# Sensitivity of SARS-CoV-2 Life Cycle to IFN Effects and ACE2 Binding Unveiled with a Stochastic Model

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## Abstract

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## 1. Introduction

- The cell-to-cell variability in SARS-CoV-2 progeny production;
- The multiplicity of single cell infection;
- The probability of infection;
- The local sensitivity of the virus production to specific life-cycle steps;
- The impact of type I IFN; and
- The effect of the RBD-ACE2 binding affinity.

## 2. Methods

#### 2.1. Deterministic Equations of SARS-CoV-2 Single Cell Infection

**Entry.**This phase is split into four stages:

- (i)
- Binding of the receptor-binding domain (RBD) of the viral spike (S) protein to ACE2 receptor (Equation (1));
- (ii)
- Priming of the virus S protein at the host cell surface by the transmembrane protease serine 2 (TMPRSS2), which leads to cleavage of the S proteins at the S1/S2 and S2 sites (Equation (2));
- (iii)
- Fusion at the cellular or endosomal membrane (Equation (3)); and
- (iv)
- Release and uncoating of viral genomic RNA (Equation (4)).

- ${x}_{1}=\left[{\mathrm{V}}_{\mathrm{free}}\right]$ is the number of free virions outside the cell membrane;
- ${x}_{2}=\left[{\mathrm{V}}_{\mathrm{bound}}\right]$ is the number of virions bound to ACE2 and activated by TMPRSS2;
- ${x}_{3}=\left[{\mathrm{V}}_{\mathrm{endosome}}\right]$ is the number of virions in endosomes; and
- ${x}_{4}=\left[{\mathrm{gRNA}}_{(+)}\right]$ is the number of ss-positive sense genomic RNA.

**Genome transcription and replication.**This phase is split into three stages:

- (v)
- The translation of the released genomic RNA into viral polyproteins (pp1a, pp1ab) which generate a number of non-structural proteins (nsp1-16), including nsp-12, which encodes the RNA-dependent RNA polymerase (RdRp) (Equation (5));
- (vi)
- The RdRp-dependent transcription of a negative sense subgenomic and genomic RNAs (Equation (6)); and
- (vii)
- The RdRp-dependent transcription of a positive sense subgenomic and genomic RNAs (Equation (7)).

- ${x}_{5}=\left[\mathrm{NSP}\right]$ is the number of non-structural proteins;
- ${x}_{6}=\left[{\mathrm{gRNA}}_{(-)}\right]$ is the number of negative sense genomic and subgenomic RNAs; and
- ${x}_{7}=\left[\mathrm{gRNA}\right]$ is the number of positive sense genomic and subgenomic RNAs.

**Translation of structural and accessory proteins.**This phase is split into two major stages:

- (viii)
- The translation of the structural nucleocapsid protein N from subgenomic RNAs by cytosolic ribosomes (Equation (8));
- (ix)
- The translation of the structural proteins S, envelope E, and membrane M proteins characterised in the model by their total abundance $\left[\mathrm{SP}\right]$, which takes place in the endoplasmic reticulum (ER) (Equation (9)).

- ${x}_{8}=\left[\mathrm{N}\right]$ is the number of N proteins per virion and
- ${x}_{9}=\left[\mathrm{SP}\right]$ is the total abundance of the structural proteins S, envelope E, and membrane M proteins.

**Assembly and release of virions.**This final phase is split into three major stages:

- (x)
- The binding of N proteins and gRNA, resulting in nucleocapsid formation (viral RNA genome coated with N protein) (Equation (10));
- (xi)
- The assembly of virions via encapsulating N-RNA complexes at the ER–Golgi compartment (Equation (11)); and
- (xii)
- The release of the assembled new virions by the infected cell via exocytosis, budding, or cell death (Equation (12)).

- ${x}_{10}=[\mathrm{N}-\mathrm{gRNA}]$ is the number of ribonucleoprotein molecules;
- ${x}_{11}=\left[{\mathrm{V}}_{\mathrm{assembled}}\right]$ is the number of assembled virions; and
- ${x}_{12}=\left[{\mathrm{V}}_{\mathrm{released}}\right]$ is the number of released virions.

#### 2.2. Quantification of SARS-CoV-2 Replication Parameters

#### 2.3. The Stochastic Model

#### 2.4. Stochastic Modelling Algorithm

- At every interval between the reactions, two uniformly distributed random numbers ${r}_{1},{r}_{2}$ on $(0,1)$ are generated. The first number gives the next timestep $\delta t=-(ln{r}_{1})/A$, where $A={\sum}_{m=1}^{M}{a}_{m}$; M is the number of reactions in the Markov chain; and ${a}_{m}$ is the propensity of the mth reaction: ${a}_{m}\mathrm{d}t$ is the probability that the mth reaction occurs in time-interval $\mathrm{d}t$. The second random number determines the next reaction index p: the smallest integer satisfying ${A}_{p}\ge A{r}_{2}$, where ${A}_{p}={\sum}_{m=1}^{p}{a}_{m}$. As we have to search among $M=26$ reactions, a binary search is employed to accelerate finding the reaction index p (see [49]). At the end of the step, the pth transition is performed, i.e., the state vector $\mathbf{x}$ is updated in accordance with Table 2.
- After updating the state vector, the propensities should be updated as well. Here, to accelerate the computation, the propensities are updated only for those reactions in which ${a}_{m}$ depends on the updated components in the given step. For this purpose, a special array is prepared in which propensities to be updated are indicated for given component ${x}_{n}$ and another similar array for every reaction m.
- The process is terminated as soon as the current time exceeds the maximal time ${t}_{\mathrm{final}}$ set in advance. To decrease the amount of stored information, the values of the state vector are stored at a uniform time-grid with the preset timestep $\Delta t$.

#### 2.5. Local Stochastic Sensitivity Analysis

## 3. Results

#### 3.1. Deterministic Versus Stochastic Dynamics of SARS-CoV-2 Replication

#### 3.2. Variability in Net Virus Production

#### 3.3. Variability of the Individual Reaction Products

#### 3.4. Probability of Productive Infection

#### 3.5. Efficiency of Life Cycle

#### 3.6. Sensitivity Analysis of the Model Parameters

- Degradation of the positive-sense vRNA in cytoplasm (${d}_{\mathrm{gRNA}}$);
- Threshold number of non-structural proteins enhancing vRNA transcription (${K}_{\mathrm{NSP}}$);
- Translation rate of non-structural proteins (${k}_{\mathrm{transl}}{f}_{\mathrm{ORF}1}$);
- Degradation rate of extracellular virions (${d}_{\mathrm{V}}$); and
- Assembly rate of structural proteins (${k}_{\mathrm{assembl}}{n}_{\mathrm{SP}}$).

#### 3.7. Inhibitory Effect of Type I IFN

- (a)
- The rate constant of translation of released genomic RNA into viral polyproteins pp1a and pp1ab (${k}_{\mathrm{transl}}{f}_{\mathrm{ORF}1}$);
- (b)
- The degradation rate constant of RNA (${d}_{\mathrm{gRNA}},\phantom{\rule{4pt}{0ex}}{d}_{{\mathrm{gRNA}}_{(-)}}$); and
- (c)
- = (a) + (b): simultaneous variation in both parameters.

#### 3.8. Effect of Binding Affinity

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

RT | Reverse Transcription |

ODE | ordinary differential equation |

MC | Markov chain |

MCMC | Markov chain Monte Carlo (method) |

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**Figure 1.**Biochemical scheme of the SARS-CoV-2 replication cycle. Targets of type I IFN-mediated inhibition of virus replication are marked.

**Figure 2.**Examples of stochastic realisations for $\left[{\mathrm{V}}_{\mathrm{free}}\right]\left(0\right)=10$. The black curves indicate the solution of the deterministic model.

**Figure 3.**Normalised histograms for the released virions number for $\left[{\mathrm{V}}_{\mathrm{free}}\right]\left(0\right)=5$ (

**left**) and $\left[{\mathrm{V}}_{\mathrm{free}}\right]\left(0\right)=10$ (

**right**). The red line shows the approximation of the histogram by the Gamma distribution fitted to the histogram by the least squares method.

**Figure 4.**Evolution of the confidence intervals for all 12 species participating in the SARS-CoV-2 replication for $\left[{\mathrm{V}}_{\mathrm{free}}\right]\left(0\right)=5$. The green, red, and black lines indicate the mean, median, and the deterministic solution, respectively.

**Figure 5.**Evolution of the confidence intervals for all 12 components participating in the SARS-CoV-2 replication for $\left[{\mathrm{V}}_{\mathrm{free}}\right]\left(0\right)=10$. The green, red, and black lines indicate the mean, median, and the deterministic solution, respectively.

**Figure 6.**(

**Left**) Dependence of confidence intervals, sample mean (green) and median (red) estimates, and the deterministic solution (black) on the initial number of free virions per cell $\left[{\mathrm{V}}_{\mathrm{free}}\right]\left(0\right)$ at $t=24$ h. (

**Right**) Probability for productive infection of the target cell in relation to the initial number of free virions per cell (MOI).

**Figure 7.**(

**Left**) Kinetics of the total number of new virions secreted by an infected cell (dotted lines) for different MOIs (explained by the colour code) and the kinetics of virions release (solid lines) computed by the deterministic model [19]. (

**Right**) The life cycle efficiency computed by the deterministic model (the red curve with circles). The mean, median, and the confidence intervals of the life cycle efficiency computed by the stochastic model (explained in the legend).

**Figure 8.**Local sensitivity indices for the number of released virions at 24 h computed with the stochastic model. The significant indices with greater values than the self-distance for the sets with baseline model parameters are marked by red.

**Figure 9.**Type I IFN-mediated effects on the probability of non-degenerate infection (

**left**) and the efficient reproduction number for MOI ranging from 1 to 15 (

**right**). The ascending/descending arrows in the legend show the four-fold increase/decrease in the indicated parameters.

**Figure 10.**Effect of binding of SARS-CoV-2 to ACE2 on the probability of non-degenerate infection (

**left**) and the efficient reproduction number (

**right**) for MOI ranging from 1 to 15. The ascending/descending arrows in the legend show the 10-fold increase/decrease in the indicated parameters.

${\mathit{k}}_{\mathrm{bind}}$ | = 12 h${}^{-1}$ | ${\mathit{d}}_{\mathrm{V}}$ | $=0.12$ h${}^{-1}$ | ${\mathit{K}}_{\mathrm{NSP}}$ | = 100 |

${k}_{\mathrm{diss}}$ | $=0.61$ h${}^{-1}$ | ${d}_{\mathrm{endosome}}$ | $=0.06$ h${}^{-1}$ | ${K}_{\mathrm{N}}$ | $=5\times {10}^{6}$ |

${k}_{\mathrm{fuse}}$ | $=0.5$ h${}^{-1}$ | ${d}_{\mathrm{gRNA}}$ | $=0.2$ h${}^{1}$ | ${K}_{{\mathrm{V}}_{\mathrm{rel}}}$ | = 1000 |

${k}_{\mathrm{uncoat}}$ | $=0.5$ h${}^{-1}$ | ${d}_{\mathrm{NSP}}$ | $=0.069$ h${}^{-1}$ | ${f}_{\mathrm{ORF}1}$ | = 1/21,000 |

${k}_{{\mathrm{tr}}_{(-)}}$ | = 3 h${}^{-1}$ | ${d}_{{\mathrm{gRNA}}_{(-)}}$ | $=0.1$ h${}^{-1}$ | ${f}_{\mathrm{N}}$ | $=1/1200$ |

${k}_{{\mathrm{tr}}_{(+)}}$ | = 1000 h${}^{-1}$ | ${d}_{\mathrm{N}}$ | $=0.023$ h${}^{-1}$ | ${f}_{\mathrm{SP}}$ | = 1/10,000 |

${k}_{\mathrm{complex}}$ | $=0.4$ h${}^{-1}$ | ${d}_{\mathrm{SP}}$ | $=0.044$ h${}^{-1}$ | ||

${k}_{\mathrm{transl}}$ | $=4.536\times {10}^{4}$ h${}^{-1}$ | ${d}_{\mathrm{N}-\mathrm{gRNA}}$ | $=0.2$ h${}^{-1}$ | ${n}_{\mathrm{N}}$ | = 456 |

${k}_{\mathrm{assemb}}$ | = 1 h${}^{-1}$ | ${d}_{\mathrm{assemb}}$ | $=0.06$ h${}^{-1}$ | ${n}_{\mathrm{SP}}$ | = 2000 |

${k}_{\mathrm{release}}$ | = 8 h${}^{-1}$ |

**Table 2.**The Markov chain: the list of individual reactions, the corresponding state transitions, and the propensities of reactions.

m | Reaction (Transition) | Propensity, ${\mathit{a}}_{\mathit{m}}$ | Equations |
---|---|---|---|

1 | $\phantom{\rule{3.0pt}{0ex}}{x}_{1}\to \phantom{\rule{3.0pt}{0ex}}{x}_{1}-1,\phantom{\rule{0.277778em}{0ex}}{x}_{2}\to {x}_{2}+1$ | ${k}_{\mathrm{bind}}{x}_{1}$ | (1), (2) |

2 | $\phantom{\rule{3.0pt}{0ex}}{x}_{1}\to \phantom{\rule{3.0pt}{0ex}}{x}_{1}+1,\phantom{\rule{0.277778em}{0ex}}{x}_{2}\to {x}_{2}-1$ | ${k}_{\mathrm{diss}}{x}_{2}$ | (1), (2) |

3 | $\phantom{\rule{3.0pt}{0ex}}{x}_{1}\to \phantom{\rule{3.0pt}{0ex}}{x}_{1}-1$ | ${d}_{\mathrm{V}}{x}_{1}$ | (1) |

4 | $\phantom{\rule{3.0pt}{0ex}}{x}_{2}\to \phantom{\rule{3.0pt}{0ex}}{x}_{2}-1,\phantom{\rule{0.277778em}{0ex}}{x}_{3}\to {x}_{3}+1$ | ${k}_{\mathrm{fuse}}{x}_{2}$ | (2), (3) |

5 | $\phantom{\rule{3.0pt}{0ex}}{x}_{2}\to \phantom{\rule{3.0pt}{0ex}}{x}_{2}-1$ | ${d}_{\mathrm{V}}{x}_{2}$ | (2) |

6 | $\phantom{\rule{3.0pt}{0ex}}{x}_{3}\to \phantom{\rule{3.0pt}{0ex}}{x}_{3}-1,\phantom{\rule{0.277778em}{0ex}}{x}_{4}\to {x}_{4}+1$ | ${k}_{\mathrm{uncoat}}{x}_{3}$ | (3), (4) |

7 | $\phantom{\rule{3.0pt}{0ex}}{x}_{3}\to \phantom{\rule{3.0pt}{0ex}}{x}_{3}-1$ | ${d}_{\mathrm{endosome}}{x}_{3}$ | (3) |

8 | $\phantom{\rule{3.0pt}{0ex}}{x}_{4}\to \phantom{\rule{3.0pt}{0ex}}{x}_{4}-1$ | ${d}_{\mathrm{gRNA}}{x}_{4}$ | (4) |

9 | $\phantom{\rule{3.0pt}{0ex}}{x}_{5}\to \phantom{\rule{3.0pt}{0ex}}{x}_{5}+1$ | ${k}_{\mathrm{transl}}{f}_{\mathrm{ORF}1}{x}_{4}$ | (5) |

10 | $\phantom{\rule{3.0pt}{0ex}}{x}_{5}\to \phantom{\rule{3.0pt}{0ex}}{x}_{5}-1$ | ${d}_{\mathrm{NSP}}{x}_{5}$ | (5) |

11 | $\phantom{\rule{3.0pt}{0ex}}{x}_{6}\to \phantom{\rule{3.0pt}{0ex}}{x}_{6}+1$ | ${k}_{{\mathrm{tr}}_{(-)}}{\theta}_{\mathrm{RdRp}}{x}_{4}$ | (6) |

12 | $\phantom{\rule{3.0pt}{0ex}}{x}_{6}\to \phantom{\rule{3.0pt}{0ex}}{x}_{6}-1$ | ${d}_{{\mathrm{gRNA}}_{(-)}}{x}_{6}$ | (6) |

13 | $\phantom{\rule{3.0pt}{0ex}}{x}_{7}\to \phantom{\rule{3.0pt}{0ex}}{x}_{7}+1$ | ${k}_{{\mathrm{tr}}_{(+)}}{\theta}_{\mathrm{RdRp}}{x}_{6}$ | (7) |

14 | $\phantom{\rule{3.0pt}{0ex}}{x}_{7}\to \phantom{\rule{3.0pt}{0ex}}{x}_{7}-1,\phantom{\rule{0.277778em}{0ex}}{x}_{10}\to {x}_{10}+1$ | ${k}_{\mathrm{complex}}{\theta}_{\mathrm{complex}}{x}_{7}$ | (7), (10) |

15 | $\phantom{\rule{3.0pt}{0ex}}{x}_{8}\to \phantom{\rule{3.0pt}{0ex}}{x}_{8}-1$ | ${n}_{\mathrm{N}}{k}_{\mathrm{complex}}{\theta}_{\mathrm{complex}}{x}_{7}$ | (8) |

16 | $\phantom{\rule{3.0pt}{0ex}}{x}_{7}\to \phantom{\rule{3.0pt}{0ex}}{x}_{7}-1$ | ${d}_{\mathrm{gRNA}}{x}_{7}$ | (7) |

17 | $\phantom{\rule{3.0pt}{0ex}}{x}_{8}\to \phantom{\rule{3.0pt}{0ex}}{x}_{8}+1$ | ${k}_{\mathrm{transl}}{f}_{\mathrm{N}}{x}_{7}$ | (8) |

18 | $\phantom{\rule{3.0pt}{0ex}}{x}_{8}\to \phantom{\rule{3.0pt}{0ex}}{x}_{8}-1$ | ${d}_{\mathrm{N}}{x}_{8}$ | (8) |

19 | $\phantom{\rule{3.0pt}{0ex}}{x}_{9}\to \phantom{\rule{3.0pt}{0ex}}{x}_{9}+1$ | ${k}_{\mathrm{transl}}{f}_{\mathrm{SP}}{x}_{7}$ | (8) |

20 | $\phantom{\rule{3.0pt}{0ex}}{x}_{9}\to \phantom{\rule{3.0pt}{0ex}}{x}_{9}-1$ | ${n}_{\mathrm{SP}}{k}_{\mathrm{assemb}}{\theta}_{\mathrm{assemb}}{x}_{10}$ | (9) |

21 | $\phantom{\rule{3.0pt}{0ex}}{x}_{9}\to \phantom{\rule{3.0pt}{0ex}}{x}_{9}-1$ | ${d}_{\mathrm{SP}}{x}_{9}$ | (9) |

22 | ${x}_{10}\to {x}_{10}-1,\phantom{\rule{0.277778em}{0ex}}{x}_{11}\to {x}_{11}+1$ | ${k}_{\mathrm{assemb}}{\theta}_{\mathrm{assemb}}{x}_{10}$ | (10), (11) |

23 | ${x}_{10}\to {x}_{10}-1$ | ${d}_{\mathrm{N}}{x}_{10}$ | (10) |

24 | ${x}_{11}\to {x}_{11}-1$,${x}_{12}\to {x}_{12}+1$ | ${k}_{\mathrm{release}}{x}_{11}$ | (11), (12) |

25 | ${x}_{11}\to {x}_{11}-1$ | ${d}_{\mathrm{assemb}}{x}_{11}$ | (11) |

26 | ${x}_{12}\to {x}_{12}-1$ | ${d}_{\mathrm{V}}{x}_{12}$ | (12) |

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## Share and Cite

**MDPI and ACS Style**

Sazonov, I.; Grebennikov, D.; Meyerhans, A.; Bocharov, G. Sensitivity of SARS-CoV-2 Life Cycle to IFN Effects and ACE2 Binding Unveiled with a Stochastic Model. *Viruses* **2022**, *14*, 403.
https://doi.org/10.3390/v14020403

**AMA Style**

Sazonov I, Grebennikov D, Meyerhans A, Bocharov G. Sensitivity of SARS-CoV-2 Life Cycle to IFN Effects and ACE2 Binding Unveiled with a Stochastic Model. *Viruses*. 2022; 14(2):403.
https://doi.org/10.3390/v14020403

**Chicago/Turabian Style**

Sazonov, Igor, Dmitry Grebennikov, Andreas Meyerhans, and Gennady Bocharov. 2022. "Sensitivity of SARS-CoV-2 Life Cycle to IFN Effects and ACE2 Binding Unveiled with a Stochastic Model" *Viruses* 14, no. 2: 403.
https://doi.org/10.3390/v14020403