Modeling Latently Infected Cell Activation: Viral and Latent Reservoir Persistence, and Viral Blips in HIV-infected Patients on Potent Therapy

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1 Moeling Latently Infecte Cell Activation: Viral an Latent Reservoir Persistence, an Viral Blips in HIV-infecte Patients on Potent Therapy Libin Rong, Alan S. Perelson* Theoretical Biology an Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico, Unite States of America Abstract Although potent combination therapy is usually able to suppress plasma viral loas in HIV-1 patients to below the etection limit of conventional clinical assays, a low level of viremia frequently can be etecte in plasma by more sensitive assays. Aitionally, many patients experience transient episoes of viremia above the etection limit, terme viral blips, even after being on highly suppressive therapy for many years. An obstacle to viral eraication is the persistence of a latent reservoir for HIV-1 in resting memory CD4 + T cells. The mechanisms unerlying low viral loa persistence, slow ecay of the latent reservoir, an intermittent viral blips are not fully characterize. The quantitative contributions of resiual viral replication to viral an the latent reservoir persistence remain unclear. In this paper, we probe these issues by eveloping a mathematical moel that consiers latently infecte cell activation in response to stochastic antigenic stimulation. We emonstrate that programme expansion an contraction of latently infecte cells upon immune activation can generate both low-level persistent viremia an intermittent viral blips. Also, a small fraction of activate T cells revert to latency, proviing a potential to replenish the latent reservoir. By this means, occasional activation of latently infecte cells can explain the variable ecay characteristics of the latent reservoir observe in ifferent clinical stuies. Finally, we propose a phenomenological moel that inclues a logistic term representing homeostatic proliferation of latently infecte cells. The moel is simple but can robustly generate the multiphasic viral ecline seen after initiation of therapy, as well as low-level persistent viremia an intermittent HIV-1 blips. Using these moels, we provie a quantitative an integrate prospective into the long-term ynamics of HIV-1 an the latent reservoir in the setting of potent antiretroviral therapy. Citation: Rong L, Perelson AS (2009) Moeling Latently Infecte Cell Activation: Viral an Latent Reservoir Persistence, an Viral Blips in HIV-infecte Patients on Potent Therapy. PLoS Comput Biol 5(10): e oi: /journal.pcbi Eitor: Rustom Antia, Emory University, Unite States of America Receive April 30, 2009; Accepte September 15, 2009; Publishe October 16, 2009 This is an open-access article istribute uner the terms of the Creative Commons Public Domain eclaration which stipulates that, once place in the public omain, this work may be freely reprouce, istribute, transmitte, moifie, built upon, or otherwise use by anyone for any lawful purpose. Funing: Portions of this work were one uner the auspices of the US Department of Energy (DOE) uner contract DE-AC52-06NA This work was supporte by NIH grants AI28433 an RR The funers ha no role in stuy esign, ata collection an analysis, ecision to publish, or preparation of the manuscript. Competing Interests: The authors have eclare that no competing interests exist. * asp@lanl.gov Introuction Following initiation of highly active antiretroviral therapy (HAART) the plasma viral loa eclines with a rapi first phase, followe by a slower secon phase (Figure 1, see reviews in [1 3]). After several months of treatment, most patients attain a level of plasma HIV-1 RNA below the etection limit (e.g., 50 copies/ml) of current stanar assays [4 6]. However, this oes not imply that viral replication has been completely suppresse by therapy. On the contrary, even in patients with unetectable plasma viral loas for many years, a low level of virus can be etecte in plasma by supersensitive assays [7 9]. This phase with HIV-1 RNA below 50 copies/ml has been referre to as the thir phase of viral ecline after treatment [1] (Figure 1), although whether virus eclines or persists at a constant level is still unresolve [9,10]. The factors influencing this low-level viral persistence an their relative contributions have not been fully eluciate. It is possible that current HAART regimens are not completely suppressive an HIV- 1 continues to replicate, particularly in some rug sanctuary sites such as the brain an testes, where rugs have poor penetration (see [11,12] an reviews in [13,14]). A secon explanation is that HIV-1 establishes a state of latent infection in resting memory CD4 z T cells [15,16], an virus is release when these cells encounter their relevant antigens an are reactivate [17]. The latent reservoir persists in patients on HAART [18 20] an ecays slowly, with the estimate half-life up to 44 months [21,22]. It is more likely that both factors contribute to viral persistence. The latent reservoir releases virus that fuels ongoing viral replication, an ongoing viral replication replenishes the latent reservoir. We still lack a quantitative unerstaning of the relative contributions from resiual ongoing viral replication an latent cell activation to the observe sustaine low-level viremia. Another line of evience for HIV-1 persistence is the observation of transient episoes of viremia ( blips ) above the etection limit in patients on HAART (Figure 1) [7,23]. Because viral blips are relatively rare events, their occurrence time, frequency, uration an amplitue are not well known. Di Mascio et al. [24] stuie viral loa time series with samples obtaine approximately one month apart from 123 patients, an foun that the mean blip frequency was 0:09+0:11=sample, an the mean blip amplitue was RNA copies/ml. They also suggeste that a viral blip was not an isolate event but rather an extene transient episoe of viremia with a uration of approximately 3 weeks [25]. In another stuy, Nettles et al. [26] examine the ynamics of blips PLoS Computational Biology 1 October 2009 Volume 5 Issue 10 e

2 Author Summary Current combination therapy can suppress viral loas in HIV-1-infecte iniviuals to below the etection limit of stanar commercial assays. However, it cannot eraicate the virus from patients. HIV-1 can generally be ientifie in resting memory CD4 + T cells an persists in patients on potent treatment for a long time. These latently infecte cells ecay slowly, but can prouce new virions when activate by relevant antigens. Many patients experience transient episoes of viremia, or blips, even though they have unetectable plasma viral loas for many years. Here, we evelop a new mathematical moel escribing latently infecte cell activation upon ranom antigenic stimulation. Using the moel, we show that programme expansion an contraction of latently infecte cells upon activation can generate both low viral loa persistence an viral blips. Occasional replenishment of the latent reservoir may explain the ifferent ecay kinetics of the reservoir observe in clinical practice. We also show that a moel with homeostatic proliferation of latently infecte cells can explain persistence of low-level virus, stability of the latent reservoir, an emergence of viral blips. These results provie novel insights into the long-term virus ynamics an coul have implications for the treatment of HIV-1 infection. with more intensive sampling over a shorter perio in a cohort of 10 patients. They foun that blips were brief with a mean uration of less than 3 ays an ha a mean amplitue of 79 copies/ml. Moreover, viral blips were not concorant on inepenent testing, inicating that ranom biological or statistical variation aroun a mean viral loa less than 50 copies/ml might be responsible for the aberrant viral loa measurements [26]. The observations by these stuies may represent ifferent phenomena, with Nettles et al. [26] observing the effects of assay variation an Di Mascio et al. [24] observing higher amplitue blips generate by occasional immune activation events [27]. Figure 1. Multiphasic viral ecline after potent treatment. After initiation of HAART, the plasma viral loa unergoes a multiphasic ecay an eclines to below the etection limit (e.g., 50 RNA copies/ ml) of stanar assays after several months. A low level of viremia below 50 copies/ml may persist in patients for many years espite apparently effective antiretroviral treatment. Intermittent viral blips with transient HIV-1 RNA above the limit of etection are usually observe in well-suppresse patients. oi: /journal.pcbi g001 The management of HIV-1 infection requires a further unerstaning of the mechanisms unerlying low viral loa persistence, stability of the latent reservoir, an occurrence of intermittent viral blips, as well as the relationships between them. We approach this through mathematical moeling. Many moels, as surveye in [28], are not capable of realistically accounting for viral loa persistence since the presence of low-level replication is extremely sensitive to small changes of rug efficacy. Stuies of the ynamics of the latent reservoir an viral blips are also ifficult because latently infecte cells are very rare [15] an blips appear to emerge ranomly [24,26]. Consiering the heterogeneity of the pool of latently infecte cells, a simple moel was evelope to stuy the ecay characteristics of the latent reservoir [29]. Kim an Perelson [30] extene the moel an showe that the latent reservoir persistence coul be explaine by bystaner proliferation of latently infecte cells. The relationship between low-level viral replication an the ecay of the latent reservoir was examine in a recent stuy by Seaghat et al. [31]. They evelope a simple moel consiering the transition between latently infecte an activate T cells. The results emonstrate that viral ynamics in patients uner HAART might be consistent with low-level viral replication but the replication i not have much impact on the ecay rate of the latent reservoir, which confirms their earlier moeling preictions [32]. Mathematical moels have also been propose to test possible mechanisms for the generation of viral blips. Jones an Perelson showe that activation of either target T cells [33] or latently infecte cells [27] coul result in a burst of virus prouction. Asymmetric ivision of activate latently infecte cells may explain the variable ecay kinetics of the latent reservoir an intermittent viral blips [34]. In this paper, we further stuy latently infecte cell activation in response to antigenic stimulation by extening the moels in [27,30,33]. We examine the hypothesis that stochastic activation of latently infecte cells can generate intermittent viral blips an maintain low-level plasma viremia, without seriously epleting the latent reservoir in patients uner HAART. The moel focuses on the response of latently infecte cells when they encounter their relevant antigens. We show that programme expansion an contraction of latently infecte cells can generate intermittent viral blips with realistic amplitue an uration. During the latent T cell response, part of the resultant activate T cell population reverts back to a resting state, proviing a mechanism to replenish the latent reservoir. An interesting result of our moel is that ifferent potentials of activate T cells to proliferate uring the response or ifferent uration or frequency of antigenic stimulation can explain the ifferences between the ivergent estimates of the half-life of the latent reservoir ecay in HAART-treate patients [21 23,35 38]. Using this moel, we stuy the influence of ongoing viral replication on both the ecay of the latent reservoir an persistence of low-level viremia. We perform sensitivity tests on a number of moel parameters. Finally, we evelop a phenomenological moel that postulates ensity-epenent homeostatic proliferation of resting memory CD4 z T cells. A recent experimental stuy supports the iea that homeostatic proliferation of latently infecte cells may ensure the latent reservoir persistence without any emonstrable evience for viral prouction [39]. The moel can robustly escribe the multiphasic viral ecline following initiation of potent antiretroviral treatment. The ifferent self-renewal potentials of latently infecte cells are also able to reconcile the variable ecay kinetics of the latent reservoir. Our moels provie a new perspective into the possible mechanisms for viral an the latent reservoir persistence an emergence of intermittent viral blips. 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3 Methos A basic moel of latent cell activation A basic moel of latent cell activation was initially evelope to examine the cell populations contributing to the secon-phase viral ecline after aministration of both reverse transcriptase (RT) an protease inhibitors [4]. Using an overall rug efficacy,, the basic moel can be reuce to the simpler form [34]: t T(t)~l{ TT{(1{ )kvt, t L(t)~g(1{ )kvt{ 0L{a L L, ð1þ t T (t)~(1{g)(1{ )kvt{t za L L, t V(t)~NT {cv, where T represents CD4 z T cells that are susceptible to HIV-1 infection, T represents prouctively infecte cells that can prouce virus particles, L represents latently infecte cells that cannot prouce virus but are reay to o so once they are activate by their recall antigens, an V represents the total viral loa. l is the recruitment rate of susceptible T cells an T is their mortality rate. The constant k is the infection rate. an 0 are the eath rate of prouctively an latently infecte cells, respectively. c is the clearance rate of free virus. N is the burst size, the total number of virions prouce by an infecte cell uring its life span. g is the fraction of infections that lea to latency. a L is the transition rate at which latently infecte cells become prouctively infecte cells. is the total rug efficacy, which is efine as ~1{(1{ RT )(1{ PI ) where RT an PI are the rug efficacy of RT an protease inhibitor, respectively. There is only one positive steay state viral loa of Eq. (1): { V~ Nl c (1{ 0 g){ T 0 za L (1{ )k : ð2þ It is biologically plausible if an only if is less than a critical efficacy, given by c T c~1{ knl(1{ : ð3þ 0 g) 0 za L If c, then the only steay state is the uninfecte steay state, with { V~ { T ~ { L~0, { Tw0. The steay state viral loa (2) can theoretically achieve any positive value close to zero. However, it remains very sensitive to small changes of rug efficacy, particularly when approaches c [28,34]. Therefore, the basic moel an its various variations [28] are not realistic to escribe the persistence of low-level viremia in patients on HAART. Furthermore, the moel cannot maintain the latent reservoir size unless the eath rate of latently infecte cells ( 0 ) an the transition rate (a L ) are both chosen to be very small [31]. If transient episoes of viremia also come from activation of latently infecte cells as suggeste in [40], then the latent reservoir will be eplete more quickly than observe in clinical stuies. Moel with programme expansion an contraction of latently infecte cells upon activation Both CD4 z an CD8 z T cell responses to infectious agents (for example, lymphocytic choriomeningitis virus (LCMV) [41] an Listeria monocytogenes [42]) can be broken own into three istinct phases: expansion/activation, contraction/eath an maintenance/memory [43]. Upon initial exposure to antigen, specific T cells unergo consierable antigen-riven expansion an ifferentiation into effector cells, whose major function is to kill infecte cells. A contraction or eath phase then ensues, in which the majority of activate T cells ie quickly by apoptosis or activationinuce cell eath. The thir phase is characterize by a stable [44] or slowly ecaying pool [45] of memory cells, which are forme uring the response an are maintaine for long perios of time. By eveloping mathematical moels consiering these phases, De Boer et al. stuie the ynamics of the CD8 z T cell response to LCMV [44] an compare them with the CD4 z T cell response to LCMV [45]. Fitting moels to experimental ata, they obtaine the T cell oubling time uring the expansion phase an the T cell half-life uring the contraction phase. These results suggest that the CD8 z T cell response has faster kinetics in almost every aspect than CD4 z T cells [45]. Jones an Perelson [33] evelope a moel that accounts for both HIV infection an the programme cascae of ivisions uring the expansion of the CD8 z T cell response to a concurrent opportunistic infection. Using the moel, they showe that target cell activation [33] or latent cell activation [27] cause by opportunistic infections was able to explain the transient low-level viremia observe in well-suppresse patients on potent treatment. Here, we reexamine the moel in [27] an evelop a new one in which latently infecte cells are hypothesize to experience programme expansion an contraction in response to their specific antigens, an in which a small portion of activate cells revert back to the resting state by the process that normally generates memory CD4 z T cells (Figure 2). We investigate whether repeate latent cell activation through this type of programme response can generate intermittent viral blips with reasonable amplitue an uration, an whether the replenishment of latently infecte cells can control the ecay of the latent reservoir. Let L 0 represent the concentration of resting latently infecte CD4 z T cells. These cells on encounter with their relevant antigens may enter the class of activate cells, L a. One moel escribing the programme expansion an contraction of latently infecte cells upon antigenic stimulation is as follows: t T(t)~l{ TT{(1{ )kvt, t L 0(t)~g(1{ )kvt{ 0 L 0 {f (t)al 0 z(1{f (t))rl a, t L a(t)~f (t)(al 0 zpl a ){(1{f (t))(azr)l a {a L L a, ð4þ t T (t)~(1{g)(1{ )kvt{(t )T za L L a, t V(t)~p vt {cv: When the antigen is present, resting latently infecte cells, L 0, are activate into the activate class L a with rate constant a. The function f (t) etermines the times at which antigen is present at concentration sufficiently high to activate cells. Activate cells proliferate at rate p. Once the antigen concentration falls, we assume there is a contraction phase, in which activate cells ie or apoptose at rate a, or revert to the resting state at rate r. In aition, activate latently infecte cells transition into prouctively infecte cells at rate a L uring the entire response. As suggeste by [28,46], we use a ensity-epenent eath rate of prouctively infecte cells in orer to reuce the sensitivity of the steay state viral loa to changes of rug efficacy. The biological justification for the ensity-epenent cell eath rate is as follows: prouctively infecte cell can be kille at a rate that epens on the ensity of effector cells. The population size of effector cells can be PLoS Computational Biology 3 October 2009 Volume 5 Issue 10 e

4 Figure 2. Schematic representation of the moel with latently infecte cell activation (Eq. (4)). Following encounter with cell-specific antigens, latently infecte cells are activate an unergo programme clonal expansion an contraction. A number of activate latently infecte cells transition to the prouctive class an prouce virions, whereas another small fraction of activate cells revert back to the latent state, proviing a mechanism to replenish the latent reservoir. oi: /journal.pcbi g002 further assume to be proportional to the ensity of infecte cells. Thus, the eath rate of prouctively infecte cells can be assume to be a function of the infecte cell ensity. We choose a simple powerlaw function, (T )~ T v, as use in [28,46], where v controls the size of the immune effect on the eath rate. Holte et al. [46] obtaine estimates of v by fitting the ensity-epenent ecay moel to patient ata. Because the moel inclues a ensity-epenent infecte cell eath rate, we have to ecouple the viral prouction rate from the cell eath rate. We assume virus is prouce at a constant rate, p v,per prouctively infecte cell, T. For simplicity, we assume p v ~N. A moification of this moel will be given later to stuy viral persistence without the assumption of ensity-epenent infecte cell eath. We employ a basic on-off moel, which has previously been use to escribe the CD4 z an CD8 z T cell responses to viral infection [44,45], to approximate the antigenic stimulation of latently infecte cells instea of explicitly moeling the interaction between naive T cells an their specific antigens as was one in [27,33]. The activation function, f (t), is antigen-epenent an takes on only two values: 0 if there is no activation, an 1 if there is full activation. If T on is the time at which the stimulation switches on an T off is the time at which the stimulation is off, then f (t) assumes the following expression: 8 >< 0 if tvt on, f (t)~ 1 if T on ƒtvt off, >: 0 if t T off : We enote by Dt~T off {T on the uration that each activation lasts. Although CD4 z an CD8 z T cells both commit to clonal expansion after antigenic stimulation, CD8 z T cells typically have a ð5þ higher proliferative potential both in vitro an in vivo compare with CD4 z T cells [41,47]. It has been estimate that CD8 z T cells ivie about 15{20 times uring an acute infection with LCMV, while CD4 z T cells ivie approximately 9 times [41]. Choosing the proliferation rate of CD4 z T cells as in [45], p~1:4 ay {1, CD4 z T cells can ivie 8 to 12 times if the expansion phase lasts 4{6 ays. If p~0:8 ay {1,thenCD4 z T cells only ivie 5 to 7 times over the same perio. With various proliferation rates, we will show that the number of times that activate latently infecte cells ivie upon stimulation is an influential factor that not only controls the ecay of the latent reservoir but that also affects the amplitue of viral blips. Because only a small fraction of latently infecte cells are specific for any given antigen, we choose a to be 0:03 ay {1.We will perform sensitivity tests on a few parameters incluing a. The eath rate of activate cells uring the contraction phase, a, is not well-known. In [45], about 0:2 ay {1 was estimate for this parameter. Because the cells activate from latently infecte cells are usually not observe, here we choose a larger eath rate, a~0:8 ay {1, such that activate latently infecte cells will ecline to low levels after a relatively short perio. We will iscuss the effect of a smaller a later. We also assume that a small fraction of activate cells revert back to the resting state, with rate r~0:01 ay {1 [27], whereas another portion of them transition into the prouctive stage with rate a L ~0:1 ay {1. We will test our moel preictions with ifferent values of r an a L. We choose the overall rug efficacy ~0:85 as the baseline value so that viral loa can be suppresse to below the etection limit after several months of treatment. In fact, as we will show below, specific values of the rug efficacy o not strongly impact viral an the latent reservoir persistence once it excees a threshol calle the critical efficacy. The ynamics of viral loa, PLoS Computational Biology 4 October 2009 Volume 5 Issue 10 e

5 the latent reservoir an viral blips will also be compare with ifferent rug efficacies. Similar arguments can be applie to the choice of the value of g, the fraction of infections resulting in latency. As long as it represents a small fraction of infections, the value of g has only a minor effect. Here we choose g~0:001 [27] as an example. The form of the activation function f (t) will be further iscusse below. The viral burst size, N, can affect the amplitue of blips generate from activation of latently infecte cells. Here we use N~2000 [48], although recently higher values of N have been estimate for SIV [49]. It is not known if these higher burst sizes apply to HIV. However, if higher values of N are use, then other parameters in Table 1 nee to be ajuste, such as the viral clearance rate, which recent work suggests may be higher in tissue than has been estimate in bloo (De Boer R., Ribeiro R. an Perelson AS, unpublishe results). The other parameter values are chosen base on previously publishe reports an are summarize in Table 1. Since we are intereste in the ynamics of the thir-phase viral ecline uring treatment, we choose the initial viral loa to be V(0)~50 RNA copies=ml. With an assumption of quasisteay state between virions an prouctively infecte cells, we obtain the initial conition for prouctively infecte cells, T (0)~0:3 cells=ml. We set T(0)~600 cells=ml as the initial conition for target T cells [30]. The total number of latently infecte cells with replication-competent viral genomes is assume to be cells [15], 98% of which are in the lymphoi tissue an the rest are in the bloo. Assuming the bloo volume is 5 L, the concentration of latently infecte cells with replicationcompetent provirus is 2 cells/ml, i.e., L 0 (0)~2 cells=ml. We assume there are no activate latently infecte cells initially, i.e., L a (0)~0. Moel with a biphasic contraction phase Homann et al. [41] suggeste a multiphasic contraction phase in the CD4 z T cell response to acute LCMV infection. De Boer et al. [45] evelope a mathematical moel that fits the Homann ata using two istinct phases of activate cell eath after the peak of the response. Here, we moify moel (4) by aopting a biphasic contraction phase in the latently infecte CD4 z T cell response. More motivations will be aresse in the Results section after we present the results of moel (4). In the moifie moel, the T an L 0 equations remain the same, while the other equations change to Table 1. Variables, parameters an values use in moels an simulations. Variable/Parameter Value Description Reference T(t) - Target T cells - L(t) - Latently infecte cells - L 0 (t) - Resting latently infecte cells - L a (t) - Activate latently infecte cells - T (t) - Prouctively infecte cells - V(t) - Viral loa - l 10 4 ml {1 ay {1 Recruitment rate of susceptible cells [28] T 0:01 ay {1 Death rate of susceptible cells [90] k 2:4 10 {8 ml ay {1 Infection rate [50] 0.85 Overall rug efficacy see text g 0:001 Fraction resulting in latency [27] 0 0:001 ay {1 Death rate of latently infecte cells [28] a L 0:1 ay {1 Rate of transition from latently to see text prouctively infecte cells 1 ay {1 Death rate of prouctively infecte cells [91] N 2000 Burst size [48] c 23 ay {1 Clearance rate of free virus [92] 0:7863 ay {1 (ml=cell) v Density-epenent mortality [28] v 0:44 Power in ensity-epenent mortality function [46] p v 2000 ay {1 Viral prouction rate [48] p varie Proliferation rate of activate cells see text a 0:03 ay {1 Activation rate of latent cells see text a 0:8 ay {1 Death rate of activate cells see text r 0:01 ay {1 Reversion rate to latency [27] a 0:02 ay {1 Base eath rate of activate cells [45] r varie Maximum proliferation rate of latent cells see text L max varie Carrying capacity ensity of latent cells see text f (t) see text Expansion function - g(t) see text Rapi contraction function - oi: /journal.pcbi t001 PLoS Computational Biology 5 October 2009 Volume 5 Issue 10 e

6 t L a(t)~f (t)(al 0 zpl a ){(1{f (t))( a zr)l a {g(t)al a {a L L a, t T (t)~(1{g)(1{ )kvt{t za L L a, ð6þ t V (t)~n T {cv, where f (t) is the expansion function efine by Eq. (5). Following the expansion phase, there is a two-phase contraction: a rapi contraction phase of length Dt c, where activate cells ie rapily by apoptosis or activation-inuce cell eath, at a rate a, an a slower phase where activate cells ie at their base mortality rate a. For simplicity, we assume the rapi contraction phase has the same length as the expansion phase (i.e., Dt~Dt c ). g(t) represents the contraction function. During the rapi contraction phase, g(t)~1, otherwise, g(t)~0. A phenomenological moel with homeostasis of latently infecte cells A recent experimental stuy by Chomont et al. [39] shows that the HIV-1 latent reservoir size may be maintaine by homeostatic proliferation of latently infecte cells. Thus, we incorporate a logistic term representing homeostatic proliferation of latently infecte cells into the basic moel (1). The L equation becomes L(t)~g(1{ )kvtzrl(1{ L ){ 0 L{a L L: ð7þ t L max The other equations for T, T an V are the same as those in moel (1). In Eq. (7), r represents the maximum proliferation rate an L max represents a threshol latent cell ensity, beyon which proliferation shuts off. Whether there is such a strict maximum is unclear an thus other forms of ensity-epenent proliferation coul also be explore, such as rl=(kzl), where K is a constant. We choose a small base value for the transition rate, a L ~0:01 ay {1 [27], because only a small fraction of latently infecte cells are specific for any given antigen. We will increase the value of a L when we stuy latently infecte cells encountering their specific antigens, which is use to moel emergence of viral blips uring treatment. In orer to maintain the latent cell pool uring potent rug therapy, we choose the proliferation rate r to be greater than 0 za L, i.e., 0:011 ay {1. In fact, it can be prove that in the case of 100% rug effectiveness, the infecte steay state exists an is locally asymptotically stable if an only if rw 0 za L. The carrying capacity (i.e., the maximum sustainable population) of latently infecte cells uring therapy is unknown. Assuming total boy lymphocytes, Chun et al. [15] reporte a total boy loa of resting CD4 z T cells with integrate HIV-1 DNA of *10 7 cells uring the asymptomatic phase of infection. Here we allow the total boy carrying capacity of latently infecte cells to vary from *10 6 cells to *10 7 cells but then convert these numbers to a cell ensity in bloo so as to be in the same units as the target cells, T. For example, if there are maximally latently infecte cells per patient uner HAART, then the maximum ensity of latently infecte cells in bloo is L max ~ %=5000~32 cells=ml since *2% of CD4 z T cells are in bloo an the typical 70 kg iniviual has about 5L of bloo. We will iscuss the effects of ifferent values of r an L max on the final moel preictions. ThesimulationwithaninitialTcellcountT(0)~10 6 cells=ml [50] an an arbitrary initial viral loa value V(0)~10 {3 virions=ml yiels a set of steay state values in the absence of rug treatment: { T~479 cells=ml, { L~123 cells=ml, { T ~5: cells=ml an { V~4: virions=ml. These values are set as the initial conitions when performing simulations of the moel uring HAART. Results Intermittent viral blips an ecay of the latent reservoir Numerical simulations of moel (4) show that programme expansion an contraction of latently infecte cells upon occasional antigenic stimulation can robustly generate intermittent viral blips with reasonable amplitue an uration, without seriously epleting the latent reservoir (Figure 3). We assume that latently infecte cells encounter their specific antigens ranomly. As an example, we assume the interval between two ajacent activations, DT, obeys a normal istribution with a mean of 50 ays an a stanar eviation of 10 ays. If we use a Poisson process to moel the encounter between latently infecte cells an antigen, then we get a similar pattern of viral blips an the latent reservoir ecay when the average waiting time between two encounters is assume to be *50 ays. The uration of activate T cell proliferation uring the latent cell response remains unknown. In fact, the mechanisms that control the rate an extent of T cell ifferentiation are complicate [47]. It may involve the amount of antigen an other types of cytokines that are present in vivo, the uration of antigen exposure, as well as whether T cell proliferation continues in the absence of further antigenic stimulation [51 54]. In our simulation, we assume the uration of activate T cell proliferation, Dt, obeys a uniform istribution over the interval of 4 to 6 ays such that CD4 z T cells ivie approximately 5 to 12 times with appropriate proliferation rates (see below) [41]. A transient episoe of viremia is observe every time an activation occurs. Thus, the timing an frequency of viral blips are etermine by when an how often latently infecte cells encounter their recall antigens. The uration of transient viremia is etermine by how long the antigen is present (Figure 3). Another important characteristic of viral blips, the amplitue, ranges from 50 RNA copies/ml to roughly 500 copies/ml in our simulations. This is consistent with observations in clinical trials [24,26,55]. An interesting result is that the amplitue of viral blips is inversely correlate with the ecay of the latent reservoir. Base on moel (4), viral blips originate from activation of latently infecte cells into the prouctive class. It was initially thought that this activation woul eplete the latent reservoir quickly in HAART-treate patients because e novo infection of susceptible T cells is maximally inhibite by potent antiretroviral rugs an prouctively infecte cells have a fast turnover rate. However, if the activation inuces a substantial proliferation of activate latently infecte cells, it can simultaneously resee the latent reservoir as a small fraction of activate cells revert to the resting state in the formation of memory T cells. To what extent the activation replenishes the latent cell pool epens heavily on the proliferative potential of activate cells, i.e., how many aughter cells are erive from the activation of latently infecte cells. In Figure 3A, the proliferation rate of activate cells is chosen to be p~1:4 ay {1, which implies that cell ivisions occur 8{12 times over an interval of 4 to 6 ays. In this case, the activation inuces a high level of activate T cells. As a consequence, a large number of prouctively infecte cells are generate. Thus, the amplitue of viral blips remains relatively high, an the latent reservoir is largely replenishe since more activate latently infecte cells revert back to the resting state. In our simulation, we i not observe a PLoS Computational Biology 6 October 2009 Volume 5 Issue 10 e

7 Figure 3. Stochastic simulations of the moel with programme expansion an contraction (Eq. (4)). The moel with programme expansion an contraction of latently infecte cells can generate viral blips with reasonable amplitue an uration. DT*N(50,10), Dt*U(4,6). Column A: p~1:4 ay {1. Activate latently infecte cells ivie about 8{12 times over an interval [4,6] ays. No statistically significant ecay of the latent reservoir is observe. Column B: p~1:35 ay {1. The latent reservoir ecays at a very slow rate. This realization shows a half-life of *44 months. Column C: p~0:8 ay {1. Activate cells ivie about 5{7 times over the same time interval. The latent reservoir ecays more quickly than it oes in B, corresponing to a half-life of roughly 6 months. The other parameter values use are liste in Table 1. The blue horizontal line represents the etection limit of 50 RNA copies/ml. oi: /journal.pcbi g003 statistically significant ecay of the latent reservoir (we performe the simulation over 3 years, but only plotte the first 300 ays in Figure 3A), suggesting that the viral reservoir can be extremely stable even with effective treatment for years. This may explain the remarkable stability of the latent reservoir in some patients on HAART [21]. In Figure 3B, we show an example with a slightly smaller proliferation rate, p~1:35 ay {1. In this situation, although occasional activation can replenish the latent reservoir, the size of the latent cell pool iminishes graually. However, the ecay is very slow, with a half-life of approximately 44 months, which is consistent with some estimates [21,22]. In Figure 3C, we choose the proliferation rate to be p~0:8 ay {1, so that activate T cells ivie 5{7 times over an interval of 4 to 6 ays. In this case, a lower level of activate cells are prouce, resulting in lower amplitue viral blips. The latent reservoir is eplete relatively quickly because cell activation consumes latently infecte cells an the replenishment of the reservoir from activate cells is minor. Figure 3C shows a realization of moel (4) in which the ecay half-life of the latent reservoir is about 6 months, which is in agreement with the estimates in some clinical stuies [23,38]. We ran stochastic simulations of the moel 30 times, recore the number an amplitues of viral blips, an calculate the halflife of the ecay of the latent reservoir base on the change in the latent reservoir size in 300 ays. The summary statistics on our simulations is given in Table 2. As the proliferation rate of activate cells (p) ecreases, we observe that both the frequency an the average amplitue of viral blips ecrease. With a smaller p, the latent reservoir size unergoes a larger ecrease, corresponing to a shorter half-life of the reservoir ecay. Thus, we expect an inverse relationship between the ecay of the latent reservoir an the frequency (or amplitue) of viral blips. This is consistent with the experimental observations in Ramratnam et al. [23]. The fraction of resting latently infecte cells that are activate by antigenic stimulation remains largely unknown. Due to the heterogeneity of latently infecte cells with respect to the antigens they respon to, it is likely that a very small fraction of latently infecte cells are activate by a particular antigen. We teste moel preictions (Eq. (4)) with ifferent activation rates a. The proliferation rate of activate cells, p~1:35 ay {1, is fixe. With a ecreasing from 0:03 ay {1 (re soli) to 0:003 ay {1 (black otte) in Figure 4A, fewer activate latently infecte cells are generate, which results in a more rapi ecay of the latent reservoir an lower viral loas. In fact, when a~0:003 ay {1,5 PLoS Computational Biology 7 October 2009 Volume 5 Issue 10 e

8 Table 2. Summary of stochastic simulations of the moel, Eq. (4), with programme expansion an contraction of latently infecte cells. Parameter value Ave number of blips over [0, 300] ays Min blip amplitue (copies/ml) Max blip amplitue (copies/ml) Ave blip amplitue (copies/ml) Change in the latent reservoir size over 300 ays Half-life of the latent reservoir ecay (months) p~1:4 ay { [140, 362] 693 [541, 877] 394 [298, 522] 20.5% [219%, +26%] 1: [33, -] p~1:35 ay { [113, 308] 524 [346, 680] 334 [263, 446] 214% [232%, +5.5%] 46 [18, -] p~0:8 ay {1 3.7 [2, 5] 61 [50, 94] 93 [71, 111] 74 [63, 98] 265% [267%, 262%] 6.6 [6.3, 7.3] Abbreviations: ave (average), min (minimum), max (maximum). Values above brackets are the average values over 30 simulation runs. Values in brackets are the ranges. There are 5 antigenic activations within 300 ays. When p~1:4 or 1:35 ay {1, viral blip (w50 RNA copies=ml) emerges each time activation occurs. When p~0:8 ay {1, not every activation generates a viral blip. In some simulations with p~1:4 or 1:35 ay {1, the latent reservoir size is preicte to increase an hence has no half-life. oi: /journal.pcbi t002 activations only generate 3 viral rebouns, in which only 2 rebouns can be regare as viral blips (w50 RNA copies=ml). With a smaller a (e.g., 0:002 ay {1 ), antigenic activation cannot generate viral blips with the parameter values we use. We also teste the sensitivity of generating blips to the transition rate a L (Figure 4B). When a L ecreases from 0:1 ay {1 (re soli) to 0:01 ay {1 (black otte), the viral loa oes not change significantly. However, a smaller transition rate leas to more substantial replenishment of the latent reservoir. Viral loa also epens on the viral prouction rate p v. For example, when Figure 4. Sensitivity tests on the activation rate a an the transition rate a L in Eq. (4). The proliferation rate of activate cells, p~1:35 ay {1, is fixe. Column A: the transition rate a L ~0:1 ay {1 is fixe an the activation rate a varies: a~0:03 ay {1 (re soli), a~0:01 ay {1 (blue ashe) an a~0:003 ay {1 (black otte). p v ~2000 ay {1 is fixe. Column B: the activation rate a~0:03 ay {1 is fixe an the transition rate varies: a L ~0:1 ay {1 (re soli), a L ~0:05 ay {1 (blue ashe) an a L ~0:01 ay {1 (black otte). p v ~2000 ay {1 is fixe. Column C: a~0:003 ay {1 an a L ~0:1 ay {1 are fixe. The viral prouction rate varies: p v ~2500 ay {1 (re soli) an p v ~2000 ay {1 (black otte). The other parameter values use are the same as those in Figure 3. The blue horizontal line represents the etection limit of 50 RNA copies/ml. oi: /journal.pcbi g004 PLoS Computational Biology 8 October 2009 Volume 5 Issue 10 e

9 a~0:003 ay {1, only 2 viral blips are observe with p v ~2000 ay {1 (black otte in Figure 4C). If we increase the viral prouction rate to p v ~2500 ay {1 (re soli), then all the 3 rebouns are greater than 50 RNA copies/ml an thus generate observable viral blips. In aition to changing the proliferation rate (p) uring expansion, it woul also be interesting to stuy the effects of varying the uration (Dt) an frequency (etermine by DT) of antigenic stimulation. As an example, we showe in Figures 5A an 5B the latent reservoir ecay an viral blips with ifferent istributions of Dt. Specifically, we assume Dt*U(8,11) in Figure 5A an Dt*U(8,10) in Figure 5B. We fixe p~0:8 ay {1 an DT*N(50,10) as use in Figure 3C. No statistically significant ecay of the latent reservoir is observe in Figure 5A, while the latent resevoir ecays at a very slow rate (with a half-life of approximately 44 months) in Figure 5B. This is not surprising since shorter uration of activation results in generation of less activate latently infecte cells, an thus less replenishment of the latent reservoir. In Figure 5C, we assume Dt*U(4,6) as in Figure 3C, but increase the frequency of activation by assuming DT*N(30,10). In this realization, there are 8 activations in 300 ays, more than the 5 activations in Figure 3C. We observe that the latent reservoir ecays more quickly than in Figure 3C. In fact, for a large proliferation rate of activate cells (e.g., p~1:35 ay {1 in Figure 3B), increasing the frequency of activation will replenish the latent reservoir more frequently an thus ecrease the ecay rate of the latent reservoir, whereas for a small proliferation rate (e.g., p~0:8 ay {1 ) an short uration of activation (e.g., Dt*U(4,6)), increasing the frequency of activation will accelerate the latent reservoir ecay (see Figures 3C an 5C). In summary, occasional activation of latently infecte cells upon stochastic antigen encounter is able to prouce a large quantity of activate T cells temporarily, an thereby generate intermittent viral blips. The blip amplitue/frequency is inversely correlate with the ecay of the latent reservoir. Using ifferent potentials of activate T cells to ivie uring the initial clonal expansion phase or ifferent uration or frequency of antigenic stimulation enables us to generate the ifferent ecay characteristics of the latent reservoir observe in ifferent clinical stuies [21 23,35 38]. Low-level viral persistence We have assume a ensity-epenent mortality rate for prouctively infecte cells in the moelgivenbyeq.(4)inorer to maintain a low steay state viral loa when antigen is absent. The reason that viral loas ecrease very quickly in the absence of activation in this moel is that activate cells ecline quickly to Figure 5. Numerical simulations of Eq. (4) with ifferent uration an frequency of activation. We fixe the proliferation rate of activate cells to be p~0:8 ay {1. Column A: DT*N(50,10), Dt*U(8,11). No statistically significant ecay of the latent reservoir is observe. Column B: DT*N(50,10), Dt*U(8,10). The latent reservoir ecays at a very slow rate. Column C: DT*N(30,10), Dt*U(4,6). In this realization, there are 8 activations in 300 ays. The latent reservoir ecays more quickly than in Figure 3C. The other parameter values use are the same as those in Figure 3. The blue horizontal line represents the etection limit of 50 RNA copies/ml. oi: /journal.pcbi g005 PLoS Computational Biology 9 October 2009 Volume 5 Issue 10 e

10 an extremely low level uring the contraction phase, with not enough cells entering the prouctive stage. Even when we choose a smaller eath rate of activate cells, for example, a~0:2 ay {1 [45], activate cells still quickly ecline to a very low level. If activate cells can be maintaine at a low level rather than ecreasing to zero quickly uring the contraction phase, then low steay state viral loa persistence is possible without assuming ensity-epenent infecte cell eath. A stuy by Chun et al. [56] reveale that a high level of HIV-1 proviral DNA persists in the activate CD4 z T cell compartment in infecte iniviuals on effective antiretroviral therapy with no etectable viremia in plasma for extene perios of time. Although some of the proviruses might be efective, spontaneous release of virus was etecte without any activating stimuli uring overnight culture [56]. This observation argues for the persistence of infectious virus in activate CD4 z T cells in patients uner effective treatment. Here we moify moel (4) (i.e., remove the assumption of ensity-epenent infecte cell eath an aopt a biphasic contraction phase, see Eq. (6) in Methos) an examine whether viral an the latent reservoir persistence, as well as intermittent viral blips, can be generate solely by occasional activation of latently infecte cells upon encounter with relevant antigen. With a ~0:02 ay {1 [45], DT*N(50,10), Dt*U(4,6), we perform numerical simulations of the moel with a biphasic contraction phase. As before, we choose ifferent proliferation rates, i.e., (A) p~1:4 ay {1, (B) p~1:35 ay {1, (C) p~0:8 ay {1, to characterize ifferent potentials of activate cells to proliferate uring the phase of expansion. Similar to Figure 3, the simulation results shown in Figure 6 exhibit three istinct ecay profiles of the latent reservoir: (A) there is almost no ecay; (B) the latent reservoir ecays at a very slow rate; (C) the reservoir ecays at a faster rate. The ecay of the latent reservoir is inversely correlate with the amplitue or frequency of viral blips. The viral loa oes not ecline to an unreasonably low level in the absence of antigenic stimulation. This low-level viremia is primarily maintaine by a small number of activate cells that transition into the prouctive class uring the secon slower contraction phase. However, the absence of antigenic stimulation over a long time (more than 4 months in our simulation, figure not shown) will eplete activate cells, an the viral loa will ecrease to an extremely low level (below 5 10 {5 virions=ml, a level that can be interprete as viral extinction [28]). Therefore, in orer to obtain a low level of viremia solely maintaine by latently infecte cell activation, there cannot exist a very long perio in which no antigenic stimulation occurs. Figure 6. Simulations of the moel with a biphasic contraction phase (Eq. (6)). The moel is able to generate viral blips as well as low-level persistent viremia. The low-level viral loa is maintaine by a low level of activate latently infecte cells uring the secon slower contraction phase in the latent cell response. In the first row, f (t) is the expansion function (re) an g(t) is the rapi contraction function (blue). Different proliferation rates, i.e., p~1:4 ay {1 (Column A), p~1:35 ay {1 (Column B), an p~0:8 ay {1 (Column C), result in ifferential ecay characteristics of the latent reservoir as in Figure 3. The other parameter values use are liste in Table 1. The blue horizontal line represents the etection limit of 50 RNA copies/ ml. oi: /journal.pcbi g006 PLoS Computational Biology 10 October 2009 Volume 5 Issue 10 e

11 The contribution of ongoing viral replication The ecay of the latent reservoir, the amplitue of viral blips, an the viral loa below the limit of etection are not largely influence by the effectiveness of the treatment as long as the overall rug efficacy is beyon a threshol value, c, the critical rug efficacy. For this moel we coul not obtain a close-form solution for c but it is numerically similar to that efine in (3). In Figures 7A an 7B, we explore the effects of HAART potency on the latent reservoir an low-level viremia by using ifferent rug efficacies: ~0:8 (re ashe line) an ~1 (blue soli line). Although 100% effectiveness may not be clinically feasible, we use an extreme case to illustrate the effect of latent cell activation. We observe that for the lower rug efficacy, the latent reservoir an the viral loa are both at slightly higher levels. However, the ifference is minuscule. This shows that both the stability of the latent reservoir an low-level persistent viral loas are principally ue to latently infecte cell activation rather than ongoing active viral replication, provie that the rug efficacy is above a certain threshol value. We further compare the relative contributions of ongoing viral replication an latent cell activation to the latent reservoir an viral persistence. In Figure 7C, we plot the ratio of g(1{ )kvt to {f (t)al 0 z(1{f (t))rl a, which represent the contributions to the latent cell pool coming from ongoing viral replication an the net effect of latent cell activation an return to latency, respectively. We fin that the ratio is very small, inicating that the contribution of ongoing viral replication to the latent reservoir size is very small. In Figure 7D, we plot the ratio of (1{g)(1{ )kvt to a L L a, which represent the contributions to the viral loa by e novo viral infection an the transition from activate latently infecte cells into prouctively infecte cells, respectively. The ratio is less than 1 except a few blips where latently infecte cell activations occur. Thus, in the absence of activation, e novo viral infection is a minor factor contributing to the viral loa, whereas viral blips are mainly ue to e novo viral infection. However, we notice that the virus causing e novo viral infection is mainly release from latent cell activation (Figure 7C). Therefore, viral persistence an the stability of the latent reservoir arise primarily from occasional activation of latently infecte cells upon antigen encounter. Resiual active viral replication uring HAART is only a minor factor. We have also performe sensitivity tests on several parameters when stuying the relative contributions. The ratio of g(1{ )kvt to {f (t)al 0 z(1{f (t))rl a increases when we increase the activation rate of latently infecte cells, a, or the fraction of infections that result in latency, g, or ecrease the reversion rate to latency, r. In Figure 8, we examine the effects of ifferent parameter values of a, g an r on the ratio of relative contributions. As the activation rate a increases, more latently infecte cells are activate, leaing to more substantial replenishment of the latent reservoir an higher amplitues of viral blips. However, even when a has a 10-fol increase (notice that in this case the transient viral loa can reach above 10 3 RNA copies=ml, which is normally not regare as a viral blip), we observe that the ratio of the relative contributions remains almost the same (Figure 8A). When the fraction of Figure 7. Relative contributions of ongoing viral replication an latent cell activation. A an B: the effects of ongoing viral replication (influence by the overall rug efficacy) on the latent reservoir an viral loa in the moel given by Eq. (6). Different rug efficacies are use: ~0:8 (re ashe line) an ~1 (blue soli line). Ongoing viral replication is only a minor contributor to the stability of the latent reservoir an low-level persistent viremia, as inicate by the minor effect of changing rug efficacy from ~0:8 to ~1. C an D: relative contributions of ongoing viral replication ( ~0:8 was fixe) an latent cell activation to the latent reservoir an viral persistence. C: the ratio of g(1{ )kvt to {f (t)al 0 z(1{f (t))rl a, an D: the ratio of (1{g)(1{ )kvt to a L L a. We chose p~1:4 ay {1. The other parameter values use are liste in Table 1. oi: /journal.pcbi g007 PLoS Computational Biology 11 October 2009 Volume 5 Issue 10 e