INTRODUCTION

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The physical state of a fluid L-phase lipid bilayer is mirrored by the organization and motions of the acyl chains comprising its hydrocarbon core (HC). This is clearly revealed by the decreases in HC thickness (Levine and Wilkins, 1971; Torbet and Wilkins, 1976) and ²H-NMR alkyl-chain order parameters (Boden et al., 1991; Koenig et al., 1997) that accompany increases in hydration. Similar effects are seen when peptides partition into bilayer interfaces and thereby increase the area per lipid (Jacobs and White, 1987, 1989; Wu et al., 1995). Measures of the structure of the HC are thus useful for understanding molecular interactions that depend upon the structure and stability of bilayers. We show here that x-ray diffraction measurements of the transbilayer distribution of bromine-labeled double-bonds in 1,2-dioleoyl-sn-glycero-3-phosphocholine (dioleoylphosphocholine; DOPC) bilayers provide a useful measure of HC structure that is remarkably sensitive to structural changes, induced in the present case by changes in hydration.

Double-bonds in phospholipid acyl chains cause bilayers to be in a fluid state at biologically relevant temperatures [reviewed by Small (1986)], especially when located in the middle of the chain (Barton and Gunstone, 1975), which is the usual case for naturally occurring monounsaturated phospholipids. For DOPC bilayers at 66% relative humidity (RH) (5.4 waters/lipid), Wiener and White (1992) showed that the transbilayer distribution of the thermally disordered double-bonds provide a measure of the thickness of the HC as well as its thermal disorder. This distribution, defined as the time-averaged positions of the double-bonds projected on to the bilayer normal, was determined exactly by neutron diffraction using DOPC specifically deuterated at the double-bonds (Wiener et al., 1991) and approximately by x-ray diffraction using DOPC specifically brominated at the double-bonds of the sn-2 chain (Wiener and White, 1991c). The latter distribution differs from the true one only by being slightly broader (~0.7 Å) due to the size of the bromines. Here we demonstrate the feasibility of using the bromine labeling method as a means of monitoring changes in HC structure. Specifically, we have determined the time-averaged transbilayer distribution of the bromine labels in bilayers formed from isomorphous mixtures (Wiener and White, 1991c) of DOPC and 1-oleoyl-2-(9, 10-dibromostearoyl)-sn-glycero-3-phosphocholine (OBPC) over a hydration range of 5.4 to 16 waters per lipid. We show that the transbilayer distributions of the bromines over the full range of hydration can be described by a pair of Gaussian functions of 1/e half-width A_Br located at z = ±Z_Br relative to the bilayer center, and that these distributions are sensitive measures of the state of the HC.

This work is part of an ongoing investigation of the use of so-called "liquid-crystallography" (Wiener and White, 1991a, b) for the determination of the structure of liquid-crystalline bilayers using refinement methods commonly used in protein crystallography [reviewed by White and Wiener (1995, 1996)]. Therefore, an additional goal was to extend the liquid-crystallographic method to high hydrations where its application can be problematic (see below). The structural images of fluid bilayers obtained by liquid-crystallography account for all of the mass of the unit cell by subdividing the phospholipids and water within it into a series of "quasimolecular fragments" (King and White, 1986) such as the carbonyls, phosphates, cholines, etc. The "structure" of each fragment consists of the time-averaged projection of the three-dimensional motion of the fragment onto the bilayer normal. Because of the central-limit theorem (Barlow, 1989), these projections are invariably Gaussian distributions, as observed experimentally (Wiener et al., 1991; Wiener and White, 1991c). The complete bilayer structure consists of the full set of these distributions. In the present work, we have determined only one of these distributions, the double-bonds.

Liquid-crystallography was originally developed using experimental data from highly oriented lipid multilayers that form nearly perfect one-dimensional lattices at low hydrations. At high hydrations, orientational disorder, thermal motion, and membrane undulations (Sirota et al., 1988; Nagle et al., 1996; Zhang et al., 1996) can decrease the number of observable diffraction orders h_obs and thereby limit the use of the method (Wiener and White, 1992). Indeed, we found in the course of the present studies that h_obs dropped from 8 at 5.4 waters/lipid to an impractical 3 orders for more than 16 waters/lipid. A modification of the original x-ray data-scaling method (Wiener and White, 1991c), described in detail in Methods, allowed determination of the distribution of the bromine labels with as few as 4 orders of data.

For perfect crystals, one's ability to resolve the atoms of the unit cell is limited only by the thermal motions of the atoms which are taken into account in the refined structural model by means of the Debye-Waller formalism (Warren, 1969). These thermal motions limit the number of diffraction orders that can be observed to a value defined as h_max. The characteristic spatial extent of thermally disordered atoms or small clusters of atoms is approximately d/h_max, where d measures the unit cell size (Wiener and White, 1991a). For example, atoms smeared over a space of ~2 Å in a bilayer unit cell with d = 50 Å will produce ~25 orders of diffraction (Sakurai et al., 1977; Suwalsky and Duk, 1987). Such a unit cell is intrinsically a high-resolution structure. In contrast, the unit cell of the liquid-crystalline bilayer is so highly thermally disordered that only a few diffraction orders are possible (typically h_max = 5-10) because the atoms are smeared together into large quasimolecular clusters with spatial extents of 5-10 Å. Such unit cells are intrinsically low-resolution structures. However, no matter what the intrinsic resolution of a unit cell is, collection of all of the h_max diffraction orders will produce a fully resolved (accurate) image of the structure. For high thermal disorder, the image will be a fuzzy one. Nevertheless, an accurate image of the fuzzy structure can be obtained.

The collection or analysis of fewer than the h_max possible diffraction orders will result in an under-resolved (inaccurate) image of the structure [see Wiener and White (1991a)]. Two types of disorder, orientational and lattice, can lead to under-resolved images of fluid bilayers. If lipid multilayers are highly oriented, i.e., the bilayer lamellae are flat and their normals are coincident, the diffraction peaks are essentially images of the incident x-ray beam so that the signal-to-noise ratio is maximized. But, if the multilayers have a range of orientations, i.e., the bilayer lamellae are curved, the diffraction peaks will be smeared into arcs (circles in the case of completely random orientations such as in simple multilamellar dispersions). This smearing can reduce the high-order diffraction peaks to below the noise level and thereby cause h_obs to be smaller than h_max. Typically, orientational disorder can cause h_obs  h_max/2 for randomly oriented samples. Lattice disorder also can reduce the number of observable diffraction orders because the loss of spatial coherence leads to a progressive broadening of diffraction peaks as h increases (Hosemann and Bagchi, 1962). High lattice disorder can therefore cause the intensities of high-order peaks to fall below the noise level. For highly oriented samples, one must therefore establish that h_obs = h_max in order to be certain that a structure is fully resolved. This can be done by measuring the widths of the diffracted peaks as a function of h provided that the x-ray optics are not limiting [see Wiener and White (1991a)]. Even then, there is the possibility of the existence of high-order structure factors that cannot be detected because they are below the noise level of the detector (Wiener and White, 1991b). The Monte Carlo refinement procedure of Wiener and White (1992) accounts for this possibility.

The above discussion shows that liquid-crystallography can be problematic at high hydrations because the combined effects of orientational and lattice disorder can cause h_obs < h_max. The primary cause of lattice disorder in bilayer systems at high hydrations is likely to be undulations (Helfrich, 1973) if the lamellae are sufficiently flexible (Sirota et al., 1988). Unlike the thermal fluctuations, which occur relative to a bilayer's mean position, undulations are fluctuating whole-body motions of the bilayers. Besides introducing lattice disorder, they can cause an additional smearing (broadening) of the Gaussian distributions that describe the thermal motion of the quasimolecular fragments. The presence of undulations is detected through high-resolution measurements of the shapes of the diffracted intensities (Sirota et al., 1988; Nagle et al., 1996; Zhang et al., 1996). Such measurements were not feasible for the present work. As an alternative, we used a simple modeling approach to examine the likelihood of undulations being the cause of the reduction in h_obs at higher hydrations. The analysis, presented in the Discussion, indicates that undulations are not a serious problem over the range of 5.4 to 16 waters/lipid.

	MATERIALS AND METHODS

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Materials

DOPC and OBPC were purchased from Avanti Polar Lipids (Alabaster, AL). Purity of OBPC was determined by elemental analysis to be >99.9% (Microlit Laboratories, Madison, NJ). Polyvinylpyrrolidone, M_r = 40,000 (PVP) with an average molecular weight of 40,000 and intrinsic viscosity of 28-32, designated as PVP-40, was purchased from Sigma Chemical Co. (St. Louis, MO).

Sample preparation

Oriented samples

Unoriented samples

Sample degradation

Collection of x-ray intensities and integration of peaks

X-ray diffraction experiments were performed with Ni-filtered CuK radiation on an 18 kW Siemens (Madison, WI) rotating anode x-ray generator operated at 38 kV and 40 mA (1.52 kW). The beam was collimated and focused at the 2D detector array using double-mirror optics (Charles Supper, Natick, MA). Diffraction patterns were recorded on a Siemens X-1000 xenon-filled area detector with position decoding circuit and real-time data display. A displayed data frame consisted of typical lamellar diffraction patterns appropriate for oriented and unoriented samples that consisted of curved arcs around the -axis with lamellar spacings along the 2 radial. The initial processing of the data from the position decoding circuit was performed using the Siemens General Area Detector Diffraction Software (GADDS). For each data set analyzed,  × 2 wedge-shaped "sectors" were chosen manually to include the diffracted intensities that were then summed around the -axis (i.e., along the arcs of the diffraction peaks) using the GADDS "bin" method. The result of the  integration is the total diffracted intensity versus the Bragg angle, 2. Using this procedure, the observed structure factors for both oriented and unoriented samples are given by

(1)

where I(h) is the intensity of the hth peak and A(h) is the absorption correction (see below).

For oriented samples the mosaic spreads never exceeded 30°, but were generally much smaller, 5° or less. Very long exposures demonstrated that for oriented samples h_obs = h_max. Because the first-order peak was much stronger and wider due to its very high intensity, the integration was performed in two steps. The intensities were first integrated around the -axis for a wide sector containing all the diffracted intensity to be certain that all of the very intense first order was collected. For this wide sector, the high-order peaks were lost in the noise because of the long length of the integration path. The sector was then changed to accommodate only the high-order peaks; the -axis integration was performed on a segment that did not include the first order and was much narrower. This reduced the amount of background included in the integration so that even the highest-order diffraction peaks could be easily detected above the noise level. The internal consistency of the two integration procedures was verified by comparing the integrated intensities of the relatively strong 3rd- and 4th-order peaks, which could be analyzed by either method. The intensities agreed within experimental uncertainty.

For unoriented samples, as expected, the integrated intensities did not depend on the width and the orientation of the integration sector. However, no more than four orders could be seen for unoriented samples in PVP solutions. Bulk samples prepared to have hydrations corresponding to those of 66 to 93% RH also gave only four diffraction orders because the weak high-order peaks were spread over a larger detector area and consequently had lowered signal-to-noise ratio (see Discussion).

After the  integration, the I(2) peaks were analyzed using the software package Origin^TM (MicroCal, Inc., Northampton, MA) in the following way: first, the peaks were deleted, leaving only the background, which was then fit with a polynomial function. This fitted background was subtracted from the original data leaving only the diffraction peaks. The peaks were integrated in Origin^TM using two different methods: numerical integration of the areas under the peaks or by fitting Gaussians to the diffraction peaks and integrating analytically. The average of the two areas given by the two methods yielded I(h) for use in Eq. 1. The difference between the two areas was usually smaller than the estimated uncertainty of the intensity calculated from (peak area + background)^1/2. The experimental uncertainties of oriented samples at 76% RH were typical of those observed for all experiments: 0.1% for the intense 1st-order peak, 2% for the strong 4th-order, and 20% for the very weak 2nd-order.

Absorption corrections

(2)

1

Scaling of structure factors

The experimental structure factors f(h) from a given experiment depend upon the amount of sample in the beam, precise geometry of the sample-beam intersection, x-ray beam intensity, and other experimental conditions. The true (absolute) structure factors, F*(h), are determined solely by the scattering factor of the unit cell. The experimental structure factors are related to the true structure factors by f(h) = KF*(h), in which K is the instrumental constant. Fourier reconstructions of bilayer scattering-length or electron density profiles yield only arbitrary fluctuations of scattering density along the bilayer normal if f(h) rather than F*(h) is used. Determination of the instrumental constant allows one to relate the scattering profiles obtained in diffraction experiments to the actual contents and molecular packing of the bilayer unit cell. To do this, one must 1) determine the true mean value of the scattering profile using the composition of the unit cell, and 2) calibrate the fluctuations around this mean value (Franks et al., 1978; Wiener and White, 1991c). This is done by a scaling procedure (Wiener and White, 1991c) summarized below.

The relative absolute scale

12

(3)

(4)

Scaling principles

(5)

(6)

(7)

^*

(8)

(9)

(10)

Scaling procedures

(11)

(12)

(13)

(14)

(15)

(16)

(17)

1.	The set of Eqs. (12) is used to linearize all the observed structure factors f. This process yields the instrumental constants for the mixtures of A and B as a function of the scaling constants for A and B and thus places all the data on an internally consistent scale (but not an absolute scale). In addition, the linear regression procedure included in the linearization yields the "best" statistical estimate x(h) of the structure factors for a particular fraction x of OBPC.
2.	Using the set of Eqs. (17) and the values of A(h) and B(h) obtained in step 1, kA, kB, ABr, and ZBr are determined in a single computational step by "gluing" the two profiles, *A and *B, in reciprocal space. The procedure is accurate provided that there are at least four orders of diffraction and that all the intensity under each peak is collected (see Discussion). The determination of kA and kB yields the relative absolute structure factors *A(h) and *B(h).
3.	The instrumental constants kx, the relative absolute structure factors F* = f/kx and their best estimates * = /kx are determined using Eq. 12 while keeping kA and kB fixed.

View larger version (21K): [in this window] [in a new window]   FIGURE 1   Relative-absolute structure factors as a function of mol fraction of OBPC in oriented OBPC/DOPC bilayers for one particular hydration (86% RH, 7.7 waters/lipid). Individual points are the relative absolute structure factors that are related to the arbitrary measured structure factors by instrumental scale factors. The error bars are obtained from the uncertainties in the integrated diffraction peaks. The solid lines are derived from the self-consistent fit to all the data by means of Eq. 12. The values of the solid lines at a given mol fraction OBPC are the best estimates of the relative absolute structure factors, *(h).

Estimates of experimental uncertainties in Gaussian parameters

X-ray phase determination

	RESULTS

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We examined oriented DOPC multilayers equilibrated at 34% to 93% RH (3-9.4 waters/lipid) and unoriented liposomal suspensions in 60% to 5% PVP solutions (12-26.1 waters/lipid) containing from 0 to 50 mol % OBPC (six different values for each hydration). As summarized in Table 1, the transbilayer bromine (double-bond) distributions were determined only over the hydration range of 5.4 to 15.9 waters/lipid because the scaling procedure described in Methods could not be applied outside that range. For low hydrations (3-5 waters/lipid), the incorporation of OBPC led to the appearance of two satellite off-axis peaks between the 3rd- and the 4th-order lamellar peaks, indicating that OBPC/DOPC bilayers were not isomorphous with DOPC bilayers. For hydrations above 16 waters/lipid, the scaling could not be performed because too few orders of diffraction could be observed (h_obs  3). The measured Bragg spacings for five or more waters/lipid did not depend upon the mol % of OBPC at any hydration used, consistent with isomorphous replacement. Fig. 2 shows the Bragg spacing of DOPC bilayers as a function of the number of water molecules per lipid. The solid squares () correspond to bilayer hydrations (Table 1) for which the distribution of the bromine label could be determined (h_obs  4). The open symbols (, ) indicate, respectively, the low-end and high-end hydrations whose distributions were not determined. There is a distinct break in the curve at 11.6 waters/lipid that corresponds to the point of completion of the phosphocholine hydration shell (LeNeveu et al., 1977; McIntosh et al., 1989). However, because this break coincided with the change in the method of hydration, there was a small possibility that it was an artifact of the protocol change. We tested this possibility through diffraction experiments on mechanically mixed samples of water and lipid that covered the combined hydration range of the two hydration protocols. The break at ~12 waters/lipid was observed under these conditions as well. Therefore, the break must result from a structural change accompanying the completion of the filling of the phosphocholine hydration shell. This is in complete agreement with the conclusions of studies performed in other laboratories that used a single method of hydration (LeNeveu et al., 1977; McIntosh et al., 1987, 1989).

View larger version (15K): [in this window] [in a new window]   FIGURE 2   Bragg spacing of DOPC bilayers as a function of the number of water molecules per lipid. The break, which occurs at 11.6 water/lipid, corresponds to the point of completion of the hydration shell. This number is in agreement with other studies (LeNeveu et al., 1977; McIntosh et al., 1989). The solid squares () correspond to hydrations for which the bromine distribution was determined. The open symbols correspond to hydrations for which the x-ray data do not provide sufficient information to scale the data: for 16-21 waters per lipid only three, and above 22 molecules per lipid only two, diffraction orders were observed (). For hydrations below 5 molecules per lipid (), the OBPC/DOPC bilayers were not isomorphous with the pure DOPC bilayer.

The structure factors of OBPC/DOPC bilayers with h_obs  4 were scaled and processed as described in Methods to convert them to best estimates (*) on the relative-absolute scale. These structure factors are presented in Table 2. Two examples of the resulting bilayer scattering density profiles with 0, 5, 10, 20, 25, and 50 mol % OBPC are shown in Fig. 3. Panel A is for oriented bilayers (7.7 waters/lipid; 86% RH) with h_obs = 6 and panel B for unoriented bilayers (14.2 waters/lipid; 40% PVP) with h_obs = 4. These relative-absolute scale profiles describe the scattering on a per-lipid basis (units: scattering length per length). Division of the relative-absolute density by the area per lipid S will convert the profiles to the true absolute scale. Note that the profiles for 7.7 waters/lipid (Fig. 3 A) have a "sharper" appearance than the profiles for 14.2 waters/lipid (Fig. 3 B) because of the larger h_obs. This is because the canonical resolution d/h_obs is better as a result of the larger number of structure factors available for the oriented bilayers at 7.7 waters/lipid. The two data sets were scaled independently. The fact that the two sets of profiles have about the same scattering density in the headgroup regions is a good indication of consistency in the scaling. We obtained similar results for all hydrations. The two peaks in the profiles, located at ~± 7 Å relative to the bilayer center, increase with increasing amounts of OBPC and are therefore identified as the transbilayer distribution of the bromine labels on the double-bond of the sn-2 chain.

View this table: [in this window] [in a new window]   TABLE 2   Relative-absolute structure factors, * (h) × 1012, for hydrations of 6.2 to 15.9 waters/lipid

View larger version (25K): [in this window] [in a new window]   FIGURE 3   Scattering density profiles of DOPC/OBPC bilayers on the relative absolute (per lipid) scale for 0, 5, 10, 20, 25, and 50 mol % OBPC. The Fourier series are generated from the best relative absolute structure factors (see Fig. 1). (A) Six-order reconstruction for 86% RH (7.7 waters/lipid). (B) Four-order reconstruction for 40% PVP (14.2 waters/lipid). The units are scattering length per length represented here as *(z) · S (see Eq. 4) to indicate that division by the area/lipid S will place the profiles on the absolute scattering density scale. The two peaks, located at ~7 Å from the bilayer center, increase with increasing amounts of OBPC and are easily identified as the transbilayer distribution of the bromine atoms.

The difference profiles for 7.7 and 14.2 waters/lipid relative to 0 mol % OBPC, constructed by Fourier synthesis from the difference structure factors (Eq. 7), are presented in Fig. 4, A and B, respectively. The Fourier ripples in Fig. 4 B extending from 20 to 30 Å relative to the bilayer center indicate that the profiles at 14.2 waters/lipid (h_obs = 4) are under-resolved. As discussed in Methods, Fourier noise such as this made the scaling method of Wiener and White (1991c) inaccurate because it is based upon the assumption of fully resolved profiles. Our modification of the method removes this limitation (see Methods). Note that in Fig. 4 A Fourier noise is virtually absent because the images are fully resolved (h_obs = h_max).

View larger version (28K): [in this window] [in a new window]   FIGURE 4   Difference scattering-density profiles obtained from the structure factors of the profiles presented in Fig. 3. (A) Six-order reconstruction for 86% RH (7.7 waters/lipid). (B) Four-order reconstruction for 40% PVP (14.2 waters/lipid). The peaks increase with increasing OBPC content. Note the Fourier noise extending from 20 to 30 Å from the bilayer center. This noise causes the scaling procedure of Wiener and White (1991b) to be inaccurate. The modified procedure presented in the Methods is not affected by this noise (see text)

Fig. 5, A and B shows the fully resolved Gaussian distributions of the bromine-labeled double-bonds for 7.7 and 14.2 waters/lipid, respectively, obtained from the scaling procedure described in Methods. A collection of Gaussian distributions covering the range of hydrations are compared in Fig. 6 A and the Gaussian parameters A_Br and Z_Br for all hydrations (5.4 to 15.9 waters/lipid, Table 1) are plotted against hydration in Fig. 6 B. For hydrations from 5.4 waters (66% RH) up to 9.4 waters per lipid (93% RH), the bromine position gradually decreases from Z_Br = 7.97 ± 0.27 Å to Z_Br = 6.59 ± 0.15 Å, while A_Br increases from 4.62 ± 0.62 Å up to 5.92 ± 0.37 Å. This behavior is consistent with the expected increase in thermal disorder with increasing hydration that causes a decrease in hydrocarbon thickness and increase in the area per lipid molecule. These changes in the double-bond distribution were anticipated by Wiener and White (1992). Just after the hydration shell of the phosphocholine headgroup is filled at ~12 water molecules per lipid (60% PVP), Z_Br increases to ~± 7.3 Å while A_Br decreases to 5.3 Å. This suggests that a discrete structural change takes place when the hydration shell becomes filled, consistent with NMR measurements of the order parameters of the phosphocholine methylenes (Bechinger and Seelig, 1991). In the range 12-16 waters/lipid, the bromine distributions are practically overlapping (Table 1). The average of the Gaussian parameters for the three hydrations are Z_Br = 7.33 ± 0.25 Å and A_Br = 5.35 ± 0.5 Å.

View larger version (27K): [in this window] [in a new window]   FIGURE 5   Gaussian fits of the difference Fourier profiles presented in Fig. 4 according to the scaling method developed in Materials and Methods. The principal difference between these distributions and those of Fig. 4 is the absence of Fourier noise.

View larger version (20K): [in this window] [in a new window]   FIGURE 6   Summary of the transbilayer distribution of the bromine-labeled double-bonds in OBPC/DOPC bilayers as a function of hydration. (A) Distribution of bromine labels for three characteristic hydrations: 5.4, 9.4, and 14.2 waters/lipid. (B) Positions and widths of the Gaussian bromine distributions for hydrations from 5.4 to 16 water molecules per lipid. For hydrations from 5.4 waters (66% RH) up to 9.4 waters/lipid (93% RH), the bromine position gradually decreases from ZBr = 7.97 ± 0.27 Å to ZBr = 6.59 ± 0.15 Å, while ABr increases from 4.62 ± 0.62 Å up to 5.92 ± 0.37 Å. After the hydration shell is filled at ~12 waters/lipid (60% PVP), we observe a shift in ZBr to ~7.3 Å, while ABr decreases to 5.3 Å, suggesting that some structural change takes place at the point of completion of the hydration shell.

Although the positions of the Gaussians of the bromine-label distribution correspond exactly with the double-bond distribution, the widths of the bromine-label Gaussians are slightly larger than the actual double-bond because the diameter of the bromines, which is convoluted with the thermal envelope of the double-bonds, is larger than the diameter of the double-bond hydrogens [see Wiener et al. (1991)]. The 1/e half-width of the double-bonds at 66% RH, obtained from specifically deuterated DOPC in neutron scattering experiments, is A_C=C = 4.29 ± 0.16 Å compared to A_Br = 4.96 ± 0.62 Å. The difference in the two widths is only due to differences in the hard-sphere radii and does not depend on hydration. We have estimated the widths of the double-bond distributions from A_Br using the method of Wiener et al. (1991). The widths A_C=C are included in Table 1.

The changes in the parameters of the bromine (and double-bond) distribution are modest over the range of hydrations studied, and especially above 76% RH. This finding is consistent with the idea that the changes in the bilayer with hydration are fairly small (McIntosh and Simon, 1986). Nevertheless, our measurements indicate that changes do occur, consistent with the recent NMR measurements of Gawrisch and his colleagues (Koenig et al., 1997). Fig. 6 B suggests that the changes in Z_Br and A_Br are inversely related. This may reflect volumetric constraints on HC of the lipid bilayer. Such constraints may be of value in the development of scaling methods for hydrations in excess of 16 waters/lipid for which h_obs  3 (Table 1).

	DISCUSSION

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Scaling of x-ray structure factors

We have obtained relative-absolute (per lipid) structure factors for DOPC over the hydration range of 6.2 to 15.9 waters/lipid using specific bromination of the double-bonds and a modification of the scaling procedure of Wiener and White (1991b), who obtained per-lipid structure factors for DOPC with 5.4 waters/lipid. This represents significant and encouraging progress toward determining the complete and fully resolved structure of fluid bilayers over a wide range of hydrations using joint refinement of x-ray and neutron data (Wiener and White, 1991b). Our set of correctly, and unambiguously, scaled structure factors also provides an opportunity to examine scaling methods that are traditionally used in membrane diffraction.

These traditional methods rely heavily upon the so-called continuous Fourier transform, which is the reciprocal space (structure factor) representation of a single unit cell (lipid bilayer profile). The continuous structure factor is plotted in this representation against the amplitude of the reciprocal space vector  S  = 2 sin / where  is the wavelength of the x-rays. The structure factors of order h obtained from multilamellar samples with N bilayers are derived mathematically from the continuous transform by convoluting it with a perfect-lattice function consisting of N delta functions spaced at intervals of d along the z-axis [see review by Franks and Levine (1981)]. This convolution causes the continuous transform to be sampled at  S  = h/d. The values of the continuous structure factor obtained at these sampled points correspond to the structure factors obtained from multilamellar samples.

The relative-absolute continuous Fourier transforms for DOPC calculated using the Shannon sampling theorem (Worthington et al., 1973) are shown in Fig. 7 A for 5.4 waters/lipid (solid curve) and 15.9 waters/lipid (dashed curve). The data points are the structure factors for all hydrations studied (5.4 to 15.9 waters/lipid). Presentation of the data in this manner provides an opportunity to check the consistency of the scaling at different hydrations and to understand the qualitative behavior of the continuous transform at different water contents. To address these issues, we modeled the change in the continuous transform of the unit cell of the bilayer by adding water between bilayers whose structure is known at low hydrations. Specifically, we added 10.5 waters/lipid distributed as a Gaussian to the 66% RH DOPC bilayer structure (Wiener and White, 1992) such that the d-spacing increased by 6 Å. This model is only approximate because 1) we assume that the bilayer structure does not change as a function of hydration (which is not exactly true, see Fig. 6); and 2) we have no detailed structural information about the distribution of water at hydrations above 66% RH (in principle it can be obtained from neutron diffraction experiments). Thus, the only requirement of the model was that the sum of the bilayer scattering profile at 66% RH (dotted line, Fig. 7 B, inset) and the newly added water (dashed line, Fig. 7 B, inset) be smooth and resemble qualitatively a profile at higher hydrations (solid line, Fig. 7 B, inset). This model cannot be used for exact predictions, but it does provide a qualitative idea about the behavior of the continuous transform of the bilayer as hydration is increased. In Fig. 7 B, the continuous transform of the hydrated bilayer at 66% RH (5.4 waters/lipid) is shown as the solid line and the transform of the model (15.9 waters/lipid) as a dashed line. The observed and model transforms of Fig. 7, A and B, respectively, are in excellent agreement and therefore indicative of consistent scaling.

View larger version (24K): [in this window] [in a new window]   FIGURE 7   Observed and model continuous transforms of DOPC bilayers for different hydrations. The continuous transforms, calculated using the Shannon sampling theorem, are the continuous structure factors of a bilayer unit cell plotted against the magnitude of the reciprocal space vector, |S| = 2 sin /. In diffraction from a multilamellar bilayer system, this transform is sampled at values of |S| = h/d to produce structure factors of order h (see text); d is the Bragg spacing. (A) Observed continuous relative-absolute structure factors for pure DOPC bilayers at different hydrations. The solid and the dotted lines are the continuous transforms for 66% RH (5.4 waters/lipid) and 30% PVP (15.9 waters/lipid), respectively. The data points are the observed discrete structure factors covering the same hydration range. (B) Model continuous transforms for the DOPC bilayer at two hydrations based upon the known (Wiener and White, 1992) complete structure of DOPC at 5.4 waters/lipid. The solid line is the continuous transform for DOPC at 5.4 waters/lipid (A) and the dashed line the continuous transform calculated for 15.9 waters/lipid from the model. The prediction of the model is in excellent qualitative agreement with the 15.9 water/lipid transform of (A). Inset: Summary of the model used to calculate the 10.5 waters/lipid transform. The model bilayer profile (solid curve) is the sum of the bilayer profile at 66% RH (dotted line) and additional Gaussian-distributed 11.4 waters/lipid (dashed line).

The discussion of our scaling procedure in the Methods indicates the complex nature of the scaling of membrane diffraction data. In the absence of isomorphous-labeling data, x-ray data obtained in separate experiments at different hydrations are treated by means of the so-called minus-fluid (F) bilayer model (Worthington et al., 1973; Worthington, 1981). The underlying assumption (Worthington et al., 1973) is that the bilayer unit cell can be subdivided in two parts: a bilayer with an electron density distribution _b(z) and thickness d_b and a fluid (water) layer of uniform electron density F and thickness d_w. The F model is defined as a bilayer with an electron density distribution _b(z) F so that the "water layer" of the F model now has an electron density of zero. Because the electron density of the water layer in the F model is 0, the 0th order is given by _bd_b + 0 · d_w, and thus does not depend on d_w. This procedure can be better understood in the context of Eqs. 4 and 5. In the F model, the mean electron density is given by

(18)

where V_lip and V_w are the molecular volumes of the lipid and water, respectively. From Eq. 4, the 0th diffraction order can be defined formally as

(19)

Equation 19 does not contain d_w and is therefore constant for all hydrations. Consequently, changes in d_w have no effect on the continuous transform of the unit cell. This procedure causes the amplitude of the continuous transforms for all hydrations to have the same value at the origin (|S| = 0). Conveniently, the 0th order of pure unbrominated bilayers generally has a value of approximately 0. Thus, if this "dehydrated" bilayer structure of the F model does not change with hydration, all structure factors measured at different d_w will fall on a single continuous tr