Exploring protein folding trajectories using geometric spanners.

Computer Science Department, Stanford, CA 94305, USA.
Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing 02/2005;
Source: PubMed

ABSTRACT We describe the 3-D structure of a protein using geometric spanners--geometric graphs with a sparse set of edges where paths approximate the n2 inter-atom distances. The edges in the spanner pick out important proximities in the structure, labeling a small number of atom pairs or backbone region pairs as being of primary interest. Such compact multiresolution views of proximities in the protein can be quite valuable, allowing, for example, easy visualization of the conformation over the entire folding trajectory of a protein and segmentation of the trajectory. These visualizations allow one to easily detect formation of secondary and tertiary structures as the protein folds.

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    ABSTRACT: In recent years, considerable advances have been made in the field of storing and processing moving data points. Data structures that discretely update moving objects or points are called Kinetic Data Structures. A particular type of data structure that constructs geometric spanner for moving point sets is called Deformable Spanner. This paper explores the underlying philosophy of deformable spanner and describes its rudiments. Deformable Spanners create a multi-resolution hierarchical subgraph that reflects a great deal about the original graph while maintaining all the existing proximity information in the graph. This proximity information plays a crucial role in wide variety of application areas, such as fast similarity queries in metric spaces, clustering in both dynamic and static multidimensional data, and cartographic representation of the complex networks. The motivation behind our work is twofold: first, to provide comprehensive understanding of the deformable spanner with sample illustrations without sacrificing depth of coverage; second, to analyze and improve deformable spanner's covering property for 2D case in Euclidean space.
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    ABSTRACT: Dissertation In this thesis we explore and extend the theory of persistent homology, which captures topological features of a function by pairing its critical values. The result is represented by a collection of points in the extended plane called persistence diagram. We start with the question of ridding the function of topological noise as suggested by its persistence diagram. We give an algorithm for hierarchically finding such epsilon-simplifications on 2-manifolds as well as answer the question of when it is impossible to simplify a function in higher dimensions. We continue by examining time-varying functions. The original algorithm computes the persistence pairing from an ordering of the simplices in a triangulation and takes worst-case time cubic in the number of simplices. We describe how to maintain the pairing in linear time per transposition of consecutive simplices. A side effect of the update algorithm is an elementary proof of the stability of persistence diagrams. We introduce a parametrized family of persistence diagrams called persistence vineyards and illustrate the concept with a vineyard describing a folding of a small peptide. We also base a simple algorithm to compute the rank invariant of a collection of functions on the update procedure. Guided by the desire to reconstruct stratified spaces from noisy samples, we use the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of a sampled stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample. Finally, to refine the measurement of local homology the thesis extends the notion of persistent homology to sequences of kernels, images, and cokernels of maps induced by inclusions in a filtration of pairs of spaces. Specifically, we note that persistence in this context is well defined, we prove that the persistence diagrams are stable, and we explain how to compute them. Additionally, we use image persistence to cope with functions on noisy domains.
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    ABSTRACT: Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its per- tinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in (10) computes the pairs from an ordering of the simplices in a triangulation and takes worst-case time cubic in the number of simplices. The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering. A side-effect of the algorithm's anal- ysis is an elementary proof of the stability of persistence diagrams (7) in the special case of piecewise-linear functions. We use the algorithm to compute 1-parameter families of diagrams which we apply to the study of protein folding trajectories.
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