Ultramicroscopy 148 (2015) 87–93
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Solving protein nanocrystals by cryo-EM diffraction: Multiple scattering artifacts Ganesh Subramanian a, Shibom Basu b, Haiguang Liu c, Jian-Min Zuo d, John C.H. Spence c,n a
Department of Materials Science and Engineering, Arizona State University, Tempe, AZ, USA Department of Chemistry and Biochemistry, Arizona State University, Tempe, AZ, USA c Department of Physics, Arizona State University, Tempe, AZ 85287-1504, USA d Department of Materials Science and Engineering, University of Illinois, Urbana, IL, USA b
art ic l e i nf o
a b s t r a c t
Article history: Received 20 March 2014 Received in revised form 27 August 2014 Accepted 31 August 2014 Available online 6 October 2014
The maximum thickness permissible within the single-scattering approximation for the determination of the structure of perfectly ordered protein microcrystals by transmission electron diffraction is estimated for tetragonal hen-egg lysozyme protein crystals using several approaches. Multislice simulations are performed for many diffraction conditions and beam energies to determine the validity domain of the required single-scattering approximation and hence the limit on crystal thickness. The effects of erroneous experimental structure factor amplitudes on the charge density map for lysozyme are noted and their threshold limits calculated. The maximum thickness of lysozyme permissible under the single-scattering approximation is also estimated using R-factor analysis. Successful reconstruction of density maps is found to result mainly from the use of the phase information provided by modeling based on the protein data base through molecular replacement (MR), which dominates the effect of poor quality electron diffraction data at thicknesses larger than about 200 Å. For perfectly ordered protein nanocrystals, a maximum thickness of about 1000 Å is predicted at 200 keV if MR can be used, using R-factor analysis performed over a subset of the simulated diffracted beams. The effects of crystal bending, mosaicity (which has recently been directly imaged by cryo-EM) and secondary scattering are discussed. Structure-independent tests for single-scattering and new microfluidic methods for growing and sorting nanocrystals by size are reviewed. & 2014 Elsevier B.V. All rights reserved.
Keywords: Protein crystallography Electron diffraction Multiple scattering Lysozyme Charge density map R-factor
1. Introduction The success of transmission electron diffraction (TED) for determination of the structure of two-dimensional (2D) protein crystals relies on the simplicity of two-dimensional indexing, and the relative absence of multiple elastic scattering perturbations on the Bragg scattered beam intensities, which increase with sample thickness. Reciprocal space consists of rods for 2D crystals, so that many more reflections intersect the Ewald sphere in this case. Glaeser et al. [1] have evaluated the importance of multiple elastic scattering on TED patterns from 2D protein crystals. At 200 keV, for monolayers that are typically less than 100 Å thick, they find only small perturbations. But at larger thicknesses, large perturbations generate Bragg intensities that bear no relationship to the kinematic or single-scattering values needed for structure analysis [2,3]. This extreme sensitivity of electron diffraction to sample
n
Corresponding author. Tel.: þ 1 602 965 6486 E-mail address: [email protected] (J.C.H. Spence).
http://dx.doi.org/10.1016/j.ultramic.2014.08.013 0304-3991/& 2014 Elsevier B.V. All rights reserved.
thickness has so far prevented the method from being generally extended to thicker crystals, making electron diffraction results frequently un-reproducible. Recently however, Shi et al. [4] have reported 3D electron density maps from lysozyme crystals estimated to be 0.5 μm thick, at 3 Å resolution using TED at 200 keV. In this paper, we have estimated the maximum thickness of such protein nanocrystals at various electron beam energies that can be expected to produce reliable density maps at atomic resolution. We have also considered the importance of modeling and the availability of structure factor phase information. The protein we have studied is lysozyme. Since multiply scattered TED intensities are known to agree with high accuracy with experimental intensities in convergent beam electron diffraction (CBED) (see [2,5] for reviews), we used ZMult (multislice [6] program in [3]) to simulate diffraction intensities as a function of thickness for a 200 keV beam oriented along (001) in lysozyme. As is common practice in the cryo-EM community, we assumed the Bragg beam intensities are proportional to the X-ray structure factor amplitudes |F| (however, see [3]). We have also studied the dependence of electron density maps on the |F| values using the X-ray crystallographic program
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PHENIX and generated the 3D electron density maps for lysozyme crystals. The inevitable (beneficial) effects of sample bending are also discussed.
2. Background The importance of multiple scattering in cryo-EM imaging was first investigated by Grinton and Cowley [7] for single-particles, following the first tomographic TEM imaging experiments of DeRosier and Klug [8] and others. At 100 keV, Grinton et.al. found false detail in high-resolution TEM images (not diffraction patterns) at 6 Å resolution and 30 nm thickness, and severe distortions due to multiple scattering at 50 nm thickness of protein. At 3 Å resolution they expected severe distortion for 100 Å thickness. For diffraction from 2D crystals, Glaeser and Downing [1] found that thicknesses up to 100 Å might be used for beam energies above 100 keV. An important effect of multiple scattering is to preserve sample symmetry in the diffraction pattern (violating Friedel symmetry). So, a simple test for the presence of multiple scattering is failure of Friedel symmetry. Moreover, all protein crystals are non-centrosymmetric due to the presence of helical structures. For 4 nmthick bacteriorhodopsin at 16 Å resolution and 100 keV, Glaeser and Downing found an R-factor of 11% between Friedel pairs, increasing to 40% for a 200 Å thick multi-layer. They recommend averaging the pairs to reduce perturbations, and support a thickness limit between 100 and 200 Å of protein at energies above 100 keV for high resolution. A detailed study of the validity domain of the single-scattering (kinematic) approximation for light-element inorganic crystals can be found in Anstis et al. [9]. In experiments however, while Friedel symmetry is well preserved for 2D crystals (because of their relative insensitivity to orientation), it is a less useful test for 3D crystals because of their extreme sensitivity to diffraction conditions and orientation (unless convergent-beam methods can be used). The “kinematic convergent-beam method” has been used to solve an oxide nanocrystal, using the absence of intensity variation within the convergent-beam (microdiffraction) disks as a test for the absence of multiple scattering [10]. Finally, the presence of significant intensity in reflections, which should be absent due to the presence of screw and glide symmetry elements is often used as an indication of multiple scattering. In fact, for an unbent thick perfect crystal, (conditions rarely met in protein crystallography unless a very small beam diameter is used) these reflections remain absent for all thicknesses and beam energies as a result of an elegant cancellation theorem [11]. Taken together, under the normal conditions of strong multiple scattering, great thickness sensitivity, and specimen imperfections, it is almost impossible to record the same set of Bragg beam intensity ratios on two separate occasions using an electron microscope, unless the sample consists of a monolayer, and a very small beam is used. These TED intensities (unlike XRD intensities) are, therefore, not normally sufficiently reproducible for structure analysis. Those collected from a particular region of known thickness, however, may be analyzed quantitatively in great detail using the convergent beam method [12]. At the same time, the sub-nanometer beam diameter of the electron microscope allows ‘perfect crystal’ data collection from regions smaller than one “mosaic block” for inorganic crystals. Striking images of these mosaic regions (perhaps 50 nm in width) in protein have been directly imaged at high resolution, for the first time recently, by Nederlof et al. [13]. (The thickness of these “blocks” is not clear, they may be regions of locally bent crystal, as discussed below). We note therefore that by using a sufficiently small beam diameter, many of the effects of defects and disorder could be eliminated as the beam becomes smaller than one “mosaic block” or bent region. (These regions of local orientation change have
recently been imaged directly by cryo-EM for the first time, in lysozyme [13]). However electron microdiffraction in cryo-EM would require instrumental modifications to allow the beam intensity to be reduced independently of its size (so that the total number of electrons in the beam is not held constant during focusing), while radiation damage severely limits the size of the smallest beam which can be used. (See [14] for estimates of the smallest crystal which can give useful electron diffraction data when subject to damage limits). The simulations which follow do not allow for mosaicity or bending effects in electron diffraction, and so do refer to the case where a small microdiffraction beam is used. The effects of bending are discussed separately in Section 5. (See [15] for a review of TED and an extended discussion of the effect of dynamical perturbations on attempts to solve organic structures by electron diffraction, and the history of this subject).
3. Theory A review of the various methods used to simulate multiplyscattered Bragg intensities in thin crystals, and the relationship between them, is given in Spence [2]. The electron microdiffraction method, where TED patterns using beams of subnanometer dimensions are obtained, is reviewed in Spence and Zuo [3]. Electron diffraction differs from X-ray diffraction (XRD) in many ways, the most important of which for these purposes are i) At energies above 100 keV, the electron wavelength is much smaller than that used for XRD, leading to a much ‘flatter’ Ewald sphere. ii) For a perfect crystal, the rocking curve width in TED is a significant fraction of the Bragg angle, unlike XRD, for which it is much smaller. This, and the thickness (t) of the sample, affects the ‘partiality’ of the reflections. Using the large beam divergence needed to form a nanodiffraction pattern, it is then normally possible, given the relatively large unit cell of many proteins crystals and flat Ewald sphere, to record many full reflections, which are simultaneously at the Bragg condition lying on a plane in reciprocal space normal to the beam. iii) The interaction of fast electrons is far stronger than that of X-rays, producing a higher ratio of useful elastic scattering per unit energy deposited by damaging inelastic interactions, than X-rays (Henderson [16]). iv) The intensities of Bragg reflections are, in the kinematic approximation, proportional to Fourier coefficients vg of the electrostatic potential (in volts) for a crystalline sample, not the charge density, as studied by XRD [3]. In the kinematic approximation, the intensity of a Bragg beam g diffracted by a parallel-sided slab of crystal in the transmission geometry is I g ¼ ðσvg tÞ2 ½ sin ðπsg tÞ2 =ðπsg tÞ2
ð1Þ
where sg, the excitation error, measures deviation from the Bragg condition and σ¼π/(λVo)¼2πmeλ/h is an interaction constant, with Vo, the microscope accelerating voltage and λ, the electron wavelength. This applies to the case of small beam divergence angle. We see that in this single-scattering approximation, the intensity at the Bragg condition (sg ¼0) varies as the square of sample thickness, and so can be compared with the actual thickness variation predicted by accurate multiple scattering calculations, in order to determine the thickness range of this approximation for various resolutions and beam energies. For weaker beams, where the Bragg condition is not exactly satisfied, the equation also predicts a thickness-squared dependence. A degree of convergence in the illumination will span a range of excitation errors sg.
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4. Results and discussion We perform simulations intended to identify (i) the kinematic (single-scattering) thickness limit in proteins (within which a direct interpretation of the diffracted intensities is possible) and (ii) the significance of tolerance to error in |F| as compared to phases. All our studies were done with the protein lysozyme in the P43212 space group and unit cell parameters a ¼b¼ 79.1 Å and c ¼37.8 Å. Our structure factors were obtained from the relativistic Hartree–Fock calculations of Doyle and Turner for neutral atoms [17], and so take no account of bonding effects. These have been refined in detail against experimental electron diffraction intensities using inorganic crystals [3] where bonding effects are very small, but may be larger for the light C, O, N atoms of a protein, or graphene [18]. Hydrogen atoms were not included. First, we studied the variation of diffracted beam intensity with thickness for the lysozyme crystals, which was subsequently used to identify the kinematic limit for the different diffracted beams. Similar curves for paraffin layers are shown in [15]. Fig. 1a and b shows the characteristic ‘pendellosung’ oscillations in the intensities of low (4 4) and high (23 13) resolution diffracted beams (approx. 13 Å and 3 Å respectively) and their Friedel pairs at 200 keV. The oscillations of the diffracted beam intensities allow for an oscillation period or an “extinction distance” to be defined. However only in a two-beam theory (or kinematic theory away from the Bragg condition) are the oscillations periodic. Eq. (1) shows that a different extinction distance can be associated with every excitation error or point within the diffraction disk. (A similar result holds in two-beam theory). In this way, both the convergent-beam and precession methods [19], by including a large range of orientations around the Bragg condition, average over many of these thickness periods and so reduce the rapidity of the thickness (Pendellosung) oscillations. Fig. 1(c and d) represent
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the same oscillation trends for the two diffracted beams, but at 400 and 1000 keV respectively. Allowing 10% error tolerance in the intensities, we calculated the kinematic (single-scattering) limit as tabulated in Table 1. At 3 Å resolution (23 13), this limit was observed to be 227.4 Å at 200 keV. This kinematic limit increases [2] with incident beam energy, thus indicating a possible use of higher incident beam energies to support the study of thicker protein samples that still diffract within the kinematic regime. But high beam energy increases the probability of knock-on radiation damage (especially for light elements). The reduction in ionization damage, but rapid increase in such “knock-on” ballistic damage, with increasing beam energy has been thoroughly studied and reported in the literature for beam energies up to 3 MeV (see [2,20] for reviews). In protein crystallography, it is widely accepted that about 2/3 of the information in a charge density map comes from structure factor phases (commonly obtained using the molecular replacement method, using a similar protein as model) and 1/3 from the amplitudes [21,22]. To evaluate this, we introduced random errors
Table 1 Variation of kinematic thickness in lysozyme with incident beam energy. Incident energy
Diffracted beam
Kinematic thickness (Å)
200 KeV
(4 4) (23 13)
568.5 227.4
400 KeV
(4 4) (23 13)
682.2 341.1
1 MeV
(4 4) (23 13)
720.1 644.3
400KeV
I (4 4) I (-4 -4)
100 x I(23 13)
I(4 4)
I(a.u.)
I(a.u.)
200KeV
t(A°)
t(A°)
1000KeV
I (23 13) I (-23 -13)
100 x I(23 13)
I(4 4)
I(a.u.)
I(a.u.)
200KeV
t(A°)
t(A°)
Fig. 1. Diffracted beam intensity vs thickness for lysozyme crystals. At 200 keV, (a) (4 4) and its Friedel pair ( 4 4) and (b) (23 13) and its Friedel pair ( 23 13). (4 4) and (23 13) at (c) 400 keV and (d) 1 MeV.
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Fig. 2. Omit Map for Lysozyme (residues 50–60 omitted from refinement) at 1.5σ. Each sub-plot contains charge density maps with increasing random errors introduced in the experimental diffracted intensities. The sticks in green are some of the omitted residues. Red wire mesh corresponds to ‘zero’ error omit map. (a) 20% error (pink wire-mesh) overlapped with zero error. (b) 50% Error (sky-blue wire-mesh) overlapped with zero error. (c) 70% Error (marine-blue wire-mesh) overlapped with zero error. (d) 100% Error (white wire-mesh) overlapped with zero error. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
to the x-ray intensities but maintained accurate phases. This way, when the charge density maps were obtained with the erroneous intensities, we could identify the percentage error beyond which, the real-space charge density information starts to be substantially different from ideal. X-ray diffraction intensities and phases for lysozyme were obtained from 4ET8.pdb [23]. Keeping the phases unchanged, we introduced random errors to the intensities. We then generated an omit map (omitting residues 50–60 from the refinement process), as in Fig. 2, where the charge densities (wireframes) were laid over the ideal structure (sticks) from the model. For a perfect fit these should agree. The red wire-mesh corresponds to ideal (zero-error) omit- map. Thus Figs. 2(a–d) correspond to charge density maps from 20%, 50%, 70% and 100% error in X-ray intensities (10%, 22%, 34% and 41% errors in |F| respectively) overlaid upon the density map with zero-error. With increasing error in the intensities, we can notice that the charge density (wire-mesh) does not match with the atomic positions. Also, the noise in the data increases with error. For this specific case, if we notice the regions around the residues 50, 51 and 59 (labeled in figure), we note that at 34% error in |F| and beyond (Fig. 2c and d), the electron density substantially fails to match the model. In fact, at 41% error (Fig. 2d), electron density is almost completely absent around the labeled residues. The threshold error in |F| in this case can, thus, be taken at 34%. However, since the random error was introduced to the diffracted intensities (reciprocal space), whereas the density maps are plotted in real space, different regions in the real space charge density might show different extents of match/mismatch for the same amount of random error introduced. Nevertheless, for any protein, there are
always specific regions that are characteristic of their structural/ functional behavior and the accuracy/inaccuracy in the structures of these regions typically determine the maximum error permissible in experiments. This can, for example, be understood using the following case where we identified the threshold error beyond which, similar (but not identical) amino acid sequences from hen and turkey lysozymes could not be differentiated. The hen-lysozyme possesses His15 (i.e., Histidine residue at position 15 in the sequence) amino acid while the turkey-lysozyme possesses Leu15 (i.e., Leucine residue at position 15 in the sequence), everything else remaining the same. For a model based on turkey-lysozyme and experiments based on hen-lysozyme, one would expect the overlap of the model (sticks) with the experimental charge density (wire-mesh) to not be perfect at that specific amino-acid position. In Fig. 3, an Fobs Fcalc difference map is plotted for this exact case. As expected, we observe positive charge densities (wire frame) falling outside the model in (a), (b) and (c) (corresponding to errors in |F| up to 34%). But with (d), we notice that there is no such charge density outside of the model. The absence of such positive density (at 41% error) indicates the threshold error beyond which, the ability to distinguish the hen-lysozyme from the turkey lysozyme is lost (despite using accurate phase information). Although Figs. 2 and 3 were mapped using x-ray diffracted intensities, our conclusions will also apply to similar data obtained from electron diffraction experiments. How this affects the maximum permissible crystal thickness under the kinematic limit is now discussed. The R-factor is a statistical parameter widely used in crystallography that measures how close the experimental |F| values are
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0% error in |F|
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22% error in |F|
41% error in |F|
34% error in |F|
Fig. 3. Fo Fc difference maps for Turkey (model)–Hen (experiment) lysozyme at 3.0σ. Density outside the model (wire frame) indicates distinguishability, absent with 41% error in |F|.
Fig. 4. R-factor vs thickness for lysozyme at 200 keV.
in relation to the crystallographic model. It is expressed as Rf actor ¼
∑jðjF obs j jF calc jÞj ∑jF obs j
ð2Þ
where |Fobs| and |Fcalc| are the is the experimental and model structure factor amplitudes respectively. In protein crystallography, a practically accepted limit for R-factor is 0.3. Wilson finds an
extreme upper limit in the R-value of 0.59, for non-centrosymmetric crystals such as proteins, beyond which the model can have the same symmetry but a completely random atomic arrangement [24]. As long as the protein crystal thickness is within the kinematic (single-scattering) limit, the R-values are well within the reasonable limit of 0.3. But for larger thicknesses, the multiply scattered intensities start to vary significantly from the
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single-scattered intensities. This is equivalent to artificially introducing errors in the ideal single-scattered diffracted intensities, as in Figs. 2 and 3. Thus, we can calculate R-factor as a function of the crystal thickness to establish a maximum crystal thickness, under the single-scattering assumption. For this, we used multiple diffracted beams (90 in total) of lysozyme at 200 keV. The structure factor amplitudes corresponding to 37.9 Å (1 unit cell thickness) were considered as the |Fobs|, with respect to which, the structure factor amplitudes of higher thicknesses (|Fcalc|) were evaluated. From Fig. 4, the maximum thickness of lysozyme crystal that can be studied for Rr0.3 was 830 Å. For thicknesses greater than 2300 Å, R Z0.59. Therefore, beyond 2300 Å thickness, the |F| values obtained from diffracted intensities completely fail to specify the atomic arrangement of the lysozyme crystal. This calculation was done with (10 10 0) as the highest order diffracted beam, which corresponds to roughly 6 Å resolution. A more rigorous study would include thousands of diffracted beams, up to the resolution of interest. We expect such a study to decrease the limit of acceptable thickness within the single-scattering approximation, below that presented here. Several tests for the absence of multiple-scattering perturbations exist, most of which require knowledge of the sample structure. However, the observation of uniform intensity in convergent-beam disks [10] does not, nor does the observation of Friedel symmetry (which is destroyed by multiple scattering) in samples known to be acentric, such as proteins. Also (as mentioned earlier), for samples whose space-group is known, reflections that are absent as a result of screw or glide planes may occur through multiple scattering. In fact, the Gjonnes-Moodie cancellation theorem [11] shows that these reflections remain absent for all thickness and beam energies, as long as they are oriented along certain specific beam directions, so that a test for multiple scattering based on observation of absent reflections fails, for perfect crystals. For proteins however, it is often a useful test, since the samples are usually bent, and hence a range of orientations is illuminated that is greater than the angular window over which cancellation of multiple-scattering paths occurs. Lastly, experience in other fields [19,25,26] shows that dynamical interactions are reduced in orientations away from prominent zone axes, since, in the 2-beam dynamical theory, the periodicity of beams is inversely proportional to the structure-factor magnitudes, and so increases for weak reflections at the Bragg condition. Although zone axes orientations excite the maximum number of strongly interacting reflections simultaneously, they are needed to synthesize the major high-symmetry projections of the density map. However, by contrast, as shown by Eq. (1), the short thickness periodicity of weak reflections, which are well away from the Bragg condition, is reduced to 2/sg, and so should be avoided.
5. Discussion and conclusions The importance of multiple scattering as a limiting factor in cryo-EM structure analysis of protein microcrystals has been studied. (i) From our multislice simulations for perfectly ordered hen-egg lysozyme, we find that the TED single-scattering limit at 3 Å resolution for lysozyme at 200 keV is about 230 Å at a zoneaxis orientation for 10% error in intensities. (ii) At higher incident beam energy, the kinematic limit increases (to about 650 Å at 1 MeV for 3 Å resolution) but at the risk of a high probability of knock-on radiation damage. (iii) By introducing random errors to |F| for lysozyme (while maintaining the correct phases) we find that at around 35% error in |F|, the charge densities obtained from the erroneous |F| start to substantially deviate from the ideal model, at the region of interest. The ability to distinguish hen and turkey lysozyme is lost at about 40% error in |F|. The high tolerance to error in diffracted intensities (a 40% error in |F| corresponds to almost
100% error in diffracted intensity) is a consequence of the inherently greater phase dependence of charge density maps. Although this seems to suggest an advantage in using TED, with its high tolerance to error in intensities, it emphasizes the necessity of accurate phase information (for which one is dependent on prior experiments [23]). (iv) R-factor analysis over a subset of simulated diffracted beams show that the maximum crystal thickness, for perfectly ordered protein crystals, that can be used to maintain a reasonable value (Ro0.3) is about 1000 Å. These results are, however, limited by the fact that (a) although all proteins have similar atomic density and composition, the thickness limit of the TED method, (in this case given for lysozyme) will vary depending on the particular crystal structure. Nanocrystals with a large unit cell produce weaker diffracted beams for the same crystal volume, and a more dense reciprocal lattice. We therefore expect a larger tolerable thickness for larger unit cells, as is confirmed by detailed simulations for inorganic oxides [2], and the theory [27] of scattering by nanocrystals that are a few unit cells thick in all three dimensions. (b) this paper takes no account of mosaicity or crystal bending. It applies strictly to microdiffraction patterns obtained from very small regions of perfect crystal. Theories of mosacity in electron diffraction have been published (see [15] for a review) in which coherent multiple scattering within mosaic blocks is coupled to incoherent multiple scattering of intensities between blocks. However experimental evidence [28,13] suggests that continuous bending is more likely for TEM samples than the tilted block structure assumed in the much thicker X-ray samples, combined with the “secondary scattering” of intensities between layers proposed by Cowley (see [28] and references therein). This can have beneficial effects for structure analysis by TED, since the thickness oscillation periods shown in Fig. 1 are proportional to the inverse of the deviation sg from the Bragg condition (Eq. 1). As in the precession electron diffraction method [19], by spanning a range of orientations, each with a definite value of sg, the effect is to smooth out the thickness oscillations. We note that in a recent paper [29] it is claimed that useful structure analysis can be achieved by TED using protein crystals in the 160–400 nm range (R 25% at 0.25 nm resolution). Some of this increase beyond our estimate of 83 nm may be due to this bending effect. It would seem that the data analysis algorithms used for XFEL nanocrystal data, which must take account of mosaicity when merging partial reflections, could benefit from these direct near-atomic resolution real-space cryo-EM images of the bending and misoriented domains [13]. We have not considered the effect of multiple scattering on single-particle imaging [7]. Recent results suggest that useful images at 3.6 Å resolution can be obtained from a virus as thick as 100 nm [30]. In the phase-object approximation, single scattering requires a phase-shift of less than ninety degrees as the beam traverses the sample. For a crystal, because molecules stack up in columns along the beam direction, this phase shift (relative to solvent columns) accumulates more rapidly than for singleparticle imaging, where the molecules are not ordered. So the tolerable thickness to avoid multiple scattering artifacts will be greater for single-particle imaging than for Bragg diffraction. The challenge of growing protein nanocrystals whose size falls within the domain of validity of the single-scattering approximation has been taken up by several groups. In particular the SONICC method [31] provides a crucial characterization tool that can identify protein crystals as small as one tenth of a micron, and has been used to enable the growth of protein nanocrystals for both membrane and soluble proteins through microfluidics [32]. Electrophoretic methods have also been used recently to sort nanocrystals by size in solution [33]. The success and versatility of protein nanocrystal growth and characterization techniques is thus directly linked to the development of TED for solving protein structures.
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Acknowledgments Supported by NSF STC Award 1231306. We are particularly grateful to Prof. U. Weierstall for useful discussions. References [1] R.M. Glaeser, K.H. Downing, Ultramicroscopy 52, 478 486 (1993). see also R.M. Glaeser and T.A. Ceska, Acta Crystallogr. A45, p. 620 (1989) for results at 20 kV and 120 kV. [2] J.C.H. Spence, High-Resolution Electron Microscopy, 4th ed., Oxford Univ. Press, Oxford, 2013. [3] J.C.H. Spence, J.M. Zuo, Electron Microdiffraction, Plenum Press, New York, 1992. [4] D. Shi, B.L. Nannenga, M.G. Iadanza, T. Gonen, eLife 2 (2013) (e01345). [5] J.M. Zuo, Rep. Prog. Phys. 67 (11) (2004) 2053–2103. [6] Diffraction Physics, in: J.M. Cowley (Ed.), 4th ed.,North Holland, New York, 1996. [7] G.R. Grinton, J.M. Cowley, Optik 34 (1971) 221. [8] D.J. DeRosier, A. Klug, Nature 217 (1968) 130. [9] G.R. Anstis, D.F. Lynch, A.F. Moodie, M.A. O’Keefe, Acta Crystallogr. A 29 (1973) 138. [10] J.T. McKeown, J.C.H. Spence, J. Appl. Phys. 106 (2009) 074309. [11] J. Gjønnes, A.F. Moodie, Acta Crystallogr. 19 (1965) 65–67. [12] W.J. Huang, J.M. Zuo, B. Jiang, K.W. Kwon, M. Shim, Nat. Phys. 5 (2009) 129. [13] I. Nederlof, Y.W. Li, M. van Heel, J.P. Abrahams, Acta Crystallogr. D 69 (2013) 852. [14] R. Glaeser, Electron Crystallography, J. Struct. Biol. 128 (1999) 3–14.
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Solving protein nanocrystals by cryo-EM diffraction: Multiple scattering artifacts Ganesh Subramanian a, Shibom Basu b, Haiguang Liu c, Jian-Min Zuo d, John C.H. Spence c,n a
Department of Materials Science and Engineering, Arizona State University, Tempe, AZ, USA Department of Chemistry and Biochemistry, Arizona State University, Tempe, AZ, USA c Department of Physics, Arizona State University, Tempe, AZ 85287-1504, USA d Department of Materials Science and Engineering, University of Illinois, Urbana, IL, USA b
art ic l e i nf o
a b s t r a c t
Article history: Received 20 March 2014 Received in revised form 27 August 2014 Accepted 31 August 2014 Available online 6 October 2014
The maximum thickness permissible within the single-scattering approximation for the determination of the structure of perfectly ordered protein microcrystals by transmission electron diffraction is estimated for tetragonal hen-egg lysozyme protein crystals using several approaches. Multislice simulations are performed for many diffraction conditions and beam energies to determine the validity domain of the required single-scattering approximation and hence the limit on crystal thickness. The effects of erroneous experimental structure factor amplitudes on the charge density map for lysozyme are noted and their threshold limits calculated. The maximum thickness of lysozyme permissible under the single-scattering approximation is also estimated using R-factor analysis. Successful reconstruction of density maps is found to result mainly from the use of the phase information provided by modeling based on the protein data base through molecular replacement (MR), which dominates the effect of poor quality electron diffraction data at thicknesses larger than about 200 Å. For perfectly ordered protein nanocrystals, a maximum thickness of about 1000 Å is predicted at 200 keV if MR can be used, using R-factor analysis performed over a subset of the simulated diffracted beams. The effects of crystal bending, mosaicity (which has recently been directly imaged by cryo-EM) and secondary scattering are discussed. Structure-independent tests for single-scattering and new microfluidic methods for growing and sorting nanocrystals by size are reviewed. & 2014 Elsevier B.V. All rights reserved.
Keywords: Protein crystallography Electron diffraction Multiple scattering Lysozyme Charge density map R-factor
1. Introduction The success of transmission electron diffraction (TED) for determination of the structure of two-dimensional (2D) protein crystals relies on the simplicity of two-dimensional indexing, and the relative absence of multiple elastic scattering perturbations on the Bragg scattered beam intensities, which increase with sample thickness. Reciprocal space consists of rods for 2D crystals, so that many more reflections intersect the Ewald sphere in this case. Glaeser et al. [1] have evaluated the importance of multiple elastic scattering on TED patterns from 2D protein crystals. At 200 keV, for monolayers that are typically less than 100 Å thick, they find only small perturbations. But at larger thicknesses, large perturbations generate Bragg intensities that bear no relationship to the kinematic or single-scattering values needed for structure analysis [2,3]. This extreme sensitivity of electron diffraction to sample
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Corresponding author. Tel.: þ 1 602 965 6486 E-mail address: [email protected] (J.C.H. Spence).
http://dx.doi.org/10.1016/j.ultramic.2014.08.013 0304-3991/& 2014 Elsevier B.V. All rights reserved.
thickness has so far prevented the method from being generally extended to thicker crystals, making electron diffraction results frequently un-reproducible. Recently however, Shi et al. [4] have reported 3D electron density maps from lysozyme crystals estimated to be 0.5 μm thick, at 3 Å resolution using TED at 200 keV. In this paper, we have estimated the maximum thickness of such protein nanocrystals at various electron beam energies that can be expected to produce reliable density maps at atomic resolution. We have also considered the importance of modeling and the availability of structure factor phase information. The protein we have studied is lysozyme. Since multiply scattered TED intensities are known to agree with high accuracy with experimental intensities in convergent beam electron diffraction (CBED) (see [2,5] for reviews), we used ZMult (multislice [6] program in [3]) to simulate diffraction intensities as a function of thickness for a 200 keV beam oriented along (001) in lysozyme. As is common practice in the cryo-EM community, we assumed the Bragg beam intensities are proportional to the X-ray structure factor amplitudes |F| (however, see [3]). We have also studied the dependence of electron density maps on the |F| values using the X-ray crystallographic program
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PHENIX and generated the 3D electron density maps for lysozyme crystals. The inevitable (beneficial) effects of sample bending are also discussed.
2. Background The importance of multiple scattering in cryo-EM imaging was first investigated by Grinton and Cowley [7] for single-particles, following the first tomographic TEM imaging experiments of DeRosier and Klug [8] and others. At 100 keV, Grinton et.al. found false detail in high-resolution TEM images (not diffraction patterns) at 6 Å resolution and 30 nm thickness, and severe distortions due to multiple scattering at 50 nm thickness of protein. At 3 Å resolution they expected severe distortion for 100 Å thickness. For diffraction from 2D crystals, Glaeser and Downing [1] found that thicknesses up to 100 Å might be used for beam energies above 100 keV. An important effect of multiple scattering is to preserve sample symmetry in the diffraction pattern (violating Friedel symmetry). So, a simple test for the presence of multiple scattering is failure of Friedel symmetry. Moreover, all protein crystals are non-centrosymmetric due to the presence of helical structures. For 4 nmthick bacteriorhodopsin at 16 Å resolution and 100 keV, Glaeser and Downing found an R-factor of 11% between Friedel pairs, increasing to 40% for a 200 Å thick multi-layer. They recommend averaging the pairs to reduce perturbations, and support a thickness limit between 100 and 200 Å of protein at energies above 100 keV for high resolution. A detailed study of the validity domain of the single-scattering (kinematic) approximation for light-element inorganic crystals can be found in Anstis et al. [9]. In experiments however, while Friedel symmetry is well preserved for 2D crystals (because of their relative insensitivity to orientation), it is a less useful test for 3D crystals because of their extreme sensitivity to diffraction conditions and orientation (unless convergent-beam methods can be used). The “kinematic convergent-beam method” has been used to solve an oxide nanocrystal, using the absence of intensity variation within the convergent-beam (microdiffraction) disks as a test for the absence of multiple scattering [10]. Finally, the presence of significant intensity in reflections, which should be absent due to the presence of screw and glide symmetry elements is often used as an indication of multiple scattering. In fact, for an unbent thick perfect crystal, (conditions rarely met in protein crystallography unless a very small beam diameter is used) these reflections remain absent for all thicknesses and beam energies as a result of an elegant cancellation theorem [11]. Taken together, under the normal conditions of strong multiple scattering, great thickness sensitivity, and specimen imperfections, it is almost impossible to record the same set of Bragg beam intensity ratios on two separate occasions using an electron microscope, unless the sample consists of a monolayer, and a very small beam is used. These TED intensities (unlike XRD intensities) are, therefore, not normally sufficiently reproducible for structure analysis. Those collected from a particular region of known thickness, however, may be analyzed quantitatively in great detail using the convergent beam method [12]. At the same time, the sub-nanometer beam diameter of the electron microscope allows ‘perfect crystal’ data collection from regions smaller than one “mosaic block” for inorganic crystals. Striking images of these mosaic regions (perhaps 50 nm in width) in protein have been directly imaged at high resolution, for the first time recently, by Nederlof et al. [13]. (The thickness of these “blocks” is not clear, they may be regions of locally bent crystal, as discussed below). We note therefore that by using a sufficiently small beam diameter, many of the effects of defects and disorder could be eliminated as the beam becomes smaller than one “mosaic block” or bent region. (These regions of local orientation change have
recently been imaged directly by cryo-EM for the first time, in lysozyme [13]). However electron microdiffraction in cryo-EM would require instrumental modifications to allow the beam intensity to be reduced independently of its size (so that the total number of electrons in the beam is not held constant during focusing), while radiation damage severely limits the size of the smallest beam which can be used. (See [14] for estimates of the smallest crystal which can give useful electron diffraction data when subject to damage limits). The simulations which follow do not allow for mosaicity or bending effects in electron diffraction, and so do refer to the case where a small microdiffraction beam is used. The effects of bending are discussed separately in Section 5. (See [15] for a review of TED and an extended discussion of the effect of dynamical perturbations on attempts to solve organic structures by electron diffraction, and the history of this subject).
3. Theory A review of the various methods used to simulate multiplyscattered Bragg intensities in thin crystals, and the relationship between them, is given in Spence [2]. The electron microdiffraction method, where TED patterns using beams of subnanometer dimensions are obtained, is reviewed in Spence and Zuo [3]. Electron diffraction differs from X-ray diffraction (XRD) in many ways, the most important of which for these purposes are i) At energies above 100 keV, the electron wavelength is much smaller than that used for XRD, leading to a much ‘flatter’ Ewald sphere. ii) For a perfect crystal, the rocking curve width in TED is a significant fraction of the Bragg angle, unlike XRD, for which it is much smaller. This, and the thickness (t) of the sample, affects the ‘partiality’ of the reflections. Using the large beam divergence needed to form a nanodiffraction pattern, it is then normally possible, given the relatively large unit cell of many proteins crystals and flat Ewald sphere, to record many full reflections, which are simultaneously at the Bragg condition lying on a plane in reciprocal space normal to the beam. iii) The interaction of fast electrons is far stronger than that of X-rays, producing a higher ratio of useful elastic scattering per unit energy deposited by damaging inelastic interactions, than X-rays (Henderson [16]). iv) The intensities of Bragg reflections are, in the kinematic approximation, proportional to Fourier coefficients vg of the electrostatic potential (in volts) for a crystalline sample, not the charge density, as studied by XRD [3]. In the kinematic approximation, the intensity of a Bragg beam g diffracted by a parallel-sided slab of crystal in the transmission geometry is I g ¼ ðσvg tÞ2 ½ sin ðπsg tÞ2 =ðπsg tÞ2
ð1Þ
where sg, the excitation error, measures deviation from the Bragg condition and σ¼π/(λVo)¼2πmeλ/h is an interaction constant, with Vo, the microscope accelerating voltage and λ, the electron wavelength. This applies to the case of small beam divergence angle. We see that in this single-scattering approximation, the intensity at the Bragg condition (sg ¼0) varies as the square of sample thickness, and so can be compared with the actual thickness variation predicted by accurate multiple scattering calculations, in order to determine the thickness range of this approximation for various resolutions and beam energies. For weaker beams, where the Bragg condition is not exactly satisfied, the equation also predicts a thickness-squared dependence. A degree of convergence in the illumination will span a range of excitation errors sg.
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4. Results and discussion We perform simulations intended to identify (i) the kinematic (single-scattering) thickness limit in proteins (within which a direct interpretation of the diffracted intensities is possible) and (ii) the significance of tolerance to error in |F| as compared to phases. All our studies were done with the protein lysozyme in the P43212 space group and unit cell parameters a ¼b¼ 79.1 Å and c ¼37.8 Å. Our structure factors were obtained from the relativistic Hartree–Fock calculations of Doyle and Turner for neutral atoms [17], and so take no account of bonding effects. These have been refined in detail against experimental electron diffraction intensities using inorganic crystals [3] where bonding effects are very small, but may be larger for the light C, O, N atoms of a protein, or graphene [18]. Hydrogen atoms were not included. First, we studied the variation of diffracted beam intensity with thickness for the lysozyme crystals, which was subsequently used to identify the kinematic limit for the different diffracted beams. Similar curves for paraffin layers are shown in [15]. Fig. 1a and b shows the characteristic ‘pendellosung’ oscillations in the intensities of low (4 4) and high (23 13) resolution diffracted beams (approx. 13 Å and 3 Å respectively) and their Friedel pairs at 200 keV. The oscillations of the diffracted beam intensities allow for an oscillation period or an “extinction distance” to be defined. However only in a two-beam theory (or kinematic theory away from the Bragg condition) are the oscillations periodic. Eq. (1) shows that a different extinction distance can be associated with every excitation error or point within the diffraction disk. (A similar result holds in two-beam theory). In this way, both the convergent-beam and precession methods [19], by including a large range of orientations around the Bragg condition, average over many of these thickness periods and so reduce the rapidity of the thickness (Pendellosung) oscillations. Fig. 1(c and d) represent
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the same oscillation trends for the two diffracted beams, but at 400 and 1000 keV respectively. Allowing 10% error tolerance in the intensities, we calculated the kinematic (single-scattering) limit as tabulated in Table 1. At 3 Å resolution (23 13), this limit was observed to be 227.4 Å at 200 keV. This kinematic limit increases [2] with incident beam energy, thus indicating a possible use of higher incident beam energies to support the study of thicker protein samples that still diffract within the kinematic regime. But high beam energy increases the probability of knock-on radiation damage (especially for light elements). The reduction in ionization damage, but rapid increase in such “knock-on” ballistic damage, with increasing beam energy has been thoroughly studied and reported in the literature for beam energies up to 3 MeV (see [2,20] for reviews). In protein crystallography, it is widely accepted that about 2/3 of the information in a charge density map comes from structure factor phases (commonly obtained using the molecular replacement method, using a similar protein as model) and 1/3 from the amplitudes [21,22]. To evaluate this, we introduced random errors
Table 1 Variation of kinematic thickness in lysozyme with incident beam energy. Incident energy
Diffracted beam
Kinematic thickness (Å)
200 KeV
(4 4) (23 13)
568.5 227.4
400 KeV
(4 4) (23 13)
682.2 341.1
1 MeV
(4 4) (23 13)
720.1 644.3
400KeV
I (4 4) I (-4 -4)
100 x I(23 13)
I(4 4)
I(a.u.)
I(a.u.)
200KeV
t(A°)
t(A°)
1000KeV
I (23 13) I (-23 -13)
100 x I(23 13)
I(4 4)
I(a.u.)
I(a.u.)
200KeV
t(A°)
t(A°)
Fig. 1. Diffracted beam intensity vs thickness for lysozyme crystals. At 200 keV, (a) (4 4) and its Friedel pair ( 4 4) and (b) (23 13) and its Friedel pair ( 23 13). (4 4) and (23 13) at (c) 400 keV and (d) 1 MeV.
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Fig. 2. Omit Map for Lysozyme (residues 50–60 omitted from refinement) at 1.5σ. Each sub-plot contains charge density maps with increasing random errors introduced in the experimental diffracted intensities. The sticks in green are some of the omitted residues. Red wire mesh corresponds to ‘zero’ error omit map. (a) 20% error (pink wire-mesh) overlapped with zero error. (b) 50% Error (sky-blue wire-mesh) overlapped with zero error. (c) 70% Error (marine-blue wire-mesh) overlapped with zero error. (d) 100% Error (white wire-mesh) overlapped with zero error. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
to the x-ray intensities but maintained accurate phases. This way, when the charge density maps were obtained with the erroneous intensities, we could identify the percentage error beyond which, the real-space charge density information starts to be substantially different from ideal. X-ray diffraction intensities and phases for lysozyme were obtained from 4ET8.pdb [23]. Keeping the phases unchanged, we introduced random errors to the intensities. We then generated an omit map (omitting residues 50–60 from the refinement process), as in Fig. 2, where the charge densities (wireframes) were laid over the ideal structure (sticks) from the model. For a perfect fit these should agree. The red wire-mesh corresponds to ideal (zero-error) omit- map. Thus Figs. 2(a–d) correspond to charge density maps from 20%, 50%, 70% and 100% error in X-ray intensities (10%, 22%, 34% and 41% errors in |F| respectively) overlaid upon the density map with zero-error. With increasing error in the intensities, we can notice that the charge density (wire-mesh) does not match with the atomic positions. Also, the noise in the data increases with error. For this specific case, if we notice the regions around the residues 50, 51 and 59 (labeled in figure), we note that at 34% error in |F| and beyond (Fig. 2c and d), the electron density substantially fails to match the model. In fact, at 41% error (Fig. 2d), electron density is almost completely absent around the labeled residues. The threshold error in |F| in this case can, thus, be taken at 34%. However, since the random error was introduced to the diffracted intensities (reciprocal space), whereas the density maps are plotted in real space, different regions in the real space charge density might show different extents of match/mismatch for the same amount of random error introduced. Nevertheless, for any protein, there are
always specific regions that are characteristic of their structural/ functional behavior and the accuracy/inaccuracy in the structures of these regions typically determine the maximum error permissible in experiments. This can, for example, be understood using the following case where we identified the threshold error beyond which, similar (but not identical) amino acid sequences from hen and turkey lysozymes could not be differentiated. The hen-lysozyme possesses His15 (i.e., Histidine residue at position 15 in the sequence) amino acid while the turkey-lysozyme possesses Leu15 (i.e., Leucine residue at position 15 in the sequence), everything else remaining the same. For a model based on turkey-lysozyme and experiments based on hen-lysozyme, one would expect the overlap of the model (sticks) with the experimental charge density (wire-mesh) to not be perfect at that specific amino-acid position. In Fig. 3, an Fobs Fcalc difference map is plotted for this exact case. As expected, we observe positive charge densities (wire frame) falling outside the model in (a), (b) and (c) (corresponding to errors in |F| up to 34%). But with (d), we notice that there is no such charge density outside of the model. The absence of such positive density (at 41% error) indicates the threshold error beyond which, the ability to distinguish the hen-lysozyme from the turkey lysozyme is lost (despite using accurate phase information). Although Figs. 2 and 3 were mapped using x-ray diffracted intensities, our conclusions will also apply to similar data obtained from electron diffraction experiments. How this affects the maximum permissible crystal thickness under the kinematic limit is now discussed. The R-factor is a statistical parameter widely used in crystallography that measures how close the experimental |F| values are
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0% error in |F|
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22% error in |F|
41% error in |F|
34% error in |F|
Fig. 3. Fo Fc difference maps for Turkey (model)–Hen (experiment) lysozyme at 3.0σ. Density outside the model (wire frame) indicates distinguishability, absent with 41% error in |F|.
Fig. 4. R-factor vs thickness for lysozyme at 200 keV.
in relation to the crystallographic model. It is expressed as Rf actor ¼
∑jðjF obs j jF calc jÞj ∑jF obs j
ð2Þ
where |Fobs| and |Fcalc| are the is the experimental and model structure factor amplitudes respectively. In protein crystallography, a practically accepted limit for R-factor is 0.3. Wilson finds an
extreme upper limit in the R-value of 0.59, for non-centrosymmetric crystals such as proteins, beyond which the model can have the same symmetry but a completely random atomic arrangement [24]. As long as the protein crystal thickness is within the kinematic (single-scattering) limit, the R-values are well within the reasonable limit of 0.3. But for larger thicknesses, the multiply scattered intensities start to vary significantly from the
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single-scattered intensities. This is equivalent to artificially introducing errors in the ideal single-scattered diffracted intensities, as in Figs. 2 and 3. Thus, we can calculate R-factor as a function of the crystal thickness to establish a maximum crystal thickness, under the single-scattering assumption. For this, we used multiple diffracted beams (90 in total) of lysozyme at 200 keV. The structure factor amplitudes corresponding to 37.9 Å (1 unit cell thickness) were considered as the |Fobs|, with respect to which, the structure factor amplitudes of higher thicknesses (|Fcalc|) were evaluated. From Fig. 4, the maximum thickness of lysozyme crystal that can be studied for Rr0.3 was 830 Å. For thicknesses greater than 2300 Å, R Z0.59. Therefore, beyond 2300 Å thickness, the |F| values obtained from diffracted intensities completely fail to specify the atomic arrangement of the lysozyme crystal. This calculation was done with (10 10 0) as the highest order diffracted beam, which corresponds to roughly 6 Å resolution. A more rigorous study would include thousands of diffracted beams, up to the resolution of interest. We expect such a study to decrease the limit of acceptable thickness within the single-scattering approximation, below that presented here. Several tests for the absence of multiple-scattering perturbations exist, most of which require knowledge of the sample structure. However, the observation of uniform intensity in convergent-beam disks [10] does not, nor does the observation of Friedel symmetry (which is destroyed by multiple scattering) in samples known to be acentric, such as proteins. Also (as mentioned earlier), for samples whose space-group is known, reflections that are absent as a result of screw or glide planes may occur through multiple scattering. In fact, the Gjonnes-Moodie cancellation theorem [11] shows that these reflections remain absent for all thickness and beam energies, as long as they are oriented along certain specific beam directions, so that a test for multiple scattering based on observation of absent reflections fails, for perfect crystals. For proteins however, it is often a useful test, since the samples are usually bent, and hence a range of orientations is illuminated that is greater than the angular window over which cancellation of multiple-scattering paths occurs. Lastly, experience in other fields [19,25,26] shows that dynamical interactions are reduced in orientations away from prominent zone axes, since, in the 2-beam dynamical theory, the periodicity of beams is inversely proportional to the structure-factor magnitudes, and so increases for weak reflections at the Bragg condition. Although zone axes orientations excite the maximum number of strongly interacting reflections simultaneously, they are needed to synthesize the major high-symmetry projections of the density map. However, by contrast, as shown by Eq. (1), the short thickness periodicity of weak reflections, which are well away from the Bragg condition, is reduced to 2/sg, and so should be avoided.
5. Discussion and conclusions The importance of multiple scattering as a limiting factor in cryo-EM structure analysis of protein microcrystals has been studied. (i) From our multislice simulations for perfectly ordered hen-egg lysozyme, we find that the TED single-scattering limit at 3 Å resolution for lysozyme at 200 keV is about 230 Å at a zoneaxis orientation for 10% error in intensities. (ii) At higher incident beam energy, the kinematic limit increases (to about 650 Å at 1 MeV for 3 Å resolution) but at the risk of a high probability of knock-on radiation damage. (iii) By introducing random errors to |F| for lysozyme (while maintaining the correct phases) we find that at around 35% error in |F|, the charge densities obtained from the erroneous |F| start to substantially deviate from the ideal model, at the region of interest. The ability to distinguish hen and turkey lysozyme is lost at about 40% error in |F|. The high tolerance to error in diffracted intensities (a 40% error in |F| corresponds to almost
100% error in diffracted intensity) is a consequence of the inherently greater phase dependence of charge density maps. Although this seems to suggest an advantage in using TED, with its high tolerance to error in intensities, it emphasizes the necessity of accurate phase information (for which one is dependent on prior experiments [23]). (iv) R-factor analysis over a subset of simulated diffracted beams show that the maximum crystal thickness, for perfectly ordered protein crystals, that can be used to maintain a reasonable value (Ro0.3) is about 1000 Å. These results are, however, limited by the fact that (a) although all proteins have similar atomic density and composition, the thickness limit of the TED method, (in this case given for lysozyme) will vary depending on the particular crystal structure. Nanocrystals with a large unit cell produce weaker diffracted beams for the same crystal volume, and a more dense reciprocal lattice. We therefore expect a larger tolerable thickness for larger unit cells, as is confirmed by detailed simulations for inorganic oxides [2], and the theory [27] of scattering by nanocrystals that are a few unit cells thick in all three dimensions. (b) this paper takes no account of mosaicity or crystal bending. It applies strictly to microdiffraction patterns obtained from very small regions of perfect crystal. Theories of mosacity in electron diffraction have been published (see [15] for a review) in which coherent multiple scattering within mosaic blocks is coupled to incoherent multiple scattering of intensities between blocks. However experimental evidence [28,13] suggests that continuous bending is more likely for TEM samples than the tilted block structure assumed in the much thicker X-ray samples, combined with the “secondary scattering” of intensities between layers proposed by Cowley (see [28] and references therein). This can have beneficial effects for structure analysis by TED, since the thickness oscillation periods shown in Fig. 1 are proportional to the inverse of the deviation sg from the Bragg condition (Eq. 1). As in the precession electron diffraction method [19], by spanning a range of orientations, each with a definite value of sg, the effect is to smooth out the thickness oscillations. We note that in a recent paper [29] it is claimed that useful structure analysis can be achieved by TED using protein crystals in the 160–400 nm range (R 25% at 0.25 nm resolution). Some of this increase beyond our estimate of 83 nm may be due to this bending effect. It would seem that the data analysis algorithms used for XFEL nanocrystal data, which must take account of mosaicity when merging partial reflections, could benefit from these direct near-atomic resolution real-space cryo-EM images of the bending and misoriented domains [13]. We have not considered the effect of multiple scattering on single-particle imaging [7]. Recent results suggest that useful images at 3.6 Å resolution can be obtained from a virus as thick as 100 nm [30]. In the phase-object approximation, single scattering requires a phase-shift of less than ninety degrees as the beam traverses the sample. For a crystal, because molecules stack up in columns along the beam direction, this phase shift (relative to solvent columns) accumulates more rapidly than for singleparticle imaging, where the molecules are not ordered. So the tolerable thickness to avoid multiple scattering artifacts will be greater for single-particle imaging than for Bragg diffraction. The challenge of growing protein nanocrystals whose size falls within the domain of validity of the single-scattering approximation has been taken up by several groups. In particular the SONICC method [31] provides a crucial characterization tool that can identify protein crystals as small as one tenth of a micron, and has been used to enable the growth of protein nanocrystals for both membrane and soluble proteins through microfluidics [32]. Electrophoretic methods have also been used recently to sort nanocrystals by size in solution [33]. The success and versatility of protein nanocrystal growth and characterization techniques is thus directly linked to the development of TED for solving protein structures.
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