Papers by Frederic Cazals
Reconstruire un modèleà partir d'échantillons est un problème central se posant en médecine numér... more Reconstruire un modèleà partir d'échantillons est un problème central se posant en médecine numérique, en ingénierie inverse, en sciences naturelles, etc. Ces applications ont motivé une recherche substantielle pour la reconstruction de surfaces, la question de la reconstruction de modèles plus généraux n'ayant paś eté examinée. Ce travail présente an algorithme visantà changer le paradigme de reconstruction en 3D comme suit. Premièrement, l'algorithme reconstruit des formes générales-des ensembles compacts et non plus des surfaces. Sous des hypothèses appropriées, nous montrons que la reconstruction a le type d'homotopie de l'objet de départ. Deuxièmement, l'algorithme ne génère pas une seule reconstruction, mais un ensemble de reconstructions plausibles. Troisièmement, l'algorithme peutêtre coupléà la persistance topologique, afin de sélectionner les traits les plus stables du modèle reconstruit. Enfin, en cas d'échec de la reconstruction, la méthode permet une identification aisée des régions sous-echantillonnées, afinéventuellement de les enrichir. Ces points clefs sont illustrés sur des modèles difficiles, et devraient permettre de mieux tirer parti de leurs caractéristiques dans les application sus-citées.
Intervor is a software computing a parameter free representation of macro-molecular interfaces, b... more Intervor is a software computing a parameter free representation of macro-molecular interfaces, based on the α-complex of the atoms. Given two interacting partners, possibly with water molecules squeezed in-between them, Intervor computes an interface model which has the following characteristics: (i) it identifies the atoms of the partners which are in direct contact and those whose interaction is water mediated, (ii) it defines a geometric complex separating the partners, the Voronoi interface, whose geometric and topological descriptions are straightforward (surface area, number of patches, curvature), (iii) it allows the definition of the depth of atoms at the interface, thus going beyond the traditional dissection of an interface into a core and a rim. These features can be used to investigate correlations between structural parameters and key properties such as the conservation of residues, their polarity, the water dynamics at the interface, mutagenesis data, etc. Intervor can be run from the web site , or in stand-alone mode upon downloading the binary file. Plugins are also made available for Visual Molecular Dynamics (VMD) and Pymol.
The center of mass of a point set lying on a manifold generalizes the celebrated Euclidean centro... more The center of mass of a point set lying on a manifold generalizes the celebrated Euclidean centroid, and is ubiquitous in statistical analysis in non Euclidean spaces. In this work, we give a complete characterization of the weighted p-mean of a finite set of angular values on S1, based on a decomposition of S1 such that the functional of interest has at most one local minimum per cell. This characterization is used to show that the problem is decidable for rational angular values –a consequence of Lindemann’s theorem on the transcendence of π, and to develop an effective algorithm parameterized by exact predicates. A robust implementation of this algorithm based on multi-precision interval arithmetic is also presented, and is shown to be effective for large values of n and p. We use it as building block to implement the k-means and k-means++ clustering algorithms on the flat torus, with applications to clustering protein molecular conformations. These algorithms are available in th...
Journal of Computational Physics, 2020
The Wang-Landau (WL) algorithm is a recently developed stochastic algorithm computing densities o... more The Wang-Landau (WL) algorithm is a recently developed stochastic algorithm computing densities of states of a physical system. Since its inception, it has been used on a variety of (bio-)physical systems, and in selected cases, its convergence has been proved. The convergence speed of the algorithm is tightly tied to the connectivity properties of the underlying random walk. As such, we propose an efficient random walk that uses geometrical information to circumvent the following inherent difficulties: avoiding overstepping strata, toning down concentration phenomena in high-dimensional spaces, and accommodating multidimensional distribution. Experiments on various models stress the importance of these improvements to make WL effective in challenging cases. Altogether, these improvements make it possible to compute density of states for regions of the phase space of small biomolecules.
Acm Acm Transactions on Mathematical Software, 2008
Reconstruire un modèleà partir d'échantillons est un problème central se posant en médecine numér... more Reconstruire un modèleà partir d'échantillons est un problème central se posant en médecine numérique, en ingénierie inverse, en sciences naturelles, etc. Ces applications ont motivé une recherche substantielle pour la reconstruction de surfaces, la question de la reconstruction de modèles plus généraux n'ayant paś eté examinée. Ce travail présente an algorithme visantà changer le paradigme de reconstruction en 3D comme suit. Premièrement, l'algorithme reconstruit des formes générales-des ensembles compacts et non plus des surfaces. Sous des hypothèses appropriées, nous montrons que la reconstruction a le type d'homotopie de l'objet de départ. Deuxièmement, l'algorithme ne génère pas une seule reconstruction, mais un ensemble de reconstructions plausibles. Troisièmement, l'algorithme peutêtre coupléà la persistance topologique, afin de sélectionner les traits les plus stables du modèle reconstruit. Enfin, en cas d'échec de la reconstruction, la méthode permet une identification aisée des régions sous-echantillonnées, afinéventuellement de les enrichir. Ces points clefs sont illustrés sur des modèles difficiles, et devraient permettre de mieux tirer parti de leurs caractéristiques dans les application sus-citées.
Modeling 3D objects with balls is routine for two reasons: on the one hand, the medial axis trans... more Modeling 3D objects with balls is routine for two reasons: on the one hand, the medial axis transform allows representing a solid object as a union of medial balls; on the other hand, selected shapes, and molecules in particular, are naturally represented by collections of balls. Yet, the problem of choosing which balls are best suited to approximate a given shape is a non trivial one. This paper addresses two problems in this realm. The rst one, conformational diversity selection, consists of choosing k molecular conformations amidst n, so as to maximize the geometric diversity of the k conformers. The second one, inner approximation, consists of approximating a molecule of n balls with k < n balls. On the theoretical side, we demonstrate that for both problems, a geometric generalization of max k-cover applies, with weights depending on the cells of a surface or volumetric arrangement. Tackling these problems with greedy strategies, it is shown that the 1 − 1/e bound known in combinatorial optimization applies in some cases but not all. On the applied side, we present a robust and eective implementation of the greedy algorithm for the inner approximation problem, which incorporates the calculation of the exact Delaunay triangulation of a points whose coordinates are degree two algebraic number, of the medial axis of a union of balls, and of a certied estimate of the volume of a union of balls. In particular, we show that the inner approximation of complex molecules yields accurate coarse-grain models with a number of balls 100 times smaller than the number of atoms, a key requirement to simulate crowded protein environments.
Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07, 2007
Proceedings of the twenty-fourth annual symposium on Computational geometry, 2008
The Delaunay triangulation and its dual the Voronoi diagram are ubiquitous geometric complexes. F... more The Delaunay triangulation and its dual the Voronoi diagram are ubiquitous geometric complexes. From a topological standpoint, the connection has recently been made between these cell complexes and the Morse theory of distance functions. In particular, in the generic setting, algorithms have been proposed to compute the flow complex-the stable and unstable manifolds associated to the critical points of the distance function to a point set. As algorithms ignoring degenerate cases and numerical issues are bound to fail on general inputs, this paper develops the first complete and robust algorithm to compute the flow complex. First, we present complete algorithms for the flow operator, unraveling a delicate interplay between the degenerate cases of Delaunay and those which are flow specific. Second, we sketch how the flow operator unifies the construction of stable and unstable manifolds. Third, we discuss numerical issues related to predicates on cascaded constructions. Finally, we report experimental results with CGAL's filtered kernel, showing that the construction of the flow complex incurs a small overhead w.r.t. the Delaunay triangulation when moderate cascading occurs. These observations provide important insights on the relevance of the flow complex for (surface) reconstruction and medial axis approximation, and should foster flow complex based algorithms. In a broader perspective and to the best of our knowledge, this paper is the first one reporting on the effective implementation of a geometric algorithm featuring cascading.
Geometric Modeling and Algebraic Geometry, 2008
Given a smooth surface, a blue (red) ridge is a curve along which the maximum (minimum) principal... more Given a smooth surface, a blue (red) ridge is a curve along which the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremal curvature and therefore encode important informations used in segmentation, registration, matching and surface analysis. State of the art methods for ridge extraction either report red and blue ridges simultaneously or separately-in which case a local orientation procedure of principal directions is needed, but no method developed so far topologically certifies the curves reported. In this context, we make two contributions. First, for any smooth parametric surface, we exhibit the implicit equation P = 0 of the singular curve P encoding all ridges of the surface (blue and red), we analyze its singularities and we explain how colors can be recovered. Second, we instantiate to the algebraic setting the implicit equation P = 0. For a polynomial surface, this equation defines an algebraic curve, and we develop the first certified algorithm to produce a topologically certified approximation of it. The algorithm exploits the singular structure of P-umbilics and purple points, and reduces the problem to solving zero dimensional systems using Rational Univariate Representations and isolate roots of univariate rational polynomials. An experimental section illustrates the efficiency of the algorithm on a Bezier patch.
Proceedings of the fifteenth annual symposium on Computational geometry, 1999
The Visual Computer, 2012
Establishing corresponding features on two non-rigidly deformed 3D surfaces is a challenging and ... more Establishing corresponding features on two non-rigidly deformed 3D surfaces is a challenging and well-studied problem in computer graphics. Unlike previous approaches that constrain the matching between feature pairs using isometry-invariant distance metrics, we constrain the matching using a discrete connectivity graph derived from the Morse-Smale Complex of the Auto Diffusion Function. We observed that the graph remains stable even for surfaces differing by topology or by significant deformation. This algorithm is simple to implement and efficient to run. When tested on a range of examples, our algorithm produces comparable results with state-of-art methods on surfaces with strong isometry but with greatly improved efficiency, and often gets better correspondences on surfaces with larger shape variances.
Proteins: Structure, Function, and Bioinformatics, 2009
The accurate description and analysis of protein-protein interfaces remains a challenging task. T... more The accurate description and analysis of protein-protein interfaces remains a challenging task. Traditional definitions, based on atomic contacts or changes in solvent accessibility, tend to over-or underpredict the interface itself and cannot discriminate active from less relevant parts. We here extend a fast, parameter-free and purely geometric definition of protein interfaces and introduce the shelling order of Voronoi facets as a novel measure for an atom's depth inside the interface. Our analysis of 54 protein-protein complexes reveals a strong correlation between Voronoi Shelling Order (VSO) and water dynamics. High Voronoi Shelling Order coincides with residues that were found shielded from bulk water fluctuations in a recent molecular dynamics study. Yet, VSO predicts such "dry" residues without consideration of forcefields or dynamics at dramatically reduced cost. More central interface positions are often also increasingly enriched for hydrophobic residues. Yet, this hydrophobic centering is not universal and does not mirror the far stronger geometric bias of water fluxes. The seemingly complex water dynamics at protein interfaces appears thus largely controlled by geometry. Sequence analysis supports the functional relevance of both dry residues and residues with high VSO, both of which tend to be more conserved. Upon closer inspection, the spatial distribution of conservation argues against the arbitrary dissection into core or rim and thus refines previous results. Voronoi Shelling Order reveals clear geometric patterns in protein interface composition, function and dynamics and facilitates the comparative analysis of protein-protein interactions.
Proteins: Structure, Function, and Bioinformatics, 2012
É valuation de la Reconstruction de Gros Assemblages Protéiques avec des Modèles Tolérancés Résum... more É valuation de la Reconstruction de Gros Assemblages Protéiques avec des Modèles Tolérancés Résumé : Ce travail introduit le canevas des modèles tolérancés (TOM), afin de modéliser des assemblages macro-moléculaires présentant des incertitudes tant sur la forme des protéines que sur leur position. Un modèle tolérancéétant un continuum de formes emboîtées, nous présentons une panoplie de statistiques permettant d'évaluer ces formesà diverseséchelles. Certaines des statistiques qualifient des aspects topologiques (contacts deuxà deux, sous-complexes impliquant certaines protéines spécifiques), alors que d'autres sont de nature géométrique (taille des sous-complexes). Nous validons le canevas enévaluant les modèles moyennés du pore nucléaire (NPC) reconstruits récemment par intégration de données, et confrontons nos statistiquesà divers résultats relatifsà des sous-complexes. De façon prospective, le canevas des modèles tolérances devrait permettre de gérer des incertitudes diverses, en particulier en cryo electron microscopie et crystallographie.
Proteins: Structure, Function, and Bioinformatics, 2012
Let the patch of a partner in a protein complex be the collection of atoms accounting for the int... more Let the patch of a partner in a protein complex be the collection of atoms accounting for the interaction. To improve our understanding of the structure-function relationship, we present a patch model decoupling the topological and geometric properties. While the geometry is classically encoded by the atomic positions, the topology is recorded in a graph encoding the relative position of concentric shells partitioning the interface atoms. The topological-geometric duality provides the basis of a generic dynamic programming based algorithm comparing patches at the shell level, which may favor topological or geometric features. On the biological side, we address four questions, using 249 co-crystallized heterodimers organized in biological families. First, we dissect the morphology of binding patches, and show that Nature enjoyed the topological and geometric degrees of freedom independently while retaining a nite set of qualitatively distinct topological signatures. Second, we argue that our shell-based comparison is eective to perform atomic-level comparisons, and show that topological similarity is a less stringent than geometric similarity. We also use the topological versus geometric duality to exhibit topo-rigid patches, whose topology (but not geometry) remains stable upon docking. Third, we use our comparison algorithms to infer specicity related information amidst a database of complexes. Finally, we exhibit a descriptor outperforming its contenders to predict the binding anities of the anity benchmark.
Protein Science, 2006
Describing macro-molecular interfaces is key to improve our understanding of the specificity and ... more Describing macro-molecular interfaces is key to improve our understanding of the specificity and of the stability of macro-molecular interactions, and also to predict complexes when little structural information is known. Ideally, an interface model should provide easy-tocompute geometric and topological parameters exhibiting a good correlation with important bio-physical quantities. It should also be parametric and amenable to comparisons. In this spirit, we recently developed an interface model based on Voronoi diagrams, which proved instrumental to refine state-of-the-art conclusions and provide new insights. This paper formally presents this Voronoi interface model. First, we discuss its connexion to classical interface models based on distance cutoffs and solvent accessibility. Second, we develop the geometric and topological constructions underlying the Voronoi interface, and design efficient algorithms based on the Delaunay triangulation and the α-complex. We conclude with perspectives. In particular, we expect the Voronoi interface model to be particularly well suited for the problem of comparing interfaces in the context of large-scale structural studies.
International Journal of Computational Geometry & Applications, 2005
The understanding of surfaces embedded in E3 requires local and global concepts, which are respec... more The understanding of surfaces embedded in E3 requires local and global concepts, which are respectively evocative of differential geometry and differential topology. While the local theory has been classical for decades, global objects such as the foliations defined by the lines of curvature, or the medial axis still pose challenging mathematical problems. This duality is also tangible from a practical perspective, since algorithms manipulating sampled smooth surfaces (meshes or point clouds) are more developed in the local than the global category. As a prerequisite for those interested in the development of algorithms for the manipulation of surfaces, we propose a concise overview of core concepts from differential topology applied to smooth embedded surfaces. We first recall the classification of umbilics, of curvature lines, and describe the corresponding stable foliations. Next, fundamentals of contact and singularity theory are recalled, together with the classification of poi...
IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2011
To address challenging flexible docking problems, a number of docking algorithms pre-generate lar... more To address challenging flexible docking problems, a number of docking algorithms pre-generate large collections of candidate conformers. To further remove the redundancy from such ensembles, a central question in this context is the following one: report a selection of conformers maximizing some geometric diversity criterion. In this context, we make three contributions. First, we tackle this problem resorting to geometric optimization so as to report selections maximizing the molecular volume or molecular surface area (MSA) of the selection. Greedy strategies are developed, together with approximation bounds. Second, to assess the efficacy of our algorithms, we investigate two conformer ensembles corresponding to a flexible loop of four protein complexes. By focusing on the MSA of the selection, we show that our strategy matches the MSA of standard selection methods, but resorting to a number of conformers between one and two orders of magnitude smaller. This observation is qualitatively explained using the Betti numbers of the union of balls of the selection. Finally, we replace the conformer selection problem in the context of multiple-copy flexible docking. On the systems above, we show that using the loops selected by our strategy can significantly improve the result of the docking process.
Computer Graphics Forum, 2014
Choosing balls which best approximate a 3D object is a non trivial problem. To answer it, we firs... more Choosing balls which best approximate a 3D object is a non trivial problem. To answer it, we first address the inner approximation problem, which consists of approximating an object FO defined by a union of n balls with k < n balls defining a region FS ⊂ FO. This solution is further used to construct an outer approximation enclosing the initial shape, and an interpolated approximation sandwiched between the inner and outer approximations. The inner approximation problem is reduced to a geometric generalization of weighted max kcover, solved with the greedy strategy which achieves the classical 1 − 1/e lower bound. The outer approximation is reduced to exploiting the partition of the boundary of FO by the Apollonius Voronoi diagram of the balls defining the inner approximation. Implementation-wise, we present robust software incorporating the calculation of the exact Delaunay triangulation of points with degree two algebraic coordinates, of the exact medial axis of a union of balls, and of a certified estimate of the volume of a union of balls. Application-wise, we exhibit accurate coarse-grain molecular models using a number of balls 20 times smaller than the number of atoms, a key requirement to simulate crowded cellular environments.
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Papers by Frederic Cazals