An Introduction to Hyperbolic Barycentric Coordinates and their Applications

Barycentric coordinates are commonly used in Euclidean geometry. The adaptation of barycentric coordinates for use in hyperbolic geometry gives rise to hyperbolic barycentric coordinates, known as gyrobarycentric coordinates. The aim of this article is to present the road from Einstein's velocity addition law of relativistically admissible velocities to hyperbolic barycentric coordinates along with applications.Comment: 66 pages, 3 figure

Gyrations: The Missing Link Between Classical Mechanics with its Underlying Euclidean Geometry and Relativistic Mechanics with its Underlying Hyperbolic Geometry

Being neither commutative nor associative, Einstein velocity addition of relativistically admissible velocities gives rise to gyrations. Gyrations, in turn, measure the extent to which Einstein addition deviates from commutativity and from associativity. Gyrations are geometric automorphisms abstracted from the relativistic mechanical effect known as Thomas precession

Extracellular alpha-synuclein: Sensors, receptors, and responses

Synucleinopathies are a group of progressive neurodegenerative diseases known for the accumulation of insoluble aggregates containing the protein alpha-synuclein (aSyn). Recently, it has been assumed that pathology spreads in the brain during disease progression, implying that, at some point in the process, aSyn may exist outside of cells. In this context, extracellular-aSyn (e-aSyn) might transduce signals to the inside of the cells it interacts with, and/or be internalized by different types of cells through the extracellular matrix. Both negatively charged lipids and membrane receptors have been hypothesized as modulators of the loss of cellular homeostasis and cytotoxicity, and of the internalization of e-aSyn. Internalized e-aSyn causes the disruption of multiple cellular processes such as the autophagy lysosomal pathway (ALP), mitochondrial function, endoplasmic reticulum (ER)-stress, UPR activation, or vesicular transport. These processes happen not only in neurons but also in glial cells, activating inflammatory or anti-inflammatory pathways that can affect both neuronal function and survival, thereby affecting disease progression. In this review, we explore possible effects e-aSyn, all the way from the extracellular matrix to the nucleus. In particular, we highlight the glial-neuronal relationship as this is particularly relevant in the context of the spreading of aSyn pathology in synucleinopathies

The Proportion of Fixed-Point-Free Elements of a Transitive Permutation Group

In 1990 Hendrik W. Lenstra, Jr. asked the following question: if G is a transitive permutation group of degree n and A is the set of elements of G that move every letter, then can one find a lower bound (in terms of n) for f(G) = |A|/|G|? Shortly thereafter, Arjeh Cohen showed that 1/n is such a bound. Lenstra’s problem arose from his work on the number field sieve. A simple example of how f(G) arises in number theory is the following: if h is an irreducible polynomial over the integers, consider the proportion: |{primes ≤ x | h has no zeroes mod p}| / |{primes ≤ x}| As x → ∞, this ratio approaches f(G), where G is the Galois group of h considered as a permutation group on its roots. Our results in this paper include explicit calculations of f(G) for groups G in several families. We also obtain results useful for computing f(G) when G is a wreath product or a direct product of permutation groups. Using this we show that {f(G) | G is transitive} is dense in [0, 1]. The corresponding conclusion is true if we restrict G to primitive groups

On peak phenomena for non-commutative H∞H^\infty

A non-commutative extension of Amar and Lederer's peak set result is given. As its simple applications it is shown that any non-commutative $H^\infty$-algebra $H^\infty(M,\tau)$ has unique predual,and moreover some restriction in some of the results of Blecher and Labuschagne are removed, making them hold in full generality.Comment: final version (the presentation of some part is revised and one reference added

Inhibition of human transthyretin aggregation by non-steroidal anti-inflammatory compounds: a structural and thermodynamic analysis

Transthyretin (TTR) is a homotetrameric protein that circulates in plasma and cerebral spinal fluid (CSF) whose aggregation into amyloid fibrils has been associated with at least two different amyloid diseases: senile systemic amyloidosis (SSA) and familial amyloid polyneuropathy (FAP). In SSA aggregates are composed of WT-TTR, while in FAP more than 100 already-described variants have been found in deposits. Until now, TTR-related diseases have been untreatable, although a new drug called Tafamidis has been approved only in Europe to specifically treat V30M patients. Thus, new strategies are still necessary to treat FAP caused by other variants of TTR. TTR has two channels in the dimer interface that bind to the hormone thyroxin and that have been used to accommodate anti-amyloidogenic compounds. These compounds stabilize the tetramers, rendering TTR less amyloidogenic. Here, we investigated the effects of three non-steroidal anti-inflammatory compounds-sulindac (SUL), indomethacin (IND) and lumiracoxib (LUM)-as tetramer stabilizers and aggregation inhibitors. WT-TTR and the very aggressive TTR variant L55P were used as models. These compounds were able to stabilize TTR against high hydrostatic pressure (HHP), increasing the ΔGf by several kcal. They were also effective in inhibiting WT-TTR and L55P acid- or HHP-induced aggregation; in particular, LUM and IND were very effective, inhibiting almost 100% of the aggregation of both proteins under certain conditions. The species formed when aggregation was performed in the presence of these compounds were much less toxic to cells in culture. The crystal structures of WT-TTR bound to the three compounds were solved at high resolution, allowing the identification of the relevant protein:drug interactions. We discuss here the ligand-binding features of LUM, IND and SUL to TTR, emphasizing the critical interactions that render the protein more stable and less amyloidogenic.CAPESCNPqInstituto Nacional de Ciência e Tecnologia de Biologia Estrutural e Bioimagem (INBEB)FAPERJFAPES

Harmonic analysis on the Möbius gyrogroup

In this paper we propose to develop harmonic analysis on the Poincaré ball $B_t^n$, a model of the n-dimensional real hyperbolic space. The Poincaré ball $B_t^n$ is the open ball of the Euclidean n-space $R^n$ with radius $t>0$, centered at the origin of $R^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb{R}^n$. For any $t>0$ and an arbitrary parameter $\sigma \in R$ we study the $(\sigma,t)$-translation, the $( \sigma,t)$-convolution, the eigenfunctions of the $(\sigma,t)$-Laplace-Beltrami operator, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on $B_t^n$ tends to the standard Euclidean harmonic analysis on $R^n$, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $B_t^n$

Ligand Binding and Hydration in Protein Misfolding: Insights from Studies of Prion and p53 Tumor Suppressor Proteins†

Protein misfolding has been implicated in a large number of diseases termed protein- folding disorders (PFDs), which include Alzheimer’s disease, Parkinson’s disease, transmissible spongiform encephalopathies, familial amyloid polyneuropathy, Huntington’s disease, and type II diabetes. In these diseases, large quantities of incorrectly folded proteins undergo aggregation, destroying brain cells and other tissues

Fourier transform ion cyclotron resonance mass spectrometric detection of small Ca2+-induced conformational changes in the regulatory domain of human cardiac troponin C

AbstractTroponin C (TnC), a calcium-binding protein of the thin filament of muscle, plays a regulatory role in skeletal and cardiac muscle contraction. NMR reveals a small conformational change in the cardiac regulatory N-terminal domain of TnC (cNTnC) on binding of Ca2+ such that the total exposed hydrophobic surface area increases very slightly from 3090 ± 86 Å2 for apo-cNTnC to 3108 ± 71 Å2 for Ca2+-cNTnC. Here, we show that measurement of solvent accessibility for backbone amide protons by means of solution-phase hydrogen/deuterium (H/D) exchange followed by pepsin digestion, high-performance liquid chromatography, and electrospray ionization high-field (9.4 T) Fourier transform Ion cyclotron resonance mass spectrometry is sufficiently sensitive to detect such small ligand binding-induced conformational changes of that protein. The extent of deuterium incorporation increases significantly on binding of Ca2+ for each of four proteolytic segments derived from pepsin digestion of the apo- and Ca2+-saturated forms of cNTnC. The present results demonstrate that H/D exchange monitored by mass spectrometry can be sufficiently sensitive to detect and identify even very small conformational changes in proteins, and should therefore be especially informative for proteins too large (or too insoluble or otherwise intractable) for NMR analysis