Geometric algorithms for cavity detection on protein surfaces

Macromolecular structures such as proteins heavily empower cellular processes or functions. These biological functions result from interactions between proteins and peptides, catalytic substrates, nucleotides or even human-made chemicals. Thus, several interactions can be distinguished: protein-ligand, protein-protein, protein-DNA, and so on. Furthermore, those interactions only happen under chemical- and shapecomplementarity conditions, and usually take place in regions known as binding sites. Typically, a protein consists of four structural levels. The primary structure of a protein is made up of its amino acid sequences (or chains). Its secondary structure essentially comprises -helices and -sheets, which are sub-sequences (or sub-domains) of amino acids of the primary structure. Its tertiary structure results from the composition of sub-domains into domains, which represent the geometric shape of the protein. Finally, the quaternary structure of a protein results from the aggregate of two or more tertiary structures, usually known as a protein complex. This thesis fits in the scope of structure-based drug design and protein docking. Specifically, one addresses the fundamental problem of detecting and identifying protein cavities, which are often seen as tentative binding sites for ligands in protein-ligand interactions. In general, cavity prediction algorithms split into three main categories: energy-based, geometry-based, and evolution-based. Evolutionary methods build upon evolutionary sequence conservation estimates; that is, these methods allow us to detect functional sites through the computation of the evolutionary conservation of the positions of amino acids in proteins. Energy-based methods build upon the computation of interaction energies between protein and ligand atoms. In turn, geometry-based algorithms build upon the analysis of the geometric shape of the protein (i.e., its tertiary structure) to identify cavities. This thesis focuses on geometric methods. We introduce here three new geometric-based algorithms for protein cavity detection. The main contribution of this thesis lies in the use of computer graphics techniques in the analysis and recognition of cavities in proteins, much in the spirit of molecular graphics and modeling. As seen further ahead, these techniques include field-of-view (FoV), voxel ray casting, back-face culling, shape diameter functions, Morse theory, and critical points. The leading idea is to come up with protein shape segmentation, much like we commonly do in mesh segmentation in computer graphics. In practice, protein cavity algorithms are nothing more than segmentation algorithms designed for proteins.Estruturas macromoleculares tais como as proteínas potencializam processos ou funções celulares. Estas funções resultam das interações entre proteínas e peptídeos, substratos catalíticos, nucleótideos, ou até mesmo substâncias químicas produzidas pelo homem. Assim, há vários tipos de interacções: proteína-ligante, proteína-proteína, proteína-DNA e assim por diante. Além disso, estas interações geralmente ocorrem em regiões conhecidas como locais de ligação (binding sites, do inglês) e só acontecem sob condições de complementaridade química e de forma. É também importante referir que uma proteína pode ser estruturada em quatro níveis. A estrutura primária que consiste em sequências de aminoácidos (ou cadeias), a estrutura secundária que compreende essencialmente por hélices e folhas , que são subsequências (ou subdomínios) dos aminoácidos da estrutura primária, a estrutura terciária que resulta da composição de subdomínios em domínios, que por sua vez representa a forma geométrica da proteína, e por fim a estrutura quaternária que é o resultado da agregação de duas ou mais estruturas terciárias. Este último nível estrutural é frequentemente conhecido por um complexo proteico. Esta tese enquadra-se no âmbito da conceção de fármacos baseados em estrutura e no acoplamento de proteínas. Mais especificamente, aborda-se o problema fundamental da deteção e identificação de cavidades que são frequentemente vistos como possíveis locais de ligação (putative binding sites, do inglês) para os seus ligantes (ligands, do inglês). De forma geral, os algoritmos de identificação de cavidades dividem-se em três categorias principais: baseados em energia, geometria ou evolução. Os métodos evolutivos baseiam-se em estimativas de conservação das sequências evolucionárias. Isto é, estes métodos permitem detectar locais funcionais através do cálculo da conservação evolutiva das posições dos aminoácidos das proteínas. Em relação aos métodos baseados em energia estes baseiam-se no cálculo das energias de interação entre átomos da proteína e do ligante. Por fim, os algoritmos geométricos baseiam-se na análise da forma geométrica da proteína para identificar cavidades. Esta tese foca-se nos métodos geométricos. Apresentamos nesta tese três novos algoritmos geométricos para detecção de cavidades em proteínas. A principal contribuição desta tese está no uso de técnicas de computação gráfica na análise e reconhecimento de cavidades em proteínas, muito no espírito da modelação e visualização molecular. Como pode ser visto mais à frente, estas técnicas incluem o field-of-view (FoV), voxel ray casting, back-face culling, funções de diâmetro de forma, a teoria de Morse, e os pontos críticos. A ideia principal é segmentar a proteína, à semelhança do que acontece na segmentação de malhas em computação gráfica. Na prática, os algoritmos de detecção de cavidades não são nada mais que algoritmos de segmentação de proteínas

QuickCSG: Fast Arbitrary Boolean Combinations of N Solids

QuickCSG computes the result for general N-polyhedron boolean expressions without an intermediate tree of solids. We propose a vertex-centric view of the problem, which simplifies the identification of final geometric contributions, and facilitates its spatial decomposition. The problem is then cast in a single KD-tree exploration, geared toward the result by early pruning of any region of space not contributing to the final surface. We assume strong regularity properties on the input meshes and that they are in general position. This simplifying assumption, in combination with our vertex-centric approach, improves the speed of the approach. Complemented with a task-stealing parallelization, the algorithm achieves breakthrough performance, one to two orders of magnitude speedups with respect to state-of-the-art CPU algorithms, on boolean operations over two to dozens of polyhedra. The algorithm also outperforms GPU implementations with approximate discretizations, while producing an output without redundant facets. Despite the restrictive assumptions on the input, we show the usefulness of QuickCSG for applications with large CSG problems and strong temporal constraints, e.g. modeling for 3D printers, reconstruction from visual hulls and collision detection

QuickCSG: Fast Arbitrary Boolean Combinations of N Solids

QuickCSG computes the result for general N-polyhedron boolean expressions without an intermediate tree of solids. We propose a vertex-centric view of the problem, which simplifies the identification of final geometric contributions, and facilitates its spatial decomposition. The problem is then cast in a single KD-tree exploration, geared toward the result by early pruning of any region of space not contributing to the final surface. We assume strong regularity properties on the input meshes and that they are in general position. This simplifying assumption, in combination with our vertex-centric approach, improves the speed of the approach. Complemented with a task-stealing parallelization, the algorithm achieves breakthrough performance, one to two orders of magnitude speedups with respect to state-of-the-art CPU algorithms, on boolean operations over two to dozens of polyhedra. The algorithm also outperforms GPU implementations with approximate discretizations, while producing an output without redundant facets. Despite the restrictive assumptions on the input, we show the usefulness of QuickCSG for applications with large CSG problems and strong temporal constraints, e.g. modeling for 3D printers, reconstruction from visual hulls and collision detection

Local object crop collision network for efficient simulation of non-convex objects in GPU-based simulators

Our goal is to develop an efficient contact detection algorithm for large-scale GPU-based simulation of non-convex objects. Current GPU-based simulators such as IsaacGym and Brax must trade-off speed with fidelity, generality, or both when simulating non-convex objects. Their main issue lies in contact detection (CD): existing CD algorithms, such as Gilbert-Johnson-Keerthi (GJK), must trade off their computational speed with accuracy which becomes expensive as the number of collisions among non-convex objects increases. We propose a data-driven approach for CD, whose accuracy depends only on the quality and quantity of offline dataset rather than online computation time. Unlike GJK, our method inherently has a uniform computational flow, which facilitates efficient GPU usage based on advanced compilers such as XLA (Accelerated Linear Algebra). Further, we offer a data-efficient solution by learning the patterns of colliding local crop object shapes, rather than global object shapes which are harder to learn. We demonstrate our approach improves the efficiency of existing CD methods by a factor of 5-10 for non-convex objects with comparable accuracy. Using the previous work on contact resolution for a neural-network-based contact detector, we integrate our CD algorithm into the open-source GPU-based simulator, Brax, and show that we can improve the efficiency over IsaacGym and generality over standard Brax. We highly recommend the videos of our simulator included in the supplementary materials.Comment: RSS 2023 https://sites.google.com/view/locc-rss2023/hom

An energy-conserving contact theory for discrete element modelling of arbitrarily shaped particles: Contact volume based model and computational issues

The contact volume based energy-conserving contact model is presented in the current paper as a specialised version of the general energy-conserving contact model established in the first paper of this series (Feng, 2020). It is based on the assumption that the contact energy potential is taken to be a function of the contact volume between two contacting bodies with arbitrary (convex and concave) shapes in both 2D and 3D cases. By choosing such a contact energy function, the full normal contact features can be determined without the need to introduce any additional assumptions/parameters. By further exploiting the geometric properties of the contact surfaces concerned, more effective integration schemes are developed to reduce the evaluation costs involved. When a linear contact energy function of the contact volume is adopted, a linear contact model is derived in which only the intersection between two contact shapes is needed, thereby substantially improving both efficiency and applicability of the proposed contact model. A comparison of this linear energy-conserving contact model with some existing models for discs and spheres further reveals the nature of the proposed model, and provides insights into how to appropriately choose the stiffness parameter included in the energy function. For general non-spherical shapes, mesh representations are required. The corresponding computational aspects are described when shapes are discretised into volumetric meshes, while new developments are presented and recommended for shapes that are represented by surface triangular meshes. Owing to its additive property of the contact geometric features involved, the proposed contact model can be conducted locally in parallel using GPU or GPGPU computing without occurring much communication overhead for shapes represented as either a volumetric or surface triangular mesh. A set of examples considering the elastic impact of two shapes are presented to verify the energy-conserving property of the proposed model for a wide range of concave shapes and contact scenarios, followed by examples involving large numbers of arbitrarily shaped particles to demonstrate the robustness and applicability for more complex and realistic problems

Triangle influence supersets for fast distance computation

We present an acceleration structure to efficiently query the Signed Distance Field (SDF) of volumes represented by trianglemeshes. The method is based on a discretization of space. In each node, we store the triangles defining the SDF behaviour inthat region. Consequently, we reduce the cost of the nearest triangle search, prioritizing query performance, while avoidingapproximations of the field. We propose a method to conservatively compute the set of triangles influencing each node. Given anode, each triangle defines a region of space such that all points inside it are closer to a point in the node than the triangle is.This property is used to build the SDF acceleration structure. We do not need to explicitly compute these regions, which is crucialto the performance of our approach. We prove the correctness of the proposed method and compare it to similar approaches,confirming that our method produces faster query times than other exact methods.This work has been partially funded by Ministeri de Ciència i Innovació (MICIN), Agencia Estatal de Investigación (AEI) and the Fons Europeu de Desenvolupament Regional (FEDER) (project PID2021-122136OB-C21 funded by MCIN/AEI/10.13039/501100011033/FEDER, UE). The first author gratefully acknowledges the Universitat Politècnica de Catalunya and Banco Santander for the financial support of his predoctoral grant FPI-UPC grant.Peer ReviewedPostprint (published version

Revisión de literatura de jerarquía volúmenes acotantes enfocados en detección de colisiones

(Eng) A bounding volume is a common method to simplify object representation by using the composition of geometrical shapes that enclose the object; it encapsulates complex objects by means of simple volumes and it is widely useful in collision detection applications and ray tracing for rendering algorithms. They are popular in computer graphics and computational geometry. Most popular bounding volumes are spheres, Oriented-Bounding Boxe s (OBB’ s), Axis-Align ed Bound ing Boxes (AABB’ s); moreover , the literature review includes ellipsoids, cylinders, sphere packing, sphere shells , k-DOP’ s, convex hulls, cloud of points, and minimal bounding boxe s, among others. A Bounding Volume Hierarchy is ussualy a tree in which the complete object is represented thigter fitting every level of the hierarchy. Additionally, each bounding volume has a cost associated to construction, update, and interference te ts. For instance, spheres are invariant to rotation and translations, then they do not require being updated ; their constructions and interference tests are more straightforward then OBB’ s; however, their tightness is lower than other bounding volumes. Finally , three comparisons between two polyhedra; seven different algorithms were used, of which five are public libraries for collision detection.(Spa) Un volumen acotante es un método común para simplificar la representación de los objetos por medio de composición de formas geométricas que encierran el objeto; estos encapsulan objetos complejos por medio de volúmenes simples y son ampliamente usados en aplicaciones de detección de colisiones y trazador de rayos para algoritmos de renderización. Los volúmenes acotantes son populares en computación gráfica y en geometría computacional; los más populares son las esferas, las cajas acotantes orientadas (OBB’s) y las cajas acotantes alineadas a los ejes (AABB’s); no obstante, la literatura incluye elipses, cilindros empaquetamiento de esferas, conchas de esferas, k-DOP’s, convex hulls, nubes de puntos y cajas acotantes mínimas, entre otras. Una jerarquía de volúmenes acotantes es usualmente un árbol, en el cual la representación de los objetos es más ajustada en cada uno de los niveles de la jerarquía. Adicionalmente, cada volumen acotante tiene asociado costos de construcción, actualización, pruebas de interferencia. Por ejemplo, las esferas so invariantes a rotación y translación, por lo tanto no requieren ser actualizadas en comparación con los AABB no son invariantes a la rotación. Por otro lado la construcción y las pruebas de solapamiento de las esferas son más simples que los OBB’s; sin embargo, el ajuste de las esferas es menor que otros volúmenes acotantes. Finalmente, se comparan dos poliedros con siete algoritmos diferentes de los cuales cinco son librerías públicas para detección de colisiones

Model Simplification for Efficient Collision Detection in Robotics

Motion planning for industrial robots is a computationally intensive task due to the massive number of potential motions between any two configurations. Calculating all possibilities is generally not feasible. Instead, many motion planners sample a sub-set of the available space until a viable solution is found. Simplifying models to improve collision detection performance, a significant component of motion planning, results in faster and more capable motion planners. Several approaches for simplifying models to improve collision detection performance have been presented in the literature. However, many of them are sub-optimal for an industrial robotics application due to input model limitations, accuracy sacrifices, or the probability of increasing false negatives during collision queries. This thesis focuses on the development of model simplification approaches optimised for industrial robotics applications. Firstly, a new simplification approach, the Bounding Sphere Simplification (BSS), is presented that converts triangle-mesh inputs to a collection of spheres for efficient collision and distance queries. Additionally, BSS removes small features and generates an output model less prone to false negatives

Accelerating Motion Planning via Optimal Transport

Motion planning is still an open problem for many disciplines, e.g., robotics, autonomous driving, due to their need for high computational resources that hinder real-time, efficient decision-making. A class of methods striving to provide smooth solutions is gradient-based trajectory optimization. However, those methods usually suffer from bad local minima, while for many settings, they may be inapplicable due to the absence of easy-to-access gradients of the optimization objectives. In response to these issues, we introduce Motion Planning via Optimal Transport (MPOT) -- a \textit{gradient-free} method that optimizes a batch of smooth trajectories over highly nonlinear costs, even for high-dimensional tasks, while imposing smoothness through a Gaussian Process dynamics prior via the planning-as-inference perspective. To facilitate batch trajectory optimization, we introduce an original zero-order and highly-parallelizable update rule: the Sinkhorn Step, which uses the regular polytope family for its search directions. Each regular polytope, centered on trajectory waypoints, serves as a local cost-probing neighborhood, acting as a \textit{trust region} where the Sinkhorn Step "transports" local waypoints toward low-cost regions. We theoretically show that Sinkhorn Step guides the optimizing parameters toward local minima regions of non-convex objective functions. We then show the efficiency of MPOT in a range of problems from low-dimensional point-mass navigation to high-dimensional whole-body robot motion planning, evincing its superiority compared to popular motion planners, paving the way for new applications of optimal transport in motion planning.Comment: Published as a conference paper at NeurIPS 2023. Project website: https://sites.google.com/view/sinkhorn-step

Efficient discretization of signed distance fields

A Signed distance field (SDF) is an implicit function that returns the distance to the surface of a volume given a point in the space. The sign of the field indicates if the point is inside or outside the volume. These fields are usually used to accelerate computer graphics algorithms in different areas, such as rendering or collision detection. There are many well-defined primitives and operators to model objects using these functions. For example, SDFs allow applying smooth boolean operations between primitives. Applying these operators to triangles meshes can require complex algorithms susceptible to precision problems. Even though SDFs allow modelling objects, they currently are not a used format, and not many modelling tools use it. Most of the time, we want to calculate this field from triangle meshes. If the mesh is two-manifold, the easiest way to calculate the signed distance from a point is by searching for the minimum distance at all the mesh triangles. This strategy requires iterating all the triangles for each query to the signed distance field. There are methods based on different strategies that accelerate this nearest triangle search. If the user does not require getting exact distances to the object, other strategies exist that discretize the space in some fixed sample points. Then, the queries to arbitrary points are calculated using an interpolation of the precalculated discretization. This project presents a new approach based on an octree-like subdivision to accelerate the computation of these signed distance fields queries from a triangle mesh. The main idea is to construct an octree structure in which each leaf will contain only the nearest triangles for all the points in that region. Therefore, when the user wants to calculate the distance from an arbitrary point in the space, it will only compare the triangles influencing that region. Moreover, we present a method to calculate approximated distances based on the discretization approach mentioned before. We designed and developed an octree discretization strategy and explored different interpolation techniques. The distance computation of this discretization is accelerated by the strategy developed in the project