Connolly surfaces

In contrast to the van der Waals surface, the Connolly surface [183, 184] has a smoother surface structure. The spiky and hard transition between the spheres of neighboring atoms is avoided. The Connolly surface can be obtained by rolling 

Connolly surfaces are standard in Molecular Modeling tools, and permit the quantitative and qualitative comparison of different molecules. 

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The following models describe those definitions of molecular surfaces that are most widely used. The van dcr Waals surface, the solvent-accessible surface, and the Connolly surface (sec below) based on Richards definitions play a major role [182],... 

Figure 2-119. The Connolly surface is determined by inoving a probe sphere (usually a water molecule) over the van der Waals surface. The surface thus obtained is also called the molecular or solvent-encluded surface (see Section 2,10,4 and Figure 2-120).

Tn general, the. solvent-accessible surface (SAS) represents a specific class of surfaces, including the Connolly surface. Specifically, the SAS stands for a quite discrete model of a surface, which is based on the work of Lee and Richards [182. They were interested in the interactions between protein and solvent molecules that determine the hydrophobicity and the folding of the proteins. In order to obtain the surface of the molecule, which the solvent can access, a probe sphere rolls over the van der Waals surface (equivalent to the Connolly surface). The trace of the center of the probe sphere determines the solvent-accessible surjace, often called the accessible swface or the Lee and Richards surface (Figure 2-120). Simultaneously, the trajectory generated between the probe and the van der Waals surface is defined as the molecular or Connolly surface. 

Solvent-excluded surfaces correlate with the molecular or Connolly surfaces (there is some confusion in the literature). The definition simply proceeds from another point of view. In this c ase, one assumes to be inside a molecaile and examines how the molecule secs the surrounding solvent molecules. The surface where the probe sphere does not intersect the molecular volume is determined. Thus, the SES embodies the solvent-excluded volume, which is the sum of the van der Waals volume and the interstitial (re-entrant) volume (Figures 2-119. 2-120). 

Molecular volumes are usually computed by a nonquantum mechanical method, which integrates the area inside a van der Waals or Connolly surface of some sort. Alternatively, molecular volume can be determined by choosing an isosurface of the electron density and determining the volume inside of that surface. Thus, one could find the isosurface that contains a certain percentage of the electron density. These properties are important due to their relationship to certain applications, such as determining whether a molecule will fit in the active site of an enzyme, predicting liquid densities, and determining the cavity size for solvation calculations. 

Fig. 8. Lck SH2 domain-peptide complex (Ac-cmF-Glu-Glu-Ile-OH, 12) revealing the twopronged plug engaging a two-holed socket 1 binding mode, reminiscent of the majority of SH2 domains (Protein Databank entry code 1BHF.PDB [118]). The protein is depicted in a Connolly surface mode, the ligand is given in a ball-and-stick representation. The cmF residue is deeply buried in its binding pocket (left)...

Fig. 4 The binding site for sorafenib is illustrated as a Connolly surface, with GluSOl and the displaced DFG sequence explicitly displayed. Hydrogen bonds from the aryl urea moiety are shown with dotted lines...

The Molecular Surface (MS) first introduced by Richards (19) was chosen as the 3D space where the MLP will be calculated. MS specifically refers to a molecular envelope accessible by a solvent molecule. Unlike the solvent accessible surface (20), which is defined by the center of a spherical probe as it is rolled over a molecule, the MS (19), or Connolly surface (21) is traced by the inwardfacing surface of the spherical probe (Fig. 2). The MS consists of three types of faces, namely contact, saddle, and concave reentrant, where the spherical probe touches molecule atoms at one, two, or three points, simultaneously. Calculation of molecular properties on the MS and integration of a function over the MS require a numerical representation of the MS as a manifold S(Mk, nk, dsk), where Mk, nk, dsk are, respectively, the coordinates, the normal vector, and the area of a small element of the MS. Among the published computational methods for a triangulated MS (22,23), the method proposed by Connolly (21,24) was used because it provides a numerical presentation of the MS as a collection of dot coordinates and outward normal vectors. In order to build the 3D-logP descriptor independent from the calculation parameters of the MS, the precision of the MS area computation was first estimated as a function of the point density and the probe radius parameters. When varying... 

Fig. 7. Side-by-side stereo presentation of the Lck SH2 domain (1BHF.PDB). The protein is shown as a ribbon diagram, while the ligand binding site is highlighted by a Connolly surface in line-mode. The two interaction-mediating ligand sidechains bind into the two-holed socket , the binding pockets of which are marked by an asterisk (left, pTyr-binding pocket), and a cube (right, pTyr+3 sidechain)...

The package of Connolly, named MSDOT, is widely used in molecular modeling for visualization of molecules (especially in the field of biochemistry and molecular biology), ESP fitting, and docking but it has been rarely used in combination with continuum solvation methods [10]. In its modern formulation, the Connolly surface presents a full analytical tessellation [11] but the reliability of it and of its differentiability has never been tested with PCM-like calculations. 

Figure I. Some examples of zeolite structures (the dots depict the Connolly surfaces).

Figure 24 Connolly surfaces of Cc Lac (a) and hCp (b) illustrating solvent exposure of the T1 site...

Figure 2 The crystal structure of SB203386 boimd to HIV-1 protease. The Connolly surface of the A chain monomer is shown.

One technique which is applicable for surfaces that are not everywhere differentiable is also suitable for the shape characterization of dot representations of molecular surfaces such as the Connolly surfaces [87], which are not only nondifferentiable, but are not even continuous. The method of 1-hulls [351] is based on a generalization of the concept of convex hull. The convex hull of a set A is the smallest convex set that contains A. Consider a three-dimensional body T. The T-hull of a point set A is the intersection of all rotated and translated versions of T which contain A. The T-hull method is suitable for shape comparisons with a common reference shape, chosen as that of the body T. Alternatively, when the shapes of two molecules, T and A are compared, one molecular body can be chosen as T and the T-hull of the other molecular body A provides a direct shape comparison [351]. 

The G-WHIM approach integrates the information contained in WHIM descriptors and overcomes any problems due to the alignment of the different molecules and the explosion of variables arising from traditional - grid-based QSAR techniques. In particular, the G-WHIM approach can take into account either all the points within the cut-off values, excluding only positive interactions within the inner part of the molecule, or the surface points at a cut-off value, i.e. points on an iso-potential-energy surface. G-WHIM descriptors calculated on the -> Connolly surface area are also called MS-WHIM [Bravi et al, 1997]. 

The area of the solvent-accessible surface is called the Solvent-Accessible Surface Area SASA (or Total Solvent-Accessible Surface Area, TSASA). Several algorithms were proposed that implement both the first original definition of SASA and that of Richards. One of the most popular algorithms that implements Richards solvent-accessible surface was proposed by Connolly [Connolly, 1983a]. It is an analytical method for computing molecular surface, and is based on surface decomposition into a set of curved regions of spheres and tori that join at circular arcs spheres, tori and arcs are defined by analytical expressions in terms of atomic coordinates, van der Waals radii and the probe radius. Ihe molecular surface calculated in such a way is sometimes referred to as Connolly surface area. This algorithm also allows the calculation of solvent-accessible atomic areas. 

Figure 5. Connolly surface of truncated coUagenase, complexed with 1. Structure of the catalytic domain of fibroblast coUagenase with a corresponding Connolly surface [55], shown in stereo. The active site cleft accommodating the catalytic zinc (white sphere) and the deep SI specificity pocket are clearly seen. The hydroxamate inhibitor (not included in the surface calculations) is shown in yellow. Protein residues involved in the catalysis are indicated.

Fig. 4.6 Cut-through of the Connolly surface of the DNA Gyrase B subunit active site. The non-hydrolysable ATP analogue ADNP is shov/n in mainly yellow. LUDI interaction sites...

Connolly surface area —> molecular surface (0 solvent-accessible molecular surface)... 

SURFCATS descriptors [Rermer and Schneider, 2006] are based on the spatial distance between P P P s on the —> Connolly surface area. Surface points are calculated with a spacing of 2 A and assigned to the pharmacophore type of the nearest atoms. 

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