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(a–c) Three sample frames from the 2.5 photon/frame data set with detected x-ray photons circled. (d) Occupancy histogram compared with the Poisson distribution. (e) The sum of all thresholded frames from the 2.5 photon/frame data set showing a uniform angular distribution of data.
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Single-particle imaging experiments of biomolecules at x-ray free-electron lasers (XFELs) require processing hundreds of thousands of images that contain very few x-rays. Each low-fluence image of the diffraction pattern is produced by a single, randomly oriented particle, such as a protein. We demonstrate the feasibility of recovering structural i...
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... However, for smaller particles, signal levels are much lower and extraneous background photons can severely hinder the alignment process. And while reconstruction algorithms are remarkably tolerant to low signal levels [6,11,14], background often poses a fundamental limit on the achievable resolution since the diffraction signal from a compact object falls off very quickly with increasing momentum transfer, but the background usually does not [15]. ...
Single-particle imaging (SPI) at x-ray free-electron lasers is a technique to determine the three-dimensional structure of nanoscale objects like biomolecules from a large number of diffraction patterns of copies of these objects in random orientations. The technique has been limited to relatively low resolution due to background noise and heterogeneity of the target particles. A recently introduced reference-enhanced holographic SPI methodology uses strongly scattering holographic references to improve background tolerance, and, thus, the achievable resolution, at the cost of additional latent variables beyond orientation. Here, we describe an improved reconstruction algorithm based on maximum likelihood estimation, which scales better, enabling fine sampling of latent parameters to reach high resolutions, and much better performance in the low signal limit. Furthermore, we show that structural variations within the target particle are averaged in real space, significantly improving robustness to conformational heterogeneity in comparison to conventional SPI. With these computational improvements, we believe reference-enhanced SPI is capable of reaching sub-nanometer resolution biomolecule imaging.
... However, for smaller particles, signal levels are much lower and extraneous background can severely hinder the alignment process. And while reconstruction algorithms are remarkably tolerant to low signal levels [6,11,14], background often poses a fundamental limit on the achievable resolution since the diffraction signal from a compact object falls off very quickly with increasing momentum transfer but the background usually does not [15]. ...
Single particle imaging (SPI) at X-ray free electron lasers (XFELs) is a technique to determine the 3D structure of nanoscale objects like biomolecules from a large number of diffraction patterns of copies of these objects in random orientations. Millions of low signal-to-noise diffraction patterns with unknown orientation are collected during an X-ray SPI experiment. The patterns are then analyzed and merged using a reconstruction algorithm to retrieve the full 3D-structure of particle. The resolution of reconstruction is limited by background noise, signal-to-noise ratio in diffraction patterns and total amount of data collected. We recently introduced a reference-enhanced holographic single particle imaging methodology [Optica 7,593-601(2020)] to collect high enough signal-to-noise and background tolerant patterns and a reconstruction algorithm to recover missing parameters beyond orientation and then directly retrieve the full Fourier model of the sample of interest. Here we describe a phase retrieval algorithm based on maximum likelihood estimation using pattern search dubbed as MaxLP, with better scalability for fine sampling of latent parameters and much better performance in the low signal limit. Furthermore, we show that structural variations within the target particle are averaged in real space, significantly improving robustness to conformational heterogeneity in comparison to conventional SPI. With these computational improvements, we believe reference-enhanced SPI is capable of reaching sub-nm resolution biomolecule imaging.
... Each of these intensities represents an average of aligned copies of a subset of the patterns from the whole set. In addition to the EMC algorithm being highly noise tolerant [7,28,29], one can also use it to examine the average models to understand what type of particles are in the dataset. ...
Single particle imaging at x-ray free electron lasers (XFELs) has the potential to determine the structure and dynamics of single biomolecules at room temperature. Two major hurdles have prevented this potential from being reached, namely, the collection of sufficient high-quality diffraction patterns and robust computational purification to overcome structural heterogeneity. We report the breaking of both of these barriers using gold nanoparticle test samples, recording around 10 million diffraction patterns at the European XFEL and structurally and orientationally sorting the patterns to obtain better than 3-nm-resolution 3D reconstructions for each of four samples. With these new developments, integrating advancements in x-ray sources, fast-framing detectors, efficient sample delivery, and data analysis algorithms, we illuminate the path towards sub-nanometer biomolecular imaging. The methods developed here can also be extended to characterize ensembles that are inherently diverse to obtain their full structural landscape.
... Each of these intensities represent an average of aligned copies of a subset of the patterns from the whole set. In addition to the EMC algorithm being highly noise-tolerant [7,29,30], one can also use it to examine the average models to understand what type of particles are in the dataset. ...
We report the 3D structure determination of gold nanoparticles (AuNPs) by X-ray single particle imaging (SPI). Around 10 million diffraction patterns from gold nanoparticles were measured in less than 100 hours of beam time, more than 100 times the amount of data in any single prior SPI experiment, using the new capabilities of the European X-ray free electron laser which allow measurements of 1500 frames per second. A classification and structural sorting method was developed to disentangle the heterogeneity of the particles and to obtain a resolution of better than 3 nm. With these new experimental and analytical developments, we have entered a new era for the SPI method and the path towards close-to-atomic resolution imaging of biomolecules is apparent.
... The EMC (Expansion-Maximization-Compression) algorithm 14,19 has been proven to be a noise-robust way to assign orientations to each diffraction pattern, even for sparse photon numbers. 20 This iterative algorithm assigns an orientation distribution for each diffraction pattern based on a model of the particle, which is automatically updated at each iteration, see Fig. 1(b). The orientation problem amounts to the assignment of three angles to each diffraction pattern. ...
X-ray free electron lasers (XFELs) now routinely produce millijoule level pulses of x-ray photons with tens of femtoseconds duration. Such x-ray intensities gave rise to the idea that weakly scattering particles—perhaps single biomolecules or viruses—could be investigated free of radiation damage. Here, we examine elements from the past decade of so-called single particle imaging with hard XFELs. We look at the progress made to date and identify some future possible directions for the field. In particular, we summarize the presently achieved resolutions as well as identifying the bottlenecks and enabling technologies to future resolution improvement, which in turn enables application to samples of scientific interest.
... Simulations have been performed to illustrate the data produced and to demonstrate the reconstruction algorithm. For simplicity, a 2D toy model has been used that is rotated in-plane, similar to previous experiments to test the performance of the EMC algorithm with sparse data [4,30]. There is one parameter for the angle and there are two for the shift, but the qualitative structure of the problem remains the same. ...
X-ray single-particle imaging involves the measurement of a large number of noisy diffraction patterns of isolated objects in random orientations. The missing information about these patterns is then computationally recovered in order to obtain the 3D structure of the particle. While the method has promised to deliver room-temperature structures at near-atomic resolution, there have been significant experimental hurdles in collecting data of sufficient quality and quantity to achieve this goal. This paper describes two ways to modify the conventional methodology that significantly ease the experimental challenges, at the cost of additional computational complexity in the reconstruction procedure. Both these methods involve the use of holographic reference objects in close proximity to the sample of interest, whose structure can be described with only a few parameters. A reconstruction algorithm for recovering the unknown degrees of freedom is also proposed and tested with toy model simulations. The techniques proposed here enable 3D imaging of biomolecules that is not possible with conventional methods and open up a new family of methods for recovering structures from datasets with a variety of hidden parameters.
... Simulations have been performed to illustrate the data produced and to demonstrate the reconstruction algorithm. For simplicity, a 2D toy model has been used which is rotated in-plane, similar to previous experiments to test the performance of the EMC algorithm with sparse data [4,28]. There is one parameter for the angle and two for the shift, but the qualitative structure of the problem remains the same. ...
X-ray single particle imaging involves the measurement of a large number of noisy diffraction patterns of isolated objects in random orientations. The missing information about these patterns is then computationally recovered in order to obtain a three-dimensional structure of the particle. While the method has promised to deliver room temperature structures at near-atomic resolution, there have been significant experimental hurdles in collecting data of sufficient quality and quantity to achieve this goal. This paper describes two ways to modify the conventional methodology which significantly ease the experimental challenges, at the cost of additional computational complexity in the reconstruction procedure. Both these methods involve the use of holographic reference objects in close proximity to the sample of interest, whose structure can be described with only a few parameters. A reconstruction algorithm to recover the unknown degrees of freedom is also proposed and tested with toy-model simulations.
... By sub-sampling the experimental data from PR772 viruses measured in [9], we show that the reconstruction quality is essentially same as from the full data set with as few as 135 relevant photons/pattern, corresponding to 0.087 photons/speckle at the detector corner. This approaches the limits of prior work using simulated data [1,2,10] or proof-of-principle experiments under highly controlled conditions not realistic for single particle imaging [22,23]. By way of contrast, the results here are based on data derived from experimental measurements on PR772 viruses incorporating particle variability and instrument background, demonstrating that the signal required for X-ray single particle imaging under realistic conditions is much lower than previously demonstrated especially in terms of the number of scattered photons required per frame. ...
An outstanding question in X-ray single particle imaging experiments has been the feasibility of imaging sub 10-nm-sized biomolecules under realistic experimental conditions where very few photons are expected to be measured in a single snapshot and instrument background may be significant relative to particle scattering. While analyses of simulated data have shown that the determination of an average image should be feasible using Bayesian methods such as the EMC algorithm, this has yet to be demonstrated using experimental data containing realistic non-isotropic instrument background, sample variability and other experimental factors. In this work, we show that the orientation and phase retrieval steps work at photon counts diluted to the signal levels one expects from smaller molecules or with weaker pulses, using data from experimental measurements of 60-nm PR772 viruses. Even when the signal is reduced to a fraction as little as 1/256, the virus electron density determined using ab initio phasing is of almost the same quality as the high-signal data. However, we are still limited by the total number of patterns collected, which may soon be mitigated by the advent of high repetition-rate sources like the European XFEL and LCLS-II.
... These reduced data, or 'diluted', patterns contain just a smattering of photons which often look like pure noise to the eye. While simulations [10,11] and proof-of-principle experiments [12][13][14] have previously been performed with such "homeopathic" photon counts of much less than one photon per pixel, they have been performed in relatively controlled conditions with low signal being the most significant hurdle. ...
... By sub-sampling the experimental data from PR772 viruses measured in Reddy et al. [9], we show that the reconstruction quality is essentially same as from the full data set with as few as 135 relevant photons/pattern, corresponding to 0.087 photons/speckle at the detector corner. This approaches the limits of prior work using simulated data [1] [2] or proof-of-principle experiments under highly controlled conditions not realistic for single particle imaging [12] [14]. By way of contrast, the results here are based on data derived from experimental measurements on PR772 viruses incorporating particle variability and instrument background, demonstrating that the signal required for X-ray single particle imaging under realistic conditions is much lower than previously demonstrated especially in terms of the number of scattered photons required per frame. ...
An outstanding question in X-ray single particle imaging experiments has been the feasibility of imaging sub 10-nm-sized biomolecules under realistic experimental conditions where very few photons are expected to be measured in a single snapshot and instrument background may be significant relative to particle scattering. While analyses of simulated data have shown that the determination of an average image should be feasible using Bayesian algorithms such as the EMC algorithm for near-perfect diffraction patterns, this has yet to be demonstrated using experimental data containing realistic non-isotropic instrument background, sample variability and other experimental factors. In this work, we show that the orientation and phase retrieval steps work at photon counts diluted to the signal levels one expects from smaller molecules or with weaker pulses using data from experimental measurements of 60-nm PR772 viruses. Even when the signal is reduced to a fraction as little as 1/256, the virus electron density determined using ab initio phasing is of almost the same quality as the high-signal data.
... However, the important case of sparse diffraction from a 3D object has only been solved experimentally in a setting different from the classical CDI problem. For example, one of the methods of orientation recovery, a statistical technique based on expectation maximization, the expand-maximizecompress (EMC) algorithm (Loh & Elser, 2009;Ayyer et al., 2016), has been applied successfully to real-space sparse radiographic data in two (Philipp et al., 2012) and three dimensions (Ayyer et al., 2014), to sparse crystallographic data limited to one (Wierman et al., 2016) and two rotation axes (Lan et al., 2017), and very recently also to synchrotron-based serial protein crystallographic data for random crystal orientations (Lan et al., 2018). ...
... A more quantitative definition of sufficiency for this orientation problem, beyond what is already in the literature (e.g. Loh & Elser, 2009;Elser, 2009;Philipp et al., 2012;Ayyer et al., 2014;Loh, 2014), is the subject of a future publication by some of the co-authors here. ...
The routine atomic resolution structure determination of single particles is expected to have profound implications for probing structure–function relationships in systems ranging from energy-storage materials to biological molecules. Extremely bright ultrashort-pulse X-ray sources – X-ray free-electron lasers (XFELs) – provide X-rays that can be used to probe ensembles of nearly identical nanoscale particles. When combined with coherent diffractive imaging, these objects can be imaged; however, as the resolution of the images approaches the atomic scale, the measured data are increasingly difficult to obtain and, during an X-ray pulse, the number of photons incident on the 2D detector is much smaller than the number of pixels. This latter concern, the signal `sparsity', materially impedes the application of the method. An experimental analog using a conventional X-ray source is demonstrated and yields signal levels comparable with those expected from single biomolecules illuminated by focused XFEL pulses. The analog experiment provides an invaluable cross check on the fidelity of the reconstructed data that is not available during XFEL experiments. Using these experimental data, it is established that a sparsity of order 1.3 × 10 ⁻³ photons per pixel per frame can be overcome, lending vital insight to the solution of the atomic resolution XFEL single-particle imaging problem by experimentally demonstrating 3D coherent diffractive imaging from photon-sparse random projections.
