Long-term dynamics of chromosomal instability in cancer: A transition probability model

A stochastic model of chromosomal instability has been previously developed which has included one adjustable parameter-the probability of a, segregat ion error. using computer simulations, we have previously analyzed this mod el and were able to reproduce a short-term dynamics of chromosome copy numb er distributions in clones of cancer cells. In a short run, segregation err ors provide a,continuous production of deviant cells with increasing variat ion of cell karyotypes, which depends upon the rate of segregation errors. In the long-term observations, many tumors slid cancer cell lines have been observed to maintain a stable, although abnormal, distribution of chromoso me number for hundreds of cell generations. This phenomenon of "stability w ithin instability" presents an interesting paradox, which could be addresse d mathematically. However, this would require modeling of long term growth of tumor cell clones for hundreds of generations. which has far exceeded ca pabilities of modern computer systems. In this study, we have analyzed asym ptotic behavior of our model using a semianalytical approach. A transition probability matrix was derived analytically and implemented in a recursive algorithm for computational experiments. Using this transition probability model, the expected frequencies of chromosome copy number have been calcula ted under various initial and boundary conditions. We have also tested seve ral alternative models, which describe various mechanisms of errors in segr egation of chromosomes, and found conditions for stabilization of distribut ion of chromosomes copy numbers over a large number of cell generations. St able clonal frequencies were estimated which are independent of initial con ditions, i.e., chromosome copy numbers in the initiator cells. These stable distributions were, however, dependent on the model assumptions regarding particular mechanism of errors in segregation of chromosomes. Thus, our mod eling results have suggested a possible connection between the form of stab le distribution of chromosome numbers in tumors and the underlying mechanis m of errors ill segregation of chromosomes. This new analytical approach al lows us to overcome technical impairments and limitations of computer simul ation, and. for the first time, provides mathematical insight into long-ter m evolution of chromosome numerical changes in human tumors. (C) 2001 Elsev ier Science Ltd. All rights reserved.