Parameters identification for a model of T cell homeostasis

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2015, 12(5): 917-936. doi: 10.3934/mbe.2015.12.917

Parameters identification for a model of T cell homeostasis

Houssein Ayoub ^1, , Bedreddine Ainseba ^1, , Michel Langlais ^1, and Rodolphe Thiébaut ^2,

1.	IMB UMR CNRS 5251, Bordeaux University, 3 Place de la Victoire, 33076 Bordeaux Cedex, France, France, France
2.	INSERM U897, ISPED, Bordeaux University, Bordeaux, France

Received November 2014 Revised April 2015 Published June 2015

Abstract
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In this study, we consider a model of T cell homeostasis based on the Smith-Martin model. This nonlinear model is structured by age and CD44 expression. First, we establish the mathematical well-posedness of the model system. Next, we prove the theoretical identifiability regarding the up-regulation of CD44, the proliferation time phase and the rate of entry into division, by using the experimental data. Finally, we compare two versions of the Smith-Martin model and we identify which model fits the experimental data best.

Keywords: partial differential equations, parameters identification., T cell homeostasis, CD44.

Mathematics Subject Classification: Primary: 92B05, 58J45; Secondary: 93B3.

Citation: Houssein Ayoub, Bedreddine Ainseba, Michel Langlais, Rodolphe Thiébaut. Parameters identification for a model of T cell homeostasis. Mathematical Biosciences & Engineering, 2015, 12 (5) : 917-936. doi: 10.3934/mbe.2015.12.917

References:

[1]	H. Ayoub, B. E. Ainseba, M. Langlais, T. Hogan, R. Callard, B. Seddon and R. Thiébaut, Parameter identification for model of T cell proliferation in Lymphopenia conditions,, Mathematical biosciences, 251* (2014), 63. doi: 10.1016/j.mbs.2014.03.002.*
[2]	S. Bernard, L. Pujo-Menjouet and M. C Mackey, Analysis of cell kinetics using a cell division marker: Mathematical modeling of experimental data,, Biophysical Journal, 84 (2003), 3414. doi: 10.1016/S0006-3495(03)70063-0.
[3]	R. J. Boer, V. V. Ganusov, D. Milutinovic, P. D. Hodgkin and A. S. Perelson, Estimating lymphocyte division and death rates from CFSE data,, Bulletin of Mathematical Biology, 68 (2006), 1011.
[4]	F. J. Burns and I. F. Tannock, On the existence of a go-phase in the cell cycle,, Cell Proliferation, 3 (1970), 321. doi: 10.1111/j.1365-2184.1970.tb00340.x.
[5]	W. B. Cannon, The Wisdom of the Body,, 1932., ().
[6]	A. Freitas and J. Chen, Introduction: Regulation of lymphocyte homeostasis,, Microbes and Infection, 4 (2002), 529. doi: 10.1016/S1286-4579(02)01568-X.
[7]	A. Freitas and B. Rocha, Population biology of lymphocytes: The flight for survival,, Annual Review of Immunology, 18 (2000), 83. doi: 10.1146/annurev.immunol.18.1.83.
[8]	V. V. Ganusov, D. Milutinovic and R. J. De Boer, IL-2 regulates expansion of CD4+ T cell populations by affecting cell death: Insights from modeling CFSE data,, The Journal of Immunology, 179* (2007), 950. doi: 10.4049/jimmunol.179.2.950.*
[9]	V. V. Ganusov, S. S. Pilyugin, R. J. de Boer, K. Murali-Krishna, R. Ahmed and R. Antia, Quantifying cell turnover using CFSE data,, Journal of Immunological Methods, 298* (2005), 183. doi: 10.1016/j.jim.2005.01.011.*
[10]	A. W. Goldrath, C. J. Luckey, R. Park, C. Benoist and D. Mathis, The molecular program induced in T cells undergoing homeostatic proliferation,, Proceedings of the National Academy of Sciences of the United States of America, 101* (2004), 16885. doi: 10.1073/pnas.0407417101.*
[11]	S. E. Hamilton, M. C. Wolkers, S. P. Schoenberger and S. C. Jameson, The generation of protective memory-like CD8+ T cells during homeostatic proliferation requires CD4+ T cells,, Nat Immunol, 7 (2006), 475. doi: 10.1038/ni1326.
[12]	T. Hogan, A. Shuvaev, D. Commenges, A. Yates, R. Callard, R. Thiebaut and B. Seddon, Clonally Diverse T Cell Homeostasis Is Maintained by a Common Program of Cell-Cycle Control,, The Journal of Immunology, 190* (2013), 3985. doi: 10.4049/jimmunol.1203213.*
[13]	S. C. Jameson, T cell homeostasis: Keeping useful T cells alive and live T cells useful,, Seminars in Immunology, 17 (2005), 231. doi: 10.1016/j.smim.2005.02.003.
[14]	S. C. Jameson, Maintaining the norm: T-cell homeostasis,, Nature Reviews Immunology, 2 (2002), 547.
[15]	H. Lee, E. Hawkins, M. S. Zand, T. Mosmann, H. Wu, P. D. Hodgkin and A. S. Perelson, Interpreting CFSE Obtained Division Histories of B Cells in Vitro with Smith-Martin and Cyton Type Models,, Bulletin of Mathematical Biology, 71 (2009), 1649. doi: 10.1007/s11538-009-9418-6.
[16]	H. Lee and A. S. Perelson, Modeling T Cell Proliferation and Death in Vitro Based on Labeling Data: Generalizations of the Smith-Martin Cell Cycle Model,, Bulletin of Mathematical Biology, 70 (2008), 21. doi: 10.1007/s11538-007-9239-4.
[17]	S. S. Pilyugin, V. V. Ganusov, K. Murali-Krishna, R. Ahmed and R. Antia, The rescaling method for quantifying the turnover of cell populations,, Journal of Theoretical Biology, 225* (2003), 275. doi: 10.1016/S0022-5193(03)00245-5.*
[18]	C. R. Parish, Fluorescent dyes for lymphocyte migration and proliferation studies,, Immunol Cell Biol, 77 (1999), 499. doi: 10.1046/j.1440-1711.1999.00877.x.
[19]	J. A. Smith and L. Martin, Do Cells Cycle?,, Proceedings of the National Academy of Sciences, 70 (1973), 1263. doi: 10.1073/pnas.70.4.1263.
[20]	C. Tanchot, F. A. Lemonnier, B. Pérarnau, A. A. Freitas and B. Rocha, Differential requirements for survival and proliferation of CD8 naïve or memory T cells,, Science, 276* (1997), 2057.*
[21]	I. V. Numerics, Imsl Fortran 90 Library: User's Guide,, Visual Numerics, (1996).
[22]	A. Yates, M. Saini, A. Mathiot and B. Seddon, Mathematical modeling reveals the biological program regulating lymphopenia-induced proliferation,, The Journal of Immunology, 180* (2008), 1414. doi: 10.4049/jimmunol.180.3.1414.*

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References:

[1]	H. Ayoub, B. E. Ainseba, M. Langlais, T. Hogan, R. Callard, B. Seddon and R. Thiébaut, Parameter identification for model of T cell proliferation in Lymphopenia conditions,, Mathematical biosciences, 251* (2014), 63. doi: 10.1016/j.mbs.2014.03.002.*
[2]	S. Bernard, L. Pujo-Menjouet and M. C Mackey, Analysis of cell kinetics using a cell division marker: Mathematical modeling of experimental data,, Biophysical Journal, 84 (2003), 3414. doi: 10.1016/S0006-3495(03)70063-0.
[3]	R. J. Boer, V. V. Ganusov, D. Milutinovic, P. D. Hodgkin and A. S. Perelson, Estimating lymphocyte division and death rates from CFSE data,, Bulletin of Mathematical Biology, 68 (2006), 1011.
[4]	F. J. Burns and I. F. Tannock, On the existence of a go-phase in the cell cycle,, Cell Proliferation, 3 (1970), 321. doi: 10.1111/j.1365-2184.1970.tb00340.x.
[5]	W. B. Cannon, The Wisdom of the Body,, 1932., ().
[6]	A. Freitas and J. Chen, Introduction: Regulation of lymphocyte homeostasis,, Microbes and Infection, 4 (2002), 529. doi: 10.1016/S1286-4579(02)01568-X.
[7]	A. Freitas and B. Rocha, Population biology of lymphocytes: The flight for survival,, Annual Review of Immunology, 18 (2000), 83. doi: 10.1146/annurev.immunol.18.1.83.
[8]	V. V. Ganusov, D. Milutinovic and R. J. De Boer, IL-2 regulates expansion of CD4+ T cell populations by affecting cell death: Insights from modeling CFSE data,, The Journal of Immunology, 179* (2007), 950. doi: 10.4049/jimmunol.179.2.950.*
[9]	V. V. Ganusov, S. S. Pilyugin, R. J. de Boer, K. Murali-Krishna, R. Ahmed and R. Antia, Quantifying cell turnover using CFSE data,, Journal of Immunological Methods, 298* (2005), 183. doi: 10.1016/j.jim.2005.01.011.*
[10]	A. W. Goldrath, C. J. Luckey, R. Park, C. Benoist and D. Mathis, The molecular program induced in T cells undergoing homeostatic proliferation,, Proceedings of the National Academy of Sciences of the United States of America, 101* (2004), 16885. doi: 10.1073/pnas.0407417101.*
[11]	S. E. Hamilton, M. C. Wolkers, S. P. Schoenberger and S. C. Jameson, The generation of protective memory-like CD8+ T cells during homeostatic proliferation requires CD4+ T cells,, Nat Immunol, 7 (2006), 475. doi: 10.1038/ni1326.
[12]	T. Hogan, A. Shuvaev, D. Commenges, A. Yates, R. Callard, R. Thiebaut and B. Seddon, Clonally Diverse T Cell Homeostasis Is Maintained by a Common Program of Cell-Cycle Control,, The Journal of Immunology, 190* (2013), 3985. doi: 10.4049/jimmunol.1203213.*
[13]	S. C. Jameson, T cell homeostasis: Keeping useful T cells alive and live T cells useful,, Seminars in Immunology, 17 (2005), 231. doi: 10.1016/j.smim.2005.02.003.
[14]	S. C. Jameson, Maintaining the norm: T-cell homeostasis,, Nature Reviews Immunology, 2 (2002), 547.
[15]	H. Lee, E. Hawkins, M. S. Zand, T. Mosmann, H. Wu, P. D. Hodgkin and A. S. Perelson, Interpreting CFSE Obtained Division Histories of B Cells in Vitro with Smith-Martin and Cyton Type Models,, Bulletin of Mathematical Biology, 71 (2009), 1649. doi: 10.1007/s11538-009-9418-6.
[16]	H. Lee and A. S. Perelson, Modeling T Cell Proliferation and Death in Vitro Based on Labeling Data: Generalizations of the Smith-Martin Cell Cycle Model,, Bulletin of Mathematical Biology, 70 (2008), 21. doi: 10.1007/s11538-007-9239-4.
[17]	S. S. Pilyugin, V. V. Ganusov, K. Murali-Krishna, R. Ahmed and R. Antia, The rescaling method for quantifying the turnover of cell populations,, Journal of Theoretical Biology, 225* (2003), 275. doi: 10.1016/S0022-5193(03)00245-5.*
[18]	C. R. Parish, Fluorescent dyes for lymphocyte migration and proliferation studies,, Immunol Cell Biol, 77 (1999), 499. doi: 10.1046/j.1440-1711.1999.00877.x.
[19]	J. A. Smith and L. Martin, Do Cells Cycle?,, Proceedings of the National Academy of Sciences, 70 (1973), 1263. doi: 10.1073/pnas.70.4.1263.
[20]	C. Tanchot, F. A. Lemonnier, B. Pérarnau, A. A. Freitas and B. Rocha, Differential requirements for survival and proliferation of CD8 naïve or memory T cells,, Science, 276* (1997), 2057.*
[21]	I. V. Numerics, Imsl Fortran 90 Library: User's Guide,, Visual Numerics, (1996).
[22]	A. Yates, M. Saini, A. Mathiot and B. Seddon, Mathematical modeling reveals the biological program regulating lymphopenia-induced proliferation,, The Journal of Immunology, 180* (2008), 1414. doi: 10.4049/jimmunol.180.3.1414.*

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