Modeling the Intracellular Dynamics for VifAPO Mediated
HIV1 Virus Infection
Yi Wang^{1}, Qi Ouyang^{1}, Luhua Lai^{1,2*}
^{ }
1: Center for Theoretical Biology,
2:
College of Chemistry and Molecular Engineering,
*Corresponding
author:
Luhua Lai
Tel: 861062757486; Fax: 861062751725
Email: lhlai@pku.edu.cn
Running title: Mathematical model for HIV1 virushost interactions
ABSTRACT
The viral
infectivity factor (Vif) was found to be essential for controlling HIV1 virus
infectivity. It targets cellular
antiviral proteins in APOBEC family (APO) to trigger its fast degradation and
inhibits
Keywords: Mathematical modeling; hostvirus
interaction; Vif;
INTRODUCTION
The modeling of HIV1 virus activity has attracted
much attention (1). Some models considered the dynamics of
cell populations to estimate key parameters such as virion clearance rate,
infected cell lifespan, viral generation time and turnover rate of virus, and
infected cells under drug treatment (25). Others were intracellular models that study
viral activities like provirus integration, regulations of viral RNA synthesis
and transport of viral components between nucleus and cytoplasm (69).
These intracellular models focused on the dynamics of early events of HIV1
virus life cycle, which serve as the source of virus production. Correspondingly, most of current therapeutic
strategies are targeting and blocking these early events, but are still far
from success in eradicating the virus (10,11).
Late events involving intracellular virushost interactions should also
play important roles in terms of mediating both the quantity and the quality of
virus production. However, to our
knowledge, few of them are systematically analyzed due to the complexity and lack
of experimental data.
Recent researches reveal that viral infectivity factor
(Vif) encoded by HIV1 is antagonistic to cellular antiviral enzyme APO, which
can be packaged into newly synthesized virions and induce lethal hypermutation in
nascent viral reverse transcripts if viral Vif is not presented (1215).
Consider a single infected cell, the overall infectivity of all virions
produced by the cell are correlated to both the total number of newly
synthesized virions and (theoretically) average infectivity per each nascent
virion. Hence the average number of
APO molecules packaged per each nascent virion can computationally serve as an
indicator of average virion infectivity; the more
MODEL
DESCRIPTION AND RESULTS
Fig. 1 depicts the virus production network used in
the present study. The model
describes the intracellular dynamics of
Simulation
results compared with experimental data
To justify the
mathematical model, we performed simulations to compare the computational results
with four sets of published experimental data. Table 3 is a detailed mapping of simulations
to the experiments.
To test the depletion
effect of viral Vif on cellular APO level (16,17), we calculated the variable _{} by varying the parameter
_{}. Fig.
We next compared the simulated and experimental
results on the
To verify the correlation
between the average number of packaged APO per virion (_{}) and viral infectivity, we depicts the experimentally
defined viral infectivity values under different A
Through these comparisons,
it is reasonable to assume that our model correctly describes the major dynamics
of the system.
Parameter
sensitivity and perturbation analysis
To exam the
influence of each parameter on the
First, we checked
the correlations among _{}, _{} and _{}. As expected,
negative correlations exist between the average number of packaged APO and Vif
(_{} and _{}) for most parameters since Vif targets
Next, we found the
base value of viral structure protein Gag related parameter _{} and _{} (where [_{}]=[_{}]=0) are near the peak of _{} curves (Fig.
Moreover, our
simulations demonstrate that most model parameters are suitable for virus to
proliferate only within a relative small range (Fig. S1). In particular, Fig. 4B demonstrates switchlike
behaviors on both _{} and _{}, which are exquisitely sensitive to Tat related parameter _{}, _{} and _{}. A small
variation (within 1 magnitude) of these parameters results a switch from low
level of _{} at reference state
to high level state, and reverse for _{}.
DISCUSSION
For simplicity,
we made several assumptions to simplify the network and parameter selection. We assume all substances are transported
or diffuse fast enough to exist homogenously in cytoplasm. Therefore, mass reaction law is applied
to each reaction for synthesis, degradation and incorporation processes. On the time scale of virion production, we assume rapid
binding and unbinding of VifAPO, GagVif and GagAPO. Hence these complexes are always in
quasi steady states, therefore explicit dynamics about association and
dissociation of these complexes are not taken into account. Instead, only their association constants
are formulated into the equations by steady state approximation. Moreover, we assumed that all other factors not explicitly presented
in this model do not essentially alter the behavior of this dynamic system. For instance, the selfassociation of
Vif (23), the localization of APO to ribonucleoprotein
(24) and other probable cellular or viral component
mediated APO expression were not considered. We theoretically related the viral
infectivity with the average number of packaged APO based on the editing
function of APO on viral DNA, but the editing ability may not be the whole myth
of APO (12).
All these assumptions reduce the ability of the model to capture the details
of this complex biological system.
However, the
simulations successfully reveal the downmodulation of APO by viral Vif (Fig.
We computationally
demonstrated the rate of Gag synthesis and the overhead of this structure
protein required per new virion is optimized for HIV1 virus (Fig.
Since APO can
edit nascent viral genome, it definitely contributes to the high mutation rate
of virus (25). In this context,
Our simulation also
supports that both virion production and infectivity are all highly sensitive
to the parameters related to early events of virus integration and
transcription (Fig. 3). This is due
to the preeminent activity of the Tat mediated positive feed back loop on viral
RNA synthesis, and subsequently on all viral components. Therefore, besides tuning parameters
directly downregulating viral RNA (_{}, _{} and _{}), targeting
any Tat related parameters (_{}, _{} and _{}) to lower Tat level will significantly reduce the virion
proliferation and restore the innate defense mechanism mediated by
The VifAPO
related virushost intracellular dynamic system modeled here was found to
widely exist in a number of cell types and viruses (16,2629).
APOBEC is a large family of proteins. Besides A
CONCLUSION
In the current
study, we first presented a model for the dynamic system of VifAPO related
HIV1 virus production, and quantitatively delineated the fundamental dynamics
when compared with existing experimental data. The model demonstrates the essential role
of APO plays on viral infectivity, and how viral Vif downregulates the level of
Acknowledgments
This research was supported in part by the Ministry of
Science and Technology of China and the National Natural Science Foundation of
China (No. 30490245, No. 90403001).
The authors thank Dr. Fangting Li, Kun Yang, Lu Wang, Wenzhe Ma, Daqi Yu
and Chun Qiao for helpful discussions.
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TABLE 1 Mathematical
model for cellular
Chemical Species 
Variable 
Related Equation 


Number of 





Viral RNA 
_{} 
_{} 
(1) 

Viral protein
Tat 
_{} 
_{} 
(2) 

Viral protein
Gag 
_{} 
_{} 
(3) 

Viral protein
Vif 
_{} 
_{} 
(4) 

Cellular RNA
of 
_{} 
_{} 
(5) 

Cellular
protein 
_{} 
_{} 
(6) 

VifAPO
complex 
_{} 
_{} 
(7) 

GagVif
complex 
_{} 
_{} 
(8) 

GagAPO
complex 
_{} 
_{} 
(9) 
Accumulated Number of 





Virions
produced 
_{} 
_{} 
(10) 

Vif packaged
in virions 
_{} 
_{} 
(11) 


_{} 
_{} 
(12) 
Average Number of 





Packaged Vif
per virion 
_{} 
_{} 
(13) 

Packaged 
_{} 
_{} 
(14) 
All variables
denote the number of molecules or virions.
TABLE 2 Parameter
values
Parameter Explanation 
Parameter 
Base Value (_{}) 
Reference 
Number of integrated
provirus 

1 (t >=
6h), 0 (t < 6h) 
(8) 
Basal
transcription rate for viral RNA 

15 transcripts/h 
(9) 
Increase in
viral RNA transcription by Tat transactivation 

1485
transcripts/h 
24.75
transcripts/min (9) 
Transcription
rate of APO RNA 

15 transcripts/h 
Assigned to
be the same as _{} 
(Eukaryotic)
steadystate translation rate 

270 proteins/h 
4.5 proteins/min
(9) 
Probability
of viral RNA to encode Tat 

0.01 
Fraction of
tat RNA in spliced RNA: 0.05 (9), we assigned
_{} as 5 fold of
spliced RNA (34). 
Probability
of viral RNA to encode Gag 

1 
(35,36) 
Probability
of viral RNA to encode Vif 

0.05 
(35,36) 
Number of Gag
per Virion for assembling 

2000 
(35,37,38) 
Association
constant of Tat with TAR 

5.2453×10^{5}/molecule 
28.57/μM (9). 
Association constant
of Vif and 

9.1798×10^{5}/molecule 
Assumed to be
50/μM (21,39). 
Association constant
of Gag and Vif 

2×10^{6}/molecule 
Selected to
keep steady state of _{} to be about
100. Range of _{} was reported to
be 60100 (35) in acutely
infected cells. 
Association constant
of Gag and 

2×10^{6}/molecule 
Assigned to
be same as _{}. 
Rate of Gag
export through virions budding 

0.08/h 
Selected to
keep the steady state of _{} about 3900 (40). 
Degradation
rate of viral RNA 

0.1733/h 
Half life: 4h
(9) 
Degradation
rate of cellular RNA of APO 

0.1733/h 
Selected to
be the same as _{}. 
Degradation
rate of Tat 

0.1733/h 
Half life: 4h
(9) 
Degradation
rate of Gag 
_{} 
0.1054/h 
10% Gag (p24) degradates in 1h (41) 
Degradation
rate of Vif 

0.4673/h 
Half life: 89min
(21) 
Degradation
rate of 

1.4341/h 
Half life: 29min
(21) 
Degradation
rate of VifAPO complex 

2.0794/h 
Half life: 18min
(21) 
The units of association
constants (_{}, _{}, _{} and _{}) were converted to molecule^{1} according to a
fixed T cell volume. We took the diameter
of a T cell as 12μm (42).
Sensitivity and perturbation analysis was performed for all parameters
(Fig. 3). In the analysis, the
parameters for probability of viral RNA to encode certain protein (_{}, _{} and _{}) may be larger than 1.
This is simply a mathematical treatment to increase the synthesis rate
of the corresponding protein.
TABLE
3 Variables and
parameter(s) used in simulation corresponding to experimental measurements and
conditions.
Experiment
Reference / Simulation
Result 
Measurements
in Experiment / Variables
in Simulation 
Conditions
in Experiment / Parameter(s)
in Simulation 
Fig. 3 B in (18) 
Protein level of A3G 
Vector Vif:APO3G (μg:μg) = 4:2, 1:2,
0.25:2, 0:2 
Fig. 2 A in this work 
Normalized _{} at 24h 
_{}= 4, 1, 0.25, 0. _{} 
Fig. 1 B in (19) 
Relative amount of A3G packaged into
virions at 48h postinfection 
Vector HAA3G:pNL43ΔVif (μg:μg) = 0:60,
1:60, 2:60, 5:60, 10:60, 20:60 
Fig. 2 B in this work 
_{} at 48h
simulation time 
_{}= 0, 0.5,
1, 2.5, 5, 10. _{}= 0 
Fig. 2 B in (20) 
Percentage of packaged A3G in total A3G
at 24h postinfection 
1. Constant pNL43 (expressing constant
amounts of Vif) = 2.5μg, varying pcDNAAPO3G. = 0, 0.5, or 2.5μg, total DNA
was adjusted to 5μg. 2. Constant Vifdefective pNL43Vif() =
2.5μg, varying Vif expression vector pNLA1 = 0, 0.5, or 2μg, total DNA was
adjusted to 6μg. 
Fig. 2 C in this work 
_{} at 24h simulation
time 
_{}, _{} 
Fig. 4 A in (21) 
Relative viral infectivity in subsequent
infection 
WT and ΔVif provirus 3μg and A3G = 0,
0.01, 0.02, 0.05, 0.1 or 0.2μg adjusted with an empty vector to 4μg total. 
Fig. 2 D in this work 
_{} at 48h
simulation time 
_{} or 0 (for ΔVif),
_{}=0, 0.1, 0.2, 0.5, 1 or 2 
Shaded row denotes the experiments; the row
directly below represents the corresponding simulations.
FIGURE
LEGENDS
Figure
Legends
FIGURE 1
Schematic of cellular
FIGURE
2
Comparison
between experiments and simulations.
(A) Viral Vif
downmodulates cellular protein
FIGURE
3
Parameter
sensitivity analysis. Vertical axis denotes relative parameter
sensitivity value on the steady state value of variable _{}, _{} and accumulated
number for _{} at 48h
postinfection. The locations of
bars are in descending order from left to right for sensitivity on variable _{}. The value bars
are grouped by each parameter. Note
the sensitivity values on variables _{} and _{} are scaled down
for plot convenience.
FIGURE
4
Perturbation
analysis. Each parameter was varied by 4 magnitude
to explore its influence on the steady state value of variable _{}, _{} and the accumulated
value of _{} at 48h postinfection. Values on variables _{} and _{} are scaled down
as legend indicated. (A) Gag
related parameter _{} and _{}. (B) Tat related
parameter _{}, _{} and _{}.
FIGURE 1
FIGURE
2
FIGURE
3
FIGURE
4