Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 15(2010) > Paper 43 open journal systems 


The Aldous-Shields model revisited with application to cellular ageing

Katharina Best, University of Freiburg
Peter Pfaffelhuber, University of Freiburg


Abstract
In Aldous and Shields (1988) a model for a rooted, growing random binary tree with edge lengths 1 was presented. For some $c>0$, an external vertex splits at rate $c^{-i}$ (and becomes internal) if its distance from the root (depth) is $i$. We reanalyse the tree profile for $c>1$, i.e. the numbers of external vertices in depth $i=1,2,...$. Our main result are concrete formulas for the expectation and covariance-structure of the profile. In addition, we present the application of the model to cellular ageing. Here, we say that nodes in depth $h+1$ are senescent, i.e. do not split. We obtain a limit result for the proportion of non-senesced vertices for large $h$.


Full text: PDF

Pages: 475-488

Published on: October 19, 2010
Bibliography
  1. D. Aldous, P. Shields. A diffusion limit for a class of randomly-growing binary trees. Probab. Theory Related Fields 79(4) (1988), 509-542. MR0966174 (90k:60052)
  2. T. Antal, K.B. Blagoev, S.A. Trugman, S. Redner. Aging and immortality in a cell proliferation model. J. Theor. Biol. 248(3) (2007), 411-417.
  3. O. Arino, M. Kimmel, G.B. Webb. Mathematical modeling of the loss of telomere sequences. J. Theor. Biol. 177(1) (1995), 45-57.
  4. Arkus, Natalie. A mathematical model of cellular apoptosis and senescence through the dynamics of telomere loss. J. Theoret. Biol. 235(1) (2005), 13-32. MR2139011
  5. M.A. Baxter, R.F. Wynn, S.N. Jowitt, J.E. Wraith, L.J. Fairbairn, I. Bellantuono. Study of telomere length reveals rapid aging of human marrow stromal cells following in vitro expansion. Stem Cells 22(5) (2004), 675-682.
  6. N. Berestycki. Recent progress in coalescent theory. Ensaios Matemáticos [Mathematical Surveys], 16. Sociedade Brasileira de Matemática, Rio de Janeiro, 2009. MR2574323
  7. E.H. Blackburn. Telomere states and cell fates. Nature 408 (2000), 53-56.
  8. M. Bonab, K. Alimoghaddam, F. Talebian, S. Ghaffari, A. Ghavamzadeh, B. Nikbin. Aging of mesenchymal stem cell in vitro. BMC Cell Biology 7(1) (2006), 14.
  9. D.S. Dean; S.N. Majumdar Phase transition in a generalized Eden growth model on a tree. J. Stat. Phys. 124(6) (2006), 1351-1376. MR2266447 (2008a:82027)
  10. M. Drmota. Random trees. An interplay between combinatorics and probability. Springer, Wien, NewYork, Vienna, 2009. MR2484382 (2010i:05003)
  11. J. Dyson; E. Sánchez; R. Villella-Bressan; G.F. Webb Stabilization of telomeres in nonlinear models of proliferating cell lines. J. Theoret. Biol. 244(3) (2007), 400-408. MR2293122 (2007j:92017)
  12. J. Dyson; R. Villella-Bressan; G.F. Webb Asynchronous exponential growth in an age structured population of proliferating and quiescent cells. Deterministic and stochastic modeling of biointeraction (West Lafayette, IN, 2000). Math. Biosci. 177/178 (2002), 73-83. MR1923804 (2003g:92026)
  13. S.N. Evans, D. Steinsaltz. Damage segregation at fissioning may increase growth rates: A superprocess model.. Theo. Pop. Biol. Biol. 71(4) (2007), 473-490.
  14. J. Felsenstein. Inferring Phylogenies. Palgrave Macmillan, 2002.
  15. N. Gupta, R. Taneja, A. Pandey, M. Mukesh, H. Singh, S.C. Gupta. Replicative senescence, telomere shortening and cell proliferation rate in gaddi goat's skin fibroblast cell line. Cell Biol. Int. 31 (2007), 1257-1264.
  16. L. Hayflick, P.S. Moorhead. The serial cultivation of human diploid cell strains. Exp. Cell Res. 25 (1961), 585-621.
  17. J.Z. Levy, R.C. Allsopp, A.B. Futcher, C.W. Greider, C.B. Harley. Telomere end-replication problem and cell aging. J. Mol. Biol. 225(4) (1992), 951-960.
  18. R. Neininger. Stochastische Analyse von Algorithmen, Fixpunktgleichungen und ideale Metriken. Habilitationsschrift, University of Frankfurt (2004).
  19. P. Olofsson, M. Kimmel. Stochastic models of telomere shortening. Math. Biosci. 158(1) (1999), 75-92. MR1681442
  20. A.M. Olovnikov. Telomeres, telomerase, and aging: origin of the theory. Exp. Gerontol. 31 (1996), 443-448.
  21. R.D. Portugal, M.G. Land, B.F. Svaiter. A computational model for telomere-dependent cell-replicative aging. BioSystems 91(1) (2008), 262-267.
  22. C.J. Proctor, T.B. Kirkwood. Modelling cellular senescence as a result of telomere state. Aging Cell 2(3) (2003), 151-157.
  23. I.A. Rodriguez-Brenes, C.S. Peskin. Quantitative theory of telomere length regulation and cellular senescence. Proc. Natl. Acad. Sci. 107(12) (2010), 5387-5392.
  24. J.W. Shay, W.E. Wright. Senescence and immortalization: role of telomeres and telomerase. Carcinogenesis 26 (2005), 867-874.
  25. G. Vogel, E. Pennisi, E. Blackburn, C. Greider, J. Szostak. Physiology Nobel. U.S. Researchers recognized for work on telomeres. Science 326 (2009), 212-213.
















Research
Support Tool
Capture Cite
View Metadata
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X