# Markov Chain-Based Stochastic Modelling of HIV-1 Life Cycle in a CD4 T Cell

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## Abstract

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## 1. Introduction

- Cell-to-cell variability in HIV progeny production;
- Multiplicity of single-cell infection;
- Global sensitivity of specific reaction steps on net virus production.

## 2. Results

#### 2.1. Governing Deterministic Equations

#### 2.1.1. Virus Entry

- Virion binding to CD4 receptors (the viral glycoprotein gp120 binds to CD4 receptors on the T cell surface);
- Binding to the co-receptor CCR5 or CXCR4;
- Virus membrane and cell membrane fusion, i.e., the nucleocapsid is uncoated and the viral RNA is injected into the cell.

- ${x}_{1}=\left[{\mathrm{V}}_{\mathrm{free}}\right]$ is the number of free virions outside the cell;
- ${x}_{2}=\left[{\mathrm{V}}_{\mathrm{bound}}\right]$ is the number of virions bound to CD4 and the co-receptor.

- ${k}_{\mathrm{bound}}=3.1$ h${}^{-1}$; ${k}_{\mathrm{fuse}}=0.7$ h${}^{-1}$;
- $d=0.38$ h${}^{-1}$; ${d}_{\mathrm{bound}}=0.0008$ h${}^{-1}$;
- represent the rate of virion binding to the CD4+ T cell membrane, the rate of virion fusion with the cell, the clearance rate of free mature virions, and the degradation rate of bound virions, respectively. Their reference values and admissible ranges are specified in [15].

#### 2.1.2. Reverse Transcription

- Synthesis of minus-strand DNA from viral RNA;
- Synthesis of plus-strand DNA;
- Double-strand DNA formation.

- ${x}_{3}=\left[{\mathrm{RNA}}_{\mathrm{cor}}\right]$ is the number of genomic RNA molecules in the cytoplasm;
- ${x}_{4}=\left[{\mathrm{DNA}}_{\mathrm{cor}}\right]$ is the number of proviral DNA molecules synthesized by reverse transcription.

- ${k}_{\mathrm{RT}}=0.43$ h${}^{-1}$; ${k}_{{\mathrm{DNA}}_{\mathrm{t}}}=0.12$ h${}^{-1}$;
- ${d}_{{\mathrm{RNA}}_{\mathrm{cor}}}=0.21$ h${}^{-1}$; ${d}_{{\mathrm{DNA}}_{\mathrm{cor}}}=0.03$ h${}^{-1}$; represent the reverse transcription rate, the transport rate of DNA from cytoplasm to nucleus, the degradation rate of RNA in the cytoplasm and the degradation rate of DNA in the cytoplasm, respectively. Their reference values and admissible ranges are specified in [15].

#### 2.1.3. Integration

- ${x}_{5}=\left[{\mathrm{DNA}}_{\mathrm{nuc}}\right]$ is the number of DNA molecules in the nucleus;
- ${x}_{6}=\left[{\mathrm{DNA}}_{\mathrm{int}}\right]$ is the number of integrated DNA.

- ${k}_{\mathrm{int}}=0.14$ h${}^{-1}$; ${d}_{{\mathrm{DNA}}_{\mathrm{nuc}}}=0.001$ h${}^{-1}$; ${d}_{{\mathrm{DNA}}_{\mathrm{int}}}=0.00002$ h${}^{-1}$; represent the integration rate, the degradation rate of DNA in the nucleus and the degradation rate of DNA integrated into the chromosome, respectively. Their reference values and admissible ranges are specified in [15].

#### 2.1.4. Transcription

- ${x}_{7}=[{\mathrm{mRNA}}_{\mathrm{g}}]$ is the number of HIV mRNA molecules in the nucleus: g for genomic or full-length;
- ${x}_{8}=\left[{\mathrm{mRNA}}_{\mathrm{ss}}\right]$ is the number of HIV singly spliced (ss) mRNA molecules in the nucleus;
- ${x}_{9}=\left[{\mathrm{mRNA}}_{\mathrm{ds}}\right]$ is the number of HIV doubly spliced (ds) mRNA molecules in the nucleus;
- ${x}_{10}=\left[{\mathrm{mRNAc}}_{\mathrm{g}}\right]$ is the number of HIV mRNA molecules in the cytoplasm: g for genomic or full-length;
- ${x}_{11}=\left[{\mathrm{mRNAc}}_{\mathrm{ss}}\right]$ is the number of HIV singly spliced (ss) mRNA molecules in the cytoplasm;
- ${x}_{12}=\left[{\mathrm{mRNAc}}_{\mathrm{ds}}\right]$ is the number of HIV doubly spliced (ds) mRNA molecules in the cytoplasm.
- ${f}_{\mathrm{TR}}={f}_{\mathrm{TR}}\left({x}_{16}\right)={\mathrm{TR}}_{\mathrm{cell}}+{f}_{\mathrm{Tat}}\left({x}_{16}\right){\mathrm{TR}}_{\mathrm{Tat}}$; ${f}_{\mathrm{Tat}}={f}_{\mathrm{Tat}}\left({x}_{16}\right)={x}_{16}/({\theta}_{\mathrm{Tat}}+{x}_{16})$;
- ${f}_{\mathrm{Rev}}={f}_{\mathrm{Rev}}\left({x}_{17}\right)={x}_{17}/({\theta}_{\mathrm{Rev}}+{x}_{17})$;
- ${g}_{\mathrm{Rev}}={g}_{\mathrm{Rev}}\left({x}_{17}\right)=1-\beta {f}_{\mathrm{Rev}}$.

- ${\mathrm{TR}}_{\mathrm{cell}}=15$ h${}^{-1}$; ${\mathrm{TR}}_{\mathrm{Tat}}=1500$ h${}^{-1}$;
- ${\theta}_{\mathrm{Tat}}={10}^{3}\phantom{\rule{3.33333pt}{0ex}}\mathrm{molec}.$; ${\theta}_{\mathrm{Rev}}=7.7$×${10}^{4}\phantom{\rule{3.33333pt}{0ex}}\mathrm{molec}.$; $\beta =0.9$;
- ${k}_{{\mathrm{ssRNA}}_{\mathrm{g}}}=2.4$ h${}^{-1}$; ${k}_{{\mathrm{eRNA}}_{\mathrm{g}}}=2.3$ h${}^{-1}$; ${k}_{{\mathrm{eRNA}}_{\mathrm{ss}}}=2.3$ h${}^{-1}$;
- ${k}_{{\mathrm{dsRNA}}_{\mathrm{ss}}}=2.4$ h${}^{-1}$; ${k}_{{\mathrm{eRNA}}_{\mathrm{ds}}}=4.6$ h${}^{-1}$; ${k}_{\mathrm{tp},\mathrm{RNA}}=2.8$ h${}^{-1}$;
- ${d}_{{\mathrm{RNA}}_{\mathrm{g}}}={d}_{{\mathrm{RNA}}_{\mathrm{ss}}}={d}_{{\mathrm{RNAc}}_{\mathrm{ds}}}=0.12$ h${}^{-1}$; represent the cell intrinsic rate of basal transcription, the level of transcription induced by Tat transactivation, the inhibitory effect of Rev on the splicing rates implying their $1/(1-\beta )$-fold reduction at the saturation level of Rev, the rate of splicing for full-length virus RNA, the rate of $\left[{\mathrm{mRNA}}_{\mathrm{g}}\right]$ export from the nucleus, the rate of $\left[{\mathrm{mRNA}}_{\mathrm{ss}}\right]$ export from the nucleus, the rate of splicing for singly spliced virus RNA, the rate of $\left[{\mathrm{mRNA}}_{\mathrm{ds}}\right]$ export from the nucleus, the transport rate of $\left[{\mathrm{mRNA}}_{\mathrm{g}}\right]$ to the cell membrane, and the degradation rates of $\left[{\mathrm{mRNA}}_{\mathrm{i}}\right]$, $\mathrm{i}\in \{\mathrm{g},\mathrm{ss},\mathrm{ds}\}$, respectively. Their reference values and admissible ranges are specified in [15].

#### 2.1.5. Translation

- ${x}_{13}=[{\mathrm{P}}_{\mathrm{Gag}-\mathrm{Pol}}]$ is the number of protein molecules: Gag-Pol;
- ${x}_{14}=\left[{\mathrm{P}}_{\mathrm{Gag}}\right]$ is the number of protein molecules: Gag;
- ${x}_{15}=\left[{\mathrm{P}}_{\mathrm{gp}160}\right]$ is the number of protein molecules: gp160;
- ${x}_{16}=\left[{\mathrm{P}}_{\mathrm{Tat}}\right]$ is the number of protein molecules: Tat;
- ${x}_{17}=\left[{\mathrm{P}}_{\mathrm{Rev}}\right]$ is the number of protein molecules: Rev.

- ${k}_{\mathrm{trans}}=524$ h${}^{-1}$;
- ${f}_{\mathrm{g},\mathrm{Gag}-\mathrm{Pol}}=0.05$; ${f}_{\mathrm{g},\mathrm{Gag}}=0.95$; ${f}_{\mathrm{ss},\mathrm{gp}160}=0.64$; ${f}_{\mathrm{ds},\mathrm{Tat}}=0.025$; ${f}_{\mathrm{ds},\mathrm{Rev}}=0.2$;
- ${k}_{\mathrm{tp},\mathrm{Gag}-\mathrm{Pol}}=2.8$ h${}^{-1}$; ${k}_{\mathrm{tp},\mathrm{Gag}}=2.8$ h${}^{-1}$; ${k}_{\mathrm{tp},\mathrm{gp}160}=2.8$ h${}^{-1}$;
- ${d}_{\mathrm{p},\mathrm{Gag}-\mathrm{Pol}}=0.09$ h${}^{-1}$; ${d}_{\mathrm{p},\mathrm{Gag}}=0.09$ h${}^{-1}$; ${d}_{\mathrm{p},\mathrm{gp}160}=0.02$ h${}^{-1}$;
- ${d}_{\mathrm{p},\mathrm{Tat}}=0.04$ h${}^{-1}$; ${d}_{\mathrm{p},\mathrm{Rev}}=0.07$ h${}^{-1}$; represent the rate of mRNA to proteins translation, ${f}_{ij}$ stand for the fraction of $\left[{\mathrm{mRNA}}_{\mathrm{i}}\right]$ coding $\left[{\mathrm{P}}_{\mathrm{j}}\right]$, $\mathrm{i}\in \{\mathrm{g},\mathrm{ss},\mathrm{ds}\}$, $\mathrm{j}\in \{\mathrm{Gag}\text{-}\mathrm{Pol}$, Gag, gp160, Tat, $\mathrm{Rev}\}$. Following them, the parameters define the rates of protein $\left[{\mathrm{P}}_{\mathrm{j}}\right]$ transport to membrane, $\mathrm{j}\in \{\phantom{\rule{3.33333pt}{0ex}}\mathrm{Gag}\text{-}\mathrm{Pol},\mathrm{Gag},\phantom{\rule{3.33333pt}{0ex}}\mathrm{gp}160\}$, and the degradation rates of proteins Gag-Pol, Gag, gp160, Tat and Rev, respectively. Their reference values and admissible ranges are specified in [15].

#### 2.1.6. Assembly, Budding and Maturation

- ${k}_{\mathrm{tp},\mathrm{RNA}}=2.8$ h${}^{-1}$; ${k}_{\mathrm{tp},\mathrm{Gag}-\mathrm{Pol}}=2.8$ h${}^{-1}$; ${k}_{\mathrm{tp},\mathrm{Gag}}=2.8$ h${}^{-1}$; ${k}_{\mathrm{tp},\mathrm{gp}160}=2.8$ h${}^{-1}$;
- ${k}_{\mathrm{comb}}=8.0$ h${}^{-1}$; ${N}_{\mathrm{RNA}}=2$; ${N}_{Gag-Pol}=250$; ${N}_{\mathrm{Gag}}=5000$; ${N}_{\mathrm{gp}160}=24$; represent the rates of RNA and protein $\left[{\mathrm{P}}_{\mathrm{j}}\right]$ transport to the membrane, $\mathrm{j}\in \{\mathrm{Gag}\text{-}\mathrm{Pol}$, Gag, $\mathrm{gp}160\}$, the incorporation rate of molecules into pre-virion complexes, the number of viral RNA transcripts in a new virion, the number of Gag-Pol molecules in a new virion, the number of Gag molecules in a new virion, and the number of gp160 molecules in a new virion, respectively. Their reference values and admissible ranges are specified in [15].

- ${d}_{{\mathrm{RNA}}_{\mathrm{g}}}=0.12$ h${}^{-1}$; ${d}_{\mathrm{mem},\mathrm{Gag}-\mathrm{Pol}}=0.004$ h${}^{-1}$;
- ${d}_{\mathrm{mem},\mathrm{Gag}}=0.004$ h${}^{-1}$; ${d}_{\mathrm{mem},\mathrm{gp}160}=0.014$ h${}^{-1}$; represent the degradation rates of $\left[{\mathrm{RNA}}_{\mathrm{mem}}\right]$, the membrane-anchored protein Gag-Pol, the membrane-anchored protein Gag, and the membrane-associated gp160, respectively. Their reference values and admissible ranges are specified in [15].

- ${x}_{22}=\left[{\mathrm{V}}_{\mathrm{pre}-\mathrm{virion}}\right]$ is the number of virions on the membrane;
- ${x}_{23}=\left[{\mathrm{V}}_{\mathrm{bud}}\right]$ is the number of free viruses after budding from the cell;
- ${x}_{24}=\left[{\mathrm{V}}_{\mathrm{mat}}\right]$ is the number of mature virions outside the cell.

- ${k}_{\mathrm{comb}}=8$ h${}^{-1}$; ${k}_{\mathrm{bud}}=2.0$ h${}^{-1}$; ${k}_{\mathrm{mat}}=2.4$ h${}^{-1}$;
- ${d}_{\mathrm{comb}}=0.52$ h${}^{-1}$; ${d}_{\mathrm{bud}}=0.38$ h${}^{-1}$; $d=0.38$ h${}^{-1}$; represent the incorporation rate of molecules into pre-virion complexes, the budding rate of new virions, the maturation rate, the degradation rate of the assembled pre-virion complex, the degradation rate for budded immature viral-like particles, and the clearance rate of free mature virions, respectively. Their reference values and admissible ranges are specified in [15].

#### 2.2. Stochastic Markov Chain Modelling

#### 2.2.1. Algorithm

#### 2.2.2. Stochastic Modelling Results

#### 2.3. Sensitivity Analysis

#### 2.3.1. Sensitivity Analysis of the Deterministic Model towards the Cumulative Virion Release

#### 2.3.2. Sensitivity Analysis of Early Stages of Cell Infection Using the Stochastic Model

## 3. Discussion

- Transport of genomic mRNA to membranes;
- Tolerance of transcription activation to Tat-mediated regulation;
- Degradation of free and mature virions.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

RT | Reverse Transcription |

probability density function | |

cdf | cumulative distribution function |

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**Figure 3.**Probability of cell infection (i.e., integration of at least one proviral DNA molecule $\left[{\mathrm{DNA}}_{\mathrm{int}}\right]$) at a certain time as a function of the initial number of free virions $\left[{\mathrm{V}}_{\mathrm{free}}\right]\left(0\right)$ (

**A**). Histograms of number of integrated proviral DNA molecules $\left[{\mathrm{DNA}}_{\mathrm{int}}\right]$ (

**B**), number of released matured virions $\left[{\mathrm{V}}_{\mathrm{mat}}\right]$ (

**C**) at $t=36$ h for initial condition $\left[{\mathrm{V}}_{\mathrm{free}}\right]\left(0\right)=4$.

**Figure 4.**Temporal evolution of the histograms of the stochastic model variables for the initial condition $\left[{\mathrm{V}}_{\mathrm{free}}\right]\left(0\right)=4$.

**Figure 5.**Histograms of the stochastic model variables at $t=36$ h for the initial condition $\left[{\mathrm{V}}_{\mathrm{free}}\right]\left(0\right)=4$.

**Figure 6.**Temporal dynamics of the intensities of stages of the HIV-1 replication life cycle. The propensities that constitute each stage intensity are indicated on the vertical axis. The propensities represent the average values over stochastic realizations normalized by their maximum values.

**Figure 7.**Sensitivity indices for the cumulative number of released virions up to 36 h computed with the deterministic model. The bars and the error bars correspond to the means ± SEMs obtained on ${N}_{r}=20$ resamples. The significance levels of Welch’s t-test for comparing total order indices against ${S}_{\mathrm{dummy}}^{\mathrm{total}}$ are $p<{10}^{-20}$ ($***$), $p<{10}^{-10}$ ($**$), $p<{10}^{-5}$ (*).

**Table 1.**Markov chain: the list of individual reactions m, the corresponding state transitions and the propensities of reactions which define the probabilities of the individual reactions to occur over a small time interval $\delta t$.

m | Transition | Propensity, ${\mathit{a}}_{\mathit{m}}$ | m | Transition | Propensity, ${\mathit{a}}_{\mathit{m}}$ |
---|---|---|---|---|---|

1 | ${x}_{1}^{--},{x}_{2}^{++}$ | ${k}_{\mathrm{bound}}{x}_{1}$ | 27 | ${x}_{13}^{--}$ | ${d}_{\mathrm{p},\mathrm{Gag}-\mathrm{Pol}}{x}_{13}$ |

2 | ${x}_{1}^{--}$ | $d{x}_{1}$ | 28 | ${x}_{14}^{++}$ | ${k}_{\mathrm{trans}}{f}_{\mathrm{g},\mathrm{Gag}}{x}_{10}$ |

3 | ${x}_{2}^{--},{x}_{3}^{++}$ | ${k}_{\mathrm{fuse}}{x}_{2}$ | 29 | ${x}_{14}^{--},{x}_{20}^{++}$ | ${k}_{\mathrm{tp},\mathrm{Gag}}{x}_{14}$ |

4 | ${x}_{2}^{--}$ | ${d}_{\mathrm{bound}}{x}_{2}$ | 30 | ${x}_{14}^{--}$ | ${d}_{\mathrm{p},\mathrm{Gag}}{x}_{14}$ |

5 | ${x}_{3}^{--},{x}_{4}^{++}$ | ${k}_{\mathrm{RT}}{x}_{3}$ | 31 | ${x}_{15}^{++}$ | ${k}_{\mathrm{trans}}{f}_{\mathrm{ss},\mathrm{gp}160}{x}_{11}$ |

6 | ${x}_{3}^{--}$ | ${d}_{{\mathrm{RNA}}_{\mathrm{cor}}}{x}_{3}$ | 32 | ${x}_{15}^{--},{x}_{21}^{++}$ | ${k}_{\mathrm{tp},\mathrm{gp}160}{x}_{15}$ |

7 | ${x}_{4}^{--},{x}_{5}^{++}$ | ${k}_{{\mathrm{DNA}}_{\mathrm{t}}}{x}_{4}$ | 33 | ${x}_{15}^{--}$ | ${d}_{\mathrm{p},\mathrm{gp}160}{x}_{15}$ |

8 | ${x}_{4}^{--}$ | ${d}_{{\mathrm{DNA}}_{\mathrm{cor}}}{x}_{4}$ | 34 | ${x}_{16}^{++}$ | ${k}_{\mathrm{trans}}{f}_{\mathrm{ds},\mathrm{Tat}}{x}_{12}$ |

9 | ${x}_{5}^{--},{x}_{6}^{++}$ | ${k}_{\mathrm{int}}{x}_{5}$ | 35 | ${x}_{16}^{--}$ | ${d}_{\mathrm{p},\mathrm{Tat}}{x}_{16}$ |

10 | ${x}_{5}^{--}$ | ${d}_{{\mathrm{DNA}}_{\mathrm{nuc}}}{x}_{5}$ | 36 | ${x}_{17}^{++}$ | ${k}_{\mathrm{trans}}{f}_{\mathrm{ds},\mathrm{Rev}}{x}_{12}$ |

11 | ${x}_{6}^{--}$ | ${d}_{{\mathrm{DNA}}_{\mathrm{int}}}{x}_{6}$ | 37 | ${x}_{17}^{--}$ | ${d}_{\mathrm{p},\mathrm{Rev}}{x}_{17}$ |

12 | ${x}_{7}^{++}$ | ${f}_{\mathrm{TR}}\left({x}_{16}\right){x}_{6}$ | 38 | ${x}_{18}^{--}$ | ${k}_{\mathrm{comb}}{N}_{\mathrm{RNA}}{f}_{\mathrm{c}}\left({x}_{18,\dots ,21}\right)$ |

13 | ${x}_{7}^{--},{x}_{8}^{++}$ | ${k}_{{\mathrm{ssRNA}}_{\mathrm{g}}}{g}_{\mathrm{Rev}}\left({x}_{17}\right){x}_{7}$ | 39 | ${x}_{18}^{--}$ | ${d}_{{\mathrm{RNA}}_{\mathrm{g}}}{x}_{18}$ |

14 | ${x}_{7}^{--},{x}_{10}^{++}$ | ${k}_{{\mathrm{eRNA}}_{\mathrm{g}}}{f}_{\mathrm{Rev}}\left({x}_{17}\right){x}_{7}$ | 40 | ${x}_{19}^{--}$ | ${k}_{\mathrm{comb}}{N}_{\mathrm{Gag}-\mathrm{Pol}}{f}_{\mathrm{c}}\left({x}_{18,\dots ,21}\right)$ |

15 | ${x}_{7}^{--}$ | ${d}_{{\mathrm{RNA}}_{\mathrm{g}}}{x}_{7}$ | 41 | ${x}_{19}^{--}$ | ${d}_{\mathrm{mem},\mathrm{Gag}-\mathrm{Pol}}{x}_{19}$ |

16 | ${x}_{8}^{--},{x}_{9}^{++}$ | ${k}_{{\mathrm{dsRNA}}_{\mathrm{ss}}}{g}_{\mathrm{Rev}}\left({x}_{17}\right){x}_{8}$ | 42 | ${x}_{20}^{--}$ | ${k}_{\mathrm{comb}}{N}_{\mathrm{Gag}}{f}_{\mathrm{c}}\left({x}_{18,\dots ,21}\right)$ |

17 | ${x}_{8}^{--},{x}_{11}^{++}$ | ${k}_{{\mathrm{eRNA}}_{\mathrm{ss}}}{f}_{\mathrm{Rev}}\left({x}_{17}\right){x}_{8}$ | 43 | ${x}_{20}^{--}$ | ${d}_{\mathrm{mem},\mathrm{Gag}}{x}_{20}$ |

18 | ${x}_{8}^{--}$ | ${d}_{{\mathrm{RNA}}_{\mathrm{ss}}}{x}_{8}$ | 44 | ${x}_{21}^{--}$ | ${k}_{\mathrm{comb}}{N}_{\mathrm{gp}160}{f}_{\mathrm{c}}\left({x}_{18,\dots ,21}\right)$ |

19 | ${x}_{9}^{--},{x}_{12}^{++}$ | ${k}_{{\mathrm{eRNA}}_{\mathrm{ds}}}{x}_{9}$ | 45 | ${x}_{21}^{--}$ | ${d}_{\mathrm{mem},\mathrm{gp}160}{x}_{21}$ |

20 | ${x}_{9}^{--}$ | ${d}_{{\mathrm{RNA}}_{\mathrm{ds}}}{x}_{9}$ | 46 | ${x}_{22}^{++}$ | ${k}_{\mathrm{comb}}{f}_{\mathrm{c}}\left({x}_{18,\dots ,21}\right)$ |

21 | ${x}_{10}^{--},{x}_{18}^{++}$ | ${k}_{\mathrm{tp},\mathrm{RNA}}{x}_{10}$ | 47 | ${x}_{22}^{--},{x}_{23}^{++}$ | ${k}_{\mathrm{bud}}{x}_{22}$ |

22 | ${x}_{10}^{--}$ | ${d}_{{\mathrm{RNA}}_{\mathrm{g}}}{x}_{10}$ | 48 | ${x}_{22}^{--}$ | ${d}_{\mathrm{comb}}{x}_{22}$ |

23 | ${x}_{11}^{--}$ | ${d}_{{\mathrm{RNA}}_{\mathrm{ss}}}{x}_{11}$ | 49 | ${x}_{23}^{--},{x}_{24}^{++}$ | ${k}_{\mathrm{mat}}{x}_{23}$ |

24 | ${x}_{12}^{--}$ | ${d}_{{\mathrm{RNA}}_{\mathrm{ds}}}{x}_{12}$ | 50 | ${x}_{23}^{--}$ | ${d}_{\mathrm{bud}}{x}_{23}$ |

25 | ${x}_{13}^{++}$ | ${k}_{\mathrm{trans}}{f}_{\mathrm{g},\mathrm{Gag}-\mathrm{Pol}}{x}_{10}$ | 51 | ${x}_{24}^{--}$ | $d{x}_{24}$ |

26 | ${x}_{13}^{--},{x}_{19}^{++}$ | ${k}_{\mathrm{tp},\mathrm{Gag}-\mathrm{Pol}}{x}_{13}$ |

**Table 2.**Sensitivity indices of the model parameters that have the biggest influence on cumulative virion release. The parameters are sorted in descending order by total order sensitivity indices. The means ± SEMs obtained on ${N}_{r}=20$ resamples are indicated.

Parameter | Description | Sensitivity Indices | |
---|---|---|---|

First Order | Total Order | ||

${k}_{\mathrm{tp},\mathrm{RNA}}$ | Transport of genomic mRNA to membrane | 0.099 ± 0.006 | 0.669 ± 0.004 |

${\theta}_{\mathrm{Tat}}$ | Tolerance of transcription activation to Tat-mediated regulation | 0.089 ± 0.009 | 0.450 ± 0.009 |

${k}_{\mathrm{trans}}{f}_{\mathrm{ds},\mathrm{Tat}}$ | Translation of Tat molecules | 0.019 ± 0.001 | 0.235 ± 0.001 |

d | Degradation of free and mature virions | 0.022 ± 0.002 | 0.226 ± 0.004 |

${k}_{\mathrm{trans}}{f}_{\mathrm{g},\mathrm{Gag}\text{-}\mathrm{Pol}}$ | Translation of Gag-Pol molecules | 0.0155 ± 0.0009 | 0.201 ± 0.002 |

${k}_{\mathrm{trans}}{f}_{\mathrm{g},\mathrm{Gag}}$ | Translation of Gag molecules | 0.0153 ± 0.0006 | 0.177 ± 0.006 |

${d}_{{\mathrm{DNA}}_{\mathrm{cor}}}$ | Degradation of DNA during RT | 0.019 ± 0.002 | 0.158 ± 0.005 |

${k}_{\mathrm{comb}}$ | Assembly of pre-virion complexes | 0.0028 ± 0.0001 | 0.068 ± 0.002 |

${\mathrm{TR}}_{\mathrm{Tat}}$ | Tat-induced transcription rate | 0.0049 ± 0.0004 | 0.066 ± 0.003 |

${k}_{\mathrm{comb}}{N}_{\mathrm{Gag}}$ | Gag contribution to pre-virion assembly | 0.0014 ± 0.0001 | 0.063 ± 0.002 |

${k}_{\mathrm{comb}}{N}_{\mathrm{Gag}\text{-}\mathrm{Pol}}$ | Gag-Pol contribution to pre-virion assembly | 0.00125 ± 0.00005 | 0.063 ± 0.002 |

${d}_{{\mathrm{RNA}}_{\mathrm{ds}}}$ | Degradation of doubly-spliced mRNA | 0.0027 ± 0.0003 | 0.058 ± 0.004 |

${K}_{{\mathrm{V}}_{\mathrm{rel}}}{N}_{\mathrm{Gag}}$ | Tolerance of pre-virion assembly to Gag availability on membrane | 0.0036 ± 0.0006 | 0.051 ± 0.003 |

**Table 3.**Local sensitivity indices ${\widehat{S}}_{i}$ of the stochastic model parameters having the most influence on the number of integrated proviruses $\left[{\mathrm{DNA}}_{\mathrm{int}}\right]\left(36\right)$.

Parameter | Description | Sensitivity |
---|---|---|

${k}_{\mathrm{bound}}$ | Binding rate of free virions to CD4+ T cell membrane | 0.498 |

d | Degradation rate of HIV particles | 0.752 |

${k}_{\mathrm{fuse}}$ | Rate of virion fusion into the cell | 0.988 |

${d}_{\mathrm{bound}}$ | Degradation rate of virions bound to membrane | 1.080 |

${k}_{\mathrm{RT}}$ | Rate of reverse transcription | 0.804 |

${d}_{{\mathrm{RNA}}_{\mathrm{cor}}}$ | Degradation rate of viral RNA in cytoplasm | 0.712 |

${k}_{{\mathrm{DNA}}_{\mathrm{t}}}$ | Rate of viral DNA transfer to nucleus | 0.738 |

${d}_{{\mathrm{DNA}}_{\mathrm{cor}}}$ | Degradation rate of viral DNA in cytoplasm | 0.464 |

${k}_{\mathrm{int}}$ | Rate of viral DNA integration into host chromosome | 0.446 |

${d}_{{\mathrm{DNA}}_{\mathrm{nuc}}}$ | Degradation rate of viral DNA in nucleus | 0.346 |

${d}_{{\mathrm{DNA}}_{\mathrm{int}}}$ | Degradation rate of DNA integrated into chromosome | 0.582 |

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**MDPI and ACS Style**

Sazonov, I.; Grebennikov, D.; Meyerhans, A.; Bocharov, G.
Markov Chain-Based Stochastic Modelling of HIV-1 Life Cycle in a CD4 T Cell. *Mathematics* **2021**, *9*, 2025.
https://doi.org/10.3390/math9172025

**AMA Style**

Sazonov I, Grebennikov D, Meyerhans A, Bocharov G.
Markov Chain-Based Stochastic Modelling of HIV-1 Life Cycle in a CD4 T Cell. *Mathematics*. 2021; 9(17):2025.
https://doi.org/10.3390/math9172025

**Chicago/Turabian Style**

Sazonov, Igor, Dmitry Grebennikov, Andreas Meyerhans, and Gennady Bocharov.
2021. "Markov Chain-Based Stochastic Modelling of HIV-1 Life Cycle in a CD4 T Cell" *Mathematics* 9, no. 17: 2025.
https://doi.org/10.3390/math9172025