Tools for visualizing and analyzing Fourier space sampling in Cryo-EM

Abstract

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 practitioners, 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.

Keywords: FSC; Reconstruction; Resolution; SCF; Sampling.

Figure 1.. Evaluation of Fourier space sampling for distinct orientation distributions.

Five different datasets taken from [8] are analyzed, each of which is generated from 10,000 projection images. The panels show (left) the projection distribution and (right) the surface sampling plot for a particular dataset. Values along the axes indicate directions on the unit sphere. On the surface sampling plots, the scale bar refers to a histogram of sampling values. Multiple different types of distributions are shown, ordered by the extent of sampling. The datasets correspond to: (A) uniform distribution, (B) Complement to “top-like” views, (C) pure “side-like” views, (D) “side-modulated” views, and (E) “top-like” views.

Figure 2.. Evaluation of Fourier space sampling for distinct orientation distributions, tilted by 30° and 45°.

A tilt angle was applied to each of the five datasets shown in Figure 1. The panels show (left) the projection distribution and (right) the surface sampling plot after applying either a 30° or a 45° tilt angle to each dataset.

Figure 3.. The effect of symmetry on sampling distributions – proteasome dataset from EMPIAR 10025.

The panels show (left) the projection distribution and (right) the surface sampling plot for each dataset. (A) Euler angles corresponding to a single asymmetric unit for the proteasome dataset. (B) Symmetrization of the distribution in A using D7 symmetry. (C) Euler angles corresponding to a single asymmetric unit of the proteasome, where the θ angle is restricted to the range 45° < θ < 90°, generating “side-like” (or top-complement) views. (D) Symmetrization of the distribution in C using D7 symmetry. (E) Euler angles corresponding to a single asymmetric unit of the proteasome, where the θ angle is restricted to the range 0° < θ < 45°, generating “top-like” views. (F) Symmetrization of the distribution in E using D7 symmetry.

Figure 4.. The effect of symmetry on sampling distributions – apoferritin dataset from EMPIAR 10216.

The panels show (left) the projection distribution and (right) the surface sampling plot for the deposited dataset. (A) Euler angles corresponding to a single asymmetric unit for the apoferritin dataset. (B) Symmetrization of the distribution in A using octahedral symmetry.

Figure 5.. Screenshot of the graphical user interface (GUI) to evaluate sampling and the effects of tilting.

Panels refer to the different options in the GUI and areas where the user has input: (A) the user must input a 3-column angle file and specify the number of particles, the symmetry, and the Fourier radius. (B) The “Calc Plots” button generates output plots. (C) Outputs include the surface sampling plot (top left), the original projection distribution (top right), and a histogram of sampling values for the Fourier voxels (bottom). The latter can be helpful to show, for example, the spread of sampling or how often values near zero occur. (D) A slider bar is used to assign a nominal tilt angle to the original projections. (E) The “Recalc Plots” button generates output plots after the application of a nominal tilt angle to the distribution. (F) The outputs here mimic those in C, except that each has been transformed in accordance to the specified tilt angle.