The rings of n-dimensional polytopes

Points of an orbit of a finite Coxeter group G, generated by n reflections starting from a single seed point, are considered as vertices of a polytope (G-polytope) centered at the origin of a real n-dimensional Euclidean space. A general efficient method is recalled for the geometric description of G- polytopes, their faces of all dimensions and their adjacencies. Products and symmetrized powers of G-polytopes are introduced and their decomposition into the sums of G-polytopes is described. Several invariants of G-polytopes are found, namely the analogs of Dynkin indices of degrees 2 and 4, anomaly numbers and congruence classes of the polytopes. The definitions apply to crystallographic and non-crystallographic Coxeter groups. Examples and applications are shown.Comment: 24 page

Structural puzzles in virology solved with an overarching icosahedral design principle

Viruses have evolved protein containers with a wide spectrum of icosahedral architectures to protect their genetic material. The geometric constraints defining these container designs, and their implications for viral evolution, are open problems in virology. The principle of quasi-equivalence is currently used to predict virus architecture, but improved imaging techniques have revealed increasing numbers of viral outliers. We show that this theory is a special case of an overarching design principle for icosahedral, as well as octahedral, architectures that can be formulated in terms of the Archimedean lattices and their duals. These surface structures encompass different blueprints for capsids with same numbers of structural proteins, as well as for capsid architectures formed from a combination of minor and major capsid proteins, and are conserved within viral lineages. They also apply to other icosahedral structures in nature, and offer alternative designs for man-made materials and nanocontainers in bionanotechnology

Mechanisms of Size Control and Polymorphism in Viral Capsid Assembly

We simulate the assembly dynamics of icosahedral capsids from subunits that interconvert between different conformations (or quasi-equivalent states). The simulations identify mechanisms by which subunits form empty capsids with only one morphology but adaptively assemble into different icosahedral morphologies around nanoparticle cargoes with varying sizes, as seen in recent experiments with brome mosaic virus (BMV) capsid proteins. Adaptive cargo encapsidation requires moderate cargo-subunit interaction strengths; stronger interactions frustrate assembly by stabilizing intermediates with incommensurate curvature. We compare simulation results to experiments with cowpea chlorotic mottle virus empty capsids and BMV capsids assembled on functionalized nanoparticles and suggest new cargo encapsidation experiments. Finally, we find that both empty and templated capsids maintain the precise spatial ordering of subunit conformations seen in the crystal structure even if interactions that preserve this arrangement are favored by as little as the thermal energy, consistent with experimental observations that different subunit conformations are highly similar

Mechanical and Assembly Units of Viral Capsids Identified via Quasi-Rigid Domain Decomposition

Key steps in a viral life-cycle, such as self-assembly of a protective protein container or in some cases also subsequent maturation events, are governed by the interplay of physico-chemical mechanisms involving various spatial and temporal scales. These salient aspects of a viral life cycle are hence well described and rationalised from a mesoscopic perspective. Accordingly, various experimental and computational efforts have been directed towards identifying the fundamental building blocks that are instrumental for the mechanical response, or constitute the assembly units, of a few specific viral shells. Motivated by these earlier studies we introduce and apply a general and efficient computational scheme for identifying the stable domains of a given viral capsid. The method is based on elastic network models and quasi-rigid domain decomposition. It is first applied to a heterogeneous set of well-characterized viruses (CCMV, MS2, STNV, STMV) for which the known mechanical or assembly domains are correctly identified. The validated method is next applied to other viral particles such as L-A, Pariacoto and polyoma viruses, whose fundamental functional domains are still unknown or debated and for which we formulate verifiable predictions. The numerical code implementing the domain decomposition strategy is made freely available

Increased Expression of the Auxiliary β(2)-subunit of Ventricular L-type Ca(2+) Channels Leads to Single-Channel Activity Characteristic of Heart Failure

BACKGROUND: Increased activity of single ventricular L-type Ca(2+)-channels (L-VDCC) is a hallmark in human heart failure. Recent findings suggest differential modulation by several auxiliary β-subunits as a possible explanation. METHODS AND RESULTS: By molecular and functional analyses of human and murine ventricles, we find that enhanced L-VDCC activity is accompanied by altered expression pattern of auxiliary L-VDCC β-subunit gene products. In HEK293-cells we show differential modulation of single L-VDCC activity by coexpression of several human cardiac β-subunits: Unlike β(1) or β(3) isoforms, β(2a) and β(2b) induce a high-activity channel behavior typical of failing myocytes. In accordance, β(2)-subunit mRNA and protein are up-regulated in failing human myocardium. In a model of heart failure we find that mice overexpressing the human cardiac Ca(V)1.2 also reveal increased single-channel activity and sarcolemmal β(2) expression when entering into the maladaptive stage of heart failure. Interestingly, these animals, when still young and non-failing (“Adaptive Phase”), reveal the opposite phenotype, viz : reduced single-channel activity accompanied by lowered β(2) expression. Additional evidence for the cause-effect relationship between β(2)-subunit expression and single L-VDCC activity is provided by newly engineered, double-transgenic mice bearing both constitutive Ca(V)1.2 and inducible β(2) cardiac overexpression. Here in non-failing hearts induction of β(2)-subunit overexpression mimicked the increase of single L-VDCC activity observed in murine and human chronic heart failure. CONCLUSIONS: Our study presents evidence of the pathobiochemical relevance of β(2)-subunits for the electrophysiological phenotype of cardiac L-VDCC and thus provides an explanation for the single L-VDCC gating observed in human and murine heart failure

Neurogenic mechanisms in bladder and bowel ageing

The prevalence of both urinary and faecal incontinence, and also chronic constipation, increases with ageing and these conditions have a major impact on the quality of life of the elderly. Management of bladder and bowel dysfunction in the elderly is currently far from ideal and also carries a significant financial burden. Understanding how these changes occur is thus a major priority in biogerontology. The functions of the bladder and terminal bowel are regulated by complex neuronal networks. In particular neurons of the spinal cord and peripheral ganglia play a key role in regulating micturition and defaecation reflexes as well as promoting continence. In this review we discuss the evidence for ageing-induced neuronal dysfunction that might predispose to neurogenic forms of incontinence in the elderly