Spectral density functions, molecular dynamics

For folded proteins, relaxation data are commonly interpreted within the framework of the model-free formalism, in which the dynamics are described by an overall rotational correlation time rm, an internal correlation time xe, and an order parameter. S 2 describing the amplitude of the internal motions (Lipari and Szabo, 1982a,b). Model-free analysis is popular because it describes molecular motions in terms of a set of intuitive physical parameters. However, the underlying assumptions of model-free analysis—that the molecule tumbles with a single isotropic correlation time and that internal motions are very much faster than overall tumbling—are of questionable validity for unfolded or partly folded proteins. Nevertheless, qualitative insights into the dynamics of unfolded states can be obtained by model-free analysis (Alexandrescu and Shortle, 1994 Buck etal., 1996 Farrow etal., 1995a). An extension of the model-free analysis to incorporate a spectral density function that assumes a distribution of correlation times on the nanosecond time scale has recently been reported (Buevich et al., 2001 Buevich and Baum, 1999) and better fits the experimental 15N relaxation data for an unfolded protein than does the conventional model-free approach. 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

In addition to corrections tising an appropriate spectral density function, in principle one also needs to consider an ensemble of structures. Bonvin et al. U993) used an ensemble iterative relaxation matrix approach in which the NOE is measured as an ensemble property. A relaxation matrix is built from an ensemble of structures, using averaging of contributions from different structures. The needed order parameters for fast motions were obtained fi um a 50-ps molecular dynamics calculation. The relaxation matrix is then used to refine individual structures. The new structures are used again to reconstruct the relaxation matrix, and a second new set of structures is defined. One repeats the process until the ensemble of structures is converged. The caveat espressed earlier that the accuracy of the result is limited by the accuracy of the spectral density function applies to all calculations of this typ . 

The relaxation dynamics of junctions in polymer networks have not been well known until the advent of solid-state P NMR spin-lattice relaxation measurements in a series of poly(tetrahydrofuran) networks with tris(4-isocyanatophenyl)-thiophosphate junctions (Shi et al., 1993). The junction relaxation properties were studied in networks with molecular weights between crosslinks, Me, ranging from 250 to 2900. The dominant mechanism for P nuclear spin lattice relaxation times measured over a wide range of temperatures were fit satisfactorily by spectral density functions, 7([Pg.225]

NMR relaxation and its field dependence are a very important source of experimental information on dynamics of molecular motions. This information is conveyed through spectral density functions, which in turn are related to time-correlation functions (TCFs), fundamental quantities in the theory of liquid state. In most cases, characterizing the molecular dynamics through NMR relaxation studies requires the identification of the relaxation mechanism (for example the dipole-dipole interaction between a pair of spins) and models for the spectral densities/correlation functions." During the period covered by this review, such model development was concerned with both small molecules and large molecules of biological interest, mainly proteins. 

Relaxation measurements provide another way to study dynamical processes over a large dynamic range in both thermotropic and lyotropic liquid crystals (see Sec. 2.6 of Chap. Ill of Vol. 2A). The two basic relaxation times of a spin system are the spin-lattice or longitudinal relaxation time 7] and the spin-spin or transverse relaxation time T2. A detailed description, however, requires a more precise definition of the relaxation times. For spin 7=1, for instance, two types of spin-lattice relaxation must be distinguished, related to the relaxation of Zeeman and quadrupolar order with rates 7j"2 and Jfg. The relaxation rates depend on spectral density functions which describe the spectrum of fluctuating fields due to molecular motions. A detailed discussion of spin relaxation is beyond the scope of this... 

Specific models for internal motions can be used to interpret heteronuclear relaxation, such as restricted diffusion and site-jump models. However, model-free formal methods are preferable, at least for the initial analysis, since available experimental data generally are insufficient to completely characterize complex internal motions or to uniquely determine a specific motional model. The model-free approach of Lipari and Szabo for the analysis of relaxation data has been used for proteins and even for peptides. It attempts to reproduce relaxation rates by a weighted product of spectral density functions with different correlation times The weighting factors are identified as order parameters for the molecular rotational correlation time and optional further local correlation times r. The term (1-S ) would then be proportional to the amplitude of the corresponding internal motion. However, the Lipari-Szabo approach is based on the assumption that molecular and local correlation times are not coupled, i.e. they should be distinct enough (e.g. differing by at least a factor of 10 in time) to allow for this separation. However, in small molecules the rates of these different processes are of the same order of magnitude, and the requirements of the Lipari-Szabo approach may not be fulfilled. Molecular dynamics simulation provide a complementary approach for the interpretation of relaxation measurements. 

As seen from the above theoretical developments, accessing geometrical (and stereochemical) information implies at least an estimation of the dynamical part of the various relaxation parameters. The latter is represented by spectral densities which rest on the calculation of the Fourier transform of auto- or cross-correlation functions. These calculations require necessarily a model for describing molecular reorientation... 

This approach yields spectral densities. Although it does not require assumptions about the correlation function and therefore is not subjected to the limitations intrinsic to the model-free approach, obtaining information about protein dynamics by this method is no more straightforward, because it involves a similar problem of the physical (protein-relevant) interpretation of the information encoded in the form of SD, and is complicated by the lack of separation of overall and local motions. To characterize protein dynamics in terms of more palpable parameters, the spectral densities will then have to be analyzed in terms of model-free parameters or specific motional models derived e.g. from molecular dynamics simulations. The SD method can be extremely helpful in situations when no assumption about correlation function of the overall motion can be made (e.g. protein interaction and association, anisotropic overall motion, etc. see e.g. Ref. [39] or, for the determination of the 15N CSA tensor from relaxation data, Ref. [27]). 

The above experimental developments represent powerful tools for the exploration of molecular structure and dynamics complementary to other techniques. However, as is often the case for spectroscopic techniques, only interactions with effective and reliable computational models allow interpretation in structural and dynamical terms. The tools needed by EPR spectroscopists are from the world of quantum mechanics (QM), as far as the parameters of the spin Hamiltonian are concerned, and from the world of molecular dynamics (MD) and statistical thermodynamics for the simulation of spectral line shapes. The introduction of methods rooted into the Density Functional Theory (DFT) represents a turning point for the calculations of spin-dependent properties [7],... 

Investigating molecular dynamics using NMR, in contrast to DS and LS, involves the application of several conceptually quite different techniques. For example, in spin-lattice relaxation studies one is concerned with familiar time correlation functions that are probed as spectral density point by point (Section II.D.2). In the case of line-shape analysis, usually a two-pulse echo sequence is applied, and the... 

Here C is the gas phase (uncoupled) flux autocorrelation function, Zbath is the bath partition function, J(co) is the bath spectral density (computed as described above from a classical molecular dynamics computation), Bi and B2 are combinations of trigonometric functions of the frequency a> and the inverse barrier frequency, and Anally ... 

An alternative way to obtain the spectral density is by numerical simulation. It is possible, at least in principle, to include the intramolecular modes in this case, although it is rarely done [198]. A standard approach [33-36,41] utilizes molecular dynamics (MD) trajectories to compute the classical real time correlation function of the reaction coordinate from which the spectral density is calculated by the cosine transformation [classical limit of Eq. (9.3)]. The correspondence between the quantum and the classical densities of states via J(co) is a key for the evaluation of the quantum rate constant, that is, one can use the quantum expression for /Cj2 with the classically computed J(co). This is true only for a purely harmonic system [199]. Real solvent modes are anharmonic, although the response may well be linear. The spectral density of the harmonic system is temperature independent. For real nonlinear systems, J co) can strongly depend on temperature [200]. Thus, in a classical simulation one cannot assess equilibrium quantum populations correctly, which may result in serious errors in the computed high-frequency part of the spectrum. Song and Marcus [37] compared the results of several simulations for water available at that time in the literature [34,201] with experimental data [190]. The comparison was not in favor of those simulations. In particular, they failed to predict... 

The dynamics of different modes of molecular librations (hindered rotations) and intramolecular vibrations in supercritical water can now be analyzed in terms of velocity autocorrelation functions for the corresponding projections (Eqns. 22-27) (Kalinichev and Heinzinger 1992, 1995 Kalinichev 1993). The velocity autocorrelation functions calculated for the quantities Qi (Eqns. 25-27) are shown in Figure 19 for two extreme cases of high-density and low-density supercritical water. The Fourier transforms of these functions result in the spectral densities of the corresponding vibrational modes. They are shown in Figure 20 for the supercritical thermodynamic states listed in Table 5. 

It should be noted that the JmL ( ) re quantities that are obtained from experiment without reference to any molecular dynamics model. When measured as a function of temperature and frequency, these spectral densities of motion provide the best test of motional models for liquid crystals. 

Another approach to the calculation of IR spectra of hydrogen-bonded complexes is based on linear response theory, in which the spectral density is the Fourier transform of the autocorrelation function of the dipole moment operator involved in the IR transition [62,63]. Recently Car-Parrinello molecular dynamics (CPMD) [73] has been used to simulate IR spectra of hydrogen-bonded systems [64-72]. 

Abstract Photoinduced processes in extended molecular systems are often ultrafast and involve strong electron-vibration (vibronic) coupling effects which necessitate a non-perturbative treatment. In the approach presented here, high-dimensional vibrational subspaces are expressed in terms of effective modes, and hierarchical chains of such modes which sequentially resolve the dynamics as a function of time. This permits introducing systematic reduction procedures, both for discretized vibrational distributions and for continuous distributions characterized by spectral densities. In the latter case, a sequence of spectral densities is obtained from a Mori/Rubin-type continued fraction representation. The approach is suitable to describe nonadiabatic processes at conical intersections, excitation energy transfer in molecular aggregates, and related transport phenomena that can be described by generalized spin-boson models. 

In the present contribution, the effective-mode decomposition of spectral densities is demonstrated for the case of oligomers of poly-phenylene-vinylene (PPV) type. Here, the relevant spectral density is constructed from a classical-statistical correlation function that is in turn obtained from molecular dynamics (MD) simulations. Together with the effective-mode decomposition, this provides a practicable procedure for characterizing and reducing spectral densities within the high-temperature limit. 

---