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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 2866–2873
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Phasing of coherent femtosecond X-ray diffraction from size-varying nanocrystals

John C. H. Spence, Richard A. Kirian, Xiaoyu Wang, Uwe Weierstall, Kevin E. Schmidt, Thomas White, Anton Barty, Henry N. Chapman, Stefano Marchesini, and James Holton  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 2866-2873 (2011)
http://dx.doi.org/10.1364/OE.19.002866


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Abstract

The scattering between Bragg reflections from nanocrystals is used to aid solution of the phase problem. We describe a method for reconstructing the charge density of a typical molecule within a single unit cell, if sufficiently finely-sampled “snap-shot” diffraction data (as provided a free-electron X-ray laser) are available from many nanocrystals of different sizes lying in random orientations. By using information on the particle-size distribution within the patterns, this digital method succeeds, using all the data, without knowledge of the distribution of particle size or requiring atomic-resolution data.

© 2011 OSA

1. Introduction

We consider the phase problem for scattering from a large number of nanocrystals of varying size, an important problem in the field of protein crystallography. Over the past century, in view of the importance of structure-factor phases in determining atomic structure, each new solution to the crystallographic phase problem has led to major advances in structural biology. Anomolous diffraction, isomorphous replacement, molecular replacement and direct methods are widely used, but all have their limitations (needing chemical modifications to the sample, atomic-resolution data, limits on the number of atoms, or a similar solved protein [1

1. N. Kasai, and M. Kakudo, “X-ray diffraction by macromolecules” (Springer New York 2005).

]). It has been said that the discovery of a genuinely new structure in molecular biology requires phase measurement, rather than phasing by modeling, based on the protein database, since the return of such new structures to the database may contribute to model bias in the database. With the invention of the hard X-ray laser (e.g. Linac Coherent Light Source (LCLS) [2

2. W. A. Barletta, J. Bisognano, J. N. Corlett, P. Emma, Z. Huang, K. J. Kim, R. Lindberg, J. B. Murphy, G. R. Neil, D. C. Nguyen, C. Pellegrini, R. A. Rimmer, F. Sannibale, G. Stupakov, R. P. Walker, and A. A. Zholents, “Free electron lasers: Present status and future challenges,” Nucl. Instrum. Methods A618, 69–96 (2010).

]), new types of diffraction data are becoming available [3

3. R. A. Kirian, X. Y. Wang, U. Weierstall, K. E. Schmidt, J. C. H. Spence, M. Hunter, P. Fromme, T. White, H. N. Chapman, and J. Holton, “Femtosecond protein nanocrystallography-data analysis methods,” Opt. Express 18(6), 5713–5723 (2010). [CrossRef] [PubMed]

,4

4. H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. DePonte, U. Weierstall, R. B. Doak, F. Maia, A. V. Martin, I. Schlichting, L. Lomb, N. Coppola, R. L. Shoeman, S. W. Epp, R. Hartmann, D. Rolles, A. Rudenko, L. Foucar, N. Kimmel, G. Weidenspointner, P. Holl, M. Liang, M. Barthelmess, C. Caleman, S. Boutet, M. J. Bogan, J. Krzywinski, C. Bostedt, S. Bajt, L. Gumprecht, B. Rudek, B. Erk, C. Schmidt, A. Hömke, C. Reich, D. Pietschner, L. Strüder, G. Hauser, H. Gorke, J. Ullrich, S. Herrmann, G. Schaller, F. Schopper, H. Soltau, K. Kühnel, M. Messerschmidt, J. D. Bozek, S. P. Hau-Riege, M. Frank, C. Y. Hampton, R. Sierra, D. Starodub, G. J. Williams, J. Hajdu, N. Timneanu, M. Seibert, J. Andreasson, A. Rocker, O. Jönsson, S. Stern, K. Nass, R. Andritschke, C. Schröter, F. Krasniqi, M. Bott, K. E. Schmidt, X. Wang, I. Grotjohann, J. Holton, S. Marchesini, S. Schorb, D. Rupp, M. Adolph, T. Gorkhover, M. Svenda, H. Hirsemann, G. Potdevin, H. Graafsma, B. Nilsson, and J. C. H. Spence, “Femtosecond X-ray protein nanocrystallography,” Nature . in press. [PubMed]

] which offer new possibilities for direct phasing, not limited in this way. There is a particular need for a direct phasing method which does not require atomic-resolution data.

It has long been appreciated that if scattering could be obtained at points between Bragg reflections, the additional information would greatly assist solution of this problem [5

5. D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5(6), 843 (1952). [CrossRef]

]. For protein crystals, variations in water content have been used to vary the cell dimensions for this purpose [6

6. M. F. Perutz, “The structure of haemoglobin,” Proc. Royal Soc. A264, 264–286 (1954). [CrossRef]

]. Since they contain chiral alpha-helices, protein crystals are acentric, however structure-factors may be real for projections with inversion symmetry. The complex molecular transform “sampled” at the Bragg condition is sufficient to reconstruct the molecular charge density, consistent with the Shannon sampling theorem (we assume one molecule per unit cell throughout). Sampling of intensities at the half-integral Bragg condition completely determines the autocorrelation of the molecular density, from which a unique solution of phases can be determined [7

7. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik (Stuttg.) 61, 247–262 (1982).

]. We show how the finite size of a nanocrystal can provide the inter-Bragg scattering needed to recover the continuous molecular transform between Bragg peaks, so that iterative phasing methods [8

8. J. C. H. Spence, in Science of Microscopy, eds., P. Hawkes and J. C. H. Spence, (Springer, 2006. New York).

] may then be used to solve the phase problem and recover a molecular image. If enough projections could be obtained from the same nanocrystal, one might treat an entire nanocrystal as a single, non-periodic object and phase it using these methods, however radiation damage and the difficulty of orienting sub-micron objects prevents this. We describe here a more efficient approach using large numbers of nanocrystals which does not require goniometry, and which allows for variations in crystal size.

The type of data required might be provided [3

3. R. A. Kirian, X. Y. Wang, U. Weierstall, K. E. Schmidt, J. C. H. Spence, M. Hunter, P. Fromme, T. White, H. N. Chapman, and J. Holton, “Femtosecond protein nanocrystallography-data analysis methods,” Opt. Express 18(6), 5713–5723 (2010). [CrossRef] [PubMed]

] by the new generation of free-electron hard X-ray lasers (e.g. LCLS [2

2. W. A. Barletta, J. Bisognano, J. N. Corlett, P. Emma, Z. Huang, K. J. Kim, R. Lindberg, J. B. Murphy, G. R. Neil, D. C. Nguyen, C. Pellegrini, R. A. Rimmer, F. Sannibale, G. Stupakov, R. P. Walker, and A. A. Zholents, “Free electron lasers: Present status and future challenges,” Nucl. Instrum. Methods A618, 69–96 (2010).

]) in the serial crystallography geometry [9

9. J. C. H. Spence and R. B. Doak, “Single molecule diffraction,” Phys. Rev. Lett. 92(19), 198102 (2004). [CrossRef] [PubMed]

], where a steady stream of hydrated protein nanocrystals, of varying sizes and in random orientations, is sprayed in single file across the femtosecond X-ray beam, and the detector is read out after every pulse, ideally with one nanocrystal per exposure. It is then found that a ten femtosecond X-ray pulse terminates before significant radiation damage occurs to the sample. As a result of the photoelectron cascase which occurs after the incident pulse has terminated (and detection has ceased) the sample is vaporized [4

4. H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. DePonte, U. Weierstall, R. B. Doak, F. Maia, A. V. Martin, I. Schlichting, L. Lomb, N. Coppola, R. L. Shoeman, S. W. Epp, R. Hartmann, D. Rolles, A. Rudenko, L. Foucar, N. Kimmel, G. Weidenspointner, P. Holl, M. Liang, M. Barthelmess, C. Caleman, S. Boutet, M. J. Bogan, J. Krzywinski, C. Bostedt, S. Bajt, L. Gumprecht, B. Rudek, B. Erk, C. Schmidt, A. Hömke, C. Reich, D. Pietschner, L. Strüder, G. Hauser, H. Gorke, J. Ullrich, S. Herrmann, G. Schaller, F. Schopper, H. Soltau, K. Kühnel, M. Messerschmidt, J. D. Bozek, S. P. Hau-Riege, M. Frank, C. Y. Hampton, R. Sierra, D. Starodub, G. J. Williams, J. Hajdu, N. Timneanu, M. Seibert, J. Andreasson, A. Rocker, O. Jönsson, S. Stern, K. Nass, R. Andritschke, C. Schröter, F. Krasniqi, M. Bott, K. E. Schmidt, X. Wang, I. Grotjohann, J. Holton, S. Marchesini, S. Schorb, D. Rupp, M. Adolph, T. Gorkhover, M. Svenda, H. Hirsemann, G. Potdevin, H. Graafsma, B. Nilsson, and J. C. H. Spence, “Femtosecond X-ray protein nanocrystallography,” Nature . in press. [PubMed]

]. By comparison with single-molecule diffraction, perfect nanocrystals increase peak (Bragg) intensity by N6 (for N molecules on a side), provide a solution to the molecular orientation problem through Miller indexing, and provide a path to single-molecule imaging as N is reduced by filtration, while also allowing study of proteins which fail to produce large crystals [10

10. C. Riekel. “Recent developments in microdiffraction on protein crystals,” J. Synchrontron Radiat . 11, 4–6 (2004). Also R. Von Dreele (2009) Personal communication: Results of powder X-ray diffraction from growth media. [CrossRef]

]. Using short pulses instead of freezing to reduce damage allows structural analysis and in situ “snap-shot chemistry” at room temperature, while the continuous flow of fresh material allows pump-probe study of irreversible processes. Finally, we show here that this geometry offers an efficient direct solution to the phase problem. In the simplest optical terms, we show that a finite grating solves the phase problem.

For a finite crystallite, the sharp Bragg reflections are convoluted with a crystal shape-transform function, and the result is then modulated by the molecular transform [11

11. M. von Laue, “The external shape of crystals and its influence on interference phenomena in crystalline lattices,” Ann. Phys. 26, 55–68 (1936).

]. The shape-transform therefore provides the required inter-Bragg scattering needed to solve the phase problem. (This has been used for the study of strain in inorganic nanocrystals recently [12

12. I. K. Robinson and R. Harder, “Coherent X-ray diffraction imaging of strain at the nanoscale,” Nat. Mater. 8(4), 291–298 (2009). [CrossRef] [PubMed]

]). Our simulations (Fig. 1(a)
Fig. 1 (a) Simulated snap-shot diffraction pattern for X-rays in transmission through a sub-micron protein nanocrystal in a random orientation for some inner reflections near the origin. Shapes are partial reflections formed by 2D intersection of Ewald sphere with 3D Fourier Transform of shape of nanocrystal near Bragg conditions. (b) |F(Δk)| for 1AKG protein on [001] in q-space, resolution d = 0.523 nm at side of pattern, 6 samples between Bragg spots.
) show resulting “partial” reflections [13

13. M. G. Rossmann and J. W. Erickson, “Oscillation photography of radiation-sensitive crystals using a synchrotron source,” J. Appl. Cryst. 16(6), 629–636 (1983). [CrossRef]

], in random orientations, each formed by one slice of the Ewald sphere through the three-dimensional shape transform at each lattice point [3

3. R. A. Kirian, X. Y. Wang, U. Weierstall, K. E. Schmidt, J. C. H. Spence, M. Hunter, P. Fromme, T. White, H. N. Chapman, and J. Holton, “Femtosecond protein nanocrystallography-data analysis methods,” Opt. Express 18(6), 5713–5723 (2010). [CrossRef] [PubMed]

]. As for a finite grating, the number of unit cells N between crystal facets along direction g is equal to p + 2, if there are p subsidiary fringes between Bragg reflections along direction g.

2. Phasing nanocrystals

Consider diffraction from a finite parallelepiped crystal, illuminated by a pulse of plane-polarized monochromatic incident radiation, with wavevector k i (|k i| = 1/λ) and small beam divergence. The diffracted photon flux In (photons/pulse/ pixel) at Δk = k i - k o produced by the n-th crystallite, consisting of N(n) = N1 x N2 x N3 unit cells, is given by [3

3. R. A. Kirian, X. Y. Wang, U. Weierstall, K. E. Schmidt, J. C. H. Spence, M. Hunter, P. Fromme, T. White, H. N. Chapman, and J. Holton, “Femtosecond protein nanocrystallography-data analysis methods,” Opt. Express 18(6), 5713–5723 (2010). [CrossRef] [PubMed]

]
In(Δk)=Jore2P|F(Δk)|2|Sn(Δk)|2ΔΩ.
(1)
Here F(Δk) is the continuous scattering from one unit cell (molecule). Jo is the incident photon flux density (photons/pulse/area), r2 e the electron cross section, P a polarization factor, and ΔΩ the solid angle subtended by a detector pixel at the sample. For small scattering angles Θ the resolution d = λ/Θ. Snk) is the transform of the truncated crystal lattice (an interference function resembling a sinc function laid down at each reciprocal lattice point). It has the form
i=13sin2(NiΨi)/sin2(Ψi),
where the Ψi define the scattering geometry. We assume an integral number of primitive unit cells. Equation (1) is remarkable in that it separates the result of complex interference effects into a simple product of intensity factors (unlike in-line holography), the second of which (Sn) is purely geometric and provides “samples” of |F(Δk)|2 sufficiently fine to permit phasing. Experimentally, the set of observed Δk may be determined using autoindexing software [3

3. R. A. Kirian, X. Y. Wang, U. Weierstall, K. E. Schmidt, J. C. H. Spence, M. Hunter, P. Fromme, T. White, H. N. Chapman, and J. Holton, “Femtosecond protein nanocrystallography-data analysis methods,” Opt. Express 18(6), 5713–5723 (2010). [CrossRef] [PubMed]

,13

13. M. G. Rossmann and J. W. Erickson, “Oscillation photography of radiation-sensitive crystals using a synchrotron source,” J. Appl. Cryst. 16(6), 629–636 (1983). [CrossRef]

] to find the Miller indices of all Bragg reflections, using fractional indexing between lattice points, and this determines the orientational relationship between diffraction patterns. The molecular transform F(Δk) between Bragg reflections is identical for all nanocrystals, but the lattice transform Snk) depends on the size and shape of a crystal, and differs between crystals. However Snk) is the same in the neighborhood of every lattice point for a given crystal. These characteristics allow the two intensities |F(Δk)| and |Snk)| to be disentangled.

The oversampled molecular transform may be “demodulated” from the diffracted intensity in principle by dividing the interference function in Eq. (1) into the measured intensity on the left. The continuous molecular transform modulus |F(Δk)| may then be phased by iterative methods, and so inverted to give a three-dimensional electron density map of the contents of one unit cell.

We first assemble the indexed patterns from all nanocrystals into a three-dimensional diffraction volume, and then average these oriented intensities over all snapshot diffraction patterns (from crystals of different size), to obtain the quantity
In(Δk)n=Jore2P|F(Δk)|2|Sn(Δk)|2nΔΩ.
(2)
Since F(Δk) is identical in each pattern,
In(Δk)n=Jore2P|F(Δk)|2|Sn(Δk)|2nΔΩ,
(3)
and the molecular transform may be expressed as
|F(Δk)|2=In(Δk)nJore2P|Sn(Δk)|2nΔΩ=In(Δk)nD(Δk)
(4)
It remains to determine the denominator D(Δk), the mean shape transform, which depends on the particle size distribution, and which we now discuss. We note that unlike Eq. (1), which can only be inverted for individual nanoparticles when |Snk)|2 is non-zero, <|Snk)|2>n is non-zero and therefore can be inverted over a wide range around the Bragg condition.

If we knew the particle size distribution G(N(n)) where N(n) defines the crystallite dimensions, we could write
|Sn(Δk)|2n=n|Sn(Δk)|2G(N(n))nG(N(n)),
avoiding the need to determine individual values of N(n) for each nanocrystal from the fringe systems. The denominator D in Eq. (4) (the mean shape-transform) may however be simply obtained without knowledge of the particle size distribution using the fact that a sum over corresponding pixels around different reflections will smooth out the molecular transform (which is different at each lattice point) but preserve the shape transform (which is the same around each lattice point for one nanocrystal). Qualitatively we find D as follows. First define a Wigner-Seitz cell around every reciprocal lattice point, extending half-way to the neighboring lattice point. We then perform a periodic average over all these Brillioun zones, by summing them all into one, then redistributing this summed cell periodically thoughout reciprocal space. Since the molecular transform varies between cells (see Fig. 1(b)), while the shape transform does not, the sum accumulates to give D in Eq. (4). In detail, we average all Bragg reflections in Eq. (2) located at reciprocal lattice points g hkl:
In(Δkghkl)n,hkl=Jore2P|F(Δkghkl)|2|Sn(Δkghkl)|2n,hklΔΩ.
(5)
Angle brackets denote an average over n and g hkl. We sum corresponding pixels (differing by g) in the three-dimensional diffraction volume within a small range of Δk around every reflection over all reflections, and also over all patterns from particles of different size (bring the intensity around every g hkl to the origin, and sum the result over all crystals). The summed cell at the origin is then redistributed about each lattice point.

Since the lattice transform is identical when translating by any g hkl, we have
In(Δkghkl)n,hkl=Jore2P|F(Δkghkl)|2|Sn(Δk)|2n,hklΔΩ
(6)
The molecular transform intensity is uncorrelated with the finite-lattice transform so that
In(Δkghkl)n,hkl=Jore2P|F(Δkghkl)|2n,hkl|Sn(Δk)|2nΔΩ.
(7)
Rearranging terms, we may now express the needed average finite-lattice transform as
|Sn(Δk)|2n=In(Δkghkl)n,hklJore2PΔΩ,
(8)
which we insert into Eq. (4) to obtain an expression for the molecular transform, purely in terms of the measured diffracted intensities
|F(Δk)|2=In(Δk)nIn(Δkghkl)n,hkl
(9)
This self-normalising expression does not require experimental scaling factors.

The smallest crystal which can be phased by this method will be one which is too small to be indexed. The largest crystal which can be phased will depend on both the dynamic range of the detector, and the number of detector pixels. The method fails for crystals much larger than a mosiac block, or when there is insufficient scattering between Bragg reflections. A larger number of inter-Bragg pixels than the Nyquist requirement is already needed to extract structure factors from these partial reflections by the Monte Carlo method [3

3. R. A. Kirian, X. Y. Wang, U. Weierstall, K. E. Schmidt, J. C. H. Spence, M. Hunter, P. Fromme, T. White, H. N. Chapman, and J. Holton, “Femtosecond protein nanocrystallography-data analysis methods,” Opt. Express 18(6), 5713–5723 (2010). [CrossRef] [PubMed]

], which requires the three-dimensional volume of the shape transforms to be estimated. The number of such detector pixels on a side is about 4 L /d, for resolution d, with L the largest nanocrystal's largest dimension. (For sub-micron nanocrystals, the effects of crystal size on the angular width of a Bragg reflection dominate the mosaicity, beam divergence and energy spread effects normally considered in conventional protein crystallography). Nyquist sampling requires 4 independent measurements of the molecular transform for each reflection (“half-orders” in three dimensions). However these samples need not be equally spaced [15

15. F. Maia, C. Yang, and S. Marchesini, “Compressive auto-indexing in femotsecond nanocrystallography,” Proc. SPIE 7800, 78000F (2010). [CrossRef]

]. Eq. (9) is also limited by the dynamic range of the detector, which can be effectively increased using summed data. In general, the envelope of the shape transform falls as 1/Δk [16

16. G. Xu, G. Zhou, and X. Zhang, “Phase recovery for X-ray crystallography,” Phys. Rev. B 59(14), 9044–9047 (1999). [CrossRef]

], so that midway between lattice points, as D approaches the noise limit, fluctuations in the denominator of Eq. (13) will amplify noise on the retrieved molecular transform. This may be controlled by using a Weiner filter [17

17. W. Press, B. P. Flannery, A. A. Teukolsky, and W. T. Vetterling, Numerical Recipes. C.U.P. (2007).

]. This smooth damping function applied to Eq. (9) can be obtained from interpolation of the power spectrum of the numerator of Eq. (9), and has values near unity when noise is low, and small values otherwise. The need for such a function was avoided in these simulations by collecting sufficient data. To determine the number of diffraction patterns which must be recorded we consider a sum of M diffraction patterns from one subclass belonging to the same orientation (within error). With a small background count Nb per shot per pixel, and Ns signal counts per shot per pixel (Ns<<1 at high angles), the signal to noise ratio S, assuming Poisson noise is S=MNs/M(Ns+Nb)MNs(1Nb/2Ns). By choosing the constant S (a commonly accepted values is S = 5) and using Ns from In in Eq. (1), and given Nb, this expression then gives the number of shots M in this orientation class for statistical significance of the “oversampled” intensity. Saturation of the most intense pixel may require several data sets with different gains. Signal-to-noise ratio also provides a condition on the inversion of Eq. (4) by optimization methods. Shannon interpolation (for the Fourier relationship between the autocorrelation of the molecular density and |F(Δk)|2) gives K2ak)**|F(Δk)|2 = |F(Δk)|, where K2ak) is the Fourier transform of the doubled unit cell and ** denotes convolution. Combining this with Eq. (4):
In(Δk)n=D(Δk)K2a(Δk)**|F(k)|2+ε
(10)
where ε is the noise contribution. This may be expressed in matrix notation as the linear equation
I=DKmF+ε=AF+ε
(11)
where the condition number of A must now exceed the signal-to-noise ratio S. Equation (11) may be solved using optimization methods, based on the pseudoinverse of A.

3. Simulations

Figure 1 shows simulations [18

18. A. Brunettia, M. Sanchez del Rioc, B. Golosiob, A. Simionovicic, and A. Somogyi, “A library for X-ray-matter interaction cross sections for X-ray fluorescence applications,” Spectrochim. Acta [A] B59, 1725–1731 (2004).

] at 8 kV for the protein alpha-conotoxin PnIB (1AKG), crystallized with one molecule in a cubic unit cell (a = b = c = 5.84 nm, α = β = γ = 90°). A 512 x 512 pixel detector is assumed with 100 mm camera length and 120 micron pixels. The mean number of molecules N = 20 on a side of the nanocrystal (as observed experimentally [19

19. D. A. Shapiro, H. N. Chapman, D. DePonte, R. B. Doak, P. Fromme, G. Hembree, M. Hunter, S. Marchesini, K. Schmidt, J. Spence, D. Starodub, and U. Weierstall, J. Synchrotron Radiat. 15, 593–599 (2008). [CrossRef] [PubMed]

]) with standard deviation σ = 2. Figure 1(a) shows the partial reflections expected from one nanocrystal. Figure 1(b) shows the modulus of the 3D molecular transform |F(Δk)| on one plane through the origin of q-space . The beam-stop obscures the central maximum at the direct beam. Figure 2(a)
Fig. 2 (a) Sum of many patterns (Eq. (3); crystal size N = 20, σ = 2, also on [001]. (b) The recovered molecular transform. Compare with 1(b).
shows the merged sum of M = 106 oriented diffraction patterns on this plane, assuming 1013 incident photons per shot with 0.5 micron beam diameter, giving 1.46 x 106 scattered photons per shot. The average finite lattice sum (Eq. (8)) shows a similar form to Fig. 2(a) but without intensity modulation, while Fig. 2(b) shows the molecular transform recovered using Eq. (9) for this noise level and particle size distribution. The results are evaluated using a standard goodness-of-fit R-factor R(M) [5

5. D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5(6), 843 (1952). [CrossRef]

] comparing F(Δk)model with F(Δk)recov from Eq. (9), as shown in Fig. 3(a)
Fig. 3 (a) Bold curve: overall goodness of fit index R(M), for 1 < M < 106 crystals. Upper curve: R(m) evaluated for the 50% of intensities which lie between reciprocal lattice points only. Lower: R(m) evaluated for remaining 50% which lie near lattice points. (b) Projected low-resolution electron density slice in real space, after iterative phasing, with [100] normal.
(bold curve) for M summed patterns. Our results suggest that 106 patterns would be needed for phasing under these conditions. If the incident flux is reduced to 1011 photons per pulse, we find that 108 patterns are needed.

The reconstruction of a charge- density map of the molecule then proceeds by phasing this transform, which may be based on any of the established iterative phasing methods [20

20. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci Inst. 78(1), 011301 (2007). [CrossRef]

,21

21. V. Elser, I. Rankenburg, and P. Thibault, “Searching with iterated maps,” Proc. Natl. Acad. Sci. U.S.A. 104(2), 418–423 (2007). [CrossRef] [PubMed]

] reviewed elsewhere [20

20. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci Inst. 78(1), 011301 (2007). [CrossRef]

,22

22. P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from X-ray diffraction data,” Acta Crystallogr. A 62(4), 248–261 (2006). [CrossRef] [PubMed]

], using a doubled cell. Results using shrinkwrap are shown in Fig. 3(b), and will depend on the detailed experimental conditions and size of beamstop. We note that the support of the molecule (the unit cell) is known. A large fraction of the unit cell for real protein crystals consists of water, not included here. Shannon's sampling theorem may be used as a constraint on predictions of the inter-Bragg scattering, once estimates of complex values at the reciprocal lattice points exist. The distance across the entire displayed area is 20nm, molecule enlarged for clarity.

In summary we have described the principle of a new crystallographic phasing method for data consisting of many diffraction patterns from nanocrystals of various sizes, in random orientations, containing identical molecules. No data of the required type (from untwinned nanocrystals) has thus far been collected at the LCLS at the fine pixel sampling rate required. We have assumed a transverse coherence width greater than the size of the largest nanocrystal, a requirement for this method. Calculations for higher resolution would be possible using greater computational time and memory - our purpose here is to demonstrate the principle of this method in a simple case. It is clear that this method could be combined with modelling information based on the molecular replacement technique. We provide an example of the method applied to simulated protein nanocrystal diffraction, and show that this new direct method for solving the phase problem does not require knowledge of the particle size distribution, independent measurement of experimental parameters, or atomic-resolution data.

Acknowledgements

We thank Profs. V. Elser, and J. Hadju for useful discussions. Supported by NSF award MCB 1021557.

References and links

1.

N. Kasai, and M. Kakudo, “X-ray diffraction by macromolecules” (Springer New York 2005).

2.

W. A. Barletta, J. Bisognano, J. N. Corlett, P. Emma, Z. Huang, K. J. Kim, R. Lindberg, J. B. Murphy, G. R. Neil, D. C. Nguyen, C. Pellegrini, R. A. Rimmer, F. Sannibale, G. Stupakov, R. P. Walker, and A. A. Zholents, “Free electron lasers: Present status and future challenges,” Nucl. Instrum. Methods A618, 69–96 (2010).

3.

R. A. Kirian, X. Y. Wang, U. Weierstall, K. E. Schmidt, J. C. H. Spence, M. Hunter, P. Fromme, T. White, H. N. Chapman, and J. Holton, “Femtosecond protein nanocrystallography-data analysis methods,” Opt. Express 18(6), 5713–5723 (2010). [CrossRef] [PubMed]

4.

H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. DePonte, U. Weierstall, R. B. Doak, F. Maia, A. V. Martin, I. Schlichting, L. Lomb, N. Coppola, R. L. Shoeman, S. W. Epp, R. Hartmann, D. Rolles, A. Rudenko, L. Foucar, N. Kimmel, G. Weidenspointner, P. Holl, M. Liang, M. Barthelmess, C. Caleman, S. Boutet, M. J. Bogan, J. Krzywinski, C. Bostedt, S. Bajt, L. Gumprecht, B. Rudek, B. Erk, C. Schmidt, A. Hömke, C. Reich, D. Pietschner, L. Strüder, G. Hauser, H. Gorke, J. Ullrich, S. Herrmann, G. Schaller, F. Schopper, H. Soltau, K. Kühnel, M. Messerschmidt, J. D. Bozek, S. P. Hau-Riege, M. Frank, C. Y. Hampton, R. Sierra, D. Starodub, G. J. Williams, J. Hajdu, N. Timneanu, M. Seibert, J. Andreasson, A. Rocker, O. Jönsson, S. Stern, K. Nass, R. Andritschke, C. Schröter, F. Krasniqi, M. Bott, K. E. Schmidt, X. Wang, I. Grotjohann, J. Holton, S. Marchesini, S. Schorb, D. Rupp, M. Adolph, T. Gorkhover, M. Svenda, H. Hirsemann, G. Potdevin, H. Graafsma, B. Nilsson, and J. C. H. Spence, “Femtosecond X-ray protein nanocrystallography,” Nature . in press. [PubMed]

5.

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5(6), 843 (1952). [CrossRef]

6.

M. F. Perutz, “The structure of haemoglobin,” Proc. Royal Soc. A264, 264–286 (1954). [CrossRef]

7.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik (Stuttg.) 61, 247–262 (1982).

8.

J. C. H. Spence, in Science of Microscopy, eds., P. Hawkes and J. C. H. Spence, (Springer, 2006. New York).

9.

J. C. H. Spence and R. B. Doak, “Single molecule diffraction,” Phys. Rev. Lett. 92(19), 198102 (2004). [CrossRef] [PubMed]

10.

C. Riekel. “Recent developments in microdiffraction on protein crystals,” J. Synchrontron Radiat . 11, 4–6 (2004). Also R. Von Dreele (2009) Personal communication: Results of powder X-ray diffraction from growth media. [CrossRef]

11.

M. von Laue, “The external shape of crystals and its influence on interference phenomena in crystalline lattices,” Ann. Phys. 26, 55–68 (1936).

12.

I. K. Robinson and R. Harder, “Coherent X-ray diffraction imaging of strain at the nanoscale,” Nat. Mater. 8(4), 291–298 (2009). [CrossRef] [PubMed]

13.

M. G. Rossmann and J. W. Erickson, “Oscillation photography of radiation-sensitive crystals using a synchrotron source,” J. Appl. Cryst. 16(6), 629–636 (1983). [CrossRef]

14.

A. G. Leslie, “The integration of macromolecular diffraction data,” Acta Crystallogr. D Biol. Crystallogr. 62(1), 48–57 (2006). [CrossRef]

15.

F. Maia, C. Yang, and S. Marchesini, “Compressive auto-indexing in femotsecond nanocrystallography,” Proc. SPIE 7800, 78000F (2010). [CrossRef]

16.

G. Xu, G. Zhou, and X. Zhang, “Phase recovery for X-ray crystallography,” Phys. Rev. B 59(14), 9044–9047 (1999). [CrossRef]

17.

W. Press, B. P. Flannery, A. A. Teukolsky, and W. T. Vetterling, Numerical Recipes. C.U.P. (2007).

18.

A. Brunettia, M. Sanchez del Rioc, B. Golosiob, A. Simionovicic, and A. Somogyi, “A library for X-ray-matter interaction cross sections for X-ray fluorescence applications,” Spectrochim. Acta [A] B59, 1725–1731 (2004).

19.

D. A. Shapiro, H. N. Chapman, D. DePonte, R. B. Doak, P. Fromme, G. Hembree, M. Hunter, S. Marchesini, K. Schmidt, J. Spence, D. Starodub, and U. Weierstall, J. Synchrotron Radiat. 15, 593–599 (2008). [CrossRef] [PubMed]

20.

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci Inst. 78(1), 011301 (2007). [CrossRef]

21.

V. Elser, I. Rankenburg, and P. Thibault, “Searching with iterated maps,” Proc. Natl. Acad. Sci. U.S.A. 104(2), 418–423 (2007). [CrossRef] [PubMed]

22.

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from X-ray diffraction data,” Acta Crystallogr. A 62(4), 248–261 (2006). [CrossRef] [PubMed]

OCIS Codes
(100.5070) Image processing : Phase retrieval
(260.1960) Physical optics : Diffraction theory
(290.5840) Scattering : Scattering, molecules
(320.7100) Ultrafast optics : Ultrafast measurements

ToC Category:
Image Processing

History
Original Manuscript: December 15, 2010
Revised Manuscript: January 26, 2011
Manuscript Accepted: January 27, 2011
Published: January 31, 2011

Citation
John C. H. Spence, Richard A. Kirian, Xiaoyu Wang, Uwe Weierstall, Kevin E. Schmidt, Thomas White, Anton Barty, Henry N. Chapman, Stefano Marchesini, and James Holton, "Phasing of coherent femtosecond X-ray diffraction from size-varying nanocrystals," Opt. Express 19, 2866-2873 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-2866


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References

  1. N. Kasai, and M. Kakudo, “X-ray diffraction by macromolecules” (Springer New York 2005).
  2. W. A. Barletta, J. Bisognano, J. N. Corlett, P. Emma, Z. Huang, K. J. Kim, R. Lindberg, J. B. Murphy, G. R. Neil, D. C. Nguyen, C. Pellegrini, R. A. Rimmer, F. Sannibale, G. Stupakov, R. P. Walker, A. A. Zholents, “Free electron lasers: Present status and future challenges,” Nucl. Instrum. Methods A618, 69–96 (2010).
  3. R. A. Kirian, X. Y. Wang, U. Weierstall, K. E. Schmidt, J. C. H. Spence, M. Hunter, P. Fromme, T. White, H. N. Chapman, J. Holton, “Femtosecond protein nanocrystallography-data analysis methods,” Opt. Express 18(6), 5713–5723 (2010). [CrossRef] [PubMed]
  4. H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter, J. Schulz, D. P. DePonte, U. Weierstall, R. B. Doak, F. Maia, A. V. Martin, I. Schlichting, L. Lomb, N. Coppola, R. L. Shoeman, S. W. Epp, R. Hartmann, D. Rolles, A. Rudenko, L. Foucar, N. Kimmel, G. Weidenspointner, P. Holl, M. Liang, M. Barthelmess, C. Caleman, S. Boutet, M. J. Bogan, J. Krzywinski, C. Bostedt, S. Bajt, L. Gumprecht, B. Rudek, B. Erk, C. Schmidt, A. Hömke, C. Reich, D. Pietschner, L. Strüder, G. Hauser, H. Gorke, J. Ullrich, S. Herrmann, G. Schaller, F. Schopper, H. Soltau, K. Kühnel, M. Messerschmidt, J. D. Bozek, S. P. Hau-Riege, M. Frank, C. Y. Hampton, R. Sierra, D. Starodub, G. J. Williams, J. Hajdu, N. Timneanu, M. Seibert, J. Andreasson, A. Rocker, O. Jönsson, S. Stern, K. Nass, R. Andritschke, C. Schröter, F. Krasniqi, M. Bott, K. E. Schmidt, X. Wang, I. Grotjohann, J. Holton, S. Marchesini, S. Schorb, D. Rupp, M. Adolph, T. Gorkhover, M. Svenda, H. Hirsemann, G. Potdevin, H. Graafsma, B. Nilsson, J. C. H. Spence, “Femtosecond X-ray protein nanocrystallography,” Nature. in press. [PubMed]
  5. D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5(6), 843 (1952). [CrossRef]
  6. M. F. Perutz, “The structure of haemoglobin,” Proc. Royal Soc. A264, 264–286 (1954). [CrossRef]
  7. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension,” Optik (Stuttg.) 61, 247–262 (1982).
  8. J. C. H. Spence, in Science of Microscopy, eds., P. Hawkes and J. C. H. Spence, (Springer, 2006. New York).
  9. J. C. H. Spence, R. B. Doak, “Single molecule diffraction,” Phys. Rev. Lett. 92(19), 198102 (2004). [CrossRef] [PubMed]
  10. C. Riekel. “Recent developments in microdiffraction on protein crystals,” J. Synchrontron Radiat. 11, 4–6 (2004). Also R. Von Dreele (2009) Personal communication: Results of powder X-ray diffraction from growth media. [CrossRef]
  11. M. von Laue, “The external shape of crystals and its influence on interference phenomena in crystalline lattices,” Ann. Phys. 26, 55–68 (1936).
  12. I. K. Robinson, R. Harder, “Coherent X-ray diffraction imaging of strain at the nanoscale,” Nat. Mater. 8(4), 291–298 (2009). [CrossRef] [PubMed]
  13. M. G. Rossmann, J. W. Erickson, “Oscillation photography of radiation-sensitive crystals using a synchrotron source,” J. Appl. Cryst. 16(6), 629–636 (1983). [CrossRef]
  14. A. G. Leslie, “The integration of macromolecular diffraction data,” Acta Crystallogr. D Biol. Crystallogr. 62(1), 48–57 (2006). [CrossRef]
  15. F. Maia, C. Yang, S. Marchesini, “Compressive auto-indexing in femotsecond nanocrystallography,” Proc. SPIE 7800, 78000F (2010). [CrossRef]
  16. G. Xu, G. Zhou, X. Zhang, “Phase recovery for X-ray crystallography,” Phys. Rev. B 59(14), 9044–9047 (1999). [CrossRef]
  17. W. Press, B. P. Flannery, A. A. Teukolsky, and W. T. Vetterling, Numerical Recipes. C.U.P. (2007).
  18. A. Brunettia, M. Sanchez del Rioc, B. Golosiob, A. Simionovicic, A. Somogyi, “A library for X-ray-matter interaction cross sections for X-ray fluorescence applications,” Spectrochim. Acta [A] B59, 1725–1731 (2004).
  19. D. A. Shapiro, H. N. Chapman, D. DePonte, R. B. Doak, P. Fromme, G. Hembree, M. Hunter, S. Marchesini, K. Schmidt, J. Spence, D. Starodub, U. Weierstall, J. Synchrotron Radiat. 15, 593–599 (2008). [CrossRef] [PubMed]
  20. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci Inst. 78(1), 011301 (2007). [CrossRef]
  21. V. Elser, I. Rankenburg, P. Thibault, “Searching with iterated maps,” Proc. Natl. Acad. Sci. U.S.A. 104(2), 418–423 (2007). [CrossRef] [PubMed]
  22. P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, D. Sayre, “Reconstruction of a yeast cell from X-ray diffraction data,” Acta Crystallogr. A 62(4), 248–261 (2006). [CrossRef] [PubMed]

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