Skip to main content

Advertisement

SpringerLink
Log in

Differential geometry based solvation model II: Lagrangian formulation

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

Solvation is an elementary process in nature and is of paramount importance to more sophisticated chemical, biological and biomolecular processes. The understanding of solvation is an essential prerequisite for the quantitative description and analysis of biomolecular systems. This work presents a Lagrangian formulation of our differential geometry based solvation models. The Lagrangian representation of biomolecular surfaces has a few utilities/advantages. First, it provides an essential basis for biomolecular visualization, surface electrostatic potential map and visual perception of biomolecules. Additionally, it is consistent with the conventional setting of implicit solvent theories and thus, many existing theoretical algorithms and computational software packages can be directly employed. Finally, the Lagrangian representation does not need to resort to artificially enlarged van der Waals radii as often required by the Eulerian representation in solvation analysis. The main goal of the present work is to analyze the connection, similarity and difference between the Eulerian and Lagrangian formalisms of the solvation model. Such analysis is important to the understanding of the differential geometry based solvation model. The present model extends the scaled particle theory of nonpolar solvation model with a solvent–solute interaction potential. The nonpolar solvation model is completed with a Poisson–Boltzmann (PB) theory based polar solvation model. The differential geometry theory of surfaces is employed to provide a natural description of solvent–solute interfaces. The optimization of the total free energy functional, which encompasses the polar and nonpolar contributions, leads to coupled potential driven geometric flow and PB equations. Due to the development of singularities and nonsmooth manifolds in the Lagrangian representation, the resulting potential-driven geometric flow equation is embedded into the Eulerian representation for the purpose of computation, thanks to the equivalence of the Laplace–Beltrami operator in the two representations. The coupled partial differential equations (PDEs) are solved with an iterative procedure to reach a steady state, which delivers desired solvent–solute interface and electrostatic potential for problems of interest. These quantities are utilized to evaluate the solvation free energies and protein–protein binding affinities. A number of computational methods and algorithms are described for the interconversion of Lagrangian and Eulerian representations, and for the solution of the coupled PDE system. The proposed approaches have been extensively validated. We also verify that the mean curvature flow indeed gives rise to the minimal molecular surface and the proposed variational procedure indeed offers minimal total free energy. Solvation analysis and applications are considered for a set of 17 small compounds and a set of 23 proteins. The salt effect on protein–protein binding affinity is investigated with two protein complexes by using the present model. Numerical results are compared to the experimental measurements and to those obtained by using other theoretical methods in the literature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Abrams JB, Rosso L, Tuckerman ME (2006) Efficient and precise solvation free energies via alchemical adiabatic molecular dynamics. J Chem Phys 125(7): 074115

    Article  Google Scholar 

  • Amovilli C, Mennucci B (1997) Self-consistent-field calculation of Pauli repulsion and dispersion contributions to the solvation free energy in the polarizable continuum model. J Phys Chem B 101(6): 1051–1057

    Article  Google Scholar 

  • Antosiewicz J, McCammon JA, Gilson MK (1996) The determinants of pK a s in proteins. Biochemistry 35(24): 7819–7833

    Article  Google Scholar 

  • Ashbaugh HS (2000) Convergence of molecular and macroscopic continuum descriptions of ion hydration. J Phys Chem B 104(31): 7235–7238

    Article  Google Scholar 

  • Azuara C, Lindahl E, Koehl P, Orland H, Delarue M (2006) PDB_Hydro: incorporating dipolar solvents with variable density in the Poisson–Boltzmann treatment of macromolecule electrostatics. Nucl Acids Res 34: W38–W42

    Article  Google Scholar 

  • Baker NA (2004) Poisson–Boltzmann methods for biomolecular electrostatics. Methods Enzymol 383: 94–118

    Article  Google Scholar 

  • Baker NA (2005) Biomolecular applications of Poisson–Boltzmann methods. In: Lipkowitz KB, Larter R, Cundari TR (eds) Reviews in computational chemistry, vol 21. Wiley, Hoboken

    Google Scholar 

  • Baker NA (2005) Improving implicit solvent simulations: a Poisson-centric view. Curr Opin Struct Biol 15(2): 137–143

    Article  Google Scholar 

  • Baker NA, Bashford D, Case DA (2006) Implicit solvent electrostatics in biomolecular simulation. In: Leimkuhler B, Chipot C, Elber R, Laaksonen A, Mark A, Schlick T, Schutte C, Skeel R (eds) New algorithms for macromolecular simulation. Springer, Berlin

    Google Scholar 

  • Baker NA, McCammon JA (2003) Electrostatic interactions. In: Bourne P, Weissig H (eds) Structural bioinformatics. Wiley, New York, pp 427–440

    Google Scholar 

  • Banavali NK, Im W, Roux B (2002) Electrostatic free energy calculations using the generalized solvent boundary potential method. J Chem Phys 117(15): 7381–7388

    Article  Google Scholar 

  • Barone V, Cossi M, Tomasi J (1997) A new definition of cavities for the computation of solvation free energies by the polarizable continuum model. J Chem Phys 107: 3210–3221

    Article  Google Scholar 

  • Bashford D, Case DA (2000) Generalized Born models of macromolecular solvation effects. Annu Rev Phys Chem 51: 129–152

    Article  Google Scholar 

  • Bashford D, Karplus M (1990) pK a ’s of ionizable groups in proteins: atomic detail from a continuum electrostatic model. Biochemistry 29(44): 10219–10225

    Article  Google Scholar 

  • Bates PW, Chen Z, Sun YH, Wei GW, Zhao S (2009) Geometric and potential driving formation and evolution of biomolecular surfaces. J Math Biol 59: 193–231

    Article  MathSciNet  MATH  Google Scholar 

  • Bates PW, Wei GW, Zhao S (2006a) The minimal molecular surface. arXiv:q-bio/0610038v1 [q-bio.BM]

  • Bates PW, Wei GW, Zhao S (2006b) The minimal molecular surface. Midwest quantitative biology conference. Mission Point Resort, Mackinac Island, MI, September 29–October 1

  • Bates PW, Wei GW, Zhao S (2008) Minimal molecular surfaces and their applications. J Comput Chem 29(3): 380–391

    Article  Google Scholar 

  • Beglov D, Roux B (1996) Solvation of complex molecules in a polar liquid: an integral equation theory. J Chem Phys 104(21): 8678–8689

    Article  Google Scholar 

  • Beglov D, Roux B (1997) An integral equation to describe the solvation of polar molecules in liquid water. J Phys Chem B 101(39): 7821–7826

    Article  Google Scholar 

  • Berger M, Gostiaux B (1988) Differential geometry: manifolds, curves, and surfaces. Springer, Berlin

    MATH  Google Scholar 

  • Bergstrom CAS, Strafford M, Lazorova L, Avdeef A, Luthman K, Artursson P (2003) Absorption classification of oral drugs based on molecular surface properties. J Med Chem 46(4): 558–570

    Article  Google Scholar 

  • Bertonati C, Honig B, Alexov E (2007) Poisson–Boltzmann calculations of nonspecific salt effects on protein–protein binding free energy. Biophys J 92: 1891–1899

    Article  Google Scholar 

  • Bertozzi AL, Greer JB (2004) Low-curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes. Commun Pure Appl Math 57(6): 764–790

    Article  MathSciNet  MATH  Google Scholar 

  • Blomberg N, Gabdoulline RR, Nilges M, Wade RC (1999) Classification of protein sequences by homology modeling and quantitative analysis of electrostatic similarity. Proteins 37(3): 379–387

    Article  Google Scholar 

  • Blomgren P, Chan T (1998) Color TV: total variation methods for restoration of vector-valued images. IEEE Trans Image Process 7(3): 304–309

    Article  Google Scholar 

  • Boschitsch AH, Fenley MO (2004) Hybrid boundary element and finite difference method for solving the nonlinear Poisson–Boltzmann equation. J Comput Chem 25(7): 935–955

    Article  Google Scholar 

  • Bostrom M, Tavares FW, Bratko D, Ninham BW (2005) Specific ion effects in solutions of globular proteins: Comparison between analytical models and simulation. J Phys Chem B 109(51): 24489–24494

    Article  Google Scholar 

  • Cai W, Deng SZ (2003) An upwinding embedded boundary method for Maxwell’s equations in media with material interfaces: 2d case. J Comput Phys 190: 159–183

    Article  MathSciNet  MATH  Google Scholar 

  • Cances E, Mennucci B, Tomasi J (1997) A new integral equation formalism for the polarizable continuum model: Theoretical background and applications to isotropic and anisotropic dielectrics. J Chem Phys 107: 3032–3041

    Article  Google Scholar 

  • Canham PB (1970) The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J Theoret Biol 26: 61–81

    Article  Google Scholar 

  • Carstensen V, Kimmel R, Sapiro G (1997) Geodesic active contours. Int J Comput Vis 22: 61–79

    Article  Google Scholar 

  • Cecil T (2005) A numerical method for computing minimal surfaces in arbitrary dimension. J Comput Phys 206(2): 650–660

    Article  MathSciNet  MATH  Google Scholar 

  • Cerutti DS, Baker NA, McCammon JA (2007) Solvent reaction field potential inside an uncharged globular protein: A bridge between implicit and explicit solvent models?. J Chem Phys 127(15): 155101

    Article  Google Scholar 

  • Chang Q, Tai X, Xing L (2010) A compound algorithm of denoising using second-order and fourth-order partial differential equations. Numer Math Theory Methods Appl 2: 353–376

    MathSciNet  Google Scholar 

  • Chen D, Chen Z, Chen C, Geng WH, Wei GW (2010) MIBPB: a software package for electrostatic analysis. J Comput Chem (in press)

  • Chen D, Wei GW, Cong X, Wang G (2009) Computational methods for optical molecular imaging. Commun Numer Methods Eng 25: 1137–1161

    Article  MathSciNet  MATH  Google Scholar 

  • Chen J, Brooks CL III (2008) Implicit modeling of nonpolar solvation for simulating protein folding and conformational transitions. Phys Chem Chem Phys 10: 471–481

    Article  Google Scholar 

  • Chen L, Holst MJ, Xu J (2007) The finite element approximation of the nonlinear Poisson–Boltzmann equation. SIAM J Numer Anal 45(6): 2298–2320

    Article  MathSciNet  MATH  Google Scholar 

  • Chen T, Strain J (2008) Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems. J Comput Phys 16: 7503–7542

    Article  MathSciNet  Google Scholar 

  • Chen YG, Weeks JD (2006) Local molecular field theory for effective attractions between like charged objects in systems with strong Coulomb interactions. Proc Natl Acad Sci USA 103(20): 7560–7565

    Article  Google Scholar 

  • Chen Z, Baker NA, Wei GW (2010) Differential geometry based solvation models I: Eulerian formulation. J Comput Phys 229: 8231–8258

    Article  MathSciNet  MATH  Google Scholar 

  • Chen Z, Wei GW (2010) Differential geometry based solvation models III: quantum formulation. J Comput Phys (submitted)

  • Chen Z, Wei GW (2011) Differential geometry based solvation models IV: apolar formulation (in preparation, to be submitted)

  • Cheng LT, Dzubiella J, McCammon AJ, Li B (2007) Application of the level-set method to the implicit solvation of nonpolar molecules. J Chem Phys 127(8)

  • Cheng Y, Suen JK, Radi Z, Bond SD, Holst MJ, McCammon JA (2007) Continuum simulations of acetylcholine diffusion with reaction-determined boundaries in neuromuscular junction models. Biophys Chem 127(3): 129–139

    Article  Google Scholar 

  • Cheng Y, Suen JK, Zhang D, Bond SD, Zhang Y, Song Y, Baker NA, Bajaj CL, Holst MJ, McCammon JA (2007) Finite element analysis of the time-dependent Smoluchowski equation for acetylcholinesterase reaction rate calculations. Biophys J 92(10): 3397–3406

    Article  Google Scholar 

  • Chern IL, Liu J-G, Weng W-C (2003) Accurate evaluation of electrostatics for macromolecules in solution. Methods Appl Anal 10(2): 309–328

    MathSciNet  MATH  Google Scholar 

  • Chiba M, Fedorov DG, Kitaura K (2008) Polarizable continuum model with the fragment molecular orbital-based time-dependent density functional theory. J Comput Chem 29: 2667–2676

    Article  Google Scholar 

  • Chopp DL (1993) Computing minimal surfaces via level set curvature flow. J Comput Phys 106(1): 77–91

    Article  MathSciNet  MATH  Google Scholar 

  • Chorny I, Dill KA, Jacobson MP (2005) Surfaces affect ion pairing. J Phys Chem B 109(50): 24056–24060

    Article  Google Scholar 

  • Chu VB, Bai Y, Lipfert J, Herschlag D, Doniach S (2007) Evaluation of ion binding to DNA duplexes using a size-modified Poisson–Boltzmann theory. Biophys J 93(9): 3202–3209

    Article  Google Scholar 

  • Connolly ML (1983) Analytical molecular surface calculation. J Appl Crystallogr 16(5): 548–558

    Article  Google Scholar 

  • Corey RB, Pauling L (1953) Molecular models of amino acids, peptides and proteins. Rev Sci Instr 24: 621–627

    Article  Google Scholar 

  • Cossi M, Barone V, Cammi R, Tomasi J (1996) Ab initio study of solvated molecules: a new implementation of the polarizable continuum model. Chem Phys Lett 255: 327–335

    Article  Google Scholar 

  • Crowley PB, Golovin A (2005) Cation–pi interactions in protein–protein interfaces. Proteins Struct Funct Bioinform 59(2): 231–239

    Article  Google Scholar 

  • David L, Luo R, Gilson MK (2000) Comparison of generalized Born and Poisson models: energetics and dynamics of HIV protease. J Comput Chem 21(4): 295–309

    Article  Google Scholar 

  • Davis ME, Madura JD, Sines J, Luty BA, Allison SA, McCammon JA (1991) Diffusion-controlled enzymatic reactions. Methods Enzymol 202: 473–497

    Article  Google Scholar 

  • Davis ME, McCammon JA (1990) Electrostatics in biomolecular structure and dynamics. Chem Rev 94: 509–521

    Article  Google Scholar 

  • De Rienzo F, Gabdoulline RR, Menziani MC, De Benedetti PG, Wade RC (2001) Electrostatic analysis and Brownian dynamics simulation of the association of plastocyanin and cytochrome F. Biophys J 81(6): 3090–3104

    Article  Google Scholar 

  • Dietrich C, Scheidegger CE, Schreiner J, Comba JLD, Nedel LP, Silva CT (2009) Edge transformations for improving mesh quality of marching cubes. IEEE Trans Vis Comput Graph 15(1): 150–159

    Article  Google Scholar 

  • Dominy BN, Brooks CL III (1999) Development of a generalized Born model parameterization for proteins and nucleic acids. J Phys Chem B 103(18): 3765–3773

    Article  Google Scholar 

  • Dong F, Olsen B, Baker NA (2008) Computational methods for biomolecular electrostatics. Methods Cell Biol 84: 843–870

    Article  Google Scholar 

  • Dong F, Vijaykumar M, Zhou HX (2003) Comparison of calculation and experiment implicates significant electrostatic contributions to the binding stability of barnase and barstar. Biophys J 85(1): 49–60

    Article  Google Scholar 

  • Dong F, Wagoner JA, Baker NA (2008) Assessing the performance of implicit solvation models at a nucleic acid surface. Phys Chem Chem Phys 10: 4889–4902

    Article  Google Scholar 

  • Dong F, Zhou HX (2006) Electrostatic contribution to the binding stability of protein–protein complexes. Proteins 65(1): 87–102

    Article  Google Scholar 

  • Dragan AI, Read CM, Makeyeva EN, Milgotina EI, Churchill ME, Crane-Robinson C, Privalov PL (2004) DNA binding and bending by HMG boxes: energetic determinants of specificity. J Mol Biol 343(2): 371–393

    Article  Google Scholar 

  • Dzubiella J, Swanson JMJ, McCammon JA (2006) Coupling hydrophobicity, dispersion, and electrostatics in continuum solvent models. Phys Rev Lett 96: 087802

    Article  Google Scholar 

  • Dzubiella J, Swanson JMJ, McCammon JA (2006) Coupling nonpolar and polar solvation free energies in implicit solvent models. J Chem Phys 124: 084905

    Article  Google Scholar 

  • Edinger SR, Cortis C, Shenkin PS, Friesner RA (1997) Solvation free energies of peptides: comparison of approximate continuum solvation models with accurate solution of the Poisson–Boltzmann equation. J Phys Chem B 101(7): 1190–1197

    Article  Google Scholar 

  • Elcock AH, Gabdoulline RR, Wade RC, McCammon JA (1999) Computer simulation of protein–protein association kinetics: acetylcholinesterase-fasciculin. J Mol Biol 291(1): 149–162

    Article  Google Scholar 

  • Fedkiw RP, Aslam T, Merriman B, Osher S (1999) A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J Comput Phys 152: 457–492

    Article  MathSciNet  MATH  Google Scholar 

  • Fedorov MV, Goodman JM, Schumm S (2009) To switch or not to switch: the effects of potassium and sodium ions on α-poly-l-glutamate conformations in aqueous solutions. J Am Chem Soc 131: 10854–10856

    Article  Google Scholar 

  • Fedorov MV, Kornyshev AA (2007) Unravelling the solvent response to neutral and charged solutes. Mol Phys 105(1): 1–16

    Article  Google Scholar 

  • Feig M, Brooks CL III (2004) Recent advances in the development and application of implicit solvent models in biomolecule simulations. Curr Opin Struct Biol 14: 217–224

    Article  Google Scholar 

  • Feng X, Prohl A (2004) Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math Comput 73: 541–567

    MathSciNet  MATH  Google Scholar 

  • Fixman M (1979) The Poisson–Boltzmann equation and its application to polyelectrolytes. J Chem Phys 70(11): 4995–5005

    Article  Google Scholar 

  • Fogolari F, Brigo A, Molinari H (2002) The Poisson–Boltzmann equation for biomolecular electrostatics: a tool for structural biology. J Mol Recogn 15(6): 377–392

    Article  Google Scholar 

  • Forsman J (2004) A simple correlation-corrected Poisson–Boltzmann theory. J Phys Chem B 108(26): 9236–9245

    Article  Google Scholar 

  • Fries PH, Patey GN (1985) The solution of the hypernetted-chain approximation for fluids of nonspherical particles. a general method with application to dipolar hard spheres. J Chem Phys 82: 429–440

    Article  Google Scholar 

  • Gabdoulline RR, Wade RC (1998) Brownian dynamics simulation of protein–protein diffusional encounter. Methods a Companion to Methods in Enzymology 14(3): 329–341

    Article  Google Scholar 

  • Gallicchio E, Kubo MM, Levy RM (2000) Enthalpy-entropy and cavity decomposition of alkane hydration free energies: Numerical results and implications for theories of hydrophobic solvation. J Phys Chem B 104(26): 6271–6285

    Article  Google Scholar 

  • Gallicchio E, Levy RM (2004) AGBNP: an analytic implicit solvent model suitable for molecular dynamics simulations and high-resolution modeling. J Comput Chem 25(4): 479–499

    Article  Google Scholar 

  • Gallicchio E, Zhang LY, Levy RM (2002) The SGB/NP hydration free energy model based on the surface generalized Born solvent reaction field and novel nonpolar hydration free energy estimators. J Comput Chem 23(5): 517–529

    Article  Google Scholar 

  • Geng W, Yu S, Wei GW (2007) Treatment of charge singularities in implicit solvent models. J Chem Phys 127: 114106

    Article  Google Scholar 

  • Geng WH, Wei GW (2010) Multiscale molecular dynamics via the matched interface and boundary (MIB) method. J Comput Phys (in press)

  • Georgescu RE, Alexov EG, Gunner MR (2002) Combining conformational flexibility and continuum electrostatics for calculating pKas in proteins. Biophys J 83(4): 1731–1748

    Article  Google Scholar 

  • Gilboa G, Sochen N, Zeevi YY (2004) Image sharpening by flows based on triple well potentials. J Math Imaging Vis 20: 121–131

    Article  MathSciNet  Google Scholar 

  • Gilson MK, Davis ME, Luty BA, McCammon JA (1993) Computation of electrostatic forces on solvated molecules using the Poisson–Boltzmann equation. J Phys Chem 97(14): 3591–3600

    Article  Google Scholar 

  • Gomes J, Faugeras OD (2001) Using the vector distance functions to evolve manifolds of arbitrary codimension. Lect Notes Comput Sci 2106: 1–13

    Article  Google Scholar 

  • Grant JA, Pickup BT, Nicholls A (2001) A smooth permittivity function for Poisson–Boltzmann solvation methods. J Comput Chem 22(6): 608–640

    Article  Google Scholar 

  • Grant JA, Pickup BT, Sykes MT, Kitchen CA, Nicholls A (2007) The Gaussian Generalized Born model: application to small molecules. Phys Chem Chem Phys 9: 4913–4922

    Article  Google Scholar 

  • Greer JB, Bertozzi AL (2004) H-1 solutions of a class of fourth order nonlinear equations for image processing. Discret Contin Dyn Syst 10: 349–366

    MathSciNet  MATH  Google Scholar 

  • Greer JB, Bertozzi AL (2004) Traveling wave solutions of fourth order pdes for image processing. SIAM J Math Anal 36: 38–68

    Article  MathSciNet  MATH  Google Scholar 

  • Grochowski P, Trylska J (2007) Continuum molecular electrostatics, salt effects and counterion binding. A review of the Poisson–Boltzmann theory and its modifications. Biopolymers 89(2): 93–113

    Article  Google Scholar 

  • Helfrich W (1973) Elastic properties of lipid bilayers: theory and possible experiments. Zeitschrift für Naturforschung Teil C 28: 693–703

    Google Scholar 

  • Holm C, Kekicheff P, Podgornik R (2001) Electrostatic effects in soft matter and biophysics. NATO science series. Kluwer, Boston

    Google Scholar 

  • Holst MJ (1993) Multilevel methods for the Poisson–Boltzmann equation. University of Illinois at Urbana/Champaign, Numerical Computing Group, Urbana/Champaign

    Google Scholar 

  • Honig B, Nicholls A (1995) Classical electrostatics in biology and chemistry. Science 268(5214): 1144–1149

    Article  Google Scholar 

  • Hori T, Takahashi H, Nakano M, Nitta T, Yang W (2006) A qm/ mm study combined with the theory of energy representation: Solvation free energies for anti/syn acetic acids in aqueous solution. Chem Phys Lett 419(1–3): 240–244

    Article  Google Scholar 

  • Huang DM, Geissler PL, Chandler D (2001) Scaling of hydrophobic solvation free energies. J Phys Chemi B 105(28): 6704–6709

    Article  Google Scholar 

  • Husowitz B, Talanquer V (2007) Solvent density inhomogeneities and solvation free energies in supercritical diatomic fluids: a density functional approach. J Chem Phys 126(5): 054508

    Article  Google Scholar 

  • Im W, Beglov D, Roux B (1998) Continuum solvation model: electrostatic forces from numerical solutions to the Poisson–Boltzmann equation. Comput Phys Commun 111(1–3): 59–75

    Article  MATH  Google Scholar 

  • Improta R, Barone V, Scalmani G, Frisch MJ (2006) A state-specific polarizable continuum model time dependent density functional theory method for excited state calculations in solution. J Chem Phys 125(054103)

  • Iwamoto M, Liu F, Ou-Yang ZC (2006) Shape and stability of two-dimensional lipid domains with dipole–dipole interactions. J Chem Phys 125: 224701

    Article  Google Scholar 

  • Jackson RM, Sternberg MJ (1995) A continuum model for protein–protein interactions: application to the docking problem. J Mol Biol 250(2): 258–275

    Article  Google Scholar 

  • Jakalian A, Bush BL, Jack DB, Bayly CI (2000) Fast, efficient generation of high-quality atomic charges. am1-bcc model: I. Method. J Comput Chem 21(2): 132–146

    Article  Google Scholar 

  • Jayaram B, Sprous D, Beveridge DL (1998) Solvation free energy of biomacromolecules: parameters for a modified generalized Born model consistent with the AMBER force field. J Phys Chem B 102(47): 9571–9576

    Article  Google Scholar 

  • Jinnouchi R, Anderson AB (2008) Electronic structure calculations of liquid–solid interfaces: combination of density functional theory and modified Poisson–Boltzmann theory. Phys Rev B 77(245417)

  • Kamerlin SCL, Haranczyk M, Warshel A (2009) Progress in ab initio qm/ mm free-energy simulations of electrostatic energies in proteins: Accelerated qm/ mm studies of pk(a), redox reactions and solvation free energies. J Phys Chem B 113: 1253–1272

    Article  Google Scholar 

  • Kirkwood JG (1934) Theory of solution of molecules containing widely separated charges with special application to zwitterions. J Comput Phys 7: 351–361

    Google Scholar 

  • Koehl P (2006) Electrostatics calculations: latest methodological advances. Curr Opin Struct Biol 16(2): 142–151

    Article  Google Scholar 

  • Kuhn LA, Siani MA, Pique ME, Fisher CL, Getzoff ED, Tainer JA (1992) The interdependence of protein surface topography and bound water molecules revealed by surface accessibility and fractal density measures. J Mol Biol 228(1): 13–22

    Article  Google Scholar 

  • Lai MC, Peskin CS (2000) An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J Comput Phys 160: 705–719

    Article  MathSciNet  MATH  Google Scholar 

  • Lamm G (2003) The Poisson–Boltzmann equation. In: Lipkowitz KB, Larter R, Cundari TR (eds) Reviews in computational chemistry.. Wiley, Hoboken, pp 147–366

    Chapter  Google Scholar 

  • Lebard DN, Matyushov DV (2008) Redox entropy of plastocyanin: developing a microscopic view of mesoscopic polar solvation. J Chem Phys 128(15): 155106

    Article  Google Scholar 

  • Lee B, Richards FM (1971) The interpretation of protein structures: estimation of static accessibility. J Mol Biol 55(3): 379–400

    Article  Google Scholar 

  • Lee MS, Salsbury JFRm, Olson MA (2004) An efficient hybrid explicit/implicit solvent method for biomolecular simulations. J Comput Chem 25(16): 1967–1978

    Article  Google Scholar 

  • Lee TS, York DM, Yang W (1996) Linear-scaling semiempirical quantum calculations for macromolecules. J Chem Phys 105(7): 2744–2750

    Article  Google Scholar 

  • LeVeque RJ, Li ZL (1994) The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J Numer Anal 31: 1019–1044

    Article  MathSciNet  MATH  Google Scholar 

  • Levy RM, Zhang LY, Gallicchio E, Felts AK (2003) On the nonpolar hydration free energy of proteins: surface area and continuum solvent models for the solute–solvent interaction energy. J Am Chem Soc 125(31): 9523–9530

    Article  Google Scholar 

  • Li H, Robertson AD, Jensen JH (2004) The determinants of carboxyl pKa values in turkey ovomucoid third domain. Proteins 55(3): 689–704

    Article  Google Scholar 

  • Li H, Robertson AD, Jensen JH (2005) Very fast empirical prediction and rationalization of protein pka values. Proteins 61(4): 704–721

    Article  Google Scholar 

  • Li J, Fisher CL, Chen JL, Bashford D, Noodleman L (1996) Calculation of redox potentials and pKa values of hydrated transition metal cations by a combined density functional and continuum dielectric theory. Inorg Chem 35(16): 4694–4702

    Article  Google Scholar 

  • Li Y, Santosa F (1996) A computational algorithm for minimizing total variation in image restoration. IEEE Trans Image Process 5(6): 987–995

    Article  Google Scholar 

  • Li ZL, Ito K (2001) Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J Sci Comput 23: 339–361

    Article  MathSciNet  MATH  Google Scholar 

  • Licata VJ, Allewell NM (1997) Functionally linked hydration changes in Escherichia coli aspartate transcarbamylase and its catalytic subunit. Biochemistry 36(33): 10161–10167

    Article  Google Scholar 

  • Liu XD, Fedkiw RP, Kang M (2000) A boundary condition capturing method for Poisson’s equation on irregular domains. J Comput Phys 160: 151–178

    Article  MathSciNet  MATH  Google Scholar 

  • Livesay DR, Jambeck P, Rojnuckarin A, Subramaniam S (2003) Conservation of electrostatic properties within enzyme families and superfamilies. Biochemistry 42(12): 3464–3473

    Article  Google Scholar 

  • Livingstone JR, Spolar RS, Record MT Jr (1991) Contribution to the thermodynamics of protein folding from the reduction in water-accessible nonpolar surface area. Biochemistry 30(17): 4237–4244

    Article  Google Scholar 

  • Lorensen WE, Cline HE (1987) Marching cubes: a high resolution 3D surface reconstruction algorithm. Comput Graph 21: 163–169

    Article  Google Scholar 

  • Lu Q, Luo R (2003) A Poisson–Boltzmann dynamics method with nonperiodic boundary condition. J Chem Phys 119(21): 11035–11047

    Article  Google Scholar 

  • Luo R, David L, Gilson MK (2002) Accelerated Poisson–Boltzmann calculations for static and dynamic systems. J Comput Chem 23(13): 1244–1253

    Article  Google Scholar 

  • Luty BA, Davis ME, McCammon JA (1992) Solving the finite-difference non-linear Poisson–Boltzmann equation. J Comput Chem 13: 1114–1118

    Article  MathSciNet  Google Scholar 

  • Lysaker M, Lundervold A, Tai XC (2003) Noise removal using fourth-order partial differential equation with application to medical magnetic resonance images in space and time. IEEE Trans Imaging Process 12: 1579–1590

    Article  Google Scholar 

  • MacDermaid CM, Kaminski GA (2007) Electrostatic polarization is crucial for reproducing pKa shifts of carboxylic residues in turkey ovomucoid third domain. J Phys Chem B 111(30): 9036–9044

    Article  Google Scholar 

  • MacKerell J, Bashford ADD, Bellot M, Dunbrack J, Evanseck RLJD, Field MJ, Fischer S, Gao J, Guo H, Ha S, Joseph-McCarthy D, Kuchnir L, Kuczera K, Lau FTK, Mattos C, Michnick S, Ngo T, Nguyen DT, Prodhom B, Reiher I, Roux WEB, Schlenkrich M, Smith JC, Stote R, Straub J, Watanabe M, Wiorkiewicz-Kuczera J, Yin D, Karplus M (1998) All-atom empirical potential for molecular modeling and dynamics studies of proteins. J Phys Chem B 102(18): 3586–3616

    Article  Google Scholar 

  • Madura JD, Briggs JM, Wade RC, Davis ME, Luty BA, Ilin A, Antosiewicz J, Gilson MK, Bagheri B, Scott LR, McCammon JA (1995) Electrostatics and diffusion of molecules in solution—simulations with the University of Houston Brownian Dynamics program. Comput Phys Commun 91(1–3): 57–95

    Article  Google Scholar 

  • Marenich AV, Cramer CJ, Truhlar DG (2008) Perspective on foundations of solvation modeling: the electrostatic contribution to the free energy of solvation. J Chem Theory Comput 4(6): 877–887

    Article  Google Scholar 

  • Massova I, Kollman PA (1999) Computational alanine scanning to probe protein–protein interactions: a novel approach to evaluate binding free energies. J Am Chem Soc 121(36): 8133–8143

    Article  Google Scholar 

  • Matousek WM, Ciani B, Fitch CA, Garcia-Moreno BE, Kammerer RA, Alexandrescu AT (2007) Electrostatic contributions to the stability of the GCN4 leucine zipper structure. J Mol Biol 374(1): 206–219

    Article  Google Scholar 

  • Mayo A (1984) The fast solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J Numer Anal 21: 285–299

    Article  MathSciNet  MATH  Google Scholar 

  • Mikula K, Sevcovic D (2004) A direct method for solving an anisotropic mean curvature flow of plane curves with an external force. Math Methods Appl Sci 27(13): 1545–1565

    Article  MathSciNet  MATH  Google Scholar 

  • Miller JL, Kollman PA (1996) Solvation free energies of the nucleic acid bases. J Phys Chem 100(20): 8587–8594

    Article  Google Scholar 

  • Mobley DL, Dill KA, Chodera JD (2008) Treating entropy and conformational changes in implicit solvent simulations of small molecules. J Phys Chemi B 112(3): 938–946

    Article  Google Scholar 

  • Mohan V, Davis ME, McCammon JA, Pettitt BM (1992) Continuum model calculations of solvation free energies: accurate evaluation of electrostatic contributions. J Phys Chem 96(15): 6428–6431

    Article  Google Scholar 

  • Mongan J, Simmerling C, McCammon JA, Case DA, Onufriev A (2007) Generalized Born model with a simple, robust molecular volume correction. J Chem Theory Comput 3(1): 159–169

    Article  Google Scholar 

  • Mu Y, Yang Y, Xu W (2007) Hybrid hamiltonian replica exchange molecular dynamics simulation method employing the Poisson–Boltzmann model. J Chem Phys 127(8)

  • Mumford D, Shah J (1989) Optimal approximations by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 42(5): 577–685

    Article  MathSciNet  MATH  Google Scholar 

  • Netz RR, Orland H (2000) Beyond Poisson–Boltzmann: fluctuation effects and correlation functions. Eur Phys J E 1(2–3): 203–214

    Article  Google Scholar 

  • Nicholls A, Mobley DL, Guthrie PJ, Chodera JD, Pande VS (2008) Predicting small-molecule solvation free energies: an informal blind test for computational chemistry. J Med Chem 51(4): 769–779

    Article  Google Scholar 

  • Nielsen JE, Andersen KV, Honig B, Hooft RWW, Klebe G, Vriend G, Wade RC (1999) Improving macromolecular electrostatics calculations. Protein Eng 12(8): 657–662

    Article  Google Scholar 

  • Nielsen JE, Vriend G (2001) Optimizing the hydrogen-bond network in Poisson–Boltzmann equation-based pK(a) calculations. Proteins 43(4): 403–412

    Article  Google Scholar 

  • Nina M, Im W, Roux B (1999) Optimized atomic radii for protein continuum electrostatics solvation forces. Biophys Chem 78(1–2): 89–96

    Article  Google Scholar 

  • Oevermann M, Klein R (2006) A cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces. J Comput Phys 219: 749–769

    Article  MathSciNet  MATH  Google Scholar 

  • Okur A, Wickstrom L, Layten M, Geney R, Song K, Hornak V, Simmerling C (2006) Improved efficiency of replica exchange simulations through use of a hybrid explicit/implicit solvation model. J Chem Theory Comput 2(2): 420–433

    Article  Google Scholar 

  • Onufriev A, Bashford D, Case DA (2000) Modification of the generalized Born model suitable for macromolecules. J Phys Chem B 104(15): 3712–3720

    Article  Google Scholar 

  • Onufriev A, Case DA, Bashford D (2002) Effective Born radii in the generalized Born approximation: the importance of being perfect. J Comput Chem 23(14): 1297–1304

    Article  Google Scholar 

  • Osher S, Fedkiw RP (2001) Level set methods: an overview and some recent results.. J Comput Phys 169(2): 463–502

    Article  MathSciNet  MATH  Google Scholar 

  • Osher S, Rudin LI (1990) Feature-oriented image enhancement using shock filters. SIAM J Numer Anal 27(4): 919–940

    Article  MATH  Google Scholar 

  • Osher S, Sethian JE (1988) Fronts propagating with curvature-dependent speed: algorithms based on the Hamilton–Jacobi formulation. J Comput Phys 79: 12–49

    Article  MathSciNet  MATH  Google Scholar 

  • Ou-Yang ZC, Helfrich W (1989) Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Phys Rev A 39: 5280–5288

    Article  Google Scholar 

  • Page CS, Bates PA (2006) Can MM-PBSA calculations predict the specificities of protein kinase inhibitors?. J Comput Chem 27(16): 1990–2007

    Article  Google Scholar 

  • Palmer DS, Frolov AI, Ratkova EL, Fedorov MV (2010) Towards a universal method for calculating hydration free energies: a 3D reference interaction site model with partial molar volume correction. J Phys Condens Matter 22(492101)

  • Palmer DS, Sergiievskyi VP, Jensen F, Fedorov MV (2010) Accurate calculations of the hydration free energies of druglike molecules using the reference interaction site model. J Chem Phys 133(044104)

  • Penfold R, Nordholm S, Jnsson B, Woodward CE (1990) A simple analysis of ion–ion correlation in polyelectrolyte solutions. J Chem Phys 92(3): 1915–1922

    Article  Google Scholar 

  • Peskin CS (1977) Numerical analysis of blood flow in the heart. J Comput Phys 25(3): 220–252

    Article  MathSciNet  MATH  Google Scholar 

  • Petrey D, Honig B (2003) GRASP2: visualization, surface properties, and electrostatics of macromolecular structures and sequences. Methods Enzymol 374: 492–509

    Article  Google Scholar 

  • Pierotti RA (1976) A scaled particle theory of aqueous and nonaqeous solutions. Chem Rev 76(6): 717–726

    Article  Google Scholar 

  • Ponder JW, Case DA (2003) Force fields for protein simulations. Adv Protein Chem 66: 27–85

    Article  Google Scholar 

  • Prabhu NV, Panda M, Yang QY, Sharp KA (2008) Explicit ion, implicit water solvation for molecular dynamics of nucleic acids and highly charged molecules. J Comput Chem 29: 1113–1130

    Article  Google Scholar 

  • Prabhu NV, Zhu P, Sharp KA (2004) Implementation and testing of stable, fast implicit solvation in molecular dynamics using the smooth-permittivity finite difference Poisson–Boltzmann method. J Comput Chem 25(16): 2049–2064

    Article  Google Scholar 

  • Quina FH, Alonso EO, Farah JPS (1995) Incorporation of nonionic solutes into aqueous micelles: a linear solvation free energy relationship analysis. J Phys Chem 99: 11708–11714

    Article  Google Scholar 

  • Ratkova EL, Chuev GN, Sergiievskyi VP, Fedorov MV (2010) An accurate prediction of hydration free energies by combination of molecular integral equations theory with structural descriptors. J Phys Chem B 114(37): 12068–12079

    Article  Google Scholar 

  • Reddy MR, Singh UC, Erion MD (2007) Ab initio quantum mechanics-based free energy perturbation method for calculating relative solvation free energies. J Comput Chem 28(2): 491–494

    Article  Google Scholar 

  • Richards FM (1977) Areas, volumes, packing, and protein structure. Annu Rev Biophys Bioeng 6(1): 151–176

    Article  Google Scholar 

  • Roux B, Simonson T (1999) Implicit solvent models. Biophys Chem 78(1–2): 1–20

    Article  Google Scholar 

  • Rudin LI, Osher S, Fatemi E (1992) Nonlinear total variation based noise removal algorithms. In: Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics: computational issues in nonlinear science. Elsevier North-Holland, Inc., Amsterdam, pp 259–268

  • Sanner MF, Olson AJ, Spehner JC (1996) Reduced surface: an efficient way to compute molecular surfaces. Biopolymers 38: 305–320

    Article  Google Scholar 

  • Sapiro G, Ringach DL (1996) Anisotropic diffusion of multivalued images with applications to color filtering. IEEE Trans Image Process 5(11): 1582–1586

    Article  Google Scholar 

  • Sarti A, Malladi R, Sethian JA (2002) Subjective surfaces: a geometric model for boundary completion. Int J Comput Vis 46(3): 201–221

    Article  MATH  Google Scholar 

  • Savelyev A, Papoian GA (2007) Inter-DNA electrostatics from explicit solvent molecular dynamics simulations. J Am Chem Soc 129(19): 6060–6061

    Article  Google Scholar 

  • Sbert C, Solé AF (2003) 3D curves reconstruction based on deformable models. J Math Imaging Vis 18(3): 211–223

    Article  MATH  Google Scholar 

  • Schaefer M, Karplus M (1996) A comprehensive analytical treatment of continuum electrostatics. J Phys Chem 100(5): 1578–1599

    Article  Google Scholar 

  • Sept D, Elcock AH, McCammon JA (1999) Computer simulations of actin polymerization can explain the barbed-pointed end asymmetry. J Mol Biol 294(5): 1181–1189

    Article  Google Scholar 

  • Sept D, McCammon JA (2001) Thermodynamics and kinetics of actin filament nucleation. Biophys J 81(2): 667–674

    Article  Google Scholar 

  • Sethian JA (2001) Evolution, implementation, and application of level set and fast marching methods for advancing fronts. J Comput Phys 169(2): 503–555

    Article  MathSciNet  MATH  Google Scholar 

  • Sham YY, Muegge I, Warshel A (1998) The effect of protein relaxation on charge–charge interactions and dielectric constants of proteins. Biophys J 74(4): 1744–1753

    Article  Google Scholar 

  • Sharp KA, Honig B (1990) Calculating total electrostatic energies with the nonlinear Poisson–Bottzmann equatlon. J Phys Chem 94: 7684–7692

    Article  Google Scholar 

  • Sharp KA, Honig B (1990) Electrostatic interactions in macromolecules—theory and applications. Annu Rev Biophys Biophys Chem 19: 301–332

    Article  Google Scholar 

  • Simonson T (2001) Macromolecular electrostatics: continuum models and their growing pains. Curr Opin Struct Biol 11(2): 243–252

    Article  Google Scholar 

  • Simonson T (2003) Electrostatics and dynamics of proteins. Rep Prog Phys 66(5): 737–787

    Article  Google Scholar 

  • Simonson T, Brunger AT (1994) Solvation free energies estimated from macroscopic continuum theory: an accuracy assessment. J Phys Chem 98(17): 4683–4694

    Article  Google Scholar 

  • Smereka P (2003) Semi-implicit level set methods for curvature and surface diffusion motion. J Sci Comput 19(1): 439–456

    Article  MathSciNet  MATH  Google Scholar 

  • Smereka P (2006) The numerical approximation of a delta function with application to level set methods. J Comput Phys 211(1): 77–90

    Article  MathSciNet  MATH  Google Scholar 

  • Sochen N, Kimmel R, Malladi R (1998) A general framework for low level vision. IEEE Trans Image Process 7(3): 310–318

    Article  MathSciNet  MATH  Google Scholar 

  • Song Y, Zhang Y, Bajaj CL, Baker NA (2004) Continuum diffusion reaction rate calculations of wild-type and mutant mouse acetylcholinesterase: adaptive finite element analysis. Biophys J 87(3): 1558–1566

    Article  Google Scholar 

  • Song Y, Zhang Y, Shen T, Bajaj CL, McCammon JA, Baker NA (2004) Finite element solution of the steady-state Smoluchowksi equation for rate constant calculations. Biophys J 86(4): 2017–2029

    Article  Google Scholar 

  • Spolar RS, Ha JH, Record MT Jr (1989) Hydrophobic effect in protein folding and other noncovalent processes involving proteins. Proc Natl Acad Sci USA 86(21): 8382–8385

    Article  Google Scholar 

  • Stillinger FH (1973) Structure in aqueous solutions of nonpolar solutes from the standpoint of scaled-particle theory. J Solut Chem 2: 141–158

    Article  Google Scholar 

  • Sun YH, Wu PR, Wei GW, Wang G (2006) Evolution operator based single-step method for image processing. Int J Biomed Imaging 83847: 1–27

    Article  Google Scholar 

  • Swanson JMJ, Henchman RH, McCammon JA (2004) Revisiting free energy calculations: a theoretical connection to MM/PBSA and direct calculation of the association free energy. Biophys J 86(1): 67–74

    Article  Google Scholar 

  • Swanson JMJ, Mongan J, McCammon JA (2005) Limitations of atom-centered dielectric functions in implicit solvent models. J Phys Chem B 109(31): 14769–14772

    Article  Google Scholar 

  • Takano Y, Houk KN (2005) Benchmarking the conductor-like polarizable continuum model (cpcm) for aqueous solvation free energies of neutral and ionic organic molecules. J Chem Theory Comput 1(1): 70–77

    Article  Google Scholar 

  • Tan C, Tan YH, Luo R (2007) Implicit nonpolar solvent models. J Phys Chem B 111(42): 12263–12274

    Article  Google Scholar 

  • Tan C, Yang L, Luo R (2006) How well does Poisson–Boltzmann implicit solvent agree with explicit solvent? A quantitative analysis. J Phys Chem B 110(37): 18680–18687

    Article  Google Scholar 

  • Tan JJ, Chen WZ, Wang CX (2006) Investigating interactions between HIV-1 gp41 and inhibitors by molecular dynamics simulation and MM-PBSA/GBSA calculations. J Mol Struct Theochem 766(2–3): 77–82

    Article  Google Scholar 

  • Tan ZJ, Chen SJ (2005) Electrostatic correlations and fluctuations for ion binding to a finite length polyelectrolyte. J Chem Phys 122: 044903

    Article  Google Scholar 

  • Tanaka M, Grosberg AY (2001) Giant charge inversion of a macroion due to multivalent counterions and monovalent coions: Molecular dynamics study. J Chem Phys 115(1): 567–574

    Article  Google Scholar 

  • Tang CL, Alexov E, Pyle AM, Honig B (2007) Calculation of pKas in RNA: on the structural origins and functional roles of protonated nucleotides. J Mol Biol 366(5): 1475–1496

    Article  Google Scholar 

  • Tawa GJ, Topol IA, Burt SK, Caldwell RA, Rashin AA (1998) Calculation of the aqueous solvation free energy of the proton. J Chem Phys 109(12): 4852–4863

    Article  Google Scholar 

  • Terekhova I, Romanova AO, Kumeev RS, Fedorov MV (2010) Selective Na+/K+ effects on the formation of α-cyclodextrin complexes with aromatic carboxylic acids: competition for the guest. J Phys Chem B 114(37): 12607–12613

    Article  Google Scholar 

  • Tjong H, Zhou HX (2007) GBr6NL: a generalized Born method for accurately reproducing solvation energy of the nonlinear Poisson–Boltzmann equation. J Chem Phys 126: 195102

    Article  Google Scholar 

  • Tomasi J, Mennucci B, Cammi R (2005) Quantum mechanical continuum solvation models. Chem Rev 105: 2999–3093

    Article  Google Scholar 

  • Tsui V, Case DA (2000) Molecular dynamics simulations of nucleic acids with a generalized Born solvation model. J Am Chem Soc 122(11): 2489–2498

    Article  Google Scholar 

  • Tsui V, Case DA (2001) Calculations of the absolute free energies of binding between RNA and metal ions using molecular dynamics simulations and continuum electrostatics. J Phys Chem B 105(45): 11314–11325

    Article  Google Scholar 

  • Tully-Smith DM, Reiss H (1970) Further development of scaled particle theory of rigid sphere fluids. J Chem Phys 53(10): 4015–4025

    Article  Google Scholar 

  • Vitalis A, Baker NA, McCammon JA (2004) ISIM: a program for grand canonical Monte Carlo simulations of the ionic environment of biomolecules. Mol Simul 30(1): 45–61

    Article  MATH  Google Scholar 

  • Vitalis A, Pappu RV (2009) ABSINTH: a new continuum solvation model for simulations of polypeptides in aqueous solutions. J Comput Chem 30(5): 673–699

    Article  Google Scholar 

  • Wade RC, Gabdoulline RR, De Rienzo F (2001) Protein interaction property similarity analysis. Int J Quant Chem 83(3–4): 122–127

    Article  Google Scholar 

  • Wagoner JA, Baker NA (2006) Assessing implicit models for nonpolar mean solvation forces: the importance of dispersion and volume terms. Proc Natl Acad Sci USA 103(22): 8331–8336

    Article  Google Scholar 

  • Wallquist A, Berne BJ (1995) Computer-simulation of hydrophobic hydration forces stacked plates at short-range. J Phys Chem 99: 2893–2899

    Article  Google Scholar 

  • Warshel A, Papazyan A (1998) Electrostatic effects in macromolecules: fundamental concepts and practical modeling. Curr Opin Struct Biol 8(2): 211–217

    Article  Google Scholar 

  • Warshel A, Sharma PK, Kato M, Parson WW (2006) Modeling electrostatic effects in proteins. Biochim Biophys Acta (BBA) Proteins Proteomics 1764(11): 1647–1676

    Article  Google Scholar 

  • Warwicker J, Watson HC (1982) Calculation of the electric potential in the active site cleft due to alpha-helix dipoles. J Mol Biol 157(4): 671–679

    Article  Google Scholar 

  • Weeks JD, Chandler D, Andersen HC (1971) Role of repulsive forces in determining the equilibrium structure of simple liquids. J Chem Phys 54(12): 5237–5247

    Article  Google Scholar 

  • Wei GW (1999) Generalized Perona–Malik equation for image restoration. IEEE Signal Process Lett 6(7): 165–167

    Article  Google Scholar 

  • Wei GW (2010) Differential geometry based multiscale models. Bull Math Biol 72: 1562–1622

    Article  MathSciNet  MATH  Google Scholar 

  • Wei GW, Jia YQ (2002) Synchronization-based image edge detection. Europhys Lett 59(6): 814

    Article  Google Scholar 

  • Wei GW, Sun YH, Zhou YC, Feig M (2005) Molecular multiresolution surfaces, pp 1–11. arXiv:math-ph/0511001v1

  • Weinzinger P, Hannongbua S, Wolschann P (2005) Molecular mechanics PBSA ligand binding energy and interaction of efavirenz derivatives with HIV-1 reverse transcriptase. J Enzyme Inhib Med Chem 20(2): 129–134

    Article  Google Scholar 

  • Willmore TJ (1997) Riemannian geometry. Oxford University Press, USA

    Google Scholar 

  • Wolfgang K (2002) Differential geometry: curves-surface-manifolds. American Mathematical Society, Providence

  • Xu G, Pan Q, Bajaj CL (2006) Discrete surface modeling using partial differential equations. Comput Aided Geom Des 23(2): 125–145

    Article  MathSciNet  MATH  Google Scholar 

  • Xu M, Zhou SL (2007) Existence and uniqueness of weak solutions for a fourth-order nonlinear parabolic equation. J Math Anal Appl 325: 636–654

    Article  MathSciNet  MATH  Google Scholar 

  • Yang AS, Gunner MR, Sampogna R, Sharp K, Honig B (1993) On the calculation of pK(a)s in proteins. Proteins Struct Funct Genet 15(3): 252–265

    Article  Google Scholar 

  • Yu S, Geng W, Wei GW (2007) Treatment of geometric singularities in implicit solvent models. J Chem Phys 126: 244108

    Article  Google Scholar 

  • Yu S, Wei GW (2007) Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities. J Comput Phys 227: 602–632

    Article  MathSciNet  MATH  Google Scholar 

  • Yu S, Zhou Y, Wei GW (2007) Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces. J Comput Phys 224(2): 729–756

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang D, Suen J, Zhang Y, Radic Z, Taylor P, Holst M, Bajaj C, Baker NA, McCammon JA (2005) Tetrameric mouse acetylcholinesterase: continuum diffusion rate calculations by solving the steady-state Smoluchowski equation using finite element methods. Biophys J 88(3): 1659–1665

    Article  Google Scholar 

  • Zhang Y, Xu G, Bajaj C (2006) Quality meshing of implicit solvation models of biomolecular structures. Comput Aided Geom Des 23(6): 510–530

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao S (2010) High order matched interface and boundary methods for the helmholtz equation in media with arbitrarily curved interfaces. J Comput Phys 229: 3155–3170

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao S, Wei GW (2004) High-order FDTD methods via derivative matching for Maxwell’s equations with material interfaces. J Comput Phys 200(1): 60–103

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou YC, Feig M, Wei GW (2008) Highly accurate biomolecular electrostatics in continuum dielectric environments. J Comput Chem 29: 87–97

    Article  Google Scholar 

  • Zhou YC, Wei GW (2006) On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method. J Comput Phys 219(1): 228–246

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou YC, Zhao S, Feig M, Wei GW (2006) High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J Comput Phys 213(1): 1–30

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou Z, Payne P, Vasquez M, Kuhn N, Levitt M (1996) Finite-difference solution of the Poisson–Boltzmann equation: complete elimination of self-energy. J Comput Chem 17: 1344–1351

    Article  Google Scholar 

  • Zhu J, Alexov E, Honig B (2005) Comparative study of generalized Born models: Born radii and peptide folding. J Phys Chem B 109(7): 3008–3022

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Michigan State University, Lansing, MI, 48824, USA

    Zhan Chen & G. W. Wei

  2. Pacific Northwest National Laboratory, P.O. Box 999, MS K7-28, Richland, WA, 99352, USA

    Nathan A. Baker

  3. Department of Electrical and Computer Engineering, Michigan State University, Lansing, MI, 48824, USA

    G. W. Wei

Authors
  1. Zhan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nathan A. Baker
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. W. Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. W. Wei.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, Z., Baker, N.A. & Wei, G.W. Differential geometry based solvation model II: Lagrangian formulation. J. Math. Biol. 63, 1139–1200 (2011). https://doi.org/10.1007/s00285-011-0402-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-011-0402-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation