Helical Reconstruction, Again

Abstract

Many protein and nucleoprotein complexes exist as helical polymers. As a result, much effort has been invested in developing methods for using electron microscopy to determine the structure of these assemblies. With the revolution in cryo-electron microscopy (cryo-EM), it has now become routine to reach a near-atomic level of resolution for these structures, and it is the exception when this is not possible. However, the greatest challenge is frequently determining the correct symmetry. This review focuses on why this can be so difficult and the current absence of a better approach than trial-and-error.

A huge number of assemblies exist in biology that are helical polymers. These range from bacteriophage tails [1–4] to F-actin filaments [5,6••] with myriad other examples in between. All amyloid filaments involved in various pathologies are helical and are now being intensively studied at near-atomic resolution by cryo-EM [7••]. Ironically, a solid-state NMR study of “the nonhelical filament of the Alzheimer’s disease tau core” [8] was actually looking at helical filaments with exactly two subunits per turn (a 2₁ screw in crystallographic notation). While almost 40 years ago [9] x-ray fiber diffraction was seen as the only way to achieve near-atomic resolution in determining the structure of a native helical assembly (tobacco mosaic virus), cryo-EM has emerged over the past 10 years as the main technique in this area. In fact, it is now routine to reach a near-atomic level of resolution using cryo-EM on biological polymers, and the instances where this is not possible are becoming the exception and not the rule.

The first application of three-dimensional reconstruction in electron microscopy was to a helical phage tail [10], so the biological community has become quite aware of the central role that helical structures have played in the development of electron microscopic techniques. What has now become apparent is that researchers in chemistry and materials science have been relatively unaware of the power of cryo-EM in determining the structure of helical assemblies formed from small peptides or small molecules, relying instead on methods such as SAXS (small angle x-ray scattering) or MD (molecular dynamics) to “validate” atomic models. This is beginning to change, with atomic structures of widely-studied cyanine dye [11•] or dipeptide [12•] nanotubes recently determined by cryo-EM. Whether one is looking at a protein polymer or a small-molecule nanotube, some of the same challenges still remain when using cryo-EM, with the main one being the determination of helical symmetry. So this short review will focus on the problems in determining helical symmetry.

Determining Helical Symmetry

There are a number of approaches for estimating the helical symmetry of a polymer, but the most powerful one involves analysis of the power spectrum [13–16]. Any helical polymer will have axial periodicities, and these give rise to layer lines in the power spectrum where diffraction only occurs on these layer lines and is zero off them. Fourier-Bessel analysis [17,18] can be used to “index” such a power spectrum, assigning Bessel orders to each layer line, and these assignments give one the symmetry, i.e., the twist and rise of the asymmetric unit, as well as a possible point group rotational symmetry, C_n. There are many good presentations explaining the relation between the peaks seen in the power spectrum and the underlying helical symmetry [19]. The classical formalism for indexing power spectra involved representing helical symmetry as integer ratios. For example, tobacco mosaic virus can be described as a helix with 49 subunits in 3 turns, or F-actin can be described as a helix with 13 subunits in 6 turns, or 28 subunits in 13 turns. It is now widely accepted that there is no reason for such symmetries to be described as ratios of integers, and helical symmetry is best thought of in terms of the twist and rise, real numbers that are continuously variable [20]. There are a number of tools readily available that simplify the process of trying to index such power spectra. One has been available on-line for a number of years: https://rico.ibs.fr/helixplorer/helixplorer. Another is a Python-based Helical Indexer which provides a graphical user interface for indexing power spectra [21•].

Nevertheless, for over 50 years people have struggled with the problem of determining the helical symmetry of filaments from x-ray fiber diffraction (which is used infrequently today) or from power spectra obtained by cryo-EM, due to intrinsic ambiguities that exist [16,22]. While a helical structure consisting of atoms all at the same radius would generate a power spectrum that is quite simple to index [13], no biological polymer is described or even approximated by such an arrangement of atoms at the same radius.

For a nucleoprotein complex, the SIRV2 virus that infects archaea living in almost boiling acid [23], the indexing problem can be readily seen. A cross-section of the atomic model for the protein wrapping around the DNA (Fig. 1a) shows how the structure extends over a range of radii, from an inner 35 Å to an outer 90 Å. The consequence of this is that the peak intensities on the layer lines in the power spectrum cannot be indexed as arising from a single radius, and the effective radius where contrast is generated may be different on different layer lines. Imagine that a structure gives rise to a layer line with n=4 and another with n=7. Imagine also that there is a very strong 4-start helical feature at a large radius, and a very strong 7-start feature at a small radius when the density is projected down the axis along these helical families. So the effective radius for the n=4 layer line will be large, and the effective radius for the n=7 layer line will be small. Thus, the radius of contrast will be quite different for these two different layer lines.

Figure 1.

(a) A cross section of the SIRV2 virus [23], shown in ribbon representation. The DNA is in red, and the protein is in yellow. The inner radius is ~ 35 Å and the outer radius is ~ 90 Å. The large range between the inner and outer radius means that contrast for different helical families can be coming from different radii. (b) The averaged power spectrum from ~ 100,000 overlapping segments of SIRV2 virions. The averaged power spectrum is not the same as the power spectrum of a class average, and typically contains much more information. The log of the intensities is shown to allow for a larger dynamic range. Three layer lines are indexed with their Bessel orders.

This problem can be seen in the averaged power spectrum generated from segments of the virus particles (Fig. 1b) where the distance of peaks from the meridian on different layer lines cannot simply be explained by their Bessel orders. In the simplest case, where all atoms are at the same radius, the amplitude along a layer line will be described by a Bessel function of order n, J_n(2πRr), where r is the radius of the atoms, and R is the distance in reciprocal space from the meridian. Left-handed helices give rise to a negative of n, while right-handed helices are positive. The power spectrum gives no information as to whether a particular layer line arises from a left-handed or a right-handed helix, it only contains information about the absolute value of n. For n≥10, the first maximum of this function is when the argument 2πRr=n+2. For this simplest case, the distance from the meridian is thus proportional to n. Knowing the radius r, one can then estimate the Bessel order n on a given layer line simply by measuring the distance R of the first maximum from the meridian. But what one can immediately see in Fig. 1b is that the first diffraction peak on a layer line indexed as n=−14 is actually closer to the meridian than the first peak on a layer line indexed as −12. This is the opposite of what would be expected in the simplest case (a single radius of atoms) and shows that the contrast for the n=−12 helix is at a smaller radius than where contrast is arising from the n=−14 helix. If they were arising from the same radius, then the peak from a Bessel order 14 would certainly be further from the meridian than a peak from a Bessel order 12. This complicated the indexing of this power spectrum, and as discussed in the original publication [23], at least 10 different helical symmetries were possible and consistent with the observed power spectrum. How does one determine which is correct? Simply by trying all of them and finding the one that yields recognizable and interpretable density.

In addition to the problem of a single helical radius not being applicable to most biological structures, the simplest formalism for helical diffraction assumes that filaments are strictly perpendicular to the x-ray beam (in the case of fiber diffraction) or electron beam (in the case of electron microscopy). While negative stain EM, with filaments adsorbed to a substrate, essentially constrains filaments to be in a plane perpendicular to the electron beam, the relatively thin films used in cryo-EM can still allow for a substantial out-of-plane tilt. Of course, the thinner the ice the less out-of-plane tilt will be possible. Assuming no out-of-plane tilt when there is a substantial amount present can lead to an assignment of the wrong helical symmetry, as was done in the case of the MAVS filament [22,24–26]. Unfortunately, there is no simple and reference-free way to determine the out-of-plane tilt for a particular filament or segment without using tomography, and the use of references can be problematic: an incorrect reference can give an incorrect assignment of tilt. In many cases, this can be used as an indication that the symmetry being used is wrong. For example, if one has long rigid filaments in relatively thin ice and there is a mean absolute out-of-plane tilt of 15–20°, this is a good clue that the chosen symmetry is wrong. What is happening is that by using a very large tilt one can get a better alignment of the projected reference volume (having the wrong symmetry) with the real segments. While this may be hard to prove, the expectation would be that the correct symmetry generates a tilt distribution with the minimum variance. But one would never want to depend upon this for confirmation of the correct symmetry, and interpretable high resolution features remain the true gold standard for the correct symmetry. The highest measured resolution, surprisingly, is not the gold standard, as will be discussed further below.

The combination of out-of-plane tilt with a finite range of atomic radii are typically both present. Even for peptides or small molecules that may assemble into thin-walled hollow tubes, such as Lanreotide [27], the finite thickness of the walls means that approximating these as atoms at a single radius rapidly breaks down. When combined with the ambiguities due to out-of-plane tilt, for the small Lanreotide octapeptide that assembles into large (~275 Å) diameter tubes, there were ~50 possible helical symmetries that were tested [27].

This problem can be illustrated in another rather thin-walled structure that has recently been described [12], self-assembled from a dipeptide (Fig. 2). The dipeptide is two phenylalanine residues (l and d stereoisomers) with a naphthalene double-ring attached. When the phenylalanines are both in the l conformation (l,l), the tubes formed are ~ 80 Å in diameter [12]. However, when the phenylalanines are l,d, the tubes formed are ~ 300 Å in diameter. The small subunit combined with the large diameter, with an unknown amount of out-of-plane tilt, again leads to a plethora of possible solutions, with ~50 tested. For example, there is a layer line seen at 1/(3.0 Å), well beyond the water ring at ~ 1/(3.7 Å) (Fig. 2b). The simplest assumption was that this was a meridional (n=0), corresponding to the reciprocal of the axial rise per asymmetric unit. It actually turned out that this was n=9; the peaks on that layer line were shifted towards the meridian due to small out-of-plane tilt, as nothing constrains these tubes to lie perfectly perpendicular to the electron beam in the ice layer. The asymmetric unit contains two dipeptides (Fig. 2a), with one forming the outer wall and the other the inner wall.

Figure 2.

(a) A cross section of a tube formed by a napthalene-conjugated dipeptide, containing an l-phenylalanine and a d-phenylalanine [12•]. A close-up view of the tube wall (center) shows that the asymmetric unit is two dipeptides, one forming the outer wall and the other forming the inner wall. The aromatic rings of the phenylalanine residues are packed in the center of the wall. (b) The averaged power spectrum from these tubes. The water ring is seen at ~ 1/(3.7 Å), which shows that individual water molecules are clearly resolved in the raw images. A layer line is indicated at a spacing of 1/(3.0 Å). While this layer line could be n=0, it was actually determined to be n=9.

As with all single-particle cryo-EM studies, conformational heterogeneity readily arises when looking at helical structures and can add complications to determining a simple helical symmetry. This heterogeneity can manifest itself as variability in the twist and/or axial rise, as well as by filament flexibility. For example, a structure of the Vibrio cholerae toxin coregulated pilus [28] was done asymmetrically, as the imposition of helical symmetry led to a very limited resolution of ~ 5.8 Å due to the extreme flexibility of these filaments. In contrast, an asymmetric reconstruction of a curved segment led to a resolution of ~ 3.8 Å.

A Better Method for Determining Helical Symmetry?

A recent paper [29] proposes a method that is almost breathtaking in its simplicity: generate an ab initio 3D reconstruction without any imposed symmetry, and then use cross-correlations to analyze this reconstructed volume for the helical symmetry parameters (rise and twist per subunit). However, there are two main problems with what is proposed here. This approach is actually one of the suggested paths for doing helical reconstruction in cryoSPARC. This is fully explained in the cryoSPARC documentation: https://guide.cryosparc.com/processing-data/all-job-types-in-cryosparc/helical-reconstruction-beta/helical-symmetry-in-cryosparc. There is a section entitled “Using asymmetric reconstructions and symmetry search utility job to explore symmetry parameters”, which explains how to do an ab initio asymmetric reconstruction for a helical filament, and then how to run the symmetry search utility which generates 2D cross-correlation plots for rise and twist. With that being said, the new paper does not seem to offer much beyond what already exists.

The problem with this approach of using an asymmetric reconstruction is that the correct or incorrect helical symmetry has already been locked in when one generates the asymmetric reconstruction, and the asymmetric reconstruction is not unique. That is, just as one can take an ensemble of images and generate different 2D averages from this same ensemble (including one that will look like Albert Einstein if Einstein is used as a reference), one can generate different 3D volumes from this ensemble with the assignment of different Euler angles and translational parameters to each image. If one uses a reference volume that has a particular helical symmetry, then this symmetry will “appear” in the asymmetric reconstruction. If one uses a featureless initial reference, assignments of Euler angles to each segment will be stochastic but one can converge on different symmetries in different reconstructed volumes using the same image stack. Thus, I was very impressed when cryoSPARC determined the correct helical symmetry from a data set of segments using this approach. But rerunning everything with the same parameters, the correct “solution” was not even in the top 20 found. This is obviously because the correct symmetry had been locked into the first asymmetric reconstruction, while an incorrect symmetry had been locked into the second volume generated by an asymmetric reconstruction. Once the incorrect symmetry gets locked in, there is no algorithm or search procedure that can find the correct symmetry in this volume. It is impossible to imagine that an asymmetric ab initio reconstruction in Relion will be any better. Unless there is some magic that can be employed to assign Euler angles to helical segments correctly, all such ab initio asymmetric volumes will have the potential to have an incorrect symmetry.

The reason that there is no simple solution to this problem of finding the correct symmetry is not a failure of current methods, it is simply a mathematical ambiguity such that there are multiple solutions that are indistinguishable (in terms of residuals, etc.) at some finite resolution. This is therefore different from what people have described as “getting trapped in local minima” when programs such as Relion are unable to converge to the correct helical symmetry. First, the function that one is trying to minimize is usually never defined. But if we imagine that it is some form of χ² looking at the squared differences between the images and the projections of the reconstruction, then in many cases there are global minima at some finite resolution that cannot be distinguished. Obviously, the higher the resolution the fewer such minima may exist.

The paper [29] shows four applications that result in the correct symmetry being found. While they raise the caveat that “Different ab initio asymmetric reconstructions from the same dataset can yield different apparent helical symmetries, and none may be correct” it is not clear from the paper the extent to which this may be the case. I therefore used their web version to calculate the helical symmetry for three different structures. In each case, we generated two asymmetric volumes from the same set of image segments. Due to the stochastic elements in this reconstruction process (discussed above), these two volumes are never identical. The dismal results are shown in Table 1. If one takes any of these parameters and uses them to initiate a helical reconstruction in either cryoSPARC or Relion they will never converge to the correct structure. Nevertheless, the correct symmetry can sometimes be found with this approach, and thus inexperienced users may find this approach a useful tool that does not require prior knowledge of Bessel functions and helical indexing.

Table 1.

Three different helical samples were used to test the utility of symmetry estimation from asymmetric reconstructions. In all three cases (A,B, and C) the true symmetry was known from high resolution reconstructions. In each case, the procedure of generating an asymmetric reconstruction was run twice (volume 1 and volume 2). Due to the stochastic elements in this reconstruction procedure, volumes 1 and 2 are always different. In all three cases, the helical parameters returned from examining these asymmetric volumes were so far from the true parameters that they would never converge to the correct solution. The images used for B have been deposited in EMPIAR with accession code #####.

		point group	rise (Å)	twist (°)	pitch (Å)
A	volume 1	C1	4.14	−35.95	41.5
	volume2	C1	4.75	−41.37	41.3
	true symmetry	C1	5.75	38.46	53.8
B	volume 1	C1	8.17	−9.51	309.3
	volume 2	C1	8.21	13.67	216.2
	true symmetry	C2	4.11	−83.41	17.7 (for 2-start)
C	volume 1	C1	5.47	19.61	1,290.6
	volume 2	C1	5.93	19.57	1,188.1
	true symmetry	C1	0.305	22.59	4.86

Symmetry and Resolution

At some finite resolution one may actually find that two different symmetries yield almost identical reconstructions. This was shown for a bacterial mating pilus [30] where, at 5 Å resolution, two symmetries yielded almost the same maps with nice rod-like density for α-helices. If one were unable to extend the resolution, then the true symmetry would have been ambiguous. But one of the two symmetries led to a 3.9 Å map, while the incorrect symmetry would never improve beyond 5 Å. This might lead to the suggestion that one can find the correct symmetry by choosing the reconstruction that leads to the highest measured resolution. One would thus have a simple metric that could be used in fully automated searches over a large number of potential symmetries to find the correct one. Unfortunately, the standard method of determining resolution, using the Fourier Shell Correlation (FSC) between two independent half-maps, is not really a measure of resolution but a measure of reproducibility [31]. For example, when looking at microtubules that contain a seam [32], averaging α- and β-tubulin subunits properly to take account of the seam (a helical discontinuity) can actually lead to a slightly worse FSC than when these are averaged together improperly (ignoring the seam). However, the map with the proper seam was fully interpretable in terms of atomic models, while the map generated by ignoring the seam had artifacts [32]. Similarly, for the diphenylalanine nanotubes [12•], all symmetries, whether correct or incorrect, yielded a map:map resolution estimate of 3–4 Å.

This immediately leads to the question of what is the precision to which resolution can be judged? It is still common in the literature to see the resolution of cryo-EM volumes described to one-hundredth of an Å (e.g., 3.22 Å), as if the last digit has some significance. To illustrate the absurdity of this, let us look (Fig. 3a) at using the “gold standard” FSC for a tad pilus reconstruction [28]. Using an extremely tight mask one can get a “gold standard” FSC resolution of 2.2 Å. But the artifacts produced by such a mask are obvious. Using a less tight mask one gets 2.7 Å with no obvious artifacts. With an even looser mask, one can get 3.7 Å. But the visual appearance of the map, and the map:model FSC, both suggested that the “true” resolution was ~ 3.3–3.4 Å [28]. Thus, trying to claim that the resolution of this volume, or any cryo-EM volume for that matter, has been determined to a precision of one-hundredth of an Ångstrom is simply ludicrous. Similarly, using the map:map FSC the resolution of the L,L tubes [12•] was judged to be 2.7 Å, while the map:model estimated a more reasonable 3.3 Å. Further, independent of this precision, no cryo-EM map has a uniform resolution. So while the best estimate may be ~ 3.2 Å in one region, it may very well be ~ 3.4 Å in another region.

Figure 3.

(a) The “Gold Standard” FSC computed by cryoSPARC for a helical volume, the tad pilus [28], using three different masks. At the top, an excessively tight mask yields a GSFSC of 2.7 Å, but one can immediately see that this is artifactual due to the fact that the tight masked curve (red) never falls to 0.0 (showing residual correlations between the two half-maps out to the Nyquist frequency), and by the large deviation of the Corrected curve (purple) from the tight masked curve. Using a less tight mask (center), the GSFSC is given as 2.7 Å, and there are no obvious artifacts. But using a much less restrictive mask (bottom) the GSFSC is given as 3.7 Å. Based upon the visual appearance of the map and the map:model FSC, a resolution of 3.3–3.4 Å was reported. The spike at 1/(4.9 Å) in many of these curves is due to the extremely strong signal in the images at 1/(4.9 Å) arising from the packing of tilted α-helices. Thus, the signal-to-noise ratio (SNR) at this frequency is anomalously high. (b) The published map:map FSC and the map:model FSC (C_ref) for a cryo-EM structure of the P53 dimer [33]. Notice that the x-scale only covers the very narrow range from 0.22 to 0.255. The assumption would be that both the FSC and C_ref curves would be 1.0 in the much larger interval from 0.0 to 0.22. Copyright Wiley-VCH GmbH. Reproduced with permission. (c) The map:map FSC for the P53 dimer (EMD-28816) calculated using the deposited half-maps. Not only is this curve completely different from the FSC curve in (b), but the curve never crosses a threshold of 0.143, and the two half-maps retain significant correlation out to the Nyquist frequency. (d) The map:model FSC for the P53 dimer, using EMD-28816 and PDB 8F2H. Not only is this curve completely different from the C_ref plot in (b), but there is no significant correlation between the map and model at any spatial frequency.

This next leads to the question of how much confidence can be placed in published statements about resolution? In Fig. 3b I reproduce a published map:map and map:model FSC for a P53 dimer structure determined by cryo-EM [33]. Theoretical arguments [34] suggested that if one uses a recommended FSC=0.143 threshold for judging the resolution in comparisons of two half-maps, then 0.5 should be the threshold when comparing the full map to a model. Remarkably, the resolution where the map:map curve crosses 0.143 is almost exactly the same resolution where the map:model curve crosses 0.5 in this stunningly beautiful plot. However, actual map:map (Fig. 3c) and map:model (Fig. 3d) comparisons using the deposited data look nothing like Fig. 3b. These blatant inconsistencies have previously been raised online: https://pubpeer.com/publications/087884F390E53FCF60DE6BD9086CC4.

The conclusion must be that the reviewers of this paper [33] never looked at the deposited data, or lacked the expertise to judge it. As the number of deposited cryo-EM structures continues to grow exponentially, it is imperative that minimum standards be maintained and that all cryo-EM papers be reviewed critically.

Future Outlook

In this very brief review I have discussed some of the challenges in symmetry determination, one of the most problematic aspects of helical reconstruction. Will these difficulties always be with us? I suspect not, given the huge advances over the past several years in machine learning. While I have discussed that searching for the correct symmetry maximizing a metric such as estimated resolution or minimizing out-of-plane tilt will likely lead to false solutions, recognition of protein structural features is something that computer programs can do as part of automated methods to test many alternative symmetries. I thus think that “expert” programs will ultimately remove most of the present need for human expertise.