The use of connected masks for reconstructing the single particle image from X-ray diffraction data. III. Maximum-likelihood based strategies to select solution of the phase problem

N. L. Lunina ; T. E. Petrova ; A. G. Urzhumtsev ; V. Yu. Lunin

Matematičeskaâ biologiâ i bioinformatika, Tome 12 (2017) no. 2, pp. 521-535.

Résumé

The main experimental limitation of biological crystallography is associated with the need to prepare the object under study in the form of a single crystal. New powerful X-ray sources, namely free-electron X-ray lasers, makes it possible to raise the question of the determination of the structure of isolated biological macromolecules and their complexes in practice. An additional advantage of working with isolated particles is the possibility to obtain information about scattering in all directions, and not only in those limited by the Laue-Bragg diffraction conditions. This significantly facilitates the solution of the phase problem of X-ray diffraction analysis. This paper is devoted to two lines of development of the method for solving the phase problem, proposed earlier by the authors, which is based on the random scanning of the configuration space of potential solutions of the phase problem. The paper suggests a new criterion for the selection of "candidates" for solving the phase problem in the process of scanning. It involves the maximization of statistical likelihood, and its effectiveness is shown in test calculations. The second line concerns the choice of the optimal scanning strategy. It is shown that the gradual expansion of the set of experimental data used in the work allows obtaining solutions of a higher quality than those obtained with all available data included into the work simultaneously from the beginning.

Bibliographie

[1] Barends T. R. M., Foucar L., Botha S., Doak R. B., Shoeman R. L., Nass K., Koglin J. E., Williams G. J., Boutet S., Messerschmidt M., Schlichting I., “De novo protein crystal structure determination from X-ray free-electron laser data”, Nature, 505 (2014), 244–247 | DOI

[2] Chapman H. N., Fromme P., Barty A., White T. A., Kirian R. A., Aquila A., Hunter M. S., Schulz J., DePonte D. P., Weierstall U., et al., “Femtosecond X-ray protein nanocrystallography”, Nature, 470 (2011), 73–77 | DOI

[3] Boutet S., Lomb L., Williams G. J., Barends T. R., Aquila A., Doak R. B., Weierstall U., DePonte D. P., Steinbrener J., Shoeman R. L., et al., “High-resolution protein structure determination by serial femtosecond crystallography”, Science, 337 (2012), 362–364 | DOI

[4] Redecke L., Nass K., DePonte D. P., White T. A., Rehders D., Barty A., Stellato F., Liang M., Barends T. R. M., Boutet S., et al., “Natively Inhibited Trypanosoma brucei Cathepsin B Structure Determined by Using an X-ray Laser”, Science, 339 (2013), 227–230 | DOI

[5] Lomb L., Barends T. R. M., Kassemeyer S., Aquila A., Epp S. W., Erk B., Foucar L., Hartmann R., Rudek B., Rolles D., et al., “Radiation damage in protein serial femtosecond crystallography using an x-ray free-electron laser”, Physical Review B, 84 (2011), 214111 | DOI

[6] Johansson L. C., Arnlund D., White T. A., Katona G., DePonte D. P., Weierstall U., Doak R. B., Shoeman R. L., Lomb L., Malmerberg E., et al., “Lipidic phase membrane protein serial femtosecond crystallography”, Nature Methods, 9 (2012), 263–265 | DOI

[7] Kern J., Alonso-Mori R., Hellmich J., Tran R., Hattne J., Laksmono H., Glöckner C., Echols N., Sierra R. G., Sellberg J., et al., “Room temperature femtosecond X-ray diffraction of photosystem II microcrystals”, Proceedings of the National Academy of Sciences of the USA, 109 (2012), 9721–9726 | DOI

[8] Lunin V. Yu., Lunina N. L., Petrova T. E., “Biologicheskaya kristallografiya bez kristallov”, Matematicheskaya biologiya i bioinformatika, 12:1 (2017), 55–72 | DOI

[9] Thibault P., Elser V., Jacobsen C., Shapiro D., Sayre D., “Reconstruction of a yeast cell from X-ray diffraction data”, Acta Crystallographica Section A: Foundations of Crystallography, 62 (2006), 248–261 | DOI

[10] Rodriguez J. A., Xu R., Chen C. C., Huang Z., Jiang H., Chen A. L., Raines K. S., Pryor A. Jr, Nam D., Wiegart L., Song C., Madsen A., Chushkin Y., Zontone F., Bradley P. J., Miao J., “Three-dimensional coherent X-ray diffractive imaging of whole frozen-hydrated cells”, IUCr Journal, 2 (2015), 575–583 | DOI

[11] Takayama Y., Yonekura K., “Cryogenic coherent X-ray diffraction imaging of biological samples at SACLA: a correlative approach with cryo-electron and light microscopy”, Acta Crystallographica Section A: Foundations and Advances, 72 (2016), 179–189 | DOI

[12] Munke A., Andreasson J., Aquila A., Awel S., Ayyer K., Barty A., Bean R. J., Berntsen P., Bielecki J., Boutet S. et al., “Coherent diffraction of single Rice Dwarf virus particles using hard X-rays at the Linac Coherent Light Source”, Sci. Data, 3 (2016), 160064 | DOI

[13] Ekeberg T., Svenda M., Abergel C., Maia F. R. N. C., Seltzer V., Claverie J. M., Hantke M., Jonsson O., Nettelblad C., van der Schot G. et al., “Three-Dimensional Reconstruction of the Giant Mimivirus Particle with an X-Ray Free-Electron Laser”, Physical Review Letters, 114 (2015), 098102 | DOI

[14] Song C., Jiang H., Mancuso A., Amirbekian B., Peng L., Sun R., Shah S. S., Zhou Z. H., Ishikawa T., Miao J., “Quantitative Imaging of Single, Unstained Viruses with Coherent X Rays”, Physical Review Letters, 101 (2008), 158101 | DOI

[15] Seibert M. M., Ekeberg T., Maia F. R. N. C., Svenda M., Andreasson J., Jönsson O., Odic D., Iwan B., Rocker A., Westphall D., “Single mimivirus particles intercepted and imaged with an X-ray laser”, Nature, 470 (2011), 78–82 | DOI

[16] Van der Schot G., Svenda M., Maia F. R. N. C., Hantke M., DePonte D., Seibert M. M., Aquila A., Schulz J., Kirian R., Liang M. et al., “Imaging single cells in a beam of live cyanobacteria with an X-ray laser”, Nature Communication, 6 (2015), 5704 | DOI

[17] Fienup J. R., “Reconstruction of an object from the modulus of its Fourier transform”, Optics Letters, 3:1 (1978), 27–29 | DOI

[18] Urzhumtseva L., Klaholz B., Urzhumtsev A., “On effective and optical resolutions of diffraction data sets”, Acta Crystallographica Section D: Biological Crystallography, 69 (2013), 625–634 | DOI

[19] Sayre D., “Some implications of a theorem due to Shannon”, Acta Crystallographica, 5 (1952), 843 | DOI

[20] Lunin V. Y., Lunina N. L., Petrova T. E., Baumstark M. W., Urzhumtsev A. G., “Mask-based approach to phasing of single-particle diffraction data”, Acta Crystallographica Section D: Structural Biology, 72 (2016), 147–157 | DOI

[21] Marchesini S., He H., Chapman H. N., Hau-Riege S. P., Noy A., Howells M. R., Weierstall U., Spence J. H. C., “X-ray image reconstruction from a diffraction pattern alone”, Phys. Rev. B, 68 (2003), 140101(R) | DOI

[22] Maia F. R. N. C., Ekeberg T., Spoel D., Hajdu J., “Hawk: the image reconstruction package for coherent X-ray diffractive imaging”, J. Applied Crystallography, 43 (2010), 1535–1539 | DOI

[23] Bricogne G., “Geometric sources of redundancy in intensity data and their use for phase determination”, Acta Crystallographica Section A: Foundations of Crystallography, 30 (1974), 349–405 | DOI

[24] Rodriguez J. A., Xu R., Chen C.-C., Zou Y., Miao J., “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities”, J. Applied Crystallography, 46 (2013), 312–318 | DOI

[25] Miao J., Kirz J., Sayre D., “The oversampling phasing method”, Acta Crystallographica Section D: Biological Crystallography, 56 (2000), 1312–1315 | DOI

[26] He H., Su W. P., “Direct phasing of protein crystals with high solvent content”, Acta Crystallographica Section A: Foundations of Crystallography, 71 (2015), 92–98 | DOI

[27] Bricogne G., “Methods and programs for direct-space exploitation of geometric redundancies”, Acta Crystallographica Section A: Foundations of Crystallography, 32 (1976), 832–847 | DOI

[28] Marchesini S., “A unified evaluation of iterative projection algorithms for phase retrieval”, Rev. Sci. Instrum., 78 (2007), 011301 | DOI

[29] Zhang K. Y. J., Cowtan K. D., Main P., “Phase improvement by iterative density modification”, International Tables for Crystallography, v. F, Second Edition, eds. Arnold E., Himmel D. M., Rossmann M. G., John Wiley and Sons, Chichester, 2012, 385–400 | DOI

[30] Millane R., Lo V. L., “Iterative projection algorithms in protein crystallography. I. Theory”, Acta Crystallographica Section A: Foundations of Crystallography, 69 (2013), 517–527 | DOI | MR

[31] Wang B. C., “Resolution of phase ambiguity in macromolecular crystallography”, Methods in Enzymology, 115 (1985), 90–112 | DOI

[32] Loh N.-T. D., Elser V., “Reconstruction algorithm for single-particle diffraction imaging experiments”, Physical Review E, 80 (2009), 026705 | DOI

[33] Elser V., “Solution of the crystallographic phase problem by iterated projections”, Acta Crystallographica Section A: Foundations of Crystallography, 59 (2003), 201–209 | DOI

[34] Lunin V. Y., Urzhumtsev A. G., Skovoroda T. P., “Direct low-resolution phasing from electron-density histograms in protein crystallography”, Acta Crystallographica Section A: Foundations of Crystallography, 46 (1990), 540–544 | DOI

[35] Lunin V. Y., Lunina N. L., Urzhumtsev A. G., “Connectivity properties of high-density regions and ab initio phasing at low resolution”, Acta Crystallographica Section A: Foundations of Crystallography, 56 (2000), 375–382 | DOI

[36] Petrova T. E., Lunin V. Y., Podjarny A. D., “Ab initio low-resolution phasing in crystallography of macromolecules by maximization of likelihood”, Acta Crystallographica Section D: Biological Crystallography, 56 (2000), 1245–1252 | DOI

[37] Lunin V. Yu., Lunina N. L., Petrova T. E., “Ispolzovanie svyaznykh masok v zadache vosstanovleniya izobrazheniya izolirovannoi chastitsy po dannym rentgenovskogo rasseyaniya”, Matematicheskaya biologiya i bioinformatika, 9:2 (2014), 543–562 | DOI

[38] Baker D., Krukowski A. E., Agard D. A., “Uniqueness and the ab initio phase problem in macromolecular crystallography”, Acta Crystallographica Section D: Biological Crystallography, 49 (1993), 186–192 | DOI

[39] Lunin V. Y., Urzhumtsev A. G., “Improvement of protein phases by coarse model modification”, Acta Crystallographica Section A: Foundations of Crystallography, 40 (1984), 269–277 | DOI

[40] Lunin V. Y., Skovoroda T. P., “R-free likelihood-based estimates of errors for phases calculated from atomic models”, Acta Crystallographica Section A: Foundations of Crystallography, 51 (1995), 880–887 | DOI

[41] Urzhumtsev A. G., Skovoroda T. P., Lunin V. Y., “A procedure compatible with X-PLOR for the calculation of electron-density maps weighted using an R-free-likelihood approach”, J. Applied Crystallography, 29 (1996), 741–744 | DOI

[42] Read R. J., “Improved Fourier Coefficients for Maps Using Phases from Partial Structures with Errors”, Acta Crystallographica Section A: Foundations of Crystallography, 42 (1986), 140–190 | DOI

[43] Broser M., Gabdulkhakov A., Kern J., Guskov A., Müh F., Saenger W., Zouni A., “Crystal structure of monomeric Photosystem II from Thermosynechococcus elongatus at 3.6 Å resolution”, J. Biol. Chem., 285 (2010), 26255–26262 | DOI

[44] Berman H. M., Westbrook J., Feng Z., Gilliland G., Bhat T. N., Weissig H., Shindyalov I. N., Bourne P. E., “The Protein Data Bank”, Nucleic Acids Research, 28 (2000), 235–242 | DOI

[45] Matthews B. M., “Solvent content of protein crystals”, Journal of Molecular Biology, 33 (1968), 491–497 | DOI

[46] Weichenberger C. X., Rupp B., “Ten years of probabilistic estimates of biocrystal solvent content: new insights via nonparametric kernel density estimate”, Acta Crystallographica Section D: Biological Crystallography, 70 (2014), 1579–1588 | DOI

[47] Urzhumtsev A., Afonine P. V., Adams P. D., “On the use of logarithmic scales for analysis of diffraction data”, Acta Crystallographica Section D: Biological Crystallography, 65 (2009), 1283–1291 | DOI