zbMATH — the first resource for mathematics

Mutation frequencies in a birth-death branching process. (English) Zbl 1404.60122
Summary: First, we revisit the stochastic Luria-Delbrück model: a classic two-type branching process which describes cell proliferation and mutation. We prove limit theorems and exact results for the mutation times, clone sizes and number of mutants. Second, we extend the framework to consider mutations at multiple sites along the genome. The number of mutants in the two-type model characterises the mean site frequency spectrum in the multiple-site model. Our predictions are consistent with previously published cancer genomic data.

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J28 Applications of continuous-time Markov processes on discrete state spaces
92D10 Genetics and epigenetics
92C50 Medical applications (general)
92D20 Protein sequences, DNA sequences
Full Text: DOI Euclid arXiv
[1] Angerer, W. P. (2001). An explicit representation of the Luria–Delbrück distribution. J. Math. Biol.42 145–174. · Zbl 1009.92028
[2] Antal, T. and Krapivsky, P. L. (2010). Exact solution of a two-type branching process: Clone size distribution in cell division kinetics. J. Stat. Mech. Theory Exp.7 P07028.
[3] Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover Publications, Inc., Mineola, NY. Reprint of the 1972 original [Springer, New York; MR0373040]. · Zbl 1070.60001
[4] Bozic, I., Gerold, J. M. and Nowak, M. A. (2016). Quantifying clonal and subclonal passenger mutations in cancer evolution. PLoS Comput. Biol.12 e1004731.
[5] Bozic, I. et al. (2013). Evolutionary dynamics of cancer in response to targeted combination therapy. eLife2 e00747.
[6] Diaz Jr., L. A. et al. (2012). The molecular evolution of acquired resistance to targeted EGFR blockade in colorectal cancers. Nature486 537.
[7] Dingli, D. et al. (2007). The emergence of tumor metastases. Cancer Biol. Ther.6 383–390.
[8] Durrett, R. (2013). Population genetics of neutral mutations in exponentially growing cancer cell populations. Ann. Appl. Probab.23 230–250. · Zbl 1377.92061
[9] Durrett, R. (2015). Branching Process Models of Cancer. Mathematical Biosciences Institute Lecture Series. Stochastics in Biological Systems1. Springer, Cham. · Zbl 1328.92004
[10] Durrett, R., Foo, J., Leder, K., Mayberry, J. and Michor, F. (2010). Evolutionary dynamics of tumor progression with random fitness values. Theor. Popul. Biol.78 54–66. · Zbl 1403.92172
[11] Durrett, R. and Moseley, S. (2010). Evolution of resistance and progression to disease during clonal expansion of cancer. Theor. Popul. Biol.77 42–48. · Zbl 1403.92171
[12] Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge Univ. Press, Cambridge. · Zbl 1165.05001
[13] Foo, J. and Leder, K. (2013). Dynamics of cancer recurrence. Ann. Appl. Probab.23 1437–1468. · Zbl 1272.92023
[14] Haeno, H. and Michor, F. (2010). The evolution of tumor metastases during clonal expansion. J. Theoret. Biol.263 30–44.
[15] Hamon, A. and Ycart, B. (2012). Statistics for the Luria–Delbrück distribution. Electron. J. Stat.6 1251–1272. · Zbl 1295.92022
[16] Iwasa, Y., Nowak, M. A. and Michor, F. (2006). Evolution of resistance during clonal expansion. Genetics172 2557–2566.
[17] Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl.110 177–245. · Zbl 1075.60109
[18] Janson, S. (2006). Limit theorems for triangular urn schemes. Probab. Theory Related Fields134 417–452. · Zbl 1112.60012
[19] Jones, S. et al. (2008). Comparative lesion sequencing provides insights into tumor evolution. Proc. Natl. Acad. Sci. USA105 4283–4288.
[20] Keller, P. and Antal, T. (2015). Mutant number distribution in an exponentially growing population. J. Stat. Mech. Theory Exp.1 P01011.
[21] Kendall, D. G. (1960). Birth-and-death processes, and the theory of carcinogenesis. Biometrika47 13–21. · Zbl 0089.36601
[22] Kessler, D. A. and Levine, H. (2013). Large population solution of the stochastic Luria–Delbrück evolution model. Proc. Natl. Acad. Sci. USA110 11682–11687. · Zbl 1292.92016
[23] Kessler, D. A. and Levine, H. (2015). Scaling solution in the large population limit of the general asymmetric stochastic Luria–Delbrück evolution process. J. Stat. Phys.158 783–805. · Zbl 1315.35223
[24] Komarova, N. (2006). Stochastic modeling of drug resistance in cancer. J. Theoret. Biol.239 351–366.
[25] Komarova, N. L., Wu, L. and Baldi, P. (2007). The fixed-size Luria–Delbruck model with a nonzero death rate. Math. Biosci.210 253–290. · Zbl 1129.92037
[26] Kuipers, J., Jahn, K., Raphael, B. J. and Beerenwinkel, N. (2017). Single-cell sequencing data reveal widespread recurrence and loss of mutational hits in the life histories of tumors. Genome Res.27 1885–1894.
[27] Lea, D. E. and Coulson, C. A. (1949). The distribution of the numbers of mutants in bacterial populations. J. Genet.49 264–285.
[28] Luria, S. E. and Delbrück, M. (1943). Mutations of bacteria from virus sensitivity to virus resistance. Genetics48 419–511.
[29] Michor, F., Nowak, M. A. and Iwasa, Y. (2006). Stochastic dynamics of metastasis formation. J. Theoret. Biol.240 521–530.
[30] Nicholson, M. D. and Antal, T. (2016). Universal asymptotic clone size distribution for general population growth. Bull. Math. Biol.78 2243–2276. · Zbl 1357.92049
[31] Ohtsuki, H. and Innan, H. (2017). Forward and backward evolutionary processes and allele frequency spectrum in a cancer cell population. Theor. Popul. Biol.117 43–50. · Zbl 1393.92035
[32] Rényi, A. (1953). On the theory of order statistics. Acta Math. Acad. Sci. Hung.4 191–231.
[33] Rosche, W. A. and Foster, P. L. (2000). Determining mutation rates in bacterial populations. Methods20 4–17.
[34] Williams, M. J., Werner, B., Barnes, C. P., Graham, T. A. and Sottoriva, A. (2016). Identification of neutral tumor evolution across cancer types. Nat. Genet.48 238–244.
[35] Zheng, Q. (1999). Progress of a half century in the study of the Luria–Delbrück distribution. Math. Biosci.162 1–32. · Zbl 0947.92024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.