12

X-Ray and Neutron Scattering as

Probes of the Dynamics of

Biological Molecules

Jeremy C. Smith

Interdisziplina¨res Zentrum fu¨r Wissenschaftliches Rechnen der Universita¨t Heidelberg, Heidelberg, Germany

I.INTRODUCTION

One of the major uses of molecular simulation is to provide useful theoretical interpretation of experimental data. Before the advent of simulation this had to be done by directly comparing experiment with analytical (mathematical) models. The analytical approach has the advantage of simplicity, in that the models are derived from first principles with only a few, if any, adjustable parameters. However, the chemical complexity of biological systems often precludes the direct application of meaningful analytical models or leads to the situation where more than one model can be invoked to explain the same experimental data.

Computer simulation gets around this problem by allowing more complicated, detailed, and realistic models of biological systems to be investigated. However, the price to pay for this is a degree of mathematical defeat, as the extra complexity prevents analytical solution of the relevant equations. We must therefore rely on numerical methods and hope that when allied with sufficient computer power they will lead to converged solutions of the equations we wish to solve. The simulation methods used involve empirical potential energy functions of the molecular mechanics type, and the equations of motion are solved using either stepwise integration of the full anharmonic function by molecular dynamics (MD) simulation or, for vibrating systems, by normal mode analysis involving a harmonic approximation to the potential function. In the case of molecular dynamics simulations of the dynamic properties of proteins, convergence is particularly difficult to achieve, because motions occur in proteins on time scales that are the same as or longer than the presently accessible time scale (nanoseconds). In the future things should improve, due to more efficient algorithms and the continuing rapid increase in computer power. Moreover, many dynamical phenomena of physical and biological interest occur on the subnanosecond time scale and thus can already be well sampled. The examples given in this chapter are mainly on this time scale.

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Computer simulation can be used to provide a stepping stone between experiment and the simplified analytical descriptions of the physical behavior of biological systems. But before gaining the right to do this, we must first validate a simulation by direct comparison with experiment. To do this we must compare physical quantities that are measurable or derivable from measurements with the same quantities derived from simulation. If the quantities agree, we then have some justification for using the detailed information present in the simulation to interpret the experiments.

The spectroscopic techniques that have been most frequently used to investigate biomolecular dynamics are those that are commonly available in laboratories, such as nuclear magnetic resonance (NMR), fluorescence, and Mossbauer spectroscopy. In a later chapter the use of NMR, a powerful probe of local motions in macromolecules, is described. Here we examine scattering of X-ray and neutron radiation. Neutrons and X-rays share the property of being found in expensive sources not commonly available in the laboratory. Neutrons are produced by a nuclear reactor or ‘‘spallation’’ source. X-ray experiments are routinely performed using intense synchrotron radiation, although in favorable cases laboratory sources may also be used.

The X-ray and neutron scattering processes provide relatively direct spatial information on atomic motions via determination of the wave vector transferred between the photon/neutron and the sample; this is a Fourier transform relationship between wave vectors in reciprocal space and position vectors in real space. Neutrons, by virtue of the possibility of resolving their energy transfers, can also give information on the time dependence of the motions involved.

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of ‘‘derived’’ quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities.

Having made the comparison with experiment one may then make an assessment as to whether the simulation agrees sufficiently well to be useful in interpreting the experiment in detail. In cases where the agreement is not good, the determination of the cause of the discrepancy is often instructive. The errors may arise from the simulation model or from the assumptions used in the experimental data reduction or both. In cases where the quantities examined agree, the simulation can be decomposed so as to isolate the principal components responsible for the observed intensities. Sometimes, then, the dynamics involved can be described by a simplified concept derived from the simulation.

X-Ray and Neutron Scattering as Dynamics Probes

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In this chapter we present some basic equations linking scattering to dynamics that are of interest to computer simulators. Their derivations from first principles are quite long, and for these readers are referred to appropriate textbooks [1,2]. Examples of different properties that can be probed are given here from work involving our group. For more details on this work, readers are referred to the original references and to two reviews [3,4].

II.BASIC EQUATIONS RELATING ATOMIC POSITIONS TO X-RAY AND NEUTRON SCATTERING

Scattering experiments involve processes in which incident particles (X-rays or neutrons)

with wave vector ki (energy Ei) interact with the sample and emerge with wave vector kf (energy Ef), obeying the conservation laws for momentum and energy,

ki kf Q	(1)
Ei Ef ω	(2)

The vectors ki, kf, and Q define the scattering geometry as illustrated in Figure 1.

We first examine the relationship between particle dynamics and the scattering of radiation in the case where both the energy and momentum transferred between the sample and the incident radiation are measured. Linear response theory allows dynamic structure factors to be written in terms of equilibrium fluctuations of the sample. For neutron scattering from a system of identical particles, this is [1,5,6]

ω

Scoh(Q, )

ω

Sinc(Q, )

1

2π

1

2π

	r ωt)G(r, t)	(3)
∫∫dt d3rei(Q
		(4)
∫∫ dt d3rei(Q r ωt)Gs(r, t)

ω

where Q is the scattering wave vector, the energy transfer, and the subscripts coh and inc refer to coherent and incoherent scattering, discussed later. Gs(r, t) and G(r, t) are

Figure 1 Scattering vector triangle.

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van Hove correlation functions, which, for a system of N particles undergoing classical dynamics, are defined as follows:

	1					(0))
G(r, t) N				δ(r Ri	(t) Rj		(5)
				i,j
		1
Gs(r, t) N				δ(r Ri(t) Ri(0))			(6)
				i

	. . . indicates an
where Ri(t) is the position vector of the ith scattering nucleus and
ensemble average.

G(r, t) is the probability that, given a particle at the origin at time t 0, any particle (including the original particle) is at r at time t. Gs(r, t) is the probability that, given a particle at the origin at time t 0, the same particle is at r at time t.

Equation (3) has an analogous form in X-ray scattering, where the scattered intensity is given as [2]

	2		1	3
						i(Q r ωt)
|F(Q, ω)|				∫∫ d	r dtP(r, t)e
			2π

where P(r, t) is the spatiotemporal Patterson function given by

∫∫ ρ ρ τ

P(r, t) dR dt (R, t) (r R, t )

(7)

(8)

and ρ(r, t) is the time-dependent electron density. Unfortunately, X-ray photons with wavelengths corresponding to atomic distances have energies much higher than those

˚

associated with thermal fluctuations. For example, an X-ray photon of 1.8 A wavelength has an energy of 6.9 keV corresponding to a temperature of 8 107 K. Until very recently X-ray detectors were not sensitive enough to accurately measure the minute fractional energy changes associated with molecular fluctuations, so the practical exploitation of Eq.

(7) was difficult. However, the use of new third-generation synchrotron sources has enabled useful inelastic X-ray scattering experiments to be performed on, for example, glassforming liquids and liquid water (see, e.g., Ref. 7).

Although inelastic X-ray scattering holds promise for the investigation of biological molecular dynamics, it is still in its infancy, so in the subsequent discussion X-ray scattering is examined only in the case in which inelastic and elastic scattering are indistinguishable experimentally.

III. SCATTERING BY CRYSTALS

In an X-ray crystallography experiment the instantaneous scattered intensity is given by [2]

		N	N
Ihkl |Fhkl|	2		*		(9)
		fi f j		exp[iQ (Ri Rj)]
		i 1	j 1

where Fhkl is the structure factor, Ri is the position vector of atom i in the crystal, and fi is the X-ray atomic form factor. For neutron diffraction fi is replaced by the coherent scattering length.

X-Ray and Neutron Scattering as Dynamics Probes

241

In an experimental analysis it is not feasible to insert into Eq. (9) the atomic positions for all the atoms in the crystal for every instant in the time of the experiment. Rather, the intensity must be evaluated in terms of statistical relationships between the positions. A convenient approach is to consider a real crystal as a superposition of an ideal periodic structure with slight perturbations. When exposed to X-rays the real crystal gives rise to two scattering components: the set of Bragg reflections arising from the periodic structure, and scattering outside the Bragg spots (diffuse scattering) that arises from the structural perturbations:

Ihkl IhklB IhklD

(10)

where IhklB is the Bragg intensity found at integer values of h, k, and l, and IhklD is the diffuse scattering, not confined to integer values of h, k, and l.

In terms of structure factors the various intensities are given by [2]

Ihkl |Fhkl|2		(11)
IhklB	|Fhkl|2	(12)
and
IhklD	|∆Fhkl|2	(13)

A.Bragg Diffraction

The Bragg peak intensity reduction due to atomic displacements is described by the well-

known ‘‘temperature’’ factors. Assuming that the position ri can be decomposed into an
average position, ri and an infinitesimal displacement, ui δRi Ri Ri then the
X-ray structure factors can be expressed as follows:
N
			(14)
Fhkl fi(Q)exp[iQ Ri ]exp[Wi(Q)]

i 1

where Wi(Q) (1/3) u2i,Q Q2 and u2i,Q is the mean-square fluctuation in the direction of Q. Wi(Q) is the Debye–Waller factor.

Example: Low Temperature Vibrations in Acetanilide

Temperature factors are of interest to structural biologists mainly as a means of deriving qualitative information on the fluctuations of segments of a macromolecule. However, X- ray temperature factor analysis has drawbacks. One of the most serious is the possible presence of a static disorder contribution to the atomic fluctuations. This cannot be distinguished from the dynamic disorder due to the aforementioned absence of accurate energy analysis of the scattered X-ray photons. For quantitative work this problem can be avoided by choosing a system in which there is negligible static disorder and in which the harmonic approximation is valid. An example of such a system is acetanilide, (C6H5ECONHECH3), at 15 K. The structure of acetanilide is shown in Figure 2. In recent work [8] a molecular mechanics force field was parametrized for this crystal, and normal mode analysis was performed in the full configurational space of the crystal i.e., including all intramolecular and intermolecular degrees of freedom. As a quantitative test of the accuracy of the force field, anisotropic quantum mechanical mean-square displacements of the hydrogen atoms were calculated in each Cartesian direction as a sum over the phonon normal modes of the crystal and compared with experimental neutron diffraction

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Figure 2 Crystalline acetanilide.

temperature factors [9]. The experimental and theoretical temperature factors are presented in Table 1. The values of the mean-square displacements are in excellent agreement.

B. X-Ray Diffuse Scattering

Any perturbation from ideal space-group symmetry in a crystal will give rise to diffuse scattering. The X-ray diffuse scattering intensity IhklD at some point (hkl) in reciprocal space can be written as

D		(15)
Ihkl N (Fn F )(Fn m F )* exp( Q Rm)

m

where Fn is the structure factor of the nth unit cell and the sum ∑m runs over the relative

positionvectors Rm between the unit cells. The correlation function (Fn F )(Fn m

F )* is determined by correlations between atomic displacements.

If the diffuse scattering of dynamic origin contributes significantly to the measured scattering, it may provide information on the nature of correlated motions in biological macromolecules that may themselves be of functional significance. To examine this possibility it is necessary to construct dynamic models of the crystal, calculate their diffuse scattering, and compare with experimental results. The advent of high intensity synchrotron sources and image plate detectors has allowed good quality X-ray diffuse scattering images to be obtained from macromolecular crystals.

SERENA (scattering of X-rays elucidated by numerical analysis) is a program for calculating X-ray diffuse scattering intensities from configurations of atoms in molecular crystals [10]. The configurations are conveniently derived from molecular dynamics simulations, although in principle any collection of configurations can be used. SERENA calculates structure factors from the individual configurations and performs the averaging required in Eq. (11).

Displacements correlated within unit cells but not between them lead to very diffuse scattering that is not associated with the Bragg peaks. This can be conveniently explored

X-Ray and Neutron Scattering as Dynamics Probes					243
Table 1 Mean-Square Displacements of Hydrogen Atoms in Crystalline Acetanilide at 15 Ka
I	II	III	I	II	III
Methyl H a	0.0359	0.0352	Phenyl Hpara a	0.0126	0.0118
b	0.0366	0.0438	b	0.0273	0.0278
c	0.0253	0.0258	c	0.0279	0.0258
isotropic	0.0326	0.0349	isotropic	0.0226	0.0218
Methyl H a	0.0150	0.0160	Phenyl Hmeta a	0.0215	0.0220
b	0.0482	0.0510	b	0.0189	0.0194
c	0.0328	0.0336	c	0.0265	0.0246
isotropic	0.0320	0.0335	isotropic	0.0223	0.0220
Methyl H a	0.0301	0.0286	Phenyl Hmeta a	0.0154	0.0162
b	0.0162	0.0172	b	0.0218	0.0210
c	0.0483	0.0606	c	0.0296	0.0256
isotropic	0.0315	0.0354	isotropic	0.0222	0.0209
Amide H a	0.0178	0.0192	Phenyl Hortho a	0.0159	0.0160
b	0.0110	0.0120	b	0.0189	0.0194
c	0.0258	0.0300	c	0.0286	0.0250
isotropic	0.0182	0.0204	isotropic	0.0211	0.0201
			Phenyl Hortho a	0.0203	0.0200
			b	0.0170	0.0164
			c	0.0264	0.0254
			isotropic	0.0212	0.0206

aColumn I: Hydrogen atoms of acetanilide. Phenyl hydrogens are named according to their position relative to the N substitution site. a, b, c refer to the crystallographic directions. Column II: Anisotropic (a, b, c crystallo-

˚ 2

graphic directions) and isotropic mean-square displacements (A ) from neutron diffraction data [9]. Column

˚ 2

III: Anisotropic (a, b, c crystallographic directions) and isotropic mean-square displacements (A ) from harmonic analysis.

using present-day simulations of biological macromolecules. However, motions correlated over distances larger than the size of the simulation model will clearly not be included.

Example: Correlated Motions in Lysozyme

Ligand binding and cooperativity often require conformational change involving correlated displacements of atoms [11]. A simple model for long distance transmission of information across a protein involves the activation and amplification of correlated motions that are present in the unperturbed protein. Although long-range correlated fluctuations are required for functional, dynamic information transfer, it is not clear to what extent they contribute to equilibrium thermal fluctuations in proteins. It is therefore important to know whether equilibrium motions in proteins can indeed be correlated over long distances or whether anharmonic and damping effects destroy such correlations.

To examine the dynamic origins of X-ray diffuse scattering by proteins, experimental scattering was measured from orthorhombic lysozyme crystals and compared to patterns calculated using molecular simulation [12]. The diffuse scattering was found to be approximately reproduced by both normal modes and molecular dynamics. More recently, a molecular dynamics analysis was performed of the dynamics of a unit cell of orthorhombic lysozyme, including four protein molecules [13]. Diffuse scattering calculated from the MD trajectory is compared in Figure 3 with that calculated from trajectories of rigid