On the total mean curvature of non-rigid surfaces

Using Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.Comment: 4 page

Some extensions of the class of convex bodies

We introduce and study a new class of \eps-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how \eps-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, and find applications to geometric tomography. We introduce the notion of a circular projection and investigate the problem of determination of \eps-convex bodies by their projection-type images. The results generalize corresponding stability theorems by H. Groemer

Bilayer thickness and curvature influence binding and insertion of a pHLIP\u3csup\u3e®\u3c/sup\u3e peptide

The physical properties of lipid bilayers, such as curvature and fluidity, can affect the interactions of polypeptides with membranes, influencing biological events. Additionally, given the growing interest in peptide-based therapeutics, understanding the influence of membrane properties on membrane-associated peptides has potential utility. pH low insertion peptides (pHLIPs) are a family of water-soluble peptides that can insert across cell membranes in a pH-dependent manner, enabling the use of pH to follow peptide-lipid interactions. Here we study pHLIP interactions with liposomes varying in size and composition, to determine the influence of several key membrane physical properties. We find that pHLIP binding to bilayer surfaces at neutral pH is governed by the ease of access to the membrane’s hydrophobic core, which can be facilitated by membrane curvature, thickness, and the cholesterol content of the membrane. After surface binding, if the pH is lowered, the kinetics of pHLIP folding to form a helix and subsequent insertion across the membrane depends on the fluidity and energetic dynamics of the membrane. We showed that pHLIP is capable of forming a helix across lipid bilayers of different thicknesses at low pH. However, the kinetics of the slow phase of insertion corresponding to the translocation of C-terminal end of the peptide across lipid bilayer, vary approximately twofold, and correlate with bilayer thickness and fluidity. Although these influences are not large, local curvature variations in membranes of different fluidity could selectively influence surface binding in mixed cell populations

Signatures of two-dimensionalisation of 3D turbulence in presence of rotation

A reason has been given for the inverse energy cascade in the two-dimensionalised rapidly rotating 3D incompressible turbulence. For such system, literature shows a possibility of the exponent of wavenumber in the energy spectrum's relation to lie between -2 and -3. We argue the existence of a more strict range of -2 to -7/3 for the exponent in the case of rapidly rotating turbulence which is in accordance with the recent experiments. Also, a rigorous derivation for the two point third order structure function has been provided helping one to argue that even with slow rotation one gets, though dominated, a spectrum with the exponent -2.87, thereby hinting at the initiation of the two-dimensionalisation effect with rotation.Comment: An extended and typos-corrected version of the earlier submissio

pHLIP Peptide Interaction with a Membrane Monitored by SAXS

The pH (Low) Insertion Peptides (pHLIP® peptides) find application in studies of membrane-associated folding, since spontaneous insertion of these peptides is conveniently triggered by varying pH. Here we employed small angle X-ray scattering (SAXS) to investigate WT pHLIP® peptide oligomeric state in solution at high concentrations and monitor changes in liposome structure upon peptide insertion into the bilayer. We established that even at high concentrations (up to 300 μM) WT pHLIP® peptide at pH 8.0 does not form oligomers higher than tetramers (which exhibit concentration-dependent transfer to monomeric state as it was shown previously). This finding has significance for medical applications, when high concentration of the peptide is injected into blood and diluted in blood circulation. The interaction of WT pHLIP® peptide with liposomes does not alter the unilamellar vesicle structure upon peptide adsorption by lipid bilayer at high pH or upon insertion across the bilayer at low pH. At the same time, SAXS data clearly reflect the insertion of the peptide into the membrane at low pH, which opens the possibility to investigate kinetic process of a polypeptide insertion and exit from the membrane in real time by time-resolved SAXS

BRST approach to Lagrangian formulation for mixed-symmetry fermionic higher-spin fields

We construct a Lagrangian description of irreducible half-integer higher-spin representations of the Poincare group with the corresponding Young tableaux having two rows, on a basis of the BRST approach. Starting with a description of fermionic higher-spin fields in a flat space of any dimension in terms of an auxiliary Fock space, we realize a conversion of the initial operator constraint system (constructed with respect to the relations extracting irreducible Poincare-group representations) into a first-class constraint system. For this purpose, we find auxiliary representations of the constraint subsuperalgebra containing the subsystem of second-class constraints in terms of Verma modules. We propose a universal procedure of constructing gauge-invariant Lagrangians with reducible gauge symmetries describing the dynamics of both massless and massive fermionic fields of any spin. No off-shell constraints for the fields and gauge parameters are used from the very beginning. It is shown that the space of BRST cohomologies with a vanishing ghost number is determined only by the constraints corresponding to an irreducible Poincare-group representation. To illustrate the general construction, we obtain a Lagrangian description of fermionic fields with generalized spin (3/2,1/2) and (3/2,3/2) on a flat background containing the complete set of auxiliary fields and gauge symmetries.Comment: 41 pages, no figures, corrected typos, updated introduction, sections 5, 7.1, 7.2 with examples, conclusion with all basic results unchanged, corrected formulae (3.27), (7.138), (7.140), added dimensional reduction part with formulae (5.34)-(5.48), (7.8)-(7.10), (7.131)-(7.136), (7.143)-(7.164), added Refs. 52, 53, 54, examples for massive fields developed by 2 way

Modulation of the pHLIP Transmembrane Helix Insertion Pathway

The membrane-associated folding/unfolding of pH (low) insertion peptide (pHLIP) provides an opportunity to study how sequence variations influence the kinetics and pathway of peptide insertion into bilayers. Here, we present the results of steady-state and kinetics investigations of several pHLIP variants with different numbers of charged residues, with attached polar cargoes at the peptide\u27s membrane-inserting end, and with three single-Trp variants placed at the beginning, middle, and end of the transmembrane helix. Each pHLIP variant exhibits a pH-dependent interaction with a lipid bilayer. Although the number of protonatable residues at the inserting end does not affect the ultimate formation of helical structure across a membrane, it correlates with the time for peptide insertion, the number of intermediate states on the folding pathway, and the rates of unfolding and exit. The presence of polar cargoes at the peptide\u27s inserting end leads to the appearance of intermediate states on the insertion pathway. Cargo polarity correlates with a decrease of the insertion rate. We conclude that the existence of intermediate states on the folding and unfolding pathways is not mandatory and, in the simple case of a polypeptide with a noncharged and nonpolar inserting end, the folding and unfolding appears as an all-or-none transition. We propose a model for membrane-associated insertion/folding and exit/unfolding and discuss the importance of these observations for the design of new delivery agents for direct translocation of polar therapeutic and diagnostic cargo molecules across cellular membranes

Beam coupling in hybrid photorefractive inorganic-cholesteric liquid crystal cells: impact of optical rotation

We develop a theoretical model to describe two-beam energy exchange in a hybrid photorefractive inorganic-cholesteric cell. A cholesteric layer is placed between two inorganic substrates. One of the substrates is photorefractive (Ce:SBN). Weak and strong light beams are incident on the hybrid cell. The interfering light beams induce a periodic space-charge field in the photorefractive window. This penetrates into the cholesteric liquid crystal (LC), inducing a diffraction grating written on the LC director. In the theory, the flexoelectric mechanism for electric field-director coupling is more important than the LC static dielectric anisotropy coupling. The LC optics is described in the Bragg regime. Each beam induces two circular polarized waves propagating in the cholesteric cell with different velocities. The model thus includes optical rotation in the cholesteric LC. The incident light beam wavelength can fall above, below, or inside the cholesteric gap. The theory calculates the energy gain of the weak beam, as a result of its interaction with the pump beam within the diffraction grating. Theoretical results for exponential gain coefficients are compared with experimental results for hybrid cells filled with cholesteric mixture BL038/CB15 at different concentrations of chiral agent CB15. Reconciliation between theory and experiment requires the inclusion of a phenomenological multiplier in the magnitude of the director grating. This multiplier is cubic in the space-charge field, and we provide a justification of the q-dependence of the multiplier. Within this paradigm, we are able to fit theory to experimental data for cholesteric mixtures with different spectral position of cholesteric gap relative to the wavelength of incident beams, subject to the use of some fitting parameters