American Institute of Mathematical Sciences

Advanced
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.  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.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  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.  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.  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.   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.  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.  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.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  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.  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.  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.   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.  doi: 10.4049/jimmunol.180.3.1414.  Google Scholar

[1]

Liancheng Wang, Sean Ellermeyer. HIV infection and CD4+ T cell dynamics. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1417-1430. doi: 10.3934/dcdsb.2006.6.1417
[2]

Angelo Favini. A general approach to identification problems and applications to partial differential equations. Conference Publications, 2015, 2015 (special) : 428-435. doi: 10.3934/proc.2015.0428
[3]

Zhixing Hu, Weijuan Pang, Fucheng Liao, Wanbiao Ma. Analysis of a CD4$^+$ T cell viral infection model with a class of saturated infection rate. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 735-745. doi: 10.3934/dcdsb.2014.19.735
[4]

Loïc Barbarroux, Philippe Michel, Mostafa Adimy, Fabien Crauste. A multiscale model of the CD8 T cell immune response structured by intracellular content. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3969-4002. doi: 10.3934/dcdsb.2018120
[5]

Yuchi Qiu, Weitao Chen, Qing Nie. Stochastic dynamics of cell lineage in tissue homeostasis. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3971-3994. doi: 10.3934/dcdsb.2018339
[6]

Tayel Dabbous. Identification for systems governed by nonlinear interval differential equations. Journal of Industrial & Management Optimization, 2012, 8 (3) : 765-780. doi: 10.3934/jimo.2012.8.765
[7]

Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems & Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025
[8]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481
[9]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[10]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[11]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[12]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[13]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032
[14]

Scott R. Pope, Laura M. Ellwein, Cheryl L. Zapata, Vera Novak, C. T. Kelley, Mette S. Olufsen. Estimation and identification of parameters in a lumped cerebrovascular model. Mathematical Biosciences & Engineering, 2009, 6 (1) : 93-115. doi: 10.3934/mbe.2009.6.93
[15]

Rinaldo M. Colombo, Andrea Corli. Dynamic parameters identification in traffic flow modeling. Conference Publications, 2005, 2005 (Special) : 190-199. doi: 10.3934/proc.2005.2005.190
[16]

Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345
[17]

Cliburn Chan, Andrew J.T. George, Jaroslav Stark. T cell sensitivity and specificity - kinetic proofreading revisited. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 343-360. doi: 10.3934/dcdsb.2003.3.343
[18]

Lisette dePillis, Trevor Caldwell, Elizabeth Sarapata, Heather Williams. Mathematical modeling of regulatory T cell effects on renal cell carcinoma treatment. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 915-943. doi: 10.3934/dcdsb.2013.18.915
[19]

Rafael Potrie. Partial hyperbolicity and foliations in $\mathbb{T}^3$. Journal of Modern Dynamics, 2015, 9: 81-121. doi: 10.3934/jmd.2015.9.81
[20]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences