A clustering approach to multireference alignment of single-particle projections in electron microscopy.

Abstract

Two-dimensional analysis of projections of single-particles acquired by an electron microscope is a useful tool to help identifying the different kinds of projections present in a dataset and their different projection directions. Such analysis is also useful to distinguish between different kinds of particles or different particle conformations. In this paper we introduce a new algorithm for performing two-dimensional multireference alignment and classification that is based on a Hierarchical clustering approach using correntropy (instead of the more traditional correlation) and a modified criterion for the definition of the clusters specially suited for cases in which the Signal-to-Noise Ratio of the differences between classes is low. We show that our algorithm offers an improved sensitivity over current methods in use for distinguishing between different projection orientations and different particle conformations. This algorithm is publicly available through the software package Xmipp.

Full text

Page 1

A clustering approach to multireference alignment of single-particle projections
in electron microscopy
C.O.S. Sorzanoa,*, J.R. Bilbao-Castrob, Y. Shkolniskyc, M. Alcorloa, R. Meleroa, G. Caffarena-Fernándezd,
M. Lie, G. Xue, R. Marabinif, J.M. Carazoa
aBiocomputing Unit, National Center of Biotechnology (CSIC), Cantoblanco, Madrid, Spain
bComputer Architecture and Electronics Dept., Univ. de Almería, Almería, Spain
cDepartment of Applied Mathematics, Tel Aviv University, Tel Aviv, Israel
dDept. Ingeniería de Sistemas, Electrónicos y de Telecomunicación, Univ. San Pablo-CEU, Boadilla del Monte, Madrid, Spain
eInstitute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, China
fComputer Science Dept., Univ. Autónoma de Madrid, Cantoblanco, Madrid, Spain
a r t i c l e i n f o
Article history:
Received 1 February 2010
Received in revised form 12 March 2010
Accepted 23 March 2010
Available online 31 March 2010
Keywords:
Single-particle analysis
2D analysis
Multireference analysis
Electron microscopy
a b s t r a c t
Two-dimensional analysis of projections of single-particles acquired by an electron microscope is a useful
tool to help identifying the different kinds of projections present in a dataset and their different projec-
tion directions. Such analysis is also useful to distinguish between different kinds of particles or different
particle conformations. In this paper we introduce a new algorithm for performing two-dimensional mul-
tireference alignment and classification that is based on a Hierarchical clustering approach using corren-
tropy (instead of the more traditional correlation) and a modified criterion for the definition of the
clusters specially suited for cases in which the Signal-to-Noise Ratio of the differences between classes
is low. We show that our algorithm offers an improved sensitivity over current methods in use for dis-
tinguishing between different projection orientations and different particle conformations. This algo-
rithm is publicly available through the software package Xmipp.
? 2010 Elsevier Inc. All rights reserved.
1. Introduction
Electron microscopy of single-particles is a powerful technique
to analyze the structure of a large variety of biological specimens.
Cryo-electron microscopy has proved able to visualize macromo-
lecular complexes at nearly native state. However, due to the low
electron dose used to avoid radiation damage as well as the low
contrast between the macromolecular complex and its surround-
ing, images exhibit very low Signal-to-Noise Ratios (SNR, below
1/3) and very poor contrast. We can distinguish between two dif-
ferent types of analysis depending on their final goal: two-dimen-
sional (2D) and three-dimensional (3D). The aim of 3D analysis is
to recover the 3D structure compatible with the projection images
recorded by the electron microscope. This goal requires the acqui-
sition of many thousands of projections from the same object in
different projection directions. 2D analysis is less ambitious, since
it addresses a substantially lower dimensionality problem. Its aim
is to analyze 2D images with the goal set at helping to recognize
different conformations or different populations of macromolecu-
lar complexes through this limited approach based on 2D images.
In spite of its stated limitations, 2D analysis is very powerful both
as a data exploratory tool and as a first step in a classical 3D
reconstruction.
Since the different classes of images are unknown a priori, this
problemisoneofunsupervisedclassificationorclustering.However,
distinguishing among different image classes usually requires that
these classes are aligned first, unless they are classified using shift
invariants, rotational invariants, or both (Schmid and Mohr, 1997;
Flusser et al., 1999; Lowe, 1999; Tuytelaars and Van Gool, 1999;
Chong et al., 2003; Pun, 2003; Lowe, 2004). Rotational invariants
havealreadybeenusedinelectronmicroscopy(SchatzandvanHeel,
1990, 1992; Marabini and Carazo, 1994) but usually they are not
powerful enough to discover subtle differences between images.
Alternatively, a multireference alignment can be performed in
which the alignment and classification steps are iteratively alter-
nated until convergence (van Heel and Stoffler-Meilicke, 1985).
Briefly explained, let us assume that we already have a set of class
representatives:
1. Each image in the experimental dataset is aligned to each class
representative and the similarity between the aligned image
and the representative is measured.
1047-8477/$ - see front matter ? 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.jsb.2010.03.011
* Corresponding author. Address: Biocomputing Unit, National Center of Bio-
technology (CSIC), c/Darwin, 3 (Campus Univ. Autónoma de Madrid), Cantoblanco,
Madrid 28049, Spain. Fax: +34 915854506.
E-mail address: coss@cnb.csic.es (C.O.S. Sorzano).
Journal of Structural Biology 171 (2010) 197–206
Contents lists available at ScienceDirect
Journal of Structural Biology
journal homepage: www.elsevier.com/locate/yjsbi

Page 2

2. The image is assigned to the class with maximum similarity.
3. Finally, the class representatives are recomputed as the average
of the images assigned to it.
This sequence is repeated until some convergence criterion is
met. By far, the most widely used similarity criterion in electron
microscopy is least-squares or, equivalently, cross-correlation (it
can be easily proved that minimizing the least-squares distance
is equivalent to maximizing the correlation).
A possible drawback of this classification method is its depen-
dencyon the initializationoftheclass representativesandthepossi-
bilityofgettingtrappedinalocalminimum.Toamelioratethesetwo
problems, (Scheres et al., 2005) devised a multireference alignment
algorithm based on a maximum likelihood approach so that images
areassignedatthesametimetoallclassesinallpossibleorientations
andtranslations,butwithdifferentprobabilities.Theclassrepresen-
tativesarethen computedasa weightedaverage ofallimagesgiving
moreweighttothoseimagesthataremorelikelytocomeinapartic-
ularorientationandtranslationfromthat class.Thegoalofthisalgo-
rithmistofindtheclassrepresentativesthatmaximizethelikelihood
of observing the experimental dataset at hand. This is done using an
expectation–maximization approach (Dempster et al., 1977).
In this paper, we show that a drawback of a family of multirefer-
ence alignment algorithms based on the minimization of the
squarederrorbetweentheexperimentalimagesandtheclassrepre-
sentatives(amongwhichweencountermaximumlikelihoodaswell
as the standard multireference alignment) is that images tend to be
misclassified depending on the Signal-to-Noise Ratio of the differ-
ence between the different classes and the number of images as-
signed to each class. We propose another algorithmic tool able to
address ‘‘details” and small differences between classes. In this
way, a datasetcouldbe splitinto manyclassesat the sametimethat
the misclassification error is minimized. The main ideas behind the
new algorithm are the following. The first one is the substitution of
thecorrelationbythecorrentropy(Santamarfaetal.,2006;Liuetal.,
2007). This is a similarity measure recently introduced in the signal
processingfieldwhichhasbeenprovedtobegoodfornon-linearand
non-Gaussian signal processing. The second idea is to assign images
toclassesbyconsideringhowwelltheyfittotheclassrepresentative
compared to the rest of the experimental images. This comparison
avoids the problem of comparing an experimental image to class
averageswithdifferentnoiselevels(usuallytheclasswithlessnoise
is favoured in a comparison searching for the maximum similarity).
Theseideasareusedina standarddivisivevectorquantizationalgo-
rithm(Gray,1984),inthisway,weguaranteethatasmanyclassrep-
resentatives can be generated as desired, and at the same time the
amount of averaging performed by each class is strongly reduced
since each class representative will only average a relatively small
number of images from the original dataset. We will refer to the
new algorithm as CL2D (Clustering 2D).
CL2D can be thought of as a standard clustering algorithm aim-
ing at subdividing the original dataset into a given number of sub-
classes (this number can be relatively large so that classes are as
pure as possible in terms of conformation and/or projection direc-
tion). Our most innovative feature lies on how we measure the dis-
tance between an image and a cluster using a robust clustering
criterion based on correntropy similarity instead of correlation.
2. Methods
We start our methodological presentation by introducing the
different pieces of our algorithm: (1) how we measure the similar-
ity between two images; (2) how the principles of divisive cluster-
ing can be used in electron microscopy; (3) our new clustering
criterion that is capable of handling images with low SNR. We fi-
nally integrate these pieces into our new algorithm CL2D.
2.1. Similarity between two images: correntropy
The cross correntropy between two random variables ðX;YÞ is
defined as
VrðX;YÞ ¼ E jrðX ? YÞ
where Ef?g is the expectation operator, and jrðxÞ is a one-dimen-
sional symmetric ðjrðxÞ ¼ jrð?xÞÞ, non-negative kernel. In practice,
the true distribution of X ? Y is unknown and the cross correntropy
can be approximated by its empirical estimate
fg;
ð1Þ
VrðX;YÞ ¼1
N
X
N
i¼1
jrðxi? yiÞ;
ð2Þ
where xiand yiare actual measurements of the X and Y variables.
Before comparing images X and Y, both images should be
aligned. For doing this, we first translationally align image X with
respect to Y using the correlation property of the Fourier transform
in cartesian coordinates producing a new image X0. Then, we rota-
tionally align X0with respect to Y using the correlation property of
the Fourier transform in polar coordinates producing a new image
X00. We repeat this process twice before computing the correntropy
between both images. Since the Shift + Rotation sequence is not
guaranteed to reach the correct alignment between the two images
(the global minimum), we also perform two iterations of the Rota-
tion + Shift sequence, and pick the aligned image that maximizes
the correntropy between the two images.
Interesting properties of the correntropy are that it is symmet-
ric ðVrðX;YÞ ¼ VrðY;XÞÞ, positive, bounded, it is the argument of
Renyi’s quadratic entropy (from which the name correntropy
comes from), it is related to robust M-estimates (Liu et al., 2007),
and it involves all the even moments of the variable X ? Y (not just
its second order moment, as is the case of correlation):
VrðX;YÞ ¼ E jrðX ? YÞ
f g ¼
X
1
n¼0
a2nE ðX ? YÞ2n
no
;
ð3Þ
where the coefficients a2n depend on the Taylor expansion of the
kernel (note that the Taylor expansion has only even terms because
the kernel is even symmetric and by definition cannot have odd
terms in its Taylor expansion). It is this latter property which makes
it specially suited to non-Gaussian signal processing and our appli-
cation. Let us assume that X and Y are two identical images except
for the noise (assumed to be white and Gaussian). It can be easily
seen that the two images are correctly aligned when the variance
of the difference image, EfðX ? YÞ2g, is minimum (if it is not mini-
mum it means that there is still some misalignment between X
and Y yielding a difference in the signals themselves, not only the
noise). At the point of minimum variance, the correntropy is also
minimum. However, thanks to the higher order terms in the Taylor
expansion of the correntropy, these differences in X ? Y due to the
misalignment contribute not only to the 2nd power (as in the var-
iance), but also to the 4th, 6th, ...(each term with less and less
weight, a2n, so that the whole series is convergent; note also that
misalignments make the distribution of the difference image to
be non-Gaussian). The result is that if we plot the correntropy (or
variance) landscape around its optimal alignment, the landscape
of the correntropy has a much sharper optimum than that of the
variance, meaning that it is more sensitive to small misalignments.
The same reasoning applies when there are small differences be-
tween X and Y, not only because of the noise and misalignments,
but because X and Y are slightly different images (they belong to
different classes).
In our algorithm we use the unnormalized Gaussian kernel
jrðxÞ ¼ exp ?1
2
r
x
? ?2
??
;
ð4Þ
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which is bounded between 0 (no similarity between X and Y) and 1
(X and Y are identical).
A key point of kernel algorithms is the estimation of the kernel
width r. In our algorithm we first estimate the noise variance r2
from the background of the experimental images (defined as the
region outside the maximum circle inscribed in the projection im-
age). This estimation is performed over the whole dataset. Let us
assume that X is an experimental image and Y its corresponding
true class which has been calculated without noise. Then the var-
iance of the variable X ? Y is r2
calculated without noise. Let us assume that Y has been calculated
from NYexperimental images. Then, the noise in Y has a variance of
r2
N
NY, and the variance of image difference X ? Y is r2
we set for our kernel
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NY
N
N. However, in practice, Y is never
Nþr2
N
NY. Therefore,
r ¼
r2
Nþr2
N
s
:
ð5Þ
2.2. Divisive vector quantization
For completeness we present here the standard multireference
classification in the framework of vector quantization (or cluster-
ing). Then, we present the standard divisive vector quantization
algorithm, which is known to produce much more robust results
in vector quantization, and show how this divisive algorithm can
be applied to multireference classification in electron microscopy.
Considering each experimental image, Xi, as a vector in a P
dimensional space (being P the total number of pixels), multirefer-
ence classification with K class centers aims at finding the vectors
bXkðk ¼ 0;1;...;K ? 1Þ such that
mental image. This is exactly the same goal as the one of vector
quantization with K code vectors.
A divisive clustering algorithm starts with N0class centers (in
our algorithm we have chosen averages of random subsets from
the experimental set of images). Then, the algorithm assigns each
image to the closest class center, and recomputes the class centers
as the average of the images assigned (see Fig. 1). This process is
iterated until the number of images that change their assignment
from iteration to iteration is smaller than a certain threshold (we
used 0.5% of the total number of images in the experimental data-
set) or a maximum number of iterations is reached. Once the N0
class averages have been computed, the divisive clustering algo-
rithm chooses the class center with the largest number of images
assigned to it, and splits this subset of images into two new classes.
The split method computes the similarity (either through corren-
P
ikXi?bXkðiÞk2
is minimized,
being kðiÞ the index of the class center assigned to the ith experi-
tropy or correlation) of each image in the cluster with the class
center of that cluster. Then, it splits the whole set of images into
two halves: the one with the 50% highest similarities, and the
one with the 50% worse similarities. With this initial split, we com-
pute two new class centers and apply the same clustering method-
ology as for the whole dataset (of the two clustering criteria
discussed in the next section, for the split in two classes we found
more useful the newly proposed robust criterion). After splitting
the largest class, the algorithm proceeds to split the second largest
class, and this process is iterated until N0 splits have been per-
formed. Then, the clustering algorithm is run again with 2N0class
centers. Split phases and clustering phases are alternated until the
desired number of class centers is reached.
Alternatively, we could have started directly with the final
number of class centers and let the clustering optimize the initial
class centers (this is exactly how multireference alignment is usu-
ally applied in electron microscopy). However, this algorithm is
known to get more easily trapped into local minima. Divisive clus-
tering is less prone to local minima (Gray, 1984), although it is not
guaranteed that the global minimum will be reached (this is a gen-
eral drawback of K-means algorithms).
Our algorithm is an adaptation of this standard clustering algo-
rithm to the cryo-EM setting. Since our main goal is recognizing
themostcommonprojectionsinthedataset,ifatanymomentaclass
represents less than a user given percentage of the total number of
images(intheorderof0:2
and K the current number of classes), we remove that class from the
quantization and split the class with the largest number of images
assigned.Notethatwedonotremovethecorrespondingexperimen-
tal images from the dataset. Instead, in the next quantization itera-
tion, these images will be assigned to one of the remaining clusters.
Nimg
K,beingNimgthetotalnumberofimages
2.3. Clustering criterion
Instead of using the L2norm of the error to measure the similar-
ity between an experimental image and a class representative, we
propose to use the correntropy introduced earlier. Correntropies
closer to 1 mean a larger agreement between the experimental im-
age and the class representative.
Unfortunately, simply assigning an image to the class with
highest correntropy does not result in a good classification of the
images. The problem is that the absolute value of the correntropy
is meaningless when comparing an experimental image to several
classes (a similar problem has already been reported in electron
microscopy using the correlation index (Sorzano et al., 2004a)).
The reason for this is that at low Signal-to-Noise Ratios, the differ-
Fig. 1. Divisive clustering: Experimental images are assigned to the current number of classes as in a standard multireference algorithm (note that in our algorithm we have
changed the distance metric and the clustering criterion). When this process has converged, we split the current classes starting by the largest cluster. Then, we reassign the
experimental images to the new set of classes. This process is iterated until the desired number of classes (not necessarily a power of 2) has been reached.
C.O.S. Sorzano et al./Journal of Structural Biology 171 (2010) 197–206
199

Page 4

ences between two images may be absolutely masked by the noise,
and the wrong assignment may be done simply because one class
average is less noisy than another one. It could be said in this sit-
uation that the cleanest class ‘‘attracts” many experimental images
even if they belong to some other class.
Let us prove this statement with a simple example. Let us pre-
sume that our experimental data comes from only two underlying
classesA0andA1,andthat,afteralignment,theexperimentalimages
aresimplynoisyversionsofthesetwoclasses:Xi¼ Ajþ Ni,whereXi
is the ith experimental image, Ajis either A0or A1, and Niis a white
noise image added during the experimental acquisition. Let us as-
sume that at a given moment we have perfectly aligned and classi-
fied the experimental images in these two classes and that we
have M0images in the class generated by A0and M1images in the
class generated by A1. Note that A0and A1 are unknown (in fact,
the multireference classification problem consists in estimating
them), and all we have are estimates obtained by averaging the
imagesassignedtoeachclass,i.e.,bAj¼ AjþbNj,wherebAjareouresti-
age obtained as the average of all the noise images of the
experimental images assigned to this class. If the variance of the
experimental images isr2, then the variance ofbNjisr2
Xithe class Ajthat minimizes
1
P
p¼1
wherePisthe totalnumberofpixels,whileXipandbAjpdenotethepth
Let us assume that Xireally belongs to class A1, and let us see under
which conditions it would be correctly assigned to class 1:
(
p¼1
1
P
p¼1
1
P
p¼1
1
P
p¼1
1
P
p¼1
)1
PM0
)1
P
M0
1
PA1? A0
r2
M0
1
M1?
matesofthetrueunderlyingclassaveragesandbNjissomenoiseim-
Mj.
The classical multireference clustering criterion is ‘‘Choose for
E
X
P
ðXip?bAjpÞ2
()00
;
pixel of the experimental image and the class average, respectively.
E
1
P
X
(
(
(
(
P
p¼1
ðXip?bA0pÞ2
X
X
X
X
A1? A0
)
> E
1
P
X
)
P
ðXip?bA1pÞ2
> E
P
)
()
) E
P
ðA1pþ Nip?bA0pÞ2
ðA1pþ Nip? A0p?bN0pÞ2
ðA1p? A0pÞ2
1
X
> E
P
p¼1
ðA1pþ Nip?bA1pÞ2
1
P
p¼1
)
()
)
) E
P
X
P
ðNip?bN1pÞ2
(
) E
P
)
þ E
)
1
P
X
P
p¼1
ðNip?bN0pÞ2
(
> E
P
ðNip?bN1pÞ2
k2þr21 þ
k2>r2
k2
>
M1?
k
1
??
> r21 þ
1
M1
??
A1? A0
k
k
M1?r2
1
)
1
) SNRdifference>
1
M0:
ð6Þ
In other words, the experimental image will fail to be correctly
classifiedunlesstheSNRofthedifferencebetweenthetwounderly-
ing classes is larger than a certain threshold (in electron microscopy
images this is not always the case, as we try to capture very subtle
differences in very noisy images). Let us further assume that there
are many more images in A0than in A1, i.e.,
1
M1, moreover M1tends to be a small number and
high. In other words, if one of the classes is large ðM0? M1Þ;Xiwill
failtobe assignedtoits correctclassandthe classwithmoreimages
assigned, A0, ‘‘attracts” many images from the other class.
1
M0is ‘‘negligible” versus
1
M1is particularly
Although we have presented the problem with the variance, this
problem also occurs when using correntropy or maximum likeli-
hood(Scheresetal.,2005).Weproposeanalternativeclusteringcri-
terion that we will refer to as the robust criterion.Instead of looking
onlyattheenergyofthedifferencebetweentheexperimentalimage
and the two estimates of the class averages, we propose to compare
thisenergytotheenergyoftherestofimagescomparedtoeachclass
average. For instance, consider an experimental image whose cor-
rentropy with the representative of class A is 0.8, and with the rep-
resentative of class B is 0.79. Because of the noise, these numbers
cannotbeblindlytrustedandsimplyassigntheexperimentalimage
to class A. Let us considerthe correntropiesof all images assigned to
class A and of all images assigned to class B. If most correntropies of
imagesinclassAisintheorderof0.9,ourimagewouldbeaverybad
memberofclassA.However,ifmostcorrentropiesofimagesinclass
B is in the order of 0.7, our image would be a very good member of
class B (despite the fact that in absolute terms, the correntropy to
class B is a little bit smaller than that to class A).
How good a correntropy is with respect to each one of these sets
can be measured through the distribution functions
VðX;bAkÞ < v
ler than a given value v within the set of images assigned to class k.
The second one measures the probability of having a correntropy
smaller than a given value v within the set of images not assigned
to class k. The class to which an image Xi is assigned should be
the one maximally fulfilling both goals
n
The classical clustering criterion is good for large differences be-
tween the class averages, while the robust clustering criterion is
specially designed for subtle differences embedded in very noisy
images. That is why we prefer to use this second criterion when
splitting the nodes during the divisive clustering algorithm.
Prb Ak
no
andPrb Ak
VðX;bAkÞ < v
no
:
ð7Þ
The first one measures the probability of having a correntropy smal-
kðiÞ¼argmax
k
Prb Ak
VðX;bAkÞ<VðXi;bAkÞ
o
Prb Ak
VðX;bAkÞ<VðXi;bAkÞ
no
:
ð8Þ
2.4. Final algorithm: CL2D
With the pieces introduced so far (correntropy, divisive cluster-
ing, and robust clustering criterion) we propose to use the follow-
ing clustering algorithm that we will refer to as CL2D.
Algorithm 1. CL2D
Input:
S: Set of images
K0: Number of classes in the first iteration
KF: Number of classes in the last iteration
Output:
CKF: Set of KFclass representatives
begin
// Initialize the classes
k ¼ K0
Ck¼ Randomly split S into k classes
Update the distributions of Eq. (7)
r2= Measure the noise power in the background of the
experimental images
// Refine the classes
Ck= Refine Ckwith the data in S
while k < KFdo
Cminð2k;KFÞ= Split the largest minðk;KF? kÞ classes of Ck
k ¼ minð2k;KFÞ
Ck¼Refine Ckwith the data in S
end
end
200
C.O.S. Sorzano et al./Journal of Structural Biology 171 (2010) 197–206

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The refinement of the current classes is performed according to the
following algorithm.
Algorithm 2. CL2D refinement of the current class
representatives ðCkÞ
Input:
S: Set of images
Ck: Set of k class representatives
r2: Noise power in the background of experimental images
Itermax: Maximum number of iterations
Nmin: Minimum number of images in a class
Output:
Ck: Refined set of k class representatives
begin
Iter ¼ 0
repeat
// Assign all images to a class
foreach image 2 S do
foreach classAverage 2 Ckdo
image ¼Align image to classAverage
Compute argument of Eq. (8)
end
Assign image to the class maximizing Eq. (8)
end
// Remove small classes and split the largest
ones
while there are classes with less than Nminimages do
Remove the class with the smallest number of images
assigned
Split the largest class of Ck
end
Update the distributions of Eq. (7)
Iter ¼ Iter þ 1
until No: of Changes < 0:5% and Iter < Itermax;
end
The split of a class is performed in a way very similar to that
of the refinement of the current classes. The main difference is
how the two subclasses are initially calculated. While in the stan-
dard algorithm these are computed by a random split, in the
CL2D split we separate the images with the largest correntropies
to the class average from the images with the smallest corren-
tropies:
Algorithm 3. CL2D split of a class
Input:
SðcÞ: Set of images assigned to class c
c: Class representative of class c
Output:
c1;c2: Two subclasses derived initially from class c
begin
// Split the images into two initial subclasses
c1= Average of all images assigned to c with the largest 50%
correntropies
c2= Average of all images assigned to c with the smallest
50%
correntropies
Update the distributions of Eq. (7) for these two subclasses
// Refine the two subclasses
Refine fc1;c2g with the data in SðcÞ
end
3. Results
To validate our algorithm we first performed a number of sim-
ulated experiments with the aim of testing the clustering approach
to the multireference alignment problem. We used two different
datasets for assessing the properties of the algorithm: one with
the bacteriorhodopsin monomer to assess its clustering properties
with respect to the projection point of view; and another with the
Escherichia coli ribosome in two different conformational states to
assess its clustering properties with respect to the conformational
state. We then applied the algorithm to experimental data. In all
the experiments we compared our results to those of ML2D
(Scheres et al., 2005), SVD/MSA (script refine2d.py) in EMAN
(Ludtke et al., 1999), Diday’s method, Hierarchical clustering and
K-means from Spider (Frank et al., 1996) (scripts cluster.spi,
hierarchical.spi, and kmeans.spi). Algorithms in Spider are
preceded by a filtering and dimensionality reduction by Principal
Component Analysis (PCA).
3.1. Simulated data: bacteriorhodopsin
To test our algorithm we generated 10,000 projections ran-
domly distributed in all projection directions from the bacteriorho-
dopsin monomer (PDB entry: 1BRD) and added white Gaussian
noise with a SNR of 1/3 and 1/30 (see Fig. 2). We then applied
our algorithm to compute 256 class averages with a maximum
iteration count of 20. We found that this parameter is not critical
as long as it is not too small so that the algorithm is stopped when
the classes are not well established yet.
In our first experiment we tried to assess the effectiveness of
each one of the three main modifications to the clustering algo-
rithm (correntropy vs. correlation, divisive clustering vs. multire-
ferenceclustering,classicalclustering
clustering criterion). For doing so, we conducted four experiments
with the dataset with SNR = 1/3: the first one with the three new
features (see its results in Supp. Fig. 1), and other three in which
one of the new features was missing (correntropy, divisive cluster-
ing, or robust clustering). For each one of the experiments we eval-
uated each cluster by computing the angle between the projection
directions of any pair of images assigned to that cluster. Finally, we
computed the histogram of these angular distances in order to
compare the different choices of the algorithm. Fig. 3 shows these
histograms. It can be seen that the best combination is the one
using two of the three new ideas introduced in this paper: corren-
tropy and divisive clustering. As has already been mentioned be-
fore, this simply means that the SNR of the differences between
classes is large enough to overcome the attraction effect (in the
next subsection we present a case in which the robust clustering
criterion performs better than the classical one because the input
images have much lower SNR).
We repeated this experiment using ML2D (see Supp. Fig. 2). The
algorithm could not produce more than 20 different classes that
accounted for 97.1% of the particles (the algorithm was run with
50 classes, and the remaining 30 classes accounted only for 58
images, existing even empty classes). This result illustrates the
‘‘attractive” effect described during the introduction of the robust
clustering criterion. Moreover, there was at least one class average
(with 129 images assigned) in the ML2D approach that did not
really represent any of the true projection images (see Fig. 4). In
fact, if we analyze the 129 images assigned to this node with our
algorithm, it can be seen that the ML2D class is actually a mixture
of at least four different classes with 29, 35, 35, and 30 images,
respectively.
criterionvs.robust
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We also repeated this experiment with SVD/MSA (classes not
shown but we did not observe the ‘‘attraction” effect), PCA/Diday,
PCA/Hierarchical, and PCA/K-means (with both dataset SNR = 1/3
and SNR = 1/30). As suggested in the multivariate data analysis
web page of Spider (http://www.wadsworth.org/spider_doc/
spider/docs/techs/MSA), we filtered our images with a Butter-
worth filter between the digital frequencies 0.25 and 0.33 (these
frequencies are normalized in such a way that 1 corresponds to
the Nyquist frequency). We also prealigned the images and re-
duced their dimensionality using a PCA with nine components.
This preprocessing was applied to the data before using the three
algorithms of Spider. In Fig. 5 we show the different estimates of
the probability density function of the angular distance between
the projection directions of images assigned to the same cluster
for the six algorithms. It can be clearly seen that CL2D produces
the tightest clusters in terms of projection directions.
The execution times in a single CPU (Intel Xeon 2.6 GHz) for
PCA/Diday, PCA/Hierarchical and PCA/K-means was about 0.3 h,
50 h for SVD/MSA, 100 h for CL2D, and 74 h for ML2D (with only
64 classes; CL2D took 41 h in computing 64 classes). However,
CL2D has been parallelized with MPI and has been run in parallel
with 32 nodes, reducing the computation time to 3.5 h.
The experiment was repeated in a more realistic setup by intro-
ducing the effect of the microscope Contrast Transfer Function
(CTF) with an acceleration voltage of 200 kV, a defocus of
?5.5 lm, and a spherical aberration of 2.26 mm. Results were sim-
ilar to the previous case (results not shown).
3.2. Simulated data: E. coli ribosome
For this test we used the public dataset of simulated ribosomes
available at the Electron Microscopy Data Bank (http://www.ebi.ac.
uk/pdbe/emdb/singleParticledir/SPIDER_FRANK_data,(Baxteretal.,
2009)). This dataset contains 5000 projections from random direc-
tions of a ribosome bound with three tRNAs at A, P, and E sites,
and other 5000 projections from random directions of an EF-
G(GDPNP)-bound ribosome with a deacylated tRNA bound in the
hybrid P/E position. Besides these differences in the ligands, these
two ribosomes also have different ribosomal conformations: the
first ribosome is in the normal conformation, while the second ribo-
some is in a ratcheted configuration (30S subunit rotated counter-
clockwise relative to the 50S subunit). Fig. 6 shows some sample
images from this dataset.
The goal of this experiment is to characterize the capability of
the new algorithm to separate the two conformational states, i.e.,
to produce classes in which the majority of the images assigned
comes from the same conformational state.
Fig. 2. Simulated data: Sample noisy projections of the bacteriorhodopsin. Top row: the SNR is 1/3. Bottom row: the SNR is 1/30.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 10 20 30 40 50 60 70 80 90
Probability Density Function
Angular distance in degrees
Correntropy+Divisive+Classical
Correlation+Divisive+Classical
Correntropy+Non-divisive+Classical
Correntropy+Divisive+Robust
Fig. 3. Clustering quality: probability density function estimate of the angular distance between projection directions of images assigned to the same cluster for the different
execution modes of our clustering algorithm. Ideally, these functions should be concentrated at low angular distances meaning that the projections assigned to a cluster have
similar projection directions. The maximum angular distance is 90? since the worse situation is that two orthogonal views (e.g. top and side views) are assigned to the same
cluster.
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With our previous dataset we already showed the superiority of
the correntropy over correlation, and of the divisive clustering over
a classical multireference clustering. In the previous dataset it also
appeared that the classical clustering criterion was superior to the
robust clustering criterion. However, as was already discussed in
theMethodsSection,thisholdsforsufficientlyhighSNRs.Thisdata-
set has a much lower SNR than the previous one. We ran our algo-
rithm to partition the input data into 256 classes using the
classical clustering criterion and the new robust criterion. For each
class we tested the hypothesis that the two ribosomes were equally
Fig. 4. Example of a class wrongly identified by ML2D: The first image on the left is the class average found by ML2D (it represented 129 images from the original dataset).
The other four images are classes found by CL2D applied to these 129 images. It reveals the existence of at least four different classes (with 29, 35, 35, and 30 images,
respectively) within the set of 129 images assigned to a single class by ML2D.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0 10 20 30 40 50 60 70 80 90
Probability Density Function
Angular distance in degrees
CL2D
PCA/K-means
PCA/Hierarchical
SVD/MSA
ML2D
PCA/Diday
0
0.005
0.01
0.015
0.02
0.025
0 10 20 30 40 50 60 70 80 90
Probability Density Function
Angular distance in degrees
CL2D
PCA/K-means
PCA/Hierarchical
SVD/MSA
ML2D
PCA/Diday
Fig. 5. Clustering quality: probability density function estimate of the angular distance between projection directions of images assigned to the same cluster for six different
clustering algorithms (top, SNR = 1/3; bottom SNR = 1/30).
C.O.S. Sorzano et al./Journal of Structural Biology 171 (2010) 197–206
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represented (the proportion of images of each type was 0.5). The
classicalclusteringcriteriononlyfound67classes(outof256)where
the proportion of the majority of the images were significantly dif-
ferent(witha95%ofconfidence)from50%.Theamountofimagesas-
signed to these classes was 2822. However, the new robust
clustering criterion was able to identify 98 classes where the major-
ity of images significantly came from a single ribosome type. The
amount of images assigned to these 98 classes was 4130. The differ-
ence between the proportions of classes with a majority of ribo-
somes of one type using the classical and the robust clustering
criteria was significantly different with a confidence of 95%.
We repeated the same experiment with SVD/MSA, ML2D, PCA/
Diday’s clustering, PCA/Hierarchical clustering and PCA/K-means.
SVD/MSA identified 107 classes with a statistically significant
majority of images of one type (4534 images were assigned to these
classes). However, the difference between the proportions of SVD/
MSA and CL2D was not statistically significant with a confidence le-
vel of 95%. ML2D identified 9 classes with a statistically significant
majority of images of one type (1439 images were assigned to these
classes).Imageswerepreprocessedinexactlythesamewayasinthe
previous experiment before applying any of Spider’s methods. PCA/
Diday method produced 11 classes with a statistically significant
majority of images of one type (899 images were assigned to these
classes). The PCA/Hierarchical clustering produced 74 classes with
a statistically significant majority of images of one type (with 3049
images assigned to them). The PCA/K-means produced 53 classes
with a statistically significant majority of images of one type (with
2114 images assigned to them).
3.3. Experimental data: p53
We applied our algorithm to a dataset that contained 7600 pro-
jections of a mutant of p53 without the C-terminal domain bound
to the GADD45 DNA sequence (Ma et al., 2007; Tidow et al., 2007).
The sample was negatively stained. The pixel size was 4.2 Å/pixel
and the image size was 60 ? 60 pixels (see Fig. 7). We applied
our algorithm to generate 256 class averages (20 iterations per
Hierarchical level). Supp. Fig. 3 shows the resulting classes. In an
experimental setting it is difficult to validate the 2D classes pro-
duced by an algorithm. In this experiment at least we tested that
the classes produced captured enough information from the origi-
nal dataset. For doing so, we built a reference volume based on the
common lines of the classes produced by CL2D (see Fig. 8a). This
reference volume was refined in two different ways: first, using
the CL2D classes themselves as the only projections in the dataset
(the refined volume is shown in Fig. 8b); second, using the whole
dataset of projections (the refined volume is shown in Fig. 8c).
Interestingly, the volumes in Fig. 8b and c are consistent with each
other up to 32 Å (at this frequency the Fourier Shell Correlation
(Harauz and van Heel, 1986) between the two volumes drops be-
low 0.5). Considering that the resolution of the volume in Fig. 8c
is around 27 Å, the approximation up to 32 Å of the volume recon-
structed using only the CL2D classes is not a bad approximation.
The same projection dataset was refined using a reference volume
obtained independently through Random Conical Tilt (Scheres
et al., 2009).
4. Discussion
In this paper we have introduced a new multireference align-
ment algorithm that can be briefly explained as follows. If we
think of projection images of size N ? N as vectors in a N2-
dimensional space, we can define our algorithm as one that
looks for N2-dimensional points trying to be close to the distri-
bution of points corresponding to the original images in this
space. This is exactly the goal of vector quantization or cluster-
ing. Close or distant images are defined using the correntropy,
a similarity function that has been proved to be useful for
non-Gaussian noise sources; while the clustering criterion has
been modified to a robust version in order to be able to over-
come the ‘‘attraction” effect affecting the any clustering criterion
based on a term similar to that of minimum variance. Summa-
rizing, the four key ideas of our algorithm are:
? Use of correntropy as a similarity measure between images
instead of the standard least-squares distance or, its equiva-
lent, cross-correlation. Correntropy has proven to be an useful
similarity measure in non-linear, non-Gaussian signal process-
ing and it has been related to the M-estimates of robust
statistics.
? Use of clustering as a means of producing many classes with a
small amount of images in each class. In this way we have a bet-
ter angular coverage of the projection sphere and avoid averag-
ing too many images in the same cluster.
? Use of a divisive algorithm for performing the clustering as a
means of avoiding getting trapped in a local minimum of the
quantization problem. Although global convergence is not guar-
anteed, it has been shown (Gray, 1984) that experimentally,
divisive clustering is more robust than a clustering attempting
to directly produce the final number of class averages.
? The assignment of an image to one class is not performed by
simply choosing the class with maximum correntropy. We
found that this simple strategy leads to a reduced number of
‘‘attractive” classes if the SNR of the differences to be detected
is not sufficiently large. Alternatively, we propose to compare
for each class, the correntropy of the image at hand, the set of
Fig. 6. Simulated data: Sample noisy projections of the ribosome in two different conformations.
Fig. 7. Examples of the experimental images: Sample projections of the p53 mutant.
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correntropies of all the images assigned to that class, and the set
of correntropies of all the images not assigned to that class.
Then, we choose the class that maximizes the probability of
being better than all the images assigned and being better than
all the images not assigned.
We have tested this algorithm with a number of simulated and
experimental datasets. In the simulated datasets we found that our
algorithm is much better suited than ML2D, SVD/MSA, PCA/Diday,
PCA/Hierarchical and PCA/K-means to discover many different pro-
jection directions. ML2D suffered from ‘‘attractor” classes, i.e., the
algorithm was unable of working with many output reference clas-
ses (many of the classes are empty or nearly empty). This cannot
happen in our algorithm by its construction: when a class is too
small (represents less than a user-specified percentage of the input
images), the class is removed and the largest class will be splitted
in two. The few images assigned to the removed representative
have to find a new representative in the next iteration. Using a suf-
ficiently large number of class representatives, it is expected to
better represent the input dataset. In fact, this is what we actually
observed using the histogram of angular differences between pairs
of images assigned to the same cluster (the angular difference be-
tween two images is defined as the difference in degrees between
their projection directions). In our first experiment with simulated
data, we observed that the combination of correntropy, divisive
clustering, and classical clustering criterion was the one producing
the best results. Changing any of these three features resulted in
lower performance. However, in our second experiment we
showed that the classical clustering criterion is not always the
most successful. The second dataset had a much lower SNR and,
as expected, the newly introduced robust clustering criterion over-
performed the classical criterion.
In our first simulated dataset we also discovered that some of
the classes returned by ML2D could be understood as a mixture
of several other classes. The problem of handling many classes
seems to be attenuated in the SVD/MSA, PCA/Hierarchical and
PCA/K-means approaches. However, we have already shown that
our algorithm is able of producing tighter clusters in terms of pro-
jection directions and, therefore, it is more likely to achieve good
common line reconstructions. With our experiment with the p53
mutant we have shown that reference volumes constructed by
common lines from our CL2D classes are capable of being refined
to yield a volume that is similar to one obtained by starting from
a Random Conical Tilt data collection. Moreover, the refinement
of the initial volume constructed with common lines using only
our CL2D classes is able to produce a volume that is compatible
with the best reconstruction of this dataset up to 32 Å (the resolu-
tion of the best reconstruction is around 27 Å).
When clustering images with different projection directions
and belonging to different conformational states (the ribosomal
simulated data), our algorithm was able to generate a proportion
of classes with a majority of images from a single ribosome type
that was not significantly different from the same proportion in
case of using SVD/MSA, and significantly better than ML2D, PCA/
Diday, the PCA/Hierarchical and the PCA/K-means. We could not
evaluate the angular spread of each one of the classes because this
information is missing in this standard benchmark.
5. Conclusions
In this paper we have introduced a new algorithm for two-
dimensional multireference alignment. It is based on the idea of
creating many classes with a small number of images assigned to
each class in order to avoid too much averaging. Moreover, we
have used correntropy as the similarity function, a recently intro-
duced alternative to cross-correlation which is sensitive to non-
Gaussian noise sources and is related to robust statistics. We have
also introduced a new clustering criterion that in our experiments
did not suffer from the ‘‘attraction” problem even at low SNR. We
have proven that this algorithm can be successfully applied to elec-
tron microscopy images of single-particles producing higher-qual-
ity results than those of the most widely used algorithms in the
field. This algorithm is freely available from the Xmipp software
package (Sorzano et al., 2004b; Scheres et al., 2008) (http://
xmipp.cnb.csic.es).
Acknowledgments
The authors are thankful to Dr. Scheres for revising the final
manuscript and for insightful discussions, and to Dr. Martín-Benito
for his help running the Spider algorithms. This work was funded
by the European Union (projects FP6-502828 and UE-512092),
the 3DEM European network (LSHG-CT-2004-502828) and the
ANR(PCV06-142771), the Spanish Ministerio de Educación
(CSD2006-0023, BIO2007-67150-C01 and BIO2007-67150-C03),
the Spanish Ministerio de Ciencia e Innovación (ACI2009-1022),
the Spanish Fondo de Investigación Sanitaria (04/0683) and the
Comunidad de Madrid (S-GEN-0166-2006). The authors wish to
thank support from grants MCI-TIN2008-01117 and JA-P06-
TIC01426. J.R. Bilbao-Castro is a fellow of the Spanish ‘‘Juan de la
Cierva” postdoctoral contract program, co-financed by the Euro-
pean Social Fund. C.O.S. Sorzano is a recipient of a ‘‘Ramón y Cajal”
fellowship financed by the European Social Fund and the Ministe-
rio de Ciencia e Innovación. G. Xu was supported in part by NSFC
under the grant 60773165, NSFC key project under the grant
10990013. The project described was supported by Award Number
R01HL070472 from the National Heart, Lung, And Blood Institute.
The content is solely the responsibility of the authors and does
not necessarily represent the official views of the National Heart,
Fig. 8. Reconstructions of p53: Isosurface representation containing 100% of the protein mass of a mutant of p53 without the N-terminal domain bound to DNA.
(a) Reconstructed from the CL2D classes using common lines. (b) Refinement of the volume in a) using only the CL2D classes. (c) Reconstruction performed from the whole
dataset using the volume obtained in (a) as the starting reference. (d) Reconstruction performed from the whole dataset but using a volume obtained by Random Conical Tilt
as the starting reference.
C.O.S. Sorzano et al./Journal of Structural Biology 171 (2010) 197–206
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Lung, And Blood Institute or the National Institutes of Health. The
funders had no role in study design, data collection and analysis,
decision to publish, or preparation of the manuscript.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, in
the online version, at doi:10.1016/j.jsb.2010.03.011.
References
Baxter, W.T., Grassucci, R.A., Gao, H., Frank, J., 2009. Determination of signal-to-
noise ratios and spectral snrs in cryo-em low-dose imaging of molecules. J.
Struct. Biol. 166 (2), 126–132.
Chong, C.W., Raveendran, P., Mukundan, R., 2003. Translation invariants of zernike
moments. Pattern Recogn. 36, 1765–1773.
Dempster, A., Laird, N., Rubin, D., 1977. Maximum likelihood from incomplete data
via the EM algorithm. J. Roy. Stat. Soc., Ser. B 39, 1–38.
Flusser, J., Zitova, B., Suk, T., 1999. Invariant-based registration of rotated and
blurred images. In: Proceedings of the Geoscience and Remote Sensing
Symposium, vol. 2. pp. 1262–1264.
Frank, J., Radermacher, M., Penczek, P., Zhu, J., Li, Y., Ladjadj, M., Leith, A., 1996.
SPIDER and WEB: processing and visualization of images in 3D electron
microscopy and related fields. J. Struct. Biol. 116, 190–199.
Gray, R.M., 1984. Vector quantization. IEEE Acoust. Speech Signal Process. Mag. 1,
4–29.
Harauz, G., van Heel, M., 1986. Exact filters for general geometry three dimensional
reconstruction. Optik 73, 146–156.
Liu, W., Pokharel, P.P., Prfncipe, J.C., 2007. Correntropy: properties and applications
in non-gaussian signal processing. IEEE Trans. Signal Process. 55, 5286–5298.
Lowe, D., 1999. Object recognition from local scale-invariant features. In:
Proceedings of the ICCV. pp. 1150–1157.
Lowe, D.G., 2004. Distinctive image features from scale-invariant keypoints. Int. J.
Comput. Vis. 60, 91–110.
Ludtke, S.J., Baldwin, P.R., Chiu, W., 1999. EMAN: semiautomated software for high-
resolution single-particle reconstructions. J. Struct. Biol. 128, 82–97.
Ma, B., Pan, Y., Zheng, J., Levine, A.J., Nussinov, R., 2007. Sequence analysis of p53
response-elements suggests multiple binding modes of the p53 tetramer to dna
targets. Nucleic Acids Res. 35, 2986–3001.
Marabini, R., Carazo, J.M., 1994. Practical issues on invariant image averaging using
the bispectrum. Signal Process. 40, 119–128.
Pun, C.M., 2003. Rotation-invariant texture feature for image retrieval. Comput. Vis.
Image Underst. 89, 24–43.
Santamarfa, I., Pokharel, P.P., Prfncipe, J.C., 2006. Generalized correlation function:
definition, properties, and application to blind equalization. IEEE Trans. Signal
Process. 54, 2187–2197.
Schatz, M., van Heel, M., 1990. Invariant classification of molecular views in electron
micrographs. Ultramicroscopy 32, 255–264.
Schatz, M., van Heel, M., 1992. Invariant recognition of molecular projections in
vitreous ice preparations. Ultramicroscopy 45, 15–22.
Scheres, S.H., Melero, R., Valle, M., Carazo, J.M., 2009. Averaging of electron
subtomograms and random conical tilt reconstructions through likelihood
optimization. Structure 17, 1563–1572.
Scheres, S.H.W., Nus ´ez-Ramfrez, R., Sorzano, C.O.S., Carazo, J.M., Marabini, R., 2008.
Image processing for electron microscopy single-particle analysis using xmipp.
Nat. Protoc. 3, 977–990.
Scheres, S.H.W., Valle, M., Núñez, R., Sorzano, C.O.S., Marabini, R., Herman, G.T.,
Carazo, J.M., 2005. Maximum-likelihood multi-reference refinement for
electron microscopy images. J. Mol. Biol. 348, 139–149.
Schmid, C., Mohr, R., 1997. Local greyvalue invariants for image retrieval. IEEE
Trans. Pattern Anal. Mach. Intell. 19, 530–535.
Sorzano, C.O.S., Jonic, S., El-Bez, C., Carazo, J.M., De Carlo, S., ThTvenaz, P., Unser, M.,
2004a. A multiresolution approach to pose assignment in 3-D electron
microscopy of single particles. J. Struct. Biol. 146, 381–392.
Sorzano, C.O.S., Marabini, R., Velázquez-Muriel, J., Bilbao-Castro, J.R., Scheres,
S.H.W., Carazo, J.M., Pascual-Montano, A., 2004b. XMIPP: a new generation of an
open-source image processing package for electron microscopy. J. Struct. Biol.
148, 194–204.
Tidow, H., Melero, R., Mylonas, E., Freund, S.M.V., Grossmann, J.G., Carazo, J.M.,
Svergun, D.I., Valle, M., Fersht, A.V., 2007. Quaternary structures of tumor
suppressor p53 and a specific p53 dna complex. Proc. Natl. Acad. Sci. USA 104,
12324–12329.
Tuytelaars, T., Van Gool, L., 1999. Content-based image retrieval based on local
affinely invariant regios. Vis. Inform. Inform. Syst., 493–500.
van Heel, M., Stoffler-Meilicke, M., 1985. Characteristic views of E. coli and B.
staerothermophilus 30s ribosomal subunits in the electron microscope. EMBO J.
4, 2389–2395.
206
C.O.S. Sorzano et al./Journal of Structural Biology 171 (2010) 197–206