INTRODUCTION

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The process of evolution is a consequence of the interplay of mutation, selection, and chance on a population of organisms, leading to an observable change in its genetic makeup. Since the time of Darwin, the influence of these factors on the evolution of organisms ranging from bacteria to humans has been intensively studied, both experimentally and theoretically, leading to a very large body of literature. Only recently, however, has attention been turned toward special problems in the evolution of viruses. Virus evolution is of particular interest and importance for three reasons. First, we desire to gain an understanding (usually in the absence of a fossil record) of how modern viruses have arisen from their earlier forms, both in recent times and in parallel with the evolution of their hosts. Second, the evolution of a virus during the course of infection of a single host, or along a short transmission chain, is of great importance in creating new populations with properties altered in important ways, such as evasion of the immune response, resistance to antiviral therapy, or altered virulence. Third, because of their high replication rates, simple genomes, large population sizes, and high mutation rates, viruses make good models for studying and testing evolutionary theory.

Particular attention has focussed on understanding the evolutionary forces that act on human immunodeficiency virus (HIV) during the course of infection of a single human host. HIV displays a remarkable extent of genetic variation concurrent with a high speed of evolution: in the most variable region of the genome (env), individual genomes within a population from an infected person can vary by as much as 3 to 5% (2, 43, 78); substitutions in env accumulate at a rate of approximately 1% per year (71), 50 million times faster than in the small subunit of rRNA (61). This variation has important consequences. It allows the virus to evolve to infect different cell types (9, 20, 30) and to rapidly become resistant to otherwise highly effective antiviral drugs (10, 47, 50); it may play a role in evading the immune system (4, 56, 73, 79). Furthermore, its high mutation rate (estimated to average about 3 × 10⁵ per nucleotide site per replication cycle [49]), large population size (variously estimated from about 10⁷ to 10⁸ productively infected cells), and continuous steady state, in which the large majority of virions and productively infected cells turns over every day (25, 77), create a situation which, at least in principle, is amenable to (and requires) mathematical modeling.

To date, a number of modeling approaches have been applied to understand the evolution of HIV in vivo. These approaches use either population genetic (mutation frequency distribution) or phylogenetic inference using virus sequences obtained from HIV-infected individuals. In general, they are based on one of two different theoretical frameworks to the evolution problem. Deterministic approaches, including quasispecies theory (15, 26), assume that the population size is very large, such that the frequency of a given mutation at any given time is completely predictable if one knows the initial frequency, the mutation rate, and the selection coefficient (i.e., the differential growth rate conferred by the different alleles). At first glance, such approaches would seem justified by the large number of infected cells at each generation (21); however, a number of factors, such as variation in the replication potential and generation times among infected cells, may lead to an effective population size much smaller than the actual number of infected cells. Stochastic models, as applied to HIV (to this point), proceed from the opposite assumption: that the effective population size is so small (or that selective forces are so weak) that random drift dominates over selection. The hypothesis of selectively neutral mutations has a long, successful history in describing the evolution of organisms where populations are small (and not uniformly distributed) and mutation rates are very low (36). Their applicability to virus populations remains to be established. Many of the assumptions that underlie neutral theory are not appropriate for virus populations, and a number of characteristics of HIV genetic variation in vivo, such as the uneven ratio of synonymous to nonsynonymous changes in different regions of the genome (5, 44, 48), argue against simple application of neutral theory. However, inclusion of selection effects in evolutionary analysis (for example, the coalescent method) presents a mathematical challenge that has not yet been fully solved in a practical fashion, although progress toward this goal has been made recently (42, 55).

As an example of the difference between deterministic and stochastic models, consider the question of the frequency in a population of a mutation that is slightly deleterious to virus replication. In a deterministic system, it can be easily calculated that the frequency of such a mutation in the population will come to equilibrium at a point equal to the mutation rate divided by the selection coefficient (24). In a stochastic system, the population will usually be completely uniform in one variant or the other (76), switching rarely but rapidly from one form to the other. This theoretical experiment is of great practical importance in that it describes the appearance of a mutation that can confer resistance to an antiviral drug even before treatment.

To solve this problem and many others, it is clear that a more general theoretical framework is needed: one that takes into account both selection and drift under a set of assumptions more appropriate to viruses than is found in theoretical works published to date. Our aim in this work was to develop, from first principles, a general theory that includes the effects of both selection and drift on a population. We use a set of assumptions appropriate to virus populations, focusing on the interplay between deterministic and stochastic behavior in the context of virologically realistic experiments. We apply these to the simplest possible model: mutation at a single site with only two alleles, replicating in a steady-state system (that is, a constant number of infected cells) under the influence of constant selective pressure in a single isolated population. Because we are dealing with a single locus, we do not consider recombination explicitly; because we are dealing with haploid populations, we do not have to consider allelic dominance. It should be noted that although we do not consider recombination explicitly, the presence of strong recombination must be, in fact, implied for the one-locus approximation to be quantitatively correct. Also, nonconserved loci must be spaced sufficiently far apart in the genome, depending on the recombination rate. Even in the absence of recombination, the one-locus approximation is a useful starting point for understanding interactions between selection and stochastic factors at a qualitative level. We present a complete model that considers the full range of possible values for population size, mutation rate, and selection effects. Despite its simplicity, the model is surprisingly rich in its descriptive power. At the extremes, the results of this model correspond to the standard results of deterministic or neutral theory; however, we have found that there is a large range of values for the key parameters in which the system behaves in an intermediate fashion: under some conditions its evolution is dominated by stochastic factors, whereas at other times it behaves in a nearly deterministic fashion. We refer to this range of parameter values as the "selection-drift" regime and describe its properties in detail.

This work is divided into two major parts. In the first, we present all the principal results in qualitative terms, using language appropriate for a reader trained in biology and with a moderate level of mathematical sophistication. This part is accompanied by a number of illustrative examples obtained by computer simulation. Although keyed to the mathematical formalism of the second part, it is designed to be read independently and to provide the reader with an understanding of the principal results and their biological significance, particularly in the context of virus populations. The second part is a formal mathematical derivation of the principal results of the model. These results are listed at the beginning of each section and derived in the following subsections. Although some of the derivation presented is not novel, in that it parallels classic work of a number of population biologists (18, 19, 23, 24, 31, 37, 81, 82), its formal application specifically to virus systems is, to the best of our knowledge, a new approach, and we present it in full for this reason, as well as to provide a thorough and self-contained review. Although some of our mathematical methods differ from the classic methods, the final results are identical.

The presentation in both parts of this work proceeds in parallel. We first develop the basic evolution equation, which describes, at least in a statistical sense, the change in frequency of a mutant allele as a function of time and the key parameters: mutation rate, selection coefficient, and population size. We then present the predicted results, for all three regimes, of a set of virological experiments: accumulation and reversion of deleterious mutations, competition between mutant and wild-type viruses, gene fixation, mutation frequencies at the steady state, divergence of two populations split from one population, and genetic turnover within a single population. Next, we discuss sampling statistics and the application of this theory to some specific real-world experimental issues of virus and organismal evolution. Finally, we discuss the application and extension of this theoretical framework to other problems, including multilocus evolution and phylogenetic analysis.

QUALITATIVE DISCUSSION AND COMPUTER SIMULATIONS

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Description of the Model and the Evolution Equation

Virus population model. First, we choose a basic model of virus evolution. For the purposes of simplicity, we consider the evolution of one nucleotide position at a time, and we assume that each nucleotide has a choice between only two alleles. (Such a model applies directly to multiple loci if the evolving loci are sufficiently distant and the recombination rate is sufficiently high. Evolution at closely situated loci or in the absence of efficient recombination is not independent [see "Many loci and other aspects" below].) Conventionally, we denote the better-fit allele as wild type and the less-fit allele as mutant. A deleterious mutation event (from wild type to mutant) will be referred to as forward mutation, and an advantageous mutation event will be referred to as reverse mutation. Each separate nucleotide will be characterized by two parameters, both of which are assumed to be much less than unity: the mutation cost (or selection coefficient), s, which is the relative difference in fitness between the two alleles, and the mutation rate per base per replication cycle, µ. We assume that mutations at different nucleotides have a weak additive effect on virus fitness. In doing so, we neglect epistasis (coselection) arising due to biological interaction between nucleotides at both the nucleotide and protein levels. We also ignore linkage disequilibrium between loci due to random drift, so that different nucleotides evolve independently (see the Introduction). The mutation rate is set, in our work, to be the same in the forward and reverse directions. For example, for HIV in infected cells the mutation rate per base is in the range of 5 × 10⁶ to 5 × 10⁵, depending on the type of substitution (49, 68). The selection coefficient will vary over a wide range according to the specific base and to the specific conditions of replication, but it is assumed to be constant over the period of observation; in other words, there is no selection for diversity.

View larger version (17K): [in this window] [in a new window]   FIG. 1.   (a) Drift of genetic composition due to random sampling of infecting virions. Circles denote infected cells, and small diamonds show free virus particles. Black and white denote virus genetic variants. (b) Full virus population model including random drift, selection, and mutation. Two consecutive generations of infected cells are shown. Lines radiating from a cell denote virions, some of which, as shown by arrows, infect new cells. Mutant cells yield fewer progeny per cell. A small fraction of infecting virions, m1 and m2, mutate to the other variant.

Stochastic equation of evolution. Different meanings can be assigned to the word "evolution." For the task at hand, evolution of the population is characterized by the dependence of the frequency of cells infected with mutant virus on time. In deterministic dynamics, which applies only in very large populations of infected cells, if one knows the initial mutant frequency and has the appropriate equations, one can, in principle, predict the mutant frequency at later times with arbitrary precision. (In practise, the equations are never known exactly, since there are many different factors in play, but this is a separate issue [68].) By contrast, in the presence of random factors, the time dependence of the mutant fraction cannot be predicted even in principle. Even if one knows its precise initial value, the error with which one can predict its value later grows with time. If random factors are strong, the error in the mutant frequency and its value become eventually comparable. Evolution of the mutant frequency, in other words, is a random process.

View larger version (25K): [in this window] [in a new window]   FIG. 2.   Illustration of the stochastic evolution equation. (a) The equation shown is derived from equations 1 and 2 in the particular case where the probability density, (f,t), is a narrow peak. Its right-hand side is a composite of three terms describing the effects of random drift, selection, and mutation on the change in the probability density with time. (b to d) The lower panels show the local changes in (f,t), corresponding to each part of the right-hand side of equation in panel a, and the upper panels show the resulting effects: spread (b) and shift (c and d) of the peak. Solid and dashed lines show the peak at two adjacent moments in time.

View larger version (26K): [in this window] [in a new window]   FIG. 3.   Stochastic evolution equation (equations 1 and 2) and its boundary conditions viewed through the formal analogy between the probability density and the local gas density. The walls at f = 0,1 correspond to the two monomorphic states. (a) Gas particles subject to diffusion and a directed force when far from the walls. (b) Boundary conditions at large population sizes: gas particles bounce off the walls; the total flux at a wall is 0 (equation 2). (c) Small population sizes: gas particles can condense on or evaporate from a wall; the total flux at a wall does not need to be 0 (equations 5 and 6).

Boundary conditions: properties of almost monomorphic populations. In the real world, the mutant frequency cannot be less than 0 or greater than 1, yet the master equation has no such restriction. Thus, the stochastic equation in Fig. 2a (and equations 1 and 2) is incomplete without describing what happens near ends of the allowed interval for the mutant frequencies, 0 and 1. The analysis shown in Fig. 2 is for the case where there is a large number of minority allele copies (that is, f is not near 0 or 1) and treats the mutant frequency (f) as a continuous variable. In many important cases, one also needs to describe the evolution of a population with only a few copies of the minority variant. The boundary conditions where f is near 0 and 1 have to be derived independently from the virus population model described in Subsection A. The derivation given in the mathematical section of this review shows that the conditions differ depending on the interval of population size, as follows.

(iv) Gene fixation (this experiment, which has received a lot of attention in population biology [19, 24, 34, 38, 80] and which is very useful for understanding other stochastic experiments, is defined only in small populations in which the total mutation rate per population, µN, is much less than 1; suppose that a single advantageous allele is introduced into an otherwise monomorphic population [f = 1/N]the allele will have one of two fates: either it will be lost due to random drift [Fig. 1a] or it will spread to the entire population, i.e., become "fixed"; the questions are: what is the fixation probability, and, if the allele is fixed and does not become extinct, how much time will it take, counting from the moment it appeared? One can also ask a more general question: what is the probability of having a new allele to grow into a subpopulation of a given size before it becomes extinct?).

(v) Steady state. Whatever the initial condition, after a sufficient time, the system passes to the stochastic steady state, in which the probability density no longer depends on time; we consider this relatively simple case separately.

(vi) Genetic divergence. One splits a steady-state population into two isolated parts. Initially, both populations have a random but identical genetic composition, from which they independently diverge. As time goes on, their respective random compositions correlate less and less. The question is, what is the characteristic time at which the loss of correlation occurs?

(vii) Genetic turnover? This experiment studies the average timescale associated with random fluctuations of the mutant frequency in the steady state.

The probability density () of the mutant frequency predicted by the stochastic equation is the main observable parameter. Unfortunately, to measure it directly, one would have to generate a histogram of mutant frequencies for a very large ensemble of populations. More amenable for experimental testing are the average (expectation) values (equation 36) and the standard deviations or variances (equation 37) of different stochastic parameters, which require a smaller number of populations to measure. Below we introduce some useful parameters whose statistics can be measured in the different experiments we outlined above. At the same time, their predicted statistics can be expressed via the probability density, as shown in the mathematical section of this review. In what follows, we assume that each parameter, for each given population, is measured with a high precision from a sufficiently large sample of sequences. The sampling effects will be discussed separately below.

The first parameter is the mutant frequency itself (f), which is self-explanatory. Its value can be compared directly with the experimental value, provided that the wild-type (best-fit) nucleotide is known.

The second is the intrapopulation genetic distance (T), defined as the proportion of sequence pairs (randomly sampled from the virus population) which differ at the base of interest. Although there are other ways to measure intrapopulation variability, we will use this definition, known in population biology as Nei's nucleotide diversity. It is equivalent to the standard definition of the genetic distance in virology as the average number of pairwise differences among randomly selected genomes, except that it applies to a single base rather than to a long genomic segment. By definition, T is calculated as 2f(1  f), and varies between 0 (at f = 0 or 1) and 0.5 (at f = 0.5). The genetic distance is usually a more convenient measure of population diversity than the mutant frequency itself since it does not require knowledge of the wild type sequence.

The third is the interpopulation genetic distance (T₁₂), which is defined in the same way as the intrapopulation genetic distance, except that the two sequences of each pair are sampled from two different populations (equation 40). The interpopulation distance is 0 when the two virus populations consist uniformly of the same genetic variant and 1 (100%) when the two virus populations are composed entirely of opposite genetic variants. The interpopulation distance, as one can show, cannot be smaller than the average of the two intrapopulation distances. Therefore, it is sometimes more convenient to consider instead the relative genetic distance between two populations (D), defined as the difference between the interpopulation distance and the average of the two intrapopulation distances [T₁₂  (T₁ + T₂)/2]. This parameter (equation 41) varies between 0 (two populations have an identical genetic composition) and 1 (one population is pure mutant, another is pure wild type). There are alternative definitions of the relative distance (54). We find this definition more clear intuitively; also, its statistical moments (average, variance) are relatively easy to calculate.

All the previous parameters can be measured at one time point, both for dynamic experiments (the first three experiments in the beginning) and in the steady state. Since all of them are, in general, stochastic, an average and standard deviation has to be calculated for each. The next parameter is more complex: it requires measurement at two different times. We define it on average and for a steady state population only.

The fourth parameter, the time correlation function of mutant frequency [K(t)], describes how quickly the system "forgets" the preceding random fluctuation of the mutant frequency (equation 45). The time correlation function usually has a maximum when the time difference is 0 and vanishes at large time differences. The characteristic time at which it decays by 50% (or, say, by a factor of e = 2.78... ) from its maximum gives the timescale of random fluctuations. The form of this decay (e.g., exponential or negative power) may be a good fingerprint of a virus population model or, within a given model, of a particular population size.

In the mathematical section of this review, we calculate these parameters for different gedanken experiments and different intervals of population size. In this section of the review, we discuss these results qualitatively and illustrate them, when possible, with Monte Carlo simulations.

View larger version (17K): [in this window] [in a new window]   FIG. 4.   The steady-state probability density in the neutral case. The curves show ss(f) when s  µ at different population numbers, N. Numbers on the curves show the corresponding values of µN.

Case with selection: µ  s 1. The situation when the selection coefficient is less than 1 but still much larger than the mutation rate is more relevant for RNA viruses and more interesting theoretically. As in the neutral limit, the larger the population size the smaller the fluctuations.

The selection factor can be neglected only if a population is very small, much smaller than the inverse selection coefficient (Ns 1), a case that has the same properties as the above-described drift regime. At larger population sizes, selection is crucial and causes the probability density (equations

View larger version (8K): [in this window] [in a new window]   FIG. 5.   Schematic plot of the steady-state probability density in the selection-drift regime. The curve shows ss(f) for the case when 1/s N 1/µ. Note the very narrow peaks at  = 0 and 1, together with the tail extending from  = 0.

View larger version (20K): [in this window] [in a new window]   FIG. 6.   Dependence of the observable parameters at steady state on the population number. N varies over the three main intervals. (a) Average mutation frequency, and genetic distance, . (b) Relative standard deviations of the same two parameters. (c) Fragments of representative Monte Carlo simulations in the respective intervals of N (see Fig. 10 to 12 for details).

Deterministic dynamics. Deterministic and stochastic theories operate with different dynamic variables. The former considers the time dependence of the frequency of mutants, and the latter uses a more complex object, the time-dependent probability density of the mutant frequency. It is important to ensure that the two approaches converge to the same result in the limit of infinite population, when they are expected to describe deterministic evolution, albeit in a different way. For this purpose, in the mathematical section of this review we solve the dynamic stochastic equation (equation 1) for the case of large populations. The resulting probability density, as expected, is a very narrow peak located at the time-dependent mutant frequency (Fig. 7b), which satisfies the deterministic equation of evolution (equations 60 and 61).

View larger version (10K): [in this window] [in a new window]   FIG. 7.   Probability density of the mutant frequency in the deterministic limit. (f) is represented by the mathematical expression for µN 1 (a) and a schematic plot (b).

View larger version (15K): [in this window] [in a new window]   FIG. 8.   Schematic dependence of the mutant frequency, f, on time in the deterministic limit. The three curves correspond to three different initial values of f(0): accumulation of mutations [f(0) = 0], growth competition [f(0) = 1/2], and reversion of a mutation [f(0) = 1]. The value of the ratio µ/s used in the figure is unrealistically high for viruses and is used for clarity of plot only. Dashed lines show initial slopes.

Boundaries of deterministic approximation. Random drift, always present even in very large populations, causes the frequency of mutants to fluctuate around its deterministic value. As the population size decreases, the magnitude of fluctuations becomes comparable to the average frequency of the minority allele (either mutant or wild type), and the deterministic description breaks down. The corresponding condition on the population size varies significantly depending on the initial conditions of the experiment (equation 65). When the population starts from a monomorphic state (reversion or accumulation), the deterministic criterion is met when µN is much larger than unity. A population that is strongly diverse to start with, as in the growth competition experiment, is already deterministic at a much smaller population size in the selection-drift regime. (The criterion for diversity is that the mutant frequency must be higher than its characteristic "tail" at steady state [Fig. 5] ). The reason for this difference is that a small polymorphism is influenced by rare and random mutation events while a strongly polymorphic population is controlled by selection alone.

Decay of the polymorphic state and gene fixation. We start our discussion from the population that is initially polymorphic, somewhere in the middle between 0 and 100%. As already discussed (see "Description of the model and the evolution equation"), mutations are not important in a polymorphic population, since they occur in the population with a frequency, µN, much less than 1 per generation. Therefore, random drift remains the only factor causing variation of the mutant frequency in time. As time passes, the mutant frequency drifts until the population accidentally ends up in either monomorphic state (cf. Fig. 1a). A representative random process is illustrated by computer simulation in Fig. 9b. The average time (the number of generations) it takes for a population to become monomorphic (i.e., for either variant to be fixed) is on the order of the population size (equations 81 and 82) (32, 80). The fixation time is quite random: its representative fluctuations are on the order of its average value. The same process can be understood in another way, from the time evolution of probability density. Figure 9a shows how the probability density, initially a narrow peak located, e.g., at 50% composition, gradually spreads out to the entire interval and then decays.

View larger version (21K): [in this window] [in a new window]   FIG. 9.   Decay of polymorphism in the drift regime. A growth competition experiment for the initial condition f0 = 0.5 and Ns 1 is shown. (a) Change in the probability density in time (equations 81 and 82). (b) The two stochastic dependences f(t) were obtained by random runs of a Monte Carlo simulation program written for the virus population model described in the text. Parameters are shown in the figure.

View larger version (30K): [in this window] [in a new window]   FIG. 10.   Time dependence of the mutant frequency in the drift regime on a long timescale. (a) Change in the probability density in time (equations 85 to 87). Sharp peaks at f = 0 and 1 correspond to the monomorphic states; their probabilities are shown by the relative peak heights (arbitrary units). (b) One Monte Carlo run is shown for Ns 1 and the initial condition f0 = 0. Parameters are shown in the figure.

Divergence of populations which have been separated and the time correlation function. The longer timescale, 1/µ, also appears in the time correlation function of mutant frequency, which characterizes the timescale of random fluctuation in the steady state and the divergence of populations which have been separated (see "Experiments on evolution and observable parameters" above). The value of the relative genetic distance, D, gradually changes from 0 to a constant value corresponding to statistically independent populations (equation 90). (Note that some other measures of interpopulation genetic distance used in population biology do not have an upper limit [54].) As it turns out, the time of this transition, the half time of the correlation function decay (equation 91), and the time in which the probability density becomes symmetric (above) are on the same order, the inverse mutation rate. Indeed, all three times are determined by the waiting time for a successful gene fixation.

Here we consider nonequilibrium experiments in the most interesting interval of population sizes (Table 1). The relative role of selection and stochasticity in population dynamics, as derived from the evolution equation in the mathematical section of this review, depends on the initial genetic composition. The dynamics of growth competition is almost deterministic (see "Deterministic dynamics and its boundaries" above), so that this experiment need not be discussed again. In the accumulation experiment, the overall dynamics is stochastic, except for the average values of the mutant frequency and the intrapopulation distance, which are, remarkably, the same as in the corresponding deterministic conditions.

Accumulation. As in the drift regime (see above), accumulation can be described as a spread of the peak of the probability density initially located at 0 (uniform wild type) into the interval between 0 and 1. However, unlike in the drift regime, the resulting steady state is not symmetric of a large peak (Fig. 5) (equation 48 or 49 to 51). The process of accumulation is reduced to generation of a small tail describing rarely occurring weakly polymorphic states (Fig. 5). As a result, the initial peak at 0 does not decay greatly and the steady state is reached in the same time as in deterministic selection (see "Deterministic dynamics and its boundaries" above) given by the inverse selection coefficient (1/s), i.e., faster than all timescales in the drift regime (equations 103 and 104).

View larger version (26K): [in this window] [in a new window]   FIG. 11.   Simulated accumulation of mutants in the selection-drift regime. One random Monte Carlo run is shown for 1/s N 1/µ and the initial condition: f0 = 0. The double-pointed arrows and dashed line show predicted scales in time and in the mutant frequency. The solid smooth line shows the deterministic dependence for comparison. Parameters are shown in the figure.

View larger version (26K): [in this window] [in a new window]   FIG. 12.   Simulated accumulation of mutants in the selection regime. Dashed lines show the average and the standard deviation at steady state for µN 1 calculated using equations 58. Parameters are shown in the figure.

Divergence of separated populations and the time correlation function. The characteristic times of divergence of separated populations (Eq. 105) and the decay time of the correlation function (Eq. 106) are on the order of the inverse selection coefficient, 1/s. Both experiments show for how long, on average, the system "remembers" its previous random fluctuation. The answer: for the half-life of a typical mutant clone, before it becomes extinct. This is because separate clones appear, due to mutation, at independent random times.

Reversion (fixation of an advantageous variant). A reversion experiment, in which the initial population is uniformly mutant, behaves rather differently. Although the same scales for time and the minority allele frequency appear in this case, they have different meaning. As in accumulation, random drift and selection dominate in smaller and larger wild-type colonies, respectively. However, in this case, selection accelerates rather than hinders the growth of a new clone. The probability that a single wild-type allele will manage to grow to a size equal to the inverse selection coefficient, 1/s, is low, s. However, above this critical size, the rest of its growth will be carried out by selection in a deterministic manner, i.e., with a probability close to 1 and over the deterministic timescale, 1/s (see "Deterministic dynamics and its boundaries" above). Hence, the bottleneck of reversion is in reaching the critical size despite random drift; after that, a clone is likely to be fixed in the population. Stochastic dynamics below the critical size is the same as in the accumulation regime (selection is not important). The average waiting time for reversion to start is determined by the fixation probability, s, and by the frequency at which single alleles are generated in a population at each generation, µN, which gives the time ~1/(µNs), i.e., the same scale as the waiting time for a high peak in accumulation regime (Fig. 11) (equation 107) (51). A few examples of reversion curves are shown in Fig. 13. Evolution of the probability density is shown in Fig. 14, including evolution of the density of polymorphic states (Fig. 14a) (equation 108) and of the two probabilities of monomorphic states (Fig. 14b) (equation 107).

View larger version (32K): [in this window] [in a new window]   FIG. 13.   Simulated reversion (fixation of advantageous variant) in the selection-drift regime, 1/s N 1/µ. The reversion curve in the deterministic limit, N = , is shown by the dashed curves for comparison. Parameters are shown in the figures. (a) Beginning of the reversion curves. Two random Monte Carlo runs are shown for each of two population sizes. (b) Full reversion curve at a smaller population size. Three random runs are shown. Solid lines show the average and the standard deviation of the mutant frequency calculated using equation 107.

View larger version (12K): [in this window] [in a new window]   </