Biophysical aspects of neutron scattering from vibrational modes of proteins.

Prog.Biophys.molec.Biol.,Vol. 57, pp. 129-179, 1992.

0079-6107/92 $15.00 © 1992PergamonPressLid

Printed in Great Britain. All rightsreserved.

BIOPHYSICAL ASPECTS OF NEUTRON SCATTERING FROM VIBRATIONAL MODES OF PROTEINS P . MARTEL

Chalk River Laboratories, AECL Research, Chalk River, Ontario, KOJ lJO, Canada Abstract--This review describes a major portion of the published work on neutron scattering experiments aimed at measuring large scale motions in proteins. The importance of these motions for enzyme function and oxygen transport is indicated. The theory applicable to each type of neutron scattering measurement is given and results are discussed with a view to biological relevance. New experiments are suggested and a comparison of neutron scattering data is made with results from other techniques such as raman scattering, infrared absorption, photolysis and molecular dynamics simulations. CONTENTS I. INTRODUCTION II. HISTORICALPERSPECTIVE

1. Early Work with Neutrons 2. A Brief History of Related Experiments with Other Techniques III. BIOPHYSICALRELEVANCE

1. The Match between Inelastic Neutron Scattering and Molecular Dynamics Calculations 2. Spectral Features that Depend on Secondary Structure 3. Examples of the Dependence of Function on Motion (a) Segmental mobility (b) Domain motion (c) Allosterism (d) Free energy considerations IV. THE NEUTRON METHOD IN BIOPHYSICS

1. General Remarks on Neutron Scattering Theory 2. Elastic Scattering (a) Coherent elastic scattering (i) Review of theory (ii) Vibrational amplitudes (iii) Hydrogen-deuterium exchange (b) Incoherent elastic scattering 3. Inelastic Scattering (a) Coherent inelastic scattering (i) The phonon r~gime (ii) Low frequency stochastic motion (b) Incoherent inelastic scattering (i) General comments on frequency domains (ii) The diffusive quasielastic r~aime: scattering from the water near protein (iii) Quasielastic scattering from heme protein in the presence of water (iv) The glass-like behaviour of proteins at low temperatures (v) Review of the theory of incoherent scattering in the vibrational rdgime (vi) Comparison of theory with experiment on pancreatic trypsin inhibitor (vii) Hinge-like oscillations (viii) The incoherent scattering approximation (ix) Enzyme-substrate binding (x) Myoalobin at higher frequencies (xi) Scattering from protein in cells: hemoglobin in situ (xii) Water-hemoglobin interactions in red blood cells (xiii) The Frohlich mode (xiv) Miscellaneous measurements reported and proposed V. SUMMARYAND FUTURE PROSPECTS ACKNOWLEDGEMENTS REFERENCES

129 131 131 132 133 133 133 133 133 134 135 136 137 137 139 139 139 139 140 140 141 141 141 144 145 145 146 148 150 152 155 156 157 158 161 165 169 171 171 173 176 176

I. INTRODUCTION

The present review summarizes neutron scattering experiments that provide information on the dynamical properties of proteins and polypeptides. The main emphasis will be on inelastic neutron scattering where a neutron either loses (gains) energy to (from) vibrational 129

130

P. MARTEL

states of a protein. Although structural studies focus on the locations of mean atomic positions, they too can yield valuable information on the amplitudes of atomic displacements from these mean positions. By means of coherent elastic neutron scattering much can be learned about the magnitudes of displacements of hydrogen atoms in proteins and associated water of hydration (Raghavan and Schoenborn, 1984). The amplitude associated with the atomic displacements of an amino acid is a "marker" for the dynamics of its motion. In this role it can signal changes in the frequency distribution of the protein under varying external forces associated with crystal packing (Phillips, 1990) and other environmental effects. However, to measure the actual changes in frequency other methods must be employed. Experiments aimed primarily at structural studies will be referred to when they touch directly on dynamical properties. There has recently been increased emphasis on dynamical measurements because of the possibility of comparing low frequency neutron spectra with molecular dynamics calculations (Loncharich and Brooks, 1990; Smith et al., 1990a). This new development stems from pioneering work on the molecular dynamics of proteins by Karplus and collaborators (see Karplus and McCammon, 1981). Because these reviews are directed at students in biology and medicine as well as experts in the field, an attempt will be made not only to report on various physical aspects but also on possible biological relevance. Non-specialists should be able to read this review up to Section IV with ease. Mathematical details will be kept to the minimum necessary to illustrate different facets of neutron scattering theory. It may be noted that " c m - 1,, is the frequency unit most commonly employed in neutron scattering studies of proteins. Occasionally T H z and meV units are utilized. For future reference the reader may recall that 33.3 cm-1 = 1 T H z = 4 . 1 4 meV. Also, heavy or deuterated water will be referred to as D 2 0 . For reference in later discussion, the structural arrangement of amino acid residues making up polypeptides and proteins is recalled in Fig. 1. One important element of the microscopic structure is a relatively rigid planar unit made up of the four atoms, O, C, N and H. Figure 1 illustrates two typical amino acids covalently joined at a carbon atom designated, C~. There are 20 common amino acids, all having this four-atom peptide group. The different amino acids are characterized by different out-of-plane "side chains" designated as S. The simplest S consists of one hydrogen atom and the resultant amino acid is called glycine. One of the more complex side chains is an aromatic, C 1oH sN, and forms part of the amino acid tryptophan. Torsional degrees of freedom associated with the polypeptide structure are designated as qS, ~b, and X. The C, atoms are the pivotal centers about which torsional motions commonly occur. Such motions can be very large at room temperature, and result in changes in the internal configuration of a protein. As the temperature is lowered, the torsional motion can become frozen into any one of many conformational substrates (Frauenfelder et al., 1988).

FIG. 1. Schematicillustration of two amino acid residuesjoined at the carbon atom, Ca. The dashed triangles delineatethe planesin whichthe rigid peptideunits (OCNH) lie. Differentamino acids have different side chains, S. Rotations about the N-C~ and C~--C bonds are denoted by ck and 4, respectively.Side chain rotations are labelled X-

Biophysicalaspectsof neutron scattering

131

Like DNA, polypeptides are linear molecules. Depending on their amino acid composition, polypeptides are configured into various secondary structures such as ~-helices, fl-sheets and fl-barrels by specialized networks of hydrogen bonds (Brooks et al., 1988). Water bridges, salt links, Van der Waals forces and disulphide bonds link adjacent portions of the same or different polypeptide strands to make up compact forms known as tertiary protein structures. Groups of similar tertiary structures make up the so-called quaternary structure of some of the more complicated proteins. The various tertiary units that make up the proteins with quaternary structure are called subunits or monomers. These subunits may communicate with one another and enhance the function of the protein as a whole; this cooperative feature of multimeric proteins is known as allosterism. Other background information necessary to understand the neutron scattering results quoted in this review may be found in standard university textbooks on biochemistry, e.g. the one by Stryer (1988). II. HISTORICAL PERSPECTIVE 1. Early Work with Neutrons

Inelastic neutron scattering from polypeptides dates from the 1960s. Much of this early work was exploratory and often involved comparison of spectra from simple polypeptides in different conformational states (Gupta et al., 1968; Boutin and Yip, 1968). In many cases infrared and raman scattering results were invoked to help identify the origins of peaks in the inelastic neutron spectra. With the simpler polypeptides such as polyglycine it was fairly easy to calculate the allowable vibrations associated with different crystalline forms and these calculations were used to identify features in the spectra. An early paper on crystalline polyglycine (Gupta et al., 1968) indicates how comparison of neutron spectra from polyglycine I and polyglycine II can be accounted for by extra hydrogen bonding in the latter. Another early experiment on poly-L-glutamic acid (Whittemore, 1968) capitalized on calculations for polyglycine to make frequency assignments in the range from 350 to 1535 cm -1. A difference spectrum derived from the wet and dry polypeptide spectra indicated a peak at 80 cm- ~due to absorbed water. Librational modes of water near 50 cmwere also detected. An interesting experiment by Drexel and Peticolas (1975) revealed dramatic changes in the neutron spectrum for poly-L-alanine on going from the ~-helical to the/~-sheet form. These measurements also illustrated how good instrumental resolution could provide dramatic improvement over earlier measurements (Gupta et al., 1967). A peak at 230 cm- ~ common to both forms, was ascribed to torsional vibrations of the methyl side chains of poly-L-alanine and a potential barrier height of 14 kJ mol- ~was calculated based on this frequency. (These torsional vibrations would be associated with the angle ~ in Fig. I.) The radical difference in the spectra for frequencies other than 230 cm- 1, suggests the possibility of using inelastic neutron scattering as a probe of secondary structure in analogy with current infrared methods (Byler and Susi, 1986). However the complexity of neutron spectra from proteins, rather than simple polypeptides, renders this virtually impossible with present neutron flux levels and state-of-the-art techniques. There were other inelastic neutron scattering measurements in the 1970s where the principal focus was not on proteins but the state of water in hydrated proteins. In many ways, because of the omnipresence of water in all live biological systems these measurements are as important as measurements on proteins themselves. Many of these experiments involved very small shifts in neutron energy and such experiments are usually termed quasielastic, although they are inelastic in the sense that the neutron loses or gains energy. Among other things the results can be used to calculate diffusion constants. Randall et al. (1978) collected data on fully deuterated C-phycocyanin wetted with various small percentages of light water. Their results indicated that the diffusion coefficient of this water was at least one order of magnitude less than the coefficient of diffusion of bulk water. These experiments involved small neutron momentum transfers by low energy neutrons. Using higher energy neutrons to probe the librational modes of water, Martel (1980)

132

P. MARTEL

concluded that water in rat muscle behaved largely like bulk water. Another inelastic neutron scattering experiment (Hecht and White, 1975) with low energy neutrons indicated no difference between the diffusion coefficients of buffer and interparticle water in tobacco mosaic virus solutions. It may be noted that in the last two experiments there was sufficient dilution that only a small percentage of the water was in contact with protein. For more information on early neutron scattering measurements aimed at elucidating the state of water in contact with protein, the reader is referred to several detailed reviews by Middendorf (Middendorf, 1984a, 1986; Middendorf and Randall, 1985). 2. A Brief History of Related Experiments with Other Techniques While inelastic neutron scattering (other than low-frequency quasielastic scattering) from proteins was in a somewhat quiescent state during the 1970s, there were developments in other fields that inspired a resurgence of this technique. An awareness of these other developments is necessary in order to better understand the inelastic neutron measurements that were to follow in the 1980s. During the 1970s several photodissociation (Austin et al., 1975) and X-ray diffraction measurements (Frauenfelder et al., 1979) by Frauenfelder and collaborators indicated that a globular protein molecule could sample many different conformational states at physiological temperatures and that as a result the final configuration of any given protein molecule in a crystal could be slightly variable on cooling to lower temperatures. Certain types of inelastic neutron scattering can be sensitive to the fluctuations between structures that occur with great rapidity between different conformational "substrates". Evidence for this will be presented below. A review of work by Frauenfelder et al. (1988) discusses the temperature dependence of configurational substrates and how these impinge on the rebinding kinetics of heme proteins. Raman scattering experiments pioneered by Peticolas and collaborators (Brown et al., 1972) led directly to inelastic neutron scattering measurements. An experiment on at-chymotrypsin (Brown et al., 1972) yielded the first evidence that low-frequency bands could exist in globular proteins. A broad peak in the raman spectrum was found at 29 cm- 1 in this protein at room temperature and it was shown that the peak disappeared upon denaturation. The same paper reported a strong raman band at about 32 cm - 1in pepsin, but none in carboxypeptidase. It was concluded that for those proteins with low-frequency bands, large portions of these molecules were constantly undergoing "coherent periodic vibration" at frequencies near the peak frequency associated with the band. Peticolas was aware of the relationship between raman and inelastic neutron scattering (cf. Drexel and Peticolas, 1975) and attempted to verify his raman results on ~-chymotrypsin by neutron scattering measurements (private communication) but was unsuccessful in an attempt in the 1970s. Many years later, an inelastic neutron scattering peak was observed in ~chymotrypsin (Martel and Ahmed, 1988) at about 20 cm- 1. The peak was weak and long counting times were required. The difference in observed frequencies was ascribed to differences in selection rules which are more restrictive for raman scattering. A few years after the discovery of Brown et al. (1972) a low-frequency band was discovered in lysozyme by raman scattering (Genzel et al., 1976). This band was centered near 25 cm- 1 and was observed in crystalline lysozyme but not in aqueous solution. Consequently the authors concluded that this 25 cm-~ band should not be attributed to internal molecular vibrations, but to intermolecular modes. Another raman scattering experiment followed later on the important molecule, bovine immunoglobulin (Painter and Mosher, 1979). At ambient temperature and humidity this protein displayed two low-frequency bands centered at 28 and 39 c m - 1 Upon hydration these bands became "almost lost in the background scatter". In this case Painter and Mosher ascribed the highly diminished intensity and broadening to damping by the aqueous solvent. A summary of low-frequency raman measurements for a dozen different proteins has been published (Painter et al., 1982). The purpose of this summary was to show that frequencies below 37 cm- ~could not be correlated with a theory (Suezaki and G6, 1975) that postulated a vibrational mode made up solely of symmetrical expansions and contractions of globular

Biophysicalaspects of neutron scattering

133

proteins modelled as spheres. Interestingly enough, Painter and his collaborators also measured a strong raman band centered at 30 cm-1 in DNA (Painter et al., 1980). The early interpretations of low-frequency modes, before the advent of high-powered molecular dynamics calculations, were summarized by Peticolas (1978). A simple, but elegant, "quasi-continuity" theory of low-frequency protein dynamics in terms of secondary structures has been given recently by Chou (1983, 1984a,b, 1985, 1987, 1988). Molecular dynamics is an approach involving sophisticated computer techniques necessitated by the large numbers of atoms in protein systems. This latter technique is the best method for taking account of the inherent complexity that arises because of the many slightly different conformations which the protein structure may adopt at physiological temperatures. III. BIOPHYSICAL RELEVANCE

1. The Match between Inelastic Neutron Scattering and Molecular Dynamics Calculations There are many reports in the literature that deal with the technical aspects of inelastic neutron scattering from condensed matter in terms of cross sections, mode frequencies and amplitudes, etc. It is more difficult to find papers that discuss the biological relevance of neutron scattering experiments such as we will describe in this review. This is partly because biological matter is so complex that experimental analysis becomes bogged down in details. With large proteins there are many frequencies crowded together in the low-frequency regime, often separated by less than I cm- 1. Now, however, with molecular dynamics and normal mode calculations made possible by supercomputers, inelastic neutron scattering has become a more viable tool in helping to understand how protein motion affects function. Currently it is possible to use supercomputers to follow the trajectories of thousands of atoms over periods of time typically of the order of 0.2 to 500 psec. For these times the frequency range one is probing is 0.07 to 167 cm- 1. This is just the range to which inelastic neutron scattering is best suited. Thus the technique is useful as a guide for honing the interatomic potentials and calculational procedures necessary to predict how proteins will behave in the low-frequency range where large scale atomic movement occurs. 2. Spectral Features that Depend on Secondary Structure At low frequencies the amplitudes of atomic oscillation tend to be large and non-local and a globular protein molecule behaves like a continuous elastic body (G6 et al., 1983). The coordinated motions of different secondary structures are expected to be different according to function. Even if most of the internal motions in the secondary structures are small relative to those required for physiological movement, they can serve as specialized "lubricants" (Karplus, 1986) that make possible specific larger scale displacements of these structures. It is now known from inelastic neutron scattering studies at temperatures near those necessary for biological activity that some proteins with a large percentage of ~-helical structure, such as myoglobin (dry or in solution) do not exhibit peaks in their inelastic neutron scattering response (Doster et al., 1990; Martel and Lin, 1989), Others with very little ~-helical structure such as bovine pancreatic trypsin inhibitor (BPTI) and chymotrypsin (Martel and Ahmed, 1988; Smith et al., 1989) do show such peaks near physiological temperatures albeit at low levels of hydration, The functions of these different proteins are of course different. Myoglobin transports oxygen in muscle tissue whereas BPTI and chymotrypsin are involved in proteolytic processes in digestion. Since there are many different types of secondary structure, often mixed together in any given protein, it remains to be seen what other correlations there are with secondary structure. As more data accumulates this may be an interesting avenue for further study.

3. Examples of the Dependence of Function on Motion (a) Segmental mobility Large amplitude motion will result when large segments of protein sub-structure are weakly attached to a central core. At the lower limits of the inelastic neutron scattering

134

P. MARTEL

frequency range protein motions of especially large amplitude should be detectable and these motions could be directly related to physiological processes. An example of this type of measurement is a neutron "spin echo" experiment carried out by Alpert et al. (1985) on immunoglobulin G (IgG) in an aqueous environment. In this experiment the F~b arms of IgG were found to wobble within a cone of + 50 °, about apices located at the points of attachment of these arms to the F¢ stem (see Fig. 2). The anharmonic "frequencies" deduced for this motion were of the order of 10- 5 cm- 1. It is not difficult to envisage these arms milling about in order to grasp at nearby antigens. Immunoglobulin G is an example of "segmental mobility" in proteins (Janin and Wodak, 1983). Other examples are fibronectin and laminin (Bennett and Huber, 1984) and histocompatibility antigens. Because various segments of these proteins are extended and spread out, their movements tend to be very dependent on viscous damping by the aqueous environment and this was the rationale used to explain higher frequency raman results (Painter and Mosher, 1979) where highly diminished intensity and line broadening were observed on dissolution of IgG. /

\

/

/ /

\\ \ \ \ \ \ / \ \ I \

/

/ / I

t

Fzb

\ \

/ / / / /

~

F~

\

\ \ \.

hi"

\ \

/ FIG. 2. Schematic of the maximum amplitude,/~, of the cone of "wobble" of the F,b arms of IgG. Neutron spin echo measurements (Alpert et al., 1985) found that fl ~50 °.

(b) D o m a i n motion At somewhat higher frequencies there is the possibility of observing domain motion where large regions of a protein vibrate cooperatively with respect to one another. Domains are regions of sub-structure that can be visualized in the most simple cases by inspection of threedimensional models. In more subtle cases they can be delineated by specialized computer techniques (Janin and Wodak, 1983). An example of a simple case is lysozyme and among the more subtle cases there are globular proteins like chymotrypsin and concanavalin A. Because the domain structure of lysozyme is so evident it has been the focus of much theoretical and experimental work. A simple approximation of the structure of hen lysozyme is two lobes or domains joined by a hinge. One can picture a librational mode of these domains such that they open and close about the hinge region where the active cleft is located. This motion would facilitate accommodation of a substrate sugar in the cleft. Rupture of bacterium cell walls by lysozyme is actually brought about by cleavage of polysaccharides in the cell wall. A theoretical calculation (McCammon et al., 1976) indicated that the two domains of lysozyme could beat at a "hinge-bending" frequency of about 4.2 cm- 1 if hydrodynamic damping were neglected. To arrive at this frequency McCammon et al. (1976) rotated one of the domains about an axis in the hinge region and calculated the resultant changes in the protein's conformational energy in order to find an overall force constant associated with the


135

motion of the domains. A later calculation (Brooks and Karplus, 1985) yielded a frequency of 3.6 cm-1 and analysis of the mode indicated that bond angle and energy changes were distributed over many residues. When hydrodynamic terms were introduced by means of the Langevin equation (Karplus, 1986) it was found that domain motion became heavily damped. These calculations inspired considerable experimental activity. Before relating the results of inelastic neutron measurements it should be noted that X-ray diffraction measurements of the structures of hen and human lysozyme (Artymiuk et al., 1979) indicated that the active site of the enzyme was in a zone of high atomic displacement. This suggested the possibility of large scale intramolecular motion. Interest in displacements measureable by X-ray diffraction has extended to the present where it has been found (Faber and Mathews, 1990) that at least five different conformational substrates can exist in lysozyme. Such conformational mobility offers alternative pathways for approach of the substrate to the active site. In one of the early experiments on lysozyme Bartunik et al. (1982) measured neutron spectra from partly deuterated hen lysozyme in concentrated heavy water solution with and without N-acetyl-glucosamine inhibitor bound to the cleft. There was no direct indication of a hinge-bending mode, but at low frequencies between 1 and 40 cm- 1 their spectra decreased in intensity when inhibitor was bound. The authors attributed this decrease in intensity to the absence of transitions between different configurational states upon binding. These experimenters also observed the mode at 25 cm- ~ first seen by raman scattering (Genzel et al., 1976). Much more work on more dilute (and hence physiologically relevant) lysozyme solutions has been carried out by Middendorf, and references to these experiments may be found in one of his more recent reviews (Middendorf, 1986). Among the features reported (Middendorf, 1986) was a band in the difference spectra between bound and unbound lysozyme. This band was found to lie between 5 and 100 cm -~ but this feature was not ascribed to the hinge-bending mode. More recently, Smith et al. (1987) found hydration-related spectral changes in lysozyme which could be related to large scale movement. Their results were consistent with an increase in the number of low-frequency global modes between 0.4 and 4 cm- ~ upon increasing hydration from 0.07 g D20/g protein to 0.20 g D20/g protein. The latter hydration marks the level where dried lysozyme recovers biological activity. Hexokinase is another protein with easily identifiable domains similar to those of lysozyme. Neutron scattering experiments (Jacrot et al., 1982) yielded a similar result to that obtained by Bartunik et al. (1982) on lysozyme. Jacrot et al. found that on binding to a substrate (glucose) there was a decrease in intensity in the low-frequency region below 40 cm- ~ and this was attributed to a decrease in the number of low-frequency modes. There was also evidence of the disappearance of a weak band centered at 30 cm- 1 when the cleft closed around the substrate. Chymotrypsin has domains that blend into one another and one might expect domain motion to be higher in frequency than in lysozyme because of the larger amount of contact between the three domains defined by computer algorithms (Janin and Wodak, 1983). Chymotrypsin is a digestive enzyme. The active site of chymotrypsin is in a cleft of this roughly spherical protein (Mathews et al., 1967). Utilizing the theory of elasticity for a freely vibrating sphere, an inelastic neutron scattering peak observed at 20 cm- 1 was ascribed to a shear mode in which the top and bottom hemispheres of the protein rotate in opposing directions (Martel and Ahmed, 1988). These authors speculated that when a polypeptide enters the active cleft, strong shear forces are available to aid in the chemical cleavage process. It remains to be seen if simple interpretations such as this one are valid; more experiments on many different enzymes are obviously desirable. An examination of the computer-generated results in Fig. 5 of the paper by Janin and Wodak (1983) shows that there are very many candidates available for such studies. (c) Allosterism Often when some of the subunits in a multimeric protein have undergone an interaction

136

P. MARTEL

with an external molecule the remaining subunits of the protein are promoted into a state of enhanced activity vis-d-vis further interactions with more of the external molecules. This is characteristic of a phenomenon known as allosterism. Conventional theories tend to regard allosteric effects solely in terms of sudden changes in structure not related to any precursor motion involving large scale vibrations. In the case of enzymes, a sudden change in the structure of one subunit is said to exert pressure on adjacent subunits and thus the latter become alerted and conditioned to the presence of substrate. Many undergraduate textbooks offer hemoglobin as an example of the phenomenon. Mammalian hemoglobin is commonly composed of two a and two fl-subunits. More than 80% of the amino acid residues in hemoglobin are in c~-helices. Older textbooks state that the four equilibrium constants, Ki(i= l, 4), for successive addition of 0 2 molecules to the four subunits can be ranked: K 4 > K 3 > K 2 > K 1 with K 3 -,~K 2, in order of magnitude. In actual fact (Chou, 1988) K 3 is an order of magnitude less than K 2. Making use of his "quasi-continuity" theory of vibrations in or-helices, Chou (1988) has advanced a simple, yet plausible explanation for the observed magnitudes of the Kis. The main idea is that subunit displacement is driven by resonant energy transfer. If there is resonant energy transfer associated with ligand binding then this energy transfer will tend to take place between like subunits having the same accordion-like modes. For the a-subunits the frequencies for the principal helices are found to be, 9.4, 43.3, 40.2, 44.7 and 7.9 c m - 1. The corresponding values for the fl-subunits are 10.7, 44.2, 38.0, 44.4 and 8.0 cm -1. The differences in these two sets of frequencies fall within 2.2 c m - 1 but according to Chou they are different enough for resonance effects to be confined to like subunits. Since the induced activation process proceeds more easily between two similar subunits it follows according to Chou (1988) that it is now possible to have K 2 > K 1 and K 4 > K 3. However, because the second subunit (a2) and the third subunit (ill) are different, K3,~K 2, as is observed experimentally. The basic idea behind Chou's method is to make use of known values for hydrogen bond force constants and to apply these to equivalent topologically transformed systems of vibration. Figure 3 illustrates how the method has been applied to a-helices. The average frequency calculated by Chou for both types of subunit was 29 c m - 1. A single broad peak in the neutron response at 27 _+3 cm - 1 has been observed for dry hemoglobin at 77 K (Martel et al., 1991) but at room temperature no peaks occur in this region, presumably because of damping. However, it seems plausible because of the pairwise similarity of the subunit frequency responses, that the subunits will maintain similar overall relationships in frequency under similar damping conditions. This possibility should be explored in future experiments employing high resolution and very low-frequency capability. Another example of energy channelling by resonance transfer involves energy exchange among the 12 fl-barrels that make up most of the structure of IgG. This antibody molecule exhibits "trigger" and "chelate" effects (Chou, 1987) that suggest energy transfer between the four fl-barrels in the F c stem and the four fl-barrels in each of the Fab arms. Since the Fab arms of IgG can be detached by enzymatic treatment it would be interesting to see if the various parts of the protein do indeed have a similar frequency spectrum. It may be noted that there are many more modes than the accordion-like mode in structures such as the a-helix (Levy and Karplus, 1979). If there are interactions due to anharmonic terms in the force constants governing the frequencies there will be energy transfer from one mode to another. For a completely harmonic system there is no mechanism for energy dissipation. (d) Free energy considerations Many calculations and experiments indicate peaks in the raman and neutron scattering responses of different proteins at frequencies below 60 cm - 1. According to Chou (1988) these peaks could be due to "dominant" modes that are involved in the energetics associated with protein function. In an illustrative example involving the binding of insulin to its receptor, it has been shown (Chou, 1988) that there is a deficit in the Gibb's free energy of - 8 kcal/mol, in the binding process. However if each of six vibrational degrees of freedom generated after

Biophysical aspects of neutron scattering

137

(a)

(b)

FIG. 3. Schematicof an ~t-helix(a) showinghydrogenbonds approximatelyparallel to the helixaxis. Chou (1984a)has shown how an equivalentplanar structure (b) can be derivedfrom (a) to set up an effectivehydrogen bond spring model for accordion-likemodes of vibration. binding has a frequency between 10 and 50 c m - 1, they will contribute a free energy (termed AGpho,) of - 1 4 . 7 to - 8 . 7 kcal/mol. This will then balance the energy equation. As the frequency is lowered, AGpho, can become exceedingly large. Because myoglobin and hemoglobin may have very low-frequency "dominant" modes at physiological temperatures there is the possibility of considerable energy exchange upon the binding of oxygen to these proteins. It is known that large displacements do occur in the heme region when oxygen is bound to myoglobin (Ansari et al., 1985). The propagation of these displacements has been depicted as occurring along the 0t-helices of this molecule. We conclude this section with the observation that besides the well-known raman mode at ~ 50 c m - 1 there is a newly discovered mode in water below 50 c m - * (Rousset et al., 1990; Walrafen, 1990) that could supply energy to help break weak bonds between protein strands and surfaces. An interesting feature of this newly discovered low-frequency band is that its frequency decreases as the temperature falls. At 40°C its frequency is ,,~ 15 c m - 1 and at 10°C, ,~ 9 c m - i . Given adequate coupling between protein and water and such an uncommon temperature dependence, it might be speculated that a feedback mechanism exists for enhancing phonon contributions to free energy so that proteins and/or enzymes can retain some functionality during swings to low temperature. IV. T H E N E U T R O N

M E T H O D IN B I O P H Y S I C S

1. General Remarks on Neutron Scattering Theory The purpose of this section is not to give a detailed account of neutron scattering theory but rather to point out basic principles necessary to better understand recent developments

138

P. MARTEL

in neutron scattering from proteins. As the theory unfolds examples will be cited to show how it can be applied in biophysics experiments. Possibilities for new experiments will be suggested. References will be given to make more detailed information available to the reader. For those readers who have no previous experience with the method the following should allow them to enter into a knowledgeable collaboration with scientists at most neutron scattering centres located throughout the world. These centres usually have personnel with a biophysical background who have the skills and computer programs necessary to help in interpreting the data. Neutron scattering from the jth atom in a given sample depends on a quantity called the cross section, a j, which is a measure of the strength of the interaction between the nucleus of that atom and an incoming neutron. The cross section is equal to 4n (b j) 2 where bj is the scattering length of the jth nucleus (Dolling, 1975). The quantity, b j, is a function of the internal structure of the nucleus and can vary from a value nearly zero, to 25 fm (1 f m = 10 -15 m). For hydrogen bj is 25 fm. Unlike X-ray scattering cross sections, neutron scattering cross sections do not show a monotonic variation as a function of atomic number. Tables of values of bj can be found in the literature (Sears, 1986). The total scattering cross section for a given nucleus is usually made up of coherent and incoherent components with their corresponding bjs. The coherent cross section of the nucleus allows the experimenter to study, and make use of, interference effects due to the order found in organized matter such as crystals. The incoherent cross section arises because of the presence of different nuclear spin states and isotope effects (Sears, 1986) and this component of the cross section cannot be utilized to determine crystal structures by neutron diffraction. It is customary to express various derivatives of the cross section, a, in terms of structure factors designated by the letter S. Before going into the details of differential cross sections it is important to note that in practice, a great deal of the neutron scattering method can be understood in terms of simple equations that are based on conservation of momentum and energy. In the case where only one quantum of energy of a crystal vibration (a phonon) is created or destroyed these conservation conditions are (Dolling, 1975): •

k o - k 1= Q

(la)

x+q=Q

(lb)

E o - E 1 = + hog/(q)

(lc)

where ko and k 1 are the wavevectors (magnitudes =2n/neutron wavelength) of an incident and outgoing neutron respectively, Q is the wavevector transfer, • a reciprocal lattice vector and q the wavevector associated with a vibrational mode, i, through a dispersion relation 09= Col(q). Multiplying both sides ofeqns (la) and (lb) by h (Planck's constant divided by 2n) yields equations that simply express conservation of momentum. Equation (lc) expresses conservation of energy involving the energy of the incoming neutron, E o, the energy of the departing neutron, E 1, and + h~oi(q), the energy of a vibrational mode or phonon that has been created ( + sign) or destroyed ( - sign). The signs imply that the neutron has lost energy ( + ) or gained energy ( - ) . Note that T has the dimensions of inverse distance and if the separation of lattice planes in any given direction is d, then T= 2n/d. The latter relation applies to crystals only. If the scattering is elastic, k o = k 1, and the magnitude of Q is (4n/2) • sin(0), where 20 is the scattering angle and 2 is the neutron wavelength. Equation (lb) is useful for describing coherent scattering arising from interference effects that depend on structure. The incoherent component of the scattering cannot be interpreted directly in terms of specific ~s and qs and less detailed information is obtained than in a coherent scattering experiment where we can take full advantage of the fact that a protein is fixed in a given orientation in a crystal lattice. When the latter obtains, we have the possibility of studying details of intramolecular and intermolecular vibration as a function of q. Such measurements are sensitive to all components of inter- and intramolecular potentials. Note that ~o can be zero, and in that case we have either incoherent or coherent elastic scattering.


139

Useful information about biological systems requires some use of the cross section, tr, but more especially its derivatives which account for the intensity of the neutron scattering as a function of structure. The structural dependence is usually expressed in terms of structure factors denoted by the letter S. Thus elastic scattering is described by the equation: dtr ~-~ = S(Q)

(2a)

where df~ designates a differential of the solid angle of scattering and S(Q) is the static structure factor. Inelastic scattering is specified by the equation: datr - - = (kl/k2) • S(Q, co) df~dE 1

(3a)

where S(Q, o9) denotes the dynamic structure factor. (Equations applicable to elastic scattering and inelastic scattering have alphanumeric designations beginning with the numbers 2 and 3, respectively.)

2. Elastic Scattering (a) Coherent elastic scattering (i) Review of theory. The S function in eqn (2a) will assume different forms depending on whether we are focusing on coherent or incoherent scattering. For coherent scattering from a structure made up of a repetition of a unique unit cell, i.e. a crystalline material, the structure factor for each cell is: S¢oh(Q = z)= ~ b~°h' exp[iz • rs] . e x p [ -

I,V~] 2.

(2b)

In this equation b~°h and rs, are the coherent scattering length and the equilibrium position of the sth atomic nucleus in the unit cell, respectively. The first exponential depends on the atomic arrangement in the unit cell and the last exponential in (2b) takes account of a dimunition in elastic peak intensity as Q increases; 2 w~ = Q2. (Ax if) (Hartmann et al., 1982) where (Ax 2) is the mean square displacement of atoms s from an equilibrium position in the lattice. By means of eqn (2b) neutron scattering can be used to determine information on the location of atoms in materials--in particular hydrogen atoms in proteins. These are detected by X-ray scattering with great difficulty, especially in large molecules. The locations of hydrogen atoms by neutron diffraction is often accomplished by exchanging deuterium (D) for protium (H) in a given protein and then searching for large localized peaks in differences of the two fourier maps corresponding to data sets for the undeuterated and deuterated protein. Because b~°h(D)= + 6.674 fm, whereas b~°h(H)= -3.740 fm, prominent intensity differences are found at hydrogen sites. For example, such intensity differences permitted verification of the suspected existence of a hydrogen bond binding the "distal" histidine in myoglobin to oxygen (Phillips, 1984). This important bond stabilizes the heme-oxygen complex in oxymyoglobin. (ii) Vibrational amplitudes. Many other coherent elastic neutron scattering experiments by Schoenborn and collaborators have provided information on dynamic properties through the Debye Waller factor, exp [ - W,], in eqn (2b). For example, coherent elastic scattering results on carbomonoxymyoglobin (Hanson and Schoenborn, 1981) indicated that the averages of mean square displacements, (Ax2~, associated with heavy atoms in the backbone and amino acid side chains were 0.144 and 0.158 A 2, respectively (1/~ = 0.1 nm). For hydrogen atoms in the side chains the corresponding value was 0.161/~2. The larger amplitudes suggest that the residues in side chains are less strongly bound than those in the backbone. They also indicate that hydrogen atoms tend to move with amplitudes similar to the heavy atoms to which they are attached. As we shall see below, the latter result is

140

P. MARTEL

important for the interpretation of incoherent inelastic neutron scattering data from proteins. (iii) Hydrogen-deuterium exchange. Diffusive motion can be probed by following the time dependence of H/D exchange in proteins soaked in D20 by means of coherent elastic scattering measurements (Kossiakoff, 1984). It has been shown by Kossiakoffthat exchange results from localized conformational mobility, involving "regional melting" in which small numbers of hydrogen bonds break and reform. It is found that exchange is not much greater in open regions of the protein where one might expect easier access for water. Small angle neutron scattering (SANS) is another form of coherent elastic scattering that we shall only mention in passing although it also could be used to monitor very low frequency dynamics in a way that has not been exploited as yet. This type of scattering can be shown (Jacrot, 1976) to be proportional to (p~-po) exp ( - Q 2 . R 2 ) , where p~ and Po are the neutron scattering length densities (~ b~°h/volume) of the sample and its solvent, respectively. In this formula, Rg can be taken to be a measure of the overall size of a solvated globular protein. If an undeuterated protein is dissolved in a bath of D20, analogous, but less detailed measurements similar to those of Kossiakoff, can be carried out because the signal will change as deuterons from the bath exchange with protons in the protein. In other words, (p~- po) can change drastically especially in the first hour after dissolution (see Fig. 4). One 2

"

5200

4800 Z 0

t,400 t,O00

10

' 20

30

L0

0

60

HOURS AFTER DISSOLVING IN D20

Fzo. 4. Decrease in small angle neutron scattering intensitybecause of hydrogenexchangeeffects followingdissolutionof superoxidedismutasein heavywater (D20).

would expect that there would be different rates of exchange depending on the number and sampling times of different conformational states in various proteins. Initially, these SANS measurements would therefore be comparative, and might also involve studies as a function of pH. (b) Incoherent elastic scattering For incoherent elastic scattering, Si,¢(Q) is no longer an explicit function of structure and so the reciprocal lattice vector z does not appear in the expression for the structure factor which therefore simplifies to: S~,c(Q) = 4rt ~(b~nC)2- e x p [ - 2 W~]

(2c)

where b~"¢ is the incoherent scattering length of the sth atomic nucleus. In applying eqn (2c) to proteins it is important to note that the incoherent scattering cross section (4n • (b~"~)2) of hydrogen is an order of magnitude greater than that of most atomic nuclei. Because approximately half of the atoms in a typical protein are hydrogens, the elastic scattering from an undeuterated protein is predominantly incoherent. Furthermore, for large scale


141

displacements the hydrogen atoms "ride" on the heavier C, N or O atoms to which they are attached. Thus (Smith et al., 1990b) for a powder specimen the elastic incoherent scattering is simply related to the isotropically averaged mean square displacements, (Ax2), by the equation:

Si,c(Q ) = C" e x p ( - Q 2 ( A x 2 ) )

(2d)

where Q is now a scalar and C a constant which depends on the cross section and the number of hydrogen atoms in the sample. As pointed out by Smith et al. (1990a) measurements analysed with eqn (2d) yield (Ax 2) values that contain no direct static disorder contribution, in contradistinction to coherent neutron scattering measurements and X-ray measurements which contain an exp(ir • L] term (see eqn 2b). This is an important result for protein studies at low temperatures where these molecules can exist in many slightly different structural configurations (Frauenfelder et al., 1988). Differences in (Ax 2) values from coherent and incoherent elastic scattering are a measure of the magnitude of configurational differences. It has been found in experiments on myoglobin (Doster et al., 1989) that eqn (2d) is only applicable between 4 K and 180 K, the temperature interval where the dynamics are purely vibrational. At higher temperatures there appears to be a dynamic transition arising from the excitation of non-vibrational motion which the authors have interpreted as torsional jumps between two states of different energy. If these states have probabilities for hydrogen occupancy of Pl and P2, and if the jump distance of the hydrogen atom is d, then it has been suggested that the following modification of (2d) applies:

S(Q ) = [ e x p ( - Q2 ( Ax2) )]{1-2plp2(1 -sin(Qd)/Qd)}.

(2e)

By applying this formula to their results, Doster et al. concluded that the enthalpy change, AH, associated with jumps between the states 1 and 2 is of the order of 12 kJ mol-1 and the jump distance ~ 1.5 A. This large value of d suggests involvement of torsional degrees of freedom or large dihedral angle fluctuations. Such events were predicted some time ago by molecular dynamics calculations (Levitt, 1983) and have been confirmed by more recent work (Loncharich and Brooks, 1990). In view of these results it seems very likely at this time that many, if not all, of the substrates, first observed by Frauenfelder and collaborators, are associated with a variety of dihedral angles that place certain residues at different separations greater than those suggested by (Ax E) values for purely vibrational modes. It is interesting that certain residues such as Asn, Glu, Lys, Pro and Thr are favourable loci for such jumps (Loncharich and Brooks, 1990). It may be noted that other measurements on dry myoglobin and hemoglobin (Martel and Lin, 1989; M artel et al., 1991) are consistent with a variation in Q of the form given by eqn (2d). Wet hemoglobin at high temperatures no longer shows a simple exponential variation in Q consistent with this equation. Instead, plots of the logarithm of the intensity vs Q2 indicate curvature consistent with eqn (2e).

3. Inelastic Scattering (a) Coherent inelastic scattering (i) The phonon r~girne. Neglecting delta functions that express the conservation conditions in eqns (1a), (1b) and (1c) above, the coherent inelastic structure factor involving the creation of one phonon mode, j, with wavevector q is described (Dolling, 1975) by: S¢oh(Q, 09) = ({nj + 1}/coj) ~ b~°h e - Wsms 1/2[Q . ej(s, q)-lcxp[i(x • rs)-]12 $

(3b)

I

where nj is the Bose-Einstein distribution function, [exp(hco~/KT)-1]-1, ej(s, q) is the polarization or normalized eigenvector of atom s in the mode (q, j), rn~ the mass of atom s in the unit cell and the other symbols have the same meanings defined above. At room temperature {nj + 1} varies approximately as 1/coj for modes where coj is less than 100 cm- 1.

142

P. MARTEL

For phonon annihilation {nj + 1} is replaced by {ni}. W~= (h/4ms).~qj]Q.ei(s, q)12(2nj+ 1)/~oj, but in practice e-w, is usually approximated by unity especially in the preliminary stages of dynamic or inelastic structure factor calculations. For any scatterer with more than one atom per unit cell, the most difficult, yet important, portion of (3b) to evaluate is [ Q . ej(s, q)]. This factor also contains the symmetry properties that make coherent inelastic scattering such a rich source of information. In order to determine the e~(s, q) for a complex structure with many atoms per unit cell it is customary to carry out lattice dynamical calculations (Cochran and Cowley, 1967). These calculations assume a "perfect" lattice with well-defined interatomic potentials, a condition that may be approached, but never completely met in protein crystals. It seems possible that such calculations could be useful in identifying different modes, but since no coherent inelastic measurements have yet been carried out on protein crystals, such a possibility cannot be accepted with certainty at present. In principle, another approach to interpreting the experimental observation of coherent inelastic peaks involves the determination of S(Q, 09) by molecular dynamics simulations with Q sampled at values such that Q = ~ + q (cf. eqn lb). So far such a calculation has not been attempted for even small proteins like BPTI, possibly because of the large number of lattice cells that would be required to calculate the frequencies at small qs. In carrying out coherent inelastic measurements on native protein the experimenter is faced with high backgrounds because of the large incoherent scattering cross section of hydrogen. Besides high general background levels, spurious peaks (Dolling, 1975) are often produced. In order to increase the ratio of coherent to incoherent signal, high experimental resolution is necessary. An alternative is deuteration which can be effected for certain proteins derived from algae (Bellisent-Funel et al., 1989) and lower life forms grown in deuterated medium. Having obtained deuterated protein one is still faced with the necessity of growing large protein crystals with dimensions typically 3 mm or more. Of course demands on minimal crystal size depend on neutron source intensities and if these were to increase, smaller crystal sizes would be acceptable. When a deuterated sample is obtained, as it was for glycine (Powell and Martel, 1979) dispersion curves can be measured. The dispersion curves for modes propagating in a direction approximately perpendicular to layers in which the glycines are hydrogen bonded are shown in Fig. 5. This direction is designated b*. Pairs ofintermolecular hydrogen bonds also exist along b*, forming anti-parallel "double layers" (Jonsson and Kvick, 1972). Dispersion curves such as those in the figure can yield much detailed information. Individual dispersion curves extending across the full range ofwavevectors are referred to as "branches". When q ~ 0 the phonons are of very long wavelength and long range forces between molecules are being sampled. When q --~qmaxthe phonon wavelengths correspond to distances that are interatomic and short range forces are important. It is remarkable how much more difficult it is to utilize force constant models or interatomic potentials to account for the whole of a dispersion curve rather than just the portion at q = 0, which can often be measured by either infrared absorption (IR) or raman scattering. Frequently, potentials or force constants that give excellent fits at q = 0, do not do so at finite q. Quite often one finds the frequencies of many low-lying branches dipping to zero when q = 0 models are extended to finite q-values. Thus dispersion curves offer very critical tests of interatomic potentials. Even without detailed model calculations, dispersion curves can yield a surprising amount of dynamical information. Coherent inelastic scattering also provides information about optical mode frequencies. The branch labelled (d) has a frequency of ~ 0.5 THz (16.5 cm - 1) at q = 0. Because raman measurements did not detect a mode there, it was possible for the authors to deduce that this frequency belongs to a previously unobserved IR-active mode. Various symmetry properties of the lattice can allow other deductions. For example, modes propagating along b* may be classified as symmetric or antisymmetric (Powell and Martel, 1979) and branches of the same symmetry cannot cross. This mutual exclusivity of symmetric and antisymmetric modes is often found in solid state physics, Further, for this structure, any branch and its folded continuation (from q=q~ax to q=O) have opposite symmetry character. Making use of these rules and the shapes of the dispersion curves it was


3

I

I

I

143

I

-r

b

>-

e

w

d

]

--

__

d

0 //I

0.2

I

I

0,4

0.6

I

0.8

1.0

q/qmax FxG. 5. Experimental intermolecular mode dispersion curves in a deuterated glyeine crystal (Powell and Martel, 1979) for modes propagating in a direction approximately perpendicular to layers in which the glycines are hydrogen bonded. The open circles represent raman scattering data.

possible to predict that yet another IR-active mode must exist with a frequency around 30 c m - 1.

The branches that decrease in frequency to zero as q approaches zero, are acoustic branches and their slopes are determined by the velocity of sound. The slope of branch (c) yields the longitudinal velocity of sound in the b* direction. Branch (a) yields the velocity of one of the shear modes of sound propagation. Elastic constants can be determined from these velocities and these can be used to compare interlayer forces in various biological structures. If one has a lattice dynamical model then detailed identification of the various modes measured by coherent scattering can be undertaken (Martel et al., 1990). There is an additional bonus in having such a model as it can also be used to identify peaks semiquantitatively in the incoherent inelastic scattering. This possibility arises when the frequencies of certain branches have little or no dispersion (i.e. dependence on q). This lack of dispersion produces peaks in the density of vibrational states that can be identified with peaks in the incoherent inelastic scattering (Peticolas, 1978). There has been some success in carrying out coherent inelastic neutron measurements on undeuterated crystals of various DNA base derivatives having hydrogen present in abundances typical of proteins (Martel et al., 1990). The difficulties involved in these experiments have been assessed in terms of the possibility of carrying out similar measurements on crystallized proteins. These authors see no insurmountable difficulties in carrying out similar experiments on acoustic modes in undeuterated proteins. Such measurements would be relevant in view of diffuse X-ray scattering measurements (Caspar et al., 1988) indicating that typical protein molecules like insulin have a fluid-like consistency in the crystalline state where water is usually present at the 50 wt% level. Since both the protein and the water are now fluid these crystals should be unable to support shear modes. However, analysis of mean square vibrational amplitudes and Young's modulus measurements (Morozov and Morozova, 1986) suggests that protein molecules rotate and vibrate, in large measure, as rigid bodies. If proteins in crystals are rigid and if there is considerable intermolecular contact there must be considerable resistance to shear. The values of the elastic constants associated with this shear (be it small or zero) could be obtained from coherent inelastic measurements of transverse acoustic modes even on undeuterated single JPB 57:3-8

144

P. MARTEL

crystals. Such measurements would obviously help to determine whether protein molecules have sufficient rigidity to participate in "lock and key" mechanisms or serve merely as liquid globules that act as amoeba-like sites for biochemical reactions. There is no doubt that much remains to be done in applying coherent inelastic scattering to protein dynamics. In particular, the nature of the low-frequency peaks (near 29 cm-1) observed by raman and incoherent inelastic neutron scattering may never be completely understood until coherent inelastic neutron scattering measurements are carried out on fully deuterated single crystals. Low-frequency modes in deuterated protein samples have been measured by coherent inelastic scattering from concentrated solutions containing up to 0.5 g D20/g protein (Bellissent-Funel et al., 1989). The protein was fully deuterated C-phycocyanin isolated from Synechococcus lividus (Crespi, 1979). The coherent scattering was weak in dry samples and diminished considerably with 50 wt% water present. The inelastic peaks were tentatively interpreted as due to short-lived coherent acoustic mode excitations with velocities that depended on water content in a manner quite unlike that observed in Brillouin light scattering measurements on DNA (Maret et al., 1979) and rat-tail tendon fibers (Cusack and Lees, 1984). As the water content in the latter samples was increased, the sound velocity tended to that of pure water (~ 1500 m/sec). With C-phycocyanin the addition of water raised the velocity from 2200 m/sec for the dry protein to a value consistent with 3300 m/sec determined in previous neutron work on pure D20 (Teixeira et al., 1985). However, the modes measured by Teixeira et al. (1985) were not those of ordinary sound and have been interpreted as propagation within hydrogen-bonded "patches". It may be noted that C-phycocyanin does have a central solvent channel but it is a compact heterodimer having the overall shape of an oblate ellipsoid. On the other hand, B-DNA is a long double-helical molecule with ordered water bridge structure (Kopka et al., 1983) and rat-tail tendons are made up largely of long triple-helix collagen fibers that can accommodate water between the strands of the helix. The authors of the work on C-phycocyanin offered a qualitative interpretation based on the existence of short-lived "patches" of contiguously hydrogen-bonded water molecules in the water of hydration surrounding the protein. They suggested that this water of hydration was more solid-like than bulk water and therefore capable of sustaining collective modes of higher frequency. It may be noted that the velocity of longitudinal propagation in these "patches" is similar to that in ice (Teixeira et al., 1985). (ii) Low-frequency stochastic motion. In biological systems, because of the abundance of hydrogen with its large incoherent cross section, very low frequency diffusive motions are usually studied by incoherent scattering as outlined below in Section (IV.3.b.ii). However "spin-echo" spectrometers now permit coherent scattering studies of these motions. These spectrometers can be found at the Institut Laue-Langevin (Grenoble) and the Laboratoire L6on Brillouin (Saclay) in France. A simplified version of the theory of the method has been given by B6e (1988). These instruments can sample frequencies in the MHz range. A spinecho experiment (Alpert et al., 1985) on IgG was described above. The response of this instrument is in real time, and the results are often analyzed in terms of the diffusional properties of the system under study. Other test experiments have been reported on the diffusion of hemoglobin (Hb) in cells and in aqueous solution (Alpert, 1980). It has been found that the diffusion constant of lib in O20 is less than in H20. It was concluded from the Q-dependence of the scattering from Hb in red blood cells that there is organized structure of the protein within the erythrocytes. An interesting "spin echo" experiment on deuterated C-phycocyanin has indicated the onset of spatial correlations resulting from hydrationinduced inter-subunit motions in this multimeric protein (Middendorf et al., 1984). The method of hydration (with H20) in this experiment was the "inverse" of that commonly used when incoherent quasielastic measurements (see below) are made on native protein hydrated with DzO. This type of experiment provides an opportunity to study the activation of inter-subunit motions like those that may be associated with the allosteric effects described earlier.


145

There are other coherent quasielastic experiments that could yield information on interdiffusional as well as cooperative motion in the same system (Borsali et al., 1989). These would involve contrast variation and spin-echo measurements on binary systems. Such a system is the 30S subunit of ribosomes. Although the 30S subunit is not a 50-50 combination of RNA and protein, (it is approximately 65-35) recent measurements on diblock polymers (Borsali et al., 1989) do suggest a two-stage experimental approach. In the first stage, relaxation oscillations of the subunit as a whole, the so-called cooperative mode, could be measured in pure D20. Then in analogy with the experiment of Borsali et al. (1989) when sufficient H 2 0 is added on-line to make the subunit as a whole "invisible" (i.e. po = Ps, see above) the relaxation oscillations of the RNA and protein beating against each other would be seen at Q-dependent frequencies that would be a measure of the binding forces between these two components. We calculate that the 30S subunit of E. coli becomes "invisible" when the aqueous solvent contains 55% D 2 0 by volume. (b) Incoherent inelastic scattering (i) General comments on frequency domains. Because approximately 50% of the atoms in native proteins are hydrogens, and because hydrogen has an incoherent scattering cross section an order of magnitude greater than that of the other atoms in proteins (in units of 10-24 cm 2, tri, c for H = 79, D = 2.0, C = 0, N = 0) most inelastic neutron scattering studies have focused on incoherent inelastic scattering techniques. This field of study may be somewhat arbitrarily divided into two r6gimes (cf. Fig. 6): (1) Very low-frequency "quasielastic" scattering. (2) The vibrational r6gime.

,/ELASTIC A ,.D r.nD w )-.

Z I,--

z

j

O.UASIELASTIC

iI I

i

/

/i

f

"1"

I

-100

f

/

I

x ,,ll / /

/

-50

50

100

w (tin-1) FIG. 6. The three regions characterizing neutron scattering from proteins. When the elastic intensity decreases due to changes in vibrational amplitude, the quasielastic and inelastic regions usually increase in intensity.

146

P. MARTEL

The concept of a pure vibrational r6gime is difficult to define because molecular motions can also involve translations and rotations of the protein molecule as a whole. As analysis shows (Morozov and Morozova, 1986) the division of S(Q, ~o) into three classes (rotation, translation, vibration) is somewhat arbitrary, especially in the case of "soft" protein molecules in solution at physiological temperatures. A formulation of normal mode molecular dynamics in terms of rigid molecules that can rotate and translate as rigid entities (Pawley, 1986) is likely to be a poor approximation for "soft" protein molecules. In recent developments of the theory (Smith et al., 1986, 1990b) explicit contributions from rotation and translation of the protein molecules as a whole are neglected in the vibrational r6gime. In the quasielastic r6gime, translations and rotations of the molecule are often described in terms of diffusional constants, D T and D r, that refer to translation and rotation, respectively. The idea that D T could be associated with periodic motion of the protein molecule as a whole is difficult to visualize unless one conceives of these vibrations as randomized or "stochastic". The same qualification applies to rotational oscillations, although these may be more periodic especially if the molecule is spherical since viscous damping is probably small in this instance (Persson, 1986). In any case the boundary between the quasielastic and "true" vibrational r6gimes is ill-defined and there is little doubt that the latter often contributes to the neutron scattering observed at very low frequencies near, or less than, 1 cm-1. In practice, the scattering is arbitrarily compartmentalized by experimentalists in search of tractable methods to present their data. In principle, all inelastic scattering can be described in terms of fourier transforms of space time correlation functions, G(r, t), (van Hove, 1954) but a detailed knowledge of this aspect of the theory is not necessary for an appreciation of experimental results, except possibly if one is seeking biophysical mechanisms by means of molecular dynamics calculations with computers (Smith et al., 1990b). Figure 6 presents a visual summary of the three types of neutron scattering often referred to by experimentalists. As noted above, the boundary between the quasielastic and "true" vibrational r6gimes is ill-defined, even with the best instrumental resolutions available, because complex systems like proteins may have very low frequency oscillatory features. Therefore characterization of the scattering as in Fig. 6 is qualitative. At low temperatures, the elastic intensity is very strong because the mean square displacements, (AXE), are small relative to values at room temperature. Recall of eqn (2d) confirms this. Separation of the elastic intensity from the quasielastic also presents a problem in some materials, but in proteins this problem is less serious because the elastic scattering tends to dominate markedly over the other two types, especially at low temperature. The experiments described in the next section depend in large part on careful measurements of linewidth. In some cases the line broadenings range from 0.005 to 1 cm- 1. The motions that are monitored here are very slow and diffusive. Good experimental resolution is required. In the following section dealing with the low-frequency dynamics of heme proteins it is the frequency dependence of the scattering that is sought and in experiments of this type the resolution may be relaxed somewhat to increase the observed intensity on the flanks of peaks that are broad at temperatures above 200K. (ii) The diffusive quasielastic r~gime: scattering from the water near protein. Here, the simplest formulation of S(Q, ~o) is in terms of the translational diffusion constant, D T, of a particle located at r--0 when t = 0 . According to Fick's law, the space time correlation function, G~(r, t), is then given by: Gs(r, t)=(4nDTt)-3/Eexp(--rE/4DTt). In this case Sine(Q, ~o) becomes: Sine(Q, to) = [ WtQ)/rr]/(co 2 + W2(Q)).

(3c)

Equation (3c) is the equation for a Lorentzian peak profile with a full width at half maximum, equal to 2W(Q), where W(Q)=DTQ 2. A plot of the full width at half maximum of the scattering intensity vs Q2 yields a slope equal to 2D T. The linear dependence of W(Q) on Qz is only observed for small Q-values. At larger Q-values there is often the onset of a downward curvature which may be accounted for by an


147

assumption that involves the existence of a residency time, z, between jumps in a so-called "jump diffusion" model (Springer, 1972). In this case W(Q) becomes: W(Q) = DTQ2/(1 + DTQ:r ).

(3d)

Typical values for D T and z are 2.4 × 10- 5 cm2/sec and 1.2 psec respectively in pure water at 20°C. Equation (3d) applies strictly to point particles only, but is utilized often in cases where rotational contributions to diffusion are small. For water at 21°C the diffusion constant can contain a rotational contribution of 4 to 15% (Trantham et al., 1984). These percentages are arrived at by assuming that the translational and rotational modes are uncoupled. If there is coupling, Sine(Q, co) can be expressed as a series expansion in spherical Bessel functions (Martel and Powell, 1981). In order to account for decreases in intensity as Q increases, eqn (3c) is often multiplied by an effective Debye-Waller factor of the form exp(-Q2 (Ax 2)) and the equation describing translation diffusion then becomes: Sine(Q, e~)=exp(-Q2(Ax2)) • [ W ( Q ) / T z ] / ( o 2 + W2(Q)).

(3e)

The inclusion of exp(-Q2(~,x2)) permits accurate extrapolations of intensities to Q =0.0 and fractions of different components (such as "free" and bound water) can be evaluated when a Debye-Waller factor is found for each component (Martel and PoweU, 1981; Trantham et al., 1984). Values for e x p ( - Q2(Ax2)) can be obtained from incoherent elastic scattering measurements evaluated by means of eqn (2d); however the instrumental resolution must be very good in order to ensure that the measured scattering is truly elastic. There must also be careful identification of background and coherent scattering contributions to the elastic scattering in order to obtain the net incoherent scattering of interest. With large protein molecules there is also a problem due to possible overlap between the quasielastic and vibrational r6gimes. Many subtleties connected with the analysis of quasielastic data are discussed in detail by Trantham et al. (1984). These authors succeeded in showing that at a hydration level of 1.2 g H20/g dry solids, some 10% of the water in the cyst cells of brine shrimp is "bound" with a diffusion constant more than an order of magnitude less than that of bulk water. For the determination of bound and free water fractions associated with protein, it may be noted that quasielastic incoherent neutron scattering, complicated as it seems, is superior in some ways to many other methods including certain nuclear magnetic resonance (NMR) techniques (Trantham et al., 1984). This is so because NMR measurements often require milliseconds or more of measurement time, t, and during this time "free" water molecules can diffuse distances equal to (2Dvt) 1/2 ~ 1/~m. On such large excursions the water molecule can collide with many protein boundaries and thereby lose the diffusional characteristics of"free" or bulk water. This effect does not occur in neutron scattering because the scattering event takes place in times much less than picoseconds and fast motions are thus observed directly on a very short time-scale. The problem of identifying different quasielastic components of neutron spectra was dealt with quite elegantly by Middendorf and collaborators (Randall et al., 1978; Middendorf and Randall, 1980; Middendorf et al., 1984) by using fully deuterated C-phycocyanin. Deuteration permitted study of very low hydration levels with light water. The hydration levels varied from 110 to 310 H 2 0 molecules per subunit mass (29,000). Monolayer coverage would require 500 to 600 H20 molecules. The quasielastic lines were very narrow and plots of W(Q) vs Q displayed a solid-like behaviour that was analysed with a lattice diffusion model (Chudley and Elliot, 1961) commonly applied to solids. The diffusion constant was very small and the residency time, very large. Plots of W(Q) vs Q exhibited minima, with a Qdependence that depended on hydration level. The existence of minima follows naturally from the Chudley and Elliot model. At low H20 concentrations it seems likely that the diffusion mechanism involves very slow exchange. Analysis of the data on phycocyanin (Middendorf et al., 1984) yielded residence times of the order of 5 to 30 nsec, and jump distances of the order of 5 to 9 A for water molecules assumed to migrate between patches of

148

P. MARTEL

hydration on the protein surface. These results suggest that water does not just stick to protein and remain localized after contact with the surface at ambient temperatures. Another approach to solving the problem of component indentification in incoherent quasielastic scattering is to subtract the spectrum of a protein dissolved in D 2 0 from that measured for an identical protein solution in HzO. It is claimed (Fletcher et al., 1988) that the spectrum due to water can then be obtained to an accuracy of 3%; this in spite of exchange that could cause the aqueous scattering to be "contaminated" by exchanged hydrogens from the protein. On the basis of the measured intensity ratios of the various components in the scattering from a well characterized aqueous solution of ~-chymotrypsin, these authors showed that the signal arising as a result of hydrogen exchange was small. However, because of scatter in the narrow spectral component due to "bound" water it was not possible to make a clear assignment of the motion as translational or rotational in the manner outlined by Trantham et al. (1984). Assuming pure translational diffusion, a Dv for the bound water ,--14% that of bulk water was obtained by Fletcher et al. (1988) for 21.7 wt% of ~-chymotrypsin in water. One of the merits of the approach of Fletcher et al. (1988) is that they were able to identify a diffusion coefficient of~-chymotrypsin itself ~ 3% that of pure water. They were also able to show for their 21.7 wt% protein samples that 450_+ 100 water molecules were bound per ~-chymotrypsin molecule. This represents 0.33 _+0.07 g of water per g of protein. The authors noted from chemical considerations, that the total number of hydrogen-bonding sites in the surface of ~-chymotrypsin is 393 and that only 50 or so water molecules are sufficiently well bound and intrinsic to the protein structure to show up in crystallographic studies. They therefore concluded that most of the waters sampled were attached to the surface. They pointed out that the surface area of chymotrypsin is sufficient for monolayer coverage of ,-~ 1000 water molecules. Fletcher et al. also carried out experiments on ~-chymotrypsin solubilized in the aqueous cores of small water-in-oil microemulsion droplets having a radius of 35 A. Since the overall dimesions of the molecules are ~51 x 40 x 40 A 3 the droplet walls were not far from the protein surface. The "oil" was the surfactant, sodium bis(2-ethyl)sulphosuccinate (AOT). It was found that the close proximity of the surfactant surface had very little effect on the mobility of the caged water molecules. It was concluded that there was therefore no evidence for any biologically relevant lipid effects of a dynamical nature that might be simulated by the presence of AOT. A recent paper (Giordano et al., 1990) describes measurements carried out on solvated lysozyme at the pulsed neutron source (ISIS) at the Rutherford Appleton Laboratory in England. Incoherent quasielastic scattering from an aqueous solution containing 10 wt% lysozyme indicated that 25% of the water in the solution had the low mobility typical of bound water. In this case the protein component of the scattering was subtracted by calculation. (iii) Quasielastic scatterin9 from heine protein in the presence of water. Many recent incoherent quasielastic measurements have concentrated on the low frequency motions in protein itself. Most of these measurements have been carried out on myoglobin (Mb). Molecular dynamics simulations have been employed (Smith et al., 1990a; Loncharich and Brooks, 1990) to analyse very detailed measurements carried out with very good resolution (Doster et al., 1989). There has also been a recent trend towards interpretation of the low temperature quasielastic neutron scattering from protein in the light of physical theories of liquids and glasses. Some biologists may object to these interpretations from a purist point of view, but such comparisons may be fruitful in providing a framework that could lead to a unified theory of these very complex phenomena. In order to describe their experimental results at energy transfers less than 25 cm - 1 Doster et al. (1989) used a Cole-Davidson function (Davidson and Cole, 1950). Si,c(Q, ~o) is then expressed as: S~.¢(Q, to)=A(Q) Im {(1 + itozo)-b}/~O

(3f)


149

where A (Q) depends on the cross sections of the scattering nuclei as well as on Q, "Ira" stands for "imaginary part of", i = ( - 1)1/2, and 3o is a time constant that is sensitive to long-term relaxations, so-called s-relaxations, with ToS greater than 50 psec. (A second time-scale r6gime with much smaller ~s of the order of picoseconds was identified with/~-relaxations due to local motions of atoms in cages formed by their neighbours.) Equation (3f) is somewhat more general than the lorentzian approximation and provides a compact general formulation that is convenient for comparing the scattering with molecular dynamics calculations and current theories of liquid-glass transitions. However explicit identification of the scattering into rotational and translational components of diffusion is no longer possible. Equation (3f) implies a power-law dependence at high frequencies with exponent - ( 1 +b), (for a lorentzian b= 1). In the original analysis of Mb data (Doster et al., 1989) a value of b of 0.25(+0.1) was found. This value was temperature-independent between 250 K and 350 K. The authors claimed that such a temperature-independent result argues against an explanation of s-relaxation processes in terms of a distribution of activation energies. Such a conclusion would provide support for a two-state model, consistent with their incoherent elastic scattering results, as discussed above. They also indicated that the shape of the total spectrum, produced by • plus/~ processes, is qualitatively consistent with results derived from a mode-coupling theory of the liquid-glass transition in a hard-core liquid (Bengtzelius et al., 1984; G6tze and Sj6gren, 1987, 1988). The main ingredient of mode-coupling theory, of interest for understanding protein dynamics, is that a-relaxation processes are collective effects resulting from non-linear coupling between density fluctuations. In some sense, therefore, this is a return to theories of large-scale modes of protein motion based on the theory of elasticity. Another conclusion arrived at by Doster et al. (1989) was that the diffusive motions in a protein resemble those in simple close-packed systems dominated by hard-core interactions. This is surprising in view of the fact that protein, like DNA, is in the first instance a linear heterogenous assembly of different structural units. However, because the b-value of 0.25 is unusually small (Doster et al., 1989) the resulting non-lorentzian form of S(Q, o9), indicates jumps between distinct energy states, and suggests that the geometry of the local motion is highly constrained by the strong interactions along the peptide chain. Thus the resemblance to simple close-packed systems was judged to be qualitative at best. It is interesting to note that the "~" scattering data of Doster et al. compares favourably with molecular dynamics simulations (Smith et al., 1990a) which suggest that the "~" response between 0.3 and 7.0 cm-1 can be approximated by a lorentzian (i.e. b = 1). Other molecular dynamics calculations (Loncharich and Brooks, 1990) suggest that b can have values anywhere between 0.11 and 0.27 in the temperature range 260 < T< 340 K. It has been suggested by Loncharich and Brooks (1990) that their variable results for b may be simply due to simulations that are not continued for sufficiently long times. Low energy response in molecular dynamics simulations is determined by the length of the simulations and these can be costly as computer time increases. Even if these termination errors are taken into account the two simulations cannot be easily reconciled. Several reasons have been suggested (Loncharich and Brooks, 1990) for the differences in the results of the molecular dynamics simulations designed to account for the quasielastic data from Mb. Among these Loncharich and Brooks suggest that their simulations are longer, have "explicit solvent" and use different potential functions. At higher frequencies these authors obtain a low temperature inelastic peak at 9 cm- 1 instead of 25 cm- 1 as obtained by other simulations (Smith et al., 1990a) and by experiment (Cusack and Doster, 1990). The fact that cold protein behaves like a glass may not be important per se as a biological phenomenon. What is important is the fact that temperature dependent transitions between configurational states of a protein may be studied at temperatures below 273 K, and categorized in a consistent manner by invoking recent theories of liquid-glass transitions. A blend of these configurations should characterize a portion of the protein dynamics at higher, more physiological, temperatures. From this point of view it seems worthwhile to study

150

P. MARTEL

temperature-dependent changes in proteins in the light of liquid-glass transitions to see how well these changes can be explained by mode-coupling theories (GStze and SjOgren, 1988). At low temperatures below 210 K, in myoglobin samples with 0.36 g DEO/g protein, the average atomic fluctuations are such that the protein may be considered to behave harmonically with (AX2) values derivable from harmonic theory (Loncharich and Brooks, 1990). At ~210 K, a glass-like transition in atomic fluctuations is seen in molecular dynamics calculations and at higher temperatures there is an increase in anharmonic atomic fluctuations. At physiological temperatures multiple atom transitions such as large z-rotations of side chains or large dihedral angle changes involving 0 and ~b (see Fig. 1) can occur (Loncharich and Brooks, 1990). As a result of a thermally-induced mix of local structural changes at high temperatures certain portions of the protein acquire a fluid-like character that seems to be important in myoglobin since it is known that the rate of oxygen diffusion in the molecule drops significantly at temperatures below 210 K (Austin et al., 1975). Above this temperature diffusion is assisted by density fluctuations (Doster et al., 1990). The theory used to account for diffusion in melted glasses may then be taken over to account for the escape of molecular groups out of "cages" formed by near-neighbours (Doster et al., 1990). The method used by Doster et al. (1990) to obtain the anharmonic component of the neutron scattering (due to large Z, 0 and ¢ angle changes) was based on the assumption that there were different length scales and temperature dependences for the harmonic and anharmonic components. Based on these assumptions it was possible to subtract a harmonic vibrational background from S(Q, co) and thus obtain a difference spectrum AS(Q, co) which has significant intensity only in the quasielastic region. It is AS(Q, co) that describes the glasslike properties of the protein. At high temperatures the simulations of Loncharich and Brooks (1990) suggest that AS(Q, co) accounts for ~60% of the atomic fluctuations in myoglobin. Noting that their value for b in eqn (3f) was definitely less than 1, and that this is the signature of a "stretching phenomenon" in glass-forming liquids, Doster et al. (1990) went on to evaluate the imaginary part of the dynamic compressibility defined as z"(Q, co) = coAS(Q, co)/kT. They showed that this quantity displays a minimum characterizing the crossover between low and high frequency power law expansions of AS(Q, co) and then using mode-coupling theory they obtained fits to their data that indicated that part of the protein behaved like a Lennard-Jones liquid at temperatures above 200 K. The variance in the fitting did not rule out a hard-core model (Barrat et al., 1989) as an alternative. The adequacy of mode-coupling theory is surprising and suggests that cage effects (largely independent of details of the force field) dominate the average short-time behaviour of protein dynamics. On the other hand one might argue that the success of this theory is not unexpected because it is being applied to a neutron scattering method (incoherent quasielastic) that is, by and large, nonspecific in much the same sense as specific heat measurements yield values that can often be accounted for by a variety of condensed matter theories. The lack of specificity arises in part because of the large number of overlapping contributions to the incoherent scattering in multi-atom systems. (This will be illustrated below in connection with incoherent scattering calculations, cf. Fig. 7.) If a more specific method such as coherent inelastic scattering were to be applied as outlined earlier, the details of the force field would be very important. Nevertheless it should be noted that a significant feature of the analysis in terms of mode-coupling theory is that it provides a compact description of spectral features that can be compared readily with molecular dynamics simulations. (iv) The glass-like behaviour of proteins at low temperatures. On the subject of the glass-like behaviour of proteins, it may be appropriate to consider other references bearing on this concept even though many of these are not linked directly to neutron scattering. Among these there is a paper suggesting that the hydration shell of myoglobin may play an important role in the coupling between the protein and the solvent so that the transition near 200 K may be considered as "slaved" (Iben et al., 1989). A related paper (Doster, 1987) indicates that


151

S (O.,w)

I

0

I

IlJlll lit i,,,k,,,....,. ...... : .................................. !

I

[

50

I

I

I

1

100

W (cm-1) FIG. 7. The neutron response, S(Q, 09) for an isolated molecule of bovine pancreatic trypsin inhibitor (adapted from Cusack, 1985). Curve B is the expected experimental result when the calculated line intensities in A are convolved with the spectrometer resolution.

there is a similar temperature dependence of (1) the O-D stretch frequency shifts for the infrared absorption of water in Mb crystals, (2) the dielectric relaxation of water in Mb crystals, and (3) the Lamb-Mrssbauer factor for the heme iron in Mb. The latter author concludes from these results that "freezing of hydrogen bonds" imposes severe constraints on conformational relaxation of the protein molecule. Other papers of a more theoretical nature tend to treat anhydrous protein as a spin glass system. The analogies between the temperature dependence of spin glasses and some of the static and dynamic properties of globular proteins have been explored (Stein, 1985, 1986). In particular the temperature dependence of the recombination kinetics of hemoglobin with CO has been accounted for (Stein, 1985) in terms of an activation energy distribution based on the spin glass theory of metastable states. Molecular dynamics simulations (Elber and Karplus, 1987) have indicated liquid-like regions, the loops and side-chain clusters at interhelix contacts in Mb, that sample various energy minima as the ~-helices move. These authors have suggested that "freezing" of these liquid-like regions could result in transitions to the glassy state. Some experimentalists (Martel and Lin, 1989) have interpreted their neutron scattering results in terms of this "freezing" phenomenon. It has been known for some time that for hydration levels of 0.3 to 0.5 g of water per gram of protein that "bound" water is not "ice-like" but mobile enough to produce narrow proton magnetic resonance signals at temperatures near 250 K (Kuntz et al., 1969). More recent nuclear magnetic resonance measurements (Andrew et al., 1983) have pushed the ice transition to temperatures below 200 K. These measurements have shown that water of hydration in contact with lysozyme is only completely frozen below 190 K and gains mobility above this temperature. Infrared absorption measurements indicate that amorphous ice in myoglobin films displays a broad glass transition between 180 and 270 K (Doster et al., 1986). Higher transition temperatures occur at hydrations (h) of 0.46 g of water per g of Mb, or greater. For h of the order of 0.22 g/g or lower, there is no observable infrared absorption indicating a transition. Presumably this is because the water clusters on the surface of the protein are too small to produce a cooperative glass transition. Measurements on crystals of Mb indicated that crystal water has a glass transition temperature near 220 K. JPB 57:3-C

152

P. MARTEL

In spite of their elegant measurements the authors (Doster et al., 1986) did not propose any definite conclusions about the microscopic nature of hydrate-protein interactions. While noting that freezing of conformational transitions might be related to freezing of hydration water, they also noted that melting involves breaking of hydrogen bonds in both water and protein. Therefore it might be possible that mobility changes can occur independently in both systems, albeit at similar temperatures. Another interesting observation is that M6ssbauer experiments with dry Mb samples yield a nearly temperature independent M6ssbauer fraction (Krupyanskii et al., 1982). This fact is consistent with observations (Doster et al., 1986) that the infrared spectrum of water associated with Mb at low humidities is independent of temperature. These results suggest a dynamic correlation between the heme iron and water of hydration, mediated by the protein matrix. It is interesting that enzymes only begin to be physiologically active for h-values greater than about 0.2 g/g (Rupley et al., 1980; Poole and Finney, 1983, 1984). This seems to be the critical water concentration for myoglobin as well. An extension of the pioneering measurements on Mb (Doster et al., 1990) as a function of h, above and below 0.2 g/g, should shed more light on the effects that hydration has on function. As indicated above (Section IV.2.b) there is evidence (Martel and Lin, 1989) that dry myoglobin does not show the same Q-dependence as the wetted protein; the Q-dependence of the latter has been interpreted with a model that postulates torsional jumps between two states (Doster et al., 1989). It seems that these states may not be well defined in the dry molecule. The suppression of ice formation at low temperatures by proteins depends on what amino acids are present (Doster et al., 1986). Alanine with its methyl side-chain is much more effective in this regard than glycine. Neutron scattering studies on H 2 0 solutions of deuterated alanine and deuterated glycine might yield valuable quantitative information on the nature of this effect. Such studies could utilize the quasielastic method described above, and/or an inelastic scattering method that monitors hindrance of librational water modes (Martel, 1982). (v) Review of the theory of incoherent scattering in the vibrational r~gime. Smith, Cusack and collaborators (Smith et al., 1986, 1990b; Cusack and Doster, 1990) have recently adapted the theory of incoherent inelastic neutron scattering (Zemach and Glauber, 1956) for the analysis of measurements on proteins. The general expression involving scattering by all combinations ofphonons (single plus multiphonon) is very complicated (Smith et al., 1986) and the corresponding Sine(Q, ~0) includes modified Bessel functions as well as summations over the various combinations. Therefore use is made of the fact that for small Q-values (in practice, Q < 2 ,~-1) the one-phonon contribution dominates and in this case analysis is carried out in terms of the one-phonon structure factor, Silnc(Q,_+o~): N

h[Q ej(s)[ 2 Sil (Q, _l_~)=e±h~j/-kr ~" (b~) i,¢ 2 e -2ws 2= 1 4o~im ~ sinh(hto~/2kT)

(3g)

where the c~(s) are normalized mass-weighted amplitude vectors for the sth atom vibrating along thejth normal mode direction in a molecule made up of N atoms. Because the ei(s) are now defined in a molecule fixed coordinate system, interference scattering between molecules is not accounted for, and no explicit description is given of either dispersion (q-dependence) or of modes that depend on intermolecular forces. The (+, - ) signs in eqn (3g) refer to neutron energy (loss, gain). Experiments using time-of-flight techniques (Cusack et al., 1988) usually involve neutron energy gain and those with the triple-axis technique (Martel and Ahmed, 1988) usually involve neutron energy loss. The value of Ws is given by W~=Z3N=f6[Aj(s) • QI 2 with: A j ( s ) =- [h/2egjms)coth(he~/2k T)] 1/2ei(s)

(3h)

where 3 N - 6 is the number of modes summed over in the molecule and the other symbols in (3g, 3h) have the same meanings defined earlier. The six modes omitted in the summation are

Biophysicalaspects of neutron scattering

153

three modes of translation and three modes of rotation of the molecule as a whole. Aj(s) is the amplitude of vibration of atom s in the jth mode. 1 Typical mode contributions (vertical lines) to Si,c (Q = 2 A - 1, o) for an isolated molecule of bovine pancreatic trypsin inhibitor (BPTI) at 298 K are shown in Fig. 7 (adapted from Cusack, 1985). These results were computed in the harmonic approximation using a normal mode analysis (Brooks and Karplus, 1983). The spectrum was averaged over six orientations of the molecule with respect to the incident beam. This simulates the scattering expected from molecules randomly oriented in a powder. If the IQ" cj(s)l a terms in (3g) do not vary greatly with 0, it can be shown that the overall frequency dependence of Sine(Q, 1 0) varies as 1/o 2 for ( h o / k T ) ~ . l . This dependence is discernible in spectrum (A) although there is evidence of some variations due to anisotropy of the ej(s). For harmonic oscillators the underlying reason for the 1/o 2 dependence of Silo(Q, 0) is that mean square displacements vary as 1/o I in the low frequency r~gime. Spectrum (B) is the result when the line intensities in (A) are convolved with a spectrometer resolution of 4.75 cm- t. By comparing (A) and (B) it can be seen that peaks in the incoherent inelastic neutron spectrum arise from two sources: (1) strong isolated modes such as the two lowest ones near 7 cm-1 and (2) a high density of fairly intense modes, discernible near 16 cm- 1. It may also be concluded that for a protein there is little likelihood of studying individual modes by incoherent scattering measurements in this low frequency region unless one possesses a spectrometer having extremely good resolution of the order of a cm-x. Unfortunately the neutron fluxes of present day reactors place a practical limit on resolution which make such measurements extremely difficult if not impossible. In the limit as Q approaches zero it may be shown (Smith et al., 1986) that the orientationally averaged value of Sine(Q, a 0), viz. (S(Q, o)), satisfies the relation:

lim{(S(Q,o)) -6o 9 } 3N-6 ~ ~ [ e x p ( h o / k T ) - 1] = Z

o~o

j=l

s= 1

~nez[cj(s)l2 6 ( o - o ~ ) .

(bs)

(3i)

ms

The quantity on the right hand side of (3i) is known as the amplitude weighted frequency distribution, G(o), and it is dominated by hydrogen atom contributions because of their large value of (b sine)2/ m s. The true frequency distribution, g(o), of the isolated molecule is given by: 3N-

g(o)= ~ J

6

5(o-oi).

(3j)

By averaging ~s(b~ne)21ci(s)12/m s over all modes in BPTI and scaling g(o) with this factor, it has been shown the g(o) and G(o) are quite similar in structure at low frequencies (Smith et al., 1986). This similarity is likely to be even closer for large proteins because these have many hydrogen atoms, distributed rather uniformly, that sample individual modes involving approximately the same overall atom motion (G6 et al., 1983) at low frequencies. In what follows the superscript and subscript will be dropped from Si~e(Q, 0). Methods for subtracting multiphonon contributions to the spectra at high Q-values can be found in the recent literature (Cusack and Doster, 1990; Smith et al., 1990b). It has been pointed out (Martel et al., 1991) that model calculations for low frequencies are best compared with S(Q, 0) rather than G(o) or g(o) since S(Q, 0) is the experimentally measured quantity. These authors showed how the factor o [ e x p ( h o / k T - 1 ) ] tends to suppress low frequency features in S(Q, 0) from appearing in G(o). On the other hand, very small fluctuations in S(Q, 0) tend to be magnified in G(o) at high frequencies. Although S(Q, o) presents the best test for calculations, the evaluation of g(o) is important for comparison with thermodynamic measurements. Also G(o), and more especially, g(o) can be used for comparisons with fractons, Debye behaviour, etc. Because there are so many atoms in a typical protein, calculations are often carried out under two assumptions. (1) The large incoherent scattering by hydrogen will dominate the neutron scattering response. This follows because half of the atoms in protein are hydrogens

154

P. MARTEL

and the incoherent cross section of hydrogen is an order of magnitude greater than that of the heavy atoms (principally, carbon and oxygen). (2) At low frequencies the hydrogen atoms follow the motion of the heavy atoms to which they are attached. This is the basis of the "extended atom" approximation (Brooks and Karplus, 1983). A caveat to the first assumption follows from elementary considerations. It turns out that although a dry hydrogenous, D-exchanged protein still has a total (coherent plus incoherent) cross section per atom some six times greater than that of D20, the protein signal gets "diluted" quickly upon addition of D20 (Middendorf, 1986). At a concentration of 10 g of D20 per g of protein, for example, ~ 2/3 of the total scattering will be due to buffer solution. For experiments with h =0.33 g of D20 per g of protein (Cusack and Doster, 1990) it has been estimated that some 98% of the incoherent scattering comes from unexchanged hydrogens in the protein. However there has been a suggestion that "self-coherent" dynamic structure factor contributions (Smith et al., 1990b) should be taken into account at low Q-values (for BPTI, Q < 0.8 A - x). Self-coherent scattering arises from self-correlations seen through the coherent cross section. For proteins the coherent cross sections are small relative to the incoherent cross sections and these effects are usually ignored. The "extended atom" approximation also is not without its caveats. One of these stems from the fact that certain hydrogens, like those of the methyl group, can rotate. More recent calculations have shown that when all atoms, including hydrogens, are treated explicitly the results are in better accord with experiment (Cusack et al., 1988). In the case of BPTI, inclusion of separate hydrogens leads to an increase in the number of atoms from 580 to over 900. For a larger protein like chymotrypsin (over 3000 atoms) the "extended atom" approximation might be a necessary approximation. So far the discussion in this section has assumed harmonic oscillation. The quasielastic results discussed in Section 3.b.i indicated the possibility of anharmonic effects at low frequencies. These effects require special attention. If there is damping due to anharmonic components in the potential function these can be accounted for by incorporating the classical Langevin oscillator model into what is known as the intermediate scattering function, F(Q, t). The temporal fourier transform ofF(Q, t) yields S(Q, co). Calculations of this type have taken account of frictional damping due to solvent and have yielded realistic spectra (Smith et al., 1990b). Another, and possibly more satisfactory method of utilizing F(Q, t) to account for anharmonic effects has also been described recently (Loncharich and Brooks, 1990; Smith et al., 1990a). This method makes use of molecular dynamics simulations and accounts in a natural way for enhancements in anharmonic events as the temperature of a protein is raised. A computer is used to solve the Newtonian equations of motion for the atoms of the protein and, if desired, the surrounding solvent (Karplus, 1986). The weak point in this method is that it is only as good as the atomic potentials that are employed. Unlike small molecules, globular proteins have many internal non-bonded interactions that play a key role in the folding and unfolding of the protein structure during biological function. The potential functions used to describe these interactions are taken from studies on smaller molecules and are most often used in an omnibus computer program called CHARMM (Brooks et al., 1983). However, as mentioned earlier, such potentials are very seldom completely satisfactory in accounting for the dispersion curves obtained from coherent neutron scattering studies of simple molecular crystals. Also, the spatial range of validity of these potentials is limited. A wide spatial range of validity of interatomic potentials is obviously needed to account for unfolding if not for large localized structural changes entailing large changes in atomic separation. For harmonic models, use is often made of internal coordinates (bond stretchings, bendings and torsions) as basis coordinates which, it is claimed (Vergoten, 1989), are usually not independent. According to Vergoten (1989) this means that not all linear terms in any potential expansion are zero so that the equilibrium configuration may not be at a true energy minimum. Nevertheless the methodology associated with calculations using programs like CHARMM remains state-of-the-art. An in-depth review of molecular dynamics simulations is beyond the scope of the present


155

paper. The power of the method becomes obvious when the results of recent calculations are examined (Smith et al., 1990a). An important feature of other work by Smith et al. (1990b) is the calculation of both the one phonon and full multiphonon spectra of S(Q, o9). It may also be noted that the numerical fourier-transform method permits calculation of S(Q, 09) with a Langevin oscillator model having a Gaussian distribution of frequency dependent friction coefficients (Smith et al., 1990b). (vi) Comparison of theory with experiment on pancreatic trypsin inhibitor. A numerical evaluation of S(Q, co) for BPTI has been published recently (Smith et al., 1990b). The method involved evaluation of fast fourier transforms of F(Q, t), convoluted with a fouriertransformed gaussian instrumental resolution function. Comparisons of these calculations with measurements on BPTI indicate several points of interest. (1) The agreement of g(~o) calculations with the experimental g(og) is markedly better for 09> 80 cm- 1 if all atoms are explicitly included. This is to be expected since the "extended atom" approximation is known to be best at low frequencies. (2) To obtain good overall values for 9(o9) the electrostatic contributions to the potential are "switched off" at a distance ranging between 6.5 and 7.5 A. This cut-off in the range of electrostatic interactions seems somewhat arbitrary. Noting a large discrepancy in the calculated and observed neutron response at low frequencies near 9.6 cm-1, the authors point out that this could originate from the nature of long range electrostatic interactions. (3) The intensity maxima in S(Q, o9) have similar frequencies, near 30 cm- 1, for either a methyl hydrogen in alanine (Alal in BPTI) or for all non-labile hydrogens in the protein. The main difference between these two harmonic normal mode results is the presence in the S(Q, 09) for the entire protein of a prominent peak at 9.6 cm- 1 due to the lowest frequency mode in the molecule. Such a mode could be heavily damped in the presence of solvent. The similar response near 30 cm- 1 raises questions about the specificity in incoherent inelastic scattering measurements on proteins. On the other hand such a result may indicate that all residues vibrate as a whole at about the same frequency; this will be discussed again later. (4) Time-of-flight spectra are distorted (see below) but sensitive to changes in scattering intensity in the low frequency r6gime. Solvation can induce large intensity changes at these frequencies especially if the protein has a non-spherical shape. The damped Langevin oscillator model seems to be quite effective in accounting for solvation-induced differences in time-of-flight spectra for BPTI (Smith et al., 1990b). At this point it is perhaps appropriate to speculate about the relevance of these and other results in relation to the biological function of BPTI. We shall discuss results on BPTI in this context. Other experiments will also be discussed similarly in terms of the theory outlined in this section. One of the functions of BPTI is to prevent attack on the pancreas by the digestive enzyme trypsin which may be present in small amounts during production by the pancreas of the proenzyme, trypsinogen. Inhibition of stray trypsin molecules is achieved by attachment of BPTI to the trypsin molecule. BPTI is a compact monodomain molecule (Janin and Wodak, 1983) with little s-helical structure. Six of the 56 amino acids in BPTI are located in a short ~t-helixnear one end of the peptide strand (Deisenhofer and Steigemann, 1975). There are no other s-helices. Trypsin also has little s-helical structure but its amino acids are located in two domains. Inhibition of trypsin is achieved by insertion of BPTI between the two domains of the protease. The Lys 15 of the inhibitor binds very tightly to the active site centered on Asp89 of trypsin (R/ihlmann et al., 1973). In addition to this specific bond there are many hydrogen bonds formed between trypsin and its inhibitor upon contact. The structures of the combined species complement each other so well that bound BPTI probably loses its conformational flexibility. An aid in maintaining conformational "conformity" would obviously stem from a match in the frequencies and non-local amplitudes of oscillation of trypsin and its inhibitor. This could help to explain the very slow dissociation rate of the complex (Stryer, 1988). There is indirect evidence of matching dynamics from a comparison of neutron scattering measurements on chymotrypsin (a homolog of trypsin) and BPTI. The experimental S(Q, o9) for dry BPTI exhibits a peak near 20 cm- x at room temperature

156

P. MARTEL

(Smith et al., 1989). Dry chymotrypsin also exhibits such a peak at about 0.6 THz, or 20 cm- 1 (Martel and Ahmed, 1988). The fact that the neutron scattering response of both proteins yielded a peak at room temperature is significant. On cooling to 77 K, it was found that the peak frequency for dry chymotrypsin rose to ,-~29 cm- 1. Measurements on BPTI at low hydration (h = 0.33 g D20/g BPTI) also yielded a peak frequency of 29 cm- 1 at 180 K (Doster and Cusack, 1987). Since other S(Q, co) measurements on other proteins (see below) showed very little shift in frequency between 77 and 180 K, these results indicate agreement in the temperature dependent responses of the protease and the inhibitor. In dilute solution (h,-~ 7) (Smith et al., 1989) the room temperature S(Q, co) for BPTI no longer exhibited a peak at finite frequency. Most of the neutron response was seen to have moved to the quasielastic region. This effect has been ascribed to increased frictonal damping from solvent-protein atom collisions and has been accounted for quite well (Smith et al., 1989) by calculations involving the Langevin formalism. Such solvent-protein collisions would likely cause similar effects in any soluble protein, and assuming this, we are led to postulate that the frequency responses of the protease and inhibitor would likely remain similar under common damping conditions. In other words non-local frequencies and displacements in the two proteins remain matched so as to help maintain the relative configuration of the two molecules. Further experiments and calculations would obviously be desirable to test this hypothesis. (vii) Hinge-like oscillations. Some of the results of combined normal mode and Langevin oscillator calculations on lysozyme (McCammon et al., 1976; Brooks and Karplus, 1985) have already been discussed from the standpoint of the hinge-like oscillations which this bilobed molecule might undergo. As is usual in these structures, the active site of this enzyme is located in the cleft between the lobes. Lysozyme is found in tears and protects the eye against certain bacteria. It accomplishes this by attaching itself to polysaccharide sugars (the substrate) found in the cell walls of these bacteria. The mechanical support offered by these sugars is destroyed and the bacterial cell wall bursts. It has been found by X-ray diffraction that when a substrate sugar is bound, the cleft of lysozyme closes down somewhat because the lobes tend to come together (Imoto et al., 1972). Other measurements on crystallized lysozyme (Artymiuk et al., 1979) have suggested that the "lips" of the active-site cleft, and the region of the enzyme that changes most on ligand binding, are regions that have the greatest displacements. These effects suggest flexibility about a hinge joining the lobes and some calculations have obtained an associated frequency between 3 and 5 cm- 1 for this scissor-like motion in dry lysozyme. As a point of interest it may be noted that similar movements should occur in kinases, dehydrogenases and citrate synthase (Karplus, 1986). Experimental S(Q, co) measurements have been reported (Doster and Cusack, 1987) on lysozyme at 180 K with a low level of D20-hydration (h = 0.33). These spectra are not very different in overall appearance from those of myoglobin and BPTI at low temperature. Whereas the peak in S(Q, co) for BPTI was located at 29 cm-1, the peak for lysozyme occurred near 27.3 cm - 1. There was no evidence in the S(Q, co) spectra of lysozyme of any extra inelastic peaks at, or above, 10 cm- 1. The onset of a strong quasielastic component occurred below this frequency and masked any weak peaks that might be present in the low frequency region. In order to study differences in S(Q, co) at low frequences it is often advantageous to utilize the time-of-flight (TOF) technique. This technique "distorts" S(Q, co) in such a way as to accentuate the scattering in the low frequency region. The physical reason for this is that the observed spectra are given by an inelastic differential cross section that is proportional to (l/z4) • S(Q, co) where ~ is the time of flight per unit path length (Cusack, 1985). Thus for long flight times corresponding to low energy transfers, the observed scattering becomes progressively more magnified as ~ increases. All of the neutron scattering results on lysozyme have been obtained with the TOF technique. No doubt part of the reason for this was the incentive offered by the possibility of finding a hinge-bending mode. One of the earliest neutron measurements was carried out on


157

lysozyme with and without N-acetyl-glucosamine (NAG) inibitor (Bartunik et al., 1982). In spite of the low frequency sensitivity of the TOF method, there was no indication in any of the spectra of a very low frequency band that might be ascribed to the hinge-bending mode. However, at very small energy transfers, between 1 and 40 cm-1, the spectra did show a dependence on inhibitor binding and on crystalline environment. The total inelastic scattering at energy transfers below 50 cm- 1 increased significantly on going from crystalline lysozyme, to lysozyme in solution with NAG, and then finally to lysozyme in solution without NAG. The authors interpreted these results in terms of "conformers" (different conformations) of the enzyme. They felt that different conformers would be favoured by binding of the inhibitor or, even more, by stereochemical constraints in a crystalline environment. To support this view they pointed out that hydrophilic side chains, because they are in contact with water, become more disordered and flexible than hydrophobic residues on going from the crystalline form to solution. Although both types of residues contain, in total, a comparable number of non-labile hydrogens, the increase in inelastic scattering below 50 cm- 1must arise from "liberated" side chains on going from crystalline to free lysozyme in solution. The effect of NAG could then be explained by stabilization of the side chains when they become "tethered" to the inhibitor in the "lip" region. Although a low frequency peak was not found, a peak at 25 cm-1 was identified by Bartunik et al. (1982) in polycrystals of lysozyme at room temperature. This is presumably related to the peak found at low temperatures at 27.3 cm- t (Doster and Cusack, 1987). The persistence of a 25 cm- 1 peak in polycrystalline lysozyme at room temperature is worthy of note. This enzyme has less than a third of its residues in a-helices and although this is a greater fraction than that for BPTI (~ 1/10) it is much less than for myoglobin where 85% of the amino acids are in a-helices. The S(Q, o~) for myoglobin does not exhibit a peak in this frequency region at room temperature. These facts will be discussed in more detail below. Intensity decreases on binding to inhibitor have also been seen in other experiments on lysozyme (Middendorf, 1984b, 1986; Middendorf et al., 1984). These experiments showed that decreases in intensity become more pronounced at low Q-values. From the Q-dependence of this effect Middendorf (1984b) was led to ascribe the decreases to fluctuations of the free enzyme over distances, d, in the 10 to 15/k range. Analysis indicated that the strong Q-dependence observed at low Q-values pointed to coherent scattering contributions due to cooperative effects. The magnitudes of the quoted d-values suggest coordinated motion of the lobes of lysozyme as a possibility. The favourable low frequency characteristics of the TOF method have been exploited more recently (Smith et al., 1987) to measure hydration-related dynamic changes in lysozyme at 293 K. Upon hydration from h = 0.07 to h = 0.2 g D20/g lysozyme, the inelastic scattering between 0.8 and 4.0 cm- t was observed to increase while the elastic scattering decreased. The biological significance of this finding is that h = 0.2 is the threshold value at which the enzyme recovers activity after being dried. The two changes in the scattering are complementary because decreased elastic scattering suggests increased atomic amplitudes resulting from low frequency motion, while the increased inelastic scattering intensity suggests additional vibrations in this frequency range. Although these frequencies overlap with predicted values for the hinge-bending mode (3-5 cm- 1) the authors caution that the calculations are based on quasiharmonic models, and that there could be considerable "anharmonic and damping effects modifying the low frequency modes". They conclude, however, that the inelastic intensity differences in the spectra occur in a frequency region where large amplitude, global modes could occur in lysozyme (Levitt et al., 1985). (viii) The incoherent scattering approximation. The usual situation in biophysical studies with neutrons is to have a significant percentage of 1H present. Therefore a rather unique experiment was carried out some time ago on fully-deuterated phycocyanin, both dry and D20-hydrated (Middendorf et al., 1984). Phycocyanin is a light-harvesting protein involved in Photosystem II of algae where it attaches to the surfaces of photosynthetic lamellae. Although the scattering was now largely coherent the results could still be analysed

158

P. MARTEL

assuming the "incoherent scattering approximation" for randomly oriented systems (Marshall and Lovesy, 1971). One result of the experiment on deuterated phycocyanin was that hydration generally decreased the observed intensities for high energy transfers > 200 cm - 1. These results were verified with light water (H20). Since these intensity changes occurred at low hydration levels it was concluded that hydration of ionizable side chains on the protein surface led to restriction of the vibrations of polar groups (e,g. hydroxyl, amide) associated with such amino acids as Asp, Glu, Lys and Arg. The "quenching" of the motion of these small groups should lower the intensities at high frequencies. (ix) Enzyme-substrate bindin9. Hexokinase is an enzyme active in metabolic processes. It catalyses the transfer of the phosphoryl group from ATP to the hydroxyl group on carbon-6 of glucose. One of the earliest difference spectra to be published was on the hexokinaseglucose system (Jacrot et al., 1982). The impetus for this experiment was the finding that hexokinase contracts when it surrounds the glucose molecule during the binding process (McDonald et al., 1979). An excess scattering was reported when the spectrum from the hexokinase-glucose complex was subtracted from the hexokinase spectrum in concentrated solution. Details of the interpretation are quite interesting in that concepts ofenthalpy and entropy were invoked to account for the changes in the vibrational spectrum upon binding of the substrate. Similar ideas were discussed in Section III.3.d but without the subtleties entering in the interpretation of the hexokinase experiment. Here the decrease in neutron scattering intensity was ascribed to a reduction in vibrational enthalpy, even though differential scanning calorimetry (Takahashi et al., 1981) indicated no overall change in enthalpy on binding. This discrepancy between the two types of measurements was explained on the basis of compensating effects. It was suggested that a reduction in the enthalpy of low frequency vibrational modes was compensated for by the "burial" of Asp189, removed from contact with water as a result of conformational changes deduced from X-ray diffraction measurements (McDonald et al., 1979). Thus the basic idea was that the closing of the cleft near Asp189 changed the environment in such a way as to compensate for changes in enthalpy due to vibrational effects. It may be noted that more recent scattering measurements (Cusack, 1985) have failed to confirm the results of Jacrot et al. (1982). Some of the results from neutron scattering measurements on another enzyme, chymotrypsin, have already been discussed, but details of the measurements were not given. In this experiment (Martel and Ahmed, 1988) a triple axis spectrometer (TAS) was utilized. With this method the energies of neutrons incident on the specimen are measured by diffraction from a monochromator crystal and the energies of neutrons scattered from the specimen are measured similarly by an analyser crystal. The differences in these energies are the energies of the vibrational modes. Comparisons of neutron scattering techniques from biomolecules by TOF and TAS may be found in the literature (Middendorf and Randall, 1985). TOF methods usually employ pulsed neutron sources (B6e, 1988; Windsor, 1981). TAS makes use of a steady-state reactor source. The advantage of TAS for S(Q, 09) measurements is that it permits utilization of the constant-Q, variable incident energy method of neutron scattering (Brockhouse, 196 I) which yields S(Q, co) directly in those cases where broadening due to instrumental resolution is negligible. This is of some importance since the quantity in incoherent inelastic neutron scattering that is most directly related to calculations (Martel et al., 1991 ) is S(Q, 09). It may be recalled that the TOF method yields a distorted S(Q, 09) that must be corrected before comparison can be made with calculations of this quantity. It may also be noted, however, that both methods have their strengths and weaknesses with multi-detector TOF being superior for the extremely low frequency measurements described above where there was insufficient intensity to make accurate measurements by the TAS technique. The TAS measurements on dry chymotrypsin powder have been compared (Martel and Ahmed, 1988) with earlier raman light scattering measurements (Brown et al., 1972). Figure 8 shows some typical constant-Q neutron scattering measurements at 77 K. The solid lines through the data points are the results of gaussian fits. These fits yielded mean

Biophysical aspects of neutron scattering i

I

t

I

I

0.=1.0

I

t

159

I

I

+

I

600

500

'l

L A:

Selective gating to vibrational modes through resonant X-ray scattering.

Retraction notice to "Protein dynamics by neutron scattering" [Biophysical Chemistry 182 (2013) 16-22].

Neutron scattering studies of chromatosomes.

Outline of Neutron Scattering Formalism.

Neutron scattering on nuclei.

The interpretation of polycrystalline coherent inelastic neutron scattering from aluminium.

Multi-component modeling of quasielastic neutron scattering from phospholipid membranes.

Temperature dependence of the low frequency dynamics of myoglobin. Measurement of the vibrational frequency distribution by inelastic neutron scattering.

Vibrational density of states and elastic properties of cross-linked polymers: combining inelastic light and neutron scattering.

Retraction of the article "Protein dynamics by neutron scattering" published in Biophysical Chemistry 2013, 182, 16-22.

Neutron scattering for the analysis of membranes.

Low-angle neutron scattering from chromatin subunit particles.

Investigating Structure and Dynamics of Proteins in Amorphous Phases Using Neutron Scattering.

Symposium report. Biophysical aspects of Auger processes.

Surface vibrational modes in disk-shaped resonators.

Subfemtosecond steering of hydrocarbon deprotonation through superposition of vibrational modes.

Terahertz vibrational modes of the rigid crystal phase of succinonitrile.

Ultra-High Resolution Inelastic Neutron Scattering.

Neutron scattering and energy deposition spectra.

Vibrational modes of aminothiophenol: a TERS and DFT study.

Vibrational modes of hemoglobin in red blood cells.

Elastic Scattering Spectroscopy (ESS): an Instrument-Concept for Dynamics of Complex (Bio-) Systems From Elastic Neutron Scattering.

Measurement of the Coherent Neutron Scattering Length of (3) He.

Magnetic small-angle neutron scattering of bulk ferromagnets.