Age-structured multiscale modeling of HCV infection

To describe the intracellular replication dynamics of HCV viral RNA, we used the following mathematical model: (1) where R(a) is the amount of intracellular viral RNA in cells that have been infected for time a. The intracellular viral RNA replicates at rate k per day, degrades at rate μ per day, and is released to extracellular space at rate ρ per day (Fig 1B). Note that if viruses have small or large ρ, then they tend to stay inside or leave the cell, respectively (see later). In our virus experiments (see Methods), the released viruses could infect other target cells. To describe multiround virus transmission (i.e., de novo infection), we needed to couple intracellular viral replication with a standard mathematical model for intercellular virus infection in cell culture [8,9] (Fig 1C). In S1 Text, we derived the following multiscale ordinary differential equation (ODE) model for HCV infection from the corresponding age-structured partial differential equation (PDE) model [10,11]: (2) (3) (4) (5) (6) Here, the intercellular variables T(t) and I(t) represent the numbers of uninfected and infected target cells (cells/well), respectively, and V(t) and V_θ(t) represent the total amount of extracellular viral RNA (copies/well) and the extracellular viral infectious titer expressed (focus formation unit [ffu]/well), respectively. The intracellular variable A(t) represents the total amount of intracellular viral RNA. The parameters g and K represent the growth rate (per day) and the carrying capacity of target cells, respectively, and β_θ and f_θ are the converted infection rate constant ([ffu/well・day]^-1) and the fraction of infectious virus (RNA copies・ffu^-1), respectively. Note that we assume no cellular loss induced by infection because HCV is a noncytopathic virus [4]. We assumed that progeny viruses were cleared at rate c per day via degradation at rate c_RNA per day and washing at rate c_w per day (i.e., c = c_RNA+c_w), and that infectious virions lose infectivity at rate r. Because of the different RNA sequence and the resulting different RNA modification status as well as the different physical property of virions, specific stability/degradation rate of each virus strain is expected. Separately, we directly estimated g, K, c, μ, and r for both HCV JFH-1 and Jc1-n in S1 Fig. Detailed explanations of Eqs 2–6 are given in S1 Text.

To assess the variability of kinetic parameters and model predictions, we performed Bayesian estimation for the whole dataset using Markov chain Monte Carlo (MCMC) sampling (see Methods). Simultaneously, we fitted Eqs 2–6 to the experimentally determined numbers of uninfected and infected cells (cells/well), extracellular viral RNA (copies/well) and infectious titer (ffu/well), and intracellular viral RNA (copies/well). These figures were derived from infection experiments using different numbers of plated cells for either HCV JFH-1 or Jc1-n as described previously [8,9,12,13]. The remaining free model parameters (i.e., β_θ, k, ρ, f_θ) along with initial values for variables (i.e., T(0), I(0), A(0), V_θ(t), V(0)) were estimated from the MCMC sampling. Experimental measurements below the detection limit were excluded in the fitting. The estimated parameters and initial values are listed in Table 1 and S1 Table. The typical behavior of the model using these best-fit parameter estimates is shown together with the data in Fig 2 for HCV JFH-1 (orange) and Jc1-n (green) (see Methods for HCV strains) and indicated that Eqs 2–6 described the in vitro data very well. The shadowed regions corresponded to 95% posterior predictive intervals, the solid and dashed lines gave the best-fit solution (mean) for Eqs 2–6, and the orange circles and green triangles showed the experimental datasets.

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Fig 2. Dynamics of HCV JFH-1 and Jc1-n infection in cell culture.

Fitting of the mathematical model to the experimental data of HCV JFH-1 and Jc1-n infection in cell culture. Three different numbers of Huh-7 cells infected with either HCV JFH-1 or Jc1-n 1 day after inoculation were seeded (experiment A: 1,000, experiment B: 2,000, and experiment C: 4,000 cells per well) and chased to detect the following values at days 0, 1, 2, 3, and 4 post seeding (log₁₀ scale): numbers of uninfected and infected cells, amount of intracellular and extracellular viral RNA (copies/well), and extracellular viral infectivity (ffu/well) (orange circle: JFH-1, green triangle: Jc1-n). The shadowed regions correspond to 95% posterior intervals and the solid curves give the best-fit solution (mean) for Eqs 2–6 to the time-course dataset. All data for each strain were fitted simultaneously. The underlying data for this figure can be found in S1 Data. ffu, focus formation unit.

https://doi.org/10.1371/journal.pbio.3000562.g002

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Table 1. Parameter values estimated from the cell culture infection experiment.

https://doi.org/10.1371/journal.pbio.3000562.t001

Using the estimated parameters shared between the original PDE model in S1 Text and the transformed ODE model (i.e., Eqs 2–6), we successfully reconstructed age information for intracellular viral RNA in infected cells of infection age a, which cannot be obtained through conventional experiments alone (S2A Fig). S2B Fig shows the differences in intracellular JFH-1 and Jc1-n viral RNA levels in cells of infection age a. At the beginning of the experiment, intracellular viral RNA increased faster under Jc1-n infection than under JFH-1 infection (shown in green). However, intracellular JFH-1 viral RNA gradually accumulated to higher levels than Jc1-n at later time points after infection (shown in yellow to brown). These data illustrated the different dynamics of these two strains and the impact of these dynamics on intracellular viral RNA production, all resulting from different strategies to transmit the viral genome (see below).

Dynamics of HCV JFH-1 and Jc1-n strain replication

Our model (Eqs 2–6) applied to time-course experimental data allowed us to extract the following kinetic parameters: the distribution of the rate constant for infection, β_θ; the release rate of intracellular viral RNA, ρ; the converted fraction of infectious viral RNA, f_θ; and the replication rate of intracellular viral RNA, k (Fig 3 and Table 1). Comparing these parameters for JFH-1 and Jc1-n showed a significant difference between the rate constant for infections, β_θ, of JFH-1 (1.26×10⁻⁴ [ffu/well・day]^-1, 95% CI 0.788–1.89×10⁻⁴ [ffu/well・day]^-1) and Jc1-n (2.29×10⁻⁴ [ffu/well・day]^-1, 95% CI 1.32–3.64×10⁻⁴ [ffu/well・day]^-1) (p = 1.48×10⁻⁴ by repeated bootstrap t test) (Fig 3A). In addition, the release rates of intracellular viral RNA, ρ, for JFH-1 and Jc1-n were 2.36×10⁻² per day (95% CI 1.73–3.15×10⁻²) and 6.38×10⁻² per day (95% CI 4.20–9.34×10⁻²), respectively (p<0.01 by repeated bootstrap t test) (Fig 3B). These estimates indicated that Jc1-n infects cells 1.82 times faster and produces progeny viruses from infected cells 2.70 times faster than JFH-1. The estimate was further validated by independent experiments, in which Jc1-n entry and virus production were indeed significantly higher than those of JFH-1, although early replication was similar for the two strains (S2 Text and S3A–S3C Fig). The cell line used in our in vitro model was deficient in interferon induction by HCV infection (S3D Fig), as reported previously [14]. No significant difference was apparent in the converted fraction of infectious virus, f_θ (Fig 3C). Because JFH-1 and Jc1-n have identical nonstructural regions essential for RNA replication (NS3–NS5B), we estimated the same viral RNA replication rate, k, for these two viruses (Fig 3D). Hence, our parameter estimation captured the characteristics of the two strains well and was able to quantitatively describe viral infection dynamics.

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Fig 3. Characterization of viral dynamics of HCV JFH-1 and Jc1-n.

The distributions of the rate constant for infection, β_θ; the release rate of intracellular viral RNA, ρ; the converted fraction of infectious viral RNA, f_θ; and the replication rate of intracellular viral RNA, k, inferred by MCMC computations are shown in (A), (B), (C) and (D), respectively, for HCV JFH-1 (orange) and Jc1-n (green). Parameters β_θ and ρ for Jc1-n were significantly larger than for JFH-1, whereas there was no significant difference in f_θ between the two strains as assessed by repeated bootstrap t test. JFH-1 and Jc1-n stains had identical viral RNA replication rates. The distributions of accumulation rates of intracellular viral RNA, k−μ−ρ, and the Malthusian parameter, M, calculated from all accepted MCMC parameter estimates are shown in (E) and (F), respectively, for HCV JFH-1 (orange) and Jc1-n (green). These indices were significantly larger for JFH-1 than for Jc1-n as assessed by the repeated bootstrap t test. The underlying data for this Figure can be found in S2 Data. HCV, hepatitis C virus; MCMC, Markov chain Monte Carlo.

https://doi.org/10.1371/journal.pbio.3000562.g003

In our multiscale model (Eqs 2–6), the accumulation rate of intracellular viral RNA was defined as the difference between the replication rate and the sum of the degradation rate and the release rate (i.e., k−μ−ρ). The distributions of calculated intracellular RNA accumulation rates for JFH-1 (1.19 per day, 95% CI 1.07–1.30) and Jc1-n (0.936 per day, 95% CI 0.819–1.05) are shown in Fig 3E (p = 8×10⁻⁶ by bootstrap t test) (Table 1). The preferential accumulation of JFH-1 RNA inside cells was consistent with its tendency toward gradual increased levels of intracellular RNA at later time points (S2B Fig). The difference in the site of virion assembly and the trafficking pathway as well as in the viral genome RNA stability may explain the higher accumulation of JFH-1 RNA in the cells. To further evaluate the total viral RNA level considering multiround virus transmission, the Malthusian parameter, M, was used as an indicator of the initial growth rate of intracellular viral RNA for each HCV strain [8,12,15]. Here, the Malthusian parameter was given by (7) (see S3 Text for derivation of the Malthusian parameter). The Malthusian parameters for JFH-1 and Jc1-n were calculated as 1.19 (95% CI 1.07–1.30) and 0.936 (95% CI 0.819–1.05), respectively, and were significantly different from one another (p = 8×10⁻⁶ by bootstrap t test) (Fig 3F and Table 1). Interestingly, even if Jc1-n had a larger infection rate, β_θ, and release rate, ρ, compared with JFH-1, the initial growth rate of total JFH-1 RNA was higher than that of Jc1-n. This result demonstrated that the capacity to accumulate viral RNA inside cells predominantly determines the initial growth rate rather than release of progeny viruses.

Stay-at-home strategy or leaving-home strategy for “optimizing” HCV proliferation

We investigated how differences between the two strains, JFH-1 and Jc1-n, might be interpreted in an evolutionary perspective. As mentioned above, we considered two opposing strategies: the “stay-at-home strategy” and the “leaving-home strategy”: if viruses have smaller ρ, they preferentially stay inside the cell, but if they have larger ρ, they leave the cell. To quantitatively characterize these different strategies, we defined the fraction of viral RNA remaining in the cells ((k−μ−ρ)/k), released from the cells (ρ/k), and degraded in the cells (μ/k) within the total intracellular viral RNA produced (Fig 4A). Using all accepted MCMC parameter estimates from the time-course experimental datasets, we calculated that the fractions of viral RNA remaining were 31.6% and 25.0%, the fractions of viral RNA degraded were 67.7% and 73.3%, and the fractions of viral RNA released were 0.629% and 1.70% for JFH-1 and Jc1-n, respectively (Fig 4B). Comparing the lattermost fractions, Jc1-n used intracellular viral RNA 2.70 times more for virus release than JFH-1, explaining the rapid transmission of Jc1-n (S2B Fig). These results indicate the preferential “leaving-home” strategy of Jc1-n as compared with JFH-1, which adopts a “stay-at-home” strategy.

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Fig 4. Different strategies adopted by JFH-1 and Jc1-n for viral proliferation.

(A) Schematic representation of the fate of replicated intracellular viral RNA. Viral RNA is used either for driving RNA replication in cells or for producing progeny viruses for release outside cells or is degraded. (B) Percentage of replicated intracellular HCV JFH-1 and Jc1-n viral RNA that remains inside cells, is released outside cells, and is degraded. (C) Change in the Malthusian parameter (Eq 7) with various release rates of intracellular viral RNA. The orange and green curves show Malthusian parameters calculated using 100 parameter sets sampled from MCMC parameter estimates as functions of ρ for JFH-1 and Jc1-n, respectively. The gray vertical lines are the corresponding release rates estimated from the actual experimental data. (D) Change in the cumulative number of newly infected cells with the various release rates (Eq 8). (Left panels) The orange and green curves represent the cumulative numbers of newly infected cells until 3, 5, 7, 9, and 11 days post infection calculated using the means of estimated parameters as a function of ρ for JFH-1 and Jc1-n, respectively. The gray vertical line represents the mean release rate estimated from the experimental data. (Right panels) Enlarged views of the gray zones in left panels. Finely calculated cumulative numbers of newly infected cells are shown. The underlying data for this figure can be found in S3 Data. HCV, hepatitis C virus; MCMC, Markov chain Monte Carlo.

https://doi.org/10.1371/journal.pbio.3000562.g004

To further investigate these two opposing strategies, we addressed the relevance of viral RNA release rates for viral proliferation using in silico analysis. With various values of the release rate of intracellular viral RNA, ρ, we calculated the Malthusian parameter (Eq 7) for each strain as an indicator of viral fitness (Fig 4C). Each curve shows Malthusian parameters calculated using 100 parameter sets sampled from MCMC parameter estimates as functions of ρ, and each gray vertical line is the corresponding estimated release rate. Interestingly, the smaller the release rate, the larger the Malthusian parameter HCV achieves. This is because intracellular viral RNAs can be amplified faster compared with viral RNAs outside of cells that are degraded or enter new cells. This result showed that the JFH-1 strain is more optimized in terms of its Malthusian parameter compared with Jc1-n because of the smaller estimated values of ρ. That is, HCV JFH-1 adopts the stay-at-home strategy for acquiring a higher initial growth rate.

Next, we defined the cumulative number of newly infected cells at time t to evaluate viral transmissibility: (8) We also calculated the cumulative number of newly infected cells for each strain using the means of the estimated parameters as functions of ρ (Fig 4D). Each curve shows the calculated cumulative numbers of infected cells until 3, 5, 7, 9, and 11 days post infection, and the gray vertical line represents the mean release rate estimated from the infection experiment. The value of the release rate, which maximized the cumulative number of newly infected cells, was between 0.1 and 0.5. This is because an intermediate release rate effectively increases extracellular viral RNA for new infection: lower release rates do not effectively produce new infections, whereas higher release rates decrease intracellular viral RNA levels and thus diminish future new infections. Thus, it appears that Jc1-n is more optimized for producing newly infected cells. This implies that HCV Jc1-n adopts the leaving-home strategy to acquire an advantage in producing newly infected cells.

Taken together, our theoretical investigation based on viral infection experiments revealed that the JFH-1 strain optimizes its initial growth rate, but the Jc1-n strain optimizes de novo infection. Ours is the first report to quantitatively evaluate these opposing evolutionary strategies and to show their significance for virus proliferation at the intracellular and intercellular levels.

Cell culture and HCV infection

Huh-7.5.1 (kindly provided by Dr. Francis Chisari, The Scripps Research Institute) and Huh7-25 cells were cultured in Dulbecco’s Modified Eagle’s Medium (Invitrogen) supplemented with 10% fetal bovine serum (Sigma), 10 units/mL penicillin, 10 mg/mL streptomycin, 0.1 mM nonessential amino acids (Invitrogen), 1 mM sodium pyruvate, and 10 mM HEPES (pH 7.4) at 37°C under a humidified atmosphere containing 5% CO₂. We used HCV strains JFH-1, a genotype 2a clinical isolate from a patient with fulminant hepatitis [4], and Jc1-n, a J6/JFH-1 chimeric laboratory strain [13]. JFH-1 and Jc1-n have 96.7% amino acid identity over the whole genome. HCV inoculum for infection experiments was recovered from the culture supernatants of Huh-7.5.1 cells transfected with the corresponding HCV RNA as described [4]. Huh-7.5.1 cells were inoculated with JFH-1 or Jc1-n at a multiplicity of infection (MOI) of 0.05 for 4 hours and then passaged to seed a new 96-well plate at different cell densities (1,000, 2,000, or 4,000 cells/well). Under these conditions, the starting number of HCV-infected cells varied. At days 0, 1, 2, 3, and 4 post seeding, culture supernatants and cell lysates were recovered to quantify HCV RNA by real-time RT-PCR as previously described [13]. The infectivity of HCV in culture supernatants was measured using a focus-forming assay as described [13]. To quantify the number of uninfected and infected cells, cells were fixed and stained with anti-HCV core antibody by immunofluorescence assay as described [13]. We prepared three replicate samples for these experiments and used the average values of replicates for analysis.

Data fitting and parameter estimation

The parameters g and K were separately estimated (see S4 Text) and fixed at 0.670 per day and 4.12×10⁴ cells, respectively, for the JFH-1 strain and 0.665 per day and 3.75×10⁴ cells, respectively, for the Jc1-n strain. A statistical model adopted from Bayesian inference assumed that measurement error followed a normal distribution with mean zero and constant variance (error variance). The noninformative prior distributions with upper and lower limits were used as the prior distributions of parameters. The posterior predictive parameter distribution as an output of MCMC computation represented parameter variability. Distributions of model parameters (i.e., β_θ, k, ρ, f_θ) and initial values (i.e., T(0), I(0), A(0), V_θ(t), V(0)) in Eqs 2–6 were inferred directly by MCMC computations. We also tested structural identifiability for all parameter estimates by calculating profile likelihood [23,24] using package dMod [25] in R Statistical Software [26]. All parameter estimates were confirmed to be structural identifiable (see S4 Fig). Distributions of derived quantities were calculated from the inferred parameter sets (Fig 3E and 3F for graphical representation). A set of computations for Eqs 2–6 with estimated parameter sets gives a distribution of outputs (the number of cells and the intra- and extracellular viral loads) as model predictions. To investigate variation in model predictions, global sensitivity analyses were performed. The ranges of possible variation are shown in Fig 2 as 95% posterior predictive intervals. Technical details of MCMC computations are summarized below.

Statistical analysis

Package FME [27] in R Statistical Software [26] was used to infer posterior predictive parameter distributions. The delayed rejection and Metropolis method [28] was used as a default computation scheme for FME to perform MCMC computations. MCMC computations for parameter inference were implemented using the predefined function modMCMC() in package FME as introduced in Methods. Convergence of Markov chains to a stationary distribution was required to ensure parameter sets were sampled from a posterior distribution. Only the last 90,000 of 100,000 chains were used as burn-in. The convergence of the last 90,000 chains was manually checked with figures produced by package coda [29], a collection of diagnostic tools for MCMC computation. The 95% posterior predictive intervals shown as a shadowed region in each panel of Fig 2 were produced from 100 randomly chosen inferred parameter sets and corresponding model predictions. We employed a bootstrap t test [30] to quantitatively characterize differences in parameters and derived quantities between HCV JFH-1 and Jc1-n. In total, 100,000 parameter sets were sampled with replacement from the posterior predictive distributions to calculate the bootstrap t-statistics. To avoid potential sampling bias, the bootstrap t test was performed 100 times repeatedly. The averages of the computed p-values were used as indicators of differences.