Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Algorithmes pour le (dés)assemblage d'objets complexes et applications à la biologie structurale

La compréhension et la prédiction des relations structure-fonction de protéines par des approches in sillico représentent aujourd'hui un challenge. Malgré le développement récent de méthodes algorithmiques pour l'étude du mouvement et des interactions moléculaires, la flexibilité de macromolécules reste largement hors de portée des outils actuels de modélisation moléculaire. L'objectif de cette thèse est de développer une nouvelle approche basée sur des algorithmes de planification de mouvement issus de la robotique pour mieux traiter la flexibilité moléculaire dans l'étude des interactions protéiques. Nous avons étendu un algorithme récent d'exploration par échantillonnage aléatoire, ML-RRT pour le désassemblage d'objets articulés complexes. Cet algorithme repose sur la décomposition des paramètres de configuration en deux sous-ensembles actifs et passifs, qui sont traités de manière découplée. Les extensions proposées permettent de considérer plusieurs degrés de mobilité pour la partie passive, qui peut être poussée ou attirée par la partie active. Cet outil algorithmique a été appliqué avec succès pour l'étude des changements conformationnels de protéines induits lors de la diffusion d'un ligand. A partir de cette extension, nous avons développé une nouvelle méthode pour la résolution simultanée du séquençage et des mouvements de désassemblage entre plusieurs objets. La méthode, nommée Iterative-ML-RRT, calcule non seulement les trajectoires permettant d'extraire toutes les pièces d'un objet complexe assemblé, mais également l'ordre permettant le désassemblage. L'approche est générale et a été appliquée pour l'étude du processus de dissociation de complexes macromoléculaires en introduisant une fonction d'évaluation basée sur l'énergie d'interaction. Les résultats présentés dans cette thèse montrent non seulement l'efficacité mais aussi la généralité des algorithmes proposés. ABSTRACT : Understanding and predicting structure-function relationships in proteins with fully in silico approaches remain today a great challenge. Despite recent developments of computational methods for studying molecular motions and interactions, dealing with macromolecular flexibility largely remains out of reach of the existing molecular modeling tools. The aim of this thesis is to develop a novel approach based on motion planning algorithms originating from robotics to better deal with macromolecular flexibility in protein interaction studies. We have extended a recent sampling-based algorithm, ML-RRT, for (dis)-assembly path planning of complex articulated objects. This algorithm is based on a partition of the configuration parameters into active and passive subsets, which are then treated in a decoupled manner. The presented extensions permit to consider different levels of mobility for the passive parts that can be pushed or pulled by the motion of active parts. This algorithmic tool is successfully applied to study protein conformational changes induced by the diffusion of a ligand inside it. Building on the extension of ML-RRT, we have developed a novel method for simultaneously (dis)assembly sequencing and path planning. The new method, called Iterative-ML-RRT, computes not only the paths for extracting all the parts from a complex assembled object, but also the preferred order that the disassembly process has to follow. We have applied this general approach for studying disassembly pathways of macromolecular complexes considering a scoring function based on the interaction energy. The results described in this thesis prove not only the efficacy but also the generality of the proposed algorithm

(Dis)assembly path planning for complex objects and applications to structural biology

Understanding and predicting structure-function relationships in proteins with fully in silico approaches remain today a great challenge. Despite recent developments of computational methods for studying molecular motions and interactions, dealing with macromolecular flexibility largely remains out of reach of the existing molecular modeling tools. The aim of this thesis is to develop a novel approach based on motion planning algorithms originating from robotics to better deal with macromolecular flexibility in protein interaction studies. We have extended a recent sampling-based algorithm, ML-RRT, for (dis)-assembly path planning of complex articulated objects. This algorithm is based on a partition of the configuration parameters into active and passive subsets, which are then treated in a decoupled manner. The presented extensions permit to consider different levels of mobility for the passive parts that can be pushed or pulled by the motion of active parts. This algorithmic tool is successfully applied to study protein conformational changes induced by the diffusion of a ligand inside it. Building on the extension of ML-RRT, we have developed a novel method for simultaneously (dis)assembly sequencing and path planning. The new method, called Iterative-ML-RRT, computes not only the paths for extracting all the parts from a complex assembled object, but also the preferred order that the disassembly process has to follow. We have applied this general approach for studying disassembly pathways of macromolecular complexes considering a scoring function based on the interaction energy. The results described in this thesis prove not only the efficacy but also the generality of the proposed algorithm