Folding and Unfolding of Polygonal Linkages, with Applications to Structural Biology

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ABSTRACTWe are studying folding and unfolding processes for polygonal linkages (and other partially rigid structures with many degrees of freedom) in dimensions two and three, with particular emphasis on applications in molecular biology. Our primary focus is on fundamental mathematical properties (topologic, geometric, algebraic) and novel algorithmic approaches, with the long run goal of understanding the fundamental laws of the protein folding simulation problem. We are proposing a novel Monte Carlo approach for move generators in energy minimization simulations in two dimensions, based on mechanisms which simultaneously change many angles and are guaranteed to be collision free, and we investigate the feasibility of extending the approach in three dimensions. The mathematics involved in this research includes techniques from algebraic topology, algebraic geometry and rigidity theory, while the computational effort is expected to lead to developing efficient data structures and algorithms for planning, analyzing, approximating, and tracking such motions. Besides its intrinsic scientific merit, we anticipate this project to have broader impact through a novel way of integrating research, teaching, inter-institutional and interdisciplinary collaborations. The grant will allow one of the PIs, who teaches at the undergraduate level at a women's college, to establish an integrated research group including (besides undergraduates) graduate students. By bringing top research resources and powerful collaborations to an undergraduate institution devoted to promoting the technical education of women in a liberal arts environment, we expect to influence the interdisciplinary curriculum of the future, enhance the infrastructure for research and education at the only women's college with an engineering program, and generate models for training a diverse body of students for graduate work, research and teaching.