Figure 2 Internal RMSF of residues (average over heavy atoms) determined for human lysozyme by the X-ray normal mode refinement method applied to real X-ray data (heavy curve), in comparison with results from a normal mode analysis on a single isolated lysozyme molecule (lightweight curve). (From Ref. 33.)

ties in terms of the normal mode eigenvectors is not so natural. However, they still can be approximately expressed in terms of the variance and covariances of internuclear vectors of spin pairs, which are in turn expressed in terms of the lowest frequency normal modes. The method then proceeds in much the same way as in the X-ray case. The method was demonstrated on a solvated 25-residue zinc finger peptide on which a molecular dynamics simulation was performed. Low frequency normal modes were used to fit the order parameters for backbone CEH, NEH spin pairs calculated directly from the simulation. The predictive power of the method was tested by calculating the order parameters of spin pairs not included in the refinement. The results showed general agreement with the directly calculated quantities, although some large discrepancies were seen in individual cases. Applying the method to NOESY intensities allows one also to refine the predicted averaged structure and gain information on correlated motion in analogy to the X-ray refinement method.

C. Validity of the Concept of a Normal Mode Important Subspace

As already mentioned in Section I, normal mode analysis is based on a physical model that is quite far from reality for a biological molecule under physiological conditions. Although some studies found partial overlap between some of the lowest frequency modes and the functional mode determined from two X-ray conformers, in general it would be fanciful to expect anything more than a moderate correspondence to individual normal modes. However, the expectation that there exists a larger subspace spanned by the first M lowest frequency normal modes (where M may be between 10% and 20% of 3N) that is also spanned by the modes with the largest fluctuation in the real molecule is more realistic. If this were the case, then the low frequency normal modes would contain information on the modes of largest fluctuation in the real molecule, which would also be largely determined by barrier-crossing motions. The success of the normal mode refinement method itself is the ultimate test of the validity of this concept.

V.NORMAL MODE ANALYSIS AND REALITY

A number of studies have compared normal mode analysis predictions with results from more realistic simulation techniques or experiments. These studies shed light on the nature of the conformational energy surface and the effect of solvent.

A. The Solvent Effect

Normal mode analyses are usually performed in a single minimum using vacuum force fields. Molecular dynamics simulations of proteins in vacuum, however, reveal not one minimum but multiple minima in the energy surface. One effect of adding solvent is to increase the density of local minima over that found in vacuum [52,53]. Extended RISM calculations have shown for the protein melittin that adding the solvation free energy to its conformational potential energy results in two minima [54] along the direction of the first normal mode, rather than one. In addition to the change in the conformational energy surface, the solvent causes damping and other hydrodynamic effects. One can attempt to incorporate these into a normal mode analysis by using a variant of the normal mode analysis developed by Lamm and Szabo [55], called the Langevin mode analysis. The resulting generalized eigenvalue problem involves the simultaneous diagonalization of the Hessian and the friction matrix. The resulting modes display the extra feature of damping. Time correlation functions and spectral densities can be calculated directly from the Langevin modes. This approach was applied to the protein crambin and a DNA duplex to reveal that a number of modes had overdamped motions [56]. One obstacle to performing a Langevin mode analysis is the accurate determination of the off-diagonal hydrodynamic terms in the friction matrix. The spectral density determined from inelastic neutron scattering experiments for BPTI shows a shallower rise from zero frequency than normal mode calculations predict [57,58]. It has been shown that this is directly due to frictional damping of the low frequency modes and can be reproduced by the Langevin mode analysis [52,53].

B. Anharmonicity and Normal Mode Analysis

The emerging model for protein dynamics is one that incorporates the dual aspects of motion within minima, combined with transitions between minima. The normal mode analysis can be seen as addressing directly only one of these two features. Ironically, however, one variant of normal mode analysis can be used to help address the other feature, namely the transition of energy barriers. To determine barrier heights for transitions occurring in a molecular dynamics simulation, instantaneous normal modes can be determined by diagonalizing instantaneous force matrices at selected configurations along the trajectory. Negative eigenvalues indicate local negative curvature possibly arising from energy barriers in the multiple-minima surface. Simulations performed at different temperatures can give information on the distribution of barrier heights [59,60].

The physical model of protein dynamics indicated above would be greatly simplified if all minima were identical. Janezic et al. [10] performed 201 normal mode analyses starting minimizations from frames along a 1 ns vacuum molecular dynamics simulation of BPTI. Comparing normal modes by taking inner products revealed that in general the normal modes remained stable, indicating similar minima. Using this assumption, a normal

harmonic mode number for BPTI. Only 12% of the modes were found to be anharmonic; the remaining 88% were harmonic modes. This analysis was also applied to a 1 ns molecular dynamics simulation of human lysozyme in water, where it was found that only the first 5% of modes were anharmonic [61]. These results suggest that the harmonic component calculated from a single normal mode analysis may be separable from the anharmonic component. Under the assumption that all minima are identical, and using the ‘‘dual aspect model’’ of protein dynamics, contributions arising from transitions between minima can be separated from contributions from harmonic motion within minima in an expression for the variance–covariance matrix of atomic fluctuations [61]. This model, the jumping-among-minima, or JAM, model, can be used to determine many features of the multiple-minima energy surface, including the distribution of minima and barrier heights.

VI. CONCLUSIONS

One of the main attractions of normal mode analysis is that the results are easily visualized. One can sort the modes in terms of their contributions to the total MSF and concentrate on only those with the largest contributions. Each individual mode can be visualized as a collective motion that is certainly easier to interpret than the welter of information generated by a molecular dynamics trajectory. Figure 4 shows the first two normal modes of human lysozyme analyzed for their dynamic domains and hinge axes, showing how clean the results can sometimes be. However, recent analytical tools for molecular dynamics trajectories, such as the principal component analysis or essential dynamics method [25,62–64], promise also to provide equally clean, and perhaps more realistic, visualizations. That said, molecular dynamics is also limited in that many of the functional motions in biological molecules occur in time scales well beyond what is currently possible to simulate.

Various techniques exist that make possible a normal mode analysis of all but the largest molecules. These techniques include methods that are based on perturbation methods, reduced basis representations, and the application of group theory for symmetrical oligomeric molecular assemblies. Approximate methods that can reduce the computational load by an order of magnitude also hold the promise of producing reasonable approximations to the methods using conventional force fields.

Evidence exists that some of the softest normal modes can be associated with experimentally determined functional motions, and most studies apply normal mode analysis to this purpose. Owing to the veracity of the concept of the normal mode important subspace, normal mode analysis can be used in structural refinement methods to gain dynamic information that is beyond the capability of conventional refinement techniques.

Ironically, the normal mode analysis method can be used to determine properties of the multiple-minima energy surface of proteins.

Although not discussed in detail here, the normal mode analysis method has been used to calculate the electron transfer reorganization spectrum in Ru-modified cytochrome c [65,66]. In this application the normal mode analysis fits comfortably into the theory of electron transfer.

Despite its obvious limitations, normal mode analysis has found varied and perhaps unexpected applications in the study of the dynamics of biological molecules. In many