A competition model for viral inhibition of host cell
proliferation
Sarah Holte
a,*, Michael Emerman
b aDivision of Public Health Science, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave North MW-500, Seattle, WA 98109, USA
b
Division of Human Biology, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave North MW-500, Seattle, WA 98109, USA
Received 11 January 2000; received in revised form 27 April 2000; accepted 5 May 2000
Abstract
Some viruses encode proteins that promote cell proliferation while others, such as the human immun-ode®ciency virus (HIV), encode proteins that prevent cell division. It has been hypothesized that the se-lective advantage determining which strategy evolves depends on the ability of the virus to induce a cellular environment which maximizes both virus production and cell life span. In HIV, the protein that causes cell cycle arrest is Vpr. In this paper, we develop a mathematical model, based on dierence equations, to study the competition between two genotypes of HIV ± one that encodes this protein (Vpr+) and one that does not (Vpr)). In particular, we are interested in parameters that could be dierent between the in vitro condition, where the Vpr) genotype dominates, and the in vivo condition, where the Vpr+ genotype dominates. Our model indicates that the infected cell death-rate, the viral half-life, and the dynamics of the target cell population all eect the competition dynamics between the Vpr+ and Vpr) viral genotypes. Perturbing any of these parameters from the in vitro estimates while holding the others ®xed has no aect on the competition outcome, i.e., the Vpr)genotype dominates. Perturbing the infected cell death-rate and the target cell source causes a switch in competitive outcome, although not necessarily at values, which represent the in vivo condition. Adding a perturbation in the viral half-life from in vitro to in vivo condition results in a switch of the competitive advantage from the Vpr) genotype to the Vpr+ genotype with pa-rameters for all three mechanisms set to estimated in vivo values. Ó 2000 Published by Elsevier Science
Inc. All rights reserved.
Keywords:HIV; Competition dynamics; Vpr protein
*Corresponding author. Tel.: +1-206 667 6975; fax: +1-206 667 4812.
E-mail address:sarah@hivnet.fhcrc.org (S. Holte).
0025-5564/00/$ - see front matter Ó 2000 Published by Elsevier Science Inc. All rights reserved.
1. Introduction
Viruses are obligate parasites with diverse ways of interacting with their hosts. Some viruses initiate lytic infections, which result in high levels of virus replication followed by clearance of the virus. On the other hand, other viruses establish life-long persistent infections in which the virus has established a quasi-homeostasis with the organism. In general, the mechanisms underlying persistent infections are poorly understood and require a complex relationship between levels of replication that are bene®cial to the virus, but not so detrimental to the host that viral trans-mission is compromised [1]. One way in which persistent infections are maintained is by making the viral life cycle dependent on pathways that induce cellular proliferation. For example, human T cell leukemia virus (HTLV), the causative agent of adult T cell leukemia is a retrovirus that incorporates its genetic material into the host cell genome and promotes cell growth. Other vi-ruses, such as the human papilloma vivi-ruses, also encode proteins that promote cell proliferation and are propagated within the host essentially by replication of the infected cell. Not surprisingly, these viruses can be oncogenic. Other viruses use additional means of driving infected cells into replication [2].
The human immunode®ciency virus (HIV), the causative agent of acquired immune de®ciency syndrome (AIDS), also establishes a persistent infection and incorporates its genetic material into the host cell chromosome. However, HIV infected cells have a short half-life after infection [3]. Moreover, not only does HIV not promote cell proliferation, it also encodes a protein, called Vpr, that prevents cells from progressing through the cell cycle and dividing [4]. Expression of Vpr is able to delay or arrest cells in the phase of the cell cycle just before mitosis occurs (called G2 phase) [4].
generations. Examples include models for HIV pathogenesis after treatment with anti-retroviral drugs [6] and mathematical descriptions of epidemics [7,8]. We chose to use dierence equations due to the discrete nature of the data we plan to collect, and the ease with which stochastic variability can be incorporated. Our model contains compartments representing free virus that is either Vpr+ or Vpr), infected cells that are producing virus that is either Vpr+ or Vpr), and uninfected target cells. By varying parameters in the model which correspond to biological pa-rameters that dier between in vivo and in vitro conditions, we are able to simulate a variety of conditions, and theoretically predict how the two genotypes interact in the setting of co-infection by the two genotypes. In this work, we focus on infected cell death-rates, the dynamics of the uninfected source population, and the half-life of infectious viral RNA. Other parameters in the model could also aect the competition outcome between the two genotypes of HIV, however, we chose to focus on these three parameters since they are known to dier between the in vivo and in vitro conditions. They are also parameters that can be perturbed in an experimental setting.
Stability of a zero equilibrium for one of the population compartments in a model is the mathematical equivalent of extinction, e.g., when the zero equilibria for the population of Vpr+ infected cells is stable, the Vpr+ genotype becomes extinct and the Vpr) genotype is the com-petitive winner. Thus, we focus on bifurcations in stability of zero equilibria for the compartments representing cells that are infected with viruses that are either Vpr+ or Vpr). An understanding of these bifurcations, i.e., changes in stability, helps us to predict when viral genotypes that inhibit host cell proliferation attain the competitive advantage and which biological factors may aect the competitive outcome. Such model-based predictions can be used to guide laboratory experiments to further understand the interaction of these two populations.
2. The dierence equation model
In this section, we develop a mathematical model to study the competition dynamics of the viral genotypes of HIV that either do or do not cause cell cycle arrest (Vpr+ and Vpr), respectively).
We modeled mixed populations of virus and infected cells containing both Vpr+ and Vpr)
ge-notypes of HIV-1. When cells become infected by HIV, the infection status is characterized as
acutelyinfected (those subject to extensive cytopathic eects of infection, high levels of proviral DNA, and superinfection of cells). Infected cells are initially in the acute state. Cells which survive
acuteinfection becomechronicallyinfected (characterized by immunity to superinfection, no signs of cytopathic eect, and small number of provirus per cell [9]). We dierentiate between acutely and chronically infected cell compartments, with ¯ow from acutely to chronically infected pop-ulations. We chose to include compartments, for both chronically and acutely infected cells, since it seems likely that when the infected cell death-rate is accelerated (in vivo condition), the dy-namics of the viral competition is dominated by the acutely infected cells. In the in vitro condition, where the infected cell death-rate is extended, the chronically infected cells may play the dominant role in determining the competitive outcome. A compartment for uninfected cells is the ®nal compartment included in this model.
Table 1
In vitro model parameters
c Infectious viral clearance rate 0.12 per virion per houra
n
a Infectious viral production rate by cells acutely
infected with the Vpr+ genotype
0.028 virions per hourb
n
c Infectious viral production rate by cells
chronically infected with the Vpr+ genotype
0.0028 virions per cell per hourb
nÿ
a Infectious viral production rate by cells acutely
infected with the Vpr)genotype
0.014 virions per cell per hourb
nÿ
c Infectious viral production rate by cells
chronically infected with the Vpr)genotype
0.0014 virions per cell per hourb
c Rate at which acutely infected cells become chronically infected
0.01 cells per hourc
r Birth-rate for Vpr+ infected cells 0.019 per cell per hourd
rÿ Birth-rate for Vpr)infected cells 0.035 per cell per hourd
r Birth-rate for uninfected cells 0.035 per cell per hourd
da Death-rate for Vpr+ acutely infected cells 0.05 per cell per houre
dc Death-rate for Vpr+ chronically infected cells 0.025 per cell per houre
dÿa Death-rate for Vpr)acutely infected cells 0.05 per cell per houre
dÿc Death-rate for Vpr)chronically infected cells 0.01 per cell per houre
du Death-rate for uninfected cells 0.01 per cell per houre
d Density dependent overall cell death-rate 10ÿ11f
i Probability of infection 210ÿ6per virion per target cell per hourg
s Constant rate of target cell replacement 0 cells per hourh a
Corresponds to viral half-life in vitro of 6 h [10].
bSee text. c
Based on data in [11].
d
Corresponds to doubling time of 36 h for Vpr+ infected cells and 20 h for Vpr)and uninfected cells [5]. e
Corresponds to life expectancies (inverse of the death-rate) of 20 h for both Vpr+ and Vpr)acutely infected cells,
100 h for Vpr)chronically infected and uninfected cells, (Figs. 1 and 2 [11]) and 40 h for Vpr+ chronically infected cells
[12].
f
Corresponds to maximal carrying capacity of 2:4109. Set to allow larger carrying capacity than expected in vivo to
account for divisions of culture that occurs in experimental setting.
g
Estimate of probability of infection in 1 h could vary from 0.05 to less that 510ÿ7 with viral clearancecin the
ranges we consider in this paper. Fig. 5 [13]. We assumed infectivity to be the same for both Vpr+ and Vpr)genotypes
[14].
h
In vitro condition, no additional target cells added after start of experiment.
Model compartments
Vt infectious Vpr+ viral population at time t
Vÿ
t infectious Vpr) viral population at time t
At cells acutely infected with the Vpr+ genotype at timet
Aÿ
t cells acutely infected with the Vpr)genotype at time t
Ct cells chronically infected with the Vpr+ genotype at time t
Cÿ
t cells chronically infected with the Vpr) genotype at timet
Tt uninfected target cells at timet
The model equations are as follows:
In developing the dierence equation model we chose the time unit to be 1 h. In Eqs. (2.1) and (2.4) the ®rst terms represent the clearance of free virus. We assume that virus decays exponen-tially with decay parametercso that eÿc
represents the proportion of viral RNA remaining after 1 h. The second terms in Eqs. (2.1) and (2.4) represent the number of new virions produced in the unit of time fromttot1. Note that we distinguish between the number of virions produced by
each infected cell depending on whether or not the cell is infected with the Vpr+ or Vpr)
genotype, and whether the cell is acutely or chronically infected.
Eqs. (2.2), (2.3), (2.5), and (2.6) describe the dynamics of the four types of infected cells, acutely or chronically infected, and infected with the Vpr+ or Vpr) genotype. Note that we include the density dependent death factor,dXt, in the equations for all cell populations. This term limits the
size of the total population of cells to some `carrying capacity' and is used frequently to more realistically describe population sizes since continued exponential growth is not possible [7]. In laboratory experiments, this issue is addressed by dividing culture every few days so that carrying capacity is not reached, although we include it here as a necessity for conducting equilibrium analysis. The ®rst term in each of the equations represents the density in those compartments at timet1 based on the size of those compartments at time t, and the combined birth and death-rates [e.g., rÿd
a ÿdXtfor acutely infected Vpr+ cells]. For acutely infected cells we also allow
for ¯ow to the chronically infected compartment at ratec. The second terms in the equations for acutely infected cells account for new infections in one unit of time. To derive these terms, we compute the probability of a cell not becoming infected by the Vpr+ genotype in 1 h using the binomial distribution as 1ÿiVt eÿiVt so that the number of infections by the Vpr+ genotype
betweentandt1 is approximately 1ÿeÿiV
t Tt. A similar derivation is applied for infections by
the Vpr)genotype. For chronically infected cells, the second term represents the contribution to that compartment from the acutely infected cell populations.
The calculations to determine the amount of infectious viral RNA produced by acutely and chronically infected cells and cells infected with the Vpr+ or Vpr) genotype n
c;na;nÿc; andnÿa
50 virions per hour, based on continuous release, which is demonstrated in [14]; (2) the amount of viral RNA produced in the G2 phase is four times the amount produced in the G1 phase per viral DNA [5] (with twice as much viral DNA in G2 as in G1); (3) the amount of viral RNA produced acutely infected cells is 10 times the amount produced by chronically infected cells [11]; and (4) both genotypes remain in the G1 phase of the cell cycle for 18 h, the Vpr)genotype remains in the G2 phase for 2 h, and the Vpr+ genotype remains in the G2 phase for 18 h [5]. We also assumed that 1/1000 of the virions produced are actually infectious. For example,n
a is calculated as
fol-lows: Ifn1is the rate of viral production in the G1 phase of the cell cycle andn2 is the rate of viral production in the G2 phase of the cell cycle, thenn1average viral production rate combined
time in G1 and G2)/(time in G124time in G2) 5036=f1000 182418g
and n24n1. Then the average rate of viral production for cells infected with the Vpr+
genotype is (n1 time in G1n2 time in G2= time in G1time in G2 0:011
180:04418= 1818 0:028.
Eq. (2.7) describes the dynamics of the uninfected cells. The expressionÿiVtÿiÿVtÿ in the ®rst term represents the loss of cells from the population of uninfected cells to the various infected cell compartments. The remaining terms in the exponential,rÿduÿdXt, represent the combined
birth and death-rate for uninfected cells, as in Eqs. (2.3)±(2.6) for infected cell compartments. Finally, the term s in Eq. (2.7) is the parameter we will vary to simulate the eects of various sources of uninfected cells into the system.
As mentioned in Section 1, we investigate via the mathematical model three factors that could dier between in vivo and in vitro conditions, and their in¯uence on genotype selection:
· the infected cell death-rate,
· dynamics of the uninfected target cell population, and
· the clearance-rate of viral RNA.
Recall that the idea behind the relevance of the infected cell death-rate is that it is higher in vivo than in vitro. Therefore, since viral production is signi®cantly higher in the G2 phase of the cell cycle, the ability to arrest the cell cycle in G2 (due to Vpr) would give the virus a selective advantage in vivo since the accelerated death-rate limits the number of cell divisions. By varying
da;dÿa;dc and dÿc we can use the mathematical model to make predictions about the eect that the rate of infected cell death has on the viral dynamics. We will be most interested in variations ofdc anddÿc, since these rates are most dramatically altered with accelerated in vivo infected cell death, although our approach is to add a single additional increment to all four death-rates, in order to simulate additional cell killing in vivo, most likely due to the immune response. Mathematically, we will add an increment, , to each of da;daÿ;dc, and dÿc in Eqs. (2.2), (2.3), (2.5) and (2.6).
The second variable we consider is the dynamics of the target cell population. In this work, we consider only two possibilities, which could be easily replicated in laboratory experiments: no additional source of target cells after initiation of the experiment or a constant replenishment of additional target cells throughout the experiment at speci®ed time intervals. Other possibilities for the dynamics of the target cell population may include activation or deactivation of target cells which might be dependent on the amount of virus present or other immunological factors,
al-though we do not consider those mechanisms in this work. The parameter s in the model
replacement, which is the mechanism most likely to be carried out in in vitro experiments. If other mechanisms for target cell replacement are proposed, then they could be incorporated into the model through more detailed modeling of the uninfected target cell compartment.
The third factor we consider is the half-life of infectious virus. This has been estimated as 6 h in vitro [10] and 20 min in vivo [3]. We assess the eects of these two values for viral clearance in combination with the parameters described in the previous two paragraphs. The parametersc
in the model corresponds to the clearance-rate for infectious viral RNA.
3. Results
3.1. Temporal results
To begin, we simulated the system dynamics by iterating the dierence equations 600 times, to assess the cellular trajectories for 25 days (600 h) after initiation of the experiment. At time zero, we assumed that the density of uninfected cells is 106 per ml, with 5000 cells infected with HIV-1
that is Vpr+ and an equal amount infected with HIV-1 that is Vpr). There is an initial spike in the percentage of cells infected with HIV-1 that is Vpr+, but after about 10 days that genotype is essentially extinct (Fig. 1).
Next, we perturbed each of the three parameters of interest in this work separately while holding the other two ®xed: the infected cell death-rate, the uninfected cell source rate, and the viral death-rate. Perturbing each of these parameters separately did not result in any change in the competition outcome. However, for cells chronically infected with the Vpr) genotype, if the death-rate is perturbed to be less than the birth-rate,s0, then both populations die out.
Fig. 1. Simulated competition between Vpr)and Vpr+ genotypes under standard in vitro conditions: no source of
uninfected target cells, chronically infected Vpr) cell life expectancy 100 h, viral half-life 6 h. Trajectories in bold
represent Vpr+ genotype. Cells infected with the Vpr)genotype attain the competitive advantage approximately 15
Fig. 2. Simulated competition between Vpr)and Vpr+ genotypes under perturbed conditions. (a) represents trajectories
when uninfected cell source is manipulated so that 100 000 uninfected cells are added to the system every hour, Vpr)
infected cell life expectancy is 100 h and viral half-life is 6 h. (c) represents trajectories when 100 000 uninfected cells are added to the system every hour and Vpr)infected cell life expectancy is 25 h and viral half-life is 6 h. (e) represents
trajectories when 100 000 uninfected cells are added to the system every 24 h and the Vpr)infected cell life expectancy is
25 h and viral half-life is 6 h, and (g) represents trajectories when 100 000 uninfected cells are added to the system every hour, Vpr)infected cell life expectancy is 25 h, and viral half-life is 20 min. Trajectories in bold represent Vpr+ genotype.
Fig. 2 shows the cellular trajectories for a variety of combinations of perturbations of these three mechanisms. All trajectories shown in Fig. 2 include a source of uninfected target cells. Fig. 2(a) and (b) depict the results of adding 105uninfected cells every hour without perturbing the
infected cell death-rate or the infectious viral half-life. In this case, the cells infected with the Vpr) genotype are the competitive winners.
When we manipulated the infected cell death-rate so that the life expectancy (the inverse of the linear contribution to the death-rate) of chronically infected cells decreased from 100 to 25 h (approximate in vivo condition) while adding 105 cells per hour and keeping the infectious viral
half-life set at 6 h, we obtain the results shown in Fig. 2(c) and (d). In this scenario, the two genotypes coexist. A more signi®cant decrease in the life expectancy of the cells infected with the Vpr)genotype does result in complete domination by cells infected with Vpr+ genotype, but not until the life expectancy of Vpr) infected cells is less than 20 h.
The results of perturbing the infected cell life expectancy as described in the previous para-graph keeping the infectious viral half-life set at 6 h, and adding uninfected cells every 24 h rather than every hour are shown in Fig. 2(e) and (f). In this scenario, the model predicts that by decreasing the source from once an hour to once a day, the population of cells infected with Vpr+ has the selective advantage. This suggests that when the populations of target cells is in some way limited but maintained, the Vpr+ genotype wins the competition for the resource of uninfected cells.
Finally, we considered a perturbation of all three parameters of interest simultaneously. We included a source of 105 uninfected target cells per hour, infected cell death-rates so that the life
expectancy of cells chronically infected with the Vpr)genotype is 25 h, and a perturbation to the viral half-life by reducing it from 6 h to 20 min (approximate in vivo condition). The results are shown in Fig. 2(g) and (h). These ®gures indicate that in this scenario, the most similar to the in vivo situation, the population of cells infected with the Vpr+ genotype dominates completely after 25 days.
3.2. Bifurcation results
Our mathematical model can be used to predict parameter values where bifurcations in the stability of the zero equilibria of cells infected with HIV-1 that is either Vpr+ or Vpr)occur, i.e., to determine values of the biological parameters where the competitive advantage switches from one genotype to the other. Thus, to further assess the Vpr+/)competition, we produce bifurcation diagrams for these equilibria values as functions of the life expectancy of cells chronically infected with the Vpr)genotype and the source of uninfected cells. We assessed these bifurcations in the situations when viral half-life is constant at 6 h (in vitro condition) and 20 min (in vivo condition). To perform the bifurcation analysis, we iterated the function de®ning the model 50 000 times, and used the last 50 iterates for each value of the parameter of interest to construct the diagrams. In all cases, we obtained equilibrium conditions for the system after 50 000 iterations.
indicate that when the viral half-life is 6 h (in vitro condition) and the life expectancy for cells
chronically infected with the Vpr) genotype is less than 20 h, then the Vpr+ genotype
domi-nates. If the life expectancy for cells chronically infected with the Vpr) genotype is between 20 and 30 h, then our model suggests that the two genotypes would coexist. When that life
ex-pectancy is greater than 30 h the Vpr) genotype dominates. Fig. 3(c) and (d) represent the
situation where the viral half-life is set to 20 min (in vivo condition). The competition outcome is quite similar to the situation where viral half-life is 6 h, although the window of coexistence is signi®cantly smaller.
A similar analysis is conduced by varying the source of uninfected cells and the results are depicted in Fig. 4. For that analysis, we assume that the life expectancy for cells chronically in-fected with the Vpr)genotype is 25 h, which is close to the estimates of life expectancy for infected cells in vivo [3]. We vary the infected cell source from 103 to 106 per hour. We assessed the
sit-uation when the viral half-life was set to 6 h (Fig. 4(a) and (b)) and when it was set to 20 min (Fig. 4(c) and (d)). Fig. 4(b) indicates that when the viral half-life is 6 h (in vitro condition) and a small <(6104 number of uninfected cells are added each hour, the Vpr+ genotype wins the
com-petition. As the amount of cells added each hour is increased, the two genotypes coexist. The
Fig. 3. Bifurcation diagram for chronically infected Vpr)infected cell life expectancy with target cell source of 100 000
overall situation is similar when the viral half-life is 20 min, although the dominance of the cells infected with the Vpr+ genotype persists at much higher levels of the source of uninfected cells.
3.3. Comparison of the model predictions with experimental data
There is a limited amount of data regarding the dynamics of selection for Vpr in infected primates. However, we know of two works, which present data from co-infections experiments in the in vivo setting. Our modeling is focused on in vitro experiments, however, to assess the validity of our model under in vivo conditions, we compared model-predicted trajectories to two sets of data on percentages of the Vpr+ genotype in mixed populations of HIV and the simian immuno de®ciency virus (SIV) [15,16].
Co-infection experiments have been conducted in macaques to study the ®tness of the Vpr+ and Vpr)genotypes of SIV [16]. These experiments involved infecting macaques either intravenously or intrarectally to test transmission parameters of Vpr or a related protein, Vpx. In these ex-periments, macaques were infected with mixed genotypes of virus, which contained 50% of the Vpr+ genotype, and 50% of the Vpr)genotype. We compared our model predictions to their data
Fig. 4. Bifurcation diagram for uninfected target cell source with infected cell death ®xed so that life expectancy of chronically infected Vpr)cells is 25 h and viral half-life is 6 h (a) and (b) and 20 min (c) and (d). Bold plots represent
with parameters for the target cell source set to 105at time zero, constant replacement of 5104
cells per hour, the infectious viral half-life to 20 min, and infected cell death-rates so that the Vpr) infected cell life expectancy is 22 h. In addition, we used a density dependent decay parameter 10ÿ7
which was chosen as to produce a reasonable in vivo carrying capacity, and is somewhat lower than the value we used in the previous simulations, since density dependent decay in vivo is more likely to be a factor than in in vitro laboratory experiments, where division of culture is performed to avoid carrying capacity restrictions on population sizes. The results are shown in Fig. 5 for both intravenous and intrarectal infection. While the general `shape' of the predicted ®tness percentage curve is in agreement with the data for intravenous infection, apparently our model predicts faster attainment of the ®tness advantage by cells infected with the Vpr+ genotype. This could be a result of the fact that we have not accounted for delay in production of free virions by infected cells, or dierences in parameters between those we used in the model and the true bi-ological parameters for this genotype of SIV in macaques. The model-based trajectories for the intrarectal infection appear to be in agreement with the data. However, for both intravenous and intrarectal infections, limited data was available, and so statistical model ®tting was not possible. In addition, for the intravenous data, there is no data for times greater than seven days, and for intrarectal infection here is no data available prior to four days post-infection. Thus, we cannot conclude that the model we present here accurately represents data throughout the course of infection.
In order to assess which parameters were most sensitive to perturbations, we performed a sensitivity analysis of the infected cell death-rate, density dependent clearance parameter,
infec-Fig. 5. Comparison of model-based predictions and observed data from macaques experimentally infected with SIV. Model parameters set as follows: target cell source to 105 at time zero, with constant replacement of 5´104cells per
hour, the viral half-life to 20 min, and the linear portion of the infected cell death-rates so that the Vpr)infected cell life
expectancy is 22 h. We also increased the density dependent death parameters from 10ÿ11to 10ÿ7 to more accurately
tivity parameter, rate of target cell source, and initial target cell size. We varied each parameter by a factor of 10, and found dramatic dierences in the `®t' to the data when we varied target cell source rate, infectivity constant, and density dependent decay constant. We found less dramatic, yet signi®cant dierences when we considered variation in the infected cell death-rate, somewhat in contrast to our simulated ®ndings. However, this dierence could be explained by dierences in density dependent decay parameter between our analysis of simulations from the mathematical model and this attempt to see how well our model describes data collected from in vivo experi-ments in macaques. Finally, we found essentially no dierence based on variability in the size of the initial uninfected target cell source.
In chimpanzees infected with HIV that was defective in thevprgene (Vpr)), reversion of thevpr
gene occurred between six weeks and two years after infection [15]. We compared our model projection to data collected on two chimpanzees 6±8 weeks post-infection, and 2 years post-in-fection. In both cases, we obtained good predictions from the model with realistic biological parameters. However, for each chimpanzee only two data points were available (results not shown).
4. Discussion
Our mathematical model allows evaluation of a number of scenarios which could in¯uence competition outcome between the isogenic genotypes of HIV-1 that dier in their ability to cause cell cycle arrest. We believe that this analysis sheds light on the complex selective forces under-lying evolution of viral interactions with the host cell cycle.
The temporal analysis suggests that with parameters similar to those found in vivo, a virus which causes a G2 arrest (the Vpr+ genotype) has the competitive advantage. When we used parameters in the model that are similar to those found in vitro, the competitive advantage is attained by a virus which does not cause a G2 arrest (the Vpr) genotype). In particular, infected cell death-rates, infectious viral half-life, and target cell source are all parameters, which dier between the in vivo and in vitro conditions, and simultaneously aect the competition dynamics between the two genotypes. Thus, our model suggests that several biological parameters can aect the dierential ®tness advantage for Vpr genotypes in the in vivo and in vitro conditions. As conjectured in [5], an accelerated infected cell death-rate is a crucial component aecting the competition outcome, but a sustained source of target cells is also required for a switch in
competitive advantage from the Vpr) genotype to the Vpr+ genotype. This eect is further
modi®ed by the infectious viral half-life. In addition, our model suggests that in the situation of a sustained but limited resource of uninfected target cells, the Vpr+ genotype attains the compet-itive advantage as long as the life expectancy of the cell infected with a Vpr) virus is suciently short. As the target cell source becomes more abundant, our model suggests that the two geno-types could coexist. Eventually the Vpr+ genogeno-types dominate in vivo [5] and previous models of HIV infection [17] and experimental data [14] have suggested that target cells are limiting in vivo. If target cells are in fact limited, it could oer a further explanation for why the Vpr+ genotype dominates in vivo.
experi-ments, which would involve perturbations of associated biological parameters in the laboratory. Predictions based on a mathematical model and associated bifurcations for population dynamics of ¯our beetles were tested experimentally [18] with impressive results. By controlling birth and death-rate parameters experimentally, population sizes and dynamics were predicted quite ac-curately. Similar experiments in HIV population biology are possible and could be used to test some of the hypothesis and predictions motivated by this model.
We plan to test our predictions with the following experiments: viral strains have been con-structed that dier in Vpr genotype and allows us to easily distinguish the wild-type genotype from the mutant genotype by PCR analysis. Co-infected cell populations are allowed to evolve, and desired perturbations will be implemented by varying the life span of the infected cell and source of uninfected cells. Cells are infected for a ®xed period of time (for example, 3 days). The infected cells are then killed with mitomycin C at a dose that leads to 90% cytotoxicity in 24 h. Immediately after treatment, fresh uninfected cells are added to the culture and virus spread is allowed to occur by cell±cell contact. This is repeated every 2 days. Thus, in this experiment the life span of the infected cell can be no more than 2 days. This will be repeated by varying the time-span of the infected cells from a maximum where no mitomycin C is used to using 3 days, 2 days, and 16 h. These conditions will bracket the estimates for the half-life of infected cells in vivo. We expect to be able to determine from these experiments the point at which the infected cell's life span becomes a force in selection for Vpr.
Our comparison of the model with experimental data suggests that the model may be able to predict the behavior of the interacting genotypes for in vivo co-infection dynamics. It should be noted that we ®t our model to the experimental data in the loosest sense of the word; i.e., we determined parameter values within reasonable biological ranges so that the model trajectories agree to some extent with the data. Thus, we are able to demonstrate that biologically plausible parameters in our model produce trajectories that are in agreement with experimental data. However, further laboratory experiments are needed to assist with more accurate estimation of many of the model parameters. In addition, we have ignored the delay between infection and initiation of viral production, which could be a key factor, especially for short-term dynamic behavior. This could be addressed by using partial dierential equations or delay equations to account for the age structure of infected cells. For in vivo modeling, the target source population dynamics and other immunological mechanisms will need to be more carefully assessed. These topics will be investigated in later work. In addition, our sensitivity analysis suggests that pa-rameter perturbations in both experimentally controlled or incorrect of papa-rameters not controlled could dramatically aect outcomes. Thus, as we embark on the analysis of experimental data, we will need precise methods for parameter estimation.
stochastic model could possibly be used to describe the data on reversion of mutantvpr(Vpr)) to the wild-type (Vpr+) presented in [23].
The model developed in this paper or similar models could be used more generally to study competition between various types of HIV. These methods are applicable in any situation where there are discrete and well-identi®ed populations of viruses and cells. For example, if single genes can be identi®ed with drug resistant phenotypes, this model could be used to explore ®tness be-tween resistant and naive populations of HIV in the presence or absence of treatment. Or it could be used to assess ®tness of viral populations, which use dierent co-receptors with associated modeling of the target cell populations. In these situations, as well as the conditions described in this work, the most eective use of mathematical modeling of viral ®tness will be achieved when the models are combined with carefully designed experiments and rigorous statistical techniques to estimate model parameters and assess the validity of the models and their predictive abilities. Finally, we will address in future work other competitive situations, as well as the ®tness of thevpr
gene through designed experiments, mathematical modeling, and statistical estimation.
Acknowledgements
The authors would like to thank Mario Stevenson for supplying data from the co-infection experiments in macaques. We would also like to thank Steve Self, Tim Randolph, John Mittler, Wei Chun Goh, and Rahm Gummuluru for review and helpful suggestions on the manuscript. This work was supported by NIH grants R01 AI42522 and R01 AI30927.
References
[1] M.B. Oldstone, Viral persistence: mechanisms and consequences, Curr. Opinion Microbiol. 1 (1998) 436. [2] A. Op De Beeck, P. Caillet-Fauquet, Viruses and the cell cycle, Prog. Cell Cycle Res. 3 (1997) 1.
[3] A.S. Perelson, A.U. Neumann, M. Markowitz, J.M. Leonard, D.D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life span, and viral generation time, Science 271 (1996) 1582.
[4] M. Emerman, HIV-1 Vpr and the cell cycle, Current Biol. 6 (1996) 1096.
[5] W.C. Goh, M.E. Rogel, C.M. Kinsey, S.F. Michael, P.N. Fultz, M.A. Nowak, B.H. Hanh, M. Emerman, HIV-1 Vpr increases viral expression by manipulation of the cell cycle: a mechanism for selection of Vpr in vivo, Nat. Med. 4 (1998) 65.
[6] W. Tan, Z. Xiang, Some state space models of HIV pathogenesis under treatment by anti-viral drugs in HIV-infected individuals, Math. Biosci. 156 (1999) 69.
[7] F.C. Hoppensteadt, Mathematical Methods of Population Biology, Cambridge University Press, New York, 1982. [8] L. Allen, A. Burgin, Comparison of deterministic and stochasitic SIS and SIR models in discrete time, Math.
Biosci. 163 (2000) 1.
[9] S.K. Weller, A.E. Joy, H.M. Temin, Correlation between cell killing and massive second-round superinfection by members of some subgroups of avian leukosis virus, J. Virol. 33 (1980) 494.
[10] D. Rocancourt, C. Bonnerot, H. Jouin, M. Emerman, J.F. Nicolas, Activation of a betagalactosidase recombinant provirus: application to titration of human immunode®ciency virus HIV and HIV-infected cells, J. Virol. 64 (1990) 2660.
[12] S.A. Stewart, B. Poon, J.B. Jowett, I.S. Chen, Human immunode®ciency virus type 1 Vpr induces apoptosis following cell cycle arrest, J. Virol. 71 (1997) 5579.
[13] S.P. Layne, J.L. Spouge, M. Dembo, Quantifying the infectivity of human immunode®ciency virus, Proc. Nat. Acad. Sci. 86 (1989) 4644.
[14] S. Gummuluru, M. Emerman, Cell cycle- and Vpr-mediated regulation of human immunode®ciency virus type 1 expression in primary and transformed T-cell lines, J. Virol. 73 (1999) 5422.
[15] P.N. Fultz et al., Vaccine protection of chimpanzees against challenge with HIV-1-infected peripheral blood mononuclear cells, Science 256 (1992) 1687.
[16] V.M. Hirsch et al., Vpx is required for dissemination and pathogenesis of SIVSM PBj: evidence of
macrophage-dependent viral amplication, Nat. Med. 4 (1998) 1401.
[17] A.N. Phillips, Reduction of HIV concentration during acute infection: independence from a speci®c immune response, Science 272 (1996) 1960.
[18] R.F. Contantino, J.M. Cushing, B. Dennis, R.A. Desharnais, Experimentally induced transitions in insect population dynamic behavior, Nature 275 (1995) 138.
[19] A.J. Brown, Analysis of HIV-1 env gene sequences reveals evidence for a low eective number in the viral population, Proc. Nat. Acad. Sci. USA 94 (1997) 1862.
[20] R. Ribeiro, S. Bonhoeer, A stochastic model for primary HIV infection: optimal timing of therapy, AIDS 13 (1998) 351.
[21] W. Tan, H. Wu, Stochast modeling of the dynamics of CD4+ T-cell infection by HIV and some Monte Carl studies, Math. Biosci. 147 (1998) 173.
[22] H. Tuchwell, E. Le Corfec, A stochastic model for early HIV-1 population dynamics, J. Theroet. Biol. 195 (1998) 451.
