Modelling Telomere Length Dynamics Subject to the Action of Telomerase

Abstract

All eukaryotic organisms, single-celled or multi-celled, have linear chromosomes, and all linear chromosomes have telomeres at their ends. All eukaryotes have genes which code for the enzyme telomerase. These entities vary amongst species, but telomeres are always guanine rich tandem repeats of DNA, and telomerase always contains a complimentary strand of RNA. Telomeres are specialized structures at the end of linear chromosomes. Through the formation of t-loops and the binding of shelterin complexes, they make chromosomes structurally and chemically stable. Due to the inability of DNA polymerase to fully replicate a chromosome during mitosis, in a process termed the end replication problem, each replicative event results in a shortened chromosome end. This shortening allows for telomeres to act as an intrinsic mitotic clock, a measure of a cell's generational age. After a fixed number of mitotic cycles, known as the Hayflick limit, a cell enters replicative senescence and ceases to divide. The enzyme telomerase is a reverse transcriptase which enables and catalyzes the lengthening of chromosome ends by \emph{de novo} DNA synthesis. In healthy somatic and germ-line cells telomerase is absent, whereas in stem cells and the large majority of cancerous cells it is present. In this thesis, we derive deterministic equations of motion from a chemical master equation which describe the behaviour of a cell population. They are a coupled finite set of first-order ordinary differential equations which incorporate mitosis, acute death, stem cell differentiation, telomerase activity, and apoptotic death. Their solutions allow for the derivation of characteristic functions which predict the number of cells, and average telomere length in a population of cells. By modelling the dynamics of telomere shortening due to the end replication problem and telomere elongation via the action of telomerase, we investigate the relationship between aging and cancer. Specifically, we find that if the rate of telomerase is above a critical value it has the effect producing a population with a divergent number of cells. This unrestrained proliferation is one of the hallmarks of cancer. Further, consistent with some experiments, we find that telomerase can lead to shorter average telomere lengths.