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

1

Poisson-Boltzmann Molecular Poisson-Boltzmann Molecular Dynamics:Dynamics:

Theory and AlgorithmsTheory and Algorithms

Ray LuoMolecular Biology and Biochemistry

University of California, Irvine

2

Different levels of Different levels of abstraction: Approximations abstraction: Approximations

of biomoleculesof biomolecules

• Quantum description: electronic & covalent structureQuantum description: electronic & covalent structure

• Atom-based description: non-covalent interactionsAtom-based description: non-covalent interactions

• Residue-based/coarse-grained description: overall Residue-based/coarse-grained description: overall motion/properties of a biomolecule motion/properties of a biomolecule

3

Intermolecular forcesIntermolecular forces

Intermolecular Forces, A.J. Stone

4

Biomolecules on computer: Biomolecules on computer: Classical molecular mechanicsClassical molecular mechanics

20 )( K

)cos(1 0nn nV 2K

Bonded

Electrostatic

Repulsion-dispersion

Nonbonded

20 )( bbKb

ij

ji

r

qq

612ijij r

B

r

A

Potential Energy

5

Challenges in biomolecular Challenges in biomolecular simulations:simulations:

Atomistic representationAtomistic representation• Realistic water environmentRealistic water environment• Long-range interactionsLong-range interactions

• Periodic boundaryPeriodic boundary• How to avoid O(nHow to avoid O(n22)?)?

6

Challenges in biomolecular Challenges in biomolecular simulations:simulations:

Time scales are in the 10Time scales are in the 109 9 time time stepssteps

1 2 3 4 5 6 7 8 90.1

0.2

0.3

0.4

0.5

Sal

t brid

ge p

opul

atio

nTimes (x10ns)

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

7

Explicit solvent and implicit Explicit solvent and implicit solvent:solvent:

Removing solvent degrees of Removing solvent degrees of freedomfreedom

exp[ ( , )]( , )

exp[ ( , )]u v

u v

u v u v

UP

d d U

r r

r rr r r r

exp[ ( )]

( )exp[ ( )]

uu

u u

WP

d W

r

rr r

exp[ ( )] exp[ ( , )]u v u vW d U r r r r

ru: solute coordinates; rv: solvent coordinates

8

Continuum solvation Continuum solvation approximationsapproximations

• Homogenous structureless solvent distributionHomogenous structureless solvent distribution

• Solute geometry (shape/size) influence in solvent Solute geometry (shape/size) influence in solvent density is weak in solvation free energy calculationdensity is weak in solvation free energy calculation

• Solvation free energy can be decomposed into Solvation free energy can be decomposed into different componentsdifferent components

pol npolW W W

npol rep attW W W

9

Polar solvationPolar solvation

p

+

+-

-

++

-

-

s

Dielectric constant

Electrostatic potential

Charge density

Charge of salt ion in solution

(r)(r) (r) n

i0q

iexp[ q

i(r)]

10

Nonpolar solvation Nonpolar solvation

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

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

Uattuv neff

i1

Nu

Vatt (r)d 3r

npol rep attW W W

Wrep A c

Wrep pV c

Uattuv

11

Is Continuum Approximation Is Continuum Approximation Sufficient?Sufficient?

I. Polar SolvationI. Polar Solvation

12

Explicit solvent (TI)Explicit solvent (TI)• TIP3P water model. Periodical TIP3P water model. Periodical

Boundary Condition. Particle Mesh Boundary Condition. Particle Mesh Ewald, real space cutoff 9Å.Ewald, real space cutoff 9Å.

• NPT ensemble, 300K, 1bar. Pre-NPT ensemble, 300K, 1bar. Pre-equilibrium runs at least 4 ns and until equilibrium runs at least 4 ns and until running potential energy shows no running potential energy shows no systematic drift.systematic drift.

• All atoms restrained to compare with All atoms restrained to compare with PB calculations on static structuresPB calculations on static structures

• 25 25 λλ’s with simulation length doubled ’s with simulation length doubled until free energies change less than until free energies change less than 0.25kcal/mol (up to 320ps 0.25kcal/mol (up to 320ps equilibration/production per equilibration/production per λλ needed). needed).

• Thermodynamic Integration:Thermodynamic Integration:

dd

)dH(G

1

0

13

Implicit solvent (PB)Implicit solvent (PB)

• Final grid spacing 0.25 Å. Two-level focusing Final grid spacing 0.25 Å. Two-level focusing was used. Convergence to 10was used. Convergence to 10-4-4..

• Solvent excluded surface. Harmonic Solvent excluded surface. Harmonic dielectric smoothing was applied at dielectric dielectric smoothing was applied at dielectric boundary.boundary.

• Charging free energies were computed with Charging free energies were computed with induced surface charges. induced surface charges.

• (110+110 snapshots) × 27 random grid (110+110 snapshots) × 27 random grid origins were used. origins were used.

• Cavity radii were refitted before comparisonCavity radii were refitted before comparisonLinearized Poisson-Boltzmann Linearized Poisson-Boltzmann Equation:Equation:

wherewhere

ε= 80

14

Fitting quality: Polar solvation Fitting quality: Polar solvation free energiesfree energies

-100 -80 -60 -40 -20 0

-100

-80

-60

-40

-20

0

Gel

ec b

y PB

(kca

l/mol

)

Gelec

by TI (kcal/mol)

Correlation Coefficient: Correlation Coefficient:

0.999950.99995

Root Mean Square Deviation: Root Mean Square Deviation:

0.33 kcal/mol 0.33 kcal/mol

AMBER/TIP3P Error (wrt Expt):AMBER/TIP3P Error (wrt Expt):

1.06 kcal/mol1.06 kcal/mol

AMBER/PB Error (wrt Expt):AMBER/PB Error (wrt Expt):

0.97 kcal/mol0.97 kcal/mol

(neutral side chain analogs)(neutral side chain analogs)

Tan et al, JPC-B, 110, 18680-18687, 2006

15

Salt-bridge charging free Salt-bridge charging free energiesenergies

(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.

16

Salt-bridge charging free Salt-bridge charging free energiesenergies

-120 -100 -80 -60 -40 -20-120

-100

-80

-60

-40

-20

Gel

e_ex

p(kc

al/m

ol)

Gele_imp

(kcal/mol)

PEPp53

PEPenh

ENH

P53

Tan and Luo, In Prep

17

Is Continuum Approximation Is Continuum Approximation Sufficient?Sufficient?

II. Nonpolar SolvationII. Nonpolar Solvation

18

• TIP3P water model. Periodical Boundary Condition. Particle TIP3P water model. Periodical Boundary Condition. Particle Mesh Ewald, real space cutoff 9Å.Mesh Ewald, real space cutoff 9Å.

• NPT ensemble, 300K, 1bar. Pre-equilibrium runs with neutral NPT ensemble, 300K, 1bar. Pre-equilibrium runs with neutral molecules for at least 8 ns and until running potential energy molecules for at least 8 ns and until running potential energy shows no systematic drift.shows no systematic drift.

• All atoms restrained to compare with single-snapshot All atoms restrained to compare with single-snapshot calculations in implicit solvent. calculations in implicit solvent.

• Thermodynamic Integration:Thermodynamic Integration:

• 60 60 λλ’s with simulation length doubled until free energies ’s with simulation length doubled until free energies change less than 0.25kcal/mol (160ps equilibration or change less than 0.25kcal/mol (160ps equilibration or production per production per λλ needed). needed).

Explicit solvent (TI)Explicit solvent (TI)

dd

)dH(G

1

0

Tan et al, JPC-B, 111, 12263-12274, 2007

19

Fitting Quality:Fitting Quality:Nonpolar repulsive free Nonpolar repulsive free

energiesenergies(A) SES

CC: 0.997RMSD: 0.30kcal/mol RMS Rel Dev: 0.026

(B) SEVCC: 0.985. RMSD: 0.69kcal/mol RMS Rel Dev: 0.082

(C) SASCC: 0.997RMSD: 0.30kcal/mol RMS Rel Dev: 0.026

(D) SAVCC: 0.998. RMSD: 0.27kcal/mol RMS Rel Dev: 0.022

05

10152025

D:SAVC:SAS

B:SEVA:SES

0 5 10 15 2005

101520

Gre

p_ex

p (k

cal/m

ol)

Grep_imp

(kcal/mol)0 5 10 15 20 25

Tan et al, JPC-B, 111, 12263-12274, 2007

20

Fitting quality:Fitting quality:Nonpolar attractive free Nonpolar attractive free

energiesenergies

CC: 0.9995RMSD: 0.16kcal/molRMS Rel Dev: 0.01

-25 -20 -15 -10 -5 0-25

-20

-15

-10

-5

0

Gat

t_ex

p (k

cal/m

ol)

Gatt_imp

(kcal/mol)

Error bars too small to be seenTan et al, JPC-B, 111, 12263-12274, 2007

21

Nonpolar solvation free Nonpolar solvation free energies of TYRenergies 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.

22

Nonpolar attractive free Nonpolar attractive free energiesenergies

CC: 0.983RMSD: 0.29 kcal/molRMS Rel Dev: 0.035

-12 -10 -8 -6 -4 -2-12

-10

-8

-6

-4

-2

Gat

t_ex

p (k

cal/m

ol)

Gatt_imp

(kcal/mol)

PEP

PEP

PGB

P53

Error bars too small to be seen

Tan and Luo, In Prep.

23

Nonpolar repulsive free Nonpolar repulsive free energiesenergies

(A) SASCC: 0.975RMSD: 2.42kcal/mol.RMS Rel Dev: 0.55

(B) SAVCC: 0.984RMSD: 0.53kcal/molRMS Rel Dev: 0.053

-4 0 4 8 12 16-4

0

4

8

12

PEP

PEP

PGB P53

Gre

p_ex

p (k

cal/m

ol)

Grep_imp

(kcal/mol)

-4

0

4

8

12

16

B

A

Tan and Luo, In Prep.

24

Behaviors of Two Estimators for Behaviors of Two Estimators for TYR Side-Chain ConformationsTYR Side-Chain Conformations

SAS SAV

0 2 4 6 8 10 12-0.2

-0.1

0.0

0.1

0.2

0.3

(kc

al/m

ol/Å

2 )

Conformations

PEP

PEP

PGB

P53

AVG_HLX

0 2 4 6 8 10 120.02

0.03

0.04

0.05

0.06

p (k

cal/m

ol/Å

3 )

Conformations

PEP

PEP

PGB

P53

AVG_HLX

Tan and Luo, In Prep.

25

Continuum solvation Continuum solvation approximationapproximation

• Conformation dependent energetics is consistent Conformation dependent energetics is consistent between implicit and explicit solvents.between implicit and explicit solvents.

• Both polar and nonpolar attractive component correlate Both polar and nonpolar attractive component correlate very well with TI from short peptides up to proteins of very well with TI from short peptides up to proteins of typical sizes.typical sizes.

• Repulsive nonpolar component works well from tested Repulsive nonpolar component works well from tested peptides to proteins if the volume estimator is used.peptides to proteins if the volume estimator is used.

26

Going beyond Fixed Charge Going beyond Fixed Charge Models withModels with

Continuum Electronic Continuum Electronic PolarizationPolarization

27

How to include How to include polarizationpolarization

in implicit solvents? in implicit solvents?

• Explicit treatmentExplicit treatment

Maple, Cao, et al., J Chem Theo Comp, 1:694, 2005.J Chem Theo Comp, 1:694, 2005.

Schnieders, Baker, et al., J Chem Phys, 126:124114, 2007.Schnieders, Baker, et al., J Chem Phys, 126:124114, 2007.

• Implicit treatment Implicit treatment

28

• Relation between P and ERelation between P and E

• Relation between Relation between and and εε

Solute dielectric constant Solute dielectric constant εε is optimized is optimized

• P is defined within the molecular volume (solvent P is defined within the molecular volume (solvent excluded volume).excluded volume).

Continuum polarizable force Continuum polarizable force fieldfield

P

29

Continuum polarizable force Continuum polarizable force filedfiled

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

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

30

Continuum polarizable force Continuum polarizable force fieldfield

• Advantage: gives us an efficient and self-consistent Advantage: gives us an efficient and self-consistent approach in treating polar interactions in biomolecular approach in treating polar interactions in biomolecular simulations more satisfactory than existing additive force simulations more satisfactory than existing additive force fields with implicit solvents. fields with implicit solvents.

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

31

Charge derivation procedure: Charge derivation procedure: RESPRESP

Convergence

No

Yes

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

32

Quantum mechanical fieldQuantum mechanical field

• Computation of quantum mechanically Computation of quantum mechanically electrostatic field:electrostatic field:

1) Optimization with HF/6-31G* 1) Optimization with HF/6-31G*

2) Single point with B3LYP/cc-pVTZ2) Single point with B3LYP/cc-pVTZ

• PCM was used for modeling polarization PCM was used for modeling polarization responses to different environments.responses to different environments.

33

Quality of fit: dielectric Quality of fit: dielectric constantconstant

2 3 4 50.6

0.9

1.2

1.5 largest error rmsd

rmsd

and

larg

est e

rror

(D

)

solute dielectric constant

2 3 4 50.0

0.3

0.6

0.9

largest error rmsd

rmsd

and

larg

est e

rror

(D

)

solute dielectric constant

monomers dimers

Left: 12 monomers in three environments (vacuum, ε = 4, water) Left: 12 monomers in three environments (vacuum, ε = 4, water)

Right: 4 dimers in three environmentsRight: 4 dimers in three environments

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

34

Fitting statistics for Fitting statistics for monomers monomers

in vacuoin vacuo εε = 4.0 = 4.0 εε = 78.4 = 78.4

rmsdrmsd 0.14250.1425 0.21060.2106 0.17590.1759

uavguavg 0.11760.1176 0.18480.1848 0.13450.1345

correlationcorrelation 0.99790.9979 0.99950.9995 0.99960.9996

Dipole moments of monomer with charges fitted simultaneously in three environments Dipole moments of monomer with charges fitted simultaneously in three environments

0 2 4 6 80

2

4

6

8

polarizable nonpolarizable

in vacuo

B3L

YP

/cc-

pVT

Z (

D)

MM (D)

0 2 4 6 80

2

4

6

8

polarizable nonpolarizable

solvent = 4.0

B3L

YP

/cc-

pVT

Z (

D)

MM (D)

0 2 4 6 8 100

2

4

6

8

10

polarizable nonpolarizable

solvent = 78.4

B3L

YP

/cc-

pVT

Z (

D)

MM (D)

Unit: DebyeUnit: Debye

35

0 3 6 90

3

6

9

B3L

YP

/cc-

pVT

Z (

D)

MM (D)

in vacuo in =4 in =78.4

Transferability among Transferability among conformationsconformations

rmsd: 0.2799 uavg: 0.2413 correlation: 0.9922charges fitted simultaneously for both alphaL and c7eq in three environments

36

Continuum electronic Continuum electronic polarizationpolarization

• Electronic polarization with a continuum dipole moment Electronic polarization with a continuum dipole moment density. The uniform solute dielectric constant is the only density. The uniform solute dielectric constant is the only parameter. parameter.

• Performance comparable to ff02 explicit polarizable force Performance comparable to ff02 explicit polarizable force field for tested dipole moments in vacuum.field for tested dipole moments in vacuum.

• A single set of charges can be used in different A single set of charges can be used in different environments and different conformations. The model environments and different conformations. The model transfers well from monomers to dimers. transfers well from monomers to dimers.

37

Poisson-Boltzmann Molecular Poisson-Boltzmann Molecular DynamicsDynamics

38

Singular Charges in PBESingular Charges in PBE

• function in the PBEfunction in the PBE

• ChallengesChallenges

- 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

( ) i ii

q r r r

0( ) ( ) 4 ( ) 4 exp[ ( )]i i in q q r r r r

39

Removal of Charge SingularityRemoval of Charge Singularity

• Solve the Laplace’s equation for reaction field potential inside and Solve the Laplace’s equation for reaction field potential inside and simultaneously solve Poisson-Boltzmann equation for total potential simultaneously solve Poisson-Boltzmann equation for total potential outside. outside.

• Reaction potential is the difference between the total potential Reaction potential is the difference between the total potential

• Coulombic potential, which is defined asCoulombic potential, which is defined as

204C

RF C

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

RF

40

Removal of Charge SingularityRemoval of Charge Singularity

2

2

0

0f

C

C

n n n

inside

outside

On the dielectric boundary

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

41

Discontinuous InterfaceDiscontinuous Interface

• Boundary conditions on the discontinuous interface of the Boundary conditions on the discontinuous interface of the PBE (uniform potential)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 - Integrating the PBE and then using the Gauss’s law give the flux conditiongive the flux condition

n n

0( ) ( ) 4 ( ) 4 exp[ ( )]i i in q q r r r r

42

Harmonic Average (HA)Harmonic Average (HA)

• This method enforces the flux conditions in the three This method enforces the flux conditions in the three orthogonal directions on the physical interface, i.e., orthogonal directions on the physical interface, i.e.,

• The dielectric constant between two grid points that are The dielectric constant between two grid points that are in two different regions is a harmonic average of the in two different regions is a harmonic average of the two dielectric constants of the two regions.two dielectric constants of the two regions.

x x

Davis and McCammon, Journal of Computational Chemistry. 1991, 12, 909.

y y

z z

1, ,

2

hi j k

a b

43

Immersed Interface Method Immersed Interface Method (IIM)(IIM)

• A more accurate method for interface treatment for FDMA more accurate method for interface treatment for FDM• IIM proposes new equations involving 27 points instead of the IIM proposes new equations involving 27 points instead of the

original 7-point finite-difference equations at the points close to the original 7-point finite-difference equations at the points close to the interface. interface.

• IIM tries to minimize the local truncation error with the help of IIM tries to minimize the local truncation error with the help of interface conditions.interface conditions.

27

1

( , , ) ( , , ) ( , , )m m m mm

i i j j k k f i j k C i j k

LeVeque and Li. SIAM Journal Numerical Analysis. 1994, 31, 1019.

27

1

( , , ) ( , , ) ( , , ) ( , , )m m m mm

T i j k i i j j k k f i j k C i j k

44

IIM + Removal of SingularityIIM + Removal of Singularity

Tested in the Poisson equation:single particle system, dielectric boundary force

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

dd 1/h1/hIIM−SingularityIIM−Singularity HA−SingularityHA−Singularity HA+SingularityHA+Singularity

Max ErrorMax Error Max ErrorMax Error Max ErrorMax Error

0.250.25 44 0.0000050.000005 0.0002870.000287 0.0071360.007136

0.250.25 88 0.0000020.000002 0.0001370.000137 0.0034020.003402

0.250.25 1616 0.0000000.000000 0.0000640.000064 0.0008450.000845

1.001.00 44 0.0044580.004458 0.0040060.004006 1.5722241.572224

1.001.00 88 0.0009400.000940 0.0012640.001264 0.0190430.019043

1.001.00 1616 0.0002230.000223 0.0003760.000376 0.0093090.009309

1.501.50 44 0.9165790.916579 1.1482971.148297 59.29063859.290638

1.501.50 88 0.1002210.100221 0.0967070.096707 6.4334416.433441

1.501.50 1616 0.0093130.009313 0.0104320.010432 0.0755640.075564

45

Dielectric boundary force: Dielectric boundary force: TheoryTheory

bnd i of P e P e

2

2

2

2 2

1( ) ( ) ( )

21 1

( ) ( ) ( ) 04 2

01( ) ( ) ( )

2

1 1(

4 2

i i i i i i i i i i i

i i i i i i i i i i i i

i i i i i i i i i i i

i i i i

E E E e e E E e e E E e ee

P e E E e e E E E e e E E e e

E E e e E E e e E E E e e

E E

1 1

) ( ) ( )4 4i i i i i ie E E e E E e

2 21 1 1 1( ) ( ) ( )

4 2 4 4o o o o o o o o o o oP e E E e E E e E E e

i i o o

i o

i o

E E

E E

E E

2 2 2 21 1 1( ) ( )

4 2 2bnd i i i i o o o of E E E E

46

(0) ( ) (0) ( ) 0bnd i i o of P e P e P e P e

(0) ( ) (0) ( ) 0bnd i i o of P e P e P e P e

2 2 2 21 1 1 1( ) ( ) ( )

4 2 2 8bnd bnd i i i i o o o o o i o if f E E E E E E

Dielectric boundary force: Dielectric boundary force: TheoryTheory

Davis and McCammon, Journal of Computational Chemistry. 1990. 11. 401.Xiang et al, Journal of Chemical Physics. 2009. submitted.

47

Dielectric boundary force: Dielectric boundary force: Newton’s third lawNewton’s third law

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

48

AcknowledgementsAcknowledgements

Profs. David Case, Michael Gilson, Hong-Kai Zhao and Zhilin LiProfs. David Case, Michael Gilson, Hong-Kai Zhao and Zhilin Li

Drs. Jun Wang, Siang YipDrs. Jun Wang, Siang Yip

Chuck Tan, Yuhong Tan, Qiang LuChuck Tan, Yuhong Tan, Qiang Lu

Qin Cai, MJ HsiehQin Cai, MJ Hsieh

Gabe Ozorowski, Seema D’SouzaGabe Ozorowski, Seema D’Souza

Morris Chen, Emmanuel ChancoMorris Chen, Emmanuel Chanco

NIH/GMSNIH/GMS