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New mathematical techniques might allow for X-ray nanocrystallography

The discovery of the double-helix structure of DNA revealed key insights on how life and heredity works, helping kick off the modern era of genetics. This and many other breakthroughs were achieved via a technique known as X-ray crystallography, but there are untold numbers of molecules this method cannot discover the structures of. Now a new technique could help improve an emerging successor to conventional X-ray crystallography known as X-ray nanocrystallography. These findings are detailed in the Proceedings of the National Academy of Sciences.

For decades, scientists have discovered the vast majority of the structures of vital molecules such as DNA and hemoglobin using X-ray crystallography. This method involves growing a crystal of the molecule one wants to investigate and then shining X-rays on it. By analyzing the way in which the crystal scatters or diffracts the X-rays, researchers can deduce the 3-D structure of the molecule.

A problem with conventional X-ray crystallography is that it is limited to molecules that scientists can form into large crystals at least 10 microns wide, or roughly a tenth of the average diameter of a human hair. It can be difficult or impossible to create large crystals of a number of key macromolecules, including critical proteins found in the membranes of all living cells. However, tinier crystals require larger X-ray doses that can damage the crystals before researchers can record the way they diffract the X-rays.

Recent advances in X-ray technology have made it possible to use smaller crystals for crystallography, ones typically less than a micron large, usually at least 100 nanometers or billionths of a meter in size. This method, X-ray nanocrystallography, relies on an ensemble of nanocrystals. Intense, ultra-fast X-ray pulses are then used, ones just femtoseconds or quadrillionths of a second long, which are brief and strong enough to reveal useful details of a molecule’s structure before before the nanocrystals are destroyed.

X-ray nanocrystallography is not without challenges. For instance, the images are noisy, and the variations in the size and orientation of the crystals and the magnitude of the X-ray doses they receive add uncertainty in the data.

Now applied mathematicians Jeffrey Donatelli and James Sethian at Lawrence Berkeley National Laboratory and the University of California, Berkeley, have developed algorithms that can help resolve this quandary. Their method can determine crystal size and orientation as well as the size of the X-ray doses they receive from noisy data.

“We were able to solve many of the issues currently plaguing x-ray nanocrystallography for our simulated data sets,” Donatelli says.

First the researchers employ a technique commonly used in conventional crystallography known as autoindexing to determine crystal orientations. Basically, the two-dimensional diffraction patterns one gets from X-ray crystallography are sets of bright spots, each of which is ultimately related to the distribution of a molecule’s electron density in three dimensions. Indexing involves finding out how spots correspond to the underlying arrangement of molecules in the crystal. (The spots are technically known as Bragg peaks.)

The scientists next determine crystal size by carrying out a Fourier analysis of the intensity of the signals around spots. A Fourier analysis essentially involves breaking down complex signals into a combination of simple waves to better understand the signals — kind of like taking a meal and figuring out its ingredients.

In many cases, autoindexing does not completely determine the way crystals are oriented. “It’s kind of like knowing the orientations of a bunch of coins, with the exception that you don’t know if each one is facing heads up or heads down,” Donatelli says. However, the researchers were able to develop a series of data analysis techniques, building on methods from clustering analysis and graph theory, which utilize knowledge of the crystal sizes to resolve this indexing ambiguity, Donatelli adds — “determining which ones are face up and which ones are face down with the coin analogy.”

Knowing crystal size and orientation can then help yield details about their structure without the need for more complex experiments, assumptions regarding the structure of the molecules, or knowledge of similar structures required by current techniques. They verified their method works on simulated diffraction images of the enzyme PuuE Allantoinase, whose structure is already known.

“The most likely criticism others might have is that we have not yet tested our approach on real data,” Donatelli says. “Our ultimate goal is to work with crystallographers and computational scientists to apply our approach to real data sets and eventually develop the ideas from our framework into software which can be used by experimentalists to increase the fidelity of their collected data.”

Categories: Applied Mathematics
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