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Anisotropic dynamics in molecularsystemsstudied by NMR relaxation

Doctoral Thesis

Author(s):Lienin, Stephan Frank

Publication date:1998

Permanent link:https://doi.org/10.3929/ethz-a-001990510

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Diss. ETH «*• 3Diss. ETH No. 12871

Anisotropic Dynamics in Molecular Systems

Studied by NMR Relaxation

Dissertation

for the degree of

Doctor of Natural Sciences

submitted to the

Eidgenossische Technische Hochschule

Zurich

presented by

Stephan Frank Lienin

Dipl. Chem. ETH

born March 23, 1970

citizen of Germany

accepted on the recommendation of

Prof. Dr. Richard R. Ernst, examiner

Prof. Dr. Rafael Bruschweiler, co-examiner

Zurich, 1998

Fur meine Eltern

Contents 1

Contents

Contents 1

Symbols and Abbreviations 5

Abstract 11

Zusammenfassung 13

1 Introduction 15

2 Spin Relaxation and Molecular Motion 21

2 1 Fundamentals of spin relaxation 21

211 Equation of motion 21

2 1 2 Semiclassical relaxation theory 22

2 1 3 Relaxation-active interactions 24

2 14 Operator representation of the relaxation superoperator 27

2 2 Relaxation-active molecular motion 28

2 2 1 Correlation functions for molecular motion 28

222 Overall rotational diffusion 30

2 2 2 1 Isotropic rotational diffusion in a homogeneous

environment 30

2 2 2 2 Rotational diffusion in a heterogeneous environment 34

2 2 3 Intramolecular motion 35

2 2 3 1 Intramolecular backbone motion in proteins 35

2 2 3 2 Internal correlation functions and model-free approach 38

2 2 3 3 Extraction of amsotropy of intramolecular motion 41

2 2 3 4 Analytical treatment of the 3D GAF model 45

2 3 Calculation of relaxation-rate constants 47

2 4 Molecular dynamics simulation 53

24 1 The force field 53

2 4 2 Calculation of correlation functions 54

2 4 3 Extraction of 3D GAF fluctuation amplitudes 54

2 5 Motion-induced fluctuations of the CSA 57

2 5 1 Analysis of CSA trajectory 58

25 2 CSA averaging due to 3D GAF motion 60

3 Anisotropic Intramolecular Backbone Dynamics of Ubiquitin 63

3 1 Introduction 63

3 2 Ubiquitin 64

3 3 Experimental 67

3 3 1 Sample preparation 67

2 Contents

3 3 2 NMR experiments 68

3 3 2 1 General setup and techniques 68

3 3 2 2 2D N relaxation experiments 68

3 3 2 3 2D C relaxation experiments 72

3 3 3 Extraction of relaxation data 73


3 4 1 Generation of molecular dynamics trajectory 75

3 4 2 Processing of trajectory 75

3 4 2 1 Determination of equilibrium peptide plane 75

3 4 2 2 Extraction of fluctuation amplitudes from the trajectory 76

3 4 3 Analysis of MD trajectory 77

3 4 3 1 Selection of peptide planes with 3D GAF motion 77

3 4 3 2 Results for the extracted fluctuation amplitudes 80

3 4 3 3 Dihedral angles and plananty of the peptide plane 81

3 4 3 4 Orientation of the peptide plane frame 82

3 5 Analysis of experimental relaxation data 83

3 5 1 Raw data analysis 83

3 5 2 Spin relaxation mechanisms in the peptide plane 87

35 3 Which parameters can be extracted from the experimental data9 91

354 Fit results for 3D GAF model 95

3 5 5 Uncertainty estimates of fit parameters 97

3 5 6 Comparison between the 3D GAF analysis and a 15N

model-free analysis 102

3 6 Discussion 104

36 1 Amsotropy of peptide-plane dynamics 104

3 6 2 Correlation with secondary structure 107

3 7 Conclusions 110

4 CSA Fluctuations Studied by MD simulation and DFT Calculations 111

4 1 Introduction 111

4 2 MD simulation and DFT calculations 112

4 3 Average 15N and 13C CSA tensors 116

43 1 Processing of the CSA trajectory 117

432 Analysis of 15N CSA tensors 118

4 3 3 Analysis of 13C CSA tensors 121

4 3 4 Comparison of different averaging approaches 123

4 4 Fluctuations of the CSA tensors 126

4 4 1 Fluctuation of CSA amsotropy and asymmetry 126

4 4 2 Onentational fluctuations of the CSA tensors 128

4 5 15N and 13C CSA relaxation of the protein backbone 131

4 5 1 Implications of CSA fluctuations on CSA relaxation 131

4 5 2 CSA relaxation based on average CSA tensors 137

Contents 3

4 5 3 Role of the antisymmetric part of CSA tensor 140

4 6 Conclusions 142

5 Rotational Motion of a Solute Molecule in a Highly Viscous Liquid 145


5 2 Experimental 148

52 1 Charactenzation of the system and sample preparation 148

5 2 2 NMR relaxation measurements 150

5 3 Results and Discussion 152

53 1 Test of a single-correlation-time model 152

5 3 2 Distribution of correlation times 154

5 3 3 Analysis of the temperature-dependent relaxation data 159

5 4 Conclusions 162

6 Appendix 163

References 179

Acknowledgements 191

Curriculum Vitae 193

4 Contents


Symbols and Abbreviations

Symbols

''„ <v<7?V

A Eigenoperator of the static Hamiltoman H0

AM) Scalar vanable of stochastic process it

A Angstrom, 10 10m

a^ Coefficients for the expansion of the tensor operator T^ mof the

interaction (i in terms of eigenoperators

B, Bm, Bn Spin operators

B0 External magnetic field vector

B0 Strength of the magnetic field along z direction

,(?) General cross-correlation function between the irreducible tensor

operators of the interactions it and v

C„v(0 , Cnv(t) Liquid state cross-correlation functions for rank 2 interactions |i and v

in the laboratory and molecular frame, respectively

ct(t) Time-dependent coefficient for expansion of state vector |\|/(f))

cmnO Time-dependent coefficient for expansion of probability /(<P, t)

c, cv Strengths of the interactions it, v, respectively

D Diffusion constant

D Axially symmetric dipolar coupling tensor

Dmn Wigner rotation matrix element of rank /

dmn Reduced Wigner rotation matrix element of rank /

Et Eigenvalue of the static Hamiltoman H

ew en, e Pnncipal axis system of the 3D GAF model

e Unit vector along the direction of the interaction pnncipal axis \i

/(O, f) Probability of finding a director at orientation O at time t

H Static Hamiltoman of a quantum mechanical system

H(t) Time-dependent Hamiltoman of the spin system including the

interaction with the environment

HCSA Hamiltoman of the antisymmetric part of the CSA interaction of the

spin system

6 Symbols and Abbreviations

Ts\m

HCSA Hamiltoman of the symmetnc part of the CSA interaction of the spin

system

HD Dipolar Hamiltoman of the spin system

Hz Zeeman Hamiltoman of the spin system

H0 Static Hamiltoman of the spin system

H, (t) Time-dependent Hamiltoman representing a stochastic perturbation

h Planck constant, h = 6 6261 10~34 Is

/ Spin operator

^11 a a({0) General cross-spectral density function between irreducible tensor

compounds of the interactions \i and v

k Boltzmann constant, k = 1 3806 10"23JK l

L Rank of the irreducible tensor operator of interaction |0.

P/(Q) Legendre polynomial of rank /

P(O0|<I>(f)) Conditional probability to find a director at the orientation c£> at time t

if it was at position <t>0 at time t = 0

p(%<_) Distribution function for overall tumbling correlation times Tc

Rh Hydrodynamic radius

R Correlation coefficient

rls Vector between the / and the S spin

rIS Distance between the 7 and the S spin

/ff Effiwt^o 15W_1

J\i\

Effective N-'H distance including averaging over NH stretching and

bending vibrations

£ Spin operator

2S Generalized rank 2 order parameter of the interactions |i and v

Absolute temperature in Kelvin

T^ Irreducible tensor operator of interaction \i

T, Longitudinal spin-lattice relaxation time

T| p

Rotating-frame transverse relaxation time

T2 Transverse spin-spin relaxation time

Tr{A) Trace of matrix A

1 Time

tJ Spin operator

P* Spin operator


Yt Normalized spherical harmonic of rank /

a, p, y Fluctuation angles about the principal axes ew e», e of the 3D GAF

model

PCD Parameter of Cole-Davidson spectral density

$kkw Parameter of the Kohlrausch-Wilhams-Watts stretched exponential

function

f Relaxation superoperator

Tmn Matnx element of the relaxation superoperator f" representing a

transfer rate constant between the base operators Bm and Bnis

^lons Matnx containing the rate constants describing longitudinal relaxation

processes of a spin system IS

is

Ttrans Matrix containing the rate constants descnbing transverse relaxation

processes of a spin system IS

y7, ys Gyromagnetic ratios of spins / and 5, respectively

Ac Amsotropy of the CSA tensor g

2

^anti Quadratic sum over antisymmetnc matrix elements of the CSA tensor

5(f) Delta function

ti Macroscopic viscosity

T| Asymmetry of the CSA tensor g

9„v(0 Angle between the interaction pnncipal axes |i and v

Xc Scaling factor for the 13C CSA principal values

XN Scaling factor for the 15N CSA pnncipal values

(i Global multiplier for all interaction strengths in the peptide plane

|i0 Permeability of vacuum, |i0 = 4ti 10

% Rotational fnction coefficient

p (t) Density operator of the spin system

p0 Equihbnum density operator of the spin system

g Chemical shielding amsotropy tensor

a1,0|| Pnncipal values of an axially symmetnc CSA tensor The

corresponding pnncipal axes are orthogonal and parallel to the

symmetry axis of the tensor

gsym Symmetnc part of the chemical shielding amsotropy tensor g

0'

Antisymmetnc part of the chemical shielding amsotropy tensor g

Symbols and Abbreviations

ax Difference between <5XX and azz

G„ Difference between o,„ and a„y yy zz

axx, a , azz Pnncipal values of the chemical shielding anisotropy tensor g

o, axz, o~VT Matnx elements of the antisymmetric part of the chemical shielding

anisotropy tensor g

oa, Oq, ay Fluctuation amplitudes for fluctuations about the axes ea, eB, e of

the 3D GAF model

aj,

Variance of the normalized spherical harmonic Y2 r

t Time

xc Overall tumbling correlation time

x^Jr Effective correlation time for cross-correlation function of interactions

(i and v

Tmf Internal correlation time

v^: Correlation time for internal cross-correlation function of interactions

(X and v

xm Mixing time in relaxation experiment

<t> Set of Euler angles relating the molecular and the laboratory frame

cp( Backbone dihedral angle of the fragment C\ _,- N, - C" - C\

2

X Least squares fitting error

\\lL Backbone dihedral angle of the fragment Nl-Cl - C\ - N, + ,

\\\i(t)) State function for a closed quantum mechanical system

1^) Eigenvectors of the static Hamiltoman H

Q (?) Time-dependent set of polar angles of the direction of the interaction

pnncipal axis (X in the defined reference system

ii°

(0 Time-dependent set of polar angles of the direction of the interaction

principal axis |x in the laboratory frame

O!"" (t) Time-dependent set of polar angles of the direction of the interaction

pnncipal axis |U in the molecular frame

co. Backbone dihedral angle of the fragment C, _, -C'(_ , -r^-C,

co Angular frequency in rads-l

co Eigenfrequency of Hamiltoman H0 in rads

COy Larmor frequency of spin X in rads


Abbreviations

2D Two-Dimensional

3D Three-Dimensional

CHARMM Chemistry at HARvard MacroMolecular dynamics

CSA Chemical Shielding Amsotropy

D Dipole-dipole interaction

GAF Gaussian Axial Fluctuation

HSQC Heteronuclear Single-Quantum Correlation

INEPT Insensitive Nuclei Enhanced by Polanzation Transfer

LRM Long-Range Motion

lab Laboratory frame

MD Molecular Dynamics

mol Molecular frame

NMR Nuclear Magnetic Resonance

NOE Nuclear Overhauser Enhancement factor

PP Peptide plane

ppf Peptide plane frame

ppm Parts per million

rf Radio frequency

SRM Short-Range Motion

10 Symbols and Abbreviations

Abstract 11

Abstract

In this thesis, experimental NMR relaxation methods and theoretical

techniques are combined to extract information about anisotropic dynamics

of molecular systems in the liquid state.

In the first part, the anisotropy of rapid fluctuations of the peptide planes15 13

in ubiquitin is explored by combined N and C nuclear spin relaxation

measurements and molecular dynamics computer simulation. T{, T2, and

NOE data are collected at B0 -field strenghts corresponding to 400 and

600 MHz proton resonance. A 1.5 ns simulation of ubiquitin in an explicit

water environment is performed using CHARMM 24. The simulation

suggests that, for 76% of the peptide planes, the relaxation-active motion of15 13

the backbone N and C spins is dominated by anisotropic Gaussian

axial fluctuations of the peptide planes about three orthogonal axes. The

dominant fluctuation axes are nearly parallel to the C{ _ Y- Cj axes. The

remaining peptide planes belong to more flexible regions of the backbone

and cannot be described by this type of motion alone. Based on the results

of the computer simulation, an analytical 3D GAF motional model is

applied to the experimental relaxation data. The fluctuation amplitudes of

the peptide planes show a significant anisotropy of the internal motion. This

15 13analysis demonstrates that a combined interpretation of N and C

relaxation data by a model derived from a computer simulation may provide

detailed insight into the fast time-scale backbone dynamics that goes

beyond the results of a standard model-free analysis.

In the second part, density functional theory calculations of chemical

shielding anisotropics (CSA) and a molecular dynamics simulation are

combined to a CSA trajectory which contains the fluctuations of the CSA15 13

tensor induced by intramolecular motion. For the N and C CSA

tensors, located in three different peptide planes in the a helix, a ($ strand,

and a loop region in ubiquitin, it is found that the fluctuations characterized

as standard deviations of the chemical shielding anisotropy distributions do

not exceed 10%. The anisotropics of the calculated average CSA tensors

differ only slightly between the different secondary structure elements. An

analysis of the CSA trajectories indicates that the fluctuations of the CSA

12 Abstract

tensors do not have to be explicitly taken into account for a description of

CSA relaxation and thus the use of a locally averaged CSA tensor is

sufficient for the back-calculation of relaxation rate constants. The effect of

motional averaging of the CSA tensors, which should be considered when

CSA tensors obtained by solid state NMR experiments at room temperature

are used for the interpretation of CSA relaxation in the liquid state, is

estimated for different values of the motional parameters. This analysis is a

first step towards an improved understanding of CSA interaction strengths

of backbone spins in proteins which is important to reduce uncertainties of

extracted motional parameters in NMR relaxation studies.

In the third part, it is investigated whether the viscosity-dependent

retarding effect of a polymeric solvent on the rotation of solute molecules

can be used to shift the NMR observation window for the timescale of

intramolecular motion. It is found that the 13C NMR relaxation

measurements of the model system 1,3-dibromoadamantane in highly

viscous polymeric chlorotrifluoroethene can be explained neither by

isotropic nor by realistic anisotropic tumbling in a single environment. The

experimental data are rationalized in terms of fast exchange between at least

two environments with correlation times differing by up to two orders of

magnitude. This demonstrates that a uniform retardation of molecular

tumbling by a polymeric solvent is not always feasible.

Zusammenfassung 13

Zusammenfassung

In der vorliegenden Arbeit werden NMR-Relaxationsexperimente und

theoretische Methoden miteinander kombiniert, um die anisotrope Dynamik

in molekularen Systemen in fliissiger Phase zu charakterisieren.

Im ersten Teil wird die Anisotropic schneller Fluktuationen der

Peptidebenen im Protein Ubiquitin untersucht, indem man 15N- und 13C-

Kernspinrelaxationsmessungen und eine Molekuldynamik-Computer-

simulation miteinander verkniipft. Hierbei werden Tl, T2 und NOE-

Relaxationsdaten bei Z?0 -Feldstarken entsprechend 400 und 600 MHz fur

die Protonenresonanz gemessen. Desweiteren wird eine 1.5 ns lange

Simulation von Ubiquitin einschlieBlich expliziter Wasserumgebung mittels

CHARMM24 durchgefuhrt. Die Simulation zeigt, daB bei 76% aller15 13

Peptidebenen die relaxationsaktive Bewegung der N - und C -Kernspins

im Backbone durch anisotrope Fluktuationen der Peptidebenen um drei

orthogonale Achsen beschrieben werden kann, wobei die Verteilung der

Fluktuationswinkel um jede der Achsen einer GauB-Veiteilungsfunktion

geniigt (3D GAF-Modell). Dabei liegen die Achsen, um welche die groBten

Fluktuationen stattfinden, nahezu parallel zu den C1 _ t- C; -Achsen. Die

restlichen Peptidebenen gehoren zu flexibleren Regionen des Backbone und

konnen nicht mittels dieser Art von Bewegung beschrieben werden.

Aufgrund der Resultate der Computersimulation wird eine analytische Form

des 3D GAF-Modells verwendet, um die experimentellen Relaxationsdaten

zu interpretieren. Die extrahierten Fluktuationsamplituden der Peptidebenen

spiegeln eine signifikante Anisotropic der internen Bewegung wider. Die15 13

Analyse zeigt, daB die kombinierte Interpretation von N- und C-

Relaxationsdaten mittels eines Modells, welches anhand einer

Computersimulation entwickelt wurde, detaillierte Einsicht in die schnelle

Backbonedynamik ermoglicht und so iiber die Resultate einer modellfreien

Analyse hinausgeht.

Im zweiten Teil werden Berechnungen der chemischen Verschiebungs-

anisotropie (CSA) mittels Dichtefunktional-Theorie und eine Molekiil-

dynamik-Simulation kombiniert. Es resultiert eine CSA-Trajektorie, welche

die durch die intramolekulare Bewegung induzierten Fluktuationen des

14 Zusammenfassung

CSA-Tensors enthalt. Im Falle der N - und C -CSA-Tensoren von drei

Peptidebenen in der a-Helix, in einem P-Faltblattstrukturelement und in

einer Schleifenregion von Ubiquitin iiberschreiten die Fluktuationen,

welche durch die Standardabweichungen der Verteilungen der chemischen

Verschiebungsanisotropie gekennzeichnet sind, nicht die 10%-Marke. Die

Anisotropien der berechneten, durchschnittlichen CSA-Tensoren

unterscheiden sich kaum in den unterschiedlichen Sekundarstruktur-

elementen. Eine Analyse der CSA-Trajektorien zeigt, daB die Fluktuationen

der CSA-Tensoren nicht explizit fiir die Beschreibung der CSA-Relaxation

beriicksichtigt werden miissen. Stattdessen geniigt es, lokal gemittelte CSA-

Tensoren fur die Berechnung der Relaxationsratenkonstanten zu verwenden.

Weitergehend wird die Bewegungsmittelung von CSA-Tensoren fiir

verschiedene Werte der Bewegungsparameter abgeschatzt. Dieser Effekt

sollte beriicksichtigt werden, wenn die bei Raumtemperatur mittels

Festkorper-NMR ermittelten CSA-Tensoren fiir die Interpretation von CSA-

Relaxation in fliissiger Phase verwendet werden. Die vorgelegte Analyse

stellt einen ersten Schritt in Richtung eines verbesserten Verstandnisses der

CSA-Interaktionsstarke fiir Kernspins im Backbone von Proteinen dar. Dies

ist wichtig, urn die Unbestimmtheit der extrahierten Bewegungsparameter

in NMR-Relaxationstudien zu verringern.

Im dritten Teil wird untersucht, ob der viskositatsabhangige

Verlangsamungseffekt eines polymeren Lbsungsmittels auf die Rotation

eines gelosten Molekiils dazu verwendet werden kann, das NMR-

Beobachtungsfenster fiir die Zeitskala intramolekularer Bewegung zu

verschieben. Es zeigt sich, daB die C-NMR-RelaxationsmeBdaten des

Modellsystems 1,3-Dibromadamantan in hochviskosem Polychlortrifluor-

ethylen weder mit isotroper noch realistisch anisotroper rotatorischer

Diffusion in einer Umgebung erklart werden konnen. Die experimentellen

Daten werden durch einen schnellen Austausch zwischen mindestens zwei

Umgebungen rationalisiert, wobei die Korrelationszeiten sich um bis zu

zwei GroBenordnungen unterscheiden. Dies zeigt, daB die einheitliche

Verlangsamung der rotatorischen Diffusion durch polymere Losungsmittel

nicht immer moglich ist.

Introduction 15

1 Introduction

For a complete description of a molecule and its interaction partners it is

crucial to characterize both structure and dynamics. Molecules are normally

described and visualized by a static arrangement of atoms in space, which

however represents only an average structure or a single snapshot of a

molecule on the move. Intramolecular motion and, in liquid systems, also the

overall tumbling motion are inherently present and have to be taken into

account for the interactions between molecules. This is exemplified best for

proteins which play a fundamental role in biological systems [1,2]. Proteins

show intramolecular dynamics within a wide time window ranging from sub-

picoseconds to seconds. Their functional role relies on intramolecular

flexibility as for example in molecular recognition processes to achieve an

optimum match [3] or in biochemical transformations where the enzymes

adopt different conformational states [1].

Some highly successful experimental methods have been established for

the elucidation of a molecular structure. In particular, X-ray crystallography

[4,5], neutron diffraction [6], and nuclear magnetic resonance (NMR) [7,8]

provide accurate "images" of molecules. Intramolecular motion is only

indirectly contained in the elucidated structures which, in fact, result from

averaging over the ensemble of molecules and different conformations

present during the experiment. For studying the molecular motion

experimentally, NMR is at present the most versatile and powerful tool [7,9-

11]. Fast motional processes in the picosecond to microsecond range can be

sampled by relaxation experiments. Here, a certain state of polarization or

coherence [7] is generated first and its decay towards an equilibrium state is

monitored subsequently.

On a microscopic level, nuclear spin relaxation is the consequence of a

weak coupling of the observed spins to their surrounding which is often

termed the "lattice" and includes essentially all other degrees of freedom

within the molecular system. Stochastic motional processes of the lattice as

the overall tumbling and intramolecular motion of a molecule lead to

16 Introduction

reorientation of internuclear vectors and molecular axes [12] and therefore

modulate magnetic interactions as, for example, the dipole-dipole interaction

or the chemical shielding anisotropy (CSA) interaction [7] The resulting

time-dependent magnetic fields induce transitions between populations or

lead to dephasing of coherences of the observed spin system The measured

relaxation rate constants therefore contain information about the frequencies

by which the local magnetic fields are modulated and can be analyzed in

terms of an appropnate motional model for the underlying stochastic

processes

NMR studies of the intramolecular motion in proteins concentrate

normally on a local description of the reorientation of internuclear vectors in

either the backbone domain or in the amino acid sidechams Improved

labeling techniques [13] enable the investigation of an increasing amount of13 15

proteins by C and N nuclei as "spies" for the motional modes

Information about backbone motion can therefore be extracted from N,

C and C relaxation data by descnbing the reorientation of mainly the1 N 15 1 a 13 a

H - Nor H - C bond vectors Hereby, the so-called "model-free"

approach of Lipan and Szabo [14,15] is often used where intramolecular

motion is parametrized by a measure for the time scale of the process and an

"order parameter" which is an abstract measure for the spatial restnction of

the relaxation-active interaction at the site of the monitored nucleus This

simple approach does not provide a very detailed physical insight into the

reorientation processes but circumvents some intrinsic problems connected

with the application of a detailed motional model First, the correctness of

such a motional model cannot be proved by relaxation measurements alone

Second, it is often not possible to discriminate between different motional

models, such as rotational diffusion or jump-like processes, which often fit

equally well to the relaxation data

In Chapter 3 of this thesis, it is tried to describe the backbone motion of the

protein ubiquitin in a way that goes beyond the description of the

reorientation of local vectors in terms of the standard model-free analysis

The proposed analysis [16] takes into account that the peptide bond, which

Introduction 17

is the central building block of the protein backbone, defines, to a good

approximation, a planar and rigid fragment The extraction of the motional

parameters, which characterize the peptide plane motion, from experimental

relaxation data is the mam goal of this chapter In the rigid fragment, the

dipolar and CSA interaction tensors, which dominate the relaxation of the15 13N and C peptide bond spins, have fixed relative orientations Since these

interaction principal axes are generally not parallel, they probe

reonentational processes about different directions in space A combined

15 13

interpretation of N and C relaxation data should therefore allow for the

exploration of locally anisotropic motion of the whole peptide plane

fragment Although it is well known that intramolecular motion of

biomolecules is generally anisotropic, which can be seen from anisotropic

crystallographic B-factors [17], the magnitude of amsotropy for the motion

of the peptide plane has never been extracted from experimental data before

The peptide plane reonentation is discussed in detail using a motional model

which is defined by the orientation of three orthogonal principal axes about

which the peptide plane fluctuates and the corresponding distribution

functions for the fluctuation angles about each principal axis Such a

motional model cannot be extracted from the expenmental relaxation data

itself but must be provided by molecular dynamics (MD) computer

simulations [18,19] which introduce independent knowledge in form of an

empirical molecular force field parametrized based on quantum chemical

calculations and mainly optical spectroscopic data Bremi and Bruschweiler

[20] have recently proposed a 3D Gaussian Axial Fluctuation (GAF) model

for the anisotropic reonentation of the peptide plane based on the analysis of

a MD trajectory of a decapeptide The pnncipal axes of reonentation consist

of the Cj _ [- C, axis and two orthogonal axes lying in the peptide plane and

orthogonal to the plane The fluctuations are charactenzed by Gaussian

distnbutions In Chapter 3, it is tested by an extended MD simulation

whether the 3D GAF model applies for a protein Furthermore, it is analyzed

under which conditions the 3D GAF model is appropnate for fitting the

relaxation data

The extraction of the amsotropy of the peptide plane motion relies on a

18 Introduction

detailed knowledge of the magnitude and orientation of the relaxation-active13 15

dipolar and the C and N CSA interaction tensors. X-ray

crystallography, neutron scattering, and optical spectroscopy can provide

experimental data on the peptide plane geometry [21] defining the dipolar

interaction strengths. CSA parameters can be extracted from solid-state

NMR experiments of small peptides [22]. The set of interaction strengths

used in the analysis of Chapter 3 is based on these experimentally determined

interaction parameters which leads to several problems. First, it is not clear

whether the interaction strengths, which have often been measured for model

compounds, can be transferred to proteins and how much these parameters

vary from residue to residue. Second, the use of interaction strengths with

fixed magnitude and orientation may not be appropriate since intramolecular

motion induces changes in the molecular geometry and in the electronic

environment which results in fluctuations of the dipolar and CSA interaction

strengths altering thereby spin relaxation. These problems are addressed in

Chapter 4, where a novel approach is presented for the interpretation of CSA

relaxation in proteins. It is based on the combination of MD simulation for

the description of the dynamics in the protein ubiquitin and a quantum

chemical method, the density functional theory (DFT) [23], for the13 15

calculation of CSA parameters. CSA tensors of C and N backbone spins

are calculated for each snapshot of a MD trajectory leading to a "CSA

trajectory" which directly contains the fluctuations of the CSA interaction

tensor due to intramolecular motion. In Chapter 4, these fluctuations are

characterized and their contribution to CSA relaxation of protein backbone

spins is assessed. It is checked whether CSA relaxation can be described with

CSA parameters averaged over the time-dependent fluctuations. The

averaged CSA parameters are compared to experimental results by solid state

NMR.

The conceptual framework underlying for Chapters 3 and 4 is summarized

in Fig. 1.1. It describes how MD simulation techniques, analytical motional

models, and quantum chemical methods have to be combined for a detailed

interpretation of NMR relaxation data in terms of intramolecular motion.

Introduction 19

NMR Relaxation Experiment

X-ray diffraction

solid-state NMR

Interaction

StrengthsQuantum Chemical

Methods

i '

,.

Order Parameters

Time Scales

f Intra- \molecular

y MotionJ

Analytical Molecular

M(3d els Dynamics

Fig 1 1 Investigation of intramolecular motion by interpretation of experimental NMR

relaxation data using a combination of quantum chemical methods molecular

dynamics computer simulation and analytical motional models

In liquid systems, the intramolecular motion is often masked by the overall

tumbling motion of the molecule The decomposition of both relaxation-

active processes could be improved if the timescale of the overall tumblingmotion can be changed selectively without affecting the internal motional

processes In Chapter 5 it is investigated [24] by NMR relaxation

experiments whether a highly viscous, polymeric solvent is an appropriatetool for the isotropic retardation of the overall tumbling motion of solute

molecules

20 Introduction

2.1 Fundamentals of spin relaxation 21

2 Spin Relaxation and Molecular Motion

In this chapter, the theoretical framework for the following chapters is

presented. First, the fundamentals of spin relaxation in terms of semiclassical

relaxation theory are shortly summarized followed by a description of the

treatment of the overall tumbling and intramolecular motion in liquid

systems as relaxation-active stochastic processes. Special emphasis is laid on

the methodology to extract information about the anisotropy of

intramolecular motion from NMR relaxation data which is exemplified for

the anisotropic peptide plane fluctuations in the protein backbone. In

addition, MD simulation techniques and quantum chemical DFT methods

are shortly introduced.

2.1 Fundamentals of spin relaxation

2.1.1 Equation of motion

The time evolution of a quantum mechanical system, e.g. a single nuclear

spin system, under the influence of a static Hamiltonian H is given by the

time-dependent Schrodinger equation [25,26]:

ijfNit)) = ff|i|/(0>, (2-1)

where H is expressed in units of h/27t and the time-dependent state function

\y(t)) contains all information about the system. It can be decomposed into

a linear combination of time-dependent coefficients c((f) and eigenvectors

III/,) of the static Hamiltonian H which obey the time-independent

Schrodinger equation:

M

IV(0> = X ^(f)IV,). with H\y) = E,ty,), (2.2)

i= l

where El is the energy of the eigenstate |t|/(). Since NMR is an ensemble

spectroscopy, it is convenient to describe an ensemble of spin systems with

its time-dependent density operator p(f) [7]:


P(0 = lY(0><V(OI, (2-3)

where the bar indicates averaging over the ensemble. The density operator is

2one element of the M -dimensional Liouville space and can be represented

as M x M matrix with respect to a basis of the Hilbert space of dimension M

[27]. Then, the diagonal elements of the density matrix correspond to the

populations of the different energy levels Et of the Hamiltonian H whereas

the off-diagonal elements represent coherences resulting from phase

correlations of pairs of different eigenstates averaged over the ensemble [7].

The time evolution of the density operator under the influence of the

Hamiltonian H is given by the Liouville-von Neumann equation [7]

jtp(t) = -i[H, p(0] , (2.4)

which can be derived by combination of the time-dependent Schrodinger

equation and the definition of the density operator for an ensemble assuming

that there are no interactions among the subsystems in the ensemble.

2.1.2 Semiclassical relaxation theory

A closed quantum mechanical system shows a unitary and reversible time

evolution. The concept of relaxation comes in when a subsystem is regarded

which is coupled to the environment treated as a perturbation in the case of

weak coupling. A microscopic semiclassical theory of spin relaxation was

formulated by Bloch, Wangsness, and Redfield (BWR) [28-31] and has

proved to be useful for the description of experimental relaxation data in

many cases [7,9]. In the semiclassical approach the spin system is treated

quantum-mechanically, and the surroundings are treated classically. This

means that the density of energy levels of the lattice is assumed to be quasi-

continuous with populations that are described by a Boltzmann distribution.

Furthermore, the lattice is assumed to be in thermal equilibrium at all times.

The total spin Hamiltonian of the spin system is described by the sum of

the static Hamiltonian H0 and a stochastic Hamiltonian HAt) which

2 1 Fundamentals of spm relaxation 23

represents the coupling of the spin system to the motional degrees of freedom

in the lattice (see Chapter 1)

H{t) = HQ + H{{t), (2 5)

where Hq = Hz + Hj consists of a Zeeman term for the coupling of the

nuclear spins to the external static magnetic field and a term for the time-

independent scalar spin-spin coupling [7] Chemical exchange processes,

rendering these terms time-dependent, and a coupling of the spin system to a

radio frequency field shall not be considered at this place The stochastic part

#j(f) represents a stochastic process with a time average of zero and is

regarded as a weak perturbation with respect to H0

The derivation of the master equation of spin relaxation in an ensemble of

spin systems by second-order perturbation theory is described in detail in the

textbook of Abragam [9] Starting by inserting Eq (2 5) into the Liouville-

von Neumann equation, the final form of the master equation is the main

result of the BWR relaxation theory and takes the form

jtP(t) = - i[H0, p(t)] - f{p(0 - p0} , (2 6)

with the relaxation superoperator i defined by the operation

-iHn1

dx (2 7)fB = J Hx{t),[e°

Hl{t-x)e°

,B]

where the bar indicates averaging over the ensemble of spin systems It is

important to note that the validity of the master equation relies on several

assumptions First, it is only valid in the so-called weak collision regime

implying that the correlation times of the involved stochastic processes (see

Section 2 2 1) are significantly shorter than the time-scale on which the

density operator p(t) evolves [9] It is also required that the magnitude of the

perturbation Hamiltoman \HA is much smaller than the inverse of the

corresponding correlation times [9] Thus, Eq (2 6) is not appropnate for


strong perturbations that are slowly modulated. In the derivation of Eq. (2.6)

the density operator p(f) has to be replaced by the difference p(f) - p0 [9]

to ensure a relaxation towards the thermal equilibrium represented by the

density operator p0. Therefore, the given master equation is an

inhomogeneous differential equation only valid in the high temperature

approximation [32].

2.1.3 Relaxation-active interactions

Spin relaxation results from the stochastically modulated reorientation of

an interaction tensor inducing a time-dependent local magnetic field at the

locus of the observed nucleus. There are a variety of relaxation-active

interactions like the electric quadrupole interaction, random field interaction,

spin rotation interaction, and interactions leading to scalar relaxation of the

first and second kind [7,9,33]. In this work, however, the discussion is

restricted to the chemical shielding anisotropy (CSA) interaction and the

dipolar interaction for spin systems consisting of spin 1 /2 nuclei.

The stochastic Hamiltonian H^ is given as the sum over the relaxation-

active interactions present, each of them expressed in terms of irreducible

tensor operators for the spin part and irreducible representations of the 3D

rotation group for the spatial part [34]:

V

"i(0 = 1^(0 I (-»\,-,«y0)7? , (2.8)

where it is assumed that each interaction \i is described by one rank / only.

|X runs over the interactions and c is a characteristic constant measuring the

strength of the interaction n. The angular momentum number / is also the

rank of the tensor and q is the magnetic quantum number. The spatial part is

represented by the normalized spherical harmonics Y, (Q..At)), where

D, (t) is a time-dependent set of polar angles defining the orientation of the

interaction u. in a chosen coordinate frame. T, are the irreducible tensor

operators for the spin part of the interaction |i.

2.1 Fundamentals of spin relaxation 25

The dipole-dipole interaction. The Hamiltonian of the magnetic dipole-

dipole interaction between the spins ) and $ connected by the vector rIS is

given in its bilinear form by HD = -)£>$, where D is the rank 2 dipolar

coupling tensor which is axially symmetric with its symmetry axis parallel to

rJS. The Hamiltonian for the laboratory frame can be decomposed according

to Eq. (2.8):

HD = -!~Vs4jf £ (-1)V«(<W)^(U), (2.9)

rIS q = -1

where the time-dependent orientation of r]S in the laboratory frame is given

by Q.D (t). The normalized second rank spherical harmonics Y2 „

and the

irreducible spin tensor operators T2 „of rank 2 are given by [34]:

r2,0(e><P) = Jii^3^086)2-1) •>r2,o(tf.^) = \{3Uzvz-^y>

r2i±1(9,<p) = T^sinecose/^; r2i±1(^, t) = t\(u±vz+ UzV±);

y2,±2(q,<p) = J§(sine)2e±2,ip;r2,±2(^^) = ^±y±) • <2-10)

The chemical shielding anisotropy interaction. The Hamiltonian for an

interaction between a spin S and the static magnetic field BQ is given by

HCSA = ysSgB0. For nuclei situated at sites with low local symmetry (e.g.

in biomolecules), the chemical shielding tensor a reflects the anisotropic

and antisymmetric nature of its electronic environment. As a general second

rank tensor, g can be decomposed into a sum of irreducible tensors of rank

0,1, and 2 [35]:

g = g(0) + g(1) + g(2), (2.11)

where g is the isotropic chemical shift which is not relaxation-active and

can be neglected here [36]. g and g corresponds to the antisymmetric

component g = (g-g )/2 and the symmetric (but not necessarily

axially symmetric) component qsym = (g + g )/2, respectively. In the


principal axis frame of g they take the form:

0 a*y axZ

cxy

0 °>*

°« -°vZ 0

,and (2.12)

sym

°xx 0 0

0 oyy 01

=

3°*

2 0 0

0-101

+ 3°.

-10 0

0 2 0

0 o °«J 0 0 -1 0 0-1

(2.13)

where ox = oxx-ozz and a = a -az,, and it has been used that any

non-axially symmetric tensor can be split into two axially symmetric,

orthogonal parts [37]. The time-dependent Hamiltonian for the rank 2

interaction of the symmetric part can be decomposed according to Eq. (2.8)2

Hsym

CSA (t) = -JsL

q = -2

+ ay X (-\)qYx_q(Q!ya\t))T2q{lBQ)q = -2

(2.14)

where Q (t), £2X (t) denote the time-dependent orientations of the CSA

principal axes x and y in the laboratory frame, respectively. For the

antisymmetric part, which cannot be diagonalized, it is convenient to

transform a first from the principal axis frame of the symmetric part to

the laboratory frame using a Wigner rotation matrix with a set of time-

dependent Euler angles D. = (9, <|), %). The Hamiltonian is then given by

[35]

l/l \

*0') = YsX ("1)1 I Si.XVW) k*&^>' <2'15>

q = -1 Vm = -1 /

where Dm_ (Q.) denote Wigner rotation matrix elements of rank 1 [34]. The

rank 1 parameter olm and the irreducible rank 1 spin tensor operators Txare given by [34,35]:

2 1 Fundamentals of spin relaxation 27

cl0 = -,j2cxy,Tl0($,B0) = 0,

°i ±1= °« ± "^ • Ti ±A Bo) = -\s±B0 (2 16)

2.1.4 Operator representation of the relaxation superoperator

Taking into account the symmetry properties Yl = (-\)qYl * and

Tl = {-\)qTl + [34], an explicit form of the relaxation superoperator

can be calculated by inserting Eq (2 8) into Eq (2 7)

= xi(-ir

x J c^)cv(? - t)y1 _?(iy o)y/y; _9.*("v(? -T))^' (217)

0

where it is assumed that each interaction \i is descnbed by one rank / only

This is possible for the CSA interaction since the contributing rank 1 and 2

interactions of the antisymmetric and the symmetric part, respectively, can be

treated independently due to missing cross-correlation between both parts of

different rank (see Section 2 3)

The inner commutator in Eq (2 17) can be transformed by expanding the

tensor operators for the interaction (J. in terms of eigenoperators A of the

static Hamiltoman H0 [9]

Ti,q = ^UA> (2 18)

p

where the eigenoperators are defined by the operation [H0, A ] = (oA

Replacing both tensor operators in Eq (2 17) yields the final form for the

relaxation superoperator in the laboratory frame

V*'-in0i v t 0X

,B


fHii.'-'^'z.-UA.*/U, Vq, q' p, p

-t

x/rj/v>-,,^K)K,tA;,,5]], (2.19)

where the power spectral density functions J, , , have beenV' v' 9' "

introduced:

J'X-^'{(0)= lcZ',-^{x)e"mdx' (2-20)

—oo

which are the Fourier transform of the ensemble averaged correlation

functions of the spherical harmonics

<Vv,_,,^(t) = c^t)cv(t -

t)fv ^(fyo)^,VW'- *))

(2.21)

For ii = v, C, , „ _„.defines an autocorrelation function, for (i ^ v, a

cross-correlation function.

In Eq. (2.19), all combinations between the quantum numbers p and p' were

included. Terms with p £ p' are non-secular in the sense of perturbation

theory and do not affect the long-time behavior of p (r) since they show a fast

oscillation in the rotating frame which averages to zero much more rapidly

than relaxation occurs [9]. In the "secular approximation" terms with p * p'

are neglected.

2.2 Relaxation-active molecular motion

2.2.1 Correlation functions for molecular motion

The overall tumbling and intramolecular motion of a dissolved molecule

with N atoms can be described by an ensemble of dynamic trajectories in a

conformational space of dimension 3N. From an experimental point of view,

however, it is not possible to characterize the reorientation of a single

molecule in a time-resolved manner, since the measurement results from an

2.2 Relaxation-active molecular motion 29

ensemble average and the variations with time contained in each of these

trajectories show a complicated and stochastic behavior [38] on a time scale

which is too rapid to be sampled experimentally. One has to rely on the fact

that the stochastic processes can be described by some averaged motional

constants. This information can be extracted by averaging over the ensemble

which is done by calculating a set of correlation functions. In principle, one

can calculate a set of 3iV autocorrelation functions

^uu(r'T) = AJt)AM + x), 3N(3N-l)/2 pair correlation functions

C v(f, x) = A (t)Av(t + x), and a large number of higher-order correlation

functions [38], where the bar indicates the average over the ensemble of

molecules in the system and {Aj(f),..., A^N(t)} represent the 3N

stochastic scalar variables describing the molecule's motion in its

configurational space. However, such a complete set of correlation functions

cannot be extracted from experimental data but only few auto- and pair-

correlation functions can be characterized. In studies of intramolecular

mobility, one concentrates on motional modes which can be described

locally for a subunit of the molecule in terms of a motional model with only

few dynamical parameters. Other simplifications comprise the assumption

that the overall tumbling motion of the molecule is independent from the

intramolecular motion. It is assumed that the molecule has a certain rigid

shape when investigating the overall rotational diffusion.

Rapid reorientational processes, which modulate relaxation-active

interactions on time scales accessible for laboratory frame relaxation

experiments, enter the relaxation superoperator via Eq. (2.19) in form of

spectral density functions, which are Fourier transforms of the ensemble

averages of the angular correlation functions in Eq. (2.21). Since the

stationarity condition applies for a system in the equilibrium, there is no

origin in time and the correlation function does only depend on the time

difference X. The correlation functions are then given as ensemble averages

over the initial times A (0)Av(x) = A (t)Av(t + x). In addition, it is

usually assumed that the reorientation processes are ergodic, which means

that a property of a reorientational process is equally given by the time

average over one molecule's trajectory and the average over the whole


ensemble of molecules at the same time and vice versa:

(A (0)Av(t)) = A (0)Av(t). Therefore, the ensemble average in the

correlation function of Eq. (2.21) can be replaced by an average over the

time-behavior of one single molecule

<vv,W> = <M0>MT>%ViV0))yw*<£W)>- (2-22)

This is an important result needed for the calculation of a correlation function

from the MD trajectory of a single molecule or from a model describing a

motional process in a single molecule.

2.2.2 Overall rotational diffusion

2.2.2.1 Isotropic rotational diffusion in a homogeneous environment

The Debye small-step rotational diffusion model [39,40] is an appropriate

description for an overall reorientation process of a molecule which is far

greater in size than the solvent molecules representing a quasi-continuous

and homogeneous medium. This is fulfilled, to a good approximation, for

proteins dissolved in water. It is often found that the model holds well even

for small solute molecules in case of low viscous solvents [41]. The model

views the reorientational motion of a molecule as being impeded by a

viscosity-related frictional force. If the rotational friction coefficient

operating at the surface of the molecule regarded as a sphere with radius Rhis that represented by the macroscopic solvent viscosity r|, then the rotational

friction coefficient has the Stokes value [42,43] \ = inRhi\ and the

rotational diffusion rate constant for isotropic rotational diffusion is given by

the Stokes-Einstein-Debye (SED) relationship [39]

D = —£i—, (2.23)

%nRhT\

with D in rads.The rotational diffusion of a molecule in a homogeneous

solvent medium has been described by several authors [9,39,40,44-48]. The

derivation of the correlation function for this overall tumbling process shall


be given in the following. The orientation of the molecular frame, exclusively

affected by the overall rotational diffusion, with respect to the laboratory

frame shall be given by the angular term $ which represents a set of three

Euler angles. For /(O, t) which is the probability of finding the molecular

frame at the orientation 4> at time / the following diffusion equation holds

with /($, 0) = 8(4> - 4>0) as initial condition. V2 is the angular part of the

Laplacian operator in spherical coordinates. Following Favro [44], the

solution of Eq. (2.24) is given by /($, t) = P(<I>0|<t>(/)), which is the

conditional probability of finding the molecule at orientation <&(£) at time t,

if it was at orientation 4>0 at time t = 0. In the following an explicit solution

for P(O0|$(r)) shall be derived. /(<!>, 0 can be expanded in terms of

Wigner rotation matrix elements Dmn

/(*.'>= I i(Oi(*)' (2-25)/, m, n

By substituting this expression into Eq. (2.24) and taking into account

V2Dlmn(^) = -1(1+ \)Dlmn{Q>) [34] as well as the orthogonality

relationships of the Wigner matrices [34]

2

K»*(*)Dm(0) = 27778"5^V (2-26)

it follows

pmn(t) = -Dl(l+ 1)4(0 with 4(0 = 4(0)*-fl/(/+1)'. (2.27)

Introducing of the initial condition by expanding the 8 function in Wigner

rotation matrix elements [34]


5($-<D0) = X ^DL*(*o>0L(*). (2.28)

/, m, n oil

/ 21+ I I *

yields cmn(0) = r-Dmn (<J>0), and finally871

P(*0|O(0)= I ^^L*(*o)^L(^"D/(/+1)r- (2-29)

/, m, n oTC

This result has now to be incorporated in the correlation function of

Eq. (2.22). Assuming statistical independence of the overall tumbling and

the intramolecular motion, Eq. (2.22) can be transformed taking into account

the transformation properties of the spherical harmonics:

civd) = <^(0)cv(T)y,i0(flJlflfc(0))y/i0*(or('t))>= I<^(*o)^o*(*(^))>

r, r'

x (^(0)^(1)7, r(nJ""(0))y,F/(jC'(t)» , (2.30)

where the condition I = I = lv has been used since cross-correlations

between interactions of different rank / are not considered in this work. In

addition, C (x) is no longer dependent on the index q, which was set to

zero, since the correlation function is invariant under an arbitrary rotation in

the laboratory frame [49]. Q denotes polar angles (9 q> ) of the

principal axis of the interaction tensor |i in the molecular frame. The overall

tumbling and the intramolecular motion are averaged separately, and the

overall tumbling average can now be expressed in terms of the conditional

probability />($0|$(t)) [9]:

(Dlr0(%)Dl,0\®{T)))

= |Ji'(*0)D'0(*0)/'(*0|«&(x))£)J,0*(*(T))d*0<«>I (2.31)

2where P(®0) = 1/(8n ) is the probability of orientation <&Q at time

t = 0. Substitution of Eq. (2.29) into (2.31) and taking into account Eq.

(2.26) yields


(Dlr0^0)Dlr,0*mx))) = ^-/(U l)°\f. (2-32)

Substitution of Eq. (2.32) into Eq. (2.30) leads to the correlation function

; _l -z(z+i)dt

SvW -

2l+\e

x £ {c^)cv{T)Ylr{n\0))Yl*(Qol{T))). (2.33)

For a rigid molecule affected by isotropic overall tumbling only,

c, cv, Q , Qv are time-independent. The interaction strengths c

, cv

can then be bracketed out from the correlation function and appear in the

constant prefactor when calculating relaxation rate constants (see Section

2.3). With the addition theorem of spherical harmonics [34]

^tlp^cose^) = X hWW^> (2-34)

q = -l

where P^cosQ^) = J4n/(2l+ l)Yj o(9nV) are Legendre polynomials

and 9 is the angle between the principal axes of the interactions \i and v,

one obtains for the correlation functions of rank /

-x/x(0

C^v(x) = e PticosQ^), (2.35)

with the definitions of the rotational tumbling correlation times

xlcl) = l/(2D) for rank 1 and T^xf1 = 1/(6D) = x^/3 forrank2. A

factor 4tc has been introduced in Eq. (2.35) to normalize the correlation

function for x = 0 to start at 1 for all auto-correlated processes (ft = v).

The corresponding spectral density function can be calculated as a Fourier

transform according to Eq. (2.20):


./uv(a>) = P;(cos9^v;2i(')

(/)21 + (COT*0)

(2.36)

This spectral density function will be needed for the calculation of the

relaxation due to the antisymmetric part of the CSA (rank 1 interaction) in

Section 4.5.3 and of the dipolar relaxation (rank 2 interaction) in Chapter 5.

2.2.2.2 Rotational diffusion in a heterogeneous environment

It is well known that the experimental data of, in particular, highly viscous

systems, often cannot be explained by a mono-exponential correlation

function (see Eq. (2.35)) for isotropic rotational diffusion in a homogeneous

solvent medium. The non-exponential character of the correlation function

can be explained in two fundamentally different ways [50-53]. On the one

hand, local density fluctuations in the liquid influence the rotational

correlation function since they cause fluctuations in the environment of a

rotating molecule [54]. Rapid local density fluctuations, whose correlation

function decays on a short time-scale compared to the rotational correlation

time, would, again, result in an exponential correlation function. However,

local density fluctuations with a correlation time slower than the rotational

correlation time (but still faster than the relaxation times, see Chapter 5),

which occur in highly viscous liquids, might give rise to a distribution of

correlation times which results in the following correlation function for an

auto-correlated rank 2 interaction:

C(x) = jp(xc)e~Z/Xcdxc, (2.37)

o

where p(ic) defines the distribution function of the correlation times. On the

other hand, the assumption of independently reorienting molecules might not

be realistic in a highly viscous liquid. If the reorientational processes of

neighbored molecules are correlated with each other, one has to consider

cooperative motion of entire subsystems. A description of such motional

processes is extremely difficult and only very simplifying theoretical


approaches exist [55]. This intrinsic non-exponential behavior is often

characterized by the empirically found Kohlrausch-Williams-Watts (KKW)

or stretched-exponential correlation function:

C(x) = exp(-(x/tc)Pw), (2.38)

with 0 < $KKw < 1. To decide on the question whether the description of

independently reorienting solute molecules in a heterogeneous environment

or of complex reorientation processes with a collective character (or both) is

realistic, it would be required to directly sample the correlation function of

small sub-ensembles in the system. This is possible for extremely viscous

liquids due to the slow time-scale of the reorientation processes and the local

density fluctuations [56]. However, for molecular systems with reorientation

processes faster than milliseconds which are studied by NMR relaxation

spectroscopy and other methods like dielectric relaxation, it is not possible to

decide on this question. Therefore, most of these studies rely on Eq. (2.37)

simply due to the fact that many (mainly empirical) models have been

developed to describe the distribution function of the correlation times p(lc)

[53]. Some of these models have been applied in Chapter 5.

2.2.3 Intramolecular motion

2.2.3.1 Intramolecular backbone motion in proteins

The entire backbone motion in proteins results from a superposition of a

large number of motional modes on different time-scales. For the fastest

time-scales with correlation times <50ps the contributions of different

modes can be classified based on normal mode analysis [57-59]. High

frequency modes ( go > 500cm ), such as bond stretching and bending, have

small amplitudes and a very local character. These very rapid fluctuations can

be included in an effective dipolar interaction strength resulting from

motional averaging of the equilibrium interaction strength (see Section

2.2.3.2). Collective low frequency modes (co<250c/n ,see ref. [59])

exhibit larger amplitudes. According to Briischweiler [59], two kinds of the

collective modes can be distinguished for the reorientation of the interaction


principal axis under consideration (e.g. the NH vector in the peptide plane i):

Long-range motion (LRM) causes angular modulations of the interaction

principal axis by affecting a whole protein segment, which contains the

principal axis, but leaves the adjacent local backbone dihedral angles <p;,

V,_i (for definition see Fig. 2.1a and ref. [1]) unchanged. Short-range

motion (SRM) leads to fluctuations of the adjacent local dihedrals. In case of

compact proteins with lots of tertiary interactions it is this short-range motion

in dihedral angle space {(pr a>(, ij/;} which represents the dominant

relaxation-active motion for backbone spins in the sub-50 ps correlation-

time window [59].

Since the fluctuations about the dihedrals co are relatively small in

amplitude (see Section 3.4.3.3), peptide planes are well-defined fragments. It

is desirable to find a local description for the reorientation of a peptide plane

which results from the cumulative effect of projections of a large number of

the collective modes on the individual peptide-plane fragments. A local

description of peptide plane / in terms of fluctuations of the dihedral angles

vj/; jand <p(, however, is not appropriate since the correlation coefficients

p show large deviations from -1 (this is shown for the protein ubiquitin

in Fig. A.2a in the appendix) which demonstrates the dependence of the

dihedral angular motion also on the motion of adjacent peptide planes

[60,61]. Bremi and Briischweiler [20,62] have concluded from the analysis

of a MD trajectory of the decapeptide antamanide that it is preferable to

describe the local small-amplitude motion of a peptide plane by a

reorientational principal axis system fixed at the equilibrium peptide plane

and characterize the motion in terms of Gaussian distribution functions for

the fluctuations about and time constants with respect to these axes. This

results in the 3D Gaussian Axial Fluctuation (3D GAF) model [20,62] whose

principal axis frame is depicted in Fig. 2.1b. The three orthogonal principal

axes of reorientation are defined by the Cl_i-Cl axis, named e,the in-

plane principal axis ea, and the principal axis eg which is orthogonal to the

equilibrium peptide plane.


Fig 2 1 (a) Definition of dihedral angles along the protein backbone (b) Definition of

the reference coordinate system ea, eo, e fixed to the peptide plane It defines the

principal axis system for the peptide plane reorientation according to the 3D GAF

model. The principal axes directions of the relevant spin interactions, expressed in this

frame, are given in Table 3 2

ct B ct o.The reonentation of the backbone C - C and C -H vectors cannot

be descnbed by the 3D GAF model since they are affected by the motion of

both adjacent peptide planes. A detailed motional model describing their

reorientation is missing so far. Intramolecular motion of the backbone on

time-scales with correlation times between 50 ps and 0.1 jxs occur mainly in


unfolded proteins, or in loop and end regions of globular proteins In those

more flexible regions additional jump processes between different

conformations add on the underlying 3D GAF motion of die peptide planes

In pnnciple, these peptide planes can be descnbed by combined 3D GAF and

jump models Motional processes involving the backbone on even slower

time-scales in the fis to 10s range as e g hinge-bending motion, hydrogen

bond dynamics and exchange, helix-coil transitions, and cis-trans isomensm

of the peptide plane shall not be discussed at this place

2.2.3.2 Internal correlation functions and model-free approach

First, it is assumed that the intramolecular motion can be separated from

the isotropic overall tumbling motion

C^x) = Cltum\x) O), (2 39)

where according to Eq (2 33) the overall tumbling correlation function is

given by C"m

(x) = e The internal correlation function takes

the form

/

C^n'(x) =

27TT I <S<°>cvC)r7 r(a;o,(0))yl r*(<o/(t))>r = -l

= £-<^(0)cv(T)P;(cose^v(T))>, (2 40)

where the addition theorem for spherical harmonics in Eq (2 34) has been

used and 9„v(0 is the time-dependent angle between the pnncipal axes In

the following, only rank 2 interactions will be considered and the interaction

strengths c,c will be regarded as time-independent and therefore will be

bracketed out from the correlation function This leads to

2

c|Tv(T) =

y 2 <r2 r("7'(0))F2 ,*(nT'(T))> = (/V^Vx))>'r = -2

(2 41)

where u. and v refer to the pnncipal axes ofrank 2 interactions, e g a dipolar


director or a CSA tensor principal axis, and P2(x) = (3x2- l)/2 is the

second order Legendre polynomial. A factor 4k has been introduced in

Eq. (2.41) to normalize the correlation function for x = 0 to start at 1 for all

auto-correlated processes (|X = v), since />2(cos(0)) = 1, and at

P2(cos9 ) with 8 s 0„V(O) for all cross-correlated processes. In case of

cross-correlation between two principal axes of the same CSA tensor, one

hasP2(cos(7t/2)) = -1/2.

The internal correlation function C (x) can be calculated explicitly for a

motional model as described in Section 2.2.3.4 for the 3D GAF model of the

peptide plane reorientation. Alternatively, an abstract, "model-free"

approach which corresponds to a simple parametrization of the spectral

density function can be used (see also Chapter 1). If the intramolecular

motion is Markovian (e.g., diffusive or jump-like), then C (x) can be

expressed as a series of exponentials Cuv(x) = Va;exp(-x/xj), where

x0-»°°, x1>x2>...>0, and 0<a(<l for all ;'= 1,..., Z with Z

dependent on the type of motion. It was shown by Lipari and Szabo [14,15]

that in case of isotropic overall tumbling motion, and internal motions with

ny Llv 2

correlation times %int m the extreme narrowing regime ((x;nfco) « 1) much

faster than the overall tumbling motion, the internal correlation function is

exactly given by

Ox) = Slv + {^(cose^-sjj/1 T"". (2.42)

2

5„v is the so-called generalized order parameter and corresponds to the

plateau-value of the correlation function. In case of auto-correlation (i = v,2

S„„ is a model-independent measure for the spatial restriction of the motion

which modulates the relaxation-active interaction \l. Absence of internal2

motion leads to S = 1, large-amplitude internal motion to small order

parameters. The plateau value can be calculated as

i = jl <y2,r(Q"0,)){y2jr*(<0/)>. (2.43)


For an auto-correlated process \i = v, Briischweiler and Wright [63] have

derived an alternative formula in terms of variances of the spherical

harmonics

Sl=l-f I42/ (2-44)

r = -2

This formula is useful for the calculation of order parameters in case of

Gaussian axial fluctuations (see Section 2.2.3.3). A similar formula for the

calculation of the cross-correlated order parameter including the time-

dependence of the interaction strengths c, cv is given in Section 2.5.1.

The auto-correlated and cross-correlated spectral density functions can be

calculated from Eq. (2.42) taking into account Eq. (2.20) and the independent

overall tumbling with correlation time xc [64]:

r2 l%c. id ,^,c ^ e2 i

ZV/

1 +W 1 + (CO

where the effective correlation time xe^ is given by

(\ff) = lc + (^n/) •In case of fast intramolecular motion, e.g. 3D

GAF motion, the correlation times xint are in the sub-50-ps range, the second

term of Eq. (2.45) becomes very small and thus the Xm( values have only

little influence on the relaxation parameters.

The spectral density function of the model-free approach can be used for

the calculation of the relaxation rate constants (see Section 2.3). Then, the

order parameter and the internal correlation time can be extracted from

experimental relaxation data and represent an abstract measure for the

amount of intramolecular motion present. On the other hand, the extracted

order parameters can also deserve as connecting link between the

experimental raw data and the motional model, since often the order

parameters are determined in a first step, subsequently interpreted using a

specific model analytically describing a motional mode (see Section 2.2.3.4).

However, this approach does only work if motional processes which are not


included in the motional model but, in pnnciple, do affect the overall order

parameter can be treated separately In Chapter 3, for instance, the order

parameters for the reorientation of the backbone ^N^H bond are descnbed

solely in terms of peptide plane reorientation according to the 3D GAF

model Such a description ignores fluctuations of the NH vector with respect

to the peptide plane frame which, in this case, consist of the stretching and

the in-plane and out-of-plane bending motion of the NH vector However,

these fluctuations are not correlated with the peptide plane reorientation and

act on a much faster timescale Thus, they can be included in an effective

dipolar interaction strength which is then defined by the effective distance

rfH [57,65]

eff -3 ~1/3

rNH = iSbend^rNH^stretch} >(2 46)

where rNH is the equilibrium NH distance Eq (2 46) results from an

independent treatment of the stretching and bending modes [65] Stretching-3 -1/3

vibrations lead to an averaged distance (rNH) t,= (rNH) The

averaging effect depends on the underlying potential A strictly harmonic

potential leads to a decreasing of the effective bond length This effect is only

partly compensated by anharmonic contributions to the potential [65]

According to the analysis of Henry and Szabo [65], the averaging effect of

the stretching vibration is almost completely independent of the environment

and the structure of the molecule which is a consequence of the fact that zero-_3

point N-H stretches are primary responsible for averaging (rNH) The

bending modes, which lead to a reorientation of the NH vector, add on the

faster stretching mode and can be taken into account by the order parameter

Sbencj leading to an increased effective NH distance The overall scaling

effect is rather small [65] due to the partial compensation of the averaging

effects of stretching and bending modes

2.2.3.3 Extraction of anisotropy of intramolecular motion

The anisotropy of intramolecular motion descnbed by reonentation about

three orthogonal pnncipal axes can be assessed expenmentally and by


computer simulations It can be extracted from NMR relaxation data if the

relaxation-active spin interactions have different directions in space In the

following, this shall be illustrated by an extended version of the "correlated

order parameter" approach of Bremi [62] For Gaussian axial fluctuation of

the interaction pnncipal axis e about a single axis e,the autocorrelation

2 2order parameter 5^ = S

ucan be denved from Eq (2 44) In case of

fluctuations much smaller than % it is approximately given by [63]

s\ = \ 3<Vln 9ucx' (2 47)

where 9 is the angle between the axes e and ea, and ca is the variance

of the fluctuation distribution

Fig 2 2 Fluctuation ellipsoid describing anisotropic intramolecular motion The

orientation of the interaction principal axis eu in the principal axis frame ew ea, e for

reorientation is given by the polar angles (9 <p )

In case of fluctuations about the three orthogonal pnncipal axes ea, ea, e,

the auto-correlated order parameter S is given by

sl=l~3(°asm V + °psin enp + Vin 9HY)2 22 2 22 22

= l-3(aa(l-sin 6 cos (p ) + Oq(1 - sin G^sin (p^) + oysin 9^),

(2 48)


where (0 , q> ) are the polar coordinates of the principal axis e in the

principal axis frame ea, ea, e which is shown in Fig. 2.2.

2 2 2If the order parameters S

, Sv, S^ for the three interaction principal axes

eu' ev eX> wnich are au" affected by three-dimensional reorientation about

the principal axes ea, en, e are known, it is possible to extract the

fluctuation amplitudes and therefore the anisotropy information by applying

the following transformation:

[o2a,al,G2/ = l-A~\{l-sl),(l-S2v),{l-S2x)]T, (2.49)

where the matrix A contains the orientation information of the three

interaction principal axes:

2 2 2 2 21 - sin 0,.cos cp.. 1 - sin 0„sin (p., sin 0..

2 2 2 2 21 - sin 0vcos cpv 1 - sin 0vsin cpv sin 0V

2 2 2 2 21 - sin 0^cos q>^ 1 - sin 9^ sin (p^ sin 0^

(2.50)

The three fluctuation amplitudes can be extracted in case of a regular matrix

A, which means that e, ev, ex have to be linear independent. The presented

approach might be useful to generally decide whether a set of spin

interactions is well-conditioned enough to extract the fluctuation amplitudes

and to estimate their errors given the errors contained in the order parameters.

However, it has to be considered that Eq. (2.48) represents a good

approximation only in case of fluctuation amplitudes smaller than 10° which

prevents a general usage of this equation.

In principle, the fluctuation amplitudes can be obtained from relaxation

data of one single spin. However, such an approach does not apply in many

practical cases since the contributing spin interactions are often not "well-

conditioned". For example, the relaxation of a 15N nucleus of the protein

backbone is governed by the (axially symmetric) magnetic dipolar ^N-'H


coupling and by the N chemical shielding anisotropy (CSA) interaction.

However, the CSA is approximately axially symmetric with the symmetry

axis nearly parallel to the ^N-1!! vector. Thus, fast intramolecular motion

scales the dipolar ^N-1!! and the 15N CSA interactions by nearly the same

2order parameter S

,the matrix A becomes nearly singular and the fluctuation

amplitudes cannot be accurately extracted from experimental data.

Fig 2 3 Orientations of the dipolar 15N-'H interaction and chemical shift anisotropy

tensors of 13C and 15N spins in the peptide plane.

The most promising approach for monitoring three-dimensional

anisotropic motion is the combination of relaxation data of different observer

nuclei fixed at a rigid molecular subunit in order to use their complementary

information to extract the anisotropy of the fragment's motion. This was first

shown by Bremi and Briischweiler [20] for the anisotropic peptide plane

reorientation using synthetic 15N and 13C relaxation data calculated from a

MD trajectory. The proposed analysis takes advantage of the fact that the

peptide-bond geometry remains, to a good approximation, planar at all times.

Thus, the relaxation-active CSA and dipolar interaction tensors have fixed

relative orientations shown in Fig. 2.3 and probe reorientational processes

about different directions in space.


2.2.3.4 Analytical treatment of the 3D GAF model

The internal correlation function for interactions modulated by the 3D

GAF peptide plane reorientation can be calculated by transforming

Eq. (2.41) [12,66]:

/, m, rri = -2

(2.51)

where the transformation properties of spherical harmonics [34] were taken

into account

Y2J(n;°'{T)) = £ D^(Q(x))Y2tm(ep/). (2.52)

m = -2

The polar coordinate set e]f ~ (6 , <p ) defines the time-independent

orientation of the interaction principal axis Li in the instantaneous peptide

plane frame (pp) ea(t), eM), eAt) rigidly attached to the peptide plane

according to Fig. 2.1b. Thus, the motional behavior of this interaction

principal axis in the molecular frame is described by the transformation

Dml(Q.(%)) where Q. represents a set of Euler angles which describes the

time-dependent transformation between the instantaneous peptide plane

frame and the molecular frame. Without loss of generality, the molecular

frame is set to be the peptide plane frame ea, en, e of the equilibrium

peptide plane. Then, the transformation from the instantaneous peptide plane

frame en(t), eM), eJt) to the molecular frame can be carried out by three

subsequent rotations with angles -a, -(3, -y about ea, e», e, respectively.

However, when using Wigner matrices it is convenient to decompose the

transformation into a series of six rotations: (i) with angle -y about e, (ii)

with rc/2 about e», (iii) with n/2 about ey, (iv) with a about e, (v) with

n/2 about eg, and (vi) with -0 about e . Taking into account this series of

rotations one obtains from Eq. (2.52)


.mol. (2)( 7t r,W2)f 11 K

y2, /(Q7(x)) = X *>${". \ -Pf^. £•^m(<") • (2-53)

k, m = -2

The Wigner rotation matrix elements can be decomposed [34]:

D?M^X)--e^d?Me-'lX, (2.54)

(2)where dkl (9) are the reduced Wigner matrix elements [34]. Substituting

Eqs. (2.53) and (2.54) into Eq. (2.51) yields the internal correlation function

2

C<*> =

y Zl2(kk)

, -i*'a(T) + i*a(0)v / i/p(x)-i/p(0Xe <e ){e )

I, k, k', m, rri = -2

, im,y(z)-imy(0\,(2)fn^,(2)fK\,{2)fn\,(2)fTt\v , p/\,7* , p/\

12

(2.55)In the next step the dihedral correlation functions of the type

(e'm ) have to be calculated for Gaussian axial fluctuation. Based

on the treatment by Chandrasekhar [67], Szabo [68] derived the dihedral

correlation function:

, im'y{x)-imy(0\ I 1 2

(e ')gaf = exp<j--a7

-

2 2 -Dx/a

m + rri -Imm'e (2.56)

where D is the diffusion constant for a diffusion process in a harmonic

potential and a is the standard deviation of the Gaussian distribution of

fluctuation angles y. Finally, the order parameter for the 3D GAF motion of

the peptide plane is given as the plateau value S = lim C'v(x) [20]:'Hv 'UA^

£.- % X'u.v exp

/ 2,,

2 .,2. 2,2 ,2,\i

oa(k +k' ) 22 cAm +m' y—~ <y -^—y—

/, k, k', m, rri

x <-oJ -'&\!>[?,(f>2f*>2i(?h Xp>rl j4">. asn\i

2.3 Calculation of relaxation-rate constants 47

where ca, cB, oy are the standard deviations (expressed in rad) of the

fluctuations about the principal axes ea, en, e„.

Equations (2.45) and (2.57) provide a convenient way to determine the

influence of 3D GAF motion on auto- and cross-correlated relaxation

parameters or, conversely, they allow the determination of the 3D GAF

parameters from experimental data as described in Chapter 3.

2.3 Calculation of relaxation-rate constants

For the calculation of relaxation rate constants of the different relaxation

pathways within a network of spins the master equation given in Eq. (2.6) is

transformed to the rotating frame [49]

j/(t) = -fr{pr(0-pj}, (2.58)

and then converted to the matrix form [49,69]

Jt =~lrnmK> (2-59)

m = 1

where it has been taken into account that T = F in case of restriction to

secular contributions only. Eq. (2.59) represents a linear system of first order

differential equations. The base coefficients bm result from decomposing the

deviation of the density operator from equilibrium in the rotating frameT T

|p (?) - p0) in terms of an orthogonal basis {Bm} in Liouville space:

<BJ(pT(0-pJ)>

(A\B) is the scalar product in Liouville space defined by the trace

(A\B) = Tr{A^B}. One has to select a basis {Bm} which is suitable for the

spin system under consideration. For X spins 1/2 the Liouville space of the

base operators has a dimension 2.In this thesis, a spin system IS of two

spins 1/2 has to be described and the 16 operators forming the shift-operator


basis are convenient [49]:

E, 21z, 2SZ, 4IzSz, I+S_, I_S+, I+, S+, I, S_, 2I+SZ, 2IvS+,

2I_Sz,2IzS_,I+S+,LS_. (2.61)

The matrix elements Tnm of the superoperator T are the transfer rate

constants between the base operators Bn and Bm, and are given by

r-=(vJ' (2-62)

Diagonal elements Tnn are the rate constants for an auto-relaxation process

of Bn, off-diagonal elements Tnm represents the rate constants for an cross-

relaxation process between operators Bn and Bm. For a single relaxation

mechanism, the relaxation matrix can be simplified and a block-diagonal

matrix is obtained [7], since many cross-relaxation rate constants are zero. In

particular, cross-relaxation between operators with different coherence

orders is precluded as a consequence of the restriction to secular

contributions (see Section 2.1.4); for example, cross-relaxation does not

occur between zero- and single-quantum coherence. Furthermore, only

coherences that are degenerate or nearly degenerate show cross-relaxation

between off-diagonal elements of the density operator in the laboratory

frame. Thus, "transverse" cross-relaxation does not occur in a system of two

unlike spins IS [69]. In case of more than one relaxation-active interaction,

cross-correlation between the interactions can lead to relaxation pathways,

which are forbidden for a single relaxation mechanism [69]. In biomolecules,

interference effects between dipolar and CSA interactions or between dipolar

interactions of different pairs of spins have to be taken into account.

Longitudinal relaxation of a heteronuclear IS spin system. The

regarded system consists of a spin S with a non-axially symmetric CSA

tensor (e.g. N or C backbone spins) and a spin / without chemical shift

anisotropy (to good approximation represented by a proton). The dipolar

interaction between the spins / and S depends on the fixed distance ris. It

has been shown [69] that the longitudinal relaxation pathways resulting from


dipolar and CSA interactions can be described by the subset of operators

{lz, S , 2/zSz} which corresponds to one block in the block-diagonalrelaxation matrix:

js

long

0 rsr2/lSzr2Vi,2/zSz

(2.63)

where relaxation described by the rate constant Fs 2j sresults only in the

case of cross-correlation between the dipolar and CSA interactions. The

auto-relaxation rate constant Fs scan be extracted from Tl experiments of

the spin S and is given as the sum of the dipolar (D) and CSA relaxation

contributions [37]:

sesz1,5

1

\,S^D

1

l,SJCSA(2.64)

with the dipolar relaxation rate constant

\,SJD

=lf^2 h_20U7t7 Un

2 2 -X2Y/Ys<>7s> {3/ (co5) + 7 (cd,-©^

+ 6/„ (a>7 + a>s)}, (2.65)

where y/( ys are the gyromagnetic ratios of the spins I and S, and (a,, co5

are the corresponding Larmor frequencies, h is Planck's constant and |X0 is

the magnetic field constant. /,,,.(«>) (with \i = {IS)) is the auto-correlated

spectral density function of the internuclear vector r]S. The longitudinal

CSA relaxation contributions of the symmetric part of the CSA can be

described as the sum of two auto-correlated CSA relaxation terms and one

cross-correlated CSA relaxation term according to Eq. (2.14)

V-)^ 1, S'CSA

(4i°x2jxx((as) + Oy^yy^s) + 2°xayJxy^s)} • (2-66)15


The parameters ov and o„ are defined by cv = cv -a,T andx y ~ x xx ZZ

oy = a -

ozz, and oxx, o , azz are the pnncipal values of the CSA

tensor Jxx{<&), Jyy((d) are the auto-correlated spectral densities of the CSA

pnncipal axes x and y, and Jxy((0) is the corresponding cross-correlated

spectral density

The cross-relaxation rate constant r; sresults solely from the dipolar

interaction and is given by

F'^ = 55® it) Y?Ys<^3>2{6Vffl/ + (n5)-V0/-^^

(2 67)

T[ sis contained in the heteronuclear steady-state NOE (Nuclear

Overhauser Effect) of spin 5 which is obtained after presaturation of spin /

NOE = 1 + — —2—\ (2 68)Ys T5

Mi

where Tx sis the total relaxation time of the spin S including all

contnbutions The other relaxation rate constants r; j , Ts 2j $ >m^

T2/ i 21 scan ^e monitored selectively by different expenments and are

given elsewhere [70] These rate constants have not been measured dunng

this work but their existence had to be considered in the design of the

relaxation expenments In particular, the cross-relaxation descnbed by the

rate constant Ts 2/ shad to be suppressed in T^ expenments (see

Chapter 3) to yield a mono-exponential decay when monitonng the auto-

relaxation process of Sz

Transverse relaxation of a IS spin system. The transverse relaxation of

the IS spin system can be descnbed by the operator subset {/+, 5, 25+/,}

[36] with the following block diagonal of the relaxation matnx


JS

_rv+ ° 0

o rs+;S+ TS+,2S+Iz0 r5+, 2S+IZ r2S+lz,2S+Il

(2.69)

The auto-relaxation rate constant Fs scan be extracted from T2

experiments of the spin S and can be decomposed into its dipolar and CSA

relaxation contributions:

s., s.

1

P2,S

1 1

T2,S*D ^T2,S'CSA

1 A lfVa\2fh\2 2 2, -3.2

+ /^((D; - cos) + 6y^(co7) + 67^(0); + ©s)} ,

t2,sjd 4ou«; wY/Y5<r's) {4V0)+37^(<°5)

(2.70)

(2.71)

r-)1^{^[4^(0) + 37^)] +

90

+ a/[4Jyy(0) + 3Jyy«os)] + 20,0^47^(0) + 3/^(a>5)]} . (2.72)

The cross-relaxation rate constant Ts SI= Fs 2s i

results from cross-

correlation between the dipolar and CSA interactions and takes the form [64]

*s,si = -^%S)f2^W<'-«>{^(4^/,*(0) + 3ys/>je(a)s))

+ cy(4JSIy(0) + 3JSIy((Os))} (2.73)

JS1 xand JSI are the spectral densities related to the cross-correlation

functions between the SI dipolar director and the principal axes x and y of

the CSA tensor of spin S, respectively. Cross-relaxation along this pathway

has to be suppressed in T2 experiments (see Chapter 3).

Recently, there was a revival of interest in dipole-CSA cross-correlated


cross-relaxation experiments in biomolecules [64,71,72]. One of their most

interesting applications might be the identification of conformational

exchange processes in the (xs to ms range [73]. The cross-relaxation rate

constant rs SIleads to differential transverse relaxation of spin S of the two

multiplet components belonging to the a and p states of the J-coupled /

spin. Modulations of the chemical shift of spin 5 equally affect both

multiplet components and therefore do not lead to differential relaxation,

provided that the 1JSI coupling is constant which often can be safely assumed

[74]. On the other hand, it is well-known that conformational exchange, that

modulates the isotropic chemical shifts, also contributes to T2 [75,76].

Thus, conformational exchange is identified by comparison between the

T2 srate constants and the cross-relaxation rate constant Ts S].

In

Chapter 3, the cross-relaxation rate constant TN NHinduced by cross-

correlated relaxation between the ^N-1!! dipolar interaction and the 15N

chemical shift anisotropy tensor is used to identify peptide planes which are

involved in conformational exchange processes.

Relaxation due to antisymmetric part of CSA. In Section 2.1.3 the

antisymmetric part of the CSA, which is a rank 1 interaction, has been

separated from the symmetric part. Due to the orthogonality relationships of

the Wigner rotation matrix elements (see Eq. (2.26)) there is no cross-

correlation between the two interactions of different rank and the

antisymmetric part gives rise to a completely independent relaxation

mechanism. T{ relaxation of the spin S due to overall tumbling modulation

of the antisymmetric part of the CSA can be calculated by taking into account

Eq. (2.15) [35]:

1 \anti 1 2 2 (I)

r-j = Wsi^LA^s)' (2-74)1\,SJCSA J

2 2 2 2 f 1)where Aoantl = G + Gxz + Gvz and / (co) is the rank 1 spectral density

function for a rigid molecule given in Eq. (2.36).

2.4 Molecular dynamics simulation 53

2.4 Molecular dynamics simulation

2.4.1 The force field

Molecular dynamics (MD) computer simulations describe the classical

motion of atoms or molecular fragments in a force field which has been

developed taking into account experimental data, quantum chemical

calculations and also empirical considerations [77-80]. A MD trajectory is

calculated by starting at a given initial conformation and solving Newton's

equations of motion for all the atoms in the system:

mrV,(0 = " ^V(rv r2,...,rs), (2.75)dt 3r,

where S is the number of atoms in the system, m[ and r, the mass and

position of particle i. The force field of the simulation package CHARMM

[77,78] which has been used in Chapters 3 and 4 is given by the following

potential energy function V :

V({rl})= l\Kb(b-bQ)2+ £ \kq(Q-%)2Bonds Angles

2 -

Improper Dihedrals

Dihedrals

pairs(i,j)

Cn{i,j) C6(i,jj12 6

+ V ^,(2.76)— ,47ten£r„

pairsxi, j) U r U

where b, Kb, and b0 are the bond length, the bond stretching force constant,

and the equilibrium distance parameter, respectively; 0, Kq ,and G0 are the

bond angle, the angle bending force constant, and the equilibrium bond

angle, respectively; \, K^, and £0 are the improper dihedral, its force

constant, and the corresponding equilibrium value, respectively; (p, K , n,

and 8 are the dihedral angle, its force constant, multiplicity, and phase,

respectively; r, ql, e0 and er are the non-bonded distance between atoms

i and j, the charge of atom i, the electric field constant, and a dielectric


parameter, respectively; Ci2(i, j) and C6(/, j) parametrize the Lennard-

Jones potential for the non-bonded interaction between atoms i and ;'.

MD simulations are of increasing importance for studying biomolecular

systems [18,19]. They represent the best theoretical technique to draw a

qualitative picture of biomolecular motions and to allow the derivation of

realistic motional models for intramolecular motion [37]. However, a

quantitative agreement between simulation and experiment is often not

observed. The differences partly result from the fact that liquid state NMR

relaxation measurements always reflect ensemble averages over a huge

number of molecules, whereas relaxation parameter extracted from a MD

trajectory result from a time average over a trajectory in the nanosecond

range. Results from MD simulation will be discussed in detail in Chapters 3

and 4.

2.4.2 Calculation of correlation functions

The internal correlation function c'"(t)= {P2(cos(6 (?)))) with the

angle 9 (f) between two axially symmetric interactions u. and v can be

calculated from the MD trajectory consisting of N snapshots with the time

increment At:

K 'k=\

where e^j. defines the orientation of the symmetry axis of the axially

symmetric interaction tensor ]i at snapshot k in the molecular frame. If the

correlation time of the observed motional process is sufficiently smaller than2

the simulation time, a plateau value of the internal correlation function, S ,

can be calculated by a single loop over the trajectory according to Eq. (2.87)

if o, avv are set to 1.

2.4.3 Extraction of 3D GAF fluctuation amplitudes

The principal axis directions and fluctuation amplitudes of the fluctuation

ellipsoid representing 3D GAF motion of a peptide plane can be determined


from the trajectory by an analytical method which was introduced by Bremi

and Briischweiler [20] and shall be derived in the following. A principal axis

frame consisting of three orthonormal vectors e{, e2, e3 is rigidly attached

to the equilibrium peptide plane. First, we assume the special case with axes

e[ = (1,0, 0), (t2 = (0, 1,0), (t3 = (0,0, 1) which are collinear to the

principal axes of the reference frame ea,en,e, respectively, defined for the

3D GAF model in Fig. 2.1b. Next, the peptide plane is reoriented by a

rotation with the matrix Rppf = Re (Y)^e(3(P)^ea(a) representing three

successive rotations, with angle a about ^, then with angle p about en, and

finally with angle y about e '

V/(0C'P'Y)

cos P cosysin a sin P cosy cos a sin P cosy

-cosasiny + sinasiny

„ .sin a sin 8 siny cosasinPsiny

cosPsiny+ cosacosy -sinacosy

-sinP sinacosP cosacosp

. (2.78)

The three orthogonal unit vectors are then given by et t= R Ja, P, y)e(

with i = 1,2, 3. Linear averaging over Gaussian distributions of the

reorientation fluctuations a, P, y yields the three vectors with respect to the

average peptide plane frame:

^ / -cl/2-al/2 \

(elt) = (<cospcosy)3OGAF,0,0) = [ep y

,0,0j,^ / -a2a/2-a2/2 \

<e2(> = (0, <cosacosy)3DGAf.,0) = \0,e ,0J,

(e~^t) = (0,0,<cosacosp)3DGAF) = (o, 0, e'^2~ °p/2j, (2.79)

where the Gaussian averaging is performed according to ref. [63]:

<cosmq>>GflIIM =

J(2ito(p2)

2-1/2 -<P /(2a ) -m*a;/2^

e cosm(paq> = e

_^ _^ _^

(2'80)

Note, that the linear averaged vectors (et t), (e2 (), (e3 ) ^^ neither

orthogonal to each other nor normalized to 1. In addition, these vectors do


not depend on the order of rotations R„ (a), /? (B), R (y) which is a

consequence of the Gaussian averaging Hence, the descnption with Rppt is

possible without loss of generality A diagonal product matnx

M,, = (e, ,) (e, ,) (i, J = 1,2,3) of these averaged vectors can beIJ 11 J t

calculated

M =

2 2

0T0 0

0 e

2

-v2

°y0

0 0 e

2 2

°P

(2 81)

and the fluctuation amplitudes are finally given by a linear combination ofthe

diagonal elements Aj, A2, A3 of the matrix M

2v A^Ata

Klogd[A,3

2 1A A3^ =

2ll0gA^ (2 82)

Rea,e2= Re$>In the general case with the orthogonal unit vectors e^

e3 = Re not colhnear to the principal axes ea, ea, e,where the rotation

by the matrix R describes the onentational offset from colhneanty and the

columns of R contain the coordinates of e^, e2, e3 with respect to the

equilibrium peptide plane frame, a product matnx M' is obtained which is no

longer diagonal However, it can be shown by a few transformations that theT

diagonal matrix M is given by the operation RM'R

This leads to the following procedure for extracting the fluctuation

amplitudes oa, o», a from the MD trajectory First, a frame of three

orthogonal unit vectors Cj, e2, £3 is fixed at the peptide plane of each

snapshot The onentation of this frame with respect to the peptide plane has

to be the same for all snapshots but can be arbitrarily chosen Diagonahzation

of the calculated product matrix M' yields the fluctuation amplitudes from

Tthe eigenvalues of the matrix M = RM'R via Eq (2 82) The orientations

for the principal axes of the peptide-plane reonentation are given by R e,,

R e1,R e-.


2.5 Motion-induced fluctuations of the CSA

The quantitative knowledge of chemical shielding anisotropy (CSA)

tensors is important in the context of many biomolecular NMR applications

CSA tensors provide important information on onentations of molecular

fragments and on the electronic environment of the nuclei, which depends on

the molecular geometry Moreover, the CSA contribution to spin relaxation

in liquids gives umque insight into overall motion as well as anisotropic

internal dynamics A quantitative interpretation of relaxation data greatly

benefits from the accurate knowledge of CSA tensors

CSA parameters can be extracted from solid-state NMR experiments [22]

In case of biological molecules, however, those studies are restricted to small

peptides Recent progress in quantum chemical methods for the calculation

of CSA tensors allows applications to molecules of moderate size with

increasingly quantitative agreement with experimental data [81-85] Density

functional theory (DFT) [23,86] has proven to be particularly useful to

calculate CSA tensors of model systems that represent essential fragments of

larger biomolecules

The CSA parameters of nuclear spins are functions of the molecular

geometry and its environment which are modulated by intramolecular

motion including vibrations, angular fluctuations, and fluctuations of near

contacts The effect of intramolecular motion on the CSA tensor might be

described by a two-level approximation In a first step, the CSA tensor can

be regarded as fixed at the molecular fragment of interest and reorients

according to the fragment's motional modes On a second level, the

magnitude and the orientation of the tensor with respect to the molecular

fragment are modified due to the changes in the chemical and electronic

environment induced by the intramolecular motion Rapid CSA tensor

modulations are sensed differently depending on the NMR experiment

While solid-state NMR experiments yield a linearly time-averaged CSA

tensor, NMR relaxation experiments reflect the variance of CSA tensor

fluctuations via second order pertubation theory

[87]:4n)factoraintroducing(by(2.40)Eq.from

calculatedbecanfunctioncorrelationtheCmt(-x),=Cint(x)accountinto

Takingterms.cross-correlatedallandauto-allofsumtheiswhichfunction

correlationrank2ndabydescribedbethencanframemoleculartheinvalues

principalandorientationstensorCSAtheoffluctuationstime-dependentThe

(2.83)

2

-1

-1

^ozzdiag+

-1

^oyydiag+

-1

-1

2

\f

diag1

vvdiag

tensors:symmetricaxiallyfree,

trace-"orthogonal",threeofsumaasrepresentedbenowcantensorCSA

symmetric)axiallynecessarilynot(butsymmetricThek°zz,k>°yy,k'axx,

eigenvaluestracelessandkez,

k'

c>,

k>

x,ek,ek,exeigenvectorsofseries

atoleadingdiagonalizedandsymmetrizedisframe,moleculartheingiven

trajectory,theof{Gk}tensorCSAEachfollowing:theinexplainedbewill

procedureThisfunction.correlationacalculatingbycharacterizedbetohave

trajectoryCSAtheincontainedparametersCSAtheoffluctuationsThe

trajectoryCSAofAnalysis2.5.1

analyzed.be

willubiquitinproteintheinspinsbackbone15NandCoftrajectoriesCSA

4,ChapterInmolecule.theofmotionintramoleculartheondependdirectly

fluctuationswhose...,N)I,=(k{ck}tensorsCSAofseriesatoleadsThis

trajectory.MDtheincontained"snapshot"everyforcalculatedisnucleus

certainafortensorCSAAimportance:centralofistrajectoryCSAthe

approach,thisIntensor.CSAtheoneffectsdynamicalstudytopossibilities

largeoffercalculationsDFTandsimulationMDofcombinationThe

MotionMolecularandRelaxationSpin58


Cin,(T) = (oxx(0)Oxx{t)P2(ex(0) ex(x)))

+ (a (0)ayy(x)P2(e (0) • e (x))>

+ <c77(0)c77(T)P2(e7(0) • e,(x))>

+ 2<o„(0)ovv(T)P2(^(0)-ev(T))>

+ 2<oJCC(0)ozz(x)P2(eJt(0)-ez(x))>

+ 2(ayy(0)azz(T)P2(ey(0) ez(x))). (2.84)

Due to the functional form of Eq. (2.84) it is not necessary to assume any

ordering of the eigenvalues. They can be taken in arbitrary order, e.g., in the

order they are provided by the diagonalization subroutine. In other words, for

each snapshot the axes x, y, z together with the eigenvalues can be arbitrarily

permuted. The individual terms of Eq. (2.84) are rather meaningless, since

only their total sum gives rise to an experimentally observable relaxation

effect. The initial value of the internal correlation function contains the

interaction strength and is not normalized to 1:

c,„/(0) = <<4> + (o2yy) + (a2zz) - (axxoyy) - (axxazz) - (ayyozz).

(2.85)

2The plateau value of the correlation function, the order parameter 5

,can be

calculated as a sum of order parameters which correspond to the auto- and

cross-correlated terms in Eq. (2.84):

S2 = S2XX + S2yy + 4 + 2S% + 2S2XZ + 2S2yz, (2.86)

2where the 5„

vcan be calculated according to ref. [62] including the principal

value fluctuation:


V = 4<0unV^0vvZv>-4(<annV^0vv)+<V^0vvZv))

+ \<V)(°w> + 3«VW<°vv(Zv*v)>

+ <VZHVKvW + <Wu»<°vvW)

+ 4(<VXJ)(°vvX5> + (V^^vv^v)

-<o^xJ><ovvyJ> - <aw^)(ovvxJ)), (2.87)

where (I, V = x, y, z are labels for the vectors of the corresponding CSA

principal axes and X, Y, Z are the cartesian coordinates of these vectors in

the molecular frame.

In case of very fast CSA fluctuations with correlation times below 20 ps the

CSA relaxation rate constants can be calculated based on the calculated order

parameters. This will be described in Chapter 4.5.1.

2.5.2 CSA averaging due to 3D GAF motion

CSA tensors which are extracted from solid-state NMR experiments at

room temperature are affected by motional averaging due to the

intramolecular motion present in the solid state. The motional averaging

effect can be calculated analytically for the 15N and 13C backbone spins,

assuming that the CSA tensor is fixed at a peptide plane undergoing 3D GAF

motion and that the motional reorientation does not lead to a modification of

the tensor's principal values and principal axis orientations with respect to

the peptide plane frame. The calculation is somehow analogous to the

procedure of Briischweiler and Case [88] who calculated the effect of

harmonic motion on the Karplus equation for spin-spin coupling.

A CSA tensor Z with an arbitrary orientation with respect to the

equilibrium peptide plane frame (ppf) ea, ea, e,can be calculated as

? = Rppf(cC, P', •f)qR~pf(a\ P', y'), where g = diag(axx, a azz) and


the rotation matnx R * (see Eq (2 78)) descnbes a rotation with fixed

angles a', P', y' about the pnncipal axes ea, en, e, respectively The

motionally averaged CSA tensor 1! is given by averaging over the ensemble

of CSA tensors reonented by 3D GAF motion

r = <J?^a,p,Y)*w/^P\Y)gJ?^(2 88)

3DGAFwhere ( )a r Y

defines die average over the Gaussian distnbution of

reonentation angles a, P, y For an CSA tensor ? with an onentation

corresponding to (a1, p', y') £ (0, 0, 0) the explicit solution for X' is quite

lengthy (not shown) Here, the solution for the trivial case

(a', P', y') = (0, 0, 0) with the pnncipal axes ex, e, ez of the CSA tensor

£ collmear to the pnncipal axes ea, en, e of the peptide plane frame, shall

be given With the matnx R <• defined m Eq (2 78) and the relations

2 GAF 1 1 -2a2<(cos(p) >(p = - + -e Tsec((p),and

2, GAF 1 1 -2a.2.((sincp)2^ = V~y ^es(<p), (2 89)

where ec((p), es(<p) with (p = a, p, y are functions of the fluctuation

amplitudes aa, oR, a descnbing the peptide plane reonentation, the

pnncipal values of the diagonalized averaged CSA tensor S' are finally given

by

a'xx = <yyy(es(a)es{$)ec(Y) + ec(a)es{y))

+ 0zz(ec(oc)es(p)ec(y) + es(a)es(y)) + oxxec($)ec(y),

+ azz(ec(a)es($)es(y) + es(a)ec(y)) + axxec($)es(y),

°\z = ^xx^(P)+ <V»ec(P) + <VC(«)*C(P) (2 90)

The onentation of the averaged CSA tensor S' is not modified compared to


the reference tensor Z The trivial case with (a', (3', y') = (0, 0,0) is useful

since the 13C CSA principal axes are approximately colhnear to the peptide

plane frame pnncipal axes (see Fig 2 3) An analogous calculation leads to

the "removal" of the 3D GAF motion from the averaged CSA tensor Z' in

order to obtain the tensor Z A detailed discussion of motional averaging of

CSA tensors using also this "3D GAF averaging" approach is given in

Chapter 4

3 1 Introduction 63

3 Anisotropic Intramolecular Backbone

Dynamics of Ubiquitin

3.1 Introduction

In this chapter, it is demonstrated experimentally that anisotropic

intramolecular backbone motion can be characterized by a combined

interpretation of relaxation data of 13C and 15N nuclei belonging to the same

peptide plane The procedure follows the general protocol originally

developed for the characterization of side-chain dynamics and subsequently

applied to the characterization of backbone dynamics, exemplified on the

cyclic decapeptide antamanide [20,37] The protocol uses information

gained from molecular dynamics (MD) computer simulations and from

analytical treatments to find a suitable parametnzation of the motion causing

nuclear spin relaxation From the detailed analysis of a 1 5 ns MD trajectory

of ubiquitin solvated in a box of water, the basic motional processes affecting

backbone spin relaxation are determined and expressed in terms of an

analytical motional model It is found that 76% of the backbone peptide

planes are not involved in conformational exchange processes and show

predominantly small-amplitude motion Their relaxation behavior can be

described by the 3D Gaussian axial fluctuation (GAF) model which has been

introduced in Section 2 2 3 Each peptide plane is treated as a rigid entity

exhibiting rapid reonentational motion about three orthogonal principal axes

with a Gaussian fluctuation distribution The axis of maximum angular

fluctuation turns out to be nearly parallel to the C, _ j- C; direction The

model, expressed in analytical terms, is applied to experimental T! and T2

data of the 13C and 15N backbone spins, and to {^HJ-^N NOE data

collected at two magnetic field strengths (400 and 600 MHz proton

resonance frequency)

The approach in this chapter differs from procedures previously proposed

for the dynamical interpretation of backbone relaxation data of proteins

Most studies focused on the interpretation of N relaxation data in terms of

the standard Lipan-Szabo approach Recently, experiments were proposed

64 Anisotropic Intramolecular Backbone Dynamics of Ubiquitm

for monitoring C relaxation. Engelke and Ruterjans [89] used a model-

free description assuming identical order parameters for the three C CSA

principal axes which is equivalent to assuming isotropic internal motion at

1 ^the C sites. They found for ribonuclease Tl significant discrepancies

between the derived C and N order parameters belonging to the same

peptide bond, but no physical explanation was provided. Allard and Hard

[90] recognized the importance of dipolar contributions of the neighboring

protons to 13C relaxation and the need for separate order parameters for the

different interactions. They modeled the backbone motion of the

thermostable Sso7d protein by two order parameters for each 13C, one

describing motion at the backbone c site and one describing the

(effective) motion of the internuclear vectors to the neighboring protons.

Similar to the treatment by Engelke and Ruterjans, their approach assumes

isotropic intramolecular motion at the C site. Dayie and Wagner [91,92]

derived information on the spectral densities of the carbonyl CSA interaction

in villin 14T. No interpretation in terms of anisotropic intramolecular motion

was undertaken. Zuiderweg and coworkers [93] measured numerous auto-

and cross-correlated relaxation rate constants involving 15N and 13C

backbone atoms of flavodoxin and determined the associated motional order

parameters. Differences in these order parameters were ascribed to

anisotropic peptide-plane motion, which was modeled in terms of uniaxial

restricted diffusion. No attempt was made to interpret the set of relaxation

data by a unified motional model.

In Section 3.2, the biological role and the structure of the studied protein

ubiquitin is characterized. In Section 3.3, details of the 15N and 13C NMR

relaxation experiments are given, whereas the generation, processing and

analysis of the MD trajectory is described in Section 3.4. In Section 3.5, the

extraction of the motional 3D GAF parameters from the NMR relaxation

data is outlined in detail, followed by a discussion in Section 3.6.

3.2 Ubiquitin

Ubiquitin is a highly conserved, 76-residue protein (molar mass of

8565 gmol ) which is present universally in eukaryotic cells. It is involved

3 2 Ubiquitin 65

in a variety of cellular processes, most importantly protein degradation

Proteins that are selected for degradation are covalently linked to multiple

molecules of ubiquitin The linkage is an isopeptide bond between the C-

terminal glycine in ubiquitin and side-chain amino groups of lysine on the

target protein The conjugates are then recognized by special proteases

within the cell Most m-vivo- degradation of abnormal or short-lived proteins

is mediated by the ubiquitin-dependent pathway More details concerning the

biochemical function of ubiquitin can be found in refs [94,95]

The primary structure of ubiquitin is given in Fig 3 la Ubiquitin lacks

cystem and tryptophan residues and includes only one histidme, one tyrosine,

and one methionine at its N terminal The isoelectric point is at 6 7 due to

11 acidic and 11 basic residues (plus one histidme) The crystal structure of

human ubiquitin was determined and refined to 1 8 A by Vijay-Kumar et al

[96] The structure in solution determined by NMR [98-100] is virtually

identical to the crystal structure As illustrated in Fig 3 lb, the structure is

rich in secondary structure elements, including a mixed five-strand (3 sheet,

which contains two strands (residues 1 to 7 and 64 to 72) in parallel and three

strands (residues 10 to 17, 40 to 45, 48 to 50) in antiparallel direction, an a

helix (residues 23-34), a short 310 helix (residues 56-59), and seven reverse

turns Nearly 90% of the backbone is involved m a hydrogen-bonded

secondary structure The curved p sheet and the flanking a helix enclose a

single core of densely packed hydrophobic side chains, which is likely to

contribute to the high stability of ubiquitin towards denaturation by heat,

extreme values of pH, and denaturing agents [101,102]

66 Anisotropic Intramolecular Backbone Dynamics of Ubiquitin

b)

1 5 10

Met-Gln-lle-Phe-Val-Lys-Thr-Leu-Thr-Gly-Lys-Thr-lle-15 20 25

Thr-Leu-Glu-Val-Glu-Pro-Ser-Asp-Thr-lle-Glu-Asn-Val-30 35

Lys-Aia-Lys-lle-Gln-Asp-Lys-Glu-Gly-lle-Pro-Pro-Asp-40 45 50

Gln-Gln-Arg-Leu-lle-Phe-Ala-Gly-Lys-Gln-Leu-Glu-Asp-55 60 65

Gly-Arg-Thr-Leu-Ser-Asp-Tyr-Asn-lle-Gln-Lys-Glu-Ser-70 75

Thr-Leu-His-Leu-Val-Leu-Arg-Leu-Arg-Gly-Gly

G76

Fig. 3.1 (a) Primary sequence and (b) schematic representation of the native state of

human ubiquitin [96] drawn by using Molscript [97],

3.3 Experimental 67

Ubiquitin has been subject of numerous NMR studies. H, C, and N

resonance assignments are available at pH 4.7 and 30°C [103], and at pH 5.7

and 30°C [104]. The backbone dynamics of ubiquitin at pH 5.7 and 30°C

was investigated by Schneider et al. [105] in terms of a standard analysis of

15N relaxation data. They found a correlation time of 4.1 ns for isotropic

overall tumbling motion and no apparent correlation between secondary2

structure and order parameters SNH for the NH vectors. The rotational

tumbling of ubiquitin at pH 4.7 and 30° C was analyzed in more detail by

Tjandra et al. [106]. An axially symmetric rotational diffusion tensor with a

diffusion anisotropy of ~ 17 % was extracted from N relaxation data

together with the order parameters for the NH vectors. Wand et al. [104] used

fractionally 13C-labeled ubiquitin to extract order parameters for the

oc ex

dynamics of backbone C - H vectors and methyl groups. They found, for

those residues within an element of secondary structure, the order parameters2 2

Sa a usually larger than the order parameters SNH of the same residue. A

C M

detailed investigation of the backbone dynamics of the partially folded A

state of ubiquitin was presented by Brutscher et al. [73]. The analysis of

cross-correlation relaxation data of the backbone of native ubiquitin was

subject of several NMR studies [64,71,107]. Ubiquitin was also used as

model compound in some molecular dynamics simulation studies. Braatz et

al. [108] compared a time-averaged simulated structure with the

experimental crystal structure. Abseher et al. [109] characterized partially

unfolded states of ubiquitin with MD simulation.

3.3 Experimental

3.3.1 Sample preparation

Fully 13C, 15N-labeled ubiquitin was kindly provided by A. J. Wand

(Buffalo, NY) and purchased from VLI (Southeastern, PA). For optimal

consistency, all C and N relaxation data used for the analysis were

measured on a single ubiquitin sample containing 2 mM ubiquitin in 90%

H20 and 10% D20 at pH 4.7 with a 45 mM sodium acetate buffer. The

sample was deoxygenated and sealed in a standard 5 mm NMR sample tube.


3.3.2 NMR experiments

3.3.2.1 General setup and techniques

NMR experiments were performed on Bruker AMX-600 and DMX-400

spectrometers, both with triple-resonance (1H, 15N, 13C) equipment

including shielded z gradients. The sample temperature was set to 300 K.

Special care was taken to achieve a temperature variation of less than 0.5 K

between different relaxation experiments and different spectrometers by

using the temperature dependence of differential cross-peak shifts in the 2D

HSQC correlation spectra as an internal probe for calibration of the actual

sample temperature. This is illustrated in Fig. 3.2 for a HSQC spectrum

resulting from a 15N Tt experiment. For all NMR experiments,

WATERGATE [110] and water flip back pulses [111] were inserted in the

pulse sequences to obtain a relaxed spin state of the water magnetization

before acquisition. Pulsed field z gradients were applied for coherence-

transfer-pathway selection [112]. Quadrature detection in the indirect

dimensions of the two-dimensional experiments was obtained by the TPPI-

States method [113]. The spectral widths and the carrier frequencies (in

parentheses) were set for *H to 12 ppm (4.76 ppm) and for 15N to 22 ppm

(118.76 ppm), the carrier frequencies for 13Ca to 57.2 ppm and for 13C to

173.9 ppm. The frequency for the selective 13C and 13Ca pulses was

switched by time-proportional phase incrementation during rf pulses.

Selective rectangular 90° pulses were used with pulse lengths of 74 u.s (400

MHz) and 57 |0.s (600 MHz). The tip angle was changed by adjusting the

power level.

3.3.2.2 2D 1SN relaxation experiments

Tj and NOE experiments were performed at 400 and 600 MHz, T2

experiments in the rotating frame at 600 MHz proton resonance frequency.

The used pulse sequences are shown in Fig. 3.3. Since a fully I3C, 15N -

labeled sample of ubiquitin was used, the Ca and C nuclei had to be

broadband decoupled during the evolution period tx. lH 180° pulses and

selective Ca and C 180° pulses were applied during the T{ and T2 mixing

3.3 Experimental 69

time delays to suppress cross-relaxation and effects caused by cross-

correlation between 15N CSA and ^N^H dipolar or 15N-13C dipolar (DD)

interactions. During the T2 relaxation period, chemical shift and JNH-

coupling evolution were suppressed by applying spin-lock rf fields with

17^2^1/271= 1.9 and 2.6 kHz. In these experiments the magnetization is

locked along an effective field vector with an orientation depending on the

frequency offset Av of the 15N nucleus. The measured relaxation rates jeas

were corrected for these offset effects using the relation

Tcorr = r^fli(riSin20)/(ri_r^ascos2e) with the angle 6 defined by

tan 9 = |yArB1|/2jtAv.

Two sets of Tj experiments were carried out both at 400 MHz and at

600 MHz to estimate statistical errors. The Trrelaxation decay was sampled

at ten time points: 20, 60, 140, 240 (2x), 340,480,660, 800, and 1000 ms at

400 MHz and 20 (2x), 60, 140, 240, 360, 480 (2x), 660, and 800 ms at

600 MHz. The relaxation decay of the two sets of T2 measurements in the

rotating frame was sampled at eight different time points: 8, 24, 48, 72, 96,

120, 160, and 200 ms at 600 MHz. The matrix size of the acquired complex

2D data was 128 x 512 with acquisition times of 143 ms (tj) and 107 ms (t2)

at 400 MHz and 96 ms (tj) and 71 ms (t2) at 600 MHz using 16 scans per

complex tj increment. For the {1H}I5N-NOE measurements, two spectra

with H saturation and one without were recorded in an interleaved manner.

Different H saturation schemes of 5 s length were applied during the recycle

delay of 7 s: either a windowless WALTZ-16 sequence or a train of 120°

pulses applied every 10 ms. The long recovery delay of 7 s was chosen to

allow the initial 15N magnetization to reach its equilibrium value despite

saturation transfer by water exchange [114]. The three acquired matrices

were 75x512 (400 MHz) and 128x512 (600 MHz), and 32 scans per

complex tj increment were used for each of the three data sets.


7 7 [ppm]

Fig 3 2 'H 15N correlation spectrum resulting from a 2D N 7", relaxation

experiment (see Fig 3 3a) with mixing time xm = ~^[ls Four pairs of two overlapping

peaks are sui rounded by circles The corresponding peptide planes were excluded from

further analysis (see Section 3 5 1) The region of three peaks surrounded by a square

belongs to the iesidues Val 5 Leu 8 and He 44 The measurement of differential

chemical shifts between the peaks of Val 5/Ile 44 and the reference peak (Leu 8) allows

to contiol the sample temperature which varies by less than 0 5 K between different

experiments at different spectrometers

3.3 Experimental 71

a) y y

1h m, n p -r.n.io„

-y o.

15nI n lAHAl Tm lAlv2 e I ;aHa| waltz

13C

PFG

JL GARP GARP

b) y y

1h IaQai. n

13,

I<£iy

15N I n U^ha^ W^ I lAHAl WALTZ

JL GARP GARP

PFG

C)Saturation: 5 s

1h nnnnnnnnnnnn

15N

13C

PFG

<J>1

JL

|Aptl/2; e | | Af| A | WALTZ

GARP GARP

Fig. 3.3 Pulse sequences used for (a) 15N T,, (b) 15N T2, and (c) {'H}-'5N-NOEmeasurements. Delays are set to A=2.2 ms. e=A+t|/2. Phase cycling: <!>] = [x, (-x)];

C>rec = [x, (-x)]. The C carrier is set between the 13Ca and C resonances to

decouple both nuclei during the tj evolution of N.


3.3.2.3 2D C relaxation experiments

2D 13C Tj and T2 relaxation experiments were carried out at 400 and 600

MHz using HNCO-type experiments [90-92,115]. The pulse sequences are

depicted in Fig. 3.4. With the evolution period tj on N one has the

possibility to directly compare the actual sample temperature in N and C

relaxation experiments by extracting differential chemical shifts in the

corresponding 1H- N correlation spectra (see Fig. 3.2). During the constant

time evolution period tt on 15N the protons are decoupled with WALTZ-16

and 13Ca with one selective 180° pulse; during the mixing time, *H and 15N

180° pulse trains were applied in both Tj and T2 experiments and a train of

selective Gaussian 180° pulses of 400 (is pulse length every 100 ms in the Tl

experiment to suppress both cross-relaxation between C and surrounding

spins and cross-correlation effects between c CSA and dipolar

interactions (15N-13C\ 1H-13C, 13Ca-13C). During the T2-relaxation period

a spin-lock with field strengths of 1.7 kHz (400 MHz) and 2.7 kHz

(600 MHz) was applied on 13C, but no pulses were applied on 13Ca. At

600 MHz the spin-lock experiment was compared with a CPMG experiment

using a train of selective C 180° pulses every 500 [is. No systematic

deviation between the two experiments was found.

Statistical errors were estimated from two sets of Tj experiments at

400 MHz and three sets of T] experiments at 600 MHz. The T{ relaxation

decay was sampled at 15 and 11 time points: 4,200, 300,400,500,600,700,

800,1000,1200,1500,1800,2200,2600, and 3000 at 400 MHz and 20,200,

400,600, 800,1000,1200,1400,1800,2200, and 2600 ms at 600 MHz. The

T2 relaxation decay was sampled at 11 and 12 different time points: 25, 50,

75, 100, 125, 150, 200, 250, 300 (2x), and 400 ms at 400 MHz and 25, 50,

75, 100, 125, 150, 200, 250, 300 (2x), 350, and 400 ms at 600 MHz. The

matrix sizes of the acquired complex 2D data were 25 x 512 (400 MHz) and

38x512 with acquisition times of 28 ms(ti)and 107 ms (t2) at 400 MHz and

28 ms (tt) and 71 ms (t2) at 600 MHz using 64 (400 MHz) and 32 (600 MHz)

scans per complex ti increment.

3 3 Experimental 73

a)y v

x 0n

1H |a[1aL|2A[ WALTZ I PI Pi l~l PI I WALTZ I2A, Jb'-Xn-X ^rec

£>0-,|

y <t>4

i5n| n i .a.n n n n n u^î AnAÂLTzio2 y ^3

13C. |GARP|

I :v2fl13ca [GARP|

pfgOI I n 11 «

b)y y

'H I41aL|2A| WALTZ | innn I WALTZ |2A|

isn| n T a\\4 n n n pW-X

-V2|l--"'

,A||A|WALTZ|

«c- n lî^&H n |GARP|

13C« ,V2fl |GARP|

pFGnn « n n n

Fig 3 4 Pulse sequences used for (a) nC T, and (b) l3C T2 relaxation measurements

Delays are set to A=2 2 ms, A[=8 5 ms, A2=14 ms, and A^=15 ms Constant time

evolution TN=A2 Phase cycling <1>|= [(-y), y], <t>2= !.4x,4( x)], <J>3= [16(-y), I6y],04= [8x, 8(-x)], Orec= [x, -x, x, -x, 2(-x, x, -x, x), x, -x, x, -x, -x, x, -x, x, 2(x, -x, x, -x),

-x, x, -x, x]

3.3.3 Extraction of relaxation data

The data were processed with the FELIX program version 95 0 (Biosym

Technologies) Prior to 2D Fourier transformation, the time-domain data

were zero-filled in the t2 dimension to 2048 complex points and multipliedwith a cosine-bell window For the

' 5N relaxation experiments, zero-filling


was applied in the ^ dimension to 1024 real points followed by

multiplication with a cosine-bell window. For the C relaxation

experiments, a mirror-image linear prediction procedure [116] was applied

along the constant-time ti dimension, which was followed by zero-filling to

1024 real points and multiplication with a Kaiser window. The resonance

assignments were taken from Wang et al. [103]. Peak intensities were

extracted from 2D spectra using a local grid search routine for each cross

peak. Tj and T2 values were determined by fitting the measured peak heights

to the mono-exponential function I(tm) = /0exp(-xm/7,1 2) with two fit

parameters, Tt 2and the peak intensity /0 at mixing time %m = 0.

Representative decay curves for C T[ and T2 measurements are shown in

Fig. 3.5. The { H}-15N steady-state NOE values were determined from the

ratios of the measured cross-peak intensities in the presence (Isal) and

absence (Iunsat) of proton saturation: NOEmeas = Isa/1unsat.

Fig. 3 5 Typical magnetization decay curves for peptide planes Lys 6 and Asp 32 in a

P strand and in the central a helix of ubiquitin, respectively Peak intensities were

measured as peak heights (a) The 13C Tj fit is based on 11 experimental data points

measured with the experiment of Fig 3 4a at 600 MHz proton resonance frequency

with mixing times of 20, 200, 400, 600, 800, 1000, 1200, 1400, 1800, 2200, and

2600 ms (b) The 13C T2 fit is based on 12 experimental data points measured with the

experiment of Fig 3 4b at 600 MHz proton resonance frequency with mixing times of

25, 50, 75, 100, 125, 150, 200, 250, 300 (2x), 350, and 400 ms


3.4 Molecular dynamics simulation

3.4.1 Generation of molecular dynamics trajectory

The coordinates of the X-ray structure of ubiquitin [96] in the Brookhaven

data bank (file lubq) were used for the starting conformation of the MD

simulation. All protons were added in their standard geometric positions

using the CHARMM program [77,78]. The resulting structure was then

energy-minimized in vacuo and immersed in a cubic box of a side length of

46.65 A containing a total of 2909 explicit water molecules. The simulation

was performed with the CHARMM force field version 24b2 under periodic

boundary conditions with an integration time step in the Verlet algorithm of

1 fs. The SHAKE algorithm [117] was applied to all bond lenghts involving

a hydrogen atom. A cutoff of 8 A was used for non-bonded interactions.

Truncation was done with a shifting function for electrostatic interactions

with a dielectric constant e = 1 and a switching function for van-der-Waals

interactions. The temperature was set to 300 K, and after an equilibration of

500 ps, snapshots were stored in intervals of 1 ps, leading to a total of 1500

conformations for the 1.5 ns simulation time. The snapshots were then

postprocessed by a mass-weighted least-square difference rotation and

translation of the protein backbone atoms with respect to a reference

conformation at 750 ps simulation time. In this way overall rotational and

translational diffusion of the molecule, that occurs during the simulation, is

eliminated yielding atomic coordinates for each snapshot in the same

molecular reference frame.

3.4.2 Processing of trajectory

3.4.2.1 Determination of equilibrium peptide plane

The following cluster analysis was performed to determine the equilibrium

orientation of a peptide plane:

1. For peptide plane i, the corresponding peptide plane frame with the

orthonormal axes ea k,e»

k, ey k,defined in Fig. 2.1b, is calculated for each

snapshot k = 1, ...,N by applying the following procedure:


i- (x (x- The axis e

kis given by the normalized Cx _ j Cl vector.

a „ax

The two vectors a =,

'

'—r and t> =

N.C',.! , C'.^C,., C^N,

N.C',.,, c ca CaNdefine

the peptide plane. The influence of the dihedral angle co( is averaged out

due to this definition. The axis enkwhich is orthogonal to the plane is

then given by normalization of the vector resulting from axb.

- Finally, the in-plane principal axis is given by ea k= en

kX ey k.

2. A matrix K of dimension N x N is build where the matrix element Kkl is

a measure for the difference in orientation of the two peptide plane frames of

snapshots k and I: Ku = (e^kx^) +(«j^x^) + (^xe~^{) .

3. Summation over each row of the matrix K yields N row sums, the smallest

of which corresponds to the snapshot with the equilibrium peptide plane.

4. In case of a 3D GAF motion with its Gaussian distribution of peptide plane

fluctuation amplitudes there are many peptide planes which lie very close to

the one with a minimal row sum. This was used to select an equilibrium

peptide plane which is nearly planar (o)( = 180°).

3.4.2.2 Extraction of fluctuation amplitudes from the trajectory

The fluctuation amplitudes of the 3D GAF model can be determined from

the trajectory by two different methods which should be regarded as

complementary. The first one (the analytical "matrix method") has been

derived in Section 2.4.3 and allows a fast extraction of the fluctuation

amplitudes and the orientations of the principal axes for the peptide plane

reorientation. However, it does not allow to check if the distributions of the

fluctuations about the three principal axes are Gaussian or not. This can be

achieved with a much more time-consuming "alignment method" [62]. Here,

the fluctuation amplitudes are calculated for each of the 1500 snapshots by

aligning the instantaneous peptide-plane axes ea k,«» fcand e

k by a


transformation with their corresponding equilibrium directions which were

calculated according to Section 3 4 2 1 The transformation is accomplished

by three successive rotations (l) rotation by the angle a about the axis ea,

(u) rotation by |3 about the axis en, and (in) rotation by y about the axis e

Due to the smallness of each of these rotations, the order of their application

is not crucial The analysis of the resulting distributions of the fluctuations

about all three axes is a main criterion for deciding whether a peptide plane

shows 3D GAF motion or not In case of Gaussian distributions the

fluctuation amplitudes are given as the corresponding standard deviations

However, this method is not appropriate for determmg the orientations of the

reonentation pnncipal axes which reveals the complementary character of

the "matnx" and the "alignment" method For all peptide planes with

dominant 3D GAF motion, the fluctuation amplitudes obtained by the two

methods are in good agreement Conversely, agreement between the two

procedures is a useful indicator for the dominance of the 3D GAF motion of

the peptide plane under consideration

3.4.3 Analysis of MD trajectory

3.4.3.1 Selection of peptide planes with 3D GAF motion

Throughout the whole analysis the peptide planes are labeled and

numbered by the amino acid residue that contnbutes the mtrogen atom The

plane of (a non-prolme) peptide bond i contains the atoms N:, H; ,

"expand O,.!

For each of the 72 non-prohne peptide planes in ubiquitin, the

reonentational probability distnbutions were obtained from the 1500

snapshots of the trajectory by using the "alignment method" desenbed in

Section 3 4 2 2 It is found that the motion of 57 non-prohne peptide planes

involves predominantly 3D Gaussian axial fluctuations In Fig 3 6a results

are shown for the representative plane of He 30 Nearly all peptide planes

located in the a helix or in one of the (3 strands belong to this group The

effective internal correlation times xint of the 3D GAF motion is for most of

the 57 peptide planes well below 20 ps


The remaining 15 peptide planes belonging to residues in the loop regions

including Thr 7 to Lys 11, Gly 35, He 36, Gly 47, Gly 53, Arg 54, and the

flexible C-terminus, Arg 72 to Gly 76, show additional jump processes on

slower time scales. This either leads to bimodal angular distributions with

two dominant conformations (see Fig. 3.6b for peptide plane Leu 8) or to

asymmetric distributions due to exchange between a larger number of sites.

They were excluded from further analysis which can be seen in Table 3.1.

Table 3 1 Exclusion of peptide planes whose motion cannot be described by a 3D GAF

model both in experiment and simulation

Reason for exclusion Excluded planesRemaining

planes

Proline residues Pro 19, Pro 37, Pro 38 72

Slow motion apparent in

MD trajectory

Thr 7 to Lys 11, Gly 35, He 36, Gly 47,

Gly 53, Arg 54, Arg 72 to Gly 76

57

NMR relaxation data

not available"

He 13, Asp 21, Glu 24, Ala 28, Gin 31,

Gly 53, Leu 67, Leu 69, Arg 72, Leu 73

50

Slow motion apparent in

relaxation datab

Leu 8, Thr 9, Gly 10, Lys 11, Asn 25,

Asp 52, Gin 62, Arg 74, Gly 75, Gly 76

47

Poor fits by 3D GAF

model

Gin 2, He 23 45

a Excluded peptide planes show either spectral overlap or very weak signals (see

Section 3 5 1)

Excluded peptide planes show either low NOE values or short T2 relaxation times

due to conformational or chemical exchange (see Section 3 5.1)


-40

-40

-105 -65 -25 15

\|/29 Ide9]

a30

40

[cleg]

P30

0 40

[deg]

-40 0 40

[deg]

-80 -40 0

-80 -40 0

125 175 225 275

V|/7 [deg]

-80 -40 0

oc8

40 80

[deg]

40 80

[deg]

Y8

40 80

[deg]

Fig. 3.6 Backbone dihedral angle distributions and peptide-plane reorientational

probability distributions about the axes ea, e=, e in ubiquitin derived from the 1.5 ns

molecular dynamics simulation in water at 300 K. (a) Probability distributions of

He 30, exhibiting a unimodal, nearly Gaussian behavior, (b) Probability distributions

of Leu 8, exhibiting multimodal and asymmetric behavior. Peptide plane motion of

Leu 8 cannot be described solely by a 3D GAF model.


3.4.3.2 Results for the extracted fluctuation amplitudes

The directions of the principal axes of the 3D GAF motion and the

fluctuation amplitudes were extracted according to Eq (2.82). Oa, 0g, and

o are plotted in Fig 3.7 for the 57 peptide planes. Numerical values are

given in Table A.l in the appendix. All 57 peptide planes show a significant

degree of intramolecular motional anisotropy with the largest fluctuation

about the axis e.Axial symmetry of the fluctuation ellipsoid with equal

fluctuation amplitudes about axes ea and Co (aa = Op = aa«) is fulfilled in

good approximation for nearly all 57 peptide planes as is visible in Fig. 3.7.

20

16

-o

12

en.

D

S

0

Mlllllllll iiiiiiiii«i

10 60 7020 30 40 50

Peptide-plane number

Fig 3 7 MD reonentational fluctuation amplitudes of the peptide planes of ubiquitin

about the principal axes ea, ea, e as functions of the peptide-plane number The

3D GAF fluctuation amplitudes aa (open circles), Op (stars), and o\, (filled circles)

were exti acted from the 1 5 ns MD trajectory by averaging over 1500 snapshots (see

Section 3 4.2 2). The secondaiy structure is indicated at the top


3.4.3.3 Dihedral angles and planarity of the peptide plane

The degree of non-planarity expressed by the dihedral angle co;, defined by

the atoms C,_ t

- C,_ j

- N: - C, of each peptide bond /, was determined

from the MD trajectory. The average value of co is for all peptide planes near

180° (177°±5°) and the average standard deviation is 6.8° ± 0.8°, which is

largely independent of the amino-acid type and the secondary structure.

Numerical values for average and standard deviation of co; of each peptide

plane are given in Table A.l in the appendix. The (dl fluctuations are in good

approximation statistically independent of the superimposed peptide plane

reorientation. The correlation coefficients for the correlation between the

dihedral angles co( and V,_i»<P,» which are plotted in Fig. A.2b in the

appendix for all residues in ubiquitin, are on average significantly smaller

than 0.5.

20 40 60 80 100"~

500 1000 1500 2000

time [fs] time [fs]

Fig. 3 8 Dihedral angle correlation functions of peptide plane 32 in ubiquitin. (a) Initial

decay of correlation functions for dihedrals a>i2, ty^ ¥31 extracted from a CHARMM

MD simulation of 4 ps length taking snapshots every 2fs. (b) Illustration of the damped

harmonic oscillation which is present in the dihedral angle correlation function

Ca (t) = ((a>32(0)-co32)(cfl32(r)-co32)) The oscillation has a wavelength of32

_|

approximately 200 cm.

Correlation functions of the dihedral angles co(, cp , i|/( _ lwere extracted

from a MD simulation of4 ps length taking snapshots every 2 fs. In Fig. 3.8a,

typical results are shown for peptide plane 32 in the a helix of ubiquitin. Due

0.5

_n r


to the shortness of the trajectory only the very fast motional modes about the

dihedral angles are sampled. Longer trajectories would falsify the

comparison since predominantly the correlation function of the dihedrals

<P,> V, _ iwould be affected by slower motion. The correlation times of the

dihedral angle ft)( are, on average, a factor 2 shorter than the ones for the

dihedral angles (p(, \y[ _ 1.The statistical independence and the shorter

correlation times for co; suggest, that an appropriate reference frame for the

3D GAF amplitudes would be the effective peptide plane with an averaged

co( angle. In Fig. 3.8b, it is illustrated that a harmonic oscillation about the

dihedral angle (Hl is still visible in the correlation function Cw (t) since me

coupling to the environment is weaker than in case of the other dihedrals. The

extracted wavelength ofapproximately 200 cm is in good agreement with

results from optical spectroscopy [118].

3.4.3.4 Orientation of the peptide plane frame

The orientations of the principal axes for the peptide-plane reorientation

were extracted for each peptide plane with the "matrix method" according to^ ^ x

Section 2.4.3. The calculated principal axes ea ,ea

pp,e differ

slightly from the principal axes ea, ea, e of the 3D GAF model defined in

Fig. 2.1b. The orientations of the principal axes e with respect to the

peptide plane frame ea, en, e are depicted in Fig. 3.9 for all 57 peptide

planes which show 3D GAF motion. On average, the e axis lies—^ (X ot

approximately in the plane and shows an angle to the e axis (C, _ j- C,

axis) of about 7°. Consequences of this small but systematic offset will be

discussed in Section 3.5.5.


Fig 3 9 Polar coordinate plot of

the orientations of the

calculated principal axes eywith respect to the principal axes

ew erj, ey, defined for the 3D

GAF model in Fig. 2.1b. The

orientations were extracted

according to Section 2 4.3 for all

57 peptide planes, which show a

3D GAF motion.

3.5 Analysis of experimental relaxation data

3.5.1 Raw data analysis

The experimental relaxation data of 62 peptide planes were analyzed. No

relaxation data were extracted for the remaining 13 peptide bonds belonging

(l) to Pro 19, Pro 37, and Pro 38 due to the absence of NH protons, (ii) to

Glu 24 and Gly 53 exhibiting peaks with low sensitivity caused by line

broadening, and (iii) to He 13, Asp 21, Ala 28, Gin 31, Leu 67, Leu 69,

Arg72, and Leu 73 due to cross-peak overlaps in the ^N-'H HSQC

spectrum (see Table 3.1). The relaxation data for the two B0-field strengths

are shown in Fig. 3.10 and Fig. 3.11 as functions of the peptide-plane

number. Numerical values for the experimental N Ti, T2, and NOE, and

13C Tt and T2 values at 400 and 600 MHz are given in Table A.2 in the

appendix. Comparison of repeated relaxation measurements yields the

following estimates for the statistical uncertainties: 1.5% for 15N Tj's at 400

and 600 MHz, 2% for 15N T2's at 600 MHz, 4% for 15N NOE's at 400 MHz,

2.5% for 15N NOE's at 600 MHz, 2.5% for 13C Tfs at 400 MHz, 2% for

13C Tys at 600 MHz, 5% for 13C T2's at 400 MHz, and 3% for 13C T2's


at 600 MHz

The peptide planes Leu 8, Thr 9, Gly 10, Lys 11, and Gin 62, which belong

to loop regions, and the peptide planes Arg 74, Gly 75, and Gly 76 forming

the C-terminus of ubiquitin show 15N NOE values below 0 52 at 400 MHz

and below 0 68 at 600 MHz, which are significantly smaller than those of

other peptide planes, which are in the range between 0 52 and 0 67 at

400 MHz and between 0 68 and 0 79 at 600 MHz This reflects the presence

of additional large-amplitude internal motions at slower time scales which

cannot be modeled solely by a 3D GAF motion of their peptide planes These

eight peptide planes were excluded from further analysis (see Table 3 1)

Peptide planes involved in conformational exchange processes were also

excluded (Table 3 1) These comprise peptide planes 25 and 52 which show

significantly reduced 13C T2 values (see Fig 3 10) The same effect is also

apparent when comparing the 15N T2 values at 600 MHz with ^N^H dipole-

15N CSA cross-correlation data obtained from CT-HSQC experiments [71]

(see Section 2 3), which is shown in Fig 3 11 The absence of a similar

anomaly for the peptide plane Asn 25 in 1 /TN NHindicates that the

anomaly in 5N T2 is caused by conformational exchange This leaves a

remainder of 47 peptide planes whose relaxation-active small-amplitude

motion can be modeled by a 3D GAF motion

3.5 Analysis of expenmental relaxation data 85

0.80

0.75

0.70

111

O 0.65

JF 0.60

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

Peptide plane number

{1H}-15N NOE

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75


Fig. 3.10 First part.


10 15 20 25 30 35 40 45 50 55 60 65 70 75


10 15 20 25 30 35 40 45 50 55 60 65 70 75


Fig 3 10 Backbone (a) l5N T,, (b) {'H}-15N NOE, (c) 13C Th and (d) 13C T2

relaxation data of ubiquitin as functions of the backbone peptide-plane number

Relaxation data were measured at two B0-field strengths corresponding to the proton

resonance frequencies 400 MHz (open circles) and 600 MHz (filled circles) at 300 K


0 25

^ 020<N

I-

zin

015

0105 10 15 20 25 30 35 40 45 50 55 60 65 70 75


Fig 3 11 Significant conformational exchange contnbutions to 15N T2 measured at

600 MHz (filled circles) can be identified by comparison with the inverse ^N-'H

dipole-CSA cross correlation rate constants FN m at 600 MHz proton frequency (see

Section 2 3) that are also given in the figure (stars)

3.5.2 Spin relaxation mechanisms in the peptide plane

The dominant relaxation-active interactions and their principal axis

onentations with respect to the peptide-plane frame are depicted m Fig 2 3

Since the interactions probe different directions, their relaxation

contributions yield complementary information on the ngid-body motion of

the peptide plane The relative onentations of the interactions relevant for

15N and C relaxation and the relative magnitude of their contnbutions are

given in Table 3 2

With regard to 15N relaxation, the standard model-free analysis considers

only the dipolar ^N-1!! and the 15N CSA relaxation and descnbes the2

intramolecular motion by the order parameter SNH One finds, however, that

also the 15N-13C and 15N-13Ca dipolar relaxation lead to measurable


contributions (see upper part of Table 3 2) They can ngorously be taken into

account within the more comprehensive 3D GAF model

For the backbone 13C spins, CSA relaxation is dominant but dipolar

contributions have also to be considered [90] Even at 600 MHz proton

resonance the dipolar contributions are non-negligible, as is shown in the

lower part of Table 3 2 The largest dipolar contributions to 13C relaxation

originate from the directly bonded Ca and from close protein protons

While the motion of the in-plane dipolar 13C'-13Ca and 13C'-1HN

interactions can be descnbed by the 3D GAF model, the internuclear out-of-

plane 13C- 1Ha and 13C- 'Fr vectors are influenced also by other types of

motion The influence of all protons, except H of the same peptide plane,

was accounted for by an isotropic dipolar leakage term, which can be

expressed [90] as the dipolar relaxation contribution of a virtual proton at an

effective distance r - This effective distance was calculated for all 15N and

13C nuclei in ubiquitin from the energy-minimized X-ray structure The rg^values for 15N range from 1 72 to 1 87 A and for 13C from 1 69 to 1 90 A

The combination of all the relaxation-active interactions summarized in

Table 3 2 leads to the dependence of the overall relaxation rate constants Tlof the 15N and 13C backbone spins on the fluctuation amplitudes aag and

a depicted in Fig 3 12 Both 15N and 13C T{ show a larger dependence

on cag than on ay In case of N (see Fig 3 12a), the dependencies on can

and ay differ only slightly since the dominant dipolar ^N-'H interaction is

almost not affected by the motion about the ea axis On the other hand, in

case of C (see Fig 3 12b) the dependence on a is 3 to 4 times weaker

than the dependence on Gag This can be explained by the onentation of the

13C CSA pnncipal axes (see Fig 2 lb and Fig 2 3) which are nearly

colhnear to the principal axes ea, ea, e of the peptide plane frame Motion

about ea fully affects the modulation of the large CSA principal values oxx

and Gzz, and leads to the largest dependence of the C Tl on aa whereas

motion about the axes en and e both affect the smallest pnncipal value a

beside axx and Gzz, respectively


Table 3 2 Relevant parameters for dipolar and CSA interaction strengths and their

contributions to relaxation of the backbone 15N and 13C spins assuming isotropic overall

tumbling with a correlation time tc = 4 03 ns and intramolecular 3D GAF motion of the

peptide plane

Relaxation-

active

interaction

Geometrical

and CSA

parameters

(e,<p)'

[deg]

Contnbutions

tor;1at 400 MHz

Contnbutions

tor;1at 600 MHz

S2 '

"N-'H 102AC (101 3, 180) 81 l%g 70 4% s 081

15N.13C, 135 A (003)" (138 4, 0) 2 2% * 18%* 0 86

15xj 13r>a 1 47 A (0 03)d (14 4,0) 14%* 1 1%* 0 91

15NJHrest 1 80 A (003)' 3 3%« 2 9%« (1)'

15N CSA "

axx

°yy

°zz

58 3 ppm

-513 ppm

109 6 ppm

(33 3, 180)

(90, 90)

(123 3,180)

12 0% * 23 8% *

0 88

080*

0 83

13c-13c<x_, 152 A (003)d (159 6,180) 12 4% * 118%* 0 90

13C. 13c« imkd (12 3,180) 0 6%* 0 6%* 091

13C. 15N 1 35 A (0 03)d (138 4, 0) 5 5%* 3 5%* 0 86

13C,.1HN 2 06A*' (66 9,180) 8 0%* 4 6%* 0 82

13p> lxrrest 1 82 A (0 03)' 213%* 118%* (1)'

13C'CSA*

°xx

ay>

-74 4 ppm

-7 4 ppm

81 8 ppm

(2 3, 0)

(92 3, 0)

(90, 90)

52 2% * 67 7% *

091

0 80'

0 80

"The pnncipal values and pnncipal axis onentations of the N CSA tensor were

taken from ref [119] The angle between the CSA z axis and the ^N-'H bond

vector is 22° (see also Fig 2 3)bThe pnncipal values and pnncipal axis onentations of the C CSA were taken

from ref [120] The angle between the CSA y axis and the 13C=0 bond is 13°

(see also Fig 2 3)


cThe l5N-lK bond length was set to the standard value of 1 02 AdThe average bond lengths and their standard deviations given in brackets were

determined from the MD trajectory (see Section 3 5 3) as well as the six bond

angles defining the peptide plane 120 3° (NHNC), 115 7° (NCaHN), 124 0°

(NC'Ca), 120 9° (CON), 118 0° (C'NCa), 121 1° (C'CaO)eA virtual distance between the 15N and C nuclei and all protons outside of the

peptide plane was calculated for all peptide planes using the energy-minimized

X-ray structure [96] The average value with standard deviation is given

f 6, tp are the polar angles of the corresponding spin interaction in the peptide

plane frame defined in Pig 2 3

s The contnbutions are given as percentage of the 15N overall relaxation rate

constants 1/7, = 3 21 s1 (400 MHz) and 2 22 s1(600 MHz) assuming a 3D

GAF motion with ca^ = 7° and cy = 14°, Tc = 4 03ns, and xm, = 2 ps

The contnbutions are given as percentage of the C overall relaxation rate

constants 1/7, = 0 88 s'(400 MHz) and 0 73 s

'(600 MHz) assuming a 3D

GAF motion with oap = 7° and c = 14°, xe = 4 03 ns, and \mt = 2 ps

' The order parameter S^ has been computed for oap = 7° and o = 14°

according to Eq (2 57)

1 A static approximation (S =1) was assumed to calculate the contribution of the

protons outside of the peptide plane since these interactions cannot be modeled

by a 3D GAF motion

The order parameter for the cross-correlation function between the y and

2z principal axes is Su= -0 38

'The order parameter for the cross-correlation function between y and z principal

axes is S = -0 35

3.5 Analysis of experimental relaxation data 91

10 15

aap, ay

20 25

[deg]20 25

[deg]

10 15

aap, ay

Fig 3 12 Dependence of (a) 15N 7", and (b) 13C T\ on the fluctuation amplitudes

aaa (solid line, a is set to ay = 0°) and cy (dashed line, aaa is set to aan = 0°).

The parameters for the interaction strengths were taken from Table 3 2. The correlation

times for the overall tumbling and the internal motion were set to xc = 4.03 ns and

xmr= 20 ps, respectively

3.5.3 Which parameters can be extracted from the

experimental data?

In principle, it is conceivable to extract all motional parameters and all

parameters defining the interaction strengths of each peptide plane in the

protein by fitting a sufficiently large number of relaxation measurements.

However, the presently available measurements do not permit such a general

approach and the extraction of the motional parameters for each peptide

plane becomes the primary goal. Several problems arise concerning the

interaction strengths. The geometry of the peptide plane has to be extracted

from results by different techniques (as X-ray, neutron scattering, and

quantum chemical calculations). The CSA tensors have to be taken from

solid-state NMR measurements of some model compounds, and it is not clear

whether these results can be transferred to proteins. In addition, it has to be

assumed that the interaction strengths do not vary significantly between

different peptide planes. In doing so, a "picture" is created where differences

between residues only show up in the motional parameters. Significant

residue-wise variation of the interaction strengths is possible and would

deteriorate the fit results for the motional parameters (see Section 3.5.5). In

case of a peptide-plane study the most critical parameters are the 'H-^N


dipolar interaction strength and the N and C CSA tensors. It was tried to

fit scaling factors for all of these interaction strengths but this was only

possible in case of the 13C CSA tensor (see below).

Motional parameters. The experimental data of the considered 47 peptide

planes were evaluated to determine their 3D GAF fluctuation amplitudes.

Based on the MD analysis that showed nearly axially symmetric fluctuation

ellipsoids for almost all peptide planes, it was assumed throughout the

2

analysis that oa = o» = aag. The order parameters 5 of Eq. (2.57) are

then determined for each peptide plane by a pair of aag, a values.

Additional parameters entering the spectral densities (Eq. (2.45)) are the

overall tumbling correlation time xc and the correlation times xwt. For

simplicity, an isotropic internal correlation time is assumed, i.e. in a given

peptide plane xint = xint for all pairs \i, v = a, (3, y. This leads to a

description of the motion of each peptide plane by the four model parameters

aap, a , xc, xint which can be extracted from the experimental relaxation

data.

For each peptide plane i, an overall tumbling correlation time xc ,was

extracted from the 15N data alone by a standard model-free analysis, fitting

the parameters xc ,, xint ,,and 5r to the experimental 15N Tl, NOE (400

and 600 MHz), and T2 (600 MHz) data. Averaging over all 47 residues (see

above) yields xc = 4.03 ns, which is well comparable with a previously

determined value of 4.09 ns obtained under similar experimental conditions

[106].

Parameters defining dipolar interaction strengths. The average bond

lengths and bond angles were extracted for a number of representative

peptide planes from the MD trajectory by averaging over the 1500 snapshots.

The results are given in Table 3.2. The standard deviations for the bond

lengths and bond angles for the ensemble of representative peptide planes do

not exceed 3%. The N-H distance was set to the standard value of

rNH = 1^2 A which has often been used in previous analyses [105,106]. It

is significantly higher than the value obtained from the MD trajectory:

(rNH) = 0.997 A. Note that reNH is an effective distance that should reflect


the averaging by rapid local stretching and bending motion of the N - H

bond, rNH = {Sbend(r'^H)Uretch} , including also zero-pointeff

vibrational effects (see Section 2.2.3.2). The value for rNH might need to be

modified in the future when more precise information from solid-state NMR

and from quantum chemical calculations is available.

Parameters defining CSA interaction strengths. It is possible that the

CSA tensors vary with the amino-acid type, with the presence of hydrogen

bonds, and with the local backbone \|/( _ jand (p( dihedral angles. Since there

are no experimental CSA values of ubiquitin available, the CSA tensors were

taken from solid-state NMR studies of small peptide fragments: The 15N

CSA tensor of Boc-Gly-Gly-[15N]Gly-OBz was determined by Hiyama et al.

[119] and the 13C CSA tensor of [l-13C]Glycyl-Gly-HCl by Stark et al.

[120]. The selected 13C CSA tensor is similar to other experimental tensors

[121-124], while for 15N CSA tensors larger variations are observed [124-

128]. It should be noted that the CSA tensors determined by solid-state NMR

correspond to tensors that are partially averaged due to intramolecular

motion. Considering this fact, 15N and 13C CSA scaling factors XN and Xcare introduced, which isotropically upscale the experimental CSA tensors.

The exact knowledge of the N CSA tensor is less crucial than the one of the

13C tensor, since at field strengths corresponding to 400 and 600 MHz

proton resonance 15N relaxation is dominated by dipolar relaxation. The

experimental value for the I5N CSA tensor of Ac = ah - o± = 164 ppm

[119] (assuming an axially symmetric tensor) was upscaled by a fixed value

of XN = 1.07 to compensate for the motional averaging in the solid-state

NMR study. This corresponds to a rigid-molecule value of Aa = 176 ppm

which is used in the following for all peptide planes. Similar values have

been applied in other NMR studies [107,129].

If the c CSA tensor is regarded as known and fast internal motion,

xmt < 20 ps, is assumed, it is possible to extract the fluctuation amplitudes

Gaa, a from N T{ and C T{ relaxation data at one magnetic field

strength alone. This is illustrated in Fig. 3.13. For fast internal motion, a

certain relaxation rate constant 15N Tx corresponds to an overall order


parameter S2NH which is almost equal (see also Section 3.5.6) to the order

parameter SNH Nfifor the dominant dipolar interaction. A certain order

parameter SNH NHcan be described in terms of combinations of the 3D

GAF fluctuation amplitudes aan,a according to Eq. (2.57). This is

2illustrated in Fig. 3.13a for different order parameters SNH NH ranging from

0.6 to 1.0. The a

ellipsoid:

ap,a combinations define, to a good approximation, an

A sinp, a = B cosp. (3.1)

The parameters A and B define the form of the ellipsoid and the parameter

p represents an angle between 0 and n/2. In Fig. 3.13b, the 13C relaxation

rate constant Tx is calculated for combinations (aan,a ) defined by the

ellipsoid which corresponds to SNH NH= 0.80. In contrary to the

corresponding constant 15N Tj ,the calculated 13C Tx cover a large range

from 0.86s"1 (for p = 0°) to 0.69s"1 (for p = 90°). The very distinct

dependence of 15N and 13C T\ on the CTap,o combinations enables a

fitting procedure with a unique solution for the fluctuation amplitudes.

a) b) 0 90

25^3NH, NH0 60^

D) 20

-0 65

-0 70

JO, -0 75

«i15-0 80

ea U0 85

10-0 90

5

"0 95

10 15 20 25 300 65,

10 20 30 40 50 60 70 80 90

oY [deg] P [deg]2

Fig. 3.13 (a) Description of the order parameter SNH NHin terms of the 3D GAF

fluctuation amplitudes aa a and a according to Eq (2 57) Different values of the

order parameter, for which the solutions (Oa n,ay) are given, are indicated (b)

Dependence of relaxation rate constant nC T\ on the parameter p which defines the

2

position on the ellipsoid which corresponds to SNH NH= 0 8 in Fig 3.13a (bold

line). The ellipsoid of the (aaa,ay) solutions is defined according to Eq (3 1) with

A = 15 5° and B = 16 5° Xc is set to 1 083 (see Section 3 5 4)


Relaxation data at multiple B0 fields enable a separation of the CSA and

the dipolar contributions based on their distinct B0-field dependence which

can be used to extract additional information about the CSA tensor from the

relaxation data. First, it was tried to individually scale the different C CSA

tensors by scaling factors Xc (for each peptide plane i. This would allow

one to partially account for the structural dependence of the CSA tensors.

Although a sufficient number of relaxation measurements was available, the

fitting procedure turned ou. to be unstable due to a strong correlation of Xc t

with the motional parameters o"ag, a .On the other hand, a stable fitting

procedure (see Section 3.5.5) results when fitting a global scaling factor Xc,which uniformly scales the assumed C CSA tensor for all peptide planes,

together with the motional parameters.

3.5.4 Fit results for 3D GAF model

The experimental data of the 47 peptide planes considered were evaluated

to determine their 3D GAF fluctuation amplitudes. The fluctuation

amplitudes oaB, <3 and the internal correlation time xmt were fitted to the

nine experimental parameters 15N Tj, T2, and NOE and 13C Tl5 and T2 at

400 and 600 MHz for each of the 47 peptide planes. The peptide planes of

Gin 2 at the N terminal of ubiquitin and He 23 were excluded (see Table 3.1)

due to a large least-square fit error (four times larger than the average one).

The relaxation data of Gin 2 reflect increased flexibility of the N terminus

which cannot be described by 3D GAF motion. The scaling factor Xc for the

13C CSA tensor was determined as a global fit parameter for the remaining

45 peptide planes. A value Xc = 1.083 ± 0.004 was obtained that reflects

the motional averaging of the CSA tensor in the solid-state measurement

[120], since Xc> 1.0. The overall tumbling correlation time xc was kept

fixed at 4.03 ns. The fit results for the fluctuation amplitudes are shown in

Fig. 3.14 as functions of the peptide-plane number. Numerical values for all

fit parameters are given in Table A.3 in the appendix.


fflllllllfflilfflll

20 30 40 50


Fig 3 14 Fit of the 3D GAF model using Eqs (2 45) and (2 57) to the experimental

data consisting of nine auto relaxation parameters (see Fig 3 10 and Fig 3 11) for the

45 peptide planes which can be described by a 3D GAF motion The optimized

parameter set consists of a global scaling factor Xc for the principal values of the C

CSA tensor and three parameters o"ag, o ,and imt for each peptide plane The

overall correlation time T was set to 4 03 ns and the principal values of the N CSA

tensor, given in Table 3 2, were upscaled with XN = 1 07 (see Section 3 5 3) The

optimum scaling factor is Xc = 1 083 The optimum values o"ag (open circles) and

ov (filled circles) are given in the figure for each peptide plane The error limits of the

fitted parameters were determined by a Monte Carlo procedure consisting of 60 fits

with random Gaussian errors added to the relaxation parameters according to the

experimental standard deviations The secondary structure elements are indicated at

the top

Nearly all of the 45 peptide planes exhibit 3D GAF motion with a

substantial degree of anisotropy with ay > aan Thus the dominant axial

fluctuation of the peptide planes takes place about the e axis connecting

-l-land Cj On average, the experimentally determined fluctuation

amplitudes, given in Fig 3 14, are about a factor 1 4 (oan) and a factor 1 6


(o ) larger than the ones observed in the MD trajectory (see Fig. 3.7). As can

be seen in Fig. 3.15, the internal correlation time xint converges for all

peptide planes to small values xint < 30 ps, consistent with the MD results,

rendering the relaxation data largely insensitive to xr

35

30 o

25

•

•

Xmt 20

tps]15 • •

•

••

•

•

10

5

n

••

•

• •

<

.•• •

•

• •

•

• •• •

•

•

• • •

10 7020 30 40 50 60


Fig 3.15 Results for the fitted internal correlation times xmt of the 45 peptide planes

which can be described by a 3D GAF motion The two representative error bars of

peptide planes 12 and 30 were determined by a Monte Carlo procedure consisting of

60 fits with random Gaussian errors added to the relaxation parameters with the

experimentally determined standard deviations

The fitted fluctuation amplitudes were correlated with the experimental

raw data. All N relaxation parameters are highly correlated with ay

(correlation coefficient R ranges from 0.5 to 0.65) but show no significant

correlation with aag (/?<0.1). On the other hand, the 13C relaxation

parameters are highly correlated with o~ag (R ranges from 0.5 to 0.8) and less

correlated with ay {R ranges from 0.2 to 0.5). This behavior reflects the

strong dependence of the C relaxation parameters on aan as illustrated in

Fig. 3.12b.

3.5.5 Uncertainty estimates of fit parameters

Experimental random errors. To estimate the influence of experimental

random errors on the fitted fluctuation amplitudes oao and ay, a Monte

Carlo error analysis was performed. In a series of 60 runs, random Gaussian

errors with the experimentally determined standard deviations (see Section


3 5 1) were added to the experimental relaxation data For each of the 60 data

sets, a simultaneous fit of the individual fluctuation amplitudes of the 45

peptide planes and of a global scaling factor Xc for the 13C CSA tensors was

performed The resulting errors of the fluctuation amplitudes are indicated as

error bars in Fig 3 14, the numerical values are given in Table A 3 in the

appendix

Dependence of 13C'-CSA tensors on secondary structure. To further

investigate the structural dependence of the 13C CSA tensor, the 45 peptide

planes are divided into the three categories "oc-hehx", "p-sheet", and "other"

The individual scaling factors were fitted for each category

jhdix= t 076±0008) x<geet= i 070±0010, X°cher= 1 096±0010 The

resulting fluctuation amplitudes of the 45 peptide planes are shown in

Fig A 3 in the appendix The similarity of the three scaling factors seems to

confirm that the 13C CSA tensors of ubiquitin vary only slightly This result

is supported by the density functional theory (DFT) calculations in Chapter 4

indicating the limited variation of the anisotropy of the C CSA located in

the a helix and in a (3 strand

Effect of interaction strengths on fit results. In Fig 3 16a, the

dependence of the fitted fluctuation amplitudes caa and o on the global

scaling factor Xc is illustrated for the peptide plane He 30 The figure shows

that Xc has a strong influence on the motional anisotropy of the peptide

plane (for definition see Eq (3 2)) Xc is uniquely defined as can be seen

from the error function of the fit The interplay between interaction strengths

and intramolecular motion is illustrated in Fig 3 16b for fit results of peptide

plane 30 A global multiplier |i which uniformly scales all dipolar and CSA

interaction strengths of the N and C spins is vaned between 0 96 and

1 09 (ft = 10 corresponds to the interaction strengths which were used in

Section 3 5 4) Increasing interaction strengths are compensated by an

increasing amount of intramolecular motion However, the very distinct

dependencies of the C relaxation rate constants on oag, a (see

Fig 3 12b) lead to a non-linear dependence of aan on |j. This prevents the

introduction of reduced variables oao = oao /u. In Fig 3 16c, the NH


bond length rNH is varied between 1.00 and 1.04 A (rj{/H = 1-02 A was used

in Section 3.5.4). An increasing NH bond length leads to a strictly decreasing

fluctuation amplitude a and a moderately increasing fluctuation amplitude

cao. As the 15N relaxation rate constants show a similar dependence on

°afl' °v (see Fig- 3.12a), the overall effect compensates the increasing rNH.

On the other hand, there is no overall effect on the C relaxation rate

constants due to the dominant dependence on oao (see Fig. 3.12b).

a)

2,

*"^»^fc.-

-* cy-

13*10 °u^__

~"

CO.

aO

:

5 x\-

610

590

'570 5C

550

1 04 1.06 1 08 1 10 1.12

0 96 1 00

H

1 04 1 08 1 00 1 01 1 02

rNH [A]1 03 1 04

Fig 3.16 Dependence of the fit parameters aag (solid line) and oy (dashed line) of

peptide plane He 30 on (a) The global scaling factor Xc for the 13C CSA tensor The2

corresponding overall fitting error function % (sum of the error contributions of all 45

peptide planes) for the global fit of Xc is given as dotted line, (b) The global multiplier

u which scales all dipolar and CSA interaction strengths of the 15N and 13C

backbone spins (c) The NH bond length The scaling factor \g was kept fixed at

1 083 in (b) and (c)


Offset of peptide plane frame from assumptions in the 3D GAF model.

In Section 3.4.3.4, the offset of the principal axis frame for peptide plane

reorientation from the assumptions in the 3D GAF model was extracted from

the MD trajectory. Test calculations which estimate the dependence of the

fitted fluctuation amplitudes on such an offset are summarized in Table 3.3.

First, a relaxation data set was calculated assuming fluctuation amplitudes of

aan = 7° and o = 14° for reorientation about the principal axis frame

.For an axially symmetric 3D GAF fluctuation ellipsoid,

'a, pp'e^PP'' y,pp

the offset of this principal axis frame from assumptions in the 3D GAF model

is uniquely defined by the polar coordinates of the assumed symmetry axisi —^ —x _j^

e with respect to the peptide plane frame ea, en, e (see Table 3.3).

Table 3 3 Estimation of the systematic error in the fitted fluctuation amplitudes aao, aywhich results from an offset of the fluctuation tensor orientation from assumptions in the

3D GAF model For details of the calculation procedure see text

orientation of e

(9, cp) [deg]a

fitted aap [deg]*

fitted ay [deg]*

(0,0) 70 14 0

(10, 0) 75 130

(10,45) 73 134

(10,90) 70 14 0

(10, 135) 70 14 3

(10, 180) 71 14 3

a Polar coordinates of assumed principal axis e with respect to the equilibrium

peptide plane frame ew ea, e

In the fitting procedure isotropic overall tumbling with a correlation time

xc = 4 03 ns, and 15N and 13C CSA scaling factors XN = 1.07 and

Xc = 1 083 are assumed

In the next step, fluctuation amplitudes were fitted to this relaxation data

set assuming the principal axis frame ea, en, e with onentations defined in

the 3D GAF model. Comparison of the fitted fluctuation amplitudes with


°aB = 7° anc^ Gy = ^° yi^^s the magnitude of the systematic error due

to the offset of the principal axis frame. The largest effect on o is observed

if e lies in the plane and is slightly oriented towards ea (corresponds to

0 > 0°, cp = 0°). The resulting systematic error might be relevant since this

case corresponds approximately to the average orientation of e extracted

from the MD trajectory (see Fig. 3.9). A slight modification of the orientation

of the principal axes ea, en, e in the 3D GAF model might be appropriate if

MD simulations of other proteins confirm the results which were found for

ubiquitin (see Fig. 3.9).

Anisotropic overall tumbling. In the present analysis it was assumed that

ubiquitin undergoes isotropic overall tumbling (see Section 3.5.4). The

tumbling anisotropy of ubiquitin has been determined by Tjandra et al. [106]

to be = 18%. An analysis of the influence of an 20% anisotropy of tumbling

on the fitted 3D GAF fluctuation amplitudes is summarized in Fig. 3.17.

First, a synthetic relaxation data set is calculated assuming an axially

symmetric rotational tumbling tensor with the correlation times

xc xx= xc = 4.29 ns and xc zz= 3.58 ns for the tumbling about the three

principal axes (please mind that (xc xx+ xc> + xc zz)/3 = 4.03 ns) and the

3D GAF fluctuation amplitudes aa(3 = 7° and ay = 14° for the

intramolecular motion. The relaxation data set is then fitted as described in

Section 3.5.4 assuming isotropic tumbling with the correlation time

xc= 4.03 ns. The fitted fluctuation amplitudes can be compared with the

values aan = 7° and c = 14° to estimate the systematic error connected

with the assumption of isotropic overall tumbling. In Fig. 3.17, it is shown

that an effect of less than 1.2° on the fitted fluctuation amplitudes aaa and

ay occurs depending on the orientation of the overall tumbling diffusion

tensor with respect to the equilibrium peptide plane where the orientation is

defined by the polar coordinates (9, (p) of the symmetry axis of the rotational

diffusion tensor in the equilibrium peptide plane frame. The maximal effect

corresponds to a symmetry axis of the rotational diffusion tensor that is

perpendicular to the average peptide plane (corresponds to

(6, cp) = (90°, 90°) for the solid line in Fig. 3.17).


aap, ay[deg]

Fig 3 17 Dependence of the fitted fluctuation amplitudes 0"ag, Gy on the orientation

of the symmetry axis of the anisotropic (but axially symmetric) rotational diffusion

tensor with respect to the equilibrium peptide plane frame The orientation is given by

the polar coordinate 0 which is varied between 0° and 180° while (p is kept fixed at

0° (dashed line) and 90° (solid line) For details of the calculation procedure see text

3.5.6 Comparison between the 3D GAF analysis and a 15N

model-free analysis

Based on the results of the 3D GAF analysis and by using Eq. (2.57), it is2

possible to calculate order parameters S for arbitrary pairs of spin

interactions within the peptide plane. In Table 3.2, the order parameters for

different interactions are given for the following values of the fluctuation

amplitudes: aag = 7,a = 14

.The calculated order parameters are

largest for the dipolar i5N-13Ca and 13C'-13Ca interactions which are lessot ex

modulated by the dominant fluctuation about the Ct _,- Ct axis than, for

example, the 15N-]H interaction.

The order parameters SNH NHof the dipolar N- H interaction were

calculated for all 45 evaluated peptide planes using Eq. (2.57) together with

the fitted 3D GAF fluctuation amplitudes from Fig. 3.14. In Fig. 3.18 these

values are compared with order parameters obtained directly from a standard

model-free analysis of the 15N Tt, T2, and NOE data at both field strengths.


These order parameters are termed here SNH. Fig. 3.18 shows a good overall

agreement between the order parameters SNH and SNH NHcalculated by the

two approaches. It illustrates the compatibility of the 3D GAF analysis with

the model-free description. The systematic offset between the two sets is due

to the assumption in the model-free analysis of an equal order parameter

SNH for the dipolar NH and for the 15N CSA interactions. This is, however,

only justified for isotropic internal motion or for an axially symmetric CSA

tensor with the unique axis collinear to the NH vector. These assumptions are

not realistic (see Table 3.2) and are not required for the 3D GAF analysis.

0.85

20 30 40 50 60


Fig. 3 18 Compatibility of 3D GAF model with a standard 15N model-free analysis

The SNH NHorder parameters for the dipolar 15N-'H interaction derived from the 3D

GAF model data of Fig 3 14 and Eq (2.57) (filled circles) are compared with the 15N

SNH order parameters determined by a model-free analysis from 15N relaxation data

at 400 and 600 MHz, (open circles). xc was set to 4.03 ns and a symmetric 15N CSA

tensor with Aa=176ppm was assumed Additional relaxation contributions from

other protons (assuming a rigid molecular frame), from the 15N-13C dipolar

interaction (assuming an averaged order parameter SNC = 0 83), and from the 15N-

13Ca dipolar interaction (assuming an averaged order parameter SNC = 0 89) were

also included in the model-free analysis


A model-free analysis based exclusively on N relaxation data neither

yields axial fluctuation amplitudes nor information on the amsotropy of

intramolecular peptide-plane motion Such additional information is

provided by the complementary C relaxation data that form an integral

part of the 3D GAF analysis

3.6 Discussion

3.6.1 Anisotropy of peptide-plane dynamics

The 3D GAF fluctuation amplitudes extracted from the MD analysis and

from the relaxation data yield insight into the degree of anisotropy of the

peptide plane motions A convement measure for the degree of anisotropy is

the "anisotropy factor"

Ac=

ay-°ap(3 2)

c (rjY + 2aap)/3'

reflecting the difference between reonentational motion about the

C, _ j- Cj axis and motion about the orthogonal axes, weighted by the

average fluctuation amplitude The anisotropy factors for all peptide planes

obtained from the MD trajectory and from the expenmental results are shown

in Fig 3 19 Numencal values are given in Table A 3 in the appendix

It shall be mentioned at this place that there are qualitative and quantitative

differences between the results of the expenmental NMR relaxation study

and the MD simulation The quantitative differences in the fluctuation

amplitudes become apparent by companng Fig 3 7 and Fig 3 14 The

expenmental fluctuation amplitudes are on average larger than in the MD

simulation Reasons for this discrepancies may be suspected in the choice of

not accurately known geometnc parameters in the data evaluation On the

other hand it is also conceivable that the used CHARMM force field is

slightly too stiff, leading to low fluctuation amplitudes

The sets of peptide planes showing large-amplitude, non-3D GAF motion

apparent in the expenmental data and the MD simulation are not identical

3 6 Discussion 105

This particularly concerns the peptide planes of Asn 25 and Glu 62 showing

non-3D GAF motion in the experimental data, but 3D GAF motion in the

simulation The peptide planes of Thr 7, Gly 35, He 36, Gly 47, and Arg 54

exhibit non-3D GAF motion only in the simulation Due to the restncted

length of the trajectory, there is some uncertainty concerning the statistical

significance of jump processes that lead in Section 3 4 3 to the exclusion of

peptide planes from the 3D GAF analysis There is also little correlation

between the residue-specific 3D GAF fluctuation amplitudes in the

expenmental and the MD data In this context, one should remember that

even a rather long MD trajectory, covering more than 1 ns, represents in

essence an "extended snapshot" of the long-term motion of the protein and

does not comprehensively cover the ensemble conformations relevant for the

expenmental results For this reason, no agreement in all details can be

expected, even for a "perfect" force field


1.5

1.0

Aa— 0.5o

0.0

-0.5

b) 3.0

2.5

• «w«

••

•

• • •

MD results

10 20 30 40 50 60


70

10 20 30 40 50


60 70

Fig. 3.19 Anisotropy of peptide-plane motions of ubiquitin defined in Eq. (3.2). Panel

(a) gives the anisotropies calculated from the MD trajectory with the fluctuation

amplitudes given in Table A.l in the appendix using caa = (o"a + Oa)/2 .Panel (b)

gives the anisotropies determined by the experimental aag, a values of Table A.3 in

the appendix.

3.6 Discussion 107

3.6.2 Correlation with secondary structure

An attempt was made to correlate the fluctuation amplitudes with the

secondary structure elements of native ubiquitin in Fig. 3.1, which are

indicated also at the top of Fig. 3.7 and Fig. 3.14. Average fluctuation

amplitudes were calculated separately for a helical, P sheet, and other

regions from the MD trajectory: {<*a#)hehx = 4.1°, (<*y)hellx = 8-0°,<°«B>

u ,

= 4-7°. <<0u ,

= 110°> <<W ,u= 5-3°- and

» ap' sheet N T sheet x Up'other

(a) ,= 11.0

.In Fig. 3.20 it can be seen that the central a helix shows

a rather homogeneous behavior with fluctuation amplitudes that are

somewhat lower than for the rest of the protein. The loop regions and the p*sheet regions are more flexible but do not show significant differences when

compared to each other. As described in Section 3.4.3 and Table 3.1, only

peptide planes are taken into account that exhibit an exclusive 3D GAF

behavior.

The differences of the experimentally determined fluctuation amplitudes

between the three categories are even smaller. The average fluctuation

amplitudes are (c^)^ = 5-5°' <°y>aw« = 164°' (ct«P>sheet = 1J°'

<°T>'sheet = 16-5°' Mother = 58°'• «* Mother « 16.8°. Fig. 3.21

visualizes the (almost) absent correlation of the experimental parameters

Gaa and o with the secondary structure elements. Ellipses are shown in the

aaa,G plane that contain the pairs of values of the planes located in one of

the three categories. It is apparent that the ellipses strongly overlap and miss

a characteristic structure-related behavior. On the other hand, there is a

significant anticorrelation between aan and ay leading to similarly inclined

ellipses for all three structural elements.

As mentioned above, the absence of a significant correlation of the

fluctuation amplitudes with the secondary structure reflects to some extent

the fact that the peptide planes with large-amplitude motion have been

excluded from the analysis. On the other hand, it demonstrates that the most

rigid parts of a globular protein, like ubiquitin, show similar fluctuation

amplitudes irrespective of the secondary structure.


P sheets central a helix

5 10 15

°a,p [deg]20

other

(%')W

'G'o.^'other = 53°.

^Y other11 oL

5 10 15

<Vp tde9]20

5 10 15

<Vp [deg]

superposition of ranges

20

5 10 15

<Vp [deg]20

Fig 3 20 Loci of the pairs of MD fluctuation amplitudes aag = (aa + oB)/2, cy for

the different peptide planes and their assignment to the three categories "a helix", "P

sheet", and "others", indicated by ellipses

3.6 Discussion 109

P sheets central a helix

10 15 20 25

<Vp[deg]

25other

20 ft +

a? 15CDD

\ +

\ +

DM0

5 (°a$other ~ 5,8°

^Y other~ 16.8U

5 10 15 20 25

°"a,(3 tde9]

superposition of ranges

10 15 20 25 10 15 20

°a,p [deg]

Fig. 3.21 Loci of the pairs of experimental fluctuation amplitudes oag, aY for the

different peptide planes and their assignment to the three categories "a helix", "|3

sheet", and "others", indicated by ellipses.


3.7 Conclusions

So far mostly local order parameters have been used to characterize

intramolecular mobility in biopolymers. The author is convinced that future

investigations will concentrate increasingly on the description of the

anisotropic motion of entire molecular subunits, such as secondary structural

elements, aromatic rings, purine or pyrimidine bases, peptide planes, or

methylene groups. Considering these subunits as rigid objects, it is possible

to deduce the motional anisotropy of their motion from the auto- and cross-

relaxation properties of several observer nuclei rigidly attached to the same

fragment.

In this chapter, the focus was lied on fast-time-scale motional processes

exhibited by the rigid backbone peptide planes which are reflected in 13C

and N relaxation data. By comparison with an extended MD simulation,

those peptide planes were identified and characterized which are dominated

by anisotropic 3D Gaussian axial fluctuations. A significantly anisotropic

motional behavior was found that depends little on the secondary structure

elements. The extracted motional parameters, however, depend critically on

the magnitude and orientation of the dipolar and CSA interaction tensors,

responsible for relaxation. It is hoped that in the future solid-state NMR

studies of labeled proteins and refined quantum-chemical calculations [81-

85] will provide more accurate information on these tensors and will lead to

a better understanding of their dependence on the local environment. This

will lead to more accurate and more realistic descriptions of the dynamics of

peptide planes and other molecular subunits. Supported by the further

developments of NMR and MD methodologies, studies of this type will

provide information for a better understanding of the relation between

dynamics and biomolecular function.

4.1 Introduction 111

4 CSA Fluctuations Studied by MD

simulation and DFT Calculations

4.1 Introduction

In this chapter, a combination of quantum chemical CSA calculations and

molecular dynamics simulation is presented to gain more insight into the

magnitude and fluctuation of CSA tensors of the backbone 15N and 13C

spins in proteins and to discuss their relevance for nuclear spin relaxation.

Density functional theory (DFT) has proven to be useful to calculate CSA

tensors of essential fragments of larger biomolecules (see Section 2.5). For

the peptide-bond moiety, it is N-methylacetamide (NMA) which has been

used as a substitute. Despite its relative small size, NMA has allowed one to

gain valuable insight into the origins of the chemical shielding properties of

nuclei within the peptide plane [82,85]. The strategy for the CSA DFT

calculations presented in this chapter relies on results of a recent study [130]

where it was demonstrated that DFT methods yield CSA tensors for 15N

nuclei in the side chain of crystalline asparagine and in the peptide bond of

crystalline alanine-alanine dipeptide with an accuracy comparable to solid-

state NMR.

In Chapter 3, the interpretation was based on several ad hoc assumptions

for the 15N and 13C CSA tensors in order to calculate the significant CSA

relaxation contributions. The CSA tensors were taken from solid-state NMR

studies of small peptides and were assumed to be the same for all peptide

planes which show a 3D GAF motion. A scaling factor for the CSA tensor

was introduced to compensate for the intramolecular motion already

contained in the CSA tensors obtained by solid-state NMR. The uncertainties

connected with such a procedure and the need for additional fitting

parameters - the scaling factors \N and %c - resulted in increased

uncertainties for the extracted motional parameters, in particular the

fluctuation amplitudes for the peptide plane reorientation. With improved

knowledge about the CSA tensors of the backbone spins in the protein a

characterization of the protein's intramolecular motion would be more

112 CSA Fluctuations Studied by MD simulation and DFT Calculations

accurate. As a first step towards this goal, this analysis is concerned with the

investigation of the magnitudes and the fluctuation of the 15N and 13C CSA

tensors located in three different peptide planes of the protein ubiquitin.

To investigate assumptions that are commonly used in relaxation studies

including the one in Chapter 3, DFT calculations are combined with a MD

trajectory of ubiquitin. CSA tensors are calculated for selected "snapshots"

of the MD trajectory with a constant time increment. This leads to a series of

CSA tensors, the "CSA trajectory", whose fluctuations directly depend on

the molecular motion. The analysis of the extracted CSA parameters and

their fluctuations allow a critical test of several assumptions. This concerns

the transferability of CSA tensors obtained by solid-state NMR for small

peptides to liquid-state NMR relaxation experiments on proteins, the

uniformity of CSA tensors located in different secondary structure elements

of the protein, the influence of intramolecular motion on the CSA tensor, the

effect of CSA fluctuations on relaxation, and the contribution of the

antisymmetric part of CSA tensors to relaxation.

In Section 4.2, the DFT calculations and their combination with the MD

trajectory are described. In Section 4.3, averaged CSA tensors are calculated

which can be compared with results obtained by solid-state NMR, and in

Section 4.4, the fluctuations of the CSA tensors during the trajectory are

characterized. Finally, the implications of magnitude and fluctuation of the

calculated CSA's on NMR relaxation are discussed in Section 4.5.

4.2 MD simulation and DFT calculations

MD simulation. The MD simulation of ubiquitin solvated in a box of

water has been described in Section 3.4.1. For the DFT calculation an

extended MD trajectory of 2.5 ns has been used and 625 snapshot were

selected with a time increment of 4 ps.

General method of DFT calculations. For the chemical shielding

calculations, the Sum-Over-States Density Functional Perturbation Theory

(SOS-DFPT) [131,132] as implemented in the deMon-NMR program [133-

135] was used. In this method, Kohn-Sham orbitals are applied to the

4 2 MD simulation and DFT calculations 113

Ramsey formula [136] for chemical shielding, and the energy denominators

are approximated by differences in Kohn-Sham orbital energies, corrected

for changes upon excitation in the exchange correlation potential The

shieldings were calculated using the Loc 1 SOS-DFPT approximation [131],

with the molecular orbitals localized by the method of Boys [137] The

gradient-corrected Perdew-Wang-91 (PW91) exchange-correlation potential

[138] was used Numencal quadrature was earned out on FINE RANDOM

[133,135] angular gnds with 64 radial shells Calculations were performed

using the IGLO-II basis sets [139] Explicitly included fragments that

represent the environment were treated by the less expensive Gaussian

orbital basis set DZVP [140,141] In some calculations water solvent

molecules within a certain cutoff distance were explicitly included For more

details see ref [130]

Strategy for CSA DFT calculations in ubiquitin First, an appropriate

substitute, containing the nucleus (or nuclei) for which the CSA tensor is

calculated, is defined with a bonding topology that resembles the original

molecule as much as possible and that at the same time is small enough to be

treated by DFT The IGLO-II basis sets are used for the atoms of interest and

for all neighbonng atoms belonging to this fragment If the atom for which a

CSA is calculated is part of a delocahzed electron system, such as an amide

plane, all atoms in this system as well as their nearest neighbors are treated

in the same way For all other atoms a DZVP basis set is used In a second

step the close contacts are identified In biomolecules these are mainly

"strong" hydrogen bonds that involve the atom(s) of interest either directly

or via directly bonded neighbors A typical hydrogen bond of the type

C = O H - N can be considered as "strong" if the distance between the

heavy atoms O and N is below 3 2 A and if the angle N-H-0 is near 180°

The close contacts influence both the amsotropy and the onentation of the

CSA The contact partners are then substituted by small molecules In

particular, peptide planes are replaced by N-methyl-acetamides (NMA)

Close water molecules and other small fragments are treated by the DZVP

basis set In a third and final step the partial charge distnbution of the

environment is included Its effect is particularly important in the presence of


surrounding fragments that carry net charges like deprotonated carboxyl- or

protonated amino-groups A 10-15 A cutoff distance for the surrounding,

which should be chosen such that the total charge inside the sphere is zero,

appears to be sufficient

The 15N and 13C CSA tensors of three different peptide planes in the

backbone of ubiquitm were examined with DFT calculations (a) peptide

plane 10 between Thr9 and Gly 10, located in a loop region, (b) peptide

plane 30 between Lys 29 and He 30, located in the central a helix, and (c)

peptide plane 43 between Arg 42 and Leu 43, located in a (3 strand The

geometry of the molecular fragments which were used to calculate the CSA

tensors in these three peptide planes are depicted in Fig 4 1 for one

representative snapshot of the MD trajectory which was modified as

descnbed in the following In all three cases, the central fragment consists of

the studied peptide plane i defined by the atoms -Cl _ j -C, _ jO-NH-C -

with an acetylated N terminus and an amide group for its C terminus The C^

centers are always replaced by a methyl group Peptide plane 10 (see

Fig 4 la), which is exposed to the solvent, shows, on average, for every

fourth snapshot one close contact to a water molecule The corresponding

water molecules were included in the DFT calculation In case of peptide

planes 30 and 43 (see Fig 4 lb and c, respectively) close contacts with

peptide planes 33 and 70, respectively, due to hydrogen bonding have to be

taken into account Peptide plane 30 shows no close contacts with water

molecules whereas the carbonyl of peptide plane 43 is involved in a hydrogen

bond with one water molecule which is trapped throughout the trajectory

CSA trajectory. DFT calculations were performed for all of the 625

modified snapshots and the resulting 15N and 13C CSA tensors were stored

as a "CSA trajectory", where each snapshot contains the full CSA tensors

expressed in the coordinate frame of the reoriented CHARMM coordinate

files (see Section 3 4 1)

4.2 MD simulation and DFT calculations 115

a)

<

®-& Gly10

water

water

b)

Asp 32

Fig. 4.1 First part.


c)

'^9f Gin 41

Arg42^ Qj)

Val70

Fig 4 1 Molecular fragments which were used in the CSA DFT calculations for (a)

peptide plane 10 (loop region), (b) peptide plane 30 (a helix), and (c) peptide plane 43

(two antiparallel (3 strands) The shown geometries correspond to a single snapshot of

the MD trajectory The central fragment is modified by introducing methyl groups

replacing all C» centers, an amide group as C terminus and an acetylated N terminus

Hydrogen bonds are indicated as dashed lines

15 13,4.3 Average 13N and 1JC CSA tensors

In this section, different approaches for calculating average CSA tensors

from the CSA trajectory are outlined They all refer to very rapid fluctuations

of the CSA tensors with correlation times in the sub-100 ps range The

"global frame averaging" approach comprises a linear averaging over the

CSA tensors of the trajectory which contains the intramolecular motion of

the peptide plane If the amount of motion contained in the trajectory and

present in the solid state at room temperature is comparable, this approach

4.3 Average 15N and 13C CSA tensors 117

corresponds to a solid-state NMR measurement which also yields a CSA

tensor as a linear time-average over the motional degrees of freedom.

Obviously, such an effective CSA tensor should not be used to define the

CSA interaction strength entering the CSA relaxation rate constants. Two

other approaches are introduced to compensate this motional scaling. In the

"local frame averaging" approach the peptide planes with the fixed CSA

tensors are reoriented in order to remove the intramolecular motion. Only

those fluctuations of the CSA tensor remain which are induced by the

changing chemical and electronic environment during the trajectory.

Averaging over these fluctuations yields the "local frame average CSA

tensor". The "3D GAF averaging" which has already been introduced in

Section 2.5.2 is a procedure for calculating the motional scaling of a CSA

tensor which results exclusively from peptide plane reorientation by 3D GAF

motion. It is either possible to calculate a "3D GAF average" tensor starting

from the CSA tensor not affected by intramolecular motion or vice versa. In

this approach, it is assumed that the changing environment has no influence

on the CSA tensor. Whether this is a good assumption or not can be tested by

comparison with the "local frame averaging" approach.

4.3.1 Processing of the CSA trajectory

Global frame averaging. The CSA tensors of the CSA trajectory {g^}(k - l,...,N), given in the molecular frame, were averaged over all

snapshots (N = 625):

N

k= 1

The averaged CSA tensor was rendered traceless and the symmetric part was

separated from the anti-symmetric part: qav = g^m + g"" ', where

sym ,T

.-,

anti,

T ..„ sym ..

°av =(°av + °avy2 d °av = (°«y-Sflv)/2- <2av W3S then

diagonalised yielding the CSA principal values au (I = x, y, z) as

eigenvalues and the principal axes in the molecular frame as eigenvectors.

The antisymmetric part aav was transformed into the principal axis system


of the symmetric tensor by the transformation V gav'V where the V matrix

columns contain the calculated eigenvectors.

Local frame averaging. First, the intramolecular motion of the peptide

plane is removed from the CSA trajectory by aligning the instantaneous

peptide plane of each snapshot with the equilibrium peptide plane. The CSA

tensor {ak} of each snapshot k is transformed accordingly by using the

rotation matrix extracted from the alignment procedure. The such modified

CSA trajectory is then processed in the same way as described for the "global

frame averaging" approach.

4.3.2 Analysis of 15N CSA tensors

Characteristic parameters of the averaged 15N CSA tensors of the peptide

planes 10, 30, and 43 are given in Table 4.1. The orientations of these CSA

tensors with respect to the equilibrium peptide planes can be found in Table

A.6 in the appendix. The equilibrium peptide plane belongs to one single

snapshot with an approximately planar structure (co=180°) and was

selected according to Section 3.4.2.1. However, the orientation of the NH

bond of this snapshot was modified by averaging over its out-of-plane and in-

plane bending motion contained in the trajectory.

The "global frame average" 15N CSA tensors of peptide planes 30 and 43

located in the central a helix and a P strand, respectively, are very similar

concerning their anisotropy and asymmetry whereas the calculated tensor

located in the loop region (peptide plane 10) shows a strongly reduced

anisotropy and an increased asymmetry. In this case, the motionally averaged

tensor reflects large-amplitude internal motion including jumps between

different conformations (see Section 4.4.1). The z principal axis (see Table

A.6 in the appendix) lies approximately in the plane in all three cases. The

angle 9^ 7between the z principal axis and the NH vector ranges from 16

to 22° (see Table 4.1). The y principal axis, normally assumed to be

orthogonal to the peptide plane, shows a significant deviation from this

orientation (see Table 4.1: angle 0j_ between 8 and 19°).


Table 4 1 Results for the pnncipal values (traceless), amsotropy Ao, asymmetry r\, and

the parameter Aa~nII of the 15N CSA tensors of the peptide planes (pp) 10, 30, and 43 in

ubiquitin using three different approaches of averaging, namely the "global frame

averaging", the "local frame averaging", and the "3D GAF averaging" approaches (see

text)

PP

no

averaging

approach [ppm] [ppm] [ppm]

Ao"

[ppm]

r\b °JVtf z

[deg] [deg] [ppm2]

10

global -59 4 -13 4 72 8 109 2 0 63 219 199 358

local -59 8 -38 0 97 8 146 7 0 22 20 4 10 6 457

30

global -64 3 -28 5 92 8 139 3 0 39 16 7 12 4 547

local -64 9 -29 6 94 5 1418 0 37 163 82 546

3D GAF/ -65 5 -29 9 95 4 143 1 0 37 16 7 124

43

global 65 3 -28 5 93 8 140 7 0 39 19 6 81 516

local -65 8 -29 8 95 6 143 3 0 38 180 72 527

3D GAF g -661 -30 6 96 7 145 1 0 37 19 6 81 —

flArj = Gzz-(oxx + Gyv)/2

r\ (°w-c«)/oz.°

9ww *ls tne angle between the z pnncipal axis and the NH bond

8j_ is the angle between the y principal axis and the normal to the peptide

plane

..

2 2 2 2

f Starting from the "global frame average" CSA tensor of this peptide plane, the

corresponding tensor not containing the 3D GAF motion is calculated under the

assumptions of the "3D GAF averaging approach" (see text) using the fluctuation

amplitudes aa = 3 90°, Op = 4 13°, g = 7 53° extracted for peptide plane

30 from the CSA trajectory

g Asf, but using the fluctuation amplitudes aa = 4 33°, Op = 2 87°, rjy = 8 95°

extracted for peptide plane 43 from the CSA trajectory

A comparison of the "global frame average" CSA tensors with results

obtained by solid-state NMR is difficult since exclusively small peptides with

mostly glycine residues as earner of the CSA tensor have been studied


experimentally (see refs [119,124-128] and Table A 4 in the appendix) The

calculated anisotropics < 141ppm are smaller than the experimental ones,

which range from 144 to 165ppm The calculated CSA tensors show, on

average, a larger asymmetry than the experimental ones Interestingly, the

smallest experimentally determined anisotropy of 144ppm belongs to the

non-glycine peptide Ala-Ala [124] Preliminary DFT calculations (not

shown) indicate that glycine residues tend towards larger 15N CSA

anisotropics In the experimental studies, it is often assumed that the y

principal axis lies orthogonal to the plane, which is a rather crude

approximation according to DFT calculations Expenmental 0^ values

range between 12 and 24° which is comparable with the range found for the

calculated values The significance of differences between calculated and

expenmental CSA tensors for spin relaxation will be discussed in Section

452

There might be several reasons that lead to the discrepancy between

calculated and experimentally determined CSA tensors First, the

environment of the peptide planes in ubiquitin with its water environment

differs significantly from the one in small and crystallized peptides studied

experimentally Second, the structure of the peptide plane has a large

influence on the CSA tensor If the geometrical setup defined by the

CHARMM force field does not compare quantitatively with the

experimentally studied peptide plane structure significant differences

between experiment and DFT calculation would result In case of 15N CSA

tensors, it might be the NH bond length which is a most cntical parameter

Since the SHAKE algorithm [117] was applied to all bond lenghts involving

a hydrogen atom, the NH bond length is fixed at rNH = 0 997 A throughout

the MD trajectory The dependence of CSA parameters on the NH bond

length was estimated by test calculations for one single snapshot of peptide

plane 43 The results are summarized in Fig 4 2 A nearly perfect linear

dependence of the anisotropy and asymmetry of the 15N CSA tensor on the

NH bond length results This linear scaling for one snapshot can be directly

transferred to the ensemble parameters of the average CSA tensor in Table

4 1 In this case, an increased NH bond length would lead to a further

4.3 Average 15N and 13C CSA tensors 121

decreased anisotropy of the calculated CSA tensor. Finally, it is not known if

the intramolecular motion present in the solid state at room temperature is

comparable to the amount of motion contained in the CSA trajectory.

Different motional scaling of the CSA tensor would lead to a discrepancy

when comparing experimental results with CSA tensors obtained by

averaging over the trajectory.

145 0.56

130

0.54

r\

0.52

0.50

0 98 0.99 1.00 1.01 1.02 1.03 1.04 1.05

rNH [A]Fig 4.2 Dependence of the anisotropy Ao and the asymmetry n of the calculated I5N

CSA tensor on the NH bond length rNH. The NH bond length was systematically

vaned for a single snapshot of peptide plane 43. For an increasing bond length the

anisotropy decreases linearly by 151 ppm/A, while the asymmetry grows linearly by

0.54/A.

4.3.3 Analysis of 13C CSA tensors

Characteristic parameters of the averaged 13C CSA tensors of the peptide

planes 10, 30, and 43 are given in Table 4.2. The orientations of these CSA

tensors with respect to the equilibrium peptide planes can be found in Table

A.7 in the appendix.

The "global frame average" C CSA tensors of peptide planes 30 and 43

show a very similar anisotropy but a significant difference in their asymmetry2

and a large difference in Aa, describing the antisymmetric part of the


CSA. The CSA anisotropy and asymmetry of peptide plane 10 reflects, again,

the large motional averaging in the loop region but the difference of the

anisotropy compared to peptide planes 30/43 is less pronounced than in the

case of the N CSA. The x principal axis is in all cases approximately

orthogonal to the peptide plane, the deviation from that direction is less than

7° (see angle 0± xin Table 4.2). The orientation of the y principal axis with

respect to the CO director shows different results for all three peptide planes:

The axis of plane 43 is nearly collinear to the CO bond whereas the axis of

planes 10 and 30 lie in the sectors defined by the atoms OC'Ca and NC'O,

respectively.

1 ^

Experimental results for the C CSA tensor of several peptides are given

in refs. [120-124] and are listed in Table A.5 in the appendix. The

anisotropics lie in the interval between 106 and 126 ppm, which includes the

calculated values of Table 4.2. Interestingly, it is again the CSA tensor

determined for the peptide Ala-Ala [124], whose anisotropy of

Aa = 112ppm fits best to the calculated values 111.7ppm and 113.7ppm

for peptide planes 30 and 43, respectively. The experimentally determined

orientations differ in nearly all cases from the calculated ones in so far as the

x and z principal axes have interchanged their orientations. This is not the

case for the peptide AcGlyTyrNH2 [121] and Ala-Ala [124] which shows an

asymmetry of x\ = 1 leading to underdetermined orientations of the x and z

principal axes in the plane orthogonal to the y principal axis. In the

experimental studies, the y principal axis lies exclusively in the sector

OC'Ca with angles 0CO ranging from 0 to 13°. The differences between

calculated and experimental CSA tensors and their significance for

relaxation will be discussed in Section 4.5.2.


Table 4 2 Results for the principal values (traceless), anisotropy Ao", asymmetry t|, and

the parameter Acanri (for definitions see Table 4 1) of the C CSA tensors of the

peptide planes 10, 30, and 43 in ubiquitin using three different approaches of averaging,

namely the "global frame averaging", the "local frame averaging", and the "3D GAF

averaging'

approaches (see text)

PP

no

averaging

approach [ppm] [ppm] [ppm]

Ao

[ppm]

T\A "

^CO y

[deg]

S.x"[deg]

AoL,[ppm2]

10

global -510 -20 0 710 106 5 0 44 101 68 41

local -67 6 113 78 9 118 3 071 39 13 57

30

global -72 0 -2 5 74 5 1117 0 93 -4 8 68 127

local 73 6 -2 1 75 7 1135 0 95 -2 5 25 129

3D GAFc -74 0 -17 75 7 1136 0 96 48 68

43

global -67 8 -8 0 75 8 1137 0 79 12 21 15

local -70 2 71 77 3 1160 0 82 21 24 15

3D GAFd -70 0 -7 0 77 0 115 5 0 82 12 21 —

"

9C0 is the angle between the y principal axis and the CO bond For

8C0 > 0°, the y principal axis lies m the sector defined by the atoms OC'Ca,

for 9C0 }< 0° in the sector NC'O

"i xls me atlS^s between the x principal axis and the normal to the peptide

planec

Starting from the "global frame average" CSA tensor of this peptide plane, the

corresponding tensor not containing the 3D GAF motion is calculated under the

assumptions of the "3D GAF averaging approach" (see text) using the fluctuation

amplitudes oa = 3 90°, op = 4 13°, oy = 7 53° extracted from the CSA

trajectory

dAsc but using the fluctuation amplitudes ca = 4 33°, Op = 2 87°,oY = 8 95°

extracted from the CSA trajectory

4.3.4 Comparison of different averaging approaches

The calculated "local frame average" CSA tensors for 15N and 13C are

given in Table 4 1 and Table 4 2, respectively The removal of the

intramolecular motion contained in the CSA trajectory leads in all cases to


increased values for the amsotropy For the N tensor, the strongest effect is

found for the very flexible peptide plane 10 where the asymmetry is

significantly reduced and the amsotropy increases by a factor 1 34 This

indicates that the differences of the "global frame average" CSA tensors of

peptide plane 10 and 30/43 result mainly from the large motional averaging

in case of peptide plane 10 The peptide planes 30 and 43, which can be both

described by a 3D GAF motion (see Chapter 3), show a much smaller scaling1 ^

of the amsotropy by a factor of 1 02 For the c tensor of peptide plane 10,

the effect of motional averaging is smaller than in the N case, the

amsotropy increases by a factor 1 1 whereas a factor of 1 02 is found in case

of the peptide planes 30/43 with 3D GAF motion

The calculated "local frame average" CSA tensors of the peptide plane 30

and 43 can be compared with results for the "3D GAF averaging" approach

also contained in Table 4 1 and Table 4 2 Here, the "global frame average"

CSA tensor was used as the reference tensor from which the contained

3D GAF motion was "removed" using the fluctuation amplitudes for the 3D

GAF motion extracted from the CSA trajectory Differences between a CSA

tensor calculated this way and the "local frame average" CSA are only due

to CSA fluctuations caused by a changing chemical and electronic

environment during the trajectory, which appear exclusively in the "local

frame averaging" approach Different behavior can be seen for N and C

1 ^tensors In case of c, the averaging due to 3D GAF reorientation is

dominant and the tensors of the two approaches are nearly identical (see

Table 4 2) On the other hand, the 15N CSA tensors of the "local frame

averaging" approach show a significantly lower amsotropy than the tensors

calculated when assuming averaging due to reorientation only It can be

concluded that the N tensor shows a larger sensitivity on changes in the

environment than the 13C CSA tensor (see also Section 4 4)

4 3 Average 15N and 13C' CSA tensors 125

Table 4 3 Subtraction of 3D GAF motional scaling from 15N and 1JC CSA tensors

Based on the "global frame average" CSA tensor a CSA tensor not containing the

peptide plane reorientation is calculated from a given set of fluctuation amplitudes

according the assumptions of the "3D GAF averaging" approach (see text)

spin(o«p,o7)

[deg] [ppm] [ppm] [ppm]

Ac

[ppm]

Tl

15N

(0,0)" 64 3 -28 5 92 8 139 3 0 39

(5,5) 65 9 -29 2 95 1 142 7 0 39

(10,10) 70 6 -315 102 1 153 2 0 39

(7,14) -67 8 33 4 1012 1518 0 34

(10 20) -715 -39 5 1110 166 5 0 29

13C,

(0,0)b -67 8 -8 0 75 8 1137 0 79

(5,5) -69 4 -8 0 77 4 116 1 0 79

(10,10) -74 0 -9 0 83 0 124 5 0 78

(7,14) 73 9 -5 5 79 4 119 1 0 86

(10,20) -812 -2 2 83 4 125 1 0 95

" The given CSA tensor corresponds to the "global frame average" CSA tensor of

peptide plane 30 given in Table 4 1 The results given in the rows below are

calculated by using this tensor as a reference

The given CSA tensor corresponds to the "global frame average" CSA tensor of

peptide plane 43 given in Table 4 2 The results given in the rows below are

calculated by using this tensor as a reference

The "3D GAF averaging approach" can be useful to remove motional

scaling effects from an experimentally obtained CSA tensor in order to

calculate an amsotropically upscaled CSA tensor which is appropriate for the

interpretation of CSA relaxation in the liquid state Results for sets of

different fluctuation amplitudes are illustrated in Table 4 3 It can be seen that

anisotropic intramolecular motion has an anisotropic scaling effect on the

CSA tensor The asymmetry of the CSA tensor is increasingly modified with

increasing anisotropy of the peptide plane motion The anisotropy of the


CSA tensor is significantly affected in case of fluctuation amplitudes

°aB Y> 10°

•In cases of additional jump motion the effects are even larger.

Here, also the orientation of the calculated CSA might differ significantly

from the reference tensor due to averaging between two (or more) distinct

conformations.

4.4 Fluctuations of the CSA tensors

In a next step, the fluctuations of the CSA parameters during the CSA

trajectory shall be described. The fluctuations of the orientation and of the

principal values of the tensor are described independently here. Both effects

come together and result in either motional averaging of the CSA tensor in

solid-state NMR measurements (see Section 4.3) or CSA relaxation (see

Section 4.5.1). For the analysis, the CSA tensor gk of each snapshot k is

symmetrized and diagonalised. The antisymmetric part is transformed for

each snapshot as described for the averaged CSA tensor in Section 4.3.1.

4.4.1 Fluctuation of CSA anisotropy and asymmetry

For the I5N and 13C CSA tensors of all peptide planes the average values

and their standard deviations for the anisotropy and asymmetry of the2

symmetric part and for the parameter Ac fof the antisymmetric part are

given in Table 4.4. An example for a distribution of Ac and r\ are depicted

in Fig. 4.3 for the N tensor of peptide plane 30. In all cases, the mean values

for the anisotropy are slightly increased when compared to the results for the

"local frame average" CSA (see Table 4.1 and Table 4.2), since the "local

frame average" approach includes averaging over orientational fluctuations

with respect to the molecular fragment. The 15N CSA parameters show

generally larger fluctuations than the 13C CSA parameters. The standard

deviations for the anisotropy values range from 7 to 15% (15N) and 4 to 6%

(C). The parameter Aaantl shows standard deviations between 30 and

50% of the average values.

4.4 Fluctuations of the CSA tensors 127

Table 4.4 Fluctuations of the !5N and 13C CSA parameters during the CSA trajectories.The average value with standard deviation of the anisotropy, the asymmetry, and Aoan/|for the antisymmetric part, which were calculated from the ensemble of 625 snapshots,are given for the peptide planes 10, 30, and 43.

spinpeptide

plane no.

anisotropy

Act [ppm]asymmetry r\ Ao„ [PPffl2]

15N

10 148.7 ±14.8 0.31 ±0.15 561± 239

30 142.7 ±10.1 0.39 ±0.10 595 ±148

43 144,2 ±10.5 0.39 ±0.14 569 ±157

13C

10 119.0 ±7.5 0.72 ±0.17 68 ±37

30 114.0 ±5.5 0.95 (±0.14)" 137 ±41

43 116.3 ±5.0 0.82 (±0.11)" 21 ±13

1The distribution of the CSA asymmetry values is strongly asymmetric.

100

80w*-»

c

<Den

> 60

CD

£40n

E

=j 20

a) (142.7 ±10.1) ppm100

80w

§ 600

o

&-40-Q

£3 20

b)

0.39 ±0.10

_JM1JU I uwin _J L^. . ,

^i-200 -180 -160 -140 -120 -100 -1 0 1

Ao [PPm] T|

Fig. 4.3 Fluctuation of I5N CSA parameters of peptide plane 30. The distributions of

(a) the anisotropy Ac and (b) the asymmetry r\ of the 15N CSA tensor are shown for

the ensemble of 625 snapshots of the CSA trajectory.


4.4.2 Orientational fluctuations of the CSA tensors

The orientational fluctuations of the CSA tensor during the trajectory are

mainly given by the peptide plane reonentation Additional fluctuations of

the CSA pnncipal axes with respect to the peptide plane frame are induced

by the changing chemical and electronic environment and can be extracted

by removing the internal motion of the peptide plane from the trajectory

which has been descnbed for the "local frame averaging" approach in

Section 4 3 1 The onentational distnbutions of the principal axes (with

respect to the equilibnum peptide plane) before and after the removal of the

internal motion are depicted in Fig 4 4 and Fig 4 5 for the N and 13C

CSA tensors, respectively In all cases, the onentational fluctuation due to the

peptide plane reonentation is dominant Fluctuations due to a changing

environment (see nght sides of Fig 4 4 and Fig 4 5) are generally larger in

case of the 15N CSA tensors Here, the distnbutions of the x and y pnncipal

axes are broader than that one of the z pnncipal axis This reflects a strong

influence of the asymmetry of the tensor on the pnncipal axis onentations

The N tensor of peptide plane 10 shows, on average, the smallest

asymmetry values during the CSA trajectory (see Table 4 4) In extreme

cases of r| —> 0, the x and y pnncipal axes he arbitrarily in a plane

orthogonal to the z pnncipal axis Correlations between the asymmetry r| of

the tensor and the orientations of the principal axes are an important result

indicating that onentation and pnncipal value fluctuations must not be

treated separately when calculating their influence on CSA relaxation (see

Section 2 5 1) In case of 13C and all three peptide planes, the onentational

distributions in the "local frame" are similar for the three pnncipal axes of

the CSA tensor reflecting its high asymmetry In addition, similar

distributions are found for all peptide planes including plane 10 This

confirms that the c tensor is less sensitive on changes of the chemical

environmental which is also visible m its relatively small amsotropy

fluctuations (see Table 4 4)

4.4 Fluctuations of the CSA tensors 129

global frame local frame

Fig. 4.4 Orientation fluctuations of the N CSA principal axes during the CSA

trajectory. Results are shown in (a,b,c) for peptide planes 10, 30, and 43. The

projections of the principal axis unit vectors on a unit sphere are given with respect to

the equilibrium peptide plane frame ew e„, e .In each row the left part corresponds to

the fluctuations including the internal motion of the peptide plane, whereas the right

part corresponds to the fluctuations left after removing the internal motion by

alignment of the instantaneous peptide plane in the equilibrium orientation.


global frame local frame

Fig. 4.5 Orientation fluctuations of the 13C CSA principal axes during the CSA

trajectory. Results are shown in (a,b,c) for peptide planes 10, 30, and 43. The

projections of the principal axis unit vectors on a unit sphere are given with respect to

the equilibrium peptide plane frame ea, e», e .In each row the left part corresponds to

the fluctuations including the internal motion of the peptide plane, whereas the right

part corresponds to the fluctuations left after removing the internal motion by

alignment of the instantaneous peptide plane in the equilibrium orientation.


4.5 iaN and UC CSA relaxation of the protein back¬

bone

In Section 4 5 1, the calculated CSA trajectory is used as a model system

to estimate systematic errors on CSA relaxation rate constants which result

from simplifying assumptions on the CSA interaction strength Since it is not

possible to measure the CSA fluctuations in a time-resolved manner it has to

be tested whether CSA fluctuations have to be explicitly taken into account

for CSA relaxation or whether the use of averaged CSA interaction strengths

is sufficiently accurate In Section 4 5 2, CSA relaxation is regarded from a

different point of view, and it is asked which assumptions on the CSA tensors

are reasonable when extracting motional parameters from NMR relaxation

measurements A discussion of the role of the antisymmetric part of the CSA

is given in Section 4 5 3

4.5.1 Implications of CSA fluctuations on CSA relaxation

CSA relaxation in proteins in the liquid state is normally calculated using

a motionally averaged CSA interaction strength obtained by solid-state NMR

measurements Fluctuations of CSA tensor principal values and orientation

(with respect to the molecular fragment) due to a changing chemical

environment induced by intramolecular motion have not been taken into

account in NMR relaxation studies so far However, these fluctuations

directly enter the autocorrelation function which determines CSA relaxation

The systematic error connected with the assumption of a constant CSA

interaction strength is estimated by comparing four different approaches for

analyzing the CSA trajectory In the "exact" approach, the time-dependent

fluctuations of the CSA tensor principal values and onentations in the

molecular frame are descnbed by a 2nd rank correlation function which can

be calculated from the CSA trajectory according to Eq (2 84) Assuming

isotropic overall tumbling and independence of internal and overall tumbling

motion, the correlation function and spectral density function in the

laboratory frame are given by


C(x) = e Cml(x) and /(co) = j C(t)cos(cox)rfx. (4.2)

The CSA contribution to T{ relaxation of 15N and 13C spins can then be

calculated

(V-l = T-AJ^x)- (4-3)

In the "global frame average" approach, the corresponding CSA tensors,

listed in Table 4.1 and Table 4.2 for the 15N and 13C backbone spins, are

fixed at the equilibrium peptide plane. Then, a CSA trajectory is calculated

by aligning the equilibrium peptide plane with each instantaneous peptide

plane of the series of 625 snapshots. The corresponding autocorrelation

function is, again, given by Eq. (2.84). This approach corresponds to the

normally applied procedure of using a constant CSA tensor determined by

solid-state NMR to calculate CSA relaxation. The "local frame average"

approach and the "3D GAF averaging" approach (in case of peptide planes

30/43) are analogous to the "global frame averaging" approach using the

"local frame average" and "3D GAF average" CSA tensors instead which

have been calculated in Section 4.3.

In Fig. 4.6, the correlation functions of the four approaches are compared

with each other for the 15N and 13C CSA tensors of the peptide planes 10,

30, and 43. Note, that the functions are not normalized and contain the

interaction strength. The resulting relaxation rate constants (Tj )CSA are

listed in Table 4.5. In all cases, the "exact" approach corresponds to the

largest rate constants, the "global frame averaging" approach to the lowest

rate constants. The ratio y of the relaxation rate constants of the "exact" and

the "global frame averaging" approach is a measure for the systematic error

when neglecting the CSA fluctuations contained in the CSA trajectory. For

peptide planes 30 and 43, which show 3D GAF motion, y takes about the

same value of approximately 1.05 in all cases. A large systematic error

results in the flexible loop region with y = 1.75 and 1.27 for the N and

13C CSA tensors of peptide plane 10, respectively. However, the correlation

functions of the "exact" and the "global frame averaging" approach show a


large correlation (constant offset between the correlation functions in

Fig 4 6) Thus, the fluctuations have not to be taken into account explicitly

but the systematic error can be compensated by an upscahng of the "global

frame average" CSA tensor Using the "local frame average" CSA tensor or

the "3D GAF average" CSA tensor as reference CSA tensor is appropriate in

case of 13C and peptide planes 30/43 which show no large-amplitude

intramolecular motion For N, the "3D GAF average" CSA tensor comes

closer to the "exact" approach than the "local frame average" CSA tensor

which reflects the averaging effect in the "local frame" In the loop region,

this effect is even larger and the "local frame average" approach only partly

compensates the systematic offset between the "exact" and the "global frame

averaging" approach However, peptide plane 10 represents a "worst case"

and it is not likely that a CSA tensor of a very flexible loop region measured

by solid-state NMR would be used for NMR relaxation calculations

Table 4 5 I5N and 13C CSA relaxation rate constants at field strength corresponding to

600 MHz proton resonance frequency for the peptide planes 10, 30, and 43 (TA )CSAwere calculated according to Eqs (4 2) and (4 3) using the correlation functions in

Fig 4 6 for three different approaches namely the "exact", the "local frame average",

and the "global frame averaging" approach The overall tumbling correlation time xc

was set to 4 03 ns which corresponds to ubiqmtm dissolved in water at 300 K

spinPP

no

(ri')«A [* ']

'exact'

approach

"local frame

average" approach

"global frame

average' approach

la

15N

10 0315 0 280 0 180 175

30 0 474 0 465 0 451 105

43 0 484 0 476 0 463 105

13C.

10 0 374 0 362 0 295 127

30 0 499 0 495 0 478 104

43 0 484 0 483 0 460 105

"

y is the ratio of the relaxation rate constants (T] )c&4of the "exact" and the

"global frame averaging" approach


500

time [ps]500

time [ps]

f)

EQ.

1 65

LL 1 55

o

500

time [ps]1000

1 45500

time [ps]

1000

1000

1000

Fig 4 6 Auto-correlation functions for CSA interaction in the molecular frame

extracted from the CSA trajectory according to Eq (2 84) for the "exact" (e), "local

frame average" (1), "3D GAF average" (f), and "global frame average" (g) approach

Results for N and peptide planes 10, 30, and 43 are given in panels (a, c, e), for C

and peptide planes 10, 30, 43 in panels (b, d, f) Note the different scales of the Cmt(t)ordmates


Dipole-CSA cross-correlated relaxation is sensitive to fluctuations of CSA

pnncipal values and angles between the CSA principal axes and the dipolar

director (it is assumed that the intemuclear distance is constant) Hence, the

relaxation rate constants do also depend on fluctuations of the dipolar

director with respect to the peptide plane The JH-15N dipole- N CSA cross-

correlation was studied following the strategy applied for the auto¬

correlation case The fluctuation distributions of the angles 0^ and

®nh zbetween the NH bond and the y and z 15N CSA pnncipal axes are

shown in Fig 4 7a and b, respectively, for peptide plane 43 The standard

deviations of the angle fluctuations of 4 2° for 6^ and 2 6° for 0^ z

reflect the larger fluctuation in case of the y pnncipal axis due to the low

asymmetry of the tensor (see also Fig 4 4) The mean value of 18 3° for

®NH zcomes cl°se t0 tne result for the "local frame averaging" approach

given in Table 4 1

The cross-conelation function in the molecular frame takes the form

Cn.nhW = (ox(0)(P2(ex(0) eNH(x))))

+ (ay(0)(P2(ey(0) eNH(T)))) + (oz(0)(P2(ez(0) eNH(x)))) (4 4)

The principal value fluctuations linearly enter the correlation function On

the other hand, the onentational fluctuations appear in the Legendre

polynomial terms which leads to non-linear averaging In case of isotropic

overall tumbling of the molecule the corresponding correlation function and

spectral density function in the laboratory frame are given by

JN NH(<o) = j CNfNH(x)cos(m)dx (4 5)

The cross-correlation relaxation rate constant FN NHcan be calculated

^ NH= -^N^NH(4JN, NH®) + 3JN, NH«°n))> <4 6)


with^ = ^^WfA'W assuming a constant NH bond length rm

The cross-correlation functions CN NH(t) were calculated according to

Eq (4 4) for all three peptide planes using, again, the "exact", the "local

frame averaging", and the "rigid averaging" approach Internal correlation

functions for the different approaches are plotted in Fig 4 7c and d for

peptide planes 10 and 43, respectively It can be seen, that even m case of

peptide plane 10 the fluctuations have not to be taken explicitly into account

and it is appropriate to use the "local frame average" CSA tensor for the

calculation of the rate constants TN NHAt field strengths corresponding to

600 MHz proton resonance the calculated ratios for the rate constants of the

"exact" and the "global frame averaging" approach are 1 33, 1 06, and 1 05

in case of peptide planes 10,30, and 43, respectively Thus, a systematic error

of at least 5% has to be considered when calculating the rate constants with

the "global frame average" CSA tensors

4.5 15N and 13C CSA relaxation of the protein backbone 137

110

200 400 600 800 1000

time [ps]

200 400 600 800

time [ps]

1000

Fig. 4.7 Fluctuation distributions of (a) the angle BNH zbetween the NH bond and die

z principal axis of the 15N CSA tensor and (b) the angle 9^ between the NH bond

and the y principal axis. Mean values and their standard deviations for the ensemble of

625 snapshots are indicated in the figure. H- N dipole- N CSA cross-correlation

functions are calculated according to Eq. (4.4) for peptide plane 10 (see panel c) and 43

(see panel d) using the "exact" (e), "local frame average" (1), and "global frame

average" (g) approach.

4.5.2 CSA relaxation based on average CSA tensors

The last chaper raises the question which average CSA tensor should be

used when motional parameters are extracted from NMR relaxation

measurements. In Table 4.6 different CSA tensors, either obtained by the


CSA trajectory of peptide planes 30 and 43 with the "global frame

averaging" approach or by solid-state NMR experiments, are translated into

the auto- and cross-correlated relaxation rate constants (Tx N)CSA and

FN NHfor 15N, (T{ C)CSA and rc cc

for 13C assuming anisotropic 3D

GAF motion of the peptide plane. A comparison of the calculated rate

constants for the CSA tensors DFT pp 30, DFT pp 43, and Ala-Ala shows

that the cross-correlation relaxation rate constants are much more sensitive

to even slight differences in the orientation of the CSA tensors since the

relative orientation of the CSA tensor with respect to the dipolar director

enters the cross-correlation function. Hence, assumptions which are

reasonable for the analysis of auto-correlation data cannot be simply

transferred to the analysis of cross-correlation data.

For the auto-correlation rate constants (7, x)cSA' tne calculated CSA

tensors located in the a helix and a (3 strand of ubiquitin yield very similar

results for bothlN and C. This supports the assumption of uniform CSA

tensors used in Chapter 3. On the other hand, the rate constants calculated

from the two different experimental CSA tensors show quite large

differences. The rate constants calculated from N and C CSA tensors of

the peptide Ala-Ala compare well with the results from the calculated

tensors. However, the large anisotropy Aa = I64.4ppm of the N CSA

tensor of Hiyama et al. [119] (for monoclinic Boc(Gly)3OBz), most often

used in NMR relaxation studies of proteins, results in a rate constant

(Tl N)CSA which is by a factor of 1.30 larger than the one for the 15N CSA

tensor of Ala-Ala. In case of 13C, the discrepancy is smaller, but with the

CSA tensor of Stark et al. [120] a relaxation rate (Ty C)CSA by a factor of

1.13 larger than calculated for the 13C CSA ofAla-Ala is obtained. Whether

the relatively small anisotropics of the calculated CSA tensors are

appropriate for the interpretation of protein relaxation data, cannot be

decided from experimental relaxation data alone since an extraction of CSA

parameters from the relaxation data -needed for comparison- does rely on

assumptions for the dipolar interaction strength affecting both the analysis of

auto- and cross-correlated relaxation data. This illustrates that the study

presented in this chapter is only a first step on the way to an accurate


interpretation of CSA relaxation data in proteins Further DFT calculations

for other peptide planes in ubiquitin or different proteins, solid-state NMR

expenments for non-glycme peptides, but also more reliable data on the

dipolar interaction strengths are necessary to gain more information about

the magmtude of the CSA tensors

Table 4 6 Calculated auto- and cross-correlated relaxation rate constants at 600 MHz

proton resonance frequency using different CSA tensors obtained either by averaging

over the CSA trajectories or by solid-state NMR The relaxation rate constants are

calculated assuming isotropic overall tumbling with correlation time %c = 4 03 ns and

intramolecular 3D GAF motion of the peptide plane with fluctuation amplitudes

°«p = 7° and Gv= 14°

15N CSA tensor"

(^1 N^CSA Is 1 *N NH *-S 1

solid-state NMR Boc(Gly)3OBz [119] 0 58 4 27

solid-state NMR AlaAla[124] 0 44 4 46

DFT pp 30 "global frame averaging" 0 43 4 08

DFT pp 43 "global frame averaging" 0 44 3 76

13C CSA tensor* (7"l c)c\SA [* ] rcc [sV

solid-state NMR GlyGlyHCl [120] 0 52 -135

solid state NMR AlaAla [124] 0 47 -1 19

DFT pp 30 'global frame averaging" 0 46 101

DFT pp 43 "global frame averaging" 0 45 -103

"For the CSA tensors obtained by solid-state NMR see also Table A 4 in the

appendixbFor the CSA tensors obtained by solid-state NMR see also Table A 5 in the

appendix

C

T/v nhls tne 15N-LH dipole 15N CSA cross-correlation relaxation rate constant

and was calculated according to Eq (2 73)dTc c

is the 13C'-13Ca dipole-13C CSA cross-correlation relaxation rate

constant and was calculated according to Eq (2 73)

In case of cross-correlated cross-relaxation, the situation is even more

difficult Most published studies are based on the simple assumption of an


axially symmetric N CSA tensor which is approximately true for the tensor

determined by Hiyama et al [119] with r\ = 0 06 The DFT calculations

and other experimental results (see Table A 4 in the appendix) indicate that

this assumption might not be appropriate since asymmetries up to r| = 0 45

have been found This may have considerable influence on the interpretation

of cross-correlation relaxation data In addition, the assumption of uniform

CSA tensors for different peptide planes seems to be much more cntical than

in the case auto-correlated relaxation (see above)

In this section, motionally averaged CSA tensors were compared In the

last section, it was shown that an average CSA tensor which has been

upscaled to compensate for the motional scaling would be appropriate for

describing NMR relaxation In principle, one could use "local frame

average" CSA tensors extracted from CSA trajectories for which it has been

shown that the amount of intramolecular motion in the liquid state is properly

reflected in the trajectory If the relaxation calculations are based on

experimentally determined CSA tensors, the upscahng of the tensor could be

done using the "3D GAF averaging" approach described in Section 4 3 4

The magnitude of the intramolecular motion present in the solid state has

then to be estimated

4.5.3 Role of the antisymmetric part of CSA tensor

2The antisymmetric part Aoantl is not manifested in standard solid-state

NMR experiments, since it is non-secular with respect to the Zeeman

interaction However, it contributes to CSA relaxation (see Section 2 3)

although it is normally neglected when calculating relaxation parameters of

proteins in solution The DFT calculations provide data for the

antisymmetric part of the CSA tensors In Table 4 1 and Table 4 2 the average2 7 2 2

values for the parameter Aaantl = a~v + axz + a in the principal axes

system of the symmetric CSA are given for N and C, respectively The

relaxation contribution of the antisymmetric and symmetric part can be

calculated with Eqs (2 36), (2 66), and (2 74) considering only overall

tumbling motion In Fig 4 8, the relative contribution of the antisymmetric

part at 600 MHz proton resonance frequency, expressed as ratio


_1 anti _j sym

(^l x)csa//(^1 x)c5A' *s pl°tte(i versus the rank 2 correlation time of the

overall tumbling motion for the 15N and 13C CSA's of peptide plane 30 and

43. The relative contribution of the antisymmetric part is much larger in case

of N. For larger molecules with tumbling correlation times tc > 4ns a

lower plateau is reached. Here, the antisymmetric CSA contributions of

about 6% for 15N and less than 2% for 13C are rather small. For smaller

molecules with tumbling correlation times xc < 300ps much larger

contributions of up to 40% and 10% in case of 15N and 13C, respectively, are

predicted.

_1 anti

(Tl,x)cSA_1 sym

(Tl,x)cSA

15N, pp 30~~-0\

0.3 N\ 15N, pp43

0.2 - \

13C, pp 30\

0.1 /

v^13C, pp43^\/

10-11 1Q-10 10-9 -i 0-8

Fig. 4 8 Ratio of relaxation rate constants of the antisymmetric and the symmetric

CSA relaxation pathways at 600 MHz proton resonance frequency as a function of the

rank 2 correlation time xc for isotropic overall tumbling motion. Results are shown for

N and 13C CSA tensors of peptide planes (pp) 30 and 43 in ubiquitin (indicated m

the figure) No intramolecular motion is included. The CSA parameters were taken

from the "local frame averaging" approach (see Table 4.1 and Table 4.2).

Relaxation due to the antisymmetric part of the 15N CSA tensor should be

measurable according to the shown results. An ideal test system would

consist of a small peptide dissolved in a deuterated solvent in order to


suppress the dominant H- N dipolar relaxation. Tl Nmeasurements at

different field strengths, temperatures or viscosities would be required to

achieve a decomposition of the symmetric and antisymmetric CSA

relaxation contributions due to their distinct dependence on the rank 2

tumbling correlation time for the overall tumbling.

4.6 Conclusions

By combination of DFT calculations and MD simulation, a CSA trajectory

is created which can be used as a model system to estimate systematic errors

in the calculation of relaxation rate constants which result when introducing

simplifying assumptions on the CSA interaction strength. It was found that

for the studied peptide planes in ubiquitin a description with an average CSA

tensor, which does not contain the reorientational averaging of the peptide

plane, is sufficient and the principal value and orientational fluctuations of

the CSA tensor have not to be taken explicitly into account. Results for the

calculated average CSA tensors located in the a helix and a P strand of

ubiquitin support the assumption of uniform CSA tensors in different peptide

planes which has been made use of in Chapter 3 when extracting motional

parameters from NMR relaxation data.

It is difficult to decide whether the absolute values of the calculated CSA

tensor parameters are realistic or not. On the one hand, there is a good

agreement of both 15N and 13C CSA tensors with a single experimental

study ofthe peptide Ala-Ala [124] which corresponds to an anisotropy at the

lower end of all experimental studies. On the other hand, the calculated

anisotropics are significantly smaller than what is normally used in NMR

relaxation studies (see Chapter 3) or, recently, has been extracted from cross-

correlation relaxation experiments ( N anisotropies of about 170ppm, see

e.g. ref. [107]). However, such NMR relaxation studies do not rely on a self-

consistent parameter set. In particular, the assumption on the NH bond length

is very critical, and it significantly affects the results for the extracted

anisotropies. Similar problems are contained in the calculation of the CSA

trajectory since the DFT calculations rely on geometric parameters

determined by the MD force field which was not optimized for this purpose.

4.6 Conclusions 143

The DFT calculations have shown that the relaxation contribution of the

antisymmetric part with respect to the symmetric part of the CSA tensor is

less than 10% in case of proteins dissolved in water. For these systems, it

seems to be unlikely that the antisymmetric part can be accurately extracted

from standard NMR relaxation measurements. On the other hand, this

relaxation contribution might be wrongly attributed to the symmetric CSA

tensor resulting in increased scaling factors for the symmetric part of the

CSA (i. e. scaling factors XN and Xc in Chapter 3).

5.1 Introduction 145

5 Rotational Motion of a Solute Molecule

in a Highly Viscous Liquid

5.1 Introduction

NMR measurement parameters are sensitive to an extremely wide range of

motional time scales from picoseconds to seconds [7]. The fastest

intramolecular dynamical processes in the picosecond to microsecond range

can be monitored by various types of relaxation experiments in the laboratory

frame. However, the inherent power of NMR relaxation spectroscopy for

studying mobility in the liquid phase is hampered by the fact that only the

composite effect of intramolecular and overall tumbling motion is sensed by

the relaxation parameters. It is impossible to measure intramolecular

mobility alone. This becomes apparent from the Lipari-Szabo approach

which has been introduced in Section 2.2.3.2. The relaxation-active motional

correlation function for an auto-correlated process of a rank 2 interaction

\l = v in the laboratory frame is then given by a combination of Eqs. (2.39)

and (2.42):

C (T) = YS + (1 -S )e ]e , (5.1)

where the definitions S = S^ and xint = x\„t have been used. The

rotational tumbling with the correlation time xc acts as a multiplicative

"masking" process of the intramolecular mobility with correlation time xmt

that is ofprimary interest. As a result, xtcan be determined with reasonable

accuracy only in a certain range, which defines the "observation window" of

intramolecular motion. This is a severe practical restriction in motional

studies. The size of the observation window depends on the type of

measurement, the type of internal motion, the overall tumbling correlation

time and the magnetic field strength. In Fig. 5.1, the dependence of the

C Tl of a dipolar ^C-1!! system on the internal correlation time xint is

calculated for different values of the overall tumbling correlation time. As

can be seen in Fig. 5.1a and b, the size of the observation window increases

with increasing overall tumbling correlation time. The center of the

146 Rotational Motion of a Solute Molecule in a Highly Viscous Liquid

observation window is approximately given by xint = xc in cases of

tumbling correlation times xc<lns. For xc>lns, the functional

dependence of 13C T{ on %int is no longer monotonous and the center is

given by xint < xc. The center is slightly shifted towards increasing internal

correlation times for increasing xc. However, this effect which is not linear

in xc decreases for increasing x .Outside of this observation window,

intramolecular motion leads to xt -independent averaging effects. In the

extreme narrowing regime tmf«xc, only the order parameter remains active

(see Eq. (5.1)) and the anisotropic relaxation-active interactions, such as

2

dipolar interaction or CSA interaction, are scaled by S .In the case of slow

internal motion (xint » xc) the relaxation rate constants which belong to

different conformations of the molecule are averaged.

It would be highly desirable to shift the observation window for xint by a

selective change of xc [142] while the intramolecular motion is not affected.

This might be illustrated by two examples. In case of the cyclic peptide

antamanide with an overall tumbling correlation time of approximately

150 ps for a solution in CDC13 (for details see refs. [70,142]), the internal

correlation times for discrete jumps between different conformations in the

phenylanaline sidechains of antamanide, which are much larger than xc [70],

might be extracted when increasing xc by a factor of at least 10 as shown in

Fig. 5.1a. In case of larger molecules as the protein ubiquitin with a overall

tumbling correlation time xc = 4.0 ns for a solution in water (see Section

3.5.3), the internal correlation times for very fast intramolecular motion with

xint < 50 ps (see Section 3.5.4) might be accessible when going to correlation

times xc in the range of 10 ns as shown in Fig. 5.1b.

Within the framework of classical hydrodynamic theory for a continuous

solvent medium, the rotational correlation time for an interaction of rank 2 in

a spherical molecule obeys the Stokes-Einstein-Debye (SED) relationship

(see Section 2.2.2.1):

<<" ?rwhere V is the effective spherical molecular volume, T| is the shear viscosity,

T is the absolute temperature, and k is the Boltzmann constant. This relation


has frequently been used in NMR since the pioneering work of Bloembergen,

Purcell, and Pound [143] Based on Eq (5 2), it is tempting to shift xc and

the observation window of xint by varying the viscosity T) of the solvent,

while hoping that the intramolecular mobility of the solute is not significantly

affected

tint NFig 5 1 Illustration of the observation window (indicated as rectangle) for the

correlation time of intramolecular motion The dependence of C T, at 600 MHz

proton resonance frequency on the internal correlation time xmt is given for (a) the

overall tumbling correlation times xc = 150ps and 1 5 ns, and (b) for xc = 4ns and

10 ns assuming the spectral density approach of Lipan-Szabo in Eq (5 1) The C T,

was calculated for an ^C-'H bond with bond length rCH=\ 08 A considering only

dipolar relaxation


Williamson and Williams [144] used polychlorotrifluoroethene (PCTFE)

as a polymeric solvent for NMR studies. Its viscosity can be varied in a large

range by changing the temperature and by mixing fractions of different

degrees of polymerization. PCTFE was subsequently used in a number of

NMR investigations [145-148] for adjusting the viscosity in structural

studies of solute molecules measuring the nuclear Overhauser effect (NOE)

which depends on xc. However, no quantitative study of the dynamical

aspects of solute molecules in PCTFE has been reported so far.

The present experimental study explores the potential of PCTFE as a

solvent to control the overall tumbling correlation time of a simple solute

molecule by varying the viscosity. The rigid and symmetrical molecule 1,3-

dibromoadamantane (see Fig. 5.2) was selected as a probe and the relaxation

properties of its methine carbon-13 spins were studied with the goal of

developing a motional model for this solute in the accessible temperature-

dependent viscosity range between 10 cP and 10 cP.

5.2 Experimental

5.2.1 Characterization of the system and sample preparation

Voltalef 10S (Atochem, France) is a highly viscous, non-polar, aprotic oil

consisting of polychlorotrifluoroethene (PCTFE) with 6 to 7 monomer units

(see Fig. 5.2) and a molecular mass of approximately 800 Da.

Cl-

F

I-C-

Cl

-C-I-ClI

n = 6-7

PCTFEJ3C

H

Fig 5 2 Components of the studied system 1,3-Dibromoadamantane is dissolved in

polychlorotnfluoroethylene (PCTFE)

5.2 Experimental 149

Viscosity measurements were performed using a dynamic stress rheometer

(Rheometrics) with a Couette geometry (concentric cylinders). At

temperatures above 282 K, the viscosity of PCTFE is within experimental

accuracy not dependent on the shear rate in the range from 10"2 to 102 s.

The shear viscosity at a shear rate of 1 s, given in Fig. 5.3, covers more

than two orders of magnitude from 59 cP at 344.1 K to 8920 cP at 283.3 K.

In this range a small but systematic deviation from an Arrhenius temperature

dependence is found, similar to other highly viscous liquids [50,149], with

an activation energy varying from Ea ~ 87kJmol at 283 K to

Ea ~ 55kJmol at 344 K. The measured temperature dependence is shown

in Fig. 5.3. On the other hand, the temperature dependence can be fitted

within experimental accuracy by a power law [149]:

r\{T) = c(T-TQ)q, (5.3)

where c = 7.119- 1012cPK_<?, TQ = 241.8K, and q = -5.501. Below

282 K, PCTFE becomes opaque and the viscosity dependent on the shear

rate. At 281 K, for example, viscosities of r| = 1.25 • 10 cp at a shear rate

9-1 4 -1of 10 s and rj = 3.7 10 cp at a shear rate of 10 s were measured (see

Table 5.1).

It is known that fluorinated organic compounds have exceptional abilities

to dissolve oxygen [150,151], which could cause paramagnetic relaxation of

guest molecules. Therefore, the polymeric oil was degassed under high

vacuum and handled in an argon atmosphere. 1,3-Dibromoadamantane

(Aldrich) was dissolved in PCTFE with a concentration of 85 mM. Benzene-

dg, enclosed in a coaxial capillary, was used for field-frequency locking.


104

57,

& 103

'53oo

.«2 102>

'"

280 290 300 310 320 330 340 350

Temperature T [K]

Fig. 5 3 Temperature dependence of the viscosity of PCTFE. The experimental data

points measured at a shear rate of 1 s are given for a temperature range from 283 to

344 K. The curve corresponds to the fitted power law of Eq.(5.3)

5.2.2 NMR relaxation measurements

The NMR measurements were performed on Bruker DMX-400 and AMX-

600 spectrometers, operating at 400 and 600 MHz proton resonance

frequency, respectively. Tj, NOE ,Ty, and T2 have been measured for the

methine carbons in 1,3-dibromoadamantane at temperatures of 274, 281,

290, 300, 324, and 339 K and a proton resonance frequency of 600 MHz.

Additional measurements at 400 MHz proton resonance were carried out at

a temperature of 300 K.

The 13C spin-lattice relaxation times T{ were measured by the standard

inversion-recovery method, after an initial polarization transfer from protons

by refocused INEPT and with proton decoupling during mixing and

acquisition times. Eight or more mixing times %m in the range

3(is < Tm < 5Tj were used. The inversion-recovery data were fitted by a

three-parameter mono-exponential function, employing the simplex

algorithm from the toolbox of MATLAB [152], The covariance matrix of the

fitted parameters was calculated for uncorrelated measurement errors of

5.2 Experimental 151

equal variance.

The 13C NOEs were measured after presaturation of the protons by a

series of 120° pulses with 10-ms interpulse delays, applied for at least

5 Tj [153]. The peak areas were integrated and the NOEs were calculated

as ratios with and without proton saturation. The experimental error was

estimated by repeating the measurement for some of the NOEs.

The 13C spin-spin relaxation times T2 were determined by the Carr-

Purcell-Meiboom-Gill (CPMG) pulse sequence after an initial INEPTIT 1 J

transfer. C rotating-frame relaxation times T{ were measured for a C

spin-locking field of 1.2 kHz. It proved unnecessary to eliminate the cross-

correlation between dipolar and chemical shift anisotropy relaxation by

applying pulses to the protons during the mixing time. A quantitative

estimate of the cross-correlated cross-relaxation effect, using an experiment

similar to the one of ref. [71], showed that the contribution to T2 or Tl is

below 1%, indicating a negligible anisotropy of the CH carbon shift. This is

in agreement with quantum chemical calculations of the CSA tensor based

on density functional theory (DFT) which showed that Ac < lOppm. In the

T2 and Ti measurements, the carrier frequency was set 100 Hz off-

resonance. The magnetization decays were fitted by a two-parameter mono-

exponential function. It was found that at 300 K and 600 MHz T{ is

independent of the spin-locking field strength set to 0, 0.6, 1.2, 1.8, and

2.8 kHz. This shows that no slow exchange processes with Texch > 10 |xs

occur.

Decoupling of the 19F spins of the solvent by strong rf irradiation during

the mixing time of NOESY and lH-T{ inversion-recovery experiments did

not noticeably affect relaxation, verifying that there is no measurable cross-

relaxation between the 19F spins of PCTFE and the spins of the guest

molecules. However, a small contribution of 19F to dipolar T^ and T2relaxation of C, not exceeding a few percent, cannot be excluded.

A mono-exponential relaxation behavior at all temperatures (even at 281 K

and 274 K where the solvent is very viscous and opaque) was observed. The


measured C relaxation parameters are listed in Table 5 1 The errors for T{and T2 vary between 0.5 and 6%, and the standard deviations for the

heteronuclear NOEs at 290 and 300 K were found to be 1%.

Table 5.1 Experimental C relaxation data for CH groups in 1,3-dibromoadamantane

dissolved in PCTFE and viscosity r| of the solvent PCTFE at the correspondingtemperatures.

COff/27I

[MHz]

T

[K]

11

[cP]

7",

[ms]

?2

[ms]

NOE

600

274 791 ± 16rf 162

281

125000"

50000 *

37000c

771 +9 d n±2d 166

290 3928' 731 ±14 e n±i,e 1 75 + 0.02'

300 1392* 750 ± 3 e 19611' 1.82 ± 0.02e

324 208' 1058 ±5 d 784 ± 4' 2 19

339 83* \5\5±\1d 133418'' 2 43

400 300 1392' 535 ± 3 e 188 ±2 d 191+002'

"

Viscosity measured at a shear rate of 0 01 s

bViscosity measured at a shear rate of 1 s

c

Viscosity measured at a shear rate of 10 s

Error is the standard deviation obtained from fitting the time course of the peakintegrals

eError is the standard deviation obtained by repeating the experiment

5.3 Results and Discussion

5.3.1 Test of a single-correlation-time model

The relaxation of the 13CH carbon spins is governed by the 13C-'H dipolar

interaction with the standard relations for 7^ and T2 according to Eqs.

(2.65) and (2.71), respectively, the cross relaxation rate constant

a = Tc Hand the heteronuclear NOE according to Eqs. (2.67) and

5.3 Results and Discussion 153

(2.68), respectively.

For simplicity it is often assumed that the correlation function of the

random process causing relaxation is mono-exponential with the correlation

time xc. This is justified for a rigid molecule tumbling isotropically in a

homogeneous medium (see Section 2.2.2.1) which leads to the Lorentzian(2)

spectral density function/(co) = 7L(co) of Eq. (2.36):

7(q>) = c—2. (5.4)1 + (coxc)

However, an analysis of the experimental data shows that the three 13C

parameters T{, T2, and NOE contradict each other if a Lorentzian spectral

density is assumed. This is illustrated in Fig. 5.4 for the experimental values

at 300 K and 600 MHz. The measured T, and T2 values were corrected by

subtracting the ~10% contribution of intramolecular dipolar relaxation

induced by remote protons, assuming isotropic tumbling and internuclear

distances rc//=1.08 A, rcc=1.54A, and tetrahedral bond angles as

geometric parameters. The observed Tl value may be rationalized by one of

the two correlation times xc « 60 ps or xc~ 7 ns .

Both are in contradiction

with the value xc = 400 ps deduced from the measured T2 and NOE values.

Fitting of the experimental data by a non-axially symmetric rotational

diffusion tensor and by applying the Woessner equations [154] would lead to

an anisotropy of rotational tumbling of at least two orders of magnitude. This

is, however, in contrast to the moment of inertia tensor of 1,3-

dibromoadamantane whose principal values have a ratio of 1:2.6:3.1,

rendering this interpretation unlikely.


0.001

Fig 5 4 Illustration of the failure of a single-correlation-time model The theoretical

dependencies of C T{, T2, and NOE on a single correlation time xc are plotted,

assuming rCH-\ 08 A and cow/2it= 600 MHz The experimental values for the data

at 300 K corrected for the dipolar relaxation by i emote protons, are given as solid

circles It is not possible to find a single correlation time x( for which all three

experimental values he on the corresponding theoretical curves The xc "unceitainty

range" is marked in gray

5.3.2 Distribution of correlation times

It is attempted to rationalize the experimental data by a distribution of

rotational correlation times (see Section 2 2 2 2) which may be the result of

an exchange between environments with different effective microviscosities

or rotational hindering potentials The strictly exponential time course in the

relaxation experiments suggests rapid exchange on time scales much shorter

than Tn If each environment has a characteristic correlation time T and an

occupational probability p{x[), the mean relaxation rate constants are given

by


(T~x% = jP^fJT^c <°K = Jp(Tc)c(Tc)rflc, (5.5)

0x c

0

with x = 1, 2, or 1 p; and the mean iVOZs is given by

In (°\(NOE)T = 1 + -2- -£-. (5.6)

Because of the linear dependence of the relaxation rate constants on 7(a)), it

is possible to express them by a mean spectral density function

(7(0))), = \p{xc) X-£—1dxc, (5.7)

which is the Fourier transform of Eq. (2.37). Based on this equation, the

relaxation data of Table 5.1 measured at 300 K for 400 and 600 MHz proton

resonance were fitted by various distribution functions p(ic).

The first models to be explored are those with n= 2 or 3 discrete

environments [155,156] leading to the mean spectral density function

"2t

<J(a»>T = Xp* ^—. (5.8)

At first, n = 2 is assumed. Prior to fitting, the experimental values were

corrected as described below Eq. (5.4). The corrected values are given in

Table 5.2. The resulting fit parameters p{ = 1 - p2, xcl, and xc2, and the

back-calculated relaxation parameters T{, T2, and NOE are included in

Table 5.2 (row A). As expected, the fit is significantly better than that for a

single correlation time. But the differences between the back-calculated

values and the corrected experimental values are still significantly larger than

the estimated measurement errors. It is remarkable that the two resulting

correlation times tc1 = 37ps and xc2 = 3.3ns differ by nearly two orders


of magnitude and indicate two vastly different environments. Environment 1

with a population of 76% seems to approximate freely moving solute

molecules, while environment 2 with a population of 24% suggests strong

association with PCTFE molecules or rotational hindrance by the PCTFE

molecules. Interestingly, for certain parameters p{ = 1 -p7, xcl, and xc2

of the bimodal model the calculated NOE parameters show an anomalous

field dependence where, in contrast to the single-correlation-time model, the

NOE may increase with increasing magnetic field strength. This contradicts

the experimental findings. In a further attempt to improve the fit for all six

relaxation parameters, an adjustable rCH bond length was used. Although2

the error of the fit (not shown) is reduced to % =53, this led to a physically

unreasonable bond length of rCH = 1.28 A.

The inclusion of a third environment (n = 3 in Eq. (5.8)) with correlation

time xc3 significantly improved the fitting quality. The fit, documented in

Table 5.2 (row B), leads to three correlation times that differ from each other

by more than an order of magnitude. Such a fitting procedure should be

considered rather as an attempt of mathematically modeling the measured

data than to imply a specific physical model. Nevertheless, it shows that the

environment experienced by the solute molecules is highly inhomogeneous,

requiring more than just two discrete correlation times.

NMR relaxation times in highly viscous liquids have previously been

successfully fitted by assuming a Cole-Davidson spectral density function

[53, 157]:

(jm\-l(

sin(pCDatan(<BTCD))

2 Pcd/2(i + (cotCD) ) )

(5.9)

with the distribution parameter (3CD, 0 < Pcz) < 1, and the effective

correlation time xCD. For (3C0= 1, the spectral density merges into the one

for a single correlation time. A simple physical interpretation of this

empirical spectral density function is not apparent. Fitting the corrected

relaxation data by the Cole-Davidson spectral density function leads to the


results in Table 5.2 (row C). The fitting error lies between the ones for two

and three discrete correlation times (Table 5.2, rows A and B). The very low

value pC£) = 0.094 indicates a significant deviation from a single-

environment situation. The fitted Cole-Davidson spectral density function is

compared with those of the bimodal and trimodal models in Fig. 5.5. While

the values (7(0)) are nearly identical for the three spectral density

functions, their forms differ at the other relevant frequencies, in particulary

for 300 and 750 MHz.

10

i ' I ' I '

0'9

— bimodal— trimodal

- t\• - - - Cole-Davidson :

\-\ .

\'\ -

\\.

-10:

::-^:::'

I L I , I

0 100 200 300 400 500 600 700 800

00/271 [MHz]

Fig 5 5 Comparison of three model spectral density functions fitted to the

experimental C data Tt, T2, and NOE at 300 K, measured at a>H/2it= 400 and

600 MHz Solid line, bimodal spectral density function; dashed line, trimodal spectral

density function, dotted line, Cole-Davidson spectral density function


Table 5 2 Model parameters obtained by fitting various motional models to the

experimental data and back-calculated relaxation data for 1,3-dibromoadamantane at

300 K and (0H/2n = 400 and 600 MHz

Motional

Model

(0H/2n

[MHz] [ms]

T2

[ms]

NOE Model Parameters" 24X

A Bimodal

model

400 560 202 1 81p, =0 757 ±0 002

p2 = 0 243 ± 0 002

Tcl =(36 7±0 5)ps

xc2 = (3 32 ± 0 03) ns

326

600 843 217 2 10

B Tnmodal

model

400 590 201 1 90p, =0 693 ±0 012

p2 =0 181 ±0010

p3 =0 126 ±0 012

xc1 =(169±2 2)ps

xc2 = (528 ± 44) ps

Tc3 = (5 75 ± 0 30 ns

8

600 825 217 1 83

C Cole

Davidson

model

400 560 200 172 (3CD =0 094 ±0 001

xCD = (8 90 ± 0 06) ns249

600 846 218 1 70

Experimentaldata at

400 589

±3

207

±2

191

±0 02

300 K c

600 826

+ 3

216

±1

182

±0 02

"The error limits of the model parameters are determined by a Monte Carlo

procedure consisting of 100 fits with random errors corresponding to the

experimental standard deviations added to all relaxation parameters

Fitting error

2_

Tl calc

A.

-T

(Ar, exp)

i exp)2+(T2cak-T2 eJ

+{NOELalc-NOE )2

^^

(Ar2 exp) (AW^exp)standard deviations AJ, , A7"2 and ANOE

The values are experimental measurements corrected for dipolar relaxation byremote protons (see text)


5.3.3 Analysis of the temperature-dependent relaxation data

The temperature-dependent relaxation data of Table 5.1 were used to study

the viscosity dependence of the dynamical behavior of the solute molecules.

At 290 K and 600 MHz a Tx minimum with T{ min= 731 ms is found. A

comparison with the theoretical value of Tl min= 224 ms for a single-

correlation-time model and rCH=l.08 A provides another indication for a

multiple-site dynamics of 1,3-dibromoadamantane. Because it is difficult to

relate the deduced correlation times to a specific physical model of the

inhomogeneous polymer environment of the solute molecules, it might be

appropriate to analyze the data at a single proton-resonance frequency of

600 MHz in terms of the bimodal model. The three model parameters

Pj = 1 - p2, xcl, and xc2 were deduced from the three measurements Tl,

T2, and NOE, again corrected for the dipolar relaxation of remote protons.

They are given in Table 5.3.

The slight discrepancy between the values at 300 K in Table 5.2 and Table

5.3 is caused by the fact that additional 400 MHz measurements are used in

the fitting procedure leading to the values in Table 5.2. Throughout the

viscosity range investigated two significantly different correlation times

result which monotonously increase for increasing viscosity. It is remarkable

that at the highest measured viscosity (281 K) the environment with the

longer correlation time is populated to 78%, whereas at the lowest measured

viscosity (339 K) the environment with the shorter correlation time

dominates with 95% population. It appears that the equilibrium is shifted

toward the "free" form of the solute at high temperature.

In Fig. 5.6, the products xc]T and xc2T are plotted in a double logarithmic

form against the viscosity r\. Based on the SED relationship of Eq. (5.2), a

linear dependence with unit slope is expected. For both components the slope

is significantly smaller than 1, following a relationship

m,

Txci = const r\ ,i= 1,2. (5.10)

A least-squares fit leads to the values /n, = 0.12 for the rapidly tumbling


population and m2 = 0 45 for the slowly tumbling population The

functional form of Eq (5 10) is in agreement with earlier findings for other

systems using NMR and dielectric relaxation measurements [158,159]

Table 5 3 Parameters of the bimodal model computed from the experimental values of

r,, T2, and NOE (corrected for dipolar relaxation by remote protons) at various

viscosities and co„/2jt= 600 MHz

T

[K] [cp]Populations

* Correlation times*

281p, =0221+0023

p2 = 0 779 ± 0 023

tcl =(74 2 + 10 7)ps

Xc2 =(7 80±0 30)ns

290 3928p, =0 505 ±0 012

p9 =0495+0012

Tc, =(359±16)ps

Tc2 = (4 79 ± 0 13) ns

300 1392p, =0 70910 003

p, =0 291 ±0 003

Tcl =(276±08)ps

ic2 = (2 84 ± 0 03) ns

324 208p, =0 909 ±0 002

p, =0 091 ±0 002

xcl =(22 5+0 5) ps

xc2 =(1 11 ±003) ns

339 83p, =0 953 ±0 003

p2 = 0 047 + 0 003

Tcl =(18 2±04)ps

tc2 = (748 + 56) ps

"

Viscosity measured at a shear rate of 1 s The viscosity at 281 K depends on the

shear rate (see Table 5 1) and was not used

bThe error limits of the model parameters are determined by a Monte Carlo

procedure consisting of 100 fits with random errors corresponding to the

experimental standard deviations added to all relaxation parameters

The violation of the SED relation implies that the motion of the solute

molecules is less influenced by the solvent viscosity than expected This is

not unreasonable for the rapidly tumbling population as the small molecules

may move with little hindrance in cavities formed by the polymenc solvent

For the derivation of the SED relation, on the other hand, it was assumed that

the solute molecules move in a homogeneous environment of very small

solvent molecules which can be approximated to be a continuum (see

5 3 Results and Discussion 161

Section 2.2.2.1) where obviously the formation of long-lived cavities is

unlikely. An Arrhemus-type approximation of the correlation times xcl and

Tc2, Tcl 2= kexp(Eal 2/(i?r)), leads to the activation energies

Eal = llkJmol'

and£a2 = 33kJmor'.

5x10

2x10

TlIcP]

11 [cP]

Fig 5 6 Viscosity dependence of xcXT for the fast and xc2T for the slow tumbling

population in the bimodal model. Only data points with shear-rate-independent

viscosities are included (see Table 5 3) The linear fits corresponds to Eq (5 10) with

the fitting parameters m,


5.4 Conclusions

This study demonstrates that 1,3-dibromoadamantane experiences PCTFE

as an inhomogeneous solvent medium The observations can formally be

modeled by a rapid exchange between at least two different environments

Similar situations have been found for solutes in other highly viscous

solvents [50,51,157] For the bimodal model, the apparent correlation times

differ by two orders of magnitude, one environment corresponding to

virtually free guest molecules, and the other environment exhibiting a strong

solute-solvent interaction The temperature and viscosity dependence of the

tumbling correlation times of the two populations is much weaker than would

be expected from the Stokes-Einstein-Debye relationship The exchange of

the guest molecules between the different environments appears to be rapid,

but is difficult to quantify based on NMR measurements alone, as no rf field

strength dispersion is observed mlj experiments The results presented in

this chapter show that the tumbling behavior of small solute molecules in

solvents such as PCTFE is not as simple as desired for shifting the rotational

correlation time in view of NMR studies of intramolecular motion

6 1 Appendix to Chapter 3 163

6 Appendix

6.1 Appendix to Chapter 3

6.1.1 Analysis of molecular dynamics trajectory

6.1.1.1 Extraction of fluctuation amplitudes from MD trajectory

Table A 1 MD reonentational fluctuation amplitudes about the pnncipal axes ea, en, e

and dihedral angles co of the peptide planes in ubiquitin The values for aa, o«, and cywere extracted from the 1 5 ns MD trajectory according to Eq (2 82) Only those 57

peptide planes which can be descnbed by a 3D GAF motion are listed

peptide

plane nooa [deg] Op [deg] oY [deg] co [deg]

a

2 4 883 4 208 9 806

3 4 282 4 133 10718 167 7 ±7 3

4 4 516 3 844 8 304 179 3 + 6 7

5 3 946 3 388 10 539 176 1 ±6 8

6 4 704 3 910 8 472 175 7 ±7 2

12 4 986 5 223 14 232 172 617 8

13 4 342 5 689 12 764 1813 + 79

14 4 855 3 916 11893 176 9 + 8 7

15 4 414 6 089 14 358 176 8 + 7 9

16 5 079 5 282 10 863 178 2±8 3

17 4 753 4 954 14 817 181 1+70

18 7 115 3 789 9 800 168 8 + 7 2

20 6 019 6 519 12 496 175 1 ±7 2

21 7 093 5 495 9 348 1806±7 6

22 4 436 4 500 10 960 181 6±7 7

23 4 465 3 064 7 765 175 0 + 6 0

24 3 887 3 837 6 946 167 9 ±60

25 4 498 3 029 7 373 1764±5 8

26 4 201 3 718 8 114 172 9±60

164 Appendix

peptide

plane noCa [de§] Op [deg] Oy [deg] ro [deg]°

27 4 279 3 231 5 973 175 5 ± 5 4

28 4 379 2 921 6 964 174 1 ±5 5

29 4 069 3 638 7 557 173 6 + 5 7

30 3 700 3 146 7 322 1709 + 5 6

31 4 291 3 670 6 997 170 7161

32 4 380 5 574 8 607 1760 + 6 3

33 5 693 4 755 10 180 179417 6

34 5 091 5 078 11 612 1865 + 8 1

39 5 006 5 985 9 305 173 2169

40 5 690 4 467 11740 1780169

41 6 345 3 994 11566 176 317 8

42 6 937 3 553 9 426 161 2168

43 4 096 4 134 9211 184 516 8

44 4 097 3 357 8 888 178 816 8

45 4 641 3 672 9717 172 616 8

46 7 776 6 985 12 565 187 217 7

48 7 992 7 574 15 645 183 9184

49 6 097 4 821 11 825 176 617 7

50 4 173 5 141 8 823 179 318 8

51 4 774 4 655 10 755 1816180

52 4 888 4 450 11588 190 1 + 8 7

55 4213 3 857 11676 174017 9

56 4 363 4 612 10 955 173 5166

57 5 216 4 229 9 408 175 116 8

58 6 199 3 650 8 772 1763 + 7 2

59 5 964 4 252 9 624 175 6 + 6 5

60 4 659 5 304 15 217 184 417 5

61 4 401 7 269 13 748 1786178


peptide

plane noo« tdeg] Op [deg] cy [deg] to [deg]"

62 4 294 7 840 11572 1816±68

63 6 486 4 431 9 288 1792 + 78

64 4 852 6 018 8 075 178 2167

65 4 229 6 131 14 147 1785 + 7 7

66 5 317 4 406 9 896 182 3 + 72

67 4 464 4 280 10 315 174 3 ±7 0

68 4 675 3 584 11836 1765 + 75

69 4 935 3 100 8 873 176 5 + 7 1

70 3 790 3 440 11934 1779 + 77

71 5 768 3 868 10 236 1789 + 63

"

Average value and standard deviation are given for the ensemble of 1500

snapshots

166 Appendix

6.1.1.2 Analysis of dihedral angles in ubiquitin

a)

10 20 30 40 50 60 70

Residue number i

140 -

100 -

60 * H

20

-20 -

-60

-100

-140

ion

'/$**»t10 20 30 40 50 60

Residue number i

70

Fig A 1 Average dihedral angles (a) \\rt and (b) (p, in ubiquitin as function of the

residue number i The standard deviation of the dihedral angle distribution resultingfrom the ensemble of 1500 snapshots is indicated in the figure (see bars) Glycineresidues are given as open circles The only non-glycine residues in a left-handed

conformation (cp, > 0°) are Ala 46, Asn 60, and Glu 64 (see also ref [96])

6.1 Appendix to Chapter 3 167

10 20 30 40 50 60 70

Residue number i

10 20 30 40 50

Residue number i

60 70

Fig. A.2 Correlation between dihedral angles. In Panel (a) the correlation coefficients

for the dihedral angles y, _,, <p, (filled circles) of the same peptide plane and for

y,, cp; (open circles) of the same residue i are compared with each other. In Panel (b)

the correlation coefficients for the dihedral angles \|/(_|,a>; (filled circles) and for

cpj, co, (open circles) are given. The coefficients are calculated from the ensemble of

1500 snapshots.

168 Appendix

6.1.2 Experimental results

Table A 2 N and C relaxation data of the backbone of human ubiquitin measured at

400 and 600 MHz proton resonance frequency and at a temperature of 300 K The data is

listed according to the peptide plane number For error bars see Section 3 5 1

pp

no"

15N

r,

400

MHz

[s]

15N

r,

600

MHz

[s]

15N

NOE

400

MHz

15N

NOE

600

MHz

15N

T2

600

MHz

[s]

13C,

Tx

400

MHz

[s]

13C.

T\

600

MHz

[s]

13C,

T2

400

MHz

[s]

13C.

T2

600

MHz

[s]

2 0 345 0 480 0 612 0 721 0 160 1057 1227 0 257 0 149

3 0 328 0 443 0 635 0 762 0171 1074 1248 0 228 0 131

4 0 322 0 438 0 639 0 769 0 168 1 111 1321 0 232 0 141

5 0 337 0 464 0 619 0 752 0 171 1088 1308 0217 0 131

6 0 317 0 445 0 633 0 752 0 169 1067 1263 0 213 0127

7 0 331 0455 0 595 0 746 0 174 0 975 1 188 0 225 0 139

8 0 351 0 469 0517 0 672 0 179 1043 1 170 0 247 0 143

9 0 349 0510 0 483 0 635 0195 1064 1 155 0 265 0 152

10 0 372 0 488 0 471 0 607 0 195 1 193 1306 0 294 0 179

11 0 399 0518 0 465 0 608 0 205 1 177 1321 0 286 0 179

12 0 357 0 493 0 556 0 689 0 186 1 120 1253 0 244 0 142

14 0 343 0 469 0 643 0 764 0 171 1072 1268 0 222 0 136

15 0 329 0 451 0613 0 762 0 166 1 113 1303 0 226 0 134

16 0 358 0 490 0 623 0 736 0 188 1095 1271 0 222 0130

17 0 326 0 452 0 633 0 769 0 163 1062 1236 0 219 0 132

18 0 334 0 478 0 653 0 772 0 168 1049 1 256 0 217 0 133

20 0 337 0 464 0 626 0 761 0 180 1061 1215 0 221 0 129

22 0318 0 447 0 637 0 763 0 178 1080 1306 0 203 0 127

23 0317 0 430 0 641 0 786 0 149 1002 1287 0 201 0 124

25 0 314 0 441 0 665 0 776 0 126 1050 1203 0 172 0 090

26 0 325 0 436 0 628 0 768 0 164 1098 1283 0 224 0 129

27 0315 0 429 0 645 0 793 0 164 1027 1 242 0 205 0 130


PP

noa

,5N

r,

400

MHz

[s]

15N

7",

600

MHz

[s]

15N

NOE

400

MHz

15N

NOE

600

MHz

15N

600

MHz

[s]

13C

r.

400

MHz

[s]

13C

T{

600

MHz

[s]

13C,

T2

400

MHz

[s]

13C.

T2

600

MHz

[s]

29 0 326 0 438 0 637 0 786 0 163 1039 1226 0 231 0132

30 0 325 0 437 0 621 0 768 0164 1053 1254 0 225 0 128

32 0 330 0 438 0 644 0 776 0164 1031 1211 0 223 0130

33 0 341 0 456 0611 0 764 0 173 1 112 1348 0 234 0135

34 0 338 0 459 0 617 0 752 0 175 1050 1260 0 216 0 134

35 0 329 0 454 0 665 0 756 0 162 1059 1250 0 231 0 144

36 0 375 0516 0 669 0 779 0178 1090 1328 0 252 0152

39 0 341 0 447 0 613 0 750 0179 1031 1 194 0216 0124

40 0 337 0 444 0 634 0 766 0 167 1 131 1312 0 230 0137

41 0 334 0 450 0 603 0 756 0 177 1 127 1328 0 237 0 139

42 0 333 0 458 0 628 0 750 0174 1059 1265 0 201 0124

43 0 332 0 465 0 639 0 758 0171 1034 1268 0 207 0130

44 0 328 0 446 0 625 0 764 0 176 1057 1272 0 210 0 120

45 0 331 0446 0 627 0 762 0 164 0 992 1222 0186 0 124

46 0 323 0 459 0 603 0 741 0 171 1036 1211 0 219 0 124

47 0 354 0 473 0618 0 738 0 178 1 112 1306 0 249 0137

48 0 349 0 466 0 645 0 750 0 171 1 165 1360 0 251 0155

49 0 372 0 484 0 567 0 691 0 189 1082 1294 0 236 0139

50 0 335 0 450 0 587 0 748 0170 1 117 1310 0 228 0133

51 0 335 0 483 0615 0 762 0171 1053 1261 0 225 0136

52 0 356 0 497 0 637 0 746 0179 1030 1 163 0118 0 063

54 0 346 0 471 0 655 0 768 0 167 1 110 1348 0 218 0 134

55 0 335 0 450 0 635 0 749 0166 1 127 1322 0 221 0131

56 0315 0 430 0 627 0 792 0 172 1019 1295 0 201 0 122

57 0 328 0 445 0 620 0 782 0 164 1043 1220 0 236 0122

58 0 321 0 436 0 673 0 787 0 161 1060 1233 0 214 0 128

170 Appendix

PP

no"

15N

r,

400

MHz

[s]

15N

r,

600

MHz

[s]

15N

NOE

400

MHz

15N

NOE

600

MHz

15N

T2

600

MHz

[s]

13C.

r,

400

MHz

[s]

13C.

Tl

600

MHz

[s]

13C,

T2

400

MHz

[s]

13C.

T2

600

MHz

[s]

59 0 335 0 446 0 602 0 762 0 179 1 171 1333 0 232 0 131

60 0 331 0 439 0 643 0 756 0 176 1 132 1343 0 226 0 137

61 0 336 0 441 0 661 0 764 0 171 1 176 1385 0 242 0 141

62 0 379 0 521 0 493 0 626 0 189 1 157 1317 0 237 0139

63 0 350 0 474 0 617 0 756 0167 1083 1256 0 230 0 137

64 0 324 0 440 0 643 0 765 0 169 1092 1268 0 222 0 129

65 0 331 0 459 0 638 0 790 0 173 1 175 1345 0 248 0 135

66 0 332 0 471 0 629 0 755 0179 1 119 1349 0 220 0 142

68 0331 0 458 0 661 0 756 0 179 1080 1289 0 221 0 131

70 0 331 0 442 0 641 0 748 0 154 1084 1222 0 226 0134

71 0 355 0 465 0 600 0 710 0 180 1020 1228 0219 0 143

74 0 540 0 623 -0 25 0 107 0 355 1 114 1043 0 474 0 281

75 0 706 0 819 -0 91 -0 35 0 540 1251 1 106 0 678 0 408

76 1 163 1242 -144 -0 92 0 804 2 021 1764 1099 0 662

aThe peptide planes are labeled and numbered by the amino acid residue that

contributes the nitrogen atom The plane of (a non proline) peptide bond i

contains the atoms N,, H, , C, ,,and O, _,


6.1.3 Analysis of relaxation data

6.1.3.1 Calculation of the order parameter of the 3D GAF model

2The calculation of the order parameters S„v according to Eq (2 57) is the most

expensive step when fitting the fluctuation amplitudes to the relaxation data The

MATLAB code [152] for the order parameter calculation which was compiled with an

ANSI C compiler and linked with the MATLAB C library is given below

Function Squad.m

function ret=Squad(fhetam,thetan,sigmal,sigma2,sigma3),%

% This function calculates the order parameter for a certain

% auto-correlated (thetam=thetan) or cross correlated (thetan.thetam)

% spin interaction in the peptide plane exhibiting 3D GAF motion

%

% The peptide plane frame is defined as follows

% principal axis gamma colhnear to C_alpha-C_alpha axis

% principal axis alpha in plane axis, perpendicular to gamma-axis, in direction C - > O

% principal axis beta axis perpendicular to the plane%=> right-handed coordinate system with alpha (x), beta (y), gamma (z)

%

% Input parameters

%thetam polar angles (theta,phi) of the interaction m (in rad)

%thetan polar angles (theta.phi) of the interaction n (in rad)

%sigmal fluctuation amplitude sigma_alpha (in rad)

%sigma2 fluctuation amplitude sigma_beta (in rad)

%sigma3 fluctuation amplitude sigma_gamma (in rad)

%

% Output%ret order parameter

% Wigner rotation matrix elements ace to R N Zare, Angular Momentum, Table 3 1, p 89

% The matnx elements are given for an angle pi/2

rmn=[[0 25 0 5 sqrt(3/8) 0 5 0 25],[ 0 5-0500505],

[sqrt(3/8) 0-050 sqrt(3/8)],[-0 5050-0505],

[0 25 -0 5 sqrt(3/8) -0 5 0 25]],

% List of normalized spherical harmonics (see R N Zare Angular Momentum, p 10)

%see function Ym.m below

Ymm=[Ym(-2 thetam(l),thetam(2)) Ym(-1 thetam(l),thetam(2)) Ym(0,thetam(l),thetam(2))

Ym( 1 ,fhetam( 1 ),thetam(2)) Ym(2,thetam( 1 ),fhetam(2))],

Ymn=[Ym(2,thetan( 1 ),thetan(2)) -Ym( 1 ,thetan(l),thetan(2)) Ym(0,thetan(l),thetan(2))

Ym( l,thetan(l),thetan(2)) Ym( 2,thetan(l),thetan(2))],

%%%%%%%%%%%%%%%%%%

%Calculation of the order parameter

%%%%%%%%%%%%%%%%%%

sigmalqh=0 5*sigmal*2,sigma2q=sigma2A2,

sigma3qh=0 5*sigma3A2

172 Appendix

pih=pi/2,

expsig2=zeros(size(rmn)),expsig3=expsig2,

for m=-2 2

mq=mA2,for md=-2 2

expsigl(m+3,md+3)=exp(-sigmalqh*(mq+mdA2))*exp(-j*pih*(m-md)),

expsig3(m+3,md+3)=exp(-sigma3qh*(mq+mdA2))*exp(j*alpha*(md-m)),end

end

% Loop to get the Order Parameter

Squad=0,for l=-2 2

a=exp(-sigma2q*lA2),for k=l 5

forkd=15

b=a*expsig 1 (k,kd)*rran(k,l+3)*rmn(kd,l+3),

for m=l 5

for md=l 5

Squad=Squad+b*expsig3(m,md)*rmn(m,k)*rmn(md,kd)*Ymm(m)*Ymn(md),end

end

end

end

end

ret=rea](Squad)*4*pi/5,

Function Ym.m:

function ret=Ym(m,theta,phi),%

% This function calculates the normalized spherical harmonics Ym

% of rank 2 for the angle theta and phi% ace to R N Zare, Angular Momentum, p 10

%

% Input%m index

%theta, phi set of angles (in rad)

%

% Output%ret Complex number'

ifm=0

ret=sqrt(5/(16*pi))*(3*cos(theta)A2-l),elseif (m==l)

ret=-sqrt(15/(8*pi))*sin(theta)*cos(theta)*exp(j*phi),elseif (m==-l)

ret=sqrt(15/(8*pi))*sin(theta)*cos(theta)*exp(-j*phi)elseif (m==2)

ret=sqrt(15/(32*pi))*sin(theta)A2*exp(2*j*phi),elseif (m==-2)

ret=sqrt(15/(32*pi))*sm(theta)A2*exp(-2*j*phi),else

fpnntf( 'Wrong input-index"')end


6.1.3.2 Fit results of 3D GAF motional model

Table A.3 Results of the least-squares fits of the 3D GAF motional model to the

experimental relaxation data of Table A.2. In addition to the three parameters aaa, Cy,

xmt for each peptide plane a global scaling parameter Xc was optimized to

Xc = 1.083 ± 0.004. Only those 45 peptide planes whose motion can be described by a

3D GAF model are considered. The error limits of the fitted parameters are determined

by a Monte Carlo procedure consisting of 60 fits with random errors of the experimental

standard deviations added to all relaxation parameters.

no °a|3 tdeg] Oy [deg] Aa/aa \n, [Ps] X?'

3 8.03 ± 1.16 15.53 ±0.97 0.71 ±0.23 11.2 5.6

4 11.45 ±0.85 11.12 ±1.48 -0.03 ±0.20 8.5 5.4

5 8.24 ±0.88 16.58 ±0.80 0.76 ±0.17 12.2 9.4

6 8.24 ±1.21 14.64 ±1.29 0.62 ± 0.26 13.4 3.4

12 7.47 ±1.23 19.97 ±0.84 1.07 ±0.25 30.1 6.2

14 6.97 ±1.28 17.71 ±0.85 1.02 ±0.27 7.7 8.2

15 9.25 ±1.10 14.56 ±1.13 0.48 ± 0.22 11.8 8.5

16 4.68 ±1.73 21.03 ±0.79 1.61 ±0.43 14.3 3.6

17 6.55 ±1.13 16.06 ± 0.92 0.98 ± 0.25 8.4 7.2

18 6.99 ±1.27 17.42 + 0.93 1.00 ±0.26 3.5 8.4

20 1.00 ±1.31 19.58 ± 0.48 2.58 ±0.40 12.0 2.4

22 6.88 ±1.40 16.03+0.96 0.92 ±0.33 9.4 18.1

26 7.61 ± 1.29 14.94 ±1.10 0.73 ±0.28 9.4 11.6

27 6.25 ±1.71 14.77 ±1.05 0.94 ± 0.46 2.0 8.4

29 5.73 ±1.59 16.18 ±0.90 1.13 ±0.41 4.2 11.4

30 7.58 ±1.28 15.24 ±1.14 0.76 ± 0.28 10.5 10.4

32 1.00 ±1.43 17.62 ±0.58 2.54 ± 0.46 7.0 12.8

33 7.87 ±1.18 16.92 ±0.96 0.83 ± 0.24 10.6 15.2

34 2.78 ±1.66 19.22 ±0.69 1.99 ±0.48 14.5 8.8

39 1.00 ±0.40 19.33 ±0.37 2.58 ±0.11 17.8 16.5

40 9.39+1.03 14.73 ±1.11 0.48 ± 0.20 8.8 16.3

174 Appendix

no °ap [deg] Oy [deg] Ao/Oa

\nt M2 b

X,

41 9 82 ± 0 86 14 8910 95 0 44 + 016 15 7 86

42 5 49 11 62 18061088 1 3010 40 126 10 8

43 6 37 ± 1 42 17 45 ± 0 98 1 1010 32 92 67

44 6 16+136 17 03 10 90 1 11 10 32 10 3 16 6

45 1 00 ± 1 53 18 1310 63 2 55 10 47 11 1 218

46 1 00 ± 1 45 18 6010 54 2 5610 44 20 0 34

48 12 7310 73 12 9311 10 002 + 0 14 119 20 3

49 5 85 ± 1 61 213410 82 141+038 27 2 15 8

50 8 62 ±1 14 15 8110 99 0 6510 23 18 8 13 3

51 4 6011 77 19051073 1531047 10 3 10 8

55 10 04 10 84 14 23 11 00 0 371017 117 16 8

56 6 2211 46 15 521108 100 + 035 20 26 2

57 3 26+183 17391081 1 77 + 0 54 70 13 5

58 3 76+178 16 151091 157+051 20 83

59 8 801101 15 9810 99 0 6410 20 13 5 19 1

60 9 34 ± 0 94 14 7910 95 0 491018 11 1 14 9

61 11 08 10 67 12 981101 0161014 58 165

63 6 55 + 131 18 4410 79 1 1310 28 118 20 5

64 7 7210 87 15 4110 86 0 75+019 86 59

65 9 371088 14 89 10 94 0 49 + 017 20 45

66 10 2710 86 15 2510 95 0 421016 10 6 44

68 7 6511 12 16 67 + 0 90 0 85 10 23 88 48

70 7 341 1 42 15 40+102 0 801031 14 5 313

71 5 4911 72 19 9310 81 1 4010 40 24 7 22 2

a

Amsotropy of peptide plane fluctuation as defined in Eq (3 2)

*The overall fitting error % of the global fit is the sum of the fitting error

2contributions % of each peptide plane i


25

20

a15

CCL

£10o

nwMiiiinMTnnnMnMninmTiMW

A •

V •

O A A &

a a

O a

an a O

#A

*

^O

O AA°

AAA

° °°

oo

10 20 60 7030 40 50


Fig. A.3 Results of an alternative fitting procedure. The 45 peptide planes are divided

into three categories: "a helix", "(3 sheets", and "other". Three global scaling factors

are extracted for the three categories: )£"'= 1.070 + 0.010, X*e,,Jt=1.076 ± 0.008,

and X°c er=1.096 ±0.010. The fitted fluctuation amplitudes are labeled as follows:

helix, x

helix .~.. . . sheet, i \

sheet,,,„ ,

a„R (open squares), ay (rilled squares), aaB (open circles), aY (filled

other . . , .other

Ja(5

circles), aan (open triangles), ay (filled triangles).

176 Appendix

6.2 Appendix to Chapter 4

6.2.1 15N and 13C CSA tensors of several peptides measuredby solid state NMR

*

Table A 4 Results for the principal values, the amsotropy Acs, asymmetry r\, and

orientation of the N CSA tensors of several peptides studied by different authors using

solid state NMR

model peptide[ppm] [ppm] [ppm]

A5

[ppm]

11ft a

[deg]Ref

GlyGlyHCl -55 4 -44 8 100 2 150 3 011 21 [125]

GlyGlyHCl 51 8 -49 2 1010 1515 0 03 20 [126]

AcGlyAlaNH2 -75 1 -34 6 109 7 164 6 0 37 21 [1261

AcGlyTyrNH2 -60 8 -35 7 96 5 144 8 0 26 19 [126]

AcGlyGlyNH2 -64 4 -410 105 4 158 1 0 22 21 [126]

Boc(Gly)3OBz (m) -58 3 513 109 6 164 4 0 06 22 [119]

Boc(Gly)3OBz (t) -77 0 -30 0 107 0 160 5 0 44 24 [119]

AlaAla -54 4 -415 95 9 143 9 0 13 116 [124]

[15N] alanyl3

gramicidin A

-65 -39 104 156 0 25 12-14 [127]

[15N] leucyl4-

gramicidin A

-65 -36 101 151 0 29 12-14 [127]

AlaLeu 55 4 -42 3 97 7 146 6 0 13 17 [128]

QNH is the angle between the z principal axis of the CSA and the NH bond It

is assumed that the z principal axis lies in the peptide plane and the y principalaxis is orthogonal to the plane


Table A 5 Results for the pnncipal values, the anisotropy Aa, asymmetry r\, and

onentation of the 13C CSA tensors of several peptides studied by different authors using

solid state NMR

model peptide°xx

[ppm]

°yy

[ppm] [ppm]

AG

[ppm]

Tl QCOy

"

[deg]

Ref

GlyGlyHCl -74 4 -7 4 818 122 7 0 82 13 [120]

GlyGlyHCl -73 7 -7 2 80 9 1214 0 82 12 [121]

AcGlyGlyNH2 -70 2 -114 816 122 4 0 72 0 [121]

AcGlyTyrNH2* -72 5 -2 3 74 8 1122 0 94 6* [121]

AcGlyAlaNH2 -69 7 -12 6 82 3 123 5 0 69 2 [121]

GlyAla -71 -13 84 126 0 0 69 10 [122]

ValGlyAla -75 -5 80 120 0 0 88 8 [122]

AlaAla -74 0 -0 9 74 9 1124 0 98 4 [124]

["Cjl-alanyb;gramicidin A

-73 4 -7 3 80 7 121 1 0 82 0 [123]

cyclofpCJ-Val-Pro-Gly-Val-Gly}

-60 6 -9 7 70 3 105 5 0 73 4 [123]

"The angle 6C0 is the angle between the CO bond and the y pnncipal axis of

the CSA tensor The z principal axis is assumed to he orthogonal to the peptide

plane*Please mind that the onentations of the x and z pnncipal axes are permutedwhen compared with the results from the other solid state NMR studies

178 Appendix

6.2.2 Orientation of the calculated CSA tensors

Table A 6 Orientation of the principal axes (in polar coordinates) of the calculated

"global frame averaged" 15N CSA tensors (see Table 4 1) of peptide planes 10, 30, and

43 of ubiquitin with respect to the peptide plane frame ea, ea, e of the corresponding

equilibnum peptide plane For comparison, the polar coordinates of some dipolar

directors in this peptide plane frame are given

interaction(9,tp)

peptide plane 10

(e,q»

peptide plane 30

0,9)

peptide plane 43

15Naxx (40 8, 149 1) (310,155 3) (32 2, 165 9)

15N oyy (107 8,80 9) (100 9,83 9) (98 0, 88 8)

15N0ZZ (125 3, 184 1) (118 7, 179 9) (1210, 183 6)

NHN (103 6, 180 9) (102 1, 177 1) (1017,179 6)

C'N (42 0, 180 4) (38 8, 179 8) (42 8,180 2)

NCa (14 6,13) (115,359 0) (17 3,0 6)

Table A 7 Orientation of the principal axes (in polar coordinates) of the calculated

"global frame averaged" 13C CSA tensors (see Table 4 2) of peptide planes 10, 30, and

43 of ubiquitin with respect to the peptide plane frame ea, e«, ey of the corresponding

equilibnum peptide plane For comparison, the polar coordinates of some dipolar

directors in this peptide plane frame are given

interaction(e,q>)

peptide plane 10

(6,9)

peptide plane 30

(6,9)

peptide plane 43

13C'<TXX (93 0, 83 9) (94 8, 85 1) (89 5, 88 0)

13C ayy (90 7, 353 8) (76 9, 356 3) (79 0, 357 0)

13C azz (3 1,70 7) (14 0,155 4) (110,180 6)

CO (817,358 6) (79 6, 0 4) (78 5, 359 0)

C'N (42 0, 180 4) (38 8, 179 8) (42 8, 180 2)

C'C (160 4, 179 6) (159 1,180 2) (160 8, 179 8)

References 179

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191

Acknowledgements

During three fascinating years at the ETH Zurich I had the opportunity

to collaborate with excellent scientists and to meet numerous interesting

people who supported me with their advice, knowledge, and friendship.

They all created a very special and exciting atmosphere which will be

missed. I would like to express my vivid thanks to

- Prof. Dr. Richard R. Ernst for providing excellent research conditions,

for teaching me how to think thoroughly about scientific problems, for his

very inspiring guidance in many discussions and his valuable advice in

various situations. His continuous support of my work has been very

motivating for me.

- Prof. Dr. Rafael Briischweiler for introducing me to the exciting field

of NMR, for sharing his deep understanding of science with me, for his

most stimulating advice in numerous scientific discussions and other

questions of life, and for the very pleasant atmosphere during our

collaboration over the last years.

- Dr. Tobias Bremi for introducing me to many different techniques, for

the exciting collaboration in the 3D GAF project, for being a great office-

mate in F34, and for being a friend.

- Dr. Christoph Scheurer for providing the DFT calculations on

ubiquitin, for first-class computer management, and for his friendship

during the last months of my work.

- Dr. Nikolai Skrynnikov for the fruitful collaboration in different

projects and for always helping when help was needed.

- Dr. Bernhard Brutscher for introducing me to advanced experimental

NMR techniques.

- Dr. Thomas Schulte-Herbriiggen for his support in the maintenance of

the AMX-600 and for valuable discussions about many questions of life.

192

- all other members of the group of Prof. Ernst for creating a very

stimulating and friendly atmosphere: Ute Drexler, Prof. Dr. Edward B.

Fel'dman, Dr. Zhehong Gan, Dr. Sabine Hediger, Dr. Hongbiao Le, Dr. Tilo

Levante, Dr. Zoltan Madi, PD Dr. Rolf Meyer, Dr. Pierre Robyr, Dr. Martin

Schick, Patrick Sommer, Dr. Suzana Straus, Dr. Marco Tomaseili, Dr.

Marcel Utz, and Dr. Rico Wiedenbruch.

- Dr. Serge Boentges, Dr. Gerald Hinze, and Dr. Bettina Wolff for their

technical support and helpful discussions in the "viscosity project".

- Prof. Dr. Beat H. Meier and Dr. Matthias Ernst for their kind support

during the last months of my work.

- our secretary Irene Miiller and many other people in the LPC for

providing an excellent infrastructure.

- Dr. Alexander Ernst, Thomas Gilbert, Dr. Felix Graf, and Dr. Michael

Wilier for many interesting discussions and their friendship during the last

years.

- my parents for their constant encouragement and support which made

my stay in Zurich possible.

193

Curriculum Vitae

Personal data

Name Stephan Frank Lienin

Date of birth March 23,1970

Place of birth Rheinfelden (Baden), Germany

Citizenship German

Parents Rosemarie and Heinz Lienin

Education

1976-1980 Primary school in Lorrach (Germany).1980-1989 Hans-Thoma-Gymnasium, Lorrach.

1989 Abitur

Studies

1990-1992

1992-1995

Aug 1995

Sept 1998

Undergraduate studies in chemistry at the university of

Mainz (Germany). Best leaving exam (Vordiplom) of the

year.

Undergraduate studies in chemistry at the EidgenossischeTechnische Hochschule (ETH) in Zurich. Degree in

chemistry in 1995. Diploma thesis in nuclear magneticresonance in the group of Prof. Dr. Richard R. Ernst.

Graduate studies in physical chemistry in the research

group of Prof. Dr. Richard R. Ernst at ETH Zurich.

Professional and Teaching Experience

1995-1998 Teaching assistant for various lectures at the Laboratoryof Physical Chemistry.

1996-1998 Responsible for maintenance of a Bruker AMX-600

high-resolution NMR spectrometer.

Scholarships

1992-1995 Studienstiftung des Deutschen Volkes

1992-1995 Deutscher Akademischer Austauschdienst (DAAD)

eth-22701-02 research collection

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