Skip to main content
SpringerLink
Log in

The Adaptive Cartesian Grid-Based Poisson–Boltzmann Solver: Energy and Surface Electrostatic Properties

  • Chapter
  • First Online:
Computational Electrostatics for Biological Applications

Abstract

The adaptive Cartesian grid-based Poisson–Boltzmann solver (CPB) employs the adaptive Cartesian grid and a singularity-free decomposition of the solution. One of the challenges in implicit solvent-based biomolecular electrostatics modeling is the evaluation of the dielectric-weighted Laplacian at the complex molecular surface where the dielectric constant, and thus also the normal electrostatic gradient, are discontinuous due to the use of dielectric regions with different dielectric permittivities. As a result, the electrostatic field required to evaluate the induced charge and dielectric pressure contributions to the atomic forces, do not converge with grid resolution unless appropriate steps are taken to explicitly resolve these discontinuities. The particular approach implemented in CPB uses standard least squares reconstruction (LSR) methods to accurately estimate the electrostatic potential and its gradients at the surface. The continuity requirements of potential and its gradients at the dielectric boundary are explicitly incorporated into the estimation together with the Poisson–Boltzmann equation itself. The approach is exercised on a collection of systems ranging from a simple sphere model problem and small biomolecules to larger biomolecular assemblies such as viruses. For smooth surfaces such as the solvent excluded surface, the LSR is shown to provide improved estimates of the energy and surface properties. At the re-entrant crevices associated with van der Waals surfaces, however, the Taylor series expansion upon which the reconstruction is based does not apply. Accordingly the reconstruction is locally not valid and, in practice, usually provides little improvement to PBE predictions.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

a :

Atomic radius; also convergence radius in Taylor series expansion

{a}:

Vector defined in (4.17b)

[A]:

Matrix whose jth row is {a} evaluated at the jth local node (grid point)

\({a}_\mathrm{p }\) :

Solvent probe radius

e :

Electronic charge

{e \(_{j}\)}:

Unit vector whose jth entry is unity and all other entries are zero

F :

Signed function used to implicitly define the molecular surface (\(F=0\))

\({G}_{\mathrm{el}}\) :

Electrostatic energy

\({k}_\mathrm{B}\) :

Boltzmann constant (\(1.3806488 \times 10^{-23}\,\mathrm{m}^{2}\,\mathrm{kg}\,\mathrm{s}^{-2}\,\mathrm{K}^{-1})\)

n :

Outward surface normal vector

N :

Number of grid points or mesh elements

\({Q}_{k }\) :

kth fixed charge

R :

Coordinate in space

s :

Distance from surface

S(R):

Switching function that is 0 when inside the molecule and 1 when outside

x, y, z :

Coordinates in surface-aligned reference frame

T :

Temperature

\({w}_{j}\) :

Neighbor weights used to evaluate the dielectric-weighted Laplacian and defined in (4.23a, b)

\(\beta \) :

Smoothing parameter used in Gaussian surface definition

\(\delta \)(R):

Dirac or delta function

\(\Delta {G}_{\mathrm{el} }\) :

Electrostatic solvation free energy

\(\Delta \Delta {G}_{\mathrm{el} }\) :

Electrostatic binding free energy

\(\Delta \Delta {G}\) :

Total binding free energy

\(\Delta {s}\) :

Grid spacing (also \(\Delta _{0}\) when used at surface)

\(\varepsilon \) :

Dielectric ratio, \({\varepsilon }_{2}/{\varepsilon }_{1}\);

\(\varepsilon \)(R):

Dielectric constant at location, R

\({\varepsilon }_{i}\) :

Dielectric constant in region \(\varOmega _{i}\)

\(\phi \) :

Interior electrostatic potential

\(\varPhi \) :

Electrostatic potential

{\({\varvec{\upmu }}\)}:

Vector of weights whose jth entry is \({\varvec{\upmu }}\!{_{j}}=({Sz}^{2}/2)_{{j}}\)

\(\tilde{\upmu }\) :

Potential quantity defined in (4.20b, c)

\(\rho \) :

Volume charge density defined in (4.2a)

\({\rho }^\mathrm{f }\) :

Fixed charge density

\({\rho }_{k }\) :

Location of kth charge

\({\rho }^\mathrm{m }\) :

Mobile charge density

\(\mathrm{I}_{1:1 }\) :

Ionic strength of 1:1 salt

\(\kappa \) :

Debye–Hückel screening parameter

{\({\varvec{\uptheta }}\)}:

Vector defined of surface potential gradients defined in (4.17a)

\(\varPsi \) :

Exterior electrostatic potential

\(\tilde{\varPsi }_0 \) :

Potential quantity defined in (4.20b, c)

\(\varOmega _{k }\) :

Domains comprising the interior (\(\varOmega _{1})\), exterior (\(\varOmega _{2})\) and Stern layer (\(\varOmega _{3})\) regions

(\({\bullet })_{0 }\) :

Refers to quantity evaluated at the edge-surface intersection point

(\({\bullet })^\mathrm{a }\) :

Refers to an analytically evaluated quantity

(\({\bullet })^\mathrm{c }\) :

Refers to numerically computed quantity

(\({\bullet })_\mathrm{s }\) :

Refers to quantity evaluated at the surface

(\({\bullet })^\mathrm{T }\) :

Denotes matrix transpose

(\({\bullet }),_\mathrm{x}\) :

Comma notation: e.g., \(\phi ,_{x}={\partial }{\phi }/\partial {x}\)

ACG:

Adaptive Cartesian grid

BEM:

Boundary element method

CPB:

Cartesian grid-based Poisson Boltzmann solver

GB:

Generalized Born

LSR:

Least squares reconstruction

MD:

Molecular dynamics

MM-PBSA:

Molecular mechanics Poisson–Boltzmann surface area

PDB:

Protein Data Bank

PBE:

Poisson–Boltzmann equation

SE:

Solvent excluded (molecular surface)

vdW:

van der Waals

References

  1. Bardhan JP (2012) Biomolecular electrostatics—I want your solvation (model). Comput Sci Discov 5:013001

    Google Scholar 

  2. Li C et al (2013) Progress in developing Poisson–Boltzmann equation solvers. Mol Based Math Biol 1:42–62

    Article  Google Scholar 

  3. Bashford D, Case DA (2000) Generalized born models of macromolecular solvation effects. Annu Rev Phy Chem 51:129–152

    Article  CAS  Google Scholar 

  4. Lu B et al (2008) Recent progress in numerical methods for the Poisson–Boltzmann equation in biophysical applications. Commun Comput Phys 3:973–1009

    Google Scholar 

  5. Boschitsch AH, Fenley MO (2011) A fast and robust Poisson–Boltzmann solver based on adaptive Cartesian grids. J Chem Theory Comput 7:1524–1540

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  6. Baker NA et al (2001) The adaptive multilevel finite element solution of the Poisson–Boltzmann equation on massively parallel computers. IBM J Res Dev 45:427

    Article  CAS  Google Scholar 

  7. Madura JD et al (1995) Electrostatics and diffusion of molecules in solution: simulations with the University of Houston Brownian dynamics program. Comp Phys Commun 91:57–95

    Google Scholar 

  8. Jo S et al (2008) PBEQ-Solver for online visualization of electrostatic potential of biomolecules. Nucl Acids Res 36:W270–W275

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  9. Bashford D, Gerwert K (1992) Electrostatic calculations of the pka values of ionizable groups in bacteriorhodopsin. J Mol Biol 224:473–486

    Article  CAS  PubMed  Google Scholar 

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

    Article  CAS  Google Scholar 

  11. 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 110:18680–18687

    Article  CAS  PubMed  Google Scholar 

  12. Rocchia W, Alexov E, Honig B (2001) Extending the applicability of the nonlinear Poisson–Boltzmann equation: multiple dielectric constants and multivalent ions. J Phys Chem B 105:6507–6514

    Article  CAS  Google Scholar 

  13. Gilson MK, Sharp KA, Honig BH (1988) Calculating the electrostatic potential of molecules in solution: method and error assessment. J Comput Chem 9:327–335

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  15. Geng W, Wei GW (2011) Multiscale molecular dynamics using the matched interface and boundary method. J Comput Phys 230:435–457

    Article  CAS  PubMed Central  PubMed  Google Scholar 

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

    Article  PubMed  Google Scholar 

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

    Article  Google Scholar 

  18. Zhou YC et al (2006) High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources. J Comput Phys 213:1–30

    Article  Google Scholar 

  19. Wang J et al (2009) Achieving energy conservation in Poisson–Boltzmann molecular dynamics: accuracy and precision with finite-difference algorithms. Chem Phys Lett 468:112–118

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  20. Cortis CM, Friesner RA (1997) Numerical solution of the Poisson–Boltzmann equation using tetrahedral finite-element meshes. J Comput Chem 18:1591–1608

    Article  CAS  Google Scholar 

  21. Hao X, Varshney A (2004) Efficient solution of Poisson–Boltzmann equation for electrostatics of large molecules. In: High performance computing symposium. Arlington, VA

    Google Scholar 

  22. Holst M, Baker N, Wang F (2000) Adaptive multilevel finite element solution of the Poisson–Boltzmann equation I: algorithms and examples. J Comput Chem 20:1319–1342

    Article  Google Scholar 

  23. Yu Z, Holst MJ, McCammon JA (2008) High-fidelity geometric modeling for biomedical applications. Finite Elem Anal Des 44:715–723

    Article  Google Scholar 

  24. Bajaj CL, Xu G, Zhang Q (2009) A fast variational method for the construction of resolution adaptive \({\rm c}^2\)-smooth molecular surfaces. Comput Methods Appl Mech Eng 198:1684–1690

    Article  PubMed Central  PubMed  Google Scholar 

  25. Bharadwaj R et al (1995) The fast multipole boundary element method for molecular electrostatics: an optimal approach for large systems. J Comput Chem 16:898–913

    Article  CAS  Google Scholar 

  26. Purisima EO (1998) Fast summation boundary element method for calculating solvation free energies of macromolecules. J Comput Chem 19:1494–1504

    Article  CAS  Google Scholar 

  27. Zauhar RJ, Varnek A (1996) A fast and space efficient boundary element method for computing electrostatic and hydration effects in large molecules. J Comput Chem 17:864–877

    Article  CAS  Google Scholar 

  28. Boschitsch AH, Fenley MO, Olson WK (1999) A fast adaptive multipole algorithm for calculating screened Coulomb (Yukawa) interactions. J Comput Phys 151:212–241

    Google Scholar 

  29. Boschitsch AH, Fenley MO, Zhou H-X (2002) Fast boundary element method for the linear Poisson–Boltzmann equation. J Phys Chem B 106:2741–2754

    Article  CAS  Google Scholar 

  30. Greengard LF, Huang J (2002) A new version of the fast multipole method for screened Coulomb interactions in three dimensions. J Comput Phys 180:642–658

    Google Scholar 

  31. Geng W, Krasny R (2013) A treecode-accelerated boundary integral Poisson–Boltzmann solver for electrostatics of solvated biomolecules. J Comput Phys 247:62–78

    Article  CAS  Google Scholar 

  32. Li P, Johnston H, Krasny R (2009) A Cartesian treecode for screened Coulomb interactions. J Comput Phys 228:3858–3868

    Google Scholar 

  33. Bajaj C, Chen S, Rand A (2011) An efficient higher-order fast multipole boundary element solution for Poisson–Boltzmann-based molecular electrostatics. SIAM J Sci Comput 33:826–848

    Article  PubMed Central  PubMed  Google Scholar 

  34. Manzin A, Bottauscio O, Ansalone DP (2011) Application of the thin-shell formulation to the numerical modeling of Stern layer in biomolecular electrostatics. J Comput Chem 32:3105–3113

    Article  CAS  PubMed  Google Scholar 

  35. Altman MD et al (2009) Accurate solution of multi-region continuum biomolecule electrostatic problems using the linearized Poisson–Boltzmann equation with curved boundary elements. J Comput Chem 30:132–153

    Article  CAS  PubMed Central  PubMed  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  37. Helgadóttir Á, Gibou F (2011) A Poisson–Boltzmann solver on irregular domains with Neumann or Robin boundary conditions on non-graded adaptive grid. J Comput Phys 230:3830–3848

    Article  Google Scholar 

  38. Mirzadeh M, Theillard M, Gibou F (2011) A second-order discretization of the nonlinear Poisson–Boltzmann equation over irregular geometries using non-graded adaptive Cartesian grids. J Comput Phys 230:2125–2140

    Article  CAS  Google Scholar 

  39. Yerry MA, Shephard MS (1984) Automatic three-dimensional mesh generation by the modified-octree technique. Int J Num Methods Eng 20:1965–1990

    Article  Google Scholar 

  40. Shephard MS, Georges MK (1991) Automatic three-dimensional mesh generation by the finite octree technique. Int J Num Methods Eng 32:709–749

    Article  Google Scholar 

  41. Samet H (1990) The Design and Analysis of Spatial Structures. Addison-Wesley, Boston

    Google Scholar 

  42. Berger MJ, MJ Aftosmis Progress Towards a Cartesian Cut-Cell Method for Viscous Compressible Flow. AIAA, 2012:1301

    Google Scholar 

  43. Aftosmis MJ, Berger MJ, Melton JE (1997) Robust and efficient Cartesian mesh generation. AIAA 97:0196

    Google Scholar 

  44. Fenley MO et al (1996) Fast adaptive multipole method for computation of electrostatic energy in simulations of polyelectrolyte DNA. J Comput Chem 17:976–991

    Article  CAS  Google Scholar 

  45. Boschitsch A, Fenley M (2007) A new outer boundary formulation and energy corrections for the nonlinear Poisson–Boltzmann equation. J Comput Chem 28:909–921

    Article  CAS  PubMed  Google Scholar 

  46. Boschitsch A, Danilov P (2012) Formulation of a new and simple non-uniform size-modified Poisson–Boltzmann description. J Comput Chem 33:1152–1164

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  47. Bredenberg JH, Boschitsch AH, Fenley MO (2008) The role of anionic protein residues on the salt dependence of the binding of aminoacyl-tRNA synthetases to tRNA: a Poisson–Boltzmann analysis. Commun Comput Phys 3:1051–1070

    Google Scholar 

  48. Fenley MO et al (2010) Revisiting the association of cationic groove-binding drugs to DNA using a Poisson–Boltzmann approach. Biophys J 99:879–886

    Google Scholar 

  49. Bredenberg JH, Russo C, Fenley MO (2008) Salt-Mediated electrostatics in the association of TATA binding proteins to DNA: a combined molecular mechanics/Poisson–Boltzmann study. Biophys J 94:4634–4645

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  50. Harris RC et al (2011) Understanding the physical basis of the salt dependence of the electrostatic binding free energy of mutated charged ligand-nucleic acid complexes. Biophys Chem 156:79–87

    Article  CAS  PubMed  Google Scholar 

  51. de Carvalho SJ, Fenley MrO, da Silva FLsB (2008) Protein-Ion binding process on finite macromolecular concentration. A Poisson–Boltzmann and Monte Carlo study. J Phys Chem B 112:16766–16776

    Article  PubMed  Google Scholar 

  52. Silalahi ARJ et al (2010) Comparing the predictions of the nonlinear Poisson–Boltzmann equation and the ion size-modified Poisson-Boltzmann equation for a low-dielectric charged spherical cavity in an aqueous salt solution. J Chem Theory Comput 6:3631–3639

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  53. Xu D et al (2007) The electrostatic characteristics of G\(\cdot \)U wobble base pairs. Nucleic Acids Res 35:3836–3847

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  54. Srinivasan AR et al (2009) Properties of the nucleic-acid bases in free and Watson-Crick hydrogen-bonded states: computational insights into the sequence-dependent features of double-helical DNA. Biophys Rev 1:13–20

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  55. Harris, RC et al (2012) Opposites attract: shape and electrostatic complementarity in protein-DNA complexes. In: Schlick T (ed) Innovations in biomolecular modeling and simulations, RSC Biomolecular Sciences, pp 53–80

    Google Scholar 

  56. Min D et al (2008) Efficient sampling of ion motions in molecular dynamics simulations on DNA: variant Hamiltonian replica exchange method. Chem Phys Lett 454:391–395

    Article  CAS  Google Scholar 

  57. Chan SL, Purisima EO (1998) Molecular surface generation using marching tetrahedra. J Comput Chem 19:1268–1277

    Article  CAS  Google Scholar 

  58. Boschitsch AH, Fenley MO (2007) A new outer boundary formulation and energy corrections for the nonlinear Poisson–Boltzmann equation. J Comput Chem 28(5):909–921

    Article  CAS  PubMed  Google Scholar 

  59. Bruccoleri RE et al (1997) Finite difference Poisson–Boltzmann electrostatic calculations: increased accuracy achieved by harmonic dielectric smoothing and charge antialiasing. J Comput Chem 18:268–276

    Article  CAS  Google Scholar 

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

    Article  PubMed  Google Scholar 

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

    Article  CAS  PubMed  Google Scholar 

  62. Anderson E et al (1999) LAPACK Users’ Guide—Third Edition. SIAM

    Google Scholar 

  63. Scharstein RW (1993) Mellin transform solution for the static line-source excitation of a dielectric wedge. IEEE Trans Antennas Propag 41:1675–1679

    Article  Google Scholar 

  64. Scharstein RW (2004) Green’s function for the harmonic potential of the three-dimensional wedge transmission problem. IEEE Trans Antennas Propag 52:452–460

    Article  Google Scholar 

  65. Bladel Jv (1985) Field singularities at the tip of a dielectric cone. IEEE Trans Antennas Propag AP–33:893–895

    Article  Google Scholar 

  66. Dolinsky TJ et al (2004) PDB2PQR: an automated pipeline for the setup of Poisson–Boltzmann electrostatics calculations. Nucleic Acids Res 32(suppl 2):W665–W667

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  67. Weiner SJ et al (1986) An all atom force field for simulations of proteins and nucleic acids. J Comput Chem 7:230–252

    Article  CAS  Google Scholar 

  68. Ma C et al (2002) Binding of aminoglycoside antibiotics to the small ribosomal subunit: a continuum electrostatics investigation. J Am Chem Soc 124:1438–1442

    Article  CAS  PubMed  Google Scholar 

  69. Harris RC, Boschitsch AH, Fenley MO (2013) Influence of grid spacing in Poisson–Boltzmann equation binding energy estimation. J Chem Theory Comput 9:3677–3685

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  70. Cheung AS et al (2010) Solvation effects in calculated electrostatic association free energies for the C3d-CR2 complex and comparison with experimental data. Biopolymers 93:509–519

    CAS  PubMed  Google Scholar 

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

    Article  CAS  Google Scholar 

  72. Feig M et al (2004) Performance comparison of generalized born and Poisson methods in the calculation of electrostatic solvation energies for protein structures. J Comput Chem 25:265–284

    Article  CAS  PubMed  Google Scholar 

  73. Rizzo RC et al (2005) Estimation of absolute free energies of hydration using continuum methods: accuracy of partial charge models and optimization of nonpolar contributions. J Chem Theory Comput 2:128–139

    Article  Google Scholar 

  74. Kollman PA et al (2000) Calculating Structures and free energies of complex molecules: combining molecular mechanics and continuum models. Acc Chem Res 33:889–897

    Article  CAS  PubMed  Google Scholar 

  75. Nicholls A et al (2008) Predicting small-molecule solvation free energies: an informal blind test for computational chemistry. J Med Chem 51:769–779

    Article  CAS  PubMed  Google Scholar 

  76. Shen J, Quiocho FA (1995) Calculation of binding energy differences for receptor-ligand systems using the Poisson–Boltzmann method. J Comput Chem 16:445–448

    Article  Google Scholar 

  77. Moreira IS, Fernandes PA, Ramos MJ (2005) Accuracy of the numerical solution of the Poisson–Boltzmann equation. J Molec Struct Theo chem 729:11–18

    Google Scholar 

  78. Baker NA et al (2001) Electrostatics of nanosystems: application to microtubules and the ribosome. Proc Natl Acad Sci USA 98:10037–10041

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  79. Li C et al (2012) Highly efficient and exact method for parallelization of grid-based algorithms and its implementation in DelPhi. J Comput Chem 33:1960–1966

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  80. Devkota B et al (2009) Structural and electrostatic characterization of pariacoto virus: implications for viral assembly. Biopolymers 91:530–538

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  81. Trylska J et al (2004) Ribosome motions modulate electrostatic properties. Biopolymers 74:423–431

    Article  CAS  PubMed  Google Scholar 

  82. Konecny R et al (2006) Electrostatic properties of cowpea chlorotic mottle virus and cucumber mosaic virus capsids. Biopolymers 82:106–120

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  83. Tjong H, Zhou H-X (2007) GBr 6NL: a generalized Born method for accurately reproducing solvation energy of the nonlinear Poisson–Boltzmann equation. J Chem Phys 126:195102–195105

    Article  PubMed  Google Scholar 

  84. Cai Q et al (2011) Dielectric boundary force in numerical Poisson–Boltzmann methods: theory and numerical strategies. Chem Phys Lett 514:368–373

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  85. Wang C et al (2013) Exploring accurate Poisson–Boltzmann methods for biomolecular simulations. Comput Theor Chem 1024:34–44

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  86. Lu B, Zhang D, McCammon JA (2005) Computation of electrostatic forces between solvated molecules determined by the Poisson–Boltzmann equation using a boundary element method. J Chem Phys 122:214102

    Article  PubMed  Google Scholar 

  87. Gilson MK et al (1993) Computation of electrostatic forces on solvated molecules using the Poisson–Boltzmann equation. J Phys Chem 97(14):3591–3600

    Article  CAS  Google Scholar 

  88. Zauhar RJ (1991) The incorporation of hydration forces determined by continuum electrostatics into molecular mechanics simulations. J Comput Chem 12(5):575–583

    Article  CAS  Google Scholar 

  89. Gilson MK et al (1993) Computation of electrostatic forces on solvated molecules using the Poisson–Boltzmann equation. J Phys Chem 97:3591–3600

    Article  CAS  Google Scholar 

Download references

Acknowledgments

This work was supported by NIH grant numbers 5 R44 GM57764-03 and 5R44GM073391-03. One of us (MOF) would like to acknowledge the invaluable contributions of Dr. Robert C. Harris, Dr. Alexander Silalahi, and Mr. Travis Mackoy in the development, validation, testing, and application of the CPB software in a variety of biophysical applications.

Author information

Authors and Affiliations

  1. Continuum Dynamics, Inc., Ewing, NJ, 08618, USA

    Alexander H. Boschitsch

  2. Institute of Molecular Biophysics, Florida State University, Tallahassee, FL, 32306, USA

    Marcia O. Fenley

Authors
  1. Alexander H. Boschitsch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcia O. Fenley
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Alexander H. Boschitsch or Marcia O. Fenley .

Editor information

Editors and Affiliations

  1. CONCEPT Lab - CompuNet, Istituto Italiano di Tecnologia, Genova, Italy

    Walter Rocchia

  2. Institute of Applied Mathematics and Inf, National Research Council, Genova, Italy

    Michela Spagnuolo

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Boschitsch, A.H., Fenley, M.O. (2015). The Adaptive Cartesian Grid-Based Poisson–Boltzmann Solver: Energy and Surface Electrostatic Properties. In: Rocchia, W., Spagnuolo, M. (eds) Computational Electrostatics for Biological Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-12211-3_4

Download citation

Publish with us

Policies and ethics

Search

Navigation