A Markov chain for numerical chromosomal instability in clonally expanding populations

Cancer cells frequently undergo chromosome missegregation events during mitosis, whereby the copies of a given chromosome are not distributed evenly among the two daughter cells, thus creating cells with heterogeneous karyotypes. A stochastic model tracing cellular karyotypes derived from clonal populations over hundreds of generations was recently developed and experimentally validated, and it was capable of predicting favorable karyotypes frequently observed in cancer. Here, we construct and study a Markov chain that precisely describes karyotypic evolution during clonally expanding cancer cell populations. The Markov chain allows us to directly predict the distribution of karyotypes and the expected size of the tumor after many cell divisions without resorting to computationally expensive simulations. We determine the limiting karyotype distribution of an evolving tumor population, and quantify its dependency on several key parameters including the initial karyotype of the founder cell, the rate of whole chromosome missegregation, and chromosome-specific cell viability. Using this model, we confirm the existence of an optimal rate of chromosome missegregation probabilities that maximizes karyotypic heterogeneity, while minimizing the occurrence of nullisomy. Interestingly, karyotypic heterogeneity is significantly more dependent on chromosome missegregation probabilities rather than the number of cell divisions, so that maximal heterogeneity can be reached rapidly (within a few hundred generations of cell division) at chromosome missegregation rates commonly observed in cancer cell lines. Conversely, at low missegregation rates, heterogeneity is constrained even after thousands of cell division events. This leads us to conclude that chromosome copy number heterogeneity is primarily constrained by chromosome missegregation rates and the risk for nullisomy and less so by the age of the tumor. This model enables direct integration of karyotype information into existing models of tumor evolution based on somatic mutations.Author summary: Chromosomal instability (CIN) is a hallmark of cancer and it results from persistent chromosome segregation errors during cell division. CIN has been shown to play a key role in drug resistance and tumor metastasis. While our understanding of CIN on the cellular level has grown over the past decade, our ability to predict the behavior of tumors containing billions of cells remains limited due to the paucity of adequate mathematical models. Here, we develop a Markov-chain model that is capable of providing exact solutions for long-term chromosome copy number distributions during tumor growth. Using this model we confirm the presence of optimal chromosome missegregation rates that balance genomic heterogeneity required for tumor evolution and survival. Interestingly, we show that chromosome copy number heterogeneity is primarily influenced by the rate of chromosome segregation errors rather than the age of the tumor. At chromosome missegregation rates frequently observed in cancer, tumors can acquire maximal genomic heterogeneity after a few hundred cell divisions. This model enables the integration of selection imparted by CIN into existing models of tumor evolution based on somatic mutations to explore their mutual effects.