Integrating NOE and RDC using sum-of-squares relaxation for protein structure determination.

Area of Science:

• Structural Biology
• Computational Chemistry
• Biophysics

Background:

• Protein structure determination from Nuclear Magnetic Resonance (NMR) data is crucial for understanding biological function.
• Traditional methods like Nuclear Overhauser Effect (NOE) restraints are often sparse and imprecise.
• Residual Dipolar Coupling (RDC) measurements offer additional geometric information but lead to non-convex optimization problems.

Purpose of the Study:

• To develop novel computational methods for accurate protein structure determination using NMR data.
• To address the challenges posed by sparse NOE and noisy RDC measurements.
• To introduce a robust optimization framework for complex structural biology problems.

Main Methods:

• Modeling protein backbones as articulated structures of rigid units.
• Applying the sum-of-squares (SOS) hierarchy, a convex relaxation technique, to solve non-convex optimization problems.
• Developing two algorithms, RDC-SOS and RDC-NOE-SOS, with polynomial time complexity.

Main Results:

• The proposed SOS-based algorithms efficiently determine protein structures with high accuracy.
• These methods achieve results close to the theoretical Cramér-Rao bound (CRB) even with noisy RDC data.
• Successful application to ubiquitin structure determination, outperforming current state-of-the-art methods in speed and accuracy.

Conclusions:

• The SOS hierarchy provides a powerful and efficient approach for solving non-convex optimization problems in structural biology.
• RDC-SOS and RDC-NOE-SOS offer significant improvements in protein structure determination accuracy and speed.
• This work represents a novel application of SOS relaxation in structural biology, paving the way for future advancements.