genetic theory pak sham sgdp, iop, london, uk. theory model data inference experiment formulation...
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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
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