1. 23 rd August 2005BioGeometry – Duke University1 Packing Quality of Atoms in Proteins Madhuwanti Vaidya Advisor: Herbert Edelsbrunner

2. 23 rd August 2005BioGeometry – Duke University 2 Introduction Packing Density. p = Covering Density. c = Volume of union of spheres Volume of space Sum of Volume of spheres Volume of space

3. 23 rd August 2005BioGeometry – Duke University 3 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.

4. 23 rd August 2005BioGeometry – Duke University 4 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.

5. 23 rd August 2005BioGeometry – Duke University 5 Current methods Voronoi volume method. (Richards F. M. 1974) Packing Density = Occluded surface method. (Fleming et. al. 1995) Packing Density = 1 - PP p Volume of atom Volume of Voronoi cell

6. 23 rd August 2005BioGeometry – Duke University 6 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.

7. 23 rd August 2005BioGeometry – Duke University 7 Drawbacks Dependent on surface solvent molecule. Do not simultaneously capture over-packing and under-packing. Without water With water

8. 23 rd August 2005BioGeometry – Duke University 8 Background Voronoi Diagram. Delaunay Triangulation. Circumsphere. Power Distance  p (x) =  x – p  2 – r p 2

9. 23 rd August 2005BioGeometry – Duke University 9 Background (contd) Weighted Voronoi Diagram. Weighted Delaunay Triangulation. Orthospheres: The unique sphere orthogonal to all four balls of the tetrahedron. || p – q || 2 = r p 2 + r q 2

10. 23 rd August 2005BioGeometry – Duke University 10 Background (contd) Positive OrthoradiusNegative Orthoradius ABO A B O

11. 23 rd August 2005BioGeometry – Duke University 11 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

12. 23 rd August 2005BioGeometry – Duke University 12 Local crowdedness (contd) Local crowdedness for each atom (a). S = {t | t  star of a}  a = (   t of t  S) / | S |

13. 23 rd August 2005BioGeometry – Duke University 13 Local crowdedness for Lattices RadiusPacking DensityCovering densityLocal crowdedness 1.0000.907 -0.143 1.1501.0001.2100.000 Hexagonal Lattice – 2D

14. 23 rd August 2005BioGeometry – Duke University 14 Local crowdedness for Lattices RadiusPacking DensityCovering densityLocal crowdedness 0.8660.680 -0.214 1.0000.9401.048-0.200 1.1181.0001.4640.000 BCC Lattice – 3D

15. 23 rd August 2005BioGeometry – Duke University 15 Local crowdedness for protein

16. 23 rd August 2005BioGeometry – Duke University 16 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.)

17. 23 rd August 2005BioGeometry – Duke University 17 Methods Establish Standard:  Consider only protons since they are sensitive to the measure.  Group them by four groups: Methyl (CH3) – VAL, ILE, LEU Methylene (CH2) – LEU, ILE, PHE, TYR Methanyl (CH) – PHE, TYR Methanyl (CH) – VAL, ILE, LEU  Calculate local crowdedness values for high-resolution data and NMR data.  Plot this as density estimates and compare.

18. 23 rd August 2005BioGeometry – Duke University 18 Results

19. 23 rd August 2005BioGeometry – Duke University 19 Methyl Hydrogens (VAL, ILE, LEU)

20. 23 rd August 2005BioGeometry – Duke University 20 Methylene Hydrogens (LEU, ILE, PHE, TYR)

21. 23 rd August 2005BioGeometry – Duke University 21 Methanyl Hydrogens (PHE, TYR)

22. 23 rd August 2005BioGeometry – Duke University 22 Methanyl Hydrogens (VAL, ILE, LEU)

23. 23 rd August 2005BioGeometry – Duke University 23 THANK YOU !!