238 research outputs found
Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations
Cubic polyhedra with icosahedral symmetry where all faces are pentagons or
hexagons have been studied in chemistry and biology as well as mathematics. In
chemistry one of these is buckminsterfullerene, a pure carbon cage with maximal
symmetry, whereas in biology they describe the structure of spherical viruses.
Parameterized operations to construct all such polyhedra were first described
by Goldberg in 1937 in a mathematical context and later by Caspar and Klug --
not knowing about Goldberg's work -- in 1962 in a biological context. In the
meantime Buckminster Fuller also used subdivided icosahedral structures for the
construction of his geodesic domes. In 1971 Coxeter published a survey article
that refers to these constructions. Subsequently, the literature often refers
to the Goldberg-Coxeter construction. This construction is actually that of
Caspar and Klug. Moreover, there are essential differences between this
(Caspar/Klug/Coxeter) approach and the approaches of Fuller and of Goldberg. We
will sketch the different approaches and generalize Goldberg's approach to a
systematic one encompassing all local symmetry-preserving operations on
polyhedra
Detecting Repetitions and Periodicities in Proteins by Tiling the Structural Space
The notion of energy landscapes provides conceptual tools for understanding
the complexities of protein folding and function. Energy Landscape Theory
indicates that it is much easier to find sequences that satisfy the "Principle
of Minimal Frustration" when the folded structure is symmetric (Wolynes, P. G.
Symmetry and the Energy Landscapes of Biomolecules. Proc. Natl. Acad. Sci.
U.S.A. 1996, 93, 14249-14255). Similarly, repeats and structural mosaics may be
fundamentally related to landscapes with multiple embedded funnels. Here we
present analytical tools to detect and compare structural repetitions in
protein molecules. By an exhaustive analysis of the distribution of structural
repeats using a robust metric we define those portions of a protein molecule
that best describe the overall structure as a tessellation of basic units. The
patterns produced by such tessellations provide intuitive representations of
the repeating regions and their association towards higher order arrangements.
We find that some protein architectures can be described as nearly periodic,
while in others clear separations between repetitions exist. Since the method
is independent of amino acid sequence information we can identify structural
units that can be encoded by a variety of distinct amino acid sequences
Combinatorics of a fractal tiling family
In this paper, we propose to enumerate all different configurations belonging
to a specific class of fractals: A binary initial tile is selected and a finite
recursive tiling process is engaged to produce auto-similar binary patterns.
For each initial tile choice the number of possible configurations is finite.
This combinatorial problem recalls the famous Escher tiling problem [2]. By
using the Burnside lemma we show that there are exactly 232 really different
fractals when the initial tile is a particular 2x2 matrix. Partial results are
also presented in the 3x3 case when the initial tile presents some symmetry
properties
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