The Use of Angular Correlations to Extract Three-Dimensional Structural Information from X-Ray Solution Scattering

  • Sebastian DoniachEmail author


In this chapter we focus on scattering from non-crystalline solutions of molecules or nanoparticles in which the scattering objects are rotationally disordered.



Thanks to Derek Mendez, Shenglan Qiao, Kevin Raines, Gundolf Schenk and Herschel Watkins who contributed to the work reported here. Partial support came from NIH grant 1R01GM09746301A1.


  1. 1.
    Ansari, A., Berendzen, J., Bowne, S.F., Frauenfelder, H., Iben, I. E. T., Sauke, T. B., et al. (1985). Protein states and protein quakes. Proceedings of the National Academy of Sciences of the United States of America, 82(15), 5000–5004.CrossRefGoogle Scholar
  2. 2.
    Arnlund, D., Johansson, L. C., Wickstrand, C., Barty, A., Williams, G. J., Malmerberg, E., et al. (2014). Visualizing a protein quake with time-resolved X-ray scattering at a free-electron laser. Nature Methods, 11(9), 923–926.CrossRefGoogle Scholar
  3. 3.
    Aroian, L. A., Taneja, V. S., & Cornwell, L. W. (1978). Mathematical forms of the distribution of the product of two normal variables: Mathematical forms of the distribution. Communications in Statistics - Theory and Methods, 7, 165–172.CrossRefGoogle Scholar
  4. 4.
    Brinkmann, L. U. L., & Hub, J. S. (2016). Ultrafast anisotropic protein quake propagation after CO photodissociation in myoglobin. Proceedings of the National Academy of Sciences of the United States of America, 113(38), 10565–10570.CrossRefGoogle Scholar
  5. 5.
    Chacon, P., Moran, F., Diaz, J. F., Pantos, E., & Andreu, J. M. (1998). Low-resolution structures of proteins in solution retrieved from X-ray scattering with a genetic algorithm. Biophysical Journal, 74(6), 2760–2775.CrossRefGoogle Scholar
  6. 6.
    Donatelli, J. J., Sethian, J. A., & Zwart, P. H. (2017). Reconstruction from limited single-particle diffraction data via simultaneous determination of state, orientation, intensity, and phase. Proceedings of the National Academy of Sciences of the United States of America, 114(28), 7222–7227.CrossRefGoogle Scholar
  7. 7.
    Donatelli, J. J., Zwart, P. H., & Sethian, J. A. (2015). Iterative phasing for fluctuation X-ray scattering. Proceedings of the National Academy of Sciences of the United States of America, 112(33), 10286–10291.CrossRefGoogle Scholar
  8. 8.
    Doniach, S. (2001). Changes in biomolecular conformation seen by small-angle x-ray scattering. Chemical Reviews, 101(6), 1763–1778.CrossRefGoogle Scholar
  9. 9.
    Fienup, J. R. (1978). Reconstruction of an object from modulus of its Fourier-transform. Optics Letters, 3(1), 27–29.CrossRefGoogle Scholar
  10. 10.
    Guinier, A. (1939). la diffraction des rayons x aux tr\(\grave `e\)s petits angles: application \(\grave a\) l’\(\acute e\)tude de phenom\(\grave e\)nes ultramicroscopiques. Annals of Physics, 12, 161–237.Google Scholar
  11. 11.
    Howie, A., & Marks, L. D. (1984). Elastic Strains and the energy-balance for multiply twinned particles. Philosophical Magazine A-Physics of Condensed Matter Structure Defects and Mechanical Properties, 49(1), 95–109.Google Scholar
  12. 12.
    Kam, Z. (1977). Determination of macromolecular structure in solution by spatial correlation of scattering fluctuations. Macromolecules, 10(5), 927–934.CrossRefGoogle Scholar
  13. 13.
    Kam, Z., Koch, M. H. J., & Bordas, J. (1981). Fluctuation X-ray-scattering from biological particles in frozen solution by using synchrotron radiation. Proceedings of the National Academy of Sciences of the United States of America-Biological Sciences, 78(6), 3559–3562.CrossRefGoogle Scholar
  14. 14.
    Kirian, R. A., Schmidt, K. E., Wang, X., Doak, R. B., & Spence, J. C. H. (2011). Signal, noise, and resolution in correlated fluctuations from snapshot small-angle x-ray scattering. Physical Review E, 84(1), 011921.CrossRefGoogle Scholar
  15. 15.
    Kratky, O., & Sekora, A. (1943). Regulation of the shape and size of separated particles from the smallest lower angle diffuse inflected x-rays. Naturwissenschaften, 31, 46–47.CrossRefGoogle Scholar
  16. 16.
    Levantino, M., Schiro, G., Lemke, H. T., Cottone, G., Glownia, J. M., Zhu, D., et al. (2015). Ultrafast myoglobin structural dynamics observed with an X-ray free-electron laser. Nature Communications, 6, Article number: 6772.Google Scholar
  17. 17.
    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(14), 140101–140104CrossRefGoogle Scholar
  18. 18.
    Mendez, D., Lane, T. J., Sung, J., Sellberg, J., Levard, C., Watkins, H., et al. (2014). Observation of correlated X-ray scattering at atomic resolution. Philosophical Transactions of the Royal Society B-Biological Sciences, 369(1647). CrossRefGoogle Scholar
  19. 19.
    Mendez, D., Watkins, H., Qiao, S., Raines, K. S., Lane, T. J., & Schenk, G. (2016). Angular correlations of photons from solution diffraction at a free-electron laser encode molecular structure. IUCRJ, 3(6), 420–429.CrossRefGoogle Scholar
  20. 20.
    Qiao, S., Hilger, D., Fonseca, R., Mendez, D., & Doniach S. (submitted). Structural details of a protein in a non-crystalline solution revealed by angular correlations of scattered xFEL photons. Optics Express.Google Scholar
  21. 21.
    Svergun, D. I. (1999). Restoring low resolution structure of biological macromolecules from solution scattering using simulated annealing. Biophysical Journal, 76(6), 2879–2886.CrossRefGoogle Scholar
  22. 22.
    Svergun, D. I., & Stuhrmann, H. B. (1991). New developments in direct shape determination from small-angle scattering . 1. Theory and model-calculations. Acta Crystallographica A, 47, 736–744.CrossRefGoogle Scholar
  23. 23.
    Walther, D., Cohen, F. E., & Doniach, S. (2000). Reconstruction of low-resolution three-dimensional density maps from one-dimensional small-angle X-ray solution scattering data for biomolecules. Journal of Applied Crystallography, 33(2), 350–363.CrossRefGoogle Scholar
  24. 24.
    Yang, C. Y. (1979). Crystallography of decahedral and Icosahedral Particles .1. Geometry of twinning. Journal of Crystal Growth, 47(2), 274–282.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departments of Applied Physics, Physics, and Photon ScienceStanford UniversityStanfordUSA
  2. 2.SLAC National Accelerator CenterMenlo ParkUSA

Personalised recommendations