PolarâNonpolar Interfaces of Inverse Bicontinuous Cubic Phases in

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Polar-nonpolar interfaces of inverse bicontinuous cubic phases in phytantriol/water system are parallel to triply periodic minimal surfaces. Toshihiko Oka, Noboru Ohta, and Stephen T Hyde Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b03320 • Publication Date (Web): 14 Nov 2018 Downloaded from http://pubs.acs.org on November 19, 2018

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Polar-nonpolar interfaces of inverse bicontinuous cubic phases in phytantriol/water system are parallel to triply periodic minimal surfaces.

Toshihiko Oka1,2,*, Noboru Ohta3 and Stephen Hyde4 1Department

of Physics, Faculty of Science and 2Nanomaterials Research Division, Research

Institute of Electronics, Shizuoka University, Shizuoka 422-8529, Japan 3SPring-8/JASRI,

4Department

1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo 679-5198, Japan

of Applied Mathematics, Research School of Physics and Engineering, Australian

National University, Canberra ACT 2601, Australia

ACS Paragon Plus Environment

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Abstract We investigated the two distinct lyotropic liquid crystal inverse bicontinuous cubic phases of phytantriol/water mixtures by small angle X-ray crystallography of single crystal regions. Reconstructed electron density maps revealed hydrophilic head and hydrophobic tail regions of the phytantriol bilayer membranes and water regions. The bilayer membranes are shown to be located on the D and Gyroid triply periodic minimal surfaces. In order to investigate the structures of the polar-nonpolar interfaces, we optimized two models: a parallel surface model and a constant mean curvature surface model. The parallel surface model agreed well with the X-ray data, and the R factors, which show the degree of agreement between those structural models and data, were less than 0.04. In stark contrast, the constant mean curvature surface model deviated significantly from the data, and the R factors were around 0.15. We therefore conclude that the polar-nonpolar interface of the inverse bicontinuous cubic phase of the phytantriol/water system is close to a parallel surface to a triply periodic minimal surface.


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Introduction A triply periodic minimal surface (TPMS) extends infinitely in all directions and separates space into two interwoven labyrinths1 (Figure 1). At any point on TPMS, the mean curvature (H) is zero, and the Gaussian curvature (K) is negative, except at isolated flat points. The Schwarz D surface and Schoen’s Gyroid (G) surface are representative of TPMS, and have cubic symmetries with the crystallographic space groups 𝑃𝑛3𝑚 and 𝐼𝑎3𝑑, respectively. Both surfaces have been observed in the inverse bicontinuous cubic (QII) phases of lyotropic liquid crystals with so-called QIID and QIIG phases.2–4 In the process of self-assembly of amphiphilic molecules, it causes microphase separation to form these structures. Similar structures have been found in various systems, such as diblock copolymers5, and living systems.6,7 And the structure in which the polarnonpolar interface is large relative to the volume is used in various fields, membrane protein crystallization8, drug delivery9, and synthesis of nanoporous silica.10 A pioneering X-ray diffraction study by Luzzati’s group showed bilayer membranes of amphiphilic molecules are located on TPMS in the QII phases.11,12 On the other hand, finer structural details, particularly the structures of polar-nonpolar interfaces in the bilayer membranes, remain unknown. Clarifying the detailed structure of the bilayer membrane leads to understanding the principle of structure formation in the QII phases. It is also important in creating molecules that form a structure similar to TPMS by self-assembly. Considering what is important in the formation of the QII phase structure, two models have been proposed to represent the interface structure: the parallel surface (PS) model13,14 and the constant mean curvature surface (CMCS) model15. In the PS model, the polar-nonpolar interface in the bilayer membrane is parallel to the TPMS: that is, the distances between the interfaces and TPMS are constant everywhere. On the other hand, in the CMCS model, the polar-nonpolar interface has constant H. Consequently, the distance between the interface and


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TPMS varies, roughly depending on K: The distance is a minimum at the flat points on the TPMS where K = 0, and a maximum where K is the most negative.16 Until now, direct observations of the interface structure have been unavailable, so the validity of the models has remained largely unknown.17 Theoretical evaluation of relative energies of a curved membrane filled with ideal polymeric chains supports the PS model, since the stretching energy of the hydrocarbon chain is larger than the bending energy of the membrane introducing constant membrane width.18 On the other hand, another theoretical result supports the CMCS model, because the CMCS model could explain the phase transition order of the QII phases better than the PS model in their calculation when the water fraction increased.19,20 To observe the structure of the QII phases directly, Luzzati and coworkers developed a method of structural reconstruction using X-ray powder diffraction.11,12 However, it has been used rarely to reconstruct spatial distribution of electron density. There are two obstacles to the reconstruction: the recovery of the lost phase information and the acquisition of accurate intensity data from overlapped X-ray powder diffraction peaks. However, the analysis based on the electron density is very important as many examples show21–24. Recently, one of us developed an improved method that we call small-angle X-ray crystallography, in which a single crystal region of the QII phase was used.25 We showed that the accuracy of X-ray diffraction measurements in the single crystal region was good and that the phase could be determined relatively easily. Here, we use the same method to analyze the QII phases of phytantriol/water system, and compared experimental data with the PS and CMCS models.

Materials and Methods


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Phytantriol (purity > 95%, mixture of isomers) (Tokyo Chemical Industry Co., Tokyo, Japan) was used without further purification. The QII phases of phytantriol were single crystallized in borosilicate glass capillaries (Mark-tube, Hilgenberg GmbH, Germany) with an inner diameter of 0.5 mm. About 1-2 μl of distilled water was introduced into the capillary, then about 4-6 mg of phytantriol was transferred into the capillary and it was centrifuged at an acceleration of 1500 g for 180 s. The open end of the capillary was sealed with a gas torch. The capillary was incubated at 25°C for ca. one month. X-ray diffraction were measured at the small angle X-ray scattering beamline BL40B2 of the synchrotron facility SPring-8 (Hyogo, Japan). The wavelength of X-rays was set to 0.1000 nm. Measurements were conducted similarly to those described in previous papers25– 27.

X-ray diffraction patterns were measured using a two-dimensional detector (PILATUS

100K, Dectris, Baden, Switzerland). Measurements were performed at room temperature (26°C). X-ray diffraction from a capillary sample was measured sequentially during a rotation of the sample from 0 to 180°. X-ray exposure time during the rotation was 180 s and the frame number of X-ray measurement was 1800. Thus, exposure time and rotation angle per frame was 0.1 s and 0.1º, respectively. Aluminum plates of 0.5 mm and 1.5 mm thickness were used as an X-ray attenuator to avoid sample damage and adjust intensity. The data set with the thick attenuator were used to make corrections when the intensities in the pixels measured with the thin attenuator exceeded 2×104 counts. The detector was placed on a z-axis stage and images at different z-positions were merged into one large image. X-ray diffraction images were


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indexed and integrated with the XDS program package.28 The space groups of the QII phases were used as already known in the XDS processes. The electron density in a crystal is obtained by Fourier transformation of the structure factors F(h): 𝜌(𝑟) = ∑𝒉𝑭(𝒉)exp ( ― 2𝜋𝑖 𝒓 ⋅ 𝒉) , where r is a vector in real space, and h = (h k l) is a Miller index. The structure factor can be written with its amplitude F and phase α: 𝑭(𝒉) = |𝑭(𝒉)|exp (𝑖 𝛼(𝒉)) = 𝐹(𝒉)exp (𝑖 𝛼(𝒉)). The diffraction intensity is the square of the structure factor: 𝐼(𝒉) = |𝑭(𝒉)|2 = (𝐹(𝒉))2. The normalized dimensionless density ρ(r) 11,25,29 is used as the electron density in this paper. Two models were compared with the X-ray data. One was based on PS model, the other was CMCS model. The method of creating the PS model was similar to the method of the previous paper.25 Electron density of the both models were constructed numerically in unit cells with 64×64×64 voxels. As the first step, TPMS and CMCS were created using Surface Evolver30. To generate the CMCS, the method published by Shearman et al. was used31. Since the volume fractions of hydrophobic regions were required in advance to calculate the CMCS, those were fixed at 0.60 in the QIID and 0.66 in the QIIG. We also tested other values of volume factions (see Results). As the second step, the minimum distance from each voxel to TPMS or CMCS was calculated. In the PS model, the distance to the TPMS was calculated simply at each voxel. On the other hand, in the CMCS model, the distance to the CMCS was calculated and distances to all points within the volume containing the TPMS (viz. hydrophobic region) were assigned as negative to distinguish the points on either side of the CMCS. Since the generated TPMS or


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CMCS were not large enough to cover the whole unit cell, the distances were calculated using the symmetry in the unit cell region without the TPMS or CMCS. As the third step, the electron densities of the models at each voxel was calculated. In both models, the middle surface of phytantriol bilayer is located on the TPMS. Moving outwards from the TPMS, one passes first through the hydrophobic tail region, then the hydrophilic head region of phytantriol and lastly the water region. We, therefore, simplified the electron density distribution into three regions. The electron densities of the head and tail regions are the highest and lowest. The electron density distribution of the Lα phase lipid bilayer membranes has been studied in detail by others.21 In the lipid bilayer membrane, the lipid head region has a high electron density while the tail region has a low electron density. Within the lipid tail region, the electron density is the lowest at the terminal methyl group.32,33 Here, our model divides the bilayer membrane of phytantriol into just two regions: the head and tail regions. In the phytantriol molecule, the nonpolar tail is an isoprene chain, containing branched methyl groups. Therefore, we assume that the influence of those methyl groups, with low electron density, is averaged throughout the region of the chains. Cross sectional electron density are therefore assumed to changes stepwise, according to the distance from the TPMS or CMCS (Figure S2). The step function of a distance,

x, for the PS model was 𝜌𝑠𝑡𝑒𝑝𝑃𝑆(𝑥) = (𝜌1 ― 𝜌2)𝑈(𝑤1 ―𝑥) +𝜌2𝑈(𝑤1 + 𝑤2 ―𝑥) , and for the CMCS model was 𝜌𝑠𝑡𝑒𝑝𝐶𝑀𝐶𝑆(𝑥) = (𝜌1 ― 𝜌2)𝑈( ― 𝑥) +𝜌2𝑈(𝑤2 ―𝑥), where U(x) = 1 (x >= 0) or 0 (x < 0), ρ1 and ρ2 are densities, and w1 and w2 are widths at hydrophobic and hydrophilic part of a phytantriol molecule, respectively. The densities of water regions were set to 0 in the functions. The voxel density of each model was Fourier transformed, and multiplied in


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reciprocal space with an isotropic three-dimensional Gaussian function, 𝑓𝐺𝑎𝑢𝑠𝑠(𝒉) = exp( ― 2 𝜋2𝜎2ℎ2) as a Debye-Waller factor.25 According to the convolution theorem, the multiplication in the reciprocal space corresponds to the convolution in real space. The calculated voxels correspond to the structure factors of the both models, FPS(h) and FCMCS(h). Thus, we derived effective structure factors for both models assuming step functions for density, bound by relevant surfaces, smeared by the Gaussian function. As the final step, we optimized the structural factors of the models into amplitudes of structure factors obtained from experiments, |Fexp(h)|, to give the best structure fits our data. The values ∆𝐹2 = ∑𝒉(|𝑭𝑒𝑥𝑝(𝒉)| ― |𝑭𝑚𝑜𝑑𝑒𝑙(𝒉)|)2 were minimized in the optimization. Parameters, ρ1, ρ2, w1, w2 and σ in the PS model, and ρ1, ρ2, w2 and σ in the CMCS model were optimized by the minimization. Since the volume fraction of the hydrophobic region was fixed in the CMCS model, the parameter w1 was not included in the model. We used the Nelder-Mead method34, which allowed optimizations without using derivatives. In the method, a calculation cycle was repeated until convergence while changing parameters gradually according to the procedure of the Nelder-Mead method; the cycle was as follows, 1) calculation of electron density using the parameters, 2) calculation of structure factors from the density, 3) calculation of ∆𝐹2. The optimization was performed 10 times or more with changing the initial value, and the parameters with the smallest ΔF2 were adopted as the optimized parameters. After the optimizations, the R factors in crystallography were calculated, which are defined as 𝑅 = ∑𝒉(|𝑭𝑒𝑥𝑝(𝒉)| ― 𝑘𝑠𝑐𝑎𝑙𝑒|𝑭𝑚𝑜𝑑𝑒𝑙(𝒉)|)/| 𝑭𝑒𝑥𝑝(𝒉)|. The structure factor of the model Fmodel(h) contains a phase factor, since Fmodel(h)=


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|Fmodel(h)|exp(iα(h)). The phase α of the model was calculated in the Fourier transformation of the electron density. The obtained phase sets were the same from the PS and CMCS model. Electron

density

maps

were

calculated

with

|Fexp(h)|

and

α(h)

as

𝜌𝑒𝑥𝑝(𝒓) =

∑ℎ|𝑭𝑒𝑥𝑝(𝒉)|exp (𝑖𝛼(𝒉))exp ( ― 2𝜋𝑖𝒓 ⋅ 𝒉) . The half-width in Figure 6 was defined as the minimum distance from a point on the TPMS primitive patches to an isodensity surface that satisfied ρ(r) = constant. The 11×11 points on the primitive patches of TPMSs were created using polynomial approximations of TPMS.25,35 We used isodensity values of 0.449 for QIID, and 0.586 for QIIG. The volume fractions of the regions less than the isodensity values are 0.57 for QIID and 0.66 for QIIG, which correspond to the volume fractions of hydrophobic regions. Mathematica 11.2 (Wolfram Research, Champaign, IL, USA) was used to construct and optimize the models.

Results and Discussion To make single crystal regions, phytantriol (Figure 2a) and water were placed adjacent to each other in a capillary tube at 25ºC. They mixed gradually due to diffusion. After a few days, QII, Lα, and L2 phases of phytantriol were observed in order from the edge in contact with water (Figure 2b and 2c). Lα phase exhibits birefringence, but QII and L2 phases do not. The order of phases was consistent with that expected from the phase diagram36: those with the highest volume fraction of water were observed in regions closest to water. Since the QIID phase forms at higher volume fraction of water than QIIG phase, the QIID lies closer to the water end of the capillary in


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the QII region. The Lα phase was observed next to the QII phase. The L2 phase, which exhibits no birefringence, seemed to be mixed with Lc phase with birefringence under the lowest water condition. Each phase was also identified using X-ray diffraction (data not shown). As time passed, the region of the QII phases grew. We measured X-ray diffraction of the QIID and QIIG phases of the sample after one-month incubation using the rotating crystal method (Figure S1). The diffraction images were processed and the intensities were determined (Figure 3 and Table S1). The X-ray diffraction intensity is equal to the square of the structure factor, but since the phase of the structure factor is indeterminable from X-ray data, electron densities cannot be calculated directly. Therefore, we recovered the lost phases by tuning them to give the best fit to PS and CMCS models as described in the previous section. This process, along with the phase recovery, also serves as a determination of which model is appropriate for describing the QII phase. Figure 3 shows the results of optimizing each model with experimental data. The amplitudes of structure factors in the PS model matches well the experimental data, whereas the deviations are larger in the CMCS model and significant at high angles. Optimized parameters are shown in Table 1. The values of ΔF2 and R, which are indicators of optimization, are considerably smaller in the PS model than in the CMCS model. The R factors of the PS model were less than 0.04, while that of the CMCS were around 0.15; the low R factor of the PS model indicates that this model fits experimental data well. Note that in the PS model the volume fractions of hydrophobic region depended on the width parameter, w1, which is adjusted to finish the model fit to experimental data. After optimizations, the volume fraction were equal to 0.57 in the QIID and 0.66 in the QIIG. These are close to the expected values (Supporting Information) and the fixed values of the CMCS model (0.60 in the QIID and 0.66 in the QIIG). To check the validity of our assumed volume fraction in the CMCS model, we also optimized the


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model with volume fractions between 0.50 and 0.65 in the QIID and between 0.52 and 0.72 in the QIIG. But we found R factors larger than 0.1 regardless of volume fractions. In the PS model, the molecular lengths of phytantriol were obtained from the optimized parameters listed in Table 1; they are equal to 1.36 nm in the both phases, close to the expected lengths, 1.37 nm in the QIID phase and 1.39 nm in the QIIG phase (Supporting Information). The phases of the structural factors, calculated from Fourier transformations of the model electron densities, were the same in the PS and CMCS models (Table S1). Electron densities of the QIID and QIIG phases were then calculated in real space using those phases of the models and experimental amplitudes of the structure factors. Figure 4a and 4b shows the resulting sliced twodimensional electron density maps. High electron density regions of phytantriol head part and low regions of tail are clearly distinguished, and intermediate electron density of water regions are surrounded by high electron density of head regions. Figure 4c and 4d shows the three-dimensional electron density maps. As expected for the QII phase, the tail regions of phytantriol with low electron densities form bilayer membranes centered on the TPMSs. The electron density distributions in the direction perpendicular to the TPMSs are shown in Figure 5. The densities expected from the PS model almost overlap with the experimental data. On the other hand, deviations from the experimental data are large in the CMCS model. In the 〈100〉 direction, the widths of the low-electron density regions of the CMCS model are wider than those of the experimental data in two phases, while those are narrower in the 〈111〉 direction. The straight lines in the 〈111〉 direction shown in Figure 1 pass through the flat points (where K = 0) on the TPMS, whereas those in the 〈100〉 direction pass through the points where K are minimum. In the CMCS model, the hydrophobic region is the thickest at the points where K of TPMS is the most negative, and the thinnest where K = 0.16 As shown in Figure 5, it seems that the density


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distributions of the CMCS model deviate from those of the experimental data by the thickness change. We examined the widths of the low electron density regions in detail. We define a halfwidth as the minimum distance from a point on the TPMS to an isodensity surface with ρ(r) = constant. The isodensitiy of QIID was 0.449 and that of QIIG was 0.586. Figure 6 shows the K dependencies of the half-widths. The PS model shows good agreement with experimental data in the QIID and QIIG phases. However, the half-widths of the PS model vary depending on K at first sight. This is due to the effect of the convolution of the Gaussian function and the PS model. Even if the membrane thickness is constant in the PS model, the apparent thickness in the electron density map changes depending on the K once Gaussian fluctuations are present (Figure S3). On the other hand, half-widths of the CMCS model, which also have K dependence, deviate largely from those of the experimental data: the half-widths are larger than the experimental ones when K is close to the most negative, and smaller when K is close to 0. In the CMCS model, the thickness of the hydrophobic region is the thinnest at the point where K = 0 on TPMS, and it is the thickest at the point where K is the most negative.37 Figure 6 indicate that these variations cause significant deviations from experimental data, and resulted large R factors of the CMCS model. We therefore conclude that the both QIID and QIIG phases of phytantriol have structures which are better described within the PS model, rather than the CMCS model.

The structure of the polar-nonpolar interface of the lyotropic liquid crystal QII phase is mainly determined by the sum of the following two free energies: bending energy and stretching energy of hydrocarbon chain.15,18 The bending energy of one amphiphile monolayer is most often described by the Canham-Helfrich Hamiltonian38


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𝐸𝑏 = ∫d𝐴{2𝜅(𝐻 ― 𝑐0)2 + 𝜅𝐾} where c0 is spontaneous curvature, κ and 𝜅 are bending rigidity and saddle-splay modulus. The stretching energy can be assumed to be harmonic about the average chain length l0, 𝐸s = ∫𝑑𝐴 𝑘𝑠(𝑙 ― 𝑙0)2 where l and ks are the local chain length and the stretching modulus. If the structure of the polar-nonpolar interface is close to the CMCS, Eb >> Es, and if it is close to PS, Eb l, so

Es > Eb is satisfied for amphiphiles with polymeric hydrophobic chains. The CMCS is therefore likely not a good approximation for the hydrophobic-hydrophilic interface. Indeed, a selfconsistent field study of diblock copolymers with longer chain lengths has also concluded that the CMCS is insufficient to describe the interface structure.39 It is therefore reasonable to infer


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that the interfacial structure of the QII phases of lipids with long hydrocarbon chains would, like our phytantriol system, be close to PS rather than CMCS. It is, however, necessary to verify experimentally these theoretical expectations. The relative dominance of Es over Eb in the QII phases also explains why the type II hexagonal phase is not observed in the phytantriol/water system at around room temperature.36 The molecular length is not constant in the type II hexagonal phase, unless the interfaces are facetted which requires large bending energy.40,41 The monoolein/water system, with a similar phase diagram4, may be governed by similar physics as well. The absence of the type II hexagonal phase also causes an interesting phenomenon. As the water fraction increases, the phase transition is observed in the order of the Lα, QIIG, and QIID phases in phytantriol36 and monoolein.4 Schwarz and Gommper had explained the order theoretically.18 The electron density maps obtained by X-ray crystallography represent the temporal and spatial average structures within the crystal. Since Gaussian smearing, presumably due to spatial and temporal fluctuations, modulates the width, it is difficult to infer a “frozen” structure from the isoelectric density surface (Figure 6 and S3). The isodensity surface is therefore not an accurate guide to distinguish different structure models, due to fluctuations. In particular, there is no way to determine the width of the hydrophobic region directly from electron density map. Similarly, it is also impossible to calculate H of the interface directly from the electron density map. However, since the experimental data are in good agreement with the PS model, the interfacial curvatures H and K of polar-nonpolar interfaces resemble those of the PS model allowing us to infer from the chain length indirectly. The H and K values


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at PS which is a distance l from TPMS, assumed to have the optimal values of 1.02 nm and 1.01 nm in the QIID and QIIG phases, are𝐻𝑙 = ―𝑙𝐾/(1 + 𝑙2𝐾) and 𝐾𝑙 = 𝐾/(1 + 𝑙2𝐾), respectively.1 X-ray single crystal structure analyses are used to study the structure of materials in many fields. However, most analyses invoke assumed structural models to deduce an ultimate result. That is, the structural model is indispensable for interpretation of experimental data in X-ray crystallography. In this work, we too are bound by this limitation, though our degrees of freedom are few: we assume only region thicknesses and Gaussian fluctuations of the bilayer membrane. In particular, we used an isotropic Gaussian function as a Debye-Waller factor without position dependence, assuming that the magnitude of fluctuation is the same at any point on the membrane. The model agreed well with the experimental data. Since one membrane is continuously connected throughout a single crystal region, it is considered that the position-dependent fluctuations are small in our experiments. In contrast, a previous study of monoolein/water system found a Debye-Waller factor to be dependent on K.25 In that case, membrane fluctuations were large probably due to the presence of organic solvent. In the work reported here, deviations from X-ray data were large in CMCS model using positionindependent Gaussian fluctuation. Even if a position-dependent Gaussian function is introduced, X-ray data cannot be explained using the CMCS model. The electron density distributions at the point of K = 0 are shown in Figure 5b and 5d. Compared with the electron density distribution using X-ray data, the negative peaks in the CMCS model are shallower


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and narrower than X-ray data. If the width of the convolution Gaussian function is adjusted, it is impossible to deepen and widen the negative peak at the same time.

Conclusion Deeper understanding of bicontinuous mesophase structures have until now been hampered by ignorance of the structural details of the polar-nonpolar interfaces. Two distinct models, PS and CMCS models, have been assumed, with little experimental support for either. We show here that small angle X-ray crystallography25 is a useful new method in the structural research of the QII phases, and clarified the structure of the phytantriol bilayer and the water region via the electron density map. Since the both PS and CMCS models have been widely used to explain the structure of the polar-nonpolar interface of the lyotropic liquid crystal QII phase, we optimized our X-ray data to the both models. The phases of the structural factor were the same for both models, but there were large differences in the amplitudes. The PS model showed good agreement with the X-ray data, but the deviations were significant in the CMCS model. In the PS model, the crystallographic R factors were as low as 0.04, whereas in the CMCS these were around 0.15. Therefore the structure of polar-nonpolar interfaces of the QII phases in the phytantriol/water system can be explained well by the PS model, and not by the CMCS model. This is likely due to a high energy cost for chain stretching, compared with bending energy. Their relative cost is possibly due to the short, branched nature of the hydrocarbon chain of phytantriol. The Q phases of other amphipathic molecules, including


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lipids, may have similar structures. However future experiments like those reported here are necessary, due to variable stretching and bending energy contributions in different molecules.


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Table 1. Optimized parameters of model functions.

QIID

QIIG

model

ρ1

ρ2

w1**

w2**

σ**

∆𝐹2

R factor

PS

-10.9

27.7

0.158

0.0519

0.0820

0.00058

0.037

CMCS*

-11.1

37.0

-

0.0301

0.0723

0.0258

0.144

PS

-7.95

29.4

0.115

0.0409

0.0540

0.00027

0.032

CMCS*

-11.0

31.3

-

0.0295

0.0519

0.0235

0.147

*Volume fractions of hydrophobic regions are 0.60 in the QIID and 0.66 in the QIIG. **An edge length of a unit cell is 1.


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FIGURES

Figure 1. Two of triply periodic minimal surfaces (TPMS), (a) Schwarz D surface, and (b) Schoen’s Gyroid (G) surface. Colors on the TPMSs indicate Gaussian curvature; red regions are close to 0 while blue regions are close to minimum. Two straight lines are perpendicular to the TPMS at the point of intersection. Black lines, which are parallel to one of the 〈111〉 directions, intersect the TPMSs at the flat points where Gaussian curvatures are 0. Red lines, which are parallel to one of the 〈100〉 directions, intersect at maximally negative Gaussian curvatures. All points with Gaussian curvature of 0 on the TPMS are crystallographically symmetric, and that with maximally negative are the same.


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Figure 2. Molecular structure of phytantriol (a) and close-up of a capillary sample after 5 days (b-c). The sample observed under a normal light condition (b). The sample placed between two orthogonal polarizing plates (c). The diameter of the capillary was about 0.25 mm.


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Figure 3. Amplitudes of structure factors, |𝐹ℎ𝑘𝑙|, of (a) QIID, and (b) QIIG. The (h k l) is a Miller index. Black circles are amplitudes determined by X-ray diffraction measurements. Orange pluses and skyblue crosses are amplitudes calculated from parallel surface and constant mean curvature surface models, respectively.


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Figure 4. Two-dimensional electron density maps sliced at x = 0, 0.1a, 0.2a, 0.3a, 0.4a, and 0.5a (from top left to right and from down left to right, where a is the lattice constant) in unit cells of (a) QIID, and (b) QIIG phases. Three-dimensional electron densities in the unit cells of (c) QIID, and (d) QIIG. Regions with density less than -1 and larger than 1 are shown.


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Figure 5. Electron density distributions on straight lines perpendicular to the TPMSs at the point of intersection for (a-b) QIID, and (c-d) QIIG. The direction of the straight line is 〈100〉 in (a) and (c), and 〈111〉 in (b) and (d). The straight lines correspond to those in Figure 1. The horizontal axis is the distance from TPMS. Black solid lines are density distributions from experimental data, while orange dashed and skyblue dotted lines are parallel surface and constant mean curvature surface models, respectively.


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Figure 6. Gaussian curvature (K) dependence of the half-width for (a) QIID, and (b) QIIG. The half-width was defined as the distance from a point on the TPMS to an isodensity surface. The isodensitiy of QIID was 0.449 and that of QIIG was 0.586. Black filled circles are those of experimental data, while orange pluses and skyblue crosses are parallel surface and constant mean curvature surface models, respectively. The K values correspond to that on the TPMS.


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ASSOCIATED CONTENT Supporting Information. The Supporting Information is available free of charge on ACS Publications website at DOI:. Amplitudes of structure factors, phases, electron density profile of a step function model, Xray diffraction image, explanation of curvature dependent width change in parallel surface model, estimation volume fraction and chain length (PDF)

AUTHOR INFORMATION Corresponding Author *[email protected] Notes The authors declare no competing financial interests. ACKNOWLEDGMENT This work was supported by JSPS KAKENHI Grant Number 15K05243 and 18K03557. Part of this research is based on a Cooperative Research Project of the Research Institute of Electronics, Shizuoka University. The synchrotron radiation experiments were performed using BL40B2 at SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2016A1174, 2016B1339, 2017A1352). Preliminary SAXS


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experiments were done at Molecular Structure Analysis Section of Shizuoka University RIGST.


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REFERENCES (1)

Hyde, S.; Blum, Z.; Landh, T.; Lidin, S.; Ninham, B. W.; Andersson, S.; Larsson, K. The Language of Shape: The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology; Elsevier, 1996.

(2)

Seddon, J. M.; Templer, R. H. Polymorphism of Lipid-Water Systems. In Structure and Dynamics of Membranes; North-Holland, 1995; Vol. 1, pp 97–160.

(3)

Yamazaki, M. Transformation Between Liposomes and Cubic Phases of Biological Lipid Membranes Induced by Modulation of Electrostatic Interactions. In Advances in Planar Lipid Bilayers and Liposomes; Academic Press, 2009; Vol. 9, pp 163–209.

(4)

Hyde, S. T.; Andersson, S.; Bodil, E.; Kåre, L. A Cubic Structure Consisting of a Lipid Bilayer Forming an Infinite Periodic Minimum Surface of the Gyroid Type in the Glycerolmonooleat-Water System. Z. Für Krist. - Cryst. Mater. 1984, 168 (1–4), 213–220.

(5)

Hajduk, D. A.; Harper, P. E.; Gruner, S. M.; Honeker, C. C.; Kim, G.; Thomas, E. L.; Fetters, L. J. The Gyroid: A New Equilibrium Morphology in Weakly Segregated Diblock Copolymers. Macromolecules 1994, 27 (15), 4063–4075.

(6)

Murakami, S.; Yamada, N.; Nagano, M.; Osumi, M. Three-Dimensional Structure of the Prolamellar Body in Squash Etioplasts. Protoplasma 1985, 128 (2–3), 147–156.

(7)

Almsherqi, Z. A.; Kohlwein, S. D.; Deng, Y. Cubic Membranes: A Legend beyond the Flatland* of Cell Membrane Organization. J. Cell Biol. 2006, 173 (6), 839–844.

(8)

Caffrey, M.; Cherezov, V. Crystallizing Membrane Proteins Using Lipidic Mesophases. Nat. Protoc. 2009, 4 (5), 706–731.

(9)

Drummond, C. J.; Fong, C. Surfactant Self-Assembly Objects as Novel Drug Delivery Vehicles. Curr. Opin. Colloid Interface Sci. 1999, 4 (6), 449–456.


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Page 28 of 32

(10) Attard, G. S.; Glyde, J. C.; Göltner, C. G. Liquid-Crystalline Phases as Templates for the Synthesis of Mesoporous Silica. Nature 1995, 378 (6555), 366–368. (11) Mariani, P.; Luzzati, V.; Delacroix, H. Cubic Phases of Lipid-Containing Systems: Structure Analysis and Biological Implications. J. Mol. Biol. 1988, 204 (1), 165–189. (12) Luzzati, V.; Mariani, P.; Delacroix, H. X-Ray Crystallography at Macromolecular Resolution: A Solution of the Phase Problem. Makromol. Chem. Macromol. Symp. 1988, 15 (1), 1–17. (13) Hyde, S. T. Microstructure of Bicontinuous Surfactant Aggregates. J. Phys. Chem. 1989, 93 (4), 1458–1464. (14) Hyde, S. T. Swelling and Structure. Analysis of the Topology and Geometry of Lamellar and Sponge Lyotropic Mesophases. Langmuir 1997, 13 (4), 842–851. (15) Anderson, D. M.; Gruner, S. M.; Leibler, S. Geometrical Aspects of the Frustration in the Cubic Phases of Lyotropic Liquid Crystals. Proc. Natl. Acad. Sci. 1988, 85 (15), 5364– 5368. (16) Große-Brauckmann, K. Gyroids of Constant Mean Curvature. Exp. Math. 1997, 6 (1), 33– 50. (17) Rappolt, M. The Biologically Relevant Lipid Mesophases as “Seen” by X-Rays. In Advances in Planar Lipid Bilayers and Liposomes; Academic Press, 2006; Vol. 5, pp 253– 283. (18) Schwarz, U. S.; Gompper, G. Bending Frustration of Lipid−Water Mesophases Based on Cubic Minimal Surfaces. Langmuir 2001, 17 (7), 2084–2096. (19) Templer, R. H. Thermodynamic and Theoretical Aspects of Cubic Mesophases in Nature and Biological Amphiphiles. Curr. Opin. Colloid Interface Sci. 1998, 3 (3), 255–263.


28

Page 29 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(20) Templer, R. H.; Seddon, J. M.; Duesing, P. M.; Winter, R.; Erbes, J. Modeling the Phase Behavior of the Inverse Hexagonal and Inverse Bicontinuous Cubic Phases in 2:1 Fatty Acid/Phosphatidylcholine Mixtures. J. Phys. Chem. B 1998, 102 (37), 7262–7271. (21) Nagle, J. F.; Tristram-Nagle, S. Structure of Lipid Bilayers. Biochim. Biophys. Acta BBA Rev. Biomembr. 2000, 1469 (3), 159–195. (22) Yang, L.; Huang, H. W. Observation of a Membrane Fusion Intermediate Structure. Science 2002, 297 (5588), 1877–1879. (23) Miyasaka, K.; Bennett, A. G.; Han, L.; Han, Y.; Xiao, C.; Fujita, N.; Castle, T.; Sakamoto, Y.; Che, S.; Terasaki, O. The Role of Curvature in Silica Mesoporous Crystals. Interface Focus 2012, 2 (5), 634–644. (24) Nakazawa, Y.; Yamamura, Y.; Kutsumizu, S.; Saito, K. Molecular Mechanism Responsible for Reentrance to Ia3d Gyroid Phase in Cubic Mesogen BABH(N). J. Phys. Soc. Jpn. 2012, 81 (9), 094601. (25) Oka, T. Small-Angle X-Ray Crystallography on Single-Crystal Regions of Inverse Bicontinuous Cubic Phases: Lipid Bilayer Structures and Gaussian Curvature-Dependent Fluctuations. J. Phys. Chem. B 2017, 121 (50), 11399–11409. (26) Oka, T. Transformation between Inverse Bicontinuous Cubic Phases of a Lipid from Diamond to Primitive. Langmuir 2015, 31 (10), 3180–3185. (27) Oka, T. Transformation between Inverse Bicontinuous Cubic Phases of a Lipid from Diamond to Gyroid. Langmuir 2015, 31 (41), 11353–11359. (28) Kabsch, W. XDS. Acta Crystallogr. D Biol. Crystallogr. 2010, 66 (2), 125–132.


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Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 32

(29) Mariani, P.; Rivas, E.; Luzzati, V.; Delacroix, H. Polymorphism of a Lipid Extract from Pseudomonas Fluorescens: Structure Analysis of a Hexagonal Phase and of a Novel Cubic Phase of Extinction Symbol Fd--. Biochemistry 1990, 29 (29), 6799–6810. (30) Brakke, K. A. The Surface Evolver. Exp. Math. 1992, 1 (2), 141–165. (31) Shearman, G. C.; Khoo, B. J.; Motherwell, M.-L.; Brakke, K. A.; Ces, O.; Conn, C. E.; Seddon, J. M.; Templer, R. H. Calculations of and Evidence for Chain Packing Stress in Inverse Lyotropic Bicontinuous Cubic Phases. Langmuir 2007, 23 (13), 7276–7285. (32) Armen, R. S.; Uitto, O. D.; Feller, S. E. Phospholipid Component Volumes: Determination and Application to Bilayer Structure Calculations. Biophys. J. 1998, 75 (2), 734–744. (33) Harper, P. E.; Gruner, S. M.; Lewis, R. N. A. H.; McElhaney, R. N. Electron Density Modeling and Reconstruction of Infinite Periodic Minimal Surfaces (IPMS) Based Phases in Lipid-Water Systems. II. Reconstruction of D Surface Based Phases. Eur. Phys. J. E Soft Matter Biol. Phys. 2000, 2 (3), 229–245. (34) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in C: The Art of Scientific Computing, Second Edition; Cambridge University Press: Cambridge ; New York, 1992. (35) Harper, P. E.; Gruner, S. M. Electron Density Modeling and Reconstruction of Infinite Periodic Minimal Surfaces (IPMS) Based Phases in Lipid-Water Systems. I. Modeling IPMS-Based Phases. Eur. Phys. J. E Soft Matter Biol. Phys. 2000, 2 (3), 217–228. (36) Barauskas, J.; Landh, T. Phase Behavior of the Phytantriol/Water System. Langmuir 2003, 19 (23), 9562–9565.


30

Page 31 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(37) Anderson, D. M.; Davis, H. T.; Scriven, L. E.; Nitsche, J. C. C. Periodic Surfaces of Prescribed Mean Curvature. In Advances in Chemical Physics; Prigogine, I., Rice, S. A., Eds.; John Wiley & Sons, Inc., 1990; pp 337–396. (38) Helfrich, W. Elastic Properties of Lipid Bilayers: Theory and Possible Experiments. Z. Für Naturforschung C 1973, 28 (11–12), 693–703. (39) Matsen, M. W.; Bates, F. S. Origins of Complex Self-Assembly in Block Copolymers. Macromolecules 1996, 29 (23), 7641–7644. (40) Perutková, Š.; Daniel, M.; Dolinar, G.; Rappolt, M.; Kralj‐Iglič, V.; Iglič, A. Stability of the Inverted Hexagonal Phase. In Advances in Planar Lipid Bilayers and Liposomes; Academic Press, 2009; Vol. 9, pp 237–278. (41) Chen, H.; Jin, C. Competition Brings out the Best: Modelling the Frustration between Curvature Energy and Chain Stretching Energy of Lyotropic Liquid Crystals in Bicontinuous Cubic Phases. Interface Focus 2017, 7 (4), 20160114.


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Table of Contents Graphic


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