Traveling waves of selective sweeps. (English) Zbl 1219.92037

Durrett, Rick; Mayberry, John

Ann. Appl. Probab. 21, No. 2, 699-744 (2011).

Summary: The goal of cancer genome sequencing projects is to determine the genetic alterations that cause common cancers. Many malignancies arise during the clonal expansion of a benign tumor which motivates the study of recurrent selective sweeps in an exponentially growing population. To better understand this process, N. Beerenwinkel et al. [PLoS Comput. Biol. 3, 2239–2246 (2007)] considered a Wright-Fisher model in which cells from an exponentially growing population accumulate advantageous mutations. Simulations show a traveling wave in which the time of the first \(k\)-fold mutant, \(T_k\), is approximately linear in \(k\) and heuristics are used to obtain formulas for \(ET_k\). We consider the analogous problem for the Moran model and prove that as the mutation rate \(\mu \rightarrow 0\), \(T_k \sim c_k \log (1 / \mu )\), where the \(c_k\) can be computed explicitly. In addition, we derive a limiting result on a log scale for the size of \(X_k(t) =\) the number of cells with \(k\) mutations at time \(t\).

MSC:

92C50	Medical applications (general)
60J85	Applications of branching processes
92C40	Biochemistry, molecular biology
65C20	Probabilistic models, generic numerical methods in probability and statistics

Keywords:

Moran model; rate of adaptation; stochastic tunneling; branching processes; cancer models

References:

[1]	Armitage, P. and Doll, R. (1954). The age distribution of cancer and a multi-stage theory of carcinogenesis. Br. J. Cancer 8 1-12.
[2]	Athreya, K. B. and Ney, P. E. (1972). Branching Processes . Springer, New York. · Zbl 0259.60002
[3]	Beerenwinkel, N., Antal, T., Dingli, D., Traulsen, A., Kinzler, K. W., Velculescu, V. E., Vogelstein, B. and Nowak, M. A. (2007). Genetic progression and the waiting time to cancer. PLoS Comput. Biol. 3 2239-2246.
[4]	Brunet, E., Rouzine, I. M. and Wilke, C. O. (2008). The stochastic edge in adaptive evolution. Genetics 179 603-620.
[5]	Calabrese, P., Mecklin, J.-P., Järvinen, H. J., Aaltonen, L. A., Tavaré, S. A. and Shibata, D. (2005). Numbers of mutations to different types of colorectal cancer. BMC Cancer 5 126.
[6]	Desai, M. M. and Fisher, D. S. (2007). Beneficial mutation selection balance and the effect of linkage on positive selection. Genetics 176 1759-1798.
[7]	Durrett, R. (2008). Probability Models for DNA Sequence Evolution , 2nd ed. Springer, New York. · Zbl 1311.92007
[8]	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.
[9]	Durrett, R., Schmidt, D. and Schweinsberg, J. (2009). A waiting time problem arising from the study of multi-stage carcinogenesis. Ann. Appl. Probab. 19 676-718. · Zbl 1219.92038
[10]	Frank, S. A. (2007). Dynamics of Cancer : Incidence , Inheritance and Evolution . Princeton Univ. Press, Princeton, NJ.
[11]	Haeno, H., Iwasa, Y. and Michor, F. (2007). The evolution of two mutations during clonal expansion. Genetics 177 2209-2221.
[12]	Iwasa, Y., Nowak, M. A. and Michor, F. (2006). Evolution of resistance during clonal expansion. Genetics 172 2557-2566.
[13]	Jones, S., Zhang, X., Parsons, D. W., Lin, J. C.-H., Leary, R. J., Angenendt, P., Mankoo, P., Carter, H., Kamiyama, H., Jimeno, A., Hong, S.-M., Fu, B., Lin, M.-T., Calhoun, E. S., Kamiyama, M., Walter, K., Nikolskaya, T., Nikolsky, Y., Hartigan, J., Smith, D. R., Hidalgo, M., Leach, S. D., Klein, A. P., Jaffee, E. M., Goggins, M., Maitra, A., Iacobuzio-Donahue, C., Eshleman, J. R., Kern, S. E., Hruban, R. H., Karchin, R., Papadopoulos, N., Parmigiani, G., Vogelstein, B., Velculescu, V. E. and Kinzler, K. W. (2008). Core signaling pathways in human pancreatic cancers revealed by global genomic analyses. Science 321 1801-1806.
[14]	A. G. Knudson, J. (1971). Mutation and cancer: Statistical study of retinoblastoma. Proc. Natl. Acad. Sci. USA 68 820-823.
[15]	Knudson, A. G. (2001). Two genetic hits (more or less) to cancer. Nat. Rev. Cancer 1 157-162.
[16]	Kurtz, T. G. (1970). Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 7 49-58. JSTOR: · Zbl 0191.47301
[17]	Luebeck, E. G. and Moolgavkar, S. H. (2002). Multistage carcinogenesis and the incidence of colorectal cancer. Proc. Natl. Acad. Sci. USA 99 15095-15100.
[18]	Muller, H. J. (1932). Some genetic aspects of sex. Am. Natur. 66 118-138.
[19]	Parsons, D. W., Jones, S., Zhang, X., Lin, J. C.-H., Leary, R. J., Angenendt, P., Mankoo, P., Carter, H., Siu, I.-M., Gallia, G. L., Olivi, A., McLendon, R., Rasheed, B. A., Keir, S., Nikolskaya, T., Nikolsky, Y., Busam, D. A., Tekleab, H., Luis A. Diaz, J., Hartigan, J., Smith, D. R., Strausberg, R. L., Marie, S. K. N., Shinjo, S. M. O., Yan, H., Riggins, G. J., Bigner, D. D., Karchin, R., Papadopoulos, N., Parmigiani, G., Vogelstein, B., Velculescu, V. E. and Kinzler, K. W. (2008). An integrated genomic analysis of human glioblastoma multiforme. Science 321 1807-1812.
[20]	Rouzine, I. M., Brunet, E. and Wilke, C. O. (2008). The traveling-wave approach to asexual evolution: Muller’s ratchet and speed of adaptation. Theor. Popul. Biol. 73 24-46. · Zbl 1202.92066
[21]	Rouzine, I. M., Wakeley, J. and Coffin, J. M. (2003). The solitary wave of asexual evolution. Proc. Natl. Acad. Sci. USA 100 587-592.
[22]	Schweinsberg, J. (2008). The waiting time for m mutations. Electron. J. Probab. 13 1442-1478. · Zbl 1191.60100
[23]	Sjoblom, T. et al. (2006). The consensus coding sequences of human breast and colorectal cancers. Science 314 268-274.
[24]	The Cancer Genome Altas Research Network (2008). Comprehensive genomic characterization defines human glioblastoma genes and core pathways. Nature 455 1061-1068.
[25]	Wodarz, D. and Komarova, N. L. (2005). Computational Biology of Cancer . World-Scientific, Singapore. · Zbl 1126.92029
[26]	Wood, L. et. al. (2007). The genomic landscape of human breast and colorectal cancers. Science 318 1108-1113.
[27]	Yu, F. and Etheridge, A. (2008). Rate of Adaptation of Large Populations. Evolutionary Biology from Concept to Application 3-27. Springer, Berlin.
[28]	Yu, F., Etheridge, A. and Cuthbertson, C. (2010). Asymptotic behavior of the rate of adaptation. Ann. Appl. Probab. 20 978-1004. · Zbl 1193.92079