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There is a growing understanding of the structural dynamics of biological molecules fueled by x-ray crystallography experiments. Time-resolved serial femtosecond crystallography (TR-SFX) with x-ray Free Electron Lasers allows the measurement of ultrafast structural changes in proteins. Nevertheless, this technique comes with some limitations. One major challenge is the quality of data from TR-SFX measurements, which often faces issues like data sparsity, partial recording of Bragg reflections, timing errors, and pixel noise. To overcome these difficulties, conventionally, large volumes of data are collected and grouped into a few temporal bins. The data in each bin are then averaged and paired with the mean of their corresponding jittered timestamps. This procedure provides one structure per bin, resulting in a limited number of averaged structures for the entire time interval spanned by the experiment. Therefore, the information on ultrafast structural dynamics at high temporal resolution is lost. This has initiated research for advanced methods of analyzing experimental TR-SFX data beyond the standard binning and averaging method. To address this problem, we use a machine learning algorithm called Nonlinear Laplacian Spectral Analysis (NLSA), which has emerged as a promising technique for studying the dynamics of complex systems. In this work, we demonstrate the power of this algorithm using synthetic x-ray diffraction snapshots from a protein with significant data incompleteness, timing uncertainties, and noise. Our study confirms that NLSA is a suitable approach that effectively mitigates the effects of these artifacts in TR-SFX data and recovers accurate structural dynamics information hidden in such data. 

A promising new route for structural biology is single-particle imaging with an X-ray Free-Electron Laser (XFEL). This method has the advantage that the samples do not require crystallization and can be examined at room temperature. However, high-resolution structures can only be obtained from a sufficiently large number of diffraction patterns of individual molecules, so-called single particles. Here, we present a method that allows for efficient identification of single particles in very large XFEL datasets, operates at low signal levels, and is tolerant to background. This method uses supervised Geometric Machine Learning (GML) to extract low-dimensional feature vectors from a training dataset, fuse test datasets into the feature space of training datasets, and separate the data into binary distributions of “single particles” and “non-single particles.” As a proof of principle, we tested simulated and experimental datasets of the Coliphage PR772 virus. We created a training dataset and classified three types of test datasets: First, a noise-free simulated test dataset, which gave near perfect separation. Second, simulated test datasets that were modified to reflect different levels of photon counts and background noise. These modified datasets were used to quantify the predictive limits of our approach. Third, an experimental dataset collected at the Stanford Linear Accelerator Center. The single-particle identification for this experimental dataset was compared with previously published results and it was found that GML covers a wide photon-count range, outperforming other single-particle identification methods. Moreover, a major advantage of GML is its ability to retrieve single particles in the presence of structural variability.