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The traveling wave approach to asexual evolution: Muller’s ratchet
and speed of adaptation
Igor M. Rouzinea,*, Éric Brunetb, and Claus O. Wilkec
a Department of Molecular Biology and Microbiology, Tufts University, 136 Harrison Avenue, Boston, MA
02111
b Laboratoire de Physique Statistique, École Normale Supérieure, 24 rue Lhomond, 75230 Paris Cédex 05,
France
c Section of Integrative Biology, Center for Computational Biology and Bioinformatics, and Institute for Cell
and Molecular Biology, University of Texas, Austin, TX 78712, USA
Abstract
We use traveling-wave theory to derive expressions for the rate of accumulation of deleterious
mutations under Muller’s ratchet and the speed of adaptation under positive selection in asexual
populations. Traveling-wave theory is a semi-deterministic description of an evolving population,
where the bulk of the population is modeled using deterministic equations, but the class of the highest-
fitness genotypes, whose evolution over time determines loss or gain of fitness in the population, is
given proper stochastic treatment. We derive improved methods to model the highest-fitness class
(the stochastic edge) for both Muller’s ratchet and adaptive evolution, and calculate analytic
correction terms that compensate for inaccuracies which arise when treating discrete fitness classes
as a continuum. We show that traveling wave theory makes excellent predictions for the rate of
mutation accumulation in the case of Muller’s ratchet, and makes good predictions for the speed of
adaptation in a very broad parameter range. We predict the adaptation rate to grow logarithmically
in the population size until the population size is extremely large.
1 Introduction
“I was observing the motion of a boat which was rapidly drawn along a narrow channel
by a pair of horses, when the boat suddenly stopped—not so the mass of water in the
channel which it had put in motion; it accumulated round the prow of the vessel in a
state of violent agitation, then suddenly leaving it behind, rolled forward with great
velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-
defined heap of water, which continued its course along the channel apparently
without change of form or diminution of speed.” (Russell, 1845)
One of the fundamental models of population genetics is that of a finite, asexually reproducing
population of genomes consisting of a large number of sites with multiplicative contribution
to the total fitness of the genome. This model has been studied for decades, and has presented
substantial challenges to researchers trying to solve it analytically. Even in its most basic
formulation, where each site is under exactly the same selective pressure, the model has not
*Corresponding author. email: igor.rouzine@tufts.edu. phone: 617-636-6759. fax: 617-636-4086.
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Published in final edited form as:
Theor Popul Biol. 2008 February ; 73(1): 24–46.
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been fully solved to this day. For the special case of vanishing back mutations, the model
reduces to the problem of Muller’s ratchet (Muller, 1964; Felsenstein, 1974). A tremendous
amount of research effort has been directed at this problem (Haigh, 1978; Pamilo et al.,
1987; Stephan et al., 1993; Higgs and Woodcock, 1995; Gordo and Charlesworth, 2000a,b;
Rouzine et al., 2003). Other special cases of this model are mutation–selection balance when
the forward and back mutation rates are equal (Woodcock and Higgs, 1996), and the speed of
adaptation under various conditions (Tsimring et al., 1996; Kessler et al., 1997; Gerrish and
Lenski, 1998; Orr, 2000; Rouzine et al., 2003; Wilke, 2004).
In 1996, Tsimring et al. pioneered a new approach to studying the multiplicative multi-site
model. They described the evolving population as a localized traveling wave in fitness space,
using partial differential equations developed to describe wave-like phenomena in physical
systems. A traveling wave is a localized profile traveling at near-constant speed and shape
(physicists refer to such phenomena also as solitary waves). We can envision a population as
a traveling wave of the distribution of the mutation number over genomes if the relative mutant
frequencies in the population stay approximately constant while the population shifts as a
whole. For example, a population may have specific abundances of sequences at one, two,
three, or more mutations away from the least loaded class at all times, but the least-loaded class
moves at constant speed by one mutation every ten generations.
Encouraging results by Tsimring et al. (1996) were based on two strong approximations.
Firstly, all fitness classes, including the best-fit class, were described deterministically,
neglecting random effects due to finite population size. Finite population size was introduced
into the problem as a cutoff of the effect of selection at the high-fitness edge, when the size of
a class becomes less than one copy of a genome. Secondly, Tsimring et al. (1996) approximated
the traveling wave profile with a function continuous in fitness (or mutation load). In these
approximations, Tsimring et al. (1996) demonstrated the existence of a continuous set of waves
with different speeds. The cutoff condition determined the choice of a specific solution and
the dependence of the speed on the population size.
Rouzine et al. (2003) confirmed the qualitative conclusions by Tsimring et al. (1996) and
refined their quantitative results in two ways, by taking into account the random effects acting
on the smallest, best-fit class, and showing that, in a broad parameter range, approximating the
logarithm of the wave profile as a smooth function of the fitness is a much better approximation
than approximating the wave profile itself as a smooth function. It was shown that the
substitution rate increases logarithmically with the population size, until the deterministic
single-site limit is reached at extremely large population sizes and the theory breaks down.
The purpose of the present paper is to show that the results by Rouzine et al. (2003) can still
be improved regarding both the treatment of the high-fitness edge and the deterministic part
of the fitness distribution. Because the original work was presented in an extremely condensed
format, we will here re-derive the general theory in detail. Then, we will present improved
treatments of the stochastic edge that lead to accurate predictions for the substitution rate
defined as the average gain in beneficial alleles per genome per generation. We consider in
detail two opposite parametric limits, Muller’s ratchet (when beneficial mutation events are
not important) and adaptive evolution (when deleterious mutations are not important).
Because the range of validity of our approach caused some confusion in the literature, we
discuss it in detail in the main text and Appendix. Briefly, both in Muller’s ratchet and in the
adaptation regime, we assume that the total number of sites is large, and the selection coefficient
s is small. The population size N should be sufficiently large, so that the difference in the
mutational load between the least-loaded and average genomes is much larger than 1. In other
words, a large number of sites are polymorphic at any time. For the adaptive evolution, the
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condition corresponds to the average substitution rate V being much larger than s/ln(V/Ub),
where Ub is the beneficial mutation rate, or population sizes being much larger than
s/Ub
adaptation occurs by isolated selective sweeps at different sites, and one-site models apply.
Two-site models of adaptation, such as the clonal interference theory (Gerrish and Lenski,
1998;Orr, 2000;Wilke, 2004), can be used to describe the narrow transitional interval in N.
Further, if the deleterious mutation rate per genome is much larger than s, and the average
population fitness is sufficiently high, an additional broad interval of N appears, where
deleterious mutations accumulate (Muller’s ratchet). Using numeric simulations and analytic
estimates, we demonstrate good accuracy of our results in a very broad parameter range relevant
for various asexual organisms, including asexual RNA and DNA viruses, yeast, some plants,
and fish.
3/ln (V/Ub). We also assume that V is much larger than Ub. In smaller populations,
The manuscript is organized as follows. In Section 2, we describe the model and the general
method to derive evolutionary dynamics in terms of the fitness distribution. In Section 3 and
4, we consider in detail two particular cases, Muller’s ratchet and the process of adaptation,
respectively, and test analytic results with computer simulations. In Section 4, we discuss our
findings.
2 Traveling-wave theory
2.1 Model assumptions
We consider a multi-site model of L sites, where each site can be in two states, i.e., carry one
of two alleles, either advantageous or deleterious. The deleterious allele reduces the overall
fitness of a genome by a factor of 1 − s, where s ≪1 is the selective disadvantage per site. We
assume that there are no biological interactions between sites (epistasis), so that the fitness of
a genome with k deleterious alleles (mutational load) is (1−s)k ≈e−ks. We refer to the frequency
of sequences with mutational load k in the population as fk, and write the population-average
fitness as wav = Σk e−sk fk. We introduce kav, which is mutational load for which a sequence’s
fitness is exactly equal to the population mean fitness, as given by wav = e−skav. For small s,
kav is approximately the average mutational load in the population, kav ≈ Σk kfk. We denote the
mutational load of the best-fit sequence in the population as k0. Note that in general k0 ≠ 0, that
is, the best-fit sequence in the population is not the sequence with the overall highest possible
fitness.
We assume that an allele can mutate into an opposite allele with a small probability μ. For the
sake of simplicity of the derivation, we assume that the mutation rate is low, so that there is,
at most, one mutation per genome per generation. The case when multiple mutations per
genome are frequent takes place at very large population sizes when the average substitution
rate is high, larger than one new beneficial allele per generation. Rouzine et al. (2003)
considered this more general case and showed that there is no essential change in the final
expression for the substitution rate. Thus, for finite population size, the assumption of a single
mutation per genome per round of replication is not limiting (see also the next subsection).
In real genomes, s varies between sites. Moreover, Gillespie (1983, 1991) and Orr (2003)
argued that the distribution of s differs between sites with beneficial and deleterious alleles. A
viable genome represents a highly-fit, non-random selection of alleles, so that deleterious
mutations should generally have larger effects than beneficial mutations. For the same reason,
these authors predicted that the effective distribution of s for beneficial alleles should have a
universal exponential form. In the present work, we do not consider variation in s. Instead, we
use a simplified model including only those sites into the total number of sites L —with either
deleterious and beneficial alleles— whose selection coefficient is on the order of the same
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typical value s, and approximate all selection coefficients at these sites with a constant s. The
choice of s and, hence, of the set of included sites depends on the time scale of evolution under
consideration. Strongly deleterious mutations with effects much larger than s are cleared
rapidly from a population. Strongly beneficial mutations are fixed at the early stages of
evolution. Note that in the “clonal interference” approximation (Gerrish and Lenski, 1998),
which considers competition between two beneficial clones emerging at two sites, variation of
s must be taken into account to make continuous adaptation possible. In contrast, in the present
theory, which allows new beneficial clones to grow inside of already existing clones, the
importance of variation in s is less obvious. We hope to address this matter elsewhere.
We discuss the validity of our approach in detail for the limits of Muller’s ratchet and adaptation
and give a summary of the central simplifications—numbered 1 through 6 and referenced
throughout the text—in the Appendix. Note that we can verify the validity of the various
assumptions only after the fact, once we have obtained our final results. All simplifications are
asymptotically exact, i.e., are based on the existence of small dimensionless parameters. The
most limiting requirements are that the high-fitness tail of the distribution is long, and that the
distribution is far from the unloaded and fully loaded (possible best-fit and less-fit) genomes.
2.2 General approach
The first idea underlying the approach of ”solitary wave” is to classify all genomes in a
population according to their fitness (mutational load), regardless of specific locations of
deleterious alleles in a genome, and focus on evolution of fitness classes. The second idea is
to describe evolution of most fitness classes deterministically. To take into account the effects
of finite population size, such as genetic drift and randomness of mutation events, only one
class with the highest fitness is described stochastically using the standard two-allele diffusion
approach. The best-fit class is considered a minority ”allele” in a population, and all other
sequences are considered the majority ”allele”. The reason why stochastic effects can be
neglected already for the next-to-best class is that, in a broad parameter range, the fitness
distribution decreases exponentially towards the stochastic edge. Hence, the next-to-best class
is large enough to neglect stochastic effects, especially in the adaptation regime (see estimates
in Sections 3 and 4).
We note that neglecting stochastic effects completely and considering the limit of infinite
population is not correct. As we show below, stochastic processes acting on the best-fit class
limit the overall evolution rate and make it dependent on the population size. Even for a modest
number of sites (L = 15–20), the substitution rate is predicted to reach the true deterministic
limit only in astronomically large populations not found in nature (Tsimring et al., 1996;
Rouzine et al., 2003; Desai and Fisher, 2007). For the same reason, the assumption of one
mutation per genome we made in our model is not a limiting factor and, as shown by Rouzine
et al. (2003), does not change much in the final expression for the average substitution rate.
The formal procedure consists of several steps, as follows.
i.
The frequencies of all fitness classes excluding the best-fit class are described by a
deterministic balance equation.
ii.
The equation is shown to have a traveling wave solution with an arbitrary speed (the
average substitution rate).
iii. The leading front of the wave (high-fitness tail of the distribution) is shown to end
abruptly at a point, expressed in terms of the wave speed.
iv. The difference between the values of the fitness distribution at the center and the edge
is expressed in terms of the wave speed.
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v.
The value at the center is found from the normalization condition.
vi. Because the biological justification for the lack of genomes beyond the high-fitness
cutoff is finite size of population, the cutoff point is identified with the stochastic
edge.
vii. To determine the wave speed as a function of the population size, the average
frequency of the least-loaded class is estimated from the classical diffusion result and
matched to the deterministic cutoff value.
2.3 Equation for the deterministic part of the fitness distribution
We proceed with the first step. On the basis of our model assumptions and neglecting multiple
mutation events per generation per genome, we can write the deterministic time evolution of
the frequency fk(t) of genomes with mutational load k as
1
fk(t + 1) =
wav(t)(e−s(k−1)μ(L − k + 1)fk−1(t)
+e−s(k+1)μ(k + 1)fk+1(t) + e−sk(1 − μL )fk(t)),
(1)
where k runs from 0 to L and f−1(t) ≡ fL+1(t) ≡ 0. By definition, Σk fk(t) = 1 for all times t. Now
we introduce the total per-genome mutation rate U = μL, and the ratio of beneficial mutation
rate per genome to the total mutation rate, αk = μk/U = k/L. Inserting these expressions into Eq.
(1), expressing wav in terms of kav, expanding e−sx ≈ 1 − sx (which we are allowed to do under
the condition that s|k −kav| ≪ 1, Simplification 1), and neglecting all terms proportional to
sU, we arrive at:
fk(t + 1) = U(1 − αk−1)fk−1(t) + Uαk+1fk+1(t) + 1 − U − s(k − kav) fk(t). (2)
As mentioned before, we consider the case when the traveling wave is far from the sequence
with the highest possible fitness, k = 0, as given by the condition |k −kav|≪ kav. Therefore,
αk depends only slowly (linearly) on k, we are allowed to replace αk by α ≡ αkav (Simplification
2), and find
fk(t + 1) − fk(t) = U(1 − α)fk−1(t) + Uα fk+1(t) − U + s(k − kav) fk(t).(3)
Our goal is to turn this expression into a continuous differential equation. As the mutation rates
are low, fk(t) evolves very slowly in time, and we can write fk(t + 1) ≈ fk(t) + ∂fk(t)/∂t
(Simplification 3).
We need to be more careful when making a continuous approximation for fk(t) as a function
of k. As we show below, fk(t) changes rapidly with k in the important high fitness tail. Therefore,
the Taylor expansion of fk(t) is not justified. However, the logarithm of fk(t) is a smooth function
of k in a broad parameter range, provided the ”lead” of the distribution is large (Simplification
4). [The lead is the difference in number of mutations from the population center to the least-
loaded class in the population (Desai and Fisher, 2007).] Therefore, a better approximation,
which represents an improvement on the work by Tsimring et al. (1996), is to do the Taylor
expansion on ln fk and write fk+1(t) = fk(t) exp[∂ln fk(t)/∂k]. With these approximations, and
after introducing a rescaled time dτ = Udt and a rescaled selection coefficient σ= s/U, we find
∂ ln fk(t)/∂τ = (1 − α)e
This nonlinear partial differential equation describes the deterministic movement of the
population in fitness space over time.
−∂lnfk(t)/∂k
+ αe
∂lnfk(t)/∂k
− σ(k − kav) − 1.(4)
(Note that, for sufficiently large N, the assumptions of Simplifications 1 and 3 may be violated,
so that technically, we can neither expand fitness in k nor replace discrete time with continuous
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time in this regime. Nevertheless, Rouzine et al. (2003) showed that the continuous equation
for the fitness distribution, Eq. (4), can be derived in a more general way, without assuming
∂ ln fk(t)/∂t≪ 1, nor expanding fitness in k, nor neglecting multiple mutations per genome per
generation [see the transition from Eq. (1) to Eq. (11) in the Supplementary Text of the quoted
work]. In the present work, we use these approximations only to simplify our derivation.)
In the equation above, α and σ depend, strictly speaking, on time. The dependence is, however,
very weak in the limit of a large genome size L. In the remainder of this paper, we shall assume
that the observation time is very small compared to the time in which kav changes by L and,
thus, α and σ can be considered constant. At the same time, because the distribution is very far
from the class without deleterious alleles, |k −kav| ≪ kav, a considerable shift of the distribution
can occur during the observation time.
2.4 Traveling wave solution
A broad class of partial differential equations affords solutions in the form of traveling waves.
A wave is a fixed shape, described by a function φ(x), that moves through space without
changing its form. In the case of an evolving population, space corresponds to the mutational
load, k. A traveling wave solution to Eq. (4) can be expressed as the movement of the center
of the population over time, kav(τ), and the shape of the wave around this center, which we
write as φ[k−kav(τ)]. We therefore make the ansatz (Simplification 5) that
ln fk(t) = φ k − kav(τ)(5)
After inserting Eq. (5) into Eq. (4) and writing x = k −kav(τ), we obtain
σx = (1 − α)e−φ′(x)+ αeφ′(x)+ vφ′(x) − 1, (6)
where φ′(x) = ∂φ(x)/∂x, and we have introduced the wave velocity v, that is, the speed of
movement of the population center kav over time, v ≡ ∂kav(τ)/∂τ.
For a given wave velocity v, Eq. (6) and φ(0) fully specify the shape of the wave φ(x). Although
it is not possible to solve Eq. (6) in the general form, analytically, below we will be able to
find the velocity of the wave. The first step is to derive φ(0) from the normalization condition.
2.5 Normalization condition. Width and speed
The validity of the approach by Rouzine et al. (2003) requires that the logarithm of the fitness
distribution, φ(x), can be approximated with a smooth function of x. In other words, the
characteristic scale in x given by the length of the high-fitness tail is much larger than unit. The
condition is always met when population is evolving at many sites at a time. Because the fitness
distribution itself, exp[φ(x)], is not replaced with a smooth function of x, it does not have to be
broad: Whether the half-width of the wave is small or large is irrelevant for the validity of the
approach. In the treatment of adaptation limit (Section 4), the wave can be either broad or
narrow, depending on the range of population sizes. The two cases, however, have different
normalization conditions, as follows.
If the wave is narrow, Var[k] = Var[x] ≪ 1, most of the distribution is localized at k ≈ kav, and
from the normalization sum ∑k=0
L
fk(t) we have
φ(0) ≈ 0. (7)
(The exception is the case when kav is nearly half-integer, and the two adjacent classes have
comparable sizes.)
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If the wave is broad, Var[k] ≫ 1, the normalization condition is less trivial. The normalization
sum ∑k=0
main contribution to the integral comes from the interval of x such that |x| is much larger than
1 but much smaller than the high-fitness tail length, so that |φ′(x)| ≪ 1. Hence, we can expand
the exponential terms in Eq. (6) to the first order (Simplification 6), and find
L
fk(t) can be replaced with an integral, ∫eφ(x)dx. As we will see momentarily, the
φ′(x) = −
σx
1 − 2α − v.
(8)
Clearly, |φ′(x)| ≪ 1 when |x| ≪ (1 −2α − v)/σ, and therefore this latter condition determines
the validity of Eq. (8). Integrating in x, taking into account the normalization condition, we
obtain the expression for φ(x) near the wave center
φ(x) = ln
σ
2π(1 − 2α − v)−
σx2
2(1 − 2α − v).
(9)
In particular, the desired expression for φ(0) has a form
φ(0) = ln
σ
2π(1 − 2α − v).
(10)
Note that φ(x) was defined as φ(x) = ln fk(t). Therefore, Eq. (9) implies that the distribution of
genomes in the vicinity of kav is approximately Gaussian, with variance
Var k = (1 − 2α − v)/σ. (11)
Because the variance and the argument of the square root in Eq. (10) must be positive, the wave
velocity has to satisfy v < 1 − 2α. In other words, if deleterious mutations accumulate, the rate
of their accumulation cannot exceed 1 − 2α if time is measured in units of U. Also, as we
already stated, the Gaussian expression for the central part of the wave is valid if the width of
the wave Var k is much larger than 1. For moderate or small wave speeds, |v| ~ 1 or |v| ≪
1, this condition implies that σ is much smaller than 1. Whether the wave is actually narrow or
broad, given the population size and other parameters, will be discussed below for Muller’s
ratchet and the adaptation regime.
Eq. (11), which is known as Fisher Fundamental Theorem, links the width of the population’s
mutant distribution, Var k , to the velocity v at which the wave is traveling. We emphasize
that the validity of this expression is not restricted to the case of broad fitness distribution and
can be derived directly from Eq. (3) (Appendix) even if Var[k] ≪ 1. Qualitatively, the theorem
states that, the broader the wave, the larger the characteristic difference in fitness between two
representative genomes, and the stronger the effect of positive selection on the substitution
rate.
2.6 High-fitness edge of distribution
In the previous subsections, we have derived a continuous wave equation that gives a
deterministic description of how the bulk of the population moves over time in fitness space.
However, the speed of the wave remains undefined. As it turns out, the behavior of a small
class of best-fit genomes determines the speed.
Below we show that the deterministic wave has a cutoff in the high-fitness tail of the wave
whose position is defined by the wave speed and model parameters. The biological reason
behind the deterministic cutoff are stochastic factors acting on finite populations. Highly-fit
genomes are either gradually lost (Muller’s ratchet) or gradually gained (adaptation) by a
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Fig. 4.
Schematic illustration of correction for discreteness. Our continuous treatment of fitness
classes predicts that ln fk(t) − ln fk0 (t) grows linearly in k − k0 near the high-fitness edge, k −
k0 ≪ |x0|. Discreteness of k introduces a correction near the edge, approximately equal to
ln (2/e)(k − k0+ 5/6) .
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Fig. 5.
Normalized ratchet speed as a function of population size: analytic results versus simulation.
Symbols correspond to simulation results, and lines correspond to theoretical predictions. The
simulation results were obtained as described (Rouzine et al., 2003). Beneficial mutations are
absent (α = 0). (A) Results for σ = s/U = 0.1. The solid blue and red lines follow from the
present work, Eq. (36). The dotted green lines follow from Gordo and Charlesworth (2000a),
Eqs. (3a) and (3b). The dashed purple lines follow from Lande (1998), Eq. (2c) times NU.
Parameters are shown. (B) As (A), but for σ = s/U = 0.01. The dashed purple lines follow again
from Lande (1998); the equivalent of the green lines in (A) falls outside of the axis range. (C)
Effect of discreteness correction. The red squares and circles are identical to those in (A) and
(B). The solid lines correspond to the prediction of traveling-wave theory without the
discreteness correction, i.e., without a factor of σ inside of the logarithm on the left-hand-side
of Eq. (36), and without the denominator 1 − v ln(e/v) + 5σ/6 inside the logarithm on the right-
hand side of Eq. (36). The dashed lines correspond to Eq. (36) without the entire second term
on the right-hand-side. The arrow shows the value of N at which the size of the least-loaded
class n0 = Ne−U/s (Haigh, 1978) in the equilibrium distribution is 1 (at s/U = 0.1).
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Fig. 6.
Schematic illustration of establishment of a new best-fit class. Once a beneficial mutant has
survived drift, its frequency grows approximately exponentially in time. A further beneficial
mutation is likely to arise and survive drift only in a short time interval of length 1/S around
the time when the currently least-loaded class has grown to its maximal value fmax.
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Fig. 7.
Normalized substitution rate as a function of population size, in the absence of deleterious
mutations. Symbols correspond to simulation results, and thick solid lines correspond to
analytic predictions obtained (A,B) from Eq. (51) or (C) from Eq. (52). Thin lines in (A,B)
indicate the analytic prediction without discreteness correction, i.e., without term ln[(V/s) ln
(V/Ub)] on the right-hand side of Eq. (51). Dotted lines in (B,C) indicate the analytic result by
Desai and Fisher (2007),Eqs. 36,38, and 39. Their result shown in (B) is outside of its designed
validity range and is shown for reference only. The simulations were carried out (A,B) in
discrete time as described (Rouzine et al., 2003) or (C) in continuous time. V was measured in
simulation as the average slope of kav(t) in the time interval [0.85t0, 1.15t0], where the time
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t0 of the interval center was determined from the condition kav(t0) = 250. The initial kav(0) was
set to 1.5kav(t0). The per-site beneficial mutation rate μ was chosen such that the genomic
beneficial mutation rate Ub had the value shown on the plot at time t0, Ub = μkav(t0). The size
of the symbols roughly corresponds to the largest standard deviation of the mean speed
estimates.
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Fig. 8.
Test of the accuracy of the stochastic edge treatment, Eq. (46). Points are obtained by simulation
using a simplified two-class model. Only points corresponding to s|x0| < 0.2 are shown.
Parameter values are shown.
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Table 1
Variables used in this work.
SymbolDefinition
αk
α
ratio of the beneficial mutation rate to the total mutation rate in class k; αk = μbk/U
effective ratio of the beneficial mutation rate to the total mutation rate; α = μbkav/U
frequency of a class of genomes with mutation number k at time t
logarithm of genome frequency; φ = ln f
the number of less-fit alleles in a genome, as compared to the best possible genome
minimum value of k in the population
effective mutational load generating a fitness equivalent to the mean fitness of the population,
kav= −1
length of genome (number of sites)
mutation rate per site
haploid population size (number of genomes)
probability that a class of genomes has frequency f at time t
selection coefficient; relative fitness gain/loss per mutation
effective coefficient of selection against the best-fit class in a population; S = U + s(k0 − kav)
rescaled selection coefficient; σ = s/U
time (in generations)
effective mutation rate per genome per generation; U = μL
beneficial mutation rate per genome per generation; Ub = (k/L)U
u = eφ′(x0)
average substitution rate of beneficial mutations; V=−dkav(t)/dt
normalized ratchet rate (substitution rate of deleterious mutations); v = (1/U)dkav(t)/dt
variance of k in the population
relative mutational load of a class; x = k − kav
minimum value of x for highest-fitness class
fk(t)
φ
k
k0
kav
sln (∑ke−ksfk)
L
μ
N
ρ(f, t)
s
S
σ
t
U
Ub
u
V
v
Var[k]
x
x0
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