NMR, two-dimensional

The two-dimensional Fourier transformation is a straightforward generalization of the one-dimensional Fourier transformaton. The complex two-dimensional Fourier transformation of the time domain signal s ti, t ) is defined by 

A two-dimensional NMR spectrum is often displayed in the form of a map. One axis, V2, contains the conventional chemical shifts. The other axis, Vi or t, may contain chemical shift or /-coupling information or both. The position of a signal on the map tells us the chemical shift of that signal and the effect on that signal due to the presence of other nuclei. 

Probably the greatest recent change in the practice of NMR has been the explosive growth in techniques and applications of 2D NMR. 

2D NMR is an extension of ordinary NMR. The basic principle was invented by Jenner (27) and covers essentially all 2D experiments in NMR, as well as other areas. The general 2D NMR experiment is characterized by up to four time periods  

Preparation time is necessary to bring the system to a known state, e.g., equilibrium magnetization, and is usually a fixed delay time. At the beginning of the Evolution period, the spins are perturbed and 

The result is a collection of FID s, the number of which equals the number of different values of the Evolution time period. All FID s are then transformed in the usual way, resulting in a collection of spectra (the familiar inversion-recovery Ti experiment is an example of this stage of the process). A new collection of FID s is assembled by taking data points from each spectrum at given frequency values. 

For example, values of spectral intensity in each spectrum found at a frequency f are taken and arranged in a data table. This is repeated for every f value in the spectrum, resulting in N FID s when N is the number of points in the original spectrum. These N FID s are then fourier transformed, resulting in N spectra. These N spectra, presented on a two-dimensional plot, are now characterized by two frequency axes. One frequency (F2) is always that of the normal observe spectrum since it results from FT of normal FID s. The significance of the remaining (FI) frequency is determined by the pulse sequence used. 

Fourier series transformation of time domain data, Snoffi), into frequency domain signal intensity data Inmr(Ei) then yields a characteristic classical ID NMR spectrum. 

On the basis of this, two-dimensional (2D) NMR spectroscopy experiments can be defined as experiments that involve one time domain for signal evolution (fi) and an additional time domain for detection and signal acquisition (t2) together with appropriate radio-frequency 

Experiments shown are homonuclear (a) COSY, (b) TOCSY and (c) NOESY experiments. Typical heteronuclear correlation experiment (d) is also shown comprising complex sequence of multiple 90° and 180° pulses involving source nuclei I and destination nuclei S. 

ATpairs, and is the basis of many so-called double-resonance experiments used for the structural determination of proteins and of other biological macromolecules, as we shall see later. A variation on the HSQC experiment is the heteronuclear multiple bond correlation (HMBC) experiment. This is a sensitive technique that maybe used to identify heteronuclear and spin-spin coupled nuclei. 

Since the advent of pulsed NMR spectroscopy, a number of advanced two-dimensional techniques have been devised. These methods afford valuable information for the solution of complex structural problems. The technical detail behind multi-dimensional NMR is beyond the scope of this book. 

Two-dimensional spectra have the appearance of surfaces, generally with two axes corresponding to chemical shift and the third (vertical) axis corresponding to signal intensity. 

It is usually more useful to plot two-dimensional spectra viewed directly from above (a contour plot of the surface) in order to make measurements and assignments. 

The most important two-dimensional NMR experiments for solving stmctural problems are COSY (Correlation SpectroscopY), NOESY (Nuclear Overhauser Enhancement SpectroscopY), HSC (Heteronuclear Shift Correlation) and TOCSY (Total Correlation SpectroscopY). Most modem high-held NMR spectrometers have the capability to routinely and automatically acquire COSY, NOESY, HSC and TOCSY spectra. 

The COSY spectrum shows which pairs of protons in a molecule are coupled to each other. The COSY speetrum is a symmetrieal speetrum that has the NMR spectrum of the substanee as both of the chemical shift axes (Fi and F2). A schematic representation of COSY spectmm is given below. 

For an instant their magnetization is forced to lie in the xz plane, but then it continues to precess about the z axis at Jv or Jvg, the residual rotating frame frequencies. A period of typically 0.05 s now ensues during which two things occur. The magne zation decays via T2 (and to a much smaller extent via Tj) onto the z axis. 

An NMR spectrum contains a great deal of information and, if many protons are present, is very complex. The complexity would be reduced if we could use two axes to display the data, with resonances belonging to different groups lying at different locations on the second axis. This separation is essentially what is achieved in two-dimensional NMR. 

Although information from two-dimensional NMR spectroscopy is trivial in an AX system, it can be of enormous help in the interpretation of more complex spectra, leading to a map of the couplings between spins and to the determination of the bonding network in complex molecules. Indeed, the spectrum of a biological macromolecule that would be impossible to interpret in one-dimensional NMR can often be interpreted reasonably rapidly by two-dimensional NMR. In Case study 13.1 we illustrate the procedure by assigning the resonances in the COSY spectrum of an amino acid. 

We have seen that the nuclear Overhauser effect can provide information about internuclear distances through analysis of enhancement patterns in the NMR spectrum before and after saturation of selected resonances. In nuclear Overhauser effect spectroscopy (NOESY) a map of all possible NOE interactions is obtained by again using a proper choice of radiofrequency pulses and Fourier transformation techniques. Like a COSY spectrum, a NOESY spectrum consists of a series of diagonal peaks that correspond to the onedimensional NMR spectrum of the sample. The off-diagonal peaks indicate which nuclei are close enough to each other to give rise to a nuclear Overhauser effect. NOESY data reveal internuclear distances up to about 0.5 nm. 

Bovey (1988). Nuclear Magnetic Resonance Spectroscopy (2nd edn). Academic Press, New York, p. 653. 

Tonelli (1989). NMR Spectroscopy and Polymer Microstructure The Conformational Connection, VCH Publishers, New York, p. 252. 

Cais and J. Kometani (1984). NMR and Macromolecules, ACS Symposium series no. 247, American Chemical Society, Washington, DC. 

Mehring (1983). High Resolution NMR in Solids, Springer-Verlag, Berlin. 

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Time-resolved spectroscopy has become an important field from x-rays to the far-IR. Both IR and Raman spectroscopies have been adapted to time-resolved studies. There have been a large number of studies using time-resolved Raman [39], time-resolved resonance Raman [7] and higher order two-dimensional Raman spectroscopy (which can provide coupling infonuation analogous to two-dimensional NMR studies) [40]. Time-resolved IR has probed neutrals and ions in solution [41, 42], gas phase kmetics [42] and vibrational dynamics of molecules chemisorbed and physisorbed to surfaces [44]- Since vibrational frequencies are very sensitive to the chemical enviromnent, pump-probe studies with IR probe pulses allow stmctiiral changes to... 

The key for optimally extracting infonnation from these higher order Raman experiments is to use two time dimensions. This is completely analogous to standard two-dimensional NMR [136] or two-dimensional 4WM echoes. As in NMR, tire extra dimension gives infonnation on coherence transfer and the coupling between Raman modes (as opposed to spins in NMR). 

Bax A and Freeman R 1981 Investigation of complex networks of spin-spin coupling by two-dimensional NMR J. Magn. Reson. 44 542-61... 

Barbate G, Ikura M, Kay L E, Pastor R W and Bax A 1992 Backbone dynamics of calmodulin studied by N relaxation using inverse detected two-dimensional NMR spectroscopy the central helix is flexible S/oefrem/sf/ y 31 5269-78... 

Jeener J, Meier B H, Bachmann P and Ernst R R 1979 Investigation of exchange processes by two-dimensional NMR spectroscopy J. Chem. Rhys. 71 4546-53... 

Di stefano D L and Wand A J 1987 Two-dimensional NMR study of human ubiquitin a main chain directed assignment and structure analysis Biochemistry 26 7272-81... 

A good introductory treatment of the density operator formalism and two-dimensional NMR spectroscopy, nice presentation of Redfield relaxation theory. 

The standard monograph for those seeking an introduction to EPR spectroscopy. Frieboiin H 1993 Basic One- and Two-Dimensional NMR Spectroscopy (New York VCH) A basic introduction to NMR spectrai anaiysis. 

Abel E W, Coston T P J, Orrell K G, Sik V and Stephenson D 1986 Two-dimensional NMR exohange speotrosoopy. Quantitative treatment of multisite exohanging systems J. Magn. Reson. 70 34-53... 

These two reviews (Orrell et al and Perrin and Dwyer) cover the modem pulse and two-dimensional NMR teclmiques for studying exchange. 

The amount of computation necessary to try many conformers can be greatly reduced if a portion of the structure is known. One way to determine a portion of the structure experimentally is to obtain some of the internuclear distances from two-dimensional NMR experiments, as predicted by the nuclear Over-hauser effect (NOE). Once a set of distances are determined, they can be used as constraints within a conformation search. This has been particularly effective for predicting protein structure since it is very difficult to obtain crystallographic structures of proteins. It is also possible to define distance constraints based on the average bond lengths and angles, if we assume these are fairly rigid while all conformations are accessible. 

Two-dimensional nmr spectroscopy has led to a much better understanding of biopolymers and their function (151). This technique has been apphed to polyacrylamide for absolute assignments of proton and carbon spectra at the tetrad level (152). 

Another technique often used to examine the stmcture of double-heUcal oligonucleotides is two-dimensional nmr spectroscopy (see AfAGNETiC SPIN resonance). This method rehes on measurement of the nuclear Overhauser effects (NOEs) through space to determine the distances between protons (6). The stmcture of an oligonucleotide may be determined theoretically from a set of iaterproton distances. As a result of the complexities of the experiment and data analysis, the quality of the stmctural information obtained is debated. However, nmr spectroscopy does provide information pertaining to the stmcture of DNA ia solution and can serve as a complement to the stmctural information provided by crystallographic analysis. 

MI Sutcliffe, CM Dobson, RE Oswald. Solution structure of neuronal bungarotoxm determined by two-dimensional NMR spectroscopy Calculation of tertiary structure using systematic homologous model building, dynamical simulated annealing, and restrained molecular dynamics. Biochemistry 31 2962-2970, 1992. 

Figure 18.17 Two-dimensional NMR spectnim of the C-terminal domain of a cellulase. The peaks along the diagonal correspond to the spectrum shown in Figure 18.16b. The off-diagonal peaks in this NOE spectrum represent interactions between hydrogen atoms that are closer than 5 A to each other in space. From such a spectrum one can obtain information on both the secondary and tertiary structures of the protein. (Courtesy of Per Kraulis, Uppsala.)...

Two-dimensional NMR spectra of proteins are interpreted by the method of sequential assignment... 

Figure 18.20 The two-dimensional NMR spectrum shown in Figure 18.17 was used to derive a number of distance constraints for different hydrogen atoms along the polypeptide chain of the C-terminal domain of a cellulase. The diagram shows 10 superimposed structures that all satisfy the distance constraints equally well. These structures are all quite similar since a large number of constraints were experimentally obtained. (Courtesy of P. Kraulis, Uppsala, from data published in P. Kraulis et ah. Biochemistry 28 7241-7257, 1989, by copyright permission of the American Chemical Society.)...

Wright, P. What can two-dimensional NMR tell us about proteins Trends Biochem. Sci. 14 255-260,... 

Englander, S. W., and Mayne, L., 1992. Protein folding studied using hydrogen exchange labeling and two-dimensional NMR. Annual Review of Biophysics and Biomolecular Structure 21 243—265. 

The program will be demonstrated with poly(vinyl alcohol) for tacticity analysis and with copolymer vinylidene chloride isobutylene for monomer sequence analysis. Peak assignments in C-13 spectra were obtained independently by two-dimensional NMR techniques. In some cases, assignments have been extended to longer sequences and confirmed via simulation of the experimental data. Experimental and "best-fit" simulated spectra will be compared. 

Numerous organisms, both marine and terrestrial, produce protein toxins. These are typically relatively small, and rich in disulfide crosslinks. Since they are often difficult to crystallize, relatively few structures from this class of proteins are known. In the past five years two dimensional NMR methods have developed to the point where they can be used to determine the solution structures of small proteins and nucleic acids. We have analyzed the structures of toxins II and III of RadiarUhus paumotensis using this approach. We find that the dominant structure is )9-sheet, with the strands connected by loops of irregular structure. Most of the residues which have been determined to be important for toxicity are contained in one of the loops. The general methods used for structure analysis will be described, and the structures of the toxins RpII and RpIII will be discussed and compared with homologous toxins from other anemone species. 

Figure 1.39 Representation of an inversion-recovery 7) experiment. (Reprinted from W. R. Croasmun and R. M. K. Carlson, Two-dimensional NMR spectroscopy applications for chemists and biochemists, copyright 1987, p. 13, with permission, from VCH Publishers Inc., 220 East, 23rd Street, New York 10010-4606.)...

Two-dimensional NMR spectroscopy may be defined as a spectral method in which the data are collected in two different time domains acquisition of the FID tz), and a successively incremented delay (tj). The resulting FID (data matrix) is accordingly subjected to two successive sets of Fourier transformations to furnish a two-dimensional NMR spectrum in the two frequency axes. The time sequence of a typical 2D NMR experiment is given in Fig. 3.1. The major difference between one- and two-dimensional NMR methods is therefore the insertion of an evolution time, t, that is systematically incremented within a sequence of pulse cycles. Many experiments are generally performed with variable /], which is incremented by a constant Atj. The resulting signals (FIDs) from this experiment depend... 

Figure 3.1 The various time periods in a two-dimensional NMR experiment. Nuclei are allowed to approach a state of thermal equilibrium during the preparation period before the first pulse is applied. This pulse disturbs the equilibrium ptolariza-tion state established during the preparation period, and during the subsequent evolution period the nuclei may be subjected to the influence of other, neighboring spins. If the amplitudes of the nuclei are modulated by the chemical shifts of the nuclei to which they are coupled, 2D-shift-correlated spectra are obtained. On the other hand, if their amplitudes are modulated by the coupling frequencies, then 2D /-resolved spectra result. The evolution period may be followed by a mixing period A, as in Nuclear Overhauser Enhancement Spectroscopy (NOESY) or 2D exchange spectra. The mixing period is followed by the second evolution (detection) period) ij.

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