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 segregation error. Using computer simulations, we have previously analyzed this model and were able to reproduce a short-term dynamics of chromosome copy number distributions in clones of cancer cells. In a short run, segregation errors provide a continuous production of deviant cells with increasing variation of cell karyotypes, which depends upon the rate of segregation errors. In the long-term observations, many tumors and cancer cell lines have been observed to maintain a stable, although abnormal, distribution of chromosome number for hundreds of cell generations. This phenomenon of ‘stability within instability’ presents an interesting paradox, which could be addressed mathematically. However, this would require modeling of long term growth of tumor cell clones for hundreds of generations, which has far exceeded capabilities of modern computer systems. In this study, we have analyzed asymptotic behavior of our model using a semianalytical approach. A transition probability matrix was derived analyatically and implemented in a recursive algorithm for computational experiments. Using this transition probability model, the expected frequencies of chromosome copy number have been calculated under various initial and boundary conditions. We have also tested several alternative models, which describe various mechanisms of errors in segregation of chromosomes, and found conditions for stabilization of distribution of chromosomes copy numbers over a large number of cell generations. Stable clonal frequencies were estimated which are independent of initial conditions, 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 modeling results have suggested a possible connection between the form of stable distribution of chromosome numbers in tumors and the underlying mechanism of errors in segregation of chromosomes. This new analytical approach allows us to overcome technical impairments and limitations of computer simulation, and, for the first time, provides mathematical insight into long-term evolution of chromosome numerical changes in human tumors.