Modeling segregation distortion for viability selection I. Reconstruction of linkage maps with distorted markers

Abstract

Molecular markers have been widely used to map quantitative trait loci (QTL). The QTL mapping partly relies on accurate linkage maps. The non-Mendelian segregation of markers, which affects not only the estimation of genetic distance between two markers but also the order of markers on a same linkage group, is usually observed in QTL analysis. However, these distorted markers are often ignored in the real data analysis of QTL mapping so that some important information may be lost. In this paper, we developed a multipoint approach via Hidden Markov chain model to reconstruct the linkage maps given a specified gene order while simultaneously making use of distorted, dominant and missing markers in an F₂ population. The new method was compared with the methods in the MapManager and Mapmaker programs, respectively, and verified by a series of Monte Carlo simulation experiments along with a working example. Results showed that the adjusted linkage maps can be used for further QTL or segregation distortion locus (SDL) analysis unless there are strong evidences to prove that all markers show normal Mendelian segregation.

Appendix

We use here the F₂ design as an example to infer the maximum likelihood estimate of recombinant fraction between two markers.

One-gene model

Provided that only one marker, M ₁, displays zygotic viability selection, the viabilities of genotypes M ₁ m ₁ and m ₁ m ₁ relative to M ₁ M ₁ are s ₁ and s ₂, respectively. Thus, the frequencies for the three genotypes among the survival individuals after selection are 1/D for M ₁ M ₁, 2s ₁/D for M ₁ m ₁, and s ₂/D for m ₁ m ₁, respectively, where D =  1 +  2s ₁ +  s ₂. If another marker, M ₂, is linked to the marker M ₁ with recombinant fraction r. The expected frequencies of nine F₂ genotypes are a function of the viability coefficients and the recombination fraction, arrayed by

The MLE of r is obtained by the EM algorithm for a normal F₂. This is because the first derivative of the likelihood function with respect to recombination fraction contains no information about the viability coefficients s ₁ and s ₂. It indicates that the estimate of r is not affected by the viability coefficients. Thus we can estimate r directly using the familiar formula in the M step of Jansen and Stam (1994)

$$\hat{r}= \frac{1}{{2n}}\left[n_{{12}} + 2n_{{13}} + n_{{21}} + \frac{{2r^{2} }}{{r^{2} + (1 - r)^{2}}}n_{{22}} + n_{{23}} + 2n_{{31}} + n_{{32}} \right]$$

(A2)

where n =  ∑ ³ _i=1 ∑ ³ _j=1 n _ij , these n _ij were the number of the nine genotypes above in matrix A1. Parameters s ₁ and s ₂ are obtained by

$$\hat{s}_{1} = \frac{{n_{{21}} + n_{{22}} + n_{{23}}}}{{2(n_{{11}} + n_{{12}} + n_{{13}})}}\quad\hat{s}_{2} = \frac{{n_{{31}} + n_{{32}} + n_{{33}}}}{{n_{{11}} + n_{{12}} + n_{{13}}}}$$

(A3)

Based on the Fisher’s information matrix, the sample variance of MLE of the recombination fraction can be indicated by

$$V(\ifmmode\expandafter\hat\else\expandafter\^\fi{ r}) = \frac{{Dr(1 - r)(1 - 2r + 2r^{2})}}{{2n\left[D(1 - 2r)^{2} (1 - 2r + 2r^{2}) + 4(1 + s_{2})r(1 - r)(1 - 2r + 2r^{2}) + 4s_{1} r(1 - r)(1 - 2r)^{2} \right]}}$$

(A4)

It is obvious that the sample variance of the recombination fraction is affected by the viability coefficients.

Two-gene model

Provided that two linked markers with recombinant fraction r, say M ₁ and M ₂, display zygotic viability selection, so the viabilities of genotypes M ₁ m ₁ and m ₁ m ₁ relative to M ₁ M ₁ are s _1,1 and s _1,2, respectively; for the marker M ₂, similarly, they are s _2,1 and s _2,2, respectively. Thus, the expected frequencies of nine F₂ genotypes are a function of the viability coefficients and the recombination fraction, arrayed by

where ${ \circ }$ stands for the component-wise product between the two matrices, the first (F _r ) only associated with r and the second with ${s_{{i,j}} (i, j=1,2)\;\hbox{and}\;D = (1 + s_{{1,2}} s_{{2,2}})(1 - r)^{2} + 2r(1 - r)[s_{{1,1}} (1 + s_{{2,2}}) + s_{{2,1}} (1 + s_{{1,2}})] + r^{2} (s_{{1,2}} + s_{{2,2}}) + 2 (1 - 2r + r^{2})s_{{1,1}} s_{{2,1}}.}$ The EM algorithm can be used to obtain the MLE of r based on matrix B1, but this will be difficult to derive because the coefficients within each cell of this matrix contain r. By dividing matrix B1 into two component matrices in B2, however, we can simplify this derivation process. Based on the results of Wu et al. (2005), similarly, the MLE of r can be expressed by

$$\ifmmode\expandafter\hat\else\expandafter\^\fi{r} = \frac{1}{{2n}}\left[n_{{12}} + 2n_{{13}} + n_{{21}} + \frac{{2r^{2} }}{{r^{2} + (1 - r)^{2}}}n_{{22}} + n_{{23}} + 2n_{{31}} + n_{{32}} \right] - \frac{{r(1 - r)}}{{2D}}\frac{{\partial D}}{{\partial r}} $$

(B3)

where

$$\frac{{\partial D}}{{\partial r}}=2\left\{ r(s_{{1,2}} + s_{{2,2}}) - (1 - r)(1 + s_{{1,2}} s_{{2,2}}) + (1 - 2r)\left[s_{{1,1}} (1 - s_{{2,1}} + s_{{2,2}}) + s_{{2,1}} (1 - s_{{1,1}} + s_{{1,2}})\right]\right\} $$

The four viability coefficients ${\ifmmode\expandafter\hat\else\expandafter\^\fi{s}_{{i,j}} } (i, j =1, 2)$ can be estimated simultaneously

$$\begin{aligned} \hat{s}_{{1,1}} &= \frac{{D(n_{{21}} + n_{{22}} + n_{{23}})}}{{2n\left[(1 + s_{{2,2}})r(1 - r) + s_{{2,1}} (1 - 2r + 2r^{2})\right]}}\;\hat{s}_{{1,2}} = \frac{{D(n_{{31}} + n_{{32}} + n_{{33}})}}{{n\left[r^{2} + 2r(1 - r)s_{{2,2}} + s_{{2,1}} (1 - r^{2})\right]}} \\ \hat{s}_{{2,1}} &= \frac{{D(n_{{12}} + n_{{22}} + n_{{32}})}}{{2n\left[(1 + s_{{1,2}})r(1 - r) + s_{{1,1}} (1 - 2r + 2r^{2})\right]}}\; \hat{s}_{{2,2}} = \frac{{D(n_{{13}} + n_{{23}} + n_{{33}})}}{{n\left[r^{2} + 2r(1 - r)s_{{1,2}} + s_{{1,1}} (1 - r^{2})\right]}} \\ \end{aligned} $$

(B4)

As for multiple viability loci, the viability coefficients are affected by its adjacent marker loci (i.e. the left and the right viability loci). Thus the coefficients of viability can be expressed as presented above.