## Abstract

Human immunodeficiency virus can spread through target cells by transmission of cell-free virus or directly from cell-to-cell via formation of virological synapses. Although cell-to-cell transmission has been described as much more efficient than cell-free infection, the relative contribution of the two transmission pathways to virus growth during multiple rounds of replication remains poorly defined. Here, we fit a mathematical model to previously published and newly generated *in vitro* data, and determine that free-virus and synaptic transmission contribute approximately equally to the growth of the virus population.

## 1. Introduction

Human immunodeficiency virus (HIV) can spread through its target cell population by different mechanisms. On the one hand, infected cells can produce free viruses, which infect susceptible cells. On the other hand, the virus can spread directly from cell-to-cell, presumably through the formation of virological synapses [1–6]. This typically involves the transfer of many virus particles from the producer cell to the target cell, a fraction of which successfully integrate into the genome of the target cell, resulting in its infection [7]. Attempts at measuring the relative contributions of cell-to-cell and cell-free transmission have been made by comparing single-round infections performed by producer cells or by the culture medium into which the cells have released virions [1,8]. Based on these observations, synaptic transmission is considered much more efficient on a per-cell basis [1], and by extension, it has been suggested to be the main mechanism of virus spread *in vivo*. The relative contribution of the two transmission pathways to virus growth through multiple rounds of replication has been examined by Sourisseau *et al.* [6], but it has not yet been quantified rigorously. Here, we fit a mathematical model to these data as well as newly generated experimental data. We find that the two transmission pathways make approximately equal contributions to virus growth *in vitro*.

## 2. Material and methods

Jurkat cells (1 × 10^{6}) in 2 ml were infected in six-well plates with the CXCR4 co-receptor utilizing GFP reporter virus NLENG1-IRES [9] on day 0. Cells were infected under static conditions for 2 h, then washed to remove unbound virions, then divided into equal cultures that were either maintained under static conditions or placed on a Nutator platform with continuous motion for the duration of infection. Titres of the virus inocula were determined by real-time PCR for HIV–1 virion RNA released into the culture medium as described in Gelderblom *et al*. [10]. Virions per volume were calculated as 1/2 the RNA titres. Each day samples of the cultures were analysed for GFP expression by flow cytometry as described in Gelderblom *et al*. [10]. To examine the efficiency of cell-free infection with and without shaking in a single round of infection, an envelope mutant virus (NLENG1-ES-IRES, [10]) that was pseudotyped with HIV-1 X4 tropic envelope was used to infect the cells. Forty-eight h after infection the cells were analysed by flow cytometry for GFP expression.

## 3. Results

### (a) Data and models

In a previous study [6], virus growth was compared in two culture conditions. In a first set of experiments, cultures were kept under gentle shaking conditions, preventing the formation of viral synapses. Thus, only cell-free transmission occurred. A second set of experiments was performed under static conditions, in which both transmission pathways were likely to occur. Data on two types of cells (the transformed CD4^{+} T cell line Jurkat and primary CD4^{+} lymphocytes) were obtained, each for two different concentrations of initial infection. Time-series of the percentage of infected cells were obtained.

Fitting an appropriate mathematical model to such data allows us to calculate the relative contribution of the two transmission pathways to the rate of virus spread through the population of target cells. Difficulties arise, however, due to the limited number of data points available in Sourisseau *et al.* [6]. Hence, we performed a new and analogous set of experiments, the details of which are described in§2. We focused on Jurkat cells, because the kinetics of cell proliferation and cell death are better defined than in primary CD4 T cells. Four experiments were performed with two different inoculation doses.

Consider the following mathematical model, based on ordinary differential equations: 3.1

The number of uninfected and infected cells is denoted by *x* and *y*, respectively. Target cells divide with a rate *r* and become infected by free virus and via synapses with rates *β*^{free} and *β*^{syn}, respectively. Free virus is assumed to be in a quasi-steady state because its turnover is much faster than that of infected cells [11]. The infection term is a saturating function of the number of cells in the system. Owing to the growth of target cells, lack of saturation (as assumed in many virus dynamics models [11]) would lead to a super-exponential growth, rather than describing straightforward exponential growth that is typically observed. Finally, infected cells die with a rate *a*. They are assumed not to divide owing to Vpr-mediated cell-cycle arrest. These equations are derived in the electronic supplementary material. Using this model, the predicted percentage of infected cells, 100*y/*(*x* + *y*), is fitted to the observed percentages, using standard nonlinear least-squares procedures provided by mathematical software packages. First, the application of the model to our new dataset on infection of Jurkat cells is described. Subsequently, the model is applied to the previously published data for comparison.

### (b) Model application to the *in vitro* replication of HIV-1 in Jurkat cells

Here, model application to our newly generated data on the growth of HIV-1 in Jurkat cells under shaking and static conditions is described. Both types of cultures were inoculated under static conditions and the virus was subsequently allowed to grow in the different settings. Two repeats with inocula of 25 and 75 virions per cell were performed (called A-25, A-75, B-25, B-75). During the first 2 days of the experiment, GFP+ cells (a measure of infected cells) are generated from the inoculum only, not by virus spread from infected to susceptible cells. This is seen in figure 1*a* where the virus in shaking and static conditions shows a similar percentage of GFP+ cells on day 2, after which the curves diverge owing to differences in viral transmission pathways. The first round of infection is also more synchronous than later rounds simply because most cells become infected at around the same time from the initial inoculum. Hence, the percentage of infected cells at day 2 can be roughly considered the initial fraction of infected cells in the model, *y*_{0}/(x_{0} + y_{0}), and day 2 is presented as day 0 in subsequent plots. Conditions were such that the number of cells did not approach levels where space became limiting during the time frame of the experiments. Jurkat cells are characterized by a doubling time of 18–24 h [12], and hence we performed two sets of model fits, assuming *r* = 0.95 day^{−1} and *r* = 0.7 day^{−1}. The death rate of infected Jurkat cells was assumed to be *a* = 0.2 day^{−1}, i.e. a lifespan of about 5 days. Because we fit the fraction of infected cells over time, *y/*(*x* + *y*), results are independent of the absolute initial number of cells. The replication rate of the virus was estimated through model fitting. Under static culture conditions, we estimate the replication rate *β*_{st} = *β*^{syn} + *β*^{free}. Under shaking conditions, we estimated *β*_{sh} = *β*^{free}. The initial fraction of infected cells was allowed to vary in the fits ±10% around the average value observed in the two culture conditions at day 2 of the experiment (day 0 of fit). Only the growth phase of the dynamics was fit, because this is sufficient for parameter estimation and the factors influencing the decline phase are not defined sufficiently for model formulation.

From the estimated values of *β*_{st} and *β*_{sh}, we can calculate the ratio of the viral replication rate for synaptic and free-virus transmission, *β*^{syn}/*β*^{free}. For a precise calculation, we needed to determine whether shaking had an impact on the rate of free-virus transmission. Shaking could impair free-virus infection, because the motion could disrupt attachment of the virus to the cell. Alternatively, shaking could enhance infection because it promotes mixing of viruses and cells. Using single-round infection experiments, we found that shaking increases the efficiency of free-virus infection on average by a factor of *f* = 1.33 (figure 1*b*). Thus, we can calculate *β*^{syn}*/ β*

^{free}= (

*β*_{st}−

*β*_{sh}

*/f*)

*/*(

*β*_{sh}

*/f*).

Figure 2*a* shows the best fit of the model to the data assuming a cellular division rate of *r* = 0.95 day^{−1}. Table 1*a* shows the estimates for *β*^{syn}*/ β*

^{free}for both

*r*= 0.95 day

^{−1}and

*r*= 0.7 day

^{−1}. On average the ratio

*β*^{syn}

*/*

*β*^{free}is 0.91 ± 0.1 for

*r*= 0.95 day

^{−1}and 1.05 ± 0.12 for

*r*= 0.7 day

^{−1}(table 1

*a*). These estimates indicate that synaptic and free-virus transmission contribute approximately equally to virus spread through the target cells.

For comparison purposes, we also performed these estimates with the previously published datasets on Jurkat cells [6], where cells were infected with 0.1 and 1 ng p24/10^{6} cells (*X*-0.1, *X*-1). The same approaches were used, since experimental conditions were identical (figure 2*b*). Although the number of data points is limited, the estimated ratios of *β*^{syn}*/ β*

^{free}are consistent with those obtained from our own data (table 1

*a*)

### (c) Model application to previously published data using primary CD4^{+} lymphocytes

Sourisseau *et al.* [6] also presented data on virus replication in primary CD4^{+} T lymphocytes, using inocula of 0.1 and 1 ng p24/10^{6} cells (*Y*-0.1, *Y*-1). They are more difficult to interpret owing to heterogeneity of the cell population and variations in cell division rates over time. After about a day, cells start dividing with a doubling time of 4–12 h, subsequently slowing down and ceasing to divide after roughly 5 days. Because this complexity and the heterogeneity of the cell population cannot be meaningfully described in the model with the information at hand, the ratio *β*^{syn}*/ β*

^{free}was estimated assuming a variety of cellular division rates and for infected cell lifespans of 2 and 5 days, the lower and upper boundaries in this cell population (table 1

*b*, sample fits shown in figure 2

*c*). For faster cell division and cell death rates, we find that synaptic transmission only contributed about half as much to virus growth as free-virus spread. For longer doubling times, the contribution of the two transmission pathways to virus growth is comparable, similar to the result in Jurkat cells.

## 4. Discussion and conclusion

Fitting mathematical models to data documenting virus growth over multiple rounds of replication under static and gently shaking conditions, we found that synaptic and free-virus transmission contribute about equally to virus spread. Hence, free-virus transmission is not negligible, as might be inferred from the prior conclusion that uptake of virus by cells is 2–4 orders of magnitude more efficient for synaptic compared with cell-free transmission [1].

This calculation has important implications for understanding *in vivo* virus dynamics, and for evaluating the effectiveness of anti-viral immune responses and drug therapy. Different neutralizing antibodies were shown to target free-virus and synaptic transmission [13], and it needs to be determined whether both pathways or only one dominant pathway have to be targeted in vaccination approaches. If only one pathway is dominant, it is sufficient to target this pathway. However, if both pathways contribute equally, as suggested by our calculations, then both free-virus transmission and synaptic transmission must be targeted with equal effort. Measuring the relative contribution of synaptic transmission also has implications for resistance against innate, intracellular anti-viral defence mechanisms. Synaptic transmission could lead to the saturation of such anti-viral factors induced by the simultaneous transfer of many viruses, enhancing the ability of the virus to infect the cell. While the identity of such factors remains uncertain [14], experiments do indicate the presence of saturable targets that inhibit infection [15]. Similarly, it is possible that multiple infection through synapses can saturate anti-viral drugs, such as reverse transcriptase inhibitors, which could contribute to ongoing viral replication during drug therapy [16]. How much this influences treatment responses is currently unclear [17], and depends on the relative importance of synaptic transmission, a first estimate of which has been provided here. Further discussion of the *in vivo* relevance is given in the electronic supplementary material.

Parameter estimates depend on the experimental conditions and the model assumptions. A crucial experimental assumption is that shaking indeed prevents the majority of synaptic transmission events, and if this was violated, we would underestimate the role of synaptic transmission. Similarly, modelling assumptions can influence results, and a number of alternatives are explored in the electronic supplementary material, showing that results remain robust. We conclude that in our system, the contributions of synaptic and free-virus transmission are comparable. Future work should verify this result using alternative experimental approaches and more complex cell cultures that mimic conditions which could be relevant *in vivo*.

## Acknowledgements

This work was supported by NIH grant no. 1R01AI093998-01.

- Received November 7, 2012.
- Accepted December 4, 2012.

- © 2012 The Author(s) Published by the Royal Society. All rights reserved.