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

## TL;DR

This paper presents a quadratic programming method to approximate the placement of protein complexes into electron density maps, addressing the NP-hard problem with convex relaxations for better solutions.

## Contribution

It introduces a quadratic programming framework with convex relaxations to efficiently approximate protein placement in cryoEM density maps.

## Key findings

- Quadratic programming effectively models the protein fitting problem.
- Convex relaxations help find near-optimal solutions for NP-hard problem.
- Method improves accuracy of protein complex localization in density maps.

## Abstract

The paper investigates the problem of fitting protein complexes into electron density maps. They are represented by high-resolution cryoEM 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 introduce quadratic programming approaches to the problem. To find an approximate solution, we convert 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.