Low-frequency Normal Modes Research Articles

Estimation of frequencies and Q-factors of Earth’s low-frequency toroidal normal modes using strain data

In the study, estimations of the parameters of three low-frequency toroidal modes 0T2, 0T3, and 0T4, excited by the 13 largest earthquakes of the 21st century, were carried out. The etimations were obtained using the data from the Baksan Laser Interferometer–Strainmeter of SAI MSU. Using an adaptive spectral algorithm, estimates of the modes’ frequencies were obtained, and they were compared with the PREM model. Significant deviations in the frequency estimates of the 0T4 mode for a series of earthquakes were found, showing their possible connection with local inhomogeneities in the values of the speed of shear waves and the Earth’s density in the upper mantle. The splitting of the 0T2 mode due to Earth’s ellipticity and rotation was studied, with frequency estimates of singlets excited after catastrophic earthquakes in Sumatra, Japan, and Chile obtained. An evaluation of the Q-factors of the low-frequency toroidal modes was performed using three different methods.

Deciphering the Influence of Ground-State Distributions on the Calculation of Photolysis Observables.

Nonadiabatic molecular dynamics offers a powerful tool for studying the photochemistry of molecular systems. Key to any nonadiabatic molecular dynamics simulation is the definition of its initial conditions (ICs), ideally representing the initial molecular quantum state of the system of interest. In this work, we provide a detailed analysis of how ICs may influence the calculation of experimental observables by focusing on the photochemistry of methylhydroperoxide (MHP), the simplest and most abundant organic peroxide in our atmosphere. We investigate the outcome of trajectory surface hopping simulations for distinct sets of ICs sampled from different approximate quantum distributions, namely harmonic Wigner functions and ab initio molecular dynamics using a quantum thermostat (QT). Calculating photoabsorption cross-sections, quantum yields, and translational kinetic energy maps from the results of these simulations reveals the significant effect of the ICs, in particular when low-frequency (∼ a few hundred cm-1) normal modes are connected to the photophysics of the molecule. Overall, our results indicate that sampling ICs from ab initio molecular dynamics using a QT is preferable for flexible molecules with photoactive low-frequency modes. From a photochemical perspective, our nonadiabatic dynamics simulations offer an explanation for a low-energy tail observed at high excitation energy in the translational kinetic energy map of MHP.

Effective-mode analysis of the dynamics of weakly bound molecular systems by an example of hydrogen-bonded water clusters

A method for the analysis of internal dynamics of nonlinear weakly bound polymolecular systems based on the effective-mode approach is proposed. The method enables one to estimate the number of the governing collective degrees of freedom of the system of interest at a preset accuracy under particular conditions and analyze the character of the modes depending on the activation energy of the system and the duration of its dynamic propagation, which provides qualitative and quantitative information about the coupling of diverse motions and the respective energy redistribution. The method is applied to the analysis of the dynamics of small water clusters stabilized by hydrogen bonds, which are unique spectacular examples of the systems with a pronounced coupling between the intramolecular and substantially delocalized intermolecular oscillations. The dynamic trajectories were generated in the adiabatic approximation at the Born-Oppenheimer level with the use of the quantum chemical description of selected clusters at the $\mathrm{MP}2/6\ensuremath{-}311++G(d,p)$ level. The initial conditions corresponded to different variants of the excitation of low-frequency normal modes, and the dynamic runs were carried out at a time step of 0.5 fs and the whole duration of 50 to 100 ps. Different prevailing characters of the cluster dynamics were identified depending on the molecular size, the total activation energy, and the mean potential-energy-increment to kinetic-energy-increment ratio, from an efficient accumulation of the excess kinetic energy on the effective modes of the cluster to the dissociation of the cluster into constituting fragments. The signs of the corresponding processes in the overlap matrices of the effective-mode vectors, kinetic-energy distribution over the modes, and the correlation between the number of the modes and the mean kinetic energy of the cluster are distinguished.

Unveiling mutation effects on the structural dynamics of the main protease from SARS-CoV-2 with hybrid simulation methods

The main protease of SARS-CoV-2 (called Mpro or 3CLpro) is essential for processing polyproteins encoded by viral RNA. Several Mpro mutations were found in SARS-CoV-2 variants, which are related to higher transmissibility, pathogenicity, and resistance to neutralization antibodies. Macromolecules adopt several favored conformations in solution depending on their structure and shape, determining their dynamics and function. In this study, we used a hybrid simulation method to generate intermediate structures along the six lowest frequency normal modes and sample the conformational space and characterize the structural dynamics and global motions of WT SARS-CoV-2 Mpro and 48 mutations, including mutations found in P.1, B.1.1.7, B.1.351, B.1.525 and B.1.429+B.1.427 variants. We tried to contribute to the elucidation of the effects of mutation in the structural dynamics of SARS-CoV-2 Mpro. A machine learning analysis was performed following the investigation regarding the influence of the K90R, P99L, P108S, and N151D mutations on the dimeric interface assembling of the SARS-CoV-2 Mpro. The parameters allowed the selection of potential structurally stable dimers, which demonstrated that some single surface aa substitutions not located at the dimeric interface (K90R, P99L, P108S, and N151D) are able to induce significant quaternary changes. Furthermore, our results demonstrated, by a Quantum Mechanics method, the influence of SARS-CoV-2 Mpro mutations on the catalytic mechanism, confirming that only one of the chains of the WT and mutant SARS-CoV-2 Mpros are prone to cleave substrates. Finally, it was also possible to identify the aa residue F140 as an important factor related to the increasing enzymatic reactivity of a significant number of SARS-CoV-2 Mpro conformations generated by the normal modes-based simulations.

Potential allosteric sites captured in glycolytic enzymes via residue-based network models: Phosphofructokinase, glyceraldehyde-3-phosphate dehydrogenase and pyruvate kinase

Likelihood of new allosteric sites for glycolytic enzymes, phosphofructokinase (PFK), glyceraldehyde-3-phosphate dehydrogenase (GADPH) and pyruvate kinase (PK) was evaluated for bacterial, parasitic and human species. Allosteric effect of a ligand binding at a site was revealed on the basis of low-frequency normal modes via Cα-harmonic residue network model. In bacterial PFK, perturbation of the proposed allosteric site outperformed the known allosteric one, producing a high amount of stabilization or reduced dynamics, on all catalytic regions. Another proposed allosteric spot at the dimer interface in parasitic PFK exhibited major stabilization effect on catalytic regions. In parasitic GADPH, the most desired allosteric response was observed upon perturbation of its tunnel region which incorporated key residues for functional regulation. Proposed allosteric site in bacterial PK produced a satisfactory allosteric response on all catalytic regions, whereas in human and parasitic PKs, a partial inhibition was observed. Residue network model based solely on contact topology identified the ‘hub residues’ with high betweenness tracing plausible allosteric communication pathways between distant functional sites. For both bacterial PFK and PK, proposed sites accommodated hub residues twice as much as the known allosteric site. Tunnel region in parasitic GADPH with the strongest allosteric effect among species, incorporated the highest number of hub residues. These results clearly suggest a one-to-one correspondence between the degree of allosteric effect and the number of hub residues in that perturbation site, which increases the likelihood of its allosteric nature.

Why are large conformational changes well described by harmonic normal modes?

Low-frequency normal modes generated by elastic network models tend to correlate strongly with large conformational changes of proteins, despite their reliance on the harmonic approximation, which is only valid in close proximity of the native structure. We consider 12 variants of the torsional network model (TNM), an elastic network model in torsion angle space, that adopt different sets of torsion angles as degrees of freedom and reproduce with similar quality the thermal fluctuations of proteins but present drastic differences in their agreement with conformational changes. We show that these differences are related to the extent of the deviations from the harmonic approximation, assessed through an anharmonic energy function whose harmonic approximation coincides with the TNM. Our results indicate that mode anharmonicity is more strongly related to its collectivity, i.e., the number of atoms displaced by the mode, than to its amplitude; low-frequency modes can remain harmonic even at large amplitudes, provided they are sufficiently collective. Finally, we assess the potential benefits of different strategies to minimize the impact of anharmonicity. The reduction of the number of degrees of freedom or their regularization by a torsional harmonic potential significantly improves the collectivity and harmonicity of normal modes and the agreement with conformational changes. In contrast, the correction of normal mode frequencies to partially account for anharmonicity does not yield substantial benefits. The TNM program is freely available at https://github.com/ugobas/tnm.

Tropical Rainfall Variability Accompanying Global Normal Mode Oscillations

AbstractUsing global precipitation datasets (GSMaP, TRMM) and the latest reanalysis data (ERA5), we performed a comprehensive analysis of the tropical rainfall variability that accompanies global-scale, low-frequency normal modes: Rossby, Rossby–gravity, and Kelvin modes. Cross-spectral analysis and lag-regression analysis both showed that coherent rainfall variations accompany not only the wavenumber-1 gravest Rossby mode (“5-day” wave) but other low-frequency modes. The normal mode rainfall variations are enhanced in regions such as the Amazon basin, but also include circumglobally traveling structures with substantial amplitude over the open ocean. These results are remarkably consistent among the three datasets including even ERA5 rainfall data. The circumglobal rainfall signals may be considered primarily as a response to the normal mode dynamical variations. We found that the phase relationship between rainfall and dynamical field variability is strongly dependent on the type of mode and even on the zonal wavenumber. We suggest that this is explained by the difference in relative importance of two underlying processes: 1) moisture-flux convergence and 2) rainfall enhancement associated with adiabatic cooling. Our determined rainfall signals are the response to quasi-monochromatic, periodic waves that have a simple vertical structure and represent one special case of tropospheric wave–rainfall coupling. Implications for the mechanism of 12-h rainfall oscillations believed to be forced by the atmospheric tide are also considered.

Eigenstates of quasi-Keplerian self-gravitating particle discs

ABSTRACT Although quasi-Keplerian discs are among the most common astrophysical structures, computation of secular angular momentum transport within them routinely presents a considerable practical challenge. In this work, we investigate the secular small-inclination dynamics of a razor-thin particle disc as the continuum limit of a discrete Lagrange–Laplace secular perturbative theory and explore the analogy between the ensuing secular evolution – including non-local couplings of self-gravitating discs – and quantum mechanics. We find the ‘quantum’ Hamiltonian that describes the time evolution of the system and demonstrate the existence of a conserved inner product. The lowest-frequency normal modes are numerically approximated by performing a Wick rotation on the equations of motion. These modes are used to quantify the accuracy of a much simpler local-coupling model, revealing that it predicts the shape of the normal modes to a high degree of accuracy, especially in narrow annuli, even though it fails to predict their eigenfrequencies.

Conformational Changes of Protein Analyzed Based on Structural Perturbation Method

Proteins perform their biological functions through their conformational changes due to ligand-binding. Though atomistic simulations have allowed for understanding the conformational dynamics of proteins, they are computationally restrictive in revealing the conformational transition pathway of proteins. In this work, we consider an elastic network model (ENM) with introducing the structural perturbation method, which mimics the breakage or formation of native contacts during the conformational changes, for gaining insight into the conformational transition pathway of proteins. It is shown that ENM with structural perturbation method enables the characterization of the conformational transition of adenylate kinase as a model protein. In addition, the low-frequency normal modes of adenylate kinase are found to play a role in its conformational transition. Our study sheds light on ENM with structural perturbation method for studying the conformational transitions of large protein complexes.

Exploring the conformational binding mechanism of fibrinogen induced by interactions with penicillin β-lactam antibiotic drugs

Herein, we present an integrated computational and experimental study to tackle the interactions between recognized β-lactam antibiotics (cloxacillin and dicloxacillin) with the fibrinogen blood plasma protein. For this purpose, molecular docking simulation with elastic network based on collective low-frequency normal modes and perturbation response scanning maps were proposed to evaluate the conformational binding mechanism of fibrinogen under the unbound and bound states with the cited β-lactam antibiotics. Aiming to theoretically explore the hidden biochemical mechanisms and structural attributes leading failures in therapy success with β-lactam antibiotics. The computational results pointing that despite these conformational differences, both antibiotics exhibit very similar affinity-based free energies of binding as FEB (cloxacillin/E-region) = − 8.7 kcal/mol and FEB (dicloxacillin/E-region) = −7.7 kcal/mol. We theoretically suggest that the semi-synthetic incorporation of an additional halogen CL-atom in the dicloxacillin, respect to cloxacillin molecule, and its relative docking-pose orientation in the fibrinogen E-region could significantly reduce the appearance of potential fibrinolytic off-target effects usually associated to parenterally administered β-lactam antibiotics. Besides, the performed interactions diagrams revealed that the most relevant antibiotic binding interactions with the fibrinogen E-region (pocket 1) are mainly based on hydrophobic (C···C)-backbone-side-chain non-covalent interactions, acceptor/donor interactions with critical regulatory E-region residues SER50:Q > SER50:N associated to allosteric modulation based long-distance-based perturbations (dicloxacillin > > cloxacillin) in the E-region (Q-chain > N-chain) with remarkable conformational rigidification by decreasing the intrinsic collectivity, and leading different pattern of perturbations as allosteric signal propagation in the intrinsic conformational dynamics under bound state from both β-lactam antibiotics. An experimental validation was carried out by using calorimetric (ITC and DSC) and spectroscopic (Raman and fluorescence) methods. These methods corroborated the computational results, adding quantitative information to explain the binding process. Finally, the obtained results open new perspectives for the “de novo rational drug-design” of new derivatives of β-lactam antibiotics with high pharmacodynamic selectivity/specificity to avoid side-effects toward to achieve optimal benefit/risk rates beyond the antibiotic drug resistance phenomena, favoring the implementation of rigorous criteria for a more personalized antibiotic therapy.

Complexity of plastic instability in amorphous solids: Insights from spatiotemporal evolution of vibrational modes.

It has been accepted that low-frequency vibrational modes are causally correlated to fundamental plastic rearrangement events in amorphous solids, irrespective of the structural details. But the mode-event relationship is far from clear. In this work, we carry out case studies using atomistic simulations of a three-dimensional Cu50Zr50 model glass under athermal, quasistatic shear. We focus on the first four plastic events, and carefully trace the spatiotemporal evolution of the associated low-frequency normal modes with applied shear strain. We reveal that these low-frequency modes get highly entangled with each other, from which the critical mode emerges spontaneously to predict a shear transformation event. But the detailed emergence picture is event by event and shear-protocol dependent, even for the first plastic event. This demonstrates that the instability of a plastic event is a result of extremely complex multiple-path choice or competition, and there is a strong, elastic interaction among neighboring instability events. At last, the generality of the present findings is shown to be applicable to covalent-bonded glasses.

Evaluation of the substitution structures of two partially fluorinated cyclopentanes C5H3F7 and C5H2F8

The pure rotational spectra of the title compounds have been recorded and analyzed. For both species the spectra from all of the singly substituted carbon-13 isotopologues were observed in natural abundance. Ground state rotational constants together with the results of quantum chemical calculations have been used to investigate the ground state structures of each molecule. The substitution structures show carbon-carbon bond lengths which are clearly wrong. As a rationale for this finding it is proposed that the presence of very low frequency normal modes cause an isotopologue-dependent rovibrational interaction and, hence, differences between the ground state geometries of the parent isotopologues and those of the singly substituted carbon-13 rings. These differences preclude satisfactory substitution structural analyses.

Dynamics-function relationship in the catalytic domains of N-terminal acetyltransferases

N-terminal acetyltransferases (NATs) belong to the superfamily of acetyltransferases. They are enzymes catalysing the transfer of an acetyl group from acetyl coenzyme A to the N-terminus of polypeptide chains. N-terminal acetylation is one of the most common protein modifications. To date, not much is known on the molecular basis for the exclusive substrate specificity of NATs. All NATs share a common fold called GNAT. A characteristic of NATs is the β6β7 hairpin loop covering the active site and forming with the α1α2 loop a narrow tunnel surrounding the catalytic site in which cofactor and polypeptide meet and exchange an acetyl group.We investigated the dynamics-function relationships of all available structures of NATs covering the three domains of Life. Using an elastic network model and normal mode analysis, we found a common dynamics pattern conserved through the GNAT fold; a rigid V-shaped groove formed by the β4 and β5 strands and splitting the fold in two dynamical subdomains. Loops α1α2, β3β4 and β6β7 all show clear displacements in the low frequency normal modes. We characterized the mobility of the loops and show that even limited conformational changes of the loops along the low-frequency modes are able to significantly change the size and shape of the ligand binding sites. Based on the fact that these movements are present in most low-frequency modes, and common to all NATs, we suggest that the α1α2 and β6β7 loops may regulate ligand uptake and the release of the acetylated polypeptide.

A minimal coarse-grained model for the low-frequency normal mode analysis of icosahedral viral capsids.

The main goal of this work is the design of a coarse-grained theoretical model of minimal resolution for the study of the physical properties of icosahedral virus capsids within the linear-response regime. In this model the capsid is represented as an interacting many-body system whose composing elements are capsid subunits (capsomers), which are treated as three-dimensional rigid bodies. The total interaction potential energy is written as a sum of pairwise capsomer-capsomer interactions. Based on previous work [Gomez Llorente et al., Soft Matter, 2014, 10, 3560], a minimal and complete anisotropic binary interaction that includes a full Hessian matrix of independent force constants is proposed. In this interaction model, capsomers have rotational symmetry around an axis of order n > 2. The full coarse-grained model is applied to analyse the low-frequency normal-mode spectrum of icosahedral T = 1 capsids. The model performance is evaluated by fitting its predicted spectrum to the full-atom results for the Satellite Tobacco Necrosis Virus (STNV) capsid [Dykeman and Sankey, Phys. Rev. Lett., 2008, 100, 028101]. Two capsomer choices that are compatible with the capsid icosahedral symmetry are checked, namely pentamers (n = 5) and trimers (n = 3). Both subunit types provide fair fits, from which the magnitude of the coarse-grained force constants for a real virus is obtained. The model is able to uncover latent instabilities whose analysis is fully consistent with the current knowledge about the STNV capsid, which does not self-assemble in the absence of RNA and is thermally unstable. The straightforward generalisability of the model beyond the linear regime and its completeness make it a promising tool to theoretically interpret many experimental data such as those provided by the atomic force microscopy or even to better understand processes far from equilibrium such as the capsid self-assembly.

Can Conformational Changes of Proteins Be Represented in Torsion Angle Space? A Study with Rescaled Ridge Regression.

Torsion angles are the natural degrees of freedom of protein structures. The ability to determine torsional variations corresponding to observed changes in Cartesian coordinates is highly valuable, notably to investigate the mechanisms of functional conformational changes or to develop computational models of protein dynamics. This issue is far from trivial in practice since the impact of modifying one torsion angle strongly depends on all other angles, and the compounding effects of small variations in bond lengths and valence angles can completely disrupt a protein fold. We demonstrate that naive strategies, such as directly comparing torsion angles between structures without correcting for variations in bond lengths and valence angles or fitting torsional variations without a proper regularization scheme, fail at producing an adequate representation of conformational changes in internal coordinates. In contrast, rescaled ridge regression, a method recently introduced to regularize multidimensional regressions with correlated explanatory variables, is shown to consistently identify a minimal set of torsion angles variations that closely reproduce changes in Cartesian coordinates. This torsional representation of conformational changes is shown to be robust to the choice of experimental structures. It also provides a better agreement with theoretical models of protein dynamics than the Cartesian representation, regarding notably the predominance of low-frequency normal modes in functional motions and the presence, in predicted equilibrium dynamics, of hints of natural selection for specific functional motions. The software is available at https://github.com/ugobas/tnm .

X-ray diffraction data as a source of the vibrational free-energy contribution in polymorphic systems.

In this contribution we attempt to answer a general question: can X-ray diffraction data combined with theoretical computations be a source of information about the thermodynamic properties of a given system? Newly collected sets of high-quality multi-temperature single-crystal X-ray diffraction data and complementary periodic DFT calculations of vibrational frequencies and normal mode vectors at the Γ point on the yellow and white polymorphs of di-methyl 3,6-di-chloro-2,5-di-hydroxy-terephthalate are combined using two different approaches, aiming to obtain thermodynamic properties for the two compounds. The first approach uses low-frequency normal modes extracted from multi-temperature X-ray diffraction data (normal coordinate analysis), while the other uses DFT-calculated low-frequency normal mode in the refinement of the same data (normal mode refinement). Thermodynamic data from the literature [Yang et al. (1989), Acta Cryst. B45, 312-323] and new periodic ab initio DFT supercell calculations are used as a reference point. Both approaches tested in this work capture the most essential features of the systems: the polymorphs are enantiotropically related, with the yellow form being the thermodynamically stable system at low temperature, and the white form at higher temperatures. However, the inferred phase transition temperature varies between different approaches. Thanks to the application of unconventional methods of X-ray data refinement and analysis, it was additionally found that, in the case of the yellow polymorph, anharmonicity is an important issue. By discussing contributions from low- and high-frequency modes to the vibrational entropy and enthalpy, the importance of high-frequency modes is highlighted. The analysis shows that larger anisotropic displacement parameters are not always related to the polymorph with the higher vibrational entropy contribution.

Normal mode analysis and beyond.

Normal mode analysis provides a powerful tool in biophysical computations. Particularly, we shed light on its application to protein properties because they directly lead to biological functions. As a result of normal mode analysis, the protein motion is represented as a linear combination of mutually independent normal mode vectors. It has been widely accepted that the large amplitude motions throughout the entire protein molecule can be well described with a few low-frequency normal modes. Furthermore, it is possible to represent the effect of external perturbations, e.g., ligand binding, hydrostatic pressure, as the shifts of normal mode variables. Making use of this advantage, we are able to explore mechanical properties of proteins such as Young’s modulus and compressibility. Within thermally fluctuating protein molecules under physiological conditions, tightly packed amino acid residues interact with each other through heat and energy exchanges. Since the structure and dynamics of protein molecules are highly anisotropic, the flow of energy and heat should also be anisotropic. Based on the harmonic approximation of the heat current operator, it is possible to analyze the communication map of a protein molecule. By using this method, the energy transfer pathways of photoactive yellow protein were calculated. It turned out that these pathways are similar to those obtained via the Green-Kubo formalism with equilibrium molecular dynamics simulations, indicating that normal mode analysis captures the intrinsic nature of the transport properties of proteins.

Normal mode analysis calculation for a full-atom model with a smaller number of degrees of freedom for huge protein molecules.

The number of degrees of freedom (DOF), N, in normal mode analysis (NMA) calculations of proteins is a crucial problem in huge systems because the eigenvalue problem of an N-by-N matrix must be solved. If it were possible to perform the analysis with a smaller number of DOF for the same system with minimal deterioration in accuracy, this would make a significant impact on the computational study of protein dynamics. We examined two models in which the number of DOF was reduced. Both of them adopted a full-atom model with dihedral angles as independent variables. In one model, side-chain dihedral angles, χ’s, and a main-chain dihedral angle, ω, were fixed and only the main-chain dihedral angles, ϕ and ψ, were variable. In another model, the dihedral angles around virtual bonds that connect neighboring Cα atoms were tested. The number of DOFs for the two models was two and one per residue, respectively. The residue-by-residue fluctuation profiles for atoms and dihedral angles were well reproduced in both models. The motion of atoms in the individual lowest-frequency normal modes of the two models was also very similar to those of the original model in which all rotatable dihedral angles were variable. Consequently, these models could predict large-amplitude concerted motion. These results also imply that proteins in a full-atom model can undergo only limited large-scale conformational changes around the native conformation, and consequently, NMA results do not strongly depend on the independent variables adopted.

Dynamic properties of oligomers that characterize low-frequency normal modes.

Dynamics of oligomeric proteins (one trimer, two tetramers, and one hexamer) were studied by elastic network model-based normal mode analysis to characterize their large-scale concerted motions. First, the oligomer motions were simplified by considering rigid-body motions of individual subunits. The subunit motions were resolved into three components in a cylindrical coordinate system: radial, tangential, and axial ones. Single component is dominant in certain normal modes. However, more than one component is mixed in others. The subunits move symmetrically in certain normal modes and as a standing wave with several wave nodes in others. Secondly, special attention was paid to atoms on inter-subunit interfaces. Their displacement vectors were decomposed into intra-subunit deformative (internal) and rigid-body (external) motions in individual subunits. The fact that most of the cosines of the internal and external motion vectors were negative for the atoms on the inter-subunit interfaces, indicated their opposing movements. Finally, a structural network of residues defined for each normal mode was investigated; the network was constructed by connecting two residues in contact and moving coherently. The centrality measure “betweenness” of each residue was calculated for the networks. Several residues with significantly high betweenness were observed on the inter-subunit interfaces. The results indicate that these residues are responsible for oligomer dynamics. It was also observed that amino acid residues with significantly high betweenness were more conservative. This supports that the betweenness is an effective characteristic for identifying an important residue in protein dynamics.

Testing performances of the optimal sequence estimation and autoregressive method in the frequency domain for estimating eigenfrequencies and zonal structure coefficients of low-frequency normal modes

Testing performances of the optimal sequence estimation and autoregressive method in the frequency domain for estimating eigenfrequencies and zonal structure coefficients of low-frequency normal modes