3,931 research outputs found
Inverse Statistical Physics of Protein Sequences: A Key Issues Review
In the course of evolution, proteins undergo important changes in their amino
acid sequences, while their three-dimensional folded structure and their
biological function remain remarkably conserved. Thanks to modern sequencing
techniques, sequence data accumulate at unprecedented pace. This provides large
sets of so-called homologous, i.e.~evolutionarily related protein sequences, to
which methods of inverse statistical physics can be applied. Using sequence
data as the basis for the inference of Boltzmann distributions from samples of
microscopic configurations or observables, it is possible to extract
information about evolutionary constraints and thus protein function and
structure. Here we give an overview over some biologically important questions,
and how statistical-mechanics inspired modeling approaches can help to answer
them. Finally, we discuss some open questions, which we expect to be addressed
over the next years.Comment: 18 pages, 7 figure
Spectral graph theory : from practice to theory
Graph theory is the area of mathematics that studies networks, or graphs. It arose from the need to analyse many diverse network-like structures like road networks, molecules, the Internet, social networks and electrical networks. In spectral graph theory, which is a branch of graph theory, matrices are constructed from such graphs and analysed from the point of view of their so-called eigenvalues and eigenvectors. The first practical need for studying graph eigenvalues was in quantum chemistry in the thirties, forties and fifties, specifically to describe the HĂŒckel molecular orbital theory for unsaturated conjugated hydrocarbons. This study led to the field which nowadays is called chemical graph theory. A few years later, during the late fifties and sixties, graph eigenvalues also proved to be important in physics, particularly in the solution of the membrane vibration problem via the discrete approximation of the membrane as a graph. This paper delves into the journey of how the practical needs of quantum chemistry and vibrating membranes compelled the creation of the more abstract spectral graph theory. Important, yet basic, mathematical results stemming from spectral graph theory shall be mentioned in this paper. Later, areas of study that make full use of these mathematical results, thus benefitting greatly from spectral graph theory, shall be described. These fields of study include the P versus NP problem in the field of computational complexity, Internet search, network centrality measures and control theory.peer-reviewe
Statistical Mechanical Treatments of Protein Amyloid Formation
Protein aggregation is an important field of investigation because it is
closely related to the problem of neurodegenerative diseases, to the
development of biomaterials, and to the growth of cellular structures such as
cyto-skeleton. Self-aggregation of protein amyloids, for example, is a
complicated process involving many species and levels of structures. This
complexity, however, can be dealt with using statistical mechanical tools, such
as free energies, partition functions, and transfer matrices. In this article,
we review general strategies for studying protein aggregation using statistical
mechanical approaches and show that canonical and grand canonical ensembles can
be used in such approaches. The grand canonical approach is particularly
convenient since competing pathways of assembly and dis-assembly can be
considered simultaneously. Another advantage of using statistical mechanics is
that numerically exact solutions can be obtained for all of the thermodynamic
properties of fibrils, such as the amount of fibrils formed, as a function of
initial protein concentration. Furthermore, statistical mechanics models can be
used to fit experimental data when they are available for comparison.Comment: Accepted to IJM
Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings
This paper is a study of the interaction between the combinatorics of
boundaries of convex polytopes in arbitrary dimension and their metric
geometry.
Let S be the boundary of a convex polytope of dimension d+1, or more
generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a
polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained
from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs
of boundary faces isometrically. Our existence proof exploits geodesic flow
away from a source point v in S, which is the exponential map to S from the
tangent space at v. We characterize the `cut locus' (the closure of the set of
points in S with more than one shortest path to v) as a polyhedral complex in
terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the
wavefront consisting of points at constant distance from v on S produces an
algorithmic method for constructing Voronoi diagrams in each facet, and hence
the unfolding of S. The algorithm, for which we provide pseudocode, solves the
discrete geodesic problem. Its main construction generalizes the source
unfolding for boundaries of 3-polytopes into R^2. We present conjectures
concerning the number of shortest paths on the boundaries of convex polyhedra,
and concerning continuous unfolding of convex polyhedra. We also comment on the
intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo
Generalized Tonnetze and Zeitnetze, and the topology of music concepts
The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the 12 pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. This article examines how these duplications can relate to important musical concepts such as key or pitch height, and details a method of removing such redundancies and the resulting changes to the homology of the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth chords, scalar tetrachords, scales, etc., as well as Zeitnetze on common cyclic rhythms or timelines. Their different topologies â whether orientable, bounded, manifold, etc. â reveal some of the topological character of musical concepts.Accepted manuscrip
Spontaneous symmetry breaking and the formation of columnar structures in the primary visual cortex II --- Local organization of orientation modules
Self-organization of orientation-wheels observed in the visual cortex is
discussed from the view point of topology. We argue in a generalized model of
Kohonen's feature mappings that the existence of the orientation-wheels is a
consequence of Riemann-Hurwitz formula from topology. In the same line, we
estimate partition function of the model, and show that regardless of the total
number N of the orientation-modules per hypercolumn the modules are
self-organized, without fine-tuning of parameters, into definite number of
orientation-wheels per hypercolumn if N is large.Comment: 36 pages Latex2.09 and eps figures. Needs epsf.sty, amssym.def, and
Type1 TeX-fonts of BlueSky Res. for correct typo in graphics file
- âŠ