Preservation of duplicate genes by originalization

Abstract

Neofunctionalization, subfunctionalization and increasing gene dosage were proposed to be the possible ways to explain duplicate-gene preservation in previous studies. However, in some natural populations, such as yeast Saccharomyces cerevisiae, a considerable proportion of the duplicate genes originated from ancient whole genomic duplication (WGD) is preserved till now, which cannot be sufficiently explained by these mechanisms. In this article, we present another possible way to explain this conundrum—originalization, by which duplicate genes are both preserved intact at a high frequency in the population under only purifying selection. With approximate equal rates of mutation at the two duplicated loci, analytical, numerical and simulation results consistently show that the mean time to nonfunctionalization for unlinked haploinsufficient gene duplication might become markedly prolonged, which results from originalization. These theoretical results imply that originalization might be an alternative effective and temporary way of preserving duplicate genes.

Appendix

Let x ₀, x ₁, x ₂, x ₃ be the frequencies of chromosomal haplotypes, “00”, “01”, “10” and “11”, respectively; and r be the recombination rate; mutation rates at duplicated loci are μ ₁ and μ ₂; for the DNR selective model, s ₁ = 0; for the HI model, s ₁ = 1. Fitness of individuals with various genotypes under the DNR and HI selective models are shown in Table 1. At every generation, mean population fitness and differential changes of chromosomal haplotype frequencies are given by

$$ w = 1 - x_{3}^{2} - 2s_{1} x_{1} x_{3} - 2s_{1} x_{2} x_{3} $$

(A1)

$$ \begin{gathered} x^{\prime}_{0} = \left[ {x_{0} x_{2} /2 + x_{1} x_{0} /2 + rx_{2} x_{1} /2} \right. \hfill \\ + x_{2} x_{0} /2 + rx_{1} x_{2} /2 + x_{0} x_{1} /2 \hfill \\ + \left( {1 - r} \right)x_{0} x_{3} /2 + \left( {1 - r} \right)x_{3} x_{0} /2 \hfill \\ \left. { + x_{0}^{2} } \right]/w - x_{0 } - \left( {\mu_{1} + \mu_{2} } \right)x_{0} \hfill \\ = \left( {x_{0} x_{3}^{2} + rx_{1} x_{2} - rx_{0} x_{3} + 2s_{1} x_{0} x_{1} x_{3} } \right. \hfill \\ \left. { + 2s_{1} x_{0} x_{2} x_{3} } \right)/w - \left( {\mu_{1} + \mu_{2} } \right)x_{0} \hfill \\ \end{gathered} $$

$$ \begin{gathered} x^{\prime}_{1} = \left[ {\left( {1 - s_{1} } \right)x_{3} x_{1} /2 + x_{1} x_{0} /2} \right. \hfill \\ + \left( {1 - r} \right)x_{2} x_{1} /2 + \left( {1 - r} \right)x_{1} x_{2} /2 \hfill \\ + x_{0} x_{1} /2 + rx_{0} x_{3} /2 + \left( {1 - s_{1} } \right)x_{1} x_{3} /2 \hfill \\ \left. { + rx_{3} x_{0} /2 + x_{1}^{2} } \right]/w - x_{1} + \mu_{1} x_{0} - \mu_{2} x_{1} \hfill \\ = \left( {x_{1} x_{3}^{2} - rx_{1} x_{2} + rx_{0} x_{3} - s_{1} x_{1} x_{3} } \right. \hfill \\ \left. { + 2s_{1} x_{1}^{2} x_{3} + 2s_{1} x_{1} x_{2} x_{3} } \right)/w + \mu_{1} x_{0} - \mu_{2} x_{1} \hfill \\ \end{gathered} $$

$$ \begin{gathered} x^{\prime}_{2} = \left[ {x_{0} x_{2} /2 + \left( {1 - r} \right)x_{2} x_{1} /2 + x_{2} x_{0} /2} \right. \hfill \\ + \left( {1 - s_{1} } \right)x_{2} x_{3} /2 + \left( {1 - r} \right)x_{1} x_{2} /2 \hfill \\ + \left( {1 - s_{1} } \right)x_{3} x_{2} /2 + rx_{0} x_{3} /2 + x_{2}^{2} \hfill \\ \left. { + rx_{3} x_{0} /2} \right]/w - x_{2} + \mu_{2} x_{0} - \mu_{1} x_{2} \hfill \\ = \left( {x_{2} x_{3}^{2} - rx_{1} x_{2} + rx_{0} x_{3} - s_{1} x_{2} x_{3} } \right. \hfill \\ \left. { + 2s_{1} x_{1} x_{2} x_{3} + 2s_{1} x_{2}^{2} x_{3} } \right)/w + \mu_{2} x_{0} - \mu_{1} x_{2} \hfill \\ \end{gathered} $$

$$ \begin{gathered} x^{\prime}_{3} = \left[ {\left( {1 - s_{1} } \right)x_{3} x_{1} /2 + rx_{2} x_{1} /2 + \left( {1 - s_{1} } \right)x_{2} x_{3} /2} \right. \hfill \\ + rx_{1} x_{2} /2 + \left( {1 - s_{1} } \right)x_{3} x_{2} /2 + \left( {1 - r} \right)x_{0} x_{3} /2 \hfill \\ \left. { + \left( {1 - s_{1} } \right)x_{1} x_{3} /2 + \left( {1 - r} \right)x_{3} x_{0} /2} \right]/w - x_{3} \hfill \\ + \mu_{2} x_{1} + \mu_{1} x_{2} = \left( {rx_{1} x_{2} - rx_{0} x_{3} - s_{1} x_{1} x_{3} - s_{1} x_{2} x_{3} - x_{3}^{2} } \right. \hfill \\ \left. { + 2s_{1} x_{1} x_{3}^{2} + 2s_{1} x_{2} x_{3}^{2} + x_{3}^{3} } \right)/w + \mu_{2} x_{1} + \mu_{1} x_{2} . \hfill \\ \end{gathered} $$

(A2)

Because w ≈ 1, Eq. A2 can be approximately given by

$$ \begin{gathered} x^{\prime}_{0} \approx x_{0} x_{3}^{2} + rx_{1} x_{2} - rx_{0} x_{3} + 2s_{1} x_{0} x_{1} x_{3} \hfill \\ + 2s_{1} x_{0} x_{2} x_{3} - \left( {\mu_{1} + \mu_{2} } \right)x_{0} \hfill \\ x^{\prime}_{1} \approx x_{1} x_{3}^{2} - rx_{1} x_{2} + rx_{0} x_{3} - s_{1} x_{1} x_{3} \hfill \\ + 2s_{1} x_{1}^{2} x_{3} + 2s_{1} x_{1} x_{2} x_{3} + \mu_{1} x_{0} - \mu_{2} x_{1} \hfill \\ x^{\prime}_{2} \approx x_{2} x_{3}^{2} - rx_{1} x_{2} + rx_{0} x_{3} - s_{1} x_{2} x_{3} \hfill \\ + 2s_{1} x_{1} x_{2} x_{3} + 2s_{1} x_{2}^{2} x_{3} + \mu_{2} x_{0} - \mu_{1} x_{2} \hfill \\ x^{\prime}_{3} \approx rx_{1} x_{2} - rx_{0} x_{3} - s_{1} x_{1} x_{3} - s_{1} x_{2} x_{3} - x_{3}^{2} \hfill \\ + 2s_{1} x_{1} x_{3}^{2} + 2s_{1} x_{2} x_{3}^{2} + x_{3}^{3} + \mu_{2} x_{1} + \mu_{1} x_{2} . \hfill \\ \end{gathered} $$

Thus, Eq. 1 in the text is obtained.