# Mathematical Models of HIV-1 Dynamics, Transcription, and Latency

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. History of Mathematical Models of HIV-1

#### 2.1. Viral Dynamics Model

#### 2.1.1. The Basic Viral Dynamics Model

#### 2.1.2. Models of ART

#### 2.2. Mechanisms of Latent Reservoir Persistence

## 3. Molecular Features Contributing to HIV-1 Latency and Reactivation

#### 3.1. Integration Site

#### 3.2. Chromatin

#### 3.3. Transcription Machinery

## 4. Transcriptional Bursting and Mathematical Modeling

#### 4.1. Transcriptional Bursting and Gene Expression Noise

#### 4.2. A Stochastic Model to Describe the HIV-1 Transcriptional Circuit

#### 4.3. Incorporating Host and Viral Phases to Model the HIV-1 Transcriptional Circuit

#### 4.4. Two-State HIV-1 Transcriptional Model

#### 4.5. Three-State HIV-1 Transcriptional Model

## 5. Discussion and Future Perspective

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Chun, T.W.; Carruth, L.; Finzi, D.; Shen, X.; DiGiuseppe, J.A.; Taylor, H.; Hermankova, M.; Chadwick, K.; Margolick, J.; Quinn, T.C.; et al. Quantification of latent tissue reservoirs and total body viral load in HIV-1 infection. Nature
**1997**, 387, 183–188. [Google Scholar] [CrossRef] [PubMed] - Finzi, D.; Hermankova, M.; Pierson, T.; Carruth, L.M.; Buck, C.; Chaisson, R.E.; Quinn, T.C.; Chadwick, K.; Margolick, J.; Brookmeyer, R.; et al. Identification of a reservoir for HIV-1 in patients on highly active antiretroviral therapy. Science
**1997**, 278, 1295–1300. [Google Scholar] [CrossRef] [PubMed] - Finzi, D.; Blankson, J.; Siliciano, J.D.; Margolick, J.B.; Chadwick, K.; Pierson, T.; Smith, K.; Lisziewicz, J.; Lori, F.; Flexner, C.; et al. Latent infection of CD4+ T cells provides a mechanism for lifelong persistence of HIV-1, even in patients on effective combination therapy. Nat. Med.
**1999**, 5, 512–517. [Google Scholar] [CrossRef] [PubMed] - Kumar, A.; Abbas, W.; Herbein, G. HIV-1 latency in monocytes/macrophages. Viruses
**2014**, 6, 1837–1860. [Google Scholar] [CrossRef] [PubMed] - Mitchell, B.I.; Laws, E.I.; Ndhlovu, L.C. Impact of myeloid reservoirs in HIV cure trials. Curr. HIV/AIDS Rep.
**2019**, 16, 129–140. [Google Scholar] [CrossRef] [PubMed] - Veenhuis, R.T.; Abreu, C.M.; Costa, P.A.; Ferreira, E.A.; Ratliff, J.; Pohlenz, L.; Shirk, E.N.; Rubin, L.H.; Blankson, J.N.; Gama, L.; et al. Monocyte-derived macrophages contain persistent latent HIV reservoirs. Nat. Microbiol.
**2023**, 8, 833–844. [Google Scholar] [CrossRef] [PubMed] - Wallet, C.; De Rovere, M.; Van Assche, J.; Daouad, F.; De Wit, S.; Gautier, V.; Mallon, P.W.; Marcello, A.; Van Lint, C.; Rohr, O.; et al. Microglial cells: The main HIV-1 reservoir in the brain. Front. Cell. Infect. Microbiol.
**2019**, 9, 362. [Google Scholar] [CrossRef] - Wei, X.; Ghosh, S.K.; Taylor, M.E.; Johnson, V.A.; Emini, E.A.; Deutsch, P.; Lifson, J.D.; Bonhoeffer, S.; Nowak, M.A.; Hahn, B.H.; et al. Viral dynamics in human immunodeficiency virus type 1 infection. Nature
**1995**, 373, 117–122. [Google Scholar] [CrossRef] - Ho, D.D.; Neumann, A.U.; Perelson, A.S.; Chen, W.; Leonard, J.M.; Markowitz, M. Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature
**1995**, 373, 123–126. [Google Scholar] [CrossRef] - Ramratnam, B.; Bonhoeffer, S.; Binley, J.; Hurley, A.; Zhang, L.; Mittler, J.E.; Markowitz, M.; Moore, J.P.; Perelson, A.S.; Ho, D.D. Rapid production and clearance of HIV-1 and hepatitis C virus assessed by large volume plasma apheresis. Lancet
**1999**, 354, 1782–1785. [Google Scholar] [CrossRef] - Perelson, A.S.; Essunger, P.; Cao, Y.; Vesanen, M.; Hurley, A.; Saksela, K.; Markowitz, M.; Ho, D.D. Decay characteristics of HIV-1-infected compartments during combination therapy. Nature
**1997**, 387, 188–191. [Google Scholar] [CrossRef] [PubMed] - McMyn, N.F.; Varriale, J.; Fray, E.J.; Zitzmann, C.; MacLeod, H.; Lai, J.; Singhal, A.; Moskovljevic, M.; Garcia, M.A.; Lopez, B.M.; et al. The latent reservoir of inducible, infectious HIV-1 does not decrease despite decades of antiretroviral therapy. J. Clin. Investig.
**2023**, 133, e171554. [Google Scholar] [CrossRef] [PubMed] - Rong, L.; Perelson, A.S. Modeling latently infected cell activation: Viral and latent reservoir persistence, and viral blips in HIV-infected patients on potent therapy. PLoS Comput. Biol.
**2009**, 5, e1000533. [Google Scholar] [CrossRef] [PubMed] - Kim, H.; Perelson, A.S. Viral and latent reservoir persistence in HIV-1–infected patients on therapy. PLoS Comput. Biol.
**2006**, 2, e135. [Google Scholar] [CrossRef] [PubMed] - Perelson, A.S.; Ribeiro, R.M. Modeling the within-host dynamics of HIV infection. BMC Biol.
**2013**, 11, 96. [Google Scholar] [CrossRef] - Conway, J.M.; Meily, P.; Li, J.Z.; Perelson, A.S. Unified model of short-and long-term HIV viral rebound for clinical trial planning. J. R. Soc. Interface
**2021**, 18, 20201015. [Google Scholar] [CrossRef] - Conway, J.M.; Coombs, D. A stochastic model of latently infected cell reactivation and viral blip generation in treated HIV patients. PLoS Comput. Biol.
**2011**, 7, e1002033. [Google Scholar] [CrossRef] - Hill, A.L. Mathematical Models of HIV Latency. Curr. Top. Microbiol. Immunol.
**2018**, 417, 131–156. [Google Scholar] - Kermack, W.O.; McKendrick, A.G. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. Contain. Pap. Math. Phys. Character
**1927**, 115, 700–721. [Google Scholar] - Brauer, F. Mathematical epidemiology is not an oxymoron. BMC Public Health
**2009**, 9, S2. [Google Scholar] [CrossRef] - Anderson, R.M.; May, R.M. Infectious Diseases of Humans: Dynamics and Control; Oxford University Press: Oxford, UK, 1991. [Google Scholar]
- Nowak, M.; May, R.M. Virus Dynamics: Mathematical Principles of Immunology and Virology: Mathematical Principles of Immunology and Virology; Oxford University Press: Oxford, UK, 2000. [Google Scholar]
- Nowak, M.A.; Bonhoeffer, S.; Hill, A.M.; Boehme, R.; Thomas, H.C.; McDade, H. Viral dynamics in hepatitis B virus infection. Proc. Natl. Acad. Sci. USA
**1996**, 93, 4398–4402. [Google Scholar] [CrossRef] [PubMed] - Perelson, A.S.; Ribeiro, R.M. Hepatitis B virus kinetics and mathematical modeling. In Seminars in Liver Disease; Thieme Medical Publishers, Inc.: New York, NY, USA, 2004; Volume 24, pp. 11–16. [Google Scholar]
- Dahari, H.; Shudo, E.; Ribeiro, R.M.; Perelson, A.S. Modeling complex decay profiles of hepatitis B virus during antiviral therapy. Hepatology
**2009**, 49, 32–38. [Google Scholar] [CrossRef] [PubMed] - Neumann, A.U.; Lam, N.P.; Dahari, H.; Gretch, D.R.; Wiley, T.E.; Layden, T.J.; Perelson, A.S. Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-α therapy. Science
**1998**, 282, 103–107. [Google Scholar] [CrossRef] [PubMed] - Chatterjee, A.; Guedj, J.; Perelson, A.S. Mathematical modeling of HCV infection: What can it teach us in the era of direct antiviral agents? Antivir. Ther.
**2012**, 17, 1171. [Google Scholar] [CrossRef] - Murillo, L.N.; Murillo, M.S.; Perelson, A.S. Towards multiscale modeling of influenza infection. J. Theor. Biol.
**2013**, 332, 267–290. [Google Scholar] [CrossRef] - Nuraini, N.; Tasman, H.; Soewono, E.; Sidarto, K.A. A with-in host dengue infection model with immune response. Math. Comput. Model.
**2009**, 49, 1148–1155. [Google Scholar] [CrossRef] - Clapham, H.E.; Tricou, V.; Van Vinh Chau, N.; Simmons, C.P.; Ferguson, N.M. Within-host viral dynamics of dengue serotype 1 infection. J. R. Soc. Interface
**2014**, 11, 20140094. [Google Scholar] [CrossRef] - Schiffer, J.T.; Swan, D.A.; Magaret, A.; Corey, L.; Wald, A.; Ossig, J.; Ruebsamen-Schaeff, H.; Stoelben, S.; Timmler, B.; Zimmermann, H.; et al. Mathematical modeling of herpes simplex virus-2 suppression with pritelivir predicts trial outcomes. Sci. Transl. Med.
**2016**, 8, 324ra15. [Google Scholar] [CrossRef] - Butterworth, E.; Jardine, B.E.; Raymond, G.M.; Neal, M.L.; Bassingthwaighte, J.B. JSim, an open-source modeling system for data analysis. F1000Research
**2013**, 2, 288. [Google Scholar] [CrossRef] - Phillips, A.N. Reduction of HIV concentration during acute infection: Independence from a specific immune response. Science
**1996**, 271, 497–499. [Google Scholar] [CrossRef] - Stafford, M.A.; Corey, L.; Cao, Y.; Daar, E.S.; Ho, D.D.; Perelson, A.S. Modeling plasma virus concentration during primary HIV infection. J. Theor. Biol.
**2000**, 203, 285–301. [Google Scholar] [CrossRef] [PubMed] - Burg, D.; Rong, L.; Neumann, A.U.; Dahari, H. Mathematical modeling of viral kinetics under immune control during primary HIV-1 infection. J. Theor. Biol.
**2009**, 259, 751–759. [Google Scholar] [CrossRef] - Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A. On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol.
**1990**, 28, 365–382. [Google Scholar] [CrossRef] [PubMed] - Ribeiro, R.M.; Qin, L.; Chavez, L.L.; Li, D.; Self, S.G.; Perelson, A.S. Estimation of the initial viral growth rate and basic reproductive number during acute HIV-1 infection. J. Virol.
**2010**, 84, 6096–6102. [Google Scholar] [CrossRef] - McLean, A.R.; Nowak, M.A. Competition between zidovudine-sensitive and. Aids
**1992**, 6, 71–79. [Google Scholar] [CrossRef] - Frost, S.; McLean, A.R. Quasispecies dynamics and the emergence of drug resistance during zidovudine therapy of HIV infection. Aids
**1994**, 8, 323–332. [Google Scholar] [CrossRef] [PubMed] - Pialoux, G.; Raffi, F.; Brun-Vezinet, F.; Meiffrédy, V.; Flandre, P.; Gastaut, J.A.; Dellamonica, P.; Yeni, P.; Delfraissy, J.F.; Aboulker, J.P. A randomized trial of three maintenance regimens given after three months of induction therapy with zidovudine, lamivudine, and indinavir in previously untreated HIV-1–infected patients. N. Engl. J. Med.
**1998**, 339, 1269–1276. [Google Scholar] [CrossRef] - Ribeiro, R.M.; Bonhoeffer, S.; Nowak, M.A. The frequency of resistant mutant virus before antiviral therapy. Aids
**1998**, 12, 461–465. [Google Scholar] [CrossRef] - Colgrove, R.; Japour, A. A combinatorial ledge: Reverse transcriptase fidelity, total body viral burden, and the implications of multiple-drug HIV therapy for the evolution of antiviral resistance. Antivir. Res.
**1999**, 41, 45–56. [Google Scholar] [CrossRef] - Li, M.; Oliveira Passos, D.; Shan, Z.; Smith, S.J.; Sun, Q.; Biswas, A.; Choudhuri, I.; Strutzenberg, T.S.; Haldane, A.; Deng, N.; et al. Mechanisms of HIV-1 integrase resistance to dolutegravir and potent inhibition of drug-resistant variants. Sci. Adv.
**2023**, 9, eadg5953. [Google Scholar] [CrossRef] - Shekhar, K.; Ruberman, C.F.; Ferguson, A.L.; Barton, J.P.; Kardar, M.; Chakraborty, A.K. Spin models inferred from patient-derived viral sequence data faithfully describe HIV fitness landscapes. Phys. Rev. E
**2013**, 88, 062705. [Google Scholar] [CrossRef] [PubMed] - Biswas, A.; Haldane, A.; Levy, R.M. Limits to detecting epistasis in the fitness landscape of HIV. PLoS ONE
**2022**, 17, e0262314. [Google Scholar] [CrossRef] [PubMed] - Biswas, A.; Haldane, A.; Arnold, E.; Levy, R.M. Epistasis and entrenchment of drug resistance in HIV-1 subtype B. Elife
**2019**, 8, e50524. [Google Scholar] [CrossRef] [PubMed] - Anderson, R.W.; Ascher, M.S.; Sheppard, H.W. Direct HIV cytopathicity cannot account for CD4 decline in AIDS in the presence of homeostasis: A worst-case dynamic analysis. J. Acquir. Immune Defic. Syndr.
**1998**, 17, 245–252. [Google Scholar] [CrossRef] [PubMed] - Ribeiro, R.M.; Mohri, H.; Ho, D.D.; Perelson, A.S. In vivo dynamics of T cell activation, proliferation, and death in HIV-1 infection: Why are CD4+ but not CD8+ T cells depleted? Proc. Natl. Acad. Sci. USA
**2002**, 99, 15572–15577. [Google Scholar] [CrossRef] - Yates, A.; Stark, J.; Klein, N.; Antia, R.; Callard, R. Understanding the slow depletion of memory CD4+ T cells in HIV infection. PLoS Med.
**2007**, 4, e177. [Google Scholar] [CrossRef] - Nowak, M.A.; Anderson, R.M.; McLean, A.R.; Wolfs, T.F.; Goudsmit, J.; May, R.M. Antigenic diversity thresholds and the development of AIDS. Science
**1991**, 254, 963–969. [Google Scholar] [CrossRef] - Regoes, R.R.; Wodarz, D.; Nowak, M.A. Virus dynamics: The effect of target cell limitation and immune responses on virus evolution. J. Theor. Biol.
**1998**, 191, 451–462. [Google Scholar] [CrossRef] - Perelson, A.S.; Neumann, A.U.; Markowitz, M.; Leonard, J.M.; Ho, D.D. HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time. Science
**1996**, 271, 1582–1586. [Google Scholar] [CrossRef] - Markowitz, M.; Louie, M.; Hurley, A.; Sun, E.; Di Mascio, M.; Perelson, A.S.; Ho, D.D. A novel antiviral intervention results in more accurate assessment of human immunodeficiency virus type 1 replication dynamics and T-cell decay in vivo. J. Virol.
**2003**, 77, 5037–5038. [Google Scholar] [CrossRef] - Andrade, A.; Rosenkranz, S.L.; Cillo, A.R.; Lu, D.; Daar, E.S.; Jacobson, J.M.; Lederman, M.; Acosta, E.P.; Campbell, T.; Feinberg, J.; et al. Three distinct phases of HIV-1 RNA decay in treatment-naive patients receiving raltegravir-based antiretroviral therapy: ACTG A5248. J. Infect. Dis.
**2013**, 208, 884–891. [Google Scholar] [CrossRef] [PubMed] - Ollerton, M.T.; Berger, E.A.; Connick, E.; Burton, G.F. HIV-1-specific chimeric antigen receptor T cells fail to recognize and eliminate the follicular dendritic cell HIV reservoir in vitro. J. Virol.
**2020**, 94, 10–1128. [Google Scholar] [CrossRef] [PubMed] - Cohn, L.B.; Chomont, N.; Deeks, S.G. The biology of the HIV-1 latent reservoir and implications for cure strategies. Cell Host Microbe
**2020**, 27, 519–530. [Google Scholar] [CrossRef] - Siliciano, J.D.; Kajdas, J.; Finzi, D.; Quinn, T.C.; Chadwick, K.; Margolick, J.B.; Kovacs, C.; Gange, S.J.; Siliciano, R.F. Long-term follow-up studies confirm the stability of the latent reservoir for HIV-1 in resting CD4+ T cells. Nat. Med.
**2003**, 9, 727–728. [Google Scholar] [CrossRef] - Davey, R.T., Jr.; Bhat, N.; Yoder, C.; Chun, T.W.; Metcalf, J.A.; Dewar, R.; Natarajan, V.; Lempicki, R.A.; Adelsberger, J.W.; Miller, K.D.; et al. HIV-1 and T cell dynamics after interruption of highly active antiretroviral therapy (HAART) in patients with a history of sustained viral suppression. Proc. Natl. Acad. Sci. USA
**1999**, 96, 15109–15114. [Google Scholar] [CrossRef] - Ruiz, L.; Martinez-Picado, J.; Romeu, J.; Paredes, R.; Zayat, M.K.; Marfil, S.; Negredo, E.; Sirera, G.; Tural, C.; Clotet, B. Structured treatment interruption in chronically HIV-1 infected patients after long-term viral suppression. Aids
**2000**, 14, 397–403. [Google Scholar] [CrossRef] [PubMed] - White, J.A.; Simonetti, F.R.; Beg, S.; McMyn, N.F.; Dai, W.; Bachmann, N.; Lai, J.; Ford, W.C.; Bunch, C.; Jones, J.L.; et al. Complex decay dynamics of HIV virions, intact and defective proviruses, and 2LTR circles following initiation of antiretroviral therapy. Proc. Natl. Acad. Sci. USA
**2022**, 119, e2120326119. [Google Scholar] [CrossRef] - Spivak, A.M.; Rabi, S.A.; McMahon, M.A.; Shan, L.; Sedaghat, A.R.; Wilke, C.O.; Siliciano, R.F. dynamic constraints on the second phase compartment of HIV-infected cells. Aids Res. Hum. Retroviruses
**2011**, 27, 759–761. [Google Scholar] [CrossRef] - Arnaout, R.A.; Martin, A.N.; Wodarz, D. HIV–1 dynamics revisited: Biphasic decay by cytotoxic T lymphocyte killing? Proc. R. Soc. Lond. Ser. Biol. Sci.
**2000**, 267, 1347–1354. [Google Scholar] [CrossRef] - Zhang, J.; Perelson, A.S. Contribution of follicular dendritic cells to persistent HIV viremia. J. Virol.
**2013**, 87, 7893–7901. [Google Scholar] [CrossRef] - Eriksson, S.; Graf, E.H.; Dahl, V.; Strain, M.C.; Yukl, S.A.; Lysenko, E.S.; Bosch, R.J.; Lai, J.; Chioma, S.; Emad, F.; et al. Comparative analysis of measures of viral reservoirs in HIV-1 eradication studies. PLoS Pathog.
**2013**, 9, e1003174. [Google Scholar] [CrossRef] [PubMed] - Crooks, A.M.; Bateson, R.; Cope, A.B.; Dahl, N.P.; Griggs, M.K.; Kuruc, J.D.; Gay, C.L.; Eron, J.J.; Margolis, D.M.; Bosch, R.J.; et al. Precise quantitation of the latent HIV-1 reservoir: Implications for eradication strategies. J. Infect. Dis.
**2015**, 212, 1361–1365. [Google Scholar] [CrossRef] - Cory, T.J.; Schacker, T.W.; Stevenson, M.; Fletcher, C.V. Overcoming pharmacologic sanctuaries. Curr. Opin. HIV AIDS
**2013**, 8, 190. [Google Scholar] [CrossRef] [PubMed] - Martinez-Picado, J.; Deeks, S.G. Persistent HIV-1 replication during antiretroviral therapy. Curr. Opin. HIV AIDS
**2016**, 11, 417. [Google Scholar] [CrossRef] - Lorenzo-Redondo, R.; Fryer, H.R.; Bedford, T.; Kim, E.Y.; Archer, J.; Kosakovsky Pond, S.L.; Chung, Y.S.; Penugonda, S.; Chipman, J.G.; Fletcher, C.V.; et al. Persistent HIV-1 replication maintains the tissue reservoir during therapy. Nature
**2016**, 530, 51–56. [Google Scholar] [CrossRef] [PubMed] - Bachmann, N.; Von Siebenthal, C.; Vongrad, V.; Turk, T.; Neumann, K.; Beerenwinkel, N.; Bogojeska, J.; Fellay, J.; Roth, V.; Kok, Y.L.; et al. Determinants of HIV-1 reservoir size and long-term dynamics during suppressive ART. Nat. Commun.
**2019**, 10, 3193. [Google Scholar] [CrossRef] - Brodin, J.; Zanini, F.; Thebo, L.; Lanz, C.; Bratt, G.; Neher, R.A.; Albert, J. Establishment and stability of the latent HIV-1 DNA reservoir. Elife
**2016**, 5, e18889. [Google Scholar] [CrossRef] - Jones, L.E.; Perelson, A.S. Transient viremia, plasma viral load, and reservoir replenishment in HIV-infected patients on antiretroviral therapy. J. Acquir. Immune Defic. Syndr.
**2007**, 45, 483. [Google Scholar] [CrossRef] - Sedaghat, A.R.; Siliciano, R.F.; Wilke, C.O. Low-level HIV-1 replication and the dynamics of the resting CD4+ T cell reservoir for HIV-1 in the setting of HAART. BMC Infect. Dis.
**2008**, 8, 2. [Google Scholar] [CrossRef] - Conway, J.M.; Perelson, A.S. Residual viremia in treated HIV+ individuals. PLoS Comput. Biol.
**2016**, 12, e1004677. [Google Scholar] [CrossRef] - Kirchhoff, F. HIV life cycle: Overview. Encycl. AIDS
**2013**, 1–9. [Google Scholar] [CrossRef] - Vargas, B.; Sluis-Cremer, N. Toward a Functional Cure for HIV-1 Infection: The Block and Lock therapeutic Approach. Front. Virol.
**2022**, 2, 917941. [Google Scholar] [CrossRef] - Schröder, A.R.; Shinn, P.; Chen, H.; Berry, C.; Ecker, J.R.; Bushman, F. HIV-1 integration in the human genome favors active genes and local hotspots. Cell
**2002**, 110, 521–529. [Google Scholar] [CrossRef] [PubMed] - Ruess, H.; Lee, J.; Guzman, C.; Malladi, V.S.; D’Orso, I. Decoding Human Genome Regulatory Features That Influence HIV-1 Proviral Expression and Fate Through an Integrated Genomics Approach. Bioinform. Biol. Insights
**2022**, 16, 11779322211072333. [Google Scholar] [CrossRef] - Chen, H.C.; Martinez, J.P.; Zorita, E.; Meyerhans, A.; Filion, G.J. Position effects influence HIV latency reversal. Nat. Struct. Mol. Biol.
**2017**, 24, 47–54. [Google Scholar] [CrossRef] [PubMed] - Jordan, A.; Defechereux, P.; Verdin, E. The site of HIV-1 integration in the human genome determines basal transcriptional activity and response to Tat transactivation. EMBO J.
**2001**, 20, 1726–1738. [Google Scholar] [CrossRef] - Burnett, J.C.; Miller-Jensen, K.; Shah, P.S.; Arkin, A.P.; Schaffer, D.V. Control of stochastic gene expression by host factors at the HIV promoter. PLoS Pathog.
**2009**, 5, e1000260. [Google Scholar] [CrossRef] - Einkauf, K.B.; Osborn, M.R.; Gao, C.; Sun, W.; Sun, X.; Lian, X.; Parsons, E.M.; Gladkov, G.T.; Seiger, K.W.; Blackmer, J.E.; et al. Parallel analysis of transcription, integration, and sequence of single HIV-1 proviruses. Cell
**2022**, 185, 266–282. [Google Scholar] [CrossRef] - Van Lint, C.; Emiliani, S.; Ott, M.; Verdin, E. Transcriptional activation and chromatin remodeling of the HIV-1 promoter in response to histone acetylation. EMBO J.
**1996**, 15, 1112–1120. [Google Scholar] [CrossRef] - Verdin, E.; Paras, P., Jr.; Van Lint, C. Chromatin disruption in the promoter of human immunodeficiency virus type 1 during transcriptional activation. EMBO J.
**1993**, 12, 3249–3259. [Google Scholar] [CrossRef] - Shukla, A.; Ramirez, N.; DLOrso, I. HIV-1 proviral transcription and latency in the new era. Viruses
**2020**, 12, 555. [Google Scholar] [CrossRef] [PubMed] - Lusic, M.; Siliciano, R.F. Nuclear landscape of HIV-1 infection and integration. Nat. Rev. Microbiol.
**2017**, 15, 69–82. [Google Scholar] [CrossRef] [PubMed] - Struhl, K.; Segal, E. Determinants of nucleosome positioning. Nat. Struct. Mol. Biol.
**2013**, 20, 267–273. [Google Scholar] [CrossRef] [PubMed] - Bartha, I.; Carlson, J.M.; Brumme, C.J.; McLaren, P.J.; Brumme, Z.L.; John, M.; Haas, D.W.; Martinez-Picado, J.; Dalmau, J.; López-Galíndez, C.; et al. A genome-to-genome analysis of associations between human genetic variation, HIV-1 sequence diversity, and viral control. Elife
**2013**, 2, e01123. [Google Scholar] [CrossRef] [PubMed] - Zhang, Z.; Klatt, A.; Gilmour, D.S.; Henderson, A.J. Negative elongation factor NELF represses human immunodeficiency virus transcription by pausing the RNA polymerase II complex. J. Biol. Chem.
**2007**, 282, 16981–16988. [Google Scholar] [CrossRef] [PubMed] - Jadlowsky, J.K.; Wong, J.Y.; Graham, A.C.; Dobrowolski, C.; Devor, R.L.; Adams, M.D.; Fujinaga, K.; Karn, J. Negative elongation factor is required for the maintenance of proviral latency but does not induce promoter-proximal pausing of RNA polymerase II on the HIV long terminal repeat. Mol. Cell. Biol.
**2014**, 34, 1911–1928. [Google Scholar] [CrossRef] [PubMed] - Kao, S.Y.; Calman, A.F.; Luciw, P.A.; Peterlin, B.M. Anti-termination of transcription within the long terminal repeat of HIV-1 by tat gene product. Nature
**1987**, 330, 489–493. [Google Scholar] [CrossRef] - Bacon, C.; D’Orso, I. CDK9: A signaling hub for transcriptional control. Transcription
**2019**, 10, 57–75. [Google Scholar] [CrossRef] - Tantale, K.; Garcia-Oliver, E.; Robert, M.C.; L’hostis, A.; Yang, Y.; Tsanov, N.; Topno, R.; Gostan, T.; Kozulic-Pirher, A.; Basu-Shrivastava, M.; et al. Stochastic pausing at latent HIV-1 promoters generates transcriptional bursting. Nat. Commun.
**2021**, 12, 4503. [Google Scholar] [CrossRef] - Mancebo, H.S.; Lee, G.; Flygare, J.; Tomassini, J.; Luu, P.; Zhu, Y.; Peng, J.; Blau, C.; Hazuda, D.; Price, D.; et al. P-TEFb kinase is required for HIV Tat transcriptional activation in vivo and in vitro. Genes Dev.
**1997**, 11, 2633–2644. [Google Scholar] [CrossRef] - Kuzmina, A.; Krasnopolsky, S.; Taube, R. Super elongation complex promotes early HIV transcription and its function is modulated by P-TEFb. Transcription
**2017**, 8, 133–149. [Google Scholar] [CrossRef] [PubMed] - Chou, S.; Upton, H.; Bao, K.; Schulze-Gahmen, U.; Samelson, A.J.; He, N.; Nowak, A.; Lu, H.; Krogan, N.J.; Zhou, Q.; et al. HIV-1 Tat recruits transcription elongation factors dispersed along a flexible AFF4 scaffold. Proc. Natl. Acad. Sci. USA
**2013**, 110, E123–E131. [Google Scholar] [CrossRef] [PubMed] - Lu, H.; Li, Z.; Zhang, W.; Schulze-Gahmen, U.; Xue, Y.; Zhou, Q. Gene target specificity of the Super Elongation Complex (SEC) family: How HIV-1 Tat employs selected SEC members to activate viral transcription. Nucleic Acids Res.
**2015**, 43, 5868–5879. [Google Scholar] [CrossRef] [PubMed] - Leyes Porello, E.A.; Trudeau, R.T.; Lim, B. Transcriptional bursting: Stochasticity in deterministic development. Development
**2023**, 150, dev201546. [Google Scholar] [CrossRef] - Tantale, K.; Mueller, F.; Kozulic-Pirher, A.; Lesne, A.; Victor, J.M.; Robert, M.C.; Capozi, S.; Chouaib, R.; Bäcker, V.; Mateos-Langerak, J.; et al. A single-molecule view of transcription reveals convoys of RNA polymerases and multi-scale bursting. Nat. Commun.
**2016**, 7, 12248. [Google Scholar] [CrossRef] - Sanchez, A.; Golding, I. Genetic determinants and cellular constraints in noisy gene expression. Science
**2013**, 342, 1188–1193. [Google Scholar] [CrossRef] - Dar, R.D.; Razooky, B.S.; Singh, A.; Trimeloni, T.V.; McCollum, J.M.; Cox, C.D.; Simpson, M.L.; Weinberger, L.S. Transcriptional burst frequency and burst size are equally modulated across the human genome. Proc. Natl. Acad. Sci. USA
**2012**, 109, 17454–17459. [Google Scholar] [CrossRef] - Suter, D.M.; Molina, N.; Gatfield, D.; Schneider, K.; Schibler, U.; Naef, F. Mammalian genes are transcribed with widely different bursting kinetics. Science
**2011**, 332, 472–474. [Google Scholar] [CrossRef] - Senecal, A.; Munsky, B.; Proux, F.; Ly, N.; Braye, F.E.; Zimmer, C.; Mueller, F.; Darzacq, X. Transcription factors modulate c-Fos transcriptional bursts. Cell Rep.
**2014**, 8, 75–83. [Google Scholar] [CrossRef] - Li, C.; Cesbron, F.; Oehler, M.; Brunner, M.; Höfer, T. Frequency modulation of transcriptional bursting enables sensitive and rapid gene regulation. Cell Syst.
**2018**, 6, 409–423. [Google Scholar] [CrossRef] - Raser, J.M.; O’Shea, E.K. Control of stochasticity in eukaryotic gene expression. Science
**2004**, 304, 1811–1814. [Google Scholar] [CrossRef] - Dey, S.S.; Foley, J.E.; Limsirichai, P.; Schaffer, D.V.; Arkin, A.P. Orthogonal control of expression mean and variance by epigenetic features at different genomic loci. Mol. Syst. Biol.
**2015**, 11, 806. [Google Scholar] [CrossRef] - Wong, V.C.; Bass, V.L.; Bullock, M.E.; Chavali, A.K.; Lee, R.E.; Mothes, W.; Gaudet, S.; Miller-Jensen, K. NF-κB-chromatin interactions drive diverse phenotypes by modulating transcriptional noise. Cell Rep.
**2018**, 22, 585–599. [Google Scholar] [CrossRef] [PubMed] - Bartman, C.R.; Hamagami, N.; Keller, C.A.; Giardine, B.; Hardison, R.C.; Blobel, G.A.; Raj, A. Transcriptional burst initiation and polymerase pause release are key control points of transcriptional regulation. Mol. Cell
**2019**, 73, 519–532. [Google Scholar] [CrossRef] [PubMed] - Cavallaro, M.; Walsh, M.D.; Jones, M.; Teahan, J.; Tiberi, S.; Finkenstädt, B.; Hebenstreit, D. 3
^{′}-5^{′}crosstalk contributes to transcriptional bursting. Genome Biol.**2021**, 22, 1–20. [Google Scholar] [CrossRef] - Bullock, M.E.; Moreno-Martinez, N.; Miller-Jensen, K. A transcriptional cycling model recapitulates chromatin-dependent features of noisy inducible transcription. PLoS Comput. Biol.
**2022**, 18, e1010152. [Google Scholar] [CrossRef] [PubMed] - Brouwer, I.; Kerklingh, E.; van Leeuwen, F.; Lenstra, T.L. Dynamic epistasis analysis reveals how chromatin remodeling regulates transcriptional bursting. Nat. Struct. Mol. Biol.
**2023**, 30, 692–702. [Google Scholar] [CrossRef] [PubMed] - Mbonye, U.; Karn, J. The molecular basis for human immunodeficiency virus latency. Annu. Rev. Virol.
**2017**, 4, 261–285. [Google Scholar] [CrossRef] - Richter, W.F.; Nayak, S.; Iwasa, J.; Taatjes, D.J. The Mediator complex as a master regulator of transcription by RNA polymerase II. Nat. Rev. Mol. Cell Biol.
**2022**, 23, 732–749. [Google Scholar] [CrossRef] - Spudich, J.L.; Koshland, D.E. Non-genetic individuality: Chance in the single cell. Nature
**1976**, 262, 467–471. [Google Scholar] [CrossRef] - McAdams, H.H.; Arkin, A. Stochastic mechanisms in gene expression. Proc. Natl. Acad. Sci. USA
**1997**, 94, 814–819. [Google Scholar] [CrossRef] [PubMed] - Becskei, A.; Serrano, L. Engineering stability in gene networks by autoregulation. Nature
**2000**, 405, 590–593. [Google Scholar] [CrossRef] [PubMed] - Elowitz, M.B.; Levine, A.J.; Siggia, E.D.; Swain, P.S. Stochastic gene expression in a single cell. Science
**2002**, 297, 1183–1186. [Google Scholar] [CrossRef] [PubMed] - Arkin, A.; Ross, J.; McAdams, H.H. Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected Escherichia coli cells. Genetics
**1998**, 149, 1633–1648. [Google Scholar] [CrossRef] - Elowitz, M.B.; Leibler, S. A synthetic oscillatory network of transcriptional regulators. Nature
**2000**, 403, 335–338. [Google Scholar] [CrossRef] - Weinberger, L.S.; Burnett, J.C.; Toettcher, J.E.; Arkin, A.P.; Schaffer, D.V. Stochastic gene expression in a lentiviral positive-feedback loop: HIV-1 Tat fluctuations drive phenotypic diversity. Cell
**2005**, 122, 169–182. [Google Scholar] [CrossRef] - Gillespie, D.T. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem.
**1977**, 81, 2340–2361. [Google Scholar] [CrossRef] - Morton, E.L.; Forst, C.V.; Zheng, Y.; DePaula-Silva, A.B.; Ramirez, N.G.P.; Planelles, V.; D’Orso, I. Transcriptional circuit fragility influences HIV proviral fate. Cell Rep.
**2019**, 27, 154–171. [Google Scholar] [CrossRef] - Lu, Y.; Bohn-Wippert, K.; Pazerunas, P.J.; Moy, J.M.; Singh, H.; Dar, R.D. Screening for gene expression fluctuations reveals latency-promoting agents of HIV. Proc. Natl. Acad. Sci. USA
**2021**, 118, e2012191118. [Google Scholar] [CrossRef] - Rouzine, I.M.; Razooky, B.S.; Weinberger, L.S. Stochastic variability in HIV affects viral eradication. Proc. Natl. Acad. Sci. USA
**2014**, 111, 13251–13252. [Google Scholar] [CrossRef] - Damour, A.; Slaninova, V.; Radulescu, O.; Bertrand, E.; Basyuk, E. Transcriptional Stochasticity as a Key Aspect of HIV-1 Latency. Viruses
**2023**, 15, 1969. [Google Scholar] [CrossRef] [PubMed] - Chavali, A.K.; Wong, V.C.; Miller-Jensen, K. Distinct promoter activation mechanisms modulate noise-driven HIV gene expression. Sci. Rep.
**2015**, 5, 17661. [Google Scholar] [CrossRef] [PubMed] - Bass, V.L.; Wong, V.C.; Bullock, M.E.; Gaudet, S.; Miller-Jensen, K. TNF stimulation primarily modulates transcriptional burst size of NF-κB-regulated genes. Mol. Syst. Biol.
**2021**, 17, e10127. [Google Scholar] [CrossRef] [PubMed] - Miller-Jensen, K.; Dey, S.S.; Pham, N.; Foley, J.E.; Arkin, A.P.; Schaffer, D.V. Chromatin accessibility at the HIV LTR promoter sets a threshold for NF-κB mediated viral gene expression. Integr. Biol.
**2012**, 4, 661–671. [Google Scholar] [CrossRef] [PubMed] - Razooky, B.S.; Cao, Y.; Hansen, M.M.; Perelson, A.S.; Simpson, M.L.; Weinberger, L.S. Nonlatching positive feedback enables robust bimodality by decoupling expression noise from the mean. PLoS Biol.
**2017**, 15, e2000841. [Google Scholar] [CrossRef] [PubMed] - Cao, Y.; Lei, X.; Ribeiro, R.M.; Perelson, A.S.; Liang, J. Probabilistic control of HIV latency and transactivation by the Tat gene circuit. Proc. Natl. Acad. Sci. USA
**2018**, 115, 12453–12458. [Google Scholar] [CrossRef] - Boettiger, A.N.; Levine, M. Synchronous and stochastic patterns of gene activation in the Drosophila embryo. Science
**2009**, 325, 471–473. [Google Scholar] [CrossRef] - Shao, W.; Zeitlinger, J. Paused RNA polymerase II inhibits new transcriptional initiation. Nat. Genet.
**2017**, 49, 1045–1051. [Google Scholar] [CrossRef] - Gressel, S.; Schwalb, B.; Cramer, P. The pause-initiation limit restricts transcription activation in human cells. Nat. Commun.
**2019**, 10, 3603. [Google Scholar] [CrossRef]

**Figure 1.**The basic viral dynamics model. (

**A**) Cell population processes that are simulated in the mathematical model (Equation (1)). (

**B**,

**C**) Example trajectory of viral load (V) and uninfected target cells (CD4 T cells, T) when ${R}_{0}>1$ (red) and ${R}_{0}<1$ (blue). Graphs were generated by numerically integrating Equation (1) with parameters $\lambda =100$ cells/µL, $\beta =7\times {10}^{-5}$ or 0/day/(virus/mL), $k\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}150$ virus/cell, ${d}_{T}=0.05/$day, ${d}_{I}=0.7/$day, $c=1.7$/day using JSim v2.21 with standard integration parameters [32]. (Figure adapted from Hill [18] and created by part (

**A**) with BioRender.com).

**Figure 2.**Schematic of a viral dynamics model involving multiple populations of infected cells. (

**A**) A flow diagram between two populations of uninfected cells (T, T${}_{2}$), virus infected cells (I, I${}_{2}$), a latently infected cell population (L) and free HIV-1 (V) according to the ODE shown in Equation (4). Red X’es indicate the complete interruption of viral infection during fully effective therapy. (

**B**) The decay of the viral load in multiple stages is shown. (

**C**) The decay of distinct host-cell populations, as predicted by the model, are depicted. Time-series were integrated using JSim v2.21 with standard integration parameters [32]. (Figure adapted from Hill [18] and created by part (

**A**) with BioRender.com).

**Figure 3.**Schematic of latent reservoir dynamics. The latent reservoir involves long-lived resting memory CD4 cells, with potentially integrated HIV-1 provirus. At subcritical viral replication rate (${R}_{0}<1$), the persistence of virus represents the maintenance of the latent reservoir. Infected host cells within this reservoir may occasionally die (marked by skull and bones), proliferate, or reactivate. A large proliferation rate leads to a decrease in viral diversity within the latent reservoir. New infections (bursts) are either completely blocked (Reactivation blocked) or may occasionally occur by stochastic processes. But continuous chains of replication are inhibited in the ${R}_{0}<1$ regime (Infection controlled). After treatment interruption (${R}_{0}>1$), reactivated cells can produce virus that infects other host cells yielding to exponential growth in viral load (see Section 4, in particular Figure 4C). (Figure adapted from Hill [18] and created with BioRender.com).

**Figure 4.**Mathematical model of the host and viral phases of the HIV-1 transcriptional program [121]. (

**A**) Simplified model of the host and viral phases of the HIV-1 transcriptional program. (

**B**) Experimental data and fitted stochastic computer simulation of a host cell infected by HIV-1 in the host phase without feedback by Tat. (

**C**) Experimental data and fitted stochastic computer simulation of a host cell infected by HIV-1 in the viral phase with feedback by Tat.

**Figure 5.**Positive feedback on the RNAPII pause release rate (PPRR) activation does not influence bimodality of the mRNA and protein distributions in the three-state transcriptional cycling model [109]. Updated three-state promoter system with HIV-1 nucleosome remodeling, RelA recruitment, and Tat-mediated transcript elongation, which is amplified via positive feedback. Positive feedback is modeled as a saturating function with an amplitude, A, and half-max, K.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

D’Orso, I.; Forst, C.V.
Mathematical Models of HIV-1 Dynamics, Transcription, and Latency. *Viruses* **2023**, *15*, 2119.
https://doi.org/10.3390/v15102119

**AMA Style**

D’Orso I, Forst CV.
Mathematical Models of HIV-1 Dynamics, Transcription, and Latency. *Viruses*. 2023; 15(10):2119.
https://doi.org/10.3390/v15102119

**Chicago/Turabian Style**

D’Orso, Iván, and Christian V. Forst.
2023. "Mathematical Models of HIV-1 Dynamics, Transcription, and Latency" *Viruses* 15, no. 10: 2119.
https://doi.org/10.3390/v15102119