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

Vesicle computers: Approximating Voronoi diagram on Voronoi automata

Irregular arrangements of vesicles filled with excitable and precipitating chemical systems are imitated by Voronoi automata --- finite-state machines defined on a planar Voronoi diagram. Every Voronoi cell takes four states: resting, excited, refractory and precipitate. A resting cell excites if it has at least one excited neighbour; the cell precipitates if a ratio of excited cells in its neighbourhood to its number of neighbours exceed certain threshold. To approximate a Voronoi diagram on Voronoi automata we project a planar set onto automaton lattice, thus cells corresponding to data-points are excited. Excitation waves propagate across the Voronoi automaton, interact with each other and form precipitate in result of the interaction. Configuration of precipitate represents edges of approximated Voronoi diagram. We discover relation between quality of Voronoi diagram approximation and precipitation threshold, and demonstrate feasibility of our model in approximation Voronoi diagram of arbitrary-shaped objects and a skeleton of a planar shape.Comment: Chaos, Solitons & Fractals (2011), in pres

Statistics of cross sections of Voronoi tessellations

In this paper we investigate relationships between the volumes of cells of three-dimensional Voronoi tessellations and the lengths and areas of sections obtained by intersecting the tessellation with a randomly oriented plane. Here, in order to obtain analytical results, Voronoi cells are approximated to spheres. First, the probability density function for the lengths of the radii of the sections is derived and it is shown that it is related to the Meijer $G$-function; its properties are discussed and comparisons are made with the numerical results. Next the probability density function for the areas of cross sections is computed and compared with the results of numerical simulations.Comment: 10 pages and 6 figure

MGOS: A library for molecular geometry and its operating system

The geometry of atomic arrangement underpins the structural understanding of molecules in many fields. However, no general framework of mathematical/computational theory for the geometry of atomic arrangement exists. Here we present "Molecular Geometry (MG)'' as a theoretical framework accompanied by "MG Operating System (MGOS)'' which consists of callable functions implementing the MG theory. MG allows researchers to model complicated molecular structure problems in terms of elementary yet standard notions of volume, area, etc. and MGOS frees them from the hard and tedious task of developing/implementing geometric algorithms so that they can focus more on their primary research issues. MG facilitates simpler modeling of molecular structure problems; MGOS functions can be conveniently embedded in application programs for the efficient and accurate solution of geometric queries involving atomic arrangements. The use of MGOS in problems involving spherical entities is akin to the use of math libraries in general purpose programming languages in science and engineering. (C) 2019 The Author(s). Published by Elsevier B.V

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

Evolution of Complexity in Out-of-Equilibrium Systems by Time-Resolved or Space-Resolved Synchrotron Radiation Techniques

Out-of-equilibrium phenomena are attracting high interest in physics, materials science, chemistry and life sciences. In this state, the study of structural fluctuations at different length scales in time and space are necessary to achieve significant advances in the understanding of structure-functionality relationship. The visualization of patterns arising from spatiotemporal fluctuations is nowadays possible thanks to new advances in X-ray instrumentation development that combine high resolution both in space and in time. We present novel experimental approaches using high brilliance synchrotron radiation sources, fast detectors and focusing optics, joint with advanced data analysis based on automated statistical, mathematical and imaging processing tools. This approach has been used to investigate structural fluctuations in out-of-equilibrium systems in the novel field of inhomogeneous quantum complex matter at the crossing point of technology, physics and biology. In particular, we discuss how nanoscale complexity controls the emergence of high temperature superconductivity (HTS), myelin functionality and formation of hybrid organic-inorganic nanostructures. The emergent complex geometries, opening novel venues to quantum technology and to development of quantum physics of living systems, are discussedComment: 18 pages, 7 figure