A geometric knowledge-based coarse-grained scoring potential for structure prediction evaluation

International audienceKnowledge-based protein folding potentials have proven successful in the recent years. Based on statistics of observed interatomic distances, they generally encode pairwise contact information. In this study we present a method that derives multi-body contact potentials from measurements of surface areas using coarse-grained protein models. The measurements are made using a newly implemented geometric construction: the arrangement of circles on a sphere. This construction allows the definition of residue covering areas which are used as parameters to build functions able to distinguish native structures from decoys. These functions, encoding up to 5-body contacts are evaluated on a reference set of 66 structures and its 45000 decoys, and also on the often used lattice ssfit set from the decoys'R us database. We show that the most relevant information for discrimination resides in 2- and 3-body contacts. The potentials we have obtained can be used for evaluation of putative structural models; they could also lead to different types of structure refinement techniques that use multi-body interactions

Modeling Macro-Molecular Interfaces with Intervor

Intervor is a software computing a parameter free representation of macro-molecular interfaces, based on the alpha-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 http://cgal.inria.fr/abs/Intervor , or in stand-alone mode upon downloading the binary file. Plugins are also made available for Visual Molecular Dynamics (VMD) and Pymol

Computing the exact arrangement of circles on a sphere, with applications in structural biology

revision de la version de Decembre 2006Given a collection of circles on a sphere, we adapt the Bentley-Ottmann algorithm to the spherical setting to compute the {\em exact} arrangement of the circles. The algorithm consists of sweeping the sphere with a meridian, which is non trivial because of the degenerate cases and the algebraic specification of event points. From an algorithmic perspective, and with respect to general sweep-line algorithms, we investigate a strategy maintaining a linear size event queue. (The algebraic aspects involved in the development of the predicates involved in our algorithm are reported in a companion paper.) From an implementation perspective, we present the first effective arrangement calculation dealing with general circles on a sphere in an exact fashion, as exactness incurs a mere factor of two with respect to calculations performed using {\em double} floating point numbers on generic examples. In particular, we stress the importance of maintaining a linear size queue, in conjunction with arithmetic filter failures. From an application perspective, we present an application in structural biology. Given a collection of atomic balls, we adapt the sweep-line algorithm to report all balls covering a given face of the spherical arrangement on a given atom. This calculation is used to define molecular surface related quantities going beyond the classical exposed and buried solvent accessible surface areas. Spectacular differences w.r.t. traditional observations on protein - protein and protein - drug complexes are also reported

Computing the Volume of a Union of Balls: a Certified Algorithm

Balls and spheres are amongst the simplest 3D modeling primitives, and computing the volume of a union of balls is an elementary problem. Although a number of strategies addressing this problem have been investigated in several communities, we are not aware of any robust algorithm, and present the first such algorithm. Our calculation relies on the decomposition of the volume of the union into convex regions, namely the restrictions of the balls to their regions in the power diagram. Theoretically, we establish a formula for the volume of a restriction, based on Gauss' divergence theorem. The proof being constructive, we develop the associated algorithm. On the implementation side, we carefully analyse the predicates and constructions involved in the volume calculation, and present a certified implementation relying on interval arithmetic. The result is certified in the sense that the exact volume belongs to the interval computed using the interval arithmetic. Experimental results are presented on hand-crafted models presenting various difficulties, as well as on the 58,898 models found in the 2009-07-10 release of the Protein Data Bank

On the Characterization and Selection of Diverse Conformational Ensembles, with Applications to Flexible Docking

To address challenging flexible docking problems, a number of docking algorithms pre-generate large collections of candidate conformers. To remove the redundancy from such ensembles, a central problem in this context is to report a selection of conformers maximizing some geometric diversity criterion. We make three contributions to this problem. First, we resort 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 afore-mentioned systems, we show that using the loops selected by our strategy can improve the result of the docking process

Robust and Efficient Delaunay triangulations of points on or close to a sphere

We propose two approaches for computing the Delaunay triangulation of points on a sphere, or of rounded points close to a sphere, both based on the classic incremental algorithm initially designed for the plane. The space of circles gives the mathematical background for this work. We implemented the two approaches in a fully robust way, building upon existing generic algorithms provided by the cgal library. The effciency and scalability of the method is shown by benchmarks

Arrangements de cercles sur une sphère: Algorithmes et Applications aux modèles moléculaires representés par une union de boules

Since the early work of Richard et al., geometric constructions havebeen paramount for the description of macromolecules and macro-molecularassemblies. In particular, Voronoï and related constructions have beenused to describe the packing properties of atoms, to compute molecularsurfaces, to find cavities. This thesis falls in this realm, andafter a brief introduction to protein structure, makes fourcontributions.First, using the sweep line paradigm of Bentley and Ottmann, wepresent the first effective algorithm able to construct the exactarrangement of circles on a sphere. Moreover, assuming the circlesstem from the intersection between spheres, we present a strategy to reportthe covering list of a face of the arrangement---that is the list ofspheres covering it. Along the way, we ascertain the fact thatexactness of the arrangement can be achieved with a smallcomputational overhead.Second, we develop the algebraic and geometric primitives required by the sweepalgorithm, so as to make it generic and robust. These primitives are integrated ina broader context, namely the CGAL 3D Spherical Kernel.Third, we use the aforementioned machinery to tackle a computational structuralbiology problem, namely the selection of diverse conformations from a large redundant set.We propose to solve this selection problem by computingrepresentatives maximizing the surface area or the volume of theselection. From a geometric standpoint, these questions can be handledresorting to arrangements of circles and spheres.The validation is carried out along two lines. On the geometric side,we show that our selections match the molecular surface area ofselections output by standard strategies but using a smaller numberof conformers by one and two orders of magnitude. Onthe docking side, we show that our selections can significantlyimprove the results obtained for a flexible-loop docking algorithm.Finally, we discuss the implementation issues and the design choices,in the context of the best practices underlying the development ofCGAL.Depuis les travaux précurseurs de Richard et al., les constructions géométriquesoccupent une place importante dans la description des macro-molécules et leurs assemblages.En particulier, certains complexes cellulaires liés au diagramme de Voronoïont été utilisés pour décrire les propriétés de compacité des empilement atomiques,calculer des surfaces moléculaires, ou encore détecter des cavités à la surface des molécules.Cette thèse se positionne dans ce contexte, et après une brève introduction à lastructure des protéines, détaille quatre contributions.Premièrement, en utilisant le principe de balayage introduit par Bentley et Ottmann, cette thèse présente le premier algorithme effectifpour construire l'arrangement exact de cercles sur une sphère.De plus, en supposant que les cercles proviennent de l'intersection entresphères, une stratégie pour calculer les listes couvrantes d'une face del'arrangement (i.e. la liste des boules qui la recouvrent) est proposée.L'exactitude n'étant pas une fin en soi, mais plutôt une façon de rendre l'algorithmiquerobuste, nous montrons expérimentalement que le surcoût induit est modeste.Deuxièmement, cette thèse développe les primitives algébriques et géométriquesrequises par l'algorithme de balayage afin de le rendre générique et robuste. Ces primitivessont intégrées dans une contexte plus général, à savoir le noyau CGAL pour les objets sphériques.Troisièmement, la machinerie introduite est utilisée pour traiter un problèmede biologie structurale computationelle : la sélection d'un sous-ensemble variéà partir d'un ensemble redondant de conformations de boucles.Nous proposons de résoudre ce problème de sélection en retenant les représentants qui maximisent l'aire ou le volume de la sélection.Ces questions peuvent être traitées géométriquement à l'aide d'arrangements de cercles sur une sphère.La validation est faîte sur deux fronts.D'un point de vue géométrique, nous montrons que notre approche génère des sélectionsdont l'aire de la surface moléculaire équivaut à celle de sélections obtenues par des stratégies classiques, mais qui sont de taille nettement inférieure.Du point de vue amarrage de protéines, nous montrons que nos sélections améliorent demanière significative les résultats obtenus à l'aide d'un algorithme manipulant desparties flexibles.Pour finir, nous discutons les problèmes et choix d'implémentation, enles replaçant dans le contexte de la librairie CGAL

Arrangements de cercles sur une sphère: Algorithmes et Applications aux modèles moléculaires representés par une union de boules

Since the early work of Richard et al., geometric constructions havebeen paramount for the description of macromolecules and macro-molecularassemblies. In particular, Voronoï and related constructions have beenused to describe the packing properties of atoms, to compute molecularsurfaces, to find cavities. This thesis falls in this realm, andafter a brief introduction to protein structure, makes fourcontributions.First, using the sweep line paradigm of Bentley and Ottmann, wepresent the first effective algorithm able to construct the exactarrangement of circles on a sphere. Moreover, assuming the circlesstem from the intersection between spheres, we present a strategy to reportthe covering list of a face of the arrangement---that is the list ofspheres covering it. Along the way, we ascertain the fact thatexactness of the arrangement can be achieved with a smallcomputational overhead.Second, we develop the algebraic and geometric primitives required by the sweepalgorithm, so as to make it generic and robust. These primitives are integrated ina broader context, namely the CGAL 3D Spherical Kernel.Third, we use the aforementioned machinery to tackle a computational structuralbiology problem, namely the selection of diverse conformations from a large redundant set.We propose to solve this selection problem by computingrepresentatives maximizing the surface area or the volume of theselection. From a geometric standpoint, these questions can be handledresorting to arrangements of circles and spheres.The validation is carried out along two lines. On the geometric side,we show that our selections match the molecular surface area ofselections output by standard strategies but using a smaller numberof conformers by one and two orders of magnitude. Onthe docking side, we show that our selections can significantlyimprove the results obtained for a flexible-loop docking algorithm.Finally, we discuss the implementation issues and the design choices,in the context of the best practices underlying the development ofCGAL.Depuis les travaux précurseurs de Richard et al., les constructions géométriquesoccupent une place importante dans la description des macro-molécules et leurs assemblages.En particulier, certains complexes cellulaires liés au diagramme de Voronoïont été utilisés pour décrire les propriétés de compacité des empilement atomiques,calculer des surfaces moléculaires, ou encore détecter des cavités à la surface des molécules.Cette thèse se positionne dans ce contexte, et après une brève introduction à lastructure des protéines, détaille quatre contributions.Premièrement, en utilisant le principe de balayage introduit par Bentley et Ottmann, cette thèse présente le premier algorithme effectifpour construire l'arrangement exact de cercles sur une sphère.De plus, en supposant que les cercles proviennent de l'intersection entresphères, une stratégie pour calculer les listes couvrantes d'une face del'arrangement (i.e. la liste des boules qui la recouvrent) est proposée.L'exactitude n'étant pas une fin en soi, mais plutôt une façon de rendre l'algorithmiquerobuste, nous montrons expérimentalement que le surcoût induit est modeste.Deuxièmement, cette thèse développe les primitives algébriques et géométriquesrequises par l'algorithme de balayage afin de le rendre générique et robuste. Ces primitivessont intégrées dans une contexte plus général, à savoir le noyau CGAL pour les objets sphériques.Troisièmement, la machinerie introduite est utilisée pour traiter un problèmede biologie structurale computationelle : la sélection d'un sous-ensemble variéà partir d'un ensemble redondant de conformations de boucles.Nous proposons de résoudre ce problème de sélection en retenant les représentants qui maximisent l'aire ou le volume de la sélection.Ces questions peuvent être traitées géométriquement à l'aide d'arrangements de cercles sur une sphère.La validation est faîte sur deux fronts.D'un point de vue géométrique, nous montrons que notre approche génère des sélectionsdont l'aire de la surface moléculaire équivaut à celle de sélections obtenues par des stratégies classiques, mais qui sont de taille nettement inférieure.Du point de vue amarrage de protéines, nous montrons que nos sélections améliorent demanière significative les résultats obtenus à l'aide d'un algorithme manipulant desparties flexibles.Pour finir, nous discutons les problèmes et choix d'implémentation, enles replaçant dans le contexte de la librairie CGAL

On the Characterization and Selection of Diverse Conformational Ensembles, with Applications to Flexible Docking

International audienceTo 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