How round is a protein? Exploring protein structures for globularity using conformal mapping.

Area of Science:

• Computational biology
• Geometric analysis
• Biophysics

Background:

• Protein structure analysis often relies on geometric descriptors.
• Quantifying the deviation of protein surfaces from ideal shapes like spheres is challenging.
• Conformal mapping is a potential tool for protein shape parametrization.

Purpose of the Study:

• To introduce a novel algorithm for computing the geometric difference between protein surfaces and spheres.
• To develop a quantitative measure of protein surface roundness and globularity.
• To explore the application and limitations of conformal mapping in protein shape analysis.

Main Methods:

• Developed an algorithm to compute discrete conformal maps between triangulated genus zero surfaces (protein and sphere).
• Minimized a symmetric elastic energy (E S (f)) to measure the map's deviation from isometry.
• Applied the method to Platonic solids and diverse protein structures.

Main Results:

• The E S (f) energy successfully quantified the roundness of Platonic solids, correlating with traditional sphericity measures.
• The method captured both global and local geometric differences, identifying protruding regions in proteins.
• E S (f) quantifies how these features deviate protein shapes from globularity.

Conclusions:

• The symmetric elastic energy (E S (f)) provides a robust measure of protein surface roundness and geometric deviation.
• The approach highlights limitations of conformal mapping for protein shape parametrization.
• The method shows potential for automatic registration of protein structures based on surface geometry.