Figure 1.
Mechanism of HIV infection including viral entry, reverse transcription, integration of viral DNA, virion assembly and release of viral particles is schematically shown. A3G, a host protein and a restriction factor, binds to viral mRNA and gets encapsulated into the viral capsid. If viruses carrying A3G infect other cells, the packaged A3G will exert several antiviral activities, which include inducing G-to-A mutations into viral reverse transcripts by deaminating C to U on the minus strand, blocking multiple steps in reverse transcription and causing integration defects. Vif, a viral protein, binds to A3G and inhibits encapsulation of A3G into virions by facilitating degradation of this protein through the proteasomal pathway.
Figure 2.
The basic HIV model: schematic diagram and simulations.
(A) The model consists of three entities: Free viruses, “uninfected” and “infected” CD4+ T cells. Before infection, only uninfected cells are present with the production rate of λ and the death rate of . In the model, infection occurs by introducing an initial amount of viruses to the body. Free viruses infect uninfected cells and give rise to infected cells with k representing infectivity rate constant. Infected cells die at a rate of
; before death, these cells produce and release N new free virions per day. The in vivo clearance rate of viruses is denoted by
. (B) The basic reproductive ratio, R0, is defined as the number of new infections that arise from a single infected cell when almost all the other cells are uninfected. This important metric determines whether the infection spreads (R0>1) or dies out (R0<1) in the body. In the numerical simulation, R0 = 20 initially and a hypothetical treatment is administered on day 100 which reduces the value of reproductive ratio by σ = 5, 15, 20, 30, and ∞. Although σ = ∞ results in the fastest decline in the viral load, the gap between curves associated with σ = 30 and ∞ is small.
Table 1.
Parameter values used for simulations and calculations.
Figure 3.
Schematic diagram of the extended models of HIV infection for WT and A3G-augmented cells.
Submodels (A) and (B) show schematic diagrams of HIV infection in A3G-augmented and WT cells, represented by dark and light blue large circles, respectively. In all the submodels, HIV infection occurs by a mixed population of A3G(+) and A3G(−) viruses, represented by dark and light red small circles, respectively. Large circles with a small circle inside them represent infected cells with the color of large and small circles demonstrating the type of cell and the type virus, which caused the infection. Submodel (C) shows an extended model drawn in (A), where the apoptosis pathway is activated in A3G-augmented cells upon their infection. The deformed blue shapes represent infected cells that have undergone apoptosis. Parameters and
denote the fraction A3G(−) viruses released from infected A3G-augmented and WT cells, respectively. The reduction in the number of released viruses from cells infected by A3G(+) viruses is denoted by c. The failure rate of the apoptosis pathway in (C) is represented by r. Model I is solely described by submodel (A) while Models IIa, IIb, and IIc consist of submodels in (A) and (B). Model III is comprised of submodels in (B) and (C).
Figure 4.
Effects of A3G-free virus release ratio on HIV replication and reproductive ratio in Model I.
Infection occurs on day 0 and A3G-SCT begins on day 100 (Model I assumes that all the cells overexpress A3G). The impact of the therapy on the total concentration of viruses and cells is shown for (A) = 0.1 and (B) 0.01. Light and dark red lines represent
and
variables, respectively, while light and dark blue lines represent
and
variables, respectively. The left and right axes show the virus and cell concentrations. Parameter c is given the value of 0.001. Although (A)
= 0.1 decreases the viral load and increases the T cell count, it cannot eradicate the virus (R1 = 2.02). However, (B)
= 0.01 successfully reduces R1 to 0.22 and the infection dies out. (C) shows the level of reduction in the reproductive ratio that can be achieved by A3G-SCT for different values of
and c. The green and black dashed lines represent the minimum level of reduction needed to stop HIV replication for R0 = 20 and 70, respectively. Note that from left to right on the bottom axis, the A3G-free virus release ratio decreases from 10−1 to 10−4. Simulation results suggest that for small values of
and c, up to three orders of magnitude reduction can be achieved.
Figure 5.
Effects of percentage of transfected cells, death rate ratios, and auto-apoptosis failure rate on HIV replication in Models IIa, IIb, IIc, and III.
In all simulations, infection occurs on day 0 and A3G-SCT begins on day 100. In all the subfigures, light and dark red lines represent and
variables, respectively, while light and dark blue lines represent
and
variables, respectively. The left and right axes show the virus and cell concentrations. Parameter c is given the value of 0.001. f = 99% for all the subfigures except (A) and (B) where f = 90% and 94%, respectively. For the first three rows,
= 0.01 (Models IIa, IIb, and IIc), while it is set to 0.1 for the last row (Model III). (A–C, R2a = 1.86, 1.2, and 0.38) simulation results for Model IIa suggest that high percentage of A3G-augmented cells is required to stop in vivo HIV replication. (D–F, R2b = 0.38, 0.6, and 1.25) Model IIb assumes lower death rates for infected A3G-augmented cells compared to infected WT cells, i.e., t >1. Simulation results suggest that the efficacy of the therapy is degraded as the value of t increases. (G–I, R2c = 0.38, 0.46, and 0.66) A3G(+) viruses are assumed to be less toxic in Model IIc. Therefore, cells infected by these viruses die more slowly compared to cells infected by A3G(−) viruses, i.e., w >1. Model IIc predicts that lower death rates for cells infected by A3G(+) viruses have a diminishing effect on the performance of the therapy. (J–L, R3 = 2.16, 0.87, and 0.37) In Model III, cells are equipped with an additional gene circuit that activates apoptosis pathway upon infection; however, the circuit has a failure rate of r. Simulation results indicate that providing cells with this additional gene circuit enhances the performance of the therapy and can reduce the reproductive ratio to values less than one even in cases that the A3G-free virus release ratio does not take very small values.
Figure 6.
Effects of percentage of transfected cells, death rate ratios, and auto-apoptosis failure rate on reproductive ratio in Models IIa, IIb, IIc, and III.
The level of reduction in the reproductive ratio that can be achieved by A3G-SCT for different values of is shown for each model. In all the subfigures, the green and black dashed lines represent the minimum level of reduction needed to stop HIV replication for R0 = 20 and 70, respectively. Note that from left to right on the bottom axis, the A3G-free virus release ratio decreases from 10−1 to 10−4. Parameter c is given the value of 0.001. (A) Model IIa suggests that f = 95% is required to block HIV replication for R0 = 20. Higher values of f are needed to block HIV replication for larger values of R0. (B) Simulation results of Model IIb predict that the performance of the therapy will be degraded if infected A3G-augmented cells die more slowly compared to infected WT cells, i.e., when t >1. (C) Model IIc also suggests that the therapy achieves lower efficacy if cells infected by A3G(+) viruses die more slowly than cells infected by A3G(−) viruses, i.e., when w >1. However, the performance degradation is less severe than that of Model IIb. (D) Finally, Model III indicates that A3G-SCT can achieve better efficacy if infected A3G-augmented cells activate apoptosis pathway upon their infection.
Figure 7.
Effects of A3G and A3GΔVif overexpression on reproductive ratio in Models IIa and III.
The level of reduction in the reproductive ratio that can be achieved by overexpression of A3G (red) and A3GΔVif (blue) is shown. In the two subfigures, the green and black dashed lines represent the minimum level of reduction needed to stop HIV replication for R0 = 20 and 70, respectively. Parameter c is given the value of 0.001. (A) Simulation results for Model IIa show that almost two orders of magnitude lower production of A3GΔVif compared to that of A3G is required to achieve the same performance. (B) Model III predicts that by decreasing the apoptosis failure rate, lower production rate of A3G and A3GΔVif is required to stop in vivo HIV replication.
Figure 8.
Effects of , A3G-free virus release ratio, on reproductive ratio.
The level of reduction in the reproductive ratio that can be achieved by A3G-SCT for different values of is shown when
= 0.001 (red) and 0.01 (blue). In our simulations,
<
<
. The green and black horizontal dashed lines represent the minimum level of reduction needed to stop HIV replication for R0 = 20 and 70, respectively. Note that from left to right on the bottom axis,
decreases from 0.83 to 10−3. Parameter c is given the value of 0.001.