Introduction

Many proteins are composed of multiple domains that realize complex and sophisticated functions by rearranging their domains flexibly. For example, each domain works together to introduce DNA supercoiling in DNA topoisomerases and each domain cooperates to refold substrate proteins in some foldases1,2,3,4. However, revealing their dynamic processes in solution remains challenging.

Small-angle X-ray scattering (SAXS) is a representative experimental technique to study such behavior of biomolecules5,6,7,8,9. The SAXS profiles include structural information in the range of approximately 1–100 nm, which is usually wide enough to study structural range of multi-domain proteins or their complexes. SAXS intensity I(Q) is calculated as a function of scattering vector (Q):

$$ {\text{Q}} = 4{\uppi }\sin {\uptheta }/{\lambda } $$
(1)
$$ {\text{I}}\left( {\mathbf{Q}} \right) = \mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{N}}} \mathop \sum \limits_{{{\text{j}} = 1}}^{{\text{N}}} {\text{f}}_{{\text{i}}} \left( {\mathbf{Q}} \right){\text{f}}_{{\text{j}}} \left( {\mathbf{Q}} \right)\frac{{\sin \left( {{\mathbf{Q}}{\text{r}}_{{{\text{ij}}}} } \right)}}{{{\mathbf{Q}}{\text{r}}_{{{\text{ij}}}} }} $$
(2)

In Eq. (1), the θ and λ are the scattering angle and the wavelength of the incident X-ray, respectively. Equation (2) is termed Debye’s equation9: N is the number of atoms in a system, rij is the distance between the i-th and j-th atoms, and fi (Q) is an atomic form factor of i-th atom, which is regarded as a constant in the measured small-angle range.

Since all molecules in a sample solution contribute an experimental SAXS profile, the profile includes information on their structural ensemble. There could be two different types of structural sets of which ensembles reproduce the same SAXS profile. One is a “homogeneous structural set” in which all molecules have a similar structure. The other is a “heterogeneous structural set”. The latter includes the diverse structures, but their averaged SAXS profile reproduces the experimental one. It is difficult to determine which set should be adopted as the state of the molecule in solution. In other words, we cannot judge whether all molecules have similar or diverse structures only from an experimental SAXS profile.

To address this issue, two main criteria are considered in the context of structural modeling; they are reviewed as “maximum parsimony” and “maximum entropy”10. In the “maximum parsimony” approach, a structural set composed of a small number of models is selected among possible structural sets matching an experimental SAXS profile. For example, the Akaike information criterion or Bayesian information criterion is calculated for possible structural sets, and a structural set minimizing the criteria is chosen11,12. Many algorithms, such as ensemble optimization, minimal ensemble search, sparse ensemble selection, and maximum occurrence, have been proposed and utilized13,14,15,16,17.

In the “maximum entropy” approach, a free energy landscape derived from a simulation force field is adopted as the prior distribution. The resultant ensemble should match an experimental SAXS profile and be least inconsistent with the force field. A force field does not always reproduce a real structural ensemble of a molecule. Therefore, correction of a free energy landscape to match an experimental SAXS profiles is often effective. There are two ways to correct a free energy landscape while satisfying entropy maximization. One direct approach is reweighting the free energy landscape to match an experimental SAXS profile after structural sampling by molecular dynamics (MD) simulations or Monte Carlo simulations18,19,20,21,22. The other approach is to perform parallel simulations with additional potential to reproduce an experimental SAXS profile23,24,25,26,27.

Although both methods are effective, they are not necessarily sufficient to model any biomolecular systems. The structural set composed of a small number of models implies that the resulting structures discretely distribute in a structural space. In a largely fluctuating system, such as a multi-domain protein with intrinsic disordered regions, it is more reasonable to model a structural set which continuously distributes in a structure space.

Maximum entropy approaches are useful in that they can construct a physically reasonable structural set. It is necessary to sufficiently explore possible molecular structures in these approaches. However, sufficient stuructural sampling is often difficult for atomistic MD simulations. Coarse-grained (CG) MD simulation is a useful alternative to overcome this difficulty19,26. In many CG models, each CG bead reproduces net charge and hydrophilicity of their corresponding atom set, resulting in roughly reasonable inter- and intra-molecular interfaces in the simulations28,29,30. However, parameter adjustment using experimental data is often required31,32. A general-purpose CGMD potential does not necessarily guarantee accurate dissociation constant for the interfaces of a biomolecule. Therefore, it may not always be suitable to perform entropy maximization using a free energy landscape from a given CGMD potential as the prior distribution.

Here, we propose another approach to enumerate possible behaviors of a biomolecule using an experimental SAXS profile and CGMD simulations. In this method, CGMD simulations are first performed to obtain trajectories that efficiently cover their conformational space. Then, time series that match the SAXS profile are extracted from the trajectories. If the resulting time series are long enough, they reflect information on possible preferred states of the biomolecule; the molecule stays in the stable states for a longer time than unstable ones.

We tested our method on a multi-domain protein ER-60, which is a member of protein disulfide isomerase family3. ER-60 is composed a, b, b′, a′ domains and possesses reaction Cys-Gly-His-Cys (CGHC) motifs in both the a and a′ domains (Fig. 1). The a and a′ domains are respectively connected to the b and b′ domains via short hinge regions3,33. We focused on the domain dynamics of ER-60. We could extract multiple time series with our method. By examining domain conformation of ER-60 in each of the time series, we got overview of possible structural ensembles of the multi-domain protein. Distance between the a and a′ domains almost linearly correlated with I(Q) at each Q value. Therefore, diversity in scattering curves among structures was explained by open-close motion of ER-60. In addition, this linear relationship was indicated as two isosbestic points in Q-I(Q) plot. The actual structural ensemble of ER-60 in solution can be a mixture of these possible domain dynamics. Our method provides "elements of biomolecular motion”, which should be useful in the context of an integrated structural biology including SAXS, MD simulations, and other experiments.

Figure 1
figure 1

Multi-domain protein ER-60. (a) Four domains constituting ER-60. (b) Structural model of ER-60 based on a crystal structure (PDBID: 3F8U)33. PyMOL39 was used to model the missing C-terminal residues and replace the mutated C60A with cysteine. In the ribbon model, the a domain in shown in blue, the b domain in green, the b′ domain in yellowish green, and the a′ domain in red. The other parts are gray. The reactive cysteines in the CGHC motif are orange. Gly133 and Leu365 are hinge regions between the a and the b domain, and the b′ and the a′ domain, respectively.

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Methods

Modeling strategy

To compose a series of structural models that reproduces a SAXS profile, we made the following two assumptions about CGMD:

  1. 1.

    When stable inter- or intra- protein interfaces appear in CGMD simulations, they are regarded as candidates of actual interfaces, and

  2. 2.

    Since atomistic-scale interactions, such as hydrogen bonds and hydrophobic interactions, cannot be expressed precisely in CGMD simulations, the accurate affinity of the interfaces is not guaranteed.

Based on these assumptions, we devised a method to collect a series of structural models that include possible stable states. This is composed of two steps. In step 1, CGMD simulations changing a possible parameter are performed. In step 2, from each of the trajectories, the longest continuous time series that reproduces the SAXS profile is extracted. In summary, we clip a part of a CGMD trajectory that matches a SAXS profile and approximates the actual behavior of the biomolecule as a repetition of the clipped time series. We designate this method “SAS-CLIP”. Although only a single region is clipped from each trajectory, we can expect that possible structural ensembles can be enumerated by applying the SAS-CLIP to multiple trajectories. This point is studied in the “Results and discussion” section below. Using the multi-domain protein ER-60 as a model system, we examined feasibility and investigated the resulting structural series of SAS-CLIP.

MD simulations

All simulations were performed using GROMACS 2020.434,35. Before CGMD simulations, it was necessary to clarify the structural stability of three folded regions, from Ser25 to Pro134 (system a), that from Ala132 to Lys366 (system b-b′), and from Tyr364 to Glu493 (system a′). For this purpose, single atomistic simulations for the three systems were first performed.

CGMD simulations were performed using the Martini 3 open-beta version. In the CGMD simulations, the Lennard–Jones potential between water and ER-60 was scaled, as in previous reports19,20. The Lennard–Jones potential between i-th and j-th particles is described as:

$$ {\text{E}}_{{{\text{Lennard}} - {\text{Jones}}}} = 4{\upvarepsilon }_{{{\text{ij}}}} \left[ {\left( {\frac{{{\upsigma }_{{{\text{ij}}}} }}{{{\text{r}}_{{{\text{ij}}}} }}} \right)^{12} - \left( {\frac{{{\upsigma }_{{{\text{ij}}}} }}{{{\text{r}}_{{{\text{ij}}}} }}} \right)^{6} } \right] $$

In this study, we regarded the εij as a parameter. When i-th and j-th bead pair is water-protein bead pair, the εij is treated as λWP εij,default. Here, the εij,default is the default value in the Martini 3 open-beta version, and λWP is the scaling factor for water-protein interactions. The σijs are constants depending on bead type, and rij is distance between i-th and j-th particles. First, we performed three 5000-ns production runs with λWP values of 1.0, 1.01, 1.02, 1.03, 1.04, 1.05, and 1.06. According to the results (described in “Results and discussion” section), we additionally performed two 5000-ns production runs with λWP of 1.035, 1.043, 1.045, 1.046, 1.049, 1.052, and 1.055. We also performed two 10,000-ns production runs with λWP values of 1.04, 1.043, 1.046, 1.049, 1.052, and 1.055. Snapshots taken every 2 ns were used for analysis. For simulation detail, please see the Supplementary Information.

Analysis

SAXS profiles of snapshots in the CGMD simulations were calculated by Pepsi-SAXS36. Since the Pepsi-SAXS requires an atomistic model, the CGMD snapshots were reverse-mapped with the software backward37 before running the Pepsi-SAXS. Detailed description of the reverse-mapping is in the Supplementary Information.

The χ2 value is given as follows:

$$ {\upchi }^{2} = \frac{1}{{{\text{N}} - 1}}\mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{N}}} \left( {\frac{{{\text{I}}_{{{\text{exp}}}} \left( {{\text{Q}}_{{\text{i}}} } \right) - {\text{I}}_{{{\text{sim}}}} \left( {{\text{Q}}_{{\text{i}}} } \right)}}{{{\upsigma }\left( {{\text{Q}}_{{\text{i}}} } \right)}}} \right)^{2} $$
$$ {\text{I}}_{{{\text{sim}}}} \left( {{\text{Q}}_{{\text{i}}} } \right) = {\text{c}}\mathop \sum \limits_{{{\text{j}} = 1}}^{{\text{M}}} {\text{I}}_{{{\text{sim}},{\text{ j}}}} \left( {{\text{Q}}_{{\text{i}}} } \right) - {\text{offs}} $$

where N numbers of data points and M is the number of models in a structure set. Iexp(qi) and Isim,j (qi) are the scattering intensities of experimental SAXS profile and calculated scattering intensity of j th model, respectively. The theoretical profiles are normalized by Isim,j(0). c and offs are adjustment parameters. The latter parameter explains possible mismatch between buffer solution and sample solution in the experimental SAXS profile. The parameters are determined by the condition: \(\frac{{\partial {\upchi }^{2} }}{{\partial {\text{c}}}} = 0\) and \(\frac{{\partial {\upchi }^{2} }}{{\partial {\text{offs}}}} = 0\).

The distance distribution function p (r) was calculated using the reverse-mapped atomistic model. For the calculation, all electrons were considered to be in the center of an atom, and hydrogen atoms were ignored.

Since snapshots were acquired every 2 ns for analysis, the duration time of the time series Lt was calculated by multiplying the number of snapshots in the time series by 2 ns/frame.

The structure of ER-60 was analyzed based on the domain positions and orientations. Domains of ER-60 are mainly defined based on domain database Pfam38, where ER-60 consists of three domains: from Ser26 to Lys130 (a domain), from Phe160 to Asp355 (b-b′ domains), and from Pro377 to Arg482 (a′ domain). In previous reports, ER-60 was treated as four domain protein3,4, where the b-b′ domain was further divided into b and b′ domains. The θa-b-b′, θb-b′-a′, and φa-b-b′-a′ are angles formed by centers of mass (COMs) of the three domains or dihedral angle formed by COMs of the four domains (Supplementary Fig. S1). φb′-b-a-CGHC(a) and φb-b′-a′-CGHC(a′) are angles formed by COMs of the three domains and the COM of the CGHC motif of either the a or a′ domain. The Da-a′ is distance between COMs of a and a′ domains. The COM of each domain or CGHC motif is defined as the averaged coordinates of backbone beads of the corresponding residues (Supplementary Fig. S1). Probability distributions were calculated by binning a variable space and counting the number of data points in each bin.

The difference in the shape of the (θa-b-b′, θb-b′-a′) probability map between a clipped time series with the SAS-CLIP and its original trajectory (i.e., the entire trajectory before performing the SAS-CLIP) was evaluated using the following Kullback–Leibler divergence:

$$ {\text{KL}}_{{{\text{SAS}} - {\text{CLIP}}}} = \mathop \sum \limits_{{\left( {{\uptheta }_{{{\text{a}} - {\text{b}} - {\text{b}}^{\prime}}} ,{{\uptheta }}_{{{\text{b}} - {\text{b}}^{\prime} - {\text{a}}^{\prime}}} } \right) \subset {\mathbf{C}}1 \cup {\mathbf{C}}2}} {\text{P}}_{{{\text{SAS}} - {\text{CLIP}}}} \left( {{\uptheta }_{{{\text{a}} - {\text{b}} - {\text{b}}^{\prime} }} ,{{\uptheta }}_{{{\text{b}} - {\text{b}}^{\prime} - {\text{a}}^{\prime} }} } \right)\log \frac{{{\text{P}}_{{{\text{SAS}} - {\text{CLIP}}}} \left( {{\uptheta }_{{{\text{a}} - {\text{b}} - {\text{b}}^{\prime} }} ,{{\uptheta }}_{{{\text{b}} - {\text{b}}^{\prime} - {\text{a}}^{\prime} }} } \right)}}{{{\text{P}}_{{{\text{original}}}} \left( {{\uptheta }_{{{\text{a}} - {\text{b}} - {\text{b}}^{\prime} }} ,{{\uptheta }}_{{{\text{b}} - {\text{b}}^{\prime} - {\text{a}}^{\prime} }} } \right)}} $$

Here, PSAS-CLIP (θa-b-b′, θb-b′-a′) is the probability at (θa-b-b′, θb-b′-a′) of a clipped time series, and Poriginal (θa-b-b′, θb-b′-a′) is the probability of the original trajectory. KLSAS-CLIP only considers the region where PSAS-CLIP (θa-b-b′, θb-b′-a′) > 0 and their vicinity. In other words, the sum was calculated only for \({\mathbf{C}}1 \cup {\mathbf{C}}2\), where C1 and C2 are defined as follows (a schematic is presented in Supplementary Fig. S1d):


C1. {(θa-b-b′, θb-b′-a′) | PSAS-CLIP (θa-b-b′, θb-b′-a′) > 0}


C2. {(θa-b-b′, θb-b′-a′) | Adjacent to C1}These conditions allow KLSAS-CLIP to be small when ER-60 stays only in a few (or one) of several stable states. Considering that the KLSAS-CLIP ignores outside the \({\mathbf{C}}1 \cup {\mathbf{C}}2\), the Poriginal (θa-b-b′, θb-b′-a′) is normalized to meet the following relationship:

$$ \mathop \sum \limits_{{\left( {{\uptheta }_{{{\text{a}} - {\text{b}} - {\text{b}}\prime }} ,{{\uptheta}}_{{{\text{b}} - {\text{b}}\prime - {\text{a}}\prime }} } \right) \subset {\mathbf{C}}1 \cup {\mathbf{C}}2}} {\text{P}}_{{{\text{original}}}} \left( {{\uptheta }_{{{\text{a}} - {\text{b}} - {\text{b}}\prime }} ,{{\uptheta}}_{{{\text{b}} - {\text{b}}\prime - {\text{a}}\prime }} } \right) = 1 $$

Graphs and figures were created using gnuplot and inkscape. Images of protein structures were created using PyMOL39.

Results and discussion

Origin of differences between crystal and solution structures of ER-60

Our previous study showed that the structure of ER-60 in solution differs from the crystal structure4 because the SAXS profile calculated from crystal structure did not reproduce the experimental structure. To clarify the origin of this discrepancy, we first examined the dynamics of each folded structure. We performed atomistic MD simulations of three parts of ER-60, including the a, b-b′, and a′ domains. In any of the three simulations, root mean square deviation (RMSD) of the folded region between a simulation snapshot and the crystal structure distributed at approximately 1.5 Å (Supplementary Figs. S2, S3, S4). In the a-, b-b′-, and a′-part simulations, the prevalence of simulation snapshot with RMSD > 2.0 Å were 12.8%, 0.25%, and 21.2%, respectively (Supplementary Fig. S4; all supplementary data are provided). These findings suggest that the structure of each four domain of ER-60 in solution is almost the same as the crystal structure. Therefore, the discrepancy between crystal and solution structures should originate from the domain conformation or domain dynamics in solution.

Feasibility of SAS-CLIP

First, we examined whether a time series with a small χ2 could be clipped with SAS-CLIP. Three 5000-ns CGMD simulations were performed for each condition of λWP = 1.0, 1.01, 1.02, 1.03, 1.04, 1.05, and 1.06. The SAS-CLIP was applied to each trajectory with “χ2 < 3.0” as the criterion for reproducing an experimental SAXS profile which we previously reported4. Figure 2 shows the three longest time series, #A-1, #A-2, and #A-3 provided by the SAS-CLIP. Each averaged scattering curve matched the experimental one, suggesting that SAS-CLIP worked well (Fig. 2b). The duration time (Lt) of #A-1, #A-2, and #A-3 were 2690, 2208, and 1462 ns, respectively (Table 1). The times were long enough to elucidate several preferred conformations originating from the CGMD force field (Supplementary Fig. S5). The finding indicated that SAS-CLIP can capture physically reasonable structural series. Interestingly, χ2 values for each simulation snapshot in the series were distributed broadly in the range below 350 (Supplementary Fig. S6). Nevertheless, the entire structure series reproduced the experimental SAXS profile. These structural sets could not be obtained by simply collecting individual structures with small χ2.

Figure 2
figure 2

Extracting time series satisfying χ2 < 3.0 from CGMD simulation trajectories by the SAS-CLIP. (a) Trajectories of χ2 during CGMD simulations. The regions corresponding to the clipped time series #A-1, #A-2, and #A-3 are shown in red. The other regions are shown in gray. (b) Averaged SAXS profiles of the time series #A-1, #A-2 #A-3 (red lines) are compared with the experimental SAXS profile (black circle with error bars of s.d.). (c) The squared residuals of the averaged scattering intensities.

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Table 1 λWP, Lt, χ2, and KLSAS-CLIP of time series obtained by the SAS-CLIP.
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Criteria for a time series to be consistent with both CGMD simulations and a SAXS profile

First, the criteria for a clipped time series to be consistent with CGMD simulations were examined in more detail. When Lt is too small, the conformational distribution of ER-60 in the clipped time series can be quite different from that of its original trajectory. To visualize this, several time series satisfying χ2 < 3.0 were clipped from the same original trajectory as #A-2. The conformational distributions of the time series with Lt ≤ 200 ns were clearly sparse and hardly reproduced the distribution of their original 5000-ns trajectory (Fig. 3a, b). Therefore, these conformational distributions did not reproduce the distribution that naturally arises from the CGMD force field. We do not regard such a time series as reflecting a force field. Therefore, we should establish criteria to eliminate the time series. The difference in conformational distribution between a clipped time series and its original trajectory can be a measure of the relationship between the distribution and CGMD force field. Based on this idea, we defined KLSAS-CLIP, indicating the difference in (θa-b-b′, θb-b′-a′) probability distribution between a clipped time series and its original trajectory (detailed definition is described in “Methods” section). As expected, KLSAS-CLIPs were larger for the time series with smaller Lt (Fig. 3a). To examine the relationship between KLSAS-CLIP and Lt, time series with χ2 < 3.0 were extensively collected from the same original trajectories as #A-1, #A-2, and #A-3. In Fig. 3c, a monotonically decreasing curve is clearly observed. The slope of this curve decreased as Lt increased, and KLSAS-CLIP remained at ~ 0.5 for the region where Lt > 700 ns. Consequently, two conditions “KLSAS-CLIP is approximately 0.5 or less” and “Lt is larger than 700 ns” can be criteria for identifying a clipped time series that reflects the CGMD potential function well. Note that #A-1, #A-2, and #A-3 satisfy both the conditions (Table 1). The Kullback–Leibler divergence-based evaluation would be generally applied to other molecules or systems.

Figure 3
figure 3

Relationship between Lt and KLSAS-CLIP. (a) Domain conformations of several clipped time series with various Lt. These were clipped from the same CGMD trajectory as the #A-2. All satisfy χ2 < 3.0. The colors in the heatmaps show the appearance probability in the (θa-b-b′, θb-b′-a′) space. The pixel with the probability > 0.035 is presented in the same color as that with the probability of 0.035. (b) The distribution of the entire CGMD trajectory before clipping time series in (a). (c) The distribution of the KLSAS-CLIP for the total of 19,315 time series. They were clipped from the same CGMD trajectories as the #A-1, #A-2, and #A-3. Each satisfies χ2 < 3.0. The colors show the appearance probability of the KLSAS-CLIP for each Lt.

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Second, the criterion for reproducing the experimental SAXS profile, which was initially χ2 < 3.0, was reconsidered. Although the #A-1, #A-2, and #A-3 satisfy the criterion, the squared residuals \(\left( {\frac{{{\text{I}}_{{{\text{exp}}}} \left( {\text{Q}} \right) - {\text{I}}_{{{\text{sim}}}} \left( {\text{Q}} \right)}}{{{\upsigma }\left( {\text{Q}} \right)}}} \right)^{2}\) exceeded 20.0 in the region 0.1 Å−1 ≤ Q ≤ 0.2 Å−1 (Fig. 2c, and residuals are shown in Supplementary Fig. S7a). This Q corresponds to correlation length between 31.4 Å and 62.8 Å. Considering that each of the a, b, b′, and a′ domains is a globular structure with a diameter of 20–30 Å, the I(Q) of this Q contains information on domain conformation. That is, the SAS-CLIP could extract structural series with incorrect domain conformation even with small χ2. According to the observations, we improved the criteria: satisfying both “χ2 < 3.0” and “the \(\left( {\frac{{{\text{I}}_{{{\text{exp}}}} \left( {\text{Q}} \right) - {\text{I}}_{{{\text{sim}}}} \left( {\text{Q}} \right)}}{{{\upsigma }\left( {\text{Q}} \right)}}} \right)^{2}\) < 12.5 for Q < 0.25 Å−1”.

A condition for obtaining time series with large L t

To enumerate the possible structural ensembles of ER-60, it is important to efficiently collect ensembles with a large Lt. We focused on the relationship between λWP and Lt. When λWP was small, Lt of the clipped time series tended to be small (Supplementary Fig. S8). Therefore, it was preferable to perform simulations with λWP ≥ 1.03.

Enumerating possible ensemble of ER-60 with SAS-CLIP

The question whether SAS-CLIP can enumerate various ensembles of ER-60 was addressed. According to the results above, we additionally performed CGMD simulations with λWP between 1.03 and 1.06 and increased available trajectories. The SAS-CLIP was re-executed with the improved criteria of both “χ2 < 3.0” and “\(\left( {\frac{{{\text{I}}_{{{\text{exp}}}} \left( {\text{Q}} \right) - {\text{I}}_{{{\text{sim}}}} \left( {\text{Q}} \right)}}{{{\upsigma }\left( {\text{Q}} \right)}}} \right)^{2}\) < 12.5 for Q < 0.25 Å−1”. We obtained six time series with Lt > 700 ns. In particular, the Lt was greater than 1 μs for each of the six time series (Table 1), which were designated as #B-1, #B-2, #B-3, #B-4, #B-5, and #B-6, respectively. KLSAS-CLIP values were reasonably small (Table 1). Their calculated SAXS profile reproduced the experimental one quite well; The new criteria reduced not only the squared residuals (Fig. 4 and Supplementary Fig. S9, and the residuals are presented in Supplementary Fig. S7b) but also the χ2 values (Table 1). We noted that the extracted time series included many individual snapshots with large χ2 even with the stricter criteria (Supplementary Fig. S10).

Figure 4
figure 4

Time series of the SAS-CLIP with the improved criteria. The calculated SAXS profiles of three time series with long Lt are shown. (a) Averaged SAXS profiles of the time series #B-1, #B-2 #B-3 (red lines) are compared with the experimental ones (black circle with error bars of s.d.). (b) The squared residuals of the averaged scattering intensities are shown.

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Next, the domain structures obtained with the SAS-CLIP were examined. First, to overview the architecture of ER-60, the two angles θa-b-b′ and θb-b′-a′ were plotted (Fig. 5a). Among the six time series from #B-1 to #B-6, the θa-b-b′ was mainly distributed ranging from 70° to 110° and θb-b′-a′ ranged from 75° to 150°. The distributions were similar for the other time series that satisfied Lt > 400 ns (Supplementary Fig. S11 and Table S1). The distribution in Fig. 5a could be roughly classified into two groups. In the first group, the θa-b-b′ distributed at approximately 90° and several clusters were observed in the two-dimensional map (#B-1 and #B-2). In the second group, θa-b-b′ was distributed at < 90° (#B-3, #B-4, #B-5, #B-6). While the structural trends were common, the detailed distributions differed from each other. Additionally, we examined the dihedral angle φa-b-b′-a′. φa-b-b′-a′ distributed between − 15° and 75° (Supplementary Figs. S12, S13). We could not find a clear correlation between φa-b-b′-a′ and θa-b-b′, and between φa-b-b′-a′ and θb-b′-a′.

Figure 5
figure 5

Structure of ER-60 in the clipped time series. (a) Domain conformations of ER-60 in the time series #B-1, #B-2, #B-3, #B-4, #B-5, and #B-6 are shown. The color in heatmaps show the appearance probability in the two-dimensional space. (b) Structures of ER-60 correspond to the I–VIII. The a domain is shown in blue, the b domain in green, the b′ domain in yellowish green, and the a′ domain in red. The other parts are gray. The reactive cysteines are yellow.

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Subsequently, we examined whether the a and a′ domains prefer a particular orientation in their motion. Supplementary Fig. S14 displays distributions of φb′-b-a-CGHC(a) and φb-b′-a′-CGHC(a′) for the six time series. In terms of the domain orientations, one or two preferred ones were observed for each time series. The preferred orientations also differed between the time series. In other words, the SAS-CLIP enumerated the possible distributions of the orientations of the a- and a′-domains.

Finally, the tertiary structures of ER-60 in the clipped ensembles were overviewed. Several structures featuring θa-b-b′ and θb-b′-a′ with high appearance probability are shown in Fig. 5b. All structures are U-shaped. In structures III and IV, the flexible C-terminal region (Glu483-Leu505) bridged a and a′ domain. These were temporary bridging in the #B-2. Similar bridging was observed in #B-1, #B-3, #B-4, and #B-5, with lifetimes varying in the range 40–160 ns. The C-terminal loop might contribute to the functional motion of ER-60 via the domain bridging.

In summary, we confirmed that SAS-CLIP can enumerate a variety of ensembles of ER-60. The four-domain architecture could be roughly classified into two classes, whereas the detailed conformational distribution differed among the clipped ensembles. The orientations of the a- and a′-domains were more diverse among the clipped ensembles.

Analysis of isosbestic points in the set of scattering curves

Reflecting the structural diversity in each clipped ensemble, their calculated scattering profiles were also diverse (Fig. 6, Supplementary Fig. S15). We next discussed significance of isosbestic points appearing in the set of scattering curves. According to a previous study40, an isosbestic point suggests existence of a variable having a approximate linear relationship with the scattering intensity as follows;

$$ {\text{I}}\left( {{\text{Q}},{\text{X}}} \right)\sim {\text{I}}\left( {{\text{Q}},{\text{X}}_{0} } \right) + \left( {{\text{X}} - {\text{X}}_{0} } \right)\frac{{\partial {\text{I}}\left( {{\text{Q}},{\text{X}}} \right)}}{{\partial {\text{X}}}} $$

Here, X is the variable and X0 is a parameter. The difference in I(Q) between molecular structures corresponds to difference in the X. Therefore, the X is a conformational coordinate.

Figure 6
figure 6

Variety of calculated SAXS profiles in each clipped time series. (a) Calculated SAXS profiles of each ER-60 structure. Ten profiles are shown for each time series. Five of them are shown by the red, orange, green, blue, and black dotted lines, respectively. The other five profiles are shown by the solid red, orange, green, blue, and black lines, respectively. For each graph, two approximate isosbestic points are shown by arrows. (b) Distributions of I(Q)s over all structures in each time series. The distribution is calculated for each value of log10Q. Therefore, for each log10Q, sum of the probabilities of log10I(Q) is 1. The color shows the appearance probability of log10I(Q) values at each log10Q. (c) Distribution of p(r) for each value of r. Distributions of p(r)s over all structures in each time series are shown. For each r, sum of the appearance probabilities of p(r) is 1. The color shows the appearance probability of p(r) at each r. Here, data are shown for the #B-1, #B-2, and #B-5. Data for the other time series are shown in the Supplementary Figure S15.

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In other words, a single molecular motion along the X can explain diversity of calculated SAXS profiles in a structural ensemble when an isosbestic point is found in Q-I(Q) plot. We verified this linear relationship for the ensembles of ER-60 obtained with SAS-CLIP. In Fig. 6a, the calculated scattering curves of 10 structures in the #B-1, #B-2, and #B-5 are shown, respactively. The I(Q) varied with the structure, especially around Q = 0.07 Å−1 and Q = 0.2 Å−1. In addition, intersections of the curves were concentrated at two regions around Q = 0.12 Å−1 (shown by arrows in Fig. 6a). Distribution of I(Q) at each Q value was also calculated to get an overview of all the scattering curves (Fig. 6b). Again, sharp I(Q) distributions were observed around Q = 0.12 Å−1. These results indicated that there were two approximate isosbestic points around Q = 0.12 Å−1. Corresponding to the I(Q) distribution, p(r) showed sharp distribution between 50 and 60 Å (Fig. 6c). Similary, two isosbestic points were also observed in each of #B-3, #B-4, and #B-6 (Supplementary Fig. S15). The two isosbestic points were not Q values at which only intra-domain scattering appear (Supplementary Fig. S16). Based on the results, a single conformational coordinate was expected to explain the diversity of the SAXS profiles.

Indeed, the distance between centers of a and a′ domain (Da-a′) approximately linearly correlated with I(Q) (Fig. 7). Here, the relationship between I(Q) and the Da-a′ was shown for three points, Q = 0.07 Å−1, 0.12 Å−1, and 0.2 Å−1, respectively. Although the linearity depending on the time series (e.g. The relation was clear in #B-5, but that was relatively weak in #B-1.), the relationship was common to the six ensembles (Supplementary Figs. S17, S18, S19).

Figure 7
figure 7

Relationship between I(Q) and domain conformation. a–c Correlation between the distance Da-a′ and scattering intensity. (a) Appearance probabilities of pairs of values the Da-a′ and I(0.07 Å−1). The color shows the probability. (b) The probabilities of pairs of values the Da-a′ and I(0.12 Å−1). (c) The probabilities of pairs of values the Da-a′ and I(0.2 Å−1). Here, the heatmaps for the time series #B-1 and #B-5 are shown. (d) Two typical structures in the #B-5. The a domain in shown in blue, the b domain in green, the b′ domain in yellowish green, and the a′ domain in red. The other parts are in gray. In panels a, b, and c, the pixels to which the structure I and II belong are marked.

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In summary, our result supports that a single molecular motion can explain diversity of I(Q)s in a structural ensemble when an isosbestic point is found in Q-I(Q) plot. In ER-60, that was open-close motion of the a and a′ domains.

Interpretation and application of SAS-CLIP

SAS-CLIP does not provide a unique solution of the structural ensemble of a biomolecule. The structural ensemble obtained by this method differ from each other, but they all reproduce the same SAXS profile. This also means that any combination of these time series also reproduce the experimental SAXS profile. Additionally, the structural distribution of each clipped time series approximates that of a long-term MD simulation. Therefore, each time series obtained by SAS-CLIP can be an element of motion of a biomolecule, and any combination of these are candidates for the real structural ensemble.

Here, we propose a method to identify a structural ensemble of a biomolecule using clipped time series and another experimental data. With X as a quantity obtained in an experiment other than SAXS, X will be expressed as follows:

$$ {\mathbf{X}} = \mathop \sum \limits_{{{\text{i}} = 1}}^{{\text{N}}} {\text{c}}_{{\text{i}}} {\mathbf{X}}_{{{\text{SAS}} - {\text{CLIP}},{\text{ i}}}} $$

where N is the number of structural series obtained from SAS-CLIP. The XSAS-CLIP,i is the quantity for i-th structure series of the SAS-CLIP, which is averaged over all structures in the i-th series. ci is a weight factor. A reasonable structural ensemble that is consistent with any of SAXS, CGMD, and another experiment can be obtained by simply determining the cis. We can assume a variety of data are the quantity X. In fact, many experimental data represent the average quantity over all molecules in solution. In addition, many kinds of experimental data can be calculated from a given atomistic structures. For example, the profile of small-angle neutron scattering (SANS)41,42,43,44, chemical shift for nuclear magnetic resonance (NMR)45,46, and efficiency of fluorescence resonance energy transfer (FRET)47 can be calculated from tertiary structures. The proposed method has at least two advantages. First, the method is advantageous for constructing realistic ensembles of highly flexible biomolecules; Combination of clipped ensembles with SAS-CLIP results in an ensemble with a very large number of structures. Second, resultant ensembles are consistent with a force field of MD simulation regardless of ci values. The linear combination approach can also avoid the possible effect of artificial free energy minimum derived from an incorrect simulation force field, which is often a problem in the entropy maximization approach. Sufficient ensembles should be clipped to perform such an analysis.

SAS-CLIP can be applied to atomistic MD simulations. It is easier for the atomistic simulations to compare a simulation step with real time. When the SAS-CLIP is applied to atomistic simulations, extracted time series will be useful to analyze experimental data including temporal information, such as neutron spin echo.

When Lt is large, KLSAS-CLIP is sufficiently small (Table 1). If we regard an original CGMD trajectory as a subspace of the free energy landscape, each clipped time series reproduces “a subspace of the subspace”. In other words, the SAS-CLIP provides time series that roughly trace a subspace of a free energy landscape of a biomolecule. This contrasts with the entropy maximization approach, where an entire free energy landscape is reproduced as much as possible. We designed our method not to narrow down the behavior of a biomolecule based on two facts. First, an SAXS profile does not contain enough information to identify structural ensemble of a biomolecule with high resolution. Second, simulation force fields contain incorrectness. Instead of narrowing an ensemble down, SAS-CLIP is designed to be easily combined with other experimental data.

In practice, there are three major advantages of SAS-CLIP. First, time series can be extracted from relatively short MD simulations with low computational cost. We can obtain many time series at the same time when multiple simulations are performed in parallel. Second, obtained structural sets reflect a force field of MD simulations, and thus they are candidates of “element of motions”. Third, this method makes it easy to obtain a structural ensemble which matches SAXS, MD, and another experiment with simple linear model. This method will provide a new way to study biomolecules by integrating various type of experiments.