A hidden Markov model approach to multilocus linkage analysis in a full-sib family

Abstract

Statistical packages for constructing genetic linkage maps in inbred lines are well developed and applied extensively, while linkage analysis in outcrossing species faces some statistical challenges because of their complicated genetic structures. In this article, we present a multilocus linkage analysis via hidden Markov models for a linkage group of markers in a full-sib family. The advantage of this method is the simultaneous estimation of the recombination fractions between adjacent markers that possibly segregate in different ratios, and the calculation of likelihood for a certain order of the markers. When the number of markers decreases to two or three, the multilocus linkage analysis becomes traditional two-point or three-point linkage analysis, respectively. Monte Carlo simulations are performed to show that the recombination fraction estimates of multilocus linkage analysis are more accurate than those just using two-point linkage analysis and that the likelihood as an objective function for ordering maker loci is the most powerful method compared with other methods. By incorporating this multilocus linkage analysis, we have developed a Windows software, FsLinkageMap, for constructing genetic maps in a full-sib family. A real example is presented for illustrating linkage maps constructed by using mixed segregation markers. Our multilocus linkage analysis provides a powerful method for constructing high-density genetic linkage maps in some outcrossing plant species, especially in forest trees.

Appendix A

Following Amstrong's deriving procedure (Armstrong 2001), we define

$$ Q\left( {r,\;r\prime } \right) = \sum\limits_y {P\left( {\left. {y,\;O} \right|r} \right)\log P\left( {\left. {y,\;O} \right|r\prime } \right)} . $$

(10)

Then, we have

$$ \begin{gathered} Q\left( {r,\;r\prime } \right) = {E_r}\left[ {\log P\left( {\left. {y,\;O} \right|r\prime } \right)} \right] \hfill \\= {E_r}\left[ {\sum\limits_{k = 1}^n {\log P\left( {\left. {{y^k},\;{O^k}} \right|r\prime } \right)} } \right] \hfill \\= \sum\limits_{k = 1}^n {\sum\limits_{{y^k}} {P\left( {\left. {{y^k}} \right|{O^k},\;r} \right)\log P\left( {\left. {{y^k},\;{O^k}} \right|r\prime } \right)} } \hfill \\= \sum\limits_{k = 1}^n {\sum\limits_{{y^k}} {P\left( {\left. {y_1^k, \cdots, y_T^k} \right|{O^k},\;r} \right)\left[ {\log P\left( {y_1^k} \right) + \log P\left( {\left. {y_2^k} \right|y_1^k,r_1^\prime } \right) + \cdots } \right.} } \hfill \\\;\;\;\;\;\left. { + \log P\left( {\left. {y_T^k} \right|y_{T - 1}^k,r_{T - 1}^\prime } \right) + \log P\left( {\left. {O_1^k} \right|y_1^k} \right) + \cdots + \log P\left( {\left. {O_T^k} \right|y_T^k} \right)} \right] \hfill \\\end{gathered} $$

(11)

In terms of $r_t^\prime$, the above can be expressed as

$$ \begin{gathered} \sum\limits_{k = 1}^n {\sum\limits_{y_t^k,y_{t + 1}^k} {P\left( {\left. {y_t^k,y_{t + 1}^k} \right|{O^k},\;{r_t}} \right)\log P\left( {\left. {y_{t + 1}^k} \right|y_t^k,r_t^\prime } \right)} } \hfill \\= \sum\limits_{k = 1}^n {\sum\limits_{i,j = 1}^4 {P\left( {\left. {y_t^k = i,\;y_{t + 1}^k = j} \right|{O^k},\;{r_t}} \right)\log \left[ {r\prime_t^{{n_t}\left( {i,j} \right)}{{\left( {1 - r_t^\prime } \right)}^{2 - {n_t}\left( {i,j} \right)}}} \right]} } \hfill \\= \sum\limits_{k = 1}^n {\left[ {\sum\limits_{i,j} {\xi_t^k\left( {i,j} \right){n_t}\left( {i,j} \right)\log \frac{{r_t^\prime }}{{1 - r_t^\prime }}} + 2\log \left( {1 - r_t^\prime } \right)} \right]} \hfill \\\end{gathered} $$

(12)

By Baum's lemma, to maximum the likelihood (Eq. 4) is equivalent to maximizing (Eq. 12) with respect to $r_t^\prime$. Therefore, differentiating (12) and setting it to zero, we obtain the likelihood estimate:

$$ {\hat{r}_t} = \frac{1}{{2n}}\sum\limits_{k = 1}^n {\sum\limits_{i,j = 1}^4 {\xi_t^k\left( {i,j} \right){n_t}\left( {i,j} \right)} \;} $$

(13)