# Competition of SARS-CoV-2 Variants in Cell Culture and Tissue: Wins the Fastest Viral Autowave

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## Abstract

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## 1. Introduction

- Can these experiments be described by the mathematical model (the simplest possible one)?
- Could the situations of one strain overtaking another strain after some prolonged time interval be observed qualitatively if the spatial effects are taken into account?
- Could spatial effects be very important: for example, could they result in any new information that cannot be obtained in the homogeneous system?

**Figure 2.**Viral replication kinetics: (

**A**,

**B**) in bronchus and lung cells [1]; (

**C**,

**D**) in nasal epithelial and human lung cells [2] (data were extracted from Figure 1 in preprint of [1] and from Figure 1 in preprint of [2]). (

**C**) Open markers show E gene concentration (crossed markers) divided by 1000 to approximately match the corresponding PFU concentration.

## 2. Methods

#### 2.1. Model Description and Governing Equations

#### 2.2. Steady Travelling Solution

#### 2.3. Two-Strain Model with Competition for Cells

#### 2.4. Model Parameters and Comparison with Experimental Data

- Arbitrary non-dimensional values (typically, of the order of 1) for plotting the explanatory graphs when deriving analytical formulas in the Appendices. For this set, in numerical calculations we take $L=200,\phantom{\rule{4pt}{0ex}}{U}_{0}=1$.
- Parameters which correspond to the SARS-CoV-2 variants Delta and Omicron replicating in human nasal epithelial cultures (hNECs) in the in vitro experiment [2] (see Table 1). We estimated them from physical reasons (e.g., D), derived/estimated from experimental results and kept fixed (e.g., ${V}_{0}$, ${\tau}_{a}$), or varied under the strict limitations as described below:
- The diffusion coefficient D was estimated using the Stockes–Einstein formula at 300 K assuming virus diameter 100 nm [16] and water viscosity.
- The experimental in vitro system in [2] is considered as homogeneous, because (a) only average concentrations are presented in that article, (b) the full size of the experimental system is 6.5 mm [17], which is comparable to the width of the front in our numerical calculations (about 1–3 mm, see Figure 5 below), and (c) substantial convection should be expected during inoculation and everyday sampling.
- Since the virus concentration determined by RT-qPCR for the E gene in [2] (see Figure 2C,D) was approximately 1000 times greater than the PFU concentration for all available time points and cell lineages (crossed vs. filled markers), that is only 1 out of 1000 viral particles was able to effectively infect a cell and reproduce with its help, we compared the model variable $V\left(t\right)$ with the united data for $\frac{\mathrm{PFU}}{\mathrm{mL}}$ (filled markers) and $\frac{\mathrm{E}\phantom{\rule{4.pt}{0ex}}\mathrm{genes}}{\mathrm{mL}}/1000$ (open markers). In particular, we estimated ${V}_{0}$ as $\frac{\mathrm{E}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{genes}}{\mathrm{mL}}(t=0)/1000$.
- Initial cell concentration ${U}_{0}$ was limited in the range $[{10}^{4},{10}^{6}]\phantom{\rule{4pt}{0ex}}\frac{\mathrm{cells}}{\mathrm{mL}}$.
- The characteristic time of infection (${\tau}_{a}$) was assumed to be 1 h for both strains. Characteristic times if $E\to I$ transition ($1/{\gamma}_{1,2}$) and cell death ($1/{\beta}_{1,2}$) were limited as 2 h and 0.1 h from below, respectively. Virus death times ($1/{\sigma}_{1,2}$) were limited in the range [0.1, 100] h.
- To avoid large differences between the corresponding parameters of Delta and Omicron variants, the rations $\frac{{\beta}_{1}}{{\beta}_{2}}$, $\frac{{\gamma}_{1}}{{\gamma}_{2}}$, $\frac{{\sigma}_{1}}{{\sigma}_{2}}$ and $\frac{{N}_{1}}{{N}_{2}}$ were limited to the range of from $\frac{1}{10}$ to 10.
- After the fitting, all parameters were rounded up to 1 decimal digit, and those that were close to each other were set as equal.
- For the obtained set of kinetic parameters, in spatial numerical calculations (with diffusion) we set L = 2 cm or more to be able to track the transition of autowave to the steady propagation regime.

#### 2.5. Numerical Methods

## 3. Results

#### 3.1. Virus Replication Number Provides the Condition for the Infection Progression, ${R}_{v}>1$

#### 3.2. Estimates of the Steady Wave Speed

#### 3.3. Equations for the Final Concentration of Intact Cells and the Total Spatial Viral Load

#### 3.4. Homogeneous Case without Competition: Omicron Is “Quick” and Wins the Start but Delta Can Overtake It after 1–2 Days’ Lag

#### 3.5. Spatially Distributed Case without Competition: Omicron Can Win the Race despite Low Concentration and ${R}_{v}$

#### 3.6. Spatially Distributed Case with Competition: Omicron Can Win and Completely Suppress Delta

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

URT | upper respiratory tract |

LRT | lower respiratory tract |

hNECs | culture of human nasal epithelial cells |

KPP or KPPF equation | Kolmogorov–Petrovskii–Piskunov–Fisher equation |

## Appendix A

#### Appendix A.1. Condition for Existence of a Positive Solution to Equation (5)

#### Appendix A.2. Estimates of the Steady Wave Speed

**Figure A4.**Solution to Equation (A7) (solid lines) and two its approximations in the ${c}_{2}\ge 0$ region giving ${c}_{2}^{*}$ (dotted line) and ${c}_{2}^{**}$ (dashed line). Parameters are the same as in Figure A2. Actual differences between solutions are hardly distinguished at these parameter values, however at larger ${R}_{v}$ all differences increase (see Figure A5).

#### Appendix A.3. Derivation of Equations for the Final Concentration of Intact Cells and the Total Spatial Viral Load

## References

- Hui, K.P.; Ho, J.C.; Cheung, M.c.; Ng, K.c.; Ching, R.H.; Lai, K.l.; Kam, T.T.; Gu, H.; Sit, K.Y.; Hsin, M.K.; et al. SARS-CoV-2 Omicron variant replication in human bronchus and lung ex vivo. Nature
**2022**, 603, 715–720. [Google Scholar] [CrossRef] [PubMed] - Peacock, T.P.; Brown, J.C.; Zhou, J.; Thakur, N.; Newman, J.; Kugathasan, R.; Sukhova, K.; Kaforou, M.; Bailey, D.; Barclay, W.S. The SARS-CoV-2 variant, Omicron, shows rapid replication in human primary nasal epithelial cultures and efficiently uses the endosomal route of entry. bioRxiv (version posted January 3)
**2022**. preprint. [Google Scholar] [CrossRef] - Murray, J. Mathematical Biology. I. An Introduction, 3rd ed.; Springer: New York, NY, USA, 2002. [Google Scholar] [CrossRef]
- Murray, J. Mathematical Biology. II. Spatial Models and Biomedical Applications, 3rd ed.; Springer: New York, NY, USA, 2003. [Google Scholar] [CrossRef]
- Flerlage, T.; Boyd, D.F.; Meliopoulos, V.; Thomas, P.G.; Schultz-Cherry, S. Influenza virus and SARS-CoV-2: Pathogenesis and host responses in the respiratory tract. Nat. Rev. Microbiol.
**2021**, 19, 425–441. [Google Scholar] [CrossRef] [PubMed] - V’kovski, P.; Kratzel, A.; Steiner, S.; Stalder, H.; Thiel, V. Coronavirus biology and replication: Implications for SARS-CoV-2. Nat. Rev. Microbiol.
**2021**, 19, 155–170. [Google Scholar] [CrossRef] [PubMed] - Hou, Y.J.; Okuda, K.; Edwards, C.E.; Martinez, D.R.; Asakura, T.; Dinnon, K.H.; Kato, T.; Lee, R.E.; Yount, B.L.; Mascenik, T.M.; et al. SARS-CoV-2 Reverse Genetics Reveals a Variable Infection Gradient in the Respiratory Tract. Cell
**2020**, 182, 429–446.e14. [Google Scholar] [CrossRef] [PubMed] - Ga̧secka, A.; Borovac, J.A.; Guerreiro, R.A.; Giustozzi, M.; Parker, W.; Caldeira, D.; Chiva-Blanch, G. Thrombotic Complications in Patients with COVID-19: Pathophysiological Mechanisms, Diagnosis, and Treatment. Cardiovasc. Drugs Ther.
**2021**, 35, 215–229. [Google Scholar] [CrossRef] [PubMed] - WHO. Tracking SARS-CoV-2 Variants. Available online: https://www.who.int/en/activities/tracking-SARS-CoV-2-variants (accessed on 13 June 2022).
- Dashkevich, N.M.; Ovanesov, M.V.; Balandina, A.N.; Karamzin, S.S.; Shestakov, P.I.; Soshitova, N.P.; Tokarev, A.A.; Panteleev, M.A.; Ataullakhanov, F.I. Thrombin activity propagates in space during blood coagulation as an excitation wave. Biophys. J.
**2012**, 103, 2233–2240. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ataullakhanov, F.; Guriia, G. Spatial aspects of the dynamics of blood coagulation. I. Hypothesis. Biophysics
**1994**, 39, 89–96. [Google Scholar] - Tokarev, A.; Ratto, N.; Volpert, V. Mathematical modeling of thrombin generation and wave propagation: From simple to complex models and backwards. In BIOMAT 2018 International Symposium on Mathematical and Computational Biology; Springer: Cham, Switzerland, 2019. [Google Scholar] [CrossRef]
- Bocharov, G.; Meyerhans, A.; Bessonov, N.; Trofimchuk, S.; Volpert, V. Interplay between reaction and diffusion processes in governing the dynamics of virus infections. J. Theor. Biol.
**2018**, 457, 221–236. [Google Scholar] [CrossRef] [PubMed] - Mahiout, L.; Mozokhina, A.; Tokarev, A.; Volpert, V. Virus replication and competition in a cell culture: Application to the SARS-CoV-2 variants. Appl. Math. Lett.
**2022**, 113, 108217. [Google Scholar] [CrossRef] [PubMed] - Sender, R.; Bar-On, Y.M.; Gleizer, S.; Bernshtein, B.; Flamholz, A.; Phillips, R.; Milo, R. The total number and mass of SARS-CoV-2 virions. Proc. Natl. Acad. Sci. USA
**2021**, 118, 1–9. [Google Scholar] [CrossRef] [PubMed] - Zhu, N.; Zhang, D.; Wang, W.; Li, X.; Yang, B.; Song, J.; Zhao, X.; Huang, B.; Shi, W.; Lu, R.; et al. A Novel Coronavirus from Patients with Pneumonia in China, 2019. N. Engl. J. Med.
**2020**, 382, 727–733. [Google Scholar] [CrossRef] [PubMed] - Epithelix. FAQ, Epithelix’s Tissues. Available online: https://www.epithelix.com/faq (accessed on 13 June 2022).
- Hoops, S.; Sahle, S.; Gauges, R.; Lee, C.; Pahle, J.; Simus, N.; Singhal, M.; Xu, L.; Mendes, P.; Kummer, U.; et al. COPASI—A COmplex PAthway SImulator. Bioinformatics
**2006**, 22, 3067–3074. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tokarev, A.A. Velocity–Amplitude Relationship in the Gray–Scott Autowave Model in Isolated Conditions. ACS Omega
**2019**, 4, 14430–14438. [Google Scholar] [CrossRef] [PubMed] - Hairer, E.; Nørsett, S.; Wanner, G. Solving Ordinary Differential Equations I. Nonstiff Problems (Russian Translation, 1990); Mir: Moscow, Russia; Springer: Berlin/Heidelberg, Germany; New York, NY, USA; London, UK; Paris, France; Tokyo, Japan, 1987. [Google Scholar]

**Figure 1.**Schemes depicting the transitions of cells between different infected states (black arrows), virus production (red arrows), and death of cells and virus (gray arrows).

**U**—uninfected cells;

**E**—“exposed” cells (infected but still not shedding the virus);

**I**—infected cells shedding the virus;

**V**—free virus particles. (

**A**) One-strain model. (

**B**) Two-strain model with competition of strains for the uninfected cells.

**Figure 3.**Graphical solution to Equation (8a) for two ${R}_{v}$ values. Thin line—$\mathrm{ln}\phantom{\rule{4pt}{0ex}}w$, bold lines—${R}_{v}\xb7(w-1)$.

**Figure 4.**

**Left**: Replication of the SARS-CoV-2 variants Delta (green + red) and Omicron (blue + cyan) in hNECs in the experiment [2] (markers, data are taken from Figure 2C as described in Section 2.4) and in the homogeneous version of the model (1) (lines).

**Right**: concentrations of cells in the same solutions. Figures are screenshots from COPASI. Model parameters are listed in Table 1.

**Figure 5.**Spatial replication and propagation of Delta (dashed lines) and Omicron (solid lines) variants without competition for cells (

**A**,

**B**) and with competition (

**C**,

**D**). (

**A**) Two uncoupled models given by Equation (1) were solved simultaneously. (

**C**) Equation (3) were solved. (

**A**,

**C**) Solutions corresponding to the same distance (1.5 mm) travelled by Omicron autowave ${V}_{2}\left(x\right)$ are shown. (

**B**,

**D**) Logarithms of integrated virus concentrations vs. time are shown. $\rho =1$, all other parameters are listed in Table 1.

Parameter | Dimension | Delta | Omicron |
---|---|---|---|

Equal for both strains: | |||

${V}_{0}$ | $\frac{\mathrm{virions}}{\mathrm{mL}}$ | $5\times {10}^{3}$ | |

${U}_{0}$ | $\frac{\mathrm{cell}}{\mathrm{mL}}$ | $1\times {10}^{6}$ | |

${\tau}_{a}$ | h | 1 | |

$a\equiv \frac{1}{{\tau}_{a}\xb7{U}_{0}}$ | $\frac{\mathrm{mL}}{\mathrm{cell}\xb7\mathrm{h}}$ | $1\times {10}^{-6}$ | |

D | $\frac{{\mathrm{cm}}^{2}}{\mathrm{h}}$ | $1\times {10}^{-4}$ | |

$\gamma ,\beta $ | ${\mathrm{h}}^{-1}$ | ${\gamma}_{1}={\beta}_{1}=0.04$ | ${\gamma}_{2}={\beta}_{2}=0.2$ |

$\sigma $ | ${\mathrm{h}}^{-1}$ | ${\sigma}_{1}=0.01$ | ${\sigma}_{2}=0.06$ |

N | $1\equiv \frac{\mathrm{virions}}{\mathrm{cell}}$ | ${N}_{1}=50$ | ${N}_{2}=10$ |

${R}_{v}=\frac{N-1}{{\tau}_{a}\sigma}$ | 1 | ${R}_{v,1}=4900$ | ${R}_{v,2}=150$ |

**Table 2.**Autowave speed $c\phantom{\rule{4pt}{0ex}}\left(\frac{\mathrm{cm}}{\mathrm{h}}\right)$, spatially integrated viral concentration $\tilde{v}\phantom{\rule{4pt}{0ex}}\left({10}^{6}\frac{\mathrm{cells}}{{\mathrm{cm}}^{2}}\right)$ and maximal viral concentration ${v}_{max}\phantom{\rule{4pt}{0ex}}\left({10}^{6}\frac{\mathrm{cells}}{\mathrm{mL}}\right)$ for Delta and Omicron steady spatial autowave propagation.

Delta | Omicron | Delta | Omicron | |
---|---|---|---|---|

$\mathbf{\rho}=\mathbf{1}$ | $\mathbf{\rho}=\mathbf{0}$ | |||

Numerical c | 0.0049 | 0.0062 | 0.0094 | 0.011 |

${c}_{min}$ as $\sqrt{min\left[F\right(\mu \left)\right]}$, Equation (6) | 0.0038 | 0.0058 | 0.0093 | 0.011 |

c from Equation (7b) | 0.0069 | 0.0088 | 0.016 | 0.015 |

Numerical $\tilde{v}$ | 24 | 0.93 | 46 | 1.88 |

$\tilde{v}=\frac{\sqrt{min\left[F\right(\mu \left)\right]}}{a}{R}_{v}$, Equations (6) and (9b) | 19 | 0.87 | 45 | 1.65 |

Numerical ${v}_{max}$ | 26 | 4.3 | 27 | 4.9 |

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**MDPI and ACS Style**

Tokarev, A.; Mozokhina, A.; Volpert, V. Competition of SARS-CoV-2 Variants in Cell Culture and Tissue: Wins the Fastest Viral Autowave. *Vaccines* **2022**, *10*, 995.
https://doi.org/10.3390/vaccines10070995

**AMA Style**

Tokarev A, Mozokhina A, Volpert V. Competition of SARS-CoV-2 Variants in Cell Culture and Tissue: Wins the Fastest Viral Autowave. *Vaccines*. 2022; 10(7):995.
https://doi.org/10.3390/vaccines10070995

**Chicago/Turabian Style**

Tokarev, Alexey, Anastasia Mozokhina, and Vitaly Volpert. 2022. "Competition of SARS-CoV-2 Variants in Cell Culture and Tissue: Wins the Fastest Viral Autowave" *Vaccines* 10, no. 7: 995.
https://doi.org/10.3390/vaccines10070995