23 rd august 2005biogeometry – duke university1 packing quality of atoms in proteins madhuwanti...
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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
23rd August 2005 BioGeometry – Duke University3
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.
23rd August 2005 BioGeometry – Duke University4
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.
23rd August 2005 BioGeometry – Duke University5
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
23rd August 2005 BioGeometry – Duke University6
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.
23rd August 2005 BioGeometry – Duke University7
Drawbacks
Dependent on surface solvent molecule.
Do not simultaneously capture over-packing and under-packing.
Without waterWith water
23rd August 2005 BioGeometry – Duke University8
Background
Voronoi Diagram.Delaunay Triangulation.Circumsphere.
Power Distance
p (x) = x – p 2 – rp 2
23rd August 2005 BioGeometry – Duke University9
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
23rd August 2005 BioGeometry – Duke University10
Background (contd)
Positive Orthoradius Negative Orthoradius
A BOA BO
23rd August 2005 BioGeometry – Duke University11
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
23rd August 2005 BioGeometry – Duke University12
Local crowdedness (contd)
Local crowdedness for each atom (a).
S = {t | t star of a}
a = ( t of t S) / | S |
23rd August 2005 BioGeometry – Duke University13
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
23rd August 2005 BioGeometry – Duke University14
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
23rd August 2005 BioGeometry – Duke University15
Local crowdedness for protein
23rd August 2005 BioGeometry – Duke University16
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.)
23rd August 2005 BioGeometry – Duke University17
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.
23rd August 2005 BioGeometry – Duke University18
Results
23rd August 2005 BioGeometry – Duke University19
Methyl Hydrogens (VAL, ILE, LEU)
23rd August 2005 BioGeometry – Duke University20
Methylene Hydrogens (LEU, ILE, PHE, TYR)
23rd August 2005 BioGeometry – Duke University21
Methanyl Hydrogens (PHE, TYR)
23rd August 2005 BioGeometry – Duke University22
Methanyl Hydrogens (VAL, ILE, LEU)
23rd August 2005 BioGeometry – Duke University23
THANK YOU !!