DENSS-multiple: A structure reconstruction method using contrast variation of small-angle neutron scattering based on the DENSS algorithm

Highlights

• •DENSS-Multiple provides a straightforward method to recreate structure density of biomolecules from neutron contrast variation dataset.
• •The different density from a multiphase complex can be better distinguished.
• •The method was implemented with open-source code that can be used and further developed by other researchers.

Abstract

The 3D structure of biomacromolecules, such as protein and DNA/RNA, provide keys to understanding their biological functions. Among many structural biology techniques, small-angle scattering techniques with ab initio methods have been widely used to reveal biomolecular structures in relevant solution conditions. Recently, a method called DENsity from Solution Scattering (DENSS) was developed to reconstruct the scattering density directly from biological small-angle X-ray and neutron scattering data instead of using a dummy atom modeling approach. Here, a method named DENSS-Multiple was developed to work simultaneously on multiple datasets from small-angle neutron scattering (SANS) contrast variation data. The easily manipulable neutron contrast has been widely exploited to study the structure and function of biological macromolecules and their complexes in solution. This new method provides a single structural result that includes all the information represented by different contrasts from SANS. The results from DENSS-Multiple generally have better resolution than those from DENSS, and more subtle features are represented by density variations from different phases of a structure. DENSS-Multiple was tested on various examples, including simulated and experimental data. These results, along with DENSS-Multiple's applications and limitations, are discussed herein.

Graphical abstract

Introduction

Small-angle scattering (SAS) with X-rays or neutrons has been used to understand the structure of a wide range of materials [1,2]. Compared with other structural biology techniques such as crystallography, cryogenic electron microscopy (cryoEM), and nuclear magnetic resonance, the SAS technique does not require biomolecules, such as protein and DNA/RNA, to be crystallized, frozen at cryogenic temperatures, or present at high concentrations. SAS is a valuable tool for providing insight on the structure–function relationship in biologically relevant conditions [3]. This becomes more intriguing as more protein structures become readily available through machine learning methods such as AlphaFold [4] and RoseTTaFold [5], yet the discrepancy between crystallized/frozen structures and solution structures needs further analysis to elucidate the structure-function relationship [6].

Dilute biomolecules in solution are usually isotropic in all orientations, so the resulting small-angle x-ray scattering (SAXS) or small-angle neutron scattering (SANS) data are typically a 1D curve of scattering intensity vs. momentum transfer q (i.e., q = 4πsinθ/λ, where θ is half of the scattering angle, and λ is the wavelength of the x-ray photon or neutron). The loss of information from a 3D structure to 1D intensity data presents a unique challenge in solving the molecular structure. In SAS data analysis, ab initio structure reconstruction methods are used to create a 3D structure from a 1D curve. Several programs, including DAMMIF, DAMMIN, and MONSA from the ATSAS suite [7] and DENsity from Solution Scattering (DENSS) [8] can perform ab initio reconstruction. DAMMIF and DAMMIN use beads, or dummy atoms, packed within a search volume to search for the shape that most closely resembles the scattering biomolecule. MONSA is similar to DAMMIN but can be used for reconstructing multiphase systems. DENSS, an ab initio iterative structure factor retrieval algorithm, can resolve the electron density within a biomolecule from SAXS data [9,10]. This method is also applicable for a single SANS dataset for neutron scattering length density reconstruction. Based on DENSS, the DENSS-Multiple method was developed to reconstruct a single, synthesized structure from the multiple datasets in a neutron contrast-variation (NCV) series that are typical in SANS experiments.

In SAXS, x-ray photons interact with electrons in molecules to reveal a structural representation in electron density. The electron density variation between the different atoms found in biomolecules as well as in solvent water molecules is usually small and not easy to manipulate. In comparison, neutrons scatter off nuclei of atoms in SANS and scattering cross-sections vary greatly especially in light elements such as C, H, O, P etc. Owing to compositional differences between different types of biomolecules (e.g., proteins, nucleic acids, and lipids), each type has a distinct scattering length density (SLD) in neutron scattering compared to x-ray scattering [11,12] (Fig. 1). As shown in Fig. 1, varying the D₂O:H₂O ratio (v/v) easily covers the SLD range of typical biomolecules. The intersecting point of a biomolecule on the water line in Fig. 1 indicates the same SLD as the water in that particular D₂O:H₂O ratio, viz. contrast match point (CMP), at which SANS cannot distinguish a molecule from its surrounding solvent. For example, the CMP of proteins is usually about 42% D₂O, while the CMP of DNA is around 65% D₂O. At other D₂O:H₂O ratios, the SLD difference between a biomolecule and its solvent, shown as Δρ, is referred to as its SANS contrast, which gives rise to the coherent SANS intensity containing structural information. The SLD can be further manipulated by hydrogen isotope replacement on hydrogen-rich biomolecules between protium (H) and deuterium (D), which have very distinct scattering lengths. In SANS studies, an NCV series takes advantage of the adjustable SLD difference between components by systematically tuning the solvent D₂O:H₂O ratio to better elucidate the biomolecular structure [11,13]. For example, each sub-component, or phase, in a molecular complex can be selectively masked by using a buffer solution at its CMP [11,13,14]. The other part of the complex can be studied selectively ‘in-situ’. In addition, an overall structure can be measured at D₂O:H₂O ratios that don't match any of the components. Together, with structures of each sub-component, one can piece together biomolecular complexes. Since its inception in 1970s [15], [16], [17], NCV experiments have been known to contain more information than a single SAS measurement, and this technique is widely used to understand the internal structure and organization of multi-component biological assemblies [18], [19], [20], [21].

Fig. 1.

Scattering length density of biological macromolecules at different D₂O ratios. The intersecting points on the water line indicate a CMP for a particular macromolecular class. The difference in SLD between a molecule and its water solvent, e.g., shown as Δρ or contrast varies greatly at different D₂O ratios to provide NCV measurements.

Ab initio methods using SANS data provide structural models in real space that can be further analyzed and compared with, e.g., models from crystallography, cryoEM, etc. Previously, dummy atom approaches such as the one implemented in MONSA were used for multiphase structure reconstruction [22]. The dummy atom approach inherently assumes a uniform density within a phase. The DENSS method reconstructs the density instead, and the density variation within a structure can distinguish different phases well. Therefore, DENSS-Multiple was developed to allow for reconstruction of a single structure using multiple NCV SANS datasets simultaneously. DENSS-Multiple provides a straightforward tool to use NCV datasets, requiring less a priori information such as the volume fraction of different phases, which can be difficult to determine when studying a novel system. With parallel iterative reconstructions from multiple datasets, DENSS-Multiple can improve the overall structure and internal SLD, thereby taking advantage of the distinct shape and density information contained in each SANS dataset.

DENSS-Multiple has been tested on several single- and multiphase biological systems, using simulated and experimental data to assess its efficacy and performance. This manuscript presents the method, the test results, and discussions on how to apply it more effectively with NCV datasets.

Results and discussion

Overall description of iterative phase retrieval with contrast variation data

Briefly we describe the DENSS algorithm here before discussing DENSS-Multiple. DENSS uses an iterative process to retrieve phase information from the scattering data by first creating a random 3D density distribution of voxels. The structure factor (Note: here we use the term as typically defined in crystallography, following the convention in [8] for consistency - not the inter-particle structure factor typically referred to in SAS), including phase and amplitude, is obtained by a Fourier transform of the 3D model density which represents the contrast (either in electron density or neutron SLD) between a molecule and its surrounding solvent background. Iteratively, the structure factor is scaled to the experimental 1D scattering curve and then converted to a new 3D density in real space via an inverse Fourier transform. Several real-space constraints are imposed on the new density at every iteration, such as flattening small values to zero for noise reduction and restraining the outer bounds of the shape via the shrinkwrap algorithm [23], etc. The iterative process terminates when the standard deviation of the χ² values dips below a set cutoff, indicating the scattering from the reconstructed density has reached a local minimum in its approximation of the experimental scattering.

For NCV data with more than one curve, the information from all datasets with different contrasts must be combined properly to generate a single synthetic structure. Assuming the SLD of biomolecule is ρ(r), where vector r represents the spatial distribution of the 3D density which is modeled in voxel arrays. In a NCV experiment, the contrasts from each solvent background with SLD ${\rho }_1,{\rho }_2,\dots {\rho }_N$(N is number of contrasts used in the experiment) are

$$\Delta {\rho }_1\left(\mathit{r}\right)=\rho \left(\mathit{r}\right)-{\rho }_1$$

$$\Delta {\rho }_2\left(\mathit{r}\right)=\rho \left(\mathit{r}\right)-{\rho }_2$$

$$\stackrel{⃛}{\Delta }{\rho }_N\left(\mathit{r}\right)=\rho \left(\mathit{r}\right)-{\rho }_N$$ (1)

The contrasts $\Delta {\rho }_1\left(\mathit{r}\right),\Delta {\rho }_2\left(\mathit{r}\right),\dots \Delta {\rho }_N\left(\mathit{r}\right)$ produce complex structure factors with phases and amplitudes by 3D Fourier transform from the real space to the reciprocal space. Further structure factors (SF) are measured as coherent intensity I₁(q), I₂(q) … I_N(q) but the phase information is lost and 3D orientational information is reduced to 1D in the measurement:

$$F\left(\Delta {\rho }_N\left(\mathit{r}\right)\right)\to S{F}_N$$ (2)

$$<{\left|S{F}_N\right|}^2>\to I_N\left(q\right)$$ (3)

where F() represents Fourier transform and the angle brackets represent averaging over all orientations. Experimentally we cannot solve structure factors from measured I_exp(q), therefore model densities at different contrasts are used to generate model structure factors using Eq. (2). Concurrently, for each contrast data set the experimental intensities I_exp(q) are used to scale model structure factors with scale factors:

$$\sigma \left(q\right)=\sqrt{\frac{I_{exp}\left(q\right)}{I_m\left(q\right)}}$$ (4)

where I_m(q) is calculated from model structure factors by spherical averaging and binning along q. The iterative process, as described in DENSS, is used to recover $\Delta {\rho }_N\left(\mathit{r}\right)$ for each contrast by inverse Fourier transform with the rescaled structure factors. Then the common SLD ${\rho }_1^{\prime }\left(\mathit{r}\right)$, ${\rho }_2^{\prime }\left(\mathit{r}\right)$,… ${\rho }_N^{\prime }\left(\mathit{r}\right)$ can be obtained by correction of solvent backgrounds within the particle envelop:

$${\rho }_1^{\prime }\left(\mathit{r}\right)=\Delta {\rho }_1\left(\mathit{r}\right)+{\rho }_1$$

$${\rho }_2^{\prime }\left(\mathit{r}\right)=\Delta {\rho }_2\left(\mathit{r}\right)+{\rho }_2$$

$$\stackrel{⃛}{{\rho }_N^{\prime }}\left(\mathit{r}\right)=\Delta {\rho }_N\left(\mathit{r}\right)+{\rho }_N$$ (5)

Note we number the common SLD ${\rho }_1^{\prime }\left(\mathit{r}\right)$, ${\rho }_2^{\prime }\left(\mathit{r}\right)$,…${\rho }_N^{\prime }\left(\mathit{r}\right)$ according to different contrasts used. Ideally, they are the same and equal to the biomolecular ρ(r), but experimentally and numerically they will come out differently as each contrast contains some unique information from the actual data. In DENSS-Multiple, the aim is to combine all information from a NCV data set to generate one single synthetic structure instead of individual ones. The most obvious option was a simple mathematical average across all contrasts in the 3D array. Averaging combines the unique values generated from each contrast without biasing the outcome toward a specific contrast. We also provide weighting factors w_N that can be used to bias the result towards one or a few particular datasets:

$$\rho \left(\mathit{r}\right)=\frac{1}{N}\sum _N{w}_N{\rho }_N^{\prime }\left(\mathit{r}\right)$$ (6)

In our implementation and tests, we treat each contrast equally with a weighting factor $w\equiv 1$. With this step the contrast variations and their spatial correlations are incorporated in individual contrasts, the iterative process, and the final structure.

It is important to mention that we assume the biomolecular SLD is not changed at solutions with different D₂O:H₂O ratios. However, the H/D exchange on dissociating hydrogens is a known effect [16]. Also the hydration shells surrounding biomolecules create slightly different contrasts at different D₂O:H₂O ratios [24]. Those are some of the reasons ${\rho }_N^{\prime }\left(\mathit{r}\right)$ varies at different contrasts and will contribute to lowering the accuracy of the final reconstructed density, but such detail is usually beyond SAS resolution limit and is ignored in our tests.

A schematic of the algorithm as we implemented in DENSS-Multiple code is presented in Fig. 2. It shows the iterative process that manipulates multiple real-space 3D SLDs and structure factors in reciprocal space to properly reconstruct ab initio models. The required inputs for DENSS-Multiple are NCV multiple datasets and the solvent background SLDs for each contrast. The DENSS-Multiple iterative process begins with a common randomly generated starting model density in a 3D array. The modeled 3D density is duplicated with the number of contrasts. For each contrast, the duplicated 3D density is then corrected for the solvent background SLD of each contrast by subtracting the SLD value uniformly from the starting density. This step creates different contrasts, Δρ = ρ_sample – ρ_solvent in vacuo as in Eq. (1), for generating the structure factors in the reciprocal space. Data from all contrasts go through the steps of the phase retrieval process in parallel (Fig. 2), including fast Fourier transform (FFT) and structure factor scaling. Once the structure factors are rescaled to each experimental curve, they are transformed back into real-space model densities by inverse FFT (IFFT). The user can apply the real-space constraints to each contrast independently. The resultant updated model density from each contrast is then rectified with the addition of the solvent background SLDs within the support boundary, which means the reconstructed SLD from different contrasts effectively remains in vacuo without surrounding solvent backgrounds and can be combined into a single overall model density by averaging all contrasts, for the next iteration, as in Eq. (6).

Fig. 2.

Flow chart of the DENSS-Multiple method implemented. The cubes represent 3D arrays and square represent 1D array in scattering intensity. See main text for detailed descriptions.

Averaging densities from each contrast

In the DENSS-Multiple algorithm, one of the most important considerations was how to combine the resultant model densities from each contrast in the iterations in an efficient and logical manner. In addition to the averaging process described previously, optimal timing of the averaging within the iterative process was also considered. It was determined that the densities of each contrast should be averaged after the real space restraints (via the support) are applied to the densities. The support discriminates between what is and what is not part of the reconstruction and is optimized using the shrinkwrap algorithm [25]. This was done to maximize the heterogeneities between different contrasts. Therefore, structure factor averaging in reciprocal space was ruled out because doing so potentially cancels out some structure factors, thereby negatively affecting density reconstruction.

Another important factor to consider with averaging the densities of each contrast together is the frequency of averaging. Averaging all structures with each iteration would inherently affect the next iteration for all contrasts, thereby diminishing possible unique features or information that each contrast could provide. However, the orientations and locations of the densities from different contrasts and iterations drift significantly in 3D real space during consecutive iterations of the algorithm. Our solution is to average the densities after each step, effectively enforcing the alignment of densities generated in different contrasts and iterations. Otherwise, waiting multiple steps to average contrasts requires a computationally expensive alignment process to avoid completely losing the relative alignment across different contrasts.

The support used for the shrinkwrap algorithm, employed as a real-space constraint in the program, is kept as a separate array for each contrast. After the iterative phase retrieval algorithm converges on a final structure, the final support is created by using an OR operator to combine all given supports, thereby possibly overestimating the support to capture the full extent of the final structure instead of possibly underestimating it. This choice differs from the original DENSS program because our final support is a combined support from each contrast instead of a support determined from a single dataset.

Termination of the DENSS-Multiple algorithm depends on the convergence of χ² values from each contrast. When the average variance of the previous 100 χ² values for each contrast reaches a point below the break condition, the algorithm has converged, and the resulting model is the reconstruction for all the given NCV data. This approach ensures a stable fit to all datasets instead of one preferential dataset.

Single-phase tests

DENSS-Multiple's ability to resolve single-phase artificial geometric shapes was tested. These scattering data from these geometric shapes were constructed using analytical functions in SasView 5.0 (sasview.org) (e.g., sphere, hollow sphere, cylinder disc, cube) to generate simulated scattering curves (Figs. S1 and S2 in Supplementary Information (SI)). Usually, a single-phase structure is well represented by one SAS curve from a single contrast; however, contrast variation provides additional data and increases the reliability of the result. The DENSS-Multiple and DAMMIF (using only 100% D₂O contrast) results are presented in Fig. S1, and more details with DENSS using only 100% D₂O contrast data are presented in Fig. S2. Experimental datasets from human serum albumin (HSA) protein measured from different D₂O ratios were also tested (Figs. S3 and S4). The results improved slightly from a single dataset, especially on structures with more complexity, such as a hollow sphere (Fig. S2(B)). Visually, DENSS-Multiple results have improved integrity and more accurate density variation, especially regarding structures with an enclosed cavity. This improvement indicates that using NCV data increases information content and provides more constraints for a better result. Overall, DENSS-Multiple can reconstruct shapes at least as well as, and often better than, single dataset reconstructions by other algorithms.

Multiphase tests

Multiphase systems—a focus of the new method—were extensively tested. The geometric models used were created with regions of vastly different densities resembling real complex systems (e.g., lipid nanodiscs, protein–DNA complex). The multiphase tests on geometric models in the solvent D₂O ratios of 0%, 65%, and 100% D₂O are presented in Fig. 3, and additional detailed views are shown in Fig. S5. DENSS-Multiple yielded well-defined overall envelopes matching the overall dimension of the geometric models for the two-layer onion (Fig. S5(A)), the nanodisc-like cylinder (Fig. S5(C)) and the two-layer long cylinder (Fig. S5(D)), except for the three-layer onion model (Fig. S5(B)). Within the shapes, the density distributions show significant variation that relates to different phases of the models. The density values in an arbitrary scale do not represent the absolute SLD assigned, as discussed in the next section. The dimensions of different phases are similar to the sizes in the models. The boundary between different phases in the density results is more diffuse than in the models because of the resolution limit of SAS, which is usually higher than about 20 Å. These results show the effectiveness of DENSS-Multiple on NCV data.

Fig. 3.

Multiphase geometric testing of DENSS-Multiple. The DENSS-Multiple results (i and ii) and the original model structures are shown. The diameter of all simulated objects is 150 Å. All samples were analyzed and visualized using PyMOL. The first shape tested was a sphere with two density layers. The sphere had an overall radius of 75 Å; the innermost core sphere had a neutron SLD equivalent to 65% D₂O and a radius of 25 Å, and the outermost density layer had a density equivalent to 42% D₂O and a layer thickness of 50 Å. Other models are in the same scale. The data were generated with solvent background SLD values equivalent to 0%, 65%, and 100% D₂O concentrations. The percentage label on the model true structure is the SLD equivalent to D₂O concentrations.

To compare DENSS-Multiple to the current state of the art in multiphase reconstruction, MONSA [7], we used MONSA through the ATSAS Online server with the same simulated data in Fig. 3. We found that MONSA was largely unable to recapitulate the phase morphologies from the same simulated NCV datasets. Of the simulated data, MONSA was only able to reconstruct the two-layer onion model (Fig. S6) when using the same dataset as in Fig. 3. In response to this, we increased the contrasts in the simulated datasets to include a range of contrasts in which no contrast matching occurs (0, 10, 20, 80, 90, and 100% D₂O). MONSA was then able to reconstruct sensible shapes for the simulated models except the three-layer onion model which is a challenge for both DENSS-Multiple and MONSA, but the quality of the phase reconstructions is noticeably less than that of DENSS-Multiple, in both correct phase morphology and location (Fig. S7). From these results, we show DENSS-Multiple utilizes NCV data to obtain a defined structure which includes its different phases in a manner that is often more effective than the current state-of-the-art.

The result with the three-layer onion model (the second row in Figs. 3 and S5(B)) showed that DENSS-Multiple had some difficulty using all the NCV data to generate this structure and highlighted an important point to consider when using NCV data. In a typical SANS NCV experiment, some contrasts are often designed to completely cancel out a certain phase (contrast matching). A whole structure complex can be solved piece by piece, and then the pieces can be combined using methods, such as rigid body modeling or molecular docking, that can be constrained by an SAS dataset with the overall structure. The contrast matching point data remove a portion of the total structure, thereby making it a more difficult test case for DENSS-Multiple. For example, in the case of the three-layer onion model with the component CMP equivalent to 42%, 65%, and 100% D₂O, from outside to inside. The NCV data was collected in solvent backgrounds of 0%, 65%, and 100% D₂O, because the NCV coincides with a significant number of the phases (two out of three), the result is not good. This result occurs because less structural similarity occurs between the individual reconstructions of different phases and averaging of these vastly different phases distorts the reconstruction. However, if the contrast matching occurs in only one or none of the phases, like in other cases in Fig. 3, then DENSS-Multiple is still quite effective. Including specific contrasts must be considered carefully when using DENSS-Multiple. A test using probing contrast inclusion/exclusion (especially those at the CMP for various phases) can be found in the SI with Fig. S8.

The next test used an experimental dataset of a lipid nanodisc made with membrane scaffold protein (MSP) cNW9 which has a CMP at ∼42% (Fig. 4) [26]. The average lipid SLD was approximately 60% D₂O (a mixture of chain-deuterated d54-DMPC(1,2-dimyristoyl-sn-glycero-3-phosphocholine)/DMPG(1,2-dimyristoyl-sn-glycero-3-phospho-(1′-rac-glycerol))=1.65:1 in molar ratio). The experimental SANS NCV series consisted of four contrasts at 0%, 42%, 80%, and 100% D₂O (Fig. 4(A)). The DENSS-Multiple reconstruction of this lipid nanodisc had a structure with noticeably less noise than the single DENSS reconstructions, as shown in Fig. 4(C). Additionally, the resolution for the DENSS-Multiple reconstruction (based on the value at which the Fourier shell correlation falls below 0.5) is better (22.3 Å) compared with the single DENSS reconstructions (∼30–40 Å). This value indicates that the DENSS-Multiple reconstructions effectively and consistently combine information from different contrasts to highlight different parts of the overall structure and subsequently improve the result. Additional scattering curves from various contrasts provide more constraints on reconstruction and potentially cancel out noise from different measurements. Additional tests with simulated protein multiphase data are also presented in Fig. S9.

Fig. 4.

The cNW9 lipid nanodisc SANS NCV series and DENSS-Multiple reconstruction. (A) Experimental SANS NCV scattering curves (B) P(r) curves from scattering data in (A), generated using GNOM. (C) Normalized and averaged density maps generated by DENSS and DENSS-Multiple with calculated Fourier shell correlation resolution shown in parentheses. The DENSS-Multiple reconstruction was created using the 0%, 42%, 80%, and 100% SANS data combined. The density is overlaid with a model of lipid nanodisc with the MSP in ribbon form for reference only. The density is in arbitrary units (a.u.).

Density value of different phases

DENSS-Multiple was further tested for its ability to resolve the absolute value of neutron SLD in a multiphase system. Using buffer SLD values and SANS scattering profiles in absolute scale of the scattering cross section (cm⁻¹) for each contrast as input, the final result can theoretically converge to absolute density. This approach was tested using the aforementioned three-layer onion sphere model. NCV data was simulated for 0%, 10%, and 20% D₂O contrasts and was reconstructed using DENSS-Multiple. The three contrasts were chosen because they represent a significant contrast compared to the SLD in the structure, and they do not contrast match any of the phases to avoid the problem we discussed previously. The three phases are easily distinguished, as shown in Fig. 5. On the density histogram (Fig. 5(A)), the phases are represented by the value ranges of approximately 0.16–0.6 for 42% D₂O phase; 0.6–1.1 for 65% D₂O phase; and 1.1–1.3 (max) for 100% D₂O phase. The result and the density distribution histogram allow the different phases to be easily determined (Fig. 5(B) and (C)). However, the density value is not a realistic SLD value and must be rescaled manually, likely due to the scaling factors applied at each iteration vary greatly among different contrasts. Therefore, the final combined density is off the absolute scale. Furthermore, the frequency of density values near zero was quite high for all tests (e.g., near zero in Fig. 5(A)); this is likely caused by noise and the Gaussian smoothing applied by the DENSS algorithm in the final step of reconstruction. For each phase, a distribution of density values is apparent instead of a discrete distribution. It is partly caused by the resolution limitations of SAS and by the aforementioned Gaussian smoothing employed by DENSS and DENSS-Multiple.

Fig. 5.

: Three-layer onion sphere model reconstruction showing different phases via absolute density output from DENSS-Multiple. (A) Density distrobution histogram showing the arbitrary density value output from DENSS-Multiple and the number of occurances of that bin for each voxel of the reconstruction. Phase ranges were determined from knowledge of the three-layer onion sphere's phase diameters. (B) Front and (C) orthogonal view of the three-layer onion sphere reconstruction (above) and slices to the center (below); phases are colored according to the ranges in (A).

Once again, this example shows that choosing the contrasts and parameters carefully with DENSS-Multiple can improve the reconstruction results. The three-layer onion model had a dissatisfactory reconstruction when using the contrasts and parameters in Fig. 3. However, when contrasts were used which avoided the CMP of the phases, DENSS-Multiple could accurately reconstruct the well-defined density shown in Fig. 5. We also found increasing the ‘NE’ value, which scales the density values of the reconstruction at various points, can help with reconstruction accuracy for our simulated systems. Other parameters, such as the shrinkwrap density threshold, change the support which determines which voxels’ density is included in subsequent steps and can help with reconstruction quality in some cases. We would like to encourage users to carefully consider both the contrasts and parameters used with DENSS-Multiple. While not required, doing so does increase the accuracy of reconstruction.

Experimental datasets collected from two different lipid nanodisc systems were used to probe the similarity between the density values given by DENSS-Multiple to the actual NSLD values in experimental biomolecules. The first lipid nanodisc, as presented in the last section, had a protiated circularized MSP protein (CMP ∼42% D₂O) with a mixture of chain-deuterated d54-DMPC/DMPG=1.65:1 in molar ratio (CMP ∼60% D₂O). It is called the ‘protiated’ nanodisc in the discussion. The other nanodisc, as called the ‘deuterated’ nanodisc, contained partially deuterated MSP protein (CMP ∼80% D₂O) with chain per-deuterated d54-DMPC phospholipids with its lipid chain CMP above 100% D₂O and its headgroup CMP ∼ 35% D₂O. The deuterated nanodisc was measured in 0%, 20%, 40%, 60% and 100% D₂O. The natural variation of SLD between different components inside the nanodiscs make them good tests for the effectiveness of resolving the SLD from multiple phases by DENSS-Multiple.

The DENSS-Multiple results for both cNW9 nanodiscs are shown in Fig. 6(A) and (B) respectively. DENSS-Multiple successfully reconstructed discoidal structures in both cases that closely resembles the dimensions of a generated cNW9 nanodisc model. However, in contrast to the previous test, the reconstruction of perfect phase boundaries was not achieved and the NSLD density values was different from the absolute NSLD in the nanodiscs. Close examination of both structures and their density distribution histogram reveals some interesting aspects. Given their lipid compositions, the maximum SLD of the deuterated nanodisc should be much higher than the lipid part of the protiated nanodisc, 6.83 × 10⁻⁶ Å⁻² vs. 3.59 × 10⁻⁶ Å⁻². The maximum histogram density value in the deuterated nanodisc indeed is higher but to a lesser degree (∼0.12 a.u./ų vs. ∼0.9 a.u./ų). Again, the frequency of density values near zero was quite high for both. Overall, the average density for the volume is 0.03753 a.u./ų and 0.02894 a.u./ų for the deuterated and protiated nanodisc, respectively, indicating that DENSS-Multiple can resolve and scale different phases qualitatively.

Fig. 6.

(A) Deuterated and (B) protiated cNW9 nanodisc with corresponding density distribution histogram of density shown in reconstruction. Density histograms were created using Matplotlib [7,8]. (i) Diagonal view showing the full extent of the nanodisc and reconstructed density. (ii) Horizontal slice along the mid-plane of the nanodisc to show internal density. (iii) Vertical slice along the midplane of the nanodisc to show internal density. Color scale is in arbitrary units, but the density here is not normalized between two structures, and native density values from reconstructions are shown. The density is overlaid with a model of lipid nanodisc with the MSP in ribbon form for reference only.

In the deuterated nanodisc, higher density in the central lipid region of the nanodisc is well presented as the deuterated lipid chains have much higher SLD than the surrounding headgroup and MSP protein. In the protiated nanodisc, similar high SLD in lipid than the belt MSP protein can also be resolved. Interestingly, a low-density spot is seen within the overall higher density in the lipid region suggesting the inhomogeneity of d54-DMPC/DMPG lipid mixture. This inhomogeneity, however, should be taken with caution. In scattering experiments, a higher q region usually carries higher resolution structural information which can resolve internal features of a particle. But in an experiment, such as the data shown in Fig. 4 (A) for example, the high q data is noisier than low q data, which decreases the confidence in the reconstruction of smaller features. Therefore, due to the SAS resolution limitations and data noise, we are unsure if the spot is an artifact of the reconstruction, or it actually reflects the inhomogeneity among DMPC and DMPG that have been found in biomembranes [27,28]. But from the resolution shown in Fig. 4(C), using NCV datasets potentially allow for resolving additional features that are difficult for a single dataset.

Conclusion

In summary, the significant contrast manipulation in SANS provides a great opportunity for density reconstruction with improved results. Recently, additional contrast variation schemes were exploited in SAXS [29]. DENSS-Multiple provides a practical use for both single- and multiphase contrast variation data that are applicable to a wide range of biological single-particle systems.

Methods

All experimental SANS data were collected at room temperature on either the Bio-SANS beamline at the High Flux Isotope Reactor, or the EQ-SANS beamline at the Spallation Neutron Source at Oak Ridge National Laboratory [30]. The data were reduced using facility software with background correction and normalization. All SAXS data were collected by an in-house Rigaku Bio-SAXS 2000 instrument.

The simulated SAXS or SANS data from geometric models that can be described by analytical functions were generated in SasView 5.0 (sasview.org). This program offers a variety of different preset models to be used, where size and neutron scattering length density can be manually set. All simulated protein SAXS and SANS scattering data were generated using CRYSOL and CRYSON [24], respectively. Simulated data up to q_max = 0.5 Å⁻¹, respectively, were interpolated to mimic typical SAS data from the in-house Bio-SAXS and SANS instruments. The statistical error of the simulated intensity was generated as $\pm \sqrt{I\left(q\right)}$, similar to the error generated from detector counting at a pixel or q-binning in experimental data. The errors were used in calculating χ². However, no additional incoherent scattering background was added to the simulated curves. The number of spherical harmonics used to simulate data using CRYSON and CRYSOL depended on the radius of gyration of the protein being simulated. All Protein Data Bank models used to simulate the scattering data were obtained from the Research Collaboratory for Structural Bioinformatics Protein Data Bank (www.RCSB.org). GNOM (ATSAS, European Molecular Biology Laboratory) was used to generate pair–distance distributions via BioXTAS RAW, the output of which was used to run DENSS and DENSS-Multiple [22,31].

CRediT authorship contribution statement

S.Q. conceptualized the study. J.S. implemented the computer code. Both authors performed the testing and validation of the method. Both authors wrote and reviewed the manuscript.

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The code is available through https://doi.org/10.11578/csmb/dc.20220328.2 under GNU General Public License v3.0.