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Transformation between inverse bicontinuous cubic phases of a lipid from diamond to primitive Toshihiko Oka Langmuir, Just Accepted Manuscript • DOI: 10.1021/la504295v • Publication Date (Web): 26 Feb 2015 Downloaded from http://pubs.acs.org on March 3, 2015
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Transformation between inverse bicontinuous cubic
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phases of a lipid from diamond to primitive
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Toshihiko Oka*,†,§
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†Department of Physics, Graduate School of Science, §Nanomaterials Research Division,
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Research Institute of Electronics, Shizuoka University, Shizuoka 422-8529, Japan
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ABSTRACT. I studied the transformation between inverse bicontinuous cubic phases (QII) from
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diamond (QIID) to primitive (QIIP) in a single-crystal region of monoolein. X-ray diffraction data
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reveal that the crystallographic orientation of QIIP rotates 55° around the [0 1 1] axis from QIID.
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This indicates that one direction of the four-branched water channels in the QIID phase is
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preserved in the six-branched water channels of the QIIP phase. I, therefore, built a transformation
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model that would keep the direction of the water channels fixed in both phases and cause the
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water channels along other direction in QIID to shrink and disappear.
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INTRODUCTION
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A triply periodic minimal surface (TPMS) has an infinitely extending structure and separates
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space into two interwoven regions1,2 (Figure 1). Three TPMSs, viz., diamond (D), primitive (P),
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and gyroid (G), which have cubic symmetry with the space groups Pn3 m , Im3m , Ia3 d ,
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respectively, are frequently encountered in real systems, such as lyotropic liquid crystals3,4,
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diblock copolymers5, prolamellar bodies of plants6, and organized smooth endoplasmic
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reticulum7. Inverse bicontinuous cubic (QII) phases formed by lipid-water systems are typical
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examples. QII phases having a diamond (QIID), primitive (QIIP), or gyroid (QIIG) structure have
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been observed3.
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The three TPMS can be transformed into each other mathematically by Bonnet
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transformation8, but the transformation requires that parts of the minimal surfaces pass through
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each other. The transformation also requires tearing the membrane of the physical system, thus
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the Bonnet transformation between QII phases is physically unrealistic3,9. Another mechanism
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has been proposed that simplifies the transformations as mergence and separation of junctions of
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a water network labyrinth 10,11. This mechanism indicated that transformation without tearing is
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feasible. Moreover, it has been shown that transformations between all three TPMS structure can
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preserve both topology and zero mean curvature throughout12, and corresponding intermediate
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structures have been proposed12–14. In the transformation between D and P, an intermediate with
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rhombohedral symmetry has been proposed. This transformation between the D and P surfaces is
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obtained by pushing or pulling along one of the directions12. The resulting changes in
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surface structure have been illustrated13. Thus, it has been predicted that the crystallographic
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orientation would rotate 60° around one of the axes during transformation in a real
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system9.
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Exploring the orientational (or epitaxial) relationships in transformations between the QII phases
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requires a technique to control the crystal orientation. Several such techniques have been
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reported15–18 and employed to study the phase transition between QIIG and QIID 16,19. However,
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only one direction of the crystal orientation can be aligned through these techniques. Thus, it
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seems difficult to elucidate orientational relationships in transformations between QII phases. My
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group, therefore, has developed a method to make a single crystal domain of the QIID phase20,
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which clearly exhibited a crystal orientation of the QIID phase. Using this method, I studied the
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QIID-QIIP phase transition in the single crystal region.
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MATERIALS AND METHODS
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A single crystal region was prepared according to a procedure reported previously20 except that
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the additive was replaced by t-butyl alcohol (tBA). A 80:20 v/v solution of water/ tBA (Wako
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Pure Chemical Industries, Osaka, Japan) was prepared. The solution was mixed with dried
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monoolein (1-oleoyl-rac-glycerol) (Sigma Chemical Co., St. Louis, MO, USA) in a 60:40 w/w
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solvent/monoolein ratio. The sample was loaded into a polyimide capillary (PIT-S, Furukawa
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Electric Co., Tokyo, Japan), whose inner diameter was 0.5 mm. The length of the loaded sample
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was about 30 mm. One end of the sample was sealed with a tube sealing compound (Chaseal,
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Chase Scientific Glass, TN, USA). The other end of the sample was soaked in water at 25°C.
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After about one month, the sample in the capillary separates into two parts: a powder region and
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a single crystal region of the QIID phase20. The powder region was cut off, and the remaining
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portion was transferred into a 20 mM L-arginine (Arg) (Wako Pure Chemical Industries, Osaka,
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Japan) solution in a horseshoe-shaped aluminum frame. The thickness of the frame was 1.0 mm,
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and its inner width and height were 20 and 35 mm, respectively. Both sides of the frame were
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sealed with 20 µm polyimide films. The volume of the 20 mM Arg in the frame was about 500
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µl.
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X-ray diffraction measurements were conducted in almost the same manner as in my previous
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paper20. The rotating crystal method was used. The sample-detector distance was set to 495 mm.
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The sample capillary in the horseshoe-shaped frame was mounted on a goniometer head. The
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sample capillary axis was set vertically and adjusted to the rotating axis. Measurements were
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performed at 25°C. The sample was rotated from 0° to 180° in increment steps of 2°, and X-rays
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for 10 s in each step. To measure the phase transition at the same sample position, measurements
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were repeated at intervals of 1 or 2 h. The results of the diffraction data analysis were almost the
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same as those reported previously20. I determined the peak positions of the diffraction spots with
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the display software ALBULA (Dectris, Baden, Switzerland). Using these peak positions,
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rotation matrices of the crystal orientations of QIID (RD) and QIIP (RP) were calculated from the
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base coordinate. Of the data for the QIID phase, diffraction spots from the {110}, {111}, {200},
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{211}, {220}, {221}, and {222} planes were used in the calculation. In the case of the QIIP
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phase, spots from {110}, {200}, {211}, and {222} planes were used in the calculation.
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Diffraction spots from the {220} planes of the QIIP phase, which were not included in the
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extinction rules, could not be observed. Some of the spots listed above were not observed
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because they were located outside the detection area or at a blind angle with respect to the
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horseshoe-shaped frame. The rotation matrices RD and RP were used to calculate the peak
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positions of unobservable diffraction spots in the extinction rules, and then the phase of the
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sample was identified. The rotation matrix from QIID to QIIP, RDP, was calculated via RDP = RD-
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RDP is an eigenvector of the matrix. The rotation angle, θ, around the rotation axis was calculated
RP. A rotation matrix is defined by a rotation axis and a rotation angle21. The rotation axis of
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from the trace of RDP via 1+2cosθ = TrRDP. The detailed calculations are in the Supporting
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Information.
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I constructed the model of a hypothetical intermediate in Figure 5 as follows. I aligned
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orientations of two four-way junctions of the QIID phase and one six-way junction of the QIIP
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phase. Next, I calculated the midpoint of the corresponding junctions in the QIID and QIIP phases
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as the hypothetical intermediate. Then, I calculated the periodical structure of the intermediate
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using the periodicity of the QIID and QIIP phases.
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RESULTS AND DISCUSSION
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Figure 2 illustrates the sample preparation process. To obtain a long single crystal region of
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monoolein in a capillary, I used a method reported previously by my group20, with the
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modification of using tBA instead of 1,4-butanediol. The length of the single crystal region was
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about 5 mm. Arg solution is known to convert the QIID phase of monoolein into QIIP 22. The open
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end of the capillary sample was soaked in 20 mM Arg to change the phase of the single crystal.
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Thus, the slow penetration of Arg solution led to a gradual QIID-QIIP phase transition from the
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open end. Figure 3 shows the time evolution of the X-ray diffraction patterns of the sample 3.5
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mm from the edge (Supporting Information Figure S1). The X-ray diffraction patterns of the
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capillary sample just after soaking (data not shown) and up to 7.5 h after soaking (Figures 3a and
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3d) indicated that only one orientation of the crystal existed in the exposure region. Analysis of
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the data from all angles confirmed that the phase of the lipid was QIID, which agreed with
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previous results20. After 12 h (Figures 3b and 3e), the spots of the QIID phase were still observed,
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but new diffraction spots had appeared as well. After 16 h (Figures 3c and 3f), the spots due to
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the QIID phase disappeared completely and only the new diffraction spots seen in Figures 3b and
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3e were observed. Analysis of these new spots indicated that the phase was QIIP. The lattice
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constants of the QIID and QIIP phases are 10.8 and 13.8 nm, respectively. The ratio of these lattice
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constant is 1.28, which agrees exactly with the theoretical value of 1.288. Diffraction patterns of
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the QIID phase at 7.5 h and 12 h show clear spots, and I could pick up the spot positions and
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easily determine the orientation of the single crystal. The diffraction spots of the QIIP phase at 12
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h and 16 h have circular tails, but I could still determine the orientation of the single crystal.
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Some weak spots were observed at positions where no spots appeared from the single crystal.
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Thus, I concluded there was one large single crystal and some small crystals of the QIIP phase. In
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Figure 3d, a diffraction spot from the (0 1 1) plane of QIID is observed. In Figure 3e, a new
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diffraction spot appeared immediately to the left of the (0 1 1) plane of QIID, which was assigned
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to the (0 1 1) plane of the QIIP phase. In Figure 3b, a diffraction spot from the (111) plane of QIID
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adjoins one from the (200) plane of QIIP.
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From the positions of the diffraction spots in the two-dimensional images, I could calculate the
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positions of the spots in reciprocal space (see Figure 4). Figure 4a is the view along the [0 1 1]
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direction of the QIID phase, which almost corresponds to the [0 1 1] direction of the QIIP phase.
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Figure 4b is the view along the [111] direction of the QIID phase, which almost corresponds to
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the [200] direction of the QIIP phase. I calculated the rotation matrix RDP from QIID to QIIP. A
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rotation matrix is defined by a rotation axis and a rotation angle21. The normalized rotation axis
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of the rotation matrix RDP was [0.143 -0.658 0.740], and the calculated rotation angle of the
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matrix was 54°. Visual inspection in reciprocal space indicated that the [0 1 1] directions of the
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two phases were nearly the same, as were the [111] direction of QIID and the [200] direction of
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QIIP (Figure 4). The rotation matrix satisfying these relationships must have [0 1 1] as its rotation
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axis and a rotation angle of 55°. The rotation axis calculated from the rotation matrix RDP was
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close to the [0 1 1] direction, and the angle difference was 9°. The difference between the
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rotation angles is very small. I, therefore, concluded that the crystallographic orientation rotated
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55° around the [0 1 1] axis in the phase transition from QIID to QIIP.
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Using this orientational relation, I propose a model for the phase transition from QIID to QIIP.
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Figure 5 shows water channels of QIID, a hypothetical intermediate, and QIIP, viewed along the
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[0 1 1] direction. The lipid surface changes with the water channels, however, to simplify the
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model, I only treat the transformation of water channels explicitly. The phase transition starts
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from the QIID phase (Figure 5a). The QIID phase has simplified water channels along four
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directions: [111], [1 1 1 ] , [ 1 1 1 ] , and [ 1 1 1] . At the start of the transition, water channels along
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the [1 1 1 ] direction start to shrink. In the hypothetical intermediate, water channels along the
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[1 1 1 ] direction in QIID shorten (Figure 5b). But one direction of the water channels is fixed,
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namely, the [111] direction of the water channels in the QIID phase. Then, the [1 1 1 ] direction of
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water channels in the QIID phase disappear, and the QIIP phase is formed (Figure 5c). The [111]
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direction of water channels in the QIID phase is still fixed in the QIIP phase, but the Miller indices
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of the direction become [200] because the crystallographic orientation changes during the
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transition.
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The transformation from D to P has been studied theoretically10–14, but the orientational
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relationships observed in a single crystal during the D-P transformation remains poorly studied.
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Squire et al. only predicted that one of the axes in the crystallographic orientation of both
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phases was fixed during the phase transition according to the rhombohedral pathway model9. I
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expected that the direction of the water channels of the QIID phase, which disappear in the phase
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transition, would be fixed. But my data showed that the [0 1 1] axis is fixed as the rotation axis of
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crystallographic orientation, and one other direction of the water channels is conserved. On the
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other hand, my model seems to agree well with the theoretical model for the surface
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transformation. The concept of merging junctions developed by Sadoc and Charvolin10 was used
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in my transformation model. Hyde and coworkers have reported a rhombohedral pathway model
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that preserves the topology and zero mean curvature throughout12–14. The hypothetical
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intermediate in Figure 5b has rhombohedral symmetry. Thus, my transition model seems to
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agree well with the rhombohedral pathway model. Because the rhombohedral pathway model
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was developed on the basis of a unit cell, the model tolerate the rotation of the unit cell.
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Moreover, the transformation model of Squire et al. is in agreement with my model, except for
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the orientational relationship. Thus, my model and previous theoretical models by other groups
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differ only in terms of the direction preserved during the transformation.
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The QIID-QIIP phase transition occurs at the interface between the two phases in the single-crystal
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region. Thus, the newly formed QIIP phase contacts with the existing QIID phase. This contact
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presumably affects the crystallographic orientation in the phase transition. Thus, having a
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membrane forming adequate connection between the QIID and QIIP phases is important for
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decreasing the transition energy and hence preserving the direction of the water channels. The
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preservation, therefore, would be energetically favorable in a macroscopic phase transition.
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Epitaxial relationships involving the type I cubic phase also seem to minimize the movement of
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the lipid and water during phase transition23. Further experimental and theoretical studies are
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required in which the effect of the interface between the QIID and QIIP phases are considered.
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In the rhombohedral pathway model, the rPD surface was considered as a pathway for the
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transformation from D to P surfaces12,13. In the transformation, the c axis length of the
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rhombohedral unit cell of the rPD surface decreases by 36% and the a axis length increases by
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28%24. This indicates that the length of the [1 1 1 ] direction of D shrinks by 36% in P, while the
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length of the direction perpendicular to [1 1 1 ] expands. My model in Figure 5 also shows the
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disappearance of the water channel of the [1 1 1 ] direction in the QIIP phase. The single crystal of
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the QIID phase is a liquid crystal, and thus, the membrane surface and water channels are
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movable during the phase transition. But a large length change in one direction would damage
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the single crystal. Indeed, X-ray diffraction reveals deterioration of the single crystal during the
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phase transition.
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I concluded that one direction of the water channels is preserved: the [111] direction of the QIID
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phase corresponds to [200] of QIIP. But there are eight choices because the set of axes
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includes eight directions. Thus, there is reason to restrict the crystallographic orientation in the
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transition. In this experiment, Arg solution was used to induce phase transition in the lipid in the
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capillary. Thus, the phase transition advanced in one direction along the capillary axis from the
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open end of the capillary located at the bottom. This would affect the choice of axis. Indeed, my
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data indicates that the preserved directions of the water channels were almost perpendicular to
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the capillary axis. The direction in which the phase transition advances might restrict the
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preserved directions of the water channels.
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The phase of monoolein is QIID under excess water conditions. In this report, I used Arg to obtain
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the QIIP phase from the QIID phase22. But the phase transition mechanism is unclear. Yamazaki
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and coworkers have been reported that the phase of monoolein with charged lipids such as
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dioleoylphosphatidyc acid depended on the surface charge, and observed the QIIP phase4,25. Thus,
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Arg might affect the membrane surface and stabilize the QIIP phase.
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The X-ray powder diffraction technique has been widely used in the analysis of QII phase
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transitions9,26,27. However, data obtained by this technique does not include information on
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sample orientation. By contrast, the single-crystallization method20 and alignment technique15–18
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of the QII phase allow us to study the orientation of the phases during the transition. These
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techniques promise to yield high-quality structural information on the QII phases.
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CONCLUSION
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By using the single crystal region of the QII phase of monoolein, I revealed orientaional
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relationships in the QIID-QIIP phase transition. I also proposed a model for the transition in which
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one direction of the water channels was preserved while the other direction shrank and
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disappeared. The change of crystallographic orientation in my model did not agree with those
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reported in previous theoretical works, but in all other respects, my model did agree well with
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previous models. This result is one example of the benefits produced by the single crystallization
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technique of the QII phase. Moreover, the techniques used in this paper would help us to obtain
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acquire further information about the transformation between QII phases.
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FIGURES
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Figure 1. TPMS structures of (a) D and (b) P. 2×2×2 unit cells are drawn. The TPMS separates
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space into two interwoven regions. The two different tones of pipes indicate the centers of the
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two interwoven regions. In the structure of the QII phases of the lipid, the positions of the TPMS
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are occupied by lipid bilayer membranes, and the two interwoven regions are occupied by water.
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(c) Adjacent two four-way junctions of interwoven regions in D. (d) One six-way junction of
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interwoven regions in P. The two four-way junctions of D merge into the one six-way junction of
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P in the D-P transformation model with rhombohedral symmetry. In this model, the
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transformations are obtained by pushing along one of the directions in D, and a pipe of
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the direction in D shrinks and disappears in P.
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Figure 2. Schematic of the sample preparation process. (a) Monoolein sample before soaking in
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water. Concentration of t-butyl alcohol (tBA) was 20%. The phase of monoolein was L3. (b)
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Sample soaked in water for several days. The tBA in the capillary diffused out gradually from
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the open edge, and a concentration gradient of tBA was formed in the capillary20. Monoolein at
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the open edge formed polycrystals of the QIID phase. (c) Sample soaked in water for one month.
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The concentration gradient of tBA was lower than in (b). A single crystal region of QIID was
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formed far from the open edge20. (d) The polycrystal region was cut off; the remaining part
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consisted of only the single crystal region. (e) The single crystal region was soaked in 20mM L-
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arginine (Arg). Arg diffused slowly into the capillary from the open edge, and the single crystal
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region of QIIP was formed gradually. (f)(g) X-ray diffraction patterns of the sample in (e) were
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measured from just after the Arg soak to about 1 day with roughly about 1-2 hour intervals. The
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capillary axis was adjusted to the rotating axis and perpendicular to the X-ray direction. The
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crystallographic orientation of the single crystal region was not aligned with the capillary axis.
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The sample was rotated from 0° to 180° in increment steps of 2°, and exposed to X-rays for 10 s
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at each step. The phase of the sample in the X-ray exposed area was QIID, as in (f). As time
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passed, the QIIP grew, and the boundary between QIID and QIIP passed through the X-ray exposed
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area. The phase of the sample under X-rays was QIIP, as in (g).
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Figure 3. X-ray diffraction images of the capillary sample, 3.5 mm from the edge immersed in 20
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mM Arg solution. The images are cumulative data of rotation angles (a-c) from 30° to 40°, (d-f)
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from 110° to 120°. Only the right halves of the images are shown because the original images are
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centrosymmetric. (a), (d) After 7.5 h of soaking in the Argsolution. (b), (e) After 12 h. (c), (f)
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After 16 h. Diffraction spots of the QIID phase are seen in (a) and (d), and those of the QIIP phase
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are in (c) and (f). Diffraction spots from both phases can be seen in (b) and (e). Some strong
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peaks from the QIID and QIIP phases are surrounded by dashed circles of red and blue,
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respectively. Some Miller indices of the QIID phase are assigned in (a) and (d), and those of the
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QIIP phase are assigned in (c) and (f).
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Figure 4. Observed positions of diffraction spots of the QIID phase (red circles) and the QIIP phase
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(blue crosses) in reciprocal space. Diffraction spots that satisfy |h|, |k|, |l| ≤ 2 (h, k, l are the Miller
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indices) are shown. Diffraction spots from {200} planes and the origin are drawn with lines:
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those of QIID are red solid, and those of QIIP are blue dashed. (a) Viewed along roughly the
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[0 1 1] direction of both phases, (b) along roughly the [111] direction of the QIID phase, and
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along roughly the [200] direction of the QIIP phase.
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Figure 5. Model of the transition pathway from the QIID phase to the QIIP phase. Pipes indicate
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the centers of the water channels. Different tones of pipes indicate two different water channels.
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Models are viewed along the [0 1 1] direction of QIID or QIIP. (a) The QIID phase. The [111]
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direction of the water channels is fixed during the phase transition. Water channels of the [1 1 1 ]
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direction begin to shrink (red dashed arrows) at the start of the phase transition. (b) Hypothetical
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intermediate. Water channels of one direction (the [1 1 1 ] direction in the QIID phase) is short. (c)
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The QIIP phase of the corresponding orientation. The [111] direction of water channels of the
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QIID phase (green dotted line) is fixed and corresponds to the [200] direction of water channels of
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the QIIP phase. All directions marked with arrows or lines in the figures are perpendicular to the
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[0 1 1] viewing direction. The angle between the [200] and [111] (or [1 1 1 ] ) directions is 55°,
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which is the rotation angle around the [0 1 1] direction in the transformation. The water channels
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of the [1 1 1 ] direction in the QIID phase (a) (red dashed line) shrink (b) and disappear (c) during
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the transformation.
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ASSOCIATED CONTENT
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Supporting Information. Sequential X-ray diffraction images for Figures 3 and two movies for
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Figure 5. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
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Corresponding Author
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*
[email protected] 7
Notes
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The authors declare no competing financial interests.
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ACKNOWLEDGMENT
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I am grateful to Prof. S. Hyde for helpful discussions. I appreciate the critical reading of the
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manuscript by Prof. M. Yamazaki.
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