Hawkes P.W., Spence J.C.H. (Eds.) Science of Microscopy. V.1 and 2

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535

Cryoelectron Tomography (CET)

Juergen M. Plitzko and Wolfgang Baumeister

Clearly, if recent improvements in electron microscopes are to be fully utilized

in biology, contrast must be enhanced without drastic molecular alteration of

the specimen or obscuration by extraneous material. . . . Thus the polytropic

montage seems to offer a means of determining the three-dimensional struc-

tures of low-contrast biological specimens at a resolution of 3 Å, or the best

resolution attainable with existing electron microscopes. (Hart, 1968)

1 Introduction

More than 70 years have now passed since the invention of the ﬁ rst

transmission electron microscope (Knoll and Ruska, 1932; Ruska, 1987).

Many reports and publications have been written covering the techno-

logical achievements in electron microscopy (EM) and more important

the scientiﬁ c breakthroughs related to the information disclosed by

EM studies. EM, in its various “ﬂ avors” (see below), is now a well-

established method in life as well as in material science. Therefore,

almost every laboratory in the ﬁ eld of structural analysis is equipped

with one or several low or intermediate voltage microscopes for routine

use and in some instances for high-end applications. However, com-

pared to light microscopy, EM is still a “young” method but with great

potential for further improvements. This has been shown, especially

in the past decade: among the highlights are the incorporation of

energy ﬁ lters, monochromators, aberration correctors, and above all

the almost complete replacement of negative plates by charge coupled

device (CCD) cameras and imaging plates. By all means, these techni-

cal achievements have been supported by the development of the com-

puter and its availability throughout science. The various techniques

in EM, such as bright ﬁ eld (BF) or dark ﬁ eld (DF) imaging, weak beam

imaging, conventional transmission EM (CTEM), high-resolution TEM

or EM (HRTEM or HREM), scanning transmission electron microscopy

(STEM), energy-ﬁ ltered TEM (EFTEM), and many others, have been

improved and, moreover, expanded with the computer power on hand.

Today’s most modern techniques are all based on automated ac quisition

536 J.M. Plitzko and W. Baumeister

and alignment procedures, reconstruction algorithms, and especially

on elaborate image processing routines.

Here, we address the advances made in biological EM and especially

in cryo-EM. Biological structures can be, by and large, characterized

as pleiomorphic. Just as every human being has a different face, cells,

proteins, and macromolecular complexes have different shapes and

forms, designed for a higher functional purpose. It is far from random

and instead of addressing them as “amorphic” or “amorphous,” as in

physics for randomly ordered solids, they are called “pleiomorphic” or

“pleiomorph.” Moreover, the inside of a cellular structure resembles

the image of a giant factory, where the single constituents act together,

building highly speciﬁ c molecular machines and, if necessary,

change their purpose (Alberts, 1998). Therefore it is a highly variable

and dynamic environment. Every intrusion into this fragile system can

lead to changes. The suitable preparation for a ﬁ nal characterization

with the EM is a major challenge. And it is likewise a challenge

in terms of the environment inside EMs: an ultrahigh vacuum and

electron radiation.

Despite the fact that biologists were impressed by the resolving

power of the EM, they remained very sceptical about the usefulness of

the EM in structural biology. The major drawback was and still is the

sensitivity to radiation of the biological samples. After a few minutes

exposure to the electron beam the biological substance in question is

literally “incinerated.” However, the present knowledge about the

ultrastructure of cells, viruses, and other biological substances was

accumulated by EM investigations with, at that time, suitable prepara-

tion techniques. Higher vacuum resistance was achieved by dehydra-

tion and water-substitution methods, beam resistance was increased

by staining the samples with heavy metals (Brenner and Horne, 1959),

and, in addition, enhanced contrast and transparency were obtained

by sectioning, e.g., “big” cells in slices about 200 nm thick, with ultra-

microtomes (Porter and Blum, 1953). These preparation techniques

enabled biologists to establish the basis of a common image of the cell’s

interior. Nevertheless, this “image” remained incomplete and was still

at a resolution far above that potentially available with the EM. Addi-

tionally the samples were altered by these preparation methods, thus

complicating and limiting image interpretation. In some instances the

artifacts introduced even misled scientists. Although these preparation

techniques are still used in various ways, but there was an obvious

need for far more reliable and less harmful procedures. In 1981

Dubochet and McDowall introduced a new means of sample preserva-

tion for EM investigations: the cryotechnique. Without doubt, cryo-

preparation is one of if not the greatest development in biological EM.

Instead of replacing the water or dehydrating the whole system, the

biological substance is embedded within its original buffer solution or

simply water by rapidly freezing at very low temperatures, namely the

temperature of liquid nitrogen (∼90 K). This way the water was trans-

ferred into an amorphous state, inhibiting crystallization, and thus

disruption of the cell due to the volume increase of crystalline water.

This plunge-freezing technique revolutionized the ﬁ eld of structural

Chapter 7 Cryoelectron Tomography (CET) 537

biology, because for the ﬁ rst time it was possible to investigate biologi-

cal samples in their native state, without introducing any artifacts,

corroborating image interpretation.

Excluding artifacts was clearly one achievement in obtaining higher

resolved images of biological structures, but since these structures are

pleiomorphic, single two-dimensional (2D) images are clearly insufﬁ -

cient for a complete structural characterization, which can only be done

three-dimensionally (3D). Additionally, due to the large depth of focus

of the EM, images represent only 2D projections of the specimen, in

which almost all information about the third dimension is lost. As

mentioned, sectioning can help, and especially serial sectioning, where

the structure of interest, e.g., big cells, is serially sliced. However, serial

sectioning was previously done only on epoxy resin-embedded sam-

ples, limited in resolution to the section slice thickness (∼50 nm) and

having lost any information about the molecular organization. But it

was evident that information retrieval down to the supramolecular

level for all three dimensions is absolutely crucial for an in-depth

structural analysis of native biological samples.

In the late 1960s and early 1970s three groups independently experi-

mented with the possibility of 3D microscopy, namely electron tomo-

graphy (Hart, 1968; DeRosier and Klug, 1968; Hoppe et al., 1968; Hoppe,

1974). Perhaps inspired by the tomographic methods invented for

medical examinations (Cormack, 1963, 1980; Hounsﬁ eld, 1972) they

developed acquisition schemes to access the 3D information by record-

ing images from different orientations or from differently oriented

particles. DeRosier and Klug (1968) studied the helical tails of T4-

bacteriophages and with a single EM they were able to present a com-

plete 3D reconstruction. By selecting an object with the a priori

knowledge of its helical symmetry they avoided a complicated data

acquisition procedure. In the same year Hart (1968) reported very pre-

cisely on his idea: the “polytropic montage.” He investigated the rod-

shaped tobacco mosaic virus (TMV). Hart and later Hoppe have chosen

a more complicated and cumbersome acquisition process: the manual

rotation of the object perpendicular to the incoming electron beam and

the acquisition of single projections from different viewing angles.

They reported on the 3D representation from a series of images acquired

over a large angular regime. In the late 1960s, when the average elec-

tron microscopist used photographic plates, these procedures were

inevitably very cumbersome and time consuming and of course recon-

struction calculations were possible only with large supercomputers.

Nevertheless, with the introduction of CCD cameras for image record-

ing in EM in the late 1980s (Mochel and Mochel, 1986) and the intro-

duction of computer-controlled EMs attempts (Figure 7–1) were made

to improve the pioneering work with the prospect of implementing it

for routine and fast use (Typke et al., 1991; Koster et al., 1992, 1997).

With the addition of suitable preparation techniques such as cryoﬁ xa-

tion of biological specimens, cryo-EM promised to be the method of

choice for further structural work in molecular biology. Nevertheless,

it was not until the mid-1990s that the ﬁ rst reports on applications on

automated cryo-electron tomography appeared (Horowitz et al., 1994;

538 J.M. Plitzko and W. Baumeister

Dierksen et al., 1995; Grimm et al., 1996a; Walz et al., 1997a; Nitsch

et al., 1998), now abbreviated cryo-ET or, for the extended community,

three-dimensional electron microscopy (3DEM).

Today, cryo-EM is established in the ﬁ eld of structural biology and

three different techniques are building the base for 3D characteriza-

tion: electron crystallography (cryo-EC), single-particle EM (cryo-EM),

and electron tomography (cryo-ET). In the following we will describe

and explain one of these tomographic approaches, namely cryo-ET

(Frank, 1992, 1996), in detail.

2 Three-Dimensional Cryoelectron Microscopy

2.1 Principles of 3D Imaging

The main contrast mechanism in cryo-EM is phase contrast. In ﬁ rst-

order approximation, the micrograph is an interference pattern of the

primary beam and the focused scattered beam. In this so-called weak

phase approximation, the electrostatic potential of the sample is the

quantity that is imaged. To create a phase delay between the primary

and scattered beams, the objective lens has to be defocused. The

Figure 7–1. Transmission electron microscopes (TEM) in the past and today. a) 1939; First commercial

TEM (‘Siemens Super Microscope’) from Siemens & Halske Ltd., Berlin, Germany. b) 2002; Tecnai F30

Helium (‘Polara’) from FEI company, Eindhoven, The Netherlands as installed at the MPI of Biochem-

istry in Martinsried (near Munich), Germany. A computer controlled TEM dedicated for cryo-electron

microscopy, which allows cooling to liquid nitrogen (90 K) and helium temperature (∼10 K).

Chapter 7 Cryoelectron Tomography (CET) 539

imaging process of weakly scattering objects in a TEM can be sum-

marized elegantly by introducing a contrast transfer function (CTF) in

Fourier space (Reimer, 1993). In this notation, the image I

of an 3D

object with an electrostatic potential V

xyz

can be written as

I CTF dzV

xyz

=⋅

()

[]

−

∫

Here, F denotes the Fourier transform and k

and k

denote the recip-

rocal vectors. The CTF is an oscillating function of the defocus value

∆z, which is damped by an envelope function. The envelope function

ﬁ nally limits the resolving power of a TEM. To achieve high-resolution

data in biological EM, the use of a ﬁ eld emission gun (FEG) instrument

with a high coherence and brightness is mandatory (Baumeister and

Typke, 1993), since the increased temporal and spatial coherence

reduces the damping of the envelope function (Frank, 1973). The oscil-

latory behavior of the CTF leads to contrast inversion of I

for certain

frequency bands. Provided the signal-to-noise ratio (SNR) of I

is suf-

ﬁ cient, I

can be corrected for these CTF effects, but CTF correction

will not be feasible close to the zeros of the CTF where hardly any

signal is transferred. In single particle analysis or cryoelectron crystal-

lography, images of different focus levels are recorded to avoid that

problem; the data of different focus levels can then be combined in a

so-called focus series (Schiske, 1968, 1973; Typke, 1992), which ensures

data coverage in Fourier space without gaps around CTF zeros.

Through-focus series reconstruction is particularly suitable in material

science studies where high magniﬁ cations combined with a high dose

can be utilized almost without restrictions to determine the atomic

positions even for very light elements in crystalline specimen (Thust

et al., 1994, 1996; Coene et al., 1996; Kisielowski et al., 2001). However,

in low-dose EM, the CTF is often not easily accessible due to the low

SNR, in particular if a CCD camera is used as a detecting device.

Therefore, the ﬁ rst zero crossing of the CTF often limits the attainable

resolution in a cryoelectron micrograph, for example, in cryo-ET.

Since an EM image is essentially a projection, features from different

z-levels within the object overlap in the resulting micrograph and cannot

be separated. The traditional way to extract meaningful information

from an electron micrograph is to reduce the z-dimension of the object,

i.e., to image almost 2D objects. In biological imaging of cells this was

accomplished by sectioning the previously ﬁ xed object using an ultrami-

crotome (Porter and Blum, 1953). By means of serial sectioning even 3D

data could be obtained by imaging successive sections belonging to the

same cell. In this case, the achievable resolution in z depends on the thick-

ness of the sections; the best resolution attained was in the range of

50 nm. However, the combination of serial sectioning with tomographic

data acquisition (see below) offers the possibility of improving the

resolution (Soto et al., 1994; Mueller-Reichert et al., 2003).

A different approach is to combine different views of the specimen

to derive 3D information (Figure 7–2). The cryo-EM methods for obtain-

ing 3D data can be summarized as tomographic or at least quasitomo-

graphic. They all rely on the fact that the parallel projection of a 3D

object corresponds to a slice in the 3D Fourier space of the object. This

540 J.M. Plitzko and W. Baumeister

“projection-slice theorem” and the possibility of restoring the 3D data

using their projections were ﬁ rst discovered by Radon (1917). An

English translation of this paper can be found in Deans (1983). To

obtain the 3D information, different slices of the Fourier space, i.e.,

projections in real space, have to be gathered to sample the entire

information. Different orientations of the sample can be realized by

changing the orientation of the specimen. For this purpose the speci-

men has to be rotated, as is the case in electron crystallography or ET,

or identical copies of a specimen that occur in different orientations

can be reconstructed in a 3D model as in “single particle” analysis. In

the ﬁ eld of EM, those tomographic methods for 3D structure determi-

nation were proposed in the late 1960s (Hart, 1968; Hoppe et al., 1968)

and realized in the pioneering work of DeRosier and Klug (1968).

2.2 The Single-Particle Approach

One powerful advantage of EM is the fact that it produces projection

images, unlike X-ray crystallography, which yields diffraction data

with no information on the position; this is known as the “phase

problem.” Thus, images may be treated in real space as well as in

Fourier space and there is no difference between those that contain

information about a single molecule without any internal symmetry

and those generated by a 2D crystalline array. This fact is exploited in

the so-called single-particle approach, which enables the electron

microscopist to determine 3D density maps of individual macromole-

cules (Frank, 1996). Depending on a variety of conditions the 3D models

show near-atomic, or, in the more characteristic case, intermediate

resolution (≥7 Å) (Jiang et al., 2003a; Spahn et al., 2001).

Figure 7–2. The Figure 7-shows a single axis tilt series data acquisition scheme. The object is repre-

sented by a pleiomorphic object (in this case a knot) to emphasize the fact that electron tomography

can retrieve 3D information from non-repetitive structures. a) A set of 2D projection images is recorded

while the specimen is tilted incrementally around an axis perpendicular to the electron beam, and

projection images of the same area are recorded on a CCD camera at each tilt angle. Tilt range is typi-

cally ±70° with tilt increments between 1.5–3°. b) The backprojection method explains the principle of

3D reconstruction in a fairly intuitive manner. For each weighted projection, a backprojection body is

calculated, and the sum of all projection bodies represents the density distribution of the original

object—the tomogram.

Chapter 7 Cryoelectron Tomography (CET) 541

Projection images from macromolecules contain information about

the 3D structure as 2D density proﬁ les, characteristic of the particular

orientation of the particles in the electron beam (Figure 7–3). To disen-

tangle molecular substructures in the third dimension a number of

different images of known projection geometry have to be combined.

Protein complexes not too large in size are usually oriented arbitrarily

in vitriﬁ ed ice, so that even one EM image contains many different

projections of the isolated protein species (Figure 7–3A). Once the rela-

tive orientations of the particles are known, i.e., the three Eulerian

angles (j, Y, q) deﬁ ning the rotation of a particle around the axes in

3D space, their projections can be placed on the surface of a sphere

centered on the origin of a common coordinate system. The Eulerian

angles exactly deﬁ ne the positions of the projections that can now be

backprojected centrally in such a way as to superimpose the 2D densi-

ties into a common 3D volume of the particle. The most popular and

utilized reconstruction algorithm in real space is the so-called weighted

backprojection (Radermacher, 1992; see Section 3.5.2).

Very large macromolecular complexes, ﬁ lamentous aggregates, or

proteins embedded in biological membranes usually occur in preferred

orientations in frozen samples. Here, and in cases where nonredundant

structures such as singular protein assemblies are to be investigated,

the specimen is tilted around a ﬁ xed axis at a certain angular incre-

Figure 7–3. Single particle investigation of the giant protein complex TPP II from Drosophila mela-

nogaster embedded in vitriﬁ ed ice. In eukaryotes, tripeptidyl peptidase II (TPPII) is a crucial compo-

nent of the protein degradation pathway. The 150-kDa subunits of Drosophila TPPII assemble into a

giant proteolytic complex of 6 MDa with a remarkable architecture consisting of two segmented and

twisted strands that form a spindle-shaped structure (length 56 nm, width 24 nm). a) Cryo-electron

micrograph of isolated TPP II complexes illustrating the very weak image contrast and the high level

of noise. b) Averaging and classiﬁ cation of a large number of equivalent projections of separate mole-

cules. Once a large set of views is available, a preliminary 3D reconstruction can be computed and

reﬁ ned iteratively. c) The 3D model obtained by cryo-electron microscopy, reveals details of the

molecular architecture and, in conjunction with biochemical data, provides insight into the assembly

mechanism. The building blocks of this complex are apparently dimers, within which the 150 kDa

monomers are oriented head to head. Stacking of these dimers leads to the formation of twisted single

strands, two of which comprise the fully assembled spindle (Rockel et al., 2002 and 2005). (See color

plate.)

542 J.M. Plitzko and W. Baumeister

ment in the EM to obtain a series of projections. This approach of

electron tomography is applicable almost universally if the specimen

is not too thick for the electrons to penetrate.

Since the electron exposure must be restricted to a value low enough

not to destroy the sensitive biological material, the images contain a

high level of so-called “shot noise” (statistical variation in the number

of electrons detected at each pixel in the image). Additionally electron

scattering from elements low in atomic number (Z), like the constitu-

ents of most biological materials, e.g., carbon, oxygen, and hydrogen,

produces very weak image contrast. Based on these facts biological

structures are barely visible in single cryo-EM images. Furthermore the

exact alignment of the particles becomes a challenge, which is tackled

by averaging a large number of equivalent projections of separate mol-

ecules, thus reducing the noise level. For this purpose, the images of

some 10

to 10

particles are sorted into distinct classes of views by

multivariate statistical analysis (Van Heel and Frank, 1981). Once a large

set of views is available, a preliminary 3D reconstruction can be com-

puted and reﬁ ned iteratively. Typically, molecular complexes must be

large in size (>200 kDa; 1 Da = 1 U = 1.66 × 10

−24

g) to provide sufﬁ cient

signal for the alignment at high resolution. If the particles possess a

high internal symmetry, e.g., virus capsids or regular structures like 2D

crystals, the averaging and reconstruction process is simpliﬁ ed and the

number of images necessary and the computation time are reduced.

To determine the structural basis of the working mechanism of

molecular machines we have to study the structures, positions, and

interactions of their building blocks, i.e., their distinct subunits. Since

the structure of protein–protein contacts is resolved only at near-atomic

resolution it is hardly possible to identify molecular borders in 3D

models of cryo-EM data by visual inspection. But the ability to identify

elements of secondary structure, and α-helices in particular, at a reso-

lution of about 6–8 Å, makes it possible to ﬁ t a previously determined

(atomic) model of protein subunits or subcomplexes into 3D density

maps of larger assemblies obtained by cryo-EM (Li et al., 2002; Bottcher

et al., 1997). This approach, called “docking,” enables us to unravel the

organization of macromolecular machines that cannot be structurally

characterized otherwise. Right now researchers continue to develop

quantitative criteria that can guide this operation (Volkmann and

Hanein, 1999).

Single–particle cryo-EM is a powerful but still slow method com-

pared to X-ray crystallography, provided that for the latter well-

diffracting crystals are available. The data collection and processing

can extend over several months or longer even for specialists in the

ﬁ eld. However, with the advent of computer-controlled electron micro-

scopes data acquisition and analysis have become subject to the devel-

opment of elaborate automation procedures that promise to reduce the

time needed to create high-resolution 3D density maps of large mac-

romolecular complexes considerably.

The real bottleneck for investigations of known or still unknown

molecular machines arose from another experimental task. Puriﬁ ca-

tion procedures commonly used in biochemistry tend to select stable

Chapter 7 Cryoelectron Tomography (CET) 543

and abundant protein complexes only. Very large molecular machines,

rare or transient macromolecular assemblies, complexes consisting of

membrane spanning and soluble components, and all those held

together by forces too weak to withstand the puriﬁ cation procedures,

escape isolation or even detection. The lesson is twofold: We need new

approaches for isolating or reconstituting fragile assemblies, and cryo-

EM of isolated single particles will probably not give us the whole

truth. To address the full complexity of functional macromolecular

assemblies noninvasive 3D imaging techniques have to take over to

fulﬁ ll the task of studying the supramolecular architecture in its native,

cellular context (Plitzko et al., 2002). The technique of cryo-ET is a

“nouvelle route” with a great potential in the ﬁ eld of structural biology

that promises to provide noninvasive and deep insight into the func-

tional organization of the cellular proteome (Sali et al., 2003).

2.3 Cryoelectron Tomography

CET is unique in the sense that it can image nonpuriﬁ ed macromole-

cules three-dimensionally. However, in terms of resolution, cryo-EC

and single-particle analysis are currently superior, which makes cryo-

ET a complementary method to the aforementioned. In contrast to

these methods, cryo-ET does not involve implicit or explicit averaging

of the specimen, which makes it a suitable method for imaging of

pleiomorphic objects. Therefore it is suitable to image entire cells

(Grimm et al., 1998; Medalia et al., 2002; Kuerner et al., 2005; Nicastro

et al., 2000, 2005; Scheffel et al., 2005) or nonsymmetric viruses

(Gruenewald et al., 2003) in situ (Figure 7–4). Furthermore it has the

Figure 7–4. Cellular cryo-electron tomography of the magnetotactic microorganism Magnetospirillum

griphiswaldense. The entire bacterium is oriented like a compass needle inside the magnetic ﬁ eld in its

search for optimal living conditions. The miniature cellular compass is made by a chain of single

nano-magnets, called magnetosomes (the scale bar represents 200 nm). a) The two-dimensional image

represents one projection (at 0°) from an angular tilt-series. b) x–y slices along the z axis through a

typical three-dimensional reconstruction (tomogram). c) Surface-rendered representation of the inside

of the cell showing the membrane (blue), vesicles (yellow), magnetite crystals (red) and a ﬁ lamentous

structure (green). Until now, it was not clear how the cells organise magnetosomes into a stable chain,

against their physical tendency to collapse by magnetic attraction. However, the biochemical analysis

revealed a protein responsible for the chain formation and the 3D investigation a cytoskeletal struc-

ture, which aligns the magnetosomes like pearls on a string (Scheffel et al., 2005). (See color plate.)

544 J.M. Plitzko and W. Baumeister

potential to unravel not only the structure but also the molecular inter-

actions of the various macromolecules inside a cell (Beck et al., 2004).

Whereas tomography of plastic embedded specimens (Marsh et al.,

2001; Murk et al., 2003) reveals primarily data on the ultrastructure and

morphology of a cell cryo-ET aims to resolve the cell at a molecular

level. Although the SNR of a tomogram of a frozen hydrated specimen

is generally worse than that of a plastic embedded specimen, more

biologically signiﬁ cant information is contained in the high spatial

frequencies since the preparation technique does not limit the inter-

pretable information.

A cryo-electron tomogram essentially depicts the entire proteome of

a cell. Therefore, cryo-ET can make signiﬁ cant contributions to the

larger enterprise of structural proteomics (Sali et al., 2003). It has the

potential to unravel the interactions of the various proteins that reside

not only in the cytoplasm but also in the cell membrane. Moreover, it

can provide medium resolution structural characterization of com-

plexes that are largely undescribed to date. A prerequisite for those

contributions is sufﬁ cient resolution to recognize macromolecules in a

cellular tomogram.

The ﬁ rst published cryo-electron tomogram of a eukaryotic cell

(Figure 7–5), the slime mould Dictyostelium discoideum, demonstrated

that cryo-ET is able to resolve individual macromolecular complexes

such as ribosomes or the 26S proteasome in the context of a cell (Medalia

et al., 2002). Currently, cellular cryo-ET is effectively conﬁ ned to rela-

tively thin samples. Cells of a thickness beyond 1 µm are not transpar-

ent to the electron beam at a voltage of 300 kV. These cells are typically

prokaryotic cells or unusually thin eukaryotic cells. Electron tomogra-

phy of frozen hydrated sections of cells (Hsieh et al., 2002; Leis et al.,

2005) can possibly extend the specimen class to thicker cells and even

tissue cells. Unfortunately, the sectioning process still introduces severe

artifacts; in particular, the forces that arise during the sectioning

process tend to compress the cells strongly. The use of vibrating

cryoknifes can possibly offer some remedy from these shortcomings

(Al-Amoudi et al., 2003). The technique is promising, but it is far from

being a routine preparation technique, as yet.

To date, cryo-ET is effectively the only 3D imaging technique that

can image cells or organelles in a close-to-native state at molecular

resolution. It is of vital importance to have an imaging method that can

image biological macromolecules since many of the complexes present

in a cell tend to be fragile or transient. The detection of statistically

signiﬁ cant distribution patterns can provide unprecedented informa-

tion about the interaction of macromolecules, in transient complexes,

for example. The basic problem that cryo-ET has to face is to derive

statistically signiﬁ cant information from the initially very noisy tomo-

grams. Interpretation of the raw data can be performed only by low-

pass ﬁ ltering the data to a resolution that is regarded as signiﬁ cant.

Application of so-called “denoi si ng” algorithms can extend this limit

gradually and greatly improve the visualization of tomograms. These

techniques mainly perform nonlinear ﬁ ltering of the data (Frangakis