Internal motions in proteins and interference effects in nuclear magnetic resonance

Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324 www.elsevier.com/locate/pnmrs

Internal motions in proteins and interference effects in nuclear magnetic resonance Dominique Frueh* ICMB, Ecole Polytechnique Fe´de´rale de Lausanne, BCH, 1015 Lausanne, Switzerland Accepted 5 September 2002

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Cross-correlated relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The master equation of relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Relaxation of observable quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Relaxation due to dipolar interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Relaxation due to chemical shift anisotropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Molecular motions and relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Overall tumbling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Internal motions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Slow internal motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Relaxation due to chemical shift modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Other influences of exchange phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

305 306 307 309 310 310 312 312 313 320 320 322 323 324

Keywords: Protein dynamics; NMR relaxation; Cross-correlation effects; Chemical shift modulation

1. Introduction The characterization of the structures and dynamics of biomolecules plays a central role in the study of biological systems. The determination of * Address: Department of Biological Chemistry and Molecular Pharmacology, 240 Longwood Avenue, Harvard Medical School, Boston, MA 02115, USA. Tel.: þ 1-617-432-3211; fax: þ 1-617432-4383. E-mail address: [email protected] (D. Frueh).

dynamic properties can provide information on the activity of the molecule, and on the change of activity upon binding with other molecules. Clearly the comparison between the flexibilities of active and non-active complexes provides information on the active site of the relevant complex [1]. Similarly changes of the dynamics of a complex with respect to a non-complexed molecule can provide evidence of the existence of such complexes and give insight into the localisation of the binding sites. An example is the binding of ‘structural’ water, which was shown to

0079-6565/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 7 9 - 6 5 6 5 ( 0 2 ) 0 0 0 5 1 - 1

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increase the flexibility of proteins [2]. It was also suggested that flexible regions can act as a gate such that some conformations have lower activation energies than others [1,3]. In nuclear magnetic resonance (NMR), spin relaxation provides information on motions occurring in molecules. The time-scales of dynamic events can be separated between internal motions that occur faster and those that are slower than the molecular tumbling. Fast dynamics have mainly been studied through relaxation measurements of various nuclei, the most popular being the application of the Lipari – Szabo ‘model free analysis’ to 15N autorelaxation [4]. The information is then principally expressed by a socalled order parameter that reflects fast motional averaging. However, in large biological molecules, the dynamics of the system is complicated, so that one needs multiple probes to obtain a clear picture of the dynamic processes. This is often achieved by measuring not only the relaxation due to single mechanisms (e.g. dipole– dipole interactions), but also the interference between two relaxation mechanisms, a phenomenon called cross-correlated relaxation. Cross-correlated relaxation measurements have thus progressively been introduced to study fast internal motions [5 –11]. Since the earliest relaxation studies, a number of different models have been proposed to correlate the relaxation rates with molecular motional processes [12 – 15]. Here, we will make use of the one-dimensional Gaussian amplitude fluctuation model (1D-GAF) [16], where each molecular fragment experiences a rotation of a given amplitude around a single axis. The influence of both the amplitude of the internal motion and the orientation of the axis of rotation with respect to the interactions involved in a particular interference effect will be discussed in detail. Slow processes have also been shown to correlate with biological activity [17]. In NMR, they can be probed through transverse relaxation measurements. They are traditionally studied by comparing longitudinal and transverse relaxation rates [4,18], by spin-locking experiments [19,20], or by Carr – Purcell – Meiboom – Gill (CPMG) techniques [19, 21 –23]. All of these methods rely on extracting the effect of exchange from the autocorrelation rates of the nuclei studied and can only indicate the presence and the location of these slow internal motions in

the molecule. In small molecules the motions can be modelled based on the knowledge of the system. In large biological systems, such as proteins, the residues are likely to be subject to complicated changes of conformations, so that the information provided by autocorrelation measurements alone is insufficient to enable a clear characterization of these processes. We will describe here how correlated modulations of the chemical shifts of nuclei involved in multiplequantum coherences can lead to cross-correlation effects. The measured cross-correlation rates are not only sensitive to the presence of slow motions, but also give information on the extent of correlation between the modulations of the chemical shifts of the two nuclei. There are a number of comprehensive reviews of NMR spin relaxation in the literature [15,24 –31]. This work will focus on a specific aspect, namely the influence of both fast and slow internal motions on cross-correlated relaxation.

2. Cross-correlated relaxation Relaxation is the process that restores a spin system to its equilibrium. In NMR of liquids, relaxation is mediated by random processes that interact with the spin system. These are typically due to molecular reorientation, internal motions or chemical exchange involving the nuclei of interest.1 A first consequence of motion is that anisotropic spin –spin interactions are averaged out to first order and that the fine structure of the spectrum only shows the effect of isotropic interactions. The time dependence of anisotropic interactions does however contribute to relaxation, and is often the dominant effect. The return to equilibrium can be described from a macroscopic point of view by relaxation rates, which account for the decay of the detected signal as a function of time.

1

In this article, chemical exchange specifically denotes phenomena where the breaking and making of bonds occur, excluding therefore conformational exchanges, which are examples of slow internal motions.

D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

always close to s~ð0Þ; one has:

2.1. The master equation of relaxation We consider a spin system with a Hamiltonian consisting of a time-independent part, H0, describing evolution under coherent processes, and a stochastic perturbation Hamiltonian H1 ðtÞ describing the spin – lattice coupling: HðtÞ ¼ H0 þ H1 ðtÞ:

ð1Þ

The evolution of the density operator is given by: d sðtÞ ¼ 2i½H0 þ H1 ðtÞ; sðtÞ: dt

ð2Þ

The effect of the random Hamiltonian can be isolated by expressing Eq. (2) in the interaction frame. The influence of H0 is accounted for explicitly by transforming each operator Q into: ~ ¼ eiH0 t Q e2iH0 t : Q

d s~ðtÞ ¼ 2i½H~ 1 ðtÞ; s~ðtÞ: dt

ð4Þ

We can then calculate s~ðtÞ from its value at t ¼ 0 by integrating Eq. (4) iteratively: ðt 0



ðt0 0

ð1 d s~ðtÞ ¼ 2 dt½H~ 1 ðtÞ; ½H~ 1 ðt 2 tÞ; s~ðtÞ: dt 0

½H~ 1 ðt0 Þ; s~ð0Þdt0 2

ðt

dt 0

H1 ðtÞ ¼

k X X

Aqk Fkq ðtÞ ¼

X

þ

p

Aqk Fkq ðtÞ

ð8Þ

q

k q¼2k

where the Aqk represent spin operators of order q and of rank k, and the Fkq ðtÞ are random functions of the time. þ The second equality, p where Aqk is the Hermitian conjugate of Aqk and Fkq ðtÞ is the complex conjugate of Fkq ðtÞ; is a consequence of the hermiticity of H1 ðtÞ: The operators Aqk can be expanded in terms of basis operators Aqk;p X Aqk ¼ Aqk;p ; ð9Þ p

0

~ 1 ðt0 Þ; ½H ~ 1 ðt00 Þ; s~ð0Þ þ · · · dt00 ½H

ð7Þ

The validity of these assumptions has been discussed elsewhere [32,33]. We will just remind the reader that ~ 1 ðt 2 tÞ do not commute, the time t has if H~ 1 ðtÞ and H to be chosen within the limits tc p t p T; where (1/T ) is the relative rate of change of s~ðtÞ; and tc is the correlation time of the random process, i.e. the average time for a random event to occur. We may then replace s~ðtÞ by s~ðtÞ: To have a better characterization of the nature of the relaxation mechanisms, one may express the spin – lattice coupling Hamiltonian in terms of irreducible tensors:

ð3Þ

It is easy to show that the evolution of the density operator in the interaction frame simply becomes:

s~ðtÞ ¼ s~ð0Þ 2 i

307

ð5Þ

where higher order terms can be neglected since small correlation times are associated with the relaxation processes [32]. Averaging over all molecules in the sample, and taking the time derivatives leads to: ðt d s~ðtÞ ¼ 2i½H~ 1 ðtÞ; s~ð0Þ 2 dt0 ½H~ 1 ðtÞ; ½H~ 1 ðt0 Þ; s~ð0Þ: dt 0 ð6Þ We will further assume that H~ 1 ðtÞ and s~ð0Þ can be averaged independently. Since H~ 1 ðtÞ is stochastic with zero mean, the first term on the right hand side of Eq. (6) vanishes. The time variable can be substituted by t ¼ t 2 t0 : If t in the upper bound of the integral in Eq. (6) can be replaced by infinity, and if s~ðtÞ is

with the property ½H 0 ; Aqk;p  ¼ vqp Aqk;p

ð10Þ

where the subscript p stands for operators of the same order q that are associated with different frequencies vqp : Examples of the operators Aqk;p and their related frequencies are given in Sections 2.3 and 2.4. The spin operators in the interaction representation are then: X q A~ q ¼ eiH0 t Aq e2iH0 t ¼ eivp t Aq : ð11Þ k

k

k;p

p

For the Hermitian conjugate: X q þ þ þ A~ qk ¼ eiH0 t Aqk e2iH0 t ¼ e2ivp t Aqk;p

ð12Þ

p q due to the property v2q p ¼ 2vp : Using Eqs. (11) and (12), and within the limits of the conditions previously

308

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mentioned, Eq. (7) can be rewritten as: 0 X ð1 0 d iðvq þvq Þt s~ðtÞ ¼ 2 dt e p p0 ½Aqk0 ;p0 ;½Aqk;p ; s~ðtÞ dt k;k0 p;q;p0 ;q0 0 q

0

 Fkq0 ðtÞFkq ðt 2 tÞ e2ivp t :

ð13Þ

We will retain only the interactions between operators of same ranks and opposite orders. This is justified because the spatial random functions Fkq ðtÞ are proportional to spherical harmonics that have the property: Ykq ð0ÞY 0k0 q0 ðtÞ ¼ dk;k0 dq;2q0 ð21Þq Yk0 ð0ÞY 0k0 ðtÞ: Furthermore we will make use of the secular approximation, i.e. neglect contributions of rapidly oscillating terms,0 and thus only retain those that fulfil the condition vqp0 ¼ 2vqp : In other words we can insert a Kronecker delta dp;p0 : Eq. (13) thus simplifies to: X 2q q d s~ðtÞ ¼ 2 ½Ak;p ;½Ak;p ; s~ðtÞ dt k;p;q ð1 q  dtFkq ðtÞFkq ðt 2 tÞ e2ivp t : ð14Þ 0

the spectral densities: ð1 Cm;n ðtÞe2ivt dt; jm;n ðvÞ ¼ 0

Jm;n ðvÞ ¼

ð1 21

¼2

ð1 0

km;n ðvÞ ¼

ð1 0

Cm;n ðtÞe2ivt dt ð17Þ Cm;n ðtÞcosðvtÞdt;

Cm;n ðtÞsinðvtÞdt:

Hence, Eq. (15) becomes: X d q q s~ðtÞ ¼ 2 ½A2q k;pm ;½Ak;pn ; s~ðtÞjm;n ðvp Þ dt k;p;q;m;n ¼2

1 X ½A2q ;½Aq ; s~ðtÞJm;n ðvqp Þ 2 k;p;q;m;n k;pm k;pn

þi

X

q q ½A2q k;pm ;½Ak;pn ; s~ðtÞkm;n ðvp Þ:

ð18Þ

p;q;m;n

At this point, Eq. (14) does not reveal possible differences in the nature of the interactions described by the operators Aqk;p and A2q k;p ; as occurs in crosscorrelation effects involving two mechanisms m and n. To make allowance for such cases, additional subscripts have to be added: X d q s~ðtÞ ¼ 2 ½A2q k;pm ;½Ak;pn ; s~ðtÞ dt k;p;q;m;n ð1 q  dtFkpm ðtÞFkpn ðt 2 tÞ e2ivp t : ð15Þ

The imaginary term in the second equality in Eq. (18) results in a small frequency shift that can be accounted for by redefining the unperturbed Hamiltonian H0 [32, 34]. The evolution of the density operator in the interaction frame is thus given by: d 1 X s~ðtÞ ¼ 2 ½A2q ;½Aq ; s~ðtÞJm;n ðvqp Þ: ð19Þ dt 2 k;p;q;m;n k;pm k;pn

Note that these operators are subject to the same conditions as before, i.e. they must have the same rank, opposite orders and the same frequency. We now introduce the definition of correlation functions:

s ¼ e2iH0 t s~ eiH0 t :

0

Cm;n ðtÞ ¼ Fkqm ðtÞFkqn ðt 2 tÞ:

ð16Þ

One speaks of autocorrelation functions when m ¼ n; and of cross-correlation functions when m – n (for convenience, we will from now on only keep the subscript k in the spin part of the interaction Hamiltonians). The characterization of these functions and their relation with molecular motions will be the subject of Section 3. The following Fourier transforms of the correlation functions lead to

The density operator in the laboratory frame is obtained from its counterpart in the interaction frame by using the reciprocal form of Eq. (3): ð20Þ

The relaxation of the density operator in the laboratory frame is thus simply obtained by means of Eqs. (19) and (20) and by using the property for unitary operators U U½A;BU þ ¼ ½UAU þ ;UBU þ 

ð21Þ

which leads to: d 1 X sðtÞ ¼ 2i½H0 ; sðtÞ 2 ½A2q ;½Aq ; sðtÞ dt 2 k;p;q;m;n k;pm k;pn  Jm;n ðvqp Þ:

ð22Þ

Eq. (22) however does not correctly describe the

D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

return to equilibrium of longitudinal magnetization (it would predict equal populations), and one has to replace sðtÞ by sðtÞ 2 seq where seq is equal to

seq ¼

e2H0 =kB T ; Tr½e2H0 =kB T 

ð23Þ

reformulate Eq. (28) in terms of a complete set of orthogonal basis operators. The density operator is then given by [36]

sðtÞ ¼

d sðtÞ ¼ 2 i½H0 ; sðtÞ dt 1 X ½A2q ;½Aq ; sðtÞ 2 seq  2 2 k;p;q;m;n k;pm k;pn  Jm;n ðvqp Þ:

bi ðtÞBi

ð29Þ

ð24Þ

where the dimension of the Liouville space is N ¼ 4n for an n spin system, and where bi ðtÞ ¼ kBi lsðtÞl: Eq. (28) can then be rewritten in matrix form X d br ðtÞ ¼ { 2 iVrs bs ðtÞ 2 Rrs ½bs ðtÞ 2 beq ð30Þ s } dt s where the characteristic frequency Vrs is given by [37]

Vrs ¼ 2.2. Relaxation of observable quantities Observable quantities can be derived from the density operator by calculating their expectation values. The latter are given by the well-known relationship kQlðtÞ ¼ Tr{QsðtÞ}

kBr l½H0 ; Bs l kBr lBr l

ð26Þ

By using the property Tr{A½B; C} ¼ Tr{½A; BC}

ð27Þ

together with the result of Eq. (24), one obtains [35]: d kQlðtÞ dt 1 2

X

Jm;n ðvqp Þ

k;p;q;m;n q q A lðtÞ 2 k½½Q; A2q {k½½Q; A2q k;pm k;pn k;pm Ak;pn leq }:

ð28Þ The double commutators in Eq. (28) yield operators that might be different from Q. In this case, one will speak of cross-relaxation processes. When on the other hand an operator Q is conserved under a double commutator, one speaks of autorelaxation mechanisms. To best describe these effects, it is useful to

ð31Þ

and where the relaxation matrix element Rrs that expresses the interconversion between the operators Bs and Br is given by: 1 Rrs ¼ 2

ð25Þ

and their time evolution is given by:   d d kQlðtÞ ¼ Tr Q sðtÞ : dt dt



N X i¼1

which leads to the final form of the master equation in operator form:

¼ k 2 i½Q; H0 lðtÞ 2

309

X

Jm;n ðvqp Þ

k;p;q;m;n

q kBr l½A2q k;pm ½Ak;pn ; Bs l

kBr lBr l

:

ð32Þ

When r ¼ s; Rrs is an autorelaxation rate, and when r – s it is a cross-relaxation rate. These indices are not to be mistaken with autocorrelation ðm ¼ nÞ and cross-correlation ðm – nÞ: Cross-relaxation denotes the interconversion between different operators, regardless of the nature of the relaxation mechanism. On the other hand, cross-correlation effects indicate the interference between mechanisms of different nature (e.g. between two different dipolar interactions (DD/DD), between a dipole –dipole and the anisotropy of a chemical shift (DD/CSA), etc.). It is convenient to write Eq. (32) as X Rrs ¼ Rrs;m;n ; ð33Þ m;n

which underlines the fact that a number of interference effects might contribute to the same relaxation rate. The auto- and cross-correlated rates Rrs;m;n can be expressed by X ai Jm;n ðvi Þ; ð34Þ Rrs;m;n ¼ krs;m;n i

where krs;m;n are constants that depend on the nature of the interactions involved in the relaxation process

310

D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

(krs;m;n can have the values dImSm dInSn ; dImSm cIn or cIm cIn ; where dIS and cI are given by Eqs. (39) and (46) of Sections 2.3 and 2.4). The constants ai reflect the weighting of contributions due to mechanisms associated with frequencies vi, which correspond to the frequencies vqp of Eq. (32). 2.3. Relaxation due to dipolar interactions We will now consider relaxation due to dipole – dipole (DD) interactions modulated by random molecular motions. The energy of a magnetic dipole ~ S is m~I in a magnetic field B ~S E ¼ 2m~I ·B

ð35Þ

~ S is the field produced by a second magnetic where B dipole, m~S : ~ S ðtÞ ¼ m~S 2 3 ðm~S ·~rIS ðtÞÞ~rIS ðtÞ : B 3 5 rIS ðtÞ ðtÞ rIS

ð36Þ

~rIS ðtÞ is a unit internuclear vector describing the orientation of the dipolar interaction. Its time dependence arises from molecular motions, and ~rIS ðtÞ thus represents the stochastic function. rIS ðtÞ is the internuclear distance. In the current work, we will only consider cases where the time dependence of rIS ðtÞ is solely due to vibrations. It has been shown [38] that given the time scale of such vibrations, they do not contribute to relaxation and the internuclear distance can safely be replaced by an effective distance rIS (for clarity no additional subscript will be added here). The reader should however remember that in the case of long-range dipole – dipole interactions, such an approximation is often not valid (e.g. a dipolar interaction between an amide proton HN and the proton on an alpha carbon Ha in a protein). The Hamiltonian for a dipolar interaction can be written as [29] ! ~ rIS ðtÞÞ m0 gI gS ð~I·~rIS ðtÞÞðS·~ DD ~ ~ HIS ðtÞ ¼ " 3 3 2 I·S ð37Þ 2 4p rIS rIS where I and S denote the operators of spin angular momentum, gI and gS are the gyromagnetic ratios of the corresponding nuclei, " ¼ h=2p and m0 is the permittivity constant in vacuum. By transforming ~rIS ðtÞ into polar coordinates and by introducing irreducible tensors [39] the Hamiltonian can be

written as [32,37] DD ðtÞ ¼ dIS HIS

2 X X

Y~ 2q ðuIS ðtÞ; wIS ðtÞÞAq2pIS

ð38Þ

q¼22 p

wIS Þ are modified spherical harmonics, where Y~ 2q ðuIS ; p ffiffiffiffiffiffi Y~ 2q ðuIS ; wIS Þ ¼ 4p=5Y2q ðuIS ; wIS Þ and where the Aq2pIS are related to the second rank irreducible spherical tensors T2q : The dipole –dipole interaction constant dIS is: pffiffi m g g ð39Þ dIS ¼ 6 0 " I 3 S : 4p rIS Since molecular motions in isotropic liquids average out dipolar couplings, the relevant part of the static Hamiltonian H0 is H0 ¼ vI Iz þ vS Sz ;

ð40Þ

which enables the calculation of the frequencies associated with the operators of the interaction Hamiltonian by using Eq. (20). Table 1 gives the values of the operators Aq2pIS and the corresponding frequencies [37]. 2.4. Relaxation due to chemical shift anisotropy The magnetic environment of a spin is seldom isotropic. As a consequence, its chemical shift must be Table 1 Tensors operators for a dipole–dipole interaction. The associated frequencies entering the corresponding spectral densities are also given q

p

Aq2pIS

A2q 2pIS

v^q p

0

0

2 pffiffi Iz Sz 6

2 pffiffi Iz Sz 6

0

0

1

1 2 pffiffi Iþ S2 2 6

1 2 pffiffi I2 Sþ 2 6

^ðvI 2 vS Þ

1

0

1 2 Iþ Sz 2

1 I S 2 2 z

^vI

1

1

1 2 Iz Sþ 2

1 IS 2 z 2

^vS

2

0

1 I S 2 þ þ

1 I S 2 2 2

^ðvI þ vS Þ

D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

represented by a tensor, V. The Hamiltonian for nuclear shielding in the principal axis system of the chemical shift tensor is [29]: ~ 0 ·V·~I: HICS ¼ gI B

the Hamiltonian for the CSA is: HICSA ðtÞ ¼ cI

2 X X

Y~ 2q ðuk ðtÞ; wk ðtÞÞAq2pI

ð45Þ

q¼22 p

ð41Þ

The shielding tensor is a rank 2 tensor that can therefore be expressed as the sum of a symmetric rank 2 tensor, an antisymmetric rank 1 tensor, and a scalar tensor of rank zero:

V ¼ Vð0Þ þ Vð1Þ þ Vð2Þ :

311

ð42Þ

where the same modified spherical harmonics as for the dipolar interaction have been used. The polar angles here denote the orientation of the unique axis of symmetry of the CSA tensor. The CSA interaction constant is: qffiffi ð46Þ cI ¼ 23 DsgI B0 : The relevant part of the static Hamiltonian is given by:

We will not consider here the contribution of the antisymmetric component. The scalar contribution, called the isotropic chemical shift and denoted by siso, is combined with the Zeeman Hamiltonian: H ¼ gI ð1 2 siso ÞB0 Iz :

ð43Þ

Since this term corresponds to the rank 0 component of the chemical shift interaction, it is not modulated by molecular tumbling and will thus not contribute to relaxation by this means. However, as will be presented in Section 4, the modulation of this term by slow conformational fluctuations can produce a contribution to relaxation. The anisotropic rank-2 component, called the chemical shift anisotropy tensor (CSA tensor) can be rewritten in its principal axis system, where it has a diagonal form 3 2 1 ð1 þ 2 h Þ 0 0 7 6 2 7 6 7 2 6 ð2Þ 7 1 ð44Þ VPAS ¼ Ds6 6 0 2 ð1 2 hÞ 0 7 3 6 7 2 5 4 0 0 1 where Ds ¼ szz 2 1=2ðsxx þ syy Þ and h ¼ 3=2ððsyy 2 sxx Þ=DsÞ: The components sxx, syy and szz are called the principal values of the CSA tensor. In the case of axial symmetry sxx ¼ syy ¼ s’ and szz ¼ sk : Therefore h ¼ 0 and Ds ¼ s’ 2 sk : It has been shown [40] that an anisotropic CSA tensor can be rewritten as the sum of two axially symmetric tensors. We can hence reduce the analysis to relaxation due to an axially symmetric CSA tensor. From Eqs. (41) and (44), and changing the frame to the laboratory frame,

H0 ¼ vI Iz :

ð47Þ

Since the field is parallel to the z-axis of the laboratory frame, the operators of order 2 vanish. The remaining operators along with their frequencies are given in Table 2. We would like to attract the reader’s attention to the choice of the parameterisation of the interaction tensors and their associated constants. Depending on whether some factors are included in the spectral densities, or if a factor is accounted for by the coefficients of the tensors rather by the interaction constants, the expressions of these quantities may vary. In fact the literature is rather chaotic and caution should be taken when comparisons between different articles are undertaken. In particular when the rates are expressed as linear combinations of spectral densities, the coefficients are obviously different when a factor (e.g., the fraction 2/5) is included in the expression of the spectral density or not. The choice we made is compatible with the textbook of Cavanagh et al. [37] and the same expressions of the spectral densities are used in Section 3. Table 2 Tensor operators for the chemical shift anisotropy interaction. The frequencies of the corresponding spectral densities are also given q

p

Aq2pI

A2q 2pI

v^q p

0

0

2 pffiffi Iz 6

2 pffiffi Iz 6

0

1

0

1 2 Iþ 2

1 I 2 2

^vI

312

D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

3. Molecular motions and relaxation As previously shown, relaxation rates are proportional to spectral densities, the Fourier transforms of correlation functions. These account for the extent of correlation between the fluctuating spatial functions. These time fluctuations arise from molecular motions. To relate relaxation and dynamics, one must first calculate the correlation function for a given motional processes, and then calculate the resulting spectral density. In this section we will establish what types of molecular motions influence relaxation and how they can be modelled. As will be shown, these include rotational diffusion of the molecule and internal motions. We will focus on the special case of globular proteins, which will allow some simplifications. More general treatments can be found in recent reviews [15,29]. Internal motions occurring at a slower time scale than the tumbling of the molecule, the so-called conformational exchange phenomena, will not be treated in this section. Their contribution to relaxation is of a different nature and will be treated in Section 4.

3.1. Overall tumbling For anisotropic interactions, such as DD and CSA relaxation mechanisms, the expression of the correlation function can be rewritten in terms of rank 2 normalized spherical harmonics 4p p LF q Cmn kY ðV ð0ÞÞY2q ðVLF ðtÞ ¼ n ðtÞÞl 5 2q m

ð48Þ

where VLF n ðtÞ denotes the polar angles u and f of the relevant interaction vector (e.g. a vector subtended by the two atoms of a given dipole – dipole interaction, or the principal component of a CSA tensor) in the laboratory frame (note that the time dependence is now represented by t instead of t as it was in Eq. (16)). The fluctuations of these angles are due to both internal motions and overall tumbling of the molecule. Fig. 1a shows a vectorial representation of this situation where each interaction i is associated with a vector m~i : These two contributions can be separated by using

Fig. 1. Separation of overall tumbling and internal motions. (a) The orientation of the vector m~m is described with respect to the laboratory frame (LF). (b) The same orientation is described with respect to a molecular fixed frame (MF), while the orientation of this frame with respect to the laboratory frame is represented by the three Euler angles VLM :

Wigner matrix elements [39,41] X 2p LM p p ðVLF Dqk ðVm ð0ÞÞY2k ðVMF Y2q m ð0ÞÞ ¼ m ð0ÞÞ; k

Y2q ðVLF n ðtÞÞ ¼

X

ð49Þ MF D2qk0 ðVLM n ðtÞÞY2k0 ðVn ðtÞÞ

k0

where V LM represent the three Euler angles a, b and g relating the molecular coordinate frame to

D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

the laboratory frame. V LM(t ) thus accounts for the rotational diffusion of the molecule. V MF denotes the two polar angles of the interaction vector m~i in the molecular frame, and its time dependence therefore reflects internal motions (Fig. 1b). If overall and local motions are independent, Eq. (48) can be rewritten as: 4p X 2p LM p kD ðV ð0ÞÞY2k ðVMF m ð0ÞÞ 5 k;k0 qk m

q Cmn ðtÞ ¼

MF  D2qk0 ðVLM n ðtÞÞY2k0 ðVn ðtÞÞl 4p X 2p LM kD ðV ð0ÞÞD2qk0 ðVLM ø n ðtÞÞl 5 k;k0 qk m p MF  kY2k ðVMF m ð0ÞÞY2k0 ðVn ðtÞÞl

ð50Þ

where one can now estimate the ensemble average of each contribution independently. The first average in Eq. (50) denotes motional averaging due to rotational diffusion of the whole molecule. It can be calculated by LM 2 LM kD2p qk ðVm ð0ÞÞDqk0 ðVn ðtÞÞl

¼

ðð

LM LM LM LM dVLM m0 dVn PðVm0 ÞPðVn ;tlVm0 ;0Þ

LM 2 LM  D2p qk ðVm ð0ÞÞDqk0 ðVn ðtÞÞ

ð51Þ

where PðVm0 Þ is the probability density for the initial orientation Vm0 ; and PðVn ;tlVm0 ;0Þ is the conditional probability for having the vector m~n with the orientation Vn at time t if the vector m~m had the initial orientation Vm0 at time t ¼ 0: The probability density can be calculated by means of Green functions (see for instance Refs. [29,42]). We will focus on the case of a spherical molecule tumbling isotropically, in the diffusion limit. For this case, Eq. (51) simplifies to [29,43] LD 2 LD 1 26Dt kD2p ¼ dk;k0 15 e2t=tc qk ðVm ð0ÞÞDqk0 ðVn ðtÞÞl ¼ dk;k0 5 e

ð52Þ where tc ¼ 1=6D is the rotational correlation time of the spherically symmetric molecule with an isotropic rotational diffusion coefficient D. The correlation function of Eq. (50) can now be re-parameterised as

313

follows 1 4p X p MF kY ðV ð0ÞÞY2k ðVMF Cmn ðtÞ ¼ e2t=tc n ðtÞÞl 5 5 k 2k m ¼ CO ðtÞCI;mn ðtÞ

!

ð53Þ

where CO ðtÞ denotes the overall correlation function and CI ðtÞ the internal correlation function. This parameterisation is rigorous for a spherical molecule, assuming only that internal motions and overall tumbling are independent. In the absence of internal MF motions, i.e. when Y2k ðVMF n ðtÞÞ ¼ Y2k ðVn ð0ÞÞ; and by making use of the spherical harmonics addition theorem, Eq. (53) becomes Cmn ðtÞ ¼ 15 e2t=tc P2 ðcos um;n Þ

ð54Þ

where um;n denotes the angle between the two (rigid) vectors m~m and m~n : P2 ðcos um;n Þ is the second order Legendre polynomial. The spectral density is, according to Eq. (17): Jmn ðvqp Þ ¼ P2 ðcos um;n Þ

2 tc : 5 1þðvqp tc Þ2

ð55Þ

For autocorrelated relaxation, um;n ¼ 0 and P2 ðcos  um;n Þ ¼ 1; and the relaxation rate only depends on the correlation time. For cross-correlated processes, the rate however depends on the angle between the vectors. In a rigid molecule, the measurement of cross-correlation rates thus provides structural information on the molecule. 3.2. Internal motions To address internal motions, one could describe the averaging in a similar way as has been done for overall tumbling in Eq. (51) and then model the motions in order to find an appropriate probability function. We will take a slightly different approach here. We will first make use of the Lipari –Szabo model free method [4] and then use a model to see the influence of motions on the parameters of this method. This has the advantage that one can interpret a posteriori results obtained by the very popular model free technique. In the Lipari – Szabo approach the correlation function is described by two parameters. A so-called order parameter, denoted by the symbol S 2, is defined

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as the limit toward infinity of the internal correlation function: S2m;n ¼ CI;mn ð1Þ ¼ lim

t!1

4p X p MF kY ðV ð0ÞÞY2k ðVMF n ðtÞÞl: 5 k 2k m

ð56Þ

The order parameter reflects spatial restrictions of the internal motions. One should note that the superscript 2 was introduced for autocorrelated mechanisms. It has been maintained for cross-correlation, but an order parameter for cross-correlated mechanisms can be negative. In the absence of internal motions, S2;rigid ¼ m;n P2 ðcos um;n Þ: An effective correlation time for internal motions, te, is defined such that the internal correlation function is exact at t ¼ 0 and t ¼ 1 CI;mn ðtÞ ¼ S2m;n þ ðCI;mn ð0Þ 2 S2m;n Þe2t=te ð57Þ

where the second equality arises from the spherical harmonic addition theorem. The total correlation function is then obtained from Eqs. (53) and (57) Cmn ðtÞ ¼ ¼

þ þ

ðP2 ðcos um;n Þ 2 S2m;n Þe2t=te Þ 2 2t=t 1 5 ðP2 ðcos um;n Þ 2 Sm;n Þe ð58Þ

21 in which t is defined such that t21 ¼ t21 c þ te the sum of the overall and internal rates. Taking the Fourier transform of Eq. (58) according to Eq. (17) leads to the spectral density: " # S2m;n tc ðP2 ðcos um;n Þ 2 S2m;n Þt 2 Jmn ðvÞ ¼ : þ 5 1 þ ðvtc Þ2 1 þ ðvtÞ2

ð59Þ In the absence of internal motions, S2m;n ¼ ¼ P2 ðcos um;n Þ S2;rigid m;n

In this work we will limit the discussion to cases where one can apply this approximation, which is sometimes called the fast internal motion limit. In this case, internal motions are only reflected by their spatial averages, which are described by the order parameters S2m;n : From Eq. (56), it can be rewritten as: 4p X p MF kY2k ðVm ð0ÞÞY2k ðVMF S2m;n ¼ lim n ðtÞÞl t!1 5 k ¼

¼ S2m;n þ ðP2 ðcos um;n Þ 2 S2m;n Þe2t=te

1 2t=tc 2 ðSm;n 5e 2t=tc 1 2 5 Sm;n e

detect them through the modulation of the isotropic interactions involving the nuclei (see Section 4). On the other hand, when te is negligible, Eq. (59) simplifies to: " # 2 S2m;n tc Jmn ðvÞ ¼ : ð61Þ 5 1 þ ðvtc Þ2

S2;rigid m;n

with ð60Þ

and Eq. (59) reduces to Eq. (55). We underline here that for slow internal motions te @ tc ; and hence t @ tc : The contribution of the order parameter vanishes from Eq. (59) and one obtains again Eq. (55). In other words, relaxation measurements are insensitive to motions on a time scale that is much slower than the overall tumbling of the molecule. In fact one can only

4p X p MF kY ðV ðtÞÞlkY2k ðVMF n ðtÞÞl: 5 k 2k m

ð62Þ

One often finds in the literature expressions that relate cross-correlation order parameters to autocorrelation parameters: S2m;n ¼ P2 ðcos um;n ÞS2m;m or S2m;n ¼ P2 ðcos um;n ÞS2n;n :

ð63Þ

While these equations are useful in estimating the magnitude of a relaxation rate, we underline that they are only valid in the case of isotropic internal motions or when the angle um;n is close to zero [7], so that the two vectors m~m and m~n are subject to the same motional averaging. To estimate the order parameter of Eq. (62) we have to find the probability distribution PðVÞ in order to evaluate the mean values of the spherical harmonics in a similar fashion as in Eq. (51). To gain insight on how motions lead to spatial averaging, we will make use of the 1D-GAF model [16] and expand it to the case of cross-correlation. We consider the influence of a Gaussian fluctuation about a single axis. The probability density distribution of the angle of rotation g is assumed to be Gaussian with variance s2g : 2 2 1 PðgÞdg ¼ qffiffiffiffiffiffiffi e2g =ð2sg Þ dg: 2 2psg

ð64Þ

Let us treat the case of two vectors m~m and m~n that are rigidly related to each other, and that both experience

D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

the same motion of amplitude sg (Fig. 2). This corresponds for example to a good approximation to cross-correlations occurring in the peptide plane, such as the interference between the CSA of the carbonyl C0 carbon and the N – HN [9,11] dipole– dipole interaction, or between the CSA of the C0 carbon and the C0 – HN dipolar interaction [9,10,44]. For convenience, we choose the Z-axis of the molecular frame to coincide with the axis of the motion. The polar angles VMF ¼ ðu; wÞ are then such that um and un are the time independent angles between each vector and the axis of rotation. The time dependence of the motion is described by the fluctuations of the angles wm ðtÞ ¼ wm ð0Þ þ gðtÞ and wn ðtÞ ¼ wn ð0Þ þ gðtÞ; with Pðwn Þ ¼ Pðwm; Þ ¼ PðgÞ: Introduction of the probability distribution given by Eq. (64) in Eq. (62) leads to kY2k ðVMF n Þl ¼

ð1 21

ð1

¼ Q2k ðun ÞF2k ðwn ð0ÞÞe

dg eikg PðgÞ

21 2ð1=2Þk2 s2g

parameter for a one dimensional GAF model applied to an interference involving two rigidly attached interactions S2m;n ¼ P2 ðcos um ÞP2 ðcos un Þ h 2 þ 3 sin um sin un cos um cos un cosðwm 2 wn Þe2sg i 2 þ 14 sin um sin un cos 2ðwm 2 wn Þe24sg ð66Þ which can be rewritten as a function of the rigid angle relating the two vectors um;n ; by using trigonometric relations: S2m;n ¼ P2 ðcos um;n Þ

h 2 3 sin um sin un cos um cos un cosðwm 2 wn Þ 2

 ð1 2 e2sg Þ þ 14 sin um sin un cos 2ðwm 2 wn Þ i 2 ð67Þ  ð1 2 e24sg Þ :

MF MF dVMF n Y2k ðVn ÞPðVn Þ

¼ Q2k ðun ÞF2k ðwn ð0ÞÞ

315

ð65Þ

where Q2k ðun Þ and F2k ðwn Þ are the polar and azimuthal parts of the spherical harmonics. This leads by using Eq. (62) to the expression of the order

Fig. 2. Interaction vectors subject to the same motion. Both vectors experience a rotation around the z-axis by an angle gðtÞ: The vectors can represent a dipole–dipole interaction, or one of the axes of a chemical shift anisotropy tensor.

For autocorrelated mechanisms the second term on the right hand side of Eq. (67) is always negative and the order parameter monotonically decreases with increasing amplitudes of the internal motions, in agreement with the general property 0 , S2m;m , 1: Internal motions in a molecule are thus manifested by a decrease of the measured order parameters for autocorrelated mechanisms. For cross-correlated relaxation on the other hand, the situation critically depends on the orientations of the interaction vectors m~m and m~n relative to each other and relative to the axis of rotation. The effect of a single fluctuation can be described by five parameters: the angle between the two vectors, the three Euler angles depicting the orientation of the axis of rotation with respect to the vectors, and the amplitude of the motion. In a molecule, many internal motional processes may occur simultaneously, e.g. correlated or anticorrelated rotations around dihedral angles, collective motion of a region of the molecule, multiple fluctuations of dihedral angles (e.g. in side-chains of a protein). Thus, several axes of rotations and amplitudes have to be considered simultaneously. This can be achieved by extending the previously described model to a 3D-GAF model, as has been proposed by Bru¨schweiler et al. [45]. Such an analysis will not be carried out in the current work. We will

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merely look at the effect that a single motion has on a measured order parameter. Let us first consider a general case. One of the vectors involved in the cross-correlation is maintained at a fixed position and the order parameter is calculated for different orientations of the second vector when both of them are experiencing a motion around the same axis. Fig. 3 shows the variation of a cross-correlation order parameter S2m;n as a function of the orientation of the second vector. The first vector m~m has fixed coordinates um ¼ 308 and wm ¼ 08: The polar coordinates of the second vector are varied between 0 and 1808. The meshed surface represents the order parameter for two rigid vectors ðsg ¼ 0Þ; and thus depicts P2 ðcos um;n Þ as a function of the polar coordinates. The filled surface describes the order parameter in the presence of a motion of amplitude sg ¼ 608 (i.e. a fluctuation of ^ 308). The comparison of the two surfaces shows that the existence of internal motional averaging not only results in a damping of the order parameter but also modifies its dependence upon the vectors orientations. As a consequence the measured rates are less sensitive toward the angle um;n ; and an uncertainty in the measurement of the rate leads to a higher error in the determination of this angle. Moreover, neglecting the effects of internal motions leads to erroneous

Fig. 3. Simulation of the order parameter S2m;n by using Eq. (67), for um ¼ 308 and wm ¼ 08: The coordinates of the second vector un and wn are varied between 0 and 1808. The surface is periodic with periods of 1808 for the polar angle and 3608 for the azimuthal angle. It is furthermore symmetric with respect to w ¼ 1808:

results. This is best shown in Fig. 4 that represents the distribution of order parameters as a function of um;n : The solid curve stands for the order parameter of a rigid molecule and the dashed and dotted lines for the one calculated for sg ¼ 608: The dashed lines represent different polar angles and the dotted lines different azimuthal angles of the second vector. The first vector still has the polar coordinates ðum ; wm Þ ¼ ð308; 08Þ: This figure underscores the necessity to account for motional averaging in the interpretation of the relaxation rates. For instance, a measured order parameter of 0.5 would lead to a calculated angle um;n of ca. 358 if one neglects the existence of internal motion. If a motion with sg ¼ 608 is present and if for example um ¼ 308; then in reality the angle um;n is about 288. The figure shows that the deviation can be up to ca. 208! Fig. 3 shows the difficulty of predicting the effect of motional averaging on cross-correlation rates. To discuss the effect of such processes, we will look at special cases. In a first example, let us consider the case of two vectors related to each other by a given angle um,n and which are subject to rotations around an axis equidistant to both vectors, and let us compare the influence of an internal motion as a function of the orientation of the axis of this motion. In order to be

Fig. 4. Simulation of the order parameter S2m;n by using Eq. (67) as in Fig. 3 (ðum ; wm Þ ¼ ð308; 08Þ; sg ¼ 608), but represented as a function of the angle um;n between the two interaction vectors. The value of the order parameter depends on the azimuthal angle of the second vector (dotted contours, outer contour for wn ¼ 08) and on its polar angle (dashed line, from 0 to 908 starting from above).

D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

allowed to apply Eq. (65), one has to change the coordinate frame such that the Z-axis is collinear with the axis of rotation. The order parameter can then be computed by Eq. (62) with X ð2Þ 0 kY2k ðVMF kDik ða1 ; a2 ; a3 ÞY2i ðVMF n ðtÞÞl ¼ n ðtÞÞl k;i

ð68Þ 0

where kY2k ðVMF n ðtÞÞl is the spherical harmonic in the new frame, where the axis of rotation is collinear with the Z-axis. We start with two vectors in the 0YZ plane that have the axis of rotation as a bisector. The axis is then rotated by an angle a2 around 0Y ða1 ¼ a3 ¼ 0Þ: Fig. 5 shows the result for an angle um;n ¼ 1098; as is found in a tetrahedral environment. The solid lines represent the order parameters calculated as a function of the angle a2 at which the axis is rotated. Each line corresponds to a different

317

amplitude of fluctuation sg. We clearly can see extrema of the order parameters S2m;n when a2 ¼ 0 or 1808, and when a2 ¼ 908: These correspond to situations where the axis of rotation is the bisector of the two vectors, and, respectively, where it is normal to the plane subtended by the two vectors. One should note in particular that for sg ¼ 10 – 408; and when the axis of rotation is close to the bisector, the order parameters show minima with magnitudes that exceed P2 ðcos um;n Þ (which has a negative sign). This again underscores the difference between crosscorrelated and autocorrelated mechanisms, since in the latter case the order parameter never exceeds S2;rigid m;m ¼ 1 that is the value in the absence of internal motion. To help understanding the variations of the order parameter one can describe it in terms of covariances of spherical harmonics [16]. The covariance s2fg of two stochastic functions f and g is related to their cross-correlation function CðtÞ by

s2fg ¼ Cð0Þ 2 Cð1Þ ¼ kf p gl 2 kf p lkgl

ð69Þ

hence, p MF MF s2Y p ðVMF MF ¼ kY2k ðVm ð0ÞÞY2k ðVn ð0ÞÞl m ÞY2k ðVn Þ 2k

p MF ðVMF 2 kY2k m ðtÞÞlkY2k ðVn ðtÞÞl

ð70Þ and the order parameter can be calculated by using Eqs. (62) and (70): 4p X 2 S2m;n ¼ P2 ðcos um;n Þ 2 s p MF ð71Þ MF : 5 k Y2k ðVm ÞY2k ðVn Þ

Fig. 5. Simulation of the order parameter S2m;n for an axis of rotation equidistant to both vectors. The horizontal axis represents the angle a2 by which the axis is tilted with respect to the case where it is collinear with the bisector of the two vectors. S2m;n has been calculated by means of Eqs. (62) and (68) for a1 ¼ 08; a3 ¼ 08; 08 , a2 , 1808 and 08 , sg , 1808: The last two parameters were incremented by steps of 108. Also shown are the positions of the vectors in the frame where the rotation occurs around the z-axis.

The covariance of two functions can be negative when they are anticorrelated, which explains why S2m;n does not necessarily decrease in the presence of internal motions, in contrast to the case of autocorrelated relaxation. The first term on the right-hand side of Eq. (71) describes the order parameter in the absence of internal motions, S2;rigid (Eq. (60)), and thus represents m;n the extent of correlation between the two vectors only under molecular reorientation. The second term on the right-hand side expresses internal motional averaging. For convenience, we define the contribution of internal motional averaging as 4p X 2 s2m;n ¼ s p MF ð72Þ MF 5 k Y2k ðVm ÞY2k ðVn Þ

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which can also be expressed by using Eq. (71) as:

s2m;n ¼ S2;rigid 2 S2m;n : m;n Since, by using Eq. (34), a cross-correlation rate can be written as:2 X Rrs;m;n ¼ krs;m;n ai Jm;n ðvi Þ; i

or by using Eq. (61) as X 2 tc Rrs;m;n ¼ krs;m;n S2m;n ai ; 5 1 þ ðvi tc Þ2 i

ð73Þ

the contribution of motional averaging can be rewritten by using Eqs. (71) –(73) as:

s2m;n ¼ P2 ðcos um;n Þ 2 krs;m;n

X i

Rrs;m;n : 2 tc ai 5 1 þ ðvi tc Þ2

ð74Þ

We now go back to the characterisation of s2m;n in the context of the 1D-GAF model. The covariances in Eqs. (71) and (72) can be regrouped according to their orders. Since Y20 does not depend on w, its covariance vanishes and Eq. (72) can be rewritten as

s2m;n ¼

4p 2 ðs1 þ s22 Þ 5

ð75Þ

where: 2 s21 ¼ s2Y p ðVMF MF þ s p Y m ÞY21 ðVn Þ 21

221

2 s22 ¼ s2Y p ðVMF MF þ s p Y m ÞY22 ðVn Þ

MF MF 222 ðVm ÞY222 ðVn Þ

22

ð76aÞ

MF ðVMF m ÞY221 ðVn Þ

:

ð76bÞ

The sums of covariances s21 and s22 can be calculated by using the 1D-GAF probability density given in Eq. (64):

s21 ¼ 3 sin um sin un cos um cos un cosðwm 2 wn Þ  ð1 2 e

s22 ¼

3 4

2s2g

Þ

ð77aÞ 2

sin2 um sin2 un cos 2ðwm 2 wn Þð1 2 e24sg Þ: ð77bÞ

2

When one of the interactions is a CSA tensor, it is assumed that it is expressed as the sum of two axially symmetric tensors. The rate Rm;n then expresses the interference between one of the components of the CSA tensor and the other interaction.

Fig. 6. Opposites of the covariances s21 (a) and s22 (b). The covariances were calculated by means of Eqs. (68), (70), (76a) and (76b) for a1 ¼ 08; a3 ¼ 08; 08 , a2 , 1808 and 08 , sg , 1808: The last two parameters were varied in steps of 108. Also shown are the positions of the vectors in the frame where the rotation occurs around the z-axis.

We can thus reinterpret the results of Fig. 5 in terms of these covariances. Fig. 6 shows the contributions of 2s21 and 2s22 to the order parameter calculated as described for Fig. 5, i.e. when the axis of rotation lies in the bisecting plane of two vectors that subtend an angle um;n ¼ 1098: The curves in Fig. 5 are thus obtained by summing the curves of Fig. 6a and b (scaled by a factor 4p/5) to P2 ðcos um;n Þ: We now understand the origins of the extrema previously mentioned: The maximum at

D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

a2 ¼ 908 arises only from s22 since at this orientation u ¼ 908; where 2s22 has a maximum and where 2s21 vanishes. For internal motions of low amplitudes, the minimum at b ¼ 908 also comes from the contribution of s22 : At this position, wm 2 wn ¼ 1808 which implies that 2s22 has a negative minimum, while 2s21 has a positive maximum. When the amplitude of the motion increases, s21 acquires a higher weight compared to s22 and the order parameter becomes positive at this orientation. In a general case, s22 dominates for small sg and the relative contribution of s21 increases with sg. This is a simple consequence of the factors in Eqs. (77a) and (77b). As a second example, let us now compare vectors that lie in the same plane as the axis of rotation, with different orientations with respect to the axis but subtending a fixed angle um,n. Once again we start by assuming that the axis of rotation is the bisector of the two vectors m~m and m~n ; and we then apply a rotation of an angle a2 around the 0X axis (a1 ¼ 908; a3 ¼ 08 in Eq. (68)). Fig. 7 shows the result for um;n ¼ 1098: The contributions of 2s21 and 2s22 are shown in Fig. 7b and c. Besides the previously mentioned extrema at a2 ¼ 0 and 1808, that correspond again to rotating axes that are collinear with the bisector of the two vectors, one sees another maximum at a2 ¼ 908: This corresponds to an axis that lies in the plane of the two vectors and that is perpendicular to their bisector. Then un ¼ 1808 2 um and 2s21 has the same value as for a2 ¼ 0: For um;n ¼ 1098 however, 2s22 is much smaller for a2 ¼ 908 than for a2 ¼ 08 and consequently the corresponding maximum of S2m;n is larger. An important observation is that when a2 ¼ um;n =2; the values of the order parameters are equal to the rigid case. Thus, internal motional averaging is ineffective when the axis of the motion is collinear with one of the interaction vectors. This is easily understood, since the vector that is collinear with the axis does not experience any internal motion. As a consequence there is no correlation between the motions of the vectors and the covariance vanishes. One also notes the presence of minima and roots that depend on the amplitude of the motion. These minima result from the fact that the minima of 2s21 do not match with extrema of 2s22 : The minimum values of S2m;n for different orientations of the rotation axis therefore occur for different amplitudes of internal motions.

319

Fig. 7. Simulations of the order parameter S2m;n (a) and of the covariances s21 (b) and s22 (c) for an axis of rotation coplanar with the two vectors. These parameters were calculated by means of Eqs. (62), (68), (70), (76a) and (76b) for a1 ¼ 908; a3 ¼ 08; 08 , a2 , 1808 and 08 , sg , 1808: The last two parameters were incremented by steps of 108. Also shown are the positions of the vectors in the frame where the rotation occurs around the z-axis.

Indeed, the nulls of motional averaging (when S2m;n ¼ P2 ðcos um;n Þ) occur for s21 ¼ 2s22 ; a condition that clearly depends on the amplitude of the motion, except for the special case where s21 ¼ 2s22 ¼ 0 as previously discussed. To summarize this section: † For a molecule that is tumbling isotropically, the correlation function of a given pair of interactions can be factorised into a contribution from overall tumbling and a contribution from internal motion.

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† Internal motions occurring on a time scale that is much shorter than overall tumbling can be accounted for by order parameters. † For cross-correlated relaxation, the order parameter S2m;n does not have to be positive and can increase through the effects of internal motion beyond its value for a rigid body. Internal motional averaging can invert the sign of the order parameter. † Motional averaging leads to extrema of S2m;n when the rotation axis coincides with the bisector of the two vectors m~m and m~n ; when it is perpendicular to this bisector while lying in the same plane as the two vectors, or when the rotation axis is normal to both vectors. † Other extrema of S2m;n might occur depending on the angle subtended between the vectors m~m and m~n and depending on the amplitude of the internal motion. † Motional averaging is ineffective when the axis of the internal motion is collinear with one of the two vectors.

4. Slow internal motions 4.1. Relaxation due to chemical shift modulation

HðtÞ ¼ gI ð1 2 siso ðtÞÞB0 Iz ¼ vI ðtÞIz

ð78Þ

This Hamiltonian describes the influence of the magnetic environment of a nucleus on its precession frequency. If this environment changes with time, due to changes of conformation or chemical exchange,

ð79Þ

which can be rewritten as: HðtÞ ¼ H0CS þ HICSM ðtÞ;

HICSM ðtÞ ¼ dvI ðtÞIz ð80Þ

where dvI ðtÞ represents the time-dependent deviation from the average frequency vI ¼ kvI ðtÞl; so that kdvI ðtÞl ¼ 0: Note that we only consider the limit of ‘fast exchange’, i.e. where only one average signal is observed for each nucleus. The superscript CSM stands for chemical shift modulation, which denotes the effect of slow motions or exchange phenomena on the spin system. HICSM ðtÞ is a random Hamiltonian, like H1 ðtÞ defined in Eq. (1) and thus may give rise to a relaxation mechanism. The random function is dvI ðtÞ; and the spin operator of the Hamiltonian is Iz : Its eigenfrequency is v ¼ 0 since: ½vI Iz ; Iz  ¼ 0:

ð81Þ

Since this interaction has a different rank than dipolar and CSA interactions, no cross-correlation may occur involving these mechanisms. The contributions of CSM to the relaxation of a coherence expressed in terms of basis operators as in Eq. (32) is hence RCSM=CSM ¼ rs

As shown in Section 3.2, internal motions on a time-scale slower than the overall tumbling of the molecule are too slow to bring about a motional averaging of relaxation processes. They do however influence the relaxation behaviour of coherences through different means. In particular, isotropic interactions that are not affected by the tumbling of the molecule should be sensitive to such processes. In this section, we will therefore show how modulation of the isotropic shifts of nuclei can lead to relaxation. We have seen in Section 2.4 that the isotropic chemical shift denoted by siso determines the Zeeman Hamiltonian: H0CS ¼ gI ð1 2 siso ÞB0 Iz ¼ vI Iz :

the Hamiltonian becomes time dependent [20]

1 X CSM=CSM kBr l½Izm ½Izn ;Bs l J ð0Þ 2 m;n m;n kBr lBr l

ð82Þ

where the indices m and n allow for the description of cross-correlation effects between the modulations of the chemical shifts of different nuclei. The spectral density associated with the chemical shift modulation expresses the extent of correlation between the modulations of the two chemical shifts ðþ1 CSM=CSM Jm;n ð0Þ ¼ dvm ðtÞdvn ðt 2 tÞdt ð83Þ 21

where the limits of the integral could be extended to infinity, because we restrict the analysis to slow events that occur faster than the range of isotropic shifts that result from the motional process. Note that when m – n; the spectral density can be negative if the modulations of the two nuclei are anticorrelated. The magnitude of the rate of Eq. (82) depends on the extent of correlation between the two modulations on the one hand, and on the amplitudes of the modulations of the chemical shifts of the nuclei Im

D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

321

and In on the other hand. When these amplitudes can clearly be defined, this fact may be emphasized by expressing the spectral density as CSM=CSM Jm;n ð0Þ ¼ Dvm Dvn

ðþ1 21

Cm;n ðtÞdt

ð84Þ

where Dvm and Dvn represent the mean amplitudes of the modulations and Cm;n ðtÞ is the correlation function of the modulations. Clearly CSM does not affect the relaxation of longitudinal magnetization, since the double commutator in Eq. (82) vanishes if Bs ¼ Iz : For the same reason CSM/CSM cross-correlation effects cannot affect single-quantum coherences. However, multiple-quantum coherences involving two spins Im and In might be affected by CSM interference effects. Cross-correlated chemical shift modulation gives rise to different relaxation rates for zero- and doublequantum coherences RCSM=CSM ðZQCÞ Im =In ¼

1 CSM=CSM kIþm I2n l½Izm ½Izn ; Iþm I2n l J ð0Þ 2 m;n kIþm I2n lIþm I2n l

1 CSM=CSM ¼ 2 Jm;n ð0Þ 2

ð85Þ

and likewise for the complex conjugate I2m Iþn ; while ðDQCÞ RCSM=CSM Im =In ¼

1 CSM=CSM kIþm Iþn l½Izm ½Izn ; Iþm Iþn l J ð0Þ 2 m;n kIþm Iþn lIþm Iþn l

1 CSM=CSM ¼ þ Jm;n ð0Þ 2

ð86Þ

and likewise for the complex conjugate I2m I2n : Fig. 8 shows the effect of chemical shift modulations on the spectra of single-, zero- and doublequantum coherences involving two nuclei. The chemical shifts are allowed to fluctuate randomly between given boundaries during the course of evolution. Dashed lines correspond to spectra in the absence of chemical shift modulation, whereas solid lines depict the effect of such a modulation. The spectra of single-quantum coherences exhibit an increased line broadening due to CSM, as predicted by Eq. (82) for autocorrelation. When the fluctuations

Fig. 8. Simulations of chemical shift modulations. The chemical shifts of each of the two nuclei Im and In experience fluctuations vI ðtÞ ¼ vI þ dvðtÞ; with dvðtÞ ¼ DvrðtÞ where rðtÞ is a random number taken between 21/2 and þ1/2 and where Dv ¼ 2pDn is the amplitude of the modulation. The average precession frequencies are nIm ¼ 250 Hz and nIn ¼ 150 Hz; and the magnitudes of the fluctuations are DnIm ¼ 5 Hz and DnIn ¼ 7 Hz: The columns show from left to right the SQC spectra of the two nuclei, the ZQC spectra and the DQC spectra. (a) Snapshots of the chemical shifts variations. The dashed lines represent the signals in the absence of modulations with frequencies vI ; whereas the solid lines display the signals with a frequency vI ðtÞ that varies randomly during the evolution (from top to bottom). (b) Line broadening due to CSM when the two fluctuations are correlated. The ZQC spectrum displays almost no CSM line broadening, since the cross-correlated CSM balances the autocorrelated CSM and since the two amplitudes DvIm and DvIn are similar. (c) Line broadening due to CSM when the two fluctuations are uncorrelated. The line-widths of the ZQC and DQC spectra are now equal since the cross-correlated contribution vanishes.

are fully correlated for the two nuclei, the linebroadening effect is increased for the double-quantum coherence and decreased for the zero-quantum coherence. Note that if in addition the amplitudes of

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the CSM were equal, no CSM line-broadening effect would be observed for the ZQC spectrum, as can be predicted from Eqs. (84) and (85). When the fluctuations are not correlated, the relaxation rates of ZQC and DQC only contain autocorrelated contributions of the CSM and the spectra thus display the same line broadening. The fact that correlations between exchange phenomena influence the relaxation of a MQC has indeed been mentioned very often [46 –48]. However, no detailed study of the interference effects had been pursued until recently [49,50]. Slow internal motions are often modelled in terms of an instantaneous exchange between different sites. We report here the case of an exchange between two sites A and B with populations pA and pB ; which is often encountered in the literature [47,49,51]. Note that a perfect correlation between the interchange between the sites for all nuclei involved follows from the two-site exchange model. The relaxation due to autocorrelated CSM is then given by: 2 RCSM=CSM ¼ pA pB tex ðDvm Im =Im AB Þ

4.2. Other influences of exchange phenomena Chemical shift modulation is not the only manifestation of exchange phenomena. If one considers for instance exchange between different sites i, the dipolar and CSA interactions might be different in these sites. In this case, Wennerstro¨m proposed to rewrite the Hamiltonian HðtÞ of Eq. (1) as [51]: X fi ðtÞHi ðtÞ ð90Þ HðtÞ ¼ i

where instantaneous jumps between the different sites are assumed, i.e. fi ðtÞ ¼ 1 if the nuclei are in the site i and fi ðtÞ ¼ 0 otherwise. These functions are related to the populations of each sites by: fi ðtÞ ¼ pi : The discussion below closely follows the description of Wennerstro¨m, albeit taking into account possible cross-correlation effects. The Hamiltonian of Eq. (90) can be expressed as the sum of a timeindependent Hamiltonian and two time-dependent Hamiltonians: X X HðtÞ ¼ pi v i I z þ ðfi ðtÞ 2 pi Þvi Iz

ð87Þ

i

X X

þ Dvm AB

vmA

vmB

where ¼ 2 is the change in precession frequency of the nucleus Im when changing from site B to site A, tex ¼ 1=ðk1 þ k21 Þ is the exchange lifetime, while k1 and k21 are the forward and backward exchange rates. The cross-correlated CSM relaxation rates for ZQC and DQC of Eqs. (85) and (86) become: n RCSM=CSM ðZQCÞ ¼ 22pA pB tex Dvm Im =In AB DvAB

ð88Þ

n RCSM=CSM ðDQCÞ ¼ þ2pA pB tex Dvm Im =In AB DvAB :

ð89Þ

These relations have often been used to model the effect of exchange phenomena. In this special case, it is easy to predict whether the cross-term would be positive or negative, since the sign only depends on the relative signs of the changes of the chemical shifts n Dvm AB and DvAB : Clearly in a protein, such two-site processes are seldom encountered. A combination of various slow motions is more likely to occur, involving multiple sites or diffusive processes. In the absence of other information, it is wiser to make use only of the more general Eq. (83).

p;q;m

i

fi ðtÞFiq ðtÞAqpm

ð91Þ

i

P where i pi vi ¼ vI represents the mean (observed) frequency of nucleus I. The evolution of the density matrix is given by [51]: d 1X sðtÞ ¼ 2i½H0 ; sðtÞ 2 ½I ; ½I ; sðtÞ 2 seq  dt 2 m;n zm zn ð1 X X n ðfim ðtÞ 2 pi ÞDvm ðfj ðt 2 tÞ 2 pj ÞDvnj dt i 21 i

j

1 X 2 ½A2q ; ½Aqpn ; sðtÞ 2 seq  2 p;q;m;n;i;j pm ð1 q fi ðtÞfj ðt 2 tÞFiqm ðtÞFjqn ðt 2 tÞe2ivp t dt

ð92Þ

21

where the definition Dvi ¼ vI 2 vi has been used. The first term is the usual evolution under coherent processes, and the second term corresponds to the effect of CSM described in Section 4.1. This term can be identified with the previous description by the relationship: X dvI ðtÞ ¼ ðfi ðtÞ 2 pi ÞDvi ð93Þ i

The third term represents the effect of exchange on

D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

relaxation rates due to anisotropic processes. Since the jumps between different sites i and j occur on a timescale that is much slower than the changes in the spatial functions F, the functions f can be rewritten: fim ðtÞ ¼ fim ðt 2 tÞ ¼ fjn ðt 2 tÞ

ð94Þ

Moreover, since f can only be equal to zero or one, ðfi ðtÞÞ2 ¼ fi ðtÞ ¼ pi : If one assumes that the functions Fiqm ðtÞ and Fjqn ðt 2 tÞ of the two interactions (e.g. two dipoles rigidly related to each other) are affected simultaneously by the jumps between the sites i, one can drop the subscript j and relaxation due to anisotropic interactions in the presence of exchange is: X d 1 X 2q q sðtÞ¼ 2 ½Apm ;½Apn ; sðtÞ2 seq  pi Jm;ni ðvqp Þ dt 2 p;q;m;n i ð95Þ Thus, the apparent relaxation rate is related to the relaxation rates in each site i by: X Rapparent ¼ pi Ri : ð96Þ i

We will finish this section by briefly mentioning the effects of chemical exchange. In a protein, these typically occur between solvent protons and the labile amide and hydroxyl protons. They lead to a loss of coherences involving the labile proton: N 2Nx HN z !ð12 xðtÞÞ2Nx Hz

ð97Þ

where xðtÞ is the fraction of protons that have exchanged with water at time t. In this example, exchange increases the relaxation rate of antiphase coherences but does not affect the decay of in-phase coherences.

5. Concluding remarks We have seen how various motional processes affect spin – spin relaxation in NMR and how these processes can be related to measured cross-correlation rates. For fast internal motions, we showed that anisotropic motional averaging can increase or decrease the magnitude of a rate, or that it can have no effect on the rate. By using a simple model, we also saw which parameters influence internal motional averaging. To enable a clearer discussion, we made

323

use of a number of simplifications. A full analysis of internal motions should also account for the exact time-scale of all motions involved. For some crosscorrelation rates, various interactions leading to interference effects may each be sensitive to different motions. One should then consider the extent of correlation between motional events. Complications will also arise for a molecule tumbling anisotropically as the effects of internal and overall motional averaging cannot easily be separated. For instance, the global effect of motional averaging will then depend on the orientation of the anisotropic internal motion with respect to the axes of the diffusion tensor. Clearly the complexity of dynamic processes at these time scales requires a combination of several measurements to enable their accurate description. Similarly, the characterisation of slow internal motions requires different sources of information. We showed how cross-correlated chemical shift modulation could provide a way to improve the understanding of slow internal motions. The measurement of the associated rates might be used when other techniques such as CPMG or spin-locking experiments are not applicable. The interference effect provides additional information, since it also reflects the extent of correlation between the fluctuations of the magnetic environments of the two nuclei involved. As mentioned elsewhere [50], correlated modulations of chemical shifts might occur through various phenomena. Fluctuations of dihedral angles could lead to both correlated or anticorrelated chemical shift modulations. Slow collective motions of protein domains, or dynamic intramolecular interactions, such as the motion of an aromatic side-chain of a protein or the existence of transient hydrogen bonds, could also lead to this cross-correlation mechanism. It could also provide information on the presence of long-lived bound solvent molecules. This mechanism has indeed recently been studied in different biomolecules such as RNA [52,53] and proteins [49,50,54].

Acknowledgements The author would like to thank Prof. Geoffrey Bodenhausen, Dr J.R. Tolman, Prof. S. Grzesiek and Dr J. Boyd for carefully reading the manuscript and

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D. Frueh / Progress in Nuclear Magnetic Resonance Spectroscopy 41 (2002) 305–324

for stimulating discussions. This work was supported by the Swiss National Science Foundation.

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