Biometrical%20Genetics

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Biometrical genetics
Pak C Sham The University of Hong Kong Manuel
AR Ferreira Queensland Institute for Medical
Research
23rd International Workshop on Methodology of
Twin and Family Studies 2nd March, 2010
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ADE Model for twin data
1/0.25
MZ/DZ
1/0.5
E1
A1
D1
A2
D2
E2
e
a
c
e
c
a
Y1
Y2
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Biometrical Genetics
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Outline
1. Basic molecular genetics
2. Components of genetic model
3. Biometrical properties for single locus
4. Introduction to linkage analysis
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1. Basic molecular genetics
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DNA
A DNA molecule is a linear backbone of
alternating sugar residues and phosphate groups
Attached to carbon atom 1 of each sugar is a
nitrogenous base A, C, G or T
Complementarity A always pairs with T, likewise
C with G
A gene is a segment of DNA which is translated to
a peptide chain
nucleotide
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Human genome
23 chromosome pairs 22 autosomes, X,Y
33,000,000,000 base pairs
25,000 translated genes
Other functional sequences non-translated
RNA binding sites for regulatory molecules
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DNA sequence variation (polymorphisms)
Microsatellites gt100,000 Many alleles, eg. (CA)n
repeats, very informative, easily automated
Single nucleotide polymorphims (SNPs) 11,883,685
(build 128, 03 Mar 08) Most with 2 alleles (up
to 4), not very informative, easily automated
A
Copy Number polymorphisms Large-scale insertions
/ deletions
B
A
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Fertilization and mitotic cell division
22 1
2 (22 1)
2 (22 1)
2 (22 1)
?
?
?
A -
A -
A -
?
B -
?
?
?
?
Mitosis
B -
B -
chr1
A -
A -
A -
A -
- A
- A
?
?
?
B -
B -
B -
B -
- B
- B
A -
- A
- A
B -
- B
chr1
- B
G1 phase
S phase
M phase
Haploid gametes
Diploid zygote 1 cell
Diploid somatic cells
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Meiosis / Gamete formation
22 1
22 1
A -
NR
(?)
B -
A -
- A
chr1
2 (22 1)
2 (22 1)
B -
- B
?
- A
Meiosis
R
chr1
(?)
(?)
?
?
- B
A -
A -
- A
- A
chr1
B -
B -
- B
- B
A -
R
chr1
chr1
chr1
chr1
(?)
A -
- A
B -
chr1
Diploid gamete precursor cell
B -
- B
- A
chr1
NR
- B
Haploid gamete precursors
chr1
Hap. gametes
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2. Components of genetic model
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A. Transmission model
Mendels law of segregation
Mother (A3A4)
Segregation (Meiosis)
Gametes
A3 (½)
A4 (½)
A1 (½)
A1A3 (¼)
A1A4 (¼)
Father (A1A2)
Offspring
A2 (½)
A2A4 (¼)
A2A3 (¼)
Note 5050 segregation can be distorted
(meiotic drive)
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A. Transmission model two unlinked loci
Phase A1B1 / A2B2
Locus B (B1B2)
Segregation (Meiosis)
B2 (½)
B1 (½)
A1 (½)
A1B1 (1/4)
A1B2 (1/4)
Locus A (A1A2)
Gametes
A2 (½)
A2B1 (1/4)
A2B2 (1/4)
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A. Transmission model two linked loci
Phase A1B1 / A2B2
Locus B (B1B2)
Segregation (Meiosis)
B2 (½)
B1 (½)
A1 (½)
A1B1 ((1-?)/2)
A1B2 (?/2)
Locus A (A1A2)
Gametes
A2 (½)
A2B1 (?/2)
A2B2 ((1-?)/2)
? Recombination fraction, between 0 (complete
linkage) and 1/2 (free recombination)
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B Population model
Frequencies
aa
AA
Aa
aa
AA
AA
Aa
AA
aa
AA
Aa
AA
Genotype frequencies AA P Aa Q aa R
Allele frequencies A PQ/2 a RQ/2
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B Population model
Hardy-Weinberg Equilibrium (Hardy GH, 1908
Weinberg W, 1908)
AA Aa aa
AA P2 PQ PR
Aa PQ Q2 QR
aa PR QR R2
Random mating
AA Aa aa
AA AA AAAa 0.50.5 Aa
Aa AAAa 0.50.5 AAAaaa 0.250.50.25 Aaaa 0.50.5
aa Aa Aaaa 0.50.5 aa
Offspring genotypic distribution
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B Population model
Hardy-Weinberg Equilibrium (Hardy GH, 1908
Weinberg W, 1908)
Offspring genotype frequencies
Genotype Frequency
AA P2PQQ2/4 (PQ/2)2
Aa 2PRPQQRQ2/2 2(PQ/2)(RQ/2)
aa R2QRQ2/4 (RQ/2)2
Offspring allele frequencies
Allele Frequency
A (PQ/2)2 (PQ/2)(RQ/2) PQ/2
a (RQ/2)2 (PQ/2)(RQ/2) RQ/2
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B. Population model
Panmixia (Random union of gametes)
Maternal allele
A (p)
a (q)
P (AA) p2
P (Aa) 2pq
A (p)
AA (p2)
Aa (pq)
Paternal allele
P (aa) q2
a (q)
aA (qp)
aa (q2)
Deviations from HWE Assortative mating
Imbreeding Population stratification
Selection
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C. Phenotype model
Classical Mendelian (Single-gene) traits
Dominant trait - AA, Aa 1 - aa 0
Huntingtons disease (CAG)n repeat, huntingtin
gene
Recessive trait - AA 1 - aa, Aa 0
Cystic fibrosis 3 bp deletion exon 10 CFTR gene
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C. Phenotype model
Polygenic model
Central Limit Theorem ? Normal Distribution
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C. Phenotype model
Quantitative traits
e.g. cholesterol levels
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D. Phenotype model
Aa
Fishers model for single quantitative trait
locus (QTL)
P(X)
AA
aa
X
d
a
Genotypic effects
a
2pq
p2
Genotype frequencies
q2
Assumption Effect of allele independent of
parental origin (Aa aA) Violated in genomic
imprinting
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3. Biometrical properties of single locus
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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)
m a(p2) d(2pq) a(q2)
Mean (X)
(p-q)a 2pqd
Note If everyone in population has genotype
aa then population mean -a ? change in mean
due to A ((p-q)a 2pqd) (-a) 2p(aqd)
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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)
2pq(a(q-p)d)2 (2pqd)2 VQTL
Broad-sense heritability of X at this locus
VQTL / V Total
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Biometrical model for single biallelic QTL
2. Partitioning of QTL variance additive
component
Maternal allele
A (p)
a (q)
Average
A (p)
a
d
paqd
Paternal allele
a (q)
d
-a
pd-qa
Average
paqd
pd-qa
(p-q)a 2pqd
Variance due to a single allele
p(q(da)-2pqd)2q(p(d-a)-2pqd)2
pq(a(q-p)d)2
For both alleles, additive variance
2pq(a(q-p)d)2
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Biometrical model for single biallelic QTL
2. Partitioning of QTL variance dominance
component
Genotype
Effect
Additive effect
AA (p2)
a
2(paqd)
Aa (2pq)
(paqd)(pd-qa)
d
-a
aa (q2)
2(pd-qa)
Dominance variance due to QTL p2(a-2(paqd))2

2pq(d-(paqdpd-qa))2

q2(-a-2(pd-qa)
(2pqd)2
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Biometrical model for single biallelic QTL
a
0
Genotypic effects
-a
aa
Aa
AA
aa
Aa
AA
aa
Aa
AA
Additive
Dominant
Recessive
Var (X) Regression Variance Residual
Variance
Additive Variance Dominance Variance
VAQTL VDQTL
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Statistical definition of dominance is scale
dependent
4
4
0.7
0.4
log (x)
aa
Aa
AA
aa
Aa
AA
No departure from additivity
Significant departure from additivity
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Practical
H\ferreira\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
1. Contribution of the QTL to the Mean (X)
2. Contribution of the QTL to the Variance (X)
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
Cov(X,Y)
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 (X,Y)
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 (X,Y)
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 (X,Y)
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 so far
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Biometrical model predicts contribution of a QTL
to the mean, variance and covariances of a trait
Association analysis
Mean (X)
a(p-q) 2pqd
Linkage analysis
VAQTL VDQTL
Var (X)
VAQTL VDQTL
Cov (MZ)
On average!
½VAQTL ¼VDQTL
Cov (DZ)
For a sib-pair, do the two sibs have 0, 1 or 2
alleles in common?
0 or 1
0, 1/2 or 1
IBD estimation / Linkage
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4. Introduction to Linkage Analysis
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For a heritable trait...
Linkage
localize region of the genome where a QTL that
regulates the trait is likely to be harboured
Family-specific phenomenon Affected individuals
in a family share the same ancestral
predisposing DNA segment at a given QTL
identify a QTL that regulates the trait
Association
Population-specific phenomenon Affected
individuals in a population share the same
ancestral predisposing DNA segment at a given QTL
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Linkage Analysis Parametric vs. Nonparametric
Gene
Chromosome
Recombination
Genetic factors
Q
M
A
Mode of inheritance
Dominant trait 1 - AA, Aa 0 - aa
Correlation
D
Phe
C
E
Environmental factors
Adapted from Weiss Terwilliger 2000
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Approach
Parametric genotypes marker locus genotypes
trait locus (latter inferred from phenotype
according to a specific disease model) Parameter
of interest ? between marker and trait loci
Nonparametric genotypes marker locus
phenotype If a trait locus truly regulates the
expression of a phenotype, then two relatives
with similar phenotypes should have similar
genotypes at a marker in the vicinity of the
trait locus, and vice-versa. Interest
correlation between phenotypic similarity and
marker genotypic similarity
No need to specify mode of inheritance, allele
frequencies, etc...
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Phenotypic similarity between relatives
Squared trait differences
Squared trait sums
Trait cross-product
Trait variance-covariance matrix
Affection concordance
T2
T1
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Genotypic similarity between relatives
IBS Alleles shared Identical By State look the
same, may have the same DNA sequence but they
are not necessarily derived from a known common
ancestor
M3
M1
M2
M3
Q3
Q1
Q2
Q4
IBD Alleles shared Identical By Descent are
a copy of the same ancestor allele
M1
M2
M3
M3
Q1
Q2
Q3
Q4
IBS
IBD
M1
M3
M1
M3
2
1
Q1
Q3
Q1
Q4
0
0
0
1
1
Inheritance vector (M)
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Genotypic similarity between relatives -
Number of alleles shared IBD
Proportion of alleles shared IBD -
Inheritance vector (M)
M2
M3
M1
M3
0
0
0
0
1
1
Q2
Q4
Q1
Q3
M1
M3
M1
M3
0.5
0
0
0
1
1
Q1
Q3
Q1
Q4
M1
M1
M3
M3
2
1
0
0
0
0
Q1
Q1
Q3
Q3
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Genotypic similarity between relatives -
A
B
C
D
22n
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VAQTL VDQTL
Var (X)
VAQTL VDQTL
Cov (MZ)
½VAQTL ¼VDQTL
Cov (DZ)
On average!
For a given twin pair
Cov (DZ)
VAQTL VDQTL
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Cov (DZ)
VAQTL
Cov (DZ)
VAQTL
Slope VAQTL
Phenotypic similarity
0.5
1
0
Genotypic similarity ( )
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Statistics that incorporate both phenotypic and
genotypic similarities to test VQTL
Regression-based methods
Haseman-Elston, MERLIN-regress
Variance components methods
Mx, MERLIN, SOLAR, GENEHUNTER