Emergence of recombinant forms of HIV: dynamics and scaling

Abstract

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 approximately 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.

Figure 1. Schematic Representation of Viral Genomes and Recombination

Viral genomes 1 and 2 employed at the onset of infection (A) and the four genomes resulting from the recombination of genomes 1 and 2 (B).

Figure 2. Model Predictions of the Overall Cell and Viral Dynamics

The time evolution of the number of uninfected cells, T, the total number of infected cells, T^*, and the total viral load, V, following the onset of infection obtained by the solution of Equations 1–6 with the following parameter values: the initial target cell number, T ₀ = 10⁶; the initial viral load, 2V ₀ = 10⁸; the birth and death rates of uninfected 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; the clearance rate of free virions, c = 0.35 d⁻¹; the infection rate constant of uninfected cells, k ₀ = 2 × 10⁻¹⁰ d⁻¹; the CD4 down-modulation timescale, t _d = 0.28 d; the recombination rate, ρ = 8.3 × 10⁻⁴ crossovers per position; and the separation between the mutations on genomes 1 and 2, l = 408 base pairs.

Figure 3. Model Predictions of the Dynamics of Different Infected Cell and Viral Subpopulations

The time evolution of the various singly (solid lines) and doubly (dashed lines) infected cell (left panels) and homozygous (solid lines) and heterozygous (dashed lines) viral subpopulations (right panels) following the onset of infection. Note that T ₁ = T ₂, T ₁₁ = T ₂₂, T ₃ = T ₄, T ₃₃ = T ₄₄, T ₁₃ = T ₂₃ = T ₁₄ = T ₂₄, V ₁₁ = V ₂₂, V ₃₃ = V ₄₄, and V ₁₃ = V ₂₃ = V ₁₄ = V ₂₄. The parameter values employed are the same as those in Figure 2 except that t _d = 2.8 d in (C) and (D) and ρ = 10⁻³ crossovers per position in (E) and (F).

Figure 4. Model Predictions of Scaling Patterns

Parametric plots of (A) the percentage of cells coinfected with genomes 1 and 2, p ₁₂, versus the total percentage of infected cells, p^*, and (B) the percentage of cells infected with the recombinant 4, p ₄, versus p ₁₂, obtained by solving Equations 1–6 for different initial viral loads, 2V ₀ = 10⁶ (green), 10⁷ (cyan), 10⁸ (blue), 10⁹ (purple), and 10¹⁰ (red). The dashed lines are scaling patterns predicted by Equation 7 . The insets show the parametric plots for the individual cases, 2V ₀ = 10⁶ (green) and 10⁷ (cyan).

Figure 5. Comparisons of Model Predictions with Data from Single-Round Infection Experiments

(A) The ratio of the percentage of cells infected with the recombinant 4, f, and the theoretical maximum percentage, f _max, as a function of the separation, l, between the mutations on genomes 1 and 2 (see Figure 1) determined by Rhodes et al. [14] (circles) and by Equation 8 (line) with ρ = 8.3 × 10⁻⁴ crossovers per position. (B) The percentage of GFP⁺ cells as a function of the crossover frequency, n, determined by Equation 9 (line), on which are mapped the experimental percentages (circles) obtained by Levy et al. [7] with HeLa CD4, Jurkat, and primary T cells (PBL). The inset shows the prediction of Equation 9 over a larger range of values of n.

Figure 6. Comparisons of Model Predictions with Experimental Scaling Relationships

Model predictions (thick lines) obtained by solving Equations 1–6, but withEquation 5d replaced by Equations 9a and 9b, compared with experimental scaling relationships (symbols) between (A) the percentage of coinfected cells (YFP⁺/CFP⁺) and the total percentage of infected cells, and (B) the percentage of GFP⁺ cells and the percentage of coinfected cells. The different symbols represent experiments conducted with cells from different donors [7]. Parameters employed for calculations are identical to those in Figure 2 except that for the red lines t _d = 2.8 d in (A) and ρ = 10⁻³ crossovers per position in (B). The thin black line in (A) is the experimental best-fit line [7].