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- PMC1752299

*Escherichia coli* Adenylate Kinase Dynamics:
Comparison of Elastic Network Model Modes with Mode-Coupling ^{15}N-NMR
Relaxation Data

^{1}Center for Computational Biology & Bioinformatics, Department of Biochemistry and Molecular Genetics, School of Medicine, University of Pittsburgh, Pittsburgh, Pennsylvania

^{2}Faculty of Life Sciences, Bar-Ilan University, Ramat-Gan 52900, Israel

## Abstract

The dynamics of adenylate kinase of *Escherichia coli*
(AKeco) and its complex with the inhibitor AP_{5}A, are characterized by
correlating the theoretical results obtained with the Gaussian Network Model
(GNM) and the anisotropic network model (ANM) with the order parameters and
correlation times obtained with Slowly Relaxing Local Structure (SRLS) analysis
of ^{15}N-NMR relaxation data. The AMPbd and LID domains of AKeco
execute in solution large amplitude motions associated with the catalytic
reaction Mg^{+2}*ATP + AMP
→ Mg^{+2}*ADP + ADP. Two
sets of correlation times and order parameters were determined by NMR/SRLS for
AKeco, attributed to slow (nanoseconds) motions with correlation time
τ_{} and low order parameters, and fast
(picoseconds) motions with correlation time τ_{||} and high
order parameters. The structural connotation of these patterns is examined
herein by subjecting AKeco and AKeco*AP_{5}A to GNM
analysis, which yields the dynamic spectrum in terms of slow and fast modes. The
low/high NMR order parameters correlate with the slow/fast modes of the backbone
elucidated with GNM. Likewise, τ_{||} and
τ_{} are associated with fast and slow GNM
modes, respectively. Catalysis-related domain motion of AMPbd and LID in AKeco,
occurring *per* NMR with correlation time
τ_{}, is associated with the first and
second collective slow (global) GNM modes. The ANM-predicted deformations of the
unliganded enzyme conform to the functional reconfiguration induced by
ligand-binding, indicating the structural disposition (or potential) of the
enzyme to bind its substrates. It is shown that NMR/SRLS and GNM/ANM analyses
can be advantageously synthesized to provide insights into the molecular
mechanisms that control biological function.

**Keywords:**Gaussian network model, Slowly Relaxing Local Structure, collective modes, conformational changes

## INTRODUCTION

NMR spin relaxation measurements can be translated into microdynamic
parameters, thereby providing important information on protein dynamics.^{1}^{–}^{3} The N-H bond is a particularly useful probe, relaxed predominantly by
dipolar coupling of the ^{15}N nucleus to the amide proton and
^{15}N chemical shift anisotropy (CSA).^{4}^{15}N relaxation data in proteins are commonly analyzed with
the model-free (MF) approach, where N-H bond dynamics is represented by two types of
motions assumed to be decoupled^{5}^{–}^{7}: the global
tumbling of the protein and the local motion of the N-H bond. In the extended
version of the MF approach, a slow internal motion is also included in the
formalism.^{7}

We recently applied the Slowly Relaxing Local Structure (SRLS) approach^{8}^{,}^{9} to NMR
spin relaxation in proteins.^{10} SRLS accounts
for the dynamical coupling between local and global motions within the scope of a
stochastic model in which the global tumbling (**R ^{C}**), the
local diffusion (

**R**), the local ordering (

^{L}**S**), and the magnetic interactions are represented by asymmetric tensors. The spectral density is obtained by solving a two-body (N-H bond associated with

**R**, and protein associated with

^{L}**R**) Smoluchowski equation. The mode-coupling SRLS approach can be considered the generalization of the mode-decoupling MF approach. We found that the SRLS picture of N-H bond dynamics is significantly more accurate than, and in some cases qualitatively different form, the MF picture.

^{C}^{10}

^{–}

^{14}

NMR spin relaxation results from an ensemble of modes of motion that
determine the concerted reorientation of the N-H bond and its surroundings. Methods
based on principal components analysis such as normal mode analysis (NMA),^{15}^{–}^{17} and essential dynamics analysis (EDA),^{18} have been widely used to dissect protein dynamics into its
contributing modes. Molecular dynamics (MD) simulations have been used in the
context of NMR spin relaxation in proteins to study local motions.^{19} The recently developed isotropic Reorientational
Eigenmode Dynamics (iRED) approach^{20}
associates MD-based internal modes with spin relaxation data in proteins. The
Gaussian network model (GNM)^{21}^{,}^{22} and its recent extension accounting for
anisotropic effects (ANM)^{23} efficiently
elucidate the spectrum of motions given the set of topological constraints
(inter-residue contacts) in the folded state. GNM (ANM) has the advantage of
yielding an analytical solution for a set of N-1 (3N-6) collective modes defined
uniquely for the examined structure. In particular, the lowest frequency modes
predicted by GNM/ANM, or elastic network (EN) models in general, have been shown in
numerous studies^{24}^{–}^{45} to be directly relevant to biologically
functional motions.

In this study, we focus on the backbone dynamics of adenylate kinase from
*Escherichia coli* (AKeco), a 23.6-kDa enzyme made of three
domains: CORE, AMPbd, and LID. AKeco catalyzes the reaction
Mg^{+2}*ATP + AMP →
Mg^{+2}*ADP + ADP.^{46}^{,}^{47} The CORE
structure is largely preserved during catalysis whereas the domains AMPbd and LID
execute large amplitude motions to configure the active site for substrate binding
and disassemble it toward product release.^{48}^{–}^{54} The structures
of the ligand-free enzyme^{52} and its complex
with the two-substrate-mimic inhibitor AP_{5}A^{51} are shown in Figure
1. The crystal structure of AKeco represents the
“open” conformation of the enzyme [Fig. 1(a)],^{52} and that of AKeco*AP_{5}A
represents the “closed” conformation [Fig. 1(b)]. The latter was shown to
be a mimic of the catalytic transition state.^{55}

**a**) AKeco

^{52}and (

**b**) AKeco in complex with the two-substrate-mimic inhibitor AP

_{5}A.

^{51}The figures were drawn with the program Molscript

^{76}using the PDB

**...**

A substantial amount of dynamic information relevant to the solution state is
currently available for AKeco and AKeco*AP_{5}A. The finding
that the domains AMPbd and LID are engaged in large-amplitude catalysis-related
displacements was first set forth by X-ray crystallographic studies.^{50}^{–}^{52}^{,}^{54} Subsequently, domain motion
was proven to take place *in solution* by time-resolved fluorescence
energy transfer studies.^{53} However, the
optical studies did not provide information on motional rates, spatial restrictions
implied by the segmental nature of domain motion, and the dynamic model *per
se*. Toward the important goal of acquiring this information, we applied
^{15}N spin relaxation methods to AKeco and
AKeco*AP_{5}A.^{13}^{,}^{14}^{,}^{56} In our first study,^{56} we used the MF approach to analyze the experimental data. Despite the
fact that the experimental NOE’s of AKeco were significantly
depressed^{14} within AMPbd and LID, which
is a clear indication of slow motions, the full-fledged MF analysis yielded
practically flat order parameter patterns for both enzyme forms.^{56} The low performance of MF was ascribed to the neglect
of mode-coupling, which is unjustified should domain motion occur on the same time
scale as the global motion.^{8} Within the scope
of the MF analysis, we determined the global diffusion tensor using the common
MF-based procedures. Although the anisotropy of the inertia tensor of the elongated
crystal structure^{52} [Fig.1(a)] is 1.49, the global
diffusion tensor was found to be practically isotropic^{56}^{,}^{57} with a
correlation time τ_{m} = 15.1 ± 0.1 ns,
in agreement with AKeco prevailing in solution as an ensemble-averaged
structure.^{13}^{,}^{14}^{,}^{53}^{,}^{56} This value of τ_{m} was
confirmed by our subsequent studies.^{13}^{,}^{14}

To test the effect of mode-coupling and local geometry, and eventually
improve the analysis, we re-analyzed our data^{13}^{,}^{14} using our recently
developed SRLS approach.^{10} SRLS detected very
clearly domain motions in AKeco.^{13} The order
parameters differed significantly between the mobile domains AMPbd and LID on the
one hand, and the structurally preserved domain CORE on the other hand. For the
first time, the rate of domain motion in kinases was quantified and set at 8.2
± 1.3 ns. As anticipated, this correlation time is on the order of the
global motion correlation time of 15.1 ns. Hence the overall and domain motions are
necessarily coupled, explaining the significantly higher performance of
mode-coupling SRLS as compared to mode-decoupled MF. Details on the MF fitting
process that generated nearly the flat S^{2} profiles can be found in
reference^{13}. SRLS also detected
nanoseconds motions experienced by specific loops of
AKeco*AP_{5}A,^{14}
which could be related to the dissociation of the catalytic transition state
mimicked^{55} by this complex.

Although the dynamic properties of AKeco and
AKeco*AP_{5}A are thus known in considerable detail, further
investigation of several key issues can be quite enlightening. For example, it would
be insightful to correlate the microdynamic parameters derived with SRLS analysis
with the modes computed using a different, and on some aspects complementary,
physical perspective. Domain mobility renders AKeco particularly well suited to be
explored with the GNM. We pursue here a combined experimental (NMR/SRLS) and
theoretical (GNM/ANM) investigation of the dynamics of AKeco and its complex with
AP_{5}A. The questions addressed are: How do the order parameters
predicted by GNM correlate with those deduced from SRLS (or MF) analysis of the
experimental NMR data? What is the structural connotation of the observed NMR
relaxation behavior? What are the dominant mechanisms of backbone motion, and how
are they reflected in the experimentally determined order parameters? What is the
relation between the observed and computed low frequency (global) GNM modes and the
functional motions of the enzyme?

## THEORY

### The Slowly Relaxing Local Structure Approach

SRLS is an effective two-body model for which a Smoluchowski equation
representing the rotational diffusion of two coupled rotors is solved.^{8}^{–}^{10} In SRLS, the coupling between the global diffusion
frame (C) and local diffusion frame (M) rigidly attached to the N-H bond is
accounted for by a potential U(Ω_{CM}), where
Ω_{CM} denotes the Euler angles between the two frames.
U(Ω_{CM}) can be expanded in the full basis set of
Wigner rotation matrix elements, D_{KM}^{L}
(Ω_{CM}). If only the lowest order (L = 2)
terms are preserved, the potential becomes

where the coefficients ${c}_{0}^{2}$ and ${c}_{2}^{2}$ account for the strengths of the axial and rhombic
contributions, respectively. The local ordering at the N-H bond is described by
the ordering tensor, **S**, the principal values of which are the
ensemble averages

and

Axial potentials feature only the first term of eq 1, $({\text{S}}_{2}^{2}=0)$.

In the absence of an ordering potential, the solution of the
Smoluchowski equation yields three distinct eigenvalues (or correlation times
τ_{K}; K = 0, 1, 2) for the local motion

where ${\text{R}}_{\parallel}^{\text{L}}$ and ${\text{R}}_{\perp}^{\text{L}}$ are the relaxation rate constants parallel and perpendicular
to the ^{15}N-^{1}H bond vector, with ${\text{R}}_{\parallel}^{\text{L}}=1/(6{\tau}_{\parallel})$ and ${\text{R}}_{\perp}^{\text{L}}=1/(6{\tau}_{\perp})$. Each K value leads to its own spectral density component
j_{K=0}(ω),
j_{K=1}(ω), and
j_{K=2}(ω).^{57}^{,}^{58} For magnetic tensors
that are axially symmetric in the M frame, only
j_{K=0}(ω) enters the measurable spectral
density (defined below). Otherwise all three components
j_{K=0}(ω),
j_{K=1}(ω), and
j_{K=2}(ω) determine the spectral density. When
the potential U(Ω_{CM}) is infinitely strong, and if the
protein is approximated by a spherical top, then the measurable spectral density
reduces to ${\tau}_{\text{m}}/(1+{\omega}^{2}{\tau}_{\text{m}}^{2})$ where τ_{m} =
(6R^{C})^{−1} is the correlation time for
overall tumbling and R^{C} is the isotropic diffusion rate.

In the general case, the solution consists of multiple modes,
(*j*), expressed in terms of the eigenvalues
1/τ(*j*) and weighing factors
c_{i}(*j*) such that^{8}^{–}^{10}

The eigenvalues 1/τ(*j*) refer to pure or
mixed dynamic modes, in accordance with the parameter range considered. Concise
expressions for the SRLS spectral density for dipolar auto-correlation,
J^{dd}(ω), ^{15}N CSA auto-correlation,
J^{cc}(ω), and ^{15}N CSA –
^{15}N-^{1}H dipolar cross-correlation,
J^{cd}(ω), are

where the coefficients A(x), B(x), and C(x), with x denoting cc, dd, or
cd, feature the trigonometric expressions obtained by the frame
transformations.^{8}^{,}^{9}^{,}^{57}^{,}^{58} The measurable spectral densities are
calculated as a function of *J*(0), *J*(ω* _{N}*),

*J*(ω

*),*

_{H}*J*(ω

*+ω*

_{H}*) and*

_{N}*J*(ω

*− ω*

_{H}*) (obtained from*

_{N}*J*

*(ω) by including the magnetic interactions) using standard expressions for NMR spin relaxation.*

^{x}^{4}

^{,}

^{59}Details on the implementation of SRLS in a data-fitting scheme featuring axial potentials were outlined previously.

^{10}

### The Model-Free Approach.

In the MF approach^{5}^{–}^{7}, the overall
tumbling of the protein and an effective local N-H motion are assumed to be
decoupled. Consequently, the correlation function for N-H bond motion is the
product of the correlation functions corresponding to these two types of
motions, i.e.,

Here C^{C}(t) =
1/5exp(−t/τ_{m}) is the correlation function
for isotropic overall tumbling and C^{L}(t) is the correlation function
for local motions, expressed as C^{L}(t) = S^{2}
+ (1 −S^{2})exp(−t/τ_{e}),^{5}^{,}^{6} where
τ_{e} denotes the effective local motion correlation
time, and S^{2} is the squared generalized order parameter defined as
S^{2} = C^{L}(∞). Mode-decoupling is
implied by τ_{e} τ_{m}.
The measurable spectral density, J(ω), is given by^{5}^{,}^{6}

where ${\tau}_{eff}^{-1}={\tau}_{m}^{-1}+{\tau}_{e}^{-1}$. If the equilibrium distribution of N-H orientations is
axially symmetric, then ${\text{S}}^{2}={\langle [3/2{\text{cos}}^{2}{\beta}_{\text{CM}}-\frac{1}{2}]\rangle}^{2}$ where β_{MC} is defined by
Ω_{MC} = (0, β_{MC}, 0).
When eq 8 cannot fit the
experimental data, the extended MF spectral density^{7}

is used, where $1/{{\tau}^{\prime}}_{f}\equiv 1/{\tau}_{f}+1/{\tau}_{m}\mathrm{\hspace{0.17em}\u200a\u200a}\text{and\hspace{0.28em}}1/{{\tau}^{\prime}}_{s}\equiv 1/{\tau}_{s}+1/{\tau}_{m},{\tau}_{f}$ is the correlation time for a fast internal motion associated
with a squared generalized order parameter ${\text{S}}_{\text{f}}^{2}$, and τ_{s} the correlation time for a slow
internal motion associated with a squared generalized order parameter, ${\text{S}}_{\text{s}}^{2}$. The slow internal motion occurs on the same time scale as the
global tumbling whereas τ_{f} is much shorter.

### The Gaussian Network Model

In the GNM, the protein is viewed as an elastic network, the nodes of
which are the amino acids represented by their C^{α} atoms. All
residue pairs located within a cutoff distance of r_{c} are assumed to
be coupled (or connected) via a harmonic potential (or a spring) with a uniform
force constant (γ), which stabilizes the native structure.^{21}^{,}^{25} The equilibrium correlation Δ**R*** _{i}* · Δ

**R**

*between the fluctuations Δ*

_{k}**R**

_{i}and Delta;

**R**

_{k}of the α-carbons i and k is given by

where k_{B} is the Boltzmann constant, T the absolute
temperature, **Γ**^{−1} the inverse of
Kirchhoff matrix of contacts characteristic of the examined structure, and the
subscript ik denotes the ikth element of the matrix. The off-diagonal elements
of **Γ** are given by
**Γ**_{ij} = −1 if residues
i and j are connected, and are zero otherwise. The diagonal elements of
**Γ** are found from the negative sum of the elements
in the corresponding column (or row), such that
**Γ**_{ij} is equal to the number of
inter-residue contacts that the ith residue makes.^{21}^{,}^{25} A cutoff
distance r_{c} = 10 Å is adopted here, which is
long enough to include all bonded and non-bonded neighbors within a first
coordination shell.^{60}

A major attribute of the GNM is its ability to assess the contribution
of individual modes to the observed dynamics. The cross-correlation
Δ**R*** _{i}* · Δ

**R**

*may be expressed as a sum over N-1 collective modes found from the eigenvalue decomposition of*

_{k}**Γ**, ranging from fast and localized motions to slow and highly cooperative motions, i.e.,

Here λ_{j} is the jth eigenvalue of
**Γ**, **u**_{j} is the jth
eigenvector, and ${A}_{ik}^{(j)}={[{\lambda}_{j}^{-1}{u}_{j}{u}_{j}^{T}]}_{ik}$ represents the contribution of the jth mode to
Δ**R*** _{i}* · Δ

**R**

*. Equation 11 reduces to the autocorrelation or mean-square fluctuation (ΔR*

_{k}_{i})

^{2}when k = i.

The GNM theory permits us to evaluate the *profile* of
the residue-specific correlation times, τ_{i,GNM}, for each
residue i. τ_{i,GNM} scales as^{22}

which enables us to determine the relative contribution of the individual modes. Equation 12 yields the relative values of the correlation times of individual residues, rather than their absolute values.

### Calculation of Squared Order Parameters Using GNM

The local geometry near the i_{th} N-H bond is depicted in Figure 2. Figure 2(a) shows the virtual bond representation of the protein
backbone. The structure is represented by a sequence of rigid planes defined by
the *trans* peptide bond and the two flanking backbone bonds. The
ith N-H bond lies within the peptide plane that contains the atoms ${\text{C}}_{\text{i}-1}^{\alpha},{({\text{C}}^{\prime}\text{)}}_{\text{i}-1},{(\text{N)}}_{\text{i}},\mathrm{\hspace{0.17em}\u200a\u200a}\text{and\hspace{0.28em}}{\text{C}}_{\text{I}}^{\alpha}$. **l**_{i} is the virtual bond ${\text{C}}_{\text{i}-1}^{\alpha}-{\text{C}}_{\text{i}}^{\alpha}$ about which the torsional fluctuation
Δ_{i} occurs. As shown in Figure 2(b), the bond (N-H)_{i} makes an angle
ε_{i} with **l**_{i}. The angular
change, Δα_{i}, in the orientation of the bond
(N-H)_{i} from its original position **m**(0) to the
position **m**(t) at time t is determined by the change in the virtual
bond dihedral angle, _{i.}, provided all the other bond
lengths and bond angles are kept fixed. The largest contribution to
Δα_{i} comes from the rotation
Δ_{i} of the ith virtual bond, with the
effect of virtual bond rotations, Δ_{j},
decreasing with increasing separation between i and j.

**a:**Schematic of the virtual bond of the GNM theory. The chain is made of consecutive peptide planes comprising the atoms . The bond vector N

_{i}-H

_{i}makes an angle ε

_{i}with

**...**

In the absence of coupling between adjacent bond rotations, the GNM
order parameter ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ for (N-H)_{i} is fully determined by
Δ_{i} and given by^{61}

with

using the equality
cosΔ_{i}
= 0, and the Gaussian approximation $\langle \mathrm{\Delta}{\phi}_{i}^{4}\rangle \approx (5/3){\langle \mathrm{\Delta}{\phi}_{i}^{2}\rangle}^{2}$ for small fluctuations. The problem of evaluating ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ thus reduces to determining the autocorrelation $\langle \mathrm{\Delta}{\phi}_{i}^{2}\rangle $ given by

Here **a**_{ij} is the transformation vector that
operates on the dihedral angles and transforms them into position vectors,
according to the relationship $\mathrm{\Delta}{R}_{\text{i}}={\sum}_{j=3}^{i-1}{a}_{i,j}\mathrm{\Delta}{\phi}_{j}$.^{62}

Neighboring dihedral angles are interdependent due to chain connectivity
and the need to localize the translational motions of the backbone. In
particular, bonds i and i±2 are strongly correlated and undergo
coupled counter rotations.^{62}^{–}^{64} These
couplings are accounted for by correcting eq 13 as

where ΔS^{2}(Δ_{i},
Δ_{k}) is the contribution of
Δ_{k} to the reorientation of
(N-H)_{i} defined as

The cross-correlation
Δ_{i}Δ_{k}
is given by

In the present calculations, cross-correlations up to second neighboring
bonds (|*k* − *i*| ≤ 2,
*k* ≠ *i*) were included. The
contributions of the bonds k = i±1 and k =
i±2 to the order parameter of (N-H)_{i} depend on the size
of the cross-correlations
Δ_{i}Δ_{k},
hence the use of the scaling term
|Δ_{i}
Δ_{k}| in eq 17. We note that equation 17 vanishes, and ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ reduces to
S^{2}(Δ_{i}), in the case of
uncorrelated torsions, i.e., when
Δ_{i}Δ_{k}
= 0. The factor $\frac{1}{2}$ in eq 17
accounts for the equal distribution of the effect of bond rotation on both sides
of the rotating bond. Our previous work showed that correlations up to the
second neighbors have a significant effect on ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$.^{61}

## RESULTS AND DISCUSSION

### Comparison of X-Ray Crystallographic and GNM B Factors

X-ray crystallographic temperature factors provide a measure for the
mobilities of individual residues in folded protein structures. Figures 3(a) and (b) show the B-factors predicted by
GNM (solid curve) superimposed on the crystallographic B-factors (dashed curve)
for the ligand-free^{52} and
inhibitor-bound^{51} forms of AKeco,
respectively. The resolution of the crystal structure of the unliganded form is
2.2 Å with an R-value of 0.183 (PDB code: 4ake).^{52} The resolution for the inhibitor bound structure
is 1.9 Å with an R-value of 0.196 (PDB code: 1ake).^{51} GNM force constants γ of 0.127 and
0.133 kcal/(mol.Å^{2}), derived from the comparison of the
predicted values with experimental data, were used for AKeco and
AKeco*AP_{5}A, respectively. γ rescales
uniformly the magnitudes of the GNM B-factors for a given protein without
affecting the relative residue-specific B-factors of residues or their
fluctuation profiles in different modes.

^{51}

^{,}

^{52}(---) and GNM-predicted (—) B-factors for AKeco (

**a**) and AKeco*AP

_{5}A (

**b**).

The correlation coefficient between experimental and theoretical results
is 0.72 for AKeco [Fig.
3(a)] and 0.59 for AKeco*AP_{5}A
[Fig. 3(b)].
The boxes along the upper abscissa depict the domains AMPbd and LID. The high
B-factors within the AMPbd and LID domains of the ligand-free enzyme point out
high mobility [Fig.
3(a)]. This property was detected in previous
crystallographic studies^{52} as well as
spectroscopic studies in solution.^{13}^{,}^{14}^{,}^{53}^{,}^{56} Theory
and experiment agree for AKeco*AP_{5}A as well, except for
the loop α_{4}/β_{3} (residues Q74-G80),
where the crystallographic data^{51} show
higher mobility [Fig.
3(b)]. This loop features the sequence AQEDCRNG,^{51} which includes quite a few long side
chains. The enhanced mobility of such long side chains may be overlooked by the
GNM.^{25} Comparison of the results for
AKeco [Fig. 3(a)]
and AKeco*AP_{5}A [Fig. 3(b)] reveal the significant decrease in the
mobility of the AMPbd and LID domains upon inhibitor binding.

### Comparison of NMR/SRLS and GNM Squared Order Parameters

Figure 4 compares the NMR-derived
(open circles) and GNM-derived (curves) squared order parameters for Akeco
[Fig. 4(a,b)],
and AKeco*AP_{5}A [Fig. 4(c,d)]. The NMR-derived ${({\text{S}}_{0}^{2})}^{2}$ values in Figure 4(a,c)
were obtained previously^{56} by fitting
with MF the experimental data acquired at 303K, and 14.1/18.8 T, with
τ_{m} = 15.1 ns. The corresponding ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ values were computed from the superposition of all the N-1 GNM
modes (solid curves). We note that the GNM results provide information on the
*distribution* of order parameters rather than their absolute
values. The correlation coefficient is in this case a good measure for
comparison with the NMR data. We calculated the correlation coefficient between
the two sets of data in Figure 4(a), taking
running averages over three consecutive residues to minimize the noise, which
yielded a correlation coefficient of 0.37. Thus, little correlation is observed
even qualitatively between the theoretical GNM results and the results of the MF
analysis. Moreover, Figure 4(a) shows that
the known^{14}^{,}^{52}^{,}^{53} mobility
of the domains AMPbd and LID is practically undetected with the MF analysis,
which generated a nearly flat order parameter profile, while it is detected
conspicuously with the simple predictive GNM analysis by significantly depressed ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ values within AMPbd and LID.

**a:**NMR order parameters obtained with MF analysis (open circles)

^{56}and theoretical GNM order parameters (solid curve), as a function of residue number for AKeco.

**b:**NMR order

**...**

The ${({\text{S}}_{0}^{2})}^{2}$ SRLS values (open circles) obtained^{13}^{,}^{14} by
fitting with SRLS the same experimental data as used in the MF analysis^{56} are shown for AKeco in Figure 4(b) and AKeco*AP_{5}A in
Figure 4(d). The GNM order parameters
obtained form the N/4 slowest modes (solid curve), and the N/4 fastest modes
(dashed curve) are shown separately. The ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ order parameters profile obtained using the complete set of
N-1 modes [shown in Fig.
4(a,c)] is very similar to that obtained from the N/4 slowest
modes, apart from a general decrease due to the disorder contributed by the fast
modes. This close similarity emphasizes the dominant role of the slow modes. The
correlation coefficient between ${({\text{S}}_{0}^{2})}^{2}$ SRLS of Figure 4(b) and
all-mode ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ data is 0.65, which is a significant improvement over 0.37
obtained with the Figure 4(a) data. We note
that this correlation coefficient is comparable to those recently obtained by
Zhang and Bruschweiler^{65} for a series of
other proteins. The results in that study were found using a simple empirical
expression based on the contacts made by the N-H hydrogen atom and the preceding
carbonyl oxygen with heavy atoms. While this empirical expression was successful
in reproducing a set of experimental data (for other proteins), physical
insights on the origin and mechanisms of the molecular motions and correlations
that give rise to the observed relaxation are provided by the GNM, as will be
further elaborated below.

Comparison of the ${({\text{S}}_{0}^{2})}^{2}$ SRLS and ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ profiles is more meaningful than the magnitude of the
empirical correlation factor. The ${({\text{S}}_{0}^{2})}^{2}$ SRLS profile shown in Figure
4(b) (open circles) singles out unequivocally the mobile domains
AMPbd and LID. Except for a few outliers within CORE, low ${({\text{S}}_{0}^{2})}^{2}$ values on the order of 0.35 are encountered exclusively within
AMPbd and LID, whereas much higher values, on average 0.86, are encountered
within CORE. Inasmuch as GNM predicts the *relative* values, it
is meaningful to examine the ratio of the computed squared order parameters at
the rigid (CORE) and the mobile (AMPBd and LID) domains. An average ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ value of 0.69 is found from the slow modes for CORE while
AMPbd and LID exhibit ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ values as low as 0.35–0.40, which leads to a ratio
of about 0.69:0.40. This ratio is smaller than that (0.86:0.35) indicated by the
SRLS analysis. Interestingly, the high ${({\text{S}}_{0}^{2})}^{2}$ values of CORE are reproduced by considering the N/4 fastest
modes exclusively (dashed curve). The superposition of the remaining $\frac{3}{4}\text{N}$ GNM modes depresses the order parameters within AMPbd and LID
to significantly lower values, consistent with the involvement of these domains
in the slow modes.

There are several regions of the AKeco backbone that show high mobility
according to ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$. These include residues G7-P9 of the P-loop, residues
Q16–Q18 of helix α_{1}, the loop
α_{4}/β_{3}, residues T175-P177 linking
helices α_{7} and α_{8}, residues G144 and
R156 of LID, and residues G198-P201 of the loop
β_{9}/α_{9}. The N-terminal chain
segment comprising the first 30 residues, which includes the P-loop and residues
Q16-Q18, has been identified as a major structural block required for the
stability of the native state.^{66} This
chain segment also plays a functional role through the P-loop binding motif.
Residues T175–P177 comprise the joint IV,^{51} which plays a key role in the catalysis-related
movements of LID.^{49}

On the whole, the ${({\text{S}}_{0}^{2})}^{2}$ SRLS and ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ profiles of AKeco agree, with extra flexibility predicted by
GNM at specific chain positions within CORE, as outlined above. Perfect
agreement between SRLS and GNM across the board is not to be expected. The basic
tenets of these methods are different. SRLS solves the stochastic rotational
diffusion equation for every N-H site in the protein.^{10} GNM is based on topological considerations related
to the alpha carbons.^{21}^{,}^{22} Despite this, the simple predictive GNM method
clearly detects catalysis-related domain motion in AKeco, in agreement with the
SRLS analysis. As pointed out above, domain motion was proven to occur in
*solution* by optical studies^{53} and NMR/SRLS.^{10}^{,}^{14} The GNM order parameters
[solid curve in Figure
4(a)] concur with these results whereas the MF order
parameters [open circles in Figure
4(a)] do not.

${({\text{S}}_{0}^{2})}^{2}$ values obtained previously^{14} by fitting with SRLS the experimental data obtained at 303K,
14.1/18.8T, with τ_{m} = 11.6 ns^{13} are shown in Figure
4(d) for AKeco*AP_{5}A, along with the ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ curves calculated for the slowest N/4 modes (solid curve), and
the fastest N/4 modes (dashed curve). The noise in the two sets of data
precludes comparison in terms of a correlation coefficient. However, the fact
that domain motion is discontinued upon inhibitor binding is borne out by both
the ${({\text{S}}_{0}^{2})}^{2}$ and ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ profiles. The AKeco*AP_{5}A backbone was
shown in previous work^{14} to be quite
rigid in solution, as shown by the high ${({\text{S}}_{0}^{2})}^{2}$ values. Only selected residues within the loops
α_{2}/α_{3},
α_{4}/β_{3}, and
α_{5}/β_{4}, and the C-terminal segment
of domain LID, are flexible according to SRLS [Fig. 4(d)]. The loops
α_{2}/α_{3} and
α_{4}/β_{3} show some mobility
according to both GNM and SRLS. GNM indicates mobility in the LID domain, and at
several additional positions along the chain. Excluding the flexible residues
mentioned above, the average ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ value obtained from the N/4 slowest GNM modes is 0.74 for
AKeco*AP_{5}A, and the average ${({\text{S}}_{0}^{2})}^{2}$ value is 0.93. The agreement improves if the fastest N/4 modes
are used to calculate ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$, but it is not as good as the GNM-SRLS agreement obtained for
the CORE domain of AKeco. We recently found that unduly high ${({\text{S}}_{0}^{2})}^{2}$ values may arise from using axial potential to fit the data,
while the actual potentials are asymmetric.^{11} For practical reasons, the ${({\text{S}}_{0}^{2})}^{2}$ values in Figure 4(d)
were obtained using axial potentials. This also applies to the AKeco ${({\text{S}}_{0}^{2})}^{2}$ values shown in Figure
4(b) but for AKeco the effect of potential asymmetry is apparently
smaller.^{11} A fitting scheme for SRLS
allowing for asymmetric potentials, the development of which is underway, is
expected to yield lower ${({\text{S}}_{0}^{2})}^{2}$ order parameters for AKeco*AP_{5}A in
better agreement with their GNM counterparts.^{11}

### ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ Versus GNM B-Factors

Some chain segments are singled out as flexible by ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ (Fig. 4) but not by the B
factors (Fig. 3). This behavior is rooted
in the compositions of these variables. The B-factors depend on the $\langle \mathrm{\Delta}{\text{R}}_{i}^{2}\rangle $ values associated with α-carbons.^{22}^{,}^{25}^{,}^{67}
${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ depends on the rotational autocorrelations,
Δ_{i}^{2}, and cross-correlations, $\langle \mathrm{\Delta}{\phi}_{1}^{2}\rangle $, derived from
Δ**R**_{i} · Δ
**R**_{k}.^{61} Translational and orientational fluctuations are usually correlated,
but need not be identical.^{25}
${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ provides a measure for the rotational mobility of the
backbone, which may in some cases tend to localize the translational motions of
the backbone. The loop α_{2}/α_{3} (residues
S44-Q48) of AKeco*AP_{5}A is a typical example shown by both ${({\text{S}}_{0}^{2})}^{2}$ and ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ to experience high rotational mobility [Fig. 4(d)], but confined to
relatively restricted spatial displacements according to crystallographic and
GNM-derived B-factors [Fig.
3(b)]. This chain segment is part of the
α_{2} helix in AKeco and represents a loop in
AKeco*AP_{5}A, constituting the only secondary structure
element which is altered upon AP_{5}A binding.

### NMR/SRLS and GNM Correlation Times

The SRLS squared order parameters of AKeco are clustered into two
distinct ranges as can be clearly seen in Figure
4(b).^{13} High order parameters
have been associated with “ps regime” dynamics
(correlation times τ_{||} on the order of picoseconds) and
low order parameters with “ns regime” dynamics
(correlation times τ_{} on the order of
nanoseconds, and τ_{||} -
τ_{}).^{10}^{,}^{13}^{,}^{14} Comparison with GNM results lends support to the
association of low NMR/SRLS order parameters with slow GNM modes and high
NMR/SRLS order parameters with fast GNM modes. It is of interest to find out
whether τ_{} and τ_{||} may
similarly be associated with the correlation times corresponding to the GNM slow
and fast modes, respectively.

The GNM correlation times calculated for AKeco using eq 12 are shown in Figure 5 (solid and dashed curves). The open circles
are the NMR/SRLS τ_{} values previously
calculated, which appear predominantly within AMPbd and LID.^{13} The τ_{i,GNM} values are found
from the five slowest GNM modes, with a proportionality constant of
τ_{0} = 2.5 ns. These modes (solid curve)
make a fractional contribution of ${\mathrm{\Sigma}}_{k=1}^{5}{\lambda}_{k}^{-1}/{\mathrm{\Sigma}}_{k=1}^{N-1}{\lambda}_{k}^{-1}=0.3$ to the observed dynamics. The proportionality constant
τ_{0} = 2.5 ns rendered these data
comparable in magnitude to τ_{}. The dashed
curve shows τ_{iGNM} values computed from the superposition
of *all* the GNM modes. The correlation times corresponding to
slower modes (solid curve) are longer than those resulting from
“all” modes, because of the contribution of a larger
number of modes to relaxation in the “all-modes”
scenario. The τ_{iGNM} profile based on the five slowest
modes shows broad peaks at the AMPbd and LID domains. This suggests that the
SRLS τ_{} values that represent the correlation
time for domain motion^{13} can be
associated with the GNM slow modes.

_{};, obtained by fitting the experimental

^{15}N T

_{1}, T

_{2}, and

^{15}N-

^{11}NOE data of AKeco

**...**

The previous NMR/SRLS analysis set τ_{||} in the
range of 7–200 ps.^{13} The
fastest N/4 modes (not shown) yielded an average correlation time of 115 ps with
a relatively uniform distribution over the chain (similar to the fast-mode-
based ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ values) [dashed curve in Fig. 4(b)], in agreement with the median
τ_{||} value. This indicates that it is possible to
associate τ_{||} from SRLS/NMR with fast GNM modes.

### Fluctuation Distributions of the Slowest GNM Modes

Figure 6 displays the mobility
profile of individual residues in the slowest GNM modes. The ordinates in Figure 6(a,b), respectively, show the
normalized distribution of squared fluctuations driven by the first
[Fig. 6(a)] and
second [Fig. 6(b)]
slowest modes, also called the first and second *global mode
shapes*. The curves are directly found from the diagonal elements of the
matrices ${u}_{1}{u}_{1}^{\text{T}}$ and ${u}_{2}{u}_{2}^{\text{T}}$ (eq 11) for
AKeco (solid curve) and AKeco*AP_{5}A (dotted curve). The
residue ranges of the AMPbd and LID domains are depicted by the black boxes on
the upper abscissa [Fig.
6(a,b), top]. The insets in Figure 6(a,b) show the ribbon diagrams of AKeco and
AKeco*AP_{5}A color-coded according to the magnitudes of
the fluctuations associated with the first [Fig. 6(a)] and second [Fig. 6(b)] global modes. The
color code is cyan-blue-red-yellow-green in the order of increasing mobility.

_{5}A (- - -) (

**a**), and second slowest global GNM mode shapes for AKeco (—) and AKeco*AP

_{5}A (- -

**...**

The slowest GNM mode of AKeco [solid curve in Fig. 6(a)] shows a broad peak in the
region corresponding to LID residues, suggesting that this mode activates the
catalysis-related movement of the LID domain. Interestingly, the solid curve in
Figure 6(b) indicates that the second
slowest GNM mode of AKeco activates the functional movement of the AMPbd domain.
Thus, the domain motions of AMPbd and LID in the ligand-free enzyme are induced
by different GNM modes. In the inhibitor-bound enzyme, the first global GNM mode
induces notable mobility in the C-terminal segment of domain LID
[dotted curve in Fig.
6(a)] and the second global GNM mode induces mobility in the
N-terminal segment of domain AMPbd [dotted curve in Fig. 6(b)]. As pointed out
previously,^{14} and shown in Figure 4(d), NMR/SRLS analysis of in
AKeco*AP_{5}A detected nanosecond motions in the
α_{2}/α_{3} loop of AMPbd and the
C-terminal segment of domain LID. These apparently important dynamic elements
associated with the catalytic transition state^{14} can be now associated with the second and first global GNM modes,
respectively. We also note that a crystallographic study of a yeast adenylate
kinase mutant bound to an ATP analogue pointed out independent motions of AMPbd
and LID,^{68} in agreement with the GNM
results.

In addition to the activation of the LID domain, the slowest GNM mode of
AKeco*AP_{5}A [dotted curve in Fig. 6(a)] induces enhanced (as
compared to AKeco) mobility in the loops
α_{4}/β_{3} (residues Q74-G80) and
α_{5}/β_{4} (residues G100-P104),
whereas in the ligand-free enzyme, the active site is flexible and the loops
α_{4}/β_{3} and
α_{5}/β_{4} are rigid. This is in
agreement with the counterweight balancing of substrate binding hypothesis.^{52} The second global GNM mode
[Fig. 6(b)]
activates the domain AMPbd and the C-terminal segment of the chain in both
enzyme forms. There is a shift in mobility within AMPbd from the C-terminal part
to the N-terminal part upon inhibitor binding. A crystallographic study of yeast
adenylate kinase bound to AP_{5}A showed large B-factors within the
C-terminal part of domain LID.^{69} Finally,
the chain segment A176-K195 is mobile in AKeco*AP_{5}A
according to the first global GNM mode [dotted curve in Fig. 6(a)].

### Hinge Centers and the Slowest GNM Modes

In general, hinge regions act as anchors about which domain (or loop) motions occur in opposite directions. In terms of GNM eigenvectors, hinge residues are located at the crossovers between the segments undergoing positive and negative fluctuations along the dominant principal/normal axes, and therefore form minima in the global mode shapes that refer to mean square fluctuations.

Based on the “fit-all” method applied to AKeco
and AKeco*AP_{5}A eight hinges H1–H8, centered
at residues T15, S30, L45, V59, L115, D159, T175, and V196, were identified by
Muller et al.^{52} It can be seen that
residues T15, T175, and V196 coincide with (or closely neighbor) minima in the
first global mode of AKeco, and residues T15, S30, L45, V59, T175, and V196
coincide with (or closely neighbor) minima in the first global mode of
AKeco*AP_{5}A [Fig. 6(a)]. Residues T175 and V196 (S30 and D159)
represent hinges in both enzyme forms according to the first (second) global GNM
mode. Residues T15, S30, L115, D159, and T175 form minima in the second global
mode of AKeco*AP_{5}A [dotted curve in Fig. 6(b)]. The first global GNM
mode singles out the region around D110 as a hinge element for the flexing of
the LID domain in both enzyme forms [Fig. 6(a)]. Hinges are typically associated with high
order parameters. Seven out of the eight hinges of AKeco, the exception being
hinge H7 (T175), feature high ${\text{S}}_{\text{i\hspace{0.28em}GNM}}^{2}$ values [Fig.
4(a), solid curve]. High SRLS ${({\text{S}}_{0}^{2})}^{2}$ values were observed for five out of six (data are not
available for residues S30 and D159) hinge residues in
AKeco*AP_{5}A and AKeco. Order parameters are high for
many residues. On the other hand, minima in the GNM global mode shapes single
out hinges with high discrimination.

### Mechanism/Biological Relevance of the Slowest Collective Modes Predicted by ANM

Figure 7 illustrates the most
probable deformations, or global reconfigurations, obtained with ANM^{23} for Akeco [Fig. 7(a)] and
AKeco*AP_{5}A [Fig. 7(b)]. The ribbon diagrams I and II represent the
two alternative (fluctuating) conformations driven by the slowest ANM mode using
the PDB structures of the two enzyme forms^{51}^{,}^{52} as starting
conformations. For visual clarity, the LID (AMPbd) domain is colored red (blue),
the inhibitor is colored yellow, and the fluctuations are amplified by a factor
of 2.

**a**) and AKeco*AP

_{5}A (

**b**) fluctuate based on the first global mode according to ANM analysis. The LID and AMPbd domains are colored red

**...**

Figure 7(a) indicates that the
first global mode of the ligand-free enzyme drives a large-scale reconfiguration
of the LID domain, which fluctuates between the “closed”
(I) and “open” (II) forms. This tendency of LID to close
over AMPbd, while the latter moves closer to LID, reflects the pre-disposition
of the free enzyme to assume its inhibitor-bound form. This is consistent with
crystallographic,^{54} optical,^{53} high temperature MD^{70} studies, and a recent elastic normal mode
analysis.^{71}

Figure 7(b) and the dotted curve in
Figure 6(a) indicate that the LID
domain exhibits some mobility also in the inhibitor-bound enzyme. However, this
motion is significantly more limited than that in AKeco, affecting the short
solvent-exposed strands β_{7} and β_{8}
and the connecting loop D147-G150 [Fig. 1(b)], while leaving the remaining portions of the
LID docked onto the CORE. Proximity of the strands β_{7} and
β_{8} to the ATP-binding site and the salt bridges
R123-D159 and R156-D158 involved in ATP binding^{51} suggest involvement of these structural elements in transition
state dissociation.

For the ligand-free form, the dominant motions elucidated with GNM are
likely to be associated with “capturing” the substrates
and properly positioning them for phosphoryl transfer. For the inhibitor-bound
form, the dominant motions are likely to be associated with movements that
initiate the dissociation of the transition state. Transition state dissociation
appears to start with the C-terminal LID segment “opening
up” the active site. Binding studies of AKeco also suggest that the
active site region is flexible.^{72} An
optical study pointed out that AMP binding may modify the ATP binding site.^{73} These are among the conclusions reached
in a recent NMR study of μs-ms conformational exchange processes in
AKeco and AKeco*AP_{5}A.^{74}

The contact model proposed by Zhang and Bruschweiler^{65} suggests that the local contacts experienced by
the amide protons and carbonyl oxygens are the major determinant of
S^{2} values. The MD-based iRED approach^{20} yields similar results. The GNM is also based on contact
topology, which further supports the importance of the distribution of contacts
in the native state. We note, however, that the contact model is usually
suitable for estimating the fast time-scale dynamics of globular proteins, as
contributions from long-range domain motions are not included. AKeco is a
typical example of an enzyme featuring mobile domains engaged in large amplitude
motions, and such cooperative motions cannot be adequately represented by a
model based on local contacts only. The GNM takes rigorous account of the
long-range coupling of all native contacts, and permits us to identify the
effect of a broad range of collective modes. The order parameters of the mobile
domains important for function are indeed shown to be associated predominantly
with slow (global) modes.

### Validity of GNM Interpretation of NMR Data

GNM and SRLS are based on different models. GNM does not account for
overall tumbling, but it yields both slow cooperative modes that activate entire
domains and fast localized modes that relate to isolated N-H sites along the
protein backbone. These are all “collective” modes in
the sense that they depend on the coordinates of *all* the
α-carbons and they are uniquely defined for the given topology of
inter-residue contacts. SRLS accounts for the coupling of each N-H site to the
overall tumbling of the protein. As shown by the results of the present study,
GNM and SRLS can be compared meaningfully. Note that except for
τ_{0} = 2.5 ns in eq 12, and the force constant γ
in eq 10, GNM was used in this
study as a predictive theory. Therefore, the agreement with the experiment-based
SRLS analysis is quite rewarding.

The GNM calculations are based on the crystal structures of AKeco in the
ligand-free and -bound forms. While the GNM satisfactorily reproduces the
mobilities (B-factors) in the crystal forms, the fluctuations in the NMR
solution structures could arguably be different, given that the proteins enjoy a
higher flexibility in solution. On the other hand, previous applications of the
GNM^{21} to other proteins determined by
both X-ray and NMR show that the fluctuation spectrum predicted by the GNM for
the two groups of structures retain a large number of common features, because
the fluctuations are essentially dominated by the topology of native contacts
that are maintained in the different forms. Whether this feature also holds for
adenylate kinase (AK) could be tested using the NMR structure of
*Mycobacterium tuberculosis* adenylate kinase (AKmt) newly
deposited in the PDB (PDB code: 1P4S).^{75}

AKmt is 181 residues long while AKeco is 214 residues. The two sequences
have 45% identity. There is a 4-residue insertion between
α5 and β4 in AKmt and a 7-residue insertion between
α8 and β5 in AKeco. AKeco LID domain is 27 residues longer
than that of AKmt. The structure of AKmt matches the
AKeco*AP_{5}A structure better than the apo enzyme AKeco.
Superposition of the structures of AKmt and AKeco*AP_{5}A
using CORE domain backbone atoms gives 2.8 Å rms deviation. This
disagrees with optical studies that showed that ligand-free AK prevails
*in solution* as a distribution of conformations,^{53} peaked at the
“open” conformation, rather than the closed form. An
explanation for this discrepancy lies in the method of determining the NMR
structure. The latter is based on NOE’s that depend on 1/r^{6}, where r denotes inter-proton distances on
the order of 2.5–5.0 Å. Due to fast conformational
averaging the experimentally measured NOE is a weighted average.^{56} Because r is small and the NOE depends on it to
the sixth power, the experimental NOE will be strongly biased toward short
r-values associated with the “closed” form, leading to
the latter structure instead of the weighted average structure.

Figure 8 compares the distribution
of mean-square fluctuations predicted by the GNM for
AKeco*AP_{5}A (crystal structure) and AKmt (solution
structure). The agreement between the two sets of predicted results is
remarkable, despite the sequence and structure differences between the two
enzymes. This agreement lends support to the robustness of the GNM results, and
suggests that many dynamic features elucidated in the present study are
conceivably generic, functional properties of adenylate kinases, conserved in
different organisms. The unique potential of GNM to elucidate functional
dynamics is thus highlighted.

## CONCLUSIONS

N-H bond motion in adenylate kinase from *E. coli* is
characterized by high NMR/SRLS order parameters and fast local motions
(“ps regime” dynamics), or low NMR/SRLS order parameters and
slow local motions (“ns regime” dynamics). It is shown
herein that the former model correlates with fast stability-related localized GNM
modes, and the latter with slow functional collective GNM modes. Catalysis-related
motion of the domains LID and AMPbd in AKeco is activated by the first and second
global GNM modes, respectively. The ANM analysis predicts functional dynamics
configuring the active site in AKeco and bringing about transition state
dissociation in AKeco*AP_{5}A. The rationalization of the
mechanisms underlying the observed NMR relaxation behavior points out prospects for
future GNM/ANM and NMR/SRLS studies aimed at relating structural dynamics to
function.

## Acknowledgments

This work was supported in part by the NIGMS grant number 065805-01A1 (I.B.), the Israel Science Foundation (grant 520/99-16.1) and the Damadian Center for Magnetic Resonance research at Bar-Ilan University, Israel (E.M.). We acknowledge Dr. Y.E. Shapiro for carrying out the experiments of references 13, 14, and 56.

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- Domain flexibility in ligand-free and inhibitor-bound Escherichia coli adenylate kinase based on a mode-coupling analysis of 15N spin relaxation.[Biochemistry. 2002]
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*Ping J, Hao P, Li YX, Wang JF.**BioMed Research International. 2013; 2013628536* - Event Detection and Sub-state Discovery from Bio-molecular Simulations Using Higher-Order Statistics: Application To Enzyme Adenylate Kinase[Proteins. 2012]
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Comparison of El...Escherichia coli Adenylate Kinase Dynamics: Comparison of Elastic Network Model Modes with Mode-Coupling 15N-NMR Relaxation DataNIHPA Author Manuscripts. Nov 15, 2004; 57(3)468PMC

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