Limitations of the iterative electron density reconstruction algorithm from solution scattering data

Small-angle X-ray scattering (SAXS) is widely utilized to study biological macromolecules in solution and methods are established to recover ab initio low resolution shapes from the scattering data. In a recent publication by T.Grant¹ an iterative algorithm named DENSS is described, which is claimed to reconstruct ab initio inhomogeneous electron-density maps from small-angle X-ray scattering (SAXS) data. We are concerned that this approach is not capable of reconstructing internal structure and we show examples of models that do not fit the experimental data. We therefore urge users to be cautious when interpreting DENSS results.

DENSS¹ employs averaging of multiple three-dimensional (3D) densities, each iteratively restored from the isotropic SAXS data. In our own tests of DENSS, the maps deduced from the averaged density agree at low resolution with the shapes of macromolecules, making the method a valuable contribution to the difficult task of shape analysis from SAXS data. Grant, however, further states that the approach is “capable of reconstructing accurate electron density maps […] of particles containing multiple different molecular species with distinct electron densities”. In other words, Grant claims that DENSS provides not only the overall particle shapes like the established ab initio methods (e.g. DAMMIN² or DAMMIF³) but also delineates different particle densities within inhomogeneous objects.

In support of this statement, Grant presents an example using a theoretical SAXS curve for a complex of endophylin A1-BAR protein with two arachidonyl–CoA lipid micelles (computed from an atomistic model available from the SASBDB archive, www.sasbdb.org, ID: SASDAX5). In our view, however, this complex is an unsuitable example of a multi-component system because the electron contrast of the lipid head groups is much higher than that of the protein moiety. The two CoA micelles dominate the intensity (by a factor of fifteen in forward scattering), and the model is therefore well approximated by a hollow dumbbell shape. We calculated the theoretical SAXS curve from SASDAX5 with CRYSOL⁴, and the ab initio shapes restored from this curve by bead modelling program DAMMIN reproducibly reconstruct the hollow dumbbell (Supplementary Figure 1). The CoA micelles can therefore not serve as an example of a multi-component system.

A natural and practically important test of the ability to reconstruct multi-component systems is to use scattering data of protein-nucleic acid complexes. In aqueous solutions, the X-ray contrast of DNA (~0.22 electron/ų) is about twice that of proteins (~0.10 electron/ų), i.e. such complexes are classical two component particles. We downloaded several protein-DNA models from the Protein Data Bank (PDB, www.wwpdb.org), and calculated the theoretical SAXS curves from these models with CRYSOL⁴. We ran DENSS on this data and superimposed the results onto the atomic models with SUPALM⁵ (Figure 1a). Whereas the overall shapes resemble those of the crystal structures, none of the maps shows higher density features at the location of the DNA moieties. Instead, independently of the position of the DNA, higher density parts in the averaged maps are located close to their geometrical centers.

Figure 1.

DENSS reconstructions from theoretical scattering data computed up to a nominal resolution of 12 Å. Averaging was done over 100 reconstructions by the ‘superdenss’ script. Two-dimensional slices are coloured from red (higher density, relative value of 20) to blue (lower density, value 5). The threshold to display was selected to ensure the best shape overlap between the initial models and DENSS maps; bar length, 50 Å. (A) Protein-DNA complexes (PDB codes 1B3T, 4E54, 5Y0D and 6FTX), and two protein models, with PDB codes 6PWC (A and R chains) and 6KYK (A and D chains), with artificially doubled contrast of one domain (R and D chains, respectively). The crystal structures are displayed as cartoons (protein, blue; DNA and doubled domains, red). (B) Ellipsoids of revolution: top row, model particles (left, uniform ellipsoid; middle and right, non-uniform ellipsoids with densities 1 (blue) and 2 (red)). Bottom row, restored densities.

As a note, attempts to further increase the DNA contrast e.g. by artificially “doubling” the DNA in 5Y0D (Figure 1a) would yield a hollow disk-like model where the scattering is fully dominated by the DNA signal. Similar to the CoA micelles shown in Supplementary Figure 1, such a hollow shape is restored both by DENSS and DAMMIN.

To avoid speculations on differences in packing, compactness, etc between proteins and DNA we also computed the scattering from several two-domain protein models where one domain was repeated twice in the structure, i.e. had twice the contrast compared to the other. Two typical DENSS restorations using such data are presented in Figure 1a (6PWC and 6KYK), and, again, we see that the elevated densities are proximal to the particle centers and not to the location of the repeated domains.

We further verified the ability of DENSS to reveal yet more clear inhomogeneities. Using MONSA⁶ we computed the scattering from three particles, all having the same simple shape (ellipsoids of revolution with half axis ratio 1:2, Figure 1b). In one case the ellipsoid was uniform. In another case it was divided into two halves with distinct densities (1 and 2) separated along the long axis. In the third case the separation was along the short axis. The scattering from these models was computed (Supplementary Figure 2) and the maps restored by DENSS from these curves are presented in Figure 1b, bottom row. These maps display similar density distributions, bearing no resemblance to the true inhomogeneities and showing instead higher densities in the middle, similar to those in Figure 1a (and also to Figure 1 in T.Grant’s paper¹)

When analyzing these and other data we found another striking fact, namely that the scattering computed from the averaged maps failed to fit the initial scattering data. The DENSS publication¹ did not present any fits computed from the averaged models. Clearly, all (meaningless) individual reconstructions do fit the data, but the averaged models do not, and the misfits computed from them are grave, for all cases we tested (Supplementary Figures 2 and 3).

We believe these findings are related to the key procedure in the DENSS method, namely, averaging of multiple loosely constrained 3D real space densities. This procedure generally yields elevated density in the central part of the average map, simultaneously leading to misfits to the data in reciprocal space. These effects are similar to a low-pass filtering of the particle density while accounting not only for the coherently scattering volume of the particle itself but also for the incoherent part due to the surrounding solvent (e.g. using programs like EMAN2⁷). Even for correct structures, such filtering always tends to yield artificially higher density in the particle center and provides misfits to the data.

Averaging over multiple reconstructions is a standard procedure in ab initio SAXS analysis⁸ to filter outliers, assess ambiguity and build consensus models. The averaging is an optional post-processing step for methods like DAMMIN/DAMMIF where all models are physically reasonable. The DENSS method heavily relies on averaging, and the elevated density close to the map center is a consequence. Further, shape determination algorithms employ a constant term subtraction to ensure the asymptotic behavior of the intensity expected for a uniform particle⁹. This procedure is physically justified, whereas in our view, the averaging in Grant’s method lacks such justification.

Analysis of spherically averaged SAXS data in terms of 3D models is a notoriously difficult problem and the paper by T.Grant¹ is undoubtedly a very interesting contribution. However, in our view, the claim of depicting the internal structure is incorrect, the systematically higher density in the central part is an artifact and the averaged models yield misfits to the data.

Methods

The scattering curves from the protein-DNA complexes (atomistic models) were calculated by CRYSOL. The scattering curves from non-uniform ellipsoids (bead models) were calculated by MONSA. For these theoretical curves, ab initio models were restored by DAMMIN and DENSS. In the latter case, the averaging was performed over 100 individual reconstructions using EMAN2. The ab initio and high resolution models were superimposed with each other by SUPALM. Electron density maps were visualized with Chimera (https://www.cgl.ucsf.edu/chimera). The scattering from the density maps was calculated using the modified version of SUPALM.

Supplementary Material

The online version contains supplementary material available at https://doi.org/10.1038/s41592-021-01082-x.

Data Availability Statement

All models, computed scattering curves and reconstructions presented in the manuscript are available as Supplementary Data file.

Code Availability Statement

The program DENSS is open source available at https://github.com/tdgrant1/denss. EMAN2 software is avalable at https://www.eman2.org. The programs DAMMIN, MONSA, CRYSOL and SUPALM are included into the ATSAS software available for academic users at https://www.embl-hamburg.de/biosaxs/software.html.