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Telomeres are repetitive elements of DNA which are located at the ends of chromosomes. During cell division, telomeres on daughter chromomeres shorten until the telomere length falls below a critical level. This shortening restricts the number of cell divisions. In this thesis, we use mathematical modelling to study dynamics of telomere length in a cell in order to understand normal ageing (telomere shortening),Werner’s syndrome (a disease of accelerated ageing) and the immortality of cells caused by telomerase (telomere constant length maintenance). In the mathematical models we compared four possible mechanisms for telomere shortening. The simplest model assumes that a fixed amount of telomere is lost on each replication; the second supposes that telomere loss depends on telomere length; for the third case the amount of telomeres loss per division is fixed but the probability of dividing depends on telomere length; the fourth cases has both telomere loss and the probability of division dependent on telomere length. We start by developing Monte Carlo simulations of normal ageing using these four cases. Then we generalize the Monte Carlo simulations to consider Werner’s syndrome, where the extra telomeres are lost during replication accelerate the ageing process. In order to investigate how the distribution of telomere length varies with time, we derive, from the discrete model, continuum models for the four different cases. Results from the Monte Carlo simulations and the deterministic models are shown to be in good agreement. In addition to telomere loss, we also consider increases in telomere length caused by the enzyme telomerase, by appropriately extending the earlier Monte Carlo simulations and continuum models. Results from the Monte Carlo simulations and the deterministic models are shown to be in good agreement. We also show that the concentration of telomerase in cells can control their proliferative potential.

Plant growth regulators have a vital role in plant growth and development. The cellular response to these regulators depends on the presence and the action of specific receptors. The plant growth regulators and their receptors act together in complexes which determine the final effects of the plant growth regulators. In the research reported here, emphasis has been given to the regulation of the activity of the receptors themselves. The regulation of the N-l-naphthylphthalamic acid (N~A) receptor through phosphorylation and dephosphorylation and the regulation of the auxin binding protein (ABP) through gene manipulation have been investigated. NPA, an auxin transport inhibitor, was found to bind specifically to a crude membrane preparation from sugar beet seedling leaf cell suspension cultures. The in vitro binding was optimal at pH 4.5 and 4?C. Binding parameters for NP A binding were determined by Scatchard analysis. The dissociation constant (Kd) and binding protein concentration were found to be 1.71 x 10-7 mol dm-3 and 220 pmoles g-I membrane protein respectively. It was found that the amount of specific 3H-NPA binding was significantly increased by adding Mg2+ A TP to the binding assay solution; treatment of membrane preparations with acid phosphatase, prior to the NP A binding assay, resulted in lower specific binding. A TP activation and phosphatase inactivation were culture stage dependent. Although a considerable effect could be detected when using cells from day 8 (representing the linear phase), the same treatment did not alter the binding if cells from day I (representing lag phase) or day 14 (representing the stationary phase) were used. These observations have strongly highlighted the possible involvement of a phosphorylation and dephosphorylation mechanism in vivo in the regulation of the activity of the NP A receptor. High phosphatase activity was found in the supernatant, but not in the membrane pellet, after 50,000 g centrifugation. The presence of a membrane-bound auxin receptor, ABP, was demonstrated by Scatchard analysis in sugar beet seedlings. The Kd value and the receptor concentration were found to be 2.15 x 10-6 mol dm-3 and 68 pmoles g-I membrane protein. The protein could be solubilised either with the detergent Triton X-I 00 or by acetone-washing, with a recovery of about 40%. An acetone-solubilised ABP preparation could be partially purified by DEAE-Sephacel ion exchange chromatography, NAA-linked AH-Sepharose 4B affmity chromatography or Sephadex G-200 gel filtration. The recovery after any of these chromatographic treatments was very low so that successive chromatography for further purification was unsuccessful. The low level of detectable binding after purification resulted mainly from the low abundance of ABP in the plant material. Non-radioactive labelling and detection techniques were used to show that an ABP-probe hybridized to sugar beet genomic DNA during dot blotting. The present study has indicated that receptor activity could be regulated by a phosphorylation and dephosphorylation mechanism in plants. The investigation has also suggested that the effect of plant growth regulators on plant development could be regulated through the manipulation of the expression of their receptor genes.