Infectio: a Generic Framework for Computational Simulation of Virus Transmission between Cells

Toward a hybrid model—cellular automaton.Initially infected virus-spreading cells have received one plaque forming unit (PFU) and undergo a time- and PFU-dependent process of cell-free and cell-associated virus production according to the mathematical model with a ratio of cell-free to cell-associated virus (scaling factor) as discussed below (see equation 1; also, see Fig. S1 in the supplemental material). The amount of neighboring cells infected near an initially infected cell is determined by the plaque growth speed, measured in micrometers per hour. Each cell is represented by a hexagonal object, and in a monolayer, a hexagonal lattice represents biological cells, similar to the model described earlier (21). CA cells can be in a noninfected state (default), infected, dead, or lysed, where infected cells emerge from noninfected and dead cells or are lysed from infected cells. A single infected cell located at the center of the grid represents an initial infection, and adjacent noninfected cells may become infected by cell-free or cell-associated viruses (secondary infections). The probability of infection was measured experimentally and designated P_{cell-free infection}, which attributes probabilistic cell infection values to the cellular automata. In this setting, we chose the cell-to-cell infection spread to be dependent on cell-associated virus and the microinfectivity of the neighboring cells.

Particle strength exchange representing diffusive behavior of infectious agents in the extracellular medium.Analyses of experimentally determined parameters of viral spreading in plaque assay indicated a contribution of cell-free adenovirus to infection (21, 31). It is, however, not known whether other viruses also freely diffuse through the agarose gel medium, as diffusion is dependent on virus size. We hence simulated free and limited diffusion conditions by coupling the advection-diffusion systems to our probabilistic CA representing the host cell population. The probability of cell-free infection is proportional to the amount of virus a cell receives from the PSE module. We introduced deterministic PSE methods mimicking diffusion-advection of extracellular virus (34). This is a continuum particle method discretizing the advection-diffusion PDE and is suitable for simulating the mass transfer of large numbers of virus particles typically present in the extracellular medium during cell-free spread. PSE was considered in a rectangular volume with an area equal to the dimensions of the CA cell, located above the cell and filled with equally spaced computational particles representing extracellular virus. The state of a CA cell is influenced by the amount of virus within the element above the cell, the movement of virus elements by advection under free-space boundary conditions allowing virus mass to exchange with neighboring PSE elements by diffusion. The method thus represents both bulk currents and local diffusion and accounts for the formation of comet-shaped and circular plaques, respectively (21).

Algorithmic implementation of Infectio.The algorithmic hybrid implementation of the cellular automaton and particle strength exchange (CA-PSE) methods is largely identical to the one published before (21). Here, we simulated the cellular monolayer using probabilistic CA (47, 48) on a hexagonal lattice. The latter is implemented in a spatiotemporal hybrid with PSE, a particle-based advection-diffusion numerical approximation method (34). PSE is based on replacing the Laplace operator with an integral operator that is subsequently discretized using particle locations as quadrature points (49). In the current implementation, an additional feature was implemented: the infection state is switched from “uninfected” to “infected” based on the parameter “speed of plaque growth,” allowing simulation of the cell-to-cell spreading mechansim. Further details can be obtained and discussed on the GitHub page under http://infectio.github.io or in the project help under http://infectio.github.io/help.html.

Software implementation of Infectio.The proposed method has been implemented as a software framework in the form of a stand-alone package. The main programming language was MATLAB with a well-structured modular design using MATLAB name spaces, which is reflected, for example, in the folder structures of the source code. Following this approach, it is easy to extend the framework by adding, setting files, or enumerating the model parameter space, each of which is different from a default scenario. Introduction of further model parameters or changes in the logics can be made using setting files. Additionally, to ensure user-friendliness of the software, we implemented a MATLAB GUID-based graphical user interface (GUI). Performance-critical parts were implemented as cross-platform C code, which is integrated using the MATLAB MEX technology. Application performance was further enhanced using parallel computing techniques through the shared memory model based on OpenMP 4.0 extensions. To ensure sufficient performance, the recommended hardware requirements include a high-end multicore workstation, or a multipurpose high-performance computer cluster.

The Infectio software is designed to take in a number of custom parameters defined by the end-user based on, e.g., experimental measurements as discussed below. An overview of input parameters, flags, and their default values are provided in Table 1. As a default output Infectio software provides frames of a time-lapse graphic representation of cells and final values of all the variables in the workspace saved as a .mat file. Additionally, one may save a graphical rendering of the PSE particles with color-coded particle strength (see the flags in Table 1). For experimental and troubleshooting purposes, one may save the state of any variable at any point of the simulation using “sensor” variables, which is deactivated by default but may be activated through a specific flag (see the flags in Table 1).

Experimentally measured cell-free virus infection probability and average cell size.To measure the probability of infection by cell-free VACV strains WR and IHD-J, we used the time-lapse titration data from a published data set (31). From these data, we estimated the fractions of infected cells depending on the amount of input virus at 12.3 h p.i., i.e., before the observable occurrence of the second round of infection. Both VACV-WR and IHD-J showed a dose-dependent increase of the fraction of infected cells. Fractions of infected cells were used as a look-up table, which allowed us to compute the probability of infection as a function of virus concentration using linear interpolation. We averaged the diameters of 40 BSC40 cells based on transmission light microscopy images (31) and found the average to be 22 ± 8 µm.

Experimentally determined speed of cell-to-cell spread.The speed of cell-to-cell spread was measured from the speed of radial growth (in micrometers per hour), detected by the emergence of the GFP signal around the initially infected cell. Although this measurement is probably less than the speed of virus transition between cells, we took it as an approximation of virus transmission from the originally infected cell to the neighboring cells.

Experimentally determined cell-free and estimated cell-associated virus production.VACV-WR produces 10 to 100 times less cell-free virus than IHD-J and 100 to 1000 times less cell-free than cell-associated virus (26, 31). Both strains produce comparable amounts of total infectious virus, and IHD-J produces about 10 times less cell-free virus than cell-associated virus. The amount of cell-free virus for VACV-WR is negligible within the time frame of our experiments, and hence we estimated the VACV-WR spreads between cells by cell-cell route only. To simulate the production of cell-free VACV-IHD-J progeny, we fitted the data from the supernatant titration experiment with a mathematical model of virus production. VACV growth curves have a sigmoidal shape, similar to the widely used one-step growth curves for other viruses (50). We hence fitted our experimental data with a sigmoidal model (see equation 1) using the least-squares fitting method. Since a negative concentration is nonphysiological, we constrained the parameter Bottom to be larger than or equal to 0. The fitting results and the residual plot are shown in Fig. S1A and B in the supplemental material. The best parameters obtained and the goodness of the fit are shown in Table 1. Y=Bottom+(Top−Bottom)1+10(log EC50 − t) × Slope(1) where Y is the virus amount, t is time after infection, and “Bottom,” “Top,” “EC₅₀,” and “Slope” are the fitting parameters.

We compared the output of the model with infection spread of different viruses and obtained closely matching results. Next, we estimated the production rate of cell-associated viruses by conjecturing that VACV-IHD-J produces cell-free virus according to the model we fitted (see Fig. S1 in the supplemental material) using the best-fit parameters (see Fig. S1C). We also assumed that the production rate of other infectious forms follows the same model in either a scaled-up or scaled-down manner. The scaling factor for VACV IHD-J cell-associated virus production was 10. For VACV-WR, we assumed and used scaling factors of 0.1 and 0.01 (both conditions were tested) for cell-free virus and 10 for cell-associated virus. We assumed that both viruses produce the same amount of cell-associated virus (scale factor, 10).

Estimated virus diffusion coefficient.To estimate the diffusion coefficient we used the Einstein-Stokes relation (equation 2) D=kBT6πηr(2) where k_B is Boltzmann’s constant, T is temperature in kelvin, η is the dynamic viscosity of the medium, and r is the spherical radius of the diffusing particle.

As previously shown (21), the Reynolds number for our experimental setting is much less than 1, which justifies the use of the Einstein-Stokes relation under the conditions of our experiment. For the calculations with VACV, we took a spherical diameter of 270 nm, which corresponds to the longest dimension of a VACV particle. The computed diffusion constant for a spherical particle of this size is 1.218 µm²/s. Although VACV particles are barrel shaped and not spherical (23), we assumed in a first approximation that this difference in shape does not significantly influence the value of the diffusion coefficient.