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

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Biological Basis: Viral Genome Replication Cycle

#### 2.2. The Realistic Reconstructed Geometry

#### 2.2.1. The Surface Reconstruction

#### 2.3. The Surface Mesh for the Model Development

#### 2.3.1. The Volume Mesh

#### 2.4. PDE Models

#### 2.4.1. The Surface PDE Model

- concentration of vRNA: $R(x,t)$;
- concentration of the viral polyprotein: $P(x,t)$;
- concentration of the Web Protein (i.e., the NSP which arises from the cleaved polyprotein and which accumulates to form the web regions) at the predefined webs: $W(x,t)$;
- host factor concentration: $H(x,t)$.

- ${r}_{1}$ describes the rate of the polymerization of new vRNA in terms of the multilinear concentrations of vRNA, web protein and host factor;
- ${r}_{2}$ describes the rate of the polyprotein translation in terms of the linear concentration of vRNA;
- ${r}_{3}$ describes the rate of the cleavage of the polyprotein into web (accumulating) protein;
- ${r}_{4}$ describes the rate of host factor depletion during vRNA polymerization in terms of the multilinear concentrations of vRNA, web protein and host factor. Note that the vRNA polymerization and the host factor depletion are proportional with the only difference of the ratio of ${r}_{1}$ to ${r}_{4}$ to enable a different amount of host factor depletion compared to vRNA polymerization while vRNA gets copied.

#### Degrees of Freedom (DoFs):

#### Parameter Set and Variations

#### 2.4.2. The “Volume” PDE Model

- ${r}_{1}$ describes the rate of the polymerization of new vRNA in terms of the multilinear concentrations of vRNA, web protein and host factor;
- ${r}_{2}$ describes the rate of the translation of NSPs in terms of the linear concentration of vRNA;
- ${r}_{3}$ describes the binding rate the NSPs to the web regions forming web (accumulating) protein;
- ${r}_{4}$ describes the rate of host factor depletion during vRNA polymerization in terms of the multilinear concentrations of vRNA, web protein and host factor. Note that as in the case of the sPDE model before, the vRNA polymerization and the host factor depletion are proportional with the only difference of the ratio of ${r}_{1}$ to ${r}_{4}$ to enable a different amount of host factor depletion compared to vRNA polymerization while vRNA gets copied.

#### 2.5. Comparison to State-of-the-Art ODEs

#### 2.6. Technical Evaluation

#### 2.7. Remarks upon the Numerical Stability

## 3. Results

#### 3.1. Surface PDE Model of vRNA, webProtein and Host Factor

#### 3.1.1. Quantitative Spatially Resolved Evaluation

#### 3.1.2. Variation of the Parameters and Web Numbers

- Figure 8a: For the initial dynamical phase, the variation of the diffusion coefficient of the RNA has a major influence upon the RNA dynamics and the host factor depletion, while it has no major influence upon the long term level. The web growth however is only affected in a minor manner by the RNA diffusion coefficient.
- Figure 8b: The diffusion constant of the host factor has a very important influence upon the complete kinetics.
- Figure 8c: The reaction of the translation of NSPs (i.e., web protein, since intermediate polyprotein not considered) has an influence at the short term level each time that the RNA reaches a new web, but even in the middle-term level, this influence is nearly negligible.
- Figure 8d: The influence of a pure change of the diffusion coefficient of the RNA inside the webs (keeping the diffusion on the ER as before) has nearly no influence upon the kinetics as long as the host factor may diffuse with a substantial diffusion coefficient.
- Figure 8e: The initial value of the host factor has strong influence on the overall dynamics, but this influence may be considered to be even less important compared to the afore described influence of the transport of the host factor.
- Figure 8f: Once the host factor diffusion is switched off, it is interesting to note that at a short-term level, the change of the RNA diffusion coefficient inside the webs shows substantial impact at the short term level, but not at the middle term level.

#### 3.2. Surface PDE Model Incorporating the Polyprotein

#### 3.3. The Volume PDE Model

## 4. Discussion

#### 4.1. The Suface PDE Models

#### 4.2. The “Volume” PDE Model

#### 4.3. Comparison of Our PDE Models to ODE Models—Relationship Form/Function

#### 4.4. Critical Summary and Potentials of Our Models

#### 4.5. Multiscale Modeling

#### 4.6. The vRNA Transport and the Web Movement

#### 4.7. Future Perspectives of Spatial Virus Modeling

#### 4.8. Context of ER Related Research

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

HCV | Hepatitis C virus |

vRNA | viral RNA |

NSP | non structural viral protein |

SP | structural protein |

ODE | Ordinary Differential Equation |

PDE | Partial Differential Equation |

sPDE | surface Partial Differential Equation |

vPDE | (“volume”) Partial Differential Equation |

FD | Finite Differences |

FV | Finite Elements |

FV | Finite Volumes |

MG | Multi Grid |

GMG | Geometric Multi Grid |

UG4 | Unstructured Grids version 4 [34,36] |

NeuRA2 | Neuron reconstruction algorithm, version 2.3 [40,43] |

## Appendix A. S1 Video: 3D Surface PDE Model, 3 Concentrations (vRNA, webProtein, Host Factor), Rotating View

## Appendix B. S2 Video: 3D Surface PDE Model 4 Concentrations (vRNA, Polyprotein, webProtein, Host Factor), Static View

## Appendix C. S3 Video: 3D “Volume” PDE Model

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**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.

**Table 1.**Geometry info of the 3D surface ER representation and the sPDE model of 4 concentrations Equations (2a)–(2d). DoF number at base level and one refinement.

Type | Number |
---|---|

vertices | 5645 |

edges | 17,124 |

faces (base) | 11,467 |

faces (1 ref) | 45,868 |

volumes | 0 |

DoF (base) | 22,580 |

DoF (1 ref) | 91,076 |

**Table 2.**Basic parameter set for the sPDE model evaluations. We emphasize that our code is not restricted to these parameters in any way and the stability does not depend on the use of the parameters reported in the table. Note: Web number parameter given by microscopy data.

Parameters | Value | Unit |
---|---|---|

${D}_{R}$ | 0.3 | $\frac{{\left(\mu m\right)}^{2}}{s}$ |

${D}_{P}$ | 0.3 | $\frac{{\left(\mu m\right)}^{2}}{s}$ |

${D}_{W}$ | 0.5 | $\frac{{\left(\mu m\right)}^{2}}{s}$ |

${D}_{H}$ | 0.05 | $\frac{{\left(\mu m\right)}^{2}}{s}$ |

${r}_{1}$ | 15 | ${s}^{-1}$ |

${r}_{2}$ | 100 | ${s}^{-1}$ |

${r}_{3}$ | 90 | ${s}^{-1}$ |

${r}_{4}$ | 1.5 | ${s}^{-1}$ |

${R}_{0}$ | 10 | $\frac{1}{{\left(\mu m\right)}^{2}}$ |

webs | 7 | # |

**Table 3.**Geometry info of the 3D continuum ER representation which serves as basis for the PDE model Equations (10a)–(10d). DoF number at base level and with one refinement.

Type | Number |
---|---|

vertices | 16,331 |

edges | 104,282 |

faces | 169,565 |

volumes | 81,619 |

DoFs (base) | 65,324 |

DoFs (1 ref) | 482,452 |

Parameters | Value | Unit |
---|---|---|

${D}_{R}$ | 1 | $\frac{{\left(\mu m\right)}^{2}}{s}$ |

${D}_{N}$ | 3 | $\frac{{\left(\mu m\right)}^{2}}{s}$ |

${D}_{W}$ | 5 | $\frac{{\left(\mu m\right)}^{2}}{s}$ |

${D}_{H}$ | 0.1 | $\frac{{\left(\mu m\right)}^{2}}{s}$ |

${r}_{1}$ | 0.2 | ${s}^{-1}$ |

${r}_{2}$ | 2 | ${s}^{-1}$ |

${r}_{3}$ | 20 | ${s}^{-1}$ |

${r}_{4}$ | 0.067 | ${s}^{-1}$ |

${R}_{0}$ | 10 | $\frac{1}{{\left(\mu m\right)}^{3}}$ |

${H}_{0}$ | 1 | $\frac{1}{{\left(\mu m\right)}^{3}}$ |

webs | 7 | # |

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## Share and Cite

**MDPI and ACS Style**

Knodel, M.M.; Reiter, S.; Targett-Adams, P.; Grillo, A.; Herrmann, E.; Wittum, G.
3D Spatially Resolved Models of the Intracellular Dynamics of the Hepatitis C Genome Replication Cycle. *Viruses* **2017**, *9*, 282.
https://doi.org/10.3390/v9100282

**AMA Style**

Knodel MM, Reiter S, Targett-Adams P, Grillo A, Herrmann E, Wittum G.
3D Spatially Resolved Models of the Intracellular Dynamics of the Hepatitis C Genome Replication Cycle. *Viruses*. 2017; 9(10):282.
https://doi.org/10.3390/v9100282

**Chicago/Turabian Style**

Knodel, Markus M., Sebastian Reiter, Paul Targett-Adams, Alfio Grillo, Eva Herrmann, and Gabriel Wittum.
2017. "3D Spatially Resolved Models of the Intracellular Dynamics of the Hepatitis C Genome Replication Cycle" *Viruses* 9, no. 10: 282.
https://doi.org/10.3390/v9100282