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 content at a quantitative trait locus (backcross, RIL, DH). Can be deduced by observation of marker groups. In the figure, the observation is for a marker coinciding with QTL.

     m 1 m 2 … m 3 … m i … m j … m k The simplest genetic model: single marker analysis Dihaploid mapping population, two homozygotes at each locus For marker m i coinciding with Q/q X mm =  - ½ d X MM =  + ½ d X MM - X mm = d Q qQ  qM mM  mQ qQ  qM mM  m d x f(x)f(x) qq QQ  f mm (x)=(1-r) f qq (x) + r f QQ (x) f MM (x)=r f qq (x) +(1-r) f QQ (x) X  - ½d)  + ½d) X mm = (1-r)(  - ½d) + r(  + ½d) X  - ½d)  + ½d) X MM = r(  - ½d) +(1-r)(  + ½d) X MM - X mm = d X MM - X mm = (1-2r) d For marker m j apart from Q/q 

QTL Interval Mapping – DH (F 1 =M 1 QM 2 /m 1 qm 2  meiosis  DH Expected distributions of the trait in the 4 marker groups look like f M 1 M 2 = [(1-r 1 ) (1-r 2 )f QQ + r 1 r 2 f qq ]/(1-r) f M 1 m 2 = [(1-r 1 )r 2 f QQ + r 1 (1-r 2 ) f qq ]/r f m 1 M 2 = [r 1 (1-r 2 )f QQ + r 2 (1-r 1 ) f qq ]/r f m 1 m 2 = [r 1 r 2 f QQ + (1-r 1 ) (1-r 2 ) f qq ]/(1-r)

ML-estimation in QTL interval analysis L (r, m, d,  )=     f i (r, m, d,  x ij ) = L (  | data) max 4 N i i=1 j=1 ML-estimates of QTL parameters:  *={ r*, m*, d*,  The model of QTL effect For additive QTL effect: x = m + dg q +  where g q = -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)f(x) qq QQ  i=1,…,4 M 1 M 2 M 1 m 2 m 1 M 2 m 1 m 2

ML analysis L (m,  ) =  f (m,  x j ) N j=1j=1 m* = x =  x j / N, max lnL (m,  ) =  ln f (m,  x j ) N j=1   =  ( x j – x)  1 N - 1 to correct the bias m  radius of convergence, i.e. we need a good initial point 00 ** 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 ± m d )  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) = log 10 = L(θ= 0.5) L(θ =0.5 | data) Logarithm of the Odds 2 ln [L(  ) / L(0.5)] = 4.6  Z ~  2 1 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. H 1 (d  0) versus H 0 (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: H 1 (θ< 0.5) versus H 0 (θ=0.5) 

Lod Score and Testing Significance If there is such connection (i.e., H 1 is true), we are supposed to get larger values of LOD compared to situations of no connection (i.e., when H 0 is true).            m 1 m 2 … m 3 … m i … m j … m k …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 H 1 or H 0 ? The answer can be reached by building artificially the situation of H 0, i.e. when there is no connection of the trait values and markers. By permutation test Reshuffling markers and trait values How ?

Reshuffling trait values relative to markers Markers Genotypes M … M … ……..…………………….………... M … ……..……………………… M … Traits tr tr … …………………………………………………………………. d A B C D E Q F G H a b c d e q f g h

Testing Significance The 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 H 0 For each reshuffled sample # i calculate LOD=LOD i. Repeat the last two steps N times (N runs). If H 1 is correct, then for the predominate majority of reshuffled samples, LOD i <<LOD*. The proportion of runs, , with LOD i  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 LOD i  LOD* are “false positive”). The higher the LOD* the lower the chance of LOD i  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 the threshold values of LOD under H 0

Calculating the threshold values of LOD under H 0 The proportion of randomized runs, , with LOD i  LOD* is called significance. It means the probability to declare a QTL that does not exist (cases LOD i  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 H 0 (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 H 0