The Geometry of Biomolecular Solvation

1. Hydrophobicity

Patrice Koehl

Computer Science and Genome Center

http://www.cs.ucdavis.edu/~koehl/

The Importance of Shape

KKAVINGEQIRSISDLHQTLKKWELALPEYYGENLDALWDCLTGVEYPLVLEWRQFEQSKQLTENGAESVLQVFREAKAEGCDITI

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-factorsinduce conformationalChange: allostery

Ligand binds

Biomolecular Solvation

Stability of Protein Structures

Geometric Measures of Protein Structures

ApplicationsAccessibilityBinding sites

Biomolecular Solvation

Stability of Protein Structures

Geometric Measures of Protein Structures

ApplicationsAccessibilityBinding 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

SolventExplicit or Implicit ?

Potential of mean force

( )

( )

∫∫−

−

=dXdYe

eYXP

kT

YXU

kT

YXU

,

,

),(

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

),()()(),( YXUYUXUYXU PSSP ++=

The potential energy U can be decomposed as: UP(X): protein-protein interactions

US(X): solvent-solvent interactions

UPS(X,Y): protein-solvent interactions

Potential of mean force

∫= dYYXPXPP ),()(

We study the protein’s behavior, not the solvent:

PP(X) is expressed as a function of X only through the definition:

∫−

−

=dXe

eXP

kT

XW

kT

XW

P T

T

)(

)(

)(

WT(X) is called the potential of mean force.

Potential of mean force

The potential of mean force can be re-written as:

)()()( XWXUXW solPT +=

Wsol(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

Wnp

Wsol

VacchW−

SolchW

( ) ( )cavvdWvac

chsol

chnpelecsol WWWWWWW ++−=+=

The SA model

Surface area potential

∑=

=+N

kkkvdWcav SAWW

1

σ

Eisenberg and McLachlan, (1986) Nature, 319, 199-203

Surface area potentialsWhich surface?

MolecularSurface

Accessiblesurface

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

ApplicationsAccessibilityBinding sites

Representations of Biomolecules

Space-filling ModelCartoon

Computing the Surface Areaand Volume of a Union of Balls

Computing the Surface Areaand Volume of a Union of Balls

Power Diagram:

Computing the Surface Areaand Volume of a Union of Balls

Decomposition of theSpace-filling diagram

Computing the Surface reaand Volume of a Union of Balls

∑=

=N

iiiA

1

24 σρπ

i

i

∑=

=N

iiiV

1

3

3

4βρ

π

i

VolumeSurface Area

Computing the Surface reaand Volume of a Union of Balls

The weighted Delaunay triangulation is the dual of the power diagram

Computing the Surface reaand Volume of a Union of Balls

The dual complex K is the dual of the decomposition of the space-filling diagram

http://www.cs.ucdavis.edu/koehl/ProShape/

Protein Delaunay Complex

K complex

Pocket

Computing the Surface Areaand Volume of a Protein

Delaunay Complex

K complex

Pocket

Computing the Surface Areaand Volume of RNA

P4-P6 domainGroup I intron

Biomolecular Solvation

Stability of Protein Structures

Geometric Measures of Protein Structures

ApplicationsAccessibilityBinding sites

H1’

HO2’

H2’H4’

H3’

H5’

H5’’

Experimental measures of accessibilities

Hydroxyl radical footprinting:

Residue number

Fo

otp

rin

tin

g c

ou

nt

/ R

ibo

se H

acc

essi

bil

ity

BINDING POCKETS IN 16S RIBOSOMAL RNA

PDB structure: 1HZN

Hygromycin B

Probe Size

1.4 Å

8 Å

BINDING POCKETS IN 16S RIBOSOMAL RNA

BINDING POCKETS IN 16S RIBOSOMAL RNA