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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

Poisson-Boltzmann Molecular Dynamics:Theory and Algorithms

Ray Luo

Molecular Biology and Biochemistry

University of California, Irvine


Different levels of abstraction approximations of biomolecules
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


Intermolecular forces
Intermolecular forces biomolecules

Intermolecular Forces, A.J. Stone


Biomolecules on computer classical molecular mechanics
Biomolecules on computer: biomoleculesClassical molecular mechanics

Bonded

Potential

Energy

Electrostatic

Nonbonded

Repulsion-dispersion


Challenges in biomolecular simulations atomistic representation
Challenges in biomolecular simulations: biomoleculesAtomistic representation

  • Realistic water environment

  • Long-range interactions

    • Periodic boundary

    • How to avoid O(n2)?


Challenges in biomolecular simulations time scales are in the 10 9 time steps
Challenges in biomolecular simulations: biomoleculesTime scales are in the 109 time steps

Multiple trajectories, often as many as 10s to 100s, are needed


Explicit solvent and implicit solvent removing solvent degrees of freedom
Explicit solvent and implicit solvent: biomoleculesRemoving solvent degrees of freedom

ru: solute coordinates; rv: solvent coordinates


Continuum solvation approximations
Continuum solvation approximations biomolecules

  • 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


Polar solvation
Polar solvation biomolecules

Dielectric

constant

Charge density

-

ep

Charge of salt ion in solution

+

-

-

Electrostatic potential

+

+

+

-

s


Nonpolar solvation
Nonpolar solvation biomolecules

Wrep: Estimated with surface (SES/SAS) or volume (SEV/SAV)

Watt: Approximated by (D. Chandler and R. Levy)


Is continuum approximation sufficient i polar solvation

Is Continuum Approximation Sufficient? biomoleculesI. Polar Solvation


Explicit solvent ti
Explicit solvent (TI) biomolecules

  • 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:


Implicit solvent pb
Implicit solvent (PB) biomolecules

  • 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


Fitting quality polar solvation free energies
Fitting quality: Polar solvation free energies biomolecules

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


Salt bridge charging free energies
Salt-bridge charging free energies biomolecules

  • 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.


Salt bridge charging free energies1
Salt-bridge charging free energies biomolecules

Tan and Luo, In Prep


Is continuum approximation sufficient ii nonpolar solvation

Is Continuum Approximation Sufficient? biomoleculesII. Nonpolar Solvation


Explicit solvent ti1
Explicit solvent (TI) biomolecules

  • 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


Fitting quality nonpolar repulsive free energies
Fitting Quality: biomoleculesNonpolar 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


Fitting quality nonpolar attractive free energies
Fitting quality: biomoleculesNonpolar 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


Nonpolar solvation free energies of tyr
Nonpolar solvation free energies of TYR biomolecules

  • 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.


Nonpolar attractive free energies
Nonpolar attractive free energies biomolecules

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


Nonpolar repulsive free energies
Nonpolar repulsive free energies biomolecules

  • 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.


Poisson boltzmann molecular dynamics theory and algorithms

Behaviors of Two Estimators for biomoleculesTYR Side-Chain Conformations

SAS

SAV

Tan and Luo, In Prep.


Continuum solvation approximation
Continuum solvation approximation biomolecules

  • 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.


Going beyond fixed charge models with continuum electronic polarization

Going beyond Fixed Charge Models with biomoleculesContinuum Electronic Polarization


How to include polarization in implicit solvents
How to include polarization biomolecules 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


Continuum polarizable force field
Continuum polarizable force field biomolecules

  • 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


Continuum polarizable force filed
Continuum polarizable force filed biomolecules

Tan and Luo, J Chem Phys, 126:094103, 2007.

Tan, Wang, and Luo, J Phys Chem, 112:7675. 2008.


Continuum polarizable force field1
Continuum polarizable force field biomolecules

  • 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.


Charge derivation procedure resp
Charge derivation procedure: RESP biomolecules

Yes

Convergence

No

Tan and Luo, J Chem Phys, 126:094103, 2007.


Quantum mechanical field
Quantum mechanical field biomolecules

  • 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.


Quality of fit dielectric constant
Quality of fit: dielectric constant biomolecules

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


Fitting statistics for monomers
Fitting statistics for monomers biomolecules

Dipole moments of monomer with charges fitted simultaneously in three environments

Unit: Debye


Transferability among conformations
Transferability among conformations biomolecules

rmsd: 0.2799 uavg: 0.2413 correlation: 0.9922

charges fitted simultaneously for both alphaL and c7eq

in three environments


Continuum electronic polarization
Continuum electronic polarization biomolecules

  • 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.



Singular charges in pbe
Singular Charges in PBE biomolecules

  • function in the PBE

  • Challenges

    - Large error in potential near singular charges

    - Large error in dielectric boundary force

    - Self energy between redistributed charges


Removal of charge singularity
Removal of Charge Singularity biomolecules

  • 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.


Removal of charge singularity1
Removal of Charge Singularity biomolecules

inside

outside

On the dielectric boundary

Cai, Q. et al. Journal of Chemical Physics. 2009, 130, 145101.


Discontinuous interface
Discontinuous Interface biomolecules

  • 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


Harmonic average ha
Harmonic Average (HA) biomolecules

  • 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.


Immersed interface method iim
Immersed Interface Method (IIM) biomolecules

  • 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.


Iim removal of singularity
IIM + Removal of Singularity biomolecules

Tested in the Poisson equation:

single particle system, dielectric boundary force

Wang, J. et al. Chemical Physics Letters. 2009, 468, 112.



Poisson boltzmann molecular dynamics theory and algorithms

Dielectric boundary force: Theory biomolecules

Davis and McCammon, Journal of Computational Chemistry. 1990. 11. 401.

Xiang et al, Journal of Chemical Physics. 2009. submitted.


Poisson boltzmann molecular dynamics theory and algorithms

Dielectric boundary force: biomoleculesNewton’s third law

Xiang et al, Journal of Chemical Physics. 2009. submitted.


Acknowledgements
Acknowledgements biomolecules

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