BEH.420 Fall 2001 Final Examinationweb.mit.edu/beh.420/www/FinalExam.pdfBEH.420 Fall 2001 Final...

BEH.420

Fall 2001

Final Examination

Answer both problems, which are worth equal credit. There is no limit to the amount of time that can be spent on the exam, but it must be returned to the Room 6–135 by noon on Monday, December 10, 2001 (if no one is in, leave the exam in the mail slot in the door). Please staple your answer to Problem 1 separately from your answer to Problem 2 and put your name on both. This is an open-book exam with free access to books and notes. Students are expected to work alone and may not collaborate with each other or receive help from anyone else.

Problem 1. Increasing Ligand Lifetime Purpose: Ligands in the bloodstream can be removed from circulation by many mechanisms, thereby decreasing their potency. The purpose of this question is to investigate one such mechanism, namely endocytosis by epithelial cells lining blood vessels. Depending on the binding and trafficking kinetics, substantial ligand degradation may occur through lysosomal processing. Proper implementation of this model will allow predictions of ligand degradation rate as a function of binding in the low pH endosome. Model assumptions (see figure and parameter values on next page):

1. The rate of receptor synthesis at the surface and the rate of free receptor endocytosis are negligible.

2. Ligand at the cell surface is transported to the endosome through two mechanisms: fluid phase endocytosis of free ligand from the surface with a rate constant of kfp and surface RL complex endocytosis with a rate constant kec.

3. The unbound ligand in the endosome is degraded with a rate constant kdeg. The complex in the endosome is recycled back to the surface with a rate constant krec.

4. The total receptor concentration Rt is the same at the surface and in the endosome. 5. The values of the concentrations are on a per cell basis; thus, we can ignore any volume

balance between surface and endosome. Questions:

A. Write down the mass-action kinetics equations for the system and the ODE governing the total concentration of ligand present in the system (CLtot).

B. To calculate the rate of degradation of the ligand initially (for this part of the question only), assume that:

a. The pseudo steady state approximation for the free receptor concentration in the endosome and at the surface is valid

b. The pseudo steady state approximation for the endosomal complex is valid c. The ligand concentration is much higher than the total receptor concentration at

the surface (Cls>>Rt) d. The ligand concentration is much lower than the total receptor concentration in

the endosome (Cli<<Rt) Show arithmetically under these conditions that the rate of total ligand degradation is equal to:

lsCClsKdsKdi

kreckeck

+

deg

Explain in words and using idea from the cartoon the biological significance of each term.

C. Relaxing the assumptions from part B, compute with numerical simulation the time for the total amount of ligand (CLtot, including free ligand and the RL complexes in the endosome and at the surface) to be depleted to 10% of the initial value as a function of Kdi for 10-3 µM ≤Kdi ≤ 103 µM. Plot t90% as a function of log(Kdi). Explain the behavior of the plot at high and low Kdi’s. Remake the plot for the same range of Kdi, but this time vary Kds also so that (Kds)/(Kdi) = 10. Explain the biological significance of your result.

Increasing Ligand Lifetime

Nomenclature: Variables Descriptions Cls Ligand concentration on the surface Ccs Complex concentration on the surface Crs Receptor concentration on the surface Cli Ligand concentration in the endosome Cci Complex concentration in the endosome Cri Receptor concentration in the endosome CLtot Total concentration of ligand in the system = Cls + Ccs + Cli + Cci Parameters Values Units Descriptions CLtot0 100 µM Total ligand concentration at t=0 kec 30 day-1 Rate of surface complex endocytosis Kds 1 µM Equilibrium constant on the surface Kdi 0.1 µM Equilibrium constant in the endosome kdeg 100 day-1 Degradation rate constant in the endosome krec 24 day-1 Recycled rate constant in the endosome Rt 20 µM Concentration of total receptors in a cell assumed to be

the same both at the surface and in the endosome. kfs 2 104 µM-1 day-1 Forward binding rate constant on the surface kfi 2 104 µM-1 day-1 Forward binding rate constant in the endosome kfp 50 Day-1 Fluid phase free ligand endocytosis rate constant * The concentrations are per cell basis so the concentrations are additive.

Problem 2. Alternative Model Formulations There are many different ways to mathematically describe a given biological process. Consider the HIV replicative life cycle; two different groups (those of Perelson & Nowak) have examined HIV models in depth, but with somewhat different approaches. Consider the following two papers, which are attached: Kepler & Perelson, PNAS 95:11514, ’98; Bonhoeffer et al., PNAS 94:6971, ’97. The basic HIV infectious life cycle is described in equations 1-3 of the former paper, and equation set [1] of the latter paper. a) Compare and contrast the different basic model formulations (i.e. eqns 1-3 vs. equation set [1]). What terms are the same, and which differ? What biological justification is there for eliminating or including each given term? b) Compare how the two groups mathematically describe potential mechanisms for emergence of drug resistance. Which mechanism do you believe is better supported by the experimental evidence? What experiments might be performed to discriminate between the alternative models?

Proc. Natl. Acad. Sci. USAVol. 95, pp. 11514–11519, September 1998Applied Mathematics, Medical Sciences

Drug concentration heterogeneity facilitates the evolutionof drug resistance

(sanctuary sites / HIV)

Thomas B. Kepler* and Alan S. Perelson†

*Biomathematics Program, Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203; and †Theoretical Division,Los Alamos National Laboratory, Los Alamos, NM 87545

Edited by Burton H. Singer, Princeton University, Princeton, NJ, and approved July 9, 1998 (received for review January 15, 1998)

ABSTRACT Pathogenic microorganisms use Darwinianprocesses to circumvent attempts at their control throughchemotherapy. In the case of HIV-1 infection, in which drugresistance is a continuing problem, we show that in one-compartment systems, there is a relatively narrow windowof drug concentrations that allows evolution of resistantvariants. When the system is enlarged to two spatially dis-tinct compartments held at different drug concentrationswith transport of virus between them, the range of averagedrug concentrations that allow evolution of resistance is sig-nificantly increased. For high average drug concentrations,resistance is very unlikely to arise without spatial heterogene-ity. We argue that a quantitative understanding of the roleplayed by heterogeneity in drug levels and pathogen trans-port is crucial for attempts to control re-emergent infectiousdisease.

A tremendous range of pest organisms and parasites includingbacteria, protozoans, fungi, macroparasites, insects, and weedshave used Darwinian processes to evade chemical control. Theresurgence of infectious disease that has become a focal pointof global health efforts is due in large part to the evolutionof resistance to antibiotic drugs. Here, we concentrate ourattention on the important example of drug resistance in HIVtype 1 (HIV-1) infection.

Some progress in the quantitative understanding of the evo-lution of drug resistance by HIV-1 has been made by using rel-atively simple mathematical models, which consider the bodyas a single compartment (1–8). Using a protypic model, weshow that the range of drug concentrations to which resis-tance is predicted to arise is unrealistically narrow. For mu-tants to arise, a parental drug-sensitive virus must replicate atsome non-negligible rate. This occurs only at drug concentra-tions below some threshold level. On the other hand, the mu-tant must have a distinct advantage to overtake the parentalstrain. We will see that this occurs only for drug concentra-tions above some second threshold. The evolution of resis-tance takes place only between these two thresholds, which ina single compartment system is a narrow window of drug con-centrations wherein both production of resistant mutants fromtheir nonresistant precursors, and their subsequent selectioncan occur.

We suggest that, in nature, the window of opportunity forthe generation of resistance is widened by having one com-partment in which mutants are generated, such as a “sanctu-ary” or region of low drug concentration that HIV-1 can enter,and a second compartment in which the drug concentration ishigh enough to give mutants a selective advantage. For HIV-1,sanctuaries may be physiologically distinguished sites, such asthe brain (9, 10) or testes or cell populations susceptible to

The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked “advertisement” inaccordance with 18 U.S.C. §1734 solely to indicate this fact.

0027-8424/98/9511514-6$0.00/0PNAS is available online at http://www.pnas.org.

infection in which intracellular bioavailability of active drug ispoor. The process of generating mutants in one compartmentand selecting them in another may be repeated several timesin the step-wise accumulation of resistance mutations.

The notion that heterogeneity influences drug resistanceis not new. Examination of the role of spatial heterogene-ity in the spread of insecticide resistance (11, 12) has shownthat, under appropriate conditons, spatial transport can giverise to an enhanced rate of spread of the resistant alleles.In the case of antibiotic resistance to tuberculosis, the effectsof both temporal and spatial heterogeneity have been mod-eled (13, 14). Those models suggest that noncompliance toantibiotic regimes, but not spatial heterogeneity, is an impor-tant cause of treatment failure. These results cannot simplybe applied to the case of HIV-1 infection in which the host–pathogen interactions are different and are described by dif-ferent models.

Resistance to the protease inhibitor indinavir provides aninstructive example because measurable resistance is foundonly in virus that has acquired at least three amino acid sub-stitutions in HIV-1 protease (15). Granted that one- or eventwo-base substitutions might be found in the pretreatment vi-ral quasispecies (16, 17), these intermediate strains do nothave the ability to grow in the presence of drug at therapeu-tic concentrations. Therefore, for uniform concentrations ofdrug in the therapeutic range, these strains cannot producethe three-base mutants that finally do show resistance. But aswe show, sanctuaries make it possible.

Approach. Our model will assume a parental virus popu-lation, at equilibrium, from which mutants arise. We examinethe process by which a strain, j mutations from wild type, pro-duces a more resistant strain with m additional substitutions,where all m are required to produce decreased drug suscep-tibility. For illustrative purposes, we use the example of in-dinavir resistance and take j = 1 and m = 2. This choice isnot critical because the number of substitutions required willsimply scale the overall time to appearance of the resistantstrain. We use j = 1 because given the rapid replication rateof HIV-1, one-base change mutants almost certainly preexist(16, 17).

Here, we consider the simplest nontrivial case: two com-partments, differing only in size and drug concentration, withmovement of virus but not target cells between them. If thesanctuary is a subpopulation of target cells, then movement oftarget cells is not a concern. If the sanctuary is the central ner-vous system, then movement is possible but highly restricteddue to the blood–brain barrier. Assuming no movement oftarget cells, this model is sufficiently simple that we can ob-tain analytical results on the rate at which viable mutants takehold and show that the range of average drug concentrationsthat allow for the evolution of drug resistance is significantlywidened when sanctuaries exist.

This paper was submitted directly (Track II) to the Proceedings office.Abbreviation: HIV-1, HIV type 1.

11514

Applied Mathematics, Medical Sciences: Kepler and Perelson Proc. Natl. Acad. Sci. USA 95 (1998) 11515

MODELS AND RESULTS

The model we use has been adapted from a simple modelof HIV-1 dynamics (6, 18) but is quite general and applicableto a variety of other systems (cf. 19, 20).

Our aim is to compute the waiting time for the arrival ofthe first phenotypically resistant virus that survives and foundsthe resistant population. We do not model the growth of theresistant population once it has been established; by defini-tion, it will grow to take over. By focusing on the produc-tion and initial establishment of the resistant variant, we needonly consider the dynamics of the parental strain and the veryearly stages of the growth of the mutant population. At thesevery early times, the resistant population does not affect theparental population.

Dynamics of Virus Production. As in ref. 18, let T , T ∗and V designate the population densities of uninfected targetcells (i.e., cells susceptible to HIV-1 infection), productivelyinfected cells, and free virions, respectively. We assume thattarget cells are generated by density-dependent proliferationas suggested by recent studies (21, 22) and ignore the possi-ble input of cells from a thymic source. Then this system maybe described by the equations

dT

dt= rT

(1− T

T0

)− kVT; [1]

dT ∗

dt= kVT − δT ∗; and [2]

dV

dt= NδT ∗ − cV − kVT; [3]

where T0 is the equilibrium density of T cells in the absenceof virus, r is the maximum rate of T cell population growth, kis the viral infectivity, δ is the per cell rate of productively in-fected cell death, and c is the rate constant for virus clearance.Death of uninfected target cells is incorporated into the logis-tic growth term so that r presents the maximum proliferationrate minus the death rate.

Critical Infectivity. Eqs. 1–3 have the property that thereis a critical infectivity, kc , given by kc�N − 1� = c/T0, suchthat for k + kc , the only stable equilibrium is the noninfectedstate, V = T ∗ = 0, T = T0, where an overbar represents anequilibrium value. (The condition k + kc is equivalent to re-quiring that the basic reproductive number R0 be + 1 (4).) Fork , kc , the stable equilibrium is an infected state with a T celldensity less than T0, i.e., V = r�1− T /T0�/k, T = c/�N − 1�k,and T ∗ = kV T /δ. In this model, if therapy reduces k belowkc , the virus will be eradicated, whereas less effective therapywill simply establish a new (lower) steady–state viral load.

We can compute kc as a fraction of the infectivity k of theparental virus from measurement of the equilibrium T celllevel and knowledge of its virus-free equilibrium value. Thisgives§

kc =T

T0k: [4]

During the asymptomatic phase of infection, quasi-steady–states are established in which the CD4+ T cell count maytypically vary between 200 and 500 cells/mm3 (23) giving ra-tios T /T0 between 0.2 and 0.5 under the assumption that thenormal CD4 count T0 = 1;000/mm3 and that the T cell countmeasured in blood is reflective of the T cell levels in tissue.Thus, we would expect—if the assumptions going into the for-mulation of Eqs. 1–3, including that of spatial homogeneity,

§In some models (5, 6, 19, 20), Eq. 1 is replaced by dT/dt = λ−dT−kVT ,and T0 A λ/d is the equilibrium level of target cells in the absence ofvirus. This model, as well as more general ones in which dT/dt = f �T �−kVT , with f �T0� = 0 and f ′�T0� + 0, leads to an identical equationfor kc .

are reasonable—that cutting the infectivity k by 50% to 80%,using antiretroviral drug therapy, should be sufficient to com-pletely eliminate the virus. These predictions are not borneout by clinical practice. Monotherapy with zidovudine, whichshould have the required efficacy, does not eliminate the virus,even though resistance develops gradually by the stepwise ac-crual of mutations (24, 25). Similarly, monotherapy with themuch more potent protease inhibitors (26) and zidovudine–lamivudine combination therapy (27), cases in which preexist-ing high level drug resistance is unlikely, do not agree withthe prediction of viral elimination.

Thus, even before we consider the evolution of resistance,some assumptions of the model appear to need reconsider-ation. Several possibilities exist including the role of long-lived productively infected cells (28) and latently infected cells(29, 30, 31), but we will here focus on the assumption of spa-tial homogeneity.

Production and Selection of Mutants. The parental popu-lations will be considered at equilibrium. The rate at whichmutant viral strains are produced from the parental strain isthen �µkV T , i.e., the product of the specific mutation rate,µ, the rate of infection, kV T , and, since the populations aregiven as densities, a volume factor � to give absolute num-bers.

Here, we consider mutations that occur because of errorsin reverse transcription. Thus, after a virus infects a cell, theDNA copy of the viral genome that is made, the provirus, maycarry a drug resistant mutation. Not only must the resistantprovirus be produced, however, it must also “take root.” Resis-tant proviruses may be produced but fail to produce progenyfor purely stochastic reasons (32–34). The probability, p, thata provirus will propagate is related to the probability, q, thata free virion will propagate by

p = 1− �1− q�n; q = κp; �5a; b�where κ is the probability that a given resistant virion produc-tively infects a cell and n is the number of progeny producedin the lifetime of an infected cell. The term �1− q�n in Eq. 5ais the probability that all n of the progeny of the founderprovirus fail to propagate, so 1 − �1 − q�n is the probabilitythat at least one of the progeny propagates. Eq. 5b says thata virion propagates if and only if it infects (probability κ) andthe provirus propagates (probability p).¶

If n is considered random rather than fixed, we take theexpectation of Eq. 5a over n. For n, a Poisson random variablewith mean N and using Eq. 5b, this becomes

p = 1− exp�−Nκp�: [6]

Alternatively, if n is fixed at N , with N large, then Eq. 6 is anexcellent approximation to Eq. 5.

The infection probability κ is related to the kinetic param-eters through

κ = krT /�c + krT �; [7]

where kr is the infectivity of the resistant virion. This equa-tion comes about from considering the two possible fates ofa virion: clearance and infection, and comparing their rates cand krT , respectively.

If Nκ + 1, on average less than one cell is infected bythe progeny of a productively infected cell, and p = 0 is theonly non-negative solution of Eq. 6.� By similar logic, one can

¶Here, p and q are the probabilities of a nonterminating line of descent.However, if we instead require that the resistance mutant only propagatefor a large number of generations, p and q will be approximated by thesolutions of Eq. 5a–5b.�One can deduce this result by considering the graphs of the functionsy = p and y = 1 − exp�−Nκp� and observing that if Nκ + 1, the latterfunction is below the line y = p except when p = 0.

11516 Applied Mathematics, Medical Sciences: Kepler and Perelson Proc. Natl. Acad. Sci. USA 95 (1998)

deduce that before drug is given, wild-type virus must haveNκ = 1 to establish a quasi-steady–state level, where κ hererefers to the infection by wild-type virus. Lastly, for Nκ , 1there is a positive solution, i.e., the resistant strains have apositive probability of surviving and of replacing the wild-typevirus. (Interestingly, the condition Nκ = 1 for wild-type virusis equivalent to the condition k = c/T �N − 1�, i.e., that k isequal to its critical value).

An immediate result is that, before the administration ofdrug, the drug-resistant mutants that preexist in the parentalquasi-species are not self-sustaining. Assuming that there is acost of resistance, either the infectivity or rate of replication ofmutants in the absence of drug will be smaller than that of thewild-type, i.e., Nκ for the mutant must be less than Nκ = 1for the wild-type, and so the probability of propagation forthese mutants is zero; they are only maintained by continualproduction from the wild-type.

The mean time τ to appearance of propagating mutants, orfounders, is the inverse of their production rate:

τ = �p�µkV T �−1: [8]

Effect of Drug. For simplicity, we assume the drug is a re-verse transcriptase inhibitor and affects the infectivity of thevirus. If the drug is a protease inhibitor, noninfectious virionsare produced, which to a good approximation can be modeledby a change in infectivity (35). Let ε be the plasma (effective)drug concentration. Then, we may write

k�z� = k0/�1+ ε/IC50� = k0/�1+ z�; [9]

where k0 is the viral infectivity in the absence of drug, z is ascaled drug concentration, z = ε/IC50, and IC50 is the plasmadrug concentration at which k is reduced to 50% of its drug-free value.

For the resistant strain, we use the form

kr�z� = ρk0/�1+ βz�; [10]

where ρ � 1 is a factor by which the resistant infectivity is de-creased relative to the wild-type in the absence of drug, i.e.,a cost of resistance, and β � 1 is a factor by which the drugconcentration is effectively reduced for the resistant strain rel-ative to the wild-type. The other viral parameters c and N areassumed unchanged in the presence of drug. In other viral dis-eases (20), it might be more appropriate to fix k and allow Nto change due to drug, or to allow both to change; the resultsobtained in this alternative approach are qualitatively similar.

Window of Opportunity. One can now see why there isa problem in the generation of propagating resistant mu-tants. The production rate of resistant viruses, p��µkV T �,is a product of two functions: the first, p, the probabilityof propagation, is an increasing function of z that is zerobelow some positive value of z, which we call zL (whereNκ�zL� = 1), while the second, the overall mutant produc-tion rate, is a decreasing function of the drug concentrationz that becomes zero at some finite value of z, which we callzU (where k�zU� = kc and hence V = 0). As shown in Fig. 1,the product of these functions generates a curve that has asingle maximum and which goes to zero at finite values of zon either side of the hump. Richman (36) has presented astrikingly similar figure for the production of drug resistantmutants at different drug concentrations based on qualita-tive arguments. We now have explicitly calculated the shapeof this curve for a simple one-compartment model. What isinteresting about our theoretical result is that it shows that,only within a window in z, between zL and zU , can mutantsbe generated. This window is surprisingly narrow with zU cor-responding to a concentration of the order of the IC50 of thedrug. By going to a two-compartment model with drug con-centration differences between the two, the viral window ofopportunity is widened considerably, as we now show.

Fig. 1. In a one-compartment model the mean production rate ofresistant virions (solid line), equal to the inverse of the mean timeto the arrival of the founding resistant virus and labeled 1/τ, is non-zero only over a finite window of drug concentrations, z, from zL tozU . According to our theory, the production rate is the product of twofunctions (dashed lines). The curves were computed assuming Eqs. 1–3 were at steady-state and that k0 = 1:5 3 10−5 mm3 day−1, c = 3day−1, T0 = 1000/mm3, r = 0:01 day−1, N = 1000, β = 0:1, ρ = 0:95.With these values, the equilibrium values of T and V are 200/mm3

and 5:3 3 105/ml, respectively. Note that above z = 4, k + kc andthe production rate falls to zero. The scaling factors for the productionrate, µ� are µ = 2 3 10−10, corresponding to the acquisition of twoindependent mutations, and � = 2:5 3 108 mm3, assuming a 5 literblood capacity and that the T cell count in the blood needs to bemultiplied by 50 to account for the 98% of CD4+ T lymphocytes thatare in tissues. Note that the production rate curves scale simply withthe mutation rate µ, so that if only one mutation were required thewaiting time would be orders of magnitude shorter.

Two Compartment Model. Now consider virus occupyingtwo compartments and moving between them by passive trans-port. The first compartment will be considered the bulk com-partment with larger volume and larger drug concentration.The second will represent the sanctuary, with lower drug con-centrations and smaller volume. Let subscripts 1 and 2 desig-nate T cells or virus particles per unit volume specific to eachof the two compartments, with u1 = 1 − u and u2 = u, beingthe relative volumes of the two compartments.

The generalization of Eqs. 1–3 for this case is then

dTidt= rTi

(1− Ti

T0

)− kiViTi; [11]

dT ∗idt= kiViTi − δT ∗i ; [12]

dVidt= NδT ∗i − cVi − kiViTi +Di�Vı − Vi�; [13]

where the circumflexed subscript means “the other one”; e.g.,2 = 1.

In this model, we neglect, for simplicity, the passage of tar-get cells between compartments but allow passive transportof virus between the two compartments characterized by thetransport coefficients Di. This form of the transport coefficientarises from requiring that there is no net flow of virus betweencompartments when the respective concentrations are equal.Requiring that transport itself does not lead to an increase ordecrease in the total amount of virus yields

0= d

dt�u1V1+u2V2�=u1D1�V2 − V1�+u2D2�V1−V2�; [14]

where only the transport terms have been considered. This issatisfied for all V1; V2 only when D1 = u2D2/u1. Therefore, weassume this condition for our model and write D2 = D andD1 = uD/�1 − u�. We expect that Di changes with the vol-ume of the compartments in a way that is dependent on the


specific transport mechanisms involved and on geometric de-tails. For example, if transport is by diffusion, the surface areaof the interface between the compartments (possibly u2/3) en-ters. Alternatively, if transport is by convection, Di in simplemodels depends on the volume flow rate divided by the com-partment volume. Here D is a parameter, but we caution thatits value can depend on u.

Although we have very little knowledge of the “true” valueof D, we find that the system functions as if it has a “sanctu-ary” only if neither D1 or D2 is significantly larger than the vi-ral clearance rate. Thus, we will take D2, which is the larger ofthe two (because we are taking compartment 2 as the smallercompartment), to be of the order of c.

It may eventually be possible to make inferences about themagnitude of viral transport from studies of genetically dis-tinguishable viruses isolated in different compartments, e.g.,brain isolates vs. spleen isolates (10). However, such data isconfounded by differential selection in the two areas, a fac-tor about which very little is known. Here, we focus on theeffect of spatial heterogeneity per se and neglect differentialselection between compartments. Thus, the infectivities ki aregiven as in the single compartment model, but now the drugconcentrations in the two compartments differ, so we have zirather than z.

Production and Selection of Mutants. A virus in compart-ment i can do one of three things: (i) productively infect,thereby producing an average of N progeny, (ii) perish with-out leaving any progeny, or (iii) move to the other compart-ment. Let the probability that the virus infects productivelyin compartment i be denoted κi and the probability that itmoves to the other compartment be denoted mi. If pi andqi are the probabilities that a provirus and virus in compart-ment i propagate, respectively, they will be described by theequations

p1 = 1− exp�−q1N�; q1 = m1q2 + κ1p1; �15a; b�p2 = 1− exp�−q2N�; q2 = m2q1 + κ2p2: �15c; d�

Eqs. 15a and 15c have the same interpretation as Eq. 6, i.e.,in order for a provirus in compartment i to propagate (prob-ability pi), at least one of its progeny virions must propagate(again, we are assuming a large fixed number or a Poissondistribution of offspring numbers). In order for a virion incompartment 1 to propagate (Eq. 15b), it must either infecta cell and propagate as a provirus (probability κ1p1) or moveto compartment 2 and propagate (probability m1q2).

Eqs. 15a–d are always satisfied by the trivial solution�p1; q1; p2; q2� = �0; 0; 0; 0�. For any fixed value of the prod-uct m1m2, we can define a critical curve in the κ1, κ2 plane,on one side of which, in addition to the trivial solution, a non-trivial solution is produced:** This curve is the lower branchof the hyperbola given by �Nκ1 − 1��Nκ2 − 1� = m1m2.For �κ1; κ2� values to the right of this curve, a newly pro-duced virion has a positive probability of propagating. Notethat when either m1 or m2 vanishes, this condition becomesNκi , 1 for either i.

The stochastic parameters are related as before to the dy-namic parameters through

mi =Di

Di + c + kriTiand κi =

kriTiDi + c + kriTi

: [16]

For this system, the waiting time for production of prop-agating mutant virus is the reciprocal of the sum of the net

**Near the critical curve, pi and qi are small. Thus, Eqs. 15a–d canbe linearized about the origin and written, to first order, in the formA�p1; q1; p2; q2�T = 0, where A is a matrix. This equation has a non-trivial solution if and only if the determinant of A vanishes. The equa-tion for the hyperbola results from setting the determinant of A to zero.

production rates in the two compartments,

τ = ��µ��1− u�k1T1V1p1 + uk2T2V2p2��−1: [17]

Below we provide numerical solutions to the model for aparticular choice of parameter values, described in the cap-tion to Fig. 1, which may be viewed as characteristic of amid-stage AIDS patient. The figures showing these solutionsare meant to be illustrative of various general principles andtrends that occur as parameters characterizing the sanctuaryand drug regime are varied. While, for simplicity, we call τthe mean time to resistance, it is only the time for the ini-tial production of a propagating mutant, not the time untilphenotypic resistance would be observed in a patient.

A Widened Window of Opportunity. In a sanctuary, wherethe drug penetrance is small, partially resistant strains can pro-liferate and produce more resistant mutants, which can thenleave the sanctuary and proliferate in the bulk compartment.Thus, in the presence of a sanctuary, the window of opportu-nity is dramatically widened. For example, as shown in Fig. 2,when the relative sanctuary volume u2 = 0:001, the windowis widened 810-fold from an upper threshold of z1 = 4 forthe one-compartment model to approximately z1 = 40 for thetwo-compartment model. Above this new upper threshold, re-sistance is unlikely. This result is more in line with experiencein which drug concentrations of 50- to 100-fold greater thanthe IC50 have therapeutic value.

There are now two distinct regimes to the window. In thefirst, corresponding to the original window where the infectiv-ity kr�z� is above its critical value, the step-wise acquisition ofmutants is very rapid. In the second regime, where the bulkinfectivity kr�z� is below the critical value, the waiting time be-tween subsequent stages in the acquisition of resistance maybe quite long but in the presence of the sanctuary, finite.

In Fig. 2, the sanctuary was completely drug-free. Fig. 3shows the waiting time between mutations as the drug pen-etrance, z2/z1, is varied. For very high bulk concentrations,z1, the effect of heterogeneity is essentially lost. The drugconcentration in the bulk is too high even for the resistantstrain. The sanctuary now acts as a single compartment inwhich both production and selection must occur. For verysmall values of z2, the drug concentration in the sanctuary,we again have increasingly large waiting times (see the curveslabeled z1 = 16; 32, or 64 in Fig. 3). This effect, in whichevolution of drug resistance is prevented by lowering the con-centration of drug in the sanctuary (i.e., lowering the selectiveadvantage of a resistant mutant), is likely particular to the

Fig. 2. Mean time to resistance, τ, vs. z1, the drug concentrationin the bulk compartment for various relative volumes, u2, of the sanc-tuary, assumed to be drug-free, i.e., z2 = 0. The curve labeled 0.0 hasno sanctuary and represents the reciprocal of the quantity shown inthe solid line in Fig. 1. Parameters are as in Fig. 1, and D = 5 day−1.

11518 Applied Mathematics, Medical Sciences: Kepler and Perelson Proc. Natl. Acad. Sci. USA 95 (1998)

Fig. 3. Mean time to resistance, τ, vs. the drug penetrance z2/z1.z1 is held constant at the indicated value for each of the curves shownand parameters are as in Fig. 2. The sanctuary has relative volumeu = 0:01. For average concentrations above z1 = 4, a homogeneoussystem has infinite waiting time (Fig. 1). As seen here, heterogeneoussystems are able to produce resistant strains even when z1 , 4.

two-compartment model. In a model with many compartmentsor a continuum within which gradients of drug concentrationcan be found, facilitation of the evolution of resistance wouldbe found even at small sanctuary drug concentrations (seebelow).

The rate at which virus is transported between compart-ments also affects the mean time needed for drug resistanceto arise (Fig. 4). There is an “optimal” transport coefficientat which the heterogeneity between compartments is maximaland for which the mean time to resistance is at a minimum.This optimum occurs for D of order 5–10 days−1 for the drugconcentrations shown in Fig. 4. For z1 = 4, the mean waitingtime is short and virtually independent of the transport coef-ficient. For higher drug concentrations, the waiting time risesquite sharply as the larger transport coefficient D becomeslarger than the clearance rate c and acts like an enhancementof this clearance. In the other limit, in which D is very small,the compartments are approaching isolation, and the resistantmutants produced in the sanctuary cannot easily move to thehigh-drug compartment where they have an advantage. Thiseffect is not as dramatic when the sanctuary has a drug con-

Fig. 4. The effect of increasing transport coefficient D on themean waiting time until resistance, τ. Compartment 2 is drug-free(z2 = 0) and of relative size u = 0:01. Other parameters are as inFig. 1. The drug concentration z1 in the larger compartment labelsthe appropriate curves.

centration sufficiently high that the resistant mutants can stillpropagate without moving to the other compartment, thoughmoving would increase their chances.

DISCUSSION

The evolution and spread of drug-resistant pathogens isknown to occur with great robustness in a wide variety of sit-uations. We have shown that simple models that fail to con-sider heterogeneity of drug concentration underestimate therange of mean drug concentrations that support the estab-lishment of drug resistance (Fig. 1). The existence of evenquite small sanctuaries, places where the drug concentrationis much smaller than in the bulk compartment, can greatlyenhance the probability of generating resistant mutants. Thesanctuaries provide a place where ongoing replication of theparental strain continues and therefore allows mutants to beproduced. Once produced, they may then migrate to the bulkareas where the drug concentration is higher and where theycan exercise their advantage and replicate.

Our analysis is based upon a relatively simple two-compart-ment model with fixed drug concentrations in either compart-ment. Further improvements to this approach can be antici-pated. First, a spatially continuous model is likely to revealfurther effects of concentration gradients. Our preliminaryanalyses of models based on partial-differential equations inone spatial dimension (T.K., Babai, and A.S.P., unpublished re-sults) show that the step-wise accumulation of resistance mu-tations is facilitated by the presence of continuous gradients ofdrug concentration. Within these gradients, there are locationswithin which the conditions for growth of any given level ofdrug-resistance are ideal. As new strains are produced, theymigrate and preferentially replicate at their ideal locations,producing the next level of resistant mutants. The process con-tinues with subsequent levels of resistant strains climbing thegradient of drug concentration. Here, we have viewed com-partments as being spatially distinct. Another possibility is thatdrug penetrance differs within different cell populations andthus different cell populations may comprise different com-partments. If there were many such populations, it would beanalogous to having a gradient in drug concentration.

Another improvement would be to include temporal fluctu-ations. Drugs are administered at discrete times, so that theconcentration of drug fluctuates temporally. We have founda facilitation of drug resistance evolution due to spatial inho-mogeneities, and we expect that there may be a similar en-hancement due to temporal inhomogeneities.

The role of sanctuaries in the evolution of drug resistance islikely to be even more central for multi-drug therapies. Whenthree or more drugs are used, many mutations are typicallyrequired to confer resistance to the therapy as a whole. Buteven long times on these therapies with no detectable serumvirus should be greeted with caution. Waiting times for fullyresistant strains in the presence of sanctuaries can be quitelong. However, in some cases it is infinite. Thus, emergenceof drug resistance is not inevitable; the conditions under whichwe expect emergence are given by Eqs. 15–17.

An effective strategy for reducing human sickness and mor-tality caused by infectious microorganisms will necessitate afar more complete understanding of the large-scale patterns ofdrug resistance evolution than is presently available. More so-phisticated mathematical models that account for spatial andtemporal structure, in conjunction with improved experimen-tal measurments of drug and virus levels in multiple compart-ments, will likely be a tool of great importance in this ongoingendeavor.

We thank J. Mittler, S. Blower, and L. Segel for useful commentsand D. Babai for programming help. Portions of this work were per-formed under the auspices of the U.S. Department of Energy andsupported by National Science Foundation Grant MCB–9357637 (to


T.B.K.), National Institutes of Health Grants RR06555 and AI40387(to A.S.P.), the Santa Fe Institute, and the Joseph P. and Jeanne M.Sullivan Foundation.

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Proc. Natl. Acad. Sci. USAVol. 94, pp. 6971–6976, June 1997Medical Sciences

Virus dynamics and drug therapy

SEBASTIAN BONHOEFFER†, ROBERT M. MAY†, GEORGE M. SHAW‡, AND MARTIN A. NOWAK†§

†Department of Zoology, University of Oxford, South Parks Road, OX1 3PS, Oxford, United Kingdom; and ‡Division of HematologyyOncology, University ofAlabama at Birmingham, 613 Lyons–Harrison Research Building, Birmingham, AL 35294

Contributed by Robert M. May, April 14, 1997

ABSTRACT The recent development of potent antiviraldrugs not only has raised hopes for effective treatment ofinfections with HIV or the hepatitis B virus, but also has led toimportant quantitative insights into viral dynamics in vivo.Interpretation of the experimental data depends upon mathe-matical models that describe the nonlinear interaction betweenvirus and host cell populations. Here we discuss the emergingunderstanding of virus population dynamics, the role of theimmune system in limiting virus abundance, the dynamics ofviral drug resistance, and the question of whether virus infectioncan be eliminated from individual patients by drug treatment.

Several anti-HIV drugs are now available that act by inhibitingspecific viral enzymes. Reverse-transcriptase inhibitors pre-vent infection of new cells; protease inhibitors stop already-infected cells from producing infectious virus particles. Anti-viral drug treatment of HIV-infected patients leads to a rapiddecline in the abundance of plasma virus (virus load) and anincrease in the CD4 cells that represent the major target cellsof the virus. The decline of virus load is roughly exponentialand occurs with a half-life of around 2 days (1–8). Unfortu-nately, the effect of single-drug therapy is often only short-lived, as the virus readily develops resistance (9–16). Thiscauses virus load to rise and CD4 cell counts to fall. Multiple-drug treatment is more successful. A combination of zidovu-dine (AZT) and 29-deoxy-39-thiacytidine (3TC) can maintaina roughly 10-fold reduction of virus load for at least a year (17,18). Triple-drug therapy—using AZT, 3TC, and a proteaseinhibitor—can lead to a more than 10,000-fold reduction ofvirus load and can in many patients maintain plasma virusbelow detection limit for the whole duration of treatment.

In chronic hepatitis B virus (HBV) carriers, single-drug ther-apy with the reverse-transcriptase inhibitor lamivudine leads to anexponential decline of plasma virus load, with a half-life of aboutone day (19). Plasma virus is below detection limit for theduration of treatment (up to 6 months) (20). But when treatmentis withdrawn, virus load rapidly resurges to pretreatment levels.

Analyzing the dynamics of decline in virus load during drugtherapy andyor the rate of emergence of resistant virus canprovide quantitative estimates of the values of crucial rate con-stants of virus replication in vivo. In this way, it has been shownfor HIV-1 that the observed decay of plasma virus impliesvirus-producing cells have a half-life of about 2 days, whereas forHBV the rate of plasma virus decay suggests that free virusparticles have a half-life of about 1 day. Such analyses can provokefurther questions. In what follows, we will suggest explanations forthe puzzling observation that there is little variation in theturnover rate of productively infected cells among HIV-infectedpatients, while there is large variation in the turnover rate of suchcells in HBV carriers. We characterize the dynamics of differenttypes of infected cells, including productively infected cells,latently infected cells, and cells harboring defective HIV provirus.

We explore the rate of emergence of resistant virus, in relationto the frequency of resistant virus mutants before therapy isbegun. Finally, we analyze the kinetics of multiple-drug therapy,exploring the crucial question of whether HIV can be eradicatedfrom patients (and if so, how long it will take).

A Basic Model

We begin with a very simple model, which captures some of theessentials. This basic model of viral dynamics has three variables:uninfected cells, x; infected cells, y; and free virus particles, v (Fig.1A). Uninfected cells are produced at a constant rate, l, and dieat the rate dx. Free virus infects uninfected cells to produceinfected cells at rate bxv. Infected cells die at rate ay. New virusis produced from infected cells at rate ky and dies at rate uv. Theaverage life-times of uninfected cells, infected cells, and free virusare thus given by 1yd, 1ya, and 1yu, respectively. The averagenumber of virus particles produced over the lifetime of a singleinfected cell (the burst size) is given by kya. These assumptionslead to the differential equations:

x 5 l 2 dx 2 bxv

y 5 bxv 2 ay

v 5 ky 2 uv. [1]

Before infection (y 5 0, v 5 0), uninfected cells are at theequilibrium x0 5 lyd. An intuitive understanding of the proper-ties of these equations can be obtained, along lines familiar toecologists and epidemiologists (22, 23). A small initial amount ofvirus, v0, can grow if its basic reproductive ratio, R0, defined as theaverage number of newly infected cells that arise from any oneinfected cell when almost all cells are uninfected, is larger thanone (Fig. 1B). Here R0 5 blky(adu). The initial growth of freevirus is exponential, given roughly by v(t) 5 v0 exp[a(R0 2 1)t]when u .. a. Subsequently the system converges in dampedoscillations to the equilibrium x* 5 (au)y(bk), y* 5 (R0 21)(du)y(bk), v* 5 (R0 2 1)(dyb). At equilibrium, any one infectedcell will, on average, give rise to one newly infected cell. Thefraction of free virus particles that manage to infect new cells isthus given by the reciprocal of the burst size, ayk. The probabilitythat a cell (born uninfected) remains uninfected during itslifetime is 1yR0. Hence the equilibrium ratio of uninfected cellsbefore and after infection is x0yx* 5 R0.

Virus Decline Under Drug Therapy

In HIV infection, reverse-transcriptase inhibitors prevent in-fection of new cells. Suppose first, for simplicity, that the drugis 100% effective and that the system is in equilibrium beforethe onset of treatment. Then we put b 5 0 in Eq. 1, and thesubsequent dynamics of infected cells and free virus are givenby y 5 2ay and v 5 ky 2 uv. This leads to y(t) 5 y*e2at andv(t) 5 v*(ue2at 2 ae2ut)y(u 2 a). Infected cells fall purely asan exponential function of time, whereas free virus fallsexponentially after an initial shoulder phase. If the half-life of

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© 1997 by The National Academy of Sciences 0027-8424y97y946971-6$2.00y0

Abbreviations: HBV, hepatitis B virus; CTL, cytotoxic T cell; PBMC,peripherial blood mononuclear cells.§To whom reprint requests should be addressed.

6971

free virus particles is significantly shorter than the half-life ofvirus-producing cells, u .. a, then (as illustrated in Fig. 2A)plasma virus abundance does not begin to fall noticably untilthe end of a shoulder phase of duration Dt ' 1yu [moreprecisely, from Fig. 2 A, Dt 5 2(1ya)ln(1 2 ayu)]. Thereaftervirus decline moves into its asymptotic phase, falling as e2at.Hence, the observed exponential decay of plasma virus reflectsthe half-life of virus-producing cells, while the half-life of freevirus particles determines the length of the shoulder phase.[Note that the equation for v(t) is symmetric in a and u, andtherefore if a .. u the converse is true.]

In the more general case when reverse-transcriptase inhibitionis not 100% effective, we may replace b in Eq. 1 with b 5 sb, withs , 1 (100% inhibition corresponds to s 5 0). If the time-scale forchanges in the uninfected cell abundance, 1yd, is longer thanother time-scales (d ,, a, u), then we may approximate x(t) by x*.It follows that the decline in free virus abundance is still describedby Fig. 1a, except now the asymptotic rate of decay is exp[2at(1 2s)] for u .. a; the duration of the shoulder phase remains Dt '1yu (exact expressions for arbitrary ayu and s , 1 are given in thelegend to Fig. 2). Thus the observed half-life of virus producingcells, T1y2 5 (ln 2)y[a(1 2 s)], depends on the efficacy of the drug.

Protease inhibitors, on the other hand, prevent infected cellsfrom producing infectious virus particles. Free virus particles,which have been produced before therapy starts, will for ashort while continue to infect new cells, but infected cells willproduce noninfectious virus particles, w. The equations be-come y 5 bxv 2 ay, v 5 2uv, w 5 ky 2 uw. The situationis more complex, because the dynamics of infected cells and

free virus are not decoupled from the uninfected cell popu-lation. However, we can obtain analytic insights if we againassume that the uninfected cell population remains roughlyconstant for the time-scale under consideration. This gives thetotal virus abundance as v(t) 1 w(t) 5 v*[e2ut 1 {(e2at 2e2ut)uy(a 2 u) 1 ate2ut}uy(a 2 u)] (ref. 7). As in Fig. 1A,for u .. a this function describes a decay curve of plasma viruswith an initial shoulder (of duration Dt 5 2(2ya)ln(1 2 ayu)' 2yu) followed by an exponential decay as e2at. The situationis very similar to reverse-transcriptase inhibitor treatment. Themain difference is that the virus decay function is no longersymmetric in u and a, and therefore a formal distinctionbetween these two rate constants can be possible; if a . u, theasymptotic behavior is no longer simply exponential, but ratherv(t)yv* 3 [ay(a 2 u)]ute2ut.

Sequential measurements of virus load in HIV-1-infectedpatients treated with reverse-transcriptase or protease inhib-itors usually permit a good assessment of the slope of theexponential decline, which reflects the half-life of infectedcells, (ln 2)ya. This half-life is usually found to be between 1and 3 days (1, 2, 6, 7). Very frequent and early measurements(Fig. 2B) also have provided a maximum estimate of thehalf-life of free virus particles, (ln 2)yu, of the order of 6 hr (7).

Half-life of Infected Cells and CTL Response

In all HIV-1-infected patients analyzed so far the half-life ofvirus-producing cells is roughly the same, around 2 days. Thisrough constancy of half-lives seems puzzling. If the lifespan ofproductively infected cells is determined by CTL-mediated lysis(24, 25), then it is surprising to find so little variation in theobserved half-life in different patients. Alternatively, if we assumethat all cell death is caused by virus-induced killing, then CTL-mediated lysis could not limit virus production.

We see two potential explanations why different levels ofCTL activity may not result in different half-lifes of infectedcells: (i) Measurements of turnover rates based on plasma virusdecay strictly imply only that those cells producing most of theplasma virus have a half-life of about 2 days (26). In anindividual with a strong CTL response, many infected cells maybe killed by CTL before they can produce large amounts ofvirus (27). A small fraction of cells, however, escapes fromCTL killing; these produce most of the plasma virus and arekilled by viral cytopathicity after around 2 days. In a weakimmune responder, on the other hand, most infected cells mayescape from CTL-mediated lysis, produce free virus, and dieafter 2 days. Thus in both kinds of infected individuals, thehalf-life of virus-producing cells is the same, but in the strongCTL responder a large fraction of virus production is inhibitedby CTL activity. (ii) A second possible explanation is based onthe fact that the virus decay slope actually reflects the slowestphase in the viral life cycle (26). If, for example, it takes onaverage 2 days for an infected cell to become a target for CTLand to start production of new virus, but soon afterwards thecell dies (either due to CTL or virus-mediated killing), then theobserved decay slope of plasma virus may reflect the firstphase of the virus life cycle, before CTL could attack the cell.In this event, variation in the rate of CTL-mediated lysis wouldhave no effect on the virus decay slope during treatment.

Long-Lived Infected Cells

Only a small fraction of HIV-1-infected cells have a half-life of2 days. These short-lived cells produce about 99% of theplasma virus present in a patient. But most infected peripherialblood mononuclear cells (PBMC) live much longer (Fig. 1C).We estimated the half-life of these cells by measuring the rateof spread of resistant virus during lamivudine therapy (2, 21).While it takes only about 2 weeks for resistant virus to grow toroughly 100% frequency in the plasma RNA population, ittakes around 80 days for resistant virus to increase to 50%frequency in the provirus population. This suggests that most

FIG. 1. Models of virus dynamics are based on ordinary differentialequations that describe the change over time in abundance of freevirus, uninfected, and infected cells. (A) In the simplest model, weassume that uninfected cells, x, are produced at a constant rate, l, anddie at rate dx. Free virus is produced by infected cells at rate ky anddies at rate uv. Infected cells are produced from uninfected cells andfree virus at rate bxv and die at rate ay. This leads to Eq. 1 in the text.(B) The average number of virus particles (burst size) produced fromone infected cell is kya. The basic reproductive ratio of the virus,defined as average number of newly infected cells produced from anyone infected cell if most cells are uninfected (x 5 lyd), is given by R05 (lbk)y(adu). (C) In HIV-1 infection, there are different types ofinfected cells. Productively infected cells have half-lifes of about 2 daysand produce most (.99%) of plasma virus. Latently infected cells canbe reactivated to become virus-producing cells; their turnover rate andabundance may vary among patients depending on the rate of immuneactivation of CD4 cells. Preliminary results suggest half-lifes of 10–20days (21). Infected macrophages may be long-lived, chronic producersof virus particles. More than 90% of infected PBMC contain defectiveprovirus. Their half–live was estimated to be around 80 days (2, 21).(D) The simplest models of drug resistance include a sensitive wild-type virus and a resistant mutant (Eq. 3). Mutation between wild typeand mutant occurs at rate m.

6972 Medical Sciences: Bonhoeffer et al. Proc. Natl. Acad. Sci. USA 94 (1997)

HIV-1-infected PBMC have half-lives of around 80 days. Infact more than 90% of infected PBMC seem to containdefective provirus (21). It is likely that most of these cells arenot targets for CTL-mediated lysis and their lifespan is similarto the lifespan of uninfected CD4 cells. The large fraction ofdefectively infected cells is mostly a consequence of their longlifetime, not because they are produced so frequently.

Less than 10% of infected PBMC harbor replication com-petent provirus (21). Some of these cells are actively producingnew virus particles (and have a half-life of 2 days) while othersharbor latent provirus, which can be reactivated to enter virusproduction. In two patients we estimated the half-life oflatently infected cells to be around 10–20 days (21). Thissuggests that in an HIV-infected patient 50% of latentlyinfected CD4 cells get reactivated on average after 10–20 days.In addition, there may be long-lived, infected macrophages intissue, which may chronically produce new virus particles.

During triple-drug therapy, plasma virus load declines dur-ing the first 1 or 2 weeks with a half-life of less than 2 days andsubsequently enters a second phase of slower decline. Thissecond phase has a half-life of around 10 to 20 days (28, 29)and reflects the decay of either latently infected cells or slow,chronic producers. Triple-drug therapy also has led to animproved estimate for the half-life of defectively infected cellsby direct observation of HIV-1 provirus decline; an averagehalf-life of around 140 days was observed (28).

Dynamics of Resistance

A main problem with antiviral therapy is the emergence ofdrug-resistant virus (Fig. 1D). Several mathematical models

have been developed to describe the emergence of drug-resistant virus (30–35). An appropriate model that capturesthe essential dynamics of resistance is:

x 5 l 2 dx 2 b1xv1 2 b2xv2

y1 5 b1~1 2 m!xv1 1 b2mxv2 2 ay1

y2 5 b1mxv1 1 b2~1 2 m!xv2 2 ay2

v1 5 k1y1 2 uv1

v2 5 k2y2 2 uv2. [2]

Here y1, y2, v1, and v2, denote, respectively, cells infected bywild-type virus, cells infected by mutant virus, free wild-typevirus, and free mutant virus. The mutation rate between wild-type and mutant is given by m (in both directions). For small m,the basic reproductive ratios of wild-type and mutant virus are R15 b1lk1y(adu) and R2 5 b2lk2y(adu). If we assume R1 . R2, thenthe equilibrium abundance of cells infected by wild-type virus, y*1,is roughly given as earlier (following Eq. 1), while the corre-sponding value of y*2 is smaller by a factor of order m:

y*2yy*1 5 my@1 2 R2yR1#. [3]

Suppose drug treatment reduces the rates at which wild-typeand mutant virus infect new cells, b1 and b2, to b91 and b92, andcorrespondingly reduces the basic reproductive ratios to R91and R92.

An important question is whether mutant virus is likely tobe present in a patient before drug treatment begins (5, 36).Let s denote the selective disadvantage of resistant mutantvirus compared to wild-type virus before therapy. In terms ofthe basic reproductive ratios, we have R2 5 R1(1 2 s). If wildtype and mutant differ by a single point mutation, then thepretreatment frequency of mutant virus is given by mys. Usinga standard quasi-species model and assuming that all (inter-mediate) mutants have the same selective disadvantage, wefind that the approximate frequency of two- and three-errormutant is, respectively, 2(mys)2 and 6(mys)3 (R. M. Ribeiro,S.B., and M.A.N., unpublished work). If the point mutationrate is about 3 3 1025 (38) and if, for example, the selectivedisadvantage is s 5 0.01, then for a one-error mutant thepretreatment frequency is about 3 3 1023, for a two-errormutant 2 3 1025 and for a three error mutant 2 3 1027. Thusthe expected pretreatment frequency of resistant mutantdepends on the number of point mutations between wild-typeand resistant mutant, the mutation rate of virus replication,and the relative replication rates of wild-type virus, resistantmutant, and all intermediate mutants. Whether or not resistantvirus is present in a patient before therapy will crucially dependon the population size of infected cells (39).

Pre-existing Mutant. First, we study the consequences of drugtherapy in situations where resistant mutants are present beforetherapy begins. This will usually apply to drugs (or drug combi-nations) where one- or two-point mutations confer resistance.Suppose R1 . R2 . 1 before therapy. There are now fourpossibilities (Fig. 3 A–D): (i) A very weak drug (low dose or lowefficacy) may not reduce the rate of wild-type reproduction belowmutant reproduction, i.e., R91 . R92 . 1. In this case the mutantvirus will not be selected. Nothing much will change. (ii) Astronger drug may reverse the competition between wild type andmutant such that R92 . R91 . 1. Here the mutant virus willeventually dominate the population after long-term treatment,but the initial resurgence of virus can be mostly wild type. (iii) Astill stronger drug may reduce the basic reproductive ratio of wildtype below one, R92 . 1 . R91. Here wild-type virus should declineroughly exponentially after start of therapy and be maintained inthe population at very low levels (only because of mutation). (iv)

FIG. 2. Short-term dynamics of virus decline during anti-HIV-1treatment. (A) For a drug that reduces cell infection rates from b tosb (s , 1), we find the amount of free virus, v(t)yv*, declinesasymptotically as [L2y(L2 2 L1)]exp(2L1t), where 2L1,2 5 (u 1 a)6 =(u 2 a)2 1 4sau (providing d ,, a, u). Thus the asymptotic slopeof ln[v(t)] versus t gives a measure of L1, while the duration of theshoulder phase, Dt, can be assessed from Dt 5 (1yL1)ln(L2y(L2 2L1)]. In the limit s 3 0 (drug 100% effective), we have L1 5 a andDt 5 (1ya)ln[uy(u 2 a)], for u . a. More generally, for positive s ,1 and u .. a, we see that the asymptotic decay rate is approximatelyL1 ' a(1 2 s)[1 2 (sayu) 1 O (a2yu2)] and the shoulder phaseduration is correspondingly Dt ' (1yu)[1 1 (ay2u)(1 2 3s) 1 O(a2yu2)]. (B) Plasma virus decline in a patient treated with theprotease inhibitor ritonavir. Data are from ref. 7. Virus load starts tofall exponentially around 30 to 40 h after initiating treatment. Thistime span is a combination of the shoulder phase described in A, apharmacological delay of the drug and the intracellular phase of thevirus life-cycle (7, 8).

Medical Sciences: Bonhoeffer et al. Proc. Natl. Acad. Sci. USA 94 (1997) 6973

In this happy case, a very effective drug may reduce the basicreproductive ratio of both wild type and mutant below one, 1 .R1 . R2, and eliminate the virus population.

Some interesting points emerge from the above dynamics. (i)The resurgence of (wild-type or mutant) virus during treatmentis a consequence of an increasing abundance of target cells (31).There is experimental evidence that rising target cell levels canlead to a rebound of wild-type virus during zidovudine treatment(16). Preventing target cell increase therefore could maintain thevirus at low levels (40). (ii) The more effective the drug, the moreintense the selection, and thus the faster the emergence ofresistant virus (41) (Fig. 3E). (iii) The eventual equilibriumabundance of (resistant) virus under drug therapy will usually bevery similar to that of wild-type virus before therapy, even if thedrug markedly reduces the basic reproductive ratio of the viruspopulation (42) (the abundances for m ,, 1, are (lya)(1 2 1yR1)and (lya)(1 2 1yR9i) with i 5 1 or 2, respectively; these are bothroughly lya, unless either R1 or R9i gets close to unity, much lessbelow it). (iv) Finally, the total benefit of drug treatment—asmeasured by the total reduction of virus load over time (or by theincrease of uninfected cells)—is roughly constant, independent ofthe efficacy of the drug (Fig. 3F).

Non-Pre-Existing Mutant. Second, we consider the situationwhere three or more point mutations are necessary for the virusto escape from drug treatment. Here the equilibrium abundanceof resistant mutant, y*2, as given by Eq. 3, may be smaller than one.That is, the deterministic model represented by Eq. 2 leads to theconclusion that, on average, less than one cell infected withmutant virus is present before therapy. In this case, the deter-ministic description is no longer valid; a stochastic model isnecessary (41). Assuming a standard Poisson process, the prob-ability that no mutant virus exists before therapy is exp(2y*2).

If the drug is strong enough to eliminate wild-type virus, R91 ,1, then we can calculate the probability that mutant virus isproduced by the declining wild-type population, even if it isinitially absent. We find that this probability is roughly sy*2, underthe reasonable assumption that uninfected cells live noticeablylonger than infected ones, which in turn live longer than free virus(u .. a .. d). Here s is the efficacy of the drug on wild-type virus,defined as s 5 R91yR1. Thus, if it is unlikely that mutant virus existsbefore therapy (y*2 , 1), then for small s it is even less likely thatit will be produced by the declining wild-type population. Thisconclusion, based on an analytic approximation, is supportedmore generally by numerical studies (Fig. 4).

If, on the other hand, the drug is unable to eliminate wild type,R91 . 1, then mutant virus will certainly be generated after sometime, and will dominate the population provided R92 . R91.

If resistant virus is not present before therapy, then astronger drug can reduce the chance that a resistant mutantemerges and can prolong the time until resistant virus isgenerated. (On the other hand, if resistant virus is present thena stronger drug usually leads to a faster rise of resistant virus.)

The above model can be expanded to include a large numberof virus mutants with different basic reproductive ratios anddifferent susceptibilities to a given antiviral therapy. Some ofthese mutants may pre-exist in most patients, while others maynot be present before therapy. The basic question is whethera given antiviral drug combination manages to supress thebasic reproductive ratios of all pre-existing variants to belowone or not. This question is central to hopes for effectivelong-term treatment against viral infections.

Why Treatment Should Be as Early as Possible and as Hardas Possible. The outcome of therapy should depend on thevirus population size before treatment. The lower the virusload, the smaller the probability that resistant virus is present.Consequently, treatment will be more succesful in patientswith lower virus load. Therefore treatment should start earlyin infection as long as virus load is still low.

The above models also suggest that antiviral therapy shouldimmediately start with as many drugs as clinically possible.

Using several drugs at once reduces the probability thatresistant virus is present in a patient before therapy. Startingwith one drug and then adding other drugs, or cycling betweendifferent drugs, creates an evolutionary scenario, which favorsthe emergence of multiple-drug resistant virus, because at anytime virus mutants will be present with basic reproductiveratios larger than one. Similarily, drug holidays or irregulardrug consumption are very disadvantageous.

FIG. 3. Dynamics of drug treatment if resistant virus is presentbefore therapy. Before treatment, the basic reproductive ratios ofwild-type and mutant virus are given by R1 and R2, respectively. Drugtherapy reduces the basic reproductive ratios to R91 and R92. There arefour possibilities depending on dosage and efficacy of the drug: (A) IfR91 . R92, then mutant virus is still outcompeted by wild type.Emergence of resistance will not be observed. Equilibrium virusabundance during treatment is similar to the pretreatment level. (B)If R92 . R91 . 1, resistance will eventually develop, but the initialresurgence of virus can be due to wild type. (C) If R92 . 1 . R91,resistant virus rises rapidly. In B and C the exponential growth rate ofresistant virus is approximately given by a(R92 2 1), thus providing anestimate for the basic reproductive ratio of resistant virus duringtreatment. (D) If 1 . R91, R92, then both wild-type and resistant viruswill disappear. (E) A stronger drug will lead to a faster rise of resistantvirus, if it exerts a larger selection pressure. (F) The total benefit ofdrug treatment, as measured by the reduction of virus load duringtherapy integrated over time, *t (v(t) 2 v*)dt, is largely independentof the efficacy of the drug to inhibit wild-type replication. A strongerdrug leads to a larger initial decline of virus load, but causes fasteremergence of resistance. Parameter values: l 5 107, d 5 0.1, a 5 0.5,u 5 5, k1 5 k2 5 500, b1 5 5 3 10210, b2 5 2.5 3 10210. Hence,R1 5 10 and R2 5 5. Treatment reduced b1 and b2 such that: (A) R915 3, R92 5 2.5; (B) R91 5 1.5, R92 5 2.25; (C) R91 5 0.5, R92 5 2; (D)R91 5 0.1, R92 5 0.5; and (E and F) R91 5 3, R92 5 4.5 (continuous line)and R91 5 1.5, R92 5 4.5 (broken line). In A–D the continuous line iswild-type virus, whereas the broken line denotes mutant.


Virus Eradication

Consider a combination treatment that reduces the basicreproductive ratio of all virus variants in a given patient tobelow one. For how long do we have to treat in order toeliminate HIV infection? Latently infected cells have half-lifesof about 10 to 20 days. Treatment for 1 year may thus reducethe initial population of latently infected cells by a factor of1028, which could mean extinction (Fig. 4).

There is one problem, however. Suppose the average half-lifeof infected PBMC is around 140 days. Most of these cells carrydefective provirus, but some may contain replication-competentprovirus integrated in a CD4 cell that has not been stimulatedsince it became infected. Such unstimulated latently infected cellmay have half-lifes equivalent to cells carrying defective provirus.With respect to eliminating this cell population, the relevanthalf-life is therefore about 140 days. Treatment for one year willreduce this cell population to about 16% of its initial value;treatment for 2 years to 3%. Extinction seems difficult. It mightbe important to develop treatment strategies that reactivatelatently infected CD4 cells, so as to reduce their half-life. Inaddition, it is possible that virus replication persists in specific siteswhere drugs do not acchieve high concentration.

HBV

In the life cycle of HBV, the virus-encoded reverse transcriptaseis responsible for transcribing the unspliced viral mRNA into theDNA genome of new virus particles. Therefore the reverse-transcriptase inhibitor, lamivudine, stops already-infected cellsfrom producing new virus particles. During drug therapy, thedynamics of infected cells and free virus are given by y 5 bxv 2ay and v 5 2uv. Thus plasma virus, v, simply falls as anexponential function of time. Hence the slope of the virus decayreflects the half-life of free virus particles, which turns out to beabout 24 hr (19).

The half-life of infected cells in HBV infection has beenestimated from the decay of virus production (comparing the rateof virus production before and after therapy) or from the declineof hepatitis E antigen levels during therapy. In contrast to HIV,virus producing cells in HBV are long-lived. There is also greatvariation in turnover rates in different patients, ranging fromabout 10 days to more than 100 days (19). HBV is considered tobe noncytopathic, and the difference in infected cell half-lives canbe attributed to different CTL activities. In HBV infection it isalso possible that infected cells lose their HBV DNA and can thusbecome uninfected. CTL may accelerate this process (43, 44).Thus our estimated turnover rates of infected cells may not simplydescribe cell death, but rather the time span a cell remainsinfected or in the state of virus production.

Emergence of resistance to lamivudine in HBV infection isslower and rarer than in HIV infection. There was no indica-tion of resistance in 50 chronic HBV carriers treated for 24weeks (19, 20), whereas the same drug usually induces HIVresistance in a few weeks (4). HBV resistance, however, ispossible and was observed after about 30 weeks in threepatients receiving liver transplantation (45, 46). The 10- to100-day half-life of HBV-producing cells suggests that thegeneration time is 5 to 50 times longer in HBV than in HIV,which could explain the slower adaptive response.

What Next?

A combination of experimental techniques and mathematicalmodels has provided new insights into virus population dynamicsin vivo. The effect of antiviral treatment on the decline of plasmavirus and infected cells, and on the emergence of drug-resistantvirus, can be largely understood in quantitative terms. This hasconsequences for interpreting success or failure of long-termtherapy and for designing optimal treatment schedules. Essen-tially all mathematical models so far suggest that HIV should behit as early as possible and as hard as possible.

Measurement of changes in plasma virus during therapyshould ideally be complemented by quantification of infected

FIG. 4. Decay of plasma virus and different types of infected cells andprobability to produce resistant virus during anti-viral treatment. (A) Thefirst phase of plasma virus decay occurs with a half-life of about 2 days andreflects the turnover rate of productively infected cells. The second phaseoccurs with a longer half-life (10–30 days) reflecting the decay of latentlyinfected cells or long-lived chronic producers. (B) Productively infectedcells disappear rapidly, followed by latently infected cells and chronicproducers. Most infected cells may carry defective provirus and declineslowly with a half-life of about 80 days. (C) What is the probability thatresistant virus is generated during therapy? Assume treatment reduceswild-type reproduction by a factor s 5 R91yR1. Using the simplified model(2), we can calculate the total amount of mutant virus, Y2, which isproduced by mutation from wild-type virus after onset of therapy: Y2 5*0

` b91mx(t)v1(t)dt. If u .. a, we have v1(t) ' k1y1yu. If a .. d, we havex(t) ' x*. During therapy y1(t) ' y*1exp(2at). Using these approximations,we obtain Y2 5 smy*1. A more accurate approximation can be obtained byassuming x ' l 2 dx during treatment (41). The probability that resistantvirus exists before treatment is P0 5 1 2 exp(2y*2), which is for small y*2approximately P0 5 y*2. Similarily the probability that resistant mutant isgenerated during therapy is P 5 1 2 exp(2Y2) ' Y2. Because y*2 . my*1,we have P , sP0. For an effective drug (small s), the probability thatresistant mutant is generated during therapy is much smaller than theprobability that it already existed before therapy. We can also take intoaccount the possibility that resistant virus is generated by mutation eventsduring production of free virus from infected cells. If this mutation rateis given by v, we find with a similar calculation that Y2 5 svy*1. If bothmutations are possible, then Y2 5 s(m 1 v)y*1. The mutation rates simplyadd up. The final result, P , sP0, remains the same. The dark shaded areashows Y2(t) for a simple model ignoring latency. The lighter shaded areashows the same quantity computed from a model with latency. Modelequations are: uninfected cells x 5 l 2 dx 2 bxv; productively infectedcells y1 5 bq1xv 2 a1y1 1 a2y2; latently infected cells y2 5 bq2xv 2 a2y2;chronic producers y3 5 bq3xv 2 a3y3; defectively infected cells y4 5bq4xv 2 a4y4; free virus v 5 k1y1 1 k3y3 2 uv. Parameter values: l 5 107,d 5 0.1, a1 5 0.5, a2 5 0.05, a3 5 0.03, a4 5 0.008, u 5 5, k1 5 500, k35 10, q1 5 0.55, q2 5 q3 5 0.025, q4 5 0.4, b 5 5 3 10210 before therapyand b9 5 0.01b during therapy.

Medical Sciences: Bonhoeffer et al. Proc. Natl. Acad. Sci. USA 94 (1997) 6975

cells in those tissues that contain most of the virus population[the lymph-system for HIV (47, 48) and the liver for HBV].Determining the abundance of infected cells in those tissuesbefore and during therapy should lead to a more directassessment of their turnover rates. It also may provide esti-mates of further parameters of virus dynamics, such as the rateof infection of new cells, b, and the rate of virus productionfrom infected cells, k (49). It will also be important toilluminate the spatial dynamics of virus infections (50).

Another important step will be to monitor changes inimmune cell populations (specific CTL or B cells) during drugtherapy, to gain insights into the rates of turnover of variouscells of the immune system and their rates of proliferation inresponse to antigenic stimulation in vivo. In addition, it wouldbe helpful to have experimental techniques that determine thefraction of infected cells (or free virus) eliminated by specificimmune responses in a given length of time. Such informationis essential for a quantitative understanding of immune re-sponse dynamics in vivo.

The dynamics of drug resistance also provides further insightsinto theories of viral population genetics and antigenic variation(51–61). The main difference between escape from drug treat-ment and escape from immune responses is that drugs provide aconstant selection pressure, while the immune response is sensi-tive to changes in the antigenic structure of the virus population(37) and also may shift between different viral epitopes (53).

The approach developed here is, of course, not limited toHIV or HBV, but can easily be adapted to other persistantinfections with replicating parasites (viruses, bacteria, proto-zoa, helminths) and also to various kinds of drug treatmentsuch as interferons, chemokines, or antibiotics. The ultimateaim is to derive a detailed understanding of the dynamics of theinteractions between populations of viruses (or other infec-tious agents) and populations of immune system cells. Suchnonlinear population dynamics often can defy any intuitionbased on interactions between individual cells and virus.

Support from the Wellcome Trust (M.A.N. and S.B.) and the RoyalSociety (S.B. and R.M.M.) is gratefully acknowledged.

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