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Turing Patterns from Dynamics of Early HIV Infection

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Abstract

We have developed a mathematical model for in-host virus dynamics that includes spatial chemotaxis and diffusion across a two-dimensional surface representing the vaginal or rectal epithelium at primary HIV infection. A linear stability analysis of the steady state solutions identified conditions for Turing instability pattern formation. We have solved the model equations numerically using parameter values obtained from previous experimental results for HIV infections. Simulations of the model for this surface show hot spots of infection. Understanding this localization is an important step in the ability to correctly model early HIV infection. These spatial variations also have implications for the development and effectiveness of microbicides against HIV.

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References

  • Boukari, H., Brichacek, B., Stratton, P., Mahoney, S. F., Lifson, J. D., Margolis, L., & Nossal, R. (2009). Movements of HIV-virions in human cervical mucus. Biomacromolecules, 10(9), 2482–2488.

    Article  Google Scholar 

  • Chen, H. Y., Di Mascio, M., Perelson, A. S., Ho, D. D., & Zhang, L. (2007). Determination of virus burst size in vivo using a single-cycle SIV in rhesus macaques. Proc. Natl. Acad. Sci. USA, 104(48), 19079–19084.

    Article  Google Scholar 

  • Cooper, S. B., & Maini, P. K. (2012). The mathematics of nature at the Alan Turing centenary. Interface Focus, 2(4), 393–396.

    Article  Google Scholar 

  • De Boer, R. J. (2007). Understanding the failure of CD8+ T-cell vaccination against simian/human immunodeficiency virus. J. Virol., 81(6), 2838–2848.

    Article  Google Scholar 

  • Edelstein-Keshet, L. (2005). Classics in applied mathematics: Vol. 46. Mathematical models in biology. Philadelphia: SIAM.

    Book  MATH  Google Scholar 

  • Ewers, H., Jacobsen, V., Klotzsch, E., Smith, A. E., Helenius, A., & Sandoghdar, V. (2007). Label-free optical detection and tracking of single virions bound to their receptors in supported membrane bilayers. Nano Lett., 7(8), 2263–2266.

    Article  Google Scholar 

  • Haase, A. T. (2010). Targeting early infection to prevent HIV-1 mucosal transmission. Nature, 464, 217–223.

    Article  Google Scholar 

  • Haase, A. T., Henry, K., Zupancic, M., Sedgewick, G., Faust, R. A., Melroe, H., Cavert, W., Gebhard, K., Staskus, K., Zhang, Z. Q., Dailey, P. J., Balfour, H. H., Erice, A., & Perelson, A. S. (1996). Quantitative image analysis of HIV-1 infection in lymphoid tissue. Science, 274(5289), 985–989.

    Article  Google Scholar 

  • Harris, T. H., Banigan, E. J., Christian, D. A., Konradt, C., Tait Wojno, E. D., Norose, K., Wilson, E. H., John, B., Weninger, W., Luster, A. D., Liu, A. J., & Hunter, C. A. (2012). Generalized Levy walks and the role of chemokines in migration of effector CD8+ T cells. Nature, 486, 545–548.

    Google Scholar 

  • Hillen, T., & Painter, K. J. (2009). A user’s guide to PDE models for chemotaxis. J. Math. Biol., 58, 183–217.

    Article  MathSciNet  MATH  Google Scholar 

  • Horstmann, D. (2003). From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I. Jahresber. Dtsch. Math.-Ver., 105(3), 103–165.

    MathSciNet  MATH  Google Scholar 

  • Hurwitz, A. (1964). On the conditions under which an equation has only roots with negative real parts. In Selected papers on mathematical trends in control theory (pp. 72–82). New York: Dover.

    Google Scholar 

  • Jin, T., Xu, X., & Hereld, D. (2008). Chemotaxis, chemokine receptors and human disease. Cytokine, 44(1), 1–8.

    Article  Google Scholar 

  • Keele, B. F., Giorgi, E. E., Salazar-Gonzalez, J. F., Decker, J. M., Pham, K. T., Salazar, M. G., Sun, C., Grayson, T., Wang, S., Li, H., Wei, X., Jiang, C., Kirchherr, J. L., Gao, F., Anderson, J. A., Ping, L.-H., Swanstrom, R., Tomaras, G. D., Blattner, W. A., Goepfert, P. A., Kilby, J. M., Saag, M. S., Delwart, E. L., Busch, M. P., Cohen, M. S., Montefiori, D. C., Haynes, B. F., Gaschen, B., Athreya, G. S., Lee, H. Y., Wood, N., Seoighe, C., Perelson, A. S., Bhattacharya, T., Korber, B. T., Hahn, B. H., & Shaw, G. M. (2008). Identification and characterization of transmitted and early founder virus envelopes in primary HIV-1 infection. Proc. Natl. Acad. Sci. USA, 105(21), 7552–7557.

    Article  Google Scholar 

  • Klasse, P. J., Shattock, R. J., & Moore, J. P. (2006). Which topical microbicides for blocking HIV-1 transmission will work in the real world? PLoS Med., 3(9), e351.

    Article  Google Scholar 

  • Lin, F., & Butcher, E. C. (2006). T cell chemotaxis in a simple microfluidic device. Lab Chip, 6(11), 1462–1469.

    Article  Google Scholar 

  • Liu, Q. X., & Zhen, J. (2007). Formation of spatial patterns in an epidemic model with constant removal rate of the infectives. J. Stat. Mech. Theory Exp., 2007(05), P05002.

    Article  Google Scholar 

  • Maini, P. K., Woolley, T. E., Baker, R. E., Gaffney, E. A., & Lee, S. S. (2012). Turing’s model for biological pattern formation and the robustness problem. Interface Focus, 2(4), 487–496.

    Article  Google Scholar 

  • Mandl, J. N., Regoes, R. R., Garber, D. A., & Feinberg, M. B. (2007). Estimating the effectiveness of simian immunodeficiency virus-specific CD8+ T cells from the dynamics of viral immune escape. J. Virol., 81(21), 11982–11991.

    Article  Google Scholar 

  • Meinhardt, H. (2012). Turing’s theory of morphogenesis of 1952 and the subsequent discovery of the crucial role of local self-enhancement and long-range inhibition. Interface Focus, 2(4), 407–416.

    Article  MathSciNet  Google Scholar 

  • Miller, M. J., Wei, S. H., Parker, I., & Cahalan, M. D. (2002). Two-photon imaging of lymphocyte motility and antigen response in intact lymph node. Science, 296(5574), 1869–1873.

    Article  Google Scholar 

  • Miller, M. J., Wei, S. H., Cahalan, M. D., & Parker, I. (2003). Autonomous T cell trafficking examined in vivo with intravital two-photon microscopy. Proc. Natl. Acad. Sci. USA, 100(5), 2604–2609.

    Article  Google Scholar 

  • Murray, J. D. (2002). Biomathematics: Vol. 19. Mathematical biology: an introduction. Berlin: Springer.

    Google Scholar 

  • Murray, J. M., Kaufmann, G., Kelleher, A. D., & Cooper, D. A. (1998). A model of primary HIV-1 infection. Math. Biosci., 154(2), 57–85.

    Article  MATH  Google Scholar 

  • Murray, J. M., Emery, S., Kelleher, A. D., Law, M., Chen, J., Hazuda, D. J., Nguyen, B.-Y. T., Teppler, H., & Cooper, D. A. (2007). Antiretroviral therapy with the integrase inhibitor Raltegravir alters decay kinetics of HIV, significantly reducing the second phase. AIDS, 21, 2315–2321.

    Article  Google Scholar 

  • Murray, J. M., Kelleher, A. D., & Cooper, D. A. (2011). Timing of the components of the HIV life cycle in productively infected CD4+ T cells in a population of HIV-infected individuals. J. Virol., 85(20), 10798–10805.

    Article  Google Scholar 

  • Nelson, P. W., Murray, J. D., & Perelson, A. S. (2000). A model of HIV-1 pathogenesis that includes an intracellular delay. Math. Biosci., 163(2), 201–215.

    Article  MathSciNet  MATH  Google Scholar 

  • Nowak, M. A., & May, R. (2000). Virus dynamics: mathematical principles of immunology and virology. London: Oxford University Press.

    MATH  Google Scholar 

  • Nowak, M. A., Bonhoeffer, S., & Hill, A. M. (1996). Viral dynamics in Hepatitis B virus infection. Proc. Natl. Acad. Sci. USA, 93(9), 4398–4402.

    Article  Google Scholar 

  • Ouyang, Q., & Swinney, H. L. (1991). Transition from a uniform state to hexagonal and striped Turing patterns. Nature, 352, 610–612.

    Article  Google Scholar 

  • Perelson, A. S., Neumann, A. U., Markowitz, M., Leonard, J. M., & Ho, D. D. (1996). HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science, 271, 1582–1586.

    Article  Google Scholar 

  • Perelson, A. S., Essunger, P., Cao, M., Vesanen, Y., Hurley, K., Saksela, A., Markowitz, M., & Ho, D. D. (1997). Decay characteristics of HIV-1-infected compartments during combination therapy. Nature, 387, 188–191.

    Article  Google Scholar 

  • Ramratnam, B., Bonhoeffer, S., Binley, J., Hurley, A., Zhang, L., Mittler, J. E., Markowitz, M., Moore, J. P., Perelson, A. S., & Ho, D. D. (1999). Rapid production and clearance of HIV-1 and Hepatitis C virus assessed by large volume plasma apheresis. Lancet, 354(9192), 1782–1785.

    Article  Google Scholar 

  • Reilly, C., Wietgrefe, S., Sedgewick, G., & Haase, A. (2007). Determination of simian immunodeficiency virus production by infected activated and resting cells. AIDS, 21(2), 163.

    Article  Google Scholar 

  • Ribeiro, R. M., Qin, L., Chavez, L. L., Li, D., Self, S. G., & Perelson, A. S. (2010). Estimation of the initial viral growth rate and basic reproductive number during acute HIV-1 infection. J. Virol., 84(12), 6096–6102.

    Article  Google Scholar 

  • Turing, A. M. (1952). The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B, Biol. Sci., 237(641), 37–72.

    Article  Google Scholar 

  • Velasquez, T. J. L. (2004). Point dynamics for a singular limit of the Keller–Segel model. SIAM J. Appl. Math., 64(4), 1198–1223.

    Article  MathSciNet  Google Scholar 

  • Wei, X., Ghosn, S. K., Taylor, M. E., Johnson, V. A., Emini, E. A., Deutsch, P., Lifson, J. D., Bonhoeffer, S., Nowak, M. A., Hahn, B. H., Saag, M. S., & Shaw, G. M. (1995). Viral dynamics in human immunodeficiency virus type 1 infection. Nature, 373, 117–122.

    Article  Google Scholar 

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Acknowledgements

We greatly acknowledge discussions with T.A.M. Langlands and P.J. Klasse on aspects of this work. This research was assisted through the support from the Australian Commonwealth Government (ARC DP1094680) and the UNSW Goldstar Scheme.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    O. Stancevic, C. N. Angstmann, J. M. Murray & B. I. Henry

  2. The Kirby Institute, University of New South Wales, Sydney, NSW, 2052, Australia

    J. M. Murray

Authors
  1. O. Stancevic
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  2. C. N. Angstmann
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  3. J. M. Murray
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  4. B. I. Henry
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Correspondence to B. I. Henry.

Appendix: Showing Conditions for Turing Instability

Appendix: Showing Conditions for Turing Instability

Proposition 1

Consider the characteristic polynomial (11). Provided a 1,a 2,b 1,b 2,b 3,c 1,c 2,c 3, and c 4 are all positive, conditions (S1)–(S4) hold.

Proof

We demonstrate each of the conditions in turn below.

  1. (S1)
  2. (S2)
  3. (S3)
  4. (S4)

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Stancevic, O., Angstmann, C.N., Murray, J.M. et al. Turing Patterns from Dynamics of Early HIV Infection. Bull Math Biol 75, 774–795 (2013). https://doi.org/10.1007/s11538-013-9834-5

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