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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Effect of interference on distribution of successful mutations.Solid curves and points show the probability of fixation of a mutation as a function of its selective coefficient, . Histograms and dashed curves show the distribution of selective coefficients of fixed mutations. The left panel shows results for moderate interference (), while the right panel shows high interference (). Mutations with small effects are strongly affected by interference, while large-effect mutations are nearly unaffected; this biases the distribution of successful mutations towards larger effects. The distribution of mutational effects, , is exponential with mean . Solid curves show the analytical approximation Eq. (12), corrected for the effect of unlinked loci and the saturation of fixation probability as  approaches 1 (see Text S4). Dashed curves show the predicted distribution of selective coefficients of fixed mutations in the absence of interference, , with  set to the width of the histogram bins. Parameters are , , and . Points and histograms are averages over 5000 simulated generations; error bars show the standard error. Only a few mutations in the simulated populations had very high values of , so the estimated probabilities of fixation for these high values are noisy. Note that the horizontal scales of the left and right panels are different.
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pgen-1002740-g008: Effect of interference on distribution of successful mutations.Solid curves and points show the probability of fixation of a mutation as a function of its selective coefficient, . Histograms and dashed curves show the distribution of selective coefficients of fixed mutations. The left panel shows results for moderate interference (), while the right panel shows high interference (). Mutations with small effects are strongly affected by interference, while large-effect mutations are nearly unaffected; this biases the distribution of successful mutations towards larger effects. The distribution of mutational effects, , is exponential with mean . Solid curves show the analytical approximation Eq. (12), corrected for the effect of unlinked loci and the saturation of fixation probability as approaches 1 (see Text S4). Dashed curves show the predicted distribution of selective coefficients of fixed mutations in the absence of interference, , with set to the width of the histogram bins. Parameters are , , and . Points and histograms are averages over 5000 simulated generations; error bars show the standard error. Only a few mutations in the simulated populations had very high values of , so the estimated probabilities of fixation for these high values are noisy. Note that the horizontal scales of the left and right panels are different.

Mentions: Figure 8 shows that alleles with small selective advantages are much more affected by interference than those with large selective advantages. To understand this, consider the probability of fixation of an allele with advantage , , given the distribution of mutational effects. (For the exponential distribution we consider here, .) If the effects of multiple interfering sweeps are additive, then following the argument given in Text S4 , we can write the probability of fixation as(10)where the factor depends only on the ratio of the selective coefficients. Eq. (10) approaches 0 at some ; alleles with selection coefficients are nearly unaffected by interference, while those with lower are strongly affected. (Obviously, the Eq. (10) only applies to values of above this cutoff ; we discuss weakly-selected alleles below.) can be understood as the rate at which the focal allele is knocked back by interfering sweeps [79].


Limits to the rate of adaptive substitution in sexual populations.

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

Effect of interference on distribution of successful mutations.Solid curves and points show the probability of fixation of a mutation as a function of its selective coefficient, . Histograms and dashed curves show the distribution of selective coefficients of fixed mutations. The left panel shows results for moderate interference (), while the right panel shows high interference (). Mutations with small effects are strongly affected by interference, while large-effect mutations are nearly unaffected; this biases the distribution of successful mutations towards larger effects. The distribution of mutational effects, , is exponential with mean . Solid curves show the analytical approximation Eq. (12), corrected for the effect of unlinked loci and the saturation of fixation probability as  approaches 1 (see Text S4). Dashed curves show the predicted distribution of selective coefficients of fixed mutations in the absence of interference, , with  set to the width of the histogram bins. Parameters are , , and . Points and histograms are averages over 5000 simulated generations; error bars show the standard error. Only a few mutations in the simulated populations had very high values of , so the estimated probabilities of fixation for these high values are noisy. Note that the horizontal scales of the left and right panels are different.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3369949&req=5

pgen-1002740-g008: Effect of interference on distribution of successful mutations.Solid curves and points show the probability of fixation of a mutation as a function of its selective coefficient, . Histograms and dashed curves show the distribution of selective coefficients of fixed mutations. The left panel shows results for moderate interference (), while the right panel shows high interference (). Mutations with small effects are strongly affected by interference, while large-effect mutations are nearly unaffected; this biases the distribution of successful mutations towards larger effects. The distribution of mutational effects, , is exponential with mean . Solid curves show the analytical approximation Eq. (12), corrected for the effect of unlinked loci and the saturation of fixation probability as approaches 1 (see Text S4). Dashed curves show the predicted distribution of selective coefficients of fixed mutations in the absence of interference, , with set to the width of the histogram bins. Parameters are , , and . Points and histograms are averages over 5000 simulated generations; error bars show the standard error. Only a few mutations in the simulated populations had very high values of , so the estimated probabilities of fixation for these high values are noisy. Note that the horizontal scales of the left and right panels are different.
Mentions: Figure 8 shows that alleles with small selective advantages are much more affected by interference than those with large selective advantages. To understand this, consider the probability of fixation of an allele with advantage , , given the distribution of mutational effects. (For the exponential distribution we consider here, .) If the effects of multiple interfering sweeps are additive, then following the argument given in Text S4 , we can write the probability of fixation as(10)where the factor depends only on the ratio of the selective coefficients. Eq. (10) approaches 0 at some ; alleles with selection coefficients are nearly unaffected by interference, while those with lower are strongly affected. (Obviously, the Eq. (10) only applies to values of above this cutoff ; we discuss weakly-selected alleles below.) can be understood as the rate at which the focal allele is knocked back by interfering sweeps [79].

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