In Situ Pulsed-Field Gradient NMR Determination of the Size of Oil

Jul 26, 2007 - We report a pulsed-field gradient NMR study of the size of the oil bodies in lettuce seeds. The pulsed-field gradient spin−echo metho...
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Anal. Chem. 2007, 79, 6718-6726

In Situ Pulsed-Field Gradient NMR Determination of the Size of Oil Bodies in Vegetable Seeds. Analysis of the Effect of the Gradient Pulse Length Armel Guillermo*,† and Michel Bardet‡ aStructures

et Proprie´ te´ s d’Architectures Mole´ culaires, UMR5819, CEA-CNRS-Universite´ Joseph Fourier, DRFMC/SPrAM, and Service de Chimie Inorganique et Biologique, UMR_E3 CEA-Universite´ Joseph Fourier, DRFMC/SCIB, Cea-Grenoble, 38054 Grenoble Cedex 9, France

We report a pulsed-field gradient NMR study of the size of the oil bodies in lettuce seeds. The pulsed-field gradient spin-echo method (PFGSE) was applied to measure the self-diffusion coefficient of triacylglycerol molecules (TAG) inside the oil bodies. The confined nature of TAG diffusion is used to determine the size dispersion of the oil bodies. At long diffusion time, we measure a spin-echo attenuation that is related to the form factor of the confining volumes in the reciprocal q space, where q is proportional to the product of the gradient intensity and the length of the pulse gradient. Specific care was taken in analyzing the influence of the gradient pulse length δ on the shape of the PFGSE decay in order to construct the function corresponding to the short gradient pulse approximation (SGP). The SGP model gives an analytical framework for the PFGSE signal that enables the size distribution of the oil bodies to be determined. The SGP function was unambiguously obtained by varying the gradient pulse length δ in order to linearly extrapolate at δ ) 0 the SGP limit. In this work, we also consider the Gaussian phase distribution (GPD) assumption that is often used to analyze confined diffusion experiments. Although the GPD assumption is known to be inaccurate in predicting the fine structure of the PFGSE function in q space, we point out that in the present case it can be used to take into account the finite value of δ. A log-normal distribution of the radius values was assumed in simulating the PFGNMR experiments since this type of distribution is observed in vegetable seeds by transmission electronic microscopy. From a practical and experimental standpoint, the NMR measurements reported here require no specific treatment of the seeds and the size of oil bodies is determined “in situ” on seeds poured into the NMR tube. In the field of vegetable seed science, initial interest in highresolution solid-state NMR studies has undoubtedly increased, owing to the possibility of rapid noninvasive in situ characterization * Corresponding author. E-mail: [email protected]. † Structures et Proprie´te´s d’Architectures Mole´culaires. ‡ Service de Chimie Inorganique et Biologique.

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of seeds with 13C or 1H NMR.1-3 It is a particularly valuable technique for investigating the lipid moiety inside seeds.4-9 The large number of studies on oil composition of seeds is a clear illustration of this affirmation. Interest in studies of lipids certainly comes from the fact that they are a major constituent of seeds, in oleaginous crops, for instance. Lipids are an essential class of metabolites not only from a biochemical point of view but also for economic reasons. They are a major source of renewable energy and do not contribute to CO2 enrichment of the atmosphere. In vegetable seeds, the majority of lipids consist of triacyglycerols (TAG), which, at room temperature, behave like liquids in NMR experiments. Using a CP/MAS probe designed for solid-state NMR, it has been shown that they can readily be analyzed under magic angle spinning (MAS) conditions using classical multipulse experiments normally employed in liquidphase NMR.10 In addition to the above-mentioned structural aspects, the morphological features of the oil bodies that contain TAG are also of importance. Indeed, the average diameter of the majority of vegetables seeds among common crops is variable but lies in the range 0.2-2 µm.11 The range of their size distribution may also be variable and depends on taxonomy.12 Most of data have been obtained by electron microscopy. Although the results are reliable, preparation of samples is very time-consuming and not straightforward. Moreover, for quantitative measurements, handling many serial sections are required to obtain meaningful measurements. For many applications, either in fundamental or in applied research (1) Odonnell, D. J.; Ackerman, J. J. H.; Maciel, G. E. Service Chim. Inorg. Biol. 1981, 29, 514-518. (2) Bardet, M.; Foray, M. F.; Bourguignon, J.; Krajewski, P. Magn. Reson. Chem. 2001, 39, 733-738. (3) Bardet, M.; Foray, M. F.; Guillermo, A. In Modern Magnetic Resonance. Part 3: Applications in Material Science and Food Science; Webb, G. W., Ed.; Springer: Dordrecht, 2006; Vol. 3, pp 1755-1759. (4) Haw, J. F.; Maciel, G. E. Anal, Chem, 1983, 55, 1262-1267. (5) Wollenberg, K. J. Am. Oil Chem. Soc. 1991, 68, 391-400. (6) Hutton, W. C.; Garbow, J. R.; Hayes, T. R. Lipids 1999, 34, 1339-1346. (7) Rutar, V. J. Agric. Food Chem. 1989, 37, 67-70. (8) Rutar, V.; Kovac, M.; Lahajnar, G. J. Am. Oil Chem. Soc. 1989, 66, 961965. (9) Sayer, B. G.; Preston, C. M. Seed Sci. Technol. 1996, 24, 321-329. (10) Bardet, M.; Foray, M. F. J. Magn. Reson. 2003, 160, 157-160. (11) Huang, A. H. C. Annu. Rev. Plant Physiol. Plant Mol. Biol. 1992, 43, 177200. (12) Tzen, J. T. C.; Cao, Y. Z.; Laurent, P.; Ratnayake, C.; Huang, A. H. C. Plant Physiol. 1993, 101, 267-276. 10.1021/ac070416w CCC: $37.00

© 2007 American Chemical Society Published on Web 07/26/2007

Figure 1. PFGNMR sequences: (a) Hahn spin-echo sequence; (b) stimulated spin-echo sequence. The gradient field axis z is along the main magnetic field axis and the NMR tube axis.

such that performed by seed producers, assessing these morphological parameters with a robust and fast technique could provide a breakthrough in this field. The aim of the present work is to show that the pulsed-field gradient (PFG) spin-echo method certainly represents a promising path to reach this goal. A typical PFG spin-echo sequence consists of two short magnetic field gradient pulses (intensity g, length δ) and of a longer diffusion time delay ∆ between the pulses (Figure 1).13,14 The two main parameters of this experiment are the product δg and the delay ∆, which, respectively, define the spatial and time scales in the PFGNMR analyses of the molecular self-diffusion. This experiment commonly characterizes molecular diffusion on a spatial scale of 1-30 µm with diffusion times between 1 ms and 1 s. The order of magnitude of the measurable self-diffusion coefficient (Ds) ranges from 10-5 to 10-10 cm2/s. For unrestricted diffusion in homogeneous solutions, the molecules never encounter a physical barrier that limits their diffusion on the NMR time scale. The value of the self-diffusion coefficient is then independent of the choice of the experimental parameters (g, δ, ∆) employed to attenuate the echo amplitude. A fine application of this NMR method is found in the domain of liquid molecular mixtures. So-called diffusion-ordered two-dimensional NMR spectroscopy (DOSY) has excited increasing interest for its ability to measure the diffusion coefficient and to isolate the resonance spectrum of each component of a mixture.15 In contrast, restricted diffusion is clearly identified when Ds is no longer independent of the diffusion time ∆. Since the pioneer work of Stejskal and Tanner,16 much effort has been devoted to relating the properties of the pulsed-field gradient spin-echo (PFGSE) decay to the shape of the regions containing the molecular (13) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288-292. (14) Tanner, J. E. J. Chem. Phys. 1970, 52, 2523-&. (15) Morris, K. F.; Johnson, C. S. J, Am, Chem, Soc, 1992, 114, 3139-3141. (16) Tanner, J. E.; Stejskal, E. O. J. Chem. Phys. 1968, 49, 1768-1777.

probes.17,18 However, so far, only a few studies involving NMR and magnetic gradient, pulsed-gradient, or permanent stray-field gradient have been performed to characterize the morphology of oil bodies in seeds.19-21 In the present work, we show that both average size of oil bodies and their size distribution can be determined from the diffusion properties of TAG molecules inside the oil bodies. In the case of restricted diffusion, interpretation of the experimental data, i.e., the variation of the spin-echo amplitude versus values of the parameters of the NMR sequence, depends strongly upon the experimental conditions: according to the diffusion times that are used, specific information on the molecular dynamics (Ds) or on the morphology and the size of the confining volume is available. Moreover, in the specific case of seeds, the fact that TAG diffuse inside submicrometer-sized oil bodies (very small domains at the PFGNMR scale) implies that the gradient pulse length appears to be a crucial parameter. Below, we will go over the background theory underlying the experimental conditions of this work. The discussion of the experimental results is presented in such way that the knowledge of the size distribution of the oil bodies is gradually increased. This part is thus intended as a guide for the NMR spectroscopists in defining the working conditions to obtain the desired information. Nowadays, pulsed-field gradient sequences can be easily implemented on almost any commercial spectrometer. For specific applications, more powerful current amplifiers and the corresponding diffusion probes can be purchased. EXPERIMENTAL SECTION Materials. Seeds of lettuce (Latuca sativa) and tomato (Ailsa craig), both in a dormant physiological state, were used for these experiments. They were put directly into a 5-mm-diameter NMR tube with no other preparation. The height of seeds in the tube was ∼35 mm. NMR. The pulsed-field gradient NMR experiments (PFGNMR) were performed on a Bruker AVANCE spectrometer operating for protons at 400 MHz and equipped with the Bruker DIFF30 diffusion probe. The maximum intensity that can be delivered by the gradient unit is 1200 G/cm (12 T/m), the magnetic field gradient axis being parallel to the main steady field. Two basic experiments were usedsthe Hahn spin-echo sequence and the stimulated echo sequencesdepending on the range of diffusion time ∆ to be explored. Both sequences with their parameters are shown in Figure 1. The lengths δ of the magnetic field gradient pulses and the diffusion delays ∆ were set in intervals from 0.6 to 6 ms and from 4 to 200 ms, respectively. In the stimulated echo sequence, the evolution time τ of the transverse magnetization was chosen to be as short as possible, taking into account the recovery time of the diffusion probe (∼1 ms). In this work, τ lays between 6 and 10 ms. Typically, 16 acquisition scans were performed for each of the 16 gradient values used to measure the echo attenuation, the experiments (17) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Oxford University Press: Oxford, 1991. (18) Price, W. S. Concepts Magn. Reson. 1997, 9, 299-336. (19) Fleischer, G.; Skirda, V. D.; Werner, A. Eur. Biophys. J. 1990, 19, 25-30. (20) Zakhartchenko, N. L.; Skirda, V. D.; Valiullin, R. R. Magn. Reson. Imaging 1998, 16, 583-586. (21) Carlton, K. J.; Halse, M. R.; Maphossa, A. M.; Mallett, M. J. D. Eur. Biophys. J. Biophys. Lett. 2001, 29, 574-578.

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lasting for ∼15-20 min. The amplitude of the PFGNMR signal was measured by integrating the Fourier transform of the echo. Relaxation experiments were performed using a standard inversion-recovery sequence for the longitudinal relaxation and a CPMG sequence for the transverse relaxation. Unless, otherwise indicated in the text, all measurements were made at room temperature (T ) 298 K).

From Free Diffusion to Restricted Diffusion. The Gaussian nature of P (r0, r1, ∆) in the case of Brownian diffusion in an infinite homogeneous isotropic medium leads to the following attenuation of the echo amplitude:

E(g,δ,∆) E(q,∆) ) ) exp(-γ2g2δ2Ds∆) ) E(0,δ,∆) E(0,∆) exp(-(2πq)2Ds∆) (2)

THEORETICAL BASIS Physical Principles. In a typical PFG experiment (Figure 1), the first gradient pulse creates the relation between spatial position and resonance frequency. In the NMR rotating frame at the Larmor frequency of the spectrometer, the linear spatial variation of the magnetic field is turned into a helix of the transverse magnetization phase. Then, the diffusion process during the diffusion time alters the phase helix in such a way that the mean square displacement of molecules determines the amplitude of the echo refocused by the second gradient pulse. The attenuation of the echo amplitude, due to an increase of g, δ or ∆, is generally analyzed according to the two main models used for PFGNMR experiments. The first approach assumes a Gaussian distribution for the phase of the transverse magnetization (the Gaussian phase distribution (GPD) assumption), while in the second, gradient pulses are considered as being infinitely short (the short gradient pulse (SGP) approximation). The GPD assumption is valid only for free, unrestricted self-diffusion. In the case of a diffusion process that is spatially confined inside micrometer-scale domains, the presence of reflecting walls prevents the distribution of the magnetization phase from being Gaussian. In such a case, the SGP approximation is more convenient for calculating the echo attenuation function, also called the PFGNMR profile. The SGP approximation presumes that the molecular displacement during the gradient pulses can be neglected. It predicts the properties of the PFGSE function whatever the diffusion delay ∆. These include when ∆ is short enough, the ∆1/2 time dependence of the apparent diffusion coefficient yielding the characterization of the surface to volume ratio of domains. The validity of both approximations were analyzed in the literature on the basis of a comparison between values of pulse sequence delays and characteristic times resulting from the molecular diffusivity and the confining size.18,22,23 The attenuation of the spin-echo amplitude due to molecular diffusion, which is predicted by the SGP approximation, is24

E(g,δ,∆) )

∫∫F(r )P(r ,r ,∆)e 0

0 1

iγgδ(r1-r0)

dr1 dr0

E(q,a,∆) ) Eo(q,a) + E1(q,a,∆)

(22) Balinov, B.; Jonsson, B.; Linse, P.; Soderman, O. J. Magn. Reson., Ser. A 1993, 104, 17-25. (23) Price, W. S.; Soderman, O. Isr. J. Chem. 2003, 43, 25-32. (24) Stejskal, E. O. J. Chem. Phys. 1965, 43, 3597-3603.

Analytical Chemistry, Vol. 79, No. 17, September 1, 2007

(3)

where E(q, a ∆) stands for Eo (q, a) is the structure factor |S(q)|2 for a sphere of radius a:

[

Eo(q,a) ) 3

(1)

where γ is the gyromagnetic ratio of the nuclear spin. P (r1, r0, ∆) is the probability density for a displacement (r1 - r0) of a given molecule in time ∆, γ δ g (r1 - r0) is the phase shift of its transverse magnetization due to molecular diffusion, and F (r0) is the equilibrium spin density. In the SGP model, the product δg is assumed to take a finite value. The pitch of the transverse magnetization helix created by the first gradient pulse allows defining a PFGNMR wave vector q ) γδg/2π.

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where Ds is the self-diffusion coefficient of the molecules bearing the nuclear spins and E (0, δ, ∆) is the reference amplitude of the echo determined by transverse (Hahn echo) or transverse and longitudinal (stimulated echo) relaxation properties. Equation 2 assumes that the steady gradient of the main magnetic field is negligible. The diffusion time term ∆ on the right-hand side of eq 2 is the limit at infinitely short δ of its equivalent term (∆ δ/3) in the Stejskal-Tanner equation, which takes into account the effect of diffusion during the gradient pulse.13,14 The specificity of confined diffusion in a closed volume is that the maximum mean square displacement is restricted to the typical size a of the confining domain. Consequently, at infinite diffusion time, the attenuation function is no longer time dependent but size dependent: E (q, ∆) becomes E (q, a).17 Its analytical expression depends on the shape of the confining domain. The crucial point is that E (q, a) looks like a diffraction pattern and is identified to the power spectrum |S(q)|2 of the reciprocal lattice of the confining structure.17,25-28 In the following, E stands for diffusive process only; relaxation properties are taken into account in experiments by normalizing the echo amplitude to its value at g ) 0. Mathematical Formalisms for Confined Diffusion. To analyze the experimental data obtained on oil bodies in vegetable seeds, we focus on the particular case of spherical volumes. Some analytical expressions proposed in the literature22,29,30 were recently reinvestigated,31 leading to the general expression for E (q, a, ∆):

]

j1(2πqa)

2

) (2πqa) 9 (sin(2πqa) - (2πqa) cos(2πqa))2 (3a) (2πqa)6

and

E1(q,a,∆) ) ∞

6

∞

∑∑

n)1 k)1

(2n + 1)Rnk2

[

2πqa‚j′n(2πqa)

] (

Rnk2 - n(n + 1) Rnk2 - (2πqa)2

2

exp -Rnk2

)

Ds∆ a2

(3b)

In these equations, jn are spherical Bessel functions, Rnk is the kth root of the equation j′n(x) ) djn(x)/dx ) 0 and Ds is free self(25) Hakansson, B.; Pons, R.; Soderman, O. Langmuir 1999, 15, 988-991.

diffusion coefficient of the confined fluid. In order to be precise about the analogy between PFGNMR and scattering experiments, note that this last equation was first established to analyze water motion in nanometric confinements investigated by quasi-elastic neutron scattering.32 As the GPD assumption will be introduced in the discussion about the effect of the finite duration of gradient pulses, let us now consider the echo attenuation expression predicted by this model for confined diffusion inside a sphere. The equation is known as the Murday-Cotts equation.33,34 The Gaussian distribution of the transverse magnetization phase inside the confining sphere yields a Gaussian dependence of the echo amplitude instead of the diffraction pattern predicted by the SGP model for monodisperse restricting volumes (see Figure 2). The Murday-Cotts (M-C) equation can be written as follows:

Figure 2. PFGSE structure factor Eo(q, a)seq 3a, continuous lines and its Gaussian approximation exp(-(2πqa)2/5), dashed linessfor a sphere of radius a. The three first minima occur at qa ) 0.715, 1.230, and 1.733. The Gaussian approximation is better than 1% when qa < 0.20, i.e., for Eo(q, a) > 0.7.

ln(E(g,δ,∆,a)) )

( ) ∑ 2

∞

2 2 2

-2γ g a

k)1

Rk

Ds

-2

a2

Rk2(Rk2 - 1)

[Ak(g,δ,a) + Bk(g,δ,a,∆)] (4)

with

approximation exp[-(2πqa)2/5], or at high q with minima of the diffraction pattern (Figure 2). (2) At intermediate diffusion times, an apparent diffusion coefficient, i.e., a time-dependent diffusivity, can be measured at low q values. When qa is small, we identify q2 expansions of eqs 3 and 4 to the q2 expansion of the standard self-diffusion equation:

E(qa , 1,∆) ) 1 - (2πq)2Dapp(∆)∆ Dsδ 2

Ak(g,δ,a) ) e-R k Dsδ/a + Rk 2

2

a2

-1

(4a) in order to define an apparent diffusion coefficient Dapp(∆) by

and

Dapp(∆)∆ ) a

Bk(g,δ,a,∆) )

(

(2 - e-Rk Dsδ/a - e-Rk Dsδ/a ) exp -Rk2 2

2

2

2

(5)

)

D s∆ a2

2

[

1

5

-2

∞

e-Rk Ds∆/a

k)1

Rk2(Rk2 - 2)

∑

2

2

]

(6)

(4b)

where Rk are the roots of the equation of Bessel functions λmJ′3/2(Rk) - 1/(2)J3/2(Rk) ) 0 or also, according to the properties of the Bessel functions, roots of the equation j′1(Rk) ) 0. It implies that the roots Rk are equal to the roots R1k of the SGP equation (eq 3). The M-C equation is the Gaussian approximation of the diffraction pattern predicted by the SGP model (Figure 2). Three main results of the PFGNMR experiments can be derived from these previous equations. (1) At long diffusion times, ∆, the contribution to the echo attenuation, describes in eqs 3b and 4b becomes negligible and the echo amplitude no longer depends on the diffusion time but on the confining geometry (eqs 3a and 4a). The sphere radius is then determined either, at low q values, following the Gaussian (26) Topgaard, D.; Malmborg, C.; Soderman, O. J. Magn. Reson. 2002, 156, 195-201. (27) Topgaard, D.; Soderman, O. Magn. Reson. Imaging 2003, 21, 69-76. (28) Torres, A. M.; Michniewicz, R. J.; Chapman, B. E.; Young, G. A. R.; Kuchel, P. W. Magn. Reson. Imaging 1998, 16, 423-434. (29) Callaghan, P. T. J. Magn. Reson., Ser. A 1995, 113, 53-59. (30) Mitra, P. P.; Sen, P. N. Phys. Rev. B 1992, 45, 143-156. (31) Veeman, W. S. Annu. Rep. NMR Spectrosc. 2003, 50, 201-216. (32) Volino, F.; Dianoux, A. J. Mol. Phys. 1980, 41, 271-279. (33) Murday, J. S.; Cotts, R. M. J. Chem. Phys. 1968, 48, 4938. (34) Neuman, C. H. J. Chem. Phys. 1974, 60, 4508-4511. (35) Malmborg, C.; Topgaard, D.; Soderman, O. J. Magn. Reson. 2004, 169, 85-91.

We would underline two main practicable consequences of eq 6. First, Dapp(∆) tends toward the free diffusion coefficient Ds when the diffusion time tends to ∆ ) 0. Second, the experimental product ∆Dapp(∆) tends toward a2/5 for long enough diffusion times. A simple expression of eq 6 can be derived by neglecting terms k > 1 of the sum. The numbers Rk>12 are much larger than R12 and consequently, eq 6 can be simplified (in a 1% approximation) to

Dapp(∆)∆ ≈

2 2 a2 [1 - e-R1 Ds∆/a ] 5

(7)

∞ with R12 ≈ 4.333 (the limit value of ∑k)1 1/(Rk2(Rk2 - 2)) is 34 1/10). (3) At very short diffusion times, the behavior law of Dapp(∆) illustrates a property of the conditional probability P (r1, r0, t) in the case of diffusion of confined molecules. The probability of moving by a distance | r1 - r0 | in time t depends on the proximity of the molecules to the walls of the confinement volume. This is not the case in an infinite volume where P (r1, r0, t) is independent of r0 and depends only on the difference r1 - r0. Mitra et al., by considering the specific behavior of molecules enclosed in a surface layer of thickness xDst, predicted that the apparent diffusion coefficient should vary as ∆1/2.36 A numerical application

(36) Mitra, P. P.; Sen, P. N.; Schwartz, L. M. Phys. Rev. B 1993, 47, 8565-8574.

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of eq 6 (with 1 e k e 12) shows that this square root behavior is observed in a sphere if DS∆/a2 e 1/10. All the previous equations, established for a single value of the radius, are now considered in the case of a distribution of sphere size. For a given sphere, the contribution of its molecules to the total magnetization is proportional to the volume of this sphere. Thus, the observed PFGNMR signal is a volume-weighted average of the elementary functions E (q, a, ∆). The PFGSE decay of the whole sample is given by

E(q,∆) )

∫

∞

0

a3P(a)E(q,a,∆) da

∫

∞

0

(8)

a3P(a) da

where P (a) is the probability density of the radius. Consequently, the structure of the diffraction pattern is averaged and tends to vanish as the distribution width increases. The mean size directly determined by the PFGNMR experiment is < a2 >v, the volumeaverage mean square radius calculated with the distribution of the molecule number N (a) ∼ a3P (a),

∫a [a P(a)] da ) ∫a P(a) da 2

2

< a >v

3

(9)

3

Thus, for polydisperse systems, the Gaussian approximation of the echo attenuation at small q is

(

E(q) ) exp -(2πq)2

)

< a2 >v 5

(10)

In this work, we consider the log-normal distribution P (a) of width σ centered on a0

[

]

(ln(a) - ln(a0))2

P(a) ) exp -

2σ2

(11)

This last equation is an unnormalized, but more convenient, expression of the log-normal distribution: the maximum probability occurs at a0, which allows direct comparison with radius histograms obtained from electron microscopy.12 For example, the discrete histograms obtained by these authors for rape and mustard seeds can be satisfactorily simulated by the log-normal distribution (eq 11) in which a0 ) 0.14 µm and σ ) 0.58 for rape seeds, a0 ) 0.15 µm and σ ) 0.60 for mustard seeds. Note that < a2 >v is very sensitive to the distribution width and may be much larger than the mean square radius < a2 >. A last point that has to be considered in practice is the finite length of the gradient pulses. The SGP approximation ignores diffusion processes during gradient pulses by assuming infinitely short gradient pulses. This approximation is valid only if the gradient pulse duration δ is much smaller than a2/2Ds, the characteristic time of the mean size of the molecular random walk inside confining domains. To illustrate this point, let us consider TAG diffusing in small oil bodies of seeds (i.e. Ds ≈ 10 µm2/s and a ) 0.15 µm); the characteristic time a2/2Ds is roughly equal 6722

Analytical Chemistry, Vol. 79, No. 17, September 1, 2007

to 1 ms, which typically corresponds to delays used for gradient pulses in PFGSE sequences. It is thus necessary to take into account the actual duration of the gradient pulse, otherwise a SGP analysis of experiments yileds an underestimate of the confinement volume.37-40 In the present work, we quantified the influence of the finite length of gradient pulses experimentally. The PFGSE echo attenuation was measured with different values of the gradient pulse width δ, for a given diffusion time delay ∆. The actual SGP function was established by linear extrapolation of the experimental results to δ ) 0. Moreover, these experiments were used as a test of the ability of the GPD approach for describing the effect of long gradient pulses.22,35 EXPERIMENTAL RESULTS AND DISCUSSION NMR Spectrum and Relaxation Properties of TAG in Lettuce Seeds. The PFGNMR signal is a spin-echo of the transverse relaxation. Prior to any self-diffusion measurement in an heterogeneous sample, it is necessary to clearly identify molecules contributing to the spin-echo. First, the transverse relaxation function of the proton magnetization of seedssMx(t)s was measured with the standard CPMG sequence. Mx(t) presents a bicomponent behavior. Its first part is a fast relaxing component that is assigned to protons of the solid domains of the seeds. The second component, observed for t > 0.25 ms, is a slow relaxing component assigned to protons belonging to the mobile domains in the seeds. The proton spectrum associated with a spin-echo measured at 2τ > 0.25 ms presents only the chemical shifts found in a pure TAG reference sample (olive oil, see spectra in the Supporting Information). This test is of the greatest importance in order to make sure that the PFGNMR measurements really concern the mobile TAG moiety of seeds. The NMR relaxation properties of TAG in seeds were also compared with those of olive oil. For TAG in seeds, the longitudinal relaxation time T1 was found equal to 585 ( 8 ms and the transverse relaxation times T2 was equal to 96 ( 9 ms. In the olive oil sample, these NMR relaxation times were found equal to 566 ( 3 and 79 ( 9 ms, respectively. These very similar values clearly indicate that there is no significant difference in the local dynamics properties of TAG molecules either in bulk or included inside the oil bodies of the seeds. We may therefore assume that the translational diffusivity of TAG is also equivalent in both samples. We measured the TAG diffusion coefficient in olive oil for different sets of gradient pulse lengths δ and diffusion times ∆ representative of the PFGNMR sequences used in the study of the seeds; we obtained Ds (TAG in olive oil) ) 1.04 ( 0.08 10-7 cm2/s at 298 K (experimental curves available as Supporting Information). Effect of the Diffusion Time on PGSE Decay Properties: Evidence for a Restricted Diffusion Process. To identify the restricted nature of the self-diffusion process, we refer to eqs 3 and 4 showing that, at long diffusion time, the attenuation of the echo amplitude tends to a limiting value defined by the confinement size (eqs 3a and 4a). The time dependent behavior of E (q, ∆) is shown in Figure 3: whatever the intensity of the magnetic (37) Blees, M. H.; Geurts, J. M.; Leyte, J. C. Langmuir 1996, 12, 1947-1957. (38) Caprihan, A.; Wang, L. Z.; Fukushima, E. J. Magn. Reson., Ser. A 1996, 118, 94-102. (39) Mitra, P. P.; Halperin, B. I. J. Magn. Reson., Ser. A 1995, 113, 94-101. (40) Linse, P.; Soderman, O. J. Magn. Reson., Ser. A 1995, 116, 77-86.

Figure 3. Proton echo attenuations (TAG molecules) measured versus the diffusion time for lettuce seeds at 298 K. The gradient pulse length δ is equal to 1.2 ms. Wave vector values (q ) γgδ/2π): (~) 610, (O) 1430, (0) 2550, (]) 3600, (4) 4600, and (3) 5620 cm-1. Figure 5. PFGNMR profiles E(q) for lettuce seeds recorded at long diffusion time (∆ ) 100 ms) with δ ) 0.6 (+), 1.2 (O), 2.4 ([0]), 3.2 ([]]), 4 (4), 4.8 (3), and 6 ms (~). Dashed lines are guides for the eyes. The procedure used to determine the true SGP function is indicated (dotted lines). Plots of experimental values q against gradient pulse lengths at E (q) ) 0.60 (O) and 0.30 (~) are shown in the inset; q values extrapolated to δ ) 0 are the wave vector magnitudes qeff in agreement with the SGP approximation.

Figure 4. Plot of ∆Dapp(∆) used to determine the number average < a2 >v of the oil bodies radii in lettuce seeds. The plateau value observed at ∆ g 0.1 s is < a2 >v/5 (eq 6). The continuous line was calculated with the radius distribution obtained from the SGP analysis; see below Figure 6. In the inset, the straight line shows that the free self-diffusion coefficient Ds of TAG inside oil bodies (data extrapolated to ∆ ) 0) is in agreement with the value found for TAG in pure olive oil, Ds ≈ 10-7 cm2/s.

field gradient, the echo intensity decays toward a plateau value for ∆ g 100 ms. Thus, 1 order of magnitude of the size of the oil bodies is given by a2 ≈ Ds∆ ) 10-8 cm2 with Ds ) 10-7 cm2/s and ∆ ) 0.1 s. The presence of a quite flat plateau means, moreover, that oil bodies are perceived as static. At a given diffusion time ∆, an apparent diffusion coefficient Dapp(∆) was measured from the low q part of the attenuation functions E (q, ∆). In this q domain, the echo attenuation depends linearly on (2π q)2∆ and Dapp(∆) is obtained by fitting experimental data to eq 5. The computed values Dapp(∆) were used to determine the size of the oil bodies by plotting the product ∆Dapp(∆) versus ∆ (Figure 4). At long diffusion time, ∆Dapp(∆) reaches a plateau value as predicted by eqs 6 and 7. The height of the plateau is assigned to the mean square value < a2 >v/5, which yields < a2 >v ) 0.36 ( 0.01 µm2. An estimate of the diffusion coefficient of TAG in seed oil bodies is obtained; in small diffusion time domain, by extrapolating to ∆ ) 0 the linear dependence of 1/(Dapp(∆)), we find Ds ≈ 10-7 cm2/s. This result validates the earlier assumption equating the diffusivity of TAG in seeds and in crude oil. At this stage of the analysis, we know that 10-7 cm2/s is an accurate value for the self-diffusion coefficient of TAG in oil bodies and that the mean radius of the molecule number distribution is

0.60 ( 0.01 µm. It must be emphasized that this value may be quite larger than the mean size of the radius because of the strong contribution of the largest vesicles to the total magnetization. In the following, the whole attenuation curve will be analyzed in order to determine the most probable parameters (ao, σ) of the radius distribution. PFGSE Experiments at Long Diffusion Times, Effect of Gradient Pulse Lengths. Determination of the SGP Function. To generate larger echo attenuations, the wave vector q was increased, either by lengthening the gradient pulse or by raising the field gradient intensity. In the case of free diffusion, these two experimental choices are equivalent. However, it is no longer true for restricted diffusion, as is seen in Figure 5. The attenuation curves at long diffusion time (here ∆ ) 100 ms) were measured for a gradient pulse length δ in the range 0.6-6 ms and a gradient strength varying between 10 and 1200 G/cm. It is worth noting in Figure 5 that, for a given value of the wave vector q, the echo attenuation depends on the length of the gradient pulse: a short pulse of a high gradient induces stronger attenuation than a longer pulse of a lower gradient. The diffusion time used in these experiments was long enough to consider that the mean square displacement of TAG keeps unchanged when δ increases from 0.6 to 6 ms. Moreover, it has been already pointed out that the standard correction for diffusion time, ∆ - δ/3, used for unrestricted diffusion, is no longer valid for a confined diffusion process.23 Consequently, the effect of the pulse width has to be related to the actual magnitude of the wave vector q itself. The molecular displacement during gradient pulses ranges between 0.1 and 0.35 µm, which is comparable to the size of oil bodies; we presume that the dephasing of the magnetization generated by more and more long gradient pulses is then spoiled in such a way that the pitch of the magnetization phase helix (q-1), at the end of the gradient pulse is more and more larger than its expected value 2π/γgδ. The actual SGP function was determined from the analysis of the dependence of the echo attenuation on the duration of the gradient pulse. For this purpose, the observed q values correAnalytical Chemistry, Vol. 79, No. 17, September 1, 2007

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dynamic simulation that the GPD assumption can be accurate enough in some cases.22 The interest of the GPD approach lies in the fact that, even at long diffusion time, the duration of gradient pulses and the free diffusion coefficient appear explicitly in the equation for the echo amplitude. It is thus fruitful to compare the analyses of the experimental data with SGP and GPD models. For this purpose, we rewrite the M-C equation (eq 4) with variables suited to the gradient pulse length influence: ∞

ln(E(g,δ,∆,a)) ) -2γ2g2a2

∑ k)1

Figure 6. Simulations of SGP(qeff) (b) and the experimental data E(q) measured with δ ) 2.4 (0), 3.2 (]), and 6 ms (~). The continuous line is the SGP profile calculated according to eqs 3a, 8, and 11 with a0 ) 0.11 ( 0.01 µm and σ ) 0.595 ( 0.015; dashed lines are GPD simulations using the same radius distribution in eqs 8, 11, and 12 with δ ) 2.4, 3.2, and 6 ms. The dotted line is a GPD calculus for δ ) 6 ms and a0) 0.08 µm and σ ) 0. 65.

sponding to a given attenuation level were plotted against the experimental delays δ used to generate these wave vector values (see insert in Figure 5). A linear dependence between δ and q was found for each attenuation level and extrapolations to δ ) 0 determine the actual values of the wave vector q corresponding to the SGP approximation. These values are denoted qeff. The function SGP (qeff) is then constructed using the experimental echo attenuation versus the magnitude of the wave vector qeff (Figure 6). As can be seen in Figure 6, the consequences of the finite length of the gradient pulse are not negligible. For instance, at low echo amplitude (E (q) ≈ 0.15), the effective q is almost half of its experimental value for δ ) 6 ms. It means that a direct SGP analysis of the PFGSE decay corresponding to this last gradient pulse would underestimate noticeably the size of confinement domains. This experiment demonstrates that analysis of PFGNMR results with the SGP model necessitates varying the gradient pulse width as it is shown in Figure 5. The SGP function E (qeff) is analyzed using a log-normal distribution of the radius (eqs 3a, 8, and 11) knowing that < a2 >v ) 0.36 µm2 as was found in the previous section. The best calculated SGP functions are obtained with a0 ) 0.11 ( 0.01 µm and σ ) 0.595 ( 0.015 (Figure 6). We note that this distribution is close to those obtained for rape and colza seeds by TEM experiments.12 PFGSE Results Analyzed with the Gaussian Phase Distribution Assumption. To take into account the finite length of gradient pulses in a more direct way, we consider in this section a GPD analysis of the experimental results. In studies of droplet size in seeds21 or in food emulsions; 41,42 the authors refer to the Murday-Cotts equation for analyzing the echo attenuations.33,34 Although this model does not predict the diffraction pattern for monodisperse restricting volumes, it has been shown by Brownian (41) Goudappel, G. J. W.; van Duynhoven, J. P. M.; Mooren, M. M. W. J. Colloid Interface Sci. 2001, 239, 535-542. (42) van Duynhoven, J. P. M.; Goudappel, G. J. W.; van Dalen, G.; van Bruggen, P. C.; Blonk, J. C. G.; Eijkelenboom, A. Magn. Reson. Chem. 2002, 40, S51S59.

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1 Rk2(Rk2 - 2)

[Ak + Bk] (12)

with

[

Ak ) 2tck2 e-δ/tck +

δ -1 tck

]

(12a)

and

Bk ) -tck2e-∆/tck[eδ/tck + e-δ/tck - 2]

(12b)

in which tck stands for a2/Rk2Ds. At long diffusion times, when ∆/tck is large, Bk tends toward zero and the GPD echo attenuation reduces to

ln(E(g,δ,a)) ) -4γ2g2a2

∞

tck2

k)1

Rk2(Rk2 - 2)

∑

[

e-δ/tck +

δ tck

]

- 1 (13)

Terms of the sum determine the effective length of the gradient pulse taking in account ratios δ/tck. The first term k ) 1 is the main term of the sum and, when δ/tc1 e 1, eq 13 is reduced to

(

1 δ ln(E(g,δ,a)) ≈ - γ2g2a2δ2 1 5 3tc1

)

(14)

In this last equation, we recognize the well-known GPD long time limit for confined diffusion in a sphere, ln(E(g,δ,a)) ) -γ2g2δ2a2/ 5, valid for short gradient pulses.16 For finite δ values up to δ(R12Ds)/a2 ≈ 1, an effective pulse length δeff must be considered in the PFGNMR experiment. It depends on the molecular diffusion rate and the size of the confinement:

(

δeff ) δ 1 -

R12Ds 6a2

δ

)

(15)

Let us note that this GPD prediction is qualitatively in agreement with the observed experimental linear behavior q/qeff(δ) (see inset in Figure 5). To highlight the pulse length effect, the PFGNMR profiles were measured at two different temperatures, 298 and 280 K (Figure 7). The pulse gradient length is 4.8 ms, and the diffusion time is long enough to measure at each temperature the long diffusion time limit of E (q). The self-diffusion coefficient of the pure olive

Figure 7. PFGNMR profiles of lettuce seeds recorded at 280 K (1) with δ ) 4.8 ms and ∆ ) 128 ms and at 298 K (3) with δ ) 4.8 ms and ∆ ) 96 ms. At 280 K, the echo attenuation was found independent of ∆ for ∆ g 128 ms. Lines are GPD profiles (eqs 8, 11, and 12) calculated with ao) 0.08 µm, σ ) 0.65, Ds (280 K) ) 3.5 × 10-8 cm2/s and Ds (298K) ) 1 × 10-7 cm2/s.

oil was measured at 280 K (Ds ) 3.5 × 10-8 cm2/s) and at 298 K (Ds ) 1 × 10-7 cm2/s). At low q, the two curves coincide, which means that the mean square < a2 >v remains unchanged. At high q, the shift of the attenuation function must be analyzed according to the effective length of the gradient pulse: the ratio Ds (T)/a2 decreases with the temperature, which yields a more efficient gradient pulse for measurements performed at low temperature (eq 15). The GPD simulations in Figure 7 were performed considering the experimental temperature dependence of Ds only; the radius distribution was unchanged. In consequence of which, a SGP analysis, neglecting the gradient pulse length effect, which would attribute the change of the shape of PFGSE signals to a variation of parameters of the radius distribution with the temperature, would be erroneous. To check the ability of the GPD assumption to describe the observed gradient pulse length effect, eqs 8, 11, and 12 were used to simulate the PFGNMR profiles measured at room temperature (Figure 6). The GPD simulations were performed with Ds ) 10-7 cm2/s and with the radius distribution determined with the previous SGP analysis. The calculated functions are close to the experimental profiles. The effect of the finite length of the gradient pulse is fairly well described by the M-C equation. However, an increasing gap between the simulation curves and the experimental data is observed when δ increases. For the longest gradient pulse (δ ) 6 ms), the best simulation of the experimental profile is obtained with a0 ) 0.08 µm and σ ) 0.65 (dotted line in Figure 6). Note that this distribution lies slightly outside the error bars of the distribution obtained with the SGP analysis. In summary, analysis of the PFGSE decay using the GPD assumption gives the same radius distribution as the function SGP (qeff) if the gradient pulse δ is not too long. For oil bodies in lettuce seeds, the limit value is δ ≈ 3 ms; i.e., Dsδ/< a2 >v ≈ 1/10. When the gradient pulse is longer than this limit, the GPD analysis involves a slight underestimate of the size of the oil bodies (dotted line in Figure 6). So, the treatment of the gradient pulse length with the assumption of a Gaussian phase distribution cannot be yet considered as fully satisfactory. Compared Results for Two Species, Lettuce and Tomato. To conclude this section, we present PFGNMR results obtained

Figure 8. Long diffusion time PFGSE decays at room temperature for tomato seeds (9) and lettuce seeds (0). These decays were recorded with δ ) 2.4 ms. Lines are GPD functions calculated (eqs 8, 11, and 12) with a0 ) 0.23 µm, σ ) 0.48 for tomato seeds and a0 ) 0.11 µm, σ ) 0.595 for lettuce seeds.

in similar experimental conditions for tomato seeds and lettuce seeds. The PFGSE decays recorded in the long diffusion time regime are compared in Figure 8; the gradient pulse duration is 2.4 ms. Echo attenuation is stronger for tomato seeds, which means that the tomato oil bodies are larger than those of lettuce. The validity of the GPD analysis of PFGNMR experiments performed with δ ) 2.4 ms was previously established for lettuce seeds (Figure 6); it means that the condition Dsδ/< a2 >v < 1/10 is also verified for larger oil bodies of tomato seeds. The GPD analysis with eqs 8, 11, and 12 yields that the log-normal distribution of the oil bodies radius of tomato is characterized by a with a0 ) 0.23 µm and σ ) 0.48 (Figure 8). CONCLUSION In this work, we have presented an illustration of the capabilities of PFGNMR experiments in the field of diffusion-diffraction. This application to the in situ determination of the size of the oil bodies requires no specific preparation of seed samples, which is a considerable advantage over TEM approaches. Analysis of restricted diffusion properties of TAG inside oil bodies underlies the determination of size of these lipid vesicles. For such submicrometric spheres, particular care must be taken for the effects of gradient pulse lengths δ. We have shown that the determination of the SGP function, necessary to determine the radius distribution, is achieved by a linear extrapolation to δ ) 0 of PFGSE decays measured in function of gradient pulses width. It must be emphasized, however, that size dispersion of the oil bodies may alter the intrinsic effect of the delay δ on the shape of the PFGSE decay. Note that an analysis using a matrix formalism to simulate the PFGNMR sequence as a multiplet of infinitesimally short narrow pulses was recently proposed in the framework of the SGP model.38,43-45 We have also pointed out that a single experiment, analyzed in the GPD framework, may be sufficient to estimate the size distribution. If the mean square radius of the volume distribution < a2 >v and the gradient pulse (43) Callaghan, P. T. J. Magn. Reson. 1997, 129, 74-84. (44) Codd, S. L.; Callaghan, P. T. J. Magn. Reson. 1999, 137, 358-372. (45) Price, W. S.; Stilbs, P.; Soderman, O. J. Magn. Reson. 2003, 160, 139-143.

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width verify Dsδ/< a2 >v e 1/10 (Ds is the free self-diffusion coefficient of TAG), the distribution is identical to that determined with the SGP analysis. The radius distributions found in lettuce and tomato seeds are in good agreement with those obtained from electronic microscopy for vegetable seeds. In this work, a NMR probe dedicated to diffusion measurements was required to generate short, high gradient pulses. However, it must be emphasized that the oil bodies investigated in this study were among the smallest ones. In fact, curves in Figure 8 show that magnitudes of the reciprocal vector q such as qao ) 0.15-0.25 (ao is the center of radius distributions) are sufficient to identify the radius distribution. So, it is conceivable to determine the radius distribution of micrometric vesicles with a standard liquid-phase NMR probe equipped with the gradient coils used for coherence pathway selection in multipulse experiments (with g ≈ 60 G. cm-1, and δ ) 5 ms, q ≈ 1.3 × 103 cm-1, i.e., 0.13 µm-1).

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ACKNOWLEDGMENT Experiments were performed with the CGRM facilities (Magnetic Resonance Center of Grenoble); we thank Mrs. M. F. Foray and Dr. G. Gerbaud for their technical assistance. We address particular acknowledgements to Dr. S. Hediger, Dr. E. Geissler, and Dr. F. Volino for their comments about the presentation and corrections of the manuscript. This study was partly supported by a Grant from GENOPLANTE-VALOR. SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review March 1, 2007. Accepted June 15, 2007. AC070416W