Low-dose cryo electron ptychography via non-convex Bayesian optimization

www.nature.com/scientificreports OPEN Received: 17 February 2017 Accepted: 27 June 2017 Published: xx xx xxxx Low-dose cryo electron ptychography via non-convex Bayesian optimization Philipp Michael Pelz 1,2, Wen Xuan Qiu3, Robert Bücker1, Günther Kassier1 & R. J. Dwayne Miller1,3 Electron ptychography has seen a recent surge of interest for phase sensitive imaging at atomic or nearatomic resolution. However, applications are so far mainly limited to radiation-hard samples, because the required doses are too high for imaging biological samples at high resolution. We propose the use of non-convex Bayesian optimization to overcome this problem, and show via numerical simulations that the dose required for successful reconstruction can be reduced by two orders of magnitude compared to previous experiments. As an important application we suggest to use this method for imaging single biological macromolecules at cryogenic temperatures and demonstrate 2D single-particle reconstructions from simulated data with a resolution up to 5.4 Å at a dose of 20e−/Å2. When averaging over only 30 low-dose datasets, a 2D resolution around 3.5 Å is possible for macromolecular complexes even below 100 kDa. With its independence from the microscope transfer function, direct recovery of phase contrast, and better scaling of signal-to-noise ratio, low-dose cryo electron ptychography may become a promising alternative to Zernike phase-contrast microscopy. The advent of direct electron detectors has led to a resolution revolution in the field of cryo electron microscopy in the last few years. The technique is now producing three-dimensional atomic potential maps of biological macromolecules of a few 100 kDa or lower with a resolution better than 3.5 Å1–3, such that individual amino acid side-chains can be resolved. An important role in this revolution play new image processing algorithms based on a Bayesian approach, which infer important parameters without user intervention4. Also the correction of beam-induced motion has become possible mostly due to the new generation of detectors5, 6. However, several challenges remain to be overcome in order to routinely reach 3 Å resolution also for small complexes7, 8: Firstly, beam-induced specimen charging and subsequent motion currently still render the high resolution information of the first few frames of a high repetition rate movie recorded with a direct electron detector unusable9, because the motion is too fast to efficiently correct for it. Secondly, the Detective Quantum Efficiency (DQE) of detectors is still imperfect at high spatial frequencies8, 10. Thirdly, the contrast of single images can still be improved to enable reconstructions with fewer particles and increase the throughput10. The last of these challenges has recently been addressed with a new phase plate model11, 12, which is comparatively simple to use and provides excellent contrast at low spatial frequencies. In addition to this hardware-based approach to achieve linear phase contrast in the measured amplitudes, discovered by Zernike in the 1930s13, it is also possible to algorithmically retrieve the phase information from a set of coherent diffraction measurements. One such technique, commonly known as ptychography or scanning coherent diffractive microscopy14, is becoming increasingly popular in the field of materials science due to experimental robustness and the possibility to obtain quantitative phase contrast over an essentially unlimited field of view15, 16. The use of ptychography for imaging radiation sensitive samples with electrons at high resolution is however precluded so far by its high dose requirements. Here, we show how the use of non-convex Bayesian optimization to solve the ptychographic phase retrieval problem fulfills the dose requirements for imaging biological macromolecules and makes it possible to obtain 2D images from single particles with sub-nanometer resolution. After a short introduction into the technique, we will also mention how ptychography offers improvements for the other two challenges discussed above. 1 Max Planck Institute for the Structure and Dynamics of Matter, Center for Free Electron Laser Science, Luruper Chaussee 149, 22761, Hamburg, Germany. 2Department of Physics, University of Hamburg, Hamburg, 22761, Germany. 3Departments of Chemistry and Physics, University of Toronto, 80 St. George Street, Toronto, M5S 1H6, Canada. Correspondence and requests for materials should be addressed to P.M.P. (email: ) SCIEntIFIC REPOrts | 7: 9883 | DOI:10.1038/s41598-017-07488-y 1 www.nature.com/scientificreports/  Figure 1. Experimental geometry in ptychography. The coherent electron wave function ψ(r ) illuminates   several regions (centered at r1 … K) across the sample, which is characterized by the transmission function T (r ). For each position, a 2D diffraction pattern I(1 … K ) is recorded in the far field at distance Δz. The sample thickness t can be neglected for biological macromolecules in the reconstruction at the resolutions presented in this paper. Reference resolution e−/Å2 D’Alfonso et al.30 ~1.5 Å 1.77 × 104 Yang et al.29 atomic 1.3 × 104 ~1 Å 9.2 × 106 ~2.3 Å 3.33 × 103 Putkunz et al. 28 Humphry et al. 32 Table 1. List of previously published electron ptychography experiments and used electron dose. Despite being initially proposed as a solution to the phase problem for electrons17, 18, ptychography has seen its biggest success in X-ray imaging, due to the less stringent sample requirements and the experimental need for lensless imaging techniques. Recent developments include the introduction of iterative algorithms to enable the reconstruction of datasets collected with an out-of focus probe19, 20, which decreases the memory requirements of the method dramatically. The algorithms also have the capability for the correction of experimental difficulties such as unknown scan positions21–23, partial coherence24, probe movement during exposure25, 26, intensity fluctuations during the scan24, 27 and reconstruction of background noise15, 27. In recent years, some of these advances have been applied in the context of electron microscopy and yielded atomic resolution reconstructions of low-atomic number materials28–30 and quantitative phase information15. Figure 1 shows the experimental set-up for an out-of focus ptychography experiment. A ptychographic dataset is collected by scanning a spatially confined, coherent beam, subsequently called ‘probe’, over the specimen and recording far-field diffraction patterns at a series of positions such that the illuminated regions of neighboring positions overlap. The diffraction-limited resolution rd (half-period) of the final image is given by rd = λ ⋅ ∆z , Npix ⋅ d pix where Npix is the number of detector pixels, dpix is the detector pixel size and λ is the de-Broglie-wavelength of the  electrons. Given the set of positions r1 … K and a realistic forward model for th (...truncated)