Population structure at QTL

d

A B C D E Q F G H

a b c d e q f g h

The population contentat a quantitative trait locus(backcross, RIL, DH).

Can be deduced by observationof marker groups. In the figure,the observation is for a markercoinciding with QTL.

m1 m2 … m3 … mi … mj … mk

The simplest genetic model: single marker analysis

Dihaploid mapping population, two homozygotes at each locus

For marker mi coinciding with Q/q

XXmm mm = = - - ½½ d d

XXMM MM = = + + ½½ d d XXMM MM -- XXmm mm = d= d

QQ q q

MM m m

d x

f(x)

qq QQ

fmm (x)=(1-r) fqq(x) + r fQQ(x)

fMM (x)=r fqq(x) +(1-r) fQQ(x)

XXmm = (1-r)(- - ½½dd)) + r(+ + ½½dd))

XXMM = r(- - ½½dd)) +(1-r)(+ + ½½dd))

XXMM MM -- XXmm mm = = (1-2r)dd

For marker mj apart from Q/q

QTL Interval Mapping – DH

(F1=M1QM2/m1qm2 meiosis DH

Expected distributions of the trait in the 4 marker groups look like

fM1M2= [(1-r1) (1-r2)fQQ + r1r2fqq]/(1-r)

fM1m2= [(1-r1)r2fQQ + r1 (1-r2) fqq]/r fm1M2=

[r1(1-r2)fQQ + r2 (1-r1) fqq]/r fm1m2

= [r1r2fQQ + (1-r1) (1-r2) fqq]/(1-r)

ML-estimation in QTL interval analysis

L (r, m, d, )= fi (r, m, d, xij) = L ( | data)

max

4 Ni

i=1 j=1

ML-estimates of QTL parameters: *={ r*, m*, d*,

The model of QTL effect

For additive QTL effect: x = m + dgq+

where gq = -1 for qq, and +1 for QQ; E= .

The QTL-effect is d=(QQ- qq)/2, qq=m-d, QQ=m+d

d x

f(x)qq QQ

i=1,…,4 M1M2 M1m2 m1M2 m1m2

ML analysis

L (m, ) =f (m, xj)N

j=1

m* = x = xj /N,

max

lnL (m, ) = ln f (m, xj) N

j=1

= (xj – x) 1N-1

to correct the biasm

radius of convergence,i.e. we need a good initial point

0

* 0

What do one expect from the analytical tools ?

To extract maximum mapping information from the experimental data

The main questions in QTL analysis:

High QTL detection power (detect QTL when it exists) Minimum “false positive” (high significance) Mapping resolution (e.g., two vs. one QTL in a region) Accuracy of parameter estimates (e.g., d±md )

Discrimination between alternative models of the trait “genetic architecture” (e.g., additive vs. heterotic)

analytical tools

Lod Scores for marker mapping

L(θ< 0.5) L(θ <0.5 | data) Z=LOD score (θ) = log10 = L(θ= 0.5) L(θ =0.5 | data)

Logarithm of the Odds 2 ln [L( ) / L(0.5)] = 4.6 Z ~ 21

Now we use LOD score to compare another pair of alternatives:

whether or not the interval of interest carries a QTL affecting our target trait, i.e. H1 (d0) versus H0 (d=0),

where d (effect) is one of parameters of our vector θ=(r, m, d, .

In other words, we are about to check whether there is a connection between the trait values and the marked interval (or chromosome), i.e., whether the effect of the chromosome is significant.

Here LOD was to compare 2 alternative hypotheses about linkage: H1 (θ< 0.5) versus H0 (θ=0.5)

Lod Score and Testing Significance

If there is such connection (i.e., H1 is true), we are supposed to get larger values of LOD compared to situations of no connection(i.e., when H0 is true).

m1 m2 … m3 … mi … mj … mk

…We are about to check whether there is a connection between

the trait values and the marked interval (or chromosome).

Let us take the data and calculate the LOD for each interval of the chromosome. How to decide whether the obtained level of LOD is indicative for H1 or H0? The answer can bereached by building artificially the situation of H0 , i.e. when there is no connection of thetrait values and markers.

By permutation test Reshuffling markers and trait values

How ?

Reshuffling trait values relative to markers

Markers Genotypes

M1 1 1 3 3 2 3 1 2 …M2 1 2 3 1 3 3 2 3 ………..…………………….………...M99 2 2 3 1 3 3 2 3 ………..………………………...........M200 1 2 3 1 1 3 2 1 …

Traitstr1 21 28 32 26 19 27 30 25 ...tr2 7.6 6.3 5.8 7.3 6.6 7.9 7.0 5.5 …………………………………………………………………….

d

A B C D E Q F G H

a b c d e q f g h

Testing SignificanceThe algorithm may look like this:

• Calculate max LOD value (LOD=LOD*) for the chromosome.

• Reshuffle trait values relative to markers build a sample that fits H0

• For each reshuffled sample # i calculate LOD=LODi.

• Repeat the last two steps N times (N runs). If H1 is correct, then for the predominate majority of reshuffled samples, LODi <<LOD*. The proportion of runs, , with LODi LOD* is called significance. It means the probability to declare a QTL that does not exist (indeed, reshuffling destroys any connection between the chromosome and the trait, thus cases LODi LOD* are “false positive”).

The higher the LOD* the lower the chance of LODi LOD*.

Calculating the QTL detection power • For the declared QTL, the significance is a measure of a “false positive” risk (i.e. declaring an effect that does not exist).

• We would like also to know the risk of “false negative”, i.e. the probability of not detecting the effect that does exist. Of course, will depend on the chosen level of .

The score 1- (probability to detect a QTL that does exist is called “detection power”. How to calculate it ? by Bootstrap analysis

It can be conducted only after calculating thethreshold values of LOD under H0

Calculating the threshold values of LOD under H0

The proportion of randomized runs, , with LODi LOD* is called significance. It means the probability to declare a QTL that does not exist (cases LODi LOD* are “false positive”). We need also to know the value of the LOD that will be over in just 5% (or 1%) of the runs.

99%

95%

LOD values under H0 (in runs)

The algorithm looks like this:

• Take a series of re-sampling steps, with returning• Conduct interval analysis • Repeat this procedure for many (e.g., N=1000) such samples• Check the proportion of samples where maxLOD exceeds the threshold (= QTL detection power = 1-)• Calculate the confidence intervals of the parameters

Calculating the QTL detection power

• The score 1- (probability to detect a QTL that does exist is called “detection power”. How to calculate it ? - by Bootstrap analysis after getting the thresholds

of LOD under H0