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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Simulation of evolution with strong interference.The figure shows data from simulated populations with mutation supply . The total genetic map length is  and mutations provide selective advantage . The baseline density of sweeps is , corresponding to interference strong enough that our approximation Eq. (8) for the rate of adaptation is beginning to break down. Top panels: Trajectories of 1000 example selective sweeps in a population of size  (left), and 713 sweeps in a population of size  (right). Frequencies are plotted on a logit scale, so that the deterministic trajectory in the absence of interference is a straight line (shown in black). While the distributions of trajectories differ between the two populations at very low and high frequencies, they are similar in the frequency range  (between the dashed lines) at which sweeps cause the most interference. For each sweep,  is set to be halfway between its origin and fixation, and time is scaled by . Most of the trajectories take longer to increase to high frequency than the deterministic trajectory in the absence of interference; on average, the sweeps are slowed down by interference. Most trajectories lie below frequency 1/2 at , i.e., they take longer to go from frequency  to 1/2 than from 1/2 to 1. At very low and high frequencies, the trajectories are dominated by drift and are far from the deterministic trajectory. At the intermediate frequencies at which they cause the most interference, most trajectories increase at a roughly steady rate, albeit more slowly than they would in the absence of interference. Bottom panel: Sojourn times (scaled by ) of the simulated sweeps shown in the top panels. Simulation results are compared to the distribution expected under the diffusion approximation with an effective population size of either the actual size, , or scaled by the reduction in fixation probability, . Points show mean sojourn times, while the error bars show the standard deviation of the sojourn time. (Note that this is not the standard error of the mean, which is smaller by a factor of .) The mean and standard deviation of the sojourn times at intermediate frequencies are approximately the same for  and . Strong interference greatly increases the variance in sojourn times. The mean increases as well, but by no more than a factor of two, much less than might be suggested by the 15-fold decrease in fixation probability. In contrast to the results in the absence of interference, the sojourn time distribution of the simulations is asymmetric about frequency 1/2. For the diffusion approximation, mean sojourn time is found from Eq. 5.53 of [50], and the standard deviation of the sojourn time is found from Eq. 27 of [105].
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pgen-1002740-g005: Simulation of evolution with strong interference.The figure shows data from simulated populations with mutation supply . The total genetic map length is and mutations provide selective advantage . The baseline density of sweeps is , corresponding to interference strong enough that our approximation Eq. (8) for the rate of adaptation is beginning to break down. Top panels: Trajectories of 1000 example selective sweeps in a population of size (left), and 713 sweeps in a population of size (right). Frequencies are plotted on a logit scale, so that the deterministic trajectory in the absence of interference is a straight line (shown in black). While the distributions of trajectories differ between the two populations at very low and high frequencies, they are similar in the frequency range (between the dashed lines) at which sweeps cause the most interference. For each sweep, is set to be halfway between its origin and fixation, and time is scaled by . Most of the trajectories take longer to increase to high frequency than the deterministic trajectory in the absence of interference; on average, the sweeps are slowed down by interference. Most trajectories lie below frequency 1/2 at , i.e., they take longer to go from frequency to 1/2 than from 1/2 to 1. At very low and high frequencies, the trajectories are dominated by drift and are far from the deterministic trajectory. At the intermediate frequencies at which they cause the most interference, most trajectories increase at a roughly steady rate, albeit more slowly than they would in the absence of interference. Bottom panel: Sojourn times (scaled by ) of the simulated sweeps shown in the top panels. Simulation results are compared to the distribution expected under the diffusion approximation with an effective population size of either the actual size, , or scaled by the reduction in fixation probability, . Points show mean sojourn times, while the error bars show the standard deviation of the sojourn time. (Note that this is not the standard error of the mean, which is smaller by a factor of .) The mean and standard deviation of the sojourn times at intermediate frequencies are approximately the same for and . Strong interference greatly increases the variance in sojourn times. The mean increases as well, but by no more than a factor of two, much less than might be suggested by the 15-fold decrease in fixation probability. In contrast to the results in the absence of interference, the sojourn time distribution of the simulations is asymmetric about frequency 1/2. For the diffusion approximation, mean sojourn time is found from Eq. 5.53 of [50], and the standard deviation of the sojourn time is found from Eq. 27 of [105].

Mentions: Since our analytical approximation Eq. (8) become inaccurate for very strong interference, we further investigated this regime by simulation. Figure 5 shows the results of a typical simulation run with parameters chosen such that there is very strong interference: , , . In the absence of interference, the fixation probability would be , slightly lower than the weak-selection approximation of , so the density of sweeps would be . In the simulations, interference reduces the average fixation probability to , which is roughly twice as large as the fixation probability predicted from Eq. (8) . Our analytical approximations are thus beginning to break down, but the general features are still roughly correct. In particular, our basic assumption that alleles are safe from loss once they reach appreciable frequency is still true. For these parameters, loss becomes unlikely once the number of copies exceeds , which is well below the frequencies at which the allele begins to interfere with others for . Our scaling argument assumes not only that common alleles are certain to be fixed, but also that their trajectory on the way to fixation is affected by interference in a way that depends only on the density of sweeps, . Figure 5 shows that this assumption is roughly accurate even at high interference; the distributions of sweep trajectories and sojourn times between 10% frequency and 90% frequency (the range in which sweeps cause the most interference) are similar for , and , .


Limits to the rate of adaptive substitution in sexual populations.

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

Simulation of evolution with strong interference.The figure shows data from simulated populations with mutation supply . The total genetic map length is  and mutations provide selective advantage . The baseline density of sweeps is , corresponding to interference strong enough that our approximation Eq. (8) for the rate of adaptation is beginning to break down. Top panels: Trajectories of 1000 example selective sweeps in a population of size  (left), and 713 sweeps in a population of size  (right). Frequencies are plotted on a logit scale, so that the deterministic trajectory in the absence of interference is a straight line (shown in black). While the distributions of trajectories differ between the two populations at very low and high frequencies, they are similar in the frequency range  (between the dashed lines) at which sweeps cause the most interference. For each sweep,  is set to be halfway between its origin and fixation, and time is scaled by . Most of the trajectories take longer to increase to high frequency than the deterministic trajectory in the absence of interference; on average, the sweeps are slowed down by interference. Most trajectories lie below frequency 1/2 at , i.e., they take longer to go from frequency  to 1/2 than from 1/2 to 1. At very low and high frequencies, the trajectories are dominated by drift and are far from the deterministic trajectory. At the intermediate frequencies at which they cause the most interference, most trajectories increase at a roughly steady rate, albeit more slowly than they would in the absence of interference. Bottom panel: Sojourn times (scaled by ) of the simulated sweeps shown in the top panels. Simulation results are compared to the distribution expected under the diffusion approximation with an effective population size of either the actual size, , or scaled by the reduction in fixation probability, . Points show mean sojourn times, while the error bars show the standard deviation of the sojourn time. (Note that this is not the standard error of the mean, which is smaller by a factor of .) The mean and standard deviation of the sojourn times at intermediate frequencies are approximately the same for  and . Strong interference greatly increases the variance in sojourn times. The mean increases as well, but by no more than a factor of two, much less than might be suggested by the 15-fold decrease in fixation probability. In contrast to the results in the absence of interference, the sojourn time distribution of the simulations is asymmetric about frequency 1/2. For the diffusion approximation, mean sojourn time is found from Eq. 5.53 of [50], and the standard deviation of the sojourn time is found from Eq. 27 of [105].
© Copyright Policy
Related In: Results  -  Collection

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

pgen-1002740-g005: Simulation of evolution with strong interference.The figure shows data from simulated populations with mutation supply . The total genetic map length is and mutations provide selective advantage . The baseline density of sweeps is , corresponding to interference strong enough that our approximation Eq. (8) for the rate of adaptation is beginning to break down. Top panels: Trajectories of 1000 example selective sweeps in a population of size (left), and 713 sweeps in a population of size (right). Frequencies are plotted on a logit scale, so that the deterministic trajectory in the absence of interference is a straight line (shown in black). While the distributions of trajectories differ between the two populations at very low and high frequencies, they are similar in the frequency range (between the dashed lines) at which sweeps cause the most interference. For each sweep, is set to be halfway between its origin and fixation, and time is scaled by . Most of the trajectories take longer to increase to high frequency than the deterministic trajectory in the absence of interference; on average, the sweeps are slowed down by interference. Most trajectories lie below frequency 1/2 at , i.e., they take longer to go from frequency to 1/2 than from 1/2 to 1. At very low and high frequencies, the trajectories are dominated by drift and are far from the deterministic trajectory. At the intermediate frequencies at which they cause the most interference, most trajectories increase at a roughly steady rate, albeit more slowly than they would in the absence of interference. Bottom panel: Sojourn times (scaled by ) of the simulated sweeps shown in the top panels. Simulation results are compared to the distribution expected under the diffusion approximation with an effective population size of either the actual size, , or scaled by the reduction in fixation probability, . Points show mean sojourn times, while the error bars show the standard deviation of the sojourn time. (Note that this is not the standard error of the mean, which is smaller by a factor of .) The mean and standard deviation of the sojourn times at intermediate frequencies are approximately the same for and . Strong interference greatly increases the variance in sojourn times. The mean increases as well, but by no more than a factor of two, much less than might be suggested by the 15-fold decrease in fixation probability. In contrast to the results in the absence of interference, the sojourn time distribution of the simulations is asymmetric about frequency 1/2. For the diffusion approximation, mean sojourn time is found from Eq. 5.53 of [50], and the standard deviation of the sojourn time is found from Eq. 27 of [105].
Mentions: Since our analytical approximation Eq. (8) become inaccurate for very strong interference, we further investigated this regime by simulation. Figure 5 shows the results of a typical simulation run with parameters chosen such that there is very strong interference: , , . In the absence of interference, the fixation probability would be , slightly lower than the weak-selection approximation of , so the density of sweeps would be . In the simulations, interference reduces the average fixation probability to , which is roughly twice as large as the fixation probability predicted from Eq. (8) . Our analytical approximations are thus beginning to break down, but the general features are still roughly correct. In particular, our basic assumption that alleles are safe from loss once they reach appreciable frequency is still true. For these parameters, loss becomes unlikely once the number of copies exceeds , which is well below the frequencies at which the allele begins to interfere with others for . Our scaling argument assumes not only that common alleles are certain to be fixed, but also that their trajectory on the way to fixation is affected by interference in a way that depends only on the density of sweeps, . Figure 5 shows that this assumption is roughly accurate even at high interference; the distributions of sweep trajectories and sojourn times between 10% frequency and 90% frequency (the range in which sweeps cause the most interference) are similar for , and , .

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