Protein self-association is fundamental for many processes in biology (Ali and Imperiali, 2005; Marsh and Teichmann, 2015), and it has been estimated that around half of all proteins form dimers or higher-order complexes (Lynch, 2012). One such protein is Speckle-type POZ protein (SPOP), a substrate adaptor in the ubiquitin proteasome system, which recruits substrates for the Cullin3-RING ubiquitin ligase (CRL3) (Hernández-Muñoz et al., 2005; Kent et al., 2006; Kwon et al., 2006). SPOP targets a range of substrates for degradation, including proteins involved in hormonal signalling, epigenetic modification, and cell-cycle control, such as the androgen receptor (AR) (An et al., 2014) and death-associated protein 6 (DAXX) (Kwon et al., 2006; Cuneo and Mittag, 2019). SPOP is thus an important regulator of cellular signalling, and mutations in SPOP are associated with a variety of cancers (Le Gallo et al., 2012; Kim et al., 2013; Cuneo and Mittag, 2019).

The 374-residue SPOP monomer consists of three domains. From N- to C-terminus, these are the MATH domain (i.e., the meprin and TRAF-C homology domain), the BTB domain (i.e. the broad-complex, tramtrack, and bric-a-brac domain), and the BACK domain (i.e. the BTB and C-terminal Kelch domain). MATH is the substrate binding domain, while the BTB domain mediates interaction with CRL3 (Zhuang et al., 2009; Bosu and Kipreos, 2008). Both the BTB and BACK domains can homodimerize, resulting in the formation of polydisperse, linear higher-order SPOP oligomers with alternating BTB-BTB and BACK-BACK interfaces (Errington et al., 2012; van Geersdaele et al., 2013; Marzahn et al., 2016). The BTB-mediated dimer is formed with nanomolar affinity, and this dimer acts as the unit of higher-order oligomerization, which occurs through micromolar affinity BACK dimerization. Thus, only even-numbered SPOP oligomers are substantially populated (Marzahn et al., 2016; Figure 1).

Chemical crosslinking experiments have shown that SPOP oligomers form inside cells (Marzahn et al., 2016), and analysis of SPOP homologues shows sequence co-variation across both the BTB-BTB and BACK-BACK interfaces (Bouchard et al., 2018), together suggesting that self-association has physiological relevance. By presenting multiple MATH domains for substrate binding, SPOP oligomers can simultaneously bind to multiple low-affinity binding motifs in a single substrate, resulting in an overall increased affinity through avidity effects (Pierce et al., 2016). The longer lifetimes of these complexes enable effective polyubiquitination (Pierce et al., 2016). This suggests that tuning SPOP’s oligomerization state could act as a mechanism to regulate substrate degradation (Errington et al., 2012). SPOP oligomerization is also involved in phase separation. SPOP localizes to nuclear speckles in cells (Marzahn et al., 2016), and upon overexpression of certain substrates, SPOP and substrate co-localize to condensates which recruit CRL3 and display active substrate ubiquitination (Bouchard et al., 2018). This process requires both SPOP oligomerization and substrate binding (Marzahn et al., 2016; Bouchard et al., 2018), and it has been proposed that SPOP oligomers function as scaffolds that enable binding of substrates both within and between oligomers, resulting in filament-formation at low substrate concentrations and condensate formation at higher substrate concentrations (Bouchard et al., 2018; Schmit et al., 2020).

The higher-order self-association of SPOP follows the isodesmic model (Marzahn et al., 2016), in which the equilibrium between oligomer i and i+1 is described by a single equilibrium constant independently of oligomer size (Oosawa and Kasai, 1962). In the case of SPOP, the BTB-mediated dimer acts as the protomer of higher-order self-association, and the isodesmic KD thus describes BACK-BACK self-association. The isodesmic model can be used to calculate the equilibrium concentration of every oligomeric species as a function of the total protomer concentration (Figure 1a and c). For SPOP, a low micromolar isodesmic KD has been determined from composition gradient multi-angle light scattering experiments (CG-MALS; Marzahn et al., 2016). While these insights describe the heterogeneity in oligomer sizes, the conformational heterogeneity of the higher-order oligomers has not been characterized. Previous work revealed that constitutive SPOP dimers, created via deletion of the BACK domain, have considerable conformational heterogeneity in the position of their MATH domains. The MATH domains are seen docked onto the BTB dimer in the structure, but small-angle X-ray scattering (SAXS) experiments showed that they could undock from the BTB domains, enabled by a long flexible linker (Zhuang et al., 2009). This may enable binding of multivalent substrates with different spacing between SPOP binding motifs. Whether this conformational flexibility also exists in higher-order SPOP oligomers is unclear.

Here, we aimed to determine simultaneously both the distribution of oligomeric states of SPOP and the conformational ensemble of each SPOP oligomer by combining SAXS experiments and MD simulations. SAXS can provide low-resolution information on protein structure in solution, but reports on an ensemble average, which in the case of SPOP is both an average over different oligomeric states and the structural heterogeneity of each oligomeric state. Therefore, SAXS experiments are often combined with MD simulations to provide a full structural model of the system (Thomasen and Lindorff-Larsen, 2022). In the case of polydisperse systems, it is sometimes possible to deconvolute the information into contributions from a small number of individual species and analyse these individually (Herranz-Trillo et al., 2017; Meisburger et al., 2021). We took a different approach and aimed to explicitly model every relevant configuration of SPOP in its range of oligomeric states along with the associated thermodynamic weight of each configuration. We collected SAXS data on SPOP at a range of protein concentrations and constructed initial conformational ensembles of every substantially populated oligomeric state using coarse-grained MD simulations. We then developed an approach to simultaneously and self-consistently optimize the distribution of oligomeric states, given by the isodesmic model (Oosawa and Kasai, 1962; Shemesh et al., 2021), and refine the conformational ensemble of each oligomer against the SAXS data using Bayesian/maximum entropy (BME) reweighting (Bottaro et al., 2020; Figure 2). Our results show that SPOP forms rigid, helical oligomers in solution, and that the linker connecting the MATH and BTB domains is likely flexible, allowing for repositioning of the MATH domains during substrate binding. Our results also provide further evidence that SPOP self-association follows the isodesmic model, and we find an isodesmic KD in good agreement with the previously determined value (Marzahn et al., 2016). Using SAXS experiments of a cancer variant of SPOP we also show how our approach can be used to determine changes in the level of self-association.

We collected a concentration series of SAXS data on a previously used truncated version of SPOP, SPOP^28–359 (full length is 374 residues), with total protein concentrations ranging from 5 to 40 µM. In order to build structural models to refine against the SAXS data, we first needed to decide which oligomeric species to include in our modelling. To this aim, we used the isodesmic self-association model, which has previously been shown to describe SPOP oligomerization well (Marzahn et al., 2016). We fitted previously measured CG-MALS data (Marzahn et al., 2016) to obtain an isodesmic KD of 1.6±0.3 µM (Figure 1—figure supplement 1). Based on the isodesmic model fitted to the CG-MALS data, the population of oligomers larger than ~30-mersshould be very low at the concentration range used in our SAXS experiments. As scattering intensity is proportional to particle size squared, larger oligomers, however, make a considerable contribution to the SAXS signal despite their low concentrations (Figure 1c). Given the concentrations from the isodesmic model and taking into account the increased scattering of larger oligomers, we decided that constructing models of oligomers up to the 60-mer should be sufficient to capture all substantial contributions to the SAXS data.

There are no crystal structures of SPOP^28–359 available, so we constructed a model of the SPOP^28–359 BTB-dimer using the crystal structure of the isolated BACK domain (4HS2) (van Geersdaele et al., 2013) and the crystal structure of a truncated construct containing only the MATH and BTB domains (3HQI) (Zhuang et al., 2009; Figure 1b). We used this model of the BTB-dimer to construct SPOP^28–359 oligomers, which we used as starting structures for MD simulations. We ran 60 µs MD simulations of oligomers ranging from the dimer to the dodecamer; we used a coarse-grained representation of SPOP modelled using the Martini 3 force field (Souza et al., 2021) further modified by increasing protein-water interactions by 6% (Thomasen et al., 2022). It would be computationally prohibitive to run simulations of large oligomers up to the 60-mer. Instead, we relied on the assumption that the dodecamer behaves similarly to a segment of an arbitrarily long oligomer, and constructed conformational ensembles of oligomers up to the 60-mer by joining together conformers from the simulations of the dodecamer at the BACK-BACK interface (Figure 1d).

We calculated SAXS intensities from our conformational ensembles and, given the relative population of each oligomer from the isodesmic model with KD=1.6 µM, determined from CG-MALS, we calculated SAXS profiles averaged over all the oligomeric species. We found that the SAXS data calculated in this way from the ensembles generated by MD simulations convoluted with the isodesmic model were in good agreement with the experimental SAXS data, giving a reduced χ2 to the concentration series of SAXS data (χr,global2) of 1.53 (Figure 3). Despite the overall good agreement, the residuals revealed some systematic deviations to the experimental SAXS profiles. These deviations could potentially arise from inaccuracies in the distribution of oligomeric states given by the isodesmic model, from inaccuracies in the modelled conformational ensembles, or from both. As a first step, we wanted to see if we could eliminate the deviations by only tuning the distribution of oligomeric states. We globally optimized the KD of the isodesmic model against the concentration series of SAXS data, which gave KD=0.9±0.4 µM, in good agreement with KD=1.6±0.3 µM determined from CG-MALS, and resulted in a χr,global2 of 1.24 to the SAXS data (Figure 3). However, this did still not fully eliminate the systematic deviations from the experimental SAXS profiles.

To improve the agreement with the experimental SAXS data further, we aimed to simultaneously refine the conformational ensemble of each oligomer and optimize the distribution of oligomeric states. We developed a self-consistent optimization scheme, in which the isodesmic KD is optimized globally to the entire concentration series of SAXS data followed by reweighting of the conformations of each oligomer against a SAXS profile deconvoluted from the experimental SAXS data (Figure 2; see Methods section for details). To reweight the ensembles, we used BME reweighting, in which the population weights of the conformational ensemble are minimally perturbed with respect to the prior ensemble (generated by the MD simulations) to improve the agreement with a set of experimental data (Bottaro et al., 2020). This approach resulted in excellent agreement with the experimental SAXS data, giving a χr,global2 of 0.69, while only small deviations remained (Figure 3). The isodesmic KD was fitted to 1.3±0.5 µM, and thus also remained in good agreement with the previously determined value (Marzahn et al., 2016). To validate our approach and to examine the possibility of overfitting, we left out one SAXS profile recorded with 15 µM protein from the optimization. The optimized KD and ensemble weights did not substantially affect the fit to this SAXS profile, suggesting that we had avoided overfitting (Figure 3—figure supplement 1). These results show that SAXS data on SPOP can be explained well by conformational ensembles of linear oligomers with populations given by the isodesmic model, and thus provide further evidence that SPOP self-association follows a simple isodesmic mechanism (Marzahn et al., 2016).

The previously published CG-MALS data on SPOP clearly precludes a simple dimer–tetramer, dimer–hexamer, dimer–octamer, or dimer–decamer equilibrium in favor of an isodesmic self-association model (Marzahn et al., 2016). To determine whether the SAXS data also favors the isodesmic model, we used our conformational ensembles to examine whether monodisperse oligomers, ranging in size between an octamer and 60-mer, as well as corresponding dimer-oligomer equilibria, would be compatible with the SAXS concentration series (Figure 3—figure supplement 2). For each dimer–oligomer equilibrium, we fitted the KD globally to the SAXS data. Thus, the isodesmic model and dimer–oligomer models are of comparable complexity, with only a single free parameter. The results show that the SAXS concentration series is in better agreement with an isodesmic distribution of oligomers than with any of the tested single oligomers or dimer–oligomer equilibria.

We also wished to examine whether the conformational ensembles of SPOP generated by the MD simulations described the SAXS data better than static structures. As a first comparison, we calculated SAXS profiles from the initial SPOP oligomer structures constructed based on crystal structures. To make the results comparable with our optimized ensembles, we fitted the isodesmic KD to the SAXS data for the static structures. This resulted in a worse agreement with the SAXS data (χr,global2=4.03 with KD=0.43 µM) than what we obtained using the ensembles both before and after reweighting. We also investigated the agreement with the SAXS data for individual structures drawn from the ensembles of the oligomers, again fitting the isodesmic KD for each set of structures. We found that some of the single structures from the ensembles could fit the SAXS data as well as the entire ensembles before reweighting, but no set of static structures fit the SAXS data as well as the reweighted ensembles (Figure 3—figure supplement 3). This result highlights that, while an ensemble of multiple conformers is likely necessary to produce the best agreement with the SAXS data, SPOP oligomers have a relatively rigid structure overall, allowing for reasonable agreement with the SAXS data without modelling the conformational heterogeneity for each oligomer. The improvement in agreement with the SAXS data over the starting structures, also for individual conformers, suggests that the MD simulations contribute, not only by modelling the conformational heterogeneity, but also by simply relaxing the structure to a more accurate state.

The results described above show that the SAXS data fit well to an isodesmic model with a KD value close to that determined from CG-MALS. We wished to validate our approach further by comparing the SAXS-derived model of self-association with the CG-MALS data more directly. We therefore calculated the average molecular weight given by the isodesmic model with the KD of 1.3 µM that we obtained by fitting to the SAXS data and compared the results with the CG-MALS data (Figure 3—figure supplement 4). This analysis confirmed that the model of self-association derived from our analysis of the SAXS data is fully consistent with the independently measured CG-MALS data.

Having generated a conformational ensemble of each SPOP oligomer in agreement with the SAXS data, we proceeded to analyze the structures. Reweighting resulted in an increase in the radius of gyration (Rg) for almost all oligomeric species, suggesting that slightly more expanded conformations than those sampled with our modified version of Martini are more consistent with the SAXS data (Figure 4a–b and Figure 4—figure supplement 1). This expansion can be attributed both to a slight increase in the end-to-end distance for most oligomers (Figure 4c–d and Figure 4—figure supplement 2), as well as a slight increase in the average distance between the MATH and BTB/BACK domains for all oligomers upon reweighting (Figure 4e–f).

In order to investigate the global flexibility and compaction of SPOP oligomers, we fitted a power law to the average end-to-end distance (RE-E) as a function of the number of subunits in the oligomer (N), RE-E=R0⁢Nν, where R₀ is the subunit segment size and ν is a scaling exponent (Figure 4c). The fit gave R₀=~3.1 nm and ν=0.99, showing a linear growth of the end-to-end distance with the number of subunits. This result is consistent with no significant curvature or compaction of the oligomers and, along with the narrow distribution of end-to-end distances for each oligomer (Figure 4—figure supplement 2), suggests that the SAXS data is compatible with a distribution of straight and relatively rigid SPOP oligomers, at least on length scales up to the ~180 nm of the 60-mer. The helical structure of larger oligomers, with ~16 subunits per turn, is evident as small periodic deviations from the fit (Figure 4c).

The MATH and BTB domains are connected through a ~20 residue long linker region (Figure 1b). We hypothesized that this linker may be flexible, allowing for reconfiguration of the MATH domains with respect to the crystal structure (Zhuang et al., 2009). We calculated the distances between the center-of-mass (COM) of the MATH domain and the COM of the BTB/BACK domains in the ensembles for every subunit of every oligomer. The distribution of these MATH-BTB/BACK distances reveal two populations overlapping with the two crystal structure configurations (Figures 4e, i ,, 5c), where the MATH domains are in close proximity to the BTB/BACK domains (Zhuang et al., 2009). However, there is also a third population in which the MATH domains are extended away from the BTB/BACK domains, suggesting that the MATH-BTB linker is flexible and allows for movement out of the configurations observed in the crystal structure. Reweighting slightly increased the population of this extended state (Figure 4e–f). This flexibility in the configuration of the MATH domains gives rise to a broad distribution of distances between the substrate binding sites in neighbouring MATH domains, which is also slightly increased upon reweighting for all oligomers (Figure 4g–h). Both the overall rigidity of the oligomers and the flexibility of the MATH domains are also evident from visual inspection of the conformational ensemble of the 60-mer (Figure 4j).

To examine further whether the SAXS data support the observed flexibility of the MATH-BTB linker and repositioning of the MATH domains, we generated new ensembles of SPOP oligomers following the same protocol as above, but this time restraining the MATH domains to the BTB-BACK domains based on the configuration in the crystal structure using the elastic network model implemented in Martini. We calculated SAXS data from the generated ensembles and again fitted the isodesmic KD globally to the SAXS data, resulting in KD=0.2±0.2 µM. The agreement with the SAXS data was substantially worse than for the original ensembles with the MATH domains unrestrained (χr,global2=4.38 and χr,global2=1.24 respectively), and the systematic deviations from the experimental SAXS profiles were clearly exacerbated (Figure 5). These results suggest that, first, the resolution of the SAXS data is high enough to distinguish between different configurations of the MATH domains and, second, that the SAXS data are indeed in better agreement with a model where the MATH-BTB linker is flexible. Taken together, our results support a model where, in solution, SPOP oligomers behave as rigid, helical structures with flexible MATH domains that can extend away from the BTB/BACK domains.

The comparison between the conformational ensembles generated with the MATH domains free or restrained also suggests that accurate conformational ensembles are necessary for accurate determination of the isodesmic KD. Fitting the SAXS data using ensembles with the MATH domains restrained resulted in a lower isodesmic KD, and calculating the agreement with SAXS for a range of isodesmic KD values revealed that there was no clear minimum in the χr,global2 for KD values > 0, which is also reflected in the large error range for the fitted KD (Figure 5—figure supplement 1). In line with this observation, validation with SAXS data at 15 µM protein revealed that improving the accuracy of the ensembles by reweighting also improved the accuracy of the fitted isodesmic KD independently of the fitted ensemble weights (Figure 3—figure supplement 1d).

To explore further how changes in the conformational ensemble would affect the agreement with the SAXS data, we used subsampling to generate ensembles with specific properties based on the ensembles with unrestrained MATH domains. To keep the comparison of ensembles unbiased by our previous fitting to the SAXS data, we used the isodesmic KD=1.6 µM from CG-MALS for all comparisons with SAXS. First, we selected frames with lower average MATH-BTB/BACK COM distances (Figure 5—figure supplement 3). In line with the results from simulations with restrained MATH domains, this worsened the agreement with the SAXS data. In contrast, selecting frames with higher average MATH-BTB/BACK COM distance slightly improved the agreement with the SAXS data, in line with the results from reweighting (Figure 5—figure supplement 4). We also wished to test how sensitive the agreement with the SAXS data was to the overall shape of the oligomers. However, the conformational space that we could explore by subsampling the ensembles was limited by the rigidity of the oligomers. Despite this limitation, we subsampled ensembles with slightly higher and lower end-to-end distances than the original ensembles, corresponding to oligomers that are more or less extended than the original ensembles. Again consistent with the results from reweighting, ensembles with lower end-to-end distance resulted in slightly worse agreement with the SAXS data (Figure 5—figure supplement 5), while ensembles with higher end-to-end distance did not substantially change the agreement with the SAXS data (Figure 5—figure supplement 6). This result suggests that the SAXS data is less consistent with more compact SPOP oligomers, at least within the local part of conformational space explored here.

Self-association allows SPOP to bind disordered substrates that contain multiple SPOP binding motifs through multivalent interactions (Pierce et al., 2016). We hypothesized that the spacing between MATH domains in SPOP oligomers could be related to the spacing between SPOP binding motifs in disordered substrates. To investigate this, we selected five SPOP substrates with multiple SPOP binding motifs located in IDRs (SETD2 Zhu et al., 2017, SCAF1 Theurillat et al., 2014, SRC3 Li et al., 2011; Geng et al., 2013; Janouskova et al., 2017, Gli2, and Gli3 Zhang et al., 2006; Zhang et al., 2009) and ran coarse-grained simulations of their IDRs (seven IDRs in total) using CALVADOS, a one-bead-per-residue implicit solvent model that has been optimized to reproduce accurate global dimensions and transient interactions in IDPs (Tesei et al., 2021). We calculated the distances between neighbouring SPOP binding motifs in the simulations, and compared these with the distances between substrate-binding sites in neighbouring MATH domains given by our ensembles of SPOP oligomers (Figure 4g). This revealed substantial overlap between the two distributions, with a similar average distance between neighbouring binding sites in SPOP and in substrates, suggesting that the spacing of SPOP binding motifs in substrates may be evolutionarily optimized for multivalent binding to MATH domains.

Having analyzed the conformational properties of wild type SPOP and shown that the SAXS data are sensitive to the degree of self-association, we next wished to test whether our approach could capture the effects of mutations on SPOP self-association. We collected a concentration series of SAXS data on the SPOP mutant R221C, which has been identified in melanoma (Krauthammer et al., 2012) and colorectal cancer (Giannakis et al., 2016). R221C is located in the BTB-BTB interface, so we hypothesized that it may affect SPOP’s propensity to self-associate. We used the same approach as for wild type to fit the isodesmic KD globally to the SAXS data, but without reweighting the conformational ensembles. For R221C, the isodesmic KD was fitted to 8.2±2.3 µM, which resulted in a reasonable fit to the SAXS data with χg⁢l⁢o⁢b⁢a⁢l2=1.79 (Figure 3—figure supplement 7), suggesting that the mutation results in a decreased propensity to self-associate compared with wild type (KD=0.9 µM using a comparable approach or 1.3 µM when also reweighting the ensemble). Because R221C is located at the BTB-BTB interface, the 6–9 fold increase of the isodesmic KD (which relates to BACK-BACK dimerization) is perhaps surprising. While a long-range effect of R221C cannot be ruled out, an alternative mechanism may involve shifting the equilibrium of the BTB-BTB dimer, thus effectively decreasing the concentration of dimeric species available for self-association.

The ability of SPOP, a cancer-associated substrate adaptor in the ubiquitination machinery, to self-associate is important for its role in biology and disease. Characterizing the conformational ensemble of flexible and self-associating proteins such as SPOP from ensemble-averaged experiments is, however, difficult due to conformational and compositional heterogeneity. In one approach, SAXS data of mixtures may be attempted to be decomposed into contributions of individual components that may then be analysed separately (Herranz-Trillo et al., 2017; Meisburger et al., 2021). Here, we have developed an alternative ‘forward modelling’ approach to characterize proteins that undergo polydisperse oligomerization by self-consistently and globally fitting the distribution of oligomeric species and reweighting the conformational ensembles of the oligomers against SAXS data. A similar idea has recently been applied to study the self-association of tubulin using static structures as input (Shemesh et al., 2021). We recorded a concentration series of SAXS data on SPOP, which is known to form linear higher-order oligomers, and combined MD simulations with our approach to simultaneously refine conformational ensembles of thirty oligomeric states of SPOP along with the relative populations.

Our results suggest that SPOP oligomers are rigid, helical structures in solution and that the MATH-BTB linker is flexible, allowing for the extension of MATH domains away from the oligomer core. This is consistent with SPOP’s proposed role in phase separation, as reconfiguration of the MATH domains could facilitate binding of substrates across multiple MATH domains and between different SPOP oligomers (Pierce et al., 2016; Bouchard et al., 2018). Indeed, the spacing of the MATH domains in our model of SPOP oligomers is consistent with the distances between motifs in ensembles of disordered SPOP substrates, based on coarse-grained simulations of disordered SPOP substrates. It has been suggested previously that rigidity could play an important role in the phase separation of SPOP oligomers by ensuring a low conformational entropy penalty upon stacking linear oligomers with cross-bound substrates in the dense phase (Schmit et al., 2020). This is also consistent with the rigid structural model of SPOP oligomers proposed here. Our results also provide orthogonal evidence that SPOP self-association is described well by the isodesmic model, and that the isodesmic KD for BACK-BACK mediated self-association is in the low micromolar range, in agreement with previous measurements by CG-MALS (Marzahn et al., 2016). We also collected SAXS data and fitted the isodesmic KD for the SPOP mutant R221C. Our results suggest that SPOP R221C has a six- to ninefold decreased propensity to self-associate.

While the analysis of the SAXS data presented here does not strictly exclude the possibility that SPOP forms branched or otherwise non-linear oligomers, our results show that linear oligomers based only on the self-association interfaces known from existing crystal structures are consistent with SAXS data. Thus, linear oligomers seem to be the most plausible model based on this and other existing experimental evidence, for example that removal or mutation of either the BACK-BACK or BTB-BTB interface results in abolishment of higher-order self-association and that higher-order oligomers are formed through the self-association of SPOP dimers with every step of subunit-addition populated (Marzahn et al., 2016). Finally, as shown here, linear isodesmic self-association with the same KD provides a good fit to both SAXS and CG-MALS data (Marzahn et al., 2016).

The approach presented here to study SPOP can be extended to other polydisperse systems to characterize the distribution of oligomeric states and their conformational properties. However, there are a few limitations to be aware of; SAXS is a low-resolution technique, and may not be able to distinguish between all relevant conformations, a problem that is likely exacerbated here, as the contribution of many species to the SAXS signal may average out distinct features in the profile. One way to mitigate this problem is to construct multiple structural models, and test whether they show any difference in the agreement with the SAXS data. In the case of SPOP we used this approach to examine the flexibility of the MATH domain in SPOP^28–359.

Another limitation of the approach is the correlation between the fitted distribution of oligomeric states and the conformational properties of the oligomers. Here, we observed that a low isodesmic KD with large uncertainty was fitted when using more compact structures (MATH domains restrained), which suggests that the model can compensate for the underestimated dimensions of the proteins by increasing the populations of larger oligomers. Therefore, it is important to use prior conformational ensembles that are as accurate as possible. Additionally, it is important to include all the oligomeric species that make a substantial contribution to the SAXS data in the modelling. In the future, it might be relevant to include independent data reporting on the distribution of oligomeric species, such as from CG-MALS, when fitting SAXS data.

In the case of SPOP, we described the distribution of oligomers using the isodesmic self-association model, but this can be replaced by any model that describes the populations of the species in solution — with the caveat that there should not be too many free parameters to fit to the SAXS data. Similarly, the approach to generate prior conformational ensembles is not limited to MD simulations, and can be varied based on the system at hand. This flexibility in the modelling approach will make it useful to study other polydisperse systems in the future.