Hydrophobicity maps


Hydrophobicity maps are a graphical representation of the binding energy of a nonpolar probe sphere rolling over the surface of the receptor. The binding energy includes both the electrostatic and nonpolar contributions to the association of a hydrophobic compound at the surface of a receptor. A continuum approach is used for the electrostatic component, whereas the van der Waals interaction describes the nonpolar contribution. The binding energy is displayed by color-rendering on the surface ofthe receptor. This yields a precise visualization of the surface hydrophobicity as well as a clear distinction between hydrophobic and hydrophilic zones in close proximity.

Binding energy of a nonpolar probe sphere at the surface of a receptor The solvent accessible surface (SAS) is spanned by the center of a probe sphere rolling over the van der Waals surface of a molecule [20]. A number of points are distributed uniformly on the SAS of the receptor to describe in a discrete manner the different positions of the center of the probe sphere. On each of these points the binding energy of the nonpolar probe sphere (AE) is approximated, as explained in the next subsection, by the sum of van der Waals interaction energy (EvdW) and electrostatic desolvation of the receptor (A Edesolv):

Parameters for the van der Waals energy and partial charges from standard force fields can be used. In the applications presented in this work, the all-hydrogen MSI CHARMm22 parameter set [21,22] was used.

The evaluation of A E for about 55 000 positions of the probe sphere on the thrombin surface requires about 35 s on a 195 MHz R10000 processor.

Van der Waals interaction energy The nonelectrostatic contributions to binding consist of the solute-solute van der Waals energy (favorable to binding), the loss of solute-solvent van der Waals energy (unfavorable), and the disruption of water structure which is a favorable entropic effect at room temperature [23]. A number of approaches have been proposed to evaluate these contributions [23-28]. Here it is assumed that solute-solvent van der Waals interactions and disruption of water structure compensate each other (see Reference 28 and Figure 6 of Reference 29), and that the solute-solute van der Waals energy can account for the nonelectrostatic component of the binding energy. Therefore, the van der Waals energy between the probe sphere and the receptor atoms (EvdW) is assumed to account for all the nonelectrostatic contributions to the association ofthe probe sphere to the receptor. It is calculated as:

where ri is the distance between the receptor atom i and the probe sphere. ei and Rt are the van der Waals energy minimum and radius of atom i. The probe sphere van der Waals radius ( Rprobe) and energy minimum (Eprobe) are input values. Since the probe sphere is rolled over the receptor van der Waals surface, it does not clash with it and is always at optimal distance from at least one receptor atom.

Electrostatic desolvation energy The electrostatic desolvation of the receptor accounts for the loss of receptor-solvent favorable electrostatic interactions due to the removal of part of the highly polarizable solvent to accommodate a nonpolarizable probe sphere. This contribution always disfavors association and can be calculated within the assumption of continuum electrostatics [13,14,30-35]. The system is partitioned into solvent and solute regions and two different dielectric constants are assigned to each region. The electrostatic energy E of the receptor in solution can be expressed in terms of the electric displacement vector D (x) and of a location dependent dielectric constant e (x) as an integral over the three-dimensional (3D) space R3 [36]:

Since D (x) is additive, for point charges it can be rewritten as a sum over all charges i ofthe receptor:

Concerning the electrostatics, docking a nonpolar sphere at the surface of the receptor has the only effect of modifying the dielectric properties in the space occupied by the sphere. Over this volume the dielectric constant changes from the solvent value ( e w) to the solute value ( ep). Usually, e w is set to 78.5 which is the value of water at room temperature, while the value of e p can range from 1 to 4. In the limit in which D (x) does not change significantly upon docking of the sphere, the variation of the electrostatic energy of the receptor (i.e., the desolvation) can be written according to Equation 3 as an integral over the volume occupied by the probe sphere (Vprobe):

where t = 1/ ep - 1/ ew. The volume occupied by the probe sphere is assumed to be a sphere of 1.7 A, radius, i.e., the van der Waals radius ofthe probe sphere augmented by 0.3 A, to include small voids between the probe and receptor surfaces. A 3D grid is built around the receptor and Equation 5 becomes:

0 0

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