# Quantitative Analysis of Hepatitis C NS5A Viral Protein Dynamics on the ER Surface

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

- Geometric setup: We use previously published confocal microscopic microscopy z-stack data [5] of cells labeled with ER markers which allow for reconstructions of realistic ER surfaces. These fine level data provide the geometric constraints for NS5A movement.
- A model and corresponding simulations: Our previous model of NS5A dynamics [35] has not been adapted to biological data so far. In this study, we perform simulations using an extended version of the model and fit the simulation parameters in order to match the experimental data.

#### 2.1. FRAP Experiments—Basics

#### 2.2. Expermimental Data and Cell types

#### 2.3. NS5A Movement Properties

#### 2.4. Modeling FRAP Experiments

#### 2.5. Pseudo Reaction Constant Fit

#### 2.6. ER Geometry Reconstruction

#### 2.7. Comparing Experiment and Simulation

## 3. Results

#### 3.1. Realistic Simulation of FRAP Experiments

#### 3.2. Estimation of the NS5A Diffusion Constant

#### 3.3. Influence of Geometry and Time Series

#### 3.4. Refinement Stability

#### 3.5. Influence of the Measurement Process

#### 3.6. Comparative 2D Simulations

#### 3.7. Final Averaged Results

## 4. Discussion

#### 4.1. Interpretation of the Diffusion Constant Values

#### 4.2. The Context of Spatial HCV Models

#### 4.3. Related Work

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

FRAP | Fluorescence recovery after photobleaching |

FLIP | Fluorescence loss in photobleaching |

ROI | region of interest |

HCV | Hepatitis C virus |

vRNA | viral RNA |

NSP | non structural viral protein |

NS5A | HCV non structural protein number 5 |

SP | structural protein |

DMV | Double membrane vesicle |

TMS | time series |

ER | Endoplasmatic Reticulum |

ODE | Ordinary Differential Equation |

PDE | Partial Differential Equation |

sPDE | surface Partial Differential Equation |

FV | Finite Volumes |

MG | Multi Grid |

GMG | Geometric Multi Grid |

UG4 | Unstructured Grids version 4 [50,51] |

NeuRA2 | Neuron reconstruction algorithm, version 2.3 [35,68] |

geo(m) | geometry |

psr | pseudo reaction |

## Appendix A. Supplemental Movies, Short Description

#### Appendix A.1. S1 Video: Movie of FRAP Simulation at ER Geometry I

#### Appendix A.2. S2 Video: Movie of FRAP Simulation at ER Geometry IV

#### Appendix A.3. S3 Video: Movie of Classical FRAP Simulation at 2D Continuum Plane

## Appendix B. Realistic Reconstructed ER Geometries—Further Details

**Figure A1.**The reconstructed ER geometries ${\mathcal{E}}_{i}$, $i=2,3,4,5$ and exemplar FRAP regions. FRAP ROIs indicated in red, covering a surface of about 38 $\mathsf{\mu}$m${}^{2}$ enabling the reader to estimate the total size of the ERs.

## Appendix C. Simple Planar 2D Geometry and Simulations—Details

## Appendix D. Neglecting the Measurement Process Induced Intensity Reduction

**Table A1.**DoF number and total face number of the ER geometries at level L, extracted from our former paper [35].

Geo | L | DoF $\mathcal{E}$ | DoF $\mathcal{U}$ | DoF $\mathcal{F}$ | Faces |
---|---|---|---|---|---|

${\mathcal{E}}_{1}$ | 0 | 815,111 | 794,128 | 20,983 | 1,636,803 |

1 | 3,270,924 | 3,187,566 | 83,358 | 6,547,212 | |

2 | 13,092,959 | 12,761,035 | 331,924 | 26,188,848 | |

${\mathcal{E}}_{2}$ | 0 | 1,212,622 | 1,174,204 | 38,418 | 2,430,181 |

1 | 4,861,079 | 4,708,431 | 152,648 | 9,720,724 | |

2 | 19,448,536 | 18,840,730 | 607,806 | 38,882,896 | |

${\mathcal{E}}_{3}$ | 0 | 170,209 | 140,022 | 30,187 | 340,108 |

1 | 680,786 | 560,740 | 120,046 | 1,360,432 | |

2 | 2,722,264 | 2,243,664 | 478,600 | 5,441,728 | |

${\mathcal{E}}_{4}$ | 0 | 601,706 | 591,336 | 10,370 | 1,208,661 |

1 | 2,414,802 | 2,373,711 | 41,091 | 4,834,644 | |

2 | 9,666,977 | 9,503,511 | 163,466 | 19,338,576 | |

${\mathcal{E}}_{5}$ | 0 | 728,636 | 699,338 | 29,298 | 1,463,597 |

1 | 2,924,907 | 2,808,345 | 116,562 | 5,854,388 | |

2 | 11,708,240 | 11,243,894 | 464,346 | 23,417,552 |

**Table A2.**DoF number and total face number of the simple planar 2D geometry at level L, extracted from our former paper [35].

L | DoF $\mathcal{E}$ | DoF $\mathcal{U}$ | DoF $\mathcal{F}$ | faces |
---|---|---|---|---|

0 | 37 | 24 | 13 | 36 |

1 | 133 | 96 | 37 | 144 |

2 | 505 | 384 | 121 | 576 |

3 | 1,969 | 1,536 | 433 | 2,304 |

4 | 7,777 | 6,144 | 1,633 | 9,216 |

5 | 30,913 | 24,576 | 6,337 | 36,864 |

6 | 123,265 | 98,304 | 24,961 | 147,456 |

7 | 492,289 | 393,216 | 99,073 | 589,824 |

8 | 1,967,617 | 1,572,864 | 394,753 | 2,359,296 |

## Appendix E. Refinement Stability

**Figure A2.**Classical FRAP analysis based upon the 2D computations explained in Section 3.6: Averages for diffusion constant estimation ${D}_{\mathrm{ns}5\mathrm{a}}$—analyzed separately for NS5A/Alone (red—${D}_{\mathrm{ns}5\mathrm{a}}^{\mathrm{A}}$), NS5A/OtherNSPs (green—${D}_{\mathrm{ns}5\mathrm{a}}^{\mathrm{N}}$) as well as for NS5A/Alone setting ${r}_{p}=0$ (blue—${D}_{\mathrm{ns}5\mathrm{a}}^{\mathrm{A}}{|}_{{r}_{p}=0}$), i.e., assuming that the measurement process induces no intensity reduction. Thin points and error bars correspond to estimated values for ${D}_{\mathrm{ns}5\mathrm{a}}$ (shown on x axis) for a single TMS (indicated on the left y axis). Aggregating over all TMS yields distributions (continuous lines, scale shown on right y axis). Thick symbols (shown on top) correspond to the averaged values ${\overline{D}}_{\mathrm{ns}5\mathrm{a}}$ reported in Table 1 for the 2D planar case and in Table A3 for the 2D planar case using ${r}_{p}^{\mathrm{A}}=0$ for the NS5A/Alone cell case.

**Table A3.**Averaged final NS5A diffusion constant ${\overline{D}}_{\mathrm{ns}5\mathrm{a}}^{\mathrm{A}}$ (for the NS5A/Alone cell case) as described in Section 3.7, but assuming vanishing signal reduction due to the measurement process, i.e., using ${r}_{p}^{\mathrm{A}}=0$ in (8). The final values are computed by means of the averaging process of the single results (as described in Section 2.7) which is shown graphically within Figure A3. We give results for the case of the use of the ER geometry setups as described in Section 3.1 and Section 3.2, but also for a simplified classical 2D planar consideration, cf. Section 3.6. Values only given for the NS5A/Alone case, since reliable estimations were not possible for the NS5A/OtherNSPs case once we used ${r}_{p}^{\mathrm{N}}=0$.

Geoms | $\mathit{D}\left[{\left(\mathsf{\mu}\mathbf{m}\right)}^{2}/\mathbf{s}\right]$ | $\mathit{\sigma}\left[{\left(\mathsf{\mu}\mathbf{m}\right)}^{2}/\mathbf{s}\right]$ |
---|---|---|

${\mathit{r}}_{\mathit{p}}^{\mathbf{A}}=\mathbf{0}$, NS5A/Alone cells | ||

plane 2D | 0.010328 | 0.001274 |

ER surface | 0.022915 | 0.000887 |

**Figure A3.**Averages for NS5A diffusion constant ${D}_{\mathrm{ns}5\mathrm{a}}^{\mathrm{A}}$ estimation on the ER surface as described in Section 2.2 and Section 3.3 for the NS5A/Alone cell case, neglecting the measurement process induced signal reduction, i.e., setting ${r}_{p}^{\mathrm{A}}=0$ in (8). Simular analysis as in Figure 7. Thin points and error bars correspond for (

**a**,

**b**) to estimated values for ${D}_{\mathrm{ns}5\mathrm{a}}$ (shown on x axis) for the combination of single TMS with single geometries (indicated on the left y axis, note different combinations for geometry and TMS). Aggregating over all TMS and ER geometries yields distributions (thin continuous lines, scale shown on right y axis) which are identical in both cases. (

**a**) Each “row” corresponds to one time series combined with all ER geometries. (For example, the left y axis value 2.5 corresponds to the combination of TMS # 2 and ER geometry ${\mathcal{E}}_{6}$). Half thick symbols (shown in the middle of each time series region) correspond to the averaged values over all geometries ${\overline{D}}_{\mathrm{ns}5\mathrm{a}}{|}_{G}$ for the respective TMS. Aggregating these averages over all TMS yields distributions (thick continuous lines, scale shown on right y axis). Thick symbols (shown on top) correspond to the averaged values ${\overline{D}}_{\mathrm{ns}5\mathrm{a}}$. (

**a**) Each “row” corresponds to one ER geometry combined with all TMS. (For example, the left y value 2.5 corresponds to ER geometry ${\mathcal{E}}_{3}$ and TMS # 5). Half thick symbols (shown in the middle of each ER geometry region) correspond to the averaged values over all TMS ${\overline{D}}_{\mathrm{ns}5\mathrm{a}}{|}_{T}$ for the respective ER geometry as reported in Section 2.6. Aggregating these averages over all ER geometries yields distributions (thick continuous lines, scale shown on right y axis). Thick symbols (shown on top) correspond to the averaged values ${\overline{D}}_{\mathrm{ns}5\mathrm{a}}^{\mathrm{A}}$. Note: The total averages are identical for (

**a**,

**b**) and are reported in Table A3.

**Table A4.**Refinement stability: Averaged NS5A diffusion constant for the simplified planar 2D case and the ER surface manifold computations (using only one FRAP ROI per ER geometry. Nota bene: the afore reported final results cover all geometric setups, i.e., the final results reported in Table 1 are based upon the use of two FRAP ROIs per reconstructed ER geometry). Evaluation for different spatial refinement levels R of base geometry and relative change C in comparison to level before in percentage. Values for the NS5A/Alone cell case without pseudo reaction, i.e., ${r}_{p}^{\mathrm{A}}=0$ in (8) and for the NS5A/OtherNSPs cell case with pseudo reaction, ${r}_{p}^{\mathrm{N}}\ne 0$.

Geometries | R | $\mathit{D}\left[{\left(\mathsf{\mu}\mathbf{m}\right)}^{2}/\mathbf{s}\right]$ | $\mathit{\sigma}\left[{\left(\mathsf{\mu}\mathbf{m}\right)}^{2}/\mathbf{s}\right]$ | C $[\%]$ |
---|---|---|---|---|

NS5A/Alone Cells, ${r}_{p}^{\mathrm{A}}=0$ | ||||

2D planar | 5 | 0.012133 | 0.001415 | — |

6 | 0.011093 | 0.001334 | 9.378 | |

7 | 0.010582 | 0.001294 | 4.832 | |

8 | 0.010328 | 0.001274 | 2.452 | |

5 ERs | 1 | 0.023701 | 0.001309 | — |

2 | 0.023239 | 0.001288 | 1.989 | |

NS5A/OtherNSPs Cells, ${r}_{p}^{\mathrm{N}}\ne 0$ | ||||

5 ERs | 1 | 0.007837 | 0.000513 | — |

2 | 0.007602 | 0.000493 | 2.996 |

## Appendix F. Variation of Pseudo Reaction

## Appendix G. Additional Simulation Screenshot

**Table A5.**Fit parameters in (A1) of linear regression dependency of diffusion constant value on pseudo reaction constant, on ER surface.

Cells | ${\mathit{D}}_{0}\left[{\left(\mathsf{\mu}\mathbf{m}\right)}^{2}/\mathbf{s}\right]$ | $\mathit{f}\left[{\left(\mathsf{\mu}\mathbf{m}\right)}^{2}\right]$ |
---|---|---|

NS5A/Alone | 0.0228726 | 9.62539 |

NS5A/OtherNSPs | 0.0012175 | 4.14657 |

**Table A6.**Fit parameters in (A1) of linear regression dependency of diffusion constant value on pseudo reaction constant on 2D planar geometry.

Cells | ${\mathit{D}}_{0}\left[{\left(\mathsf{\mu}\mathbf{m}\right)}^{2}/\mathbf{s}\right]$ | $\mathit{f}\left[{\left(\mathsf{\mu}\mathbf{m}\right)}^{2}\right]$ |
---|---|---|

NS5A/Alone | 0.0103104 | 4.15337 |

NS5A/OtherNSPs | 0.0007236 | 1.89964 |

**Figure A4.**Dependency diffusion constant of NS5A on ER surface on pseudo reaction constant, linear fit of (A1) for numerical stability test reasons. “estim”: estimated values, “fit”: linear fit.

**Figure A5.**Dependency diffusion constant on pseudo reaction constant, linear fit of (A1) for 2D planar geometry. “estim”: estimated values, “fit”: linear fit.

**Figure A6.**Screenshot of NS5A FRAP experiment simulation on ER surface, Geometry IV (suppl. movie, “S2 Video in supplementary material”).

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**Figure 1.**Intensity changes in unbleached control region $\mathcal{C}$ according to Equation (10): Experimental data for time series (TMS) #2 (NS5A/OtherNSPs) and the corresponding fit. The determination of the exponential decay rate of the signal itself (far away from the FRAP region, where no diffusion takes place) allows for a more precise evaluation of the diffusion constant.

**Figure 2.**Averages for pseudo reaction rate ${r}_{p}$—analyzed separately for NS5A/Alone (red—${r}_{p}^{\mathrm{A}}$) and NS5A/OtherNSPs (green—${r}_{p}^{\mathrm{N}}$). Thin points and error bars correspond to estimated values for ${r}_{p}$ (shown on x axis) for a single TMS (indicated on the left y axis). Aggregating over all TMS yields distributions (continuous lines, scale shown on right y axis). Thick symbols (shown on top) correspond to the averaged values ${\overline{r}}_{p}$ reported in Section 2.5. The final averaged values ${\overline{r}}_{p}^{\mathrm{A}}$, ${\overline{r}}_{p}^{\mathrm{N}}$ enter the diffusion equation of NS5A on the ER surface for the corresponding TMS of the two cell lines.

**Figure 3.**Surface mesh of reconstructed ER geometry ${\mathcal{E}}_{1}$. (

**a**) Computational domain used for the simulations of the FRAP experiments of NS5A on (intra)cellular level. Dark blue: unbleached region $\mathcal{U}$, Red: FRAP region $\mathcal{F}$ used for bleaching (covering a surface of 38 $\mathsf{\mu}$m${}^{2}$ in the 2D projection plane as in experiment); cf. Section 2.1 and Section 2.6. (

**b**) Magnification around FRAP ROI $\mathcal{F}$.

**Figure 4.**Simulation of NS5A concentration at the ER surface during a FRAP experiment, screenshot of supplemental movie “S1 Video in supplementary material”. The movie shows the simulation of the diffusion of NS5A on the ER surface, (8). At the beginning, the NS5A concentration is small within the (bleached) FRAP ROI $\mathcal{F}$. During the simulation, the diffusion of NS5A enhances the FRAP ROI concentration again. (The complete equilibrium is not reached within the time which corresponds to the time of the FRAP experiments as within the experimental case [34]). Red indicates high NS5A concentration, blue low concentration. Right hand side: zoom of the zone around the FRAP ROI.

**Figure 5.**FRAP region intensity evaluation: experiment and simulation (computed with $\mathcal{I}$ of (12)), example of NS5A/Alone cell case. The curves depict the uprise of the concentration within the (bleached) FRAP ROI $\mathcal{F}$ for the in vitro and the in silico case. The in silico case inherits two different ways of theoretical description: The case ${r}_{p}^{\mathrm{A}}=0$ (“no psr”) neglects the measurement process induced signal reduction within (8), whereas the other case incorporates the afore estimated non-zero value of ${r}_{p}^{\mathrm{A}}$ (“psr on”, value cf. (11)). The in silico curves are that curves which arise for the estimated optimal value of ${D}_{\mathrm{ns}5\mathrm{a}}^{\mathrm{A}}$ for TMS # 0 of the NS5A/Alone cell case adapted to the reconstructed ER geometry ${\mathcal{E}}_{4}$.

**Figure 6.**FRAP region intensity evaluation: experiment and simulation (computed with $\mathcal{I}$ of (12)), example of NS5A/OtherNSPs cell case. The curves depict the uprise of the concentration within the (bleached) FRAP ROI $\mathcal{F}$ for the in vitro and the in silico case. The in silico case inherits only the ${r}_{p}^{\mathrm{N}}\ne 0$ case ("psr on", value cf. (11)) which models the measurement process induced signal reduction within (8). The in silico curves are that curves which arise for the estimated optimal value of ${D}_{\mathrm{ns}5\mathrm{a}}^{\mathrm{N}}$ for the time series # 5 of the NS5A/OtherNSPs cell case adapted to the reconstructed ER geometry ${\mathcal{E}}_{1}$.

**Figure 7.**Averages for NS5A diffusion constant ${D}_{\mathrm{ns}5\mathrm{a}}$ estimation on the ER surface as described in Section 2.2 and Section 3.3: Analyzed separately for NS5A/Alone (red—${D}_{\mathrm{ns}5\mathrm{a}}^{\mathrm{A}}$) and NS5A/OtherNSPs (green—${D}_{\mathrm{ns}5\mathrm{a}}^{\mathrm{N}}$). Thin points and error bars correspond for (

**a**,

**b**) to estimated values for ${D}_{\mathrm{ns}5\mathrm{a}}$ (shown on x axis) for the combination of single TMS with single geometries (indicated on the left y axis, note different combinations for geometry and TMS). Aggregating over all TMS and ER geometries yields distributions (thin continuous lines, scale shown on right y axis) which are identical in both cases. (

**a**) Each “row” corresponds to one time series combined with all ER geometries. (For example, the left y axis value 2.5 corresponds to the combination of time series #2 and ER geometry ${\mathcal{E}}_{6}$). Half thick symbols (shown in the middle of each time series region) correspond to the averaged values over all geometries ${\overline{D}}_{\mathrm{ns}5\mathrm{a}}{|}_{G}$ for the respective TMS. Aggregating these averages over all TMS yields distributions (thick continuous lines, scale shown on right y axis). Thick symbols (shown on top) correspond to the averaged values ${\overline{D}}_{\mathrm{ns}5\mathrm{a}}$. (

**b**) Each “row” corresponds to one ER geometry combined with all TMS. (For example, the left y value 2.5 corresponds to ER geometry ${\mathcal{E}}_{3}$ and TMS # 5). Half thick symbols (shown in the middle of each ER geometry region) correspond to the averaged values over all TMS ${\overline{D}}_{\mathrm{ns}5\mathrm{a}}{|}_{T}$ for the respective ER geometry as reported in Section 2.6. Aggregating these averages over all ER geometries yields distributions (thick continuous lines, scale shown on right y axis). Thick symbols (shown on top) correspond to the averaged values ${\overline{D}}_{\mathrm{ns}5\mathrm{a}}$. Note: The total averages are identical for (

**a**,

**b**) and are reported in Table 1.

**Figure 8.**Classical 2D simulation of the FRAP process: Screenshot of supplemental movie “S3 Video in supplementary material”. Simulation of FRAP experiment on simple 2D planar continuum geometry. (Red high concentration, blue low concentration). At the beginning, the bleached region has low concentration, but the 2D diffusion refills it again during the process, i.e. the color shifts slowly to red again because from the high concentration unbleached region around, fluorescating NS5A is diffusing inside. The ER structure is neglected.

**Table 1.**Averaged final NS5A diffusion constant ${\overline{D}}_{\mathrm{ns}5\mathrm{a}}$ as described in Section 3.7. The final values are computed by means of the averaging process of the single results (as described in Section 2.7) which was shown graphically within Figure 7. We give results for the case of the use of the ER geometry setups as described in Section 3.1 and Section 3.2, but also for a simplified classical 2D planar consideration, cf. Section 3.6.

Geoms | $\mathit{D}\left[{\left(\mathsf{\mu}\mathbf{m}\right)}^{2}/\mathbf{s}\right]$ | $\mathit{\sigma}\left[{\left(\mathsf{\mu}\mathbf{m}\right)}^{2}/\mathbf{s}\right]$ |
---|---|---|

NS5A/Alone cell type | ||

plane 2D | 0.014815 | 0.001546 |

ER surface | 0.033307 | 0.001142 |

NS5A/OtherNSPs cell type | ||

plane 2D | 0.003873 | 0.000695 |

ER surface | 0.007696 | 0.000353 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Knodel, M.M.; Nägel, A.; Reiter, S.; Vogel, A.; Targett-Adams, P.; McLauchlan, J.; Herrmann, E.; Wittum, G.
Quantitative Analysis of Hepatitis C NS5A Viral Protein Dynamics on the ER Surface. *Viruses* **2018**, *10*, 28.
https://doi.org/10.3390/v10010028

**AMA Style**

Knodel MM, Nägel A, Reiter S, Vogel A, Targett-Adams P, McLauchlan J, Herrmann E, Wittum G.
Quantitative Analysis of Hepatitis C NS5A Viral Protein Dynamics on the ER Surface. *Viruses*. 2018; 10(1):28.
https://doi.org/10.3390/v10010028

**Chicago/Turabian Style**

Knodel, Markus M., Arne Nägel, Sebastian Reiter, Andreas Vogel, Paul Targett-Adams, John McLauchlan, Eva Herrmann, and Gabriel Wittum.
2018. "Quantitative Analysis of Hepatitis C NS5A Viral Protein Dynamics on the ER Surface" *Viruses* 10, no. 1: 28.
https://doi.org/10.3390/v10010028