Figure 3 Calculated X-ray diffuse scattering patterns from (a) a full molecular dynamics trajectory of orthorhombic hen egg white lysozyme and (b) a trajectory obtained by fitting to the full trajectory rigid-body side chains and segments of the backbone. A full description is given in Ref. 13.

bodies obtained by fitting to the full simulation. The full simulation scattering is reproduced by the approximate representation to an agreement factor (R-factor) of 6%.

IV. NEUTRON SCATTERING

In contrast to X-rays, the mass of the neutron is such that the energy exchanged in exciting or deexciting picosecond time scale thermal motions is a large fraction of the incident

˚

energy and can be measured relatively precisely. A thermal neutron of 1.8 A wavelength has an energy of 25 meV corresponding to kbT at 300 K. To further examine the neutron scattering case, we perform space Fourier transformation of the van Hove correlation functions [Eqs. (3) and (4)]:

Neutrons are scattered by the nuclei of the sample. Because of the random distribution of nuclear spins in the sample, the scattered intensity will contain a coherent part

arising from the average neutron–nucleus potential and an incoherent part arising from fluctuations from the average. The coherent scattering arises from selfand cross-correla- tions of atomic motions, and the incoherent scattering, from single-atom motions. Each isotope has a coherent scattering length bi,coh and an incoherent scattering length bi,inc that define the strength of the interaction between the nucleus of the atom and the neutron. For more details on the origin of incoherent and coherent scattering see Ref. 1. We see from Eqs. (16) and (18) that the coherent and incoherent dynamic structure factors are time Fourier transforms of the coherent and incoherent intermediate scattering functions,

ω

Icoh(Q, t) and Iinc(Q, t); these are time-correlation functions [14]. Sinc(Q, ) and Scoh(Q, ω) may contain elastic (ω 0) and inelastic (ω ≠ 0) parts. Elastic scattering probes correlations of atomic positions at long times, whereas the inelastic scattering process probes position correlations as a function of time.

A. Coherent Inelastic Neutron Scattering

The use of coherent neutron scattering with simultaneous energy and momentum resolution provides a probe of time-dependent pair correlations in atomic motions. Coherent inelastic neutron scattering is therefore particularly useful for examining lattice dynamics in molecular crystals and holds promise for the characterization of correlated motions in biological macro-molecules. A property of lattice modes is that for particular wave vectors there are well-defined frequencies; the relations between these two quantities are the phonon dispersion relations [1]. Neutron scattering is presently the most effective technique for determining phonon dispersion curves. The scattering geometry used is illustrated in Figure 4. The following momentum conservation law is obeyed:

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships. (a) Scattering in real space; (b) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample and the phonon wave vector, q. Heavy dots represent Bragg reflections.

The vibrational excitations have a wave vector q that is measured from a Brillouin zone center (Bragg peak) located at , a reciprocal lattice vector.

If the displacements of the atoms are given in terms of the harmonic normal modes of vibration for the crystal, the coherent one-phonon inelastic neutron scattering cross section can be analytically expressed in terms of the eigenvectors and eigenvalues of the harmonic analysis, as described in Ref. 1.

Example: Lattice Vibrations in L-Alanine

Zwitterionic L-alanine ( H3NEC(CH3)ECO2E) is a dipolar molecule that forms large well-ordered crystals in which the molecules form hydrogen-bonded columns. The strong interactions lead to the presence of well-defined intraand intermolecular vibrations that can usefully be described using harmonic theory.

Coherent inelastic neutron scattering experiments have been combined with normal mode analyses with a molecular mechanics potential function to examine the collective vibrations in deuterated L-alanine [15]. In Figure 5 are shown experimental phonon frequencies νi (q)(ν ω/2π) for several modes propagating along the crystallographic direction b*. The solid lines represent the most probable paths for the dispersion curves νi(q). The theoretical dispersion curves are also given. The comparison between theory and experiment can be used to assess the accuracy with which the theory reproduces longrange interactions in the crystal.

B. Incoherent Neutron Scattering

Neutron scattering from nondeuterated organic molecules is dominated by incoherent scattering from the hydrogen atoms. This is largely because the incoherent scattering cross section (4πb2inc) of hydrogen is approximately 15 times greater than the total scattering cross section of carbon, nitrogen, or oxygen. The measured incoherent scattering thus essentially gives information on self-correlations of hydrogen atom motions. A program for calculating neutron scattering properties from molecular dynamics simulations has been published [16].

In practice, the measured incoherent scattering energy spectrum is divided into elastic, quasielastic, and inelastic scattering. Inelastic scattering arises from vibrations. Quasielastic scattering is typically Lorentzian or a sum of Lorentzians centered on ω 0 and arises from diffusive motions in the sample. Elastic scattering gives information on the self-probability distributions of the hydrogen atoms in the sample.

A procedure commonly used to extract dynamic data directly from experimental incoherent neutron scattering profiles is described in Ref. 17. It is assumed that the atomic position vectors can be decomposed into two contributions, one due to diffusive motion,

where Sd(Q, ω) and Sv(Q, ω) are obtained by Fourier transformation of Id(Q, t) and Iv(Q,

t) and the symbol denotes the convolution product. Appropriate descriptions of ri,d(t)

where Gd(r, t) is the contribution to the van Hove self-correlation function due to diffusive

motion. A0(Q) is thus determined by the diffusive contribution to the space probability distribution of the hydrogen nuclei.

Direct experiment–simulation quasielastic neutron scattering comparisons have been performed for a variety of small molecule and polymeric systems, as described in detail in Refs. 4 and 18–21. The combination of simulation and neutron scattering in the analysis of internal motions in globular proteins was reviewed in 1991 [3] and 1997 [4].

A dynamic transition in the internal motions of proteins is seen with increasing temperature [22]. The basic elements of this transition are reproduced by MD simulation [23]. As the temperature is increased, a transition from harmonic to anharmonic motion is seen, evidenced by a rapid increase in the atomic mean-square displacements. Comparison of simulation with quasielastic neutron scattering experiment has led to an interpretation of the dynamics involved in terms of rigid-body motions of the side chain atoms, in a way analogous to that shown above for the X-ray diffuse scattering [24].

Example: Change in Dynamics on Denaturing

Phosphoglycerate Kinase

In this example we examine the change in the experimental dynamic neutron scattering signal on strong denaturation of a globular protein, phosphoglycerate kinase (PGK) [25]. Evidence for this comes from the EISF plotted in Figure 6. The main difference in the EISF is in the asymptote as Q → ∞, which is significantly lower in the case of the denatured protein. The asymptotic value can be shown to correspond to a nondiffusing fraction of the hydrogen atoms in the protein. Whereas the nondiffusing fraction is 40% in the native protein, it is reduced to 18% in the denatured protein.

Inelastic Incoherent Scattering Intensity. For a system executing harmonic dynamics, the transform in Eq. (4) can be performed analytically and the result expanded in a power series over the normal modes in the sample. The following expression is obtained [26]: