3D Spatially Resolved Models of the Intracellular Dynamics of the Hepatitis C Genome Replication Cycle

Abstract

Mathematical models of virus dynamics have not previously acknowledged spatial resolution at the intracellular level despite substantial arguments that favor the consideration of intracellular spatial dependence. The replication of the hepatitis C virus (HCV) viral RNA (vRNA) occurs within special replication complexes formed from membranes derived from endoplasmatic reticulum (ER). These regions, termed membranous webs, are generated primarily through specific interactions between nonstructural virus-encoded proteins (NSPs) and host cellular factors. The NSPs are responsible for the replication of the vRNA and their movement is restricted to the ER surface. Therefore, in this study we developed fully spatio-temporal resolved models of the vRNA replication cycle of HCV. Our simulations are performed upon realistic reconstructed cell structures-namely the ER surface and the membranous webs-based on data derived from immunostained cells replicating HCV vRNA. We visualized 3D simulations that reproduced dynamics resulting from interplay of the different components of our models (vRNA, NSPs, and a host factor), and we present an evaluation of the concentrations for the components within different regions of the cell. Thus far, our model is restricted to an internal portion of a hepatocyte and is qualitative more than quantitative. For a quantitative adaption to complete cells, various additional parameters will have to be determined through further in vitro cell biology experiments, which can be stimulated by the results deccribed in the present study.

Keywords: (surface) partial differential equations; 3D spatio-temporal resolved mathematical models; Finite Volumes; computational virology; hepatitis C virus (HCV); massively parallel multigrid solvers; mathematical models of viral RNA cycle; realistic geometries; viral dynamics; within-host viral modelling.

Figure 1

The HCV viral replication cycle: Virus endocytosis, vRNA uncoating, vRNA translates polyprotein, polyprotein splitting into structural and nonstructural proteins (SP/NSP), NSPs create membranous web at ER, vRNA replication inside web, new vRNA and SPs assembled to new virus particles, exocytosis of complete new viruses which infect other cells.

Figure 2

Reconstruction of ER surface (channel of cell data stained with calnexin)—special example: single steps of reconstruction process: (a) Raw confocal z-stacks, Calnexin ER-marker (b) deblured based on Huygens SVI (c) Surface mesh of (with NeuRA2) reconstructed ER surface.

Figure 3

Reconstructed ER surface and web regions—choice of the cutout region for the 3D spatio-temporal resolved model development. Surface of ER (blue) and membranous web (red). Green frame marks for cutout choice. (a) Front view of complete cell (b) Back view of complete cell (c) Small cutout part of ER ($\mathcal{E}$ in blue, $\mathcal{W}$ web: in red).

Figure 4

Subdomains of the small geometry for the development of the surface model. (a,b) Front view and back view; (c,d) ER rotated slightly; (c) webs unvisible, ribosomes open; (d) webs visible, ribosomes hidden. All surfaces together form the computation domain $\mathcal{D}$. Middle blue: $\mathcal{E}$, other colors: single web regions ${\mathcal{W}}_i$ and ribosomal regions ${\mathcal{R}}_i$, $i=1,2,\dots 7$.

Figure 5

(a): ER geometry of the cutout enriched by the ribosomic belt. (b): Surface grid enriched by the sphere where the RNA start concentration will be located. (c): Clip plane of the tetrahedralized domain $\mathrm{\Omega }$. Subdomains of the tetrahedral volume element, i.e., part of the cell where the ER lumen is excluded. Subdomains: cytosol dark blue, ER surface cyan, the colored “blobs” are the web subdomains. The complete region $\mathrm{\Omega }$ represents the computational domain.

Figure 6

Screenshot of the movie S1 Video in Supplementary Material: Simulation of vRNA, NSP, web protein and host factor interplay dynamics on cutting plane version of rotating ER surface. Simulation of model Equations (8a)–(8c), described in detail in Section 3.1.

Figure 7

Spatially resolved sPDE model evaluation (Equations (8a)–(8c)) of vRNA (rna), Web Protein (web) and host factor (host) concentrations separated by subdomains (a–c) and complete computational domain (d), examples using heuristic values for diffusion constants and initial concentrations. Note: The complete computational domain refers as the cutout of the cell rather then the complete cell.

Figure 8

Variation of parameters in Equations (8a)–(8c). Note: Always only one parameter gets varied, all others are kept fixed and take the values as reported in Table 2, besides in case f. Variations: (a) Diffusion constant of the RNA $D_R$; (b) Diffusion constant of the host $D_H$; (c) NSP (i.e., web protein) translation reaction rate $r_2$; (d) Diffusion constant of the RNA inside the webs $D_R{|}_{\mathcal{W}}$; (e) initial value of Host, $H(x,0)$ at all spatial points of the computational domain; (f) Switching off host factor diffusion $D_H=0$ and varying diffusion constant of RNA inside the webs $D_R{|}_{\mathcal{W}}$. All parameters not denoted are as indicated in the standard parameter set, cf., Table 2.

Figure 9

Switching off two of seven webs (web number 3 and 5) to test for the influence of the web density. Indicates effective change of the experimental data as base of the geometric setup. This case is shown separately since it is only based in part upon the geometry as derived from experiment. All other parameters from standard parameter set, cf. Table 2.

Figure 10

Screenshot of the attached movie S2 Video in Supplementary Material of the simple surface model including the intermediate polyprotein state. Simulation of model Equations (9a)–(9d), description cf. Section 3.2.

Figure 11

Screenshot of the attached supplemental movie S3 Video in Supplementary Material aiming the (volume) PDE model simulation (Equations (10a)–(10d) at the cell geometry where the ER lumen is excluded). For details, cf. Section 3.3. Brief description: The RNA diffuses away from the starting ball-like region, produces NSPs at the ribosome belt, the NSPs diffuse away and bind to the pre-defined web regions (as indicated by the reconstructions) as web protein. Inside the webs, the web (bound) protein “waits” for the RNA which diffuses there. The bound web protein replicates the vRNA at the membranous web region, the RNA diffuses back to the ribosome belt and the cycle is closed and continues.

Figure 12

Evaluation of concentrations of viral RNA, NSP, web protein (webP) and host factor (host) in subdomains of the spatio-temporal resolved volume model which is based on a tetrahedral volume grid with the exclusion of the ER lumen volume.