INTRODUCTION

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Human immunodeficiency virus (HIV) shows a high level of genetic diversity compared to other viruses (45). Genetic variation in antigenically important regions complicates attempts at vaccine development (3, 29, 48, 49), and drug resistance mutations limit the efficacy of treatment with replication inhibitors (6, 25). From another perspective, a pattern of genetic evolution contains important information about biological factors acting on the virus population. In particular, the role of the immune response in HIV infection is difficult to assess experimentally. It remains a subject of intensive debate, and the mechanism of HIV persistence in vivo in the face of an apparently vigorous immune response is still unknown. At the same time, the presence of an at least partly functional immune response can be inferred from the genetic diversity data. The large proportion of nonsynonymous substitutions, e.g., in env reveals a powerful selection for diversity (4, 33, 41), presumably caused by the immune response and by adaptation to different cell types.

The average genetic distances and relative proportions of different types of substitutions vary strongly both between and within different HIV genes, implying that there are different dominant factors of evolution at different loci. The highest degree of diversity, 3 to 5% within an individual (1, 21, 47) and 8 to 17% between individuals from the same geographical location (1, 20, 26), is observed in the variable regions of the env gene. A somewhat smaller but still impressive level of intrapatient diversity has been reported for gag (1 to 2%) (52) and for pol and pro (0.4 to 1%) (22, 31). In pro, synonymous and conservative substitutions dominate overall variation (22), implying a stronger role for purifying selection than in env.

The aim of this work was to reconstruct the mechanism of evolution in a relatively conserved HIV gene, such as pro, by using data on its genetic variation (22). Genetic evolution of a virus population may be affected by a multitude of different factors (19): mutation, selection forces, stochastic factors (random drift), recombination, transmission between individuals, spread of the virus between infected organs, etc. The main difficulty, as is always the case with mathematical theories of real experimental systems, is that it is not known in advance which factors, among many, are of the greatest importance to the behavior of the system. The quality of approximations cannot be evaluated until the analysis is over, and it depends on the parameters one is trying to predict. We thus face a time paradox: on the one hand, a well-defined set of approximations must be introduced in the beginning to make the analysis possible and intuitively clear; on other hand, finding the right set of approximations (the biological model) is the final aim of such research. The only escape from the time paradox that we know of is a dynamic interplay between experiment and theory (38). One starts from a simple model to match several important experimental features. After calculating its predictions, one searches for a contradiction to the experimental data and, when such is found, checks initial assumptions by changing them, one by one, and estimating what had changed in the predictions. This process has to be repeated several times. The resulting model, when all available information is used up, is considered a working model subject to further experimental tests.

In the present work, we applied this strategy as follows. After several attempts, we chose a set of experimental features from the database of Lech et al. (22) that are difficult to explain within a mathematically and biologically consistent model. This allowed us to select a working model (or models) from a large number of possibilities. We found a model of deterministic evolution in an individual under conditions of purifying selection that agrees with several experimental features from those we have chosen and used it as a starting model of the evolution of pro. Next, to describe genetic variation of the initial virus variant in infecting inocula and to predict the average frequency of variable sites in individuals, we had to take transmission between individuals into consideration. We have determined that such a model can explain a high frequency of variable sites in an individual, which is a very important experimental feature. However, this theoretical result turned out to be entirely dependent on our initial assumption that an animal in the natural host population is, in 99% of cases, infected from a single source. Even if coinfection occurred as rarely as in 5% of cases, this would bring the frequency of variable sites many times below the observed value. We did not find independent facts that would confirm such a low frequency of coinfection. Therefore, we tried to find another explanation for the high frequency of variable sites by searching for a weak spot in our approximations. We have estimated (approximately) quantitative contributions from a number of potential factors of HIV evolution to the frequency of the variable sites and found most of these contributions to be very small. The factors we considered included the possibility that the genetic variation was dominated by random drift (39), temporal changes in the virus population size, adaptation of the virus sequence to different cell types, nonuniformity of virus distribution in the body and virus spread between infection sites, the possibility that the coinfection sources are not independent, coselection between loci, and the immune response. Finally, after excluding all apparent possibilities, we could find a plausible explanation, alternative to a very low coinfection rate in the natural host: in the presence of the immune response, the best-fit sequence of virus will differ strongly between individuals due to individual variation in major histocompatibility complex class I (MHC-I) subtypes, and selection for a variant unrecognized by the immune system leads to a cascade of compensatory mutations, both synonymous and nonsynonymous. This mechanism is also seen following the appearance of drug-resistant mutants.

To make the work available to a broad audience, we parallel qualitative arguments in Results and Discussion with mathematical derivations, given in Materials and Methods. We consider in detail four principal models: deterministic evolution in an individual, the same model with transmission between individuals, the same model including coinfection of an infected individual from independent sources, and the model based on variation in MHC-I subtypes. The other, dead-end models and all approximations are discussed.

The reader should keep in mind that the present work is about the coarse-grained picture of evolution of pro in a typical patient; given the complexity of the problem, we did not attempt to explain the diversity of all HIV genes or all individual variations. Neither did we expect that models and predictions obtained in this work were necessarily final and seek to substitute mathematical analysis for experimental data. The hypotheses selected by this work eventually have to be tested experimentally.

	MATERIALS AND METHODS

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Calculation of genetic distances. The genetic distance at a chosen base, between or within patients, is mathematically defined as the probability that two sequences differ at a given base, if sampled randomly from the same or two different patients, respectively. This definition is equivalent to the standard definition of the genetic distance as the average number of pairwise differences between two samples of a genome segment, except it deals with a single base and makes sense only after averaging over many such pairs of sequences. Both genetic distances can be expressed in terms of the frequency of substitutions. Let f_ki be the frequency of substitutions at site i and for patient k, with respect to any chosen reference sequence, such as a database consensus sequence, a best-fit sequence, etc. Then, the intrapatient distance for each patient, T_ki^intra, and the interpatient genetic distances for each pair of patients, T_k1k2i^inter, are given by the respective expressions

(1)

The intrapatient distance, as follows from its definition, has a maximum, T_ki^intra = 1/2, at f_ki = 1/2. The maximum interpatient distance is T_k1k2i^inter = 1, which corresponds to either f_k1i = 1 and f_k2i = 0 or vice versa; the two populations are genetically uniform in opposite alleles. Note that the right-hand sides of equations 1 do not change upon replacement of f_ki with 1  f_ki; i.e., both genetic distances are independent of the choice of consensus, as they should be.

Evolution in an individual. The initial set of deterministic equations describing the evolution over time of two variants of a virus has the form

(2)

(3)

Here n₁ and n₂ are the numbers of mutant and wild-type proviruses, respectively; ₁ and ₂ are the replication coefficients; t_rep is the replication cycle time; and µ_f and µ_r are the forward and reverse mutations rates, respectively. We assume that t_rep is the same for the two variants and that cell death is a random Poisson process. Neither assumption is important for a long-term selection. The selection coefficient s is defined as the relative difference in fitness, s = (₂/₁)  1, and is assumed to be constant and to fulfill the double inequality µ_f(r)  s  1. The ratio ₂/₁ agrees with the standard experimental definition of the relative fitness measured by comparing the exponential growth rates, at a low multiplicity of infection, of the virus variant under study and the reference variant. Using the additional condition that the total population size is constant, n₁ + n₂ = N, as is the case during the asymptomatic stage of HIV infection (see the section on Verifying approximations), equations 2 and 3 can be replaced by the single equation

(4)

where f  n₁/N is the mutant frequency. Equation 4 is nonlinear since the replication coefficients, _i, must depend on the mutant frequency, f. It is implied that _i depends on the number of available target cells, which adjusts to keep the total provirus population constant. Although the clearance rate of infected cells is assumed to be constant and to be equal for the two genetic variants, the same equation, equation 4, can be shown to follow in a more general case as well, except that the definition of the selection coefficient must include the ratio of the clearance rates. Solving equation 4 for the arbitrary initial condition f(0) = f₀, we obtain

(5)

where _f(r)  µ_f(r)/s and error of the order of (µ_f(r)/s)² is allowed in the right-hand side. Two particular initial conditions in equation 5 yield reversion and accumulation dependences: f_rev(t) = f(t,1) and f_acc(t) = f(t,0). Plots of f_rev(t) and f_acc(t) for a realistic set of parameters are given below (see Fig. 4). We define the reversion time, t₅₀, as given by the equation f_rev(t₅₀) = 0.5; one obtains t₅₀ = (t_rep/s)ln(s/µ_r). In the limit of large t, the mutant frequency converges at the steady-state value, f_eq = µ_f/s 1.

Chain of single-clone transmission. Let us assume that each individual infects the next individual at time t* after the moment of his/her own infection with a single genetic variant. Let f_n^* be the probability that the virus variant which infects person n is mutant. The expected value of the mutant frequency in person n at time t is given by

(6)

The probability f_n^* is equal to the expectation value of the mutant frequency in the source of infection at the time of transmission, as given by

(7)

Together, equations 6 and 7 fully describe the evolution of a base along the chain of infection. If transmission occurs late, t* > t₅₀ (t*  t₅₀  t_rep/s), probability f_n^* can be shown to converge, after a few transmissions, to the individual steady-stage value µ_f/s. If transmission occurs prior to the reversion time, t* < t₅₀, f_n^* changes slowly with n, and equations 6 and 7 can be simplified to the differential equation

(8)

where

(9)

(10)

Solving equation 8 at the arbitrary initial condition f₀^*, we obtain

(11)

As follows from equation 11, the probability of occurrence of a mutant base converges, after approximately n_eq transmissions, to a "chain steady-state" value, f_n^*  f^*. The dependence of parameters f^* and n_eq on the transmission time t*, which is exponentially fast, is shown below (see Fig. 5a and b). In the range of transmission times t_rep/s < t* < t₅₀, the two parameters are simply related, f^* = µ_fn_eq/s.

Coinfection from independent sources. Let us assume that a pair of genomes is transmitted from two different persons or animals with a probability q and that a single genome is transmitted with a probability 1  q. Equation 6 is replaced by

(12)

where f_com(t) is the growth competition curve defined as f(t,1/2) in equation 5 (the dashed line in Fig. 4) and f_n^* meets recursive equation 7. Equation 12 assumes that the two infection sources are epidemiologically distant and, therefore, statistically independent. The chain steady-state value, f^*, is given by the smaller of two zeros of the quadratic equation

(13)

where f_acc, f_rev, and f_com are taken at t = t*. The dependence f^* versus t*, q = 0.05, is shown below (see dashed line in Fig. 5a). The maximum probability of a mutant base, f^*(t* = 0), is plotted below versus parameter q (see Fig. 5c). At intermediate values of q, such that µ_f + µ_r  q  1, we obtain the asymptotics f^* (t* = 0) = 2µ_f/qs. As expected, at 1/q  n_eq, this expression matches the formula f^* = µ_fn_eq/s obtained above for a singel-clone transmission.

Coinfection from the same source. Let us assume that transmission always occurs from a single source and that one and two virus variants are transmitted with probabilities 1  q and q, respectively. (Similar considerations apply to the case of two different sources which are very close epidemiologically.) When two variants are transmitted, the initial steady-state population is assumed to be 50% mutant. In this case, the two sequences sampled are not statistically independent and equation 12 does not apply. Instead, one can write two separate equations for expectation values of the mutant frequency, f_n(t), and of its square, y_n(t), respectively:

and

(14)

where f_rev(t), f_acc(t), and f_com(t) are given by equation 5 with f₀ = 1, 0, and 1/2, respectively; parameters f_n^* and y_n^*, are both defined by recursive equation 7. The chain steady-state values, f^* and y^*, are found from equation 14 and the usual steady-state conditions f_n^* = f_n1^* = f^* and y_n^* = y_n1^* = y^*. The general expressions are cumbersome, and we give them only for the limit t*  0 in which f^* is maximum. In this case, the dependence on q cancels out, and we get the same result as in the case of single-clone transmission (cf. equation 9 at t*  0):

(15)

Value of q predicted for HIV. We consider positions at which the wild type is A or C since such positions are predicted to dominate variable sites (see Fig. 5c). The average observable frequency of a variable base, f^exp, is given by the general expression

(16)

where (s) is the distribution density of s over different bases, and f^*(s) is the probability of a mutant base being inserted for wild-type A or C (see Fig. 5c and d). The limits of the integrals in s implied in equation 16 are the boundaries of the variable zone in s, which will be specified below. The factor g(s) is the fraction of patients tested during the time interval in which a base with selection coefficient s is observably variable (see Fig. 3), given by

(17)

where (t) is the distribution density of the time of testing among patients, L = ln(s/µ_r) = 7.8 (s = 1%), and coefficient   1 depends on the boundaries of the interval in f when the base is considered variable. At an average sample size of 20 clones, a site with 0.05 < f < 0.95 is observably diverse, which yields   3. Assuming that (t) is constant in the interval between 1.25 and 8.75 years, and treating L as a large parameter, from equation 17 we obtain

(18)

if 0.57 < s <4, and nearly zero otherwise. Here is defined as given by the equation t_repL/==5 years. The interval in s specified in equation 18 sets the limits of the integrals in equation 16. Approximating the distribution density (s) by a constant in this interval of s, and using the function f^*(s) calculated from equation 13, we find that equation 16 matches the observed value of variable sites, f^exp = 0.16, at q  0.085.

	RESULTS

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Two types of selection. There are several groups of factors which can shape the evolution and diversity of a virus genome at a given locus: mutation, random drift (13, 50), selection, transmission between individuals, and virus spread between different infection sites within an individual. Transmission and spread between infection sites create genetic bottlenecks, which modify the action of other factors, creating founder effects due to random sampling from the infecting population.

Sequence diversity in pro. To obtain a more detailed picture of genetic variation in pro, we analyzed the database of sequences described by Lech et al. (22). Sequences were obtained at a single time point from protease-inhibitor-naive patients. The 265 clones of proviral DNA were derived from peripheral blood mononuclear cells of 13 individuals by 70 cycles of nested PCR amplification, and the consensus sequence for each patient's virus was obtained. Of the 13 patients, two (06 and 12) had stop codons in their consensus sequences and were excluded from further analysis. For the remaining 11 patients, 6% of the sequences contained deletions, insertions, or stop codons; these sequences were also filtered out, since the present work addresses point mutations, which do not render the virus nonviable. The remaining database comprised 213 clones of proviral DNA. We determined the common consensus sequence by noting the most frequent variant for each base. The consensus sequence starts from the conserved sequence CCTCAgATCACTCTT (PQITL), where g shows a variable base, and is 297 bases long. It is identical to the subtype B consensus in the LANL database (19a). Of the total number of substitutions, 25% were transversions with respect to the consensus sequence. To simplify further analysis, we ignored transversions, replacing them by the consensus sequence. As a result, each base was represented by two genetic variants: either A/G or C/T. (A theory taking into consideration both types of mutation would have to contain more parameters, namely, the forward and reverse mutation rates for transversions, which are very low and known with poor accuracy. This would complicate the theory without much gain in information.) Finally, to partially correct for errors accumulated during PCR, we ignored mutations present in a single copy in the entire database. These sporadic mutations occurred with a frequency of 0.06% per base, with respect to the consensus, and were distinctly overrepresented in nonsynonymous changes relative to mutations appearing two or more times (Fig. 1; Table 1). This value, as well as the very small number of mutations which appear exactly twice in the database, is consistent with a PCR error rate of about 1 per 10⁵ bases per cycle, which does not exceed reported values (2, 44).

View larger version (21K): [in this window] [in a new window]   FIG. 1.   Intrapatient (a) and interpatient (b) genetic distances, averaged over patients, at different positions in pro. The upper and lower histograms in each figure correspond to synonymous and nonsynonymous sites, respectively. Dots on the upper horizontal line in panel a show the positions of sporadic mutations (seen only once in the data set).

View larger version (40K): [in this window] [in a new window]   FIG. 2.   Gray-scale diagram of intrapatient genetic distance, Tki, in different patients at different variable sites in pro. The intrapatient genetic distance is indicated by the degree of shading, as shown on the scale on the right. Letters and numbers under the diagram show consensus nucleotides and positions in pro, respectively.

View larger version (27K): [in this window] [in a new window]   FIG. 3.   Model of evolution of a nucleotide along the chain of infected individuals under purifying selection. The numbers at the top denote cycles of transmission. X signs denote mutants. A mutant base appears spontaneously in person n, who infects person A with the mutant and person B with the wild-type variant. Person A passes the mutant down the chain to person C, after which his/her own virus population slowly reverts to the wild type. In person D, who is stably coinfected with a pair of sequences polymorphous at that base, selection clears the mutant virus rapidly.

Model 1: evolution in an individual. Neglecting transversions (which we do in both theory and experiment), a base can assume one of two variants, either A/C or G/T. One of the two variants (by definition the wild type) is better fit, in the sense defined above. We assume that the wild-type sequence is the same for all individuals and all cell types involved. (This and other assumptions are discussed in detail in Discussion.) Each base is characterized by the relative difference in fitness between the two variants (the selection coefficient, s) and the mutation rate (µ). The best estimate of the HIV mutation rate is µ = 4 · 10⁵ for AG and CT transitions, and 5 to 10 times lower for the opposite transitions (24). The selection coefficient, s, varies strongly among different bases and is almost never known in vivo; therefore, it will be treated as a fitting parameter (see its formal definition in Materials and Methods).

View larger version (35K): [in this window] [in a new window]   FIG. 4.   Time dependence of the mutant frequency at different initial populations: purely mutant (reversion) (thicker line in upper panel), purely wild type (accumulation) (lower panel), or 50% mutant (growth competition) (dashed line). The thinner curve shows the intrapatient genetic distance T(t) during reversion. The horizontal bar is the time interval during which the base would be classified as variable (5 to 95%). Values shown in the upper left corner correspond to parameters as follows: the forward (wild type  mutant) and reverse mutation rates, µf and µr, corresponding to either A or C wild type; the replication cycle time, trep; and selection coefficient, s. The reversion half-time, t50, is given by the equation t50 = (trep/s)ln(s/µr).

Model 2: chain of single-clone transmission. In the absolute sense, the average proportion of patients sharing the same variable site (the frequency of variable sites in individuals) is still quite high, 16%, contributing to the high average intrapatient distance in pro. Can the reversion model explain such a high value? To answer this question, we have to explicitly consider the evolution of the virus along the chain of infection (Fig. 3). Suppose that each individual infects the next individual at a fixed time since the moment of his/her own infection with, as we still assume, a single genetic variant, either mutant or wild type at a particular base. A predictable parameter, related to the frequency of variable sites in an individual, is the probability that a base is mutant in the inoculum for person n in the transmission chain.

View larger version (38K): [in this window] [in a new window]   FIG. 5.   (a) Probability of a mutant base being in the inoculum at the chain steady state, f*, as a function of the transmission time, t*. Solid line, strictly single-clone infection; dashed line, coinfection from two sources with probability q = 0.05. (b) Number of cycles required to reach the chain steady state, neq, as a function of transmission time, t*. (c) Probability of a mutant base for a very short transmission time, t*  0, versus the probability of dual infection, q. Upper and lower curves correspond to wild-type A/C and G/T, respectively. (d) The same probability versus s, for a fixed value q = 0.01. trep = 2 days; µf = 4 · 105, µr = 4 · 106 for wild type A/C and vice versa for G/T.

Model 3: coinfection. Thus, if a single clone of virus is transmitted each time, any base with a small selection coefficient will eventually become highly diverse. (The selection coefficient has to be less than the inverse of the number of virus replication cycles between two transmissions [Fig. 5a]) Is this model a realistic explanation for the large number of variable sites in pro? To answer this question, we reexamined the approximations that were incorporated in the model and found that the theoretical explanation critically depends on the assumption that a recipient is always infected from a single source. As we show below, even very rare coinfections from two or more independent sources within a short time interval can prevent accumulation of mutants and sharply decrease the number of variable sites.

Value of q estimated for HIV. We now return to the experimental data to find out how small the probability of coinfection, q, must be to explain the high degree of diversity observed in pro. Since, as follows from our calculation, the probability of a mutant base is much less for sites with wild-type G or T (Fig. 5c), most variable bases are expected to have A or C as the wild type. If the test time and the replication cycle time did not vary between patients, the probability of receiving a mutant base in the inoculum would be equal to the observed frequency at which a variable site appears in individuals, which is 16%. Since both times, in fact, vary between patients, the probability of a mutant being in the inoculum must be higher than 16%, because if an infected person is tested too early or too late (compared to an average patient), given a limited sample size (~20), a reverting base will not be detectably diverse (Fig. 4). The variation in replication time, as follows from statistical analysis of data from 20 patients (17), is relatively small and can be neglected. Still, we have to take into account the probable broad distribution of the test time among patients. The time of progression to AIDS for HIV-infected individuals is almost uniformly distributed between 2 and 14 years (42). We assume that the test time is somewhat past the middle of the infection course and is distributed uniformly as well, between 1.3 and 8.7 years, with the average being 5 years. The distribution density of the selection coefficient around the value s  0.01 (which is, of course, unknown) also enters our calculation. Fortunately, as we had checked, the final answer happens to be rather insensitive to the shape of the distribution density; as an estimate, we assume a uniform distribution in the region of interest, which happens to be (see Materials and Methods) s = 0.005 to 0.05. Under these approximations, we estimate that the observed experimental frequency of a variable site, 16%, requires a q value of 0.01 or less (see Materials and Methods). At this value of q, the probability of a mutant being in the inoculum is close to 100% if the wild type is A or C and around 10% if it is G or T (Fig. 5c).

	DISCUSSION

Top Abstract Introduction Materials and Methods Results Discussion References

In this paper, we present and evaluate a model designed to explain the distribution of mutations observed in a relatively conserved region of the HIV genome. In its simplest form, the model describes the accumulation of diversity at individual sites as a consequence of passing of slightly deleterious mutations upon relatively rapid transmission of the virus from individual to individual, accompanied by a slow reversion within infected individuals. As described in more detail below, the model is robust to variation in most of the assumptions we initially made, with one exception. The accumulation of mutations during the chain of transmission is exquisitely sensitive to coinfection of infected individuals prior to further transmission from epidemiologically distant sources. To sustain the observed variability in pro, either the frequency of coinfection must be 1% or less or one of the other assumptions must be changed radically. We present an alternative explanation at the end of this section.

The scenario we describe here is also a possible mechanism for the accumulation of deleterious mutations in frequently passaged viruses, as observed in vitro for vesicular stomatitis virus (5, 10, 11, 43). Another possible explanation (10) is the Muller's ratchet effect, in which the processes of reversion at different loci interfere with each other. This effect of stochastic evolution theory, which applies to populations that are on the order of the inverse mutation rate, leads to enhanced accumulation of slightly deleterious mutations (12, 14, 30). The testable differences between the more simple effect described in this analysis and Muller's ratchet are follows. (i) The "simple" ratchet acts on an individual locus, while Muller's ratchet involves two or more loci and can, therefore, be suppressed if recombination events are frequent (12, 14, 30). (ii) While the Muller's ratchet effect is restricted to small virus populations (12, 14, 30), the simple ratchet is expected to work even if the average virus population size between transmissions is large. (iii) Simple ratchet implies that the reversion processes at different loci do not have sufficient time to occur, making their interference (Muller's ratchet) redundant.

As we have found, a high frequency of variable sites in HIV could be explained as a combined effect of a transmission bottleneck and frequent passaging between individuals, provided coinfection from independent sources is extremely rare. The rate of superinfection of individuals depends on both virological and epidemiological factors and is hard to estimate, in general. A clue to the rate in modern times is given by the frequency of coinfection with multiple HIV subtypes in geographical regions where no single subtype dominates. Infections with two different HIV type 1 (HIV-1) subtypes (or with HIV-1 and HIV-2) have been directly observed and are estimated to occur in 10 to 15% of such cases (35, 36). This figure does not include reports of intersubtype recombination (15, 40), which imply coinfection at some indeterminate time in the past.

The fact that coinfection is relatively frequent despite the effect of superinfection protection detected in SIV-positive animals (51) suggests either that a narrow time interval between primary infection and establishment of protection exists or that the protection is only partial in a probabilistic sense; data (51) allow for either interpretation. The two-subtype data can be used to estimate the overall frequency of coinfection, q. Assuming that the two subtypes circulate in equal quantities, and taking into account the fact that cases of double infection with the same subtype are not included in the above percentage, we obtain, as a lower bound, q = 0.2 to 0.3, much higher than our prediction of q  0.01. We have no way of knowing whether this value is representative of conditions in ancient animal transmission, which, according to the model, determine the mutant frequency in an inoculum in the modern-day pandemic; however, there is no obvious barrier to the coinfection rate which would keep its value as low as 1%. Although we cannot dismiss the possibility of a very low coinfection rate in the natural host, we do not know any independent facts to support this model.

Verifying approximations. The above considerations suggest that we may have to look elsewhere for an explanation for the high frequency of variable sites. To find another possibility, we will reexamine the simplifications common for all models discussed so far.

(i) Ignoring transversions. We ignored transversions in both experiment and theory. The danger of this approximation is that a double transversion may appear as a transition and thus interfere with our count of transitions. Given the small relative weight of single transversions in the net variation (25%), this effect is expected to be but a relatively small correction to the numbers of transitions, on the order of (0.25)² (~0.06).

(ii) Neglecting stochastic effects in steady state. Stochastic effects include randomness of mutation times and random drift of the genetic composition. The linkage disequilibrium test (39) implies that the effective HIV population is larger than 10⁵ infected cells (P = 0.05) and that, correspondingly, stochastic effects on the evolution of separate bases within an individual are relatively small. In the same work, we considered the effect of recombination on the test results and found that it cannot decrease the lower estimate of the effective population size by more than a factor of ~2.

(iii) Constant HIV population size. In fact, in a typical patient, the virus load expands sharply within 1 to 2 weeks and then drops by 1 to 2 orders of magnitude over 2 to 3 months during the acute phase of infection. Then, during the asymptomatic phase of infection, the virus load climbs back slowly, over a scale of a few years, before a large expansion at the final stage, leading to AIDS. In this work, we consider the evolution of bases with a small selection coefficient, which occurs slowly during the asymptomatic phase of infection. The ongoing slow change in the population size is not important for the almost deterministic evolution (which we found is the case), since it does not greatly depend on the population size.

(iv) Same wild type for all cells and individuals. In fact, HIV adapts to new cell types. The magnitude of this effect on the HIV genome after a thousand replication cycles, the relevant time scale, is hard to infer from the existing literature. It is certain that adaptation on a scale comparable to the fraction of variable sites in HIV pro, 16%, can be caused within 1 year by a very adverse factor, such as an efficient protease inhibitor (8). It could be also forced by the immune response, which we have not considered to this point.

(v) "Well-stirred pot" approximation. We implicitly replaced the complex topography of virus distribution between and within different lymphoid organs by an imaginary infected organ with well-mixed cells infected with different genetic variants. Direct visualization of two env variants of HIV in the spleen by selective labeling shows that although infected cells are nonuniformly distributed into islands, most of these islands are shared by different variants (37). This result implies that virions mix well between these islands; i.e., a significant part of the de novo infection is due to far-travelling virus particles (38). The conclusion is supported (38) by the large number of virion particles removed daily from peripheral blood of SIV-infected animals (28).

(vi) Independent sources of coinfection. When calculating the purifying effect of coinfection, we implied that two typical sources of coinfection in the host population are epidemiologically distant, so that the probabilities of receiving mutant bases from the two sources are independent. For epidemiologically close sources, the purifying effect of coinfection is expected to become weaker. Transmission of two clones from the same source is a limiting case of nonindependent coinfection; it does not have a purifying effect (see Materials and Methods). Whether two typical animals in a population are epidemiologically distant depends on the coinfection frequency q and on the size of the animal population. Coinfections randomize the distribution of virus variants between animals and make them more independent. If coinfections are very rare (i.e., q is much less than 1), a mutant base can pass through a long chain of animals without converting to the opposite variant. Then, two animals from even a large population are likely to have the same initial variant. If, however, the value of q is not too small, say ~0.2 to 0.3, then the two typical animals are statistically independent even in a small herd comprising just a few animals. Therefore, invoking the epidemiological proximity does not really help to explain the high mutant frequency in inocula; one still has to postulate that q is very small.

(vii) Absence of group selection. The possibility of group selection (as opposed to selection of individual genomes) has been raised (27). However, at this time, there is no compelling evidence that such an effect exists for RNA viruses.

(viii) Neglecting coselection. This approximation is rather crude for a particular pair of sites. When studying linkage disequilibrium (39) at close pairs of variable sites, we found that some haplotypes are 2.5-fold more frequent than one would expect for independently evolving sites, implying a noticeable coselection effect. However, when obtaining an average pattern of variation in a segment of genome as long as pro, containing over 40 variable sites, contributions from negatively and positively coselected pairs of bases to the net variation are expected to cancel out, at least partially. For example, if coselection is responsible for 50% of the relative fitness of haplotypes at a separate pair of sites, and there are 40 variable sites with a random sign of coselection between each pair, then the effect of coselection on the relative fitness at an average site is expected to be, by the laws of statistics, ~50%/39 (~8%), i.e., a small correction. This argument, however, applies only under certain restrictions (see below).

(ix) Neglecting selection for diversity. The predominance of synonymous and conservative substitutions in pro in patients at late stages of asymptomatic infection was the reason why we initially neglected selection for diversity. This argument does not imply that immune pressure is completely absent from pro but rather states that its selective pressure is smaller than the effect of purifying selection at a typical observation time.