1. Quantitative Trait Loci, QTL An introduction to quantitative genetics and common methods for mapping of loci underlying continuous traits:

2. 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:

3. 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;

4. A pedigree representation

5. 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;

6. A simple model for the phenotype
Y = X + e
where
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;

7. Distribution of Y : a single biallelic locus

8. 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.

9. Example: The pygmy gene, pg
From data we have the following mean values of weight:
X++ = 14g, X+pg = 12g, Xpgpg = 6g,
2a = = 8 implies a =4,
(1 + k)a = = 6 implies k = 0.5.
Data suggest recessivity (although not complete) of the pygmy gene.

10. 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 behaves 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;

11. Least-squares linear regression

12. Model 1

14. Interpretations in the linear regression
is the heritable component of the genotype,
δis the non-heritable part;
the sum of an individuals additive allelic effects, αi+αj is called the breeding value and is denoted Λij
under random mating αican be interpreted as the average excess of allele Ai
this is defined as the difference between the expected phenotypic value when one allele (e.g. the paternally transmitted) is fixed at Ai and the population average, μ;

15. Linear Regression

16. Graphically

17. Linear Regression Model solvingX
prob.
a(1+k) 1 2a 2

20. average excesses

21. Interpretations under random mating
α= a [1+ k (p1-p2)] ;
α= - p2 α;
α= p1 α,
Population parameters for k≠0
α is called the average effect of allelic substitution:
substitute A1 A2for a randomly chosen A1 –allele
then the expected change in X is,
(X12 -X11) p1 + (X22 -X12) p2 ;
which equals α. (simple calculations).

22. : Average effect of allelic substitution

23. α is a function of p2 and k :

24. 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;

25. VA

26. V (X); VA; VD are functions of p2 and k:

27. Example: The Booroola gene, (Lynch and Walsh, 1998)

28. In summary
The homozygous effect a, and the dominance coefficient 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.

29. 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?

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

31. 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

32. Neglecting V (ε)
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;

33. 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)

34. 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.

35. Joint distribution of sibling trait values
Single biallelic, dominant (k =1 ) model. Correlation 0.46.

36. Measures of relatedness
N = the number of alleles shared IBD by two relatives at a given locus;
the kinship coefficient, θ , is given by
2 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).

37. 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:

38. Covariance formula for a single locus
Under the assumed model

39. A single locus; perfect marker data

40. Covariance formula for multiple loci
n independently segregating loci assuming no
epistatic interaction, i.e. putting V (ε) = 0

41. Covariance formula for multiple loci
n independently segregating loci assuming no
epistatic interaction, i.e. putting V (ε) = 0

42. Covariance... continued Define for every pair of relatives
For two related individuals we then have,

43. 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;

44. 0.5 Proportion of marker alleles identical by decent
HE: an example
0.5 Proportion of marker alleles identical by decent
Solid line is the tted regression line;
Dotted line indicates true underlying relationship

45. HE: motivation Assume strictly additive gene action at each locus,
i.e.VD = 0. Then, for a putative QTL at x,

46. HE: linkage test

47. HE: examples with simulated data
simulated data from n = 200 sib-pairs;
top to bottom: h2 = 0:50; 0:33; 0:25.

48. 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.

49. HE: two-point analysis
where is the expected proportion of marker alleles shared IBD.
depends on the type of relatives considered;
for sib pairs
recombination fraction (θ) and effect size (VA;l )
are confounded and cannot be separately
estimated;

50. 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

51. VCA Yi=m+bTai+gi+qi+ei QTL Polygenes Mathematically: Trait value
Independent
environment
QTL
Trait value
Mathematically:
Yi=m+bTai+gi+qi+ei
where m 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.

52. VCA: an additive model

53. VCA: major assumption
The 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;

54. 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.

55. VCA vs HE: LOD score proles
From Pratt et al.; Am. J. Hum. Genet. 66: , (2000)

56. 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;

57. 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 either
linear regression (HE) or
variance components analysis (VCA);