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2018 | OriginalPaper | Chapter

9. Structure Determination by Continuous Diffraction from Imperfect Crystals

Authors : Kartik Ayyer, Oleksandr M. Yefanov, Henry N. Chapman

Published in: X-ray Free Electron Lasers

Publisher: Springer International Publishing

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Abstract

The coherent diffraction pattern of a non-periodic finite object does not consist of Bragg peaks but is continuously and smoothly varying. Such patterns do not suffer from the well-known phase problem of crystallography. In this case, robust iterative algorithms exist to determine the electron density of the object from the diffraction pattern alone. Continuous diffraction is accessible from ensembles of aligned molecules, including disordered protein crystals. We discuss the application of the concepts of coherent diffractive imaging to such cases and describe the experimental considerations to adequately measure the weak continuous diffraction signals.

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previous chapter Phasing Serial Crystallography Data

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Throughout this chapter we will use the indices abc to uniquely distinguish atom a of rigid body b in unit cell c. When it is not needed to report on which body or cell an atom is part of, we just use the index i.

Although \({\mathbf {C}}_{ac\,a'c'}\) runs over four subscripts, this is really two dimensional, since any given atom in the crystal is specified by the indices ac (or a′c′) specifying which atom a in the molecule and which unit cell c in the crystal.

Ayyer, K., Yefanov, O. M., Oberthür, D., Roy-Chowdhury, S., Galli, L., Mariani, V., et al. (2016). Macromolecular diffractive imaging using imperfect crystals. Nature, 530, 202–206.CrossRef

Bates, R. H. T. (1982). Fourier phase problems are uniquely solvable in more than one dimension: 1. Underlying theory. Optik, 61, 247–262.

Bernal, J. D., Fankuchen, I., & Perutz, M. (1938). An X-ray study of chymotrypsin and hæmoglobin. Nature, 141, 523–524.CrossRef

Bragg, W. L., & Perutz, M. F. (1952). The structure of hæmoglobin. Proceedings of the Royal Society of London, 213, 425–435.

Bruck, Y., & Sodin, L. (1979). On the ambiguity of the image reconstruction problem. Optics Communication, 30, 304–308.CrossRef

Caleman, C., Tîmneanu, N., Martin, A. V., Jönsson, H. O., Aquila, A., Barty, A., et al. (2015). Ultrafast self-gating Bragg diffraction of exploding nanocrystals in an X-ray laser. Optics Express, 23, 1213–1231.CrossRef

Chapman, H. N., Barty, A., Marchesini, S., Noy, A., Hau-Riege, S. P., Cui, C., et al. (2006). High-resolution ab initio three-dimensional X-ray diffraction microscopy. Journal of the Optical Society of America A, 23, 1179–1200.CrossRef

Chapman, H. N., Yefanov, O. M., Ayyer, K., White, T. A., Barty, A., Morgan, A., et al. (2017). Continuous diffraction of molecules and disordered molecular crystals. Journal of Applied Crystallography, 50, 1084–1103.CrossRef

Clarage, J. B., Clarage, M. S., Phillips, W. C., Sweet, R. M., & Caspar, D. L. D. (1992). Correlations of atomic movements in lysozyme crystals. Proteins: Structure, Function, and Bioinformatics, 12(2), 145–157.CrossRef

10.

Cowley, J. M. (1981). Diffraction physics. Amsterdam: North-Holland.

11.

Cowtan, K. (1998). Introduction to density modification. In Direct methods for solving macromolecular structures. Dordrecht: Springer.CrossRef

12.

Crimmins, T. R., Fienup, J., & Thelen, B. J. (1990). Improved bounds on object support from autocorrelation support and application to phase retrieval. Journal of the Optical Society of America A, 7, 3–13.CrossRef

13.

Crowther, R., DeRosier, D., & Klug, A. (1970). The reconstruction of a three-dimensional structure from its projections and its applications to electron microscopy. Proceedings of the Royal Society of London, 317, 319–340.

14.

Elser, V. (2003). Phase retrieval by iterated projections. Journal of the Optical Society of America A, 20, 40–55.CrossRef

15.

Elser, V. (2013). Direct phasing of nanocrystal diffraction. Acta Crystallographica Section A, 69, 559–569.CrossRef

16.

Elser, V., & Millane, R. P. (2008). Reconstruction of an object from its symmetry-averaged diffraction pattern. Acta Crystallographica Section A, 64, 273–279.CrossRef

17.

Fienup, J. R. (1978). Reconstruction of an object from the modulus of its Fourier transform. Optics Letters, 3, 27–29.CrossRef

18.

Fienup, J. R. (1982). Phase retrieval algorithms: a comparison. Applied Optics, 21, 2758–2769.CrossRef

19.

Flewett, S., Quiney, H. M., Tran, C. Q., & Nugent, K. A. (2009). Extracting coherent modes from partially coherent wavefields. Optics Letters, 34, 2198–2200.CrossRef

20.

French, S., & Wilson, K. (1978). On the treatment of negative intensity observations. Acta Crystallographica Section A, 34, 517–525.CrossRef

21.

Gerchberg, R. W., & Saxton, O. (1972). Practical algorithm for determination of phase from image and diffraction plane pictures. Optik, 35, 237–246.

22.

He, H., & Su, W.-P. (2015). Direct phasing of protein crystals with high solvent content. Acta Crystallographica Section A, 71, 92–98.CrossRef

23.

He, H., Fang, H., Miller, M. D., Phillips, G. N. Jr., & Su, W.-P. (2016). Improving the efficiency of molecular replacement by utilizing a new iterative transform phasing algorithm. Acta Crystallographica Section A, 72, 539–547.CrossRef

24.

Hensley, C. J., Yang, J., & Centurion, M. (2012). Imaging of isolated molecules with ultrafast electron pulses. Physical Review Letters, 109, 133, 202.

25.

Howells, M. R., Beetz, T., Chapman, H. N., Cui, C., Holton, J. M., Jacobsen, C. J., et al. (2009). An assessment of the resolution limitation due to radiation-damage in X-ray diffraction microscopy. Journal of Electron Spectroscopy and Related Phenomena, 170, 4–12.CrossRef

26.

Kabsch, W. (2010). Integration, scaling, space-group assignment and post-refinement. Acta Crystallographica Section D, 66, 133–144.CrossRef

27.

Kewish, C. M., Thibault, P., Bunk, O., & Pfeiffer, F. (2010). The potential for two-dimensional crystallography of membrane proteins at future X-ray free-electron laser sources. New Journal of Physics, 12, 035,005.CrossRef

28.

Marchesini, S. (2007). A unified evaluation of iterative projection algorithms for phase retrieval. The Review of Scientific Instruments, 78, 011301.CrossRef

29.

Marchesini, S., He, H., Chapman, H. N., Hau-Riege, S. P., Noy, A., Howells, M. R., et al. (2003). X-ray image reconstruction from a diffraction pattern alone. Physical Review B, 68, 140,101.CrossRef

30.

Miao, J., Sayre, D., & Chapman, H. N. (1998). Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects. Journal of the Optical Society of America A, 15, 1662–1669.CrossRef

31.

Miao, J., Charalambous, P., Kirz, J., & Sayre, D. (1999). Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens. Nature, 400, 342–344.CrossRef

32.

Millane, R. P. (1990). Phase retrieval in crystallography and optics. Journal of the Optical Society of America A, 7, 394–411.CrossRef

33.

Millane, R. P. (2017). The phase problem for one-dimensional crystals. Acta Crystallographica Section A, 73, 140–150.CrossRef

34.

Millane, R. P., & Lo, V. L. (2013). Iterative projection algorithms in protein crystallography. I. Theory. Acta Crystallographica Section A, 69, 517–527.CrossRef

35.

Mizuguchi, K., Kidera, A., & Gō, N. (1994). Collective motions in proteins investigated by X-ray diffuse scattering. Proteins: Structure, Function, and Bioinformatics, 18, 34–48.CrossRef

36.

Moore, P. B. (2009). On the relationship between diffraction patterns and motions in macromolecular crystals. Structure, 17, 1307–1315.CrossRef

37.

Oberthuer, D., Knoška, J., Wiedorn, M. O., Beyerlein, K. R., Bushnell, D. A., Kovaleva, E. G., et al. (2017). Double-flow focused liquid injector for efficient serial femtosecond crystallography. Scientific Reports, 7, 44628.CrossRef

38.

Oszlanyi, G., & Suto, A. (2004). Ab initio structure solution by charge flipping. Acta Crystallographica Section A, 60, 134–141.

39.

Peck, A., Poitevin, F., & Lane, T. J. (2018). Intermolecular correlations are necessary to explain diffuse scattering from protein crystals. IUCrJ, 5, 211–222.CrossRef

40.

Rees, D. C. (1980). The influence of twinning by merohedry on intensity statistics. Acta Crystallographica Section A, 36, 578–581.CrossRef

41.

Sayre, D. (2002). X-ray crystallography: The past and present of the phase problem. Structural Chemistry, 13, 81–96.CrossRef

42.

Schmidt, E., & Neder, R. B. (2017). Diffuse single-crystal scattering corrected for molecular form factor effects. Acta Crystallographica Section A, 73, 231–237.CrossRef

43.

Shannon, C. E. (1949). Communication in the presence of noise. Proceedings of the IRE, 37, 10–21.CrossRef

44.

Shapiro, D., Thibault, P., Beetz, T., Elser, V., Howells, M., Jacobsen, C., et al. (2005). Biological imaging by soft X-ray diffraction microscopy. Proceedings of the National Academy of Sciences, 102, 15343–15346.CrossRef

45.

Simonov, A., Weber, T., & Steurer, W. (2014). Experimental uncertainties of three-dimensional pair distribution function investigations exemplified on the diffuse scattering from a tris-tert-butyl-1,3,5-benzene tricarboxamide single crystal. Journal of Applied Crystallography, 47, 2011–2018.CrossRef

46.

Simonov, A., Weber, T., & Goodwin, A. (2017). Single crystal diffuse scattering—A solution to the phase problem? Acta Crystallographica Section A, 73, C1045.CrossRef

47.

Spence, J. C. H., Weierstall, U., Fricke, T. T., Glaeser, R. M., & Downing, K. H. (2003). Three-dimensional diffractive imaging for crystalline monolayers with one-dimensional compact support. Journal of Structural Biology, 144, 209–218.CrossRef

48.

Spence, J. C. H., & Doak, R. B. (2004). Single molecule diffraction. Physical Review Letters, 92, 198102.CrossRef

49.

Spence, J. C. H., Kirian, R. A., Wang, X., Weierstall, U., Schmidt, K. E., White, T. et al. (2011) Phasing of coherent femtosecond X-ray diffraction from size-varying nanocrystals. Optics Express, 19, 2866–2873.CrossRef

50.

Stroud, R. M., & Agard, D. A. (1979). Structure determination of asymmetric membrane profiles using an iterative Fourier method. Biophysical Journal, 25, 495–512.CrossRef

51.

Szoke, A. (1999). Time-resolved holographic diffraction at atomic resolution. Chemical Physics Letters, 313, 778–788.CrossRef

52.

Szoke, A. (2001). Diffraction of partially coherent X-rays and the crystallographic phase problem. Acta Crystallographica Section A, 57, 586–603.CrossRef

53.

von Laue, M. (1936). The external shape of crystals and its influence on interference phenomena in crystalline lattices. Annales de Physique, 26, 55–68.CrossRef

54.

Waasmaier, D., & Kirfel, A. (1995). New analytical scattering-factor functions for free atoms and ions. Acta Crystallographica Section A, 51, 416.

55.

Welberry, T. R. (1985). Diffuse X-ray scattering models of disorder. Reports on Progress in Physics, 48, 1543–1593.CrossRef

56.

White, T. A., Kirian, R. A., Martin, A. V., Aquila, A., Nass, K., Barty, A., et al. (2012). CrystFEL: a software suite for snapshot serial crystallography. Journal of Applied Crystallography, 45, 335–341.CrossRef

57.

White, T. A., Mariani, V., Brehm, W., Yefanov, O., Barty, A., Beyerlein, K. R., et al. (2016). Recent developments in CrystFEL. Journal of Applied Crystallography, 49, 680–689.CrossRef

58.

Whitehead, L. W., Williams, G. J., Quiney, H. M., Vine, D. J., Dilanian, R. A., Flewett, S., et al. (2009). Diffractive imaging using partially coherent X rays. Physical Review Letters, 103, 243902.CrossRef

59.

Wilson, A. J. C. (1949). The probability distribution of X-ray intensities. Acta Crystallographica, 2, 318–321.CrossRef

60.

Yefanov, O., Gati, O., Bourenkov, G., Kirian, R. A., White, T. A., Spence, J. C. H., et al. (2014). Mapping the continuous reciprocal space intensity distribution of X-ray serial crystallography. Philosophical Transactions of the Royal Society B, 369, 1647.CrossRef

61.

Yefanov, O., Mariani, V., Gati, C., White, T. A., Chapman, H. N., & Barty, A. (2015). Accurate determination of segmented X-ray detector geometry. Optics Express, 23, 28459–28470.CrossRef

Title: Structure Determination by Continuous Diffraction from Imperfect Crystals
Authors: Kartik Ayyer
Oleksandr M. Yefanov
Henry N. Chapman
Publisher: Springer International Publishing
Book: X-ray Free Electron Lasers
Print ISBN: 978-3-030-00550-4

Electronic ISBN: 978-3-030-00551-1

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-030-00551-1_9