A Rapid-Mutation Approximation for Cell Population Dynamics

How mitotic errors contribute to karyotypic diversity in cancer

Aneuploidy is a common feature of cancer cells, and is believed to play a critical role in tumorigenesis and cancer progression. Most cancer cells also exhibit high rates of mitotic chromosome mis-segregation, a phenomenon known as chromosomal instability, which leads to high variability of the karyotype. Here, we describe the nature, nuances, and implications of cancer karyotypic diversity. Moreover, we summarize recent studies aimed at identifying the mitotic defects that may be responsible for inducing chromosome mis-segregation in cancer cells. These include kinetochore attachment errors, spindle assembly checkpoint dysfunction, mitotic spindle defects, and other cell division inaccuracies. Finally, we discuss how such mitotic errors generate karyotypic diversity in cancer cells.

Eradication-resolution dynamics with stochastic flare-ups

In infectious disease as well as in cancer, the ultimate outcome of the curative response, mediated by the body itself or through drug treatment, is either successful eradication or a resurgence of the disease ("flare-up" or "relapse"), depending on random fluctuations that dominate the dynamics of the system when the number of diseased cells has become very low. The presence of a low-numbers bottle-neck in the dynamics, which is unavoidable if eradication is to take place at all, renders at least one phase of the dynamics essentially stochastic. However, the eradicating agents (e.g. immune cells, drug molecules) generally remain at high numbers during the critical bottle-neck phase, sufficiently so to warrant a deterministic treatment. This leads us to consider a hybrid stochastic-deterministic approach where the infected cells are treated stochastically whereas the eradicating agents are treated deterministically. Exploiting the fact that the number of eradicating agents typically decreases monotonically during the resolution phase of the response, we derive a set of coupled first-order differential equations that describe the probability of ultimate eradication as a function of the system's state, and we consider a number of biomedical applications.