Normal Mode Approximation Research Articles

Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: Comparison of POD and asymptotic nonlinear normal modes methods

The aim of the present paper is to compare two different methods available for reducing the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD), and an asymptotic approximation of the nonlinear normal modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the partial differential equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed. The response is investigated also for a fixed excitation frequency by using the excitation amplitude as bifurcation parameter for a wide range of variation. Bifurcation diagrams of Poincare maps obtained from direct time integration and calculation of the maximum Lyapunov exponent have been used to characterize the system. © 2007 Elsevier Ltd. All rights reserved.

Non-linear modal properties of non-shallow cables

A non-linear mechanical model of non-shallow linearly elastic suspended cables is employed to investigate the non-linear modal characteristics of the free planar motions. An asymptotic analysis of the equations of motion is carried out directly on the partial-differential equations overcoming the drawbacks of a discretization process. The direct asymptotic treatment delivers the approximation of the individual non-linear normal modes. General properties about the non-linearity of the in-plane modes of different type—geometric, elasto-static and elasto-dynamic—are unfolded. The spatial corrections to the considered linear mode shape caused by the quadratic geometric forces are investigated for modes belonging to the three mentioned classes. Moreover, the convergence of Galerkin reduced-order models is discussed and the influence of passive modes is highlighted.

Interpreting Correlated Motions Using Normal Mode Analysis

With the increased popularity of normal mode analyses in structural biology, it is important to carefully consider how to best utilize the results for gaining biological insights without over interpretation. The discussion in this article argues that for the purpose of identifying correlated motions in biomolecules, a case separate from concomitant conformational changes of structural motifs, it is generally important to use a large number of normal modes. This is illustrated through three increasingly complex examples. The simplest case includes two bilinearly coupled harmonic oscillators and serves as a straightforward problem where the important considerations are explicit and transparent. The argument is then generalized to include a system of N-coupled harmonic oscillators and finally to a realistic biomolecule. Although a small number of normal modes are useful for probing structural flexibility, it is clear that a much larger number of modes are required for properly investigating correlated motions in biomolecules.

Energetics and Dynamics of Constrained Actin Filament Bundling

The formation of filopodia-like bundles from a dendritic actin network has been observed to occur in vitro as a result of branching induced by Arp2/3 complex. We study both the energetics and dynamics of actin filament bundling in such a network to evaluate their relative importance in bundle formation processes. Our model considers two semiflexible actin filaments fixed at one end and free at the other, described using a normal-mode approximation. This model is studied by both Brownian dynamics and free-energy minimization methods. Remarkably, even short filaments can bundle at separations comparable to their lengths. In the dynamic simulations, we evaluate the time required for the filaments to interact and bind, and examine the dependence of this bundling time on the filament length, the distance between the filament bases, and the cross-linking energy. In most cases, bundling occurs in a second or less. Beyond a certain critical distance, we find that the bundling time increases very rapidly with increasing interfilament separation and/or decreasing filament length. For most of the cases we have studied, the energetics results for this critical distance are similar to those obtained from dynamics simulations run for 10 s, suggesting that beyond this timescale, energetics, rather than kinetic constraints, determine whether or not bundling occurs. Over a broad range of conditions, we find that the times required for bundling from a network are compatible with experimental observations.

Ab initio nonadiabatic quantum dynamics of cyclohexadiene/hexatriene ultrafast photoisomerization

Reaction mechanisms of the ultrafast photoisomerization between cyclohexadiene and hexatriene have been elucidated by the quantum dynamics on the ab initio potential energy surfaces calculated by multireference configuration interaction method. In addition to the quantum wave-packet dynamics along the two-dimensional reaction coordinates, the semiclassical analyses have also been carried out to correctly estimate the nonadiabatic transition probabilities around conical intersections in the full-dimensional space. The reaction time durations of radiationless decays in the wave-packet dynamics are found to be generally consistent with the femtosecond time-resolution experimental observations. The nonadiabatic transition probabilities among the ground (S0), first (S1), and second (S2) excited states have been estimated by using the semiclassical Zhu-Nakamura formula considering the full-dimensional wave-packet density distributions in the vicinity of conical intersections under the harmonic normal mode approximation. The cyclohexadiene (CHD) ring-opening process proceeds descending on the S1(1 1B) potential after the photoexcitation. The major part of the wave-packet decays from S1(1 1B) to S1(2 1A) by the first seam line crossing along the C2-symmetry-breaking directions. The experimentally observed ultrafast S1-S0 decay can be explained by the dynamics through the S1-S0 conical intersection along the direction toward the five-membered ring. The CHD: hexatriene (HT) branching ratio is estimated to be approximately 5:5, which is in accordance with the experiment in solution. This branching ratio is found to be mainly governed by the location of the five-membered ring S1-S0 conical intersection along the ground state potential ridge between CHD and HT.

Comparison of Mode Analyses at Different Resolutions Applied to Nucleic Acid Systems

More than two decades of different types of mode analyses has shown that these techniques can be useful in describing large-scale motions in protein systems. A number of mode analyses are available and include quasiharmonics, classical normal mode, block normal mode, and the elastic network model. Each of these methods has been validated for protein systems and this variety allows researchers to choose the technique that gives the best compromise between computational cost and the level of detail in the calculation. These same techniques have not been systematically tested for nucleic acid systems, however. Given the differences in interactions and structural features between nucleic acid and protein systems, the validity of these techniques in the protein regime cannot be directly translated into validity in the nucleic acid realm. In this work, we investigate the usefulness of the above mode analyses as applied to two RNA systems, i.e., the hammerhead ribozyme and a guanine riboswitch. We show that classical normal-mode analysis can match the magnitude and direction of residue fluctuations from the more detailed, anharmonic technique, quasiharmonic analysis of a molecular dynamics trajectory. The block normal-mode approximation is shown to hold in the nucleic acid systems studied. Only the mode analysis at the lowest level of detail, the elastic network model, produced mixed results in our calculations. We present data that suggest that the elastic network model, with the popular parameterization, is not best suited for systems that do not have a close packed structure; this observation also hints at why the elastic network model has been found to be valid for many globular protein systems. The different behaviors of block normal-mode analysis and the elastic network model, which invoke similar degrees of coarse-graining to the dynamics but use different potentials, suggest the importance of applying a heterogeneous potential function in a robust analysis of the dynamics of biomolecules, especially those that are not closely packed. In addition to these comparisons, we briefly discuss insights into the conformational space available to the hammerhead ribozyme.

Comparative estimation of vibrational entropy changes in proteins through normal modes analysis

We compare the vibrational entropy changes of proteins calculated using a full and a number of approximate normal modes analysis methods. The vibrational entropy differences for three conformational changes and three protein binding interactions were computed. In general, the approximate methods yield good estimates of the vibrational entropy change in a fraction of the time required by full normal modes analysis. The absolute entropies are either overestimated or greatly underestimated, but the difference is sufficiently accurate for some methods. This indicates that some of the approximate methods can give reasonable estimates of the associated vibrational entropy changes, making them suitable for inclusion in free energy calculations.

Diffusion-assisted long-range reaction between the ends of a polymer: Effective sink approximation

We report a Brownian dynamics (BD) simulation study of the Förster energy transfer in a dye-labeled Rouse polymer chain. The simulation method is based on the normal mode BD propagation and numerical path integration of the survival probability. It is shown that a properly constructed truncated normal-mode approximation (TNMA) can speed up the simulations considerably, without essential loss of accuracy. In particular, an effective-sink TNMA scheme is found to be quite efficient. The idea is based on a standard time scale separation ansatz, where all the normal modes are separated into slow and fast, in terms of the corresponding relaxation times. The fast normal modes are assumed to be equilibrated in the course of reaction and thus can be integrated out. Their effect is to modify the reaction sink for the slow modes. The first-order approximation can be handled most easily, without a simulation. Even this simple approximation can be preferable to the well-known Wilemski–Fixman approximation, if the reaction sink is wide, i.e., when the Förster radius exceeds the polymer mean bond length, the condition often chosen in experiments on polymer folding.

Quantum calculation of highly excited vibrational energy levels of [formula omitted] on a new empirical potential energy surface and semiclassical analysis of 1:2 Fermi resonance

We report a refined potential energy function for the ground electronic state of CS 2 based on a least-squares fitting to several low-lying experimental vibrational frequencies. Energy levels up to 20,000 cm −1 have been obtained on this empirical potential using the Lanczos algorithm and potential optimized discrete variable representation. Among them, 329 levels below 10,000 cm −1 are assigned with approximate normal mode quantum numbers (n 1, n 2 0, n 3) , based on expectation values of one-dimensional (1D) reference Hamiltonians. An effective Hamiltonian is extracted from these assigned levels. The agreement with experimental data, including those of several isotopically substituted species, is excellent. In addition, some Fermi and anharmonic resonances are analyzed. The nearest neighbor level spacing and Δ 3 distributions indicated that the vibrational spectrum of CS 2 is largely regular in the energy range up to 20,000 cm −1. Semiclassical phase space analysis, including bifurcation analysis of the spectroscopic Hamiltonian, is used to interpret subtle anomalies signaled by expectation values used in normal mode assignments. The meaning of Fermi resonance is clarified by contrasting the semiclassical analysis of CS 2 and CO 2.

Vibrational dephasing of an anharmonic solute strongly coupled to solvent

We present new results for an old model: an anharmonic solute linearly coupled to a harmonic bath. A comparison of numerical simulations of the classical mechanical absorption spectrum to the results of conventional perturbation theory and the instantaneous normal mode (INM) approximation shows that the INM performs poorly in the low-friction regime, but yields reasonable results in the high-friction regime, in which perturbation theory is inappropriate. An analytical theory for the line shape is formulated to describe the regime of intermediate friction, in which neither perturbation theory nor the INM approximation works well.

Analysis of domain motions by approximate normal mode calculations.

The identification of dynamical domains in proteins and the description of the low-frequency domain motions are one of the important applications of numerical simulation techniques. The application of these techniques to large proteins requires a substantial computational effort and therefore cannot be performed routinely, if at all. This article shows how physically motivated approximations permit the calculation of low-frequency normal modes in a few minutes on standard desktop computers. The technique is based on the observation that the low-frequency modes, which describe domain motions, are independent of force field details and can be obtained with simplified mechanical models. These models also provide a useful measure for rigidity in proteins, allowing the identification of quasi-rigid domains. The methods are validated by application to three well-studied proteins, crambin, lysozyme, and ATCase. In addition to being useful techniques for studying domain motions, the success of the approximations provides new insight into the relevance of normal mode calculations and the nature of the potential energy surface of proteins.

Hydrogen bonded clusters in the liquid phase: I. Analysis of the velocity correlation function of water triplets

We present an analysis of hydrogen bonded triplet clusters of water in terms of velocity projections on locally defined molecular coordinates. This makes it possible to perform an approximate normal mode analysis and to establish the relative motions of hydrogen bonded molecules. We find that the interpretation of the velocity autocorrelation function (VACF) of water in terms of stretching and bending motions of hydrogen bonded neighbours is an oversimplification due to cooperative effects in the hydrogen bonded network. Especially, for the peak at we find that it is not only due to O-O-O bending motions in hydrogen bonded groups but also to librational motions of the whole cluster.

High Overtones of C−H Stretching Vibrations in Isoxazole, Thiazole, and Related Methyl and Dimethyl Derivatives

The high overtone spectra of liquid isoxazole and thiazole, two five-membered heterocyclic molecules with different aromaticity characters, have been investigated in the C−H stretching region by visible absorption spectroscopy. The local-mode model describes satisfactorily the C−H stretching vibrations at the fifth and sixth quanta of excitation, where the localized character is increased. Each overtone band generally displays a number of peaks corresponding to the different types of C−H oscillators. Comparison between the overtone spectra of the parent molecules and those of the methyl derivatives provided sufficient information for a conclusive assignment of the progressions. Deconvolution of the fifth overtone bands gave further evidence regarding the shape and the width of the distinct peaks. Anharmonicity values of the overtone vibrations, C−H bond lengths, and force constants were obtained from the local-mode analysis and discussed in comparison with the results obtained from the normal-mode approximation. A correlation between overtone frequencies and chemical shifts in proton magnetic resonance, generally observed in aromatic compounds, has been found only for thiazole, whose aromaticity is responsible for a significant electron current in the ring.

Frequency and wave-vector dependent dielectric function of water: Collective modes and relaxation spectra

The longitudinal frequency and wave-vector dependent complex dielectric response function χ(k,ω)=1−1/ε(k,ω) is calculated in a broad range of k values by means of molecular dynamics computer simulation for a central force model of water. Its imaginary part, i.e., Im{ε(k,ω)}/|ε(k,ω)|2, shows two main contributions in the region of small k values: Debye-like orientational relaxation in the lower frequency part of the spectrum and a damped librational resonance at the high frequency wing. The Debye relaxation time does not follow a de Gennes-like pattern: τ(k) goes through a maximum at k≈k*≈1.7 Å−1, while the static polar structure factor S(k) peaks at k≈3 Å−1. The resonance frequency ω(k) and the decay decrement γ(k) show a dispersion law, indicative of a decaying optical-like mode, the libron. With an approximate normal mode approach, we analyze the origin of this mode on a molecular level which shows that it is due to a damped propagation of molecular orientational vibrations through the network of hydrogen bonds. At high k the decay, due to dissipation of collective into single particle motions, dominates. The static dielectric function is calculated on the basis of the response function spectra via the Kramers–Kronig relation. In the small k region ε(k) decreases from the macroscopic value ε≈80 to a value ≈15, i.e. it exhibits a Lorentzian-type behavior. This behavior is shown to be determined by higher order multipole correlation functions. In the intermediate and high k range, our results on ε(k) and χ(k) are in excellent agreement with data extracted from experimental partial pair correlation functions: ε(k) exhibits two divergence points on the k axis with a range of negative values in between where a maximum in χ(k) is found with χmax(k)≫1, indicative of overscreening. Consequences of quantum corrections to χ(k) with respect to a purely classical calculation are discussed and consequences are shown for the interaction energy between hydrated ions.

Untangling the physical contributions to instantaneous normal mode approximations: Inhomogeneous broadening, motional narrowing, and energy relaxation

An instantaneous normal mode (INM) approach to vibrational lineshapes, including motional narrowing, is presented. Simulations and calculations are carried out for a diatomic in Lennard-Jones solvent as a function of vibrational frequency, with an emphasis on determining the contributions of different physical relaxation mechanisms. The velocity correlation of a bond is easily related to a bond-weighted INM density of states, containing both resonant energy relaxation (ER) and unnarrowed inhomogeneous broadening. An effective weighted density of states or static spectrum, the distribution of an effective time-dependent frequency Ω(t), is introduced and proposed as a measure of the inhomogeneous linewidth only. It is found that the vibrational INM are in the motionally narrowed or fast modulation limit; motional narrowing of INM cannot be ignored. A dynamic spectrum containing only the motionally narrowed inhomogeneous spectrum and corresponding pure dephasing relaxation is isolated. Reintroducing energy relaxation results in excellent agreement with simulation. The validity of INM approximations and the relative importance of different relaxation mechanisms as a function of vibrational frequency is analyzed. It is suggested that, through INM, a role may be found for motional narrowing in intermolecular dynamics.

Analysis of electron correlation and vibronic coupling in the reaction center of bacterial photosynthetis

Approximate normal modes for the special pair dimer are calculated on the PM3 level. Their couplings to the lowest electronically excited states of the dimer and its cation are deduced from results based on INDO-CI calculations including electron correlation effects. Several modes are found which couple strongly to the internal charge-transfer states and are important for the internal relaxation after photoexcitation and the consecutive formation of the dimer cation.

Acoustic normal modes below 4 kHz for a rigid, closed violin-shaped cavity

Numerical calculations of standing wave patterns ≤4 kHz in a closed (no f-holes), rigid, violin-shaped cavity excited by a movable point source were used to investigate internal resonant pressure mode shapes. Approximate normal mode shapes were extracted by appropriate choice of excitation point and tested for orthogonality. Excellent agreement was found with published experimental mode shapes and frequencies below 2 kHz, and with mode frequencies up to 4 kHz when no mode shapes were available. Calculations performed without the end and corner blocks (interior outline same as exterior) from 0.4–4 kHz showed little sensitivity of mode frequency and shape to this cavity shape change. The frequency f(A1) of the lowest closed cavity mode A1 at ∼490 Hz was observed to decrease by 24 Hz when the arch was reduced to zero (flat top and back plate) with rib height held constant. On the other hand when the rib height was reduced (arch still zero), f(A1) increased, but more slowly. The effects of arch and rib height changes are consistent with experimental data from actual instruments for the A1 mode.

A study of the influence of an off-shore rise on low-frequency modal propagation with Arctic surface loss.

Simple up-slope propagation of underwater acoustic energy can usually be modeled by an adiabatic normal mode approximation, provided the slope is not too severe. Bottom interacting paths in such an environment are usually sufficiently attenuated to make consideration of mode coupling unnecessary. An environment further complicated by an off-shore rise can cause additional acoustic energy to be introduced into lower-order modes due to bottom interaction. These low-order modes will then propagate in deep water until encountering the continental shelf. Experimental results from the Arctic Ocean between 25 and 45 Hz suggest such propagation conditions. These results, and their interpretation in the context of a coupled mode study, will be presented. Environmental parameters and source depth will be varied to explore under what conditions mode coupling can be ignored when considering detection and localization problems. Arctic surface loss will be applied to the modeling results to show how mode varying attenuation will affect those results. The coupled mode model (COUPLE) [R. B. Evans, J. Acoust. Soc. Am. 74, 188–195 (1983)], is used as the vehicle for this study. [This work is sponsored by SPAWAR PMW 182-2.]

Static and dynamic vibrational dephasing in a dense fluid

We present a theory of the statically broadened vibrational line shape of a molecule in liquid solution. In this limit of static broadening, the molecule vibrates in a static potential posed by fixed solvent molecules in a configuration chosen from the equilibrium distribution of fluid configurations. The line shape is calculated within the instantaneous normal mode approximation, in which the solute’s potential is approximated by a harmonic surface whose curvature agrees with that of the exact potential at the solute’s initial configuration. Within this approximation, the line shape is related to a configuration-averaged phonon Green’s function, which is calculated approximately with an analytical procedure. This theory represents a modification of our previous treatment of vibrational line shapes [J. Chem. Phys. 102, 2326 (1995)], in which the solvent dynamics were included. Comparison of the line shapes for static and dynamic solvents permits determination of the relative importance of static (inhomogeneous) and dynamic (homogeneous) contributions to line broadening. We carry out such comparisons for a harmonic diatomic in a Lennard-Jones solvent over a wide range of temperature and density.

Matched-field processing in a range-dependent shallow water environment in the Northeast Pacific Ocean

This paper describes matched-field processing (MFP) of data collected in shallow water off the western coast of Vancouver Island in the Northeast Pacific Ocean. The data were collected from a vertical line array (VLA) as part of the PACIFIC SHELF trial carried out on the continental shelf and slope during September 1993, sensors in the 16-element VLA were evenly spaced at depths between 90 and 315 m, while the sound source was towed along radial paths or arcs. In this paper, we present results of the analysis of data from a continuous wave (CW) source which was towed downslope at a depth of 30 m in water from 150 to 375 m deep, in order to model the range-dependence of the acoustic propagation efficiently, the replica fields were calculated using the adiabatic normal mode approximation. This approximation was considered appropriate for the bottom slopes of the environment. Using sparse bathymetric data, a water sound speed profile and estimates of bottom properties, MFP correlations on individual ambiguity surfaces were found to be greater than 0.9 for the strongest signals. On account of environmental mismatch, the source position could not be determined unambiguously from most of the ambiguity surfaces even at high signal-to-noise ratios. Nevertheless, when an efficient linear tracker was applied to the ambiguity surfaces to find tracks, the source track was recovered at both low and high signal-to-noise ratios, this tracker performs the analysis at a constant depth and reports the track with the highest estimated track signal-to-noise ratio.