HIV Coreceptor Usage and Drug Treatment

J. theor. Biol. (2002) 217, 443–457 doi:10.1006/yjtbi.3049, available online at http://www.idealibrary.com on

HIV Coreceptor Usage and Drug Treatment Roland R. Regoes*w and Sebastian Bonhoefferw wEcology and Evolution, Swiss Federal Institute of Technology Zurich, ETH Zentrum NW, CH-8092 Zurich, Switzerland (Received on 8 January 2002, Accepted in revised form on 4 April 2002)

The human immunodeficiency virus (HIV) infects a wide range of human cells. Cell entry is mediated through the CD4 receptor and a variety of coreceptors, most importantly the chemokine receptors CCR5 and CXCR4. Some antiretroviral agents selectively inhibit different HIV phenotypes depending on their coreceptor usage. Here, we analyse mathematical models, which describe the in vivo interaction of HIV phenotypes, differing in their coreceptor usage, with two target cell types (naive and memory CD4+ T cells). In particular, we investigate how the selection pressures on CCR5- and CXCR4-using HIV variants change as a result of treatment with coreceptor-specific antiretroviral agents. Our main result is that CXCR4 inhibitors increase the selection pressure in favor of the emergence of CCR5-using variants, thus selecting for coexistence of CXCR4- and CCR5-using variants, whereas CCR5 inhibitors increase the selection pressure against CCR5-using variants, thus selecting against coexistence. These results shed new light on the potential risks and benefits of coreceptor inhibitors. r 2002 Elsevier Science Ltd. All rights reserved.

Introduction To enter its target cells the human immunodeficiency virus (HIV) utilizes the surface protein CD4 as a receptor. In addition to CD4, HIV requires coreceptors for successful infection, the most important of which are the chemokine receptors CCR5 and CXCR4. The coreceptor usage represents a criterion according to which variants of the HIV quasispecies can be classified: HIV variants which use CCR5 are referred to as R5 viruses, variants which use CXCR4 are referred to as X4 viruses, and variants which are able to use both coreceptors are referred to as *Corresponding author. Current address: Department of Biology, Emory University, 1510 Clifton Rd., Atlanta, GA 30322, U.S.A. Tel.: +1-404-727-17-65; fax: +1-404727-2880. E-mail addresses: [email protected] (R.R. Regoes), [email protected] (S. Bonhoeffer). 0022-5193/02/$35.00/0

X4R5 viruses (Berger et al., 1998). The classification with regard to their coreceptor usage correlates with other important phenotypic properties of HIV variants. CXCR4 using variants typically display fast replication kinetics and are of the syncytium-inducing phenotype (Bjorndal et al., 1997). Most importantly, the emergence of X4 or R5X4 virus variants correlates with accelerated progression towards disease and death (Asjo et al., 1986; Fenyo et al., 1988; Tersmette et al., 1988, 1989a,b; ChengMayer et al., 1990; Schuitemaker et al., 1992; Connor et al., 1993; Keet et al., 1993; Koot et al., 1993a,b; Karlsson et al., 1994; Katzenstein et al., 1996; Spijkerman et al., 1998; Xiao et al., 1998). The main target cells of HIV are the CD4+ T cells (Zhang et al., 1999). Until recently the prevailing view was that mainly activated/ memory CD4+ T cells are susceptible to virus r 2002 Elsevier Science Ltd. All rights reserved.

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infection (Schnittman et al., 1990; Helbert et al., 1997; Spina et al., 1997; Chou et al., 1997; Roederer et al., 1997). This has been challenged by studies which show that HIV does infect naive CD4+ T cells, and is capable of reproducing in these cells (Ostrowski et al., 1999; Zhang et al., 1999; Blaak et al., 2000; Eckstein et al., 2001). The coreceptor expression profile of CD4+ T cells depends on their activation status: activated/memory CD4+ T cells express both, CCR5 and CXCR4, whereas resting/naive CD4+ T cells express mainly CXCR4 (Bleul et al., 1997; Ostrowski et al., 1998; Lee et al., 1999). Therefore, R5 variants primarily infect activated/memory CD4+ T cells, while X4 and R5X4 variants can infect both, memory and naive CD4+ T cells. The expansion of the target cell range, which goes hand in hand with a switch from CCR5-using to CXCR4-using variants, together with the capacity of HIV to replicate in naive CD4+ T cells, could explain the increased pathogenesis associated with the emergence of X4 and X4R5 virus (Blaak et al., 2000). Some antiretroviral agents have differential effect on R5 and X4 HIV variants. Currently several drugs are in development based on modifications of the natural ligands (MIP-1a; MIP-1b; RANTES, SDF-1) to the CCR5 and CXCR4 coreceptors. Moreover, coreceptorspecific agents based on small molecule inhibitorsFsuch as the CCR5 inhibitors TAK 779, PRO 140 and SCH-C (Baba et al., 1999; Trkola et al., 2001; Strizki et al., 2001), and the CXCR4 inhibitors AMD3100, ALX40-4C, T22, T134 and T140 (Schols et al., 1997; O’Brien et al., 1996; Murakami et al., 1997; Arakaki et al., 1999; Tamamura et al., 1998)Fare currently used in clinical trials. There is concern that inhibitors that preferentially inhibit R5 viruses would lead to a switch to the more pathogenic SI viruses and thus lead to faster disease progression. Conversely, a CXCR4-specific inhibitor that could induce a switch back to CCR5 using viruses may be highly beneficial. Many mathematical models have been developed to study the effect of drug treatment on viral evolution (Frost & McLean, 1994; Bonhoeffer et al., 1997a,b; Bonhoeffer & Nowak, 1997; Nowak et al., 1997; Kepler & Perelson,

1998; Ribeiro et al., 1998), and the evolution of HIV tropism (Wodarz et al., 1998a; Wodarz & Nowak, 1998; Callaway et al., 1999). However, to our knowledge there are no models which address the effect of antiretroviral treatment on HIV tropism. Here, we present a mathematical framework that allows us to investigate the change of HIV coreceptor usage under different treatment regimens. In the first part, we introduce a model describing the interaction of the HIV quasispecies with naive and memory CD4+ T cells, and study the competition of R5 and X4 variants for target cells. We find that an R5 variant can be outcompeted by an X4 variant if the affinity of the X4 variant for CXCR4 exceeds a threshold, whereas an X4 variant cannot be outcompeted by an R5 variant regardless of how high the affinity of the R5 variant for CCR5 is. Thus, due to the differing target cell ranges of R5 and X4 variants, X4 variants appear to be more persistent than R5 variants. In the second part, we investigate how drug treatment changes the selection pressure on R5 and X4 variants with special focus on the conditions for a treatment-induced HIV phenotype switch. Considering the effect of coreceptor inhibitors, we find that CCR5 inhibitors select against R5 variants and likely lead to their suppression, while CXCR4 inhibitors may result in the emergence of R5 variants, but will not result in suppression of the CXCR4-using variants. Model We start by introducing a simple population dynamical model which describes the interaction of HIV with CD4+ T cells. We assume two target cell populations, N and M; which are referred to as naive and memory CD4+ T cells in the following, but which in a stricter sense denote CD4+ T cells which at the time point of infection have coreceptor expression patterns that are characteristic of naive and memory CD4+ T cells, respectively. Furthermore, we assume n HIV variants, vi ; i ¼ 1; y; n; which differ in their infectivities for naive and memory CD4+ T cells, bni and bmi ; respectively. After infection with HIV variant i; naive and memory T cells become infected CD4+ T cells, Ni and

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Mi : The model is given by the following set of equations: X bnj vj ; ð1Þ N’ ¼ ln  dn N  N j

’ ¼ lm  dm M  M M

X

bmj vj ;

ð2Þ

j

N’ i ¼ bni Nvi  an Ni ;

ð3Þ

’ i ¼ bmi Mvi  am Mi ; M

ð4Þ

v’i ¼ cm Mi þ cn Ni  uvi :

ð5Þ

Note that the variable Ni does not denote the concentration of infected naive CD4+ T cells, but the concentration of CD4+ T cells that were naive at the time of infection. This more subtle definition is crucial to our model, since we assume that the cells denoted by the variable Ni release virions at a rate cn Ni : (Naive cells, however, do not release virions at high rates before activation.) Our subtle definition serves the purpose to avoid the explicit incorporation of the activation process into the model which would complicate the equations considerably. Instead, the activation of CD4+ T cells is implicitly incorporated into the parameters of our model. This affects in particular the parameters lm ; dn ; an and cn which we define to contain the activation rate. Firstly, lm containsFbesides the proliferation rate of memory CD4+ T cellsFthe rate at which memory CD4+ T cells arise from naive CD4+ T cells. Secondly, the clearance rates dn and an of uninfected and infected cells that displayed a naive coreceptor expression profile at the time of infection, are higher than the clearance rates of uninfected and infected naive CD4+ T cells. This is because some of the cells that were naive at the time of infection may have been activated and become memory CD4+ T cells which have higher clearance rates. Lastly, we assume that HIV-infected CD4+ T cells which were naive at the time of infection release virions at the rate cn Ni : One could conceive the rate constant cn as a composite rate of two processes: first, the lowlevel production of virions by infected naive CD4+ T cells, and second, the differentiation of naive CD4+ T cells into memory CD4+ T cells

and subsequent release of virions at increased levels. A description of the parameters of Model 1–5 and biologically plausible values are given in Table 1. A diagrammatic representation of this model can be found in Fig. 1. Since we are interested in HIV coreceptor usage, the parameters of primary interest are the infectivities, bmi and bni ; which measure the preference of the i-th HIV variant for memory and naive CD4+ T cells, respectively. The infectivity, bmi ; is determined by the affinities of a certain HIV variant for CCR5 and CXCR4 (since memory cells express both coreceptors), whereas the infectivity, bni ; is determined by the affinity of the respective HIV variant for CXCR4 only: bCXCR4 þ nCCR5 bCCR5 ; bmi ¼ nCXCR4 m m i i

ð6Þ

bni ¼ nCXCR4 bCXCR4 : n i

ð7Þ

and nCCR5 denote the abundance of Here nCXCR4 m m CXCR4 and CCR5 on memory CD4+ T cells, denotes the abundance respectively, and nCXCR4 n of CXCR4 on naive CD4+ T cells. We assume that there is a continuous spectrum of coreceptor preference ranging from viruses that exclusively use CCR5, to HIV variants which exclusively use CXCR4. Thus in our model, an HIV variant is given by its affinities for the coreceptors CCR5 and bCXCR4 ; respectively. and CXCR4, bCCR5 i i For the sake of simplicity, we assume approximately equal levels of CCR5 and CXCR4 on the surface of memory CD4+ T cells and equal levels of CXCR4 on naive and memory CD4+ T cells. Our conclusions, however, do not change qualitatively if we assume different levels of CCR5 and CXCR4 on naive and memory CD4+ T cells. Moreover, we restrict our analysis to CD4+ T cells because these cells are the cell type predominantly infected by HIV (Zhang et al., 1999), and because we still lack evidence for a substantial contribution of other cell types, such as macrophages and dendritic cells, to viral replication. Furthermore, we exclude the possibility that cells are coinfected. We also assume that the HIV variants which appear at any stage during the infection are present throughout the infection albeit at very low concentrations. This

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Table 1 Parameters of model 1–5 Parameter

Meaning

Value

ln lm

Production and proliferation rate of naive CD4+ T cells Activation rate of CD4+ T cells and proliferation rate of memory CD4+ T cells Clearance rate of naive CD4+ T cells due to apoptosis Clearance rate of activated memory CD4+ T cells due to apoptosis Clearance rate of infected naive CD4+ T cells due to activation, apoptosis, immune-mediated lysis, and viral cytopathicity Clearance rate of infected activated memory CD4+ T cells due to apoptosis, immune-mediated lysis, and viral cytopathicity Clearance rate of free virus Reproduction rate of HIV in naive CD4+ T cells (comprises activation rate) Reproduction rate of HIV in activated memory CD4+ T cells Rate of infection of naive CD4+ T cells by HIV variant i Rate of infection of activated memory CD4+ T cells by HIV variant i

E15=ðml  dayÞ E40=ðml  dayÞ

dn dm an am u cn cm bni bmi

production, proliferation

0.015 day1 (corr. half-life of 46 days) 0.2 day1 (corr. half-life of 3.5 days) 0.1 day1 (corr. half-life of 7 days) 0.5 day1 (corr. half-life of 1.4 days) 5.0 day1 (corr. half-life of 3.5 h) E200 day1 E500 day1 Eð104  103 Þml day1 Eð104  103 Þml day1

activation, proliferation

CXCR4 apoptosis

resting/naive

activated/ memory

CXCR4

CD4 T cells CCR5

infection

clearance

CD4 T cells

infection

infected

infected

CD4 T cells

CD4 T cells

activation & virus production

apoptosis

clearance

HIV virus production

clearance

Fig. 1. Diagrammatic representation of the model in eqns (1–5). The arrows correspond to processes which we incorporated into the model. Multiple layers of circles for HIV and infected cells indicate that there are many variants that can infect the target cells, giving rise to different classes of infected cells.

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ANALYTICAL RESULTS

The equilibria of the above model can be solved analytically. The dynamics allow at most two HIV variants coexisting at equilibrium (see Appendix A). Assume that variant r; characterized by infectivities bnr and bmr ; is present at equilibrium. If this equilibrium is ‘‘challenged’’ with a rare variant i; one of the following three outcomes can be observed: (i) the challenging variant i invades and outcompetes the resident variant r; (ii) the challenging variant i invades and coexist with the resident variant r in a stable equilibrium; (iii) the challenging variant i cannot invade and goes extinct. Which of the outcomes is observed depends on and bCXCR4 of the challenging the affinities bCCR5 i i variant for CCR5 and CXCR4. By considering mutual invasibility, we can derive the conditions under which a particular outcome will occur (see Appendix B). Figure 2 shows for which affinities and bCXCR4 of the challenging variant a bCCR5 i i particular outcome occurs, if the resident variant preferentially uses CCR5.

0.0004

0.0003 CXCR4 affinity

view is supported by case reports that documented transmission and infection by SI viruses (which use mainly CXCR4) but reversion to NSI phenotype (which use mostly CCR5) early after infection (Lathey et al., 1997; Cornelissen et al., 1995). Consequently, it is not mutation which causes the emergence of a certain HIV variant, but changes in the selection pressure on the HIV quasispecies. Lastly, we assume that the selection pressure exerted by the immune system changes only slowly compared with the changes in target cell availability due to antiretroviral treatment. This last assumption enables us to study the effect of coreceptor-specific drugs on the HIV quasispecies without considering the immune response to the infection by the virus explicitly. [For models addressing the potential impact of the immune response on the evolution of HIV tropism see Wodarz & Krakauer (2000), Callaway et al. (1999), and Wodarz et al. (1998).]

challenging strain outcompetes resident strain

0.0002 coexistence

0.0001 resident strain (R5) challenging strain cannot invade

0

0

0.0002

0.0004

0.0006

0.0008

CCR5 affinity

Fig. 2. Whether a variant i can coexist or even outcompete a resident R5 variant depends on its affinity for CCR5 and CXCR4. (Parameters as in Table 1.)

CONDITIONS FOR A CORECEPTOR SWITCH

Let us first consider a switch from a preference for CCR5 to a preference for CXCR4. Suppose that a CCR5-using HIV variant is present at equilibrium which is characterized by the corand bCXCR4 ; whereby the eceptor affinities bCCR5 r r subscript r denotes the resident variant at bbCXCR4 : There are two equilibrium and bCCR5 r r CXCR4 affinity thresholds which a challenging X4 variant has to overcome in order to take over the system. Firstly, in order to coexist with the resident R5 variant, the affinity of the X4 variant has to exceed bCXCR4 coexist E

g ; 1þa

ð8Þ

where a ¼ lm cm an bnr =ðln cn am bmr ) and g ¼ bnr ðlm cm an =ðln cn am Þ þ 1Þ: Secondly, if the affinity of the challenging X4 variant exceeds bCXCR4 takeover E

Zþx z

ð9Þ

whereby x ¼ ln cn bnr =an ; z ¼ ln cn =an þ lm cm =am and Z ¼ lm cm bmr =am ; it will take over the system. and The CXCR4-affinity thresholds, bCXCR4 coexist CXCR4 btakeover ; are illustrated in Fig. 3(a). For a derivation of these thresholds see Appendix C. The main point here is that if an X4 variant has

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R. R. REGOES AND S. BONHOEFFER

resident strain (X4)

CXCR4 affinity

βCXCR4 takeover

CXCR4

β sat CXCR4

β coexist

resident strain (R5)

CCR5

CCR5 affinity

β coexist

(b)

(a)

Fig. 3. Illustration of the affinity thresholds for (a) an R5 resident variant and (b) an X4 resident variant. Note that the two scenarios are not symmetrical. (Parameters as in Table 1.)

an affinity for CXCR4 which exceeds bCXCR4 takeover it will outcompete the resident R5 variant. Let us now consider the conditions for the reverse switch, i.e. from a preference for CXCR4 to a preference for CCR5. Thus, we suppose that the resident variant is an X4 virus which is characterized by the coreceptor affinities bCCR5 r and bCXCR4 ; whereby bCCR5 5bCXCR4 : In the r r r present case there is also an affinity threshold which the R5 variant has to exceed in order to coexist with the resident X4 variant given by bCCR5 coexist E

g a

ð10Þ

whereby a ¼ lm cm an bnr =ðln cn am bmr ) and g ¼ bnr ðlm cm an =ðln cn am Þ þ 1Þ as above. This affinity threshold is derived in Appendix C and illustrated in Fig. 3(b). However, unlike in the case where the resident variant was an R5 virus, there is no takeover threshold in the affinity for CCR5 which could correspond to bCXCR4 takeover : Instead, the upper bound of the set of affinities for which a variant coexists with the X4 resident variant Ex=z saturates at the CXCR4 affinity bCXCR4 sat whereby x ¼ ln cn bnr =an and z ¼ ln cn =an þ lm cm =am as above. Thus, even with very high CCR5-affinity an R5 variant cannot outcompete the resident X4 variant. A resident X4 variant can only be outcompeted by a variant which has sufficiently high affinities for both coreceptors, i.e. it has to be of the R5X4 phenotype. The conclusions with regard to a switch from X4 to R5 are in opposition to findings in some patients who were infected with X4 virus but

switched to R5 during primary infection (Lathey et al., 1997; Cornelissen et al., 1995). This discrepancy suggests that factors which are not taken into consideration in our model, such as the effect of HIV-specific immune responses, are responsible for the natural switch from X4 to R5 virus in primary infection. The analysis in this section reveals two important points. Firstly, the widely used term ‘‘switch’’ is misleading because it suggests that the virus population in patients consists either of exclusively R5 variants or of exclusively X4 variants, and thus neglects that R5 and X4 variants can coexist. It is more appropriate to define the following three, mutually exclusive states of the viral quasispecies: (i) a state in which the virus population is dominated by R5 virus variants, (ii) a state in which R5 and X4 variants coexist, and (iii) a state in which X4 dominates the virus population (see Fig. 5). Our analysis above determines the conditions under which transitions between these three states take place. Secondly, in our model the dynamics of a switch from the dominance of R5 variants to the dominance of X4 variants is not symmetrical to the reverse switch. The transition from an R5dominated virus population to a population in which R5 and X4 variants coexist is irreversible, i.e. once an X4 variant has emerged it will not be suppressed again. The asymmetry of the dynamics of coreceptor switching is based on the asymmetry of the coreceptor expression profile on resting/naive and activated/memory CD4+ T cells (see Fig. 1), and has important implications

449

HIV CORECEPTOR USAGE

when we compare the effect of CCR5 and CXCR4 inhibitors in the next section.

we re-calculate the coexistence set and the CXCR4 CXCR4 CXCR4-affinity thresholds b% coexist and b% takeover ; where the bar over the symbols indicates that these CXCR4-affinity thresholds are calculated under treatment. We find that under treatment both CXCR4-affinity thresholds are reduced considerably. In Appendix C, we show that

The Effect of Coreceptor Inhibitors Having solved Model 1–5 enables us to consider the effects of antiretroviral agents on the evolution of the HIV quasispecies. We can incorporate the effect of a coreceptor inhibitor into Model 1–5 by reducing the affinity of the virus variants for the respective coreceptor. The main question we intend to investigate is whether under a regimen of coreceptor inhibition some HIV variants are able to coexist with or even to outcompete the wild-type variant although they were suppressed before treatment. Let us first consider the effect of a CCR5 inhibitor on a patient who harbors R5 virus. We approach our question by considering how the and two CXCR4-affinity thresholds bCXCR4 coexist ; change as a result of inhibiting the bCXCR4 takeover binding of HIV to CCR5. Assume that the binding of HIV to CCR5 is inhibited by a factor 1  s; where 0oso1: This inhibition equally reduces the CCR5-affinities of the resident R5 variant and of all other variants which the patient harbors, but does not change their CXCR4-affinities. With the reduced CCR5affinity of the resident variant under treatment,

CXCR4 b% coexist E sbCXCR4 coexist ;

14s4s;

CXCR4 b% takeover E sbCXCR4 takeover :

ð11Þ

ð12Þ

The factor of reduction in the coexistence threshold bCXCR4 coexist is smaller than that of the takeover threshold bCXCR4 takeover : Figure 4(a) illustrates how the coexistence sets change as a result of CCR5 inhibition. The conclusion from this analysis is that a large majority of X4 variants which prior to treatment could coexist with the resident R5 variant, are able to outcompete the resident R5 variant as a result of CCR5 inhibition. We also find some X4 variants which are able to coexist with the resident R5 variant as a result of CCR5 inhibition although they were suppressed before treatment. Coexistence between the resident R5 variant and X4 variants, however, is less likely under treatment as can be seen by comparing the size of the coexistence sets in Fig. 4(a). In short,

CXCR4 relative CXCR4 affinity β i / βr

CXCR4

7 6

resident strain (X4)

1

5 4 3

0.5

2 resident strain (R5)

1 0

0

0.5

1

1.5

0

0

5

relative CCR5 affinity β i

CCR5

(a)

10

15

20

/ β CCR5 r

(b)

Fig. 4. Coexistence sets before (black) and during treatment (gray) with a coreceptor inhibitor that reduces the affinity for the respective coreceptor by 75%: Plot (a) shows the effect of an CCR5 inhibitor. Plot (b) shows the effect of an CXCR4 inhibitor. The dashed line represents the lower bound of the coexistence set without treatment which is partly covered by the coexistence set during treatment (gray). To be able to compare the coexistence sets we plotted the relative affinities, bCCR5 =bCCR5 and bCXCR4 =bCXCR4 : (Parameters as in Table 1.) i r i r

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CCR5 inhibitors select against coexistence between R5 and X4 variants: if an X4 variant coexisted with an R5 variant before treatment, CCR5 inhibition will lead to the suppression of the R5 variant, and the X4 variant will take over; if the only variant present before treatment was using CCR5, however, a CCR5 inhibitor is unlikely to lead to the emergence of X4 virus. Thus, CCR5 inhibitors are very likely to lead to the suppression of the resident R5 variants if an X4 variant coexisted with the resident R5 variant before treatment. Let us now consider the effect of an CXCR4 inhibitor on a patient who harbors X4 virus. Analogously to our approach above, we consider how the affinity threshold bCCR5 coexist changes as a result of CXCR4 inhibition [see Fig. 4(b)]. The main difference to the inhibition of CCR5 is that in the present case there is no takeover threshold. Inhibition of CXCR4 with efficacy 1  s; 0oso1 leads to a reduction in the coexistence threshold bCCR5 coexist by approximately a factor s (see Appendix C): CCR5 b% coexist E sbCCR5 coexist :

ð13Þ

Here, b% coexist denotes the coexistence threshold during treatment. Moreover, the relative level at which the upper bound of the coexistence set =bCXCR4 ; remains approxisaturates, bCXCR4 sat r mately constant. Thus, a CXCR4 inhibitor increases considerably the probability of the emergence of R5 variants which subsequently coexist with the resident X4 variant. However, regardless of how effective the inhibition of CXCR4 is, an R5 variant will never outcompete the resident virus variant. The main conclusion of our analysis is that, CXCR4 inhibition increases the selection pressure in favor of the coexistence between R5 and X4 variants, whereas CCR5 inhibitors select against coexistence between R5 and X4 virus. Figure 5 summarizes these conclusions. CCR5

Discussion In this paper, we investigate a simple model that describes the competition of different HIV variants for its primary target cells, the CD4+ T cells. In particular, we study the effect of

R5 dominant X4 suppressed CXCR4 inhibition has no effect

CCR5 inhibition

R5 and X4 coexist

CXCR4 inhibition

CCR5 inhibition

X4 dominant R5 suppressed Fig. 5. Diagrammatic representation of our results. The viral quasispecies within an HIV-infected patient can be classified into one of three categories: (i) a state in which R5 virus dominates and X4 variants are suppressed, (ii) a state of coexistence between R5 and X4 virus variants, and (iii) a state in which X4 virus dominates the quasispecies. The arrows and their thickness illustrate how and to what extent transitions between these states are affected by coreceptor inhibitors. If R5 and X4 variants coexist prior to treatment then CCR5 inhibitors lead to the suppression of R5 variants and therefore select in favor of the dominance of X4 variants. However, if R5 variants dominate prior to treatment CCR5 inhibitors only marginally facilitate the emergence of X4 variants. CXCR4 inhibitors, on the other hand, select in favor of the emergence of R5 variants, but will not induce the suppression of X4 variants.

antiretroviral treatment on the coreceptor preference of the HIV quasispecies. In our model, we consider only CD4+ T cells and disregard other potential target cells, such as dendritic cells and macrophages. The rationale of this assumption is that CD4+ T cells are the predominant cell type which are found to be infected in vivo (Zhang et al., 1999). Moreover, we assume that the HIV variants which appear at any stage during the infection are present throughout the infection, albeit at very low concentrations. Consequently, it is not mutation which causes the emergence of a certain HIV variant, but changes in the selection pressure on the HIV quasispecies. Finally, we do not consider the

HIV CORECEPTOR USAGE

influence of immune responses on HIV coreceptor usage. This last simplification renders our model inappropriate to study the natural course of HIV coreceptor usage which may be due to changes in the immune responses. To explain changes of coreceptor usage during primary infection, or to understand the switch HIV coreceptor preference during the asymptomatic phase, the detailed interaction network of the target and immune cells has to be taken into consideration. Some of these aspects have already been studied elsewhere (Wodarz & Krakauer, 2000; Callaway et al., 1999; Wodarz & Nowak, 1998). Nevertheless, the target-celloriented approach we adopted here is well-suited to investigate the impact of antiretroviral treatment on HIV coreceptor usage since treatment represents a fast change in the susceptibility of the target cell population to viral infection, during which the immune responses are unlikely to change drastically. Our target-cell-based model is used to investigate the effect of coreceptor inhibitors on HIV coreceptor usage. We determine how coreceptor inhibitors change the selection pressures on R5 and X4 virus and whether this change of selection pressures leads to the emergence or the suppression of either R5 or X4 variants. We find that a CCR5 inhibitor is likely to lead to the suppression of R5 variants if an X4 variant coexisted with the R5 variant before treatment. In contrast, a CXCR4 inhibitor may lead to the emergence of R5 variants which will coexist with X4 variants, but will not suppress X4 variants which are already present. Roughly speaking, CCR5 inhibitors select for a switch to X4 variants by selecting against coexistence of R5 and X4 variants, whereas CXCR4 inhibitors select for coexistence of R5 and X4 virus, but do not lead to the suppression of X4 variants (see Fig. 5). This asymmetry in the effects of CCR5 and CXCR4 inhibitors is based on the asymmetry of coreceptor expression profiles on resting/ naive and activated/memory CD4+ T cells. With regard to treatment with CCR5 inhibitors, these results are encouraging since they indicate that the main effect of CCR5 inhibitors is the suppression of R5 variants and not the induction of the emergence of X4 variants. However, since our model also indicates that,

451

to a limited degree, CCR5 inhibitors can lead to the emergence of X4 virus, caution is advised. In a more quantitative study, one should try to estimate the probability of inducing the emergence of X4 variants by applying CCR5 inhibitors. This could help to minimize the risk of selecting for the emergence of X4 variants by CCR5 inhibition. Concerning treatment with CXCR4 inhibitors, however, our results are discouraging since they suggest that the suppression of X4 variants by CXCR4 inhibition is highly unlikely. Thus, the potential benefit of CXCR4 inhibitors, i.e. the reversion of the HIV quasispecies from a preference for CXCR4 to a preference for CCR5, may not be observed in vivo. Our findings are experimentally testable in competition experiments between R5 and X4 variants in vitro, if the experimental setup reflects that the target cell population is composed of naive/resting and activated/memory CD4+ T cells. In such an experiment, our analysis predicts the suppression of R5 variants by CCR5 inhibitors, and the emergence of R5 variants under a CXCR4 inhibitor regimen. Unfortunately, the few experimental studies on HIV phenotype switches which have been conducted so far (Trkola et al., 2002; Maeda et al., 2000; Este et al., 1999; Schols et al., 1998) are either not designed as competition experiments between R5 and X4 virus variants, or do not use target cell populations which consist of naive/ resting and activated/memory CD4+ T cells. In four studies (Trkola et al., 2002; Maeda et al., 2000; Mosier et al., 1999; Schols et al., 1998), the virus stock used consist entirely of either R5 or X4 virus. Thus, these studies do not investigate the competition between, but the evolution of R5 or X4 variants under the selection pressure exerted by a coreceptor inhibitor. The pattern that emerges from three of these studies (Trkola et al., 2002; Maeda et al., 2000; Schols et al., 1998), is that the virus develops resistance to the coreceptor inhibitor without changing its coreceptor preference. Interestingly, in the study by Mosier et al. (1999) which was conducted in SCID mice, CCR5 inhibition lead to a coreceptor switch to CXCR4. The virus population which patients harbors, however, is usually heterogeneous and are believed to contain X4

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and R5 variants throughout the infection. To account for the heterogeneity of the virus population in vivo, these experiments would need to be repeated with mixtures of R5 and X4 virus, or with clinical virus isolates. The study by Este et al. (1999), on the other hand, is designed as a competition experiment between R5 and X4 virus variants, but the target cell population consists of activated CD4+ T cells only, which restores the symmetry between coreceptor expression pattern. Thus, the main result of the study by Este et al. (1999)Fthe shift from X4 to R5 virusFis based on an experimental setup which is not only incompatible with our model assumptions, but presumably also does not reflect the in vivo situation. The general conclusion from our analysis is that the complex composition of the target cell population is essential when assessing the risks or benefits of coreceptor inhibitors and other antiretroviral agents which have a coreceptorspecific effect. The asymmetrical coreceptor expression profiles of resting/naive and activated/memory CD4+ T cells turn out to be of particular importance since they inherit their asymmetry to the effect of CCR5 and CXCR4 inhibitors. Thus, this aspect should be taken into account when designing experiments which aim to determine the effect of coreceptor inhibitors on the HIV quasispecies. We thank Angelique van’t Wout and Alexandra Trkola for advice regarding HIV tropism, Vincent Jansen and four anonymous reviewers for commenting on the manuscript, and Georg Funk and Rustom Antia for fruitful discussions. This work was supported by the Boehringer Ingelheim Fonds (R.R.R.) and the Novartis Research Foundation (R.R.R.,S.B.).

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Appendix A Equilibrium Solutions of Model 1–5 Here, we derive all equilibrium solutions of the basic model for HIV coreceptor usage 1–5 and give criteria for their stability. At equilibrium the growth rate of the i-th variant is given by lm bmi cm ln bni cn þ  u; ðA:1Þ an ðdn þ on Þ am ðdm þ om Þ P P where on ¼ j bnj v#j and om ¼ j bmj v#j and v#j denote the equilibrium abundances of the j-th variant. The growth rate of the resident variant vanishes at equilibrium. If the equilibrium is stable, the growth rates of all other variants are negative. In many other systems, the basic reproductive rate (Dietz, 1975, 1976; Anderson & May, 1981, 1991; May & Anderson, 1983; Bremermann & Thieme, 1989; Diekmann et al., 1990) is taken as a measure of fitness of a particular variant. This is justified if the target cell population is homogeneous, i.e. if there is only one type of target cell. In systems with a heterogeneous host population, the basic reproductive rate represents an appropriate measure of parasite only in exceptional cases (Regoes et al., 2000). Unfortunately, in this system, the basic reproductive rate, gi ðon ; om Þ ¼

Ri ¼

ln bni cn lm bmi cm þ dn an u dm am u

ðA:2Þ

is not an appropriate measure of the fitness of the i-th variant at equilibrium. In our model 1–5, there cannot be more than two variants present in an equilibrium. Assume, we have three variants in a stable equilibrium, say variants k; l and m: For these three variants the growth rates vanish at equilibrium: gk ðon ; om Þ ¼ 0;

ðA:3Þ

gl ðon ; om Þ ¼ 0;

ðA:4Þ

gm ðon ; om Þ ¼ 0:

ðA:5Þ

Equations (A.3) and (A.4) for variants k and l give rise to a pair of solutions for on and om : The remaining eqn (A.5) leads to relations between the parameters of our system, which are too limiting to be fulfilled in a realistic caseF a ‘‘conspiracy of parameters’’. Essentially, the three eqns (A.3–A.5) over-determine our system, which has only two dynamically independent variables, on and om : Thus, at most two variants can be present in an equilibrium of the basic model for HIV tropism 1–5. ONE-VARIANT EQUILIBRIA

If only one variant is present in an equilibrium, say variant k; then we have gk ðon ; om Þ ¼   0; with on ¼ bnk v k and om ¼ bmk vk ; where vk denotes the abundance of the resident variant. Solving for v k yields: dn dm ln cn lm cm  þ þ 2bnk 2bmk 2uan 2uam #2 (" dn dm ln cn lm cm þ   þ þ 2bnk 2bmk 2uan 2uam )1=2 dn dm ln cn dm lm cm dn  þ þ : bnk bmk uan bmk uam bnk

v k ¼

ðA:6Þ

If this equilibrium is stable, the growth rate of all other variants is negative.

TWO-VARIANT EQUILIBRIA

If two variants are present in an equilibrium, say variant k and l; their growth rates vanish, gk ðon ; om Þ ¼ 0 and gl ðon ; om Þ ¼ 0: In the case of a two-variant equilibrium, we have on ¼ bnk vnk þ bnl vnl and om ¼ bmk vnk þ bml vnl ; where vnk and vnl denote the equilibrium abundance of the resident variants. This can be solved for the equilibrium abundances of the coexisting variants k and l: vnk ¼

bnl om  bml on ; bnl bmk  bnk bml

ðA:7Þ

vnl ¼

bnk om  bmk on ; bnk bml  bnl bmk

ðA:8Þ

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where ln cn ðbnk bml  bnl bmk Þ  dn ; on ¼ an uðbml  bmk Þ

ðA:9Þ

lm cm ðbnl bmk  bnk bml Þ  dm : am uðbnl  bnk Þ

ðA:10Þ

om ¼

dn dm ln cn dm bnj lm cm dn bmj þ þ  bni bmi uan bmi bni uam bni bmi

:

ðB:4Þ

The challenging variant i will outcompete the resident variant r; if

Again, if this particular two-variant equilibrium is stable, the growth rates of all other variants are negative.

vj ¼0 ð jarÞ 4gr jvr ¼v r ; vj ¼0 ð jarÞ

ðB:5Þ

gr jvi ¼vi ;

vj ¼0 ð jaiÞ ogi jvi ¼v i ; vj ¼0 ð jaiÞ

ðB:6Þ

which is equivalent to  v r oinvr’i and vi 4invi’r :

Stability of Equilibria Invasion criteria enable us to determine which equilibrium is stable, i.e. which variant or variants will persist for a given set of parameters. Assume variant r resides in the system with the equilibrium abundance v r : The system cannot be invaded by any variant i; if the growth rate of the i-th variant is less than the growth rate of the resident variant r in this equilibrium, i.e. if for all iar: gi jvr ¼vr ; vj ¼0 ðjarÞ ogr jvr ¼vr ; vj ¼0 ðjarÞ ¼ 0: In this case, the equilibrium is stable, and the resident variant r persists. If, however, there is a variant i; which can invade then either this variant outcompetes variant r or both variants coexist. For coexistence, mutual invasibility is required: vj ¼0 ð jarÞ 4gr jvr ¼v r ; vj ¼0 ð jarÞ

gi jvr ¼vr ; and

Appendix B

gi jvr ¼vr ;

)1=2

Equations (B.3) and (B.7) allow us to determine whether a variant can coexist or even outcompete the resident variant. The outcome depends on the infectivities of the challenging variant bni and bmi; as well as the infectivities of the resident variant, bnr and bmr: In Fig. 2 we determined whether a variant can coexist or outcompete an R5 resident variant in depenand dence of the coreceptor affinities bCCR5 i of the challenging variant. bCXCR4 i Condition (B.3) can be written in terms of the parameters only, and enables us to derive approximate expressions for the upper and lower bounds of the set of infectivities for which resident and challenging variant coexist. Lower and upper bounds are implicitly given by

ðB:1Þ

and

ðB:7Þ

v r ¼ invr’i

ðB:8Þ

v i ¼ invi’r

ðB:9Þ

and gr jvi ¼vi ;

: vj ¼0 ð jaiÞ 4gi jvi ¼v i ; vj ¼0 ð jaiÞ

ðB:2Þ

Mutual invasibility is equivalent to the conditions:  v r oinvr’i and vi oinvi’r ;

ðB:3Þ

where the invasion threshold is defined as dn dm ln cn bnj lm cm bmj invi’j ¼   þ þ 2bni 2bmi 2uan bni 2uam bmi 8" #2 < dn dm ln cn bnj lm cm bmj þ   þ þ : 2bni 2bmi 2uan bni 2uam bmi

respectively. If ln cn =ðan uÞbdn =bni and lm cm =ðam uÞbdm =bmi Fwhich is fulfilled for those parameters that are likely to characterize HIV infection (see Table 1)Fwe can approximate eqns (B.8) and (B.9). The lower bound is given by a linear relation between bni and bmi : bni ¼ abmi þ g;

ðB:10Þ

where a ¼ lm cm an bnr =ðln cn am bmr ) and g ¼ bnr ðlm cm an =ðln cn am Þ þ 1Þ: This is equivalent to the following relation between bCXCR4 i

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and bCCR5 : i bCXCR4 ¼ i

a CCR5 g b : þ 1þa i 1þa

ðB:11Þ

The upper bound can be approximated by a hyperbolic relation between the infectivities: bni ¼

xbmi ; zbmi  Z

ðB:12Þ

where x ¼ ln cn bnr =an ; z ¼ ln cn =an þ lm cm =am and Z ¼ lm cm bmr =am : This is equivalent to the followand bCCR5 : ing relation between bCXCR4 i i bCCR5 ¼ i

xbCXCR4 i zbCXCR4  i

Z

:  bCXCR4 i

ðB:13Þ

as a function (The inverse relation giving bCXCR4 i is less straightforward.) of bCCR5 i The intersections of the boundaries of the ¼ 0 and coexistence set with the axes bCCR5 i ¼ 0; give rise to affinity thresholds. bCXCR4 i Appendix C Affinity Thresholds The set of affinities for which a variant can coexist with an R5 resident variant can be characterized by two CXCR4-affinity thresholds which are given by the intersection of the lower and the upper bound of the coexistence set with the CXCR4-affinity axis [see Fig. 3(a)]. From eqns (B.11) and (B.13) it follows that: bCXCR4 coexist E

g ; 1þa

Zþx bCXCR4 : takeover E z

ðC:1Þ

the main text. For our parameters (see Table 1) ¼ and a R5 resident variant given by bCXCR4 r CCR5 4 4 ¼ 5  10 ; the thresholds 0:5  10 and br CXCR4 and are approximately bCXCR4 coexist E 2:1  br CXCR4 CXCR4 : btakeover E 8:2  br Analogously, we can determine a coexistence threshold for a system in which the resident variant is of the X4 phenotype. From eqn (B.11) it follows that: g bCCR5 coexist E : a

Since the upper bound of the coexistence set does not intersect with the CCR5-affinity axis, there is no analogon to bCXCR4 takeover : Instead of intersecting with the CCR5-affinity axis, the upper bound of the coexistence set saturates at x E : bCXCR4 sat z

These CXCR4-affinity thresholds are illustrated in Fig. 3(a), and their interpretation is given in

ðC:4Þ

CXCR4 Figure 3(b) illustrates bCCR5 : coexist and bsat It is straightforward to derive that under treatment with a CCR5-inhibitor with efficacy 1  s; 0oso1; the CXCR4-affinity thresholds change approximately to CXCR4 bCXCR4 coexist - sbcoexist ;

s :¼

sð1 þ aÞ 4s; sþa

CXCR4 bCXCR4 takeover - sbtakeover :

ðC:5Þ ðC:6Þ

As a result of CXCR4 inhibition with efficacy 1  s; 0oso1; the CCR5-affinity threshold bCCR5 coexist becomes: CCR5 bCCR5 coexist - sbcoexist ;

ðC:2Þ

ðC:3Þ

ðC:7Þ

whereas the relative level of the CXCR4-affinity at which the upper bound of the coexistence set =bCXCR4 remains approximately saturates, bCXCR4 sat r constant.