| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
Genetics, Vol. 176, 1759-1798, July 2007, Copyright © 2007
doi:10.1534/genetics.106.067678
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
,1
,2
* Department of Physics, Department of Molecular and Cell Biology and Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138
1 Corresponding author: Lewis-Sigler Institute for Integrative Genomics, Carl Icahn Laboratory, Princeton University, Princeton, NJ 08544.
E-mail: mmdesai{at}princeton.edu
 |
ABSTRACT |
|---|
When beneficial mutations are rare and selection is strong, positive selection results in a succession of selective sweeps. A mutation occurs, spreads through the population due to selection, and soon fixes. Some time later, another such event may occur. This situation is sometimes called the strong-selection weak-mutation regime. To make its character clear, we refer to it as the successional-mutations regime: between sweeps, there is a single "ruling" population. In this regime, the effect of positive selection on patterns of genetic variation is reasonably well understood. A selective sweep reduces the genetic variation in regions of the genome linked, over the timescale of the sweep, to the site at which a beneficial mutation occurs: other mutations in these regions hitchhike to fixation.
Successional-mutations behavior typically occurs in small- to moderate-sized populations in which beneficial mutations are sufficiently rare. However, a different regime occurs in larger populations, in which beneficial mutations occur frequently. When beneficial mutations are common enough that many mutant lineages can be simultaneously present in the population, selective sweeps will overlap and interfere with one another (i.e., different beneficial mutations will grow in the population concurrently). If, in addition, selection is strong enough that it is not dominated by random drift (except while mutants are very rare), we have a "strong-selection strong-mutation" regime. For clarity, we refer to this as the concurrent-mutations regime. The effects of concurrent mutations in asexual populations are the focus of this article. As we will see, the concurrent-beneficial-mutations regime is not an unusual special case: many viral, bacterial, and simple eukaryotic populations likely experience evolution via multiple concurrent mutations.
In populations that contain many different beneficial mutants, there will be substantial variation in fitness within the population. This variation will be acted on by selection. But in the absence of new mutations, the variation will soon disappear. Thus the traditional approach to evolution of quantitative traits—to assume that genetic variation always exists (as for traits not subject to selection)—fails badly. New mutations are crucially needed to maintain the variation on which further selection can act. Thus to understand adaptation when multiple mutations are involved, it is essential to analyze the interplay between selection and new beneficial mutations, especially how the latter maintains the variation acted on by the former. Understanding this beneficial mutation–selection balance and the resulting dynamics is the primary goal of this article.
Both the successional- and the concurrent-mutations regimes require that selection dominates drift except while mutants are very rare. A qualitatively different regime occurs with weakly beneficial mutations: these do not sweep in the traditional sense because drift dominates their dynamics. This weakly beneficial regime most readily occurs in small populations, where selective forces cannot overcome drift, or when considering mutations of very small effect, such as those that affect synonymous codon usage (LI 1987; COMERON et al. 1999; PRZEWORSKI et al. 1999; MCVEAN and CHARLESWORTH 2000). In this article we are interested in beneficial mutations in moderate to large populations, so we focus exclusively on the strong-selection regimes for which drift is important for beneficial mutant lineages only while they are a tiny minority of the population.
The essential difference between the successional-mutations and concurrent-mutations regimes is presented in Figure 1, which depicts beneficial mutations in an asexual population. In a small enough population, or one whose beneficial mutation rate (Ub) is low, beneficial mutations occur rarely enough that they are well separated in time and one can sweep before another arises (Figure 1a). This is the successional-mutations regime, in which the beneficial mutations all behave independently. However, in a larger population or at higher Ub, multiple mutant populations exist concurrently and they are no longer independent (Figure 1b). Mutations that occur in different lineages cannot both fix in the absence of recombination: at least one of them must be "wasted."
|
The dynamics of evolution in the concurrent-mutations regime are important to understand. At the very least, this is essential for forming sensible null expectations about experimental, observational, and genomic data from large populations. Knowing how the effects of beneficial mutations depend on mutation rate and population size is crucial for making meaningful comparisons between different populations. Most important, in our view, is developing an intuition for how large populations evolve. The simple picture of successive selective sweeps in the successional-mutations regime is a valuable guide to thinking about positive selection. Yet we have little intuitive guidance when the successional-mutations approximation does not apply. This is a serious shortcoming in our understanding of the evolution of a wide array of populations, including viruses and most unicellular organisms.
Although it is not as well understood as the successional-mutations regime, the concurrent-mutations regime has been the subject of substantial interest since the 1930s. FISHER (1930) and MULLER (1932) first noted the potential importance of interference between beneficial mutations (Muller drew diagrams very similar to our Figure 1). They proposed what has come to be known as the Fisher–Muller hypothesis for the advantage of sex: sexual populations can recombine beneficial mutations in competing lineages into the same individual. This prevents mutational events from being wasted, as they often are in asexual populations.
Much subsequent work on positive selection in the concurrent-mutations regime has focused on the implications for the evolution of sex. CROW and KIMURA (1965), BODMER (1970), and MAYNARD SMITH (1971) attempted to quantify the Fisher–Muller effect in the late 1960s and the early 1970s. However, their analysis was incomplete—it did not fully account for stochastic behavior, ignored triple and higher mutations, and did not correctly account for the effects of sex. Contemporaneously, HILL and ROBERTSON (1966) looked at this problem from the perspective of the linkage disequilibrium generated by multiple linked beneficial mutations segregating simultaneously. This has become known as the Hill–Robertson effect. It is essentially equivalent to the Fisher–Muller effect (see FELSENSTEIN 1974 for a detailed discussion). In recent years, BARTON (1995), OTTO and BARTON (1997, 2001), and BARTON and OTTO (2005) have analyzed the Fisher–Muller effect from the Hill–Robertson perspective. Their work focuses on the buildup of linkage disequilibrium due to mutations and selection and the average effect of recombination on the variance in fitness and the destruction of disequilibrium. This provides useful insight into the effects of sex, but does not explain the full evolutionary dynamics or population genetic structure created by this type of positive selection.
In this article, we step back from the long tradition of studying the implications of concurrent mutations for the evolution of sex and focus instead on the basic dynamics shown schematically in Figure 1b. We show how an asexual population in the concurrent-mutations regime accumulates many beneficial mutations, what the fitness distribution looks like, how it develops, and how quickly selected substitutions occur via collective sweeps. We develop a framework for thinking more generally about positive selection and its effects that is applicable to large populations of asexuals or any other case where linkage between mutations is important.
We do not analyze the questions about sex or patterns of diversity in this article. However, these questions should be informed by our results; some can be studied within the framework we present in this article. For example, when recombination is rare, the average effects of sex may be irrelevant—instead all that matters is whether or not it creates rare individuals that are much more fit than the majority of the population. To study this, we must first understand the full distribution of genetic diversity within the population. Similarly, before analyzing the patterns of genetic variation exhibited by populations in which multiple linked beneficial mutations have occurred—or are occurring—one must understand the rate of beneficial substitutions and typical interference patterns between these within the linked regions.
To understand the concurrent-mutations dynamics in detail, it is essential to start with a specific model that focuses on some subset of the important effects. Features can then be added after enough understanding has been gleaned to enable predictions of which effects are model specific and which are more general. Positive selection can involve various complications, including epistasis (interactions between effects of mutations), conditionally beneficial mutations, frequency-dependent benefits, and changing environments, among others. Many different scenarios are possible. At present we have little understanding of which, if any, of these situations are biologically "typical" and which ones are unusual. In this article, we do not attempt to catalog all possible complications; this is an impossibly broad subject. Instead we look at the simplest possible situation involving positive selection of concurrent mutations. We suppose that a variety of beneficial mutations are available to a population and ask how the population acquires them. We assume these mutations interact in a simple multiplicative way (additive for the growth rates) with no epistasis, frequency dependence, or changing environment of any kind. In short, we ask how the population climbs a single smoothly sloped "hill" in fitness space.
This simple scenario is probably common. Populations often find themselves in an environment where they can accumulate quite a few different beneficial mutations that each roughly independently help them adapt. Even when this simple hill-climbing scenario does not apply, it is an important null model. Some more complex forms of positive selection may also prove tractable within the framework we describe, while others will not; these leave open many avenues for future work.
Various other authors have studied the dynamics of multiple concurrent beneficial mutations under the simple assumptions outlined above. GERRISH and LENSKI (1998) analyzed clonal interference between mutations of different strengths; this has since been extended by various authors (ORR 2000; GERRISH 2001; JOHNSON and BARTON 2002; KIM and STEPHAN 2003; CAMPOS and DE OLIVEIRA 2004; WILKE 2004). This work focuses on the interference between mutations of different strengths that occur in the same lineage, while neglecting the competition between mutations that arise in different lineages—in particular multiple mutants. Yet we show below that if population parameters are such that clonal interference is important, the effects of multiple mutants are usually at least of comparable importance. Thus there is some inconsistency in focusing on clonal interference alone. Our analysis in this article starts instead with the other concurrent-mutation effect, multiple mutants, initially in a model in which clonal interference is absent. In any real situation, the two effects will both occur. We thus discuss the interplay between clonal interference and multiple mutations in a later section. KIM and ORR (2005) have also recently analyzed a model that combines some aspects of clonal interference and multiple mutations.
To focus on the effects of multiple mutants without clonal interference, two additional simplifying approximations are useful. For most of this article, we study a model in which each beneficial mutation has the same effect, s, on fitness (i.e., each step uphill is of the same size). Furthermore, to focus on the effects of positive selection, we neglect deleterious mutations in the primary analysis. Even though neither assumption will typically be true, these turn out to be reasonable approximations in many circumstances. Situations in which they are not appropriate are more complicated scenarios for positive selection, some of which, especially the effects of a distribution of fitness increments, we discuss briefly.
Remarkably, even the simplest possible model with many equal-strength beneficial mutations available is only partially understood. KESSLER et al. (1997) and RIDGWAY et al. (1998) analyzed a similar simple model, but their initial work did not handle random drift correctly. More recently, they have developed a sophisticated although somewhat unwieldy moment-based approach (D. KESSLER and H. LEVINE, unpublished results) from which it is unfortunately hard to understand the essential aspects of the dynamics. ROUZINE et al. (2003) also studied a model similar in its essential aspects to our simplest model (although also including deleterious mutations of the same magnitude). They were concerned with viral evolution, and their results are primarily valid for very large mutation rates appropriate for many viruses; we focus instead on regimes primarily applicable to single-celled organisms (and some viruses). Nevertheless, if worked out more fully from Rouzine et al.'s analysis, several results can be obtained that are closely related to ours. But our analysis involves a less mathematically formal approach—we believe it is both clearer and a better basis for further development (some of which is included herein). We discuss the relationship between our analysis and that of ROUZINE et al. (2003) in more detail below.
The outline of this article is as follows. We begin by describing in the next section a heuristic approach to the dynamics. This analysis gets the behavior roughly correct and illustrates the ideas underlying our approach. We then describe the simplest model more precisely and analyze it in the following section. We next discuss transient behavior before the population has reached its steady-state fitness distribution and address the effects of deleterious mutations. In the next section, we make comparisons between our analytic results and simulations. We then relax our assumption that all mutations have the same effect and discuss the relationship between our theory and clonal interference analysis. Finally, we summarize our results and discuss future directions.
|
HEURISTIC ANALYSIS AND INTUITION |
|---|
1 + r). That is, we use "fitness" to mean what is sometimes called log fitness. Thus in the absence of epistasis, which we generally assume, two mutations of magnitude s1 and s2 increase fitness by s1 + s2. We call the rate of increase, d
r
/dt, of the average fitness of a population the speed of evolution and denote it v.
To focus on the effects of multiple mutants in a situation in which clonal interference does not occur, we initially restrict consideration to the approximation that all beneficial mutations have the same effect. A k-tuple mutant thus has fitness ks greater than the original wild type. The speed of evolution is then simply 
.
We begin by reviewing the successional-mutations regime where beneficial mutations are sufficiently separated in time for them to sweep independently, as in Figure 1a. Although this is exactly solvable and well known, it is instructive to consider it from a heuristic perspective. We then turn to a heuristic analysis of the more complex concurrent-mutations dynamics illustrated in Figure 1d.
Successional-mutations regime and the establishment of mutants:
Small asexual populations evolve by accumulating beneficial mutations sequentially. Beneficial mutations occur in the population at a total rate NUb. The probability that a particular mutant will survive random drift is proportional to its selective advantage s (provided 
). The constant of proportionality depends on the specific model for the stochastic dynamics; for our model it is 1 and we discuss in the SIMPLEST MODEL section below the minor modifications of our results that are needed for other stochastic dynamics. We call the process by which the lineage of a beneficial mutant that survives drift becomes large enough for the population of its descendants to grow deterministically the establishment of the mutant clone. As we show below in the section on the fate of a single mutant, a mutant population becomes established when its size reaches of order 1/s individuals. Roughly speaking, this is because a mutant lineage of size n takes n generations to change by of order n individuals due to random drift. Since selection adds on average ns individuals to the lineage per generation, in this time selection has an average effect of adding n2s individuals. So selection dominates drift provided n2s > n or 
. Thus the mutant lineage must reach a size 
before it becomes "safe" from extinction and begins to grow mostly deterministically.
We show in the section on the fate of a single mutant that if a mutant is destined to become established, it will reach this size 1/s very quickly. Thus new beneficial mutations are established at a rate roughly NUbs per generation (other mutant populations die out due to random drift), so a new mutation will become established about once every 
generations. Once established on reaching size of order 1/s, the mutant lineage grows roughly exponentially at rate s and hence takes of order 
generations to fix (we loosely call "fixed" a mutant lineage that has grown to represent a large fraction of the population; the conventional definition corresponds to fully fixed, which takes about twice as long).
When the population size or mutation rate is small enough, fixation will happen more quickly than establishment. This occurs when
 | (1) |
. When this condition holds, we are in the successional-mutations regime, in which the establishment rate is limiting: a mutation A that arises and fixes will do so long before the next mutation destined to survive drift, B, is established. Thus mutation B occurs in a population that has already fixed A, yielding AB, and B fixes well before mutation C is established. Beneficial mutations continue to accumulate in this simple way. New mutations arise and fix at average rate NUbs, each one increasing the fitness by s. Thus fitness increases at a speed
 | (2) |
Concurrent-mutations regime:
In larger populations, the behavior is more complex, as illustrated by Figure 1b. In this case, the establishment times of new mutants are shorter than their fixation times, corresponding to
 | (3) |
As noted in the Introduction, two types of interference are important. First, competition occurs when two mutations that have different strengths occur independently in individuals with similar initial fitness (clonal interference). We focus in the bulk of this article on the other type of interference: a mutation that arises in a fitter background (e.g., one with an earlier beneficial mutation) will outcompete another mutation of similar effect that occurs in a less fit background. In the constant-s model clonal interference is explicitly absent, and we thus initially focus exclusively on this latter effect. In this constant-s approximation, two different mutants that occur among those with the same fitness (in particular members of the same clone) will compete equally and sweep together, each becoming only partially fixed. Unless we are interested in the neutral genetic variability of the population, all subpopulations with the same fitness can be considered as a single subpopulation: we do this except in the DISCUSSION at the end of this article. Also, we postpone discussion of the interplay between clonal interference and multiple mutants (i.e., going beyond the constant-s model) to a later section below.
First consider starting from a monoclonal population. Mutations initially give rise to a subpopulation with fitness increased by s (Figure 2a). The size of this mutant subpopulation drifts stochastically, but eventually becomes large enough, 
1/s individuals, to become deterministic. This takes a (stochastic) establishment time, 
1. After its establishment but before its fixation, mutations can occur in the still-small mutant subpopulation to create double mutants with fitness 2s (Figure 2b). This typically happens well before the single mutants have fixed (else we are by Equation 1 in the successional-mutations regime). We assume the double mutants never arise before the single-mutant subpopulation has established; as we discuss below and in APPENDIX G, this will be true unless mutation rates are extremely high or selection is very weak. A double-mutant population thereby becomes established a time 2 after the establishment of the single-mutant population. Triple mutants then begin to arise and become established after an additional time 3. This interval is typically shorter than 2, primarily because double mutants grow faster than single mutants and hence generate more mutations and, in addition, because the triple mutants are more fit than double mutants and hence survive drift more easily (with probability 3s rather than 2s).
|
It is useful to consider this process in more general terms. The key to the behavior is the balance between mutation, which increases the variation in fitness within the population, and selection, which decreases the variation by eliminating all but the fittest individuals. If we were discussing deleterious mutations, mutation would also oppose the tendency of selection to increase the mean fitness, leading to a steady-state distribution of fitness (ignoring Muller's ratchet, which for large populations only matters on extremely long timescales). This deleterious mutation–selection balance, which is independent of population size for large N, has long been understood (GILLESPIE 1998). In our case, the dynamics are more subtle because the important mutations are beneficial. The basic idea of mutation–selection balance, however, is unchanged. Mutations broaden the fitness distribution while selection narrows it, creating a steady-state variance around an increasing mean fitness. But unlike the deleterious case, the dynamics of the rare individuals near the most-fit tail of the fitness distribution (the "nose") control the behavior. We show below that selection moves the distribution toward higher fitness at a rate very close to the steady-state variance in fitness—the classic result in the absence of mutations (the "fundamental theorem of natural selection") (FISHER 1930). But new beneficial mutations at the nose are essential to maintain this variance: in their absence the fitness distribution would collapse to a narrow peak near the most-fit individual and evolution would grind to a halt.
The crucial dependence on new mutations in the nose makes the analysis of the beneficial mutation–selection balance more complex than in the deleterious case. It is now essential to account properly for random drift in the small populations near the nose. In the case of deleterious mutation–selection balance, rare new mutants are less fit than the rest of the population. They will die out soon anyway, so failing to account properly for the stochastic dynamics by which they do so has no serious consequences. Random drift is important with solely deleterious mutations only if Muller's ratchet is operating, i.e., if the most-fit individuals are rare enough that they can die out due to random drift. The beneficial mutation–selection balance is quite analogous to this Muller's ratchet case. Here too the subpopulations that are more fit than average control the long-term behavior of the population, and these are small enough that correct stochastic treatment is essential. As is the case with Muller's ratchet, infinite-N deterministic approximations are not even qualitatively correct. Indeed, with a large supply of beneficial mutations, deterministic analysis incorrectly predicts a rapid acceleration of the nose toward an infinite speed of evolution. This nonsense result is because of the creation in the deterministic approximation of (what are effectively) fractional numbers of new much fitter mutants that then grow exponentially, unhampered by drift, and dominate the behavior soon after (we describe this in more detail in APPENDIX A).
There are two factors that determine the dependence of the speed of evolution on the population size. The first is the dynamics of already established subpopulations, which is dominated by selection. The second is the new mutations that occur in the fittest subpopulation. We define the lead of the fitness distribution, Q, as the difference between the fitness of the most-fit individual and the mean fitness of the population (more precisely, Q – s is the difference between the mean fitness and that of the most-fit established mutant class). We define q by Q = qs, so that if the lead is Q, the most-fit individuals have q more beneficial mutations than the average individual: they have a "lead" Q in the race to higher fitness. Once it is established, this fittest population grows exponentially. In the time this population took to become established, in steady state the mean fitness must have increased by s, so the newly established population will initially grow exponentially at rate (q – 1)s and later more slowly as the mean continues to advance. Growing from its establishment upon reaching size 1/qs until it reaches a large fraction of N will thus take time 
, since 
is its average growth rate during the period between establishment and fixation. In this time the mean fitness will increase by (q – 1)s. Therefore v [(q – 1)s]2/[2 ln(Nqs)]. One can show that this v is equal to the variance in fitness, as expected if mutation is indeed negligible compared to selection in the bulk (i.e., away from the nose) of the distribution, so that the fundamental theorem of natural selection applies.
The other factor is the dynamics of the nose, where mutations are essential. A more-fit mutant that moves the nose forward by s will be established some time q after the previous most-fit mutant. Thus the nose advances at a speed v = s/q, where q is the average q. After it is established, the fittest established population nq–1 will grow exponentially at rate (q – 1)s and produce mutants at a rate Ubnq–1 Ube(q–1)st/qs. Many new mutants will establish soon after the time q at which 
becomes equal to one, so the time it takes a new mutant to establish is 
. This means the nose advances at rate v s/q (q – 1)s2/ln(s/Ub). Significantly, the behavior of the nose depends only on mutations from the most-fit subpopulation; it is almost independent of the less-fit populations and thus can depend on N only via the lead, qs. As far as the nose is concerned, the majority of the population—destined to die out shortly—is important only to ease the competition for the fittest few. Yet we argued above that the bulk of the population fixes the speed of the mean via the selection pressure: 
. In steady state, the speed of the mean must equal the speed of the nose—the mutation–selection balance. This implies that
 | (4) |
 | (5) |
For large NUb, we have found that v depends logarithmically on N and Ub, much slower than the linear dependence on NUb that holds for smaller populations. This reduction occurs because at large NUb, almost all beneficial mutations occur in individuals far from the nose of the fitness distribution (i.e., in a bad genetic background) and are therefore wasted, since these subpopulations are doomed to extinction. Thus increasing N does not directly increase the supply of important mutations, as these occur in the relatively few individuals at the nose. Rather, the effect of increasing N is to increase the time required for selection to move the mean fitness, which increases the lead, which makes individuals at the nose more fit relative to the mean fitness, which speeds the establishments at the nose. Similarly, increasing Ub does not directly affect the dynamics of most of the fitness distribution. Rather, it decreases the time for new mutations to occur at the nose, which means that more mutations can occur before the mean moves, which increases the lead and speeds the evolution.
This also explains why v is not a function of NUb: N directly affects only selection timescales, while Ub directly affects only the mutation supply rate, so v depends on N and Ub separately. It is not a function of the commonly used parameter 
= 2NUb. Instead, it is a function of the parameters Ns (which describes selective forces) and 
(which describes the strength of selection relative to mutation), and it is valid in the regime where both are large. The expression for q above is of order the basic selective timescale, 
divided by the basic mutation timescale, 
, which makes sense since the lead is set by the balance between these two forces. More generally, the two factors that determine the timescales of the multiple mutation dynamics are
 | (6) |
Although these are both logarithmic in the population parameters and thus never huge, they can be large enough to be considered as large parameters. Many of our more detailed results are valid in the limit that both L and 
are large, with corrections (some of which we include) smaller by powers of 1/ or 1/L.
We show below that our result for v is consistent with the fundamental theorem of natural selection. Viewed in this light, our result for the speed of evolution is not in itself novel: the speed is just the variance in fitness, as usual. What our analysis does is to obtain what this variance is. In many aspects of quantitative genetics, the variance of a quantitative trait (such as fitness here) is taken as some external parameter. When the variance has accumulated during a period when it was neutral and is only starting to be selected on, this may be appropriate. But beyond that, it is surely not. Our analysis deals with the case when variance is accumulating while being selected on. That is, when variance in fitness is increasing due to mutations while at the same time it is being acted on by selection, then, even if the adaptation speed is only indirectly related to new mutations, it is essentially dependent on them: without mutations the variance will rapidly collapse to zero.
However, neither our heuristic analysis above nor our more careful work described below ever explicitly involves the fitness variance. Rather, the natural measure of the width of the fitness distribution is the lead. It is the lead, not the variance or the standard deviation, that can be most productively thought of as a balance between mutation and selection. It is true, of course, that the variance is also increased by mutation and decreased by selection. However, this is not the clearest way to understand the behavior. The increase in the variance from mutations is delayed and indirect. The new mutations that occur at the nose will only increase the variance after they have grown enough—and by then the important new mutations that will keep the variance high later are happening further out in the nose. This is not to say that a variance (and higher-moment)-based approach is impossible, but it is unwieldy and prone to hard-to-understand errors when any approximations are made. We discuss such moment-based approaches in APPENDIX A.
|
SIMPLEST MODEL |
|---|
In addition to the more innocuous simplifications, we make two essential biological assumptions: that there is no frequency-dependent selection and that there is no epistasis, so that the fitness of an individual with k mutations is (k – )s greater than the fitness of an individual with mutations. When either of these conditions fails, the evolutionary dynamics can be very different from our predictions.
Key approximations:
There are two primary difficulties in analyzing the multiple subpopulations that occur even in the simplest model. The first is the stochastic aspects: when a subpopulation with a given fitness is rare, stochastic drift plays a crucial role and must be handled correctly. The second is the interactions between the subpopulations: the constraint of fixed total population size means that there is effectively a frequency dependence to the growth of a subpopulation—albeit a simple one.
To model the stochastic effects, we assume that the basic process of birth and death is a continuous-time branching process. All individuals have the same constant death rate 1, which means that the average lifetime of an individual is 1 (i.e., the units of time are generations) and that the lifetimes are exponentially distributed. Each individual in the population has some number, y, of beneficial mutations. We define 
to be the average value of y across the population (i.e., the average number of beneficial mutations per individual). An individual with y beneficial mutations has a birth rate 
. This ensures that the average birth rate in the population is 1, so the population stays at a constant size N. We assume all individuals give rise to mutant offspring at rate Ub, independent of their birth rate (i.e., mutants arise at a constant rate per unit time). If mutations instead occur at a constant rate per birth event, our assumption underestimates the mutation rate for the most-fit individuals. However, we always assume 
for all individuals (i.e., the lead, Q 
), so that the two definitions are almost equivalent.
The branching process model allows one to calculate simple analytic expressions for a number of important quantities that are not readily available in diffusion approximations of the standard Wright–Fisher model. However, branching process models cannot easily deal with the nonlinear saturation effects required to maintain a constant population size. By "saturation" effects, we refer to when a mutant subpopulation has become large enough to influence the mean fitness of the population and hence begins to compete with itself, slowing its growth: this is the essential effect of the fixed total population size. To handle the saturation effects, we make use of a simple observation: stochastic effects are important only when a subpopulation is rare, while saturation is important only when a subpopulation is common. Thus we use the stochastic branching process model, ignoring saturation effects, to describe the dynamics of a subpopulation while it is small. Conversely, when it is large, we ignore random drift and treat it with the correctly saturating deterministic equations. Our use of both deterministic and stochastic analyses requires an appropriate way of linking the two together. In this article, we describe a method for doing so. This method accounts for all of the important aspects of genetic drift and is simple and intuitive. It should be of broad applicability to related evolutionary problems.
This approach works as long as the stochastic regime and the saturation regime are different. That is, a subpopulation must become large enough to neglect random drift before it is too large to ignore saturation. We can treat a subpopulation of size n deterministically so long as 
. On the other hand, saturation can be ignored when 
. Thus to separate the stochastic and the saturating phases of growth of a subpopulation, we require 
. Throughout this article, we assume this condition holds. Unless s is extremely small (s Ub), a population small enough that 
will usually be too small for clonal interference or multiple mutation effects to matter, so this is not a serious limitation.
A situation in which there are multiple subpopulations of varying sizes is illustrated in Figure 3: this shows the logarithm of a typical fitness distribution within a steadily evolving population. Where the subpopulations are small, at the front of the distribution, stochastic analysis is necessary but nonlinearities can be ignored. When a subpopulation represents a substantial fraction of the total, nonlinear saturation is important but stochasticity is not. As long as , there is an intermediate regime where neither matters. We can thus use a nonlinear deterministic analysis in the bulk of the distribution and a linear stochastic analysis near the front and match the two in the intermediate regime in which both are valid. These approximations are fully controlled and any corrections to our results will be small for .
|
) will become established is proportional to s in both models, it is cs with the coefficient c = 2 in the Wright–Fisher model and c = 1 in ours. Since it is likely that the population dynamics in any real population are not well represented by either of these models, there is no one "correct" model [e.g., for populations dividing by binary fission, as in many experimental studies of evolution, the establishment probability is closer to 2.8s (JOHNSON and GERRISH 2002)]. Fortunately, in our analysis of the behavior of large populations, these differences cause only negligible corrections in the arguments of logarithms [e.g., replacing ln(Ns) with ln(cNs) when ]. For smaller populations, however, the speed of evolution is proportional to the probability of establishment and thus does depend on more details of the model: in particular, the successional-mutation result for the speed is v cNUbs2.
It would in principle be possible to use a diffusion approximation to the Wright–Fisher model instead of our branching process model. This would have the advantage of being able to handle saturation and drift at the same time and thus cases where . Such a model could in principle treat all the different subpopulations stochastically, including all mutations between these populations. However, this would lead to a complex and difficult to analyze infinite-dimensional diffusion process. There is, however, a controlled approximation—valid for large Ns—to the full diffusion process that is exactly equivalent to ours; as it would add little, we do not discuss this explicitly here.
|
ANALYSIS |
|---|
The fate of a single mutant individual:
We begin by considering the fate of a single mutant individual. We assume that in a large clonal population of size N, at time t = 0 there is a single mutant individual with a beneficial mutation conferring fitness advantage s. We denote the size of the subpopulation carrying this beneficial mutation at time t as n(t) [by assumption, n(0) = 1]. We study the effects of selection and drift on this population by calculating the probability distribution of future n(t), 
, assuming that no further mutations occur. This provides an essential building block for all the subsequent analysis and also illustrates our basic approach in a simple context.
Throughout this analysis, we assume that the number of individuals with the beneficial mutation is small relative to the total population size, . Thus the mutants do not interfere with one another. Naturally, if the mutant becomes established it will supplant the wild-type population and this condition will cease to be true. By this time, however, the mutant subpopulation will be large enough that we can switch from the stochastic analysis described here to a correctly saturating deterministic analysis.
Because the mutant subpopulation is too small to affect the mean fitness, mutant individuals have a birth rate 1 + s and death rate 1. We define g(n, n0, t) to be the probability of having n descendants at time t, starting from n0 descendants at t = 0. We are interested in calculating g(n, 1, t). The probability of a birth or a death event in a unit of time dt is (2 + s)dt, and this event is a birth with probability 
and a death with probability 
. This means that
 | (7) |
n,0 = 1 if n = 0 and is 0 otherwise. This is a standard birth–death process (ALLEN 2003). Assuming that individual lineages are independent and defining the generating function
 | (8) |
 | (9) |
We can now determine 
from G(z, t). A standard inversion yields
 | (10) |
 | (11) |
We are interested primarily in understanding the distribution of n given that the mutant population is not destined to go extinct. This is given approximately by
 | (12) |
, the regime of primary interest. Note, however, that although the crucial features are more apparent in the approximate expression, all the results below follow from the exact equations.
At this stage, the above results merely reproduce classical analysis, but it is useful to pause to compare them with various intuitive predictions. We first compute the average number of mutant individuals at time t,
 | (13) |
 | (14) |
, this is n | not extinct 1 + t. At long times, , the extinction probability becomes 
, and 
. Note that short times correspond to 
, while long times mean 
. (Note also that none of these expressions saturate as n approaches N; they are valid for , as discussed above.)
It is useful to ignore mutations that are destined to go extinct due to drift and focus only on those that are destined to become established. We do this for the remainder of this section; all results are thus implicitly conditional on nonextinction. However, some care is required. If a mutation occurs at time t = 0 and survives drift to become established, it may seem that on average it will grow as n(t) = est, because it started from one individual at t = 0 and grows on average exponentially. However, this is incorrect. Given that it survived drift, it is likely to have grown faster than est in the early stochastic phase of its growth during which drift is faster than selection (OTTO and BARTON 1997; BARTON 1998). This is apparent from the expressions above: for , n | not extinct 1 + t, which is much faster than n = est 1 + st. Once the population is large and stochastic effects can be neglected, it naturally grows as est. However, because it grew faster than this in the early stochastic phase, it will on average be larger than if it had grown this fast through its entire history. As is clear from the expression for the average n at long times, , the behavior can be crudely approximated by assuming that it started at size 
(rather than size 1) at t = 0 and then grew exponentially as est thereafter. This approximation is of course not valid during the early phase of growth. Note that the above also implies that, given that a mutation is not destined to go extinct due to drift, it will fix in a time of order , not 
, as is sometimes seen in the literature. For s 0.01, this is a difference of 500 generations. To be more precise, the fixation time is a random variable with a distribution of width 1/s and mean close to , rather than the naive .
For much of the subsequent analysis, we are concerned with the size of a subpopulation only after it is big enough to be essentially deterministic. Yet as the above discussion makes clear, the stochastic phase of growth affects the later deterministic dynamics. Thus we are interested in "summing up" the stochastic effects in terms of their impact on later deterministic growth.
Focusing only on the effects of stochasticity on later deterministic dynamics allows us to make a key simplification. Once the subpopulation is large enough to grow deterministically, but still small enough that saturation can be ignored (i.e., 
), its dynamics can be described by n = 
est. The value of is a random variable that depends on how fast the population grew in its stochastic phase. However, the only effect of this stochasticity on the later deterministic growth is to create random variation in . As almost all this stochasticity accumulates at short times, at large t (after the population has become deterministic) we can describe the overall effects of stochasticity in terms of a probability distribution 
. This is a big simplification, because the full probability distribution conditioned on nonextinction, A(n, t), depends on both n and t, while for large t is independent of t, as we show below. This simplification is possible because at large t the only time dependence is the deterministic exponential growth.
We can justify the above heuristic argument rigorously. The definition of is just a transformation of n, 
ne–st. This is valid in the early stochastic phase of growth as well as in the later deterministic phase. However, in the stochastic phase we do not expect that will be independent of t. As we have the probability distribution A(n, t), it is straightforward to transform this to the distribution 
. When we take the large-t limit of , it becomes independent of t. This justifies our expectation that at large t, we have , independent of time.
Rather than using the probability distribution of , it will prove useful to define a related variable by
 | (15) |
The random variable is simply related to : 
. Since is a simple transformation of n, we can immediately calculate 
(with 
the probability density as we are treating as a continuous variable) from A(n, t). We find
 | (16) |
As with , this describes the distribution of n both in the deterministic and in the stochastic phase. Since n depends on t, so does the distribution of . However, as expected from the previous discussion, the distribution of becomes independent of t for large t. We define est as (t 
) and find
 | (17) |
est can be easily computed from this distribution. We have
 | (18) |
is Euler's constant = 0.577216.
We see from Equation 16 that the large-t condition required for the distribution of to become independent of t is . This is the time at which 
. This indicates that our choice of as the size at which a population becomes established is appropriate. After a time 
, when the population on average reaches this size provided it has not gone extinct, the probability distribution of begins to become independent of t, indicating that the behavior of the population crosses over from mostly stochastic to mostly deterministic.
The variable est has an intuitive interpretation: est is the time at which n would have reached size had it always grown deterministically, as calculated by looking at n(t) at large t and extrapolating backward. This is illustrated in Figure 4a. We can therefore approximate the destined-to-be-established subpopulation as drifting randomly for a time est, at which time it reaches size and then grows deterministically thereafter. With this simplification, the only important stochasticity is the duration of the drift period. This is the key simplification that allows us to smoothly connect the branching process with the nonlinear dynamics once the subpopulation is no longer rare. It jibes with our intuitive expectation that the subpopulation is dominated by drift when rarer than and then behaves deterministically once it exceeds this size. Note, however, that in addition to telling us nothing about n(t) before time est, it also gives a slightly inaccurate picture immediately after est when n(t) is . The time est is not in fact the time at which the subpopulation reaches size (see Figure 4a). Rather, it is the time at which n(t) would have reached size if we assumed that it always behaved deterministically, but it gets the large-t behavior right. In fact, some small drift does take place after reaching size ; our approximation does not ignore this drift, but rather adds up all the drift that takes place through all the time and rolls it into a change in est. This can thus be thought of as the time at which the mutation establishes. In asking how quickly beneficial mutations accumulate, this is the most natural variable.
|
est < 0; the distribution B(est) above shows that this is not even particularly improbable. This reflects the fact that, given that a mutant subpopulation is not going to go extinct, it is reasonably likely to grow remarkably fast in the early stochastic phase. A est < 0 simply indicates that the mutant subpopulation grew so fast when rare that if we look at the subpopulation size much later and assume it always grew exponentially at rate s, the subpopulation would have had a size > at t = 0.
We note that 
, while 
for large t (as always, conditional on nonextinction). This may naively seem inconsistent, since 
for large t. However, it merely reflects the fact that eX 
eX. The difference between these two averages is in fact the essential reason that est will prove to be such a useful variable to focus on. This is because the value of n(t) depends much more sensitively on the tails of than does est.
Mutants generated by a changing population:
The above analysis of the population size of a clone founded by a single mutant individual is an important building block. However, it does not address the full problem. We must now ask how the mutants arise in the first place. In the simplest case, we might imagine a wild-type population of size N, starting with 0 mutants at time t = 0. This population generates mutants at rate NUb. Each mutant follows the dynamics given in the above section, beginning at the time it was created, but now we have multiple such initial mutants that are created at random times.
Generally, the relevant process is even more complex. Starting from a wild-type population, a single-mutant subpopulation is generated, experiences a stochastic period, and then begins to grow deterministically. Then double mutants are created by mutation within the single-mutant population while it is still growing (i.e., before it fixes). The rate at which these double mutants are generated increases with time because the single-mutant subpopulation is growing. Later, the double mutants may themselves generate mutants before they fix (and possibly before the single mutants fix), and so on.
We therefore must tackle a more general problem: the distribution of the population size n(t) of a mutant subpopulation that starts with 0 individuals and is "fed" by mutants from a less-fit subpopulation of (growing) size f(t). If this less-fit clone is small enough that its growth is s
TOP
ABSTRACT
HEURISTIC ANALYSIS AND INTUITION
, and saturation is strong only when a subpopulation size is large, n 
, extrapolating backward from its long-time deterministic behavior. This includes both the time to generate a mutant destined to establish and the time for it to drift to substantial frequency.