FHtR卜s|NN1上4k SPE'TROSCO)Y LSI in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 Technical aspects of nmr spectroscopy with biological macromolecules and studies of hydration in solution erhard Wider Institut fur Molekularbiologie und Biophysik, Eidgenossische Technische Hochschule, Honggerberg, CH-8093 Zurich, Switzerland Received 1 December 1997 Contents I. Introduction 2. Basic principles 2. 1. Theoretical aspects 2.1. 1. Magnetization, precession and Bloch equations 2.1.2. Operators, coherence, and product operator formalism 2. 1.3. Descriptive representations of experimental schemes 2.1. 4. Relax 2. 1.5. Through-bond correlations 203 2. 1.6. Through-space correlations 204 2.2. Radiofrequency pulses 2.2.1. Rectangular pulse 2.2.2. Amplitude-modulated pulses 2.2.3. Amplitude- and phase-modulated puls 2.3. Magnetic field gradients 210 2.4. Data acquisition 2. 1. Digitizing the signal 2.4.2. Handling the water resonance 2.4.3. Decoupling during acquisition 2.4.4. Oversampling and digital filter 214 5. Multidimensional NMR 215 2.6. Data processing 2.6. 1. Transforming the time domain data into a spectrum 217 2.6.2. Referencing the chemical shift 3. NMR instrumentation 3. 1. Layout of a high-resolution NMR spectrometer 3. 2. Spectrometer configuration for biomolecular NMR 3.3 116333455;fax:+41163315l 0022-2860/98/$19.00@ 1998 Elsevier Science B V. All rights reserved PS0079-6565(98)00014-4
G wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-2 3.3. 1. The transmitting path 3.3. 2. The receiving path 3.3.3. The lock system 3.4. The magnet 3.5. The prol 3.5. 1. The radiofrequency coil 3.5. 2. The magnetic field gradient coil 228 3.5.3. The variable temperature operation 3.6. Stability of the sys stel 4. Basic segments of pulse sequences 230 4. 1. Evolution segments 4.2. Transfer segments 235 4.2. 1. Homonuclear through-bond transfer 4.2.2. Homonuclear through-space transfer 4.2.3. Heteronuclear transfer 43 Decoupling sequences 4.4. Pulsed magnetic field gradients 4.5. Combinations of basic segments 4. 5. 1. The HsQC and the hmQc scheme 4.5.2. Concatenating basic segment: 4. 6. Artefact reduction 5.2. NMR and hydration 5.3. Basic experiments 5.3. 1. NOEs between water and protein protons 53. 2. HYDRA 5.3.,3. Measurement of exchange rates using diffusion filter experiments 5.3. 4. Relaxation dispersion experiments 262 5.4. Artefacts in hydration studies at high magnetic fields 5.4.1. Radiation damping and demagnetizing field effect 5.4.2. Minimizing artefacts in hydration measurements 266 5.4.3. Consequences of radiation damping and demagnetizing field effects Acknowledgements Appendix A: The Bloch equations Appendix B: The product operator formalism 270 References Keywords: Protein NMR: Instrumentation; Hydration; Multidimensional NMR 1. Introduction macromolecules has been a growing field in research and applications [2]. The capability to observe signals The first nuclear magnetic resonance (NMR) from individual atoms in complex biological macro- spectrum of a protein was published some forty molecules in solution makes possible the measure- years ago [l] and ever since, NMR of biological ments of parameters that can be analysed in terms of
in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 molecular structure, conformation and dynamics dynamics of biological macromolecules by NMR are Complete assignments of signals in an NMR spectrum established techniques. The rapid expansion of NMr to individual atoms in the molecule are a prerequisite techniques for applications to biological macro- for such studies; a problem that cannot generally be molecules increases the number of interested users solved on the basis of one-dimensional (ID) NMr with little technical background in NMR spectro- spectra. Only the application of two-dimensional scopy. These newcomers find it increasingly difficult (2D)NMR spectroscopy [3, 4], which spreads signals to follow and make use of the myriad of NMR experi into two frequency dimensions, allowed the develop ments available today. The applications and theoreti ment of a general strategy for the assignment of pro- cal foundation of biomolecular NMR are described in on signals in protein spectra using two types of 2D many excellent books, e. g [15-27 however, often petra[5-7 In['H,HJ-COSY spectra [3, 4 protons only a few experimental schemes are discussed and are correlated which are separated by up to three the common features of different pulse sequences are chemical bonds. In [H, HI-NOESY spectra [8, 91 not always made transparent. In addition, important correlations between protons which are closer than technical details of experimental implementations are 0.5 nm through space are detected. The combination either not discussed or may be missed in the over of these two techniques allows the assignment of most whelming amount of information. This review is proton NMR signals to individual protons in small intended to address the need for an introduction to proteins [6, 10,11]. In a further step all distances general technical and methodological aspects of obtainable from NOESY spectra provide the data for modern NMR experiments with biological macro- the calculation of protein structures [12, 13]. These molecules to these newcomers. The basis for this elatively simple techniques allow the determination presentation is not to provide complete experimental of structures of proteins with a molecular weight up to schemes, which change rather rapidly, but instead to 10 kDa whereas for larger proteins extensive signal present the underlying basic technical methods and overlap and increasing resonance linewidths prevent the basic segments from which individual experi complete assignments of all signals, This barrier can ments are constructed and which change much more be overcome with three-dimensional (3D)NMR slowly. The understanding of the basic segments techniques [14] and uniformly"C-and N-labelled should provide the basis for clarifying the functioning proteins. With these methods, systems with molecular of existing and newly developed experiments and weights up to 30 kDa can be studied. However, not assist the reader in making their own adjustments to only overlapping signals limit the size of the macre experiments or even in developing new methods. molecules that can be investigated; in addition faster This review was written with a reader in mind who relaxation of the signals with increasing molecular is interested in technical and methodological aspects weight leads to a substantial sensitivity loss in experi- of NMR with macromolecules in solution, who has ments. The molecular weight limit can be increased to had first contact with spectra of proteins in one and about 50 kDa using deuteration of the protein to two dimensions and knows the principles of their reduce relaxation. Simultaneously with the methodo- analysis. In addition, knowledge of the product logical developments, technical advances revolutio- operator formalism [28] is an advantage since the di nized the design of NMR spectrometers making it cussion of the basic segments requires the application possible to implement the complex experimental of this formalism. This text should help such readers schemes needed for multidimensional NMR experi- to quickly become familiar with the technicalities of ments. The increased complexity of the instrumen- multidimensional NMR experiments ation has been more and more hic om the user The main text starts with Section 2 where some by complex software control theoretical aspects are discussed briefly, followed by selection of all modes of operation from a software an introduction of technical principles starting with interface radiofrequency pulses and ending with multidimen- In parallel to the methodological and technical sional NMR and data processing. Not only in this velopments nmR has become an accepted tool in section but throughout the text, mathematics is kept ructural biology and investigations of structure and to the minimum necessary for the presentation of the
G wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 technical aspects of NMR spectroscopy. Section 3 Hertz(Hz) the symbols w or f are used for the former introduces those parts of a modern NMR spectrometer and v for the latter which critically influence the performance of NMR The basis of all NMR experiments is the nuclear experiments. Section 4 concentrates on the basic ich can be interpreted as a magnetic momer experimental segments from which most of the vast A spin i nucleus in this view forms a small number of experiments available today are con- This dipole orients either parallel(a state)or structed. Section 5 discusses hydration studies with allel(B state) to a magnetic field leading to a small NMR and serves two purposes. First, it gives exam- energy difference Ae between the two states ples of experiments using the segments introduced in Section 4 and the les discussed in Section 2. AE=:Bo=hyBo Second, it introduces the technical aspects of a ver interesting application of NMR which allows detaile where Bo is a large externally applied homogeneous studies of individual water molecules in the hydration magnetic field, h is Planck's constant (h h/(2T), shell of a protein. and y the gyromagnetic ratio which is a property of the nucleus and can have a positive or a negative value. Table I lists the gyromagnetic ratio and some 2. Basic principles other properties of nuclei important in NMR of bio- logical macromolecules. from the two states of a L. Th favourable and thus possesses a higher population This section presents some basic theoretical aspects than the p state. Transitions between adjacent energy of nMR which are relevant to a technical discussio levels can be induced by small additional magnetic of the principles and experimental procedures used fields perpendicular to B, which oscillate with a fre- when studying biological macromolecules in solution quency vo fulfilling the resonance condition v,= AE/ by NMR. A rigorous discussion of the theoretical h. The frequency vo typically lies in the radio- foundation of NMR can be found in many textbooks, frequency range and is often referred to as the Larmor e.g. [16, 19, 24, 26, 29]. Only a very limited theoretical frequency. In the equilibrium state the boltzmann foundation is necessary for the technically oriented distribution favours the lower energy states. Thus, discussion of NMR methods in the following sections, the sum of all contributing nuclear magnetic moments The concept of energy levels, the Bloch equations and of the individual nuclei leads to a resulting macro- the product operator formalism are sufficient in most scopic magnetization M along the homogeneous cases. NMR is intimately related with frequencies and external field Bo. In the framework of classical to obtain a clear distinction between angular frequen- physics the behaviour of this magnetization under cies with units rad s and technical frequencies in the action of time-dependent magnetic fields can be Properties of selected nuclei Nucleus y710 Ts Natural abundance/% Relative sensitivit H 99985 H 4.106625 0015 9.65×10 H 6.72828 1.59x10 193378 0.037 ×10-2 10.8394 6.64×10 y, gyromagnetic ratio: 2=1/(2 For an equal number of nuclei to protons
G Wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 described by the Bloch equations [30]. Because the In the rotating frame where b becomes static th spin is a quantum mechanical phenomenon this main magnetic field B. vanishes for nuclei with reso- description has a very limited scope, but it proves nance frequency vo. Hence, Eq(2)in this rotating ery useful for the description of single resonance frame contains only a transversecomponent hes under the action of radiofrequency(rf) pulses w=-y(B1, 0,0)with BI chosen along and thus for the characterization of the effect of RF Consequently M precesses by an angle p around the magnetic field B applied as a pulse for the short 2.1. Magnetization, precession and bloch equations In a classical description the macroscopic magneti- zation M created by the spins is described by a vector where B is called the flip angle of the pulse. The fip M parallel to the magnetic field vector Bo M is forced angle is often indicated in degrees, for example during move away from the direction of Bo by an a 90 pulse M can precess from the z axis to the x axis additional linearly polarized magnetic field BI per- The angle B depends on y and is negative for positive pendicular to Bo. BI must fulfil the resonance con- y values such as for protons(Table 1). For positive y ition and oscillate with the resonance frequency vo. values a magnetic field B pointing along the positive The magnetization vector M precesses about the x axis turns M towards the negative y axis. a B, field resulting magnetic field B=B.+BI with an angular along the +y axis turns M towards the +x axis velocity vector w pointing in the opposite direction [16, 31]. When applying an rF pulse with a frequency ind the components as described in Eq (2) differing from vo the action of the rf pulse becomes more complex as described in Appendix A, a situation r(2B, cos(2xv,(+6), 2B, sin(2 vol +o), B,) often referred to as non-ideal behaviour of the RF resonance The oscillating magnetic field 2B, coS(wRFt) used where w=(wr, Wy, Wo), B. is chosen along the z for excitation is linearly polarized in the laboratory ind o describes the angle between the x axis and B frame. The transformation into the rotating frame The magnetic field B is often applied only for short can best be followed when this is thought of as a time periods as RF pulses. The discussion of the superposition of two counter-rotating, circular polar motion of the magnetization vector M in space due zed fields with an amplitude B. When transforming to RF pulses is usually based on a rotating frame of into the rotating frame one component matches the reference which has the same z axis along the static Larmor frequency whereas the other oscillates at magnetic field B as the laboratory frame but rotates twice the Larmor frequency and does not fulfil the around the z axis with a frequency which is often resonance condition. Bloch and Siegert [32] calcu- hosen equal to the resonance frequency vo. In this lated the effect of the non-resonant field and found rotating frame of reference, the relevant component that it slightly shifts the frequency of the observed of the applied oscillating field B, appears static resonance lines away from the disturbing field by making the discussion and visualization much easier the small amount vB=(yB,)/Aw where y=Y/(2T) To fulfil the physical requirement that the magnetiza. and Av stands for twice the resonance frequency tion vector M returns to its equilibrium position in a The Bloch-Siegert shift is small and amounts, for finite period of time after a disturbance, a longitudinal example, to O5 Hz for a frequency of 600 MHz dur elaxation time T,(spin-lattice relaxation)is intro. ing an RF pulse with duration T of 10 us and a fip duced. The loss of coherent precession is described angle of 3(Eq (3)or 90. The shift disappears as by a transverse relaxation time T2(spin-spin relaxa- soon as b is switched off. An effect similar to the tion). The motion of the magnetization vector M Bloch-Siegert shift occurs whenever an RF field is under the action of the magnetic field B and hence applied with a frequency Av off-resonance for the under w can be described by the Bloch equations nuclear spins. Although first described by Ramsey [30]. These equations are presented in Appendix a [33] it is still very often referred to as the bloch- for further reference Siegert effect. To better distinguish it from the effect
G. wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 lue to the counter- rotating field the term"non-reso- than their mutual scalar coupling J; this situation nant effect was introduced [34]. Since this additional often referred to as weak spin-spin coupling whereas field may be rather strong and close to the resonance strong spin-spin coupling specifies the case where frequency, this effect can become quite large and is close to or even smaller than J. With these assump- needs to be compensated [35](Section 2.2.1) tions simple rules can be calculated which describe the evolution of spin operators under the action of 2. 1. 2. Operators, coherence, and product operator chemical shift, J coupling and RF pulses. The form formalism alism combines the exact quantum mechanical treat- The description of NMr experiments by the Bloch ment with an illustrative classical interpretation and is equations and by magnetization vectors in the rotating the basis for the development of many NMR experi- frame has significant limitations particularly for the ments. However, some parts of experimental schemes description of multipulse experiments. On the other for example ToCSY sequences(Section 4.2.1), can hand, a full quantum mechanical treatment which only be described with a full quantum mechanical describes the state of the system by calculation of treatment. Calculations with the formalism are not the time evolution of the density operator under the fficult, but many terms may have to be treated an action of the appropriate Hamiltonian can be cumber aplementations of the formalism within computer some. In a quantum mechanical description an R programs are very helpful in such situations [36,37 pulse applied to the equilibrium state creates a coher- Although most of the experiments applied in bio- ent superposition of eigenstates which differ in their molecular NMR correlate three or more spins, the magnetic quantum number by one, often simply majority of interactions can still be understood based referred to as a coherence. In more complex experi- on an analysis of two spins. For two spins I and S the ments the magnetic quantum numbers between states operator basis for the formalism contains 16 elements differ by a value g different from one, leadi Two sets of basis operator, cartesian and shift opera a q quantum coherence with at least q spins involved. tors, have proven very useful for the description of However, only in-phase single quantum coherences experimental schemes and are used in parallel. The (Table 2)are observable and correspond to the classi- two basis sets and the nomenclature used to character- al magnetization detected during the acquisition of ize individual states are summarized in Table 2 [28] an NMR experiment. Multiple quantum coherences The operators /, and S are identical in the two basis cannot be observed directly, but they influence the sets and a simple relationship exists between the two spin state and this information can be transferred to other cartesian and shift operators bservable magnetization In a step towards a full quantum mechanical treat- lx=(7++/)/2 I=l+il (4) ment, the product operator formalism for spin , nuclei as introduced [28]. In this approach it is assumed I-ilv at there is no relaxation and that the difference in the Three operators, which represent the action of the resonance frequency Ar of two nuclei is much larger Hamiltonians for chemical shift, scalar coupling and Table 2 Product operator basis for a two-spin system Cartesian operator basis Nomenclature Shift operator basis Longitudinal magnetization 、l,SS In-phase transverse magnetization 1,,S+,S 2/S:2/S Anti-phase I spin magnetization 2/+S,2S 25/252 Anti-phase S spin magnetization 1, 2S I. 2/Sx2Sx21/S,2/.S 2/*5+,2/S, 2/S,2/"S 21.S ongitudinal two-spin order
G. Wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)793-275 RF pulse, act on these basis operators and may trans- cartesian product operator may be associated with form them into other operators within the basis set. several coherence orders, for example 21_S, describes this way the spin states created during an NMr a linear combination of both DQC and zero quantum experiment can be described and the observable coherence(ZQC). The cartesian x and y components magnetization calculated. The operator formalism of a multiple quantum coherence are given by linear can be summarized by simple rules [28] which are combinations of the shift or cartesian operators. For listed for both basis systems in Appendix B for further example, the ZQCs and DQCs of a two-spin system reference. Operators transform individually under can be described as follows these rules even in products of operators except for (ZQC) =(2/.+21,S,)=(+s"+/s+)(5) and transformed accordingly; however, they can be (DQC)=(21Sx-21S)=(+S++I S) treated consecutively when different couplings the same nucleus exist The cartesian operators transform more easily ZQC)=(2lS-21S2)=i+S-S+) under pulses and their single operators have a direct classical interpretation as magnetization vectors. The DQC)=(2IS, +21 S)=-i(I*S+-I-S) shift basis provides a useful alternative for the On the basis of the operator formalism the selection of description of the evolution due to chemical shift particular states using phase cycling of Rf pulses in and/or the influence of magnetic field gradients an NMR experiment can be rationalized Coherences (Section 2.3)and is better suited for the description present in an experiment can be classified into their of coherence orders and coherence pathways(Fig. 1) different orders or coherence levels which can be Their single operators describe a transition from the a represented in a pictorial way(Fig. 1)to visualize to the B state 1, or from the B to the a state, I the coherence transfer pathways in an experimental Hence, shift product operators are uniquely associated scheme [38,39). The order of coherence p corresponds with one coherence order, for example /*S* describes to the change q in the magnetic quantum number only double quantum coherence(DQC), whereas the between the two connected states [28]. Hence for n coupled spins the maximal coherence level that can be reached is n Free precession conserves the coherence order whereas pulses may cause coherences to be transferred from one order to another(Fig. 1). The pulse is proportional to its order a q quantum coher ence will experience a phase shift Ao of an rf pulse Fig. 1. Coherence level diagram. The coherence levels p are for as q4d. If the pulse results in a change in the coher mally represented by products of the shift operators/and/ which ence order of Ap, the corresponding phase shift are conserved during periods of free evolution. The application of radiofrequency pulses may transfer coherences from one order xperienced by the affected coherence will be (level)to another. The positions of three pulses are indicated in △φAp. Proper phase cycling of consecutive RF the figure by vertical arrows labelled rf rf, and rf. Thick lines pulses allows for selection of a specific succession represent the coherence pathway starting at the equilibrium state of coherence levels that define a coherence pathway (p= 0) passing through single quantum coherences after the first pulse(pl= 1), reaching double quanturn coherences after the second (Fig. I). The concept of coherence transfer pathways pulse (pl= 2)and ending as observable magnetization (p clarifies the role of phase cycling in NMR experi after the third pulse Only single quantum coherences (p!= 1)can ments and describes their action with a simple set of be observed. The spectrometer detects only one of these two rules [21, 24. 38, 40 A particular RF pulse can be coherence levels which is usually assumed to be p=-[38] and designed to select a certain difference in coherence hence all other coherence orders after the third pulse cannot be order Ap t nN(n=0. 1, 2,..)with a phase cycle tected and therefore are not drawn thin lines indicate alternative pathways which have to be suppressed if only the pathway indicated comprising N phase steps Ap of the same size equal to by thick lines should contribute to the signal measured at the end of 360/N. The N signals obtained must be summ together with the proper receiver phase -kApAp to
G. Wider/Progress in Nuclear Magnetic Resonance Spectroscopy 32(1998)193-275 compensate for the phase change experienced by the The experiment in Fig. 2 starts at time point a in coherence,k takes on values (0, 1, 2, 3, .. An thermal equilibrium(Fig 3)where the populations on example for the design of a phase cycle using the upper Pu and the lower energy level P, across system is given in Section 4.2.1 with the discussion transition fulfil the Boltzmann distribution(Eq. (6) of the double quantum filter. a detailed discussion of Because AE in Eq (1) is typically much smaller than phase cycling can be found in most textbooks on kT we can approximate this exponential distribution NMR,e.g.[6,21,24,261 by the first term in a Taylor expansion 2. 1.3. Descriptive representations of experimental △E △E Pu/P=exp( h kT An NMr experiment can be graphically described to a limited extent based on a classical physical model where k is Boltzmann's constant and T the absolute using populations and magnetization vectors in the temperature. With Eq (1)the energies E1,E,E3,and rotating frame or based on quantum mechanical E4 of the four different energy levels in the system can principles using the product operator formalism. be calculated Both descriptions find widespread applications for =h(-n-Bs+J/2)/2 the discussion and development of NMR experiments The different representations are discussed on the E2=h(-v1+us -J/2)/2 basis of the scheme shown in Fig. 2. The application of this experiment to a system of two scalar coupled E3=h(n-s-J2)/2 spins I and s is described with four different represen tations in Fig 3 for each of the five time points a to e. E4=h(1+s +J/2)/2 Fig.3 represents energy level diagrams(E), the where vi and vs stand for the resonance frequencies of observable spectra( S), magnetization vector diagrams the I and S nuclei, respectively. The resonance fre V)and the notation in the product operator formalism quencies of nuclei with positive gyromagnetic ratio (O)showing the cartesian and the shift operator y such as protons and carbons are negative(Eq(2), basis. In Fig. 3 spins I and s are assumed to be proton and, hence, E, becomes the highest and E4 the lowest and carbon nuclei, respectively, with the size of energy(Fig 3). For nuclei with a positive y value the representative vectors proportional to the correspond- a state(spin 2) has lower energy than the B state(spin ing populations, but qualitatively the figure applies ). The polarizations Mf and Mi are proportional to all nuclei with spin and positive gyromagnetic to the energy differences(E+- E2) and(E3-ED ratIo respectively, and they determine the intensity of the orresponding transitions 2+ 4(24)and 1+3(13). The consistent use of signs and transformation proper ties as presented in Fig. 3 may seem not to be of great importance and, indeed, has very often no direct experimental consequences. But there are situations where inconsistencies occur and the interpretation of Fig. 2. Sketch of the experimental scheme used for the discussion of data becomes confusing or wrong [31, 411 different representations shown in Fig. 3. The black narrow bars from the consistent illustration of different indicate 90 RF pulses. five time-points on the time axis I are representations for the description of an NMR experi denoted by the letters a, b. c, d and e. The RF pulses applied on- ment, Fig. 3 demonstrates that the scheme shown in sonance to the two species of nuclei I and S are indicated on the ig. 2 transfers polarization from proton to carbon nes marked with the corresponding letters, a particular pulse acts uclear species. The scalar coupling between the two pins. At time point d the proton polarization Mi is separated by the time inverted. As a consequence the populations across the rod(2). The phases of the RF pulses are indicated by x or y at arbon transitions 12 and 34 acquire a larger differ he top of the pulses, where x or y stand for the application of the B nce than at thermal equilibrium at time point a and held in the rotating frame along the positive r or y axis, respectively. hence, the experimental scheme allows measurement
G Wider/Progress in Nuclear MaGnetic Resonance Spectroscopy 32(1998)193-275 巴 5巴a E4点 E3 gmv变 三8 3J a
ider/Progress in Nuclear Resonance Spectroscopy 32(1998)193-275 of carbon spectra with higher sensitivity, Such polar- Fig. 4 shows a plot of J(a)as a function of the fre ization transfer experiments are extremely important quency w for the three correlation times T of 5, 10 and in heteronuclear NMR experiments and are discussed 20 ns. These correlation times represent the motion of further in Section 4.2.3 small, medium and large globular proteins in In addition, Fig ustrates that low fre 2.1. 4. Relaxation motions are especially effective in NMR rel processes for proteins. Using the concept of spectral Relaxation processes re-establish an equilibrium density functions the different behaviour of longitudi distribution of spin properties after a perturbation After a disturbance, the non-equilibrium state decays nal relaxation and relaxation of transverse magnetize the simplest case exponentially characterized by the tion can be rationalized. When considering onl spin-lattice relaxation time T. Re-establishing relaxation due to fluctuating dipolar interactions thermal equilibrium requires changes in the popu- caused by stochastic motion, the relaxation rate TI lation distribution of the spin states and lowers the is proportional to J(wo)since only stochastic magnetic energy of the spin system. Thus, it involves energy fields in the transverse plane at the resonance fre transfer from the spin system to its surroundings quency w, are able to interact with the transverse which are usually referred to as the lattice. Micro- magnetization components, bringing them back to scopically, relaxation is caused by fluctuating values J( o decrease for the three increasing values the z axis. For frequencies larger than 25 MHz the magnetic fields. Dynamical processes such as atomic of Te represented in Fig 4. The longitudinal relaxation actions and facilitate spin-lattice relaxation. The times, therefore, increase for increasing molecular extent of the overlap between the frequency spectrum very small Te the values J (wo)values get smaller aga of the motional process and the relevant resonance leading to an increase of T, compared to the value for lap is described by th density function J(w). Since J()is the Fourier trans- obtained when woTe=l, e.g. at 600 MHz for a Te of form of the time correlation function describing the 0.26 ns. Transverse relaxation shows a different notion, its functional form depends on the mechanism dependence on the molecular weight of the molecule of motion. An exponential correlation function with T2 relaxation not only depends on Jwo)but also on correlation time Te results in the spectral density func J(O) since the z components of stochastic magnetic tion[16,24,29 fields(zero frequency)reduce the phase coherence of transverse magnetization components which con 1(0)=31+ ( 8) sequently sum up to a smaller macroscopic magneti zation. Since J(o) increases monotonously with J() 0°s]|=20ms 4tc=lOns T. 5n 0.001 001 0102051250[109rads] 63280160320800v[MHz] 4. Plot of the spectral density function /(o)(Eq (8))versus the frequency w on a logarithmic scale, Three correlation times 5, 10 and 20 ns indicated which represent small, medium and large proteins. The frequency scale is given in units of rad s and in MHz
