Orientador: Carlile Campos Lavor / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T18:42:28Z (GMT). No. of bitstreams: 1 Rezende_GermanoAbudde_D.pdf: 1418033 bytes, checksum: 61d29b02274278ede5ffca797e26371a (MD5) Previous issue date: 2014 / Resumo: A geometria de distâncias é o estudo da geometria baseado no conceito de distância. Ela é útil em várias aplicações, onde os dados de entrada consistem de um conjunto incompleto de distâncias, e a saída é um conjunto de pontos no espaço euclidiano, que realiza as distâncias dadas. No Problema de Geometria de Distâncias (DGP), é dado um inteiro K > 0 e um grafo simples, não direcionado, G = (V,E,d), cujas arestas são ponderadas por uma função não negativa d. Queremos determinar se existe uma função (realização) que leva os vértices de V em coordenadas no espaço euclidiano K-dimensional, satisfazendo todas as restrições de distâncias dadas por d. Um DGPk (com K fixado) está fortemente relacionado a um outro tipo de problema, que trata dos possíveis completamentos de uma certa matriz de distâncias euclidianas. Este último pode ser visto, em um certo sentido, como o "dual do primeiro problema". Neste trabalho, exploramos essa dualidade com a finalidade de propor melhorias no método Branch-and-Prune aplicado a uma versão discreta do DGPk / Abstract: Distance Geometry is the study of geometry based on the concept of distance. It is useful in many applications where the input data consists of an incomplete set of distances, and the output is a set of points in some Euclidean space which realizes the given distances. In the Distance Geometry Problem (DGP), it is given an integer K > 0 and a simple undirected weighted graph G = (V,E,d), whose edges are weighted by a non-negative function d. We want to determine if there is a (realization) function that associates the vertices of V with coordinates of the K-dimensional Euclidean space satisfying all distance constraints given by d. A DGPk (with K fixed) is closely related to another type of problem, which treats the possible completions of a certain Euclidean distance matrix. In some sense, this is the "dual" of the first problem. We explore this duality in order to improve the Branch-and-Prune method applied to a discrete version of the DGPk / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Orientador: Carlile Campos Lavor / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T22:04:46Z (GMT). No. of bitstreams: 1 Fidalgo_FelipeDelfiniCaetano_D.pdf: 5383479 bytes, checksum: 8f7bf5142b44fa99ea2742f6183ee1c6 (MD5) Previous issue date: 2015 / Resumo: Motivado por estudos em estruturas 3D de proteínas, biomoléculas imprescindíveis no estudo da vida, surgiu um problema chamado Discretizable Molecular Distance Geometry Problem (DMDGP) que provou ser NP-Difícil. Para resolvê-lo, existe um algoritmo da literatura, Branch & Prune (BP), que utiliza uma estratégia combinatória de exploração de uma árvore binária de soluções associada ao problema. Além disso, foram descobertas relações de simetria que permitem obter uma solução, a partir de outra, através de reflexões nos chamados vértices de simetria. Alguns pesquisadores passaram a realizar este trabalho em paralelo (ParallelBP), dividindo uma instância em sub-instâncias, resolvendo localmente com o BP (o que pode ser feito em duas direções) e unindo as sub-soluções com movimentos rígidos, com o intuito de determinar as soluções em menor tempo. Nossa proposta é fornecer uma estratégia Dividir-e-Conquistar para resolver o DMDGP, de modo a melhorar a abordagem em paralelo. Ela possui três estágios. Inicialmente, dividimos uma instância em sub-instâncias duas-a-duas sobrepostas através dos vértices de simetria. Depois, utiliza-se os chamados gaps para decidir a direção em que o BP deve fornecer a solução local. Por fim, utilizamos rotações baseadas na Álgebra de Quatérnios para combinar as sub-soluções em soluções factíveis / Abstract: Motived by studies in 3D structures of proteins, essential biomolecules for Life, arised a problem called Discretizable Molecular Distance Geometry Problem (DMDGP) which proved to be NP-Hard. Aiming to solve it, there is an algorithm in the literature, Branch & Prune (BP), which uses a combinatorial strategy of exploring a binary tree of solutions that is associated to the problem. Moreover, some symmetry relations have been discovered which allows the obtainance of one solution from the other one by means of reflections in the so-called symmetry vertices. Some researchers started to do this work using parallel computing (ParallelBP), dividing one instance into sub-instances, solving the problem locally with the BP (what can be done in two directions) and joining the sub-solutions with rigid movements, with the objective of determining the solutions in a smaller time. Our purpose, thus, is to provide a Divide-and-Conquer strategy to solve the DMDGP in order to improve the parallel version. It has three stages. Initially, the instance is divided into sub-instances two-by-two overlapping by means of the symmetry vertices. After, the so-called gaps are used to decide the direction that the BP ought to provide the local solution. Finally, we propose to use Quaternion Rotations to combine sub-solutions into feasible solutions / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

En bioinformatique structurale, une partie des méthodes computationnelles qui calculent les structures de protéines à l'aide de données expérimentales, effectuent une optimisation de la position des atomes sous les contraintes expérimentales mesurées sur le système étudié, ainsi que sous des contraintes provenant de la connaissance générique de la stéréochimie organique. Ces méthodes d'optimisation présentent l'inconvénient de ne pas garantir la détermination de la meilleure solution. De plus, la validation de l'optimisation se fait en comparant les résultats obtenus pour des calculs répétés, et le résultat d'un calcul est accepté dans la mesure où le même résultat est obtenu plusieurs fois. Par cette approche, on rend plus difficile la détection de conformations alternatives de protéines, qui sont pourtant le sujet d'un vif intérêt dans la littérature. En effet, le développement de la sensibilité des techniques de résonance magnétique nucléaire (RMN) a permis de mettre en évidence plusieurs cas d'échange conformationnel reliés à la fonction des protéines. Dans ce projet de thèse, nous avons étudié une nouvelle approche pour le calcul de structures des protéines et l'exploration de leurs espaces conformationnels, basée sur la résolution du problème de Géométrie de Distance associé aux contraintes de distances dans une protéine par l'algorithme "interval Branch and Prune". Le logiciel implémentant cette méthode est appelée iBPprot, il incarne l'une des premières tentatives d'échantillonnage exhaustive des espaces conformationnels des protéines. Dans un premier temps, on s'est intéressé à l'application de la méthode en utilisant exclusivement des constraintes de distances exactes. Les résultats ont démontré que iBPprot était capable de reconstruire des structures références en s'appuyant seulement sur quelques contraintes à courte portée. De plus, la reconstruction a été d'une précision telle que la conformation générée présentait un RMSD de 1 Angstrom maximum avec la structure référence. L'exploration exhaustive de l'espace conformationnel a été possible pour une bonne partie des protéines cibles. Les temps de calcul pour l'exploration des espaces conformationnels ont été très variables allant de quelques secondes pour quelques protéines jusqu'à des semaines pour d'autres. L'évaluation de la qualité des structures obtenues a démontré qu'au moins 68% des valeurs de phi et psi sont localisées dans la zone 'core' du diagramme de Ramachandran. Cependant, des clash stériques ont été détectées dans plusieurs conformations mettant en jeu jusqu'à 7% d'atomes dans quelques unes de ces conformations. Dans un deuxième temps, on s'est intéressé à l'application de la méthode en incluant des intervalles de distances comme contraintes dans les calculs. Dans ce cas de figure, la méthode a réussi a reconstruire des structures références avec un RMSD inférieur à 5 Angstrom pour plus de la moitié des protéines cibles. En contre partie, le parcours complet de l'espace conformationnel n'a été possible que pour la plus petite protéine de l'ensemble des protéines étudiées. Pour la moitié des autres protéines, plus de 70% des atomes ont vu leurs positions échantillonnées. La qualité des structures obtenues a regressé en comparaison avec les simulations faites avec des distances exactes. En effet, seulement 53% des valeurs de phi et psi étaient localisées dans la zone 'core' du diagramme de Ramachandran, et le pourcentage d'atomes impliqués dans un clash stérique s'élevait jusqu'à 22% pour quelques protéines. Concernant le temps de calcul, le taux de génération de conformations a été déterminé pour chaque protéine cible, et il s'est avéré que globalement sa valeur etait compétitive par rapport aux valeurs des taux observables dans la littérature... / Structural biology has allowed us expand our knowledge of living organisms. It is defined as the investigation of the structure and function of biological systems at the molecular level. Studying a biomolecule's structure offers insight into its geometry, as angles and distances between the biomolecule's atoms are measured in order to determine the biomolecular structure. The values of these geometrical parameters may be obtained from biophysical techniques, such as X-ray crystallography or nuclear magnetic resonance (NMR) spectroscopy. One of the most used methods to calculate protein structures from geometric restraints is simulated annealing. This method does not guarantee an exhaustive sampling of protein conformational space, which is a shortcoming as one protein may adopt multiple functional conformations, and it is important to determine them exhaustively. In this PhD project, the efficiency of a new method - derived from operations research and computational geometry - is studied in order to answer this question: How does this method explore the conformational spaces of small proteins? This method - implemented within the iBPprot software framework - treats protein structure determination as a distance geometry problem, which the interval branch-and-prune algorithm tries to solve by the full exploration of its solutions space. The results obtained by iBPprot on a set of test proteins, with sizes ranging from 24 to 120 residues and with known structures, are analyzed here. Using short-range exact distance restraints, it was possible to rebuild the structure of all protein targets, and for many of them it was possible to exhaustively explore their conformational spaces. In practice, it is not always possible to obtain exact distance restraints from experiments. Therefore, this method was then tested with interval data restraints. In these cases, iBPprot permitted the sampling of the positions of more than 70% of the atoms constituting the protein backbone for most of the targets. Furthermore, conformations whose r.m.s. deviations closer than 6 Angstrom to the target ones were obtained during the conformational space exploration. The quality of the generated structures was satisfactory with respect to Ramachandran plots, but needs improvement because of the presence of steric clashes in some conformers. The runtime for most performed calculations was competitive with existing structure determination method...

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