Limits...
Limits to the rate of adaptive substitution in sexual populations.

Weissman DB, Barton NH - PLoS Genet. (2012)

Bottom Line: Heritable variance v in log fitness due to unlinked loci reduces Λ by e⁻⁴(v) under polygamy and e⁻⁸ (v) under monogamy.We also consider the effect of sweeps on neutral diversity and show that, while even occasional sweeps can greatly reduce neutral diversity, this effect saturates as sweeps become more common-diversity can be maintained even in populations experiencing very strong interference.Our results indicate that for some organisms the rate of adaptive substitution may be primarily recombination-limited, depending only weakly on the mutation supply and the strength of selection.

View Article: PubMed Central - PubMed

Affiliation: Institute of Science and Technology Austria, Klosterneuburg, Austria. dbw@ist.ac.at

ABSTRACT
In large populations, many beneficial mutations may be simultaneously available and may compete with one another, slowing adaptation. By finding the probability of fixation of a favorable allele in a simple model of a haploid sexual population, we find limits to the rate of adaptive substitution, Λ, that depend on simple parameter combinations. When variance in fitness is low and linkage is loose, the baseline rate of substitution is Λ₀ = 2NU , where N is the population size, U is the rate of beneficial mutations per genome, and is their mean selective advantage. Heritable variance v in log fitness due to unlinked loci reduces Λ by e⁻⁴(v) under polygamy and e⁻⁸ (v) under monogamy. With a linear genetic map of length R Morgans, interference is yet stronger. We use a scaling argument to show that the density of adaptive substitutions depends on s, N, U, and R only through the baseline density: Λ/R = F (Λ₀/R). Under the approximation that the interference due to different sweeps adds up, we show that Λ/R ~(Λ₀/R) / (1 +2Λ₉/R) , implying that interference prevents the rate of adaptive substitution from exceeding one per centimorgan per 200 generations. Simulations and numerical calculations confirm the scaling argument and confirm the additive approximation for Λ₀/R ~ 1; for higher Λ₀/R , the rate of adaptation grows above R/2, but only very slowly. We also consider the effect of sweeps on neutral diversity and show that, while even occasional sweeps can greatly reduce neutral diversity, this effect saturates as sweeps become more common-diversity can be maintained even in populations experiencing very strong interference. Our results indicate that for some organisms the rate of adaptive substitution may be primarily recombination-limited, depending only weakly on the mutation supply and the strength of selection.

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The density of sweeps as a function of the baseline density.The rate of sweeps per unit map length , plotted against the baseline rate, . The solid line shows , the dashed curve shows the additive approximation given by the solution to Eq. (8) , and the points show simulation results. Different kinds of points represent different values of ; as predicted by the scaling argument,  depends on  only through .  until interference becomes strong at , after which  increases only slowly. While the simulated values of  continue to increase above Eq. (8) 's “upper limit” of 0.5, they do so only very slowly, remaining  even for . (Note that even when Eq. (8) underestimates , it appears that our scaling argument still holds.) Selection and map length are held constant at  and  while population size  and mutation rate  are varied. The points show simulation results averaged over  generations for  (circles),  (squares),  (diamonds),  (upward-pointing triangles), and  (downward-pointing triangles). For each value of , values of  are shown up the point at which the strength of interference at which the probability of fixation falls to  and the neutral accumulation of mutations becomes important (see Figure S7). The standard errors in the simulation results are less than the size of the points.
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pgen-1002740-g004: The density of sweeps as a function of the baseline density.The rate of sweeps per unit map length , plotted against the baseline rate, . The solid line shows , the dashed curve shows the additive approximation given by the solution to Eq. (8) , and the points show simulation results. Different kinds of points represent different values of ; as predicted by the scaling argument, depends on only through . until interference becomes strong at , after which increases only slowly. While the simulated values of continue to increase above Eq. (8) 's “upper limit” of 0.5, they do so only very slowly, remaining even for . (Note that even when Eq. (8) underestimates , it appears that our scaling argument still holds.) Selection and map length are held constant at and while population size and mutation rate are varied. The points show simulation results averaged over generations for (circles), (squares), (diamonds), (upward-pointing triangles), and (downward-pointing triangles). For each value of , values of are shown up the point at which the strength of interference at which the probability of fixation falls to and the neutral accumulation of mutations becomes important (see Figure S7). The standard errors in the simulation results are less than the size of the points.

Mentions: We still face a difficulty, however, in that the locations and times of sweeps are not independent: because the amount of interference varies stochastically over the genome and through time, we expect them to be overdispersed. The scaling argument will still hold if the effects of different sweeps add up (the approximation developed below), or if the distribution in scaled time and map length is non-uniform but still depends on the population parameters only through . We show by simulation that the heuristic scaling argument is in fact accurate (Figure 3 and Figure 4), and that distribution of sweeps is close to uniform even for very strong interference (Figure 2). This may seem somewhat puzzling – sweeps should preferentially begin at loci and times that are experiencing less interference. However, when sweeps are rare, most of the genome experiences almost no interference in most generations, and thus little variation in the amount of interference. Conversely, when sweeps are common, most of the genome experiences substantial interference from multiple sweeps in most generations, and the stochastic variations in the amount of interference experienced from locus to locus and generation to generation are small compared to this average effect.


Limits to the rate of adaptive substitution in sexual populations.

Weissman DB, Barton NH - PLoS Genet. (2012)

The density of sweeps as a function of the baseline density.The rate of sweeps per unit map length , plotted against the baseline rate, . The solid line shows , the dashed curve shows the additive approximation given by the solution to Eq. (8) , and the points show simulation results. Different kinds of points represent different values of ; as predicted by the scaling argument,  depends on  only through .  until interference becomes strong at , after which  increases only slowly. While the simulated values of  continue to increase above Eq. (8) 's “upper limit” of 0.5, they do so only very slowly, remaining  even for . (Note that even when Eq. (8) underestimates , it appears that our scaling argument still holds.) Selection and map length are held constant at  and  while population size  and mutation rate  are varied. The points show simulation results averaged over  generations for  (circles),  (squares),  (diamonds),  (upward-pointing triangles), and  (downward-pointing triangles). For each value of , values of  are shown up the point at which the strength of interference at which the probability of fixation falls to  and the neutral accumulation of mutations becomes important (see Figure S7). The standard errors in the simulation results are less than the size of the points.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3369949&req=5

pgen-1002740-g004: The density of sweeps as a function of the baseline density.The rate of sweeps per unit map length , plotted against the baseline rate, . The solid line shows , the dashed curve shows the additive approximation given by the solution to Eq. (8) , and the points show simulation results. Different kinds of points represent different values of ; as predicted by the scaling argument, depends on only through . until interference becomes strong at , after which increases only slowly. While the simulated values of continue to increase above Eq. (8) 's “upper limit” of 0.5, they do so only very slowly, remaining even for . (Note that even when Eq. (8) underestimates , it appears that our scaling argument still holds.) Selection and map length are held constant at and while population size and mutation rate are varied. The points show simulation results averaged over generations for (circles), (squares), (diamonds), (upward-pointing triangles), and (downward-pointing triangles). For each value of , values of are shown up the point at which the strength of interference at which the probability of fixation falls to and the neutral accumulation of mutations becomes important (see Figure S7). The standard errors in the simulation results are less than the size of the points.
Mentions: We still face a difficulty, however, in that the locations and times of sweeps are not independent: because the amount of interference varies stochastically over the genome and through time, we expect them to be overdispersed. The scaling argument will still hold if the effects of different sweeps add up (the approximation developed below), or if the distribution in scaled time and map length is non-uniform but still depends on the population parameters only through . We show by simulation that the heuristic scaling argument is in fact accurate (Figure 3 and Figure 4), and that distribution of sweeps is close to uniform even for very strong interference (Figure 2). This may seem somewhat puzzling – sweeps should preferentially begin at loci and times that are experiencing less interference. However, when sweeps are rare, most of the genome experiences almost no interference in most generations, and thus little variation in the amount of interference. Conversely, when sweeps are common, most of the genome experiences substantial interference from multiple sweeps in most generations, and the stochastic variations in the amount of interference experienced from locus to locus and generation to generation are small compared to this average effect.

Bottom Line: Heritable variance v in log fitness due to unlinked loci reduces Λ by e⁻⁴(v) under polygamy and e⁻⁸ (v) under monogamy.We also consider the effect of sweeps on neutral diversity and show that, while even occasional sweeps can greatly reduce neutral diversity, this effect saturates as sweeps become more common-diversity can be maintained even in populations experiencing very strong interference.Our results indicate that for some organisms the rate of adaptive substitution may be primarily recombination-limited, depending only weakly on the mutation supply and the strength of selection.

View Article: PubMed Central - PubMed

Affiliation: Institute of Science and Technology Austria, Klosterneuburg, Austria. dbw@ist.ac.at

ABSTRACT
In large populations, many beneficial mutations may be simultaneously available and may compete with one another, slowing adaptation. By finding the probability of fixation of a favorable allele in a simple model of a haploid sexual population, we find limits to the rate of adaptive substitution, Λ, that depend on simple parameter combinations. When variance in fitness is low and linkage is loose, the baseline rate of substitution is Λ₀ = 2NU , where N is the population size, U is the rate of beneficial mutations per genome, and is their mean selective advantage. Heritable variance v in log fitness due to unlinked loci reduces Λ by e⁻⁴(v) under polygamy and e⁻⁸ (v) under monogamy. With a linear genetic map of length R Morgans, interference is yet stronger. We use a scaling argument to show that the density of adaptive substitutions depends on s, N, U, and R only through the baseline density: Λ/R = F (Λ₀/R). Under the approximation that the interference due to different sweeps adds up, we show that Λ/R ~(Λ₀/R) / (1 +2Λ₉/R) , implying that interference prevents the rate of adaptive substitution from exceeding one per centimorgan per 200 generations. Simulations and numerical calculations confirm the scaling argument and confirm the additive approximation for Λ₀/R ~ 1; for higher Λ₀/R , the rate of adaptation grows above R/2, but only very slowly. We also consider the effect of sweeps on neutral diversity and show that, while even occasional sweeps can greatly reduce neutral diversity, this effect saturates as sweeps become more common-diversity can be maintained even in populations experiencing very strong interference. Our results indicate that for some organisms the rate of adaptive substitution may be primarily recombination-limited, depending only weakly on the mutation supply and the strength of selection.

Show MeSH
Related in: MedlinePlus