Tools for Visualizing and Analyzing Fourier Space Sampling in Cryo-EM

A complete understanding of how an orientation distribution contributes to a cryo-EM reconstruction remains lacking. It is necessary to begin critically assessing the set of views to gain an understanding of its effect on experimental reconstructions. Toward that end, we recently suggested that the type of orientation distribution may alter resolution measures in a systematic manner. We introduced the sampling compensation factor (SCF), which incorporates how the collection geometry might change the spectral signal-to-noise ratio (SSNR), irrespective of the other experimental aspects. We show here that knowledge of the sampling restricted to spherical surfaces of sufficiently large radii in Fourier space is equivalent to knowledge of the set of projection views. Moreover, the SCF geometrical factor may be calculated from one such surface. To aid cryo-EM researchers, we developed a graphical user interface (GUI) tool that evaluates experimental orientation distributions. The GUI returns plots of projection directions, sampling constrained to the surface of a sphere, the SCF value, the fraction of the empty region of Fourier space, and a histogram of the sampling values over the points on a sphere. Finally, a fixed tilt angle may be incorporated to determine how tilting the grid during collection may improve the distribution of views and Fourier space sampling. We advocate this simple conception of sampling and the use of such tools as a complement to the distribution of views to capture the different aspects of the effect of projection directions on cryo-EM reconstructions.


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Cryo-EM reconstructions using single-particle methods are appearing at an ever-increasing 3 rate, and the number of map depositions into the electron microscopy database is growing 4 accordingly [1][2][3]. Advances to technology are continuing to propel the field and open up 5 directions for numerous biological applications [4]. However, there remains a persistent 6 problem. Current vitrification technologies for preparing samples for imaging lead to 7 adherence of macromolecular specimens to one of the two air-water interfaces [5,6]. This 8 seems to affect the vast majority of specimens that are currently analyzed using cryo-EM [6].

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The interfaces attract hydrophobic patches on macromolecular surfaces, and 10 macromolecules therefore adopt "preferred orientations" with respect to the electron beam.

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One, or several, preferred orientations can exist within a given dataset. This results in a 12 non-uniform projection orientation distribution within a cryo-EM dataset.

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The orientation of a macromolecular object embedded within a vitreous ice layer and imaged 15 using an electron microscope can be described by three Euler angles, f, q, and y. In 16 conventional notation, the three Euler angles describe: f, the rotation of the object around the 17 Z axis, or the longitudinal position around the zenith of a sphere; q, the rotation of the object 18 around the new Y axis, or the latitudinal position about a sphere; y, the rotation of the object 19 around the new Z-axis, or the relative in-plane rotation [7]. For each image, the angles f and q 20 uniquely describe the relative position of the object on a sphere. A scatter plot of f versus q 21 angles provides a complete description of the set of orientations assigned to the object 22 within a cryo-EM dataset. Therefore, the standard manner by which to represent an 23 orientation distribution of a set of images in cryo-EM is to plot Euler angles f and q, and to 24 assess their relative distribution.

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We and others described efforts to provide a quantitative measure to a given orientation 2 distribution [8,9]. An uneven set of projection views compromises the cryo-EM 3 reconstruction. During the reconstruction procedure, each projection is inserted as a central 4 slice through the 3D Fourier transform of the object orthogonal to the direction of the 5 projection [10]. Therefore, non-uniformity in the set of projection distributions also leads to 6 non-uniformity in Fourier space sampling [8]. We refer to "sampling" simply as the number of 7 times that a representative point in Fourier space has been measured. There are two typical 8 cases to consider during the reconstruction process: (1) the case where all Fourier voxels 9 receive adequately high sampling and (2) the case where certain sets of contiguous voxels 10 have not been sampled at all.

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The first case occurs, for example, when projections are distributed azimuthally around the 13 imaged object. For a C/D-fold symmetric object, or for helical assemblies, a regular 14 distribution of projections about the object and perpendicular to the symmetry axis is 15 commonly referred to as "side-like" views. Modulations in the regularity of these views (i.e. 16 modulating the f angle) produces a set of side-like distributions, where there are less 17 well-sampled pockets of Fourier space. A typical second case occurs when projections do 18 not (approximately) form any great circle around the imaged object and there are no 19 additional orientations. In cryo-EM, the second scenario is commonly observed for 20 asymmetric objects that have a single preferential orientation; it can also be observed for 21 symmetric objects, but depends on the symmetry and the location of preferred orientation 22 with respect to the symmetry axis. This scenario leads to unsampled zero values in the 23 transform. Numerous synthetic case studies of both fully sampled distributions and those 24 containing missing information have been recently described [8]. For all the above cases, the 25 set of projection orientations are by definition non-uniform, leading to uneven angular 26 sampling of Fourier voxels. The non-uniformity of orientation distributions, and the 27 concomitant irregularity of the sampling, was shown to attenuate global resolution. For cases 1 characterized by zeros, there will also be a limit to the maximum attainable resolution, even 2 as the number of particles increases.

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Attenuation of global resolution due to non-uniformity can be accounted for by a geometrical 5 factor that we previously termed the sampling compensation factor (SCF) [8]. The name 6 derives from the idea that the non-uniformity in sampling leads (approximately) to a 7 non-uniformity in effective noise variance, but otherwise does not affect the envelopes 8 describing the reconstruction. The SCF is related to the average of the reciprocal sampling 9 over shells in Fourier space. The reciprocal sampling arises naturally in calculations where 10 noise is regrouped and effectively decremented in performing direct Fourier reconstructions.

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An underlying idea is that Fourier space filling more directly affects the signal-to-noise ratio; 12 however, it is harder to interpret the effect on the SSNR directly from the set of projection 13 views. Also, we pointed out that when there is missing data, it is necessary to assign 14 appropriate variance to unmeasured values of the transform. This leads to a more 15 appropriately defined version of the SSNR, which accounts for zeros, and is termed SCF*.

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This SCF* thus accounts for the decrement of the corrected SSNR due to the collection.

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Under control experiments, and assuming that orientations are perfectly established, the 18 SCF and SCF* appear to be adequate estimates for how resolution is attenuated due to 19 uneven sampling geometry. qualitative orientation distribution and a number, such as the SCF/SCF*, used to evaluate its 25 "goodness" is not intuitive to the cryo-EM practitioner. Indeed, most users are hard-pressed 26 to explain whether the set of views they have acquired are adequate for recapitulating 27 structural details of the imaged object at a defined resolution.

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To address current limitations, we developed a set of diagnostic tools to visualize Fourier 3 space sampling. In this work we introduce a plot, which describes the sampling on the 4 surface of a sphere as a representation of Fourier space coverage. We also developed a 5 graphical user interface (GUI), which helps users visualize both the orientation distribution 6 and Fourier space coverage side-by-side. The GUI facilitates understanding, in a more 7 intuitive manner, how missing data and/or sampling modulations can affect the 8 reconstruction. The GUI reports both SCF and SCF* values, which quantitatively describe the 9 sampling distributions. The SCF* values, as opposed to SCF values, are adjusted for the 10 presence of zeros in the transform and provide a more intuitive understanding of the effects 11 of sampling on resolution attenuation. We present multiple case studies to illustrate the 12 ideas. These tools should be generally relevant to the cryo-EM community and lead to a 13 better and more quantitative understanding of the effects of preferred specimen orientation 14 and non-uniform projection distributions. We wished to provide a framework that will allow the user to directly visualize the effects of a 5 projection distribution on Fourier space sampling. To this end, we developed surface 6 sampling plots, which provide a quantitative description of Fourier space coverage. Our 7 conception of sampling is as follows. Every point of Fourier space may or may not be 8 measured by a given projection. For each lattice site included in the sampling map, we 9 measure the total number of times that the region indicated by the point is measured by the 10 set of projections. We can use a set of lattice sites that are constrained to a surface of a given 11 Fourier radius, which is our sampling map constrained to a (hemi)sphere. Since the direction 12 of a projection is orthogonal to the 2D slice within the 3D transform of the object, an increase 13 in the number of projections along a particular view accordingly populates the surface 14 sampling plots orthogonal to the direction of the projection. The specific quantitative 15 considerations for the projection plots and the surface sampling plots are discussed in 16 sections 3.1 -3.6. The considerations when applying symmetry or an arbitrary tilt angle are 17 described in sections 3.7 -3.8. The effects of different orientation distributions on surface 18 sampling plots will be evident in the subsequent section.

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To determine how different projection orientation distributions contribute to surface sampling 23 plots, we examined multiple cases that arise in experimental settings. Five different 24 orientation distributions are demonstrated: (1) uniform, (2) top-complement, (3) side-like, (4) 1 side-modulated, and (5) top-like. We previously examined these distributions when we 2 introduced the SCF [8]. The first four cases represent fully sampled distributions, wherein 3 every Fourier voxel has been assigned some value during the reconstruction procedure. The 4 last case, the "top-like" distribution, results in a missing cone of information, the presence of 5 zeros in the transform, limits the attainable global resolution (as reported by the SSNR/FSC), 6 and as described previously, must be treated distinctly [8]. These distributions will now be 7 examined using surface sampling plots.

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This can be observed within the surface plots. For example, for an object characterized by a 6 60°-cone-complement (i.e. the q-angle is randomly modulated to 60°<q<120° or +/-30° 7 above the equator), there is an approximately two-fold increase in sampling toward the 8 Z-axis in comparison to the X/Y axes ( Figure 1B). The top-complement distribution is 9 characterized by SCF/SCF* values that range from 8/p 2 (~0.81) toward a maximum of 1, 10 where 8/p 2 is the theoretical value that has been described previously for perfect side views The side-like distribution describes a scenario wherein the q angle is restricted to a value of 16 90°, or along the equator, but with an otherwise uniform f distribution. For the case of side 17 views, it is well known that the sampling is complete [11]. In an experimental setting, the 18 side-like distribution is most closely associated with helical specimens [10]. For helical 19 specimens (especially long filaments), there is often a very small or negligible deviation of the 20 q angle from the equator. The side-distribution may also be associated with rotationally 21 symmetric single particles that adopt a preferred orientation to their side, but not their top 22 views. However, such cases typically have some amount of tilt angle modulation about the 23 equator, and therefore may be better characterized by a top-complement distribution with a 24 large cone. Again, there is a large accrual of sampling along the Z-axis compared to the X/Y 25 plane. However, the extent is more pronounced than for the top-complement case. For a 26 perfect side distribution, the accrual of sampling can range by up to an order of magnitude 1 ( Figure 1C). The SCF for the perfect side distribution evaluates to the theoretical value 2 of 8/ $ = .81 reported in [8] and the SCF* is identical to SCF, because there are no zeros. In 3 the side-like cases, the residual non-uniformity compromises the SSNR by ~20% in 4 comparison to uniform, and would therefore have a small effect on experimental resolution 5 attenuation. The side-modulated distribution is an extension of the side-like distribution and describes a 9 scenario wherein the q-angle is restricted to a value of 90°, but the f-angle is modulated to 10 varying extents. In an experimental setting, the side-modulated distribution can arise for 11 objects characterized by low rotational symmetry (e.g. C3, C4, etc.) and adhered to the 12 air-water interface along their side view (sampling for a C2-symmetric object, adhered to its 13 side view, behaves like a top-like distribution described below, as for example the HIV 14 intasomes) [12]. The higher the rotational symmetry, the closer the behavior will be to perfect 15 side views. Similar to the side-like distribution, there is an increased sampling along the 16 Z-direction at the expense of the X/Y plane. In addition, there are also pockets of incomplete 17 sampling around the perimeter. Depending on the modulation of the f-angle, these pockets 18 can be substantial. For example, when there are three full periods of f-angle modulation 19 (corresponding to C3 symmetry), the values within the gaps may closely approach zero 20 ( Figure 1D). In the most extreme cases, especially when the dataset is small (<10,000 21 particles), there could be zeros within the gaps in the transform. The side-modulated cases 22 are characterized by SCF values ranging from 0 < X < 8/p 2 . X depends on the symmetry of the

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Assuming that the dataset size is sufficiently large, the side-modulated case would almost 1 always be fully sampled. Nonetheless, the resolution may be attenuated by up to an order of 2 magnitude, depending on the amount of modulation.
The top-like distribution is a pathological distribution that describes a scenario characterized 6 by a single dominant view. It is one of the most common distributions encountered within 7 experimental cryo-EM datasets, due to the problem of preferred orientation [6]. It arises 8 because the specimen adheres to one of two air-water interfaces (AWI) in a single orientation 9 (with a random in-plane distribution), giving rise to a conical distribution of projection views.

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The top-like distribution is characteristic of any asymmetric object that adheres to the AWI in 11 a single orientation. It can also describe the sampling for rotationally symmetric objects that 12 adhere to the AWI along their top, but not their side interfaces, or for C2-symmetric objects 13 adhered to their side view. This is a pathological distribution that is, in the absence of other 14 views, characterized by zeros in the transform. For a 30° cone, one can readily observe 15 several features to such distributions ( Figure 1E). First, there is increased sampling closer to 16 the X/Y plane. The magnitude can vary significantly, depending on the size of the cone.

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Second, there are zeros in the transform, indicated by red regions on the plots. As described 18 previously, the zeros need to be treated differently; unit variances must be assigned to One of the motivations for developing sampling plots as a readout of Fourier space coverage 6 was to provide a quantitative framework with which to evaluate the effects of altering an 7 experimental orientation distribution. For example, it is known that an orientation distribution 8 can be altered through physical tilting of the microscope stage during the imaging 9 experiment [13,14]. To this end, we examined how different orientation distributions, and the 10 corresponding SCF values, would be affected by applying a nominal tilt angle. For each 11 distribution described in figure 2, we applied a transform to simulate a 30°-tilt or a 45°-tilt 12 angle inside the electron microscope (see section 3.8 for further details).

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The uniform distribution is not affected by the application of a tilt angle, as would be 15 expected. Accordingly, no changes are observed to the sampling distribution or the SCF* 16 value ( Figure 2A). For the top-complement case, the application of a tilt angle spreads out the 17 distribution to a nearly uniform Fourier coverage. The SCF* increases from 0.84 to 0.91 for 18 the 30° tilt and to 0.96 for the 45° tilt. The increased uniformity is evident both in the new 19 sampling plot and the new projection distribution ( Figure 2B). For the perfect side case, a tilt 20 angle converts the distribution into one that behaves similar to the top-complement scenario.

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The projections are pushed from a single thin band around the equator to a thicker band, due 22 to the tilt. The peak of the sampling down the z-axis is pushed to a ring encircling the z-axis, 23 which directly relates to the tilt angle. A 30° tilt angle improves the SCF from 0.81 to 0.86, and 24 a 45° tilt angle improves to 0.92 ( Figure 2C). A similar situation is encountered in the side-modulated case. However, now the SCF improves more substantially, from 0.07 to 0.81 1 and 0.90 for the 30° and 45° tilts, respectively ( Figure 2D). This implies that, with certain 2 low-symmetry distributions, the application of a small tilt angle may have substantial effects 3 to improve the quality of the map. The largest improvements to sampling arise for the cases 4 with missing data. Notably, when there is missing data, the SCF can be very small, ranging by 5 up to three orders of magnitude [8]. The introduction of additional orientations can 6 substantially improve the sampling. For example, in the case of a 60-degree missing cone, 7 the application of a 30° tilt angle improves the SCF* by over five-fold, while a 45° tilt angle 8 improves by fifteen-fold ( Figure 2E). This means that simply tilting the grid can have 9 substantial benefits to improving the reconstruction. Indeed, many experimental cases have Symmetry has the effect of increasing the number of views. For single-particle datasets, 19 symmetry multiplies the effective dataset size, and can also improve the SCF* by increasing 20 Fourier space coverage. It is worthwhile to distinguish these effects. The first effect is 21 straightforward: an n-fold symmetry effectively multiplies the dataset size by n. The second 22 needs to be considered more closely. Whereas preferred orientation affects both symmetric 23 and asymmetric samples, symmetry can mitigate the sampling problem. An example is the 24 20S proteasome, (EMPIAR-10025), displayed in Figure 3. This specimen has two preferred orientations within a region of the asymmetric unit -one along its "side" view and one along 1 its "top" view. Due to D7 symmetry, both orientation are equally likely to be observed in one 2 of fourteen different positions in the usual representation of projections on the sphere.

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Consequently, these two preferred orientations become repeated multiple times when the 4 object is reconstructed, and the resulting SCF* value improves from 0.04 to 0.84, by a factor 5 of ~20 ( Figure 3A-B). However, just because an object has symmetry does not mean that 6 symmetrization will lead to a more isotropic reconstruction. The isotropy depends on the axis 7 of preferred orientation and its relationship to the molecular symmetry. This is demonstrated 8 when the symmetry for the proteasome is split between its constituent side and top views. If 9 the axis of preferred orientation is positioned perpendicular to the seven-fold molecular 10 symmetry axis (i.e. along the "side-like" views), then symmetrization improves the SCF* by

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The presence of symmetry also simplifies the angular search space under experimental 20 conditions. Specifically, consider two equivalent scenarios: (1) assigning angles to a single 21 asymmetric unit, then symmetrizing the reconstruction; (2) assigning angles to the entirety of 22 the (hemi)sphere, then performing an asymmetric reconstruction. The resulting maps would 23 be identical. However, the first scenario is computationally less expensive. Therefore, many 24 algorithms will only search orientations within a confined space characterized by the point 25 group symmetry. For such cases, it is necessary to symmetrize the orientations in order to 26 properly determine the sampling and the related SCF values. Just like in the case with the 1 20S proteasome (Figure 3), particle orientations for apoferritin (EMPIAR-10216) are only 2 assigned to one asymmetric unit, out of its eight symmetry-related copies. If symmetry is not 3 specified, the resulting SCF* value is 0.03; however, when the proper octahedral symmetry is 4 taken into account, the SCF is correctly determined at near unity (0.99), which is consistent 5 with the isotropic reconstruction ( Figure 4). Therefore, it is necessary for users to properly 6 specify the appropriate symmetry to completely sample Fourier space when analyzing 7 orientation distributions for their datasets.  18 they agree at endpoints, and differ everywhere by less than 5%.

Projection-sampling transform 21
This sub-section discusses the linear relationship between the set of projection views and the 22 sampling from a formal perspective. By casting this relationship into the appropriate basis, 23 we argue that the set of projection views and the sampling restricted to a set of points on a 24 sphere, contain equivalent information, by showing that the relationship is invertible (up to the 1 accuracy determined by the spacing on this sphere). This is one of the main concepts 2 underlying the current work. We achieve this by showing that the linear mapping has 3 spectrum bounded in magnitude away from zero.

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The sampled points must be divided by the 4 $ available points. The ratio of these 21 quantities is 1/2k.

Invertibility of the transform 1
The equations of the form (3.1) can be brought to diagonal form, by means of a 2 transformation to spherical harmonics (essentially the so-called Funk-Hecke theorem [28]).

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The resulting equation in the harmonic basis is: where is the angular frequency and must be even, and is the standard orientational 6 parameter expressing the coefficients with respect to a special axis (commonly, ).

Thinning of the Sampling Function and Investigating Zeros 17 18
In supplemental Figure 2, we plot the product D b (k) as a function of k, for the kernel given 19 by (B.10). For the delta function approximant, given above, the product is just P b (0)/2 which 20 is a constant independent of k. The full expression for D b (k) asymptotes to this quickly 21 and for all l (although only l=4 is shown), meaning that the kernel, which maps projections to sampling, is seen to depend almost exactly on 1/Fourier radius. This implies that the patterns 1 should be minimally affected, with the only nontrivial aspect being the possible appearance 2 of zeros occupying Fourier space as the Fourier radius is increased. This idea is reinforced by 3 Supplemental Figure 2B where the sampling plots are generated at Fourier radii (FR) 20, 60, 4 and 180. For example, at thrice the distance from the origin, there are 9 times as many 5 sampling points on the sphere, but the number of points in a plane distant from the origin is 6 only thrice the original. Therefore, we expect that the sampling decreases as a factor of 3.

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Indeed, the patterning of the sampling for FR=60 is identical to the patterning for FR=20, but 8 with only one third the sampling across the surface of the sphere. Likewise, a histogram of 9 the sampling values over the points on a sphere shows a similar pattern, but with only one 10 third the sampling (value along the X-axis). Analogous behavior is observed for the case with 11 FR=180. For "top-like" views, a similar behavior is observed (Supplemental Figure 2C). The 12 one behavior that we observed is, when there is an excessive number of zeros in the 13 transform, the SCF* values tend to increase slightly with increasing Fourier radii. The detailed 14 behavior of SCF as a function of Fourier radii may depend slightly on lattice type and other 15 non-universal features, and is therefore beyond the scope of this work. The user is able to 16 re-enter the Fourier radius at which the sampling is determined via entry box on the GUI. in 17 this way, the patterning of the sampling can be compared as the Fourier radius is increased.

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As a default, the user can begin with a Fourier radius that is ½ the boxsize of the dataset, then

Sampling in a Fibonacci spiral on the sphere 2
For the purposes of plotting the sampling restricted to a sphere of Fourier radius, , a 3 Fibonnaci spiral is created with a number of independent points that should be attributed to 4 such a hemisphere: namely, 2 $ . If the uncertainty of angular assignment is , this should 5 correspond to an approximate maximum angular frequency of: hi = 1/( ) . Now the 6 maximum (even) angular frequency, jkl , that are supported on a sphere with 2 $ points, 7 can be found by equating this last number by the number of points that are described by all 8 even frequencies up to jkl , which is ( jkl + 1)( jkl + 2)/2. Ideally, we would like The last equality is the approximate solution for for the previous quadratic when large.

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Simply put, we should ideally like to sample the sphere more finely than the uncertainty in the

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For a good value of analysis for the Fourier radius, a final inequality emerges from the 20 consideration that we would like the amount of sampling at that radius to still be large on 21 average. Therefore

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(which we term SSNR* and has variance assigned to unmeasured values), is decremented 18 due to the projection views, as described in [8]. This decrement is termed SCF * . If the average 19 of the non-zero values of the reciprocal sampling is augmented by the ratio of the number of 20 zero to non-zero values, then the product with the average sampling will again be a number 21 that can be no less than unity. This product is 1/SCF * , and SCF * therefore is lower than one.

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The inverse of SCF* can be written as the average sampling of the occupied sites, times the 23 weighted average of the inverse sampling and unity. In formulae: , SCF * = 1/(1 + 3 − 1 ) . Note that SCF > 1, but SCF * ≤ 1. Indeed, we could have a situation where only a 4 small part of space is sampled ( << 1), but that part is sampled very highly ( ≫ 1) a 5 situation that we have termed "hyper-sampling". In that situation, SCF ≫ 1, but SCF * ≪ 1.

Sampling and Symmetry 10
In this sub-section, we describe how to formulate an expression for the sampling, when a 11 symmetry is specified. If the rotation of a map acting on a model is given by = † iˆ , then 12 the position on the projection sphere may be argued to be 13 14 which is the application of the transpose of the map transform (see Baldwin,Lyumkis [B.13]) 15 to the unit vector . Note that the last Euler angle, , is irrelevant to the product, which is why 16 we can parameterize a point on the projection sphere by the first two Eulers.

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Euler angles may fall within the asymmetric unit of some Platonic symmetry. Therefore, we 19 might wish a single projection to represent itself as well as those projections associated to it 20 by symmetry. Therefore, on the GUI there is a dropdown menu to select symmetry C, D or 21 Platonic solid symmetries. If a representative symmetry is given by a matrix operation Sym,

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then the original rotation, R, yields equivalent copies, R Sym, with Sym being applied first 23 because the symmetry pertains to the map, and not to the situation after rotation. Now, 1 points on the projection sphere are given by: Since the full set of symmetries is given by 2 where E is the number of edges, we get a 4 proliferation of unit normal by a factor of 60, 24 and 12 for icosahedral, octahedral and 5 tetrahedral symmetries respectively. Once the symmetry is chosen, the extra unit vectors are 6 created, and the sampling is created as described above in sub-section 3.4. Typical 7 application of these symmetries give much more complete sampling and higher values of 8 SCF (or SCF * ).

Geometry of Tilting 11
In this sub-section, we describe how tilting the grid gives rise to new sampling, and a 12 heuristic derivation of the type of patterning that is observed. The application of a tilt is to 13 apply an additional Euler angle to the string of operations listed in Eq (3.5), and which we can 14 assume to be with respect to the y axis: ''b' . This will change the effective Euler angle of a 15 given particle to a rotation given by: ''b' , with the ultimate point on the projection sphere 16 given by: This is how the projection is plotted after choosing a tilting angle with the tilt slider within the To understand the patterning that results from tilting, it is an instructive exercise to show that 5 tilting maps points on the projection sphere to circles centered around these points, under 6 the assumption that the last in plane rotation is uniformly distributed before the action of 7 tilting. Although, this is never strictly the case in practice, this reflects the general appearance 8 that is observed (points mapped to circles, due to tilting). To see this mathematically, we can 9 take the inner product of these points which will yield determining the effect of tilting on the sampling and on the same metrics. We envision that 13 future improvements will build upon this baseline framework established here.
14 15 16 There is a remaining problem in the current application of the SCF to experimental data.

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Since the orientations are not known de facto, we currently assume that they are properly 18 and correctly assigned during refinement. In other words, the SCF is calculated from a given   The delta function approximation to the integral is -(0)/(2 ), and is equivalent to (B.10) in 5 the large k limit. As shown in Supplementary Figs 1 and 2, the kernel, -( ), stays bounded 6 away from zero, and is well approximated by a functional dependence on , as 1/ , which is 7 the appropriate form for the thinning.

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In the context of tilting, for the expressions (3.9) to (3.10), the kernel in (3.10) becomes

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This particular expression is typically non-zero, although not strictly so: the implication being, 12 that one can still approximately reconstruct the original sampling, from its tilted version.