) 0
• For heterozygous locus, Var(S) ?2/4
• Given random mating, average SV 2pq?2/4 VA/4
• As VA is fixed to 1 in the path model, SV ¼ ¼
  ½