Problem box 9.3

Derive the expected marker-class means for a backcross mating design

A commonly employed QTL mapping design is the backcross of an F1 individual to one of the P1 individuals shown in Fig. 9.15. One drawback of a backcross mating design for QTL mapping is that the resulting estimates of a depend on the value of d. To see that this is the case, consider the backcross A1Q1B1 A1Q1B1

and the marker-class means a2q2b2 a1q1b1

in the population of progeny. Derive the expected marker-class means for A1A1B1B1 and A1A2B1B2 marker genotypes (call these

PwlSl and pa;aab2 respectively) and then compute the expected value of a = Pba1a1b1b1 - Pba1a2b1b2. To work through this problem, first notice f w;

that the P1 individual will produce v A1Q1B1

only one type of gamete whereas an F1 f A1Q1B1 ^

individual will produce eight


types of gametes with the frequencies given in Fig. 9.16. This problem is made easier by the commonly invoked assumption that the marker loci are close enough on the chromosome such that double recombination events are so rare they can be ignored. Therefore, the F1 parent gametes A2Q1B2 and A1Q2B1 can be left out of the marker-class means.

Start by constructing a table with two rows for the two F1 gametes produced by no recombination and four rows for the F1 gametes produced by a single recombination event. Then, following the model of Tables 9.5 and 9.6, fill in columns for the six categories of backcross progeny. The column headings are F1 parent gamete, expected F1 gamete frequency, backcross progeny genotype, backcross progeny genotypic value, frequency-weighted genotypic value, backcross progeny marker genotype expected frequency, and marker-class genotype mean. When adding the two terms that make up the A1A1B1B2 or A1A2B1B1 frequency-weighted genotypic values, any terms that contain rArB can be crossed out because of the assumption that double recombination events are so rare that they can be ignored. The marker class means are obtained by dividing the frequency-weighted genotypic value by the marker genotype expected frequency.

A hypothetical example of data produced by interval QTL mapping in an F2 design is shown in Fig. 9.17. This example illustrates the difference in the value of the homozygous marker-class means (e.g. the difference between GA A and GA A ) for each of

A1A1D1D1 A2A2B2B 2'

17 genetic marker loci. The difference in marker-class means is given on the y axis while the position of each marker locus on a chromosome is given on the x axis. The marker-class mean differences are near zero for marker loci not in gametic disequilibrium with a QTL. Marker loci near QTLs show some difference in marker-class means, but the marker loci and the QTLs experienced frequent recombination and so are not strongly associated. As marker loci closer to the QTLs are considered, the amount of gametic disequilibrium between a marker locus and a QTL increases, producing a greater difference in marker-class means. In this hypothetical example, the genetic markers at 33 and 85 map units lie closest to QTLs, as indicated by peak values for marker-class mean differences. The marker-class means at these marker loci differ by over one phenotypic standard deviation, indicating a statistically meaningful difference given sampling error and multiple statistical tests. (The threshold for statistical significance of marker-class mean differences is often judged by a log of odds or LOD score and differs depending on the details of each QTL study; see Van Ooijen (1999).) The QTL near 33 map units increases the mean value while the QTL near 85 map units decreases the mean value. These two hypothetical QTLs have opposite effects on the trait, a situation sometimes called dispersion. Dispersion can lead to downwardly biased estimates of QTL effects because each marker locus is associated with two QTLs with opposite phenotypic effects. The perceived marker-class mean difference is therefore the net phenotypic effect caused by association with two QTLs.

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