Section: New Results

FFT-accelerated methods for fitting molecular structures into Cryo-EM maps

Participants : Alexandre Hoffmann, Sergei Grudinin.

We have developed a set of new methods for fitting molecular structures into Cryo-EM maps. The problem can be formally written as follows, We are given two proteins ${𝒫}_1$ and ${𝒫}_2$, and we also have $d_1:{ℝ}^3\to ℝ$, the electron density of ${𝒫}_1$ and ${\left(Y_k\right)}_{k=0\cdots N_{atoms}-1}$, the starting positions of the atoms of ${𝒫}_2$. Assuming we can generate an artificial electron density $d_2:{ℝ}^3\to ℝ$ from ${\left(Y_k\right)}_{k=0\cdots N_{atoms}-1}$, our problem is to find a transformation of the atoms $T:{ℝ}^3\to {ℝ}^3$ that minimize the $L^2$ distance between $d_1$ and $d_2$.

In image processing this problem is usually solved using the optimal transport theory, but this method assumes that both of the densities have the same $L^2$ norm which is not necessarily the case for the fitting problem. To solve this problem, one instead starts by splitting $T$ into a rigid transformation $T_{rigid}$ (which is a combination of translation and rotation) and a flexible transformation $T_{flexible}$. Two classes of methods have been developed to find $T_{rigid}$ :

• the first one uses optimization techniques such as gradient descent, and

• the second one uses Fast Fourier Transform (FFT) to compute the Cross Correlation Function (CCF) of $d_1$ and $d_2$.

The NANO-D team has already developed some algorithms based on the FFT to find $T_{rigid}$ and we have been developing an efficient extension of these to find $T_{flexible}$.