Spatial Moran models, II: cancer initiation in spatially structured tissue

Journal of Mathematical Biology

Volumn 72, Issue 5, 2016, Pages 1369-1400

  Durrett, R ^a   Foo, J ^b   Leder, K ^c  

^a DUKE UNIVERSITY   (United States)

^b UNIVERSITY OF MINNESOTA   (United States)

^c University of Minnesota   (United States)

Author keywords

Asymptotics for waiting times; Biased voter model; Cancer field effect; Shape theorem

Indexed keywords

BIOLOGICAL MODEL; COMPUTER SIMULATION; GENETICS; HUMAN; MARKOV CHAIN; MATHEMATICAL PHENOMENA; MUTATION; NEOPLASMS; PATHOLOGY; TIME FACTOR;

COMPUTER SIMULATION; HUMANS; MATHEMATICAL CONCEPTS; MODELS, BIOLOGICAL; MODELS, GENETIC; MUTATION; NEOPLASMS; STOCHASTIC PROCESSES; TIME FACTORS;

EID: 84959252466     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-015-0912-1     Document Type: Article

References (33)

1. Bozic I et al (2010) Accumulation of driver and passenger mutations during tumor progression. Proc Natl Acad Sci 107:18545–18550
2. Bramson M, Cox T, Le Gall J (2001) Super-Brownian Limits of Voter Model Clusters. Ann Probab 29:1001–1032
3. Bramson M, Griffeath D (1980a) Asymptotics for Interacting Particle Systems on Zd. Z fur Wahr 53:183–196
4. Bramson M, Griffeath D (1980b) On the Williams-Bjerknes tumour growth model. II Math Proc Cambridge Philos Soc 88:339–357
5. Bramson M, Griffeath D (1981) On the Williams-Bjerknes tumour growth model. I Ann Probab 9:173–185
6. Chatterjee S, Durrett R (2011) Asymptotic Behavior of Aldous’ Gossip Process. Ann Appl Probab 21:2447–2482
7. Cox JT, Durrett R, Perkins EA (2000) Rescaled voter models converge to super-Brownian motion. Ann Probab 28:185–234
8. Durrett R, Moseley S (2014) Spatial Moran models. I. tunneling in the neutral case. Ann Appl Probab (to appear)
9. Durrett R, Zähle I (2007) On the width of hybrid zones. Stoch Proc Appl 117:1751–1763
10. Durrett R (1995) Ten lectures on particle systems. In: St. Flour lecture notes. Lecture notes in math 1608. Springer-Verlag, New York, pp 97–201
11. Durrett R (2008) Probability models for DNA sequence evolution. Springer, New York
12. Durrett R, Schmidt D (2008) Waiting for two mutations: with applications to regulatory sequence evolution and the limits of Darwinian evolution. Genetics 180:1501–1509
13. Ethier S, Kurtz T (1986) Markov processes: characterization and convergence. Wiley, New York
14. Foo J, Leder K, Ryser MD (2014) Multifocality and recurrence risk: a quantitative model of field cancerization. J Theor Biol 355:170–184
15. Griffeath DS (1978) Additive and cancellative interacting particle systems. In: Lecture notes in mathematics, vol 724. Springer, New York
16. Harris TE (1976) On a class of set-valued Markov processes. Ann Probab 4:175–194
17. Jones S et al (2008) Comparative lesion sequencing provides insights into tumor evolution. Prov Natl Acad Sci 105:4283–4288
18. Kimura M, Weiss GH (1964) The stepping stone model of population structure and the decrease of genetic correlation with distance. Genetics 49:561–576
19. Komarova NL (2007) Spatial stochastic models of cancer initiation and progression. Bull Math Biol 68:1573–1599
20. Komarova NL (2013) Spatial stochastic models of cancer: fitness, migration, invasion. Math Biosci Eng 10:761–775
21. Lieberman E, Hauert C, Nowak MA (2005) Evolutionary dynamics on graphs. Nature 433:312–316
22. Martens EA, Hallatschek O (2011) Interfering waves of adaptation promote spatial mixing. Genetics 189:1045–1060
23. Martens EA, Kostadinov R, Maley CC, Hallatschek O (2011) Spatial structure increases the waiting time for cancer. New J Phys 13:115014
24. Maruyama T (1970) On the fixation probability of mutant genes in a subdivided population. Genet Res 15:221–225
25. Maruyama T (1974) A simple proof that certain quantities are independent of the geographical structure of population. Theor Pop Biol 5:148–154
26. Merle M (2008) Hitting probability of a distant point for the voter model started with a single 1. Ann Probab 36:807–861
27. Nowak MA (2006) Evolutionary dynamics: exploring the equations of life. Belknap Press, Cambridge
28. Parzen E (1999) Stochastic processes. In: Classics in Applied Mathematics, vol 24. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (Reprint of the 1962 original)
29. Revuz D, Yor M (1991) Continuous Martingales and Brownian motion. Springer, New York
30. Sprouffske K, Merlo L, Gerrish P, Maley C, Sniegowski P (2012) Cancer in light of experimental evolution. Curr Biol 22:R762–R771
31. Weinberg R (2013) The biology of cancer. Garland Science, New York
32. Williams T, Bjerknes R (1972) Stochastic model for abnormal clone spread through epithelial basal layer. Nature 235:19–21
33. Wodarz D, Komarova NL (2014) Dynamics of cancer: mathematical foundations of oncology. World Scientific Publishing Company, Singapore