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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 18 — Aug. 29, 2011
  • pp: 17318–17335
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Reconstructing an icosahedral virus from single-particle diffraction experiments

D. K. Saldin, H.-C. Poon, P. Schwander, M. Uddin, and M. Schmidt  »View Author Affiliations


Optics Express, Vol. 19, Issue 18, pp. 17318-17335 (2011)
http://dx.doi.org/10.1364/OE.19.017318


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Abstract

The first experimental data from single-particle scattering experiments from free electron lasers (FELs) are now becoming available. The first such experiments are being performed on relatively large objects such as viruses, which produce relatively low-resolution, low-noise diffraction patterns in so-called “diffract-and-destroy” experiments. We describe a very simple test on the angular correlations of measured diffraction data to determine if the scattering is from an icosahedral particle. If this is confirmed, the efficient algorithm proposed can then combine diffraction data from multiple shots of particles in random unknown orientations to generate a full 3D image of the icosahedral particle. We demonstrate this with a simulation for the satellite tobacco necrosis virus (STNV), the atomic coordinates of whose asymmetric unit is given in Protein Data Bank entry 2BUK.

© 2011 OSA

1. Introduction

The free electron lasers (FELs) now beginning to come online produce radiation many orders of magnitude brighter than than any existing source, and enable experiments previously the domain only of science fiction. One such proposed experiment [1

1. R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond X-ray pulses”, Nature 406, 752–757 (2000). [CrossRef] [PubMed]

] envisages reconstructing the 3D structure of a microscopic entity such as a virus from many ultrashort diffraction patterns of many identical copies of the particles in random orientations from single pulses of FEL radiation. Although the particles will undoubtedly suffer catastrophic radiation damage, the ultrashort nature of FEL radiation is expected to produce diffraction patterns of the particles before significant disintegration. An experiment on individual mimivirus particles was reported recently [2

2. M. M. Seibert, T. Ekeberg, F. R. N. C. Maia, M. Svenda, J. Andreasson, O. Jonsson, D. Odic, B. Iwan, A. Rocker, D. Westphal, M. Hantke, D. P. DePonte, A. Barty, J. Schulz, L. Gumprecht, N. Coppola, A. Aquila, M. Liang, T. A. White, A. Martin, C. Caleman, S. Stern, C. Abergel, V. Seltzer, J.-M. Claverie, C. Bostedt, J. D. Bozek, S. Boutet, A. A. Miahnahri, M. Messerschmidt, J. Krzywinski, G. Williams, K. O. Hodgson, M. J. Bogan, C. Y. Hampton, R. G. Sierra, D. Starodub, I. Andersson, S. Bajt, M. Barthelmess, J. C. H. Spence, P. Fromme, U. Weierstall, R. Kirian, M. Hunter, R. Bruce Doak, Stefano Marchesini, Stefan P. Hau-Riege, Matthias Frank, Robert L. Shoeman, Lukas Lomb, Sascha W. Epp, Robert Hartmann, Daniel Rolles, A. Rudenko, C. Schmidt, L. Foucar, N. Kimmel, P. Holl, B. Rudek, B. Erk, A. Homke, C. Reich, D. Pietschner, G. Weidenspointner, L. Struder, G. Hauser, H. Gorke, J. Ullrich, I. Schlichting, S. Herrmann, G. Schaller, F. Schopper, H. Soltau, K.-U. Kuhnel, R. Andritschke, C.-D. Schroter, F. Krasniqi, M. Bott, S. Schorb, Da. Rupp, M. Adolph, T. Gorkhover, H. Hirsemann, G. Potdevin, H. Graafsma, B. Nilsson, H. N. Chapman, and J. Hajdu, “Single mimivirus particles intercepted and imaged with an X-ray laser,” Nature 470, 78–81 (2011). [CrossRef] [PubMed]

]. The paper illustrates convincing diffraction patterns of the virus particle in two different orientations, from which 2D projections of the particles are reconstructed using an iterative phasing algorithm. Although such particles are known to be largely icosahedral, little evidence of the icosahedral shape is evident in the reconstructed projections. Several algorithms have been proposed for reconstructing a full 3D image of the particle from an ensemble of many such diffraction patterns from randomly oriented particles. The methodology followed by some of these approaches [3

3. V. L. Shneerson, A. Ourmazd, and D. K. Saldin, “Crystallography without crystals I: the common-line method for assembling a 3D diffraction volume from single-particle scattering,” Acta Crystallogr. 64, 303–315 (2008). [CrossRef]

5

5. N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E 80, 026705 (2009). [CrossRef]

] is to find the likely orientation of the measured diffraction patterns in the 3D reciprocal space of the particle.

Another approach [6

6. D. K. Saldin, V. L. Shneerson, R. Fung, and A. Ourmazd, “Structure of isolated biomolecules from ultrashort x-ray pulses: exploiting the symmetry of random orientations,” J. Phys: Condens. Matter 21, 134014(2009). [CrossRef]

] dispenses with finding the likely orientations of the individual diffraction patterns by integrating over orientations, in an attempt to find the spherical harmonic representation of the 3D diffraction volume of a single particle from the averages of the angular correlations of the intensities on the measured diffraction patterns. This method of analysis is even applicable to individual diffraction patterns from multiple identical particles [7

7. Z. Kam, “Determination of macromolecular structure in solution by spatial correlation of scattering fluctuations,” Macromolecules 10, 927–934 (1977). [CrossRef]

]. The particles need to be frozen in space or time while the scattering is taking place. If the scattering is from a single FEL pulse of radiation, the particles will be essentially frozen in time for the duration of the scattering even if not frozen in space. This opens this method to the analysis of scattering from particles in random orientations within a droplet. With such an approach, the “hit rate” in an experiment with a FEL can become 100%, whereas a low hit rate is to be expected when attempting to hit submicron particles with a submicron pulsed laser beam. The signal-to-noise ratio from such snapshot patterns is independent of the number of particles per shot, but increases with the square root of the number of shots [8

8. R. A. Kirian, K. E. Schmidt, and J. C. H. Spence, “Signal, noise, and resolution in correlated fluctuations from snapshot SAXS,” submitted for publication (2011).

]. This approach also has the advantage that it operates on a compressed version of the voluminous data produced by a FEL.

We point out here another advantage of this approach: it is easily amenable to simplifications resulting from any known point-group symmetry of the particles under study. This is a powerful advantage for the study of virus structure, which is dominated by that of its protein coat which encloses the genetic material, DNA or RNA, which contain the instructions for the replication of the virus. To quote from Caspar and Klug [9

9. D. L. D. Caspar and A. Klug, “Physical principles in the construction of regular viruses,” Cold Spring Harbor Symp. Quant. Biol. 27, 1–24 (1962). [CrossRef] [PubMed]

] “there are only a limited number of efficient designs possible for a biological container which can be constructed from a large number of identical protein molecules, The two basic designs are helical tubes and icosahedral shells”. Viruses have regular shapes since they are formed by the self assembly of identical protein subunits which are coded by the limited quantity of genetic material capable of being stored within the small volume enclosed by its protein coat. An icosahedron, for example can be formed by the self assembly of at least 60 identical subunits. The genetic material needs to code for just one of these subunits, a factor of at least 60 smaller than the entire structure.

2. Icosahedral Harmonics

The first aim of this approach is to find the spherical harmonic representation of the intensity distribution of any resolution shell in the reciprocal space of a single particle. Any prior information about the nature of this distribution may be incorporated by limiting the set of spherical harmonics over which the summation is performed and by any relationship amongst the amplitudes of the different spherical harmonics which are a consequence of any known point-group symmetry.

An obvious restriction of the form of the intensity distribution
I(q,θ,φ)=lmIlm(q)Ylm(θ,φ)
(1)
is its known inversion (or Friedel) symmetry. Since
Ylm(πθ,π+φ)=(1)lYlm(θ,φ)
(2)
it follows that a spherical harmonic expansion of an intensity distribution may contain only even values of the angular momentum quantum number l. The fact that the intensity distribution is real, allows the restriction to a summation over just the so-called real spherical harmonics (RSHs) Slm(θ,φ) defined by the combinations of spherical harmonics:
Slm(θ,φ)={12[Ylm(θ,φ)+(1)mYlm(θ,φ)]m>0Yl0(θ,φ)m=01i2[Ylm(θ,φ)(1)mYlm(θ,φ)]m<0
(3)
where the set of RSH’s with m ≥ 0 form a set, whose ϕ dependence is of the form cos (), and the set with m ≤ 0 likewise a set with ϕ dependence of the form sin (). If the reconstructed intensity distribution has a mirror plane, this may be chosen to be the xz plane, or the plane for which ϕ = 0. Then (1) may be replaced by a summation over only the subset of RSHs for which m ≥ 0, and we may take
I(q,θ,φ)=l,m0Rlm(q)Slm(θ,φ).
(4)
Since both the right hand side (RHS) and the left hand side (LHS) of the above equation are real, the coefficients Rlm(q) may also be taken as real.

The 3D polar plots of Fig. 1 display the familiar forms of the RSHs for the values of l = 0, 1, 2. Further point group symmetries of I(q, θ, ϕ) result in still further restrictions on allowed terms of the general expansion (1) above. When the 3D intensity distribution has icosahedral symmetry, for example, (1) may be replaced by
I(q,θ,φ)=lgl(q)Jl(θ,φ),
(5)
where the quantities Jl (θ, ϕ) are known as icosahedral harmonics (IHs), specified up to and including l =30 by only the angular momentum quantum number l. Since the orientation of our reconstructed 3D intensity distribution in the frame of reference of the particle may be chosen arbitrarily, the xz plane may be chosen to be the mirror plane, allowing the IHs (5) to be constructed from just the RSHs of positive m, i.e.
Jl(θ,φ)=m0almSlm(θ,φ)
(6)
where the coefficients alm are the real numbers for normalized RSHs tabulated by e.g. Jack and Harrison (1975) [10

10. A. Jack and S. C. Harrison, “On the interpretation of small-angle x-ray solution scattering from spherical viruses,” J. Mol. Biol. 99, 15–25 (1975). [CrossRef] [PubMed]

], the ones for the lowest allowed even values of l being reproduced in Table 2. Note the rows are characterized by l and the columns by m. Since the IHs involve a sum over the magnetic quantum number, at least up to l = 30, they depend on the quantum number l only. The forms of the icosahedral harmonics of lowest even degree, l=0, 6, 10, 12, and 16 are illustrated in Fig. 2 using the same 3D polar plots.

Fig. 1 Visualization of real spherical harmonics (RSHs) of angular momentum quantum numbers l = 0, 1, and 2. The plots are made with MATLAB by an adaptation of software by Denise L. Chan (avilable from the Mathworks web site).
Fig. 2 Icosahedral harmonics of angular momentum quantum number l = 0, 6, 10, 12 and 16. Each is a linear combination of the RSH’s of the magnetic quantum numbers indicated. Visualization by the same software as Fig. 1

Note that since the RSH’s Slm(θ,φ) are orthonormal with respect to integrations over spherical shell, the icosahedral harmonics Jl (θ, ϕ) will also be orthonormal with respect to the same integration provided
malm2=1,l.
(7)
This condition is clearly satisfied by the coefficients in Table 2.

3. Reconstructing the Diffraction Volume

The average angular correlations amongst the resolution rings of the different measured diffraction patterns contain information about the 3D diffraction volume of a single particle. Such angular correlations are defined by
C2(q,q,Δφ)=1Nppn=0N1Ip(q,φn)Ip(q,φn+Δφ)=1Npp1Nm=0N1Im(q)Im(q)*exp(imφn)
(8)
where Ip is the intensity on diffraction pattern p, Np is the number of available diffraction patterns from random orientations of the particle, ϕn is the n-th of N discrete values of ϕ, and
Im(q)=n=0N1Ip(q,φn)exp(imφn)
(9)
the angular Fourier tranform of each resolution ring of the p-th diffraction pattern. Indeed the fastest way to calculate the average angular correlation C 2(q,q′,Δϕ) is to exploit the cross-correlation theorem by performing the angular Fourier transform (9) of each individual diffraction pattern, take the product of the Fourier transform and its complex conjugate followed by the inverse transform, and to average the results over the diffraction patterns ((9) plus the second equality of (8)).

However, finding the correct Rlm(q)’s from known Bl (q,q′)’s is still a formidable task since it involves taking a matrix square root [6

6. D. K. Saldin, V. L. Shneerson, R. Fung, and A. Ourmazd, “Structure of isolated biomolecules from ultrashort x-ray pulses: exploiting the symmetry of random orientations,” J. Phys: Condens. Matter 21, 134014(2009). [CrossRef]

]. Such a square root is necessarily ambiguous by an orthogonal matrix which cannot be found from the Bl (q,q′)’s alone. In principle, such an orthogonal matrix may be found from the so-called angular triple correlations [11

11. Z. Kam, “The reconstruction of structure from electron micrographs of randomly oriented particles,” J. Theor. Biol. 82, 15–39 (1980). [CrossRef] [PubMed]

] or by an iterative phasing algorithm that alternately satisfies constraints to the meaured angular correlations and in the 3D space of the reconstructed intensity disrtibution [12

12. D. K. Saldin, H. C. Poon, V. L. Shneerson, M. Howells, H. N. Chapman, R. Kirian, K. E. Schmidt, and J. C. H. Spence, “Beyond small angle x-ray scattering: exploiting angular correlations,” Phys. Rev. B 81, 174105 (2010). [CrossRef]

].

When the particle under study is known to have a high degree of symmetry, like an icosahedral virus, this problem is greatly simplified. Comparing (4) and (5) with definition (6), one can deduce that, for a diffraction volume with icosahedral symmetry,
Rlm(q)=gl(q)alm
(15)
Substituting (15) into (14) we see that one may write
Bl(q,q)=gl(q)gl(q)malm2
(16)
and using (7) this may be simplified further to
Bl(q,q)=gl(q)gl(q)
(17)

A knowledge of the expansion coefficients for all the resolution shells should enable a reconstruction of the 3D diffraction volume via (5). If this intensity distribution is interpolated onto an oversampled [13

13. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallohgraphy to allow imaging of mikcrometer-sized non-crstalline specimens,” Nature 400, 342–344 (1999). [CrossRef]

] 3D Cartesian reciprocal-space grid, (qx, qy, qz), say, an iterative phasing algorithm [14

14. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 049901 (2007). [CrossRef]

] may be applied to reconstruct the 3D electron density of the scattering particle.

4. Numerical Tests

A central thesis of this paper is that the the scattered intensity from an icosahedral particle may be represented by a sum of icosahedral harmonics. We first sought to verify this proposition by calculating first the spherical harmonic expansion coefficients of a simple icosahedral particle via the expresion
Alm(q)=iljfj(q)jl(qrj)Ylm(r^j),
(22)
where fj(q) is the form factor of the jth atom, r j is its coordinate, and jl is a spherical Bessel function of order l.

For our initial tests we simulated the scattering from an artificial molecule of identical atoms at the vertices of a regular icosahedron (Fig. 3) of edge length 2 [15

15. See, e.g., the Wikipedia entry at http://en.wikipedia.org/wiki/Icosahedron.

] (which we take to be to be in Å units, with Cartesian coordinates (also assumed to be in Å):
  • (0, ±1, ±Φ)
  • (± 1, ±Φ, 0)
  • (±Φ, 0, ± 1)
where Φ is the golden ratio (1+5)/2.

Fig. 3 Regular icosahedron [15]

The resulting calculated values of the amplitudes Alm (arbitrarily taking fj(q) = 1, ∀j) for all possible values of of l and m are listed in Fig. 4. Note that the amplitudes Alm are all real, and that, for the values listed, they are non-zero only for l=0 and 6. The values for l=1, 2, 3, 4, 5, and 7 are all seen to be zero, corresponding to non-existing icosahedral harmonics for these values of l. Here too, all coefficients are zero except those for which l=0 or 6, and all non-zero coefficients are real. Note that this result will be true for any icosahedral orientation since the a rotation matrix (Wigner D-matrix) will mix only amplitudes of different magnetic quantum number corresponding to the same angular momentum l. The z-axis of the simple icosahedron used for this test is a 2-fold rotation axis, not 5-fold, unlike e.g. Fig. 2 above, or Table 1 below. This is why the amplitudes corresponding to every other value of m are non zero for l=6 rather than every integer multiple of 5, when z is chosen to be a 5-fold axis.

Fig. 4 Calculated values of the Alm coefficients (arbitrarily taking fj(q) = 1, ∀j) assuming 12 identical atoms at the vertices of a regular icosahedron. The first two entries in each column in each line are the l and m values. The next two are the real and imaginary parts of Alm(q). It will be seen that all coefficients are zero except those for which l=0 or 6, and that all non-zero coefficients are real.

Table 1. Expansion Coefficients of the Lowest Even-degree Icosahedral Harmonics with z-axis Chosen to be the 5-fold Rotation Axis

table-icon
View This Table

Of greater interest for our method are the allowed values of L for the coefficients, ILM, of the spherical harmonic expansions of the scattered intensity. Since
I(q)=|A(q)|2,
(23)
it must follow that if A(q) has icosahedral symmetry, so must I(q). However, this is not entirely obvious from the relationship between the two sets of coefficients
ILM(q)=lm;lmAlm(q)Alm*(q)Ylm(q^)Ylm*(q^)YLM*(q^)dq^=lm;lmAlm(q)Alm*(q)Ylm*(q^)Ylm(q^)YLM(q^)dq^=lm;lmAlm(q)Alm*(q)(2l+1)(2L+1)4π(2l+1)Cl0L0l0ClmLMlm
(24)
where ClmLMlm is a Clebsch-Gordan coefficient [16

16. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, 1988), p.148.

]. According to the usual theory of the vector addition of angular momenta, the allowed values of L are all integers in the range from |ll′| to l + l′, with no obvious indication that L=1, 2, 3, 4, 5, and 7, for instance, are forbidden. However, a straightforward evaluation of the ILM (q) coefficients via (24) reveals this to be the case, as is seen by the tabulated values of these coefficients in Fig. 5.

Fig. 5 Same as Fig. 4, except for values of the ILM (q) coefficients calculated by Eq. (24) from the Alm(q) values in Fig. 4.

Fig. 6 Top part of the structure of the satellite tobacco necrosis virus (STNV) viewed down its 5-fold axis (from structure data in PDB entry: 2BUK)
Fig. 7 Real and imaginary parts of the Ilm(q) coefficients calculated from the computed diffraction volume of STNV. Each dot represents a value of the (lm) pair. Note that these coefficients are largely absent for l=1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 15.

Since this result is a consequence of the icosahedral symmetry of the diffraction volume I(q), it is to be expected of the diffraction volume of all icosahedral viruses (assuming the protein coat to be the dominant scatterer). In view of (13) this must mean that the Bl (q,q′) coefficients computed from the data of diffraction patterns of random orientations of all icosahedral particles must all have vanishing values for l = 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 15,.., thus providing a very simple test of whether the diffraction patterns measured in a “diffract and destroy” experiment with a FEL are from an icosahedral particle.

Assuming this is indeed found to be approximately true in practice (even the so-called icosahedral viruses may have appendages which break the icosahedral symmetry of the protein coat, and of course the genetic material inside the protein coat would not be expected to have this symmetry. However, if the bulk of the material of the virus may be assumed to constitute the protein coat, this must be approximately the case). The icosahedral structure of the protein coat may be found by an analysis of the large l = 0, 6, 10, 12, 16,... Bl (q,q′) coefficients extactable from the average angular correlations of the diffraction data.

5. Reconstruction of STNV from Simulated Diffraction Patterns

The procedure described in section 3 was then followed to reconstruct a 3D diffraction volume, consisting of set of scattered intensities I(qx, qy, qz) over 3D reciprocal space as a function of the reciprocal-space coordinate q⃗ ≡ (qx, qy, qz). In our simulations, we took this to be a 61×61×61 array of real numbers. The computer time for this process was almost ridiculously short, amounting to no more than a few seconds on a single-processor desktop computer.

The final step is the recovery of a 3D electron density of the particle. This may be done by a standard iterative phasing algorithm. We used the “charge flipping” algorithm of Oszlányi and Süto [22

22. G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. 60, 134–141 (2004). [CrossRef]

, 23

23. G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping. II. Use of weak reflections,” Acta Crystallogr. 61, 147–152 (2005).

]. In order to judge the accuracy of the our algorithm in recovering the 3D diffraction volume, we performed this recovery of the 3D electron density from both the diffraction volume I(q) calculated directly from the STNV structure factors (Fig. 8), and also by our algorithm from the Bl (q,q′) coefficients (Fig. 9), which may be computed from the measured data of the FEL diffraction patterns from random particle orientations. The similarity of the reconstructed images of Figs. 8 and 9 was a further indication of the validity of the method of image reconstruction from the quantities Bl (q,q′) derivable from the average angular correlations. The fact that the reconstructed image consists of a thin protein shell is also seen from the slice perpendicular to a 5-fold rotation axis through the reconstructed image of Fig. 9 depicted in Fig. 10. In the case of all three figures, a ribbon representation of the structure from the biological assembly of the STNV virus from the same PDB structure used to simulate the diffraction patterns is superimposed on the semi-transparent electron density to show the excellence of the reconstruction. It should be emphasized that nowhere in our theory is it assumed that the structure consists of a thin protein shell, unlike the so-called shell model that has been used in the SAXS analysis of virus capsids [24

24. Y. Zheng, P. C. Doerschuk, and J. E. Johnson, “Determination of three-dimensional low-resolution viral structure from solution X-ray scattering data,” Biophys. J. 69, 619–639 (1995). [CrossRef] [PubMed]

]. In our case, the existence of a shell is deduced by an iterative phasing algorithm from the anaysis of data from diffraction patterns of random particle orientations without any assumptions on our part.

Fig. 8 Reconstructed image from the diffraction volume of a single STNV particle computed directly from a structure factor calculation. STNV is about 20 nm in diameter. The figure depicts a view of the icosahedron close down its 5-fold rotation axis. The reconstruction assumed a maximum value of q, qmax, of about 4.7 nm−1, implying a resolution of ∼ 1.3 nm. Both the outer and inner surfaces of the virus capsid are apparent in this representation. A ribbon diagram of the structure in PDB entry 2BUK is seen to fit within this capsid.
Fig. 9 (Media 1) Same as Fig. 8 except that the diffraction volume was reconstucted from the average of angular correlations on 10,000 diffraction patterns of STNV from uniformly distributed directions over SO(3). The reconstructed electron density is seen to be remarkably similar to that in Fig. 8.
Fig. 10 Same as Fig. 9 except that image displayed is a cut perpendicular to the 5-fold axis of the virus. The 5-fold symmetry of both the external and internal surfaces of the capsid in this projection are clearly visible.

6. Beyond the Icosahedral Approximation

Satellite tobacco necrosis virus (STNV) is an example of a virus with a perfectly icosahedral protein coat [25

25. VIPERdb, a database for icosahedral virus capsid structures, may be accessed at http://viperdb.scripps.edu. As of May 5, 2011, it contained 298 icosahedral viruses, of which the structures of 265 has been determined by x-ray diffraction, and 33 by cryo electron microscopy.

]. A host cell gets access to the genetic material of this virus by ingesting it whole and dissolving its protein coat.

Many of the larger viruses are only approximately icosahedral: they often have appendages, such as a neck sticking out of the coat that is used to inject the genetic material inside the coat into a host cell whose protein making capability is hijacked by the virus DNA or RNA.

An ultimate reconstruction algorithm should be able to reconstruct these non-icosahedral parts of the structure in addition to the icosahedral part. The above procedure has determined the icosahedral harmonic expansion coefficients gl (q) that best fit the measured quantities Bl (q,q′). Any deviations from these values are due to the non-icosahedral parts of the structure. Any differences between the experimental values of Bl (q,q′) and gl (q)gl (q′) may be written
δBl(q,q)=malm{gl(q)δRlm(q)+δRlm(q)gl(q)}+δRlm(q)δRlm(q),
(28)
in terms of δRlm(q), the extra contribution to the RSH expansion coefficients due to deviations from icosahedral symmetry. Note that for (l,m) combinations not associated with icosahedral harmonics, e.g. those for which there is no entry in a list like Table 1, the terms alm will be zero, and only the quadratic terms in δRlm will survive in (28). Determination of the δRlm(q) coefficients which optimize the agreement the theoretical expression (28) and the measured values will enable the construction of a better estimate of a single-particle diffraction volume via
I(q)=lm{gl(q)alm+δRlm(q)}Slm(q^).
(29)
The presence of the correction terms δRlm(q), which have no symmetry restrictions (apart from Friedel symmetry) will allow the diffraction volume calculated by this formula to include deviations from icosahedral symmetry.

Application of an interative phasing algorithm to an oversampled diffraction volume calculated by this expression will enable the determination of the full structure of the virus, including any appendages that break the approximate icosahedral symmetry.

7. Discussion

It should be emphasized this is not necessarily an absolute limit of the resolution obtainable with the use of icosahedral harmonics. The higher order harmonics, at least up to l=44, have been tabulated by Zheng et al. [24

24. Y. Zheng, P. C. Doerschuk, and J. E. Johnson, “Determination of three-dimensional low-resolution viral structure from solution X-ray scattering data,” Biophys. J. 69, 619–639 (1995). [CrossRef] [PubMed]

]. At least up to this value, the degeneracy of the icosahedral harmonics characterized by a particular value of l is no more than two. Although the algorithm for recovering the expansion coefficients of such degenerate hamonics from the experimental data is a little more complicated, it seems far from an insuperable problem.

The images in Figs. 8 to 9 were computed by an iterative phasing algorithm [22

22. G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. 60, 134–141 (2004). [CrossRef]

, 23

23. G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping. II. Use of weak reflections,” Acta Crystallogr. 61, 147–152 (2005).

] from a reciprocal-space distribution of intensities oversampled [13

13. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallohgraphy to allow imaging of mikcrometer-sized non-crstalline specimens,” Nature 400, 342–344 (1999). [CrossRef]

] by a factor of ∼ 2 with respect to the size of STNV, up to a qmax value of ∼ 0.47 Å−1 (a 61×61×61 array), implying a resolution of about 13 Å, and a d/R ratio closer to 1/8. Further, the images of Fig. 810 reveal this coat to be hollow. The slice (Fig. 10) through the reconstructed image perpendicular to the 5-fold axis reveals both external and internal surfaces of 5-fold rotational symmetry. The revelation of the hollow nature of the protein coat is of course an extra feature contained in the 3D intensity distribution above and beyond the assumed icosahedral symmetry. It is revealed by the iterative reconstruction algorithm used [22

22. G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. 60, 134–141 (2004). [CrossRef]

, 23

23. G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping. II. Use of weak reflections,” Acta Crystallogr. 61, 147–152 (2005).

] due to the paricular variation of the Bl (q,q′) coefficients with the radial reciprocal-space coordinates q and q′.

Some the advantages of this method of analysis of single particle diffraction patterns from unknown particle orientations compared with other proposed algorithms [4

4. R. Fung, V. L. Shneerson, D. K. Saldin, and A Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. 5, 64–67 (2009). [CrossRef]

,5

5. N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E 80, 026705 (2009). [CrossRef]

] should be pointed out. Since it has been shown [7

7. Z. Kam, “Determination of macromolecular structure in solution by spatial correlation of scattering fluctuations,” Macromolecules 10, 927–934 (1977). [CrossRef]

, 12

12. D. K. Saldin, H. C. Poon, V. L. Shneerson, M. Howells, H. N. Chapman, R. Kirian, K. E. Schmidt, and J. C. H. Spence, “Beyond small angle x-ray scattering: exploiting angular correlations,” Phys. Rev. B 81, 174105 (2010). [CrossRef]

, 19

19. H. C. Poon and D. K. Saldin, “Beyond the crystallization paradigm: structure determination from diffraction patterns of ensembles of randomly oriented particles,” Ultramicroscopy 111, 798–806 (2011). [CrossRef]

, 27

27. D. K. Saldin, V. L. Shneerson, M. Howells, S. Marchesini, H. N. Chapman, R. Kirian, K. E. Schmidt, and J. C. H. Spence, “Structure of a single particle from scattering of many particles randomly oriented about an axis: towards structure solution without crystallization?” New J. Phys. 12, 035014 (2010). [CrossRef]

, 28

28. D. K. Saldin, H. C. Poon, M. Bogan, S. Marchesini, D. A. Shapiro, R. A. Kirian, U. Weierstall, and J. C. H. Spence, “New light on disordered ensembles: ab-inito structure determination of one particle from scattering fluctuations of many copies,” Phys. Rev. Lett. 106, 115501 (2011). [CrossRef] [PubMed]

], that the angular correlations of multiple identical particles in arbitrary orientations are essentially identical to those from a single particle, the method we have described is equally applicable to droplets containing multiple particles injected into the XFEL [29

29. D. De Ponte, R. B. Doak, P. Fromme, G. Hembree, M. Hunter, S. Marchesini, K. E. Schmidt, U. Weierstall, and J. C. H. Spence, “Powder diffraction from a continuous microjet of submicrometer protein crystals,” J. Synch. Rad. 15, 593–599 (2008). [CrossRef]

] as to the injection of single particles in random orientations. Thus there is no need to discard diffraction patterns from multiple particle hits.

Since the inputs to our algorithm are not the direct photon counts, but rather the average of the angular correlations between intensities of the same diffraction patterns, it is insensitive to shot-to-shot fluctuations between the diffraction patterns, as may be caused by intensity variations of the incident X-ray beam or, for example, by the number of particles scattering a particular X-ray pulse.

The raw experimental data is likely to consist of ∼ 106 diffraction patterns, each of ∼ 106 pixels. Thus the raw experimental data will require TB of storage. Of course, this is very noisy data, and the structural information content is much less than this. The averaging of the angular correlations that we perform may be regarded as a form of data averaging that results in infomormation concentration and noise reduction. Even if the number of values of q chosen is, say, 61, and these coefficeints are evaluated for, say, 30 values of l, the total number of these (real) coefficients will be only of the order of 100,000, requiring less than a MB of storage. This data reduction is best performed at the site of the data to allow the tranference of a million times less data over the internet to the site of image reconstruction. The reconstruction of 3D images of the quality of Figs. 810 from a properly constructed set of Bl (q,q′) coefficients is extraordinarily rapid. In our calculations, reconstruction of an array of I(qx, qy, qz) values representing a 3D diffraction volume at reciprocal-space coordinates q⃗ = (qx, qy, qz) on a 61×61×61 Cartesian grid took just a few seconds on a single Intel Q6600 processor, using an Intel Fortran compiler. The reconstruction of real-space images of the quality of Figs. 810 from this array by means of a “charge flipping” algorithm [22

22. G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. 60, 134–141 (2004). [CrossRef]

, 23

23. G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping. II. Use of weak reflections,” Acta Crystallogr. 61, 147–152 (2005).

] took a further 4 minutes for 200 iterations on a laptop PC.

Of course, the averaging of the data from the different diffraction pattterns assumes they all arise from copies of the particle in different orientations (as does the technique of small angle X-ray scattering, SAXS, for example). In order to distinguish between different conformations of the individual molecules, it may be necessary to operate on the entire ensemble of all the measured diffraction patterns One of the disadvantages of such methods is the need to operate on single-particle diffraction patterns and thus, unlike with our method, diffraction patterns from multiple hits need to be removed. Such methods also face the problem of the uncertain normalization of incident intensities between successive pulses of incident radiation. Also, in contrast to our method, such techniques may require the tranferance of perhaps TB of data over the internet to the site of the data performing the analysis, which needs to be equipped with a cluster of computers performing parallel computations.

8. Conclusions

When reconstructing the structure of a virus from “diffract and destroy” type single-particle diffraction experiments proposed for the free electron laser [1

1. R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond X-ray pulses”, Nature 406, 752–757 (2000). [CrossRef] [PubMed]

], one may exploit the dictum of Caspar and Klug [9

9. D. L. D. Caspar and A. Klug, “Physical principles in the construction of regular viruses,” Cold Spring Harbor Symp. Quant. Biol. 27, 1–24 (1962). [CrossRef] [PubMed]

] that “there are only a limited number of efficient designs possible for a biological container which can be constructed from a large number of identical protein molecules, The two basic designs are helical tubes and icosahedral shells”. We offer here a solution for the case of icosahedral viruses. For those viruses which are substantially, though not completely icosahedral, the method proposed is expected to be useful nontheless for initially reconstructing the approximate icosahedral structure. The deviations from this structure can then be found by a perturbation theory which does not impose this symmetry.

The input to the algorithm is data from diffraction patterns of randomly oriented identical particles (where the particle orientations are unknown) in the form of the average of the angular correlations. As a consequence of any approximate icosahedral symmetry of the scattering particles, the angular momentum decomposition of the angular correlations contains only a few dominant contibutions from low values of the angular momenta.

This immediately suggests a simple test of whether the experimentally measured data are from the scattering by an icosahedral object. If so, the components of the quantities Bl (q,q′), derivable from the angular correlations, should have much smaller values for l=1, 2, 3, 4, and 5 than for l=0 and l=6, for example. What is more, as we have shown in this paper, the coefficients gl (q) of the icosahedral harmonic expansion of the 3D diffraction volume of the particle may be derived from the Bl (q,q′) data and a positivity condition on the intensities of the 3D diffraction volume. This will be the case even if the individual diffraction patterns are a result of scattering from more than one particle, so there will be no need to discard the diffraction patterns from multiple particles.

Having obtained the coefficients of an icosahedral harmonic expansion, the 3D diffraction volume may be reconstructed as a sum over these icosahedral harmonics. By definition, the resulting diffraction volume will have icosahedral symmetry. If this is constructed at a grid that is oversampled by a factor of 2 in each dimension, we have shown that a “charge flipping” algorithm with no fixed support contraint is able to reconstruct a 3D image of the particle. We find that this procedure not only reconstructs an icosahedral shape for the particle, in simulations for the satellite tobacco necrosis virus (STNV) it even reveals the hollow nature of the protein coat.

Acknowledgments

We acknowledge helpful discussions with Profs. Abbas Ourmazd and John Spence, and financial support from DOE grant No. DE-SC0002141.

References and links

1.

R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond X-ray pulses”, Nature 406, 752–757 (2000). [CrossRef] [PubMed]

2.

M. M. Seibert, T. Ekeberg, F. R. N. C. Maia, M. Svenda, J. Andreasson, O. Jonsson, D. Odic, B. Iwan, A. Rocker, D. Westphal, M. Hantke, D. P. DePonte, A. Barty, J. Schulz, L. Gumprecht, N. Coppola, A. Aquila, M. Liang, T. A. White, A. Martin, C. Caleman, S. Stern, C. Abergel, V. Seltzer, J.-M. Claverie, C. Bostedt, J. D. Bozek, S. Boutet, A. A. Miahnahri, M. Messerschmidt, J. Krzywinski, G. Williams, K. O. Hodgson, M. J. Bogan, C. Y. Hampton, R. G. Sierra, D. Starodub, I. Andersson, S. Bajt, M. Barthelmess, J. C. H. Spence, P. Fromme, U. Weierstall, R. Kirian, M. Hunter, R. Bruce Doak, Stefano Marchesini, Stefan P. Hau-Riege, Matthias Frank, Robert L. Shoeman, Lukas Lomb, Sascha W. Epp, Robert Hartmann, Daniel Rolles, A. Rudenko, C. Schmidt, L. Foucar, N. Kimmel, P. Holl, B. Rudek, B. Erk, A. Homke, C. Reich, D. Pietschner, G. Weidenspointner, L. Struder, G. Hauser, H. Gorke, J. Ullrich, I. Schlichting, S. Herrmann, G. Schaller, F. Schopper, H. Soltau, K.-U. Kuhnel, R. Andritschke, C.-D. Schroter, F. Krasniqi, M. Bott, S. Schorb, Da. Rupp, M. Adolph, T. Gorkhover, H. Hirsemann, G. Potdevin, H. Graafsma, B. Nilsson, H. N. Chapman, and J. Hajdu, “Single mimivirus particles intercepted and imaged with an X-ray laser,” Nature 470, 78–81 (2011). [CrossRef] [PubMed]

3.

V. L. Shneerson, A. Ourmazd, and D. K. Saldin, “Crystallography without crystals I: the common-line method for assembling a 3D diffraction volume from single-particle scattering,” Acta Crystallogr. 64, 303–315 (2008). [CrossRef]

4.

R. Fung, V. L. Shneerson, D. K. Saldin, and A Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. 5, 64–67 (2009). [CrossRef]

5.

N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E 80, 026705 (2009). [CrossRef]

6.

D. K. Saldin, V. L. Shneerson, R. Fung, and A. Ourmazd, “Structure of isolated biomolecules from ultrashort x-ray pulses: exploiting the symmetry of random orientations,” J. Phys: Condens. Matter 21, 134014(2009). [CrossRef]

7.

Z. Kam, “Determination of macromolecular structure in solution by spatial correlation of scattering fluctuations,” Macromolecules 10, 927–934 (1977). [CrossRef]

8.

R. A. Kirian, K. E. Schmidt, and J. C. H. Spence, “Signal, noise, and resolution in correlated fluctuations from snapshot SAXS,” submitted for publication (2011).

9.

D. L. D. Caspar and A. Klug, “Physical principles in the construction of regular viruses,” Cold Spring Harbor Symp. Quant. Biol. 27, 1–24 (1962). [CrossRef] [PubMed]

10.

A. Jack and S. C. Harrison, “On the interpretation of small-angle x-ray solution scattering from spherical viruses,” J. Mol. Biol. 99, 15–25 (1975). [CrossRef] [PubMed]

11.

Z. Kam, “The reconstruction of structure from electron micrographs of randomly oriented particles,” J. Theor. Biol. 82, 15–39 (1980). [CrossRef] [PubMed]

12.

D. K. Saldin, H. C. Poon, V. L. Shneerson, M. Howells, H. N. Chapman, R. Kirian, K. E. Schmidt, and J. C. H. Spence, “Beyond small angle x-ray scattering: exploiting angular correlations,” Phys. Rev. B 81, 174105 (2010). [CrossRef]

13.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallohgraphy to allow imaging of mikcrometer-sized non-crstalline specimens,” Nature 400, 342–344 (1999). [CrossRef]

14.

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 049901 (2007). [CrossRef]

15.

See, e.g., the Wikipedia entry at http://en.wikipedia.org/wiki/Icosahedron.

16.

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, 1988), p.148.

17.

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Dover Publications, 1984).

18.

L. Lovisolo and E. A. B. Da Silva, “Uniform distribution of points on a hyper-sphere with applications to vector bit-plane encoding,” IEE Proc. Vision Image Signal Process. 148, 187–193 (2001). [CrossRef]

19.

H. C. Poon and D. K. Saldin, “Beyond the crystallization paradigm: structure determination from diffraction patterns of ensembles of randomly oriented particles,” Ultramicroscopy 111, 798–806 (2011). [CrossRef]

20.

C. Xiao, P. R. Chipman, A. J. Battisti, V. D. Bowman, P. Renesto, D. Raoult, and M. G. Rossmann, “Cryo-electron microscopy of the giant mimivirus,” J. Mol. Biol. 353, 493–496 (2005). [CrossRef] [PubMed]

21.

M. V. Cherrier, C. A. Xiao, V. D. Bowman, A. J. Battisti, X. D. Yan, P. R. Chipman, T. S. Baker, J. L. van Etten, and M. G. Rossmann, “An icosahedral algal virus has a complex unique vertex decorated by a spike,” Proc. Natl. Acad. Sci. U.S.A. 106, 11085–11089 (2009). [CrossRef] [PubMed]

22.

G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. 60, 134–141 (2004). [CrossRef]

23.

G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping. II. Use of weak reflections,” Acta Crystallogr. 61, 147–152 (2005).

24.

Y. Zheng, P. C. Doerschuk, and J. E. Johnson, “Determination of three-dimensional low-resolution viral structure from solution X-ray scattering data,” Biophys. J. 69, 619–639 (1995). [CrossRef] [PubMed]

25.

VIPERdb, a database for icosahedral virus capsid structures, may be accessed at http://viperdb.scripps.edu. As of May 5, 2011, it contained 298 icosahedral viruses, of which the structures of 265 has been determined by x-ray diffraction, and 33 by cryo electron microscopy.

26.

J. B. Pendry, Low Energy Electron Diffraction, (Academic Press, 1974).

27.

D. K. Saldin, V. L. Shneerson, M. Howells, S. Marchesini, H. N. Chapman, R. Kirian, K. E. Schmidt, and J. C. H. Spence, “Structure of a single particle from scattering of many particles randomly oriented about an axis: towards structure solution without crystallization?” New J. Phys. 12, 035014 (2010). [CrossRef]

28.

D. K. Saldin, H. C. Poon, M. Bogan, S. Marchesini, D. A. Shapiro, R. A. Kirian, U. Weierstall, and J. C. H. Spence, “New light on disordered ensembles: ab-inito structure determination of one particle from scattering fluctuations of many copies,” Phys. Rev. Lett. 106, 115501 (2011). [CrossRef] [PubMed]

29.

D. De Ponte, R. B. Doak, P. Fromme, G. Hembree, M. Hunter, S. Marchesini, K. E. Schmidt, U. Weierstall, and J. C. H. Spence, “Powder diffraction from a continuous microjet of submicrometer protein crystals,” J. Synch. Rad. 15, 593–599 (2008). [CrossRef]

OCIS Codes
(100.3200) Image processing : Inverse scattering

ToC Category:
Image Processing

History
Original Manuscript: June 7, 2011
Revised Manuscript: July 24, 2011
Manuscript Accepted: July 27, 2011
Published: August 18, 2011

Virtual Issues
Vol. 6, Iss. 9 Virtual Journal for Biomedical Optics

Citation
D. K. Saldin, H.-C. Poon, P. Schwander, M. Uddin, and M. Schmidt, "Reconstructing an icosahedral virus from single-particle diffraction experiments," Opt. Express 19, 17318-17335 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17318


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References

  1. R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential for biomolecular imaging with femtosecond X-ray pulses”, Nature 406, 752–757 (2000). [CrossRef] [PubMed]
  2. M. M. Seibert, T. Ekeberg, F. R. N. C. Maia, M. Svenda, J. Andreasson, O. Jonsson, D. Odic, B. Iwan, A. Rocker, D. Westphal, M. Hantke, D. P. DePonte, A. Barty, J. Schulz, L. Gumprecht, N. Coppola, A. Aquila, M. Liang, T. A. White, A. Martin, C. Caleman, S. Stern, C. Abergel, V. Seltzer, J.-M. Claverie, C. Bostedt, J. D. Bozek, S. Boutet, A. A. Miahnahri, M. Messerschmidt, J. Krzywinski, G. Williams, K. O. Hodgson, M. J. Bogan, C. Y. Hampton, R. G. Sierra, D. Starodub, I. Andersson, S. Bajt, M. Barthelmess, J. C. H. Spence, P. Fromme, U. Weierstall, R. Kirian, M. Hunter, R. Bruce Doak, Stefano Marchesini, Stefan P. Hau-Riege, Matthias Frank, Robert L. Shoeman, Lukas Lomb, Sascha W. Epp, Robert Hartmann, Daniel Rolles, A. Rudenko, C. Schmidt, L. Foucar, N. Kimmel, P. Holl, B. Rudek, B. Erk, A. Homke, C. Reich, D. Pietschner, G. Weidenspointner, L. Struder, G. Hauser, H. Gorke, J. Ullrich, I. Schlichting, S. Herrmann, G. Schaller, F. Schopper, H. Soltau, K.-U. Kuhnel, R. Andritschke, C.-D. Schroter, F. Krasniqi, M. Bott, S. Schorb, Da. Rupp, M. Adolph, T. Gorkhover, H. Hirsemann, G. Potdevin, H. Graafsma, B. Nilsson, H. N. Chapman, and J. Hajdu, “Single mimivirus particles intercepted and imaged with an X-ray laser,” Nature 470, 78–81 (2011). [CrossRef] [PubMed]
  3. V. L. Shneerson, A. Ourmazd, and D. K. Saldin, “Crystallography without crystals I: the common-line method for assembling a 3D diffraction volume from single-particle scattering,” Acta Crystallogr. 64, 303–315 (2008). [CrossRef]
  4. R. Fung, V. L. Shneerson, D. K. Saldin, and A Ourmazd, “Structure from fleeting illumination of faint spinning objects in flight,” Nat. Phys. 5, 64–67 (2009). [CrossRef]
  5. N.-T. D. Loh and V. Elser, “Reconstruction algorithm for single-particle diffraction imaging experiments,” Phys. Rev. E 80, 026705 (2009). [CrossRef]
  6. D. K. Saldin, V. L. Shneerson, R. Fung, and A. Ourmazd, “Structure of isolated biomolecules from ultrashort x-ray pulses: exploiting the symmetry of random orientations,” J. Phys: Condens. Matter 21, 134014(2009). [CrossRef]
  7. Z. Kam, “Determination of macromolecular structure in solution by spatial correlation of scattering fluctuations,” Macromolecules 10, 927–934 (1977). [CrossRef]
  8. R. A. Kirian, K. E. Schmidt, and J. C. H. Spence, “Signal, noise, and resolution in correlated fluctuations from snapshot SAXS,” submitted for publication (2011).
  9. D. L. D. Caspar and A. Klug, “Physical principles in the construction of regular viruses,” Cold Spring Harbor Symp. Quant. Biol. 27, 1–24 (1962). [CrossRef] [PubMed]
  10. A. Jack and S. C. Harrison, “On the interpretation of small-angle x-ray solution scattering from spherical viruses,” J. Mol. Biol. 99, 15–25 (1975). [CrossRef] [PubMed]
  11. Z. Kam, “The reconstruction of structure from electron micrographs of randomly oriented particles,” J. Theor. Biol. 82, 15–39 (1980). [CrossRef] [PubMed]
  12. D. K. Saldin, H. C. Poon, V. L. Shneerson, M. Howells, H. N. Chapman, R. Kirian, K. E. Schmidt, and J. C. H. Spence, “Beyond small angle x-ray scattering: exploiting angular correlations,” Phys. Rev. B 81, 174105 (2010). [CrossRef]
  13. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallohgraphy to allow imaging of mikcrometer-sized non-crstalline specimens,” Nature 400, 342–344 (1999). [CrossRef]
  14. S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 049901 (2007). [CrossRef]
  15. See, e.g., the Wikipedia entry at http://en.wikipedia.org/wiki/Icosahedron .
  16. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, 1988), p.148.
  17. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration , 2nd ed. (Dover Publications, 1984).
  18. L. Lovisolo and E. A. B. Da Silva, “Uniform distribution of points on a hyper-sphere with applications to vector bit-plane encoding,” IEE Proc. Vision Image Signal Process. 148, 187–193 (2001). [CrossRef]
  19. H. C. Poon and D. K. Saldin, “Beyond the crystallization paradigm: structure determination from diffraction patterns of ensembles of randomly oriented particles,” Ultramicroscopy 111, 798–806 (2011). [CrossRef]
  20. C. Xiao, P. R. Chipman, A. J. Battisti, V. D. Bowman, P. Renesto, D. Raoult, and M. G. Rossmann, “Cryo-electron microscopy of the giant mimivirus,” J. Mol. Biol. 353, 493–496 (2005). [CrossRef] [PubMed]
  21. M. V. Cherrier, C. A. Xiao, V. D. Bowman, A. J. Battisti, X. D. Yan, P. R. Chipman, T. S. Baker, J. L. van Etten, and M. G. Rossmann, “An icosahedral algal virus has a complex unique vertex decorated by a spike,” Proc. Natl. Acad. Sci. U.S.A. 106, 11085–11089 (2009). [CrossRef] [PubMed]
  22. G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping,” Acta Crystallogr. 60, 134–141 (2004). [CrossRef]
  23. G. Oszláyi and A. Süto, “Ab initio structure solution by charge flipping. II. Use of weak reflections,” Acta Crystallogr. 61, 147–152 (2005).
  24. Y. Zheng, P. C. Doerschuk, and J. E. Johnson, “Determination of three-dimensional low-resolution viral structure from solution X-ray scattering data,” Biophys. J. 69, 619–639 (1995). [CrossRef] [PubMed]
  25. VIPERdb, a database for icosahedral virus capsid structures, may be accessed at http://viperdb.scripps.edu . As of May 5, 2011, it contained 298 icosahedral viruses, of which the structures of 265 has been determined by x-ray diffraction, and 33 by cryo electron microscopy.
  26. J. B. Pendry, Low Energy Electron Diffraction , (Academic Press, 1974).
  27. D. K. Saldin, V. L. Shneerson, M. Howells, S. Marchesini, H. N. Chapman, R. Kirian, K. E. Schmidt, and J. C. H. Spence, “Structure of a single particle from scattering of many particles randomly oriented about an axis: towards structure solution without crystallization?” New J. Phys. 12, 035014 (2010). [CrossRef]
  28. D. K. Saldin, H. C. Poon, M. Bogan, S. Marchesini, D. A. Shapiro, R. A. Kirian, U. Weierstall, and J. C. H. Spence, “New light on disordered ensembles: ab-inito structure determination of one particle from scattering fluctuations of many copies,” Phys. Rev. Lett. 106, 115501 (2011). [CrossRef] [PubMed]
  29. D. De Ponte, R. B. Doak, P. Fromme, G. Hembree, M. Hunter, S. Marchesini, K. E. Schmidt, U. Weierstall, and J. C. H. Spence, “Powder diffraction from a continuous microjet of submicrometer protein crystals,” J. Synch. Rad. 15, 593–599 (2008). [CrossRef]

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