Section: New Results

Flexible molecular fitting

Participants : Alexandre Hoffmann, Sergei Grudinin.

We have started a PhD on flexible molecular fitting. The first part of the PhD aims at developing a new method for non-rigid molecular fitting. The problem is the following : We have two proteins ${𝒫}_1$ and ${𝒫}_2$ and we know $d_1:{ℝ}^3\to ℝ$, the electron density of ${𝒫}_1$ and ${\left(Y_k\right)}_{k=0\cdots N_{atoms}-1}$, the average 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 minimizes 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 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 $d_2$.

We have already developed several algorithms based on the FFT to find $T_{rigid}$ and we now want to develop an efficient algorithm to find $T_{flexible}$.

The majority of algorithms first finds the best $T_{rigid}$ and then use Normal Mode Analysis (NMA) to improve their fitting, the problem with such a method is that one can miss the optimal solution. We aim at developing a method that uses convex optimization to find the best $T_{flexible}$ for each $T_{rigid}$ sampled on a grid, and therefore find the best $T$ possible on a grid.

The rest of the PhD will be focused on the improvement of the modeling of the atom's motion, by using machine learning algorithms and methods that go beyond linear NMA. We hope that such an improvement can improve the quality of the fitting method.