An Order on Circular Permutations

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.

Modelling structural rearrangements in proteins using Euclidean distance matrices

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.