Application of the level-set method to the implicit solvation of nonpolar molecules

arXiv:0809.0181v1 [physics.chem-ph] 1 Sep 2008 Application of the level-set method to the implicit solvation of nonpolar molecules Li-Tien Cheng,1, ∗Joachim Dzubiella,2, 3, 4, † J. Andrew McCammon,3, 5, ‡ and Bo Li1, § 1Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112 2Physics Department (T37), Technical University Munich, James-Franck-Straße, 85748 Garching, Germany 3NSF Center for Theoretical Biological Physics (CTBP) 4Department of Chemistry and Biochemistry, University of California, San Diego, La Jolla, California 92093-0365 5Department of Chemistry and Biochemistry, and Department of Pharmacology, University of California, San Diego, La Jolla, California 92093-0365 (Dated: November 4, 2018) A level-set method is developed for numerically capturing the equilibrium solute-solvent interface that is defined by the recently proposed variational implicit solvent model (Dzubiella, Swanson, and McCammon, Phys. Rev. Lett. 104, 527 (2006) and J. Chem. Phys. 124, 084905 (2006)). In the level-set method, a possible solute-solvent interface is represented by the zero level-set (i.e., the zero level surface) of a level-set function and is eventually evolved into the equilibrium solute-solvent interface. The evolution law is determined by minimization of a solvation free energy functional that couples both the interfacial energy and the van der Waals type solute-solvent interaction energy. The surface evolution is thus an energy minimizing process, and the equilibrium solute-solvent interface is an output of this process. The method is implemented and applied to the solvation of nonpolar molecules such as two xenon atoms, two parallel paraffin plates, helical alkane chains, and a single fullerene C60. The level-set solutions show good agreement for the solvation energies when compared to available molecular dynamics simulations. In particular, the method captures solvent dewetting (nanobubble formation) and quantitatively describes the interaction in the strongly hydrophobic plate system. PACS numbers: 61.20.Ja, 68.03-g., 82.20.Wt, 87.16.Ac, 87.16.Uv I. INTRODUCTION The correct description of solvation free ener- gies and detailed solution structures of biomolecules is crucial to our understanding of molecular pro- cesses in biological systems. Efficient theoretical ap- proaches to such descriptions are typically given by so-called implicit (or continuum) solvent models of the aqueous environment [1, 2]. In those models, the solvent molecules and ions (e.g., as in physio- logical electrolyte solutions) are treated implicitly and their effects are coarse-grained. In particular, the description of the solvent is reduced to that of the continuum solute-solvent interface and related macroscopic quantities, such as the surface tension and the position-dependent dielectric constant serv- ∗e-mail address:lcheng@math.ucsd.edu †e-mail address:jdzubiel@ph.tum.de ‡e-mail address:jmccammon@ucsd.edu §e-mail address:bli@math.ucsd.edu ing as input or fitting parameters. Most of the existing implicit solvent models are built upon the concept of solvent accessible surface area (SASA) defined in several ways [3, 4]. In these models, the solvation free energy is proportional to the SASA for the nonpolar contribution, comple- mented by the Poisson-Boltzmann (PB) [5, 6, 7] or Generalized Born (GB) [8, 9] description of electro- statics, i.e., the polar contribution. Although suc- cessful in many cases, the general applicability of these rather empirical models with many system- dependent, adjustable parameters (e.g., individual atomic surface tensions) is often questionable, when compared to more accurate but computationally ex- pensive explicit molecular dynamics (MD) simula- tions or experimental results. It is believed that the key issues here are the decoupling and separate anal- ysis of surface area, dispersion and polar parts of the free energy, and the inaccurate free energy estima- tion due to a predefined solvent-solute interface, an ad hoc input. It is additionally well established by now that cavitation free energies do not scale with 2 surface area for high curvatures [10, 11], a fact of critical importance in the implicit modeling of hy- drophobic interactions in biomolecular systems [12]. Recently, Dzubiella, Swanson, and McCammon [13, 14] have developed a variational implicit solvent model. The basic idea of this approach is to intro- duce a free energy functional of all possible solute- solvent interfaces, coupling both the nonpolar and polar contributions of the system, and allowing for curvature correction of the surface tension to ap- proximate the length-scale dependence of molecular hydration. Minimizing the functional leads to a par- tial differential equation whose solution determines the equilibrium solute-solvent interface and the min- imum free energy of the solvated system. This stable solute-solvent interface is an output of the theory. It results automatically from bala

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