Long Proteins with Unique Optimal Foldings in the H-P Model

It is widely accepted that (1) the natural or folded state of proteins is a global energy minimum, and (2) in most cases proteins fold to a unique state determined by their amino acid sequence. The H-P (hydrophobic-hydrophilic) model is a simple combinatorial model designed to answer qualitative questions about the protein folding process. In this paper we consider a problem suggested by Brian Hayes in 1998: what proteins in the two-dimensional H-P model have unique optimal (minimum energy) foldings? In particular, we prove that there are closed chains of monomers (amino acids) with this property for all (even) lengths; and that there are open monomer chains with this property for all lengths divisible by four.Comment: 22 pages, 18 figure

The Size of the Open Sphere of Influence Graph in L ∞ Metric Spaces

Let V be a set of distinct points in some metric space. For each point x ∈ V, let r x be the distance from x to its nearest neighbour, and let s x be the open ball centered at x with radius equal to the distance from x to its nearest neighbour. We refer to these balls as the spheres of influence of the set V. The open sphere of influence graph on V is defined as the graph where (x,y) is an edge if and only if s x and s y intersect

Geometric and Computational Aspects of Molecular Reconfiguration

We examine geometric problems of reconfiguring molecules modeled by polygons and polygonal chains in two and three dimensions. The molecule can be continuously reconfigured as long as the edge lengths are maintained and the object does not self-intersect.We begin with a treatment of polygons in chapters 3 and 4. We prove that in two dimensions all convex configurations of a given polygon are reachable from any other and provide an efficient algorithm for convexifying planar monotone polygons. We also describe an algorithm to convexify three-dimensional polygons with simple projections. [...]Nous étudions les problèmes géométriques liés à la reconfiguration de molécules modelées par des polygones et des chaînes polygonales en deux et trois dimensions. Une molécule peut être reconfigurée de façon continue si elle ne s'auto-intersecte pas et si les longueurs des arcs sont préservées.Nous commençons par une étude sur les polygones dans les chapitres 3 et 4. Nous montrons qu'en deux dimensions toutes les configurations convexes d'un polygone sont accessibles depuis toutes autres et nous présentons un algorithme efficace pour convexifier tout polygone monotone planaire. Nous décrivons aussi un algorithme pour convexifier tout polygone tridimensionnel admettant une projection simple. […

On the size of the sphere of influence graph

Let V be a set of distinct points in some metric space. We draw the following proximity graph on V: For each point x∈V , let sx be the open ball centered at x with radius from x to its nearest neighbour. Then (a, b) is an edge if and only if sa and sb intersect. This graph is known as the sphere of influence graph of the point set V. In this thesis, we demonstrate that in the d-dimensional infinite-order Minkowski metric, no sphere of influence graph of n vertices contains ( 22d-1-2d-1 )n edges or more. We also prove an asymptotic lower bound of ( 2d+2-3d-4 )n/9 on the maximum size of the graph. Lastly, we demonstrate an upper bound of 15n on the size of the sphere of influence graph of n vertices in the Euclidean plane