An Adaptive Grid Based Method for Computing Molecular Surfaces and Properties

We present an adaptive grid based algorithm to compute a family of relevant molecular surfaces. Molecular interfaces are important in simulations and visualization involving biomolecules. The Richards surface has traditionally been used as a good approximation to the surface, and defined as the surface formed by the inner facing part of a solvent probe atom rolling along the van der Waals surface of the molecule. Computing and representing this surface has traditionally involved complex geometrical data structures like alpha shapes. Adaptive and uniform trilinear grids are commonly used in various simulations involving interactions of molecules or computation of electrostatics and other energy terms. We make use of this grid directly to compute the Molecular Surface and properties like area, volume, curvatures, surface atoms and other surfaces. We compare geometrical and biochemical properties with other methods as a validation. 1 Molecular Surface Definitions Explicit surface definitions as the interface between the solvent and proteins have been given since 1970s. Since it is easier to handle implicitly defined models mathematically, different implicit approximations to these surfaces have been developed. 1.1 van der Waals and Lee Richards Surface Definitions The most common model for molecules is as a collection of atoms represented by spheres, with radii equal to their van der Waals radii. The surface of the set of spheres is known as the van der Waals surface. Lee and Richards introduced the concept of accessibility to the solvent. Proteins are not isolated, but commonly present in solutions, especially water. Also, the van der Waals surface contained too many internal atoms and patches which are not accessible by the solvent. Hence, Lee and Richards gave a new definition for the molecular surface as the surface accessible to the solvent [32]. They modeled water molecules as spheres with radius 1.4Å, and considered the locus of the center of one such ‘probe’, as it rolled along the protein surface as the Solvent Accessible Surface (SAS). Richards then gave a more commonly used definition for molecular surface as a set of contact and reentrant patches in [42]. A probe solvent sphere, rolling over the atoms of a protein defines a region in which none of its points pass through. The boundary of this volume is continuous and defines a new molecular surface. This surface is composed of convex patches where the probe touches the atom surfaces, concave spherical patches when the probe touches more than 2 atoms simultaneously and toroidal patches when the probe rolls between two atoms. Connolly called this as an alternative definition of the SAS surface in [18], but is now commonly known as the Solvent Contact Surface (SCS), or Solvent Excluded Surface (SES) or simply the Molecular Surface. These surfaces, for a 3 atom example is shown as a 2D cross section in figure 1. We also provide analogous volume definitions for each surface.