Elastic Normal Modes Research Articles

Rayleigh waves and cyclotron surface modes of gyroscopic metamaterials

We investigate the elastic normal modes of two-dimensional media with broken time-reversal and parity symmetries due to a Lorentz term. Our starting point is an elasticity theory that captures the low-energy physics of a diverse range of systems such as gyroscopic metamaterials, skyrmion lattices in thin-film chiral magnets and certain Wigner crystals. By focusing on a circular disk geometry we analyze finite-size effects and study the low-frequency shape oscillations of the disk. We demonstrate the emergence of the Rayleigh surface waves from the bottom of the excitation spectrum and investigate how the curvature of the disk-boundary modifies their propagation at long wavelengths. Moreover, we discover a near-cyclotron-frequency wave that is almost completely localized at the boundary of the disk, but is distinct from the Rayleigh wave. It can be distinguished from the latter by a characteristic excitation pattern in a small region near the center of the disk.

Optimising Elastic Network Models for Protein Dynamics and Allostery: Spatial and Modal Cut-offs and Backbone Stiffness

The family of coarse-grained models for protein dynamics known as Elastic Network Models (ENMs) require careful choice of parameters to represent well experimental measurements or fully-atomistic simulations. The most basic ENM that represents each protein residue by a node at the position of its C-alpha atom, all connected by springs of equal stiffness, up to a cut-off in distance. Even at this level a choice is required of the optimum cut-off distance and the upper limit of elastic normal modes taken in any sum for physical properties, such as dynamic correlation or allosteric effects on binding. Additionally, backbone-enhanced ENM (BENM) may improve the model by allocating a higher stiffness to springs that connect along the protein backbone. This work reports on the effect of varying these three parameters (distance and mode cutoffs, backbone stiffness) on the dynamical structure of three proteins, Catabolite Activator Protein (CAP), Glutathione S-transferase (GST), and the SARS-CoV-2 Main Protease (M pro ). Our main results are: (1) balancing B-factor and dispersion-relation predictions, a near-universal optimal value of 8.5 Å is advisable for ENMs; (2) inhomogeneity in elasticity brings the first mode containing spatial structure not well-resolved by the ENM typically within the first 20; (3) the BENM only affects modes in the upper third of the distribution, and, additionally to the ENM, is only able to model the dispersion curve better in this vicinity; (4) BENM does not typically affect fluctuation-allostery, which also requires careful treatment of the effector binding to the host protein to capture.

Using resonance to measure the independent elastic constants of problematic materials such as elastomers

The determination of the elastic constants for a solid material may be conveniently accomplished by measuring resonances corresponding to the elastic normal modes of a sample. Isotropic materials should be particularly straightforward, since they involve only two independent elastic constants, the shear modulus and the longitudinal modulus. In practice one typically measures the shear modulus and Young's modulus, but for some problematic materials these are not truly independent. Probing the microscopic processes of solid mechanical behavior requires knowledge of independent moduli. In this paper, a resonance method for directly measuring the independent moduli for problematic materials is described.

Dynamic Transmission of Protein Allostery without Structural Change: Spatial Pathways or Global Modes?

We examine the contrast between mechanisms for allosteric signaling that involve structural change, and those that do not, from the perspective of allosteric pathways. In particular we treat in detail the case of fluctuation-allostery by which amplitude modulation of the thermal fluctuations of the elastic normal modes conveys the allosteric signal, and address the question of what an allosteric pathway means in this case. We find that a perturbation theory of thermal elastic solids and nonperturbative approach (by super-coarse-graining elasticity into internal bending modes) have opposite signatures in their structure of correlated pathways. We illustrate the effect from analysis of previous results from GlxR of Corynebacterium glutamicum, an example of the CRP/FNR transcription family of allosteric homodimers. We find that the visibility of both correlated pathways and disconnected sites of correlated motion in this protein suggests that mechanisms of local elastic stretch and bend are recruited for the purpose of creating and controlling allosteric cooperativity.

Mechanical Resonance Dispersion Revisited

Mechanical resonance dispersion is the inelastic response of a solid to a periodic shear stress. Instead of the elastic Young's Modulus, the phenomenon is described by both a real J', and an imaginary J'' component of complex shear compliance, corresponding to in phase and out of phase strain responses, respectively. The experimental results are plots of J' and J'' vs. frequency, which are typically in the audiofrequency range of 10 - 5600 Hz. Resonances are observed as maxima in J'' and inversions in J' at frequencies corresponding to modes of plastic deformation, which are much lower frequencies (audiofrequency range) than elastic normal modes. The theoretical explanation of Edwin R. Fitzgerald involves particle waves and momentum transfer and leads to a particle-in-a-box frequency formula for these inelastic modes. Unfortunately, most of his and other published raw data were never analyzed by this model. The purpose of this article is to apply this formula to previously uninterpreted resonance dispersion curves and to address some of the earlier criticism of Fitzgerald's work. Results of these calculations support the Fitzgerald Theory to a high degree, demonstrate the importance of impurities and chemical analysis, largely mollify previous criticisms, and suggest the possibility of a new particle wave mass spectroscopy at great distances.

Normal Modes of Prion Proteins: From Native to Infectious Particle

Prion proteins (PrP) are the infectious agent in transmissible spongiform encephalopathies (i.e., mad cow disease). To be infectious, prion proteins must undergo a conformational change involving a decrease in α-helical content along with an increase in β-strand content. This conformational change was evaluated by means of elastic normal modes. Elastic normal modes show a diminution of two α-helices by one and two residues, as well as an extension of two β-strands by three residues each, which could instigate the conformational change. The conformational change occurs in a region that is compatible with immunological studies, and it is observed more frequently in mutant prions that are prone to conversion than in wild-type prions because of differences in their starting structures, which are amplified through normal modes. These findings are valuable for our comprehension of the conversion mechanism associated with the conformational change in prion proteins.

Flexibility of beta-sheets: principal component analysis of database protein structures.

Protein folds are built primarily from the packing together of two types of structures: alpha-helices and beta-sheets. Neither structure is rigid, and the flexibility of helices and sheets is often important in determining the final fold (e.g., coiled coils and beta-barrels). Recent work has quantified the flexibility of alpha-helices using a principal component analysis (PCA) of database helical structures (J. Mol. Bio. 2003, 327, pp. 229-237). Here, we extend the analysis to beta-sheet flexibility using PCA on a database of beta-sheet structures. For sheets of varying dimension and geometry, we find two dominant modes of flexibility: twist and bend. The distributions of amplitudes for these modes are found to be Gaussian and independent, suggesting that the PCA twist and bend modes can be identified as the soft elastic normal modes of sheets. We consider the scaling of mode eigenvalues with sheet size and find that parallel beta-sheets are more rigid than antiparallel sheets over the entire range studied. Finally, we discuss the application of our PCA results to modeling and design of beta-sheet proteins.

Exploring Global Distortions of Biological Macromolecules and Assemblies from Low-resolution Structural Information and Elastic Network Theory

A theory of elastic normal modes is described for the exploration of global distortions of biological structures and their assemblies based upon low-resolution image data. Structural information at low resolution, e.g. from density maps measured by cryogenic electron microscopy (cryo-EM), is used to construct discrete multi-resolution models for the electron density using the techniques of vector quantization. The elastic normal modes computed based on these discretized low-resolution models are found to compare well with the normal modes obtained at atomic resolution. The quality of the normal modes describing global displacements of the molecular system is found to depend on the resolution of the synthetic EM data and the extent of reductionism in the discretized representation. However, models that reproduce the functional rearrangements of our test set of molecules are achieved for realistic values of experimental resolution. Thus large conformational changes as occur during the functioning of biological macromolecules and assemblies can be elucidated directly from low-resolution structural data through the application of elastic normal mode theory and vector quantization.

Comparative study of superconducting gravimeters and broadband seismometers STS-1 / Z in seismic and subseismic frequency bands

Superconducting gravimeters and broadband seismometers (vertical component) both measure gravity, but whereas the former are most sensitive to very long period signals (gravity tides with periods longer than 6 h), the latter are designed for recording the seismic band (elastic normal modes with periods shorter than 1 h). We investigate here the behaviour of each type of instrument in the spectral band where it is not generally used. More precisely, we compare the French superconducting gravimeter, located at Station J9 near Strasbourg, and the vertical component of an STS-1 seismometer located in a mine at Echery (ECH) in the Vosges, about 70 km away. Two different frequency bands are considered: the seismic band (frequencies between 0.2 and 1.667 mHz), for the study of normal modes after the Bolivian earthquake of 9 June 1994, and the subseismic band (frequencies lower than 0.2 mHz), including the study of gravity tides. The analysis of Fourier amplitude spectra and power spectral densities shows the obvious result that the broadband seismometer is more sensitive than the superconducting gravimeter in the seismic band because of a lower noise level, whereas the reverse is true in the subseismic band. The poorer quality of the gravimeter record in the seismic band is probably due to site effects (sediments vs. bedrock) rather than of instrumental origin. In contrast, the higher noise level of the seismometer in the subseimic band is probably due to the temperature response of the instrument. It is expected that operating the STS-1 isothermally, or recording on-site temperature changes for correction will considerably improve its signal-to-noise ratio in the subseimic band. In comparison with recent mean models of high and low seismic background noise levels, both instruments nevertheless indicate low noise levels at all frequencies.

Acoustic properties of paraelectric and incommensurate phases of monoclinic piezoelectric crystal NH4HSeO4

There in the paper are given the results of ultrasonic studies of the phase transitions succession: initial-incommensurate-commensurate polar phases in the monoclinic piezoelectric crystal NH4HSeO4 at frequency of 10 MHz. It is shown that NH4HSeO4 crystal is a strong piezoelectric. A large contribution of piezoeffect into elastic properties of crystal is revealed. In accounting the piezoeffect, the elastic moduli CEij are calculated. The anomalous behaviour of acoustic properties due to peculiarities of interaction between elastic waves and normal modes of incommensurate phase is discussed.

Scattering of elastic normal modes in a plate with a two-dimensional surface roughness distribution: Merkulov, L.G., Firsov, I.P. Soviet Physics-Acoustics, Vol 16, No 1 (July– September 1970) pp 86–90

Scattering of elastic normal modes in a plate with a two-dimensional surface roughness distribution: Merkulov, L.G., Firsov, I.P. Soviet Physics-Acoustics, Vol 16, No 1 (July– September 1970) pp 86–90

Dynamic Elasto-Plastic Response of Rigid Frames

The procedure for determining the elastic response of structures to transient loads in terms of the elastic normal modes is well known. Assuming ideal plasticity, Bleich and Salvadori recently proposed a method for obtaining the response beyond the elastic range in terms of elasto-plastic normal modes. They applied the method to free-free beams acted on by impulsive loads. In this paper, the method of Bleich and Salvadori is applied in general to two hinged and fixed rigid frames acted upon by a time dependent concentrated force at the knee and in particular to a two hinged frame (for which the elastic normal modes are known) and a linearly decaying forcing function.