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Nuclear magnetic resonance

Introduction

From a purely intellectual viewpoint, one of the fascinating things about nuclear magnetic resonance (NMR) is the complexity of the subject. This complexity can be the source of much frustration for those wishing to understand and use NMR. As with other physical techniques used in studies of biological systems, NMR may be used in an empirical mode; for example, simply noting variations in an NMR parameter with alteration of an experimental variable.

A better understanding of the NMR phenomenon is often rewarded with additional elucidation of the system under study. Although there may be thresholds of knowledge of NMR necessary to read the literature critically or to conduct NMR studies, there is a continuum of knowledge to be gained about NMR . This topic is intended to provide an introduction to the NMR phenomenon. A good introductory text was written by Derome (Derome, 1987) With an orientation primarily towards protein structure determination, a good description of background and applications is found in the monograph of Cavanagh et al (Cavanagh et al., 1996).

The nuclear magnetic resonance phenomenon can be described in a nutshell as follows. If a sample is placed in a magnetic field and is subjected to radiofrequency (RF) radiation (energy) at the appropriate frequency, nuclei in the sample can absorb the energy. The frequency of the radiation necessary for absorption of energy depends on three things. First, it is characteristic of the type of nucleus (e.g., 1H or 13C). Second, the frequency depends on chemical environment of the nucleus. example, the methyl and hydroxyl protons of methanol absorb at different frequencies, and amide protons of two different tryptophan residues in a native protein absorb at different frequencies since they are in different chemical environments. The NMR frequency also depends on spatial location in the magnetic field if that field is not everywhere uniform. This last variable provides the basis for magnetic resonance imaging (MRI), for self-diffusion coefficient measurements, and for coherence selection – topics which will not be discussed further in this introductory chapter. For diffusion coefficient measurements and for imaging, the magnetic field is made to vary linearly over the sample.

Nuclear Magnetic Resonance

Magnetic Resonance

Nuclear spine

Nuclei have positive charges. Many nuclei behave as though they were spinning. Charged and moves has a magnetic moment and produces a magnetic field. This tiny magnet is often called a nuclear spin. If we put this small magnet in the field of a much larger magnet, its orientation will no longer be random. if the tiny magnet is oriented precisely 180° in the opposite direction, that position could also be maintained In scientific jargon the most favorable orientation would be the low-energy state and the less favorable orientation the high-energy state.

The charged nucleus rotating with angular frequency w ( = 2pn) creates a magnetic field B and is equivalent to a small bar magnet whose axis is coincident with the spin rotation axis.

The Resonance Phenomenon

The small nuclear magnet may spontaneously "flip'' from one orientation (energy state) to the other as the nucleus sits in the large magnetic field. Energy equal to the difference in energies of the two nuclear spin orientations is applied to the nucleusmore flipping between energy levels is induced . The absorption of energy by the nuclear spins causes transitions from higher to lower energy as well as from lower to higher energy.

The energy required to induce flipping and obtain an NMR signal is just the energy difference between the two nuclear orientations and is shown in to depend on the strength of the magnetic field Bo in which the nucleus is placed:

DE = ghBo/2p where h is Planck's constant (6.63 x 10-27 erg sec). The gyro magnetic ratio g is a constant for any particular type of nucleus and is directlyproportional to the strength of the tiny nuclear magnet. lists the gyromagnetic ratios for severalnuclei of biologic interest.

Sensitivity and the Boltzmann Equation

We noted earlier that our small bar magnet (nuclear spin) could be oriented in one of two ways.

The extent to which one orientation (energy state) is favored over the other depends on the strengthof the small nuclear magnet (proportional to gyromagnetic ratio) and the strength of the strongmagnetic field Bo in which it is placed. In practice, we do not put one nucleus in a magnetic field.

Rather a huge number (approaching Avogadro's number) of nuclei are in the sample that isplaced in a magnetic field. The distribution of nuclei in the different energy states (i.e., orientations of nuclear magnets) under conditions in which the nuclear spin system is unperturbed by application.

Magnetization

The above description of nuclear magnetization is based essentially on quantum mechanics.

Another way of viewing the NMR phenomenon is to use a classical mechanical Description.The angular frequency of precession is wo=2pno, where no is the same linear frequency so wo=gBo. In a real sample, we will have a tremendous number of nuclear spins all precessing about the z-axis. We know that the nuclei may be oriented either parallel or antiparallel to the direction of the magnetic field.

Some spins precess about the positive z-axis and some about the negative z-axis. The magnetization emanating from a real sample is simply the sum of all the individual nuclear magnetic moments (spins). As we know from the previous discussion, there will be a slight exces of nuclei oriented with the magnetic field, i.e., in the lower energy state, so the sum will yield a magnetization along the positive z-axis. It will be the total magnetization that determines an NMR signal – not the magnetic moment of an individual nucleus. stationary magnetic field Bo that defines the z-axis. In reality, the angle between the vectors and thez-axis is much smaller than is shown for illustrative purposes. The arrows over M o and B o signify that they are vectors, i.e., that they have direction as well as magnit.

The Nuclear Magnetic Resonance Experiment

It is possible to generate a proton NMR spectrum by slowly sweeping either the magnetic field or frequency such that the resonance condition expressed is achieved by all the protons, with signals in the sample occurring successively depending on chemical environment.

This method is used for special purposes. However, most spectrometers utilize pulses of radiationwhich cover the frequency range for all nuclei of a given type, e.g., 13C.

Fourier Transform Nuclear Magnetic Resonance

The above discussion on pulsed NMR is sufficient if there is only one frequency for the nuclei of interest, e.g., monitoring the single 19F signal from fluorouracil bound to thymidylate synthase or the proton signal of H2O in a tissue (because it is present in great excess over other observable proton signals). However, the FID is a time-domain signal with contributions typically from manydifferent nuclei, say the various 15N nuclei in a protein. The usual frequency-domain spectrum canbe obtained by computing the Fourier transform of the signal-averaged FID.

Relaxation Basis

The process by which a nuclear spin system returns to thermal equilibrium after absorption of RF energy. We can consider any of the spin systems in as a starting point. For example, for protons in a T field, there will be an excess of 128 of two million in the lower energy state at thermal equilibrium. If RF energy is applied to the nuclear spin system at the resonance frequency the probability of an upward transition is equal to that of a downward transition. Because there is a greater number of nuclei in the lower energy state, there will be more transitions from the lower energy state to the upper state than vice versa resulting in a nonequilibrium distribution of nuclear spins. For the protons in the example, the difference will be reduced to <128, even to 0, at which point the NMR signal disappears and the system is "saturated."

Relaxation processes, which neither emit nor absorb radiation, permit the nuclear spin system to redistribute the population of nuclear spins. Some of these processes lead to the nonequilibrium spin distribution (N lower – N upper) exponentially approaching the equilibrium distribution (N lower – N upper) equal:

(N lower – N upper) = (N lower – N upper) equal (1  e-t/T1) .where the time constant for the exponential relaxation is T1, the spin-lattice relaxation time. Such a relaxation curve is illustrated in Figure 1.8 for the example used above. T1 is the length of time required for the perturbed system to return 63% of the way toward thermal equilibrium. The lattice is the environment around the nucleus, including other molecules in the sample as well as the remainder of the molecule containing the nucleus of interest. Obviously, different 13C nuclei with in the same molecule could have different rates of relaxation, i.e., different T1 values.

There are additional relaxation processes that adiabatically redistribute any absorbed energy among the many nuclei in a particular spin system without the spin system as a whole losing energy. Therefore, the lifetime for any particular nucleus in the higher energy state may be decreased, but the total number of nuclei in that state will be unchanged. This also occurs exponentially and has a time constant T2, the spin-spin relaxation time. Under some circumstances, the line width of an NMR signal at half-height, W1/2, can be related to T2 by

W1/2 = 1/(pT2)

Causes

For spin-1/2 nuclei, the relaxation processes occur by interaction of the nuclear spin with magnetic fields, produced by magnetic dipoles which are fluctuating due to random molecular motions, both rotational and translational). The nature and the rate of the molecular motions affect the T1 and T2 relaxation times. Molecular motions that occur at a rate comparable to the resonance frequency no for the nucleus are most effective in promoting spin-lattice relaxation, i.e. yield the lowest values for T1. T2 values can be decreased even further as the molecular motion becomes slower than no, but T1 values will begin to increase. These relationships can be expressed generally as follows:

1

T1

= g2H2 tc

1 + (2pnotc )2

Measurement

Several different pulse sequences have been utilized to measure T1 and T2. The most common (and probably best) technique for determining T1 is to employ the inversion recovery sequence. After an initial 180° pulse inverts the spin populations the spin system begins relaxing toward thermal equilibrium as discussed earlier. After time t, a 90° pulse is applied, and the FID following the pulse is acquired. As t becomes longer, the magnetization more closely approaches the equilibrium population distribution rather than the inverted magnetization . A 90° pulse must be applied to be able to detect any magnetization because detection is possible only in the transverse plane may be determined by measuring the magnitude M of the FID as a function of t using the sfollowing expression:

M = M¥ (1 - 2e-t/T1) where M¥ is the magnitude of the FID when t = infinity (i.e., at thermal equilibrium).Determination of T2 is best carried out using a CarrPurcell pulse sequence with the Meiboom-Gill modification. Illustrates the process in the rotating coordinate frame. An initial 90° pulse rotates the magnetization into the y' direction. Because the Bo magnetic field is not perfectly homogeneous, the individual magnetic spins will have slightly different precessional frequencies (Consequently, the individual spins will fan out in the xy' plane with a loss of phase coherence (C). After a time t, a 180° pulse is applied (D). Because the 90°-t-180° T2 experiment. Radiofrequency pulses of strength B1 are applied along the x'-axis in the rotating coordinate system. Detection is along the y'-axis so a signal can bedetected after the 90° pulse (B) and at formation of the spin-echo (F).

General Features of Nuclear Magnetic Resonance Spectrum

The various NMR spectral parameters to be discussed subsequently are illustrated a one-dimensional spectrum is represented. However, as we encounter two-, threeCopyright or four-dimensional spectra, it should be apparent how the features mentioned here may be manifest in those multidimensional spectra.

Chemical Shift

V0= YB0/2π

Nuclei of different elements, having different gyro magnetic ratios, will yield signals at different frequencies in a particular magnetic field. However, it also turns out that nuclei of the same type can achieve the resonance condition at different frequencies. This can occur if the local magnetic field experienced by a nucleus is slightly different from that of another similar nucleus; for example, the two 13C NMR signals of ethanol occur at different frequencies because the local field that each carbon experiences is different.

The reason for the variation in local magnetic fields can be understood. If a molecule containing the nucleus of interest is put in a magnetic field Bo, simple electromagnetic theory indicates that the Bo field will induce electron currents in the molecule in the plane perpendicular to the applied magnetic field. These induced currents will then produce a small magnetic field opposed to the applied field that acts to partially cancel the applied field, thus shielding the nucleus. In general, the induced opposing field is about a million times smaller than the applied field. Consequently, the magnetic field perceived by the nucleus will be very slightly altered from the applied field, so the resonance condition of Equation will need to be modified:

n = (g/2p)Blocal = (gBo/2p)(1-s) .

where Blocal is the local field experienced by the nucleus and s is a nondimensional screening or shielding constant. The frequency n at which a particular nucleus achieves resonance clearly depends on the shielding which reflects the electronic environment of the nucleus. Electron currents around a nucleus are induced by placing the molecule in a magnetic field Bo. These electron currents, in turn, induce a much smaller magnetic field opposed to the applied magnetic field B0.

Spin-Spin Coupling (Splitting)

A nucleus with a magnetic moment may interact with other nuclear spins resulting in mutual splitting of the NMR signal from each nucleus into multiplets. The number of components into

which a signal is split is 2nI+1, where I is the spin quantum number) and n is the number of other nuclei interacting with the nucleus. For example, a nucleus interacting with three methyl protons will give rise to a quartet. To a first approximation, the relative intensities of the multiplets are given by binomial coefficients: for a doublet, for a triplet, and 1:3:3:1 for a quartet. The difference between any two adjacent components of a multiplet is the same and yields the value of the spin-spin coupling constant J (in hertz).

One important feature of spin-spin splitting is that it is independent of magnetic field strength. So increasing the magnetic field strength will increase the chemical shift difference between two peaks in hertz (not parts per million), but the coupling constant J will not change. To simplify a spectrum and to improve the S/N ratio, decoupling) is often employed, especially with 13C and 15N NMR. Strong irradiation of the protons at their resonance frequency will cause a collapse of the multiplet in the 13C or 15N resonance into a singlet.

Linewidth

The relationship between the minimum linewidth W1/2 and T2 relaxation has already been given In samples with macromolecules the linewidth may be broader than the scalar splitting, so the latter may not necessarily be apparent. It should also be noted that rapidly exchanging –OH and –NH protons do not cause splitting. However, if coupling constants can be determined, they can sometimes reveal details about molecular geometry.

Peak intensity

The area of an NMR signal (the peak intensity), but not the height (the peak amplitude), is directly proportional to the number of nuclei contributing to the signal under suitable experimental conditions. Those conditions are that the delay between acquiring free induction decays for signal averaging purposes should be ³ 4T1. Consequently, if the concentration of nuclei is known for a particular peak, it can be used as a standard. This delay is required for complete relaxation of all nuclei. As already noted, to achieve the best S/N for a given time of signal acquisition, spectroscopists typically do not wait for full relaxation.

For some types of NMR experiments, a comparison of peak intensities is required (e.g., for nuclear Overhauser effect measurements.The practitioner should recognize that the different peak intensities may not be compared rigorously but approximately when the pulse delay is not sufficient for full relaxation.

Nuclear Overhauser Effect (NOE)

The nuclear Overhauser effect or NOE is a relaxation parameter which has been used as the primary tool for determining three-dimensional molecular structure .

When two nuclei are in sufficiently close spatial proximity, there may be an interaction between the two dipole moments. The interaction between a nuclear dipole moment and the magnetic field generated by another was already noted to provide a mechanism for relaxation. The nuclear dipole-dipole coupling thus leads to the NOE as well as T1 relaxation. If there is any mechanism other than from nuclear dipole-dipole interactions leading to relaxation, e.g., from an unpaired electron, the NOE will be diminished – perhaps annihilated.

While it is possible to measure an NOE from a 1D NMR spectrum, usually 2D NOE (or sometimes 3D NOE) experiments are performed. the 2D NOE pulse sequence and a schematic 2D NOE spectrum. The intensities of the cross-peaks in the spectrum depend on the distance between the interacting nuclei; it is this relationship that provides structural information. While the situation is actually more complicated 1991 as long as the mixing time is kept short (in practice <100 msec), to a first approximation the cross-peak intensity decreases with increasing internuclear distance r according to r-6. The 2D NOE spectrum in thus would imply that protons 1 and 2 are in close proximity, 2 and 3 are further apart, while 2 and 4 are separated yet further. With current high field instruments, it is possible to detect crosspeaks from protons up to 5-6 Å apart.(possibly 7 Å if methyl protons are involved); this is applicable to molecules which have a correlation time larger than ca. one nanosecond. Basic pulse sequence to obtain a 2D NOE spectrum., we can assume that we are dealing only with protons. The relaxation delay simply denotes the time period to allow for nuclear relaxation before application of the pulse sequence for signal averaging purposes.The initial 90° pulse rotates the magnetization into the x'y' plane. The spins of nuclei with different chemical shifts have different precessional frequencies. So after time t1 the extent of magnetization in the x'y' plane (sum up all individual nuclear spin vectors) for nuclei with different chemical shifts will differ. The second 90° pulse rotates the transverse magnetization onto the z-axis and during the mixing time period, cross-relaxation can occur. The effect of cross-relaxation during the mixing time is to transfer magnetization between neighboring protons. The third 90° pulse rotates the magnetization again into the x'y' where it can be detected during the acquisition time t2. Fourier transformation (FT) of the free induction decay acquired during t2 yields a 1D spectrum which will depend upon the length of the mixing time tm and the value of t1. The acquisition and Fourier transformation can be repeated several times with incrementally varying values of t1. The intensity of any signals in the resulting 1D spectra will be modulated at a frequency depending upon chemical shift. A second Fourier transformation can be carried out as a function of t1, just as the individual spectra were generated via FT as a function of t2. The result is a 2D NOE spectrum with frequency axes corresponding to the two separate Fourier transformations. If nuclei which are frequency-labeled according their chemical shifts during t1 interact with nuclei with a different chemical shift during tm, in addition to the peaks along the diagonal, cross-peaks occur in the 2D NOE spectrum corresponding to the two different chemical shifts. This is illustrated schematically in the lower figure for signals from five protons. Note that a projection along either axis yields the same information as the 1D NMR spectrum, but the 2D NOE spectrum gives us the additional data about relative proximity of protons in the molecule.

Chemical Exchange

The subject of chemical exchange is vital to biological NMR. A detailed discussion of the subjecthas been presented . Nuclear magnetic resonance parameters can be affected by the monitored nucleus spending part of its time in one environment and part of itstime in another environment). If the exchange between the environments is sufficiently rapid, the observed NMR parameter will be a weighted average of the parameter values in each of the different environments or sites, the weighting taking account of the relative number of nuclei in each of the sites. One `often-encountered example of exchange is provided by the observation of the 1H NMR of molecules containing hydroxyl, amide or amino groups in aqueous solution Generally, the protons on these functional groups exchange so quickly with H2O protons that only a single averaged resonance signal is seen for these "exchangeable" protons; because H2O is usually present in great excess, the properties of this averaged resonance signal are weighted such that bulk H2O protons dominate. It should be noted, however, that in proteins and nucleic acids, the complicated structure precludes much of that exchange thus enabling signals from these hidden "exchangeable" protons to be observed. Observation of signals from amide protons of proteins, for example, is invaluable for ascertaining parts of the molecule which or do not become exposed to solvent water. Observation of signals from imino (and sometimes amino) protons in nucleic acids likewise demonstrate stability of base pair formation. Another example of exchange is that of a small molecule that may spend part of its time free in solution and part bound to a macromolecular structure. It's properties will reflect that averaging long as the exchange is sufficiently fast.

Chemical exchange processes. A proton which contributes to the observed H2O proton dominate.

With rapid exchange between the two sites A and B, a single resonance signal centered at frequency

n = fAnA + fBnB

with line width W1/2 [= 1/(1/T2)] such that

1/T2 = 1/T2A + 1/T2B

and

1/T1 = 1/T1A + 1/T1B

where fA and fB represent the fractions of nuclei at sites A and B, and the subscripts A and B refer to the values of those parameters when the rapidly exchanging nuclei are in sites A or B.

CONCLUSION

The general phenomenon of nuclear magnetic resonance is introduced using both a classical and a quantum mechanical perspective. This provides the basis for understanding measurable and informative NMR parameters: chemical shift, spin-spin splitting, linewidths, relaxation, the nuclearOverhauser effect and chemical exchange. The emphasis here is on molecules in solution but muchof the fundamentals pertain to molecules in the gas or solid phase as well.

REFERENCE

1. James, T. L. 1995. Nuclear Magnetic Resonance and Nucleic Acids. Methods in Enzymology. Academic Press, New York. 644.

2. Academic Press, New York. 644 James, T. L., and N. J. Oppenheimer. 1994. Nuclear Magnetic Resonance, Part C. Methods inEnzymology. Academic Press, New York. 813.