Analytical calculation of the solvent-accessible surface area and its nuclear gradients by stereographic projection: A general approach for molecules,polymers, nanotubes,helices, and surfaces

In this article, we explore an alternative to the analytical Gauss–Bonnet approach for computing the solvent-accessible surface area (SASA) and its nuclear gradients. These two key quantities are required to evaluate the nonelectrostatic contribution to the solvation energy and its nuclear gradients in implicit solvation models. We extend a previously proposed analytical approach for finite systems based on the stereographic projection technique to infinite periodic systems such as polymers, nanotubes, helices, or surfaces and detail its implementation in the Crystal code. We provide the full derivation of the SASA nuclear gradients, and introduce an iterative perturbation scheme of the atomic coordinates to stabilize the gradients calculation for certain difficult symmetric systems. An excellent agreement of computed SASA with reference analytical values is found for finite systems, while the SASA size-extensivity is verified for infinite periodic systems. In addition, correctness of the analytical gradients is confirmed by the excellent agreement obtained with numerical gradients and by the translational invariance achieved, both for finite and infinite periodic systems. Overall therefore, the stereographic projection approach appears as a general, simple, and efficient technique to compute the key quantities required for the calculation of the nonelectrostatic contribution to the solvation energy and its nuclear gradients in implicit solvation models applicable to both finite and infinite periodic systems.     

SMART: A solvent-accessible triangulated surface generator for molecular graphics and boundary element applications

Summary An algorithm is presented for generating a representation of the solvent-accessible molecular surface as a smooth triangulated manifold. The algorithm, called SMART (SMooth moleculAR surface Triangulator), divides the contact and reentrant portions of the solvent-accessible molecular surface into curvilinear three-sided elements. In contrast to the author's earlier implementation of this general approach [Zauhar, R.J. and Morgan, R.S., J. Comput. Chem., 11 (1990) 603], the SMART algorithm defines elements directly on the appropriate geometric surface types (rather than using interpolation over cubic elements), and has special features to handle highly distorted regions which often appear in deep crevices and internal cavities. While the method is designed for use with boundary element techniques in continuum electrostatics, it can also be applied to the accurate computation of molecular surface areas and volumes, and the generation of shaded surfaces for display with interactive computer graphics. Availability: Programs (in C) for surface generation, area and volume computation are available from the author. Also available is a graphics display program which runs on Silicon Graphics workstations.     

Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation

A hybrid approach for solving the nonlinear Poisson-Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion-exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.     

Enveloping triangulation method for detecting internal cavities in proteins and algorithm for computing their surface areas and volumes

Detection and quantitative characterization of the internal cavities in proteins remain an important topic in studying protein structure and function. Here we propose a new analytical method for detecting the existence of cavities in proteins. The method is based on constructing the special enveloping triangulation enclosing the cavities. Based on this method, we develop an algorithm and a fortran package, CAVE, for computing volumes and surface areas of cavities in proteins. We first test our method and algorithm in some artificial systems of spheres and find that the calculated results are consistent with exact results. Then we apply the package to compute volumes and surface areas of cavities for some protein structures in the Protein Data Bank. We compare our calculated results with those obtained by some other methods and find that our approach is reliable.     

A new analytical method for computing solvent-accessible surface area of macromolecules and its gradients

In the calculation of thermodynamic properties and three-dimensional structures of macromolecules, such as proteins, it is important to have an efficient algorithm for computing the solvent-accessible surface area of macromolecules. Here, we propose a new analytical method for this purpose. In the proposed algorithm we consider the transformation that maps the spherical circles formed by intersection of the atomic surfaces in three-dimensional space onto the circles on a two-dimensional plane, and the problem of computing the solvent-accessible surface area is reduced to the problem of computing the corresponding curve integrals on the plane. This allows to consider only the integrals along the circular trajectories on the plane. The algorithm is suitable for parallelization. Testings on many proteins as well as the comparison to the other analogous algorithms have shown that our method is accurate and efficient.     

Application of the thin-shell formulation to the numerical modeling of Stern layer in biomolecular electrostatics

In this article, the thin-shell formulation is applied to efficiently modeling the Stern layer within computational algorithms oriented toward the boundary element solution of the linearized Poisson-Boltzmann equation. The attention is focused on the calculation of the electrostatic potential in proximity to a biomolecule immersed in an electrolyte medium. Following the proposed approach, the Stern layer is made to collapse to a zero-thickness region (two-dimensional surface) with interface conditions linking the electrostatic potential over the molecular and the bulk ion accessible surfaces. Advantages lie in the limitation of divergent integral problems and in the halving of the unknown number, with a significant impact on computational time and memory requirements when modeling large biomolecules.     

Fast summation boundary element method for calculating solvation free energies of macromolecules

We present a boundary element method (BEM) for calculating the reaction field energy of a macromolecule embedded in a high-dielectric medium such as water. In a BEM calculation, the key computational task is the calculation of the induced surface charge distribution at the dielectric boundary. This is obtained by solving a system of linear equations whose dimension can run into the tens of thousands for a macromolecule. In this work, we use a fast summation hierarchical multipole method to solve for the induced surface charge densities. By careful analysis of the levels of approximation required for the various terms in the calculation, we avoid the unnecessary computation of terms that contribute negligibly to the final outcome and, consequently, achieve high computational efficiency. For a protein such as BPTI with 890 atoms, the calculation of the induced surface charge density distribution and the reaction field energy was completed in 7.9 s on an SGI workstation with an R10000 CPU. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1494–1504, 1998     

Use of surface area computations to describe atom-atom interactions

Accessible surface (ASA) and atomic contact (ACA) areas are powerful tools for protein structure analysis. However, their use for analysis purposes could be extended if a relationship between them and protein stability could be found. At present, this is the case only for ASAs, which have been used to assess the contribution of the hydrophobic effect to protein stability. In the present work we study whether there is a relationship between atomic contact areas and the free energy associated to atom-atom interactions. We utilise a model in which the contribution of atomic interactions to protein stability is expressed as a linear function of the accessible surface area buried between atom pairs. We assess the validity of this hypothesis, using a set of 124 lysozyme mutants (Matthews, 1995, Adv Protein Chem, 249–278) for which both the X-ray structure and the experimental stability are known. We tested this assumption for residue representations with increasing numbers of atom types. Our results indicate that for simple residue representations, with only 4 to 5 atom types, there is not a clear linear relationship between stability and buried accessible area. However, this relationship is observed for representations with 6 to 9 atom types, where gross heterogeneities in the atom type definition are eliminated. Finally, we also study a version of the linear model in which the atom- atom interactions are represented utilising a simple function for the buried accessible area, which may be useful for protein structure prediction studies.     

Theoretical pKa calculations of proteins; the tyrosine and lysine residues of b-elicitin

Elicitins are small proteins that are secreted by plant pathogenic fungi. In this work we have used a computer program that utilizes the boundary element method for heterogeneous dielectrics with ionic strength to calculate the pK _a of all titrating groups in the 98-residue protein β-cryptogein. Our results are in reasonable agreement with the experimentally determined pK _a values for the Tyr residues in the protein. We find that the functionally important Lys13 residue has a normal pK _a of 10.3. Our work also shows that there is no direct correlation between the exposure of an amino acid sidechain and its pK _a. Received: 24 April 1998 / Accepted: 4 August 1998 / Published online: 11 November 1998     

A Symmetry adapted tessellation of the GEPOL surface: applications to molecular properties in solution

A procedure to partition the GEPOL molecular surface into tesserae that respects the molecular symmetry constraints is presented. Using this method it is possible to build a solvent reaction field for the Polarizable Continuum Model with the same symmetry of the nuclear potential. Several applications are reported and discussed to evaluate the performance of this new procedure. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1262–1272, 2001     

Calculating the electric potential of macromolecules: A simple method for molecular surface triangulation

An improved version of the “marching-cube” method¹ is proposed for molecular surface triangulation. This new algorithm involves fewer and simpler basic building blocks and avoids the artificial gaps of the original one. Moreover, to make it applicable to the boundary element method, the procedures for the protein cavity identification and triangle reduction are also presented. The triangulation procedure was tested by incorporating it into the boundary element method (BEM) to estimate the pK_a values of subtilisin BPN′ and bovine trypsin inhibitor (BPTI). © 1995 by John Wiley & Sons, Inc.     

On the transferability of hydration-parametrized continuum electrostatics models to solvated binding calculations

Using molecular mechanics force field partial atomic charges, we show the nonuniqueness of the parametrization of continuum electrostatics models with respect to solute atomic radii and interior dielectric constant based on hydration (vacuum-to-water transfer) free energy data available for small molecules. Moreover, parameter sets that are optimal and equivalent for hydration free energy calculations lead to large variations of calculated absolute and relative electrostatic binding free energies. Hence, parametrization of solvation effects based on hydration data, although a necessary condition, is not sufficient to guarantee its transferability to the calculation of binding free energies in solution.     

Evaluation of the protein solvent-accessible surface using reduced representations in terms of critical points of the electron density

The aim of our study is the development of a method for calculating the interface of dimerization of protein-protein complexes based on simplified medium-resolution structures. In particular, we wished to evaluate if the existing concepts for the computation of the Solvent-Accessible Surface Area (SASA) of macromolecules could be applied to medium-resolution models. Therefore, we selected a set of 140 protein chains and computed their reduced representations by topological analysis of their electron density maps at 2.85 A crystallographic resolution. This procedure leads to a limited number of critical points (CPs) that can be identified and associated to backbone and side-chain parts. To evaluate the SASA and interfaces of dimerization of the reduced representations, we chose and modified two existing programs that calculate the SASA of atomic representations, and tested (1) several radii tables of amino acids, (2) the influence of the backbone and side-chain points, and (3) the radius of the solvent molecule, which rolls over the surface. The results are shown in terms of relative error compared to the values calculated on the corresponding atomic representations of the proteins.     

A fast adaptive multigrid boundary element method for macromolecular electrostatic computations in a solvent

A fast multigrid boundary element (MBE) method for solving the Poisson equation for macromolecular electrostatic calculations in a solvent is developed. To convert the integral equation of the BE method into a numerical linear equation of low dimensions, the MBE method uses an adaptive tesselation of the molecular surface by BEs with nonregular size. The size of the BEs increases in three successive levels as the uniformity of the electrostatic field on the molecular surface increases. The MBE method provides a high degree of consistency, good accuracy, and stability when the sizes of the BEs are varied. The computational complexity of the unrestricted MBE method scales as O(N_at), where N_at is the number of atoms in the macromolecule. The MBE method is ideally suited for parallel computations and for an integrated algorithm for calculations of solvation free energy and free energy of ionization, which are coupled with the conformation of a solute molecule. The current version of the 3-level MBE method is used to calculate the free energy of transfer from a vacuum to an aqueous solution and the free energy of the equilibrium state of ionization of a 17-residue peptide in a given conformation at a given pH in ∼ 400 s of CPU time on one node of the IBM SP2 supercomputer. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 569–583, 1997     

Statistical investigation of surface bound ions and further development of BION server to include pH and salt dependence

Ions are engaged in multiple biological processes in cells. By binding to the macromolecules or being mobile in the solvent, they maintain the integrity of the structure of macromolecules; participate in their enzymatic activity; or screen electrostatic interactions. While experimental methods are not always able to assign the exact location of ions, computational methods are in demand. Although the majority of computational methods are successful in predicting the position of ions buried inside macromolecules, they are less effective in deciphering positions of surface bound ions. Here, we propose the new BION algorithm ( http://compbio.clemson.edu/bion_server_ph/ ) that predicts the location of the surface bound ions. It is more efficient and accurate compared to the previous version since it uses more advanced clustering algorithm in combination with pairing rules. In addition, the BION webserver allows specifying the pH and the salt concentration in predicting ions positions. © 2015 Wiley Periodicals, Inc.     

Parameterization of the approximate valence bond (AVB) method to describe potential energy surface in the reaction catalyzed by HIV‐1 protease

The approximate valence bond (AVB) method was parameterized to obtain the potential energy surface for ongoing classical and further quantum classical molecular dynamics simulations describing the enzymatic reaction of the human immunodeficiency virus type 1 protease (HIV‐1 PR). The parameterized AVB method allows to describe the complete enzymatic reaction, including a proton transfer between a water molecule and an aspartic acid, nucleophilic attack of a hydroxy anion on the substrate peptide bond, conformational changes of an intermediate, two proton transfers between the intermediate and aspartic acids, and cleavage of the intermediate into reaction products. The AVB method was parameterized based on density functional theory (DFT) calculations performed for small molecular systems. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 86–103, 2001     

A new approach to describe the skin surface physical properties in vivo

In the present paper, we describe a new mechanical method characterising the physico-chemical properties of human skin and their variations along with liquid exposure scenario to the skin surface. A specific bio-tribometer has been developed to study the physical properties of the skin in vivo by measuring the maximum adhesion force between the skin and the bio-tribometer. We showed that the lipidic film present on skin surface was responsible for skin adhesion due to capillary phenomena. The measure of pull-off force between skin and bio-tribometer has permitted to estimate the liquid/vapour surface tension of the lipidic film (γ_LV ≈ 6.3 mJ/m² in 30-year-old volunteer). The kinetic of sorption/desorption (sorption means indifferently adsorption and absorption process) of distilled water from the skin has been observed through the variation of the indenter/skin pull-off force versus time after distilled water application to the skin surface. This permits to follow in real time the variation of the skin physico-chemical properties after liquid application onto the skin surface. Finally, the increasing of skin friction coefficient after distilled water application onto skin surface was explained by the capillary adhesion force between the probe and the skin.