optimal determination of particle orientation, absolute hand...

Optimal Determination of Particle Orientation,Absolute Hand, and Contrast Loss in Single-particleElectron Cryomicroscopy

Peter B. Rosenthal* and Richard Henderson

MRC Laboratory of MolecularBiology, Hills Road, CambridgeCB2 2QH, UK

A computational procedure is described for assigning the absolute handof the structure of a protein or assembly determined by single-particleelectron microscopy. The procedure requires a pair of micrographs of thesame particle field recorded at two tilt angles of a single tilt-axis specimenholder together with the three-dimensional map whose hand is beingdetermined. For orientations determined from particles on one micro-graph using the map, the agreement (average phase residual) betweenparticle images on the second micrograph and map projections is deter-mined for all possible choices of tilt angle and axis. Whether the agree-ment is better at the known tilt angle and axis of the microscope or itsinverse indicates whether the map is of correct or incorrect hand. Anincreased discrimination of correct from incorrect hand (free hand differ-ence), as well as accurate identification of the known values for the tiltangle and axis, can be used as targets for rapidly optimizing the searchor refinement procedures used to determine particle orientations.Optimized refinement reduces the tendency for the model to match noisein a single image, thus improving the accuracy of the orientation determi-nation and therefore the quality of the resulting map. The hand determi-nation and refinement optimization procedure is applied to image pairsof the dihydrolipoyl acetyltransferase (E2) catalytic core of the pyruvatedehydrogenase complex from Bacillus stearothermophilus taken by low-dose electron cryomicroscopy. Structure factor amplitudes of a three-dimensional map of the E2 catalytic core obtained by averaging untiltedimages of 3667 icosahedral particles are compared to a scattering referenceusing a Guinier plot. A noise-dependent structure factor weight is derivedand used in conjunction with a temperature factor (B ¼ 21000 A2) torestore high-resolution contrast without amplifying noise and to visualizemolecular features to 8.7 A resolution, according to a new objectivecriterion for resolution assessment proposed here.

q 2003 Elsevier Ltd. All rights reserved.

Keywords: electron cryomicroscopy; single particle reconstruction; absolutehand; pyruvate dehydrogenase; tilt pairs*Corresponding author

Introduction

Electron cryomicroscopy can be used to deter-mine the structure of unstained proteins andmacromolecular assemblies in rapidly frozen thinfilms of amorphous ice without the need for

crystals. For thin specimens, particle images inelectron micrographs after correction for the phasecontrast transfer function (CTF) are projections ofthe structure, and three-dimensional maps of thestructure can be calculated by recording all theunique projections required to a given resolution.Different projections can be obtained by tilting thespecimen as in electron tomography. However,radiation damage prevents the recording of even asingle image with high signal-to-noise ratio froman unstained protein molecule in ice, and it is

0022-2836/$ - see front matter q 2003 Elsevier Ltd. All rights reserved.

E-mail address of the corresponding author:[email protected]

Abbreviations used: CTF, contrast transfer function;MTF, modulation transfer function.

doi:10.1016/j.jmb.2003.07.013 J. Mol. Biol. (2003) 333, 721–745

necessary to align and average noisy, low-contrastimages of many particles in many orientationsrecorded under low-dose conditions (5–20 e2/A2).A theoretical estimate suggests that averaging lessthan 106 images may in principle be sufficient todetermine the structure of a protein to atomicresolution, provided the position and orientationof the particles can be determined from low-contrast images.1 Improvements in the quality ofimages as well as in the computational proceduresfor determining structures from images arerequired to realize this in practice.

Methods have been described for calculating low-resolution maps de novo from untilted projectiondata by classifying and averaging similar imagesand then determining the relative orientation ofthe average projections in three dimensions,termed angular reconstitution.2 Iterative model-based refinement, which compares the agreementbetween projections of the model and the experi-mental images, can then be used to improve theassignment of single-particle orientations andmicroscope parameters leading to better andhigher-resolution models. Because images ofproteins in ice have low contrast, noise can lead toincorrect determination of particle orientations.The difficulty in assessing whether a model hasaligned to signal or noise leads to a danger ofmodel bias.3

Untilted projection data alone cannot determineabsolute hand, since exactly the same views canbe produced by a three-dimensional model of theopposite hand. Thus, knowledge of the absoluteorientation of two views of the particle is required,otherwise the arbitrary hand of a starting modelwill persist during refinement. If a map is at highenough resolution, the recognition of individualmolecular features or interpretation with a high-resolution X-ray model of a component can indi-cate which hand is correct. Klug & Finch4 – 7 firstdescribed tilting experiments for the determinationof the absolute hand of negatively stained icosa-hedral viruses, and more recently a computationalprocedure for hand determination has beendescribed.8

In this study, we describe an extension of thesehand determination procedures which is straight-forward to perform and can be used as a tool tooptimize the determination of particle orientationsand to improve the fidelity of the resulting struc-ture determination. The computational proceduredepends on comparison of particle orientations onmicrographs recorded from tilted and untiltedspecimens. We apply it to high-resolution, low-dose images of the dihydrolipoyl acetyltransferase(E2) catalytic core of the pyruvate dehydrogenasecomplex of Bacillus stearothermophilus. We thenexamine the factors that influence the determi-nation of particle orientations and which thereforeaffect the confidence with which correct hand canbe discriminated from incorrect hand. We showthat the known tilt angle and tilt axis in the experi-ment can be used as unbiased targets for rapidly

optimizing the computational parameters thatinfluence the accuracy of orientation determi-nation. Using these optimized parameters leads tobetter particle orientations and to better three-dimensional maps from untilted data.

In order to visualize the highest-resolutionfeatures once such a map is obtained, it is necess-ary to correct for the loss of contrast in the map athigh resolution from causes such as radiationdamage, imaging imperfections, and errors in thereconstruction procedure. We estimate the decayin the structure factor amplitudes with resolutionby comparing structure factors on an absolutescale with an objective scattering reference. Wedetermine the temperature factor (B ¼ 21000 A2)and derive noise-dependent structure factorweights required to correct the structure factoramplitudes without amplifying noise. Featuresof the potential map of the E2 catalytic corethus obtained at 8.7 A resolution using 3667 icosa-hedral particle images (220,020 asymmetric units)are similar to those of a structural modelobtained by X-ray crystallography. Comparison ofthe map and X-ray model allows an inde-pendent assessment of the resolution of themap and confirms the success of the restorationprocedures.

Theoretical Background

Guinier analysis

Proteins contain a hierarchy of structuralfeatures. Spherically averaging structure factoramplitudes in resolution shells emphasizes com-mon resolution-dependent features of all proteinsand facilitates comparison with reference structurefactors or theoretical scattering criteria. Analysisof the structure factors of a three-dimensionalpotential map determined by electron microscopymay be treated in the same framework as thatused to study the spherically averaged structurefactor data obtained in solution scattering experi-ments using X-rays or neutrons. In addition,spherical averages of structure factor correlationsbetween independent maps are the basis ofresolution assessment and signal-to-noiseestimation in single-particle analysis.

Reciprocal space structure factor amplitudes of aprotein span several orders of magnitude as a func-tion of resolution. A Guinier plot shows the naturallogarithm of the average structure factor as afunction of resolution ð1=d2Þ: Figure 1 presents aschematic Guinier plot of a protein consisting oftwo regions: (1) a steeply descending shape/solvent-dependent region at low resolution; and(2) a relatively flat secondary structure/randomatom region at high resolution.

At the lowest resolution, the amplitudes aredetermined by the shape of the protein contrastedagainst solvent. In the Guinier approximation,9

applicable to resolutions d , 2pRg; where Rg is

722 Single-particle Cryomicroscopy

the radius of gyration, the scattering amplitude isdescribed by:

F ¼ ðr2 r0ÞVproteine2ð2p2R2g=3d2Þ

where (r 2 r0) is the difference in scatteringdensity between protein (r) and solvent (r0).When written in terms of protein atomic scatteringfactors fi:

F ¼ ððr2 r0Þ=rÞðX

i

nifiÞ e2ð2p2R2g=3d2Þ

The scattering maximum occurs at zero angle ðd ¼1Þ where all atoms scatter in phase and is equalto

Pi nifi multiplied by the solvent contrast term

ðr2 r0Þ=r: The mean structure factor amplitude islarge but decays rapidly with resolution and con-tinues to reflect shape and solvent to about 10 A.

Beyond 10 A, the mean scattering amplitudedepends to some extent on protein-specific featuresincluding fold and secondary structure, but onaverage is determined by the essentially randomposition of atoms in the interior of the protein.According to Wilson statistics,10 the average scat-tering amplitude from randomly positioned atoms

is given by F ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPi nif 2

i

q: This average amplitude

decreases slowly with resolution roughly in line

with individual atomic form factors, but is essen-tially constant when compared to decay in theshape/solvent region at lower resolution. Predomi-nantly a-helical structures tend to have strongerdiffraction around 10 A resolution, whereas thosewith larger amounts of b-structure diffract morestrongly around 4.5 A, producing small deviationsfrom Wilson scattering.

Experimental structure factor amplitudes may beplaced on an absolute scale by setting the zeroangle scattering equal to ððr2 r0Þ=rÞNatoms andaverage scattering amplitude in the Wilson regimeequal to

ffiffiffiffiffiffiffiffiffiffiffiffiffiNatoms

p; where Natoms is the number of

equivalent atoms of identical scattering factor

such that Natoms=ffiffiffiffiffiffiffiffiffiffiffiffiffiNatoms

p¼ ð

Pi nifi=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPi nif 2

i

qÞ: For

a typical protein, Natoms may be determined byassuming the protein molecular mass is made up ofNatoms equivalent carbon atoms (12 £ Natoms Daltons).In this simplified model, the scattering amplitudeat both low and high resolution may be expressedin terms of a single parameter derived from themolecular mass. The molecular mass of a proteinis usually known in advance of structural study.

The solvent contrast factor ðr2 r0Þ=r is equal toabout 0.28 for X-rays (assuming protein density ris 0.42 e2/A3 in proteins and solvent r0 is 0.30 e2/A3

for ice of mass density 0.92 g/cm3). When calculated

Figure 1. Schematic Guinier plot shows the natural logarithm of the spherically averaged structure factor amplitude(F) for a protein against 1/d 2, where d is the resolution (A). Zero angle scattering is equal to Natoms carbon equivalentsof the molecular mass multiplied by the solvent contrast (0.28) and places the scattering on an absolute scale. Theprotein scattering curve (red line) consists of a low-resolution region (d . 10 A) determined by molecular shape andsolvent contrast, and a high-resolution region (d , 10 A) which approaches the scattering of randomly placed atomsdescribed by Wilson statistics, which decreases only slightly with resolution and may be approximated by the hori-zontal line of amplitude

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiNatoms

p: The high-resolution region may also have structure corresponding to fold-specific

features, including a-helix and b-sheet. The average noise amplitude is FNoise(1) for a single image or FNoiseð1Þ=ffiffiffiffiN

pafter

averaging N images. Low-resolution structure factor amplitudes are also shown for a large structure that might bestudied by tomography and a small molecular mass particle which has a low-resolution scattering amplitude belowthe noise level for one image (blue lines). The experimental contrast loss for structure factors at high resolution dueto imperfect images is indicated by a dotted red line labeled by its slope, the temperature factor Bimage. Additional con-trast lost due to imperfect computations gives a line with slope Boverall, which is the sum of temperature factors Bimage

and Bcomputation. The resolution limit is indicated where the structure factor curve equals the noise level, which in thisexample occurs at 106 particles for Boverall, but at 105 particles if Bcomputation ¼ 0.

Single-particle Cryomicroscopy 723

using neutral atom electron-scattering factors, ðr2r0Þ=r is approximately 0.42.

Averaging particle images

The average noise level of a low-dose imagelimits the ability to record structure factors to aresolution given by the intersection of the proteinamplitude curve with the average noise amplitudefor one particle image. The amplitude of theaverage noise spectrum for one particle image canbe placed on the absolute scale of the proteinstructure factors and is represented in Figure 1 asa line with amplitude FNoiseð1Þ: The ability toaverage images of identical particles increases thesignal-to-noise ratio and makes high-resolutioninformation accessible. When the amplitude of thenoise after averaging N particle images is plottedon an absolute scale, this corresponds to a noiselevel FNoiseðNÞ ¼ ðFNoiseð1Þ=

ffiffiffiffiN

pÞ, where FNoise(1) is the

noise level for one image. FNoise(N) is drawn forvarious numbers of images in Figure 1. The resol-ution achieved for a given experiment and a givennumber of images is therefore the point at whichthe protein structure factor amplitude equals theamplitude of the average noise level.

Coherent averaging of structure factors requiresdetermination of particle orientations and pos-itions on the film. Structure factors (reflecting bothshape and higher-resolution features) above thenoise level provide the signal used to determineorientations of individual particles. Since the struc-ture factor is proportional to molecular mass, aparticle of higher molecular mass will intersect thenoise level at higher resolution than a smallerparticle. A lower molecular mass threshold exists,perhaps around 50 kDa,1 where the low-resolutionsignal does not reach above the noise level for asingle low-dose image. In this case it will beimpossible to determine the orientation parametersfor the particle in order to average images of manyidentical particles.

The structure factor amplitudes at resolutionsessential for alignment must not be attenuated bythe microscope CTF. Defocus is used to increasecontrast by enhancing low-resolution structurefactor frequencies that are essential for alignment.Effects of high defocus include the introduction ofzeroes in the structure factor transform and spatialdisplacement of structure factor spacings in theimage.11 The image must be corrected for theseeffects of the CTF, requiring coherent illuminationand an accurate knowledge of the defocus andastigmatism.

Tomography

For specimens in which every particle structureis different, averaging many particles is notpossible, and multiple projections must beobtained by tomography. The orientation of eachprojection relative to the others in the tomographicseries is known. Therefore, the particle orientation

parameters need not be determined in each micro-graph, and a lower dose can be used to recordeach projection, but the number of images in theseries is still limited by radiation damage. Accord-ing to the dose fractionation theorem, the signal-to-noise ratio of a projection of the final reconstruc-tion is not greater than if the total dose were usedto record just the single projection.12 The ultimateresolution of tomography is therefore limited tothe intersection of the average structure factoramplitude with the noise level in a single imagerecorded with the total dose (See Figure 1, inter-section of the structure factor amplitude with thenoise level FNoise(1) for one image). A highermolecular mass object will intersect the noiselevel at higher resolution than a smaller particle.Tomographic reconstruction to ,20 A for highmolecular mass specimens is an important goal.13

The limiting resolution of tomography may be esti-mated by extrapolation from the resolution andnoise level obtained when averaging single-particleimages to that obtained from one image. If low-resolution features tolerate a greater electron dosethan high-resolution features, then use of a greatertotal dose than optimum for high-resolution,single-particle analysis may result in a furthersmall improvement in the resolution predicted forcryotomography.

Loss of contrast

Experimental density maps determined bysingle-particle electron microscopy show a loss ofcontrast at high resolution. As elsewhere,14 the con-trast loss can be measured as the ratio of structurefactor amplitudes of the experimental map andreference amplitudes. The reference amplitudescan be derived from a high-resolution X-ray struc-ture or theoretical scattering criteria such as thosedescribed above.

The loss of contrast at high resolution is aproperty of the images, and therefore recordingthe best possible experimental images is essential.Factors which may degrade image contrast includespecimen movement and charging, radiationdamage, inelastic electron-scattering events, partialmicroscope coherence, and particle flexibility andheterogeneity.14 The combined effect of thesefactors may be modeled by a Gaussian fall-off ofstructure factors at high resolution given bye2ðBimage=4d2Þ with temperature factor Bimage asdrawn in Figure 1.

The contrast may be decreased further by theinability of the computational procedures to extractand average the signal from the micrograph. Thedensitometer modulation transfer function (MTF)may contribute to contrast loss in the digitizedimage. The poor signal-to-noise ratio of the imagesmakes computational determination of orientationparameters and microscope parameters difficult.Inaccurate or completely incorrect assignment ofthese parameters will blur features in the image inproportion to resolution or add noise in more


complex ways. This incoherent averaging causes adecline in contrast, which may be described bye2ðBcomputation=4d2Þ as shown in Figure 1. The relativecontribution of this computational loss of contrastmay be difficult to distinguish from the experimen-tal loss of contrast in the images themselves. Thestructure will have an overall temperature factor,Boverall, the sum of Bimage and Bcomputation, whichreflects all the factors which reduce contrast.Because the resolution limit of the structure isdetermined by the intersection of the averagestructure factor with the average noise level, alarger temperature factor requires averaging moreparticles to achieve a given resolution.

Contrast restoration

It is necessary to restore the high-resolutionstructure factor amplitudes of a map so that lowand high-resolution terms have correct relativescaling (also called sharpening). The down-weighting of the high-resolution data due tocontrast loss will make the map look relativelysmooth and featureless. The experimental(positive) temperature factor describing the con-trast loss can be measured with respect to a scatter-ing reference on a Guinier plot. The amplitudes canthen be corrected by applying a negative tempera-ture factor (Brestore) in the form e2ðBrestore=4d2Þ:15

However, this type of simple sharpening willamplify the noise as well as the signal (signal-to-noise ratio unchanged). At high resolution wherethe signal has become weak, this will simply makethe map noisier, so the amount of sharpening mustbe appropriately weighted according to the signal/noise present in the map at a given resolution.

The signal-to-noise ratio of the map may bederived from a common criterion for resolutionassessment, the Fourier shell correlation (FSC).16,17

In the Appendix we argue for a resolution criterionbased on the estimated correlation between a mapcalculated from a full dataset and a perfect refer-ence map containing no errors. This correlation,Cref, may be calculated for a resolution shell asCref ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2FSC=ð1 þ FSCÞÞ

p; where FSC is the corre-

lation between two independent maps, each calcu-lated from half the data. Also shown in theAppendix, Cref is equal to the cosine of the averagephase error and is equivalent to the crystallo-graphic figure-of-merit,18 a common measure ofmap interpretability in X-ray crystallography.

Because an experimental structure factor is thesum of both signal and noise components,

we show in the Appendix that Cref ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS2=ðS2 þ ðN2=2ÞÞ

pwhere S is the average signal

amplitude and N=ffiffiffi2

pis the average noise ampli-

tude in the full dataset. At low resolution, wherethe structure factors are free of noise, Cref , 1. Atthe resolution limit of the map, where S , N; Cref

becomes less than 1. Noise-weighted structurefactors that are the product of Cref and F willdown-weight resolution shells containing noise.

Thus, as with the figure-of-merit weighting usedin X-ray crystallography,18 a map calculated usingamplitudes multiplied by Cref e2ðBrestore=4d2Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2FSC=ð1 þ FSCÞp

e2ðBrestore=4d2Þ will be the best

map.

Results

Microscopy and image analysis of E2CD

We chose the E2 catalytic core of the B. stearo-thermophilus pyruvate dehydrogenase complex asa test specimen for high-resolution, single-particlecryomicroscopy. Pyruvate dehydrogenase is a com-plex of up to 10 MDa containing three enzymes(E1, E2, E3) that catalyze the conversion ofpyruvate to acetyl CoA and CO2. The E2 protein isa dihydrolipoyl acetyltranferase and, dependingon the organism, forms a cubic or icosahedral coreof the complex through the oligomerization of itsC-terminal catalytic domain (28 kDa). E2 also con-tains one or more N-terminal lipoyl domains andan E1/E3 binding domain connected by flexiblepolypeptide linkers.19 X-ray studies20 and electronmicroscopy studies21 of a truncated form of theB. stearothermophilus E2 protein containing only the28 kDa catalytic core domain (E2CD) show anicosahedral complex with 60 subunits and mass of1.5 MDa.

High-resolution images of E2CD in ice werecollected under low-dose conditions (15–20 e2/A2) at 300 kV on a microscope with a field emissiongun. A range of defocus values (2.3–7.7 mm) wereincluded to fill in reciprocal space zeroes resultingfrom the CTF. All the defocus values were large toenhance low-resolution contrast helpful in findingparticles. A similar strategy of using large defocusvalues will be essential in microscopy of lowermolecular mass particles. We included the carbonsupport surrounding the ice holes in the image toreduce specimen charging22 and to provide asource of strong scattering for an initial estimateof the defocus for the film. Particle coordinates ona digitized micrograph were selected manuallyand individual particle images were cut out inboxes. Typical particle images are shown inFigure 2(a) and (b).

We used a previously described 14.5 Aresolution map,21 calculated by angular reconstitu-tion from untilted images with subsequent refine-ment, as a starting point for model-baseddetermination of orientation and microscope para-meters of the single-particle images. The programFREALIGN23 was used to refine parameters forindividual images, correct for the effects of themicroscope CTF, and to calculate three-dimen-sional electron potential maps. The orientation ofa particle is described by three Euler angles (c, u,f) which give the relative orientation of the three-dimensional model corresponding to the imageprojection and the translation ðx; yÞ describing the


location of the center of the particle projection.Microscope parameters for individual particlesinclude the magnification and three parametersdescribing defocus (DF1, DF2, and the angle ofastigmatism). The quality of the reconstructionultimately depends on the correct assignment ofthe particle orientations and microscope par-ameters. During iterative model refinement, theimages are first corrected for the effects of theCTF, and then particle orientations or microscopeparameters are refined by improving the agree-ment between calculated model projections andCTF-corrected images. A better model shouldtherefore lead to more accurate determination ofparticle orientations and microscope parameters.When more accurate microscope parameters areused in CTF correction, the subsequent refinementof particle orientations will be more accurate.

CTF-correction is performed on raw images by

first multiplying image transforms by the CTFwhich includes both phase and amplitude contrastterms (see the equation given in the legend toTable 1), and is a function of three image-specificparameters DF1, DF2, and the angle of astigmatism.An effect of defocus (DF) is to displace imagespacings of resolution d by a distance lDF=d;where l is the electron wavelength.24 In order toinclude all the information, the raw image boxsize was chosen to be D þ 2R; where D is theparticle diameter and R is the displacementexpected for the most defocused image at themaximum resolution spacing expected in thereconstruction. In reciprocal space, the effect ofmultiplying by an accurately determined CTF is toinvert image phases which are of the wrong signdue to the microscope transfer function, enhancefrequencies near CTF maxima, and attenuatefrequencies near zeroes of the CTF. The effect in

Figure 2. Images of B. stearothermophilus E2CD in ice at 59,000£ magnification and a defocus of 5.9 mm. (a) Raw par-ticle images selected from a film recorded with an untilted specimen holder and (b) the same particles from a secondfilm recorded with the specimen holder tilted by 108. (c) Model used to determine particle orientations and to calculateprojections matching the particle orientations shown in the bottom rows of (a) and (b).


real space is to convolute image densities with theFourier transform of the CTF, which moves dis-placed image spacings back to their true locationin the specimen projection. A real space circularmask with a cosine edge is applied to the CTF-corrected image and the additional non-centeredparticles beyond the reconstruction radius seen inFigure 2 do not influence subsequent processing.The transform of the three-dimensional reconstruc-tion is computed by interpolating and summingthe two-dimensional transform of each image atlattice points determined by the particle orientation(c,u,f), and normalizing each lattice point bydividing by the sum of (CTF)2 for all images contri-buting to the lattice point.23 After averaging imagesat a variety of defocus values, the resulting struc-ture factor profile should be directly comparableto that expected in a theoretical Guinier plot forthe specimen.

The agreement between a projection calculated

from a model and an image is given by the ampli-tude-weighted phase residual between the Fouriertransform of the CTF-corrected image and theFourier transform of the calculated model projec-tion, PR ¼

Pi lDfFil=

Pi lFil; where Df is the

phase difference and Fi is the amplitude of theimage structure factor. The value of the phaseresidual depends on the accuracy of the orientationand microscope parameters, the resolution rangeover which it is calculated, and also on the noiselevel and particular view of the particle in theimage. However, a low value for the phase residualcan reflect a fit between the model projection andnoise, and is not in itself a guarantee of accuracyin orientation refinement.

For a given model, orientations and microscopeparameters were refined by Powell minimizationwith the phase residual as a target.23 Refinementparameters that influence the determination ofparticle orientations are listed in Table 1. Theseinclude the resolution range of the data, a refine-ment temperature factor Br which determines therelative weight of high and low-resolution dataaccording to e2ðBr=4d2Þ; the reconstruction radius,and the relative weight of amplitude and phasecontrast terms in CTF correction. Additional par-ameters control further details of the refinementstrategy. In particular, we chose a refinementstrategy in which the particle orientations wererandomized and then minimized a specifiednumber of times to avoid local minima and tospeed convergence. In this study, the same micro-scope parameters were used for all the particlesrecorded on the same film.

A map based on 1681 icosahedral particles from14 micrographs recorded independently of thoseused to make the starting model contained signalto 11 A and is shown in Figure 2(c). Our initialchoice of parameters controlling the refinementcame from reasonable assumptions about theireffects and the desire to avoid systematic bias inthe refinement. The maximum data resolution forrefinement was between the first and secondzeroes of the CTF. Reconstructions were alwayscalculated to a resolution higher than the resol-ution used in refinement. Signal in the map inthese unbiased resolution shells not used forrefinement was used to judge map improvement.

Absolute hand determination

The three-dimensional model of the E2 core com-plex described above was derived entirely fromimages of untilted specimens, and was thereforeof unknown hand. Although the hand of the mapcould be compared to the hand of the X-ray model(see Comparison with X-ray model), we sought todevelop an automatic procedure to determine theabsolute hand using a pair of micrographsrecorded with the specimen at two tilt anglesrelative to the electron beam. The application of atilt transformation to the orientations determinedfor untilted particles gives a predicted orientation

Table 1. Effect of orientation refinement parameter onabsolute hand determination

Refinementparameter Film 1 score

Film 2 score (data 100–35 A)at phase residual minimum

Data resolution(A)

funtilt Tilt angle ffreehand Dffreehand

Min Max

100 35 35.7 7.5 48.53 6.5080 25 44.2 11.5 43.86 16.3280 20 51.1 10.6 43.40 16.7280 15 60.5 10.8 43.25 16.6580 35 34.8 7.0 45.70 5.9045 20 62.0 - 46.44 14.48300 20 49.8 10.8 43.41 16.3280 a15 46.8 10.0 43.05 17.04Map qualityMap 006 49.5 10.0 42.97 15.46Map 031 49.5 10.0 42.99 15.63Map (Figure 1(c)) 46.3 10.0 40.76 17.06

Refinement parameter (data resolution range) was varied indetermining orientations of 50 particle images from untiltedfilm 1 and scored with average phase residual funtilt.Orientations on film 1 were then scored for agreement against50 particle images from tilted film 2. The scores on film 2include the tilt angle, average phase residual at the minimumcorresponding to the correct hand (ffreehand, the “free handphase residual”), and the difference between the average phaseresidual at the correct hand and the incorrect hand (Dffreehand,the “free hand difference”). Different maps were comparedafter optimizing all refinement parameters below. Values forrefinement parameters influencing orientation determinationafter optimization: reconstruction radius: 146 A. Resolutionrange: 100–15 A. Number of cycles of randomization andrefinement: 200. PBC ¼ 100, BOFF ¼ 35, DANG ¼ 200.Fraction amplitude contrast WGH: 0.07 (see following CTFequation). CTF ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 WGH2

psin x2 WGH cosx; where x ¼

2p=l�DFu2

22

Csu4

4

�; Cs is the spherical aberration of the objec-

tive lens (2.0 mm), u is the diffraction angle l/d, and the defocusDF in direction Fobs is DF ¼ DF1cos2Fþ DF2sin2F, where DF1

and DF2 are the amounts of defocus in two orthogonal direc-tions, F ¼ Fobs 2Fangast; and Fangast is the angle with defocusDF1.

a Refinement temperature factor expð2Br=4d2Þ Br, 3000 A2.


for the tilted particles. If the model is of correcthand, its projection should agree with the imageof the tilted particle after rotating by the tilt angleused in the experiment. If the model is of oppositehand, its projection will agree with the imagesafter rotation by the negative of the tilt angle. Theagreement can be scored by the phase residual.

Micrographs were recorded first from anuntilted specimen and then from the same speci-men tilted by 10.08. Fifty particles present in bothmicrographs were selected, several of which areshown in Figure 2. The orientations of untiltedparticles were determined as described in the lastsection. Tilt transformations {rotx; roty} (seeMethods) in 18 increments up to ^158 covering allpossible directions and tilt angles for thegoniometer were applied to the orientation of eachuntilted particle to produce a set of predictedorientations for each particle corresponding toeach one of the 961 tilt transformations. The phaseresidual for each of these tilt-transformed orien-tations (after re-optimizing the particle translationx,y) was determined on the tilted particle image.The phase residual for each tilt transformationwas then averaged for 50 particles and shown inthe contour plot in Figure 3(a). Though a tilt pairfor one particle can show a phase residual mini-mum sufficient to determine the absolute hand,some particle images may give an incorrect con-clusion, and it is therefore better to average astatistically significant number of particles.

Since the specimen was tilted by þ108 to obtainthe second micrograph, a minimum should occurat a tilt angle of either þ108 (if the model has thecorrect hand), or 2108 (for the incorrect hand).Only an approximate knowledge of the tilt axisdirection on a micrograph at a given magnificationis necessary for determining absolute hand in thisprocedure (see Methods for a detailed protocol fordetermining which tilt transformation correspondsto the microscope tilt axis direction and polarity).The position of the minimum phase residual,49.58, at co-ordinates of (þ7, þ2) corresponds tothe known direction and polarity of the microscopetilt axis, and demonstrates that the handedness ofour three-dimensional model is correct. Theaverage phase residual for the opposite hand at(27, 2 2) is higher by 7.88 than the correct hand(called the “free hand difference”). However, theminimum in the plot corresponds to a tilt angle of78, which differs from the known tilt angle of 108used in the experiment. Because the goniometerrotation is accurate to 0.18, the fact that the mini-mum is not precisely at 108 indicates that theEuler angle determinations may not be correct forall the particles.

Use of absolute hand to optimize thedetermination of particle orientations

We next tested whether the systematic under-estimate of the true tilt angle in absolute handdetermination was caused by errors in the deter-

mination of particle orientations. Inaccurate orincorrect particle orientations could result fromnon-optimal choices of the refinement parametersthat influence orientation determination. Some

Figure 3. Determination of the absolute hand of E2CD.Average phase residuals for 50 particle images recordedat a tilt of 108 were determined using the 3D model andtilt-transformed orientation parameters of the corre-sponding untilted particle images. The contours showphase residuals for tilt transformations up to 158 alongthe x and y axes. A vector from the origin to any pointin the map corresponds to the direction of the tilt axisand its length is the tilt angle ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p: The direction

of the known tilt axis of the microscope is shown as ablack diagonal line. (a) Black squares indicate that thephase residual for the model shown in Figure 2(c) has aminimum (49.58) near the rotation angle of the experi-ment (78) and a lower value than for the negativerotation (57.38), indicating that the hand of the model iscorrect. (b) Same as (a) after parameter optimizationaccording to Table 1. The minimum (43.28) is lower andoccurs at a rotation angle of 108. The discrimination ofcorrect from incorrect hand or “free hand difference” is16.78 (compared to 7.88 in (a)).


choices for refinement parameters were better ableto discriminate the correct hand of the map fromthe incorrect hand. We systematically varied therefinement parameters when determining orien-tations for the untilted particles only (Table 1 andFigure 4). The accuracy of the orientations of theuntilted particles for a given choice of refinementparameter was then assayed on the tilted particleimages. For the tilted particles, phase residuals forall tilt transformations were calculated with a con-stant choice of parameters using image data tolow resolution (rmin ¼ 100 A and rmax ¼ 35 A) sothat the resulting phase residuals could be directlycompared with each other.

The difference between the phase residual at thecorrect and incorrect hand (the “free hand differ-

ence”) as well as the depth of the minimum at thecorrect hand (the “free hand phase residual”)were used as targets for optimizing all the par-ameters influencing orientation determination. Inaddition, the true tilt angle and tilt axis are knownduring the experiment and are the same for all par-ticles on the pair of micrographs, making themobjective targets for optimizing refinement par-ameters which influence orientation determinationand which affect the calculated tilt axis and tiltangle given by the location of the minimum phaseresidual. All parameters listed in the legend toTable 1 were varied in determination of theorientations of the untilted particles.

As shown in Table 1, our results demonstratethat data resolution limits (rmin and rmax) are critical.

Figure 4. Optimization of parameters influencing orientation determination. (a) Refinement temperature factor (Br).Free hand phase residual minimum and free hand difference maximum occur at a Br of 3000 A2. (b) Effect of variationof defocus DF on free hand difference showing a maximum at 0.3 mm change from starting value of 5.9 mm.


When rmin ¼ 100 A and rmax ¼ 35 A, the free handphase residual is 48.58, the free hand difference is6.58, and the minimum occurs at an angle of 7.58.When rmin ¼ 80 A and rmax ¼ 20 A, the free handphase residual drops to 43.48, and the free handdifference increases to 16.78. In this case the calcu-lated tilt angle from the data is 10.68, much closerto the actual tilt angle of 108. The better agreementwith the target values and its correlation withbetter free phase residuals suggests betterorientation determination when including higherresolution data.

In Figure 4, we show further optimization of twoparameters. The refinement B-factor, Br , whichincreases or decreases the weighting of higher-resolution data in the image, was varied between

23000 A2 and þ12,000 A2. A minimum occurs at3000 A2 as shown in Figure 4(a). This indicatesthat optimal orientations are found when the dataat 15 A (rmax) are attenuated by a factor of 20,reducing the effect of noise at high resolution inthe images. In Figure 4(b), the optimization of amicroscope parameter, the defocus, is also shownto influence the free hand difference and theaccuracy of orientation determination.

Figure 3(b) shows the phase residual map afterapplying the optimization procedure for all refine-ment parameters in Table 1. A much lower andmore symmetrical phase residual minimum (43.28)occurs at coordinates (þ10, þ2) and correspondsto a 108 rotation around the known direction andpolarity of the microscope tilt axis. The free hand

Figure 5. (a) Absolute hand determination as in Figure 3, but with a relative tilt angle of 308 (recorded at ^158). Theþ158 tilted film was recorded first, so the phase residual minimum occurs at 2308. (b) Hand determination for 108 tiltusing the second exposed tilted film to determine the hand on the first untilted film. (c) Phase residual for modelagainst a pair of images of the same particle field recorded without tilting. The tilt-transformed oriention parametersfor each film (labelled ORI 1, ORI 2) were scored against either film 1 or film 2. Tilt angles up to 78 about a singleaxis are shown.


difference (16.78) is much higher than with theinitial parameters used in Figure 3(a). Using theseoptimized refinement parameters, the scores forabsolute hand determination also depend on thequality of the three-dimensional map, as shown inTable 1 for several maps calculated over the courseof the structure determination. The improvementin a map at high resolution can also be demon-strated by evaluating the minimum phase residualat the correct hand as a function of resolution.

Hand determination using large tilt angles islikely to experience all the problems associatedwith images of tilted specimens, such as defocusgradients and greater charging. We have shownthat our procedure works successfully for tiltangles of 5–108, where these problems areexpected to be minimal. For large tilt angles, how-ever, the change in particle view is more dramaticgiving a larger free hand difference and, inprinciple, a greater sensitivity for the optimizationof parameters. A large tilt angle also makes thehand determination procedure a sensitive measureof particle flattening. We have recorded images ata variety of tilt angles and in Figure 5(a) we showthe hand determination using the same optimizedrefinement parameters for particle images fromtwo micrographs recorded with a relative tiltangle of 30(^158). The tilt angle calculated throughhand determination is 29.48 and was 30.88 for asecond independent experiment. We therefore con-clude there is no evidence for flattening of particlesduring sample preparation for cryomicroscopy, norfrom the effects of irradiation and likely mass loss,both of which would result in an apparent under-estimate of the tilt angle.

The same hand determination procedure can beperformed by first determining orientations forthe tilted particles and then scoring the orien-

tations against the untilted particles. The resultingcontour plot (using the same data as Figure 3(b))is shown in Figure 5(b). Now the minimum shouldoccur for a 2108 rotation (since changing the orderof the films is equivalent to changing the sense ofrotation). However, using optimized parameters,the calculated rotation angle is 27.58, rather than2108, indicating less accuracy in orientation deter-mination. In other cases where both micrographswere recorded from tilted specimens, first filmorientations as scored by phase residuals on thesecond film were as good as those where the speci-men was initially untilted. This suggests that poororientations on the second film are not the resultof tilting per se, but are the result of radiationdamage. To further demonstrate the effect of radi-ation damage on orientation determination, werecorded two consecutive micrographs of thesame particle field without tilting, in which caseorientations on the two films should be identical,excluding particle movement.

After determining orientations for particles onthe first film, we calculated the effect of applyinga single-axis rotation to the orientations and scor-ing the average phase residual for particle imageson the same film in the data range rmin ¼ 100 Aand rmax ¼ 35 A (Figure 5(c)). For orientationsdetermined on the second film and scored againstthe second film, the average phase residual ishigher at the minimum (53.48) than for first filmorientations scored against the first film (45.48),and the variation of phase residual with tilt angleis shallower, suggesting that orientations are onaverage less well-defined. When orientations deter-mined on the second film are scored against thefirst film, the phase residual is lower on the firstfilm (50.98) than the second (53.48), but is not aslow as with orientations determined on the first

Figure 6. Determination of tilt axis for each particle pair. Particle orientations were determined separately forparticles on the untilted and 108 tilted films. The best tilt axis relating orientations on the untilted and tilted film areplotted for each particle before (a) and after (b) optimization according to Table 1. Tilt axes within the red circle havean RMS deviation of 3.28.


film (45.48). The increased phase residual for orien-tation determination on the second film reflects thefact that the images themselves have poorer agree-ment with the model even at low resolution, butalso that orientation determination on the secondfilm is less accurate, probably because of thedamage to higher-resolution features.

As a final demonstration of the improvement inparticle orientation resulting from parameteroptimization, we used an entirely differentapproach requiring the separate determination ofthe orientations of the untilted and tilted particles.For each particle, the best tilt axis was found thatrelates the untilted and tilted particle orientationswith the constraint that it lies in the plane normalto the electron beam (see Methods). We plot thetilt axis direction and tilt angle for each particlepair before (Figure 6a) and after (Figure 6b) opti-mization using the same data and refinementparameters as used in Figure 3(a) and (b). Asshown by the clustering at the known target

values, the tilt axis and angle are also more accu-rately determined in Figure 6b on an individualparticle basis. Orientations are therefore moreaccurately determined after optimization. TheRMS deviation for the well-clustered tilt axes(within the small circle in Figure 6b) is 3.28. Theerror for the tilt axis reflects errors in orientationon both films, but the error in orientation on asingle film is the most important quantity to knowfor most data collection strategies. For example, iforientations have two times worse RMS error onthe second film than on the first because of radi-ation damage, then the RMS orientation error inorientations on the first film is approximately 1.48.Using Figure 6 it is possible to identify individualimage pairs for which the tilt angle is incorrect.Removing images that are clearly outliers due tonoise features, damage, or particle heterogeneitymay also improve hand discrimination by the freehand difference.

In summary, the ability to determine absolute

Figure 7. Data for the final E2CD map. (a) Distribution of orientation angles for particles, with 5, 3, and 2-fold sym-metry axes labeled. (b) Fourier shell correlation (FSC), Cref, and CXRAY for the final dataset.


hand depends on accurate orientation determi-nation. By using the free hand difference, tiltangle, and tilt axis as targets for rapidly optimizingthe parameters that influence orientation determi-nation, the accuracy of the orientations improved.Orientation determination is influenced by noise,and each image in a tilt pair will have differentnoise features. With the refinement parametersoptimized by hand determination, the minimumphase residual discriminates between the realimage features and the noise. Better orientationsproduce a better map at higher resolution byreducing the computational temperature factor(Bcomputation).

Contrast loss and restoration

We have applied the optimized procedures forthe refinement of particle parameters to a datasetof 3667 untilted images of icosahedral E2CD andcalculated a map. The orientations for all the par-ticles are shown in Figure 7(a). We plot the Fouriershell correlation (FSC) for the map in Figure 7(b).The FSC is the correlation between two indepen-dent maps, where each map is calculated fromhalf the images. In the Appendix, we argue thatthe resolution of the map should be assigned atthe point where the FSC crosses a threshold of0.143. This corresponds to the resolution at whichthe estimated correlation between a density mapcalculated from all the data and a perfect referencemap (Cref ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2FSC=ð1 þ FSCÞÞ

p; also plotted in

Figure 7(b)) is 0.5. Cref is equivalent to the crystallo-graphic figure-of-merit, a common measure of map

interpretabiliy in X-ray crystallography. The resol-ution of the map is thus 8.7 A.

We now assess high-resolution contrast loss inthe experimental map by comparing sphericallyaveraged structure factor amplitudes to a scatter-ing reference as shown in the Guinier plot inFigure 8. The reference structure factors are com-puted from an X-ray model of the complex (seethe next section for details on the model) assuminga background solvent density of 1.0 g/cm3. Bothexperimental and calculated structure factors areplaced on an absolute scale by setting the zeroangle scattering equal to the product of the sol-vent-contrast term (0.28) and Natoms, the number(136,100) of carbon atom equivalents correspond-ing to a molecular mass of the complex (1.5 MDa).The experimental structure factor amplitudes havea similar average value to the reference at lowresolution where scattering is dominated by theshape and solvent contrast of the complex. Smalldifferences likely result from the inability tomodel solvent adequately in the reference. At highresolution the amplitudes decay considerablyfrom the relatively flat profile of the reference.Also shown is a horizontal line of amplitudeffiffiffiffiffiffiffiffiffiffiffiffiffi

Natoms

prepresenting the average scattering ampli-

tude of randomly positioned atoms given byWilson statistics, a reference criterion for structurefactor amplitudes at high resolution.

In Figure 8, we also plot noise-weighted struc-ture factors, CrefF, where Cref is calculated from theFSC after smoothing (noise-weighted amplitudescan be calculated up to the resolution at which theweighting function Cref becomes random and theamount of signal in the data becomes negligible).

Figure 8. Guinier plot showing the natural logarithm of the spherical average of F versus 1/d 2 for the experimentalmap (thick red line), the experimental map amplitudes weighted by Cref (thin red line), weighted amplitudes aftersharpening (broken red line), Wilson statistics (horizontal black line), and the X-ray model (blue line). Linear fit ofdata for 1/d 2 between 0.005 and 0.015 yields B ¼ 1200 A2 for the experimental map and slope 200 A2 for the modelstructure factors (dotted black lines). The difference, B ¼ 21000 A2, is used to sharpen the experimental map.


CrefF better represents the average signal in themap, particularly at and beyond the resolutionlimit of the map where signal becomes pro-portional to noise (see Theoretical Background).The weighted curve shows a more linear, mono-tonic fall-off over a wider resolution range, consist-ent with the high-resolution amplitude decaybeing modelled by a single temperature factor. Itis therefore possible to measure more accuratelythe temperature factor at the resolution limit ofthe map. The straight line fitted to the linear partof the noise-weighted amplitude curve at highresolution indicates a temperature factor of1200 A2 compared to the flat line representingWilson statistics. It may be noted that the linear fitto the decay intersects zero angle at

ffiffiffiffiffiffiffiffiffiffiffiffiffiNatoms

p; the

amplitude given by Wilson statistics. This is alsoconsistent with modelling the high-resolutionstructure factors by Wilson statistics and a singletemperature factor.

Over the same resolution range, the X-ray modelstructure factors decay by an amount equivalent toa temperature factor of approximately 200 A2 com-pared to Wilson statistics, due to features of themolecular fold and the decline of scattering factors.The amplitude decay compared to the X-ray modelis therefore best described by a temperature factorof 1000 A2. For resolutions greater than 10 A, theextent to which the true structure factors for anygiven protein deviate from Wilson statistics islikely to be small, and a general correction for theobserved decay in all proteins may suffice. Theestimate of the temperature factor can therefore bemade for proteins for which no known referencestructure or scattering curve exists.

To observe molecular details at the resolutionlimit of the map, it is necessary to correct the struc-ture factor amplitudes for the loss of contrast. Wemultiply the noise-weighted structure factor ampli-tudes CrefF by e2ðBrestore=4d2Þ, where Brestore is atemperature factor of 21000 A2. Figure 8 alsoshows the spherically averaged structure factoramplitudes of the resulting sharpened map.Figure 9 shows a comparison of the unsharpenedmap (Figure 9(a)), the sharpened map with noise-weighted structure factor amplitudes (Figure 9(b)),and a map obtained by applying the same tem-perature factor to unweighted structure factors(Figure 9(c)), all calculated to 8.0 A.

The sharpened map shows detailed molecularfeatures not present in the unsharpened map asfurther analyzed in the next section. In the mapcalculated without weighted structure factors,noise is amplified to an extent that producesextraneous detail. A map calculated to the nominalresolution of 8.7 A without noise weighting lookssimilar to the 8.0 A map with noise weighting,

Figure 9. Effects of sharpening (Brestore ¼ 21000 A2) onmaps calculated to 8.0 A: (a) unsharpened; (b) shar-pened, noise-weighted structure factors; (c) sharpened,unweighted.


since the application of the precise resolutionthreshold removes terms where there is little signalcontributing to the map. With noise-weightedstructure factors the map is insensitive to thechosen resolution cut-off of the map andpotentially contains molecular features beyond thenominal resolution limit of the map. In addition,noise-weighted structure factors will be lesssensitive to over-sharpening due to errors in tem-perature factor estimate, and little difference wasobserved in maps calculated with a Brestore of21000 A2 or 21200 A2. Given an electron potentialmap and a measure of the signal as a function ofresolution (Cref), the appropriate calculation torestore contrast is determined without bias by thedata itself.

Comparison with X-ray model

The potential map of E2CD determined byelectron cryomicroscopy was compared to high-resolution crystallographic models of the catalyticcore. A 2.3 A X-ray structure has been reported forthe E2 catalytic domain of Azotobacter vinlandii(PDB 1eea), which is a cubic form of the enzymecontaining eight copies of a basic trimer structuralunit.25,26 A. vinlandii and B. stearothermophilus share40% sequence identity in the catalytic core.A structure for the dodecahedral B. stearothermophi-lus catalytic core has been reported based on 4.2 Acrystallographic data and molecular replacementusing the A. vinlandii trimer (pdb 1b5s), demon-strating that the trimer can form quasi-equivalentinteractions in cubic and icosahedral symmetry.20

However, in that study, the limited resolution orquality of the data prevented the refinement of theB. stearothermophilus model, and the resulting struc-ture is a homology model and does not containside-chains. Because the available icosahedralB. stearothermophilus X-ray model lacked side-chains, we superimposed the A. vinlandii monomeron each of the 60 monomers of B. stearothermophilusand calculated structure factors, spherical averagesof structure factors, and map density for compari-son with the results of cryomicroscopy. At theresolutions under investigation, the massdistribution is important, but the identity of theside-chains is unimportant.

The experimental map (Figure 10(a)) and X-raymodel were compared by aligning icosahedralsymmetry elements which uniquely define themodel location without further fitting, except for a3% adjustment of the magnification of the map.Overlap with the model and density is shown inFigure 10(b) and (c). The model and map possessthe same absolute hand. Helices are clearlyresolved at this resolution as shown in the stereodiagram of the density covering a single monomerin Figure 10(d). b-Sheets are apparent butindividual strands are not differentiated. The mapalso differs from the model in loop regions, andsome portions of the model may require refittingfor optimum placement in the map density.

Density for residues N-terminal to Gly204 is absentor appears at very low contour (Figure 10(d)). Inthe A. vinlandii structure these residues extendaway from the monomer and interact with adjacentmonomers in the trimer. These residues are theleast conserved between A. vinlandii and B. stear-othermophilus and could be in a different confor-mation or disordered. This observation isconsistent with these residues being part of anextended mobile linker connected to the E1/E2binding domain.21

Comparison of the appearance of the cryo-microscopy potential map and electron densitymaps computed from the model suggests that thefeatures of the map are in agreement with corre-sponding X-ray maps computed between 8 A and9 A resolution and confirms the resolution assess-ment. The correlation between the map and X-raymodel (CXRAY) is shown in Figure 7(b) and has simi-lar values whether X-ray or electron form factorsare used. In principle, CXRAY should be similar tothe correlation (Cref) between the full image datasetand a perfect reference. CXRAY is 0.46 at a resolutionof 8.7 A where Cref is 0.5. When structure factorscalculated from the X-ray model includes scatter-ing from a flat bulk solvent of 0.3 e2/A3, CXRAY is0.51.

Discussion

The structure of the icosahedral E2 catalytic coreof the pyruvate dehydrogenase complex fromB. stearothermophilus has been determined to 8.7 Aresolution by electron cryomicroscopy and imageanalysis of 3667 untilted icosahedral particles(220,020 asymmetric units). Images were recordedat high defocus on a field emission gun microscopeat 300 kV and corrected for the effects of the CTF.Model-based refinement was used to determinethe orientation parameters of the particles andmicroscope parameters for each image.

An advantage of our choice of E2CD as a testobject for a critical analysis of single-particle tech-niques is that an X-ray structure is known, allow-ing direct confirmation of the success of themethods for reconstruction, contrast restoration,and resolution assessment. The structure lookssimilar to an electron density map computed froman X-ray model for E2CD from the 40% sequenceidentical A. vinlandii. Some differences likely reflectreal features of B. stearothermophilus E2CD. Struc-tural studies of multi-enzyme complexes such aspyruvate dehydrogenase are essential to under-stand how the geometric arrangement of theprotein subunits determines their complex reactionkinetics.19,21

In the course of the work, we have sought todevelop unbiased procedures for addressing twofundamental issues in the image analysis of singleparticles. The first concerns how best to determinethe particle orientation in an image. Our initialattempts at hand determination of E2CD using the


best images showed that more accurateorientations were required. We showed that handdetermination could be used to optimize refine-ment parameters leading to better orientations andtherefore a better map. Once a map is obtained,the second issue is how the decay of structurefactor amplitudes at high resolution can berestored.

Absolute hand determination is important ininterpreting structures determined by high-resolution electron microscopy of single particlesbecause the chirality of a biological molecule is thebasis of its function. We have developed a semi-automatic procedure for hand determinationrequiring a model and two image stacks corre-sponding to particles from a single pair of low-dose images recorded at two angles of a tilt axis.We used the procedure on a molecule of known

hand, demonstrating that we have chosen theright conventions throughout the image processingsteps that are capable of inverting the hand.

The hand determination procedure maps out theaverage phase residual for all possible tilt axesbased on the agreement of tilted particle imageswith orientations predicted from untilted particles.A minimum is expected at the known tilt angleand axis of the experiment. The confidence of thehand determination is measured by the differencebetween the score at the tilt angle correspondingto the correct hand and the score at the negativetilt angle corresponding to the opposite hand(here called the free hand difference). A confidentresult for hand determination depends onaccurately determining particle orientations.A given refinement strategy based on the mini-mization of the phase residual between model

Figure 10 (a) and (b) (legend opposite)


projection and image may be influenced by noisefeatures in the image rather than signal. For aknown tilt angle and tilt axis, the particle orien-tation should change in a predictable way, but thenoise should be uncorrelated. The success of orien-tation determination on the first film can beassayed on the second film, independent of thenoise or spurious features on the first film. Theknown tilt angle, tilt axis, and free hand differenceare unbiased targets for optimizing the parameters

that influence the accuracy of orientation determi-nation. Since the accuracy of microscopeparameters is shown to influence orientations,microscope parameters may also be refined by thismethod. All refinement parameters can thus berapidly and systematically tested on a single pairof images, but the results may be applied to allimages in the dataset. In the absence of such aprocedure, the arduous task of repeated structuredeterminations to high resolution with variation of

Figure 10. Experimental map density for E2CD. (a) Final map density viewed down a 2-fold axis. (b) Final mapdensity with superimposed coordinate model viewed down the 5-fold axis. (c) Close-up view of map density on thecoordinate model. (d) Stereo view of monomer density on the model with the location of Gly204 labelled.


all parameters would be required. Because handdetermination is ultimately a test of the consistencyof the model with the data, it can be used to testthe quality or validity of a model, and possiblyidentify particle heterogeneity in a dataset bydetecting image pairs that together are inconsistentwith a model. A second tilted image couldroutinely be included in an automatic datacollection strategy to identify particle hetero-geneity or damage.

We have shown that individual particle orien-tations are better determined after the parameteroptimization using absolute hand. Particle orien-tations need to be well-determined, becauseaveraging images inaccurately or incoherentlyleads to blurring of high-resolution features or tocontrast decay. It is still an open question whetherthe orientations are well enough determined forstructure determination to atomic resolution.27 Ifthe particle orientations are determined with per-fect accuracy, then the remaining contrast loss isan intrinsic property of the images. Because themeasured tilt angle is the same as the known tiltangle of the experiment for large relative tilt anglesup to 308, there is no evidence for flattening duringsample preparation and exposure in the electronbeam, suggesting that cryomicroscopy can providehigh-resolution structures without artifacts. Never-theless, radiation damage does destroy high-resolution features and reduces the accuracy oforientation determination. In the future, recordingparticle images on electronic detectors with high-resolution pixels may lead to more accurateorientations through fractionation of the accumu-lated dose on separate frames and determinationof resolution-dependent weights for each frame.

We have shown that contrast restoration is essen-tial to reveal the highest-resolution molecularfeatures of the electron potential map. Weestimated the structure factor amplitude decay bycomparison of spherically averaged structurefactors with reference structure factors using aGuinier plot. The structure factors were put on anabsolute scale by setting the zero angle scatteringequal to the molecular mass times a solvent con-trast factor. Structure factor averages at resolutionsbeyond the shape/solvent scattering region areroughly flat and approach Wilson statistics. There-fore, an approximate theoretical scattering curvemay be used as a reference without the need toperform a small-angle X-ray scatteringexperiment.28,29 The large temperature factormeasured for maps obtained from single particlesis unlikely to be sensitive to small deviations inthe true structure factors from the averagereference scattering curve for typical proteins. Wehave modeled the loss of contrast by a single tem-perature factor. However, a more complex decayfunction may ultimately be required.

The application of a negative temperaturefactor will amplify both signal and noise. Weapplied a correction to amplitudes of the formCrefF e2Brestore=4d2

; where Cref is the estimated corre-

lation of the map with a perfect reference, andBrestore is the negative temperature factor. In theAppendix we argue that Cref is the best objectiveweight to apply to the electron potential map andis equivalent to the crystallographic figure-of-merit. Cref can be expressed in terms of the FSCbetween data half-sets and is a measure of thedata quality as a function of resolution. Noise-weighted structure factor amplitudes CrefF directlyrepresent the signal in the data, give a less noisyestimate of the temperature factor, and down-weight noise during contrast restoration. The con-trast restoration is therefore determined in anobjective way by the observed amplitude decayand the quality of the data as a function ofresolution.

Extrapolating the number of particles requiredto high resolution

To reach the ultimate goal of a high-resolutionmap that may be interpreted in terms of an atomicmodel, many more images are required. The num-ber may be extrapolated from the experimentalstructure factor amplitude decay curve. Near theresolution limit of the map, this decay may bemodelled by a single temperature factor Boverall. Asdescribed earlier (see Theoretical Background), theintersection of the average structure factor ampli-tude with the noise level determines the resolutionof the features that can be observed, and the num-ber of particle images determines the noise level.The amplitude decay curve reaches the noise levelat 8.7 A (FSC ¼ 0.15). The structure factor ampli-tude (F ¼ 100.5) at this resolution is equal to theamplitude of the noise after averaging 3667 experi-mental particle images. The noise amplitude forone image or for N images may then be deter-mined by observing that FNoiseðNÞ ¼ ðFNoiseð1Þ=

ffiffiffiffiN

pÞ:

In this sense, any amplitude value on the structurefactor decay curve at any resolution may beexpressed in terms of the number of particlesrequired to reach that resolution.

In Figure 11, the straight-line fit of the amplitudedecay curve with B ¼ 1000 A2 is plotted againstresolution with the structure factor amplitudeexpressed in equivalent particle numbers. BecauseBoverall was calculated by comparing referencestructure factor amplitudes with the final experi-mental map amplitudes, which include all the fac-tors that degrade contrast in the experiment andin the calculations, the estimate of the number ofparticles required at each resolution reflects intrin-sic image and densitometry defects as well aserrors in particle parameters and in reconstructionalgorithms. Improvement of any of the experimen-tal or computational factors that contribute to con-trast loss will lead to a smaller temperature factorand will produce higher resolution with fewerimages. Also shown in Figure 11 (dotted lines) arethe estimated number of particles assuming tem-perature factors with two, four, and eight timesslower fall-off with resolution. These curves have


an identical intercept at zero angle ðd ¼ 1Þ wherethe temperature factor has no effect and which cor-responds to the amplitude expected for Wilsonstatistics.

The extrapolated noise level (F ¼ 6028.8) for oneparticle image intersects the average structurefactor amplitude curve in Figure 8 at about 30 A,well above the average amplitude given byWilson statistics, in the part of the amplitudecurve dependent on the overall shape of themolecule and contrast against solvent. In theory,this also represents the limiting resolution thatwould be obtained from studying a similar struc-ture by tomography and fractionation of the totaldose. Tomography is likely to provide informationon overall shape, but averaging similar structuresis essential to obtain higher resolution features.

The number of particle images used to reach8.7 A resolution in this study may be comparedwith recent theoretical estimates.1,30 In one suchestimate,1 the number of images, Npart, required to

a resolution d is given by NinprojðpD=NasymdÞwhere Ninproj is the number of images needed forthe average structure factor intensity in eachprojection to reach a threshold signal-to-noiseratio, and pD=Nasymd is the number of unique pro-jections for a particle of diameter D containingNasym asymmetric units. Because the signal-to-noise ratio of the average structure factor in animage is the same as in the corresponding electrondiffraction pattern,14 Ninproj may be estimated byrequiring a minimum signal-to-noise ratio for theaverage structure factor intensity of the electrondiffraction pattern. Ninproj is given byðkSl2=kNl2Þ=ððkIobsl=IoÞD

2NeÞ, where kSl2=kNl2 is theaverage number of electrons required in eachstructure factor (noting that a single electronrecorded in a structure factor has a signal-to-noiseratio of 1), and the denominator is equal to thenumber of electrons in the average structure factorin a single low-dose image. The incident numberof electrons is D 2Ne where Ne ¼ 5 e2=A2 is theallowed dose and D 2 is the area of the particle (orunit cell), and kIobsl=Io ¼ se=30D is the mean struc-ture factor intensity as a fraction of the incidentnumber of electrons,1 where se ¼ 0.004 A2 is theelastic cross-section for carbon at 300 kV.31 Thusthe total number of particles Npart ¼ð1=NasymÞððkSl2=kNl2Þ30p=NesedÞ: Previously, a 3scriterion for the signal-to-noise ratio on amplitudes(kSl2=kNl2 ¼ 10; where kSl and kNl are the signaland noise for the map computed from all the data)was chosen as a conservative criterion for estimat-ing Ninproj, and the images were assumed to haveperfect contrast. In the present study, we haveshown that a resolution criterion related to mapinterpretability has kSl=kNl ¼ 1=

ffiffiffi3

pat the

resolution limit of the map. This reduces the num-ber of images required to any given resolution bya factor of 30. In addition, the effect of contrastloss described by Gaussian decay leads to anincrease in the required number of images by afactor ðeðBoverall=4d2ÞÞ2: For B ¼ 1000 A2 measured inthis study, the theoretical estimate of the numberof particles for 8.7 A is 2200 particles, slightlymore than half the number used in the experiment,3667 (Figure 11).

The theoretical calculation assumes an electrondose of (5 e2/A2) which is likely to be the limitingdose due to radiation damage for features nearatomic resolution. The electron dose used in theexperiment (18 e2/A2) is three to four times thislimiting dose. While the extra dose does not con-tribute to structure factor amplitudes near atomicresolution, it may have enhanced the signal up tothe resolution limit of the final map and madedetermination of particle parameters easier. If theexperimental electron dose is used in the calcu-lation, the number of particles required by theoryto 8.7 A is decreased by a factor of 3–4, and thenumber of experimental images then exceeds thetheoretical number by a factor of 6–8, even afteraccounting for the loss of contrast by a Gaussianfall-off. The best way to determine the more

Figure 11. Estimate of the number of icosahedralimages required as a function of resolution. The continu-ous line with B ¼ 1000 A2 is the fit of the amplitudedecay in Figure 8, but with the amplitude expressed inparticle numbers. A single point (white circle) representsthe resolution of the map obtained in this study with3600 particle images. Calculated curves (dotted lines)for 500, 200, and 125 A2 assume the same zero angleintercept in agreement with Wilson statistics. Theoreticalestimate of the number of particle images Npart ¼ð1=NasymÞ½ðkSl2=kNl2Þ30p=Nesed� eB=2d2

required to 8.7 Aresolution assuming kSl=kNl ¼ 1=

ffiffiffi3

pand B ¼ 1000 A2,

se ¼ 0.004 A2, Ne ¼ 5 e2/A2, and Nasym ¼ 60 is 2200 par-ticles (black circle).


complex function that models experimental con-trast loss is to measure the build-up of imageamplitude as a function of time, dose, and resol-ution by fractionation of the accumulated doseduring image acquisition.

Though still a large assembly, E2CD is smallerthan a number of icosahedral complexes solved tobetter than 10 A by electron microscopy.32,33 Wehave used highly defocussed images, such as willbe required to extend higher-resolution studies tosmaller particles. The large temperature factormay be a consequence of the large defocus(coherence envelope), but there may also be otherimaging imperfections that depend inversely onparticle size, such as beam-induced movement orimage blurring. Particle heterogeneity could alsocontribute to the temperature factor. Given theexcellent quality of the map, we have so far notattempted classification of the E2CD particles withthe intent of identifying any conformational hetero-geneity, such as has been reported for themammalian enzyme34 nor have we assessed itscontribution to the overall temperature factor.

A 4 A resolution map in which a protein modelcan be constructed cannot be achieved withrealistic amounts of data of a quality comparableto that used here. It will therefore be necessary toobtain better quality micrographs with a decreasedoverall temperature factor. More accurate compu-tation of particle orientations and microscopeparameters will contribute by reducing the compu-tational portion of the temperature factor(Bcomputation). It will be essential to reduce theintrinsic amplitude decay of images (Bimage) byreducing movement and charging. Such improve-ments will lead to higher-resolution structures andwill open a wide range of membrane proteins andmacromolecular assemblies that are difficult tostudy in the crystalline state to structural anddynamical analysis by single-particle electronmicroscopy.

Methods

Electron cryomicroscopy

Purified B. stearothermophilus dihydrolipoyl acetyl-transferase inner core, a 60-mer of the 28 kDa acetyl-transferase domain comprising residues 173–427 of theE2 chain,35 was a gift from Gonzalo J. Domingo andRichard Perham. Samples (3 mg/ml) in 50 mM sodiumphosphate buffer, pH 7.0 were stored at 270 8C, andwere diluted 1:10 (v/v) in PBS (phosphate-bufferedsaline) immediately prior to use. Holey carbon filmswere prepared as described36 on 400-mesh Cu/Rh gridsand glow-discharged for ten seconds in air: 2 ml of pro-tein solution was applied to the carbon film in a high-humidity chamber,37 blotted for 20 seconds with filterpaper to remove excess sample, and plunged into liquidethane at 2180 8C. Grids were stored in liquid nitrogenand transferred to a Gatan cold stage. Images of particlesover holes were recorded under low-dose conditions(15–20 e2/A2) on an FEI F30 microscope operating at300 kV with 70 mm condenser and 50 mm objective aper-

tures, 59,000£ magnification, and a defocus rangebetween 2.3 mm and 7.7 mm.

Image analysis

Images were digitized on a Zeiss-SCAI flatbed densi-tometer using a 7 mm step and processed using pro-grams from the MRC package.38 Particles were selectedmanually using Ximdisp, boxed using LABEL and storedas one image stack per film. Image densities were floatedto zero average. For images recorded at 60,000£ mag-nification, adjacent pixels were averaged 4 £ 4 with aresulting resolution of 4.98 A/pixel in 128 £ 128 pixelsquares or averaged 2 £ 2 with a resolution of 2.49 A/pixel: 1681 particles from 14 films were included in thefirst dataset and 3667 particles from 26 films recorded at300 kV were used in the final dataset.

Map refinement

Particle orientation parameters were determined firstusing a previously described 3D model21 but wereimproved with successive rounds of refinement usingFREALIGN.23 Orientation and translation parametersfor each particle image were determined each cycle byfinding the minimum phase residual (see the text) withthe model calculated using parameters from the lastcycle. Two hundred cycles of particle orientation ran-domization followed by Powell minimization were usedto determine the new orientation parameters. Parameterscontrolling the refinement are given in Table 1. To speedrefinement, the five-orientation parameter search wasconducted on particle images in a 128 £ 128 pixel box at4.98 A/pixel, and the maximum resolution of data usedin refinement was 15 A. For maps calculated to resol-utions higher than 15 A, the x; y parameters only werere-refined against particle images in a 256 £ 256 pixelbox at 2.49 A per pixel. Initial defocus parameters (DF1,DF2, and angle of astigmatism) for the CTFs used to cor-rect the image structure factors were determined usingCTFFIND2.38 Both the CTF and magnification wererefined using the same values for all particles on a film.Particle orientations were given by the Euler angles(c,u,f) according to a described convention.11

Procedure for recording and digitizing pairs ofimages of tilted specimens

Each tilt pair was recorded at two goniometer anglestaking care to observe the sense of rotation of the speci-men tilt axis. A positive value of the relative tilt anglecorresponded to a clockwise rotation of the specimenholder viewed along the direction of specimen insertionfrom the nitrogen dewar toward the tip. Tilt pairs wererecorded at 21,000£ , 29,000£ , 39,000£ and 59,000£magnification.

The orientation and polarity of the F30 microscope tiltaxis on micrographs were determined by either of twomethods: (1) recording the movement of markers on agrid as the specimen stage was translated into the micro-scope; or (2) observing whether Thon rings from themicrograph indicated the relative position of the speci-men as up (overfocus) or down (underfocus) on eitherside of the tilt axis. The tilt axis may therefore bedescribed as a right-hand rotation about a vectororiented with respect to the film number recorded onthe micrograph, a feature of fixed geometry common toall micrographs on a given microscope.


Because the orientation of the tilt axis varies withmagnification, the relative orientation of the microscopetilt axis on the film at different magnifications wasfurther checked by comparing images of the samecatalase crystal (a gift from John Berriman). The axiswas found to be rotated by 908 in the film plane at59,000£ compared to other magnifications.

Films were scanned with a consistent orientation on aZeiss SCAI film densitometer as judged by the markingnumbers on the microscope negative. Correspondingparticles were selected on both micrographs of the tiltpair, boxed, and stored in register in separate stacks (pro-gram LABEL). Inspection of the full image containingfilm numbers in our display software (Ximdisp) allowedthe known direction of the tilt axis to be assigned as anapproximate vector direction in our software coordinatesystem. The tilt transformation corresponding to thisknown tilt direction and polarity is then described by{rotx; roty} (transformation defined below). We confirmedthat all rotation transformations of three-dimensionalvolumes (program FREALIGN, program ANGPLOT)gave the expected effect when calculated projectionswere displayed using Ximdisp.

Image analysis of tilt pairs

Particle orientations were then determined separatelyfor the untilted and tilted image stacks with FREALIGNusing the same procedure described above for untilteddata. We analyzed pairs of images from tilted specimensin two different ways.

(1) Phase residual maps for all possible tilt axes. Theapplication of a tilt transformation {rotx; roty} tothe orientation determined for an untilted particlegives a predicted orientation for the tilted particles.The tilt transformation {rotx; roty} is a rotation by

an angle of magnitude tiltangle ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirot2

x þ rot2y

qabout an axis lying in the film plane in an orien-tation described by an angle tiltaxis ¼ arctanðroty=rotxÞ: This transformation may be writtenas the Eulerian rotation (c ¼ tiltaxis, u ¼ tiltangle,f ¼ 2 tiltaxis) (program ANGPLOT). The agree-ment between the projection of the model in thepredicted orientation with the tilted particleimage was scored by the phase residual afterrefining the translation parameters ðx; yÞ usingdata from 100–35 A. Tilt transformations in 18increments of both rotx and roty; corresponding toall possible directions and tilt angles for the micro-scope goniometer, were applied to the untiltedparticle orientations. The average phase residualon the image of tilted particles was determinedfor 50–100 particles. The minimum phase residualin the resulting maps of {rotx; roty} identifies thecalculated tilt angle and tilt axis direction. The pro-cedure can be run automatically given tilted anduntilted image stacks. The location and polarity ofthe laboratory axis on the micrograph must beidentified as described above. Programs areprovided as part of the MRC Image ProcessingPackage.38 These results are shown in Figures 3and 5.

(2) Best tilt axis given untilted and tilted orientations.A tilt axis and angle were assigned to each particlepair (program TILTDIFF) using the Euler anglesassigned separately to the untilted and tiltedparticles by searching for the minimum angular

error in transforming the untilted particle orien-tation onto the tilted particle orientation. The tiltaxis was constrained to lie in the plane of themicrograph with a maximum tilt angle of 308. Thetransformation was represented by an Eulerianrotation of the form (c ¼ tiltaxis, u ¼ tiltangle, f ¼2tiltaxis). For a particle with symmetry, it is essen-tial to test all symmetry-related orientations to findthe tilt axis with minimum error. The result isshown in Figure 6.

Comparison to X-ray models

The X-ray models for B. stearothermophilus (PDB code1b5s) or A. vinlandii (PDB code 1eea) E2CD were com-pared with the electron potential map density using theprogram O,39 confirming the experimental assignmentof absolute hand. Spherically averaged structure factorsfor the model (Figure 8) including solvent were com-puted using CRYSOL.40 Additional structure factor cal-culations on maps were performed using programsfrom the CCP4 suite.41 X-ray model structure factorswere calculated with and without a flat bulk solventmodel using XPLOR.42 The magnification of the experi-mental map was decreased by 3% to match the X-raymodel as determined by visual inspection and maximi-zation of the correlation with the map calculated fromthe X-ray model.

Figures 2(c) and 9 were made using Surf.43 Figure 10was made using O.

Acknowledgements

P.B.R gratefully acknowledges the support of aHuman Frontier Science Program Long TermFellowship and an Agouron Institute Fellowship(Research Grant AI-SA1-99.3). We thank GonzaloDomingo and Richard Perham for providing theB. stearothermophilus pyruvate dehydrogenase E2catalytic domain used in this work, John Berrimanfor help with microscopy, and members of the Lab-oratory of Molecular Biology for discussions. Wethank Tony Crowther, Richard Perham, and JohnRubinstein for a critical reading of this manuscript.

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Appendix

An Objective Criterion for Resolution Assessment inSingle-particle Electron Microscopy

The resolution of a three-dimensional densitymap calculated from single-particle images can beestimated by the Fourier shell correlation (FSC)between two independent maps, where each mapis calculated from half the data.A1,A2 However,disagreement exists as to what threshold value ofthe FSC, a function of resolution, should be usedto define the limiting resolution of the map. Here,we propose a threshold criterion based on theestimated correlation between a density map calcu-lated from a full set of image data and a perfectreference map, and identify the correspondingthreshold for the FSC between maps calculatedfrom data half-sets. This threshold criterion forsingle-particle electron microscopy is the same asa common measure of map interpretability inX-ray crystallography and provides a unifiedcriterion for assessing the resolution of a structuredetermined by either technique.

The Fourier shell correlation is the complexcorrelation of structure factors (F1,F2) of two mapscalculated from the data half-sets as follows:

FSC ¼

PF1Fp

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPlF1l

2PlF2l2

q ðA1Þ

where the sum runs over a resolution shell ofreciprocal space voxels. The FSC includes bothamplitude and phase information and its advan-tages over other resolution criteria have beendiscussed.A3 Because the correlation over all ofreciprocal space will be dominated by low-resol-ution terms, calculating the FSC in reciprocalspace shells provides a more sensitive measure ofits fall-off with resolution. When the value of theFSC curve is near 1.0, then the information con-tained in each reciprocal space voxel, on average,is a nearly perfect determination of the structurefactor and has little noise.

Ideally, the FSC should be examined as a func-tion of resolution to identify regions of reciprocalspace with poor correlation, such as those due tozeroes of the CTF. However, identifying a single-value resolution for the map where the FSC dropsbelow a threshold provides a rapid measure ofmap improvement during refinement and a com-mon standard for reporting progress in the litera-ture in the same way that common measures ofstructure quality are reported in X-ray crystallo-graphy. Such a standard resolution criterion shouldreflect the interpretability of a map in terms ofmolecular features but should not be dependenton irrelevant details of map calculation.

One commonly proposed threshold for identify-ing the resolution from the fall-off of the FSC is toobserve where the FSC reaches the level expectedfrom the FSC calculated for pure noise with aGaussian distribution. This FSC for pure noise is

given by 1=ffiffiffiffiN

p; where N represents the number of

voxels in a resolution shell, and the threshold istaken as either two or three times this figure. Sucha threshold can be used as a relative measure ofresolution for maps calculated in an identical way.Because it depends on the number of voxels in anarbitrary resolution shell of the map, it cannot pro-vide an absolute resolution criterion. Suppose thestructure of a molecule is determined both aloneand in the context of a larger structure suchthat both maps are equally interpretable. Eachreciprocal space voxel will have the same signal-to-noise ratio so that the plot of the FSC againstresolution will be virtually identical. However,since the number of voxels between resolutionshells will be different, the resolution assessmentwill be different because the noise correlationfigure depends on the number of voxels. It hasalso been suggested that the noise figure for a sym-metric structure should be ð1=

ffiffiffiffiN

pÞ

ffiffiffiffiffiffiffiffiffiffiNAU

p; where

NAU is the number of asymmetic units, becausesymmetry-related pixels are not independent.A4

However, a map with or without symmetry willbe equally interpretable when the FSC is the same.Any threshold criterion that depends on the num-ber of pixels in the map is not an absolute criterionfor the evaluation of resolution.

Another common threshold criterion is tonominate the point where the FSC between datahalf-sets equals 0.5 as the resolution. This is anabsolute criterion with a signal-to-noise ratio sub-stantially greater than zero,A5 and is not dependenton how the map is calculated. The argumentsbelow suggest that it is an underestimate of the res-olution, and that a better value for the thresholdcan be proposed, related to the interpretability ofthe map.

Individual structure factors for each half datasetcan be represented by a common signal term witha noise term added, F1 ¼ S þ N1 and F2 ¼ S þ N2;and their correlation by:

C ¼

PðS þ N1ÞðS þ N2ÞpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPlS þ N1l2

PlS þ N2l2

q ðA2Þ

Assuming signal and noise are uncorrelated, andfor data on the same scale, the above expressionmay be written as follows:

FSC ¼

PlSl2P

lS2 þ N2lðA3Þ

When the correlation FSC is 0.5, kS 2l is approxi-mately equal to kN 2l and half the power ðF2Þ ineach data half-set is signal ðS2Þ and half is noiseðN2Þ:

The threshold FSC ¼ 0.5 is an underestimate ofthe resolution of the structure, because it describesthe resolution of a map calculated from only halfthe data, and a map computed from more imageswill be of higher resolution. By definition there arenever two independent full datasets to compare,


but the resolution of the full dataset can beestimated from the FSC as follows.

When the two halves of the data are averaged toproduce the best estimate of the structure, thesignal-to-noise ratio for the full average isincreased by

ffiffiffi2

prelative to the half-set average,

or equivalently, the noise term N ! N=ffiffiffi2

pin

expression (A3) and the FSCfull becomes:

FSCfull ¼

PlSl2P

S2 þN2

2

��¼

2FSC

1 þ FSCðA4Þ

By substituting (A3), the FSC, into the left part of(A4), the resolution estimate for a map computedfrom all the data can be written in terms of theFSC between maps calculated from half thedata.A6 When half the power is signal and half isnoise, the FSC is 0.5 but the FSCfull is higher at 2/3or 0.67. If we choose as the threshold FSCfull ¼ 0.5,this occurs when the FSC is 0.33.

The choice of FSCfull ¼ 0.5 is still too pessimistic.Ideally we desire a measure of the agreementbetween the best map we can calculate from a fulldataset and a perfect map containing no errors.Though a perfect map is unknown, we can esti-mate the correlation from signal-to-noise argu-ments. Such a correlation can be obtained fromexpression (A2) by setting the noise term N2 ¼ 0in expression (A2) so that F2 ¼ S represents a per-fect map without noise, setting N1 ¼ N=

ffiffiffi2

pfor the

noise in a full dataset, and again assuming signaland noise are uncorrelated.

Cref ¼

PS þ

Nffiffiffi2

p

Sp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPS þ

Nffiffiffi2

p

��2P

lSl2s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS2

S2 þN2

2

vuuut

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2FSC

1 þ FSC

rðA5Þ

In this case, when kS2l equals kN2l; Cref ¼ 0:816:Again, substitution of FSC (A3) into the left-handside of (A5) gives an expression for the correlationof the full dataset against a perfect reference interms of the FSC. If we choose a threshold for Cref

of 0.5, this corresponds to an FSC of 0.143.The choice of Cref ¼ 0.5 as a threshold can be

justified for two reasons. First, we can write anexpression for the correlation between a full data-set and a perfect reference which is equivalent to(A5) but in terms of structure factors as follows:

Cref ¼

PF1FrefffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

lF1l2PlFrefl

2q ¼

PlF1llFreflcosðDfÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

lF1l2PlFrefl

2q ðA6Þ

Here, the known structure factor of the perfectreference provides a perfect phase and the corre-lation with the experimental map is written interms of Df, the phase error. Expression (A6) istherefore the amplitude weighted average of cos

ðDfÞ which is equivalent to the figure-of-merit mused in X-ray crystallography.A7 A map with anaverage m ¼ 0.5, corresponding to a phase error of608, is commonly regarded as interpretable, i.e. amolecular structure can be built.A8 Cref is thereforeequivalent to a figure-of-merit, and an identicalthreshold may be adopted.

Second, the FSC is also related to the real spacecorrelation coefficient, R; a common measure forthe agreement of two density maps:

R ¼kr1r2l2 kr1lkr2lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kr21l2 kr1l

2q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kr22l2 kr2l

2q ðA7Þ

When the map density is expressed as structurefactors, krl ¼ ð1=VÞFð0Þ; kr2l ¼ ð1=V2Þ

PlF2l; and

kr1r2l ¼ ð1=V2ÞP

F1ðhÞF2ð�hÞ: When an experimentalmap rexp is compared with a perfect reference rref,and assuming no amplitude errors in theexperimental structure factors, krexprrefl ¼ð1=V2Þ

PlFrefl

2cosðDfÞ. Then for maps on the same

scale (F(0) ¼ 0), the real space correlation coeffi-cient R is identical with equation (A6):

R ¼krexprrefl2 krexplkrreflffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kr2expl2 krexpl

2q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

kr2refl2 krrefl

2q

¼

PlFrefl

2cosðDfÞP

lFrefl2

ðA8Þ

where the summation runs over all of reciprocalspace.A8 It has been shown that a sufficient con-dition for including an additional resolution shellthat will improve the overall map correlation coef-ficient is that the additional resolution shell have acorrelation of 0.5 against the perfect reference.A8

However, if the amplitudes of the extra resolutionshell are adequately large, a resolution shell witha lower correlation coefficient can still improve themap.

Therefore, choosing Cref equal to 0.5 represents asingle objective threshold criterion for the FSCindependent of the way the maps are calculated. Itis equivalent to a common measure of mapinterpretability for X-ray crystallography, thefigure-of-merit, and identifies the highestresolution shell that is guaranteed to improve themap quality. In practice, a perfect reference map isunavailable, but a Cref of 0.5 is equivalent to anFSC of 0.143 between data half-sets.

Table A1 below shows corresponding values fora given map quality for each of the correlationcoefficients outlined above. The proposedthreshold criterion unifies common measures of

Table A1. Correlation coefficients

FSC FSCfull Cref Phase error (deg.) S/N1/2 S/NFULL

0.50 0.67 0.82 35 1.00 1.410.33 0.50 0.71 45 0.71 1.000.14 0.25 0.50 60 0.41 0.58


map quality used in both single particlemicroscopy and crystallography, including theFourier shell correlation, the figure-of-merit, andthe real space correlation coefficient.

Assignment of an appropriate weight to a struc-ture factor based on its probability of being thetrue structure factor should improve the mapcalculation as with a figure-of-merit. Cref is ameasure of the average signal-to-noise ratio in agiven reciprocal space resolution shell and theerror in the structure factors. Noise-weightedstructure factors can be calculated as CrefF whereCref ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2FSCÞ=ð1 þ FSCÞ

pfor the resolution shell.

Cref will have a value near 1.0 in well-determinedshells, but will decrease at high resolution wherethe structure is less certain. In the main text weapply this weighting scheme to maps to determinemore accurately the fall-off of structure factoramplitudes with resolution and to suppress noisewhen applying a positive temperature factorduring map sharpening.

Acknowledgements

Peter Rosenthal, R. A. Crowther and RichardHenderson are co-authors of this appendix.

References

A1. Saxton, W. O. & Baumeister, W. (1982). The corre-lation averaging of a regularly arranged bacterialcell envelope protein. J. Microsc. 127, 127–138.

A2. van Heel, M. & Stoffler-Meilicke, M. (1985). Charac-teristic views of E. coli and B. stearothermophilus 30 Sribosomal subunits in the electron microscope.EMBO J. 4, 2389–2395.

A3. van Heel, M. (1987). Similarity measures betweenimages. Ultramicroscopy, 21, 95–100.

A4. Orlova, E. V., Dube, P., Harris, J. R., Beckman, E.,Zemlin, F., Markl, J., van Heel, M. et al. (1997).Structure of keyhole limpet hemocyanin type 1(KLH1) at 15 A resolution by electron cryomicro-scopy and angular reconstitution. J. Mol. Biol. 271,417–437.

A5. Penczek, P. (1998). Appendix: measures of resolutionusing Fourier shell correlation. J. Mol. Biol. 280,115–116.

A6. Grigorieff, N. (2000). Resolution measurement instructures derived from single particles. ActaCrystallog. sect. D, 56, 1270–1277.

A7. Blow, D. M. & Crick, F. H. C. (1959). The treatmentof errors in the isomorphous replacement method.Acta Crystallog. 12, 794–802.

A8. Lunin, V. Y. & Woolfson, M. M. (1993). Phase errorand the map correlation coefficient. Acta Crystallog.sect. D, 49, 530–533.

Edited by W. Baumeister

(Received 9 April 2003; received in revised form 24 July 2003; accepted 24 July 2003)


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