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Review
. 2010 Nov;57(4):381-419.
doi: 10.1016/j.pnmrs.2010.07.001. Epub 2010 Jul 30.

Radial sampling for fast NMR: Concepts and practices over three decades

Brian E Coggins  1 , Ronald A Venters, Pei Zhou
Affiliations
Review

Radial sampling for fast NMR: Concepts and practices over three decades

Brian E Coggins et al. Prog Nucl Magn Reson Spectrosc. 2010 Nov.
No abstract available

PubMed Disclaimer

Figures

Figure 1
Figure 1. Ambiguity from Discrete Sampling of a Continuous Signal
(a) In NMR, a continuous sinusoidal signal (solid line) is normally measured as discrete samples (points). (b) The actual information available about the original signal after sampling is limited to the values at the sampling points, introducing two forms of ambiguity. (c) One form of ambiguity is the lack of information about what happens between sampling points. In the case of regularly spaced samples, as plotted here, an infinite number of sinusoidal signals of various frequencies can be considered to fit the data equally well. (d) The second form of ambiguity concerns what happens after the end of the sampling period. The data do not allow one to distinguish between a signal that stops abruptly and a signal that continues infinitely.
Figure 2
Figure 2. Comparison of Spectra for Continuous and Sampled Signals
(a) The spectrum of a continuous signal contains a single, infinitely sharp peak at the signal frequency. (b) The spectrum of the same signal sampled at regular intervals over a finite period of time shows both aliasing and truncation artifacts, reflecting the ambiguities of the sampling.
Figure 3
Figure 3. Point Responses, Convolution and Discrete Sampling
The consequences of discretely sampling a continuous signal can be understood through the convolution theorem of the Fourier transform. In the time domain, the sampling process can be written mathematically as a multiplication of the continuous signal with functions describing the sampling—in this case, one function specifying evenly distributed samples, and a second function specifying the limited duration of the sampling interval. Each of these sampling functions has a Fourier transform, shown below, which is called its point response. According to the convolution theorem, the effects of sampling in the frequency domain are described by convolving (in the commonly-accepted convention of Bracewell, indicated by the operator “*”) the continuous spectrum with the point responses from the two sampling functions, yielding the discrete spectrum, with its aliasing and truncation artifacts.
Figure 4
Figure 4. Comparison of Conventional and Radial Sampling
(a) Conventionally, multidimensional NMR experiments have been sampled on regular grids. In this example, we depict an experiment with 8 × 8 = 64 sampling points. (b) In radial sampling, data are collected at evolution times falling on radial spokes. Because it is often possible to determine spectral information from a small number of radial spokes, each of those spokes can be measured to longer evolution times than for a conventional experiment of the same duration. Here, we show a radial experiment with four spokes each of 16 points; the number of samples and resulting measurement time are the same as in (a), but the resolution has been doubled. (c) Radial sampling is in fact a sampling in polar coordinates.
Figure 5
Figure 5. The Projection-Slice Theorem
The projection-slice theorem of the Fourier transform states that a slice through the time domain yields, upon Fourier transformation, a projection in the frequency domain. (a) Measuring slices at a 30° angle through two different multidimensional sinusoidal signals. The frequency observed on such a slice is a linear combination of the signal’s original x and y frequencies, with coefficients depending on the angle of the slice. Note that two signals with very different x and y frequencies can happen to appear identically on a slice, as shown here. (b) The Fourier transform of such a slice shows a projection of the original spectrum. In this example, the two signals coproject at the chosen angle.
Figure 6
Figure 6. The Point Response of a Single Radial Spoke
(a) A radial spoke at an angle of 30°, sampled uniformly at an interval of Δtr. (b) The point response for the radial spoke. A ridge of intensity is observed in the frequency domain passing through the origin and running perpendicular to the direction of the radial spoke, reflecting the complete lack of information about the modulation of the signal in the 120° direction. Additional ridges are observed with a spacing of 1/Δtr; these are aliases, resulting from the discrete sampling of the spoke.
Figure 7
Figure 7. Identification of a Multidimensional Signal Position from Projections
Given a set of projections of a multidimensional signal, it is often possible to determine the position of the signal in the full spectrum. In this example, by extending lines back from each projected peak, one finds a single intersection point, which must be the location of the original signal. This geometric logic has been exploited in a number of approaches: explicitly for reconstruction, as well as implicitly in calculations done directly from projected peak positions.
Figure 8
Figure 8. Radial Spokes and Projections in a 3-D Space
(a) A radial spoke tr can be measured in a time domain with three indirect dimensions tx, ty and tz. (b) The resulting point response shows a plane of intensity, perpendicular to the sampling direction, reflecting the complete lack of information about the location of the signal within this planar region.
Figure 9
Figure 9. Scaled Wavelengths in the Time Domain Determine Projected Peak Positions in the Frequency Domain
(a) Consider a signal with modulation only in the x dimension, at a frequency of ωx,1. In the time domain, this signal would appear as a plane wave with a wavelength λx,1 of 1/ωx,1. The wavelength λr,1 that would be observed by a slice at a 30° angle would be 1/(ωx,1 cos 30°). The wave therefore appears to be scaled by a factor of 1/(cos 30°) when measured by the slice. (b) In the 2-D frequency domain, the peak from the signal would be located along the ωx axis at position ωx,1. The Fourier transform of the 30° slice would show a projection of this peak, appearing at ωx,1 cos 30°.
Figure 10
Figure 10. Convolution Interpretation of Radial Spoke Peak Positions, and Conversion of Hypercomplex to Complex Data
(a) The frequency domain result of measuring a radial spoke can be understood geometrically as taking a projection; it can also be understood, however, as a convolution (denoted by *). In this example, a spoke is measured at an angle of 30° in a 2-D time domain. From the perspective of the spoke, the x signal would appear scaled by a factor of cos 30°. The y signal would likewise appear scaled by a factor of sin 30°. The actual time domain observation is the product of these two scaled signals, meaning that the resulting frequency domain signal is their convolution, a single peak at the position ωx cos 30° + ωy sin 30°. (b) If one measured a slice at an angle of −30°, the direction of the y modulation in the time domain would appear reversed. In the frequency domain, the y signal would appear at the position −ωy sin 30° instead of ωy sin 30°, and the result after convolution would be ωx cos 30° − ωy sin 30°. Measuring a slice at a negative angle would mean recording data at negative evolution times, which is not physically meaningful. However, equivalent data can be obtained by taking linear combinations of hypercomplex components. (c) NMR data collection is normally hypercomplex, and can only be carried out for positive evolution times. Once collected, however, that hypercomplex data can be converted to complex data, to produce slices at both positive and negative angles. The conversion process preserves all of the information in the data; in this example, four independent hypercomplex measurements become two complex data values in the +tx +ty quadrant, and two complex data values in the +tx −ty quadrant.
Figure 11
Figure 11. Radial Spoke Peak Positions in the Absence of Full Quadrature Detection
Fourier transformation of a radial spoke measured with quadrature information for only one dimension leads to a frequency domain multiplet. That can be understood by the convolution argument, shown here for the example of a slice at angle θ through a 2-D time domain, with quadrature detection in x and real detection in y. (a) Because complex data are available for x, a radial slice of the x signal alone would show as a single peak at the position ωx cos θ. (b) Since imaginary components are not available for the y signal, the Fourier transform of a slice of it alone would show a doublet. (c) The actual time domain observation is the product of the x and y signals, yielding a convolution of (a) and (b) in the frequency domain. The result is a doublet centered on the scaled x frequency, with a splitting equal to twice the scaled y frequency.
Figure 12
Figure 12. Geometric Explanation for Why Multiple Projections are Obtained from a Single Hypercomplex Spoke
(a) In this example, a 2-D spectrum contains a single peak, in the upper right corner. If hypercomplex components are measured for a radial spoke at angle θ, after conversion to complex components and Fourier transformation one obtains both the projection at angle θ (top) and a second projection at angle −θ (bottom). This can be explained by convolution, as in Fig. 10b. It can also be explained geometrically as follows. (b) An individual hypercomplex component cannot distinguish which quadrant contains the true peak, showing a mirror image duplicate in each quadrant. Projections of hypercomplex components at angle θ likewise show the true peak as well as its duplicates, each projection containing a quartet. (c) Taking a linear combination of the hypercomplex components selects for one of the four peaks. Depending on the particular linear combination, this may or may not be the true peak. Here, the duplicate peak in the lower right was selected. After taking a linear combination of the projections at angle θ of the hypercomplex components, one obtains a projection at angle θ with a single peak. In this case, the linear combination has selected for the projection of the lower-right duplicate peak. (d) The projection of the true peak at angle −θ obtained in panel (a) is in fact the projection at angle θ of the lower-right duplicate peak.
Figure 13
Figure 13. Identification of Multidimensional Signal Positions from Projections When Multiple Signals are Present
(a) Two peaks are present on each of these two projections (one projection parallel to x, and the other at an angle of 30° to x). By extending lines back across the 2-D space of interest from the projected peaks, one can determine the locations of the original signals, which are found where the backprojection lines intersect. In this case, there are only two possible intersection points for the backprojection vectors within the spectral region, meaning that the original positions of the two peaks can be identified unambiguously. (b and c) In a less favorable case, there are two possible interpretations for the projection data, both equally likely. The original peaks could be located as in (b) or they could be located as in (c); these two configurations give identical projection data, and one needs additional information to resolve the ambiguity. Note that this ambiguity would plague any method trying to interpret this data, regardless of whether it is automatic or manual, and regardless of whether it attempts to reconstruct a spectrum or merely attempts to calculate frequencies from the projected peak positions. (d) The relative intensities of the projected peaks might provide a means for resolving ambiguities. In this case, one possible answer is that there are three peaks of equal height, A, B and C, positioned as shown. However, without additional information one could not exclude another possibility, namely a configuration like that of (b), but with a peak A twice as strong as B.
Figure 14
Figure 14. Backprojection and Filtered Backprojection Reconstructions
(a) The backprojection reconstruction of a Lorentzian signal from 16 simulated projections. One observes ridges perpendicular to the projection directions; where these ridges intersect, the peak is formed, albeit broadened. The merging of the ridges also leads to an elevated baseline. (b) The backprojection reconstruction of a signal from 128 projections. Here, the ridges have completely merged. The peak is still broadened, however, and the baseline is still elevated. (c) By applying the filter function shown in the inset to each time domain radial spoke, the lineshapes on the projections are altered from the Lorentzian shape at left to the modified shape at right. (d) The filtered backprojection reconstruction of the same signal, from 128 projections. The peak is of the correct width, and the baseline is not elevated. Panel (c) is adapted, with permission, from [44]. © 2006 Springer.
Figure 15
Figure 15. The Lower-Value, Backprojection and Hybrid Backprojection/Lower-Value Reconstruction Algorithms
In these three panels, the stacked plots show reconstructions by the three methods of a plane extracted from the (3,2)-D HNCO of GB1. Four projections were used, at the angles 0°, 45°, 90° and 135°. The reconstructions have been normalized to allow for the comparison of signal and noise levels. The diagrams at top illustrate how each method calculates an output given the four inputs A, B, C and D, representing the four projections. (a) In the lower-value (LV) algorithm, the value assigned to a point in the spectrum is the minimum of the corresponding values found on the projections. The resulting reconstruction shows the two peaks found on this plane. (b) In backprojection (BP) reconstruction, the value assigned to a point in the spectrum is the sum of the corresponding values found on all of the projections. This produces reconstructions with backprojection ridges. The two true peaks are formed in locations where the ridges intersect; there are additional intersection points between ridges, however, which lead to spurious peaks of various heights. The signal level is eight times higher in this reconstruction than in the lower-value reconstruction, because of the additive nature of the backprojection process. (c) The hybrid (HBLV) method involves computing the sums of all possible combinations of k projections, and then assigning to the reconstruction point the smallest value encountered from among the set of sums. For four projections and a bin size k = 2, there are six combinations to be compared, which are shown in the diagram at top. The resulting spectrum shows stronger signals than in lower-value reconstruction, reflecting the partially additive nature of the process, without introducing artifacts. Reprinted, with permission, from [72]. © 2005 American Chemical Society.
Figure 16
Figure 16. Weighting of Data Points in the Polar Fourier Transform
(a) With conventional grid sampling, the area occupied by each sampling point, ΔA, is the same, and no special weighting is required during the Fourier transform. (b) In polar coordinates, sampling points that are closer to the origin are spaced more closely together. To correct for this, one must weight the points during the Fourier transform according to their area, ΔA. The appropriate weighting factor for 2-D is tr, the distance from the origin. Note that this weighting factor is identical to the filter function used in filtered backprojection.
Figure 17
Figure 17. Data Reflection and Lineshapes
(a) When complex data are available for only one quadrant, the result is a mixed-phase lineshape, as shown here in a contour plot (positive values are blue contours; negative values are pale red contours). (b) By reflecting the time domain data into a second quadrant, the dispersive terms are made to cancel, producing a purely absorptive lineshape.
Figure 18
Figure 18. The Radial Sampling Point Response, and Polar Fourier Transform Results
(a) The point response for radial sampling, plotted here for 25 radial spokes, can be separated into a ripple pattern and a ridge pattern. The ripples are the result of truncation from the finite duration of the sampling, and can be smoothed out by apodization. The ridges, which do not begin right at the peak but rather some distance from it, are the result of the radial configuration of the sampling points and are essentially a form of aliasing. The size of the “clear zone” that is free of ridge artifacts has been found to depend on the maximum evolution time in the time domain and the number of radial spokes. (b) Polar Fourier transforms from simulated radial data with 16 and 64 spokes is compared to the Fourier transform of simulated conventional data. With 16 spokes, the clear zone extends only just beyond the peak, and the ridges are seen over most of the spectrum. With 64 spokes, the clear zone extends beyond the edge of the spectrum, and thus no artifacts are seen. Reprinted from [37].
Figure 19
Figure 19. Bessel Functions and the Radial Sampling Point Response
The radial sampling point response can be derived analytically as the sum of a set of terms generated by the individual rings of sampling points. (a) A plot of the Bessel functions of orders zero to four, traditionally designated J0 to J4. (b) The Fourier transform of a single ring of sampling points is a Bessel function with respect to radius, and a sinusoid with respect to angle. This case corresponds to 10 radial spokes. (c) The case of a ring of sampling points corresponding to 16 radial spokes. Increasing the number of sampling points increases the frequency of the sinusoidal oscillation with respect to angle, and increases the order of Bessel function with respect to radius. (d) The case of a ring of sampling points corresponding to 16 radial spokes, but with a larger radius in the time domain. The result in the frequency domain is the same as (c), except scaled to have a smaller radius in the frequency domain, and therefore a smaller clear zone. Since each concentric ring of sampling points in a radial pattern has the same number of points, the terms they generate are of the same order but with different scaling, as in (c) and (d). The sum of terms like (c) and (d) for many radii generates the point response shown in Fig. 17.
Figure 20
Figure 20. Accordion Spectroscopy
(a) In an accordion experiment, the evolution time, t1, and the mixing time, τm, are increased simultaneously. The expanding pulse sequence can be likened to the bellows of an accordion. (b) The sampling points in an accordion experiment trace out a radial spoke, but unlike most radial experiments, one of the two dimensions is a mixing time rather than a chemical shift. The proportionality constant relating the simultaneous increases in the two experimental parameters is κ, which determines the slope (and therefore the angle) of the radial spoke. (c) A schematic representation of the data that would result from the accordion experiment. The lineshapes of the diagonal and cross-peaks reflect the dynamics of the exchange process observed during the experiment. (d) The peaks in an accordion spectrum can be inverse-Fourier-transformed to reveal the buildup curves. Shown here are the transforms of one diagonal peak (top) and one crosspeak (bottom) from cis-decalin, measured at 240 K in a Bruker 300 MHz spectrometer. Panels (c) and (d) are reprinted from [82].
Figure 21
Figure 21. Reduced Dimensionality NMR
Contour plot of a plane from the first reduced dimensionality experiment, the multiple quantum (3,2)-D HACANH of the mixed disulfide of E coli glutaredoxin (C14S) and glutathione, recorded on a 600 MHz spectrometer. The plane is taken at an Hα chemical shift of 4.28 ppm. Reprinted, with permission, from [47] (© 1993 ESCOM Science Publishers B.V.).
Figure 22
Figure 22. Reduced Dimensionality Beyond Three Dimensions
(a) Coevolving three dimensions in a (4,2)-D reduced dimensionality experiment would produce quartets. To reduce spectral crowding, Ding and Gronenborn used TPPI to introduce large frequency offsets between the multiplet components. (b) These offsets result in a spectrum in which the multiplet components appear grouped together as four subspectra. This example is from their HN(CO)CAHA of GB1, recorded on a 500 MHz spectrometer. (Note that these are the same as the four subspectra one would obtain if one used a full quadrature radial sampling approach.) (c) For a (5,3)-D experiment, Löhr and Rüterjans grouped the indirect dimensions into pairs, with each pair separately coevolved. This produces rectangular 2-D quartets. (d) Representative planes from their experiment on the protein flavodoxin, recorded on a 500 MHz spectrometer. Panel (b) is reprinted from [51]. Panel (d) is reprinted, with permission, from [108] (© 1995 ESCOM Scientific Publishers B.V. ).
Figure 23
Figure 23. Full Quadrature Reduced Dimensionality
When quadrature detection is used in all dimensions, it is possible to separate the multiplet components onto independent subspectra. (a) Data from the first full quadrature reduced dimensionality experiment, a (4,3)-D 13C/15N-filtered NOESY experiment recorded on a 600 MHz spectrometer, reported in 1995 by Brutscher and coworkers for the Rhodobacter capsulatus ferrocytochrome c2. Crosspeaks are produced at coordinates (HC, C + N, HN) and (HC, C − N, HN), with the former appearing in the subspectrum at left and the latter in the subspectrum at right. Data are shown for HC = 4.44 ppm. (b) An example of spectra obtained by Kozminski and Zhukov from (3,2)-D HN(CO)CA (top) and HNCA (bottom) of ubiquitin, the slices showing the sequential connectivity between residues I36 and G35. The spectra were recorded on a 500 MHz spectrometer. Panel (a) is reprinted from [49]. Panel (b) is reprinted, with permission, from [55] (© 2003 Kluwer Academic Publishers).
Figure 24
Figure 24. (5,2)-D GFT. The first GFT experiment was the (5,2)-D HACACONH of ubiquitin
(a) The pulse sequence. (b) Data for residue S20, showing the full hierarchical splitting pattern and the separation of multiplet components onto subspectra. At left are the strips for the 8 basic spectra, followed by the four first-order central peak spectra, the two second-order central peak spectra, and finally the one third-order central peak spectrum. In each case, the top half of the panel shows the transforms of the hypercomplex components prior to application of the G matrix, showing the full multiplet patterns, while the bottom half shows the result after application of the G matrix, with multiplet components separated onto independent subspectra. Reprinted, with permission, from [52] (© 2003 American Chemical Society).
Figure 25
Figure 25. G2FT
(a) Echoing the previous work of Löhr and Rüterjans, Atreya and coworkers introduced GFT experiments with multiple groups of coevolved dimensions. (b) Because they use full quadrature detection, however, Atreya and coworkers were able to separate the multiplet components onto independent subspectra. Any individual subspectrum will show only one of the muiltiplet components (filled circle) and omit the others (unfilled circles). (c) Atreya and coworkers designed their experiments to facilitate sequential assignment, grouping together the N and CO dimensions as one coevolved group, to produce a reduced dimensionality “fingerprint” for each residue, and grouping together Cα and Cβ as a second group, giving each connectivity its own unique signature.
Figure 26
Figure 26. Geometric Relationship Between Projections and the Full Spectrum
A 3-D contour plot of the lower-value reconstruction of the (3,2)-D HNCO of GB1 is shown here, along with three of the projections used in its reconstruction. Residue F52 is highlighted.
Figure 27
Figure 27. The First Projection-Reconstruction NMR Experiment
Comparison of the 2-D planes at HN=8.77 ppm for the projection-reconstruction (left) and conventional (right) HNCO spectra of ubiquitin. Reprinted, with permission, from [53] (© 2003 Kluwer Academic Publishers).
Figure 28
Figure 28. (4,2)-D PR Sequential Assignment Experiments on the 29 kDa HCA II
(a) The TROSY HN/N correlation spectrum of HCA II, recorded at 800 MHz. (b) Enlargement of the region containing residues K80, D110, A152 and E233. (c,d,e) Cα/Cβ planes at the (HN,N) position of A152 for the intra-HNCACB, HNCACB and HN(CO)CACB experiments, respectively. Each spectrum was measured as eight radial spokes, becoming 23 projections after conversion from hypercomplex to complex data, and then reconstructed using the HBLV algorithm with a bin size of eight projections. (f,g,h) Comparison of lower-value, backprojection and HBLV (k=8) reconstructions of the Cα/Cβ plane of the HNCACB experiment for residue K126. The lower-value reconstruction is free from artifacts, but shows only the strong intra-residue correlation. The backprojection reconstruction detects the inter-residue correlation to T125 as well, but its intensity is equal to that of the backprojection ridges. Additionally, all nearby crosspeaks are obscured through the broadening of the intraresidue peak. The HBLV reconstruction shows the K126 intra- and interresidue correlations, as well as all nearby crosspeaks, clearly, with no visible artifacts and no line broadening. Reprinted, with permission, from [72] (© 2005 American Chemical Society).
Figure 29
Figure 29. Projection-Reconstruction Methyl/Amide NOESY of HCA II
Data are shown for residue S50. (a) Strip from the 3-D conventional experiment at the (HN,N) coordinates of residue S50. (b) Contour plot of the equivalent HM/CM plane from the (4,2)-D reconstruction, computed from 100 projections using FBP. (c) Comparison of peak volumes between the conventional 3-D experiment and the (4,2)-D reconstruction. (d) Stacked plot. (e) Stacked plot from a BP reconstruction. The comparison with (d) shows the advantages of using the filter function. Panels (a)–(d) reprinted, with permission, from [71] (© 2005 American Chemical Society).
Figure 30
Figure 30. The Algebraic Reconstruction Technique
ART is based on the fact that projection values are integrals of the spectrum along lines. The spectrum is divided into discrete elements, here 2-D pixels numbered S(x, y). For an individual observed projection value, such as point A at left, one can write a linear equation describing the projection intensity as the sum of weighted contributions from the pixels of the spectrum. To determine the weight for a specific pixel on a specific projection point, one extends vectors from the projection across the spectrum (dashed lines) and determines the overlap integral. For example, the weight of pixel S(2, 2) on the orange projection point is determined by the area of the region U, while that of S(3, 3) on point B is determined by the area of region V. After defining this system of linear equations, ART proceeds by adjusting the values of S iteratively until the calculated sums agree with the observed data.
Figure 31
Figure 31. NMR Spectra Computed from Radial Samples with the Polar Fourier Transform
(a) C/N plane at HN=6.28 ppm from the HNCO spectrum of ubiquitin recorded on a 500 MHz spectrometer, as calculated from radial spokes at 4.5°, 45° and 85.5° by a hypercomplex Fourier transform. Note that the hypercomplex transform supplied with data for the +tx +ty quadrant automatically produces the symmetric ridges one would expect using a complex transform with data for both the +tx +ty and +tx −ty quadrants. (b) C/N plane at HN=8.54 ppm from the HNCO spectrum of ubiquitin recorded at 600 MHz, calculated in subpanel (1) from 18 radial spokes with artificially added noise, in (2) from 18 radial spokes with the natural experimental noise, and in (3) from a subset of 6 out of the 18 radial spokes, in all cases by a hypercomplex transform. Panel (a) is reprinted from [14]. Panel (b) is reprinted, with permission, from [75] (© 2006 Springer).
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