Quantitative Trait Loci, QTL

An introduction to quantitative geneticsand common methods for mapping of

loci underlying continuous traits:

Why study quantitative traits?

• Many (most) human traits/disorders are complex in the sense that they are governed by several genetic loci as well as being influenced by environmental agents;

• Many of these traits are intrinsically continuously varying and need specialized statistical models/methods for the localization and estimation of genetic contributions;

• In addition, in several cases there are potential benefits from studying continuously varying quantities as opposed to a binary affected/unaffected response:

For example:

• in a study of risk factors the underlying quantitative phenotypes that predispose disease may be more etiologically homogenous than the disease phenotype itself;

• some qualitative phenotypes occur once a threshold for susceptibility has been exceeded, e.g. type 2 diabetes, obesity, etc.;

• in such a case the binary phenotype (affected/unaffected) is not as informative as the actual phenotypic measurements;

A pedigree representation

Variance and variability

• methods for linkage analysis of QTL in humans rely on a partitioning of the total variability of trait values;

• in statistical theory, the variance is the expected squared deviation round the mean value,

• it can be estimated from data as:

• the square root of the variance is called the standard deviation;

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A simple model for the phenotype

Y = X + ewhere• Y is the phenotypic value, i.e. the trait value;• X is the genotypic value, i.e. the mean or

expected phenotypic value given the genotype;• e is the environmental deviation with mean 0.• We assume that the total phenotypic variance is

the sum of the genotypic variance and the environmental variance, V (Y ) = V (X ) + V (e), i.e. the environmental contribution is assumed independent of the genotype of the individual;

Distribution of Y : a single biallelic locus

A single biallelic locus: genetic effects

Genotype

Genotypic value

• a is the homozygous effect,• k is the dominance coeffcient• k = 0 means complete additivity,• k = 1 means complete dominance (of A2),• k > 1 if A2 is overdominant.

Example: The pygmy gene, pg

• From data we have the following mean values of weight:

X++ = 14g, X+pg = 12g, Xpgpg = 6g,

• 2a = 14 -6 = 8 implies a =4,

• (1 + k)a = 12 - 6 = 6 implies k = 0.5.

Data suggest recessivity (although not complete) of the pygmy gene.

Decomposition of the genotypic value, X

• Xij is the mean of Y for AiAj-individuals;• when k = 0 the two alleles of a biallelic locus be

haves in a completely additive fashion: X is a linear function of the number of A2-alleles;

• we can then think of each allele contributing a purely additive effect to X ;

• this can be generalized to k ≠ 0 by decomposition of X into additive contributions of alleles together with deviations resulting from dominance;

• the generalization is accomplished using least-squares regression of X on the gene content;

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Partitioning the genetic variance

• the variance, V (X ), of the genotypic values in a population is called the genetic variance:

• is the additive genetic variance, i.e. variance associated with

additive allelic effects;• dominance genetic

variance, i.e. due to dominance deviations;

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In summary• The homozygous effect a, and the dominance co

efficient k are intrinsic properties of allelic products.

• The additive effect αi, and the average excess αi* are properties of alleles in a particular population.

• The breeding value is a property of a particular individual in reference to a particular population. It is the sum of the additive effects of an individual's alleles.

• The additive genetic variance, VA, , is a property of a particular population. It is the variance of the breeding values of individuals in the population.

Multilocus traits

• Do the separate locus effects combine in an additive way, or do there exist non-linear interaction between different loci: epistasis?

• Do the genes at different loci segregate independently?

• Do the gene expression vary with the environmental context: gene by environment interaction?

• Are specic genotypes associated with particular environments: covariation of genotypic values and environmental effects?

Example: epistasis

Average length of vegetative internodes in the lateral branch(in mm) of teosinte. Table from Lynch and Walsh (1998).

Two independently segregating loci

• Extending the least-squares decomposition of X :

• Λk is the breeding value of the k'th locus,

δk is the dominance deviation of the k'th locus,

ε is a residual term due to epistasis;• if the loci are independently segregating

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• the epistatic variance components contributing to V (ε) are often small compared to VA and VD;

• in linkage analysis it is this often assumed that V (ε) = 0;

• note however: the relative magnitude of the variance components provide only limited insight into the physiological mode of gene action;

• epistatic interactions, can greatly inflate the additive and/or dominance components of variance;

Resemblance between relatives

A model for the trait values of two relatives:

Yk = Xk + ek, k = 1 , 2,

where for the k’th relative• Yk is the phenotypic value,• Yk is the genotypic value,• ek is the mean zero environmental deviation.• the ek’s are assumed to be mutually independent

and also independent of k. Hence, the covariance of the trait values of two relatives is given by the genetic covariance, C(X1; X2), i.e.

C(Y1; Y2) = C(X1; X2)

A (preliminary) formula for C(X1 ,X 2)

For a single locus trait

C(X1; X2) = c1VA + c2VD

• c1 and c2 are constants determined by the type of relationship between the two relatives.

• same formula applies for multilocus traits if no epistatic variance components are included in the model, i.e. V (ε) = 0.

• in this latter case and are given by summation of the corresponding locus-specific contributions.

Joint distribution of sibling trait values

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Measures of relatedness

• N = the number of alleles shared IBD by two relatives at a given locus;

• the kinship coefficient, θ , is given by2 θ = E(N) / 2;

i.e. twice the kinship coefficient equals the expected proportion of alleles shared IBD at the locus.

• The coefficient of fraternity, Δ, is defined as

Δ = P(N = 2).

Some examples

• Siblings

(z0; z1; z2) = (1/4; 1/2; 1/4) implying E(N) = 1.Thus θ= 1/4 and Δ = 1/4:

• Parent-offspring

(z0; z1; z2) = (0; 1; 0) implying E(N) = 1.Thus θ = 1/4 and Δ = 0:

• Grandparent - grandchild

(z0; z1; z2) = (1/2; 1/2; 0) implying E(N) = 1=2.Thus θ = 1/8 and Δ = 0:

Covariance formula for a single locus

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Haseman-Elston method

• Uses pairs of relatives of the same type: most often sib pairs;

• for each relative pair calculate the squared phenotypic difference: Z = (Y1 –Y2)2;

• given MDx regress the Z's on the expected proportion of alleles IBD, π(x) = E [Nx |MDx]/2, at the test locus;

• a slope coefficient β< 0, if statistically significant, is considered as evidence for linkage;

HE: an example

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HE: motivation

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Heritability and power

• for a given locus we may define the locus-specific heritability as the proportion of the total variance 'explained' by that particular site, e.g. (in the narrow-sense),

• the locus-specific heritability is the single most important parameter for the power of QTL linkage methods;

• heritabilities below 10% leads, in general, to unrealistically large sample sizes.

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HE: in summary

Simple, transparent and comparatively robust but:

• poor statistical power in many settings;• different types of relatives cannot be mixed;• parents and their offspring cannot be used in HE;• assumptions of the statistical model not generally

satisfied;

• Remedy:• use one of several suggested extensions of HE;• alternatively, use VCA instead

Mathematically:

Yi=+Tai+gi+qi+ei

where is the population mean, a are the “environmental” predictor variables, q is the major trait locus, g is the polygenic effect, and e is the residual error.

Polygenes Independent environment

QTL

Trait value

VCA

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VCA: major assumptionThe joint distribution of the phenotypic values in a pedigree is assumed to be multivariate normal with the given mean values, variances and covariances;

• the multivariate normal distribution is completely specified by the mean values, variances and covariances;• the likelihood, L, of data can be calculated and we can estimate the variance components VA;x; VD;x ; VA;R; VD;R;

VCA: linkage test

The linkage test of

H0 : VA;x = VD;x = 0

uses the LOD score statistic

When the position of the test locus, x, is varied over a chromosomal region the result can be summarized in a LOD score curve.

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From Pratt et al.; Am. J. Hum. Genet. 66:1153-1157, (2000)

Linkage methods for QTL

• Fully parametric linkage approach is difficult;• Model-free tests comprise the alternative choice; • We will discuss

Haseman-Elston Regression (HE);Variance Components Analysis (VCA);

Both can be viewed as two-step procedures:1. use polymorphic molecular markers to extract information on inheritance patterns;2. evaluate evidence for a trait-influencing locus at specified locations;

Similarities and differences

• HE and VCA are based on estimated IBD-sharing given marker data;

• both methods require specification of a statistical model!('model-free' means 'does not require specification of genetic model')

• similarity in IBD-sharing is used to evaluate trait similarity using eitherlinear regression (HE) orvariance components analysis (VCA);