Abstract
The duration of the eclipse phase, from cell infection to the production and release of the first virion progeny, immediately followed by the virusproduction phase, from the first to the last virion progeny, are important steps in a viral infection, by setting the pace of infection progression and modulating the response to antiviral therapy. Using a mathematical model (MM) and data for the infection of HSCF cells with SHIV in vitro, we reconfirm our earlier finding that the eclipse phase duration follows a fattailed distribution, lasting 19 h (18–20 h). Most importantly, for the first time, we show that the virusproducing phase duration, which lasts 11 h (9.8–12 h), follows a normallike distribution, and not an exponential distribution as is typically assumed. We explore the significance of this finding and its impact on analysis of plasma viral load decays in HIV patients under antiviral therapy. We find that incorrect assumptions about the eclipse and virusproducing phase distributions can lead to an overestimation of antiviral efficacy. Additionally, our predictions for the rate of plasma HIV decay under integrase inhibitor therapy offer an opportunity to confirm whether HIV production duration in vivo also follows a normal distribution, as demonstrated here for SHIV infections in vitro.
Introduction
In this study, we sought to determine the distributions that describe the amount of time simian CD4^{+} T cells (HSCF cells) spend in the eclipse and virusproducing phases, upon successful infection by the SHIVKS661 strain of the simianhuman immunodeficiency virus (SHIV). The eclipse phase is defined as the time which elapses between the entry of the first virion which successfully infects the cell and the release of the first virion produced by that newly infected cell. The virusproducing phase, which the cells enter immediately upon leaving the eclipse phase, is defined as the time which elapses between production of the first and last virions by an infected cell, often followed by apoptosis of that cell.
In previous work by our group^{1}, we determined that the eclipse phase in this system is ~24 h in length. But most importantly, we determined that the time spent in the eclipse phase by SHIVinfected HSCF cells was distributed according to a Gamma distribution with a shape parameter n_{E} ≈ 3.5, a finding we reconfirm herein using a different approach. This shape parameter, which we determine herein for both the eclipse and virusproducing phases, is important for a number of reasons. With respect to the eclipse phase, a shape parameter of n_{E} = 1 indicates that cells spend an exponentiallydistributed amount of time in the eclipse phase, whereas (e.g., n_{E} = 20) indicates they spend a normallydistributed time in that phase. An exponentially distributed eclipse length implies that some cells can theoretically produce virus immediately upon infection, which is not biologically possible^{2}. On the other hand, assuming that the eclipse length is normally distributed amounts to assuming that all steps or successive processes involved in this phase — from successful virion entry into cells up until successful budding and release of the first virions — are of somewhat similar durations, and that said durations are short relative to the duration of the entire eclipse phase. Under these conditions, the eclipse phase, which corresponds to the total duration of all these processes, will follow a normal distribution, as per the central limit theorem.
Interestingly, in our previous investigation^{1}, we found that, in fact, while this distribution for the eclipse phase is definitely not exponential (n_{E} ≠ 1), it is not normal either (nE 1). Instead, it lies in between (n_{E} = 4), i.e. a distribution that does enforce a delay before initiation of virus production, unlike the exponential distribution, but still keeps an exponentiallike distribution for longer waiting times, unlike the very symmetric normal distribution. In other words, it allows for some cells to take a much longer than average time to produce their first virion progeny, referred to as a fattailed distribution. We believe this is because one of the processes involved in early SHIV replication is significantly longer than others, and possibly accounts for most of the eclipse phase’s duration. We suspect this step to be the integration of the reversetranscribed SHIV DNA into the host cell’s genome. In contrast, we have shown that a virus like influenza A which replicates its segmented, negativestrand vRNA using its own polymerase, i.e. following a process devoid of this particular bottleneck, has an eclipse phase duration that follows a normal distribution^{2,3,4}.
While we reconfirm this results for the eclipse phase distribution herein, our main focus in the present work is on determining, for the first time, the shape parameter which describes the phase immediately following the eclipse phase, namely the virusproducing phase, i.e., the duration for which cells infected with SHIV will produce and release virion progeny before this process is shut down, possibly as cells undergo apoptosis. An exponentiallydistributed virusproducing phase (n_{I} = 1) would suggest that the process which leads to the termination of SHIV production in vitro is stochastic, completely independent of the time elapsed since cell infection. It would imply that cessation of virus production by SHIVinfected cells in vitro is not the result of accumulation of toxicity or exhaustion of intracellular host resources by virus replication, nor is it a virus production shutdown built into the virus replication cycle. It is biologically unlikely, and possibly unrealistic, that cessation of virus production not be the result of at least one of these processes. Yet, all mathematical models (MMs) of SHIV to date have invariably assumed the virusproducing phase, i.e. the duration of virus progeny production by SHIVinfected cells, to be exponentially distributed. Herein we show that, at least in SHIVinfected HSCF cellsin vitro, the duration of the virusproducing phase decisively does not follow an exponential distribution, but rather a normallike distribution. We also evaluate the impact of assuming an exponentially vs a normally distributed virusproducing phase on the analysis of viral load decays observed in HIV patients treated with various antiviral regimens.
Results
Determining the distribution of the eclipse and virusproducing phases
Our goal is to determine the shape of the distributions describing the amount of time SHIVinfected cells spend in the eclipse phase, i.e. from successful virus entry to the release of the first virion, and then in the virusproducing phase, i.e. the duration of SHIV production from the release of the first to the last virion progeny by an infected cell. We use an Erlang distribution — a special case of the Gamma distribution in which the shape parameter must be an integer — to describe the celltocell variability in the amount of time spent by SHIVKS661 infected cells in each of these two phases. By varying the value of the Erlang distribution’s shape parameter, one can shift the distribution from an exponential (=1) to a normal (1) distribution, as illustrated in Fig. 1. As such, determination of the Erlang shape parameters for the eclipse and virusproducing phases will enable us to identify the shape of the distributions describing the time spent by SHIVinfected cells in these phases.
In previous work^{1}, we used in vitro data from the infection of HSCF cells infected with SHIVKS661 at a concentration of 4.2 TCID_{50}/cell to identify this shape parameter. Specifically, given the virus concentration in the inoculum, we assumed that all cells were simultaneously infected, and we used the cumulative fraction of cells that have entered the virusproducing phase — i.e., all cells that were either positive for the presence of the SHIV Nef protein or were no longer viable (have presumably died as a result of infection) — to identify this shape parameter for the duration of the eclipse phase. This nondynamical approach is very attractive because the shape of the eclipse phase length distribution can be directly observed from this data alone, with the mathematical analysis providing a quantitative confirmation of what can already be seen.
Herein, we use a more indirect, dynamical approach by explicitly representing the kinetics of SHIV infection with MM (1), presented in the Methods section. This different approach enables us to relax the assumption that all cells were simultaneously infected by the initial inoculum, and allows us to also determine the distribution of the virusproducing phase duration, which until now has been assumed to be exponentially distributed. The experimental data used in the present analysis includes that used in our previous work^{1}, as well as additional data collected as part of the previous experiment, but unused until now. Briefly, the experiment consisted in the infection of HSCF cells with an inoculum containing SHIVKS661 at concentrations of 4.2, 2.1, 1.1, 0.53, or 0.26 TCID_{50}/cell. The total virus concentration (vRNA/mL), the fraction of viable HSCF cells, and the fraction of virusproducing (i.e., SHIV Nefpositive cells) were determined at regular intervals over the course of the infection.
The complete experimental data set is presented in Fig. 2, alongside simulated infection time courses from MM (1). The two solid lines in Fig. 2 correspond to the two bestfits of MM (1) to the data when we assume that the duration of the virusproducing phase is distributed either exponentially (n_{I} = 1), as is typically assumed in all existing MMs for (S)HIV, or that it is normally distributed (n_{I} = 20, chosen based on a more thorough bestfit analysis described below). While both distributions visually seem to provide reasonable fit to the data, the fit which assumes that SHIVinfected cells produce virus over a normally distributed amount of time is better (lower sumofsquared residuals or SSR of 115, computed as per Eqn. (2)) than that assuming it is exponentially distributed (SSR of 135).
To investigate this further, we used a Markov chain Monte Carlo (MCMC) approach to determine the posterior likelihood distribution (PLD) for each of the MM parameters. In particular, we sought to determine the likelihood that the virusproducing phase is exponentially (n_{I} = 1) versus normally () distributed, given the set of observed experimental data. These results are presented graphically in Fig. 3, and quantitatively in Table 1. We find that all parameters are robustly extracted (see Supplementary Fig. S1 for 2parameter PLDs), with little to no correlations resulting in narrow PLDs. The MCMC analysis determined that newly infected HSCF cells spent 19 h (95% credible region or CR of τ_{E} = 18–20 h) in the eclipse phase before they begin producing SHIVKS661 progeny. Once underway, virus production proceeds at a rate of 5,000 (95% CR: p_{RNA} = 4,000–6,300) SHIV virion per cell per day which infected cells maintain for 11 h (95% CR: τ_{I} = 9.8–12 h) before presumably undergoing apoptosis. This results in a total burst size of 2,500 (95% CR: p_{RNA} · τ_{I} = 1,600–3,200) SHIVKS661 virion produced by each SHIVinfected HSCF cell over its virusproducing lifespan. Despite this high virus yield, infection progresses slowly with 18 h (95% CR: t_{inf} = 14–25 h) elapsed between the start of virion progeny production and infection of the first cell by that progeny, resulting in a modest basic reproductive number of 1.7 (95% CR: R_{0} = 0.91–2.9). We also confirm our earlier finding that the duration of the eclipse phase follows a fattailed distribution that is somewhere between exponential and normal, with a shape parameter value of 4 (95% CR: n_{E} = [3–5]), in agreement with the value of 3.5 obtained previously using a different approach^{1}.
Of greatest interest, however, is the PLD of the parameter characterizing the shape of the distribution for the duration of the virusproduction phase, n_{I}. Because values of are statistically equivalently likely, we constrained our MCMC process to values of n_{I} ∈ [1, 100]. As such, the PLD for n_{I} is not a true posterior likelihood density. Nonetheless, this enables us to identify that the mode of the PLD is 12, with a 95% CR of [6, 97]. Furthermore, out of the >7,000,000 MCMCaccepted parameters, corresponding to roughly 100,000 independent parameter estimates (given our autocorrelation length of ~70), not a single one had n_{I} = 1. Therefore, based on Feldman and Cousins^{5}, we can state with 95% confidence that the likelihood of an exponentiallydistributed virusproducing phase (n_{I} = 1) is no more than 3 in 100,000 or <10^{−4}.
In Fig. 3(k), we supplement the MCMC analysis by performing a nonlinear regression to the data presented in Fig. 2, while holding n_{I} fixed to values ∈ [1, 100]. Leaving parameter n_{I} as a free parameter to be fitted was not appropriate given that values of are all equivalently likely (causing the fitter to diverge) and the fact that n_{I} can only take on integer values (which causes the fitter to misbehave). We find that fits for n_{I} ∈ [9, 100] have a bestfit likelihood (BFL) >85% that of the very best BFL, obtained here for n_{I} = 20. In particular, we find the BFL increases monotonically from n_{I} = 1 to n_{I} = 20, with the BFL for n_{I} = 1 being <10^{−4} that for n_{I} = 20. In other words, both the PLD and BFL for parameter n_{I} unambiguously, statistically significantly exclude the possibility that SHIVinfected cells could produce virus progeny for an exponentially distributed (n_{I} = 1) amount of time. Instead, the duration of the virusproducing phase is consistent with a mildly fattailed or more normallike distribution.
Impact of the eclipse and virusproducing phase distributions on predicted viral load decays under antiviral therapy
The shape of the distribution for the time spent by cells in the eclipse and virusproducing phases can appear to be a theoretical concern of little relevance to our understanding of, and therapeutic approach to, HIV. However, the increasing prevalence of MM analyses, and the ability they confer to determine antiviral efficacy based solely on early viral load decays observed in treated patients, has made them an essential part of the toolkit used in developing, evaluating, and optimizing novel HIV therapies. All MMs must invariably make assumptions and simplifications which, occasionally, can lead to misinterpretations or inaccurate quantification. Here, we tackle one such simplification in the context of our findings regarding the shape parameters of the eclipse and virusproducing phase distributions.
To date, almost all MMs applied to the analysis of HIV infection kinetics have assumed that the durations of the eclipse phase and of virus production by HIVinfected cell are exponentially distributed. In other words, these MMs assume a cell that has just been infected can immediately begin producing virus, and that a cell which has just begun to produce and release HIV progeny is as likely to cease production and undergo apoptosis as one that has been producing virus for hours or even days. In the previous section, we established in vitro that the duration of the SHIV eclipse phase is fattailed distributed, and there is little reason to believe that this would not be the case in vivo. Though we have also established the duration of the virusproducing phase is normally distributed in vitro, this might not be the case in vivo where host factors and immune responses could abrogate or otherwise significantly affect the actual duration of virus production by HIVinfected cells. Therefore, it is important to understand how the decays predicted using the traditional MM (Exp,Exp), differ from those of the more biologically correct MMs with a fattailed distributed eclipse phase and either an exponential (Fat,Exp) or normally (Fat,Norm) distributed eclipse phase. Herein, we explore the impact of this finding on interpretations of antiviral efficacy based on observed patterns of early plasma viral load decay upon therapy initiation in HIV patients.
In Fig. 4, we compare the rate of decay of plasma HIV viral loads decays under antiviral therapy with an integrase inhibitors (INIs) such as raltegravir, as predicted by MMs. We explore how the MMpredicted HIV decay rates are impacted by four key infection kinetic parameters: the in vivo HIV clearance rate, antiviral efficacy, and the durations of the eclipse and virusproducing phases. We investigate how that impact in turn depends on the assumptions made regarding the distributions of the eclipse and virusproducing phases (Exp,Exp or Fat,Exp or Fat,Norm). The last row of Fig. 4 shows how, for given parameter values, the MMpredicted decay rates under INI therapy differ between these three assumptions. For example, the predicted total vRNA viral load under INI therapy differs by one to two orders of magnitude (10 to 100fold) between the MMs with an Exp vs Normal virusproducing lifespan. We expect such a difference would be experimentally measurable.
The traditional (Exp,Exp) MM and the two variants considered here all predict a biphasic decay under monotherapy with a INI for most parameter values, as is typically observed in INItreated HIV patients, without the need to resort to assuming one shortlived and one longlived infected cell population^{6}. For a given eclipse phase duration, the traditional MM predicts a lesser rate of viral load decay under therapy. As such, the traditional MM would likely underestimate the true eclipse phase duration. More generally, we find that having observed a given decay in a patient, use of the traditional MM (Exp,Exp) to analyse said decay will overestimate the true antiviral efficacy and/or the HIV clearance rate, and/or it will underestimate the true duration of the eclipse or virusproducing phase. The over/underestimation is even greater if one further assumes that the virusproducing duration is normally (Fat,Norm) rather than exponentially (Fat,Exp) distributed. For example, the decay predicted under INI therapy at 90% efficacy by the (Fat,Norm) MM (yellow line) is predicted at an antiviral efficacy of ~99% by the traditional (Exp,Exp) MM (blue line). The 10fold difference (10% vs 1% antiviral escape) in the estimates between these two different MMs would likely be statistically significant and could impact other MM predictions such as time to antiviral resistance emergence.
The MM which assumes a normally distributed duration of virus production (Fat,Norm) predicts a very sharp decay at a rate proportional to the viral clearance rate, which stops in a shoulder whose depth depends on the antiviral efficacy, followed by a second phase of decay whose rate depends on the duration of the eclipse phase. This shape is observed provided the viral clearance rate (c_{body}) is greater than ~5 h^{−1} and the duration of virus production is less than ~11 h. While it is now believed the viral clearance rate is greater than 5 h^{−1}, and likely as high as 23 h^{−1}^{7}, it is not clear whether or not the duration of virus production by HIVinfected cell in vivo is indeed less than 11 h. If it is not, the MM (Fat,Norm) predicts the duration of virus production would affect both the first and second phase of viral load decay, but not the depth of the shoulder.
Discussion
Herein we have made use of a mathematical model (MM) combined with a Markov chain Monte Carlo approach to determine the shape of the distribution describing the time spent by SHIVinfected HSCF cells in the eclipse and virusproducing phases. In previous work^{1}, we determined that a Gamma shape parameter of 3.5 best describes the distribution of time spent by cells in the eclipse phase, i.e. the time elapsed between cell infection and the release of the first SHIV progeny by that cell. Herein, we reconfirm this finding (we found a value of 4 with a 95% credible region (CR) of [3–5]) using a more extensive data set and a different approach.
However, the primary aim of the present work was to extract the shape parameter for the phase which follows the eclipse phase, namely the virusproducing phase, i.e. the time elapsed between production of the first and last virus progeny by a productively SHIVinfected cell. MMs for (S)HIV infection kinetics have, up until this point, almost always assumed that the virusproducing phase is exponentially distributed. Herein, our experimental data, MM, and analysis, enable us to statistically exclude (probability < 10^{−4}) the hypothesis that the duration of virus production by SHIVinfected HSCF cells follows an exponential distribution. Instead, we find that the Gamma shape parameter for the distribution of time SHIVcells spend in the virusproducing phase is ~12, consistent with a mildly fattailed or normallike distribution (n_{I} = [6, 97], 95% CR). To our knowledge, this is the first time the virusproducing lifespan of SHIVinfected cells in vitro has ever been shown to follow a normallike distribution, rather than an exponential distribution as has been assumed in all MMs for (S)HIV until now. This finding is consistent with earlier work by Petravic et al.^{8} which also suggests that the eclipse phase duration and virusproducing lifespan of HIVinfected cells in vitro are inconsistent with an exponential distribution. Due to the nature of the data used in their analysis, Petravic et al. assumed, rather than identified, that the duration of both the eclipse and virusproduction phases followed a fattailed, lognormallike distribution. Such a distribution is consistent with that identified here for the duration of the eclipse phase, but differs from the more normallike distribution favoured herein by our MCMC analysis for the virusproduction phase. However, since Petravic et al. assumed rather than determined the distribution, it is likely that a normally distributed virusproduction phase would also successfully reproduce their data.
Use of a normally rather than an exponentially distributed virusproducing cell lifespan can also impact estimates of the key SHIV kinetic parameters from MM analysis. Our results establish that SHIVinfected cells begin progeny virus production and release approximately 19 h (τ_{E} ∈ [18, 20] h) postinfection, and cease approximately 11 h (τ_{I} ∈ [9.8, 12] h) after they have begun, or 30 h (τ_{E} + τ_{I} ∈ [28, 31] h) after the successful entry of the virion which infected them. This appears to be consistent with the distributions parametrised by Petravic et al.^{8} for the duration of the eclipse and virusproduction phases of HIVinfected cells in vitro whose mode we visually estimate to be about 28 h (τ_{E}) and 12 h (τ_{I}), respectively. In contrast, our estimates for the duration of the virusproducing cell lifespan (τ_{I}) obtained herein is 1.7–2.1× longer than that obtained in previous work^{9} wherein an exponentiallydistributed virusproducing cell lifespan was assumed. Our estimate of the basic reproductive number, namely the number of secondary cells infected by the virus progeny produced by a single SHIVinfected cells over its virusproducing lifespan in vitro, was 1.7 (R_{0} ∈ [0.91, 2.9]), lower than previously estimated in vivo ranges of 4.3–13 assuming no eclipse phase or 5.4–54 assuming an exponential eclipse phase, both estimated in ref. 10, or 4–11 assuming no eclipse phase in ref. 11 for HIV1 infections in human patients, and 2.2–4.6 assuming no eclipse phase in macaques infected with simian immunodeficiency virus (SIV)^{12}.
MMs are widely used to analyse and interpret the decay of plasma HIV load in patients treated with antivirals. Hence, we also investigated the impact of assuming the eclipse and virusproducing phases are exponentially distributed, as is typically done, in analysing and interpreting said decays. Assuming the eclipse phase duration is exponentiallydistributed, which is inconsistent with our findings herein, can cause an overestimation of the antiviral efficacy or an underestimation of the duration of the eclipse or virusproducing phases. The over/underestimation is even more significant if the true duration of virusproduction is normally, rather than exponentially distributed. Having found a 10fold difference in the antiviral efficacy estimated by MMs with an exponential vs normally distributed virusproducing cell lifespan, we expect these parameter over/underestimation from viral decay data observed under INI therapy would be statistically significant. In turn, these statistically significant differences in parameter value estimates could potentially have an impact on therapeutic decisions. Actual analysis of viral load decays under therapy using these different MMs is needed to confirm this possibility, and to identify which parameter(s) would be most significantly affected. Previous works analysing plasma viral load decays under antiviral therapy in vivo, such as that by Althaus et al. for HIV^{13} and by Rong et al. for hepatitis C virus^{14}, suggest that such MM differences can lead to statistically significantly different parameter estimates.
We furthermore find that under antiviral therapy with an integrase inhibitor (INI), the shape of the MMpredicted decay differs drastically between the assumption of a normally vs exponentially distributed duration of virus production. The true decays observed in HIV patients under INI therapy^{6} thus far can be reproduced under either assumptions (not shown here), albeit for different parameter values. As such, it is not currently possible to use this distinction to confirm the true duration of virus production by HIVinfected cells in vivo. However, under the assumption of a normallydistributed virusproduction phase, and when that phase is less than ~11 h in duration, the MMpredicted viral load decay is very sharp, occurring over a few hours, with a drop proportional to the efficacy of the INI therapy. This pattern cannot be reproduced under the assumption of an exponential duration of virusproduction. Observing such a pattern under INI therapy would require more frequent sampling early after therapy initiation, and would provide a strong confirmation that the duration of virus production by HIVinfected cells in vivo follow a normallike distribution.
A normally distributed duration of virus production indicates that the time of cessation of virus production by SHIVinfected cells in vitro occurs some relatively fixed amount of time after the start of virus production and release. This implies that the cause of this cessation is likely a consequence of that production and release process, such as the accumulation of damage (e.g., cytotoxicity, host cell resource exhaustion) or signalling (e.g., downregulation due to virus replicationtriggered cytokines). Recently, measuring intracellular HIV1 replication kinetics alongside gene expression, with high temporal resolution, has become possible^{15}. Such methodology would enable investigation of the detailed apoptotic gene expression profile during HIV1 infection. This, in turn, would permit accurate quantification of the duration of virus progeny production and the mechanisms behind its cessation.
Our results are based on analysis of in vitro infection of HSCF cells with SHIVKS661. However, in the context of an in vivo HIV infection, the host immune response, absent in our in vitro analysis, might alter the distribution and duration of the virusproduction phase. For example, killing of virusproducing cells by the host immune response might lead to an exponentially distributed virusproducing phase, but only if HIVinfected cells which have just begun virus production are killed at the same rates as those which have been producing HIV progeny for a while. It would furthermore require that a majority of HIVproducing cells cease virus production because of that host immune response so that cells exhibiting an exponentially distributed infectious lifespan dominates over those exhibiting their ‘natural’, normallike duration of virus production. Viral load decay rates — believed to be proportional to the duration of the virus production phase — were reported to be the same in the presence and absence of CD8^{+} T cells in macaques chronically infected with SIV under combination antiretroviral therapy^{16,17,18}. This suggests that at least the killing of SIVinfected cells by CD8^{+} T cell in vivo would not significantly affect virusproduction duration. Additionally, while the rate of hostmediated killing of HIVinfected cells is high in the acute phase of an HIV infection, this rate is likely much lower in the chronic phase of the disease, when antiviral therapy is administered and studied. Using a mathematical modelling analysis, Ganusov et al.^{19} found a large decrease in the rate of viral escape from cytotoxic T lymphocyte (CTL) killing between the acute and chronic phases of HIV1 infections and suggest this would be consistent with a decrease in the magnitude of epitopespecific CTL responses. For all these reasons, it is not inconceivable that the duration of virus production by HIVinfected T cells in vivo also follows a normal distribution, like that characterized herein for the SHIVinfection of HSCF cells in vitro.
Methods
Quantification of viable and infected cells
These experimental procedures were outlined in ref. 1, but are repeated here for completeness. Virus infection of the HSCF cells was measured by FACS analysis using markers for surface CD4 and intracellular SIV Nef antigen expression. The number of total and viable cells were first determined using an automated blood cell counter (F820; Sysmex, Kobe, Japan). Viable HSCF cells (gated by forward and sidescatter results) were examined by flow cytometry to measure the surface CD4 and intracellular SIV Nef antigen expression. Cells were permeabilized with detergentcontaining buffer (Permeabilizing Solution 2, BD Biosciences, San Jose, CA). The permeabilized cells were stained with phycoerythrin conjugated antihuman CD4 monoclonal antibody (Clone NuTH/I; Nichirei, Tokyo, Japan) and antiSIV Nef monoclonal antibody (04001, Santa Cruz Biotechnology, Santa Cruz, CA) labelled by Zenon Alexa Fluor 488 (Invitrogen, Carlsbad, CA), and analyzed on FACSCalibur (BD Biosciences, San Jose, CA).
Quantification of viral load
These experimental procedures were outlined in ref. 1, but are repeated here for completeness. The total viral load was measured via realtime PCR quantification, as described previously^{9,20}. Briefly, total RNA was isolated from the 100 fold diluted culture supernatants (140 μL) of virusinfected HSCF cells with a QIAamp Viral RNA Mini kit (QIAGEN, Hilden, Germany). RT reactions and PCR were performed by a QuantiTect probe RTPCR Kit (QIAGEN, Hilden, Germany) using the following primers for the gag region; SIV2696F (5′GGA AAT TAC CCA GTA CAA CAA ATAGG3′) and SIV2784R (5′TCT ATC AAT TTT ACC CAGGCA TTT A3′). A labelled probe, SIV2731T (5′FamTGTCCA CCT GCC ATT AAG CCC GTamra3′), was used for detection of the PCR products. These reactions were performed with a Prism 7500 Sequence Detector (Applied Biosystems, Foster City, CA) and analyzed using the manufacturer’s software. For each run, a standard curve was generated from dilutions whose copy numbers were known, and the RNA in the culture supernatant samples was quantified based on the standard curve.
Mathematical model (MM)
In order to compare distributions ranging from exponential, to normallike, to Dirac deltalike for the duration spent by cells in the eclipse phase (E) and in the subsequent virusproducing infectious phase (I), we follow previous work^{1,2,3} and adopt a set of ordinary differential equations which make use of an Erlang distribution to represent these phases, namely
Herein, susceptible, target HSCF cells are infected at rate β by infectious virus, , and upon successful infection, they enter the eclipse phase, wherein they are SHIVinfected, but are not yet producing infectious virus. After an average time τ_{E} has elapsed since becoming infected, cells in the eclipse phase become infectious, , wherein they are assumed to produce infection () and total (V_{RNA}) virus at a constant rates and p_{RNA}, respectively. Ultimately, virusproducing infectious cells cease virus production, possibly undergoing apoptosis, after spending an average time τ_{I} in the virusproducing state. Total virus, V_{RNA}, quantified via realtime quantitative PCR in units of vRNA/mL, was measured experimentally over the course of the infection. Infectious virus, , typically quantified via tissue culture infectious dose in units of TCID_{50}/mL was not measured experimentally. The rate of cell infection is dependent on the concentration of infectious virus, not total virus. The ratio of infectious to total virus is not constant over the course of an in vitro SHIV infection due to the differing rates at which infectious virus loses infectivity compared to the rate at which total virus degrades or breaks down, i.e., for and c_{RNA} for V_{RNA} where . Therefore, in MM (1), both the total and infectious virus population is accounted for: the former for comparison to experimental measurements, the latter to appropriately capture infection kinetics.
The time spent by newly infected cells in each of the eclipse and virusproducing phases is represented in MM (1) by the Erlang distribution, a special case of the Gamma distribution in which the shape parameter (here n_{E} and n_{I}) can only take on integer values. In Eqn. (1), the eclipse (or infectious) phase is separated into n_{E} (or n_{I}) separate compartments, each lasting an exponentiallydistributed time of equal average length τ_{E}/n_{E} (or τ_{I}/n_{I}). As such, cells will spend on average τ_{E} (or τ_{I}) time in the eclipse (or infectious) state prior to transitioning to the infectious (or dead) state.
We also define the following, additional, derived parameters which are computed from the base MM parameters introduced above. The infecting time, which corresponds to the time elapsed between the production of the first virion by an infected cell and the infection of the first cell by that progeny^{2}, is computed as
where N = 10^{6} is the number of cells in the culture. The basic reproductive number, which corresponds to the number of secondary infections resulting from a single infected cell over its virusproducing lifespan in a fully susceptible cell population is computed as
Experimental and mathematically simulated in vitro SHIV infections
Experimentally, HSCF cells were incubated with SHIVKS661 and centrifuged at 4,000 rpm for 1 h at 25 °C, then rinsed three times as described in ref. 1. In order to computationally reproduce the experimental infection of HSCF cells with SHIVKS661, we used MM (1) with the following initial conditions. We consider the rinse time to correspond to t = 0 such that the infection, and therefore the mathematical simulation, begins at time t = −1 h, followed by a rinse at time t = 0, after which the infection is allowed to proceed without further intervention. This is because each sample (time point measurements of virus or cell viability or cell infection status) are taken from separate wells (independent replicates) such that each time point corresponds to an infection left undisturbed since rinsing, up until sampling. Thus, at time t = −1 h, we assume that no cell is initially infected (E_{i} = I_{j} = D = 0). We further assume that not all HSCF cells are susceptible to SHIV infection (T(−1 h) = 0.9 × 10^{6}), because ~10% of cells appear to have remained uninfected by the end of the experiment (see Fig. 2). We found that this assumption (90% rather than 100% cell susceptibility to infection) was better supported (significantly better SSR) by the data.
The experimental infections were preformed at five different virus concentrations, namely at an undiluted infectious SHIVKS661 concentration of 4.2 TCID_{50}/cell (Base MOI), and at serial 2fold dilutions down to 2.1, 1.1, 0.53, and 0.26 TCID_{50}/cell in four additional experiments. Mathematically, this is represented by setting the initial infectious virus concentration to , where dil = 0, 1, 2, 3, 4 is the dilution factor of the initial inoculum which is used undiluted (dil = 0), or diluted by 2fold in four additional experiments. As such, our Base MOI should correspond to an initial virus inoculum of 4.2 × 10^{6} TCID_{50}/mL. However, since there is poor correspondence between virus infectiousness across separate experiments^{3}, i.e. TCID_{50} is a relative rather than an absolute measure of virus infectiousness^{21,22}, the true Base MOI was left as a free parameter to be determined. The initial total virus concentration was set to based on the ratio of 6,000 vRNA per TCID_{50} in our SHIVKS661 samples, specifically, the ratio of the initial (t = 0) data points in Fig. 2 of Iwami et al.^{9}.
After the 1 h incubation period, the cells are rinsed three times to remove excess virus. Based on the experimental measurements presented in Fig. 2, it would appear that at high virus inocula (MOI = 4.2, and 2.1 TCID_{50}/cell), the residual virus postrinse was not proportional to the inoculum, but rather seemed to reduce to the residual virus down to ~1.4 × 10^{9} vRNA/mL, whereas at the lower virus inocula, the rinse did seem to result in residual virus proportional to the inocula. Therefore, in our MM, we simulate the rinse by multiplying both and V_{RNA}(0) at time t = 0 by which is proportional to the true rinse. Since only total virus is measured (V_{RNA}), yet has no impact on our simulated infection kinetics which instead depends on infectious virus , application of a proportional rather than an absolute rinsing factor merely means that the units of the rates of infectious virus production and infectivity β will be relatively, rather than absolutely accurate, which is always the case^{21}.
In addition to its initial conditions, MM (1) contains a total of 9 parameters: β, , p_{RNA}, , c_{RNA}, τ_{E}, τ_{I}, n_{E}, and n_{I}. From our previous work reported in Iwami et al.^{9}, we have determined the rate of loss of infectious and total SHIVKS661 virions to be and , respectively, and the rate of infectious virus production to be , which we use since infectious virus is not explicitly measured in the experiments analyzed herein. This leaves a total of 6 parameters (β, p_{RNA}, τ_{E}, τ_{I}, n_{E}, n_{I}) and one initial condition (Base MOI) to be estimated by our analysis. In performing the nonlinear leastsquare fits (Figs 2 and 3), we fixed n_{E} and n_{I} to various integer values ∈ [1, 100], while allowing the remaining 5 quantities to vary. The bestfit parameters obtained for the exponential (n_{I} = 1) and best fit for a normallike distribution (n_{I} = 20) are presented in Table 2. In constructing the Markov chain Monte Carlo (Fig. 3), we allowed all 7 quantities to vary. The parameters obtained through the MCMC process are presented in Table 1.
In matching the experimental data to variables of our MMs, we compared the experimentally quantified total virus concentration via realtime quantitative PCR (vRNA/mL) to MM variable V_{RNA}, compared the fraction of virusproducing cells tagged via antiSIV Nef monoclonal antibodies and quantified via FACS to the sum of all MM variables corresponding to the fraction of cells in the infectious, virusproducing phase , and compared the fraction of nonviable cells quantified using FACS gated by forward and sidescatter to MM variable D corresponding to the fraction of dead cells. In our nonlinear leastsquare regressions to identify the bestfit, 6parameter set for two different values of n_{I}, we aimed to maximize the goodnessoffit by minimizing the sumofsquared residuals (SSR) between the experimental data, weighted for each of V_{RNA}, I, and D as follows:
where , σ_{I} = 0.047, and σ_{D} = 0.080 correspond to the standard deviations of the pooled residuals for each of , and , respectively, for 100 best fits performed for n_{I} ∈ [1, 100]. I_{ii} and D_{di} in the above correspond to fraction rather than number of cells. The SSR calculation directly uses the fraction of cells, and no transformation (e.g. logit function) is applied in calculating the SSR for the fractions of infectious or dead cells. This is because we found that the distribution of the bestfit residuals of the cell fraction data resembles a Gaussian while that for the residuals of the logittransformed cell fraction data does not (not shown).
The nonlinear regression was performed using successive applications of python’s scipy implementations of the LevenbergMarquardt (scipy.optimize.leastsq) and NelderMead downhill simplex (scipy.optimize.fmin) methods. Additionally, we used emcee, a python module implementation of the Markov chain Monte Carlo (MCMC) method^{23}, to identify the posterior distributions for all seven MM parameters from a total of >7,000,000 MCMCaccepted parameter sets. In constructing the MCMC chains, exp(−SSR/2) was used as the likelihood function for each set of parameters (dimensions), where the SSR is computed as per Eqn. (2). The ‘Bestfit likelihood’ (BFL) graph which appears in Fig. 3(k) was computed as exp(−SSR/2) of the minimum SSR obtained via a bestfit for each individual value of n_{I} ∈ [1, 100], divided by the likelihood of the very best fit (n_{I} = 1) so that the latter’s likelihood is one.
Simulation of in vivo antiviral therapies with various modes of action
To simulate therapy with an integrase inhibitor (ε_{IN}), we modified MM (1) as follows
where is the steadystate level of CD4^{+} T cells in patients chronically infected with HIV, i.e. we set and simulate only the early viral decay post therapy initiation. The new parameter is the rate of clearance of HIV from the plasma of chronically infected HIV patients, as previously estimated^{7}. Because , the rates of loss of infectivity and viral degradation can be safely neglected.
The steadystate of MM (3) prior to therapy initiation is given by
where we set τ_{E} = 19 h, τ_{I} = 11 h, n_{E} = 1 or 4 (exponential or fattailed eclipse phase), and n_{I} = 1 or 90 (exponentially vs normally distributed virusproducing lifespan), based on Table 1. Additionally, we set and based on refs 24,25. We use since the value of this variable is not important to the MMpredicted kinetics and is equivalent to rescaling the viral load to its starting, steady state value. As in MM (1), we set . Following the steadystate equations above, we have
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How to cite this article: Beauchemin, C. A. A. et al. Duration of SHIV production by infected cells is not exponentially distributed: Implications for estimates of infection parameters and antiviral efficacy. Sci. Rep. 7, 42765; doi: 10.1038/srep42765 (2017).
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Acknowledgements
This research was supported in part by the JST RISTEX program (to S.I.); the Japan Agency for Medical Research and Development, AMED (H27ShinkoJitsuyokaGeneral016) (to S.I.); Mitsui Life Social Welfare Foundation (to. S.I.); JST CREST program (to S.I.); JST PRESTO program (to S.I.); JSPS KAKENHI Grant Numbers 15KT0107, 16H04845, 16K13777, and 26287025 (to S.I.), with additional support through the iTHES research programme at RIKEN (to C.A.A.B.), Discovery Grant 3558372013 from the Natural Sciences and Engineering Research Council of Canada (to C.A.A.B.), and Early Researcher Award ER1309040 (to C.A.A.B.) from the Ministry of Research and Innovation of the Government of Ontario.
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T.M. and S.I. conceived and designed the experiments. T.M. performed the experiments. C.A.A.B. and S.I. analyzed the data. T.M. contributed reagents/materials/analysis tools. C.A.A.B., T.M., and S.I. wrote the paper. C.A.A.B. and S.I. developed the modelling framework.
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Beauchemin, C., Miura, T. & Iwami, S. Duration of SHIV production by infected cells is not exponentially distributed: Implications for estimates of infection parameters and antiviral efficacy. Sci Rep 7, 42765 (2017). https://doi.org/10.1038/srep42765
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DOI: https://doi.org/10.1038/srep42765
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