A Rapid-Mutation Approximation for Cell Population Dynamics
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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.
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.
- 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
- Duesberg, P., Li, R., Fabarius, A., Hehlmann, R., 2005. The chromosomal basis of cancer. Cell. Oncol. 27, 293–318. Google Scholar
- 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
- 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
- Gatenby, R.A., Vincent, T.L., 2003. An evolutionary model of carcinogenesis. Cancer Res. 63, 6212–20. Google Scholar
- Griffiths, D.J., 2004. Introduction to Quantum Mechanics. Pearson/Prentice Hall, Upper Saddle River. Google Scholar
- 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
- Weinberg, R.A., 2007. The Biology of Cancer. Garland Science, New York. Chaps. 13 and 15. Google Scholar