Abstract
Knowledge of the structure of biological macromolecules, especially in their native environment, is crucial because of the close structure–function relationship. Xray smallangle scattering is used to determine the shape of particles in solution, but the achievable resolution is limited owing to averaging over particle orientations. In 1977, Kam proposed to obtain additional structural information from the crosscorrelation of the scattering intensities. Here we develop the method in two dimensions, and give a procedure by which the singleparticle diffraction pattern is extracted in a modelindependent way from the correlations. We demonstrate its application to a large set of synchrotron Xray scattering images on ensembles of identical, randomly oriented particles of 350 or 200 nm in size. The obtained 15 nm resolution in the reconstructed shape is independent of the number of scatterers. The results are discussed in view of proposed ‘snapshot’ scattering by molecules in the liquid phase at Xray freeelectron lasers.
Introduction
The majority of macromolecular structures currently available at atomic resolution has resulted from Xray cryocrystallography at synchrotrons, which requires crystals larger than a few micrometres to avoid excessive radiation damage^{1}. Threedimensional (3D) molecular shapes may also be extracted from solution smallangle Xray scattering (SAXS)^{2,3}, whereby the few nanometers limit in the resolution^{4,5} is owing to the fact that only the isotropic component of the singleparticle scattering intensity is accessible.
The advent of Xray freeelectron lasers (XFELs)^{6} is stimulating the emergence of new imaging methodologies^{7,8}. Among these is serial crystallography on easiertogrow submicrometer crystals, in the ‘diffractanddestroy’ mode, which was recently demonstrated to provide nearatomic resolution^{9}. However, accessing structural information on noncrystalline and nonoriented samples beyond the capabilities of SAXS remains challenging. Snapshot diffraction from single particles has been considered by several groups^{10,11,12}, motivated by the highintensity of the fsduration XFEL pulses. But despite the enormous increase in peak brilliance, a typical XFEL singleshot diffraction image of an isolated biomolecule is predicted to yield much less than one photon per coherent pixel at the relevant resolution, requiring sophisticated statistical and computational methods for establishing the model of the 3D diffraction intensity, which is in best agreement with the noisy experimental data^{10,11}.
A remarkable alternative, which exploits the same features of XFEL radiation, is the method proposed already in 1977 by Kam^{13}, who suggested evaluating the crosscorrelations (CCs) of the scattered intensity from identical particles in solution. These correlations are sensitive to angular fluctuations, which are averaged out in the traditional SAXS analysis. The disadvantage of the CC method, that the singleparticle diffraction pattern must be derived indirectly, is largely compensated by the fact that multiparticle scattering is implicitly allowed, that experimental complications such as background scattering and nonuniform beam profile are easily accounted for, and, most importantly, that dealing with the large number of scattering images necessary to counteract the low photon count rates is absolutely straightforward.
For our demonstration experiment at a synchrotron source, the reduced Xray beam fluence was made up for by using strongly scattering nanostructures, and by immobilizing them to permit arbitrarily long acquisition. Liquidstate behaviour was mimicked by illuminating different random configurations of particles. Reducing the number of effective dimensions in the problem from three to two simplified both sample preparation and CC data analysis. Such an experiment was recently proposed^{14,15} and realized with nanocylinders^{16}. However, the reconstructed structure is obscured by a prominent ballshaped feature, and the claimed resolution is of the order of the small cylinder dimension, both of which limit the image quality. We report here 2D structural determinations of gold nanoparticles exhibiting threefold symmetry, using a rigorous CCbased protocol. Several innovations allow for the first time the full and modelindependent calculation of the singleparticle Xray diffraction patterns, and the resulting reconstructions fit extremely well to scanning electron microscopy (SEM) images of the particles. We also confirm experimentally the key prediction, that the CC signaltonoise ratio is essentially independent of the average number of scatterers^{17}.
Results
CC method in 2D
The workflow of the CC method is illustrated schematically in Fig. 1. The input consists of a large set of scattering images with intensities I acquired on a 2D liquid, meaning that each exposure is taken on a random ensemble of multiple identical particles. The crucial step is the indirect determination of the diffraction pattern of a single particle in reciprocal space
from which the 2D electron density of the particle is reconstructed, via phase retrieval, as in standard coherent diffractive imaging^{18,19}. A polar coordinate system (q,φ) centred at the reciprocal space origin is the natural choice for the CC analysis. are azimuthal Fourier components, which vanish for odd n due to Friedel's law, and for which , because S is a real quantity.
We define the SAXS or powder pattern as the 1point CC,
which depends only on the absolute value of the momentum transfer q. With ‹...› we denote an average over the images, and with the rotational average over the azimuthal angle (Fig. 2a). The 1point CC is related to the singleparticle scattering intensity by
where N is the effective average number of particles in the Xray beam, and, at this point, is simply a scaling factor.
Additional structural information can be extracted from higherorder correlations. The 2point CC results from the average of the product of two scattering intensities. It depends only on the absolute values of the two momentum transfers and the angle between them (Fig. 2b). The definition is^{14,15,16}
where δI(q,φ)=I(q,φ)−C^{(1)}(q) is the difference of the measured intensity from its average value. The Fourier components are related to the singleparticle scattering intensity by
κ_{(2)}≤1 is a factor that depends only on the Xray beam shape (see Methods). Equation (5) forms the basis of the CC method. Provided that the experimental data on the lefthand side are undisturbed, one can find solutions s_{n}(q) that are unique to within the scaling factor and phasors exp(−iχ_{n}) (see Methods). In contrast to previous works^{15,20}, to resolve these ambiguities we use the 3point CC, which is defined in an analogous way as the 2point CC and represents a generalization of previous suggestions^{21,22},
and whose Fourier components yield
κ_{(3)} is an analogue of κ_{(2)}. These equations are used to fix the values of N and χ_{n}. Thus the remaining ambiguity in the Fourier coefficients s_{0}(q) and {s_{n}(q)}_{n≠0} is eliminated, and the singleparticle diffraction pattern S(q,φ) can be fully determined. Moreover, the number of independent equations (7) is much larger than the number of unknowns, which allows validation of the computed set of Fourier coefficients.
Experimental 2D structure determinations
The samples for the 2D CC demonstration experiment consisted of a large number of nominally identical 2D gold nanostructures, anchored to a membrane, with random positions and orientations (Fig. 1d). The membrane was scanned through a focused Xray beam of 2 Å wavelength, and the scattering patterns were recorded with a lownoise pixel detector^{23} positioned 7.2 m from the sample. We measured samples with threefold symmetric particles, with either 350 (Fig. 1e) or 200 nm size, and three different densities of 1.25, 10 or 40 particles per 100 μm^{2} area. For each sample, 3,751 scattering images were acquired at different positions on the membrane, simulating instantaneous exposures of a 2D liquid. Such a set of images was analysed following the CC protocol, and an example of the final 2D nanoparticle structure is shown at the end of the workflow diagram in Fig. 1c, for 350 nm nanostructures and reconstructed from the data set taken on the sample with an average density of 10 particles per 100 μm^{2}. Panel (e) of the figure demonstrates the excellent agreement of the recovered 2D shape with the shape from an SEM micrograph of an individual nanostructure.
In the following, we go through the steps described in the previous section that lead to the final reconstruction, presenting the intermediate results in more detail. Figure 2 shows three examples of scattering images. The inset in (a) shows the 1point CC after subtraction of the background signal taken from exposures of an empty membrane. In all three images, subtle azimuthal intensity fluctuations are visible. They are filtered out by the C^{(1)}azimuthal averaging, but they provide precisely the additional information that contributes to the higherorder correlations. Figure 3 is devoted to the 2point CCs. Panels (c) and (d) show examples of C^{(2)}(q_{1},q_{2},ψ), while panels (a) and (b) display the coefficients and , respectively, where the 2D representation as a function of q_{1} and q_{2} serves to emphasize the importance of the nondiagonal (q_{1}≠q_{2}) CCs in our protocol. The encoded information was then used, in combination with that from the 3point CC, to extract the Fourier components {s_{n}(q)}_{n≠0} of the singleparticle diffraction intensity. In the analysed range of transferred momentum (0.009≤q≤0.24 nm^{−1}), only the coefficients with n=6, 12, 18, 24, 30, displayed in Fig. 4a, could be calculated consistently and validated. Figure 4b displays ratios of 3point CC components to the singleparticle coefficients s_{n}, as points in the complex plane (see equation (13) in the Methods for a precise formulation) and validity is inferred by the fact that almost all such points fall at the same realpositive value, which is proportional to . This determines the effective average number of scattering particles to be N≈20. Other coefficients were not considered further, because they could neither be determined selfconsistently nor validated (see Supplementary Figs S1 and S2b–e). Their contribution is in any case negligible (Supplementary Fig. S2a), with the exception of n=2, and the motivations for considering s_{2}(q) as an artifact are described in detail in the Supplementary Methods.
The singleparticle diffraction intensity S(q,φ), shown in Fig. 4c, is obtained by merging the valid coefficients {s_{n}(q)}_{n=}_{6,12,18,24,30} together with s_{0}(q), and obviously shows sixfold symmetry. Results from simulations suggest that the small regions with negative intensity are owing to nonperfectly identical scatterers (see Supplementary Fig. S3). S was subsequently used as input to a phase retrieval algorithm for reconstructing the 2D electron density ρ, shown in Fig. 5a, normalized to be on average unity in the interior of the structure. We estimate the spatial resolution to be ~17.5 nm from the average sharpness of the edge of the structure, defined as the mean separation between the 0.1 and 0.9 contour levels. The high reliability of the phase retrieval reconstruction was assessed from the phase retrieval transfer function (PRTF)^{24}, shown as a function of the momentum transfer q in Fig. 5b, whose value is well above 0.5 in the relevant range in reciprocal space (q<0.18 nm^{−1}).
For the smaller 200 nm particles, we obtained similar results, illustrated in Fig. 6. In this case, the resolution is estimated to be ~13.5 nm (corresponding to q=0.23 nm^{−1}). For both 350 and 200 nm particle sizes, the experimental data measured at the three different surface densities imply average numbers of scattering particles of N≈2.5/20/80, which are consistent with the known sample densities, beam size and beam shape (see Methods). Moreover, as can be seen in Fig. 7a, the data taken on the different samples demonstrate that the 2point CCs scale properly with N, as required by equation (5), with some deviations at larger q. Equivalently, Fig. 7b show that the extracted s_{0}(q) and s_{n}(q) are in general independent of N. Only at large q is there a noticeable suppression of s_{n} with decreasing N, an observation that we cannot explain at present.
Discussion
In the framework of Xray diffraction, the results obtained represent the first modelindependent experimental proofofprinciple of the CC method applied to 2D structure determination. From the low singleparticle signaltobackground ratio highlighted in Fig. 7b, it is manifest that standard 2D structure determination by phase retrieval from a unique singleparticle diffraction image would be unsuccessful. The difference between the reconstructed shape and the real shape of one representative particle is much smaller than the achieved resolution of ~15 nm, which is not an intrinsic limitation of the method, but rather reflects the difficulty of fabricating identical but randomly oriented nanostructures. Characteristic features of the protocol, described in detail in the Methods, are the ‘diagonalization’ procedure to consistently solve the 2point equation (5), and the use of the 3point equation (7) for validation of the singleparticle diffraction Fourier components. In this way, the singleparticle diffraction pattern is determined unambiguously and in a modelindependent fashion. This distinguishes our protocol from previous work. Saldin et al.^{16} derived the phases of such Fourier components from previous knowledge of the elongated shape of the particles under investigation and did not explicitly discuss how the overall ratio between s_{0} and {s_{n}}_{n>0} was determined. We found this ratio to be decisive for the quality of the reconstructed 2D structures. As s_{0} and {s_{n}}_{n>0} scale as 1/N and , respectively (equations (3) and (7)), underestimating the average number of scatterers N or setting κ_{(2)}=1, as for a flattop beam profile, results in a decrease of contrast in the calculated singleparticle diffraction pattern. Using such reduced contrast images as input for the phasing algorithm, we were able to produce prominent, ballshaped artefacts, of the sort present in the final reconstruction presented in ref. 16.
The basic equations (5) and (7) are strictly valid in the limit of an infinite number of scattering images, N_{im}, which is necessary for averaging out the contributions to the CCs arising from pairs of particles in the same image (Supplementary Note 1). In practice, this means that a ‘sufficient’ number of images has to be acquired, which leads one to consider the fluctuations, or signaltonoise, of the CC at large N_{im} and as a function of the number of scatterers N. The signaltonoise of the 2point CC was evaluated analytically for the 3D case^{17}, and found to be independent of N, unless the background scattering is comparable to that from an individual particle, in which case increasing N becomes advantageous. These findings are supported by an appropriate analysis of our data. We take the coefficients {s_{n}(q)}_{n>0} to be the signal, which is Nindependent, hence estimating the signaltonoise Δs_{n}(q) becomes equivalent to estimating the noise. Figure 7d displays Δs_{n}(q) derived from the experimental data for N_{im}=100 (Methods), and for three different N=2.5, 20 and 80. For the 350 nm particles, Δs_{n}(q) is found to be independent of N, while for the case of the smaller, 200nmsized structures, which have a less favourable signaltobackground scattering ratio (Fig. 7b), a decrease in Δs_{n}(q) at large q is evident as N is increased from 2.5 to 20.
A multiparticle diffraction image is the sum of the images of each particle plus an interference term from each pair of particles, if these are closer to one another than the transverse coherence length. As this term is phasemodulated by the interparticle distance, random particle positions cause its contribution to C^{(2)}(q_{1},q_{2},ψ) to average out in the limit of an infinite number of images, with the exception at q_{1}=q_{2} and ψ=0 or ψ=π (see Supplementary Note 2). This explains the two spikes of the same height present in the diagonal CCs, which are proportional to the square of the average number of particles in a coherence area and which are marked with orange and red dots in Figs 3d and 7a. The optimal size of the coherence area is thus twice the size of a single particle, in order to achieve Nyquist sampling and at the same time to minimize the number of interfering particles.
It is worth mentioning the other factors that modify the CC identities. Poisson noise and (uncorrelated) background scattering result in a spike at ψ=0 in C^{(2)}(q_{1},q_{2},ψ) for q_{1}=q_{2} (Supplementary Fig. S4 and Supplementary Note 3), marked with orange dots in Figs 3d and 7a. Because perturbations from noise and interference effects both appear as localized spikes, it is not difficult to evaluate their importance, and in our case we could also easily verify that they were filtered out during the procedure to obtain the coefficients s_{n} (Methods). A preferred orientation of the identical particles also requires corrections of (5) and (7) (Supplementary Note 1). Finally, a brief discussion should be made of the general case of nonidentical scattering particles. If the situation can be reduced to a finite number of 2D particle species with uncorrelated orientation, the validity of equation (5) is restored by replacing the righthand side with a sum over the species , and a similar modification is necessary in (3) and (7). This opens the door to studying systems of different 2D particle species in ‘chemical equilibrium’, or of identical 3D particles presenting a finite number of different 2D projections with respect to the incoming beam. For the simplest case of a twocomponent particle/solvent system, the separately measured contribution to the CCs of the solvent alone can be subtracted. In our experiments, the solvent is represented by the membrane on which the nanoparticles were anchored, and thus we determined solvent contributions using the scattering data from an empty membrane. The clear contribution to the 1point CC, which we attribute to membrane thickness fluctuations, was taken into account by background subtraction (inset of Fig. 2a). For the 2 and 3point CCs, however, we infer an effectively vanishing contribution.
There is a number of 2D static systems that are not strictly included in the framework described above, and which are studied or characterized by SAXS measurements at synchrotron sources. Evaluating the higherorder CCs of the scattered images is trivial, and could already at present disclose additional information^{25,26}, for example the average anisotropy of the domains in magnetic storage media or of the pores in fuel cell membranes. The CC analysis may also be useful to access shortrange structural properties in disordered, slowly changing systems. Wochner et al.^{27} have used the diagonal 2point CC, derived from synchrotron Xray coherent diffraction data from highly concentrated suspensions of PMMA nanospheres, to identify the preferential symmetry of 3D clusters^{25} that form temporarily within the sample. Regarding possible future investigations on 3D structural features of macromolecules or other particles in solution, which exploit the brilliance of XFEL radiation sources, it appears that applying the CC method in the spirit of its original formulation^{13} is a suitable approach; dynamical information may be accessed through the straightforward generalization of the CC analysis to laser pump/Xray probe scattering images, or even to scattering data obtained with Xray splitanddelay data^{28,29}. The present 2D demonstration experiment and its discussion have direct relevance to the 3D case, regarding the effects of interparticle interference and of the unavoidably nonuniform beam intensity profile, as well as the independence of the signaltonoise from the average number of scattering particles. Increasing the dimensionality from 2D to 3D, however, calls for some additional considerations. The 3D singleparticle diffraction pattern can be decomposed into spherical harmonics
and there are clearly more components than for the 2D case. In contrast, the number of 2point CC identities^{13,30} in 3D is the same as in 2D:
Here, are the components of C^{(2)}(q_{1,}q_{2,}ψ) with respect to the nth order Legendre polynomials in cosψ. Elser^{31} has recently shown that without additional assumptions, these equations cannot be solved for the coefficients s_{nm}(q) to the same extent as in 2D (that is, to within the parameters N and χ_{n}). These assumptions may include a priori knowledge of the particle symmetries, which reduce the number of independent parameters in S(q,θ,φ). For axially symmetric particles, only the s_{n0} coefficients are nonvanishing^{13}. Similarly, if the particles exhibit icosahedral symmetry, as is often the case in viruses^{21}, for each n there is at most one component proportional to an icosahedral harmonic^{32}. In both cases, the problem becomes equivalent to the 2D case. In general, equation (9) may be interpreted as constraints on S, to be used in addition to the 3D equivalent of (3):
for modelling a 3D electron density of the particles at higher resolution than is possible with equation (10) alone. Regarding the use of the 3point CC, the definition (6) is applicable also in the 3D case. It represents a generalization of that put forward in ref. 21, in that it implies access to three instead of two reciprocal space vectors. Although the experimental geometry in Xray diffraction experiments limits these triples to be coplanar, the new definition provides additional constraints that can be exploited for fixing the s_{nm}(q) coefficients, in a similar manner to that described in ref. 21.
At large scattering angles, straightforward corrections are necessary for the anisotropic scattering owing to linear polarization^{33} of the incoming beam and for the curvature of the Ewald sphere. However, the bottleneck to achieving subnanometer resolution is that the key parameter determining the precision of the CC method is the number of scattered photons per coherent pixel and per particle^{17}, which especially for macromolecules may be well below unity. There is no doubt that the method, as any other singleparticle approach, would greatly profit from increases in the Xray beam fluence. The adavantage of using CCs is that the low scattering power can be compensated simply by drastically increasing the number of acquired images, although even weak solvent scattering may become a significant, additional issue. For this reason, the next round of experiments, which will attempt to identify nmscale features of 3D molecules in liquid or aerosol jets, is considered crucial for the development of the CC method.
Methods
Samples
The samples were prepared on a 2 × 2 mm^{2} Si_{3}N_{4} membrane of 200 nm thickness, held by a Si wafer and coated with a Cr/Au (5/20 nm) seed layer. The gold nanoparticles were fabricated by electroplating into a 500nmthick PMMA resist mould, patterned by electron beam lithography. More details on the nanofabrication procedures can be found elsewhere^{34}. The particles with designed threefold symmetry were randomly oriented about the axis perpendicular to the membrane, and placed in random positions within an 800 × 800 μm^{2} field, with the restriction of a 500 nm minimum interparticle distance. The measurements were performed on several samples, which differed in the diameter/thickness of the nanostructures (350/350 nm, respectively, 200/250 nm), and in the average particle surface density (1.25/10/40 particles per 100 μm^{2} area).
Beamline setup and data acquisition
The Xray measurements were performed at the cSAXS beamline (X12SA) of the Swiss Light Source synchrotron at the Paul Scherrer Institute. The beamline was set to a photon energy of 6.2 keV, with a relative bandwidth of ~10^{−4} obtained with a Si(111) doublecrystal monochromator, and with a flux of ~2·10^{11} photons per second. The beam was focused onto the sample; the horizontal and vertical focusing were achieved by bending appropriately the second monochromator crystal and the higherharmonic rejection mirror, located 5.5 and 4.5 m upstream of the sample, respectively. The beam size was 14 × 6 μm^{2} FWHM (see below), and the transverse coherence length was at least twice the dimension of a nanostructure, that is, at least 700 nm. The scattered Xrays were transported through an evacuated flight tube to the detector, which was placed 7.2 m behind the sample. The detector^{23} was a Pilatus 2M, with 1675 × 1475 square pixels of 172 μm size, and was protected from the direct beam by a central beamstop. The Xray beam diameter at the detector position, measured with an attenuated beam and without inserted sample, corresponded to 3–4 detector pixels. For each sample, 3,751 scattering images of 1 s acquisition time were taken, scanning the membrane on a 121 × 31 point rectangular grid, which covered an area of 600 × 600 μm^{2}.
Xray beam shape parameters and beam shape assessement
The correction parameters κ_{(2)} and κ_{(3)} and the relation between the effective parameter N and the particle density are determined by the 2D intensity illumination function at the sample, Ω(x,y), according to κ_{(j)}=(∬dxdy Ω^{j})/(∬dxdy Ω) (Supplementary Note 4). Ω(x,y) was determined experimentally as described in Supplementary Fig. S5 and the Supplementary Methods, and resulted in κ_{(2)}≈0.26, κ_{(3)}≈0.14 and N/N_{FWHM}≈π^{2}/4, where N_{FWHM} is the number of particles in the FWHM area of the beam.
Discretization of the diffracted intensity
The position of the scattering centre on the pixel detector was established by averaging a number of centreofgravity points of pairs of opposite Bragg reflections produced by a 2D hexagonal periodic array (Supplementary Fig. S5c). A polar ‘spiderweb’ grid was defined around this centre as (q_{a,}φ_{k}), with a{1,…,N_{q}} and φ_{k}=k 2π k/N_{φ}, for k{0,…,N_{φ}−1}, which also prescribes the binning of the detector pixels. The scattering intensities in polar coordinates I(q_{a},φ_{k}) were then calculated as the average over the pixels belonging to the same polar bin. For the 350nmsized structures, q_{a} was defined to be spaced by the equivalent of two detector pixels (0.0015, nm^{−1}), and three different q_{a} ranges were considered: 0.009–0.006 nm^{−1} (N_{q}=35), 0.015–0.12 nm^{−1} (N_{q}=71) and 0.03–0.24 nm^{−1} (N_{q}=141), with N_{φ}=32, 64 and 128 azimuthal sectors, respectively. For the 200 nm structures, q_{a} was defined to be spaced by the equivalent of four detector pixels (0.003 nm^{−1}), and four different q_{a} ranges were considered: 0.009–0.06 nm^{−1} (N_{q}=18), 0.015–0.12 nm^{−1} (N_{q}=36), 0.03–0.24 nm^{−1} (N_{q}=71) and 0.06–0.37 nm^{−1} (N_{q}=106), with N_{φ}=32, 64, 128 and 256 azimuthal sectors, respectively.
Computation of the singleparticle diffraction pattern: theory
The core of the presented work concerns the calculation of the singleparticle diffraction pattern S(q,φ) from the CCs, that is, the solution of equations (3), (5) and (7) for s_{n}(q). Let q_{a} be a finite, discrete set of q values. Inspired by Kam and coworkers^{22}, and also by Saldin and coworkers^{30}, for n≠0 we use the 2point CC coefficients to define the Hermitian matrix , which can be diagonalized to yield real eigenvalues λ_{n,i} (i{1,…,N_{q}}). We designate λ_{n,1} to be the largest positive eigenvalue and write the eigenvector decomposition as
with and (i{2,…,N_{q}}). Note that the vector s_{n} is defined to within an overall phase χ_{n}. According to (5), one expects a single nonvanishing, positive eigenvalue λ_{n,1}. Therefore, a signature of the various perturbations, such as finite number of images, shot noise, uncorrelated background noise, interparticle interference, preferred particle orientation and inequivalent particles, being small is that
which is equivalent to requiring the second term on the righthand side of (11) to be negligible. If this is the case, one uses the eigenvectors s_{n} to compute the ratios
The last equality arises from the 3point CC equation (7) in combination with equation (15) below, and enables fixing χ_{n} and the effective number of particles N. These parameters are actually overdetermined, which allows for an additional validation. Finally, the desired singleparticle diffraction components are set to
and
Computation of the singleparticle diffraction pattern: practice
The 2D reciprocal space discretization described above was used for all calculations, whereby each relevant equation translates in a straightforward way to the discretized form. Before the computation of the CC, the average intensity was subtracted from each individual diffraction image, in order to eliminate eventual beamline artefacts that result in configurationindependent contributions to the scattering intensity. Diagonalization along the lines of equation (11) was performed for even n in the three or four q_{a} ranges for 350 or 200 nm structures, respectively (Supplementary Methods). Requirement (12) was translated into , and, if not fulfilled, the lower limit of the q_{a} range was raised (due to the high sensitivity to disturbances of C_{n,ab}, when s_{n}≈0 at low q_{a}). The s_{n} from the q_{a} ranges were then merged by matching their phase at overlapping q_{a} intervals. The phase relations of equation (13), for the relevant n and m and for a number of different q_{a}, q_{b} and q_{c}, represent an overconstrained system of firstorder equations in the overall correction phases χ_{n}, which is straightforwardly solved following the procedure described in the Supplementary Methods. Because of the mirror symmetry of the nanostructures considered in the experiments, the s_{n} can be chosen real, so that the phase ambiguity is effectively only a sign ambiguity. However, the method to fix the overall phases of the s_{n} is absolutely general, as demonstrated in Supplementary Fig. 6. Finally, the parameter N was chosen to best fulfill the amplitude relation of (13), for various n, m. The final values of s_{0}(q_{a}) and s_{n}(q_{a}) were then determined using (14) and (15), and the coefficients with n6N were used to calculate the singleparticle diffraction pattern S(q,φ) with equation (1) on a polar grid with N_{φ}=128 azimuthal points for the 350 nm structures (n≤30) and N_{φ}=256 for the 200 nm structures (n≤48).
Phase retrieval
For reconstruction of the structures, the polar expressions for singleparticle diffraction intensities S(q,φ) were interpolated onto a Cartesian grid (q_{x},q_{y}), suitable for fast Fourier transform algorithms. For the 350 nm particles, the grid was 512 × 512 points with q_{max}=0.24 nm^{−1}, which corresponds to a realspace pixel size of 13.0 nm. For the 200 nm structures, the grid was 500 × 500 points with q_{max}=0.37 nm^{−1}, giving a realspace pixel size of 8.4 nm. A highpass version of the structure autocorrelation^{35}, given by the inverse Fourier transform of S(q_{x},q_{y}), was used to determine the size of the autocorrelation support. The support region of the 2D electron density was then defined as a rectangle of half that size, enlarged by one pixel in each direction to avoid artificial sharpening of the reconstruction edges. The reconstruction of the 2D structures was performed with an iterative transform algorithm using the abovementioned support as realspace constraint and as reciprocal space modulus constraint. The procedure^{36} consisted of a series of 45 hybrid input–output^{37} iterations followed by five errorreduction steps, repeated until a total of 1,000 iterations was reached. In the reciprocal space projection steps, the algorithm was allowed to fill the regions obscured by the beamstop (q<0.009 nm^{−1}) and those with negative S(q_{x},q_{y}) values. As the polar reconstruction of S(q,φ) is limited to a diskshaped region, the Cartesian representation has large regions of missing data at the corners. To avoid artifacts from the abrupt interruption of data at the boundary of this disk, we used Fourierweighted projections^{36} with a nonbinary mask, suitable to allow the algorithm to slightly extrapolate the measured data by a few pixels without introducing a large number of free parameters in the reconstruction. Starting with different random phase sets, 50 reconstructions were performed, which were then registered within a small fraction of a pixel^{38}, and averaged to obtain the final 2D structure shown in Fig. 5a. The reliability of the reconstruction was evaluated from the azimuthal average of the PRTF^{24}, calculated from the 50 reconstructions.
Signaltonoise estimates
For a given number of scattering images, N_{im}, we estimate the noise Δs_{n}(q) of the Fourier coefficients of the singleparticle diffraction pattern as follows:
Here, N_{sets} is the number of sets of scattering images used for the estimate, each consisting of N_{im} randomly chosen images from the full set of 3,751. s_{σ,n} is the coefficient calculated from the images in the set σ, and s_{ref,n} is the reference value, calculated from all 3,751 images.
Additional information
How to cite this article: Pedrini, B. et al. Twodimensional structure from random multiparticle Xray scattering images using crosscorrelations. Nat. Commun. 4:1647 doi: 10.1038/ncomms2622 (2012).
References
Henderson, R. The potential and limitations of neutrons, electrons and Xrays for atomic resolution microscopy of unstained biological molecules. Q. Rev. Biophys. 28, 171–193 (1995).
Svergun, D. I. Restoring low resolution structure of biological macromolecules from solution scattering using simulated annealing. Biophys. J. 76, 2879–2886 (1999).
Chacón, P., Morán, F., Díaz, J. F. & Andreu, J. M. Lowresolution structures of proteins in solution retrieved from Xray scattering with a genetic algorithm. Biophys. J. 74, 2760–2775 (1998).
Lipfert, J. & Doniach, S. Smallangle Xray scattering from RNA, proteins, and protein complexes. Ann. Rev. Biophys. Biomol. Struct. 36, 307–327 (2007).
Mertens, H. D. T. & Svergun, D. I. Structural characterization of proteins and complexes using smallangle Xray solution scattering. J. Struct. Biol. 172, 128–141 (2010).
Emma, P. et al. First lasing and operation of an ångstromwavelength freeelectron laser. Nat. Photon. 4, 641–647 (2010).
Chapman, H. et al. Femtosecond diffractive imaging with a softXray laser. Nat. Phys. 2, 839–843 (2006).
Seibert, M. M. et al. Single mimivirus particles intercepted and imaged with an Xray laser. Nature 470, 78–82 (2011).
Chapman, H. et al. Femtosecond Xray protein crystallography. Nature 470, 73–77 (2011).
Fung, R., Shneerson, V., Saldin, D. K. & Ourmazd, A. Structure from fleeting illumination of faint spinning objects in flight. Nat. Phys. 5, 61–64 (2009).
Loh, N.T. D. & Elser, V. Reconstruction algorithm for singleparticle diffraction imaging experiments. Phys. Rev. E 80, 026705 (2009).
Loh, N.T. D. et al. Cryptotomography: reconstructing 3d Fourier intensities from randomly oriented singleshot diffraction patterns. Phys. Rev. Lett. 104, 225501 (2010).
Kam, Z. Determination of macromolecular structure in solution by spatial correlations of scattering fluctuations. Macromolecules 10, 927–934 (1977).
Saldin, D. K. et al. Structure of a single particle from scattering by many particles randomly oriented about an axis: toward structure solution without crystallization? N. J. Phys. 12, 035014 (2010).
Saldin, D. K. et al. Beyond smallangle Xray scattering: exploiting angular correlations. Phys. Rev. B 81, 411705 (2010).
Saldin, D. K. et al. New light on disordered ensembles: ab initio structure determination of one particle from scattering fluctuations of many copies. Phys. Rev. Lett. 106, 115501 (2011).
Kirian, R. A., Schmidt, K. E., Wang, X., Doak, R. B. & Spence, J. C. H. Signal, noise, and resolution in correlated fluctuations from snapshot smallangle Xray scattering. Phys. Rev. E 84, 011921 (2011).
Chapman, H. N. & Nugent, K. A. Coherent lensless Xray imaging. Nat. Photon. 4, 833–839 (2010).
Nugent, K. A. Coherent methods in the Xray sciences. Adv. Phys. 59, 1–99 (2010).
Elser, V. Threedimensional structure from intensity correlations. N. J. Phys. 13, 123014 (2011).
Kam, Z. & Gafni, I. Threedimensional reconstruction of the shape of human wart virus using spatial correlations. Ultramicroscopy 17, 251–262 (1985).
Kam, Z., Gafni, I. & Kessel, M. Enchancement of twodimensional projections from electron microscope images using spatial correlations. Ultramicroscopy 7, 311–320 (1982).
Henrich, B. et al. Pilatus: a single photon counting pixel detector for Xray applications. Nucl. Instrum. Meth. Phys. Res. A 607, 247–249 (2009).
Shapiro, D. et al. Biological imaging by soft Xray diffraction microscopy. Proc. Natl Acad. Sci. USA 102, 15343–15346 (2005).
Altarelli, M., Kurta, R. P. & Vartanyants, I. A. Xray crosscorrelation analysis and local symmetries of disordered systems: general theory. Phys. Rev. B 82, 104207 (2010).
Kurta, R. P., Altarelli, M., Weckert, E. & Vartanyants, I. A. Xray crosscorrelation analysis applied to disordered twodimensional systems. Phys. Rev. B 85, 184204 (2012).
Wochner, P. et al. Xray cross correlation analysis uncovers hidden local symmetries in disordered matter. Proc. Natl Acad. Sci. USA 106, 11511–11514 (2009).
Gutt, C. et al. Measuring temporal speckle correlations at ultrafast Xray sources. Opt. Exp. 17, 55–61 (2009).
Kam, Z. & Rigler, R. Crosscorrelation light scattering. Biophys. J. 39, 7–13 (1982).
Saldin, D. K., Shneerson, V. L., Fung, R. & Ourmazd, A. Structure of isolated biomolecules obtained from ultrashort Xray pulses: exploting the symmetry of random orientations. J. Phys.: Cond. Matter 21, 134014 (2009).
Elser, V. Strategies for processing diffraction data from randomly oriented particles. Ultramicroscopy 111, 788–792 (2011).
Saldin, D. K., Poon, H. C., Schwander, P., Uddin, M. & Schmidt, M. Reconstructing an icosahedral virus from singleparticle diffraction experiments. Opt. Exp. 19, 17318–17335 (2011).
Nielsen, J. A. & McMorrow, D. Elements of Modern Xray Physics Wiley (2001).
Gorelick, S., Guzenko, V., VilaComamala, J. & David, C. Direct ebeam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating. Nanotechnology 21, 295303 (2010).
Thibault, P., Elser, V., Jacobsen, C., Shapiro, D. & Sayre, D. Reconstruction of a yeast cell from Xray diffraction data. Acta Cryst. A 62, 248–261 (2006).
GuizarSicairos, M. & Fienup, J. R. Phase retrieval with Fourierweighted projections. J. Opt. Soc. Am. 25, 701–709 (2008).
Fienup, J. R. Phase retrieval algorithms: a comparison. Appl. Opt. 21, 2758–2769 (1982).
GuizarSicairos, M., Thurman, S. T. & Fienup, J. R. Efficient subpixel image registration algorithms. Opt. Lett. 33, 156–158 (2008).
Acknowledgements
We acknowledge valuable discussions with J. F. van der Veen and S. Flewett. We are grateful to A. Käch, H. Dietsch and A. Mihut for their assistance during the early stages of the project. Finally, we thank M. Kalisch and H.R. Künsch for collaboration concerning statistical aspects. Financial support was provided by the NCCRMUST program.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figures S1S6, Supplementary Notes 14 and Supplementary Methods (PDF 792 kb)
Rights and permissions
About this article
Cite this article
Pedrini, B., Menzel, A., GuizarSicairos, M. et al. Twodimensional structure from random multiparticle Xray scattering images using crosscorrelations. Nat Commun 4, 1647 (2013). https://doi.org/10.1038/ncomms2622
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/ncomms2622
This article is cited by

Modelindependent particle species disentanglement by Xray crosscorrelation scattering
Scientific Reports (2017)

XFEL data analysis for structural biology
Quantitative Biology (2016)

The role of the coherence in the crosscorrelation analysis of diffraction patterns from twodimensional dense monodisperse systems
Scientific Reports (2015)

Imaging live cell in microliquid enclosure by Xray laser diffraction
Nature Communications (2014)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.