Instantaneous normal modes and the protein glass transition.
ABSTRACT In the instantaneous normal mode method, normal mode analysis is performed at instantaneous configurations of a condensed-phase system, leading to modes with negative eigenvalues. These negative modes provide a means of characterizing local anharmonicities of the potential energy surface. Here, we apply instantaneous normal mode to analyze temperature-dependent diffusive dynamics in molecular dynamics simulations of a small protein (a scorpion toxin). Those characteristics of the negative modes are determined that correlate with the dynamical (or glass) transition behavior of the protein, as manifested as an increase in the gradient with T of the average atomic mean-square displacement at approximately 220 K. The number of negative eigenvalues shows no transition with temperature. Further, although filtering the negative modes to retain only those with eigenvectors corresponding to double-well potentials does reveal a transition in the hydration water, again, no transition in the protein is seen. However, additional filtering of the protein double-well modes, so as to retain only those that, on energy minimization, escape to different regions of configurational space, finally leads to clear protein dynamical transition behavior. Partial minimization of instantaneous configurations is also found to remove nondiffusive imaginary modes. In summary, examination of the form of negative instantaneous normal modes is shown to furnish a physical picture of local diffusive dynamics accompanying the protein glass transition.
- SourceAvailable from: Jennifer A. Hayward[show abstract] [hide abstract]
ABSTRACT: Molecular dynamics simulations are performed of bovine pancreatic trypsin inhibitor in a cryosolution over a range of temperatures from 80 to 300 K and the origins identified of elastic dynamic neutron scattering from the solution. The elastic scattering and mean-square displacement calculated from the molecular dynamics trajectories are in reasonable agreement with experiments on a larger protein in the same solvent. The solvent and protein contributions to the scattering from the simulation model are determined. At lower temperatures (< approximately 200 K) or on shorter timescales ( approximately 10 ps) the scattering contributions are proportional to the isotopic nuclear scattering cross-sections of each component. However, for T > 200 K marked deviations from these cross-sections are seen due to differences in the dynamics of the components of the solution. Rapid activation of solvent diffusion leads to the variation with temperature of the total elastic intensity being determined largely by that of the solvent. At higher temperatures (>240 K) and longer times ( approximately 100 ps) the protein makes the only significant contribution to the scattering, the solvent scattering having moved out of the accessible time-space window. Decomposition of the protein mean-square displacement shows that the observed dynamical transition in the solution at 200-220 K involves activation of both internal motions and external whole-molecule rotational and translational diffusion. The proportion that the external dynamics contributes to the protein mean-square displacement increases to approximately 30 and 60% at 300 K on the 10- and 100-ps timescales, respectively.Biophysical Journal 08/2003; 85(2):679-85. · 3.67 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: We present two methods to probe the energy landscape and motions of proteins in the context of molecular dynamics simulations of the helix-forming S-peptide of RNase A and the RNase A-3'-UMP enzyme-product complex. The first method uses the generalized ergodic measure to compute the rate of conformational space sampling. Using the dynamics of nonbonded forces as a means of probing the time scale for ergodicity to be obtained, we argue that even in a relatively short time (< 10 psec) several different conformational substrates are sampled. At longer times, barriers on the order of a few kcal/mol (1 cal = 4.184 J) are involved in the large-scale motion of proteins. We also present an approximate method for evaluating the distribution of barrier heights g(EB) using the instantaneous normal-mode spectra of a protein. For the S-peptide, we show that g(EB) is adequately represented by a Poisson distribution. By comparing with previous work on other systems, we suggest that the statistical characteristics of the energy landscape may be a "universal" feature of all proteins.Proceedings of the National Academy of Sciences 02/1993; 90(3):809-13. · 9.74 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: Two onsets of anharmonicity are observed in the dynamics of the protein lysozyme. One at T approximately 100 K appears in all samples regardless of hydration level and is consistent with methyl group rotation. The second, the well-known dynamical transition at T approximately 200-230 K, is only observed at a hydration level h greater than approximately 0.2 and is ascribed to the activation of an additional relaxation process. Its variation with hydration correlates well with variations of catalytic activity suggesting that the relaxation process is directly related to the activation of modes required for protein function.Physical Review Letters 08/2005; 95(3):038101. · 7.94 Impact Factor
Instantaneous Normal Modes and the Protein Glass Transition
Roland Schulz,†Marimuthu Krishnan,†* Isabella Daidone,‡and Jeremy C. Smith†
†University of Tennessee/ORNL Center for Molecular Biophysics, Oak Ridge National Laboratory, Oak Ridge, Tennessee;
and‡Interdisciplinary Center for Scientific Computing, Ruprecht Karls University-Heidelberg, Heidelberg, Germany
a condensed-phase system, leading to modes with negative eigenvalues. These negative modes provide a means of character-
izing local anharmonicities of the potential energy surface. Here, we apply instantaneous normal mode to analyze temperature-
dependent diffusive dynamics in molecular dynamics simulations of a small protein (a scorpion toxin). Those characteristics of
the negative modes are determined that correlate with the dynamical (or glass) transition behavior of the protein, as manifested
as an increase in the gradient with T of the average atomic mean-square displacement at ~220 K. The number of negative eigen-
values shows no transition with temperature. Further, although filtering the negative modes to retain only those with eigenvectors
corresponding to double-well potentials does reveal a transition in the hydration water, again, no transition in the protein is seen.
However, additional filtering of the protein double-well modes, so as to retain only those that, on energy minimization, escape
to different regions of configurational space, finally leads to clear protein dynamical transition behavior. Partial minimization of
instantaneous configurations is also found to remove nondiffusive imaginary modes. In summary, examination of the form of
negative instantaneous normal modes is shownto furnish a physical picture of local diffusive dynamicsaccompanying the protein
In the instantaneous normal mode method, normal mode analysis is performed at instantaneous configurations of
The importance of internal dynamics in the function of
proteins is widely recognized (1–3). Proteins exhibit
a wide spectrum of dynamical processes ranging from
localized femtosecond-timescale harmonic vibrations to
collective diffusive movements with timescales up to the
millisecond range or longer (4).
Among the computational methods used to characterize
protein dynamics are normal mode analysis (NMA) and
molecular dynamics simulation (MD). NMA makes the
harmonic approximation to the potential energy around an
equilibrium configuration. In the realm of validity of this
approximation, the dynamics is determined completely by
the second-derivative (Hessian) matrix of the potential
energy with respect to the mass-weighted atomic coordi-
nates. The vibrational dynamics of the system around the
local minimum is described by the eigenvectors and eigen-
values of the Hessian matrix.
Standard NMA cannot be used to characterize anharmonic
dynamics. Instead, MD simulation, involving stepwise inte-
gration of the equations of motion with the full, anharmonic
potential, can be used. However, the analysis of MD trajec-
tories to extract useful information on anharmonic motion is
nontrivial. Here, we examine the application to proteins of
the Instantaneous Normal Mode (INM) method, which has
proven to be a useful analysis tool for diffusive dynamics
in condensed phase systems (5–8). INM is a statistical
mechanical theory connecting dynamic properties (e.g.,
diffusion constants) to equilibrium averages of the curva-
tures of the potential energy surface at sample configura-
tions. As in NMA, INM is based on the second derivative
of the potential energy. However, unlike in NMA, the
Hessian matrix is computed for snapshot configurations
generated, for example, during MD or Monte Carlo simula-
tion, instead of at fully energy-minimized configurations.
Negative eigenvalues thus result.
INM theory, which has been developed based on the
concept of inherent structure (9–11) and on a random energy
model (6,12–14), predicts that the self-diffusion constant
and the configurational entropy are proportional to the
average fraction of diffusive instantaneous normal modes
(7,8,15–19). However, although much progress has been
made, present theoretical derivations rely on significant
approximations and still have open questions (6,8,20–28).
It has therefore been necessary to employ computational
techniques to examine the relation between diffusive modes
and dynamical properties of liquids (15,28). Instantaneous
normal modes with negative curvature are called unstable
modes. To relate INM to diffusive properties filtering
methods have been proposed for separating diffusive from
nondiffusive unstable modes (see later) (19).
Computational INM studies have been hitherto performed
mainly on simple, supercooled liquids, with a focus on glass
transition, mode-coupling, and crystallization behavior
(6,15,23,25,27,29). These results have demonstrated that it
is possible to calculate useful temperature-dependent dynam-
ical properties, such as the self-diffusion constant, from INM
on a small number of sampled configurations. For proteins,
an integral equation connecting the fraction of unstable
INM modes with the barrier height distribution has been
derived and shown to aid in exploring the energy landscape
Submitted August 15, 2008, and accepted for publication October 15, 2008.
Editor: Nathan Andrew Baker.
? 2009 by the Biophysical Society
476Biophysical JournalVolume 96 January2009 476–484
(30,31). Also, vibrational energy relaxation rates of selected
vibrational modes of proteins have been computed using
INM analysis (32). However, research connecting dynamical
properties to unstable INM modes has not been documented
for proteins. The goal in this work is to investigate this
connection, and we examine which INM properties are
related to diffusive dynamics in proteins.
We concentrate on the dynamical or glass transition in
internal protein dynamics. Many experiments and simula-
tions (33–39) have shown that, for proteins at 180–220 K,
the average mean-square displacement exhibits a change in
gradient with temperature, involving the excitation of anhar-
monic displacements (40,41). The role of hydration in the
dynamical transition is an area of active research (37,42,43).
Calculations are performed here for a small protein as a func-
tion of temperature to determine which INM properties are
related to the physical nature of the transition. As for liquids,
it is essential to accurately compute the subset of unstable
modes from the set of negative modes, and so the filtering
methods proposed for liquids are evaluated here for proteins.
It is found that the number of unfiltered negative-eigenvalue
modes does not correlate with dynamical transition behavior.
However, selection of double-well modes that escape to dis-
tinct regionsofconfigurationalspace doesmirrorthe behavior
of the temperature-dependent mean-square displacement.
We first provide a brief overview of INM filtering methods. INM studies on
simple liquids have shown that some negative modes are not connected to
diffusion (26). Therefore, the nondiffusive negative modes should be filtered
out. In an early study on a Lennard-Jones liquid (6), the filtering was per-
formed by subtracting the number of negative modes observed for a crystal
(none of which are diffusive) from the number of negative modes for the
liquid. After this work, a cutoff-based filtering method was proposed in
which those unstable modes with frequency juj above a cutoff value were
considered to be diffusive (19). However, it was found that both of the above
filtering methods are material-specific. Subsequently, improved filtering
methods weredeveloped, basedon the topologyof the potentialenergyland-
scape (16,22), and these will now be outlined.
In the double-well filtering method, one-dimensional potential energy
profiles along the negative modes are calculated and used to classify them
into double-well or shoulder modes (22,44). For the double-well modes,
the one-dimensional potential energy profiles consist of two minima sepa-
rated by a barrier, while the shoulder modes exhibit single wells with shoul-
ders. It has been suggested that double-well modes are diffusive whereas
shoulder modes are not, and therefore the diffusive dynamics should be
associated only with the former (27). The filtering was further refined by
classifying the double-well modes as extended or localized depending on
the fraction of atoms that participate in these modes (22), the extended
double-well modes being assumed to make the primary contribution to the
diffusive dynamics. However, it was later demonstrated that the double-
well filtering method failed to provide a satisfactory description of the diffu-
sive dynamics of various materials, particularly ionic melts (45). Moreover,
it was shown that, for some double-well modes, quenching from either side
of the barrier converged to the same inherent structure, thus suggesting that
these double-well modes do not play any special role in diffusion (26).
A further problem arises when the one-dimensional potential energy
profile is computed along a straight-line in the direction of the eigenvector
computed at the instantaneous configuration. Because of the anharmonicity
of the energy landscape, the frequency, and thus curvature, changes quickly
along this line, due to a fast increase of the potential energy arising from van
der Waals repulsion between neighboring atoms (16). This fast increase
prohibits finding more distant saddles, and hence the evaluated number of
double-well modes corresponds to only those double wells with close-by
The finding that double-well modes do not always minimize to different
inherent structure basins led to the definition of escape modes (16,46). In
escape filtering, the configurations corresponding to the two one-
dimensional minima along the mode on either side of the saddle are mini-
mized. An inherent structure basin is defined as the region of the potential
energy surface that minimizes to the same point. By quenching the config-
urations corresponding to one-dimensional minima, it is possible to deter-
mine whether or not these minima belong to the same inherent structure
basin. Those modes for which the minima do not belong to the same basin
are escape or true double-well modes, the others being false double-well
modes. In an analysis of liquid water, the temperature dependence of the
number of escape modes was found to exhibit a behavior similar to that
of the self-diffusion constant (15,21).
As well as the energy landscape-based methods discussed above, two
minimization-based methods have been suggested: the saddle-order (SO)
and the partial quenching (PQ) methods. The SO method suggests that by
minimizing the squared gradient of the potential energy (jVUj2) the system
can be driven to the nearest stationary point (which can be a minimum or
a saddle). The number of imaginary frequency modes calculated at the
stationary point gives an estimate of the saddle order and thus the number
of diffusive modes (47).
In the PQ approach the instantaneous configuration is partially quenched
and the INM analysis performed at the partially quenched configuration so
as to remove spurious, nondiffusive, negative modes. It has been found
that the negative nondiffusive modes in a Lennard-Jones crystal vanish after
the first minimization steps, while only the diffusive modes persist in the
liquid state (47).
For all the filtering methods described above, linear proportionality to the
self-diffusion constant was found for at least some materials. The computa-
tional methods used here to examine the relation between the self-diffusion
constant and the number of diffusive modes for proteins will be discussed in
the next section.
MD simulations were performed on Toxin II fromthe ScorpionAndroctonus
australis Hector (PDB (48) id: 1PTX), solved at 1.3 A˚resolution (49), in the
is small enough (64 residues) to permit ready evaluationof a large number of
conditions. Toxin II crystallizes in an orthorhombic unit cell with the
following lattice parameters: a ¼ 45.94 A˚, b ¼ 40.68 A˚, and c ¼ 29.93 A˚.
The unit cell consists of four proteins. Explicit water molecules that were
resolved in the crystallographic analysis were included, i.e., 107 water mole-
talline form, although experimentally unrealizable, provides an appropriate
on the dynamical transition and corresponding simulations (50).
Calculations were performed with the CHARMM program (51) using
the CHARMM27 force field (52). Water molecules were represented by
the TIP3P model (53). Electrostatic interactions were computed using the
particle mesh Ewald method (54) for which the direct sum cutoff was
14 A˚and the reciprocal space interaction were computed on a 48 ? 48 ?
32 grid using sixth-degree B-splines.
Simulationswere carried out in the isothermal-isobaric (NPT)ensemble at
temperatures starting at 20 K and increasing to 300 K in steps of 20 K. The
system at each temperature was equilibrated for 1 ns followed by a produc-
tion run of 1 ns during which atomic coordinates were written out every
50 fs. The starting structure for each temperature was the final structure from
the preceding temperature.
Biophysical Journal 96(2) 476–484
Instantaneous Normal Modes of Proteins477
For each temperature, 40 equally spaced configurations (separated by
5 ps) were extracted from the last 200-ps segment of MD trajectory and
were used for the INM analysis. For mean-square displacement (MSD)
computations, the full production run was used. This procedure determines
whether the diffusion constant calculated using the INM analysis from a few
configurations over a short timescale can furnish information pertaining to
the diffusion constant computed from the MSD of a longer trajectory.
The computation of the Hessian matrix followed by its diagonalization is
computationally demanding for a complex system studied here (with long-
range electrostatic interactions treated with PME). In this investigation,
the Hessian matrices were computed with a finite difference method using
CHARMM on a single processor. The finite difference method computes
the first derivative of the energy function (the force) for a small displacement
to either side of the minimum for each of the 3N directions (where N is the
difference method followed by diagonalization), each requiring ~6.5 CPU
The landscape-based filtering methods were implemented in the
CHARMM program as follows. The eigenvectors and the atomic coordi-
nates were read and one-dimensional energy profiles calculated by displac-
ing the atomic coordinates along the eigenvector directions. An energy
threshold of 0.6 kcal/mol was used for exploring the potential energy profile.
The energy profiles were used to classify the imaginary modes into shoulder
or double-well types. In total, there were ~400,000 negative modes (evalu-
ated at 600 instantaneous configurations) and all modes were thus classified.
To identify escape modes, the configurations corresponding to the two
one-dimensional minima along each double-well mode were energy-mini-
mized. The adopted basis Newton-Raphson (ABNR) method was used for
these minimizations. After the minimization, the RMS distance between
the pair of quenched configurations was computed. It was assumed that
the minimized structures converge to a single minimum if the RMS distance
is less thana threshold value.The numberof minimizationsteps(1000steps)
and the distance cutoff (0.01–0.02 A˚) were chosen after a few convergence
tests. The same procedure was applied to both the double-well and shoulder
modes so as to determine which of these types behave like escape modes.
For shoulder modes, the starting structures were chosen either with energy
or distance criteria.
The two minimization-based methods were implemented in CHARMM.
In the PQ method, instantaneous configurations were partially minimized
for 4–16 steps and the Hessian matrices computed and diagonalized so as
to determine the fraction of unstable modes.
The SO method uses generalized stationary points on the potential energy
landscape to describe the diffusive dynamics (47). In this method, the
squared gradientof the potential energyis minimized,leading to minimaand
saddles around the instantaneous configurations. Here again the second-
derivative matrix was required, to minimize the squared gradient, thus
making this procedure computationally intensive.
The mean-square displacement (MSD) averaged over the
atoms as a function of temperature was calculated from the
MD trajectories, and the self-diffusion constant computed
from the long-time slope of the time-dependent MSD. In
Fig. 1 is shown the temperature dependence of the diffusion
constant for both the whole system (including the solvent)
and the protein. A significant change in the diffusion
constant at ~220 K is evident, originating from the dynam-
ical transition as observed in many biological systems
(1,2,33,37,42,55–67). The temperature dependence of the
diffusion constant in Fig. 1 was used as a benchmark with
which to compare the INM results.
As a first necessary step for the INM method, the Hessian
matrices were computed. This was performed from the simu-
lations of 15 temperatures, for 40 configurations each, taken
from the last 200 ps of the MD production runs in steps of
5 ps. In Fig. 2 is shown the number of imaginary frequency
modes (Fu) as a function of temperature. As expected, Fu
increases with temperature, as anharmonic regions of the
potential energy surface become explored. However, unlike
the diffusion constant, Fudoes not exhibit any slope change
at ~220 K and instead shows an almost constant slope. Thus,
it is evident that Fucannot be equated with D and does not
adequately describe internal diffusion in proteins.
In the landscape-based filtering methods the energy
profiles are explored by evaluating the potential energy along
the modes. Inspection of the profiles revealed that a small
scorpion toxin at 15 temperatures on an ~2 GHz CPU (see text
for number of configurations)
Approximate CPU times for the INM analysis of
Minimization for escape filtering
Partial minimization (45 matrices)
the full system (circles) as a function of temperature.
Diffusion constant of the protein (squares, scaled by 10?) and
The total number of modes (negative and positive) is 15,348.
Number of negative modes (Fu) as a function of temperature.
Biophysical Journal 96(2) 476–484
478Krishnan et al.
number of negative modes correspond to double wells while
no double wells were found for positive frequency modes (as
described in Methods, the potential energy profile can only
reveal saddles close to the instantaneous position; because
no saddle can be close in the direction of a positive frequency
mode, no double well can be found for positive frequencies).
Representative double-well profiles are shown in Fig. 3.
The temperature dependence of the number of diffusive
modes (FuDW) after double-well filtering, and the tempera-
ture dependence of the average negative frequency of the
double-well modes huuDWi, are shown in Fig. 4, panels
a and b, respectively. In this analysis, the contribution to
the number of negative modes by the protein was computed
by decomposing the normalized eigenvectors (~ n) into protein
(~ np) and water (~ nw) components such that~ n ¼~ npþ~ nw. We
compute the projection r of protein atoms as
r ¼ ~ n,~ np:
The negative modes for which r ~ 1 are assumed to be domi-
nated by diffusive protein dynamics while modes with
r ~ 0 are dominated by solvent dynamics. Fig. 4 a exhibits
a small slope change in the number of double-well modes
contributing to the motion of the water. However, similar
to the unfiltered case, no change of slope is observed for
the protein atoms, and thus the dynamical transition seen
in the temperature-dependent protein MSD is not seen in
the double-well filtered modes. The average frequency is
proportional to the temperature without inflection. The
variation of huuDWi as a function of temperature is one of
the reasons for the weak T-dependence of the D f furelation
observed for simple liquids (19).
The double-well modes were further classified into escape
modes and false double-well modes. The two one-dimen-
sional minima of an escape mode belong to different inherent
structure basins, while for false double wells they belong to
the same inherent structure. To check whether a given
double-well mode is an escape mode, the pair of configura-
tions corresponding to two minima of the double well was
energy-minimized (see Methods). This requires an efficient
the minimized configurations belong to the same inherent
structure. To determine a satisfactory minimization scheme,
a few modes were selected and escape-mode classification
was performed using different minimization methods (steep-
est descent, conjugate gradient, and Adopted Basis Newton-
Raphson (ABNR)) and numbers of steps. ABNR was found
to be the most suitable method for this classification.
To select the optimal number of minimization steps
required to filter escape modes, the following procedure
was carried out: Let Rpertdenote a perturbed configuration,
obtained by displacing the instantaneous configuration, Rinst
along arandomly-chosendouble-wellmode suchthat Rpert¼
Rinstþ DRinitudw, where udwis the eigenvector correspond-
ing to a double-well mode and DRinitdenotes the RMSD
between Rpertand Rinst. Let Rpertminand Rinstmindenote
configurations obtained by energy minimizing Rpert and
Rinst, respectively. The RMSD between Rpertminand Rinstmin
is denoted as DRmin. For various values of DRinitand the
number of minimization steps, DRminwas calculated and
the results are presented in Fig. 5. It is evident that DRmin
is sensitive to the number of minimization steps: increasing
the number of steps leads to more highly-converged results.
For instance, the variations in DRmin(for ?0.8 A˚< DRinit<
?0.2 A˚and 0.2 A˚< DRinit< 0.8 A˚) decrease as the number
ated at a randomly-chosen instantaneous configurations sampled at 300 K.
Representative double-well potential energy profiles evalu-
(FuDW). (b) Average frequency huuDWi of double-well modes.
Temperature dependence of (a) number of double-well modes
Biophysical Journal 96(2) 476–484
Instantaneous Normal Modes of Proteins479
of minimization steps increases, indicating that increasing
the number of minimization steps leads to more unique
inherent structures. Moreover, the diffusive nature of double-
well modes is evident in Fig. 5: a small displacement along
the negative direction of the mode leads to an inherent struc-
ture (after 1000 steps of minimization) with DRmin~ 0.04 A˚
while a small displacement along the positive direction leads
to a different inherent structure with DRmin¼ 0.0 A˚. This
analysis suggests that 1000 minimization steps are sufficient
to filter out escape-modes.
In addition to selecting the optimal number of minimiza-
tion steps, a suitable cutoff for the RMSD between the
inherent structures is required, so as to allow a decision to
be made as to whether perturbations to either side of
a double-well mode lead to the same inherent structure. To
estimate this cutoff, a set of perturbed configurations were
generated by displacing randomly-chosen instantaneous
configurations (sampled at 300 K) by DRinit¼ 0.014 A˚along
the double-well modes. The value of DRinit¼ 0.014 A˚was
obtained by averaging the distances along the double-well
modes at which the potential energy is 0.6 kcal/mol (KBT
at 300 K) higher than that at the instantaneous configuration.
minimized (with 1000 ABNR steps) resulting in an ensemble
of inherent structures. The RMSD (DRmin) between these
quenched configurations and Rinstminwas calculated as
described above and the histogram of DRminis shown in
Fig. 6. Two most-probable inherent structures are found:
one at DRmin~ 0.0 A˚and the other at DRmin~ 0.038 A˚: these
values are consistent with the data presented in Fig. 5.
The first minimum seen at DRmin~ 0.01 A˚clearly sepa-
rates the probable inherent structures, suggesting that DRmin
~ 0.01 A˚is a suitable cutoff for distinguishing between a pair
of inherent structures. Consequently, for each double-well
mode, a pair of configurations corresponding to two minima
of the double well was minimized with 1000 ABNR steps
and the double-well mode was classified as an escape
mode if the Euclidean distance between the two minimized
configurations thus obtained was >0.01 A˚.
increases gradually with temperature until ~200 K and an
abrupt change is observed at ~220 K. The large increase in
the population of escape modes at ~220 K is concomitant
with the onset of large amplitude displacements observed in
the MSD (Fig. 1). This observation indicates that the number
the protein increases at the dynamical transition temperature.
To further characterize the escape modes, we have calcu-
lated for each escape mode the participation ratio (P) using
with uibeing the component of the eigenvector contributed
by atom i. P ~ 1 for an extended eigenmode while P ~ 0
for strongly localized modes (68).
Fig. 8 shows the frequency dependence of P together with
the density of states of the escape modes. The escape modes
perturbed configurations were further energy-
minimizing the instantaneous structure and a perturbed structure obtained by
displacing (by DRinit) along a randomly-chosen double-well mode for three
different numbers of minimization steps.
Distance (DRmin) between a pair of configurations obtained by
FIGURE 6Histogram of (DRmin).
ysis performed at 10 (squares) and 40 (circles) configurations as a function
of temperature. A distance cutoff of 0.01 A˚was used for the escape filtering
Number of escape modes (Fuesc) calculated from INM anal-
Biophysical Journal 96(2) 476–484
480Krishnan et al.