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

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

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3 Intermolecular forces Intermolecular Forces, A.J. Stone

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

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5 Challenges in biomolecular simulations: Atomistic representation Realistic water environment Realistic water environment Long-range interactions Long-range interactions Periodic boundary Periodic boundary How to avoid O(n 2 )? How to avoid O(n 2 )?

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6 Challenges in biomolecular simulations: Time scales are in the 10 9 time steps Multiple trajectories, often as many as 10s to 100s, are needed

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7 Explicit solvent and implicit solvent: Removing solvent degrees of freedom Explicit solvent and implicit solvent: Removing solvent degrees of freedom r u : solute coordinates; r v : solvent coordinates

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

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9 Polar solvation pp ss Dielectric constant Electrostatic potential Charge density Charge of salt ion in solution

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

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11 Is Continuum Approximation Sufficient? I. Polar Solvation

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12 Explicit solvent (TI) TIP3P water model. Periodical Boundary Condition. Particle Mesh Ewald, real space cutoff 9Å.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.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 structuresAll 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).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:Thermodynamic Integration:

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

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14 Fitting quality: Polar solvation free energies Fitting quality: Polar solvation free energies Correlation Coefficient: 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, , 2006

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

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

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17 Is Continuum Approximation Sufficient? II. Nonpolar Solvation

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18 TIP3P water model. Periodical Boundary Condition. Particle Mesh Ewald, real space cutoff 9Å.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.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.All atoms restrained to compare with single-snapshot calculations in implicit solvent. Thermodynamic Integration: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).60 λ’s with simulation length doubled until free energies change less than 0.25kcal/mol (160ps equilibration or production per λ needed). Explicit solvent (TI) Tan et al, JPC-B, 111, , 2007

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19 Fitting Quality: Nonpolar repulsive free energies (A)SES CC: RMSD: 0.30kcal/mol RMS Rel Dev: (B) SEV CC: RMSD: 0.69kcal/mol RMS Rel Dev: (C) SAS CC: RMSD: 0.30kcal/mol RMS Rel Dev: (D) SAV CC: RMSD: 0.27kcal/mol RMS Rel Dev: Tan et al, JPC-B, 111, , 2007

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

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

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

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

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

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25 Continuum solvation approximation Conformation dependent energetics is consistent between implicit and explicit solvents. 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. 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. Repulsive nonpolar component works well from tested peptides to proteins if the volume estimator is used.

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26 Going beyond Fixed Charge Models with Continuum Electronic Polarization

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

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

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

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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. 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. Limitation: lack of atomic-detailed polarization within a molecular environment. This may be overcome by use of functional-group-specific dielectric constants.

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

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

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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:1.385 Å atomic radii: UA0 probe radius:1.385 Å

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34 Fitting statistics for monomers in vacuo ε = 4.0 ε = 78.4 rmsd rmsd uavg uavg correlation correlation Dipole moments of monomer with charges fitted simultaneously in three environments Unit: Debye

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

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36 Continuum electronic polarization Electronic polarization with a continuum dipole moment density. The uniform solute dielectric constant is the only parameter. 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. 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. A single set of charges can be used in different environments and different conformations. The model transfers well from monomers to dimers.

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37 Poisson-Boltzmann Molecular Dynamics

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

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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. 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 Reaction potential is the difference between the total potential Coulombic potential, which is defined as Coulombic potential, which is defined as Cai, Q. et al. Journal of Chemical Physics. 2009, 130,

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

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

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42 Harmonic Average (HA) This method enforces the flux conditions in the three orthogonal directions on the physical interface, i.e., 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. 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.

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43 Immersed Interface Method (IIM) A more accurate method for interface treatment for FDM 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 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. IIM tries to minimize the local truncation error with the help of interface conditions. LeVeque and Li. SIAM Journal Numerical Analysis. 1994, 31, 1019.

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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.d1/hIIM−SingularityHA−SingularityHA+Singularity Max Error

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45 Dielectric boundary force: Theory

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

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

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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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