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Persistence and emergence of X4 virus in HIV infection.

Approximately 50% of late-stage HIV patients develop CXCR4-tropic (X4) virus in addition to CCR5-tropic (R5) virus. X4 emergence occurs with a sharp decline in CD4+ T cell counts and accelerated time to AIDS. Why this phenotypic switch to X4 occurs is not well understood. Previously, we used numerical simulations of a mathematical model to show that across much of parameter space a promising new class of antiretroviral treatments, CCR5 inhibitors, can accelerate X4 emergence and immunodeficiency. Here, we show that mathematical model to be a minimal activation-based HIV model that produces a spontaneous switch to X4 virus at a clinically-representative time point, while also matching in vivo data showing X4 and R5 coexisting and competing to infect memory CD4+ T cells. Our analysis shows that X4 avoids competitive exclusion from an initially fitter R5 virus due to X4v unique ability to productively infect nave CD4+ T cells. We further justify the generalized conditions under which this minimal model holds, implying that a phenotypic switch can even occur when the fraction of activated nave CD4+ T cells increases at a slower rate than the fraction of activated memory CD4+ T cells. We find that it is the ratio of the fractions of activated nave and memory CD4+ T cells that must increase above a threshold to produce a switch. This occurs as the concentration of CD4+ T cells drops beneath a threshold. Thus, highly active antiretroviral therapy (HAART), which increases CD4+ T cell counts and decreases cellular activation levels, inhibits X4 viral growth. However, we show here that even in the simplest dual-strain framework, competition between R5 and X4 viruses often results in accelerated X4 emergence in response to CCR5 inhibition, further highlighting the potential danger of anti-CCR5 monotherapy in multi-strain HIV infection.


Introduction
Without antiretroviral therapy, human immunodeficiency virus type-1 (HIV) generally depletes an infected individual's immunologically critical CD4 + T cell population, leading to AIDS onset and death after approximately 10-12 years [1,2,3]. HIV targets CD4 + T cells, as the virus binds and infects cells displaying the CD4 receptor. Yet, binding to CD4 alone is insufficient for HIV to enter a target cell: a coreceptor is required. HIV strains that utilize CCR5 as a coreceptor are called R5 viruses while those that bind CXCR4 are known as X4 viruses [4,5,6,7].
X4 viruses are rarely seen during early infection, where R5 viruses predominate, whatever the route of infection [5,6,8,9,10]. Moreover, individuals homozygous for a 32 base pair deletion in the allele for CCR5 (CCR5Δ32/Δ32) are almost entirely immune to HIV infection [5], implying that X4 viruses are at a severe selection disadvantage during early infection. However, in approximately 40-50% of progressing patients, X4 viruses emerge late in infection, often becoming the dominant strain [5]. X4 emergence is correlated with a steep decline in CD4 + T cell counts, which explains earlier work noting a "phenotypic switch" to a more virulent viral phenotype in many late-stage HIV patients [11,12,13,14,15].
In vitro competition assays between R5 and X4 virus usually result in X4 dominance [5]. Since about fivefold more lymphocytes are CXCR4 + rather than CCR5 + [16], one wonders why X4 is unable to dominate in vivo. A compelling explanation for R5's in vivo dominance and the basis for our models is CCR5's disproportionate presence on activated and recently activated memory CD4 + T cells. Memory CD4 + T cells can often be distinguished from their naïve precursor cells, because memory cells display the cell surface receptor CD45R0 [12]. Naïve cells generally display the receptor CD45RA, which is modified to its isoform CD45RO after an antigen 'naïve' CD4 T cell encounters its cognate antigen, thereby activating it into an effector memory cell.
Using the distinct cell surface receptors of naive and memory cells as well as antibodies that specifically bind to CCR5 and CXCR4, respectively, Lee et al. estimated the per-cell concentrations of CCR5 and CXCR4 molecules on naïve and memory T cells, respectively [16] ( Table 1). The authors went further, dividing both naïve and memory cell populations into activated and quiescent subsets, based on whether the cells also expressed the receptor CD62L, which is displayed by naïve and memory cells in quiescent states [17]. Using quantitative fluorescence-activated cell sorting (QFACS), they found an average of 4741 R5 antibody-binding sites on CD62L + CD45RO + quiescent memory cells with only 1,013 X4 binding sites on this cell population. Among highly activated memory CD62L − CD45RO + CD4 + T cells the difference is even more pronounced, with 9,576 R5 binding sites and only 505 X4 binding sites (Table 1). Conversely, the authors measured virtually no R5 antibody binding sites on naïve CD45RA + CD4 + T cells on which X4 binding sites dominate. In general, as Table 1 shows, CXCR4 is more common on naïve and quiescent cells, while CCR5 dominates in the effector memory population.
As a result of CCR5's higher per-cell density among memory cells, which are more likely to be activated than naive cells [18,19], R5 viruses may have an advantage on the whole over X4 viruses. Comparative snapshots of CD4 + T cells during SIV infection show approximately five times as many virions surround infected, activated CD4 + T cells as surround infected, phenotypically-quiescent CD4 + T cells [20]. Moreover, phenotypicallyactivated (Ki67 + ) CD4 + T cells produce over 90% of the virions during the chronic phase of SIV infection [21].
The relevant question is then: how do X4 viruses emerge late in infection if R5 viruses are simply better at infecting the all-important subset of memory CD4 + T cells? Previous mathematical models have analyzed several hypotheses for this emergence [22,23,24,25,26,27,28]. Specifically, Regoes and Bonhoeffer [27] pursued a model where antiretroviral treatment disproportionately inhibits R5 virus, precipitating a switch to X4. This cannot explain the documented emergence of X4 virus in treatment-naïve individuals [29]. Other models [23,24,26] analyzed the impact of differential immune responses on phenotypic switching, but these immune-based models utilize specific assumptions that current data argue against. Wodarz et al. [24] neglects the fact that over 90% of productive R5 infection occurs in CD4 + T cells, not macrophages [30]. The model by Wodarz and Nowak [23] cannot explain the disproportionate increase in X4 viral loads (VLs) after CD8 depletion [6]. Finally, the model by Callaway et al. [26] appears inconsistent with the fact that the greatest correlate of disease progression in HIV patients, and the only consistent difference between pathogenic and non-pathogenic lentiviral infections, is increased immune activation, including increased cytotoxic T-lymphocyte (CTL) activation [31,32]. Since X4 onset is strongly correlated with disease progression, an active cytotoxic immune response is more likely a cause or consequence of X4 emergence than an inhibitor.
Here, the co-occurrence of X4 emergence and immune cell activation is explored in mathematical detail, using current data to derive conditions under which increased targetcell activation over the course of dual R5, X4 HIV infection drives a late-stage switch to X4 virus. As in the above studies, the switch to X4 in our models is the result of progressive HIV infection altering the fitness landscape in favor of X4. Yet, the mechanism altering the fitness landscape is different here, as we use in vivo derived data to justify conditions showing changing T cell activation rates directly changing the fitness landscape in favor of X4. Building upon previous studies arguing that target-cell activation drives the switch to X4 [28,33], we derive a minimal target-cell activation-based model for understanding multitropism and its attendant immunodeficiency in HIV.

Generalized conditions for a phenotypic switch
Curve fitting a data set measured in vivo [34], which determined the fractions of activated T cells using the cell-cycle activation marker (Ki67), Ribeiro et al. [25] found that the fractions of naïve cells that are activated (a n ) and the fractions of memory cells that are activated (a m ) obey the following inverse relationships with respect to the total CD4 + T cell count: Here, CD4 denotes the total number of uninfected and infected CD4 + T cells per microliter of blood.
During HIV infection, CD4 + T counts decline, causing both a n and a m to increase. The increase in a n lets X4 virus benefit from CXCR4's strong presence on activated naïve CD4 + T cells [16] (Table 1), allowing for a switch.
In a previous paper [33], we claimed that one can generalize (1) to show that a switch can occur even if the fraction of activated naïve CD4 + T cells increases at a slower rate than the fraction of activated memory CD4 + T cells. We then chose a minimal model in this more general setting, and used simulations to show that CCR5 inhibitors, i.e. drugs that bind to CCR5 and prevent X5 viruses from entering cells, can accelerate X4 emergence, a prediction supported by recent data [35,36].
Weinberger et al. [33] examined the possibility of a phenotypic switch when Eq. (2) To justify these conditions, we note that throughout infection a far greater fraction of CD4 + memory cells are activated than naive CD4 + lymphocytes [18,19]. Thus, we set a n < a m . Furthermore, increased immune activation is strongly correlated with CD4 + T cell decline in HIV patients [32,37,38,39] and this increased activation is manifested in both naïve and memory CD4 + T cells [19,34], so a n ′(CD4)< 0 and a m ′(CD4)< 0. To justify the final condition in Eq. (2), we note that This derivative is negative if and only if Because a m ′(CD4) < 0, this is true if and only if: Clearly, Eq. (1) is a particular system satisfying Eq. (2), because in Eq. (1) we have a n < a m and a n ′ = a m ′ < 0. So in justifying Eq. (2) we are allowing for a larger class of models.
To justify Eq. (3) and thus the final condition of Eq. (2), we note that if a n ′ is a larger fraction of a m ′ than a n is of a m , Eq. (3) holds. We have already shown a n < a m ., implying that: Thus, the fraction of naïve cells that is quiescent is greater than the fraction of memory cells that is quiescent. We let n t and m t represent the total numbers (i.e., activated + nonactivated) of naïve and memory CD4 + T cells, respectively. Because naïve and memory cell counts are initially similar and because R5 virus disproportionately depletes memory CD4 + T cells [40,41], we assume that n t > m t during R5 infection, implying: Eq. (5) Furthermore, many of the newly activated memory CD4 + T cells were previously quiescent naïve CD4 + T cells activated by interaction with antigen. These additions to a m also increase a n by reducing the number of quiescent naive CD4 + T cells. Thus, given the large measured differences between the fractions of activated naïve and memory CD4 + T cells, we argue that discrepancies between the rates of increase of activated naïve and memory CD4 + T cells will often be relatively small. That is, we claim that in many cases (a n ′(CD4)/a m ′(CD4)) > (a n (CD4)/a m (CD4)), which is equivalent to Eq. (3). Data sets such as the one from which Eq. (1) was derived, give us evidence that this is reasonable. We note that Eq. (3) clearly holds when a n ′(CD4) ≤ a m ′(CD4) < 0, that is, when the fraction of activated naïve CD4 + T cells increases at least as quickly as the corresponding fraction of memory cells, as CD4 + T cells decline. Such a scenario is obviously to the increasing benefit of X4 virus in an activation-based model. In fact, this idea was used to explain the switch in Ribeiro et al. [28]. Yet, because a n < a m , Eq. (3) is even satisfied in certain cases in which 0 > a n ′(CD4) > a m ′(CD4) (i.e., when a n increases at a slower rate than a m in response to CD4 + T cell decline). Such a broadened scenario would occur if a n ≪ a m and a n ′ only slightly less negative than a m ′. Of course, in situations where a n increases far slower than a m in response to CD4 + T cell decline, Eq. (3) would likely not hold.

Switch
In our preceding paper [33], we began by extending the basic model of viral dynamics [1,2] to the simplest dual-strain framework, denoted Model 1 there and below. Through simulations, we showed that R5-to-X4 switches arise from this model, but claimed that such switches are beset by competitive exclusion, given the single-compartment nature of that model. Competitive exclusion is not consistent with in vivo data, which show X4 and R5 coexisting post-switch [36]. Here, we analytically show that competitive exclusion is the result of Model 1 and further show that accelerated emergence of X4 virus due to anti-CCR5 treatment is a basic result of strain competition for target-cells and is present in even the simplest of competitive models.

(Model 1)
In this model, all variables (capitalized) are concentrations per microliter (1/μl), λ has the units cells/(μl•day), k 4 and k 5 have the units μl/(virions•day), and the remaining parameters have units 1/day. Specifically, T represents the concentration of uninfected CD4 + T cells, and (without loss of generality) is given an initial value of 1000 CD4 + T cells/μl. I 4 and I 5 reflect the concentrations of CD4 + T cells abortively, latently, and productively infected by X4 and R5 viruses, respectively; V 4 and V 5 , describe X4 and R5 virus concentrations. λ is the rate of production of CD4 + T cells and k 4 and k 5 are the respective infection rate coefficients for X4 and R5 infection of CD4 + T cells. Also, d T is the death rate of uninfected CD4 + T cells and is set equal to λ/T 0 to allow for steady-state pre-infection, δ is the death rate of infected CD4 + T cells, p is the rate of viral production by activated infected cells, and c is the viral clearance rate. a n and a m are required to satisfy Equation (2) and represent the fractions of activated naïve and memory CD4 + T cells for a given value of CD4. Since CD4 represents the total number of uninfected and infected CD4 + T cells per microliter, CD4 = T + I 4 + I 5 .
We assume that when activated cells become infected they produce virus at rate p per cell. In our model, it is only these activated infected cells that produce virus. We thus multiply a n I 4 and a m I 5 by p, to obtain the total concentrations of virions produced each day. In a more complex model, one could allow a small amount of viral production from infected resting cells. Importantly, the products a n I 4 and a m I 5 assume that infected cells are no more likely to be activated than uninfected cells. This is because infection in our model is not necessarily productive, and in general most infections have been measured to be non-productive [30].
Given the per-cell concentrations of CCR5 and CXCR4 recorded shown in Table 1, we assume that X4 virus only productively infects naïve CD4 + T cells and thus make X4's viral production dependent on a n , but not a m . Conversely, we use the same dataset to justify making R5's production dependent on a m , but not a n . Because CXCR4's median cell surface density is almost three times as high as that of CCR5 across all lymphocytes [16], we also assume k 4 > k 5 . As above, this does not imply that X4 productively infects more target cells than R5 at the beginning of infection, since very few naïve cells are activated early in infection [34].

Deriving a Switch Threshold for Model 1
In analyzing Model 1, we first determine how many productively infected cells each strain has at a given point in time. Let R eff4 and R eff5 be time-dependent functions for the average number of infected cells that an average X4 and R5 infected cell produces. R eff4 and R eff5 are thus "effective reproductive ratios," in contrast to the "basic reproductive ratios," R 04 and R 05 , which evaluate R eff4 and R eff5 at the initial time point. The equations for R eff4 and R eff5 are (Eq. 6) We note that while R eff4 and R eff5 are functions of t (time), the explicit time-dependencies of R eff4 and R eff5 cancel in the quotient R eff4 /R eff5 . This allows us to explore and subsequently differentiate R eff4 /R eff5 as a function of CD4 alone.
Initially, we assume that R eff4 < R eff5 , because at the large CD4 + T cell counts prevalent during early infection a m ≫ a n , implying that a m ·k 5 > a n ·k 4 despite the fact that k 4 >k 5 (i.e., we assume that initially the disparity between a n and a m is greater than the disparity between k 4 and k 5 ). With a higher effective reproductive ratio, R5 virus is more efficient and dominates early, consistent with observation. As infection progresses, Eq. (2) shows that the relative fraction of activated naïve cells increases as CD4 + T cells decrease. This yields Eq. (7) In other words, if Eq. (2) holds, lowering CD4 + T cell counts preferentially benefits X4 by increasing its fitness relative to that of R5 virus. This accounts for the possibility of a switch at low CD4 + T cell counts. Here we show that when CD4 counts decrease enough for R eff4 / R eff5 to go above 1, a switch to X4 virus occurs at a future time point. Conversely, if R eff4 / R eff5 never increases beyond 1, a switch to X4 cannot occur, potentially explaining why 50% of patients do not exhibit a switch to X4 Virus during HIV infection.
In order for X4 virus to overtake R5 virus at time t*, the following conditions are necessary and sufficient: d/dt(V 4 (t*)) > d/dt (V 5 (t*)) and V 4 (t*) = V 5 (t*). Solving these switch equations simultaneously yields a necessary and sufficient switch condition for this model: Equation (8) describes the threshold at which the switch to X4 occurs. We can find an earlier necessary and sufficient threshold for a n /a m above which a future switch to X4 is guaranteed to occur. To do so, we note that because a n /a m < 1 for all time, the right-hand side of Eq. (8) must be less than one. Thus, I 4 (t*) > I 5 (t*) is a necessary condition for a switch. In biological terms, when X4's advantage, manifested in a greater number of infected cells, outweighs R5's advantage, manifested in a higher target-cell activation level (i.e., a higher probability that its infected cells are productively infected), the switch occurs.
For I 4 to overtake I 5 at t*, a necessary condition is that at some earlier time point, t**<t*, the rate of growth of X4-infected cells was higher than that of R5-infected cells. Hence, R eff4 (t**) > R eff5 (t**) is a necessary condition for the R5 to X4 switch to occur. In fact, R eff4 (t**) > R eff5 (t**) is also a sufficient condition for the R5 to X4 switch. Because Eq. (8) implies that R eff4 is always increasing relative to R eff5 as CD4 + T cells decline, R eff4 (t**) > R eff5 (t**) means that R eff4 (t) > R eff5 (t) for all t ≥ t**. Thus, if R eff4 (t**) > R eff5 (t**), X4 will eventually overtake R5.
An R5 to X4 switch always results in the eventual extinction of R5 in Model 1. This is because coexistence at steady state means: implying that Eq. (11) Plugging Eq. (10) into Eq. (11) tells us that a necessary and sufficient coexistence condition is (k 4 /k 5 ) · (a n /a m ) = 1. By Eq. (6), this is equivalent to the coexistence iff R eff4 = R eff5 .
Thus, a necessary and sufficient condition for a switch to occur at some point t*>t** and for R5 to approach extinction is: R eff4 (t**) > R eff5 (t**) means that a n /a m increases beyond k 5 /k 4 . Because a n /a m increases as CD4 + T cells decline, it is the level of CD4 + T cell depletion engendered by HIV that is directly implicated in the model's switch. To quantify this for the measured functions of a n and a m given in Eq. (1), we substitute Eq. (1) into Eqs. (6) and (12) to yield the following necessary and sufficient threshold beyond which a switch is guaranteed to eventually occur: Eq. (13) Since k 4 > k 5 , the quotient on the right hand side is positive. Hence, at a CD4 count below a threshold, the switch condition is satisfied, guaranteeing that X4 will eventually take over. With the exception of changes to k 4 and k 5 , it is clear from Eq. (13) that all changes to the model's parameters that accelerate CD4 + T cell depletion accelerate an R5 to X4 switch. Conversely, mitigating the level of infection and consequent CD4 + T cell depletion lengthens the time until the switch occurs. Because the partial derivative of the right side of Eq. (13) is positive with respect to k 4 , increasing k 4 also accelerates the switch by increasing the right hand side while decreasing CD4 counts through heightened X4 infection (Fig. 1). Yet, the partial derivative of Eq. (13) with respect to k 5 is negative, meaning that both right and left sides of the equation decrease in response to higher levels of k 5 , making it initially unclear as to whether increasing k 5 promotes a switch to X4 virus.

CCR5 Inhibitors Can Promote Switches to X4 virus in a Single Compartment Model
In general, reducing k 5 -as occurs in CCR5 inhibitor treatments-increases k 4 /k 5 , but it also decreases a n /a m by increasing CD4 + T cell counts through decreased R5 infection. By Eq. (6), R eff4 /R eff5 is the product of k 4 /k 5 and a n /a m , so the question is whether the increase to k 4 / k 5 is greater than the decrease to a n /a m . If so, R eff4 /R eff5 increases in response to lowering k 5 , implying that anti-CCR5 treatments can accelerate switches to X4.
We examined how modulating k 5 affects the switch to X4 virus. When a n and a m are defined as in Eq. (1), increasing k 5 from 1•10 −4 μl/(virions•day) to 1.5•10 −4 μl/(virions•day) accelerates the time at which X4 emerges (i.e., it increases R 4 /R 5 ) (Fig. 2a, upper panel). However, increasing k 5 even further to 3•10 −4 μl/(virions•day) prevents a switch (Fig. 2a,  lower panel). In fact, the model predicts a steady state with high X4 viral loads only at intermediate values of k 5 : increasing k 5 beyond a threshold blocks X4 emergence (Fig. 2b). To understand why increasing k 5 beyond a threshold prevents a switch to X4 Virus, we note that large values of k 5 (e.g. k 5 =3•10 −4 ) allow R5 to infect the vast majority of CD4 + T cells, leaving few uninfected R5 target cells. This causes diminishing returns in the number of new CD4 + T cells that can be infected through further increases to k 5 . As a result, when k 5 is initially large and k 5 is further increased, the increase to a n /a m from further CD4 T cell declines is unlikely to outweigh the decrease to k 4 /k 5 , causing a decrease in R 4 /R 5 and inhibiting X4 emergence. Thus, if k 5 is initially large and a CCR5 inhibitor only partially decreases k 5 -keeping us in the high k 5 diminishing returns regime-the increase to k 4 /k 5 from decreasing k 5 can outweigh the decrease to a n /a m from the small increase in CD4 + T cell counts, increasing R eff4 /R eff5 and promoting a switch to X4 (Fig. 2c). Significantly, these switches to X4 are prevented by combination therapies such as HAART (Fig. 2d, left  panel) or combined CCR5, CXCR4 inhibition (Fig. 2d, right panel), which combat both R5 and X4 virus equally.
Model 1 is thus a simplified model in which we can rigorously see that competition for target cells may make anti-CCR5 treatment a risky proposition. And while this simplified model seems to imply that CCR5 inhibitors do not do any damage insofar as the steady-state CD4 + T cell count is not lowered as a result of X4 emergence (Fig. 2c), the reality, as described more faithfully in Model 3 below, is that X4's emergence uniquely depletes the naïve CD4 + T cell population, which serves as the pipeline for new memory CD4 + T cells.

Model 2: Coexistence, but no competition
Having analyzed a simplified one-compartment switch-inducing model in detail, we are left with the problem of competitive exclusion. This all-or-nothing result is inconsistent with data that shows the possibility of coexistence after a phenotypic switch [42]. We previously claimed that maintaining distinct target-cell populations for R5 and X4 viruses is sufficient to produce coexistence [33]. Here we rigorously show this.

(Model 2)
The equations in this system are analogous to those in Model 1 but the uninfected CD4 + T cell population is now split into uninfected naïve (N) and memory (M) subpopulations. The target cell death rates, d n and d m , are defined analogously to d T in Model I, ensuring that both subsets of the uninfected CD4 + T cell population are in equilibrium pre-infection. Additionally, f is defined to be the fraction of naïve cells activated by antigen, which then divide and differentiate into CD45RO + memory cells. The rest of the activated cells are assumed to have been upregulated via cytokines or other antigen-T cell receptor independent processes and thus remain phenotypically naïve (CD45RA + ) [43,44,45]. Again, for simplicity it is assumed that X4 virus solely infects naive cells and that R5 virus only infects memory cells. Since the target cell population is now split, the effective reproductive ratios of R5 and X4 become functions of distinct target cell populations:

Eq. (14)
An immediate result is that k 4 > k 5 is no longer required for a switch to occur. In fact, if R5 depletes most of its target cells, X4 virus will have an advantage even when X4 has a lower infection rate coefficient. That is, a n *k 4 *N > a m *k 5 *M is possible even when a n *k 4 < a m *k 5 .
Because of the differential target-cell compartments, after a phenotypic switch R eff5 can rebound and increase relative to R eff4 , a fact that could not occur in the above singlecompartment model. This occurs because when X4 viral loads burgeon during a switch, X4 encounters an untapped naïve target cell pool, while most memory target cells have already been depleted by R5 infection. This means that the naïve CD4 + T cell population will decrease more rapidly than the corresponding memory cell population after an R5-to-X4 switch. If the resulting decrease in N/M is greater than the increase in a n /a m that results from the lowered CD4 + T cell counts post-switch, then R eff4 /R eff5 decreases by Eq. 14. Since R eff4 / R eff5 can decrease after a switch in a two-compartment model, coexistence is now possible (Supporting Figure 1).
The equations for V 4 and V 5 in the two compartment model are identical to those in the single-compartment model, so the same switch condition persists (found by setting V 4 =V 5 and d/dt(V 4 )>d/dt(V 5 )). Thus, as in Model 1, a switch occurs if and only if a n /a m goes above the threshold in (8), or, equivalently, if and only if there is sufficient CD4 + T cell depletion.
Thus, modulating parameters to increase CD4 + T cell decline accelerates an R5 to X4 switch, while down-regulating infection, for example via drug intervention, inhibits X4 incidence. This result clearly extends to changes in k 5 , as X4 and R5 are independent viruses here so that X4 receives no advantage from a weakened R5 virus. Moreover, having a CD4 + T cell threshold for an R5 to X4 switch means that despite R5's ability to increase after X4 depletes the naïve CD4 population post-switch, R5 is not likely to overtake X4 post-switch, because doing so requires an increase in CD4 + T cell counts.
While this two-compartment model can produce switching and coexistence, it is oversimplified in assuming that X4 cannot infect any memory CD4 + T cells. In fact, despite being outcompeted by R5 for CCR5 + , CXCR4 + CD4 + T cells, X4 productively infects certain memory CD4 + T cells, predominantly those that are resting, CD62L + [16,46].

Model 3: Two Compartments with Competition
In order to account for the observed competition of X4 and R5 viruses for the infection of memory CD4 + T cells, which allows X4 to increase in response to CCR5 inhibition as seen in in vivo experiments [36], and in order to prevent competitive exclusion of the less fit viral strain, we combine Models 1 and 2, allowing X4's infection of both naïve and memory CD4 + T cells: In this model, k N4 , k M4 , and k M5 , are the infection rate coefficients of X4 and R5 for naïve (N) and memory (M) CD4 + T cells, while N 4 , M 4 , and M 5 are the infected cell concentrations corresponding to the originating target cell and infecting viral strain. All other parameters, variables, and initial conditions have been previously defined. Because CCR5 is far more strongly expressed on memory CD4 + T cells than is CXCR4 [16] (Table  1), we set k M5 ≫ k M4 . Conversely, CXCR4 is more highly expressed on naïve CD4 + T cells than on memory CD4 + T cells [16], making k N4 ≫ k M4 .
Given that X4 and R5 were already shown to coexist in Model 2 due to decreases in N/M post-switch, coexistence results in this extended dual compartment model as well [33]. To show this graphically, we note that the relative fitness of X4 to R5 is given by: Simulations show R eff4 /R eff5 rising to a local maximum before the R5 to X4 switch only to decrease post-switch due to an X4-driven decrease in N/M. R eff4 /R eff5 eventually fixates at the value 1, allowing for coexistence of R5 and X4 strains post-switch (Supporting Figure  2).
In our previous paper [33], we derived the following switch conditions for Model 3: Eq. (15b) (16) As in the preceding models, Equation (16) implies that, with the exception of changes to k M5 , modulating parameters to accelerate CD4 + T cell decline hastens an R5 to X4 switch while changing parameters to mitigate CD4 + T cell decline hinders a phenotypic switch. Thus, successful antiretroviral therapy will generally inhibit X4's emergence. However, as one might predict from our results in Model 1, because R5 and X4 are in competition, CCR5 inhibitors can generate more complicated kinetics. In fact, the utility of CCR5 inhibitors depends on the strength of the competition between X4 and R5 virus, which is modulated by X4's infection rate coefficient for memory CD4 + T cells, k M4 . In the "non-competitive" regime in which k M4 ≪ k M5 , X4's viral set-point is a monotonically decreasing function of the CCR5 inhibitor's efficacy (Fig. 3a, left panel). This is because, due to the low value of k M4 , X4 is unable to infect the majority of target cells blocked from R5 infection, allowing CD4 counts to rise and causing a drop in activation levels. Conversely, in the "competitive" regime where k M4 is closer in value to k M5 but still less than k M5 to remain consistent with FACS data [44], X4's viral set-point is a monotonically increasing function of the CCR5 inhibitor's efficacy (Fig. 3a, right panel).
In the "competitive regime" the steady-state CD4 + T cell count is not decreased by CCR5 inhibition (Fig. 3a, right panel), which might lead one to suspect that these treatments are safe in this regime as well. Critically, however, the CD4 + T cell count crashes far sooner in the "competitive regime" when anti-CCR5 treatment is employed (Fig. 3b). Thus, CCR5 inhibitors may accelerate immunodeficiency in patients with a competitive X4 virus.
Why is this the case? CCR5 inhibitors decrease k M5 , causing R5's viral load to decline, and memory CD4 + T cell counts to increase. X4 is now able to infect some of these newly generated memory CD4 + T cells, but X4's ability to do so depends on k M4 . With k M4 sufficiently large (the "competitive regime"), X4 infects a non-negligible fraction of newly generated memory CD4 + cells, increasing X4's viral load. But with a larger viral load, X4 is now better able to infect its main target cell pool: naïve CD4 + T cells, the untapped target cell reserve where CXCR4 is highly present. The slight increase in memory CD4 + T cell counts due to CCR5 inhibition thus causes severe and accelerated depletion of the naïve CD4 + T cell population in this "competitive regime" (Fig. 3c). Thus, a single parameter, k M4 , controls the efficacy of anti-CCR5 therapy in dually infected HIV patients, highlighting the need for circumspection in prescribing these treatments.

DISCUSSION
In this paper a set of mathematical models for dual R5, X4 infection in HIV has been rigorously derived from a multi-strain version of the basic model of viral dynamics. The models were analyzed to show how an increase in the ratio of the fractions of activated naïve and memory CD4 + T cells (a n /a m ) can trigger an R5 to X4 "phenotypic" switch in dually infected individuals. Importantly, this allows for phenotypic switching even when the fraction of activated naïve CD4 + T cells increases at a slower rate than the corresponding fraction of memory CD4 + T cells (as long as the ratio of the two fractions increases beyond a threshold). Our models also help explain why 50% of patients do not manifest a noticeable switch to X4 virus: their relevant parameter regimes may simply keep a n /a m below the threshold. Finally, anti-CCR5 treatment is shown to promote X4 virus even in the simplest competitive framework (Model 1).
While we predict that R5 blockers can promote X4 emergence in dual infection, we find that non-CCR5 specific, antiretroviral therapies such as HAART have the opposite effect. This prediction is in fact supported by a recent clinical trial on 15 women with X4 virus prior to undergoing HAART [42]. During HAART, the patients showed marked increases in CD4 + T cell counts as well as a correlated reversion in viral tropism toward CCR5. Other groups have also found that antiretroviral treatment inhibits X4 virus [47,48,49]. However, one group claims that HAART promotes R5 to X4 switching [50]. Delobel and colleagues' conclusion arises from an analysis of the genotypes of peripheral blood mononuclear cells of patients on HAART for more than three years with viral loads below detection. Notwithstanding the overall utility of testing coreceptor usage when viremia levels are below detection, the genotypic algorithms used by Delobel et al. [50] do not always correctly predict the actual coreceptor usage [51]. To avoid such errors, the Philpott study, which found preferential suppression of X4 strains in patients on HAART, used a phenotypic MT-2 cell line characterization and a direct HOS-CD4 coreceptor binding assay in addition to a genotypic V3 analysis [42].
An activation-induced switch and its deleterious clinical effects are also consistent with the proposed new paradigm of lentiviral pathogenicity, which argues that immune overactivation is the distinguishing characteristic of symptomatic lentiviral infection in new hosts as opposed to asymptomatic lentiviral infection in natural hosts [32,34,37,38,52,53,54,55,56]. Evidence of a correlation between immune activation and disease progression is also supported by the fact that T cell activation levels are lowered almost immediately following successful HAART [34].
One might argue that phenotypic switching in HIV has little to do with target-cell activation levels and is instead the result of cumulative mutations that occur over the course of HIV. One would have two reasons for such an argument. First, given its status as a retrovirus, HIV is extremely prone to mutation [57]. Second, it takes very few mutations to go from R5 to X4 virus. For example, Ho et al. showed that in rhesus macaques ten amino acid changes in the V3 loop of an X4-tropic virus are sufficient to modify viral coreceptor usage to CCR5 [58]. That said, many V3 mutations yield viruses with lower fitness, implying that fitness troughs exist between R5 and X4 variants [59]. Perhaps as a result, mutation between R5 and X4 strains does not seem to be common in vivo [60,61]. When drugs are employed to selectively block CCR5 in cases of R5-only infection, HIV's method of escape is not to evolve tropism for CXCR4 but to find a novel way of binding CCR5 despite the blockage [60,62]. Finally, X4 is simply outcompeted by R5 during early dual-infection, arguing in favor of an early exogenous selection pressure toward R5, which is mitigated over the course of infection in switching patients.
Current data are insufficient to test our conclusion that the efficacy of CCR5 blockers in dual infection depends on k M4 , because k M4 has an unknown value. The importance of testing whether CCR5 inhibitors have only partial regimes of utility stems from the fact that these treatments, in contrast to HAART and traditional antiretroviral drugs, are mainly nontoxic [63]. In fact, CCR5Δ32/Δ32, a natural deletion mutation that prevents CCR5 expression, is found in 1% of humans with no known side-effects. The question associated with CCR5 blockers is whether they promote X4 in R5's stead. This is because X4 virus quickly depletes R5-immune naïve CD4 + T cells, compounding the earlier immunodeficiency that R5 engendered in the memory compartment. Naïve CD4 + T cells are the source for new memory cells and a prime defense against unseen infections: hence, the victim of a phenotypic switch gets the worst of both worlds-memory CD4 + T cell loss by R5 followed by naïve CD4 + T cell loss by X4-greatly lessening the chance of survival.

Methods
Models 1-3 were first solved numerically using the program Berkeley Madonna. We used the Rosenbruck algorithm for solving stiff ODEs with the parameters given in the Figures.  Other than k N4 , k M4 , k M5 and f, these parameters have been estimated from in vivo measurements. Further, V 4 and V 5 were each given initial values of 1000 virions/ml, as in [28,33], which reflects experiments in macaques in which high viral doses are given to ensure infection [36].
Since the purpose of the first two models is to motivate the added complexity in Model 3 and since our main conclusions are taken from Model 3, we offer a justification of the parameter values for Model 3. In particular, λ, the rate at which naïve CD4 + T cells emigrate from the thymus, has been shown to remain relatively constant during HIV infection [64]. Following a recent theoretical analysis [65], we set λ to the constant value of 33 cells/ (μl•day). The viral clearance rate, c, has been directly measured to have an average value of 23/day [66]. The rate of virion production by productively infected cells, p, was set to 2100/ day, which is line with the in vivo measure in [67] but smaller than the value reported in [68]. Finally, we set the infected cell death rate δ to 0.5/day, following the measurements in [69], although values as high as 1.0/day are also feasible [70]. The final four parameters k N4 , k M4 , k M5 and f have unknown values, but can nonetheless be substantially restricted. The fraction of naïve CD4 + T cells that are activated by antigen, f, is between 0 and 1. Furthermore, the FACS data summarized in Table 1 leads us to restrict the infection rate coefficients as follows: k N4 ≫ k M4 , k M5 ≫ k M4 . We chose exact values for these 4 parameters, subject to the above constraints, by repeated simulations of our Models so as to produce the general dynamics of long-term HIV infection, including the common phenotypic switch.
We also note that because simulations require an exact form for a n and a m , we used the particular form fit in (1). Of course, the analysis throughout this paper shows that we can apply any equations which satisfy (2), with obvious parameter adjustments.
Subsequent to these simulations, we reproduced our work in MATLAB (with the stiff solver, ode23s) so that we could generate three-dimensional plots and show that the switch is accelerated when CCR5 is blocked in "competitive" regimes (i.e., those situations in which k M4 is relatively large). All code is available upon request.

Supplementary Material
Refer to Web version on PubMed Central for supplementary material. shows, parameter changes-with the exception of k 5which enhance infection accelerate the switch (bottom panels of (b), (c), (d)), while those that dampen infection hinder switching (top panels of (b), (c), (d)). In each top panel (i.e., decreased infection regime), X4 stays below detection for all 5000 days of the simulation. While Fig. 1 implies that increasing the level of infection accelerates R5 to X4 switches, changes to k 5 present more complicated kinetics. In (a) the model is simulated at two increased levels of k 5 relative to (1a). As expected from Fig. 1, an initial increase of the R5 infection rate coefficient to k 5 = 1.5•10 −4 μl/(virions•day) accelerates X4 emergence (a, top panel). However, increasing k 5 more significantly to 3•10 −4 μl/(virions•day)-prevents X4's onset (a, bottom panel). In (b), X4 is shown to go extinct as k 5 crosses a threshold. In (c), where k 5 =3•10 −4 μl/(virions•day), anti-CCR5 therapy with efficacy below ~70% actually promotes X4 emergence. This occurs because k 5 is not decreased beyond the diminishing returns regime. In (d) combining reverse transcriptase and protease inhibitors (representing the anti-HIV drug cocktail known as HAART), which target R5 and X4 strains equally, is shown to prevent X4 emergence and increase CD4 counts in Model 1. Similarly, in (e) combining anti-CCR5 and anti-CXCR4 therapy is shown to prevent X4 emergence and increase CD4 counts in Model 1. Model 3 is simulated in two representative parameter regimes, labeled "non-competitive" and "competitive," respectively. In the non-competitive regime, the parameter values are: λ =33 cells/(μl•day), c=23/day, p=2100/day, f=0.8, δ=0.5/day, k N4 =0.00108 μl/(virions•day), k M4 = 4•10 −5 μl/(virions•day), and k M5 =0.0068 μl/(virions•day). In the competitive regime, k M4 is increased to 5•10 −4 μl/(virions•day) and k N4 is correspondingly decreased to 0.001 μl/(virions•day) to keep X4 in check, but all other parameters are held constant. The crucial distinction between the two regimes, which display similar dynamics in the absence of anti-CCR5 treatment, is that in the non-competitive regime k M4 is relatively small, preventing X4 from infecting many CCR5 + , CXCR4 + memory CD4 + T cells after CCR5 inhibition. Thus, in the "non-competitive regime" (a, left panel) as we increase the efficacy of a permanent CCR5 inhibitor first given at t=180 days and maintained for all future time, steady-state CD4 + T cell counts rise and steady-state X4 levels fall. However, in the competitive regime (a, right panel), anti-CCR5 treatment does not depress steady-state X4 levels and it does not increase steady-state CD4 + T cell counts, either. While steady-state X4 levels and CD4 + T cell counts remain unchanged under this treatment schedule, in (b) the time at which this steady-state occurs is shown to be earlier. That is, in the competitive regime, CCR5 inhibitors accelerate CD4 + T cell depletion. In (c), this accelerated outcome is seen to be the result of a temporal gain in memory CD4 + T cell counts triggered by an initially effective anti-CCR5 treatment. The "competitive" X4 virus then increases by infecting newly generated memory cells. This strengthens X4 infection, after which it can severely deplete naïve CD4 + T cells, and counteract any initial gains in memory CD4 + T cells.