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Characterization of a biomimetic mesophase composed of non-ionic surfactants and an aqueous solvent. Vladimir Adrien, Gamal Rayan, Myriam Reffay, Lionel Porcar, Amir Maldonado, Arnaud Ducruix, Wladimir Urbach, and Nicolas Taulier Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b02744 • Publication Date (Web): 12 Sep 2016 Downloaded from http://pubs.acs.org on September 13, 2016
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Characterization of a biomimetic mesophase composed of non-ionic surfactants and an aqueous solvent. V. Adrien1,2*, G. Rayan1, M. Reffay1, L. Porcar3, A. Maldonado4, A. Ducruix2, W. Urbach1,5, N. Taulier5 1
Laboratoire de Physique Statistique, Ecole Normale Supérieure, PSL Research University; Université Paris Diderot Sorbonne Paris Cité; Sorbonne Universités UPMC Univ Paris 06, CNRS; 24 rue Lhomond, 75005 Paris, France 2 Univ Paris Descartes, Sorbonne Paris Cité. Laboratoire de Cristallographie et RMN Biologiques, CNRS UMR 8015, Paris, France 3 Institut Laue-Langevin, 71 avenue des Martyrs, CS 20156, 38042 Grenoble Cedex 9, France 4 Departamento de Física, Universidad de Sonora, Apdo Postal 1626, 83000 Hermosillo, Sonora, Mexico 5 Sorbonne Universités, UPMC Univ Paris 06, CNRS, INSERM, Laboratoire d’Imagerie Biomédicale, F75006, Paris, France * To whom correspondence should be addressed. E-mail :
[email protected] ABSTRACT We have investigated the physical and biomimetic properties of a sponge (L3) phase composed of pentaethylene glycol monododecyl ether (C12E5), a non-ionic surfactant, an aqueous solvent, and a co-surfactant. The following co-surfactants, commonly used for solubilizing membrane proteins, were incorporated: n-octyl-β-D-glucopyranoside (β-OG), n-dodecyl-β-D-maltopyranoside (DDM), 4cyclohexyl-1-butyl-β-D-maltoside (CYMAL-4), and 5-cyclohexyl-1-pentyl-β-D-maltoside (CYMAL-5). Partial phase diagrams of these systems were created. The L3 phase was characterized using crossed polarizers, diffusion of a fluorescent probe by fluorescence recovery after pattern photobleaching (FRAPP), and freeze fracture electron microscopy (FFEM). By varying the hydration of the phase, we were able to tune the distance between adjacent bilayers. The characteristic distance (db) of the phase was obtained from small angle scattering (SAXS/SANS) as well as from FFEM, which yielded complementary db values. These db values were neither affected by the nature of the co-surfactant nor by the addition of membrane proteins. These findings illustrate that a biomimetic surfactant sponge phase can be created in the presence of several common membrane protein-solubilizing detergents, thus making it a versatile medium for membrane protein studies.
INTRODUCTION Lipidic mesophases, such as the cubic phase, are of particular interest, especially when studying transmembrane proteins. Indeed, detergent micelles in which membrane proteins are solubilized often lead to a relatively rapid denaturation of the protein under investigation. Moreover, on a more conceptual point of view, the protein-detergent complex lacks the bi-dimensionality offered by the cell membrane. Therefore, any study dealing with membrane proteins might be better performed under conditions that mimic the planar biological membrane without affecting the structure and activity of the protein. Like the cubic phase, the sponge (or L3) phase is a bicontinuous mesophase with interconnected networks of bilayers dividing two interpenetrating but non-connecting aqueous compartments.1 A way to create an L3 phase is to start from a regular lamellar phase, then add molecules (cosurfactants hereafter) that will wedge themselves between the surfactant molecules of the bilayers. Depending upon its geometrical shape and chemical structure, the co-surfactant can induce a curvature of the bilayer that will transform a lamellar phase into a sponge phase. Globally, the sponge phase can be viewed as a distorted cubic phase, whereas locally, it is like a lamellar (Lα) phase : a superposition of flat membranes separated by a characteristic distance. The main asset of the sponge phase is that it is significantly less viscous than the cubic and lamellar phase and can be manipulated easily. Also, the L3 phase is optically isotropic (unlike the Lα phase) and transparent, thus amenable to many biophysical spectroscopic studies. Additionally, a convenient property of the sponge phase is the ability to precisely tune the distance between bilayers by varying the hydration ACS Paragon Plus Environment
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of the system. Sponge phases have been used in various applications such as membrane protein and peptide crystallization,2–6 drug delivery,7,8 the extraction of polycyclic aromatic hydrocarbons,9 as model bilayers used to study the lateral diffusion of membrane proteins and peptides,10,11 and interactions between transmembrane proteins.12 The cubic and L3 phases are mostly prepared with monoolein, a lipid that can form several phases in aqueous solution: lamellar, cubic, hexagonal and fluid isotropic/inverse micellar.13–15 The L3 phase is formed by adding some polar surfactants, such as ethanol, propylene glycol, polyethylene glycol 400, dimethyl sulfoxide.16,17 Nevertheless, the existence domain of the phase in the phase diagram is very narrow, and depends on the cosurfactant : thus, the addition of a membrane protein might move the sample into another area on the phase diagram, leading to a loss of the L3 phase. A wider sponge phase domain can be obtained by using non-ionic surfactants instead of lipids. Zapf et al.18 have demonstrated the existence of an L3 phase over a wide range of cosurfactant/surfactant ratios in a ternary system composed of calcium dodecyl sulfate, an aqueous buffer, and one of the following alcohols; pentanol, hexanol, heptanol, or octanol. However the presence of an alcohol such as hexanol affects the conformation of membrane proteins (e.g. CaATPase19), which may make this sponge phase unsuitable for membrane protein studies. Strey et al.20 demonstrated the existence of an L3 phase composed of a non-ionic surfactant frequently used in membrane protein crystallization (C12E5)21 and an aqueous buffer at temperatures over than 50°C, and high water contents (> 99% wt). This L3 phase was found in a narrow region of the phase diagram. Detailed information on the structure and properties of the L3 phase has been reviewed recently by Beck and Hoffmann.1 Since membrane proteins are solubilized in detergents, the addition of those solubilized proteins to the above buffer/C12E5 sponge phase would also result in the addition of detergents that would ultimately alter the phase, especially if proteins were solubilized in a detergent other than C12E5. Consequently, we decided to examine the effects of the addition of small amounts of n-octyl-β-Dglucopyranoside (β-OG) to the buffer/C12E5 system. This co-surfactant was chosen as it is one of the most widely used non-ionic detergents in membrane protein studies.22 The sponge phase formed by this ternary system was characterized in this work. We have used other co-surfactants that are routinely used for membrane protein solubilization : n-dodecyl-β-D-maltopyranoside (DDM), 4cyclohexyl-1-butyl-β-D-maltoside (CYMAL-4), or 5-cyclohexyl-1-pentyl-β-D-maltoside (CYMAL-5). In order to draw the phase diagram of the L3 phase, we used crossed polarizers, diffusion of a fluorescent probe by fluorescent recovery after pattern photobleaching (FRAPP), and freeze fracture electron microscopy (FFEM). We used SAXS/SANS and FFEM to measure the characteristic distance (db) of the phase. We also performed conductivity measurements on the phase, as well as membrane viscosity measurements by Fluorescence Lifetime Imaging Microscopy (FLIM). EXPERIMENTAL SECTION Materials. Sodium chloride, sodium phosphate monobasic and dibasic, and glycerol were purchased from Sigma-Aldrich (St. Louis, MO, USA). These reagents were used to make the phosphate buffered saline (PBS) with 100 mM NaCl, 50 mM sodium phosphate, 5% (v/v) glycerol, pH 7.5 buffer that was complemented with the desired detergent. The following non-ionic detergents were purchased from Affymetrix-Anatrace (Maumee, OH, USA); n-octyl-β-D-glucopyranoside (β-OG), n-dodecyl-β-Dmaltopyranoside (DDM), 4-cyclohexyl-1-butyl-β-D-maltoside (CYMAL-4), and 5-cyclohexyl-1-pentylβ-D-maltoside (CYMAL-5). Pentaethylene glycol monododecyl ether (C12E5) was obtained from Nikko Chemicals (Tokyo, Japan). 5-dodecanoylaminofluorescein (C12-AF), a fluorescent probe, was obtained from Life Technologies-Invitrogen (Grand Island, NY, USA). All phospholipids, were purchased from Avanti Polar Lipids, Alabama, USA, including the ones labelled with fluorescent probes, either 7Nitrobenz-2-oxa-1,3-diazol-4-yl (NBD) or carboxyfluorescein (CF). The molecular rotor C10-BODIPY was generously provided by Dr. M.K. Kuimova. Sample Preparation. Sponge phase samples were prepared by mixing the required amounts of surfactant (C12E5), co-surfactant (either β-OG, DDM, CYMAL-4, or CYMAL-5), and water (or buffer) into glass vials. This was done by mixing the surfactant with required amount of 3 solutions containing buffer and various co-surfactant concentrations (see details in Fig. S1), as a function of the desired surfactant volume fraction. The samples were vortexed, sonicated, and equilibrated at 20°C for at least 10 minutes prior to characterization of the resultant phases. Co-surfactants were dissolved in water or PBS at a stock concentration of 10% (w/v). To reconstitute fluorescent lipids ACS Paragon Plus Environment
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and surfactants into the bilayers, 1 μL of labeled lipid at about 1 mg/mL was deposited into the glass vial and dried under vacuum for at least one hour. The L3 phase was reconstituted over the dried lipid film before being vortexed for a few minutes. After vortexing, the sample rested for at least 5 minutes at room temperature. They could be kept for several weeks at 4°C and remained stable under these conditions. Phase Characterization. Preliminary characterization of samples was performed optically at 20°C, by placing them between crossed polarizers. The Lα (lamellar) phase is viscous, and optically anisotropic (exhibiting birefringence between crossed polarizers), while the L1 (micellar) and L3 (sponge) phases are both non-viscous, and optically isotropic. This enabled us to discriminate different domains of the phase diagram. Small Angle X-ray Scattering (SAXS). The set-up has been described previously.12,23,24 Briefly, the source of X-rays is a rotating anode producing the Cu Kα lines (1.54 Å) with a fine focus (1 mm × 0.1 mm). A gold-coated quartz mirror is used to collimate the beam. The linear detector, consisting of 512 channels, is positioned at a distance of 77 cm from the sample. The resolution of the set-up is 0.2 nm. Samples are loaded into sealed-glass capillaries and placed into a temperature-controlled holder. The SAXS experiments were performed at 23°C. Small Angle Neutron Scattering (SANS). For SANS experiments, the samples were prepared with D2O instead of water/buffer. SANS measurements were performed at the NIST Center for Neutron Research using both NG3 and NG7 instruments.25 The typical q range for these measurements was 0.003 to 0.4 Å-1. The incident neutron beam had a wavelength of 6 Å. The scattering data from each sample were corrected for background, detector efficiency, empty cell scattering, and sample transmission in the usual way, and intensities were placed on an absolute scale (cm-1) using beam flux measurements.26 The measurements were carried out at 20°C. Freeze Fracture Electron Microscopy (FFEM). The microstructures of the samples were studied by FFEM. A drop of the liquid sample was placed between two small copper holders and then quickly plunged into liquid nitrogen in order to freeze the sample. The copper holder, with the frozen sample, was transferred into a sample block of a JFD-9010 Freeze-Etching Equipment (Jeol, Japan). The sample was fractured with a knife cooled at -170°C in a vacuum chamber previously cooled at – 180°C. Right away after fracture, a solid replica of the sample was created by depositing a carbon/platinum film on the exposed surface. The replica was submerged in methanol in order to remove any remaining component of the original sample. The resulting solid replica was then collected on a copper grid, and was observed with a JEOL JEM-2010F transmission electron microscope. Fluorescence Recovery After Pattern Photobleaching (FRAPP). The FRAPP technique, described previously,10–12,27,28 was used to measure the diffusion coefficients of fluorescent probes such as C12AF or fluorescent lipids, in the isotropic phases, and discern the boundary of the L3 phase (see Figure 1). FRAPP experiments were performed at room temperature. The L3 phase was injected into capillary tubes of a 200 μm thickness (VitroCom, Mountain Lakes, New Jersey) which were sealed with wax in order to prevent evaporation. Shortly, two interfering laser beams were focused on a dot of approximately 250 μm of diameter creating a fringe pattern. The interfringe distances were tuned from i = 10 to 50 μm. The recovery curves were fitted by an exponential decay. In all experiments we observed a pure monoexponential recovery of fluorescence, indicating the diffusion of monodisperse objects. The fact that the diffusion was Brownian was verified by obtaining a linear variation of the characteristic recovery times τ plotted versus i2, and the D value was deduced from the slope of the linear regression following this equation : D = i2/4π2τ. The diffusion coefficients were measured with an accuracy of 10%. Alternatively, we measured the diffusion coefficients of fluorescent probes in the phases by FRAPP with a confocal microscope, following recently published guidelines.29 Conductivity Measurements. Measurements were carried out at 23°C. The home made cell walls are made of Teflon, except two facing walls, separated by 1 mm, composed of two gold-coated copper electrodes whose areas are both equal to 1.1 cm2. The cell is filled with a volume of 200 μL of sample. The set-up can be viewed as a parallel RC circuit. The circuit is fed by an external electric source with a resistance of R1 = 500 Ω to create a low-pass filter. The frequency dependent module |H| and phase φH of the transfer function H are measured. These values are used to calculate the conductivity: ACS Paragon Plus Environment
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1 1 cos 1 −1 + | |
where A = (1.2 ± 0.1) × 10-2 m, is a constant determined using several solutions of known conductivity. Conductivity of sponge phase samples was measured as a function of frequency, ranging from 5 Hz to 200 kHz. Viscosity measurements. We performed Fluorescence Lifetime Imaging (FLIM) experiments to measure the viscosity of bilayers,30 using a Molecular Rotor (MR), the meso-alkoxyphenyl-4,4difluoro-4-bora-3a,4a-diaza-s-indacene (or C10-BODIPY). The concentration of BODIPY was controlled so that the surfactant:dye ratio was kept between 800:1 and 5000:1. Two-photons excitation was done at 800 nm with a Vision II femtosecond laser (Coherent Inc.) on a Leica SP8 SMD system with an inverted DMI6000 microscope and a 63x NA 1.4 objective. The fluorescence decays, measured by time-correlated single-photon counting, were recorded using a band-pass filter (500-550 nm) on a hybrid detector (Hamamatsu) in non-descanned position. The image format was set to 512x512 pixels. We fitted the sum of the fluorescence decays for all the image pixels to a mono-exponential model with a characteristic decay time τ, using the Levenberg-Marquardt algorithm in TRI2 Software Version 2.7, provided by Gray Institute for Radiation Oncology and Biology.31 The viscosity was determined from the average of the measured fluorescence decay times using the Förster– Hoffmann equation :32 ln = a ln η + Cst. This linear relationship between ln τ and ln η was observed in solutions of viscosity ranging from 7.7 to 1140 mPa.s,33 which allowed for conversion of FLIM-measured decay times in bilayers to viscosity. RESULTS & DISCUSSION Surfactant Sponge Phases. Phase diagrams of ternary systems consisting of C12E5, buffer, and a cosurfactant were partially determined at 20°C. The investigated co-surfactants were β-OG, DDM, CYMAL-4 and CYMAL-5. Fig. 1 displays the phase diagram of C12E5, PBS, and β-OG. The other ternary systems, containing DDM, CYMAL-4, or CYMAL-5 instead of β-OG, exhibited similar diagrams (not shown). Those other co-surfactants were chosen because not all membrane proteins are solubilized by β-OG, or tend to be much better solubilized by other detergents. Each point reported in the phase diagram corresponds to a sample with a precise concentration of each component (C12E5, PBS, and co-surfactant). We first evaluated the optical properties of each sample and could therefore determine if the sample was monophasic, biphasic, isotropic, or non-isotropic. Approximated borders were drawn to separate the different phases. At this stage, only the lamellar phase (i.e. Lα phase) could be recognized, thanks to its characteristic birefringence. For all the investigated systems, an isotropic liquid phase was found in diagram regions ranging from 0.12 to 0.95 M C12E5 at 0.02 M of co-surfactant and from 0.12 to 0.71M C12E5 at 0.17 M of co-surfactant. The following results characterize these phase and are intended to give specific evidence that demonstrate the presence of a sponge phase (i.e. L3 phase). FRAPP was utilized to measure the diffusion coefficients of fluorescent probes in the isotropic regions of the phase diagram in order to distinguish the boundary of the L3 phase and the suspected L1 phase. We first verified that the diffusion coefficients did not depend on the location nor on the structure of the dye : we compared the mobility of the same lipid, labelled either on the headgroup or on the aliphatic chain, and they exhibit the same coefficient diffusion. Then, we compared the mobility of another lipid, labelled with 2 differents dyes, NBD and CF : they also have the same mobility (Table 1). Fluorescent probe 14:0-6:0-PE-NBD 14:0-PE-NBD (DMPE) 18:1-PE-NBD (DOPE) 18:1-PE-CF (DOPE) C12-AF
Diffusion coefficient (μm2/s) 4.2 ± 0.2 4.1 ± 0.1 5.2 ± 0.4 5.5 ± 0.1 3.7 ± 0.3
Dye location Aliphatic chain Headgroup Headgroup Headgroup Headgroup
Table 1. Diffusion coefficients of labelled lipids and surfactant in the L3 phase as measured by FRAPP. Neither the dye location on the diffusing
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molecule nor the molecular structure of the dye influences significantly the mobility of the diffusing specie. Diffusion of C12-AF and other lipids in the L3 phase was significantly slower than that in the micellar phase (L1), thus enabling us to discern the boundary of the two isotropic phases. Subsequently we focused our investigation on a dilution line with a constant co-surfactant to C12E5 surfactant molar ratio of 1/9 (as represented by the line in Fig. 1).
Figure 1. Partial, approximate phase diagrams of the ternary systems consisting of C12E5, PBS, and β-OG. Other ternanry systems formed by DDM / CYMAL-4 / CYMAL-5 instead of β-OG have similar diagrams. Each dot corresponds to an experimental sample that was investigated at 20°C. The lines drawn on the diagram represent approximate boundaries between the phases. The black line in the L3 region of the diagram corresponds to the dilution line that was investigated with the following methods; SAXS, SANS, FFEM, and FRAPP. SAXS and SANS were acquired for the phase under study, prepared in the presence of various cosurfactants. All scattering curves exhibited a broad peak whose shape and Bragg position, qB, is identical regardless of the co-surfactant present (as shown in Fig. 2). The peak position depended however on the volume fraction, Ф, of both surfactant and co-surfactant (see Fig. 3).
Figure 2. Small Angle Neutron Scattering (SANS) intensity curves of the ternary systems consisting of C12E5, PBS, and one of the following coACS Paragon Plus Environment
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surfactants; β-OG (continuous line), CYMAL-4 (dashed line), and CYMAL-5 (dash dotted line), and DDM (dotted line) at the co-surfactant/C12E5 concentration of 0.08. The scattering curves are respectively translated along the y-axis by a value of 2, otherwise the data will almost perfectly overlap, indicating the change in the co-surfactant does not alter the properties of the sponge phase. The position of the Bragg peak (qmax) is related to the characteristic distance (db) via the following relation: db = 2π/qmax.
Figure 3. The continuous lines represent SANS intensity curves corresponding to bilayers composed of co-surfactant β-OG and surfactant C12E5. The co-surfactant to surfactant ratio is equal to 0.08 for all, while the bilayer volume fractions are equal to 0.08, 0.16, 0.22, 0.27, and 0.32 respectively from the left to the right curve. The dashed lines are the best fits of the data according to the model of Lei et al. 34. These results suggest a phase that locally displays an ordered structure characterized by the length dB = 2π/qB . The SAXS/SANS scattered intensity I(q) can be expressed as I(q) = S(q)F2(q), where S(q) and F(q) are the structure and form factors, respectively. Models have been developed to calculate these factors for different phases such as micellar, hexagonal, lamellar, or sponge phases. In our case, an accurate fit of our experimental scattering data could be achieved using the model of Lei et al.,34 which has been shown to adequately fit sponge phases. In this model, the structure factor is written as: arctan 2 = 1 + + 2 1 + || −
where ξ1 is a correlation length associated with the in-out order parameter, dB = 2π/qB is the average cell-cell length (or pore size) with a cell-cell correlation length ξ2, C1 and C2 are two constants. The form factor is the one of a randomly oriented disk of radius R and (contrast) thickness δh (corresponding in our case to the hydrophobic thickness of the surfactant – co-surfactant bilayer): % sin ) 2 # √1 − % = !" & ' % * +% 3 √1 − % ) 2
where J1 is the first-order Bessel function, x = cos(θ) where θ is the angle between the normal of the disk plane and the scattering vector q. Figure 3 shows an example of the fits of the experimental data with the above model along the dilution line. Regardless of the co-surfactant, fits of all ACS Paragon Plus Environment
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scattering data at room temperature, gave a value R = 21 ± 2 Å that did not vary significantly with bilayer volume fraction Ф. For all co-surfactants, we observed that δh values vary from 25 ± 3 Å at Ф = 0.08 down to 17 ± 2 at Ф = 0.22, then δh keeps this value at larger volume fractions Ф. If we set the hydrophobic thickness to δh =17 Å for Ф < 0.22 we cannot obtain accurate fits of the scattering. The hydrophobic thickness of the C12E5 bilayer has been estimated to be equal to 17 ± 2Å.35 Since this value is very close to the one found for δh at Ф ≥ 0.22, we thus believe that the addition of β-OG does not change significantly the bilayer thickness of C12E5. In addition this suggests that the change in δh observed in the fit for Ф < 0.22 is unrealistic and might be due to the fact that the model is inadequate at low membrane fraction volumes. Other parameters depend slightly on the bilayer (i.e. surfactant and co-surfactant) volume fraction Ф. ξ1 and ξ2 vary approximately from 160 ± 10 Å and 20 ± 1 Å, respectively, at Ф = 0.08 to 70 ± 10 Å and 40 ± 1 Å, respectively, at Ф = 0.32. The good agreement of the fit with our experimental data is the first piece of evidence that the observed isotropic phase is a sponge phase. We have reported the values of the characteristic length dB extracted from the SAXS and SANS intensity curves in Fig. 4 as a function of the surfactant-co-surfactant volume fraction, Ф. We see that the dB value decreases from 290 to 60 Å as Ф increases from 0.03 to 0.44 (i.e. when water content diminishes). The good agreement between the values obtained from SANS and SAXS suggests that D2O did not affect the phase properties. Moreover when doing SAXS measurements, we measured the same dB value when the sample was made either with PBS or with water. Thus it appears that solvent properties do not alter this isotropic phase.
Figure 4. The characteristic distance (db) as a function of bilayer volume fraction (Φ) of L3 phases consisting of C12E5, β-OG and PBS, as measured by SANS (○), SAXS (●), and FFEM (□). Freeze-fracture electron micrographs were obtained along the same dilution line (see Fig. 1) but only for samples containing β-OG as co-surfactant. An example of such micrograph is given in Fig. 5A at a high surfactant-co-surfactant volume fraction (i.e. Ф = 0.33). Its appearance is similar to the one observed by Hoffmann et al.36 from micrographs made from sponge phases. In Fig. 5A, we observed domains separated by gaps. These domains should represent bilayer surfaces, as freeze fracture tends to propagate along these surfaces. The presence of gaps indicates that these surfaces curve at some point to create a complex structure. The gap size represents the solvent space between two neighbor bilayers. From micrographs, we could estimate (using Scion Image software) the characteristic size dFFEM of these gaps with a precision of 0.5 nm. The gap size distribution was plotted into a histogram for each sample (see Fig. 5B). A Gaussian fit of these histograms gave average sizes that are reported in Fig. 4. The size values, dFFEM, obtained by FFEM satisfactorily correlate to the characteristic size, dB, obtained from SAXS and SANS. This is correlated with the study of Strey et al.37 showing that the characteristic distance (dFFEM) obtained by FFEM on their sponge phase was in agreement with that obtained by SANS on the same sponge phase. ACS Paragon Plus Environment
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Figure 5. Freeze Fracture Electron Microscopy (FFEM) image of A) an L3 phase composed of C12E5, β-OG and PBS, at the membrane volume fraction (Φ) of 0.33, and B) a corresponding histogram used to calculate the mean characteristic distance (db). The insert in A shows channels that were sliced through and were used to measure the characteristic distance between the bilayers. The distribution of the characteristic distance size was plotted and fit with a Gaussian curve in B. The mean characteristic distance (db) obtained from the micrograph was found to be in agreement with that acquired by SAXS on the same sample. When diluted with its solvent, microemulsion topology induces a characteristic dilution law for the length dB when plotted as a function of Ф. S.T. Hyde has derived dilution laws for the various sponge phase topologies.38 In his approach, the shape parameter, s, of the bilayer is defined as:
/) 4 3 where M and K are the Gaussian and mean curvatures, respectively, of the bilayer that possesses a thickness δ. It appears that only those models in which the sponge phase is symmetric are compatible with our experimental data. During the dilution, the sponge phase can exhibit either a constant bilayer thickness or a constant surfactant surface area. Two models succeed to accurately fit our data. The first one describes a sponge phase of an arbitrary shape (i.e. the value s is unknown, but it should lies between 0.5 and 2/3 ≈ 0.66 for a homogeneous sponge phase) whose bilayer mean curvature is zero and bilayer thickness remains constant upon dilution. In this model, the variation of dB values obeys the following swelling relation: - = 1 + .) +
) 3 1 2 5 +
The fit of the data by the above equation (dashed line in Fig. 4) gave s = 0.51 ± 0.02 and δ = 38 ± 2 Å. The resulting value for the shape factor indicates that the system is a sponge phase but the bilayer thickness value exceeds the values reported for C12E5 bilayers.12,35 The second successful model describes a sponge phase that keeps a constant surfactant surface area upon dilution. The dilution law is written as: 1 = 5 + 6 + 57 + 67 6
where c1 and c3 are two constants. The fit of our data (solid line in Fig. 4) gives c1 = 9.7 ± 0.1 Å and c3 = 57800 ± 4200 Å3. We deduced from the dilution behavior that our sponge phase is symmetrical and it seems that it is the surfactant surface area that remains constant during the dilution rather than the bilayer thickness. Finally, we measured the conductance of samples located on the dilution line with a volume fraction Ф of surfactant and co-surfactant ranging from 0.03 to 0.33. Ion mobility should be hindered by the ACS Paragon Plus Environment
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bilayer network, thus the electrical conductivity of the phase would depend on the bilayer topology. The topology was evaluated using a reduced conductivity or obstruction factor S defined as :20 9:;3< = 7 3=>?
