William E. Schiesser - Virus Host Cell Genetic Material Transport: Computational ODE/PDE Modeling with R

An individual viral particle, called a virion, is a far simpler structure than a bacterium. It has often been questioned whether a virus is alive. It is certainly not living in the everyday sense of the word. Virions consist of genetic materialDNA or RNA enclosed in a protein coating. Many viruses, called enveloped viruses, have an additional outer membrane that encloses the protein coat. This membrane envelope is material co-opted from the cells own membrane. As the new virion buds out from an infected host cell, it is wrapped by the cells bilayer membrane and carries with it any protein that happens to be embedded in the membrane at the budding site. Enveloped viruses are then free to begin a new cycle of infection by fusing their cell-derived envelope with the cellular membrane of an uninfected cell.

The movement of the VGM across the host cell outer membrane is a spatiotemporal dynamic process that is modeled in this book as a system of ordinary and partial differential equations (ODE/PDEs). First, a ODE/PDE model is developed for a single protein transmitted from the virus through the cell membrane into the cell interior where it replicates, and possibly mutates.

The movement of the virus protein through the cell membrane is modeled as a diffusion process expressed by the diffusion PDE (Ficks second law) with dependent variable V1(x, t), and independent variables x, the position in the membrane, and t, time. Within the cell, the time variation of the VGM is modeled as an ODE with dependent variable C1(t). The subscript 1 denotes the first virus protein considered.

The single protein model is then extended to the replication, and possibly mutation, of the virus protein within the cell interior to produce additional virus proteins that diffuse out through the cell membrane where they can infect other host cells.

The model time scale in minutes is 0t240 (four hours), and the evolution of the dependent variables is computed by the numerical integration of the ODE/PDEs from zero initial conditions (IC) V1(x, t=0)=C1(t=0)=0. The departure of the dependent variables from zero is in response to the virus protein concentration at the outer membrane surface (the point at which the virus binds to the host cell).

The numerical integration of the ODE/PDE model equations is performed with routines coded (programmed) in R, a quality, open-source scientific computing system that is readily available from the Internet. Formal mathematics is minimized, e.g., no theorems and proofs. Rather, the presentation is through detailed examples that the reader/researcher/analyst can execute on modest computers. The ODE dependent variables are plotted against t and the PDE dependent variables are plotted against x and t with basic R plotting utilities. The solution is by the numerical method of lines (MOL), an established general algorithm for ODE/PDE systems.

As extensions of the ODE/PDE models,

The R routines are available from a download link so that the example models can be executed without having to first study numerical methods and computer coding. The routines can then be applied to variations and extensions of the ODE/PDE model, such as changes in the parameters and the form of the model equations.

The author would welcome comments/suggestions concerning this approach to the analysis of the virus host cell dynamics (directed to wes1@lehigh.edu).

[1] Cohen, F.S. (2016), How Viruses Invade Cells, Biophysical Journal, , pp 10261032