A major goal of this project is to develop mathematical models describing biological systems of importance for understanding the molecular mechanisms of cancer with applications to therapy. We initiated the development of two models: 1) effects of telomerase inhibitors on tumor growth and 2) kinetics of G1 to S transition in cancer cells. The telomerase is an attractive target for anticancer therapy. Unlike other anticancer drugs the telomerase inhibitor effect is delayed because cells would enter crisis and eventually die only after certain number of cell doublings related to the telomere length at the time of inhibition. In the simplest case of homogeneous telomeres the inhibitory time delay, T, equals Rtd, where R=(L-Lcr)/l is the cell replication capacity, L and Lcr being the telomere lengths at the time of inhibition and at crisis (about 1.5 kb), respectively, l is the telomere length decrease per each cell division (about 0.1 kb) and td - the doubling time of the tumor cells. Because td varies from several days to more than several months while R may be in the range of several to one hundred divisions T may range from weeks to years. We developed more sophisticated kinetic models, based on the solution of a system of differential equations, which take into account the telomere heterogeneity and the kinetics of tumor growth as described by the exponential, logistic and Gompertz equations. Two results are of particular importance: 1) The telomere heterogeneity decreases the delay time T; for example, one of the models predicted that the time at which the tumor volume in the presence of inhibitors is half of that without inhibitors equals td(R)1/2, and 2) The tumor growth kinetics affects the delay time in a complex way in dependence on the kinetic constants and the telomere size distribution, but always leads to a relative increase in the number of cells which had the longest telomeres before treatment. While these models may predict the order of magnitude of the delay time and help in organizing our knowledge about the factors affecting the tumor growth in presence of telomerase inhibitors, only further experimental results on tumor telomere lengths and the kinetics of their reduction will allow a more accurate prediction of the kinetics of tumor growth. We also initiated the development of a mathematical model of the G1 to S transition based on the molecular diagrams proposed by K. Kohn. The study of this complex system now is in progress.
