To interpret the common dynamics of the divergence and diversity within a host in the chronic phase of HIV-1 infection, we developed a sequence evolution model where each viral sequence is represented by its distance to the founder strain, d. In this model, the number of sequences, N(d,t), at a distance d from the founder strain at time t is dictated by two factors: 1) the fitness, F(d), defined as the total number of offspring sequences from sequence d generated per unit time; and 2) the probability, M(d), of sequence d to evolve to sequence d+1 per unit time. Here we assume that the unit of time is chosen such that the probability of evolving to distances greater than d+1 in one time unit is negligible.

As shown in Figure 1A, at time 0, the total number of copies of virus 0 is N(0,0). At time 1, the total number of offspring sequences from virus 0 is F(0)N(0,0). Out of this total number of offspring, the number of mutant sequences is F(0)N(0,0)M(0), and the number of non-mutant sequences is F(0)N(0,0)(1−M(0)). Hence M(d) denotes the proportion of offspring that are mutants. In general form, this process is expressed as(1)

This model simulates the growth of the true genetic distance over time. Rather than a simple Hamming distance, which for finite sequence lengths cannot grow at a constant rate, the genetic distance we emulate is the distance realistic substitution models attempt to estimate [26]–[28]. In our later tree analyses of real data, we have used a general-time-reversible model that includes rate variation across sites that has been shown to realistically describe HIV-1 nucleotide evolution [26],[29]. We show below that the evolutionary rate of the 15 patients we analyze is approximately 10⁻³ per site per month. Since we analyze about a 600 nucleotide region of the HIV-1 env gene, this implies that we expect less than one substitution per month. Thus, a time unit of approximately one month is appropriate to analyze this data. Thus, our model is not following all the point mutations that can occur due to reverse transcription but rather simulates the growth of the true genetic distance from the founder in the presence of selection. In reality, multiple variants can exist at the same distance from the founder strain. In our model those variants have the same identification index, d, the distance from the founder strain and this implies that we need a measure of diversity that does not rely directly on sequence information but rather on the distribution of genetic distances d.

Since an approximately constant number of sequences were sampled at all time points [13], we consider the normalized distribution of the distances at time t, i.e.,(2)The divergence D_divergence(t), i.e., the average number of nucleotide substitutions that accumulated along the branch from the founder strain as a function of time [13], is measured by the mean value of d, i.e.,(3)(Figure 1B). The diversity D_diversity(t), measured by Shankarappa et al. [13] as the average pairwise nucleic acid distance between all sequences at time t, is here measured with the standard deviation of P(d,t) as(4)(Figure 1B). In our model, because we do not discriminate between the variants at the same distance, we measure the level of diversity with the level of spread in the distance from the founder strain. This measure is an approximation made for consistency with our modeling approach. To examine this approximation, we calculated the measure of genetic diversity at the nucleotide level used by Shankarappa et al. and the measure of the standard deviation of the distance distribution from the founder strain, Eq. (4), for the same data in [13]. The two measures were found to be proportional to each other (Figure 2).

We assumed that the probability of mutation varied as a function of the distance from the original strain, d, according to M(d) = m if d≤d_s, m(d_max−d)/(d_max−d_s) if d_s<d<d_max, and M(d) = 0 if d≥d_max, where m is a constant, d_s is the starting point (distance) for the decline of M(d), and d_max is the distance at which M(d) = 0 (left panel in Figure 1C). The profile of M(d) directly reflects the retardation in the rate of sequence evolution as the virus evolves further from the founder strain.

To observe the effect of the profile of M(d) (left panel in Figure 1C) on the macroscopic evolution patterns, we first fixed the fitness as a constant, F(d) = f. Figure 3 shows the fit of the model to the dynamics of divergence and diversity of patients S-P1—S-P11 [13]. The fit of the model is summarized in Table 1. The method of calculating divergence and diversity dynamics is provided in Materials and Methods. Encouragingly, our model successfully quantified the dynamics of the divergence and the diversity based on first a constant evolutionary rate, then followed by a decline of the evolutionary rate (left panel in Figure 1C).

To further investigate the relationship between the profile of M(d) and the dynamics of divergence and diversity, we studied two special cases of M(d) in greater detail. Submodel 1 is defined by M(d) equal to a constant, m (middle panel in Figure 1C). For a constant fitness, F(d) = f, the normalized distribution of the distance at time t, P(d,t), satisfies(5)with P(d,0) = 1 if d = 0 and P(d,0) = 0 otherwise. The generating function, , satisfies the following equation,(6)with F(z,0) = 1. The solution of Eq. (6) is given by F(z,t) = e^m(z−1)t. This implies that P(d,t) is a Poisson distribution, P(d,t) = e^−mt(mt)^d/d!. The mean of this distribution gives the divergence, D_divergence(t) = mt, and the standard deviation gives the diversity, . In this special case, both divergence and diversity increase as a function of time rather than saturate or decrease at later time points. Thus, assuming a constant fitness and constant evolutionary rate over the period of chronic infection fails to describe well the simultaneous intrahost dynamics of divergence and diversity. The fit of submodel 1 to the divergence and the diversity dynamics in all 9 patients is summarized in Table 1.

In submodel 2, we set d_s = 0, resulting in a linear decrease of the probability of accumulated mutations per unit time, given by M(d) = (1−d/d_max) for d≤d_max and M(d) = 0 for d>d_max (right panel in Figure 1C). In this case, from Eqs. (1) and (2), the dynamics of the evolution is summarized as(7)for d≤d_max and ∂P(d,t)/∂t = 0 for d>d_max.

Now the generating function satisfies(8)with F(z,0) = 1 from P(d,0) = 1 when d = 0 and P(d,0) = 0 otherwise.

This equation can be solved using the method of characteristics. Let z and t be the functions of s. Then F(z,t) = F(z(s),t(s)) = F(s) and(9)If we choose the characteristic curve such that(10)with t(s = 0) = 0, we have t = s. By comparing Eq. (8) with (9), we obtain(11)By integrating Eq. (11), we have(12)Along this characteristic curve, by inserting Eqs. (8), (10) and (11) into Eq. (9), we obtain(13)By integrating Eq. (13), we obtain(14)where we have used the initial condition of F(z,0) = 1.

Since s = t and from Eq. (12),(15)If we insert Eq. (15) into Eq. (14), we obtain the solution of Eq. (8), (16)Then P(d,t), the coefficient of z^d of Eq. (16), is given as a binofmial distribution, . Hence, the divergence as a function of time is measured by the mean of this binomial distribution, , and the diversity is given by the standard deviation of P(d,t), . In submodel 2, the divergence first grows linearly and then saturates, and the diversity first increases and later decreases, which captures the saturation of divergence and the decline of the diversity at later stages of HIV-1 infection. The fit of submodel 2 to the divergence and the diversity dynamics of all 9 subjects is summarized in Table 1.

Comparing the full model and the two special cases using Akaike's information criterion (AIC) showed that in all the patients the full model fitted best except for S-P5, S-P7, and S-P9 (Table 1). Submodels 1 and 2 are interesting to consider because they are simpler and have analytical solutions. Comparing the two submodels to each other showed smaller or equal sum of squared errors for submodel 2 in all subjects except S-P3 (Table 1). One extra parameter in submodel 2, however, resulted in larger AIC values than for submodel 1 in all subjects except S-P6. Despite this, we prefer submodel 2 because it qualitatively captures the decrease of the diversity at the later stage of HIV-1 infection.

We next studied the impact of viral fitness on the dynamics of the divergence and diversity. Recent ex vivo experimental data have suggested that the replication rate of viruses sampled at a later stage of HIV-1 infection is greater than that of viruses at an early stage of infection [6]. Therefore, we tested how fitness would affect the viral evolutionary pattern at the sequence level. First, we let fitness grow linearly as a function of the distance from the founder strain, i.e., F(d) = f₁+f₂d, in both submodels 1 and 2. In the range of f₂/f₁ from 0 to 10, we do not find any qualitative change in the patterns of divergence and diversity with time in either submodels 1 or 2 (Figure 4). This showed that an overall increase in fitness over the disease progression did not have a large effect on the diversity and divergence dynamics.

Second, we studied the case where the fitness is reduced after a given distance, F(d) = f for d≤d_c and F(d) = f′ for d>d_c where f′ is less than f. Here the proportion of offsprings that are mutants is constant for all viruses, M(d) = 0.5. We found that reduced fitness for viruses with a distance greater than d_c reproduced the observed patterns for divergence and diversity. Figure 5 displays the calculated dynamics of divergence and diversity when we reduce the fitness of viruses having a distance greater than 50 mutations to 50% of the fitness of viruses with a distance less than 50 mutations. Although the profile of reduced fitness for the viruses after a given distance qualitatively explain the common dynamics of divergence and diversity, the reduction in the fitness of a virus population at later stages does not seem realistic considering the observation of increased fitness over the course of infection [6].

Finally, we investigated the case where only certain types of viruses may evolve to have a greater level of fitness. This situation has been described for emerging CXCR4-using viruses later in disease progression, and was found to correlate with a decline of diversity [13]. To simulate the outcome of emerging CXCR4-using viruses, potentially with greater level of fitness since they have a greater target cell range than CCR5-using viruses by infecting naïve CD4+ T-cells, we assigned a greater level of fitness to a fraction, α, of viruses having a distance larger than a critical value d_c. This process is expressed as(17)where F_high(d) = F_high for d≥d_c and F_high(d) = F otherwise. In this way, a proportion of viruses, 1−α, have fitness F and a proportion α have fitness F_high when d>d_c. When d≤d_c, the fitness is given by a constant F. We here chose d_c = 50 mutations for the following reason. As we will show below, the average overall evolutionary rate for the 15 studied patients was estimated at approximately 10⁻³ nucleotide substitutions per site per month. This corresponds to 0.012 substitutions per site per year. With around 600 nucleotides in the dataset [13], 50 mutations corresponds to the mutations one expects to accumulate during ∼7 years. Thus, we chose the emergence of CXCR4-using viruses at d = 50 from this calibration since usually X4 viruses appear at later stages of infection.

Figure 6 plots the dynamics of divergence and diversity by changing the fraction (α) of X4 viruses that have a 50% increase of fitness at distance d_c = 50 mutations. As we increase the value of α, we observe an increase in divergence, then a transient rapid increase followed again by the inital slope of linear increase. The emergence and persistency of X4 viruses in the population leads to a rapid increase of diversity followed by a decline of diversity. Then at the final stage, diversity starts to increase again. This trend is robust to both the amount of fitness increase and the value of d_c. For example, when we chose d_c = 30, the transient rapid increases in the divergence and diversity still occur, but were shifted to 4.2 years. An initial rapid increase both in diversity and divergence due to the emergence of more fit virus is not compatible with the in vivo measurements from HIV-1 infected patients (Figure 3).

Overall, these simulations suggest that the probability profile of the evolutionary rate, M(d), rather than the fitness profile, F(d), is the main component in our model that determines realistic within-patient HIV-1 evolution.

To test the prediction made by the model, i.e., a slowdown of the evolutionary rate as virus population evolves further from the founder strain, we calculated the rate of HIV-1 sequence evolution in consecutive windows over a maximum likelihood (ML) tree from each patient, starting from the root (see Materials and Methods). We used longitudinal sequence samples for 15 patients from two independent studies [13],[22]. As an example, Figure 7A shows the tree describing the HIV-1 evolution in patient S-P6. Figure 7B shows the resulting evolutionary rate as a function of the distance from the root for all patients. Interestingly, the rate is not constant but rather displays a dynamic behavior as HIV-1 evolves. In agreement with our model predictions, 13 out of the 15 patients showed a decline of the evolutionary rate as the sequence population evolved further from its founder strain. The same dynamic behavior was observed using other window sizes (Δ = 0.06 for the Shankarappa data and Δ = 0.03 for the Wolinsky data). Thus, the observed decline of the evolutionary rate was robust to the size of the window. In Figure 7B, we also plotted the evolutionary profile obtained by a fit to the divergence and diversity dynamics with the full model. The dynamics of the evolutionary rate calculated from the maximum likelihood tree was reasonably consistent with that obtained by a model fit to the divergence and diversity dynamics for each patient.

Sometimes we observed negative evolutionary rates in some patients when the distance from the root was large, mostly in later stages of infection (Figure 7B) when the sequence population hardly evolves any more. As a consequence some sequence variants may have a smaller distance from the founder stochastically, and if enough of such variants are detected then a negative evolutionary rate will be apparent. Also, the apparent negative rate of evolution may be due to the emergence of less evolved strains from latent reservoirs at later sampling time points.

When the rate of change of the evolutionary rate was compared to the rate of change of CD4+ T-cell counts (Figure 8A), a significant correlation (r = 0.68, P = 0.0014) was observed (Figure 8B). In the initial interval where CD4+ T-cell counts were relatively stable (to the left of the dashed bar in Figure 8A), the evolutionary rate stayed relatively stable too. As CD4+ T-cell counts decreased and disease progressed in the patients the evolutionary rate slowed down. However, if one compares the overall (average) evolutionary rate from the whole study period (as defined by Eq. (20) in Materials and Methods), not its slope, with the disease progression rate, no clear correlation was seen (Figure 8B inset). The overall evolutionary rate of 15 patients was 10.4±3.14×10⁻⁴ substitutions per site per month. Note that increased or stable viral RNA counts rather than contraction in viral loads were observed in 7 patients under antiretroviral therapy in [13]. Thus, the decrease in the rate of evolution seems not to be associated with the onset of therapy.

We estimated overall synonymous and nonsynonymous evolutionary rates across maximum likelihood trees based on synonymous and nonsynonymous changes only using HyPhy [30]. Similar to the overall total substitution rate, we found that neither synonymous nor nonsynonymous overall evolutionary rates correlated with the disease progression rate. For progressors with progression time less than seven years (S-P1, S-P5, S-P6, S-P7, and S-P8), the average synonymous and nonsynonymous evolutionary rates were estimated at 6.6±3.5×10⁻⁴ and 12±5×10⁻⁴ substitutions per site per month, respectively. For slow disease progressors with progression time greater than seven years (S-P2, S-P3, S-P7, S-P9 and SP-11), the average synonymous and nonsynonymous evolutionary rates were estimated at 6.8±2.3×10⁻⁴ and 13±4.5×10⁻⁴ substitutions per site per month, respectively. Lemey et al. reported lower overall synonymous evolutionary rates for these same slow disease progressors [19]. These contradictory observations may be explained by the use of different methods in the estimation of the overall evolutionary rates. While Lemey et al. used codon substitution models with a Bayesian relaxed clock model, we estimated the overall synonymous and nonsynonymous evolutionary rates in separate maximum likelihood trees based on synonymous and nonsynonymous changes [30] to allow for detecting rate changes across the trees. A common finding with Lemey et al. is that they also reported higher nonsynonymous rates (8.2±3.0×10⁻⁴) than synonymous rates (3.8±1.9×10⁻⁴). Importantly, although the overall synonymous evolutionary rate did not correlate with the disease progression rate in our calculations, we found that both synonymous and nonsynonymous evolutionary rates decline as disease progresses in 7 and 8 out of 9 patients in Ref. [13], respectively (Figure 9).

It is well known that HIV-1 recombines during its evolution. Therefore, we investigated whether recombination could have obscured our estimates of the evolutionary rates. All patient populations showed some signal for recombination (Table 2). This signal was, however, strongly correlated to the degree of homoplasy (r = 0.91). The homoplasy also grew with number of sequences per patient (R = 0.92), and all patients showed departures from neutral evolution, suggesting stochastic effects as well as selective environments rather than recombination. Most importantly, all our ML trees showed a clear time order of how sequences had been sampled through time (Figure 6), and additional trees calculated using SplitsTree showed that if recombination had occurred, then mostly samples taken close in time had been involved (data not shown). Thus, although difficult to exactly quantify, recombination had no large effect on our estimates of the evolutionary rate.

We analyzed sequence data from two independent studies, the env C2-V5 region from nine patients [13] and the V3–V5 region from another six patients [22]. Briefly, sequences from the first nine patients were collected over their entire disease progression. The follow-up time varied between 6 to 12 years, at which time seven had developed AIDS and seven of the patients received antiretroviral treatment [13]. The other six patients were followed for 3 to 10 years [22]. Three of these patients received antiretroviral therapy 2–5 years after infection. All HIV-1 sequences were downloaded from the HIV database (GenBank Accession numbers AF137629-AF138163, AF138166-AF138263, AF138305-AF138703 and U35894-U36185).

Sequences were aligned using Se-Al [53]. Trees were created using enhanced and parallelized versions of fastDNAml and Rates [54],[55], that fit a general-time-reversible substitution model (RevML) and site rate specific rates (RevRates) in an iterative way [41],[56]. Briefly, a candidate tree topology was created assuming uniform site rates and an initial random estimate of nucleotide frequencies and transition rates. RevML proceeds in a heuristic and piecewise way, starting from a small set of sequences and building up the tree topology and branch lengths while making placement decisions that maximize the tree likelihood score, similar to a stepwise addition algorithm. The resulting tree then constrains per-site rate optimization of tree likelihood as a function of global estimates of baseline nucleotide frequency and transition rates. These estimates are fit using the conjugate gradient algorithm in the RevRates program. A second RevML run was then performed using these estimates and in turn another rate estimation procedure refined from the second tree. A final tree was estimated using the twice-refined global and local site rates. Each of the trees in the refinement procedure was independently estimated from the global and site local rate parameters.

Trees based on synonymous and non-synonymous substitutions, respectively, were inferred using HyPhy with optimized MG94xREV models [30]. This model uses a codon-based substitution model (MG94) that considers substitutions involving non-stop codons, augmented with a general-time-reversible nucleotide substitution model (REV) to include the heterogeneity in nucleotide frequencies and substitution rates [57]. The total number of changes per codon is decomposed into synonymous and nonsynonymous changes according to the universal coding table. Separate synonymous and nonsynonymous rates are then fitted to each branch of the tree. Prior to model fitting and tree reconstruction, alignments were codon corrected using the HyPhy SeqAlignment procedure (with the HIV-1 25% scoring matrix).

We used SITES [46] and SplitsTree [51] to investigate potential recombination signals in each patient set of sequences, PAUP* [58] to estimate the amount of homoplasy, SNAP [41] to estimate average synonymous/non-synonymous rates, and Tajima's D to estimate neutral evolution[48].

The rate of evolution was calculated in consecutive windows over a maximum likelihood (ML) tree from each patient, starting from the root. The distance to the root for all taxa in each window [d, d+Δ] was calculated from the tree (Figure 7A), and the resulting evolutionary rate was estimated as(19)where d_i(d_j) is the distance from the root of sequence i(j) at sampling time point t_i(t_j). Here d ̅ is the average distance from the root over all the sequences within the window, [d, d+Δ]. The window size was Δ = 0.09 for the Shankarappa data set and Δ = 0.02 for the Wolinsky data set. The average evolutionary rate over the entire sampling period from a patient was calculated as(20)by averaging the rate of evolution over the sequences in reference to the founder strains which are sampled at the earliest time point in each subject. Here, N_F is the total number of founder strains and N_s is the total number of sequences in a patient.