Analysis of fast boundary-integral approximations for modeling electrostatic contributions of molecular binding
We analyze and suggest improvements to a recently developed approximate continuum-electrostatic model for proteins. The model, called BIBEE/I (boundary-integral based electrostatics estimation with interpolation), was able to estimate electrostatic solvation free energies to within a mean unsigned error of 4% on a test set of more than 600 proteins¶a significant improvement over previous BIBEE models. In this work, we tested the BIBEE/I model for its capability to predict residue-by-residue interactions in protein–protein binding, using the widely studied model system of trypsin and bovine pancreatic trypsin inhibitor (BPTI). Finding that the BIBEE/I model performs surprisingly less well in this task than simpler BIBEE models, we seek to explain this behavior in terms of the models’ differing spectral approximations of the exact boundary-integral operator. Calculations of analytically solvable systems (spheres and tri-axial ellipsoids) suggest two possibilities for improvement. The first is a modified BIBEE/I approach that captures the asymptotic eigenvalue limit correctly, and the second involves the dipole and quadrupole modes for ellipsoidal approximations of protein geometries. Our analysis suggests that fast, rigorous approximate models derived from reduced-basis approximation of boundaryintegral equations might reach unprecedented accuracy, if the dipole and quadrupole modes can be captured quickly for general shapes.
-  Matlab v.6 and R2012b, The Mathworks, Inc., Natick, MA
-  M. D. Altman. Computational ligand design and analysis in protein complexes using inverse methods, combinatorial search, and accurate solvation modeling. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, U.S.A., 2006.
-  M. D. Altman, J. P. Bardhan, B. Tidor, and J. K. White. FFTSVD: A fast multiscale boundary-element method solver suitable for BioMEMS and biomolecule simulation. IEEE T. Comput.-Aid. D., 25 (2006), 274–284.
-  M. D. Altman, J. P. Bardhan, J. K. White, and B. Tidor. An efficient and accurate surface formulation for biomolecule electrostatics in non-ionic solution. Engineering in Medicine and Biology Conference (EMBC), 7591-7595, 2005.
-  M. D. Altman, J. P. Bardhan, J. K. White, and B. Tidor. Accurate solution of multi-region continuum electrostatic problems using the linearized Poisson–Boltzmann equation and curved boundary elements. J. Comput. Chem., 30 (2009), 132–153.
-  K. E. Atkinson. The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, 1997.
-  C. Bajaj, S.-C. Chen, and A. Rand. An efficient higher-order fast multipole boundary element solution for Poisson– Boltzmann-based molecular electrostatics. SIAM J. Sci. Comput., 33 (2011), 826–848.
-  N. A. Baker, D. Sept, M. J. Holst, and J. A. McCammon. Electrostatics of nanoysystems: Application to microtubules and the ribosome. Proc. Natl. Acad. Sci. USA, 98 (2001), 10037–10041. [Crossref]
-  J. P. Bardhan. Interpreting the Coulomb-field approximation for Generalized-Born electrostatics using boundaryintegral equation theory. J. Chem. Phys., 129 (2008), Art. ID 144105. [Crossref]
-  J. P. Bardhan. Numerical solution of boundary-integral equations for molecular electrostatics. J. Chem. Phys., 130 (2009), Art. ID 094102. [Crossref]
-  J. P. Bardhan, M. D. Altman, J. K. White, and B. Tidor. Numerical integration techniques for curved-panel discretizations of molecule–solvent interfaces. J. Chem. Phys., 127 (2007), Art. ID 014701. [Crossref]
-  J. P. Bardhan, R. S. Eisenberg, and D. Gillespie. Discretization of the induced-charge boundary integral equation. Phys. Rev. E, 80 (2009), Art. ID 011906. [Crossref]
-  J. P. Bardhan and M. G. Knepley. Mathematical analysis of the boundary-integral based electrostatics estimation approximation for molecular solvation: Exact results for spherical inclusions. J. Chem. Phys., 135 (2011), Art. ID 124107. [Crossref]
-  J. P. Bardhan and M. G. Knepley. Computational science and re-discovery: open-source implementation of ellipsoidal harmonics for problems in potential theory. Computational Science and Discovery, 5 (2012), Art. ID 014006.
-  J. P. Bardhan, M. G. Knepley, and M. Anitescu. Bounding the electrostatic free energies associated with linear continuum models of molecular solvation. J. Chem. Phys., 130 (2009), Art. ID 104108. [Crossref]
-  D. Bashford and D. A. Case. Generalized Born models of macromolecular solvation effects. Annu. Rev. Phys. Chem., 51 (2000), 129–152. [Crossref]
-  C. Berti, D. Gillespie, J. P. Bardhan, R. S. Eisenberg, and C. Fiegna. Comparison of three-dimensional poisson solution methods for particle-based simulation and inhomogeneous dielectrics. Phys. Rev. E, 86 (2012), Art. ID 011912. [Crossref]
-  A. J. Bordner and G. A. Huber. Boundary element solution of the linear Poisson–Boltzmann equation and a multipole method for the rapid calculation of forces on macromolecules in solution. J. Comput. Chem., 24 (2003), 353–367.
-  A. H. Boschitsch, M. O. Fenley, and H.-X. Zhou. Fast boundary element method for the linear Poisson–Boltzmann equation. J. Phys. Chem. B, 106 (2002), 2741–2754. [Crossref]
-  B. O. Brandsdal, J. Åqvist, and A. O. Smalås. Computational analysis of binding of P1 variants to trypsin. Protein Science, 10 (2001), 1584–1595.
-  B. O. Brandsdal, A. O. Smalås, and J. Åqvist. free energy calculations show that acidic P1 variants undergo large pKa shifts upon binding to trypsin. Proteins: Structure, Function, and Bioinformatics, 64 (2006), 740–748.
-  B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and M. Karplus. CHARMM: A program for macromolecular energy, minimization, and dynamics calculations. J. Comput. Chem., 4 (1983), 187–217.
-  N. Carrascal and D. F. Green. Energetic decomposition with the Generalized-Born and Poisson–Boltzmann solvent models: Lessons from association of G-protein components. J. Phys. Chem. B, 114 (2010), 5096–5116. [Crossref]
-  D. Chen, Z. Chen, C. Chen, W. Geng, and G.-W. Wei. MIBPB: A software package for electrostatic analysis. J. Comput. Chem., 32 (2011), 756–770.
-  J. H. Chen, C. L. Brooks, and J. Khandogin. Recent advances in implicit solvent-based methods for biomolecular simulations. Curr. Opin. Struc. Biol., 18 (2008), 140–148. [Crossref]
-  Y. Chen, J. S. Hesthaven, Y. Maday, and J. Rodriguez. Certified reduced basis methods and output bounds for the harmonic maxwell’s problems. SIAM J. Sci. Comput., 32 (2010), 970–996.
-  C. J. Cramer and D. G. Truhlar. Implicit solvation models: Equilibria, structure, spectra, and dynamics. Chem. Rev., 99 (1999), 2161–2200. [Crossref]
-  G. Dassios. Ellipsoidal harmonics: theory and applications. Cambridge University Press, 2012.
-  B. Fares, J. S. Hesthaven, Y. Maday, and B. Stamm. The reduced basis method for the electric field integral equation. J. Comput. Phys., 230 (2011), 5532–5555. [Crossref]
-  P. O. Fedichev, E. G. Getmantsev, and L. I. Menshikov. O(n log n) continuous electrostatics solvation energies calculation method for biomolecu les simulations. J. Comput. Chem., 32 (2011), 1368–1376.
-  M. Feig and C. L. Brooks III. Recent advances in the development and application of implicit solvent models in biomolecule simulations. Curr. Opin. Struc. Biol., 14 (2004), 217–224. [Crossref]
-  M. Feig, A. Onufriev, M. S. Lee, W. Im, D. A. Case, and C. L. Brooks III. Performance comparison of generalized Born and Poisson methods in the calculation of electrostatic solvation energies for protein structures. J. Comput. Chem., 25 (2004), 265–284.
-  A. Ghosh, C. S. Rapp, and R. A. Friesner. Generalized Born model based on a surface integral formulation. J. Phys. Chem. B, 102 (1998), 10983–10990. [Crossref]
-  M. K. Gilson and B. Honig. Calculation of the total electrostatic energy of a macromolecular system: Solvation energies, binding energies, and conformational analysis. Proteins, 4 (1988), 7–18.. [Crossref]
-  D. F. Green and B. Tidor. Design of improved protein inhibitors of HIV-1 cell entry: Optimization of electrostatic interactions at the binding interface. Proteins: Structure, Function, and Bioinformatics, 60 (2005), 644–657.
-  P. Grochowski and J. Trylska. Continuum molecular electrostatics, salt effects, and counterion binding-a review of the Poisson–Boltzmann theory and its modifications. Biopolymers, 89 (2008), 93–113. [Crossref]
-  R. Helland, J.Otlewski, O. Sundheim, M. Dadlez, and A. O. Smalås. The crystal structures of the complexes between bovine beta-trypsin and ten P1 variants of BPTI. J. Mol. Biol., 287 (1999), 923–942.
-  Z. S. Hendsch, C. V. Sindelar, and B. Tidor. Parameter dependence on continuum electrostatic calculations: A study using protein salt bridges. J. Phys. Chem. B, 102 (1998), 4404–4410. [Crossref]
-  Z. S. Hendsch and B. Tidor. Electrostatic interactions in the GCN4 leucine zipper: Substantial contributions arise from intramolecular interactions enhanced on binding. Protein Science, 8 (1999), 1381–1392.
-  E. W. Hobson. The theory of spherical and ellipsoidal harmonics. Chelsea Pub Co., 1931.
-  G. C. Hsiao and R. E. Kleinman. Error analysis in numerical solution of acoustic integral equations. International Journal for Numerical Methods in Engineering, 37 (1994), 2921–2933.
-  L. Hu and G.-W. Wei. Nonlinear Poisson equation for heterogeneous media. Biophys. J., 103 (2012), 758–766. [Crossref]
-  A. D. MacKerell Jr., D. Bashford, M. Bellott, R. L. Dunbrack Jr., J. D. Evanseck, M. J. Field, S. Fischer, J. Gao, H. Guo, S. Ha, D. Joseph–McCarthy, L. Kuchnir, K. Kuczera, F. T. K. Lau, C. Mattos, S. Michnick, T. Ngo, D. T. Nguyen, B. Prodhom, W. E. Reiher III, B. Roux, M. Schlenkrich, J. C. Smith, R. Stote, J. Straub, M. Watanabe, J. Wiorkiewicz–Kuczera, D. Yin, and M. Karplus. All-atom empirical potential for molecular modeling and dynamics studies of proteins. J. Phys. Chem. B, 102 (1998), 3586–3616.
-  A. H. Juffer, E. F. F. Botta, B. A. M. van Keulen, A. van der Ploeg, and H. J. C. Berendsen. The electric potential of a macromolecule in a solvent: A fundamental approach. J. Comput. Phys., 97 (1991), 144–171. [Crossref]
-  J. G. Kirkwood. Theory of solutions of molecules containing widely separated charges with special application to zwitterions. J. Chem. Phys., 2 (1934), 351. [Crossref]
-  I. Klapper, R. Hagstrom, R. Fine, K. Sharp, and B. Honig. Focusing of electric fields in the active site of Cu-Zn superoxide dismutase: Effects of ionic strength and amino-acid modification. Proteins, 1 (1986), 47–59. [Crossref]
-  P. Koehl. Electrostatics calculations: latest methodological advances. Curr. Opin. Struc. Biol., 16 (2006), 142–151. [Crossref]
-  P. Kukic and J. E. Nielsen. Electrostatics in proteins and protein-ligand complexes. Future Med Chem., 2 (2010), 647–666.
-  S. Kuo, B. Tidor, and J. White. A meshless, spectrally accurate, integral equation solver for molecular surface electrostatics. ACM Journal on Emerging Technologies in Computing Systems, 4 (2008), 6.
-  L.-P. Lee and B. Tidor. Optimization of binding electrostatics: Charge complementarity in the barnase-barstar protein complex. Protein Science, 10 (2001), 362–377.
-  R. M. Levy and E. Gallicchio. Computer simulations with explicit solvent: Recent progress in the thermodynamic decomposition of free energies and in modeling electrostatic effects. Ann. Rev. Phys. Chem., 49 (1998), 531–567. [Crossref]
-  J. Liang and S. Subramaniam. Computation of molecular electrostatics with boundary element methods. Biophys. J., 73 (1997), 1830–1841. [Crossref]
-  B. Lin and B. M. Pettitt. Electrostatic solvation free energy of amino acid side chain analogs: implications for the validity of electrostatic linear response in water. J. Comput. Chem., 32 (2010), 878–885.
-  Y. Lin, A. Baumketner, W. Song, S. Deng, D. Jacobs, and W. Cai. Ionic solvation studied by image-charge reaction field method. J. Chem. Phys., 134 (2011), Art. ID 044105. [Crossref]
-  S. M. Lippow, K. D. Wittrup, and B. Tidor. Computational design of antibody-affinity improvement beyond in vivo maturation. Nature Biotechnology, 25 (2007), 1171–1176.
-  B. Z. Lu, X. L. Cheng, J. Huang, and J. A. McCammon. Order N algorithm for computation of electrostatic interactions in biomolecular systems. Proc. Natl. Acad. Sci. USA, 103 (2006), 19314–19319. [Crossref]
-  J. D. Madura, J. M. Briggs, R. C. Wade, M. E. Davis, B. A. Luty, A. Ilin, J. Antosiewicz, M. K. Gilson, B. Bagheri, L. Ridgway-Scott, and J. A. McCammon. Electrostatics and diffusion of molecules in solution: Simulations with the University of Houston Brownian Dynamics program. Comput. Phys. Comm., 91 (1995), 57–95. [Crossref]
-  J. Michel, R. D. Taylor, and J. W. Essex. The parameterization and validation of generalized Born models using the pairwise descreening approximation. J. Comput. Chem., 25 (2004), 1760–1770.
-  K. S. Midelfort, H. H. Hernandez, S. M. Lippow, B. Tidor, C. L. Drennan, and K. D. Wittrup. Substantial energetic improvement with minimal structural perturbation in a high affinity mutant antibody. J. Mol. Biol., 343 (2004), 685–701. [Crossref]
-  S. Miertus, E. Scrocco, and J. Tomasi. Electrostatic interactions of a solute with a continuum – a direct utilization of ab initio molecular potentials for the prevision of solvent effects. Chem. Phys., 55 (1981), 117–129. [Crossref]
-  M. S. Minkara, P. H. Davis, and M. L. Radhakrishnan. Multiple drugs and multiple targets: An analysis of the electrostatic determinants of binding between non-nucleoside hiv-1 reverse transcriptase inhibitors and variants of HIV-1 RT. Proteins-Structure Function and Bioinformatics, 80 (2012), 573–590.
-  A. Onufriev, D. A. Case, and D. Bashford. Effective Born radii in the generalized Born approximation: The importance of being perfect. J. Comput. Chem., 23 (2002), 1297–1304.
-  M. Orozco and F. J. Luque. Theoretical methods for the description of the solvent effect in biomolecular systems. Chem. Rev., 100 (2000), 4187–4225. [Crossref]
-  J. J. Perona, C. A. Tsu, M. E. McGrath, C. S. Craik, and R. J. Fletterick. Relocating a negative charge in the binding pocket of trypsin. J. Mol. Biol., 230 (1993), 934–949. [Crossref]
-  M. L. Radhakrishnan. Designing electrostatic interactions in biological systems via charge optimization or combinatorial approaches: insights and challenges with a continuum electrostatic framework. Theor. Chem. Acc., 131 (2012). [Crossref]
-  K. N. Rankin, T. Sulea, and E. O. Purisima. On the transferability of hydration-parametrized continuum electrostatics models to solvated binding calculations. J. Comput. Chem., 24 (2003), 954–962.
-  S. Ritter. The spectrum of the electrostatic integral operator for an ellipsoid. In R. E. Kleinman, R. Kress, and E. Martensen, editors, Inverse scattering and potential problems in mathematical physics, pages 157–167, Frankfurt/ Bern, 1995.
-  S. Ritter. On the magnetostatic integral operator for ellipsoids. J. Math. Anal. Appl., 207 (1997), 12–28. [Crossref]
-  S. Ritter. On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech., 78 (1998), 66–72. [Crossref]
-  W. Rocchia, E. Alexov, and B. Honig. Extending the applicability of the nonlinear Poisson–Boltzmann equation: Multiple dielectric constants and multivalent ions. J. Phys. Chem. B, 105 (2001), 6507–6514. [Crossref]
-  W. Rocchia, S. Sridharan, A. Nicholls, E. Alexov, A. Chiabrera, and B. Honig. Rapid grid-based construction of the molecular surface and the use of induced surface charge to calculate reaction field energies: Applications to the molecular systems and geometric objects. J. Comput. Chem., 23 (2002), 128–137.
-  B. Roux and T. Simonson. Implicit solvent models. Biophys. Chem., 78 (1999), 1–20. [Crossref]
-  S. Rush, A. H. Turner, and A. H. Cherin. Computer solution for time-invariant electric fields. J. Appl. Phys., 37 (1966), 2211–2217. [Crossref]
-  M. Sanner, A. J. Olson, and J. C. Spehner. Reduced surface: An efficient way to compute molecular surfaces. Biopolymers, 38 (1996), 305–320. [Crossref]
-  M. F. Sanner. Molecular surface computation home page. http://www.scripps.edu/ sanner/html/msms_home.html, 1996.
-  M. Scarsi and A. Caflisch. Comment on the validation of continuum electrostatics models. J. Comput. Chem., 20 (1999), 1533–1536.
-  M. Schaefer and M. Karplus. A comprehensive analytical treatment of continuum electrostatics. J. Phys. Chem., 100 (1996), 1578–1599. [Crossref]
-  P. B. Shaw. Theory of the Poisson Green’s-function for discontinuous dielectric media with an application to protein biophysics. Phys. Rev. A, 32 (1985), 2476–2487. [Crossref]
-  G. Sigalov, P. Scheffel, and A. Onufriev. Incorporating variable dielectric environments into the generalized Born model. J. Chem. Phys., 122 (2005), Art. ID 094511. [Crossref]
-  T. Simonson. Macromolecular electrostatics: Continuum models and their growing pains. Curr. Opin. Struc. Biol., 11 (2001), 243–252. [Crossref]
-  D. Sitkoff, K. A. Sharp, and B. Honig. Accurate calculation of hydration free-energies using macroscopic solvent models. J. Phys. Chem., 98 (1994), 1978–1988. [Crossref]
-  W.C. Still, A. Tempczyk, R. C. Hawley, and T. F. Hendrickson. Semianalytical treatment of solvation for molecular mechanics and dynamics. J. Am. Chem. Soc., 112 (1990), 6127–6129. [Crossref]
-  S. Varma and S. B. Rempe. Coordination numbers of alkali metal ions in aqueous solutions. Biophys. Chem., 124 (2006), 192–199. [Crossref]
-  A. Warshel, P. K. Sharma, M. Kato, and M. W. Parson. Modeling electrostatic effects in proteins. Biochimica et Biophysica Acta, 1764 (2006), 1647–1676.
-  J. Warwicker and H. C. Watson. Calculation of the electric potential in the active site cleft due to alpha-helix dipoles. J. Mol. Biol., 157 (1982), 671–679. [Crossref]
-  T. W. Whitfield, S. Varma, E. Harder, G. Lamoureux, S. B. Rempe, and B. Roux. Theoretical study of aqueous solvation of K+ comparing ab initio, polarizable, and fixed-charge models. J. Chem. Theory Comput., 3 (2007), 2068–2082. [Crossref]
-  Z. Xu and W. Cai. Fast analytical methods for macroscopic electrostatic models in biomolecular simulations. SIAM Review, 53 (2011), 683–720. [Crossref]
-  C. Xue and S. Deng. Three-layer dielectric models for generalized coulomb potential calculation in ellipsoidal geometry. Phys. Rev. E, 83 (2011), Art. ID 056709. [Crossref]
-  B. J. Yoon and A. M. Lenhoff. A boundary element method for molecular electrostatics with electrolyte effects. J. Comput. Chem., 11 (1990), 1080–1086.
-  S. N. Yu, Y. C. Zhou, and G. W. Wei. Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces. J. Comput. Phys., 224 (2007), 729–756. [Crossref]
-  R. J. Zauhar and R. S. Morgan. A new method for computing the macromolecular electric-potential. J. Mol. Biol., 186 (1985), 815–820. [Crossref]
-  R. J. Zauhar and R. S. Morgan. The rigorous computation of the molecular electric potential. J. Comput. Chem., 9 (1988), 171–187.
-  Y. C. Zhou, M. Feig, and G. W. Wei. Highly accurate biomolecular electrostatics in continuum dielectric environments. J. Comput. Chem., 29 (2008), 87–97.
-  M. Zink and H. Grubmüller. Mechanical properties of the icosahedral shell of southern bean mosaic virus: a molecular dynamics study. Biophys. J., 96 (2009), 1350–1363. [Crossref]