Table 1.
Symbol definitions.
Figure 1.
A selective sweep causes interference over a time and a genetic distance
.
Fixation probability of a new mutation with advantage occurring after an interfering sweep with the same selective advantage
. The fixation probability
, scaled by its baseline value
, is plotted against the scaled map position of the new mutation relative to the interfering sweep,
, and its scaled time of occurrence relative to the time at which the interfering sweep reaches frequency
,
. Note that the relationship between these scaled variables is independent of
, as long as
. The X marks the time when the interfering sweep is at frequency
for
; it is assumed to follow a deterministic trajectory. The sweep causes the most interference once it becomes common (frequency
), and causes little interference to common alleles (i.e., alleles that arise around the same time or earlier).
is calculated numerically using Eqs. (2) and (3) .
Figure 2.
The distribution of sweeps in time across the genome.
Points show the beginnings of simulated selective sweeps. The distribution over time and map length appears approximately uniform. Time is in generations from the beginning of the simulation, and position is map distance in Morgans from the end of the chromosome. In the right panel, the time scale is halved and the length scale is doubled compared to the left panel, illustrating the effect of a doubling of on the scaled distribution of sweeps that enters into Eq. (4) for the scaled probability of fixation
. If we consider a focal mutation occurring in the middle of the chromosome at generation 2500 (the large gold dot), the rescaling changes the interference it experiences from any given sweep (e.g., the one marked by the large purple dot), but the total expected interference from the whole distribution of sweeps remains unchanged. Simulation parameters are chosen such that there is strong interference:
,
,
,
.
Figure 3.
Reduction in fixation probability only depends on baseline density of sweeps.
The scaled probability of fixation of a beneficial mutation, , plotted as a function of the strength of selection,
.
is varied along with
, so that the ratio
(and therefore
) is held constant. Circles show simulation results and curves show the analytical approximation given by Eq. (8) . The scaled probability of fixation is nearly constant until
becomes large enough that unlinked sweeps become important
.
,
is shown in purple;
,
is shown in gold;
,
is shown in blue.
for all points and curves. Note that for
, Eq. (8) slightly overestimates the amount of interference, because the chromosome is short enough that boundary effects must be considered. All simulations were run until the rate of substitution approached a steady value, and then continued until at least 1000 substitutions accumulated. The standard error is less than the radius of the points.
Figure 4.
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.
Figure 5.
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].
Figure 6.
Differing effects of sweeps on selected and neutral alleles.
The scaled fixation probability of beneficial alleles and scaled neutral diversity as a function of the baseline density of sweeps . Points show simulation results, curves show analytical approximations. The circles and the black curve are the scaled fixation probability
, and show the same data as in Figure 4. The squares and colored curves show the scaled neutral diversity,
. At small
, beneficial alleles do not interfere with each other, but still reduce neutral diversity substantially. However, increasing
to larger values has little additional effect on neutral diversity, both because interference limits the increase in the number of sweeps (
decreases), and because the combined effect of overlapping sweeps on neutral diversity is less than the sum of their individual effects (the squares lie above the additive analytical approximation). The analytical approximations match the simulation results up to strong interference (
), at which point they begin to break down. The squares are the averages over 100 simulation runs; see the Methods for how
was measured. The colored curves show Eq. (9) for
as a function of
, with
taken empirically from the simulations. The mutation rate
is varied, with other parameters held constant at
,
, and
. For these parameter values, essentially all interference is caused by tightly-linked loci.
Figure 7.
Effect of interference among alleles with a distribution of selective advantages.
Simulation results for scaled mean probability of fixation for mutations with exponentially distributed selective advantages (blue circles) and scaled mean selective advantage for successful mutations
(green diamonds), as a function of the baseline density of sweeps
– i.e., the amount of interference. The purple squares shows
for the same parameter values, but with all mutations conferring an identical selective advantage
. Allowing for a distribution of selective effects makes little difference in the rate of sweeps,
, and the mean selective advantage of sweeps stays close to
(dashed black line), even for strong interference. The theoretical predictions Eqs. (7) and (13) (purple and blue dashed curves, respectively) are accurate for weak interference, but underestimate fixation probability with strong interference. The mutation rate
is varied, with other parameters held constant at
,
, and mean selective advantage provided by a mutation
. All points are averages over 5000 simulated generations. Error bars on the top curve show the standard deviation of
for successful mutations. The standard errors are less than the size of the points.
Figure 8.
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.