intro to quantitative genetics - vipbg.vcu.edu€¦ · intro to quantitative genetics 1/18: course...

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)

From Pearson and Lee (1903) p.378

Parent Offspring Correlations

From Pearson and Lee (1903) p.387

Sibling Correlations

Nuclear Family Correlations

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

Virginia Twin Study of Adolescent Behavioral Development

Twin Correlations

MZ Stature DZ Stature

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

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.

Building a Genetic Model

Population level

Transmission level

Phenotype level

Allele and genotype frequencies

Mendelian segregation Genetic relatedness

Biometrical model Additive and dominance components

Genetic Concepts

Population level

1. Allele frequencies

A single locus, with two alleles - Biallelic / diallelic - Single nucleotide polymorphism, SNP

Alleles A and a - Frequency of A is p - Frequency of a is q = 1 – p

Every individual inherits two alleles - A genotype is the combination of the two alleles - e.g. AA, aa (the homozygotes) or Aa (the heterozygote)

2. Genotype frequencies (Random mating)

A (p) a (q)

Allele 1 A

2 AA (p2)

aA (qp)

Aa (pq)

aa (q2)

Hardy-Weinberg Equilibrium frequencies

P (AA) = p2

P (Aa) = 2pq

P (aa) = q2

p2 + 2pq + q2 = 1

Population level

Transmission level

Pure Lines AA aa

F1 Aa Aa

AA Aa Aa aa

3:1 Segregation Ratio

Intercross

Mendel’s experiments

F1 Pure line

Back cross

1:1 Segregation ratio

Transmission level

Pure Lines AA aa

F1 Aa Aa

AA Aa Aa aa

3:1 Segregation Ratio

Intercross

F1 Pure line

Back cross

1:1 Segregation ratio

Transmission level

Segregation, Meiosis

Mendel’s law of segregation

A3 (½) A4 (½)

A1 (½)

A2 (½)

Mother (A3A4)

A1A3 (¼)

A2A3 (¼)

A1A4 (¼)

A2A4 (¼)

Gametes

Father (A1A2)

Transmission 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

Phenotype level

2. Dominant Mendelian inheritance

D (½) d (½)

D (½)

d (½)

Mother (Dd)

DD (¼)

dD (¼)

Dd (¼)

dd (¼)

Father (Dd)

Phenotype level

3. Dominant Mendelian inheritance with incomplete penetrance and phenocopies

D (½) d (½)

D (½)

d (½)

Mother (Dd)

DD (¼)

dD (¼)

Dd (¼)

dd (¼) Father (Dd)

Phenocopies

Incomplete penetrance

Phenotype level

4. Recessive Mendelian inheritance

D (½) d (½)

D (½)

d (½)

Mother (Dd)

DD (¼)

dD (¼)

Dd (¼)

dd (¼)

Father (Dd)

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

1 Gene 3 Genotypes 3 Phenotypes

2 Genes 9 Genotypes 5 Phenotypes

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?

Phenotype level

Quantitative traits

Phenotype level

AA Aa aa

Genotypic means

Biometric Model

Genotypic effect

Phenotype level

m -a m +d m +a

Very Basic Statistical Concepts

1. Mean (X)

2. Variance (X)

3. Covariance (X,Y)

4. Correlation (X,Y)

Mean, variance, covariance

1. Mean (X)

2. Variance (X)

3. Covariance (X,Y)

Mean, variance, covariance (& correlation)

4. Correlation (X,Y)

rx,y =covx,ysxsy

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 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 and (3) Covariance between individuals for this quantitative trait X

Biallelic locus - Genotypes: AA, Aa, aa - Genotype frequencies: p2, 2pq, q2

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 and (3) Covariance between individuals for this quantitative trait X

1. Contribution of the QTL to the Mean (X)

aa Aa AA Genotypes

Frequencies, f(x)

Effect, x

p2 2pq q2

a d -a

= a(p2) + d(2pq) – a(q2) Mean (X) = a(p-q) + 2pqd

2. Contribution of the QTL to the Variance (X)

aa Aa AA Genotypes

Frequencies, f(x)

Effect, x

p2 2pq q2

a d -a

= (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 = ΣVQTL / V Total

= (a-m)2p2 + (d-m)22pq + (-a-m)2q2 Var (X)

= 2pq[a+(q-p)d]2 + (2pqd)2

= VAQTL + VDQTL

d +a –a

AA aa Aa

Additive effects: the main effects of individual alleles Dominance effects: represent the interaction between alleles

= (a-m)2p2 + (d-m)22pq + (-a-m)2q2 Var (X)

= 2pq[a+(q-p)d]2 + (2pqd)2

= VAQTL + VDQTL

AA aa Aa

–a +a d

= (a-m)2p2 + (d-m)22pq + (-a-m)2q2 Var (X)

= 2pq[a+(q-p)d]2 + (2pqd)2

= VAQTL + VDQTL

AA aa Aa

–a +a d

aa Aa AA

Var (X) = Regression Variance + Residual Variance = Additive Variance + Dominance Variance

Var (X) = 2pq[a+(q-p)d]2 + (2pqd)2

VAQTL + VDQTL

Demonstrate

2A. Average allelic effect

2B. Additive genetic variance

NOTE: 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

2A. Average allelic effect (α)

The deviation of the allelic mean from the population mean

a(p-q) + 2pqd

A a αa αA

? ? Mean (X)

Allele a Allele A Population

AA Aa aa a d -a

A p q ap+dq q(a+d(q-p)) a p q dp-aq -p(a+d(q-p))

Allelic mean Average allelic effect (α)

Denote the average allelic effects - αA

= q(a+d(q-p)) - αa

= -p(a+d(q-p))

If only two alleles exist, we can define the average effect of allele substitution - α = αA - αa - α = (q-(-p))(a+d(q-p)) = (a+d(q-p))

Therefore: - αA

= qα - αa

= -pα

2B. Additive genetic variance

The variance of the average allelic effects

Additive effect

2A. Average allelic effect (α)

αA + αa

= 2qα

= (q-p)α

= -2pα

VAQTL = (2qα)2p2 + ((q-p)α)22pq + (-2pα)2q2

= 2pqα2

= 2pq[a+d(q-p)]2 d = 0, VAQTL= 2pqa2

p = q, VAQTL= ½a2

αA = qα

αa = -pα

2B. Additive genetic variance 2A. Average allelic effect (α)

3. Contribution of the QTL to the Covariance (X,Y)

2. Contribution of the QTL to the Variance (X) 1. Contribution of the QTL to the Mean (X)

AA Aa aa (a-m) (d-m) (-a-m)

(-a-m)

(a-m)2

(-a-m)

(d-m)2

(d-m) (-a-m) (-a-m)2

3. Contribution of the QTL to the Cov (X,Y)

(-a-m)

(a-m)2

(-a-m)

(d-m)2

(d-m) (-a-m) (-a-m)2

3A. Contribution of the QTL to the Cov (X,Y) – MZ twins

= (a-m)2p2 + (d-m)22pq + (-a-m)2q2 Covar (Xi,Xj)

= VAQTL + VDQTL

= 2pq[a+(q-p)d]2 + (2pqd)2

(-a-m)

(a-m)2

(-a-m)

(d-m)2

(d-m) (-a-m) (-a-m)2

pq2 q3

3B. Contribution of the QTL to the Cov (X,Y) – Parent-Offspring

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(p+q) = p3

(-a-m)

(a-m)2

(-a-m)

(d-m)2

(d-m) (-a-m) (-a-m)2

pq2 q3

= (a-m)2p3 + … + (-a-m)2q3 Cov (Xi,Xj)

= ½VAQTL = pq[a+(q-p)d]2

3B. Contribution of the QTL to the Cov (X,Y) – Parent-Offspring

(-a-m)

(a-m)2

(-a-m)

(d-m)2

(d-m) (-a-m) (-a-m)2

2pq3 q4

= (a-m)2p4 + … + (-a-m)2q4 Cov (Xi,Xj)

3C. Contribution of the QTL to the Cov (X,Y) – Unrelated individuals

Cov (Xi,Xj)

3D. Contribution of the QTL to the Cov (X,Y) – DZ twins and full sibs

¼ genome

¼ (2 alleles) + ½ (1 allele) + ¼ (0 alleles)

MZ twins P-O Unrelateds

¼ genome ¼ genome ¼ genome

# identical alleles inherited from parents

0 1 (mother)

1 (father)

= ¼ Cov(MZ) + ½ Cov(P-O) + ¼ Cov(Unrel)

= ¼(VAQTL+VDQTL

) + ½ (½ VAQTL) + ¼ (0)

= ½ VAQTL + ¼VDQTL

Biometrical model predicts contribution of a QTL to the mean, variance and covariances of a trait

Var (X) = VAQTL + VDQTL

Cov (MZ) = VAQTL + VDQTL

Cov (DZ) = ½VAQTL + ¼VDQTL

Var (X) = Σ(VAQTL) + Σ(VDQTL

) = VA + VD Multiple QTL

Cov (MZ)

Cov (DZ)

= Σ(VAQTL) + Σ(VDQTL

) = VA + VD

= Σ(½VAQTL) + Σ(¼VDQTL

) = ½VA + ¼VD

Summary

Biometrical model underlies the variance components estimation performed in Mx

Var (X) = VA + VD + VE

Cov (MZ)

Cov (DZ)

= VA + VD

= ½VA + ¼VD

Summary

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 = k2 + m2 + 2 kpm  Var B = l2 + n2  Var C = o2

Genetic Examples

 MZ Twins Reared Together  DZ Twins Reared Together  MZ Twins Reared Apart  DZ Twins Reared Apart  Parents & Offspring

MZ Twins Reared Together

Expected Covariance

Twin 1 Twin 2

Twin 1 a2+c2+e2

variance a2+c2

Twin 2 a2+c2

covariance a2+c2+e2

MZ Twins RT

DZ Twins Reared Together

Expected Covariance

Twin 1 Twin 2

Twin 1 a2+c2+e2 .5a2+c2

Twin 2 .5a2+c2 a2+c2+e2

DZ Twins RT

MZ Twins Reared Apart

DZ Twins Reared Apart

Twins and Parents

Role of model mediating between theory and data

intro to quantitative genetics - vipbg.vcu.edu€¦ · intro to quantitative genetics 1/18: course...

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