Algorithmic framework for X-ray nanocrystallographic reconstruction in the presence of the indexing ambiguity

Abstract

X-ray nanocrystallography allows the structure of a macromolecule to be determined from a large ensemble of nanocrystals. However, several parameters, including crystal sizes, orientations, and incident photon flux densities, are initially unknown and images are highly corrupted with noise. Autoindexing techniques, commonly used in conventional crystallography, can determine orientations using Bragg peak patterns, but only up to crystal lattice symmetry. This limitation results in an ambiguity in the orientations, known as the indexing ambiguity, when the diffraction pattern displays less symmetry than the lattice and leads to data that appear twinned if left unresolved. Furthermore, missing phase information must be recovered to determine the imaged object's structure. We present an algorithmic framework to determine crystal size, incident photon flux density, and orientation in the presence of the indexing ambiguity. We show that phase information can be computed from nanocrystallographic diffraction using an iterative phasing algorithm, without extra experimental requirements, atomicity assumptions, or knowledge of similar structures required by current phasing methods. The feasibility of this approach is tested on simulated data with parameters and noise levels common in current experiments.A lthough conventional X-ray crystallography has been used extensively to determine atomic structure, it is limited to objects that can be formed into large crystal samples ð>10 μmÞ. An appealing alternative, made possible by recent advances in light source technology, is X-ray nanocrystallography, which is able to image structures resistant to large crystallization, such as membrane proteins, by substituting a large ensemble of easier to build nanocrystals, typically <1 μm, often delivered to the beam via a liquid jet (1-6) ( Fig. 1). However, the beam power required to retrieve sufficient information destroys the crystal, hence ultrafast pulses (≤70 fs) are required to collect data before damage effects alter the signal. Using nanocrystals introduces several challenges. Due to the small crystal size, Bragg peaks are smeared out, and there is noticeable signal between peaks. Typically, only partial peak reflections are measured, resulting in reduced intensities. Variations in crystal size and incident photon flux density, unknown orientations, shot noise, and background signal from the liquid and detector add additional uncertainty to the data.If crystal orientations were known, noise and variation in the peak measurements could be averaged out, and the data could be inverted to retrieve the object's electron density. Although autoindexing techniques can be used to determine crystal orientation up to lattice symmetry from the location of a sufficient number of Bragg peaks, they typically face difficulties in the presence of partial and non-Bragg reflections common in nanocrystal diffraction images. Furthermore, these techniques only narrow down orientation to a list of possibilities when the diffraction pattern has less ...

Combined genomic, transcriptomic, and metabolomic analyses provide insights into chayote (Sechium edule) evolution and fruit development

“…Differential expression analysis was performed between the different sample groups. Pearson’s correlation coefficient ( r ) was used for repeatability assessment 104 . DESeq2 105 was used for differential expression analysis between sample groups to obtain the DEG sets between two biological samples.…”