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Structure Determination by Continuous Diffraction from Imperfect Crystals

  • Kartik Ayyer
  • Oleksandr M. Yefanov
  • Henry N. ChapmanEmail author
Chapter

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.

Notes

Acknowledgements

We acknowledge the Gottfried Wilhelm Leibniz Program of the DFG, and the European Research Council under the European Union’s Seventh Framework Programme ERC Synergy Grant 609920 “AXSIS.”

References

  1. 1.
    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.CrossRefGoogle Scholar
  2. 2.
    Bates, R. H. T. (1982). Fourier phase problems are uniquely solvable in more than one dimension: 1. Underlying theory. Optik, 61, 247–262.Google Scholar
  3. 3.
    Bernal, J. D., Fankuchen, I., & Perutz, M. (1938). An X-ray study of chymotrypsin and hæmoglobin. Nature, 141, 523–524.CrossRefGoogle Scholar
  4. 4.
    Bragg, W. L., & Perutz, M. F. (1952). The structure of hæmoglobin. Proceedings of the Royal Society of London, 213, 425–435.Google Scholar
  5. 5.
    Bruck, Y., & Sodin, L. (1979). On the ambiguity of the image reconstruction problem. Optics Communication, 30, 304–308.CrossRefGoogle Scholar
  6. 6.
    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.CrossRefGoogle Scholar
  7. 7.
    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.CrossRefGoogle Scholar
  8. 8.
    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.CrossRefGoogle Scholar
  9. 9.
    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.CrossRefGoogle Scholar
  10. 10.
    Cowley, J. M. (1981). Diffraction physics. Amsterdam: North-Holland.Google Scholar
  11. 11.
    Cowtan, K. (1998). Introduction to density modification. In Direct methods for solving macromolecular structures. Dordrecht: Springer.CrossRefGoogle Scholar
  12. 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.CrossRefGoogle Scholar
  13. 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.Google Scholar
  14. 14.
    Elser, V. (2003). Phase retrieval by iterated projections. Journal of the Optical Society of America A, 20, 40–55.CrossRefGoogle Scholar
  15. 15.
    Elser, V. (2013). Direct phasing of nanocrystal diffraction. Acta Crystallographica Section A, 69, 559–569.CrossRefGoogle Scholar
  16. 16.
    Elser, V., & Millane, R. P. (2008). Reconstruction of an object from its symmetry-averaged diffraction pattern. Acta Crystallographica Section A, 64, 273–279.CrossRefGoogle Scholar
  17. 17.
    Fienup, J. R. (1978). Reconstruction of an object from the modulus of its Fourier transform. Optics Letters, 3, 27–29.CrossRefGoogle Scholar
  18. 18.
    Fienup, J. R. (1982). Phase retrieval algorithms: a comparison. Applied Optics, 21, 2758–2769.CrossRefGoogle Scholar
  19. 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.CrossRefGoogle Scholar
  20. 20.
    French, S., & Wilson, K. (1978). On the treatment of negative intensity observations. Acta Crystallographica Section A, 34, 517–525.CrossRefGoogle Scholar
  21. 21.
    Gerchberg, R. W., & Saxton, O. (1972). Practical algorithm for determination of phase from image and diffraction plane pictures. Optik, 35, 237–246.Google Scholar
  22. 22.
    He, H., & Su, W.-P. (2015). Direct phasing of protein crystals with high solvent content. Acta Crystallographica Section A, 71, 92–98.CrossRefGoogle Scholar
  23. 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.CrossRefGoogle Scholar
  24. 24.
    Hensley, C. J., Yang, J., & Centurion, M. (2012). Imaging of isolated molecules with ultrafast electron pulses. Physical Review Letters, 109, 133, 202.Google Scholar
  25. 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.CrossRefGoogle Scholar
  26. 26.
    Kabsch, W. (2010). Integration, scaling, space-group assignment and post-refinement. Acta Crystallographica Section D, 66, 133–144.CrossRefGoogle Scholar
  27. 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.CrossRefGoogle Scholar
  28. 28.
    Marchesini, S. (2007). A unified evaluation of iterative projection algorithms for phase retrieval. The Review of Scientific Instruments, 78, 011301.CrossRefGoogle Scholar
  29. 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.CrossRefGoogle Scholar
  30. 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.CrossRefGoogle Scholar
  31. 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.CrossRefGoogle Scholar
  32. 32.
    Millane, R. P. (1990). Phase retrieval in crystallography and optics. Journal of the Optical Society of America A, 7, 394–411.CrossRefGoogle Scholar
  33. 33.
    Millane, R. P. (2017). The phase problem for one-dimensional crystals. Acta Crystallographica Section A, 73, 140–150.CrossRefGoogle Scholar
  34. 34.
    Millane, R. P., & Lo, V. L. (2013). Iterative projection algorithms in protein crystallography. I. Theory. Acta Crystallographica Section A, 69, 517–527.CrossRefGoogle Scholar
  35. 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.CrossRefGoogle Scholar
  36. 36.
    Moore, P. B. (2009). On the relationship between diffraction patterns and motions in macromolecular crystals. Structure, 17, 1307–1315.CrossRefGoogle Scholar
  37. 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.CrossRefGoogle Scholar
  38. 38.
    Oszlanyi, G., & Suto, A. (2004). Ab initio structure solution by charge flipping. Acta Crystallographica Section A, 60, 134–141.Google Scholar
  39. 39.
    Peck, A., Poitevin, F., & Lane, T. J. (2018). Intermolecular correlations are necessary to explain diffuse scattering from protein crystals. IUCrJ, 5, 211–222.CrossRefGoogle Scholar
  40. 40.
    Rees, D. C. (1980). The influence of twinning by merohedry on intensity statistics. Acta Crystallographica Section A, 36, 578–581.CrossRefGoogle Scholar
  41. 41.
    Sayre, D. (2002). X-ray crystallography: The past and present of the phase problem. Structural Chemistry, 13, 81–96.CrossRefGoogle Scholar
  42. 42.
    Schmidt, E., & Neder, R. B. (2017). Diffuse single-crystal scattering corrected for molecular form factor effects. Acta Crystallographica Section A, 73, 231–237.CrossRefGoogle Scholar
  43. 43.
    Shannon, C. E. (1949). Communication in the presence of noise. Proceedings of the IRE, 37, 10–21.CrossRefGoogle Scholar
  44. 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.CrossRefGoogle Scholar
  45. 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.CrossRefGoogle Scholar
  46. 46.
    Simonov, A., Weber, T., & Goodwin, A. (2017). Single crystal diffuse scattering—A solution to the phase problem? Acta Crystallographica Section A, 73, C1045.CrossRefGoogle Scholar
  47. 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.CrossRefGoogle Scholar
  48. 48.
    Spence, J. C. H., & Doak, R. B. (2004). Single molecule diffraction. Physical Review Letters, 92, 198102.CrossRefGoogle Scholar
  49. 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.CrossRefGoogle Scholar
  50. 50.
    Stroud, R. M., & Agard, D. A. (1979). Structure determination of asymmetric membrane profiles using an iterative Fourier method. Biophysical Journal, 25, 495–512.CrossRefGoogle Scholar
  51. 51.
    Szoke, A. (1999). Time-resolved holographic diffraction at atomic resolution. Chemical Physics Letters, 313, 778–788.CrossRefGoogle Scholar
  52. 52.
    Szoke, A. (2001). Diffraction of partially coherent X-rays and the crystallographic phase problem. Acta Crystallographica Section A, 57, 586–603.CrossRefGoogle Scholar
  53. 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.CrossRefGoogle Scholar
  54. 54.
    Waasmaier, D., & Kirfel, A. (1995). New analytical scattering-factor functions for free atoms and ions. Acta Crystallographica Section A, 51, 416.Google Scholar
  55. 55.
    Welberry, T. R. (1985). Diffuse X-ray scattering models of disorder. Reports on Progress in Physics, 48, 1543–1593.CrossRefGoogle Scholar
  56. 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.CrossRefGoogle Scholar
  57. 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.CrossRefGoogle Scholar
  58. 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.CrossRefGoogle Scholar
  59. 59.
    Wilson, A. J. C. (1949). The probability distribution of X-ray intensities. Acta Crystallographica, 2, 318–321.CrossRefGoogle Scholar
  60. 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.CrossRefGoogle Scholar
  61. 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.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Kartik Ayyer
    • 1
  • Oleksandr M. Yefanov
    • 1
  • Henry N. Chapman
    • 1
    • 2
    • 3
    Email author
  1. 1.Center for Free-Electron Laser Science, DESYHamburgGermany
  2. 2.Department of PhysicsUniversity of HamburgHamburgGermany
  3. 3.Centre for Ultrafast ImagingUniversity of HamburgHamburgGermany

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