Journal of Structural Biology 133, 119–131 (2001)doi:10.1006/jsbi.2001.4376, available online at http://www.idealibrary.com on

Spherical Reconstruction: A Method for Structure Determinationof Membrane Proteins from Cryo-EM Images

Qiu-Xing Jiang, David W. Chester, and Fred J. Sigworth

Department of Cellular and Molecular Physiology, Yale University, 333 Cedar Street, New Haven, Connecticut 06520

Received January 25, 2001, and in revised form May 10, 2001

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We propose a new method for single-particle re-construction, which should be generally applicableto structure determination for membrane proteins.After reconstitution into a small spherical vesicle, amembrane protein takes a particular orientationrelative to the membrane normal, and its position inthe projected image of the vesicle directly definestwo of its three Euler angles of orientation. Thespherical constraint imposed by the vesicle effec-tively reduces the dimensionality of the alignmentsearch from 5 to 3 and simplifies the detection of theparticle. Projection images of particles in vesiclescollectively take all possible orientations and there-fore cover the whole Fourier space. Analysis of im-ages of vesicles in ice showed that the vesicle den-sity is well described by a simple model formembrane electron scattering density. In fittingthis model we found that osmotically swollen vesi-cles remain nearly spherical through the freezingprocess. These results satisfy the basic experimen-tal requirements for spherical reconstruction. Acomputer simulation of particles in vesicles showedthat this method provides good estimates of the twoEuler angles and thus may improve single-particlereconstruction and extend it to smaller membraneproteins. © 2001 Academic Press

Key Words: spherical reconstruction; cryo-elec-tron microscopy; membrane proteins; reconstitu-tion; single-particle reconstruction; Euler angles.

INTRODUCTION

In the past three decades, single-particle recon-struction (SPR; Hoppe and Hegerl, 1980; Frank,1996) has evolved into a useful method for struc-tural studies of macromolecular complexes largerthan about 500 kDa in mass. SPR does not requirelarge sample quantities and avoids the crystalliza-tion step required for crystallographic studies. It hasrevealed structural information at moderate resolu-tions for high-symmetry virus particles (Zhou et al.,2000; Bottcher et al., 1997; Conway et al., 1997) as

119

well as for low-symmetry or nonsymmetric particlessuch as ryanodine receptors and ribosomes (Gabash-vili et al., 2000; Samso and Wagenknecht, 1998;

erysheva et al., 1999).In the case of ribosomes, the laboratories of Frank

and van Heel have obtained structures at moderateresolutions (Frank, 1998; Gabashvili et al., 2000;van Heel, 2000) along with valuable informationabout the locations of tRNAs and elongation factors(Stark et al., 1997a, 1997b, 2000; Agrawal et al.,2000; Frank, 2000) and the docking surface for theyeast translocon (Beckmann et al., 1997). These re-sults have demonstrated the potency of SPR for thestructural studies of large protein complexes. More-over, with the development of new techniques forspecimen preparation and data collection, and novelimage processing algorithms, it may become possi-ble to determine protein structures through SPR athigher resolution. There is also the hope that theminimum size limit (Henderson, 1995) for particlesmay be reduced below 500 kDa, where many mem-brane proteins will become accessible to SPR.

When applied to a membrane protein, the usualSPR protocol starts from cryo-EM images of thepurified protein solubilized in detergent (Samso andWagenknecht, 1998; Serysheva et al., 1999; Grigor-ieff, 1998; Orlova et al., 2000). During image pro-cessing, the five parameters required for the accu-rate description of the position and the orientation ofeach projected particle are determined through a setof classification and alignment procedures. With re-liable orientation and location determination, theprojections are then combined in Fourier space togenerate a three-dimensional structural model.

In this paper we propose an SPR approach thatimages membrane proteins reconstituted in smallspherical lipid vesicles and uses the geometrical po-sition of particles in vesicles to determine two oftheir three orientation angles. First, the rationaleunderlying the scheme is introduced. Then we showthat to satisfy the requirements for its practical

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120 JIANG, CHESTER, AND SIGWORTH

application, it is possible to preserve the sphericityof small vesicles (35–100 nm in diameter) and todescribe the electron scattering density of lipid bi-layer well enough to separate the particle imagesout of the vesicle images. Finally, a computer simu-lation of the scheme with simulated images of pro-tein particles in vesicles is presented. The resultssuggest that the experimental implementation ofspherical reconstruction on membrane protein com-plexes will be advantageous in orientation determi-nation and may facilitate SPR from images of mem-brane proteins smaller than 500 kDa in size.

MATERIALS AND METHODS

Preparation of vesicles. All lipids were purchased from Avanti(Alabaster, AL) and detergents from Anatrace Inc. (Maumee,OH). To prepare vesicles, lipids in chloroform were transferredinto a glass test tube. The solvent was evaporated by a stream ofdry N2. The lipids were then kept under vacuum for 30 min beforedetergent solution was added. The final solution contained 0.5mg/ml lipids, 16 mM CHAPS, 100 mM NaCl, 2.0 mM KCl, 0.5 mMNa2EDTA, and 10 mM HEPES at pH 7.4. The solution wassonicated under argon for about 30 min at room temperatureuntil it was completely clear.

Lipid vesicles were formed in a cold room by gel filtration ordialysis. For gel filtration, Sephadex G-50 resin (Pharmacia, Upp-sala, Sweden) was hydrated in phosphate-buffered saline (150mM NaCl, 20 mM NaPO4, pH 7.1) plus 0.02% NaN3 for at least24 h. A 1.0 3 15.0 cm column (Bio-Rad) was packed and equili-brated with 10 vol of running buffer (100 mM NaCl, 2.0 mM KCl,0.5 mM Na2EDTA, and 10 mM Hepes, pH 7.4). Two hundredmicroliters of lipid–detergent solution was loaded onto the col-umn and eluted with the same buffer. The outlet of the columnwas connected to an in-line UA-6 detector (ISCO) set at 453 nm.The fractions containing vesicles were collected and pooled to-gether.

When dialysis was used to prepare vesicles, 1.0 ml of thelipid–detergent mixture was loaded into a piece of 10-mm-diam-eter dialysis tubing and dialyzed against 4000 vol of the runningbuffer supplemented with 0.5 g Bio-Beads (Bio-Rad) for 20 h withone buffer change in the middle. About 16 h before freezing,vesicles were extruded 11 times through a 50- or 80-nm polycar-bonate filter in a miniextruder (Avanti). Twenty minutes beforefreezing, the vesicles were osmotically swollen by adding 20% volof water into the suspension. The vesicles from gel filtration ordialysis showed no difference in the membrane scattering densityprofile or in sphericity (see below).

Cryo-EM imaging. Copper–palladium grids (400-mesh) werepurchased from Ted Pella (Redding, CA). A holey carbon film(Hayat, 1989) was laid onto the polished side (front side) of thegrids. The thickness of the carbon film was in the range 20–40nm. In some experiments, Quantifoil grids (Ted Pella) with 1.2- or2-mm holes were used instead. The laboratory-made grids wereaged for at least 1 week, lightly coated on the back side withcarbon evaporation on the day before use, and glow-discharged inthe presence of n-amyl amine (Sigma) for 45 s about 1 h beforeuse. The Quantifoil grids were glow-discharged in the same way,but with no other treatment.

Quick freezing of the specimens (Dubochet et al., 1988) waserformed in a cold room at ;76% relative humidity. Five micro-iters of the swollen vesicle suspension was loaded onto a holeyrid, blotted away, and loaded again. After being blotted from theront side for 7–8 s with a slip of filter paper (Whatman No. 4), therid was plunged into liquid ethane.

Images were obtained using the Philips CM12 and CM200 FEGlectron cryo-microscopes at the Skirball Institute, New Yorkniversity. Low-dose techniques were used such that the dose for

ach exposure was about 10 e/Å2. Images were taken at 45 000 or50 000 magnification and 0.8- to 2.5-mm defocus. They were ex-posed on Kodak SO-163 film and developed for 12 min in full-strength D19 developer. Negatives were selected using a diffrac-tometer and scanned with a Zeiss SCAI film scanner.

Processing vesicle images. From single EM negatives or tiltpairs, vesicle images were picked using the software packageSPIDER plus WEB (Frank et al., 1995). For each vesicle image,CTF-filtered vesicle models of different radii were first used tolocate the vesicle and to determine its radius. These parameterswere refined by least-squares search. Two additional free param-eters describing the ellipticity of the vesicle were then introducedand the least-squares search was repeated. All the programs werewritten in MATLAB (version 5, the Math Works Inc., Natick,MA), running on Macintosh and Linux computers.

Simulations. The electron density map of Ca21-ATPase at 8-Åresolution (Zhang et al., 1998) was kindly provided by Dr. DavidStokes. After estimating and subtracting from the map a densitycorresponding to the lipid bilayer, we created a speculative dimerstructure by duplicating the monomer map in twofold symmetryabout the membrane normal. The dimer map, projected fromdifferent orientations, was added into the images of ideal vesicles.The relative scaling of vesicle and protein density was chosen tomatch that observed in the tubular crystals of Ca21-ATPase.After the images were filtered according to a CTF with a B factorof 300 Å2 and a random defocus value in the range of 2.0–4.0 mmfor 200-keV electrons, Gaussian noise was added such that theirsignal-to-noise ratio roughly matched that of the experimentalcryo-EM images of vesicles. These images were used as the inputto procedures that search for particles, determine the angles, andperform a 3D reconstruction.

RESULTS

The Rationale Underlying SphericalReconstruction

The general idea of spherical reconstruction isshown in Fig. 1, where images of individual twofoldsymmetric particles in spherical vesicles are simu-lated. The reconstruction starts from noisy images ofvesicles (Fig. 1B). The membrane density is fittedwith a simple model and then subtracted from theimages, leaving the particle images behind (Fig. 1C).In the domain of the vesicles, an exhaustive searchis then performed to find the exact positions of theparticles (Fig. 1D). Each particle’s position inside avesicle image yields estimates of its “latitude andlongitude,” defining two of its three orientation an-gles in 3D space. The third angle, determined in aparticle search by projection-matching, completesthe set of alignment parameters that are requiredfor 3D reconstruction of these particle images.

Theoretical considerations. The fundamental as-sumption underlying spherical reconstruction isthat in a liposome a membrane protein takes a spe-cific orientation relative to the local membrane nor-mal. This is expected to be the case for many mem-brane proteins. For example, a tetrameric ion

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121SPHERICAL RECONSTRUCTION FOR MEMBRANE PROTEINS

channel is expected to have its fourfold symmetryaxis very close to the membrane normal. In Fig. 2A,particles a and b at symmetrical positions in aspherical vesicle of radius R are projected along theZ-axis of real space to form images at the identicalposition D inside the vesicle image. In general, theprojection of particle a differs from that of particle b,which requires that particles in the upper and lowerhemispheres of the vesicle be distinguished. Theparticles in all possible positions on the vesicle col-lectively cover the whole Fourier space. Actually,particles at all positions along the rim in vesicleimages form a complete single-axis rotational setaround the Z*-axis in reciprocal space. Therefore,the projection images of particles at various loca-tions in vesicles cover the whole Fourier space re-dundantly.

The position of a particle in a vesicle can be usedto define its azimuthal angle f and latitudinal angle

FIG. 1. Simulated vesicle images. From the scattering densitysimulated images (180 3 180 pixels, 4 Å/pixel) of 50-nm vesicles wImages were filtered with the CTF (200 keV, 3-mm defocus, and B 5of experimental images. Noiseless (A) and noisy images (B) are showsubtraction of the vesicle density (C) were subjected to an exhaustives shown in row (D) where the white areas indicate maxima. The p

u. Search for rotation angle c about the membranenormal, along with the resolution of the ambiguityin u, can be accomplished by projection matching(Frank, 1996) or sinogram correlation (van Heel,1987) or possibly by maximum-likelihood techniques(Sigworth, 1998). Particles in different vesicles atdifferent orientations can be combined together inFourier space to reconstruct a 3D structure modelfor the particle.

Determining the orientation angles. The two an-gles f and u can be determined from the geometricalrelationship between the particle and the vesicleimage (Fig. 2A). The azimuthal angle f is deter-mined directly from the particle’s position relative tothe X-axis. The distance r from the vesicle center tothe particle’s mass center yields the value of u fromthe relationship

r 5 ~R 6 d!sin u, (1)

f the sarcoplasmic-reticulum Ca21-ATPase and the bilayer model,mputed with ATPase “dimers” incorporated at different locations.), and noise was adjusted to roughly match the signal-to-noise ratio

different values of u at f 5 0° and c 5 0°. Particle images after thecorrelation search at c 5 0°. The resulting cross-correlation function

are identified at the correct position in each case.

map oere co300 Å2

n forcross-

articles

ra

122 JIANG, CHESTER, AND SIGWORTH

where R is the radius of the vesicle, and d is theadial distance between the particle’s mass centernd the middle of the bilayer.Two possible values for u come from particles re-

siding in the upper or lower hemispheres of vesicles.The value of u must therefore be resolved during theevaluation of an image by comparison with the twopossible projections for the particle. A second ambi-guity arises from the two possible incorporation di-rections of membrane proteins in vesicles, leading tothe two possible signs for d in Eq. (1). The projectionimages of a particle in the same position in a vesiclebut with two different incorporation directions aremirror images of each other. Resolution of this am-biguity can also be achieved during the alignmentsince we need to discern only the two possibilities.The magnitude of d can be refined during the recon-struction procedure, as the position of the particle’smass center is computed during each iteration.

Practical requirements for the scheme. Forspherical reconstruction to work properly, the vesi-cles must have a spherical shape, and their mem-brane scattering density must be quantitatively welldescribed to allow its subtraction from the images ofvesicles to yield particle images alone. To testwhether we can satisfy these two requirements, weobtained cryo-EM images of small vesicles and fitted

FIG. 2. Real-space positions and Fourier-space central sectiosymmetrically related positions in a spherical vesicle. The electrprojections will be in the same position D in the vesicle image as sha are labeled by the convention of Radermacher (1991). The posittwo Euler angles (detailed in the text). (B) In Fourier space, the pa horizontal common line.

to them a model for the electron scattering density ofa lipid bilayer.

Modeling Vesicles

X-ray scattering of packed multilayer membranes(Wiener and White, 1992) shows a density profilewith peaks at the head group regions and a centraldip representing the van der Waals distance be-tween the terminal methyl groups of the two oppos-ing lipid chains (Tanford, 1991), which is about 4 Å.We expect the electron scattering density of the bi-layer to be qualitatively similar to its X-ray density(Hoppe and Hegerl, 1980; Henderson, 1995; Frank,1992). A starting model for fitting the scatteringdensity of lipid vesicles of the form shown in Fig. 3Awas chosen. Here, the average scattering density oflipid chains is assumed to be unity. The model pa-rameters are then the membrane thickness d, theheight t1 of the extra head-group density, and thedepth t2 of the central dip. The half-widths of thehead-group peaks and of the central dip are as-sumed to be 4 Å. After Gaussian filtering to describeapproximately the thermal disorder in native hy-drated bilayers, the model assumes a smooth formas shown in Fig. 3B. The convolution of the modelwith the point-spread function of the electron micro-scope will also smoothen the sharp edges in the

membrane proteins in vesicles. (A) Particles a and b are at twom is assumed to travel along the Z-direction. The two particles’t the bottom. The three orientation angles (f, u, and c) of particlethe particle images in the projected image of a vesicle can defineions of the two particles correspond to distinct planes that share

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123SPHERICAL RECONSTRUCTION FOR MEMBRANE PROTEINS

model. Since we do not see much difference in usingthe Gaussian filtered model rather than the nonfil-tered one, we have used the nonfiltered model in ourcalculations.

From the starting model one can compute thedensity profile of the projection image of an idealvesicle. Figure 3C shows the density profile D(r)from the center of a projected vesicle image (Fig. 3D)to its edge along the radial direction, computed fromthe lipid bilayer density r( x) according to

D~r! 5 Er

`

r~ x!x

Îx2 2 r2 dx. (2)

Starting from the center, there is a gradual increasein density. The peak is at the head-group region ofthe inner bilayer leaflet, and a rapid decrease indensity extends to the outer leaflet.

FIG. 3. The vesicle model. (A) The model of the membrane scead group peak height t1 5 0.2, and the central dip t2 5 0.05.

(B) The model after filtering with a Gaussian filter with a cornemage of a vesicle of 500 Å in diameter with the membrane profi

The Simple Model Fits the Vesicle Density VeryWell

To test how well the model fits the scatteringdensity of a lipid bilayer, we used vesicle imagestaken with a field-emission electron microscope at200 kV and 0.8- to 2.3-mm defocus. Figure 4A showsthe FEG image of an 87-nm-diameter vesicle madefrom egg phosphatidylcholine (PC). The best-fittingmodel for simulating the membrane scattering den-sity is presented in Fig. 4B. The goodness of the fitcan be seen in Fig. 4C through the comparison of therotationally averaged density profiles of the modeland the centered vesicle image. The close matchbetween them shows that the model describes thevesicle density very well. Subtraction of the fittedmodel from the raw data leaves only a noisy back-ground (Fig. 4D).

Fitting the images of 40 egg PC vesicles of 60–100nm in diameter at different defocus levels gave con-sistent results: the membrane thickness d 5 50 6 3

g density profile with membrane thickness d 5 50 Å, the extraalf-widths of the head group peaks and the central dip are 4 Å.ency 1/11 Å21. (C and D) The density profile and the projectionn in A.

atterinThe h

r frequle show

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124 JIANG, CHESTER, AND SIGWORTH

Å (mean 6 SD), the head group peak t1 5 0.21 60.17, the central dip t2 5 0.05 6 0.07, and theellipticity coefficients were all below 0.03. The least-squares search was not very sensitive to changes inthe head-group peak height and the central dipdepth, probably because of the small contributions ofthese peaks to the general shape of the density pro-file and the peak-broadening effect due to the ther-mal disorder in the fully hydrated bilayer. A modelwith the average d, t1, and t2 values was found to fitall the vesicles quite well and was therefore used inthe following fitting of tilted pairs of vesicle images.

The Sphericity of Small Vesicles Is Well Preserved

Since the images of small vesicles in the holes ofholey grids appeared to be nearly circular, we made

FIG. 4. Fitting of the vesicle membrane density. (A) A vesicleat 200 keV. The CTF was estimated from the fitting of two areas omm and B 5 290 Å2. Vesicles were extruded through an 80-nm fin diameter. (B) The fitted model computed according to the CTead-group peak height t1 5 0.18, depth of the central dip t2 5he elliptical coefficients e1 and e2 are related to e and f0 in Eq. (

density profile of the vesicle image (dots) and the model (solid curvregion of the inner membrane leaflet is downward instead of usubtracted from the raw image in (A).

a first-order approximation to their deviations fromcircular shape through an elliptical model of theprojected density. An ellipse can be expressed inpolar coordinates (R, f) by the equation

R~f! 5R0

Î1 1 2e cos@2~f 2 f0!#, (3)

where R0 represents the average radius, e repre-ents the ellipticity coefficient, and the orientationngle f0 is the angle between the short axis and the

abscissa. If a and b denote the length of the long andshort radii of an ellipse and are close to each other,we define e as

of egg phosphatidylcholine was imaged at nominal 2-mm defocusn film in the same micrograph, yielding the defocus value of 2.23iving final diameters of 60–110 nm. The vesicle shown is 87 nm

optimized radius r 5 434 Å, membrane thickness d 5 48 Å,and the two elliptical coefficients e1 5 0.0024 and e2 5 0.0011.5 =e1

2 1 e22 and tan(2f0) 5 e2/e1. (C) The rotationally averaged

to the CTF effect, the membrane density peak at the head-groupas in Fig. 3C. (D) The noise image after the model in (B) was

madef carbolter, gF with0.10,4) as ee). Duepward

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125SPHERICAL RECONSTRUCTION FOR MEMBRANE PROTEINS

e 5a2 2 b2

2~a2 1 b2!<

a 2 b2 R0

. (4)

To evaluate the sphericity of vesicles in vitreousice, we prepared vesicles of lipid mixtures of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphotidylcholine(POPC):1-palmitoyl-2-oleoyl-sn-glycero-3-phos-photidylethanolamine (POPE):1,2-dioleoyl-sn-glyc-ero-3-phosphotidylserine (DOPS):sphingomyelin:brain phosphatidic acid (PA):cholesterol in a weightratio of 40:20:10:3:2:5. The vesicles were 35 to 75 nmin diameter after being extruded with a 50-nm filterand were imaged at 0° and 54° tilt (Fig. 5). Compar-ing the ellipticity of these tilt pairs should tell usabout the magnitude of distortions in the vesicles.As a control, pairs of images were taken at 630°(data not shown) and showed no difference in ellip-ticity distribution.

The interpretation of the tilted images requires amodel. Suppose that a spherical vesicle is com-pressed along the Z-axis, making it an oblate spher-oid. Its surface satisfies the equation

x2 1 y2

a2 1z2

b2 5 1 ~a . b!. (5)

Its projection in the ( x, y) plane will be an ellipsewith ellipticity ez as given in Eq. (4). In the projec-tion of such a vesicle along a direction tilted from the

FIG. 5. Images of vesicles at 0° and 54° tilt. Vesicles werephingomyelin:PA:cholesterol at 40:20:10:3:2:5 (by weight) solubiameters of 35–75 nm. (A) The image was taken without tiltingngle (a) and the direction of the tilt axis were determined from987).

Z-axis by angle a, for small ez we can approximatethe short axis b9 of the projection image by

b9 <R0

Î1 2 2ezcos 2a. (6)

The long axis in the projection image is still a. Theobserved ellipse will have a coefficient e given by

e <~1 2 cos 2a!ez

2 2 2~1 1 cos 2a!ez. (7)

For 54° tilt, e ' 0.65 ez, while for 30° tilt, e ' 0.25ez. Thus if there is substantial vesicle distortion inthe vertical direction, we expect to see large elliptic-ity values in the 54° tilt images.

For 20 pairs of images with 0° and 54° tilt, theobserved ellipticity is presented in the histograms inFig. 6. At zero tilt, all of the vesicles had ellipticityvalues below 4% with a median value of 1%. With54° tilt, the maximum observed ellipticity was 4.5%and the median was 2.5%. The larger ellipticity val-ues in the tilted images mean that there is moredistortion of the vesicles in the vertical directionthan in the horizontal directions. Is the distortion ofa consistent nature, such as a vertical flattening ofthe vesicles? To test this, we plotted e and 2f0 in apolar representation (Figs. 6C and 6D). As it turnsout, there is negligible correlation between f and

red by gel filtration of a lipid mixture of POPC:POPE:DOPS:n CHAPS. They were extruded through a 50-nm filter, yieldingoughly the same field was exposed again with a 54° tilt. The tilting the positions of vesicles in the images (Radermacher et al.,

prepailized i. (B) Rmatch

0

126 JIANG, CHESTER, AND SIGWORTH

the tilt axis. Thus although there is substantial dis-tortion in the vertical direction, it is not consistentlya flattening or an extension of the vesicles.

The effect of the vertical distortions on the orien-tation determination is expected to be relativelysmall. From Eq. (7), the median ellipticity of 2.5%that we observed in 54° tilt images corresponds to anellipticity of 3.8% in the vertical direction. Suchdeviation from spherical shape will have its largesteffect on the estimation of latitudinal angle u andleads in the worst case to a 4.2° error in u. Thiscorresponds in turn to a 3.6-Å error in the location ofatoms at a radius of 50 Å from the center of aparticle. We conclude that small vesicles of 35–75nm in diameter can be kept spherical enough duringthe sample preparation for cryo-EM. Their geomet-rical properties can therefore be reliably used fordetermining the orientation of particles incorpo-rated in them.

Different membrane proteins may require specific

FIG. 6. Small vesicles preserve their sphericity. The distribumajority of the vesicles have ellipticity e smaller than 4%. (C and54° tilt. The two components of the ellipticity, e1 and e2, are plotin Fig. 5) and is plotted as a dashed line. The mean vectors are verthe tilt axis (101.8° and 14.8°, respectively) as shown by the arro

lipid compositions for stable and efficient reconsti-tution. We took images of extruded vesicles made ofegg PC plus brain phosphatidic acid (9:1 by weight),DOPS, or soybean lipids. These vesicles of differentcompositions remain nearly spherical as well. It ap-pears that as long as the lipid mixture can maintainits homogeneity under the experimental conditions,the vesicles maintain their spherical shape. Anotherpoint worth considering is that the presence of amembrane protein particle may be expected to in-troduce a local distortion in the shape of a vesicle.However, we expect that this local effect will notchange the general sphericity of the vesicle andhence will not introduce large errors into the angledetermination.

Simulations with Dimeric Ca21-ATPase

To evaluate the spherical reconstruction method,we simulated images of Ca21-ATPase dimers in48-nm spherical vesicles and subjected them to an

f ellipticity for vesicles at 0° (A) and 54° (B) tilt are shown. Thevectorial presentation of the ellipticity for the vesicles at 0° ande tilt axis is along 1230° (twice the angle of the tilt axis shownl (0.0025 and 0.0035 in length, respectively) and point away froms.

tion oD) A

ted. Thy smalwhead

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127SPHERICAL RECONSTRUCTION FOR MEMBRANE PROTEINS

alignment search and 3D reconstruction. Examplesof the simulated images are shown in Fig. 1. Fittingand subtraction of the membrane density yields im-ages containing a single particle, which can then belocated and oriented by an exhaustive cross-correla-tion search.

A cross-correlation algorithm was designed to per-form an efficient, exhaustive search of the orienta-tion parameters (Fig. 7). We found that in noisyimages the radius and the center location of a vesiclecan both be determined to very high precision, muchbetter than one pixel. Precise determination of thevesicle center allows a meaningful transformation ofthe vesicle image into polar coordinates, where thesearch in the azimuthal angle f can be carried outefficiently by a method related to the Polar FourierTransform technique developed for viral particles

FIG. 7. Composite cross-correlation of a simulated particleimage. (A) A particle image before and after subtraction of thevesicle membrane density. (B) The subtracted image after multi-plication of its Fourier transform by the CTF. The image is shownin both Cartesian coordinates (left) and in polar coordinates(right). (C) References for the cross-correlation. In the left panel isshown a Z-projection of the reference for cross-correlation of thecentral region of the image. In the right panel is the reference atf 5 0°, u 5 80°, and c 5 0° that is used to search the region of theimage corresponding to u values from 72° to 90°. (D) The final,merged cross-correlation is shown in Cartesian and polar coordi-nates. The MATLAB program performing the cross-correlationsearch of a 192 3 192 pixel image using 12 u values and 8 c valuesrequired about 30 s of CPU time on a Pentium III computer.

(Baker and Cheng, 1996). Once in polar coordinates(Fig. 7B), the position of a particle in the radialdirection r is related to its latitudinal angle u so thata simple two-dimensional cross-correlation with amodel projection in the polar (r, f) representationshould simultaneously determine both angles u andf.

What should be used as the reference for suchcross-correlation? Ideally projections with differentu values should be used as the references to searchthe vicinity of each r value. We decided to use asreferences six projections of a 3D reference map,evaluated at u values between 12° and 80°, andanother six projections with u 5 100° to 168°. One ofthese references is shown in Fig. 7C. The u valueswere chosen on the basis of the known particle struc-ture, with the criterion that the cross-correlationpeak would be reduced from the ideal by less than3% through the use of these discrete u values. Thusfor each of the 12 references, a cross-correlation isperformed through the use of 2D FFTs on the polar-transformed image, and the resulting set of correla-tion maps is merged to yield the polar map shown inFig. 7D.

The polar cross-correlation is not suitable for find-ing a particle when it is close to the center of thevesicle image. To identify such a particle, a separate2D cross-correlation is carried out on the originalimage in Cartesian coordinates. The resulting cross-correlation map is merged with the polar data toyield a complete cross-correlation map.

It now remains to determine the angle of rotationc about the membrane normal. To do this, we repeatthe calculation of the cross-correlation map usingreference projections having various c values. In thesimulations, we allowed c to take eight values be-tween 0° and 180°. The global maximum of this setof cross-correlations was identified and interpolatedto yield estimates of f, u, and c for each particleimage.

A potential problem with the determination oforientation from particle position is the singularityin the mapping [Eq. (1)] of the radial position toangle u in the vicinity of u 5 90°. In our simulatedimages with a 60-pixel vesicle radius, a 1-pixel un-certainty in particle position corresponds to a 1°uncertainty in u near u 5 0°, but a 10° uncertaintynear u 5 90°. Nevertheless, subpixel precision isften attained in determining particle positions. Fig-re 8 compares values of angles determined fromarticle positions with those obtained by projection-atching in our simulated images. The estimates of

u from particle positions indeed show the greatestdispersion in the vicinity of 90°; however, the overallerrors in these estimates (Fig. 8B) are lower than inthose obtained from projection-matching (Fig. 8D).

t

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128 JIANG, CHESTER, AND SIGWORTH

Most of the u errors in Fig. 8B are misassignments ofmirror-image projections, resulting in inversion of uin about 7% of the cases. With projection-matching,on the other hand, there is a substantial number oflarge errors, with 14% of the estimates in error bygreater than 15°.

Obtaining u values from particle positions accord-ing to Eq. (1) is expected to work as long as theparticles can be reliably detected in the images, andthe size of errors will then depend only weakly onparticle size. This contrasts with projection match-ing, where the reliability of u estimates dependsstrongly on particle size. The particles modeled hereare in a size range where the two methods givecomparable results. For such particles an improvedorientation determination can be envisioned inwhich u values from particle positions are refined byincorporating information from projection-match-ing, for example, as described in Sigworth (1998).

3D Reconstruction

We used the Wiener filter approach (Hawkes,1980; Grigorieff, 1998) for 3D reconstruction fromthe simulated particle images. The first step in theprocessing was to Fourier-transform each image and

FIG. 8. Estimation of Euler angles from simulated images. (A–maximizing the cross-correlation peak, from 500 simulated partic1 and 7 and is such that about 7% of u values were inverted, resu

ote the increased dispersion in the vicinity of u 5 90° in B. (D, Ewere located and f was determined by cross-correlation as aboveprojections; a local translational alignment was performed for eafiner search in c eliminated the “stair steps” seen in C.

multiply it by the calculated CTF. This filtered im-age was then used for the cross-correlation searchdescribed above, using a low-resolution (25 Å) mapof the true structure as the reference. Given theparticle position and orientation angles obtainedfrom the search, the particle along with its vicinitywas extracted from the filtered images, shifted to theorigin, and Fourier-transformed. This was thenadded as a plane at orientation angles v 5 {f, u, c}to the accumulated 3D transform. In parallel to this,a corresponding plane uj(v) containing the squaredCTF of the image was added to a “reference” 3Dtransform. These two transforms then embody thenumerator and denominator of the expression forthe jth voxel aj of the Fourier transform of thereconstructed model,

aj 5

Oi

cij xij~v!

g 1 Oi

cij2uj~v!

. (8)

Here, xij(v) is the jth voxel of the plane containinghe Fourier transform of the ith image, c is the CTF

ermination of f and u from particle position, and the c value fromges. The signal-to-noise ratio was the same as illustrated in Figs.rom misclassification of mirror images of the projected particles.mination of u and c from projection-matching. After the particlesc were determined by matching the particles against reference

pair. Both u and c were searched in steps of 5° in this case. The

C) Detle imalting f

) Deter, u andch u,c

ij

ai

129SPHERICAL RECONSTRUCTION FOR MEMBRANE PROTEINS

value of that image at that voxel, and g is the Wie-ner filter constant. Figure 9A shows a section of thereconstructed density map obtained from a set of2000 noisy images. Under the conditions simulatedhere, the resolution of the reconstruction is limitedmore by noise than by errors in particle orientation.The Fourier shell correlation shows that the recon-struction improved the resolution of the structurefrom 25 to about 16 Å.

DISCUSSION

Single-particle reconstruction has been pursuedfor more than two decades (Hoppe and Hegerl, 1980;Frank, 1996). The general procedures require thedetermination of the position and the orientationangles for each particle whose projection is to beincorporated into the final reconstruction. The de-termination of orientation angles is sensitive to er-rors in the alignment and classification proceduresand to noise in the raw images. The spherical recon-

FIG. 9. Reconstruction from 2000 simulated images of a Csubtracted. Central sections (A) and projections (B) are shown(25 Å) starting reference, and the reconstruction. In the righreconstruction, representing a circular membrane patch 80 Åreconstruction, the particle location and Euler angles estimareconstruction according to Eq. (8). Fourier shell correlations

bout 1/16 Å21 in the reconstructed structure (D). A reconstrundistinguishable from that shown here.

struction approach determines two of the three an-gles from the geometrical position of particles invesicles and thus simplifies the orientation determi-nation. Analysis of vesicle images shows that sphe-ricity of vesicles is well maintained in practice, andour simulations show that the directly determinedangles have errors smaller than those defined byusual projection matching (Fig. 8). With this advan-tage, spherical reconstruction shows promise as animproved SPR technique for membrane proteins.

Spherical reconstruction makes use of the imagesof proteins reconstituted in vesicles. In proteolipo-somes, membrane proteins are much more stablethan in detergent micelles, and their conformationalhomogeneity may be significantly improved as well.Our results from vesicles of various lipid composi-tions show no difficulty in keeping vesicles spherical.Our simple model for membrane scattering densityleads to accurate determination of the vesicle centerand excellent subtraction of membrane density from

Pase dimer structure from which the membrane density wasoriginal structure sampled with 4-Å pixels, the low-resolutionpanels, the membrane model density was added back to thedius. Each panel is 200 3 200 Å in extent. To perform the

om cross-correlation as in Fig. 7 were used to produce a 3Dn increase of resolution from 1/25 Å21 in the reference (C) toased on the true angles, rather than the estimated ones, was

a21-ATof thet-most

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G

A

B

B

B

C

D

F

F

F

F

F

G

G

H

H

H

H

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130 JIANG, CHESTER, AND SIGWORTH

these vesicle images. We therefore believe the basicexperimental requirements for spherical reconstruc-tion can be satisfied.

An alternative to the use of small spherical vesi-cles would be to use large, flattened vesicles contain-ing the membrane proteins of interest. Taking theflattened vesicle surfaces to be planar, the randomconical tilt procedure could then be applied for 3Dreconstruction. The disadvantage of this approach,however, is the need to tilt the specimen to obtain3D information. Tilting reduces the quality of high-frequency data due to the effects of charging, whilealso leaving a missing cone of information in Fourierspace. Alternatively, the edges of flattened vesiclescould be used to obtain “edge-on” views of membraneproteins, but in this case the curvature of the mem-brane at the vesicle edges is unknown, making mod-eling difficult. It is the spherical shape of smallvesicles that allows a simple model to be used forsubtraction of the vesicle membrane contribution tothe images presented here.

Spherical reconstruction in principle should begenerally applicable to all membrane proteins thatcan maintain their integrity during biochemical pu-rification and can be reconstituted into small vesi-cles. This method is especially promising for pro-teins smaller than the current size limit because thedetermination of two out-of-plane angles depends onthe locations of particles, not on the matching ofprojections, and therefore does not depend stronglyon particle size as long as the particles can be accu-rately located. Recently Woldin et al. (1999) havebeen able to recognize the long arms of the 350-kDaheterotetrameric insulin receptor under both deter-gent-solubilized and reconstituted conditions. Thisraises the possibility for spherical reconstruction tobe applied to small membrane proteins.

We are greatly indebted to Drs. Michael Lewis, Chen Xu, andHoward Young for their generous help and patient instruction inthe freezing of specimens and the operation of the electron mi-croscopes at NYU. Special thanks are also due to Dr. DavidStokes for making available to us the NYU facility and providingthe Ca21-ATPase density map. This work was supported by NIH

rant NS21501 to F.J.S.

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