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), 63-71. 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), 3414-3424. 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), 1011-1031.

[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.

[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), 529-530. 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), 83-111. 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), 950-957. 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), 183-200. 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), 16885-16890. 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), 475-481. 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), 3985-3993. 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), 231-237. doi: 10.1016/j.smim.2005.02.003.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

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

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.

[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.

[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.

[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.

[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), 529-530. 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), 83-111. 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), 950-957. 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), 183-200. 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), 16885-16890. 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), 475-481. 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), 3985-3993. 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), 231-237. doi: 10.1016/j.smim.2005.02.003.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

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

[1]

Liancheng Wang, Sean Ellermeyer. HIV infection and CD4+ T cell dynamics. Discrete and 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]

Loïc Barbarroux, Philippe Michel, Mostafa Adimy, Fabien Crauste. A multiscale model of the CD8 T cell immune response structured by intracellular content. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3969-4002. doi: 10.3934/dcdsb.2018120
[4]

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 and Continuous Dynamical Systems - B, 2014, 19 (3) : 735-745. doi: 10.3934/dcdsb.2014.19.735
[5]

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

Pingping Niu, Shuai Lu, Jin Cheng. On periodic parameter identification in stochastic differential equations. Inverse Problems and Imaging, 2019, 13 (3) : 513-543. doi: 10.3934/ipi.2019025
[7]

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

Linghui Yu, Zhipeng Qiu, Ting Guo. Modeling the effect of activation of CD4$^+$ T cells on HIV dynamics. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4491-4513. doi: 10.3934/dcdsb.2021238
[9]

Nguyen Thi Van Anh, Bui Thi Hai Yen. Source identification problems for abstract semilinear nonlocal differential equations. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022030
[10]

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
[11]

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
[12]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete and Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481
[13]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[14]

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

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[16]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[17]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032
[18]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[19]

Runzhang Xu. Preface: Special issue on advances in partial differential equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : i-i. doi: 10.3934/dcdss.2021137
[20]

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

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy