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

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

Rick Durrett, Deena Schmidt, and Jason Schweinsberg

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

Abstract

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.

First Page: Show

Primary Subjects: 60J99

Secondary Subjects: 92C50, 92D25, 60J85

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1241702247
Digital Object Identifier: doi:10.1214/08-AAP559
Zentralblatt MATH identifier: 05566097
Mathematical Reviews number (MathSciNet): MR2521885

back to Table of Contents

References

[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.

Mathematical Reviews (MathSciNet): MR373040

[4] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion: Facts and Formulae, 2nd ed. Birkhäuser, Boston.

Mathematical Reviews (MathSciNet): MR1912205

[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.

Mathematical Reviews (MathSciNet): MR1398879

Zentralblatt MATH: 0856.60002

[7] Durrett, R. (2005). Probability: Theory and Examples, 3rd ed. Duxbury, Belmont, CA.

Mathematical Reviews (MathSciNet): MR1609153

Zentralblatt MATH: 0709.60002

[8] Durrett, R. and Schmidt, D. (2007). Waiting for regulatory sequences to appear. Ann. Appl. Probab. 17 1–32.

Mathematical Reviews (MathSciNet): MR2292578

Digital Object Identifier: doi:10.1214/105051606000000619

Project Euclid: euclid.aoap/1171377175

[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.

Mathematical Reviews (MathSciNet): MR838085

[11] Ewens, W. J. (2004). Mathematical Population Genetics, 2nd ed. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR2026891

Zentralblatt MATH: 1060.92046

[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.

Mathematical Reviews (MathSciNet): MR2122451

Digital Object Identifier: doi:10.1016/j.jtbi.2004.09.001

[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.

Mathematical Reviews (MathSciNet): MR2067856

Digital Object Identifier: doi:10.1016/S0022-5193(03)00120-6

[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.

Mathematical Reviews (MathSciNet): MR127989

Digital Object Identifier: doi:10.1017/S0305004100033193

[20] Nowak, M. A. (2006). Evolutionary Dynamics: Exploring the Equations of Life. Belknap Press, Cambridge, MA.

Mathematical Reviews (MathSciNet): MR2252879

[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.

Mathematical Reviews (MathSciNet): MR2438813

[24] Wodarz, D. and Komarova, N. L. (2005). Computational Biology Of Cancer: Lecture Notes And Mathematical Modeling. World Scientific, Singapore.

previous :: next