Mathematical Sciences: Quantum Self-Trapping

研究者继续他对几个微分方程式的量子化的研究。特别是Ablwitz-Ladik方程和一种形式的离散非线性薛定谔方程，称为离散自陷方程。这些方程是在描述各种非线性系统的行为时产生的，其中许多来自生物学背景。他利用这些结果加深了对几个应用的理解：两个方程的量子理论的比较，特别是关注孤子结合能的问题，分子晶体的光谱数据分析，以及微管生长动力学的建模。更好地理解细胞的行为是生物学的一个中心主题。这个项目使用数学工具来考虑细胞生物学的几个相关方面，也关注工具本身。因此，研究人员研究了微管(一种蛋白质晶体)的结构，对分子晶体进行了更一般的分析，并发展了描述非线性现象的某些微分方程组的理论。微管是管状微管蛋白的排列。微管蛋白是一种在细胞保持形状和移动能力方面发挥关键作用的蛋白质。因此，更好地了解微管的生长可能会导致更清晰的细胞发育图景；这在胎儿发育和肿瘤生长等不同领域都很重要。

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.