The ability to accelerate the accumulation of favorable combinations of mutations renders recombination a potent force underlying the emergence of forms of HIV that escape multi-drug therapy and specific host immune responses. We present a mathematical model that describes the dynamics of the emergence of recombinant forms of HIV following infection with diverse viral genomes. Mimicking recent in vitro experiments, we consider target cells simultaneously exposed to two distinct, homozygous viral populations and construct dynamical equations that predict the time evolution of populations of uninfected, singly infected, and doubly infected cells, and homozygous, heterozygous, and recombinant viruses. Model predictions capture several recent experimental observations quantitatively and provide insights into the role of recombination in HIV dynamics. From analyses of data from single-round infection experiments with our description of the probability with which recombination accumulates distinct mutations present on the two genomic strands in a virion, we estimate that ∼8 recombinational strand transfer events occur on average (95% confidence interval: 6–10) during reverse transcription of HIV in T cells. Model predictions of virus and cell dynamics describe the time evolution and the relative prevalence of various infected cell subpopulations following the onset of infection observed experimentally. Remarkably, model predictions are in quantitative agreement with the experimental scaling relationship that the percentage of cells infected with recombinant genomes is proportional to the percentage of cells coinfected with the two genomes employed at the onset of infection. Our model thus presents an accurate description of the influence of recombination on HIV dynamics in vitro. When distinctions between different viral genomes are ignored, our model reduces to the standard model of viral dynamics, which successfully predicts viral load changes in HIV patients undergoing therapy. Our model may thus serve as a useful framework to predict the emergence of multi-drug-resistant forms of HIV in infected individuals.
During the reverse transcription of HIV in an infected cell, the viral enzyme reverse transcriptase switches templates frequently from one genomic RNA strand of a virion to the other, yielding a recombinant proviral DNA that is a mosaic of the two parent genomes. If one strand contains a mutation that confers upon HIV resistance to one administered drug and the other strand resistance to another drug, recombination may bring the two mutations together and give rise to progeny genomes resistant to both those drugs [1,2]. Recombination may thus accelerate the emergence of multi-drug resistance in infected individuals. A prerequisite for recombination to induce genomic diversification is the presence of heterozygous virions [3], which contain nonidentical genomic RNA strands and are formed when individual cells are infected by multiple virions. Recent experiments present evidence of the predominance of multiple infections of cells both in vitro and in vivo [4–7]: infected splenocytes from two HIV patients, for instance, were found to harbor up to eight proviruses, with three to four proviruses per cell on average [6]. The high incidence of multiple infections of cells coupled with the high recombination rate of HIV, estimated to be several times greater than the HIV point mutation rate [7–10], sets the stage for recombination to act as a powerful agent driving the emergence of multi-drug-resistant forms of HIV in patients undergoing therapy. In addition, recombination may serve to preserve diversity in genomic regions not affected by bottlenecks introduced by drug therapy or host immune responses, and improve the adaptability of HIV to new environments [11]. Indeed, in addition to several circulating recombinant forms of HIV, a large number of recombinant forms unique to individuals have been identified [12]. It is of great importance, therefore, to understand how recombinant forms of HIV arise in infected individuals. Remarkable insights into HIV recombination emerge from recent in vitro experiments, in which target cells were simultaneously exposed to two kinds of reporter viruses, and cells infected with recombinant proviruses detected [3–5,7, 13,14]. Rhodes et al. determined using single-round infection assays that the likelihood of the accumulation by recombination of distinct mutations present on the two viral genomes in a virion increases with the separation between the mutations and reaches an asymptotic maximum at a separation of ∼1,500 base pairs [14]. Further, Rhodes et al. found that the cell type employed for infection—CD4+ T cells or macrophages, for instance—does not influence the recombination rate. In contrast, Levy et al. argue that subtle virus–cell interactions cause recombination to occur at different rates in different types of cells [7]. Further, using replication-competent reporter viruses, Levy et al. investigated the dynamics of the emergence of recombinant genomes in vitro and in SCID-hu mice [7]. Interestingly, Levy et al. observed two scaling patterns. First, the percentage of cells coinfected with both the reporter genomes employed for infection was proportional to the total percentage of infected cells. Second, the percentage of cells infected with recombinant proviruses was linearly proportional to the percentage of coninfected cells and, correspondingly, proportional to the square of the total percentage of infected cells. Further, Levy et al. found that the scaling patterns were independent of the initial viral load, the time following the onset of infection, and whether the experiments were conducted in vitro or in SCID-hu mice. Standard models of viral dynamics, which successfully describe short-term (a few weeks) viral load changes in patients undergoing therapy, are predicated on the infection of individual cells by single virions and ignore recombination [15–17]. Recent modeling advances and simulations incorporate descriptions of multiple infections of cells and recombination, and present insights into the subtle interplay between mutation, recombination, fitness selection, and random genetic drift that underlies the genomic diversification of HIV in vivo [18–23]. Bretscher et al. developed a model of HIV dynamics that includes mutation, double infections of cells, and recombination and found that for infinitely large cell populations, the influence of recombination on the development of drug resistance depends sensitively on epistasis, i. e. , on the nature of fitness interactions between mutations [20]. Bretscher et al. argue that phenotypic mixing—the assortment of viral proteins arising from different proviral genomes within a multiply infected cell during the assembly of progeny virions—compromises the selective advantage of the fittest strains and enhances the relative abundance of less fit strains: less fit strains piggyback on fitter strains. At the same time, in a two-locus/two-allele model, recombination, which breaks nonrandom associations of mutations and hence lowers linkage disequilibrium, enhances the relative abundance of single mutant strains compared to wild-type and of double mutant strains when fitness interactions result in positive epistasis, i. e. , when mutations interact antagonistically in lowering viral fitness. Bretscher et al. predict, contrary to the prevalent paradigm, that phenotypic mixing and positive epistasis together result in a deceleration of the growth of drug-resistant viruses upon increasing the recombination rate [20]. Recent experimental evidence points to a mean positive epistasis underlying fitness interactions in HIV-1, which, following the predictions of Bretscher et al. [20], raises questions about the benefits of recombination to HIV-1 and, more generally, of the evolutionary origins of recombination and sexual reproduction [24]. Fraser presents a detailed model of HIV dynamics considering up to three infections of cells, mutation, recombination, fitness selection, and different dependencies of the frequency of multiple infections of cells on the viral load [22]. In agreement with Bretscher et al. [20], Fraser found that recombination inhibits the development of drug resistance during antiretroviral therapy and, further, that this effect is modulated not only by epistasis but also by the dependence of the frequency of multiple infections on viral load [22]. In a more recent study, Althaus and Bonhoeffer extend the description of Bretscher et al. [20] to finite population sizes, where the relative abundance of different mutant strains may be determined stochastically rather than deterministically [18]. Interestingly, Althaus and Bonhoeffer found that even when positive epistasis governs fitness interactions between resistance mutations, recombination may significantly accelerate the development of drug resistance when the effective population size is ∼104–105 [18]. Using bit-string simulations, Bocharov et al. found that multiple infections of cells and recombination act in synergy to enhance viral genomic diversity [19], which in turn may increase the likelihood of the emergence of drug resistance. Bocharov et al. note, however, that the time for the selection of fitter genomes that contain multiple mutations may be highly variable, indicative of the stochastic nature of viral evolution in vivo [19]. Rouzine and Coffin developed a description of viral evolution with fitness selection, recombination between multiple loci, and random genetic drift, and predict that below a critical population size, the viral population within an individual may converge to a clone in the absence of mutation, leaving little scope for recombination to introduce genomic diversification [23]. Rouzine and Coffin suggest therefore that a reduction of the viral population in an infected individual by combination antiretroviral therapy may decelerate significantly the emergence of drug resistance [23]. The effective population size in vivo remains to be established [25]. Currently available models thus make valuable predictions of the influence of recombination on HIV dynamics and the emergence of drug resistance in infected individuals undergoing therapy. The predictions, however, are diverse and have not been compared with available experimental data [3–5,7, 13,14]. An important gap thus exists in our understanding of HIV recombination. Consequently, for instance, the origins of the experimental scaling and dynamical patterns associated with HIV recombination [7] remain poorly understood. Similarly, the recombination rate, or the frequency of template switching events during reverse transcription, remains to be established [7,14]. One limitation of currently available models lies in the approximate descriptions of the dynamics of multiple infections of cells employed. For example, the frequency with which cells are doubly infected is assumed either to be constant [18,20], proportional to the viral load, or proportional to the square of the viral load [22]. Because multiple infections of cells are a prerequisite for the formation of heterozygous virions, an accurate description of HIV dynamics with recombination depends critically on the underlying description of multiple infections, as suggested also by Fraser [22]. The frequency of multiple infections depends not only on the viral load, but also on CD4 down-modulation induced by viral gene expression following the first infection of a cell [26–28], which lowers the susceptibility of the cell to further infections. In a recent study, Dixit and Perelson developed a model that explicitly accounts for CD4 down-modulation and presents a rigorous description of the orchestration of multiple infections of cells by free virions [29]. The model elucidates the origins of one of the two scaling relationships observed by Levy et al. [7]: that the number of doubly infected cells is proportional to the square of the total number of infected cells. Levy et al. note that at the onset of infection in their experiments, because equal numbers of the two kinds of reporter viruses are employed, the probability that a cell is infected with both the reporter genomes is the product of the probabilities that the cell is infected independently with each of the genomes, which explains the origin of the observed scaling early during infection. Later in the infection process, multiple infections of a cell need not be simultaneous and may be sequential. Yet, the quadratic scaling persists. Levy et al. speculate that the absence of any functional hindrance to multiple infections may underlie the persistence of the scaling throughout the infection period. The inhibition of multiple infections by CD4 down-modulation, however, may not be negligible. Dixit and Perelson consider the expected inhibition of multiple infections by CD4 down-modulation and identify conditions under which the observed scaling relationship may hold [29]. The latter model, however, does not distinguish between different viral genomes that infect cells and thereby precludes a description of recombination. Thus, for instance, the dynamics of the emergence of recombinant genomes and the origins of the second scaling relationship observed by Levy et al. —that the percentage of cells infected by recombinant genomes is linearly proportional to the percentage of cells coinfected with both the reporter genomes employed for infection—remain to be elucidated. The analysis of Dixit and Perelson [29] provides insights into the origins of the latter scaling. According to the analysis, one set of conditions under which the quadratic scaling between the population of coinfected cells and that of all infected cells holds occurs at time periods that are long compared to the characteristic viral production and clearance times, so that viral populations are in pseudo equilibrium with the infected cell populations [29]. Under the same conditions, when two distinct reporter genomes are employed for infection, the population of heterozygous virions containing a copy each of the two reporter genomes is expected be in pseudo equilibrium with, and hence proportional to, the population of cells coinfected with both the reporter genomes. Further, for time periods long compared to the lifetimes of infected cells and the characteristic timescale of the infection of uninfected cells, the production and death rates of infected cells are expected to exhibit a pseudo steady state. The number of cells infected with recombinant proviruses would then be proportional to the population of heterozygous virions and hence to the population of coinfected cells, which may explain the origins of the second scaling relationship observed by Levy et al. [7]. (We present more mathematical arguments below.) Whether currently available models of HIV dynamics that include infections by distinct viral genomes and recombination validate the above arguments and predict the observed scaling remains unclear. Dixit and Perelson predict the existence of the above scaling relationships under certain parameter regimes and, importantly, that the scaling relationships may depend on the length of time following the onset of infection [29]. In contrast, Levy et al. found that the scaling is independent of the time following the onset of infection [7]. Further, Levy et al. found that for the different initial viral loads employed, which varied well over two orders of magnitude, the parametric plots of different cell populations defining the scaling relationships superimpose remarkably tightly [7], whereas a similar superimposition at all times following the onset of infection is not apparent from the above scaling arguments, which hold after the establishment of pseudo equilibrium between viral production and clearance. A comprehensive model of HIV dynamics with recombination that quantitatively captures available experimental data is currently lacking. In this work, we develop a detailed model of HIV dynamics that considers multiple infections of cells by distinct viral genomes and describes recombination. Our model captures several recent in vitro experimental findings quantitatively and provides key insights into the mechanisms underlying the emergence of recombinant forms of HIV. At the same time, our model is consistent with the standard model of viral dynamics, which successfully captures viral load changes in patients following the onset of antiretroviral therapy, and may therefore be extended to describe HIV dynamics with recombination in vivo.
We consider in vitro experiments where a population of uninfected CD4+ cells, T, is exposed simultaneously to two populations, V11 and V22, of homozygous virions containing genomes 1 and 2, respectively. The genomes 1 and 2 are assumed to be distinct at two nucleotide positions, l1 and l2, a distance l apart on the viral genome, with genome 1 having a mutation at position l1 and genome 2 at l2 (Figure 1A). Following exposure, target cells become infected singly or multiply with one of the genomes, or coinfected with both genomes. Coinfected cells produce heterozygous virions, V12, which contain a copy each of genomes 1 and 2. Infection of target cells by the virions V12 yields two kinds of “recombinant” genomes depending on the template switching events during reverse transcription (Figure 1B). When the mutations at positions l1 and l2 are both included in the resulting proviral DNA, a recombinant genome that we denote as genome 3 is formed. When both the mutations are excluded, the other recombinant, genome 4, results. When one of the two mutations is included but not the other, genomes 1 and 2 are recovered. Thus, four kinds of viral genomes, 1,2, 3, and 4, eventually infect cells. We distinguish infected cells by the proviral genomes they contain. We denote by Ti cells containing a single provirus i, where i ∈ {1,2, 3,4} represents the four genomes above. Thus, cells T1 contain a single provirus 1, cells T2 contain a single provirus 2, and so on. We denote by Tij, where i and j ∈ {1,2, 3,4}, cells that contain two proviruses. Thus, cells T11 contain two copies of provirus 1, and cells T12 contain a copy of provirus 1 and a copy of provirus 2. Because cells Tij are indistinguishable from cells Tji, we subject i and j to the constraint i ≤ j, resulting in ten kinds of doubly infected cells: T11, T12, T13, T14, T22, T23, T24, T33, T34, and T44. Extending the description, cells Tijk are infected with three proviruses, and so on. Our aim is to describe the dynamics of recombination observed in experiments that employ two kinds of reporter viruses to infect cells. In these experiments, the number of cells infected with more than two genomes is estimated to be small [7]. Therefore, and for simplicity, we restrict our model to single and double infections of cells. Random assortment of viral RNA produced in infected cells gives rise to ten different viral populations, which we denote Vij, where i ≤ j and i and j ∈ {1,2, 3,4}, based on the viral genomes, i and j, contained in the virions. Thus, for instance, virions V34 contain a copy each of genomes 3 and 4. Cells infected with a single kind of provirus, Ti and Tii, give rise to homozygous virions, Vii. Cells coinfected with distinct proviruses, Tij, produce the homozygous virions Vii and Vjj and the heterozygous virions Vij. Below, we write equations to describe changes in the various cell and viral populations following the onset of infection. We solve Equations 1–6 (Methods) using the following parameter estimates drawn from in vitro studies [29,31]: the birth and death rate of target cells, λ = 0. 624 d−1 and μ = 0. 018 d−1; the death rate of infected cells, δ = 1. 44 d−1; the viral burst size, N = 1,000; and the clearance rate of free virions, c = 0. 35 d−1. We let an initial target cell population, T0 = 106, be exposed to two equal viral populations, V11 = V22 = V0, which we vary over the experimental range, 2V0 = 106 to 1010 [7]. The infection rate constant, k0, the timescale of CD4 down-modulation, td, and the recombination rate, ρ, are not well-established, and we vary these parameters over ranges that define their best current estimates. We choose l, the separation between the mutations on genomes 1 and 2, in accordance with experiments (see below). Available in vitro experiments, where cells are simultaneously exposed to two distinct kinds of viral genomes, may be segregated into two categories. First, single-round infection experiments employ replication-incompetent (heterozygous) virions to infect cells, and measure the fraction of cells that contain recombinant proviral genomes [3,7, 13,14]. Second, viral dynamics experiments employ replication-competent (homozygous) virions and determine the time evolution of populations of cells infected with recombinant genomes [7]. We employ our description of the recombination probability (Equation 5b) to predict data from single-round infection experiments and our entire model (Equations 1–6) to describe the latter viral dynamics experiments.
The emergence of recombinant forms of HIV that are resistant to multiple drugs often underlies the failure of current antiretroviral therapies for HIV infection. Yet, the dynamics of the emergence of recombinant genomes in individuals infected with HIV remains poorly understood. Current models of HIV dynamics are unable to explain available experimental data of the frequency of occurrence and the time evolution of recombinant HIV genomes quantitatively. We developed a model that describes the dynamics of the emergence of recombinant forms of HIV and quantitatively captures key experimental observations. Mimicking recent experiments [5,7, 14], we considered target cells exposed simultaneously to two kinds of homozygous virions. We constructed integral equations that predict the time evolution of the population of cells coinfected with both kinds of viruses. Following the first infection of a cell, viral gene expression induces CD4 down-modulation, which lowers the susceptibility of the cell to further infections. Because cells are infected asynchronously, determination of the frequency of multiple infections requires accounting for the different susceptibilities of individual cells to further infections at any given time based on the different times elapsed from their respective first infections, which is accomplished by the integral equation formalism [29]. Coinfected cells produce heterozygous progeny virions, which infect cells and yield recombinant proviral genomes. We developed a probabilistic description of template switching during reverse transcription and predicted the frequency with which heterozygous virions give rise to recombinant genomes. We integrated our descriptions of multiple infections of cells and recombination into standard models of HIV dynamics [15–17] and formulated dynamical equations that predict the time evolution of the populations of uninfected, singly infected, and multiply infected cells, and of homozygous, heterozygous, and recombinant viruses. Model predictions are in agreement with the T cell dynamics observed in vitro. Levy et al. [7] found that following the onset of infection, the infected cell subpopulations evolve in two phases, an initial rise and a subsequent fall. Further, the percentage of cells infected by recombinant genomes, p4, is a small fraction of the percentage of coinfected cells, p12, which in turn is a small fraction of the total percentage of cells infected, p*. The two-phase dynamics and the relative prevalence of various infected cell subpopulations are in agreement with our model predictions. Our model also captures the scaling patterns relating the frequency of infection, coinfection, and recombination observed experimentally. Levy et al. [7] found remarkably that p12 is proportional to (p*) 2 and that p4 is proportional to p12, independent of the initial viral load and the time following the onset of infection. Our model predicts both these scaling patterns and that the patterns are independent of the initial viral load and the time following the onset of infection. Quantitative comparison between our model predictions and the experimental scaling relationship between p12 and (p*) 2 is precluded by the poorly characterized dynamics of cells not susceptible to infection by HIV that may be present in the experimental cultures. We showed, however, that the presence of non-susceptible cells does not influence the linear scaling relationship between p4 and p12. Indeed, our model predictions are in quantitative agreement with the experimental scaling relationship between p4 and p12. The quantitative agreement indicates that our model captures the underlying dynamics of HIV recombination accurately. Our model also captures data from single-round infection experiments on the frequency of the accumulation by recombination of distinct mutations present on the two RNA strands within a virion. From comparisons of model predictions with the experiments of Rhodes et al. [14], we estimate that ∼8 template switches, or crossovers, occur on average (95% confidence interval: 6–10) during the reverse transcription of an entire HIV genome of ∼104 nucleotides. This number is in agreement with independent estimates from direct sequence analysis by Levy et al. [7], who observed ∼7. 5 crossovers (range 3–13) on average. Comparison of our model predictions with the single-round infection assays performed by Levy et al. yields crossover frequencies of ∼7. 1 in Jurkat T cells, ∼4. 6 in HeLa CD4 cells, and ∼8. 3 in primary CD4+ T cells. Whereas the crossover frequency in HeLa cells is lower, the frequency in the other two cell types is in agreement with the estimate obtained from our analysis of the experiments of Rhodes et al. [14]. Further, the scaling relationship between p4 and p12 described above is also consistent with a recombination rate of ∼8 crossovers per ∼104 nucleotides. The power law scaling that the number of doubly infected cells is proportional to the square of the total number of infected cells is also predicted by the model of HIV dynamics with multiple infections developed by Dixit and Perelson [29]. In contrast to experiments, however, the predicted scaling is dependent on the time following the onset of infection and the initial viral load. Here, by considering percentages rather than numbers of infected cells and by distinguishing between cells doubly infected by a single kind of genome and coinfected with distinct genomes, we mimic experimental quantities more accurately and find that the scaling relationship is independent of the time following infection or the initial viral load, as observed in experiments [7]. Fraser argues that the quadratic scaling observed between the percentage of doubly infected cells and the total percentage of infected cells, the latter predominantly singly infected, may imply a deviation from mass action kinetics for the second (and perhaps further) infections of cells and suggests, motivated by the scaling, that the rate of second infection of cells may be proportional to the square of the viral load (r ∼ kV2) [22]. Here, we find that the quadratic scaling emerges without deviations from mass action kinetics (r ∼ kV). Further, to address possible differences between the rates of first, second, and third infections, due, for instance, to CD4 down-modulation, Fraser postulates the use of different values of the rate constants, k, for successive infections. In our model, the differences in the rate constants for multiple infections follow naturally from our description of CD4 down-modulation (Equation 3). The latter description facilitates accurate estimation of the frequency of multiple infections under varying viral loads: when the viral load is high, for instance, the second infection of a cell may occur rapidly after its first infection, in which case the rate constants for the first and second infections are expected to be similar due to negligible CD4 down-modulation in the intervening interval. A fixed rate constant for second infection (independent of the viral load) would then tend to underestimate the frequency of double infections. The dependence on the viral load of the variation of the apparent infection rate constant with the number of infections implies that when viral load changes are rapid, the likelihood of a cell suffering multiple infections would depend on the instant of its first infection. For instance, in patients undergoing efficacious antiretroviral therapy, a cell first infected at the start of therapy, when the viral load is large, is expected to have a higher rate constant for second infection than a cell that is first infected a day after the onset of therapy, when the viral load is significantly reduced. Our integral equation formalism, which accounts for the asynchronous first infections of cells, allows accurate determination of the frequency of multiple infections and consequently the influence of recombination throughout the infection period. If the distinction between different viral genomes is ignored, our model reduces to the model of HIV dynamics with multiple infections developed by Dixit and Perelson [29] when more than two infections of cells are rare. Indeed, changes in the total viral load, target cell numbers, and total infected cell numbers predicted by our model (Figure 2) are identical to those predicted by the latter model. Importantly, the latter model reduces to the standard model of HIV dynamics [15–17] when viral production from cells is independent of the number of infections cells suffer. Our model is thus consistent with the standard model of HIV dynamics, which successfully predicts viral load changes in patients following the onset of antiretroviral therapy [15–17]. Further, infections in SCID-hu mice also show scaling patterns similar to those observed in vitro [7] and predicted by our model, reinforcing the notion that our model may be applied to describe the dynamics of recombination in vivo. Several advances of our model are essential, however, to predict the emergence of recombinant genomes in vivo. First, that infected splenocytes from two HIV patients harbored 3–4 proviruses per cell on average [6] suggests that multiple infections of cells may be more prevalent in vivo than is assumed in our model. Second, multiple infections in vivo may be orchestrated by cell–cell transmission as well as by free virions [32,33]. Third, HIV has a high mutation rate [34], which introduces genomic variations in vivo that may subsequently be accumulated by recombination. Indeed, the high mutation and turnover rates of HIV in vivo [17,35] suggest that the likelihood of the preexistence of individual drug resistance mutations in patients is high [36]: approximately half of the HIV patients in the United States are estimated to be infected with genomes that possess resistance to at least one of the currently available drugs [37]. Our model describes how preexisting mutations may become associated by recombination. Determination of the existence of individual mutations, however, requires a description of the HIV mutation process, which we ignore. Fourth, fitness interactions between mutations [24] modulate the relative prevalence of recombinant genomes, whereas we assume all viral genomes to be equally fit. We note that incorporating fitness selection enables our present model to describe additionally in vitro serial passage experiments of the emergence of drug resistance via recombination [2,38]. Finally, whereas we consider two loci, a description of recombination between more than two loci is essential in vivo, as more than two mutations are typically responsible for resistance to individual drugs [37]. With the above advances, some of which are suggested in currently available models [18,20,22,23], our model may facilitate prediction of the emergence of multi-drug–resistant strains of HIV in infected individuals.
We non-dimensionalize Equations 1–4 and 6 using the following dimensionless quantities: and obtain and We solve the dimensionless Equations 5 and 11–Equation 15 as follows. We recognize that the equations are strongly coupled because of the integral equation formalism (Equation 12) employed; for instance, evaluation of the integral in Equation 12a to determine requires knowledge of at all times from 0 to, which in turn depends on through Equation 15. Using the initial conditions, we first integrate the differential equations for and (Equations 11 and 15) for a small time step θ, i. e. , from to. Next, we integrate the differential equations for the conditional probabilities M (Equation 14) by discretizing, which can vary from 0 to θ, into intervals of length θm and determining, where α assumes integer values from 0 to θ/θm, by linear interpolation between and. We then evaluate the integrals in Equation 12 to determine, , and. We march forward in time and evaluate and by integrating Equation1 11 and 15 from time to, integrate Equation 14 by allowing to vary from 0 to 2θ, and evaluate the integrals in Equation 12 to determine, , and. We repeat the procedure until (i. e. , t = 10 d). The solution is implemented by a computer program written in Fortran 90. We derive below the scaling relationships mentioned in Equation 7. Following the onset of infection, for times small compared to the timescale of CD4 down-modulation, i. e. , t ≪ td, because k ≈ k0, we write the dynamics of singly infected cells as where the first term on the right-hand side is the rate of formation of T1 by the infection of uninfected cells, and the second and third terms are the losses of T1 due to further infections and cell death. At the start of infection, the dominant viral populations are V11 and V22 (Figure 3B), of which infection by the former alone yields T1. Further, because T1 is small (Figure 3A), the loss terms, which are linear in T1, are negligible. Equation 16 then simplifies to For lengths of time short compared to the timescales over which V and T change, we let T ≈ T0 and V11 ≈ V0 and integrate Equation 17 to obtain where we use the initial condition T1 (0) = 0. With assumptions similar to those employed in obtaining Equation 17, we find that the coinfected cell population evolves according to the following equation: where we recognize that T1 = T2 and V11 = V22 ≈ V0. Substituting for T1 from Equation 18 and integrating with the initial condition T12 (0) = 0, we get The evolution of the heterozygous viral population, V12, is given by where we note that 1/2 of the virions produced from cells T12 are heterozygous. We ignore viral clearance because V12 is expected to be small. Substituting for T12 from Equation 20 and integrating with the initial condition V12 (0) = 0, we obtain The time evolution of the cell population infected with recombinants, T4, is then given by where we recognize that because V12 ≫ V44 ≫ V14, most of the cells T4 are formed due to infection by V12 followed by recombination. Substituting for V12 from Equation 21, and integrating with the initial condition that T4 (0) = 0, we find Because the total infected cell population comprises largely cells T1 and T2, which in turn are significantly smaller in number than uninfected cells (Figures 2 and 3A), we obtain the total percentage of infected cells, Similarly, the percentage of coinfected cells, and the percentage of cells infected with recombinants, Combining Equations 24–26, we obtain We thus find that early during infection, the scaling laws p12 ∼ (p*) 2 and p4 ∼ (p12) 2 hold. We next consider times longer than the timescale over which viral production and clearance reach pseudo steady state, t > teq. The magnitudes of the viral subpopulations still follow V11 = V22 ≫ V12 ≫ V44 (Figure 3B). Similarly, for the infected cell subpopulations, we have T1 = T2 ≫ T12 ≫ T4 (Figure 3A). The relevant evolution equations may then be written as and where we let k1 be the “mean” infection rate of singly infected cells. We note that k1 is a function of the CD4 down-modulation timescale, td. If td is large, for instance, then k1 ≈ k0. Applying the pseudo steady state approximation for the viral populations yields Substituting for V11 from Equation 33 in Equation 28, we obtain Assuming that changes in the target cell population, T, and the total viral load, V, occur slowly compared to changes in T1, which is expected in the initial stages of infection, we integrate Equation 34 to obtain where and is the value of T1 when t = teq. Note that in Equation 35, t ≫ teq. Substituting for V11 and T1 in Equation 29 yields which upon integrating with the initial condition when t = teq and recognizing that gives Combining Equations 30,33, and 36, we get Integrating Equations 37 and assuming yields Again, assuming that the singly infected cells are predominant in the infected cell population, the percentage of total infected cells is where the small percentage of infected cells allows us to write T* + T ≈ T. The percentage of coinfected cells is then where the last approximation follows from the sharp rise in T1 following the establishment of pseudo steady state (Figure 3A), which according to Equations 34 implies that. Finally, the percentage of cells T4 infected with recombinants is Thus, later in the infection (t > teq), the scaling laws p12 ∼ (p*) 2 and p4 ∼ p12 hold.
Retroviral recombination, a process akin to sexual reproduction in higher organisms, may accelerate the accumulation of mutations and the development of multi-drug resistance in HIV patients. Recombination occurs when the enzyme reverse transcriptase switches between the two RNA strands of a virion, yielding a provirus that is a mosaic of the two strands. The latter strands are often distinct, thereby allowing recombinational diversification, when multiple viruses infect individual cells. The enormous HIV recombination rate and recent evidence of frequent multiple infections of cells render recombination a powerful force underlying the development of multi-drug resistance in vivo. The dynamics of the emergence of recombinant genomes, however, remains poorly understood. Recent experiments allow a closer look at HIV recombination: cells are exposed to two kinds of reporter viruses and the frequency of recombinant proviruses is detected, which enables direct quantification of the extent of recombination. The observations, however, are not described by available models, leaving a gap in our understanding of HIV recombination. We present a model that describes HIV dynamics with multiple infections of cells and recombination, captures several recent experimental observations quantitatively, provides insights into HIV recombination, and presents a framework for describing the development of multi-drug resistance in HIV patients.
viruses
infectious diseases
mathematics
in vitro
biophysics
Emergence of Recombinant Forms of HIV: Dynamics and Scaling