Transcript AA - Virginia Institute for Psychiatric and Behavioral Genetics
Intro to Quantitative Genetics
HGEN502, 2011 Hermine H. Maes
Intro to Quantitative Genetics
1/18: Course introduction; Introduction to Quantitative Genetics & Genetic Model Building 1/20: Study Design and Genetic Model Fitting 1/25: Basic Twin Methodology 1/27: Advanced Twin Methodology and Scope of Genetic Epidemiology 2/1: Quantitative Genetics Problem Session
Aims of this talk
Historical Background Genetical Principles Genetic Parameters: additive, dominance Biometrical Model Statistical Principles Basic concepts: mean, variance, covariance Path Analysis Likelihood
Quantitative Genetics Principles
Analysis of patterns and mechanisms underlying variation in continuous traits to resolve and identify their genetic and environmental causes Continuous traits have continuous phenotypic range; often polygenic & influenced by environmental effects Ordinal traits are expressed in whole numbers; can be treated as approx discontinuous or as threshold traits Some qualitative traits; can be treated as having underlying quantitative basis, expressed as a threshold trait (or multiple thresholds)
Types of Genetic Influence
Mendelian Disorders Single gene, highly penetrant, severe, small % affected (e.g., Huntington’s Disease) Chromosomal Disorders Insertions, deletions of chromosomal sections, severe, small % affected (e.g., Down’s Syndrome) Complex Traits Multiple genes (of small effect), environment, large % population, susceptibility – not destiny (e.g., depression, alcohol dependence, etc)
Genetic Disorders
Great 19th Century Biologists
Gregor Mendel (1822-1884): Mathematical rules of particulate inheritance (“Mendel’s Laws”) Charles Darwin (1809-1882): Evolution depends on differential reproduction of inherited variants Francis Galton (1822-1911): Systematic measurement of family resemblance Karl Pearson (1857 1936): “Pearson Correlation”; graduate student of Galton
Family Measurements
Standardize Measurement
Pearson and Lee’s diagram for measurement of “span” (finger-tip to finger-tip distance)
Parent Offspring Correlations
From Pearson and Lee (1903) p.378
Sibling Correlations
From Pearson and Lee (1903) p.387
Nuclear Family Correlations
© Lindon Eaves, 2009
Quantitative Genetic Strategies
Family Studies Does the trait aggregate in families?
The (Really!) Big Problem: Families are a mixture of genetic and environmental factors Twin Studies Galton’s solution: Twins One (Ideal) solution: Twins separated at birth But unfortunately MZA’s are rare Easier solution: MZ & DZ twins reared together
Twin Studies Reared Apart
Minnesota Study of Twins Reared Apart (T. Bouchard et al, 1979 >100 sets of reared-apart twins from across the US & UK All pairs spent formative years apart (but vary tremendously in amount of contact prior to study) 56 MZAs participated
Types of Twins
Monozygotic (MZ; “identical”): result from fertilization of a single egg by a single sperm; share 100% of genetic material Dizygotic (DZ, “fraternal” or “non identical”): result from independent fertilization of two eggs by two sperm; share on average 50% of their genes
Logic of Classical Twin Study
MZs share 100% genes, DZs (on avg) 50% Both twin types share 100% environment If rMZ > rDZ, then genetic factors are important If rDZ > ½ rMZ, then growing up in the same home is important If rMZ < 1, then non-shared environmental factors are important
Causes of Twinning
For MZs, appears to be random For DZs, Increases with mother’s age (follicle stimulating hormone, FSH, levels increase with age) Hereditary factors (FSH) Fertility treatment Rates of twins/multiple births are increasing, currently ~3% of all births
Zygosity of Twins
Chorionicity of Twins
100% of DZ twins are dichorionic ~1/3 of MZ twins are dichorionic and ~2/3 are monochorionic
Twin Correlations
8 3 -2 -7 -12
Virginia Twin Study of Adolescent Behavioral Development
Scatterplot for corrected MZ stature Scatterplot for age and sex corrected stature in DZ twins 20 13 r=0.924
10 0 -10 r=0.535
-10 -20 -5 0 HTDEV1 5 10 -16 -11 -6 -1 HTDEV1 4 MZ Stature DZ Stature 9 14 © Lindon Eaves, 2009
Ronald Fisher (1890-1962)
1918: On the Correlation Between Relatives on the Supposition of Mendelian Inheritance 1921: Introduced concept of “likelihood” 1930: The Genetical Theory of Natural Selection 1935: The Design of Experiments Fisher developed mathematical theory that reconciled Mendel’s work with Galton and Pearson’s correlations
Fisher (1918): Basic Ideas
Continuous variation caused by lots of genes (polygenic inheritance) Each gene followed Mendel’s laws Environment smoothed out genetic differences Genes may show different degrees of dominance Genes may have many forms (multiple alleles) Mating may not be random (assortative mating) Showed that correlations obtained by Pearson & Lee were explained well by polygenic inheritance [“Mendelian” Crosses with Quantitative Traits]
Biometrical Genetics
Lots of credit to:
Manuel Ferreira, Shaun Purcell Pak Sham, Lindon Eaves
Building a Genetic Model
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: (1) Mean (2) Variance (3) Covariance between individuals for a quantitative phenotype as a function of genetic parameters.
Genetic Concepts Population level
Allele and genotype frequencies
Transmission level
Mendelian segregation Genetic relatedness
Phenotype level
Biometrical model Additive and dominance components
G G G G G G G G G G G G G G G G G G G G G G G G P P
Population level 1. Allele frequencies
A single locus, with two alleles - Biallelic / diallelic - Single nucleotide polymorphism, SNP Alleles
A
and
a
- Frequency of
A
- Frequency of
a
is
p
is
q
= 1 –
p
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)
A a a
Population level 2. Genotype frequencies
(Random mating)
A
(
p
) Allele 1
a
(
q
)
A
(
p
)
a
(
q
)
AA
(
p 2
)
aA
(
qp
)
Aa
(
pq
)
aa
(
q 2
) Hardy-Weinberg Equilibrium frequencies
P
(
AA
) =
p 2
P
(
Aa
) =
2pq
P
(
aa
) =
q 2
p 2
+
2pq
+
q 2
= 1
Transmission level Mendel’s experiments
Pure Lines AA aa F1 Intercross Aa Aa AA Aa Aa 3:1 Segregation Ratio aa
Transmission level
F1 Aa Back cross Pure line aa Aa aa 1:1 Segregation ratio
Transmission level
Pure Lines AA F1 Intercross AA Aa aa Aa Aa Aa 3:1 Segregation Ratio aa
Transmission level
F1 Aa Back cross Pure line aa Aa aa 1:1 Segregation ratio
Transmission level Mendel’s law of segregation
Father (
A 1 A 2
)
A 1
(
½
)
A 2
(
½
)
A 3
( Mother (
A 3 A 4
)
½
)
A 4
Segregation, Meiosis
(
½
)
Gametes
A A 1 2 A A 3 3
( (
¼ ¼
) )
A 1 A 4
(
¼
)
A 2 A 4
(
¼
)
Phenotype level 1. Classical Mendelian traits
Dominant trait (
D
- presence,
R
-
AA
,
Aa aa
D R
- absence) Recessive trait (
D
- absence,
R
-
AA
,
Aa aa
D R
- presence) Codominant trait (
X, Y, Z
) -
AA
-
Aa
-
aa
X Y Z
Phenotype level 2. Dominant Mendelian inheritance
Mother (
Dd
) Father (
Dd
)
D
(
½
)
d
(
½
)
D
(
½
)
D D
(
¼
)
d D
(
¼
)
d
(
½
)
D d
(
¼
)
d d
(
¼
)
Phenotype level 3. Dominant Mendelian inheritance with incomplete penetrance and phenocopies
Mother (
Dd
) Father (
Dd
)
D
(
½
)
d
(
½
)
D
(
½
)
D D
(
¼
)
d D
(
¼
)
d
(
½
)
D d
(
¼
)
d d
(
¼
)
Incomplete penetrance Phenocopies
Phenotype level 4. Recessive Mendelian inheritance
Mother (
Dd
) Father (
Dd
)
D
(
½
)
d
(
½
)
D
(
½
)
D D
(
¼
)
d D
(
¼
)
d
(
½
)
D d
(
¼
)
d d
(
¼
)
Phenotype level Two kinds of differences
Continuous Graded, no distinct boundaries e.g. height, weight, blood-pressure, IQ, extraversion Categorical Yes/No Normal/Affected (Dichotomous) None/Mild/Severe (Multicategory) Often called “threshold traits” because people “affected” if they fall above some level of a measured or hypothesized continuous trait
Phenotype level Polygenic Traits
Mendel’s
Experiments in Plant Hybridization
, showed how discrete particles (particulate theory of inheritance) behaved mathematically: all or nothing states (round/wrinkled, green/yellow), “Mendelian” disease How do these particles produce a continuous trait like stature or liability to a complex disorder?
1 Gene 3 Genotypes 3 Phenotypes 2 Genes 9 Genotypes 5 Phenotypes 3 Genes 27 Genotypes 7 Phenotypes 4 Genes 81 Genotypes 9 Phenotypes
Phenotype level Quantitative traits
.072
.128205
g==-1 0 -3.90647
0 .128205
g==1 qt 0 -3.90647
.128205
g==-1 .128205
.128205
0 g==-1 g==0 g==1 0 0 g==1 .128205
2.7156
0 -3.90647
2.7156
qt Histograms by g g==0
AA
g==0 -3.90647
Aa
-3.90647
qt Histograms by g
aa
2.7156
qt Histograms by g 2.7156
2.7156
Phenotype level
P
(
X
)
Aa aa AA aa
-a m -a
Biometric Model
m
Aa
d m +d
X
AA
+a m +a
Genotypic effect Genotypic means
Very Basic Statistical Concepts 1. Mean
(
X
)
2. Variance
(
X
)
3. Covariance
(
X,Y
)
4. Correlation
(
X,Y
)
Mean, variance, covariance 1. Mean
(
X
) )
n
i x i
i
Mean, variance, covariance 2. Variance
(
X
) (
i
i
Mean, variance, covariance 3. Covariance
(
X,Y
)
Y
X Y
Y i
i
X y i
Y i i
y i
1
Mean, variance, covariance (& correlation) 4. Correlation
(
X,Y
)
r x
,
y
cov
x
,
y s x s y
Biometrical model for single biallelic QTL
Biallelic locus - Genotypes:
AA, Aa, aa
- Genotype frequencies:
p 2 , 2pq, q 2
Alleles at this locus are transmitted from P-O according to Mendel’s 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 individuals
for this quantitative trait and
X
(3) Covariance between
Biometrical model for single biallelic QTL
Biallelic locus - Genotypes:
AA, Aa, aa
- Genotype frequencies:
p 2 , 2pq, q 2
Alleles at this locus are transmitted from P-O according to Mendel’s 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 individuals
for this quantitative trait and
X
(3) Covariance between
Biometrical model for single biallelic QTL 1. Contribution of the QTL to the Mean
(
X
)
i x f
Genotypes Effect,
x
Frequencies,
f
(
x
)
AA a p 2 Aa d 2pq aa -a q 2
Mean
(
X
) =
a ( p 2 ) + d ( 2pq ) – a ( q 2 )
=
a ( p q ) + 2 pq d
Biometrical model for single biallelic QTL 2. Contribution of the QTL to the Variance
(
X
)
i
Genotypes Effect,
x
Frequencies,
f
(
x
)
AA a p 2 Aa d
2pq
aa -a q 2
Var
(
X
) =
( a m ) 2 p 2 + ( d m ) 2 2pq + ( a m ) 2 q 2
=
V QTL
Broad-sense heritability of
X
at this locus =
V QTL
/
V
Total Broad-sense total heritability of
X
= Σ
V QTL
/
V
Total
Biometrical model for single biallelic QTL
Var
(
X
) =
(
a m ) 2 p 2 + ( d m ) 2 2pq + ( a m ) 2 q 2
=
2 pq [ a +( q p ) d ] 2 + ( 2pq d ) 2
=
V AQTL
+
V DQTL
Additive effects: the main effects of individual alleles Dominance effects: represent the interaction between alleles
aa Aa
m
AA
d = 0 –a d +a
Biometrical model for single biallelic QTL
Var
(
X
) =
(
a m ) 2 p 2 + ( d m ) 2 2pq + ( a m ) 2 q 2
=
2 pq [ a +( q p ) d ] 2 + ( 2pq d ) 2
=
V AQTL
+
V DQTL
Additive effects: the main effects of individual alleles Dominance effects: represent the interaction between alleles
aa Aa AA
m d > 0 –a d +a
Biometrical model for single biallelic QTL
Var
(
X
) =
( a m ) 2 p 2 + ( d m ) 2 2pq + ( a m ) 2 q 2
=
2 pq [ a +( q p ) d ] 2 + ( 2pq d ) 2
=
V AQTL
+
V DQTL
Additive effects: the main effects of individual alleles Dominance effects: represent the interaction between alleles
aa Aa AA
m d < 0 –a d +a
Biometrical model for single biallelic QTL
+a d m –a aa
Aa
AA
Var (
X
) = Regression Variance + Residual Variance = Additive Variance + Dominance Variance
Biometrical model for single biallelic QTL
Var
(
X
) = 2
pq [ a +( q p ) d ] 2
+
(
2
pq d ) 2 V AQTL Demonstrate
+
V DQTL 2A. Average allelic effect 2B. Additive genetic variance
NOTE: Additive genetic variance depends on
allele frequency & additive genetic value
as well as
dominance deviation p a d
Additive genetic variance typically greater than dominance variance
1/3
Biometrical model for single biallelic QTL
2A. Average allelic effect ( α)
The deviation of the allelic mean from the population mean Mean (
X
) Allele
a
?
Population
a
(
p q
) +
2 pq d
Allele
A
?
A a AA a p Aa d q p
a
α
a
α
A
A
aa -a q
Allelic mean Average allelic effect ( α)
a p + d q d p a q q
(
a + d
(
q p
))
-p
(
a + d
(
q p
))
Biometrical model for single biallelic QTL
2/3 Denote the average allelic effects -
α A
=
q
(
a + d
(
q p
))
α a
=
-p
(
a + d
(
q p
))
If only two alleles exist, we can define the
average effect of allele substitution
-
α
=
α A α a α
=
(
q
-(
-p
))(
a + d
(
q p
)) = (
a + d
(
q p
))
Therefore: -
α A
=
q α α a
=
-p α
Biometrical model for single biallelic QTL
2A. Average allelic effect ( α) 2B. Additive genetic variance
The variance of the average allelic effects
AA Aa aa
Freq.
p
2pq
q 2 2
Additive effect 2 α
A
α
A
+ α
a
2 α
a
= 2
q
α = (
q p
) α = -2
p
α
α A
=
q α α a
=
-p α
3/3
V AQTL
=
(
2
q
α ) 2
p 2
+
((
q
-
p
)
α
) 2
2pq
+ (-
2
p
α ) 2
q 2
= 2
pq
α 2 = 2
pq
[
a
+
d
(
q
-
p
)]
2
d p
= 0, V A QTL =
2
pq a
2 =
q
, V AQTL =
½
a
2
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 ( α) 2B. Additive genetic variance
3. Contribution of the QTL to the Covariance
(
X,Y
)
Biometrical model for single biallelic QTL 3. Contribution of the QTL to the Cov
(
X,Y
) (
i
y
AA
(
a
-
m
)
Aa
(
d
-
m
)
aa
(
-a
-
m
)
AA
(
a
-
m
)
(
a
-
m
) 2
(
a
-
m
) (
d
-
m
) (
a
-
m
) (
-a
-
m
)
Aa
(
d
-
m
) (
d
-
m
) 2
(
d
-
m
)
(
-a
-
m
)
aa
(
-a
-
m
) (
-a
-
m
) 2
Biometrical model for single biallelic QTL 3A. Contribution of the QTL to the Cov
(
X,Y)
(
i
y
– MZ twins
AA
(a
-
m
)
Aa
(d
-
m)
aa
(-a
-
m)
AA
(a
-
m
)
Aa
(d
-
m)
p
2
(a
-
m
) 2
0
(a
-
m) (d
-
m)
0
(a
-
m) (-a
-
m)
2pq
(d
-
m
) 2
0
(d
-
m
)
(-a
-
m) Covar
(
X i ,X j
) =
(
a
-
m
) 2
p 2
+
(
d
-
m
) 2
2pq
+
(-
a
-
m
) 2
q 2
= 2
pq
[
a
+(
q
-
p
)
d
] 2
+
(
2pq d
) 2
q
2
aa
(-a
-
(-a m)
=
V AQTL
+
V DQTL 2 m)
Biometrical model for single biallelic QTL 3B. Contribution of the QTL to the Cov
(
X,Y
) – Parent-Offspring
AA
(a
-
m
)
Aa
(d
-
m)
aa
(-a
-
m)
AA
(a
-
m
)
p
3
(a
-
m
) 2
p 2 q
(a
-
m) (d
-
m)
0
(a
-
m) (-a
-
m)
Aa
(d
-
m)
pq
(d
-
m
) 2
pq 2
(d
-
m
)
(-a
-
m)
aa
(-a
-
q
3
(-a
-
m) 2 m)
Biometrical model for single biallelic QTL
e.g.
given an
AA AA
x
AA
or father, an
AA
x
Aa AA
offspring can come from either parental mating types
AA
x
AA
will occur
p 2
×
p 2 = p 4
and have
AA
offspring Prob()=1
AA
x
Aa
will occur
p 2
×
2pq = 2p 3 q
and have
AA
offspring Prob()=0.5
and have
Aa
offspring Prob()=0.5
therefore, P(
AA
father &
AA
offspring) =
p 4 + p 3 q
=
p 3 (p+q)
=
p 3
Biometrical model for single biallelic QTL 3B. Contribution of the QTL to the Cov
(
X,Y
) – Parent-Offspring
AA
(a
-
m
)
Aa
(d
-
m)
aa
(-a
-
m)
AA
(a
-
m
)
p
3
(a
-
m
) 2
p 2 q
(a
-
m) (d
-
m)
0
(a
-
m) (-a
-
m)
Aa
(d
-
m)
pq
(d
-
m
) 2
pq 2
(d
-
m
)
(-a
-
m)
aa
(-a
-
q
3
(-a
-
m) 2 m) Cov
(
X i ,X j
) = =
(
a
-
m
) 2
p 3
+
…
+
(-
a
-
m
) 2
q
3
pq
[
a
+(
q
-
p
)
d
] 2
= ½
V AQTL
Biometrical model for single biallelic QTL 3C. Contribution of the QTL to the Cov
(
X,Y
) – Unrelated individuals
AA
(a
-
m
)
Aa
(d
-
m)
AA
(a
-
m
)
Aa
(d
-
m)
aa
(-a
-
m)
p
4
(a
-
m
) 2
2p 3 q
(a
-
m) (d
-
m)
4p 2 q 2
(d
-
m
) 2
p 2 q 2
(a
-
m) (-a
-
m)
2pq 3
(d
-
m
)
(-a
-
m)
aa
(-a
-
q
4
(-a
-
m) 2 m) Cov
(
X i ,X j
) =
(
a
-
m
) 2
p 4
+
…
+
(-
a
-
m
) 2
q
4
= 0
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
2 1
(father)
1
(mother)
0
¼
(2 alleles) + ½ (1 allele) +
¼ (0 alleles) MZ twins P-O Unrelateds Cov
(
X i ,X j
) =
¼ Cov(MZ) + ½ Cov(P-O) + ¼ Cov(Unrel)
= ¼(
V AQTL
+
V DQTL
) + ½ (½
V AQTL ) + ¼ (0)
= ½
V AQTL +
¼
V DQTL
Summary
Biometrical model predicts contribution of a QTL to the mean, variance and covariances of a trait
1 QTL
Var
(
X
) =
V AQTL
+
V DQTL Cov
(
MZ
) =
V AQTL
+
V DQTL Cov
(
DZ
) = ½
V AQTL
+ ¼
V DQTL
Multiple QTL
Var
(
X
) = Σ(
V AQTL
) + Σ(
V DQTL
) =
V A
+
V D Cov
(
MZ
) = Σ(
V AQTL
) + Σ(
V DQTL
) =
V A
+
V D Cov
(
DZ
) = Σ(½
V A QTL
) + Σ(¼
V D QTL
) = ½
V A
+ ¼
V D
Summary
Biometrical model underlies the variance components estimation performed in Mx
Var
(
X
) =
V A
+
V D + V E Cov
(
MZ
) =
V A
+
V D Cov
(
DZ
) = ½
V A
+ ¼
V D
Path Analysis
HGEN502, 2011 Hermine H. Maes
Model Building
Write equations for means, variances and covariances of different type of relative or Draw path diagrams for easy derivation of expected means, variances and covariances and translation to mathematical formulation
Method of Path Analysis
Allows us to represent linear models for the relationship between variables in diagrammatic form, e.g. a genetic model; a factor model; a regression model Makes it easy to derive expectations for the variances and covariances of variables in terms of the parameters of the proposed linear model Permits easy translation into matrix formulation as used by statistical programs
Path Diagram Variables
Squares or rectangles denote observed variables Circles or ellipses denote latent (unmeasured) variables Upper-case letters are used to denote variables Lower-case letters (or numeric values) are used to denote covariances or path coefficients
Variables
latent variables observed variables
Path Diagram Arrows
Single-headed arrows or paths ( –>) are used to represent causal relationships between variables under a particular model - where the variable at the tail is hypothesized to have a direct influence on the variable at the head Double-headed arrows (< –>) represent a covariance between two variables, which may arise through common causes not represented in the model. They may also be used to represent the variance of a variable
Arrows
double-headed arrows single-headed arrows
Path Analysis Tracing Rules
Trace backwards, change direction at a 2 headed arrow, then trace forwards (implies that we can never trace through two-headed arrows in the same chain).
The expected covariance between two variables, or the expected variance of a variable, is computed by multiplying together all the coefficients in a chain, and then summing over all possible chains.
Non-genetic Example
Cov AB
Cov AB = kl + mqn + mpl
Expectations
Cov AB = Cov BC = Cov AC = Var A = Var B = Var C =
Expectations
Cov AB = kl + mqn + mpl Cov BC = no Cov AC = mqo Var A = k 2 + m 2 + 2 kpm Var B = l 2 + n 2 Var C = o 2
Genetic Examples
MZ Twins Reared Together DZ Twins Reared Together MZ Twins Reared Apart DZ Twins Reared Apart Parents & Offspring
MZ Twins Reared Together
MZ Twins RT
Expected Covariance Twin 1 Twin 1 Twin 2 a 2+ c 2+ e 2 variance a 2+ c 2 covariance Twin 2 a 2+ c 2 a 2+ c 2+ e 2
DZ Twins Reared Together
DZ Twins RT
Expected Covariance Twin 1 Twin 1 a 2+ c 2+ e 2 Twin 2 .5a
2+ c 2 Twin 2 .5a
2+ c 2 a 2+ c 2+ e 2