Genetics, Vol. 177, 2209-2221, December 2007, Copyright © 2007
doi:10.1534/genetics.107.078915

The Evolution of Two Mutations During Clonal Expansion

Hiroshi Haeno*, Yoh Iwasa* and Franziska Michor{dagger},1

* Department of Biology, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan and {dagger} Society of Fellows, Harvard University, Cambridge, Massachusetts 02138

1 Corresponding author: Computational Biology Center, Memorial Sloan-Kettering Cancer Center, New York, NY 10065.
E-mail: michorf{at}mskcc.org

Knudson's two-hit hypothesis proposes that two genetic changes in the RB1 gene are the rate-limiting steps of retinoblastoma. In the inherited form of this childhood eye cancer, only one mutation emerges during somatic cell divisions while in sporadic cases, both alleles of RB1 are inactivated in the growing retina. Sporadic retinoblastoma serves as an example of a situation in which two mutations are accumulated during clonal expansion of a cell population. Other examples include evolution of resistance against anticancer combination therapy and inactivation of both alleles of a metastasis-suppressor gene during tumor growth. In this article, we consider an exponentially growing population of cells that must evolve two mutations to (i) evade treatment, (ii) make a step toward (invasive) cancer, or (iii) display a disease phenotype. We calculate the probability that the population has evolved both mutations before it reaches a certain size. This probability depends on the rates at which the two mutations arise; the growth and death rates of cells carrying none, one, or both mutations; and the size the cell population reaches. Further, we develop a formula for the expected number of cells carrying both mutations when the final population size is reached. Our theory establishes an understanding of the dynamics of two mutations during clonal expansion.