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Diffusion ordered nuclear magnetic resonance spectroscopy: principles and applications
Progress in Nuclear Magnetic Resonance Spectroscopy 34 (1999) 203–256
Diffusion ordered nuclear magnetic resonance spectroscopy: principles and applications C.S. Johnson Jr.* Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290, USA Received 21 October 1998
Contents Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Previous reviews of DOSY and related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The PFG-NMR experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Magnetic field gradients and magnetization helices . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Bloch equations with diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Pulse sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. The spin-echo (SE) sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. The stimulated echo (STE) sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The FT-PFG-NMR experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Component analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Diffusion ordered NMR spectroscopy (DOSY) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Experimental requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Eddy current reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Pulse sequences for minimizing effects of eddy currents and J-modulation . . . . . . . . . . 4.1.3. Suppression of convection current effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. Dispersion and resolution enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5. Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6. Utilizing the stray field to obtain large, steady gradients . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Data inversion and display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Discrete samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Polydisperse samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Complete bandshape methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4. Analysis recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Effects of chemical exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Exchange effects in diffusion spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Artifacts from chemical shift encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Applications of 1D and 2D DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Discrete samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1. Biofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * Tel.: ⫹1-919-966-5229; fax: ⫹1-919-962-2388. E-mail address: [email protected] 0079-6565/99/$ - see front matter 䉷 1999 Elsevier Science B.V. All rights reserved. PII: S0079-656 5(99)00003-5
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6.1.2. Separation by means of hydrophobicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3. Equilibria involving sodium dodecylsulfate (SDS) and bovine serum albumin (BSA) . . . 6.1.4. Mixtures of polymer additives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Polydisperse samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Phospholipid vesicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Blood plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3. The viscoelastic CTAB/sodium salicylate/water system . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4. Molecular weight distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. 3D DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. COSY-DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. HMQC-DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. NOESY-DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. TOCSY-DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Merged sequences for PFG-DQS, PFG-NOESY, and PFG-TOCSY . . . . . . . . . . . . . . . . . . . . . 8. Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Keywords: Diffusion ordered NMR; Pulsed magnetic field gradient-NMR experiment; Pulse sequences
Nomenclature GLOSSARY ALS (computer program) ARK* a priori knowledge BPP* Bipolar Pulse Pairs CONTIN (computer program) CORE COSY-DOSY CTP DECRA DEPT DISCRETE DLS DSTE DOSY* Diffusion Ordered NMR SpectroscopY EXSY FID FIDDLE (computer program) GCSTE GCSTESL GPC-NMR GRAM (computer algorithm) HDL (lipoprotein) HMQC-DOSY HR-DOSY HSQC
INEPT ILT LDL LED*
(inverse Laplace tranform) (lipoprotein) Longitudinal Eddy current Delay or Longitudinal Encode–Decode
MaxEnt MCR (computer algorithm) MOSY* Mobility Ordered NMR SpectroscopY MWD NIPALS (computer program) NLREG (computer program) NOESY-DOSY PFG-NMR PVA (computer algorithm) RDCON SE SPLMOD STE STEP (computer program) VLDL (lipoprotein) VMAX (computer algorithm) CSJ is responsible for LED, DOSY, and MOSY and some of the hyphenated forms containing them. He may have been the first to use the abbreviations BPP and APK. The others are defined in the text.
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1. Introduction One of the most fruitful ideas in NMR spectroscopy was the introduction of a second frequency dimension [1]. This was made possible through the use of pulse sequences having two independent precession periods. In one class of two-dimensional NMR (2DNMR) experiments, the Hamiltonian is switched between the evolution period and the detection period. As a consequence of the evolution period, resonances are spread into a second dimension to reveal their origins. Examples of such 2D ‘resolved’ spectroscopies include J-resolved where the Hamiltonian is switched through spin decoupling [2] and NMR imaging where magnetic field gradient directions are switched [3]. A logical extension of these ideas is the introduction of additional NMR dimensions that depend on molecular properties such as size, shape, mass, and charge that are not explicitly included in spin Hamiltonians. These overall molecular properties are not well represented in conventional NMR as spin interactions tend to be quite local. Therefore, dispersion on the basis of such properties can provide new information as well as a means for editing NMR spectra. The problem is to identify ways that molecular properties influence NMR spectra or can be made to affect NMR spectra. Nuclear relaxation times are obvious candidates because they depend on correlation times for molecular motion, and the correlation times in turn depend on molecular sizes and shapes. However, relaxation times can be quite different for different nuclei in the same molecule because of site specific magnetic interactions and because local or segmental motions may obscure overall molecular motions. In the case of longitudinal relaxation, high frequency local segmental motion may provide the dominant relaxation mechanism, leading to T1 values that are relatively independent of molecular mass. While for transverse relaxation, local motions may contribute motional averaging effects comparable to those resulting from overall rotation. Even so, there are cases where relaxation resolved spectra of, for example, backbone 13C nuclei in rigid molecules or bilayer 1H nuclei in vesicles can provide useful information about size dependent molecular reorientation. It is clear that new NMR dimensions should be
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based on molecular properties that have the same effect on all nuclei in a given molecule. Transport properties of molecules and ions, as determined by diffusion measurements and electrophoresis meet this criterion. The connection with structural properties arises because diffusion coefficients (D) depend on friction factors and electrophoretic mobilities (m ) depend on both friction factors and effective charges. According to the Debye–Einstein theory [4]: D
kB T fT
1
where kB is the Boltzmann constant, T is the absolute temperature, and fT is the friction factor. For the special case of a spherical particle of hydrodynamic radius rH in a solvent of viscosity h , the friction factor is given by fT 6phrH . More realistic models for fT represent molecules by ellipsoids of revolution or collections of spherical subunits [5]. Electrophoresis concerns the terminal velocity v of a charged particle in an applied electric field, Edc. The relationship between the electrophoretic mobility, defined as m v=Edc , and molecular properties is not simple; but for small ions m is proportional to the overall charge, Ze, and inversely proportional to fT [6]. The implementation of transport ordered NMR is possible because information about translational motion can be encoded in NMR data sets through the use of pulsed magnetic field gradient NMR (PFG-NMR) experiments [7]. The idea, as with conventional 2D NMR, is to increment an experimental variable that modulates the detected signal and then to transform the data with respect to that variable to produce a ‘‘spectrum’’ related in this case to molecular translation. Spectra based on electrophoretic mobilities are obtained by incrementing Edc and then transforming the NMR signal amplitudes with respect Edc. This scheme is realized in mobility ordered NMR spectroscopy (MOSY) [8]. The MOSY method has not yet found widespread use because of the unusual instrumentation requirements, e.g. NMR compatible electrophoresis cells, and the restriction to conducting samples with low ionic strengths. However, the success of the MOSY concept motivated the development of the more general diffusion based spectroscopy. Diffusion spectra can be obtained by incrementing the areas of the gradient pulses (q) in PFG-NMR and
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Fig. 1. The simple Carr–Purcell spin echo (SE) often called the Hahn echo.
transforming the NMR signals amplitudes with respect to q 2. The result is diffusion ordered NMR spectroscopy (DOSY) [9]. The three basic DOSY requirements are (1) distortion free absorption mode data sets acquired with precise gradient encoding, (2) effective data inversion (transformation) procedures, and (3) algorithms for the display of the diffusion spectra. These requirements turn out to be quite severe because the signal inversion step is extremely sensitive to noise and distortions in the signals. This has necessitated significant enhancements of the original PFG-NMR experiments and experimentation with alternative data inversion methods. Even data display for DOSY is not straightforward because decisions must be made about how to generate the spectra. The contrast with the Fourier transform NMR (FTNMR) is striking. With FT-NMR, one has a unique transformation with an inverse that returns the original signal. Also, the resulting spectra are ready for display. This review is concerned with the various implementations of DOSY experiments and with
illustrations of the power of this technique. The implementations present solutions to the unique problems of data acquisition, transformation, and display. With appropriate instrumentation and software, the user can be offered menu choices for analysis methods and types of display. The result is a convenient NMR method for the analysis of mixtures that can reveal unexpected components and interactions in mixtures through useful and appealing plots.
2. Previous reviews of DOSY and related topics Transport ordered NMR [10] and diffusion measurements by magnetic field gradient methods including DOSY [11] have previously been reviewed. Related reviews of MOSY are also available [12,13]. A complete treatment of translational dynamics and its study by NMR can be found in the book by Callaghan [14]. Ka¨rger et al. [15] have reviewed the principles and applications of PFG-NMR, and Stilbs has provided a detailed review of FT diffusion studies
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[16]. The recent tutorial articles on PFG-NMR by Price are also of interest [17,18].
3. The PFG-NMR experiment 3.1. Background Time domain NMR dates from Hahn’s observations of the free induction decay (FID), the spin echo (SE), and the stimulated echo (STE) [19,20]. The effects of molecular diffusion, in the presence of magnetic field gradients, on echo amplitudes were evident from the beginning, and Hahn reported a derivation of diffusion dependent signal attenuation, which he attributed to C.P. Slichter [20]. All NMR diffusion measurements are based on the fact that the diffusion coefficient can be calculated from the echo attenuation if the amplitude and duration of the magnetic field gradient are known. The original measurements were carried out with continuous gradients, but the advantages of pulsed gradients were convincingly demonstrated by Stejskal and Tanner [7]. Here we review the principles of PFG-NMR and display selected applications of PFG-NMR to provide the background for DOSY. 3.1.1. Magnetic field gradients and magnetization helices NMR diffusion measurements can be made by means of either gradients in the main (dc) magnetic field, B0, or gradients in radio frequency fields (B1). In the following only gradients in B0 are considered. For applications of RF gradients the reader should consult the review article by Canet [21] and the monograph by Kimmich [22]. Here the z-direction is defined by the direction of B0, and we are concerned with gradients in the z component of B. Typically, a spatially constant gradient is applied externally by means of current in a coil set, either of the Maxwell pair [23] or quadrupole type [24,25]. The resulting gradient g is described by: g
2B z ^ 2B z ^ 2 B z ^ k i⫹ j⫹ 2x 2y 2z
2
where ˆi, ˆj and kˆ are unit vectors in the x, y, and z directions, respectively. Accordingly, the total
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external magnetic field at r is given by B r B0 ⫹ g·r:
3
In the following we assume that only a z-gradient of magnitude g g·k^ is present. The purpose of the gradient is to label nuclear spins with phase angles that depend on their positions in space, or in this case their displacement in the z-direction. This is, of course, possible because spins precess with the angular frequency
v r ⫺gB r
4
and the acquired phase angle depends linearly on both B(r) and the duration of the gradient. Therefore, a zgradient of duration d produces the position dependent phase angle f z ⫺gB zd. PFG-NMR experiments involving constant z-gradients can readily be visualized by imagining layers of the sample perpendicular to the z-axis that are thin enough to experience a uniform magnetic field but thick enough to contain a large number of spins. Each layer is associated with a magnetization vector (isochromat), and these vectors are assigned to the positions of the layers on the z-axis. We begin the experiment with a hard 90⬚x RF pulse that rotates all of the vectors into the y-direction to create a magnetization ribbon in the rotating coordinate frame as shown in Fig. 1. The effect of the gradient g is then to twist the ribbon into a helix [26] defined by the relative phase angles Df z ⫺ ggdz. The pitch of the helix is given by L 2p=q where q ggd is the area of the gradient pulse in units of m ⫺1. Thus, the effect of a constant gradient is to produce a magnetization pattern with a cosinusoidal projection on the yzplane. Diffusion in the z-direction will, of course, smear out this pattern, and the smaller the pitch the more rapidly this will happen. 3.1.2. Bloch equations with diffusion The magnetization of uncoupled spins is well described by the Bloch equations, and the effects of diffusion can easily be incorporated. As gradients in Bz only affect the transverse components of magnetization, it is appropriate to begin with the Bloch equation for the complex magnetization, M⫹ Mx ⫹ iMy . We assume that the magnetization has been rotated to the y-direction in the rotating frame by a 90⬚x pulse or some other set of pulses, and that only the main field
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Fig. 2. The Hahn stimulated echo (STE) with pulsed field gradients.
Bz is present for t ⬎ 0. With Bx By 0 the Bloch equation becomes [27]:
2M ⫹ M ⫺iv0 M⫹ ⫺ ⫹ ⫺ ig g·rM⫹ ⫹ D72 M⫹ 2t T2 5 where the average precession frequency in the sample is denoted by v0 gB0 . The precession frequency and the effects of T2 relaxation can be transformed away by means of the substitution: M⫹ c z; texp iv0 t ⫺ t=T2
6
to give:
2c z; t ⫺iggzc z; t ⫹ D72 c z; t 2t
7
When D 0, Eq. (7) describes the free precession of transverse magnetization in the gradient g. At the end of a gradient pulse of duration t, the isochromats define a helix as described earlier. The effect of diffusion on the amplitude (diameter) of the helix, C(t), can be obtained by substituting Zt 0 0 c z; t c texp ⫺igz g t dt 8 0
into Eq. (7). The result is: Zt lnc t ⫺D q2 t 0 dt 0 0
9
where 0
q t
Zt 0 0
gg t 00 dt 00
10
Of course, this is only an attenuation factor, and for its observation the isochromats must be refocused in the xy-plane to form an FID or echo. It should be noted that, while gradients only affect the transverse components of the magnetization, the attenuation by diffusion as described by Eq. (9) also applies to sinusoidal patterns in the z-component of magnetization, i.e. stored magnetization. 3.2. Pulse sequences Immediately after a 90⬚x pulse, the signal detected along the y-axis in the rotating frame decays in amplitude. The primary cause of the decay in liquid state NMR is dephasing that results from magnetic field inhomogeneities. This decay is rapid in the presence of the applied constant gradient g because the volume
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integral of the helix of isochromats is zero when the pitch is much smaller than the length of the sample. In order to detect a signal, the isochromats must be refocused, i.e. the helix must be unwound, so that the volume integral of the magnetization gives a nonvanishing component in the xy-plane. This requires a sequence with two matched gradient pulses. The first pulse encodes nuclear positions through position dependent phase angles, and thus sensitizes the sample to diffusion and flow. The second gradient reverses the encoding and brings the isochromats back into the yz-plane, thus forming an echo. In the following we consider only PFG-NMR experiments where the gradients are applied in the form of relatively short pulses. The major advantages of an echo sequence with pulsed gradients are (i) the gradient pulse areas can be controlled independently of the time for the echo and (ii) the signal can be read out in a homogeneous magnetic field. The disadvantage is that relatively large currents must be switched on and off. This produces mechanical forces, Joule heating, and transient eddy currents. 3.2.1. The spin-echo (SE) sequence A simple Carr–Purcell [28] sequence is shown in Fig. 1. Also shown is the ribbon of isochromats produced by the 90⬚x pulse and the ‘‘Saarinen helix’’ produced by the first gradient pulse [26]. The effect of the 180⬚y pulse is to reverse the effect of the previous gradient pulses to give the effective gradient sequence (g*) [14]. This is equivalent to defining the effective gradient at any time by g* pg where p is the coherence order. Straightforward application of Eq. (9) to this sequence shows that the echo amplitude at 2t is [7]: S 2t M0 exp ⫺2t=T2 exp⫺Dq2 D ⫺ d=3
11
where M0 is the equilibrium magnetization and the relaxation factor containing T2 has been reintroduced. The correction term d /3 is a consequence of the rectangular shape of the gradient pulses. Other shapes, such as the sine lobe, can easily be incorporated [29]. The Stejskal–Tanner attenuation factor [7] for diffusion can be isolated as c 2t S 2t=S0 2t where S0 2t is the echo amplitude in the absence of a gradient. Continuous background gradients have been neglected in deriving Eq. (11). This is usually
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adequate for experiments in modern, high homogeneity magnets; but a more complete expression is available for the case where a background gradient g0 must be considered [7]. The Carr–Purcell echo has important advantages. In particular, the maximum possible signal is recovered, in the absence of relaxation effects, and chemical shifts are refocused at the echo. The disadvantages result from the long period that the magnetization is transverse, i.e. in the xy-plane. Transverse magnetization is subject to both transverse relaxation and J-modulation effects. T2 can be short for slowly tumbling macromolecules, and this can lead to a severe loss of signal. J-modulation refers to signal modulation resulting from hard RF pulses that exchange the spin states of nuclei that are coupled to the nuclei of interest thus preventing complete refocusing. These effects present special problems for strongly coupled spin systems [30]. 3.2.2. The stimulated echo (STE) sequence Sequences containing two and three 90⬚ RF pulses were investigated by Hahn in his classic paper on spin echoes [20]. He found that the three-pulse sequence with a steady (cw) gradient can generate up to five echoes. The first echo after the third RF pulse, the socalled stimulated echo (STE), is of particular interest here. The effects of diffusion on the STE with both steady [31] and pulsed gradients [32] have been computed. We show the standard PFG-STE diffusion experiment in Fig. 2. An instructive three dimensional (spherical polar) model of the first part of this experiment including the formation of the primary echo at 2t (not shown) was presented by Hahn and attributed to E.M. Purcell [20]. The amplitude of the PFG-STE is given by S T ⫹ 2t M0 =2exp ⫺2t=T2 ⫺ T=T1 exp⫺Dq2 D ⫺ d=3
12
where the Stejskal–Tanner factor has again been derived by the application of Eq. (9). It should be recognized at the outset that the STE is quite different from the SE. First, we see that the amplitude is reduced by a factor of two. This results from two features of the sequence. The second 90⬚x pulse stores the magnetization by rotating only the y-components into the ^z-directions. The x-components remain
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Fig. 3. A 1H FT-PFG-NMR stack plot obtained at 99.6 MHz for a microemulsion sample containing sodium octylbenzenesulfonate (SOBS), n-butanol, toluene, and water (D2O). Reproduced with permission [16].
transverse and can contribute to the primary and secondary echoes. Then, after the storage period T, the third 90⬚x pulse returns the z-components to the ^y-directions where the action of the second gradient pulse refocuses the isochromats so that the STE signal appears at t T ⫹ 2t. However, the isochromats are not coplanar at the time of the echo, and in fact their projection onto the xy-plane defines a circle that is tangential to the xz-plane. The advantages of the STE sequence arise because the evolution time for transverse magnetization can be limited. With 2t p T, spin relaxation depends primarily on T1 rather than T2, and with t p 1/J, J-modulation is not significant. The advantage of T1 relaxation relative to T2 relaxation depends on the ratios R T2 =T1 and X T=T1 . Assuming that T 艑 D and T q t in the STE experiment, we find that the STE/ SE signal ratio is 0:5expR X ⫺ 1=X. For example when R 0.5, X must be greater than ln(2) to break even, but with R 0.1 and X 0.5 the enhancement factor is greater than 200. In general the advantages of STE more than compensate for the 50% smaller coefficient.
Two other points need to be mentioned. First the reduction of t to a value only slightly larger than the gradient pulse duration d means that the STE is very close to the trailing edge of the second gradient pulse. Unless appropriate provisions are made, the signal will be distorted by gradient pulse induced eddy current effects. The second point concerns the encoding of chemical shifts after the first RF pulse. For a spin with offset frequency v A and position z, the component cos vA t ⫹ ggzd will be stored in the z-direction by the second 90⬚ pulse. The third 90⬚ pulse brings this component back into the yz-plane and, after an additional time t , the component in the y-direction is cos2 vA t ⫹ ggzd neglecting the effects of diffusion. The echo signal, obtained by integration of the y-components over the sample volume, gives 1/2 as expected when g is present, but in the absence of g we encounter the modulation factor 1=2cos2 2vA t. Therefore, in STE experiments the echo amplitude S0 T ⫹ 2t for g 0, i.e. the q 0 point in plots of echo amplitude versus q 2, is dependent on the chemical shift and
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Fig. 4. 400 MHz 1H NMR spectra for human blood plasma: (a) normal spectrum with 10% maximum gradient strength, (b) spectrum obtained with 50% strength, and (c) the difference between (a) and (b). Reproduced with permission [39].
should be avoided in data analysis (see Section 5.2) [33]. 3.3. The FT-PFG-NMR experiment For analysis of spin relaxation in complex mixtures it is essential that we make use of the complete
spectral information that is contained in FID’s and half-echoes. The FT method for accomplishing this was described by Vold et al., in connection with their study of frequency resolved inversion recovery [34]. This idea can also be extended to NMR diffusion measurements when pulsed gradients are used as the FID’s can be acquired in the absence of applied
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gradients. In the following we focus on features and applications of FT-PFG-NMR that are relevant to the later discussion of DOSY. 3.3.1. Component analysis 3.3.1.1. Complex mixtures James and McDonald demonstrated the determination of diffusion coefficients for each component in a multicomponent system by means of FT-PFG-NMR [35]. In their experiment, a Carr–Purcell echo (Fig. 1) was acquired as a function of the gradient duration d with t 艑 d, and the second half of the echo was Fourier transformed to produce a set of NMR spectra. The attenuation factor, c 2t S 2t=S0 2t; for each line was then used with Eq. (11) to determine the corresponding value of D. The gradient g was calibrated in a separate experiment on a compound with a known diffusion coefficient. This groundbreaking experiment, carried out on a standard commercial NMR spectrometer, established the basic NMR diffusion measurement still in use. The authors recognized the analytical implications, because each component of a mixture is revealed by its unique diffusion coefficient, and possibilities for the study of dynamics in solutions. In particular, the effects of rapid chemical exchange were considered, and the determination of binding constants for small molecules with large molecules by means of the weighted average diffusion coefficient was suggested. The major limitation of the James and McDonald experiment was the small gradient amplitude available and the necessity of using long gradient pulses. Kida and Uedaira remedied the problem by designing a gradient driver that permitted the use of narrow gradient pulses that were compatible with the operation of a field stabilized NMR system [36]. They also introduced the stack plot display of spectral intensity versus q2 D ⫺ d=3. Other features of this early paper are the analysis of the apparent hydroxyl group proton diffusion coefficients on the basis of rapid exchange between water and methanol, and the correlation of molecular diffusion coefficients with realistic models for translational friction factors (see Eq. (1)). The FT-PFG-NMR method, primarily in the spin echo version, has been applied to a wide variety of chemical systems. We note the studies of complex mixtures, especially those containing surfactants, by
Stilbs and coworkers. That work has been reviewed, and an illustration involving a microemulsion is shown in Fig. 3 [16]. Note that the stack plots are arranged with the gradient pulse duration d increasing for the lower (front) spectra, and the effects of Jmodulation are evident for the inverted toluene resonances. 3.3.1.2. Spectral editing Stilbs has emphasized the fact that, in a PFG-NMR diffusion measurement for a mixture, the complete spectrum for each component is attenuated as the quantity q2 D ⫺ d=3 is increased; and the non-overlapping signals can be classified by their measured diffusion coefficients [37]. He also suggested that spectra with different amounts of attenuation could be scaled and subtracted to zero out a component and in favorable cases to isolate a component. A demonstration of this procedure with 1 H spectra of a 50:50 mixture of decane and 1-decanol was presented, and potential problems with Jmodulation and T2 effects were noted. This type of spectral editing was characterized as ‘‘size-resolved NMR spectrometry.’’ The pseudo-separation of different molecules in a complex mixture by spectral editing on the basis of nuclear relaxation times and molecular diffusion coefficients has recently been pursued with the aid of sophisticated, modern PFG-NMR sequences. For example, an STE experiment including the WATERGATE water elimination sequence [38] has been used to assign spectral resonances of slowly diffusing molecules in human blood plasma [39]. This experiment is illustrated in Fig. 4. The spectrum in Fig. 4(a) was acquired with a small gradient (g 59 mT/m, D 500 ms) and shows essentially no attenuation. The spectrum in Fig. 4(b), acquired with g 295 mT/m emphasizes large molecules and permits the assignment of peaks between 3.4 and 3.9 that have previously been obscured by amino acids and carbohydrates. Finally, Fig. 4(c), the difference between Fig. 4(a) and (b), shows the small rapidly diffusing molecules and their assignments. These spectral editing experiments and numerous other experiments involving diffusion filters take advantage of strong diffusion based discrimination against small rapidly moving molecules without requiring sophisticated data transformations. Editing and filtering techniques can be very valuable; but, of
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Fig. 5. The LED pulse sequence [48].
course, do not provide complete diffusion spectra of mixtures. 3.3.1.3. Affinity NMR Another variant of diffusion edited NMR, known as affinity NMR, deserves comment [40,41]. In the pharmaceutical industry combinatorial chemistry methods are now producing large numbers of compounds in mixtures for testing in drug discovery programs. It is important to be able to detect the presence of molecules with desired properties without resorting to physical separation of the mixtures. If the desired property involves complexation with a partner in solution, differences in diffusion coefficients can be important indicators. In the affinity NMR experiment, the gradient amplitude and duration are adjusted so that all signals from small molecules in a mixture just vanish. Then a potential complexing agent is added and the experiment is repeated. In a demonstration experiment, nine components with molecular weights in the range 200400 were attenuated to the noise level by adjusting the gradient duration in a PFG-NMR experiment. Hydroquinine 9-phenanthryl ether was then added and the spectra were found to contain signals from two of the small compounds in addition to the added ether [40]. The combination of PFG-NMR and TOCSY was sufficient to identify the two complexing compounds [42].
4. Diffusion ordered NMR spectroscopy (DOSY) 4.1. Experimental requirements For the standard DOSY experiment we envision automated data collection with a programmed set of gradient areas. This is followed by data inversion by
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means of one or more user selected transformations and the generation of one, two, or three-dimensional spectra. This scheme can only be successful if data collection and initial processing, i.e. Fourier transformation, phasing, baseline correction and deconvolution, yield undistorted absorption mode NMR spectra. DOSY requires high quality gradient probes that incorporate active shielding [43,44] and are designed to provide constant (flat) gradients over the NMR active sample volume. DOSY also requires computer controlled gradient drivers that can provide gradient pulses with reversible polarity and pulse areas that are matched to within at least 10 ppm [45]. The latter feature is essential for automated DOSY experiments. We assume that such equipment is commercially available or will be available in the near future and do not discuss it further. However, readers should be aware that much commercial gradient equipment currently in use does not meet these criteria. As previously discussed, the pulse sequences of choice are based on the STE sequence (see Fig. 2). The immediate problem is that gradient pulses tend to induce eddy currents in the surrounding metal structures of the probe and the magnet. These eddy currents in turn produce slowly decaying magnetic fields at the sample that lead to spectral distortions resulting from time dependent phase changes. Therefore, experiments must be designed that avoid or at least minimize the effects of eddy currents. There is also the related requirement that the NMR resolution be maximized to avoid overlap of peaks from different components in a mixture as data transformations, required to produce diffusion spectra, fail when NMR peaks for similar sized molecules overlap. Here we consider current hardware, software, and experimental designs that address these requirements. 4.1.1. Eddy current reduction The best way to avoid the effects of eddy currents is to prevent the formation of eddy currents in the first place. There are some easy but not completely effective solutions. For example eddy currents in the probe can be reduced by means of special RF coil designs, and eddy currents in the inner bore of the magnet can be reduced by using widebore magnets. Also, the rate of change of the magnetic field when a gradient pulse is switched on or off can be reduced by shaping the gradient pulse. Shapes that have been investigated
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Fig. 6. The BPP-LED pulse sequence. The phase cycle for the 90⬚ pulses is: P1: 016, P2: (0022)4, P3: 04 24 14 34, P4: 0202 2020 1313 3131, P5: 04 24 14 34, Rec: 0220 2002 3113 1331, and the 180⬚ pulses are ⫹ x or 2 throughout [53].
include sine functions, sine squared functions, and nearly rectangular functions with modified rise and fall times [29]. Gradient pulse shaping is helpful and this capability is now available on some commercial instruments. The most effective means for avoiding eddy currents is to reduce the magnetic fields of the gradient coil set outside the probe to such low values that significant disturbances do not occur. This reduction can be achieved through active shielding of the gradient coils. Imagine a wire carrying a time-dependent current close to a conducting metal sheet. A surface current distribution is induced in the sheet that screens the magnetic field and reduces it to zero inside the sheet. The idea of active shielding is to introduce a mesh of wires with an externally generated current pattern that mimics the induced surface current distribution in the conducting sheet. Mansfield and Chapman have reported an iterative procedure for determining the positions of wires in a discrete cylindrical mesh to approximate the continuous current distribution required to screen the field of a current loop [43]. Another strategy is to use a multidimensional minimization program to determine the axial coordinates and radii of a predetermined number of Maxwell pairs that will shield most effectively a specified gradient coil set. A small volume outside the shielding coils is chosen as the indicator of shielding efficiency [44]. Active shielding is now so well established that commercial imaging and gradient probes can be expected to include efficient shielding coils.
Fig. 7. NMR spectra of protons on a -carbons of alanine in D2O: (a) 5 ms after a bipolar gradient pulse pair (g ^ 1.50 T m ⫺1, d /2 1 ms, t 1.5 ms), (b) 5 ms after monopolar gradient pulse (g 1.50 T m ⫺1, d 2 ms), (c) 100 ms after a monopolar gradient pulse (g 1.50 T m ⫺1, d 2 ms). The frequency origin is arbitrary, and the dashed line (b) shows the baseline. Reproduced with permission [53].
4.1.2. Pulse sequences for minimizing effects of eddy currents and J-modulation The pulse sequences described in the following text all require phase cycling for coherence transfer pathway (CTP) selection. The principles of CTP through phase cycles are, of course, well known [1,46]; but in practice the construction of cycling procedures that are both efficient and effective is not straightforward. As complete phase cycles for coupled spin systems may be very time consuming, it is common to reduce the number of steps by removing what are thought to be the least important parts. However, combinations of incomplete phase cycles with gradient pulses are not unique and cycling procedures are often determined by trial and error without optimization. The optimum phase cycle depends on the nature of the sample but the weighting of the various CTP’s also depends on the properties of the RF pulses and the presence of transport phenomena. Jerschow and Mu¨ller have recently developed a method for evaluating the latter effects by simulating the CTP selection
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Fig. 8. Pulse sequences (a) GCSTE [Phase: P1: 0213, P2: (0828)2 (1838)2, P3: (0424)2 (1434)2, Rec: f 2 ⫹ f 3 ⫺ f 1] and (b) GCSTESL [Phases: P1: 0213, P2: (0828)2 (1838)2, P3: (0424)2 (1434)2, P4: 13 ⫹ 064264 ⫹ f 2 ⫹ f 3 ⫺ f 1, Rec:. f 2 ⫹ f 3 ⫺ f 1] [54].
process and have implemented this method in a computer program named CCCP [47]. For most of the pulse sequences illustrated here, the phase cycles actually used are listed in the figure captions. 4.1.2.1. Longitudinal eddy current delay (LED) sequence In spite of the best efforts, eddy current effects are still significant, and they depend on the strength of the gradient pulses. This, of course, can be disastrous for experiments that automatically sample a wide range of gradient values. The stimulated echo (see Fig. 2) is primarily affected by eddy currents induced by the final gradient pulse, and the problem is exacerbated by the need to keep t short in order to minimize transverse relaxation and Jmodulation. This conflict motivated Gibbs’s modification of the STE sequence as illustrated in Fig. 5 [48]. The major change is the addition of a fourth 90⬚ pulse at the center of the stimulated echo for the purpose of storing the magnetization in the longitudinal direction while the eddy currents decay. After the eddy current settling period Te, the magnetization is recalled with a fifth 90⬚ pulse and the FID is acquired. The effectiveness of this sequence is further enhanced by adding three gradient pre-pulses (not shown) to make a chain of five equally spaced pulses. This arrangement ensures
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that the transient magnetic fields resulting from previous gradient pulses have the same effect during the transverse evolution periods after the first and third RF pulses. Either phase cycling or homospoil pulses can be used to eliminate transverse components during T. However, phase cycling of the last two RF pulses is essential to remove the effects of longitudinal relaxation during Te. This can be accomplished by alternately storing the STE in the ⫹z and ⫺z directions and then taking the difference between the associated signals that are returned to the xy-plane by the fifth RF pulse. We note that the LED sequence is analogous to B1 gradient experiments. The first 90⬚x –g–90⬚⫺x composite pulse sandwich where g gz is equivalent to a single B1 gradient pulse in encoding the spatial position in longitudinal magnetization [21,49]. Then after the storage period T, the second sandwich decodes position and again stores longitudinal magnetization. Hence LED can stand for either Longitudinal Encode–Decode or Longitudinal Eddy-current Delay. The LED sequence significantly improves the quality of spectra obtained in FT-PFG-NMR experiments but still suffers from the slowly decaying eddycurrents. The consequence is that the Te period, required to obtain undistorted spectra in experiments with large gradient pulses, can be unacceptably long. There is also the problem that the gradient pre-pulses introduce additional heat. 4.1.2.2. Bi-polar LED (BPP-LED) sequence One of the best ways to diminish the eddy currents induced by a short gradient pulse (g) is to replace the pulse with two pulses of different polarity separated by a 180⬚ RF pulse, i.e. the composite bipolar gradient pulse combination (g–180⬚–( ⫺ g)). As previously noted, the effective gradient g* is equivalent in these two cases; but the composite pulse provides self-compensation of the induced eddy currents [50]. Gradient pulse sequences with alternating polarity were introduced into PFG-NMR by Karlicek and Lowe [24] to take advantage of the fact that the 180⬚ RF pulses refocus static gradients. Also, Cotts et al. [51] proposed a number of STE sequences with alternating grading polarities to minimize the effect of background gradients. More recently Fordham et al. [52] replaced all of the gradient pulses in the LED
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background gradients, and more importantly for DOSY it refocuses chemical shifts. The latter can be very important when chemical exchange [33] or spin diffusion is present [54]. At present the BPP-LED sequence is the sequence of choice for many DOSY experiments, especially those requiring maximum gradient strengths with small temperature rises. It, of course, can be combined with relaxation filters, water elimination sequences, etc. as desired.
Fig. 9. Velocity insensitive gradient pulse sequences: (a) A gradient sequence with zero first moment (see text), and (b) the double STE sequence (DSTE) [59].
sequence with bipolar pulse pairs (BPPs) to permit diffusion measurements in the presence of large background gradients. The LED experiment with BPPs shown in Fig. 6 was investigated for DOSY applications by Wu et al. [53] With short gradient pulse separations (t ⬃ 1 ms) the BPPs were found to cancel more than 95% of the eddy currents, and undistorted signals could be obtained with Te reduced by a factor of 20. The effectiveness of the BPPs in reducing signal distortions is illustrated in Fig. 7. Further, this improvement could be obtained without the need for gradient prepulses, thus reducing undesirable heating effects. The use of Eq. (9) with the effective gradient g* for the BPP-LED sequence gives the corrected attenuation factor:
c D ⫹ d ⫹ 2t exp⫺Dq2 D ⫺ d=3 ⫺ t=2:
13
Of course, eddy current compensation is more complete when both d and t are short. The extra 180⬚ pulses introduced here cause some loss of signal because of the greater sensitivity to inhomogeneities in the RF pulses. However, this turns out to be an advantage because signal acquisition is limited to the region where the gradient is constant and higher quality data result. Also, the refocusing effect of the 180⬚ pulses does eliminate the effect of steady
4.1.2.3. New sequences and comparisons The choice of a pulse sequence for DOSY depends on both the capabilities of the available instrumentation and the nature of the system under study. When very small diffusion coefficients are involved, large gradient amplitudes are required to obtain adequate signal attenuation. As disturbances to the local field and the lock signal increase with the amplitude of the gradient pulse, the LED feature is often required. However, when only modest gradients are required and an efficient, well designed probe with active shielding is available, the settling period Te may not be necessary. In such cases the BPP-STE sequence should be considered [54,55]. The direct detection of the STE, without storage and recall, is especially important when the phase of the echo must be determined as in MOSY experiments [8]. The self-compensating feature of BPPs is extremely important in eddy current reduction, but there are other benefits from the (g)–180⬚–(⫺g) composite pulse as well. The refocusing and cancellation of steady gradients in inhomogeneous systems was the initial motivation for the introduction of alternating gradients in PFG-NMR. In high resolution NMR two other effects are encountered that are also sensitive to refocusing. In the absence of a 180⬚ pulse the chemical shifts are encoded along with the position dependent phase information during the first transverse interval of the STE sequence. When g 苷 0 chemical shift information usually does not appear in the STE; however, when spin exchange or chemical exchange interchange chemical shifts during the storage interval T, the chemical shifts do affect the amplitude of the STE [33,54]. With coupled spin systems there is another consequence of unrefocused chemical shifts because the second 90⬚ RF pulse generates zero-quantum coherences (ZQCs) in
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Fig. 10. Block diagram of the stop-and-go sample spinning system. Reproduced with permission [60].
addition to z-magnetization [22,56]. In cases where the ZQCs decay more slowly than the longitudinal magnetization, phase errors become evident with long values of the storage time T [54]. Fortunately, the (g)–180⬚–(⫺g) sandwich eliminates both the exchange and ZQC problems, and as discussed earlier the extra 180⬚ pulses increase the quality of the collected data because of their tendency to restrict the excitation volume. The negative aspect is that the BPP-STE and BPP-LED sequences require considerable phase cycling. This is not a problem when signal-to-noise ratios are low and signal averaging is already necessary, but with strong signals the experimental time must be increased to accommodate the phase cycle sequences. Another potential disadvantage of sequences incorporating BPPs is that the total amount of time required to complete the composite gradient pulse pair will exceed the time required for a single gradient pulse. When T2 is very short, the extra amount of time with transverse magnetization will lead to loss of signal. Pelta et al. [54] have recently compared the STE based PFG-NMR sequences in use (STE, BPP- STE, LED, and BPP-LED) and have suggested two additional sequences with self-compensating gradient pulse pairs. The new sequences, known as gradient compensated stimulated echo (GCSTE) and gradient compensated stimulated echo spin lock (GCSTESL), are illustrated in Fig. 8(a) and (b), respectively. Both of these sequences feature BPPs in which
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one of the pulses is placed in the storage interval T. This arrangement avoids the extra 180⬚ pulses required when the BPPs are placed in the transverse intervals t and provides a homospoil effect as well. The major advantage is a reduction in the required phase cycling. The disadvantages are (1) double the amount of heating for the same q values, (2) potential phase anomalies associated with ZQCs for coupled spin systems, and (3) amplitude modulation effects associated with spinexchange and chemical exchange. The GCSTESL sequence incorporating the spin locking interval t ST corrects the line shapes for ZQCs but not for exchange effects, and provides somewhat better resolution by restricting detection to the region of constant gradient. We conclude that all of the BPP-STE sequences are useful under appropriate circumstances. In the absence of spin coupling and exchange effects and when only small gradients are required, the BPPSTE and GCSTE sequences are reasonable choices. When exchange effects are not present and modest gradients suffice, GCSTESL becomes a strong candidate, but in the general case requiring chemical shift refocusing, background gradient compensation, and strong gradients the BPP-LED sequence is indicated. These conclusions are, of course, based on current technology. 4.1.3. Suppression of convection current effects Convection currents are easily induced in nonviscous samples by temperature gradients. The well known Rayleigh–Benard instability results from temperature inversions where higher temperature, lower density layers lie below lower temperature, higher density layers. The resulting fluid flow produces a distribution of velocity components parallel to the z-gradients in typical PFG-NMR experiments, and a corresponding attenuation of the STE that interferes with diffusion measurements [57]. This effect is characterized by a downward curvature in plots of the logarithm of the STE amplitude versus q 2. However, the deviation may not be easy to distinguish from a simple increase in the diffusion coefficient. Of course, the velocity distributions found in gravity driven mass convection have less effect on diffusion measurements made with gradients in the x or y directions. The first protection against convection currents is a well designed temperature control system that can
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minimize temperature gradients in the sample. Such systems usually require a large flow rate for the heat exchange gas. Also, sample tubes with smaller diameters are less susceptible to convection. An extreme example is the stabilization of solutions of polymers in liquid and super-critical CO2 in fused silica capillaries having inner diameters of approximately 100 mm [58]. When one must deal with low viscosity solvents at temperatures far from ambient, mass convection is often difficult to avoid. In such circumstances it is advisable to use pulse sequences that are insensitive to constant velocities. Such sequences can be constructed by requiring that the first moment of effective gradient sequence, g*, vanish [14], i.e. Zt 0
g* t 0 t 0 dt 0 0:
14
A gradient pulse sequence satisfying Eq. (14) is shown in Fig. 9(a). The velocity insensitive double STE diffusion sequence (DSTE) shown in Fig. 9(b) was proposed by Jerschow and Mu¨ller [59]. (Here selection of the proper coherence pathway is essential because g* pg.) They found that the diffusion coefficient measured for test a sample of 80% glycol in DMSO-d6 (Bruker temperature-calibration sample) at 347 K with an uncompensated BPP-LED sequence showed a deviation of a factor of 9 from the correct value. This deviation was eliminated when the compensated DSTE sequence or a BPP version of it was used. 4.1.4. Dispersion and resolution enhancement A hard lesson for most experimentalists to learn is that sums of exponential functions with added noise are extremely difficult to resolve into unique sets of components. Even with two components having well separated decay constants, one pays a considerable price in accuracy and computing time compared with the analysis of a single exponential decay. Therefore, for mixtures of monodisperse components the highest accuracy in DOSY analyses will be obtained for components that are already completely resolved in the chemical shift dimension. The realization of this fact has motivated attempts to obtain resolution of NMR peaks through experimental refinements and signal processing. In the following we consider
resolution enhancements for 1D-NMR. 3D DOSY based on 2D NMR is covered in Section 7.
4.1.4.1. Stop-and-go spinner Sample spinning in NMR is often the final step in getting the maximum possible resolution. When line widths are dominated by magnetic field inhomogeneities, spinning at even modest frequencies, e.g. 20 Hz, can offer significant resolution enhancement. However, it is common knowledge that conventional sample spinning is not compatible with PFG-NMR diffusion measurements because of unavoidable sample movement in the gradient direction. Here the restrictions are severe as PFG-NMR experiments can detect displacements in the gradient direction of the order of 1 mm. With modern shim stacks, the requirement for non-spinning samples may not be a serious problem. However, if spinning is a necessity, the options are limited. It is probably possible to design a precision spinner that will limit sample excursions in the z-direction to acceptable bounds. But this is likely to be a costly venture and to require much less convenient sample tubes and handling procedures. Fortunately, a low cost alternative, based on the fact that a stationary 5 mm NMR sample tube can be spun up to the required 20 Hz speed in approximately 10 ms, has been demonstrated. Fig. 10 shows a block diagram of a stop-and-go spinner system that is compatible with high-resolution NMR spectrometers [60]. A computer controlled DC servo motor is directly coupled to the sample tube so that the sample tube can be arrested during the motion sensitive part of the experiment and can be restarted during the delay period Te when the magnetization is stored in the z-direction. The spinning sample is stopped in about 10 ms by an ‘‘active brake’’ after data acquisition, but the liquid in the sample requires 1–2 s to reach a quiescent state that is compatible with a diffusion measurement. With a standard resolution test sample in a 10 mm gradient probe mounted in the narrow bore magnet of a Bruker AC-250 spectrometer, the best nonspinning linewidth was about 2 Hz. This width was reduced to about 0.2 Hz when spinning was controlled by the DC motor. This apparatus has few disadvantages and has been used routinely for data collection in DOSY experiments requiring the best resolution [61].
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Fig. 11. The STE-INEPT pulse sequences for heteronuclear detected DOSY with coherence transfer [68].
4.1.4.2. Reference deconvolution Even with the best efforts at field shimming, eddy current avoidance, sample spinning, etc., one is left with line shapes in FT-PFG-NMR experiments that suffer from experimental artifacts. If the remaining linewidths interfere with the resolution of components, it is worthwhile to consider postprocessing of the data to enhance the resolution or to improve the line shapes. Consider a single NMR line with a frequency offset of V. The corresponding FID has the form f t g texp ⫺iVt with g t s tb t where s(t) is the ideal response obtained with a perfect instrument and b(t) accounts for all deviations from ideality. The line shape in the frequency domain, I(v ), is given by the Fourier transform of the product s tb t I v FTb t·s t
15
and according to the convolution theorem this shape is
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also equal to the convolution of B v FTb t with S v FTs t or I v B v*S v [1,62]. The ideal spectrum with a shape depending only on the intrinsic T2 can in principle be extracted from the experimental shape by simply dividing f(t) by b(t) prior to performing the Fourier transformation. One can even remove some of the intrinsic linewidth by additional filtering of the FID at the risk of adding noise and distorting the frequency domain signal. The tradeoffs between sensitivity and resolution have been investigated in detail [63], and it is now common practice to multiply the FID with a weighting function that provides some resolution enhancement without introducing unacceptable noise. Common examples are the sine–bell and Lorentz–Gauss weighting functions. The demands on postprocessing in DOSY may be especially severe as the FIDs are acquired in the presence of some level of transient magnetic fields and the strengths of the transient fields depend on the amplitude and duration of the gradient pulses. Therefore, the free induction decays may not be truly free, and the Fourier transforms of decays acquired in the presence of changing frequencies cannot produce symmetrical line shapes. A reasonable approach to this problem is to use the experimental FID itself for deconvolution. This appealing idea, now known as reference deconvolution (RDCON), has been rediscovered many times in different contexts [64]. In practice one locates a signal from a singlet
Fig. 12. The pulse sequence for shuttle based fringe field 2D-DOSY [phase cycle: P1: 08, P2: 08, P3: 08, P4: 0202 0202, P5: 0220 1331, Rec: 0022 1133] [74].
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Fig. 13. Comparison of FFT and ILT transformations.
component in an NMR spectrum, zeros the remainer of the spectrum, and performs an inverse FT to obtain the reference FID, g(t). Division of the experimental FID by g(t) corresponds to removing all linewidth and does not lead to useful results. However, multiplying the FID by s t=g t, where s(t) is a guess at the ideal FID for a single line, is practical and useful. The implementation of the RDCON method requires that a complex reference signal be isolated and that a spectral region be available that contains sufficient ranges of both absorption and dispersion mode signals. The magnitude of the reference FID must be greater than zero and the baseline must be chosen carefully. These and many other practical considerations are discussed in detail in a recent review article by Morris et al. [64]. Also, a software package named fiddle has been prepared for certain spectrometers. Impressive demonstrations of RDCON in DOSY data processing have been reported, and it is claimed that with modern actively shielded diffusion probes the improved diffusion fit gives an increase in accuracy by about a factor of three [64,65]. Differences in diffusion coefficients of 0.5% are distinguished in this work. 4.1.4.3. Heteronuclear NMR Still another approach to the problem of peak overlap is to increase the spectral dispersion. The simplest way to do this is to increase the chemical shifts by working at the largest possible magnetic fields. Unfortunately, the chemical shift range for protons is inherently small and spectral crowding may be a problem with commonly available
500 and 600 MHz spectrometers. Chemical shifts are, of course, much larger for nuclei with larger atomic numbers and heteronuclear NMR may be an alternative. Consider for example 13C NMR. In addition to the wider range of chemical shifts, resolution for 13C is aided by longer values of T2, the virtual absence of homonuclear spin-spin coupling, and lower sensitivity to magnet field inhomogeneities and eddy current effects relative to 1H because of the smaller gyromagnetic ratio. The problem with 13C for DOSY applications is that the sensitivity is quite low. One must contend with low natural abundance, and the low gyromagnetic ratio and the long relaxation times that enhance resolution also limit signal-to-noise ratios. Further, the all important q factor is proportional to the gyromagnetic ratio, and this can be a problem when available gradient amplitudes are small. Coherence transfer experiments can solve the q-factor problem while improving sensitivity. The idea is to transfer polarization information from 1H to 13C prior to detection. This can be accomplished by combining a STE based PFG sequence with INEPT [66] or DEPT [67]. A simple INEPT-DOSY sequence is shown in Fig. 11 [68]. After the first 90⬚ proton pulse, a composite BPP encodes spatial positions and the magnetization is then returned to the z-direction by the second 90⬚ pulse. The optional homospoil gradient pulses during the storage period T eliminate residual magnetization on the xy plane. Then detection is carried out with a refocussed-decoupled INEPT sequence [69] where an initial 90⬚ heteronuclear RF pulse has been added to remove the natural heteronuclear spin polarization. An example of the 13C INEPT-DOSY is shown in Fig. 14 of Section 4.2.1.1 for a test sample containing sucrose, glucose, and sodium dodecyl sulfate (SDS). 4.1.5. Data collection Assuming that provisions have been made to obtain maximum resolution and undistorted absorption mode signals for each value of q, there are still choices to be made concerning data collection. It is important to select the minimum number of q 2 values that will permit the fastest and slowest decaying components to be analyzed with sufficient accuracy. Minimizing the number of q 2 values will, of course, increase the
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Fig. 14. 13C INEPT-DOSY for a mixture containing glucose, sucrose, and SDS in D2O. Dotted lines show average diffusion coefficients of each component. The 1D 13C INEPT spectrum of the mixture is shown at the top. Reproduced with permission [68].
efficiency of signal averaging and permit samples having limited lifetimes, e.g. blood plasma, to be studied. Also, the judicious choice of sampling values can avoid the collection of useless data sets that do not provide enough information for the determination of diffusion coefficients of all the components. We assume that the attenuation of NMR peaks with increasing gradients strengths can be represented by functions of the form S x exp ⫺Gx, or weighted sums of such factors, where G D; x q2 D 0 and D 0 is the diffusion time appropriately corrected for the shape and duration of a gradient pulse. As the G values required to fit diffusion data for all the peaks in an NMR spectrum may differ by orders of magnitude, it is seldom appropriate to use linear spacing of x values. In fact, when the ratios of G values exceed three, linear spacings lead to very large errors. Logarithmic spacing of x values can be a very good choice if the parameters are properly chosen [70]. A convenient form for computing xn is shown in Eq. (16) xn x1 2 n⫺1=m :
16
We take the smallest decay ‘time’ to be tmin 1=Gmax and the largest to be tmax 1=Gmin , and it is reasonable to set xN 5tmax where N is the maximum value of n. The choice of the first data point, x1, is a bit more
tricky because it is easy to waste data points at small x values while undersampling the larger values. We suggest choosing x1 0:2tmin and then computing, or determining by trial and error, the value of m that will permit N data points to fit in the desired range (according to Eq. (16) the appropriate value is m N ⫺ 1ln 2=ln xN =xl ). Labadie et al. have investigated errors in biexponential data analysis with various sampling schemes [71]. They find that geometric spacing with xn x 1
an ⫺ 1 a ⫺ 1
17
can give low errors and a reasonable distribution of data points along the decay. Here a is the constant ratio of successive interval lengths. The reported error analysis suggests that with logarithmic spacing x1 should not be smaller than t min/5.8, while for geometric spacing the optimum value of x1 is t min/ 33.4. In some applications it is also important to collect a data point with a very small value of q. For example, systems involving chemical exchange may show large initial slopes in plots of ln(Signal) versus q2 D 0 with strong curvatures that make the extrapolation to the zero point difficult. However, one must bear in mind that with a zero gradient, the echo amplitude in a STE
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experiment will depend on the chemical shift unless bipolar gradient pulse pairs are used. Also, very small q values introduce two additional problems. For a sample of length L in the direction of the gradient if q is not much larger than 2p /L, the magnitude of the stored magnetization will depend on the phase of the second 90⬚ pulse. Also, restricted diffusion will introduce errors unless DD=L2 ⬍ 0:001 [72]. 4.1.6. Utilizing the stray field to obtain large, steady gradients NMR measurements of very small diffusion coefficients require large gradient amplitudes, but in PFGNMR large gradients invariably produce heating effects, vibrations, and eddy currents. These problems were avoided by Kimmich et al. in an elegant experiment making use of the steady gradient in the fringe field on a superconducting magnet [73]. Experiments analogous to PFG-NMR can be run in the fringe field with an STE sequence as encoding/decoding occurs only when the magnetization is transverse to the main field. There are, however, two major problems with fringe field diffusion measurements. First, data collection in the presence of a large gradient eliminates resolution and permits only the echo amplitude to be recorded. Thus, all structural information is lost, and it is not even possible to distinguish between the resonances of 1H and 19F. If resolution is not essential for some sample, there is still the problem that available RF pulses excite only a thin slice of a sample in a strong gradient field. DOSY experiments with resolution in both the chemical shift and diffusion dimensions require that the FIDs be acquired in a homogeneous magnetic field. This was accomplished in stray field DOSY [74] with a shuttle system similar to that used in zero-field NMR experiments [75]. The LED pulse sequence used in this experiment and the location of the sample as a function of time (dotted line) are shown in Fig. 12. The STE is created by the first three p /2 RF pulses, the fourth p /2 pulse stores the echo magnetization in the z-direction for the shuttling period Ts, and the fifth p /2 pulse recalls the magnetization for detection in the homogeneous field. A widebore (89 mm) magnet was equipped with a homebuilt shuttle probe containing electronics for excitation in the fringe field at 140 MHz and detection in the homogeneous field at 360 MHz. A Kel-F
(36 ml) sample cell was shuttled pneumatically in a precision bore quartz tube a distance of about 0.3 m between the two RF coils. The RF pulse widths in the homogeneous field and the fringe field were 10 and 1.4 ms, respectively. Accordingly, only a 0.3 mm thick slice was excited in the fringe field (g 53 T m ⫺1) and special phase cycling was necessary to select only the spins excited and encoded with diffusion information in the fringe field from the background of all protons excited in the homogeneous field. A DOSY experiment was performed on a test sample containing SDS, glycerol, and H2O (1:2:4 by weight) with T 5 ms, shuttle time Ts 150 ms, and d incremented from 10 to 300 ms in 18 steps [74]. This provided satisfactory resolution for the test sample even though the NMR linewidths exceeded 20 Hz because of susceptibility differences between the sample and the short cylindrical cell. This experiment demonstrates that fringe field DOSY experiments are practical for samples having sufficiently long T1s and well resolved spectra, and the advantages are considerable. Experiments can be performed with very short diffusion times and with gradient amplitudes not easily obtained in other ways. Further, this can be done without gradient drivers and coils and with relatively inexpensive, though unorthodox, instrumentation. 4.2. Data inversion and display The ability of DOSY to provide accurate distributions of diffusion coefficients for 1D analysis or for the construction of 2D and 3D DOSY displays depends on the inversion of data sets that consist of NMR spectra collected with predetermined values of q2 D 0 . In the following we assume that FT-PFG-NMR experiments of the STE or BPP-STE types provide 2D data sets of the form: X An nm exp⫺Dn D 0 q2 18 I q; nm n
where An nm is the amplitude of the 1D-NMR spectrum of the nth diffusing species when g is small but not zero (see Section 4.1.5), and Dn is the associated tracer diffusion coefficient. Also, D 0 D ⫺ d1 where 1 depends on the shape of the gradient pulse. The goal of DOSY analysis is to transform 2D data
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sets as shown in Eq. (18), which are basically stack plots of attenuated spectra, into 2D spectra with chemical shifts on one axis and the distribution of diffusion coefficients on the other. We define a ‘discrete’ system as one that can be described by small set of values of n in Eq. (18). The maximum value of n may be more than ten, but for a particular peak in a high-resolution spectrum we expect that the attenuation can be described by a few values of n, i.e. usually single exponential fits. In the event that a polydisperse component contributes to the peak at frequency n , a continuum of values of D must be considered. The 1D data set I(s) describing attenuation of this peak must then be described by: Z∞ a lexp ⫺lsdl 19 I s 0
where l D D ⫺ d1 and s q 2. (An alternative choice is l D and s q2 D ⫺ d1.) In Eq. (19) we recognize that I(s) is the Laplace transform of a(l ) and that a(l ) is the Laplace spectrum of diffusion coefficients. When only discrete components with the decay constants l i are present, a(l ) is a weighted sum of delta functions, d l ⫺ li and Eq. (18) is recovered. The hope of new comers to this field is that a unique transform, akin to the FT, exists that can produce unique diffusion spectra and can invert them to recover the decay curves. This problem is illustrated in Fig. 13. At the top an FFT converts the FID to a unique NMR spectrum including the line shapes. At the bottom the decay on the left contains two components with diffusion coefficients and amplitudes differing by a factor of two. A perfect transform would produce the Laplace spectrum of delta functions shown on the right and the inverse transformation would exist. In fact, there is no perfect transform, and in the presence of noise it may be impossible to obtain any useful spectrum. The dotted curves in the diffusion spectrum indicate broadening associated with errors in an approximate transformation that would be acceptable. Gardner et al. [76] took a significant step toward finding the Laplace spectrum for discrete sums of exponential components by introducing the transformation s exp(x) and constructing the function exp(x)a[exp(x)]. This function turns out to be the convolution of the desired spectrum of decay
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constants with the shape function exp(x) exp[⫺exp(x)]. Thus the solution is at hand if the shape function can be removed by deconvolution. The catch is that the removal of the large line widths by means of Fourier deconvolution or any other method usually introduces unacceptable errors, e.g truncation artifacts, and this scheme is not practical, though it can serve as the starting point for more extensive analyses. Actually, the method of Gardner et al., as modified by Swingler [77] was investigated for DOSY applications but was abandoned in favor of the more robust schemes described in the following text [78]. Here we confront the fact that the desired spectrum a(l ) is the inverse Laplace transform (ILT) of the decay function I(s). Computing a(l ) is an ill-conditioned problem and one that may be intractable [79]. Actually, solutions for a(l ) can usually be found that agree with I(s), but they are often not unique. This is the source of much wasted computer time and much nonsense in the literature. The same data inversion problem has been faced for many years by the dynamic light scattering (DLS) community and extensive reviews are available in the literature [80–82]. The success of DOSY analysis hinges on the more modest goal of computing the most likely spectrum of diffusion coefficients by means of an approximate ILT or some appropriate fitting algorithm. It is usually true that FT-PFG-NMR data sets do not contain enough information to permit an exact analysis. Reasonable assumptions must be made and additional information must be supplied. We know for example that NMR absorption mode signals are positive and that their Laplace spectra are also positive as are the associated decay constants, and in dealing with polydisperse samples it may be reasonable to assume that the distribution functions are in some sense smooth. There is also the possibility of combining information from peaks at different chemical shifts, and of course there are physical limits on the maximum possible diffusion coefficients and the minimum values that can be detected in any given experiment. Prior knowledge concerning non-negativity, parsimony, and other features may be essential for obtaining the most likely diffusion spectrum from an experimental data set. DOSY analysis begins with a set of N absorption mode NMR spectra each having v frequency points or
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Fig. 15. Early DOSY display for a sample containing tetramethyl ammonium ions (TMA) and mixed micelles in D2O. The 1H data set was analyzed with DISCRETE. Reproduced with permission [9].
channels. In the following we consider various data analysis schemes that have been implemented to process the channels, groups of channels, or complete spectra. The analyses that are readily available to the NMR community in the form of software packages are emphasized. 4.2.1. Discrete samples Samples are characterized as discrete if they contain N monodisperse components and can be completely described by N pairs of amplitudes (concentrations) and decay constants. The decision to use Eq. (18) as the model function involves absolute prior knowledge (APK) and assumptions because the data set alone may not suffice to distinguish between a continuous distribution and a number of discrete components at the same chemical shift. Even the knowledge that a solution was made with N monodisperse solutes does not assure that N discrete hydrodynamic entities are present because aggregation and chemical exchange may be present. When there is doubt about the nature of a solution, an analysis method that can handle continuous distributions should be applied first. The analysis programs for discrete samples report a list of diffusion coefficients and amplitudes for each chemical shift channel. In principle diffusion spectra can be constructed from these lists with delta functions having appropriate amplitudes at the specified positions on the diffusion axis. However, this
prescription is not satisfactory in practice because analysis errors cause channel to channel fluctuations in the computed diffusion coefficients, even within a single absorption peak. This means that a projection of the diffusion spectra from all the channels onto a single axis will now show clusters of peaks for each diffusing species rather than a single line. A better procedure is to construct the DOSY spectrum with normalized Gaussians having center positions and intensities equal to the diffusion coefficients and amplitudes, respectively [9,83]. This gives a DOSY data set of the form: F D; n
Nl X
Aj nGj D
20
j1
where
" # D ⫺ Dj 2 1 Gj q exp ⫺ 2s2j 2ps2j
21
Here s j can be set equal to the standard deviation reported by the analysis program, a fixed value, or some combination of the two to take into account the local error and the estimated systematic error for the complete data set. The Gaussian component method is now widely accepted for the construction of 2D and 3D DOSY displays. 4.2.1.1. Levenberg–Marquardt The Levenberg– Marquardt (L–M) algorithm is the standard
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Fig. 16. DOSY display of the SPLMOD analysis of simulated data. All diffusion peaks are correctly positioned. Reproduced with permission [83].
non-linear least squares method [84,85]. L–M requires a data set with standard deviations, the model function to be fit, and initial guesses for the parameters. The standard deviations are seldom known, but they can be estimated; and the model function for discrete samples is known in principle. However, it is our experience with typical NMR data sets that fits to sums of two or more exponential components are unreliable unless the signal-to-noise ratios and the ratios of decay constants are large. A major problem is that estimates of the parameters must be supplied, and this is a very bad idea for DOSY analysis. Subjectivity in the analysis must be avoided as far as possible. We recommend that the L–M method be reserved for situations where the model function is a single exponential, and the data set itself can be used to provide estimates of the amplitude and decay factor. Also, it is important that the data be fitted to the exponential function directly rather than fitting the logarithm of the data to a straight line. The latter procedure incorrectly weights the standard deviations and even when corrected standard deviations are supplied appears to give inferior results. With special techniques, for example using highfields, high-resolution spectra can often be obtained
that lend themselves to single component analyses such as L–M. An example is provided by 13C INEPT-DOSY of a mixture of glucose, sucrose (Aldrich Chemical Company), and sodium dodecyl sulfate (SDS) (Aldrich Chemical Company) in D2O (500 mM each) [68]. The spectrum shown in Fig. 14 was obtained with a Bruker AC-250 spectrometer, the gradient driver was home-built [45], and the gradient probe was a modified Bruker dual probe. The pulse sequence shown in Fig. 11 was used with D 0.103 s, and the q-values ranged from 11 364 to 681 880 m ⫺1 in 15 steps. For analysis, the area of each peak was fitted to the single exponential decay in Eq. (13) by non-linear regression with the L–M method, and the 2D DOSY was displayed and plotted by Felix (Hare Research, Version 1.1). The 1D 13C INEPT spectrum at the top of Fig. 14 shows that most of the peaks are well resolved, and the 2D DOSY display permits the diffusion coefficients of all components to be easily read: glucose (3.60 × 10 ⫺10 m 2/s), sucrose: (2.92 × 10 ⫺10 m 2/s), and SDS: (7.78 × 10 ⫺11 m 2/s). Note that SDS is involved in rapid exchange between monomers and the dominant micelles. The line widths for the Gaussian components (diffusion dimension) were taken to be the fitting errors reported by the L–M routine.
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Fig. 17. DOSY display of the SPLMOD analysis of simulated data. The ‘‘cross-talk’’ artifact is illustrated. The correct (input) diffusion coefficients are indicated by dotted lines. Reproduced with permission [83].
There turns out to be one significant overlap (sucrose and SDS) in the 13C NMR spectrum, and this produces a stray peak in the L–M analysis. Multi-component programs such as DISCRETE and SPLMOD can resolve the diffusion peak as the diffusion coefficients of the components differ by a factor of 3.8, but at the cost of increased errors elsewhere. 4.2.1.2. DISCRETE The Fortran program DISCRETE [86,87] has been freely distributed for many years. DISCRETE makes use of data transforms to estimate the parameters and then proceeds with a non-linear least squares analysis. It, therefore, does not require initial guesses, and it provides best fit results for a range of N values. DISCRETE was, in fact, used in the first reported DOSY analysis to determine amplitudes and decay rates for every frequency point (channel) [9]. A drawback of this procedure is that the results obtained for different frequencies within a single NMR peak seldom agree exactly. An early example of DISCRETE analysis is the stack plot DOSY display in Fig. 15 for a sample containing 10.0 mM tetramethylammonium chloride
(TMA) with mixed micelles[4.00 mM SDS and 8.00 mM octaethylene glycol dodecylether (C12E8)]. The 1H spectra were obtained at 295 K with a Bruker AC-250 spectrometer and custom built probe and gradient driver. A LED pulse sequence (Fig. 5) was used with D 100.0 ms, t ⫺ d 0.500 ms, and Te 100.0 ms; and NMR spectra were collected with qvalues ranging from 144 to 8.19 × 10 3 cm ⫺1. The 2D data set was analyzed channel by channel using DISCRETE, and the widths of the Gaussian components were chosen to be the average of the s i 0 values for the complete data set.
4.2.1.3. SPLMOD SPLMOD [88,89], the successor to DISCRETE, is also freely distributed. SPLMOD also analyzes sums of exponentials without requiring initial guesses, but is not restricted to exponential components, and permits the analysis of Nd 1D data sets simultaneously. The user supplies the data sets and specifies the maximum number of components to be searched for in an analysis. The major advantage for DOSY applications is that all of the frequency points in an NMR peak or multiplet can be fit in parallel with the restriction
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Fig. 18. DOSY displays for a mixture containing D2O, TEA, and SDS micelles. (a) 2D spectrum generated directly from SPLMOD ‘‘best fit’’ parameters. (b) 2D spectrum generated from SPLMOD parameters after processing with rejection criteria (see text). Reproduced with permission [83].
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that all of the points must share the same small set of decay constants [83]. Successful SPLMOD analysis of PFG-NMR data requires decisions concerning the setup, the application of APK, and the screening of potential solutions [83]. An input program must be prepared to take a set of spectra processed by FELIX or some other NMR software package and to arrange the data according to user specifications in a form that can be read by SPLMOD. In the implementation of Morris and Johnson [83], the input program prompts the user for a threshold value and then divides the spectrum into regions containing the data columns to be analyzed. The user also specifies the form of the decay kernels and other restrictions. For DOSY, exponentials with positive amplitudes and decay constants are selected; and the maximum number of discrete components is specified to be three. This means that SPLMOD will automatically carry out analyses with one, two, and three components. The SPLMOD output consists of a set of decay constants, a set of amplitudes, and the associated standard deviations for each region of the spectrum. These results are repeated for one to three component fits and the ‘‘best fit’’ is determined. However, there may be reasons to select one of the other results. For example, experimental artifacts such as residual eddy currents and phase errors may distort the decays in ways that cannot be represented with exponential kernels. In such cases, fits with larger numbers of components tend to give smaller deviations in spite of the lack of any physical justification. A pragmatic operating procedure is to reject fits on the basis of a list of reasonable criteria. If the solution with Nl components is rejected for any reason, the solution with Nl ⫺ 1 components is evaluated, and so on down to one component. This procedure may lose some authentic peaks, but at least it does not introduce spurious results. On the basis of experience the following criteria were selected for the rejection filter [83]. 1. Diffusion coefficients must be feasible and experimentally accessible. 2. Standard errors in diffusion coefficients must be less than 30%. 3. Pairs of diffusion coefficients must differ by a factor of two or more.
Numerous simulated data sets with added noise have been analyzed with SPLMOD to investigate the range of applicability of this program. Fits to the simulated data sets were quite successful; but, of course, these have the advantage that the decay kernels are truly exponential. For example, a data set was synthesized for two isolated Lorentzian peaks and three overlapping Lorentzian peaks with a maximum S/N of 20 and half widths 5.0 Hz as shown at the top of Fig. 16. The set contained 50 spectra with q-values ranging from 556 to 2.78 × 10 4 cm ⫺1 and the decay constants were given by lj Dj D ⫺ d=3 with D 0.1 s and d 0.0 s. In the SPLMOD analysis the NMR channels were divided into three sets and a maximum of three components were allowed for each set. The result of the analysis is shown in Fig. 16 where the 2D spectrum is projected onto the left hand diffusion axis to show the complete diffusion spectrum. The diffusion peaks reproduce quite accurately the input values of the diffusion coefficients (dotted lines). Fig. 17 illustrates a problem encountered in the analysis of a synthesized data set that was not caught by the listed criteria. The simulation contains three Lorentzian peaks with a maximum S/N of 20 and half widths of 10.0 Hz, and the central peak has two distinguishable components. In the analysis 50 spectra were used and 85 frequency points were analyzed simultaneously with a maximum of four components. Unfortunately, the SPLMOD generated 2D-spectrum shows three diffusion components for the center peak instead of the correct two components, and the peaks for the valid components are displaced from the correct values (dotted lines). The extra peak at 515 Hz is a result of ‘‘cross-talk’’ with the peak at 475 Hz. This artifact is associated with the number and magnitude of decay constants in one analysis block and does not require overlapping peaks. We find that when a decay constant is required in some frequency channels it will be used elsewhere throughout the block to improve the fit. One strong lesson is that three components at the same chemical shift with decay constants differing by factors of less than ten always lead to considerable errors. Real data may differ from simulated data because experimental artifacts can affect the decay kernels. One consequence is that the decays of isolated single components may be better fitted by mixing in a small amount of a second or third component. The effect of
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Fig. 19. DOSY display processed by CONTIN for a mixture containing D2O, glucose, and methyl cellulose. Reproduced with permission [83].
the rejection filter in this situation is illustrated in Fig. 18 for the SPLMOD analysis of a mixture containing 40 mM tetraethylammonium chloride (TEA) and 20 mM SDS [83]. An LED experiment was performed with 40 q-values ranging from 167 to 7.79 × 10 3 cm ⫺1, d from 1 to 2 ms, D 100.0 ms, Te 50.0 ms, and t 2.50 ms. Each spectral region was then fitted with a maximum of two components (NNL 2). The DOSY display in Fig. 18(a) was computed directly from the SPLMOD ‘‘best fits’’ without the benefit of the post analysis rejection filter. The real overlap of TEA and SDS peaks at 1.11 ppm is properly handled, but the other major peaks all show errors in position and artificial companion peaks here enclosed by dotted ellipses. The amplitudes of the peaks and the shifts in positions are evident in the projected diffusion spectrum on the left hand side. The remedial effect of the rejection filter is clearly shown in Fig. 18(b). Here the spurious two-component fits were rejected in favor of single-component fits on the basis of standard deviations and peak
separations. Thus, all artifacts were eliminated and consistent diffusion spectra were obtained. From the spectrum the diffusion coefficients for HOD, TEA, and SDS were found to be 1.74 × 10 ⫺5, 4.54 × 10 ⫺6, and 8.84 × 10 ⫺7 cm 2 s ⫺1, respectively. The conclusion is that filtered-SPLMOD analysis of real data, where two components occasionally overlap, can give reasonable results. It is easy to imagine extensions of the rejection criteria to take into account data from other spectral regions and our knowledge of the features of NMR spectra of complex spin systems. For example, with complex molecules it is unlikely that a decay constant would appear at only one chemical shift. Ultimately, artificial intelligence based software may provide the best route to the ‘‘most likely’’ diffusion spectra. 4.2.2. Polydisperse samples Many systems of interest are characterized by continuous distributions of diffusion coefficients, e.g. polymers and vesicles. In such cases Eq. (18) describing the 2D FT-PFG-NMR data set must be
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replaced with: Z I q; nm R T1 ; T2 a D; nm exp⫺q2 D D ⫺ d1dD 22 where R T1 ; T2 accounts for nuclear spin relaxation during intervals of the pulse sequence and a(D) is the mass weighted distribution of tracer diffusion coefficients D. The Laplace spectrum in this case is the product R T1 ; T2 a D; nm rather than the desired distribution a(D). If the relaxation times are correlated with the diffusion coefficients, the separation of a(D) may not be possible. Fortunately, in high molecular weight polymers the relaxation rates often depend on segmental motion and are approximately independent of the molecular weight thus permitting R T1 ; T2 to be replaced with a constant [90,91]. An interesting situation arises with phospholipid vesicles where T2 values for the bilayer protons depend on tumbling rates of the vesicles and hence on their sizes. The result is that the a(D) distribution obtained for bilayer protons contains an excessive contribution from the smaller vesicles that have smaller line widths. This produces a larger apparent diffusion coefficient for the bilayer protons than is measured for protons entrapped in the aqueous cavities of the vesicles [70]. In the following we assume that the relaxation factor is constant and can be removed from the integral. As analysis of polydisperse components produces distributions of diffusion coefficients rather than peaks and positions, the plotting of diffusion spectra is more straightforward. However, two points must be considered. First, the programs usually give a small number of points that barely suffice to define spectra when more than one peak is present, especially if one is narrow. Additional points can be generated by interpolation but the line shapes are likely to be distorted. The second, less obvious point, has to do with apparent peak heights and areas in plots of intensities versus D or ln(D). There is really no problem when spectra are plotted with a linear D axis as the peak areas are given by Z a DdD 23 as expected. Confusion arises when the dispersion axis is still based on D but logarithmic spacing is
used because the apparent peak widths have very different meanings in different parts of the D axis. If the diffusion axis is relabeled to be linear in the logarithm of D, e.g. to show ⫺ 4, ⫺ 5, ⫺ 6 in place of 10 ⫺4, 10 ⫺5, 10 ⫺6, it is implied that the integration is linear in ln(D). Therefore, the integration in Eq. (23) must be replaced with Z Z Da DdD=D Da Dd ln D 24 and it makes sense to plot Da(D) rather than a(D) versus ln(D) or log(D). 4.2.2.1. CONTIN with extensions The first program used for DOSY analysis of polydisperse data was CONTIN [83]. This program has been available since the early 1980s and has been extensively tested [80,92,93]. CONTIN uses constrained regularization to fit experimental data, yk and tk, to functions of the form yk
Ng X m1
cm f lm ; tk a lm ⫹
NL X
bi Lki tk
25
i1
where the cm are weights of the quadrature formula, f l; tk exp ⫺ltk are the known decay kernels, Ng is the number of grid points, and a(l ) is determined by the analysis. The second term in Eq. (25) permits a constant background to be included, e.g. NL 1 and Lk1 1 gives the constant background b 1. The problem with solving Eq. (25) for a(l ) is that an infinite number of oscillatory solutions for a(l ), that have no physical meaning, are consistent with the data set yk. CONTIN attempts to eliminate oscillatory solutions through the use of constraints based on (a) APK, (b) statistical prior knowledge, and (c) parsimony [94,95]. What this means to the user is that non-negativity is enforced for a(l ) and that, of the solutions not eliminated by (a) and (b), the simplest one is chosen. In this context the simplest solution is the one with the least detail, i.e. the smoothest one with the minimum number of peaks. Smoothness is selected by penalizing solutions on the basis of the integrated squared second derivatives, and the extent of the penalty depends on the regularization parameter chosen. It is straightforward to use CONTIN in DOSY applications because the user only has to supply a
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Fig. 20. DOSY display of a mixture of SDS, ATP, and glucose, all at 0.1 M in D2O. The diffusion axis is in units of mm 2 s ⫺1. (a) The DOSY spectrum, (b) central region of the same spectrum. Reproduced with permission [98].
threshold value, and the program output provides the Laplace spectrum a(l ) essentially ready for display. The major complaint is that the essential smoothing feature broadens all of the peaks so that even monodisperse components show considerable linewidths. For many applications this is acceptable as broader distributions are usually well represented, and in samples having monodisperse and polydisperse components the monodisperse components are usually distinguishable [96].
An example of the CONTIN analysis for a mixture containing both monodisperse and polydisperse components is shown in Fig. 19. Data was obtained with an LED sequence for 75.0 mM glucose and 1.00 wt.% methyl cellulose in D2O at 22⬚C. The 26 q-values ranged from 278 to 2.50 × 10 4 cm ⫺1, with d 1 to 5 ms, D 350.0 ms, Te 50.0 ms, and t 5.80 ms. CONTIN analysis produced 50 point diffusion coefficient distributions for each of the 500 channels in the 2D data set, and an interpolation routine
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was used to generate 512 data points from each 50 point distribution. No user input other than a threshold value was required. This experiment permitted the glucose and methyl cellulose peaks to be easily distinguished and the distribution of diffusion coefficients for the polymer to be measured; but it also demonstrated the unnatural broadening and distortion of the glucose peak and the misleading area ratios arising from logarithmic spacing on the D axis. As expected, the treatment of the monodisperse component is marginal because smoothing by the regularization procedure coupled with the small number of points produces strange peak shapes. We note that the HOD peak in this spectrum was deleted when the DOSY display was constructed. For broad distributions in the presence of noise CONTIN tends to give smaller average diffusion coefficients 具D典 and reduced standard deviations 具SD典/具D典 relative to the true values. These errors result from the tendency of CONTIN to oversmooth a(D) at the small D end while undersmoothing for large D values [93]. The effect, which is clearly evident in distributions that have a small secondary peak on the high D side, arises because the penalty for oscillatory solutions is based on a logarithmic axis. The identity a D dD a DDd ln D suggests that a(D)D rather than a(D) should be analyzed on the logarithmic axis. This change is easily accomplished in CONTIN by setting ‘‘integration off’’ with the control parameter selection IQUAD 1 so that cm 1. With this change most of the ‘‘noise’’ peak disappears, but in order to obtain the greatest accuracy in the computed a(D) function a more sophisticated adjustment of the penalty weighting factor is required [97] (see Section 6.2.4).
4.2.2.2. MaxEnt A DOSY processing module based on the maximum entropy method (MaxEnt) is now included in the GIFA software package for NMR analysis [98,99]. In MaxEnt the idea is to determine the ‘‘most probable’’ Laplace spectrum by maximizing the entropy of the spectral distribution subject to certain constraints. Accordingly, the Laplace spectrum a(l i) in Eq. (19) is associated with the most probable distribution Pi, and the
entropy of the distribution is defined by S⫺
n X
Pi =Fln Pi =F
26
i1
where F is a normalization constant. The normalization of the Pi and the relationship between Pi and the experimental data set Ii specified in Eq. (19) serve as constraints on the allowed Pi values. In MaxEnt the function Q S ⫺ lx2 is maximized for various values of l where x 2 represents the Euclidean distance between the data and the Laplace transform of Pi. MaxEnt is a general purpose analysis method that competes with CONTIN for polydisperse samples and does a reasonable job for discrete samples as well. In a comparison, Levenberg-Marquardt (L–M), CONTIN, and MaxEnt were applied to 100 synthetic data sets containing four exponential components with decay constants ranging from 0.1 to 13.0 and amplitudes between 1000 and 8000 [98]. It was found that L–M was much less accurate than either CONTIN or MaxEnt and that MaxEnt was better than CONTIN at dealing with weak components. Also, CONTIN frequently failed to find the correct number of components because of over-smoothing. MaxEnt analysis for DOSY is illustrated in Fig. 20 for a mixture containing SDS, adenosine 5 0 -triphosphate, and glucose all at 0.1 M in D2O at 35⬚C [98]. A BPP-LED experiment was performed with d 3 ms, D 100 ms, and 28 geometrically spaced gradient values ranging from 1 to 44 G/cm. The Laplace inversion was calculated by MaxEnt on 100 points with a maximum of 1000 iterations, a job requiring 50 min on an R5000 SGI workstation. The only user input was to select 50:1 as the S/N threshold for the maximum signal in each channel. The zoomed insert in Fig. 20(b) shows impressive resolution for such a robust scheme. 4.2.3. Complete bandshape methods PFG-NMR has a major advantage over scattering methods, e.g. dynamic light scattering, because of the additional information obtained from chemical shifts. The problem of resolving exponential components can often be avoided by finding isolated NMR peaks for individual species. The most desirable case is, of course, to have sufficient resolution to avoid any
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overlap. But in the real world overlap often occurs, and it is desirable to have analysis methods that can resolve components with arbitrary amounts of overlap. The methods reviewed to this point have involved analysis at a single chemical shift or a limited range of chemical shifts and have not made use of the total information available. In this section we consider methods that can analyze the complete bandshape. 4.2.3.1. Multivariate analysis Kubista has proposed a method for the analysis of correlated data sets [100]. In particular he imagined two types of spectra being obtained for a mixture with varying concentrations, and he showed that exact solutions for the ‘pure’ spectra of the components could be obtained. Schulze and Stilbs implemented the Kubista scheme (NIPALS) by designing a type of double PFG-NMR experiment with different gradient amplitudes in the two parts [101]. Unfortunately, the two data sets were compromised by this design and the resulting spectra were distorted. The idea was revived by Antalek and Windig who pointed out that, because the decays in PFG-NMR are exponential, the required pair of data sets can be obtained from a single experiment, e.g. spectra 1–10 and spectra 2–11 provide two satisfactory sets [102,103]. Consider experiments giving c spectra having v points each for a mixture with n components. The correlated data sets are represented by: A CP;
B CbP
27
where A and B are data matrices of size cv, C(cn) is the concentration matrix, P(nv) is the matrix of pure spectra, and b(nn) is a diagonal matrix. (Note that the spectra and the concentrations are completely correlated in the two data sets but that there is a scaling factor defined by b.) The Kubista scheme, reformulated in terms of the generalized rank annihilation method (GRAM) [104,105], has been applied to FT-PFG-NMR data sets for mixtures with considerable success. This method, now known as the Direct Exponential Curve Resolution Algorithm (DECRA), is direct, does not require a threshold, can deal with any amount of overlap, is not very sensitive to noise, and further, is fast. The requirement of exponential decays may be a limitation in some cases, and tests with mixtures containing large numbers of components and both
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monodisperse and polydisperse components are necessary; but DECRA appears to be well suited for generating data for DOSY displays. The software routines for DECRA, developed for the MatLab environment, are available for independent testing [103]. Recently, an analysis method known as multivariate curve resolution (MCR), another descendent of NIPALS, has been reported for DOSY and GPCNMR experiments [106]. MCR does not require a priori knowledge except that the spectral intensities and diffusion decay profiles must both be non-negative. In particular, the decay profiles are not restricted to exponentials. This method is, therefore, more generally applicable than GRAM; however, if exponentials are known to be present, it is expected that better answers can be found by making use of that knowledge. Impressive analyses of DOSY data sets by MCR have been reported, and this method deserves much wider use. The implementation involves principal value analysis (PVA) [107] to obtain eigenvectors and eigenvalues of a data matrix followed by conversion of the abstract vectors into chemical factors. The generation of ‘‘pure’’ spectra and concentration profiles is accomplished by a Varimax rotation (VMAX) [107–109] followed by alternating leastsquares optimization (ALS) [110]. Software for the MCR analysis was written in GRAMS/386 (Galactic Industries). 4.2.3.2. Component resolved NMR A global leastsquares analysis method labeled ComponentResolved FT-PGSE NMR spectroscopy (CORE) has been reported by Stilbs et al. [111]. Their procedure is to perform a total fit of the 2D raw PFG-NMR data set. This has been implemented in a FORTRAN program that includes two copies of a direct-search minimization routine (STEP). One global (higherlevel) minimization optimizes the diffusion coefficients and a second (lower-level) minimization fits the component amplitudes in each frequency channel using the higher-level results. CORE is based on modeling each frequency channel with a number of discrete exponentials (1–5), or if it is decided that a channel has contributions from polydisperse components, a sum of Kohlrauch– Williams–Watts (KWW) distributions (1–2) in addition to discrete exponentials (1–3). The operation of
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this program implies considerable knowledge of the mixture by the user and interactive processing. CORE has been applied to simulated data and to various test samples, and it is reported that global minimization ‘‘may increase the signal/noise ratio by a factor or more than 10.’’ The global analysis is computer intensive, but with the current rate of increase in affordable computing power this should not be a problem. The more serious cost is user time and effort in preparing the input model. Unfortunately, the CORE software is not generally available.
4.2.4. Analysis recommendations The philosophy of DOSY from the beginning has been to obtain the most likely distribution of diffusion coefficients at each chemical shift using any available methods or software. As unique solutions cannot be guaranteed when signals from different species overlap, it is desirable to make use of a variety of analysis methods with complementary strengths and weaknesses. This is easy to accomplish with available software and computer workstations. A reasonable analysis suite might contain the following: 1. single exponential analysis (Levenberg– Marquardt); 2. biexponential fitting routine with rejection rules (perhaps SPLMOD); 3. continuous distribution analysis (CONTIN or similar program). However, it is essential that the user of such programs be well informed about artifacts and potential pitfalls. The fact that a spectrum a(D) is computed does not by itself prove that it has any physical validity. The recently introduced MaxEnt package appears to be a good addition to the aforementioned list. It can be used to screen samples and may be able to compete with CONTIN in the analysis of broad distributions. Choices 1–3 plus MaxEnt will serve very well for high-resolution spectra of multi-component mixtures of discrete, low molecular compounds. In contrast, when there is severe overlap and a relatively small number of components, the multivariate analysis methods DECRA and MCR may be good alternatives. Also, brute force analysis by totally fitting 2D data sets with user defined models such as CORE can be useful in special cases.
5. Effects of chemical exchange Chemical rate processes have been studied by NMR methods for decades [112]. In the standard model for chemical exchange, nuclei or groups of nuclei explore a number of sites by means of a Markovian random process. These sites are characterized by spin Hamiltonians that may be associated with a chemical environment, e.g. a basic group that can accept a proton, or a molecular configuration, e.g. cis and trans isomers. Slow exchange is defined as the situation in which spectra, characteristic of the individual sites, can be observed; and fast exchange implies the observation of some kind of average spectrum. The path from slow exchange to fast exchange consists of line broadening, coalescence, and motional narrowing as the mean lifetimes for occupation of the sites decrease. Time is always the variable, and the exchange rates, i.e. inverse lifetimes, are manipulated by changes in temperature, or perhaps concentrations for intermolecular reactions. In diffusion NMR new possibilities arise because of the importance of sites that differ in hydrodynamic properties with or without differences in spin Hamiltonians. Also, in PFG-NMR for DOSY applications, the variable is q 2 rather than time, and there is nothing analogous to lifetime broadening in the diffusion dimension [113]. Time does enter through the storage delay T, and the variation of T sometimes permits the observation of slow and fast exchange limits without changing the physical condition of the sample. 5.1. Exchange effects in diffusion spectra Here we consider the effects of chemical exchange on longitudinal magnetization during the diffusion sensitive interval T in STE type experiments (Fig. 2) with t; d p T so that T ⬇ D. Taking into account diffusion, flow, and chemical exchange, the rate of change of the nuclear magnetization Mn(r,T) for the nth species with spatial coordinates r is given by [1,27]: dMn r; T 1 ⫺ M r; T ⫺ 7·Jn r; T dT T1n n X Knm Mm r; T ⫹
28
m
where the rate constants are kn ⫺Knn and
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two-site exchange, and for this case analytical solutions can be obtained. Thus Eq. (30) reduces to dM K; T LM K; T dT where L
LAA
LAB
LBA
LBB
!
32
⫺kA ⫺ RA
kB
kA
⫺kB ⫺ RB
! : 33
Fig. 21. Echo amplitude versus q 2T for two-site exchange calculated with Eq. (35) with DA 2.0, DB 0.1, kA 10, and MA 0.40.
The matrix elements [exp(LT)]AB are available in the literature [1], the required combinations, e.g. AA ⫹ AB for MA, for 1D NMR spectra are: MA0 dMA0 ⫺ kB MB0 ⫺ exp ⫺s ⫹ DT MA 2D 2 MA0 dMA0 ⫺ kB MB0 ⫹ exp ⫺s ⫹ 2 2D ⫺ DT;
knm Knm , and the flux Jn r; T is defined by: Jn r; T ⫺Dn 7Mn r; T ⫹ Mn r; Tvn r; T:
29
MB
In Eq. (29) Dn and vn r; T are the diffusion coefficient and velocity for the nth species, respectively. The calculation of the STE amplitude is simplified by taking the spatial Fourier transform of Eq. (28) to obtain the linear algebraic equations: X dMn q; T iVn ⫺ Rn Mn q; T ⫹ Knm Mm q; T dT m 30 Z
exp ⫺iq·rMn r; Tdr:
31
In Eq. (30) Rn 1=T1n ⫹ Dn q2 describes relaxation and Vn qvn is the frequency resulting from uniform flow in the z-direction. These equations and their solutions have been discussed by Ka¨rger, et al. [15], and similar equations for nuclear magnetic relaxation in multiple phase systems were previously treated by Zimmerman and Brittin [114]. Here we solve Eq. (30), neglecting 1/T1n and Vn, for the Fourier components Mn(q,T) that determine the amplitude of the STE. Real applications often involve
MB0 dMB0 ⫺ kA MA0 ⫹ exp ⫺s ⫹ DT 2D 2 MB0 dMB0 ⫺ kA MA0 ⫺ exp ⫺s ⫹ 2 2D ⫺ DT;
34b
where MA0 MA(q,0), MB0 MB(q,0), and the symbols s , d , and D are defined by:
s
1 k ⫹ kB ⫹ DA q2 ⫹ DB q2 2 A
d
1 k ⫺ kB ⫹ DA q2 ⫺ DB q2 2 A
where Mn q; T
34a
q D d2 ⫹ kA kB (The functions d and D should be distinguished from previously defined time intervals that are associated with the same symbols.) If the chemical shift difference between the sites is much larger than the exchange rate, then Eqs. (34a) and (34b) describe the dependence of the individual peak areas on exchange rates and diffusion coefficients. We consider the special case in which the
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Fig. 22. Diffusion spectra for two-site exchange with equal populations (pA 0.5). The effective rates are indicated by values of N 0:5 kA ⫹ kB T. Reproduced with permission [113].
chemical shift difference is zero or is much less than the exchange rate so that a single line is observed in the NMR spectrum. In this situation the diffusion spectrum is calculated with the sum MA q; T MA q; T ⫹ MB q; T: M0 L ⫹ exp ⫺s ⫹ DT M 2D 2 M0 L ⫺ 35 exp ⫺s ⫺ DT ⫹ 2 2D where M0 MA0 ⫹ MB0 and
L dMB0 ⫺ MA0 ⫹ MA0 kA ⫹ MB0 kB : It is should be noted that the right side of Eq. (35) is not a simple sum of exponentials because the variable q appears in the coefficients as well as the exponents. In this simple example the probability of occupation of the nth (n A,B) site is pn and the mean lifetime of a spin in the nth site is tn 1=kn . Also, k kA =pB kB =pA 1=t, pA tA = tA ⫹ tB , and for convenience we let Mn0 pn . Another important parameter is N kT/2 that can be interpreted as the mean number of times that a spin changes sites during the interval T. Also, the storage time can be written as T tA ⫹ tB where tA and tB represent the total amounts of time that a spin occupies sites A and B, respectively. It should be clear that tA and tB are related to the equilibrium constant Keq PB =PA tB =tA but
Fig. 23. Stack plot of 1H spectra of Vancomycin and DDFA mixture as t increases: (a) LED pulse sequence (t increases from 0.3 to 6.2 ms) and (b) BPP-LED pulse sequence (t increases from 0.4 to 6.4 ms). T 100 ms, g 5.8 G/cm, and Te 5 ms in both experiments. Reproduced with permission [33].
give no information about the magnitude of the exchange rate or the magnitudes of the mean lifetimes. In attempting to understand the two-site exchange problem, it is useful to plot ln(M) versus q 2T for a range of T values as shown in Fig. 21. The slow exchange limit is captured in the upper most curve where T ⬍ 2/k and N ⬍ 1. It is essentially the sum of two exponentials, one associated with DA and the other with DB. The slope of this curve in the limit of large q 2T values is Dslow, which we take to be DB, and the initial slope is given by Dav pADA ⫹ pBDB. If the S/N ratio is high, it is possible to estimate the pA and pB values as follows. The intercept, i.e. the q 2T 0 value, of the line with slope Dav is taken to be ln(MA ⫹ MB) while the intercept of the slow exchange line extrapolated from large q 2T values is equal to ln(MA). It is also possible to use the latter intercept for a set of curves near the slow exchange limit to estimate the exchange rate as intercept constant ⫺ kBT. The lowest curve, actually a straight line with slope Dav, represents the fast exchange limit obtained by setting T ⬎ 20/k so that N ⬎ 10. A DOSY experiment on this two site system would, under favorable conditions, yield a diffusion spectrum with two peaks in the slow exchange limit. Similarly, we would expect the fast exchange limit to give a diffusion spectrum with a single peak. In the
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Fig. 24. DOSY display for protons on alpha carbons in a mixture containing alanine (A), glutamine (Q), and lysine (K) in D2O. Reproduced with permission [53].
intermediate exchange regime DOSY analysis is likely to fail or to indicate extreme polydispersity. However, the exact diffusion spectra can be derived from Eq. (35) either by directly performing the ILT to obtain analytical expressions [115] or by substituting iv for q 2 and performing the inverse Fourier transformation of the resulting expressions with respect to v to obtain the diffusion spectra [113]. A two-site spectrum computed with pA pB 1/2 is shown in Fig. 22 for N values ranging from 2 to 50 [113]. For N ⬍ 1, delta functions appear at DA and DB with intensities proportional to exp(⫺kAT) and exp(⫺kBT), respectively. The line widths for the sharp peaks in Fig. 22 result from the numerical computation and are not meaningful. For N ⱖ 5, intensity with a Gaussian profile grows at the position of the average diffusion coefficient (DA ⫹ DB)/2, and as N continues to increase this peak narrows while maintaining a standard deviation that is proportional to N ⫺1/2. The calculation for pA 苷 pB is similar except that for large values of N the intensity is centered at pADA ⫹ pBDB. 5.2. Artifacts from chemical shift encoding It was previously mentioned that chemical shift encoding can seriously affect echo amplitudes in FT-STE and FT-LED experiments when chemical exchange is present [33]. To see how this comes
about consider an STE experiment (Fig. 2) for spins with the frequency offset v A. After the first 90⬚ RF pulse, the magnetization precesses with the frequency v A plus a contribution from the field gradient depending on the displacement in the z-direction. The second 90⬚ RF pulse stores the y-components of magnetization in the z-direction and the remaining x and y components are eliminated by phase cycling or homospoil pulses. For spins in the layer at z to z ⫹ dz the stored magnetization is described by cos vA t ⫹ qz where q g gd . The third 90⬚ RF pulse returns magnetization to the yz-plane and precession reduces the y-component by an additional factor of cos vA t ⫹ qz at the echo. For a sample of length L the amplitude of the peak at v A in the absence of relaxation is given by 1 ZL=2 cos2 vA t ⫹ qzdz L ⫺ L=2
36
1 1 ⫹ cos 2vA tsinc Lq=p: 2 2 The last factor on the right hand side, sinc(Lq/p ) sin(Lq)/Lq, is unity when the gradient vanishes thus permitting the amplitude of the echo to depend on the frequency offset v A. As previously noted, the q 0 point should be avoided in diffusion measurements. However, as soon as Lq exceeds 2p , i.e. the pitch of the magnetization
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Fig. 25. 500 MHz DOSY display of a perchloric acid extract of gerbil brain in D2O. Selected assignments are: ac acetate: ala alanine: cho choline: cr creatine: cre creatinine: etn ethanolamine: GABA g -aminobutyric acid: glu glutamine: GPC glycerophosphocholine: lac lactate: m-ino myo-inositol: NAA N-acetylasparate: succ succinate: and tau taurine. Reproduced with permission [118].
helix becomes less than the sample length, the modulation term becomes very small and vanishes completely with significant gradients. When chemical exchange occurs, intensity modulation reappears for the exchanging groups even in the presence of gradients. The previous discussion is easily extended to two-site chemical exchange. Suppose that the sites have different chemical shifts and that exchange can be neglected during the encode and decode intervals. During the diffusion delay T, exchanges do occur so that the spins in site A at the time of the third 90⬚ RF pulse may have occupied either site during the encode interval. In terms of the offset frequencies (v A, v B) and the mole fractions (xA, xB), the peak
amplitude at v A is then given by 1 ZL=2 x cos vA t ⫹ qz L ⫺L=2 A ⫹xB cos vB t ⫹ qzcos vA t ⫹ qzdz
37
1 x ⫹ xB cos vA t ⫺ vB t: 2 A
The modulation predicted by Eq. (37) has been observed in affinity NMR experiments where the pulse duration d was set equal to encode interval t , and spectra were obtained as a function of t . An example (Fig. 23) is provided by a mixture of Vancomycin (Sigma) and the tetrapeptide
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Fig. 26. (a) DOSY display processed with SPLMOD for a mixture containing 5.0 mM methanol, 10.0 mM iso-propanol, 10.0 mM t-butanol, and 10.0 mM neo-pentanol in D2O, and (b) DOSY display for a mixture with the same solute concentrations plus 0.150 M DTAB. Reproduced with permission [61].
Asp–Asp–Phe–Ala (DDFA) in D2O [33]. The signals for the free DDFA (1.30 ppm) and the bound DDFA (0.55 ppm) that are involved in exchange decay with increasing t because of diffusion and transverse spin relaxation, but also oscillate at 373 Hz, their chemical shift difference in the 500 MHz spectrometer. The other (non-exchanging) peaks show no oscillation. Intensity oscillations can be useful in identifying exchanging pairs and editing spectra, but can also be deceptive if not properly identified. Fortunately, the oscillations can be completely avoided by using BPP based sequences that refocus chemical shifts.
6. Applications of 1D and 2D DOSY 6.1. Discrete samples Most DOSY applications to date have concerned discrete mixtures. Selected examples are presented in this section. The reader is referred to the literature for additional DOSY examples including studies of chloroaluminate melts [116] and perturbed cis-trans isomerization of phenylalanylproline [117]. 6.1.1. Biofluids Biofluids here refer to multicomponent mixtures of amino acids and other molecules commonly found in
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Fig. 27. DOSY display for a solution containing 2 g/dl BSA, 2 g/dl SDS, and 0.01 M b -mercaptoethanol in phosphate buffer (pH 7.2, ionic strength 0.020 M), The unlabeled line represents the reaction product, HOCH2CH2SSCH2CH2OH. Reproduced with permission [145].
tissue extracts. These examples emphasize the optimization of resolution in the chemical shift dimension to minimize overlap and the incorporation of experimental design features to maximize the accuracy of the extracted diffusion coefficients to enhance resolution in the diffusion dimension. A mixture of alanine (A), glutamine (Q), and lysine (K) in D2O provides a good illustration of the necessity of high resolution in the DOSY experiment as the diffusion coefficients are much too similar to be separated when there is overlap. The display in Fig. 24 shows part of the 2D spectrum (protons on a -carbons) for this mixture [53]. As previously described, the spectrum was constructed with Gaussian components centered on the computed values of the diffusion coefficients and having widths equal to the errors reported in the data analysis. Sample spinning with the ‘‘stop-and-go’’ system and the BBP sequence were used to achieve high resolution (Dn ⬍ 0.5 Hz), and good line shapes in the chemical shift dimension were obtained without special filtering of the FIDs. With no overlap and S/N ratios greater than 100, data analysis with SPLMOD provided diffusion coefficients with errors of a few percent, but even better results would be expected from an L–M analysis limited to single components. For this work a Bruker AC250 spectrometer with a custom built 10 mm diffusion probe was used. The other experimental parameters were:d /2 1 ms, t
1.5 ms, 27 gradient amplitudes (g ^ 0.03 to ^ 0.62 T m ⫺1) and Te 20 ms. Impressive examples of mixture analysis have also been reported under the label high-resolution DOSY (HR-DOSY) [54,64,65,118]. An example is shown in Fig. 25 for a perchloric acid extract of gerbil brain [118]. Fifteen spectra were acquired with the LED sequence on a 500 MHz spectrometer using 512 transients per spectrum. Post processing with reference deconvolution (FIDDLE) was based on the reference line for DSS (sodium 3-trimethylsilypropanesulfonate), and the DOSY display was constructed from measurements on 116 peaks. The contour lengths (errors) in the diffusion dimension are somewhat larger than in test cases because of poorer S/N ratios in the 1D NMR spectra. The assignments of a number of metabolites are shown in Fig. 25, and as expected the diffusion coefficients are well correlated with molecular sizes. 6.1.2. Separation by means of hydrophobicity FT-PFG-NMR has proved a powerful tool for the study of solubilization [119]. DOSY based on highresolution 1D NMR spectra extends this work and provides ‘‘at-a-glance’’ recognition of the components in complex mixtures and their interactions. In a number of such cases complete resolution of peaks has been obtained even at 250 MHz through the use of the ‘‘stop-and-go’’ spinner. The example shown in
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Fig. 28. 1H DOSY display for a mixture containing ‘‘Tinuvin P’’ ( ⫹ ), ‘‘Irganox 1330’’ (*), and ‘‘Irganox 1098’’ (W). The solvent is labeled S and silicon grease contaminant is labeled Sg. Reproduced with permission [146].
Fig. 26(a) for a mixture of 5.0 mM methanol, 10.0 mM iso-propanol, 10.0 mM t-butanol, and 10.0 mM neo-pentanol in D2O does not require especially high chemical shift resolution, but does suffer from accidental overlap in the diffusion dimension because of the near equality of the diffusion coefficients of t-butanol and neo-pentanol [61]. Fig. 26(b) shows that the effective diffusion coefficients can be manipulated by adding the micelle forming surfactant, dodecyl trimethylammonium bromide (DTAB). This is an example of fast chemical exchange where solute molecules partition into the micelles according to their solubilities but undergo exchange between the interior of the micelle and the bulk solution. In the fast exchange limit DOSY only reports the time average diffusion coefficient Dj for the jth solute ⫹ 1 ⫺ pj Dfree Dj pj Dmic j j
38
where pj is the degree of solubilization, and Djmic and Djfree are the tracer diffusion coefficients for solubilized and free molecules, respectively. It should be recognized that Djmic is equal to the diffusion coefficient of the micelle as the average displacement of a micelle during the diffusion time D is much greater than the radius of the micelle. The experiment was performed at 250 MHz with the LED sequence. Twenty-eight spectra were acquired with q-values ranging from 208 to 8.34 × 10 3 cm ⫺1, d 1 ms, t 2.0 ms, D 100 ms, and Te 50 ms. SPLMOD (N 2) was used for the data analysis. Complete resolution of the diffusion peaks was obtained by addition of 0.150 M DTAB, and the values of pj shown for the solutes in Fig. 26(b) were calculated with Eq. (38) using Djfree values from the DTAB free solution (Fig. 26) and all Djmic values set equal to the diffusion coefficient measured for DTAB.
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Fig. 29. Structures for polymer additives. Reproduced with permission [146].
6.1.3. Equilibria involving sodium dodecylsulfate (SDS) and bovine serum albumin (BSA) SDS–protein interactions are important in a number of fields. For example protein/SDS-polyacrylamide gel electrophoresis (PAGE) is routinely used to determine protein molecular weights and to separate proteins. Also, SDS is known to be a potent protein denaturant. However, the protein–SDS complexes and their electrophoretic mobilities are not completely understood, and this has provided motivation for physical studies. The aim of a DOSY study of BSASDS equilibria was to determine the binding isotherm, i.e. the fraction of surfactant bound to a protein versus the surfactant concentration. The DOSY display for a D2O solution containing
2 g/dl BSA, 2 g/dl SDS, and 0.01 M b -mercaptoethanol in phosphate buffer (pH 7.2, ionic strength 0.020 M) at 298 ^ 1 K is shown in Fig. 27. This spectrum was obtained from 20 to 40 FIDs in an LED experiment with q-values ranging from 2.1 × 10 4 to 1.9 × 10 6 m ⫺1 and D values typically between 50 and 80 ms. The diffusion dimension reveals BSA signals from the slowly diffusing BSA–SDS complex, a single SDS peak, and weak peaks from the denaturing compound and the unlabeled reaction product, HOCH2CH2SSCH2CH2OH. The single SDS peak indicates the fast exchange limit for the SDS exchange among monomers, micelles, and the BSA–SDS complex. This limit reduces the amount of information available and requires that the analysis begin with
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Fig. 30. DOSY display for medium unilamellar vesicles in a mixture containing 30 mM total lipid (POPC) with 100 mM sucrose in D2O. Reproduced with permission [70].
an expression for the observed average diffusion coefficient 具D典 for SDS: 具D典 Ppro Dpro ⫹ Pmon Dmon ⫹ Pmic Dmic
39
with Ppro ⫹ Pmon ⫹ Pmic 1
40
where Ppro, Pmon, and Pmic are the fractions of SDS
molecules on the protein, as monomers, and in micelles, respectively; and Dpro, Dmon, and Dmic are the associated self-diffusion coefficients. The calculation of the desired quantity Ppro requires the determination of all the diffusion coefficients and either Pmon or Pmic. Both 具D典 and Dpro can be read directly from Fig. 27, but Dmon and Dmic must be determined from additional experiments and the use of the mass action
Fig. 31. The 1H DOSY display (250 MHz) for whole plasma at 40⬚C. The reference is TMA at 3.22 ppm [126].
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Fig. 32. DOSY display processed with CONTIN for a mixture containing CTAB, PVME, and NaSal (see text). Reproduced with permission [96].
model [120]. The conclusion of this study was a plot of the binding ratio [bound SDS]/(SDS) versus [total SDS]/[BSA] in the range 0–6 (g/g) showing a saturation binding of 1.9 g bound SDS per gram BSA that is reached at about 4 g total SDS per gram of protein. Also the effective hydrodynamic radius of the complex was determined as a function of the number of bound SDS molecules. 6.1.4. Mixtures of polymer additives DOSY has been used to analyze mixtures of polymer additives [146]. The diffusion coefficients facilitated spectral assignments and were especially helpful for isolated protons where no J-couplings could be resolved. 2D-DOSY of a mixture containing ‘‘Tinuvin P’’ (Mn 225) ‘‘Irganox 1098’’ (Mn 637), and ‘‘Irganox 1330’’ (Mn 775) all from Ciba in 1,1,2,2tetrachloroethane-d2 (TCE-d2) is shown in Fig. 28, and the associated molecular structures are shown in Fig. 29. The data were acquired with BPP-LED on a Bruker AM360 spectrometer equipped with a 5 mm z-gradient Bruker inverse probe. The gradient pulse duration was 2 ms, the diffusion delay was 0.5 or 1.0 s, and Te was 15 ms. Typically, 15–20 gradient values were used and the values ranged from 0.026 to 0.30 T m ⫺1. The diffusion information in Fig. 28 correlates with the structures of the additives. ‘‘Irganox 1330’’ is star
shaped while ‘‘Irganox 1098’’ has a more extended linear shape. The shapes compensate for the differences in molecular weights and produce quite similar diffusion coefficients. Also, it is observed that the exchangeable OH protons (4.96, 5.08, and 11.15 ppm) have larger diffusion coefficients than the other protons in their respective molecules. Apparently the protons are exchanging with protons of water in the solvent. Note also the well resolved resonances for the solvent and silicon grease in the diffusion dimension.
6.2. Polydisperse samples DOSY is especially useful for samples containing polydisperse components. With widely available software, diffusion spectra can be obtained that permit components with broad distributions of diffusion coefficients to be distinguished from those with narrow distributions, i.e. the monodisperse components. Also, with no additional assumptions beyond smoothness, the shape of distributions of diffusion coefficients can be measured. In this section DOSY examples are displayed that have been processed by CONTIN. Similar results can be obtained with other analysis programs. In particular we note that a recent DOSY study of polymer mixtures successfully used
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both CONTIN and the analysis program NLREG [121]. 6.2.1. Phospholipid vesicles The first DOSY paper showed a diffusion resolved spectrum of a sample containing sucrose and phospholipid vesicles in D2O [9]. Two sucrose spectra were evident in the 2D display at diffusion coefficients of 3.00 × 10 ⫺6 and 1.86 × 10 ⫺8 cm 2 s ⫺1, and it was reasonable to assign the spectra to free sucrose and sucrose entrapped in vesicles, respectively. Data analysis in this case was performed with DISCRETE and, consequently, information concerning polydispersity of the vesicles was lost. A much more extensive CONTIN based analysis was performed on palmitoyl-2-oleoyl phosphatidylcholine (POPC) vesicles with diameters ranging from 30 to 100 nm [70]. A DOSY display for sucrose with medium POPC vesicles is shown in Fig. 30. At this level of gain the signals from the vesicle bilayer are almost invisible, the exception being the small peak at 3.1 ppm. The bilayer proton signals are more pronounced for smaller vesicles and in fact dominate the spectrum for 30 nm vesicles. From the DOSY peaks it is evident that the effective diffusion coefficients of the bilayer protons are larger that the diffusion coefficients of the entrapped sucrose protons. This fact became a focus of the study. The vesicle study made use of dynamic light scattering (DLS) and electron microscopy (EM) as well as CONTIN analysis of PFG-NMR data. As the largest vesicles have diffusion coefficients in the vicinity of 2 × 10 ⫺8 cm 2 s ⫺1, it is necessary to include large q-values in the analysis. Here the q-values ranged from 2000 to 32 000 cm ⫺1 giving grating spacings of 30 to 2 mm. The LED sequence was used because BPP-LED was not available and the Te values tended to be long. Also, at the largest gradient values convective flow was a problem in 10 mm sample tubes but not in 5 mm tubes. This probably resulted from poor temperature control in the probe. The major conclusions were the following: (1) The distribution of particle diameters for large unilamellar vesicles obtained from the CONTIN analysis of PFGNMR data agrees quite well with distributions obtained from DLS and electronmicroscopy even though the width from the CONTIN analysis contains a contribution from the smoothing effect. (2) The
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apparent diffusion coefficients obtained for phospholipids in the bilayers are heavily weighted by contributions from the smaller vesicles in the distribution that tumble more rapidly and give narrower NMR signals. 6.2.2. Blood plasma Plasma lipoproteins are microemulsions-like assemblies consisting of a protein–phospholipid shell with a core of triglycerides and cholesterol. The compositions of the lipoprotein particles are related to their densities and sizes, and in general the lowest density particles have the largest radii [122]. Lipoproteins have received a lot of attention in recent years because their concentrations in blood have been correlated with the risk of coronary heart disease. Unfortunately, standard clinical procedures for measuring the low density fraction are inconvenient and subject to experimental errors. An alternative NMR method for rapid analysis has been proposed that takes advantage of differences in chemical shifts for methyl 1H signals exhibited by the various lipoprotein fractions: very low density (VLDL), low density (LDL) and high density (HDL) [123,124]. This is also a potential application for DOSY as the chemical shifts appear to be correlated with particle diameters. A DOSY CONTIN generated display for whole plasma with added D2O and tetramethylammonium chloride (TMA) as a reference compound is shown in Fig. 31 [125,126]. The experiment was performed on a 250 MHz spectrometer with ‘‘stop-and-go’’ sample spinning and the LED pulse sequence. Typically, 23 exponentially spaced values of q (84–12 540 cm ⫺1) were used with d 2 ms, t 3.5 ms, D 100 ms, and Te 50 ms. Water signals were eliminated in all experiments by continuous saturation except during data acquisition. The diffusion coefficients associated with the lipoprotein lipid signals [methyl, methylene, vinyl, and allylic] are all the same order of magnitude, and as expected the smaller molecules (lactate, EDTA, TMA) have much larger diffusion coefficients [127]. As the methyl and methylene signals for the larger lipoproteins are shifted downfield from those of smaller particles, we expect the diffusion coefficients to vary with the chemical shifts in the DOSY display. There is a trend toward smaller diffusion coefficients at the low end but not as pronounced as expected.
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Fig. 33. The simulated distribution a(D)D with M0 10 5 and s 3.2 (solid line), computed cm values (dashed line), and the effective distribution to be estimated by CONTIN with the computed cm values (dotted line). Cubic splines were used to obtain smooth displays from the 31 cm values. Reproduced with permission [97].
This unimpressive result prompted DOSY analysis of VLDL, LDL, and HDL fractions prepared by agarose gel chromatography and kindly supplied by J.D. Otvos. CONTIN reports moments Mn of peaks in the diffusion spectrum: Z 41 Mn ln a ldl Here the moments for the methyl peaks were used to calculate average diffusion coefficients, i.e. 具D典 D ⫺ d=3⫺1 M1 =M0 . The resulting mean diffusion coefficients yield uncorrected Stokes–Einstein ˚ (VLDL), 46 A ˚ (LDL), and 23 A ˚ radii of 106 A (HDL). The ratios of these numbers are in reasonable agreement with the literature [122], but the calculated radii are considerably smaller than expected. This may indicate the presence of breakdown products or the operation of relaxation effects that emphasize the contribution of smaller particles. Also, chemical exchange cannot be ruled out. Successful completion of this work demands the dispersion and sensitivity of a high field spectrometer. 6.2.3. The viscoelastic CTAB/sodium salicylate/water system The well-known viscoelastic micellar system hexadecyltrimethylammonium bromide (CTAB)/sodium salicylate/NaSal/ water has been found to undergo a remarkable polymer-induced non-Newtonian to
Newtonian transition [128]. Upon the addition of poly(vinyl methyl ether) a striking decrease in viscosity occurs that is thought to result from the breaking up of long rod-like micellar aggregates. A proposed model for the transition includes the formation of spherical micelles ‘‘wrapped around’’ by the polymer and the retention of bound Sal ⫺ ions. A DOSY study of this interesting system included the CONTIN processed display shown in Fig. 32 for a Newtonian mixture containing 25 mM CTAB, 15.0 mM NaSal, and 7.40 mg/ml PVME at 30⬚C [96]. The spectrum shows PVME peaks at 3.28 and 1.69 ppm with a broad distribution of diffusion coefficients. At approximately the same average diffusion coefficient, we find CTAB peaks (2.97, 1.28, and 0.87 ppm) with much narrower distributions. In fact, the diffusion distribution for these peaks results entirely from the smoothing effect of CONTIN. This is consistent with the model if the CTAB ions exchange rapidly between different polymer chains. Also, we find Sal ⫺ at about 8 × 10 ⫺6 cm 2 s ⫺1 which is about 10 times slower than the diffusion coefficient for Sal ⫺ in the absence of CTAB. With the help of Eq. (38) it is estimated that the fraction of bound Sal ⫺ is about 0.95, again in agreement with the proposed model. The diffusion coefficient of CTAB in the presence of PVME is greatly increased and at 40⬚C is consistent with spherical CTAB micelles.
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Fig. 34. Pulse sequence for 3D DOSY-COSY experiment [Phases: P1: 0, P2: 0, P3: 016 216, P4: 0, P5: 0, P6: 08 28, P7: 04 14, P8: 0213 1321, Rec: 0022 1133 2200 3311 0033 2200 3311 0022 1133]. Reproduced with permission [133].
6.2.4. Molecular weight distributions DOSY analysis for polydisperse samples produces a diffusion or Laplace spectrum a(D) that is actually a diffusion coefficient distribution. The question here is whether DOSY data can also be used to obtain molecular weight distributions (MWD). Other popular physical methods for the determination of MWDs (M ⬎ 25 000) include gel permeation chromatography (GPC) and DLS. As with PFG-NMR, these methods measure transport rates that depend on molecular size and require calibration. Also, hydrodynamic methods such as DOSY and DLS yield accurate parameters for individual molecules only in the limit of low concentrations [129]. In the following we assume that concentrations are sufficiently low that microaveraging effects are negligible [130,131]. The only reported DOSY MWD determination was based on a diffusion spectrum, a(D), computed with CONTIN [97]. In that work it was found that the standard application of CONTIN leads to unacceptable errors in a(D) because of D dependent undersmoothing of the distribution as discussed in Section 4.2.2.1. With CONTIN the inversion of the data set yk is handled by solving the set of linear algebraic equations shown in Eq. (25). A major improvement of the analysis for broad distributions is obtained by setting cm 1 so that a(D)D rather than a(D) is analyzed on the logarithmic axis as previously discussed. The fact that a scale change improves the results suggests that cm be varied continuously to improve the analysis in regions where the amplitude of a(D) is small. The procedure adopted was to replace cm with Dm =Dmax x
m
where xm was incremented from ⫹ 2 to ⫺ 2 as log(D) ranged from ⫺ 12 to ⫺ 9. In general the range of log(D) should be set as narrow as possible while still covering the distribution to zero amplitude at both limits. With reasonable limits the amplitude of G(D)D is typically about 10% of the maximum value at the xm ^1 points. This gives cm 1 near the maximum of the distribution (where D Dmax) and provides appropriate amplitude enhancement where the amplitude is small. The idea is that the cm values can be chosen to emphasize the region of interest, and CONTIN will then return the distribution a Dm Dm =cm from which a(Dm) can be extracted. The mass weighted distribution, W(M), can be computed directly with the equation W M a D兩dD=dM兩 if the relation between D and M is known for the particular system. Also, the number weighted MWD, n(M), can be obtained because n M W M=M. For the special case of the Gaussian random coil the scaling relation has the form D AM a
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but this cannot be assumed for real polymers. These ideas were tested by computer simulation. The MWD, W(M), was represented by log normal distributions with the reference molecular weight M0 and s values ranging from 1.25 to 3.25. The diffusion distributions a(D) were obtained from the W(M) by means of the scaling law in Eq. (42) with A 10 ⫺7.5 and a ⫺ 0.6. Finally, PFG-NMR dates sets were computed with Eq. (19) for sets of q-values and Gaussian random noise with RMS deviation of 10 ⫺3 was added. The simulated data sets were then processed with standard CONTIN, CONTIN with cm 1, and CONTIN with computed cm values to obtain
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Fig. 35. A 3D view of the DOSY-COSY data. In the F1 and F2 dimensions the chemical shifts range from 1.0 to 4.0 ppm, and the diffusion axis ranges from 3.0 × 10 ⫺10 to 9.0 × 10 ⫺10 m 2 s⫺1. Reproduced with permission [133].
‘‘experimental’’ diffusion distributions a(D). Fig. 33 shows a simulated distribution a(Dm)Dm, the computed cm values, and the function a Dm Dm =cm processed by CONTIN. From the computed a(D) distributions the
quantities 具D典, standard deviation (SD), the MWD W(M), mass average molecular weight Mw, and number average molecular weight Mn were computed as functions of s . It was found that these quantities obtained with all versions of CONTIN agreed almost perfectly with values computed directly from the initial log normal distribution for s ⬍ 2.0. However, for larger values of s , standard CONTIN showed large and erratic errors, the errors were moderate with cm 1, and with computed cm values the agreement was very good. CONTIN with computed cm values has been successfully used to determine Mn and Mw values for commercial samples of polydisperse poly(ethylene oxide) [97]. Recent work by Jerschow and Mu¨ller also demonstrates that 2D DOSY can be effective in resolving mixtures of polymers [121]. They have obtained impressive DOSY displays for mixtures of high and low molecular weight polypropylene and mixtures of polypropylene and polystyrene from experiments performed on a 600 MHz spectrometer by means of the BPP-LED sequence. The ILTs were performed
Fig. 36. (a) 2D COSY display obtained with DOSY-COSY pulse sequence having the lowest q value, and (b)–(d) the 2D COSY planes (slices) from the 3D DOSY data set that correspond to the diffusion coefficients of alanine, glutamine, and arginine, respectively. Reproduced with permission [133].
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Fig. 37. A pulse sequence for X-H COSY-DOSY or HMQC-DOSY [136].
with both CONTIN and NLREG [132]. These programs make use of regularized least squares procedures but use different methods to determine the regularization parameter. In the examples presented, the NLREG provided somewhat better separations than CONTIN. 7. 3D DOSY Here we describe 3D DOSY experiments in which a diffusion coordinate is added to conventional 2DNMR. The initial motivation for the development of 3D DOSY was to obtain additional dispersion of NMR peaks in order to avoid overlap as the highest accuracy in DOSY analysis is obtained when single component decays can be assumed. The analysis of data cluster intensity (cross-peaks in the 2D-NMR spectrum) versus q 2 then yields diffusion peaks that lie in layers corresponding to the diffusion coefficients
Fig. 38. A pulse sequence for gHMQC-DOSY. Delays t 1/4JCH, gradient pulse length is d , separation of midpoints of outer gradient pulses is D. The simple, time saving phase cycle was: first 13C 90⬚ pulse follows ( ⫹ x, ⫺ x) or 02 on successive transients, second 1H 90⬚ pulse follows 0022 1133, and for the receiver 0220 3113. Reproduced with permission [137].
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of the various species in solution. Here as elsewhere we emphasize that the acronym DOSY implies that the third dimension consists of peaks on a diffusion axis and not simple signal attenuation versus q 2. The creation of 3D DOSY requires (a) the design of a combined 2D-NMR/DOSY pulse sequence and (b) the development of algorithms for grouping data points in clusters corresponding to cross-peaks, inverting cluster intensity versus q 2 data sets, and constructing 3D spectra. In the original report it was emphasized that any 2D-NMR experiment can be used in this combination and 3D COSY DOSY was simply used as an illustration [133]. Also, HMQCDOSY was reported and NOESY-DOSY and EXSY-DOSY were suggested. In the following a variety of 3D DOSY experiments from different laboratories are reviewed. Some of the following examples demonstrate only the PFG-NMR step of obtaining q-dependent attenuation of cross-peaks, and thus do not fully implement 3D-DOSY. 7.1. COSY-DOSY A pulse sequence for COSY-DOSY constructed by linking BBP-LED and COSY sequences is shown in Fig. 34 [133]. After the eddy current delay Te, the COSY evolution time t1 and acquisition time t2 provide, through Fourier transformation, the two chemical shift dimensions. A 32-step phase cycle (Fig. 34) was devised to select the coherence transfer pathway for phase modulated 2D COSY [1] and to encode diffusion information. A FORTRAN program (DOSY3D) was used with the data set having the smallest (non-zero) value of q to assign the pixels, above a preselected threshold value, to cross-peaks. This assignment grid was then used for the remainder of the data sets. The analysis was performed by inverting either pixel intensity versus q 2 decays with SPLMOD, or integrated crosspeak integrals (volumes) versus q 2 by means of the L–M algorithm. The latter method is preferable when S/N ratios are modest and there is little crosspeak overlap. COSY-DOSY was illustrated with a mixture of three amino acids, alanine (A), glutamine (Q), and arginine (R), each at a concentration of 100 mM in D2O. Eight q values (184–1966 cm ⫺1) with D 0.104 s were used in the DOSY preparation period
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Fig. 39. Pulse sequence for NOESY-DOSY. h1 and h2 are homospoil type gradients and tm and t1 are the NOESY evolution and mixing times, respectively. Reproduced with permission [139].
to generate eight magnitude mode 2D spectra with 512 × 256 data points each. These were processed with DOSY3D and the L–M algorithm to obtain 64 points in the diffusion dimension and 512 × 256 × 64 total data points. Then Felix (Hare Research, Version 1.1) was used to produce the 3D display shown in Fig. 35. This view is not very useful, but the box can be rotated on a workstation screen to reveal the planes of points for the various species. Fig. 36(a) shows the 2D COSY spectrum of the mixture (lowest q value) and the COSY planes for the individual amino acids, obtained by summing over a few data points in the diffusion dimension, are shown in Fig. 36(b)–(d). The circle in Fig. 36(a) indicates the resolved cross-peaks associated with Qa and Ra that overlap exactly in the 1D spectrum. Note that the 2D planes appear at the correct positions in the diffusion dimension for alanine, glutamine, and arginine. Also, the COSY planes are identical with COSY spectra obtained for the separated compounds except for the diagonal peak of a -protons of
glutamine and alanine. This overlapping peak, that cannot be resolved in the diffusion dimension, was included in the planes for both glutamine and arginine through expanded integration ranges in the diffusion dimension. As expected 3D DOSY has higher effective chemical shift resolution than 2D DOSY and single exponentials are usually adequate. Resolution of planes in the diffusion dimension, of course, depends on standard deviations of the single components fits, which in favorable cases for 1H NMR are less than 2%. There is, however, a cost in time for 3D DOSY. With 124 FIDs for each of the eight q values, data acquisition required 36 h on an old 250 MHz spectrometer. With a modern 500 MHz system, the same S/N ratio could be obtained in less than an hour
7.2. HMQC-DOSY We have seen that both the use of heteronuclear ( 13C) chemical shifts and cross-peaks in 2D-NMR reduce peak overlap and enhance DOSY analysis. Even better resolution is possible through the combination of DOSY with 1H– 13C shift correlation spectroscopy. The problem, of course, is low sensitivity resulting for 13C signals compounded by extensive data acquisition required for 3D DOSY. X-H shift correlation with inverse detection, e.g. HSQC [134] or HMQC [135], may provide the best chance of success because these experiments benefit from the
Fig. 40. TOCSY-DOSY sequence. The narrow and wide filled rectangles represent 90⬚ and 180⬚ pulses and the rectangles with diagonal lines are ramped gradients. The phase cycle used was: P1: 0022; P2: 1230; P3: 222; rec: 0220 2002 2002 0220. Reproduced with permission [140].
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Fig. 41. 3D 1H TOCSY-DOSY display for a mixture of propylene and decane. Reproduced with permission [140].
large magnetic moment of the proton for both spatial encoding and data acquisition. HMQC-DOSY as implemented by Wu et al. [136] combined BPP-LED and X-H COSY is shown in Fig.
Fig. 42. (a) The PFG-DQS sequence for merged DOSY, (b) sequences for NOESY/ROESY and TOCSY diffusion experiments [141].
37. The gradient pulses after the second 1Hp /2 RF pulse are optional homospoil pulses to minimize phase cycling requirements. This experiment was considered to be impractical on our 250 MHz spectrometer, and was demonstrated on a 500 MHz spectrometer in the University of North Carolina Medical School. A sample containing alamine, glucose, sucrose, and caffeine all at 500 mM in D2O was analyzed in the chemical shift ranges 3.0–5.5 ppm for 1H and 20–100 ppm for 13C to produce a 3D display similar to the one shown in Fig. 35. However, the S/N ratios obtained with an acquisition time of 40 h were less than 10, and the resulting errors in diffusion coefficients were approximately 10%. Unfortunately the limited resolution in the diffusion dimension did not permit complete separation of the 2D spectra for all of the compounds. A successful implementation of HMQC-DOSY was recently reported by Barjat et al.[137] That work made use of the combination of GCSTE [54] and conventional gHMQC [138] shown in Fig. 38. The gradient pulses during the t D delay provide eddy current compensation, but otherwise are optional. As experimental time is crucial, the number
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Fig. 43. A representation of 3D-DOSY obtained from a series of 10 PFG-TOCSY experiments on a mixture of ATP/g -cd. A 65 mn MLEV-17 supercycle was used for spin locking. Reproduced with permission [141].
of increments for the diffusion dimension and the number of phase cycling steps were necessarily kept small. Gradient pulses were used in the coherence selection part of the HMQC sequence to reduce t1 noise and to reduce the need for phase cycling. Another, important experimental point concerns proton decoupling. The broadband adiabatic WURST method with low power was selected because the heat input from standard decoupling sequences, e.g. WALTZ, and the resulting convection currents could not be tolerated. The DOSY-gHMQC experiment was applied to a test mixture containing quinine (30 mg), geraniol (20 ml), camphene (19 mg), and TMS in deuterated methanol room temperature. Data were collected with a Varian INOVA 400 spectrometer equipped with a 5 mm indirect detect PFG probe. The storage delay t D was kept short (43 ms) to avoid T1 relaxation, and only five increments of the gradient pulse amplitude were used. Even so, data acquisition required 17 h. This experiment and the homonuclear COSYDOSY of the previous section, while impressive in resolving complex mixtures, are very time consuming even with test samples containing relatively high concentrations.
7.3. NOESY-DOSY Gozansky and Gorenstein combined LED and NOESY sequences as shown in Fig. 39 to obtain diffusion labeling of NOE cross-peaks [139]. They did not actually implement 3D DOSY as the data were not inverted and diffusion spectra were not generated. The diffusion labeling of cross-peaks was demonstrated for a D2O sample (0.6 ml) of 1.6 mg d(AG) dinucleotide and 5 mg 14-mer duplex d(ACAATATATATGR)2 in a phosphate buffer (pH 7.4). The data were collected with a 400 MHz spectrometer with a wide bore magnet and an actively shielded gradient probe. The gradient amplitudes ranged approximately 4–40 G/cm in eight equal steps, t 6.6 ms, T 20.0 ms, Te 50 ms, h1 3.0 ms, h2 97.0 ms, and t m 100 ms. Diffusion coefficients obtained from cross-peak attenuation were found to lie in the proper ranges, but there was some dependence on the delay T. 7.4. TOCSY-DOSY A TOCSY-DOSY experiment was implemented by Jerschow and Mu¨ller for the study of polydisperse polymer samples [140]. The sequence
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(linked BBP-LED and TOCSY) is shown in Fig. 40, and it should be noted that the homospoil pulses are orthogonal to the encoding and decoding gradients. It is clearly appropriate here to use an STE based experiment because T1 q T2. This 3D DOSY experiment differs from the others described here because data inversion was performed by CONTIN on data sets described by Eq. (22) with appropriate correction for BPP-pulse pairs (Eq. 13). The demonstration experiment was performed on a sample of 0.2% propylene (Mw 44 000, Mw/Mn 3.3) plus 0.8% n-decane in tetrachloroethane-d2 at 298 K with a 600 MHz spectrometer equipped with an actively shielded gradient probe. The parameters were T 120 ms, d 2.6 ms, Te 24 ms, t 1 ms, t m 30 ms, and the gradient amplitudes were incremented linearly in 32 steps to a maximum of 63 G/cm. The 2D intensity data for discrete coordinate pairs (v 1, v 2) were fed into CONTIN for analysis, and the resulting 3D data set is displayed in Fig. 41. In spite of the large peak volumes, the components are separated quite well into their respective diffusion planes. 7.5. Merged sequences for PFG-DQS, PFG-NOESY, and PFG-TOCSY Birlirakis and Guittet [141] have reported ‘‘merged’’ pulse sequences that introduce variable gradient pulses into 2D-NMR without the elongation found in the linked schemes described previously. Examples are shown in Fig. 42(a) for PFG-DQS and Fig. 42(b) for PFG-(NOESY, ROESY, or TOCSY). With PFG-DSQ no phase cycling is required and the spectra are obtained in the magnitude mode. This experiment was demonstrated with a mixture containing ATP/g -cyclodextrin (50:7 mM) in D2O at 298 K on a 600 MHz spectrometer. A series of 11 gradient values were used and data acquisition required 14 h. The resolution permitted monoexponential fitting yielding Dg -cd 2.39 × 10 ⫺10 m 2 s ⫺1 and DATP 3.08 × 10 ⫺10 m 2 s ⫺1 with a dispersion smaller that 7% and the quality of exponential fitting R ⬎ 0.997. The 2D TOCSY experiment was implemented for the same sample with ten gradient values and a total data collection time of 43 h. In this case the dispersion increased to 10%. The DOSY idea was carried out by manual construction of 2D spectra to obtain the
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separation of components shown in Fig. 43. The merged sequences described here can produce increased senstivity in some cases, but the advantages of STE based experiments should be kept in mind when short T2s and J-modulation are problems.
8. Future prospects The concept of diffusion spectra and their incorporation as a new ‘‘frequency’’ dimension in NMR is now well established. DOSY has been fully implemented in several laboratories, and interesting examples of diffusion ordered spectra are available as demonstrated in this review. With the advent of both free and commercial DOSY analysis packages and the availability of high quality diffusion accessories from major vendors, the benefits of DOSY analysis are certain to spread to a much larger audience. The result will be another NMR tool to select from a menu. We expect that DOSY will become transparent and natural to users and will no longer attract special attention any more than FT-NMR does. The success of DOSY will tend to make it invisible. The question here is whether there will be major advances in instrumentation or analysis methods that will significantly affect the power and ease of use of DOSY. We do not expect exciting advances in data analysis. The analysis problems have been around for a long time, and the available techniques are able to extract most of the information from experimental data sets. Therefore, we expect the future to hold refinements, slicker user interfaces, and perhaps the incorporation of artificial intelligence to protect users from themselves. Of course, pulse sequences and experimental design are certain to evolve. For example the use of multiple quantum NMR to enhance the effects of gradients may become more common [142]. The question of hardware is much more open. Higher sensitivity through higher frequencies and better probes is certain. One obvious area not yet explored for DOSY is the use of RF gradients in place of or in addition to the pulsed dc gradients. We note the recent development of toroid cavity detectors that permit RF imaging and diffusion measurements with respect to the radial (transverse) direction [143]. These devices have important advantages in special cases such as the measurement of
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diffusion in thin films and in high pressure samples [144]. DOSY applications of toroid cavities may also be possible, but there is the immediate problem of reduced resolution resulting from magnetic susceptibility mismatch that becomes serious in the short cylindrical cavities. Acknowledgements This work was supported in part under a grant from the National Science Foundation (CHE-9528530). It was facilitated by a Reynolds Research Leave from the University of North Carolina. The author thanks Aidi Chen, Stephen J. Gibbs, Kevin F. Morris, Konstantin Momot, and Donghui Wu for reading the manuscript and providing helpful comments. Also, the author thanks Alexej Jerschow, Cynthia K. Larive, Gareth A. Morris, and Norbert Mu¨ller for sending relevant preprints. References [1] R.R. Ernst, G. Bodenhauen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Oxford University Press, Oxford, 1987. [2] L. Mu¨ller, A. Kumar, R.R. Ernst, J. Chem. Phys. 63 (1975) 5490. [3] A. Kumar, D. Welti, R.R. Ernst, J. Magn. Reson. 18 (1975) 69. [4] H.J.V. Tyrrell, K.R. Harris, Diffusion in Liquids: A Theoretical and Experimental Study, Butterworths, London, 1984. [5] C.R. Cantor, P.R. Schimmel, Biophysical Chemistry, Part II: Techniques for the Study of Biological Structure and Function, W.H. Freeman, San Francisco, 1980. [6] C. Tanford, Physical Chemistry of Macromolecules, Wiley, New York, 1961. [7] E.O. Stejskal, J.E. Tanner, J. Chem. Phys. 42 (1965) 288. [8] K.F. Morris, C.S. Johnson Jr., J. Am. Chem. Soc. 114 (1992) 776. [9] K.F. Morris, C.S. Johnson Jr., J. Am. Chem. Soc. 114 (1992) 3139. [10] C.S. Johnson Jr., in: R. Tycko (Ed.), Nuclear Magnetic Resonance Probes of Molecular Dynamics, Kluwer Academic, Dordrecht, 1994, p. 455. [11] C.S. Johnson Jr., in: D.M. Grant, R.K. Harris (Eds.), Encyclopedia of NMR, Wiley, New York, 1996, p. 1626. [12] C.S. Johnson Jr., in: D.M. Grant, R.K. Harris (Eds.), Encyclopedia of NMR, Wiley, New York, 1996, p. 1886. [13] C.S. Johnson Jr., Q. He, in: W.S. Warren (Ed.), Advances in Magnetic Resonance, 13, Academic Press, New York, 1989, p. 131.
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Diffusion ordered nuclear magnetic resonance spectroscopy: principles and applications C.S. Johnson Jr.* Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290, USA Received 21 October 1998
Contents Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Previous reviews of DOSY and related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The PFG-NMR experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Magnetic field gradients and magnetization helices . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Bloch equations with diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Pulse sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. The spin-echo (SE) sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. The stimulated echo (STE) sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The FT-PFG-NMR experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Component analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Diffusion ordered NMR spectroscopy (DOSY) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Experimental requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Eddy current reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Pulse sequences for minimizing effects of eddy currents and J-modulation . . . . . . . . . . 4.1.3. Suppression of convection current effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4. Dispersion and resolution enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5. Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6. Utilizing the stray field to obtain large, steady gradients . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Data inversion and display . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Discrete samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Polydisperse samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Complete bandshape methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4. Analysis recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Effects of chemical exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Exchange effects in diffusion spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Artifacts from chemical shift encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Applications of 1D and 2D DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Discrete samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1. Biofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * Tel.: ⫹1-919-966-5229; fax: ⫹1-919-962-2388. E-mail address: [email protected] 0079-6565/99/$ - see front matter 䉷 1999 Elsevier Science B.V. All rights reserved. PII: S0079-656 5(99)00003-5
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6.1.2. Separation by means of hydrophobicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3. Equilibria involving sodium dodecylsulfate (SDS) and bovine serum albumin (BSA) . . . 6.1.4. Mixtures of polymer additives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Polydisperse samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Phospholipid vesicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Blood plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3. The viscoelastic CTAB/sodium salicylate/water system . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4. Molecular weight distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. 3D DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. COSY-DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. HMQC-DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. NOESY-DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. TOCSY-DOSY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Merged sequences for PFG-DQS, PFG-NOESY, and PFG-TOCSY . . . . . . . . . . . . . . . . . . . . . 8. Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Keywords: Diffusion ordered NMR; Pulsed magnetic field gradient-NMR experiment; Pulse sequences
Nomenclature GLOSSARY ALS (computer program) ARK* a priori knowledge BPP* Bipolar Pulse Pairs CONTIN (computer program) CORE COSY-DOSY CTP DECRA DEPT DISCRETE DLS DSTE DOSY* Diffusion Ordered NMR SpectroscopY EXSY FID FIDDLE (computer program) GCSTE GCSTESL GPC-NMR GRAM (computer algorithm) HDL (lipoprotein) HMQC-DOSY HR-DOSY HSQC
INEPT ILT LDL LED*
(inverse Laplace tranform) (lipoprotein) Longitudinal Eddy current Delay or Longitudinal Encode–Decode
MaxEnt MCR (computer algorithm) MOSY* Mobility Ordered NMR SpectroscopY MWD NIPALS (computer program) NLREG (computer program) NOESY-DOSY PFG-NMR PVA (computer algorithm) RDCON SE SPLMOD STE STEP (computer program) VLDL (lipoprotein) VMAX (computer algorithm) CSJ is responsible for LED, DOSY, and MOSY and some of the hyphenated forms containing them. He may have been the first to use the abbreviations BPP and APK. The others are defined in the text.
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1. Introduction One of the most fruitful ideas in NMR spectroscopy was the introduction of a second frequency dimension [1]. This was made possible through the use of pulse sequences having two independent precession periods. In one class of two-dimensional NMR (2DNMR) experiments, the Hamiltonian is switched between the evolution period and the detection period. As a consequence of the evolution period, resonances are spread into a second dimension to reveal their origins. Examples of such 2D ‘resolved’ spectroscopies include J-resolved where the Hamiltonian is switched through spin decoupling [2] and NMR imaging where magnetic field gradient directions are switched [3]. A logical extension of these ideas is the introduction of additional NMR dimensions that depend on molecular properties such as size, shape, mass, and charge that are not explicitly included in spin Hamiltonians. These overall molecular properties are not well represented in conventional NMR as spin interactions tend to be quite local. Therefore, dispersion on the basis of such properties can provide new information as well as a means for editing NMR spectra. The problem is to identify ways that molecular properties influence NMR spectra or can be made to affect NMR spectra. Nuclear relaxation times are obvious candidates because they depend on correlation times for molecular motion, and the correlation times in turn depend on molecular sizes and shapes. However, relaxation times can be quite different for different nuclei in the same molecule because of site specific magnetic interactions and because local or segmental motions may obscure overall molecular motions. In the case of longitudinal relaxation, high frequency local segmental motion may provide the dominant relaxation mechanism, leading to T1 values that are relatively independent of molecular mass. While for transverse relaxation, local motions may contribute motional averaging effects comparable to those resulting from overall rotation. Even so, there are cases where relaxation resolved spectra of, for example, backbone 13C nuclei in rigid molecules or bilayer 1H nuclei in vesicles can provide useful information about size dependent molecular reorientation. It is clear that new NMR dimensions should be
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based on molecular properties that have the same effect on all nuclei in a given molecule. Transport properties of molecules and ions, as determined by diffusion measurements and electrophoresis meet this criterion. The connection with structural properties arises because diffusion coefficients (D) depend on friction factors and electrophoretic mobilities (m ) depend on both friction factors and effective charges. According to the Debye–Einstein theory [4]: D
kB T fT
1
where kB is the Boltzmann constant, T is the absolute temperature, and fT is the friction factor. For the special case of a spherical particle of hydrodynamic radius rH in a solvent of viscosity h , the friction factor is given by fT 6phrH . More realistic models for fT represent molecules by ellipsoids of revolution or collections of spherical subunits [5]. Electrophoresis concerns the terminal velocity v of a charged particle in an applied electric field, Edc. The relationship between the electrophoretic mobility, defined as m v=Edc , and molecular properties is not simple; but for small ions m is proportional to the overall charge, Ze, and inversely proportional to fT [6]. The implementation of transport ordered NMR is possible because information about translational motion can be encoded in NMR data sets through the use of pulsed magnetic field gradient NMR (PFG-NMR) experiments [7]. The idea, as with conventional 2D NMR, is to increment an experimental variable that modulates the detected signal and then to transform the data with respect to that variable to produce a ‘‘spectrum’’ related in this case to molecular translation. Spectra based on electrophoretic mobilities are obtained by incrementing Edc and then transforming the NMR signal amplitudes with respect Edc. This scheme is realized in mobility ordered NMR spectroscopy (MOSY) [8]. The MOSY method has not yet found widespread use because of the unusual instrumentation requirements, e.g. NMR compatible electrophoresis cells, and the restriction to conducting samples with low ionic strengths. However, the success of the MOSY concept motivated the development of the more general diffusion based spectroscopy. Diffusion spectra can be obtained by incrementing the areas of the gradient pulses (q) in PFG-NMR and
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Fig. 1. The simple Carr–Purcell spin echo (SE) often called the Hahn echo.
transforming the NMR signals amplitudes with respect to q 2. The result is diffusion ordered NMR spectroscopy (DOSY) [9]. The three basic DOSY requirements are (1) distortion free absorption mode data sets acquired with precise gradient encoding, (2) effective data inversion (transformation) procedures, and (3) algorithms for the display of the diffusion spectra. These requirements turn out to be quite severe because the signal inversion step is extremely sensitive to noise and distortions in the signals. This has necessitated significant enhancements of the original PFG-NMR experiments and experimentation with alternative data inversion methods. Even data display for DOSY is not straightforward because decisions must be made about how to generate the spectra. The contrast with the Fourier transform NMR (FTNMR) is striking. With FT-NMR, one has a unique transformation with an inverse that returns the original signal. Also, the resulting spectra are ready for display. This review is concerned with the various implementations of DOSY experiments and with
illustrations of the power of this technique. The implementations present solutions to the unique problems of data acquisition, transformation, and display. With appropriate instrumentation and software, the user can be offered menu choices for analysis methods and types of display. The result is a convenient NMR method for the analysis of mixtures that can reveal unexpected components and interactions in mixtures through useful and appealing plots.
2. Previous reviews of DOSY and related topics Transport ordered NMR [10] and diffusion measurements by magnetic field gradient methods including DOSY [11] have previously been reviewed. Related reviews of MOSY are also available [12,13]. A complete treatment of translational dynamics and its study by NMR can be found in the book by Callaghan [14]. Ka¨rger et al. [15] have reviewed the principles and applications of PFG-NMR, and Stilbs has provided a detailed review of FT diffusion studies
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[16]. The recent tutorial articles on PFG-NMR by Price are also of interest [17,18].
3. The PFG-NMR experiment 3.1. Background Time domain NMR dates from Hahn’s observations of the free induction decay (FID), the spin echo (SE), and the stimulated echo (STE) [19,20]. The effects of molecular diffusion, in the presence of magnetic field gradients, on echo amplitudes were evident from the beginning, and Hahn reported a derivation of diffusion dependent signal attenuation, which he attributed to C.P. Slichter [20]. All NMR diffusion measurements are based on the fact that the diffusion coefficient can be calculated from the echo attenuation if the amplitude and duration of the magnetic field gradient are known. The original measurements were carried out with continuous gradients, but the advantages of pulsed gradients were convincingly demonstrated by Stejskal and Tanner [7]. Here we review the principles of PFG-NMR and display selected applications of PFG-NMR to provide the background for DOSY. 3.1.1. Magnetic field gradients and magnetization helices NMR diffusion measurements can be made by means of either gradients in the main (dc) magnetic field, B0, or gradients in radio frequency fields (B1). In the following only gradients in B0 are considered. For applications of RF gradients the reader should consult the review article by Canet [21] and the monograph by Kimmich [22]. Here the z-direction is defined by the direction of B0, and we are concerned with gradients in the z component of B. Typically, a spatially constant gradient is applied externally by means of current in a coil set, either of the Maxwell pair [23] or quadrupole type [24,25]. The resulting gradient g is described by: g
2B z ^ 2B z ^ 2 B z ^ k i⫹ j⫹ 2x 2y 2z
2
where ˆi, ˆj and kˆ are unit vectors in the x, y, and z directions, respectively. Accordingly, the total
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external magnetic field at r is given by B r B0 ⫹ g·r:
3
In the following we assume that only a z-gradient of magnitude g g·k^ is present. The purpose of the gradient is to label nuclear spins with phase angles that depend on their positions in space, or in this case their displacement in the z-direction. This is, of course, possible because spins precess with the angular frequency
v r ⫺gB r
4
and the acquired phase angle depends linearly on both B(r) and the duration of the gradient. Therefore, a zgradient of duration d produces the position dependent phase angle f z ⫺gB zd. PFG-NMR experiments involving constant z-gradients can readily be visualized by imagining layers of the sample perpendicular to the z-axis that are thin enough to experience a uniform magnetic field but thick enough to contain a large number of spins. Each layer is associated with a magnetization vector (isochromat), and these vectors are assigned to the positions of the layers on the z-axis. We begin the experiment with a hard 90⬚x RF pulse that rotates all of the vectors into the y-direction to create a magnetization ribbon in the rotating coordinate frame as shown in Fig. 1. The effect of the gradient g is then to twist the ribbon into a helix [26] defined by the relative phase angles Df z ⫺ ggdz. The pitch of the helix is given by L 2p=q where q ggd is the area of the gradient pulse in units of m ⫺1. Thus, the effect of a constant gradient is to produce a magnetization pattern with a cosinusoidal projection on the yzplane. Diffusion in the z-direction will, of course, smear out this pattern, and the smaller the pitch the more rapidly this will happen. 3.1.2. Bloch equations with diffusion The magnetization of uncoupled spins is well described by the Bloch equations, and the effects of diffusion can easily be incorporated. As gradients in Bz only affect the transverse components of magnetization, it is appropriate to begin with the Bloch equation for the complex magnetization, M⫹ Mx ⫹ iMy . We assume that the magnetization has been rotated to the y-direction in the rotating frame by a 90⬚x pulse or some other set of pulses, and that only the main field
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Fig. 2. The Hahn stimulated echo (STE) with pulsed field gradients.
Bz is present for t ⬎ 0. With Bx By 0 the Bloch equation becomes [27]:
2M ⫹ M ⫺iv0 M⫹ ⫺ ⫹ ⫺ ig g·rM⫹ ⫹ D72 M⫹ 2t T2 5 where the average precession frequency in the sample is denoted by v0 gB0 . The precession frequency and the effects of T2 relaxation can be transformed away by means of the substitution: M⫹ c z; texp iv0 t ⫺ t=T2
6
to give:
2c z; t ⫺iggzc z; t ⫹ D72 c z; t 2t
7
When D 0, Eq. (7) describes the free precession of transverse magnetization in the gradient g. At the end of a gradient pulse of duration t, the isochromats define a helix as described earlier. The effect of diffusion on the amplitude (diameter) of the helix, C(t), can be obtained by substituting Zt 0 0 c z; t c texp ⫺igz g t dt 8 0
into Eq. (7). The result is: Zt lnc t ⫺D q2 t 0 dt 0 0
9
where 0
q t
Zt 0 0
gg t 00 dt 00
10
Of course, this is only an attenuation factor, and for its observation the isochromats must be refocused in the xy-plane to form an FID or echo. It should be noted that, while gradients only affect the transverse components of the magnetization, the attenuation by diffusion as described by Eq. (9) also applies to sinusoidal patterns in the z-component of magnetization, i.e. stored magnetization. 3.2. Pulse sequences Immediately after a 90⬚x pulse, the signal detected along the y-axis in the rotating frame decays in amplitude. The primary cause of the decay in liquid state NMR is dephasing that results from magnetic field inhomogeneities. This decay is rapid in the presence of the applied constant gradient g because the volume
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integral of the helix of isochromats is zero when the pitch is much smaller than the length of the sample. In order to detect a signal, the isochromats must be refocused, i.e. the helix must be unwound, so that the volume integral of the magnetization gives a nonvanishing component in the xy-plane. This requires a sequence with two matched gradient pulses. The first pulse encodes nuclear positions through position dependent phase angles, and thus sensitizes the sample to diffusion and flow. The second gradient reverses the encoding and brings the isochromats back into the yz-plane, thus forming an echo. In the following we consider only PFG-NMR experiments where the gradients are applied in the form of relatively short pulses. The major advantages of an echo sequence with pulsed gradients are (i) the gradient pulse areas can be controlled independently of the time for the echo and (ii) the signal can be read out in a homogeneous magnetic field. The disadvantage is that relatively large currents must be switched on and off. This produces mechanical forces, Joule heating, and transient eddy currents. 3.2.1. The spin-echo (SE) sequence A simple Carr–Purcell [28] sequence is shown in Fig. 1. Also shown is the ribbon of isochromats produced by the 90⬚x pulse and the ‘‘Saarinen helix’’ produced by the first gradient pulse [26]. The effect of the 180⬚y pulse is to reverse the effect of the previous gradient pulses to give the effective gradient sequence (g*) [14]. This is equivalent to defining the effective gradient at any time by g* pg where p is the coherence order. Straightforward application of Eq. (9) to this sequence shows that the echo amplitude at 2t is [7]: S 2t M0 exp ⫺2t=T2 exp⫺Dq2 D ⫺ d=3
11
where M0 is the equilibrium magnetization and the relaxation factor containing T2 has been reintroduced. The correction term d /3 is a consequence of the rectangular shape of the gradient pulses. Other shapes, such as the sine lobe, can easily be incorporated [29]. The Stejskal–Tanner attenuation factor [7] for diffusion can be isolated as c 2t S 2t=S0 2t where S0 2t is the echo amplitude in the absence of a gradient. Continuous background gradients have been neglected in deriving Eq. (11). This is usually
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adequate for experiments in modern, high homogeneity magnets; but a more complete expression is available for the case where a background gradient g0 must be considered [7]. The Carr–Purcell echo has important advantages. In particular, the maximum possible signal is recovered, in the absence of relaxation effects, and chemical shifts are refocused at the echo. The disadvantages result from the long period that the magnetization is transverse, i.e. in the xy-plane. Transverse magnetization is subject to both transverse relaxation and J-modulation effects. T2 can be short for slowly tumbling macromolecules, and this can lead to a severe loss of signal. J-modulation refers to signal modulation resulting from hard RF pulses that exchange the spin states of nuclei that are coupled to the nuclei of interest thus preventing complete refocusing. These effects present special problems for strongly coupled spin systems [30]. 3.2.2. The stimulated echo (STE) sequence Sequences containing two and three 90⬚ RF pulses were investigated by Hahn in his classic paper on spin echoes [20]. He found that the three-pulse sequence with a steady (cw) gradient can generate up to five echoes. The first echo after the third RF pulse, the socalled stimulated echo (STE), is of particular interest here. The effects of diffusion on the STE with both steady [31] and pulsed gradients [32] have been computed. We show the standard PFG-STE diffusion experiment in Fig. 2. An instructive three dimensional (spherical polar) model of the first part of this experiment including the formation of the primary echo at 2t (not shown) was presented by Hahn and attributed to E.M. Purcell [20]. The amplitude of the PFG-STE is given by S T ⫹ 2t M0 =2exp ⫺2t=T2 ⫺ T=T1 exp⫺Dq2 D ⫺ d=3
12
where the Stejskal–Tanner factor has again been derived by the application of Eq. (9). It should be recognized at the outset that the STE is quite different from the SE. First, we see that the amplitude is reduced by a factor of two. This results from two features of the sequence. The second 90⬚x pulse stores the magnetization by rotating only the y-components into the ^z-directions. The x-components remain
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Fig. 3. A 1H FT-PFG-NMR stack plot obtained at 99.6 MHz for a microemulsion sample containing sodium octylbenzenesulfonate (SOBS), n-butanol, toluene, and water (D2O). Reproduced with permission [16].
transverse and can contribute to the primary and secondary echoes. Then, after the storage period T, the third 90⬚x pulse returns the z-components to the ^y-directions where the action of the second gradient pulse refocuses the isochromats so that the STE signal appears at t T ⫹ 2t. However, the isochromats are not coplanar at the time of the echo, and in fact their projection onto the xy-plane defines a circle that is tangential to the xz-plane. The advantages of the STE sequence arise because the evolution time for transverse magnetization can be limited. With 2t p T, spin relaxation depends primarily on T1 rather than T2, and with t p 1/J, J-modulation is not significant. The advantage of T1 relaxation relative to T2 relaxation depends on the ratios R T2 =T1 and X T=T1 . Assuming that T 艑 D and T q t in the STE experiment, we find that the STE/ SE signal ratio is 0:5expR X ⫺ 1=X. For example when R 0.5, X must be greater than ln(2) to break even, but with R 0.1 and X 0.5 the enhancement factor is greater than 200. In general the advantages of STE more than compensate for the 50% smaller coefficient.
Two other points need to be mentioned. First the reduction of t to a value only slightly larger than the gradient pulse duration d means that the STE is very close to the trailing edge of the second gradient pulse. Unless appropriate provisions are made, the signal will be distorted by gradient pulse induced eddy current effects. The second point concerns the encoding of chemical shifts after the first RF pulse. For a spin with offset frequency v A and position z, the component cos vA t ⫹ ggzd will be stored in the z-direction by the second 90⬚ pulse. The third 90⬚ pulse brings this component back into the yz-plane and, after an additional time t , the component in the y-direction is cos2 vA t ⫹ ggzd neglecting the effects of diffusion. The echo signal, obtained by integration of the y-components over the sample volume, gives 1/2 as expected when g is present, but in the absence of g we encounter the modulation factor 1=2cos2 2vA t. Therefore, in STE experiments the echo amplitude S0 T ⫹ 2t for g 0, i.e. the q 0 point in plots of echo amplitude versus q 2, is dependent on the chemical shift and
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Fig. 4. 400 MHz 1H NMR spectra for human blood plasma: (a) normal spectrum with 10% maximum gradient strength, (b) spectrum obtained with 50% strength, and (c) the difference between (a) and (b). Reproduced with permission [39].
should be avoided in data analysis (see Section 5.2) [33]. 3.3. The FT-PFG-NMR experiment For analysis of spin relaxation in complex mixtures it is essential that we make use of the complete
spectral information that is contained in FID’s and half-echoes. The FT method for accomplishing this was described by Vold et al., in connection with their study of frequency resolved inversion recovery [34]. This idea can also be extended to NMR diffusion measurements when pulsed gradients are used as the FID’s can be acquired in the absence of applied
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gradients. In the following we focus on features and applications of FT-PFG-NMR that are relevant to the later discussion of DOSY. 3.3.1. Component analysis 3.3.1.1. Complex mixtures James and McDonald demonstrated the determination of diffusion coefficients for each component in a multicomponent system by means of FT-PFG-NMR [35]. In their experiment, a Carr–Purcell echo (Fig. 1) was acquired as a function of the gradient duration d with t 艑 d, and the second half of the echo was Fourier transformed to produce a set of NMR spectra. The attenuation factor, c 2t S 2t=S0 2t; for each line was then used with Eq. (11) to determine the corresponding value of D. The gradient g was calibrated in a separate experiment on a compound with a known diffusion coefficient. This groundbreaking experiment, carried out on a standard commercial NMR spectrometer, established the basic NMR diffusion measurement still in use. The authors recognized the analytical implications, because each component of a mixture is revealed by its unique diffusion coefficient, and possibilities for the study of dynamics in solutions. In particular, the effects of rapid chemical exchange were considered, and the determination of binding constants for small molecules with large molecules by means of the weighted average diffusion coefficient was suggested. The major limitation of the James and McDonald experiment was the small gradient amplitude available and the necessity of using long gradient pulses. Kida and Uedaira remedied the problem by designing a gradient driver that permitted the use of narrow gradient pulses that were compatible with the operation of a field stabilized NMR system [36]. They also introduced the stack plot display of spectral intensity versus q2 D ⫺ d=3. Other features of this early paper are the analysis of the apparent hydroxyl group proton diffusion coefficients on the basis of rapid exchange between water and methanol, and the correlation of molecular diffusion coefficients with realistic models for translational friction factors (see Eq. (1)). The FT-PFG-NMR method, primarily in the spin echo version, has been applied to a wide variety of chemical systems. We note the studies of complex mixtures, especially those containing surfactants, by
Stilbs and coworkers. That work has been reviewed, and an illustration involving a microemulsion is shown in Fig. 3 [16]. Note that the stack plots are arranged with the gradient pulse duration d increasing for the lower (front) spectra, and the effects of Jmodulation are evident for the inverted toluene resonances. 3.3.1.2. Spectral editing Stilbs has emphasized the fact that, in a PFG-NMR diffusion measurement for a mixture, the complete spectrum for each component is attenuated as the quantity q2 D ⫺ d=3 is increased; and the non-overlapping signals can be classified by their measured diffusion coefficients [37]. He also suggested that spectra with different amounts of attenuation could be scaled and subtracted to zero out a component and in favorable cases to isolate a component. A demonstration of this procedure with 1 H spectra of a 50:50 mixture of decane and 1-decanol was presented, and potential problems with Jmodulation and T2 effects were noted. This type of spectral editing was characterized as ‘‘size-resolved NMR spectrometry.’’ The pseudo-separation of different molecules in a complex mixture by spectral editing on the basis of nuclear relaxation times and molecular diffusion coefficients has recently been pursued with the aid of sophisticated, modern PFG-NMR sequences. For example, an STE experiment including the WATERGATE water elimination sequence [38] has been used to assign spectral resonances of slowly diffusing molecules in human blood plasma [39]. This experiment is illustrated in Fig. 4. The spectrum in Fig. 4(a) was acquired with a small gradient (g 59 mT/m, D 500 ms) and shows essentially no attenuation. The spectrum in Fig. 4(b), acquired with g 295 mT/m emphasizes large molecules and permits the assignment of peaks between 3.4 and 3.9 that have previously been obscured by amino acids and carbohydrates. Finally, Fig. 4(c), the difference between Fig. 4(a) and (b), shows the small rapidly diffusing molecules and their assignments. These spectral editing experiments and numerous other experiments involving diffusion filters take advantage of strong diffusion based discrimination against small rapidly moving molecules without requiring sophisticated data transformations. Editing and filtering techniques can be very valuable; but, of
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Fig. 5. The LED pulse sequence [48].
course, do not provide complete diffusion spectra of mixtures. 3.3.1.3. Affinity NMR Another variant of diffusion edited NMR, known as affinity NMR, deserves comment [40,41]. In the pharmaceutical industry combinatorial chemistry methods are now producing large numbers of compounds in mixtures for testing in drug discovery programs. It is important to be able to detect the presence of molecules with desired properties without resorting to physical separation of the mixtures. If the desired property involves complexation with a partner in solution, differences in diffusion coefficients can be important indicators. In the affinity NMR experiment, the gradient amplitude and duration are adjusted so that all signals from small molecules in a mixture just vanish. Then a potential complexing agent is added and the experiment is repeated. In a demonstration experiment, nine components with molecular weights in the range 200400 were attenuated to the noise level by adjusting the gradient duration in a PFG-NMR experiment. Hydroquinine 9-phenanthryl ether was then added and the spectra were found to contain signals from two of the small compounds in addition to the added ether [40]. The combination of PFG-NMR and TOCSY was sufficient to identify the two complexing compounds [42].
4. Diffusion ordered NMR spectroscopy (DOSY) 4.1. Experimental requirements For the standard DOSY experiment we envision automated data collection with a programmed set of gradient areas. This is followed by data inversion by
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means of one or more user selected transformations and the generation of one, two, or three-dimensional spectra. This scheme can only be successful if data collection and initial processing, i.e. Fourier transformation, phasing, baseline correction and deconvolution, yield undistorted absorption mode NMR spectra. DOSY requires high quality gradient probes that incorporate active shielding [43,44] and are designed to provide constant (flat) gradients over the NMR active sample volume. DOSY also requires computer controlled gradient drivers that can provide gradient pulses with reversible polarity and pulse areas that are matched to within at least 10 ppm [45]. The latter feature is essential for automated DOSY experiments. We assume that such equipment is commercially available or will be available in the near future and do not discuss it further. However, readers should be aware that much commercial gradient equipment currently in use does not meet these criteria. As previously discussed, the pulse sequences of choice are based on the STE sequence (see Fig. 2). The immediate problem is that gradient pulses tend to induce eddy currents in the surrounding metal structures of the probe and the magnet. These eddy currents in turn produce slowly decaying magnetic fields at the sample that lead to spectral distortions resulting from time dependent phase changes. Therefore, experiments must be designed that avoid or at least minimize the effects of eddy currents. There is also the related requirement that the NMR resolution be maximized to avoid overlap of peaks from different components in a mixture as data transformations, required to produce diffusion spectra, fail when NMR peaks for similar sized molecules overlap. Here we consider current hardware, software, and experimental designs that address these requirements. 4.1.1. Eddy current reduction The best way to avoid the effects of eddy currents is to prevent the formation of eddy currents in the first place. There are some easy but not completely effective solutions. For example eddy currents in the probe can be reduced by means of special RF coil designs, and eddy currents in the inner bore of the magnet can be reduced by using widebore magnets. Also, the rate of change of the magnetic field when a gradient pulse is switched on or off can be reduced by shaping the gradient pulse. Shapes that have been investigated
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Fig. 6. The BPP-LED pulse sequence. The phase cycle for the 90⬚ pulses is: P1: 016, P2: (0022)4, P3: 04 24 14 34, P4: 0202 2020 1313 3131, P5: 04 24 14 34, Rec: 0220 2002 3113 1331, and the 180⬚ pulses are ⫹ x or 2 throughout [53].
include sine functions, sine squared functions, and nearly rectangular functions with modified rise and fall times [29]. Gradient pulse shaping is helpful and this capability is now available on some commercial instruments. The most effective means for avoiding eddy currents is to reduce the magnetic fields of the gradient coil set outside the probe to such low values that significant disturbances do not occur. This reduction can be achieved through active shielding of the gradient coils. Imagine a wire carrying a time-dependent current close to a conducting metal sheet. A surface current distribution is induced in the sheet that screens the magnetic field and reduces it to zero inside the sheet. The idea of active shielding is to introduce a mesh of wires with an externally generated current pattern that mimics the induced surface current distribution in the conducting sheet. Mansfield and Chapman have reported an iterative procedure for determining the positions of wires in a discrete cylindrical mesh to approximate the continuous current distribution required to screen the field of a current loop [43]. Another strategy is to use a multidimensional minimization program to determine the axial coordinates and radii of a predetermined number of Maxwell pairs that will shield most effectively a specified gradient coil set. A small volume outside the shielding coils is chosen as the indicator of shielding efficiency [44]. Active shielding is now so well established that commercial imaging and gradient probes can be expected to include efficient shielding coils.
Fig. 7. NMR spectra of protons on a -carbons of alanine in D2O: (a) 5 ms after a bipolar gradient pulse pair (g ^ 1.50 T m ⫺1, d /2 1 ms, t 1.5 ms), (b) 5 ms after monopolar gradient pulse (g 1.50 T m ⫺1, d 2 ms), (c) 100 ms after a monopolar gradient pulse (g 1.50 T m ⫺1, d 2 ms). The frequency origin is arbitrary, and the dashed line (b) shows the baseline. Reproduced with permission [53].
4.1.2. Pulse sequences for minimizing effects of eddy currents and J-modulation The pulse sequences described in the following text all require phase cycling for coherence transfer pathway (CTP) selection. The principles of CTP through phase cycles are, of course, well known [1,46]; but in practice the construction of cycling procedures that are both efficient and effective is not straightforward. As complete phase cycles for coupled spin systems may be very time consuming, it is common to reduce the number of steps by removing what are thought to be the least important parts. However, combinations of incomplete phase cycles with gradient pulses are not unique and cycling procedures are often determined by trial and error without optimization. The optimum phase cycle depends on the nature of the sample but the weighting of the various CTP’s also depends on the properties of the RF pulses and the presence of transport phenomena. Jerschow and Mu¨ller have recently developed a method for evaluating the latter effects by simulating the CTP selection
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Fig. 8. Pulse sequences (a) GCSTE [Phase: P1: 0213, P2: (0828)2 (1838)2, P3: (0424)2 (1434)2, Rec: f 2 ⫹ f 3 ⫺ f 1] and (b) GCSTESL [Phases: P1: 0213, P2: (0828)2 (1838)2, P3: (0424)2 (1434)2, P4: 13 ⫹ 064264 ⫹ f 2 ⫹ f 3 ⫺ f 1, Rec:. f 2 ⫹ f 3 ⫺ f 1] [54].
process and have implemented this method in a computer program named CCCP [47]. For most of the pulse sequences illustrated here, the phase cycles actually used are listed in the figure captions. 4.1.2.1. Longitudinal eddy current delay (LED) sequence In spite of the best efforts, eddy current effects are still significant, and they depend on the strength of the gradient pulses. This, of course, can be disastrous for experiments that automatically sample a wide range of gradient values. The stimulated echo (see Fig. 2) is primarily affected by eddy currents induced by the final gradient pulse, and the problem is exacerbated by the need to keep t short in order to minimize transverse relaxation and Jmodulation. This conflict motivated Gibbs’s modification of the STE sequence as illustrated in Fig. 5 [48]. The major change is the addition of a fourth 90⬚ pulse at the center of the stimulated echo for the purpose of storing the magnetization in the longitudinal direction while the eddy currents decay. After the eddy current settling period Te, the magnetization is recalled with a fifth 90⬚ pulse and the FID is acquired. The effectiveness of this sequence is further enhanced by adding three gradient pre-pulses (not shown) to make a chain of five equally spaced pulses. This arrangement ensures
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that the transient magnetic fields resulting from previous gradient pulses have the same effect during the transverse evolution periods after the first and third RF pulses. Either phase cycling or homospoil pulses can be used to eliminate transverse components during T. However, phase cycling of the last two RF pulses is essential to remove the effects of longitudinal relaxation during Te. This can be accomplished by alternately storing the STE in the ⫹z and ⫺z directions and then taking the difference between the associated signals that are returned to the xy-plane by the fifth RF pulse. We note that the LED sequence is analogous to B1 gradient experiments. The first 90⬚x –g–90⬚⫺x composite pulse sandwich where g gz is equivalent to a single B1 gradient pulse in encoding the spatial position in longitudinal magnetization [21,49]. Then after the storage period T, the second sandwich decodes position and again stores longitudinal magnetization. Hence LED can stand for either Longitudinal Encode–Decode or Longitudinal Eddy-current Delay. The LED sequence significantly improves the quality of spectra obtained in FT-PFG-NMR experiments but still suffers from the slowly decaying eddycurrents. The consequence is that the Te period, required to obtain undistorted spectra in experiments with large gradient pulses, can be unacceptably long. There is also the problem that the gradient pre-pulses introduce additional heat. 4.1.2.2. Bi-polar LED (BPP-LED) sequence One of the best ways to diminish the eddy currents induced by a short gradient pulse (g) is to replace the pulse with two pulses of different polarity separated by a 180⬚ RF pulse, i.e. the composite bipolar gradient pulse combination (g–180⬚–( ⫺ g)). As previously noted, the effective gradient g* is equivalent in these two cases; but the composite pulse provides self-compensation of the induced eddy currents [50]. Gradient pulse sequences with alternating polarity were introduced into PFG-NMR by Karlicek and Lowe [24] to take advantage of the fact that the 180⬚ RF pulses refocus static gradients. Also, Cotts et al. [51] proposed a number of STE sequences with alternating grading polarities to minimize the effect of background gradients. More recently Fordham et al. [52] replaced all of the gradient pulses in the LED
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background gradients, and more importantly for DOSY it refocuses chemical shifts. The latter can be very important when chemical exchange [33] or spin diffusion is present [54]. At present the BPP-LED sequence is the sequence of choice for many DOSY experiments, especially those requiring maximum gradient strengths with small temperature rises. It, of course, can be combined with relaxation filters, water elimination sequences, etc. as desired.
Fig. 9. Velocity insensitive gradient pulse sequences: (a) A gradient sequence with zero first moment (see text), and (b) the double STE sequence (DSTE) [59].
sequence with bipolar pulse pairs (BPPs) to permit diffusion measurements in the presence of large background gradients. The LED experiment with BPPs shown in Fig. 6 was investigated for DOSY applications by Wu et al. [53] With short gradient pulse separations (t ⬃ 1 ms) the BPPs were found to cancel more than 95% of the eddy currents, and undistorted signals could be obtained with Te reduced by a factor of 20. The effectiveness of the BPPs in reducing signal distortions is illustrated in Fig. 7. Further, this improvement could be obtained without the need for gradient prepulses, thus reducing undesirable heating effects. The use of Eq. (9) with the effective gradient g* for the BPP-LED sequence gives the corrected attenuation factor:
c D ⫹ d ⫹ 2t exp⫺Dq2 D ⫺ d=3 ⫺ t=2:
13
Of course, eddy current compensation is more complete when both d and t are short. The extra 180⬚ pulses introduced here cause some loss of signal because of the greater sensitivity to inhomogeneities in the RF pulses. However, this turns out to be an advantage because signal acquisition is limited to the region where the gradient is constant and higher quality data result. Also, the refocusing effect of the 180⬚ pulses does eliminate the effect of steady
4.1.2.3. New sequences and comparisons The choice of a pulse sequence for DOSY depends on both the capabilities of the available instrumentation and the nature of the system under study. When very small diffusion coefficients are involved, large gradient amplitudes are required to obtain adequate signal attenuation. As disturbances to the local field and the lock signal increase with the amplitude of the gradient pulse, the LED feature is often required. However, when only modest gradients are required and an efficient, well designed probe with active shielding is available, the settling period Te may not be necessary. In such cases the BPP-STE sequence should be considered [54,55]. The direct detection of the STE, without storage and recall, is especially important when the phase of the echo must be determined as in MOSY experiments [8]. The self-compensating feature of BPPs is extremely important in eddy current reduction, but there are other benefits from the (g)–180⬚–(⫺g) composite pulse as well. The refocusing and cancellation of steady gradients in inhomogeneous systems was the initial motivation for the introduction of alternating gradients in PFG-NMR. In high resolution NMR two other effects are encountered that are also sensitive to refocusing. In the absence of a 180⬚ pulse the chemical shifts are encoded along with the position dependent phase information during the first transverse interval of the STE sequence. When g 苷 0 chemical shift information usually does not appear in the STE; however, when spin exchange or chemical exchange interchange chemical shifts during the storage interval T, the chemical shifts do affect the amplitude of the STE [33,54]. With coupled spin systems there is another consequence of unrefocused chemical shifts because the second 90⬚ RF pulse generates zero-quantum coherences (ZQCs) in
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Fig. 10. Block diagram of the stop-and-go sample spinning system. Reproduced with permission [60].
addition to z-magnetization [22,56]. In cases where the ZQCs decay more slowly than the longitudinal magnetization, phase errors become evident with long values of the storage time T [54]. Fortunately, the (g)–180⬚–(⫺g) sandwich eliminates both the exchange and ZQC problems, and as discussed earlier the extra 180⬚ pulses increase the quality of the collected data because of their tendency to restrict the excitation volume. The negative aspect is that the BPP-STE and BPP-LED sequences require considerable phase cycling. This is not a problem when signal-to-noise ratios are low and signal averaging is already necessary, but with strong signals the experimental time must be increased to accommodate the phase cycle sequences. Another potential disadvantage of sequences incorporating BPPs is that the total amount of time required to complete the composite gradient pulse pair will exceed the time required for a single gradient pulse. When T2 is very short, the extra amount of time with transverse magnetization will lead to loss of signal. Pelta et al. [54] have recently compared the STE based PFG-NMR sequences in use (STE, BPP- STE, LED, and BPP-LED) and have suggested two additional sequences with self-compensating gradient pulse pairs. The new sequences, known as gradient compensated stimulated echo (GCSTE) and gradient compensated stimulated echo spin lock (GCSTESL), are illustrated in Fig. 8(a) and (b), respectively. Both of these sequences feature BPPs in which
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one of the pulses is placed in the storage interval T. This arrangement avoids the extra 180⬚ pulses required when the BPPs are placed in the transverse intervals t and provides a homospoil effect as well. The major advantage is a reduction in the required phase cycling. The disadvantages are (1) double the amount of heating for the same q values, (2) potential phase anomalies associated with ZQCs for coupled spin systems, and (3) amplitude modulation effects associated with spinexchange and chemical exchange. The GCSTESL sequence incorporating the spin locking interval t ST corrects the line shapes for ZQCs but not for exchange effects, and provides somewhat better resolution by restricting detection to the region of constant gradient. We conclude that all of the BPP-STE sequences are useful under appropriate circumstances. In the absence of spin coupling and exchange effects and when only small gradients are required, the BPPSTE and GCSTE sequences are reasonable choices. When exchange effects are not present and modest gradients suffice, GCSTESL becomes a strong candidate, but in the general case requiring chemical shift refocusing, background gradient compensation, and strong gradients the BPP-LED sequence is indicated. These conclusions are, of course, based on current technology. 4.1.3. Suppression of convection current effects Convection currents are easily induced in nonviscous samples by temperature gradients. The well known Rayleigh–Benard instability results from temperature inversions where higher temperature, lower density layers lie below lower temperature, higher density layers. The resulting fluid flow produces a distribution of velocity components parallel to the z-gradients in typical PFG-NMR experiments, and a corresponding attenuation of the STE that interferes with diffusion measurements [57]. This effect is characterized by a downward curvature in plots of the logarithm of the STE amplitude versus q 2. However, the deviation may not be easy to distinguish from a simple increase in the diffusion coefficient. Of course, the velocity distributions found in gravity driven mass convection have less effect on diffusion measurements made with gradients in the x or y directions. The first protection against convection currents is a well designed temperature control system that can
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minimize temperature gradients in the sample. Such systems usually require a large flow rate for the heat exchange gas. Also, sample tubes with smaller diameters are less susceptible to convection. An extreme example is the stabilization of solutions of polymers in liquid and super-critical CO2 in fused silica capillaries having inner diameters of approximately 100 mm [58]. When one must deal with low viscosity solvents at temperatures far from ambient, mass convection is often difficult to avoid. In such circumstances it is advisable to use pulse sequences that are insensitive to constant velocities. Such sequences can be constructed by requiring that the first moment of effective gradient sequence, g*, vanish [14], i.e. Zt 0
g* t 0 t 0 dt 0 0:
14
A gradient pulse sequence satisfying Eq. (14) is shown in Fig. 9(a). The velocity insensitive double STE diffusion sequence (DSTE) shown in Fig. 9(b) was proposed by Jerschow and Mu¨ller [59]. (Here selection of the proper coherence pathway is essential because g* pg.) They found that the diffusion coefficient measured for test a sample of 80% glycol in DMSO-d6 (Bruker temperature-calibration sample) at 347 K with an uncompensated BPP-LED sequence showed a deviation of a factor of 9 from the correct value. This deviation was eliminated when the compensated DSTE sequence or a BPP version of it was used. 4.1.4. Dispersion and resolution enhancement A hard lesson for most experimentalists to learn is that sums of exponential functions with added noise are extremely difficult to resolve into unique sets of components. Even with two components having well separated decay constants, one pays a considerable price in accuracy and computing time compared with the analysis of a single exponential decay. Therefore, for mixtures of monodisperse components the highest accuracy in DOSY analyses will be obtained for components that are already completely resolved in the chemical shift dimension. The realization of this fact has motivated attempts to obtain resolution of NMR peaks through experimental refinements and signal processing. In the following we consider
resolution enhancements for 1D-NMR. 3D DOSY based on 2D NMR is covered in Section 7.
4.1.4.1. Stop-and-go spinner Sample spinning in NMR is often the final step in getting the maximum possible resolution. When line widths are dominated by magnetic field inhomogeneities, spinning at even modest frequencies, e.g. 20 Hz, can offer significant resolution enhancement. However, it is common knowledge that conventional sample spinning is not compatible with PFG-NMR diffusion measurements because of unavoidable sample movement in the gradient direction. Here the restrictions are severe as PFG-NMR experiments can detect displacements in the gradient direction of the order of 1 mm. With modern shim stacks, the requirement for non-spinning samples may not be a serious problem. However, if spinning is a necessity, the options are limited. It is probably possible to design a precision spinner that will limit sample excursions in the z-direction to acceptable bounds. But this is likely to be a costly venture and to require much less convenient sample tubes and handling procedures. Fortunately, a low cost alternative, based on the fact that a stationary 5 mm NMR sample tube can be spun up to the required 20 Hz speed in approximately 10 ms, has been demonstrated. Fig. 10 shows a block diagram of a stop-and-go spinner system that is compatible with high-resolution NMR spectrometers [60]. A computer controlled DC servo motor is directly coupled to the sample tube so that the sample tube can be arrested during the motion sensitive part of the experiment and can be restarted during the delay period Te when the magnetization is stored in the z-direction. The spinning sample is stopped in about 10 ms by an ‘‘active brake’’ after data acquisition, but the liquid in the sample requires 1–2 s to reach a quiescent state that is compatible with a diffusion measurement. With a standard resolution test sample in a 10 mm gradient probe mounted in the narrow bore magnet of a Bruker AC-250 spectrometer, the best nonspinning linewidth was about 2 Hz. This width was reduced to about 0.2 Hz when spinning was controlled by the DC motor. This apparatus has few disadvantages and has been used routinely for data collection in DOSY experiments requiring the best resolution [61].
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Fig. 11. The STE-INEPT pulse sequences for heteronuclear detected DOSY with coherence transfer [68].
4.1.4.2. Reference deconvolution Even with the best efforts at field shimming, eddy current avoidance, sample spinning, etc., one is left with line shapes in FT-PFG-NMR experiments that suffer from experimental artifacts. If the remaining linewidths interfere with the resolution of components, it is worthwhile to consider postprocessing of the data to enhance the resolution or to improve the line shapes. Consider a single NMR line with a frequency offset of V. The corresponding FID has the form f t g texp ⫺iVt with g t s tb t where s(t) is the ideal response obtained with a perfect instrument and b(t) accounts for all deviations from ideality. The line shape in the frequency domain, I(v ), is given by the Fourier transform of the product s tb t I v FTb t·s t
15
and according to the convolution theorem this shape is
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also equal to the convolution of B v FTb t with S v FTs t or I v B v*S v [1,62]. The ideal spectrum with a shape depending only on the intrinsic T2 can in principle be extracted from the experimental shape by simply dividing f(t) by b(t) prior to performing the Fourier transformation. One can even remove some of the intrinsic linewidth by additional filtering of the FID at the risk of adding noise and distorting the frequency domain signal. The tradeoffs between sensitivity and resolution have been investigated in detail [63], and it is now common practice to multiply the FID with a weighting function that provides some resolution enhancement without introducing unacceptable noise. Common examples are the sine–bell and Lorentz–Gauss weighting functions. The demands on postprocessing in DOSY may be especially severe as the FIDs are acquired in the presence of some level of transient magnetic fields and the strengths of the transient fields depend on the amplitude and duration of the gradient pulses. Therefore, the free induction decays may not be truly free, and the Fourier transforms of decays acquired in the presence of changing frequencies cannot produce symmetrical line shapes. A reasonable approach to this problem is to use the experimental FID itself for deconvolution. This appealing idea, now known as reference deconvolution (RDCON), has been rediscovered many times in different contexts [64]. In practice one locates a signal from a singlet
Fig. 12. The pulse sequence for shuttle based fringe field 2D-DOSY [phase cycle: P1: 08, P2: 08, P3: 08, P4: 0202 0202, P5: 0220 1331, Rec: 0022 1133] [74].
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Fig. 13. Comparison of FFT and ILT transformations.
component in an NMR spectrum, zeros the remainer of the spectrum, and performs an inverse FT to obtain the reference FID, g(t). Division of the experimental FID by g(t) corresponds to removing all linewidth and does not lead to useful results. However, multiplying the FID by s t=g t, where s(t) is a guess at the ideal FID for a single line, is practical and useful. The implementation of the RDCON method requires that a complex reference signal be isolated and that a spectral region be available that contains sufficient ranges of both absorption and dispersion mode signals. The magnitude of the reference FID must be greater than zero and the baseline must be chosen carefully. These and many other practical considerations are discussed in detail in a recent review article by Morris et al. [64]. Also, a software package named fiddle has been prepared for certain spectrometers. Impressive demonstrations of RDCON in DOSY data processing have been reported, and it is claimed that with modern actively shielded diffusion probes the improved diffusion fit gives an increase in accuracy by about a factor of three [64,65]. Differences in diffusion coefficients of 0.5% are distinguished in this work. 4.1.4.3. Heteronuclear NMR Still another approach to the problem of peak overlap is to increase the spectral dispersion. The simplest way to do this is to increase the chemical shifts by working at the largest possible magnetic fields. Unfortunately, the chemical shift range for protons is inherently small and spectral crowding may be a problem with commonly available
500 and 600 MHz spectrometers. Chemical shifts are, of course, much larger for nuclei with larger atomic numbers and heteronuclear NMR may be an alternative. Consider for example 13C NMR. In addition to the wider range of chemical shifts, resolution for 13C is aided by longer values of T2, the virtual absence of homonuclear spin-spin coupling, and lower sensitivity to magnet field inhomogeneities and eddy current effects relative to 1H because of the smaller gyromagnetic ratio. The problem with 13C for DOSY applications is that the sensitivity is quite low. One must contend with low natural abundance, and the low gyromagnetic ratio and the long relaxation times that enhance resolution also limit signal-to-noise ratios. Further, the all important q factor is proportional to the gyromagnetic ratio, and this can be a problem when available gradient amplitudes are small. Coherence transfer experiments can solve the q-factor problem while improving sensitivity. The idea is to transfer polarization information from 1H to 13C prior to detection. This can be accomplished by combining a STE based PFG sequence with INEPT [66] or DEPT [67]. A simple INEPT-DOSY sequence is shown in Fig. 11 [68]. After the first 90⬚ proton pulse, a composite BPP encodes spatial positions and the magnetization is then returned to the z-direction by the second 90⬚ pulse. The optional homospoil gradient pulses during the storage period T eliminate residual magnetization on the xy plane. Then detection is carried out with a refocussed-decoupled INEPT sequence [69] where an initial 90⬚ heteronuclear RF pulse has been added to remove the natural heteronuclear spin polarization. An example of the 13C INEPT-DOSY is shown in Fig. 14 of Section 4.2.1.1 for a test sample containing sucrose, glucose, and sodium dodecyl sulfate (SDS). 4.1.5. Data collection Assuming that provisions have been made to obtain maximum resolution and undistorted absorption mode signals for each value of q, there are still choices to be made concerning data collection. It is important to select the minimum number of q 2 values that will permit the fastest and slowest decaying components to be analyzed with sufficient accuracy. Minimizing the number of q 2 values will, of course, increase the
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Fig. 14. 13C INEPT-DOSY for a mixture containing glucose, sucrose, and SDS in D2O. Dotted lines show average diffusion coefficients of each component. The 1D 13C INEPT spectrum of the mixture is shown at the top. Reproduced with permission [68].
efficiency of signal averaging and permit samples having limited lifetimes, e.g. blood plasma, to be studied. Also, the judicious choice of sampling values can avoid the collection of useless data sets that do not provide enough information for the determination of diffusion coefficients of all the components. We assume that the attenuation of NMR peaks with increasing gradients strengths can be represented by functions of the form S x exp ⫺Gx, or weighted sums of such factors, where G D; x q2 D 0 and D 0 is the diffusion time appropriately corrected for the shape and duration of a gradient pulse. As the G values required to fit diffusion data for all the peaks in an NMR spectrum may differ by orders of magnitude, it is seldom appropriate to use linear spacing of x values. In fact, when the ratios of G values exceed three, linear spacings lead to very large errors. Logarithmic spacing of x values can be a very good choice if the parameters are properly chosen [70]. A convenient form for computing xn is shown in Eq. (16) xn x1 2 n⫺1=m :
16
We take the smallest decay ‘time’ to be tmin 1=Gmax and the largest to be tmax 1=Gmin , and it is reasonable to set xN 5tmax where N is the maximum value of n. The choice of the first data point, x1, is a bit more
tricky because it is easy to waste data points at small x values while undersampling the larger values. We suggest choosing x1 0:2tmin and then computing, or determining by trial and error, the value of m that will permit N data points to fit in the desired range (according to Eq. (16) the appropriate value is m N ⫺ 1ln 2=ln xN =xl ). Labadie et al. have investigated errors in biexponential data analysis with various sampling schemes [71]. They find that geometric spacing with xn x 1
an ⫺ 1 a ⫺ 1
17
can give low errors and a reasonable distribution of data points along the decay. Here a is the constant ratio of successive interval lengths. The reported error analysis suggests that with logarithmic spacing x1 should not be smaller than t min/5.8, while for geometric spacing the optimum value of x1 is t min/ 33.4. In some applications it is also important to collect a data point with a very small value of q. For example, systems involving chemical exchange may show large initial slopes in plots of ln(Signal) versus q2 D 0 with strong curvatures that make the extrapolation to the zero point difficult. However, one must bear in mind that with a zero gradient, the echo amplitude in a STE
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experiment will depend on the chemical shift unless bipolar gradient pulse pairs are used. Also, very small q values introduce two additional problems. For a sample of length L in the direction of the gradient if q is not much larger than 2p /L, the magnitude of the stored magnetization will depend on the phase of the second 90⬚ pulse. Also, restricted diffusion will introduce errors unless DD=L2 ⬍ 0:001 [72]. 4.1.6. Utilizing the stray field to obtain large, steady gradients NMR measurements of very small diffusion coefficients require large gradient amplitudes, but in PFGNMR large gradients invariably produce heating effects, vibrations, and eddy currents. These problems were avoided by Kimmich et al. in an elegant experiment making use of the steady gradient in the fringe field on a superconducting magnet [73]. Experiments analogous to PFG-NMR can be run in the fringe field with an STE sequence as encoding/decoding occurs only when the magnetization is transverse to the main field. There are, however, two major problems with fringe field diffusion measurements. First, data collection in the presence of a large gradient eliminates resolution and permits only the echo amplitude to be recorded. Thus, all structural information is lost, and it is not even possible to distinguish between the resonances of 1H and 19F. If resolution is not essential for some sample, there is still the problem that available RF pulses excite only a thin slice of a sample in a strong gradient field. DOSY experiments with resolution in both the chemical shift and diffusion dimensions require that the FIDs be acquired in a homogeneous magnetic field. This was accomplished in stray field DOSY [74] with a shuttle system similar to that used in zero-field NMR experiments [75]. The LED pulse sequence used in this experiment and the location of the sample as a function of time (dotted line) are shown in Fig. 12. The STE is created by the first three p /2 RF pulses, the fourth p /2 pulse stores the echo magnetization in the z-direction for the shuttling period Ts, and the fifth p /2 pulse recalls the magnetization for detection in the homogeneous field. A widebore (89 mm) magnet was equipped with a homebuilt shuttle probe containing electronics for excitation in the fringe field at 140 MHz and detection in the homogeneous field at 360 MHz. A Kel-F
(36 ml) sample cell was shuttled pneumatically in a precision bore quartz tube a distance of about 0.3 m between the two RF coils. The RF pulse widths in the homogeneous field and the fringe field were 10 and 1.4 ms, respectively. Accordingly, only a 0.3 mm thick slice was excited in the fringe field (g 53 T m ⫺1) and special phase cycling was necessary to select only the spins excited and encoded with diffusion information in the fringe field from the background of all protons excited in the homogeneous field. A DOSY experiment was performed on a test sample containing SDS, glycerol, and H2O (1:2:4 by weight) with T 5 ms, shuttle time Ts 150 ms, and d incremented from 10 to 300 ms in 18 steps [74]. This provided satisfactory resolution for the test sample even though the NMR linewidths exceeded 20 Hz because of susceptibility differences between the sample and the short cylindrical cell. This experiment demonstrates that fringe field DOSY experiments are practical for samples having sufficiently long T1s and well resolved spectra, and the advantages are considerable. Experiments can be performed with very short diffusion times and with gradient amplitudes not easily obtained in other ways. Further, this can be done without gradient drivers and coils and with relatively inexpensive, though unorthodox, instrumentation. 4.2. Data inversion and display The ability of DOSY to provide accurate distributions of diffusion coefficients for 1D analysis or for the construction of 2D and 3D DOSY displays depends on the inversion of data sets that consist of NMR spectra collected with predetermined values of q2 D 0 . In the following we assume that FT-PFG-NMR experiments of the STE or BPP-STE types provide 2D data sets of the form: X An nm exp⫺Dn D 0 q2 18 I q; nm n
where An nm is the amplitude of the 1D-NMR spectrum of the nth diffusing species when g is small but not zero (see Section 4.1.5), and Dn is the associated tracer diffusion coefficient. Also, D 0 D ⫺ d1 where 1 depends on the shape of the gradient pulse. The goal of DOSY analysis is to transform 2D data
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sets as shown in Eq. (18), which are basically stack plots of attenuated spectra, into 2D spectra with chemical shifts on one axis and the distribution of diffusion coefficients on the other. We define a ‘discrete’ system as one that can be described by small set of values of n in Eq. (18). The maximum value of n may be more than ten, but for a particular peak in a high-resolution spectrum we expect that the attenuation can be described by a few values of n, i.e. usually single exponential fits. In the event that a polydisperse component contributes to the peak at frequency n , a continuum of values of D must be considered. The 1D data set I(s) describing attenuation of this peak must then be described by: Z∞ a lexp ⫺lsdl 19 I s 0
where l D D ⫺ d1 and s q 2. (An alternative choice is l D and s q2 D ⫺ d1.) In Eq. (19) we recognize that I(s) is the Laplace transform of a(l ) and that a(l ) is the Laplace spectrum of diffusion coefficients. When only discrete components with the decay constants l i are present, a(l ) is a weighted sum of delta functions, d l ⫺ li and Eq. (18) is recovered. The hope of new comers to this field is that a unique transform, akin to the FT, exists that can produce unique diffusion spectra and can invert them to recover the decay curves. This problem is illustrated in Fig. 13. At the top an FFT converts the FID to a unique NMR spectrum including the line shapes. At the bottom the decay on the left contains two components with diffusion coefficients and amplitudes differing by a factor of two. A perfect transform would produce the Laplace spectrum of delta functions shown on the right and the inverse transformation would exist. In fact, there is no perfect transform, and in the presence of noise it may be impossible to obtain any useful spectrum. The dotted curves in the diffusion spectrum indicate broadening associated with errors in an approximate transformation that would be acceptable. Gardner et al. [76] took a significant step toward finding the Laplace spectrum for discrete sums of exponential components by introducing the transformation s exp(x) and constructing the function exp(x)a[exp(x)]. This function turns out to be the convolution of the desired spectrum of decay
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constants with the shape function exp(x) exp[⫺exp(x)]. Thus the solution is at hand if the shape function can be removed by deconvolution. The catch is that the removal of the large line widths by means of Fourier deconvolution or any other method usually introduces unacceptable errors, e.g truncation artifacts, and this scheme is not practical, though it can serve as the starting point for more extensive analyses. Actually, the method of Gardner et al., as modified by Swingler [77] was investigated for DOSY applications but was abandoned in favor of the more robust schemes described in the following text [78]. Here we confront the fact that the desired spectrum a(l ) is the inverse Laplace transform (ILT) of the decay function I(s). Computing a(l ) is an ill-conditioned problem and one that may be intractable [79]. Actually, solutions for a(l ) can usually be found that agree with I(s), but they are often not unique. This is the source of much wasted computer time and much nonsense in the literature. The same data inversion problem has been faced for many years by the dynamic light scattering (DLS) community and extensive reviews are available in the literature [80–82]. The success of DOSY analysis hinges on the more modest goal of computing the most likely spectrum of diffusion coefficients by means of an approximate ILT or some appropriate fitting algorithm. It is usually true that FT-PFG-NMR data sets do not contain enough information to permit an exact analysis. Reasonable assumptions must be made and additional information must be supplied. We know for example that NMR absorption mode signals are positive and that their Laplace spectra are also positive as are the associated decay constants, and in dealing with polydisperse samples it may be reasonable to assume that the distribution functions are in some sense smooth. There is also the possibility of combining information from peaks at different chemical shifts, and of course there are physical limits on the maximum possible diffusion coefficients and the minimum values that can be detected in any given experiment. Prior knowledge concerning non-negativity, parsimony, and other features may be essential for obtaining the most likely diffusion spectrum from an experimental data set. DOSY analysis begins with a set of N absorption mode NMR spectra each having v frequency points or
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Fig. 15. Early DOSY display for a sample containing tetramethyl ammonium ions (TMA) and mixed micelles in D2O. The 1H data set was analyzed with DISCRETE. Reproduced with permission [9].
channels. In the following we consider various data analysis schemes that have been implemented to process the channels, groups of channels, or complete spectra. The analyses that are readily available to the NMR community in the form of software packages are emphasized. 4.2.1. Discrete samples Samples are characterized as discrete if they contain N monodisperse components and can be completely described by N pairs of amplitudes (concentrations) and decay constants. The decision to use Eq. (18) as the model function involves absolute prior knowledge (APK) and assumptions because the data set alone may not suffice to distinguish between a continuous distribution and a number of discrete components at the same chemical shift. Even the knowledge that a solution was made with N monodisperse solutes does not assure that N discrete hydrodynamic entities are present because aggregation and chemical exchange may be present. When there is doubt about the nature of a solution, an analysis method that can handle continuous distributions should be applied first. The analysis programs for discrete samples report a list of diffusion coefficients and amplitudes for each chemical shift channel. In principle diffusion spectra can be constructed from these lists with delta functions having appropriate amplitudes at the specified positions on the diffusion axis. However, this
prescription is not satisfactory in practice because analysis errors cause channel to channel fluctuations in the computed diffusion coefficients, even within a single absorption peak. This means that a projection of the diffusion spectra from all the channels onto a single axis will now show clusters of peaks for each diffusing species rather than a single line. A better procedure is to construct the DOSY spectrum with normalized Gaussians having center positions and intensities equal to the diffusion coefficients and amplitudes, respectively [9,83]. This gives a DOSY data set of the form: F D; n
Nl X
Aj nGj D
20
j1
where
" # D ⫺ Dj 2 1 Gj q exp ⫺ 2s2j 2ps2j
21
Here s j can be set equal to the standard deviation reported by the analysis program, a fixed value, or some combination of the two to take into account the local error and the estimated systematic error for the complete data set. The Gaussian component method is now widely accepted for the construction of 2D and 3D DOSY displays. 4.2.1.1. Levenberg–Marquardt The Levenberg– Marquardt (L–M) algorithm is the standard
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Fig. 16. DOSY display of the SPLMOD analysis of simulated data. All diffusion peaks are correctly positioned. Reproduced with permission [83].
non-linear least squares method [84,85]. L–M requires a data set with standard deviations, the model function to be fit, and initial guesses for the parameters. The standard deviations are seldom known, but they can be estimated; and the model function for discrete samples is known in principle. However, it is our experience with typical NMR data sets that fits to sums of two or more exponential components are unreliable unless the signal-to-noise ratios and the ratios of decay constants are large. A major problem is that estimates of the parameters must be supplied, and this is a very bad idea for DOSY analysis. Subjectivity in the analysis must be avoided as far as possible. We recommend that the L–M method be reserved for situations where the model function is a single exponential, and the data set itself can be used to provide estimates of the amplitude and decay factor. Also, it is important that the data be fitted to the exponential function directly rather than fitting the logarithm of the data to a straight line. The latter procedure incorrectly weights the standard deviations and even when corrected standard deviations are supplied appears to give inferior results. With special techniques, for example using highfields, high-resolution spectra can often be obtained
that lend themselves to single component analyses such as L–M. An example is provided by 13C INEPT-DOSY of a mixture of glucose, sucrose (Aldrich Chemical Company), and sodium dodecyl sulfate (SDS) (Aldrich Chemical Company) in D2O (500 mM each) [68]. The spectrum shown in Fig. 14 was obtained with a Bruker AC-250 spectrometer, the gradient driver was home-built [45], and the gradient probe was a modified Bruker dual probe. The pulse sequence shown in Fig. 11 was used with D 0.103 s, and the q-values ranged from 11 364 to 681 880 m ⫺1 in 15 steps. For analysis, the area of each peak was fitted to the single exponential decay in Eq. (13) by non-linear regression with the L–M method, and the 2D DOSY was displayed and plotted by Felix (Hare Research, Version 1.1). The 1D 13C INEPT spectrum at the top of Fig. 14 shows that most of the peaks are well resolved, and the 2D DOSY display permits the diffusion coefficients of all components to be easily read: glucose (3.60 × 10 ⫺10 m 2/s), sucrose: (2.92 × 10 ⫺10 m 2/s), and SDS: (7.78 × 10 ⫺11 m 2/s). Note that SDS is involved in rapid exchange between monomers and the dominant micelles. The line widths for the Gaussian components (diffusion dimension) were taken to be the fitting errors reported by the L–M routine.
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Fig. 17. DOSY display of the SPLMOD analysis of simulated data. The ‘‘cross-talk’’ artifact is illustrated. The correct (input) diffusion coefficients are indicated by dotted lines. Reproduced with permission [83].
There turns out to be one significant overlap (sucrose and SDS) in the 13C NMR spectrum, and this produces a stray peak in the L–M analysis. Multi-component programs such as DISCRETE and SPLMOD can resolve the diffusion peak as the diffusion coefficients of the components differ by a factor of 3.8, but at the cost of increased errors elsewhere. 4.2.1.2. DISCRETE The Fortran program DISCRETE [86,87] has been freely distributed for many years. DISCRETE makes use of data transforms to estimate the parameters and then proceeds with a non-linear least squares analysis. It, therefore, does not require initial guesses, and it provides best fit results for a range of N values. DISCRETE was, in fact, used in the first reported DOSY analysis to determine amplitudes and decay rates for every frequency point (channel) [9]. A drawback of this procedure is that the results obtained for different frequencies within a single NMR peak seldom agree exactly. An early example of DISCRETE analysis is the stack plot DOSY display in Fig. 15 for a sample containing 10.0 mM tetramethylammonium chloride
(TMA) with mixed micelles[4.00 mM SDS and 8.00 mM octaethylene glycol dodecylether (C12E8)]. The 1H spectra were obtained at 295 K with a Bruker AC-250 spectrometer and custom built probe and gradient driver. A LED pulse sequence (Fig. 5) was used with D 100.0 ms, t ⫺ d 0.500 ms, and Te 100.0 ms; and NMR spectra were collected with qvalues ranging from 144 to 8.19 × 10 3 cm ⫺1. The 2D data set was analyzed channel by channel using DISCRETE, and the widths of the Gaussian components were chosen to be the average of the s i 0 values for the complete data set.
4.2.1.3. SPLMOD SPLMOD [88,89], the successor to DISCRETE, is also freely distributed. SPLMOD also analyzes sums of exponentials without requiring initial guesses, but is not restricted to exponential components, and permits the analysis of Nd 1D data sets simultaneously. The user supplies the data sets and specifies the maximum number of components to be searched for in an analysis. The major advantage for DOSY applications is that all of the frequency points in an NMR peak or multiplet can be fit in parallel with the restriction
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Fig. 18. DOSY displays for a mixture containing D2O, TEA, and SDS micelles. (a) 2D spectrum generated directly from SPLMOD ‘‘best fit’’ parameters. (b) 2D spectrum generated from SPLMOD parameters after processing with rejection criteria (see text). Reproduced with permission [83].
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that all of the points must share the same small set of decay constants [83]. Successful SPLMOD analysis of PFG-NMR data requires decisions concerning the setup, the application of APK, and the screening of potential solutions [83]. An input program must be prepared to take a set of spectra processed by FELIX or some other NMR software package and to arrange the data according to user specifications in a form that can be read by SPLMOD. In the implementation of Morris and Johnson [83], the input program prompts the user for a threshold value and then divides the spectrum into regions containing the data columns to be analyzed. The user also specifies the form of the decay kernels and other restrictions. For DOSY, exponentials with positive amplitudes and decay constants are selected; and the maximum number of discrete components is specified to be three. This means that SPLMOD will automatically carry out analyses with one, two, and three components. The SPLMOD output consists of a set of decay constants, a set of amplitudes, and the associated standard deviations for each region of the spectrum. These results are repeated for one to three component fits and the ‘‘best fit’’ is determined. However, there may be reasons to select one of the other results. For example, experimental artifacts such as residual eddy currents and phase errors may distort the decays in ways that cannot be represented with exponential kernels. In such cases, fits with larger numbers of components tend to give smaller deviations in spite of the lack of any physical justification. A pragmatic operating procedure is to reject fits on the basis of a list of reasonable criteria. If the solution with Nl components is rejected for any reason, the solution with Nl ⫺ 1 components is evaluated, and so on down to one component. This procedure may lose some authentic peaks, but at least it does not introduce spurious results. On the basis of experience the following criteria were selected for the rejection filter [83]. 1. Diffusion coefficients must be feasible and experimentally accessible. 2. Standard errors in diffusion coefficients must be less than 30%. 3. Pairs of diffusion coefficients must differ by a factor of two or more.
Numerous simulated data sets with added noise have been analyzed with SPLMOD to investigate the range of applicability of this program. Fits to the simulated data sets were quite successful; but, of course, these have the advantage that the decay kernels are truly exponential. For example, a data set was synthesized for two isolated Lorentzian peaks and three overlapping Lorentzian peaks with a maximum S/N of 20 and half widths 5.0 Hz as shown at the top of Fig. 16. The set contained 50 spectra with q-values ranging from 556 to 2.78 × 10 4 cm ⫺1 and the decay constants were given by lj Dj D ⫺ d=3 with D 0.1 s and d 0.0 s. In the SPLMOD analysis the NMR channels were divided into three sets and a maximum of three components were allowed for each set. The result of the analysis is shown in Fig. 16 where the 2D spectrum is projected onto the left hand diffusion axis to show the complete diffusion spectrum. The diffusion peaks reproduce quite accurately the input values of the diffusion coefficients (dotted lines). Fig. 17 illustrates a problem encountered in the analysis of a synthesized data set that was not caught by the listed criteria. The simulation contains three Lorentzian peaks with a maximum S/N of 20 and half widths of 10.0 Hz, and the central peak has two distinguishable components. In the analysis 50 spectra were used and 85 frequency points were analyzed simultaneously with a maximum of four components. Unfortunately, the SPLMOD generated 2D-spectrum shows three diffusion components for the center peak instead of the correct two components, and the peaks for the valid components are displaced from the correct values (dotted lines). The extra peak at 515 Hz is a result of ‘‘cross-talk’’ with the peak at 475 Hz. This artifact is associated with the number and magnitude of decay constants in one analysis block and does not require overlapping peaks. We find that when a decay constant is required in some frequency channels it will be used elsewhere throughout the block to improve the fit. One strong lesson is that three components at the same chemical shift with decay constants differing by factors of less than ten always lead to considerable errors. Real data may differ from simulated data because experimental artifacts can affect the decay kernels. One consequence is that the decays of isolated single components may be better fitted by mixing in a small amount of a second or third component. The effect of
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Fig. 19. DOSY display processed by CONTIN for a mixture containing D2O, glucose, and methyl cellulose. Reproduced with permission [83].
the rejection filter in this situation is illustrated in Fig. 18 for the SPLMOD analysis of a mixture containing 40 mM tetraethylammonium chloride (TEA) and 20 mM SDS [83]. An LED experiment was performed with 40 q-values ranging from 167 to 7.79 × 10 3 cm ⫺1, d from 1 to 2 ms, D 100.0 ms, Te 50.0 ms, and t 2.50 ms. Each spectral region was then fitted with a maximum of two components (NNL 2). The DOSY display in Fig. 18(a) was computed directly from the SPLMOD ‘‘best fits’’ without the benefit of the post analysis rejection filter. The real overlap of TEA and SDS peaks at 1.11 ppm is properly handled, but the other major peaks all show errors in position and artificial companion peaks here enclosed by dotted ellipses. The amplitudes of the peaks and the shifts in positions are evident in the projected diffusion spectrum on the left hand side. The remedial effect of the rejection filter is clearly shown in Fig. 18(b). Here the spurious two-component fits were rejected in favor of single-component fits on the basis of standard deviations and peak
separations. Thus, all artifacts were eliminated and consistent diffusion spectra were obtained. From the spectrum the diffusion coefficients for HOD, TEA, and SDS were found to be 1.74 × 10 ⫺5, 4.54 × 10 ⫺6, and 8.84 × 10 ⫺7 cm 2 s ⫺1, respectively. The conclusion is that filtered-SPLMOD analysis of real data, where two components occasionally overlap, can give reasonable results. It is easy to imagine extensions of the rejection criteria to take into account data from other spectral regions and our knowledge of the features of NMR spectra of complex spin systems. For example, with complex molecules it is unlikely that a decay constant would appear at only one chemical shift. Ultimately, artificial intelligence based software may provide the best route to the ‘‘most likely’’ diffusion spectra. 4.2.2. Polydisperse samples Many systems of interest are characterized by continuous distributions of diffusion coefficients, e.g. polymers and vesicles. In such cases Eq. (18) describing the 2D FT-PFG-NMR data set must be
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replaced with: Z I q; nm R T1 ; T2 a D; nm exp⫺q2 D D ⫺ d1dD 22 where R T1 ; T2 accounts for nuclear spin relaxation during intervals of the pulse sequence and a(D) is the mass weighted distribution of tracer diffusion coefficients D. The Laplace spectrum in this case is the product R T1 ; T2 a D; nm rather than the desired distribution a(D). If the relaxation times are correlated with the diffusion coefficients, the separation of a(D) may not be possible. Fortunately, in high molecular weight polymers the relaxation rates often depend on segmental motion and are approximately independent of the molecular weight thus permitting R T1 ; T2 to be replaced with a constant [90,91]. An interesting situation arises with phospholipid vesicles where T2 values for the bilayer protons depend on tumbling rates of the vesicles and hence on their sizes. The result is that the a(D) distribution obtained for bilayer protons contains an excessive contribution from the smaller vesicles that have smaller line widths. This produces a larger apparent diffusion coefficient for the bilayer protons than is measured for protons entrapped in the aqueous cavities of the vesicles [70]. In the following we assume that the relaxation factor is constant and can be removed from the integral. As analysis of polydisperse components produces distributions of diffusion coefficients rather than peaks and positions, the plotting of diffusion spectra is more straightforward. However, two points must be considered. First, the programs usually give a small number of points that barely suffice to define spectra when more than one peak is present, especially if one is narrow. Additional points can be generated by interpolation but the line shapes are likely to be distorted. The second, less obvious point, has to do with apparent peak heights and areas in plots of intensities versus D or ln(D). There is really no problem when spectra are plotted with a linear D axis as the peak areas are given by Z a DdD 23 as expected. Confusion arises when the dispersion axis is still based on D but logarithmic spacing is
used because the apparent peak widths have very different meanings in different parts of the D axis. If the diffusion axis is relabeled to be linear in the logarithm of D, e.g. to show ⫺ 4, ⫺ 5, ⫺ 6 in place of 10 ⫺4, 10 ⫺5, 10 ⫺6, it is implied that the integration is linear in ln(D). Therefore, the integration in Eq. (23) must be replaced with Z Z Da DdD=D Da Dd ln D 24 and it makes sense to plot Da(D) rather than a(D) versus ln(D) or log(D). 4.2.2.1. CONTIN with extensions The first program used for DOSY analysis of polydisperse data was CONTIN [83]. This program has been available since the early 1980s and has been extensively tested [80,92,93]. CONTIN uses constrained regularization to fit experimental data, yk and tk, to functions of the form yk
Ng X m1
cm f lm ; tk a lm ⫹
NL X
bi Lki tk
25
i1
where the cm are weights of the quadrature formula, f l; tk exp ⫺ltk are the known decay kernels, Ng is the number of grid points, and a(l ) is determined by the analysis. The second term in Eq. (25) permits a constant background to be included, e.g. NL 1 and Lk1 1 gives the constant background b 1. The problem with solving Eq. (25) for a(l ) is that an infinite number of oscillatory solutions for a(l ), that have no physical meaning, are consistent with the data set yk. CONTIN attempts to eliminate oscillatory solutions through the use of constraints based on (a) APK, (b) statistical prior knowledge, and (c) parsimony [94,95]. What this means to the user is that non-negativity is enforced for a(l ) and that, of the solutions not eliminated by (a) and (b), the simplest one is chosen. In this context the simplest solution is the one with the least detail, i.e. the smoothest one with the minimum number of peaks. Smoothness is selected by penalizing solutions on the basis of the integrated squared second derivatives, and the extent of the penalty depends on the regularization parameter chosen. It is straightforward to use CONTIN in DOSY applications because the user only has to supply a
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Fig. 20. DOSY display of a mixture of SDS, ATP, and glucose, all at 0.1 M in D2O. The diffusion axis is in units of mm 2 s ⫺1. (a) The DOSY spectrum, (b) central region of the same spectrum. Reproduced with permission [98].
threshold value, and the program output provides the Laplace spectrum a(l ) essentially ready for display. The major complaint is that the essential smoothing feature broadens all of the peaks so that even monodisperse components show considerable linewidths. For many applications this is acceptable as broader distributions are usually well represented, and in samples having monodisperse and polydisperse components the monodisperse components are usually distinguishable [96].
An example of the CONTIN analysis for a mixture containing both monodisperse and polydisperse components is shown in Fig. 19. Data was obtained with an LED sequence for 75.0 mM glucose and 1.00 wt.% methyl cellulose in D2O at 22⬚C. The 26 q-values ranged from 278 to 2.50 × 10 4 cm ⫺1, with d 1 to 5 ms, D 350.0 ms, Te 50.0 ms, and t 5.80 ms. CONTIN analysis produced 50 point diffusion coefficient distributions for each of the 500 channels in the 2D data set, and an interpolation routine
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was used to generate 512 data points from each 50 point distribution. No user input other than a threshold value was required. This experiment permitted the glucose and methyl cellulose peaks to be easily distinguished and the distribution of diffusion coefficients for the polymer to be measured; but it also demonstrated the unnatural broadening and distortion of the glucose peak and the misleading area ratios arising from logarithmic spacing on the D axis. As expected, the treatment of the monodisperse component is marginal because smoothing by the regularization procedure coupled with the small number of points produces strange peak shapes. We note that the HOD peak in this spectrum was deleted when the DOSY display was constructed. For broad distributions in the presence of noise CONTIN tends to give smaller average diffusion coefficients 具D典 and reduced standard deviations 具SD典/具D典 relative to the true values. These errors result from the tendency of CONTIN to oversmooth a(D) at the small D end while undersmoothing for large D values [93]. The effect, which is clearly evident in distributions that have a small secondary peak on the high D side, arises because the penalty for oscillatory solutions is based on a logarithmic axis. The identity a D dD a DDd ln D suggests that a(D)D rather than a(D) should be analyzed on the logarithmic axis. This change is easily accomplished in CONTIN by setting ‘‘integration off’’ with the control parameter selection IQUAD 1 so that cm 1. With this change most of the ‘‘noise’’ peak disappears, but in order to obtain the greatest accuracy in the computed a(D) function a more sophisticated adjustment of the penalty weighting factor is required [97] (see Section 6.2.4).
4.2.2.2. MaxEnt A DOSY processing module based on the maximum entropy method (MaxEnt) is now included in the GIFA software package for NMR analysis [98,99]. In MaxEnt the idea is to determine the ‘‘most probable’’ Laplace spectrum by maximizing the entropy of the spectral distribution subject to certain constraints. Accordingly, the Laplace spectrum a(l i) in Eq. (19) is associated with the most probable distribution Pi, and the
entropy of the distribution is defined by S⫺
n X
Pi =Fln Pi =F
26
i1
where F is a normalization constant. The normalization of the Pi and the relationship between Pi and the experimental data set Ii specified in Eq. (19) serve as constraints on the allowed Pi values. In MaxEnt the function Q S ⫺ lx2 is maximized for various values of l where x 2 represents the Euclidean distance between the data and the Laplace transform of Pi. MaxEnt is a general purpose analysis method that competes with CONTIN for polydisperse samples and does a reasonable job for discrete samples as well. In a comparison, Levenberg-Marquardt (L–M), CONTIN, and MaxEnt were applied to 100 synthetic data sets containing four exponential components with decay constants ranging from 0.1 to 13.0 and amplitudes between 1000 and 8000 [98]. It was found that L–M was much less accurate than either CONTIN or MaxEnt and that MaxEnt was better than CONTIN at dealing with weak components. Also, CONTIN frequently failed to find the correct number of components because of over-smoothing. MaxEnt analysis for DOSY is illustrated in Fig. 20 for a mixture containing SDS, adenosine 5 0 -triphosphate, and glucose all at 0.1 M in D2O at 35⬚C [98]. A BPP-LED experiment was performed with d 3 ms, D 100 ms, and 28 geometrically spaced gradient values ranging from 1 to 44 G/cm. The Laplace inversion was calculated by MaxEnt on 100 points with a maximum of 1000 iterations, a job requiring 50 min on an R5000 SGI workstation. The only user input was to select 50:1 as the S/N threshold for the maximum signal in each channel. The zoomed insert in Fig. 20(b) shows impressive resolution for such a robust scheme. 4.2.3. Complete bandshape methods PFG-NMR has a major advantage over scattering methods, e.g. dynamic light scattering, because of the additional information obtained from chemical shifts. The problem of resolving exponential components can often be avoided by finding isolated NMR peaks for individual species. The most desirable case is, of course, to have sufficient resolution to avoid any
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overlap. But in the real world overlap often occurs, and it is desirable to have analysis methods that can resolve components with arbitrary amounts of overlap. The methods reviewed to this point have involved analysis at a single chemical shift or a limited range of chemical shifts and have not made use of the total information available. In this section we consider methods that can analyze the complete bandshape. 4.2.3.1. Multivariate analysis Kubista has proposed a method for the analysis of correlated data sets [100]. In particular he imagined two types of spectra being obtained for a mixture with varying concentrations, and he showed that exact solutions for the ‘pure’ spectra of the components could be obtained. Schulze and Stilbs implemented the Kubista scheme (NIPALS) by designing a type of double PFG-NMR experiment with different gradient amplitudes in the two parts [101]. Unfortunately, the two data sets were compromised by this design and the resulting spectra were distorted. The idea was revived by Antalek and Windig who pointed out that, because the decays in PFG-NMR are exponential, the required pair of data sets can be obtained from a single experiment, e.g. spectra 1–10 and spectra 2–11 provide two satisfactory sets [102,103]. Consider experiments giving c spectra having v points each for a mixture with n components. The correlated data sets are represented by: A CP;
B CbP
27
where A and B are data matrices of size cv, C(cn) is the concentration matrix, P(nv) is the matrix of pure spectra, and b(nn) is a diagonal matrix. (Note that the spectra and the concentrations are completely correlated in the two data sets but that there is a scaling factor defined by b.) The Kubista scheme, reformulated in terms of the generalized rank annihilation method (GRAM) [104,105], has been applied to FT-PFG-NMR data sets for mixtures with considerable success. This method, now known as the Direct Exponential Curve Resolution Algorithm (DECRA), is direct, does not require a threshold, can deal with any amount of overlap, is not very sensitive to noise, and further, is fast. The requirement of exponential decays may be a limitation in some cases, and tests with mixtures containing large numbers of components and both
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monodisperse and polydisperse components are necessary; but DECRA appears to be well suited for generating data for DOSY displays. The software routines for DECRA, developed for the MatLab environment, are available for independent testing [103]. Recently, an analysis method known as multivariate curve resolution (MCR), another descendent of NIPALS, has been reported for DOSY and GPCNMR experiments [106]. MCR does not require a priori knowledge except that the spectral intensities and diffusion decay profiles must both be non-negative. In particular, the decay profiles are not restricted to exponentials. This method is, therefore, more generally applicable than GRAM; however, if exponentials are known to be present, it is expected that better answers can be found by making use of that knowledge. Impressive analyses of DOSY data sets by MCR have been reported, and this method deserves much wider use. The implementation involves principal value analysis (PVA) [107] to obtain eigenvectors and eigenvalues of a data matrix followed by conversion of the abstract vectors into chemical factors. The generation of ‘‘pure’’ spectra and concentration profiles is accomplished by a Varimax rotation (VMAX) [107–109] followed by alternating leastsquares optimization (ALS) [110]. Software for the MCR analysis was written in GRAMS/386 (Galactic Industries). 4.2.3.2. Component resolved NMR A global leastsquares analysis method labeled ComponentResolved FT-PGSE NMR spectroscopy (CORE) has been reported by Stilbs et al. [111]. Their procedure is to perform a total fit of the 2D raw PFG-NMR data set. This has been implemented in a FORTRAN program that includes two copies of a direct-search minimization routine (STEP). One global (higherlevel) minimization optimizes the diffusion coefficients and a second (lower-level) minimization fits the component amplitudes in each frequency channel using the higher-level results. CORE is based on modeling each frequency channel with a number of discrete exponentials (1–5), or if it is decided that a channel has contributions from polydisperse components, a sum of Kohlrauch– Williams–Watts (KWW) distributions (1–2) in addition to discrete exponentials (1–3). The operation of
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this program implies considerable knowledge of the mixture by the user and interactive processing. CORE has been applied to simulated data and to various test samples, and it is reported that global minimization ‘‘may increase the signal/noise ratio by a factor or more than 10.’’ The global analysis is computer intensive, but with the current rate of increase in affordable computing power this should not be a problem. The more serious cost is user time and effort in preparing the input model. Unfortunately, the CORE software is not generally available.
4.2.4. Analysis recommendations The philosophy of DOSY from the beginning has been to obtain the most likely distribution of diffusion coefficients at each chemical shift using any available methods or software. As unique solutions cannot be guaranteed when signals from different species overlap, it is desirable to make use of a variety of analysis methods with complementary strengths and weaknesses. This is easy to accomplish with available software and computer workstations. A reasonable analysis suite might contain the following: 1. single exponential analysis (Levenberg– Marquardt); 2. biexponential fitting routine with rejection rules (perhaps SPLMOD); 3. continuous distribution analysis (CONTIN or similar program). However, it is essential that the user of such programs be well informed about artifacts and potential pitfalls. The fact that a spectrum a(D) is computed does not by itself prove that it has any physical validity. The recently introduced MaxEnt package appears to be a good addition to the aforementioned list. It can be used to screen samples and may be able to compete with CONTIN in the analysis of broad distributions. Choices 1–3 plus MaxEnt will serve very well for high-resolution spectra of multi-component mixtures of discrete, low molecular compounds. In contrast, when there is severe overlap and a relatively small number of components, the multivariate analysis methods DECRA and MCR may be good alternatives. Also, brute force analysis by totally fitting 2D data sets with user defined models such as CORE can be useful in special cases.
5. Effects of chemical exchange Chemical rate processes have been studied by NMR methods for decades [112]. In the standard model for chemical exchange, nuclei or groups of nuclei explore a number of sites by means of a Markovian random process. These sites are characterized by spin Hamiltonians that may be associated with a chemical environment, e.g. a basic group that can accept a proton, or a molecular configuration, e.g. cis and trans isomers. Slow exchange is defined as the situation in which spectra, characteristic of the individual sites, can be observed; and fast exchange implies the observation of some kind of average spectrum. The path from slow exchange to fast exchange consists of line broadening, coalescence, and motional narrowing as the mean lifetimes for occupation of the sites decrease. Time is always the variable, and the exchange rates, i.e. inverse lifetimes, are manipulated by changes in temperature, or perhaps concentrations for intermolecular reactions. In diffusion NMR new possibilities arise because of the importance of sites that differ in hydrodynamic properties with or without differences in spin Hamiltonians. Also, in PFG-NMR for DOSY applications, the variable is q 2 rather than time, and there is nothing analogous to lifetime broadening in the diffusion dimension [113]. Time does enter through the storage delay T, and the variation of T sometimes permits the observation of slow and fast exchange limits without changing the physical condition of the sample. 5.1. Exchange effects in diffusion spectra Here we consider the effects of chemical exchange on longitudinal magnetization during the diffusion sensitive interval T in STE type experiments (Fig. 2) with t; d p T so that T ⬇ D. Taking into account diffusion, flow, and chemical exchange, the rate of change of the nuclear magnetization Mn(r,T) for the nth species with spatial coordinates r is given by [1,27]: dMn r; T 1 ⫺ M r; T ⫺ 7·Jn r; T dT T1n n X Knm Mm r; T ⫹
28
m
where the rate constants are kn ⫺Knn and
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two-site exchange, and for this case analytical solutions can be obtained. Thus Eq. (30) reduces to dM K; T LM K; T dT where L
LAA
LAB
LBA
LBB
!
32
⫺kA ⫺ RA
kB
kA
⫺kB ⫺ RB
! : 33
Fig. 21. Echo amplitude versus q 2T for two-site exchange calculated with Eq. (35) with DA 2.0, DB 0.1, kA 10, and MA 0.40.
The matrix elements [exp(LT)]AB are available in the literature [1], the required combinations, e.g. AA ⫹ AB for MA, for 1D NMR spectra are: MA0 dMA0 ⫺ kB MB0 ⫺ exp ⫺s ⫹ DT MA 2D 2 MA0 dMA0 ⫺ kB MB0 ⫹ exp ⫺s ⫹ 2 2D ⫺ DT;
knm Knm , and the flux Jn r; T is defined by: Jn r; T ⫺Dn 7Mn r; T ⫹ Mn r; Tvn r; T:
29
MB
In Eq. (29) Dn and vn r; T are the diffusion coefficient and velocity for the nth species, respectively. The calculation of the STE amplitude is simplified by taking the spatial Fourier transform of Eq. (28) to obtain the linear algebraic equations: X dMn q; T iVn ⫺ Rn Mn q; T ⫹ Knm Mm q; T dT m 30 Z
exp ⫺iq·rMn r; Tdr:
31
In Eq. (30) Rn 1=T1n ⫹ Dn q2 describes relaxation and Vn qvn is the frequency resulting from uniform flow in the z-direction. These equations and their solutions have been discussed by Ka¨rger, et al. [15], and similar equations for nuclear magnetic relaxation in multiple phase systems were previously treated by Zimmerman and Brittin [114]. Here we solve Eq. (30), neglecting 1/T1n and Vn, for the Fourier components Mn(q,T) that determine the amplitude of the STE. Real applications often involve
MB0 dMB0 ⫺ kA MA0 ⫹ exp ⫺s ⫹ DT 2D 2 MB0 dMB0 ⫺ kA MA0 ⫺ exp ⫺s ⫹ 2 2D ⫺ DT;
34b
where MA0 MA(q,0), MB0 MB(q,0), and the symbols s , d , and D are defined by:
s
1 k ⫹ kB ⫹ DA q2 ⫹ DB q2 2 A
d
1 k ⫺ kB ⫹ DA q2 ⫺ DB q2 2 A
where Mn q; T
34a
q D d2 ⫹ kA kB (The functions d and D should be distinguished from previously defined time intervals that are associated with the same symbols.) If the chemical shift difference between the sites is much larger than the exchange rate, then Eqs. (34a) and (34b) describe the dependence of the individual peak areas on exchange rates and diffusion coefficients. We consider the special case in which the
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Fig. 22. Diffusion spectra for two-site exchange with equal populations (pA 0.5). The effective rates are indicated by values of N 0:5 kA ⫹ kB T. Reproduced with permission [113].
chemical shift difference is zero or is much less than the exchange rate so that a single line is observed in the NMR spectrum. In this situation the diffusion spectrum is calculated with the sum MA q; T MA q; T ⫹ MB q; T: M0 L ⫹ exp ⫺s ⫹ DT M 2D 2 M0 L ⫺ 35 exp ⫺s ⫺ DT ⫹ 2 2D where M0 MA0 ⫹ MB0 and
L dMB0 ⫺ MA0 ⫹ MA0 kA ⫹ MB0 kB : It is should be noted that the right side of Eq. (35) is not a simple sum of exponentials because the variable q appears in the coefficients as well as the exponents. In this simple example the probability of occupation of the nth (n A,B) site is pn and the mean lifetime of a spin in the nth site is tn 1=kn . Also, k kA =pB kB =pA 1=t, pA tA = tA ⫹ tB , and for convenience we let Mn0 pn . Another important parameter is N kT/2 that can be interpreted as the mean number of times that a spin changes sites during the interval T. Also, the storage time can be written as T tA ⫹ tB where tA and tB represent the total amounts of time that a spin occupies sites A and B, respectively. It should be clear that tA and tB are related to the equilibrium constant Keq PB =PA tB =tA but
Fig. 23. Stack plot of 1H spectra of Vancomycin and DDFA mixture as t increases: (a) LED pulse sequence (t increases from 0.3 to 6.2 ms) and (b) BPP-LED pulse sequence (t increases from 0.4 to 6.4 ms). T 100 ms, g 5.8 G/cm, and Te 5 ms in both experiments. Reproduced with permission [33].
give no information about the magnitude of the exchange rate or the magnitudes of the mean lifetimes. In attempting to understand the two-site exchange problem, it is useful to plot ln(M) versus q 2T for a range of T values as shown in Fig. 21. The slow exchange limit is captured in the upper most curve where T ⬍ 2/k and N ⬍ 1. It is essentially the sum of two exponentials, one associated with DA and the other with DB. The slope of this curve in the limit of large q 2T values is Dslow, which we take to be DB, and the initial slope is given by Dav pADA ⫹ pBDB. If the S/N ratio is high, it is possible to estimate the pA and pB values as follows. The intercept, i.e. the q 2T 0 value, of the line with slope Dav is taken to be ln(MA ⫹ MB) while the intercept of the slow exchange line extrapolated from large q 2T values is equal to ln(MA). It is also possible to use the latter intercept for a set of curves near the slow exchange limit to estimate the exchange rate as intercept constant ⫺ kBT. The lowest curve, actually a straight line with slope Dav, represents the fast exchange limit obtained by setting T ⬎ 20/k so that N ⬎ 10. A DOSY experiment on this two site system would, under favorable conditions, yield a diffusion spectrum with two peaks in the slow exchange limit. Similarly, we would expect the fast exchange limit to give a diffusion spectrum with a single peak. In the
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Fig. 24. DOSY display for protons on alpha carbons in a mixture containing alanine (A), glutamine (Q), and lysine (K) in D2O. Reproduced with permission [53].
intermediate exchange regime DOSY analysis is likely to fail or to indicate extreme polydispersity. However, the exact diffusion spectra can be derived from Eq. (35) either by directly performing the ILT to obtain analytical expressions [115] or by substituting iv for q 2 and performing the inverse Fourier transformation of the resulting expressions with respect to v to obtain the diffusion spectra [113]. A two-site spectrum computed with pA pB 1/2 is shown in Fig. 22 for N values ranging from 2 to 50 [113]. For N ⬍ 1, delta functions appear at DA and DB with intensities proportional to exp(⫺kAT) and exp(⫺kBT), respectively. The line widths for the sharp peaks in Fig. 22 result from the numerical computation and are not meaningful. For N ⱖ 5, intensity with a Gaussian profile grows at the position of the average diffusion coefficient (DA ⫹ DB)/2, and as N continues to increase this peak narrows while maintaining a standard deviation that is proportional to N ⫺1/2. The calculation for pA 苷 pB is similar except that for large values of N the intensity is centered at pADA ⫹ pBDB. 5.2. Artifacts from chemical shift encoding It was previously mentioned that chemical shift encoding can seriously affect echo amplitudes in FT-STE and FT-LED experiments when chemical exchange is present [33]. To see how this comes
about consider an STE experiment (Fig. 2) for spins with the frequency offset v A. After the first 90⬚ RF pulse, the magnetization precesses with the frequency v A plus a contribution from the field gradient depending on the displacement in the z-direction. The second 90⬚ RF pulse stores the y-components of magnetization in the z-direction and the remaining x and y components are eliminated by phase cycling or homospoil pulses. For spins in the layer at z to z ⫹ dz the stored magnetization is described by cos vA t ⫹ qz where q g gd . The third 90⬚ RF pulse returns magnetization to the yz-plane and precession reduces the y-component by an additional factor of cos vA t ⫹ qz at the echo. For a sample of length L the amplitude of the peak at v A in the absence of relaxation is given by 1 ZL=2 cos2 vA t ⫹ qzdz L ⫺ L=2
36
1 1 ⫹ cos 2vA tsinc Lq=p: 2 2 The last factor on the right hand side, sinc(Lq/p ) sin(Lq)/Lq, is unity when the gradient vanishes thus permitting the amplitude of the echo to depend on the frequency offset v A. As previously noted, the q 0 point should be avoided in diffusion measurements. However, as soon as Lq exceeds 2p , i.e. the pitch of the magnetization
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Fig. 25. 500 MHz DOSY display of a perchloric acid extract of gerbil brain in D2O. Selected assignments are: ac acetate: ala alanine: cho choline: cr creatine: cre creatinine: etn ethanolamine: GABA g -aminobutyric acid: glu glutamine: GPC glycerophosphocholine: lac lactate: m-ino myo-inositol: NAA N-acetylasparate: succ succinate: and tau taurine. Reproduced with permission [118].
helix becomes less than the sample length, the modulation term becomes very small and vanishes completely with significant gradients. When chemical exchange occurs, intensity modulation reappears for the exchanging groups even in the presence of gradients. The previous discussion is easily extended to two-site chemical exchange. Suppose that the sites have different chemical shifts and that exchange can be neglected during the encode and decode intervals. During the diffusion delay T, exchanges do occur so that the spins in site A at the time of the third 90⬚ RF pulse may have occupied either site during the encode interval. In terms of the offset frequencies (v A, v B) and the mole fractions (xA, xB), the peak
amplitude at v A is then given by 1 ZL=2 x cos vA t ⫹ qz L ⫺L=2 A ⫹xB cos vB t ⫹ qzcos vA t ⫹ qzdz
37
1 x ⫹ xB cos vA t ⫺ vB t: 2 A
The modulation predicted by Eq. (37) has been observed in affinity NMR experiments where the pulse duration d was set equal to encode interval t , and spectra were obtained as a function of t . An example (Fig. 23) is provided by a mixture of Vancomycin (Sigma) and the tetrapeptide
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Fig. 26. (a) DOSY display processed with SPLMOD for a mixture containing 5.0 mM methanol, 10.0 mM iso-propanol, 10.0 mM t-butanol, and 10.0 mM neo-pentanol in D2O, and (b) DOSY display for a mixture with the same solute concentrations plus 0.150 M DTAB. Reproduced with permission [61].
Asp–Asp–Phe–Ala (DDFA) in D2O [33]. The signals for the free DDFA (1.30 ppm) and the bound DDFA (0.55 ppm) that are involved in exchange decay with increasing t because of diffusion and transverse spin relaxation, but also oscillate at 373 Hz, their chemical shift difference in the 500 MHz spectrometer. The other (non-exchanging) peaks show no oscillation. Intensity oscillations can be useful in identifying exchanging pairs and editing spectra, but can also be deceptive if not properly identified. Fortunately, the oscillations can be completely avoided by using BPP based sequences that refocus chemical shifts.
6. Applications of 1D and 2D DOSY 6.1. Discrete samples Most DOSY applications to date have concerned discrete mixtures. Selected examples are presented in this section. The reader is referred to the literature for additional DOSY examples including studies of chloroaluminate melts [116] and perturbed cis-trans isomerization of phenylalanylproline [117]. 6.1.1. Biofluids Biofluids here refer to multicomponent mixtures of amino acids and other molecules commonly found in
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Fig. 27. DOSY display for a solution containing 2 g/dl BSA, 2 g/dl SDS, and 0.01 M b -mercaptoethanol in phosphate buffer (pH 7.2, ionic strength 0.020 M), The unlabeled line represents the reaction product, HOCH2CH2SSCH2CH2OH. Reproduced with permission [145].
tissue extracts. These examples emphasize the optimization of resolution in the chemical shift dimension to minimize overlap and the incorporation of experimental design features to maximize the accuracy of the extracted diffusion coefficients to enhance resolution in the diffusion dimension. A mixture of alanine (A), glutamine (Q), and lysine (K) in D2O provides a good illustration of the necessity of high resolution in the DOSY experiment as the diffusion coefficients are much too similar to be separated when there is overlap. The display in Fig. 24 shows part of the 2D spectrum (protons on a -carbons) for this mixture [53]. As previously described, the spectrum was constructed with Gaussian components centered on the computed values of the diffusion coefficients and having widths equal to the errors reported in the data analysis. Sample spinning with the ‘‘stop-and-go’’ system and the BBP sequence were used to achieve high resolution (Dn ⬍ 0.5 Hz), and good line shapes in the chemical shift dimension were obtained without special filtering of the FIDs. With no overlap and S/N ratios greater than 100, data analysis with SPLMOD provided diffusion coefficients with errors of a few percent, but even better results would be expected from an L–M analysis limited to single components. For this work a Bruker AC250 spectrometer with a custom built 10 mm diffusion probe was used. The other experimental parameters were:d /2 1 ms, t
1.5 ms, 27 gradient amplitudes (g ^ 0.03 to ^ 0.62 T m ⫺1) and Te 20 ms. Impressive examples of mixture analysis have also been reported under the label high-resolution DOSY (HR-DOSY) [54,64,65,118]. An example is shown in Fig. 25 for a perchloric acid extract of gerbil brain [118]. Fifteen spectra were acquired with the LED sequence on a 500 MHz spectrometer using 512 transients per spectrum. Post processing with reference deconvolution (FIDDLE) was based on the reference line for DSS (sodium 3-trimethylsilypropanesulfonate), and the DOSY display was constructed from measurements on 116 peaks. The contour lengths (errors) in the diffusion dimension are somewhat larger than in test cases because of poorer S/N ratios in the 1D NMR spectra. The assignments of a number of metabolites are shown in Fig. 25, and as expected the diffusion coefficients are well correlated with molecular sizes. 6.1.2. Separation by means of hydrophobicity FT-PFG-NMR has proved a powerful tool for the study of solubilization [119]. DOSY based on highresolution 1D NMR spectra extends this work and provides ‘‘at-a-glance’’ recognition of the components in complex mixtures and their interactions. In a number of such cases complete resolution of peaks has been obtained even at 250 MHz through the use of the ‘‘stop-and-go’’ spinner. The example shown in
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Fig. 28. 1H DOSY display for a mixture containing ‘‘Tinuvin P’’ ( ⫹ ), ‘‘Irganox 1330’’ (*), and ‘‘Irganox 1098’’ (W). The solvent is labeled S and silicon grease contaminant is labeled Sg. Reproduced with permission [146].
Fig. 26(a) for a mixture of 5.0 mM methanol, 10.0 mM iso-propanol, 10.0 mM t-butanol, and 10.0 mM neo-pentanol in D2O does not require especially high chemical shift resolution, but does suffer from accidental overlap in the diffusion dimension because of the near equality of the diffusion coefficients of t-butanol and neo-pentanol [61]. Fig. 26(b) shows that the effective diffusion coefficients can be manipulated by adding the micelle forming surfactant, dodecyl trimethylammonium bromide (DTAB). This is an example of fast chemical exchange where solute molecules partition into the micelles according to their solubilities but undergo exchange between the interior of the micelle and the bulk solution. In the fast exchange limit DOSY only reports the time average diffusion coefficient Dj for the jth solute ⫹ 1 ⫺ pj Dfree Dj pj Dmic j j
38
where pj is the degree of solubilization, and Djmic and Djfree are the tracer diffusion coefficients for solubilized and free molecules, respectively. It should be recognized that Djmic is equal to the diffusion coefficient of the micelle as the average displacement of a micelle during the diffusion time D is much greater than the radius of the micelle. The experiment was performed at 250 MHz with the LED sequence. Twenty-eight spectra were acquired with q-values ranging from 208 to 8.34 × 10 3 cm ⫺1, d 1 ms, t 2.0 ms, D 100 ms, and Te 50 ms. SPLMOD (N 2) was used for the data analysis. Complete resolution of the diffusion peaks was obtained by addition of 0.150 M DTAB, and the values of pj shown for the solutes in Fig. 26(b) were calculated with Eq. (38) using Djfree values from the DTAB free solution (Fig. 26) and all Djmic values set equal to the diffusion coefficient measured for DTAB.
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Fig. 29. Structures for polymer additives. Reproduced with permission [146].
6.1.3. Equilibria involving sodium dodecylsulfate (SDS) and bovine serum albumin (BSA) SDS–protein interactions are important in a number of fields. For example protein/SDS-polyacrylamide gel electrophoresis (PAGE) is routinely used to determine protein molecular weights and to separate proteins. Also, SDS is known to be a potent protein denaturant. However, the protein–SDS complexes and their electrophoretic mobilities are not completely understood, and this has provided motivation for physical studies. The aim of a DOSY study of BSASDS equilibria was to determine the binding isotherm, i.e. the fraction of surfactant bound to a protein versus the surfactant concentration. The DOSY display for a D2O solution containing
2 g/dl BSA, 2 g/dl SDS, and 0.01 M b -mercaptoethanol in phosphate buffer (pH 7.2, ionic strength 0.020 M) at 298 ^ 1 K is shown in Fig. 27. This spectrum was obtained from 20 to 40 FIDs in an LED experiment with q-values ranging from 2.1 × 10 4 to 1.9 × 10 6 m ⫺1 and D values typically between 50 and 80 ms. The diffusion dimension reveals BSA signals from the slowly diffusing BSA–SDS complex, a single SDS peak, and weak peaks from the denaturing compound and the unlabeled reaction product, HOCH2CH2SSCH2CH2OH. The single SDS peak indicates the fast exchange limit for the SDS exchange among monomers, micelles, and the BSA–SDS complex. This limit reduces the amount of information available and requires that the analysis begin with
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Fig. 30. DOSY display for medium unilamellar vesicles in a mixture containing 30 mM total lipid (POPC) with 100 mM sucrose in D2O. Reproduced with permission [70].
an expression for the observed average diffusion coefficient 具D典 for SDS: 具D典 Ppro Dpro ⫹ Pmon Dmon ⫹ Pmic Dmic
39
with Ppro ⫹ Pmon ⫹ Pmic 1
40
where Ppro, Pmon, and Pmic are the fractions of SDS
molecules on the protein, as monomers, and in micelles, respectively; and Dpro, Dmon, and Dmic are the associated self-diffusion coefficients. The calculation of the desired quantity Ppro requires the determination of all the diffusion coefficients and either Pmon or Pmic. Both 具D典 and Dpro can be read directly from Fig. 27, but Dmon and Dmic must be determined from additional experiments and the use of the mass action
Fig. 31. The 1H DOSY display (250 MHz) for whole plasma at 40⬚C. The reference is TMA at 3.22 ppm [126].
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Fig. 32. DOSY display processed with CONTIN for a mixture containing CTAB, PVME, and NaSal (see text). Reproduced with permission [96].
model [120]. The conclusion of this study was a plot of the binding ratio [bound SDS]/(SDS) versus [total SDS]/[BSA] in the range 0–6 (g/g) showing a saturation binding of 1.9 g bound SDS per gram BSA that is reached at about 4 g total SDS per gram of protein. Also the effective hydrodynamic radius of the complex was determined as a function of the number of bound SDS molecules. 6.1.4. Mixtures of polymer additives DOSY has been used to analyze mixtures of polymer additives [146]. The diffusion coefficients facilitated spectral assignments and were especially helpful for isolated protons where no J-couplings could be resolved. 2D-DOSY of a mixture containing ‘‘Tinuvin P’’ (Mn 225) ‘‘Irganox 1098’’ (Mn 637), and ‘‘Irganox 1330’’ (Mn 775) all from Ciba in 1,1,2,2tetrachloroethane-d2 (TCE-d2) is shown in Fig. 28, and the associated molecular structures are shown in Fig. 29. The data were acquired with BPP-LED on a Bruker AM360 spectrometer equipped with a 5 mm z-gradient Bruker inverse probe. The gradient pulse duration was 2 ms, the diffusion delay was 0.5 or 1.0 s, and Te was 15 ms. Typically, 15–20 gradient values were used and the values ranged from 0.026 to 0.30 T m ⫺1. The diffusion information in Fig. 28 correlates with the structures of the additives. ‘‘Irganox 1330’’ is star
shaped while ‘‘Irganox 1098’’ has a more extended linear shape. The shapes compensate for the differences in molecular weights and produce quite similar diffusion coefficients. Also, it is observed that the exchangeable OH protons (4.96, 5.08, and 11.15 ppm) have larger diffusion coefficients than the other protons in their respective molecules. Apparently the protons are exchanging with protons of water in the solvent. Note also the well resolved resonances for the solvent and silicon grease in the diffusion dimension.
6.2. Polydisperse samples DOSY is especially useful for samples containing polydisperse components. With widely available software, diffusion spectra can be obtained that permit components with broad distributions of diffusion coefficients to be distinguished from those with narrow distributions, i.e. the monodisperse components. Also, with no additional assumptions beyond smoothness, the shape of distributions of diffusion coefficients can be measured. In this section DOSY examples are displayed that have been processed by CONTIN. Similar results can be obtained with other analysis programs. In particular we note that a recent DOSY study of polymer mixtures successfully used
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both CONTIN and the analysis program NLREG [121]. 6.2.1. Phospholipid vesicles The first DOSY paper showed a diffusion resolved spectrum of a sample containing sucrose and phospholipid vesicles in D2O [9]. Two sucrose spectra were evident in the 2D display at diffusion coefficients of 3.00 × 10 ⫺6 and 1.86 × 10 ⫺8 cm 2 s ⫺1, and it was reasonable to assign the spectra to free sucrose and sucrose entrapped in vesicles, respectively. Data analysis in this case was performed with DISCRETE and, consequently, information concerning polydispersity of the vesicles was lost. A much more extensive CONTIN based analysis was performed on palmitoyl-2-oleoyl phosphatidylcholine (POPC) vesicles with diameters ranging from 30 to 100 nm [70]. A DOSY display for sucrose with medium POPC vesicles is shown in Fig. 30. At this level of gain the signals from the vesicle bilayer are almost invisible, the exception being the small peak at 3.1 ppm. The bilayer proton signals are more pronounced for smaller vesicles and in fact dominate the spectrum for 30 nm vesicles. From the DOSY peaks it is evident that the effective diffusion coefficients of the bilayer protons are larger that the diffusion coefficients of the entrapped sucrose protons. This fact became a focus of the study. The vesicle study made use of dynamic light scattering (DLS) and electron microscopy (EM) as well as CONTIN analysis of PFG-NMR data. As the largest vesicles have diffusion coefficients in the vicinity of 2 × 10 ⫺8 cm 2 s ⫺1, it is necessary to include large q-values in the analysis. Here the q-values ranged from 2000 to 32 000 cm ⫺1 giving grating spacings of 30 to 2 mm. The LED sequence was used because BPP-LED was not available and the Te values tended to be long. Also, at the largest gradient values convective flow was a problem in 10 mm sample tubes but not in 5 mm tubes. This probably resulted from poor temperature control in the probe. The major conclusions were the following: (1) The distribution of particle diameters for large unilamellar vesicles obtained from the CONTIN analysis of PFGNMR data agrees quite well with distributions obtained from DLS and electronmicroscopy even though the width from the CONTIN analysis contains a contribution from the smoothing effect. (2) The
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apparent diffusion coefficients obtained for phospholipids in the bilayers are heavily weighted by contributions from the smaller vesicles in the distribution that tumble more rapidly and give narrower NMR signals. 6.2.2. Blood plasma Plasma lipoproteins are microemulsions-like assemblies consisting of a protein–phospholipid shell with a core of triglycerides and cholesterol. The compositions of the lipoprotein particles are related to their densities and sizes, and in general the lowest density particles have the largest radii [122]. Lipoproteins have received a lot of attention in recent years because their concentrations in blood have been correlated with the risk of coronary heart disease. Unfortunately, standard clinical procedures for measuring the low density fraction are inconvenient and subject to experimental errors. An alternative NMR method for rapid analysis has been proposed that takes advantage of differences in chemical shifts for methyl 1H signals exhibited by the various lipoprotein fractions: very low density (VLDL), low density (LDL) and high density (HDL) [123,124]. This is also a potential application for DOSY as the chemical shifts appear to be correlated with particle diameters. A DOSY CONTIN generated display for whole plasma with added D2O and tetramethylammonium chloride (TMA) as a reference compound is shown in Fig. 31 [125,126]. The experiment was performed on a 250 MHz spectrometer with ‘‘stop-and-go’’ sample spinning and the LED pulse sequence. Typically, 23 exponentially spaced values of q (84–12 540 cm ⫺1) were used with d 2 ms, t 3.5 ms, D 100 ms, and Te 50 ms. Water signals were eliminated in all experiments by continuous saturation except during data acquisition. The diffusion coefficients associated with the lipoprotein lipid signals [methyl, methylene, vinyl, and allylic] are all the same order of magnitude, and as expected the smaller molecules (lactate, EDTA, TMA) have much larger diffusion coefficients [127]. As the methyl and methylene signals for the larger lipoproteins are shifted downfield from those of smaller particles, we expect the diffusion coefficients to vary with the chemical shifts in the DOSY display. There is a trend toward smaller diffusion coefficients at the low end but not as pronounced as expected.
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Fig. 33. The simulated distribution a(D)D with M0 10 5 and s 3.2 (solid line), computed cm values (dashed line), and the effective distribution to be estimated by CONTIN with the computed cm values (dotted line). Cubic splines were used to obtain smooth displays from the 31 cm values. Reproduced with permission [97].
This unimpressive result prompted DOSY analysis of VLDL, LDL, and HDL fractions prepared by agarose gel chromatography and kindly supplied by J.D. Otvos. CONTIN reports moments Mn of peaks in the diffusion spectrum: Z 41 Mn ln a ldl Here the moments for the methyl peaks were used to calculate average diffusion coefficients, i.e. 具D典 D ⫺ d=3⫺1 M1 =M0 . The resulting mean diffusion coefficients yield uncorrected Stokes–Einstein ˚ (VLDL), 46 A ˚ (LDL), and 23 A ˚ radii of 106 A (HDL). The ratios of these numbers are in reasonable agreement with the literature [122], but the calculated radii are considerably smaller than expected. This may indicate the presence of breakdown products or the operation of relaxation effects that emphasize the contribution of smaller particles. Also, chemical exchange cannot be ruled out. Successful completion of this work demands the dispersion and sensitivity of a high field spectrometer. 6.2.3. The viscoelastic CTAB/sodium salicylate/water system The well-known viscoelastic micellar system hexadecyltrimethylammonium bromide (CTAB)/sodium salicylate/NaSal/ water has been found to undergo a remarkable polymer-induced non-Newtonian to
Newtonian transition [128]. Upon the addition of poly(vinyl methyl ether) a striking decrease in viscosity occurs that is thought to result from the breaking up of long rod-like micellar aggregates. A proposed model for the transition includes the formation of spherical micelles ‘‘wrapped around’’ by the polymer and the retention of bound Sal ⫺ ions. A DOSY study of this interesting system included the CONTIN processed display shown in Fig. 32 for a Newtonian mixture containing 25 mM CTAB, 15.0 mM NaSal, and 7.40 mg/ml PVME at 30⬚C [96]. The spectrum shows PVME peaks at 3.28 and 1.69 ppm with a broad distribution of diffusion coefficients. At approximately the same average diffusion coefficient, we find CTAB peaks (2.97, 1.28, and 0.87 ppm) with much narrower distributions. In fact, the diffusion distribution for these peaks results entirely from the smoothing effect of CONTIN. This is consistent with the model if the CTAB ions exchange rapidly between different polymer chains. Also, we find Sal ⫺ at about 8 × 10 ⫺6 cm 2 s ⫺1 which is about 10 times slower than the diffusion coefficient for Sal ⫺ in the absence of CTAB. With the help of Eq. (38) it is estimated that the fraction of bound Sal ⫺ is about 0.95, again in agreement with the proposed model. The diffusion coefficient of CTAB in the presence of PVME is greatly increased and at 40⬚C is consistent with spherical CTAB micelles.
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Fig. 34. Pulse sequence for 3D DOSY-COSY experiment [Phases: P1: 0, P2: 0, P3: 016 216, P4: 0, P5: 0, P6: 08 28, P7: 04 14, P8: 0213 1321, Rec: 0022 1133 2200 3311 0033 2200 3311 0022 1133]. Reproduced with permission [133].
6.2.4. Molecular weight distributions DOSY analysis for polydisperse samples produces a diffusion or Laplace spectrum a(D) that is actually a diffusion coefficient distribution. The question here is whether DOSY data can also be used to obtain molecular weight distributions (MWD). Other popular physical methods for the determination of MWDs (M ⬎ 25 000) include gel permeation chromatography (GPC) and DLS. As with PFG-NMR, these methods measure transport rates that depend on molecular size and require calibration. Also, hydrodynamic methods such as DOSY and DLS yield accurate parameters for individual molecules only in the limit of low concentrations [129]. In the following we assume that concentrations are sufficiently low that microaveraging effects are negligible [130,131]. The only reported DOSY MWD determination was based on a diffusion spectrum, a(D), computed with CONTIN [97]. In that work it was found that the standard application of CONTIN leads to unacceptable errors in a(D) because of D dependent undersmoothing of the distribution as discussed in Section 4.2.2.1. With CONTIN the inversion of the data set yk is handled by solving the set of linear algebraic equations shown in Eq. (25). A major improvement of the analysis for broad distributions is obtained by setting cm 1 so that a(D)D rather than a(D) is analyzed on the logarithmic axis as previously discussed. The fact that a scale change improves the results suggests that cm be varied continuously to improve the analysis in regions where the amplitude of a(D) is small. The procedure adopted was to replace cm with Dm =Dmax x
m
where xm was incremented from ⫹ 2 to ⫺ 2 as log(D) ranged from ⫺ 12 to ⫺ 9. In general the range of log(D) should be set as narrow as possible while still covering the distribution to zero amplitude at both limits. With reasonable limits the amplitude of G(D)D is typically about 10% of the maximum value at the xm ^1 points. This gives cm 1 near the maximum of the distribution (where D Dmax) and provides appropriate amplitude enhancement where the amplitude is small. The idea is that the cm values can be chosen to emphasize the region of interest, and CONTIN will then return the distribution a Dm Dm =cm from which a(Dm) can be extracted. The mass weighted distribution, W(M), can be computed directly with the equation W M a D兩dD=dM兩 if the relation between D and M is known for the particular system. Also, the number weighted MWD, n(M), can be obtained because n M W M=M. For the special case of the Gaussian random coil the scaling relation has the form D AM a
42
but this cannot be assumed for real polymers. These ideas were tested by computer simulation. The MWD, W(M), was represented by log normal distributions with the reference molecular weight M0 and s values ranging from 1.25 to 3.25. The diffusion distributions a(D) were obtained from the W(M) by means of the scaling law in Eq. (42) with A 10 ⫺7.5 and a ⫺ 0.6. Finally, PFG-NMR dates sets were computed with Eq. (19) for sets of q-values and Gaussian random noise with RMS deviation of 10 ⫺3 was added. The simulated data sets were then processed with standard CONTIN, CONTIN with cm 1, and CONTIN with computed cm values to obtain
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Fig. 35. A 3D view of the DOSY-COSY data. In the F1 and F2 dimensions the chemical shifts range from 1.0 to 4.0 ppm, and the diffusion axis ranges from 3.0 × 10 ⫺10 to 9.0 × 10 ⫺10 m 2 s⫺1. Reproduced with permission [133].
‘‘experimental’’ diffusion distributions a(D). Fig. 33 shows a simulated distribution a(Dm)Dm, the computed cm values, and the function a Dm Dm =cm processed by CONTIN. From the computed a(D) distributions the
quantities 具D典, standard deviation (SD), the MWD W(M), mass average molecular weight Mw, and number average molecular weight Mn were computed as functions of s . It was found that these quantities obtained with all versions of CONTIN agreed almost perfectly with values computed directly from the initial log normal distribution for s ⬍ 2.0. However, for larger values of s , standard CONTIN showed large and erratic errors, the errors were moderate with cm 1, and with computed cm values the agreement was very good. CONTIN with computed cm values has been successfully used to determine Mn and Mw values for commercial samples of polydisperse poly(ethylene oxide) [97]. Recent work by Jerschow and Mu¨ller also demonstrates that 2D DOSY can be effective in resolving mixtures of polymers [121]. They have obtained impressive DOSY displays for mixtures of high and low molecular weight polypropylene and mixtures of polypropylene and polystyrene from experiments performed on a 600 MHz spectrometer by means of the BPP-LED sequence. The ILTs were performed
Fig. 36. (a) 2D COSY display obtained with DOSY-COSY pulse sequence having the lowest q value, and (b)–(d) the 2D COSY planes (slices) from the 3D DOSY data set that correspond to the diffusion coefficients of alanine, glutamine, and arginine, respectively. Reproduced with permission [133].
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Fig. 37. A pulse sequence for X-H COSY-DOSY or HMQC-DOSY [136].
with both CONTIN and NLREG [132]. These programs make use of regularized least squares procedures but use different methods to determine the regularization parameter. In the examples presented, the NLREG provided somewhat better separations than CONTIN. 7. 3D DOSY Here we describe 3D DOSY experiments in which a diffusion coordinate is added to conventional 2DNMR. The initial motivation for the development of 3D DOSY was to obtain additional dispersion of NMR peaks in order to avoid overlap as the highest accuracy in DOSY analysis is obtained when single component decays can be assumed. The analysis of data cluster intensity (cross-peaks in the 2D-NMR spectrum) versus q 2 then yields diffusion peaks that lie in layers corresponding to the diffusion coefficients
Fig. 38. A pulse sequence for gHMQC-DOSY. Delays t 1/4JCH, gradient pulse length is d , separation of midpoints of outer gradient pulses is D. The simple, time saving phase cycle was: first 13C 90⬚ pulse follows ( ⫹ x, ⫺ x) or 02 on successive transients, second 1H 90⬚ pulse follows 0022 1133, and for the receiver 0220 3113. Reproduced with permission [137].
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of the various species in solution. Here as elsewhere we emphasize that the acronym DOSY implies that the third dimension consists of peaks on a diffusion axis and not simple signal attenuation versus q 2. The creation of 3D DOSY requires (a) the design of a combined 2D-NMR/DOSY pulse sequence and (b) the development of algorithms for grouping data points in clusters corresponding to cross-peaks, inverting cluster intensity versus q 2 data sets, and constructing 3D spectra. In the original report it was emphasized that any 2D-NMR experiment can be used in this combination and 3D COSY DOSY was simply used as an illustration [133]. Also, HMQCDOSY was reported and NOESY-DOSY and EXSY-DOSY were suggested. In the following a variety of 3D DOSY experiments from different laboratories are reviewed. Some of the following examples demonstrate only the PFG-NMR step of obtaining q-dependent attenuation of cross-peaks, and thus do not fully implement 3D-DOSY. 7.1. COSY-DOSY A pulse sequence for COSY-DOSY constructed by linking BBP-LED and COSY sequences is shown in Fig. 34 [133]. After the eddy current delay Te, the COSY evolution time t1 and acquisition time t2 provide, through Fourier transformation, the two chemical shift dimensions. A 32-step phase cycle (Fig. 34) was devised to select the coherence transfer pathway for phase modulated 2D COSY [1] and to encode diffusion information. A FORTRAN program (DOSY3D) was used with the data set having the smallest (non-zero) value of q to assign the pixels, above a preselected threshold value, to cross-peaks. This assignment grid was then used for the remainder of the data sets. The analysis was performed by inverting either pixel intensity versus q 2 decays with SPLMOD, or integrated crosspeak integrals (volumes) versus q 2 by means of the L–M algorithm. The latter method is preferable when S/N ratios are modest and there is little crosspeak overlap. COSY-DOSY was illustrated with a mixture of three amino acids, alanine (A), glutamine (Q), and arginine (R), each at a concentration of 100 mM in D2O. Eight q values (184–1966 cm ⫺1) with D 0.104 s were used in the DOSY preparation period
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Fig. 39. Pulse sequence for NOESY-DOSY. h1 and h2 are homospoil type gradients and tm and t1 are the NOESY evolution and mixing times, respectively. Reproduced with permission [139].
to generate eight magnitude mode 2D spectra with 512 × 256 data points each. These were processed with DOSY3D and the L–M algorithm to obtain 64 points in the diffusion dimension and 512 × 256 × 64 total data points. Then Felix (Hare Research, Version 1.1) was used to produce the 3D display shown in Fig. 35. This view is not very useful, but the box can be rotated on a workstation screen to reveal the planes of points for the various species. Fig. 36(a) shows the 2D COSY spectrum of the mixture (lowest q value) and the COSY planes for the individual amino acids, obtained by summing over a few data points in the diffusion dimension, are shown in Fig. 36(b)–(d). The circle in Fig. 36(a) indicates the resolved cross-peaks associated with Qa and Ra that overlap exactly in the 1D spectrum. Note that the 2D planes appear at the correct positions in the diffusion dimension for alanine, glutamine, and arginine. Also, the COSY planes are identical with COSY spectra obtained for the separated compounds except for the diagonal peak of a -protons of
glutamine and alanine. This overlapping peak, that cannot be resolved in the diffusion dimension, was included in the planes for both glutamine and arginine through expanded integration ranges in the diffusion dimension. As expected 3D DOSY has higher effective chemical shift resolution than 2D DOSY and single exponentials are usually adequate. Resolution of planes in the diffusion dimension, of course, depends on standard deviations of the single components fits, which in favorable cases for 1H NMR are less than 2%. There is, however, a cost in time for 3D DOSY. With 124 FIDs for each of the eight q values, data acquisition required 36 h on an old 250 MHz spectrometer. With a modern 500 MHz system, the same S/N ratio could be obtained in less than an hour
7.2. HMQC-DOSY We have seen that both the use of heteronuclear ( 13C) chemical shifts and cross-peaks in 2D-NMR reduce peak overlap and enhance DOSY analysis. Even better resolution is possible through the combination of DOSY with 1H– 13C shift correlation spectroscopy. The problem, of course, is low sensitivity resulting for 13C signals compounded by extensive data acquisition required for 3D DOSY. X-H shift correlation with inverse detection, e.g. HSQC [134] or HMQC [135], may provide the best chance of success because these experiments benefit from the
Fig. 40. TOCSY-DOSY sequence. The narrow and wide filled rectangles represent 90⬚ and 180⬚ pulses and the rectangles with diagonal lines are ramped gradients. The phase cycle used was: P1: 0022; P2: 1230; P3: 222; rec: 0220 2002 2002 0220. Reproduced with permission [140].
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Fig. 41. 3D 1H TOCSY-DOSY display for a mixture of propylene and decane. Reproduced with permission [140].
large magnetic moment of the proton for both spatial encoding and data acquisition. HMQC-DOSY as implemented by Wu et al. [136] combined BPP-LED and X-H COSY is shown in Fig.
Fig. 42. (a) The PFG-DQS sequence for merged DOSY, (b) sequences for NOESY/ROESY and TOCSY diffusion experiments [141].
37. The gradient pulses after the second 1Hp /2 RF pulse are optional homospoil pulses to minimize phase cycling requirements. This experiment was considered to be impractical on our 250 MHz spectrometer, and was demonstrated on a 500 MHz spectrometer in the University of North Carolina Medical School. A sample containing alamine, glucose, sucrose, and caffeine all at 500 mM in D2O was analyzed in the chemical shift ranges 3.0–5.5 ppm for 1H and 20–100 ppm for 13C to produce a 3D display similar to the one shown in Fig. 35. However, the S/N ratios obtained with an acquisition time of 40 h were less than 10, and the resulting errors in diffusion coefficients were approximately 10%. Unfortunately the limited resolution in the diffusion dimension did not permit complete separation of the 2D spectra for all of the compounds. A successful implementation of HMQC-DOSY was recently reported by Barjat et al.[137] That work made use of the combination of GCSTE [54] and conventional gHMQC [138] shown in Fig. 38. The gradient pulses during the t D delay provide eddy current compensation, but otherwise are optional. As experimental time is crucial, the number
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Fig. 43. A representation of 3D-DOSY obtained from a series of 10 PFG-TOCSY experiments on a mixture of ATP/g -cd. A 65 mn MLEV-17 supercycle was used for spin locking. Reproduced with permission [141].
of increments for the diffusion dimension and the number of phase cycling steps were necessarily kept small. Gradient pulses were used in the coherence selection part of the HMQC sequence to reduce t1 noise and to reduce the need for phase cycling. Another, important experimental point concerns proton decoupling. The broadband adiabatic WURST method with low power was selected because the heat input from standard decoupling sequences, e.g. WALTZ, and the resulting convection currents could not be tolerated. The DOSY-gHMQC experiment was applied to a test mixture containing quinine (30 mg), geraniol (20 ml), camphene (19 mg), and TMS in deuterated methanol room temperature. Data were collected with a Varian INOVA 400 spectrometer equipped with a 5 mm indirect detect PFG probe. The storage delay t D was kept short (43 ms) to avoid T1 relaxation, and only five increments of the gradient pulse amplitude were used. Even so, data acquisition required 17 h. This experiment and the homonuclear COSYDOSY of the previous section, while impressive in resolving complex mixtures, are very time consuming even with test samples containing relatively high concentrations.
7.3. NOESY-DOSY Gozansky and Gorenstein combined LED and NOESY sequences as shown in Fig. 39 to obtain diffusion labeling of NOE cross-peaks [139]. They did not actually implement 3D DOSY as the data were not inverted and diffusion spectra were not generated. The diffusion labeling of cross-peaks was demonstrated for a D2O sample (0.6 ml) of 1.6 mg d(AG) dinucleotide and 5 mg 14-mer duplex d(ACAATATATATGR)2 in a phosphate buffer (pH 7.4). The data were collected with a 400 MHz spectrometer with a wide bore magnet and an actively shielded gradient probe. The gradient amplitudes ranged approximately 4–40 G/cm in eight equal steps, t 6.6 ms, T 20.0 ms, Te 50 ms, h1 3.0 ms, h2 97.0 ms, and t m 100 ms. Diffusion coefficients obtained from cross-peak attenuation were found to lie in the proper ranges, but there was some dependence on the delay T. 7.4. TOCSY-DOSY A TOCSY-DOSY experiment was implemented by Jerschow and Mu¨ller for the study of polydisperse polymer samples [140]. The sequence
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(linked BBP-LED and TOCSY) is shown in Fig. 40, and it should be noted that the homospoil pulses are orthogonal to the encoding and decoding gradients. It is clearly appropriate here to use an STE based experiment because T1 q T2. This 3D DOSY experiment differs from the others described here because data inversion was performed by CONTIN on data sets described by Eq. (22) with appropriate correction for BPP-pulse pairs (Eq. 13). The demonstration experiment was performed on a sample of 0.2% propylene (Mw 44 000, Mw/Mn 3.3) plus 0.8% n-decane in tetrachloroethane-d2 at 298 K with a 600 MHz spectrometer equipped with an actively shielded gradient probe. The parameters were T 120 ms, d 2.6 ms, Te 24 ms, t 1 ms, t m 30 ms, and the gradient amplitudes were incremented linearly in 32 steps to a maximum of 63 G/cm. The 2D intensity data for discrete coordinate pairs (v 1, v 2) were fed into CONTIN for analysis, and the resulting 3D data set is displayed in Fig. 41. In spite of the large peak volumes, the components are separated quite well into their respective diffusion planes. 7.5. Merged sequences for PFG-DQS, PFG-NOESY, and PFG-TOCSY Birlirakis and Guittet [141] have reported ‘‘merged’’ pulse sequences that introduce variable gradient pulses into 2D-NMR without the elongation found in the linked schemes described previously. Examples are shown in Fig. 42(a) for PFG-DQS and Fig. 42(b) for PFG-(NOESY, ROESY, or TOCSY). With PFG-DSQ no phase cycling is required and the spectra are obtained in the magnitude mode. This experiment was demonstrated with a mixture containing ATP/g -cyclodextrin (50:7 mM) in D2O at 298 K on a 600 MHz spectrometer. A series of 11 gradient values were used and data acquisition required 14 h. The resolution permitted monoexponential fitting yielding Dg -cd 2.39 × 10 ⫺10 m 2 s ⫺1 and DATP 3.08 × 10 ⫺10 m 2 s ⫺1 with a dispersion smaller that 7% and the quality of exponential fitting R ⬎ 0.997. The 2D TOCSY experiment was implemented for the same sample with ten gradient values and a total data collection time of 43 h. In this case the dispersion increased to 10%. The DOSY idea was carried out by manual construction of 2D spectra to obtain the
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separation of components shown in Fig. 43. The merged sequences described here can produce increased senstivity in some cases, but the advantages of STE based experiments should be kept in mind when short T2s and J-modulation are problems.
8. Future prospects The concept of diffusion spectra and their incorporation as a new ‘‘frequency’’ dimension in NMR is now well established. DOSY has been fully implemented in several laboratories, and interesting examples of diffusion ordered spectra are available as demonstrated in this review. With the advent of both free and commercial DOSY analysis packages and the availability of high quality diffusion accessories from major vendors, the benefits of DOSY analysis are certain to spread to a much larger audience. The result will be another NMR tool to select from a menu. We expect that DOSY will become transparent and natural to users and will no longer attract special attention any more than FT-NMR does. The success of DOSY will tend to make it invisible. The question here is whether there will be major advances in instrumentation or analysis methods that will significantly affect the power and ease of use of DOSY. We do not expect exciting advances in data analysis. The analysis problems have been around for a long time, and the available techniques are able to extract most of the information from experimental data sets. Therefore, we expect the future to hold refinements, slicker user interfaces, and perhaps the incorporation of artificial intelligence to protect users from themselves. Of course, pulse sequences and experimental design are certain to evolve. For example the use of multiple quantum NMR to enhance the effects of gradients may become more common [142]. The question of hardware is much more open. Higher sensitivity through higher frequencies and better probes is certain. One obvious area not yet explored for DOSY is the use of RF gradients in place of or in addition to the pulsed dc gradients. We note the recent development of toroid cavity detectors that permit RF imaging and diffusion measurements with respect to the radial (transverse) direction [143]. These devices have important advantages in special cases such as the measurement of
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diffusion in thin films and in high pressure samples [144]. DOSY applications of toroid cavities may also be possible, but there is the immediate problem of reduced resolution resulting from magnetic susceptibility mismatch that becomes serious in the short cylindrical cavities. Acknowledgements This work was supported in part under a grant from the National Science Foundation (CHE-9528530). It was facilitated by a Reynolds Research Leave from the University of North Carolina. The author thanks Aidi Chen, Stephen J. Gibbs, Kevin F. Morris, Konstantin Momot, and Donghui Wu for reading the manuscript and providing helpful comments. Also, the author thanks Alexej Jerschow, Cynthia K. Larive, Gareth A. Morris, and Norbert Mu¨ller for sending relevant preprints. References [1] R.R. Ernst, G. Bodenhauen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Oxford University Press, Oxford, 1987. [2] L. Mu¨ller, A. Kumar, R.R. Ernst, J. Chem. Phys. 63 (1975) 5490. [3] A. Kumar, D. Welti, R.R. Ernst, J. Magn. Reson. 18 (1975) 69. [4] H.J.V. Tyrrell, K.R. Harris, Diffusion in Liquids: A Theoretical and Experimental Study, Butterworths, London, 1984. [5] C.R. Cantor, P.R. Schimmel, Biophysical Chemistry, Part II: Techniques for the Study of Biological Structure and Function, W.H. Freeman, San Francisco, 1980. [6] C. Tanford, Physical Chemistry of Macromolecules, Wiley, New York, 1961. [7] E.O. Stejskal, J.E. Tanner, J. Chem. Phys. 42 (1965) 288. [8] K.F. Morris, C.S. Johnson Jr., J. Am. Chem. Soc. 114 (1992) 776. [9] K.F. Morris, C.S. Johnson Jr., J. Am. Chem. Soc. 114 (1992) 3139. [10] C.S. Johnson Jr., in: R. Tycko (Ed.), Nuclear Magnetic Resonance Probes of Molecular Dynamics, Kluwer Academic, Dordrecht, 1994, p. 455. [11] C.S. Johnson Jr., in: D.M. Grant, R.K. Harris (Eds.), Encyclopedia of NMR, Wiley, New York, 1996, p. 1626. [12] C.S. Johnson Jr., in: D.M. Grant, R.K. Harris (Eds.), Encyclopedia of NMR, Wiley, New York, 1996, p. 1886. [13] C.S. Johnson Jr., Q. He, in: W.S. Warren (Ed.), Advances in Magnetic Resonance, 13, Academic Press, New York, 1989, p. 131.
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