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## Section: New Results

### Quadratic Programming Approach to Fit Protein Complexes into Electron Density Maps

Participants : Alexander Katrutsa, Sergei Grudinin.

We investigated the problem of simultaneous fitting protein complexes into electron density maps of their assemblies. These are represented by high-resolution cryo-EM density maps converted into overlapping matrices and partly show a structure of a complex. The general purpose is to define positions of all proteins inside it. This problem is known to be NP-hard, since it lays in the field of combinatorial optimization over a set of discrete states of the complex. We introduced quadratic programming approaches to the problem. To find an approximate solution, we converted a density map into an overlapping matrix, which is generally indefinite. Since the matrix is indefinite, the optimization problem for the corresponding quadratic form is non-convex. To treat non-convexity of the optimization problem, we use different convex relaxations to find which set of proteins minimizes the quadratic form best.