Dynamics of latent HIV under clonal expansion

The HIV latent reservoir exhibits slow decay on antiretroviral therapy (ART), impacted by homeostatic proliferation and activation. How these processes contribute to the total dynamic while also producing the observed profile of sampled latent clone sizes is unclear. An agent-based model was developed that tracks individual latent clones, incorporating homeostatic proliferation of cells and activation of clones. The model was calibrated to produce observed latent reservoir dynamics as well as observed clonal size profiles. Simulations were compared to previously published latent HIV integration data from 5 adults and 3 children. The model simulations reproduced reservoir dynamics as well as generating residual plasma viremia levels (pVL) consistent with observations on ART. Over 382 Latin Hypercube Sample simulations, the median latent reservoir grew by only 0.3 log10 over the 10 years prior to ART initiation, after which time it decreased with a half-life of 15 years, despite number of clones decreasing at a faster rate. Activation produced a maximum size of genetically intact clones of around one million cells. The individual simulation that best reproduced the sampled clone profile, produced a reservoir that decayed with a 13.9 year half-life and where pVL, produced mainly from proliferation, decayed with a half-life of 10.8 years. These slow decay rates were achieved with mean cell life-spans of only 14.2 months, due to expansion of the reservoir through proliferation and activation. Although the reservoir decayed on ART, a number of clones increased in size more than 4,000-fold. While small sampled clones may have expanded through proliferation, the large sizes exclusively arose from activation. Simulations where homeostatic proliferation contributed more to pVL than activation, produced pVL that was less variable over time and exhibited fewer viral blips. While homeostatic proliferation adds to the latent reservoir, activation can both add and remove latent cells. Latent activation can produce large clones, where these may have been seeded much earlier than when first sampled. Elimination of the reservoir is complicated by expanding clones whose dynamic differ considerably to that of the entire reservoir.


Introduction
Estimates of the latent reservoir half-life under antiretroviral therapy (ART) give a bulk measure of how this collection of cells changes over time [1]. However the degree to which individual cell processes such as homeostatic proliferation and antigen-induced cellular activation, inhibit or accelerate this dynamic is uncertain. So too is the role they play in contributing to residual viremia during ART. Recent data describing frequency distributions of infected cell clonal numbers, provide more detailed data at the individual infected cell level that may help address these questions [2][3][4]. Infection of a cell results in integration of the viral genome into the host cell's DNA. By comparing integration positions between infected cells, researchers can determine whether these occurred through different infection events (the integration positions are different), or the cells are clones of a previously infected cell which experienced at least one round of cell division (the integration positions are identical, as are the proviral genomes). These clones can arise through homeostatic proliferation, or from antigen-induced cellular activation, abbreviated simply to activation, where the memory cells remaining after antigen mediated cellular proliferation and clearance, arise from an infected cell that has maintained latency [5]. The relatively high frequencies of some clones in samples of CD4+ T cells, point to very high expansion of these clones to levels that can reach in their millions and that can also contribute to viremia [6]. These clones can also be maintained over many years even in the presence of ART [3]. Although the structure of sampled clonal distributions has been described [7], what these profiles tell about the reservoir's dynamics at the clonal level have not. A description of clonal dynamics will provide a much more detailed analysis of processes affecting the latent reservoir, and how it may be impacted.
To investigate the impact these processes have on the latent reservoir, an agent-based model was constructed to track clones (since all integrations will be tracked through the model, the term 'clone' will also include singletons (unique integrations)), from the initial infection of an individual at primary HIV infection (PHI) through 10 years prior to the commencement of ART, and followed for a further 11.4 years after ART initiation. Simulating the reservoir both prior to and during ART estimates its impact on the reservoir's dynamics via its changing clonal structure. The model was calibrated to reproduce the general latent reservoir dynamics as well as clonal data for patient 1 of Maldarelli et al. [3], whose latent reservoir was sampled extensively at three time points, years 0.2, 4.8 and 11.4 after ART initiation (Fig 1, data available from the Retrovirus Integration Database rid.ncifcrf.gov). Additionally simulations were compared to the clone profiles of the four other adults described in Maldarelli et al. [3], as well as the three vertically infected children from Wagner et al. [8].

Results
The agent model incorporated death of latent cells at rate μ, homeostatic cellular proliferation at rate λ, and activation of clonal populations at rate α. Seeding of the reservoir occurred at PHI but also throughout the period prior to ART initiation. Separate latent populations of cells containing either defective or genetically intact provirus are modelled, the latter population contributing to plasma viremia levels (pVL, HIV RNA copies/mL) when a clone is activated and when proliferation fails to maintain latency and the cell becomes productively infected. By chance, activation may lead to expansion of a clone that remains at high numbers after antigen clearance. A graphical description of the model is shown in Fig 2. Given uncertainty in parameter values, the model for the intact reservoir component was initially run over 2000 Latin Hypercube samples (LHS) for all parameters, a subset of 382 parameter sets being chosen where the full model reasonably reproduced observed reservoir dynamics (see Materials and Methods). For all 382 parameter sets, the median total reservoir showed little increase in size (0.3 log 10 , Fig 3), over the 10 years prior to ART initiation. Although the latent reservoir can only be directly measured during ART when high numbers of productively infected cells are not contaminating the measurement, some estimate of its dynamics prior to ART has been obtained. Estimates from back-projecting the second phase decay of integrated HIV DNA for a number of individuals where their duration of infection  prior to ART were available, indicated latently infected cell numbers increased with duration of untreated infection when assessed relative to 10 6 CD4+ T cells [9]. That same analysis, when estimated over the whole body, suggests that after the initial infection the latent reservoir is generally stable in number (S1 Fig), as observed here. Change in reservoir size after commencement of ART is limited. For these simulations, it decreases with a median half-life of 15 years measured in total numbers, while still showing a median decay half-life of 2.2 years when measured relative to 10 6 CD4+ T cells (and estimated between the 0.2 and 4.8 year time points on ART, Fig 3. Conversion from number of latent cells in the body to per 10 6 CD4+ T cells was calculated assuming 5.5% of all CD4+ T cells are in peripheral blood [10], distributed through 5×10 6 μLs and using the patient's CD4+ T cell count at the time of each sample). The disparity in these changes is due to CD4+ T cells per μL increasing in number from 48 to 185 copies for this patient.
The model simulations also reasonably reproduced the clone distributions observed for the adult patients [3]. Numerically sampling the same number of latently infected cells as occurred Initial infection seeds the latent reservoir to an amount s 0 cells, and there is continual reseeding prior to ART at each timestep (0.1 years) at the annual rate s 0 = � s � . Both defective (red nucleus) and intact (green nucleus) are seeded, with the latter at 1/100 th of the rate of defective latent cells. Cells die at rate μ per cell per year (all times in years), and have the ability to undergo homeostatic proliferation to produce two new clonal cells at rate λ per cell per year, drawn at first integration from a lognormal distribution for each clone with mean μ and variance λ v . At proliferation, a cell containing genetically intact HIV may instead become sufficiently activated that it progresses to productive infection, with probability 1-p λ , and produces virions. Clones are activated at rate α per clone per year but with probability 1-p α produce memory cells from that clone after undergoing n rounds of division drawn from a Poisson distribution with mean n div . Alternatively, with probability p α activation converts all cells in that clone (up to a maximum of 2 ndiv+mact cells) into productively infected cells, in which case intact clones will produce virions and those cells will be eliminated from the reservoir. The same coloured lines in these plots connect clone size fractions generated within a simulation, displayed as the fraction of clones of that size or higher. The simulation that best reproduced the ART year 11.4 data for Patient 1 is shown as a black line. There was a median of 13 clones that appeared at the first time point (year 0.2) for Patient 1 and were also in the numerical samples at year 11.4 (Fig 4D), the same as the 13 clones cross-sampled for those times in practice (Fig 1). However the simulated high clones at year 11.4 were also mostly detected at the earlier times, which was not the case for this patient. For the only other patient with longitudinal data, Patient 3, there were no clones sampled at both times in practice and a median of 1 clone sampled numerically. It should be noted that both of these patients had complex treatment histories, with Patient 3 undergoing a 10 to 14 day treatment interruption prior to the last sample time, while Patient 1 stopped ART several times between years 4.8 and 11.4, and was diagnosed with lingual squamous cell cancer [3,6]. The treatment interruption at the last time for Patient 3 may have led to the very high clone so that simulations under-estimate the clone distributions. Omitting that size 62 clone produces simulations that encompass the remaining clone sizes (Fig 4, last plot "PT3_Y7.2, low"). Note that all simulations use the same parameter sets so the very different distributions for each patient are due to whatever reservoir changes occur over each time period but are mostly determined by the sample size. How accurately these simulations predict clone size will also have been affected by the time prior to ART and the level of CD4 depletion (unknown period but taken as 10 years for Patient 1 and these simulations (22 CD4+ T cells/μL), 14.8 years for Patient 3 (24 CD4+ T cells/μL), and 6.5 years for Patient 4 (924 CD4+ T cells/μL)).

Dynamics of latent HIV under clonal expansion
Although the reservoir size changes little over time in these simulations (Fig 3), the number of distinct clonal populations decreases for both the defective and intact clones (Fig 5A and  5B). The fast early decay soon after PHI is due to an estimated higher contribution to the reservoir at PHI than through reseeding and the greater chance of a clonal population's loss when it is of size one, which they all are at first infection. The growth of the reservoir arises from the expansion of clonal populations through proliferation but also activation and survival of those infected clones. Activation gives rise to large clone sizes, while proliferation enhances survival through low level expansion. Here clones can achieve a maximum size of around 10 8 cells, values similar to estimates of~10 7 cells from a sampled intact clone [6]. In these simulations, the lower maximum intact clone size is due to chance and the smaller reservoir from which they are drawn, rather than any lower likelihood of surviving activation.
The early loss of lower proliferative clones, that are more likely to be singletons, is also shown by their percentage of the total reservoir ( Fig 6A). In these simulations most of the   (Fig 6B and 6C).
In this model there are two cellular sources of virions, those that arise from activation of an intact latently infected clone (I α ), and those that undergo failed proliferation which instead activates the intact cell (I λ ). Scaling these values to per mL of plasma and then through estimates of the steady state virions that they produce (Materials and Methods), gives numbers of   [12]). Despite their slowly decaying dynamic, some simulations by chance give rise to blips where pVL rise intermittently above the assay detection limit, as observed in some patients [13]. The calculations here are only stored at 4 timepoints after commencing ART so will miss some of the intervening blips that reflect activation of some of the large clones.

Sensitivity analysis
A sensitivity analysis was conducted to determine the parameters most influential in terms of reservoir and pVL dynamics. Using the original 2,000 LHS parameter sets and the simulations of the intact reservoir component, which parallels the dynamics of the total reservoir (Fig 3), reservoir rates of change prior to and post ART were calculated as well as pVL values at the start of ART and its rate of change during ART. Correlations with the model parameters are listed in S1 Table. Rates of reservoir change, regardless of ART, were most significantly negatively correlated with latent cell death rate μ, and positively correlated with the mean number of divisions cells undergo during activation n div . This latter parameter also had the most significant impact leading to higher pVL at the start of ART but also to a greater rate of pVL decay.

Individual simulations
To provide finer detail on how the clones can change over time, the parameter set was analysed that best reproduced the clonal profile at the 11.4 year timepoint on ART for Patient 1 (Fig 4).
As for the multiple simulations, the total reservoir increased over time prior to ART when the larger contribution at PHI is having its greatest impact. The relatively short mean life-span of latent cells in this simulation (14.2 months), resulted in a decaying reservoir on ART, on all measures (Fig 8). The 4.6 year half-life for all clonal populations was shorter than the half-life of the whole reservoir (13.9 years), with the intact reservoir decreasing even more slowly (18 year half-life), likely due to the greater impact on this smaller reservoir of any clones being expanded to high number through activation. The maximum intact clone decreased slightly in size from 4.0×10 5 cells at 10.2 years to 1.8×10 5 cells at 21.4 years. Activation during ART ensured that not all clones would decrease however. Of the 18 intact clones of size 10% or higher of the maximum clone size at 21.4 years within this simulation, 5 of these had been small at 10.2 years (�34 cells), and underwent a mean 7,644-fold increase. Of these 5 expanding clones, 3 had been established at PHI, while one occurred in the year prior to ART initiation. The dynamics of the total reservoir will not be replicated by all individual clonal populations, and activation plays both roles of increasing some, while removing others to contribute to residual pVL (Fig 8). The shift over time to higher proliferating clones, is due to their greater likelihood to survive to expansion through either proliferation or activation (S2 Fig), as has been described by others [14], so that higher levels of detection of integrations in oncogenic sites are less likely due to HIV preferentially integrating in these locations [8].
For this calculation, residual viremia decayed over time with a half-life of 10.8 years, mostly tracking the higher contribution from proliferation than activation in this instance (Fig 8D  and 8E). As antigen-induced cellular activation is more likely to convert a large number of latent cells within an intact clone to productive infection, virion numbers produced by activation may vary dramatically over time, and in this simulation gave rise to intermittent viral blips of a clonal origin, as has been observed in practice [15]. On the other hand, proliferation will act at an average rate over the entire reservoir and hence will contribute smoother changes in pVL. Individuals for which activation is higher will have much more varied pVL profiles and experience more frequent blips.

Dynamics of latent HIV under clonal expansion
The clone size distributions for the adult patients were also reasonably reproduced, despite the simulations using a single parameter set, the stochasticity of the computations, and all assuming a 10 year duration prior to ART initiation (Fig 9). However the largest simulated clones for Patients 1 and 3 at the last sampled times, were also contained in previous samples, in contrast to the data. The larger simulated samples included intact clones, but these were much smaller than the defective clones.

Dynamics of latent HIV under clonal expansion
The other data to which model simulations were compared were for young children vertically infected with HIV [8]. These children had much shorter periods prior to ART initiation, in these simulations taken to be one year. The number of CD27+ memory CD4+ T cells significantly increase from birth to adulthood at the rate of 0.7 per year [16], so that young children will have a lower potential latent reservoir than an adult. Assuming a 10-fold lower starting latent reservoir, and a proportionately lower reseeding rate, with ART commenced one year after infection, leads to clonal distributions for these children shown in Fig 10A. These reasonably match the data for their clone distributions, as well as the number of times clones were recorded at multiple time points Fig 10B, although the simulations tended to produce larger sized clones than sampled.

The relative contributions of proliferation and activation
To further dissect the relative contributions to the latent reservoir of each process that contributes to clonal expansion, a model tracking clonal size dynamics was developed where

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Dynamics of latent HIV under clonal expansion expanding clone numbers only occurred through proliferation-activation was assumed only to remove clonal populations (S1 Text). The dynamics of the number of clones of size i, when only considering expansion through proliferation at a fixed rate λ, is approximately described by a power law D i � � D z iÀ 1 ; i ¼ 1; 2; . . .. At PHI the value of z is small since all clones are size 1, but as infection continues, some will expand under proliferation λ in spite of loss through death μ and activation α, so that some clones will grow in size and the value of z increases over time to the smaller of the values λ/μ or 1, according to the equation This dynamic of proliferating clones provides a good fit to the agent based simulations for the component driven by proliferation (Fig 11, comparison with calculations for children are displayed in S3 Fig.). Under these calculations, the largest defective clone size driven solely be proliferation will be size 232 at the last time point. The probability a clone of this size would be sampled more than once at that time from the estimated 389 million latently infected cells is p = 3.1×10 −7 . Hence the clones sampled multiple times will almost certainly have arisen

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Dynamics of latent HIV under clonal expansion through activation. Proliferation can expand these clones, but activation and survival from that process will be the driver in generating these large clones.
The mean clone size for the proliferative component is given by

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Dynamics of latent HIV under clonal expansion and since z increases over time, and especially when new infections are restricted on ART, the mean clone size will also increase. Under this calculation the mean proliferating clone size is 17.5 at the last time point. The mean size of all clones in the agent based simulation is much higher at 46.7 since it will include the very high sizes arising from activation as well. On the other hand, the median clone size is only 7 mostly determined by the larger number of clones that have expanded by proliferation (8,315,740 of size 232 or less) than activation (19,160).

Discussion
These simulations incorporate the chance events establishing and maintaining the latent reservoir. For certain parameter values they reproduce not only the gross dynamics of the reservoir, but also the more clone-specific elements as observed through longitudinal sampling. The reservoir dynamics were most affected by the average latent cell death rate μ, so that a higher death rate leads to less reservoir growth prior to ART and greater decay during ART, but where this can be offset by a lower activation rate α, and a larger number of divisions a surviving clone has undergone n div . If residual pVL is to change little during ART, then these two aspects related to activation will be low (S1 Table).
For the parameter subsets that reasonably reproduced the limited available longitudinal data [11] (Fig 3), the number of clonal populations within the reservoir always decreased with time, despite the median total number of latent cells showing little change. Both proliferation and expansion through activation compensated the loss in clone numbers. The large clone sizes within the reservoir and observed via sampling, were produced by activation, since this latter process generated the largest clone sizes while proliferation on its own produced much smaller clones (Fig 11). That antigen-induced cellular activation is mainly responsible for very large clones, is consistent with higher clonality in CMV versus Gag-responding CD4+ T cells [17]. The relativities of proliferation and activation also impacted on variability of residual pVL and the occurrence of viral blips (Fig 8).
The parameter set that best reproduced the simulated clone profile for the patient 1 data of Maldarelli et al. [3], exhibited a relatively short latent cell lifespan of 14.2 months. Despite this, activation and proliferation were able to partially compensate for reservoir loss and resulted in decay of the total reservoir with a 13.9 year half-life on ART (Fig 8). The model only includes a single latent cell phenotype, except for allowing variance of proliferation around the mean, so the best fit average lifespans of 14.2 months, much shorter than 20 year estimates [18], may be a composite of resting cells and those with a more activated phenotype. Transition between HLA-DR negative and positive [19,20], may contribute to this lower estimated average cell life-span while incorporating the processes of proliferation. The 18% annual increase in infected HLA-DR memory CD4+ T cells during ART [20], would impact the continued reservoir decay, to an eventual slower reservoir loss or even rise. This evolution in proliferation of the reservoir would also impact on its decay rate relative to a set number of CD4+ T cells or their resting component.
Homeostatic proliferation by itself will not produce the large sampled clone sizes. Under these calculations, it is responsible for the expansion and survival of the majority of clones but that differ only marginally in size (Fig 11), and when sampled will contribute almost exclusively to singletons. The large sampled clone sizes have been generated through activation, regardless of ART. The largest clone sizes sampled at year 11.4 of ART for patient 1 had not been previously sampled which would indicate that they had been expanded in number some time during ART. This patient however had stopped ART for a period prior to this time, during which the integration may have occurred. Proliferation by itself can produce high clone sizes if their proliferative capacity is very high, but then these clones would continue to expand

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Dynamics of latent HIV under clonal expansion relative to lower proliferative clones, so that they would completely dominate samples obtained at later times. Most multiply sampled clones for this patient that also occurred at all sample times (Fig 1), were relatively stable in proportion indicating that they were not driven by continuing high proliferation.
Activation, while responsible for the removal of most clonal populations, also plays a role in off-setting some of the macro loss of the latent reservoir through large expansions of other clones, which may have even been established at PHI and stayed low in number till much later. Time to latent eradication is under-estimated when based on the entire reservoir since simulations exhibiting a 2.1 year intact reservoir half-life relative to 10 6 CD4+ T cells, also contained individual clones that expanded markedly, and was not indicative of the 18 year half life of the intact reservoir in toto (Fig 8). This activation rate produces an average 27 cells per day from the intact reservoir at the final time, considerably higher than previous estimates of 4 cells/day or lower, based on time to viral rebound after ART cessation [21]. The model does not include target cells so cannot simulate time to rebound, and therefore it is uncertain how it will be impacted by this level of activation and the slower decay of the reservoir than previous 44 month estimates [1]. Nevertheless, time to viral rebound can be affected by many factors, not the least being the nonlinearity of viral dynamics [22,23]. That activation occurs continually in these simulations while not always leading to total reservoir loss during ART (Fig 3), indicates that latency reversing agents would need to be potent to add significantly to reservoir loss.
These simulations indicate that a slowly decaying latent reservoir need not indicate cells within the reservoir are long-lived. In the simulation shown in Fig 8, a reservoir decaying with  a 13.9 year half-life was established with cells described by an average 14.2 month half-life. Even with a greatly reduced annual reseeding that is 1/42 nd of that provided at PHI, proliferation and activation were sufficient to maintain the reservoir and produce the observed clonal profiles. As a result of this low reseeding value, the vast majority of the reservoir was established at PHI, 79% within the first year of infection compared to only 2.4% in the year prior to ART (S2 Fig). This is markedly different to results suggesting the majority of the reservoir is established in the year before ART initiation [24][25][26], and even to findings where it is established more uniformly during untreated infection [15]. To achieve either of these results requires a higher rate of reseeding and a higher clearance rate if the reservoir is to be relatively stable in number after PHI. The second best-fit simulation had approximately double the rate of reseeding and did contribute more to the reservoir from the last year before ART (9.7% versus 59% within the first year of infection, S4 Fig), but also produced a larger increase in reservoir size, despite latent cells exhibiting an 11.9 month mean life-span. Although the initial 2,000 simulations covered a wide range of parameter values they cannot be exhaustive, and there may be some choices that better reflect the reservoir and clonal dynamics and also the time contribution to the reservoir. For that to happen would require a balance where the higher reseeding is offset by even greater clearance. In these simulations proliferation is assumed roughly equal to the death rate, so a greater loss would then need an average proliferation rate lower than the death rate, leading to a faster overall reservoir decay at rate μ+α-λ (S1 Text). However these rates may not be independent of ART status, as was assumed here. Superinfection of latent cells prior to ART and their faster clearance than during treatment can also achieve a preferential removal of latent cells established earlier in infection, with a slower dynamic during ART [27].
Although the model and calculations have been constructed to represent as many agents in the model as cells in the latent reservoir, they include a number of simplifications that will affect their ability to truly duplicate its behaviour. All latently infected cells are assumed to have the same approximate lifespan whereas the reservoir is composed of cells with different activated and resting phenotypes [19,20,28]. No differences were assumed in the ability of

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Dynamics of latent HIV under clonal expansion intact or defective clones to survive activation to produce new memory cells from these clones. The memory cells established after clearance of their antigen can be very large, reaching sizes in excess of one million (Fig 5), which would correspond to more than 20 cell doublings from a single infected cell. This is much larger than CD4+T cell expansions that are expected to be of the order of 1,000 fold of which more than 95% would die after antigen clearance [29]. Nevertheless expansions of this magnitude have been observed for even intact clones [6]. High clonal sizes sampled from the reservoir also point on a proportional basis to expansion through activation and subsequent survival of cell doublings of this magnitude (Fig 1) [3].
In conclusion, these agent-based model simulations of the establishment and maintenance of the latent reservoir produce dynamics and clonal size profiles consistent with observations. They also produce pVL at the expected low level via activation of intact clones or proliferation. These calculations point to a reservoir where the size and dynamics of the entire reservoir is not necessarily reflected in the behaviour of individual clones, which can expand to a high degree even while the reservoir is decaying on ART. This will impact prospects for elimination.

Materials and methods
The typical ordinary differential equation model cannot suitably incorporate stochastic events such as when a particular clone will proliferate or die, but instead describe systems where parameters and the resulting simulations represent average behaviour. On the other hand, agent-based models can assign different behaviour to each agent, so that these models are more suitable for describing the generation and dynamics of clones that may have different proliferative abilities or where an event such as activation may or may not happen. The preand post-ART times over which the agent model was run, were chosen to reproduce the data for Patient 1 of Maldarelli et al. [3], who commenced ART at a CD4+ T cell count of 22 cells/ μL (indicating a long history of untreated infection), and whose infected CD4+ T cells were then sampled at years 0.2, 4.8 and 11.4 on ART. Given the low CD4+ T cell count at ART initiation, time to ART was set at 10 years to encompass the median time of 7.7 years to AIDS for an individual of this age [30].
As well as establishing integration events at PHI [31], studies of perinatally infected children undergoing periods of one or two-drug regimens prior to enrollment on ART, show the establishment of latent viral genomes with mutations resistant to the non-suppressive drugs [32]. The slow establishment of drug resistant mutations for non-suppressive regimens was also observed in adults who had initially been enrolled on a lamivudine pre-HAART regimen and who developed the resistant 184V mutation [33]. Hence, in the model, latent cells, both intact and defective, are assumed established at initial infection but with additional reseeding over the period prior to ART initiation from new infection events where the cells revert to latency (Fig 2). The model stored the establishment time of each clone. Here cells containing genetically intact virus are assumed to be capable of producing virions when reverting from latent to productive infection. Since ART essentially halts productive infection [34][35][36], no reseeding is assumed once ART is initiated so that the reservoir will decay dependent on the levels of activation, death and proliferation.
Given that incorporation of the viral genome into each cell can occur at different positions within the host cell genome, including at oncogenic sites which may lead to higher rates of proliferation [8], or that the cells themselves may inherently differ in proliferative ability relative to death [14], it is assumed that each latent cell is established with a proliferation rate λ i that is randomly chosen from a lognormal distribution with mean μ given by the average latent cell death rate (all time is in years, Fig 2). If a cell within a clone is chosen to divide (from a

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Dynamics of latent HIV under clonal expansion binomial distribution at rate λ i ) then that cell will be replaced by two identical daughter cells with the same proliferative rate. Homeostatic proliferation of a cell, which increases the reservoir by one, and death, which decreases the reservoir by one, are chosen to be approximately equal so that in the absence of HIV infection CD4+ T cell numbers are approximately constant. In these calculations, cells chosen to die are removed prior to calculations of proliferation. By chance (probability 1-p λ ), cells containing genetically intact virus may instead upregulate viral protein synthesis upon proliferation to the point where they become productively infected, thereby producing virions and with that cell being cleared from the reservoir ("failed homeostatic proliferation"). Activation is assumed to occur over the entire clone (up to a maximum of 2 ndiv+mact cells within the clone), since they will all recognise the same antigen, at the rate of α clonal populations per year. The maximum clearance amount of large clones on activation allows observations of clonal pVL over extended periods [3,6]; also when clones can reach sizes of 9 million cells [6], it is unlikely that any single event would be able to activate the entire clone. Upon activation of a clone, expansion of these memory cells will produce a subset that remains after activation to act as further memory cells for future occurrences of that antigen. By chance (probability 1-p α ) replacement memory cells will be produced from a cell containing the viral genome which will undergo n div rounds of division (Poisson distributed) producing 2 ndiv clonal copies. If this occurs for a clone containing intact virus, then the cell will instead, with probability p α , become productively infected, giving rise to virion production.
Simulations of the processes included in the model and described in the previous paragraph, are used to determine how they can produce the observed rate of change of the latent reservoir on ART as well as the distribution of clone sizes found in the longitudinal samples of this patient. However they may also explain the source of residual viremia during ART. Under these assumptions, there are two cellular sources of virions, those that arise from activation of an intact latently infected clonal population (I α ), and those that undergo failed proliferation which instead activates the intact cell (I λ ). Virion production by these cells follows the dynamic dV dt ¼ N I a þ I l ð Þ À cV, where N denotes the number of virions exported by a productively infected cell per day and c is the daily loss rate of virions; since the clearance rate of virions is much faster than the loss rate of productively infected cells, this differential equation is in a quasi-steady state so that V � N c I a þ I l ð Þ. This can be translated into HIV RNA copies per ml assuming virions are distributed in the same proportion as percentage of CD4+ T cells in a mL of peripheral blood, and where 5.5% of all CD4+ T cells are in peripheral blood [10] and distributed through 5,000 mLs. With 2 HIV RNA copies per virion, c = 23 per day [37], and N = 1150 (in the range of 1,000 to 50,000 virions per day [38]), this gives HIV RNA copies/mL � 2 � N 23 � 5:5 100 � 1 5000 � I a þ I l ð Þ ¼ 0:0011 I a þ I l ð Þ. The reservoir consists of cells containing either defective or intact virus. Here cells with intact virus are assumed capable of producing virions once activated, or via failed homeostatic proliferation. Depending on how it is measured, the intact latent reservoir is estimated to be in the range of 10 5 to 10 7 cells [39][40][41]. These measurements depend on how 'intact' is determined, for example, through the analysis of the packaging and env regions of the viral genome with the intact proviral DNA assay (IPDA), with the quadruplex PCR with four probes (Q4PCR) assay, or via a quantitative viral outgrowth assay (QVOA) [42,43]. It is also affected by whether it is only measured in resting CD4+ T cells [44], or whether cells with less of a resting phenotype are included, such as those expressing HLA-DR [20], or indeed all of CD4+ T cells [3]. As a consequence the proportion of the reservoir that is intact also varies, with intact relative to total estimates of 11.7% [45], 2.4% [2], or down to 1,000 fold lower [43]. Therefore in these calculations, the number of intact latent cells seeded initially was assumed to be 3×10 6

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Dynamics of latent HIV under clonal expansion cells, with the defective reservoir 100-fold higher. Reseeding also occurred with this ratio of intact versus defective.
Given uncertainty in parameter values, the model for the intact reservoir component was initially run over 2000 Latin Hypercube samples (LHS) for all parameters, with the uninfected CD4+ T cell death rate over a range that ran from an average lifespan of 8 months to 20 years (μ2[0.05, 1.5]), values that encompass the estimated 20 year average time to death for T cells, and close to the 22 week average time to division for memory CD4+ T cells. However simulations allowing up to a 20 year average death rate failed to reproduce a relatively stable latent reservoir prior to ART and at the same time generate large clone sizes observed at the last time for Patient 1. Hence this parameter range was restricted to μ2[0.3, 1.5]. The 100-fold lower intact reservoir calculations were considerably faster than the full model, and when rescaled to the size of the total reservoir produced estimates for their dynamics. A subset of 413 parameter sets were extracted that produced "reasonable levels" in terms of reservoir size and pVL levels. Pinzone et al. [11] had estimated levels of integrated HIV DNA per μL of blood for 2 individuals at time points during ART (Subjects 1 and 2, Supp Table 3, Pinzone et al.). Simulations were extracted that maintained reservoir levels at years 0.2 and 7.2 on ART, within 0.5 logs of two data points: log10ðReservoirSizeð10:2Þ per mLÞ � log10ð3:58Þ þ 0:5&-log10ðReservoirSizeð17:2Þ per mLÞ � log10ð2:17Þ À 0:5. Additionally parameter sets were restricted so that simulated pVL values at the start of ART were no more than 10 copies per mL [12,46]. The 413 parameter sets were run on the full model and the 382 simulations that completed within 24 hours clocktime (each running on 12 CPU nodes and a maximum of 64GB of memory) were retained for analysis.
Tracking the initial 3.03×10 8 latent cells and subsequent expansions/additions/deletions was computationally expensive. Hence comparisons with patient data was estimated from simulations evaluated at the closest times of 0.2, 4.8, 7.2 and 11.4 years on ART, the sample times for Patient 1 and the last time for Patient 3 of Maldarelli et al. [3]. For example comparisons to the last sample for R1 at year 12.3 was determined by numerically sampling the reservoir at year 11.4 on ART.
All calculations were performed with Matlab R2021a, Mathworks Inc., Natick MA, USA.