The Annals of Applied Probability

A waiting time problem arising from the study of multi-stage carcinogenesis

Rick Durrett, Deena Schmidt, and Jason Schweinsberg

Full-text: Open access


We consider the population genetics problem: how long does it take before some member of the population has m specified mutations? The case m=2 is relevant to onset of cancer due to the inactivation of both copies of a tumor suppressor gene. Models for larger m are needed for colon cancer and other diseases where a sequence of mutations leads to cells with uncontrolled growth.

Article information

Ann. Appl. Probab. Volume 19, Number 2 (2009), 676-718.

First available in Project Euclid: 7 May 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J99: None of the above, but in this section
Secondary: 92C50: Medical applications (general) 92D25: Population dynamics (general) 60J85: Applications of branching processes [See also 92Dxx]

Multi-stage carcinogenesis waiting times Moran model branching process Wright–Fisher diffusion


Durrett, Rick; Schmidt, Deena; Schweinsberg, Jason. A waiting time problem arising from the study of multi-stage carcinogenesis. Ann. Appl. Probab. 19 (2009), no. 2, 676--718. doi:10.1214/08-AAP559.

Export citation


  • [1] Armitage, P. and Doll, R. (1954). The age distribution of cancer and a multi-stage theory of carcinogenesis. Brit. J. Cancer 8 1–12.
  • [2] Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen–Stein method. Ann. Probab. 17 9–25.
  • [3] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • [4] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion: Facts and Formulae, 2nd ed. Birkhäuser, Boston.
  • [5] Calabrese, P., Mecklin, J. P., Järvinen, H. J., Aaltonen, L. A., Tavaré, S. and Shibata, D. (2005). Numbers of mutations to different types of colorectal cancer. BMC Cancer 5 126.
  • [6] Durrett, R. (1996). Stochastic Calculus: A Practical Introduction. CRC Press, Boca Raton, FL.
  • [7] Durrett, R. (2005). Probability: Theory and Examples, 3rd ed. Duxbury, Belmont, CA.
  • [8] Durrett, R. and Schmidt, D. (2007). Waiting for regulatory sequences to appear. Ann. Appl. Probab. 17 1–32.
  • [9] Durrett, R. and Schmidt, D. (2008). Waiting for two mutations: With applications to regulatory sequence evolution and the limits of Darwinian evolution. Genetics 180 1501–1509.
  • [10] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [11] Ewens, W. J. (2004). Mathematical Population Genetics, 2nd ed. Springer, Berlin.
  • [12] Iwasa, Y., Michor, F., Komarova, N. L. and Nowak, M. A. (2005). Population genetics of tumor suppressor genes. J. Theoret. Biol. 233 15–23.
  • [13] Iwasa, Y., Michor, F. and Nowak, M. A. (2004). Stochastic tunnels in evolutionary dynamics. Genetics 166 1571–1579.
  • [14] Jones, S. (2008). Comparative lesion sequencing provides insights into tumor evolution. Proc. Natl. Acad. Sci. USA 105 4283–4288.
  • [15] Knudson, A. G. (1971). Mutation and cancer: Statistical study of retinoblastoma. Proc. Natl. Acad. Sci. USA 68 820–823.
  • [16] Kolmorogov, A. N. (1938). Zur Lösung einer biologischen Aufgabe. Izv. NII Mat. Mekh. Tomsk. Univ. 2 1–6.
  • [17] Komarova, N. L., Sengupta, A. and Nowak, M. A. (2003). Mutation-selection networks of cancer initiation: Tumor suppressor genes and chromosomal instability. J. Theoret. Biol. 223 433–450.
  • [18] Luebeck, E. G. and Moolgavkar, S. H. (2002). Multistage carcinogenesis and the incidence of colorectal cancer. Proc. Natl. Acad. Sci. 99 15095–15100.
  • [19] Moran, P. A. P. (1958). Random processes in genetics. Proc. Cambridge Philos. Soc. 54 60–71.
  • [20] Nowak, M. A. (2006). Evolutionary Dynamics: Exploring the Equations of Life. Belknap Press, Cambridge, MA.
  • [21] Schinazi, R. B. (2006). The probability of treatment induced drug resistance. Acta Biotheoretica 54 13–19.
  • [22] Schinazi, R. B. (2006). A stochastic model for cancer risk. Genetics 174 545–547.
  • [23] Schweinsberg, J. (2008). The waiting time for m mutations. Electron. J. Probab. 13 1442–1478.
  • [24] Wodarz, D. and Komarova, N. L. (2005). Computational Biology Of Cancer: Lecture Notes And Mathematical Modeling. World Scientific, Singapore.