Mathematical modelling of cell cycle and telomere dynamics

The eukaryotic cell cycle primarily consists of five phases, namely a resting state, G0, and four cycling phases G1, S, G2 and M phase, with cells progressing in this order before dividing into two cells back in phase G1. Understanding how a drug affects the cell cycle can give insight into the drug's mechanism of action and may assist research into potential treatment strategies. The pentacyclic acridinium salt RHPS4 (3,11-difluoro-6,8,13-trimethyl-8H-quino[4,3,2-kl] acridinium methosulfate) is an attractive agent because it is potentially cell-cycle specific and inhibits the activity of telomerase, an enzyme known for its role in cellular immortalisation in human cancer. The precise mechanism of action of the drug on the cell cycle dynamics, however, remains unclear. We have devised experiments, collected experimental data and formulated a mathematical model describing the cell cycle dynamics of cancer cells and their time- and dose-dependent modulation by RHPS4 to investigate how the compound affects cells in each stage of the cell cycle. In addition to a control case, in which no drug was used, we treated colorectal cancer cells with three different concentrations of the drug and fitted simulations from our models to experimental observations. We have shown that the model is "identifiable", meaning that, at least in principle, the parameter values can be determined from observable quantities. Our fitting procedure also generates information on the sensitivity of parameters in the model. We found that RHPS4 caused a marked concentration-dependent cell death in treated cells, which is well modelled by allowing the rate parameters corresponding to cell death to be sigmoidal functions of time. Since the drug uptake into the nucleus is rapid (saturation within 5 hours), the observed delay effect of 5 days of the compound is unexpected and is a novel finding of our research into this compound. Our results show that, at low concentrations, RHPS4 primarily affects the cells in the G2/M phase, and that the delay decreases at larger doses. We propose that secondary effects lead to the induction of observed cell death and that changes in the molecular structure of the non-coding DNA sequences at chromosome ends, called telomeres, might be a precursor of delayed cell death. We therefore investigated the dynamics of telomere length in different conformational states, that is, t-loops, G-quadruplex structures and those being elongated by telomerase. By formulating differential equation models we studied the effects of various levels of telomerase and RHPS4 concentrations on the distribution of telomere lengths and analysed how these effects evolve over large numbers of cell generations. As well as calculating numerical solutions, we use quasicontinuum methods to approximate the behaviour of the system over time, and predict the shape of the telomere length distribution. We showed that telomere length maintenance is tightly regulated: too high levels of telomerase lead to continuous telomere lengthening, and large concentrations of RHPS4 lead to progressive telomere erosion. Our results suggest different effects of RHPS4 dependent on the drug concentration used: low concentrations reduce telomere length, but do not impair the equilibrium of the system, and high concentrations destabilise the system leading to chromosome degradation and senescence and/or cell death. Moreover, our models predict a positively skewed distribution of telomere lengths at equilibrium, and our model predictions are in good agreement with experimental data.