Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.
Proteins undergo large structural rearrangements such as circular permutations, dimerisation via domain swapping, and loss of core secondary structure elements in domain atrophy, among others. These structural changes can be naturally represented as distance matrix transformations, exploiting their conserved native residue contacts at the protein core. Here we present an homology modelling approach to formulate structural rearrangements as a Euclidean distance matrix (EDM) problem and use it to build their 3D structures. This modelling approach aims to be lightweight, flexible and fast, suitable for large-scale analyses. Models are typically coarse-grained and solely based on protein geometry. We demonstrate various applications of EDM-based modelling for protein structure analysis and release an open repository with the source code at: https://github.com/lafita/protein-edm-demo.
This chapter presents data about cyclic properties of the genetic code in its matrix forms of presentation. These cyclic properties concern cyclic changes of genetic Yin-Yang-matrices and their Yin-Yangalgebras (bipolar algebras) at many kinds of circular permutations of genetic elements in genetic matrices. These circular permutations lead to such reorganizations of the matrix form of presentation of the initial genetic Yin-Yang-algebra that arisen matrices serve as matrix forms of presentations of new Yin-Yang-algebras, as well. They are connected algorithmically with Hadamard matrices. New patterns and relations of symmetry are described. The discovered existence of a hierarchy of the cyclic changes of genetic Yin-Yang-algebras allows one to develop new algebraic models of cyclic processes in bioinformatics and in other related fields. These cycles of changes of the genetic 8-dimensional algebras and of their 8-dimensional numeric systems have many analogies with famous facts and doctrines of modern and ancient physiology, medicine, and so forth. This viewpoint proposes that the famous idea by Pythagoras (about organization of natural systems in accordance with harmony of numerical systems) should be combined with the idea of cyclic changes of Yin-Yang-numeric systems in considered cases. This second idea reminds of the ancient idea of cyclic changes in nature. From such algebraic-genetic viewpoint, the notion of biological time can be considered as a factor of coordinating these hierarchical ensembles of cyclic changes of the genetic multi-dimensional algebras.
Sorting Circular Permutations by Reversal