Diffusion of PEG Confined between Lamellae of Negatively

Dec 21, 2007 - UniVersity of Toronto at Mississauga, 3359 Mississauga Road, Mississauga, Ontario, Canada L5L 1C6. ReceiVed July 23, 2007. In Final For...
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Langmuir 2008, 24, 518-527

Diffusion of PEG Confined between Lamellae of Negatively Magnetically Aligned Bicelles: Pulsed Field Gradient 1H NMR Measurements Ronald Soong and Peter M. Macdonald* Department of Chemistry, UniVersity of Toronto, and Department of Chemical and Physical Sciences, UniVersity of Toronto at Mississauga, 3359 Mississauga Road, Mississauga, Ontario, Canada L5L 1C6 ReceiVed July 23, 2007. In Final Form: October 16, 2007 The diffusion of various molecular weight poly(ethyleneglycol)s (PEG) confined between the lamellae of magnetically aligned bicelles has been measured using stimulated echo (STE) pulsed field gradient (PFG) 1H nuclear magnetic resonance (NMR) spectroscopy. Bicelles were formulated to contain dimyristoylphosphatidylcholine (DMPC), dimyristoylphosphatidylglycerol (DMPG), and dihexanoylphosphatidylcholine (DHPC) in the proportion DMPG/ DMPC ) 0.05 and q ) (DMPC + DMPG)/DHPC ) 4.5. PEG diffusion within the interlamellar spaces between such bicelles was found to be unrestricted over diffusion distances of tens of microns. Two confinement regimes could be differentiated according to the dependence of the reduced PEG diffusivity D/D0, where D0 is the unconfined PEG diffusion coefficient, on the relative confinement Rh/H, where Rh is the unperturbed hydration radius of the particular PEG and H ≈ 60 Å is the separation between apposing lamellae of the magnetically aligned bicelles. In the regime Rh/H < 0.4, the reduced PEG diffusivity was altered only in proportion to the viscosity increase associated with the bicelle dispersion relative to bulk solution. In the regime Rh/H > 0.4, the reduced PEG diffusivity scaled as (Rh/H)-2/3, in agreement with scaling theories for confined polymers.

Introduction Diffusion of polymers in confined geometries has become a subject of increasing relevance with the advent of micro- and nanofluidic devices.1,2 Early on, de Gennes used scaling concepts to predict the properties of polymers in the “strong” confinement regime, which occurs when the confinement dimensions are small relative to the polymer’s unconfined radius of gyration.3-5 The values of the scaling exponents predicted by these scaling concepts were confirmed in a number of simulations of self-avoiding random walk (SAW) polymers in various confined geometries6-9 and using mean field theory combined with the Kirkwood approximation.10 Despite the extensive theoretical and computational studies of confined polymers, there have been relatively few experimental investigations of polymers under well-defined confinement situations (see reviews by Deen11 and Teraoka12). Recently, the characteristics of DNA confined within channels, or between plates, having confinement dimensions characteristic of micro- and nanofluidic devices, have been scrutinized both computationally and experimentally and are found to exhibit properties in accord with scaling concepts under conditions of strong confinement.13,14 * Author to whom correspondence should be addressed. Tel: 905 828 3805. Fax: 905 828 5425. E-mail: [email protected]. (1) Squires, T. M.; Quakes, S. R. ReV. Mod. Phys. 2005, 77, 977-1026. (2) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. ReV. Fluid Mech. 2004, 36, 384-411. (3) Brochard, F.; de Gennes, P. G. J. Chem. Phys. 1977, 67, 52-56. (4) Daoud, M.; de Gennes, P. G. J. Phys. (Paris) 1977, 38, 85-93. (5) de Gennes, P.-G. In Scaling Concepts in Polymer Physics; Cornell University Press: Ithica, NY, 1979. (6) Wall, F. T.; Seitz, W. A.; Chin, J. C.; de Gennes, P. G. Proc. Natl. Acad. Sci. U.S.A. 1975, 75, 2069-2070. (7) Lax, M.; Barr, R.; Brender, C. J. Chem. Phys. 1981, 75, 460-462. (8) Whittington, S. C. J. Statist. Phys. 1983, 30, 449-456. (9) Ishinabe, T. J. Chem. Phys. 1985, 83, 423-427. (10) Harden, J. L.; Doi, M. J. Phys. Chem. 1992, 96, 4046-4052. (11) Deen, W. M. AIChE J. 1987, 33, 1409-1424. (12) Teraoka, I. Progr. Polym. Sci. 1996, 21, 89-149. (13) Jendrejack, R. M.; Schwartz, D. C.; Graham, M. D.; de Pablo, J. J. J. Chem. Phys. 2003, 119, 1165-1173.

In the field of micro- and nanofluidics, there is increasing interest in creating biomimetic surfaces,15 and coating the channels of such devices with a lipid bilayer membrane has been shown to be an effective means of doing so. Furthermore, in the pursuit of nanoscale fluidic devices,16 it has been demonstrated that lipid bilayers in the form of liposomes may be manipulated to fabricate nanoscale fluidic networks.17,18 Relative to the “hard” confinement walls produced when such devices are fabricated in the normal fashion from cross-linked polydimethylsiloxane supported on glass, the confinement produced by lipid bilayer membranes must be regarded as “soft”, since lipid bilayer membranes in aqueous media are highly elastic and fluctuate about an “average” position.18 DNA, on the other hand, is a relatively stiff polymer, characterized by a persistence length on the order of 1000 Å.19,20 Thus, it is of particular interest to examine experimentally whether scaling concepts apply universally to the behavior of both highly flexible and relatively stiff polymers under confinement between both soft and hard confinement walls. As a first step toward this goal, we describe here nuclear magnetic resonance (NMR) diffusion studies of a highly flexible polymer under soft confinement between biomimetic lipid bilayer surfaces separated on a nanometer length scale. The confining surfaces we employ here are formed by bicelles, or bilayered micelles, which are a relatively new model membrane system composed of mixtures of long-chain amphiphiles, such as dimyristoylphosphatidylcholine (DMPC), and short-chain (14) Chen, Y.-L.; Graham, M. D.; de Pablo, J. J.; Randall, G. C.; Gupta, M.; Doyle, P. S. Phys. ReV. E 2004, 70, 0609011-0609014. (15) Phillips, S.; Cheng, Q. Anal. Chem. 2005, 77, 327-334. (16) Eijkel, J. C. T.; van den Berg, A. Microfluid Nanofluid 2005, 1, 249-267. (17) Karlsson, M.; Davidson, M.; Karlsson, R.; Karlsson, A.; Bergenholtz, J.; Konkoli, Z.; Jesorka, A.; Lobovkina, T.; Hurtig, J.; Voinova, M.; Orwar, O. Annu. ReV. Phys. Chem. 2004, 55, 613-649. (18) Nagle, J.F.; Tristram-Nagle, S. In Lipid Bilayers: Structure and Interactions; Katsaras, J., Gutberlet, T., Eds.; Springer: New York, 2001; pp 1-23. (19) Hagerman, P. J. Annu. ReV. Biophys. Biophys. Chem. 1988, 17, 265-286. (20) Crothers, D. M.; Drak, J.; Kahn, J. D.; Levene, S. D. Methods Enzymol. 1992, 212, 3-29.

10.1021/la7022264 CCC: $40.75 © 2008 American Chemical Society Published on Web 12/21/2007

Diffusion of PEG Confined between Lamellae

Figure 1. A schematic showing PEG as a random coil polymer confined between two negatively magnetically aligned bicelles, such that their bilayer normals are oriented perpendicular to the external magnetic field, B0, with an interlamellar spacing H. The PEG diffusion tensor is anisotropic with two independent components D⊥ and D| corresponding, respectively, to diffusion perpendicular and parallel to the bilayer normal.

amphiphiles, such as dihexanoylphosphatidylcholine (DHPC).21,22 The long-chain amphiphiles self-assemble into planar bilayer sheets, while the short-chain amphiphiles tend to segregate toward regions of high curvature. At ratios of q ) DMPC/DHPC g 2 the resulting self-assemblies consist of DMPC-rich broad ribbons or lamellar sheets perforated by toroidal holes lined with DHPC.23-25 Bicelles have the particular property that they spontaneously align in the magnetic field of a NMR spectrometer over a wide range of values of q, temperature, and water content. The alignment is driven by the interaction between the magnetic field and the magnetic susceptibility anisotropy of the bicellar self-assembly, aided by cooperative interactions between adjacent lamellae.26 For DMPC/DHPC bicelles, having a net negative magnetic susceptibility anisotropy, the spontaneous direction of alignment is such that the normal to the plane of the lipid bilayer lies perpendicular to the direction of the magnetic field. The resulting negatively magnetically aligned bicelles consist, therefore, of uniformly aligned parallel planar sheets, or ribbons of lipid bilayers, each separated by an aqueous layer. The thickness of a lipid bilayer sheet is on the order of 40 Å, while that of an aqueous layer is on the order of 60 Å for a 25 wt % lipid dispersion, the latter value increasing with increasing water content.24 Thus, negatively magnetically aligned bicelles can be regarded as a set of spontaneously self-assembled, biomimetic, parallel plates/ channels, with nanoscale separations and having transverse axes uniformly oriented perpendicular to the direction of the magnetic field, as illustrated in Figure 1. Poly(ethyleneglycol) (PEG) was chosen as the test polymer because it is highly water soluble and highly flexible (persistence (21) Sanders, C. R.; Hare, B. J.; Howard, K. P.; Prestegard, J. H. Prog. Nucl. Magn. Reson. Spectrosc. 1994, 26, 421-444. (22) Sanders, C. R.; Prosser, R. S. Structure 1998, 16, 1227-1234. (23) Nieh, M.-P.; Ginka, C. J.; Krueger, S.; Prosser, R. S.; Katasaras, J. Langmuir 2001, 17, 2629-2638. (24) Nieh, M.-P.; Ginka, C.J.; Krueger, S; Prosser, R. S.; Katasaras, J. Biophys. J. 2002, 82, 2487-2498. (25) Rowe, B. A.; Neal, S. L. Langmuir 2003, 19, 2039-2048. (26) Boroske, E.; Helfrich, W. Biophys. J. 1978, 24, 863-868.

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length on the order of 3.7 Å), has low toxicity, and is easy to handle. It is perhaps one of the most intensely studied and best understood water-soluble polymers.27 In water, PEG takes on an expanded conformation having properties conforming with theoretical predictions for a random coil polymer in a good solvent.28 In biology, biochemistry, and medicine, PEG is used widely as a precipitant, to stabilize or aggregate particles, and even as a drug delivery vehicle. One of PEG’s important properties is its ability to induce lipid vesicle aggregation and fusion by virtue of its depletion-attraction to lipid bilayer surfaces.29-32 In the studies described here, we have been careful to employ PEG concentrations far below those at which aggregation or fusion effects come into play. Another attractive feature of PEG for our purposes is that, due its simple chemical structure and high internal flexibility, its 1H NMR spectrum consists essentially of a single narrow resonance, virtually regardless of molecular weight. To measure PEG diffusion in the lamellar spaces between magnetically aligned bicelles, we employ here the pulsed field gradient (PFG) NMR diffusion method, a long established technique, widely employed in a host of applications, as described fully in a number of learned reviews.33-36 The chief advantages for our purposes are that PFG NMR diffusion measurements are nonperturbing, require no specific labeling of the molecule of interest, and permit simultaneous diffusion measurements on multiple molecular species provided their NMR resonances are resolvable. For the specific case of negatively magnetically aligned bicelles, by applying the field gradient pulses along the z-direction, parallel to the direction of the main magnetic field, diffusion along the slit-channel between the parallel plates/ribbons formed by the aligned bicellar lamellae may be measured directly. In the following we demonstrate, first, that the magnetic alignment of bicelles is readily achieved in the presence of even large molecular weight PEG within the aqueous interstices between adjacent bicellar lamellae and, second, that PEG diffusion along the channel direction is virtually unrestricted over micron scale diffusion distances. We then proceed to examine the molecular weight dependence of the PEG diffusion coefficients and the degree to which scaling predictions apply to the observed diffusivity of a highly flexible polymer such as PEG in the case of the nanoscale “soft” confinement geometry produced by magnetically aligned bicelles. Experimental Section Materials. DMPC (1,2-dimyristoyl-sn-glycero-3-phosphocholine), DHPC (1,2-dihexanoyl-sn-glycero-3-phosphocholine), and DMPG (1,2-dimyristoyl-sn-glycero-3-phosphoglycerol) were purchased from Avanti Polar Lipids, Alabaster, AL. Poly(ethylene glycol)s (PEG) of molecular weight 200, 600, 1000, 2000, 3400, 4600, 6000, 8000, 10 000, 12 000, and 20 000 were purchased from Sigma-Aldrich, Oakville, ON, Canada, as were all other biochemicals and reagents employed. (27) Branca, C.; Faraone, A.; Magazu`, S.; Maisano, G.; Migliardo, P.; Villari, V. J. Mol. Liq. 2000, 87, 21-68. (28) Devanand, K.; Selser, J. C. Macromolecules 1991, 24, 5943-5947. (29) Arnold, K.; Pratsch, L.; Gawrisch, K. Biochim. Biophys. Acta 1983, 728, 121-128. (30) Hui, S.W.; Kuhl, T.L.; Guo, Y.Q.; Israelachvili, J. Colloids Surf. B 1999, 14, 213-222. (31) Kuhl, T.; Guo, Y.; Alderfer, J. L.; Berman, A. D.; Leckband, D.; Israelachvili, J.; Hui, S. W. Langmuir 1996, 12, 3003-3014. (32) Hui, S. W.; lsac, T.; Boni, L. T.; Sen, A. J. Membr. Biol. 1985, 84, 137-146. (33) Stilbs, P. Progr. Nucl. Magn. Reson. Spectrosc. 1987, 19, 1-45. (34) Ka¨rger, J.; Pfeifer, H.; Heink, W. AdV. Magn. Opt. Reson. 1988, 12, 1-89. (35) Price, W. S. Concepts Magn. Reson. 1997, 9, 299-336. (36) Price, W. S. Concepts Magn. Reson. 1998, 10, 197-237.

520 Langmuir, Vol. 24, No. 2, 2008 Sample Preparation. Bicelles were prepared to consist of 25 wt % lipid in 450 µL of D2O, plus the prescribed weight of a particular molecular weight PEG. The ratio q, being the molar ratio of longto-short chain amphiphiles, (DMPC + DMPG)/DHPC, was kept constant at q ) 4.5. The long chain amphiphiles were taken to include DMPC plus 5 mol % DMPG, the latter expressed as a percentage of the former. A typical preparation involved adding 350 µL of D2O to 150 mg of DMPC + DHPC + DMPG in powder form, followed by a repeated cycle of gentle mixing, heating to 40 °C, and cooling to 4 °C in order to generate bicelles, as indicated by obtaining an optically clear solution. The desired quantity of a particular molecular weight PEG was then added in a further 100 µL of D2O, and the mixing, heating, and cooling cycle was repeated several times further. Such bicelle + PEG preparations were then stored at 4 °C for several hours in order to further ensure complete equilibration of the PEG throughout the sample. The samples were transferred to an NMR sample tube at 4 °C and kept on ice until placed in the magnetic field of the NMR spectrometer. Magnetic alignment of the bicelle + PEG samples was encouraged by an annealing process involving repeated cycling of the temperature between 20 and 35 °C, i.e., through the DMPC gel-to-liquid-crystalline phase transition temperature, with 10 to 15 min of equilibration at either extreme. Magnetic alignment was assessed via 31P NMR spectroscopy. NMR Spectroscopy. All NMR spectra were recorded on a Varian Infinity 500 MHz NMR spectrometer using a Varian 5 mm doubleresonance liquids probe. All spectra were recorded at a sample temperature of 35 ( 0.5 °C. 31P NMR spectra were recorded at 202.31 MHz using single pulse excitation, quadrature detection, complete phase cycling of the pulses, and WALTZ proton decoupling during the signal acquisition using a proton decoupler field strength of 2 kHz. Typical acquisition parameters are as follows: a 90° pulse length of 3.7 µs, a recycle delay of 3 s, a spectral width of 100 kHz, and an 8K data size. Spectra were processed with an exponential multiplication equivalent to 25 Hz line broadening prior to Fourier transformation and were referenced to 85% phosphoric acid. 1H NMR diffusion measurements were performed at 499.78 MHz using the stimulated echo (STE) pulsed field gradient (PFG) procedure,37,38 with constant gradient pulse duration (5 ms) and variable gradient pulse amplitude. The field gradient pulses were applied along the longitudinal (z) direction exclusively. Typical acquisition parameters were as follows: a 90° pulse length of 16 µs, a spin echo delay of 10 ms, a recycle delay of 5 s, a spectral width of 10 kHz, and a 4K data size. The phases of the radio frequency pulses were cycled as described by Fauth et al.39 to remove unwanted echoes. Spectra were processed with an exponential multiplication equivalent to 5 Hz line broadening prior to Fourier transformation and were referenced to tetramethylsilane. Gradient strength was calibrated from the known diffusion coefficient of HDO at 25 °C.40 Proton T1 relaxation times were measured using a standard inversion recovery protocol.

Results 31P NMR Assessment of Bicelle Magnetic Alignment in the Presence of PEG. Before commencing diffusion measurements, it is essential to assess the quality and direction of magnetic alignment of bicelles in the presence of PEG within the interstices. 31P NMR provides a convenient means of doing so.21 Figure 2 shows 31P NMR spectra of magnetically aligned q ) 4.5 bicelles containing 5 mol % DMPG plus various molecular weight PEG added to the aqueous solution, all at 3.33 mg of PEG per mL. The 31P NMR spectra in all cases consist of two major wellresolved narrow resonances (e.g., the DMPC width-at-half-height,

(37) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288-295. (38) Tanner, J. E. J. Chem. Phys. 1970, 52, 2523-2526. (39) Fauth, J.-M.; Schweiger, A.; Braunschweiler, L.; Forrer, J.; Ernst, R. R. J. Magn. Reson. 1986, 66, 74-85. (40) Mills, R. J. Phys. Chem. 1973, 77, 685-688.

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Figure 2. 31P NMR spectra of negatively magnetically aligned q ) (DMPC + DMPG)/DHPC ) 4.5 bicelles containing 5 mol % DMPG (25 wt % lipid, 35 °C) in the presence of 3.33 mg mL-1 PEG 600 (top), 6000 (middle), and 20 000 (bottom). The resonances, from highest to lowest frequency, are assigned to DHPC, DMPG, and DMPC. Their chemical shifts and narrow line widths are characteristic of negatively magnetically aligned bicelles having a narrow mosaic spread of orientations. Thus, the presence of PEG at this concentration and over this range of molecular weights does not perturb the bicelle magnetic alignment.

∆ν1/2, is on the order of 150 Hz). The more intense resonance at approximately -12 ppm is assigned to DMPC present within the planar regions of the bicellar assembly, since this frequency corresponds to the 90° shoulder of the corresponding powder spectrum of nonaligned DMPC bilayers, and the spontaneous direction of alignment for these bicelles is such that the normal to the plane of the bicelle lies perpendicular to the direction of the magnetic field. The less intense resonance at approximately -5 ppm is assigned to DHPC located preferentially (but not exclusively) within the toroidal regions of the bicelles and subject, therefore, to more nearly isotropic motional averaging than DMPC. A third, minor resonance is evident at approximately -9 ppm and is assigned to DMPG. The integrated intensities of

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Table 1. Summary of 31P NMR Chemical Shifts and Intensities in Negatively Magnetically Aligned Bicelles Composed of DMPC + DMPG + DHPC and in the Presence of Various Mw PEGa 31

P NMR chemical shift (ppm)b

PEG Mw (g mol-1)

DHPC

DMPG

DMPC

integrated intensityc

200 600 1000 2000 3400 4600 6000 8000 10000 12000 20000

-6.1 -5.9 -6.0 -5.9 -5.4 -5.5 -5.1 -5.4 -5.1 -5.2 -5.2

-9.2 -9.0 -9.1 -9 -8.4 -8.6 -8.3 -8.4 -8.3 -8.3 -9

-13.2 -12.9 -13.1 -12.9 -12.3 -12.5 -12.2 -12.4 -12.1 -12.2 -12.3

4.3 4.4 4.4 4.4 4.4 4.6 4.4 4.4 4.6 4.4 4.4

a PEG concentration was 3.33 mg mL-1 in all cases. b Relative to 85% H3PO4. c q ) (DMPC + DMPG)/DHPC.

these three resonances yields a calculated value of q ) (DMPC + DMPG)/DHPC very close to the target value of 4.5. Conventionally, PEG-ylated lipids would have been incorporated into such bicelle mixtures to enhance colloidal stability via steric repulsion between apposing bicellar lamellae,41 but this obviously would have been impractical for our present purposes. Instead, DMPG was incorporated in order to replace that steric stabilization with Coulombic stabilization through the resulting electrostatic repulsion between apposing bicellar lamellae. In order not to screen the DMPG surface charges, and thereby obviate the Coulombic stabilization, no salt was included in the aqueous solution. As evident from Figure 2, and as further detailed in Table 1, the chemical shifts and narrow line widths of the various 31P NMR resonances are virtually identical for all PEG molecular weights examined, apart from a slight decrease in chemical shift with increasing PEG molecular weight. Thus, at the concentration of PEG employed here, and over the chosen range of PEG molecular weights, Coulombic stabilization permits attainment of a highly uniform distribution of negatively magnetically aligned bicelles. These are, therefore, ideal for our PFG NMR diffusion studies. STE PFG 1H NMR Diffusion Measurements of BicelleConfined PEG. Given that conditions have been established under which the added PEG does not perturb the magnetic alignment of the bicelles, one may proceed confidently with measuring the PEG diffusion coefficients under bicelle confinement. The stimulated echo (STE) pulsed field gradient (PFG) NMR method used to measure the PEG diffusion coefficients is shown schematically in Figure 3.37,38 During the first gradient pulse of magnitude g (T m-1) and duration δ (s), nuclear spins are encoded with a phase shift φ ) γδgz, where γ is the particular magnetogyric ratio and z is the spin’s displacement along the direction of the applied field gradient. In the absence of diffusion during the diffusion time ∆ the second identical gradient pulse produces an equal rephasing of the magnetization and, consequently, an unattenuated echo forms. If diffusion occurs during the delay ∆, then the rephasing is incomplete and the resulting echo is attenuated proportionately. In STE PFG NMR for the case of unrestricted isotropic diffusion of a single species, the stimulated echo intensity decays according to (41) King, V.; Parker, M.; Howard, K. P. J. Magn. Reson. 2000, 142, 177182.

I ) I0 exp

( ) ( )

-2τ2 -τ1 exp exp(-γ2g2δ2D[∆ - δ/3]) T2 T1

(1)

where D is the isotropic diffusion coefficient and ∆ is the experimental diffusion time, while T1 and T2 are the longitudinal and transverse relaxation times, respectively. Experimentally, either the gradient pulse amplitude, its duration, or the diffusion time is incremented progressively. To extract diffusion coefficients from such experimental data, the resonance intensity ln(I/I0) is plotted versus k ) [(γgδ)2(∆ - δ/3)] so that the diffusion coefficient corresponds to the slope. In the STE PFG NMR sequence, ∆ ) τ1 + τ2, so that for situations where T1 > T2 the experimentally accessible diffusion time is limited by T1 rather than T2, which confers the ability to employ longer diffusion times, thereby facilitating diffusion measurements for cases of slower diffusion or lower gradient strengths or lower γ nuclei. Figure 3 also shows a series of 1H NMR spectra of negatively magnetically aligned q ) 4.5 bicelles containing 5 mol % DMPG plus 3.33 mg mL-1 PEG 8000 as a function of increasing gradient amplitude in the STE PFG 1H NMR sequence. Two intense resonances dominate the spectrum. One is attributed to HDO (4.3 ppm) and the other to the ethylene oxides of the PEG 8000 (3.4 ppm). Other expected resonances, such as the DMPC and DHPC choline methyl protons and acyl chain methyls and methylene protons, are absent due to their short T2 relative to the value of the echo delay used in the experiment (τ2 ) 10 ms). For example, control samples, in which no PEG was included with the bicelles, yield STE PFG 1H NMR spectra containing only the HDO resonance (data not shown). In the series of spectra shown in Figure 3, it is evident that the HDO resonance decays rapidly with increasing gradient pulse amplitude, as expected given the rapid diffusion of water. In contrast, the PEG resonance decays much more slowly, reflecting the relatively slow diffusion of this much larger species, all other factors being equal. Before considering in detail the diffusion of bicelle-confined PEG, it is useful and informative to examine first the behavior of the identical molecular weight PEG free in aqueous solution. The 1H NMR spectrum of PEG in D2O is essentially identical to those shown in Figure 3. Figure 4 illustrates the diffusive decays obtained using STE PFG NMR for these various molecular weight PEG in aqueous solution, all at a concentration of 3.33 mg mL-1. It is obvious that in every case a monoexponential decay is obtained, even when the intensity falls as far as 1% of its initial value. Thus, whatever polydispersity of PEG molecular weight is present is too small to be detected in this experiment.42 Moreover, clustering of PEG, which is a possibility,43 albeit a remote one,44,45 appears, therefore, not to be a concern here. For the case of a monoexponential decay, the diffusion coefficient is obtained from the slope in diffusive decays such as those in Figure 4. Figure 6 shows the manner in which log D varies with log N for these various PEG free in solution, where N is the number of ethylene oxide units per PEG chain. The slope here equals -0.473, which conforms with previous determinations of the molecular weight dependence of the diffusion coefficient of free PEG in aqueous solution.46,47 D is expected to be proportional to Mw-1/2 for the case of a flexible random coil polymer chain such as PEG in a good solvent such as water. (42) von Meerwall, E. D. J. Magnet. Reson. 1982, 50, 409-416. (43) Polverari, M.; van de Ven, T. G. M. J. Phys. Chem. 1996, 100, 1368713695. (44) Devenand, K.; Selser, J. C. Nature 1990, 343, 739-741. (45) Kinugasa, S.; Nakahara, H.; Fudagawa, N.; Koga, Y. Macromol. 1994, 27, 6889-6892. (46) Couper, A.; Stepto, R. F. T. Trans. Faraday Soc. 1969, 65, 2486-2496. (47) Masaro, L.; Zhu, X. X.; Macdonald, P. M. J. Polym. Sci. 1999, 37, 23962403.

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Figure 3. (Top) The stimulated echo (STE) pulse field gradient (PFG) NMR pulse sequence of Tanner38 used here for self-diffusion measurements of PEG confined between lamellae of negatively magnetically aligned bicelles. (Bottom) A series of 1H NMR spectra (35 °C) of magnetically aligned q ) (DMPC + DMPG)/DHPC ) 4.5 bicelles containing 5 mol % DMPG (25 wt % lipid, 35 °C) in the presence of 3.33 mg mL-1 PEG 8000 as a function of the applied gradient amplitude (g) in the STE PFG NMR sequence. The gradient pulse duration was 5 ms, with τ2 and τ1 equal to 10 and 800 ms, respectively. The gradient pulse amplitudes were, from front to back, 0.058, 0.145, 0.232, 0.319, 0.406, 0.493, 0.580, 0.667, 0.754, 0.841, 0.928, 1.01, 1.10, 1.18, 1.27, 1.36, 1.45, 1.53, 1.62, and 1.71 T m-1. The water resonance at 4.3 ppm decays rapidly due to the fast diffusion of water, while the PEG resonance at 3.4 ppm decays far more slowly, as expected given its larger hydrodynamic radius.

Furthermore, it is indicative of a non-free-draining diffusion in which the solvent molecules are carried along with the polymer, as is to be expected given PEG’s propensity to hydrogen bond with water (see Lu¨sse and Arnold48 and references within). From the diffusion coefficient, one calculates the hydration radius of the polymer chain, Rh, using the Stokes-Einstein equation

D)

kT 6πηRh

(2)

where k is the Boltzmann constant and η is the solution viscosity at the measuring temperature T (ηwater ) 0.719 cP at 35 °C). The value of Rh for each of these various PEG free in solution is listed in Table 2. Several conclusions may be drawn from this exercise. First, the bicelle confinement dimension Rh/H approaches, but never (48) Lu¨sse, S.; Arnold, K. Macromol. 1996, 29, 4251-4257.

exceeds, unity for the molecular weights under consideration. This is based on the reported thickness H ) 60 Å for the aqueous layer separating the apposed bilayers in a similar bicelle system having the same water content as employed here.24 This is significant because it explains the good magnetic alignment obtained with our bicelles in the presence of even large molecular weight PEG. Specifically, scaling arguments indicate that the free energy cost (largely entropic in origin) of polymer confinement increases as the square of the ratio of the unperturbed polymer chain dimension to the confinement dimension.5 If the energy cost of polymer confinement exceeds the bending energy of the lipid bilayers, then the magnetic alignment of bicelles will be compromised. Lipid bilayers in general, and magnetically aligned bicelles in particular, are elastic; they fluctuate, both at the level of the individual lipids, whose positions can only be described in terms of a distribution function, and at the level of the bilayer collective, which undergoes undulation fluctuations.18 In stacked lipid bilayers or bicelles, such undulations are

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Figure 4. Diffusive decay of the normalized 1H stimulated echo intensity versus k ) (γδg)2(∆ - δ/3) for various molecular weight PEG free in aqueous solution at a concentration of 3.33 mg mL-1and a temperature of 35 °C. The diffusion coefficient in such a plot of ln(I/I0) versus k is proportional to the slope. Since in all cases the diffusive decays are strictly monoexponential, the various PEG samples are relatively monodisperse. The PEG molecular weights are, from left to right, 200, 600, 1000, 2000, 3400, 4600, 6000, 8000, 10 000, 12 000, and 20 000. Hence, diffusion slows monotonically with increasing PEG molecular weight (see Figure 6 for further details).

Figure 5. Diffusive decay of the normalized 1H stimulated echo intensity versus k ) (γδg)2(∆ - δ/3) for various molecular weight PEG confined between the lamellae of negatively magnetically aligned q ) (DMPC + DMPG)/DHPC ) 4.5 bicelles containing 5 mol % DMPG (25 wt % lipid, 35 °C) in the presence of various molecular weight PEG at a concentration of 3.33 mg mL-1. Diffusive decays at any one diffusion time ∆ were measured as a function of increasing gradient amplitude g for a series of different diffusion times, as indicated (O, 100 ms; b, 200 ms; 1, 400 ms; ∆, 600 ms; 9, 800 ms; 0, 1000 ms). It is evident that the diffusive decays for different diffusion times overlap for any one PEG molecular weight. The PEG molecular weight increases, from left to right, as 200, 1000, 2000, 4600, 8000, 12 000, and 20 000.

dampened by steric, electrostatic, and hydration repulsive interactions between adjacent bilayers. The dampening increases as the interbilayer spacing decreases. Dampening of fluctuations by interactions between bilayers likewise aids and abets the magnetic alignment of lipid bilayers. Isolated unilamellar lipid vesicles, for instance, fluctuate so much that they are difficult to align in a magnetic field.26 In fact, the magnetic susceptibility anisotropy, through which the external magnetic field exerts its orienting torque on the lipid bilayer, is volume additive, so that closely apposed bilayers experience a greater torque.26 By ensuring that the polymer confinement dimension Rh/H is less than unity, and therefore that the energy cost of polymer

confinement is low, we avoid compromising the quality of the magnetic alignment of our bicelles, as is evident from the 31P NMR spectra of Figure 2. While we have not thoroughly investigated this question, we have found that in the presence of higher concentrations (10 mg mL-1) of higher molecular weight PEG, bicelles become increasingly difficult to properly magnetically align. A second conclusion is that, at the PEG concentration employed here (3.33 mg mL-1), PEG remains well below its critical overlap concentration c* for all molecular weights investigated. This is significant because as the polymer concentration approaches and exceeds c*, entanglement of overlapping polymer chains radically

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Figure 6. (A) log-log plot of the PEG diffusion coefficient versus the degree of polymerization N: b, free PEG in aqueous solution (3.33 mg mL-1, 35 °C); O, PEG confined between the lamellae of negatively magnetically aligned q ) (DMPC + DMPG)/DHPC ) 4.5 bicelles containing 5 mol % DMPG (25 wt % lipid, 35 °C) in the presence of 3.33 mg mL-1 PEG. The solid line is from a regression analysis of the free PEG data and has a slope equal to -0.473, indicating that free PEG in aqueous solution behaves like a non-free-draining random coil polymer in a good solvent. The dashed line has the same slope as the solid line but was scaled lower by 30% to account for the increased aqueous viscosity within the bicelle interlamellar aqueous space. (B) log-log plot of the reduced diffusivity D/D0, where D is the diffusion coefficient of the bicelle-confined PEG and D0 is that of the same molecular weight PEG free in aqueous solution, versus the ratio of the hydration radius to the confinement dimension Rh/H. Rh was calculated from the diffusion coefficient of PEG free in solution as per the Stokes-Einstein equation. H was take to be 60 Å.24 The dashed line shows the results for the Pawar-Anderson equation simulating a hard sphere confined between two walls.72 The solid line shows the de Gennes scaling prediction5 wherein D/D0 ∼ (Rh/H)-2/3.

alters polymer dynamics, as manifest by a sharply increasing viscosity and decreasing diffusivity with increasing concentration. The overlap concentration is calculated on the basis of the polymer’s radius of gyration, Rg, and molecular weight, Mw, according to eq 349

c* )

Mw 4 πR 3N 3 g A

(3)

where NA is Avogadro’s number. Rg is obtained from the diffusion coefficient of PEG free in solution via the Stokes-Einstein equation using Rh ) 0.655Rg. Values of c* for the various PEG (49) Doi, M.; Edwards, S. F. In The Theory of Polymer Dynamics: Oxford University Press: New York, 1986; pp 141.

investigated here are listed in Table 2 and demonstrate that our PEG concentration of 3.33 mg mL-1 lies well below the critical overlap concentration for all molecular weights considered. We turn now to the case of PEG confined between the lamellae of negatively magnetically aligned bicelles. For a diffusant confined between parallel plates, the diffusion coefficient is anisotropic. In the molecular frame defined with respect to the parallel plate geometry, as shown schematically in Figure 1, only two independent diffusion tensor elements persist, specifically, D| and D⊥, representing, respectively, diffusion parallel and perpendicular to the interplate, or transverse, direction having dimension H.50,51 When the field gradients are applied solely (50) Callaghan, P. T.; So¨derman, O. J. Phys. Chem. 1983, 87, 1737-1744. (51) Lindblom, G.; Ora¨dd, G. Prog. Nucl. Magn. Reson. Spectrosc. 1994, 26, 483-515.

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Table 2. Properties of PEG Free in Aqueous Solution from STE PFG 1H NMR Diffusion Measurements at 35 °C

a

PEG Mw (g mol-1)

hydrodynamic radius, Rh (Å)

overlap concn,b c* (mg mL-1)

200 600 1000 2000 3400 4600 6000 8000 10000 12000 20000

5.8 9.1 11.5 18.0 22.5 23.9 28.9 33.9 39.4 40.6 44.3

114 89 73 38 33 38 28 23 18 20 26

Calculated from eq 2. b Calculated from eq 3.

along the laboratory z-direction (defined by the direction of the main magnetic field B0), the g2D term in eq 1 is replaced by gz2Dzz, where Dzz is the particular element of the symmetric Cartesian diffusion tensor, as defined in the laboratory frame, which is measured under such circumstances. Transforming from the molecular frame to the laboratory frame yields the relationship between the apparent diffusion coefficient measured in the laboratory frame Dzz and the diffusion tensor elements in the relevant molecular frame, as per eq 4

Dzz ) D⊥ sin2 θ + D| cos2 θ

(4)

where θ is the polar angle between the interplate direction and that of the applied field gradient. For negatively magnetically aligned bicelles, the interplate direction is oriented uniformly at 90° relative to the magnetic field direction, so that the second term in eq 4 disappears when the field gradients are applied exclusively along the laboratory z-direction. Hence, for a perfectly aligned bicelle sample, i.e., having an infinitely narrow mosaic spread of alignments, the apparent diffusion coefficient measured in our STE PFG NMR experiment is directly equal to the diffusion coefficient along the longitudinal plate direction. Figure 5 compares the STE PFG 1H NMR diffusive intensity decays measured at 35 °C for various molecular weight PEG confined between the lamellae of negatively magnetically aligned q ) 4.5 bicelles containing 5 mol % DMPG. Evidently, and as one might expect, the diffusion coefficient of a confined PEG decreases with increasing molecular weight, as is clear from the dependence of the slope on the molecular weight. Prior to discussing the details of how confinement has influenced these diffusion coefficients, it will be useful to highlight certain salient features of the diffusive decays in Figure 5. First, only absolute intensities exhibiting signal-to-noise ratios (SNR) greater than 10:1 were deemed reliable for quantitation purposes, in accord with accepted limits of quantification.52 Second, in all cases the diffusive intensity decays were monitored until the relative intensity had decreased to less than 5% of its initial value, i.e., ln(I/I0) e -3, thereby capturing the diffusion behavior of at least 95% of the PEG population. Third, there is a gentle curvature in the diffusive decays for longer diffusion times ∆, most evident for low molecular weight PEG, but decreasing with increasing PEG molecular weight. There are several possible origins of such curvature. One is the polydispersity of the size of the diffusant.42 However, the free PEG diffusion decays in Figure 4 demonstrate that, whatever the polydispersity present in these PEG samples, the molecular (52) Long, G. L.; Winefordner, J. D. Anal. Chem. 1983, 55, 712A-724A.

weight distributions are too narrow to cause such curvature. Moreover, since in all cases the PEG concentration is well below c*, PEG overlap and/or clustering is unlikely to be a contributing factor. Another potential source of such curvature is the presence of slow or intermediate exchange of PEG between two (or more) environments characterized by different diffusion coefficients.53-56 In such circumstances the STE PFG 1H NMR diffusive intensity decays measured for different diffusion times ∆ are superimposable and, as Figure 5 shows, this is true for PEG confined between bicellar lamellae to a good approximation. The only obvious candidate for a more-slowly diffusing population would be PEG bound to the bicelle surface. However, extensive studies by Arnold and co-workers57-60 led to the conclusion that PEG experiences a depletion-attraction interaction with lipid bilayer surfaces, wherein PEG is excluded from a zone near the lipid bilayer surface as a result of the entropic cost of altering the preferred polymer chain conformation to accommodate the surface. This depletion-attraction scenario was confirmed by Israelachvili and co-workers61,62 for PEG in the molecular weight range 8000-12 000. Although lower molecular weight PEG, i.e., Mw < 1000, was able to enter the “near membrane” region, it did not bind to the membrane surface itself. In our NMR diffusion measurements on bicelles, the decay of the HDO signal is strictly linear, so that exchange of water between surface and bulk regions is fast. The same should be true for lower molecular weight PEG resident in the surface region. Only higher molecular weight PEG, i.e., Mw ≈ 20 000, was found to be sufficiently hydrophobic to bind to the membrane.61,62 One might argue that, nevertheless, the special hydrogen-bonding proclivity of both PEG48 and DMPG63 tilts the scenario in favor of PEG binding to DMPG-containing bicelles. However, in control experiments on otherwise identical bicelles lacking DMPG, we observed the same curvature of the PEG diffusive decay. Hence, we conclude that, for the molecular weight range of interest here, the gentle curvatures evident in Figure 5 are not likely to be due to binding of PEG to the bicelle’s lipid bilayer surface. Another potential source of nonlinear diffusive decays is anomalous, i.e., obstructed or restricted, diffusion.64 A characteristic property of anomalous diffusion in the STE PFG NMR experiment is a diffusion-time-dependent apparent diffusion coefficient: that is, different experimental diffusion times ∆ yield different apparent diffusive intensity decays and diffusion coefficients. Situations of particular relevance here include diffusion through porous media50,65 or within wormlike micelles.66 The anomalous behavior becomes manifest only when the characteristic experimental root-mean-square diffusion distance 〈r2〉1/2 exceeds the unobstructed mean-free-path available to the (53) Ka¨rger, J.; Pfeifer; H., Heink,W. AdV. Magn. Res.1988, 12, 1-89. (54) Ka¨rger, J. Ann. Phys. 1969, 24, 1-4. (55) Ka¨rger, J. Ann. Phys. 1971, 27, 107-109. (56) Johnson, C. S., Jr. J. Magn. Reson. A 1993, 102, 214-218. (57) Arnold, K.; Pratsch, L.; Gawrisch, K. Biochim. Biophys. Acta 1983, 728, 121-128. (58) Arnold, K.; Hermann, A.; Gawrisch, K.; Pratsch, L. Stud. Biophys. 1985, 110, 135-141. (59) Arnold, K.; Lvov, Y. M.; Szogyi, M.; Gyorgyi, S. Stud. Biophys. 1986, 113, 7-14. (60) Arnold, K.; Zschoernig, O.; Barthel, D.; Herold, W. Biochim. Biophys. Acta 1990, 1022, 303-310. (61) Kuhl, T. L.; Berman, A. D.; Hui, S. W.; Israelachvili, J. N. Macromolecules 1998, 31, 8250-8257. (62) Kuhl, T. L.; Berman, A. D.; Hui, S. W.; Israelachvili, J. N. Macromolecules 1998, 31, 8258-8263. (63) Zhao, W.; Ro´g, T.; Gurtovenko, A. A.; Vattulainen, L.; Karttunen, M. Biophys. J. 2007, 92, 114-1124. (64) Stejskal, E. O. J. Chem. Phys. 1965, 43, 3597-3603. (65) Mitra, P. P.; Sen, P. N. Phys. ReV. B 1992, 45, 143-156. (66) Angelico, R.; Olsson, U.; Palazzo, G.; Ceglie, A. Phys. ReV. Lett. 1998, 81, 2823-2826.

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diffusant. The latter is characteristic of a particular confinement geometry, while for two-dimensional diffusion the former is calculated as per eq 5

〈r2〉1/2 ) (4Dt)1/2

(5)

where D is the diffusion coefficient and t ) ∆ is the characteristic experimental diffusion time. For example, in the case of PEG 200, where D ) 4.0 × 10-10 m2 s-1, and using ∆ ) 610 ms, the rms diffusion distance equals on the order of 30 µm. As shown in Figure 5, the STE PFG 1H NMR diffusive intensity decays for PEG confined between bicelles as measured for different diffusion times ranging from 100 to 1000 ms are virtually superimposable, regardless of molecular weight. Only at the longest diffusion times is there curvature evident. Thus, diffusion of PEG confined between bicelles is unrestricted over diffusion distances of tens of micrometers, regardless of molecular weight. Only when the diffusion distances approach 30-40 µm is curvature of the diffusive decays at all evident. Such diffusion distances roughly correspond to the bicelle domain sizes observed in polarized optical microscopy (POM) studies of similar composition bicelles, at similar temperatures, albeit not magnetically aligned.67 While the correspondence may be coincidental, it suggests the possibility that diffusive crossing of PEG from one bicelle domain to another within the sample involves traversing an obstruction. Only a favorable combination of diffusion coefficient D, diffusion time ∆, and “wave vectors” (γgδ) will be capable of exploring such diffusion distances. For example, our largest PEG, Mw ) 20 000, yields D ) 2.17 × 10-11 m2 s-1, corresponding, in the case of ∆ ) 1000 ms, to a diffusion distance of roughly 9 µm and exhibits no curvature of the diffusive decay. Likewise, as we have noted already, in these NMR diffusion measurements on bicelles, the decay of the HDO signal is strictly linear. The diffusion coefficient of HDO in the bicelle preparations was found to equal 2.0 × 10-9 m2 s-1 at 35 °C, a reduction of about 30% relative to bulk water at the same temperature, and was independent of the particular PEG present. With the diffusion time of 20 ms used to test the linearity of the HDO diffusive decay, the corresponding diffusion distance is just greater than 10 µm. Additionally, in our previous studies of PEG-ylated lipids in bicelles,68-70 diffusion of such membraneanchored species was found to be unrestricted over diffusion distances of up to 5 µm. These details of the analysis of the NMR diffusion experiment results as described above serve to demonstrate that PEG diffusion, even when confined between the lamellae of magnetically aligned bicelles, is readily measured via STE PFG NMR and appears to be largely unrestricted over distances of tens of microns. We may now turn to the primary focus, the effects of such confinement on PEG diffusion. Effect of Bicelle Confinement on PEG Diffusion. Figure 6A compares the diffusion coefficients of PEG confined between the lamellae of magnetically aligned bicelles versus free in solution. As described above, the diffusion coefficients of free PEG decrease with increasing molecular weight as D ∝ Mw-1/2 as expected for a non-free-draining random coil polymer in a good solvent, water in this case, below the polymer’s critical overlap concentration. When confined between bicellar lamellae, PEG diffusion first parallels the behavior of free PEG for the (67) Harroun, T. A.; Koslowsky, M.; Nieh, M.-P.; de Lannoy, C.-F.; Raghunathan, V. A.; Katsaras, J. Langmuir 2005, 21, 5356-5361. (68) Soong, R.; Macdonald, P. M. Biophys. J. 2005, 88, 255-268. (69) Soong, R.; Macdonald, P. M. Biophys. J. 2005, 89, 1850-1860. (70) Soong, R.; Macdonald, P. M. Biochim. Biophys. Acta 2007, 1768, 18051814.

lower molecular weight species. The decrease in PEG diffusion for these smaller PEG species can be accounted for entirely by the increased viscosity of the aqueous solution separating the bicelle lamellae. Specifically, the diffusion of water in these interstices is slower by a factor of roughly 30% relative to bulk water, independent of PEG molecular weight (see above), which translates directly to a 30% increase in local viscosity, averaged over the entire interstitial volume. The dashed line in Figure 6A is the predicted diffusion coefficient if this 30% viscosity increase were the only additional factor. With increasing PEG molecular weight, diffusion is progressively hindered relative to free PEG, an effect that must be attributed to PEG confinement. Figure 6B shows the dimensionless PEG diffusivity D/D0 as a function of Rh/H, where D0 and Rh are the unconfined PEG diffusion coefficient and hydration radius, respectively, while H is taken to equal 60 Å.24 Evidently, there is a broad crossover from rather weak confinement effects when Rh/H < 0.4 to much stronger confinement effects when Rh/H > 0.4. We note that little difference in the PEG 1H NMR line widths was evident when comparing free and bicelle-confined cases, nor was any great difference in the PEG 1H NMR longitudinal relaxation time T1 discernible in the free versus bicelle-confined cases, although T1 did decrease as expected with increasing PEG molecular weight in both case (data not shown). Thus, shortrange local motions that contribute to PEG T1 and T2 relaxation are relatively unperturbed upon bicelle confinement, while longrange diffusive motions are hindered relative to free PEG. Nevertheless, even the largest molecular weight PEG under bicelle confinement was able to diffuse unrestricted over rather long distances, as discussed above. Scaling arguments predict how strong confinement, defined by de Gennes as Rh > H, influences polymer diffusion.3-5 The polymer’s radius of gyration in the direction parallel to the confining plates Rg|| is predicted to scale as

Rg|| ∼ NV(2)/HV(2)/V(3)-1

(6)

where V(d) ) 3/(d + 2) is the Flory-Edwards exponent, with d equal to 3, 2, or 1 for bulk, slit, or pore geometries, respectively. The resultant scaling, relative to the Flory radius RF, for a parallel plate confinement geometry is Rg||/RF ∼ (N/H)1/4. For a channel or pore confinement, the equivalent scaling is Rg||/RF ∼ (N/H)2/3. The validity of the scaling exponents has been confirmed in Monte Carlo simulations.6-10 The difficulty with scaling arguments is to evaluate the numerical coefficients and to define the crossover point between bulk and confined behavior. Analytical models capable of doing so have considered the case of a hard spherical particle confined within a pore71 or between parallel plates.72 Pawar and Anderson72 derived the following expression for a hard sphere of radius a diffusing between hard walls separated by 2d, where λ ) a/d, found to be valid for the case λ e 0.5,

D/D0 ) (1 - λ)-1[1 + (9/16)λ ln(λ) - 1.19358λ + 0.159317λ3] (7) where D0 is the bulk diffusivity. The dashed line in Figure 6B shows the Pawar-Anderson equation for the case in which we identify a ) Rh and d ) H/2. While such models are found to be useful for understanding polymer diffusion through porous membranes, where diffusion is much slower than any of the (71) Brenner, H.; Gaydos, L. J. J. Colloid Interface Sci. 1977, 58, 312-356. (72) Pawar, Y.; Anderson, J. L. Ind. Eng. Chem. Res. 1993, 32, 743-746.

Diffusion of PEG Confined between Lamellae

instances here,11,12 this model overestimates the hindering effects of confinement on the PEG diffusion in magnetically aligned bicelles, precisely because it treats the polymers as hard spheres when they are, in fact, flexible and nonspherical. When, instead, the native flexibility of polymers is taken into consideration, as in the Monte Carlo simulations by Van Vliet et al.,73,74 a finer parsing of confinement regimes and effects could be achieved. These authors concluded that for large values of the plate separation (i.e., H . Rg), the polymer coil was unperturbed. However, beginning at H/Rg ∼ 2.5, the polymer, being asymmetric in shape, oriented with its longest axis parallel to the plate direction, while the same average dimensions were maintained. Only below H/Rg ∼ 1.5 was the polymer squeezed, and only below H/Rg ∼ 1.0 was the two-dimensional behavior predicted by scaling arguments attained. Recently, the case of double-stranded DNA confined to square cross-section microchannels has been simulated using the “Brownian dynamics combined with hydrodynamics interactions” (BD-HI) method.75 The point is that scaling arguments fail to consider changes in hydrodynamic interactions brought on by confinement. These authors find that diffusion is perturbed relative to bulk behavior, even for channel widths (H) as large as 10 times the free-solution radius of gyration of the polymer chain, while a strongly confined Rouse-type behavior (D ∝ Mw-1) is predicted for channel widths less than twice the bulk Rg.75 In this strongly confined regime, these simulations yield D/D0 ∼ (Rg/ H)-1/2, in contrast to the -2/3 exponent predicted by scaling arguments. It is argued that this difference arises because simple scaling theory does not take into account the additional length scale involving polymer-wall interactions. (73) van Vliet, J. H.; ten Brinke, G. J. Chem. Phys. 1990, 93, 1436-1441. (74) van Vliet, J. H.; Luyten, M. C.; ten Brinke, G. Macromolecules 1992, 25, 3802-3806. (75) Jendrejack, R. M.; Schwartz, D. C.; Graham, M. D., de Pablo, J. J. J. Chem. Phys. 2003, 119, 1165-1173.

Langmuir, Vol. 24, No. 2, 2008 527

When the BD-HI method was applied to the additional case of double-stranded DNA confined between parallel plates,76 the predicted polymer dimensions under confinement agreed with scaling theory, i.e., Rg||/Rg ∼ (Rg/H)m, where m equals 2/3 for a channel and 1/4 for a parallel plate geometry. However, diffusivity scaled approximately as D/D0 ∼ (Rg/H)-2/3 identically for both confinement geometries. These predictions were compared to, and found to agree with, experimental epiflourescence diffusivity measurements of variously sized double-stranded single DNA molecules confined within rectangular polydimethylsiloxane microchannels. Comparing these findings to the PEG diffusion results in Figure 6B, it is evident that a strong confinement regime is encountered when Rh/H > 0.4, in agreement with the DNA diffusivity results.76 Furthermore, the PEG diffusivity scales with increasing confinement according to D/D0 ∼ (Rh/H)-2/3, as shown by the solid line in Figure 6B, again in accord with the DNA diffusivity results.76 Thus, we conclude that a flexible polymer confined between “soft” walls displays reduced diffusivity comparable to that exhibited by a stiff polymer confined between hard walls. Since both channel and plate confinement geometries yield identical diffusivity scaling within the strong confinement regime, it is not possible on the basis of the PEG diffusivity scaling with molecular weight to differentiate between the perforated lamellae and wormlike micelle models of bicelle morphology.24 The present results, nevertheless, will allow us to proceed with investigations of confinement effects on the diffusivity of a stiff polymer confined between soft walls. Acknowledgment. This research was supported by a grant from the Natural Science and Engineering Research Council (NSERC) of Canada. LA7022264 (76) Chen, Y.-L.; Graham, M. D.; de Pablo, J. J.; Randall, G. C.; Gupta, M.; Doyle, P. S. Phys. ReV. E 2004, 70, 0609011-0609014.