Cryo-EM Theory

Density Simulation

Atomic structures are often represented in Cryo-EM fitting as a sum of 3D Gaussians, representing the electrostatic potential or electron density. For an atom at position \(\vec{r}_i\), the density contribution at grid point \(\vec{x}\) is:

\[\rho_i(\vec{x}) = \exp\left( -\frac{|\vec{x} - \vec{r}_i|^2}{2\sigma^2} \right)\]

The total density map is the sum over all \(N\) atoms:

\[\rho_{total}(\vec{x}) = \sum_{i=1}^N \rho_i(\vec{x})\]

Cross-Correlation (CC)

The standard metric for comparing two EM maps \(\rho_A\) and \(\rho_B\) is the Pearson correlation coefficient (Rossmann, 2000):

\[CC = \frac{\sum (\rho_A - \bar{\rho}_A)(\rho_B - \bar{\rho}_B)}{\sqrt{\sum (\rho_A - \bar{\rho}_A)^2 \sum (\rho_B - \bar{\rho}_B)^2}}\]

In diff-em, this function is fully differentiable with respect to the atomic coordinates \(\vec{r}_i\), allowing for gradient-based structural refinement. This approach is consistent with modern differentiable volumes used in cryoDRGN (Zhong et al., 2021).