Modeling the Intracellular Dynamics for Vif-APO Mediated HIV-1 Virus Infection

Yi Wang1, Qi Ouyang1, Luhua Lai1,2*


1: Center for Theoretical Biology, Peking University, Beijing 100871, China

2: Beijing National Laboratory for Molecular Sciences, State Key Laboratory for Structural Chemistry of Stable and Unstable Species,

College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China



*Corresponding author:

Luhua Lai

College of Chemistry and Molecular Engineering

Peking University, Beijing 100871, China

Tel:  86-10-62757486; Fax:  86-10-62751725



Running title:  Mathematical model for HIV-1 virus-host interactions




The viral infectivity factor (Vif) was found to be essential for controlling HIV-1 virus infectivity.  It targets cellular antiviral proteins in APOBEC family (APO) to trigger its fast degradation and inhibits APO packaging into nascent virion.  In the present study, we propose a mathematical model to quantitatively study the intracellular dynamics of these typical virus-host interactions.  Four sets of published experimental data were compared with simulation results to justify the model.  Systematic parameter sensitivity and perturbation analysis showed that parameters related to APO are crucial for the infectivity of newly synthesized HIV-1 virus.  Interestingly, we discovered that the synthesis rate of the viral structure protein Gag and the required number per nascent virion are optimized to achieve high virion production with minimal level of packaged APO, and large portion of model parameters are beneficial to virus only within a relative small range.  Furthermore, minor variations in several parameters related to viral protein Tat, the activator of viral RNA synthesis, were found to induce switch-like behaviors on both incorporated Vif and APO.  These findings may provide new insights on understanding the high mutation rate of HIV-1 virus and its latency, as well as help identify key targets for therapeutic design.


Keywords: Mathematical modeling; host-virus interaction; Vif; APO; Gag; Tat




The modeling of HIV-1 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 life-span, viral generation time and turnover rate of virus, and infected cells under drug treatment (2-5).  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 (6-9).  These intracellular models focused on the dynamics of early events of HIV-1 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 virus-host 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 HIV-1 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 (12-15).  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 APO packaged, the lower the infectivity.  To investigate the dynamics of virion production and viral infectivity, we constructed a mathematical model with a focus on these Vif-APO related interactions, and compared it with published experimental data.  Extensive parameter sensitivity and perturbation analysis were subsequently performed.




Fig. 1 depicts the virus production network used in the present study.  The model describes the intracellular dynamics of APO related HIV-1 virus production in a single infected cell. In the beginning, viral RNA is transcribed from provirus integrated in cellular genome, and then viral proteins Tat, Gag and Vif are translated by utilizing the cellular resources.  Tat boosts viral transcription and results in viral RNA accumulation in the host cell, while Vif associates with APO and accelerates its degradation.  Both Vif and APO can be packaged into newly produced virion.  This biological system was formulated by a set of differential equations (listed in Table 1).  The rates of change in the concentration of each chemical species are represented by their synthesis, degradation or export from the cell to nascent virions.  Table 1 also defines the symbols of viral and cellular components, including those to represent their interactions.  The detailed model was described in the Supplemental Material.  Parameter values were either adopted from related literatures or estimated from their reasonable domain.  The explanation of symbols and parameter base values were listed in Table 2.  The base values were used as default through out this report if not otherwise explained.  We used MATLAB to perform simulations and analysis.  Function ode15s was used for solving the ordinary differential equations with the initial values of all substances being zero.


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. 2A shows the simulation results compared with the experimentally measured protein level of A3G under different Vif expressing conditions (18).  In the experiment, no other viral proteins except Vif was expressed, thus we set .  The choice of  were direct proportional to the amount of vector (Table 3).  The simulation result of relative level of APO was normalized to the case of =0.  We obtained similar APO levels between simulation and experiment: the smaller the rate of Vif synthesis, the higher level of APO.

We next compared the simulated and experimental results on the APO incorporation ability under different conditions.  Fig. 2B shows the experimentally observed relative amount of packaged A3G into nascent virion with respect to increasing cellular A3G synthesis without Vif (19) in comparison with simulated  by increasing parameter  and setting  = 0.  We selected HA-A3G:pNL4-3ΔVif (μg:μg) = 2:60 as the base case when  and other  values were assigned proportional to the quantity of vector HA-A3G.  The result shows good correlation when  is small, whereas larger  lead to higher simulated value of packaged APO than experimental ones.  This may ascribe to the transfection efficacy becomes marginally lower when too many vectors were added.  In addition, Fig. 2C shows the calculated ratios of  to the total amount of APO of all forms by continuously tuning both  and .  This is qualitatively corresponding to the percentages of packaged A3G under different Vif and A3G expressing conditions in experiment (20).  The simulation correctly shows the range of the encapsulated APO is around 10~20% of total APO as reported by the experiment (20).


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 A3G synthesis conditions with or without Vif expression (21) in comparison with the calculated  by adjusting  and  in Fig. 2D. Here we took A3G = 0.1μg as the reference state.  It qualitatively shows that a negative correlation exists between  and .

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 APO and Vif packaging efficiency and the virion production, we performed parameter sensitivity analysis and systematically varied each parameter to test their effects on the variables ,  and , respectively, at 48h post-infection, when  and  are in quasi steady states.  Fig. 3 shows the parameter sensitivity to these three variables. The relative sensitivity  of parameter (Para) on variable (Var) is defined as , where  stands for the reference state of the variable and  the base value of the parameter.   denotes the change of variable when parameter changes by .  Each parameter was varied by 0.1% at each step, i.e. .  Fig. 4 depicts the profiles of variable ,  and  at 48h simulation time by changing several Gag and Tat related parameters over 4 magnitude.  Fig. S1 in Supplemental Material shows the results for all the parameters. We defined .


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 APO to degradation (Fig. 3).  It is also obvious that there are positive correlations between  and  for most parameters because these two variables are direct defined by viral components.  But for parameter , variable ,  and  are all positively correlated due to the rates of both viral and cellular protein synthesis are directly regulated by the cellular protein translation rate.  As Fig. 3 is sorted in descending order for , increasing the left portion of parameters upregulates , while increasing the right ones downmodulates .  Considering the absolute values, the most sensitive parameters (||>15) distribute at the left most and right most part of Fig. 3.  The most sensitive ones are directly correlated to APO synthesis (), degradation () and viral early events (, ).  Parameters related to the ability for APO to be packaged into new virion () and viral Vif synthesis () are ranked immediately behind (||>10, ||>10).  The most sensitive parameters to  are related to early events of viral RNA transcription (, , ,  and ).  But many parameters do not significantly affect .  Parameters about the generation and consumption of Vif (, ) are insensitive to , this is consistent with the experiment (22).  And Vif-APO interaction related parameter  and  present little correlation with  (Fig. 3 and flat  curves in Fig. S1).  However, effects of Vif-APO interaction (, ) on APO incorporation () can not be ignored (||>5, ||>5).  These correlations reveal valuable information about therapeutic strategies, which will be discussed later.

Next, we found the base value of viral structure protein Gag related parameter  and  (where []=[]=0) are near the peak of  curves (Fig. 4A).  Either relative larger or smaller value of  and  (within 1 magnitude) will lead to lower virion number.  This implies that Gag synthesis and consumption (export) are approximately tuned to produce the highest amount of nascent virions.  A more detailed analysis about this phenomenon was given in the discussion section.  The  profiles for these two parameters show  and  are also at the edge of low to high transition of .  The average number of packaged APO will significantly increase if their values go higher (within 1 magnitude).

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 switch-like 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 .




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 Vif-APO, Gag-Vif and Gag-APO.  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 self-association 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 down-modulation of APO by viral Vif (Fig. 2A), the APO incorporation capability under various conditions (Fig. 2B and 2C) and the reverse correlation between packaged APO and viral infectivity (Fig. 2D).  Our model correctly captures the fundamental dynamics of this virus-host network by quantitative comparison or qualitative agreement with the experimental results.

We computationally demonstrated the rate of Gag synthesis and the overhead of this structure protein required per new virion is optimized for HIV-1 virus (Fig. 4A), which is not obvious.  Other parameters do not significantly demonstrate this optimization property (Fig. S1).  The underlying mechanism is that new virion export poses a negative feedback effect on the intracellular level of viral RNA (term  in Eq. 1).  This implies the virus may have evolved to best exploit the cellular resources and utilize the rate of RNA transcription and protein translation in order to produce its own structure components to the maximum level once it gets the chance to multiply.  Overall, the range of parameters for virus to survive well is relatively small, less than 1 magnitude change toward certain direction on more than half of the parameters (less than 2 magnitude for all parameters), which will result in either significantly reduced virion production or higher level of packaged APO (Fig. S1).  Under this circumstance, the virus adopts simple but effective strategy.  It produces vast volume of offspring to counteract harmful mutations, and let chance to decide their fate in the next round of infection.  Normally virus wins in that functional virions of small fraction in a large offspring pool is sufficient for lasting infection given unlimited host cells (if the host still alive).  That is why HIV-1 infected individuals shows rapid turnover of both free virus and virus-producing cells (5).

Since APO can edit nascent viral genome, it definitely contributes to the high mutation rate of virus (25).  In this context, APO is not only an antiviral component, but also beneficiary to virus in terms of helping it evolve through mutation to overcome innate defense mechanism and also evade antiviral drugs.  In fact, APO is even found to be dependent on viral enzyme RNase H to activate its deaminase function (19).  Hence there exists a subtle balance between the high mutation rate and the impaired viral infectivity.  The viral latency is such a stable balance formed and kept in a long run, accordingly the major difficulty for eradicating HIV-1 is to target the viral reservoirs established in infection (10,11).  Therefore, cellular APO contributes to viral latency by being a key player in mediating the virus-host balance.  For therapy concerns, our analysis predicts that upregulating the rate of APO RNA transcription or decreasing its degradation rate, increasing the ability of APO incorporation into virions, blocking the binding between Vif and APO and reducing the degradation rate of Vif-APO complex are all effective ways to reduce viral infectivity even under the condition of relative high viral RNA transcripts and high volume of virion production (Fig. S1).

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 APO (Fig. 4 B).

The Vif-APO related virus-host intracellular dynamic system modeled here was found to widely exist in a number of cell types and viruses (16,26-29).  APOBEC is a large family of proteins.  Besides A3G and A3F, APOBEC3B and APOBEC3DE were also identified as antiviral factors through intervening viral Vif (30-32).  They all have homologous cytidine deaminase domain(s) and show similar functionality (15).  Besides human, mice also expresses its own A3G, but HIV-1 Vif can not efficiently form complex with it (33).  APO has already been found related to innate defense mechanisms in primary human T cells (16), macrophages (26) and immature dendritic cells (27).  There are evidences for hepatitis B virus and hepatitis C virus to interact with A3G and/or A3F (28,29).  With proper modifications and parameter selection, our model may be extended to study other virus-host interactions.



In the current study, we first presented a model for the dynamic system of Vif-APO related HIV-1 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 APO.  We found that the virus evolves to maximize its ability to proliferate while escaping from the host innate defense mechanism.  The early steps of viral RNA transcription were computationally identified to be critical for both the volume of nascent virion produced and antiviral activity of cellular protein APO and could be utilized in drug design against HIV virus.  As the Vif-APO interaction is a typical virus-host relationship and contributes to viral mutation and latency, better understanding of this intriguing virus-host relationship shall help the treatment and drug design for viral diseases.



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 APO and viral Vif related virus production

Chemical Species


Related Equation


Number of





Viral RNA



Viral protein Tat



Viral protein Gag



Viral protein Vif



Cellular RNA of APO



Cellular protein APO



Vif-APO complex



Gag-Vif complex



Gag-APO complex


Accumulated Number of





Virions produced



Vif packaged in virions



APO packaged in virions


Average Number of





Packaged Vif per virion



Packaged APO per virion


All variables denote the number of molecules or virions.


TABLE 2     Parameter values

Parameter Explanation


Base Value ()


Number of integrated provirus

1 (t >= 6h), 0 (t < 6h)


Basal transcription rate for viral RNA

15 transcripts/h


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) steady-state translation rate

270 proteins/h

4.5 proteins/min (9)

Probability of viral RNA to encode Tat


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



Probability of viral RNA to encode Vif



Number of Gag per Virion for assembling



Association constant of Tat with TAR


28.57/μM  (9).

Association constant of Vif and APO


Assumed to be 50/μM (21,39).

Association constant of Gag and Vif


Selected to keep steady state of  to be about 100.  Range of  was reported to be 60-100 (35) in acutely infected cells.

Association constant of Gag and APO


Assigned to be same as .

Rate of Gag export through virions budding


Selected to keep the steady state of  about 3900 (40).

Degradation rate of viral RNA


Half life: 4h (9)

Degradation rate of cellular RNA of APO


Selected to be the same as .

Degradation rate of Tat


Half life: 4h (9)

Degradation rate of Gag


10% Gag (p24) degradates in 1h (41)

Degradation rate of Vif


Half life: 89min (21)

Degradation rate of APO


Half life: 29min (21)

Degradation rate of Vif-APO complex


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:APO-3G (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 post-infection

Vector HA-A3G:pNL4-3ΔVif (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 post-infection

1.  Constant pNL4-3 (expressing constant amounts of Vif) = 2.5μg, varying pcDNA-APO3G. = 0, 0.5, or 2.5μg, total DNA was adjusted to 5μg.

2.  Constant Vif-defective pNL4-3Vif(-) = 2.5μg, varying Vif expression vector pNL-A1 = 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



Schematic of cellular APO and viral Vif related virus production.  Viral RNA is transcribed from provirus integrated in cellular genome and yields viral proteins Tat, Gag and Vif.  Tat transactivates viral transcription to accelerate viral RNA synthesis.  Gag is a polyprotein used to form the main virus structure.  Regulatory protein Vif acts as anti cell defense factor to form complex with cellular cytidine deaminase enzyme APO and promote its degradation through ubiquitination.  Both Vif and APO are able to be incorporated into nascent formed virion.



Comparison between experiments and simulations.  (A) Viral Vif downmodulates cellular protein APO.  Note the direction of horizontal axis is reversed.  (B) Overexpressing APO increase the relative ratio of packaged APO into virions.  (C) 2D plot of the percentage of packaged APO in total APO, by varying transcription rate of APO RNA and rate of Vif expressing.  (D) Negative correlation between viral infectivity and simulated .  Note the vertical axis direction of  is reversed.  See text and Table 3 for details.



Parameter sensitivity analysis.  Vertical axis denotes relative parameter sensitivity value on the steady state value of variable ,  and accumulated number for  at 48h post-infection.  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.



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 post-infection.  Values on variables  and  are scaled down as legend indicated.  (A) Gag related parameter  and .  (B) Tat related parameter ,  and .