Mathematical Sciences: Quantum Self-Trapping

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The investigator continues his studies of quantization of several differential equations. particularly the Ablowitz-Ladik equation and a form of discrete nonlinear Schrodinger equation called the discrete self-trapping equation. The equations arise in describing the behavior of a variety of nonlinear systems, many coming from a biological context. He uses these results to enhance understanding of several applications: comparisons of quantum theories for the two equations, especially focussing on the issue of soliton binding energies, analysis of spectral data for a molecular crystal, and modelling the growth dynamics of microtubule. A better understanding of how cells behave is a central theme in biology. This project uses mathematical tools to consider several related aspects of cell biology and also concerns itself with the tools themselves. Hence the investigator studies the structure of microtubule (a kind of "protein crystal"), undertakes a more general analysis of molecular crystals, and develops the theory of certain kinds of differential equations that describe nonlinear phenomena. A microtubule is an arrangement of tubulin in a tube-like shape. Tubulin is a protein that plays a key role in the ability of cells to hold their shape and to move. So a better understanding of the growth of microtubule could lead to a clearer picture of cell development; this is important in areas as diverse as the development of fetuses and the growth of tumors.