Proc. Natl. Acad. Sci. USAVol. 93, pp. 7008-7012, July 1996Biophysics

Structure of the ripple phase in lecithin bilayers(biomembranes/phospholipids/x-ray diffraction)

W.-J. SUNt, S. TRISTRAM-NAGLEt, R. M. SUTERt, AND J. F. NAGLE*tt§Departments of tPhysics and tBiological Sciences, Carnegie Mellon University, Pittsburgh, PA 15213

Communicated by Watt W Webb, Cornell University, Ithaca, NY, April 9, 1996 (received for review December 27, 1995)

ABSTRACT The phases of the x-ray form factors arederived for the ripple (Pp.) thermodynamic phase in thelecithin bilayer system. By combining these phases withexperimental intensity data, the electron density map of theripple phase of dimyristoyl-phosphatidylcholine is con-structed. The phases are derived by fitting the intensity datato two-dimensional electron density models, which are createdby convolving an asymmetric triangular ripple profile with atransbilayer electron density profile. The robustness of themodel method is indicated by the result that many differentmodels of the transbilayer profile yield essentially the samephases, except for the weaker, purely ripple (0k) peaks. Evenwith this residual ambiguity, the ripple profile is well deter-mined, resulting in 19 i for the ripple amplitude and 100 and26° for the slopes of the major and the minor sides, respec-tively. Estimates for the bilayer head-head spacings show thatthe major side of the ripple is consistent with gel-like struc-ture, and the minor side appears to be thinner with lowerelectron density.

Lipids in water self-assemble into lamellar bilayers, whichcomprise the basic structural element of biomembranes. Thesupramolecular packing of the bilayers, in turn, forms lyotropicliquid crystals with rich phase behavior and diverse structures.Among the thermodynamic phases observed in lipid bilayersystems, the ripple (Pp,) phase in lecithin bilayers is especiallyfascinating to a broad range of researchers in condensedmatter physics and physical chemistry as an example of peri-odically modulated phases. Many theoretical papers haveattempted to explain the formation of the ripples (1-8). Suchunderstanding has been delayed because, despite many exper-imental studies (9-19), the structure has not been as wellcharacterized as have the structures for the lower temperaturegel (Lp,) phase (20) and the higher temperature liquid crys-talline (La) phase (21). When an accurate ripple structure isknown, some indication of the energetics of the ripple forma-tion may be obtained by comparing the relative size of theripple wavelength and the ripple amplitude to the size of lipidmolecules and the bilayer thickness (5). Also, correlationbetween the chirality of the constituent lipid molecules and theasymmetry of the ripples may lead to elucidation of themicroscopic origin of the macroscopic properties of the ripplephase (8, 19).

Structural information about the ripple phase can be dividedinto two categories. The first category consists of the two-dimensional lattice parameters, namely, the ripple wavelengthAr, the bilayer packing repeat distance d, and the oblique angle,y for the unit cell that are illustrated in Fig. 1. These well-established parameters have been accurately obtained byindexing low-angle x-ray scattering peaks (9, 10, 12, 16, 18). Forexample, Wack and Webb (18) reported for dimyristoyl-phosphatidylcholine (DMPC) with 25% water by weight at180C that Ar = 141.7 A, d = 57.94 A, and y = 98.40. The second

category consists of the structure within the ripple unit cell,including the shape and the amplitude of the ripples and thebilayer thickness. This second category of structural informa-tion has been quite uncertain. Many ripple profiles have beenproposed (1, 2, 4-10, 14), including sinusoidal, peristaltic, andsawtooth and estimates of the ripple trough-to-peak amplituderange widely from 15 to 50 A (9, 10, 14, 22, 23).The second category of information involves electron den-

sity maps. These require the amplitudes of the form factors,which, ignoring fluctuation corrections (21), are the squareroots of the Lorentz-corrected intensities of the x-ray scatter-ing peaks, and the correct phases for these measured formfactor amplitudes. A traditional approach to solve the phaseproblem is the "pattern recognition" method employed byTardieu and coworkers (9); this method selects those phasecombinations that will give rise to electron density mapsagreeing with known physical properties of the bilayer system.Considering that there are typically 20 or so peaks in most ofthe ripple phase x-ray data, finding the right phase combina-tion out of about 220 possibilities is a difficult task. Theapproach we employed involved modeling and nonlinear leastsquares fitting. Possibly, this method may be generally usefulwhen studying other disordered and fluctuating biologicalsystems that have intrinsically continuous electron densitymaps rather than an atomic resolution crystal structure.

METHODSThe modeling approach first creates a functional form for theelectron density that incorporates known and plausible prop-erties of the bilayer but that contains free parameters toincorporate unknown detailed structure. The success of themethod depends on this modelling step, which requires somecare and is, therefore, the emphasis of this section. Then, thevalues of the free parameters in the real space model aredetermined by routine nonlinear least squares procedures sothat the Fourier components of the model provide the best fitto the measured intensities. This last step automatically de-termines the phases of the model, which are then used with theoriginal intensity data to produce an electron density map.

Lattice Structure. It has been shown from x-ray studies (9,10, 12, 16, 18, 19) that ripples in different bilayers areregistered to form a two-dimensional oblique lattice as shownby the unit cell in Fig. 1. The unit cell vectors can be expressed(see Fig. 1) as

a= d cot y x + d z and b = Ar x. [1]

The corresponding reciprocal lattice unit cell vectors (see Fig.1) are

2,w 27ur 27ffA= 2 andA = x - zd Ar Artan y [2]

Abbreviations: DMPC, dimyristoyl-phosphatidylcholine; SDF, simpledelta function; MDF, modulated delta function; M1G, modulated 1G.§To whom reprint requests should be addressed.

The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertisement" inaccordance with 18 U.S.C. §1734 solely to indicate this fact.

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qz 1 2 3 4600fD/

I

a triangular shape as shown in Fig. 1. We shall call the longerside of the ripple the major side, and the shorter side the minorside. The triangular ripple profile is completely described bythe amplitudeA and xO, the projection of the major side ontothe x axis, with obvious relations for other quantities such asthe slope angles of the major and minor sides. Choosing theorigin at the center of the unit cell shown in Fig. 1 gives theripple profile

A tAr- _ (x + 2)ifAr XO 2)i

A-x ifXO

A_ x- 2r ifAr-XO '2)i

Ar XO

2 x < ---2- 2'

xO xo

2-~~ A

2'AXo Ar2 2X2

[7]

FIG. 1. Real space pattern (Lower) shows the asymmetric rippleprofile by bold lines for two bilayers. The q-space pattern (Upper)shows the (h,k) peak positions as the centers of the circles. Largermagnitudes of the form factors for reflections that were observed (18)are indicated by darker circles. The dot-dash lines show the perpen-diculars to the major and the minor sides of the ripple profile.

so the q-space vector qhk for the Bragg peak with Miller indices(h,k) is

qhk = hA + kB, [3]

with some examples shown in Fig. 1.Electron Density Model and Form Factor. The ripple profile

is the line z = u(x) that describes the center of the bilayer inthe (x,z) plane, such as the one shown in Fig. 1. The closelyrelated contour function is defined as

C(x,z) = S - u(x)]. [4]

The transbilayer electron density function Tq,(x,z) expressesthe actual electron density at points (x,z) that lie on a straightline with slope q, from the vertical as shown in Fig. 1. Anunderlying electron density profile will be chosen similar tomodels for flat bilayer phases. Then, the electron densitymodel for p(x,z) within the unit cell is described as theconvolution of a ripple contour function C(x,z) and thetransbilayer electron density profile T,(x,z):

p(x,z) = C(x,z)*T1p(x,z). [5]

This convolution moves the transbilayer electron density func-tion so that it is centered on the ripple profile.The form factorF(4) is the Fourier transform of the electron

density expressed in Eq. 5 and is the product

F(q) = FC (q) FT(q-) [6]of the Fourier transform Fc(q) of C(x,z) and of the Fouriertransform FT(q) of T,&(x,z).

Ripple Profile. The results of freeze fracture electron mi-croscopy (11, 13, 15) and scanning tunneling microscopy (22,23) strongly suggest that the ripple profile is asymmetric. Thisconclusion is further supported by the result of x-ray diffrac-tion, in which the ripple unit cell is oblique (y not equal to 900)(9, 10, 12, 16-19). For our model ripple profile u(x), we choose

This particular choice shows explicitly the inversion symmetryin the ripple profile, so that the form factors are real.Using Eq. 7, the contour part of the form factor is

1 A-Fc(i) = AJ2dx ei[qxx+q,u(x)]

2

xO Ar -x cos 2(qxAr+2=A srnc(( r) +-w)Ar lkr ~~cos 2 V qxAr-C)

[8]X i qzAk _ )

where the intermediate variable w is defined as

1

co(q-) = (qjxo + q_,A). [9]

The first term on the right hand side of Eq. 8 is the contributionfrom the major side of the ripple, with scattering amplitudeXo/A,, proportional to its projected length onto the x axis,whereas the last term is the contribution from the minor sideof the ripple, with amplitude (A - Xo)/Ar. Eq. 8 also shows thatthe maximum scattering from the major side occurs along thatq direction determined by = 0 in Eq. 9. In real space, this isalso the direction normal to the major side of the ripple.Similarly, the maximum scattering direction of the minor sideis given by qxAr/2 - w = 0, and this can be shown to be thedirection normal to the minor side. These directions areillustrated by the dot-dashed lines in Fig. 1. This theoreticalpicture for the location of the strong (h,k) peaks agreesqualitatively with the experimental intensity pattern of Wackand Webb (18) shown in Fig. 1 as well as other x-ray data (16,19). This general pattern would also be expected to pertain toripple profiles that are not perfectly triangular, for example,with rounded corners.

Transbilayer Electron Density Profile Tq,(x,z). In a study ofgel phase DPPC bilayers using the electron density modelingapproach (24), it was found that the precise functional form ofthe electron density model did not much affect the resultsprovided that the essential structural features of bilayers wereincorporated into the model. Therefore, we first used a verysimple model for Tq,(x,z) in which the two opposite headgroupregions consist of two identical positive 5 functions, withamplitude pH, separated by the head-head spacing 2Xh, andwith the central methyl trough consisting of a negativefunction, with amplitude PM. The methylene region of bilayershas an electron density close to that of water and hence is set

k2 -1

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equal to zero in the following model [minus fluid (25)] ofT,p(x,z):Tp(x,z) = b(x + z tan ti) {pH[8(z - Xh cos qi)

+ 8(Z + Xh cos q)] - PM6(Z)}. [10]

The form factor corresponding to this transbilayer electrondensity profile is

FT(q) = PM[RHM cos(qzXh cos q-qXh sin qi) - 1], [11]

where RHM = 2PH/PM. We will call this the simple deltafunction (SDF) model. Following the procedures in ref. 24, wealso used a more realistic model, called the 1G hybrid model,that consists of Gaussian functions for the headgroups and theterminal methyl trough and that also allows for differencesbetween the electron density of methylenes and of water. Wewill call this the simple 1G model. We also considered morecomplex models that modulated the electron density-along thedirection perpendicular to a- in real space in Fig. 1; this is alsothe direction B in reciprocal space, so this modulation stronglyaffects the (0,k) pure ripple form factors. Our best ripplemodulated models (MDF and M1G) had two additionalparameters, one to allow for the electron density across theminor side to be different by a ratiofi from the electron densityacross the major side and a second parameter f2, which ismultiplied by 8 functions 8[(x ± xo/2)] to allow for a differentelectron density near the kink between the major and theminor sides.

Determination of the Phases and the Electron Density Map.Combining Eqs. 6, 8, and 11, the Lorentz-corrected scatteringintensity for the (h,k) reflection of the SDF model is

[ WhkArX(k h- ])hk)

x [RHM cos(qhkXh cos - qhkX sin tf) - 1]2, [12]which contains six independent parameters. Two parametersare related to the ripple profile, namely the ripple amplitudeA and the horizontal projection length of the major side of theripple, xo. The transbilayer profile contains three parameters,the headgroup to methyl trough electron density ratio (RHM),the monolayer shift angle (4i), and the headgroup positionrelative to the center of the bilayer (Xh). The sixth parameteris a common scaling factor for all the I(h,k) peaks. For thesimple 1G model, all the transbilayer profile parameters weretaken from gel phase data (24) except for the headgrouppositionXh. The MDF and MiG models had the two additionalparameters,fi andf2, mentioned above. [The parameters A, d,and ry are known from indexing the (h,k) peaks.] Standardnonlinear least squares fitting procedures were employed toobtain unknown parameters by finding the best fit of theintensities given by Eq. 12 to the IF(h,k)12 reported by Wackand Webb (18). This procedure does not require advanceknowledge of any phases. However, once the best values of theparameters in the electron density model are determined, theform factors, and especially the phases, are determined. Usingthese phases and the amplitudes of the form factors IF(h,k)lfrom the intensity data (18), an experimental relative electrondensity map p(r) is obtained using

p(x,z) = E EF(h,k)cos(qhkx+qhkz).h20 k

[13]

RESULTS AND DISCUSSIONThe absolute form factors IF(h,k)I reported by Wack and Webb(18) are shown in Table 1 along with two of our many modelfits. Our most striking result is that all of our models have the

FIG. 2. Electron density map using measured absolute form factors(18) and phases from Table 1, omitting (O,k) reflections. The smallestelectron density is pure black and the largest electron density is purewhite with a linear gray scale interpolating between. The black dottedlines show two of the loci of maximum electron density. For theelectron density along dashed lines A, B, and C, see Fig. 3. Theparallelogram shows the unit cell whose width is Ar, which equals141.7 A.

same phases for all the (h,k) reflections displayed in Table 1,except for the pure ripple reflections with h = 0. Fig. 2 showsthe Fourier map of the electron density using the measured(18) absolute form factors IF(h,k)l, omitting the pure ripplereflections (O,k) and our phases, shown in Table 1. Since theF(O,k) are rather small, their inclusion with any phase combi-nation makes rather small differences to the electron densitymap and negligible differences to the ripple profile.

Table 1. Form factors

100 X q, Data* Model F(h, k)h k A-i IF(h, k)l SDF MlG0 1 4.48 5.3 -1.4 7.90 2 8.96 9.7 1.0 -4.60 3 13.4 7.8 -0.4 0.91 -2 13.0 - 5.0 6.71 -1 11.1 60.8 -42.8 -60.41 0 10.8 100.0 -96.7 -99.11 1 12.3 26.9 38.1 28.51 2 15.0 -23.6 -11.81 3 18.5 7.6 13.9 4.81 4 22.3 -6.3 -2.42 -3 23.8 8.6 10.92 -2 22.2 15.1 -9.0 -14.12 -1 21.5 71.2 -66.9 -71.72 0 21.7 39.7 -41.0 -39.92 1 22.8 33.9 31.1 30.62 2 24.6 22.7 -24.3 -22.82 3 27.1 14.2 17.1 15.22 4 30.1 7.8 -10.3 -9.13 -3 33.3 -6.1 -6.03 -2 32.5 29.3 22.3 29.03 -1 - 32.2 44.2 42.2 44.53 0 32.5 12.0 2.2 0.73 1 33.5 -11.3 -10.53 2 35.0 10.5 14.9 13.93 3 37.0 14.9 -15.0 -13.43 4 39.4 10.0 12.5 10.84 -2 43.0 -81.1 -39.2t4 -1 43.0 -63.5 -23.6t4 0 43.4 26.1 10.3t*From Wack and Webb (18).tValues are consistent with the densitometer trace shown in figure 4in ref. 18.

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Table 2. Ripple structural quantities

Model A, A xo, A P xh, A fi f2SDF 18.6 103 50 20.1 1.0 (fixed) 0 (fixed)MDF 20.0 103 90 20.4 0.7 -2MlG 19.0 103 90 19.3 0.6 -1

The ripple profile is shown in Fig. 2 by the dotted lines thatare defined as the loci of maximum intensity in the headgroupregion. This ripple profile clearly has the basic triangular shapeassumed in our modeling, but this result is not tautologicalbecause the experimental absolute form factors are employedin Fig. 2. The ripple profile shown in Fig. 2 is in excellentquantitative agreement with the fitted model ripple profiles.The parametersA andxo that describe the ripple profile, as wellas other parameters in the models, are given in Table 2 forthree models. From this we conclude that the ripple amplitudeA is about 19 A and the projection of the major side on the aaxis is about 103 A for the single DMPC sample for whichWack and Webb (18) reported absolute form factors.To obtain a more quantitative picture of the thickness of the

bilayer in various parts of the ripple, electron density profiles(using the experimental IF(h,k)l) are plotted in Fig. 3 along thethree slices shown by straight dashed lines in Fig. 2 (A, B, andC). SliceA is along the normal of the major side and is centeredin the middle of the hydrocarbon region; it indicates that thehead-head spacing is 38 A and the water spacing is 21 A on themajor side. Slice B is along the normal to the minor side andcentered in the middle of the hydrocarbon region; it indicatesthat the bilayer head-head spacing on the minor side is 31 A.Slice C is along the normal to the minor side, but centered inthe water region; it indicates that the water spacing of theminor side is 18 A. All these spacings are, of course, subject toFourier truncation errors since the intensity data only includeup to three lamellar orders (h = 1-3). Furthermore, nocorrections to the form factors have been made to account forfluctuations. For smectic liquid crystals, such corrections areconsiderably more complex than simple Debye-Waller fac-tors; they are expected to be smaller for the ripple phase thanfor the fluid L

Proc. Natl. Acad. Sci. USA 93 (1996)

Although the phases for the pure ripple (O,k) reflections arenot as well-determined by our procedure as the h f 0 phases,the latter phases, which are very robustly determined using ourmodel method, already determine the ripple profile. Indeed,the effect of different (O,k) phases on the electron density mapsis less important than the effect of including the three h = 4reflections shown in Table 1 for the MlG model. Furtherprogress toward understanding the molecular packing of lipidsin the ripple phase should employ careful analysis of wide-angle scattering data. This study establishes the frameworkinto which such additional information must be incorporated.

This research was supported by National Institutes of Health GrantGM44976.

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