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## 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 ${T}_{flexible}$.