Figure 1.
The scheme of Gag and Gag-Pol processing by HIV-1 PR.
Our model includes the 11 canonical cleavage sites at which the PR cleaves the polyproteins to release the functional enzymes and structural proteins of the virus (indicated by numbers 1–11). A cleavage site within PR itself that mediates auto-inactivation of proteolytic activity is indicated by a dashed line (12). RT is active as a heterodimer of p51 and p66 (p51 and p15 still uncleaved). Abbreviations: MA – matrix, CA – capsid, SP1 – spacer peptide 1, NC – nucleocapsid, SP2 – spacer peptide 2, TFP – transframe peptide, PR – protease, RT – reverse transcriptase, IN – integrase. Boxes representing individual domains are not drawn to scale.
Figure 2.
The time course of simulated Gag and Gag-Pol processing.
Initial concentrations of Gag and Gag-Pol were set to reflect the quantities within a single virion; cleavage rates were parameterized according to in vitro estimates (Table 1). (A) Virus maturation time (VMT) as defined by the molecular species known to govern virion maturation: Morphological maturation (indicated by dashed red line in all panels) is triggered by the decay of the CA.SP1 fragment (blue line; threshold of trans-dominant inhibition of particle maturation indicated by dashed horizontal line) and is not limited by the availability of liberated CA molecules (green line; threshold of one capsid unit corresponding to 1,500 CA molecules per particle is indicated by solid horizontal line). (B) Generation of catalytically active intermediate dimeric forms containing PR. Full-length Gag-Pol (red line) dimerizes rapidly and N-terminal auto-cleavage gives rise to enzymatically active intermediate dimeric forms (black, blue and green lines). (C) Decay of Gag substrate (black line) and accumulation of final Gag cleavage products. (D) Accumulation of final Pol cleavage products. (E) Enzyme concentrations and related metrics. The ratio PRdPR/Etot indicates the relative contribution of mature PR dimers to the proteolytic activity. The ratio Etot/Stot of the total concentration of active enzyme forms and the total concentration of uncleaved cleavage sites stays below one throughout the simulated time course, which justifies the use of Michaelis-Menten kinetics. Etot – total proteolytic activity; Stot – all uncleaved cleavage sites; IEF – all active intermediate enzyme (PR) forms; RT: p51/p66 heterodimer. All other dimers are indicated in the form M1dM2, where M1,2 are the monomers.
Table 1.
Parameters used for the simulation of the modela.
Figure 3.
The processing of canonical cleavage sites in the simulations of Gag and Gag-Pol processing.
(A) The fraction of a given cleavage site that is yet uncleaved (the total concentration of all molecular species that contain the uncleaved site, divided by the initial concentration of Gag or Gag-Pol, respectively). (B) The order of cleavage defined by the time points when a fixed threshold of processing (50% and 95%), indicated by horizontal lines, has been reached. (C) Schematic representation of the proposed order of processing events in HIV-1 Gag and Gag-Pol, respectively. The size of the arrowheads indicates different rates of cleavage, with large arrowheads representing faster cleavage. The order of processing for Gag is based on the times needed to attain 50% processing in the model, as in (B). Note that the sizes of arrowheads are not drawn to scale.
Figure 4.
The effect of parameter variation on the virion maturation time (VMT).
(A) The effect of varying each parameter individually along a geometric series between 0.1 to 10 times its default value, while fixing all other parameters at their default values. Color symbols depict VMT as a function of the parameters with the strongest effects (kcat of CA/SP1: red; rate constant of Gag-Pol auto-cleavage: blue; KM of NC/SP2: green; KM (yellow) and kcat (brown) of p6pol/PR; kd of Gag-Pol: purple); the effect of variation in all other parameters is illustrated by black symbols, which are overlaid on the horizontal line positioned at default VMT, indicating no discernible effect. (B–C) Multivariate exploration of the parameter space. All parameters were drawn randomly from lognormal distributions parameterized such that 95% of the values fell in the range of 0.1 to 10 times the default value of the parameters, or 0.01 to 100 times the default for the parameters with no direct empirical estimates. Default parameter values are indicated by solid vertical lines, the limits of the 95% range by dashed vertical lines. The catalytic rate constant of CA/SP1 cleavage (B) had a strong impact on VMT; the rate constant of Gag-Pol auto-cleavage had very similar effect. None of the other parameters had any discernible effect on VMT: (C) shows a representative example with VMT plotted against the catalytic rate constant of NC/SP2 cleavage. Results from 10,000 simulations are shown; default VMT is indicated by solid horizontal line; parameters were plotted on a log scale in all three panels, with a dimension of s−1 in B–C.
Figure 5.
Isoclines of virion maturation time (VMT).
The isoclines are drawn in the plane of the two rate parameters with the strongest effect on VMT. Generic hyperbola-like functions of the form y = a+b/(x−c)d could be fitted to the data. For VMT = 15 min: a = 10−4, b = 5×10−4, c = 0.058, d = 0.69; for VMT = 30 min: a = 10−4, b = 10−4, c = 0.027, d = 0.79; for VMT = 45 min: a = 9×10−5, b = 2×10−5, c = 0.017, d = 0.93; dimensions for parameters a–c and for both axes of the figure are s−1. The perpendicular lines indicate the default values of both parameters.
Figure 6.
Accuracy of predicting combined parameter effects on VMT assuming additive (A) and multiplicative (B) interaction.
Simulations were run with a geometric series of 20 values ranging from 0.16 to 6.25 times the default for both parameters, and with all 400 combinations of the variants. Under the additive model, predictions of combined effects were obtained by adding up absolute changes in VMT observed when changing each of the parameters separately; predictions under the multiplicative model were obtained by multiplying the relative (fold change) individual effects. Deviations (in minutes) from both predictions are plotted with color-coding; the dimensions of both parameters were s−1. The additive model performs better (smaller deviation) when both parameters are increased or both are decreased; the multiplicative model performs better when one parameter is increased and the other decreased.
Figure 7.
The effect of PR inhibitors on VMT.
(A) The dependence of VMT on the concentration of PI that binds to mature PR (red symbols) and on the concentration of a hypothetical PI that binds to full-length Gag-Pol dimers and inhibits auto-cleavage (blue symbols). The binding rate constants of both PIs were parameterized with data estimated for the PR binding of darunavir. A hyperbola-like function of the form y = a+b/(c−x)d could be fitted to the simulated VMT data, with a = 28.4 min, b = 8.75 min×mM, c = −0.91 mM, d = 1.53 for the inhibitor of PR, and with a = 29.4 min, b = 2.70 min×mM, c = −0.97 mM, d = 1.51 for the inhibitor of auto-cleavage. The vertical line indicates the theoretical maximum concentration of PR dimers at half of the initial concentration of Gag-Pol; the horizontal line indicates VMT in the absence of drug. (B) The effect of Gag-Pol content on critical drug concentration. Red dots indicate the theoretical maximum concentration of functional PR enzymes (half of Pol) at varied initial Gag-Pol content; black dots indicate the darunavir concentration required to delay virus maturation to VMT = 100 min in the model. (C) The dependence of VMT on the dissociation rate constant kd of an inhibitor of PR (red) and of an inhibitor of Gag-Pol (blue). The inhibitor of auto-cleavage requires greater binding affinity (lower log(kd)) to take effect. The vertical line indicates the estimated dissociation rate constant of darunavir. The horizontal line indicates VMT in the absence of drug; drug concentration was set to the possible maximal enzyme concentration (half of initial Gag-Pol) at 0.095 mM. (D) Isoclines of VMT = 30, in the plane of darunavir concentration against fold increase in the two catalytic parameters with the strongest effect on VMT. The same function as in (A) could be fitted with a = 0.57, b = 0.68 mM, c = −1.36 mM, d = 1.10 for the catalytic rate constant of CA/SP1 cleavage, and with a = 0.70, b = 0.66 mM, c = −1.18 mM, d = 0.94 for the catalytic rate constant of auto-cleavage. Vertical asymptotes (positioned by c values) show the limits on the compensation of drug effect by increased catalytic rates. In all panels, parameters were set as in Table 1 unless otherwise stated.
Figure 8.
The rate of change in the concentration of a given fragment Si,j.
Assuming quasi steady state for the enzyme-substrate complexes, , the rate of change consists of loss by cleavage at any of the internal cleavage sites, and production by “trimming” longer fragments that have an uncleaved site at either terminus.