Estimating the contribution of CD4 T cell subset proliferation and differentiation to HIV persistence

Study cohort

The HOPE cohort consists of 37 PWH on suppressive ART (clinical and demographic information in Supplementary Table 1), 24 of whom underwent a 45-day deuterium labeling study to measure CD4+ T cell turnover rates and were reported previously¹⁷ in a cross-sectional study. Here, we report a prospective 3-year longitudinal analysis of levels of integrated HIV DNA in distinct maturational CD4 cell subsets from all 37 HOPE participants and integrated these data with measured CD4 cell subset turnover rates. Follow up began 1–10 years after achieving viral suppression. Levels of integrated HIV DNA per million CD4+ T cells tended to be stable over time within individuals but differed between individuals by several orders of magnitude (Supplementary Fig. 1).

Quantifying HIV DNA in CD4+ T cell subsets

From these longitudinal samples, resting (HLA-DR-) CD4+ T cells were isolated and sorted by flow cytometry into six CD4 cell subsets (sort schematic in Supplementary Fig. 2): naïve (T_N), stem-cell memory (T_SCM), central memory (T_CM), transitional memory (T_TM), effector memory (T_EM) cells, and a putative terminally differentiated (T_TD) population. As we observed contamination with T_N in T_TD, the present analysis was focused on the first five sorted populations, each of which was sorted with high purity¹⁷.

CD4+ T cell subset frequency was calculated as the ratio of subset cells per resting CD4 cells (Fig. 1A). T_N and T_CM were most common, each with a median across participants and time of ~25% of all resting CD4 cells. The infection frequency was then calculated as the number of integrated HIV DNA copies per million resting cells within each subset (Fig. 1B). Typically, ~1 in 1000 resting T_TM and T_EM harbored integrated HIV DNA, whereas the other subsets less commonly harbored HIV DNA¹⁶. Finally, by multiplying the subset frequency by the infection frequency, we derived the subset HIV DNA level which reflects the relative contribution of each subset to the measured HIV DNA, i.e., the number of integrated HIV DNA copies in a given subset per million total CD4 cells (Fig. 1C). Although not the highest in infection frequency, given its high subset frequency, T_CM contributed the highest median HIV DNA levels, with ~100 infected T_CM for every million CD4 cells. Median HIV DNA levels were generally lower but not significantly different in other memory phenotypes (T_TM and T_EM). Considerable variability was noted within each subset and for each data type.

HIV infected cells decay faster than non-infected cells in T_TM and T_EM (but not other) subsets

To determine if HIV DNA cleared differently in each subset, we used a statistical framework (log-linear mixed effects model) to assess changes in subset infection frequency over the 3-year study period (Fig. 2A). Although the decay rates were heterogeneous (and even positive, i.e., growing, in certain individuals), the average integrated DNA levels within T_N, T_SCM, and T_CM did not significantly change over time (t test p > 0.05 against null hypothesis of no change), while those within T_TM and T_EM decayed slowly but significantly over time (t test p < 1e-8) (Fig. 2B). Accordingly, T_N and T_CM rates were significantly different from T_TM and T_EM rates (pairwise t tests p < 0.005 according to Bonferroni correction for multiple comparisons). Estimated median half-lives were 81 and 59 months for T_TM and T_EM, respectively. A declining subset infection frequency implies that HIV-infected cells decay faster than non-HIV-infected cells in that subset, suggesting an active process whereby HIV-infected cells are selectively removed.

Measuring cellular turnover via deuterium labeling

We used deuterated water labeling³⁷ performed on 24 of the 37 HOPE participants to estimate cellular turnover rates in each subset¹⁷. In these experiments, the cellular turnover rate is derived from modeling the proportion of cells that take up a deuterium label during a 45-day labeling period (model schematic in Supplementary Fig. 3). More specifically, the fraction of cells that divided during exposure to deuterated water is calculated^37–39. Although what is initially measured from deuterium incorporation into genomic DNA is S-phase cell division, or proliferation⁴⁰, we instead use the term turnover rate here because this rate represents the combination of all mechanisms that impact levels of deuterium in a subset, including migration/trafficking and/or differentiation. For instance, labels in a given subset can rise due to maturation of a labeled progenitor cell or fall due to further maturation⁴¹. Cellular turnover rates ranged across subsets from slowest (T_N median 0.2/year) to most rapid (T_EM median 2.6/years) (Fig. 2C). Turnover rates were generally more rapid in more differentiated subsets, with the greatest differences between T_N to T_SCM and T_CM to T_TM (pairwise t-test p-values in Fig. 2C). A turnover rate of 1 per year corresponds to a half-life of 8.3 months, so these CD4 subsets have median half-lives of 35, 5.3, 4.3, 3.4, and 3.1 months, respectively. Considerable variability was noted within each subset.

CD4+ T cell turnover is often but not always accompanied by HIV DNA turnover in certain subsets

In all subsets except T_N, the cellular turnover rate was roughly an order of magnitude faster than the rate of decay of HIV-infected cells (compare Fig. 2B, C). This suggests that cellular turnover of HIV-infected cells does not usually result in removal of HIV DNA. We therefore estimated the percentage of cellular turnover events that might also be accompanied by HIV turnover rather than HIV clearance (Methods). For the five subsets respectively, we calculated medians of 112, 94, 99, 96, and 94% (Fig. 2D). In T_N, this number is greater than 100% suggesting some increases in HIV DNA in this subset; however, there was very high variability across participants making the median less reliable. Additionally, the much lower cellular turnover rates invoke lower signal compared to noise in the deuterium labeling measurements, potentially reducing precision. In the T_CM subset, we estimate that cellular turnover almost always results in HIV turnover, so HIV DNA does not necessarily decline. Finally, in T_SCM, T_TM, and T_EM, 94–96% of cellular turnover can be associated with HIV turnover. That is, roughly 5% of cellular turnover events are accompanied by clearance of HIV DNA in these subsets (see example for T_EM in Fig. 2E). Together, these results indicate that most, but not all, events that increase cell numbers—cellular proliferation and other mechanisms contributing to turnover—are accompanied by concomitant increases in HIV DNA. Any slight imbalance towards cell number increases without HIV increases could drive decay of HIV DNA in certain CD4 subsets.

Mechanistic modeling of subset HIV DNA suggests differentiation rapidly passages HIV through CD4+ T cell subset maturation pathways

CD4+ T cell subsets are connected to one another by known steps of lineage maturation¹⁴. Previously, in this cohort, we found HIV DNA integrated into identical human chromosomal sites among T_CM and T_TM and T_TM and T_EM subsets, a strong sign that differentiation of HIV-infected cells can occur¹⁷. Moreover, HIV DNA frequencies and levels were found to correlate between certain subsets (Supplementary Fig. 4). Yet, the relative degree to which differentiation into a given CD4 cell subset versus proliferation within that subset contributes to HIV DNA persistence remains unclear. Therefore, we next sought to model HIV DNA levels with a mechanistic model that included specific rules of cellular proliferation, death, and differentiation.

We developed a variety of models inclusive of different mechanistic processes and degrees of complexity (Table 1, see Methods for equations and text describing assumptions). The list of models encodes scenarios in which HIV DNA levels are governed by one or more mechanisms including slow decay, proliferation, and cell differentiation between subsets. A schematic and table of definitions illustrates the rates we consider (Fig. 3A, B). We then tested these models for fit against levels of subset HIV DNA (e.g., Fig. 1C). Importantly, this is a different data type than in Fig. 2 and provides a common denominator of million CD4+ T cells for each subset. In our model, the levels of HIV DNA are linked across subsets, allowing proliferation and differentiation rates to be directly compared.

Models were ranked by their accuracy (fit to data) but also penalized for complexity using information criterion. The selected model (Fig. 3C, Supplementary Movie 1) ranked best by both Akaike and Bayesian information criteria⁴² (AIC and BIC, Table 1). In this best model, each subset level of HIV DNA $H_s$ has a repopulation rate ${\theta }_s$ that encapsulates the balance of cell proliferation and death. Cellular differentiation passages HIV DNA between subsets $i$ to $j$ with rate ${\varphi }_{i:j}$. Because we did not include the terminally differentiated subset (T_TD) due to T_N experimental contamination, we could not estimate T_EM clearance and differentiation rates simultaneously. Therefore, we explicitly note a combination of the two phenomena (see * in Fig. 3C). We also constrained parameter estimation to ensure rates for each subset were no larger than observed cellular turnover rates for that subset (Supplementary Fig. 5A, B). When this constraint on parameter space was relaxed, some models performed slightly better, but our initial best model remained second only to a model with the same structure but including biologically unrealistic rates (Table 1). Therefore, for the remainder of the analysis, we proceeded with this more conservative model.

Qualitative features of model selection provide several mechanistic results. First, all models lacking differentiation had significantly poorer fit compared to the optimal model (ΔAIC > 2, Table 1). A model that attempted to explain HIV levels through differentiation without cell proliferation was substantially worse than the optimal model (ΔAIC = 85). The selected model includes passaging of HIV DNA along CD4 maturation pathways (i.e., linearly from least to most differentiated subsets) but additionally was improved by the addition of “skip” differentiation from T_N to T_CM, and from T_CM to T_EM. A simpler model with purely linear differentiation T_N > T_SCM > T_CM > T_TM > T_EM was ranked 3rd but did not provide as strong a fit to data (Table 1). Together, these findings suggest differentiation is necessary but not sufficient to precisely describe HIV DNA dynamics in CD4 cell subsets over time.

To potentially broaden the applicability of this model, we provide a table of initial conditions, mean and standard deviation of population rates, and estimated variability of HIV DNA data (Supplementary Table 2).

Sensitivity analysis on model selection

To assess whether the sparse 3-year sampling could have resulted in observations favoring a model with skip differentiation, we simulated the best-fit version of the model with linear differentiation, added appropriate noise, and sampled time points per the 3-year study scheme (Supplementary Fig. 6). We then refit this model to the linear- and skip-differentiation models. As expected, the linear differentiation model fit these data better than the skip-differentiation model (ΔLL = 1.5, ΔAIC = 10 compared to skip-differentiation model). This sensitivity analysis illustrates how model selection can be self-consistent, such that data generated with a given model contains enough information to recover the same model via model selection. In addition, it supported that the skip differentiation model was not innately favored based on noise or the sampling scheme.

Estimating HIV DNA decay half-lives in the model inclusive of cellular differentiation

With some exceptions, model fits were excellent across highly variable subset trajectories (see 18 of 37 fits for participants with three time points, Fig. 4A). The overall population trends for each subset show that, notwithstanding some degree of heterogeneity, the average integrated HIV DNA level decays per million CD4+ T cells in 4/5 subsets with a half-life of: 4.3 years in T_N, 2.6 years in T_SCM, 3.2 years in T_CM and 3.7 years in T_EM (Fig. 4B). At the same time, HIV DNA levels in T_TM appeared to increase (which implies no half-life). When HIV DNA levels in all subsets were summed, the net half-life across all subsets was calculated to be 5.4 years. Although these data are not inclusive of all CD4 cell subsets capable of harboring HIV genomes, and individuals have different timeframes of ART (i.e., see trajectories in Supplementary Fig. 1), these half-life estimates are within ranges of previously-estimated HIV DNA decay^22,32,43.

Quantifying the contribution of cell proliferation, death, and differentiation to integrated HIV DNA persistence

To compare and contrast the mechanisms underlying HIV persistence in the best model, we next directly applied the cellular turnover data to estimate the absolute number of integrated HIV DNA copies (per million CD4+ T cells) that enter and leave each subset pool during a typical year due to proliferation, differentiation in and out, and death (Methods, Fig. 3A, B).

In a typical year in the average individual, we calculated (Methods) that 1–10 HIV DNA copies per million CD4 T cells are generated by proliferation of T_N and T_SCM while 100–1000 copies are generated by proliferation in T_CM, T_TM, and T_EM (Fig. 5B). Meanwhile, similar numbers of HIV DNA copies are removed by death (Fig. 5D). These numbers imply that HIV DNA persists in a rapid and dynamic near-equilibrium state (Supplementary Movie 1). At the same time, few HIV DNA copies per million CD4+ T cells enter T_N and T_SCM (Fig. 5A), and 1–10 copies exit those subsets (Fig. 5C) due to differentiation. On average, ten copies enter, and 100 copies leave T_CM due to differentiation (Fig. 5C). The unequal differentiation in and out then requires a slight imbalance favoring proliferation over death (Fig. 5B vs. Fig. 5D) to maintain T_CM near equilibrium. T_TM differentiation was almost balanced (mean ~100 copies in, ~70 copies out in Fig. 5A vs. Fig. 5C). We could not distinguish T_EM outward differentiation from death using these data since terminally differentiated cells were not studied in this analysis. Considerable variability was noted across participants within each subset.

Next, we compared mechanisms relative to one another by calculating the percentage of creation (differentiation in and proliferation) and removal (differentiation out and death) events from each mechanism and for each cell subset (Fig. 5E). Proliferation was the dominant mechanism contributing to the persistence of integrated HIV DNA in T_N, T_CM, and T_TM. However, differentiation inward may play an important role in maintaining HIV genomes in T_SCM and T_EM. Differentiation outward was an important mechanism particularly for T_N and T_CM, in which removal was projected to occur more through differentiation than death. T_EM are known to proliferate frequently and had the highest cellular turnover rates. However, the absolute contribution of proliferation estimated here was lower than differentiation in. If HIV DNA dynamics mirror cellular dynamics measured with deuterated water experiments, this suggests that cellular turnover of HIV-infected T_EM may particularly be influenced by differentiation.

In summary the model portrays typical HIV DNA levels as a rapidly proliferating, dying, and differentiating population that, in aggregate, maintains a nearly equilibrated system such that integrated HIV DNA only decays slowly and only in more mature CD4 subsets (Supplementary Movie 1). Importantly, proliferation remains the predominant mechanism in the generation of integrated HIV DNA. T_N and T_SCM contain less HIV DNA; therefore, the absolute HIV DNA creation and removal in those subsets is orders of magnitude smaller than that found in memory subsets. Proliferation is of particular impact in the context of T_CM and T_TM: when coupled with differentiation outward (to one or more subsets), these subsets contribute meaningfully to HIV DNA persistence in the rapidly dying/differentiating T_EM pool (Fig. 5F).

Modeling cell-associated HIV RNA

We also fit models with no differentiation, linear differentiation, and our favored model with skip differentiation to cell-associated HIV RNA (caRNA) levels measured in the same participants (Supplementary Fig. 7). For these data, the model without differentiation was optimal via AIC. In line with observations for HIV DNA, caRNA levels per million CD4 T cells appeared to increase slightly in T_CM and decrease in T_EM in these participants. But unlike for DNA, RNA increased in T_TM. Together, these data suggest that RNA levels are less tightly connected across subsets, potentially because RNA is generated by DNA and additional variability in this process reduces correlations.

In silico knockout demonstrates the theoretical capacity of reservoir reduction through reduced cell proliferation and/or enhanced cell differentiation

Mechanistic modeling provides the valuable ability to project the dynamics of HIV DNA persistence in the context of perturbed CD4+ T cell subset proliferation and/or differentiation. Thus, we used the model to simulate three therapeutic scenarios over a period of three years: ART alone (Fig. 6A), ART with anti-proliferative therapy that reduces cellular proliferation for all subsets by a factor of 2 (Fig. 6B), and ART with enhanced differentation therapy that increases differentiation for all subsets by a factor of 2 (Fig. 6C). We calculated changes in HIV DNA per million CD4+ T cells over time. For ART alone, (as observed in the raw experimental data) we projected a relatively minimal median change and wide variability inclusive of increases and decreases in all subsets. For ART and anti-proliferative therapy, median HIV DNA across subsets dropped by 300 copies (or ~90%) with most simulations resulting in overall decrease. For ART and enhanced differentiation therapy, median HIV DNA across subsets dropped by 200-300 copies (or ~80–90%) with slightly more simulations inclusive of no change or increase versus anti-proliferative therapy.

Inclusion and ethics

All participants were over 18 years old and provided written informed consent for inclusion before they participated in the study. The study (NCT00187512) is an observational, prospective study of HIV-1 infected volunteers designed to provide a specimen bank of samples with carefully characterized clinical data. The study was conducted in accordance with the Declaration of Helsinki, and the protocol was approved by the University of California San Francisco Committee on Human Research.

Study participant characteristics

Thirty-seven persons living with HIV (PWH) on ART were recruited between 2015 and 2019 from the clinic-based SCOPE and OPTIONS cohorts at Zuckerberg San Francisco General Hospital. Study participants returned yearly for 1-3 time points of follow up. The SCOPE cohort enrolls PWH with chronic HIV, whereas the OPTIONS cohort enrolls PWH < 12 months (before 2003) and <6 months (after 2003) following HIV antibody seroconversion. Viral suppression by ART was a requirement for study entry. Duration of viral suppression was estimated based on clinic records (typically assessed every 3–6 months). HIV acquisition timing for each participant was estimated as previously described⁷⁹.

Isolation of CD4+ T cell subpopulations

All participants underwent leukaphereses performed as outpatients. PBMC were isolated and viably cryopreserved. Frozen PBMC were thawed and CD4 T cells enriched with the EasySep™ Human CD4 + T Cell Negative Selection Enrichment Kit (Stemcell). Cells were stained with Live/Dead Fixable Aqua (Life Technologies) and the following monoclonal antibodies cocktail: anti-CD3-FITC, anti-CD4-AlexaFluor700, anti-CCR7-PE-Cyanine7, anti-CD27-APC, anti-HLA-DR-APC H7, anti-CD57-Brilliant Violet 421, and anti-CD95-PE (Becton Dickinson) as well as anti-CD45RA-ECD (Beckman Coulter). HLA-DR- CD4 + T cell subpopulations were sorted on a FACS ARIA II flow cytometer (BD Biosciences) at >97% purity. Dry pellets were snap-frozen at −80 °C. Flow cytometry data were analyzed on FACSDiva v8.0.1 (BD Biosciences) and FlowJo v8.7 (Tree Star). Sorting schema is provided in Supplementary Fig. 2.

Integrated HIV DNA quantification

Total DNA was extracted using the Allprep DNA/RNA/miRNA Universal Kit (Qiagen). Integrated HIV DNA copies were quantified with a two-step PCR reaction⁸⁰ using isolated genomic DNA for PCR amplification instead of whole cell lysates. Integrated HIV DNA was pre-amplified with two Alu primers and a primer specific for the HIV LTR region, in addition to primers specific for the CD3 gene to determine cell counts. Nested qPCR was then used to amplify HIV and CD3 sequences from the first round of amplification. Specimens were assayed with up to 500 ng cellular DNA in triplicate and copy number was determined by extrapolation against a 5-point standard curve (3–30,000 copies), using extracted DNA from ACH-2 cells.

Cell-associated HIV RNA quantification

Total RNA was extracted using the Allprep DNA/RNA/miRNA Universal Kit (Qiagen) with on-column DNase treatment (Qiagen RNase-Free DNase Set). HIV RNA levels were quantified with a qPCR TaqMan assay using LTR-specific primers F522-43 (5′ GCC TCA ATA AAG CTT GCC TTG A 3′; HXB2 522-543) and R626-43 (5′ GGG CGC CAC TGC TAG AGA 3′; 626-643) coupled with a FAM-BQ probe (5′ CCA GAG TCA CAC AAC AGA CGG GCA CA 3) on a StepOne Plus Real-time PCR System (Applied Biosystems, Inc.)⁸¹. Up to 500 ng of total RNA per sample were characterized in triplicate, and copy numbers were determined by extrapolation against a 7-point standard curve (1–10,000 copies). The input cell number in each PCR well was estimated using independent qPCR measurement of the cellular housekeeping human RPLP0 gene.

Estimating the slope of subset infection fraction

To estimate the slope of HIV subset infection frequency (per million cells of each resting subset), we assumed that the longitudinal kinetics of each subset infection frequency $f_X$ followed an independent exponential model:

So that each subset (denoted by $X$) has a rate of change per year (or log-linear slope) ${\mathrm{\Delta }}_X$. Using MONOLIX²⁵, we estimated the five values of ${\mathrm{\Delta }}_X$. Importantly, we did not assume this rate was negative, such that increases (rather than clearance) were possible. Then, for subsets with negative values of this rate, the half-life in years could be estimated as $hl=-\ln \left(2\right)/{\mathrm{\Delta }}_X$.

Calculating the percentage of cellular turnover events that result in HIV repopulation

In Fig. 2D, we used each subset infection frequency decay rate ${\mathrm{\Delta }}_X$ and its matching cellular turnover rate ${\mathcal{T}}_X$ to calculate the percentage of cellular turnover events resulting in HIV repopulation. Assuming the net decay can be accounted for as a balance of turnover and repopulation ${\mathrm{\Delta }}_X=r_X-{\mathcal{T}}_X$, the repopulation percentage is $r_X/{\mathrm{\Delta }}_X$ or:

Normalized correlations between subset levels

Further evidence for connections between subsets emerged from a correlation analysis (Supplementary Fig. 4). For both subset frequencies and subset HIV DNA data, values were normalized to each individual’s longitudinal average value (i.e., ${\tilde{f}}_X\left(t\right)=f_X\left(t\right)/{⟨f_X⟩}_t$). This procedure prevents spurious correlations (Simpson’s paradox) related to large or small absolute reservoir sizes. Then, pairwise Spearman correlations were computed using the SciPy Stats package.

Mechanistic mathematical models for subset HIV DNA

Our general model of the connected system of HIV DNA in each subset is governed by a system of differential equations that splits the kinetics of HIV DNA into the processes of proliferation, death, and differentiation between subsets. Others have used similar equations⁸². Each model can be written in vector form as:

Where subset HIV DNA in each subset is the vector $H_s\mathit{=}\left\{H_N,H_S,H_C,H_T,H_E\right\}$, and the clearance and differentiation rates are written with the vectors as ${\theta }_s$ and ${\varphi }_{i:j}$, respectively. Differentiation could be generally from different compartments into others so it is not necessarily the same sized vector in each model. The models tested are numbered as follows:

Model 1 assumes each subset is independent and decays or grows independently (similar to the model used for subset infection frequency in Eq.(1)):

Based on past observations of a net decrease in HIV DNA over years of ART, Model 2 tested the hypothesis that all subset HIV DNA decays independently by using the same structure as Model 1 but forcing ${\psi }_s<0$.

Model 3 assumes a linear differentiation model whereby each subset had a decay term and differentiation terms in and out from most proximal subsets. There are, therefore, four differentiation terms: $\mathit{\varphi }=\left\{{\varphi }_{N:S},{\varphi }_{S:C},{\varphi }_{C:T},{\varphi }_{T:E}\right\}$.

Model 4 assumed a more complex differentiation pattern derived from the significant correlations between subsets observed in Supplementary Fig. 4. In this model, there are 6 differentiation terms, the same four linear differentiation rates as in Model 3, and two additional skip terms: ${\varphi }_{N:C}$ and ${\varphi }_{C:E}$.

For models including differentiation, we generally assumed that the differentiation rate of HIV DNA into naïve cells from some unknown/unobserved compartment was zero: ${\varphi }_{?:N}=0$. This assumption is based on TREC content observations suggesting thymic emigrants are not carrying HIV DNA frequently, if at all¹⁷. We also assumed differentiation out from T_EM was zero:${\varphi }_{E:?}=0$. There may be other terminally differentiated cells that T_EM can transition into, but these were not observed in the study. Therefore, the clearance rate of T_EM effectively covers death and differentiation out and is denoted ${\psi }_E$ rather than ${\theta }_E$ to make this explicit in Fig. 3.

As another approach, Model 5 assumes each subset was independent and followed a logistic growth term with a carrying capacity. This tests the hypothesis that decay was not occurring and that HIV DNA levels in each subset had a rough equilibrium:

Yet another approach (Model 6) more explicitly tested the hypothesis that proliferation and differentiation were linked. We assumed that some fraction $\zeta \in \left[\mathrm{0,\; 1}\right]$ of repopulation events are associated with differentiation:

As a final note, multiphasic decay is well documented for HIV DNA clearance after initiation of ART^32,83. However, these phases are generally equilibrated within a year or two of ART initiation, which was irrelevant to our data.

Model fitting and selection with population non-linear mixed effects modeling (pNLME)

Model fit and selection was performed using MONOLIX²⁵ software, which employs a population nonlinear mixed-effects (pNLME) approach. We assumed assay variability (noise) was log normal. Repopulation parameters were generally assumed to be normally distributed (allowing for negative values), and differentiation rates were generally assumed to be lognormally distributed. Population parameters were found to be uncorrelated, but across individuals, certain parameters were strongly correlated (Supplementary Fig. 5C). This finding suggests that those with higher rates in one subset tend to also have higher rates in others. Individual best fit parameters for each participant using the optimal model are collected in Supplementary Data 1.

Imputing turnover rate to define mechanistic components

The underlying assumption of the repopulation rate is that it is a balance of proliferation and death, ${\theta }_s={\alpha }_s-{\delta }_s$. To estimate these component rates, we use the cellular turnover data (Fig. 3B). We begin with a general equation for the $i$-th HIV DNA subset level from the best model and use a quasistatic assumption $\stackrel{°}{H_i}=0$ to indicate that cellular turnover mitigates a balance inward and outward of each subset. Yet, this balance has an absolute value ${\mathcal{T}}_i$ such that in some subsets, although net zero change occurs, there is more inward and outward flow. We therefore set inward and outward mechanisms (from Eq. (5)) equal and split repopulation into proliferation and death, leaving:

The turnover rate ${\mathcal{T}}_i$ (per year) then can be factored out of the rhs as ${\mathcal{T}}_i={\Sigma }_k{\varphi }_{i:k}^{out}+{\delta }_i$, such that the death rate can be defined as the turnover rate minus the differentiation rate out:

And similarly solving ${{\Sigma }_j\varphi }_{j:i}^{in}{T}_j+{\alpha }_i{T}_i={\mathcal{T}}_i{T}_i$, leads to:

Which we approximate by using the values of $T_i\left(0\right)$.

Tracking equations to distinguish mechanistic contributions

After imputing the turnover rates to define the mechanistic compartments, the HIV DNA created at any time $t$ (instantaneously in the interval $\mathrm{\Delta }t$) due to each mechanism was computed. This computation occurs independently after solving the differential equations. Thus, the proliferation and death terms follow

While the differentiation terms follow:

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Peer review information

Nature Communications thanks Eugenio Montini, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.