Browsing by Author "Shah, M."

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    A Symmetry Preserving Singular Value Decomposition
    (2005-01) Shah, M.; Sorensen, D.C.
    A reduced order representation of a large data set is often realized through a principle component analysis based upon a singular value decomposition (SVD) of the data. The left singular vectors of a truncated SVD provide the reduced basis. In several applications such as facial analysis and protein dynamics, structural symmetry is inherent in the data. Typically, reflective or rotational symmetry is expected to be present in these applications. In protein dynamics, determining this symmetry allows one to provide SVD major modes of motion that best describe the symmetric movements of the protein. In face detection, symmetry in the SVD allows for more efficient compression algorithms. Here, we present a method to compute the plane of reflective symmetry or the axis of rotational symmetry of a large set of points. Moreover, we develop a symmetry preserving singular value decomposition (SPSVD) that best approximates the given set while respecting the symmetry. Interesting subproblems arise in the presence of noisy data or in situations where most, but not all, of the structure is symmetric. An important part of the determination of the axis of rotational symmetry or the plane of reflective symmetry is an iterative re-weighting scheme. This scheme is rapidly convergent in practice and seems to be very effective in ignoring outliers (points that do not respect the symmetry).
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    Simulating Nanoscale Functional Motions of Biomolecules
    (2006-05) Wriggers, W.; Zhang, Z.; Shah, M.; Sorensen, D.C.
    We are describing efficient dynamics simulation methods for the characterization of functional motion of biomolecules on the nanometer scale. Multivariate statistical methods are widely used to extract and enhance functional collective motions from molecular dynamics (MD) simulations. A dimension reduction in MD is often realized through a principal component analysis or a singular value decomposition (SVD) of the trajectory. Normal mode analysis is a related collective coordinate space approach, which involves the decomposition of the motion into vibration modes based on an elastic model. Using the myosin motor protein as an example we describe a hybrid technique termed amplified collective motions that enhances sampling of conformational space through a combination of normal modes with atomic level MD. Unfortunately, the forced orthogonalization of modes in collective coordinate space leads to complex dependencies that are not necessarily consistent with the symmetry of biological macromolecules and assemblies. In many biological molecules, such as HIV-1 protease, reflective or rotational symmetries are present that are broken using standard orthogonal basis functions. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry from the trajectory frames. Moreover, we develop an SVD that best approximates the given trajectory while respecting the symmetry. Finally we describe a local feature analysis (LFA) to construct a topographic representation of functional dynamics in terms of local features. The LFA representations are low-dimensional, and provide a reduced basis set for collective motions, but unlike global collective modes they are sparsely distributed and spatially localized. This yields a more reliable assignment of essential dynamics modes across different MD time windows.