Fast, robust, and accurate determination of transmission electron microscopy contrast transfer function.

Abstract

Transmission electron microscopy, as most imaging devices, introduces optical aberrations that in the case of thin specimens are usually modeled in Fourier space by the so-called contrast transfer function (CTF). Accurate determination of the CTF is crucial for its posterior correction. Furthermore, the CTF estimation must be fast and robust if high-throughput three-dimensional electron microscopy (3DEM) studies are to be carried out. In this paper we present a robust algorithm that fits a theoretical CTF model to the power spectrum density (PSD) measured on a specific micrograph or micrograph area. Our algorithm is capable of estimating the envelope of the CTF which is absolutely needed for the correction of the CTF amplitude changes.

Full text

Page 1

Fast, robust, and accurate determination of transmission
electron microscopy contrast transfer function
C.O.S. Sorzanoa,b,*, S. Jonicc, R. Nu ´n ˜ez-Ramı ´reza, N. Boissetc, J.M. Carazoa
aUnidad de Biocomputacio ´n, Centro Nacional de Biotecnologı ´a (CSIC), Campus Universidad Auto ´noma s/n, 28049 Cantoblanco, Madrid, Spain
bDepartment of Ingenierı ´a de Sistemas Electro ´nicos y de Telecomunicacio ´n, University of San Pablo-CEU,
Campus Urb. Monteprı ´ncipe s/n, 28668 Boadilla del Monte, Madrid, Spain
cUniversite ´ Pierre et Marie Curie, CNRS, IMPMC-UMR7590, Universite ´ Paris 7, Paris F-75005, France
Received 29 March 2007; received in revised form 18 August 2007; accepted 22 August 2007
Available online 29 August 2007
Abstract
Transmission electron microscopy, as most imaging devices, introduces optical aberrations that in the case of thin specimens are usu-
ally modeled in Fourier space by the so-called contrast transfer function (CTF). Accurate determination of the CTF is crucial for its
posterior correction. Furthermore, the CTF estimation must be fast and robust if high-throughput three-dimensional electron micros-
copy (3DEM) studies are to be carried out. In this paper we present a robust algorithm that fits a theoretical CTF model to the power
spectrum density (PSD) measured on a specific micrograph or micrograph area. Our algorithm is capable of estimating the envelope of
the CTF which is absolutely needed for the correction of the CTF amplitude changes.
? 2007 Elsevier Inc. All rights reserved.
Keywords: Contrast transfer function; Transmission electron microscope; Power spectrum density; Astigmatism
1. Introduction
Structural biology is a key tool to fully understand the
function of macromolecular complexes within living cells
(Sali et al., 2003). Transmission electron microscopy
(TEM) is a very useful device to acquire structural informa-
tion about these complexes (Frank, 2002; Henderson, 2004;
van Heel et al., 2000). However, the electron microscope
distorts the structural information by changing amplitudes
and phases of recorded electron waves. This is so due to the
aberrations that exist in the microscope as in any imaging
device, and to the particular nature of the propagation of
electron waves. These distortions can be modeled in
Fourier space by a multiplication of the Fourier transform
of an ideal two-dimensional projection of a three-dimen-
sional object with the microscope transfer function called
in the field Contrast Transfer Function (CTF). The reader
interested in a physical justification of the multiplication in
Fourier space (or the equivalent linear convolution in real
space) as well as in the physical model of the CTF is
referred to the works of De Rosier (2000), Philippsen
et al. (2007), Unwin (1973), Wade (1992), and Zou (1995).
Aware of the need to select good micrographs relying on
their CTF behavior, several works proposed a fast sorting
of micrographs with visible Thon rings devoid of astigma-
tism or drift (Gao et al., 2002; Jonic et al., 2007). Astigma-
tism itself may not be a negative issue if it can be taken into
account by CTF correction algorithms. However, astig-
matic images are traditionally rejected when using CTF
correction procedures that assume only non-astigmatic
images such as CTF correction of volumes from defocus
series (Penczek et al., 1997).
1047-8477/$ - see front matter ? 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jsb.2007.08.013
*Corresponding author. Address: Unidad de Biocomputacio ´n, Centro
Nacional de Biotecnologı ´a (CSIC), Campus Universidad Auto ´noma s/n,
28049 Cantoblanco, Madrid, Spain. Fax:+34 913724049.
E-mail address: coss.eps@ceu.es (C.O.S. Sorzano).
www.elsevier.com/locate/yjsbi
Available online at www.sciencedirect.com
Journal of Structural Biology 160 (2007) 249–262
Journal of
Structural
Biology

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The estimation of CTF parameters is usually performed
in two steps:
• Estimation of the power spectrum density. The power
spectrum density (PSD) determines the amount of
energy present at each spectral frequency. Considering
the CTF as a transfer function of a linear system that
takes an input (unknown) image and transforms it into
an output (experimentally observed) image, and with-
out taking into account noise, the PSD of the output
image is the PSD of the input image multiplied by
the square modulus of the CTF. Therefore, the PSD
of experimental images is directly related to the CTF.
The estimation of the experimental PSD can be done
with classical methods such as periodogram averaging
(Avila-Sakar et al., 1994; Ferna ´ndez et al., 1997;
Welch, 1967; Zhu et al., 1997) or parametric methods
such as AR or ARMA (Ferna ´ndez et al., 1997; Vela ´z-
quez-Muriel et al., 2003). Practically speaking, period-
ogram averaging is a simple and fast method to
estimate the PSD, although the estimates are quite
noisy (Broersen, 2000). Conversely, parametric meth-
ods are more complex and slower to compute, though
PSD estimates seem more accurate (Broersen, 2000;
Vela ´zquez-Muriel et al., 2003). The CTF estimation
procedure presented in this work can work with both
kinds of PSD estimates.
• Estimation of CTF parameters. Once the PSD has
been computed, CTF parameters corresponding to
the experimental PSD are estimated. This is usually
done by minimizing some measure of dissimilarity
between the experimental PSD and a theoretical
PSD determined from the CTF parameters. Some
works concentrate only on estimation of parameters
such as defocus in two principal directions, astigma-
tism (Mindell and Grigorieff, 2003) or on the contrast
amplitude factor (Toyoshima and Unwin, 1988; Toyo-
shima et al., 1993). This modeling allows the phase
correction of experimental images (Frank, 2006).
Some other works estimate the amplitude decay either
using a Gaussian envelope (specified by a parameter
called B-factor) (Huang et al., 2003; Saad et al.,
2001; Mallick et al., 2005; Sander et al., 2003) or
by fitting parameters
(Vela ´zquez-Muriel et al., 2003; Zhou et al., 1996;
Zhu et al., 1997). Some of the previous works mini-
mize the dissimilarity between the model PSD and
the experimental PSD as 1D functions. These 1D
functions (or radial profiles) are obtained by rotation-
ally averaging of the 2D experimental PSD (Saad
et al., 2001; Zhou et al., 1996; Zhu et al., 1997) or
by elliptical averaging (Mallick et al., 2005). A disad-
vantage of the radial averaging is that the information
about astigmatism is lost. Elliptical averaging is not
the ideal solution either since it does not take into
account anisotropy of background noise, as shown
in this paper. Some other works address a fully 2D
ofaphysicalmodel
optimization (Mindell and Grigorieff, 2003; Vela ´z-
quez-Muriel et al., 2003). The work of Huang et al.
(2003) lies somewhere in between a fully 2D optimiza-
tion and a 1D optimization, since astigmatism is esti-
mated in averaged sectors.
Once the CTF is estimated it can be corrected with any
of the available methods: Wiener filtering (Frank and Pen-
czek, 1995; Grigorieff, 1998), combination of differently
defocused volumes (Holmes et al., 2003; Penczek et al.,
1997), maximum entropy (Skoglund et al., 1996), iterative
data refinement (Sorzano et al., 2004a), direct deconvolu-
tion in Fourier space (Stark et al., 1997), Chahine’s method
(Zubelli et al., 2003), etc.
The CTF estimation method proposed in this work
differs from previous methods in many ways, although
it takes into account many of their best features. It per-
forms a fully 2D estimation and thus overcomes prob-
lemsassociated withradial/elliptical
estimates a physical model of the decaying envelope
instead of a simplified Gaussian decay. It also estimates
the local characteristics of the background noise. With
respect to our previous publication on CTF estimation,
the new method has a modified model of the background
(which allows us to make accurate CTF estimations) but
the same physical models of CTF and CTF envelope. We
allow astigmatism not only in the CTF defoci but also in
each term of the background (that is, we allow aniso-
tropic noise). In this paper, we also show that the back-
ground is effectively anisotropic, which means that any
model based on a radially/elliptically symmetric back-
ground is inaccurate. The optimization strategy presented
here is completely different from our previous one (Vel-
a ´zquez-Muriel et al., 2003). This new optimization
method relies on enhanced PSDs introduced in Jonic
et al. (2007). The use of enhanced PSDs makes optimiza-
tion robust, which is a necessary requirement when a
large number of parameters has to be estimated. Further-
more, the present optimization strategy is faster than the
one of our previous approach.
The CTF estimation method presented in this work can
be applied to a whole micrograph, its local areas, or an
individual particle image. Hence, each particle image may
have its own CTF model, and we implicitly account for
tilted micrographs whose local areas have different defoci
values.
We summarize our PSD model in Section 2.1 and
describe our fitting of the experimental PSD in Section
2.2. Results obtained with the new algorithm are reported
in Section 3. Finally, we discuss the results (Section 4)
and conclude (Section 5).
averaging.It
2. Methods
In this section, we first describe the PSD model that is
fitted, then we explain the algorithm that has been devel-
oped for fitting the PSD parameters.
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2.1. PSD model
We assume that the model of image formation in the
electron microscope is (Vela ´zquez-Muriel et al., 2003)
pexperimentalðrÞ ¼ hðrÞHðpidealðrÞ þ pnbðrÞÞ þ pnaðrÞ;
where r 2 R2denotes a spatial location, pidealis the ideal
projection of the 3D object being studied, h is the point
spread function (PSF) of the microscope, w denotes the
convolution operation, and pnbðrÞ and pnaðrÞ represent noise
terms added before and after the convolution with the PSF.
Given this image formation model, the corresponding PSD
is
ð1Þ
PSDexperimentalðRÞ ¼ jHðRÞj2ðPSDidealðRÞ þ PSDnbðRÞÞ
þ PSDnaðRÞ;
where R 2 R2is a given spatial frequency, H is the Fourier
transform of the PSF (i.e., the CTF), and PSDx is the
power spectrum density of px.
Even if only one object is imaged (as is the case in cellu-
lar tomography), we cannot not know pidealsince generally
we do not know the 3D structure of the object under study.
When imaging multiple copies of the same object (as is the
case in single-particle analysis), it is even more difficult to
know pidealsince the orientation of particles is unknown
beside their unknown 3D structure. Therefore, we assume
that PSDideal(R) = 0, which is not so far from the truth
since the noise power is much more important than the sig-
nal power in a typical electron micrograph. Moreover, we
will assume that the noise before the CTF is white,
PSDnbðRÞ ¼ K2. Under these two hypothesis, the model
simplifies to
ð2Þ
PSDtheoreticalðRÞ ¼ K2jHðRÞj2þ PSDnaðRÞ:
The structure of this PSD is formed by two terms. The
first one is the PSD of the noise colored by the CTF. The
second one is the PSD of the noise after CTF and is
referred to as ‘‘background’’ PSD. Models for these two
terms are described in Sections 2.1.1 and 2.1.2, respectively.
ð3Þ
2.1.1. CTF model
The CTF model used in this work is based on the one
already used in Vela ´zquez-Muriel et al. (2003). It is briefly
reproduced here for convenience. The interested reader is
referred to Vela ´zquez-Muriel et al. (2003), Zhou et al.
(1996) and references therein for a justification of each
term.
A typical microscope has a frequency response approx-
imated by
HidealðRÞ ¼ ?ðsinðvðRÞÞ þ QðRÞcosðvðRÞÞÞ;
where Q(R) is the fraction of electrons being scattered at
each frequency (in our model, it is assumed to be constant,
Q0) and
ð4Þ
vðRÞ ¼ pk jDfðRÞjjRj2þ1
2CsjRj4k2
??
:
ð5Þ
Csrepresents the spherical aberration coefficient, and Df(R)
is the defocus vector given by
DfðRÞ ¼ DfMcosð\R ? hÞ;Dfmsinð\R ? hÞ
\R is the angle of the 2D frequency R. The defocus vector
describes an ellipse with major and minor semi-axes DfM
and Dfm, respectively. The angle of the major semi-axis
with respect to the horizontal axis is h. k is the electron
wavelength which is computed as
ðÞ:
ð6Þ
k ¼
1:23 ? 10?9
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where V is the acceleration voltage of the microscope.
A real microscope has a frequency response similar to
the ideal one except for a damping envelope E(R), which
results in a low-pass filtering of the ideally projected 3D
object. Thus, the frequency response of a real microscope is
V þ 10?6V2
p
;
ð7Þ
HðRÞ ¼ EðRÞHidealðRÞ:
We consider three different effects hindering the maximum
achievable resolution: the beam energy spread, the beam
coherence, and the sample drift. The three effects combine
into a single envelope function as
ð8Þ
EðRÞ ¼ EspreadðRÞEcoherenceðRÞEdriftðRÞ:
The beam energy spread envelope is computed as
?
ð9Þ
EspreadðRÞ ¼ exp
?
p
4Cak
?
DV
Vþ 2DI
logð2Þ
I
????2
jRj4
!
;
ð10Þ
where Cais the chromatic aberration coefficient,DV
energy spread of the emitted electrons represented as a
fraction of the nominal acceleration voltage, andDI
lens current instability expressed as a fraction of the nom-
inal current.
We compute the beam coherence envelope as (Frank,
2006)
?
Vis the
Iis the
EcoherenceðRÞ ¼ exp
?p2a2Csk2jRj3þ jDfðRÞjjRj
?2
??
;
ð11Þ
where a is the semi-angle of aperture.
Finally, assuming the mechanical displacement perpen-
dicular to the focal plane DF and the displacement in the
focal plane (drift) DR, the envelope due to sample shift is
modeled as
EdriftðRÞ ¼ J0ðpDFkjRj2ÞsincðjRjDRÞ:
The envelope model E can be well approximated by a
Gaussian function (an envelope that has been used in other
works, Huang et al., 2003; Saad et al., 2001; Mallick et al.,
2005; Sander et al., 2003) if DF ¼ DR ¼DV
Csk2jRj3> Df (R)jRj. However, our model is not simplified
ð12Þ
V¼DI
I¼ 0 and
C.O.S. Sorzano et al. / Journal of Structural Biology 160 (2007) 249–262
251

Page 4

in this way and keeps all envelope terms modeling the
microscope physics.
Summarizing, the parameters required to fully specify
the CTF in our model are
?
We assume that V and Csare fixed for a given microscope
and known by the user. The rest of the parameters, 11 in
total, will be searched by our algorithm.
K;V ;Cs;DfM;Dfm;h;Q0;Ca;DV
V
;DI
I;a;DF;DR
?
:
ð13Þ
2.1.2. Background PSD model
We assume the noise after the CTF to be colored by the
film/scanner or CCD frequency response. Physically mod-
eling the corresponding PSD as the output of a linear sys-
tem, although possible, is out of the scope of this work.
Instead, we will model the background PSD as a linear
combination of exponential functions. The main behavior
of the background PSD depends with e?
term has to be corrected at low resolution with a couple of
Gaussians, a positive and a negative one. The formal model
for the background noise PSD used in this work is
?
? ?jGðRÞjðjRj ? jCðRÞjÞ2
? ð?jgðRÞjðjRj ? jcðRÞjÞ2Þ;
where
ffiffiffiffi
jRj
p
. However, this
PSDnaðRÞ ¼ b þ Ksexp ?jsðRÞj
?
ffiffiffiffiffiffiffi
jRj
p
?
þ KGexp
?
? Kgexp
ð14Þ
sðRÞ ¼ ðsMcosð\R ? hsÞ;smsinð\R ? hsÞÞ;
CðRÞ ¼ ðCMcosð\R ? hGÞ;Cmsinð\R ? hGÞÞ;
GðRÞ ¼ ðGMcosð\R ? hGÞ;Gmsinð\R ? hGÞÞ;
cðRÞ ¼ ðcMcosð\R ? hgÞ;cmsinð\R ? hgÞÞ;
gðRÞ ¼ ðgMcosð\R ? hgÞ;gmsinð\R ? hgÞÞ:
The first term provides a constant baseline; the second term
is a decaying exponential representating the background
PSD behavior; the third and fourth terms of the model
are intended to provide more flexibility in the PSD model-
ing process. The second term will be referred to as the
exponential term because of its dependence on the square
root of the frequency. The third and fourth terms will be
referred to as the positive and negative Gaussians, respec-
tively. To simplify the writing of equations we will use
the following notation for the background PSD
ð15Þ
ffip-
PSDnaðRÞ ¼ b þ PSDffi
All terms are assumed to be elliptically symmetric
accounting for a possible anisotropy of the noise after
convolution with the CTF. Parametrical models of the cor-
responding ellipses are given in Eq. (15) and their interpre-
tation is analogous to that of Eq. (6).
Summarizing, there are 17 parameters defining the back-
ground PSD, namely
pðRÞ þ PSDGðRÞ ? PSDgðRÞ
¼ PSDlowerðRÞ ? PSDgðRÞ:
ð16Þ
ðb;Ks;sM;sm;hs;KG;GM;Gm;CM;Cm;hG;Kg;gM;gm;cM;cm;hgÞ:
ð17Þ
2.2. CTF determination procedure
The experimental PSD must be first estimated from the
micrograph image in real space. Periodogram averaging
(Avila-Sakar et al., 1994; Frank, 2006; Zhu et al., 1997)
is a very popular algorithm to do this due to its simplicity
and computational speed. The main drawback of this esti-
mator is that it is very noisy having a standard deviation of
the same size as the quantity to be estimated (Broersen,
2000). Alternative estimators are based on parametric
models (AR, ARMA) and they have been successfully
applied in electron microscopy (Ferna ´ndez et al., 1997; Vel-
a ´zquez-Muriel et al., 2003). The main disadvantages of
these estimators are that they require a large computational
time and that they cannot reproduce regions where the
PSD is strictly zero.
The CTF determination procedure described below can
work with any PSD estimator. In the experiments pre-
sented in this work, we only used periodogram averaging
since the CTF determination algorithm introduced here is
robust enough to tolerate the high amount of noise typical
of periodogram averaging. However, in difficult specific
cases, parametric models may be used instead to provide
much cleaner and more accurate estimations of experimen-
tal PSDs.
Our CTF determination algorithm searches automati-
cally for the values of the 28 unknown parameters (11 for
the CTF and 17 for the background noise) determining
the theoretical PSD that best fits the experimental PSD.
This fit is evaluated as a fit of two 2D images. The deter-
mined CTF is therefore also a 2D image. That is, CTF
astigmatism as well as background noise anisotropy are
explicitly taken into account.
Attempting to look simultaneously for all 28 parameters
without any guidance is a formidable task for any optimi-
zation algorithm. Hence, the optimization problem is
divided into smaller subproblems that can be easily solved
either because there is an analytical solution or because
they involve the adjustment of a few parameters with
respect to the values of parameters found in a previous step
of the algorithm. Therefore, the parameters of the CTF as
well as those of the background PSD are determined in the
following four subsequent steps:
• Step 1: Determination of the theoretical PSD lower
bound.
• Step 2: Determination of the theoretical PSD upper
bound.
• Step 3: Defocus determination.
• Step 4: Final model adjustment.
As will be further explained, the fitting is always done
by minimizing a given measure of error between a 2D
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Page 5

experimental PSD and a 2D theoretical PSD computed for
the values of parameters known at that stage (parameters
are progressively estimated; thus, in the first substeps only
a few of them are known). In our algorithm, the dissimilar-
ity between two 2D images is usually computed based on
the l1-norm of the error vector. This is so since computing
the absolute value of a given quantity is much faster than
performing a multiplication (related to the more popular
l2-norm). The employed optimizer is the Powell’s conjugate
gradient algorithm (Press et al., 1992) which is known for a
fast local convergence without the need of explicit deriva-
tives of the goal function. However, there are situations in
which the problem structure is simple enough so that a
solution of the weighted l2-norm optimization problem
can be analytically computed. In these cases, we first com-
pute the analytical solution of the corresponding weighted
l2-norm optimization problem, and then input it to Pow-
ell’s algorithm as the initial solution of the l1-norm related
problem.
Except in Step 4c (see below), all optimizations are per-
formed by considering a coarse regular grid of frequencies.
That is, we do not compare all possible frequencies since
this would result in a much slower algorithm. In the last
optimization step, the coarse regular grid is made finer
and finer until all available frequencies are used for the
fitting.
For the purpose of illustration of the optimization
procedure, we performed the estimation of the CTF
for an experimental cryo-EM image recorded on a
JEOL JEM 2100F electron microscope equipped with
a field emission gun. The image was recorded under
low dose conditions with a magnification of 50,000·,
an acceleration voltage of 200 kV, and a spherical
aberration of 0.5 mm, and it was digitized with a pixel
size of 1.59 A˚· 1.59 A˚. In the succeeding sections, we
show the computed theoretical PSD for this image
after each performed optimization step. This experi-
mental image has been chosen because it is almost
non-astigmatic. The radial average can therefore be
shown for clarity of illustration. Indeed, although all
the optimization is fully performed in 2D, explicitly
considering CTF astigmatism and noise anisotropy,
we show mainly overlapped radial averages of the
experimental and theoretical PSDs. 2D images or par-
ticular 1D radial profiles are shown only at the stages
were they give more visual information than the radial
average.
2.2.1. Step 1: determination of the theoretical PSD lower
bound
The estimation of the theoretical PSD lower bound is
performed in four substeps:
• Steps 1a and 1b. Adjustment of initial values of the
exponential parameters and of the baseline.
• Steps 1c and 1d. Adjustment of initial values of the posi-
tive Gaussian parameters.
ffip-
2.2.2. Steps 1a and 1b: adjustment of initial values of the
ffi
b, Ks, sM, sm, and hs. First, an initial guess with sM= smand
hs= 0 is found so that the weighted l2-norm of the error
between the experimental PSD and the theoretical PSD is
minimized. Second, this solution is refined now letting
sM„ smand hs„ 0 so that it optimizes the error in the l1
sense. Finally, the theoretical PSD is further refined by
optimization of a penalized l1-error measure. This penaliza-
tion moves down the estimated PSDtheoretical so that it
becomes a lower bound of the experimental PSD. The
radial average of the experimental PSD and the one of
the theoretical PSD after Steps 1a and 1b, for the experi-
mental image described in Section 2.2, can be seen in Fig. 1.
p-exponential parameters and of the baseline
In this step, we compute rough estimates of parameters
1. Step 1a. Parameters Ks, sM, sm, and hsare sought with
the constraints sM= smand hs= 0 so that the l2norm
of the error between the experimental PSD and the the-
oretical PSD is minimized. This is achieved by the
weighted least-squares solution of the equation system
logðPSDexperimentalðRÞÞ ¼ logðKsÞ ? sM
where we have one equation for each R in a regular grid
X ? R2, the region in the frequency space where the two
PSDs (experimental and theoretical) are being com-
pared. In practice, X is an annular region defined by
the inner and outer radii specified by the user. It is
important to judiciously select this region since at very
low frequencies the approximation PSDideal(R) = 0 is
not valid. The weight of each one of these equations is
ffiffiffiffiffiffiffi
jRj
p
;
ð18Þ
wðRÞ ¼ 1 þ max
R02XjR0j ? jRj;
ð19Þ
That is, the goal function to be minimized is
X
L¼
Ri2X
wðRiÞ log PSDexperimentalðRiÞ
???logðKsÞþsM
ffiffiffiffiffiffiffiffi
jRij
p
??2
ð20Þ
The weighted least-squares solution of such an equation
system can be found in Lawson and Hanson (1995).
2. Step 1b. The first guess of the
obtained in the previous step is refined and pushed down
in this step. To this goal, the two constraints sM= smand
hs= 0areremoved,theparameterb(whoseinitialvalueis
0) is also estimated, and the error is penalized at frequen-
cies where the theoretical PSD is above the experimental
PSD. Thus, the functional to be minimized in this step is
X
ð1 þ WIPSDexperimental<PSDtheoreticalðRiÞÞ;
where IA(x) denotes the indicator function (this function
is 1 if x belongs to the set A, and is 0 otherwise), W is the
penalization weight and follows the sequence 0, 2, 4, 8,
16, and 32. For each W, Powell’s conjugate gradient
algorithm is used to minimize the penalized functional
ffip-exponential term
L ¼
Ri2X
PSDexperimentalðRiÞ ? b þ PSDffi
pðRiÞ
??
??????
ð21Þ
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253

Page 6

starting from the solution obtained for the previous va-
lue of W.
2.2.3. Steps 1c and 1d: adjustment of initial values of the
positive Gaussian parameters
In this step, we compute rough estimates of parameters
KG, CM, Cm, GM, Gm, and hGand refine these newly intro-
duced parameters together with the ones of the baseline
and the
Steps 1a and 1b, the positive Gaussian term of the back-
ffip-exponential term (b, Ks, sM, sm, and hs). As in
ground PSD is computed in two steps: first, a constrained
l2-error optimization is performed on a low frequency
region of the experimental PSD previously radially symme-
trized. This produces a radially symmetric Gaussian that
helps the
quency spectrum of the experimental PSD; second, the
condition of radial symmetry of the Gaussian is removed,
and a penalized l1-error optimization is performed. The
radial average of the experimental PSD and the one of
the theoretical PSD after these two steps, for the experi-
mental image described in Section 2.2, can be seen in Fig. 1.
ffip-exponential term to reproduce the low fre-
1
2
3
4
5
6
7
8
9
10
11
0.04 0.06 0.08 0.1 0.12 0.14
Frequency (1/A)
0.16 0.18 0.2 0.22 0.24 0.26
PSD Amplitude
Experimental PSD
Step 1a
Step 1b
1
2
3
4
5
6
7
8
9
0.04 0.06 0.08 0.1 0.12 0.14
Frequency (1/A)
0.16 0.18 0.2 0.22 0.24 0.26
PSD Amplitude
Experimental PSD
Step 1c
Step 1d (Lower bound)
Fig. 1. Radial average of the experimental PSD and the theoretical PSD lower bound after Steps 1a, 1b (top) and 1c, 1d (bottom). After Step 1a, the lower
bound goes amid the experimental PSD. In Step 1b, the lower bound is pushed down below the experimental PSD. In Step 1c, we correct the lower bound
found in Step 1b by first estimating a radial Gaussian centered at low frequencies. Step 1d defines the theoretical PSD lower bound.
254
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Page 7

1. Step 1c. The experimental PSD is radially symmetrized
as well as the penalized
found in Step 1. Starting from the lowest frequency, we
look for the first frequency at which the two curves are
closer. This frequency called Rmindetermines the end
of the low frequency region. Within this region, we look
for the frequency at which the two curves are more sep-
arated, Rmax. Parameters CM and Cm are set to
CM= Cm= Rmax. hGis set to 0, and GMis constrained
to be equal to Gm. Therefore, only parameters KGand
GMare left. They are chosen so that they minimize the
weighted l2-norm of the error between the experimental
PSD and the theoretical PSD. This is achieved by the
weighted least-squares solution of the equation system,
?
¼ logKG? GMðjRj ? CMÞ2:
We evaluate this equation in the same spectrial grid
points as the one of the
the same weight as in Eq. (19).
2. Step 1d. this curve is also pushed down so that it really is
a background support. The pushing down is done by
minimization of
X
þ WIPSDexperimental<PSDtheoreticalðRiÞÞ
with respect to all parameters estimated so far. The
weight W follows the sequence 0, 2, 4, 8, 16, and 32.
The output of this Step 1d is called the theoretical
PSD lower bound. It is a fully 2D lower bound although
for clarity purposes we only represent its radial average
in Fig. 1.
ffip-exponential term and baseline
log PSDexperimentalðRÞ ? b þ PSDffi
pðRÞ
???
ð22Þ
ffi p-exponential, and we give
L ¼
Ri2X
PSDexperimentalðRiÞ ? PSDlowerðRiÞ
????ð1
ð23Þ
2.2.4. Step 2: determination of the theoretical PSD upper
bound
In this step, we search for the following parameters of
the envelope: K, Ca,
unknown parameters of the envelope (Df and a) are cou-
pled in the term Ecoherence. They will be determined in Step
3 when searching for defocus parameters (DfM,Dfm,h) on
which this term depends. Therefore, we assume that
Df(R) = (0,0) and a = 0 (i.e., Ecoherence(R) = 1) at this
point. We set PSDnaðRÞ to its lower bound found in Step
1d and we only look for the envelope parameters. As in
Step 1d, the search for the upper bound of the PSD is per-
formed by minimizing a penalized goal function. The goal
function used in this step is
X
ð1 þ WIPSDexperimental>PSDtheoreticalðRiÞÞ:
The initial values of the unknown parameters in this opti-
mization step are
DV
V,
DI
I, DF, and DR. The other two
L ¼
Ri2X
PSDexperimentalðRiÞ ? PSDlowerðRiÞ þ K2E2ðRiÞ
??
????
ð24Þ
K;Ca;DV
V
;DI
I;DF;DR
?
where Cð0Þ
can be supplied by the user (by default, its value is 0). The
penalization W follows the sequence 0, 2, 4, 8, 16, and 32.
TheoutputofthisstepisreferredtoasthetheoreticalPSD
upper bound. The radial average of the experimental PSD
and those of the theoretical PSD lower and upper bounds,
for the experimental image described in Section 2.2, can be
seen in Fig. 2. The lower and upper bounds will be used in
the next two steps to weight the PSD fitting errors.
?
¼ 1;Cð0Þ
a;0;0;0;0
??;
ð25Þ
ais an initial chromatic aberration coefficient that
2.2.5. Step 3: defocus determination
In this step, we determine the defocus parameters
(DfM,Dfm,h) and the aperture semi-angle a. First, we com-
pute a rough estimate of the defocus values making use of
the estimated lower and upper bounds. Then, we refine all
parameters determined until that point.
One of the problems encountered when fitting a PSD
model is that the fitting errors committed at high frequen-
cies are of little importance because of the PSD amplitude
damping (the PSD amplitude is very small at these high fre-
quencies). Here is where the lower and upper bounds of the
PSD come into play to help us define an error measure that
is less dependant on the frequency. Given the lower bound
PSDlower(R) and the upper bound PSDlower(R) + K2E2(R),
each PSD used in this step is normalized as follows:
g
PSDðRÞ ¼
PSDðRÞ ? PSDlowerðRÞ
PSDlowerðRÞ þ K2E2ðRÞ ? PSDlowerðRÞ
¼PSDðRÞ ? PSDlowerðRÞ
K2E2ðRÞ
This normalization guarantees that any PSD within the
lower and upper bounds will be mapped between 0 and
1, and therefore all frequencies will similarly contribute
to the PSD fitting error as long as the lower and upper
bounds are accurately computed.
The goal function to be minimized at this stage is
X
? q
:
ð26Þ
L ¼
1
jXj
Ri2X
PSDenhancedðRiÞ;HidealðRiÞEðRiÞ
g
PSDexperimentalðRiÞ ?g
g
where jXj is the number of spectral grid points in the set X,
and q(x,y) is the correlation coefficient between signals x
and y defined as
PSDdefocusðRiÞ
??? ???
??
;
ð27Þ
qðx;yÞ ¼
Efxyg
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Efx2gEfy2g
p
;
ð28Þ
where E{Æ} is the expectation operator. PSDenhancedis a fil-
tered version of the experimental PSD computed as in Jon-
ic et al. (2007). The experimental PSD and its enhanced
version, both used in Step 3 for the image described in Sec-
tion 2.2, can be seen in Fig. 3. PSDdefocusis computed as
follows:
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255

Page 8

PSDdefocusðRÞ ¼ PSDlowerðRÞ þ K2jHðRÞj2:
ð29Þ
Again, this step is divided in the following two substeps:
• Step 3a. A first estimate of the defocus values is obtained
byexhaustivesearchof
(DfM,Dfm,h) on a regular grid. Each explored point is
used as the initial solution of Powell’s conjugate gradi-
ent optimizer. This algorithm computes values of the
three parameters by minimizing the goal function shown
in Eq. (27) and is quite ‘‘fast’’ despite the initial exhaus-
tive search. The best fitting parameters computed at this
step are used as the initial solution for Step 3b.
• Step 3b. We refine all parameters found so far (23
parameters) by minimizing the goal function in Eq.
(27). The only parameters that have not been found
yet are those of the negative background Gaussian
(PSDg). They will be determined in Step 4.
thethree parameters
The radial average of the experimental PSD, the one of
the theoretical PSD after Steps 3a and 3b, as well as those
of the theoretical PSD lower and upper bounds, for the
experimental image described in Section 2.2, can be seen
in Fig. 4.
2.2.6. Step 4: final model adjustment
In this step, we estimate first the parameters of PSDg
(Step 4a). Then, we refine all parameters of the model using
a coarse grid (the same grid as in all previous steps) (Step
4b). Finally, we refine all parameters using a fine evaluation
grid (Step 4c). The output of this step is the output of the
CTF determination procedure.
1
0.04
2
3
4
5
6
7
8
9
10
11
0.06 0.08 0.1 0.12 0.14
Frequency (1/A)
0.16 0.18 0.2 0.22 0.24 0.26
PSD Amplitude
Experimental PSD
Step 2 (Upper bound)
Step 1d (Lower bound)
Fig. 2. Radial average of the experimental PSD and the theoretical PSD lower (Step 1d) and upper (Step 2) bounds.
Fig. 3. Experimental PSD (top) and enhanced experimental PSD
(bottom).
256
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Page 9

1. Step 4a. We compute a first estimate of PSDgsimilarly
as in Step 1c. We assume the term to be circularly sym-
metric and therefore cM= cm, gM= gm, and hg= 0.
Thus, this first guess can be found as a weighted least-
squares solution of the equation system
logðPSDlowerðRÞ þ K2jHðRÞj2? PSDexperimentalðRÞÞ
¼ logKg? gMðjRj ? cMÞ2:
Thereisoneequationforeach2Dfrequencyforwhichthe
theoretical PSD estimated in Step 3b is larger than the
experimental PSD. We do this because at these frequen-
cies some negative term is needed in order to compensate
the difference between the experimental and the theoreti-
cal PSDs. Each equation has a weight given by Eq. (19).
2. Step 4b. All model parameters are refined on a coarse fre-
quencygrid.Bydefault,thecoarsegridisdefinedbytaking
1frequencysampleoutof4consecutiveonesineachdirec-
tion.ThegoalfunctiontobeoptimizedisshowninEq.(27).
3. Step 4c. The frequency grid is made finer and finer until
all available frequencies are used. Thus, the grid is made
finer by dividing by 2 the grid spacing until this value is
1. For each grid spacing, the model parameters are
refined again using the same goal function and optimiza-
tion algorithm as in Step 4b.
ð30Þ
TheradialaverageoftheexperimentalPSDandtheoneof
the theoretical PSD after Steps 4a, 4b, and 4c for the exper-
imental image described in Section 2.2, can be seen in Fig. 5.
3. Results
Fig. 5 shows that the proposed PSD model correctly
reproduces the experimental PSD when using the values
of parameters found by the estimation algorithm. How-
ever, it is difficult to check the accuracy of the estimated
parameters in the case of real micrographs since the true
values of parameters are unknown. Moreover, our PSD
model is much more complex than the ones used in other
programs. Therefore, our estimates cannot be directly com-
pared with the estimates found by other programs. For this
reason, we used two different quality assesment setups. In
the first setup, simulated micrographs were used to assess
the accuracy of the CTF estimation algorithm by measur-
ing the difference between the ground truth PSD and the
estimated PSD. In the second setup, experimentally
acquired micrographs were used. In this case, we compared
the defoci values of our algorithm to those determined by
CTFtilt (Mindell and Grigorieff, 2003), one of the most
widespread programs used to estimate the CTF. In the sec-
ond setup, we estimated the CTF on two different sets of
micrographs: one set from the Large T Antigen (LTag,
Go ´mez-Lorenzo et al., 2003; Valle et al., 2006) and another
one from the Glutamate synthase (Glts, Cottevieille et al.,
in preparation). Thetwo
characteristics.
sets havedifferentPSD
3.1. Simulated micrographs
Ten micrographs of size 4096 · 4096 pixels with a pixel
size of 5.6 A˚per pixel were simulated. No particle was pres-
ent in the simulated micrographs. The CTF parameters
used for the simulation were chosen to be the same as
one of the micrographs of the LTag micrographs (see
below). In this simulated experiment we exactly know the
underlying PSD and compared the estimated PSD to the
ground truth PSD. The average error of the estimated
PSD is 1.2 ± 0.3% of the true PSD. The error in the smallest
1
0.04
2
3
4
5
6
7
8
9
10
11
0.06 0.08 0.1 0.12 0.14
Frequency (1/A)
0.16 0.18 0.2 0.22 0.24 0.26
PSD Amplitude
Experimental PSD
Step 2 (Upper bound)
Step 1d (Lower bound)
Step 3a
Step 3b
Fig. 4. Radial average of the experimental PSD, the theoretical PSD lower and upper bounds, the theoretical PSD fits after Steps 3a and 3b.
C.O.S. Sorzano et al. / Journal of Structural Biology 160 (2007) 249–262
257

Page 10

defocus value was 1.0 ± 0.1%, the error of the largest defo-
cus value was 1.3 ± 0.2%, and the error in the angle of the
defocus ellipse was of 2.3 ± 1.8?.
3.2. Experimental micrographs
The two sets of experimental images (LTag and GltS)
correspond to samples embedded in ice with no carbon.
The LTag images had a sampling rate of 5.6 A˚per pixel
so that the diffraction rings occupy a wide area of the
PSD image. The GltS images were digitized with a pixel
size of 1.59 A˚. Thus, the area occupied by the same number
of diffraction rings is smaller in the GltS case than in the
LTag case. Fig. 6 shows the (enhanced) experimental
PSD and the fitted theoretical PSD for two micrographs,
one from each of these groups.
For the search of CTF parameters, we provided our
algorithm with the experimental PSD estimated by period-
ogram averaging, the microscope voltage, the spherical
aberration, the micrograph sampling rate, and a wide defo-
cus range for searching (?103A˚to ?105A˚). From both
sets of images LTag and GltS, we first removed drifted
images as well as images with no visible diffraction rings
using sorting of enhanced experimental PSD images as in
Jonic et al. (2007). Note that in this step we removed all
the images with a strong drift, but that does not mean that
1
0.04
2
3
4
5
6
7
8
9
0.06 0.08 0.1 0.12 0.14
Frequency (1/A)
0.16 0.18 0.2 0.22 0.24 0.26
PSD Amplitude
Experimental PSD
Step 4a
Step 4b
1
0.04
2
3
4
5
6
7
8
9
0.06 0.08 0.1 0.12 0.14
Frequency (1/A)
0.16 0.18 0.2 0.22 0.24 0.26
PSD Amplitude
Experimental PSD
Step 4c (Final result)
Fig. 5. Radial average of the experimental PSD and the theoretical PSD after Steps 4a, 4b (top) and 4c (bottom).
258
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Page 11

the drift parameters should not be estimated in the remain-
ing images, since some drift may still be present although it
was not so strong as to not consider the image.
The LTag data set contained 92 micrographs of good
quality (after the removal of drifted images and images
with no visible diffraction rings) taken in a Phillips Tecnai
F20 microscope at a magnification of 50,000. The acceler-
ation voltage was 200 kV, and the spherical aberration was
2.26 mm. The defocus range was between ?3.5 lm and
?7.5 lm. Neither CTFtilt nor the algorithm presented in
this paper (from now on, we will refer to it as Xmipp since
it is implemented in this software package, Sorzano et al.,
2004b) failed to find the CTF parameters. In this context,
failure means that the algorithm found defocus values that
are clearly not the true ones even if the differences are small
one (this can be easily checked by visual inspection). For
each micrograph, we considered the ellipse defined by its
minor and major defocus values. We compared the mini-
mum defocus provided by CTFtilt and Xmipp. In average,
their values in A˚differed by 2.7 ± 1.4%. The corresponding
maximum defoci differed by 3.5 ± 1.6%. In both cases, the
defocus estimated by Xmipp was slightly smaller than the
one estimated by CTFtilt (i.e., in the case of minimum
defocus, the value estimated by Xmipp was on average a
2.7% smaller than the one of CTFtilt). These differences
in defocus values turned into a small difference in the posi-
tion of the first-zero, on average, the first-zero positions
found by the two algorithms differed only by 1.6 ± 0.6%.
CTFtilt estimated an average difference between the mini-
mum and maximum defocus values of 1189 ± 734 A˚. The
same average difference in the case of Xmipp was
770 ± 610 A˚.
The GltS data set contained 125 cryo-electron micro-
graphs after the removal of drifted images and images with
no visible diffraction rings. They were taken in a JEOL
JEM 2100 F microscope at a magnification of 50,000, an
acceleration voltage of 200 kV and spherical aberration
of 0.5 mm. The defocus range was between ?1.7 lm and
?3.2 lm. CTFtilt could not find the defoci parameters in
1 occasion, while Xmipp in 3. The parameters returned
by the programs in these four occasions were not totally
off the true (unknown) parameters, but we could establish
visually that the parameters returned were not close
enough to the true ones. The hypothesis that both pro-
grams have the same failure rate cannot be rejected (confi-
dence of 95%). It should be reminded that Xmipp always
provides as part of the solution an estimation of the spec-
tral envelope that CTFtilt does not compute. This piece of
information is particularly important for any process
related to CTF amplitude correction, as will be further
commented in Section 4. The difference between the mini-
mum defocus estimated by CTFtilt and Xmipp was
?0.3 ± 1.5% (i.e., Xmipp minimum defocus was slightly
larger than the one of CTFtilt), and the corresponding dif-
ference for the maximum defocus was 0.4 ± 1.6% (i.e.,
Xmipp maximum defocus was slightly smaller than the
one of CTFtilt). The average difference in the first-zero
position between CTFtilt and Xmipp was 0.5 ± 0.4%.
The average difference between the minor and major defoci
for CTFtilt was 229 ± 163 A˚, while it was 404 ± 399 A˚for
Xmipp.
As has already been shown, our model can estimate
astigmatic CTF but also anisotropic background noise.
The following example shows that experimental PSDs cer-
tainly have anisotropic backgrounds even if the CTF defo-
cus parameters cannot be really considered astigmatic. For
the image described in Section 2.2, which is one of the
images of the GltS data set, the two defocus parameters
as estimated by Xmipp are ?22,877 A˚and ?22,462 A˚
(the results of the estimation are shown in Figs. 1–5). With
a difference of only 415 A˚, the corresponding CTF can be
considered to be non-astigmatic. However, radial plots
along two perpendicular axes show that the two back-
grounds are different (see Fig. 7), that is, the background
noise level effectively depends on the direction. If the
CTF parameters are fitted imposing a radially symmetric
background, then the estimated defoci are ?21,585 A˚and
?23,582 A˚, i.e., the estimated astigmatism appears to be
1997 A˚. This experiment shows that assuming radially sym-
metric backgrounds may result in inaccurate estimates of
the defoci, in this case with an error of about 3.5%.
In order to know whether the anisotropy comes from
the microscope or the digitizer we digitized twice the same
micrograph with a rotation of 90? between the two scanned
images. Fig. 8 shows the final fit performed on the x- and y-
axes in the micrograph at 0?, and the plot of the final fit on
Fig. 6. Top row: CTF model for a LTag micrograph. Bottom row: CTF
model for a GltS micrograph. The left image of each row represent the left
half-plane of the enhanced PSD, and the right half-plane of the CTF
model. The right image makes a similar representation over quadrants.
Representing in quadrants have an advantage when trying to visually
validate the estimation on astigmatic images.
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259

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the y-axis in the micrograph digitized at 90? (the x-axis in
the micrograph at 90? corresponds to the y-axis in the
micrograph at 0?). It can be seen that at low frequency
the fits along the two equivalent axes are similar, while at
high frequency the fit along the y-axis in the micrograph
at 90? is similar to the fit along the y-axis of the micrograph
at 0?. From this fact, it can be concluded that in the exper-
iment carried out anisotropy was due to both devices (elec-
tron microscope and digitizer).
4. Discussion
An accurate estimation of the CTF function is crucial
for its subsequent correction. If the correction used is only
1
0.04
2
3
4
5
6
7
8
9
10
11
0.06 0.08 0.1 0.12 0.14
Frequency (1/A)
0.16 0.18 0.2 0.22 0.24 0.26
PSD Amplitude
Experimental PSD along X axis
Experimental PSD along Y axis
Final fit along X axis
Final fit along Y axis
Fig. 7. Radial plots along two perpendicular axes of the experimental PSDs and the corresponding model fits. It can be seen that the experimental PSDs
are much noisier than the corresponding radial average for the same micrograph (see Fig. 5). Note that the background noise level differs between the two
directions, this effect is specially noticeable at high frequency.
1
1.5
2
2.5
3
3.5
4
0 0.05 0.1 0.15
Frequency (1/A)
0.2 0.25 0.3 0.35
PSD Amplitude
Fit along y-axis in the micrograph at 0 degrees
Fit along x-axis in the micrograph at 0 degrees
Fit along x-axis in the micrograph at 90 degrees
Fig. 8. Plot of the final fit on a micrograph along a given direction, a perpendicular direction in this micrograph, and a direction equivalent to the first plot
in the 90? rotated micrograph.
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Page 13

phase-flipping, it suffices to know the values of the defocus
parameters. However, if the amplitude decay is also to be
corrected, it is very important to accurately determine the
CTF envelope. The envelope model used in this paper is
justified by the physics of the electron microscope and sim-
plifies to the more popular Gaussian decay used in other
works (Huang et al., 2003; Saad et al., 2001; Mallick
et al., 2005; Sander et al., 2003). This is one of distinctive
features of our algorithm. An additional important feature
of our PSD model is the background noise model. It allows
for anisotropic noise and makes use of a novel negative
Gaussian term which allows a more accurate modeling in
the low frequency part of the spectrum (see Figs. 5 and
7). Fig. 7 shows that the use of anisotropic background
noise is particularly important for a correct estimation of
the amplitude parameters. Anisotropic background noise
also suggests that the use of radial averaging of the esti-
mated experimental PSD may mask differences in ampli-
tude decaysoverdifferent
estimates based on radial averages may be biased. CTF
estimates computed on elliptical averages are not safer
since the elliptical averaging is usually computed using
the CTF astigmatism that, as shown in the example of
Fig. 7, may be less important than the anisotropic noise.
To correctly find all parameters of our model, we fol-
lowed a careful optimization plan that progressively finds
the unknown parameters. Background parameters are
determined first, then envelope parameters, and finally
defocus parameters. The successive optimization steps
aim at simplifying the global optimization of CTF and
noise parameters and avoiding to get trapped in local min-
ima. However, this is not always possible and the algorithm
did not automatically find the CTF parameters in 3 out of
217 micrographs (about 1.4% of the micrographs). It is
worth to stress that in 98.6% of the cases, we are providing
not only estimates of the zeros of the CTF, but also of the
spectral envelope. This additional information is key in
recovering the amplitude of the object under study as part
of any restoration process, especially at high frequencies.
Furthermore, the error rate can be smaller if the user pro-
vides a narrower defocus range (the one used in our exper-
iment was from ?103A˚to ?105A˚). Robustness of the
optimization procedure is greatly improved by the use of
enhanced PSDs without which the failure rate increases
up to about 30%.
It must be taken into account that our algorithm opti-
mizes the fitting of a theoretical model of the PSD to the
experimentally observed PSD. Therefore, all the conclu-
sions can be done at the PSD level (for instance, which is
the amplitude of the CTF at a given frequency) but not
at the parameter level, i.e., individual parameters (chro-
matic aberration, energy spread, etc.) cannot be taken as
the parameters faithfully representing the physical configu-
ration of the microscope. All that can be said is that the
combination of parameters returned by the algorithm
faithfully explains the experimental PSD. In fact, some
cross-talking between parameters is expected: some enve-
directions.Thus, CTF
lope decay actually explained at the physical level by a
given parameter might be shifted by the algorithm into
some other envelope parameter. Although also possible,
cross-talking between envelope and background parame-
ters should not be so important.
Thanks to the use of the enhanced PSD, our algorithm is
robust enough to work with estimates of the experimental
PSD computed by periodogram averaging. Thus, the algo-
rithm described here is a much advanced version of our
previous algorithm on CTF estimation (Vela ´zquez-Muriel
et al., 2003) in terms of robustness, accuracy and speed.
Our current implementation of the algorithm takes only
5 min to estimate the CTF parameters for a PSD estimate
of size 512 · 512 pixels (and the computation of the PSD
estimate by periodogram averaging takes only 30 s). These
times have been measured on a PC with an Intel Pentium
M processor at 1 GHz and 512 Mb of RAM.
Our algorithm fits a PSD theoretical model to an esti-
mate of the experimental PSD. In a more general case of
tilted micrographs, our algorithm can be used to divide
the micrograph in small areas, to estimate the experimental
PSD for each area, and to fit the theoretical model to the
estimate of each local experimental PSD. Moreover, if very
few particles are to be used from a micrograph, the exper-
imental PSD can be actually estimated for the area that
takes into account only the selected particles. In this way,
the theoretical PSD model will be as customized as possible
to the selected particles.
5. Conclusions
In this paper we have introduced a new algorithm for
the determination of the CTF parameters including an
accurate estimate of the spectral envelope. The algorithm
is fast, accurate, and robust. These three characteristics
make it ideal for an automatic image processing workflow
in which micrographs are automatically digitized, classified
according to the quality of their Thon rings, and for those
having good frequency response, the CTF parameters
(including the parameters related to the amplitude decay)
are automatically estimated. Hence, the proposed algo-
rithm is a new step towards high-throughput electron
microscopy. The algorithm is publicly available as a part
of the Xmipp software package (http://xmipp.cnb.csic.es).
Acknowledgments
The authors are grateful to Dr. Martn Benito who sug-
gested a new way of showing in the same image the theoret-
ical PSD and the experimental PSD. We acknowledge
partial support from the ‘‘Comunidad Autnoma de Ma-
drid’’ through Grant S-GEN-0166-2006, the ‘‘Ministerio
de Educacin y Ciencia’’ of Spain through
CSD2006-00023, BFU2004-00217, the Spanish FIS Grants
PI040683and2004-2-OE-192,
through Grants UE-031688, FP6-502828 and the NIH
through Grant HL070472.
Grants
theEuropeanUnion
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261

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References
Avila-Sakar, A.J., Guan, T.L., Arad, T., Schmid, M.F., Loke, T.W.,
Yonath, A., Piefke, J., Franceschi, F., Chiu, W., 1994. Electron
cryomicroscopy of bacillus stearothermophilus 50S ribosomal subunits
crystallized on phospholipid monolayers. Journal of Molecular Biol-
ogy 239, 689–697.
Broersen, P.M.T., 2000. Facts and fiction in spectral analysis. IEEE
Transactions on Instrumentation and Measurement 49, 766–772.
Cottevieille, M., Jonic, S., Larquet, E., Petoukhov, M.V., Dossena, L.,
Svergun, D.I., Vanoni, M.A., Boisset, N., in preparation. Structural
study of bacterial glutamate synthase complex by 3D cryo-electron
microscopy, homology modelling, and small-angle X-ray scattering.
De Rosier, D.J., 2000. Correction of high-resolution data for curvature of
the Ewald sphere. Ultramicroscopy 81, 83–98.
Ferna ´ndez, J.J., Sanjurjo, J., Carazo, J.M., 1997. A spectral estimation
approach to contrast transfer function detection in electron micros-
copy. Ultramicroscopy 68, 267–295.
Frank, J., Penczek, P., 1995. On the correction of the contrast transfer
function in biological electron microscopy. Optik 98, 125–129.
Frank, J., 2002. Single-particle imaging of macromolecules by cryo-
electron microscopy. Annual Review of Biophysics and Biomolecular
Structure 31, 303–319.
Frank, J., 2006. Three-Dimensional Electron Microscopy of Macromo-
lecular Assemblies: Visualization of Biological Molecules in their
Native State. Oxford University Press, USA.
Gao, H., Spahn, C., Grassucci, R., Frank, J., 2002. An assay for local
quality in cryo-electro micrographs of single particles. Ultramicrosco-
py 93, 169–178.
Go ´mez-Lorenzo, M., Valle, M., Frank, J., Gruss, C., Sorzano, C.O.S.,
Chen, X.S., Donate, L.E., Carazo, J.M., 2003. Large T antigen on the
simian virus 40 origin of replication: a 3D snapshot prior to DNA
replication. EMBO Journal 22, 6205–6213.
Grigorieff, N., 1998. Three-dimensional structure of bovine NADH:
ubiquinone oxidoreductase (complex I) at 22 A˚in ice. Journal of
Molecular Biology 277, 1033–1046.
Henderson, R., 2004. Realizing the potential of electron cryo-microscopy.
Quarterly Review of Biophysics 37, 3–13.
Holmes, K.C., Angert, I.J.K.F., Jahn, W., Schroeder, R.R., 2003.
Electron cryo-microscopy shows how strong binding of myosin to
actin releases nucleotide. Nature 425, 423–427.
Huang, Z., Baldwin, P.R., Mullapudi, S., Penczek, P.A., 2003. Automated
determination of parameters describing power spectra of micrograph
imagesinelectronmicroscopy.JournalofStructuralBiology144,79–94.
Jonic, S., Sorzano, C.O.S., Cottevieille, M., Larquet, E., Boisset, N., 2007.
A novel method for improvement of visualization of power spectra for
sorting cryo-electron micrographs and their local areas. Journal of
Structural Biology 157, 156–167.
Lawson, C.L., Hanson, R., 1995. Solving least-squares problems. SIAM,
Philadephia, PA., USA.
Mallick, S.P., Carragher, B., Potter, C.S., Kriegman, D.J., 2005. ACE:
automated CTF estimation. Ultramicroscopy 104, 8–29.
Mindell, J.A., Grigorieff, N., 2003. Accurate determination of local
defocus and specimen tilt in electron microscopy. Journal of Structural
Biology 142, 334–347.
Penczek, P.A., Zhu, J., Schro ¨der, R., Frank, J., 1997. Three dimensional
reconstruction with contrast transfer compensation from defocus
series. Scanning Microscopy 11, 147–154.
Philippsen,A.,Engel,H.A.,Engel,A.,2007.Thecontrast-imagingfunction
for tilted specimens. Ultramicroscopy 107, 202–212.
Press, W., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992.
Numerical recipes in C, second ed. Cambridge University Press,
Cambridge.
Saad, A., Ludtke, S., Jakana, J., Rixon, F., Tsuruta, H., Chiu, W., 2001.
Fourier amplitude decay of electron cryomicroscopic images of single
particles and effects on structure determination. Journal of Structural
Biology 133, 32–42.
Sali, A., Glaeser, R., Earnest, T., Baumeister, W., 2003. From words to
literature in structural proteomics. Nature 422, 216–225.
Sander, B., Golas, M.M., Stark, H., 2003. Automatic CTF correction for
single particles based upon multivariate statistical analysis of individ-
ual power spectra. Journal of Structural Biology 142, 392–401.
Skoglund, U., Ofverstedt, L.G., Burnett, R., Bricogne, G., 1996. Maxi-
mum-entropy three-dimensional reconstruction with deconvolution of
the contrast transfer function: a test application with adenovirus.
Journal of Structural Biology 117, 173–188.
Sorzano, C.O.S., Marabini, R., Herman, G.T., Censor, Y., Carazo, J.M.,
2004a. Transfer function restoration in 3D electron microscopy via
iterative data refinement. Physics in Medicine and Biology 49, 509–
522.
Sorzano, C.O.S., Marabini, R., Vela ´zquez-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. Journal of Structural Biology 148, 194–204.
Stark, H., Orlova, E.V., Rinke-Appel, J., Junke, N., Mu ¨ller, F., Rodnina,
M., Wintermeyer, W., Brimacombe, R., van Heel, M., 1997. Arrange-
ment of tRNAs in pre- and posttranslocational ribosomes revealed by
electron cryomicroscopy. Cell 88, 19–29.
Toyoshima, C., Unwin, N.T.P., 1988. Contrast transfer for frozen-
hydrated specimens: determination from pairs of defocused images.
Ultramicroscopy 25, 279–292.
Toyoshima, C., Yonekura, K., Sasabe, H., 1993. Contrast transfer for
frozen-hydrated specimens II: amplitude contrast at very low frequen-
cies. Ultramicroscopy 48, 165–176.
Unwin, P., 1973. Phase contrast electron microscopy of biological
materials. Journal of Microscopy 98, 299–312.
Valle, M., Chen, X.S., Donate, L.E., Fanning, E., Carazo, J.M., 2006.
Structural basis for the cooperative assembly of large T antigen on the
origin of replication. Journal of Molecular Biology 357, 1295–1305.
van Heel, M., Gowen, B., Matadeen, R., 2000. Single-particle electron
cryo-microscopy: towards atomic resolution. Quarterly Review of
Biophysics 33, 307–369.
Vela ´zquez-Muriel, J.A., Sorzano, C.O.S., Ferna ´ndez, J.J., Carazo, J.M.,
2003. A method for estimating the CTF in electron microscopy based
on ARMA models and parameter adjusting. Ultramicroscopy 96, 17–
35.
Wade, R., 1992. A brief look at imaging and contrast transfer.
Ultramicroscopy 46, 145–156.
Welch, P.D., 1967. The use of fast Fourier transform for the estimation of
power spectra: a method based on time averaging over short, modified
periodograms. IEEE Transactions on Audio Electroacoustics AU-15,
70–73.
Zhou, Z.H., Hardt, S., Wang, B., Sherman, M.B., Jakana, J., Chiu, W.,
1996. CTF determination of images of ice-embedded single particles
using a graphics interface. Journal of Structural Biology 116, 216–222.
Zhu, J., Penczek, P.A., Schro ¨der, R., Frank, J., 1997. Three-dimensional
reconstruction with contrast transfer function correction from energy-
filtered cryoelectron micrographs: procedure and application to the
70S Escherichia coli ribosome. Journal of Structural Biology 118, 197–
219.
Zou, X., 1995. Electron crystallography of inorganic structures—theory
and practice. Ph.D. thesis, Stockholm University.
Zubelli, J.P., Marabini, R., Sorzano, C.O.S., Herman, G.T., 2003. Three-
dimensional reconstruction by Chahine’s method from electron
microscopic projections corrupted by instrumental aberrations. Inverse
Problems 19, 933–949.
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C.O.S. Sorzano et al. / Journal of Structural Biology 160 (2007) 249–262