- Why calculate?
- Types: A.S., M.S., H.S.
- How calculate A.S.

- Why calculate?
- Voronoi construction for calculating volumes
- How to do this?
- Problems applying it to proteins
- Richards' solution
- Problems with his solution and ways of dealing with this

- Nitty gritty of how to do calculations in 3D. How to calculate with planes, lines, and so forth. This is generally useful.
- Case study of issues in taking off-shelf mathematical methods and applying them to proteins.

- Protein is solid object. Surface is where action takes place.
- Surface useful for docking and drug-design
- Hydrophobic energy proportional to surface area

- Accessible Surface
- Molecular Surface
- Hydration Surface

- Pick an arbitrary direction from which to view the protein. Slice it into many sections perpendicular to this direction.
- In each section, cycle over all the atoms. Each atom is represented as a sphere with a radius that is the sum of its VDW radius plus that of a probe solvent -- i.e. 1.4 for water.
- For each atom determine the circle corresponding to the intersection of this sphere with the sectioning plane. Remove all parts (i.e. arcs) of this circle occluded by the circles of other atoms.
- Multiply the total amount of non-occluded arc length by the sectioning width to get the surface area for atom. Sum over all atoms and all sections to get total area.

- Surround each atom with sphere of uniformly spaced dots (e.g. 92).
- Remove dots contained in other atoms spheres. Total number of remaining dots is accessible surface.

- C.S. = points of tangency between probe sphere and protein when probe sphere is only touching one atom
- R.S. = solid angle of probe sphere when tangent to two protein atoms

- Measuring packing efficiency is equivalent to quantifying Lennard-Jones energy (U = A/r^12 - B/r^6). L-J interactions, which are also known as Van der Waals (VDW) interactions, are the most basic type of interaction in molecular systems.
- It is observed that protein interiors are tightly packed, fitting together like a jig-saw puzzle. This essentially implies that VDW interactions are very important in structuring the protein interior.
- Because of tight packing the various types of protein residues and atoms occupy well-defined amounts of space.
- Tight packing is a driving force in ligand binding and is essential in the specificity of various recognition processes (e.g. antibodies and antigens)
- Protein packing is interesting because other substances have very different types of packing. For instance, water structure is dominated by H-bonding rather than tight packing.

- Nearest neighbor problems. The nearest neighbor of a query point is center of the Voronoi diagram in which it resides
- Largest empty circle in a collection of points has center at a Voronoi vertex
- Voronoi volume of "something" often is a useful weighting factor. This fact can be used, for instance, to weight sequences in alignment to correct for over or under-representation

- Connect all centers with lines (which are perpen. bisectors to edges)
- Border of D.T. is Convex Hull
- D.T. produces "fatest" possible triangles which makes it convenient for things such as finite element analysis.

- Each polyhedra vertex sits at the center of sphere that goes
through 4 atom centers (with no other atom centers being closer). So
using 4 sets of atom center coordinates (x,y,z), solve for the four
unknowns in

(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 .

- Central atom is atom 0
- Each neighboring atom has an index number (i=1,2,3...)
- Planes are denoted by the indices of the 2 atoms that form them (01)
- Lines are denoted by the indices of 3 atoms (012)
- Vertices are denoted by 4 indices (0123)

- Create artificial shell of water positions around the protein.
- Use molecular simulation to realistically position a waters around the protein.

R S = ----- D which is the same as R s = S r R + rwhere R is radius of the large atom and S is distance of the plane from the large atom, r and s are the analogous quantities for the small atom, and D=s+S.

- Vertices are replaced by tiny "error tetrahedrons"
- Typically these are very small, less than 1 part in 500, but they nevertheless spoil the mathematical purity of the procedure

-CH2- 23.681 2.037 -CH3 36.673 3.021 -CH= 21.126 1.938 -O 15.972 1.909 -OH 17.239 1.792 =O 16.813 2.101 -NH- 15.643 1.911 -NH2 23.380 2.304 -NH3 19.976 1.562 -S- 19.349 8.296 -SH 37.801 4.105 >C= 10.275 0.729 >CH- 14.570 1.217 C 9.247 0.668 CA 13.412 1.051 N 13.885 1.133 O 15.839 1.394 Water ~30

- Lee, B. & Richards, F. M. (1971). The Interpretation of Protein Structures: Estimation of Static Accessibility. J. Mol. Biol. 55, 379-400.
- Shrake, A. & Rupley, J. A. (1973). J. Mol. Biol. 79, 351.

- Richards, F. M. (1977). Areas, Volumes, Packing, and Protein Structure. Ann. Rev. Biophys. Bioeng. 6, 151-76.
- Connolly, M. (1983). Solvent-accessible Surfaces of Proteins and Nucleic Acids. Science 221, 709-713.

- Gerstein, M. & Lynden-Bell, R. M. (1993). What is the natural boundary for a protein in solution? J. Mol. Biol. 230, 641-650.

- J O'Rourke. Computational Geometry in C. Cambridge U.P.

- Richards, F. M. (1974). The Interpretation of Protein Structures: Total Volume, Group Volume Distributions and Packing Density. J. Mol. Biol. 82, 1-14.
- Richards, F. M. (1985). Calculation of Molecular Volumes and Areas for Structures ofKnown Geometry. Methods in Enzymology 115, 440-464.

- Gellatly, B. J. & Finney, J. L. (1982). Calculation of Protein Volumes: An Alternative to the Voronoi Procedure. J. Mol. Biol. 161, 305-322.
- Gerstein, M., Tsai, J. & Levitt, M. (1995). The volume of atoms on
the protein surface: Calculated from simulation, using Voronoi
polyhedra.
*J. Mol. Biol.***249**: 955-966.

- Y Harpaz, M Gerstein & C Chothia (1994). "Volume Changes on
Protein Folding,"
*Structure***2**: 641-649. - J Tsai, M Gerstein & M Levitt (1996). "Keeping the Shape but
Changing the Charges: A Simulation Study of Urea and its
Iso-steric Analogues,"
*J. Chem. Phys.*(in press).

v1 . u = K1 & v2 . u = K2 & v3 . u = K3 |K1 v1y v1z| / |v1x v1y v1z| ux = |K2 v2y v2z| / |v2x v2y v2z| |K3 v3y v3z| / |v3x v3y v3z|

A = 0.5 | Px Py | | Qx Qy |

P . Q x R 1 | Px Py Pz | V = --------- = - | Qx Qy Qz | 6 6 | Rx Ry Rz |