III: Medium: Geometric and topological approaches to biomolecular structure and dynamics

Michigan State University, East Lansing MI

Investigators

Abstract

Experimental exploration of self-organizing biomolecular systems, such as viruses, molecular motors and proteins in Alzheimer's disease, has been a dominating driving force in scientific discovery and innovation in the past few decades. Unfortunately, quantitative understanding of biomolecular structure, function, and dynamics severely lags behind the pace of the experimental progress. An average protein in human body has about 5500 atoms, which, together with its surrounding water molecules, involve about 100,000 degrees of freedom. The dimensionality increases dramatically for complex biological processes and biomolecular systems. The real time structure optimization, dynamic simulation, and data analysis of molecular motors and/or viruses in human cells are intractable with full-atom models at present. A crucial question is how to reduce the number of degrees of freedom, while retaining the fundamental physics in complex biological systems. The proposed research may be transformative. As the first differential geometry based multiscale/ multiresolution approach to biomolecular systems, it will open a new direction and foster similar approaches in multiscale modeling of other large data systems in future research. Additionally, new persistently stable manifold strategy can be applied to other fields, such as image processing, computer aided design, and fluid mechanics. Furthermore, the proposed new coupled equations will lead to new research topics in geometry, topology, PDE analysis and mathematical biology. Finally, our new theoretical framework is directly integrated into popular software packages to ensure extensive usage by the community of researchers throughout mathematics, computer science and biology. The proposed research has a solid educational component. The project will support the training of student and junior researchers in mathematical modeling, data analysis and algorithm development. The enhancement of curricula from the proposed research is planned as a continuation of PIs teaching-research practice. Special curriculum development, outreach program and annual workshops are designed to further broaden educational and societal impacts. The proposed research addresses grand challenges in the structure, function and dynamics of self-organizing biomolecular systems due to exceptionally massive data sets. These challenges are tackled through the introduction of a new differential geometry based multiscale model, together with a multiresolution coarse grained method based on persistently stable manifolds in molecular dynamics data. This proposal offers innovative new approaches to an important area in massive data management, dimensionality reduction, computational mathematics and mathematical modeling. This project uses a number of geometric and topological approaches to address the scaling issues.. First, the multidisciplinary team will use multiscale framework which reduces the dimensionality and number of degrees of freedom by a macroscopic continuum description of the aquatic environment, and a microscopic discrete description of biomolecules. To further reduce the dimensionality of excessively large biomolecular systems, they introduce a multiresolution coarse-grained approach based on persistently stable manifolds in molecular dynamics data. A total free energy functional is introduced to bring the macroscopic surface tension and microscopic potential interactions on an equal footing. The differential geometry theory of surfaces is utilized to describe the interface between macroscopic and microscopic domains. Potential driven geometric flows are constructed to minimize the total free energy functional. Euler characteristic and total curvature are employed to analyze the topology and corresponding function of biomolecules. Frenet frames are utilized to characterize the local geometry and associated stable manifolds in dynamical data of biomolecular systems. Machine learning algorithms are proposed to extract stable manifolds. In the last step, a strategy is introduced to explore the persistence of stable manifolds, which provides the assurance for the reliability of the coarse grained model. In addition to promising and extensive preliminary results illustrating the power of this approach, extensive validation and application have been proposed to ensure that this methodology yields robust and powerful tools for biomolecular structure optimization and dynamical simulation.