^{1}

^{1}

^{2}

^{*}

Conceived and designed the experiments: PA NMD. Performed the experiments: PA. Analyzed the data: PA NMD. Contributed reagents/materials/analysis tools: PA NMD. Wrote the paper: PA NMD.

The authors have declared that no competing interests exist.

New antiretroviral drugs that offer large genetic barriers to resistance, such as the recently approved inhibitors of HIV-1 protease, tipranavir and darunavir, present promising weapons to avert the failure of current therapies for HIV infection. Optimal treatment strategies with the new drugs, however, are yet to be established. A key limitation is the poor understanding of the process by which HIV surmounts large genetic barriers to resistance. Extant models of HIV dynamics are predicated on the predominance of deterministic forces underlying the emergence of resistant genomes. In contrast, stochastic forces may dominate, especially when the genetic barrier is large, and delay the emergence of resistant genomes. We develop a mathematical model of HIV dynamics under the influence of an antiretroviral drug to predict the waiting time for the emergence of genomes that carry the requisite mutations to overcome the genetic barrier of the drug. We apply our model to describe the development of resistance to tipranavir in

The ability of HIV to rapidly acquire mutations responsible for resistance to administered drugs underlies the failure of current antiretroviral therapies for HIV infection. The recent advent of drugs that offer large genetic barriers to resistance, e.g., tipranavir and darunavir, presents a new opportunity to devise therapies that remain efficacious over extended durations. The large number of mutations that HIV must accumulate for resistance to drugs with large genetic barriers impedes the failure of therapy. Further, these drugs appear to exhibit activity against viral strains resistant to other drugs in the same drug class, thereby significantly improving options for therapy. Rational identification of treatment protocols that maximize the impact of these new drugs requires a quantitative understanding of the process whereby HIV overcomes large genetic barriers to resistance. We develop a model that describes HIV dynamics under the influence of a drug that offers a large genetic barrier to resistance and predict the time of emergence of viral strains that overcome the large barrier. Model predictions provide insights into the roles of various evolutionary forces underlying the development of resistance, quantitatively describe the development of resistance to tipranavir

Current antiretroviral therapies for HIV infection often fail to elicit lasting virological responses in patients because of the emergence of multidrug resistant strains of HIV

Current treatment guidelines for HIV infection recommend a combination of 3, but at least 2, active drugs, (i.e., drugs for which resistance has not developed) in order partly to increase the overall genetic barrier of therapy

For second-line therapy, which follows the failure of the initial regimen, a drug from a new drug class is recommended in order to minimize the risk of cross-resistance

Description of the development of resistance to a drug with a large ^{6}, or 64, distinct strains (see below) may emerge in the course of infection. Because HIV is diploid, the 64 strains yield 64 homozygous and 2016 different kinds of heterozygous virions, whose evolutionary dynamics must be followed to describe how the genetic barrier of tipranavir is overcome. Second, the population size of HIV

Here, we develop a model of HIV dynamics that quantitatively predicts the expected waiting time for the emergence of genomes that carry the requisite mutations for resistance to a drug with any given genetic barrier. Extant models of HIV dynamics assume that deterministic forces are predominant in the emergence of drug resistance

We consider uninfected cells, _{jh}_{10}, for instance, are indistinguishable from virions _{01}, we impose the constraint _{jh}_{i}_{i}_{ij}_{i}_{ii}_{ii}_{ij}_{ii}_{jj}_{ij}

Non-infectious virions are crossed.

We construct dynamical equations to predict the time-evolution of various cell and viral populations and estimate the average waiting times for the first production of each of the

We solve model equations to describe the development of resistance in _{0} uninfected cells are exposed to viruses in the presence of a known concentration of the PI. Infection is allowed to progress until time _{p}_{0} uninfected cells in the next passage. At the start of the first passage, the viral population is assumed to consist of _{00} wild type viruses, highly susceptible to the drug. Gradually, genomes with increasing levels of drug resistance emerge.

We perform calculations for a genetic barrier _{0}, and against the strain with _{n}_{m}^{*}^{*}^{*}^{*}^{*}

The inset shows the corresponding fitness ( = 1−ε_{m}

The time evolution of (A) the number of uninfected cells (red), infected cells (blue), and infectious virions (green) and (B) homozygous virions carrying wild-type genomes (pink) and single (blue), double (green), triple (orange), quadruple (red), and quintuple (black) mutants, obtained by solving Eqs. (1)–(9) with the parameters _{0} = 10^{6} cells, _{00} = 5×10^{5} virions, _{m}

In _{1}<ε_{0}; _{2}<ε_{1}<ε_{0} (

Several characteristics of drugs, viz., the genetic barrier,

To examine the influence of the genetic barrier, we vary _{0}, ε_{n}_{0} and ε_{n}

The expected waiting time for the emergence of (A) genomes with different numbers of resistance mutations for different ^{th} mutants as a function of _{0} = 0.85 and ε_{n}_{0} = 0.1 and ε_{n}

The increase of

Interestingly, recombination decreases

That recombination invariably lowers ^{th} mutants. We find interestingly that recombination increases _{0}, ε_{n}

We apply our model to describe the development of resistance to tipranavir in _{50} values for different intermediate mutants from the ranges determined experimentally (_{i}

The different mutants and the corresponding _{50} values are listed in _{i}

Current models of HIV dynamics successfully predict short-term changes in the plasma viral load in patients undergoing therapy but fail to provide a quantitative description of the emergence of drug resistance

Simulations of viral evolution, based on models of population genetics, consider finite populations and present descriptions of the stochastic emergence of drug resistant genomes

Here, we develop a model that employs the deterministic framework of models of HIV dynamics and at the same time captures the influence of stochastic effects associated with the emergence of drug resistant genomes. To accomplish this, we invoke the concept of the expected waiting time. We develop a detailed description of mutation and recombination between multiple loci, which enables calculation of the probability of the formation of resistant genomes in one replication event. Given the viral and cell populations and the efficacy of the drug, the frequency of replication events and hence the rate of formation of resistant genomes is determined. From the rate of formation, we estimate the expected waiting time for the first resistant genome to emerge. Different mutant genomes are assumed to appear first in the viral population at their respective expected waiting times. The limitation of current models of HIV dynamics, which predict the emergence of resistant genomes immediately upon the start of therapy, is thus overcome. Yet, by calculating the “expected” waiting time, our model captures the influence of stochastic effects associated with the emergence of resistant genomes in an averaged sense and retains the dynamical framework of current models. The limitations of population genetics based simulations are also thus overcome.

The waiting time for the emergence of a genome carrying a certain number of mutations depends on the times of emergence and the growth of subpopulations of genomes with fewer mutations. Our model assumes that the latter genomes emerge at their expected waiting times. Consequently, the variation in the waiting times for the emergence of higher mutants due to the variation in the times of emergence of lower mutants is suppressed in our model. Further, following emergence, particularly when the population size is small, the chance that stochastic forces cause the extinction of genomes may be significant. We assume, however, that following emergence, the growth of genomes is deterministic. The extent of the uncertainties introduced in our model predictions by these simplifying assumptions remains to be estimated. Semi-stochastic simulations, where the times of emergence of mutant genomes alone are determined stochastically, and fully stochastic simulations (see, e.g.,

Model predictions indicate that the waiting time,

When distinctions between different viral genomes are ignored, the expected waiting time vanishes and our model reduces to the basic model of HIV dynamics, which successfully captures viral load changes in patients undergoing therapy

We present equations below that describe the

_{T}_{i}^{n}_{0} = 10^{6}.

_{0}_{jh}_{jh}_{i}_{0}_{i}_{jh}_{jh}_{i}

The rate _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

_{i}_{ii}_{ii}_{1}(<_{0}) be the mean rate constant for the infection of singly infected cells

For cells infected with two different kinds of proviruses, we write_{ij}_{i}_{j}_{ii}_{ii}_{ij}_{ij}_{ii}_{ij}

_{i}_{i}_{i}_{i}

We recognize that the evaluation of the waiting times requires knowledge of the rates, e.g., _{i}

To evaluate the probability _{i}_{k}_{ik}^{d}

To determine _{k}_{1}, _{2}, etc., between the _{des}^{th} distinctive site, then

To calculate the probability of mutation, _{ik}

_{ii}_{ij}_{i}_{ii}_{ii}_{ij}_{ii}_{jj}_{ij}_{ii}_{00} alone exist at the start of the first passage. For every subsequent passage, the free virions at the end of the previous passage are employed to initiate infection.

The efficacy of a PI is the fraction of progeny virions that it renders non-infectious. We assume that the drug efficacy ε_{ii}_{ii}_{m}_{0}, and that against the strain with _{n}_{m}_{i}_{ii}_{ii}_{ij}_{ij}_{ii}_{jj}

When the efficacy is determined as a function of the drug concentration, _{ii}_{50}(_{50}(_{ii}

Equations (1) to (9) represent a model of HIV dynamics that describes the development of resistance to a PI with a genetic barrier

We employ the following parameter values based on earlier studies ^{−1} and _{T}^{−1}; the death rate of infected cells, ^{−1}; the viral burst size, ^{3}; the viral clearance rate, ^{−1}; the second order rate constants of the infection of uninfected and singly infected cells, _{0} = 10^{−8} day^{−1} and _{1} = 0.7_{0}; the mutation and recombination rates, ^{−5} per site per replication, and ^{−4} crossovers per site per replication.

Model predictions of the times of emergence of various mutants as the genetic barrier varies from

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Model predictions of the time of emergence of nth mutants for different epistatic interactions,

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Schematic representation of the production of genome _{k}(jh)_{des}(1)_{1}, the probability of which we write as _{des}(2) = P_{even}(l_{1})_{des}(2) = P_{odd}(l_{1})_{k}(jh)_{des}(m)_{even}(l) = exp(−ρl)cosh(ρl)_{odd}(l) = exp(−ρl)sinh(ρl)

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Sequences resistant to tipranavir and their _{50} values. Sequences with different combinations of resistance mutations observed experimentally, corresponding binary sequences illustrating the specific locations of mutations, marked as 1, when _{50} values employed in our model are listed. The experimental _{50} values _{50} values to genomes as follows. To each genome _{50} value equal to the _{50} of the genome below that has the maximum number of mutations in common with the genome _{50} of 101 nM, whereas the genome 000110 is assigned an _{50} of 60 nM, equal to the wild-type.

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Drug concentrations employed in the experiments

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Estimates of waiting times

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We are grateful to Elizabeth Wardrop and Michael G. Cordingley for providing details of their experimental protocol.