Over the past several decades, NMR relaxation hasplayed a major role in unraveling the molecular-level dy-namical properties of a wide range of biological and colloi-dal systems. In the motional narrowing regime, where theconventional perturbation theory of spin relaxation applies,1
the accessible information about molecular dynamics is con-tained in the spectral density function, J���, the one-sidedFourier transform of the time correlation function �TCF�,
C�t�. The multiscale structure and dynamics of complex sys-tems make the interpretation of J��� challenging unless ithas been characterized exhaustively. In favorable cases, thefrequency dependence �or dispersion� of J��� can be mappedover five orders of magnitude by varying the NMRfrequency.2 The extensive datasets provided by such mag-netic relaxation dispersion experiments make it possible todiscriminate among specific dynamical models of high com-plexity. On the other hand, in 2H, 13C, or 15N relaxationstudies of conformational dynamics in proteins and otherbiomolecules, information about J��� is usually limited to afew frequencies. To physically interpret such data, the spec-tral density function, or the associated TCF, must be ex-pressed in terms of a small number of parameters. There isa�Electronic mail: [email protected].
THE JOURNAL OF CHEMICAL PHYSICS 131, 224507 �2009�
then a considerable risk of overinterpretation since the lim-ited data may be consistent with several physically distinctmodels.
Such concerns led to the development of interpretationalframeworks based on approximations of a general nature,allowing the TCF to be parametrized in a physically mean-ingful way without commitment to a specific dynamicalmodel. This approach originated in the 1970s, when orienta-tional order parameters were used to link first-order linesplittings to second-order relaxation rates in ordered fluidssuch as liquid crystals and bilayer membranes.3–6 A formalderivation of a general TCF formula for isotropic fluids,known as the two-step model, was presented in 1981 �Ref. 7�and had then already been used in NMR relaxation studies ofmicelles8 and proteins.9
In the following year, Lipari and Szabo,10 apparentlyoverlooking the substantial body of preceding work in thisarea, presented a general TCF formula that came to beknown as the model-free �MF� approach. While the Lipari–Szabo formula has been used extensively in the proteinfield,11–15 the nearly equivalent two-step formula has usuallybeen preferred in applications to micelles, microemulsions,and other colloidal systems.16 Our focus here is on applica-tions to protein solutions, but the new results derived and theconclusions reached are valid for a wide range of complexmolecular systems. We refer to the closely related Lipari–Szabo and two-step formulas, along with their underlyingapproximations, as MF-A and MF-B, respectively. BothMF-A and MF-B address the case with one internal and oneglobal motion. A third version of the MF approach, which wecall MF-C, has been proposed for handling the case of inter-nal motions on two distinct time scales, in addition to theglobal motion.17
Although the MF approach has been almost universallyadopted, substantial disagreement remains about its physicalfoundations and range of validity.18–21 This unfortunate situ-ation has arisen, at least partly, because all three MF formu-las were originally derived7,10,17 under unduly restrictiveconditions. Furthermore, in the presentation of two of theMF formulas, formal rigor was traded for accessibility10 andbrevity.17 As a result, the wide range of validity of the threeMF formulas has not been fully appreciated. Furthermore,the differences and similarities between the MF-A and MF-Bformulas have not been fully clarified.
In a series of papers, Meirovitch, Freed, andco-workers18,20,22–27 vigorously criticized the MF approach,arguing that it is physically unreasonable and that it mayproduce a qualitatively incorrect picture of protein dynamics.According to these authors, the principal deficiency of theMF approach is its neglect of dynamical coupling betweeninternal �conformational fluctuations� and global �usuallyprotein tumbling� motions. To rectify this perceived defi-ciency, Freed and co-workers developed the so-called slowlyrelaxing local structure �SRLS� model, which is based on atwo-body Smoluchowski equation �SE�. First applied �in asimpler form� to probe motion in liquid-crystallinesolvents,28–31 the SRLS model has more recently been usedto describe protein dynamics.18,20,22–27 More sophisticatedversions of the SRLS model have also been used by Poli-
meno, Moro, Freed, and co-workers32–36 to analyze the rota-tional dynamics of a probe molecule interacting with a“cage” of solvent molecules.
The SRLS model differs fundamentally in spirit from theMF approach. While the SRLS model is restricted to small-step diffusive internal motion in a potential of mean torque�POMT� of specified form, the essence of the MF approachis its independence of any detailed specification of the dy-namical mechanism. It is clear, therefore, that the SRLSmodel cannot be regarded as a generalization of the MF ap-proach. Conversely, the MF formulas cannot be regarded aslimiting cases of the SRLS model. On the other hand, it isnot at all clear that, as invariably assumed, the SRLS modelprovides a more accurate description of protein dynamicsthan does the MF approach even for the particular choice ofPOMT and reorientation mechanism on which the SRLSmodel is based. If this is not the case, there is little incentiveto sacrifice the analytical simplicity of the MF approach forthe considerable computational burden of the SRLS model.
Our objective here is to clarify the physical basis of theMF approach and to unambiguously define its range of va-lidity. Our strategy for achieving this goal is twofold. First,in Sec. II, we present rigorous formal derivations of the threeMF formulas under significantly more general conditionsthan were imposed in the original publications.7,10,17 Specifi-cally, for MF-A, we remove the limitation to uniaxial spin-lattice interaction tensor, isotropic internal mobility tensor,and coincident principal frames for these tensors. For MF-B,we remove the restriction to weak anisotropy of the internalmotion. These generalizations do not alter the simple form ofthe MF formulas, but they enable a more general interpreta-tion of the parameters. Explicit expressions are presentedhere that allow order parameters and internal correlationfunctions to be calculated for specific dynamical models. InSec. II, we also discuss cross-TCFs and we derive a novelMF formula applicable to microcrystalline proteins and othersolids.
The second part of our scrutiny of the MF approach is adetailed analysis of dynamical coupling, which is neglectedin the MF approach but included �in some form� in the SRLSmodel. This analysis is divided into three sections. In Sec.III, we formulate and compare the two decoupling approxi-mations invoked in the MF approach: the superposition ap-proximation �MF-A and MF-C� and the adiabatic approxima-tion �MF-B and MF-C�. The issue of dynamical coupling isconceptually subtle and the SRLS model’s lack of analyticaltransparency hampers a direct comparison with the MF ap-proach. Therefore, in Sec. IV, we present a simplified versionof the SRLS model, which can be solved analytically. Thisplanar SRLS model retains the dynamical complexity of thefull SRLS model, but the geometry is simplified. In the adia-batic �time-scale separated� limit, the planar SRLS modelreduces to the planar version of the popular diffusion-in-a-cone model.37,38 In other limits, our analysis shows that theSRLS model makes counterintuitive predictions that we at-tribute to the inappropriate theoretical foundations of themodel rather than to dynamical coupling between internaland global motions. This conclusion refutes the widely heldbelief that the SRLS model is more accurate than the MF
224507-2 Bertil Halle J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
approach. Finally, in Sec. V, we explore the two principalmechanisms of dynamical coupling in proteins: torque-mediated and friction-mediated coupling. We argue by wayof specific analytically solvable models that torque-mediatedcoupling �which SRLS attempts to capture� is unimportantbecause the relatively slow internal motions that mightcouple to the global motion tend to be intermittent �jumplike�in character, whereas friction-mediated coupling �which nei-ther MF nor SRLS incorporates� may be important for pro-teins with unstructured parts or flexibly connected domains.
II. MODEL-FREE APPROACH
In the motional narrowing regime, all autocorrelatednuclear spin relaxation observables measured in isotropic so-lution report on the time autocorrelation function of a nuclearspin-lattice interaction tensor V,1,39
C�t� = �V�0�:V�t�� , �2.1�
where the angular brackets denote an ensemble average. Therank-2 traceless symmetric tensor V typically describes themagnetic dipole-dipole, magnetic shielding anisotropy, orelectric quadrupole interaction.1 The scalar contraction �indi-cated by a colon� can be evaluated in any fixed coordinatesystem provided that the components of V are expressed inthe same frame at times 0 and t. The natural choice is thelaboratory-fixed coordinate frame �L�, with the zL axis alongthe external magnetic field. Cross-correlated relaxation ob-servables report on cross-TCFs that, under certain condi-tions, can be obtained by a trivial extension of the treatmentof auto-TCFs �Sec. II D�.
If the scalar contraction is expanded in laboratory-framespherical tensor components Vm
�L�, Eq. �2.1� yields
C�t� = �m=−2
2
�Vm�L���0�Vm
�L��t�� = 5�V0�L���0�V0
�L��t�� , �2.2�
where the asterisk denotes conjugation. The equality of thefive TCFs in the sum follows from the isotropy of thesolution,39 which also implies that �V�=0. Because V�t�must be independent of V�0� in the limit t→�, it followsthat the TCF decays to 0, that is, C���=0. In Sec. II E, weconsider anisotropic systems �solids and liquid crystals�,where the TCFs in the sum of Eq. �2.2� depend on the indexm.
Ubiquitous subpicosecond vibrational �or librational�motions average the spin-lattice interaction tensor V, butthese motions are usually too fast to contribute directly tospin relaxation. Henceforth, we regard V as the vibrationallyaveraged spin-lattice interaction tensor and we define the in-teraction frame �F� as the principal frame wherein V is diag-onal. By expanding the scalar contraction in the F-framecomponents Vn
�F�, we can express the initial value of the TCFas
C�0� = �n=−2
2
�Vn�F��2 = �V0
�F��21 +�2
3 � Veff
2 , �2.3�
where � is the asymmetry parameter of the V tensor, con-ventionally defined as1
� � �6V�2
�F�
V0�F� =
Vxx�F� − Vyy
�F�
Vzz�F� . �2.4�
In the following, we shall work with a dimensionless V ten-sor normalized such that
V:V = Tr VV� = 1. �2.5�
In expressions relating relaxation observables to TCFs �orspectral density functions�, the latter should thus be multi-plied by Veff
2 . With this normalization, the TCF decays to 0from an initial value of 1,
C�0� = 1, C��� = 0. �2.6�
A TCF with these properties, known as reduced TCF, is de-noted by C�t� throughout this paper. We shall also encounterTCFs, denoted by G�t�, that do not have �both of� theseproperties. Such TCFs can always be reduced by the trans-formation,
C�t� =G�t� − G���G�0� − G���
. �2.7�
In the simplest and most general versions of the MFapproach, MF-A and MF-B, the TCF is assumed to be gov-erned by two kinds of motion, involving internal and globaldegrees of freedom. To obtain a simple parametrization ofthe TCF, two approximations must be invoked. The first is adecoupling approximation asserting that the internal and glo-bal motions are �effectively� independent. The physical basisof this approximation is elucidated in Secs. III–V. The sec-ond approximation is a symmetry requirement. Depending onhow these two approximations are formulated, one obtainsMF formulas that differ in physical content and in range ofvalidity. It is straightforward to extend the MF approach tothree or more motions. This can be done in several ways,depending on which approximations are invoked. Here, weconsider the widely used three-motion MF-C formula.
In order to apply the approximations, the time depen-dence in the laboratory-frame component V0
�L��t� in Eq. �2.2�must be formally linked to the relevant internal and globaldegrees of freedom. This is done by transforming the tensorcomponent from the laboratory frame �L� to the interactionframe �F� via three additional coordinate frames denoted as I,D, and G �Table I�. The I and G frames are the principalframes for the internal and global mobility tensors, respec-tively. The symmetry of each mobility tensor can be isotro-pic, uniaxial, or biaxial, corresponding to one, two, or threedistinct principal components. The director frame D is theexternal reference frame for the local POMT that acts on the
TABLE I. Definition of coordinate systems.
Symbola Associated with
F �D, CSA, Q� spin-lattice interaction tensorI �M� internal mobility tensorD �C�� local POMTG �C� global mobility tensorL �L� external magnetic field
aSymbols used by Freed et al. �Ref. 18� are given within parentheses.
224507-3 Protein dynamics from NMR J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
“body” that executes the internal motion. If a second internalmotion is present, as in MF-C, two additional frames, I� andD�, are needed �Sec. II C�.
The rotational transformations between these frames areconveniently expressed with the aid of the rank-2 Wignerrotation matrix,
V0�L� = �
n�
p�
q�
r
�D0n2���LG�Dnp
2���GD�Dpq2���DI�Dqr
2���IF�Vr�F�,
�2.8�
where the argument of the Wigner functions is the set ofEuler angles, �AB, that carry frame A into frame B.40 Hereand in the following, all sums run from �2 to +2. By regard-ing the vibrationally averaged principal-frame componentsVr
�F� in Eq. �2.8� as time-independent constants, we excludefrom consideration “intermolecular” relaxation mechanisms,such as magnetic dipole-dipole relaxation induced by trans-lational diffusion.1 Furthermore, the Euler angles �IF are re-garded as constant parameters that define the relative orien-tation of the principal frames for the internal mobility tensorand the vibrationally averaged spin-lattice interaction tensor.It is convenient to define the time-independent geometricalcoefficients,
�q � �r
Dqr2 ��IF�Vr
�F��. �2.9�
It follows Eqs. �2.4� and �2.5� that these coefficients may bewritten as
�q = 1 +�2
3−1/2�Dq0
2 ��IF� +�
�6�Dq2
2 ��IF� + Dq−22 ��IF��� ,
�2.10�
and that they are normalized as
�q
��q�2 = 1. �2.11�
The internal and global motions are described by thetime-dependent Euler angles �DI�t� and �LG�t�, respectively.The Euler angles �GD in Eq. �2.8� are regarded as constantparameters that define the relative orientation of the globalmobility frame and the local director frame. With the timedependence made explicit, Eq. �2.8� can now be written as
V0�L��t� = �
n�
p�
q
D0n2���LG�t��Dnp
2���GD�Dpq2���DI�t���q
�.
�2.12�
The TCF is obtained by substituting this expression into Eq.�2.2�,
C�t� = 5�n
�n�
�p
�p�
�q
�q�
Dnp2 ��GD�Dn�p�
2� ��GD��q�q��
� �D0n2 ��LG
0 �D0n�2� ��LG�Dpq
2 ��DI0 �Dp�q�
2� ��DI�� ,
�2.13�
where the superscript 0 on the Euler angles signifies the ini-tial time. In the following subsections, we show how this
expression is simplified by invoking dynamical decouplingapproximations and by imposing static or dynamic symmetryconstraints. Static symmetries constrain the L- and D-frameprojection indices m and p, respectively. For example, theisotropic symmetry of the solution allowed us to set m=0 inEq. �2.2�. Symmetries of the global and internal mobilitytensors constrain the G- and I-frame projection indices n andq, respectively.
A. Model-free version A
In MF-A, decoupling of internal and global motions isimposed by a superposition approximation, formulated as acondition of statistical independence of these motions �Sec.III A�. The ensemble average in Eq. �2.13� can then be re-placed by a product of ensemble averages over the internaland global degrees of freedom, respectively,
�D0n2 ��LG
0 �D0n�2� ��LG�Dpq
2 ��DI0 �Dp�q�
2� ��DI��
= �D0n2 ��LG
0 �D0n�2� ��LG���Dpq
2 ��DI0 �Dp�q�
2� ��DI�� .
�2.14�
The second approximation in MF-A is a requirement ofisotropic global motion, meaning that
5�D0n2 ��LG
0 �D0n�2� ��LG�� = nn� exp�− t/global�
� nn�Cglobal�t� . �2.15�
This form is valid for rotational diffusion of a compactspherical protein, in which case global=1 / �6DG� is the rank-2rotational correlation time of a spherical-top rotor with rota-tional diffusion coefficient DG.41 However, Eq. �2.15� alsodescribes exchange-mediated orientational randomization fora ligand or solvent molecule bound to a rotationally immo-bilized protein,42–44 in which case global equals the meanresidence time res of the bound species. If both of theseprocesses occur simultaneously, and if they are statisticallyindependent, the global correlation time is given by
1
global= 6DG +
1
res. �2.16�
Because the global TCF in Eq. �2.15� is independent of theG-frame projection index n, Eqs. �2.13�–�2.15� yield a fac-torized TCF,
C�t� = Cglobal�t�Ginternal�t� , �2.17�
with the internal TCF given by
Ginternal�t� = �q
�q�
�q�q�� �
p
�Dpq2 ��DI
0 �Dpq�2� ��DI�� . �2.18�
Lipari and Szabo10 did not elaborate on the meaning of “sta-tistical independence,” but simply stated �correctly� that itleads to the factorization of the TCF as in Eq. �2.17�. In Secs.III–V, we discuss the physical significance of this approxi-mation in detail.
It follows from the unitarity of the Wigner rotationmatrix40 and the normalization of the geometric coefficients�q in Eq. �2.11� that Ginternal�0�=1. However, the internalTCF does not decay to 0. The asymptotic TCF Ginternal��� is
224507-4 Bertil Halle J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
obtained from Eq. �2.18� by noting that the orientation �DI atinfinite time must be independent of the initial orientation�DI
0 . The ensemble average of the product of Wigner func-tions can then be replaced by the product of ensemble aver-aged Wigner functions, yielding
Ginternal��� = S2, �2.19�
with the generalized rank-2 orientational order parameter Sgiven by
S2 = �p��
q
�q�Dpq2 ��DI���2
. �2.20�
Combination of Eqs. �2.7�, �2.15�, �2.17�, and �2.19� nowyields the MF-A formula,
The reduced internal TCF Cinternal�t� is defined explicitly as
Cinternal�t� =Ginternal�t� − S2
1 − S2 , �2.22�
with Ginternal�t� given by Eq. �2.18�. If the internal motion isfast compared to the Larmor frequency �extreme motionalnarrowing� and compared to the global motion, then the spinrelaxation rates depend on the internal motion only via thetime integral of Cinternal�t�. This integral defines an effectivecorrelation time,
internal � �0
�
dt Cinternal�t� . �2.23�
However, if the internal motion is slower, then Eq. �2.21�must be supplemented with an assumption about the �expo-nential or otherwise� decay of Cinternal�t�.
Equations �2.18� and �2.20�–�2.22� represent the MF-Ain its most general form. We now show how these generalresults reduce to more familiar expressions when additionalsymmetry is present. The Lipari–Szabo treatment10 was re-stricted to a uniaxial interaction tensor, �=0, in which caseEq. �2.10� reduces to �q=Dq0
2 ��IF�. It was also assumed thatthe I and F frames coincide ��IF=0�, whereby �q=q0 so Eq.�2.20� reduces to
S2 = �p
��Dp02 ��DI���2. �2.24�
Equation �2.20� is the generalization of this familiar result toa biaxial spin-lattice tensor with arbitrary orientation relativeto the internal mobility tensor. If, in addition to �=0 and�IF=0, the POMT is uniaxial in the D frame �meaning thatzD is an n-fold symmetry axis with n�3�, then �Dp0
2 ��DI��vanishes unless p=0 so Eq. �2.24� reduces further to
S = �P2�cos �DI�� . �2.25�
With Eq. �2.20� it is straightforward to compare relaxationdata pertaining to different �autocorrelated� spin-lattice inter-action tensors in the same molecular fragment, such as themagnetic dipole-dipole and shielding anisotropy of a 13C or15N spin or the electric field gradient tensors of the 2H and17O spins in an internal water molecule.45 If no symmetry ispresent, the generalized order parameter S depends on 25
independent partial order parameters �Dpq2 ��DI��. If the local
POMT is uniaxial in the I frame, only the five order param-eters �Dp0
2 ��DI�� are nonzero. They are linearly related to thedirection cosines that constitute Saupe’s traceless symmetricCartesian ordering tensor. On the other hand, if the localPOMT is uniaxial in the D frame, then only the five termswith p=0 survive.
The general result for the internal TCF, Eqs. �2.18� and�2.22�, simplifies considerably in special cases. If the internalmobility tensor is uniaxial, then only terms with q=q� sur-vive in Eq. �2.18�. If the internal mobility tensor is isotropic,as assumed by Lipari and Szabo,10 then
�Dpq2 ��DI
0 �Dpq�2� ��DI�� = qq��Dp0
2 ��DI0 �Dp0
2���DI�� , �2.26�
in analogy with Eq. �2.15�. Inserting this into Eq. �2.18� andmaking use of the unitarity of the Wigner rotation matrix�which, in this case, reduces to the spherical harmonic addi-tion theorem� and the normalization �2.11�, we obtain withEq. �2.22�,
Cinternal�t� =�P2�eI�0� · eI�t��� − S2
1 − S2 , �2.27�
where eI is the unit vector along the zI axis and S is given byEq. �2.24�. More generally, group-theoretical methods can beused to implement any internal symmetry constraints on theorder parameter and the TCF.46
B. Model-free version B
Many proteins deviate substantially from sphericalshape, so the requirement of isotropic global motion limitsthe applicability of the MF-A formula �2.21�. However,MF-B does not suffer from this limitation. MF-B is alsobased on two approximations, albeit not the same ones as inMF-A. To set the stage for these approximations, we splitV0
�L��t� in two parts:
V0�L��t� = V0,s
�L��t� + V0,f�L��t� , �2.28�
with
V0,s�L��t� = �V0
�L��t��DI � V0�L��t� , �2.29a�
V0,f�L��t� = V0
�L��t� − V0�L��t� . �2.29b�
The overbar indicates that the tensor component has beenaveraged over the internal degrees of freedom �DI. Uponsubstituting Eq. �2.28� into Eq. �2.2�, we obtain two auto-TCFs and two cross-TCFs,
C�t� = Gss�t� + Gff�t� + Gsf�t� + Gfs�t� . �2.30�
In MF-B, decoupling of internal and global motions isimposed by an adiabatic approximation that requires the in-ternal motion to be much faster than the global motion �Sec.III B�. If this is the case, then V0,s
�L��t� and V0,f�L��t� fluctuate on
disjoint time scales so the cross-TCFs in Eq. �2.30� vanish,leaving
C�t� = Gss�t� + Gff�t� , �2.31�
where
224507-5 Protein dynamics from NMR J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
Gss�t� = 5�V0�L���0�V0
�L��t�� , �2.32a�
Gff�t� = 5��V0�L��0� − V0
�L��0����V0�L��t� − V0
�L��t��� . �2.32b�
Equation �2.31� may be contrasted with Eq. �2.17�, where wehave a product rather than a sum of partial TCFs. However,when the internal motion is much faster than the global mo-tion, the MF-A formula �2.21� is also a sum of terms associ-ated exclusively with global and internal dynamics.
Combination of Eqs. �2.12� and �2.29a� yields for theslowly fluctuating component
V0�L��t� = �
n�
p�
q
D0n2���LG�t��Dnp
2���GD��Dpq2���DI���q
�.
�2.33�
The second approximation in MF-B is the requirement thatthe local POMT is uniaxial in the D frame, meaning that
�Dpq2���DI�� = p0�D0q
2���DI�� . �2.34�
Using Eqs. �2.32a�, �2.33�, and �2.34�, we can express the“slow” TCF as
Gss�t� = S2Cglobal�t� , �2.35�
with the reduced global TCF Cglobal�t� defined as in Eq. �2.7�and the generalized order parameter S given by
S2 = ��q
�q�D0q2 ��DI���2
. �2.36�
This is seen to be a special case of Eq. �2.20�, where nowonly the p=0 term survives the averaging in the uniaxiallocal environment.
The explicit form of the reduced global TCF in Eq.�2.35� is
Cglobal�t� = 5�n
�n�
Dn02 ��GD�Dn�0
2� ��GD�
��D0n2 ��LG
0 �D0n�2� ��LG�� . �2.37�
From the spatial isotropy of the solution and the orthogonal-ity of the Wigner functions,40 it follows that
�D0n2 ��LG�� = 0, �2.38a�
�D0n2 ��LG
0 �D0n�2� ��LG
0 �� = nn�15 , �2.38b�
which, together with the unitarity of the Wigner rotation ma-trix, show that Cglobal�t� decays from 1 to 0, as expected for areduced TCF. However, no assumption has been made aboutthe form of this decay. For example, the decay is triexponen-tial for symmetric-top rotational diffusion,41
with the reduced internal TCF defined as Cinternal�t�=Gff�t� / �1−S2�. It follows from Eqs. �2.32b�, �2.33�, �2.34�,
�2.36�, and �2.37� and the unitarity of the Wigner rotationmatrix that Cinternal�t� is given by Eqs. �2.18� and �2.22�, justas for MF-A. The original derivation7 of the MF-B formula�2.40� differs in two respects from the present one. First,the generalized order parameter was defined asA= �1+�2 /3�1/2S. Second, an approximation of weak aniso-tropy for the internal motion was invoked, which is actuallynot needed. The MF-B formula �2.40� is thus more generalthan suggested by its original derivation.
C. Model-free version C
When slow internal motions occur, they are usually su-perimposed on faster internal motions. To handle such situ-ations, Clore and co-workers17 proposed what has come to beknown as the extended MF approach. This is actually a hy-brid of MF-A and MF-B and we refer to it as MF-C. Thetensor component V0
�L��t� is now modulated by three kinds ofmotion �global, slow internal, and fast internal�, so a generaltreatment requires two new coordinate frames in addition tothe five frames in Table I. We associate the D and I frameswith the slow internal motion and we introduce the analo-gous frames D� and I� to describe the fast internal motion.
As in MF-A, the global motion is assumed to be isotro-pic and statistically independent of the internal motions. Thetotal TCF therefore factorizes as in Eq. �2.17� with Cglobal�t�given by Eq. �2.15� and Ginternal�t�, which now representsinternal motions on two time scales, given by
Ginternal�t� = �p
�Vp�D���0�Vp
�D��t�� , �2.41�
with
Vp�D��t� = �
q�
r�
s
Dpq2���DI�t��Dqr
2���ID��Drs2���D�I��t���s
�.
�2.42�
The Euler angles �DI�t� and �D�I��t� are modulated by theslow and fast internal motions, respectively, while the fixedEuler angles �ID� define the relative orientation of the mo-bility frame for the slow internal motion and the directorframe for the fast internal motion.
Next, the tensor components Vp�D��t� are split into two
parts as in Eq. �2.28�,
Vp�D��t� = Vp,s
�D��t� + Vp,f�D��t� , �2.43�
with
Vp,s�D��t� = �Vp
�D��t��D�I� � Vp�D��t� , �2.44a�
Vp,f�D��t� = Vp
�D��t� − Vp�D��t� . �2.44b�
The overbar now indicates that the tensor component hasbeen averaged over the fast internal degrees of freedom�D�I�. Upon substituting Eq. �2.43� into Eq. �2.41�, we ob-tain two auto-TCFs and two cross-TCFs. We now assumethat the fast internal motion is much faster than the slowinternal motion �adiabatic approximation� so that the cross-TCFs vanish and
224507-6 Bertil Halle J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
Ginternal�t� = Ginternalss �t� + Ginternal
ff �t� , �2.45�
where
Ginternalss �t� = �
p
�Vp�D���0�Vp
�D��t�� , �2.46a�
Ginternalff �t� = �
p
��Vp�D��0� − Vp
�D��0����Vp�D��t� − Vp
�D��t��� .
�2.46b�
As the fourth and last approximation in MF-C, we requirethe POMT for the fast internal motion to be uniaxial in theD� frame so that
�Drs2���D�I��� = r0�D0s
2���D�I��� . �2.47�
Using Eqs. �2.42�, �2.44a�, �2.46a�, and �2.47�, we can ex-press the slow internal TCF as
Ginternalss �t� = Sfast
2 �Sslow2 + �1 − Sslow
2 �Cinternalslow �t�� , �2.48�
with the generalized order parameters Sfast and Sslow given by
Sfast2 = ��
q
�q�D0q2 ��D�I����2
, �2.49�
Sslow2 = �
p��
q
Dq02 ��ID���Dpq
2 ��DI���2. �2.50�
The reduced slow internal TCF, defined as in Eq. �2.7�, isgiven by
Cinternalslow �t� =
Ginternalslow �t� − Sslow
2
1 − Sslow2 , �2.51�
with
Ginternalslow �t� = �
q�q�
Dq02 ��ID��Dq�0
2� ��ID��
��p
�Dpq2 ��DI
0 �Dpq�2� ��DI�� . �2.52�
Combination of Eqs. �2.15�, �2.17�, �2.45�, �2.46b�, and�2.48� now yields the MF-C formula:
C�t� = exp�− t/global� Sfast2 �Sslow
2 + �1 − Sslow2 �Cinternal
slow �t��
+ �1 − Sfast2 �Cinternal
fast �t�� . �2.53�
The reduced fast internal TCF Cinternalfast �t�, defined as in Eq.
�2.7�, is given by Eqs. �2.18� and �2.22� with S replaced bySfast and �DI replaced by �D�I�. As seen from its derivation,the MF-C formula �2.53� requires all four types of approxi-mation used in the MF-A and MF-B treatments. As expected,Eq. �2.53� reduces to the MF-A result �2.21� when Sfast=1�no fast internal motion� or when Sslow=1 �no slow internalmotion�. Effective correlation times internal
slow and internalfast can be
defined as the time integral of Cinternalslow �t� and Cinternal
fast �t�, as inEq. �2.23�.
If the fast internal mobility tensor is isotropic, the fastinternal reduced TCF becomes
Cinternalfast �t� =
�P2�eI��0� · eI��t��� − Sfast2
1 − Sfast2 �2.54�
as in MF-A. If the slow internal mobility tensor is isotropic,the slow internal reduced TCF becomes
Cinternalslow �t� =
�P2�eI�0� · eI�t��� − Sslow2
1 − Sslow2 . �2.55�
This result follows Eqs. �2.26�, �2.51�, and �2.52� and theunitarity of the Wigner rotation matrix.
D. Cross correlations
Cross-correlated relaxation observables report on thecross-TCF
CAB�t� = �VA�0�:VB�t�� , �2.56�
where VA and VB are two different spin-lattice interactiontensors. Provided that these tensors have a fixed �time-independent� geometrical relationship, it is straightforward togeneralize the three MF formulas. The main difference in thederivations is that the initial value of the partial internal TCFis no longer 1. Each of the interaction tensors is taken to benormalized as in Eq. �2.5�. In expressions relating cross-correlated relaxation observables to spectral density func-tions, the latter should be multiplied by Veff
A VeffB with Veff
A andVeff
B defined as in Eq. �2.3�. The geometrical coefficients �qA
and �qB each obey the normalization �2.11�, but the cross-
TCF depends on the geometrical quantity
�AB � �q
�qA�q
B�, �2.57�
which can be expressed more explicitly with the aid of Eq.�2.10�,
�AB = �1 +�A
2
31 +
�B2
3�−1/2
��P2�cos �� +1
2sin2 ���A cos 2� + �B cos 2��
+1
6�A�B��1 + cos2 ��cos 2� cos 2�
− 2 cos � sin 2� sin 2��� , �2.58�
where �� ,� ,�� are the Euler angles that carry frame FA intoframe FB. If both interaction tensors are uniaxial so that�A=�B=0, then
�AB = P2�cos �� . �2.59�
The generalization of the MF formulas simply amountsto replacing 1 by �AB. For example, the MF-A formula �2.21�generalizes to
CAB�t� = exp�− t/global��SAB2 + ��AB − SAB
2 �CinternalAB �t�� ,
�2.60�
with SAB2 and Cinternal
AB �t� given by the obvious generalizationsof Eq. �2.20�,
224507-7 Protein dynamics from NMR J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
SAB2 = �
q�q�
�qA�q�
B��p
�Dpq2 ��DI���Dpq�
2� ��DI�� , �2.61�
and of Eqs. �2.18� and �2.22�,
CinternalAB �t� =
GinternalAB �t� − SAB
2
�AB − SAB2 , �2.62�
with
GinternalAB �t� = �
q�q�
�qA�q�
B��p
�Dpq2 ��DI
0 �Dpq�2� ��DI�� . �2.63�
The formula �2.60� has been given previously for the specialcase of uniaxial interaction tensors ��A=�B=0�.47,48
In general, there is no simple relation between the cross-TCF CAB�t� and the auto-TCF CAA�t�. However, if the inter-nal mobility tensor is isotropic, we obtain from Eqs. �2.26�,�2.57�, �2.60�, �2.62�, and �2.63�,
CAB�t� = �ABCAA�t� . �2.64�
This result was previously obtained for the special case ofuniaxial interaction tensors, where Eq. �2.59� applies, andunder the additional �but unnecessary� restriction to small-amplitude internal motion.49
E. Anisotropic systems
Although the MF approach was originally conceived inconnection with relaxation studies of anisotropic systems,3–6
specific dynamical models have usually been used to inter-pret relaxation data from solids50 and liquid crystals.51,52 Inmany cases, only local motions are important so there is noneed for dynamical decoupling. However, in situations whereboth global and internal/local motions occur, suitably modi-fied MF formulas are applicable. We refer to such “solid-adapted” versions of the MF formulas derived in Sec. II Aand II B as MF-A/S and MF-B/S. Formulas of MF-B/S type�based on the adiabatic approximation� were derived in theoriginal MF-B paper.7 For applications of MF-B/S to liquidcrystals, group-theoretical methods can be used to fully ex-ploit the rotational symmetries of mesophases and supramo-lecular structures.46
Solid-state magic-angle-spinning �MAS� NMR relax-ation studies of microcrystalline protein samples have thepotential to reveal internal motions that cannot be probed bydipole-dipole, shielding anisotropy, or electric quadrupole in-duced relaxation in a solution of freely tumblingproteins.53,54 So far, the interpretation of such data has beenbased on specific models53 or on the initial-slopeapproximation.50,54 Here, we present a rigorous derivation ofa MF-A/S formula that is applicable to such data.
For an anisotropic system, the autocorrelated relaxationobservables depend on three independent laboratory-frameTCFs �m=0,1 ,2�,
Gm�L��t� = G−m
�L��t� = �Vm�L���0�Vm
�L��t�� , �2.65�
involving the fluctuating part of the tensor component
Vm�L��t� = Vm
�L��t� − �Vm�L�� . �2.66�
Introducing a crystal-fixed frame C and a MAS frame Mwith zM as spinning axis, one obtains50
Gm�L��t� = �
l
�dml2 ��LM��2�
k�k�
dlk2 ��MC�dlk�
2 ��MC�
�exp�i�k� − k��MC�Gkk��C��t� . �2.67�
For a single-crystal sample �without MAS�, the M frame issuperfluous and we can set �LM=0. With dml
2 �0�=ml, Eq.�2.67� then shows how the relaxation observables depend onthe orientation ��LC,�LC� of the crystal with respect to theexternal magnetic field.46 For a MAS experiment with �LM
=54.74° on a polycrystalline sample, each spectral compo-nent has contributions from microcrystals with different ori-entations ��MC,�MC� relative to the spinning axis, and thedecay of the spectral density is obtained as a powder averageof the relaxation decays associated with each microcrystalorientation.50,53,54
The desired dynamical information is contained in the
crystal-frame TCFs Gkk��C��t�. There are 25 of them, but micro-
scopic time-reversal invariance ensures that at most 15 areindependent.46 This number is further reduced if rotationalsymmetry is present. Here, we shall assume that the globalPOMT is uniaxial in the C frame. Then, all off-diagonal �k�k�� TCFs vanish and Eq. �2.67� reduces to
Gm�L��t� = �
l
�dml2 ��LM��2�
k
�dlk2 ��MC��2Gk
�C��t� , �2.68�
with only three distinct crystal-frame TCFs �k=0,1 ,2�,
Gk�C��t� = G−k
�C��t� � Gk�C��t� − Gk
�C���� , �2.69�
Gk�C��t� = �Vk
�C���0�Vk�C��t�� . �2.70�
As in MF-A, we consider the case of global and internalmotions that modulate the Euler angles �CG�t� and �DI�t�,respectively. Transforming the crystal-frame tensor compo-nents Vk
�C� as in Eq. �2.8�, we can express the crystal-frameTCF Gk
�C��t� as in Eq. �2.13�, but without the factor of 5 andwith D0n
2 ��LG0 �D0n�
2� ��LG� replaced by Dkn2 ��CG
0 �Dkn�2� ��CG�.
We now introduce the two MF-A approximations. The super-position approximation allows us to factorize the ensembleaverage as in Eq. �2.14� and the requirement of an isotropic
224507-8 Bertil Halle J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
global mobility tensor implies that
�Dkn2 ��CG
0 �Dkn�2� ��CG�� = nn��Dk0
2 ��CG0 �Dk0
2���CG��
� nn�Gglobal,k�t� . �2.71�
In contrast to Eq. �2.15� for the isotropic case, the globalTCF defined here depends on the index k and it is not inreduced form. The limits of this TCF are
Gglobal,k�0� = Ak, �2.72a�
Gglobal,k��� = k0Sglobal2 , �2.72b�
where we have introduced the mean-square fluctuationamplitudes7,46
Ak = ��dk02 ��CG��2� = �
15 + 2
7Sglobal + 1835Qglobal, k = 0
15 + 1
7Sglobal − 1235Qglobal, k = 1
15 − 2
7Sglobal + 335Qglobal, k = 2
� �2.73�
and the rank-2 and rank-4 global order parameters
Sglobal = �P2�cos �CG�� , �2.74a�
Qglobal = �P4�cos �CG�� . �2.74b�
The model-independent expressions in Eq. �2.73� were ob-tained with the aid of the Clebsch–Gordan series.40 The orderparameters are readily evaluated for specific models. For ex-ample, if the global motion is modeled as diffusion in acone,37,38
Combining Eq. �2.71� with the counterpart of Eq. �2.13�and making use of the unitarity of the Wigner rotation ma-trix, we find that the crystal-frame TCF factorizes as
Gk�C��t� = Gglobal,k�t� Ginternal�t� , �2.76�
with the internal TCF Ginternal�t� given by Eq. �2.18�. Finally,we obtain the MF-A/S formula by combining Eqs. �2.69�,�2.76�, �2.72�, and �2.22�,
with the �internal� order parameter S given by Eq. �2.20�, thereduced internal TCF Cinternal�t� by Eqs. �2.18� and �2.22�,and the reduced global TCF given by
Cglobal,k�t� =�Dk0
2 ��CG0 �Dk0
2���CG�� − k0Sglobal2
Ak − k0Sglobal2 . �2.78�
In the limit of vanishing amplitude for the global motion,�CG→0, we have Ak=k0 and Sglobal=1, so Eq. �2.77� re-duces to
Gk�C��t� = k0�1 − S2�Cinternal�t� . �2.79�
If the internal motion is so fast that it does not contributedirectly to relaxation, then Eq. �2.77� yields
Gk�C��t� = �Ak − k0Sglobal
2 �S2Cglobal,k�t� . �2.80�
In the absence of a global POMT, when feq��CG�=1 / �8�2�, we have Sglobal=0 and Ak=1 /5. Furthermore, theglobal TCF in Eq. �2.78� becomes independent of k. It thenfollows Eqs. �2.68� and �2.77� that
Because this result is independent of m, Eq. �2.2� shows thatthe MF-A/S formula reduces to the isotropic MF-A formula�2.21�, as expected in this limit.
III. DECOUPLING APPROXIMATIONS
A. Superposition approximation
Theories of condensed-phase dynamics usually employ astochastic description of the relevant degrees of freedom.55
The dynamical variables �LG�t� and �DI�t� in the MF ap-proach �MF-A or MF-B� are thus regarded as stochastic pro-cesses governed by a set of time-dependent probabilitydistributions.55 The partial TCFs appearing in Eq. �2.13� canthen be expressed as
�D0n2 ��LG
0 �D0n�2� ��LG�Dpq
2 ��DI0 �Dp�q�
2� ��DI�� =� d�LG0 � d�DI
0 Peq��LG0 ,�DI
0 �D0n2 ��LG
0 �Dpq2 ��DI
0 �
�� d�LG� d�DIP��LG,�DI,t��LG0 ,�DI
0 �D0n�2� ��LG�Dp�q�
2� ��DI� . �3.1�
When multiplied by d�LG d�DI, the joint propagatorP��LG,�DI, t ��LG
0 ,�DI0 � gives the probability of having, at
time t, Euler angles to within d�LG of �LG and to withind�DI of �DI, given that they were �LG
0 and �DI0 initially. The
propagator has the limiting values
P��LG,�DI,0��LG0 ,�DI
0 � = ��LG − �LG0 ���DI − �DI
0 � ,
�3.2a�
limt→�
P��LG,�DI,t��LG0 ,�DI
0 � = Peq��LG,�DI� . �3.2b�
224507-9 Protein dynamics from NMR J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
The joint equilibrium distribution Peq��LG,�DI� is thenormalized Boltzmann distribution
Peq��LG,�DI� =exp�− U��LG,�DI�/�kBT��
�d�LG�d�DI exp�− U��LG,�DI�/�kBT��,
�3.3�
where U��LG,�DI� is the POMT that governs the orientationof the I frame relative to the L frame. It is usually a goodapproximation to assume that this potential is separable,
U��LG,�DI� = U��LG� + U��DI� . �3.4�
It then follows from Eq. �3.3� that the joint equilibrium dis-tribution factorizes,
Peq��LG,�DI� = Peq��LG�Peq��DI� , �3.5�
where the marginal equilibrium distributions on the right aredefined as
Peq��LG� � � d�DIPeq��LG,�DI� �3.6�
and similarly for Peq��DI�. When Eq. �3.5� holds, the twosets of stochastic variables �LG and �DI are said to be sta-tistically independent.55 For an isotropic solution sample,where U��LG� is a constant, Eq. �3.5� becomes
Peq��LG,�DI� =1
8�2 Peq��DI� . �3.7�
The analogous definition of statistical independence forthe stochastic processes �LG�t� and �DI�t� is that the jointpropagator factorizes,
P��LG,�DI,t��LG0 ,�DI
0 � = P��LG,t��LG0 �P��DI,t��DI
0 � ,
�3.8�
where the marginal propagators are defined as
P��LG,t��LG0 � � � d�DI
0 Peq��DI0 �
�� d�DIP��LG,�DI,t��LG0 ,�DI
0 � ,
�3.9�
and similarly for P��DI, t ��DI0 �. Because the propagators
evolve toward the corresponding equilibrium distributions�see Eq. �3.2b��, it is clear that Eq. �3.8� implies Eq. �3.5�.The condition �3.8� may be regarded as a superposition ap-proximation, stating that the total motion of I relative to L isa pure superposition of the motion of I relative to D and themotion of G relative to L. �Note that the D frame is fixedrelative to the G frame.� This is not true, in general. How-ever, when the two motions are statistically independent,then the TCF in Eq. �3.1� factorizes into a product of globaland internal TCFs, as in Eq. �2.14�. This is readily seen bycombining Eqs. �3.1�, �3.5�, and �3.8�.
To formulate the physical requirements for statistical in-dependence, we assume that the stationary multivariate pro-cess �LG�t� ,�DI�t�� is Markovian. The joint propagator thenobeys the master equation55
�
�tP��LG,�DI,t��LG
0 ,�DI0 �
= L��LG,�DI�P��LG,�DI,t��LG0 ,�DI
0 � , �3.10�
where the evolution operator L��LG,�DI� acts on �LG and�DI and also may depend parametrically on these variables.Associated with the master equation is the initial condition inEq. �3.2a� and a set of boundary conditions that we expressin terms of an operator B��LG,�DI�,
B��LG,�DI�P��LG,�DI,t��LG0 ,�DI
0 � = 0. �3.11�
Statistical independence requires that the evolution operatorand the boundary conditions are separable,
L��LG,�DI� = L��LG� + L��DI� , �3.12a�
B��LG,�DI� = B��LG� + B��DI� . �3.12b�
When these conditions are satisfied, the solution to Eq.�3.10� can be written on the factorized form �3.8�. This isreadily seen by inserting Eqs. �3.8� and �3.12a� into Eq.�3.10� to obtain
1
P��LG,t��LG0 �
� �
�t− L��LG�P��LG,t��LG
0 ��= −
1
P��DI,t��DI0 �� �
�t− L��DI�P��DI,t��DI
0 �� ,
and by noting that this equality holds generally only if boththe bracketed expressions are identically zero.
B. Adiabatic approximation
The adiabatic approximation, like the Born–Oppenheimer approximation in molecular quantum mechan-ics, is based on an assumption of time-scale separation ofdifferent degrees of freedom. The stochastic processes�LG�t� and �DI�t� are said to be time-scale separated if thereexists a time interval t� during which �LG�t� remains con-stant, while �DI�t� has become independent of its initialvalue. Then,29,33
P��LG,�DI,t��LG0 ,�DI
0 �
= ���LG − �LG0 �P��DI,t��DI
0 ;�LG� , t � t�
P��LG,t��LG0 �Peq��DI��LG� , t � t�.
� �3.13�
On the short time scale �t� t��, there is no slow motion, andon the long time scale �t� t��, the fast variables �DI arecompletely slaved to the slow variables �LG. In the absenceof external fields or, more generally, if the POMT is sepa-rable as in Eq. �3.4�, the fast propagator and the fast �condi-tional� equilibrium distribution become independent of theslow variables. The adiabatic approximation can then be ex-pressed in the stronger form56
P��LG,�DI,t��LG0 ,�DI
0 �
= ���LG − �LG0 �P��DI,t��DI
0 � , t � t�
P��LG,t��LG0 �Peq��DI� , t � t�.
� �3.14�
Because the propagator in Eq. �3.14� factorizes at all times,
224507-10 Bertil Halle J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
the stochastic processes �LG�t� and �DI�t� are, in fact, sta-tistically independent. Time-scale separation in the sense ofEq. �3.14� thus implies statistical independence, but the re-verse is not true. In other words, time-scale separation is astronger condition than statistical independence. Thus,�LG�t� and �DI�t� can be statistically independent eventhough they fluctuate on the same time scale �see Sec. V A�.
IV. TWO-BODY SMOLUCHOWSKI APPROACH
A. Reduction to two dimensions
In this section we consider a simple but nontrivial modelwhere the internal and global motions take place in a two-dimensional �2D� space. Coordinate frames and sets of Eulerangles � in three dimensions correspond to vectors andangles � in two dimensions. The orientations of the internalmobility vector I �taken to coincide with the spin-lattice in-teraction vector F� and of the global mobility vector G �takento coincide with the local director D�, both defined with re-
spect to the laboratory-fixed vector L, are denoted by �LI and�LG. The 2D orientational TCF corresponding to the TCF inEq. �2.2� is
C�t� � �exp ik��LI�t� − �LI�0����
= �0
2�
d�LI0 Peq��LI
0 � exp�− ik�LI0 �
��0
2�
d�LI P��LI,t��LI0 � exp�ik�LI� , �4.1�
where the rank index k is a non-negative integer. �In thethree-dimensional �3D� case, k=2 typically.� To exhibit theeffects of internal and global motions, we introduce an inter-nal coordinate �GI defined through
�LI = �LG + �GI. �4.2�
In place of the transformation �2.8�, we now haveexp�ik�LI�=exp�ik�LG�exp�ik�GI� and the TCF in Eq. �4.1�becomes
The relative orientation of the I and G vectors is con-strained by the local POMT, which is taken to depend onlyon the internal coordinate �GI. Furthermore, the system isassumed to be isotropic. Then, in analogy with Eq. �3.7�,
Peq��LG0 ,�GI
0 � =1
2�Peq��GI
0 � . �4.4�
It will prove convenient to introduce the modified marginalpropagator
Q��GI,t��GI0 � �
1
2��
0
2�
d�LG0 exp�− ik�LG
0 �
��0
2�
d�LG exp�ik�LG�P��LG,�GI,t��LG0 ,�GI
0 �
�4.5�
in terms of which Eq. �4.3� can be expressed as
C�t� =� d�GI0 exp�− ik�GI
0 �Peq��GI0 �
�� d�GI exp�ik�GI�Q��GI,t��GI0 � . �4.6�
B. Planar SRLS model
In this subsection, we formulate and solve a 2D versionof the SRLS model, which assumes that the I and G vectors�or the “bodies” to which they are attached� undergo inde-pendent rotational diffusion, with diffusion coefficients DI
and DG, but are coupled via their mutual interaction.18,20,32
We shall analyze this planar SRLS model for the case of a
FIG. 1. Definition of the geometrical parameters in the planar SRLS model.The G vector bisects the 2� angular range of the “wedge” and is fixed to the“protein” �shaded�.
224507-11 Protein dynamics from NMR J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
hard repulsive interaction that confines the I vector to a re-stricted angular range of width 2� centered on the G vector�Fig. 1�. Thus,
Peq��GI0 � =
1
2�. �4.7�
The SRLS model assumes that the coupled evolution of the Iand G vectors is governed by a two-body Smoluchowskiequation �SE�.32 For the excluded-volume interaction consid-ered here, there is no mean torque. The two-body SE there-fore reduces to a two-body �free� diffusion equation and thePOMT enters the problem solely via the boundary condi-tions. Apart from the effect of their mutual interaction, the Iand G vectors rotate independently relative to the laboratory-fixed L vector. The propagator P��LG,�LI , t ��LG
0 ,�LI0 � thus
obeys the bivariate diffusion equation
�
�tP��LG,�LI,t��LG
0 ,�LI0 �
= �DG�2
��LG2 + DI
�2
��LI2 �P��LG,�LI,t��LG
0 ,�LI0 � �4.8�
on the domain D defined by the shaded region in Fig. 2�a�.The initial condition is
P��LG,�LI,0��LG0 ,�LI
0 � = ��LG − �LG0 ���LI − �LI
0 � . �4.9�
The boundary conditions follow from the 2� periodicity ofthe variables �LG and �LI, from the normalization of the jointpropagator
�0
2�
d�LG�0
2�
d�LI P��LG,�LI,t��LG0 ,�LI
0 � = 1, �4.10�
and from the confinement condition
− � � �LI − �LG � � , �4.11�
imposed by the mutual interaction of the I and G vectors. Asa result of these three conditions, the propagator and its de-rivatives obey periodic boundary conditions along the dashedlines in Fig. 2�a�, while reflecting boundary conditions applyalong the solid lines in Fig. 2�a�. A reflecting boundarymeans that the normal component of the probability flux Jvanishes. For example, along the line �LI=�LG+�,
u · J = − u · �P = DG� �P
��LG�
�LI=�LG+�
− DI� �P
��LI�
�LI=�LG+�
= 0, �4.12�
where u= �−1,1� /�2 is the outward-pointing unit normal forthe line considered.
The joint propagator P��LG,�LI , t ��LG0 ,�LI
0 � evolves to-ward the joint equilibrium distribution
Peq��LG,�LI� = Peq��LG�Peq��LI��LG�
= Peq��LI�Peq��LG��LI�
= � 1
4��, �LG,�LI� � D
0, �LG,�LI��” D .� �4.13�
Because Peq��LG,�LI� cannot be written as a product of themarginal equilibrium distributions Peq��LG� and Peq��LI�, itis clear that the stochastic variables �LG and �LI are statisti-cally dependent. Therefore, the stochastic processes �LG�t�and �LI�t� cannot be statistically independent under any con-ditions. With regard to the condition �3.12�, we note that thediffusion operator in Eq. �4.8� is separable, but the boundarycondition in Eq. �4.12� is not.
While �LG,�LI� are the natural variables for formulat-ing the two-body Smoluchowski �or diffusion� equation�4.8�, they are not the most convenient variables for analyz-ing the coupled dynamics of the I and G vectors in terms ofinternal and global motions. We therefore transform the in-dependent variables from �LG,�LI� to �LG,�GI�, with theinternal coordinate �GI defined by Eq. �4.2� �see also Fig. 1�.The Jacobian of this transformation is unity, so the trans-formed diffusion equation is obtained simply by applying thechain rule to Eq. �4.8� with the result
1
�DI + DG��
�tP��LG,�GI,t��LG
0 ,�GI0 � = ��
�2
��LG2 − 2�
�2
��LG � �GI+
�2
��GI2 �P��LG,�GI,t��LG
0 ,�GI0 � , �4.14�
FIG. 2. Allowed domains for �a� the dependent variables �LG and �LI andfor �b� the independent variables �LG and �GI. The solid and dashed linesindicate reflecting and periodic boundary conditions, respectively.
224507-12 Bertil Halle J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
where we have introduced the dimensionless dynamical pa-rameter
� �DG
DI + DG. �4.15�
Equation �4.14� is obeyed in the domain defined by theshaded region in Fig. 2�b�. Periodic boundary conditions ap-ply along the dashed lines �LG=0 and �LG=2�, while re-flecting boundary conditions apply along the solid lines �GI
=−� and �GI=�. The latter conditions are obtained by trans-forming Eq. �4.12�, yielding
� �P
��GI�
�GI=��
= �� �P
��LG�
�GI=��
. �4.16�
Finally, we have the initial condition
P��LG,�GI,0��LG0 ,�GI
0 � = ��LG − �LG0 ���GI − �GI
0 � .
�4.17�
The cross term in the evolution operator of Eq. �4.14�implies that the stochastic processes �LG�t� and �GI�t� are notstatistically independent, in general. However, the jointpropagator P��LG,�GI, t ��LG
0 ,�GI0 � evolves toward a joint
equilibrium distribution that factorizes
Peq��LG,�GI� = Peq��LG�Peq��GI� =1
4��. �4.18�
The new independent variables are thus statistically indepen-dent. Therefore, the stochastic processes �LG�t� and �GI�t�may be statistically independent under certain conditions.
The diffusion equation �4.14� can also be obtained fromthe two-body SE used by Freed and co-workers by taking thelimit of vanishing POMT in the 2D version of Eq. �3� in Ref.20. For the continuous POMT employed by these authors,the diffusion equation contains additional terms, which, how-ever, do not violate the superposition approximation. Thebreakdown of the superposition approximation is caused bythe cross term, which is present in Eq. �4.14� as well as inEq. �3� of Ref. 20. The effect of the additional terms thatappear for a continuous POMT is to reduce the deviationfrom the superposition approximation. Our model, with asquare-well POMT, thus emphasizes this deviation, whichshould vanish in the strong-coupling limit.
The boundary value problem defined by Eqs.�4.14�–�4.17� does not have an analytical solution. However,as shown in Appendix A of Ref. 57, the modified marginalpropagator Q��GI, t ��GI
0 � can be obtained analytically. Byway of Eq. �4.6�, we can then also obtain the TCF analyti-cally. The result can be recast in the functional form of theMF-A formula �2.21�,
but the parameters must now be regarded as apparent quan-tities. The apparent global correlation time is
global =1
k2 1
DG+
1
DI =
1
k2DG�1 − ��. �4.20�
The apparent order parameter is
S = sinc��� , �4.21�
with sinc x�sin x /x the un-normalized cardinal sinc func-tion and
� � �1 − ��k� . �4.22�
The apparent reduced internal TCF is given by
Cinternal�t� = �n=1
�
An exp�− n�
2�2
�DI + DG�t� , �4.23�
with relative mode amplitudes
An = �2
1 − sinc2� �1 − �− 1�ncos�2���
��n�/2�2 − �2�2 . �4.24�
The sum in Eq. �4.23� converges rapidly and under mostconditions it can be truncated after the first term. For ex-ample, A1=0.90 and A2=0.09 for �=1.22, which corre-sponds to 2�=70° for �=0 and k=2. For stronger confine-ment �smaller �� or slower internal motion �larger ��, thedominance of the first term is even stronger. The decay of theapparent internal TCF Cinternal�t� is thus very nearly exponen-tial under most conditions �Fig. 3�. The effective internalcorrelation time is obtained by integrating Eq. �4.23�, as inEq. �2.23�, and then summing the resulting trigonometric se-ries,
internal =2�2
�DI + DG�� 1
�2 −sinc2�
3�1 − sinc2��� . �4.25�
C. Adiabatic limit
The adiabatic limit of the SRLS model corresponds toDI�DG and, hence, ��1. In this limit, the adiabatic ap-proximation invoked in MF-B is valid. Since, in addition, theG vector �which coincides with the director D in the planarSRLS model� is a symmetry axis for the internal motion, theTCF in Eq. �4.19� can be identified with the MF-B formula�2.40�,
FIG. 3. Internal TCF for the planar SRLS model with k=2, �=30°, and theindicated � values. The exact TCF from Eq. �4.23� �solid curves� is com-pared with the single-exponential approximation Cinternal�t�=exp�−t /internal�with the effective correlation time internal from Eq. �4.25� �solid circles�.
224507-13 Protein dynamics from NMR J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
C0�t� = S02Cglobal
0 �t� + �1 − S02�Cinternal
0 �t� , �4.26�
where the adiabatic limit is indicated by a 0 subscript orsuperscript. In the adiabatic limit, the global TCF Cglobal
0 �t�depends only on DG and the internal TCF Cinternal
0 �t� dependsonly on the internal parameters DI and �. In addition, theorder parameter S0 is a true equilibrium property, dependingon the interaction parameter �, but not on the dynamicalparameters DI and DG. These expectations are confirmed bytaking the �→0 limit of Eqs. �4.20�–�4.24�. The TCF in Eq.�4.19� then reduces to Eq. �4.26� with
S0 = sinc�k�� , �4.27�
Cglobal0 �t� = exp−
t
global0 = exp�− k2DGt� , �4.28�
Cinternal0 �t� = �
n=1
�
An0 exp�− n�
2�2
DIt� , �4.29�
An0 = � �k��2
1 − sinc2k�� �1 − �− 1�ncos�2k���
��n�/2�2 − �k��2�2 , �4.30�
and Eq. �4.25� reduces to
internal0 =
2
k2DI�1 −
sin2k�
3�1 − sinc2k��� . �4.31�
Because the global motion in the planar SRLS model is iso-tropic, the TCF in Eq. �4.26� also coincides with the MF-Aformula �2.21�. In other words, the planar SRLS model hassufficient symmetry to make the MF-A and MF-B formulasequivalent in the adiabatic limit.
The adiabatic limit of the planar SRLS model is the 2Danalog of the widely used 3D diffusion-in-a-cone model.37,38
The 3D model does not admit an analytical solution, but theorder parameter,37 S0=cos ��1+cos �� /2, and the effectivecorrelation time internal
0 �Ref. 38� can be expressed in closedform. The order parameter for the cone model is numericallyclose to the order parameter in Eq. �4.27� for k=2 �Fig. 4�.The 2D order parameter with k=2 vanishes when the I vectororientation is constrained to a half-circle ��=90°�. Similarly,
for the 3D cone model, the rank-2 order parameter S= �P2�cos ��� vanishes when uniformly averaged over ahemisphere ��=90°�. Furthermore, the effective internal cor-relation time internal
0 has almost the same dependence on �for the two models, the difference being essentially a con-stant factor that accounts for the faster sampling of angularspace in 2D than in 3D �Fig. 4�. This close correspondenceensures that the conclusions drawn here on the basis of a 2Dmodel �with rank k=2� are quantitatively relevant to real 3Dsystems.
D. Physical interpretation
Outside the adiabatic limit, that is, when � is not �1, theSRLS model cannot be fully described by the MF approach.The TCF can still be expressed on the MF-A form, as in Eq.�4.19�, but the parameters do not have the same physicalmeaning as in the MF approach and they cannot be inter-preted in terms of the explicit expressions given in Sec. II.
Since it depends on the dynamical parameter �, the ap-parent order parameter S in Eq. �4.21� is no longer a trueequilibrium property determined solely by the internalPOMT �here parametrized via the confinement angle ��.Therefore, 1−S2 cannot be interpreted as a measure of ori-entational disorder of the I vector, related to its configura-tional entropy. When the protein tumbles faster or the inter-nal motion slows down, � increases and thus 1−S2 decreases�Fig. 5�. This is a large effect. For example, for �=30° andDG=DI �corresponding to �=0.5�, 1−S2 is reduced from theadiabatic �DG�DI� value by a factor of 3.6 �from 0.316 to0.088�.
In the adiabatic limit, the effective internal correlationtime in Eq. �4.31� does not depend on DG. In contrast, theapparent effective internal correlation time in Eq. �4.25� de-creases with increasing � �Fig. 6�. Because the apparent in-ternal TCF is strongly dominated by the first exponentialterm in Eq. �4.23�, the decrease in internal in Fig. 6 can beattributed to the appearance of the relative diffusion coeffi-cient DI+DG in the exponent of Eq. �4.23� rather than to theDG dependence of the mode amplitude An. To an excellentapproximation, we can therefore obtain internal from theadiabatic-limit expression �4.31� simply by replacing DI withDI+DG �Fig. 6�.
FIG. 4. Squared order parameter S02 and effective internal correlation time
internal0 in units of 1 /DI vs confinement angle � for the planar SRLS model
with k=2 in the adiabatic limit �solid curves� and for the 3D diffusion-in-a-cone model �dashed curves�. The curve showing internal
0 DI for the planarmodel has been multiplied by a factor of 4.
FIG. 5. Apparent orientational disorder, 1−S2, for the planar SRLS modelwith k=2 and �=30° vs the dynamical parameter �. The dashed line indi-cates the true equilibrium value, 1−S0
2, reached in the adiabatic limit ��→0�.
224507-14 Bertil Halle J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
Figure 7 shows the total TCF C�t� and the internal TCFCinternal�t� for the SRLS model, as well as the correspondingquantities C0�t� and Cinternal
0 �t� in the adiabatic limit. Becausethe apparent internal motion in the SRLS model involves therelative diffusion coefficient DI+DG, Cinternal�t� decays fasterthan Cinternal
0 �t� and the difference can, to an excellent ap-proximation, be accounted for simply by replacing DI in Eq.�4.29� with DI+DG �see insets in Fig. 7�. Because of the
faster �apparent� internal motion and because of the larger�apparent� order parameter, the total TCF for the SRLSmodel is relatively less affected by the internal motion. Inother words, C�t� in Eq. �4.19� is more strongly dominatedby the first term S2 exp�−t /global� than it is in the adiabatic�MF� limit.
The apparent global correlation time global in Eq. �4.20�depends not only on the protein mobility �DG� but also on theinternal mobility �DI�. It is evident from Eqs. �4.20� and�4.25� that internal is dominated by the fastest process �thelargest of DI and DG�, whereas global is dominated by theslowest process �the smallest of DI and DG�. For this reason,internal can never exceed global in the SRLS model �Fig. 8�.In the adiabatic regime �DI�DG�, internal is determined byDI and global is determined by DG, as in the MF approach. Inthe opposite regime of slow internal motion �DI�DG�,internal is determined by DG and global is determined by DI.In this regime, S=1 so the total TCF is C�t�=exp�−k2DIt�. Inthe limit DI=0 �no internal motion�, the SRLS model thusyields C�t�=1. In the strong-coupling limit, corresponding to�=0, Eqs. �4.19�–�4.24� yield for nonzero �,
C�t� = Cglobal�t� = exp�− k2 1
DG+
1
DI−1
t� , �4.32�
that is, the TCF still depends on the rate �DI� of internalmotion even though the amplitude of the internal motion iszero.
Thus, while the SRLS model coincides with the MF ap-proach in the adiabatic regime �DI�DG�, it makes predic-tions in other limits that contradict physical intuition. Forexample, when the internal motion is much slower than theglobal motion �DI�DG�, it should have no effect on spinrelaxation and the TCF should be the same as for a rigidprotein, C�t�=exp�−k2DGt�. However, the SRLS model pre-dicts that C�t�=exp�−k2DIt� in this limit. This striking dis-crepancy cannot be attributed to a dynamical coupling be-tween the internal and global motions because such acoupling can only exist when the two motions occur on thesame time scale �Sec. III�.
The origin of the counterintuitive predictions of theSRLS model can be traced back to its foundation, the two-body Smoluchowski �or diffusion� equation �4.8�. This equa-
FIG. 6. Apparent effective internal correlation time internal in units of 1 /DI
for the planar SRLS model with k=2 and �=30° vs the dynamical param-eter �. The dashed line indicates the effective internal correlation timeinternal
0 in the adiabatic limit ��→0�. Also shown is the prediction of theadiabatic-limit result �4.31� with DI replaced by the relative diffusion coef-ficient DI+DG �solid circles�.
FIG. 7. Total TCF C�t� and internal TCF Cinternal�t� �inset� for the planarSRLS model with k=2, �=30°, and �a� �=0.2 or �b� �=0.8. The exact TCF�solid curves� is compared with the TCF in the adiabatic ��→0� limit�dashed curves�. The dotted curves show the contribution to C�t� from thefirst term S2 exp�−t /global�. Also shown in the inset is the prediction of theadiabatic-limit internal TCF Cinternal
0 �t� in Eq. �4.29� with DI replaced by therelative diffusion coefficient DI+DG �solid circles�.
FIG. 8. The ratio of the apparent internal and global correlation times for theplanar SRLS model with k=2 and �=30° vs the mobility ratio DG /DI. Theexact result �solid curve� is compared with the result for the adiabatic limit�dashed curve�, where the internal and global motions are decoupled.
224507-15 Protein dynamics from NMR J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
tion �and its associated boundary conditions� is completelysymmetrical in I and G. In particular, because it describesoverdamped motion, Eq. �4.8� ignores the fact that the pro-tein has a much larger moment of inertia than the small mo-lecular fragment that executes the internal motion. The two-body SE is an appropriate model for two similarly sizedmolecules coupled by a POMT that tends to align a vector �I�in one molecule with a vector �G� in the other molecule. Inthe strong-coupling limit, the two molecules should rotate asa single rigid body with a friction coefficient that is the sumof the friction coefficients of the individual molecules. Be-cause the diffusion coefficient is inversely related to the fric-tion coefficient, this intuition is seen to be in accord with Eq.�4.32�. We thus conclude that the counterintuitive predictionsof the SRLS model result not from dynamical coupling ofinternal and global motions but from the failure of the two-body SE to describe the inherent asymmetry of the dynami-cal problem �local conformational fluctuations in a tumblingprotein� to which the SRLS model has been applied byMeirovitch, Freed, Polimeno, and co-workers.18,20,22–27
V. DYNAMICAL COUPLING IN PROTEINS
In NMR relaxation studies of conformational motion inproteins, dynamical coupling can enter in two ways. First,because of the covalent connectivity of the backbone and thedense packing of side chains, the motions of proximal bondvectors or molecular fragments are always coupled to someextent. Examples include the rotation of the three CuHbonds in a methyl group,58 crankshaftlike coupled torsionaltransitions in a polymer chain,59 and wobbling of a planarpeptide unit carrying an 15NuH bond and a 13C� shieldinganisotropy tensor.60 This type of dynamical coupling, involv-ing spin-lattice interaction tensors at two different nuclearsites, gives rise to experimentally accessible cross-TCFs.61 Inthe limit of total correlation, where the relative orientation ofthe two interaction tensors is fixed, it is straightforward togeneralize the MF treatment to cross-TCFs �Sec. II D�.
Dynamical coupling between different sites may also af-fect the internal-motion part of the auto-TCFs. Since the MFapproach does not rely on specific dynamical models, it canaccommodate this type of dynamical coupling via Eq. �2.18�or Eq. �2.52�. Of course, the MF parameters from multiplesites are not independent if the internal motions at these sitesare coupled. To simplify the simultaneous analysis of multi-site autocorrelated relaxation data, several schemes have thusbeen proposed based on molecular dynamics trajectories,62
analytical models,63 or a combination thereof.64 Althoughthese schemes incorporate dynamical coupling, it is not pos-sible to prove the existence of dynamical coupling betweeninternal motions at different sites by analyzing autocorrelatedrelaxation data.
The second type of dynamical coupling, and the one thatwe are concerned with here, is the coupling between twomotions that modulate the same spin-lattice interaction ten-sor at the same nuclear site. Specifically, we focus on thecoupling between internal and global motions that bothmodulate the orientation of a given interaction tensor, speci-fied by the Euler angles �LF�t�. If these motions are dynami-
cally coupled, the MF approach is not rigorously valid. Asnoted in Sec. III, the internal and global motions can bedynamically coupled only if they occur on the same timescale. Therefore, only relatively slow internal motions, ontime scales of 10−10 s or longer, are relevant. On these timescales, atomic motions are overdamped and inertia plays norole. Provided that the solvent can be approximately treatedas a continuous viscous fluid, the dynamics of a protein com-prising N atoms can then be described by an N-particle SE,featuring friction �or diffusion� coefficients for each atomand a force field describing the mutual interactions of the Natoms. A familiar example of this level of description is theRouse–Zimm model of polymer dynamics.65 A reduction orcontraction of the N-particle SE to only two degrees of free-dom would produce a non-Markovian evolution equationwith time-dependent force and friction coefficients.66 TheMarkovian two-body SE on which the SRLS model is basedignores all such memory effects. As applied to protein dy-namics, the two-body SE must therefore be regarded as aphenomenological model, the validity of which must bejudged according to its ability to describe real systems overthe relevant range of parameter values. There is thus no guar-antee that the SRLS model improves upon the much simplerMF approach under the conditions where it has been applied.In fact, the analysis of the planar SRLS model in Sec. IVindicates that this is not the case.
To develop a realistic model that incorporates dynamicalcoupling, it is helpful to have a clear physical picture of theunderlying mechanism. Since the motions are overdamped�diffusive�, dynamical coupling must be either torque medi-ated or friction mediated. In Sec. V A, we argue that torque-mediated dynamical coupling, which the SRLS model at-tempts to describe, is generally not important in foldedproteins. In Sec. V B, we consider friction-mediated dynami-cal coupling, which is not included in the SRLS model.
A. Persistent and intermittent motions
As noted above, dynamical coupling can only arise if theinternal motion is relatively slow. Diffusive conformationalmotions in proteins can be slow either because they experi-ence a large friction �and thus a small rotational diffusioncoefficient DI� or because one or more potential barriers haveto be surmounted. In applications of the SRLS model to pro-teins, only the former possibility has been considered.
In the absence of barriers, DI must be smaller than DG inorder for the correlation times internal and global to be com-parable �since the internal motion is restricted�, which is nec-essary for dynamical coupling to develop. Generally, the fric-tion coefficient increases with the size of the rotating bodyand with the correlation time of the fluctuating torque ex-erted on it. While the torque on the “I-body” might decaymore slowly than the solvent-generated torque acting on theentire protein, it is not clear that this could offset the effect ofthe size difference.
On the other hand, it is clear that most slow conforma-tional motions within densely packed native proteins are ac-tivated processes, involving torsional or other barriers in thePOMT. Broadly speaking, we can classify diffusive internal
224507-16 Bertil Halle J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
motions as persistent or intermittent. Persistent or continu-ous motions, such as libration in a potential well, tend to bemuch faster than protein tumbling and therefore cannot bedynamically coupled. The reaction coordinate of an intermit-tent or infrequent motion features at least one barrier that islarge compared to kBT. Such motions are not intrinsicallyslow �DI is not small� but involve a long time scale becausea successful barrier crossing is typically preceded by manyunsuccessful attempts. If the barrier is sufficiently large, anintermittent rotational motion can be accurately modeled interms of instantaneous jumps between discrete orientations.In NMR jargon, this type of motion is usually referred to as“conformational exchange.”
Consider an intermittent internal motion where the inter-nal orientational variable �GI only can assume the discretevalues �GI
� , �=1,2 , . . . ,N. The internal orientation can thenbe represented by the state index � and the joint propagatorcan be written as P��LG,� , t ��LG
0 ,�0�. The evolution of thispropagator is described by a set of N coupled master equa-tions ��=1,2 , . . . ,N�,
�
�tP��LG,�,t��LG
0 ,�0� = ��=1
N
L��LG,�,��P��LG,�,t��LG0 ,�0� ,
�5.1�
with the evolution operators
L��LG,�,�� = L��LG� + W����� − �� ���=1
N
W������ . �5.2�
Here, W�� ��� is the transition rate from state � to state �.From the point of view of dynamical coupling, the criticalfeature in this model is the physically reasonable assumptionthat the transition rates are independent of the global motion.In other words, the probability per unit time for a jump fromone state to another is the same whether the protein is fixedor if it is tumbling. The evolution operator �5.2� is then sepa-rable as in Eq. �3.12a�, and the superposition approximation,Eq. �3.8�, is valid. If the internal motion is intermittent, it isthus statistically independent from the global motion even ifthe internal and global motions take place on the same timescale. Under such conditions, the MF-A approach is valid butnot the MF-B approach.
As an illustration of the foregoing general remarks, weshow in Appendix B of Ref. 57 that the planar model in Fig.1 with intermittent jumps between the two internal states�GI
� = �� yields a TCF on the MF-A form,
C0�t� = exp�− t/global0 ��S0
2 + �1 − S02�exp�− t/internal
0 �� . �5.3�
As before, the global correlation time is global0 =1 / �k2DG�.
The internal motion is characterized by the conformationalexchange time internal
0 , related to the mean residence times −
and + in the two states as
1
internal0 =
1
−+
1
+, �5.4�
and the order parameter S0 is related to the jump angle 2�
and the equilibrium populations P− and P+ in the two statesas
S0 = �1 – 4P−P+ sin2k��1/2. �5.5�
B. Hydrodynamic coupling
The most important mechanism for coupling the internaland global motions in a protein is a conformational changethat alters the shape of the protein-solvent interface. We referto this type of dynamical coupling as hydrodynamic cou-pling, since it is mediated by the global friction generated bythe aqueous solvent, which is usually modeled at a hydrody-namic �continuum-solvent� level. Hydrodynamic coupling isparticularly important for intrinsically unstructured or dena-tured proteins, but it may also be significant for proteins withextended flexible loops or flexibly connected rigid domains.In such cases, the analytical simplicity of the MF approachmay have to be sacrificed for computationally intensivesimulation-based interpretation schemes.62,64
Hydrodynamic coupling is only significant if all of thefollowing three conditions are satisfied: �i� the global motionis rotational diffusion of the entire protein �rather than ex-change of a bound ligand or solvent molecule�, �ii� the inter-nal motion substantially alters the shape of the protein, and�iii� the internal and global motions occur on the same timescale. The evolution operator in Eq. �3.10� is then of the form
where LLG is the angular momentum operator acting on theEuler angles �LG. Because the global rotational diffusiontensor DG��GI� depends on the internal configuration �GI,the evolution operator is not separable, as required for statis-tical independence �see Eq. �3.12a��.
While shape-preserving interdomain motions have beentreated within the MF context,67 the effect of hydrodynamiccoupling on the TCF has been considered only recently.68 Toillustrate the consequences of hydrodynamic coupling in asimple and transparent way, we analyze an extended versionof the model in Sec. V A, where now the global diffusioncoefficient depends on the internal state: DG��GI
� �=DG�. This
is essentially a 2D version of the “diagonal model” consid-ered in Ref. 68. For simplicity, we assume that the two statesare equally populated: P+= P−=1 /2. We show in Appendix Cof Ref. 57 that the TCF for this model can be recast in theMF-A form,
but with apparent parameters. The apparent correlation timesare
global = �k2DG −��1 + 2 − 1�
2internal0 �−1
, �5.8�
internal =internal
0
�1 + 2, �5.9�
with internal0 given by Eq. �5.4� and
224507-17 Protein dynamics from NMR J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
� k2�DG+ − DG
− �internal0 , �5.10�
DG � 12 �DG
+ + DG− � . �5.11�
The apparent order parameter is
S = � 12 �1 − �� + 2� cos2k�
1 + �2 �1/2
, �5.12�
with
� � + �1 + 2. �5.13�
In the fast-exchange limit, where ���1, the three appar-ent parameters in Eq. �5.7� reduce to the corresponding pa-rameters in Eq. �5.3� with P+= P−, that is, global=global
0 ,internal=internal
0 , and S=S0=cos k�. The dynamical couplingthus vanishes if ���1, which is a weaker condition than theadiabatic condition internal�global if DG
+ and DG− differ rela-
tively little.In the slow-exchange limit, where ���1, Eqs.
�5.8�–�5.13� yield 1 /global=k2DG− and 1 /global+1 /internal
=k2DG+ and S=1 /�2. The TCF in Eq. �5.7� is then a
population-weighted superposition,
C�t� = 12exp�− k2DG
− t� + 12exp�− k2DG
+ t� . �5.14�
The internal motion thus has no effect on the TCF, as ex-pected since the condition ���1 implies that internal
�global. This is in contrast to the SRLS model, where theglobal motion has no effect on the TCF in this limit �Sec.IV D�.
VI. CONCLUDING REMARKS
A. Validity of the MF approach
In Fig. 9 we summarize the decoupling approximationsand symmetries required to obtain each of the four MF for-mulas considered in Sec. II. The first three of these formulas�MF-A, MF-B, and MF-C� are more general than the originalversions, while the last one �MF-A/S� is new. The general-ized MF formulas have the same functional form as theoriginal, widely used formulas, but their range of validity isextended by the more general form of order parameters andreduced TCFs. For example, our generalized treatmentshows that the MF-A formula �2.21� is valid for any type ofinternal motion, as long as it is superimposed on, and hencedynamically decoupled from, the global motion. In particu-lar, it is not necessary to assume, as did Lipari and Szabo,10
that the internal mobility tensor is isotropic, that the spin-lattice interaction tensor is uniaxial, and that the principalframes �I and F� of these tensors coincide.
1. Time-scale separation
It is often asserted that MF-A requires the internal mo-tion to be much faster than the global motion. However, asexplained in Sec. III, time-scale separation is sufficient, butnot necessary, for statistical independence �superposition�.Moreover, slow conformational dynamics in folded proteinsare likely to be intermittent, with infrequent barrier cross-ings. The time scale of intermittent motions is related to themean lifetimes of the conformational states, which may belong. However, the actual motion, within each potential welland during the infrequent barrier crossings, is fast. For suchmotions, the superposition approximation is expected to beaccurate on all time scales �Sec. V A�. Unless the internalmotion alters the shape of the protein substantially �Sec.V B�, MF-A should therefore be accurate in the vast majorityof cases. This expectation is supported by several analysesbased on long molecular dynamics simulations ofproteins.69–71
2. Order parameters
Unlike the rank-2 spin-lattice interaction and mobilitytensors, which can only have isotropic, uniaxial or biaxialsymmetry, the POMT U��DI� is related to two frames and itssymmetry group is the direct product of the symmetry groupsassociated with the two frames.46 The direct-product groupdetermines the symmetry selection rules on the partial orderparameters �Dpq
2 ��DI��. Thus, for example, the POMT maybe uniaxial in the D frame or in the I frame or in both. AD-frame-uniaxial POMT, such as a prolate-shaped body con-fined within a circular cone, yields the selection rule p=0.
FIG. 9. Overview of the four MF formulas derived in Sec. II, showing forwhich dynamical variables decoupling through the superposition �SA�and/or adiabatic �AA� approximation is invoked and in which frames iso-tropic �iso� or uniaxial �uni� symmetry is imposed.
224507-18 Bertil Halle J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
An I-frame-uniaxial POMT, such as a cylindrical body con-fined within an elliptical cone, yields the selection rule q=0. In either of these cases, the five independent partial orderparameters are linearly related to the five independent com-ponents of a Cartesian traceless, symmetric ordering tensorof rank 2. However, in the general case, there are 25 distinctpartial order parameters.
The generalized order parameter S in Eq. �2.20� retainsall 25 partial order parameters �Dpq
2 ��DI�� since no assump-tions about the symmetry of the POMT were made in thederivation of the MF-A formula. It represents the completemodel-independent information about the POMT availablefrom autocorrelated relaxation observables. �The reduced in-ternal TCF Cinternal�t� also contains information about thePOMT, but this information cannot be separated from thekinetics of the internal motion in a model-independent way.�The order parameter S is usually interpreted as a measure ofthe spatial restriction imposed by the POMT or the angularamplitude of the internal motion. Such naive geometrical in-terpretations can be misleading, since the single number Sonly represents a particular “projection” of the orientationaldistribution function Peq��DI�. In Eq. �2.20�, the partial orderparameters �Dpq
2 ��DI�� are associated with the internal dy-namical variables �DI, but S also depends on the fixed Eulerangles �IF via the geometrical coefficients �q. More infor-mation about the POMT can thus be obtained by comparing,with the aid of Eq. �2.20�, order parameters from two ormore noncoincident spin-lattice tensors attached to the samerigidly moving fragment.45,72 By using Eqs. �2.3�–�2.5� and�2.9�, we can recast Eq. �2.20� on the form
S2 = 1 +�2
3−1
��p��Dp0
2 ��DF�� +�
�6�Dp2
2 ��DF� + Dp−22 ��DF���2
.
�6.1�
Unless the F and I frames happen to coincide ��IF=0�, theEuler angles �DF are not the natural dynamical variables andthe quantities �Dpq
2 ��DF�� in Eq. �6.1� are not the naturalpartial order parameters. The I frame is usually defined insuch a way that the POMT U��DI� is minimized when �DI
=0. Then, for example, the partial order parameter�D00
2 ��DF�� does not necessarily assume is maximum valueof 1 for complete ordering of the fragment to which the Fframe is attached. In applications to dipolar relaxation data,Eq. �6.1� with �=0 is generally used. Estimates of configu-rational entropy for the degrees of freedom associated withthe observable internal motion are commonly derived fromthe order parameter with the aid of a one-parameter repre-sentation of the POMT that assumes that the minimum-energy configuration corresponds to the bond vector �for ex-ample, the peptide NuH bond� being aligned with the localdirector zD.73–77 Of course, one can always define the Dframe so that this condition is satisfied, but the POMT thencannot be modeled in terms of a single parameter unless ithappens to be uniaxial in both the D and F frames.
3. Exponential decay of internal TCF
None of the MF formulas in Sec. II make any assump-tions about the mathematical form of the reduced internalTCF�s�. In contrast, both MF-A and MF-C were originallypresented with the additional approximation of exponentiallydecaying internal TCF�s�.10,17 As noted by Lipari and Szabo,this approximation is not needed when the internal motion isin the extreme motional narrowing regime.10 They also madethe important observation that when the exponential approxi-mation is numerically accurate, which is often the case, thenthe decay time is exactly the effective correlation time de-fined in Eq. �2.23�. Nevertheless, for conceptual and practi-cal reasons, we prefer to retain the generality of the basic MFformulas by not invoking further approximations until theyare needed. The most important advantage of the MF ap-proach is that it separates the available information about theinternal motion into an equilibrium quantity, S, and a kineticquantity, Cinternal�t�. The order parameter S is not a true equi-librium property since it is “filtered” by the global motion�internal motions much slower than the global motion are notmanifested in S�, but it is independent of the mechanisticdetails of the internal motion, such as the symmetry of theinternal mobility tensor. This advantage is most decisivewhen the internal motion is much faster than the global mo-tion, as is usually the case. Then, because the internal andglobal relaxation dispersions do not overlap, S can be deter-mined without the need for realistic modeling of Cinternal�t�.
4. Symmetry of internal motion
In the original derivation of MF-A,10 it was assumed thatthe internal mobility tensor is isotropic. As shown by thederivation in Sec. II A, this assumption is unnecessary. TheMF-A formula �2.21� is valid for any kind of internal motion,with Cinternal�t� given explicitly by Eqs. �2.18� and �2.22�. Ofcourse, a subsequent single-exponential approximation islikely to be less numerically accurate if the internal mobilitytensor has lower symmetry �for example, if two differentinternal diffusion coefficients are involved�. However, thiscomplication does not affect the validity of the basic MF-Aformula �2.21� or the accuracy of the order parameter derivedwith the aid of this formula. This general conclusion is con-sistent with the results of stochastic simulations for a particu-lar dynamical model.21
5. Anisotropic global motion
Lipari and Szabo argued that the MF-A formula, al-though no longer rigorous, remains an accurate approxima-tion also for anisotropic global motion.10 For a uniaxial�symmetric-top� rotational diffusion tensor, the factor exp�−t /global� in Eq. �2.21� can then simply be replaced by thereduced global TCF in Eq. �2.39�. However, Lipari andSzabo tested this approximation for the diffusion-in-a-conemodel with internal diffusion much faster than global diffu-sion. Both approximations in MF-B are then valid so, ineffect, they tested the MF-B formula �2.40�, which is exactunder these conditions. The good numerical agreement foundcan thus not be taken as proof for the validity of MF-A with
224507-19 Protein dynamics from NMR J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
anisotropic global motion under other conditions, notablywhen the internal motions are not time-scale separated.
B. Validity of the SRLS model
Freed and co-workers18,20,22–27 describe the SRLS modelas a more accurate generalization of the MF approach. Spe-cifically, they argue that the SRLS model is more accuratebecause it incorporates a number of features that they believeare neglected in the MF approach. These features are �intheir own words� as follows:20
�1� a proper treatment of general features of local geom-etry,
�2� rhombic local ordering,�3� axial local motion, and�4� rigorous account of mode coupling.
We now examine each of these claims.
1. Local geometry
Freed et al. asserted that the MF approach is limited tothe case where the principal frames of the spin-lattice inter-action and internal mobility tensors coincide, that is, when�IF=0. This simplifying assumption was made by Lipari andSzabo,10 but it is not required to obtain the MF-A formula�2.21�. As seen from Eq. �2.9�, the geometrical coefficient�q, which appears in the definition �2.20� of the generalizedorder parameter and in the definition �2.18� of the internalTCF, allows for arbitrary relative orientation of the I and Fframes.
2. Local ordering
In general, the POMT can be expanded in the completebasis of Wigner functions,
U��DI� = − �L=0
�
�p=−L
L
�q=−L
L
CpqL Dpq
L ��DI� . �6.2�
Freed and co-workers18,20,24–27 approximated this infinite se-ries by one or two terms. Their most general potential modelis
U��DI� = − C20D002 ��DI� − C22�D02
2 ��DI� + D0–22 ��DI�� . �6.3�
For this particular model, which presupposes cylindricalsymmetry in the D frame and D2h symmetry in the I frame,only two of the 25 partial order parameters survive,
S20 � �D002 ��DI�� = 1
2 �3�cos2 �DI� − 1� , �6.4a�
S22 � �D022 ��DI�� + �D0–2
2 ��DI��
=�6
2�sin2�DI cos2�DI� . �6.4b�
These are referred to as “axial” and “rhombic” orderparameters.18,20,24–27
Freed et al.18,20,24–27 insisted that MF-A is valid only foran “axially symmetric” POMT, by which they mean that S20
but not S22 is included in MF-A. However, as explained inSec. VI A 2, MF-A is completely general as regards the sym-
metry of the POMT. Even MF-B is based on a more generalPOMT than the SRLS model. As seen from Eq. �2.36�, theorder parameter in MF-B features 5, in general distinct, par-tial order parameters �D0q
2 ��DI��. Therefore, S cannot be ex-pressed in terms of S20 and S22 alone.
Apart from the imposed symmetries, the validity of thePOMT model in Eq. �6.3� is limited by the omission of theleading terms �of rank L=1� in the expansion �6.2�. For ex-ample, the interaction of the fragment dipole with the localelectric field gives rise to a rank-1 potential.
3. Internal mobility tensor
As explained in Sec. VI A 4, the original restriction to anisotropic internal mobility tensor10 is not required to obtainthe MF-A formula �2.21�. In fact, the internal TCF in Eq.�2.18� allows for a biaxial internal mobility tensor and is thusmore general than the SRLS model, which assumes auniaxial tensor.
4. Dynamical coupling
The principal justification for replacing the popular MFapproach with the computationally intensive and less trans-parent SRLS model is the asserted ability of the latter toaccount rigorously for dynamical coupling between internaland global motions.18,20,24–27 Freed et al. assumed that theSRLS model is more accurate than the MF approach and,therefore, when they analyzed the same data with eitherSRLS or MF and obtained different results, they attributedthe difference to the shortcomings of MF. However, thepresent analysis indicates that these differences have twoprincipal causes.
First, by not recognizing the true generality of the MFapproach, Freed et al. interpreted the MF parameters too nar-rowly. For example, they identified the generalized order pa-rameter S of MF-A with their axial order parameter S20 ratherthan with the more general equation �2.20�. Second, ouranalysis of the planar SRLS model shows that this modelproduces unphysical results that are not attributable to dy-namical coupling �Sec. IV�. This applies also to the versionof the SRLS model used by Freed et al. For example, forDG /DI=0.57 the planar SRLS model �with a hard repulsivePOMT� yields global /global
0 =1.57, not far from the value1.49 obtained with Freed’s version of the SRLS model �witha cos2�DI POMT�.25 This substantial slowing down of pro-tein tumbling by the internal motion is an artifact of theSRLS model, but Freed et al. regarded it as a real effect andthey argued that the MF approach is inaccurate because itdoes not predict such behavior.25 We conclude that the SRLSmodel lacks rigor and is less accurate than the MF approach�which neglects dynamical coupling� under the conditionswhere dynamical coupling is expected to be most important�when DI�DG�. In particular, protein tumbling times ex-tracted from SRLS analysis are likely to be inaccurate.78
Freed and co-workers20 asserted that the internal andglobal motions in the SRLS model are dynamically decou-pled only if these motions occur on different time scales�adiabatic limit� and if the local ordering is either very weakor very strong �so that the order parameter is either �1 or
224507-20 Bertil Halle J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
close to 1�. However, our analysis of the planar SRLS model�Sec. V� shows that all effects of dynamical coupling vanishin the adiabatic limit without any condition on the orderparameter �or confinement angle ��. This behavior is consis-tent with the general fact that time-scale separation impliesstatistical independence �Sec. III�.
The conclusions of the present study regarding the effectof dynamical coupling of internal and global motions on theanalysis of NMR relaxation data may be summarized in thefollowing points.
�1� If the internal motion is much faster than the globalmotion, there can be no dynamical coupling and theMF approach is rigorously valid.
�2� If the internal and global motions occur on the sametime scale, they may be dynamically coupled. However,in folded proteins, internal motions on this time scaletend to be intermittent, and the superposition approxi-mation invoked in MF-A is then valid as long as theinternal motion does not alter the shape of the proteinsubstantially �hydrodynamic coupling�.
�3� In their present forms, neither the SRLS model nor theMF approach treats hydrodynamic coupling.
�4� In the SRLS model, the internal motion can be slowonly if the diffusion coefficient DI is small, but infolded proteins slow internal motions usually involvebarrier crossings. The SRLS model is therefore not ap-plicable to typical slow internal motions in folded pro-teins.
�5� Even under the rare conditions where nonhydrody-namic coupling may be significant, the usefulness ofthe SRLS model is compromised by its unphysicalfoundation: a two-body SE that ignores the inherentasymmetry between internal and global degrees of free-dom.
�6� The SRLS model is rigorous only in the adiabatic limit�DI�DG�, where it offers no advantage over the moregeneral MF approach.
ACKNOWLEDGMENTS
I thank Jack Freed, Eva Meirovitch, Barry Hughes, andPer-Åke Malmqvist for helpful discussions. This work wassupported by the Swedish Research Council.
1 A. Abragam, The Principles of Nuclear Magnetism �Clarendon, Oxford,1961�.
2 F. Noack, Prog. Nucl. Magn. Reson. Spectrosc. 18, 171 �1986�.3 H. Wennerström, G. Lindblom, and B. Lindman, Chem. Scr. 6, 97�1974�.
4 G. W. Stockton, C. F. Polnaszek, A. P. Tulloch, F. Hasan, and I. C. P.Smith, Biochemistry 15, 954 �1976�.
5 J. Seelig, Q. Rev. Biophys. 10, 353 �1977�.6 J. H. Davis, K. R. Jeffrey, and M. Bloom, J. Magn. Reson. �1969-1992�
29, 191 �1978�.7 B. Halle and H. Wennerström, J. Chem. Phys. 75, 1928 �1981�.8 B. Halle and G. Carlström, J. Phys. Chem. 85, 2142 �1981�.9 B. Halle, T. Andersson, S. Forsén, and B. Lindman, J. Am. Chem. Soc.
103, 500 �1981�.10 G. Lipari and A. Szabo, J. Am. Chem. Soc. 104, 4546 �1982�.11 V. A. Daragan and K. H. Mayo, Prog. Nucl. Magn. Reson. Spectrosc. 31,
63 �1997�.12 D. M. Korzhnev, M. Billeter, A. S. Arseniev, and V. Y. Orekhov, Prog.
Nucl. Magn. Reson. Spectrosc. 38, 197 �2001�.13 A. G. Palmer III, Chem. Rev. �Washington, D.C.� 104, 3623 �2004�.14 V. A. Jarymowycz and M. J. Stone, Chem. Rev. �Washington, D.C.� 106,
1624 �2006�.15 T. I. Igumenova, K. K. Frederick, and A. J. Wand, Chem. Rev. �Wash-
ington, D.C.� 106, 1672 �2006�.16 I. Furó, J. Mol. Liq. 117, 117 �2005�.17 G. M. Clore, A. Szabo, A. Bax, L. E. Kay, P. C. Driscoll, and A. M.
Gronenborn, J. Am. Chem. Soc. 112, 4989 �1990�.18 V. Tugarinov, Z. Liang, Y. E. Shapiro, J. H. Freed, and E. Meirovitch, J.
Am. Chem. Soc. 123, 3055 �2001�.19 L. Vugmeyster, D. P. Raleigh, A. G. Palmer, and B. E. Vugmeister, J.
Am. Chem. Soc. 125, 8400 �2003�.20 E. Meirovitch, Y. E. Shapiro, A. Polimeno, and J. H. Freed, J. Phys.
Chem. A 110, 8366 �2006�.21 K. K. Frederick, K. A. Sharp, N. Warischalk, and A. J. Wand, J. Phys.
Chem. B 112, 12095 �2008�.22 V. Tugarinov, Y. E. Shapiro, Z. Liang, J. H. Freed, and E. Meirovitch, J.
Mol. Biol. 315, 155 �2002�.23 Y. E. Shapiro, E. Kahana, V. Tugarinov, Z. Liang, J. H. Freed, and E.
Meirovitch, Biochemistry 41, 6271 �2002�.24 E. Meirovitch, Y. E. Shapiro, V. Tugarinov, Z. Liang, and J. H. Freed, J.
Phys. Chem. B 107, 9883 �2003�.25 E. Meirovitch, Y. E. Shapiro, Z. Liang, and J. H. Freed, J. Phys. Chem. B
107, 9898 �2003�.26 E. Meirovitch, A. Polimeno, and J. H. Freed, J. Phys. Chem. B 110,
20615 �2006�.27 E. Meirovitch, Y. E. Shapiro, A. Polimeno, and J. H. Freed, J. Phys.
Chem. B 111, 12865 �2007�.28 C. F. Polnaszek and J. H. Freed, J. Phys. Chem. 79, 2283 �1975�.29 J. H. Freed, J. Chem. Phys. 66, 4183 �1977�.30 W.-J. Lin and J. H. Freed, J. Phys. Chem. 83, 379 �1979�.31 A. E. Stillman and J. H. Freed, J. Chem. Phys. 72, 550 �1980�.32 A. Polimeno and J. H. Freed, Adv. Chem. Phys. 83, 89 �1993�.33 A. Polimeno, G. J. Moro, and J. H. Freed, J. Chem. Phys. 104, 1090
�1996�.34 G. J. Moro and A. Polimeno, J. Chem. Phys. 107, 7884 �1997�.35 A. Polimeno, D. Frezzato, G. Saielli, G. J. Moro, and P. L. Nordio, Acta
Phys. Pol. B 29, 1749 �1998�.36 G. J. Moro and A. Polimeno, J. Phys. Chem. B 108, 9530 �2004�.37 K. Kinosita, S. Kawato, and A. Ikegami, Biophys. J. 20, 289 �1977�.38 G. Lipari and A. Szabo, Biophys. J. 30, 489 �1980�.39 P. S. Hubbard, Phys. Rev. 131, 275 �1963�.40 D. M. Brink and G. R. Satchler, Angular Momentum, 2nd ed. �Clarendon,
Oxford, 1968�.41 L. D. Favro, Phys. Rev. 119, 53 �1960�.42 B. Halle, Prog. Nucl. Magn. Reson. Spectrosc. 28, 137 �1996�.43 B. Halle, Magn. Reson. Med. 56, 60 �2006�.44 E. Persson and B. Halle, J. Am. Chem. Soc. 130, 1774 �2008�.45 B. Halle, V. P. Denisov, and K. Venu, in Biological Magnetic Resonance,
edited by N. R. Krishna and L. J. Berliner �Kluwer Academic/Plenum,New York, 1999�, Vol. 17, pp. 419–484.
46 S. Gustafsson and B. Halle, Mol. Phys. 80, 549 �1993�.47 L. E. Kay and D. A. Torchia, J. Magn. Reson. �1969-1992� 95, 536
�1991�.48 V. A. Daragan and K. H. Mayo, J. Magn. Reson., Ser. B 107, 274 �1995�.49 N. Tjandra, A. Szabo, and A. Bax, J. Am. Chem. Soc. 118, 6986 �1996�.50 D. A. Torchia and A. Szabo, J. Magn. Reson. �1969-1992� 49, 107
�1982�.51 B. Halle, in Encyclopedia of Nuclear Magnetic Resonance, edited by D.
M. Grant and R. K. Harris �Wiley, Chichester, 1999�, pp. 790–797.52 R. Y. Dong, Nuclear Magnetic Resonance of Liquid Crystals, 2nd ed.
�Springer-Verlag, New York, 1997�.53 N. Giraud, M. Blackledge, M. Goldman, A. Böckmann, A. Lesage, F.
Penin, and L. Emsley, J. Am. Chem. Soc. 127, 18190 �2005�.54 V. Agarwal, Y. Xue, B. Reif, and N. R. Skrynnikov, J. Am. Chem. Soc.
130, 16611 �2008�.55 N. G. van Kampen, Stochastic Processes in Physics and Chemistry
�North-Holland, Amsterdam, 1981�.56 B. Halle, H. Wennerström, and L. Piculell, J. Phys. Chem. 88, 2482
�1984�.57 See EPAPS Document No. http://dx.doi.org/10.1063/1.3269991 for deri-
vations of analytical TCF expressions for the planar SRLS model
224507-21 Protein dynamics from NMR J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
�Appendix A�, for the planar 2-state jump model �Appendix B�, and forthe planar 2-state jump model with hydrodynamic coupling �AppendixC�.
58 P. S. Hubbard, J. Phys. Chem. 52, 563 �1970�.59 G. J. Moro, J. Phys. Chem. 100, 16419 �1996�.60 S. F. Lienin, T. Bremi, B. Brutscher, R. Brüschweiler, and R. R. Ernst, J.
Am. Chem. Soc. 120, 9870 �1998�.61 D. Frueh, Prog. Nucl. Magn. Reson. Spectrosc. 41, 305 �2002�.62 J. J. Prompers and R. Brüschweiler, J. Am. Chem. Soc. 124, 4522
�2002�.63 D. Abergel and G. Bodenhausen, J. Chem. Phys. 121, 761 �2004�.64 E. Caballero-Manrique, J. K. Bray, W. A. Deutschman, F. W. Dahlquist,
and M. G. Guenza, Biophys. J. 93, 4128 �2007�.65 M. Doi and S. F. Edwards, The Theory of Polymer Dynamics �Clarendon,
Oxford, 1986�.66 T. Åkesson, B. Jönsson, B. Halle, and D. Y. C. Chan, Mol. Phys. 57,
1105 �1986�.67 Y. E. Ryabov and D. Fushman, J. Am. Chem. Soc. 129, 3315 �2007�.
68 V. Wong, D. A. Case, and A. Szabo, Proc. Natl. Acad. Sci. U.S.A. 106,11016 �2009�.
69 A. J. Nederveen and A. M. J. J. Bonvin, J. Chem. Theory Comput. 1, 363�2005�.
70 V. Wong and D. A. Case, J. Phys. Chem. B 112, 6013 �2008�.71 P. Maragakis, K. Lindorff-Larsen, M. P. Eastwood, R. O. Dror, J. L.
Klepeis, I. T. Arkin, M. Ø. Jensen, H. Xu, N. Trbovic, R. A. Friesner, A.G. Palmer, and D. E. Shaw, J. Phys. Chem. B 112, 6155 �2008�.
72 V. P. Denisov, K. Venu, J. Peters, H. D. Hörlein, and B. Halle, J. Phys.Chem. B 101, 9380 �1997�.
73 M. Akke, R. Brüschweiler, and A. G. Palmer, J. Am. Chem. Soc. 115,9832 �1993�.
74 D. Yang and L. E. Kay, J. Mol. Biol. 263, 369 �1996�.75 Z. Li, S. Raychaudhuri, and A. J. Wand, Protein Sci. 5, 2647 �1996�.76 E. Johnson, S. A. Showalter, and R. Brüschweiler, J. Phys. Chem. B 112,
6203 �2008�.77 D.-W. Li and R. Brüschweiler, J. Am. Chem. Soc. 131, 7226 �2009�.78 Y. E. Shapiro and E. Meirovitch, J. Phys. Chem. B 113, 7003 �2009�.
224507-22 Bertil Halle J. Chem. Phys. 131, 224507 �2009�
Downloaded 25 Dec 2009 to 130.235.252.29. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp
