Advertisement

Journal of Mathematical Biology

, Volume 56, Issue 6, pp 861–892 | Cite as

Determining the expected variability of immune responses using the cyton model

  • Vijay G. Subramanian
  • Ken R. DuffyEmail author
  • Marian L. Turner
  • Philip D. Hodgkin
Article

Abstract

During an adaptive immune response, lymphocytes proliferate for 5–20 cell divisions, then stop and die over a period of weeks. The cyton model for regulation of lymphocyte proliferation and survival was introduced by Hawkins et al. (Proc. Natl. Acad. Sci. USA 104, 5032–5037, 2007) to provide a framework for understanding this response and its regulation. The model assumes stochastic values for division and survival times for each cell in a responding population. Experimental evidence indicates that the choice of times is drawn from a skewed distribution such as the lognormal, with the fate of individual cells being potentially highly variable. For this reason we calculate the higher moments of the model so that the expected variability can be determined. To do this we formulate a new analytic framework for the cyton model by introducing a generalization to the Bellman–Harris branching process. We use this framework to introduce two distinct approaches to predicting variability in the immune response to a mitogenic signal. The first method enables explicit calculations for certain distributions and qualitatively exhibits the full range of observed immune responses. The second approach does not facilitate analytic solutions, but allows simple numerical schemes for distributions for which there is little prospect of analytic formulae. We compare the predictions derived from the second method to experimentally observed lymphocyte population sizes from in vivo and in vitro experiments. The model predictions for both data sets are remarkably accurate. The important biological conclusion is that there is limited variation around the expected value of the population size irrespective of whether the response is mediated by small numbers of cells undergoing many divisions or for many cells pursuing a small number of divisions. Therefore, we conclude the immune response is robust and predictable despite the potential for great variability in the experience of each individual cell.

Keywords

Immune response Expected variability Continuous time Branching processes Time dependent offspring distributions 

Mathematics Subject Classification (2000)

60J85 92D25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abate J., Choudhury G.L. and Whitt W. (1999). Computational probability. In: Grassman, W. (eds) An Introduction to Numerical Transform Inversion and its Application to Probability Models, pp 257–323. Kluwer, Boston Google Scholar
  2. 2.
    Abate J. and Whitt W. (1992). The Fourier-series method for inverting transforms of probability distributions. Queueing Syst. Theory Appl. 10(1–2): 5–87 CrossRefMathSciNetGoogle Scholar
  3. 3.
    Abate J. and Whitt W. (1992). Numerical inversion of probability generating functions. Oper. Res. Lett. 12(4): 245–251 zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ahmed R. and Gray D. (1996). Immunological memory and protective immunity: understanding their relation. Science 272(5238): 54–60 CrossRefGoogle Scholar
  5. 5.
    Athreya K.B. and Ney P.E. (2004). Branching Processes. Dover Publications, Mineola zbMATHGoogle Scholar
  6. 6.
    Billingsley P. (1995). Probability and Measure. Willey, New York zbMATHGoogle Scholar
  7. 7.
    Cantrell D.A. and Smith K.A. (1984). The interleukin-2 T-cell system: a new cell growth model. Science 224(4655): 1292–1361 CrossRefGoogle Scholar
  8. 8.
    Choudhury G.L., Lucantoni D.M. and Whitt W. (1994). Multidimensional transform inversion with applications to the transient M/G/1 queue. Ann. Appl. Probab. 4(3): 719–740 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    De Boer R.J. and Perelson A.S. (2005). Estimating division and death rates from CFSE data. J. Comput. Appl. Math. 184(1): 140–164 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    De Boer R.J., Homann D. and Perelson A.S. (2003). Different dynamics of CD4+ and CD8+ T cell responses during and after acute lymphocytic choriomeningitis virus infection. J. Immunol. 171(8): 3928–3935 Google Scholar
  11. 11.
    Deenick E.K., Gett A.V. and Hodgkin P.D. (2003). Stochastic model of T cell proliferation: a calculus revealing IL-2 regulation of precursor frequencies, cell cycle time, and survival. J. Immunol. 170(10): 4963–4972 Google Scholar
  12. 12.
    Froese G. (1964). The distribution and interdependence of generation times of HeLa cells. Exp. Cell Res. 35(2): 415–419 CrossRefGoogle Scholar
  13. 13.
    Ganusov V.V., Milutinović D. and De Boer R.J. (2007). IL-2 regulates expansion of CD4 T cell populations by affecting cell death: insights from modeling CFSE data. J. Immunol. 179: 950–957 Google Scholar
  14. 14.
    Ganusov V.V., Pilyugin S.S., De Boer R.J., Murali-Krishna K., Ahmed R. and Antia R. (2005). Quantifying cell turnover using CFSE data. J. Immunol. Methods Mar.(1–2): 183–200 CrossRefGoogle Scholar
  15. 15.
    Gett A.V. and Hodgkin P.D. (2000). A cellular calculus for signal integration by T cells. Nat. Immunol. 1(4): 239–244 CrossRefGoogle Scholar
  16. 16.
    Harris T.E. (2002). The Theory of Branching Processes, Dover Phoenix Editions. Dover Publications, Mineola Google Scholar
  17. 17.
    Hawkins E.D., Turner M.L., Dowling M.R., van Gend C. and Hodgkin P.D. (2007). A model of immune regulation as a consequence of randomized lymphocyte division and death times. Proc. Natl. Acad. Sci. USA 104: 5032–5037 CrossRefGoogle Scholar
  18. 18.
    Homann D., Teyton L. and Oldstone M.B.A. (2001). Differential regulation of antiviral T-cell immunity results in stable CD8+ but declining CD4+ T-cell memory. Nat. Med. 7: 913–919 CrossRefGoogle Scholar
  19. 19.
    Hommel M. and De Boer R.J. (2007). TCR affinity promotes CD8+ T cell expansion by regulation survival. J. Immunol. 179: 2250–2260 Google Scholar
  20. 20.
    Kao E.P.C. (1997). An Introduction to Stochastic Processes. Duxbury, NY Google Scholar
  21. 21.
    Kimmel M. and Axelrod D.E. (2002). Branching processes in biology, volume 19 of Interdisciplinary Applied Mathematics. Springer, New York Google Scholar
  22. 22.
    K. León, Faro J. and Carneiro J. (2004). A general mathematical framework to model generation structure in a population of asynchronously dividing cells. J. Theoret. Biol. 229(4): 455–476 CrossRefMathSciNetGoogle Scholar
  23. 23.
    Milutinović D. and De Boer R.J. (2007). Process noicse: an explanation for the fluctiations in the immune response during acute viral infection. Biophys. J. 92: 3358–3367 CrossRefGoogle Scholar
  24. 24.
    Nachtwey D.S. and Cameron I.L. (1968). Methods in Cell Physiology, vol. III, pp. 213–257. Academic, New York Google Scholar
  25. 25.
    Polyanin A.D. and Manzhirov A.V. (1998). Handbook of Integral Equations. CRC, Boca Raton zbMATHGoogle Scholar
  26. 26.
    Prescott D.M. (1968). Regulation of cell reproduction. Cancer Res. 28(9): 1815–1820 Google Scholar
  27. 27.
    Shields R. (1977). Transition probability and the origin of variation in the cell cycle. Nature 267(5613): 704–707 CrossRefGoogle Scholar
  28. 28.
    Smith J.A. and Martin L. (1973). Do cells cycle? Proc. Natl. Acad. Sci. USA 70(4): 1263–1267 CrossRefGoogle Scholar
  29. 29.
    Tangye S.G., Avery D.T., Deenick E.K. and Hodgkin P.D. (2003). Intrinsic differences in the proliferation of naive and memory human B cells as a mechanism for enhanced secondary immune responses. J. Immunol. 170(2): 686–694 Google Scholar
  30. 30.
    Turner, M., Hawkins, E., Hodgkin, P.D.: Manuscript in preparation (2007)Google Scholar
  31. 31.
    Veiga-Fernandes H., Walter U., Bourgeois C., McLean A. and Rocha B. (2000). Response of naïve and memory CD8+ t cells to antigen stimulation in vivo. Nat. Immunol. 1: 47–53 CrossRefGoogle Scholar
  32. 32.
    Yates A., Chan C., Strid J., Moon S., Callard R. and George A.J.T. (2007). Stark J. Reconstruction of cell population dynamics using CFSE. BMC Bioinformatics 8: 196 Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Vijay G. Subramanian
    • 1
  • Ken R. Duffy
    • 1
    Email author
  • Marian L. Turner
    • 2
  • Philip D. Hodgkin
    • 2
  1. 1.Hamilton InstituteNational University of IrelandMaynoothIreland
  2. 2.Walter & Eliza Hall Institute of Medical ResearchMelbourneAustralia

Personalised recommendations