Biometrical Genetics

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Biometrical genetics
Manuel Ferreira Shaun Purcell Pak Sham
Boulder Introductory Course 2006
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Outline
1. Aim of this talk
2. Genetic concepts
3. Very basic statistical concepts
4. Biometrical model
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1. Aim of this talk
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Revisit common genetic parameters - such as
allele frequencies, genetic effects, dominance,
variance components, etc
Use these parameters to construct a biometrical
genetic model

• Model that expresses the
• Mean
• Variance
• Covariance between individuals
• for a quantitative phenotype as a function of
  genetic parameters.

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0.25/1
0.5/1
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1
1
1
1
1
E
A
D
A
D
E
e
a
d
e
d
a
PT1
PT2
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2. Genetic concepts
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G
G
G
G
Population level
G
G
G
G
G
Allele and genotype frequencies
G
G
G
G
G
G
G
G
G
G
G
Transmission level
G
G
Mendelian segregation
Genetic relatedness
G
G
Phenotype level
Biometrical model
P
P
Additive and dominance components
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Population level
1. Allele frequencies
A single locus, with two alleles - Biallelic /
diallelic - Single nucleotide polymorphism, SNP
A
a
Alleles A and a - Frequency of A is p -
Frequency of a is q 1 p
A
a
Every individual inherits two alleles - A
genotype is the combination of the two alleles
- e.g. AA, aa (the homozygotes) or Aa (the
heterozygote)
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Population level
2. Genotype frequencies (Random mating)
Allele 1
A (p)
a (q)
A (p)
AA (p2)
Aa (pq)
Allele 2
a (q)
aA (qp)
aa (q2)
Hardy-Weinberg Equilibrium frequencies
P (AA) p2
P (Aa) 2pq
p2 2pq q2 1
P (aa) q2
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Transmission level
1. Mendels experiments
AA
aa
Pure Lines
F1
Aa
Aa
Intercross
AA
Aa
Aa
aa
31 Segregation Ratio
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Transmission level
F1
Pure line
Aa
aa
Back cross
Aa
aa
11 Segregation ratio
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Transmission level
AA
aa
Pure Lines
F1
Aa
Aa
Intercross
Aa
Aa
aa
AA
31 Segregation Ratio
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Transmission level
F1
Pure line
Aa
aa
Back cross
Aa
aa
11 Segregation ratio
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Transmission level
1. Mendels law of segregation
Mother (A3A4)
Segregation, Meiosis
Gametes
A3 (½)
A4 (½)
A1 (½)
A1A3 (¼)
A1A4 (¼)
Father (A1A2)
A2 (½)
A2A4 (¼)
A2A3 (¼)
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Phenotype level
1. Classical Mendelian traits
Dominant trait (D - presence, R - absence) -
AA, Aa D - aa R
Recessive trait (D - absence, R - presence) -
AA, Aa D - aa R
Codominant trait (X, Y, Z) - AA X - Aa
Y - aa Z
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Phenotype level
2. Dominant Mendelian inheritance
Mother (Dd)
D (½)
d (½)
D (½)
DD (¼)
Dd (¼)
Father (Dd)
d (½)
dd (¼)
dD (¼)
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Phenotype level
3. Dominant Mendelian inheritance with incomplete
penetrance and phenocopies
Mother (Dd)
D (½)
d (½)
DD (¼)
Dd (¼)
D (½)
Father (Dd)
Incomplete penetrance
dD (¼)
dd (¼)
d (½)
Phenocopies
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Phenotype level
4. Recessive Mendelian inheritance
Mother (Dd)
D (½)
d (½)
D (½)
DD (¼)
Dd (¼)
Father (Dd)
d (½)
dd (¼)
dD (¼)
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Phenotype level
5. Quantitative traits
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Phenotype level
P(X)
Aa
Biometric Model
AA
aa
X
AA
Aa
aa
m
d
a
a
Genotypic effect
m a
m d
m a
Genotypic means
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3. Very basic statistical concepts
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Mean, variance, covariance
1. Mean (X)
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Mean, variance, covariance
2. Variance (X)
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Mean, variance, covariance
3. Covariance (X,Y)
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4. Biometrical model
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Biometrical model for single biallelic QTL
Biallelic locus - Genotypes AA, Aa, aa -
Genotype frequencies p2, 2pq, q2
Alleles at this locus are transmitted from P-O
according to Mendels law of segregation
Genotypes for this locus influence the expression
of a quantitative trait X (i.e. locus is a QTL)
Biometrical genetic model that estimates the
contribution of this QTL towards the (1) Mean,
(2) Variance and (3) Covariance between
individuals for this quantitative trait X
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Biometrical model for single biallelic QTL
1. Contribution of the QTL to the Mean (X)
aa
Aa
Genotypes
AA
a
d
-a
Effect, x
p2
2pq
q2
Frequencies, f(x)
a(p2) d(2pq) a(q2)
Mean (X)
a(p-q) 2pqd
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Biometrical model for single biallelic QTL
2. Contribution of the QTL to the Variance (X)
aa
Aa
Genotypes
AA
a
d
-a
Effect, x
p2
2pq
q2
Frequencies, f(x)
(a-m)2p2 (d-m)22pq (-a-m)2q2
Var (X)
VQTL
Broad-sense heritability of X at this locus
VQTL / V Total
Broad-sense total heritability of X
SVQTL / V Total
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Biometrical model for single biallelic QTL
(a-m)2p2 (d-m)22pq (-a-m)2q2
Var (X)
2pqa(q-p)d2 (2pqd)2
VAQTL VDQTL
Additive effects the main effects of individual
alleles
Dominance effects represent the interaction
between alleles
d 0
aa
Aa
AA
m
a
d
a
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Biometrical model for single biallelic QTL
(a-m)2p2 (d-m)22pq (-a-m)2q2
Var (X)
2pqa(q-p)d2 (2pqd)2
VAQTL VDQTL
Additive effects the main effects of individual
alleles
Dominance effects represent the interaction
between alleles
d gt 0
aa
Aa
AA
m
a
d
a
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Biometrical model for single biallelic QTL
(a-m)2p2 (d-m)22pq (-a-m)2q2
Var (X)
2pqa(q-p)d2 (2pqd)2
VAQTL VDQTL
Additive effects the main effects of individual
alleles
Dominance effects represent the interaction
between alleles
d lt 0
aa
Aa
AA
m
a
d
a
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Biometrical model for single biallelic QTL
aa
Aa
AA
Var (X) Regression Variance Residual
Variance Additive Variance Dominance Variance
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Practical
H\manuel\Biometric\sgene.exe
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Practical
Aim
Visualize graphically how allele frequencies,
genetic effects, dominance, etc, influence trait
mean and variance
Ex1
a0, d0, p0.4, Residual Variance 0.04, Scale
2. Vary a from 0 to 1.
Ex2
a1, d0, p0.4, Residual Variance 0.04, Scale
2. Vary d from -1 to 1.
Ex3
a1, d0, p0.4, Residual Variance 0.04, Scale
2. Vary p from 0 to 1.
Look at scatter-plot, histogram and variance
components.
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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

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Biometrical model for single biallelic QTL
Var (X)
2pqa(q-p)d2 (2pqd)2
Demonstrate
VAQTL VDQTL
2A. Average allelic effect
2B. Additive genetic variance
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1/3
Biometrical model for single biallelic QTL
2A. Average allelic effect (a)
The deviation of the allelic mean from the
population mean
Allele a
Allele A
Population
a(p-q) 2pqd
?
?
Mean (X)
A
a
aa
aA
AA Aa aa
Allelic mean Average allelic effect (a)
a d -a
A p q apdq q(ad(q-p))
a p q dp-aq -p(ad(q-p))
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2/3
Biometrical model for single biallelic QTL
Denote the average allelic effects - 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 - aa -pa
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3/3
Biometrical model for single biallelic QTL
2A. Average allelic effect (a)
2B. Additive genetic variance
The variance of the average allelic effects
aA qa aa -pa
Additive effect
Freq.
AA
p2
2qa
2aA
aA aa
(q-p)a
2pq
Aa
aa
q2
2aa
-2pa
VAQTL
(2qa)2p2 ((q-p)a)22pq (-2pa)2q2
2pqa2
2pqad(q-p)2
d 0, VAQTL 2pqa2
p q, VAQTL ½a2
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d 0, VAQTL 2pqa2
VAQTL
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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
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-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
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Biometrical model for single biallelic QTL
1. Contribution of the QTL to the Mean (X)
2. Contribution of the QTL to the Variance (X)
2A. Average allelic effect (a)
2B. Additive genetic variance
3. Contribution of the QTL to the Covariance (X,Y)
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Biometrical model for single biallelic QTL
3. Contribution of the QTL to the Cov (X,Y)
(a-m)
AA
Aa
aa
(d-m)
(-a-m)
(a-m)
(a-m)2
AA
(d-m)
(a-m)
(d-m)
(d-m)2
Aa
aa
(-a-m)
(-a-m)
(d-m)
(-a-m)
(-a-m)2
(a-m)
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Biometrical model for single biallelic QTL
3A. Contribution of the QTL to the Cov (X,Y) MZ
twins
(a-m)
AA
Aa
aa
(d-m)
(-a-m)
(a-m)
(a-m)2
p2
AA
(d-m)
(a-m)
(d-m)
(d-m)2
Aa
0
2pq
aa
(-a-m)
(-a-m)
(d-m)
(-a-m)
(-a-m)2
(a-m)
0
0
q2
(a-m)2p2 (d-m)22pq (-a-m)2q2
Covar (Xi,Xj)
VAQTL VDQTL
2pqa(q-p)d2 (2pqd)2
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Biometrical model for single biallelic QTL
3B. Contribution of the QTL to the Cov (X,Y)
Parent-Offspring
(a-m)
AA
Aa
aa
(d-m)
(-a-m)
(a-m)
(a-m)2
p3
AA
(d-m)
(a-m)
(d-m)
(d-m)2
Aa
p2q
pq
aa
(-a-m)
(-a-m)
(d-m)
(-a-m)
(-a-m)2
(a-m)
0
pq2
q3
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• 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

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Biometrical model for single biallelic QTL
3B. Contribution of the QTL to the Cov (X,Y)
Parent-Offspring
(a-m)
AA
Aa
aa
(d-m)
(-a-m)
(a-m)
(a-m)2
p3
AA
(d-m)
(a-m)
(d-m)
(d-m)2
Aa
p2q
pq
aa
(-a-m)
(-a-m)
(d-m)
(-a-m)
(-a-m)2
(a-m)
0
pq2
q3
(a-m)2p3 (-a-m)2q3
Cov (Xi,Xj)
½VAQTL
pqa(q-p)d2
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Biometrical model for single biallelic QTL
3C. Contribution of the QTL to the Cov (X,Y)
Unrelated individuals
(a-m)
AA
Aa
aa
(d-m)
(-a-m)
(a-m)
(a-m)2
p4
AA
(d-m)
(a-m)
(d-m)
(d-m)2
Aa
2p3q
4p2q2
aa
(-a-m)
(-a-m)
(d-m)
(-a-m)
(-a-m)2
(a-m)
p2q2
2pq3
q4
(a-m)2p4 (-a-m)2q4
Cov (Xi,Xj)
0
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Biometrical model for single biallelic QTL
3D. Contribution of the QTL to the Cov (X,Y) DZ
twins and full sibs
¼ genome
¼ genome
¼ genome
¼ genome
identical alleles inherited from parents
0
1 (mother)
1 (father)
2
¼ (2 alleles) ½ (1 allele)
¼ (0 alleles)
MZ twins
Unrelateds
P-O
Cov (Xi,Xj)
¼ Cov(MZ) ½ Cov(P-O) ¼ Cov(Unrel)
¼(VAQTLVDQTL) ½ (½ VAQTL) ¼ (0)
½ VAQTL ¼VDQTL
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Summary
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Biometrical model predicts contribution of a QTL
to the mean, variance and covariances of a trait
1 QTL
VAQTL VDQTL
Var (X)
VAQTL VDQTL
Cov (MZ)
½VAQTL ¼VDQTL
Cov (DZ)
Multiple QTL
S(VAQTL) S(VDQTL) VA VD
Var (X)
Cov (MZ)
S(VAQTL) S(VDQTL) VA VD
Cov (DZ)
S(½VAQTL) S(¼VDQTL) ½VA ¼VD
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Biometrical model underlies the variance
components estimation performed in Mx
VA VD VE
Var (X)
Cov (MZ)
VA VD
Cov (DZ)
½VA ¼VD