Telomeres are guanine-rich sequences at the end of chromosomes which shorten during each replication event and trigger cell cycle arrest and/or controlled death (apoptosis) when reaching a threshold length. The enzyme telomerase replenishes the ends of telomeres and thus prolongs the life span of cells, but also causes cellular immortalisation in human cancer. G-quadruplex (G4) stabilising drugs are a potential anticancer treatment which work by changing the molecular structure of telomeres to inhibit the activity of telomerase. We investigate the dynamics of telomere length in different conformational states, namely t-loops, G-quadruplex structures and those being elongated by telomerase. By formulating deterministic differential equation models we study the effects of various levels of both telomerase and concentrations of a G4-stabilising drug on the distribution of telomere lengths, and analyse 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 find those telomerase and G4-concentrations where telomere length maintenance is successfully regulated. Excessively high levels of telomerase lead to continuous telomere lengthening, whereas large concentrations of the drug lead to progressive telomere erosion. Furthermore, our models predict a positively skewed distribution of telomere lengths, that is, telomeres accumulate over lengths shorter than the mean telomere length at equilibrium. Our model results for telomere length distributions of telomerase-positive cells in drug-free assays are in good agreement with the limited amount of experimental data available.
The pentacyclic acridinium salt RHPS4 displays anti-tumour properties in vitro as well as in vivo and is potentially cell-cycle specific. We have collected experimental data and formulated a compartmental model using ordinary differential equations 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 found that RHPS4 caused a concentration-dependent, marked cell death in treated cells, which is best modelled by allowing the rate parameters corresponding to cell death to be sigmoidal functions of time. We have shown that the model is "identifiable", meaning that, at least in principle, the parameter values can be determined from observable quantities. We find that at low concentrations RHPS4 primarily affects the cells in the G(2)/M phase, and that the drug has a delayed effect with the delay decreasing at larger doses. Since the drug diffuses into the nucleus, the observed delayed effect of the compound is unexpected and is a novel finding of our research into this compound.
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