Bulletin of Mathematical Biology

, Volume 72, Issue 2, pp 359–374 | Cite as

A Rapid-Mutation Approximation for Cell Population Dynamics

  • Rainer K. Sachs
  • Lynn Hlatky
Open Access
Original Article


Carcinogenesis and cancer progression are often modeled using population dynamics equations for a diverse somatic cell population undergoing mutations or other alterations that alter the fitness of a cell and its progeny. Usually it is then assumed, paralleling standard mathematical approaches to evolution, that such alterations are slow compared to selection, i.e., compared to subpopulation frequency changes induced by unequal subpopulation proliferation rates. However, the alterations can be rapid in some cases. For example, results in our lab on in vitro analogues of transformation and progression in carcinogenesis suggest there could be periods where rapid alterations triggered by horizontal intercellular transfer of genetic material occur and quickly result in marked changes of cell population structure.

We here initiate a mathematical study of situations where alterations are rapid compared to selection. A classic selection-mutation formalism is generalized to obtain a “proliferation-alteration” system of ordinary differential equations, which we analyze using a rapid-alteration approximation. A system-theoretical estimate of the total-population net growth rate emerges. This rate characterizes the diverse, interacting cell population acting as a single system; it is a weighted average of subpopulation rates, the weights being components of the Perron–Frobenius eigenvector for an ergodic Markov-process matrix that describes alterations by themselves. We give a detailed numerical example to illustrate the rapid-alteration approximation, suggest a possible interpretation of the fact that average aneuploidy during cancer progression often appears to be comparatively stable in time, and briefly discuss possible generalizations as well as weaknesses of our approach.

Carcinogenesis Somatic cell population diversity Proliferation Fitness alterations Evolutionary ecology Replicator-mutator equations 



refers to changes that can affect the fitness of a somatic cell and its progeny, for example, any of the following: point mutations in important genes, other comparatively small-scale DNA modifications, larger-scale DNA gains or losses, chromosome rearrangements such as translocations, changes in chromosome copy number, or persistent epigenetic changes.


  1. Bjerkvig, R., Tysnes, B.B., Aboody, K.S., Najbauer, J., Terzis, A.J., 2005. Opinion: the origin of the cancer stem cell: current controversies and new insights. Nat. Rev. Cancer 5, 899–904. CrossRefGoogle Scholar
  2. Born, M., Wolf, E., Bhatia, A.B., 1999. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Cambridge University Press, Cambridge. Google Scholar
  3. Chi, Y.H., Jeang, K.T., 2007. Aneuploidy and cancer. J. Cell. Biochem. 102, 531–8. CrossRefGoogle Scholar
  4. Chin, L., Gray, J.W., 2008. Translating insights from the cancer genome into clinical practice. Nature 452, 553–63. CrossRefGoogle Scholar
  5. Coller, H.A., Khrapko, K., Bodyak, N.D., Nekhaeva, E., Herrero-Jimenez, P., Thilly, W.G., 2001. High frequency of homoplasmic mitochondrial DNA mutations in human tumors can be explained without selection. Nat. Genet. 28, 147–50. CrossRefGoogle Scholar
  6. Duesberg, P., Li, R., Fabarius, A., Hehlmann, R., 2005. The chromosomal basis of cancer. Cell. Oncol. 27, 293–318. Google Scholar
  7. Duesberg, P., Li, R., Sachs, R., Fabarius, A., Upender, M.B., Hehlmann, R., 2007. Cancer drug resistance: The central role of the karyotype. Drug Resist. Updat. Google Scholar
  8. Eshleman, J.R., Casey, G., Kochera, M.E., Sedwick, W.D., Swinler, S.E., Veigl, M.L., Willson, J.K., Schwartz, S., Markowitz, S.D., 1998. Chromosome number and structure both are markedly stable in RER colorectal cancers and are not destabilized by mutation of p53. Oncogene 17, 719–25. CrossRefGoogle Scholar
  9. Frank, S.A., Nowak, M.A., 2004. Problems of somatic mutation and cancer. Bioessays 26, 291–9. CrossRefGoogle Scholar
  10. Gatenby, R.A., Vincent, T.L., 2003. An evolutionary model of carcinogenesis. Cancer Res. 63, 6212–20. Google Scholar
  11. Griffiths, D.J., 2004. Introduction to Quantum Mechanics. Pearson/Prentice Hall, Upper Saddle River. Google Scholar
  12. Gusev, Y., Kagansky, V., Dooley, W.C., 2001. Long-term dynamics of chromosomal instability in cancer: a transition probability model. Math. Comput. Model. 33, 1253–1273. zbMATHCrossRefMathSciNetGoogle Scholar
  13. Hofbauer, J., Sigmund, K., 1998. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge. zbMATHGoogle Scholar
  14. Iwasa, Y., Michor, F., Nowak, M.A., 2004. Stochastic tunnels in evolutionary dynamics. Genetics 166, 1571–9. CrossRefGoogle Scholar
  15. Jin, Y., Jin, C., Lv, M., Tsao, S.W., Zhu, J., Wennerberg, J., Mertens, F., Kwong, Y.L., 2005. Karyotypic evolution and tumor progression in head and neck squamous cell carcinomas. Cancer Genet. Cytogenet. 156, 1–7. CrossRefGoogle Scholar
  16. Kijima, M., 1997. Markov Processes for Stochastic Modeling. Cambridge University Press, Cambridge. zbMATHGoogle Scholar
  17. Komarova, N.L., 2005. Mathematical modeling of tumorigenesis: mission possible. Curr. Opin. Oncol. 17, 39–43. CrossRefGoogle Scholar
  18. Li, L., McCormack, A.A., Nicholson, J.M., Fabarius, A., Hehlmann, R., Sachs, R.K., Duesberg, P.H., 2009. Cancer-causing karyotypes: chromosomal equilibria between destabilizing aneuploidy and stabilizing selection for oncogenic function. Cancer Genet. Cytogenet. 188, 1–25. CrossRefGoogle Scholar
  19. Little, M.P., Vineis, P., Li, G., 2008. A stochastic carcinogenesis model incorporating multiple types of genomic instability fitted to colon cancer data. J. Theor. Biol. 254, 229–38. CrossRefGoogle Scholar
  20. Macville, M., Schrock, E., Padilla-Nash, H., Keck, C., Ghadimi, B.M., Zimonjic, D., Popescu, N., Ried, T., 1999. Comprehensive and definitive molecular cytogenetic characterization of HeLa cells by spectral karyotyping. Cancer Res. 59, 141–50. Google Scholar
  21. Marx, C., Posch, H.A., Thirring, W., 2007. Emergence of order in selection-mutation dynamics. Phys. Rev. E 75, 061109. CrossRefMathSciNetGoogle Scholar
  22. Merlo, L.M., Pepper, J.W., Reid, B.J., Maley, C.C., 2006. Cancer as an evolutionary and ecological process. Nat. Rev. Cancer 6, 924–35. CrossRefGoogle Scholar
  23. Michor, F., Frank, S.A., May, R.M., Iwasa, Y., Nowak, M.A., 2003. Somatic selection for and against cancer. J. Theor. Biol. 225, 377–82. CrossRefMathSciNetGoogle Scholar
  24. Michor, F., Hughes, T.P., Iwasa, Y., Branford, S., Shah, N.P., Sawyers, C.L., Nowak, M.A., 2005. Dynamics of chronic myeloid leukaemia. Nature 435, 1267–70. CrossRefGoogle Scholar
  25. Minc, H., 1988. Nonnegative Matrices. Wiley, New York. zbMATHGoogle Scholar
  26. Mitelman, F., Johansson, B., Mertens, F., 2007. The impact of translocations and gene fusions on cancer causation. Nat. Rev. Cancer 7, 233–45. CrossRefGoogle Scholar
  27. Moolgavkar, S.H., Luebeck, E.G., 2003. Multistage carcinogenesis and the incidence of human cancer. Genes Chromosomes Cancer 38, 302–6. CrossRefGoogle Scholar
  28. Nagy, J.D., 2004. Competition and natural selection in a mathematical model of cancer. Bull. Math. Biol. 66, 663–87. CrossRefMathSciNetGoogle Scholar
  29. Nowell, P.C., 1976. The clonal evolution of tumor cell populations. Science 194, 23–8. CrossRefGoogle Scholar
  30. Page, K.M., Nowak, M.A., 2002. Unifying evolutionary dynamics. J. Theor. Biol. 219, 93–8. MathSciNetGoogle Scholar
  31. Roschke, A.V., Stover, K., Tonon, G., Schaffer, A.A., Kirsch, I.R., 2002. Stable karyotypes in epithelial cancer cell lines despite high rates of ongoing structural and numerical chromosomal instability. Neoplasia 4, 19–31. CrossRefGoogle Scholar
  32. Sachs, R.K., Chan, M., Hlatky, L., Hahnfeldt, P., 2005. Modeling intercellular interactions during carcinogenesis. Radiat. Res. 164, 324–31. CrossRefGoogle Scholar
  33. Sachs, R.K., Shuryak, I., Brenner, D., Fakir, H., Hlatky, L., Hahnfeldt, P., 2007. Second cancers after fractionated radiotherapy: Stochastic population dynamics effects. J. Theor. Biol. 249, 518–531. CrossRefGoogle Scholar
  34. Tan, W., 2002. Stochastic Models with Applications to Genetics, Cancers, AIDS, and Other Biomedical Systems. World Scientific, Singapore. zbMATHGoogle Scholar
  35. Thompson, S.L., Compton, D.A., 2008. Examining the link between chromosomal instability and aneuploidy in human cells. J. Cell. Biol. 180, 665–72. CrossRefGoogle Scholar
  36. Vucic, E.A., Brown, C.J., Lam, W.L., 2008. Epigenetics of cancer progression. Pharmacogenomics 9, 215–34. CrossRefGoogle Scholar
  37. Weinberg, R.A., 2007. The Biology of Cancer. Garland Science, New York. Chaps. 13 and 15. Google Scholar
  38. Yuen, K.W., Desai, A., 2008. The wages of CIN. J. Cell. Biol. 180, 661–3. CrossRefGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Departments of Mathematics and PhysicsUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of MedicineTufts School of MedicineBostonUSA

Personalised recommendations