23 rd August 2005BioGeometry – Duke University1 Packing Quality of Atoms in Proteins Madhuwanti Vaidya Advisor: Herbert Edelsbrunner

23rd August 2005 BioGeometry – Duke University 1

Packing Quality of Atoms in Proteins

Madhuwanti VaidyaAdvisor: Herbert Edelsbrunner

23rd August 2005 BioGeometry – Duke University2

Introduction

Packing Density.

p =

Covering Density.

c =

Volume of union of spheres

Volume of space

Sum of Volume of spheres

Volume of space


Gap in Packing and Covering

spheres do not overlap p 1.

spheres cover entire space c 1.

Atoms in proteins overlap but do not cover the entire space. Technically neither definitions apply to proteins, though the term ‘packing density’ is commonly used.


Why measure packing density in proteins ?

Interior of a protein is tighter packed as compared to exterior.

Protein-protein interaction surface, in a protein-protein complex, consists of regions of high and low packing density.

Hot spots are more likely to be located in regions with higher packing density.


Current methods

Voronoi volume method.

(Richards F. M. 1974)

Packing Density =

Occluded surface method.

(Fleming et. al. 1995)Packing Density = 1 - PPp

Volume of atom

Volume of Voronoi cell


Current methods (contd)

Small-probe contact dot method. (Richardson et. al. 1999)

Score = w(gap) + 4*Vol(Hbond) – 10*Vol(Overlap)

Local Density. (Ban et. al. 2005)

Similar to Voronoi Volume method, but with local

analysis at atom-level.


Drawbacks

Dependent on surface solvent molecule.

Do not simultaneously capture over-packing and under-packing.

Without waterWith water


Background

Voronoi Diagram.Delaunay Triangulation.Circumsphere.

Power Distance

p (x) = x – p 2 – rp 2


Background (contd)

Weighted Voronoi Diagram.Weighted Delaunay Triangulation.

Orthospheres: The unique sphere orthogonal to all four balls of the tetrahedron.

|| p – q ||2 = rp 2 + rq2


Background (contd)

Positive Orthoradius Negative Orthoradius

A BOA BO


Local Crowdedness

Local crowdedness for each tetrahedron (t).

Range of values.

V = Volume of atoms in simplex (tetrahedron) W = Volume of orthogonal sphere

-V/4 ≤ W <

-1 ≤ t ≤ 1

t = V / (V + 2*W) - 1


Local crowdedness (contd)

Local crowdedness for each atom (a).

S = {t | t star of a}

a = ( t of t S) / | S |


Local crowdedness for Lattices

Radius Packing Density Covering density Local crowdedness

1.000 0.907 0.907 -0.143

1.150 1.000 1.210 0.000

Hexagonal Lattice – 2D


Local crowdedness for Lattices

Radius Packing Density Covering density Local crowdedness

0.866 0.680 0.680 -0.214

1.000 0.940 1.048 -0.200

1.118 1.000 1.464 0.000

BCC Lattice – 3D


Local crowdedness for protein


Application

Use local crowdedness to distinguish between well-packed and under-packed or over-packed regions in the protein.

Establish a standard using high-resolution X-ray data and evaluate NMR data based on that. (Along the lines of Andrew’s work in establishing local density as a standard.)


Methods

Establish Standard: Consider only protons since they are sensitive to the

measure.Group them by four groups:

Methyl (CH3) – VAL, ILE, LEUMethylene (CH2) – LEU, ILE, PHE, TYRMethanyl (CH) – PHE, TYRMethanyl (CH) – VAL, ILE, LEU

Calculate local crowdedness values for high-resolution data and NMR data.

Plot this as density estimates and compare.


Results


Methyl Hydrogens (VAL, ILE, LEU)


Methylene Hydrogens (LEU, ILE, PHE, TYR)


Methanyl Hydrogens (PHE, TYR)


Methanyl Hydrogens (VAL, ILE, LEU)


THANK YOU !!

Documents

23 rd August 2005BioGeometry – Duke University1 Packing Quality of Atoms in Proteins Madhuwanti Vaidya Advisor: Herbert Edelsbrunner