An analytical formula to estimate confidence interval of QTL location with a saturated genetic map as a function of experimental design

Abstract

Analytical formulae are derived for the confidence interval for location of a quantitative trait locus (QTL) using a saturated genetic map, as a function of the experimental design, the QTL allele substitution effect, and the number of individuals genotyped and phenotyped. The formulae are derived assuming evenly spaced recombination events, rather than the actual unevenly spaced distribution. The formulae are useful for determining desired sample size when designing a wide variety of QTL mapping experiments, and for evaluating a priori the potential of a given mapping population for defining the location of a QTL. The formulae do not take into account the finite number of recombination events in a given sample.

Appendix

Derivation of a formula for CI of QTL location for an F2 mapping population using only the recombinant progeny

The contrast for an F₂ mapping population is based on individuals homozygous for alternative marker alleles. Thus, to be included in the recombinant F₂ mapping population, an individual must be recombinant for at least one chromosome, and homozygous for at least one of the marker alleles. Three of the nine possible F₂ marker genotypes do not meet these criteria. These are the two homozygous parental types (M₁M₂/M₁M₂ and m₁m₂/m₁m₂) and the double recombinant type (M₁m₂/m₁M₂). The remaining six F₂ marker genotypes that meet these criteria are listed in Table 2. The genotypic value of each genotype, assuming that the QTL is located at marker M₁, and the expected number of individuals having that genotype in the mapping population are also listed in the table.

Table 2 Composition of the F₂ including only recombinant progeny

The contrast for an F₂ mapping population is composed of the expected mean value of marker genotype groups that are homozygous for the alternative markers M₁, m₁, M₂, and m₂. Each of these, however, is composed of two recombinant marker genotype groups. For example, the marker genotype group M₂M₂ is composed of the recombinant genotype groups: m₁M₂/M₁M₂ (class E in Table 2) with genotypic value h and frequency 2(1−r)r/4 in the entire F₂ population, and marker genotype group m₁M₂/m₁M₂ (class D in Table 2) with genotypic value −d, and frequency r²/4. The mean genotypic value of the M₂M₂ group including recombinants only is the mean of the genotypic values of the classes E and D, weighted by their relative frequency in the M₂M₂ recombinant group i.e., 2(1−r)r/[2(1−r)r+r²] and r²/[2(1−r)r+r²], which simplifies to 2(1−r)/(2−r) and r/(2−r), respectively. These relative frequencies are the “weighting factors” listed in column three of Table 2. The contrast for the F₂ mapping population, D′, is computed as follows:

$$D' = (\underline{{\text{M}}} _{1} \underline{{\text{M}}} _{1} - \underline{{\text{m}}} _{1} \underline{{\text{m}}} _{1} ) - (\underline{{\text{M}}} _{2} \underline{{\text{M}}} _{2} - \underline{{\text{m}}} _{2} \underline{{\text{m}}} _{2} )$$

(17)

Letting A, B, C, D, E, and F represent their respective genotypic values, and letting k₁=2(1−r)/(2−r), and k₂=r/(2−r), so that k₁+k₂=1, we have:

$$\begin{array}{*{20}l} {{\underline{{\text{M}}} _{1} \underline{{\text{M}}} _{1} } \hfill} & {{ = k_{1} {\text{A}} + k_{2} {\text{B}}} \hfill} \\ {{{\text{m}}_{1} \underline{{\text{m}}} _{1} } \hfill} & {{ = k_{1} {\text{C}} + k_{2} {\text{D}}} \hfill} \\ {{\underline{{\text{M}}} _{2} \underline{{\text{M}}} _{2} } \hfill} & {{ = k_{1} {\text{E}} + k_{2} {\text{D}}} \hfill} \\ {{\underline{{\text{m}}} _{2} \underline{{\text{m}}} _{2} } \hfill} & {{ = k_{1} {\text{F}} + k_{2} {\text{B}}} \hfill} \\ \end{array} $$

Substituting in (17) gives

$$\begin{array}{*{20}l} {{D'} \hfill} & {{ = {\left[ {{\left( {k_{1} {\text{A}} + k_{2} {\text{B}}} \right)} - {\left( {k_{1} {\text{C}} + k_{2} {\text{D}}} \right)}} \right]} - {\left[ {{\left( {k_{1} {\text{E}} + k_{2} {\text{D}}} \right)} - {\left( {k_{1} {\text{F}} + k_{2} {\text{B}}} \right)}} \right]}} \hfill} \\ {{} \hfill} & {{ = k_{1} {\text{A}} + k_{1} {\text{F}} - k_{1} {\text{C}} - k_{1} {\text{E}} + {\text{2}}k_{2} {\text{B}} - {\text{2}}k_{2} {\text{D}}} \hfill} \\ {{} \hfill} & {{ = k_{1} {\left( {{\text{A}} + {\text{F}} - {\text{C}} - {\text{E}}} \right)} + 2k_{2} {\left( {{\text{B}} - {\text{D}}} \right)}} \hfill} \\ {{} \hfill} & {{ = {\left[ {1/{\left( {2 - r} \right)}} \right]}{\left\{ {{\left[ {2{\left( {1 - r} \right)}} \right]}{\left[ {d + h + d - h} \right]} + 2r{\left[ {d + d} \right]}} \right\}}} \hfill} \\ {{} \hfill} & {{ = 4d/{\left( {2 - r} \right)}} \hfill} \\ \end{array} $$

(18)

To calculate SE(D), note that:

$$\begin{array}{*{20}l} {{\sigma ^{2}_{{\text{A}}} = \sigma ^{2}_{{\text{F}}} = \sigma ^{2}_{{\text{C}}} = \sigma ^{2}_{{\text{E}}} = 1/[2(1 - r)rN/4] = 4/2(1 - r)rN} \hfill} \\ {{\sigma ^{2}_{{\text{B}}} = \sigma ^{2}_{{\text{D}}} = 4/r^{2} N} \hfill} \\ \end{array} $$

Thus:

$$\begin{array}{*{20}l} {{{\text{SE}}^{{\text{2}}} {\left( {D'} \right)}} \hfill} & {{ = 4{\left[ {2{\left( {1 - r} \right)}/{\left( {2 - r} \right)}} \right]}^{2} 4/{\left[ {2{\left( {1 - r} \right)}rN} \right]} + 2{\left[ {2r/{\left( {2 - r} \right)}} \right]}^{2} {\left[ {4/{\left( {r^{2} N} \right)}} \right]}} \hfill} \\ {{} \hfill} & {{ = 4{\left[ {2{\left( {1 - r} \right)}} \right]}/{\left( {2 - r} \right)}^{2} {\left[ {4/rN} \right]} + 2{\left[ {4/{\left( {2 - r} \right)}^{2} } \right]}{\left[ {4/N} \right]}} \hfill} \\ {{} \hfill} & {{ = 32{\left( {1 - r} \right)}/{\left( {2 - r} \right)}^{2} rN + 32/{\left( {2 - r} \right)}^{2} N = {\left[ {32{\left( {1 - r} \right)} + 32r} \right]}/{\left[ {{\left( {2 - r} \right)}^{2} rN} \right]}} \hfill} \\ {{} \hfill} & {{ = 32/{\left( {2 - r} \right)}^{2} rN} \hfill} \\ \end{array} $$

(19)