The Geometry of Biomolecular Solvation 1. Hydrophobicity Patrice Koehl Computer Science and Genome Center

The Importance of Shape KKAVINGEQIRSISDLHQTLKK WELALPEYYGENLDALWDCLTG VEYPLVLEWRQFEQSKQLTENG AESVLQVFREAKAEGCDITI Sequence Structure Function ligand

Enzyme – Substrate Binding + Substrate (ligand) Enzyme (receptor) Induced Fit

Receptor Ligand Co-factors may induce the fit: allostery Co-factors bind Co-factors induce conformational Change: allostery Ligand binds

Energy Landscape Unfolded State Expanded, disordered Molten Globule Compact, disordered Native State Compact, Ordered Barrier Height 1 ms to 1s 1  s Barrier crossing time ~ exp[Barrier Height]

Biomolecular Solvation Stability of Protein Structures Geometric Measures of Protein Structures Applications Accessibility Binding sites

Biomolecular Solvation Stability of Protein Structures Geometric Measures of Protein Structures Applications Accessibility Binding sites

Energy of a Protein Bonded Interactions (chemistry) Bonds, Angles, Dihedral angles Non Bonded Interactions (“local” information) van der Waals interactions, Electrostatics Solvent (environment) Most difficult

Atomic interactions Torsion angles Are 4-body Angles Are 3-body Bonds Are 2-body Non-bonded pair

Forces between atoms Strong bonded interactions b   All chemical bonds Angle between chemical bonds Preferred conformations for Torsion angles: -  angle of the main chain -  angles of the sidechains (aromatic, …)

Forces between atoms: vdW interactions 1/r 12 1/r 6 R ij r Lennard-Jones potential

Forces between atoms: Electrostatics interactions r Coulomb potential qiqi qjqj

Solvent Explicit or Implicit ?

Potential of mean force A protein in solution occupies a conformation X with probability: X: coordinates of the atoms of the protein Y: coordinates of the atoms of the solvent The potential energy U can be decomposed as: U P (X): protein-protein interactions U S (X): solvent-solvent interactions U PS (X,Y): protein-solvent interactions

Potential of mean force We study the protein’s behavior, not the solvent: P P (X) is expressed as a function of X only through the definition: W T (X) is called the potential of mean force.

Potential of mean force The potential of mean force can be re-written as: W sol (X) accounts implicitly and exactly for the effect of the solvent on the protein. Implicit solvent models are designed to provide an accurate and fast estimate of W(X).

+ + Solvation Free Energy W np W sol

The SA model Surface area potential Eisenberg and McLachlan, (1986) Nature, 319,

Surface area potentials Which surface? Molecular Surface Accessible surface

Hydrophobic potential: Surface Area, or Volume? (Adapted from Lum, Chandler, Weeks, J. Phys. Chem. B, 1999, 103, 4570.) “Radius of the molecule” Volume effect Surface effect For proteins and other large bio-molecules, use surface

Biomolecular Solvation Stability of Protein Structures Geometric Measures of Protein Structures Applications Accessibility Binding sites

Representations of Biomolecules Space-filling Model Cartoon

Computing the Surface Area and Volume of a Union of Balls

Power Diagram:

Computing the Surface Area and Volume of a Union of Balls Decomposition of the Space-filling diagram

Computing the Surface  rea and Volume of a Union of Balls ii ii ii Volume Surface Area

Computing the Surface  rea and Volume of a Union of Balls The weighted Delaunay triangulation is the dual of the power diagram

Computing the Surface  rea and Volume of a Union of Balls The dual complex K is the dual of the decomposition of the space-filling diagram

Protein Delaunay Complex K complex Pocket Computing the Surface Area and Volume of a Protein

Delaunay Complex K complex Pocket Computing the Surface Area and Volume of RNA P4-P6 domain Group I intron

Biomolecular Solvation Stability of Protein Structures Geometric Measures of Protein Structures Applications Accessibility Binding sites

H1’ HO2’ H2’ H4’ H3’ H5’ H5’’ Experimental measures of accessibilities Hydroxyl radical footprinting:

Residue number Footprinting count / Ribose H accessibility

BINDING POCKETS IN 16S RIBOSOMAL RNA PDB structure: 1HZN Hygromycin B

Probe Size 1.4 Å 8 Å BINDING POCKETS IN 16S RIBOSOMAL RNA