Correlations in low-frequency atomic displacements predicted by molecular dynamics simulations on the order of 1 ns are undersampled for the time scales currently accessible by the technique. This is shown with three different representations of the fluctuations in a macromolecule: the reciprocal space of crystallography using diffuse x-ray scattering data, real three-dimensional Cartesian space using covariance matrices of the atomic displacements, and the 3N-dimensional configuration space of the protein using dimensionally reduced projections to visualize the extent to which phase space is sampled. Experimental Tests of Molecular DynamicsClassical molecular dynamics is a computational technique to simulate the behaviors, both thermodynamic and dynamic, of an atomic model. In the long time limit, the sampled trajectory yields detailed information about the approximate model Hamiltonian. Shorter trajectories yield incomplete information and confound comparison of the model with experiment. Relatively short trajectories of biological macromolecules, on the order of 1 ns, have been used with reasonable success for a number of experimental comparisons-for example, crystallographic B values (1) from x-ray scattering, incoherent structure factors from neutron scattering (2, 3), and order parameters from NMR spectroscopy (4). Because these experimental techniques probe the movement of single atoms we can conclude that the amplitude and frequency of displacement for particular individual atoms in the protein are being simulated accurately.However, on the basis of such local results we cannot also conclude that these same molecular dynamics calculations accurately simulate, or converge on, the correlation between two or more atoms in a protein (e.g., whether one part of a protein tends to act concertedly with another). The state of our present models, then, is not completely tested by our experimental methods. In this article we consider a particular test set to measure the convergence of two-body properties in protein simulations.An experimental technique that is sensitive to statistics between pairs of atoms, and thus to collective behavior in macromolecules, is diffuse x-ray scattering (5, 6). All x-ray intensity on a detector besides the sharp Bragg peaks is typically ignored in structural studies. However, this remaining intensity, Diffuse scattering experiments on several proteins and nucleic acids (7-10) demonstrate that correlations in displacements of neighboring atoms fall off roughly exponentially with the distance between the atoms, with a characteristic length of 4-8 A. That is, (8iA8) = e(Wx(J)exp(-Jr| rjl/y), where the correlation length y is less than the dimension of the protein molecule. Thus, in analogy with liquid structures, where the correlations in atomicpositions decay approximately exponentially with pair separation, the motion in proteins is often called liquid-like, in the sense that correlations in atomic displacements fall off rapidly with distance.As a test of whether the pair corr...

Together, the Bragg and diffuse scattering present a self-consistent description of the motions in the flexible linker of calmodulin. The other mobile regions of the protein are also of great interest. In particular, the high variations in the calcium-binding sites are likely to influence how strongly they bind ions. This is especially important in the N-terminal sites, which regulate the activity of the molecule.

Diffuse X-ray scattering from protein crystals provides information about molecular flexibility and packing irregularities. Here we analyse diffraction patterns from insulin crystals that show two types of scattering related to disorder: very diffuse, liquid-like diffraction, and haloes around the Bragg reflections. The haloes are due to coupled displacements of neighbouring molecules in the lattice, and the very diffuse scattering results from variations in atomic positions that are only locally correlated within each molecule. The measured intensity was digitally separated into three components: the Bragg reflections and associated haloes; the water and Compton scattering; and the scattering attributed to internal protein movements. We extend methods used to analyse disorder in membrane structures to simulate the diffuse scattering from crystalline insulin in terms of (1) the Patterson (autocorrelation) function of the ideal, ordered crystal structure, (2) the root-mean-square (r.m.s.) amplitude of the atomic movements, and (3) the mean distance over which these displacements are coupled. Movements of the atoms within the molecules, with r.m.s. amplitudes of 0.4-0.45 A, appear to be coupled over a range of approximately 6 A, as in a liquid. These locally coupled movements account for most of the disorder in the crystal. Also, the protein molecules, as a whole, jiggle in the lattice with r.m.s. amplitudes of approximately 0.25 A that appear to be significantly correlated only between nearest neighbours.

Diffuse scattering data have been collected on two crystal forms of lysozyme, tetragonal and triclinic, using synchrotron radiation. The observed diffraction patterns were simulated using an exact theory for simple model crystals which relates the diffuse scattering intensity distribution to the amplitudes and correlations of atomic movements. Although the mean square displacements in the tetragonal form are twice that in the triclinic crystal, the predominant component of atomic movement in both crystals is accounted for by short-range coupled motions where displacement correlations decay exponentially as a function of atomic separation, with a relaxation distance of approximately 6 A. Lattice coupled movements with a correlation distance approximately 50 A account for only about 5-10% of the total atomic mean square displacements in the protein crystals. The results contradict various presumptions that the disorder in protein crystals can be modeled predominantly by elastic vibrations or rigid body movements.

The singular value decomposition (SVD) provides a method for decomposing a molecular dynamics trajectory into fundamental modes of atomic motion. The right singular vectors are projections of the protein conformations onto these modes showing the protein motion in a generalized low-dimensional basis. Statistical analysis of the right singular vectors can be used to classify discrete configurational substates in the protein. The configuration space portraits formed from the right singular vectors can also be used to visualize complex high-dimensional motion and to examine the extent of configuration space sampling by the simulation.

Configuration space projections provide invaluable information about protein motions that would be extremely difficult to obtain otherwise. While the configuration space for myoglobin is quite large, it does have structure. Our analysis of this structure shows that the protein hops between a number of distinct global conformational states, much like the local behavior observed for an individual residue.

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Time-averaged structure-factor restraints have been used in two molecular dynamics refinement schemes to define ensembles of conformations for myoglobin that fit the experimentally measured Bragg scattering from P6 crystals. The geometries of the structures have been maintained to the same currently acceptable limits in all cases. Free R value analysis was used to assess the validity of the two approaches. In the first scheme, where atoms have no B values, the decrease in R value was found to be spurious as judged by a concomitant increase in the free R value. The other scheme, however, which retains individual B values, was found to yield both low R values and low free R values; thus, here the additional variables introduced by modeling the protein in terms of an evolving ensemble of states do not overfit the data. For comparison, refinements were also carried out on the system using several other techniques for isotropic and anisotropic crystallographic refinement. The time-averaged refinements with B values compare quite favorably with the standard methods, but yield additional information about substates of the system. Hence, correctly applied time-averaged refinements can yield accurate models for protein molecules; moreover, by essentially relaxing the harmonic approximation from the refinement process, these refinements allow a more detailed description of the motions of complex molecules, such as proteins, to be determined from X-ray crystallographic data.