A model of HIV drug resistance driven by heterogeneities in host immunity and adherence patterns

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1 a. Adherence attern Based on hyothesized causes and timescales Month b. Pharmacokinetics Liver TDF TFV ME ME Cell membrane c. Pharmacodynamics TDF= R relative to WT WT K65R M184V TFV MP DP Intracellular TDF-DP (mg/ml) d. Virus dynamics c k 1 k 2 k n x + v 1 v 2 v n... y 1 y 2... y n z d u a a a b e. Virus dynamics c x + k 1 k k v 2 n 1 v 2 v n... y 1 y 2... y n z d u h a h a h a b w 1 w 2... w n A model of HIV drug resistance driven by heterogeneities in host immunity and adherence atterns Bershteyn and Eckhoff BMC Systems Biology 213, 7:11

2 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 RESEARCH ARTICLE Oen Access A model of HIV drug resistance driven by heterogeneities in host immunity and adherence atterns Anna Bershteyn * and Phili A Eckhoff Abstract Background: Poulation transmission models of antiretroviral theray (ART) and re-exosure rohylaxis (PrEP) use simlistic assumtions tyically constant, homogeneous rates to reresent the short-term risk and long-term effects of drug resistance. In contrast, within-host models of drug resistance allow for more detailed dynamics of host immunity, latent reservoirs of virus, and drug PK/PD. Bridging these two levels of modeling detail requires an understanding of the levers model arameters or combinations thereof that change only one indeendent observable at a time. Using the examle of accidental tenofovir-based re-exosure rohyaxis (PrEP) use during HIV infection, we will exlore methods of imlementing host heterogeneities and their long-term effects on drug resistance. Results: We combined and extended existing models of virus dynamics by incororating harmacokinetics, harmacodynamics, and adherence behavior. We identified two levers associated with the host immune ressure against the virus, which can be used to indeendently modify the setoint viral load and the shae of the acute hase viral load eak. We roose arameter relationshis that can exlain differences in acute and setoint viral load among hosts, and demonstrate their influence on the rates of emergence and reversion of drug resistance. The imortance of these dynamics is illustrated by modeling long-lived latent reservoirs of virus, through which ast intervals of drug resistance can lead to failure of suressive drug regimens. Finally, we analyze assumtions about temoral atterns of drug adherence and their imact on resistance dynamics, finding that with the same overall level of adherence, the dwell times in drug-adherent versus not-adherent states can alter the levels of drug-resistant virus incororated into latent reservoirs. Conclusions: We have shown how a diverse range of observable viral load trajectories can be roduced from a basic model of virus dynamics using immunity-related levers. Immune ressure, in turn, influences the dynamics of drug resistance, with increased immune activity delaying drug resistance and driving more raid return to dominance of drug-suscetible virus after drug cessation. Both immune ressure and atterns of drug adherence influence the long-term risk of drug resistance. In the case of accidental PrEP use during infection, raid transitions between adherence states and/or weak immunity fortifies the memory of revious PrEP exosure, increasing the risk of future drug resistance. This model framework rovides a means for analyzing individual-level risks of drug resistance and imlementing heterogeneities among hosts, thereby achieving a crucial rerequisite for imroving oulation-level models of drug resistance. * Corresondence: abershteyn@intven.com Eidemiological Modeling Grou, Intellectual Ventures Laboratory, Washington, USA 213 Bershteyn and Eckhoff; licensee BioMed Central Ltd. This is an Oen Access article distributed under the terms of the Creative Commons Attribution License (htt://creativecommons.org/licenses/by/2.), which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited.

3 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 2 of 15 Background Quantitative analysis of the risk of HIV drug resistance is imortant on both an individual and a oulation level, esecially when the real-world use of drugs may differ significantly from the more ideal setting of randomized controlled trials. Examles of non-ideal real-world situations include oor adherence, late entry into theray, drug stockouts, unauthorized re-distribution of antiretrovirals, and interrution of multi-drug regimens in which one drug has a longer half-life, which creates an interval of effective monotheray when only this drug is resent at high levels [1]. Interest in models of drug resistance has increased recently due to the addition of re-exosure rohylaxis (PrEP) to the world s HIV revention toolkit [2]. Clinical trials have demonstrated that PrEP with tenofovir disoroxil fumarate (TDF), combined TDF and emtricitabine (FTC), and tenofovir vaginal gel can reduce the risk of HIV acquisition [3-6]. No PrEP regimen revented transmission entirely, though increased adherence correlated with increased rotection against HIV. This creates a risk of accidental PrEP use during breakthough infections, until the individual is diagnosed with HIV and PrEP is discontinued. Additionally, there is risk of accidental initiation of PrEP by infected individuals due to faulty testing or early window eriod testing. In addition to their use in PrEP, TDF and FTC are also found in two oular single-ill, once-daily fixed-dose combination theraies(atrilaandcomlera)aswellasthefixed-dose quad ill recently aroved by the FDA. Thus, there is concern as to whether resistance caused by PrEP could threaten the ability to use two of the safest available drugs available in a convenient, single-dose, once-daily regimen. Mathematical models of ART and PrEP have been used to assess the risk of drug resistance on the individual and oulation level [7]. However, state-of-the-art oulation models have failed to cature heterogeneities in the risk of drug resistance among individuals, due to the disarity in model detail between oulation-level and within-host models. Poulation-level models of PrEP and resistance tend to focus on secific conditions of HIV transmission, with a majority of recent oral PrEP models focusing on heterosexual generalized eidemics [8-1], and others on concentrated eidemics among men who have sex with men [11]. Of the small number of oulation PrEP models that include drug resistance, only one has considered reversion of resistance by assuming that resistance and reversion occur at fixed rates for the treated and ost-treated suboulations, resectively [1,11]. Recent models of combination ART that account for resistance and reversion treat these similarly [12,13]; though one model used stochastic numerical methods as a roxy for variability in acquisition and transmission of resistance, these are still assumed to occur at a fixed rate [13] with no consideration for heterogeneous host biology and behavior. Within-host models have rovided some mechanistic insights into HIV rogression, drug efficacy, and the risk of resistance. Nowak and Bangham formulated a model of virus dynamics that includes mutation [14], and has insired dozens of variations, such as mechanisms of viral escae arising from cometition among quasiseciessecific cytotoxic T lymhocyte (CTL) [15] and antibody [16] resonses. A common alication of these models is redicting the relationshi between dosage and acquisition of resistance: a quantity not well-characterized in clinical studies [17]. Very recently, a detailed within-host model similar to the one resented here was used to roose a mechanism for drug resistance during HAART that includes a rotease inhibitor, roosing a relationshi between adherence level and risk of treatment failure [18]. This and other models [19] have assumed that adherence is rimarily characterized by the ercentage, not the attern, of doses taken. Other models have incororated time-correlated field data [2] and to exlore the imact of clustering droed doses on the drug resistance outcome [21]. Measuring the true atterns of drug adherence can be challenging [22-25], but models can reresent a range of ossible behaviors in silico to understand the effects of different ossible atterns. The host immune resonse is another source of heterogeneity that, as we will show, contributes substantially to the dynamics of resistance and reversion. Though a wealth of biological studies have revealed the comlex relationshi between immunity and virologic control, few models have exlored the effect of viral relication and immunity on drug efficacy [26,27]. Ultimately, oulation models must cature the imortant outcomes of within-host heterogeneities and react aroriately to a range of assumtions about viral fitness, host immune resonse, drug harmacology, and adherence. Here, we use the examle of tenofovir-based PrEP to exlore the insights that can be gained from considering such heterogeneities in a combined model of within-host virus dynamics, drug harmacokinetics/harmacodynamics (PK/PD), and atterns of drug adherence. Methods We constructed a model that combines the harmacokinetic/harmacodynamic outcome of time-varying drug adherence with the dynamics of cometition between drug-resistant mutants. Hyotheses or data about drug adherence atterns are rovided as inut for the simulation, and the model redicts the time-varying oulations of CD4+ T cells and virus (WT and drug-resistant) in the form of lasma RNA and integrated DNA. The model is deterministic, but can be driven stochastically through the choice of adherence attern. Figure 1 illustrates the comonents of the model and the way in which they are linked. First, an adherence

4 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 3 of 15 a. Adherence attern Based on hyothesized causes and timescales Month b. Pharmacokinetics Liver TDF TFV ME Digestive comartment ME Cell membrane c. Pharmacodynamics R relative to WT TDF= WT K65R M184V TFV MP DP Intracellular TDF-DP (mg/ml) d. Virus dynamics λ c x + v 1 v 2 v n... β k 1 k 2 k n y 1 y 2... y n z d u a a a b e. Virus dynamics λ c x + k 1 k k v 2 n 1 v 2 v n β-ng... y 1 y 2... y n z d u h a h a h a b g w 1 w 2... w n Figure 1 Model schematic. (a) The model first assumes an adherence attern, which is translated into a series of taken or missed doses over time. Doses, if taken, are assumed to be taken at a secified interval, e.g., daily for TDF. (b) A harmacokinetic model of TDF translates the series of doses into a time-varying concentration of TDF-DP in the active intracellular comartment. By including the longer half-life of intracellular TDF- DP comared to TFV, this model catures the harmacologically forgiving roerties of TDF. (c) The relationshi between the TDF-DP concentration and the relication of WT or mutant virus is assumed to be a Hill function following the median-effect model. WT is better able to relicate in the absence of drug, whereas the drug-resistant mutants are able to relicate at higher drug concentrations. K65R has higher fitness than M184V at all drug concentrations, and thus is exected to redominate over M184V in all simulations. (d) The basic virus dynamics model, governed by Equations 2 5, is deicted grahically. The additional factors of forward- and back-mutation due to error in reverse transcrition, not deicted here, are shown in Equations 6 7. (e) The exanded virus dynamics model, which includes a latently infected cell comartment w, is deicted here and in Equations

5 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 4 of 15 attern is created based on a hyothesis about the timescales of adherent and non-adherent time intervals. For examle, adherence could be assumed to be random in time, eriodic with a given duration of adherence and non-adherence, or stochastic based on a Markov model described in detail below. Next, a detailed harmacokinetics model is used to calculate the time-varying concentration of the active form of the drug in the relevant comartment, given the sequence of doses that were taken over time. For oral TDF, the model includes bioavailability, artitioning across the cell membrane, and the two hoshorylation stes required to create the active dihoshate form of the drug. A harmacodynamics model is used to translate the drug concentration into an effect on the relicative caacity of the WT and mutant viral quasisecies. Finally, a dynamic model of viral relication resonds to the changes in relicative caacity caused by fluctuating drug concentrations. Here, uninfected CD4+ target cells (denoted x) are infected by a given viral quasisecies (v i for quasisecies i, e.g., v WT and v mutant ) to form a roductively infected cell (y i ). Infected cells can be eliminated by virus-secific CTLs (z). In an alternative version of this model (Figure 1b), infection can yield long-lived latently infected cells (w i ) with robability g, or roductively infected cells (y i ) with robability 1-g. Latently infected cells roduce no virus and are not suscetible to CTLs until they become activated into roductively infected cells at a slow rate, h. Model arameters All arameter values and associated references are listed in Table 1. Excet for studies of acute infection in the absence of drug, the model was initialized at resumed steady-state values of x ss, y ss, v ss, and z ss for numerical integration. Simulations of acute infection dynamics were initialized at x =1 6, v i = 1, and y i = w i = z =. Adherence atterns A hyothesis or data set related to adherence is converted to a bit string reresenting whether the dose is taken or missed at each time in the dosing schedule. The sequence could otentially be informed by field data, e.g., if a time-correlated data set such as electronic monitoring of bottle oening were used [24,25]. We begin with a simle eriodic model in which the frequency of switching between adherent and non-adherent states is altered while maintaining the same total fraction of missed doses. However, when considering otential causes of oor adherence, neither eriodic nor random/ unstructured missing of doses is fully realistic: there are various reasons for oor adherence with different intrinsic time scales, which can introduce temoral correlations in adherence atterns. As an examle of an intermediate between the fully eriodic and fully random models, we also imlemented a Markov model of discrete drug-taking and drug-missing states, which we call adherence states. Each state has a characteristic dwell time, analogous to models of acket loss in communication channels [28,29]. The time sent in the i th adherence state before switching to a different state was calculated using the Gillesie stochastic simulation algorithm [3,31] as 1 Δt ¼ InðrandÞ ð1þ r i j Xj Ji where J i reresents the set of ossible states that could be entered from i, r i j is the rate of transitioning from state Table 1 Parameter values Parameter Value Units Ref Parameter Value Units Ref v SS 51, coies ml -1 [32] m WT Unitless [46] λ 1 5 cells ml -1 day -1 [32] m K65R Unitless [46] b.27 day -1 [32] IC 5,WT.5* μg ml -1 [46] d.1 day -1 [32] IC 5,K65R 2.2* μg ml -1 [46] u 5 day -1 [32] K a day -1 [41] a.5 day -1 [32] K e 9.6 day -1 [41] c 1 1 cells ml -1 day -1 [32] K 1f 9.6 day -1 [41] β 2f 1 5 ml count -1 day -1 [32] K 1b 3.3 day -1 [41] f Variable (Equation 9) Unitless K 2f 27.7 day -1 [41] h.2 day -1 [41] K 2b 95.5 day -1 [41] w ss /y ss (for g).39 Unitless [72] K m 24 day -1 [41] k WT 1/f day -1 [32] K c 1.1 day -1 [41] k K65R 61.54/f day -1 [46] f lasma-bound.7 Unitless [41] H 18 Unitless [42] * Concentration in lasma.

6 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 5 of 15 i to state j, andrand is a random number uniformly distributed between and 1. After time Δt, the next state was chosen among ossible states J i roortionally to the relative magnitude of the transition rates r i j.finally,the states were discretized into a binary sequence of taken or missed doses by determining whether the individual was in an ART-taking or ART-missing state at the time of each dose. Virus dynamics and immune resonse We used a classic model of virus dynamics [14,32,33] extended to include immune ressure by CTLs [32-34] as shown in Figure 1b. For uninfected CD4+ target cells x, infected cells y, free virus v, and CTLs z, the cometing dynamics of multile quasisecies, indexed by i, are governed by: ẋ ¼ λ dx X i βxv i ð2þ ẏ i ¼ βxv i ay i y i z ð3þ and wild-tye (Equation 7) infected cell oulations are affected by this assumtion as follows: ẏ i; i WT ¼ ð1 qþβxv i ay i y i z þ qβxv WT ð6þ ẏ WT ¼ 1 X! q βxv WT ay WT y WT z i WT þ X qβxv i i WT ð7þ whereas the growth of target cells, free virions, and CTLs were still governed by Equations 2, 4, and 5, resectively. At steady-state ẋ ¼ ẏ i ¼ v i ¼ ż ¼ with the aroximation that βxv i when calculating ν (negligible deletion of virus due to infection of new target cells) and the assumtion of negligible mutation, Equations 2 5 can be solved to estimate the steady-state viral load of the dominant quasisecies [32]: v i ¼ k i y i uv i βxv i ð4þ λk v ss ¼ a þ c b d β ð8þ ż ¼ c bz ð5þ We assumed that the oulation of free viruses and cell-incororated viruses interact with a common ool of uninfected target cells, and that immune ressure was exerted equally on all infected cells. (This assumtion would break down if the drug resistance mutation occurred in an eitoe targeted by two CTLs that may be resent at significantly different levels or subject to different levels of immune regulation.) The steady-state frequency of HIV-secific CTLs z ss = c/b was set to 37 cells/ul, consistent with henotyic analysis of HIV-infected human eriheral blood [35,36]. We assumed z grows at rate c indeendently of z, y, ortheroductzy, thedynamicsof which are described elsewhere [32-34]. We exlored this system under varying because this is a known heterogeneity among hosts: viremia is elevated [37] or suressed [38] in the resence of certain HLArestricted CTL subtyes. Single-oint mutants form during infection of a target cell x by wild tye (WT) drug-suscetible virus v WT, due to reverse transcrition with a erosition mutation robability q. We assumed that each mutant could revert to wild-tye with the same robability. The dynamics of the mutant (Equation 6) Increasing the magnitude of the CTL resonse reduces v ss. To modulate while reserving v ss and the clearance rates of each secies, a multilier f can be alied to modify the original infection rate β into an adjusted rate β = β f, and modify the original virus roduction rate k into an adjusted rate k=k /f. The value of f is calculated from the original arameter values and the desired value of v ss using the equation! f ¼ 1 λk v ss a þ c b d u β ð9þ Changing f alone reserves the overall dynamics of the system, including the basic reroductive ratio of the virus [32]: R ¼ βλk a þ c b ð1þ du We investigated the effect of changing with and without a comensating change in f that reserves v ss at the mean value measured in discordant heterosexual coules in Zambia, 51, coies/ml [39].

7 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 6 of 15 To incororate latency into the model for a subset of the simulations, we added an otional latent comartment w with latency rate g and reactivation rate h (for clarity, shown here using the simlified equations without mutation): ẏ i ¼ ðβ gþxv i þ hw i ay i y i z ð11þ ẇ i ¼ gxv i hw i ð12þ For the dominant subtye, the steady-state level of latently infected cells is aroximately: w ss ¼ g h x ssv ss ¼ g adu λ ð13þ hβ kβ The total body load of activated CD4 + T cells containing incororated HIV DNA has been measured in lymh nodes and blood and estimated to be aroximately 3.1 x 1 7, comared to aroximately 1.2 x 1 7 resting CD4 + T cells with incororated viral DNA [4]. We fixed the ratio w ss /y ss,with λ y ss ¼ a þ c b du βk ¼ u k v ss ð14þ This relationshi allowed us to calculate g for a given value of h to ensure w ss /y s =.39 using g ¼ h βk w ss ð15þ au y ss In the latency simulations shown, we set the latently infected cell reactivation rate h to.2 days -1, reresenting the long-lived comonent of the latently infected cell reservoir [41]. The simlifying assumtions made by this imlementation of latency include (1) a constant low rate of gradual reactivation, in lieu of the wide range of re-activation rates ossible from different cell subtyes, and (2) the roduction of a single activated cell by each latently infected cell, neglecting roliferation in the rocess of re-activation. Pharmacokinetics We imlemented a reviously reorted model by Dixit and Perelson [42] that catures the more harmacologically forgiving nature of intracellular TDF-DP as comared to shorter-lived TDF in lasma. Tenofovir artitions across cell membranes and, inside cells, is hoshorlyated twice into its active antiviral form, TDFdihoshate (TDF-DP) [43]. Imortantly, the half-life of intracellular TDF-DP is 1-fold higher than that of TDF in lasma [44,45]. We used model equations and arameters ublished by Dixit and Perelson [42]. Briefly, we modeled the drug concentration in the digestive comartment (C d ), extracellular comartment (C e ), intracellular comartment (C c ), intracellular singly hoshorylated state (C c ), and the active, intracellular double hoshorylated state (C c ) using the following equations: C d ¼ K a C d ð16þ C e ¼ Ka C d K e C e ð17þ C c C c C c ¼ Km maxð; ð1 mþhc e C c Þ K c C c K 1f C c þ K 1b C c ¼ K c C c þ K 1f C c K 1b C c K 2f C c þ K 2b C c ¼ K c C c þ K 2f C c K 2b C c ð18þ ð19þ ð2þ The rate constants for this model, also taken from Dixit and Perelson [41], are listed in Table 1. Pharmacodynamics The harmacodynamics comonent of the model takes as inut the time-varying drug concentration in its active form (C c ) and determines its time-varying effect on viral relication. We modeled only the single mutants known to confer drug resistance using recently reorted dose resonse data from single-cycle relication studies [46]. Mutation is assumed to incur a fitness enalty that reduces the virus roduction rate k. In the case of the K65R mutant, this rate is reduced by 39% at zero drug concentration. Multile-osition mutants were not modeled. TDF-DP concentrations D(t) in the intracellular comartment are assumed to reduce β through their effect on reverse transcritase. Each mutant is affected according to the median-effect model [47] 1 β i ¼ β m ð21þ 1 þ Dt ðþ IC 5;i where IC 5 is the median-effect dose and m is the Hill coefficient. We used values recently reorted from an in vitro study [46] and multilied the IC 5 by the artition coefficient of TDF across the cell membrane in the harmacokinetic model [42] to reflect the intracellular activity of TDF-DP. Results Heterogeneities in infection and immunity The early transient (acute hase) and steady-state (chronic hase) trajectories of viral load are measurable and vary among individuals, otentially reflecting heterogeneity in

8 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 7 of 15 underlying viral relication arameter values. To understand the insights that may be gained by including these heterogeneities in our model of adherence/resistance (Figure 1a), we began by identifying the levers within the virus dynamics model (Figure 1a and b) that can be used to reroduce these clinically observable heterogeneities. We hyothesized that the immune ressure arameter, alone and in combination with other arameters, could be leveraged to effect these heterogeneities. Observations of viremia during early HIV infection have revealed an early eak in viral load that is 2 logs (±1 log) higher than the setoint viral load [48-52]. The time of the acute hase eak has been reorted in the range of days [53] or 5 19 days [54]. Some models lacking an immune comartment attribute the eak [55] and setoint [56] viremia exclusively to the deletion of uninfected target cells rather than immune ressure. While such models could still achieve a 2-log reduction in viremia from eak to setoint, the deletion of uninfected target cells is extreme when this dro is driven by target cell deletion alone, with nearly all CD4+ T cells ending u in the infected state. An immune resonse comonent was required to reserve a substantial fraction of uninfected target cells while roducing a characteristic acute hase, both of which are features of realistic infection dynamics. Furthermore, correlations between CTLs and viremia in humans [57-59], as well as direct studies of CD8+ T cell deletion in rimates [6-62], rovide evidence for an imortant role for CTLs in bringing viral load down to setoint. Therefore, we chose a model wherein both target cell deletion and CTLmediated killing contribute to setoint and acute hase virus dynamics. We then hyothesized that variability in immune resonse among hosts could, even in a simle model, reroduce the range of ossible viral load trajectories among individuals. Increasing the immune ressure arameter reduced setoint viral load with little influence on the acute hase (Figure 2a), which occurs during the ram-u in CTL frequency and therefore is subject to lower levels of immune ressure comared to steady-state levels. As redicted by Equation 1 and illustrated in Figure 2b, increasing reduces the basic reroductive ratio (R )of both wild-tye and mutant virus. Because mutant virus incurs a higher fitness cost in the absence of drug, the level of required to suress the mutant virus oulation (R < 1) is lower for the mutant than for the wildtye virus. The steady-state viral load declines with increasing, roviding a lever to modulate setoint viral load over the range of hysiologically relevant values. A second, indeendent lever was required to modulate the acute hase viral load trajectory. This is because viral load during the acute hase is not redictive of setoint viral load [53,63] nor rate of AIDS rogression [5,63], although AIDS rogression is correlated with setoint viral load [64-66] and symtoms such as fever, vomiting, diarrhea, and headache during acute infection [5]. Frequent longitudinal samling of atients with an estimated date of infection has revealed three equally common atterns of viral load trajectories: raid decline to setoint, a slow shoulder with gradual decline to setoint, and an initial interval of viral suression followed by a rebound of viremia, which takes over 9 days to return to setoint [53]. With this range of biological variability in mind, we searched for a lever that can vary the shae (height, timing, and shoulder ) of the acute hase viral load trajectory without changing setoint viral load. Sensitivity analysis of the virus dynamics model (Additional file 1: Table S1) found the acute hase eak time to be most sensitive to β and k, and the acute hase eak height to be most sensitive to k (closely followed by λ and u). Dividing k and multilying β by a factor f reserves the number of virions roduced with each relication cycle, but modulates the setoint viral load. To kee the setoint viral load constant, we used f to offset the change in setoint viral load caused by, changing the two arameters together as er Equation 9. We found that these counterbalanced arameters modulate the eak time, magnitude, and settling time of the eak of acute hase viremia while maintaining a constant setoint viral load (Figure 2c-d). A biological interretation of f might be ascribed to variability in non-ctl comonents of the host immune resonse, such as the relative magnitude of the neutralizing antibody resonse, which reduces β, and innate immune resonses such as antiviral factors, which reduce k. Both levers described here rely on the immune ressure arameter, an exected source of heterogeneity among hosts. We next asked how influences the resonse to antiretroviral theray and the risk of resistance, using the context of misused tenofovir-based PrEP as a relevant scenario. Resistance and reversion of mutations: effect of immune resonse Human studies of TDF monotheray have not agreed on a characteristic timescale for the emergence of resistance. Two human studies of daily TDF, lasting 21 days [67] and 28 days [68] resectively, did not find evidence of resistance. In the rhesus macaque SHIV model, TDF monotheray elicited K65R mutations with associated viral rebound in all animals within 2 to 12 weeks (median 4 weeks) of theray initiation [69]. This imlies that the human trials were likely too short to detect the emergence of resistance. We therefore used our model to simulate longer timescales of drug exosure.

9 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 8 of 15 a coies/ml or cells/μl WT viral load K65R viral load uninfected CD4 infected CD4 CTL b c R Final v WT R = 1 WT K65R x 14 x 1 5 d coies/ml or cells/μl WT viral load K65R viral load uninfected CD4 infected CD4 CTL WT VL at eak Days to eak x x x x 1 5 Figure 2 Influence of CTL efficacy on the dynamics of early infection with changing or constant setoint viral load. (a) Coies/mL of WT virus (black) and drug resistant mutant K65R (red), and number/μl of CTLs (green), uninfected CD4+ cells/ml (blue) and infected CD4+ cells/ ml (magenta), for ten values saced between and 1-4. Lighter shading corresonds to smaller values of, and arrows oint in the direction of increasing to show its influence on the trajectories. (b) Calculated estimate of the basic reroductive ratio for WT virus and K65R mutant as a function of. (c) Simulated WT viral load after 5 days, sweeing over while holding all other arameters constant, reveals that the final viral load dros toward zero as reduces the R of WT toward 1. (d) Increasing and corresondingly decreasing f according to Equation 9 maintains the steady-state viral load at 51, coies/ml, but raises the acute hase eak and leads to a longer shoulder before the viral load reaches setoint. (Same color scheme as (a), but with a timescale of only 6 days to show the details of acute hase dynamics.) The effects on the height and timing of the eak are shown in (e) and (f), resectively, over the same range of as in (b) and (c). e f Using a harmacokinetic model by Dixit and Perelson [42] to translate 3 mg daily dosing of TDF into intracellular dihoshate concentrations, our model confirmed that TDF monotheray transiently suresses viremia, but then selects for the K65R mutant. Figure 3a shows our reroduction of the Dixit and Perelson harmacokinetic model, emhasizing the imortance of modeling intracellular drug secies, which exhibit a longer half-life and therefore higher concentrations than those found in lasma. Figure 3b shows an examle of rolonged adherence to TDF monotheray initiated during HIV infection, with set to Consistent with exerimental data, resistance begins to emerge after 5 days, with the K65R mutant dominating the viral oulation at a lower viral load and a higher CD4 count than WT infection due to the mutant s reduced relicative caacity. This reduced fitness drives reversion to a WT-dominated infection after cessation of TDF monotheray. To investigate the effect of immune ressure on resistance and reversion rates, we next analyzed the effect of varying, balanced with f (er Equation 9 and similarly to Figure 2c), on resistance and reversion times for a fixed interval of TDF monotheray. We defined resistance as the first time K65R viral load reaches 5 coies/ ml, and reversion as the crossover oint between K65R and WT viremia. We analyzed a range of that ermits growth of the K65R mutant according to its R at eak and trough TDF-DP concentrations, shown in Figure 3c. Note that the harmacokinetic model with intracellular TDF-DP ensures that the eak-to-trough change in active drug concentration only brings R of the WT above 1 for a very narrow range of, illustrating the harmacological forgiveness of daily TDF for most levels of immune ressure. Increasing lengthened the delay time until resistance from 29 days to ever-increasing values as R of K65R declined toward 1, as shown in Figure 3d. The time from monotheray cessation until reversion to a WTdominated infection declined with increasing over the same range. Reversion required a maximum of 37 days in the absence of CTL activity with set to zero. As increasing drove the R of K65R toward 1, this duration droed toward a minimum of 15 days (Figure 3e).

10 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 9 of 15 a. b. TDF concentration (mg/ml) dcd dcb dcc dcc dcc viral load (RNA coies/ml) 1 x 14 5 Intracellular TDF WT VL K65R VL uninfected CD4 infected CD4 1 5 cells/µl days since TDF monotheray initiation c. R R = 1 WT eak K65R eak WT trough K65R trough d. days until mutant VL > e. days until reversion x x x 1 5 Figure 3 Effect of immune ressure on rate of develoment and reversion of drug resistance. (a) Concentration of digestive (C d ), lasma (C b ), intracellular (C c ), intracellular hoshorylated (C c ), and intracellular dihoshorylated (C c ) tenofovir using the model of Dixit and Perelson [42] that includes drug artitioning across the cell membrane. (b) Examle of viral load and CD4+ T cell resonse to TDF monotheray with erfect adherence and cessation after 3 days. TDF-DP concentration, re-scaled from (a) in units of μg/ml along the righthand axis, is overlaid for reference. (c) Basic reroductive ratio of WT and drug-resistant virus at steady-state eak and trough TDF-DP concentrations, as a function CTL efficacy varied with f to maintain a setoint viral load of 51, coies/ml. (d) Influence of, varied as in (c), on the time from HIV infection on TDF monotheray until emergence of the K65R mutant at a viral load above 5 coies/ml. The time to resistance is 29 days at =, and rises to ever-increasing durations as reduces R. (d) Influence of on the time until reversion of resistance after monotheray cessation. Reversion times range from 37 days at = to 15 days as R aroaches 1. Thus, increasing immune ressure led to slower aearance and more raid reversion of the drug-resistant mutant. Intuitively, this result can be construed as a balance between immune ressure, which favors the WT due to the mutant s fitness cost, and harmacological ressure, which favors the mutant. In the drug-free condition shown in Figure 2b, a sufficiently high value of drives the mutant R below 1, only allowing the WT to survive in the absence of drug. With daily TDF dosing, it is the mutant that can survive for an intermediate range of, and this range occurs at overall lower values of comared to the drug-free state as a result of harmacological ressure exerted on both quasisecies. The deendence of resistance/reversion rates on immune strength may have further imlications on the effect of late entry into theray, or, in the case of PrEP, whether the accidental misuse of PrEP occurred during initial or early infection (due to breakthrough infection or false negative HIV tests during the window eriod of early infection) or well into infection (due to unauthorized drug use or failure of HIV testing). Long-term effect of monotheray: latent reservoirs The net effect of increasing in the short term is a shorter interval of resistance-dominated infection, i.e., fewer viral relication cycles involving the drug-resistant mutant. However, the model as used thus far allows the system to raidly return to equilibrium after cessation of monotheray, yielding no long-term imlications of transient drug resistance. To investigate these long-term imlications, we included another key feature of HIV infection: the incororation of virus into long-lived latently infected cells caable of re-activating after months or years of suressed viral relication. For each mutant, we included a latently infected cell comartment w i, as shown in Equations 6 7 and illustrated in Figure 1c. We hyothesized that slow reactivation rates would endow this comartment with a memory of the virus subtyes that have undergone rior relication cycles, such that more intense and rolonged infection with drug-resistant virus would increase the fraction of drug-resistant mutants available for future re-activation. During non-suressed viremia, the contribution of this comartment would be negligible

11 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 1 of 15 comared to rates of forward- and back-mutation. However, during fully suressive theray, the latent comartment would become the sole source of new virions, and therefore would govern the roduction rate of drugresistant mutant virions. To test this hyothesis about the dynamics of the latency-embellished model, we simulated an interval of monotheray, followed by one decade of untreated infection and, finally, an interval of rolonged suressive theray. The delay of one decade allows the very longterm effects of ast drug resistance to be evaluated. Suressive theray is assumed to be a regimen against which the single mutations that exist in all infected individuals do not confer resistance, such as trile-drug highly active antirectroviral theray (HAART). (Any given single-osition mutant is almost guaranteed to exist in an infected individual due to the high error rate of reverse transcritase, but the re-existance of a mutant resistant to all comonents of HAART is unlikely [32]). Rather than modeling the harmacokinetics of all three drugs in detail, we used a simle caricature for suressive theray in which we reduced the burst size of the virus by 1-fold. As shown in Figure 4a, this raidly drove the free virus and the activated, roductively infected cell oulations toward extinction. When the long-lived latent comartment w was added, as in Figure 4b-d, the latent reservoir maintained a very lowlevel viremia, reresenting the source of otential reactivation of virus that is observed after treatment cessation or failure. The level of drug-resistant mutants resent in this comartment (w mutant ) grows during monotheray when relication of the mutant is favored over that of the wild-tye, and declines in the absence of monotheray while being gradually relaced with wild-tye latently infected cells (w WT ) until it reaches its equilibrium concentration. At equilibrium, both the latent and the active infected cell oulations are rimarily maintained through forward mutation of the wild-tye virus due to its error rate q. When only 3 days of monotheray are used to initially grow w mutant, this equilibrium level is reached within a decade, as shown in Figure 4b. When monotheray is alied for longer intervals, such as 9 days (Figure 4c) or 1 year (Figure 4d), w mutant requires over a decade to reach equilibrium. Thus, as exected and shown in Figure 4e (grahed at the values achieved after 2 years of suressive theray), the fraction of the latent reservoir harboring drug resistance during the final eriod of suressive theray increases with increasing duration of initial monotheray, excet for short monotheray exosures (<5 days) for which equilibrium is reached during the intervening decade. This threshold time would be shorter for shorter intervening sans between monotheray and suressive theray. The fraction of drug-resistant free virus roduced from the latent reservoir exhibits the same trend as the fraction contained in the latent reservoir, but is lower due to the reduced relicative caacity of the mutant, as shown in Figure 4f. The reviously discussed examles were conducted with immune ressure, set to As a final ste in analyzing the behavior of our extended model with latency, we varied, which reduced the fraction of drugresistant mutant in the latent comartment (Figure 4g) and in the free virus roduced by the latent comartment (Figure 4h). As with the monotheray duration, the trends were the same for latently infected cells and free virus roduced, but the fraction of mutant virus was lower in the free virus ool due to the reduced relicative caacity of the mutant. At a sufficiently large, no further reduction in the fraction of mutant was observed. Similar to the minimum duration of monotheray required for variation in Figure 4e-f, this threshold occurs because back-mutation of the wild-tye virus became the rimary source of drug-resistant mutants. We hyothesize that incororation of drug-resistant virus into the long-lived latent reservoir could exlain why ast exosure to subclinical drug concentrations (e.g., single-dose neviraine administration for revention of mother-to-child-transmission) can lead to increased risk of treatment failure long after the initial drug exosure has waned and any raidly-relicating viral oulations should have returned to equilibrium. Thus, minimizing the total exosure to drug-resistant viremia would reduce the incororation of these mutants into the latent ool. Patterns of adherence: effect on drug resistance dynamics We have already learned that increased immune ressure can reduce exosure to resistance by delaying its onset and hastening its reversion. However, the attern of adherence to a drug regimen, including accidental PrEP during infection, is another imortant source of heterogeneities that may influence the total exosure the drug resistance and thus influence the long-term outcome. We therefore finish with an exloration of drug adherence atterns and their effect on total exosure to drug-resistant viremia. Unlike suressive theray, for which auses in treatment lead to subclinical drug concentrations that drive resistance, accidental PrEP use during infection can drive resistance even with erfect adherence. Even so, we found that the dynamics of adherence determine the roortion of time sent with drug resistance. For a fixed fraction of doses taken, this deends on the duration of adherent and non-adherent sans another variable is the degree of eriodicity or stochasticity in these durations. In the examles shown, we used a simle assumtion of 5% adherence with immune ressure fixed at 1-5, a value at which

12 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 11 of 15 log 1 cells/ml a 1 5 No Latency monotheray no theray suressive ther. WT v mutant v mutant y b log 1 cells/ml 5 3 Days Monotheray mutant w WT v mutant v mutant y years years log 1 cells/ml c 5 9 Days Monotheray mutant w WT v mutant v mutant y d log 1 cells/ml 5 1 Year Monotheray mutant w WT v mutant v mutant y fraction drug resistant virus e x years latent reservoir free virus days exosure to monotheray f fraction drug resistant virus x years latent reservoir free virus x 1 5 Figure 4 Long-lived latently infected cells remember drug resistance. Exeriments begin with a variable duration of daily TDF monotheray alied with erfect adherence (ink regions), followed by a decade without treatment (yellow region), and finally, five years of suressive theray (1- fold reduction in viral relication, blue region). (a) In the model without latency, counts of free virus (v, solid line for WT and dashed line for mutant) and cells with incororated viral DNA (y, dotted line for mutant) are shown for a 15-year eriod beginning with 9 days of daily TDF monotheray. The system returns to equilibrium shortly after monotheray cessation. When a latently infected comartment w is added, the number of cells with latently incororated mutant virus grows throughout a monotheray exosure of 3 days (b),9days(c), or 365 days (d).forthedecadefollowingmonotheray exosure, the reservoir declines while being relaced with WT latently infected cells. When suressive theray is alied for the final five years, the latent reservoir rovides the dominant source of virus. In (b), the level of mutant virus lateaus aroximately 7 years after monotheray cessation because the latent reservoir becomes a smaller source of mutants than back-mutation of relicating virus. (e) Longer monotheray exosure roduces more mutant virus in the latent reservoir after two years of suressive theray. The fraction of mutant free virions, though also increased by long monotheray, is lower than the fraction in the latent comartment due to the reduced relicative caacity of the mutant. (f) For a 9-day interval of monotheray, increasing immune ressure reduces the fraction of drug-resistant free virus and latently infected cells, but loses any incremental effect at >1-5.Thisisbecause, similarly to (b), back-mutation of relicating WT virus becomes a larger source of latently incororated mutants during the untreated (yellow) interval. resistance occurs in 37 days of constant monotheray, and reversion after rolonged monotheray occurs after 24 days. We comared a erfectly eriodic schedule to the oosite extreme of a Markov rocess. For eriodic 1-day transitions between adherence states, resistance cannot develo in a single san of monotheray, but leads to eventual and sustained resistance with no oortunity for reversion (Figure 5a, left anel). Longer durations of adherence states ermit reversion of resistance, leading to a smaller roortion of time sent in the resistant state (Figure 4a).

13 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 12 of 15 a cells/μl or coies/ml days days days TDF infected CD4 uninfected CD4 WT viral load b time since infection (days) c time since infection (days) time since infection (days) K65R viral load cells/μl or coies/ml time since infection (days) Figure 5 Adherence atterns influence the dynamics of drug resistance. (a) Dynamics of drug resistance assuming 5% adherence with eriodic adherent and non-adherent sans lasting 1 days (left anel), 3 days (middle anel), or 5 days (right anel). TDF-DP concentration is overlaid in gray for reference. (b) Examle of how a Markov model for drug adherence could reresent underlying drivers of missed doses. (c) Dynamics of drug resistance using a two-state Markov model with exonentially distributed duration sent in the adherent and non-adherent states, each with an average duration of 3 days. This is the stochastic version of the eriodic model in the middle anel of (a), roviding an examle of how such a model could be driven stochastically to reflect this asect of real-life adherence atterns. In a stochastically driven model, elaborate Markov chains can be constructed with this model to reresent different causes of adherence that may lead to different durations of dose-taking and dose-missing, as shown in Figure 5b. A simle, two-state Markov model with transition rates of (3 days) -1 allows for wide stochastic variation in the time sent in adherence states, including occasional longer intervals sent in states that favor drug resistance (Figure 5c). These longer dwell times may be the main drivers of resistance in individuals whose time to resistance is long comared to the average transition rate between states (e.g., due to a higher immune ressure ). The atterns of adherence articularly, whether reversion can occur within the intervals of non-adherence can therefore influence the overall exosure to drug resistance even for the same total fraction of doses missed. Knowing that immune ressure influences the required time for develoment and reversion of resistance, the critical dwell times in the dose-taking and dose-missing states would vary among individuals, such that oulations with different immune henotyes may exhibit different resonses to articular atterns of drug adherence. Through our latency model, we further hyothesize that these changes in short-term resistance dynamics may influence treatment outcomes long after the exosure to drug resistance. Discussion Modern oulation models of HIV transmission are limited in their ability to redict the imact of drug resistance on individuals and the eidemic, because their reresentation of the develoment of drug resistance, if any, is simlistic. This simlification is chosen in art for lack of a way to model within-host dynamics in a manner that is consistent with observables such as viral load, which is heterogeneous across individuals, oulations, and time eriods [7]. We have shown how observable quantities such as setoint and acute eak viral load can be indeendently tuned in such a model over the range of values observed in atients [53,63]. Modulation of the arameters and f in our model can recaitulate common atterns of early viremia: raid decline to setoint, a slow shoulder with gradual decline to setoint, or an initial interval of viral suression followed by a rebound of viremia, which aroaches setoint after 9 days or more [53]. The kinetics of tenofovir resistance due to K65R mutation are not fully characterized in humans [67-69]. By accounting for adherence atterns, harmacokinetics, and the cometition between quasisecies at time-varying drug concentrations, we were able to model the deendence of resistance and reversion rates on heterogeneities in host immunity and atterns of drug adherence. At

14 Bershteyn and Eckhoff BMC Systems Biology 213, 7:11 Page 13 of 15 constant overall levels of adherence, we showed the relationshi between the duration of dose-taking and doseskiing intervals and the resulting roortion of time that K65R dominated the infection. We also examined the extremes of eriodic versus exonentially distributed transitions between drug-taking and drug-missing states. Reality is likely somewhere in-between: for examle, lack of a rescrition refill may cause monthly transitions into a drug-missing state, yet the transition back to a drugtaking state may be more variable, deending on the accessibility of the harmacy, availability of drugs, transortation resources, and other factors. Without a long-lived latent comartment, our model redicted raid reversion of resistance with no longerterm imact on the host. With fully suressive theray, the oulations of both WT and mutant virus dro toward extinction, as back-mutation of one quasisecies can no longer contribute to sustaining the others. In this state, reactivation of long-since-incororated latent virus becomes a significant contributor to the viral ool. By including a latent comartment as has been described [32,33], we were able to observe increasing roortions of K65R mutant in both the latent reservoir and the free virus oulation as the duration of TDF exosure increased, and as immune ressure decreased. This model feature rovides a otential link between ast monotheray exosure (e.g., misuse of PrEP or single-dose neviraine administration for revention of mother-to-child-transmission) and future suscetibility to treatment failure. It exlains why ast exosure to monotheray may increase the risk of treatment failure many years later. Further, it allows for host heterogeneities in immune resonse to be considered in evaluating the risk of this outcome. Our analysis has thus far been limited to single-osition mutations. In art, this is driven by the availability of detailed in vitro dose resonse data for single-osition mutants [46]. Additionally, the mutation rate of HIV is high enough that any given single-oint mutant is likely to exist within a atient by the end of the acute hase. In contrast, fewer than one-third of double mutants are resent during a WT-dominated infection, and a given trile-mutant is unlikely to already exist somewhere in the body [71]. In such situations, a stochastic simulation with discrete viral counts would be required to account for small oulations such as rare mutants. In the resent analysis, we deliberately chose scenarios in which our continuous deterministic model would agree with the outcome of a stochastic model. We confirmed this through stochastic simulation of a subset of the exeriment oints. Although we found that simulations of a single milliliter did exhibit secies fade-out, simulations of aroximately 1% of the total system volume fully aroached deterministic model results. Alication of stochastic modeling to new areas in which the results could deviate from deterministic models, such as multi-osition mutants, is an area of ongoing work. Conclusions To date, no model has linked the oulation-level dynamics of HIV transmission with a reresentation of withinhost interactions of HIV virus dynamics, host immune resonse, and drug PK/PD [7]. We have shown that host immunity and atterns of drug adherence are imortant drivers of individual-level risks and dynamics of drug adherence. Understanding how within-host models can cature among-host heterogeneities is an imortant ste toward the milestone of bridging within-host and oulation-level models of HIV drug resistance. Additional file Additional file 1: Table S1. Sensitivity analysis of the virus dynamics model (without antiviral drugs). Parameters of the model (left column) are systematically decreased by 1%, then increased by 1%. The ercent change in the steady-state values for WT viral load, WT-infected cell count, K65R mutant viral load, and K65R mutant-infected cell count are shown. Additionally, the ercent change in the eak viral load value during the acute hase, and of the time of this eak, are shown. Steadystate values are the most sensitive to changes in k, λ, and u, while acute hase dynamics are most sensitive to changes in β, k, and u. Parameter values are set as in Figure 1/Table 1, with = 1.5x1-5. Table S2. Sensitivity of resistance and reversion times to model arameters. The time of emergence of drug resistance is sensitive to the infection rate, viral roduction rate, and viral clearance rate (β, k, and u), as well as f, which combines changes in β and k such that steady-state viral load remains constant. In is also sensitive to the roduction and clearance of target cells (λ and d), clearance of infected cells (a), and the roduction, clearance, and killing ower of immune cells (b, c, and ). Parameter values are set as in Figure 2/Table 1, with = 1.5x1-5. Cometing interests The authors declare that they have no cometing interests. Authors contributions AB and PE constructed the model, ran the model, and analyzed results. AB drafted the manuscrit. AB and PE edited and aroved the final manuscrit. Acknowledgements We gratefully acknowledge Dr. C. Scott Wylie, Dr. Joshua L. Proctor, Dr. Daniel J. Klein, Dr. Erez Leiberman Aiden, and Dr. Aviva Presser Aiden for helful discussions during the develoment of this model. The authors thank Bill and Melinda Gates for their active suort of this work and their sonsorshi through the Global Good Fund. Received: 5 June 212 Acceted: 16 January 213 Published: 4 February 213 References 1. Taylor S, Boffito M, Khoo S, Smit E, Back D: Stoing antiretroviral theray. AIDS 27, 21: Hurt CB, Eron JJ, Cohen MS: Pre-Exosure Prohylaxis and Antiretroviral Resistance: HIV Prevention at a Cost? Clin Infect Dis 211, 53: Karim QA, Karim SSA, Frohlich JA, Grobler AC, Baxter C, Mansoor LE, Kharsany ABM, Sibeko S, Mlisana KP, Omar Z, Gengiah TN, Maarschalk S, Arulaan N, Mlotshwa M, Morris L, Taylor D, On behalf of the CAPRISA 4 Trial Grou: Effectiveness and Safety of Tenofovir Gel, an Antiretroviral