1. Introduction

The neural-network-based artificial intelligence (AI) programs AlphaFold (AF) (Jumper et al., 2021) and Rosetta­Fold (Baek et al., 2021) have revolutionized the field of protein structure prediction from sequence. In particular, the AF consortium has produced and made publicly available a database of predicted protein structures (https://alphafold.ebi.ac.uk), first for the entire UniProt database of curated protein sequences (Tunyasuvunakool et al., 2021) and more recently for the entire catalogue of genomics-derived protein sequences (UniProt Consortium, 2021). Impressively, AF-predicted structures from the CASP14 data set achieved a mean Cα r.m.s.d. accuracy of ∼1 Å (Jumper et al., 2021).

The AF database has already had a major impact in structural biology, and the predicted structures are being used as templates for solving crystal structures [see e.g. Flower & Hurley (2021), Chai et al. (2021), McCoy et al. (2022) and Oeffner et al. (2022)], as an aid in the interpretation of cryoEM maps [see e.g. Fontana et al. (2022)], in assessing the accuracy of NMR structures in solution [see e.g. Fowler & Williamson (2022)] and to infer structure–function relationships [see e.g. Ferrario et al. (2022), Urban & Pompon (2022), Akdel et al. (2022) and Heo et al. (2022)]. Conversely, AF structures usually have regions with low confidence or poor accuracy, and experimental information has the potential to improve these predictions (Terwilliger et al., 2022).

While there are many examples of AF-predicted structures having an impressive level of accuracy, there remain several challenges. The AF prediction algorithm depends on deep learning from an extensive catalogue of structures, which nonetheless is limited by training on the set of solved protein structures in the Worldwide Protein Data Bank (https://www.wwpdb.org; wwPDB Consortium, 2019). Flexibility, however, is often a necessary aspect of protein function, and the protein universe is replete with multi-domain proteins composed of structured units with flexible linkers of variable length that limit both crystallographic and cryoEM studies. Thus, we see that there are opportunities to test and complement the AF predictions with experimental techniques, notably those that are readily available in core facilities or accessible on dedicated large infrastructures.

A recent paper (Brookes & Rocco, 2022) presented a database stemming from the first two AF releases, where for each predicted structure the calculated circular dichroism (CD) spectrum, the hydrodynamic parameters, the pair-wise atomic distance distribution function P(r) versus r [hereinafter indicated simply as P(r)] and ancillary information are stored (https://somo.genapp.rocks/somoaf/). On the basis of the UniProt annotations, predicted initiator sequences and post-translationally cleaved pro-peptides were removed from the structures prior to the calculations. Calculated hydrodynamic parameters were employed to show that, within a given molecular mass interval, these parameters could effectively distinguish between structures and thus could be employed for rapid tests of the predicted conformation in solution (Brookes & Rocco, 2022).

The P(r) profile, which can be determined from experiment as the indirect Fourier transform of small-angle X-ray scattering (SAXS) data (Glatter, 1977; Svergun et al., 1988), is quite sensitive to the relative disposition of folded units, as nicely demonstrated in a multi-domain test protein (Koch et al., 2003). As such, P(r) is ideal for evaluating AF-predicted structures containing potentially flexible linkers between structured domains, noting that the solution SAXS experiment reports on the time and ensemble average of the structures present in solution. In the previous study (Brookes & Rocco, 2022) a set of solution SAXS data was selected from the Small-Angle Scattering Biological Data Bank (SASBDB) (Valentini et al., 2015; https://www.sasbdb.org) where there was a corresponding AF-predicted structure. While there were several examples of good agreement between the AF-predicted P(r) profile and that derived from the SAXS data, there were a number that showed significant differences. In this study, we have returned to consider these comparisons in the context of a careful evaluation of the SAXS profiles and their associated metadata to ensure the data were from solutions of monodisperse particles, free of aggregation and interparticle correlations, which is the fundamental requirement for reliable 3D atomistic modelling of SAXS data. The AF structures were then examined for predicted unstructured and/or low-confidence regions, which were assigned as flexible segments to obtain an ensemble of structures that gave an improved fit to the SAXS data. Our initial ensemble modelling focused on fitting in real space with P(r) as the target function. This approach facilitated rapid calculations from thousands of alternative conformations generated using the Monomer Monte Carlo (MMC) simulation tool in the SASSIE-web suite (Curtis et al., 2012; Perkins et al., 2016) (https://sassie-web.chem.utk.edu/sassie2/). Evaluation in reciprocal space (i.e. against the measured intensity data) was also possible in reasonable time using CRYSOL (Svergun et al., 1995). Each resulting ensemble model was then further evaluated in reciprocal space using the more physically advanced but computationally intensive WAXSiS program (Chen & Hub, 2014; Knight & Hub, 2015) (https://waxsis.uni-saarland.de).

2. Methods

Initial data sets for analysis were selected by identifying structures present in the US-SOMO-AF database (Brookes & Rocco, 2022) with corresponding experimental SAXS intensity data deposited in the SASBDB. Intensity profiles are presented here as I(q) versus q [hereinafter simply indicated as I(q), where q = (4πsinθ)/λ with θ being half the scattering angle and λ the wavelength of the incident radiation]. Mandatory requirements included that the experimental data were collected on complete single-chain structures with no prosthetic groups and from the same organism as the corresponding AF structure (Brookes & Rocco, 2022), which limited the initial SASBDB pool to 43 entries.

P(r) is related to I(q) by a Fourier transform and, as the finite experiment measurement range for I(q) prohibits an analytical solution, P(r) is generally calculated from SAXS data using indirect methods, for example as implemented in the programs GNOM (Svergun, 1992) or BayesApp (Larsen & Pedersen, 2021). Both methods yield a P(r) profile with associated errors estimated using Monte Carlo simulations. However, different P(r)-generating methods yield very different error estimates starting from the same I(q), and the question of their reliability is an open one. In this study, the P(r) profiles calculated from SASBDB intensity curves and used for comparison with model calculations and as targets for ensemble modelling were obtained with GNOM as implemented in PrimusQt/ATSAS 3.1 (Manalastas-Cantos et al., 2021) and, for comparison, BayesApp as implemented at https://somo.chem.utk.edu/bayesapp.

For each selected AF structure with a corresponding SAXS I(q) profile, the conformational space of the AF-predicted structure obtained from the US-SOMO-AF database was explored utilizing the Monomer Monte Carlo (MMC) program of SASSIE-web (Curtis et al., 2012; Perkins et al., 2016) (https://sassie-web.chem.utk.edu/sassie2/), where the backbone di­hedral allowed angles along chosen segments of the protein are changed in sequential discrete steps. Except for the residue ranges for flexible regions and a number of trial attempts that are detailed in Table S1 in the supporting information, the default MMC parameters were used.

MMC flexible regions were selected by visual inspection of the AF structures' low-confidence regions. MMC rejects structures with steric clashes, and it is recommended to run 10 000 to 50 000 trials to sample the conformational space adequately. From the MMC pool of accepted structures (the `original pool') we selected a subset by extracting every `sub-selection stride' from the MMC-produced multi-structure PDB file with the mdconvert program of MDTraj (Version 1.9.4; McGibbon et al., 2015) utilizing the XSEDE (Towns et al., 2014) allocated Jetstream2 (Hancock et al., 2021) cloud-computing resource. In each case, the distribution of R_g values for this final subset of structures was compared with that of the original pool to ensure that it was representative. This representative pool will hereafter be referred to simply as `the pool.'

Each multi-structure PDB file was processed by the open-source hydrodynamic and SAS data analysis and simulation program US-SOMO (Brookes & Rocco, 2018; Revision 6730+, https://somo.aucsolutions.com/) in batch mode to compute R_g, predicted P(r) profiles (normalized by the sample molecular weight and with a 1 Å bin size) and I(q) curves generated using CRYSOL (Svergun et al., 1995). The P(r) profiles were computed on the dry structures as (Brookes et al., 2013)

where b_i and b_j are the numbers of electrons of the i and j atomic groups, respectively, and the b_0i and b_0j terms account for the solvent scattering density. For SAXS, b_0i = 10 × (r_i/r_wat)³, where 10 is the number of electrons in a water molecule, r_i is the radius of the ith atom and r_wat is the radius of a bulk water molecule (1.93 Å). The Kronecker delta δ(r − r_ij) is applied to the distances r_ij between the centres of atoms i and j for every bin r. While the contribution of the hydration layer is not considered in this implementation of the P(r) calculation, its effect is relatively minor. Tests comparing the P(r) calculated on the starting dry AF structures with those derived from a WAXSiS-generated I(q) profile (see below) indicate a shift of about 1 Å of the global pattern toward shorter r values, and some local differences mainly in the amplitudes (data not shown). Schemes utilizing explicit hydration of the starting structures are computationally intensive (and not implemented within US-SOMO). The approach used here allows for fast processing of thousands of structures as a preliminary screening step to generate a pool of suitable structures that are then evaluated taking into account the contribution of hydration (see below).

For this study, the US-SOMO batch processing protocol for generating I(q) profiles of up to a few thousand structures utilized CRYSOL (Version 2.8; Svergun et al., 1995), with 25 for the maximum number of spherical harmonics, 18 for the order of Fibonacci's grid, 0.335 e Å⁻³ for the solvent electron density and 0.02 e Å⁻³ for the hydration shell contrast, using the same q grid as the experimental one. Running on an eight-core Intel Core i7-4790 CPU/16 GB RAM workstation (Linux Ubuntu 16.04.7 LTS) for the Q16543 AF structure (M_w 44 459 Da, 378 residues) it takes ∼27 and ∼424 s for every 100 structures to compute the P(r) with 170 bins and the I(q) with a q grid of 1869 points, respectively. A more recent CRYSOL release (Version 3.2) has an option to use a different hydration scheme with dummy water beads, which should in principle be more efficient in dealing with structures presenting extended non-structured segments (Franke et al., 2017). This version has now been made accessible from within US-SOMO and will be made available to the general user in the next planned release. As a check, we repeated all I(q) calculations with CRYSOL 3.2 using the dummy water beads option, and a comparison of the results obtained with CRYSOL 2.8 is presented in the supporting information (Section S1 and Tables S2–S4) with brief summary conclusions provided in the Discussion below.

GNOM P(r) curves derived from the experimental data were automatically rebinned to 1 Å steps, with errors interpolated upon loading the GNOM *.out file into US-SOMO. The NNLS (non-negatively constrained least-squares) utility of US-SOMO (Brookes et al., 2016) was then used to fit the predicted P(r) and I(q) curves to their experimentally derived counterparts. NNLS optimization minimizes ||Ax − b||₂ subject to x ≥ 0, where A is an m × n matrix, x an n vector and b an m vector. When the n columns of A are populated with predicted profiles and b with the experimental data, the algorithm, based on projections, produces a result as an x vector populated with zeros or positive numbers representing the fractions of the corresponding predicted profiles (Lawson & Hanson, 1995). There are no restrictions on the number of columns of A (number of predicted profiles) but, given a sufficiently large number of columns or solution-contributing predicted profiles that are positive linear combinations of other predicted profiles, differing collections could provide the same minimum value for ||Ax − b||₂ on the half-space x ≥ 0. NNLS produces only one such collection. The projection-set nature of the algorithm apparently tends to provide the solution with the minimum number of predictions. No rebinning was performed on the original reference SAXS I(q) curves, notwithstanding their evident oversampling at high q values. While this will lead to somewhat artificially low χ² values as a goodness-of-fit measure, here we are not concerned with their absolute values but only with changes between the starting structure profiles and the NNLS-selected composite ones. Moreover, with the logarithmic rebinning needed to reduce the oversampling effectively, important features in the error-weighted residual I(q) plots between experiment and model would be suppressed, such as oscillating features that signify differences in mean dispositions of domains.

Structures identified by NNLS as contributing to the real-space P(r) and to the reciprocal-space I(q) CRYSOL-based curves were further processed by the more computationally intensive program WAXSiS. This program uses a short explicit solvent molecular dynamics simulation to build a spatial envelope containing the structure and its solvation shell, while constraining the backbone atoms with a harmonic potential to ensure no conformational deviations from the input structure. The computation of the excluded-solvent scattering is based on a pure-water simulation and a SAXS I(q) curve is computed that accounts for the hydration contribution (Chen & Hub, 2014; Knight & Hub, 2015). WAXSiS calculations used the default options except for a thorough convergence choice and using the experimental I(q) curve to define the q range and interval to produce predicted I(q) curves that were also subsequently NNLS-fitted to the experimental I(q) curve by US-SOMO.

Root-mean-square (r.m.s.) average radii of gyration {[〈(R_g)²〉]^1/2, hereinafter indicated simply as 〈R_g〉} were calculated from the computed R_g of each dry structure and from the Guinier R_g reported by WAXSiS, weighted by their fractions in the NNLS fit (PDB and WAXSiS 〈R_g〉 values are given in Table 2). As these two values were always very close, an average between the two is quoted in Section 4 below.

Directly computing χ² for the P(r) NNLS fits presented some issues related to the limited number of points and the reliability of their associated errors when used as weights. Further, there is merit in evaluating the model fits against the measured I(q) data. Therefore, from the WAXSiS-calculated I(q) of each selected MMC structure, a sum weighted by the respective fractions from the P(r) NNLS fit was produced. The resulting composite I(q) curve was then scaled against the original data, yielding χ² values over the same data range and with the same number of points as those determined for the other NNLS fits, which then could be meaningfully compared.

Graphs were prepared using Origin (Version 6.0; Microcal) or OriginLab 2019b (https://www.originlab.com). Atomistic structure figures were prepared with UCSF Chimera (Version 1.15; Pettersen et al., 2004) using the `supersmooth' ribbon representation, and superpositions were done on all atoms in the selected residues using Chimera's MatchMaker. Figures were assembled using PaintShopPro (Version 5.3; JASC Software, now Corel, https://www.paintshoppro.com).

3. Selection of AF-predicted structures with corresponding experimental SAXS data and evaluation of SAXS data quality

An initial qualitative survey of SAXS data sets with corresponding AF-predicted structures meeting the requirements detailed above (in Section 2) revealed three candidates where the AF-predicted and SAXS-derived P(r) values are significantly different: AF-Q16543, AF-Q06187 and AF-Q9UKA9, with corresponding SAXS data SASDBP9 (Bunney et al., 2018), SASDF83 (Duarte et al., 2020) and SASDM77 (Simpson et al., 2004), respectively (Fig. 1). In each case there were extended low-confidence regions in the AF-predicted structure that might signify flexibility. Having identified these three potential candidate structures that appear to require modification to represent the solution state properly, we proceeded to assess the quality of the SAXS data and their suitability for modelling.

Figure 1 Ribbon representations of each selected AF-predicted structure, colour-coded from red to blue according to an increasing confidence level. (a) Q16543 (Homo sapiens Hsp90 co-chaperone Cdc37, residues 1–378). (b) Q06187 (H. sapiens Bruton's tyrosine kinase, mature protein residues 2–659). (c) Q9UKA9 (H. sapiens polypyrimidine tract-binding protein 2, residues 1–531). The insets show their P(r) profiles calculated from the dry structures, and SAXS-derived profiles as retrieved from SASBDB (red and black lines, respectively). The scale bar shown in (b) also applies to the other panels.

Solution SAXS data from proteins with substantial flexibility present several challenges. First, they are more susceptible to small degrees of aggregation compared with folded structures, so it is highly desirable to work with data collected using inline size-exclusion chromatography (SEC–SAXS) when available. Second, the selection of d_max for a P(r) transformation is challenging, as the P(r) from an ensemble of structures with a range of d_max values generally yields a P(r) that approaches d_max with a long, very low intensity tail with large errors that nonetheless can have a significant effect on R_g calculated from the second moment of P(r). Third, the determination of the Porod volume from the scattering invariant assumes an object of uniform scattering density and a sharp interface with the solvent. While this is an arguably valid approximation in the case of a compact fully structured protein, this is hardly relevant in the case of a flexible structure.

For each of the candidate SAXS data sets, the SAXS-derived molecular mass (M^expt) agrees reasonably with the mass calculated from the chemical composition (M^calc) (Table 1), although we note that quite different methods were used for determining M^expt (see the footnotes to Table 1). Inspection of the P(r) transforms present in SASBDB for each of these data sets revealed that the selected d_max did not yield a P(r) profile with the expected gradual approach to a near-horizontal tangent at d_max; rather there was a sharp cut-off that was most severe for SASDM77. We therefore recalculated the P(r) transforms using GNOM with a standardized approach for d_max selection: d_max was selected such that release of the constraint P(r) = 0 at d_max did not result in a significant increase in P(r) intensity at long r. The resulting P(r) profiles all showed the expected shape at long r, but with large errors as r approaches d_max. P(r) profiles obtained using BayesApp resulted in profiles that had similar R_g values, but with d_max values that were shorter by 8–20 Å. Thus, there is a degree of uncertainty in the selection of d_max for these structures. On average, Guinier-derived R_g values were ∼2 Å smaller than those derived from P(r), except for SASDM77 where the difference was ∼4 Å, but here the error in the P(r) R_g was more than four times that for the other data sets. The P(r) fits were all acceptable, as estimated by the GNOM total quality estimate values (0.72–0.79). The χ² values for the P(r) fits were in the range 1.03–1.15 with acceptable CorMAP P values (Franke et al., 2015), except for SASDF83 (χ² 1.35, P value 0.0003) (Table 1). The error-weighted residual difference plot for the P(r) fit for SASDF83 was nevertheless flat and featureless, with deviations predominantly in the range ±3. Alternative d_max values resulted in worse fitting parameters for this data set and it was a very narrow q region that was responsible for the low P value (0.134–0.139 Å⁻¹). We thus conclude that all our GNOM-derived P(r) fits to experiment are acceptable.

Table 1 Parameters derived from SAXS data SASBDB ID, AF structure ID Mcalc (kDa) Mexpt (kDa) Guinier Rg (Å) Maximum qRg P(r) Rg (Å)† dmax (Å)† P(r) fit: χ2‡ P(r) fit: P value‡ SASDBP9, Q16543 44 49§ 40.9 ± 0.5 1.0 42.4 ± 0.3 170 1.03 0.155 SASDF83, Q06187 77 67¶ 41.4 ± 0.4 1.0 43.8 ± 0.3 176 1.35 0.0003 SASDM77, Q9UKA9 57 60†† 39.1 ± 0.9 1.3 43.9 ± 1.1 170 1.11 0.95 †Values are for the GNOM-derived P(r). Values obtained using the BayesApp-calculated P(r) without inputting a dmax value were for Rg 41.8, 43.5 and 45.2, and for dmax 152, 152 and 162, for SASDBP9, SASDF83 and SASDM77, respectively. ‡χ2 and P values were calculated using the Data Comparison tool of PrimusQt (Manalastas-Cantos et al., 2021). §From I(0) relative to a standard, bovine serum albumin (BSA). ¶From Bayesian inference. ††From DAMMIN envelope volume, conversion factor not specified.

Finally, to give an accurate characterization of a structure of maximum dimension d_max, one must adequately sample I(q) in the Guinier regime and q_min should be < π/d_max. The experimental q_min values for SASDBP9 and SASDF83 are 2.48 × 10⁻³ and 8.16 × 10⁻³ Å⁻¹, respectively. There are 95 and 64 data points in their Guinier regions, respectively, noting that for SASDBP9 the first 25 data points are excluded from Guinier analysis due to an upturn indicative of parasitic scattering or some large particle contaminant that is sufficiently low level that it does not impact the SAXS-derived molecular mass. For SASDM77, q_min is only 1.41 × 10⁻² Å⁻¹ with just 17 data points in the Guinier region. Thus, while SASDBP9 and SASDF83 meet the minimal requirements for characterizing structures as large as 350–400 Å, the limit for SASDM77 is 220 Å, much closer to the experimentally derived P(r) d_max value range (170–176 Å).

In conclusion, while the SAXS data we identified as indicating that the AF-predicted structures required some modification to represent the solution conformations were not collected in the preferred SEC–SAXS mode, we can conclude that these batch-mode-acquired data are suitable for evaluating atomistic modelling as assessed by the data quality parameters in the 2017 guidelines for biomolecular small-angle scattering (Trewhella et al., 2017).

4. Modelling the SAXS data

Briefly summarized, our overall approach to modelling began by first quantitatively assessing the AF structure predicted I(q) and P(r) profiles without modification against their respective experimental SAXS profiles; χ² values for each AF structure were calculated using WAXSiS-generated I(q) profiles scaled to experiment, and Guinier and P(r)-derived structural parameters were compared. For these comparisons, and for all modelling with P(r) as the target function, the experimentally derived P(r) recalculated with the standardized approach for d_max selection target was used. Considering the observed differences plus the low-confidence predicted regions for the AF structures, the conformational space to be explored in developing an ensemble model was expanded by employing MMC with potentially flexible sequence segments to generate a pool of potential structures (`the original pool'), from which a representative subset (`the pool') was selected. Ensemble modelling was first performed using NNLS with this pool of structures to optimize the fit to the experimentally derived P(r), without and with errors generated in the indirect Fourier transform. To evaluate the resulting conformational ensembles in reciprocal space, the ensemble I(q) profile was calculated by summing WAXSiS-generated I(q) profiles for the individual conformations in each ensemble, weighted by the NNLS-reported fraction. To compare results obtained by optimizing the fit to P(r) with those obtained by directly fitting in reciprocal space against the measured I(q), NNLS optimizations were also performed starting with the same MMC pool of potential structures as for the P(r) fitting but using CRYSOL 2.8-generated individual predicted I(q) profiles. Finally, WAXSiS was used to calculate I(q) profiles for all conformations present in the ensemble fits to P(r) or I(q), and these were subjected to NNLS optimization against the experimental I(q) profile.

4.1. AF-Q16543 predicted structure and SASDBP9 data

The AF-Q16543 structure [Fig. 1(a)] is composed of two folded domains connected by an unstructured linker spanning residues 121–139, with an AF average prediction confidence level (APCL) of 66 ± 14%. There is also a long unstructured C-terminal tail (residues 343–378) with an APCL of 35 ± 10%. The P(r) calculated for the AF structure differs significantly from that deposited in the SASBDB for SASDBP9 [see Fig. 1(a), inset]. The AF-Q16543 P(r) has a double peak and a shoulder, indicative of multiple folded domains with dispositions that are, on average, relatively constrained. The SAXS-derived P(r) has a single peak with a shoulder and an extended tail approaching a d_max value that is similar to the AF-calculated value, suggesting a broad distribution of compact to highly extended structures, weighted towards the more compact ones, consistent with a flexible linker in solution and a potentially flexible C-terminal region. The high χ² value (19.044) [Table 2(a)] calculated between the computed I(q) for the AF structure and the experimental I(q) after scaling [Fig. 2(a)] indicates that there are substantial differences between the AF-predicted structure and the solution state. The large oscillations in the error-weighted residual plot for the experimental versus predicted I(q) [Fig. 2(b)] (−12 to 16) indicate that the differences are highly significant.

Table 2 Model fit parameters (χ2) and mean structural parameters [PDB 〈Rg〉, WAXSiS 〈Rg〉, P(r) 〈dmax〉, and P(r) max dmax and their percent contribution] for NNLS ensemble fits 〈Rg〉 stands for the r.m.s. average radius of gyration [〈(Rg)2〉]1/2 (see Section 2, Methods). Expt. is an abbreviation for experimental and conf. is an abbreviation for conformations. (a) Fits to SASDBP9 based on AF-Q16543, NNLS fits with flexible linkers (sequence segments 121–139 and 343–378). Fit method WAXSiS, I(q), scaled NNLS, expt. P(r) target NNLS, expt. I(q) target NNLS, expt. I(q) target     MMC pool P(r)     Structure pool AF-Q16543 No error weighting Error weighting MMC pool [CRYSOL 2.8 I(q)] All NNLS selected conf. [WAXSiS I(q)] Fit parameters (χ2) 19.044 1.399† 2.065† 1.602 1.228   PDB 〈Rg〉 (Å) 41.7 38.6 39.2 40.8 WAXSiS 〈Rg〉 (Å) 41.5 38.5 39.1 39.3 P(r) 〈dmax〉 (Å) 157.1 139.4 146.0 150.1 P(r) max dmax (Å) 201 (14%) 158 (8%) 185 (23%) 187 (6%) (b) Fits to SASDF83 based on AF-Q06187, NNLS fits with flexible linker (sequence segment 170–210). Fit method WAXSiS I(q), scaled NNLS, expt. P(r) target NNLS, expt. I(q) target NNLS, expt. I(q) target     MMC pool P(r)     Structure pool AF-Q06187 No error weighting Error weighting MMC pool [CRYSOL 2.8 I(q)] All NNLS selected conf. [WAXSiS I(q)] Fit parameters (χ2) 31.250 1.997† 2.716† 1.673 1.763   PDB 〈Rg〉 (Å) 40.7 42.7 42.3 41.1 WAXSiS 〈Rg〉 (Å) 40.6 42.4 42.0 41.0 P(r) 〈dmax〉 (Å) 133.1 138.7 136.4 132.1 P(r) max dmax (Å) 165 (15%) 184 (4%) 187 (2%) 178 (6%) (c) Fits to SASDM77 based on AF-Q9UKA9, NNLS fits with flexible linkers (sequence segments 1–54 and 273–336). Fit method WAXSiS, I(q), scaled NNLS, expt. P(r) target NNLS, expt. I(q) target NNLS, expt. I(q) target     MMC pool P(r)     Structure pool AF-Q9UKA9 No error weighting Error weighting MMC pool [CRYSOL 2.8 I(q)] All NNLS selected conf. [WAXSiS I(q)] Fit parameters (χ2) 3.674 1.279† 1.493† 1.208 1.179   PDB 〈Rg〉 (Å) 43.8 40.4 52.2 48.4 WAXSiS 〈Rg〉 (Å) 43.7 40.5 52.0 48.4 P(r) 〈dmax〉 (Å) 152.8 143.8 162.9 152.8 P(r) max dmax (Å) 192 (4%) 168 (10%) 243 (8%) 243 (5%) †The χ2 for the P(r) NNLS fits were determined by computing the I(q) of each selected MMC structure using WAXSiS, and then making a weighted sum using the respective fractions from the NNLS fit. The resulting I(q) curve was then scaled against the original data.

Figure 2 (a) I(q) versus q for SASDBP9 (black symbols with standard error bars) overlaid with the fit of the AF-Q16543 WAXSiS-calculated SAXS profile (red line) and with the NNLS ensemble fit model generated from the CRYSOL 2.8-calculated SAXS profiles of the MMC pool structures (violet line). (b) Error-weighted residual plots for the fits shown in panel (a). (c) and (d) GNOM-derived P(r) profiles (black symbols without/with standard error bars) overlaid with that from the AF-Q16543 prediction (red lines) and with the NNLS ensemble fit from the P(r) calculated on the MMC pool without and with error weighting, respectively (blue and orange lines). (e) I(q) versus q for SASDBP9 (black symbols with standard error bars) overlaid with the NNLS ensemble fit model calculated using the WAXSiS-generated I(q) versus q profiles of all NNLS-selected structures from the CRYSOL 2.8 and P(r) fits (magenta line). In the inset, four representative structures selected with a significant percentage by at least two of the four NNLS fits are shown, after superposition on the 1–120 N-terminal residues [see Table 2(a) for the full fitting results]. (f) Error-weighted residual plot for the fit shown in panel (e).

4.1.1. Ensemble modelling with P(r) as the target function

The MMC protocol was used to generate a pool of potential solution conformations for AF-Q16543 by allowing dihedral angle variations for sequence segments 121–139 and 343–378 (the run summary is given in Table S1). The original pool included 15 661 structures, of which 1740 representative ones (one in nine) formed the pool input to the NNLS tool to find the best fit to the experimentally derived P(r). Visual inspection of the P(r) fit without errors [Fig. 2(c)] shows excellent qualitative agreement. The resulting χ² value of 1.399 [Table 2(a)] for the composite WAXSiS-calculated I(q) fit to the experimental data is more than an order of magnitude improvement compared with the unmodified structure. Furthermore, the 〈R_g〉 of 41.6 Å is in excellent agreement with the SAXS-derived values [Tables 1 and 2(a)]. The P(r) maximum d_max value is within the uncertainty of the experimental P(r) d_max, with the population weight of the structure having the longest d_max being 14%. The considerably shorter 〈d_max〉 compared with the experimental P(r) d_max reflects the spread in d_max values among the selected structures. In contrast, the NNLS fit employing error weighting [Fig. 2(d)] clearly underused the contributions of structures having d_max values in the long-r range, where the experimentally derived P(r) has the largest errors. Notably, the χ² value was significantly higher (2.065) and the 〈R_g〉 and maximum d_max values were all lower than the experimentally derived ones, with an even shorter 〈d_max〉 [Table 2(a)]. These results bring into question the utility and/or accuracy of the P(r)-associated errors in this protocol.

Histograms of the percent contribution to the 〈R_g〉 of individual structures in the P(r) NNLS fits [Figs. 3(a) and 3(b)] show that a single structure with an R_g of 36 Å accounts for ∼41 and ∼62% of the ensemble population without and with error weighting, respectively (see also Table S2). As expected from the above analysis, the histogram for R_g values obtained without error weighting is skewed towards higher values [Fig. 3(a)]. The two experimentally derived Guinier and P(r) R_g values are near a cluster of contributing structures with intermediate R_g values, while that for the AF structure is near a small cluster of the most extended ones, albeit with low percentage contributions, that are altogether missing from the ensemble obtained with error weighting. Inspection of the individual P(r) profiles of the contributing structures and their percent contributions for the NNLS fits without and with error weighting [Figs. S1(a) and S1(b), respectively] reveals that the P(r) profiles can be clustered into five and four categories, respectively. The P(r) shape clustering is supported by a similar clustering observed for the ribbon representations of each structure, after superposition on the N-terminal 25–110 sequence [Figs. S1(c) and S1(d)].

Figure 3 Histogram plots of the Rg values and their percent contribution for the individual structures selected by the NNLS fits for (a)–(d) the SASDBP9/Q16543, (f)–(i) the SASDF83/Q06187 and (k)–(n) the SASDM77/Q9UKA9 systems, respectively. (a), (f) and (k) P(r) NNLS fits without error weighting. (b), (g) and (l) P(r) NNLS fits with error weighting. (c), (h) and (m) CRYSOL NNLS fits. (d), (i) and (n) WAXSiS NNLS fits on the structures selected by all the other methods. In all histogram panels, the inverted blue and red triangles indicate the Rg values derived from Guinier and GNOM P(r) analyses of the experimental data, respectively, while the dark cyan inverted triangles represent the Rg of the starting AF structures. (e), (j) and (o) Distributions of the Rg values for the structures in the original MMC pool (solid black lines) and for the sub-selected MMC pool (solid red lines) used for the NNLS fitting. (e) Q16543, (j) Q06187 and (o) Q9UKA9.

4.1.2. Ensemble modelling with I(q) as the target function

The ensemble model obtained using NNLS with CRYSOL-calculated I(q) profiles for the same MMC pool of structures as for the P(r) fits yielded an I(q) profile fit [Figs. 2(a) and 2(b)] whose quality is comparable to the P(r) fit without error weighting, though with a slightly higher χ² value [1.602, Table 2(a)]. The 〈R_g〉 value was ∼6% lower than the experimentally derived values, and the maximum d_max value (23% contribution) was ∼9% higher than that of the GNOM-derived P(r), with a 〈d_max〉 in between those of the P(r) NNLS fits [Tables 1 and 2(a)]. The corresponding histogram of the R_g values [Fig. 3(c)] shows a similar preference for more compact structures to that observed for the conformational ensemble obtained from the P(r) fit with error weighting. Three structures having R_g values in the range 36–43 Å account for more than ∼92% of the structures present, with one of them (contributing ∼22%) being close to the experimentally derived R_g values.

Finally, all the structures selected by the NNLS fits with either P(r) or I(q) as the target, plus the starting AF-predicted structure (a total of 22 structures), were taken as a set and calculated WAXSiS I(q) profiles were used in an NNLS fit to the SASDBP9 I(q) [Figs. 2(e) and 2(f)]. This calculation yielded the lowest χ² value [1.228, Table 2(a)], and the 〈d_max〉 and maximum d_max values are essentially the same as those found by the NNLS fit with CRYSOL-generated I(q) profiles with the larger MMC pool, while the 〈R_g〉 values are closer to those obtained by the NNLS fits of the P(r) without error weighting. The corresponding R_g distribution histogram [Fig. 3(d)] shows two major clusters of R_g values in the ranges 36–38 and 40–43 Å, accounting for ∼49 and ∼36% of the structures, respectively, the latter encompassing experimentally derived R_g values (see Table 1). In partial contrast with the results obtained using CRYSOL, two more elongated structures were selected for a ∼15% contribution. Moreover, the starting AF-predicted structure, which was not selected by either the two P(r)-based fits or the CRYSOL fit, contributes 4.5% to the WAXSiS-based NNLS fit (Table S2).

For completeness, overlays of the SASDBP9 experimental profile with the CRYSOL- and WAXSiS-generated individual I(q) profiles of the selected structures, with their percent contributions, are shown in Figs. S2(a) and S2(b), respectively. They are accompanied by ribbon representations of the corresponding structures aligned and grouped according to the P(r)-derived classes [Fig. S2(c)], with a lone extra single structure that contributes ∼22 and ∼13% to the CRYSOL and WAXSiS NNLS fits, respectively.

Finally, the four structures with the highest percent contributions from all the NNLS fits [inset in Fig. 2(e)] all have R_g values in the range 36–42 Å, corresponding to structures in the MMC sets at the lower end of the R_g distribution, as shown in Fig. 3(e), which also highlights the good correspondence between the original entire MMC pool and the pool employed for the NNLS fits. Overall, our analysis indicates that this protein adopts significantly more compact conformations than does the starting AF-Q16543 structure.

4.2. AF-predicted structure Q06187 and SASDF83 data

The structure of AF-Q06187 [Fig. 1(b)] comprises an N-terminal folded domain (residues 1–169) connected by an essentially unstructured linker (residues 170–210, APCL 36 ± 5%) to a larger folded C-terminal domain (residues 211–659). Parts of both domains appear close in space, although no real contact interface is observed. The experimentally derived P(r) retrieved from the SASDF83 entry is significantly more extended (by ∼60 to 70 Å) than that calculated for the structure [see inset in Fig. 1(b)]. As found for the AF-Q16543/SASDBP9 case, its WAXSiS-generated I(q) profile gives a very poor fit to the experimental curve [Fig. 4(a); χ² = 31.25, Table 2(b)] and an oscillating error-weighted residual plot (−14 to 18) [Fig. 4(b)]. These observations strongly suggest that the sequence segment 170–210 could be a flexible linker and the two domains are, on average, more separated in solution than in the AF-predicted structure.

Figure 4 (a) I(q) versus q SAXS data taken from SASDF83 (black symbols with standard error bars) overlaid with the fit of the AF-Q06187 WAXSiS-calculated SAXS profile (red line) and with the NNLS ensemble fit model generated from the CRYSOL 2.8-calculated SAXS profiles on the MMC pool structures (violet line). (b) Error-weighted residual plots for the fits shown in panel (a). (c) and (d) GNOM-derived P(r) profiles from SAXS data (black symbols without/with standard error bars) overlaid with that from the AF-Q06187 prediction (red lines) and with the NNLS ensemble fits from the P(r) calculated on the MMC pool structures without and with error weighting, respectively (blue and orange lines). (e) I(q) versus q SAXS data taken from SASDF83 (black symbols with standard error bars) overlaid with the NNLS ensemble fit model calculated using the WAXSiS-generated I(q) versus q profiles of all NNLS-selected structures from the CRYSOL 2.8 and P(r) fits (magenta line). In the inset, four representative structures selected with a significant percentage by at least two of the four NNLS fits are shown, after superposition on the 213–659 C-terminal residues [see Table 2(b) for the full fitting results]. (f) Error-weighted residual plot for the fit shown in panel (e).

4.2.1. Ensemble modelling with P(r) as the target function

A total of 14 582 accepted conformations were generated using the MMC protocol by allowing dihedral angle variations in the 170–210 linker, from which 972 were selected (one every 15; run summary provided in Table S1) as the pool for input to the NNLS tool to find the optimal fit to the SASDF83 GNOM-derived P(r). Calculations were performed without and with error weighting [Figs. 4(c) and 4(d), respectively]. As for AF-Q16543, visual inspection indicates that the NNLS fits yield improved agreement, with error weighting resulting in a worse fit, as indicated by the χ² values for the corresponding WAXSiS-based composite I(q) fit to the experimental data [1.997 and 2.716, respectively, Table 2(b)]. In both cases, however, there is still more than an order of magnitude improvement over the unmodified structure and the 〈R_g〉 values are in reasonable agreement with the Guinier and P(r)-derived experimental R_g values [Tables 1 and 2(b)]. Further, the P(r)-derived longest d_max values show deviations from experimental values [Tables 1 and 2(b)] on a similar scale to what was observed for AF-Q16543, but the 〈d_max〉 values are considerably shorter than the experimental P(r) d_max, suggesting a clustering of selected structures towards more compact ones within the pool. Indeed, histograms of the percent contribution to the 〈R_g〉 of individual structures in the P(r) NNLS fits [Figs. 3(f) and 3(g)] show a cluster with R_g values of 32–36 Å (together accounting for 53 and 55% of the contribution), close to that of the starting AF structure, with one structure being heavily selected by both fits with or without errors (see also Table S3). A single structure with an R_g value practically identical to that from the GNOM-derived P(r) was selected only by the NNLS fit without errors and with a 31% contribution. Notwithstanding the visually poorer fit at longer r values for the NNLS P(r) fit with errors, it selected structures with larger R_g values, although with relatively smaller percent contributions than for calculations without error weighting.

The individual P(r) contributions and global views of the ensembles of structures selected by the two NNLS fits (Fig. S3) show a similar P(r) clustering between the two calculations, with three clear classes, one of which can be split into two by the different orientation of the N-terminal domain that is absent from the P(r) NNLS fit with error weighting.

4.2.2. Ensemble modelling with I(q) as the target function

The experimental I(q) for SASDF83 includes 1284 data points with a non-uniform q spacing (Δq): Δq is 4.696 × 10⁻⁴ Å⁻¹ for q in the range [0.8164, 2.648] × 10⁻² Å⁻¹, and then Δq has variable steps in the range 2.3 × 10⁻⁶ to 4.696 × 10⁻⁴ Å⁻¹ up to q = 0.14398 Å⁻¹, followed by a uniform Δq = 4.713 × 10⁻⁴ Å⁻¹ to q_max = 0.49462 Å⁻¹. The US-SOMO CRYSOL implementation generated I(q) profiles for each structure from the MMC pool using a fixed grid spacing of 4.696 × 10⁻⁴ Å⁻¹, yielding 1038 I(q) points that were interpolated to match the 1284 points of the experimental data for the NNLS fitting procedure with SASDF83 I(q) as the target. The resulting ensemble model gives a significantly improved fit to the experimental data [Figs. 4(a) and 4(b)] with a χ² of 1.673 [Table 2(b)]. The 〈R_g〉 value is, within experimental error, equal to the Guinier-derived value and lies between those of the P(r) NNLS fits [Tables 1 and 2(b)]. The histogram of the R_g distribution for the selected structures [Fig. 3(h)] is similar to that obtained from the P(r) NNLS fit with error weighting, with the cluster at 32–36 Å accounting for ∼60% of the contribution. The 〈d_max〉 is similar to the P(r) NNLS values and the maximum d_max is close, within the uncertainties, to the GNOM-derived value from experimental data, with the most extended structure being present at a very low (2%) contribution [Tables 1 and 2(b)].

The WAXSiS-generated I(q) profiles from all the structures selected for any of the NNLS fits plus the starting AF-predicted structure (for a total of 21 structures) were then used as input for an NNLS fit against the SASDF83 I(q) profile [Figs. 4(e) and 4(f)]. The χ² value of 1.763 for this fit is just slightly worse than that of the fit using only the CRYSOL-generated I(q) profiles from the MMC pool. The 〈R_g〉 is again almost identical to the Guinier-derived value, the 〈d_max〉 is similar to that found for all other fits and the structure with the longest d_max (6% contribution) is very similar to the experimentally derived value [Tables 1 and 2(b)]. The histogram of R_g values for the selected structures [Fig. 3(i)] shows two major peaks, at 34 and 46 Å, with 46 and 34% contributions, respectively, reflecting significant populations of more extended conformations compared with the AF starting structure (which was not selected; see also Table S3).

The SASDF83 experimental profile is overlaid with the CRYSOL- and WAXSiS-generated individual I(q) profiles of the selected structures with their percent contributions in Figs. S4(a) and S4(b), respectively, accompanied by the corresponding structures as ribbon representations grouped according to the P(r)-derived classes [Fig. S4(c), same orientations as in Figs. S3(c) and S3(d)]. Here an additional compact structure, with a similar R_g value to the second compact class present in Fig. S3(c) but a different orientation of the N-terminal domain, is heavily selected by the CRYSOL NNLS fit (∼33%) and is also present in the WAXSiS fit (∼6%). The intermediate class having a lone structure in Fig. S3(c) is more populated in the I(q) fits [Fig. S4(c)], as is the more extended structural class.

The four structures with the highest percent contributions from all NNLS fits are shown as ribbon representations after superposition on the C-terminal 218–659 sequence [Fig. 4(e), inset]. The highest proportion of R_g values are all in the lower half of the MMC-generated structure R_g distribution (<48 Å), with smaller percent contributions in the upper range (48–62 Å), as shown in Fig. 3(j), which again highlights the good correspondence between the original entire MMC pool and the pool employed for the NNLS fits. Overall, our analyses indicate that this protein assumes a range of conformations in solution that on average are significantly more extended than the starting AF-predicted structure.

4.3. AF-Q9UKA9 predicted structure and SASDM77 data

The structure of AF-Q9UKA9 [Fig. 1(c)] comprises two small N-terminal folded domains (residues 58–159 and 177–272) connected by a relatively short predicted unstructured segment (APCL 35 ± 4%) and linked by a much longer predicted unstructured segment (residues 273–336, APCL 42 ± 13%) to a C-terminal domain (residues 337–531) where two subdomains appear to have a defined interface between them. At the N-terminal there is also a long predicted unstructured segment (residues 1–57, APCL 36 ± 4%). As for the AF-Q06187 case, the experimentally derived P(r) retrieved from the SASDM77 entry is more extended than that calculated for the structure [see inset in Fig. 1(c)], albeit to a lesser extent (by ∼20 Å). From Fig. 5(a) it is evident that the WAXSiS-generated I(q) fits poorly, although with a relatively lower χ² value (3.674) that in part reflects the larger statistical errors in this data set compared with our other two examples. Significantly, there is a clear oscillation in the error-weighted residual plot (−5 to 8) that is most evident in the intermediate q range, ∼0.04 to ∼0.15 Å⁻¹ [Fig. 5(b)]. These observations suggest that some of the potentially unstructured regions could be flexible in solution, resulting in variable spatial dispositions between the domains.

Figure 5 (a) I(q) versus q SAXS data taken from SASDM77 (black symbols with standard error bars) overlaid with the fit of the AF-Q9UKA9 WAXSiS-calculated SAXS profile (red line) and with the NNLS ensemble fit model generated from the CRYSOL 2.8-calculated SAXS profiles on the MMC pool structures (violet line). (b) Error-weighted residual plots for the fits shown in panel (a). (c) and (d) GNOM-derived P(r) profiles from SAXS data (black symbols with standard error bars) overlaid with that from the AF-Q9UKA9 prediction (red lines) and with the NNLS ensemble fits from P(r) calculated on the MMC pool structures without and with error weighting, respectively (blue and orange lines). (e) I(q) versus q SAXS data taken from SASDM77 (black symbols with standard error bars) overlaid with the NNLS ensemble fit model calculated using the WAXSiS-generated I(q) versus q profiles of all NNLS-selected structures from the CRYSOL 2.8 and P(r) fits (magenta line). In the inset, four representative structures selected with a significant percentage by at least two of the four NNLS fits are shown, after superposition on the 63–270 N-terminal residues [see Table 2(c) for the full fitting results]. (f) Error-weighted residual plot for the fit shown in panel (e).

4.3.1. Ensemble modelling with P(r) as the target function

A first test was conducted with MMC where only the N-terminal 1–57 segment was allowed to be flexible, but this resulted in very minor changes in the P(r) distribution that could not account for the observed differences from experiment (data not shown). We then included in the MMC run the predicted unstructured linker (residues 273–336) plus the N-terminal 1–57 segment, choosing not to add the potential additional short low-confidence sequence segment (residues 160–176) to limit the degrees of freedom in the modelling. The MMC run allowing the dihedral angles of the two sequence segments to vary yielded 17 284 conformations, from which 1728 conformations were selected (one in every ten; run summary provided in Table S1) for NNLS fitting to the SASDM77-derived P(r). Visually good NNLS fits were obtained, without and with error weighting [Figs. 5(c) and 5(d)]. As found for the other two examples, a better fit is obtained without error weighting, most evident in the long tail at long r where the P(r) errors are largest. The χ² values for the fits to the measured I(q) of the WAXSiS-generated composite I(q) profiles for all the selected structures, weighted by their contribution, were 1.279 and 1.493, respectively [Table 2(c)], which are 3- and 2.5-fold improvements, respectively, compared with the starting AF-predicted structure. The 〈R_g〉 for the NNLS fit with error weighting is very close to the Guinier-derived value, while when error weighting for the fit was omitted a close match with the larger GNOM-derived values was obtained [Tables 1 and 2(c)]. The P(r)-derived longest d_max value for the NNLS fit with errors was essentially the same as the GNOM-derived value, but 22 Å longer without error weighting [Tables 1 and 2(c)]. The 〈d_max〉 values are smaller than the GNOM-derived P(r) d_max, suggesting a higher proportion of more compact structures selected in the fit. Histograms of the percent contribution to the 〈R_g〉 of individual structures in the P(r) NNLS fits [Figs. 3(k) and 3(l)] indeed show a predominant cluster of structures with small R_g values (34–40 Å) comparable to that of the starting AF structure, and very close to the Guinier-derived value, accounting for 58 and 75% of the contribution, with two being strongly selected by fits with or without error weighting. Fitting without error weighting gave a single structure contributing significantly (∼10%) to a higher R_g value (see also Table S4).

The individual P(r) contributions and global views of the ensembles of structures selected by the two NNLS fits are shown in Fig. S5. A broad distribution of P(r) profiles is apparent in the NNLS fit without errors [Fig. S5(a)], which is somewhat reduced with error weighting [Fig. S5(b)]. Four and three principal structural classes could be defined, respectively [Figs. S5(c) and S5(d)], with a larger percentage of contributing structures clustering in the low-R_g range. A few structures having a wide separation between the domains are more present in the NNLS fits without errors.

4.3.2. Ensemble modelling with I(q) as the target function

As for SASDF83, the experimental I(q) for SASDM77 does not have a uniform Δq: Δq is 1.102 × 10⁻³ Å⁻¹ for q ≤ 6.9519 × 10⁻² Å⁻¹, thereafter having a variable step (Δq in the range [3.53, 4.53] × 10⁻⁴ Å⁻¹) to q_max = 0.32532 Å⁻¹. CRYSOL generated I(q) profiles on the MMC pool using a fixed grid spacing of 1.102 × 10⁻³ Å⁻¹, resulting in 295 I(q) points that were interpolated to match the 644 experimental data points for the NNLS fitting procedure. An excellent fit [Figs. 5(a) and 5(b)] was obtained with a χ² of 1.208, comparable to the value of 1.279 obtained with the P(r) NNLS fit without errors [Table 2(c)]. However, the 〈R_g〉 value of 52.1 Å is significantly larger than the R_g of both the P(r) NNLS fits and the experimental values [Tables 1 and 2(c)]. While the 〈d_max〉 value in this case is close to the GNOM-derived d_max value, the maximum d_max is also greater than the experimentally derived value (243 versus 170 Å), with a 7% contribution [Tables 1 and 2(c)]. These results are reflected in the R_g histogram [Fig. 3(m)] where, together with the cluster at R_g values smaller than or near to that of the starting AF structure or close to the Guinier-derived value, there are individual structures with quite large R_g values, up to ∼92 Å for a structure that is 7% of the ensemble population.

The WAXSiS-generated I(q) profiles from all the structures selected for any of the NNLS fits, plus the starting AF structure (a total of 24 structures), were then used as input for an NNLS fit against the SASDM77 I(q) profile [Figs. 5(e) and 5(f)], resulting in an excellent fit with the best χ² of 1.179 among all the NNLS fits performed on this sample [Table 2(c)]. The quite high 〈R_g〉 value of ∼48.4 Å, as well as the 〈d_max〉 and maximum d_max, with the most elongated structure contributing 4% to the ensemble, are in line with the values obtained with the CRYSOL/NNLS fit [Table 2(c)]. The R_g histogram [Fig. 3(n)] shows, however, that in this case the cluster at smaller R_g values (more extended than in the CRYSOL fit case, 35–45 Å) is even more predominant, accounting for 75% of the structures selected, and spanning from the AF starting structure (which was not selected; see also Table S4) to those of the GNOM-derived values.

The individual I(q) profiles selected in the CRYSOL and WAXSiS NNLS fits can be seen in Figs. S6(a) and S6(b), respectively, with the corresponding structural classes in Fig. S6(c) [same superposition and orientations as those in Figs. S5(c) and S5(d)]. The compact conformational cluster can be split into two, followed by two clusters of relatively and quite extended structures, respectively, and the most extended one can be set apart, also based on its very peculiar I(q) profile [pink lines in Figs. S6(a) and S6(b)].

As with the other two systems studied, the four structures with the highest percent contributions from all NNLS fits are shown as ribbon representations after superposition on the N-terminal 63–270 domain [Fig. 5(e), inset]. The highest proportion of R_g values are all in the ascending half of the MMC-generated pool R_g distribution (32–60 Å), with smaller percent contributions in the other half (60–92 Å), selected only by the I(q) NNLS fits as shown in Fig. 3(o), which also confirms the good correspondence between the original entire MMC pool and the pool employed for the NNLS fits.

This example is distinct from the other two in that the populations obtained by modelling with P(r) versus I(q) as the target give different results with respect to the population of extended structures present, reflected in both the 〈R_g〉 and maximum P(r) d_max values. It has already been noted that the SAXS data for this example are of poorer statistical quality than the other two data sets, and it is also the case that q_min is only 0.0141 Å⁻¹ (compared with 0.0025 and 0.0082 Å⁻¹ for SASDBP9 and SASDF83, respectively), with far fewer data points in the Guinier region (by factors of three to five). These observations raise the question of whether the low-q limit and sampling frequency in the Guinier regime for SASDM77 are sufficient for a reliable characterization of the most extended structures present in the sample. For this data set, the experimental q_min is such that structures with d_max > 220 Å⁻¹ would not be reliably characterized, and it would also limit the accuracy of d_max when calculating P(r). Modelling with P(r) as the target may thus artificially limit d_max, while modelling with I(q) as the target would probably allow for more extended structures.

With some question as to the precise nature of the population of extended structures, our analysis nevertheless suggests that the predominant confomations for this protein are just slightly more extended than that of the starting AF structure but can experience transitions to very elongated conformations in a relatively low proportion of the total population.

5. Discussion

AlphaFold has been shown to provide excellent predictions for large numbers of proteins. However, proteins with flexible segments cannot be adequately represented with a single static structure. Although this could be seen as a weakness of AF, it simply reflects what is often a necessary aspect in protein function. Coupling AF predictions with experimentally derived constraints and conformational space expansion could provide the researcher with an enhanced representation of the system under study.

Indeed, for the three examples identified here where there is an AF-predicted structure and a corresponding SAXS data set, we have shown that the AF-predicted structure cannot account for the experimental data without modifications. Using the confidence level indicators provided with each AF-predicted structure, we identified potential flexible linkers connecting the higher-confidence structured domains. Using the MMC routine, which efficiently creates tens of thousands of plausible all-atom structures with torsion angles ϕ and ψ in the allowed regions of the Ramachandran plot, we generated a pool of conformers from which a weighted population was identified that predicted the experimental SAXS data. In two cases (SASDBP9 and SASDF83), modelling with the calculated P(r) as the target gave similar results, in terms of the range and average R_g and d_max values, to modelling with I(q) as the target. In the third case, there were strong similarities but with a difference in the population of the most extended structures present in the optimized ensemble. The result obtained when modelling I(q) included significantly more extended structures than those obtained when modelling P(r). This difference appears to be attributable to a too-large q_min and Δq in the Guinier region for the SASDM77 data to characterize reliably the structures with the longest d_max values. These data were collected in 2004 on the European Molecular Biology Laboratory (EMBL) X33 beamline at the DORIS III storage ring (Hamburg, Germany) using a 1D gas detector (Blanchet et al., 2012) that has since been decommissioned. Improvements in instrumentation since then deliver data of a quality and q range more in line with the SASDBP9 data set (collected on the P12 beamline of EMBL at the storage ring PETRA-III of the Deutsches Elektronen-Synchrotron, Hamburg, using a Pilatus 2M detector) and SASDF83 data set [collected on the BM29 beamline at the ESRF (Grenoble, France) using a Dectris Pilatus 1M 2D detector (Pernot et al., 2013)], which would potentially resolve the discrepancy observed here for SASDM77 modelling with P(r) versus I(q) as the target.

Our results demonstrate that modelling with P(r) as the target can give reliable results, provided that the SAXS data meet quality metrics that ensure the data represent monodisperse proteins in solution, free of interparticle correlations, and both q_min and Δq meet the requirements for reliable characterization of the most extended structures present. Depending on the specific system, the scattering curve I(q) may be a more sensitive reporter of a global conformational change than the P(r) profile. Conversely, the P(r) can exhibit a wealth of structural features associated with domain shapes and their arrangement within the molecule that are not evident in an apparently featureless scattering curve. Computationally, the main advantage for modelling in real space is the ease and speed compared with the much more intensive I(q) calculation, especially if methods relying on explicit water all-atom molecular dynamics to account for hydration are used, e.g. as in the case for WAXSiS. While our P(r) calculations from individual structure coordinates did not account for the scattering contribution from hydration water, this effect is relatively minor compared with the difference observed between the AF structure and experimentally derived P(r), and it also becomes less significant as the protein size increases. Nevertheless, it could account for some of the differences observed between P(r) and I(q) NNLS fits, as both WAXSiS and CRYSOL consider the contribution from hydration water. On the other hand, if we compare the models pre-selected by P(r)- and I(q)-based methods (Tables S2–S4) we see that, for AF-Q16543, of the eight models selected by NNLS on the WAXSiS-generated I(q), three (with a combined 47% contribution) were also picked at the dry P(r) level. For the AF-Q06187 system, the numbers were five (with a combined 65% contribution) over six, and for the AF-Q9UKA9 system the numbers were six (with a combined 70% contribution) over ten. In any case, the development of methods to account reliably for hydration water in computing the P(r) from dry structures without sacrificing its speed advantage, such as those based on the statistical distributions of water used in US-SOMO to compute hydrodynamic properties (Rai et al., 2005), would constitute a welcome improvement.

The main challenge for modelling the real-space function lies in the fact that the experimentally derived P(r) is obtained as an indirect Fourier transform that often includes a user-selected d_max value along with assumptions that P(r) goes to zero at r = 0 and d_max. For flexible structures, depending on the nature of the population and the measured q_min, there can be significant uncertainty in d_max, with P(r) exhibiting a long low-intensity tail with large errors. Using multiple methods to calculate P(r) can provide a measure of the uncertainty in d_max, and here we compared GNOM-derived P(r) profiles obtained using a standard approach to d_max selection and a Bayesian application (BayesApp) without a user-selected d_max. The latter method generally yielded somewhat smaller d_max values, and almost an order of magnitude smaller uncertainties. We repeated the NNLS analysis for SASDBP9 and found very similar results to those observed using the GNOM-derived P(r) (data not shown), indicating that the low-intensity large-error tail on the P(r) was not very influential.

Regardless of the modelling method used, best practice is always to assess the fit against the actual measured data, which for SAXS is the I(q) profile. In all three cases tested here, the best-fit ensembles from the P(r) modelling also gave good fits to the experimental I(q) as assessed by χ² and error-weighted residual plots. While error weighting in our P(r) fitting resulted in some differences, further work would be required to understand how to account for the errors properly, as they are not true counting statistics. Indeed, we have observed differences in the magnitude of errors up to a factor of ten between different software programs computing P(r) from I(q).

The computation of the I(q) profile from a structure is also dependent, among other things, on the treatment of the hydration contribution. Several computational methods are available, and a comparison of some of the most widely used ones has been presented in a recent benchmarking study (Trewhella et al., 2022). To complement our P(r)-based pre-selections we have chosen CRYSOL, as implemented within US-SOMO (Brookes & Rocco, 2018), which has proven to be fast enough to allow batch-mode computation over a few thousand structures without resorting to high-end computing facilities. The main results presented here were based on CRYSOL 2.8, which approximates the hydration as a layer of uniform density and uniform thickness. The calculations were redone with CRYSOL 3.2 using the more recently implemented option for explicit representation of hydration as dummy beads (Franke et al., 2017), which is in principle better suited to structures presenting extended unstructured regions. However, relatively minor differences were observed (see Section S1). Moreover, when a more advanced computational approach was employed that uses a short molecular dynamics simulation within a full solvation box, namely WAXSiS (Chen & Hub, 2014; Knight & Hub, 2015), very similar results were obtained using structures pre-selected by the P(r) approach and either CRYSOL 2.8 or CRYSOL 3.2, or both (Section S1 and Tables S2–S4). Possibly, when a structure contains a mix of folded and unstructured regions, the differences in the hydration models are not as significant compared with the case of, for instance, an intrinsically disordered protein.

For this study, we were dependent on the chance co­incidence of there being a SAXS data set corresponding to an AF-predicted structure where significant differences were apparent between the predicted and experimentally derived P(r) functions. From the initial pool of 43 data sets, just three examples were identified where the experimental data generally satisfied the quality criteria presented in the SAS guidelines (Trewhella et al., 2017). While none of these were obtained using the preferred SEC–SAXS measurement mode that increases the likelihood of the sample being monodisperse, the quality assessment done demonstrates that careful measurement in batch mode can yield reliable data. Among the criteria to evaluate the quality of SAXS data, the accuracy of SAXS-derived molecular mass values is critical and not always easy to achieve. There are multiple methods available that sometimes provide differing values that should be accounted for, but there is a strong temptation simply to accept the one that gives the best agreement with expectation. For the three data sets we analysed here, three different methods were used for the reported SAXS-derived molecular mass: calculated from I(0) relative to a BSA standard (Mylonas & Svergun, 2007) for SASDBP9, from Bayesian inference (Hajizadeh et al., 2018) for SASDF83 and from a DAMMIN envelope volume (Svergun, 1999) for SASDM77. In reporting SAXS data, it is highly recommended to provide SAXS-derived M^expt values determined using multiple methods, but importantly including from I(0)/c [where c is the protein concentration of the sample (mass/volume)], with scattering intensities placed on an absolute scale (Trewhella et al., 2017). This method requires an accurate concentration measurement of the SAXS sample and knowledge of the partial specific volume, which can be calculated from the sequence [see Trewhella et al. (2022)]. The uncertainty in the concentration determination, coupled with that in the partial specific volume calculation, may be the limiting factor, but for all its shortcomings, this method for estimating M^expt is important, particularly in the case of flexible molecules for which estimates derived from the scattering curve independent of the concentration are more problematic.

We initially considered a fourth example for this study, AF-P50891 and its corresponding SASBDB data set SASDHP4, where the predicted and experimental P(r) profiles differed significantly. However, this protein presented three well characterized N-glycosylation sites (Olson et al., 2020). Using the Glycam website (https://glycam.org) we built the three high-mannose carbohydrate chains to complete the atomic description of the protein, and the differences between prediction and experiment at the I(q) profile level were substantially reduced, demonstrating the importance of accounting for post-translational modifications. However, further analysis of this system was beyond the scope of this study.

Coupling of the MMC methodology with NNLS fitting in both real and reciprocal space as presented in this work has led to interesting insights in each of the systems presented. Since automation of the entire real-space analysis and of some aspects of the reciprocal-space analysis, such as internally calculated CRYSOL profiles and externally generated WAXSiS profiles, does not seem to present major hurdles, further development with a dedicated module of the US-SOMO online website (https://somoweb.genapp.rocks) is planned. This additional useful tool for biomolecular SAXS would nicely complement the very important advances that AlphaFold has brought to the wider biostructural community.