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Chapter 7 Cryoelectron Tomography (CET) 585

mark the positions of the features in the input scene. Three different

conventional XCF, phase-only ﬁ ltering (POF), (Horner and Gianino,

1964), and mutual cross-correlation function (MCF; van Heel et al.,

1992). In all those methods a cross-correlation function is computed:

The XCF computes the unmodiﬁ ed cross-correlation function of tem-

plate and micrograph whereas the POF and MCF vary the moduli of

the template to sharpen the cross-correlation peaks. In electron micros-

copy especially, the low frequencies carry an extraordinarily strong

signal, which can be misleading (van Heel et al., 1992). The strength of

the low-frequency contributions can differ considerably within elec-

tron micrographs of amorphous specimens, such as vitriﬁ ed speci-

mens, since these are typically nonhomogeneous. The POF sets the

moduli of template and input scene to one while the MCF replaces the

moduli by their square roots to reduce the inﬂ uence of the very low-

frequency amplitudes systematically.

Another way of taking into account the strong local variations of the

image contrast has been introduced recently (Roseman, 2000). By nor-

malizing the XCF to the local cross-correlation function (LCF) variance,

the efﬁ ciency of locating local features such as macromolecules within

a micrograph can be increased signiﬁ cantly. Typically, the normaliza-

tion region should correspond to the shape of the molecule. This cal-

culation of the local variance does not lengthen the computing time

unreasonably, since it can be calculated very efﬁ ciently in Fourier space

too (Figures 7–23 and 7–24).

Figure 7–23. Mapping ribosomes in an intact S. melliferum cell. a) x–y slice from the corresponding

tomogram. Image analysis of this portion of the cell is displayed in subsequent panels. b) Locations

and orientation of all ribosomes detected by template matching. Each 70S ribosome is represented by

the averaged density (see Fig. 26) derived from the tomogram. The colour coding indicates the detec-

tion ﬁ delity; green is high, yellow is intermediate, red is low and probably represents false positives.

The brightness of the colour corresponds to correlation peaks heights. c) Final ribosome atlas after

removal of putative false positives (images courtesy of J. Ortiz, MPI of Biochemistry, Martinsried,

CCFs have been introduced in the ﬁ eld of 2D electron microscopy:

Germany; Ortiz et al., 2006). (See color plate.)

586 J.M. Plitzko and W. Baumeister

To evaluate which ﬁ lter is best for the task of locating macromole-

cules, some theoretical considerations are useful. The conventional

XCF has been shown to be optimal with respect to minimization of the

location error (Kumar et al., 1992) and high SNR of the resulting func-

features within micrographs or cryoelectron tomograms yields intoler-

ably high rates of false positives; the XCF tends to locate high-contrast

features rather than low-contrast macromolecules. The same typically

holds for the MCF, whereas the POF overampliﬁ es noise, which makes

96%73%6%detection reliability

143

174

presumably correct (~fit)

137127433

CC peaks

50%

33%

16,50%

ribosomes expected in

CC range (~fit)

CC>0.3430.343 CC>0.3

CC range

III

Figure 7–24. a) Histogram of experimentally obtained CCCs (dashed line). Ideally, the histogram

should be Gaussian, reﬂ ecting the variances in SNR of individual ribosomes in the tomogram, and

thresholding of the CCC should allow discriminating between correctly identiﬁ ed ones and false

positives. Non-speciﬁ c correlation with high contrast features such as the cell membrane, leads to a

relative large number of false positives. For an objective setting of thresholds, the Gaussian ﬁ t (con-

tinuous line) was divided into three probability sectors larger than the mean CCC value (I, dark grey);

lower than the mean value up to one standard deviation (II, intermediate grey); lower than the mean

value and between one and two standard deviations (III, light grey). The spreadsheet in b) illiustrates

the Gaussian ﬁ t of the experimental distribution of CCCs and the ﬁ delity of the experimental detection

of 70S ribosomes. About 80% of the particles are detected with more than 85% reliability (images

0.3 CC>0.257

tion (VanderLugt, 1964). However, use of the XCF for locating small

≥

courtesy of J. Ortiz, MPI of Biochemistry, Martinsried, Germany; Ortiz et al., 2006).

Chapter 7 Cryoelectron Tomography (CET) 587

it unfavorable. Therefore, local normalization is appropriate. However,

regarding the SNR of the resulting correlation function, the LCF is

inevitably worse than XCF or MCF; conﬁ nement of the correlation to

N voxels raises the noise level, which should be proportional to N

−1/2

The size of the normalization region should correspond to the envelope

of the template under scrutiny. Choosing the local region larger will

increase the SNR of the ﬁ lter output at the cost of increasing the chance

of false positives simultaneously, in particular if particles tend to be

very dense within the input scene. Whenever justiﬁ ed circular (2D) or

spherical (3D) normalization regions (Θ) will be used because this

accelerates the computation considerably compared to approaches

using asymmetric regions. If the template has one rotational degree of

freedom perpendicular to the specimen plane within the input scene,

the problem of matching a 2D template in a 2D image can be formu-

lated as determining the maxima of a cross-correlation function as a

function of spatial variable r and rotation angle ϕ:

XCF I R T

rrrr

,ϕϕ

=⋅

(

)

′′

′

∑

Here R

denotes the rotation operator. By expanding the template in

circular harmonics, exhaustive scanning of the polar angle ϕ can be

avoided (Kunath and Sack-Kongehl, 1989). In 3D, the problem of ﬁ nding

a macromolecule in a tomogram that matches a 3D template by cross-

correlation becomes six-dimensional: Using the Euler angles ϕ, ψ and

ϑ to describe the orientation, the maxima of the CCF

;ϕψϑ

are to be

determined. Again, the translational dependence can be covered using

FFT, but it is normally necessary to perform an explicit scanning in

angular space to cover those parameters. An exhaustive search is com-

putationally very time demanding. However, because of the emerging

facilities of parallel computing this task can easily be faced: Since the

CCF can be computed individually for different orientations, the entire

calculation can be distributed over different processors with hardly

any administrative overlap (Frangakis and Hegerl, 2002; Rath et al.,

2003).

In cryo-ET the partial sampling of the object’s Fourier space, the

“missing wedge,” has to be considered further. This can be done by

conﬁ ning the correlation to the sampled region in Fourier space or

equivalently by convoluting the object with the point-spread function

(Frangakis and Hegerl, 2002). In the framework of the locally normal-

ized CCF the missing wedge affects the local normalization. The

missing wedge can be taken into account in Fourier space by multipli-

cation of the template with a suitable binary function or equivalently

by convoluting with a suitable PSF in real space. In this notation, the

conﬁ ned cross-correlation function can be deﬁ ned as

CCF

IT PSF

rr r r

rrr r

ϕψϑ

⋅⊗ ⋅

()

⋅−

()

′′ ′

′

′′

′

∑

where I

′ represents the local mean value within Θ. A natural way of

avoiding the limitation of beam damage in tomography is averaging

of copies of structural elements occurring in a tomogram. Saxberg and

588 J.M. Plitzko and W. Baumeister

Saxton (1981) already concluded from their results on resolution limita-

tion in electron tomography due to beam damage that averaging is a

practical way of increasing the effective electron doses and should be

considered to achieve high resolutions. The importance of those tech-

niques to derive information on the molecular level was reemphasized

recently (Steven and Aebi, 2003).

According to the dose fractionation theorem a tomogram describes

a volume element with the same statistical signiﬁ cance as a projection

with the same dose (Hegerl and Hoppe, 1976; McEwen et al., 1995).

Therefore, tomograms should be just as well suited to posterior averag-

ing procedures for obtaining high-resolution information as compared

to single-particle averaging. In practice, both approaches have their

pros and cons, which are discussed below.

Averaging of particles from tomograms can be performed in the

same way as single-particle averaging. Projection matching is crucial

in these techniques: The individual particle projections are iteratively

aligned to projections of a reference from different angles by means of

cross-correlation. The reference is thus the average of the particles

using the orientations and displacements of the previous iteration. The

iterative reﬁ nement of orientation and displacements improves the

resolution of the average until convergence is achieved. The analogous

procedure in cryo-ET could be termed “tomogram matching.” Subto-

mograms containing the roughly centered particles of interest need to

be extracted from the entire tomogram and aligned using procedures

as sketched in Figure 7–25. The aim of those tomogram matching pro-

cedures is to optimize the orientations, i.e., the Euler angles ϕ, ψ, and

ϑ, and the displacement shifts of the individual particles. A ﬁ rst real-

ization of this approach was reported by Walz et al. (1997). Application

of this implementation on tomograms of puriﬁ ed protein complexes

resulted in the generation of a pseudo-atomic map of the thermosome

in the open conformation (Nitsch et al., 1998).

Figure 7–25. a) Averaged structure of the 70S ribosome derived from the dataset above (average from

300 individual particles to ∼4 nm resolution). The map highlights the 30S subunit in yellow. b) Docking

of high resolution structures of the 70S ribosome into the map shown in A. c) “Crown view” of B

(images courtesy of J. Ortiz, MPI of Biochemistry, Martinsried, Germany). (See color plate.)

Chapter 7 Cryoelectron Tomography (CET) 589

However, cryo-ET is conventionally not used for this purpose for

several major reasons: (1) It is much easier to acquire the data by spend-

ing the entire dose on only one projection; this makes acquisition of

large amounts of individual molecules much easier. (2) The SNR of a

projection of a particle is better than the SNR of a tomogram of the

same particle acquired with the same electron dose. (3) There is only

one level of alignment: In tomography the projections need to be

aligned to a common origin prior to aligning the individual particles

with respect to each other. Both steps inevitably introduce errors that

sum up.

But averaging of cryo-ET data can offer some advantages that are

worth the additional efforts in some cases. The most important reason

is the ability to image nonpuriﬁ ed samples (Figure 7–26). CET is cur-

rently the only technique that can be used for quaternary structure

determination of fragile or even transient complexes. The degree of

alteration that biological macromolecules undergo under physiological

conditions is largely undetermined since there was no imaging tech-

nique that could resolve them in the context of a cell. Apart from this,

averaging of subtomograms also offers one principal advantage com-

pared to averaging from projections: It is fundamentally easier to

determine the orientations of an individual copy from 3D data than

from projection data.

4 Perspectives: New Strategies and Developments

The achievements made by cryo-ET over the past decade have proven

the possibilities and the feasibility of this technique for quasi in vivo

studies of the ultrastructure and larger supramolecular assemblies

within whole cells. However, based on the initial developments in

cryo-ET, major improvements in instrumentation and sample prepara-

tion have to be made in the future to exploit its full potential.

The ultimate goal in structural biology is to investigate the

structure–function relationship of molecular complexes and supramo-

lecular assemblies in their native environment, e.g., in large cells or

even tissue, across several dimensions, from the micrometer level

to the subnanometer level, and if possible within one single experi-

ment, to realize the complete view into the inner space of cells and its

constituents. Even at the present practical level cryoelectron tomo-

grams of organelles and cells contain an imposing amount of informa-

tion. They are, essentially, 3D images of entire proteomes, and they

should ultimately enable us to map the spatial relationships of the full

complement of macromolecules in an unperturbed cellular context.

However, it is obvious that new strategies in sample preparation,

advancements in instrumentation, and innovative image analysis tech-

niques are needed to make this dream come true, at least to some

extent.

In principle there are three major routes to increase the image

quality, and especially the 3D image quality, which are intrinsically

linked to each other: (1) the technological developments comprising

590 J.M. Plitzko and W. Baumeister

advanced electron optics, highly resolved high-sensitive detection,

improved tilting devices, and the geometry of the sample holder, (2)

the sample preparation and thus the quality of the sample, and (3)

the design of new software algorithms and acquisition schemes to

guarantee a quantitative, objective, and exhaustive analysis of the

recorded data.

TEM or EM is still a very young method if compared to light micro-

scopy. While the latter was developed and improved over centuries,

EM has a history of less than 80 years. Microscopes can now easily can

reach a resolution in 2D of less than 1 Å, clearly orders of magnitude

better than what light microscopy can offer. However, the electron

optical system, determined mainly by the objective lens system and

characterized by the spherical and chromatic aberration coefﬁ cients, is

far from being ideal and is still inferior to light optical devices. Simpli-

ﬁ ed, the performance of present objective lens systems in EM can be

described as an attempt to focus through the bottom of a champagne

bottle. This way, the quality of the recorded micrograph in EM suffers

greatly from the severe shortcomings of the objective lens system, e.g.,

the higher the spherical aberration coefﬁ cient C

, the better the result-

ing image contrast, but the lower the resolution. However, the spherical

aberration coefﬁ cient is linked directly to the “spacing” inside the

objective lens system and thus determines the maximum achievable

tilt range for an angular acquisition. Recent developments in C

correc-

tion technology for TEM (and even for SEM) demonstrated the possi-

bilities and the ﬁ nal image improvement, especially in material science

studies. However, EM investigations of biological samples at cryotem-

peratures are characterized by a very low image contrast, due to the

very weak scatterers, mainly low Z elements, such as carbon, oxygen,

and hydrogen.

Thus high defocus values are more or less mandatory, to increase

the image contrast. Unfortunately, by tuning the objective lens to very

low defocus values of a couple of micrometers, low-frequency infor-

mation is enhanced, while high-frequency information is almost

completely obscured if not totally lost. Thus C

correction without

any additional equipment is not an option in any cryo-EM study,

because the actual improvement in image quality is restricted to a

defocus regime in the range of a couple of tenths of nanometers, thus

orders of magnitude higher than what would be needed for a high-

contrast (low-dose) cryoimage. Nevertheless, it might be possible to

increase the contrast without defocusing the objective lens, by using

so-called phase plates in combination with C

correctors. The beneﬁ t

of C

correction would then become clearly visible, because even high-

frequency information would be accessible. Clearly the design and the

application of phase plates are not new, but the technological prob-

lems in former times could not be addressed to enable their routine

use. Two of the major problems were contamination and stability.

While placed in the back focal plane, very close to the sample, con-

tamination caused charging and drift problems. Moreover, because

the stability was low, it was mandatory to readjust the phase plate in

respect to the central beam, which, in terms of required accuracy, is

Chapter 7 Cryoelectron Tomography (CET) 591

possible today only with piezo-controlled stages. However, present

achievements and possibilities in nanostructuring and nanotechnol-

ogy made it possible to utilize the already existing designs of phase

plates, addressing the aforementioned problems, and place them in a

normal EM environment.

The prospects are good and in the near future experiments will be

done to show not only the feasibility of this approach, but also the ﬁ nal

image improvement if, for example, used in combination with C

cor-

rection. However, the total sum of elastically scattered electrons will

be reduced and thus the large fraction of inelastically scattered elec-

trons has to be removed or even utilized for ﬁ nal image formation.

“Removal” can be done with imaging energy ﬁ lters, which are now

routinely incorporated into the EM setup. They are especially helpful

in studies in which thick cells or sections are used, because the larger

the material layer to be penetrated by the electron beam, the larger the

fraction of inelastically and multiple-scattered electrons. They will

have a different energy, and thus they will be slightly out of focus,

blurring the recorded image. However, if the chromatic electrons are

removed, the image contrast can be deﬁ nitely improved, but the number

of elastically scattered electrons (forming the image) will be just a frac-

tion of the total sum of electrons leaving the sample. In most cases the

amount is only a half or a third of the total applied electron dose to

the sample, which contributes to ﬁ nal image contrast. This way, the

detection has to be very sensitive, so that literally every electron will

count. Today, typically CCD cameras are used, which detect the signal

indirectly and which add noise to the recorded image. Moreover the

signal is spread over several pixels, decreasing the lateral resolution

and thus the transfer of the recorded image. Direct detection would be

beneﬁ cial and favorable, because every electron would contribute to

the ﬁ nal signal, with no readout noise and the best possible lateral

resolution. Developments will utilize detectors based on CMOS tech-

nology, which will guarantee a noise-free detection and, if back-

thinned, a highly resolved recording. However, the radiation hardness

of these devices, for 300-keV electrons, is at the moment insufﬁ cient for

routine use in EM, but improvements are being made to increase the

lifetime of these detection devices, so that they can be used almost

continuously.

Since we cannot increase the total applied dose to the sample, not

even by cooling to liquid helium temperature, the only option we do

have is to increase the performance of the detection device and the

electron optics. If, for example, an almost ideal detector, could be

utilized, the dose in a recorded image can be as low as possible

and thus the number of acquired projections in an angular regime

can be as large as possible. This way the ﬁ nal resolution would be

deﬁ nitely improved. Moreover, if high-end electron optical systems

are used, e.g., C

or even C

correctors (which would even utilize the

fraction of inelastically scattered electrons for image formation)

in combination with phase plates and energy ﬁ lters, not only the

low-frequency information, responsible for the ﬁ nal image contrast,

but also the high-frequency information, containing the ﬁ ne struc-

592 J.M. Plitzko and W. Baumeister

tural details, could be harvested. Moreover, with just one single 2D

image, it would be possible to obtain phase and amplitude informa-

tion separately.

In addition to the improved quality of the single 2D projection, in

phase plate C

-corrected imaging, ET is still hampered by an uneven

sampling in Fourier space and thus by the fact that information in the

direction of the tilt axis cannot be accessed. This missing information

(missing wedge), due to the limited tilt range, can be decreased by

utilizing dual-axis acquisition schemes to a missing pyramid. Clearly,

with the 90° in-plane rotation of the sample and the acquisition of two-

tilt series of the same sample area, it is possible to increase the amount

of information by more than 20% and thus reduce the resulting arti-

facts related to single-axis-acquisition schemes. The resolution is not

improved, however, it is deﬁ nitely more isotropic. Especially for sub-

sequent automatic detection of macromolecules and proteins within

the tomographic volume, e.g., by template matching, errors (false-

positives) can be minimized, and thus the quality of the detection

can be vastly improved.

At the present resolution of 4–5 nm only very large complexes (ribos-

omes, 26S proteasomes) can be detected reliably within cryoelectron

tomograms of whole cells. However, an improvement in resolution to

2 nm will allow us to detect medium sized complexes in the range of

200–400 kDa. Some of the problems in the interpretation of tomograms

will disappear once a resolution of 2 nm is obtained. At the moment,

with a resolution of 4–6 nm, docking of high-resolution structures to

yield pseudo-atomic maps of molecular complexes is computationally

demanding and time consuming. However, this procedure of template

matching and docking will become easier if a resolution of 2 nm can

be routinely obtained. The computational methods described will aid

and complement the information provided by other approaches and

information, such as localization, labeling studies, and the binding

properties of molecules. While tomograms with a resolution of 2 nm

are a realistic prospect, major technical innovations (see above) will be

required to go beyond this.

CET of whole cells allows us to investigate the structure–

function relationship of molecular complexes and supramolecular

assemblies in their native environment. It thus results in a fundamental

change in the way we approach biochemical processes that underlie

and orchestrate higher cellular functions. In the past, molecular inter-

actions were studied mostly in a collective manner, whereas now we

have the tools to visualize the interactions between individual mole-

cules in their unperturbed functional environments. Although they

share common underlying principles, no two cells or organelles are

identical, owing to the inherent stochasticity of biochemical processes

in cells as well as their functional diversity. Therefore, it will be a major

challenge to extract generic features from the maps, such as the modes

of interaction between molecular species. The ultimate goal, the dis-

covery of general rules that underlie cellular processes, has to go

beyond observing qualitative features and has to be based on stringent

analytical criteria (Lucic et al., 2005).

Chapter 7 Cryoelectron Tomography (CET) 593

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