Jean-François Sadoc

Protein secondary structure assignment through Voronoï tessellation.

We present a new automatic algorithm, named VoTAP (Voronoï Tessellation Assignment Procedure), which assigns secondary structures of a polypeptide chain using the list of α‐carbon coordinates. This program uses three‐dimensional Voronoï tessellation. This geometrical tool associates with each amino acid a Voronoï polyhedron, the faces of which unambiguously define contacts between residues. Thanks to the face area, for the contacts close together along the primary structure (low‐order contacts) a distinction is made between strong and normal ones. This new definition yields new contact matrices, which are analyzed and used to assign secondary structures. This assignment is performed in two stages. The first one uses contacts between residues close together along the primary structure and is based on data collected on a bank of 282 well‐refined nonredundant structures. In this bank, associations were made between the prints defined by these low‐order contacts and the assignments performed by different automatic methods. The second step focuses on the strand assignment and uses contacts between distant residues. Comparison with several other automatic assignment methods are presented, and the influence of resolution on the assignment is investigated. Proteins 2004.

Nonatomic Solvent-Driven Voronoi Tessellation of Proteins:An Open Tool to Analyze Protein Folds

A three‐dimensional Voronoi tessellation of folded proteins is used to analyze geometrical and topological properties of a set of proteins. To each amino acid is associated a central point surrounded by a Voronoi cell. Voronoi cells describe the packing of the amino acids. Special attention is given to reproduction of the protein surface. Once the Voronoi cells are built, a lot of tools from geometrical analysis can be applied to investigate the protein structure; volume of cells, number of faces per cell, and number of sides per face are the usual signatures of the protein structure. A distinct difference between faces related to primary, secondary, and tertiary structures has been observed. Faces threaded by the main‐chain have on average more than six edges, whereas those related to helical packing of the amino acid chain have less than five edges. The faces on the protein surface have on average five edges within 1% error. The average number of faces on the protein surface for a given type of amino acid brings a new point of view in the characterization of the exposition to the solvent and the classification of amino acid as hydrophilic or hydrophobic. It may be a convenient tool for model validation. Proteins 2002;49:446–456.

Structure of amorphous Cu5Y: Anomalous X-ray diffraction and modelling

The availability of synchrotron radiation has made it possible to perform X-ray diffraction studies using anomalous scattering near two absorption edges in order to derive partial distribution functions for structural studies of disordered material such as Cu5Y. The results are applied to a discussion of the tetrahedral packing model with comparison to the crystalline phase.

About collagen, a tribute to Yves Bouligand

Yves Bouligands analysis of the organizations of biological materials in relation to those of liquid crystals enabled the development of the idea that physical forces exerting their actions under strong spatial constraints determine the structures and morphologies of these materials. The different levels of organization in collagen have preoccupied him for a long time. We present here our recent works in this domain that we were still discussing with him a few months before his death at the age of 76 on 21 January 2011. After recalling the hierarchical set of structures built by collagen molecules, we analyse them, exploiting the properties of the curved space of the hypersphere and of the algorithm of phyllotaxis. Those two geometrical concepts can be proposed as structural archetypes founding the polymorphism of this complex material of biological origin.

Hidden order in non-crystalline structures : The curved space approach

Abstract The structure of condensed matter results from the competition between local interactions, topologic and geometric rules imposed by space filling requirements. There is one way to overcome this conflict : let the space curve in order to satisfy the local interactions without consideration to the euclidean space filling problem. This method leads to model with a local order similar to amorphous materials, but as they are described in curved space long range description needs a further step. This step consists in a change of the curvature by the mean of defect (disclination lines) which introduce fluctuation in the local order. This method has opened a large field of study in different kinds of disordered materials.

Curved Spaces and Geometrical Frustration

Regular structures are such that no contradiction exists between local and global requirements in which case the global approach (with symmetry groups) reveals to be very powerful. In less regular structures, the local configuration may be viewed in some cases as the discrete analog of a quantity which is the local curvature. Defining an ideal structure where the local configuration can propagate, is then equivalent to finding a new geometry with the appropriate distribution of curvature. If such geometry allows for a global description, this ideal model is again regular and can be studied on its own. The relation between the initial structure and the ideal one is studied under different types of mapping. This point of view is called the “curved space model” of disordered systems and will be discussed here.

Geometrical Foundation of Mesomorphic Polymorphism

The aggregation of amphiphilic molecules in water, and in particular the growth and first steps of micellar organization in the dilute region of their phase diagrams, were discussed in earlier chapters of this book, primarily the first two. In these diluted regions the problem of the formation of structures by amphiphilic molecules in aqueous solution could be treated within a rather simple geometrical framework which can be summarized as follows. First, the “micellar” topology of the organization—an infinite number of finite aggregates separated by a connected film of water—does not change during aggregation and growth. The aggregation and growth processes were therefore described as continuous deformations of aggregates of simple shapes. Second, the aggregates are sufficiently diluted for the solution of micelles to be treated as ideal, except for steric interactions in first order, so that the only interaction present between interfaces takes place within the aggregates and constrains their size along one direction to stay constant, with a value of about twice the length of a molecule. The deformations of the aggregates under concentration changes, or the evolution of their interfacial curvature, were then considered to occur subject to this one spatial constraint only.

Polytopes, non-crystalline structures and their diffractions

Abstract The {3,3,5} polytope is an efficient model for tetrahedral close packed structures. But in order to obtain realistic structures, it is needed to decurve the space in which the {3,3,5} is embedded. This has been done by iterative procedures generating large polytopes of decreasing curvature. These polytopes are tetrahedral close packed structures with hierarchical interlaced network of disclinations. In this paper we describe these structures, and we relate it to diffraction patterns.