Fluctuation X-ray scattering (FXS) is an emerging biophysical imaging technique which seeks to image multiple identical particles in solution, while avoiding the low data-to-parameter ratio commonly encountered in traditional small- and wide-angle X-ray scattering (SAXS/WAXS) experiments.  In an FXS experiment, the imaging process is conducted below rotational diffusion times, so that the particles are essentially frozen in orientation and position during imaging. The resulting diffraction images have angular intensity fluctuations which contain several orders of magnitude more structural information than can be extracted through SAXS and WAXS. This information can be efficiently extracted by computing angular correlations from several images, and then averaging these quantities over all images.

 

The average angular correlation information collected in an FXS experiment is directly related to the structure of the imaged particles. In particular, this relation can be greatly simplified by expanding the correlation information and the intensity function into appropriate bases. For 2D systems, such as macromolecules randomly oriented about a single axis, the circular harmonic coefficients of the angular cross-correlation function are equivalent to Gram matrices of the circular harmonic coefficients of the intensity function. For 3D systems, where all orientations of the macromolecules are equally likely, the Legendre polynomial expansion coefficients of the cross-correlation function is equivalent to Gram matrices of the spherical harmonic coefficients of the intensity function. Although the relation from the electron density of the structure to the measured FXS data is relatively straightforward, the process of determining structure from its associated FXS data is a challenging non-linear inverse problem. In particular, unlike standard diffractive imaging, where one measures diffraction intensities and only has to recover the missing complex phases, reconstruction from FXS also requires the recovery of the intensity information from the correlation data. The relations between the collected diffraction patterns and the FXS data and between the structure's electron density and the FXS data are both summarized below.

 

Fluctuation X-ray Scattering: Current and Future Work

Jeffrey J. Donatelli, Peter H. Zwart, and James A. Sethian
 

FXS Reconstruction via Multi-tiered Iterative Phasing (M-TIP):

 

In order to determine structure from FXS, we have developed a multi-tieried iterative phasing (M-TIP) algorithm, which alternatingly projects a model of the structure to satisfy multiple tiers of constraints, such as the intensity function being consistent with the FXS data, the Fourier magnitudes matching the intensity function, and real space constraints; such as size and shape limitations, density bounds, and/or symmetry. The M-TIP algorithm is a natural generalization of iterative phasing that allows one to efficiently and simultaneously determine intensities, complex phases, and molecular structure from FXS data on a desktop machine. The M-TIP algorithm is summarized below.