Section: New Results
FFTaccelerated methods for fitting molecular structures into CryoEM maps
Participants : Alexandre Hoffmann, Sergei Grudinin.
We have developed a set of new methods for fitting molecular structures into CryoEM maps. The problem can be formally written as follows, We are given two proteins ${\mathcal{P}}_{1}$ and ${\mathcal{P}}_{2}$, and we also have ${d}_{1}:{\mathbb{R}}^{3}\to \mathbb{R}$, the electron density of ${\mathcal{P}}_{1}$ and ${\left({Y}_{k}\right)}_{k=0\cdots {N}_{atoms}1}$, the starting positions of the atoms of ${\mathcal{P}}_{2}$. Assuming we can generate an artificial electron density ${d}_{2}:{\mathbb{R}}^{3}\to \mathbb{R}$ from ${\left({Y}_{k}\right)}_{k=0\cdots {N}_{atoms}1}$, our problem is to find a transformation of the atoms $T:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3}$ 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 NANOD 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 ${T}_{flexible}$.