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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.
ACKNOWLEDGMENTS
We have benefited significantly from conversations and correspondence with Joel Berendzen, Nobuhiro Go, G.N. Phillips, and Tom Terwilliger. We
thank J. Clarage and G.N. Phillips for providing
Figure 1. This work was supported at Los Alamos
and Cornell by the U.S. Department of Energy.
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