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Journal of Virology, January 2006, p. 802-809, Vol. 80, No. 2
0022-538X/06/$08.00+0     doi:10.1128/JVI.80.2.802-809.2006
Copyright © 2006, American Society for Microbiology. All Rights Reserved.

Naïve and Memory Cell Turnover as Drivers of CCR5-to-CXCR4 Tropism Switch in Human Immunodeficiency Virus Type 1: Implications for Therapy

Ruy M. Ribeiro,1 Mette D. Hazenberg,2 Alan S. Perelson,1* and Miles P. Davenport3

Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico 87545,1 Department of Clinical Viro-Immunology, Sanquin Research Centre at CLB, and Department of Immunology, University Medical Centre Utrecht, Utrecht, The Netherlands,2 Department of Haematology, Prince of Wales Hospital, and Centre for Vascular Research, University of New South Wales, Sydney, New South Wales, Australia3

Received 27 June 2005/ Accepted 16 October 2005


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ABSTRACT
 
Early human immunodeficiency virus infection is characterized by the predominance of CCR5-tropic (R5) virus. However, in many individuals CXCR4-tropic (X4) virus appears in late infection. The reasons for this phenotypic switch are unclear. The patterns of chemokine receptor expression suggest that X4 and R5 viruses have a preferential tropism for naïve and memory T cells, respectively. Since memory cells divide approximately 10 times as often as naïve cells in uninfected individuals, a tropism for memory cells in early infection may provide an advantage. However, with disease progression both naïve and memory cell division frequencies increase, and at low CD4 counts, the naïve cell division frequency approaches that of memory cells. This may provide a basis for the phenotypic switch from R5 to X4 virus observed in late infection. We show that a model of infection using observed values for cell turnover supports this mechanism. The phenotypic switch from R5 to X4 virus occurs at low CD4 counts and is accompanied by a rapid rise in viral load and drop in CD4 count. Thus, low CD4 counts are both a cause and an effect of X4 virus dominance. We also investigate the effects of different antiviral strategies. Surprisingly, these results suggest that both conventional antiretroviral regimens and CCR5 receptor-blocking drugs will promote R5 virus over X4 virus.


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INTRODUCTION
 
Human immunodeficiency virus type 1 (HIV-1) binding to and entry into cells are mediated by the interaction of the viral gp120 protein with cell surface CD4 molecules. Studies of HIV-1 infection have shown that the virus, in addition to binding to cellular CD4, also binds to a variety of chemokine receptors (3). Usually, in early HIV-1 infection, the virus shows a tropism for the CCR5 receptor (R5 virus). However, strains of virus that bind to the CXCR4 chemokine receptor (X4 virus) appear in late infection in at least 50% of patients (24). This change in viral phenotype is associated with rapid depletion of CD4 cells and disease progression (8). The reason for this phenotypic change in late disease is uncertain (41). It has been suggested that X4 virus may be more virulent than R5 virus. However, if this were the case, it is not clear why X4 virus should not dominate from early infection, since it must arise frequently due to the small number of mutations required to make the shift from CCR5 to CXCR4 tropism (11). It has also been suggested that R5 virus dominates early infection, because its tropism for CCR5 promotes infection of macrophages at the site of infection. However, R5 virus predominates even after parenteral infection (57), and CD4 T cells, not macrophages, are found to be the dominant cell type infected in both early and late infection (47, 54). In addition, a recent study has shown that target cell availability, as measured by coreceptor expression, cannot be the driving force for R5-to-X4 viral evolution, since higher levels of CXCR4 expression among both total and CD45+ R0– CD4+ cells ("naïve" cells) were associated with a delayed development of X4 viruses (55). Thus, there has been no consistent explanation for the observed changes in viral phenotype with disease progression in HIV.

CCR5 and CXCR4 are preferentially expressed in memory and naïve T cells, respectively (5). Thus, a recent study (9) has proposed that the observed tropism of R5 virus for memory cells and X4 virus for naïve cells (4) may drive the evolution of phenotypes with disease progression. Recent studies have shown that viral production from naïve cells is many times lower than viral production from memory cells (17, 62). It is possible that the relatively high division rate of memory T cells compared to that of naive T cells in an uninfected host (9, 23, 32, 33) provides a relative advantage for memory cell-tropic (R5) virus in early infection, since CD4+ T-cell division, even if not an absolute requirement for infection (17, 22, 52, 62), is thought to make viral production more efficient (6, 50, 61). With disease progression and depletion of CD4 cells, the fractions of both naïve and memory T cells in division increase (e.g., as measured by Ki67). However, the fraction of naïve T cells in division increases more rapidly and by late stage disease approaches that of memory T cells (23, 34, 45, 46). This is also suggested by an association between higher plasma levels of interleukin-7, a cytokine that drives naïve cell proliferation, and the presence of X4 viruses (28).

In this paper we develop a mathematical model to explore the relationship between cell turnover and the evolution of viral phenotype. We demonstrate that either an association of cell division with increased viral production or with increased infectivity is consistent with the observed phenotypic changes. We also examine the effects of conventional antiretroviral therapy using this model. Our results suggest that successful treatment, which leads to a reduction in viral load and T-cell turnover (23), should result in a reversion from the CXCR4 phenotype to the CCR5 phenotype. This reversion has been observed in treated patients, especially those with a strong treatment-induced suppression in viral load (48). Finally, we use the model to predict the effects of drugs that block the interaction of the virus with cellular chemokine receptors (CCR5-blocking drugs) on viral phenotype and disease progression. The model suggests that CCR5-blocking drugs will not promote a conversion to X4 virus, and when given early in disease may even delay the emergence of X4 virus. When given after the emergence of X4 virus, CCR5-blocking drugs may still inhibit X4 virus. If R5 virus levels remain significant, then by reducing the level of R5 virus, they may allow some CD4 cell recovery, thereby reducing naïve cell turnover and X4 virus production.


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MATERIALS AND METHODS
 
Analysis of the role of cell turnover and viral tropism in HIV-1 infection requires explicit consideration of cell division rates in naïve and memory T cells. Experimental evidence suggests that activated or proliferating cells produce higher levels of virus than do resting cells when infected (17, 22, 62). Therefore, we have developed a model that incorporates the division rates of naïve and memory T cells and their effect on viral production. In addition, the model incorporates viral phenotypes differing in their ability to bind to naïve and memory CD4+ T cells. A schematic outline of the model is presented in Fig. 1, and the equations are presented in the Appendix. The underlying assumptions of the model can be summarized briefly as follows. (i) Naïve cells are produced from the thymus at a rate {lambda}. (ii) The division of naïve and memory T cells (rN and rM, respectively) increases as the total number of CD4 cells decreases according to an experimentally observed relationship. Specifically, we assume that the division rates of the memory and naïve T-cell populations are proportional to the fraction of each population that is Ki67+ (10, 43) (see the Appendix for details). (iii) When a naïve cell divides, a daughter cell with probability f becomes a memory cell and with a probability of 1–f remains a naïve cell (16, 21, 36). (iv) Naïve and memory cells become infected with virus of type i at rates ßNi and ßMi, respectively, determined by the affinity of virus of type i for naïve (N) and memory (M) cells (KNi and KMi), where i = R5, X4, or R5X4. (v) Infected naïve and memory cells produce virus at rates pN and pM, respectively, proportional to the division rates of naïve and memory cells, rN and rM, respectively. (vi) Free virus is cleared (by immune and nonimmune mechanisms) at a rate c. In addition, free virus is removed by infection of naïve and memory cells. We do not include viral mutation and evolution in this model (38). So few mutations are necessary to mediate the switch in tropism between R5 and X4 viruses (11), that in the model when even a small amount of one is present, the other one will also be present. Thus, here the rise in X4 virus is not a question of waiting for a mutation to arise but a question of selection and dominance of X4 virus.


Figure 1
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  		FIG. 1. Schematic of the model. The model includes naïve (N), naïve infected (NI), memory (M), and memory infected (MI) CD4+ T cells. New cells emigrate from the thymus at rate {lambda} and enter the naïve cell pool. Naïve cells die at rate {delta}N, become infected with viral strain i at rate ßNiV, and divide at rate rN. When naïve cells divide, a proportion (f) convert to memory cells, and the rest maintain their naïve phenotype. Memory cells die at rate {delta}M, and replicate at rate rM. Infected cells produce virus at rate p. Free virus is cleared at rate c and binds to naïve or memory T cells (at rates bKNiN and bKMiM, respectively). Infected cells die at a higher death rate ({delta}I) than uninfected cells due either to immune-mediated killing or viral cytopathic effect.


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RESULTS
 
"Infection" is initiated in the model by the introduction of 1,000 viral particles/ml each of R5 virus, X4 virus, and dual-tropic virus. The results of the model correlate well with the observed features of disease progression, without requiring time dependence in any of the parameters (Fig. 2a). An early CD4 cell decline in acute infection is followed by a slight recovery in CD4 cell numbers and a subsequent steady decline with time. Recently, it has been shown that this early T-cell depletion is very pronounced in some lymphoid compartments, such as gut-associated lymphoid tissue (27, 31). R5 virus dominates early in the course of infection, because naïve cell division is low, and thus, the X4 tropism for naïve cells is disadvantageous. As CD4 cell numbers decline, naïve and memory cell turnover increases, as suggested experimentally by Ki67 measurements (23) (Fig. 3 and Appendix). A phenotypic switch from R5 to X4 virus occurs at low CD4 numbers, when cell division is high. This switch is accompanied by a rise in total viral levels (49) and a drop in CD4 counts (25, 29). Notably, in our model R5 virus was not eliminated after the switch to X4 virus, in agreement with experimental data (58). Indeed, the percentage of the population comprised by R5 and the time for the switch in viral tropism depend on the model parameters, providing an explanation why different patients experience the switch at different points in time. Indeed, for certain combination of parameters, the switch may occur so late (>10 years) that it is possible that some patients would succumb to opportunistic diseases before the phenotypic switch is observed (data not shown). Dual-tropic virus, capable of binding to both CCR5 and CXCR4 chemokine receptors (R5X4) may also be observed, depending on the relative affinities of the different viruses for CCR5 and CXCR4. If these viruses are not truly dual-tropic, i.e., not efficient at using both receptors, then they will be rarely seen even after the switch (Fig. 2a). Importantly, the shift from R5 to X4 is both a consequence of low CD4 counts (which promote increased naïve cell division) and a cause of rapid CD4 cell decline (since X4 virus targets the naïve cell pool) (4, 9).


Figure 2
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  		FIG. 2. Model outcome without therapy: CD4 T-cell count (in cells per microliter) (black), total viral load (red), R5 (blue), and X4 (green) viral loads. (a) Model assuming that cell division is required for efficient viral production, (b) model assuming that cell division increases the level of viral infection. Parameter values used in the simulations are given in the Appendix. When there is only R5 virus, the total virus curve (red) is hidden under the R5 curve (blue).


Figure 3
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  		FIG. 3. Relationship between CD4 count and cell division (Ki67+ cells). The experimental relationship between total CD4+ T-cell count (cells per microliter) and Ki67+ naïve (red) and memory (blue) CD4+ T cells is shown (data from reference 23). (Bottom) The statistical results of fitting the data to an inverse relationship (y = A/x + B), as well as the r2 for a linear regression (y = a + mx) are shown below the graph. SE, standard error.

Effects of cell division. The basic model outlined above is based upon the assumption that cell division is required for efficient production of virus from infected CD4+ T cells and is consistent with observations from Zhang et al. (62) that dividing cells contain about five times more HIV RNA than resting cells do. However, an alternative assumption about the role of cell division is possible on the basis of in vitro studies that suggested that division may be needed for efficient infection of CD4+ T cells (6, 50, 61). To explore the consequences of this alternative proposal, we let infection depend on cell division but assumed that viral production rates are independent of the rate of cell division. This situation is modeled by replacing assumptions iv and v above with the following. (iv) Naïve and memory cells become infected with virus at rates ßNi and ßMi, respectively, proportional to the affinity of virus of phenotype i for naïve (N) and memory (M) cells (KNi and KMi), and the replication rate of naïve and memory cells (see Appendix). In this context, faster dividing cells are more permissive for virus infection. (v) Infected naïve and memory cells produce virus at equal rates, i.e., pN = pM.

The results of this version of the model are very similar to those when division is required for viral production (Fig. 2b) and again are consistent with observed clinical features. This is not surprising, since both assumptions link increased cell turnover with increased viral production, either directly by having each cell produce more virus or indirectly by having more infected cells.

Effects of antiretroviral therapy. We tested the effects of different forms of therapy on the evolution of the viral phenotype using the model. The effects of a protease inhibitor (PI), with efficacy {varepsilon}PI, were modeled by having only a fraction, (1 – {varepsilon}PI), of the virus produced being infectious, with the remaining fraction ({varepsilon}PI) being noninfectious (39). On the other hand, reducing the infectivity of free virus (ß) by a factor {varepsilon}RT corresponds to applying a reverse transcriptase inhibitor (RTI) with efficacy {varepsilon}RT. We simulated antiretroviral therapy with a combination of protease and reverse transcriptase inhibitors with overall efficacy {varepsilon} = 1 – [(1 – {varepsilon}PI)(1 –{varepsilon}RT)] = 0.45, against both X4 and R5 viruses (Fig. 4). We note that in this model, without latency and long-lived infected cells, higher drug efficacy would lead to virus eradication (39), which is not the scenario we want to study. Treatment leads to reduction in the level of virus and the infection of CD4 cells, and CD4 counts slowly increase with time (Fig. 4). We simulated two scenarios, one in which therapy is introduced before the phenotypic switch to CXCR4-tropic virus and the other in which therapy is not commenced until after the switch.


Figure 4
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  		FIG. 4. Predicted effects of antiretroviral therapy (purple bar at the top of each panel) introduced before (1,000 days) (a) or after (1,800 days) (b) the phenotypic switch to X4 virus. Predicted effects of CCR5-blocking drugs administered prior to (c) or after (d) the switch. CD4+ T-cell count (cells per microliter) (black), total viral load (red), R5 (blue), and X4 (green) viral loads are shown. Parameter values used in the simulations are given in the Appendix. When there is only R5 virus, the total virus curve (red) is hidden under the R5 curve (blue). After antiretroviral therapy, in panels a and b, the total viral load also includes noninfectious virus, which would be measured in an assay.

Introduction of treatment early in the course of disease (prior to the phenotypic switch) reduces viral loads and slows or prevents CD4 cell decline (Fig. 4a). Since the dominance of X4 virus is related to CD4 decline and increased naïve cell turnover, treatment delays or averts the rise in X4 virus (Fig. 4a). The introduction of treatment later in disease (after the phenotypic switch to X4 virus dominance) causes a reduction in the levels of virus (Fig. 4b). However, X4 virus is more strongly affected. This occurs because treatment reduces viral loads and allows some recovery of CD4+ T-cell numbers. As CD4 cell numbers increase, naïve cell turnover decreases to a point where infection of CXCR4-positive naïve cells is no longer productive enough, leading to a reversion to R5 predominance (Fig. 4b).

Effects of chemokine receptor blocking. Drugs that block viral entry into cells by interfering with binding to chemokine receptors are in clinical trials. However, one concern about the use of these drugs is that blocking the infectivity of one viral phenotype may simply promote the other phenotype (19, 53). This is particularly of concern in the case of CCR5 receptor blocking, since conversion to X4 virus is associated with rapid CD4 depletion and disease progression. Thus, the use of CCR5 receptor-blocking agents could potentially accelerate disease progression. Theoretical models of viral tropism have the potential to provide some insight into the likely effects of these drugs.

Drugs blocking viral binding to CCR5 receptors can be simulated in the model by reducing the binding of virus to these receptors. If, for example, we assume a drug blocks viral binding to CCR5 receptors with an efficacy of 45%, this is equivalent to reducing the affinity of viruses to CCR5 receptors by 45%. It is then possible to investigate whether the use of CCR5 receptor-blocking agents would promote a shift from R5 to X4 virus. Again, this was investigated in two scenarios: (i) treatment with a CCR5 receptor blocker early in infection (before the phenotypic switch to X4 virus), and (ii) treatment with a CCR5 receptor blocker after the phenotypic switch. In our simulations CCR5 receptor blockers do not increase the viral titers of X4 virus in either case.

Administration of a CCR5 receptor-blocking drug before the switch to X4 virus reduces R5 virus load, prevents CD4 depletion, and leads to a rise in CD4 cell numbers (Fig. 4c). Rising CD4 numbers decrease naïve cell division and thus make X4 virus infection less productive. Thus, CCR5 blockers act to delay or prevent the rise of X4 virus (Fig. 4c). Administration of CCR5 receptor-blocking drugs after the phenotypic switch to X4 virus has similar effects (Fig. 4d). Since R5 virus is still present at significant levels after the phenotypic switch and thus continues to contribute to CD4 cell death, reducing its binding to CCR5 causes a slight reduction in overall viral load and a slight rise in CD4 numbers. The rise in CD4 cell numbers reduces naïve cell turnover and therefore X4 virus levels. Thus, somewhat surprisingly, in this framework, blocking CCR5 receptor binding reduces both R5 and X4 virus levels (Fig. 4d), as long as R5 virus still comprises a significant part of the viral population prior to treatment.


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DISCUSSION
 
The reasons for the phenotypic shift from CCR5-tropic to CXCR4-tropic virus in late HIV infection are unclear. We have developed a mathematical model of HIV infection that demonstrates that an explanation based on naïve and memory cell turnover and tropism is consistent with the available data on viral and T-cell dynamics. The results of the model qualitatively reproduce the observed clinical features of early HIV infection, including an early viral peak, early CD4 cell decline, and subsequent rebound, followed by progressive rises in viral load and declines in CD4 cell count and eventually a viral phenotypic switch to X4 virus in late infection. Uniquely, the long time scale associated with the phenotypic switch is an intrinsic feature of the model dynamics and does not depend on parameters varying in time. The model does not explain why X4 virus does not arise in some individuals (15). However, it seems likely that this may occur due to interindividual variations in the proliferative response of CD4 cells and/or the patient succumbing to opportunistic infection before X4 virus dominates. In this regard, it is important to note that variations in the relationship between proliferation and total CD4+ T-cell count (Fig. 3), which would be expected among patients, lead to differences in the time to the viral phenotype switch and in the relative viral load of each phenotype. For instance, lowering A, a parameter in the division of memory cells, in Fig. 3 by just 10% increases the period of R5 dominance, before the phenotype switch, by about 1 year (Fig. 5a). By contrast, varying the initial X4 inoculum between 10 copies/ml and 105 copies/ml changes the time to the phenotypic switch by only ~9 months (Fig. 5b). Two other observations not explored in this model could also contribute to variations in time for the phenotypic switch. Some reports suggest that both the relative proportion of memory cells (at least in peripheral blood) and the expression levels of CCR5 increase with time (2, 14, 37, 44). The interplay between these changes and the increase in the division rate of naïve cells (23, 46) and how they influence the phenotypic switch warrant further study.


Figure 5
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  		FIG. 5. Predicted effect on the time for the phenotypic switch of changing the parameter A for a memory cell in the relationship between T-cell numbers and division rates (Fig. 3) (a) and the initial inoculum of X4 virus (note the logarithmic x axis) (b). All other parameters were held constant at the values shown in the Appendix, and the dashed vertical line shows our original simulations (Fig. 2a).

The model was used to address the effects of antiretroviral therapy on viral tropism. Only a few studies have looked at the effects of antiretroviral therapy on viral phenotype, when introduced after the rise in X4 virus. These studies have suggested that antiretroviral therapy leads to a reversion to predominantly R5 virus (12, 18, 40) and that in treated patients X4 to R5 reversion correlates with significant reductions in viral load (48). By contrast, X4 virus was maintained in treated patients that failed to suppress viral load. Furthermore, in patients with good viral control and initial reversion to R5 phenotype, X4 virus arose again with loss of viral control (48). The reversion to R5 virus with viral control is consistent with our proposed mechanism of viral phenotype selection and the predictions of the model presented here (Fig. 4b). We should point out that these therapy results depend on the efficacy of therapy being similar in cells infected with R5 and X4 viruses. If therapy differentially inhibits those viruses, then the results may be different. For instance, if drug activity depends on intracellular phosphorylation, such as in the case of zidovudine (26), and this is more efficient in activated cells, then that drug may preferentially inhibit R5 viruses. From the clinical point of view, it is also important to note that although a reversion to R5 phenotype is observed after initiation of antiretroviral treatment, both R5 and X4 HIV variants have been shown to persist in viral reservoirs after prolonged suppression of plasma viremia below the level of detection (13, 56).

The model was also used to address the likely effects of CCR5 receptor-blocking agents on viral phenotype. The results suggest that CCR5 receptor blockers should not promote X4 viral dominance, but instead, provided they suppress R5 virus sufficiently, they should act to drive a reversion to predominantly R5 virus. This is due to the recovery of CD4+ T cells during treatment and the corresponding reduction in the proportion of cells dividing. Thus, the validity of this result depends on the relationship between cell division and CD4+ T-cell numbers (Fig. 3) being valid during treatment, which has not yet been tested. We note that another model, which did not include this mechanism, predicted a selection of X4 virus (42). However, in an important recent experiment, R5 entry blockers were tested in macaques doubly infected with R5 and X4 viruses, and a marked decrease in R5 virus (~50-fold) without a switch to a CXCR4-tropic virus was found (60). However, in two of three macaques, there was a transient increase (<15-fold) in plasma X4 virus (60). In vitro experiments have also been used to analyze the effects of CCR5 receptor blockers on viral phenotype. Even though in these in vitro experiments there was no pressure from T-cell dynamics, CCR5 receptor blockers still did not promote a shift to CXCR4 receptor tropism, rather they selected for viral mutations that increased viral affinity for the CCR5 receptor (19, 53). Here we did not address the effect of such mutations on R5 viruses. In experiments with mouse models of infection, treatment with a CCR5 antagonist led to a tropism switch in some mice in one experiment (albeit more than 3 weeks after treatment was stopped) (35) but no outgrowth of X4 virus in another study (51). Again, this is consistent with the results of our model. There is no advantage for the virus to acquire CXCR4 receptor tropism while naïve cell division rates are low. In at least one case of a human patient infected with R5 and X4 viruses and treated with a CCR5 antagonist, R5 viruses were preferentially suppressed, but there was not a measurable drop in viral load. Moreover, upon the end of treatment, R5 viruses increased to pretreatment levels. Even though we did not simulate this situation, because in our treatment results, the viral load always declines, the differential regulation of R5 and X4 viral loads implied by this study is in agreement with our model, since R5 and X4 viral load levels are regulated by cell division status (59).

In summary, we have presented a model that provides a mechanistic basis for the evolution of X4 virus with disease progression. It also provides an explanation for recent observations of the effects of antiretroviral therapy on viral phenotype. Finally, the model predicts that CCR5 receptor-blocking agents should not promote X4 viral tropism. The model is not intended to "prove" that CD4 cell turnover drives the viral phenotype switch; further clinical research is clearly required to address this. However, the model provides a testable hypothesis to explain a variety of experimental observations on R5 and X4 viruses.


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APPENDIX
 
Model equations. The model is defined by the following differential equations describing the changes in naïve cells (N), memory cells (M), naïve and memory cells infected with virus of phenotype i (NIi and MIi, respectively), and free virus of phenotype i (Vi), where i = R5, X4, or R5X4:

Formula 1(1)

Formula 2(2)

Formula 3(3)

Formula 4(4)

Formula 5(5)

The rates of viral production from infected naïve and memory cells are proportional to the division rate of the respective cell class as follows, pN = prN and pM = prM, with infectivities of ßNi = ßKNi and ßMi = ßKMi. When we consider the model where infection is dependent on cell division, pN = pM, and the rates of infection by virus of phenotype i of naïve and memory cells (ßNi and ßMi, respectively) are calculated as follows, ßNi = ßKNirN and ßMi = ßKMirM.

This model was simulated using Berkeley Madonna v. 8.0.1 (Macey & Oster). Some of the parameter values were taken from the literature. These include rN = {gamma}[10/(T – 0.0095] day–1, rM = {gamma} [10/(T + 0.05)] day–1 (see below for a derivation of these formulas), where T is the number of CD4+ T cells per microliter at time t, c = 20/day (39), {lambda} = 5 cells/(µl day) (1), and {delta}I = 0.5/day (39). {delta}N = rN(0)(1 – 2f) + {lambda}/N0 day–1 and {delta}M = 2frN(0)N0/M0 + rM (0) day–1 were calculated to ensure naïve and memory CD4+ T-cell steady state before infection (and N0 = M0 = 500 cells/µl are the initial values before infection); the remaining parameters (f = 0.75, p = 100 virions/cell, KNR5 = 0/cell, KMR5 = 0.2/cell, KNX4 = 0.2/cell, KMX4 = 0.038/cell, KNR5X4 = 0.1/cell, KMR5X4 = 0.1/cell, ß = 0.25 cell/virus, and b = 1/day) were chosen to produce the natural course of infection as in Fig. 2a. We emphasize that the choice of these last parameters is not unique, and indeed many different combinations, varying by severalfold, produce similar results. In fact, we tried several alternatives for our model to test its robustness and the impact of our assumptions. For example, in the model of equations 1 to 5, we neglect division of infected cells. This is one possibility, but we also simulated the other extreme, with infected cells dividing at the same rate as uninfected cells. Our results (not shown), both for production and infection dependent on division, were very similar to those shown in Fig. 2 and 4. In addition, here we allowed for infection of memory cells by X4 viruses (KMX4 = 0.038/cell), as suggested by Blaak et al. (4). However, in other simulations (not shown) we precluded this possibility. In those cases we again obtained very similar results to our Fig. 2 and 4, but the dual-tropic strain was more readily observed in a manner dependent on the value of KMX4. This may indicate that the observation of R5X4 virus in vivo may in part depend on the ability of X4 to also infect memory cells. In our model, this results from competition between X4 and R5X4 for memory cell infection, with the efficiency of X4 to infect these cells dictating in part the ability of the dual-tropic virus to thrive in the population. The same mechanism could be operational in vivo. However, in vivo R5X4 viruses could also fulfill other roles, such as being a stepping stone for emergence and adaptation of the X4 virus. In our model this is not the case, because we start the simulations with all strains present, and mutation and evolution of individual strains are not considered. Because many basic questions regarding the interaction between virus and target cells remain, we do not have a way to define a unique, most appropriate model (41). However, by exploring the impact of several specific assumptions, we have shown that our results are robust to a variety of model changes.

We simulated reverse transcriptase inhibitor (RTI) therapy by reducing the infectivity of both X4 and R5 virus by (1 – {varepsilon}RT), where {varepsilon}RT is the efficacy of the RT inhibitor. Likewise, we simulated protease inhibitor therapy by dividing the viral population into infectious and noninfectious viruses, with only (1 – {varepsilon}PI) of the newly produced viruses being infectious. We chose {varepsilon}RT = {varepsilon}PI = 0.25838, such that (1 – {varepsilon}RT)(1 – {varepsilon}PI) = 0.55. Thus, the overall effectiveness of therapy is ~45%. We simulated CCR5 inhibitor therapy reducing KMR5 and KMR5X4 by 45%. In our model, without long-lived or latently infected cells and with the parameters chosen above, larger drug-induced reductions in viral production or infectivity rates lead to virus eradication, a situation we are not interested in studying here.

Division rates of naïve and memory T cells. HIV-1 disease progression leads to a progressive decline in CD4 T-cell numbers. As CD4 cell numbers decline, the fraction of the remaining CD4+ T cells in division increases. This increase in fraction of cells dividing has been ascribed to both homeostatic mechanisms (20, 30) and immune activation (7, 23). Although the precise reason for this increase is unclear, it is a consistent finding in both human and animal studies (23, 34, 45, 46). Importantly, studies that have differentiated between naïve and memory cell division have consistently found that the fraction of naïve cells in division shows a greater increase following HIV infection than memory cell division (34, 45, 46). A more recent study suggests that the fraction of CD4 cell in division, as measured by Ki67+ cells, is negatively correlated with CD4 counts (23). We analyzed this experimental data on the fraction of naïve and memory cells in division (Fig. 3 and reference 23) to investigate their association with CD4 cell counts. We found that in the case of naïve CD4 cells the association is more consistent with an inverse relationship (y = A/x + B) than a linear relationship (y = mx + b), where y is the fraction of cell dividing and x is the total number of CD4+ T cells per ml (Fig. 3). It is difficult to differentiate between a linear and an inverse relationship in the case of memory cells, although the data are compatible with an inverse relationship. Here we used an inverse relationship for rN and rM, although use of a linear relationship for memory cells does not alter the qualitative outcome (data not shown). The correspondence between the fraction of cells dividing (measured by Ki67) at a given point in time and the division rate depends on the duration of the cell cycle (more specifically, the time cells remain Ki67+). For example, if we assume that cells in division are Ki67+ for an average of ~1 day, then the replication rate is numerically identical to the fraction of cell in division (Ki67+). In our model this means that {gamma} = 1 (cell cycle) day–1. If, for example, cells were Ki67+ for ~0.5 day, then numerically the replication rate would be twice the fraction of Ki67+ cells (i.e., {gamma} = 2 day–1). Here for simplicity we adopt {gamma} = 1 day–1.


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ACKNOWLEDGMENTS
 
We thank R. van Rij, H. Schuitemaker, A. Weinberger, and J. Zaunders for very helpful discussions and comments on the manuscript. R.M.R. wishes to thank the hospitality of the Department of Zoology, University of Oxford, where some of this work was performed.

Portions of this work were performed under contract W-7405-ENG-36 of the U.S. Department of Energy. R.M.R. gratefully acknowledges support from a Marie Curie Fellowship of the European Community program "Quality of Life," contract number QLK2-CT-2002-51691 and NIH grant RR18754-02. M.P.D. is supported by the James S. McDonnell Foundation 21st Century Research Awards/Studying Complex Systems. A.S.P. is supported by grants RR06555 and AI28433 from the National Institutes of Health.


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FOOTNOTES
 
* Corresponding author. Mailing address: Theoretical Biology and Biophysics, MS K710, Los Alamos National Laboratory, Los Alamos, NM 87545. Phone: (505) 667-6829. Fax: (505) 665-3493. E-mail: ruy{at}lanl.gov. Back


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Journal of Virology, January 2006, p. 802-809, Vol. 80, No. 2
0022-538X/06/$08.00+0     doi:10.1128/JVI.80.2.802-809.2006
Copyright © 2006, American Society for Microbiology. All Rights Reserved.



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