Genetic Theory

Pak Sham

SGDP, IoP, London, UK

Theory Model Data

Inference

ExperimentFormulation

Interpretation

Components of a genetic model

POPULATION PARAMETERS

- alleles / haplotypes / genotypes / mating types

TRANSMISSION PARAMETERS

- parental genotype offspring genotype

PENETRANCE PARAMETERS

- genotype phenotype

Transmission : Mendel’s law of segregation

A

A

A

A

Paternal

Maternal

AA

AA

A

A A

A

½ ½

½

½

¼

¼

¼

¼

Two offspring

AA AA AA AA

AA

AA AA

AA AA AA

AA AA

AA AA

AA AA

AA AA

AA AA

AA AA

AA AA

AA AA

AA AA

AA AA

AA AA

AA AA

AA AA

AA AA

AA AA

Sib 2

Sib1

IBD sharing for two sibs

AA AA AA AA

AA

AA AA

AA 0

0

0

0

1

1

1

1

1

1

1

1

2

2

2

2

Pr(IBD=0) = 4 / 16 = 0.25Pr(IBD=1) = 8 / 16 = 0.50Pr(IBD=2) = 4 / 16 = 0.25

Expected IBD sharing = (2*0.25) + (1*0.5) + (0*0.25) = 1

IBS IBD

A1A2 A1A3

A1A2 A1A3

IBS = 1

IBD = 0

1

2

- identify all nearest common ancestors (NCA)

X Y

- trace through each NCA and count # of meioses

via X : 5 meiosesvia Y : 5 meioses

- expected IBD proportion = (½)5 + (½)5 = 0.0625

Sib pairs

Expected IBD proportion = 2 (½)2 = ½

Segregation of two linked loci

Parental genotypes

Likely (1-)

Unlikely ()

= recombination fraction

Recombination & map distance

2

1 2me

Haldane mapfunction

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.2 0.4 0.6 0.8 1

Map distance (M)

Re

co

mb

ina

tio

n f

rac

tio

n

Segregation of three linked loci

(1-1)(1-2)

1 2

(1-1)2

1(1-2)

12

Two-locus IBD distribution: sib pairs

Two loci, A and B, recombination faction

For each parent:

Prob(IBD A = IBD B) = 2 + (1-)2 =

either recombination for both sibs,

or no reombination for both sibs

2 )1(2 2)1(

2)1(2 2)1(

)1( )1( )1(21

0

1/2

1

0 1/2 1

at QTL

at M

Conditional distribution of at maker given at QTL

Correlation between IBD of two loci

For sib pairs

Corr(A, B) = (1-2AB)2

attenuation of linkage information with increasing

genetic distance from QTL

Population Frequencies

Single locus

Allele frequencies A P(A) = p

a P(a) = q

Genotype frequencies

AA p(AA) = u

Aa p(Aa) = v

aa p(aa) = r

Mating type frequencies

u v r

AA Aa aa

u AA u2 uv ur

v Aa uv v2 vr

r aa ur vr r2

Random mating

Hardy-Weinberg Equilibrium

u+½v r+½v

A a

u+½v A

r+½v a

u1 = (u0 + ½v0)2

v1 = 2(u0 + ½v0) (r0 + ½v0)r1 = (r0 + ½v0)2

u2 = (u1 + ½v1)2

= ((u0 + ½v0)2 + ½2(u0 + ½v0) (r0 + ½v0))2

= ((u0 + ½v0)(u0 + ½v0 + r0 + ½v0))2

= (u0 + ½v0)2 = u1

Hardy-Weinberg frequencies

Genotype frequencies:

AA p(AA) = p2

Aa p(Aa) = 2pq

aa p(aa) = q2

Two-locus: haplotype frequencies

Locus B

B b

Locus A A AB Ab

a aB ab

Haplotype frequency table

Locus B

B b

Locus A A pr ps p

a qr qs q

r s

Haplotype frequency table

Locus B

B b

Locus A A pr+D ps-D p

a qr-D qs+D q

r s

Dmax = Min(ps,qr), D’ = D / Dmax

R2 = D2 / pqrs

Causes of allelic association

Tight Linkage

Founder effect: D (1-)G

Genetic Drift: R2 (NE)-1

Population admixture

Selection

Genotype-Phenotype Relationship

Penetrance = Prob of disease given genotype

AA Aa aa

Dominant 1 1 0

Recessive 1 0 0

General f2 f1 f0

Biometrical model of QTL effects

Genotypic

means

AA m + a

Aa m + d

aa m - a

0

d +a-a

Quantitative Traits

Mendel’s laws of inheritance apply to complex traits influenced by many genes

Assume: 2 alleles per locus acting additivelyGenotypes A1 A1 A1 A2 A2 A2

Effect -1 0 1

Multiple loci Normal distribution of continuous variation

Quantitative Traits

0

1

2

3

1 Gene 3 Genotypes 3 Phenotypes

0

1

2

3

2 Genes 9 Genotypes 5 Phenotypes

01234567

3 Genes 27 Genotypes 7 Phenotypes

0

5

10

15

20

4 Genes 81 Genotypes 9 Phenotypes

Components of variance

Phenotypic Variance

Environmental Genetic GxE interaction

Components of variance

Phenotypic Variance

Environmental Genetic GxE interaction

Additive Dominance Epistasis

Components of variance

Phenotypic Variance

Environmental Genetic GxE interaction

Additive Dominance Epistasis

Quantitative trait loci

Biometrical model for QTL

Genotype AA Aa aa

Frequency (1-p)2 2p(1-p) p2

Trait mean -a d a

Trait variance 2 2 2

Overall mean a(2p-1)+2dp(1-p)

QTL Variance Components

Additive QTL variance

VA = 2p(1-p) [ a - d(2p-1) ]2

Dominance QTL variance

VD = 4p2 (1-p)2 d2

Total QTL variance

VQ = VA + VD

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

Correlation in QTL effects

Since is the proportion of shared alleles,

correlation in QTL effects depends on

0 1/2

1

Additive component 0 1/2 1

Dominance component 0 0 1

Average correlation in QTL effects

MZ twins P(=0) = 0

P(=1/2) = 0

P(=1) = 1

Average correlation

Additive component = 0*0 + 0*1/2 + 1*1

= 1

Dominance component = 0*0 + 0*0 + 1*1

= 1

Average correlation in QTL effects

Sib pairs P(=0) = 1/4

P(=1/2) = 1/2

P(=1) = 1/4

Average correlation

Additive component = (1/4)*0+(1/2)*1/2+(1/4)*1

= 1/2

Dominance component = (1/4)*0+(1/2)*0+(1/4)*1

= 1/4

Decomposing variance

0AdoptiveSiblings

0.5 1DZ MZ

A

C

E

Covariance

Path analysis

allows us to diagrammatically represent linear

models for the relationships between variables

easy to derive expectations for the variances and

covariances of variables in terms of the

parameters of the proposed linear model

permits translation into matrix formulation (Mx)

Variance components

Phenotype

ACE

e ac

D

d

UniqueEnvironment

AdditiveGeneticEffects

SharedEnvironment

DominanceGeneticEffects

P = eE + aA + cC + dD

ACE Model for twin data

PT1

ACE

PT2

A C E

1

[0.5/1]

e ac eca

QTL linkage model for sib-pair data

PT1

QSN

PT2

Q S N

1

[0 / 0.5 / 1]

n qs nsq

Population sib-pair trait distribution

Under linkage

No linkage

Theory Model Data

Inference

ExperimentFormulation

Interpretation