Journal of Structural Biology 178 (2012) 152–164

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Journal of Structural Biology

journal homepage: www.elsevier .com/ locate/y jsbi

High-throughput subtomogram alignment and classification by Fourier spaceconstrained fast volumetric matching

Min Xu a, Martin Beck b, Frank Alber a,⇑a Program in Molecular and Computational Biology, University of Southern California, Los Angeles, CA 90089, USAb Structural and Computational Biology Unit, European Molecular Biology Laboratory, Meyerhofstr. 1, 69117 Heidelberg, Germany

a r t i c l e i n f o

Article history:Available online 7 March 2012

Keywords:Cryo-electron tomographySubtomogram alignment and classificationFast rotational matching

1047-8477/$ - see front matter � 2012 Elsevier Inc. Adoi:10.1016/j.jsb.2012.02.014

⇑ Corresponding author. Address: 1050 Childs Wa90089, USA. Fax: +1 213 740 2437.

E-mail address: alber@usc.edu (F. Alber).

a b s t r a c t

Cryo-electron tomography allows the visualization of macromolecular complexes in their cellular envi-ronments in close-to-live conditions. The nominal resolution of subtomograms can be significantlyincreased when individual subtomograms of the same kind are aligned and averaged. A vital step for sucha procedure are algorithms that speedup subtomogram alignment and improve its accuracy to allow ref-erence-free subtomogram classifications. Such methods will facilitate automation of tomography analy-sis and overall high throughput in the data processing. Building on previous work, here we propose a fastrotational alignment method that uses the Fourier equivalent form of a popular constrained correlationmeasure that considers missing wedge corrections and density variances in the subtomograms. The fastrotational search is based on 3D volumetric matching, which improves the rotational alignment accuracyin particular for highly distorted subtomograms with low SNR and tilt angle ranges in comparison to fastrotational matching of projected 2D spherical images. We further integrate our fast rotational alignmentmethod in a reference-free iterative subtomogram classification scheme, and propose a local featureenhancement strategy in the classification process. As a proof of principle, we can demonstrate thatthe automatic method can successfully classify a large number of experimental subtomograms withoutthe need of a reference structure.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Cryo-electron tomography (cryoET) enables the visualization ofa cell’s interior under close to live conditions (Beck et al., 2009,2011; Nickell et al., 2006). The 3D tomogram is reconstructed froma set of two-dimensional micrographs, which are collected by tilt-ing the sample around a single rotational axis. The reconstructedtomograms have typically resolutions that are sufficient to detectindividual macromolecular complexes in their cellular context(Nicastro et al., 2006; Kühner et al., 2009; Medalia et al., 2002;Komeili et al., 2006). Once detected, subtomograms of the samecomplexes can be aligned and averaged to achieve a higher nomi-nal resolution and signal-to-noise ratio (SNR) for their 3D densitymaps (Frank, 2006).

Subtomogram alignment and subsequent classification is anintegral part of this strategy. The alignment relies on the searchfor the best rigid transformation of one subtomogram withrespect to the second so that the similarity measure between themis maximized. However, several factors make the alignment ofsubtomograms challenging. Tomograms contain relatively high

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noise levels (Frangakis and Förster, 2004) and are typically of lownon-isotropic resolution (P4 nm after averaging (Lucic et al.,2005; Förster et al., 2005; Briggs et al., 2009)). Moreover, tomo-grams are subject to distortions. One source of distortions is thevariation of the Contrast Transfer Function (CTF) within and be-tween individual micrographs used in the 3D density reconstruc-tion (Förster et al., 2008). More critical are orientation specificdistortions as a result of the so-called missing wedge effect, whichis a consequence of the limited data collection due to the limitedtilt ranges when collecting individual micrographs (with a maxi-mal tilt range from �70 to +70 degrees). As a result, in Fourierspace structure factors are missing in a characteristic wedgeshaped region. This missing data leads to anisotropic resolutionand different kinds of artifacts that depend on the structure ofthe object and its orientation with respect to the direction of thetilt-axis.

Several subtomogram alignment approaches take into accountthe missing wedge effects by using a constrained similarity mea-sure (e.g., Förster et al., 2008; Bartesaghi et al., 2008; Amat et al.,2010; Volkmann, 2010). For instance, Förster et al. (2008) intro-duced a correlation metric that constrains the similarity measureonly to the structure factors (i.e., Fourier coefficients) common toboth pairs of subtomograms. Because sample thickness can bevariable in a tomogram, this method also corrects for the local

M. Xu et al. / Journal of Structural Biology 178 (2012) 152–164 153

contrast difference in individual subtomograms by normalizing thesimilarity measure with respect to the mean and variance of theintensity distributions in both subtomograms (Förster et al., 2008).

In another method the influence of noise has been reduced inthe alignment process by considering only a small percentage ofhigh magnitude Fourier coefficients when computing the cross-correlation based similarity metric and therefore this method ex-cludes those coefficients that are expected to be dominated bynoise (Amat et al., 2010).

Most subtomogram alignment methods use an exhaustive scan-ning over all rotations of one subtomogram relative to the secondto identify the orientation that maximizes the similarity metric(e.g., Förster et al., 2008; Amat et al., 2010; Volkmann, 2010). Thescanning is performed at fixed angle intervals and at each sampledrotation a fast translational search is performed using the Fast Fou-rier Transform (FFT) (Frank, 2006). For highly accurate alignments,relatively small sampling intervals are needed, which significantlyincreases the cost of the rotational search. Exhaustive rotationalscanning is computationally intensive, which limits its applicabil-ity when large data sets need to be aligned for subtomogram clas-sification. This problem becomes even more relevant withincreasing cryoET resolutions (Murata et al., 2010) and resultinglarger subtomogram volumes. Therefore development of new algo-rithms that improve speed and accuracy of subtomogram align-ment is a vital step in automation of tomography analysis andoverall high throughput in data processing and reconstruction.

To enhance computational efficiency, Bartesaghi et al. (2008)use a fast rotational matching in which the best rotational transfor-mation is computed rapidly using the convolution theorem withinthe Spherical Harmonics framework. Each subtomogram is trans-formed into a two-dimensional spherical image by integratingthe magnitude of Fourier coefficients positioned along raysthrough the Fourier space origin. However, such a projection ofthe Fourier coefficient magnitudes onto the points of a unit spheremay increase ambiguities in the alignment. Moreover, the appliedalignment dissimilarity score is not normalized with respect to themean and variance of two subtomograms, therefore does not con-sider the non-uniformness of the tomogram.

In this paper, we formulate a Fourier-space equivalent similar-ity measure for the normalized constrained correlation introducedby Förster et al. (2008) and combine it with a fast rotational align-ment based on 3D volumetric rotational matching. We adapt a 3Dvolumetric rotational matching method that was previously usedfor fitting atomic structures into density maps (Kovacs andWriggers, 2002; Garzon et al., 2007) and extend its applicabilityto subtomogram alignments by including missing wedge effectsin the matching score. This method significantly increases thealignment speed in comparison to the standard exhaustive rota-tional scanning approach (Förster et al., 2008). It also improvesthe alignment accuracy for subtomograms with low SNR and smalltilt angle ranges in comparison to the fast rotational the rotationalmatching of 2D spherical images (Bartesaghi et al., 2008). More-over, by using a Fourier equivalent score of the normalized con-strained correlation our method also corrects for the non-evenness of contrast in tomograms.

As test case for our fast alignment method we perform refer-ence-free subtomogram classifications. Reference-free classifica-tion is fundamental for providing an unbiased structuralcategorization of the macromolecular complexes in subtomogramsbecause the initial classification is directly derived from the inputdata. Several types of reference-free subtomogram classificationapproaches exist, including methods based on maximum likelihoodapproaches (e.g., Scheres et al., 2009), methods using rotationinvariant subtomogram features (e.g., Xu et al., 2009, 2011), andfinally methods that rely on iterative successive alignment andclassification steps (i.e., the alignment-through-classification

approach) (e.g., Bartesaghi et al., 2008; Winkler, 2007; Winkleret al., 2009; Hrabe et al., 2012).

Our alignment method is sufficiently fast for carrying out allpair-wise alignments even for a relatively large number of subto-mograms (e.g., a few thousands). Therefore we are able to embedthe method into the alignment-through-classification framework,which is widely popular in 2D single particle averaging. Further-more, we propose an additional generic and automatic local fea-ture enhancement step to the framework. This step emphasizesthe most discriminative local features between subtomograms,which improves the clustering performance. Moreover, we alsointegrate an automatic optimal cluster selection into the frame-work. We can demonstrate that the automated framework can suc-cessfully classify experimental subtomograms even for highlysimilar but distinct complexes.

2. Materials and methods

2.1. Fourier space equivalence of constrained correlation

Two subtomograms f and g are defined as two integrable func-tions f ; g : R3 ! R. To calculate the similarity between two subto-mograms, Förster et al. (2008) proposed a constrained correlationwith missing wedge correction. It is based on a transform of thesubtomograms that eliminates the Fourier coefficients in the miss-ing wedge region. This goal is achieved by introducing a binarymissing wedge mask function as M : R3 ! f0;1g. The missingwedge mask function M defines valid and missing Fourier coeffi-cients in Fourier space. For example, in single tilt electron tomog-raphy with tilt angle range �h, the missing wedge mask functioncan be defined as MðnÞ :¼ Iðjn3 j6jn1 j tanðhÞÞðnÞ, where I is the indicatorfunction. GivenM, the real space subtomogram then excludes anycoefficients located inside of any of the two missing wedge regionsand is defined as

f � :¼ RfF�1½ðF f Þ X�g ð1Þ

where R denotes the real part of a complex function; KR is the rota-tion operator such that ðKReÞðxÞ :¼ e½R�1ðxÞ� for any functione : R3 ! C; F is the Fourier transform operator; andX :¼MðKRMÞ ensures that only structure factors are consideredthat are defined in both subtomograms, therefore excluding anystructure factors that are located in the missing wedge region ofany of the two subtomograms. Correspondingly, the real spacefunction of the second subtomogram is defined as

g� :¼ RfF�1½ðFsaKRgÞ X�g ð2Þ

where sa is defined as the translation operator so thatðsagÞðxÞ :¼ gðx� aÞ with a 2 R3. Then the normalized constrainedcorrelation value between two subtomograms f and g that also con-siders the missing-wedge corrections can be calculated as (Försteret al., 2008)

c ¼R½f � � lðf �Þ� ½g� � lðg�Þ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR½f � � lðf �Þ�2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR½g� � lðg�Þ�2

q ð3Þ

where l is the mean operator that returns the mean value of a func-

tion h, and is defined as lh ¼R

hðxÞSh . S is the size operator, and corre-

spondingly Sh returns the size of the subtomogram h (i.e., the totalnumber of voxels in the subtomogram). Because this similaritymeasure c is normalized by the mean and variance of intensity val-ues of both subtomograms it considers the local contrast differencesfor individual regions in a tomogram.

To allow a fast rotational alignment of the local constrainedsimilarity measure, we formulate the constrained correlation

1 When q is a real valued function, the complex conjugate KRq ¼ KRq.

154 M. Xu et al. / Journal of Structural Biology 178 (2012) 152–164

measure directly in Fourier space. According to the basic propertiesof the Fourier transform, we can formulate for any given two com-plex valued functions p; q : R3 ! CZ

xpðxÞqðxÞdx ¼ lðFpFqÞ ð4Þ

where q denotes the complex conjugate of q. Here the mean opera-tor l for complex Fourier space functions is defined in the sameway as l for real space functions. In addition,

F½p� lðpÞ� ¼ Fp�Z

p ð5Þ

BecauseR

p ¼ ðFpÞð0Þ, the Fourier transform of a function p aftersubstracting its mean can always be obtained by setting ðFpÞð0Þto zero. Therefore, we assume ðF f Þð0Þ ¼ 0 and ðFgÞð0Þ ¼ 0, corre-spondingly. When the Fourier transform F is realized using FFT,and F f and Fg have the same size (i.e., SF f ¼ SFg), the Fourierspace form of c is calculated as

c ¼ R

R½ðF f ÞX�ðFsaKRgÞXffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR

½ðF f ÞX� ðF f ÞXq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR

½ðFsaKRgÞX� ðFsaKRgÞXq

0B@

1CA ð6Þ

which is the real part of a complex valued function. According to thebasic properties of the Fourier transform, we can reformulate c as

c ¼ R

RðF f ÞM2 e2pia>n KR½ðFgÞM2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR

jðF f ÞMj2 KR½M2�q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR

M2 KR½jFgMj2�q

0B@

1CA ð7Þ

When using a binary maskM2 f0;1g, thenM2 ¼M. In summary,the general outcome of this section is the formulation of the con-strained Fourier-space correlation measure c, which includes amissing-wedge correction and is normalized.

2.2. Fast rotational sampling by spherical harmonics convolution

Because we have formulated the constrained correlation mea-sure in Fourier space it is possible to apply a fast rotational match-ing, which generates a small number of candidate rotational anglesthat are used for a subsequent fast translational alignment. Findinga good approximate translation-invariant rotational alignment iskey in such a strategy. As mentioned in the last section, c is the realpart of a complex valued function where the translation differencesbetween subtomograms only lead to differences in the phase partof the complex valued function. Therefore, following the approachof (Bartesaghi et al., 2008), the fast rotational search can first beperformed by removing phase information from c. Specifically,from Eq. (7), a constrained local score can be defined so that onlymagnitudes of the Fourier coefficients are included:

c0 :

R½jF f jM2� KR½jFgjM2�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR

jðF f ÞMj2 KR½M2�q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR

M2 KR½jFgMj2�q ð8Þ

This score is then used in fast rotational matching.In the previously published approach by Bartesaghi et al. (2008)

fast rotational alignment is achieved by projecting all the Fouriercoefficient magnitudes positioned along rays through the Fourierspace origin onto the corresponding point on the unit sphere sur-face. Then 2D matching is applied on the corresponding two 2Dspherical images (derived from the projected unit sphere surfaces).However, the compression of the Fourier coefficient magnitudevalues may increase ambiguity in the alignment, which may causeproblems in the alignment especially for subtomograms with highnoise levels (Amat et al., 2010).

Instead, here we apply a fast 3D volumetric rotational matching,previously used for efficient fitting of atomic structures into cryo-

electron microscopy density maps (Kovacs and Wriggers, 2002).Here, we adapt the method for subtomogram alignment byexpanding it to include missing wedge corrections and densityvariances. It can be seen that Eq. (8) can be formulated as beingcomposed of three rotational correlation functions of the formcR :¼

Rp KRq, where p and q are component functions1. Specifically,

we can represent c0 as

c0 ¼ c0Rffiffiffiffiffi

c1R

q ffiffiffiffiffic2

R

q ¼R

p0 KRq0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRp1KRq1

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRp2 KRq2

q ð9Þ

wherep0 :¼ jF f jM2; q0 :¼ jFgjM2;p1 :¼ jðF f ÞMj2; q1 :¼M2;p2 :¼M2,and q2 :¼ jFgMj2. When represented in spherical coordinates, the pand q components of these three functions cR can be approximatedby a Spherical Harmonics (SH) expansion. Following (Garzon et al.,2007) the p and q components in Eq. (8) can be approximated as

pðr;b; kÞ �XB�1

l¼0

Xl

m¼�l

CplmðrÞYlmðb; kÞ

qðr;b; kÞ �XB�1

l¼0

Xl

m¼�l

CqlmðrÞYlmðb; kÞ

where B is the bandwidth, and ClmðrÞ are the coefficients associatedwith the complex-valued spherical harmonic function Ylmðb; kÞ withdegree l and order m; r;b and k are the radial distance, co-latitudeand longitude, respectively, which define the position in sphericalcoordinate. When a suitable parameterization of the three-dimen-sional rotational group is achieved, the rotation correlation functioncR :¼

Rp KRq of all sampled rotations R can be represented as an in-

verse FFT of a 3D array of the sum of integrals as follows (Kovacsand Wriggers, 2002; Garzon et al., 2007):

cR ¼ F�1m;n;m0

Xl

dlmnd

lnm0

ZCp

lmðrÞCqlm0ðrÞr2dr

" #ð10Þ

where dlmh are real coefficients defining the elements of the Wigner

small d-matrix evaluated at 90� (Biedenharn et al., 1981). As a con-sequence of the above formulation the cross-correlation functionscan be efficiently sampled by using FFT simultaneously over allsampled rotations (Kovacs and Wriggers, 2002). The sampling is gi-ven as twice the bandwidth used in the harmonic transformation ofthe maps. Therefore c0 can be efficiently computed over all sampledrotations R. The set of candidate rotations are then obtained byidentifying the local maxima of c0 with respect to the rotational de-grees of freedom. To obtain the optimal translation, a fast transla-tional search is performed for each candidate rotation over thefull correlation function c by using FFT. Finally, the best combina-tion of rotation and translation is chosen.

2.3. Reference-free automatic classification and averaging ofsubtomograms with automatic local feature enhancement

One of the main applications of fast subtomogram alignment isthe classification of subtomograms into distinct complexes and thesubsequent averaging of the classified subtomograms to determinea higher nominal resolution for the resulting density maps of themacromolecular complexes (e.g., Bartesaghi et al., 2008; Försteret al., 2005). Before describing in detail the automatic reference-free subtomogram classification method (Fig. 1), we first introducean automatic local feature enhancement strategy, which will im-prove considerably the classification performance.

Subtomograms

Set weightmap W to 1

Calculate distance matrixusing W

Hierarchical clustering

Optimal cluster cutoff selection

Generate averagedsubtomogram for

each cluster

Realign averaged subtomograms into common reference frame

Convergence?

Classified Subtomograms

Calculate weight map W

from aligned subtomograms

Realign all subtomograms into common reference frame

Fig.1. Reference-free iterative subtomogram classification scheme with featureenhancement.

f1

f2

.

.

.

fK

W

Alig

ned

subt

omog

ram

s

Feature enhancement weights

Fig. 2. Determining the feature enhancement mapW (For visual simplicity only 2Dimages are shown instead of the 3D density maps). Given a set of alignedsubtomograms the feature enhancementW (x) for a given voxel x is determined bya covariance measure of all voxel neighbors based on all aligned K subtomograms.The voxels are separated by the grid of gray lines.

M. Xu et al. / Journal of Structural Biology 178 (2012) 152–164 155

2.3.1. Automatic local feature enhancement strategy for subtomogramclassification

Clustering of the aligned subtomograms is one of the main com-ponents in reference-free classifications. However, clustering ischallenging when the differences between the subtomograms aresubtle, for instance when subtomograms contain variants of thesame complex. Variants may only differ in a few regions of the sub-tomogram due to conformational differences and/or additionalbinding partners in one isoform. For example, subtomograms ofGroEL and GroEL/ES (Förster et al., 2008) differ only slightly andat low SNR their subtomograms are difficult to differentiate. Toclassify complex isoforms it would be beneficial to define a dis-tance measure that automatically focuses on those subtomogramregions that are most discriminative while at the same time arenot background noise.

To achieve this goal we implement a local feature enhancementstrategy by applying a real space weight mask W to the con-strained correlation measure (Fig. 2). This weight mask reducesthe influence of noise in empty regions and enhances the featuresthat are most discriminative.

For a given voxel x in an aligned subtomogram its weight WðxÞis calculated as

WðxÞ ¼ 1jN xj

Xy2N x

COVxy ð11Þ

where N x is the set of all neighbor voxels of x and COVxy is thecovariance between voxel x and one of its neighbor voxels y. Thiscovariance is defined as

COVxy ¼1KPKk¼1ðf k

x � ~f xÞðf ky � ~f yÞ ð12Þ

where f kx is the intensity value of voxel x in the kth subtomogram

and ~f x is the mean intensity value of voxel x across all the K subto-mograms ~f x :¼ 1

K

PKk¼1f k

x . ~f y is defined accordingly. If the neighbor-hood region for voxel x is occupied predominantly by only one ofthe complex isoforms then voxels in this region tend to have highintensity values in all the subtomograms of this isoform, and lowintensity values in all other subtomograms. Therefore, this regionwould have a large mean covariance value indicating that thisneighborhood region can be effectively used to separate the subto-mograms into distinct groups. Regions that are shared in all iso-forms or regions that are highly variant in each subtomogrambecause they represent background noise have a low covariance va-lue. This covariance measure therefore allows an effective way toenhance certain regions when calculating a distance measure basedon the correlation between two maps. To incorporate the local fea-ture enhancement, the constrained pairwise correlation measurebetween two subtomograms fj and fk can then be expressed as

cjk ¼ R

RFjXjk FkXjkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR

FjXjk FjXjk

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRFkXjk FkXjk

q0B@

1CA ð13Þ

with Fj ¼ F½ðWÞðsaj KRjfjÞ�; Fk ¼ F½ðWÞðsak

KRkfkÞ� and Xjk ¼

ðKRjMÞðKRk

MÞ, where sajand KRj

are translation and rotation ofsubtomogram j obtained at a particular iteration; (sak

and KRkare

defined accordingly).

156 M. Xu et al. / Journal of Structural Biology 178 (2012) 152–164

2.3.2. Automatic iterative subtomogram classification and averagingwith local feature enhancement

After having established the local feature enhancing mapW wenow describe the reference-free subtomogram classification. Theclassification process is performed in an iterative procedure con-taining 8 main steps (Fig. 1):

Step 1: Calculation of a local feature enhancing map W. The featureenhancing map W is calculated from the aligned set ofsubtomograms in a cluster (Fig. 2). However, at the initialstep of the classification the subtomograms are not alignedand not clustered yet. In this case the elements of W areset to 1. During each of the following iterationsW is recal-culated from the aligned subtomograms.

Step 2: Calculation of a distance matrix using the feature enhancingmapW. To quantify the dissimilarity between two alignedsubtomograms fj and fk we can calculate the Euclidean dis-tance between the constrained transformation of fj and fk:

djk ¼Z ðf �j � lðf �j ÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR

ðf �j � lðf �j ÞÞ2

q � ðf �k � lðf �k ÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRðf �k � lðf �k ÞÞ

2q

��������������22

64375

12

ð14Þ

where f �j :¼ RfF�1½FðW saj KRjfjÞ Xjk�g and f �k :¼ RfF�1

½FðW sakKRk

fkÞ Xjk�g. Because of the subtomogram normali-zation, this constrained distance can be calculated from theconstrained correlation:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq djk ¼ 2� 2cjk ð15Þ

To generate a distance matrix, all distances between allpairs of aligned subtomograms are calculated.

Step 3: Hierarchical clustering. Based on the distance matrix in step2, average linkage hierarchical clustering is applied (John-son, 1967). A hierarchical tree is generated, in which verysimilar sub-tomograms are joined by short branches andincreasingly different sub-tomograms by longer branches.

Step 4: Perform automatic cluster cutoff detection. To automaticallydetermine the optimal set of clusters, we calculate theaverage Silhouette width for each clustering cutoff (Kauf-man and Rousseeuw, 1990). The average Silhouette widthis defined as follows. Assume that at a given clustering cut-off the subtomograms are partitioned into k clustersS1; S2; . . . ; Sk. Then the Silhouette width xf of a single sub-tomogram f is defined as

xf ¼minJdfJ � �df

maxðminJdfJ ;�df Þ

ð16Þ

where dfJ is the average distance between f and a subtomo-gram in a different cluster SJ that does not contain f ; �df isthe average distance between f and all subtomograms inthe same cluster as f. The Silhouette width ranges between�1 and 1. The clustering cutoff is chosen that maximizesthe average Silhouette width.

Step 5: Determine averaged subtomogram for each cluster. Afterclustering, an averaged subtomogram template is gener-ated by averaging all the aligned subtomograms in thesame cluster.

Step 6: Alignment of averaged templates. In this step, all averagedcluster templates that are similar to each other are alignedto a common reference frame. First, all the templates arepairwise aligned using our fast alignment method andclustered using the constrained correlation (as defined in

Eq. (6)) as similarity measure. After hierarchical clustering,the maximal average Silhouette method is used to deter-mine the optimal clustering cutoff, which defines the opti-mal set of template clusters. Then within each templatecluster, all the averaged templates are aligned to the aver-aged template from the largest subtomogram cluster (asdefined in step 5).

Step 7: Alignment of all subtomograms against the averaged subto-mograms of each cluster. Every subtomogram is re-alignedto each of the averaged cluster templates and only thealignment with the best similarity score is then furtherconsidered. In this step it is possible that subtomogramsare re-classified to different clusters.

Step 8: Testing for convergence. The convergence of the classifica-tion process is terminated when the maximum averageSilhouette width calculated from the clustering reaches aplateau with increasing number of iterations. If conver-gence is not yet reached the process is iteratively repeatedstarting with step 1.

2.4. Generating simulated cryo-electron tomograms

For a reliable assessment of the method, subtomograms mustbe generated by simulating the actual tomographic image recon-struction process, allowing the inclusion of noise, tomographic dis-tortions due to missing wedge, and electron optical factors such asContrast Transfer Function (CTF) and Modulation Transfer Function(MTF). We follow a previously applied methodology for simulatingthe tomographic image formation mechanism as realistically aspossible (Beck et al., 2009; Förster et al., 2008; Nickell et al.,2005; Xu et al., 2011; Xu and Alber, 2012). The electron opticaldensity of a macromolecule is proportional to its electrostatic po-tential and the density map can be calculated from the atomicstructure by applying a low pass filter at a given resolution. Here,density maps are generated at 4 nm resolution using the PDB2VOLprogram of the Situs 2.0 package (Wriggers et al., 1999) with voxellength of 1 nm. These initial density maps are then used as samplesfor simulating electron micrograph images at different tilt angles.In cryoET the sample is tilted in small increments around a sin-gle-axis. At each tilt angle a simulated micrograph is generatedfrom the sample. As a result our data contains a wedge-shaped re-gion in Fourier space for which no structure factors have beenmeasured (i.e., the missing wedge effect), similar to experimentalmeasurements. The missing wedge effect leads to distortions ofthe density maps. These distortions depend on the structure ofthe object and its orientation with respect to the direction of thetilt-axis. To generate realistic micrographs, noise is added to theimages according to a given SNR level, defined as the ratio betweenthe variances of the signal and noise (Förster et al., 2008). More-over, the CTF and MTF describe distortions from interactions be-tween electrons and the specimen and distortions due to theimage detector (Frank, 2006; Nickell et al., 2005). Therefore,the resulting image is convoluted with a CTF, which describesthe imaging in the transmission electron microscope in a linearapproximation. Any negative contrast values beyond the first zeroof the CTF are eliminated. We also consider the MTF of a typicaldetector used in tomography, and convolute the density map withthe corresponding MTF. Typical acquisition parameters that werealso used during actual experimental measurements (Beck et al.,2009) were used: voxel grid length ¼ 1 nm, the spherical aberra-tion ¼ 2� 10�3m, the defocus value ¼ �4� 10�6m, the voltage¼ 200kV, the MTF corresponded to a realistic electron detector(McMullan et al., 2009), defined as sincðpx=2Þwhere x is the frac-tion of the Nyquist frequency. Finally, we use a backprojectionalgorithm (Nickell et al., 2005) to generate a tomogram from the

Table 1Median rotational alignment error DE of all 100 pairwise alignments for phantommodels 1 and 2. DE is defined as the difference between the rotational matrixdetermined by the fast rotational matching and the ground truth: DE ¼ jjRdif � IjjF .The maximum value of DE is 8. Gray fields are considered failed alignments, whitefields are considered correct alignments.

M. Xu et al. / Journal of Structural Biology 178 (2012) 152–164 157

individual 2D micrographs that were generated at the various tiltangles (Beck et al., 2009; Xu et al., 2011; Xu and Alber, 2012). Totest the influence of increasing noise, we add different amount ofnoise to the images, so that the SNRs range between 0.1 and0.001, respectively (Fig. 3).

2.4.1. Benchmark setThe classification is tested on simulated tomograms of two

phantom models (Fig. 3) and four different protein complexes ta-ken from the Protein Data Bank (PDB) (Berman et al., 2000), as wellas experimentally determined subtomograms (see text for details).For each simulated tomogram complexes were randomly orientedand placed and tomograms simulated at six different SNRs (0.0010.005, 0.01, 0.05, 0.1, 1) and variable tilt angle ranges (Fig. 3 andTable 1).

2.5. Analyzing the computational efficiency

We tested the computational efficiency by measuring the timefor aligning 100 pairs of subtomograms of model 1 (Fig. 3) usinga computer with 2.67 GHz speed. Our fast alignment consists oftwo steps: (1) The rotational alignment step, which depends onthe volume of the subtomogram and the angle sample interval;(2) the translational alignment and evaluation step of a candidate,which depends on the volume of the subtomogram. The averagecomputation time was calculated from 100 repeated rotationalalignments. Also, we measured the average time for evaluatingthe translational alignment for each candidate rotation. The fastrotational alignments in Section 3.1 resulted in less than 100 can-didate rotations per alignment. An upper bound for the total align-

SNR

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0.05

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0.005

0.001

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8

50 o+- 40 o+-

A B

C D70 o+-Tilt angle range

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Fig. 3. Tomograms of phantom models simulated at various signal-to-noise ratios(SNR) and tilt angles ranges. Tomograms are of size 643 voxels. (A) Isosurface ofmodel 1 consisting of four Gaussian functions of the same shape. The principal axisof three Gaussian functions are in the same plane, while the principal axis of thefourth Gaussian function is positioned orthogonal to the first three Gaussianfunctions. (B) Isosurface of model 2 consisting of three Gaussian functions withdifferent intensity distributions. (C) Slices taken along the y direction through thetomograms at different levels of noise and tilt angle ranges. The Isosurface is plottedusing the Chimera software (Pettersen et al., 2004).

ment time was therefore estimated by trotation þ 100� tcandidate,where trotation is the average time spent for the rotational alignmentand tcandidate is the average time spent for translational alignment ofa given candidate rotation. For comparison we also measured thetotal computational time used for the exhaustive rotational scan-ning by determining the average time spent evaluating each ofthe scanned rotations in (Förster et al., 2008), multiplied by the to-tal number of scanned rotations in the exhaustive search (Table 3).For comparison, both methods were implemented in MATLAB.

2.6. Software

For proof of principle the fast alignment method is imple-mented in MATLAB. For fast calculation of the rotational correla-tion functions we used the DOTM library (Simons, 2001) tocompute the Spherical Harmonics expansion and the Wignerd-matrix. A standalone program of our approach is in preparationand to be released upon completion.

Table 2Ratio of the rotational alignment errors between a fast rotational alignment using 3Dvolumetric matching as described in this paper, and a fast rotational matching basedon the alignment of 2D spherical images (see text for details) (Bartesaghi et al., 2008).Gray areas indicate subtomogram conditions where both methods fail to provide thecorrect alignment. Yellow areas indicate subtomogram conditions where 3Dvolumetric matching is able to provide a correct alignment while the 2D matchingstrategy fails.

Table 3Estimated computational time in seconds, for our fast alignment method and thealignment using standard exhaustive rotational scanning (in parentheses), fordifferent subtomogram volumes (in voxel number) and rotational angle intervals(RAI, in degrees).

RAI VOL

163 323 643

10� 13.0 (3.73 � 102) 26.5 (1.96 � 103) 94.9 (1.49 � 104)5� 15.6 (3.04 � 103) 32.1 (1.64 � 104) 108 (1.21 � 105)2.5� 39.2 (2.39 � 104) 61.5 (1.30 � 105) 153 (9.67 � 105)

158 M. Xu et al. / Journal of Structural Biology 178 (2012) 152–164

3. Results

Our subtomogram alignment and reference-free classificationapproach is tested on realistically simulated cryo-electron tomo-grams of phantom models, structures of protein complexes, andcryo-electron subtomograms of purified complexes or complexesextracted from whole-cell tomograms.

3.1. Assessment of the alignment accuracy using phantom models

First, we assess the alignment accuracy on two asymmetricphantom models (Fig. 3). For each of the two phantom models100 pairs of subtomograms are generated where the phantommodels are randomly oriented and placed and the subtomogramssimulated for a variety of different SNR levels and tilt angleranges (Fig. 3). Then each pair of subtomograms is aligned andthe determined relative rotations and translations comparedwith the ground truth, which is the rotational matrix that is usedto generate the subtomogram orientation. To quantify the rota-tional alignment accuracy we calculate the difference betweenthe rotational matrix of the optimal alignment and the groundtruth. The difference between both matrices DE is measured asDE ¼ jjRT

trueRalign � IjjF , where Ralign is the rotation matrix deter-mined by the fast rotational alignment and Rtrue is the groundtruth (i.e., the rotation matrix used to generate the subtomogramorientation); jj � jjF denotes the Frobenius norm. The absolute er-ror DE can range from 0 to a maximal value of 8 indicating thatthe object is placed in the opposite direction from the groundtruth. Any DE value larger than 0.1 depicts a false alignment.

Our analysis shows that the accuracy of our fast rotationalsearch method is very robust even at relatively high noise leveland missing wedge distortions (Table 1). At SNR of 0.01 the fastrotational alignment is able to identify the correct alignment attilt angle ranges that are typical used in tomographic experi-ments. Even at an SNR level of 0.005 our method can still accu-rately detect the optimal rotation. Only at extremely low SNRlevels of 0.001 the method fails.

Next, we compare our fast rotational alignment based on 3Dvolumetric matching with the fast rotational search based onprojected 2D spherical images proposed in (Bartesaghi et al.,2008). To compare the two strategies we calculate the ratiosbetween the rotational error DE for both methods (Table 2). Weobserve that the 3D volumetric matching performs better at low-er SNR and smaller tilt angle ranges (Table 2). For instance, atSNR levels of 0.01 2D spherical image matching is unable todetermine the correct alignment, while 3D volumetric matchingis able to produce correct alignments at even lower SNR levelsof 0.005. 3D volumetric matching therefore extends the applica-bility for fast rotational matching to subtomograms with lowerSNR (Table 2).

3.2. Analysing the computational efficiency

Next, we compare the computational cost between the fastrotational matching and the exhaustive rotational scanning

(Förster et al., 2008). For comparison we have implemented bothmethods in MATLAB and performed subtomogram alignments onmodel 1 (Section 2.5).

The exhaustive rotational scanning consists of iterations overall sampled rotational angles. At each step the constrained corre-lation (Förster et al., 2008) is calculated. Its computation consistsof several computational intensive operations, such as the FFT,element wise array products, and 3D summations, as well as3D rotations, which are all efficiently implemented by theMATLAB system. Therefore we can expect that the efficiency ofour rotational scanning implementation is similar in scope tothe optimal implementations of the method in other program-ming languages.

Although the current implementation of our fast alignmentmethod is not fully optimized for computational efficiency, wecan already demonstrate that the fast rotational matching drasti-cally decreases the computational time in comparison to theexhaustive rotational scanning (Table 3). In addition the methodis also more scalable with increasing subtomogram sizes anddecreasing angle sampling intervals. For example, with a subtomo-gram size of 643 voxels our method is on average � 6000 times fas-ter, with the correct alignment found at 153 s using a MATLABimplementation. By contrast, the standard exhaustive rotationalscanning with an angle interval of 2:5

takes more than 10 days

to find the correct alignment in a MATLAB implementation. Evenwith relatively small subtomogram volumes (i.e., 163 voxels) anda large rotational angle interval of 10 degrees, the standard exhaus-tive scanning approach still takes on average 373 s, while the fastrotational matching only takes 13 s.

3.3. Reference-free alignment and classification of subtomograms

We now test our method for the reference-free alignment andclassification of subtomograms. The fast rotational matching al-lows very efficient alignments and therefore the comparison of alarge number of subtomograms. We test our approach first on sim-ulated subtomograms based on real protein complex structuresextracted from the Protein Data Bank (PDB) (Berman et al.,2000). Then we test our method on experimental subtomograms,namely tomograms of purified GroEL and GroEL/ES complexes,and finally experimental subtomograms extracted from whole-celltomography.

3.3.1. Reference-free classification of subtomograms simulated fromprotein complex structures

Structures of four protein complexes are obtained from thePDB: GroEL (PDBID: 1KP8), Carbamoyl phosphate synthase (PDBID:1BXR), octameric enolase (PDBID: 1W6T), and ClpP (PDBID: 1YG6)(Fig. 4A). For each of the four complexes 100 randomly orientedand placed subtomograms are generated at a SNR of 0.01 and a tiltangle range �60

(Fig. 4A) (Section 2.4).

The reference-free automatic classification procedure (Sec-tion 2.3.2) converges after 18 iterations and a clear classificationinto four different groups is evident (Fig. 5A, C and E). Notably,the final distance matrix calculated with feature enhancement isable to clearly differentiate the four classes of subtomograms(Fig. 5C) with 100% classification accuracy. Distances between sub-tomograms of the same complex are very small, while distancesbetween different complexes are relatively large (Fig. 5B). Averag-ing all aligned subtomograms in each class reproduces well the ini-tial density maps of the complexes (Fig. 4B).

In contrast, without feature enhancement the distinction be-tween the groups of complexes is more challenging (Fig. 5B andD). When subtomograms are aligned to their templates withoutfeature enhancement the resulting distances are relatively large,and therefore the variance of distances between subtomograms

SNR: 0.01 Tilt angle range: 60o+-

IIVII

III

Initial density maps

Simulated subtomgrams

averaged subtomograms after classification

A

B I IVII III

Fig. 4. Reference-free subtomogram classification for simulated subtomograms from four different protein complexes. (A) Density map of selected protein complexes(I: GroEL (PDBID: 1KP8), II: Carboamoyl phosphate synthase (PDBID: 1BXR), III: octameric enolase (PDBID: 1W6T), IV: ClpP (PDBID: 1YG6)) and a slice of a simulatedsubtomogram generated for SNR 0.01 and a tilt angle range �60. The slice is taken along the y direction. (B) Density maps of complexes generated by averaging all theclassified and aligned subtomograms in each class. Also shown is a slice though the averaged tomograms.

M. Xu et al. / Journal of Structural Biology 178 (2012) 152–164 159

of different complexes is relatively small (Fig. 5B and D). Clusteringinto distinct complex groups is more challenging (Fig. 5D). There-fore, the feature enhancement is clearly of great benefit for subto-mogram classification.

3.3.2. Reference-free alignment and classification of subtomograms ofpurified complexes

We now further test our reference-free classification method ona set of experimental tomograms, namely tomograms of purifiedGroEL and GroEL/GroES complexes previously published in (Försteret al., 2008). The dataset consists of two sets of subtomograms. 214subtomograms obtained from 13 cryo-electron tomograms ofpurified GroEL complexes, and 572 subtomograms obtained from11 cryo-electron tomograms containing GroEL/GroES complexes.The differences between the subtomograms of the two differentcomplexes are subtle and the classification and averaging ischallenging.

After about 14 iterations and automatic cluster cutoff detection,the iterative classification procedure (Section 2.3) can separate thesubtomograms into two distinct clusters corresponding to the twoclasses of complexes (Fig. 6A, C and F) with a classification accu-racy of 68.2%. Note that the classification and averaging of this datamight not only be affected by the subtleness of differences be-tween the two types of complexes but also the heterogeneity inthe data (Scheres et al., 2009; Hrabe et al., 2012). The subtomo-gram averages obtained from each of the two clusters show thedistinct features of the GroEL and GroEL/ES complexes (Fig. 6E).

Differences between the two complexes are clearly visible. Whenusing a distance measure without feature enhancement it is notpossible to dissect the two sets of subtomograms into two classesby hierarchical clustering (Fig. 6B and D), in agreement with a pre-vious study (Förster et al., 2008).

3.4. Reference-free alignment and classification of subtomogramsextracted from whole cell tomograms

Finally, we test our method on subtomograms extracted from awhole cell tomogram of Leptospira interrogans in undisturbed condi-tions (Beck et al., 2009). The classifications of subtomograms fromwhole-cell studies is significantly more challenging because the het-erogeneity of complexes is expected to be much higher when subto-mograms are extracted from crowded whole cell tomograms andbecause the SNR levels and resolution of these subtomograms is typ-ically much lower in comparison to the tomograms of purified com-plexes. Beck et al. (2009) have identified several types of complexesin the tomogram using template matching (Best et al., 2007). Thetemplates were generated by simulating subtomograms of struc-tures from the PDB (Berman et al., 2000) and convoluted with CTFand MTF. As a test case, we selected the two complexes with thehighest cellular abundance in the whole cell tomogram (i.e., theribosome and ATP synthase, with more than 200 instances for eachcomplex in the tomogram). To ensure high confidence for our dataset, we only identified the 100 best scoring template matches foreach of the two complexes. For each of these matches we extracted

Final distance matrix with feature enhancement after 18 iterationsI IVII III

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Fig. 5. Reference-free subtomogram classification for the simulated subtomograms of four different protein complexes as shown in Fig. 4. (A) Distance matrix after 18iterations of reference-free subtomogram classification with the feature enhancement weightW. The order of subtomograms is according to the ground truth. The distancematrix is calculated using the feature enhancement weight. (B) Distance matrix calculated without feature enhancement weight after all subtomograms are aligned to theirrespective template density map. The order of subtomograms is according to the ground truth. (C) Clustered distance matrix after 18 iterations of the reference-freeclassification process using the feature-enhancement weight W. Four different clusters can be clearly differentiated. (D) Clustering of the distance matrix from C. (E)Convergence of the classification process. The maximal average Silhouette width is calculated from the resulting subtomogram clusters at each iteration of the classificationprocess. The convergence is reached when the maximal average Silhouette width reaches a plateau with increasing number of iterations.

160 M. Xu et al. / Journal of Structural Biology 178 (2012) 152–164

Final distance matrix with feature enhancement after 14 iterationsAI II

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Fig. 6. Reference-free classification of 786 experimental subtomograms, containing GroEL and GroEL/ES complexes. (A) Distance matrix after 14 iterations of reference-freesubtomogram classification with the feature enhancement weightW. The order of subtomograms is based on the tomogram type. The distance matrix is calculated using thefeature enhancement weight. (B) Distance matrix calculated without feature enhancement weight after all subtomograms are aligned to a reference subtomogram. The orderof subtomograms is identical to those in panel A. (C) Clustered distance matrix after 14 iterations of the reference-free classification process. Two different clusters are clearlydetected by the automatic cutoff detection. (D) Clustering of distance matrix as shown in panel B. The clustering into the two subgroups fails without feature enhancementand reference-free classification. (E) Averaged subtomograms after iterative classification and alignment. (Upper panel) slices through the averaged subtomograms and(lower panel) isosurface of the averaged subtomograms of each cluster. Clearly shown are the distinct features for the GroEL complex (left panel) and the GroEL/ES complex(right panel). A slice through the difference map of both subtomograms shows the distinct differences in both complexes. (F) Convergence of the classification process. Themaximal average Silhouette width is calculated from the resulting subtomogram clusters at each iteration of the classification process. The convergence is reached when themaximal average Silhouette width reaches a plateau with increasing number of iterations.

M. Xu et al. / Journal of Structural Biology 178 (2012) 152–164 161

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Fig. 7. Reference-free classification of 200 experimental subtomograms, extracted from a whole cell cryo-electron tomogram of L. interrogans (Beck et al., 2009). (A) Distancematrix after 10 iterations of reference-free subtomogram classification with the feature enhancement weight W. The order of subtomograms is based on the result oftemplate matching. The distance matrix is calculated using the feature enhancement weight. (B) Distance matrix calculated without feature enhancement weight after allsubtomograms are aligned to the corresponding subtomograms with the highest matching score in template matching (Best et al., 2007). The order of subtomograms isidentical to those in panel A. (C) Clustered distance matrix after 10 iterations of the reference-free classification process. Two different clusters are detected by the automaticcutoff detection, differentiating the ATP synthase complexes from the ribosome complexes. (D) Clustering of distance matrix as shown in panel B. The clustering into the twosubgroups fails without feature enhancement and reference-free classification. (E) Averaged subtomograms after iterative classification. Shown are slices through theaveraged subtomograms and the isosurface of the averaged subtomograms of each cluster. (F) Averaged subtomograms when all the subtomograms in each determinedcluster are aligned to the respective ground truth template. Shown are slices through the averaged subtomograms and the isosurace of the averaged subtomograms. (G)Convergence of the classification process. The maximal average Silhouette width is calculated from the resulting subtomogram clusters at each iteration of the classificationprocess. The convergence is reached when the maximal average Silhouette width reaches a plateau with increasing number of iterations.

162 M. Xu et al. / Journal of Structural Biology 178 (2012) 152–164

M. Xu et al. / Journal of Structural Biology 178 (2012) 152–164 163

a subtomogram of size 323 voxels that was large enough to containan instance of the complex.

The reference-free classification with feature enhancementcombined with automatic cluster cutoff determination is able todetect correctly the two classes of subtomograms (Fig. 7A and C)with a classification accuracy of 99% when the subtomograms fromtemplate matching are taken as ground truth. The classificationconverges after around nine iterations (Fig. 7G). In contrast, thedistance matrix for the same subtomograms calculated withoutfeature enhancement is unable to discriminate between the twoclasses, even if all the subtomograms are first aligned to a corre-sponding reference subtomogram (Fig. 7B and D). Clustering of thisdistance matrix does not allow the distinction between the twocomplex classes (Fig. 7D).

4. Conclusion

Fast and accurate subtomogram alignments are key for reference-free subtomogram classification and averaging. However, most ofexisting alignment methods rely on exhaustive rotational scanningof one subtomogram with respect to the other. The computationalcost of such rotational scanning is often the limiting factor in refer-ence-free subtomogram classifications in particular with the increas-ing number, and size of subtomograms and alignment precision.

In this paper, we propose a fast rotational alignment methodbased on the Fourier equivalent form of a popular constrained cor-relation measure. The constrained correlation takes into accountthe mean and variance of the intensity values in the subtomograms,therefore corrects for the non-evenness of tomograms due to theimaging process. The fast rotational search is based on 3D volumet-ric matching. We demonstrate that the use of 3D volumetric match-ing improves the alignment accuracy, in particular for highlydistorted subtomograms with low SNR and tilt angle ranges, com-pared to the matching strategy based on projected 2D sphericalimages (Bartesaghi et al., 2008). To further increase alignment accu-racy, the alignment can be refined using gradient guided methods(Bartesaghi et al., 2008; Xu and Alber, 2011; Xu and Alber, 2012).

We apply a reference-free iterative classification framework withour fast alignment method, and propose a local feature enhance-ment strategy. We can demonstrate that the method can be usedto successfully classify subtomograms extracted from tomogramsof purified protein complexes and whole cell tomograms.

For proof of principles we have used the popular hierarchicalclustering approach with a simple optimal cluster selection metric(i.e., Silhouette width). However, the clustering step may further beimproved by incorporating user supervision (e.g., Castaño-Díezet al., 2012) and other methods for clustering and optimal clusterselection, especially those specifically designed for subtomogramclustering (e.g., Yu and Frangakis, 2011).

Acknowledgments

The authors thank Dr. Friedrich Förster for providing the GroELand GroEL/ES subtomograms for classification. This work is sup-ported by the Human Frontier Science Program Grant RGY0079/2009-C to F.A., Alfred P. Sloan Research Foundation Grant to F.A.;NIH Grants 1R01GM096089 and 2U54RR022220 to F.A., NSF CA-REER grant 1150287 to F.A. F.A. is a Pew Scholar in Biomedical Sci-ences, supported by the Pew Charitable Trusts.

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