Introduction

Human immunodeficiency virus (HIV-1) actively replicates in CD4⁺ T cells [1,2]. During the infection process, viral RNA is reverse-transcribed into DNA, which is then incorporated into the genome of the host cell [3]. This integrated viral DNA is referred to as a provirus. Most infections result in the rapid production of new viruses, leading to the death of these infected cells within a few days [4]. However, in a fraction of cells, HIV-1 is capable of lying in a dormant, “latent” state [5]. While antiretroviral therapy suppresses active HIV-1 replication, it is unable to eliminate latently infected cells or their integrated proviruses [6]. And while many latent proviruses are defective, some remain capable of reactivation, resulting in a quick return to active infection if antiretroviral treatment (ART) is interrupted, even in individuals who have undergone effective drug treatment for many years [7,8]. This population of long-lived, latently infected cells, known as the latent reservoir (LR), therefore presents the major barrier to an HIV-1 cure.

Understanding factors that contribute to LR persistence could greatly contribute to HIV-1 cure efforts. However, it is difficult to obtain a comprehensive picture of LR dynamics from direct measurements due to its small size. For typical HIV-1-infected individuals, roughly one in 10⁴ CD4⁺ T cells are latently infected, and active HIV-1 replication occurs in only around 1% of these latently infected cells in viral outgrowth experiments [9–11]. Thus, latently infected cells, especially ones that can readily reactivate, are rare. Subsequent studies have also found that multiple rounds of stimulation can prompt latent cells that initially remained dormant to reactivate, making it difficult to determine the total number of latently infected cells that are capable of reactivation [11,12].

Despite these challenges, recent work has provided insights into the dynamics and heterogeneity of the LR. Subsets of cells bearing integrated HIV-1 can undergo clonal expansion in patients receiving suppressive ART [13,14]. The degree of expansion of clones as well as their persistence varies greatly and is associated with the specific integration sites [13,14,14] as well as stimulation by antigens [15,16]. Levels of HIV-1 expression in latently infected cells differ, and there appears to be progressive selection for more transcriptionally silent integration sites during long-term ART [17,18]. Persistence can also occur via T cell proliferation, including both homeostatic proliferation [19] and in response to antigen [15,16,20,21]. While many highly-expanded clones contain defective proviruses [22,23], at least half of the cells carrying intact proviruses also belong to expanded clones [23–27]. A strong negative correlation has also been observed between clone size and reactivation rate in viral outgrowth assays [25]. As patients remain on ART for long times, the diversity of observed clones decreases and the proportion of HIV-1 proviruses in the largest clones progressively increases [28,29].

Mathematical modeling has also provided insights into the LR, with some model predictions validated in experiments [30]. Studies investigating the relationship between latently infected cells and plasma viremia during ART [31–36] suggest that as long as ART is marginally effective, the persistence of latent virus is most strongly influenced by the longevity of infected cells and the rate at which they reactivate. Recent modeling work has also suggested that uneven homeostatic proliferation of latently infected cells early in infection may lead to the observed spread in clone sizes in the LR [37].

To gain a deeper understanding of the persistence of the latent reservoir in HIV-infected individuals, it is important to develop mathematical models that reflect the biological mechanisms that govern its dynamics. The LR is composed of diverse clones with different T cell receptors (TCRs), which affect their activation potential and antigen specificity. Moreover, these clones have distinct viral integration sites, which influence their transcriptional activity and reactivation probability. These factors contribute to the clonal heterogeneity of the LR and its persistence. Thus, there is a need to incorporate more comprehensive and biologically motivated features of clonal heterogeneity, which are not typically incorporated in existing mathematical models of the LR.

We addressed this challenge by developing a novel stochastic model of LR dynamics that explicitly accounts for clonal heterogeneity. We consider genetic changes in HIV-1 sequences and variable probabilities of reactivation, while also incorporating the effects of antigenic stimulation on latently infected clones with different TCRs. The dynamics of these clonal populations are integrated with interactions between free viruses, susceptible cells, and cells that are actively infected. We model multiple latently infected clones, integrated into distinct T cell clones with different TCRs and thus different responses to antigen.

Our model recapitulates experimentally observed features of HIV-1 infection while also providing insights into LR structure, dynamics, and persistence. We recover the decay kinetics of HIV-1 RNA in blood, HIV-1 DNA in peripheral blood mononuclear cells (PBMCs), and latent cells that reactivate upon stimulation, which occur over widely-varying time scales (days, months, and years) following the start of ART [38–40], without the use of time-varying parameters (S1 Table). Among other findings, our model reproduces the observation of defective proviruses in highly expanded clones [22] and the negative correlation found between clone size and reactivation probability for patients who have undergone ART treatment for many years [25]. Stimulation by antigens combined with heterogeneous reactivation rates for different clones leads to a broad distribution of clone sizes, which are stratified by their reactivation rates. Over long times, we find that the LR becomes progressively more concentrated on a small number of clones with low reactivation rates, which play a key role in LR persistence. These insights could inform the development of new therapeutic approaches to reduce the size of the LR and achieve a functional HIV-1 cure.

Results

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Stochastic model incorporating LR heterogeneity.

Our model blends elements from multiple prior mathematical studies [30–32,37,38,41–46]. We consider four main populations: uninfected CD4⁺ T cells (T), productively infected activated CD4⁺ T cells (A), latently infected resting CD4⁺ T cells (L), and HIV-1 virions (V) (Fig 1A). All cells have finite lifespans determined by their respective death rates (see Methods for a complete list of parameters and supporting references). We used a constant replacement rate to approximate the replenishment of uninfected target cells from the thymus [45,47,48]. The rate of production of virions is given by the product of the death rate of actively infected cells [49] and the viral burst size n [50]. Virions are then cleared at a constant rate c [51].

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Fig 1. Model schematic.

(A) When a new cell is infected, it forms a new “clone” with a specific TCR sequence. The generation of defective proviruses, point mutations, and active or latent infection are all determined by chance. (B) Clonal expansion and reactivation are possible outcomes of the dynamics of latently infected clones upon stimulation by antigens. As a simplifying assumption, susceptible T cells all have identical dynamics in our model, and the dynamics of response to antigen are modeled only for latently infected cells.

https://doi.org/10.1371/journal.pcbi.1013433.g001

HIV-1 virions can infect susceptible CD4⁺ T cells. A small fraction of infections, , will result in defective integrated proviruses due to effects like large deletions or hypermutation. During ART, this leads to proviruses with fatal defects outnumbering intact proviruses by a factor of 10–50 to 1 [28]. HIV-1 also mutates during infection due to error-prone reverse transcription, with an estimated error rate of per base per replication cycle [52–54]. Given the length of the HIV-1 genome, this implies that approximately a third () of successful infection events will lead to a mutation. Finally, a fraction of infection events will result in latent rather than active infection. We fit such that the HIV-1 DNA per 10⁶ PBMCs at the beginning of ART was in the range 10³–10⁴, consistent with patient data [38]. Consistent with reported HIV-1 RNA levels during ART [39,55] and following previous modeling studies [35], we assume that viral replication is attenuated but not perfectly suppressed during ART.

To account for the heterogeneity of latently infected cells, we modeled each latently infected clone individually, including the expansion and antigen-driven proliferation of individual clones (Fig 1B). Clones are defined as latently infected cells with identical TCRs, integrated proviruses, and integration sites. Each time a new clone is created through a latent infection event, it is assigned a random probability of reactivation (Methods), following the observation that integration is stochastic and different integration sites can affect the capacity for reactivation [56]. In addition, each clone is stimulated by a background concentration of antigen that fluctuates in time (Methods), inspired by past models of T cell repertoire dynamics [46]. We use the same homeostatic death and proliferation rates for all clones; however, stochastic differences in antigenic stimulation drive differences in clonal proliferation. As a simplifying assumption, we model all susceptible T cells as identical, and we only consider antigen-driven proliferation dynamics for latently infected cells. Our assumption that T cell activation and recognition of antigens can drive HIV-1 reactivation, and that reactivation is not deterministic, follows experimental observations [57,58]. Prior work has also shown that reactivation during homeostatic proliferation is rare [57]. Overall, our model differs from previous ones that considered a constant rate of reactivation for latently infected cells [35] or different cell populations with different half-lives [34]. We simulated our model using a system of stochastic differential equations describing the dynamics of cells during HIV-1 infection both before and during ART (Methods).

Seeding of the reservoir and clonal proliferation pre-ART.

We first simulated the seeding and development of the latent reservoir during the first months of infection. Simulations begin with a number of virions in the system and zero active and latently infected cells (Methods). We decreased viral infectivity after one month to capture suppression of viral replication by the immune system. Even during the first weeks of infection, we observed a large number of distinct latently infected clones in the reservoir. Most clones are very small: for up to a year after infection, the average clone size is less than 10 cells, with a median clone size of 2 cells (Fig 2). During this time, the largest clone is typically smaller than 1000 cells. Most of these clones are also short-lived. The average age of a clone after one year of infection is 15 days, with a median age of 6 days. Such short-lived clones are highly likely to reactivate, with an average probability of reactivation p_R around 9%.

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Fig 2. Distribution of clones in the HIV-1 latent reservoir early in infection.

Most clones are small and have high reactivation probabilities. However, as time passes, clones with lower reactivation probabilities begin to grow in size. Stimulation from antigens drives some rare clones with very low probabilities of reactivation to large sizes. Note gaps in the heatmap occur for clones with <10 cells because we enforce integer numbers of cells for small clones (see Methods).

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During this early phase, some clones will be stimulated to proliferate by exposure to antigens. However, the effect on different clones in the reservoir differs substantially depending on how likely the latent virus is to reactivate when stimulated. In clones with low reactivation probability, proliferation due to antigenic stimulation typically results in net growth. In clones that readily reactivate, however, the reactivation of latent virus reduces the effective growth rate due to antigenic stimulation, which can ultimately lead to the elimination of these clones. These dynamics lead to a progressive increase in the number of clones with low reactivation probabilities and large clone sizes over time, consistent with recent work that has observed a progressive decrease in clonal diversity with time [29].

Kinetics of plasma viral load, HIV-1 DNA, and the inducible viral reservoir after ART initiation.

After the LR has been seeded, we simulated the response of viral populations, including both latent and actively infected cells, to long-term ART. To simulate viral kinetics during ART, we decreased viral infectivity β at month 60 due to treatment (Methods), keeping all other parameters constant. Here, we chose 5 years after infection as a start point for ART initiation following typical times from HIV-1 infection to diagnosis [60,61].

Clinical data shows that after initiating ART, HIV-1 RNA in blood decreases rapidly over the course of around 2 weeks, with observed half-lives of 0.9-1.9 days [38,62]. This is followed by a more gradual decline over the next 4 weeks ( 7.8-27.2 days [38,62]). The total number of latently infected cells, measured by HIV-1 DNA in PBMCs, decays steadily over the first few months on ART ( 99-133 days). Infectious units per million PBMCs (IUPM), measured in viral outgrowth assays (Methods) declines very slowly, with a measured half-life of approximately 44 months [59].

Our model quantitatively recovers the decay rates of HIV-1 RNA and latently infected cells (combined across all clones) spanning days, months, and years on ART (Fig 3; see also S1 Fig). In our simulations, viral load first drops sharply, which is primarily driven by the death of actively infected cells (Fig 3A and S2 Fig). At the same time, small clones are gradually eliminated through reactivation or random cell death. Due to reduced viral replication, these clones are no longer replenished at the same rate, leading to a net decline in the total number of latently infected cells. Clones with high reactivation probabilities are depleted more rapidly than those that do not readily reactivate (Fig 4A). Collectively, these factors lead to a shift in the LR toward larger clones with lower rates of reactivation, slowing the decline in viral load and HIV-1 DNA in PBMCs (Figs 3A, 3B, 4A, and S3 Fig). As larger clones are slowly eliminated, we find a decline in the inducible reservoir consistent with measurements from clinical data (Fig 3C).

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Fig 3. Latent reservoir dynamics after ART initiation in 100 replicate simulations.

(A) ART first results in the rapid decline of plasma viral load [38], first due to the death of actively infected cells and later driven by the elimination of small clones in the LR. (B) Over slightly longer times, the number of latently infected cells steadily declines [38] as small clones die out and are no longer quickly replenished by new infections. (C) Infectious units per million (IUPM), a measure of cells in the LR capable of reactivation, declines over the course of years, approximately following the 44-month half-life measured in clinical data [59].

https://doi.org/10.1371/journal.pcbi.1013433.g003

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Fig 4. Dynamics of the latent reservoir during ART.

(A) As ART begins, many clones are observed with different sizes and probabilities of reactivation. Reactivation and random fluctuations lead to the preferential loss of small clones and ones with high probabilities of activation over time. The largest clone size scales roughly with the inverse of the reactivation probability, a relationship that is stable over time. (B) Despite viral replication, few mutations accumulate in latent clones during ART.

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Long-term clonal dynamics in the latent reservoir.

During ART, clones with higher probabilities of reactivation have a shorter effective survival time than clones with lower probabilities of reactivation. We therefore find that the average probability of reactivation decreases over time. However, clones that readily reactivate are not entirely eliminated. Occasional reactivation of latent cells from large clones leads to bursts of viral replication that partially reseed the reservoir. These dynamics result in a long-term quasi-steady state, where small clones with high probabilities of reactivation turn over frequently while large, quiescent clones slowly fluctuate in frequency (S4 Fig and S5 Fig).

Over long times, we find that, for the largest clones, clone size n scales inversely with the probability of reactivation p_r, , with an exponent (Fig 4A). This finding is consistent with previous work that observed a power law relationship between clone size and probability of reactivation in viral outgrowth assays in data from multiple subjects years after ART initiation [25]. Clones that are small and/or have low probabilities of reactivation (i.e., ones occupying the lower left corners in Fig 4A) are particularly challenging to quantify in patient data because their probabilities of being sampled in sequencing HIV-1 DNA from PBMCs or viral outgrowth assays are exceedingly small. Such clones are likely to be observed only once in data if they are sampled, which is consistent with observations in clinical data [25].

Our results also show that after a year on ART, the relative size of large clones changes little over time (S6 Fig). Collectively, our simulations are consistent with longitudinal studies that found few significant changes in the proportion of different clones in the LR when sequencing proviral DNA [28,63,64], while significant changes in clonal distributions can be observed when sequencing reactivated viruses from viral outgrowth assays [63,64].

Dynamics of clone age pre- and post-ART.

To further characterize the clonal dynamics of the LR, we tracked clone age (defined as the time since the latent infection event that first established a new clone) and probability of reactivation over time. Before ART begins, most clones in the LR have been recently deposited, though we also observe a spike in the age distribution that corresponds to latent infection events during the early, acute stage of infection (S6 Fig). After ART, the age distribution progressively shifts toward older clones. This is due to the death of small and highly reactive clones during the time immediately after ART initiation, as described above. However, there remains a spike in the age distribution near zero due to new latent infection events in short bursts of viral replication during ART. Overall, this finding agrees with recent work that has found that the decay of younger clones in the LR is faster than for older ones [65,66].

Processes driving latent cell proliferation and death.

In our model, there are three ways for latent cells to be produced: homeostatic or antigen-driven proliferation of latent clones, and new latent infection events. There are two ways that latent cells can be removed from the reservoir: through reactivation or through homeostatic death. We tracked the contributions of each of these factors to latent cell proliferation and death both before and after ART (S8 Fig). Interestingly, we found that new infection events only constitute the main contribution to latent cell growth during a brief period very early in infection. After the latent reservoir has been seeded, homeostatic proliferation and antigenic stimulation are by far the dominant factors driving latent cell growth, with homeostatic proliferation making the largest contribution in our model. Both of these factors have been cited in prior experimental work as contributors to reservoir growth and persistence [15,16,19–21]. We find that homeostatic death is by far the largest contributor to latent cell death both before and after ART. However, as described above, the uneven probabilities of reactivation across clones result in a gradual shift in the composition of the reservoir (Fig 4A and S3 Fig), even though reactivation constitutes a small fraction of overall latent cell loss.

Variation in viral dynamics over multiple simulations.

We performed 100 replicate simulations to test the variation in dynamics between different stochastic realizations of the model (Fig 3). Dynamics early after ART initiation were the most repeatable in our simulations, and the dynamics of latent clones after years of ART was the most variable. This is expected because the dynamics immediately after ART initiation involve large numbers of cells and are therefore nearly deterministic, while long-term ART dynamics are driven by fluctuations of smaller numbers of latent clones.

Because our replicate simulations were performed with identical underlying parameters, they demonstrate the minimum level of stochasticity inherent in our model. However, real variation in viral dynamics between individuals are influenced by a broad range of factors, including differences in host immune responses, duration of infection before ART, and therapy adherence and effectiveness, that would best be modeled by variation in simulation parameters. Thus, these replicate simulations represent a lower bound on variation between individuals.

Choice of reactivation probability distribution and the role of heterogeneous reactivation rates.

Evidence suggests that different latent clones have different propensities for reactivation [22,25,56,67,68]. Ultimately, in our model this heterogeneity in reactivation plays a central role in concisely explaining widely differing and multiphasic decay rates of HIV-1 RNA in blood, HIV-1 DNA in PBMCs, and IUPM observed in clinical data. Heterogeneous reactivation rates are also essential to reproduce the observed association between the probability of reactivation in viral outgrowth assays and latent clone size [25].

However, there is no singular distribution of reactivation probabilities that uniquely reproduces the observed clinical values. For simplicity, we opted to use a lognormal distribution, which is a natural choice when a variable (i.e., probability of reactivation) is obtained as the product of many independent explanatory variables (e.g., virus genetic background, cell type, integration site, local chromatin context, etc. [68]). Previous modeling work has used a lognormal distribution for turnover rates to define uneven proliferation of clones in the first year of infection [37]. Modeling the probabilities of reactivation with this distribution, which is grounded in the underlying biology of latent infection and consistent with clinical data, allows us to extend our model not only to active infection but also to describe long-term dynamics of the latent reservoir during ART.

Models with reactivation probabilities that were the same for all clones were difficult to fit with constraints on short-term and long-term viral dynamics. With the probability of reactivation set to 1% for all clones, for example, we find that the dynamics shortly after ART begins are recovered well, but the long-term decay of the LR is much faster than expected (S9 Fig).

Role of heterogeneity in antigenic stimulation.

To gauge the importance of heterogeneous antigenic stimulation in our model, we considered an alternative scenario in which the level of antigenic stimulation was kept at a constant, uniform level across all latent clones. In such simulations, the level of antigenic stimulation must be carefully tuned: low values result in the rapid death of latent clones, while high ones cause unbounded proliferation. Simulations with a concentration of antigen equal to the long-term average in standard simulations successfully recovered viral kinetics immediately post-ART (S10 Fig). However, this condition also led to a more rapid decay of the LR than expected from clinical constraints. In addition, the maximum clone size was sharply limited in simulations with constant antigenic stimulation, more than an order of magnitude smaller than the largest clone sizes in our standard simulations (S10 Fig). This feature is also inconsistent with observations of highly expanded clones that can comprise substantial fractions of the overall intact reservoir within an individual [25,69]. Thus, at least in our model, we find that heterogeneous antigenic stimulation and rates of reactivation are needed to fit the constraints of clinical data.

Presence or absence of HIV-1 evolution during ART.

While ART strongly suppresses viral replication, it may not be completely effective. Past modeling work has suggested that low amounts of replication can continue after treatment intensification and may influence the level of detectable virus, but are unlikely to allow for long-term sequence evolution [32,34–36]. Others have argued that ongoing HIV-1 replication can lead to measurable viral evolution during ART [70–72], though this point is hotly debated [73,74], and multiple studies have failed to observe evolution in the reservoir during ART [75–78].

To test whether or not viral sequence evolution would occur in our model, we tracked the number of accumulated mutations in individual clones after ART initiation. In our simulations, the mean number of new infection events resulting from active infection in a single cell is smaller than one due to the suppressive effects of ART (S8 Fig). This implies that persistent, self-sustaining active replication is impossible. However, because our model is stochastic, we observe occasional “bursts” of viral replication. In phylogenetic terms, the replication dynamics in our model would thus generate star-like phylogenies (i.e., with few mutations around stable clones in the LR), rather than the ladder-like trees that can be generated from sequential evolution. Despite occasional bursts, we found no evidence for progressive accumulation of mutations or genetic divergence over time (Fig 4B), consistent with past observations [75–78] and modeling work [32,34–36].

One prominent prior study that argued for continued HIV-1 evolution during ART was based on observations made during the first year of ART [72]. As stated above, we do not find evidence for significant sequence evolution in our model. However, we do observe immense changes in individual clone sizes during early ART, especially for many small clones that are eliminated (S5 Fig). These dynamics support previous arguments that sampling of different clones could explain the appearance of evolution shortly following ART [69].

Effects of early intervention on LR structure.

Recent studies found that the composition of the LR is altered in individuals who have undergone early ART treatment [79]. To mimic early ART, we adjusted our simulations to include a sharp drop in viral infectivity 15 days after initial HIV-1 infection (Methods). Unlike previous simulations, early intervention results in a much smaller number of clones in the LR (Fig 5A), which quickly becomes dominated by just a few large clones (Fig 5B). Our simulations thus recapitulate studies showing that the LR in those with early ART are mono- or oligo-clonal, with little reactivation and background replication [79] and fewer intact proviruses [22]. While our model was not calibrated to model analytical treatment interruption (ATI, a process in which individuals temporarily cease ART in a controlled setting), exploratory simulations also showed that viral rebound occurred much slower for the early ART scenario (S11 Fig, Methods). Delayed viral rebound after early ART has been observed both in clinical data [80–82] and in animal models [83,84].

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Fig 5. Distribution of clones in the latent reservoir after early intervention.

(A) We simulated the effects of early intervention by initiating ART conditions 12 days after infection. At this time, there are few clones that are large or have small probabilities of reactivation, limiting the potential for clonal expansion. (B) After 8 months, few clones remain.

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Factors underlying long-term LR persistence.

Despite suppressive ART, the latent reservoir persists for decades in HIV-1-infected individuals. How do different components of the LR contribute to its persistence? We find that the presence of large latent clones is the most important factor in the long-term persistence of the latent reservoir, despite their low probabilities of reactivation. In fact, small p_r allows these clones to grow to large sizes with minimal decay due to reactivation (Fig 4A). Because they are large, these clones are also very unlikely to die stochastically due to fluctuations in clone size, unlike smaller clones that turn over rapidly (S5 Fig).

Interestingly, the largest clones are not the ones that are most likely to generate rebound viruses. On average, the rate of viral outgrowth is proportional to the product of the clone size and reactivation probability. In typical simulations, this is maximized by clones with intermediate sizes and reactivation probabilities. This is because clones with very small probabilities of reactivation are fairly rare, and clones with very high probabilities of reactivation tend to be small and short-lived. This can be seen quantitatively by computing the total contribution of clones of different sizes to reactivation during ART (S12 Fig). Thus, even though we find that the largest clones (with small probabilities of reactivation) are chiefly responsible for LR persistence, they are unlikely to be the typical first source of outgrowing viruses during rebound.

Relatedly, it has sometimes been challenging to identify the source of rebound viruses in the latent reservoir in ATI studies [85]. Given that we find clones of size 10²-10³ contribute most to reactivation, and considering the size of the LR in our simulations, this suggests that one would typically need to sequence roughly 8,500 intact proviruses to sample a specific clone driving rebound. In the case of early ART, fewer intact proviruses may need to be sampled to identify the rebound virus (roughly 400 in a typical case), given the smaller size of the LR.

Discussion

Here we developed a stochastic model of the latent reservoir of HIV-1 that accounts for the inherent heterogeneity of different clones in the reservoir. Our mechanistic approach and direct simulation of each clone allows us to delve deeper into the underlying processes shaping reservoir dynamics. Our model recapitulates changes in HIV-1 measurements in clinical data over the scale of days (decline in HIV-1 RNA in the blood after ART) to years (decline in IUPM over years on ART). We also quantitatively recover a “power law” relationship between clone size and reactivation for large clones, and we find realistic distributions of clones in the LR in simulations that mimic early intervention with antiretroviral drug treatment.

Our results complement several recent experimental investigations of the latent reservoir. Studies have shown preferential integration of intact proviruses into transcriptionally silent regions of the genome [17], and that most transcriptionally active proviruses were selected against during ART [18]. Less transcriptionally active proviruses may be in a deeper state of latency, and thus correspond with clones with smaller probabilities of reactivation in our model. We consistently find that more reactive latent clones are selected against during long-term ART, as reactivation limits their capacity for proliferation and self-renewal. A number of studies have also pointed to the importance of antigenic stimulation in LR persistence [15,21]. In our model, antigenic stimulation is essential for driving the growth of the largest clones; as in prior studies of the T cell repertoire [46], birth-death noise alone is not sufficient to drive very broad differences in clone sizes. Computationally, our work also connects with recent studies that have begun to explore the consequences of heterogeneity and clonal structure in the latent reservoir [37,69,86].

Beyond comparisons with experimental and clinical data, our model makes several predictions about the structure and long-term dynamics of the latent reservoir. Our study suggests that the LR consists of a very large number of clones, especially small clones. Due to their small size and (for some clones) low probability of reactivation when stimulated, they would be difficult to detect through conventional means in studies that seek to characterize the viral reservoir. Nonetheless, collectively, they contribute to the diversity of the latent reservoir and serve as a potential source of viral rebound.

Clonal heterogeneity, including the propensity of different clones for reactivation and stimulation by antigens, emerged as a critical factor to reconcile both short- and long-term dynamics of the latent reservoir after ART. Differences in probabilities of reactivation lead to a slow but progressive “coarsening” of the reservoir, as clones that readily reactivate are eliminated and larger, quiescent ones persist. The persistence of clones with low probabilities of reactivation in our model aligns with recent longitudinal studies that have reported the positive selection of proviruses with lower transcriptional activity during prolonged ART [18].

Our simulations also show nuanced effects of sporadic viral replication during ART. With zero viral replication, all small clones in the LR would ultimately be eliminated due to random clone size fluctuations. However, the level of active replication needed to sustain a population of small clones is insufficient to produce long-term sequence evolution of the virus [32,34–36]. This emphasizes an important distinction between viral replication and evolution, especially evolution within the LR. Persistent, self-sustaining viral replication will lead to the accumulation of mutations (i.e., evolution) over time. However, the same is not true for sporadic bursts of replication that cannot be sustained, and which must be restarted from the same pool of unmutated latent viruses after previous active infections die out. Our model suggests that sporadic replication during ART is consistent with experimental data, but persistent replication is not.

Past work has identified various factors that could affect clonal proliferation, including exposure to antigen and different HIV-1 integration sites [56]. Here, we found that random differences in antigenic stimulation alone are sufficient to reproduce the observed structure of the latent reservoir. This result should not be interpreted as evidence that different integration sites do not play a role in heterogeneous clonal expansion. Rather, our work shows that differences in antigenic stimulation can already lead to stratification in clone sizes and dynamics. Additional factors could also further contribute to the heterogeneity of the LR. For example, a proliferative advantage associated with specific integration sites could promote clonal expansion, potentially extending the lifetime of the reservoir.

Our study has several limitations. In repeat simulations, we chose identical underlying parameters, including the time for ART initiation. In future work, it would be ideal to sample the biologically plausible parameter space more deeply, giving a greater sense of the types of variation in viral dynamics that can or cannot be captured in our model. It could also be helpful to rigorously match simulation conditions with the details of clinical data sets to better constrain model parameters and identify areas where the model must be extended. Our model could also be further tuned to align with outcomes from ATI trials [85] or observed fluctuations in clone size [87]. Regarding differences in the timing of ART, specifically, we find that the distribution of small clones rapidly reaches quasi-equilibrium soon after acute infection (Fig 2). However, very large clones typically take more time to appear. Thus, we expect viral dynamics that depend on the existence of very large clones (or a very broad distribution of clone sizes) to be most sensitive to the timing of ART initiation.

Extensions to our model could also improve its realism and fit to data. For example, we currently model the effects of host immune responses implicitly by reducing viral infectivity. Explicitly modeling host immunity could allow us to understand how differences in immune responses affect the latent reservoir and its dynamics. This would be useful, for example, in investigating unique properties of the reservoir in elite controllers [88]. In addition, some studies have observed large latent clones that are transcriptionally active [18] and contribute to active viral replication [21]. In our current model, we would expect these instances to be rare. However, it may be possible to explain these cases through differences in the baseline proliferation or death rates of individual clones, independent from antigenic stimulation, as in other recent computational LR models [37].

Many assays have been devised to quantify the magnitude and diversity of the HIV-1 latent reservoir, but quantifying the true size of the LR remains challenging [11]. To address this, a hybrid approach combining stochastic modeling and statistical analysis that accounts for the limitations of experimental noise, similar to proposals for T cell repertoire diversity estimation [89], may offer an effective quantitative measurement of the LR. Our work could contribute to this effort by providing a way to study small clones, which are difficult to access experimentally, using a rigorous model constrained by experimental and clinical data.

Understanding the structure and dynamics of the HIV-1 latent reservoir could aid in the development of HIV-1 cure strategies that aim to eliminate or permanently suppress the LR. Our model contributes to these efforts by providing a quantitative description of the LR that is consistent with existing data, but which also extends to “unseen” areas that are difficult to characterize experimentally. One important finding relevant for HIV-1 cure strategies is that large clones that are replication-competent but relatively unlikely to reactivate play a key role in long-term persistence of the LR. In future work, our model could be used to simulate responses to different types of therapeutic interventions, evaluating plausible paths to an HIV-1 cure.

Methods

Our model consists of four main populations: uninfected CD4⁺ T cells (T), productively infected activated CD4⁺ T cells (A), latently infected resting CD4⁺ T cells (L), and HIV-1 virions (V). All cells have finite lifespans determined by their respective death rates, and virions are cleared at a clearance rate c. Uninfected cells are produced at a constant, fixed rate, and virions at a rate proportional to the number of active cells. HIV-1 virions can infect target CD4⁺ T cells, and upon infection, a small fraction will result in cells with defective proviruses due to effects like large deletions or hyper-mutations. Functional proviruses will accumulate a mutation with probability . Finally, a small fraction of infection events will result in latently infected cells. To account for stochasticity, these processes are modeled with probabilities instead of rates.

To account for the heterogeneity of the LR in our model, we define a latently infected clone as a set of cells that have the same TCR, proviral DNA sequence, and integration site. Individual clones will differ in how they are stimulated by antigens and their propensity for reactivation. Due to differences in integration sites, each new clone L_i is assigned a random probability of reactivation . Following work describing dynamics of T cell repertoires [46], we describe the stimulation of each clone by antigens with where k_ij is the interaction coefficient between clone i and antigen j (when clones are cross-reactive), and a_j(t) is the overall concentration of an antigen j as a function of time. We assume that antigen concentration decays exponentially after its introduction at random times as pathogens are encountered and cleared, either passively or through the action of the immune response.

When a latently infected cell is stimulated to divide, there is a probability of the latent provirus reactivating, converting the cell into an actively infected cell. In this case, the number of latent cells decreases by 1 and the number of actively infected cells increases by 1. If no reactivation occurs, then the latently infected cell proceeds to divide in response to the antigen interaction.

The dynamics followed by a latently infected clone L_i are then driven by its basal division rate , death rate , probability of reactivation , and its interaction with antigens f_i(t). These dynamics can be described by a stochastic process in which events (division, death, ...) are selected to occur with probabilities proportional to their rates. After each event, time then advances by a random increment that is exponentially distributed following the sum of all of the rates [90].

While mathematically exact, this explicit simulation approach becomes extremely computationally intensive when dealing with large systems (i.e., large numbers of cells or virions, in our case). However, in this limit the dynamics can be simplified. One can instead then describe the evolution of different populations according to stochastic differential equations (SDEs), which feature a deterministic component and a random term that adds noise to the dynamics [90]. This noise can represent both finite population noise (i.e., the system size is not infinite, so by chance there may be slightly more or less than the expected number of cell deaths at some point in time, for example) and stochastic variation in the environment (i.e., fluctuations in the concentration of antigens, in our case). As one prototypical example, Brownian motion can be modeled with SDEs.

Taking this limit, the dynamics of each latent clone i are described by the SDE

(1)

Here, the first term multiplying the time step dt is the deterministic term, representing changes in latent clone size from antigenic stimulation and reactivation, homeostatic proliferation, and death, respectively. The second term multiplying the Gaussian white noise, , quantifies the random fluctuations in these contributions to clone size fluctuations.

As mentioned above, the function f_i(t) encodes the fluctuating level of antigenic stimulation experienced by clone i. The stochastic process giving rise to f_i(t) is a sum of Poisson-distributed, exponentially decaying spikes. This process is not easily amenable to analytical treatment or simulations, so following the approach of Desponds et al. [46], we assume that correlations among clones are weak and replace the function with a simpler one with the same temporal autocorrelation, that is an Ornstein-Uhlenbeck process:

(2)

Here is a Gaussian white noise, is the inverse of the characteristic lifetime of antigens, and quantifies the strength of variability of the antigenic environment. The antigen concentration experienced by each clone, then, stochastically fluctuates around a baseline value over the course of the simulation.

To model the active cells and virions we need to consider what happens during an infection event, which is illustrated in Fig 1. A virion with sequence k finds and successfully infects a susceptible T cell at rate β. During infection, there is a probability that the integrated provirus will be defective due to large deletions, hypermutation, or other similar alterations. For proviruses that are not defective, we model the accumulation of point mutations with probability . Finally, we consider a probability p_L for the infection to be latent. Each latent infection defines a new latent clone, since we assume that the probability that two identical viruses integrate at the same location in two T cells with identical T cell receptors in separate infection events is essentially zero. This clone could share the same sequence as another latently infected clone but have a very different integration site and, therefore, a different probability of reactivation.

In the same limit as above, the dynamics governing actively infected cells and virions are defined by the following system of SDEs:

(3)

(4)

(5)

In other words, change in the number of actively infected cells with sequence k is driven by: 1) latent reactivation, 2) new infection events that do not result in a defective virus or latent infection (mutants have a different sequence , and thus they also do not contribute here), and 3) the death of actively infected cells. Virions are produced at what we assume for simplicity to be a constant rate from actively infected cells, and they can be lost either from clearance/degradation by the host or through new infection events. The noise term D couples the fluctuations of the virions and actively infected cells.

Finally, the susceptible T cells in our model follow the simple stochastic differential equation

(6)

That is, new susceptible T cells are produced at a constant rate by the thymus, and they are lost by either infection (in which case they become actively infected cells) or death.

Supporting information