Contractions of Filippov algebras

We introduce in this paper the contractions $\mathfrak{G}_c$ of $n$-Lie (or Filippov) algebras $\mathfrak{G}$ and show that they have a semidirect structure as their $n=2$ Lie algebra counterparts. As an example, we compute the non-trivial contractions of the simple $A_{n+1}$ Filippov algebras. By using the \.In\"on\"u-Wigner and the generalized Weimar-Woods contractions of ordinary Lie algebras, we compare (in the $\mathfrak{G}=A_{n+1}$ simple case) the Lie algebras Lie$\,\mathfrak{G}_c$ (the Lie algebra of inner endomorphisms of $\mathfrak{G}_c$) with certain contractions $(\mathrm{Lie}\,\mathfrak{G})_{IW}$ and $(\mathrm{Lie}\,\mathfrak{G})_{W-W}$ of the Lie algebra Lie$\,\mathfrak{G}$ associated with $\mathfrak{G}$.Comment: plain latex, 36 pages. A few misprints corrected. This v3 is actually v2 (v1 had been replaced by itself by error). To appear in J. Math. Phy

VarMod: modelling the functional effects of non-synonymous variants.

Unravelling the genotype–phenotype relationship in humans remains a challenging task in genomics studies. Recent advances in sequencing technologies mean there are now thousands of sequenced human genomes, revealing millions of single nucleotide variants (SNVs). For non-synonymous SNVs present in proteins the difficulties of the problem lie in first identifying those nsSNVs that result in a functional change in the protein among the many non-functional variants and in turn linking this functional change to phenotype. Here we present VarMod (Variant Modeller) a method that utilises both protein sequence and structural features to predict nsSNVs that alter protein function. VarMod develops recent observations that functional nsSNVs are enriched at protein–protein interfaces and protein–ligand binding sites and uses these characteristics to make predictions. In benchmarking on a set of nearly 3000 nsSNVs VarMod performance is comparable to an existing state of the art method. The VarMod web server provides extensive resources to investigate the sequence and structural features associated with the predictions including visualisation of protein models and complexes via an interactive JSmol molecular viewer. VarMod is available for use at http://www.wasslab.org/varmod

Topics on n-ary algebras

We describe the basic properties of two n-ary algebras, the Generalized Lie Algebras (GLAs) and, particularly, the Filippov (or n-Lie) algebras (FAs), and comment on their n-ary Poisson counterparts, the Generalized Poisson (GP) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends the familiar Whitehead's lemma to all $n\geq 2$ FAs, n=2 being the standard Lie algebra case. When the n-bracket of the FAs is no longer required to be fully skewsymmetric one is led to the n-Leibniz (or Loday's) algebra structure. Using that FAs are a particular case of n-Leibniz algebras, those with an anticommutative n-bracket, we study the class of n-Leibniz deformations of simple FAs that retain the skewsymmetry for the first n-1 entires of the n-Leibniz bracket.Comment: 11 page

The Phyre2 web portal for protein modeling, prediction and analysis

Phyre2 is a suite of tools available on the web to predict and analyze protein structure, function and mutations. The focus of Phyre2 is to provide biologists with a simple and intuitive interface to state-of-the-art protein bioinformatics tools. Phyre2 replaces Phyre, the original version of the server for which we previously published a paper in Nature Protocols. In this updated protocol, we describe Phyre2, which uses advanced remote homology detection methods to build 3D models, predict ligand binding sites and analyze the effect of amino acid variants (e.g., nonsynonymous SNPs (nsSNPs)) for a user's protein sequence. Users are guided through results by a simple interface at a level of detail they determine. This protocol will guide users from submitting a protein sequence to interpreting the secondary and tertiary structure of their models, their domain composition and model quality. A range of additional available tools is described to find a protein structure in a genome, to submit large number of sequences at once and to automatically run weekly searches for proteins that are difficult to model. The server is available at http://www.sbg.bio.ic.ac.uk/phyre2. A typical structure prediction will be returned between 30 min and 2 h after submission

Cohomology of Filippov algebras and an analogue of Whitehead's lemma

We show that two cohomological properties of semisimple Lie algebras also hold for Filippov (n-Lie) algebras, namely, that semisimple n-Lie algebras do not admit non-trivial central extensions and that they are rigid i.e., cannot be deformed in Gerstenhaber sense. This result is the analogue of Whitehead's Lemma for Filippov algebras. A few comments about the n-Leibniz algebras case are made at the end.Comment: plain latex, no figures, 29 page

From Nonspecific DNA–Protein Encounter Complexes to the Prediction of DNA–Protein Interactions

©2009 Gao, Skolnick. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.doi:10.1371/journal.pcbi.1000341DNA–protein interactions are involved in many essential biological activities. Because there is no simple mapping code between DNA base pairs and protein amino acids, the prediction of DNA–protein interactions is a challenging problem. Here, we present a novel computational approach for predicting DNA-binding protein residues and DNA–protein interaction modes without knowing its specific DNA target sequence. Given the structure of a DNA-binding protein, the method first generates an ensemble of complex structures obtained by rigid-body docking with a nonspecific canonical B-DNA. Representative models are subsequently selected through clustering and ranking by their DNA–protein interfacial energy. Analysis of these encounter complex models suggests that the recognition sites for specific DNA binding are usually favorable interaction sites for the nonspecific DNA probe and that nonspecific DNA–protein interaction modes exhibit some similarity to specific DNA–protein binding modes. Although the method requires as input the knowledge that the protein binds DNA, in benchmark tests, it achieves better performance in identifying DNA-binding sites than three previously established methods, which are based on sophisticated machine-learning techniques. We further apply our method to protein structures predicted through modeling and demonstrate that our method performs satisfactorily on protein models whose root-mean-square Ca deviation from native is up to 5 Å from their native structures. This study provides valuable structural insights into how a specific DNA-binding protein interacts with a nonspecific DNA sequence. The similarity between the specific DNA–protein interaction mode and nonspecific interaction modes may reflect an important sampling step in search of its specific DNA targets by a DNA-binding protein

PIP2-Binding Site in Kir Channels: Definition by Multiscale Biomolecular Simulations†

Phosphatidylinositol bisphosphate (PIP(2)) is an activator of mammalian inwardly rectifying potassium (Kir) channels. Multiscale simulations, via a sequential combination of coarse-grained and atomistic molecular dynamics, enabled exploration of the interactions of PIP(2) molecules within the inner leaflet of a lipid bilayer membrane with possible binding sites on Kir channels. Three Kir channel structures were investigated: X-ray structures of KirBac1.1 and of a Kir3.1-KirBac1.3 chimera and a homology model of Kir6.2. Coarse-grained simulations of the Kir channels in PIP(2)-containing lipid bilayers identified the PIP(2)-binding site on each channel. These models of the PIP(2)-channel complexes were refined by conversion to an atomistic representation followed by molecular dynamics simulation in a lipid bilayer. All three channels were revealed to contain a conserved binding site at the N-terminal end of the slide (M0) helix, at the interface between adjacent subunits of the channel. This binding site agrees with mutagenesis data and is in the proximity of the site occupied by a detergent molecule in the Kir chimera channel crystal. Polar contacts in the coarse-grained simulations corresponded to long-lived electrostatic and H-bonding interactions between the channel and PIP(2) in the atomistic simulations, enabling identification of key side chains