American Institute of Mathematical Sciences

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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

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), 63-71. doi: 10.1016/j.mbs.2014.03.002.  Google Scholar

[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), 3414-3424. doi: 10.1016/S0006-3495(03)70063-0.  Google Scholar

[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), 1011-1031. Google Scholar

[4]

F. J. Burns and I. F. Tannock, On the existence of a go-phase in the cell cycle, Cell Proliferation, 3 (1970), 321-334. doi: 10.1111/j.1365-2184.1970.tb00340.x.  Google Scholar

[5]

W. B. Cannon, The Wisdom of the Body,, 1932., ().   Google Scholar

[6]

A. Freitas and J. Chen, Introduction: Regulation of lymphocyte homeostasis, Microbes and Infection, 4 (2002), 529-530. doi: 10.1016/S1286-4579(02)01568-X.  Google Scholar

[7]

A. Freitas and B. Rocha, Population biology of lymphocytes: The flight for survival, Annual Review of Immunology, 18 (2000), 83-111. doi: 10.1146/annurev.immunol.18.1.83.  Google Scholar

[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), 950-957. doi: 10.4049/jimmunol.179.2.950.  Google Scholar

[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), 183-200. doi: 10.1016/j.jim.2005.01.011.  Google Scholar

[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), 16885-16890. doi: 10.1073/pnas.0407417101.  Google Scholar

[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), 475-481. doi: 10.1038/ni1326.  Google Scholar

[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), 3985-3993. doi: 10.4049/jimmunol.1203213.  Google Scholar

[13]

S. C. Jameson, T cell homeostasis: Keeping useful T cells alive and live T cells useful, Seminars in Immunology, 17 (2005), 231-237. doi: 10.1016/j.smim.2005.02.003.  Google Scholar

[14]

S. C. Jameson, Maintaining the norm: T-cell homeostasis, Nature Reviews Immunology, 2 (2002), 547-556. Google Scholar

[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), 1649-1670. doi: 10.1007/s11538-009-9418-6.  Google Scholar

[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), 21-44. doi: 10.1007/s11538-007-9239-4.  Google Scholar

[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), 275-283. doi: 10.1016/S0022-5193(03)00245-5.  Google Scholar

[18]

C. R. Parish, Fluorescent dyes for lymphocyte migration and proliferation studies, Immunol Cell Biol, 77 (1999), 499-508. doi: 10.1046/j.1440-1711.1999.00877.x.  Google Scholar

[19]

J. A. Smith and L. Martin, Do Cells Cycle?, Proceedings of the National Academy of Sciences, 70 (1973), 1263-1267. doi: 10.1073/pnas.70.4.1263.  Google Scholar

[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), 2057-2762. Google Scholar

[21]

I. V. Numerics, Imsl Fortran 90 Library: User's Guide, Visual Numerics, 1996. Google Scholar

[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), 1414-1422. doi: 10.4049/jimmunol.180.3.1414.  Google Scholar

show all references

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), 63-71. doi: 10.1016/j.mbs.2014.03.002.  Google Scholar

[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), 3414-3424. doi: 10.1016/S0006-3495(03)70063-0.  Google Scholar

[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), 1011-1031. Google Scholar

[4]

F. J. Burns and I. F. Tannock, On the existence of a go-phase in the cell cycle, Cell Proliferation, 3 (1970), 321-334. doi: 10.1111/j.1365-2184.1970.tb00340.x.  Google Scholar

[5]

W. B. Cannon, The Wisdom of the Body,, 1932., ().   Google Scholar

[6]

A. Freitas and J. Chen, Introduction: Regulation of lymphocyte homeostasis, Microbes and Infection, 4 (2002), 529-530. doi: 10.1016/S1286-4579(02)01568-X.  Google Scholar

[7]

A. Freitas and B. Rocha, Population biology of lymphocytes: The flight for survival, Annual Review of Immunology, 18 (2000), 83-111. doi: 10.1146/annurev.immunol.18.1.83.  Google Scholar

[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), 950-957. doi: 10.4049/jimmunol.179.2.950.  Google Scholar

[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), 183-200. doi: 10.1016/j.jim.2005.01.011.  Google Scholar

[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), 16885-16890. doi: 10.1073/pnas.0407417101.  Google Scholar

[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), 475-481. doi: 10.1038/ni1326.  Google Scholar

[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), 3985-3993. doi: 10.4049/jimmunol.1203213.  Google Scholar

[13]

S. C. Jameson, T cell homeostasis: Keeping useful T cells alive and live T cells useful, Seminars in Immunology, 17 (2005), 231-237. doi: 10.1016/j.smim.2005.02.003.  Google Scholar

[14]

S. C. Jameson, Maintaining the norm: T-cell homeostasis, Nature Reviews Immunology, 2 (2002), 547-556. Google Scholar

[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), 1649-1670. doi: 10.1007/s11538-009-9418-6.  Google Scholar

[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), 21-44. doi: 10.1007/s11538-007-9239-4.  Google Scholar

[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), 275-283. doi: 10.1016/S0022-5193(03)00245-5.  Google Scholar

[18]

C. R. Parish, Fluorescent dyes for lymphocyte migration and proliferation studies, Immunol Cell Biol, 77 (1999), 499-508. doi: 10.1046/j.1440-1711.1999.00877.x.  Google Scholar

[19]

J. A. Smith and L. Martin, Do Cells Cycle?, Proceedings of the National Academy of Sciences, 70 (1973), 1263-1267. doi: 10.1073/pnas.70.4.1263.  Google Scholar

[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), 2057-2762. Google Scholar

[21]

I. V. Numerics, Imsl Fortran 90 Library: User's Guide, Visual Numerics, 1996. Google Scholar

[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), 1414-1422. doi: 10.4049/jimmunol.180.3.1414.  Google Scholar

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