SRLS - American Chemical

12050

J. Phys. Chem. B 2009, 113, 12050–12060

Domain Mobility in Proteins from NMR/SRLS Yury E. Shapiro,* Edith Kahana, and Eva Meirovitch* The Mina and EVerard Goodman Faculty of Life Sciences, Bar-Ilan UniVersity, Ramat-Gan 52900, Israel ReceiVed: February 19, 2009; ReVised Manuscript ReceiVed: June 7, 2009

Enhanced internal mobility in proteins is typically functional. Domain motion in enzymes, necessarily related to catalysis, is a prototype in this context. Experimental 15N spin relaxation data from E. coli adenylate kinase report qualitatively on nanosecond motion experienced by the domains AMPbd and LID. Previous quantitative analysis based on the mode-coupling slowly relaxing local structure approach confirmed nanosecond mobility but yielded unduly small local ordering and local geometry not interpretable directly in terms of the local protein structure. Here, we show that these features ensue from having assumed axial local ordering and highly axial local diffusion. After eliminating these simplified second-rank tensor properties, a physically sound picture, with the local motion interpretable as domain motion, is obtained. Rhombic local ordering, with components given by ) 0.471, ) -0.952 and ) 0.481, and main ordering axis, YM, lying along CRi-1 - CRi , has been determined. The associated rhombic potential is given by axial (rhombic) coefficients of ) -3.3 ( ) 17.8). The average correlation time for domain motion is 10.4 (6.4) ns at 288 (302) K; the corresponding correlation time for global motion is 20.6 (14.9) ns. The rates for domain motion exhibit noteworthy Arrhenius-type temperature-dependence, yielding activation energies of 63.8 ( 7.0 (53.0 ( 9.1) kJ/mol for the AMPbd (LID) domain. The traditional model-free analysis ignores mode-coupling and simplifies tensor properties. Within its scope, the AKeco backbone emerges as largely rigid, ≈ 0.94; the main ordering axis, ZM, lies along N-H, ≈ 16 (c22 ) 0); and the slow local motional correlation time lies at the low end of the nanosecond time scale. I. Introduction Internal mobility in proteins is traditionally conceptualized as fast small-amplitude fluctuations. However, the kind of mobility that is typically functional consists of slow largeamplitude movements. Domain motion in enzymes is a typical example, and E. coli adenylate kinase (AKeco), comprising the domains AMPbd, LID, and CORE, is a prototype system in this context. AKeco catalyzes phosphoryl transfer from ATP to AMP in the presence of Mg2+.1 There is compelling evidence based on X-ray crystallography studies that in the course of catalysis the CORE domain is preserved structurally, whereas the AMPbd and LID domains execute large-amplitude motions to configure the active site for substrate binding and dissociate it for product release.2,3 The extreme stages of the catalytic cycle are represented by the crystal structure of AKeco4 and AKeco*AP5A.5 The ligand AP5A is a two-substrate-mimic inhibitor, and the complex AKeco*AP5A is a transition state mimic.6,7 Time-resolved fluorescence energy transfer studies showed that in solution the ligand-free enzyme also experiences internal domain motion.8,9 NMR provides powerful methods for studying internal mobility in proteins.10-17 15N relaxation dispersion17 investigations of Mg2+-, and substrate-saturated AKeco indicate that domain motion occurs on the millisecond time scale.18 Experimental 15N spin relaxation data from ligand-free AKeco show qualitatively, and our previous analysis based on the modecoupling slowly relaxing local structure (SRLS) approach has shown quantitatively, that in this system domain motion occurs * Corresponding authors. E-mail: [email protected] (E.M.), [email protected] (Y.S.); phone: +972 3 5318049 (E.M.), +972 3 5317926 (Y.S.); fax: +972 3 7384058 (Y.S.).

on the nanosecond time scale.19-21 Recent molecular dynamics (MD) studies support both findings (see below).22 In this study we are concerned with domain dynamics in ligand-free AKeco studied with 15N spin relaxation methods. The experimental 15N-{1H} NOEs are large within the CORE domain and small within the AMPbd and LID domains.19-21 The traditional method for analyzing NMR spin relaxation in proteins is the simple model-free (MF) approach, which assumes that (a) the local and global motions experienced by the NMR probe (in this case the amide bond) are decoupled, given that they occur on highly separated time scales; and (b) the properties of the second-rank tensors involved may be simplified.23,24 Shortly after the inception of the MF application to proteins it was found that data sets, which feature relatively small 15N{1H} NOEs, cannot be fit with good statistics. This led to the development of the extended model-free (EMF) approach,25 which features a slow (as compared to the global motion) local motion, besides the fast local motion featured by the original MF approach. Clearly, small 15N-{1H} NOEs are fingerprints of complex dynamics, in general, and local motions comparable to the global motion, in particular. Using the EMF method, numerous studies singled out flexible chain segments as featuring nanosecond local motions and relatively low squared order parameters.10-16 By inference, the small 15N-{1H} NOEs within AMPbd and LID are also expected to lead through EMF analysis to similar parameters. We have studied the AKeco system with 15N spin relaxation for quite some time.19-21,26 In our first attempt26 to analyze the experimental data we used MF.23-25 Instead of detecting nanosecond local motion and relatively low order parameters within the AMPbd and LID domains, the MF analysis yielded largely picosecond local motions and high order parameters throughout the AKeco backbone.26

10.1021/jp901522c CCC: $40.75  2009 American Chemical Society Published on Web 08/12/2009

Domain Motion in Proteins by NMR/SRLS This is implied by the basic MF assumptions, (a) and (b) outlined above, being not valid in this case. The EMF spectral density is consequently oversimplified yielding inaccurate (having absorbed unaccounted-for factors) best-fit parameters, however, with good statistics. We developed in recent years the stochastic two-body coupled-rotator slowly relaxing local structure (SRLS) approach27-29 for NMR spin relaxation in proteins.30 SRLS is the generalization of MF, yielding the latter in simple limits.27,31,32 It accounts rigorously for dynamical coupling between the global and local motions through a local potential, allowing for arbitrary time scale separation.27 All the tensors involved can be asymmetric and oriented arbitrarily.29 The MF restrictions (a) and (b) are thereby removed altogether.32 Moreover, by systematically decreasing tensor symmetry and increasing geometric complexity, the generality of SRLS makes possible the determination of that parameter combination which matches data sensitivity. For practical reasons, our first fitting scheme for SRLS was based on axial local potentials/local ordering.30 Also, large RL| / R⊥L ratios, where RL denotes the local diffusion tensor, were considered. This establishes formal analogy with the EMF spectral density,25 where τ| ) 1/(6RL| ) is interpreted as fast effective local motion, τf, and τ⊥ ) 1/(6R⊥L ) as slow effective local motion, τs (see eq B6 of ref 33 and the EMF spectral density given in ref 25). This fitting scheme was applied to AKeco.19-21 We obtained at 303 K ) 8.2 ( 1.3 ns (τ ) 1/(6RL) < 150 ps) for most residues within AMPbd and LID (CORE).19 This is a strong indication that R⊥L represents the rate of domain motion. The temperature dependence of the R⊥L values could be analyzed using the Arrhenius equation to yield apparent activation energies of reasonable magnitude. This cannot be accomplished with the EMF parameter 1/(6τs). Clearly, accounting for modecoupling rendered the emerging picture consistent with the experimental data and very different from the MF picture. However, several features of the SRLS analysis outlined above are unsatisfactory. The local potential/local ordering within AMPbd and LID is unduly small for tightly packed protein cores, and the orientation of the main local ordering/ local diffusion axis cannot be associated with a physical axis in the protein structure.19,20 We found in previous work semiquantitatively that rhombic ordering prevails at N-H sites in proteins.34 Also, the relation RL| . R⊥L implies infinitely fast motion of N-H about its symmetry axis, which is not quite realistic. We developed recently a general fitting scheme for SRLS where (among others) the local potential is allowed to be rhombic.32 In accordance with the traditional treatment of restricted motions in liquids, its form is given by the lowest (L ) 2) terms of the Wigner matrix element expansion.35-39 This fitting scheme is used in the present study. In the forthcoming we show that the problems outlined above have been resolved by allowing for rhombic potentials, and removing the restriction that RL| /R⊥L . 1. As expected, high local rhombic potentials are determined, and the main local ordering/local diffusion axis can R - CRi axis. This conforms to the be associated with the Ci-1 local structure and agrees with similar results obtained by other methods (e.g., refs 40 and 41). Domain motion is only slightly faster than the global tumbling, and its rate obeys the Arrhenius equation, yielding in the 288-302 K range an activation energy of approximately 58 kJ/mol. This dynamic picture cannot be improved by further reducing tensor symmetry and increasing geometric complexity. Thus,

J. Phys. Chem. B, Vol. 113, No. 35, 2009 12051 the objective of this study - devising a dynamic model, which is physically realistic, computationally tractable, matches the sensitivity of the experimental data, and does not overfit the latter - has been achieved. Materials and methods are outlined in Section II. A brief theoretical background is provided in Section III. Our results are presented and discussed in Section IV, and conclusions appear in Section V. II. Materials and Methods 1. Sample Preparation. Uniformly 15N-labeled AKeco was prepared as described earlier.26 The NMR sample contained 1 mM 15N-labeled AKeco dissolved in 40 mM sodium phosphate buffer prepared in 95% H2O/5% D2O. 2. NMR Spectroscopy. NMR experiments were carried out in the temperature range of 288-310 K on Bruker DMX-600 (14.10 T) and DRX-800 (18.79 T) spectrometers, using 5 mm 1 H-13C-15N triple resonance inverse detection probes. B-VT2000 and BTO-2000 temperature control units were used on the DMX-600 and DRX-800 spectrometers, respectively. Temperature calibration was carried out with methanol standard. The temperature stability in the course of the NMR experiments was (0.2 K. The NMR data were analyzed on Silicon Graphics workstations using the software packages nmrPipe and modelXY.42 The previously determined assignments of the 1H-15N correlations of AKeco,43 complemented and revised in ref 26, were used. 15N R1 and R2, and 15N-{1H} NOEs were measured using the generic inversion recovery, CPMG, and steady-state NOE experiments adapted to proteins.44-46 III. Theoretical Background 1. 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.27-29 In SRLS the coupling between the isotropic global diffusion frame, C, rigidly attached to the protein, and the local diffusion frame, M, rigidly attached to the probe (N-H bond in this case), is accounted for by a potential U(ΩCM). For global diffusion tensors of lower symmetry the variable will be ΩC′M instead of ΩCM, with C′ denoting a local director.30,32 U(ΩCM) can be expanded in the full basis set of Wigner rotation matrix elements, DLKM(ΩCM). If only the lowest order (L ) 2) terms are preserved, the potential is given by:32

u ) U(ΩCM)/kBT ) -c20D200(ΩCM) - c22[D202(ΩCM) + 2 (ΩCM)] (1) D0-2

where the coefficients c02 and c22 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:

S20 ) 〈D200(ΩCM)〉

(2)

2 S22 ) 〈D202(ΩCM) + D0-2 (ΩCM)〉

(3)

and

Axial potentials feature only the first term of eq 1. Hence one has S22 ) 0.

12052

J. Phys. Chem. B, Vol. 113, No. 35, 2009

Shapiro et al.

Figure 1. Ribbon diagram of the crystal structures of (a) AKeco4 and (b) AKeco in complex with the two-substrate-mimic inhibitor AP5A (ref 5). The figures were drawn with the program Molscript49 using the PDB coordinate files 4ake for AKeco and 1ake (complex II) for AKeco*AP5A.

One may convert S02 and S22 into Cartesian ordering tensor components according to:

Sxx ) (√3/2S22 - S20)/2, Syy ) -(√3/2S22 + S20)/2 , and Szz ) S20

(4)

Note that Sxx + Syy + Szz ) 0. In the case of zero potential, c02 ) c22 ) 0, and axial probe diffusion, the solution of the diffusion equation associated with the time evolution operator features three distinct eigenvalues:

1/τK ) 6R⊥L + K2(R|L - R⊥L ) for K ) 0, 1, 2

(5)

where RL| ) 1/(6τ|) and R⊥L ) 1/(6τ⊥) ) 1/(6τ0). Only diagonal jK(ω) ≡ jKK(ω) terms are nonzero, and they can be calculated analytically as Lorentzian spectral densities, each defined by width 1/τK. When the ordering potential is axially symmetric, c02 * 0, c22 ) 0, again only diagonal terms survive. They are given as infinite sums of Lorentzian spectral densities, which are defined in terms of eigenvalues 1/τi of the Smoluchowski operator, and weighing factors cK,i, such that:

jK(ω) )

c τ

∑ 1 +K,iωi2τ2 i

Due to additional symmetry, jM,K,K′ ) jM,-K,-K′, only the nine distinct couples (K,K′) ) (-2,2), (-1,1), (-1,2), (0,0), (0,1), (0,2), (1,1), (1,2), (2,2) need to be considered. For dipolar autocorrelation, where ΩMD ) (0, βMD, 0), one has the explicit expression:

(6)

i

The eigenvalues 1/τi represent eigenmodes of motion,27 in accordance with the parameter range considered. Note that although, in principle, the number of terms in eq 6 is infinite, in practice a finite number of terms is sufficient for numerical convergence of the solution. Finally, when the local ordering potential is rhombic, c02 * 0, c22 * 0, both diagonal jK(ω) and nondiagonal jKK′(ω) terms are different from zero and need to be evaluated explicitly according to expressions analogous to eq 6.

JDD(ω) ) (d200(βMD))2j00(ω) + (2d210(βMD))2j11(ω) + (2d220(βMD))2j22(ω) + 4d200(βMD)d220(βMD)j02(ω) + 2 2 (βMD)d210(βMD)j-11(ω) + 2d-20 (βMD)d220(βMD)j-22(ω) 2d-10 (7)

with only six couples s (K,K′) ) (0,0), (1,1), (2,2), (0,2), (-1,1), and (-2,2) s involved.32 The autocorrelated CSA spectral density, JCC(ω), is calculated either in analogy with eq 7,32 or from eq 7 by applying the appropriate Wigner rotation.30 The measurable 15N relaxation parameters, 15N R1, R2, and 15N{1H} NOE, are calculated as functions of JDD(ωi) with ωi ) 0, ωH, ωN, ωH - ωN, and ωH + ωN, and JCC(ωi) with ωi ) 0 and ωN, using standard expressions for NMR spin relaxation.47,48 2. Analysis of NMR Relaxation Data Based on Fitting Schemes for SRLS. The SRLS theory is fully developed in refs 27 and 28. Its application to NMR spin relaxation in proteins is developed in ref 30. The fitting scheme for SRLS/NMR, which is based on axial potentials and the relation RL| /R⊥L . 1, is described in ref 30. Our general fitting scheme for SRLS, free of these restrictions, is described in ref 32. 15N spin relaxation of AKeco was studied in refs 19-21 and 26. In this study we use our temperature-dependent rates R⊥L taken from ref 21. A fitting scheme for SRLS,30 which constitutes an enhancement of the common MF fitting schemes, is provided upon request. IV. Results and Discussion 1. E. coli Adenylate Kinase. Figure 1a is the ribbon diagram of the AKeco conformation (“open” form) captured upon

Domain Motion in Proteins by NMR/SRLS

J. Phys. Chem. B, Vol. 113, No. 35, 2009 12053

Figure 2. Experimental (a) 15N R1, (b) 15N R2, and (c) 15N-{1H} NOE values of AKeco acquired at 302 K at 14.10 T (circles) and 18.79 T (triangles), as a function of residue number.

Figure 3. Experimental (a) 15N R1, (b) 15N R2, and (c) 15N-{1H} NOE values of AKeco acquired at 14.10 T and 288 (circles), 296.5 (triangles), 302 (squares) and 310 K (diamonds), as a function of residue number.

crystallization.4 Figure 1b shows the “closed” form in the context of the X-ray structure of the transition state mimic AKeco*AP5A.5 Both AKeco and AKeco*AP5A have been studied extensively in recent years with various experimental and theoretical methods. The latter include straightforward MD,50,51 weighted masses MD,52 methods exploring the roles of the various AKeco domains for stability and catalysis,53 principal component analysis (PCA),54 a 100 ns molecular dynamics study of subdomain motion,55 hydrogel-mediated translation of substrate recognition into macroscopic motion,56 graph theory,57 essential dynamics analysis,58 the Gaussian network model,59 a plastic network model exploring large-amplitude conformational changes,60 a coarse-grained model that considers ligand interactions approximately,61 elaborate coarse-grained methods,62 an MD-based method exploring the pathways between the “open” and “closed” states of AKeco at atomic detail,22 optical methods,8,9 single molecule fluorescence resonance energy transfer,63 15N relaxation dispersion,18 studies associated with protein folding,64,65 and 15N spin relaxation.19-21,26,66 Despite the multitude of different investigations, characterization of domain motion in terms of accurate rates, activation energies, local structure, local ordering potentials, and relative equilibrium probability distribution functions has not been accomplished so far. Such characterization is the objective of the present study, using NMR spin relaxation methods. Experimental 15N R1, R2 and 15N-{1H} NOE data of AKeco acquired at magnetic fields of 14.10 and 18.79 T, and 302 K, are shown in Figure 2. Experimental 15N R1, R2 and 15N-{1H} NOE data of AKeco acquired at 14.10 T, and 288, 296.5, 302, and 310 K, are shown in Figure 3. The experimental 15N-{1H}

NOEs of AKeco are small within the AMPbd and LID domains and are large within the CORE domain (Figures 2c and 3c). This is a clear indication that AMPbd and LID, but not CORE, experience nanosecond local motions. Note that 15N spin relaxation methods detected nanosecond domain motion in ligand-free AKeco.19-21 On the other hand, 15 N relaxation dispersion investigations reported on millisecond domain motion in Mg2+- and substrate-saturated AKeco.18 A recent MD study shows that the free energy profile of AKeco is barrier-less,22 whereas the free energy profile of coexisting AKeco and its transition-state mimic, AKeco*AP5A,6,7 has a 12.5 kJ/mol barrier between the two states.22 Hence, one can have nanosecond domain motion in ligand-free AKeco and millisecond domain motion in Mg2+- and substrate-saturated AKeco. 2. Model-Free Analysis of AKeco. On the basis of assumption (a) outlined above, the total time correlation function, C(t), is factored into CC(t) × CL(t), where CC(t) is the time correlation function for global motion and CL(t) is the time correlation function for local motion. Within the scope of assumptions (a) and (b), a simple form of C(t) is obtained. The global diffusion tensor, RC, which determines CC(t), is obtained from 15N R2/R1 ratios of amino acid residues free of local motional effects.67 Strategies for filtering the complete 15N R2/R1 data set have been developed.68 In recent years methods for deriving RC with hydrodynamic calculations have been set forth.69-72 In our first attempt26 to analyze experimental data from AKeco acquired at 14.10 T and 303 K we used MF.23-25 The traditional 15 N R2/R1-based analysis67,68 yielded RC| /R⊥C ) 1.25 and τm(app) ≡ [2 × (2R⊥C + RC| )]-1 ) 15.05 ( 0.5 ns. The squared MF

12054


order parameter, S2, did not distinguish between AMPbd/LID and CORE. Slow local motional correlation times were obtained for residues not necessarily located within AMPbd and LID, and their magnitude did not exceed 1 ns. This disagrees with the experimental 15N-{1H} NOE patterns (Figures 2c and 3c), indicating that the analysis is oversimplified. Similar results were obtained using combined 14.1 and 18.8 T data and taking the global diffusion to be isotropic.19,20 The overall picture obtained recently by the authors of ref 66, who determined RC| /R⊥C to be 1.41 for an AKeco sample similar to ours, is essentially the same as the MF picture obtained by us. The relatively small differences between the various MF analyses ensue from unaccounted-for factors having been absorbed by different parameters, given that somewhat different 15 N R2/R1 filtering procedures have been used to determine RC. Thus, the RC tensor itself absorbed such factors in ref 66. the Rex term did so in the relevant parts of refs 19 and 20, and both of these quantities were distorted in ref 26. The analysis is moderately sensitive to the nature of the filtered 15N R2/R1 data set when the protein is relatively rigid. It is quite sensitive to this feature when an internally mobile protein is treated as if it were relatively rigid. Theoretical SRLS calculations have shown that moderate axiality of the global diffusion tensor has a much smaller effect on the analysis than moderate rhombicity of the local potential - see Tables 10 and 11 of ref 32. AKeco prevails in solution as an ensemble of rapidly interconverting (on the chemical shift time scale) conformations. We adopted the conservative view that this intricate solution structure is on average isotropic. In ref 66 it was assumed that the solution structure is the same as the elongated conformation trapped upon crystallization.4 If this were the case, then RC(θi) would depend linearly on P2(cos θi) (where θi denotes the angle between a given N-H bond and the ZC axis).73,74 We found that RC(θi) and P2(cos θi) are practically uncorrelated,26 in support of a solution structure which is on average isotropic. The unrealistic MF picture of AKeco dynamics stems primarily from the neglect of dynamical coupling between the local and global motions, which occur at comparable rates. This oversimplification is also inherent in recent attempts to develop models for domain motion in proteins.75-79 There is inconsistency between the time correlation functions used, based on the large time scale separation paradigm of C(t) ) CC(t) × CL(t), and the results reported. Thus, in ref 75 the ratio τs/τm (with τs interpreted as domain motion) is limited to lie within the range of 0.25-1.0, although the EMF time correlation function used to determine τs requires that τs/τm , 1 (ref 25). In refs 76 and 77 domain motion is modeled as wobble-in-a-cone. Although the EMF formula25 used applies to the large time scale separation limit, the rate of wobbling, Dw, was determined to be on the order of, or even slower than, 1/(6τm). Finally, in refs 78 and 79 the domain exchange rate, kex, obtained with a factorized time correlation function, C(t), which requires that kex . 1/(6τm), was found to be significantly smaller than 1/(6τm). All of these local motional rates are therefore parametrizing quantities. 3. SRLS Analysis of the AMPbd and LID Domains of AKeco. Considering the MF analysis inappropriate, we proceeded with SRLS analysis of the experimental data. 3.1. SRLS Analysis Based on Axial Potentials.19-21 In our first fitting scheme for SRLS it was assumed (for practical reasons) that the local potential is axially symmetric, and RL| . R⊥L (in analogy with MF model 6 (ref 80)). By applying this fitting scheme to the combined 14.10 and 18.79 data of AKeco we obtained at 303 K ) 8.2 ( 1.3 ns for practically all

Shapiro et al. the residues within the AMPbd and LID domains, and τ < 150 ps for practically all the residues with the CORE domain.19 Low local ordering, with an average squared order parameter of 0.4, was obtained for the AMPbd and LID domains. High local ordering, with an average squared order parameter on the order of 0.85, was obtained for the CORE domain. Clearly, accounting for mode-coupling rendered the emerging picture consistent with the experimental 15N-{1H} NOE profile (Figures 2c and 3c). Parameter correspondence within the domains AMPbd/LID, and conspicuous distinction from the analogous CORE domain parameters, is a strong indication that τ⊥ represents collective domain motion. The average best-fit potential coefficient corresponding to ) 0.4 is ) 3.2. These values represent unduly small spatial restrictions at N-H sites in globular proteins. The average best-fit tilt angle between the main ordering/diffusion axis and the N-H bond is ) 17.5°. There is no structural element tilted at ∼17.5° from the N-H bond that might serve as main local ordering/local diffusion axis. These results require improvement. Motivated by previous semiquantitative evidence that axial potentials are oversimplified, we developed a fitting scheme for SRLS, where rhombic potentials are allowed for.32 The application of this fitting scheme to the experimental 15N spin relaxation data from the domains AMPbd and LID, shown in Figures 2 and 3, is described below. 3.2. SRLS Analysis Based on Rhombic Potentials. We find that physically meaningful local geometry can only be obtained when the local potential is allowed to be rhombic. By doing so we obtain a local ordering/local diffusion axis tilted at approximately 100° from the N-H bond. Since the canonical angle R - CRi axis is 101.3°,40 it between the N-H bond and the Ci-1 is reasonable to assume that this structural element acts as local ordering/local diffusion axis. Within the scope of the βMD ≈ 100° geometry obtained with rhombic potentials we consider the local diffusion tensor, RL, to be either isotropic or axially symmetric. Taking RL to be isotropic is an approximation which leads to strong rhombic potentials, and a diffusion rate constant, RL, which obeys the Arrhenius equation. Allowing for axial local diffusion leads to strong rhombic potentials and RL| /R⊥L on the order of 5-10. However, none of the rates RL| , R⊥L , and (1/3)(2R⊥L + RL| ) obeys the Arrhenius equation. Taking the RL tensor to be rhombic instead of axial might yield rate constants, Rxx, Ryy, and Rzz, exhibiting Arrhenius-type temperature dependence. However, this enhancement requires additional free variables, which is an undesirable feature. Separation of the local diffusion and local ordering frames based on physical considerations might also help. This option will be pursued in future work, where SRLS will be enhanced to make this distinction possible. Herein we assumed that the local diffusion is isotropic, with c20, c22, RL, and βMD allowed to vary in the fitting process. Thereby we obtained ) 95° and average ordering tensor components of ) 0.482, ) -0.949 and ) 0.467. Table 1 shows temperature-dependent averages. It can R be seen that |Syy| is close to 1 (high ordering around Ci-1 R R R Ci ), and |Sxx| ≈ |Szz| (the symmetry around Ci-1 - Ci is nearly axial). Figure 4 illustrates the designation of CRi-1 - CRi as main ordering axis by virtue of its identification with YM, which corresponds to Syy ∼ -0.9. A useful measure for the extent of the local ordering is Syy2. Since the local ordering around CRi-1 - CRi is both nearly axially symmetric and high, one may consider Syy2 to represent the mean squared fluctuation amplitude of all the internal motions.29 The



TABLE 1: Temperature Dependence of the Average Values of the Order Parameters in Irreducible (S20 and S22) and Cartesian (Sxx, Syy, Szz) Tensor Notation for the Domains AMPbd and LID. Average Values of the Potential Coefficients c20 and c22, and of 2, are Also Shown

2

AMPbd 288 0.471 -1.174 296.5 0.464 -1.168 302 0.461 -1.160

0.484 0.483 0.480

-0.954 -0.947 -0.941

0.910 0.897 0.885

0.471 0.463 0.461

-4.45 19.10 -0.52 18.90 -4.80 15.79

LID 288 296.5 302

0.483 0.483 0.481

-0.955 -0.949 -0.948

0.912 0.901 0.899

0.472 0.467 0.467

-4.05 18.54 -1.50 18.20 -4.35 16.43

T, K

0.472 -1.174 0.467 -1.169 0.467 -1.166

average value of Syy2 is 0.9 (Table 1) which is high, but not excessively high. The average correlation time for domain motion is 10.4 (6.4) ns at 288 (302) K; the corresponding correlation time for global motion is 20.6 (14.9) ns. Thus, domain motion is only about twice faster than the global motion. This picture of domain dynamics is realistic, consistent with the local protein structure, and interesting. It is intriguing that one can have slow (relative to the global motion) local motions in the presence of high local ordering. Table 1 also comprises the average values of the potential coefficients, c20 and c22. When the potential is strong and/or highly rhombic, that is, the values of c02 and/or c22 are large, then the functional dependence of the order parameters on the potential coefficients is such that small changes in Sxx, Syy, and Syy imply large changes in c02 and c22 (e.g., see Figure 4 of ref 32). This has significant implications on the accuracy of the best-fit values of c02 and c22, convergence issues, and error estimation. On the basis of extensive computations we found that a quantum number of J ) 16 ensures convergence32 of the data-fitting calculations. The latter lead to appropriate values of Sxx, Syy and Syy, with an error margin of approximately 10%. The values of the angle βMD are approximately 100°, and the local motional rates, RL, lie at the high end of the nanosecond time scale and obey the Arrhenius equation. The errors in βMD and RL are also

Figure 4. Schematic illustration of the local ordering frame, M, in the context of the peptide bond/peptide plane and the adjacent tetrahedral CR centers. The main ordering axis is YM, which can be R associated with the Ci-1 - CiR axis.

estimated at 10%. We consider these calculations, and the emerging picture, robust and reliable. On the other hand, the errors in c02 and c22 can be quite large. Both order parameters and potential coefficients convey in essence information on local spatial restrictions. We consider the order parameters as better quantitative measures of the strength of the local spatial restrictions and their symmetry. On the other hand, the potential coefficients are instrumental in illustrating pictorially typical potential forms and equilibrium probability distribution functions (see below). The statistical criterion used in this study is the percentile value for χ2 distribution. Typical calculations are associated with two degrees of freedom (six data points and four variables). According to reference 81, χ2 has to be below 5.99 for a commonly used 5% threshold. Our results abide, in general, by this requirement. As indicated, all the scenarios considered (MF-, axial-SRLS-potential-, and rhombic-SRLS-potential-based data fitting) feature four variables. The SRLS-based calculations have not been subjected to Fisher testing because the pictures of N-H bond dynamics emerging from the former two scenarios are oversimplified (see below). The temperature dependence of the ordering tensor components is depicted in Table 1. All the parameters decrease with increasing temperature. This is reasonable provided that the symmetry of the local spatial restrictions does not change significantly, as indeed borne out by the data shown in Table 1. The other parameters associated with the P-loop show nonmonotonic variations, which we assign to more complex dynamics. It is of interest to compare the order-parameter-related information provided by SRLS and the 3D Gaussian axial fluctuations (3D GAF) model. The perspective that (1 - S2) is in direct proportion to the sum of the second moments of the spatial parts of the relevant dipolar interaction allows analytical treatment of the influence of Gaussian fluctuations about arbitrary axes on S2 (ref 82). 3D GAF, which describes anisotropic peptide plane motion in terms of 3D harmonic local reorientational fluctuations consistent with molecular dynamics simulations, is such a model.82 In its application to 15N and 13C′ spin relaxation from the relatively rigid protein ubiquitin, 3D GAF reproduced the experimental data of 76% of the peptide planes studied.40 The dominant fluctuations were found to be R - CRi axis. The more flexible regions nearly parallel to the Ci-1 of the backbone could not be described by this model. Thus, 3D GAF is not suitable for studying domain motion, or any other slow large-amplitude motion experienced by flexible segments of the protein backbone. On the other hand, for nearly rigid backbone probes, 3D GAF is likely to provide information on dominant local fluctuations about CRi-1 - CRi with higher efficiency than SRLS. 3.3. Potential Shapes and Ordering Tensor Symmetries; Equilibrium Probability Distribution Functions. It is of interest to examine the shapes of the local potential functions, and the corresponding symmetries of the local ordering tensors. The local ordering/local diffusion frame, M, is defined in SRLS with ZM parallel to ZD. The angle βMD, obtained with data fitting, specifies the relative orientation of the actual main ordering axis and ZD. SRLS has detected strong rhombic potentials. Table 1 shows that Syy ∼ -0.9 and |Sxx| ∼ |Szz|. This means that by permuting the labels of the M frame axes so that YM becomes the main local ordering/local diffusion axis, the potential should have nearly axial symmetry.

12056


Shapiro et al.

Figure 5. (a) The potential u ) 4.57(1.5 cos2 βCM - 0.5) - 16.11(3/2)1/2 sin2 βCM cos 2γCM as a function of the spherical coordinates (βCM, γCM) in units of π. (b) u ) -4.74(1.5 cos2 βCM - 0.5) as a function of the spherical coordinates (βCM, βCM) in units of π. (c) u ) -16.11(1.5 cos2 βCM - 0.5) as a function of the spherical coordinates (βCM, βCM) in units of π. (d) The relative probability distribution function Prel ) exp{[-4.57(1.5 cos2 βCM - 0.5)] + 16.11(3/2)1/2 sin2 βCM cos 2γCM}sin(βCM) ∆βCM∆γCM depicted in the local director (C) frame. (e) Prel ) exp{[4.74(1.5 cos2 βCM - 0.5)]}(sin βCM)∆βCM. (f) Prel ) exp{[16.1(1.5 cos2 βCM - 0.5)]}(sin βCM)∆βCM. XC, YC, and ZC denote the principal axes of the uniaxial local director frame,83 with ZC parallel to the equilibrium N-H orientation and XC ) YC.

Changing X, Y, and Z into Y, Z, and X yields cˆ02 ) -0.5c02 1.224c22 and cˆ22 ) 0.612c02 - 0.5c22, with the symbols with the carets representing the new potential coefficients.83 For example, the average c02 and c22 values for the domain AMPbd are -0.52 and 18.9 (-4.8 and 15.8) at 296.5 (302) K (Table 1). One permutation cycle sets XM as main ordering axis, yielding a potential form with axial and rhombic coefficients given by -23.3 and -9.8 (-17.3 and -10.8) at 296.5 (302) K. A second permutation cycle sets YM as main ordering axis, yielding a potential form with axial and rhombic coefficients given by 23.8 and -9.3 (22.1 and -5.1) at 296.5 (302) K. In the letter set one has |3c02/2| > |6c22/2|, 6c22/2 < 0 and c02 < 0.37 These relations show that YM orients preferentially perpendicular to C to a greater degree than both ZM and XM, and ZM orients along C to a greater degree than XM. The information inherent in the potential associated with the original and “permuted” M frames is clearly the same; only the geometric perspective is different. However, the physical insight gained is illuminating. The average potential in the original M frame is u ) 4.57 × (1.5 cos2 βCM - 0.5) - (3/2)1/2 × 16.11 × sin2 βCM cos(2γCM) at 302 K (Figure 5a). The corresponding relative probability distribution function, Prel ) exp(-u) (sin βCM)∆βCM ∆γCM, is shown in Figure 5d. This function depicts the N-H orientations in the infinitesimal range βCM ( ∆βCM and γCM ( ∆γCM for any R (the C frame is uniaxial), plotted as a function of the spherical coordinates (βCM, γCM). The lobes along the Y-axis R - CRi orientation) indicate that the main local ordering (Ci-1 axis, YM, orients preferentially perpendicular to the N-H bond. The factor exp(-u) in the expression for Prel is large. Since the solid angle is also large, the Prel function spans a significant R portion of the conformational space around Ci-1 - CRi (Figure 5d). This agrees qualitatively with the large distribution of protein conformations in solution reported by refs 84-87. Strong axial potentials obtained with MF yield narrow distributions (see discussion of Figure 5f below). We show in Figure 5b the average potential, u ) -4.74 × (1.5 cos2 βCM - 0.5), obtained at 302 K for the AMPbd and LID domains with axial-potential-based SRLS fitting.19 In principle a 2D figure would suffice. The 3D figure was generated to facilitate comparison with the rhombic SRLS potential shown in Figure 5a. The apparent potential is weak (small c02).

The relative probability distribution function corresponding to the potential shown in Figure 5b is depicted in Figure 5e. This function increases radially from zero to a maximum value with position determined by c20 and decreases again to zero. The width of the projection of this function onto the ZX- and ZYplanes is given by Szz2. The spatial distribution of N-H orientations is large (because c02 is small) and centered at the N-H orientation (because the potential is axial). Clearly the rhombic potential SRLS scenario (Figures 5a and 5d) is very different from the axial potential SRLS scenario (Figures 5b and 5e). Let us compare the SRLS scenarios with the MF scenario. The average squared MF order parameter, S2, obtained for AKeco at 293 K is 0.88 (Figure 3a of ref 66). This corresponds to a c02 value of 16.1,30,32 which yields the potential shown in Figure 5c. The principal axis of the local ordering frame lies (implicitly) along N-H.23,24 The factor exp(-u) is large when S2 is large, but the solid angle (sin βCM)∆βCM is small because βCM is close to zero. This leads to the void in the middle, and the overall reduced Prel values in Figure 5f as compared to Figure 5d. The N-H bond vectors in Figure 5f lie preferentially on the surface of a cone with a small vertex angle. Hence, they lay close to the equilibrium N-H orientation, spanning a limited region of the conformational space. This picture disagrees with the substantial conformational distributions reported in refs 84-87. 3.4. ActiWation Energies for Domain Motion. We proceed with the examination of RL, the rate for domain motion. Experimental data are available in the temperature range of 288-302 K.21 The parameter τ ) 1/(6RL) ranges from 1 to 12 ns, and the time scale separation RC/RL ranges from 0.7 to 0.1. The parameter RC/RL is shown as a function of temperature in Figures 6d-f. For comparison the time scale separation, RC/ R⊥L , determined with axial potentials in the range of 288-310 K,21 is shown as a function of temperature in Figures 6a-c. All the time scale separations shown in Figure 6 increase (i.e., RC/RL decreases) with increasing temperature, pointing out decreasing coupling between the global and local motions with increasing temperature. The relative changes are larger for rhombic potentials. This is implied by the activation energies being on the order of 60 kJ/mol for rhombic potentials (Table



Figure 6. Best-fit values of RC/R⊥L obtained with axial-potential-based fitting for the P-loop (a), the AMPbd domain (b), and the LID domain (c). Black, red, green, and blue symbols represent results obtained at 288, 296.5, 302, and 310 K, respectively.21 Best-fit values of RC/RL obtained with rhombic-potential-based fitting for the P-loop (d), the AMPbd domain (e), and the LID domain (f). Black, red, and green symbols represent results obtained at 288, 296.5, and 302 K, respectively. Secondary structure elements, as defined based on the crystal structure of AKeco,4 are shown on the top.

TABLE 2: Average Activation Energies, , and Pre-exponential Factors, A, Obtained with the Arrhenius Equation for the Motion of the Domains AMPbd and LID, and the P-loop , kJ/mol

correlation coefficient

Axial Local Potential P-loop AMPbd LID

30.4 ( 4.3 29.7 ( 3.3 32.1 ( 4.3

29.1 ( 1.7 29.0 ( 1.3 29.9 ( 1.7

-0.981 -0.989 -0.984

Rhombic Local Potential P-loop AMPbd LID

16.5 ( 6.4 63.8 ( 7.0 53.0 ( 9.1

23.5 ( 2.6 43.8 ( 2.9 39.3 ( 3.7

-0.930 -0.987 -0.966

2), 30 kJ/mol for axial potentials (Table 2 and ref 21), and 16.9 kJ/mol for the global tumbling (ref 21). The parameter RC/R⊥L , associated with axial potentials, decreases between 288 and 310 K from 0.53 to 0.46 for all the mobile chain segments (Figures 6a-c). The parameter RC/RL, associated with rhombic potentials, shows better resolution (Figures 6d-f). It decreases between 288 and 302 K from 0.34 to 0.19 for the AMPbd domain and from 0.27 to 0.16 for the LID domain. For the P-loop, RC/RL is on the order of 0.5 throughout the temperature range examined. Discrimination among helices, β-strands, and loops is also observed. Thus, within AMPbd the RC/RL values of helix R2 are higher than those of helix R3 and loops R2/R3 and R3/R4 (Figure 6e). The RC/RL values of the β-strands comprised within domain LID are on average higher than the RC/RL values of the intervening loops (Figure 6f). The temperature dependence of the correlation time for local motion, τ, obtained with rhombic-potential-based fitting is shown in Figures 7d-f. The information provided by this parameter differs from that provided by RC/RL (Figures 6d-f) in that the latter also contains implicitly information on activation energies. The τ-based discrimination among secondary structure elements and loops within AMPbd and LID is clearly visible, with

sensitivity higher than exhibited by RC/RL. The correlation times for local motion are on average larger for the helices R2 and R3 than for the loops R2/R3 and R3/R4 (Figure 7e). Whereas helix R2 is defined by X-ray crystallography studies to comprise residues V49 - A55, similar τ values have been obtained for the chain segment K47 - D54. This might point out differences between the solution and X-ray structures. On the basis of similar τ values, the “block” comprising β7, the loop β7/β8, β8, and the loop β8/R7 (Figure 7f), which forms the outer part of domain LID in the course of assembling enzyme-inhibitor complexes, appears to be engaged in collective motion. For axial potentials the correlation time for domain motion is given by τ⊥ (Figures 7a-c).21 In general, τ⊥ is larger than τ obtained with rhombic potentials (Figures 7d-f). Unlike τ, the parameter τ⊥ is quite uniform throughout the AMPbd and LID domains and the P-loop. Using the Arrhenius equation we obtained with rhombicpotential-based fitting activation energies of 63.8 ( 7.0, 53.0 ( 9.1, and 16.5 ( 6.4 kJ/mol for the AMPbd domain, the LID domain, and the P-loop, respectively (Table 2). The corresponding values obtained with axial-potential-based-fitting are 29.7 ( 3.3, 32.1 ( 4.3, and 30.4 ( 4.3 kJ/mol (Table 2 and ref 21). Clearly, rhombic potentials are required to obtain accurate activation energies. The pre-exponential factors are comparable in the axial-potential-based scenario and are diversified in the rhombic-potential-based scenario (Table 2). This agrees with our general finding that the rhombic-potential-based analysis is more sensitive. The site-specific activation energies for domain motion are shown in Figure 8. One can distinguish among secondary structure elements and loops. The average Ea value of helix R3 of the AMPbd domain is (79.7 ( 9.5) kJ/mol. This value is approximately three times as high as the activation energy of helix R2 of the same domain. Similar to Ea of the helix R3, the Ea values of the strands β5 and β6 are within the range of 80-90

12058


Shapiro et al.

Figure 7. Best-fit local motion correlation times, τ⊥, obtained with axial-potential-based fitting for the P-loop (a), the AMPbd domain (b), and the LID domain (c). Black, red, green, and blue symbols denote results obtained for 288, 296.5, 302, and 310 K, respectively.21 Best-fit local motional correlation times, τ, obtained with rhombic-potential-based fitting for the P-loop (d), the AMPbd domain (e), and the LID domain (f). Black, red, and green symbols denote results obtained for 288, 296.5, and 302 K, respectively. Secondary structure elements, as defined based on the crystal structure of AKeco,4 are shown on the top.

Figure 8. Activation energies based on the Arrhenius equation using local motional rates, RL, obtained with rhombic-potential-based fitting for the P-loop (a), the AMPbd domain (b), and the LID domain (c). Secondary structure motifs are shown on the top.

kJ/mol. This corresponds to typical activation energies of reactions catalyzed by multidomain enzymes (e.g., see refs 88-90). The Ea values of the loops within the AMPbd and LID domains are on the order of 20-40 kJ/mol. The P-loop has Ea ) 16.5 kJ/mol, comparable to the activation energy for global tumbling (16.9 kJ/mol).21 This indicates that the activation barrier associated with the motion of the P-loop is not augmented as compared to the activation barrier for global motion by local structural restraints. To our knowledge activation energies for local N-H motion, let alone for domain motion in proteins, have not been reported previously within the scope of MF. For various reasons the single-decay-constant (two-decay-constant) approximation in

MF (EMF) is an oversimplification.32 Therefore τe of MF,23,24 and τf and τs of EMF,25 are parametrizing quantities, which absorb unaccounted for factors in different ways at different temperatures, leading to unphysical temperature-dependence. Evena¨s et al.91 found that the average activation energy for microsecond-millisecond conformational exchange between the “open” and “closed” forms of the Ca2+-saturated E140Q mutant of the calmodulin C-terminal domain is (61 ( 11) kJ/mol. We found that the AMPbd and LID domains of AKeco execute large amplitude nanosecond motions between the “open” and “closed” forms of this enzyme, with activation energies of (63.8 ( 7.0) and (53.0 ( 9.1) kJ/mol, respectively. It is interesting that internal motions comparable in nature, occurring with rates that

Domain Motion in Proteins by NMR/SRLS differ by over 3 orders of magnitude, are associated with similar activation energies. The Eyring equation was used to estimate the Gibbs free energy, ∆G‡, associated with domain motion in AKeco. We obtained (28.3 ( 1.7) kJ/mol for the AMPbd domain and (28.6 ( 2.5) kJ/mol for the LID domain at 298 K. ∆G‡ values of the catalytic reaction may be calculated using the expression ∆G‡ ) -RT ln[(kcath)/(kBT)] (with the various constants having their usual connotation). Using the kcat values for AKeco catalysis reported in ref 92 (ref 93) we obtained ∆G‡298 ≈ 58.8 (57.9) kJ/mol. V. Conclusions 15

N spin relaxation parameters from the AMPbd and LID domains of E. coli adenylate kinase have been analyzed with the SRLS approach. The amide bonds move locally about the R - CRi axes while restricted by strong nearly axial local Ci-1 potentials. The relative equilibrium probability distribution of R - CRi is substantial. The average N-H orientations about Ci-1 rates of local motion are 10.4 and 6.4 ns at 288 and 302 K, respectively. The rates of global motion at these temperatures are 20.6 and 14.9 ns. On the basis of parameter correspondence within the mobile domains, and clear distinction from analogous parameters in the structurally preserved domain CORE, the local motions detected are associated with domain motion. This picture is comprehensive, internally consistent, physically sound, computationally tractable, and matches data sensitivity. A unique aspect of the SRLS analysis is its ability to provide activation energies for internal motion. The AMPbd and LID domains move with an average activation energy of 58 kJ/mol. Helix R3 of the AMPbd domain, and strands β5, β6, and β8, of the LID domain, are associated with activation energies of 80-100 kJ/ mol, typical of the free energy of enzymatic reactions. Helix R2 of the AMPbd domain and several loops, including the P-loop, are associated with activation energies of 20-40 kJ/ mol, to be compared with an activation energy for global motion of 16.9 kJ/mol. To our knowledge this is the first report on activation energies for domain/secondary structure motif motion in proteins. Abbreviations AK, adenylate kinase; AKeco, adenylate kinase from Escherichia coli; ATP, adenosine triphosphate; ADP, adenosine diphosphate; AMP, adenosine monophosphate; AMPbd, AMPbinding domain; AP5A, P1,P5-di(adenosine-5′)pentaphosphate; CPMG, Carr-Purcell-Meiboom-Gill; EMF, extended model-free; MD, molecular dynamics; MF, model-free; NMR, nuclear magnetic resonance; SRLS, slowly relaxing local structure. Acknowledgment. The support of the Israel Science Foundation (Grant No. 347/07 to E.M.) and of the Damadian Center for Magnetic Resonance at Bar-Ilan University are gratefully acknowledged. References and Notes (1) Noda, L. In The Enzymes, 3rd ed.; Boyer, P. D., Ed.; Academic Press: London, 1973; Vol. 8, pp 279-305. (2) Vornhein, C.; Schlauderer, G. J.; Schulz, G. Structure 1995, 3, 483– 490. (3) Mu¨ller-Dickermann, H.-J.; Schulz, G. E. J. Mol. Biol. 1995, 246, 522–530. (4) Mu¨ller, C. W.; Schlauderer, G. J.; Reinstein, J.; Schulz, G. Structure 1996, 4, 147–156. (5) Mu¨ller, C. W.; Schulz, G. J. Mol. Biol. 1992, 224, 159–177. (6) Richard, J. P.; Frey, P. A. J. Am. Chem. Soc. 1978, 110, 7757– 7758.

J. Phys. Chem. B, Vol. 113, No. 35, 2009 12059 ´ den, J.; Wolf-Watz, M. J. Am. Chem. Soc. 2007, 129, 14003– (7) A 14012. (8) Sinev, M. A.; Sineva, E. V.; Ittah, V.; Haas, E. Biochemistry 1996, 35, 6425–6437. (9) Sinev, M. A.; Sineva, E. V.; Ittah, V.; Haas, E. FEBS Lett. 1996, 397, 273–276. (10) Ishima, R.; Torchia, D. A. Nat. Struct. Biol. 2000, 7, 740–743. (11) Case, D. Acc. Chem. Res. 2002, 35, 325–333. (12) Bruschweiler, R. Curr. Opin. Struct. Biol. 2003, 13, 175–183. (13) Palmer, A. G., III. Chem. ReV. 2004, 104, 3623–3640. (14) Mittermaier, A.; Kay, L. E. Science 2006, 312, 224–228. (15) Igumenova, T. I.; Frederick, K. K.; Wand, J. A. Chem. ReV. 2006, 106, 1672–1699. (16) Jarymowycz, V. A.; Stone, M. J. Chem. ReV. 2006, 106, 1624– 1671. (17) Palmer, A. G., III. Chem. ReV. 2006, 106, 1700–1719. (18) Wolf-Watz, M.; Thai, V.; Henzler-Wildman, K.; Hadjipavlou, G.; Eisenmesser, E. Z.; Kern, D. Nat. Struct. Mol. Biol. 2004, 11, 945–949. (19) Tugarinov, V.; Shapiro, Yu. E.; Liang, Z.; Freed, J. H.; Meirovitch, E. J. Mol. Biol. 2002, 315, 171–186. (20) Shapiro, Yu. E.; Kahana, E.; Tugarinov, V.; Liang, Z.; Freed, J. H.; Meirovitch, E. Biochemistry 2002, 41, 6271–6281. (21) Shapiro, Yu. E.; Meirovitch, E. J. Phys. Chem. B 2006, 110, 11519– 11524. (22) Arora, K.; Brooks, C. L., III. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 18496–18501. (23) Lipari, G.; Szabo, A. J. Am. Chem. Soc. 1982, 104, 4546–4559. (24) Lipari, G.; Szabo, A. J. Am. Chem. Soc. 1982, 104, 4559–4570. (25) Clore, G. M.; Szabo, A.; Bax, A.; Kay, L. E.; Driscoll, P. C.; Gronenborn, A. M. J. Am. Chem. Soc. 1990, 112, 4989–4991. (26) Shapiro, Yu. E.; Sinev, M. A.; Sineva, E. V.; Tugarinov, V.; Meirovitch, E. Biochemistry 2000, 39, 6634–6664. (27) Polimeno, A.; Freed, J. H. AdV. Chem. Phys. 1993, 83, 89–210. (28) Polimeno, A.; Freed, J. H. J. Phys. Chem. 1995, 99, 10995–11006. (29) Liang, Z.; Freed, J. H. J. Phys. Chem. B 1999, 103, 6384–6396. (30) Tugarinov, V.; Liang, Z.; Shapiro, Yu. E.; Freed, J. H.; Meirovitch, E. J. Am. Chem. Soc. 2001, 123, 3055–3063. (31) Meirovitch, E.; Shapiro, Yu. E.; Liang, Z.; Freed, J. H. J. Phys. Chem. B 2003, 107, 9898–9904. (32) Meirovitch, E.; Shapiro, Yu. E.; Polimeno, A.; Freed, J. H. J. Phys. Chem. A 2006, 110, 8366–8396. (33) Lin, W. J.; Freed, J. H. J. Phys. Chem. 1979, 83, 379–401. (34) Meirovitch, E.; Shapiro, Yu. E.; Tugarinov, V.; Liang, Z.; Freed, J. H. J. Phys. Chem. B 2003, 107, 9883–9897. (35) Nordio, P.-L.; Busolin, P. J. Chem. Phys. 1971, 55, 5485–5490. (36) Polnaszek, C. F.; Bruno, G. V.; Freed, J. H. J. Chem. Phys. 1973, 58, 3185–3199. (37) Polnaszek, C. F.; Freed, J. H. J. Phys. Chem. 1975, 79, 2283– 2306. (38) Emsley, J. W.; Lindon, J. C. NMR Spectroscopy Using Liquid Crystal SolVents; Pergamon Press: Oxford, 1975. (39) The Molecular Dynamics of Liquid Crystals; Luckhurst, G. R., Veracini, C. A. Eds.; Kluwer Academic Publishers: The Netherlands, 1994. (40) Lienin, S. F.; Bremi, T.; Brutscher, B.; Bruschweiler, R.; Ernst, R. R. J. Am. Chem. Soc. 1998, 120, 9870–9879. (41) Blackledge, M. Prog. Nucl. Magn. Reson. Spectrosc. 2005, 46, 26– 61. (42) Delaglio, F.; Grzesiek, S.; Vuister, G. W.; Zhu, G.; Pfeifer, J.; Bax, A. J. Biomol. NMR 1995, 6, 277–293. (43) Burlacu-Miron, S.; Perrier, V.; Gilles, A.-M.; Mispelter, J.; Barzu, O.; Craescu, C. T. J. Biomol. NMR 1999, 13, 93–94. (44) Palmer, A. G.; Skelton, N. J.; Chazin, W. J.; Wright, P. E.; Rance, M. Mol. Phys. 1992, 75, 699–711. (45) Grzesiek, S.; Bax, A. J. Am. Chem. Soc. 1993, 115, 12593–12594. (46) Farrow, N. A.; Muhandiram, R.; Singer, A. U.; Pascal, S. M.; Kay, C. M.; Gish, G.; Shoelson, S. E.; Pawson, T.; Forman-Kay, J. D.; Kay, L. E. Biochemistry 1994, 33, 5984–6003. (47) Peng, J. W., Wagner, G. In Methods in Enzymology; James, T. L., Oppenheimer, N. J., Eds.; Academic Press: New York, 1994; Vol. 239, pp 563-595. (48) Abragam, A. Principles of Nuclear Magnetism; Oxford University Press (Clarendon): London, 1961. (49) Kraulis, P. J. J. Appl. Crystallogr. 1991, 24, 946–995. (50) Lou, H.; Cukier, R. I. J. Phys. Chem. B 2006, 110, 12796–12808. (51) Kubitzki, M.; de Groot, B. Structure 2008, 16, 1175–1182. (52) Elamrani, S.; Berry, M. B.; Phillips, G. N., Jr.; McCammon, J. A. Proteins 1996, 25, 79–88. (53) Bae, E.; Phillips, G. N., Jr. Proc. Natl. Acad. Sci. U.S.A. 2005, 103, 2132–2137. (54) Lou, H.; Cukier, R. I. J. Phys. Chem. B 2006, 110, 24121–24137. (55) Pontiggia, F.; Zen, A.; Micheletti, C. Biophys. J. 2008, 95, 5901– 5912.

12060


(56) Yuan, W.; Yang, J.; Kopecekova, P.; Kopecek, J. J. Am. Chem. Soc. 2008, 130, 15760–15761. (57) Jacobs, D. J.; Rader, A. J.; Kuhn, L. A.; Thorpe, M. F. Proteins 2001, 44, 150–165. (58) Van Aalten, D. M. F.; De Groot, B. L.; Findlay, J. B. C.; Berendsen, H. J. C.; Amadei, A. J. Comput. Chem. 1997, 18, 169–181. (59) Temiz, A.; Meirovitch, E.; Bahar, I. Proteins 2004, 57, 468480. (60) Maragakis, P.; Karplus, M. J. Mol. Biol. 2005, 352, 807–822. (61) Whitford, P. C.; Miyashita, O.; Levy, Y.; Onuchic, J. N. J. Mol. Biol. 2007, 366, 1661–1671. (62) Chu, J.-W.; Voth, G. A. Biophys. J. 2007, 93, 3860–3871. (63) Hanson, J. A.; Duderstadt, K.; Watkins, L. P.; Bhattacharyya, S.; Brokaw, J.; Chu, J.-W.; Yang, H. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 18055–18060. (64) Kumar, S.; Sham, Y. Y.; Tsai, C.-J.; Nussinov, R. Biophys. J. 2001, 80, 2439–2454. (65) Ruan, Q.; Ruan, K.; Balmy, C.; Glaser, M.; Mantulin, W. W. Biochemistry 2001, 40, 14706–14714. (66) Henzler-Wildman, K. A.; Lei, M.; Thai, V.; Kerns, S. J.; Karplus, M.; Kern, D. Nature 2007, 450, 913–916. (67) Kay, L. E.; Torchia, D.; Bax, A. Biochemistry 1989, 28, 8972– 8979. (68) Pawley, N. H.; Wang, C.; Koide, S.; Nicholson, L. K. J. Biomol. NMR 2001, 20, 149–165. (69) de la Torre, J. G.; Huertas, M. L.; Carasco, B. J. Magn. Reson. 2000, 147, 138–146. (70) Ortega, A.; de la Torre, J. G. J. Am. Chem. Soc. 2005, 127, 12764– 12764. (71) Ryabov, Y. E.; Geraghty, C.; Varshney, A.; Fushman, D. J. Am. Chem. Soc. 2006, 128, 15432–15444. (72) Barone, V.; Zerbetto, M.; Polimeno, A. J. Comput. Chem. 2009, 30, 2–13. (73) Osborne, M. J.; Wright, P. E. J. Biomol. NMR 2001, 19, 209230. (74) Copie´, V.; Tomita, Y.; Akiyama, S. K.; Aota, A.; Yamada, K. M.; Venable, R. M.; Pastor, R. W.; Krueger, S.; Torchia, D. A. J. Mol. Biol. 1998, 277, 663–682.

Shapiro et al. (75) Chen, K.; Tjandra, N. J. Am. Chem. Soc. 2008, 130, 12745–12751. (76) Baber, J. L.; Szabo, A.; Tjandra, N. J. Am. Chem. Soc. 2001, 123, 3953–3959. (77) Chang, S.-L.; Szabo, A.; Tjandra, N. J. Am. Chem. Soc. 2003, 125, 11379–11384. (78) Ryabov, Y. E.; Fushman, D. Proteins 2006, 63, 787–796. (79) Ryabov, Y. E.; Fushman, D. J. Am. Chem. Soc. 2007, 129, 3315– 3327. (80) Fushman, D.; Cahill, S.; Cowburn, D. J. Mol. Biol. 1997, 266, 173– 194. (81) Spiegel, M. R.; Liu, J. Mathematical Handbook of Formulas and Tables. Schaum’s Outline Series, 2nd ed.; McGraw-Hill: New York, 1999; p 266. (82) Bremi, T.; Bruschweiler, R. J. Am. Chem. Soc. 1997, 119, 6672. (83) Barnes, J. P.; Freed, J. H. Biophys. J. 1998, 75, 2532–2546. (84) Lindorf-Larssen, K.; Best, R. B.; DePristo, M. A.; Dobson, C. M.; Vendruscolo, M. Nature 2005, 433, 128–132. (85) Salvatella, X.; Richter, B.; Vendruscolo, M. J. Biomol. NMR 2008, 40, 71–81. (86) Lange, O. F.; Lakomek, N. A.; Fares, C.; Schroder, G. F.; Walter, K. F. A.; Becer, S.; Meiler, J.; Gru¨bmuller, H.; Griesinger, C.; de Groot, B. L. Science 2008, 320, 1471–1475. (87) Lakomek, N. A.; Walter, K. F. A.; Fares, C.; Lange, O. F.; de Groot, B. L.; Gru¨bmuller, H.; Munk, R. B. A.; Becker, S.; Meiler, J.; C. Griesinger, C. J. Biomol. NMR 2008, 41, 139–155. (88) Jencks, W. P. Catalysis in Chemistry and Enzymology; Dover: New York, 1987. (89) Hammes, G. Biochemistry 2002, 41, 8221–8228. (c) Benkovic, S. J.; Hammes-Schiffer, S. Science 2003, 301, 1196–1202. (90) Hammes-Schiffer, S. Biochemistry 2002, 41, 13335–13343. (91) Evena¨s, J.; Forsen, S.; Malmendal, A.; Akke, M. J. Mol. Biol. 1999, 289, 603–611. (92) Reinstein, J.; Vetter, I. R.; Schlichting, I.; Ro¨sch, P.; Wittinghofer, A.; Goody, R. S. Biochemistry 1990, 29, 7440–7450. (93) Tian, G.; Yan, H.; Jiang, R.-T.; Kishi, F.; Nakazawa, A.; Tsai, M.D. Biochemistry 1990, 29, 4296–4304.

JP901522C

SRLS - American Chemical

Recommend Documents