Masato Masuya

Computing and Communications Center, Kagoshima University

masatom@biocomputing.cc

In computer simulation of proteins and nucleic acids, such as MD and/or MC, conformational energy + solvation energy calculation is needed. However, solvation energy calculation (implicit water, PB, RISM, etc.) is very time consuming. Therefore it can be applied to small system, or it can be included as implicit solvation effects, such as continuum model using distance dependent dielectric constant. While, there exist two methods that calculate solvation energy using solvent accessible surface area; ASP (atomic solvation parameters) model and GB/SA. In these methods, as solvent accessible surface area calculate fast, solvation energy could be calculate fast. So I decided to develop fast algorithm for molecular surface area calculation. NSOL is a fast solvation energy calculation program using fast molecular surface area calculation algorithm.

Solvent accessible surface is that part of the surface of a
sphere centered at an atom with *r _{vdW}* +

Atom type | Radius (Å = 10^{-10}m = 10^{-1}nm) | ||||

(oons, jrf_) | (we92, sch3, sch4) | Bondi^{*} | |||

C | Aliphatic | 2.00 | 1.90 | 1.70 | |

Aromatic | 1.75 | ||||

Carbonyl/Carboxyl | 1.55 | ||||

O | 1.40 | 1.40 | 1.52 | ||

N | 1.55 | 1.70 | 1.55 | ||

S | 2.00 | 1.80 | 1.80 | ||

H | N/A | N/A | 1.20 | ||

Water | 1.40 | 1.40 | N/A |

^{*}Bondi, J.Phys.Chem., 68, 441, 1964.

Analytical:

- Complex

Two spheres case, three spheres case, four or more spheres cases are exist. - Singularity

In very near case, surface area cannot be calculated. - Approximation

To avoid singularity, several approximation algorithms have been developed. - Easy derivative

First and second derivatives can be defined mathematically, but approximately.

Numerical:

- Simple

Not depends on spatial position of spheres. - Stable

Singularity problem never occurs. - Accurate

Depends on point distribution method and numbers of points. But accurate results can obtain at many points case. - Numerical derivative

Time consuming and error accumulation.

Because of its stability, I adapt a __ numerical__ method.

Surface area of atom group *i* is obtained by

,

where *r _{j}*: solvent accessible
radius (van der Waals radius + 1.4) of atom

It is permitted up to 12 points for distributing points equally. Several methods have proposed so far:

- Tetrahedral tessellation
- Triangular tessellation
- Icosahedral tessellation

Eisenharber et al.(Journal of Computational Chemistry, vol.16, p.273, 1995) - Lattice tessellation

Shrake and Rupley(Journal of Molecular Biology, vol.79, p.351, 1973) - Latitude / longitude tessellation
- Minimum energy

Bliznyuk and Gready(Journal of Computational Chemistry,vol.17, p.962, 1996)

Saff's(The Mathematical Intelligencer, vol.19, p.5, 1997) distribution by energy minimization gives the best results.

To accelerate the computation, selected neighbors of a central atom to be removed from the computation preprocessing step. Such as;

- DCLM (Eisenharber et al.)
- NLR, BAE (Weiser et al.)

__NSOL 1.x__ adopt a method that uses artificial grid as the
same as Eisenharber et al. This is a standard approach in
computational geometry to group spatially close objects. In
difference of NSC, an implementation of DCLM, NSOL 1.x has an
optimized code for vector processor.

- Determination of the minimum and the maximum of the x-,y-,z-coordinates of all atoms.
- Make large box that contains all atoms.
- Divide large box by small cube that edge length is 2 x
r
_{max}.

Each atom has neighbors only in its own cube and in its neighboring cubes. - After making neighbor list, overlap checking is executed to each distributed points.

The DCLM method of Eisenharber et al. uses the second cube division for the points on sphere, but I used only spatial division.

__NSOL 2.x__ adopt a method by sorting algorithm.

- Sort and index atoms by x-, y-, z-coordinates, respectively, using quick sort algorithm. When number of atoms less than 8, simple insertion method is used.
- To an atom index i, the two farthest atom indexes si,n and li,n are obtained for each coordinates.
- Neighbor atoms of an atom index i are obtained by logical operations and calculations of distances.
- After making neighbor list, overlap checking is executed to each distributed points.

__NSOL 3.x__ adopt my newest algorithm using mathematical morphology. In this method, distance calculation is not needed.

- Make surface using mathematical morphology.
- In each surface grids, mark distributed points on sphere and calculate surface area.

You can download each version of NSOL from following link:

NSOL 1.7 | 2003/12/11 | Fortran
77, C^{*} |

NSOL 2.4 | 2003/12/22 | Fortran 77 |

NSOL 3.0 |

^{*} Compare to nsol.f, nsol.c is slightly slower and
does not have vector optimization code.

Assuming that solvation energy is approximated in proportion to solvent accessible surface area.

where *σ _{i}*: atomic solvation parameter,

Several atomic solvation parameters have been proposed, such as, Ooi et al., Vila et al., Wesson et al., etc. And Temperature dependent solvation energy has been also proposed by Ooi et al.

**oons**: T. Ooi, M. Oobatake, G. Némethy, and H.A. Scheraga,
Accessible surface areas as a measure of the thermodynamic
parameters of hydration of peptides, Proc. Natl. Acad. Sci. USA
(1987) vol.84, pp.3086-3090.

**jrf_**: J. Vila, R.L. Williams, M. Vásquez, and H.A. Scheraga,
Empirical solvation models can be used to differentiate native from
near-native conformations of bovine pancreatic trypsin inhibitor,
PROTEINS: Structure, Function, and Genetics (1991) vol.10,
pp.199-218.

**we92, sch3**: L. Wesson and D. Eisenberg, Atomic solvation parameters
applied to molecular dynamics of proteins in solution, Protein
Science (1992) vol.1, pp.227-235.

**sch4**: C.A. Schiffer, J.W. Caldwell, P.A. Kollman, and R.M. Stroud,
Protein structure prediction with a combined solvation free
energy-molecular mechanics force field, Molecular Simulation (1993)
vol.10, pp.121-149.

Copyright © 1999-2010 Masato Masuya. All rights reserved.