2. The detection criterion and the limits of cross-correlation alignment

The main tool for aligning two noisy EM images of a biological object is the cross-correlation function (Fig. 1; Frank, Reference Frank1970; Langer et al. Reference Langer, Frank, Feltynowski and Hoppe1970). (Even though some fancier functions have subsequently been introduced, the main arguments to be made below remain valid.) The potential of the single-particle averaging approach in EM can be inferred from the observation that the peak of the cross-correlation function, even though its width is comparable with the resolution of the images, can be located with much higher precision. The reason is that theoretically, this peak is identical with the autocorrelation function of the point spread function associated with the imaging process (Frank, Reference Frank1975a), which is a smooth centrosymmetric function whose center can be determined with a precision of approximately one-fifth of the resolution distance, if not better. This is, in fact, a corollary of Michael Rossmann's later observation (Rossmann, Reference Rossmann2000) that the precision of fitting an X-ray structure into a cryo-EM density map is much better (an estimated factor of 5) than the resolution.

Fig. 1. Cross-correlation function, showing a peak indicating the relative position of two successive images taken of the same carbon film. Data reproduced from Frank (Reference Frank1970) .

Much of the noise in the image is due to the need to image the molecule at very low doses, to mitigate radiation damage. The extent and deleterious effects of radiation damage were generally recognized (see Beer et al. Reference Beer, Frank, Hanszen, Kellenberger and Williams1974) following Bob Glaeser's earlier ‘fade-out’ diffraction studies with α-valine (Glaeser, Reference Glaeser1971) and led to Unwin and Henderson's pioneering study of the purple membrane protein (Unwin & Henderson, Reference Unwin and Henderson1975; Henderson & Unwin, Reference Henderson and Unwin1975). As the electron dose is lowered, the signal-to-noise (SNR) ratio of the image decreases and so does the SNR ratio in the cross-correlation function and with it our ability to locate the cross-correlation peak with sufficient certainty. A principal question that had to be resolved in the beginning was the range of parameters affecting the SNR of the cross-correlation function, namely particle size D, contrast c, critical dose p _crit and resolution d (expressed as length), for which particle alignment is feasible. The answer was formulated, as a relationship between these quantities, in a paper I coauthored with Owen Saxton during the time I spent at the Cavendish Laboratory (Saxton & Frank, Reference Saxton and Frank1976):

The result clearly indicated that alignment of raw images of two molecules embedded in ice presenting the same orientation was feasible for molecules in a biologically interesting size range, at a resolution that would give significant structural information and with a dose low enough (below p _crit, a value determined from experiments such as Bob Glaeser's) to avoid radiation damage. What might be called ‘biologically interesting’ is the approximate size range of molecular machines (~150 Å, e.g., the transcription complex, to ~1500 Å, e.g., the nuclear pore complex), that is, complex assemblies formed by multiple binding partners that perform a biological function in a processive manner (Alberts, Reference Alberts1998). It is the large size and function-related conformational variability of such assemblies that pose great hurdles to the application of X-ray crystallography.

This relationship, based on rough estimates, was later revisited and refined by Richard Henderson (Reference Henderson1995) as he investigated the feasibility of determining the structure of single molecules lacking symmetry using neutrons, electrons or X-rays, making use of the experimental scattering data in his estimation. However, the much more simplistic relationship I formulated together with Owen Saxton 20 years earlier gave the green light to forge ahead with the development of software that would allow the formulation and execution of complex processing paths for alignment, averaging and reconstruction of biological macromolecules from single-particle images.

3. SPIDER – software for single-particle averaging and reconstruction

In exploring and realizing the single-particle approach, the need for a versatile software system in which procedures could be formulated by stringing together basic commands for operations on images (such as ‘mask,’ ‘shift,’ ‘cross-correlate,’ ‘Fourier transform’) into scripts was obvious. Software systems existing at the time addressing the needs of the EM community, such as ‘EM’ in Martinsried (Hegerl & Altbauer, Reference Hegerl and Altbauer1982) and the MRC suite of programs (see later documentation by Crowther et al. Reference Crowther, Henderson and Smith1996), were of the ‘old-school’ kind, in the sense that they were large single-standing programs, where addition of new functionality required changes and recompilation of the entire code. The key new requirement was modularity – each module should address a single operation on the image; it should be small and easily overseeable; it should be independent from all other modules and communicate with them by means of a common database. Such a system existed at the Jet Propulsion Laboratory at the time I worked there as a postdoctoral student, in 1970. VICAR, a modular Fortran-based program (Billingsley, Reference Billingsley1970), had been created in the 1960s to process images sent back to earth from space probes (in 1970, Jupiter was the favorite subject). I wrote several subroutines as ‘piggybacks’ of the VICAR system, taking advantage of its highly developed infrastructure in which, among other features, Do-loops and execution-time changes of file names were supported. The result of my effort was, incidentally, a paper on heavy versus light atom discrimination based on a focus series of DNA supplied by Walter Stoeckenius (Frank, Reference Frank1972). The first incarnation of a truly modular system aimed at the needs of EM was SEMPER, developed by Owen Saxton in the early 1970s but not published until later (Saxton et al. Reference Saxton, Pitt and Horner1979).

The design of SPIDER (System for Processing of Image Data from Electron microscopy and Related fields), whose basic structure was developed by 1978 as a result of a brainstorming with two gifted students, Helen Dowse and Brian Shimkin, addressed the need for modularity and high versatility and included a nested hierarchical calling structure and passing of parameters by keyed document files (Frank et al. Reference Frank, Shimkin and Dowse1981a). Extensive contributions to the 3D reconstruction algorithms and the program's infrastructure were later made by Michael Radermacher (Radermacher et al. Reference Radermacher, Wagenknecht, Verschoor and Frank1987; Radermacher, Reference Radermacher1988), Pawel Penczek (Penczek et al. Reference Penczek, Radermacher and Frank1992, Reference Penczek, Grassucci and Frank1994, Reference Penczek, Zhu and Frank1996, Reference Penczek, Zhu, Schröder and Frank1997), and Ardean Leith in my group. Although written in Fortran, an ancient programming language by today's standards, SPIDER is still maintained and widely used by the structural biology community.

4. Single molecules versus crystals

Initially, the idea of obtaining structural information from single, i.e., unattached and non-crystallized molecules imaged under conditions of low dose, was greeted with little more than bemused interest. How could structures be solved if not from crystals or highly ordered aggregates of molecules? What could be the possible meaning of results obtained from individual noisy images that had to be forced into a common reference frame?

The prevailing view was that for a molecular structure to be quantitatively characterized, it had to be in crystal, highly ordered form. Indeed, as the technique of cryo-EM was perfected at the EMBL in the early 1980s, it was first exclusively applied to 2D crystals, helices, and spherical viruses. Thus, the community who enthusiastically adopted cryo-EM technology for exploration of native molecular structure made primarily use of concepts and working methods of X-ray crystallography, suitably modified to deal with EM images (e.g., Amos et al. Reference Amos, Henderson and Unwin1982).

In X-ray crystallography, information on amplitudes is concentrated in small spots or layer lines in the Fourier transform, and phases are not supplied directly by the diffractometer but have to be garnered by computation with the aid of additional experiments, employing multiple heavy atom replacements or multiple wavelengths. In contrast, the EM produces actual images, which is tantamount to saying it measures both amplitudes and phases in Fourier space, giving rise to an adage coined by Walter Hoppe's (see Hoppe, Reference Hoppe1983), of the EM as a ‘phase-measuring diffractometer.’

From this Fourier-centric perspective, the idea of applying real-space averaging to images of single molecules (Frank, Reference Frank1975b) seemed a step back, into territory ruled by subjectivity. It was therefore of crucial importance, in gaining respectability, to demonstrate the reproducibility of the results in a quantitative way, as first done in the study of the HeLa 40S ribosomal subunit (Fig. 2; Frank et al. Reference Frank, Verschoor and Boublik1981b). As the outset, two averages from independently drawn images of the negatively stained subunit looked strikingly similar, suggesting a high degree of reproducibility (Fig. 2a, b). The challenge was to measure the reproducibility as a function of spatial frequency. In fact, reproducibility turned out to be a key to the question how spatial resolution could be defined.

Fig. 2. Averaging of 40S subunits from HeLa cell ribosomes. (a) Electron micrograph of negatively stained subunits, with “L” denoting left-facing views selected for alignment and averaging. (b, c) Averages of half-sets of 40 particles each; (d) variance map; (e) total average; (f) three measures of resolution: R, R factor; Δ, Euclidean distance; δϕ, DPR indicating a resolution of 20 Å at 45°. Data reproduced from Frank et al. (Reference Frank, Verschoor and Boublik1981b); panels R and Δ in (f) are from unpublished data.

5. The meaning and measurement of resolution

The absence of diffractions spots in a non-periodic structure poses a problem in defining and measuring resolution. Unlike the Fourier transform of periodic structures, where signal-related information is sharply localized, the transform of a non-periodic structure has signal and noise components intermixed in a seemingly indistinguishable way. The answer, still underlying the principle of resolution measurements today, is to split the data set randomly in half and compare the two averages (or 3D reconstructions) over rings (or shells, respectively) with increasing radius in Fourier space using a suitable measure of reproducibility. As to the criterion to be used, I originally proposed three measures (Fig. 2f): the differential phase residual (DPR), which was familiar to X-ray crystallographers and thus had the advantage of being widely appreciated; an ‘R factor’ based on a comparison of Fourier amplitudes; and the (generalized, normalized) Euclidean distance, which is essentially an inverted Fourier ring correlation plot. Of the three measures, two were axed by the editor to enforce the length restrictions of a Science report, keeping only the DPR in the paper, and leaving it to other authors to propose the now-popular Fourier ring/shell correlation (Saxton & Baumeister, Reference Saxton and Baumeister1982; van Heel et al. Reference van Heel, Keegstra, Schutter and van Bruggen1982).

While the typical DPR (Fig. 2f, bottom) starts at 0 and rises like a parabola, indicating increasing phase residuals, the typical Fourier ring or shell correlation curve starts at 1 and falls off in a more or less sigmoidal way, reaching a value of 0 or close to zero asymptotically (Fig. 3). Thus, both curves show that the reproducibility of structural information is generally quite good at low spatial frequencies and poor at high spatial frequencies. To arrive at a single quotable figure, a criterion had to be defined for the acceptable level of reproducibility. Initial controversies (Orlova et al. Reference Orlova, Dube, Harris, Beckman, Zemlin, Markl and van Heel1997; Malhotra et al. Reference Malhotra, Penczek, Agrawal, Gabashvili, Grassucci, Jünemann, Burkhardt, Nierhaus and Frank1998) have given way to the acceptance of FSC=0·5 as criterion, which was initially introduced by Boettcher et al. (Reference Boettcher, Wynne and Crowther1997) in a study of the hepatitis B virus capsid. (This criterion is, incidentally, quite consistent with the criterion DPR=45° earlier used.)

Fig. 3. Example of Fourier shell correlation curves, with resolution criterion (FSC=0·5) indicated. Data reproduced from Sengupta et al. (Reference Sengupta, Nilsson, Gursky, Kjeldgaard, Nissen and Frank2008) .

The half-set-based measurement of resolution has two shortcomings, with opposite effects: one is that it is inherently based on the statistical properties of merely half the data set and thus leads to a pessimistic estimate of resolution, the other is that before being split in half, the entire data set has been processed in the same way, sharing intermediate references in the angular refinement all the way to the end. The halves of the data set are, therefore, not statistically independent, making the resolution estimate overly optimistic (see Grigorieff, Reference Grigorieff2000). It is unrealistic to expect that the two opposite tendencies, overestimation and underestimation, cancel each other out precisely, but it is comfortable to know that they cooperate to push the result closer to the truth. (To jump ahead in time: there is also now an emerging consensus that evidence of structural features resolved, such as RNA helices (Malhotra et al. Reference Malhotra, Penczek, Agrawal, Gabashvili, Grassucci, Jünemann, Burkhardt, Nierhaus and Frank1998), alpha-helices (Boettcher et al. Reference Boettcher, Wynne and Crowther1997; Conway et al. Reference Crowther, Henderson and Smith1997), beta-sheets (Kong et al. Reference Kong, Zhang, Baker and Ma2004; Baker et al. Reference Baker, Ju and Chiu2007), or individual phosphorus atoms showing up as ‘bumps’ along RNA strands (LeBarron et al. Reference LeBarron, Grassucci, Shaikh, Baxter, Sengupta and Frank2008), are the best criterion for resolution achieved in practice.)

6. Correlation averaging: crystals treated as ordered collections of single particles

Application of correlation functions to EM images of thin crystals proved that most crystals are far from ideal. When a small fragment of the crystal image is cross-correlated with the entire field, the resulting peaks lie on curves, not on straight lines, indicating distortions of the lattice. Occasionally, there are dislocations, as well or changes of local structure. As a result, application of algorithms that assume perfect crystals will yield a blurred average. It is straightforward to see that treatment of the crystal as a collection of single particles, one for each repeating unit, will result in a superior average. First applications of the single-particle approach on images of bacterial surface layers confirmed this expectation (W. Baumeister and J. Frank, 1981, unpublished; Baumeister and Saxton, Reference Saxton and Baumeister1982). One can demonstrate that front and back layers of a collapsed bacterial envelope can be easily separated in the real-space approach (Kessel et al. Reference Kessel, Radermacher and Frank1985). However, the most important benefit of the single-particle approach to crystal averaging is that it lends itself to the application of classification (Frank et al. Reference Frank, Chiu and Degn1988), as further elaborated below.

Another approach to exploit information in images of distorted crystals has become more prevalent, under the name of ‘unbending.’ Instead of using the data directly, boxed out from the image and properly repositioned after alignment, the idea here is to ‘unbend’ the data so that they conform to existing programs that have been designed for ideal lattices (van Heel & Hollenbeck, Reference van Heel, Hollenbeck, Baumeister and Vogell1980; Henderson et al. Reference Henderson, Baldwin, Downing, Lepault and Zemlin1986). Unfortunately, computational convenience in this approach leads to the sacrifice of valuable information relating to the clustering of the data due to local changes in conformation (see Frank et al. Reference Frank, Chiu and Degn1988; Morgan et al. Reference Morgan, Grant, Chiu and Frank1992).

In contrast, the single-particle approach has been pursued, with great success, in the characterization and 3D reconstruction of helical polymers such as actins, where changes in local structure appear to be the norm and even seem required for biological function (Egelman, Reference Egelman2007).

7. The need for classification – multivariate data analysis of projections

The need for an objective means of classification was recognized early when 2D averaging of negatively stained biological molecules produced images with somewhat disappointing resolution (Frank et al. Reference Frank, Goldfarb, Eisenberg and Baker1978; Zingsheim et al. Reference Zingsheim, Neugebauer, Barrantes and Frank1980). First of all, classification is a general prerequisite to 2D averaging as it is required to separate molecules in a field into subsets presenting the same view. Since the drum-shaped glutamine synthetase molecules, the subject of the first application of single-particle methods, had a tendency to stand on their circular bases, there was no need for this step in this particular case. However, inspection of these images showed clear evidence of conformational variability, some of it related to radiation damage (Kessel et al. Reference Kessel, Frank, Goldfarb and Baumeister1980).

A fortuitous osmosis of know-how, recounted in a recent profile article (Mossman, Reference Mossman2007), led to the introduction of multivariate statistical analysis in the form of correpondence analysis (see Lebart et al. Reference Lebart, Morineau and Warwick1984) – a standard technique in the field of laboratory medicine at the time – into single-particle analysis. The reason is that once aligned, images of a molecule can be regarded as comparable multicomponent vectors whose closeness is a reflection of their similarity. Hence, clustering in factorial space reflected the existence of cohesive subsets of molecules with identical or closely similar structure and presenting similar orientations. Hemocyanin molecules, which are oligomeric assemblies of one or more building block proteins, present a fascinating variety of views and thus were ideally suited for development and testing of algorithms of classification. Half-molecules of Limulus polyphemus hemocyanin are composed of four hexamers arranged in rhombus. In one of the first demonstrations of the power of the classification technique, images of this molecule were separated into four groups, according to their ‘flip’ versus ‘flop’ views on the grid and the finer division into two rocking positions in both of these views (Fig. 4; van Heel & Frank, Reference van Heel and Frank1981; see Frank, Reference Frank1984). The unique features of the molecule's 3D architecture, the twisted arrangement of the four hexamers, which were revealed in a surprising way from patterns of variation in a 2D factor map, were later confirmed and elaborated in much higher-resolution detail (for most recent works, see Martin et al. Reference Martin, Depoix, Stohr, Meissner, Hagner-Holler, Hammouti, Burmester, Heysd, Wriggers and Markl2007; Cong et al. Reference DeRosier2009).

Fig. 4. Factor map (factors 1 versus 2) resulting from correspondence analysis of L. polyphemus hemocyanin half molecules. Averages of images falling into demarcated clusters are shown in the four corners (a–d). (a) and (c) are interpreted as two rocking positions of molecules lying on one (“flip”) side, while (b) and (d) relate to rocking positions on the other (“flop”) side. Adapted from Frank (Reference Frank1984) .

Following this proof of concept, this and related methods of multivariate data analysis and automated computer techniques of classification based upon them have become indispensible tools in single-particle reconstruction (Frank, Reference Frank1990; van Heel, Reference van Heel1987; van Heel, Reference van Heel1989; see Frank, Reference Frank2006).

8. The jump into the third dimension

The main difficulty in realizing 3D reconstruction of single particles is to obtain an initial reliable assignment of angles to the particle images. The random–conical reconstruction method (Fig. 5a) (Frank et al. Reference Frank, Goldfarb, Eisenberg and Baker1978; Radermacher et al. Reference Radermacher, Wagenknecht, Verschoor and Frank1987; Radermacher, Reference Radermacher1988) achieved this initial assignment by taking advantage of one or more preferential orientations of the particles on the grid and by making use of an additional tilt of the whole field of particles. At once, the extensive bookkeeping required in this approach proved worth the investment, made in the years before, in the formal language aspects of the SPIDER software; in particular, its nested calling capability, its versatility in execution-time file naming, and its capability to pass keyed sets of parameters from one operation to the other. First reconstructions of the ribosome following this protocol were obtained for negatively stained specimens in 1987 (Fig. 5b–d; Radermacher et al. Reference Radermacher, Wagenknecht, Verschoor and Frank1987) and for cryo-EM specimens in 1991 (Frank et al. Reference Frank, Penczek, Grassucci and Srivastava1991; Penczek et al. Reference Penczek, Radermacher and Frank1992).

Fig. 5. Random–conical reconstruction. (a) Principle of the random–conical data collection method. Two images are taken of the same field of molecules. Only molecules are considered that present the same view on the grid. Azimuthal angles are obtained by aligning the images of the untilted micrograph. Thus, with both azimuth and tilt angles known, the Fourier transform of each projection can be properly placed into the 3D Fourier reference frame of the molecule. From J. Frank (unpublished hand-drawing on overhead transparency, 1979). (b–d) Density map of the 50S ribosomal subunit from E. coli, the first 3D reconstruction using the random–conical data collection method. (a) Surface representation of intersubunit face; (b, c) higher-threshold solid model obtained by stacking of contoured slices, viewed from front and back. The subunit was negatively stained with uranyl acetate and air-dried, which accounts for the partial flattening. The ridge of the deep groove running horizontally, termed interface canyon, is created by the helix 69 of 23S rRNA, as later recognized when the X-ray structure of the large subunit was solved. Annotations refer to morphological details; for example, pocket ‘P2’ was suggested to be the peptidyl transferase center and CP the central protuberance. Data reproduced from Radermacher et al. (Reference Radermacher, Wagenknecht, Verschoor and Frank1987) .

An alternative approach to obtaining a de novo assignment of angles is by forming class averages where each class represents particles showing the same view. Their relative angles can then be determined by finding their common lines in Fourier space, following a method first proposed for virus particles (Crowther et al. Reference Crowther, DeRosier and Klug1970), or, equivalently, by finding their common one-dimensional projections in real space (‘angular reconstitution’ – van Heel, Reference van Heel1987). Another variant of the common-lines approach was later developed by Penczek et al. (Reference Penczek, Zhu and Frank1996). A common thread underlying these closely related approaches is found in the Radon transform whose value in EM-related image processing and 3D reconstruction has been explored by Radermacher (Reference Radermacher1997).

Evidently, once angles are determined, reconstruction can proceed along paths well treaded before, except for one complication: weighted back-projection, the non-iterative, hence rapid reconstruction technique, required even-spaced angles while single-particle projections are randomly distributed in angular space. (The availability of a fast algorithm was a particularly important consideration at that time.) This problem was solved by the introduction of general weights designed ad hoc for a given geometry (Radermacher et al. Reference Radermacher, Wagenknecht, Verschoor and Frank1987; Radermacher, Reference Radermacher1988).

Both methods for de novo assignment of angles to projections of an unknown structure will yield an approximate, low-resolution estimate. For an exhaustive exploitation of the data, iterative angular refinement (Penczek et al. Reference Penczek, Grassucci and Frank1994) is subsequently used, in the course of which angle assignments are refined on progressively finer angular grid, until stabilized. One of the major problems that had to be solved is the merging of data with different contrast transfer functions (CTFs), toward a final, CTF-corrected reconstruction (Grigorieff, Reference Grigorieff1998; Penczek et al. Reference Penczek, Zhu, Schröder and Frank1997; van Heel et al. Reference van Heel, Gowen, Matadeen, Orlova, Finn, Pape, Cohen, Stark, Schmidt, Schatz and Patwardhan2000; Zhu et al. Reference Zhu, Penczek, Schröder and Frank1997).

As to the progress in the study of the ribosome structure, further milestones were reached with 25 Å (Fig. 6) (Frank et al. Reference Frank, Zhu, Penczek, Li, Srivastava, Verschoor, Radermacher, Grassucci, Lata and Agraewal1995; Stark et al. Reference Stark, Müller, Orlova, Schatz, Dube, Erdemir, Zemlin, Brimacombe and van Heel1995), 11·5 Å (Gabashvili et al. Reference Gabashvili, Agrawal, Spahn, Grassucci, Frank and Penczek2000) and eventually the range of 6–7 Å (LeBarron et al. Reference LeBarron, Grassucci, Shaikh, Baxter, Sengupta and Frank2008; Villa et al. Reference Villa, Sengupta, Trabuco, LeBarron, Baxter, Shaikh, Grassucci, Nissen, Ehrenberg, Schulten and Frank2009; Schuette et al. Reference Schuette, Murphy, Kelley, Weir, Giesebrecht, Connell, Loerke, Mielke, Zhang, Penczek, Ramakrishnan and Spahn2009). The first discoveries reflecting on the binding of tRNA (Agrawal et al. Reference Agrawal, Penczek, Grassucci, Li, Leith, Nierhaus and Frank1996; Stark et al. Reference Stark, Orlova, Rinke-Appel, Jünke, Mueller, Rodnina, Wintermeyer, Brimacombe and van Heel1997a) and elongation factors (Agrawal et al. Reference Agrawal, Penczek, Grassucci and Frank1998; Stark et al. Reference Stark, Rodnina, Rinke-Appel, Brimacombe, Wintermeyer and van Heel1997b) and the conformational changes accompanying mRNA–tRNA translocation (Frank & Agrawal, Reference Frank and Agrawal2000) – essentially snapshots of a molecular machine in motion – were made just before the X-ray structures of ribosomal subunits appeared (Ban et al. Reference Ban, Nissen, Hansen, Moore and Steitz2000; Wimberly et al. Reference Wimberly, Brodersen, Clemons, Morgan-Warren, Carter, Vonrhein, Hartsch and Ramakrishnan2000).

Fig. 6. Cryo-EM reconstruction of the (empty) E. coli ribosome from 4300 particle images, at 25-Å resolution. Cut-open density maps depict the way mRNA, tRNAs, and the polypeptide chain were thought to be positioned during protein synthesis, a model that has stood up to the test of time. Landmarks on the large subunit (50S): CP, central protuberance; St, stalk base; L1, L1 stalk; IC, interface canyon; T, polypeptide exit tunnel; T1, T2, two putative branches of the tunnel exit; E1, E2, corresponding exit sites. Landmarks on the small subunit (30S): h, head; p, platform; ch, mRNA channel. A, P, putative positions of A- and P-site tRNA. Data reproduced from Frank et al. (Reference Frank, Zhu, Penczek, Li, Srivastava, Verschoor, Radermacher, Grassucci, Lata and Agraewal1995) .

Comprehensive protocols are now available for all steps of single-particle reconstruction: specimen preparation (Grassucci et al., Reference Grassucci, Taylor and Frank2007), cryo-EM imaging (Grassucci et al., Reference Grassucci, Taylor and Frank2008), and image processing using SPIDER (Shaikh et al., Reference LeBarron, Grassucci, Shaikh, Baxter, Sengupta and Frank2008).

9. Classification of data with entangled viewing angles and class memberships

Classification in cryo-EM applications was initially concerned with two different, very specific situations: grouping into classes was to be achieved either for different copies of a presumably invariable molecule presenting different orientations – and here the objective of classification was the separation of molecule images according to view – or different conformers of a molecule coexisting in the sample and occurring in essentially the same orientation. The latter situation is typical for negatively stained samples, which often show strong preferences of orientation.

As we have seen, a powerful approach to solve either of these two separate problems was found early on by the introduction of multivariate data analysis into EM. Yet cryo-EM of molecules from a heterogeneous sample presents a more formidable problem as the molecules lack preferences for distinct orientations, so that the variation in the appearance of the projections is the result of two intermingled effects, one being the variation in view angle over the entire angular sphere, the other the coexistence of different states of the molecule (which includes binding states of functional ligands). In this case, application of multivariate data analysis is not helpful, and another solution is needed to be found.

The approaches to the solution fall in two categories, referred to as supervised and unsupervised classification. In the first case, each experimental image is compared with two or more 3D reference maps, and assignment to a class is based on the similarity to those references, as measured by the cross-correlation coefficient (see Valle et al. Reference Valle, Sengupta, Swami, Burkhardt, Nierhaus, Agrawal and Frank2002). In the second case, data are clustered according to their self-similarity and consistency, under the assumption that they originate from one of a known number of 3D structures. The second kind of approaches is obviously much preferred, as the outcome is independent of model assumptions. Among several ones recently developed, I should mention Scheres et al.'s (Reference Scheres, Goa, Valle, Herman, Eggermont, Frank and Carazo2007) maximum likelihood approach, which produced remarkable agreement with the result of supervised classification in the case of a well-characterized problem: of dividing a data set of a ribosome complex according to its conformational state (macrostate I versus II; see Frank et al. Reference Frank, Gao, Sengupta, Gao and Taylor2007) (Fig. 7). However, to develop new algorithms, one would ideally require a data set generated from a phantom, such that the class membership of each projection is known. Only in that case can the performance of the algorithm be characterized by the number of true and false positives and so forth. Generation of such a data set requires the simulation of the EM imaging and noise processes. A step in this direction has recently been done by Baxter et al. (Reference Baxter, Grassucci, Gao and Frank2009). Ultimately, unsupervised classification of data created by cryo-EM of molecular machines engaged in all steps of processing may allow us to create ‘movies’ in which ligands and other moving parts come to life.

Fig. 7. Classification of ribosome complexes with or without EF-G by two methods: supervised and unsupervised classification. The experimental data set contains two states of the ribosome, distinguished as follows: (one) contains no EF-G; the A and P sites are occupied by tRNAs in the classic state; and the ribosome is in macrostate I (i.e., unratcheted), the other has EF-G bound; there is a single tRNA in P/E position; and the ribosome is in macrostate II (i.e., ratcheted). (a) Results of classification with the maximum likelihood (ML) method, with four classes specified. (b) Cross-reference between classes from ML and those derived from dividing the correlation histogram obtained by supervised classification. (c) Results of supervised classification using two reference maps, in ribosome macrostates I and II but without EF-G. Data reproduced from Scheres et al. (Reference Scheres, Goa, Valle, Herman, Eggermont, Frank and Carazo2007) .

10. Interpretation of EM density maps by flexible fitting

Most density maps obtained in EM are of insufficient resolution to allow ab initio chain tracing or even a determination of the atomic structure as practiced in X-ray crystallography. In a few cases, this has been achieved for highly symmetric molecules where the number of repeats in a single molecule boosts the statistics (e.g., Ludtke et al. Reference Ludtke, Baker, Chen, Song, Chuang and Chiu2008; see review by Lindert et al. Reference Lindert, Stewart and Meiler2009). The situation more frequently encountered is that the resolution is in a range (~6–12 Å) where the map can be used as one of several constraints for flexible fitting, and the resulting structural model, far from being fixed with all of its side chains, must be understood as a plausible representative of an entire ensemble of structures all consistent with the cryo-EM data (see a discussion of this point by Trabuco et al. Reference Trabuco, Villa, Mitra, Frank and Schulten2008). The terms quasi-atomic or pseudo-atomic structure have been used to refer to coordinates that have this residual uncertainty, but these terms are somewhat negatively loaded, as they contrast structural models obtained through interpretation of cryo-EM maps with a purportedly ‘real’ structure obtained by interpretation of X-ray density maps. A few words are in order on the rapidly shrinking difference between models of one kind and the other.

There is frequently a situation where the building of atomic models with tentative positioning of side chains is desired to characterize the binding of a ligand to a target molecule. Even though the density map cannot pin down the side chains, the choices are drastically narrowed down since steric and chemical constraints come into play (Volkman et al. Reference Volkmann, Amann, Stollova-McPhie, Egile, Winter, Hazelwood, Heuser, Li, Pollard and Hanein2001). In a certain sense, the building of a structural model from a 3D EM density map by flexible fitting can be compared with the building of atomic models from an electron density map obtained by X-ray crystallography, since in both cases known structural primitives (i.e., entire component structures in the former case and the known structures of an enumerable set of residues in the latter) are assembled to explain the observed density, following rules of stereochemistry (see, for instance, Topf et al. Reference Topf, Lasker, Webb, Wolfson, Ciu and Sali2008).

Again, the ribosome, with its large size and complexity and existence of large-scale conformational changes such as the ratchet motion (Frank & Agrawal, Reference Frank and Agrawal2000), proves to be the most challenging testing ground for the development of algorithms for flexible fitting. As a molecular machine with large-scale conformational changes and multiple binding interactions, the ribosome seen by cryo-EM is ever-changing, proving that the underlying atomic structure possesses intrinsic instabilities and that it visits several thermally accessible states in its working cycle (Frank et al. Reference Frank, Gao, Sengupta, Gao and Taylor2007). Three approaches to flexible fitting and their application to the ribosome should be mentioned: real-space refinement (Chapman, Reference Crowther, DeRosier and Klug1995; Gao et al. Reference Gao, Sengupta, Valle, Korostelev, Eswar, Stagg, Van Roey, Agrawal, Harvey, Sali, Chapman and Frank2003), normal-mode analysis flexible fitting (Tama et al. Reference Tama, Miashita and Brooks2004) and molecular dynamics flexible fitting (Trabuco et al. Reference Trabuco, Villa, Mitra, Frank and Schulten2008). Their relative merits have been discussed by Trabuco et al. (Reference Trabuco, Villa, Mitra, Frank and Schulten2008) and Lindert et al. (Reference Lindert, Stewart and Meiler2009). Arguably, fitting tools employing molecular dynamics, such as molecular dynamics flexible fitting, appear to come closest to a structural interpretation bestowing the atomic model with realistic stereochemical properties.

One of the most challenging problems in modeling is presented by the 80S eukaryotic ribosome, for which no structural data exist except for density maps by cryo-EM. While the functional core, composed of rRNAs, is highly conserved, peripherally located expansion segments acquired in the course of evolution have largely unknown structure and function. Here the challenge is to build a model de novo conforming with RNA sequence, local engagement rules, and observed density. Second, there are proteins with eubacterial counterparts that require homology modeling. Lastly, there are a number of proteins entirely without any eubacterial counterparts, which require de novo modeling. Two models exist at the present time, one for yeast (Spahn et al. Reference Spahn, Beckmann, Eswar, Penczek, Sali, Blobel and Frank2001; Fig. 8), the other for dog (Chandramouli et al. Reference Chandramouli, Topf, Menetret, Eswar, Cannone, Gutell, Sali and Akey2008). Another, more comprehensive atomic model for yeast based on a cryo-EM map of the ribosome from the closely related species T. lanuginosus is forthcoming (Taylor et al. Reference Taylor, Devkota, Huang, Topf, Narayanan, Sali, Harvey and Frank2009). Thus, already today, the eukaryotic ribosome, recently highlighted as one of the ‘structures of desire’ by X-ray crystallographers (Bhattacharya, Reference Bhattacharya2009), presents a rich trove of structural information thanks to the improvements both of cryo-EM as a technique and of the tools for mining the resulting density maps.

Fig. 8. Homology modeling of the proteins and RNA of the 80S eukaryotic ribosome from the fungus T. lanuginosus, based on the sequence of a close relative, Saccharomyces cerevisiae, and on the X-ray structure of E. coli (Taylor et al. Reference Taylor, Devkota, Huang, Topf, Narayanan, Sali, Harvey and Frank2009). Structure in red seen in front is eEF2.

11. The potential and limits of single-particle reconstruction

Glancing over the peripatic history of the single-particle reconstruction method, from its obscure beginning in 1978 to its present wide acceptance makes me realize that the technique has succeeded in a way nobody imagined. For one, I would not have predicted that a molecule, after the harsh treatment of freeze-plunging and without support of companions in a crystal matrix, would allow us to see molecular detail at its very periphery (Fig. 9). Apparently, only the outermost alpha-helix of protein S2 has been affected in this reconstruction of the ribosome.

Fig. 9. Cryo-EM reconstruction of the 70S ribosome of E. coli, with Phe-tRNA^Phe·EF-Tu·GDP bound in the presence of kirromycin (LeBarron et al. Reference LeBarron, Grassucci, Shaikh, Baxter, Sengupta and Frank2008). Yellow: 30S subunit; blue: 50S subunit. Highlighted peripheral ribosomal protein S2 (encircled) is compared with its X-ray structure (orange) presented in the same orientation. Arrow points to an outermost helix that is missing in the density map, presumably because of disorder, while the rest of the structure is intact.

Foremost, single-particle reconstruction provides a way to image macromolecular complexes too large and too flexible to allow application of X-ray crystallography. The technique's strength lies in its ability to yield 3D images of ‘molecular machines’ in a native environment, in a variety of biochemically well-characterized states, unconstrained by intermolecular contacts characteristic for a crystal. It is difficult, however, to find recipes for improving the resolution beyond what has been achieved, except for brute-force increasing data collection in ways that severely challenge the current means of storage and processing. The fact that close to atomic resolution has been actually achieved for some molecules bearing high symmetry offers hope that the same can be done for molecules lacking it, but it gives daunting estimates on the number of particles to be collected and processed. Ultimately, what needs to happen is the establishment of a data collection and processing pipeline in which smart scrutiny based on existing knowledge is applied at the very start.