Journal of Structural Biology 157 (2007) 106–116

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Application of the iterative helical real-space reconstruction method to large membranous tubular crystals of P-type ATPases

Andrew J. Pomfret a, William J. Rice b, David L. Stokes a,b,¤

a Skirball Institute of Biomolecular Medicine, Department of Cell Biology, New York University School of Medicine, New York, NY 10016, USAb New York Structural Biology Center, New York, NY 10027, USA

Received 7 April 2006; received in revised form 10 May 2006; accepted 10 May 2006Available online 14 June 2006

Abstract

Since the development of three-dimensional helical reconstruction methods in the 1960’s, advances in Fourier–Bessel methods havefacilitated structure determination to near-atomic resolution. A recently developed iterative helical real-space reconstruction (IHRSR)method provides an alternative that uses single-particle analysis in conjunction with the imposition of helical symmetry. In this work, wehave adapted the IHRSR algorithm to work with frozen-hydrated tubular crystals of P-type ATPases. In particular, we have imple-mented layer-line Wltering to improve the signal-to-noise ratio, Wiener-Wltering to compensate for the contrast transfer function, solventXattening to improve reference reconstructions, out-of-plane tilt compensation to deal with Xexibility in three dimensions, systematic cal-culation of Fourier shell correlations to track the progress of the reWnement, and tools to control parameters as the reWnement progresses.We have tested this procedure on datasets from Na+/K+-ATPase, rabbit skeletal Ca2+-ATPase and scallop Ca2+-ATPase in order to eval-uate the potential for sub-nanometer resolution as well as the robustness in the presence of disorder. We found that Fourier–Bessel meth-ods perform better for well-ordered samples of skeletal Ca2+-ATPase and Na+/K+-ATPase, although improvements to IHRSR arediscussed that should reduce this disparity. On the other hand, IHRSR was very eVective for scallop Ca2+-ATPase, which was too disor-dered to analyze by Fourier–Bessel methods.© 2006 Elsevier Inc. All rights reserved.

Keywords: Electron microscopy; Image analysis; Helical reconstruction; Iterative helical real-space reconstruction; Tubular crystals

1. Introduction (DeRosier and Klug, 1968). This helical application oVered

Methods for 3D image reconstruction from electronmicrographs were developed in the 1960’s and 1970’s byKlug and colleagues (Klug, 1983). They recognized that a3D structure could be determined from multiple views of anobject at diVerent orientations and used Fourier transformsas a convenient way to combine the corresponding data(Crowther and DeRosier, 1970). An alternative methodemployed a Fourier–Bessel transform to produce a 3Dstructure from a single view of an object with helical sym-metry (Klug and Crick, 1958), with the Wrst successfulapplication being to the helical tail of the T4 bacteriophage

* Corresponding author.E-mail address: stokes@saturn.med.nyu.edu (D.L. Stokes).

1047-8477/$ - see front matter © 2006 Elsevier Inc. All rights reserved.doi:10.1016/j.jsb.2006.05.012

a great advantage over 2D crystallographic methods thatrequired systematic tilting of the specimen to build up a 3Ddataset (Henderson and Unwin, 1975). Basic steps in helicalreconstruction were described by DeRosier and Moore(1970) and involve indexing of the layer lines that composethe Fourier transform, extracting amplitude and phase datafrom these layer lines, and correcting phase data for out-of-plane tilt. To improve the signal-to-noise ratio, data frommultiple images were averaged after Wnding a commonphase origin and Fourier–Bessel transformation was usedto produce the 3D map. For these methods, data manipula-tions were done entirely in Fourier space.

Over the ensuing two decades, these methods became thestandard in the Weld and were used on a wide variety of bio-logical systems, including actin, tubulin, myosin, and bacte-rial Xagella. Adaptations were made to the basic algorithm

A.J. Pomfret et al. / Journal of Structural Biology 157 (2007) 106–116 107

to maximize resolution for particular applications. Notableadvances include an algorithm to account for bending inthe plane of electron micrographs of helical objects byspline-Wtting and real-space re-interpolation of curved actinWlaments (Egelman, 1986) and automated processing oflarge datasets to extract high resolution (10 Å) data whenonly low resolution (25 Å) data were visible in the Fouriertransforms from individual bacterial Xagellar Wlaments(Morgan and DeRosier, 1992).

A substantial reworking of the original algorithms wasundertaken by Toyoshima and Unwin (1990) in their workon membranous protein crystals of the nicotinic acetylcho-line receptor. New procedures included interpolation of theimage in real-space to ensure that layer-lines not only werealigned to the sampling lattice of the Fourier transform, butalso fell at an integral number of reciprocal pixels. Also,real-space averaging of 3D maps was used to combine datafrom diVerent helical symmetries. Further improvementswere made by Beroukhim and Unwin (1997) analogous tothe unbending procedures that had been developed for 2Dcrystals (Henderson and Baldwin, 1990). Helical unbendinginvolved determination of precise orientational parametersfrom segments of the tubular crystals (i.e. in-plane bending& scale changes and out-of-plane tilting & twist around thetube axis) based on comparison with a reference data set.Notable applications of these techniques include Ca2+-ATPase (Stokes and Zhang, 1998; Xu and Rice, 2002) andbacterial Xagella (Yonekura and Maki-Yonekura, 2003) atbetter than 10 Å resolution, and a recent structure of theacetylcholine receptor at 4 Å (Unwin, 2005). Finally, aniterative Fourier–Bessel algorithm was developed forreconstruction of helical structures with severe Bessel over-lap (Wang and Nogales, 2005), along with methods forcombining data from diVerent symmetry groups both byaveraging real-space densities derived from 3D maps(Toyoshima and Unwin, 1990) and by averaging Fourier–Bessel coeYcients (gn,l (r)) prior to calculation of 3D maps(DeRosier and Stokes, 1999).

Despite this tremendous progress, Fourier–Bessel recon-struction methods are limited in their ability to deal withXexible and disordered helical samples. In order to addressthese problems, a hybrid iterative helical real-space recon-struction (IHRSR) method was recently developed, whichemploys algorithms from single particle analysis in con-junction with real-space helical averaging, to overcome sig-niWcant disorder in thin helical Wlaments like RecA(Egelman, 2000). This method divides images into smalloverlapping segments, whose relative orientations aredetermined by the standard single particle technique ofprojection matching (see Jiang and Ludtke, 2005, for arecent review). After using back projection to produce a 3Dstructure, helical symmetry is empirically determined andimposed on this structure prior to the subsequent round ofreWnement. Over the last few years, these methods havefound wide application in helical assemblies, includingF-actin (Galkin and Orlova, 2003), RecA (VanLoock andYu, 2003), and bacterial pilli (Mu and Egelman, 2002).

In the current work, we have adapted and modiWed theoriginal IHRSR methodology for use on large frozen-hydrated tubular crystals of P-type ATPases, with the goalof achieving 3D reconstructions at sub-nanometer resolu-tion. Previously, we have encountered limitations in Fou-rier–Bessel reconstruction methods due to the presence ofbending, variable symmetry, indexing ambiguity, and lowsignal-to-noise ratios (SNR). To overcome these problems,we have enhanced the IHRSR algorithm by implementinglayer-line Wltering to improve the SNR, Wiener-Wltering tocompensate for the contrast transfer function, solvent Xat-tening to iteratively produce improved reference recon-structions, out-of-plane tilt to track short-range Xexibilityin three dimensions, automatic isolation of single moleculesfrom the lattice to track reWnement via Fourier Shell Corre-lation, as well as tools to control and track the reWnement.The procedure has been tested on three distinct datasetsthat demonstrate its relative strengths and weaknesses.

2. Methods

Tubular crystals of rabbit sarcoplasmic reticulum (SR)Ca2+-ATPase and duck Na+/K+-ATPase were prepared andimaged as previously reported (Xu and Rice, 2002; Riceand Young, 2001). Scallop SR Ca2+-ATPase was isolatedand crystallized as described by Castellani and Hardwicke(1985). The resultant crystals were plunge-frozen in liquidethane and imaged at 50000£ magniWcation on a PhilipsCM200 Weld-emission electron microscope (FEI, Eindhoven,Netherlands) under low-dose conditions (»15 electrons/Å2)with an Oxford CT3500 (Gatan, Pleasanton CA) cryo-holder. Images were taken on Kodak SO-163 (Rochester,NY) electron imaging Wlm and push processed in full-strength D19 developer for 12min. Optical diVraction wasused to identify good images, which were scanned at 14�mintervals with a Zeiss SCAI densitometer (Intergraph, Madi-son, AL), resulting in an eVective sampling size of 2.73Å/pixel.

Computations were performed on dual-processor, 64-bitOpteron workstations running RedHat Fedora Core 3 ver-sion of the LINUX operating system. The image processingpackage SPIDER (Frank and Radermacher, 1996) wasused for many of the steps of the reWnement cycle with theaddition of two independent programs written by Ed Egel-man for the helical symmetry search of the asymmetric 3Dvolume and, subsequently, for imposing this symmetry onthis 3D volume (Egelman, 2000). The series of operationswere controlled from a SPIDER script provided by EdEgelman and modiWed to include the additional stepsdescribed in Section 3. For the Wrst round of reWnement, weused an interactive version of the helical search program tomanually look for the most likely helical parameters.Thereafter, the cycle could be run without user interven-tion. With our computer hardware, a single cycle including600 subimages took 3 h; most of the time was spent on mul-tireference alignment, which depends on the square of thesubimage dimension and linearly on the number ofsubimages. A complete reWnement typically took 3–4 days.

108 A.J. Pomfret et al. / Journal of Structural Biology 157 (2007) 106–116

3. Results

The eight basic steps of the original IHRSR algorithmdeveloped by Egelman (2000) are illustrated in Fig. 1. Tosummarize, the input data are generated by cutting individ-ual images of tubular crystals into a series of square sub-images that typically overlap each other by »90%. ThereWnement cycle then begins by comparing these subimagesto a reference structure, which can be a 3D structure of arelated helix, a low-resolution Fourier–Bessel reconstruc-tion, or even a featureless cylinder of the same diameter.This comparison is done by generating projection imagesfrom the reference at all possible orientations about thetube axis (step 1), which are then used for multi-referencealignment of the input series of subimages (step 2). TheEuler angles specifying the relative orientation of each sub-image are determined by the orientation of the matchingreference projection. These Euler angles are used for back-projection of the input subimages to generate an asymmet-ric 3D volume (step 3). A systematic search for the helicalsymmetry in this 3D volume is then conducted (step 4), fol-lowed by imposition of this symmetry on the volume (step5) to generate a new reference for subsequent rounds ofreWnement. This cycle is iterated until convergence of thevarious parameters (Euler angles and helical symmetry) isachieved. In Fig. 1, the italicized comments in parenthesesrefer to additions that we made to this basic algorithm,each of which is described in detail below.

We have adapted the IHRSR reconstruction algorithmto helical crystals of three diVerent P-type ATPases pre-served in the frozen-hydrated state (Fig. 2). The Wrst datasetcame from tubular crystals of Na+/K+-ATPase from thesupraorbital glands of salt-adapted ducks, which were pre-viously solved using Fourier–Bessel reconstruction

methods as implemented by Beroukhim and Unwin (1997)to »12 Å resolution (Rice and Young, 2001). Recent eVortsto improve this reconstruction failed to extend the resolu-tion beyond »10 Å, even after increasing the number oftubes and applying constraints such as solvent Xattening.We hoped that the IHRSR algorithm would be successfulwhere Fourier–Bessel methods were not. The second data-set came from well-ordered tubular crystals of rabbit SRCa2+-ATPase, which are better ordered and have previ-ously been solved to 6.5 Å resolution (Xu and Rice, 2002)using these same Fourier–Bessel methods. This sampleoVered a comparison to Na+/K+-ATPase to see how thealgorithm responded to higher resolution data. The Wnaldataset came from more disordered tubular crystals of scal-lop SR Ca2+-ATPase, which have previously only beenstudied at low resolution by Fourier–Bessel reconstructionof negatively stained crystals (Castellani and Hardwicke,1985). A low SNR has made it diYcult to establish the heli-cal symmetry of these scallop Ca2+-ATPase tubes andimpossible to Wnd a common phase origin for averagingdata from multiple tubes. Ultimately, our goal is to useIHRSR to overcome the limitations of Fourier–Besselreconstructions and thereby to resolve the secondarystructural elements, in the case of Na+/K+-ATPase, and atleast the domain organization, in the case of scallop Ca2+-ATPase.

3.1. Na+/K+-ATPase tubular crystals

Compared to previous studies of skeletal SR Ca2+-ATP-ase (Xu and Rice, 2002) and nicotinic acetylcholine recep-tor (Toyoshima and Unwin, 1990), tubular crystals fromNa+/K+-ATPase are relatively short (maximum of 300–400 nm, compare Fig. 2A and C), but still suitable for

Fig. 1. Overview of the IHRSR cycle. (1) A series of 2D projections are calculated from a 3D reference model. (2) Tubular crystal segments (i.e., “single-particle” subimages) are aligned against the reference projections (i.e., multi-reference alignment). (3) Single-particle subimages are back-projected to forman initial asymmetric 3D volume. (4) A helical search is conducted on the asymmetric 3D volume to determine helical symmetry parameters. (5) Helicalsymmetry is imposed onto the 3D volume. (1) The helically symmetric 3D volume is then used as the new “reference” model for the next round, and thecycle is iterated until a stable reconstruction is obtained. SigniWcant modiWcations to the original algorithm are italicized in brackets and are described inTable 1.

Initial 3D reference volume

Helically symmetric reference reconstruction

(FSC)1. Make reference projections

(out-of-plane tilt)subimages from tubular crystals(Fourier filters &

rectangular mask)

4. Helical symmetry searchAligned single-particleswith associated Euler angles

3. 3D back-projection

2. Multi-reference alignment(constrained range based on previous round)

Asymmetric 3D volume(solvent flattening)

5. Impose helical symmetry

A.J. Pomfret et al. / Journal of Structural Biology 157 (2007) 106–116 109

Fourier–Bessel methods. More problematic is that Na+/K+-ATPase tubes are wider (»800 Å) and therefore tend toadopt a larger number of helical symmetries. In previous

Fig. 2. Three diVerent samples were used to develop and test the IHRSRalgorithm. Na+/K+-ATPase from duck salt gland (A and B). Ca2+-ATPasefrom rabbit skeletal muscle (C and D). Ca2+-ATPase from scallop catchmuscle (E and F). All three samples show layer lines and in the case ofNa+/K+-ATPase and rabbit Ca2+-ATPase, these extend to 15 Å resolu-tion. These high order reXections are near the second maximum in theCTF and the approximate position of the Wrst CTF zero is shown by thewhite arc in (D). Note that the image of skeletal Ca2+-ATPase in (C) is atlower magniWcation and that this tube is much longer than the others.Scale bars correspond to 200 nm.

BA

DC

FE

work, we followed the conventional strategies of Fourier–Bessel reconstruction, increasing the numbers of tubularcrystals in several symmetry groups and using real-spaceaveraging in order to maximize the resolution. In contrastto Ca2+-ATPase where we were able to achieve 7 Å resolu-tion with as few as 13 tubes (Stokes and Delavoie, 2005), a12 Å structure of Na+/K+-ATPase required 27 tubes (Riceand Young, 2001) and further increases in this number havenot improved this resolution signiWcantly (unpublishedresults). This suggests the presence of disorder such as shortrange bending or variable twist that cannot be compen-sated by Fourier–Bessel methods, but may be tractable byIHRSR.

Our Wrst problem occurred in the search for helical sym-metry during the initial round of reWnement, which did notproduce a clear minimum for the helical symmetry parame-ters �� and �z. We hypothesized that this problem was dueto the low SNR arising from the innately low contrast offrozen-hydrated, unstained samples and, in this case, fromthe rather thick ice required to fully embed these 800-Åthick tubes. We therefore implemented additional pre-pro-cessing steps devised to maximize the signal for initialcycles of this dataset. In particular, we implemented a Wlterto select data along the layer lines that characterize thesehelical samples (compare Figs. 3B and D), thus eliminatingconsiderable background noise. This Wlter was applied tothe Fourier transform of the original image prior to prepar-ing the subimages for multireference alignment (FourierWlters in Fig. 1) and consisted of generous masks around allthe strong layer-lines using the program MASKTRANfrom the MRC image processing suite (Crowther and Hen-derson, 1996). After applying this Wlter, a shallow, but dis-cernable, minimum was obtained at the expected values of�� and �z. This Wlter has the disadvantage of eliminatingweaker layer lines that are not visible in raw images and istherefore only used during the initial stages of reWnementuntil the reference becomes better determined. Thereafter,subimages from the original, non-layer-line-Wltered imageswere used for further reWnement of the orientation andhelical symmetry.

In order to include data beyond »20 Å resolution, datamust be compensated for the CTF. The CTF oscillates peri-odically between +1 and ¡1, causing certain spatial fre-quencies to be dampened or even to contribute withinverted contrast (Erickson and Klug, 1971). Furthermore,diVering defocus levels in individual images induce charac-teristic changes in the period of the CTF oscillation, requir-ing compensation prior to combining subimages into onelarge dataset. During initial rounds of reWnement, weapplied a low-pass Fourier Wlter to the input subimages,which limits data to the Wrst band of the CTF, but this alsolimits resolution to 20–25 Å. For subsequent rounds weconsidered several alternative approaches for a more com-prehensive solution. Ideally, the Fourier terms would sim-ply be divided by the CTF, but this approach would lead toampliWcation of noise in regions where the CTF is close tozero and where the SNR is innately low. Another approach

110 A.J. Pomfret et al. / Journal of Structural Biology 157 (2007) 106–116

would be to restore the phases in the transform, whichwould prevent inclusion of data with inverted contrast, butdo nothing about the dampened Fourier amplitudes bothnear the origin and at the nodes in the transform. The Wie-ner Wlter not only restores the phases, but also adjusts theamplitudes depending on the value of the CTF and theestimated SNR of the dataset: W(k)DCTF¤ (k)/(|CTF(k)|2 + 1/SNR) (where W (k) is the Wiener Wlter func-tion and CTF¤ (k) is the complex conjugate of the CTF atthe spatial frequency k) (Penczek and Zhu, 1997). The valueof the SNR eVectively limits the maximum correction nearthe CTF nodes and we used a value of 2.0 for our samples(Fig. 3F).

Although the original IHRSR algorithm included trans-lational and rotational alignment of subimages within theprojection plane, it did not compensate for tilt of the tubeout of this plane, which is a standard consideration forFourier–Bessel reconstruction (DeRosier and Moore,1970). Based on Fourier–Bessel analysis of these Na+/K+-ATPase tubes, this out-of-plane tilt is substantial andgenerally ranges between §5°. We have included a compen-sation in our IHRSR reWnement by simply creating addi-tional reference projections with the helical axis rotated outof the plane of the micrograph by §10° (step 1 in Fig. 1). Ofcourse, this procedure substantially increases the number ofreference projections and consequently slows the reWne-ment cycle.

Solvent Xattening is a constraint that is routinely appliedto X-ray crystallographic reWnement, and has recently alsobeen used in conjunction with real-space averaging of Fou-rier–Bessel reconstructions (Yonekura and Toyoshima,2000). Solvent Xattening involves replacing densities withinthe solvent regions of an electron density map with a single,average value, thus removing the noise and improving thereference that will be used for the next round of reWnement.We have implemented solvent Xattening in the IHRSRreWnement cycle by Wrst isolating a single unit cell from the

3D reconstruction and then deWning density thresholds thatproduce 150 and 200% of the molecular volume expectedfor Na+/K+-ATPase. A feathered mask is then applied tothe map with a Gaussian falloV between the density thresh-holds corresponding to molecular volumes of 150 and200%; a single, average density value was imposed outsidethis mask (solvent Xattening in Fig. 1). This average densityvalue depends on the radius and is determined from themean radial density distribution, which is derived as theinverse Fourier transform of the equator (e.g., Toyoshimaand Unwin, 1990; Yonekura and Toyoshima, 2000). Fur-ther constraints are applied to the map by setting densityvalues to zero both in the center of the tube and beyond theoutermost radius.

The subimages used for Na+/K+-ATPase are muchlarger than previous IHRSR applications, due to the largerdiameter of these tubular crystals and the computationalrequirement for square subimages. We reasoned that thislarge dimension limited our ability to follow bending andtwisting along the length of the tube. We therefore maskedthe subimages so that the image data only Wlled a third theheight of the subimages (i.e., 360 pixels in width and 120pixels in height; rectangular masks in Fig. 1). The remainingpixels were Wlled with a uniform background value. Thismasking reduced the SNR during projection matching andwas therefore applied only after initial rounds of the reWne-ment had created a well-deWned reference.

The eVects of these various improvements were judgedby the Fourier Shell Correlation (FSC). Calculation of theFSC requires comparison of two data sets, which is gener-ally done by splitting the data into two parts. In the currentcase, we have used a Fourier–Bessel average from ninetubular crystals as a Wxed reference for comparing theeVects of the improved IHRSR algorithm applied to a sin-gle tubular crystal (Fig. 4). In order to prevent interferencefrom the packing of molecules into a crystal lattice, weapplied a mask to select a single molecule from each of the

Fig. 3. Subimages of Na+/K+-ATPase and their Fourier transforms after application of various Fourier Wlters. (A and B) Original images. (C and D) Layerline Wltering. (E and F) Wiener Wltering. (G and H) Both Wiener and layer line Wltering.

B C

FE H

A D

G

A.J. Pomfret et al. / Journal of Structural Biology 157 (2007) 106–116 111

two data sets and, after alignment, calculated the FSC. Thismask was loose-Wtting and had gently tapered edges toavoid artifacts associated with alignment of the mask edges.It is clear from the FSC’s shown in Fig. 4, that WienerWltering, out-of-plane tilt compensation (oopt), solvent-Xat-tening (solvXt), and subimage masking (rectmsk) eachmake an incremental improvement to the resolution of thereconstruction.

A fundamental limit to all reconstructions is the accu-racy with which the orientation of a given projection isdetermined. In IHRSR, this orientation is determined byprojection matching and is constrained to the intervals at

Fig. 4. Fourier shell correlations of Na+/K+-ATPase reconstructions. (A)FSC of reconstructions resulting from the implementation of variousmodiWcations to the cycle; legend is shown directly below with columnscorresponding to (1) low pass Wlter, (2) phase reversal, (3) Wiener Wlter, (4)layer line Wlter, (5) out-of-plane tilt correction, (6) solvent Xattening, and(7) rectangular subimages. (B) FSC of the Wnal IHRSR reconstruction(open circles) versus previous Fourier–Bessel reconstruction of the samesymmetry group dataset (Wlled circles).

resolution (Å)

54.0 27.0 16.7 13.5 10.8

italerroClleh

Sreiruo

Fno

0.0

0.2

0.4

0.6

0.8

1.0

lopass phsflp wiener llflt oopt solvflt rectmsk- - + + + + +

- - + - + + +

- - + + - + +

- - + + + - +

- - + - - - +

- - + - - - -

- + - + - - -

- + - - - - -

+ - - - - - -

A

B

resolution (Å)

54.0 27.0 16.7 13.5 10.8 9.0 7.7

italerroClle h

Sr eiru o

Fno

0.0

0.2

0.4

0.6

0.8

1.0

Fourier-BesselIHRSR

which the reference projections are calculated. The intervaldepends linearly on the resolution desired and on thedimension of the object being reconstructed, according tothe Crowther formula: ��D d/D, where �� is the angularincrement of projections, d is the resolution and D is thediameter of the object. Although initial reWnement roundscan be done at a coarser increment, this increment shouldbe decreased as the reWnement progresses. However, afterincluding out-of-plane tilt and attempting to extend the res-olution by decreasing the angular increment, the number ofreference projections included in the multireference align-ment becomes computationally prohibitive. We have there-fore implemented a tracking system for each individualsubimage that deWnes a limited range for projection match-ing about the Euler angles determined in the previousreWnement round (step 2 in Fig. 1). In a similar fashion, thetracking system includes both translational alignments ofsubimages and the helical symmetry of reconstructions, andallows all of these parameters to be more Wnely sampled asthe reWnement progresses. As a further convenience, param-eters and reWnement rates can be initialized at the begin-ning and adjusted depending on progress of the reWnement,which should be judged by the FSC determined after split-ting the data set in half.

Finally, we applied this enhanced IHRSR algorithm to alarger data set of 9 Na+/K+-ATPase tubular crystalsbelonging to the ¡35,11 helical family (characterized byBessel orders for the layer lines indexed as 1,0 and 0,1,respectively). This data set comprises »7200 individual sin-gle-particle images with dimensions of 360£ 120 pixels. Weestimated the resolution by comparing the resulting recon-struction with that obtained from 35 tubular crystals pro-cessed with Fourier–Bessel methods and combined withreal-space averaging (Fig. 4B). Based on a cutoV of 0.5 inthe FSC, the resolution of this reconstruction is 15.2 Å,compared to 13.8 Å resolution obtained from the samedataset using Fourier–Bessel methods. The features of theIHRSR reconstruction are also comparable to the Fourier–Bessel reconstructions (Fig. 5), with both the catalyticdomains in the cytoplasm and the �-subunit on the extra-cellular side of the membrane clearly identiWable. However,the distribution of density normal to the membrane (i.e.,radial density distribution of the tube) is diVerent, with thetransmembrane density notably weaker in the IHRSRreconstruction. This density distribution is determined bythe equator in the Fourier transform, which is stronglyaVected by the CTF compensation at very low resolution.Thus, the gross diVerences in Figs. 5B and D are likely toreXect failure of the Wiener Wlter to fully restore equatorialdata, particularly as it approaches the origin of the Fouriertransform.

3.2. Rabbit SR Ca2+-ATPase tubular crystals

Tubular crystals of rabbit SR Ca2+-ATPase are consid-erably longer (up to 10 �m) and thinner (»600 Å) withabundant straight sections of 0.5–1�m (Fig. 2C). Furthermore,

112 A.J. Pomfret et al. / Journal of Structural Biology 157 (2007) 106–116

they are generally better ordered with layer lines frequentlyvisible to 15 Å resolution in Fourier transforms (Fig. 2D).As with Na+/K+-ATPase, tube diameters are variable dueto diVerent helical symmetries, though the underlying 2Dlattice is relatively constant. Previous work with thesetubes established not only this helical symmetry, but alsorepeat length, out-of-plane tilt, and defocus of individualtubes from a variety of diVerent data sets (e.g., Xu andRice, 2002; Toyoshima and Sasabe, 1993; Zhang andToyoshima, 1998). We tested the improved IHRSR algo-rithm described above on one of these Ca2+-ATPase tubu-lar crystals in order to compare the results with Fourier–Bessel methods. The size and overlap of the subimagesrepresent parameters that need to be customized for agiven data set. For skeletal Ca2+-ATPase tubes, we simplyused the same size as for Na+/K+-ATPase tubes (despitethe thinner diameter) and empirically determined theoverlap as follows. Initially, a series of subimages withone-pixel shifts were made, and the number of consecutivesubimages matching a given reference projection wasnoted. In this case, four consecutive subimages generallymatched the same reference projection, indicating that afour-pixel shift could be used for creating the subimagesto minimize redundancy, yet maximize angular samplingduring back-projection.

Fig. 5. Surface rendering of 3D reconstructions of Na+/K+-ATPase tubularcrystals. (A and B) Reconstruction resulting from the IHRSR reWnementshown both as an overview of the helical tube (A) and an isolated molecule(B). (C and D) Fourier–Bessel reference volume in comparable views.

A C

B D

membrane

cytoplasm

extracell

β-sub

We ran the IHRSR algorithm on a single Ca2+-ATPasetube with the helical symmetry deWned as ¡23,7. This dataset consisted of 635 subimages with dimensions of360£ 120 pixels, a relative shift of 4 pixels, and comprisinga total of »1500 unit cells. Initially, we used a referencefrom a Fourier–Bessel reconstruction from 13 tubular crys-tals with the same helical symmetry and with an estimatedresolution of »7 Å (Stokes and Delavoie, 2005). We easilyobtained a well-deWned minimum at the expected helicalparameters �� and �z during the Wrst round of reWnement,even though we did not employ layer line Wltering for thisdata set. The various Wlters, masks and corrections weregradually introduced over the Wrst 30 cycles of IHRSR,which was allowed to run for a full 100 cycles. At this point�� and �z no longer changed, indicating convergence ofthe reWnement. A calculation of FSC relative to the refer-ence reconstruction indicated a resolution of 14–15 Å (FSCof 0.5, Fig. 6A). A dip in the FSC at »17 Å comes from thenode of the CTF at this resolution (see Fig. 2D).

To test the sensitivity of IHRSR to the starting model,we also ran the cycle starting with a well deWned referencewith a diVerent helical symmetry (¡27,7). In this case, theminimum in helical symmetry parameters was not clear,

Fig. 6. Fourier shell correlation of a single rabbit Ca2+-ATPase tubularcrystal. (A) Results of reconstructions determined by Fourier–Bessel(open circles) and IHRSR (Wlled circles) methods. (B) Results of IHRSRreWnement starting with a reference model with an incorrect symmetry.ReWnements were initiated by specifying either the correct (open circles)or incorrect (Wlled circles) helical symmetry. Thereafter, the symmetry wasfreely reWned by the algorithm.

resolution (Å)

54.0 27.0 18.2 13.6 10.9 9.1 7.8

erroC lleh

S reiruoF

noital

0.0

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resolution (Å)

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1.0wrong symmetrycorrect symmetry

B

A

A.J. Pomfret et al. / Journal of Structural Biology 157 (2007) 106–116 113

even after running several rounds of reWnement withoutimposing symmetry parameters. We therefore ran tworeWnement cycles in which these symmetry parameters wereinitially constrained to correct and incorrect values andthereafter allowed to move with the reWnement. Fig. 6Bshows the FSC for these two reWnements, illustrating thefact that IHRSR converged to a plausible structure onlyafter using the correct symmetry parameters. For compari-son we also produced a reconstruction from this same tubeusing advanced Fourier–Bessel methods described byBeroukhim and Unwin (1997). These methods also involvedividing the tube into subimages—though larger and notoverlapping—and reWning orientation parameters against areference. We used the same reference for this reWnementand calculated the FSC using the same mask and algorithmas for the IHRSR reconstruction. The FSC for the Fourier–Bessel reconstruction indicates a resolution of 11 Å(Fig. 6A), which is somewhat higher than for the IHRSRreconstruction, but may be biased somewhat by the waythis data had been iteratively reWned against this very samereference.

3.3. Scallop Ca2+-ATPase tubular crystals

Tubular crystals of scallop Ca2+-ATPase are relativelyshort and considerably less ordered than those from rabbitCa2+-ATPase or Na+/K+-ATPase (Fig. 2). Layer lines arevisible to only »25 Å resolution and are often broadened inthe direction of the helical axis. We were able to determinethe helical indexing of several tubes, which diVers from rab-bit skeletal Ca2+-ATPase by having an additional set oflayer lines that suggest a doubled unit cell. However, inattempting a Fourier–Bessel reconstruction, we wereunable to Wnd a common phase origin for averaging Fou-rier data from multiple tubes due to the innately low SNR.Nevertheless, we used the Fourier–Bessel method to calcu-late a 3D reconstruction from individual tubes and usedthese structures as a reference for the IHRSR algorithm.This algorithm was independently applied to two individ-ual scallop Ca2+-ATPase tubes, which each contributed»700 subimages with »850 unit cells. Helical parameterscould not be reliably ascertained after the Wrst round ofreWnement, so we initially used values derived from theindexing of Fourier transforms, which were subsequentlyallowed to change over the course of the reWnement. Similarto rabbit skeletal Ca2+-ATPase tubes, we tested the depen-dence of the algorithm on the starting model. In this case,we produced a Fourier–Bessel reconstruction using analternative (incorrect) helical indexing (¡29,4 instead of¡31,4). Predictably, the helical symmetry search did notproduce a well-deWned minimum. Therefore we ran thereWnement twice: once using the helical parameters for theincorrect symmetry (��, �zD92.54°, 7.93 Å) and onceusing helical parameters for the correct symmetry (��,�zD272.69°, 8.40 Å). These helical parameters were con-strained for the Wrst few rounds, and then allowed to movefreely with the reWnement. Although the alternative model

converged to the expected structure when using the correctsymmetry parameters, the reWnement failed to converge toa plausible structure when using the incorrect parameters(Fig. 7A).

The improvement over the initial Fourier–Besselreconstruction is demonstrated in Fig. 8, which shows anoverview of an initial reference (Fig. 8A) compared withthe corresponding Wnal reconstruction (Fig. 8B). Also,the domain organization of scallop Ca2+-ATPase(Fig. 8C) appears similar to that of rabbit skeletal Ca2+-ATPase (Fig. 8D), though with some interesting varia-tions that will be the subject of future research. The reso-lution was judged by FSC to be »17 Å (Fig. 7A) bycomparing reconstructions from the two scallop Ca2+-ATPase tubes which had been reWned independently(starting models derived from Fourier–Bessel reconstruc-tions from the respective tubes and data separatedthroughout the reWnement). Fig. 7B illustrates the pro-

Fig. 7. Fourier shell correlation of scallop Ca2+-ATPase reconstructions.(A) FSC determined by comparing two independent IHRSR reconstruc-tions of scallop Ca2+-ATPase. For the triangles, the correct helical symme-try was used to create the initial reference and to search for helicalparameters throughout the reWnement. For the circles, the reference wasmade with an incorrect symmetry and reWnement proceeded near eitherthe correct symmetry (open circles) or the incorrect symmetry (Wlled cir-cles). (B) FSC plotted in resolution shells as a function of reWnementround for a single tube. This plot facilitates tracking of the reWnement andshows that the algorithm converged after »25 rounds. This particulartube was reWned with the correct symmetry and corresponds to the trian-gles in (A).

resolution (Å)

54.0 27.0 18.0 13.5 10.8 9.0 7.7

erroC lleh

S reiruoF

noital

0.0

0.2

0.4

0.6

0.8

1.0

cycle number

0 5 10 15 20 25 30

rroC lleh

S reiruoF

noital e

0.0

0.2

0.4

0.6

0.8

1.046Å

33Å

21Å

15Å

12Å

B

A

114 A.J. Pomfret et al. / Journal of Structural Biology 157 (2007) 106–116

gress of the reWnement of one of these scallop Ca2+-ATP-ase tubes, which converges after 25 rounds of reWnement.To calculate this running FSC, subimages from the onetube were arbitrarily divided into two equal parts andused to create two interim reconstructions; each line inFig. 7B corresponds to the correlation coeYcient fromone of the resolution shells.

3.4. Computational resources

The computational considerations required for ourIHRSR application are more intense than previous applica-tions, due primarily to the larger subimage dimension (deWnedby the tube diameter) and the greatly increased number of ref-erence-projections for considering out-of-plane tilt and poten-tial future consideration of twisting and stretching. In general,IHRSR is far more demanding than Fourier–Bessel recon-struction, which represents a trivial load on modern desktopcomputers. Although we have made a number of modiWca-tions to maximize computational eYciency, reWnements gen-erally take several days on dual-processor workstations. Thisprocessing time will increase as we add either more data ormore reference projections in the attempt to improve the reso-lution. Thus, the use of distributed processing on a cluster is

Fig. 8. Surface rendering of scallop Ca2+-ATPase reconstruction. (A) Viewdown the axis of the tube produced by Fourier–Bessel reconstruction of asingle tubular crystal; this structure served as the reference for IHRSR.(B) View down the tube axis of the reWned structure from a single tube. (C)Single molecule taken from the IHRSR reconstruction of scallop Ca2+-ATPase. (D) Single molecule taken from the Fourier–Bessel reconstruc-tion of rabbit skeletal Ca2+-ATPase characterized in Fig. 6.

A

C

B

D

membrane

cytoplasm

lumen

highly recommended, which fortunately is supported by theSPIDER image processing suite.

4. Discussion

We have adapted the IHRSR method to work with widetubular crystals of the membrane-embedded P-type ATP-ases. Compared to previous samples analyzed by this tech-nique, our tubes present new challenges, arising from theirmuch larger diameter, the presence of multiple helical sym-metry families and a lower SNR. Our use of frozen-hydrated samples and our goal of subnanometer resolutionrequired compensation for the CTF and for bending of thetubes in three dimensions. Our additions to the originalIHRSR reWnement algorithm include layer-line Wltering,Wiener-Wltering, solvent Xattening, rectangular subimages,and out-of-plane tilt. In addition, we have implementedcontrols for reWning the search of subimage orientation andfor monitoring the progress of the reWnement using FSC(summarized in Table 1). The modiWed algorithm success-fully determined structures from tubular crystals of Na+/K+-ATPase and Ca2+-ATPase from both rabbit and scallopSR. For the former two, the 15 Å resolution was onlyslightly worse than that obtained from Fourier–Besselreconstructions using the same data. For scallop SR, the17 Å resolution was excellent considering that Fourier–Bes-sel reconstruction was not possible and that only two tubeshave so far been processed.

Fourier–Bessel and IHRSR reconstructions have com-plementary strengths and weaknesses, some of which canbe seen in the current results. If a tubular crystal is wellordered, then Fourier data are concentrated in discretelocations and Fourier based analysis of relevant layer lineshas the eVect of Wltering out a tremendous amount of noisethat continues to contribute to IHRSR reconstructions.This can be seen in the noticeable diVerence between FSC’sfor rabbit skeletal Ca2+-ATPase (Fig. 6A) and in the eVectsof layer line Wltering on Na+/K+-ATPase images (Fig. 4).On the other hand, Fourier–Bessel reconstructions havelimited ability to deal with disorder. Enhancements byBeroukhim and Unwin (1997) were designed to partiallyaccount for stretching and bending along the tubular axis,but these parameters were determined for relatively longsegments (»1000 Å in length). The overlap of subimages forIHRSR is able to compensate considerably shorter rangedisorder. So far, we have implemented only bending disor-der, but it would also be possible to account for stretchingand twisting by including additional reference projectionsduring multireference alignment. Indeed, stretching couldbe compensated by applying an appropriate re-interpola-tion, and diVerent twists could be segregated into subpopu-lations of subimages, both features that we plan toincorporate into future versions of our algorithm. On theother hand, Xattening of these cylindrical structures is diY-cult to cope with. Given the large diameter of Na+/K+-ATPase tubes (800 Å) relative to a typical layer of ice on afrozen-hydrated sample support (1000–1500 Å), it is easy to

A.J. Pomfret et al. / Journal of Structural Biology 157 (2007) 106–116 115

imagine that such Xattening may be the ultimate resolutionlimit for these particular samples.

The CTF is another area where IHRSR needs furtherdevelopment. Although the Wiener Wlter is in widespreaduse in single particle analysis (Penczek and Zhu, 1997), it isless rigorous than the method used in previous Fourier–Bessel reconstructions (Beroukhim and Unwin, 1997),where a weighted average of amplitudes was obtained froma group of tubes with diVerent defocus values, followed byfull restoration using the aggregate CTF. Accurate CTFcorrection is particularly important for helical samplesbecause the radial distribution of mass in the reconstruc-tion is determined by the equatorial layer line which, inturn, is strongly aVected by the diminishing CTF as itapproaches the origin of the transform. Thus, incorrectcompensation produces artifactual density gradients as onemoves from one side of the membrane to the other (i.e., inthe radial direction of the reconstruction, see Figs. 5 and 8).A similar strategy is currently in development for thisIHRSR method, at least with respect to the low-resolutionpart of the equator.

Our results indicate that IHRSR is currently most eVec-tive when applied to relatively disordered tubular crystalsand is less eVective than Fourier–Bessel methods on rela-tively well ordered samples. In particular, resolutions pro-duced by IHRSR on tubes from Na+/K+-ATPase andskeletal Ca2+-ATPase are somewhat lower than the corre-sponding Fourier–Bessel analyses. Thus, if layer lines arewell-deWned enough to allow Fourier analysis and averag-ing of Fourier data from multiple tubes, then Fourier–Bes-sel reconstruction should be attempted. Nevertheless, thepower of IHRSR is apparent in our results from scallopCa2+-ATPase tubes. These samples have poorly deWnedlayer lines and have proven intractable by Fourier–Besselanalysis, yet IHRSR was able to produce creditable recon-structions, even when the wrong helical indexing wasapplied to the initial reference volume.

A weakness of IHRSR is its inability to objectivelydetermine helical parameters under certain conditions. Inall cases, one must have some idea of the helical parame-

ters, which can be readily deduced from the positions oflayer lines in the Fourier transform and is used to deWne theneighborhood for searching the structure produced fromthe Wrst round of reWnement. For well ordered crystals withwell-deWned starting references, we obtained clear minimafor �� and �z, but not for poorly ordered crystals withquestionable starting references. One diVerence from previ-ous work is that the one-start helix used for specifying thehelical symmetry of our wide tubular crystals represents ahigh order layer line (�z of 8 Å or less) with highly variablevalues for ��. Thus, closely related scallop Ca2+-ATPasesymmetries of ¡29,4 and ¡31,4 produce values for �� thatdiVer by »180°. The corresponding minima in a search of�� and �z are therefore far apart and separated by manylocal minima, making it impossible for the search algorithmto smoothly migrate from one symmetry to another.Although layer line Wltering is eVective in reWnement of anestablished minimum, it is not able to distinguish one sym-metry from another, especially given the absence of anobjective reference for the initial determination of subim-age orientation. Nevertheless, we have shown that reWne-ments do not converge to a meaningful structure whenusing incorrect helical symmetry parameters. This wouldsuggest a strategy of using convergence as a criterion forchoosing amongst related symmetries for a given tube.Candidate symmetries could be deduced from even a fewvisible layer lines in the Fourier transform. Although itwould be beneWcial to include subimages from multipletubes in a given reWnement, it would not seem possible toinclude tubes with diVerent symmetries, as they would pro-duce innately diVerent projection images and prevent therational application of helical averaging (step 5, Fig. 1).Given the preponderance of diVerent symmetries for ourwide tubular crystals, we plan to set up separate reWne-ments for each symmetry (which can include data frommultiple tubes) and, upon conclusion, to mask individualmolecules or unit cells from the separate reconstructionsfor alignment and averaging in real space.

In conclusion, Egelman and colleagues have previouslydemonstrated how the IHRSR algorithm can be used on a

Table 1Overview of improvements/modiWcations to the original IHRSR algorithm

Procedure Implementation EVect

Layer-line Wltering Mask layer lines in Fourier transform prior to creating subimages

Allowed reWnement of helical symmetry parameters from noisy images of frozen-hydrated samples

Wiener Wlter Applied to original image prior to creating subimages

Phase and amplitude compensation for CTF required for extending resolution beyond »15 Å

Out-of-plane tilt Created additional projections from model prior to multireference alignment

Corrects for bending of tubular crystal out of the plane of the micrograph. Standard correction for Fourier–Bessel methods

Solvent Xattening Molecular mask deWned from the symmetrized structure and used to zero solvent densities

ReWnement constraint that promotes convergence by improving the quality of the reference structure

Rectangular subimages Rectangular mask applied to the square subimages required by SPIDER

Allows detection of shorter range disorder

Restricted multireference alignment

Euler angles from previous round of reWnement used to restrict search range

Reduces computational load, thus allowing Wner increments for orientational search

ReWnement tracking Molecule masked from the reconstruction and used for running FSC calculation

Provides feedback about reWnement and allows intelligent adjustment of parameters as resolution increased

116 A.J. Pomfret et al. / Journal of Structural Biology 157 (2007) 106–116

wide range of smaller, solid helices with relatively littleknowledge of their helical symmetry (Egelman, 2000). Wecan now extend this principle to the large, hollow helicesfrom membrane proteins in the frozen-hydrated state and,after implementation of our algorithmic improvements,have the prospect of extending the resolution well beyond15 Å simply by increasing the number of tubular crystalsincluded in the reWnement.

Acknowledgments

We thank Ed Egelman for providing programs andassistance during initial phases of this work. Support wasprovided by NIH Grant R01GM56960 to D.L.S.

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