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Poisson-Boltzmann Molecular Dynamics: Theory and Algorithms

Poisson-Boltzmann Molecular Dynamics: Theory and Algorithms. Ray Luo Molecular Biology and Biochemistry University of California, Irvine. Different levels of abstraction: Approximations of biomolecules. Quantum description: electronic & covalent structure

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Poisson-Boltzmann Molecular Dynamics: Theory and Algorithms

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  1. Poisson-Boltzmann Molecular Dynamics:Theory and Algorithms Ray Luo Molecular Biology and Biochemistry University of California, Irvine

  2. Different levels of abstraction: Approximations of biomolecules • Quantum description: electronic & covalent structure • Atom-based description: non-covalent interactions • Residue-based/coarse-grained description: overall motion/properties of a biomolecule

  3. Intermolecular forces Intermolecular Forces, A.J. Stone

  4. Biomolecules on computer: Classical molecular mechanics Bonded Potential Energy Electrostatic Nonbonded Repulsion-dispersion

  5. Challenges in biomolecular simulations:Atomistic representation • Realistic water environment • Long-range interactions • Periodic boundary • How to avoid O(n2)?

  6. Challenges in biomolecular simulations:Time scales are in the 109 time steps Multiple trajectories, often as many as 10s to 100s, are needed

  7. Explicit solvent and implicit solvent:Removing solvent degrees of freedom ru: solute coordinates; rv: solvent coordinates

  8. Continuum solvation approximations • Homogenous structureless solvent distribution • Solute geometry (shape/size) influence in solvent density is weak in solvation free energy calculation • Solvation free energy can be decomposed into different components

  9. Polar solvation Dielectric constant Charge density - ep Charge of salt ion in solution + - - Electrostatic potential + + + - s

  10. Nonpolar solvation Wrep: Estimated with surface (SES/SAS) or volume (SEV/SAV) Watt: Approximated by (D. Chandler and R. Levy)

  11. Is Continuum Approximation Sufficient?I. Polar Solvation

  12. Explicit solvent (TI) • TIP3P water model. Periodical Boundary Condition. Particle Mesh Ewald, real space cutoff 9Å. • NPT ensemble, 300K, 1bar. Pre-equilibrium runs at least 4 ns and until running potential energy shows no systematic drift. • All atoms restrained to compare with PB calculations on static structures • 25 λ’s with simulation length doubled until free energies change less than 0.25kcal/mol (up to 320ps equilibration/production per λ needed). • Thermodynamic Integration:

  13. Implicit solvent (PB) • Final grid spacing 0.25 Å. Two-level focusing was used. Convergence to 10-4. • Solvent excluded surface. Harmonic dielectric smoothing was applied at dielectric boundary. • Charging free energies were computed with induced surface charges. • (110+110 snapshots) × 27 random grid origins were used. • Cavity radii were refitted before comparison ε= 80 Linearized Poisson-Boltzmann Equation: where

  14. Fitting quality: Polar solvation free energies Correlation Coefficient: 0.99995 Root Mean Square Deviation: 0.33 kcal/mol AMBER/TIP3P Error (wrt Expt): 1.06 kcal/mol AMBER/PB Error (wrt Expt): 0.97 kcal/mol (neutral side chain analogs) Tan et al, JPC-B, 110, 18680-18687, 2006

  15. Salt-bridge charging free energies • Tested salt bridge with atom ids. • PEPenh, a 16mer helix from1enh. • ENH, (1enh, ~50 aa). • P53a, (1tsr, ~200 aa) • ARG154-GLU76 on p53. • P53b, ARG178-GLU190 on p53. Tan and Luo, In Prep.

  16. Salt-bridge charging free energies Tan and Luo, In Prep

  17. Is Continuum Approximation Sufficient?II. Nonpolar Solvation

  18. Explicit solvent (TI) • TIP3P water model. Periodical Boundary Condition. Particle Mesh Ewald, real space cutoff 9Å. • NPT ensemble, 300K, 1bar. Pre-equilibrium runs with neutral molecules for at least 8 ns and until running potential energy shows no systematic drift. • All atoms restrained to compare with single-snapshot calculations in implicit solvent. • Thermodynamic Integration: • 60 λ’s with simulation length doubled until free energies change less than 0.25kcal/mol (160ps equilibration or production per λ needed). Tan et al, JPC-B, 111, 12263-12274, 2007

  19. Fitting Quality:Nonpolar repulsive free energies • SES • CC: 0.997 • RMSD: 0.30kcal/mol RMS Rel Dev: 0.026 • (B) SEV • CC: 0.985. • RMSD: 0.69kcal/mol RMS Rel Dev: 0.082 • (C) SAS • CC: 0.997 • RMSD: 0.30kcal/mol RMS Rel Dev: 0.026 • (D) SAV • CC: 0.998. • RMSD: 0.27kcal/mol RMS Rel Dev: 0.022 Tan et al, JPC-B, 111, 12263-12274, 2007

  20. Fitting quality:Nonpolar attractive free energies CC: 0.9995 RMSD: 0.16kcal/mol RMS Rel Dev: 0.01 Tan et al, JPC-B, 111, 12263-12274, 2007 Error bars too small to be seen

  21. Nonpolar solvation free energies of TYR • Tested side chain with atom ids. • PEPα, a 17mer helix from 1pgb. • PEPβ, a 16mer hairpin from 1pgb. • PGB, 1pgb, ~50 aa. • P53, 1tsr, ~200 aa. Tan and Luo, In Prep.

  22. Nonpolar attractive free energies CC: 0.983 RMSD: 0.29 kcal/mol RMS Rel Dev: 0.035 Tan and Luo, In Prep. Error bars too small to be seen

  23. Nonpolar repulsive free energies • SAS • CC: 0.975 • RMSD: 2.42kcal/mol. • RMS Rel Dev: 0.55 • (B) SAV • CC: 0.984 • RMSD: 0.53kcal/mol • RMS Rel Dev: 0.053 Tan and Luo, In Prep.

  24. Behaviors of Two Estimators for TYR Side-Chain Conformations SAS SAV Tan and Luo, In Prep.

  25. Continuum solvation approximation • Conformation dependent energetics is consistent between implicit and explicit solvents. • Both polar and nonpolar attractive component correlate very well with TI from short peptides up to proteins of typical sizes. • Repulsive nonpolar component works well from tested peptides to proteins if the volume estimator is used.

  26. Going beyond Fixed Charge Models withContinuum Electronic Polarization

  27. How to include polarization in implicit solvents? • Explicit treatment Maple, Cao, et al., J Chem Theo Comp, 1:694, 2005. Schnieders, Baker, et al., J Chem Phys, 126:124114, 2007. • Implicit treatment

  28. Continuum polarizable force field • Relation between P and E • Relation between  and ε Solute dielectric constant ε is optimized • P is defined within the molecular volume (solvent excluded volume). P

  29. Continuum polarizable force filed Tan and Luo, J Chem Phys, 126:094103, 2007. Tan, Wang, and Luo, J Phys Chem, 112:7675. 2008.

  30. Continuum polarizable force field • Advantage: gives us an efficient and self-consistent approach in treating polar interactions in biomolecular simulations more satisfactory than existing additive force fields with implicit solvents. • Limitation: lack of atomic-detailed polarization within a molecular environment. This may be overcome by use of functional-group-specific dielectric constants.

  31. Charge derivation procedure: RESP Yes Convergence No Tan and Luo, J Chem Phys, 126:094103, 2007.

  32. Quantum mechanical field • Computation of quantum mechanically electrostatic field: 1) Optimization with HF/6-31G* 2) Single point with B3LYP/cc-pVTZ • PCM was used for modeling polarization responses to different environments.

  33. Quality of fit: dielectric constant monomers dimers Left: 12 monomers in three environments (vacuum, ε = 4, water) Right: 4 dimers in three environments atomic radii: UA0 probe radius:1.385Å

  34. Fitting statistics for monomers Dipole moments of monomer with charges fitted simultaneously in three environments Unit: Debye

  35. Transferability among conformations rmsd: 0.2799 uavg: 0.2413 correlation: 0.9922 charges fitted simultaneously for both alphaL and c7eq in three environments

  36. Continuum electronic polarization • Electronic polarization with a continuum dipole moment density. The uniform solute dielectric constant is the only parameter. • Performance comparable to ff02 explicit polarizable force field for tested dipole moments in vacuum. • A single set of charges can be used in different environments and different conformations. The model transfers well from monomers to dimers.

  37. Poisson-Boltzmann Molecular Dynamics

  38. Singular Charges in PBE • function in the PBE • Challenges - Large error in potential near singular charges - Large error in dielectric boundary force - Self energy between redistributed charges

  39. Removal of Charge Singularity • Solve the Laplace’s equation for reaction field potential inside and simultaneously solve Poisson-Boltzmann equation for total potential outside. • Reaction potential is the difference between the total potential • Coulombic potential, which is defined as Cai, Q. et al. Journal of Chemical Physics. 2009, 130, 145101.

  40. Removal of Charge Singularity inside outside On the dielectric boundary Cai, Q. et al. Journal of Chemical Physics. 2009, 130, 145101.

  41. Discontinuous Interface • Boundary conditions on the discontinuous interface of the PBE (uniform potential) - The potential is continuous on the interface - Integrating the PBE and then using the Gauss’s law give the flux condition

  42. Harmonic Average (HA) • This method enforces the flux conditions in the three orthogonal directions on the physical interface, i.e., • The dielectric constant between two grid points that are in two different regions is a harmonic average of the two dielectric constants of the two regions. Davis and McCammon, Journal of Computational Chemistry. 1991, 12, 909.

  43. Immersed Interface Method (IIM) • A more accurate method for interface treatment for FDM • IIM proposes new equations involving 27 points instead of the original 7-point finite-difference equations at the points close to the interface. • IIM tries to minimize the local truncation error with the help of interface conditions. LeVeque and Li. SIAM Journal Numerical Analysis. 1994, 31, 1019.

  44. IIM + Removal of Singularity Tested in the Poisson equation: single particle system, dielectric boundary force Wang, J. et al. Chemical Physics Letters. 2009, 468, 112.

  45. Dielectric boundary force: Theory

  46. Dielectric boundary force: Theory Davis and McCammon, Journal of Computational Chemistry. 1990. 11. 401. Xiang et al, Journal of Chemical Physics. 2009. submitted.

  47. Dielectric boundary force: Newton’s third law Xiang et al, Journal of Chemical Physics. 2009. submitted.

  48. Acknowledgements Profs. David Case, Michael Gilson, Hong-Kai Zhao and Zhilin Li Drs. Jun Wang, Siang Yip Chuck Tan, Yuhong Tan, Qiang Lu Qin Cai, MJ Hsieh Gabe Ozorowski, Seema D’Souza Morris Chen, Emmanuel Chanco NIH/GMS

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