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Method for the mapping of a female partial-sterile locus on a molecular marker linkage map

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Abstract

The female gametophyte is an absolutely essential structure for angiosperm reproduction, and female sterility has been reported in a number of crops. In this paper, a maximum-likelihood method is presented for estimating the position and effect of a female partial-sterile locus in a backcross population using the observed data of dominant or codominant markers. The ML solutions are obtained via Bailey’s method. The process for the estimating of the recombination fractions and the viabilities of female gametes are described, and the variances of the estimates of the parameters are also presented. Application of the method is demonstrated using a set of simulated data. This method circumvents the problems of the traditional mapping methods for female sterile genes which were based on data from seed set or embryo-sac morphology and anatomy.

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Authors and Affiliations

  1. Faculty of Life Science, Hubei University, 430062, Wuhan, China

    Jianguo Chen

  2. Department of Ecology and Evolutionary Biology, Biosciences West, University of Arizona, Tucson, AZ, 85721, USA

    Bruce Walsh

Authors
  1. Jianguo Chen
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  2. Bruce Walsh
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Corresponding author

Correspondence to Bruce Walsh.

Additional information

Communicated by M. Sillanpaa.

Appendix

Appendix

Derivation of the formula:

$$ L(r_{1} ,r_{2} ,t) = {\frac{n!}{{n_{1} !n_{2} !n_{3} !n_{4} !}}}p_{1}^{{n_{1} }} p_{2}^{{n_{2} }} p_{3}^{{n_{3} }} p_{4}^{{n_{4} }} $$
$$ \ln L = \ln {\frac{n!}{{n_{1} !n_{2} !n_{3} !n_{4} !}}} + n_{1} \ln p_{1} + n_{2} \ln p_{2} + n_{3} \ln p_{3} + n_{4} \ln p_{4} $$
$$ \begin{gathered} {\mathbf{X}} = (x_{ij} ) = \left[ \begin{gathered} {\frac{{\partial p_{1} }}{{\partial r_{1} }}}\quad {\frac{{\partial p_{1} }}{{\partial r_{2} }}}\quad {\frac{{\partial p_{1} }}{\partial t}} \hfill \\ {\frac{{\partial p_{2} }}{{\partial r_{1} }}}\quad {\frac{{\partial p_{2} }}{{\partial r_{2} }}}\quad {\frac{{\partial p_{2} }}{\partial t}} \hfill \\ {\frac{{\partial p_{3} }}{{\partial r_{1} }}}\quad {\frac{{\partial p_{3} }}{{\partial r_{2} }}}\quad {\frac{{\partial p_{3} }}{\partial t}} \hfill \\ {\frac{{\partial p_{4} }}{{\partial r_{1} }}}\quad {\frac{{\partial p_{4} }}{{\partial r_{2} }}}\quad {\frac{{\partial p_{4} }}{\partial t}} \hfill \\ \end{gathered} \right] \hfill \\ = \left[\begin{gathered} - (1 - r_{2} )(1 - t) + r_{2} t\quad - (1 - r_{1} )(1 - t) + r_{1} t\quad - (1 - r_{1} )(1 - r_{2} ) + r_{1} r_{2} \hfill \\ (1 - r_{2} )t - r_{2} (1 - t)\quad(1 - r_{1} )(1 - t) - r_{1} t\quad r_{1} (1 - r_{2} ) - (1 - r_{1} )r_{2} \hfill \\ (1 - r_{2} )(1 - t) - r_{2} t\quad (1 - r_{1} )t - r_{1} (1 - t)\quad - r_{1} (1 - r_{2} ) + (1 - r_{1} )r_{2} \hfill \\ - (1 - r_{2} )t + r_{2} (1 - t)\quad - (1 - r_{1} )t + r_{1} (1 - t)\quad (1 - r_{1} )(1 - r_{2} ) - r_{1} r_{2} \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered} $$
$$ S_{{r_{1} }} = {\frac{\partial \ln L}{{\partial r_{1} }}} = \left( {{\frac{{n_{1} }}{{p_{1} }}} - {\frac{{n_{3} }}{{p_{3} }}}} \right)x_{11} + \left( {{\frac{{n_{2} }}{{p_{2} }}} - {\frac{{n_{4} }}{{p_{4} }}}} \right)x_{21} $$
$$ S_{{r_{2} }} = {\frac{\partial \ln L}{{\partial r_{2} }}} = \left( {{\frac{{n_{1} }}{{p_{1} }}} - {\frac{{n_{2} }}{{p_{2} }}}} \right)x_{12} + \left( {{\frac{{n_{3} }}{{p_{3} }}} - {\frac{{n_{4} }}{{p_{4} }}}} \right)x_{32} $$
$$ S_{t} = {\frac{\partial \ln L}{\partial t}} = \left( {{\frac{{n_{1} }}{{p_{1} }}} - {\frac{{n_{4} }}{{p_{4} }}}} \right)x_{13} + \left( {{\frac{{n_{2} }}{{p_{2} }}} - {\frac{{n_{3} }}{{p_{3} }}}} \right)x_{23} $$
$$ {\frac{{\partial^{2} \ln L}}{{\partial r_{1} r_{1} }}} = - \left( {{\frac{{n_{1} }}{{p_{1}^{2} }}} + {\frac{{n_{3} }}{{p_{3}^{2} }}}} \right)x_{11}^{2} - \left( {{\frac{{n_{2} }}{{p_{2}^{2} }}} + {\frac{{n_{4} }}{{p_{4}^{2} }}}} \right)x_{21}^{2} $$
$$ {\frac{{\partial^{2} \ln L}}{{\partial r_{1} r_{2} }}} = \left( {{\frac{{n_{1} }}{{p_{1} }}} - {\frac{{n_{2} }}{{p_{2} }}} - {\frac{{n_{3} }}{{p_{3} }}} + {\frac{{n_{4} }}{{p_{4} }}}} \right) + \left( {n_{2} {\frac{{x_{21} }}{{p_{2}^{2} }}} - n_{1} {\frac{{x_{11} }}{{p_{1}^{2} }}}} \right)x_{12} + \left( {n_{3} {\frac{{x_{11} }}{{p_{3}^{2} }}} - n_{4} {\frac{{x_{21} }}{{p_{4}^{2} }}}} \right)x_{32} $$
$$ {\frac{{\partial^{2} \ln L}}{{\partial r_{1} t}}} = \left( {{\frac{{n_{1} }}{{p_{1} }}} + {\frac{{n_{2} }}{{p_{2} }}} - {\frac{{n_{3} }}{{p_{3} }}} - {\frac{{n_{4} }}{{p_{4} }}}} \right) - \left( {n_{1} {\frac{{x_{11} }}{{p_{1}^{2} }}} + n_{4} {\frac{{x_{21} }}{{p_{4}^{2} }}}} \right)x_{13} - \left( {n_{2} {\frac{{x_{21} }}{{p_{2}^{2} }}} + n_{3} {\frac{{x_{11} }}{{p_{3}^{2} }}}} \right)x_{23} $$
$$ {\frac{{\partial^{2} \ln L}}{{\partial r_{2} r_{1} }}} = \left( {{\frac{{n_{1} }}{{p_{1} }}} - {\frac{{n_{2} }}{{p_{2} }}} - {\frac{{n_{3} }}{{p_{3} }}} + {\frac{{n_{4} }}{{p_{4} }}}} \right) + \left( {n_{2} {\frac{{x_{21} }}{{p_{2}^{2} }}} - n_{1} {\frac{{x_{11} }}{{p_{1}^{2} }}}} \right)x_{12} + \left( {n_{3} {\frac{{x_{11} }}{{p_{3}^{2} }}} - n_{4} {\frac{{x_{21} }}{{p_{4}^{2} }}}} \right)x_{32} $$
$$ {\frac{{\partial^{2} \ln L}}{{\partial r_{2} r_{2} }}} = - \left( {{\frac{{n_{1} }}{{p_{1}^{2} }}} + {\frac{{n_{2} }}{{p_{2}^{2} }}}} \right)x_{12}^{2} - \left( {{\frac{{n_{3} }}{{p_{3}^{2} }}} + {\frac{{n_{4} }}{{p_{4}^{2} }}}} \right)x_{32}^{2} $$
$$ {\frac{{\partial^{2} \ln L}}{{\partial r_{2} t}}} = \left( {{\frac{{n_{1} }}{{p_{1} }}} - {\frac{{n_{2} }}{{p_{2} }}} + {\frac{{n_{3} }}{{p_{3} }}} - {\frac{{n_{4} }}{{p_{4} }}}} \right) - \left( {n_{1} {\frac{{x_{12} }}{{p_{1}^{2} }}} + n_{4} {\frac{{x_{32} }}{{p_{4}^{2} }}}} \right)x_{13} + \left( {n_{2} {\frac{{x_{12} }}{{p_{2}^{2} }}} + n_{3} {\frac{{x_{32} }}{{p_{3}^{2} }}}} \right)x_{23} $$
$$ {\frac{{\partial^{2} \ln L}}{{\partial tr_{1} }}} = \left( {{\frac{{n_{1} }}{{p_{1} }}} + {\frac{{n_{2} }}{{p_{2} }}} - {\frac{{n_{3} }}{{p_{3} }}} - {\frac{{n_{4} }}{{p_{4} }}}} \right) - \left( {n_{1} {\frac{{x_{11} }}{{p_{1}^{2} }}} + n_{4} {\frac{{x_{21} }}{{p_{4}^{2} }}}} \right)x_{13} - \left( {n_{2} {\frac{{x_{21} }}{{p_{2}^{2} }}} + n_{3} {\frac{{x_{11} }}{{p_{3}^{2} }}}} \right)x_{23} $$
$$ {\frac{{\partial^{2} \ln L}}{{\partial tr_{2} }}} = \left( {{\frac{{n_{1} }}{{p_{1} }}} - {\frac{{n_{2} }}{{p_{2} }}} + {\frac{{n_{3} }}{{p_{3} }}} - {\frac{{n_{4} }}{{p_{4} }}}} \right) - \left( {n_{1} {\frac{{x_{12} }}{{p_{1}^{2} }}} + n_{4} {\frac{{x_{32} }}{{p_{4}^{2} }}}} \right)x_{13} + \left( {n_{2} {\frac{{x_{12} }}{{p_{2}^{2} }}} + n_{3} {\frac{{x_{32} }}{{p_{3}^{2} }}}} \right)x_{23} $$
$$ {\frac{{\partial^{2} \ln L}}{\partial tt}} = - \left( {{\frac{{n_{1} }}{{p_{1}^{2} }}} + {\frac{{n_{4} }}{{p_{4}^{2} }}}} \right)x_{13}^{2} - \left( {{\frac{{n_{2} }}{{p_{2}^{2} }}} + {\frac{{n_{3} }}{{p_{3}^{2} }}}} \right)x_{23}^{2} $$

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Chen, J., Walsh, B. Method for the mapping of a female partial-sterile locus on a molecular marker linkage map. Theor Appl Genet 119, 1085–1091 (2009). https://doi.org/10.1007/s00122-009-1110-6

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