Section: New Results

Towards the development of FFT-accelerated flexible fitting methods

Participants : Alexandre Hoffmann, Valerie Perrier, Sergei Grudinin.

We studied a set of new methods for non-rigid molecular fitting. The problem can be formulated as follows : Let ${𝒫}_1$ and ${𝒫}_2$ be two molecular structures (e.g. proteins). We are given $d_1:{ℝ}^3\mapsto ℝ$, the electron density of ${𝒫}_1$ and ${(Y_k\in {ℝ}^3)}_{k=1\cdots N_{atoms}}$, the average positions of the atoms of ${𝒫}_2$. Assuming we can generate an artificial electron density $d_2:{ℝ}^3\mapsto ℝ$ from ${(Y_k\in {ℝ}^3)}_{k=1\cdots N_{atoms}}$, our problem is to find a transformation of the atoms $T:{ℝ}^{3N_{atoms}}\mapsto {ℝ}^{3N_{atoms}}$ 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 $T_{flex}$. Two classes of methods have been developed to find $T_{rigid}$ :

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

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

We have developed several algorithms based on the FFT to find $T_{rigid}$ and we have developed two algorithms for flexible molecular fitting that are based on convex and non-convex optimization and the trust region methods. Our tests demonstrate that while one method gives good results for small deformations, the other gives good results for bigger deformations.

We have been also improving the current NMA method (which is essentially a model reduction technique), that is used in other tools such as the flexible fitting to small angle scattering profiles. Finally, we started the development of a method for a harder fitting/docking problem in which only electron density would be known. The basic idea would be to find the $C^1$-diffeomorphism $T:{ℝ}^3\mapsto {ℝ}^3$ that minimizes the $L^2$ distance between $d_1$ and $d_2$.

We developed several stand-alone C++ libraries to solve some of our problems including:

• a non-convex optimization library,

• a normal mode analysis library,

• a fitting library that implements our new methods.