Single-Particle Analysis (SPA)

The core computational challenge of SPA is determining the Euler angles (φ, θ, ψ) and in-plane translations (x, y) for each particle image. This is solved iteratively using Maximum Likelihood Estimation (MLE) or Expectation-Maximization (EM) algorithms. The process begins with a low-resolution initial model and refines orientation parameters as the 3D reconstruction improves.

• Projection Matching: Compares each 2D particle to re-projections of the current 3D model
• Angular Sampling: Searches orientation space with adaptive step sizes (e.g., 7.5° → 1.8°)
• Bayesian Framework: RELION's implementation uses a prior on orientations to regularize the search

Gold-Standard Fourier Shell Correlation (FSC) is the definitive method for estimating resolution without overfitting. The particle stack is randomly split into two independent half-sets before any 3D refinement begins. Both half-sets are reconstructed independently, and the FSC curve measures the correlation between their Fourier shells.

• FSC=0.143 Criterion: The spatial frequency where the FSC drops to 0.143 defines the reported resolution
• Overfitting Prevention: Processing half-sets independently prevents noise correlation
• Masked FSC: A soft mask excludes solvent noise for a more accurate resolution estimate

The Contrast Transfer Function (CTF) is a sinusoidal modulation of image contrast caused by the objective lens aberrations and defocus. Each micrograph has a unique CTF that must be computationally estimated and corrected before 3D reconstruction.

• Defocus Estimation: CTFFIND4 or Gctf fit Thon rings in the power spectrum to determine defocus values
• Phase Flipping: Corrects the alternating sign of CTF oscillations in Fourier space
• Wiener Filtering: Optimal amplitude correction accounting for the signal-to-noise ratio at each frequency
• Per-Particle CTF: Modern pipelines refine CTF parameters for individual particles, not just whole micrographs

Bayesian Polishing is a per-particle motion correction algorithm implemented in RELION that models and reverses beam-induced specimen movement. It operates on the movie frames of dose-fractionated data, tracking how each particle's position shifts during electron exposure.

• Motion Trajectories: Fits a 3D motion model to each particle's observed frame-to-frame displacements
• Radiation Damage Weighting: Optimally down-weights later frames where high-resolution information is lost
• B-factor Per Frame: Applies a relative B-factor to each frame to account for cumulative damage
• Re-exported Particles: Outputs polished particle stacks with enhanced high-resolution signal

Heterogeneous Refinement addresses the reality that cryo-EM samples often contain multiple conformational states or distinct compositions. This computational classification separates particles into structurally homogeneous subsets for independent 3D reconstruction.

• 3D Classification: RELION's implementation uses maximum-likelihood classification with a user-defined number of classes
• Multi-Class Ab-Initio: cryoSPARC generates multiple 3D models simultaneously from random initializations
• Continuous Heterogeneity: 3D Variability Analysis (3DVA) and CryoDRGN model a continuous landscape of motions rather than discrete classes
• Masked Classification: Focuses classification on a specific region of interest using a soft mask

Map Sharpening applies a negative B-factor weighting to the Fourier amplitudes of the final 3D reconstruction to restore high-frequency detail attenuated by the imaging process. This is essential for visualizing atomic features like side-chain density and bound ligands.

• B-factor Estimation: Derived from the Guinier plot of the FSC curve or by maximizing map connectivity
• Local Sharpening: Tools like DeepEMhancer and LocScale apply spatially varying B-factors using deep learning or local resolution estimates
• Amplitude Scaling: Matches the radial power spectrum to a reference (e.g., from an atomic model or X-ray crystallography data)
• Phenix Auto-Sharpen: Iteratively optimizes B-factors to maximize map interpretability