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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

Abstract

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 

Abbreviation

“Alteration”

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.

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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

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