Mathematicians build an algorithm to ‘do the twist’
Mathematicians at the Center for Advanced Mathematics for Energy Research Applications (CAMERA) at Lawrence Berkeley National Laboratory (Berkeley Lab) have developed a mathematical algorithm to decipher the rotational dynamics of twisting particles in large complex systems from the X-ray scattering patterns observed in highly sophisticated X-ray photon correlation spectroscopy (XPCS) experiments.
These experiments — designed to study the properties of suspensions and solutions of colloids, macromolecules, and polymers — have been established as key scientific drivers to many of the ongoing coherent light source upgrades occurring within the U.S. Department of Energy (DOE). The new mathematical methods, developed by the CAMERA team of Zixi Hu, Jeffrey Donatelli, and James Sethian, have the potential to reveal far more information about the function and properties of complex materials than was previously possible.
Particles in a suspension undergo Brownian motion, jiggling around as they move (translate) and spin (rotate). The sizes of these random fluctuations depend on the shape and structure of the materials and contain information about dynamics, with applications across molecular biology, drug discovery, and materials science.
XPCS works by focusing a coherent beam of X-rays to capture light scattered off of particles in suspension. A detector picks up the resulting speckle patterns, which contain several tiny fluctuations in the signal that encode detailed information about the dynamics of the observed system. To capitalize on this capability, the upcoming coherent light source upgrades at Berkeley Lab’s Advanced Light Source (ALS), Argonne’s Advanced Photon Source (APS), and SLAC’s Linac Coherent Light Source are all planning some of the world’s most advanced XPCS experiments, taking advantage of the unprecedented coherence and brightness.
But once you collect the data from all these images, how do you get any useful information out of them? A workhorse technique to extract dynamical information from XPCS is to compute what’s known as the temporal autocorrelation, which measures how the pixels in the speckle patterns change after a certain passage of time. The autocorrelation function stitches the still images together, just as an old-time movie comes to life as closely related postcard images fly by.
Current algorithms have mainly been limited to extracting translational motions; think of a Pogo stick jumping from spot to spot. However, no previous algorithms were capable of extracting “rotational diffusion” information about how structures spin and rotate — information that is critical to understanding the function and dynamical properties of a physical system. Getting to this hidden information is a major challenge. More