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Measurement of anisotropy of pore diffusion in protein crystals by PFG NMR and by CLSM
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Microporous and Mesoporous Materials 112 (2008) 474–480 www.elsevier.com/locate/micromeso
Measurement of anisotropy of pore diffusion in protein crystals by PFG NMR and by CLSM Ruslan Archipov b
a,1
, Aleksandar Cvetkovic b, Frank Stallmach
a,*
, Adrie J.J. Straathof
b
a Universita¨t Leipzig, Fakulta¨t fu¨r Physik und Geowissenschaften, Linne´str. 5, 04103 Leipzig, Germany Department of Biotechnology, Delft University of Technology, Julianalaan 67, 2628 BC Delft, The Netherlands
Received 30 August 2007; received in revised form 6 October 2007; accepted 12 October 2007 Available online 24 October 2007
Abstract Solvent (water) and solute (fluorescein and rhodamine B) diffusion in crystalline lysozyme protein, which represent a mesoporous material with an anisotropic pore structure, was studied by pulsed field gradient nuclear magnetic resonance (PFG NMR) and confocal laser scanning microscopy (CLSM), respectively. Water and fluorescent dyes reveal anisotropic intracrystalline diffusion behaviour. The averaged intracrystalline self-diffusion coefficient of water was found to be about one order of magnitude smaller than the value for bulk liquid water at the same temperature. However, the intracrystalline self-diffusion coefficients of water are about two orders of magnitude faster than the corresponding values of the fluorescent dyes. For water and fluorescein in tetragonal lysozyme crystals, the anisotropy ratio, which is defined as the ratio of the largest component of the diffusion tensor to its averaged orthogonal value, was about 0.16, whereas for orthorhombic crystals the anisotropy ratios measured by CLSM for fluorescein (0.15 ± 0.03) and rhodamine B (0.25 ± 0.05) were in the same order of magnitude but somewhat larger compared to the value for water (0.11 ± 0.04) measured by PFG NMR. 2007 Elsevier Inc. All rights reserved. Keywords: Diffusion anisotropy; Protein crystals; Lysozyme; PFG NMR; CLSM
1. Introduction Protein crystals are structures with a porosity ranging from 20% to 80% and pore diameters in the range of 0.5– 10 nm [1]. Protein crystals are usually obtained from an aqueous buffer. Although crystallization may be cumbersome, the crystal structure of about 30,000 proteins has been solved. Moreover, crystallization has become the preferred recovery procedure for some proteins that are produced on a large scale [2]. Protein crystals can be stabilized against mechanical, thermal, pH and organic solvent stresses upon cross-linking with bifunctional reagents. Then they can be used as mesoporous materials for a variety of purposes, such as biocatalysis and selective adsorption *
1
Corresponding author. Tel.: +49 341 973 2518; fax: +49 341 973 2549. E-mail address: [email protected] (F. Stallmach). Present address: Kazan State University, Russian Federation.
1387-1811/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.micromeso.2007.10.024
[3]. These materials are particularly interesting because of their very wide structural diversity, the inherent anisotropy and chirality of the pores, and, in case of enzyme crystals, the presence of a selective catalytic site. Besides, they are useful for delivery of biopharmaceuticals [4]. However, diffusion through the internal pore structure of the crystals is required for molecules of cross-linkers, enzyme substrates, adsorbates, and biopharmaceuticals. Measurement of intracrystalline diffusivities is required to develop the aforementioned applications. Preferably, techniques should be used, which measure displacements or concentration profiles inside the porous hosts, thus avoiding complications arising from macroscopic techniques, which observe the total uptake of guest molecules [5]. Recently, two such techniques have been developed for measuring anisotropic diffusion in mesoporous materials. Anisotropic solute diffusion has been studied by monitoring the uptake of solutes by lysozyme protein crystals using
R. Archipov et al. / Microporous and Mesoporous Materials 112 (2008) 474–480
confocal laser scanning microscopy (CLSM) [6]. This technique yields time-resolved three-dimensional micrographs of intracrystalline solute concentration. However, its application is restricted to fluorescent dyes as solutes, which are relatively large (>0.7 nm), and only to individual protein crystals larger than 10 lm, which are carefully selected under the microscope. In contrast, pulsed field gradient nuclear magnetic resonance (PFG NMR) yields the self-diffusion for molecules adsorbed in a bed of the porous material of roughly 106 m3 volume [7]. It is suitable to observe diffusion of the smaller water (solvent) molecules by using 1 H resonance. 1H PFG NMR studies of diffusion anisotropy have been performed in microporous MFI zeolites [8] and mesoporous MCM-41 [9]. However, since these measurements are performed in a bed of the material with random orientation of the crystals (and, thus, the pores) with respect to the direction of the diffusion measurement, which is defined by the direction of the externally applied magnetic field gradient, analysis of the experimentally observed PFG NMR spin echo attenuation curves requires model assumptions to deduce the elements of the diffusion tensor [7–9]. Thus, CLSM and PFG NMR may be considered as complimentary methods in studying solute and solvent diffusion in anisotropic micro- and mesoporous materials. The goal of the present work is to compare experimental data on water and fluorescent dye diffusion in lysozyme protein crystals, as obtained by PFG NMR and CLSM, respectively. We will show that solvent and solute diffusion in these protein crystals is anisotropic with similar values for the anisotropy ratio. 2. Theoretical section: anisotropic diffusion in PFG NMR In PFG NMR diffusion studies, the projection of the mean square displacement of the molecules on the direction of the externally applied decoding and encoding pulsed magnetic field gradients ! g is measured [7,10,11]. The time scale of this measurement is defined by the distance D between the pulsed field gradients in the spin echo NMR pulse sequence and referred to as diffusion time. Generally, for a fixed diffusion time the intensity of the spin echo NMR signal M is recorded as function the amplitude g and the width d of these pulsed field gradients. The three quantities g, d and D, which are summarized for convenience to the gradient parameter b(g, d, D), control the attenuation of the NMR signal due to self-diffusion. For example, for PFG NMR diffusion studies with two pulsed field gradients, b is simply given by b ¼ ðcgdÞ2 ðD 13 dÞ, where c denotes the gyromagnetic ratio of the nuclei under investigation. In order to evaluate the diffusion process and calculate the self-diffusion coefficients from the observed spin echo intensities M(b), the spin echo attenuation M(b)/M0 (with M0„M(b = 0)) is plotted as function of b on a semi-logarithmic scale. In this presentation, a single-component isotropic self-diffusion shows up as a linear decay. From its
475
slope, the self-diffusion coefficients D is obtained using the equation [7,10,11] MðbÞ=M 0 ¼ expðD bÞ:
ð1Þ
Any deviation from this single-exponential spin echo attenuation indicates the presence of a distribution of self-diffusion coefficients. In the case of a multi-component isotropic diffusion, one expects a superposition of exponential decays according to [7,11] X MðbÞ=M 0 ¼ pi expðDi bÞ; ð2Þ i
with pi and Di denoting the relative contribution of component i to the spin echo signal and its self-diffusion coefficient, respectively. In microporous crystalline materials, anisotropic intracrystalline diffusion may be caused by the anisotropy of the micropore structure and the diffusion process must be described by a second rank diffusion tensor D [8,9,12]. The orientation of this tensor with respect to the direction of the externally applied pulsed field gradient ! g modifies the spin echo attenuation. For example, for one-dimensional diffusion in straight channels with polar angle H between gradient and channel direction, a diffusion of Dpar along the channel direction leads to an orientation-dependent exponent of b Dpar cos2 H in the spin echo attenuation of Eq. (1). If one has a distribution of different channel orientations, as it is usually the case in beds of crystalline materials, Dpar cos2 Hi replaces Di in Eq. (2) and pi denotes the probability of finding the orientation Hi in the sample. Thus, in PFG NMR anisotropic diffusion in beds of microporous materials may be considered as special case of multi-component diffusion, where the different components are just caused by the different orientations with respect to the pulsed field gradient direction. If one has a random packing of the microporous materials in the bed, all orientation of the diffusion tensor associated with the anisotropic micropore structure are equally probable and the spin echo attenuation is obtained by the powder average over all orientations. The CLSM studies with tetragonal and orthorhombic lysozyme protein crystals showed that fluorescent dye diffusion is anisotropic with three different diagonal elements (Dii > 0) of the diffusion tensor D, each describing the diffusion along one of the three orthogonal crystallographic directions. In such a situation, the complete powder average over the polar angles H and u yields Z 2p Z p 1 MðbÞ=M 0 ¼ expfbðD11 cos2 H 4p 0 0 þ D22 sin2 H cos2 u þ D33 sin2 H sin2 uÞg sin HdHdu
ð3Þ
for the PFG NMR spin echo attenuation [11]. Qualitatively, Eq. (3) describes a non-exponentially, monotonously decaying function. Its slope decreases with increasing b-valP ues and yields at b = 0 the trace of the tensor 13 i Dii . Only
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the total shape of this function including high b-values reveals detailed information on all three elements of the diffusion tensor. It is in the nature of PFG NMR measurements that high b-values correspond to strong spin echo NMR signal attenuations. Thus, a poor signal-tonoise ratios is inherent for parts of the experimental data necessary for the detailed analysis using Eq. (3). Additionally, the integrals in Eq. (3) cannot be solved further and analysis of experimental spin echo attenuations in anisotropic systems requires numerical solutions and approximations, respectively. A first step to approach this complex situation represents the reduction of free parameters in Eq. (3) by using properties of the microporous pore space of the protein crystals. In crystallographic c-direction, the pore diameter of the tetragonal (dp = 1.38 nm) and the orthorhombic (dpc = 1.97 nm) lysozyme crystals is about twice as wide as in the two orthogonal a- and b-directions, where one has 0.77 nm and 0.84 nm, respectively, for the tetragonal and 0.74 nm and 1.01 nm, respectively, for the orthorhombic structure. These pore size data were obtained by modelling of the protein crystal structure in our previous work [13]. Compared to this major difference, the deviation in diameter between the a- and b-directions is small. If – as for the fluorescent dyes – the pore size represents also for the water molecules the major reason for diffusion anisotropy, one may assume that the difference between the diffusion coefficients in the crystallographic a and b directions is negligible compared to the much faster diffusion in c-direction. Thus, the diffusion in the ab-plane may be approximated by a single value Dperp (= D22 = D33 in Eq. (3)), which, however, is expected to be smaller than the diffusion along the c-direction Dpar (= D11), where the micropore structure exhibits the substantially larger pores. The results of this approximation correspond to the model of an axisymmetrical diffusion tensor, for which the spin echo attenuation of Eq. (3) simplifies to (see also Refs. [9–11]) Z 1 p MðbÞ=M 0 ¼ expfbðDpar cos2 H þ Dperp sin2 HÞg 2 0 sin HdH: ð4Þ The indices par and perp were used in order to distinguish between the diffusion along (parallel to) the principal axis of the tensor (Dpar) and perpendicular to it (Dperp). From Eq. (4), the special cases for one-dimensional diffusion along randomly oriented channels and two-dimensional diffusion in randomly oriented sheets are easily derived by setting Dperp and Dpar equal to zero, respectively. 3. Experimental section Lysozyme protein crystals of tetragonal (6LYT) and orthorhombic (1AKI) structures were synthesized from chicken egg-white lysozyme following a procedure as described previously [6]. Cross-linked protein crystal were
obtained by using glutaraldehyde as cross-linking agent. After synthesis, the protein crystals were filtered from the synthesis solution and rinsed in water. For the 1H NMR studies of the intracrystalline water self-diffusion, a procedure first developed for measurements of diffusion anisotropy in beds of MCM-41 particles was adapted [9]: A bed of cross-linked protein crystals immersed in water was introduced in a 7.5 mm NMR sample tube. The samples were cooled inside the NMR spectrometer to –15 C and successively warmed up to temperatures of –10 C and –5 C, respectively. At these temperatures, the diffusion measurement were performed, because – according to the Gibbs-Thomson equation dp = k/D Tm, k = 5.7 · 108 Km for water [14]), which relates the melting point depression DT m observed in small pores to the pore diameter d P – the water inside the nanometer size pores of the protein crystals is in the liquid state, while the water in the large (micrometer size) spaces between the crystals in the NMR tube is still frozen to ice. Pulsed field gradient NMR diffusion measurements were performed on the home-built NMR spectrometer FEGRIS 400NT [7] operating at 400 MHz 1H resonance frequency. The NMR signals of the intercrystalline frozen water and of the 1H nuclei of the protein backbone, which have very short transverse relaxation times, were suppressed using a delay of s = 700 ls between the succeeding 90 and 180 rf pulses in the 13-interval stimulated spin echo PFG NMR sequence [7,15] applied for these studies. The liquid intracrystalline water has a mono-exponential transverse relaxation with a relaxation time of about 2 ms. Thus, the stimulated spin echo NMR signal from this water is reduced by this setting but still detectable with a good signal-to-noise ratio. Spin echo intensities M(b) were recorded as function of the amplitude g (maximum value of up to 15 T/m) and the width d (maximum value of up to 400 ls) of the four alternating pulsed field gradients. The maximum gradient amplitude used was ±10 T/m. For the 13-interval PFG NMR sequence, the parameter b, as introduced in the Theoretical section, is given by b ¼ ðcg2dÞ2 ðD 16 d 12 sÞ. For each measurement temperature, the diffusion time D was varied between about 5 ms and 20 ms. Confocal laser scanning microscopy measurements of fluorescent dye diffusion were performed using the approach and set-up described previously [6]. In such studies, the uptake of sodium fluorescein or rhodamine B from the aqueous solution surrounding a single protein crystal is observed in time-resolved three-dimensional microscopic intensity maps. They are analysed using a 3D diffusion model [13] to yield the components of the diffusion coefficients in the three crystallographic directions (Da, Db, Dc). For comparison with the NMR data, an averaged diffusion perpendicular to the crystallographic c-axis (Dper = 1/2(Da + Db)) and an anisotropy ratio (g = Dab/Dc) were calculated, while assuming Dpar = Dc. For the present crystals, this approach corresponds to the approximation of an axisymmetrical diffusion tensor as presented in the Theoretical Section
R. Archipov et al. / Microporous and Mesoporous Materials 112 (2008) 474–480
and is justified since pores in the c-direction are about twice as wide as those in the a and b directions.
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anisotropy ratio g depends slightly on the observation time, which will be discussed below. Given the good agreement between the experimental PFG NMR data and the anisotropic diffusion model for the spin echo attenuation at different experimental conditions, it is reasonable to conclude that water diffusion inside the protein crystals is anisotropic. The component of highest water diffusivity (Dpar) is found to be about one order of magnitude faster than the diffusion in the plan perpendicular to this direction (Dperp). Since the pore structure of both lysozyme protein crystals consists of pores of about two times larger diameter in the crystallographic c-direction than in both orthogonal (a and b) directions, it is reasonable to interpret Dpar as the value for diffusion along the c-direction and Dperp as the averaged diffusion coefficient in the a,b-plane perpendicular to it. The dependence of the PFG NMR spin echo attenuation on diffusion-time, as shown in Fig. 2, refers to a time-dependent water diffusion inside the protein crystals. This is more pronounced for the bed of tetragonal lysozyme crystals which consists of large crystals but also a considerable amount of smaller pieces (see optical micrographs in Fig. 3). Additionally, the crystal extension in direction of highest water diffusivity, which is for both crystal types the c-direction, is in the order of 50 lm for the tetragonal crystals and in the order of 200 lm for the orthorhombic crystals (see Figs. 3c and d). Therefore, we attribute the observed time-dependence to restricted intracrystalline diffusion on the external crystal boundary of the smaller crystals. Although the PFG NMR data (Fig. 2) show a slight trend of decreasing diffusion coefficients with increasing diffusion time, they do not deviate significantly in their shape. Thus, a strong dependence of the anisotropy ratios on the diffusion time is not to be expected. Nevertheless, since PFG NMR measurements at short diffusion
4. Results and discussion Examples for PFG NMR spin echo attenuations observed for water in beds of tetragonal and orthorhombic lysozyme protein crystals at –5 C and –10 C are shown in Figs. 1 and 2. Fig. 1 represents the spin echo attenuation observed at –5 C and 10 ms diffusion time. It compares different approaches to analyse the experimental data. Clearly, a single-exponential decay (equal to a straight line in the semi-logarithmic presentation and representative for single-component isotropic diffusion according to Eq. (1)) does not fit the experimental data well. A better agreement is achieved by assuming a two component isotropic diffusion. According to Eq. (2), this model consists of already four independent variables (pi, Di; i = 1, 2). However, there is no reasonable explanation for a two-component isotropic intracrystalline diffusion behaviour of water in the protein crystals. A very good agreement with the experimental data provides the model of the axisymmetrical diffusion tensor according to Eq. (4), if one assumes non-zero values for its only two independent variables Dpar and Dperp. Setting one of these values to zero and requiring a good presentation of the initial decay always results in insufficient agreement between the experimental data and the model at large b-values and spin echo attenuations. This is shown by the theoretical curves labelled with 1d (for one-dimensional diffusion, Dperp = 0) and 2d (for two-dimensional diffusion, Dpar = 0) in Fig. 1. Fig. 2 shows spin echo attenuations for both types of protein crystals at –10 C for different diffusion times. Also these measurements are well presented by the anisotropic diffusion model of Eq. (4) if one assumes Dpar > Dperp > 0. However, the obtained 1
M(b)/M 0
a
b
1d
1d
2d
2d
0.1
0
+10
5x10
+11
1x10
+11
2x10
+11
2x10
+10
0
b/sm
5x10
1x10+11 2x10+11 2x10+11
-2
Fig. 1. Examples for the PFG NMR spin echo attenuation plots for water self-diffusion in orthorhombic (a) and tetragonal (b) lysozyme protein crystals at 268 K and 10 ms observation time. The lines represent fits using models of a single-component isotropic diffusion (dashed line), a two-component isotropic diffusion (dotted lines) and the anisotropic diffusion of Eq. (4) with Dpar > Dperp > 0 (thick full lines), Dpar > Dperp = 0 (full lines, label 1d) and Dperp > Dpar = 0 (full lines, label 2d). All fits were forced to represent the initial decay of the experimental data correctly, which represents the averaged self-diffusion coefficient of (3.0 ± 0.1) · 1011 m2 s1 for the tetragonal and (3.3 ± 0.1) · 1011 m2 s1 for the orthorhombic lysozyme protein crystals.
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R. Archipov et al. / Microporous and Mesoporous Materials 112 (2008) 474–480
M(b)/M0
1
a
b
0.1
0
+10
5x10
+11
1x10
+11
2x10
+11
2x10
0
+10
5x10
+11
1x10
+11
2x10
+11
2x10
b /s m-2 Fig. 2. PFG NMR spin echo attenuation plots for water diffusion in tetragonal (a) and orthorhombic (b) lysozyme protein crystals at 263 K for observation times of 5 ms (4), 10 ms (d), 15 ms (s) and 20 ms (h), respectively. The full lines represent the fits using the anisotropic diffusion model of Eq. (3).
Fig. 3. Optical micrographs of the orthorhombic (a, c) and tetragonal (b, d) lysozyme protein crystals. Figures (a) and (b) represent low-resolution micrographs of a large amounts of crystals. Figures (c) and (d) show individual crystals selected under the microscope for the CLSM studies. The black bars in the figures represent the length scales in these micrographs.
times are less influenced by these obstructions, we compare the diffusion coefficients only for a fixed, short diffusion time of 10 ms with each other and with the values of fluorescent dye diffusion in CLSM.
Table 1 summarizes the results for the diffusion studies of fluorescein, rhodamine B and water in tetragonal and orthorhombic lysozyme protein crystals. The largest component of diffusion of water in the micropores (Dpar) is
R. Archipov et al. / Microporous and Mesoporous Materials 112 (2008) 474–480
479
Table 1 Diffusion of water and fluorescent dyes as measured by PFG NMR and CLSM in lysozyme protein crystals of tetragonal and orthorhombic structure Diffusant
Method
T (K)
Dpar 1012 m2 s1
Dperp 1012 m2 s1
g
Tetragonal
Water
PFG NMR
Tetragonal orthorhombica
Fluorescein Water
CLSM PFG NMR
Orthorhombic Orthorhombic
Fluorescein Rhodamine B
CLSM CLSM
268 ± 1 263 ± 1 289 ± 1 268 ± 1 263 ± 1 289 ± 1 289 ± 1
93 ± 5 57 ± 5 0.33 ± 0.02 89 ± 5 56 ± 5 0.70 ± 0.05 0.41 ± 0.03
12 ± 3 10 ± 3 0.052 ± 0.01 8.3 ± 1.5 6±2 0.11 ± 0.02 0.10 ± 0.02
0.13 ± 0.04 0.16 ± 0.05 0.16 ± 0.03 0.09 ± 0.04 0.11 ± 0.04 0.15 ± 0.03 0.25 ± 0.05
Crystal structure a
a
Cross-linked crystals.
more than one order of magnitude smaller than that of bulk liquid (super-cooled) water at the same temperature [16], which demonstrates the restricting effect of the micropores even in the large pores in crystallographic cdirection. For the fluorescein and rhodamine B molecules, diffusion in crystallographic c-direction, is reduced by more than two orders of magnitude compared to the corresponding value in liquid water. These reductions of diffusivities reflect the interaction of the diffusant with the protein and the pore structure. They are larger for the fluorescein and rhodamine B because steric hindrances are more pronounced for larger diffusants [17]. Fluorescein, rhodamine B, and water diffusion coefficients in lysozyme protein crystals depend on the direction with respect to the crystal orientation. Perpendicular to the largest components it is almost one order of magnitude smaller than Dpar. For the tetragonal structure, the anisotropy ratios g for fluorescein and water diffusion are in good agreement with each other. For the needle-like orthorhombic structure the anisotropy is in the same order of magnitude for fluorescein, rhodamine B, and water. However compared to both fluorescent dyes, water shows the smallest values of g. The reason for the observed difference in diffusion anisotropies may be the different intracrystalline pore geometries, pore wall charges and hydrophobicities in the three crystallographic directions of the protein crystals. Additionally, a partial orientation of the needle-like orthorhombic crystals inside the NMR tube might have reduced the observed anisotropy value for the water measurements. When comparing the two crystal types, the PFG NMR data on water diffusion suggest similar diffusivities in the crystallographic c-direction (Dpar) for both types of crystals and somewhat higher diffusivities in the orthogonal abdirections in the tetragonal compared to the orthorhombic structure. This was not expected, because the CSLM data on fluorescein indicate a generally higher diffusivity in the orthorhombic structure, which could be attributed to its larger pore sizes compared to the tetragonal structure [13]. 5. Conclusions PFG NMR and CLSM are both convenient methods to measure anisotropic diffusion in nanoporous materials. With respect to the anisotropy ratio, results from both
methods are in reasonable agreement for the investigated lysozyme protein crystals of orthorhombic and tetragonal structure. Compared to CLSM, the advantage of PFG NMR is that the type of molecules, which can be investigated, is not restricted to soluble fluorescent dyes and that, therefore, also diffusion processes in smaller pore structures and of pure solvents are accessible. However, prior knowledge of pore structure and their effects on diffusion anisotropy are necessary to develop a model for the observed spin echo attenuation, which allows the determination of the elements of the diffusion tensor from the measurements. The advantages of CLSM are its much higher sensitivity compared to NMR methods, allowing smaller concentrations of the diffusants to be investigated, and the direct observation of the uptake of the fluorescent dye in timeresolved 3d maps, which at least for simple crystal morphologies, allows a fast assessment of diffusion anisotropy. Additionally, both methods are different with respect of the total volume of nanoporous materials probed. While CLSM works only with a single crystal, carefully selected under a microscope, PFG NMR measures averaged diffusion behaviour for a bed of randomly oriented crystals. Thus, both methods are complementary and their combined application might be useful for a better understanding of diffusion phenomena in nanoporous materials and, especially, in protein crystals. Acknowledgments We thank the DFG (Germany) and the NWO (The Netherlands) for financial support via the joint International Research Training Group (IRTG) ‘‘Diffusion in Porous Materials’’. References [1] [2] [3] [4]
B.W. Matthews, J. Mol. Biol. 33 (1968) 315. V. Klyushnichenko, Curr. Opin. Drug Disc. Dev. 6 (2003) 848. A.L. Margolin, M.A. Navia, Angew. Chem. Int. Ed. 40 (2001) 2205. S.K. Basu, C.P. Govardhan, C.W. Jung, A.L. Margolin, Expert Opin. Biol. Therapy 4 (2004) 301. [5] J. Ka¨rger, S. Vasenkov, Microporous Mesoporous Mater. 85 (2005) 195. [6] A. Cvetkovic, A.J.J. Straathof, D.N. Hanlon, S. van der Zwaag, R. Krishna, L.A.M. van der Wielen, Biotechnol. Bioeng. 86 (2004) 389.
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[7] F. Stallmach, P. Galvosas, Annu. Rep. NMR Spectrosc. 61 (2007) 52. [8] U. Hong, J. Ka¨rger, H. Pfeifer, U. Mu¨ller, K.K. Unger, Z. Phys. Chem. 173 (1991) 225. [9] F. Stallmach, J. Ka¨rger, C. Krause, M. Jeschke, U. Oberhagemann, J. Am. Chem. Soc. 122 (2000) 9237. [10] E.O. Stejskal, J.E. Tanner, J. Chem. Phys. 42 (1965) 288. [11] P.T. Callaghan, Principles of Nuclear Magnetic Resonance Microscopy, Clarendon Press, Oxford, 1991. [12] J. Ka¨rger, M.D. Ruthven, Diffusion in Zeolites and Other Microporous Solids, Wiley & Sons, New York, 1992.
[13] A. Cvetkovic, C. Picioreanu, A.J.J. Straathof, R. Krishna, L.A.M. van der Wielen, J. Am. Chem. Soc. 127 (2005) 875. [14] J.B.W. Webber, J.H. Strange, J.C. Dore, Magn. Reson. Imaging 19 (2001) 395. [15] R.M. Cotts, M.J.R. Hoch, T. Sun, J.T. Markert, J. Magn. Reson. 83 (1989) 252. [16] M. Holz, S.R. Heil, A. Sacco, Phys. Chem. Chem. Phys. 2 (2000) 4740. [17] A. Cvetkovic, C. Picioreanu, A.J.J. Straathof, R. Krishna, L.A.M. van der Wielen, J.Phys. Chem. B. 109 (2005) 10561.
Microporous and Mesoporous Materials 112 (2008) 474–480 www.elsevier.com/locate/micromeso
Measurement of anisotropy of pore diffusion in protein crystals by PFG NMR and by CLSM Ruslan Archipov b
a,1
, Aleksandar Cvetkovic b, Frank Stallmach
a,*
, Adrie J.J. Straathof
b
a Universita¨t Leipzig, Fakulta¨t fu¨r Physik und Geowissenschaften, Linne´str. 5, 04103 Leipzig, Germany Department of Biotechnology, Delft University of Technology, Julianalaan 67, 2628 BC Delft, The Netherlands
Received 30 August 2007; received in revised form 6 October 2007; accepted 12 October 2007 Available online 24 October 2007
Abstract Solvent (water) and solute (fluorescein and rhodamine B) diffusion in crystalline lysozyme protein, which represent a mesoporous material with an anisotropic pore structure, was studied by pulsed field gradient nuclear magnetic resonance (PFG NMR) and confocal laser scanning microscopy (CLSM), respectively. Water and fluorescent dyes reveal anisotropic intracrystalline diffusion behaviour. The averaged intracrystalline self-diffusion coefficient of water was found to be about one order of magnitude smaller than the value for bulk liquid water at the same temperature. However, the intracrystalline self-diffusion coefficients of water are about two orders of magnitude faster than the corresponding values of the fluorescent dyes. For water and fluorescein in tetragonal lysozyme crystals, the anisotropy ratio, which is defined as the ratio of the largest component of the diffusion tensor to its averaged orthogonal value, was about 0.16, whereas for orthorhombic crystals the anisotropy ratios measured by CLSM for fluorescein (0.15 ± 0.03) and rhodamine B (0.25 ± 0.05) were in the same order of magnitude but somewhat larger compared to the value for water (0.11 ± 0.04) measured by PFG NMR. 2007 Elsevier Inc. All rights reserved. Keywords: Diffusion anisotropy; Protein crystals; Lysozyme; PFG NMR; CLSM
1. Introduction Protein crystals are structures with a porosity ranging from 20% to 80% and pore diameters in the range of 0.5– 10 nm [1]. Protein crystals are usually obtained from an aqueous buffer. Although crystallization may be cumbersome, the crystal structure of about 30,000 proteins has been solved. Moreover, crystallization has become the preferred recovery procedure for some proteins that are produced on a large scale [2]. Protein crystals can be stabilized against mechanical, thermal, pH and organic solvent stresses upon cross-linking with bifunctional reagents. Then they can be used as mesoporous materials for a variety of purposes, such as biocatalysis and selective adsorption *
1
Corresponding author. Tel.: +49 341 973 2518; fax: +49 341 973 2549. E-mail address: [email protected] (F. Stallmach). Present address: Kazan State University, Russian Federation.
1387-1811/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.micromeso.2007.10.024
[3]. These materials are particularly interesting because of their very wide structural diversity, the inherent anisotropy and chirality of the pores, and, in case of enzyme crystals, the presence of a selective catalytic site. Besides, they are useful for delivery of biopharmaceuticals [4]. However, diffusion through the internal pore structure of the crystals is required for molecules of cross-linkers, enzyme substrates, adsorbates, and biopharmaceuticals. Measurement of intracrystalline diffusivities is required to develop the aforementioned applications. Preferably, techniques should be used, which measure displacements or concentration profiles inside the porous hosts, thus avoiding complications arising from macroscopic techniques, which observe the total uptake of guest molecules [5]. Recently, two such techniques have been developed for measuring anisotropic diffusion in mesoporous materials. Anisotropic solute diffusion has been studied by monitoring the uptake of solutes by lysozyme protein crystals using
R. Archipov et al. / Microporous and Mesoporous Materials 112 (2008) 474–480
confocal laser scanning microscopy (CLSM) [6]. This technique yields time-resolved three-dimensional micrographs of intracrystalline solute concentration. However, its application is restricted to fluorescent dyes as solutes, which are relatively large (>0.7 nm), and only to individual protein crystals larger than 10 lm, which are carefully selected under the microscope. In contrast, pulsed field gradient nuclear magnetic resonance (PFG NMR) yields the self-diffusion for molecules adsorbed in a bed of the porous material of roughly 106 m3 volume [7]. It is suitable to observe diffusion of the smaller water (solvent) molecules by using 1 H resonance. 1H PFG NMR studies of diffusion anisotropy have been performed in microporous MFI zeolites [8] and mesoporous MCM-41 [9]. However, since these measurements are performed in a bed of the material with random orientation of the crystals (and, thus, the pores) with respect to the direction of the diffusion measurement, which is defined by the direction of the externally applied magnetic field gradient, analysis of the experimentally observed PFG NMR spin echo attenuation curves requires model assumptions to deduce the elements of the diffusion tensor [7–9]. Thus, CLSM and PFG NMR may be considered as complimentary methods in studying solute and solvent diffusion in anisotropic micro- and mesoporous materials. The goal of the present work is to compare experimental data on water and fluorescent dye diffusion in lysozyme protein crystals, as obtained by PFG NMR and CLSM, respectively. We will show that solvent and solute diffusion in these protein crystals is anisotropic with similar values for the anisotropy ratio. 2. Theoretical section: anisotropic diffusion in PFG NMR In PFG NMR diffusion studies, the projection of the mean square displacement of the molecules on the direction of the externally applied decoding and encoding pulsed magnetic field gradients ! g is measured [7,10,11]. The time scale of this measurement is defined by the distance D between the pulsed field gradients in the spin echo NMR pulse sequence and referred to as diffusion time. Generally, for a fixed diffusion time the intensity of the spin echo NMR signal M is recorded as function the amplitude g and the width d of these pulsed field gradients. The three quantities g, d and D, which are summarized for convenience to the gradient parameter b(g, d, D), control the attenuation of the NMR signal due to self-diffusion. For example, for PFG NMR diffusion studies with two pulsed field gradients, b is simply given by b ¼ ðcgdÞ2 ðD 13 dÞ, where c denotes the gyromagnetic ratio of the nuclei under investigation. In order to evaluate the diffusion process and calculate the self-diffusion coefficients from the observed spin echo intensities M(b), the spin echo attenuation M(b)/M0 (with M0„M(b = 0)) is plotted as function of b on a semi-logarithmic scale. In this presentation, a single-component isotropic self-diffusion shows up as a linear decay. From its
475
slope, the self-diffusion coefficients D is obtained using the equation [7,10,11] MðbÞ=M 0 ¼ expðD bÞ:
ð1Þ
Any deviation from this single-exponential spin echo attenuation indicates the presence of a distribution of self-diffusion coefficients. In the case of a multi-component isotropic diffusion, one expects a superposition of exponential decays according to [7,11] X MðbÞ=M 0 ¼ pi expðDi bÞ; ð2Þ i
with pi and Di denoting the relative contribution of component i to the spin echo signal and its self-diffusion coefficient, respectively. In microporous crystalline materials, anisotropic intracrystalline diffusion may be caused by the anisotropy of the micropore structure and the diffusion process must be described by a second rank diffusion tensor D [8,9,12]. The orientation of this tensor with respect to the direction of the externally applied pulsed field gradient ! g modifies the spin echo attenuation. For example, for one-dimensional diffusion in straight channels with polar angle H between gradient and channel direction, a diffusion of Dpar along the channel direction leads to an orientation-dependent exponent of b Dpar cos2 H in the spin echo attenuation of Eq. (1). If one has a distribution of different channel orientations, as it is usually the case in beds of crystalline materials, Dpar cos2 Hi replaces Di in Eq. (2) and pi denotes the probability of finding the orientation Hi in the sample. Thus, in PFG NMR anisotropic diffusion in beds of microporous materials may be considered as special case of multi-component diffusion, where the different components are just caused by the different orientations with respect to the pulsed field gradient direction. If one has a random packing of the microporous materials in the bed, all orientation of the diffusion tensor associated with the anisotropic micropore structure are equally probable and the spin echo attenuation is obtained by the powder average over all orientations. The CLSM studies with tetragonal and orthorhombic lysozyme protein crystals showed that fluorescent dye diffusion is anisotropic with three different diagonal elements (Dii > 0) of the diffusion tensor D, each describing the diffusion along one of the three orthogonal crystallographic directions. In such a situation, the complete powder average over the polar angles H and u yields Z 2p Z p 1 MðbÞ=M 0 ¼ expfbðD11 cos2 H 4p 0 0 þ D22 sin2 H cos2 u þ D33 sin2 H sin2 uÞg sin HdHdu
ð3Þ
for the PFG NMR spin echo attenuation [11]. Qualitatively, Eq. (3) describes a non-exponentially, monotonously decaying function. Its slope decreases with increasing b-valP ues and yields at b = 0 the trace of the tensor 13 i Dii . Only
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the total shape of this function including high b-values reveals detailed information on all three elements of the diffusion tensor. It is in the nature of PFG NMR measurements that high b-values correspond to strong spin echo NMR signal attenuations. Thus, a poor signal-tonoise ratios is inherent for parts of the experimental data necessary for the detailed analysis using Eq. (3). Additionally, the integrals in Eq. (3) cannot be solved further and analysis of experimental spin echo attenuations in anisotropic systems requires numerical solutions and approximations, respectively. A first step to approach this complex situation represents the reduction of free parameters in Eq. (3) by using properties of the microporous pore space of the protein crystals. In crystallographic c-direction, the pore diameter of the tetragonal (dp = 1.38 nm) and the orthorhombic (dpc = 1.97 nm) lysozyme crystals is about twice as wide as in the two orthogonal a- and b-directions, where one has 0.77 nm and 0.84 nm, respectively, for the tetragonal and 0.74 nm and 1.01 nm, respectively, for the orthorhombic structure. These pore size data were obtained by modelling of the protein crystal structure in our previous work [13]. Compared to this major difference, the deviation in diameter between the a- and b-directions is small. If – as for the fluorescent dyes – the pore size represents also for the water molecules the major reason for diffusion anisotropy, one may assume that the difference between the diffusion coefficients in the crystallographic a and b directions is negligible compared to the much faster diffusion in c-direction. Thus, the diffusion in the ab-plane may be approximated by a single value Dperp (= D22 = D33 in Eq. (3)), which, however, is expected to be smaller than the diffusion along the c-direction Dpar (= D11), where the micropore structure exhibits the substantially larger pores. The results of this approximation correspond to the model of an axisymmetrical diffusion tensor, for which the spin echo attenuation of Eq. (3) simplifies to (see also Refs. [9–11]) Z 1 p MðbÞ=M 0 ¼ expfbðDpar cos2 H þ Dperp sin2 HÞg 2 0 sin HdH: ð4Þ The indices par and perp were used in order to distinguish between the diffusion along (parallel to) the principal axis of the tensor (Dpar) and perpendicular to it (Dperp). From Eq. (4), the special cases for one-dimensional diffusion along randomly oriented channels and two-dimensional diffusion in randomly oriented sheets are easily derived by setting Dperp and Dpar equal to zero, respectively. 3. Experimental section Lysozyme protein crystals of tetragonal (6LYT) and orthorhombic (1AKI) structures were synthesized from chicken egg-white lysozyme following a procedure as described previously [6]. Cross-linked protein crystal were
obtained by using glutaraldehyde as cross-linking agent. After synthesis, the protein crystals were filtered from the synthesis solution and rinsed in water. For the 1H NMR studies of the intracrystalline water self-diffusion, a procedure first developed for measurements of diffusion anisotropy in beds of MCM-41 particles was adapted [9]: A bed of cross-linked protein crystals immersed in water was introduced in a 7.5 mm NMR sample tube. The samples were cooled inside the NMR spectrometer to –15 C and successively warmed up to temperatures of –10 C and –5 C, respectively. At these temperatures, the diffusion measurement were performed, because – according to the Gibbs-Thomson equation dp = k/D Tm, k = 5.7 · 108 Km for water [14]), which relates the melting point depression DT m observed in small pores to the pore diameter d P – the water inside the nanometer size pores of the protein crystals is in the liquid state, while the water in the large (micrometer size) spaces between the crystals in the NMR tube is still frozen to ice. Pulsed field gradient NMR diffusion measurements were performed on the home-built NMR spectrometer FEGRIS 400NT [7] operating at 400 MHz 1H resonance frequency. The NMR signals of the intercrystalline frozen water and of the 1H nuclei of the protein backbone, which have very short transverse relaxation times, were suppressed using a delay of s = 700 ls between the succeeding 90 and 180 rf pulses in the 13-interval stimulated spin echo PFG NMR sequence [7,15] applied for these studies. The liquid intracrystalline water has a mono-exponential transverse relaxation with a relaxation time of about 2 ms. Thus, the stimulated spin echo NMR signal from this water is reduced by this setting but still detectable with a good signal-to-noise ratio. Spin echo intensities M(b) were recorded as function of the amplitude g (maximum value of up to 15 T/m) and the width d (maximum value of up to 400 ls) of the four alternating pulsed field gradients. The maximum gradient amplitude used was ±10 T/m. For the 13-interval PFG NMR sequence, the parameter b, as introduced in the Theoretical section, is given by b ¼ ðcg2dÞ2 ðD 16 d 12 sÞ. For each measurement temperature, the diffusion time D was varied between about 5 ms and 20 ms. Confocal laser scanning microscopy measurements of fluorescent dye diffusion were performed using the approach and set-up described previously [6]. In such studies, the uptake of sodium fluorescein or rhodamine B from the aqueous solution surrounding a single protein crystal is observed in time-resolved three-dimensional microscopic intensity maps. They are analysed using a 3D diffusion model [13] to yield the components of the diffusion coefficients in the three crystallographic directions (Da, Db, Dc). For comparison with the NMR data, an averaged diffusion perpendicular to the crystallographic c-axis (Dper = 1/2(Da + Db)) and an anisotropy ratio (g = Dab/Dc) were calculated, while assuming Dpar = Dc. For the present crystals, this approach corresponds to the approximation of an axisymmetrical diffusion tensor as presented in the Theoretical Section
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and is justified since pores in the c-direction are about twice as wide as those in the a and b directions.
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anisotropy ratio g depends slightly on the observation time, which will be discussed below. Given the good agreement between the experimental PFG NMR data and the anisotropic diffusion model for the spin echo attenuation at different experimental conditions, it is reasonable to conclude that water diffusion inside the protein crystals is anisotropic. The component of highest water diffusivity (Dpar) is found to be about one order of magnitude faster than the diffusion in the plan perpendicular to this direction (Dperp). Since the pore structure of both lysozyme protein crystals consists of pores of about two times larger diameter in the crystallographic c-direction than in both orthogonal (a and b) directions, it is reasonable to interpret Dpar as the value for diffusion along the c-direction and Dperp as the averaged diffusion coefficient in the a,b-plane perpendicular to it. The dependence of the PFG NMR spin echo attenuation on diffusion-time, as shown in Fig. 2, refers to a time-dependent water diffusion inside the protein crystals. This is more pronounced for the bed of tetragonal lysozyme crystals which consists of large crystals but also a considerable amount of smaller pieces (see optical micrographs in Fig. 3). Additionally, the crystal extension in direction of highest water diffusivity, which is for both crystal types the c-direction, is in the order of 50 lm for the tetragonal crystals and in the order of 200 lm for the orthorhombic crystals (see Figs. 3c and d). Therefore, we attribute the observed time-dependence to restricted intracrystalline diffusion on the external crystal boundary of the smaller crystals. Although the PFG NMR data (Fig. 2) show a slight trend of decreasing diffusion coefficients with increasing diffusion time, they do not deviate significantly in their shape. Thus, a strong dependence of the anisotropy ratios on the diffusion time is not to be expected. Nevertheless, since PFG NMR measurements at short diffusion
4. Results and discussion Examples for PFG NMR spin echo attenuations observed for water in beds of tetragonal and orthorhombic lysozyme protein crystals at –5 C and –10 C are shown in Figs. 1 and 2. Fig. 1 represents the spin echo attenuation observed at –5 C and 10 ms diffusion time. It compares different approaches to analyse the experimental data. Clearly, a single-exponential decay (equal to a straight line in the semi-logarithmic presentation and representative for single-component isotropic diffusion according to Eq. (1)) does not fit the experimental data well. A better agreement is achieved by assuming a two component isotropic diffusion. According to Eq. (2), this model consists of already four independent variables (pi, Di; i = 1, 2). However, there is no reasonable explanation for a two-component isotropic intracrystalline diffusion behaviour of water in the protein crystals. A very good agreement with the experimental data provides the model of the axisymmetrical diffusion tensor according to Eq. (4), if one assumes non-zero values for its only two independent variables Dpar and Dperp. Setting one of these values to zero and requiring a good presentation of the initial decay always results in insufficient agreement between the experimental data and the model at large b-values and spin echo attenuations. This is shown by the theoretical curves labelled with 1d (for one-dimensional diffusion, Dperp = 0) and 2d (for two-dimensional diffusion, Dpar = 0) in Fig. 1. Fig. 2 shows spin echo attenuations for both types of protein crystals at –10 C for different diffusion times. Also these measurements are well presented by the anisotropic diffusion model of Eq. (4) if one assumes Dpar > Dperp > 0. However, the obtained 1
M(b)/M 0
a
b
1d
1d
2d
2d
0.1
0
+10
5x10
+11
1x10
+11
2x10
+11
2x10
+10
0
b/sm
5x10
1x10+11 2x10+11 2x10+11
-2
Fig. 1. Examples for the PFG NMR spin echo attenuation plots for water self-diffusion in orthorhombic (a) and tetragonal (b) lysozyme protein crystals at 268 K and 10 ms observation time. The lines represent fits using models of a single-component isotropic diffusion (dashed line), a two-component isotropic diffusion (dotted lines) and the anisotropic diffusion of Eq. (4) with Dpar > Dperp > 0 (thick full lines), Dpar > Dperp = 0 (full lines, label 1d) and Dperp > Dpar = 0 (full lines, label 2d). All fits were forced to represent the initial decay of the experimental data correctly, which represents the averaged self-diffusion coefficient of (3.0 ± 0.1) · 1011 m2 s1 for the tetragonal and (3.3 ± 0.1) · 1011 m2 s1 for the orthorhombic lysozyme protein crystals.
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M(b)/M0
1
a
b
0.1
0
+10
5x10
+11
1x10
+11
2x10
+11
2x10
0
+10
5x10
+11
1x10
+11
2x10
+11
2x10
b /s m-2 Fig. 2. PFG NMR spin echo attenuation plots for water diffusion in tetragonal (a) and orthorhombic (b) lysozyme protein crystals at 263 K for observation times of 5 ms (4), 10 ms (d), 15 ms (s) and 20 ms (h), respectively. The full lines represent the fits using the anisotropic diffusion model of Eq. (3).
Fig. 3. Optical micrographs of the orthorhombic (a, c) and tetragonal (b, d) lysozyme protein crystals. Figures (a) and (b) represent low-resolution micrographs of a large amounts of crystals. Figures (c) and (d) show individual crystals selected under the microscope for the CLSM studies. The black bars in the figures represent the length scales in these micrographs.
times are less influenced by these obstructions, we compare the diffusion coefficients only for a fixed, short diffusion time of 10 ms with each other and with the values of fluorescent dye diffusion in CLSM.
Table 1 summarizes the results for the diffusion studies of fluorescein, rhodamine B and water in tetragonal and orthorhombic lysozyme protein crystals. The largest component of diffusion of water in the micropores (Dpar) is
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Table 1 Diffusion of water and fluorescent dyes as measured by PFG NMR and CLSM in lysozyme protein crystals of tetragonal and orthorhombic structure Diffusant
Method
T (K)
Dpar 1012 m2 s1
Dperp 1012 m2 s1
g
Tetragonal
Water
PFG NMR
Tetragonal orthorhombica
Fluorescein Water
CLSM PFG NMR
Orthorhombic Orthorhombic
Fluorescein Rhodamine B
CLSM CLSM
268 ± 1 263 ± 1 289 ± 1 268 ± 1 263 ± 1 289 ± 1 289 ± 1
93 ± 5 57 ± 5 0.33 ± 0.02 89 ± 5 56 ± 5 0.70 ± 0.05 0.41 ± 0.03
12 ± 3 10 ± 3 0.052 ± 0.01 8.3 ± 1.5 6±2 0.11 ± 0.02 0.10 ± 0.02
0.13 ± 0.04 0.16 ± 0.05 0.16 ± 0.03 0.09 ± 0.04 0.11 ± 0.04 0.15 ± 0.03 0.25 ± 0.05
Crystal structure a
a
Cross-linked crystals.
more than one order of magnitude smaller than that of bulk liquid (super-cooled) water at the same temperature [16], which demonstrates the restricting effect of the micropores even in the large pores in crystallographic cdirection. For the fluorescein and rhodamine B molecules, diffusion in crystallographic c-direction, is reduced by more than two orders of magnitude compared to the corresponding value in liquid water. These reductions of diffusivities reflect the interaction of the diffusant with the protein and the pore structure. They are larger for the fluorescein and rhodamine B because steric hindrances are more pronounced for larger diffusants [17]. Fluorescein, rhodamine B, and water diffusion coefficients in lysozyme protein crystals depend on the direction with respect to the crystal orientation. Perpendicular to the largest components it is almost one order of magnitude smaller than Dpar. For the tetragonal structure, the anisotropy ratios g for fluorescein and water diffusion are in good agreement with each other. For the needle-like orthorhombic structure the anisotropy is in the same order of magnitude for fluorescein, rhodamine B, and water. However compared to both fluorescent dyes, water shows the smallest values of g. The reason for the observed difference in diffusion anisotropies may be the different intracrystalline pore geometries, pore wall charges and hydrophobicities in the three crystallographic directions of the protein crystals. Additionally, a partial orientation of the needle-like orthorhombic crystals inside the NMR tube might have reduced the observed anisotropy value for the water measurements. When comparing the two crystal types, the PFG NMR data on water diffusion suggest similar diffusivities in the crystallographic c-direction (Dpar) for both types of crystals and somewhat higher diffusivities in the orthogonal abdirections in the tetragonal compared to the orthorhombic structure. This was not expected, because the CSLM data on fluorescein indicate a generally higher diffusivity in the orthorhombic structure, which could be attributed to its larger pore sizes compared to the tetragonal structure [13]. 5. Conclusions PFG NMR and CLSM are both convenient methods to measure anisotropic diffusion in nanoporous materials. With respect to the anisotropy ratio, results from both
methods are in reasonable agreement for the investigated lysozyme protein crystals of orthorhombic and tetragonal structure. Compared to CLSM, the advantage of PFG NMR is that the type of molecules, which can be investigated, is not restricted to soluble fluorescent dyes and that, therefore, also diffusion processes in smaller pore structures and of pure solvents are accessible. However, prior knowledge of pore structure and their effects on diffusion anisotropy are necessary to develop a model for the observed spin echo attenuation, which allows the determination of the elements of the diffusion tensor from the measurements. The advantages of CLSM are its much higher sensitivity compared to NMR methods, allowing smaller concentrations of the diffusants to be investigated, and the direct observation of the uptake of the fluorescent dye in timeresolved 3d maps, which at least for simple crystal morphologies, allows a fast assessment of diffusion anisotropy. Additionally, both methods are different with respect of the total volume of nanoporous materials probed. While CLSM works only with a single crystal, carefully selected under a microscope, PFG NMR measures averaged diffusion behaviour for a bed of randomly oriented crystals. Thus, both methods are complementary and their combined application might be useful for a better understanding of diffusion phenomena in nanoporous materials and, especially, in protein crystals. Acknowledgments We thank the DFG (Germany) and the NWO (The Netherlands) for financial support via the joint International Research Training Group (IRTG) ‘‘Diffusion in Porous Materials’’. References [1] [2] [3] [4]
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