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The analysis of QTL by simultaneous use of the full linkage map

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Abstract

An extension of interval mapping is presented that incorporates all intervals on the linkage map simultaneously. The approach uses a working model in which the sizes of putative QTL for all intervals across the genome are random effects. An outlier detection method is used to screen for possible QTL. Selected QTL are subsequently fitted as fixed effects. This screening and selection approach is repeated until the variance component for QTL sizes is not statistically significant. A comprehensive simulation study is conducted in which map uncertainty is included. The proposed method is shown to be superior to composite interval mapping in terms of power of detection of QTL. There is an increase in the rate of false positive QTL detected when using the new approach, but this rate decreases as the population size increases. The new approach is much simpler computationally. The analysis of flour milling yield in a doubled haploid population illustrates the improved power of detection of QTL using the approach, and also shows how vital it is to allow for sources of non-genetic variation in the analysis.

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Acknowledgments

We gratefully acknowledge the Grains Research and Development Corporation (GRDC) for support through Key Programme 3 of their National Statistics Project. We thank the Australian Winter Cereals Molecular Marker Program and it’s predecessor the National Wheat Molecular Marker Program, both funded by GRDC, for the flour milling yield data analysed in this paper. We are grateful to Simon Diffey, New South Wales Department of Primary Industries, for his excellent implementation of the approach using R and the qtl package. Lastly, we thank the Associate Editor and the referees whose comments have led to substantial improvements and clarifications being incorporated into the paper.

Author information

Authors and Affiliations

  1. School of Agriculture, Food and Wine, The University of Adelaide, PMB 1, Glen Osmond, SA, 5064, Australia

    Arūnas P. Verbyla

  2. Statistical Bioinformatics—Agribusiness, Mathematical and Information Sciences, CSIRO, PMB 2, Glen Osmond, SA, 5064, Australia

    Arūnas P. Verbyla

  3. Biometrics, Wagga Wagga Research Institute, New South Wales Department of Primary Industries, Wagga Wagga, NSW, 2650, Australia

    Brian R. Cullis

  4. School of Mathematical Sciences, Queen Mary College, University of London, Mile End Rd., London, E1 4NS, UK

    Robin Thompson

  5. Centre for Mathematical and Computational Biology, Department of Biomathematics and Bioinformatics, Rothamsted Research, Harpenden, Hertfordshire, AL5 2JQ, UK

    Robin Thompson

Authors
  1. Arūnas P. Verbyla
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  2. Brian R. Cullis
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  3. Robin Thompson
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Correspondence to Arūnas P. Verbyla.

Additional information

Communicated by J.-L. Jannink.

Appendix

Appendix

The expectation result (12) relies on Haldane’s mapping function (4). In terms of recombination frequencies,

$$ \theta_{k;j} = \frac{1}{2} (1-e^{-2d_{k;j}}), \quad 1-\theta_{k;j} = \frac{1}{2} (1+e^{-2d_{k;j}}), \quad 1-2\theta_{k;j} = e^{-2d_{k;j}} $$

Thus for example, on substituting for θ k;j in (7)

$$ \lambda_{k;j+1,j}(d_{k;j}) = \frac{1-2\theta_{k;j,j+1}} {\theta_{k;j,j+1}(1-\theta_{k;j,j+1})} \frac{e^{2d_{k;j}}-e^{-2d_{k;j}}}{4} $$

and hence assuming d k;j ∼ U[0,  d k;j,j+1] we find (using x as the dummy variable for integration of the distance)

$$ \begin{aligned} \hbox{E}\left({\lambda_{k;j+1,j}}\right) =& \int\limits_0^{d_{k;j,j+1}} \lambda_{k;j+1,j}(x) \frac{1}{d_{k;j,j+1}} dx\\ & = \frac{1-2\theta_{k;j,j+1}}{4d_{k;j,j+1}\theta_{k;j,j+1}(1-\theta_{k;j,j+1})} \left(\frac{e^{2d_{k;j,j+1}}}{2} + \frac{e^{-2d_{k;j,j+1}}}{2} - 1 \right)\\ & = \frac{1-2\theta_{k;j,j+1}}{4d_{k;j,j+1}\theta_{k;j,j+1}(1-\theta_{k;j,j+1})} \left(\frac{1}{2(1-2\theta_{k;j,j+1})} + \frac{1-2\theta_{k;j,j+1}}{2} - 1 \right)\\ &= \frac{\theta_{k;j,j+1}}{2d_{k;j,j+1}(1-\theta_{k;j,j+1})}\\ \end{aligned} $$

as given in (12). The result for λ k;j,j follows by symmetry or by repeating the integration process explicitly using (6).

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Verbyla, A.P., Cullis, B.R. & Thompson, R. The analysis of QTL by simultaneous use of the full linkage map. Theor Appl Genet 116, 95–111 (2007). https://doi.org/10.1007/s00122-007-0650-x

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