# Reconstructing 3D Molecular Structure from the Angular Correlations of Scattered X-rays

Fluctuation X-ray scattering (FXS) is an emerging imaging technique in which one collects a series of diffraction images from particles in solution, with X-ray exposures taken below rotational diffusion times of the particles. From these images, an average 3D angular cross-correlation function is computed, which can be directly related to the electron density of the underlying molecular structure, essentially independently from the number of particles in the beam. Consequently, FXS allows for high-throughput studies of the structure and behavior of biomolecules in near-native environments.

However, reconstructing molecular structure from these experiments is challenging. In particular, the reconstruction problem can be formulated as a hyper phase problem to determine a 3D intensity function from the correlation data, in addition to the classical phase problem of determining the electron density from this intensity, both of which are high-dimensional nonconvex inverse problems. Furthermore, the data is often plagued by large systematic issues, background, and noise, which can complicate the analysis if not treated properly.

In this talk, we describe the theory behind FXS and present the multi-tiered iterative phasing (M-TIP) algorithm, which is able to reconstruct *ab initio* 3D structure from FXS data by simultaneously solving both the classical and hyper phase problems. Furthermore, we discuss how to treat systematic issues in the data, and describe an iterative filtering algorithm that is able to boost the signal-to-noise ratio of the correlations by several orders of magnitude. Finally, we show the application of these techniques in determining the 3D structures of several viruses from the correlations of experimental single-particle and multiple-particle X-ray scattering data collected at the Linac Coherent Light Source (LCLS).

*Jeffrey Donatelli is a computational research scientist in the mathematics group at Lawrence Berkeley National Laboratory and the deputy director of the Center for Advanced Mathematics for Energy Research Applications (CAMERA). He received his Ph.D. in applied mathematics from UC Berkeley in 2013. His research interests include numerical analysis, computational harmonic analysis, and high-performance computing applied to problems in imaging. His recent work has focused on developing new mathematics and algorithms to solve challenging inverse problems arising from emerging X-ray experiments enabled by free-electron lasers, including fluctuation X-ray scattering, single-particle diffraction, and X-ray nanocrystallography.*