Tree Structures in Immunology

  • J. K. Percus
Conference paper
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 17)


An important component of the behavior of cell populations is often associated with branching processes in time, physical space, or more abstract configuration spaces. Several examples are presented, arising from studies in mathematical immunology. First is simple stem cell proliferation, with the hemopoietic stem cell as prototype, to introduce concepts, theorems, and approximations. Extension to multitype branching is then considered, as in macrophage production, with the analysis restricted to that of population moments. A further area of application is that of somatic mutations contributing to the fine scale diversity of antibody production, which is modeled to accommodate the machinery developed above. Finally, the immune network is introduced as an example in which branching in species space may capture a fair amount of the phenomenology.


Cayley Tree Immune Network Bethe Lattice Hemopoietic Stem Cell Point Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. (1).
    This is the framework adopted e.g. in J. K. Percus, “Combinatorial Methods,” Springer, New York, 1971.MATHCrossRefGoogle Scholar
  2. (2).
    A quite up-to-date and accessible survey is to be found in I. Roitt, J. Brostoff and D. Male, “Immunology,” Mosby, 1985.Google Scholar
  3. (3).
    See e.g., P. Jagers, “Branching Processes with Biological Applications,” Wiley, 1975.Google Scholar
  4. (4).
    See e.g., Th. L. Lentz, “The Cell Biology of Hydra,” North-Holland, 1966.Google Scholar
  5. (5).
    For an extended discussion, see K. B. Athreya and P. E. Ney, “Branching Processes,” Springer-Verlag, 1972.Google Scholar
  6. (6).
    See e.g., J. K. Percus, “Combinatorial Methods in Developmental Biology,” Courant Inst. Math. Sci, 1977.Google Scholar
  7. (7).
    L. M. Milne-Thomson, “The Calculus of Finite Differences,” MacMillan, 1951, p. 341.Google Scholar
  8. (8).
    J. K. Percus, “The Mathematics of Immunology,” Courant Inst. Math. Sci., 1988.Google Scholar
  9. (9).
    T. Suda, J. Suda, and M. Ogawa, Proc. Natl. Acad. Sci. USA 81 (1984), p. 2520.CrossRefGoogle Scholar
  10. (10).
    An extensive analysis is presented in C. A. Macken and A. S. Perelson, “A Branching Process Model of Cell Proliferation and Differentiation,” Springer-Verlag, 1988.Google Scholar
  11. (11).
    C. J. Mode, “Multiple Branching Processes,” Elsevier, 1971.Google Scholar
  12. (12).
    See reference 10 as well as C. A. Macken, A. S. Perelson, and C. C. Stewart, in “Modeling of Biomedical Systems,” (editor, J. Eisenfeld and M. Witten), IMACS Transactions, 1985.Google Scholar
  13. (13).
    See e.g., S. Tonegawa, Sci. American 253 (1985), p. 122.CrossRefGoogle Scholar
  14. (14).
    N. K. Jerne, Ann. Immuno. (Inst. Pasteur) 125C (1974), p. 373.Google Scholar
  15. (15).
    P.H. Richter, in “Theoretical Immmunology” (editors, Bell, Perelson and Pinkley), Dekker, 1978.Google Scholar
  16. (16).
    J. K. Percus, in “Sante Fe Conference on Theoretical Immunology,” (editor, A. S. Perelson), Addison-Wesley, 1988.Google Scholar
  17. (17).
    G. W. Hoffmann and A. Cooper-Willis, in “Mathematical Modeling in Immunity and Medicine,” North-Holland (editor, Marchuk and Belykh), 1983.Google Scholar
  18. (18).
    L. Segel and A. S. Perelson, in reference 16.Google Scholar
  19. (19).
    See e.g. J. K. Pecus and G. O. Williams, in “Fluid Interfacial Phenomena,” (editor, C. A. Croxton) Wiley, 1986.Google Scholar
  20. (20).
    J. L. Jackson and L. S. Klein, Phys. Fluids 7 (1964), p. 228.MathSciNetMATHCrossRefGoogle Scholar
  21. (21).
    B. Widom, J. Chem. Phys. 39 (1963), p. 2808.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • J. K. Percus
    • 1
  1. 1.Courant Institute of Math. Sci. and Physics Dept.New York UniversityNew YorkUSA

Personalised recommendations