Normal Mode Space Research Articles

Scaling properties of q-breathers in nonlinear acoustic lattices

Recently $q$-breathers - time-periodic solutions which localize in the space of normal modes and maximize the energy density for some mode vector $q_0$ - were obtained for finite nonlinear lattices. We scale these solutions together with the size of the system to arbitrarily large lattices. We generalize previously obtained analytical estimates of the localization length of $q$-breathers. The first finding is that the degree of localization depends only on intensive quantities and is size independent. Secondly a critical wave vector $k_m$ is identified, which depends on one effective nonlinearity parameter. $q$-breathers minimize the localization length at $k_0=k_m$ and completely delocalize in the limit $k_0 \to 0$.

Q-Breathers in Finite Two- and Three-Dimensional Nonlinear Acoustic Lattices

In their celebrated experiment, Fermi, Pasta, and Ulam (FPU) [Los Alamos Report No. LA-1940, 1955] observed that in simple one-dimensional nonlinear atomic chains the energy must not always be equally shared among the modes. Recently, it was shown that exact and stable time-periodic orbits, coined q-breathers (QBs), localize the mode energy in normal mode space in an exponential way, and account for many aspects of the FPU problem. Here we take the problem into more physically important cases of two- and three-dimensional acoustic lattices to find existence and principally different features of QBs. By use of perturbation theory and numerical calculations we obtain that the localization and stability of QBs are enhanced with increasing system size in higher lattice dimensions opposite to their one-dimensional analogues.

Q-Breathers and the Fermi-Pasta-Ulam Problem

The Fermi-Pasta-Ulam (FPU) paradox consists of the non-equipartition of energy among normal modes of a weakly anharmonic atomic chain model. In the harmonic limit each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q. We continue normal modes from the harmonic limit into the FPU parameter regime and obtain persistence of these periodic orbits, termed here q-breathers (QB). They are characterized by time periodicity, exponential localization in the q-space of normal modes and linear stability up to a size-dependent threshold amplitude. Trajectories computed in the original FPU setting are perturbations around these exact QB solutions. The QB concept is applicable to other nonlinear lattices as well.

Normal mode based flexible fitting of high-resolution structure into low-resolution experimental data from cryo-EM

A new method for the flexible fitting of high-resolution structures into low-resolution maps of macromolecular complexes from electron microscopy has been recently described in applications to simulated electron density maps. This method uses a linear combination of low-frequency normal modes in an iterative manner to deform the structure optimally to conform to the low-resolution electron density map. Gradient-following techniques in the coordinate space of collective normal modes are used to optimize the overall correlation coefficient between computed and measured electron densities. With this approach, multi-scale flexible fitting can be performed using all-atoms or Cα atoms. In this paper, illustrative studies of normal mode based flexible fitting to experimental cryo-EM maps are presented for three different systems. Large, functionally relevant conformational changes for elongation factor G bound to the ribosome, Escherichia coli RNA polymerase and cowpea chlorotic mottle virus are elucidated as the result of the application of NMFF from high-resolution structures to cryo-electron microscopy maps.

Protein Motions Represented in Moving Normal Mode Coordinates

The molecular dynamics trajectory of a protein molecule is frequently characterized by the strongly anharmonic dynamics on the rugged potential surface. We herein propose a model, moving normal mode coordinates, to describe the anharmonic protein motions recorded in the trajectory, particularly those occurring in the low-frequency or large-amplitude normal mode space. A set of normal mode coordinates is defined at each time instant of the trajectory by principal component analysis (PCA) with a small time window. The time evolution of the normal mode coordinates defines the moving normal mode coordinates, the axes of which are translated and rotated time-dependently along the trajectory. Because the harmonic part of the dynamics is accounted for by the PCA in a quasi-harmonic manner, the motions of the coordinate system are expected to represent the anharmonic part of the protein dynamics. We applied this model to the analyses of molecular dynamics trajectories of a small protein, myoglobin. Translation of...

Some theoretical considerations on predictability of linear stochastic dynamics

Predictability is a measure of prediction error relative to observed variability and so depends on both the physical and prediction systems. Here predictability is investigated for climate phenomena described by linear stochastic dynamics and prediction systems with perfect initial conditions and perfect linear prediction dynamics. Predictability is quantified using the predictive information matrix constructed from the prediction error and climatological covariances. Predictability measures defined using the eigenvalues of the predictive information matrix are invariant under linear state-variable transformations and for univariate systems reduce to functions of the ratio of prediction error and climatological variances. The predictability of linear stochastic dynamics is shown to be minimized for stochastic forcing that is uncorrelated in normal-mode space. This minimum predictability depends only on the eigenvalues of the dynamics, and is a lower bound for the predictability of the system with arbitrary stochastic forcing. Issues related to upper bounds for predictability are explored in a simple theoretical example.

Singular Vectors, Finite-Time Normal Modes, and Error Growth during Blocking

The evolution of finite-time singular vectors growing on four-dimensional space–time basic states is studied for cases of block development over the Gulf of Alaska and over the North Atlantic, using a two-level tangent linear model. The initial singular vectors depend quite sensitively on the choice of norm with the streamfunction norm characterized by small-scale baroclinic disturbances, the kinetic energy norm giving intermediate-scale baroclinic disturbances, and the enstophy norm typified by large-scale disturbances with large zonal flow contributions. In all cases, the final evolved singular vectors consist of large-scale equivalent barotropic wave trains across the respective blocking regions. There are close similarities between the evolved singular vectors in each of the norms, particularly for the longer time periods considered, and with corresponding evolved finite-time adjoint modes and evolved maximum sensitivity perturbations. For the longer time periods considered, each of these evolved perturbations also closely resembles some of the dominant finite-time normal mode disturbances, which are norm independent. For periods between about two weeks and a month, the convergence of the evolved leading singular vector and leading finite-time normal mode toward the leading left Lyapunov vector has been examined. The evolution of errors, represented by singular vectors, is also considered in the space of finite-time normal modes. In all cases the evolved error dynamics contracts onto a low-dimensional subspace characterized by the dominant finite-time normal modes. The growth of norms based on streamfunction, kinetic energy, or enstrophy is compared with the growth of a norm based on the projection coefficients of the disturbance onto the dominant finite-time normal modes. The prospect of ensemble prediction schemes in which the control initial conditions are perturbed by superpositions of the dominant finite-time normal modes is discussed.

Limit cycle prediction for ballooning strings

This paper uses the Hopf bifurcation and center manifold theorems to predict the limit cycles of ballooning strings observed in many textile manufacturing processes. The steady state and linearized solutions of the non-linear governing equations are reviewed. The non-linear dynamic equations are discretized and projected to a finite-dimensional linear normal mode space, resulting in a set of non-linearly coupled but linearly uncoupled ordinary differential equations. The first two equations, corresponding to the lowest normal mode, are analyzed using the Hopf bifurcation theorem. The first Lyapunov coefficient is calculated to prove the existence of stable limit cycles for double-loop balloons with small string length. Analysis of the first two modes using the center manifold method verifies the effectiveness of the two-dimensional approximation. The bifurcation theorem, however, fails to apply to the large string length Hopf bifurcation point because the first Lyapunov coefficient is indeterminable. Numerical simulation of the non-linear two-dimensional equations agrees with experimental results quantitatively for small string length and qualitatively for large string length.

Four-nuclear mixed-valence vanadium clusters: divergence of phenomenological and molecular orbital models

Abstract The electronic structure of four-nuclear vanadium cluster representing the well-known vanadyl pyrophosphate catalytic system is studied on the basis of model Hamiltonian and semi-empirical molecular orbital approaches. The possible charge redistribution resulting from the vibronic interaction is analyzed for different mixed-valence combinations of V V and V IV . The adiabatic surfaces are calculated in the space of normal modes built from the VO vibrations of vanadyl groups. It is shown that in some cases the two approaches may lead to different localization patterns of excess electron(s). The molecular orbital model is a more adequate way to describe the structural features of real compounds, whereas the phenomenological model allows to analyze the qualitative influence of the different electronic interactions.

Rate Theory and Quantum Energy Flow in Molecules: Modeling the Effects of Anisotropic Diffusion and of Dephasing

We examine the effect on energy flow and unimolecular dissociation of anisotropic diffusion in quantum number space and dephasing, using a scaling approach to describe the energy flow and localization. This approach can be applied to systems with local couplings in the quantum number space of approximate normal modes. We find that anisotropy in the coefficient of diffusion on the constant energy surface can combine with anistropy in the finite extent of state space to produce results which mimic those of a lower dimensional isotropic system, due to the early saturation of rapidly relaxing modes. We discuss the diffusion which is caused by dephasing, for both delocalized and localized systems. The results for the energy flow dynamics are used to find corrections to the RRKM reaction rate caused by returns to the region of reactive states. We find that returns are enhanced near the transition to localized eigenstates, causing a greatly diminished average reaction rate. However, we also find that dephasing can cause the average reaction rate to increase over the value for an isolated system.

Analyzing the normal mode dynamics of macromolecules by the component synthesis method.

This paper presents a general method for studying the harmonic dynamics of large biomolecules and molecular complexes. The performance and accuracy of the method applied to a number of molecules are also reported. The basic approach of the method is to divide a macromolecule into a number of smaller components. The local normal modes of the components are first calculated by treating individual components and the interactions between nearest neighboring components. The physical displacements of all atoms are then represented in the local normal mode space, in which a selected range of high-frequency local modes is neglected. The equation of motion of the molecule in the local normal mode space will then have a smaller dimension, and consequently the normal modes of the whole structure, particularly for large molecules, can be solved much more easily. The normal modes of two polypeptides--(Ala)6 and (Ala)12--and a double-helical DNA--d(ATATA).d(TATAT)--are analyzed with this method. Reductions on the dimensions of harmonic dynamic equations for these molecules have been made, with the fraction of the deleted high-frequency modes ranging from 1/2 to 5/6. The calculated low-frequency normal modes are found to be very accurate as compared to the exact solutions by standard procedure. The major advantage of the present approach on macromolecule harmonic dynamics is that the reduction on the dimensionality of the eigenvalue problems can be varied according to the size of molecules, so the method can be easily applied to large macromolecules with controlled accuracy.

Response and hub loads sensitivity analysis of a helicopter rotor

A sensitivity study of blade response and oscillatory hub loads in forward flight for a hingeless rotor with respect to structural design variables is examined. Structural design variables include nonstructural mass distribution (spanwise and chord wise), chordwise offset of center of gravity, and blade bending stiffnesses (flap, lag, and torsion). The blade is discretized into a number of beam elements, and response equations are transformed into the normal mode space. The nonlinear, periodic, steady response of the blade is calculated using a finite element in time approach. Then, the vehicle trim and blade steady response are calculated iteratively as one coupled solution using a modified Newton method. The formulation for the derivatives of blade response and hub loads is implemented, using a direct analytical approach (chain rule differentiation), and forms an integral part of the basic response analysis. For calculation of the sensitivity derivatives, a 96% reduction of CPU time is achieved by using the direct analytical approach, compared with the finite-difference approach.

Orthogonal Vertical Normal Modes for a Multilevel Model

Abstract Orthogonal vertical normal modes are found for a multilevel sigma-coordinate model, linearized about a state of rest with a nonisothermal mean temperature profile. Orthogonality permits the partitioning of energy into the vertical modes, and simplifies the application of variational techniques to normal mode initialization in the multilevel case. Improved procedures are described for the vertical transforms required during normal mode initialization. We derive relationships between the time derivatives of the energy computed in vertical normal mode space and in gold point space, and analogous relationships between the changes made by initialization to the vertical normal mode coefficients and the corresponding changes made in grid point space.