Subscriber access provided by UNIV OF MICHIGAN

The Journal of Physical Chemistry B is published by the American ChemicalSociety. 1155 Sixteenth Street N.W., Washington, DC 20036

Article

Ultrahigh Resolution Characterization of Domain Motions and Correlationsby Multialignment and Multireference Residual Dipolar Coupling NMR

Charles K. Fisher, Qi Zhang, Andrew Stelzer, and Hashim M. Al-HashimiJ. Phys. Chem. B, 2008, 112 (51), 16815-16822• DOI: 10.1021/jp806188j • Publication Date (Web): 02 December 2008

Downloaded from http://pubs.acs.org on February 18, 2009

More About This Article

Additional resources and features associated with this article are available within the HTML version:

• Supporting Information• Access to high resolution figures• Links to articles and content related to this article• Copyright permission to reproduce figures and/or text from this article

Ultrahigh Resolution Characterization of Domain Motions and Correlations byMultialignment and Multireference Residual Dipolar Coupling NMR

Charles K. Fisher, Qi Zhang, Andrew Stelzer, and Hashim M. Al-Hashimi*Department of Chemistry & Biophysics, The UniVersity of Michigan, 930 North UniVersity AVenue,Ann Arbor, Michigan 48109-1055

ReceiVed: July 14, 2008; ReVised Manuscript ReceiVed: October 12, 2008

Nuclear magnetic resonance (NMR) residual dipolar couplings (RDCs) provide a unique opportunity forspatially characterizing complex motions in biomolecules with time scale sensitivity extending up tomilliseconds. Up to five motionally averaged Wigner rotation elements, ⟨D0k

2(R�)⟩ , can be determinedexperimentally using RDCs measured in five linearly independent alignment conditions and applied to definemotions of axially symmetric bond vectors. Here, we show that up to 25 motionally averaged Wigner rotationelements, ⟨Dmk

2(R�γ)⟩ , can be determined experimentally from multialignment RDCs and used to characterizerigid-body motions of chiral domains. The 25 ⟨Dmk

2(R�γ)⟩ elements form a basis set that allows one to measuremotions of a domain relative to an isotropic distribution of reference frames anchored on a second domain(and vice versa), thus expanding the 3D spatial resolution with which motions can be characterized. The 25⟨Dmk

2(R�γ)⟩ elements can also be used to fit an ensemble consisting of up to eight equally or six unequallypopulated states. For more than two domains, changing the identity of the domain governing alignment allowsaccess to new information regarding the correlated nature of the domain fluctuations. Example simulationsare provided that validate the theoretical derivation and illustrate the high spatial resolution with which rigid-body domain motions can be characterized using multialignment and multireference RDCs. Our results furthermotivate the development of experimental approaches for both modulating alignment and anchoring it onspecifically targeted domains.

Introduction

An important challenge in biophysics is to reconstruct thedynamics of biomolecules on the basis of experimentalmeasurements.1,2 There has been great interest in recent yearsin harnessing the broad time-scale sensitivity of NMR residualdipolar couplings (RDCs)3,4 in visualizing the dynamics ofcomplex biomolecules.5-7 RDCs can be measured in moleculesthat are partially aligned, either spontaneously when they havea significant magnetic susceptibility anisotropy3,8 or, moregenerally, by dissolution in an appropriate ordering medium.4,9

The RDC observed between two nuclei depends on ⟨(3 cos 2 θ-1)/2⟩ , where θ is the angle between the internuclear bondvector and the magnetic field and the angular brackets denotea time average over all orientations that are sampled at a ratefaster than the inverse of the dipolar interaction (typically10-100 ms).

Several frameworks have been introduced that allow theinterpretation of RDCs in terms of the average orientation anddynamics of bond vectors (reviewed in refs 5 and 6). The directinterpretation of dipolar couplings (DIDC)10,11 and model-free12,13

formalisms allow the determination of five independent param-eters per bond vector based on the measurement of RDCs underfive linearly independent alignment conditions.14 The fiveparameters correspond to motionally averaged Wigner rotationelements, ⟨D0l

2(R�)⟩ , where R and � are polar angles definingthe orientation of a bond relative to a molecule-fixed chiral framethat typically corresponds to the principal axis system (PAS)of the overall alignment tensor. The five parameters describethe mean bond vector orientation, the amplitude of motion, and

the direction and degree of any motional asymmetry. Collectionsof coplanar RDCs can also be combined in the dynamic-meccano approach and used to simultaneously determine therelative orientation and dynamics of peptide planes.15 Alterna-tively, multialignment RDCs can be included along with otherexperimental and nonexperimental restraints in the simulatedannealing protocol to refine an ensemble of interconvertingconformations.16

The information that can be retrieved regarding bond vectordynamics is limited by the axially symmetric nature of thedipolar interaction, which is inherently insensitive to one ofthree rotational degrees of freedom involving rotations aroundthe bond vector itself. However, motions of chiral domains canbe complex and involve all three degrees of rotational freedom.17,18

Such motions can, in principle, be characterized with greaterspatial resolution since many RDC vectors spanning differentorientations can serve as reporters of the underlying motionaltrajectory.19 Stated differently, the alignment tensor of a chiraldomain, which can be determined experimentally using collec-tions of RDCs, can deviate from axial symmetry, allowingmotions to be characterized with more complete 3D rotationalsensitivity.

Several studies have used RDCs in characterizing rigid-bodymotions of chiral structural fragments, including secondarystructural elements19 and intact domains in proteins,20-23 sugarmoieties in carbohydrates,24 and A-form helices in RNA.25,26

The order tensor formalism27-29 and related frameworks havebeen widely used in interpreting RDCs in terms of the relativeorientation and dynamics of domains. When the overall align-ment is axially symmetric,30 the five motionally averaged ordertensor elements can easily be interpreted in terms of five

* Corresponding author. Tel.: 734-615-3361. Fax: 734-647-4865. E-mail:hashimi@umich.edu.

J. Phys. Chem. B 2008, 112, 16815–16822 16815

10.1021/jp806188j CCC: $40.75 2008 American Chemical SocietyPublished on Web 12/02/2008

parameters describing the average orientation and dynamics ofa domain, in analogy to the five parameters obtained for axiallysymmetric bond vectors using multialignment RDCs. However,the interpretation is not straightforward when the overallalignment departs from axial symmetry and the intrinsic spatialsensitivity to domain motions is extended from two to threerotational degrees of freedom. Other approaches have beendeveloped that characterize the conformational space in termsof maximum allowed probabilities for any conformation so tobe consistent with the measured RDCs.31 The analysis of domainmotions has largely been limited to RDCs measured in one ortwo alignments and is complicated by potential deviations fromthe “decoupling approximation,” the assumption that domainmotions do not affect the overall alignment.

Here, we present a theoretical framework that extracts themaximum information from RDCs in characterizing rigid-bodydomain motions. We show that a set of twenty-five motionallyaveraged Wigner rotation elements can be determined experi-mentally from RDCs measured under five linearly independentalignment conditions and used to characterize rigid-body domainmotions. Compared to the five parameter description of bondvector dynamics, the twenty-five parameters afford 3D ratherthan 2D rotational sensitivity and offer stronger restraints onthe domain orientational distribution; thus sharpening theresolution with which motions can be characterized. Further-more, we show that changing the identity of the referencedomain governing overall alignment in multidomain moleculesprovides access to new information regarding the correlatednature of the domain fluctuations that is inaccessible from simplychanging alignment.

Theory

We show in what follows that all 25 Dmn2(R�γ) elements

representing the second rank approximation to the orientationalprobability distribution function (OPDF)38 describing the dy-namics of a domain relative to a reference domain can bedetermined experimentally from analysis of collections of fiveor more independent RDCs measured under five linearlyindependent alignment conditions. We assume that domainmotions do not alter overall alignment, which is dominatedentirely by a “reference” domain. This situation applies for anydomain whose motion does not alter the overall alignment.Alternatively, a reference domain can deliberately be establishedby, for example, introducing a paramagnetic metal3,22 or tag39,40

or by other chemical modifications, such as elongating a domainso that its becomes more anisotropic.41 We consider two domains(I and II) that are linked by a flexible hinge. Unless statedotherwise, domain I will be the reference domain dominatingalignment. We ignore the effects of local motions of bond

vectors, which are assumed to lead to uniform scaling of themeasured RDCs, and also assume that the mean orientation ofRDC targeted bond vectors within each domain is known.

Let Tk2(I:PI) be elements of the domain I overall alignment

tensor expressed in its overall PAS (PI). The motionallyaveraged alignment tensors measured for domains I ⟨Tk

2(I:AI)⟩and II ⟨Tk

2(II:AII)⟩ expressed in arbitrary frames (AI and AII)are given by (Figure 1),

where Dmk2(∆) transforms domain I from its alignment tensor

PAS to the arbitrary molecular frame, AI (Figure 1a). The Tk2(I:

AI) elements are obtained experimentally by determining thedomain I alignment tensor using RDCs and establishedprotocols.25,28,42 Dmk

2(R�γ) is a time-dependent Wigner rotationthat transforms the reference domain I alignment tensor fromits PAS to its instantaneous orientation relative to the domainII arbitrary frame, AII (Figure 1b). The angular brackets denotea time average over all orientations sampled at a rate faster thanthe inverse of the dipolar interaction. In the PAS T(1

2(I:PI) )0 and the 15 ⟨D0k

2(R�γ)⟩ , ⟨D2k2(R�γ)⟩ , and ⟨D-2k

2(R�γ)⟩elements are required to relate ⟨Tk

2(II:AII)⟩ to Tm2(I:PI). For an

axially symmetric domain I alignment, T(12(I:PI) ) T(2

2(I:PI)) 0, and only five elements, ⟨D0k

2(R�)⟩ , are relevant. In thegeneral case, eq 1 is underdetermined since there are fifteenunknown Wigner elements and only five experimentally deter-

Figure 1. Molecular frames and rotations used in the analytical treatment of motions for the (a) reference domain dominating alignment and (band c) moving domain under (b) one and (c) multiple reference domain I alignments.

Figure 2. Depiction of correlated, anticorrelated, and uncorrelatedmotions between domains I and III in a three-domain system.

⟨Tk2(I:AI)⟩ ) Tk

2(I:AI) ) ∑m)-2

2

Tm2(I:PI)Dmk

2(∆)

⟨Tk2(II:AII)⟩ ) ∑

m)-2

2

Tm2(I:PI)⟨Dmk

2(R�γ)⟩(1)

16816 J. Phys. Chem. B, Vol. 112, No. 51, 2008 Fisher et al.

mined ⟨Tk2(II:AII)⟩ elements. Additional measurements can be

obtained by changing the alignment of the reference domainI14 and acquiring a new set of RDCs. Changing the orientationof alignment will, however, necessarily lead to deviations fromthe original alignment PAS, requiring a new set of ⟨Dmk

2(Rl�lγl)⟩elements relating the orientation of the lth domain I alignmentPAS relative to domain II,

To effectively take advantage of the new set of RDC measure-ments, we use the closure relationship to break down⟨Dmk

2(Rl�lγl)⟩ into two rotations,

where Dmn2(θl) transforms domain I from the PAS of the lth

alignment to an arbitrary common reference frame (e.g., in thiscase the PAS of the l ) 1 alignment) and Dnk

2(R�γ) from thereference frame to the instantaneous orientation relative todomain II in its arbitrary frame, as is diagrammatically illustratedin Figure 1c. Note that any arbitrary common reference framecan be chosen and that the elements of Dmn

2(θl) will be knownfor every lth alignment based on determination of the domainI alignment tensor and that the averaging is performed over thetime-dependent matrix elements only. This yields the followingexpression for ⟨Tk

2(II:AII)⟩ l measured under the lth alignment:

Since the domain motions no longer originate from the PASfor every alignment, all 25 ⟨Dnk

2(R�γ)⟩ Wigner matrix elementsare necessary to describe the system of equations. Theseelements can be numerically solved for using eq 4 by experi-mentally determining ⟨Tk

2(II:AII)⟩ under five linearly indepen-dent reference domain I alignments. Note that with thedetermination of ⟨Dnk

2(R�γ)⟩ , one has also determined theinverse Wigner elements ⟨Dkn

2*(R�γ)⟩ and therefore the corre-sponding dynamics of the reference domain relative to the chiraldomain.

It has recently been shown43 that a subset of elements,⟨D0k

2(R�)⟩ and ⟨Dk02*(�γ)⟩ , can be determined experimentally

under the limiting case of having axially symmetric referencealignments anchored independently onto domains I and II,

Interpreting ⟨Dnk2(r�γ)⟩

The 25 ⟨Dnk2(R�γ)⟩ elements obtained from five independent

alignments provide a more complete 3D description of domainmotions when compared to the ten subelements ⟨D0l

2(R�)⟩ and

⟨Dl02*(�γ)⟩ that have been measured previously.43 A unique

feature of the ⟨Dnk2(R�γ)⟩ elements is that they contain several

elements (n and k * 0) that are simultaneously sensitive to allthree rotational degrees freedom (R�γ) and therefore anycorrelations between them. In contrast, ⟨D0l

2(R�)⟩ and ⟨Dl02*(�γ)⟩

are insensitive to one rotational degree of freedom and, evenwhen combined, remain insensitive to any correlations betweenR and γ when � does not vary. The elements of ⟨Dnk

2(R�γ)⟩are also not subject to the 180° inversion degeneracy that arisesfor ⟨D0l

2(R�)⟩ and ⟨Dl02*(�γ)⟩ , as first demonstrated by Prest-

egard and co-workers.44

The elements of ⟨Dnk2(R�γ)⟩ constitute internal order param-

eters with 3D rotational sensitivity. We define a chiral general-ized internal order parameter, SCL

2, that ranges between 0 to 1for maximum and minimum domain motional amplitudes:

SCL2 provides a frame independent measure of domain motional

amplitudes. In contrast, conventional order parameters do notsense motions around the axially symmetric interactions andtypically probe reorientation of the bonds relative to a molecule-fixed alignment or diffusion tensor.

The 25 ⟨Dnk2(R�γ)⟩ allow one to compute motions of a given

chiral domain relative to any reference alignment frameanchored on a second domain (and vice versa), even thoughexperimental measurements are made relative to only fivedistinct alignments. This is because eq 4 and the 25 ⟨Dnk

2(R�γ)⟩elements can be used to compute motionally averaged alignmenttensors for a domain (⟨Tk

2⟩) given any reference domainalignment tensor (Tm

2Dmn2(θ)). For example, one can compute

motionally averaged alignment tensors for a domain given anisotropic distribution of axially symmetric alignment tensorsanchored onto the reference domain. In this way, one can obtainfive independent parameters describing motions of a domainrelative to every direction on the second domain.

The 25 ⟨Dnk2(R�γ)⟩ elements can also be fit to a dynamical

ensemble of n distinct interdomain conformations each definedby three Euler angles Rn�nγn. Such an ensemble based approachwas first proposed by Clore and co-workers in the analysis ofbond vector orientation.16 The added spatial sensitivity affordedby determining the full Wigner matrix is self-evident here as itallows construction of ensembles defined by a larger numberof parameters. For example, assuming equal populations forvarious members of the ensemble, a maximum of eight states(25/3 rounded down to the nearest integer) can be used to fitthe twenty-five independent Wigner elements. For unequalpopulations, an n state ensemble will require 3n spatial and n- 1 weighting parameters allowing fitting to a maximum ofsix states. Of course, it is preferable to find an ensemble thathas the smallest number of independent parameters that canexplain the data. It is possible in such analyses that more thanone degenerate solution exists that can reproduce the average⟨Dnk

2(R�γ)⟩ elements to within statistical significance, particu-larly in the presence of experimental uncertainty, and it istherefore critical to vastly explore the parameter space duringthe fitting procedure. In simulations, we were able to perfectlyreproduce equally populated discrete states for n up to 8 (datanot shown) on the basis of RDCs measured under five linearlyindependent alignment conditions. No degeneracies were ob-served in our simulations, though we did not attempt to fitmodels involving different populations and did not perform a

⟨Tk2(II:AII)⟩ l ) ∑

m)-2

2

Tm2(I:PI)l⟨Dmk

2(Rl�lγl)⟩ (2)

⟨Dmk2(Rl�lγl)⟩ ) ⟨ ∑

n)-2

2

Dmn2(θl)Dnk

2(R�γ)⟩

) ∑n)-2

2

Dmn2(θl)⟨Dnk

2(R�γ)⟩ (3)

⟨Tk2(II:AII)⟩ l ) ∑

m)-2

2

∑n)-2

2

Tm2(I:PI)lDmn

2(θl)⟨Dnk2(R�γ)⟩

(4)

⟨Tk2(II:PII)⟩ ) T0

2(I:PI)⟨D0k2(R�)⟩

⟨Tk2(I:PI)⟩ ) T0

2(II:PII)⟨Dk02*(�γ)⟩(5)

SCL2 ) 1

5 ∑m)-2

2

∑n)-2

2

⟨Dmn2*(R�γ)⟩⟨ Dmn

2(R�γ)⟩ (6)

Multialignment and Multireference RDC NMR J. Phys. Chem. B, Vol. 112, No. 51, 2008 16817

brute force search algorithm to explore all local minima of thefitting landscape.

Multiple Domains

The treatment for more than two domains readily followsfrom the framework presented above. As an example, weconsider three domains (I, II, and III) with domain I as areference. The motionally averaged alignment tensors for thethree domains are given by,

where Dnk2(i f j) transforms domain i from its arbitrary frame

to its instantaneous orientation relative to the domain j. The 50⟨Dnk

2(If II)⟩ and ⟨Dnk2(If III)⟩ elements can be experimentally

determined from 5 linearly independent domain I alignments.However, no direct information is obtained regarding the relativemotions of domains II and III, as given by ⟨Dnk

2(II f III)⟩ .If we assume that the relative motions of domain I and domain

II are completely independent of the relative motions of domainIII and domain II then, by eq 3,

and ⟨Dnk2(II f III)⟩ can be determined from ⟨Dnk

2(I f II)⟩ and⟨Dnk

2(I f III)⟩ using eq 8. If the domain motions are notindependent, all 75 Wigner rotation elements are required todescribe the dynamics. The additional 25 elements can bemeasured by changing the reference domain. The system ofequations when making domain II a reference is given by,

in which ⟨Dnk2(IIf III)⟩ can be determined in the same manner

as discussed above. Thus for n domains, RDCs need to bemeasured under 5 linearly independent alignment conditions forn - 1 different reference domains in order to solve for the 25∑i ) 1

n (n - i) Wigner elements.

Correlated Domain Motion

Changing the reference domain provides the basis forresolving the correlated nature of motions between two or moredomains. To illustrate this, we consider three domains (I, II,and III). Assuming that motions of the domains are not cor-

related to one another (Figure 2), by eq 8, the chiral orderparameters describing relative motions of the domains I and III(SCL

2(I f III)) can be expressed in terms of the product ofsuccessive order parameters describing motions of domain Irelative domain II (SCL

2(I f III)) and domain II relative todomain III (SCL

2(II f III)),

If the motions of domains I and III are correlated (Figure 2),the motional amplitude of one domain from the reference frameof the other appears exaggerated as compared to the noncor-related case and

For anticorrelated motions (Figure 2) between domains I andIII, the motional amplitude of one domain from the referenceframe of the other domain appears diminished compared to thenoncorrelated case and

The degree to which the domain motions are correlated oranticorrelated can be assessed by the deviation from noncor-related behavior with a correlation parameter, Hcorr, defined as,

The value of Hcorr is a percentage deviation from noncorre-lated motion but is dependent on the motional model andtherefore should only serve as a qualitative measure ofmotional correlations. A positive value of Hcorr impliescorrelated motion, a negative value anticorrelated motion,and Hcorr approximately equal to zero implies little motionalcorrelation.

Characterization of Motion: Examples from Simulations

We preformed several simulations using Mathematicaversion 5.2 to establish the utility of ⟨Dnk

2(R�γ)⟩ in character-izing domain motions. We present results for three illustrativedomain motions (i) continuous single axis rotation of adomain by (25° about a fixed arbitrary axis, (ii) a motionalmodel comprising collective motions of two helical domainsas derived from a previously reported 65 ns moleculardynamics simulation of the HIV-1 transactivation responseelement (TAR),25,34 and (iii) (30° correlated, anticorrelated,and uncorrelated bending motions involving three domains(Figure 2).

In these simulations, five linearly independent alignmentswere generated for a reference domain using the Gram-Schmidtorthonormalization36 procedure on five arbitrarily chosenalignment tensors. For the third motional model, fiveindependent alignments were constructed independently whenhaving domain I and II as the reference domain. The relativeorientations were defined using Euler angles and unitquaternions. The latter are 4-dimensional complex numbersthat represent an orientation as a single axis rotation.32 The

⟨Tk2(I:AI)⟩ ) Tk

2(I:AI) ) ∑m)-2

2

Tm2(I:PI)Dmk

2(∆)

⟨Tk2(II:AII)⟩ ) ∑

m)-2

2

∑n)-2

2

Tm2(I:PI)Dmn

2(θ)⟨Dnk2(I f II)⟩

⟨Tk2(III:AIII)⟩ ) ∑

m)-2

2

∑n)-2

2

Tm2(I:PI)Dmn

2(θ′)⟨Dnk2(I f III)⟩

(7)

⟨Dmk2(I f III)⟩ ) ∑

n)-2

2

⟨Dmn2(I f II)⟩⟨ Dnk

2(II f III)⟩

(8)

⟨Tk2(I:PI)⟩ ) ∑

m)-2

2

∑n)-2

2

Tm2(II:PII)Dmn

2(θ)⟨Dnk2(II f I)⟩

⟨Tk2(II:AII)⟩ ) Tk

2(II:AII) ) ∑m)-2

2

Tm2(II:PII)Dmk

2(∆′)

⟨Tk2(III:AIII)⟩ ) ∑

m)-2

2

∑n)-2

2

Tm2(II:PII)Dmn

2(θ′)⟨Dnk2(II f III)⟩

(9)

SCL2(I f III) ) SCL

2(I f II)SCL2(II f III) (10)

SCL2(I f III) < SCL

2(I f II)SCL2(II f III) (11)

SCL2(I f III) > SCL

2(I f II)SCL2(II f III) (12)

Hcorr ) -SCL

2(I f III) - SCL2(I f II)SCL

2(II f III)

SCL2(I f II)SCL

2(II f III)(13)

16818 J. Phys. Chem. B, Vol. 112, No. 51, 2008 Fisher et al.

relationship between the quaternions and the Euler angles isgiven by,

For motional models i and iii, the “real” average Wignerrotation elements ⟨Dnk

2(R�γ)⟩ real were computed by averagingover 10 001 equally spaced states along the trajectory. Formotional model ii, interhelical Euler angles were calculatedfor each picosecond in the 64.998 ns simulation of thetransactivation response element (TAR) RNA33,34 as describedpreviously,35 and ⟨Dnk

2(R�γ)⟩ real were computed by averagingover all states. The motionally averaged alignment tensorelements ⟨Tk

2(II)⟩ real were determined using eqs 4, 7, and 9.The elements of ⟨Tk

2(II)⟩ real determined under each of the fivealignment conditions were then used to solve for the“experimental” ⟨Dnk

2(R�γ)⟩ exp elements by minimizing theroot-mean-square deviation between ⟨T(II)⟩ real and ⟨T(II)⟩ exp

in the Cartesian representation using the Nelder-Meadalgorithm45 implemented in Mathematica.

As shown in Figure 3a, we observe perfect agreementbetween ⟨Dnk

2(R�γ)⟩ real and ⟨Dnk2(R�γ)⟩exp for all three motional

models examined, demonstrating that all twenty-five matrixelements can be determined experimentally using RDCs mea-sured under five linearly independent alignments. In contrast,as shown in Figure 3b, substantial deviations are observed whensolving for ⟨Dnk

2(R�γ)⟩ using five alignments, only four of whichare linearly independent as the fifth alignment was constructedfrom a linear combination of the initial four.

We explored the effects of experimental uncertainty byrandomly perturbing the alignment tensors calculated from theMD trajectory and using the resulting tensors to back-predictthe Wigner matrix. The alignment tensors were perturbed by arandom rotation and scaling in a manner analogous to thatdescribed in ref 25. Three different levels of error were explored:(small) e5° uncertainty in orientation and e5% uncertainty inmagnitude, (medium) 5°-10° uncertainty in orientation and5-10% uncertainty in magnitude, and (large) 10°-15° uncer-tainty in orientation and 10-15% uncertainty in magnitude. Thelatter error ranges account for both RDC uncertainty andstructural heterogeneity in the domain structure. For RNA

A-form helices, the small range corresponds to values expectedbased on typical RDC uncertainty and documented deviationsfrom the idealized A-form structure as described previously.25

The impact of the error on the experimental Wigner elementswas assessed by computing the rmsd between the real andexperimental values over 100 iterations.

As shown in Figure 3c, the Wigner elements can be accuratelydetermined in the presence of uncertainty levels even greaterthan those typically encountered in experimental studies. Theerrors observed at “medium” levels of alignment tensor uncer-tainty were roughly twice those observed at “small” levels ofuncertainty. However, “large” errors in the alignment tensorsled to a breakdown in the ability to resolve the Wigner matrix,resulting in very large errors and leading to mathematicallyimpossible values in our unconstrained fitting procedure. Forthese cases, a more comprehensive and thorough analysis ofthe effects of uncertainty is clearly required but is beyond thescope of the current work.

We used different approaches to interpret the ⟨Dnk2(R�γ)⟩

elements for each motional model. A model-free approach wasused to analyze the rotation around and an axis model. Here,the 25 ⟨Dnk

2(R�γ)⟩ elements are used to determine alignmenttensors for the mobile domain II (⟨Tk

2(II:AII)⟩) given anyarbitrary alignment tensor on the reference domain (Tm

2(I:PI)Dmn

2(θ)). In particular, we computed the entire family of(⟨Tk

2(II:AII)⟩) elements for an isotropic distribution of axiallysymmetric alignment tensors, with principal values of -0.0005,-0.0005, and 0.001 for the x, y, and z components, on thereference domain. Each five ⟨Tk

2(II:AII)⟩ elements provide adescription of the orientational ordering of a designated referencedomain I direction relative to the chiral domain II. In analogyto the description of bond vector dynamics, the five parametersdescribe the average orientation of that direction relative to thechiral domain, the amplitude of its motion (but not twistingmotion around the long axis in case of axial symmetry) and thedirection and degree of motional asymmetry. In this manner, afive parameter description regarding orientational ordering ofevery direction on reference domain I relative to chiral domainII can be obtained.

The 25 ⟨Dnk2(R�γ)⟩ elements were used compute the family

of ⟨Tk2(II:AII)⟩ using eq 4 for an isotropic distribution of axially

symmetric alignment tensors on the reference domain I. The⟨Tk

2(II:AII)⟩ elements were then converted into the familiarCartesian form. The value of the ϑint,29 which describes the

Figure 3. Comparison of real ⟨Dnk2(R�γ)⟩ real Wigner rotation elements with experimental ⟨Dnk

2(R�γ)⟩exp values computed using eqs 4, 7, and 9based on (a) five and (b) four linearly independent alignment conditions. Results are shown for three motional models; continuous rotation of adomain by (25° about an arbitrary axis (square); collective motions of two helical domains in a 65 ns molecular dynamics simulation of TARRNA25,34(circle); and (30° anticorrelated bending motions for domains I-II (triangle up), II-III (triangle down), and I-III (diamond). (c) Effectsof RDC error and structural noise on Wigner elements determined from the MD trajectory for three levels of uncertainty in the alignment tensors,(blue error bars) e5° uncertainty in orientation and e5% uncertainty in magnitude, (green error bars) 5°-10° uncertainty in orientation and 5-10%uncertainty in magnitude, (red error bars) 10°-15° uncertainty in orientation and 10-15% uncertainty in magnitude. Error bars represent the rmsdcalculated over 100 random iterations.

qR�γ ) [cos(�2)cos(R + γ

2 ), sin(�2)sin(-R + γ

2 ),sin(�

2)cos(-R + γ2 ), cos(�

2)sin(R + γ2 ) ] (14)

Multialignment and Multireference RDC NMR J. Phys. Chem. B, Vol. 112, No. 51, 2008 16819

amplitude of motion for a given direction (but not twistingdynamics around it) ranging between 0 and 1 for maximumand minimum motions, was computed for each starting align-ment tensor. Here, each direction relative to the domain Iarbitrary frame (AI) is represented as a point on the globe. Theamplitude of motion for each direction is then represented bycoloring the ϑint value.

We observe a symmetric distribution of decreasing ϑint valuesaround a maximum value of 1 observed for a direction inclinedat 40° and 50° relative to the domain I arbitrary frame z and xaxes, respectively (Figure 4). Thus, this direction does notchange its orientation relative to the chiral domain II. Asexpected, this direction exactly corresponds to the rotation axis

of the domain motion. The ⟨Tk2(II:AII)⟩ elements obtained for

this direction yield a Tzz direction relative to domain II that isalso in perfect coincidence with the axis of motion. As expected,maximum motional amplitudes are observed for directionsorthogonal to the rotation axis (Figure 4). In this manner, thenature of the motion i.e. rotation around a single axis, can be

Figure 4. Diagram of domain motions by multialignment RDCs. Globe showing the internal generalized degree of order (ϑint) for domain II foran isotropic distribution of axially symmetric alignment tensors centered on reference domain I, for a motional model involving rotation of domainII about an arbitrary axis direction relative to reference domain I. Each point on the globe represents a specific direction relative to referencedomain I. The value of the ϑint is color-coded for each direction with ϑint increasing in value (and motions therefore decreasing in amplitude) fromred to blue for 0.699 and 1, respectively. The ϑint value provides a measure of the extent to which a given direction undergoes reorientationaldynamics relative to the chiral domain II. The smallest degree of motion is observed for a direction around which the rotational motions occur. Theaxis of rotation is denoted by *.

Figure 5. n-State ensemble fit to ⟨Dnk2(R�γ)⟩ real computed for a motional model involving collective motions of two helical domains in a 65 ns

molecular dynamics simulation of TAR RNA.25,34 (a) �2 value as a function of the number of states in the ensemble. (b) Comparison of the MDtrajectory (shown in gray) and 2 (orange circles), 4 (green circles), and 8 (blue circles) state fits using an Euler angle representation describedpreviously.43 (c) Comparison of the fitted states to 64 998 state MD trajectory. The color coding is as in part b. Each state defined as an interhelicaldomain orientation is represented using a unit quaternion; a 4-dimensional complex number that represents an orientation as a single axis rotationfrom the z-axis. The relationship between the quaternions and the Euler angles is given in eq 14. Shown are the values for the four quaternioncomponents; Q0, Q1, Q2, and Q3. Q0 encodes the rotational amplitude, and Q1, Q2, and Q3 encode the x, y, and z components of the rotation axis,respectively. Error bars are shown for the two-state fit and represent the rmsd calculated over 100 random iterations. The color coding is as inFigure 3. The order of the two states was chosen to minimize the overall rmsd compared to the zero error ensemble.

TABLE 1: Analysis of Motional Correlations in a ThreeDomain System

correlated anti-correlated uncorrelateduncorrelated

(ideal)

SCL2(I f III) 0.542 1.000 0.723 0.702

Hcorr 0.229 -0.423 -0.029 0.000

16820 J. Phys. Chem. B, Vol. 112, No. 51, 2008 Fisher et al.

inferred de novo. Note that delineating the type of motion inthis way is only feasible with a uniform distribution of referenceframes and hence requires all 25 ⟨Dnk

2(R�γ)⟩ elements. Onecould also visualize in an analogous manner the averageorientation, degree and direction of motional asymmetry forevery direction examined and employ axially asymmetric tensorsfor the reference domain I, in which case the ⟨Tk

2(II:AII)⟩elements will also be sensitive to rotational motions around eachchosen axis.

The ⟨Dnk2(R�γ)⟩ elements obtained for the motional model

derived from the MD trajectory were analyzed using an n-stateensemble method. Here, we fit the 25 ⟨Dnk

2(R�γ)⟩ elements ton equally populated states. A �2 given by, �2 ) ∑m)-2

2 ∑n)-22

[|⟨Dnk2(R�γ)⟩ real - ⟨Dnk

2(qR�γ)⟩ensemble|2/error2], where the errorwas set to 10-7 to account for numerical imprecision based onthe accuracy with which randomly generated states could beback-predicted (data not shown) and unit quaternions were usedto specify orientation, was optimized using the DifferentialEvolution37 genetic algorithm provided in Mathematica.

As shown in Figure 5a, the �2 agreement between measuredand back-predicted ⟨Dnk

2(R�γ)⟩ elements improved continuouslywhen increasing n from 1 to 8. The Euler angles describing theinterhelical orientations for the MD and best fit n-state ensembles(where R, γ, and � define twisting about domains II and I andinterhelical bending, respectively) are shown in Figure 5b forn ) 2, 4, and 8. In each case, the distinct states draw out atrajectory similar in 3D spatial distribution to that observed bythe MD.

In Figure 5c, we compare the best fit states to 64 998 statesin the MD trajectory. To aid presentation, each state and theMD trajectory are represented using unit quaternions with a timeordering. The position of each of the best fit states along thetime axis was determined as the point in time where each statemost closely intersects the MD trajectory. As can be seen inFigure 5c, all of the 2, 4, and 8 states fall along the MD motionaltrajectory sampling distinct conformations nearly over the entire65 ns time frame. This supports the validity of using discreten-state ensembles to capture salient 3D spatial features of whatmight otherwise be a highly complex continuous motionaltrajectory.43

To examine the effects of experimental uncertainty, the fitswere also preformed after noise corrupting the alignment tensorswith three levels of uncertainty. The error bars for the 2 statefit in Figure 5c represents the rmsd calculated over 100 iterationsfor each level of alignment tensor error. Due to the highercomputational cost associated with fitting models with largernumbers of states, only 10 iterations were performed for the 4and 8 state models. To illustrate the range of variation for the4 and 8 state ensembles the points from all 10 iterations wereplotted for each level of error (Figures S1 and S2 in theSupporting Information). As might be expected, the robustnessof the analysis increases with decreasing number of states andalignment tensor error. For the relatively low alignment tensoruncertainty expected in the analysis of A-form RNA helices,the analysis appears to be robust up to 8 states (Figures 5c, S1,and S2). However, larger alignment tensor uncertainty can leadto a great deal of error particularly when fitting a large number(>4) of states. The latter likely arises from a combination ofthe lower ratio of measurements to degrees of freedom anddifficulty in sampling and converging toward a single uniquecorrect solution. Thus, the inclusion of other interdomainconstraints,16 such as the simple steric restraints implementedin ref 43 will be important in carrying out such analyses usingexperimental data.

Finally, we simulated the correlated, anticorrelated, anduncorrelated (30° bending about the y-axis motion for a threedomain system. Average Wigner matrices were constructed byaveraging over 10 001 equally spaced states along motionaltrajectory for a given domain, with the matrix elementsdescribing reorientation of domain I relative to domain III as,

For the uncorrelated case, conformations were generated overthe entire trajectory and then randomly permuted. The SCL

2(If II), SCL

2(I f III), SCL2(II f III), and Hcorr were calculated

for all three motional models. Since the motion of domain Irelative to domain II is equal in amplitude and character tomotion of domain II relative to domain III, we observe SCL

2(If II) ) SCL

2(IIf III) ) 0.838 for the correlated, anticorrelatedand uncorrelated models. However, the three models can bedistinguished by their SCL

2(I f III) and Hcorr, shown in Table1. The SCL

2(I f III) for the correlated motion is significantlyless than that of the uncorrelated case, which is significantlyless than for the anticorrelated motion. As a result, the Hcorr forthe uncorrelated motion is an order of magnitude smaller thanthat of the correlated and anticorrelated motions.

Conclusions

We have presented a theoretical approach to the analysis ofdomain motions using multialignment and multireference RDCsthat is applicable to any system of rigid domains where thedecoupling approximation is not violated. Using 5 independentalignments, 25 internal order parameters can be measuredproviding 3D rotational sensitivity. The 25 Wigner matrixelements provide a means to characterize domain motions withunprecedented resolution, as demonstrated by an eight stateensemble fit to a 65 ns molecular dynamics simulation of HIV-1TAR. In addition, we have shown that changing the identity ofthe reference domain in a multidomain molecule provides insightinto correlated domain motions that is inaccessible by simplymodulating alignment.

Our results motivate the development of novel experimentalstrategies for obtaining linearly independent alignments and forestablishing reference domains in multidomain molecules. Theinclusion of other restraints in interpreting the Wigner elementswill be important for overcoming the effects of experimentaluncertainty, which will require a more thorough exploration thanpresented here. Finally, development of new theoretical methodsfor interpreting the measured internal order parameters in termsof continuous motional models may enhance the ability tocharacterize highly complex motional trajectories.

Acknowledgment. We thank members of the Al-Hashimilaboratory for insightful comments and Dr. Ioan Andricioaei(University of California, Irvine) and Aaron Frank for providingus with the TAR MD trajectory. This work was supported byfunding from the NIH (RO1 AI066975-01) to H.M.A.

Supporting Information Available: Effect of experimentaluncertainty on the 4 and 8 state fits to the MD trajectoryillustrated in Figures S1 and S2, respectiely. This material isavailable free of charge via the Internet at http://pubs.acs.org.

References and Notes

(1) Mittermaier, A.; Kay, L. E. Science 2006, 312, 224.(2) Henzler-Wildman, K.; Kern, D. Nature 2007, 450, 964.

⟨Dmk2(I f III)⟩ ) ∑

n)-2

2

⟨Dmn2(I f II)Dnk

2(II f III)⟩

(15)

Multialignment and Multireference RDC NMR J. Phys. Chem. B, Vol. 112, No. 51, 2008 16821

(3) Tolman, J. R.; Flanagan, J. M.; Kennedy, M. A.; Prestegard, J. H.Proc. Natl. Acad. Sci. U.S.A. 1995, 92, 9279.

(4) Tjandra, N.; Bax, A. Science 1997, 278, 1111.(5) Tolman, J. R.; Ruan, K. Chem. ReV. 2006, 106, 1720.(6) Bouvignies, G.; Markwick, P. R.; Blackledge, M. Chemphyschem

2007, 8, 1901.(7) Getz, M.; Sun, X.; Casiano-Negroni, A.; Zhang, Q.; Al-Hashimi,

H. M. Biopolymers 2007, 86, 384.(8) Bothner-By, A. A. Magnetic field induced alignment of molecules.

In Encyclopedia of Nuclear Magnetic Resonance; Grant, D. M., Harris,R. K., Eds.; Wiley: Chichester, 1995; pp2932.

(9) Bax, A.; Grishaev, A. Curr. Opin. Struct. Biol. 2005, 15, 563.(10) Briggman, K. B.; Tolman, J. R. J. Am. Chem. Soc. 2003, 125,

10164.(11) Ruan, K.; Briggman, K. B.; Tolman, J. R. J. Biomol. NMR. 2008,

41, 61.(12) Meiler, J.; Prompers, J. J.; Peti, W.; Griesinger, C.; Bruschweiler,

R. J. Am. Chem. Soc. 2001, 123, 6098–6107.(13) Peti, W.; Meiler, J.; Bruschweiler, R.; Griesinger, C. J. Am. Chem.

Soc. 2002, 124, 5822.(14) Ramirez, B. E.; Bax, A. J. Am. Chem. Soc. 1998, 120, 9106.(15) Bouvignies, G.; Markwick, P.; Bruschweiler, R.; Blackledge, M.

J. Am. Chem. Soc. 2006, 128, 15100.(16) Clore, G. M.; Schwieters, C. D. Biochemistry 2004, 43, 10678.(17) Bruschweiler, R.; Liao, X.; Wright, P. E. Science 1995, 268, 886.(18) Ryabov, Y. E.; Fushman, D. J. Am. Chem. Soc. 2007, 129, 3315.(19) Tolman, J. R.; Flanagan, J. M.; Kennedy, M. A.; Prestegard, J. H.

Nat. Struct. Biol. 1997, 4, 292.(20) Fischer, M. W. F.; Losonczi, J. A.; Weaver, J. L.; Prestegard, J. H.

Biochemistry 1999, 38, 9013.(21) Skrynnikov, N. R.; Goto, N. K.; Yang, D.; Choy, W. Y.; Tolman,

J. R.; Mueller, G. A.; Kay, L. E. J. Mol. Biol. 2000, 295, 1265.(22) Bertini, I.; Del Bianco, C.; Gelis, I.; Katsaros, N.; Luchinat, C.;

Parigi, G.; Peana, M.; Provenzani, A.; Zoroddu, M. A. Proc. Natl. Acad.Sci. U.S.A. 2004, 101, 6841.

(23) Ryabov, Y.; Fushman, D. Magn. Reson. Chem. 2006, 44 Spec No,S143.

(24) Tian, F.; Al-Hashimi, H. M.; Craighead, J. L.; Prestegard, J. H.J. Am. Chem. Soc. 2001, 123, 485.

(25) Musselman, C.; Pitt, S. W.; Gulati, K.; Foster, L. L.; Andricioaei,I.; Al-Hashimi, H. M. J. Biomol. NMR 2006, 36, 235.

(26) Mollova, E. T.; Hansen, M. R.; Pardi, A. J. Am. Chem. Soc. 2000,122, 11561.

(27) Saupe, A. Angew. Chem., Int. Ed. Engl. 1968, 7, 97.(28) Losonczi, J. A.; Andrec, M.; Fischer, M. W. F.; Prestegard, J. H.

J. Magn. Reson. 1999, 138, 334.(29) Tolman, J. R.; Al-Hashimi, H. M.; Kay, L. E.; Prestegard, J. H.

J. Am. Chem. Soc. 2001, 123, 1416.(30) Mueller, G. A.; Choy, W. Y.; Skrynnikov, N. R.; Kay, L. E.

J. Biomol. NMR 2000, 18, 183.(31) Bertini, I.; Gupta, Y. K.; Luchinat, C.; Parigi, G.; Peana, M.; Sgheri,

L.; Yuan, J. J. Am. Chem. Soc. 2007, 129, 12786.(32) Blumich, B.; Spiess, H. W. J. Magn. Reson. 1985, 61, 356.(33) Musselman, C.; Al-Hashimi, H. M.; Andricioaei, I. Biophys. J. 2007,

93, 411.(34) Dethoff, E. A.; Hansen, A. L.; Musselman, C.; Watt, E. D.;

Andricioaei, I.; Al-Hashimi, H. M. Biophys. J., in press.(35) Bailor, M. H.; Musselman, C.; Hansen, A. L.; Gulati, K.; Patel,

D. J.; Al-Hashimi, H. M. Nat. Protoc. 2007, 2, 1536.(36) Bretscher, O. Linear Algebra with Applications, 3rd ed.; Pearson

Prentiss Hall: New York, 2005; pp 202.(37) Storn, R.; Price, K. J. Glob. Optim. 1997, 11, 341.(38) VanGurp, M. Colloid Polym. Sci. 1995, 273, 607.(39) Wohnert, J.; Franz, K. J.; Nitz, M.; Imperiali, B.; Schwalbe, H.

J. Am. Chem. Soc. 2003, 125, 13338.(40) Rodriguez-Castaneda, F.; Haberz, P.; Leonov, A.; Griesinger, C.

Magn. Reson. Chem. 2006, 44 Spec No., S10.(41) Zhang, Q.; Sun, X.; Watt, E. D.; Al-Hashimi, H. M. Science 2006,

311, 653.(42) Zweckstetter, M.; Bax, A. J. Am. Chem. Soc. 2000, 122, 3791.(43) Zhang, Q.; Stelzer, A. C.; Fisher, C. K.; Al-Hashimi, H. M. Nature

2007, 450, 1263.(44) Al-Hashimi, H. M.; Valafar, H.; Terrell, M.; Zartler, E. R.; Eidsness,

M. K.; Prestegard, J. H. J. Magn. Reson. 2000, 143, 402.(45) Nedler, J. A.; Mead, R. Comput. J. 1965, 7, 303.

JP806188J

16822 J. Phys. Chem. B, Vol. 112, No. 51, 2008 Fisher et al.