Fitting Voronoi Diagrams to Planar Tesselations

Given a tesselation of the plane, defined by a planar straight-line graph $G$, we want to find a minimal set $S$ of points in the plane, such that the Voronoi diagram associated with $S$ "fits" \ $G$. This is the Generalized Inverse Voronoi Problem (GIVP), defined in \cite{Trin07} and rediscovered recently in \cite{Baner12}. Here we give an algorithm that solves this problem with a number of points that is linear in the size of $G$, assuming that the smallest angle in $G$ is constant.Comment: 14 pages, 8 figures, 1 table. Presented at IWOCA 2013 (Int. Workshop on Combinatorial Algorithms), Rouen, France, July 201

A coordinate system on a surface: definition, properties and applications

Coordinate systems associated to a finite set of sample points have been extensively studied, especially in the context of interpolation of multivariate scattered data. Notably, Sibson proposed the so-called natural neighbor coordinates that are defined from the Voronoi diagram of the sample points. A drawback of those coordinate systems is that their definition domain is restricted to the convex hull of the sample points. This makes them difficult to use when the sample points belong to a surface. To overcome this difficulty, we propose a new system of coordinates. Given a closed surface $S$, i.e. a $(d-1)$-manifold of $\mathbb{R} ^d$, the coordinate system is defined everywhere on the surface, is continuous, and is local even if the sampling density is finite. Moreover, it is inherently $(d-1)$-dimensional while the previous systems are $d$-dimensional. No assumption is made about the ordering, the connectivity or topology of the sample points nor of the surface. We illustrate our results with an application to interpolation over a surface

Scutoids are a geometrical solution to three-dimensional packing of epithelia

As animals develop, tissue bending contributes to shape the organs into complex three-dimensional structures. However, the architecture and packing of curved epithelia remains largely unknown. Here we show by means of mathematical modelling that cells in bent epithelia can undergo intercalations along the apico-basal axis. This phenomenon forces cells to have different neighbours in their basal and apical surfaces. As a consequence, epithelial cells adopt a novel shape that we term “scutoid”. The detailed analysis of diverse tissues confirms that generation of apico-basal intercalations between cells is a common feature during morphogenesis. Using biophysical arguments, we propose that scutoids make possible the minimization of the tissue energy and stabilize three-dimensional packing. Hence, we conclude that scutoids are one of nature's solutions to achieve epithelial bending. Our findings pave the way to understand the three-dimensional organization of epithelial organs

Modelling disordered foams using the vertex ensemble

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Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences

Protein-water interactions studied by molecular dynamics simulations

Most proteins have evolved to function optimally in aqueous environments, and the interactions between protein and water therefore play a fundamental role in the stability, dynamics, and function of proteins. Although we understand many details of water, we understand much less about the protein-water interface. In this thesis we use molecular dynamics (MD) simulations to cast light on many structural and dynamical properties of protein hydration for which a detailed picture is lacking.We show that the 1 ms MD simulation of the bovine pancreatic trypsin inhibitor (BPTI) by Shaw \textsl{et al.} (Science 2010, 330, 341) reproduces the mean survival times from magnetic relaxation dispersion (MRD) experiments by computing the relevant survival correlation function that is probed by these experiments. The simulation validates several assumptions in the model used to interpret MRD data, and reveals a possible mechanism for the water-exchange; water molecules gain access to the internal sites by a transient aqueduct mechanism, migrating as single-file water chains through transient tunnels or pores. The same simulation was also used to reveal a possible mechanism for hydrogen exchange of backbone amides, involving short-lived locally distorted conformations of the protein whereby the amide is presolvated by two water molecules before the catalyst can approach the amide through a water wire.We perform MD simulations of several small globular proteins in dilute aqueous solution to spatially resolve protein hydration. Defining mono-molecular thick hydration shells as a metric from the protein surface, we compute structural and dynamical properties of water in these shells and show that the protein-induced water perturbation is short ranged, essentially only affecting water molecules in the first hydration shell, thus validating the model used to interpret MRD data. Compared to the bulk, the first shell is 6 \% more dense and 25-30 \% less compressible. The shell-averaged rotation of water molecules in the first hydration shell is retarded by a factor 4-5 compared to bulk, and the contributions to this retardation can be resolved based on a universal confinement index. The dynamical heterogeneity in the first shell is a result of water molecules rotating by different mechanisms on a spectrum between two extremes: a collective bulk-like mechanism and a protein-coupled mechanism where water molecules in confined sites are orientationally restricted and require an exchange event

Gauss images of hyperbolic cusps with convex polyhedral boundary

We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2\pi$ is the metric of the Gauss image of some convex polyhedral cusp. This result is an analog of the Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their Gauss images. The proof uses a variational method. Namely, a cusp with a given Gauss image is identified with a critical point of a functional on the space of cusps with cone-type singularities along a family of half-lines. The functional is shown to be concave and to attain maximum at an interior point of its domain. As a byproduct, we prove rigidity statements with respect to the Gauss image for cusps with or without cone-type singularities. In a special case, our theorem is equivalent to existence of a circle pattern on the torus, with prescribed combinatorics and intersection angles. This is the genus one case of a theorem by Thurston. In fact, our theorem extends Thurston's theorem in the same way as Rivin-Hodgson's theorem extends Andreev's theorem on compact convex polyhedra with non-obtuse dihedral angles. The functional used in the proof is the sum of a volume term and curvature term. We show that, in the situation of Thurston's theorem, it is the potential for the combinatorial Ricci flow considered by Chow and Luo. Our theorem represents the last special case of a general statement about isometric immersions of compact surfaces.Comment: 55 pages, 17 figure

Voronoi binding site models

A frequently occurring problem in drug design and enzymology is that the binding constants for several compounds to the same site are known, but the geometry and energetic interactions of the site are not. This paper presents in detail a novel approach to the problem which accurately but compactly represents the allowed conformation space of each ligand, accurately depicts their three-dimensional structures, and realistically allows each ligand to adopt the conformation and positioning in the site which is most favorable energetically. The investigator supplies only the ligand structures and observed binding free energies, along with a proposed site geometry. With no further assumptions about how the ligands bind and what parts of the ligands are important in determining the binding, the algorithm fits the observed binding energies without leaving outliers, predicts exactly how each of the given ligands binds in the site, and predicts the strength and mode of binding of new compounds, regardless of chemical similarity to the original set of ligands. The method is illustrated by devising a simple site that accounts for the binding of five polychlorinated biphenyls to thyroxine binding prealbumin. This model then predicts the binding energies correctly for an additional six biphenyls, and fails on one compound.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/38276/1/540080703_ftp.pd