GreMuTRRR: A Novel Genetic Algorithm to Solve Distance Geometry Problem for Protein Structures

Nuclear Magnetic Resonance (NMR) Spectroscopy is a widely used technique to predict the native structure of proteins. However, NMR machines are only able to report approximate and partial distances between pair of atoms. To build the protein structure one has to solve the Euclidean distance geometry problem given the incomplete interval distance data produced by NMR machines. In this paper, we propose a new genetic algorithm for solving the Euclidean distance geometry problem for protein structure prediction given sparse NMR data. Our genetic algorithm uses a greedy mutation operator to intensify the search, a twin removal technique for diversification in the population and a random restart method to recover stagnation. On a standard set of benchmark dataset, our algorithm significantly outperforms standard genetic algorithms.Comment: Accepted for publication in the 8th International Conference on Electrical and Computer Engineering (ICECE 2014

Protein structure determination via an efficient geometric build-up algorithm

Abstract Background A protein structure can be determined by solving a so-called distance geometry problem whenever a set of inter-atomic distances is available and sufficient. However, the problem is intractable in general and has proved to be a NP hard problem. An updated geometric build-up algorithm (UGB) has been developed recently that controls numerical errors and is efficient in protein structure determination for cases where only sparse exact distance data is available. In this paper, the UGB method has been improved and revised with aims at solving distance geometry problems more efficiently and effectively. Methods An efficient algorithm (called the revised updated geometric build-up algorithm (RUGB)) to build up a protein structure from atomic distance data is presented and provides an effective way of determining a protein structure with sparse exact distance data. In the algorithm, the condition to determine an unpositioned atom iteratively is relaxed (when compared with the UGB algorithm) and data structure techniques are used to make the algorithm more efficient and effective. The algorithm is tested on a set of proteins selected randomly from the Protein Structure Database-PDB. Results We test a set of proteins selected randomly from the Protein Structure Database-PDB. We show that the numerical errors produced by the new RUGB algorithm are smaller when compared with the errors of the UGB algorithm and that the novel RUGB algorithm has a significantly smaller runtime than the UGB algorithm. Conclusions The RUGB algorithm relaxes the condition for updating and incorporates the data structure for accessing neighbours of an atom. The revisions result in an improvement over the UGB algorithm in two important areas: a reduction on the overall runtime and decrease of the numeric error.Peer Reviewe

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Eigenvector Synchronization, Graph Rigidity and the Molecule Problem

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend on previous work and propose the 3D-ASAP algorithm, for the graph realization problem in $\mathbb{R}^3$, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in $\mathbb{R}^3$, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms.Comment: 49 pages, 8 figure

Geometric Build-up Solutions for Protein Determination via Distance Geometry

Proteins carry out an almost innumerable amount of biological processes that are absolutely necessary to life and as a result proteins and their structures are very often the objects of study in research. As such, this thesis will begin with a description of protein function and structure, followed by brief discussions of the two major experimental structure determination methods. Another problem that often arises in molecular modeling is referred to as the Molecular Distance Geometry Problem (MDGP). This problem seeks to find coordinates for the atoms of a protein or molecule when given only a set of pair-wise distances between atoms. To introduce the complexities of the MDGP we begin at its origins in distance geometry and progress to the specific sub-problems and some of the solutions that have been developed. This is all in preparation for a discussion of what is known as the Geometric Build-up (GBU) Solution. This solution has lead to the development of several algorithms and continues to be modified to account for more and different complexities. The culmination of this thesis, then, is a new algorithm, the Revised Updated Geometric Build-up, that is faster than previous GBU’s while maintaining the accuracy of the resulting structure

A distributed SDP-based algorithm for large noisy anchor-free graph realization

Master'sMASTER OF SCIENC

Surface Reconstruction from Scattered Point via RBF Interpolation on GPU

In this paper we describe a parallel implicit method based on radial basis functions (RBF) for surface reconstruction. The applicability of RBF methods is hindered by its computational demand, that requires the solution of linear systems of size equal to the number of data points. Our reconstruction implementation relies on parallel scientific libraries and is supported for massively multi-core architectures, namely Graphic Processor Units (GPUs). The performance of the proposed method in terms of accuracy of the reconstruction and computing time shows that the RBF interpolant can be very effective for such problem.Comment: arXiv admin note: text overlap with arXiv:0909.5413 by other author

Geometric Methods in Machine Learning and Data Mining

In machine learning, the standard goal of is to find an appropriate statistical model from a model space based on the training data from a data space; while in data mining, the goal is to find interesting patterns in the data from a data space. In both fields, these spaces carry geometric structures that can be exploited using methods that make use of these geometric structures (we shall call them geometric methods), or the problems themselves can be formulated in a way that naturally appeal to these methods. In such cases, studying these geometric structures and then using appropriate geometric methods not only gives insight into existing algorithms, but also helps build new and better algorithms. In my research, I develop methods that exploit geometric structure of problems for a variety of machine learning and data mining problems, and provide strong theoretical and empirical evidence in favor of using them. My dissertation is divided into two parts. In the first part, I develop algorithms to solve a well known problem in data mining i.e. distance embedding problem. In particular, I use tools from computational geometry to build a unified framework for solving a distance embedding problem known as multidimensional scaling (MDS). This geometry-inspired framework results in algorithms that can solve different variants of MDS better than previous state-of-the-art methods. In addition, these algorithms come with many other attractive properties: they are simple, intuitive, easily parallelizable, scalable, and can handle missing data. Furthermore, I extend my unified MDS framework to build scalable algorithms for dimensionality reduction, and also to solve a sensor network localization problem for mobile sensors. Experimental results show the effectiveness of this framework across all problems. In the second part of my dissertation, I turn to problems in machine learning, in particular, use geometry to reason about conjugate priors, develop a model that hybridizes between discriminative and generative frameworks, and build a new set of generative-process-driven kernels. More specifically, this part of my dissertation is devoted to the study of the geometry of the space of probabilistic models associated with statistical generative processes. This study --- based on the theory well grounded in information geometry --- allows me to reason about the appropriateness of conjugate priors from a geometric perspective, and hence gain insight into the large number of existing models that rely on these priors. Furthermore, I use this study to build hybrid models more naturally i.e., by combining discriminative and generative methods using the geometry underlying them, and also to build a family of kernels called generative kernels that can be used as off-the-shelf tool in any kernel learning method such as support vector machines. My experiments of generative kernels demonstrate their effectiveness providing further evidence in favor of using geometric methods