Job: Orientation Diagnostics

A new job in CryoSPARC v4.4+ to diagnose the presence of preferred orientation

Description

Orientation Diagnostics is a job (new in CryoSPARC v4.4) that can aid in diagnosing the presence of preferred orientation. It includes and builds upon 3DFSC (Tan et al., 2017) and Fourier Sampling (Baldwin & Lyumkis, 2020).

By default, Orientation Diagnostics reports a single number: the conical FSC area ratio, or cFAR. cFAR values below 0.5 generally indicate the presence of preferred orientation. cFAR accounts for both the viewing direction distribution and the signal content present within each particle by quantifying the variance of directional half-map Fourier correlations across the viewing sphere.

If particles are supplied, the job will also report the Sampling Compensation Factor or SCF* (Baldwin & Lyumkis, 2020 — see below for more information about the significance of the star). SCF* values below 0.81 generally indicate the presence of preferred orientation. SCF* characterizes the sampling of Fourier shells by considering the viewing directions of all particles, without accounting for the signal content contained therein. A junk particle is given equal weight to a real particle.

We provide a short summary of the two metrics below. Please see the detailed definitions at the end of this job guide or the Orientation Diagnostics tutorial for more information.

cFAR and SCF* at a glance...

	Range	Anisotropy Threshold	A function of…
cFAR	[0, 1] higher is better	0.5	alignments & particle images
SCF*	[0, 1] higher is better	0.81	alignments only

cFAR	SCF*	Conclusion
↑	↑	No orientation bias
↑	↓	? (inconclusive, perhaps a pathological alignment distribution)
↓	↑	Anisotropy due to junk or other contaminants
↓	↓	Anisotropy due to junk and/or preferred orientation

Input

• Volume

• [Optional] Particles

  • If supplied, Fourier sampling (and its associated metric, SCF*) will be computed along with other per-particle diagnostics

• [Optional] Mask

  • If supplied, half-maps will be masked prior to the computation of conical FSCs

Common Parameters

Conical FSC

• Number of Directions

  • The conical FSC Area Ratio (cFAR) metric is relatively robust to number of conical axis directions. However, reducing this number can speed up the job for volumes with large box sizes. Increasing this number will create denser spherical plots.

Sampling Compensation Factor

• Symmetry

  • Particle viewing directions will be expanded to account for symmetry. This parameter should be set to the symmetry applied in the upstream refinement.

Output

• 3DFSC volume

  • A volume of radial cFSC curves interpolated at each 3D voxel location

Common Next Steps

• Particle picking (to find missing views)

More details about cFAR and SCF*

cFSC Weighted Area-under-Curve (wAuC)

Given a conical Fourier shell correlation, $C_r (\hat{v})$, (where $r$ is the Fourier radius in wave number and $\hat{v}$ is the conical axis), we define the wAuC as

\begin{align}\text{wAuC}(\hat{v}) &\overset{\Delta}{=} \sum_r w_r C_r(\hat{v}) \\ &= \sum_r 4 \pi r^2 C_r(\hat{v}). \end{align}

In this weighted sum, the Fourier radius, $r$, ranges from the DC component to the radius at which $C_r(\hat{v})$ first crosses 0.143. Intuitively, for each Fourier radius $r$, the correlation between the half maps is multiplied by the surface area of the Fourier shell at $r$. The final result is proportional to the ‘mass’ of the cone in units of correlation.

cFSC Area Ratio (cFAR)

We define the cFSC as the ratio of the minimum to the maximum $\text{wAuC}(\hat{v})$ over $\hat{v}$, or

$\text{cFAR} \overset{\Delta}{=} \frac{\min_{\hat{v}} \text{wAuC}(\hat{v})}{\max_{\hat{v}} \text{wAuC}(\hat{v})}.$

Sampling Compensation Factor (SCF*) (Baldwin and Lyumkis, 2020)

SCF* measures orientation bias in particle viewing directions — it allows CryoSPARC to convert a potentially difficult-to-parse viewing distribution into a single metric.

SCF* is computed from the statistics of ‘Fourier sampling’. Given a single particle, Fourier sampling is a binary function that indicates which ‘slab’ (i.e., a stack of slices) of Fourier voxels are affected when the particle is used in back-projection. Summing this sampling over many particles allows one to see if large chunks of Fourier space are poorly sampled or missing completely. The degree to which this occurs is measured by the Sampling Compensation Factor—a single number that quantifies the extent to which anisotropic viewing direction distributions attenuate the global FSC value.

Mathematical Definition (expandable section)

To compute the SCF, we consider the sampling of Fourier bins at a particular radius, $R$. In total there are ~$2 \pi R^2$ unique bins in Fourier space. For each particle viewing direction $\hat{v}$, we compute its associated sampling via the slab condition (Baldwin and Lyumkis, 2020):

$\text{Sp}(\hat{k}, \hat{v}, R) \overset{\Delta}{=} \boldsymbol{1} ( \hat{k} \cdot \hat{v}\leq \frac{1}{2 R} ),$

where $\hat{k}$ is a unit vector that defines a Fourier voxel on the shell of radius $R$, and $\boldsymbol{1}(\cdot)$ is the indicator function. Intuitively, this function returns 1 for all Fourier voxels that belong to a ring of radius $R$, within a plane orthogonal to $\hat{v}$ (and 0 otherwise). Next, we sum $\text{Sp}(\hat{k}, \hat{v}, R)$ for all particle viewing directions to produce $\text{Sp}$, a single set of values that indicate the number of times each bin was sampled. The SCF can then be computed as

$\text{SCF} = \left( <\text{Sp}> <\frac{1}{\text{Sp}}> \right)^{-1},$

where $\frac{1}{\text{SP}}$ denotes element-wise reciprocals, and $<\cdot>$ is the arithmetic mean. This value is positive and always less than or equal to 1. Higher numbers indicate more uniform sampling distributions. If there are zeros amongst the sampling set, the above will be not defined. To account for this potential we report

$\text{SCF*} = \left(\frac{<\text{Sp}^*>}{p} \left(p <\frac{1}{\text{Sp}^*}> + q \right) \right)^{-1},$

where $q$ is the fraction of zero sampling bins, $p = 1 - q$, and $\text{Sp}^*$ are the non-zero sampling values. Note that if $q=0$, then $\text{SCF} = \text{SCF*}$. Please refer to (Baldwin and Lyumkis, 2021) for more details.

Plot Explanations

cFSC wAuC vs. conical axis

The wAuC of a cFSC curve is a proxy for directional signal content. If wAuC is relatively constant when the conical axis is varied, then the signal is isotropic in viewing direction. This plot helps illustrate the variation in cFSC wAuC and can aid in diagnosing the ‘structure’ of anisotropy.

Weighted Area under Curve (wAuC, shown via colour) vs. conical axis of cFSC (parameterized by azimuth and elevation).

cFSC resolution vs. conical axis

This plot is similar to the plot above, but visualizes the 0.143 crossing, rather than wAuC, of each cFSC curve.

cFSC resolution (i.e., the resolution at which the cFSC curve first crosses 0.143) vs. conical axis of cFSC (parameterized by azimuth and elevation).

Summary of cFSC curves

We summarize all cFSC curves in this plot, visualizing statistics as a function of spatial frequency rather than conical axis.

In blue: statistics over cFSC curves: mean, min, max, +/- one standard deviation plotted against spatial frequency. In green: histogram over 0.143 crossings of the same curves (cf., cFSC Resolution vs. conical axis plot above).

cFSC summaries within azimuth/elevation regions

To add back in a coarse notion of directionality to the above plot, we reproduce the same statistics for twelve different regions of axis space. This plot can help identify differences in cFSC variance to further elucidate the source of anisotropy.

cFSC curves for 12 different subsets of conical axes defined by azimuth / elevation limits. Inset: cFSC resolution vs. conical axis with a visual representation of the conical axis region (shaded).

3DFSC volume

The 3DFSC volume (Tan et. al, 2017) is another way to summarize cFSC curves by storing them in a volume whose voxels are interpolated from cFSC values at the nearest conical axis. In this plot, we visualize the 3DFSC volume via central slices. The ‘sphericity’ of the volume (which is output by the job) is a proxy for anisotropy.

Central slices of the 3DFSC volume (Tan et. al, 2017) composed of interpolated cFSC values. This volume can be inspected via Chimera(X).

Fourier Sampling

(Only generated if particles are connected). These plots visualize the Fourier sampling accumulated over a random subset of the particle viewing directions (default: 10000) — Fourier sampling is anti-podally symmetric (and hence only has ~ $2 \pi R^2$ bins) so we visualize only the $z>0$ hemisphere, following the original publication (Baldwin, P. R., & Lyumkis, D., 2020). N.B., the elevation / azimuth plot on the left should be visually similar to the Posterior Precision plot as posterior precision measures Fourier sampling modulated by the CTF.

The Fourier sampling $\text{Sp}$ (see equation in mathematical definition) visualized in 3D (right) and via an azimuth / elevation parameterization of the hemisphere (left).

Particle scale factor vs. viewing direction

(Only generated if particles are connected and per-particle scales are not all 1.0). This figure visualizes the average particle scale for a set of viewing directions (uniformly sampled on the viewing sphere).

(top) particle scale factors visualized by colour as function of particle viewing direction. (bottom) particle scale factor histogram (reproduced in other refinement jobs).

References

Tan et al. (2017), Addressing preferred specimen orientation in single-particle cryo-EM through tilting. Nat Methods 14(8), 793-796.

Baldwin, P. R., & Lyumkis, D. (2020). Non-uniformity of projection distributions attenuates resolution in Cryo-EM. Progress in biophysics and molecular biology 150, 160-183.