PROTEINS: Structure, Function, and Genetics 29:153–160 (1997) Variations on a Theme by Debye and Waller: From Simple Crystals to Proteins Angel E. Garcı́a,1* James A. Krumhansl,1,2 and Hans Frauenfelder3 Biology and Biophysics Group, T-10 MSK710, Los Alamos National Laboratory, Los Alamos, New Mexico 2Department of Physics, Cornell University, Ithaca, New York 3Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 1Theoretical ABSTRACT Debye and Waller showed how to adjust scattering intensities in diffraction experiments for harmonic motions of atoms about an average, static reference configuration. However, many motions, particularly in biological molecules as compared to simple crystals, are far from harmonic. We show how, using a variety of simple anharmonic, multiconformational models, it is possible to construct a variety of Generalized Debye-Waller Factors, and understand their meaning. A central result for these cases is that, in principle, intensity factors cannot be obtained from true total mean square displacements of the atoms. We make the distinction between the intensity factors for unimodal quasiharmonic motions and those for the anharmonic, multimodal (valley hopping) motions. Only the former affect the conventional B factors. Proteins 29:153– 160, 1997. r 1997 Wiley-Liss, Inc.† Key words: protein dynamics; molecular dynamics; crambin; X-ray INTRODUCTION It has been nearly a century since the foundations of x-ray diffraction, neutron scattering, and light scattering were laid. These have enabled us not only to know crystals, glasses, and biological materials from their external morphology, but to visualize their internal structures down to the atomic level. However, to accomplish the latter it is necessary to process, based on theoretical understanding, the actual experimental data to reach that visualization. Thus, the theory of radiation scattering is an essential companion to experiment. To be sure, by now the theory of diffraction itself is highly developed, as well as are a variety of data refinement methods. However, embedded within the latter, among several steps, is an accounting for the fact that over and above any average, nominally average stable structural form, the atoms move. The effect of those motions on the scattering intensity was the concern of Debye and Waller (DW) in the early part of this century. Their result is widely used, and their theory is discussed in many places; for a review see Ref. 1. The limitation of the harmonic Debye-Waller recipe, with intrinsic anharmonicity present has r 1997 WILEY-LISS, INC. †This article is a US government work and, as such, is in the public domain in the United States of America. been recognized for sometime. For moderate anharmonicity2 cumulant and many-body perturbation approaches have been developed. For crystals or molecules with fixed (centered) structure these have been quite successful. Biomolecules, specifically proteins like myoglobin, are quite a different matter because they can make transitions between conformational substates. This switching is illustrated in Figure 1, from a simulation reported by Clarage and Phillips3 (their Fig. 6b). Studies by Karplus et al.4 and by Kuriyan et al.5 have addressed several aspects of this topic. It is clear from Figure 1 that such conformational motions can never be approximated well by cumulant or quasiharmonic corrections in the thermal factors. In situations when substates are clearly separated in space and can be identified with the displacement of a small number of atoms (e.g., a side-chain rotation) it is trivial to correct the DWF. However, small collective displacements of a large number of atoms may also describe a large conformational change, and the corrections to the DWF can also be described in a simple way. It is in that context that we addressed the problems of diffraction from multimodal systems in our previous work.6–8 We cannot present at this time a general method capable of correcting both simulations and refinements for strong anharmonic effects. However, we can show why, for basic reasons, mean square displacements (MSD) derived from simulations are intrinsically larger than those derived from experimental B factors by using standard Debye-Waller interpretation. This systematic disagreement, found in a number of comparisons of simulations with experiments, is not due to errors in refinement but is due to the fundamentally unjustified use of the total MSD in the interpretation of experimental intensity factors when large nonharmonic molecular motions are present. It is our purpose here to address the motional intensity factor directly from the outset using intrin- Contract grant sponsor: U.S. Department of Energy. *Correspondence to: Dr. Angel E. Garcı́a, Theoretical and Biophysics Group, T-10 MSK710, Los Alamos National Laboratory, Los Alamos, NM 87545. E-mail: angel@tio.lanl.gov Received 25 November 1996; Accepted 29 April 1997 154 A.E. GARCÍA ET AL. cases, fl 5 fl (r) is taken as a sum of gaussians.) Then, it follows that the total scattering is given by, with slight generalization, 1 2 0o ds d V coh ~ 0 2 f *l e 2ik·rl 7e ik·ul 8 l (3) where l covers all atoms. The f *l is the Fourier transform of fl (r) and may be energy and k dependent.10 In their original study Debye11 and Waller12 considered that the motions ul were in fact those of a collection of harmonic oscillators. Then, and only for the harmonic case, the result is that2,9 7e 2ik·ul 8harmonic 5 exp [2 217k · ul 82 ] (4) and Fig. 1. Projection along the xy plane of the atomic displacements of the Cg2 atom from the mutated I68 residue of myoglobin over the course of the time-averaged refinement of B factors of Clarage and Phillips.3 Three conformational substates are sampled. sically and totally anharmonic models. A few simple pedagogical examples bring out the salient features. FORMALISM: BASIC DIFFRACTION THEORY The essential point of DW is that for a real sample the observed scattering is not from a perfectly rigid atomic structure, but from many structures near the average, due to the motions of the atoms. For structural determination, the usual experiment is the coherent elastic scattering of the radiation by the average electronic charge density 7r(r)8, given by2,9 1d V2 ds 5 coh sc 4p 0 e dr exp (ik · r) 7r(r)8 0 2 (1) d V coh ~ l 3 1 fl exp [2ik · rl ] exp 2 7(k · ul)2 8 2 o fld(r 2 Rl ) (2) l atoms where fl is an atomic structure factor and the instantaneous atomic positions are given by Rl 5 rl 1 ul (t ) where rl defines the static average position of the lth atom and ul is the motion of the lth atom with respect to the static (reference) lattice. (In practical 40 2 . (5) Within this approximation, the effect of thermal fluctuations can be characterized by a MSD 7u 2l 8 of the scattering atom. This harmonic approximation is commonly used in the interpretation of experimental data, whether justified or not. Yet the more general expression9 for modification of the theoretical rigid lattice-scattering amplitude, 7e 2ik·ul 8, is remarkably simple in form, and being based on general diffraction theory can be explored directly for situations where the harmonic approximation is not suitable (e.g., structural or biomolecular conformational instabilities or dynamics). Our purpose here is to explore several variations on this basic theme using what we now call the generalized Debye-Waller factor (GDWF), F : F 5 0 7e 2ik·ul 8 0 2 where [k 5 (ki 2 ks )] is the scattering vector, ka and ks are the incident and scattered light wavectors, respectively, and 7 8 implies a time or ensemble average. As long as the electronic distribution on each atom is not significantly changed by the atom motions, for point atoms, r(r) 5 1 2 0o ds (6) The DWF is only one special case of the GDWF. In the remainder of this section we will discuss how experimental diffraction data are handled in practice. After instrument corrections, using reference atomic scattering factors, and assuming a starting structure, a first approximation to scattering data is found. Thereafter, many refinement procedures, including stereochemical and other (energy, phase) constraints, based on variational iterations, are used to determine a set of atomic coordinates and intensity corrections to the atomic factors, Fexp, that model the experimental data and are not dependent on any dynamical model assumptions. The issue here is how are Fexp, FDW, FGDW, Fsim related? (The notation is the obvious one.) Practice has been to compare Fexp with the FDW and Fsim predictions, assuming that in fact they should agree in principle 155 FROM SIMPLE CRYSTALS TO PROTEINS (which fails badly when multimodal motions are present). It is also necessary to clarify some terminology. Motivated by the 7(k · u l )2 8 exponent in FDW, Equation (4), and the relationship that 0 k2 0 ~ 1 2 sin2 u l2 it is a matter of experimental convenience to write 3 1 24 F 5 exp 22B sin2 u l2 where u is the scattering angle. If this is done purely for data reporting convenience, then in analogy to the above one could find Bexp from refined experimental data, and BDW, BGDW, Bsim, all from dynamical models. Standard practice has been to assume that the DW model applies, whence (for isotropic motions) 3 7u 2l 8 , 8p (7) VARIATIONS ON THE GDWF THEME Example 1 As the first, perhaps simplest example, applicable to small atoms rattling classically around in a cage, in a sense the extremely anharmonic opposite of harmonic motion about the center of the cage, consider the GDWF for an atom moving classically and ergodically throughout a spherical ‘‘square well’’ with infinitely hard walls. In this case the probability density is uniform within the well and Î F 5 7e ik·u 8 can be expressed as 5 3 4pd eee d 3 0 p 0 2p 0 ÎF 5 3 0 kd 0 3 1 sin kd kd 2 cos kd 2 (9) where d is the radius of the spherical well. This and a number of related cases have been discussed some years ago by Dash, Johnson, and Visscher,13 in connection with the Mössbauer experiments. Two features are important: 1. In this classical model there is no temperature dependence of the GDWF. Moreover, there is no apparent connection with exp [221 7(k · u)2 8], or any siharmonic frequency. 2. In addition, notice that the scattering angle dependence is not exp [2B sin2 (u)/l2 ], but it is oscillatory, showing nodal structure when kd , 4.3, 7.6, . . . , etc.13 Example 2 Bl,exp. 2 It will become apparent, from several models of the generalized DWF to be presented in the next section, that for many types of anharmonic motions, particularly multimodal or multiconformational, the use of Equation (7) to infer the actual MSD is not justified. Specifically, comparing a simulation 7u 2l 8sim with (3/8p2 )Bl,exp will be quantitatively incorrect for any large scale (i.e., principal mode) motions. Various GDWF models need to be explored, as appropriate to each molecule in question. In this regard, molecular dynamics simulations can indeed be a useful guide to a choice of a model GDWF. As an application, we examine, in some detail, the implications for the interpretation of a protein molecule, crambin, as regards both the experimental diffraction and the simulation results. In the next section we explore general models of the intensity factors. ÎF which can be evaluated exactly e ik·uu2 sin u du du dc (8) Example 1 samples highly anharmonic motion of an atom, but about a single reference configuration (conformation). Suppose however, that the molecule flips intermittently between two or more conformations over long periods of time (e.g., proton tunneling ferroelectrics;14–17 proteins with bimodal conformations, i.e., moving between ‘‘landscape’’ minima.18) What, then, does the GDWF look like? To begin with consider only the flip-flop motions between two conformations, modeled by u 5 6a/2, each with probability 21. The evaluation of 7e 2ik·u 8 is trivial, ÎF 5 21(e 2ik·a/2 1 e ik·a/2 ) 5 cos k·a 2 . (10) A temperature independence of F is again noted. Example 3 As more and more complex structures are examined (cf. the following section on crambin or N. Go et al.19 ) it appears that the motions can be divided into two classes: ‘‘unimodal’’ small harmonic fluctuations (vibrations), or large conformation changing motions of a flip-flop nature. In this example it is envisioned that this flip-flop motion occurs slowly compared with the fluctuation frequencies of the many background harmonic vibrational modes. The intensity factor is now modeled by adding a set of randomphase thermal fluctuation variables, xf, to the model in Example 2 as follows: u5 a 2 (1) 1 xf u52 a 2 (2) 1 xf (11) 156 A.E. GARCÍA ET AL. each conformation with probability GDWF is ÎF 1 2. Then, the 5 7e 2ik·u 8 (1) (2) 5 21[e 2ik·a/2 7e 2ik·xf 8 1 e ik·a/2 7e 2ik·x f 8]. (12) If it is reasonable to assume that the background fluctuation probabilities are uncorrelated and gaussian, as they would be for the background motions in a lattice, and independent of the 6a/2 conformations, then Î F 5 cos 122e ka 2 27(k·xf ) 8 /2 5 cos 324e ka 2 2 2Bf (sin u/l ) 122 1 7x 8 a2 Proceeding further, we present results from an extensive series of studies by S.L. Mair17 on a variety of quantitatively realistic models. Among them the double-quadratic well potential for a chain of coupled bimodal nonlinear oscillators incorporate both large amplitude motions and fluctuations and are treated formally by using statistical mechanics. The characteristic features from the pedagogical models above are now found again in a reasonably realistic physical calculation.15 Thus, in a chain with harmonic intersite coupling, but on-site two-valley potential U (u) 5 v20 ( 0 u 0 2 a/2)2 (13) Now only the fluctuation (thermal) quasiharmonic structure factors, Bf , introduce a temperaturedependent standard DWF, while the (flip-flop) conformational motion introduces a new term, cos (ka/2), which can be called the conformational-DWF (CDWF). In this classical model the CDWF is temperature-independent, although the probability of occupying different minima is temperature-dependent when the states are not degenerate. Equation (13) is consistent with the common practice of assigning two (or more) weighted conformations and B factors to protein side-chain atoms. One may look at this in a slightly different way. The total MSD, 7u 2 8 5 Example 4 where a is the distance between the two potential energy minima, the following results are correct in the strong intersite coupling limit, (i.e., v0 # vc , that is, single systems (e.g., protein molecules) in the crystal are strongly coupled to each other): The MSD for this model is 7u 2 8 5 122 a2 1 1 2mv0vcb 1 b 5 (kBT )21 and 2 f 1 so that 122 a2 122 2 F 5 8p2 3 37u 8 2 122 4 2 a2 a2 is the overlap between harmonic oscillators at 6(a/2). On the other hand, the GDWF is given by 5 7x 2f 8 and then Bf 5 (15) (1 1 S0 ) where m is the mass of the particle, and vc is the lattice frequency, S0 5 exp 2v0vcb 7u 2 8 2 122 a2 (14) where 7x 2f 8 ~ kBT. It is now apparent that one must substract from the MSD the mean square amplitude of the flip-flop mode, (a/2)2, to find the temperature fluctuation B factors, Bf. The thermal intensity factor 7x 2f 8 is clearly less than the total MSD; however, simulation data generally provide the total MSD. Although these results come from a model that assumes independent anharmonic and fluctuational motions, they are found in other models in which the correlation between atomic motions of different atoms is taken into account. One such model, formally solvable, takes into account collective motions; we summarize it now. exp [2k2/4mv0vcb](S0 1 cos (ka/2)) 1 1 S0 . (16) In the low T limit, S0 = 0. At this limit the distribution of the fluctuations are two gaussians centered at 6a/2. The GDWF becomes F = exp 1 1 2k2 7u 2 8 2 2 12 2 2 a2 2 cos 122 ka (17) which is similar to Equation (13) and the MSD is 7u 2 8 = 122 1 2v v . a2 k BT 0 (18) c Here the MSD does not tend to zero as T = 0. This agrees with a situation in which single particle (i.e., 157 FROM SIMPLE CRYSTALS TO PROTEINS proteins) in the crystal will spend most of the time in one well or another. While the models in this section have been chosen for pedagogic simplicity, there have been many more extensive and realistic studies of the issues brought out here, for a variety of materials in condensed matter physics; Mair and coworkers15–17 formalized a number of such models and compared them with experiments.14 From examples 1–5 a general feature emerges: the experimental temperature factors are smaller than the true total MSD in the presence of multimodal, conformation-changing motions. The difference will be illustrated in quantitative detail in the following section, for crambin. It has become widely realized that in many proteins much of the energy of motion is contained in a few essentially anharmonic, large amplitude, multimodal motions. These have been referred to as molecule optimal dynamic coordinates (MODC),8,20,21 Principal Modes,22,23 and Essential Dynamics (ED).24,25 These modes have been suggested to be the principal determinants of biological functions determined by shape changes of the protein.24 We note two features relating to Equation (13): 1. For the k used for structure determination, and l0 a characteristic interatomic distance, kl0 , 1. 2. In the principal modes, large motions, a, are localized in particular parts of the protein (e.g., hinge bending regions), and ka for them may also be ,1. For these cos [ka/2] can not be approximated by 1 2 21(ka/2)2 , exp [221(ka/2)]. Thus, those parts of the molecule that swing, flip, etc., by amounts characteristic of anharmonic motion can contribute strongly reduced CDWF that is nongaussian, but oscillatory in wave vector, k. The purpose of the discussion in this and the previous section is to distill the physics of the GDWF due to arbitrary motions of atoms to the simplest features that demonstrate how different the intensities can be in systems whose motions have at least some degrees of freedom, which are large, stochastic, and nonlinear. In these cases the use of the standard DWF could be misleading, and can be quantitatively incorrect. SIMULATION ON CRAMBIN In the introductory sections we used simplified models of the molecular motions for pedagogical purposes; we now present the application to a realistic description.8 Information on the protein dynamics is obtained from a 400 ps, constant temperature (300 K), molecular dynamics simulation of crambin in a crystal environment, including water molecules. An all-atoms representation of the protein consists of 654 atoms. The unit cell contains two crambin molecules, with initial conformations satisfying a P21 symmetry, and 182 water molecules.26 The cell dimensions are a 5 40.96 Å, b 5 18.65 Å, c 5 22.52 Å, and b 5 90.77°.27 The particle mesh Ewald summation (PME) method,28 implemented in Amber,29 was used to calculate the electrostatic energies. A 64 3 32 3 32 mesh and a cubic spline interpolation were used in the PME. The potential energy was modeled with the force field of Weiner and colleagues.30 The TIP3P model was used for water.31 The initial system was first, energy minimized, and then, heated to 300 K over the first 10 ps of the MD simulation. The simulation was then extended to 400 ps. Averages are calculated from configurations saved at a rate of 20/ps. The first 100 ps of the simulation are not included in any averaging. We analyzed the MD trajectory in terms of collective modes (Principal Modes) that best represent the cartesian mean square fluctuations of the system. This method has been described in the literature.8,20,19,32 These principal modes are obtained from all saved configurations of a trajectory by diagonalizing the resulting covariance matrix (si, j ) of the cartesian displacements of 327 nonhydrogen atoms in each protein molecule. The MSD of atoms about their averaged positions over the molecule are described by 7u 2 8 5 1 N tr (s) 5 1 N ol v (19) v where lv represents the eigenvalue of the mv Principal Mode, and N 5 is the number of nonhydrogen atoms in the protein molecules. Individual atomic MSD can be found from the eigenvectors by 7u 2i 8 5 1 N olm v v 2 v,i. These eigenvalues are used to systematically rank modes with respect to their contribution to the MSD. The total MSD for the protein in the MD simulation is 0.38 Å2. The use of the covariance matrix as a generator of an alternative coordinate basis set carries no implication of quasiharmonic motion. Figure 2a shows the projection of the last 300-ps trajectory along the five modes with largest eigenvalues. Mode 1 contributes 29% of the MSD and more to the MSD than the other four modes together, 26%. The large contribution of this mode to the MSD comes from a transition (at t , 50 ps during the production period of 300 ps) from one energy valley to another. The fluctuations of the motions in each valley are comparable to the fluctuations exhibited by the other modes.2–5 Figure 2b shows a projection of the last 300 ps trajectory on a plane spanned by modes 1 and 2. At least two substates are sampled during this trajectory. The projection of the MD trajectory on this plane resembles the trajectory shown in Figure 1. However, here we are projecting on a plane spanned by the displacements along two 158 A.E. GARCÍA ET AL. Fig. 2. (a, left) Projections, in angstroms, of the final 300 ps molecular dynamics trajectory along the five modes that best represent the major motions of the system (Principal Modes) are shown on the left-hand-side plots. The labels on top of each curve shows the eigenvalue ordering (from large to small) and the corresponding eigenvalues, lv, in Å2. The right-hand-side plot principal modes while Figure 1 shows the projection of a cartesian XY plane of one atom. The conformational transitions described by modes 1 and 2 are not merely transient effects due to initial conditions. In 1.2 ns and 5.1 ns MD simulations of crambin in solution33 and the crystal environments,34 respectively, we have observed that the same kind of transitions occur for all simulation time scales, consistent with the ‘‘landscape’’ picture suggested by Frauenfelder.18 Figure 3 shows the B factors obtained from the MD simulation by Bsim 5 (8p2/3)7u 28, the B factors obtained from refinement of x-ray diffraction data and reported for crambin crystals.27 The largest variation between the experimental and the MD simulations results are near amino acids 7 and 42. Figure 4 shows the amplitude of the first two Principal Modes (with the largest eigenvalues, l 5 36.13 Å2 and 15.50 Å2 ). Note that these modes exhibit bimodal amplitude distributions (in Fig. 2). The first mode describes a motion with MSD similar in character to the differences between the experimental and MD B factors. Two factors contribute to this, first, the magnitude of the contribution of this mode to the total MSD is proportional to l1 which is more than twice as large as l2, and, second, the amplitude of the mode, 0 ml,i 0 , for some individual amino acids is twice as large than for mode 2. The discrepancy between the MSD and the experimental B factors does not mean that the protein molecules in the crystal do not shows histograms of the frequency of occurrence of the amplitude of displacements along the corresponding modes. (b, right) Projection of the last 300-ps trajectory on a plane spanned by modes 1 and 2. Arrows indicate the protein configuration at times 0, 60, and 300 ps. Fig. 3. Comparison of the simulation and experimental B factors. The simulation B factors are calculated from Bi 5 (8p2/3)7u i2 8, where 7u i2 8 is the MSD of the ith atom during the production stage of the MD simulation. The point of Equation (13) is that these B factors need to be corrected when multiple substates are sampled. The solid line shows the calculated B factors obtained from simulations when both molecules in the crystal unit cell are considered as one unit. These B factors include contributions from relative translations and rotations of the two molecules. The long-dashed line shows the calculated B factors from one molecule when conformations of each protein in the unit cell are superimposed independently. These B factors do not include contributions from relative rotations and translations of the two molecules. The heavier dotted line shows the B factors obtained by the refinement of x-ray crystallographic data27; these experimental B factors when compared to the solid line values show clear differences at several sites, particularly in the vicinity of Ca 7, 20, and 42. 159 FROM SIMPLE CRYSTALS TO PROTEINS Fig. 4. a: Normalized simulated MSD of the Ca atoms, as a function of amino acid number, along the two principal modes that showed bimodal distributions in Figure 2. The solid line shows the Ca amplitudes for mode 1, while the dashed line show the Ca amplitudes for mode 2. The contribution of these modes to the MSD is proportional to l (Eq. 19). The contribution to the B factors is proportional to (8p2/3)l, where l1 5 36.1 Å2, and l2 5 15.5 Å2. According to Equation (20), the contributions to the MSD resulting from the separation between minima should be subtracted from the MSD such that the obtained a generalized B factor that can be compared with the measured values. b: Stereo plots of the Ca displacements described by MODCs 1 and 2. show collective, multimodal motions, but, as discussed above, that the DWF does not contain much information about such multivalley dynamics as described in the simple models described in Examples 1–5. It is straight forward but algebraically somewhat tedious to show now, that if the Principal Mode motions are well approximated by conformational motions between degenerate substrates, that the complete GDWF is given by the continued product of terms like Equation (13) over mode eigenvectors; thus a given atom, i, contributes the factor 0 Fi 0 2 5 1 2 2 · exp 1 7(k · u ) 82 p exp (7(k · u p cos l 2 k · al,i l,i 2 v,i ) 8) 2 (20) v to diminishing the intensity. Here, l is summed over principal (conformational) modes with substates separated by al,i, exhibiting small displacements ul,ii around each substate, and v is summed over quasiharmonic modes. In any case, as found below, any estimate of MSD from simulations will disagree significantly with experimental 7u 2 8exp in a region with large anharmonic motions. This will be particularly pronounced in regions of the molecule at which the eigenvectors from the simulation have large amplitude. While we have used crambin simulations for illustration, it appears that Go et al.19 had found that for BPTI a principal mode with multimodal distribution needed to be discarded to fit MD MSD to normal mode fluctuation MSD. From our analysis we now know that this is also a good approximation if we were to compare experimental B factors to MD simulation results. Our analysis also reveals why normal mode analysis MSD correlate surprisingly well with experimental estimates of the MSD.35 DISCUSSION The major concern of the preceding sections has been to illustrate the distinction between the intensity factors for unimodal, quasiharmonic motions and those for anharmonic, multimodal, valley hopping, motions. Only the former strongly affect the thermal B factors determined from refinement. What then might be done to gain information about the large significant conformational modes? Reference to Examples 2, 3, and 4 above indicates what could be done. In essence, when the scattering configuration jumps between different conformations the effective diffraction pattern is a weighted superposition of that from two different structures. Model Examples 2–5 (above) are represented as being displaced from each other by a finite distance a. It is then both physically and algebraically natural that modulation factors such as cos (k · a/2) are found in the GDWF intensity factors. What is then suggested is that experiments at different wavelengths might be useful to establish the presence of multimodal, valley-hopping motions. We have pre- 160 A.E. GARCÍA ET AL. sented models for which the corrections to the DWF are simple. Other models may contain substates that are not degenerate or in which the separation between substates is small, that is, k · a 9 1. The DWF in the latter case may not be easily distinguished from the harmonic DWF. In passing, it is to be noted that the dynamic modulation effect could be similar in appearance to that from a static mixture of conformational substates as proposed by Frauenfelder36 and Petsko.37 The appearance may resemble that from static conformational disorder, but in fact it is dynamical phenomena that appears as well in simulations. Then, to distinguish between these, some method dependent on timescale is needed. Frauenfelder and colleagues38 have employed the Mössbauer effect in this respect. As illustrated in Example 1 above, the GDWF in a Mössbauer experiment can provide information on the large inter-conformational motions, while, on the other hand, in the case of diffraction, as illustrated in Example 4, Equation (13), and demonstrated in the preceding section on crambin, the conformational motions do not enter in the DWF in the same way as fluctuations in the experimentally observed B factors. Much more could be said about further theoretical and experimental issues, and even the potential effects on (inelastic) diffuse scattering, but that is not in the spirit of this paper, which is to bring out the essence of the diverse diffraction phenomena possible with the rich dynamics of biomolecules. 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