We use extremely bright and ultrashort pulses from an x-ray free-electron laser (XFEL) to measure correlations in x rays scattered from individual bioparticles. This allows us to go beyond the traditional crystallography and single-particle imaging approaches for structure investigations. We employ angular correlations to recover the three-dimensional (3D) structure of nanoscale viruses from x-ray diffraction data measured at the Linac Coherent Light Source. Correlations provide us with a comprehensive structural fingerprint of a 3D virus, which we use both for model-based and ab initio structure recovery. The analyses reveal a clear indication that the structure of the viruses deviates from the expected perfect icosahedral symmetry. Our results anticipate exciting opportunities for XFEL studies of the structure and dynamics of nanoscale objects by means of angular correlations.
Fluctuation X-ray scattering (FXS) is an extension of small-and wide-angle X-ray scattering in which the X-ray snapshots are taken below rotational diffusion times. This technique, performed using a free electron laser or ultrabright synchrotron source, provides significantly more experimental information compared with traditional solution scattering methods. We develop a multitiered iterative phasing algorithm to determine the underlying structure of the scattering object from FXS data.fluctuation scattering | iterative phasing | polar Fourier transform X -ray solution scattering of macromolecular complexes is a versatile technique, providing low-resolution structural information that, when supplemented with high-resolution crystallographic data, can result in fundamental insights into the physiological behavior and function of macromolecular machines (1). Although solution scattering has been applied successfully to many problems in the biological sciences (2, 3), the technique suffers from a significant shortcoming. Due to the 3D averaging that occurs during the X-ray snapshots, the effective information content of the data is typically only around 9-15 independent parameters (4). This lack of sufficient information in the data ultimately results in nonunique or poorly determined structural hypotheses.To overcome these issues, it was proposed to perform the solution scattering process at timescales below rotational diffusion times (5, 6). By avoiding physical rotational averaging, speckle patterns emerge from which angular correlation functions can be computed. These so-called fluctuation X-ray scattering experiments can be performed on modern synchrotrons (7) and free electron laser sources (8, 9). The angular correlation information provided from these experiments is directly related to the imaged structure (10) and contains significantly more information than standard smalland wide-angle X-ray scattering (SAXS/WAXS) data. Although the relation from real space structure to fluctuation scattering data is straightforward, the inverse problem of determining a molecular model from fluctuation scattering data is nontrivial (8).For 2D systems, such as macromolecules randomly oriented around a single axis (11), an analytical route for image reconstruction is available (12). However, this approach requires triple correlation measurements across the full resolution range of interest, which may not be fully accessible due to experimental limitations.Earlier work attempting to determine 3D shape from fluctuation scattering data was based on solving two phase problems consecutively. Although this route has shown success in cases where the scattering species has helical (13) or icosahedral symmetry (14), it fails to produce a structure for the general case. Another approach was developed in which the inverse problem was solved using a reverse Monte Carlo method (8), similar to what is used to determine macromolecular envelopes from SAXS data (15)(16)(17). Although this approach allows a real space object to be reconstru...
Constant Amplitude Zero Autocorrelation (CAZAC) waveforms u are analyzed in terms of the ambiguity function Au. Elementary number theoretic considerations illustrate that peaks in A u are not stable under small pertubations in its domain. Further, it is proved that the analysis of vector-valued CAZAC waveforms depends on methods from the theory of frames. Finally, techniques are introduced to characterize the structure of Au, to compute u in terms of Au, and to evaluate MSE for CAZAC waveforms.
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