Undulation Modes Research Articles

Effect of Undulations on Surface Tension in Simulated Bilayers

To understand the effect of the finite size of simulation cells on the equilibrium properties of bilayers, an extensive series of glycerolmonoolein bilayer molecular dynamics simulations in which the surface area and system size were systematically changed have been conducted. Systems ranging from 200 to 1800 lipids were simulated, covering length scales up to 20 nm. The dependence of the surface tension on the area per lipid is shown, although long simulation times were needed (up to 40 ns) to obtain reliable estimates. As the size of simulated patches increases, long wavelength undulatory modes appear with a concomitant increase in the area compressibility due to coupling of undulation modes to area fluctuations. Both the undulatory intensities and the peristaltic intensities of the bilayer fluctuations can be fitted in the long wavelength limit to continuum model predictions. The effect of system size on surface tension appears to depend on the stress conditions.

Fish and tetrapod trace fossils from the Upper Carboniferous of Puertollano, Spain

Well-preserved fish and tetrapod trace fossils occur in Carboniferous (Stephanian C) beds at Puertollano. They represent the first evidence of vertebrate ichnites in the Carboniferous of Spain. The fish swimming traces are assigned to Undichna britannica and Undichna unisulca. The dimensions and morphological features of the traces indicate that Orthacanthus (Xenacanthidae: Elasmobranchii), a shark very common in the basin, might have been the trace-producer. A morpho-functional analysis of the traces suggests an axial undulatory (=anguilliform) mode of locomotion of xenacanth sharks. This conclusion is consistent with the eel-like body found in all known xenacanthids. Tetrapod trackways, referred to Puertollanopus microdactylus ichnogen. et ichnosp. nov., and resting traces are described. Both walking and resting tracks are assigned to a small terrestrial tetrapod. Dimensions and proportions of the tracks, and size and morphology of the impressions of manus and pes suggest a microsaur amphibian or a small reptile as the track-maker. This represents the first evidence of these vertebrates in the basin. The superposition of fish trails and tetrapod trackways strongly suggests tidal processes in a marginal-marine depositional setting. This interpretation agrees with recent sedimentological, geochemical and palynological data from the Puertollano basin.

Dynamic Light Scattering from the Oriented Lamellar State of Diblock Copolymers: The Undulation Mode

Using a fully oriented sample of a diblock copolymer, poly(dimethylsiloxane)-block-poly(ethylene-co-propylene), we have observed, by dynamic light scattering, the undulation mode of the block copol...

First results of TXRF measurements of low‐Z elements on Si wafer surfaces at the PTB plane grating monochromator beamline for undulator radiation at BESSY II

AbstractFirst measurements of low‐Z elements on Si wafer surfaces with undulator and wiggler radiation excited total reflection x‐ray fluorescence (TXRF) spectrometry were performed at the PTB plane grating monochromator beamline for undulator radiation at the electron storage ring BESSY II. The synchrotron radiation (SR) TXRF spectrometer for low‐Z elements of the Atominstitut was used, equipped with a down‐looking Ge(HP) detector with a 300 nm NORWAR window, with the wafer positioned horizontally. Experiments were performed with excitation energies above 0.7 keV in the wiggler mode and with excitation energies below 0.7 keV in the undulator mode of the PTB insertion device U180. For droplet samples on Si wafer surfaces detection limits for Na down to C were found to be in the low‐picogram range. Copyright © 2001 John Wiley & Sons, Ltd.

Mesoscopic Undulations and Thickness Fluctuations in Lipid Bilayers from Molecular Dynamics Simulations

Molecular dynamics simulations of fully hydrated Dipalmitoylphosphatidylcholine bilayers, extending temporal and spatial scales by almost one order of magnitude, are presented. The present work reaches system sizes of 1024 lipids and times 10–60 ns. The simulations uncover significant dynamics and fluctuations on scales of several nanoseconds, and enable direct observation and spectral decomposition of both undulatory and thickness fluctuation modes. Although the former modes are strongly damped, the latter exhibit signs of oscillatory behavior. From this, it has been possible to calculate mesoscopic continuum properties in good agreement with experimental values. A bending modulus of 4 × 10 −20 J, bilayer area compressibility of 250–300 mN/m, and mode relaxation times in the nanosecond range are obtained. The theory of undulatory motions is revised and further extended to cover thickness fluctuations. Finally, it is proposed that thickness fluctuations is the explanation to the observed system-size dependence of equilibrium-projected area per lipid.

Anomalous Diffusion in Active Intracellular Transport

AbstractThe dynamic movements of tracer particles have been used to characterize their local environment in dilute networks of microtubules, and within living cells. In the former case, 300 nm diameter beads are fixed to individual microtubules, so that the movements of the bead reveal undulatory modes of the polymer. The mean square displacement shows a scaling of t3/4 in keeping with mode analysis arguments. Inside a cell, beads show a more complicated behavior that reflects internal dynamics of the cytoskeleton and associated motors.When placed near the cell edge, 3 micron diameter beads coated by proteins that mediate membrane adhesion are engulfed underneath the membrane and drawn toward the center by a contracting flow of actin. On reaching the region surrounding the nucleus, the beads continue to move but lose directionality, so that they wander within a restricted space. Measurement of the mean square displacement now shows a scaling of t1 up to times of ~1 sec. At longer times the scaling varies between t and t1/2 in the various runs. The data do not fit a crossover between ballistic (t2)and diffusive (t1) behavior. The movement is clearly driven by non-thermal interactions, as it cannot be stopped by an optical trap. Treatment of the cell to depolymerize microtubules restores ordinary diffusion, while actin depolymerization has no effect, indicating that microtubule-based motor proteins are responsible for the motion. Immunofluorescence microscopy shows that the mesh size of the microtubules is smaller than the bead diameter.We propose that the observations are related, and that the non-trivial scaling in the polymer system leads to time-dependent friction in a network, which in turn leads to a generalized Einstein relation operative in the intracellular environment. This results, in the driven system, in sub-ballistic motion at short times and sub-diffusive motion at longer times.

Undulation instability of lamellar phases under shear: A mechanism for onion formation?

We consider a lamellar phase of bilayer membranes held between two parallel plates and subject to a steady shear. Accounting for the coupling with the shear flow of the short wavelength undulation modes that are responsible for the membrane excess area, we argue that the flow generates an effective force which acts to reduce the excess area. From the viewpoint of the macroscopic lamellar whose geometric dimensions are fixed, this force translates into an effective lateral pressure. At low shear rates $\dot \gamma$ this pressure is balanced by the elastic restoring forces of the lamellar. Above a critical shear rate $\dot \gamma _c \sim d^{ - 5/2} D^{ - 1/2}$ , where d is the interlayer distance and D is the gap spacing, the lamellar buckles into a harmonic shape modulation, and we predict its wavelength λc and amplitude U o. We show that our model is isomorphic to a dilative strain, which is known to induce a similar buckling (undulation) instability. Indeed, at threshold the wavelength is $\lambda _c \sim \sqrt {Dd}$ and is identical in both cases. Using a non-linear analysis, we discuss how the wavelength and amplitude vary with shear rate away from the threshold. For $\dot \gamma \gg \dot \gamma _c$ we find $\lambda _c \sim \dot \gamma ^{ - 1/3}$ and $U_o \sim \dot \gamma ^{2/3}$ . We then focus on the coupling of the buckling modulation itself with the flow, and obtain a criterion for the limit of its stability. Motivated by experiments of D. Roux and coworkers, we assume that at this limit of stability the lamellar breakups into “onion”-like, multilamellar, vesicles. The critical shear rate $\dot \gamma *$ for the formation of onions is predicted to scale as $\dot \gamma * \sim \dot \gamma _c \sim d^{ - 5/2} D^{ - 1/2}$ . The scaling with d is consistent with available experimental data.

Undulation modes in bilayer membranes

Through the complete analysis of the solution for one dimensional bilayer membranes, it has been shown that under the Helfrich spontaneous curvature model, periodic shapes only exist in the case of Δ=0. The undulation modes under force balance which correspond to positive, negative and zero tensile stresses have been given.

Neuromuscular control of anguilliform locomotion: patterns of red and white muscle activity during swimming in the american eel anguilla rostrata

Two areas that have received substantial attention in investigations of muscle activity during fish swimming are (1) patterns of fiber type recruitment with swimming speed and (2) the timing of muscle activation in relation to muscle strain. Currently, very little is known about either of these areas in eels, which represent an extreme body form among fishes and utilize a mode of locomotion found at one end of the undulatory spectrum (anguilliform locomotion). To assess how this swimming mode and body form influence the neuromuscular control of swimming, I recorded electromyographic data from red and white muscle at four positions, 0.3L, 0.45L, 0.6L and 0.75L, where L is body length, in eels (Anguilla rostrata) simultaneously video-taped (250 fields s-1) swimming at three speeds, 0.5, 0.75 and 1.0 L s-1. As in other fish, exclusively red muscle is used at slow swimming speeds and white muscle is additionally recruited at higher swimming speeds. However, this study also revealed a novel posterior-to-anterior pattern of muscle recruitment with increasing swimming speed. At slow speeds, anteriorly located muscles are never active, muscle strain is negligible and forward thrust must be generated by posterior muscles. As speed increases, more anterior muscles are additionally recruited. Electromyogram (EMG) burst durations typically occupy between 0.2 and 0.3 undulatory cycles, irrespective of speed or position. EMG burst intensity increases significantly with swimming speed. The onset of EMG activity typically occurred near the end of muscle lengthening, whereas the offset of EMG activity occurred during shortening (typically before the muscle's return to resting length). There was a significant shift in red muscle onset times such that anterior muscles were typically active later in their strain cycle than posterior muscles. When red muscle activity patterns across various fish taxa are compared, differences in propulsive wavelength among species are related to differences in muscle activity, providing insight into the underlying neuromuscular bases of differences among undulatory swimming modes.

Surface dynamics of a freely standing smectic-Afilm

A theoretical analysis of surface fluctuations of a freely standing thermotropic smectic-A liquid crystal film is provided, including the effects of viscous hydrodynamics. We find two surface dynamic modes (undulation and peristaltic). For long wavelengths and small frequencies in a thin film, the undulation mode is the dominant mode. Permeation enters the theory only through the boundary conditions. The resulting power spectrum is compared with existing experiments. It is also shown that feasible light scattering experiments on a freely standing smectic-A film can reveal viscosity and elastic coefficients through the structure of the power spectrum of both the undulation and peristaltic modes.

Undulating Charged Fluid Membranes and their Bending Constants

We consider harmonically undulating ionic surfactant membranes, calculating a general expression for the bending modulus valid for all salt and surfactant concentrations, and surface charge densities. This is achieved through a perturbative expansion of the mean-field electrostatic potential and free energy about a planar reference state, consisting of two parallel planar membranes with intervening salt solution. For a given choice of undulation mode, the result for the bending modulus is seen to be generally the same as that obtained from considerations of membranes deformed into a cylindrical geometry. Thus, we show that the bending modulus is independent of global aggregate geometry, at a general system composition. Specializing to the limit of excess added salt (equivalent to a single undulating membrane in contact with bulk electrolyte), we are able to extend the free energy calculation to fourth order in undulation amplitude, and derive the bending constants of the curvature expansion to this order. These have been suggested previously to be of importance in explaining the stability of bicontinuous crystalline and disordered phases, and the formation of passages in lamellar phases. We also discuss the breakdown of the curvature description at shorter wavelengths.

Physics of Soap Films

Soap films constituting from two surfactant monolayers sandwiching a thin layer of water are investigated. The structure of such films is not static because thermal fluctuations induce corrugations on the monolayers which can be decompose into two principle types of modes: undulation (or bending) modes for which the monolayers oscillate in phase, and squeezing modes for which these oscillations have a 180° phase difference. The eigen-modes spectrum for both types of motions is found. It is shown that the squeezing mode (where the water is pumped back and forth) induces fluctuational (anharmontc) contributions to viscosity and diffusion coefficients which are divergent at large scales, leading to anomalous scaling behavior of the coefficients.

Undulatory Locomotion in Elongate Aquatic Vertebrates: Anguilliform Swimming since Sir James Gray

Anguilliform locomotion is the mode of undulatory swimming used by eels and other elongate vertebrates in which waves (of shorter length than the body of the animal) are propagated posteriorly along the length of the animal, propelling it forward. Although a broad array of vertebrate taxa convergently use this locomotor mode, much of what we know regarding the kinematics of anguilliform swimming derives from the classic work of Sir James Gray in the 1930s on the European eel, Anguilla anguilla. To better understand the variability within this phylogenetically widespread mode of locomotion, I have reviewed recent work on the swimming kinematics of elongate fishes, salamanders and snakes. The amplitude and relative speed of the undulatory waves propagated along the body during swimming differs among elongate vertebrate tax, and can also vary with swimming speed. In addition, the lateral velocity and orientation of tail segments can reach their maxima at different times within a tailbeat cycle (in contrast to Gray's suggestion that these two events occurred simultaneously). Finally, the angle of attack of tail segments is negative during a consistent portion of each tailbeat cycle in the elongate salamander Siren. While this has yet to be examined in other anguilliform taxa, it is possible that they will also show this pattern. Hence, although Gray's descriptions of anguilliform swimming are generally accurate, gaining a broader understanding of this mode of locomotion requires the characterization of the diversity and complexity found among animals using this undulatory mode.

Electrostatic rigidity of charged membranes in systems without added salt

The thermodynamics of surfactants in solution has largely been understood in the framework of the flexible surface model. We calculate the bending rigidity for a case where this model might be put to a strong test: ionic surfactant systems with no added salt to screen the electrostatics. In so doing, we present a perturbative solution to the Poisson-Boltzmann equation for the case of two undulating sheets with intervening counterions. The predictions to which this leads for the bending modulus are shown (for a particular choice of undulation mode) to be identical to that of the solution for the geometry of concentric cylinders. This indicates that the curvature free energy is independent of the global aggregate geometry, even in systems with counterions only.

Mechanical properties of a hyperswollen lyotropic smectic liquid crystals at low frequencies

Smectic liquid crystals are unique condensed phases in that they exhibit properties of a one-dimensional solid and of a two-dimensional liquid. There exist various types of the hydrodynamic modes, propagating and diffusive modes, such as second sound or layer undulation modes, related to the existence of the internal degree of freedom in the complex fluid. Mechanical properties have been investigated of hyperswollen lyotropic smectic phases, with focus on its low dimensionality and large intrinsic fluctuations. This layer structure is supported by the microscopic long-range intermembrane interactions such as electrostatic or steric repulsion that play an important role in the hyperswollen regime. The layer compressibility measurements tell one that the layer structure is supported by entropic repulsion, which originated in the undulation fluctuation motion of the membranes. The light scattering experiments are also shown which provide information on the relation between the static structure and the dynamics of hydrodynamic fluctuation modes.

Experience on the operation of the 2-in-1 electromagnetic undulator of FELICITA-I

The 5m-Undulator of FELICITA I has been installed at the final position in the storage ring DELTA in spring 95. Detailed magnetic measurements have been performed in place to get a complete characterization of this device in both possible modes, pure undulator and optical klystron mode, respectively. The undulator consists of 38 identic poles. Four power supplies are connected to the main coils. Three of them drive the central six poles to get the dispersive section with its matching to the outer undulator sections. This setup allows a change between the optical klystron and the pure undulator mode without changing the hardware. For measuring the magnetic field components a hall-probe was used. It was found that correction coils for compensating peak field variations were not necessary, because of errors less then 0.2 %. Only for the purpose of steering correction coils had to be used. The field integrals were also measured with the pulsed-wire technique. For fast response concerning the matching, in particular of the dispersive section this technique was found to be very useful. Because of the large period length of 25 cm the wire sag in the rather long undulator could be neglected.

Statistical mechanics of closed fluid membranes.

We discuss the statistical mechanics of systems, such as vesicle phases, that consist of many closed, disconnected fluid membranes. We show that the internal undulation free energy of a single such membrane always contains a logarithmically scale-dependent ``finite-size'' contribution, with a universal coefficient, that arises essentially from the absence of undulation modes of wavelength longer than the size of the surface. We show that the existence of this contribution is closely related to the existence of additional collective degrees of freedom, such as translations and polydispersity, that are present in a system of disconnected surfaces but absent in a system of infinite surfaces such as a lamellar or bicontinuous phase. The combination of undulation and collective mode free energy contributions found here is shown to yield thermodynamic properties (e.g., size distributions) for a vesicle phase whose parameter dependence is consistent with the scale invariance of the Helfrich bending energy. We briefly discuss the conceptually related problem of a polydisperse ensemble of fluctuating one-dimensional aggregates, such as rodlike micelles, for which we obtain the experimentally observed scaling of micelle size with concentration. We conclude by discussing the application of our results to the interpretation of recent experiments on equilibrium vesicle phases. (c) 1995 The American Physical Society

Electrostatic interactions in two-component membranes

The effect of electrostatic interactions on membrane undulations is examined. We consider a mixed membrane consisting of neutral and charged amphiphiles in an aqueous solution. In general, this system may exhibit a non-uniform surface charge density. Two limiting cases are distinguished: quenched surface-charge distribution; and annealed inplane distribution which is coupled to an undulation mode. For the former case the electrostatic free energy of an undulating membrane is calculated in the limit of strong ionic solution. Spatial modulations of the surface charge density are found to induce a local spontaneous curvature in a single mono layer. In addition, the electrostatic contribution to the elastic bending modulus is obtained for a general non-uniform surface-charge density. In the latter case of a self-adjusting (annealed) membrane, an electrostatic coupling between the inplane distribution and the membrane curvature produces an added effective interaction which stabilizes modulated phases

Scaling behavior of self-avoiding tethered vesicles.

The scaling behavior of self-avoiding tethered vesicles is analysed using shell theory and scaling arguments. For closed networks, there is a linear coupling between the out-of-plane undulation modes and the in-plane phonon modes that causes a strong suppression of out-of-plane fluctuations at iong length scales. This leads to new scaling behavior, which has important consequences for the analysis of experimental and simulation data. Molecular-dynamics simulation data for tethered vesicles are also presented and analysed using these results. We find that the exponent η, which describes the scale dependence of the bending rigidity, has the value η=0.81±0.03

Dynamics of flat membranes and flickering in red blood cells

A theory of the dynamics of polymerized membranes in the flat phase is presented. We discuss the renormalization of the kinetic coefficients for the undulation and phonon modes due to hydrodynamic “backflow”. The long time behavior of the dynamic structure factor is given by stretched exponentials. We also study the dynamics of the thickness fluctuations in red blood cells (flicker phenomenon) taking into account the underlying polymerized spectrin skeleton. Qualitatively different dynamical behavior is predicted for spectrin skeletons isolated from their natural lipid environment.