1

Scaling in Biomolecular Scaling in Biomolecular SolvationSolvation

Are Proteins Large?Are Proteins Large?

Ray LuoMolecular Biology and Biochemistry

University of California, Irvine

2

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

in Biomoleculesin Biomolecules

• Quantum description: electronic & covalent Quantum description: electronic & covalent structurestructure

• Atom-based description: non-covalent Atom-based description: non-covalent interactionsinteractions

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

3

Challenges in biomolecular Challenges in biomolecular simulationssimulations

22dUdt=−∇r

2 20 0 126bonds angles torsions atompairs() () [cos()1] ij ij ijb ij ij ijQQABUkbbk kn rrrθ φθθ ϕδ⎡ ⎤=−+−+ ++++−⎢ ⎥⎢ ⎥⎣ ⎦∑∑∑∑

Mathematical models should be Mathematical models should be as realistic as possibleas realistic as possible

• Every atom is represented as a Every atom is represented as a classical particle.classical particle.

• Potential energy is in a pairwise Potential energy is in a pairwise form only.form only.

4

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)?)?

5

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

Salt bridge population

Times (x10ns)

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

6

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

Removing solvent degrees of Removing solvent degrees of freedomfreedom

exp[(,)](,) exp[(,)]uvuv uv uvUPddUββ−=−∫rrrrrr rr exp[()]()exp[()]uu u uWPdWββ−=−∫rrr r exp[()]exp[(,)]u v uvWdUβ β−≡−∫rr rr

ru: solute coordinates; rv: solvent coordinates

7

Biomolecules in implicit Biomolecules in implicit solventssolvents

Mathematical models should be Mathematical models should be as realistic as possibleas realistic as possible

• Solute biomolecule is still in all-Solute biomolecule is still in all-atom representation.atom representation.

• Solvent molecules are now in Solvent molecules are now in continuum representation.continuum representation.

• There is an interface between There is an interface between solute and solvent.solute and solvent.

2 20 0 126bonds angles torsions atompairs() () [cos()1] ij ij ijb ij ij ijQQABUkbbk kn rrrθ φθθ ϕδ⎡ ⎤=−+−+ ++++−⎢ ⎥⎢ ⎥⎣ ⎦∑∑∑∑

d 2r

dt 2=−∇(U +W)

8

Continuum solvationContinuum solvationapproximationsapproximations

• 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

W =Wes +Wnp

Wnp =Wrep +Watt

9

Implicit electrostatic solventImplicit electrostatic solvent

p

+

+-

-

++

-

-

s

Dielectric constant

Electrostatic potential

Charge density

Charge of salt ion in solution

Wes =12

qii∑ φi

∇⋅(r)∇φ(r) =−ρ(r) − ni

0qi exp[−βqiφ(r)]∑

10

Implicit nonpolar solvents Implicit nonpolar solvents

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

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

Uattuv = niv(r)∫

i=1

Nu

∑ Vatt(r)d3r

Wnp = Wrep + Watt

Wrep =γA+ c

Wrep =pV + c

Uattuv

11

Implicit solvents: pros and Implicit solvents: pros and cons cons

Are structureless Are structureless implicit solvents sufficient?implicit solvents sufficient?

Computational efficiency for alanine dipeptideComputational efficiency for alanine dipeptide

Solvent G (kcal/mol) CPU Time (s)

EXP -13.40 3.4×105

IMP -13.38 1.0×10-1

12

Does size matter in Does size matter in biomolecular solvation?biomolecular solvation?

D. Chandler, Nature, 437, 640-647, 2005

13

How consistent are implicit and How consistent are implicit and explicit solvents explicit solvents

on conformation dependent on conformation dependent energetics?energetics?

310 Helix α Helix π Helix

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

14

Electrostatic SolvationElectrostatic Solvation

15

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:

λλλ

λ

dd

)dH(G

1

0∫=Δ

16

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

17

Accurate Atomic Radii:Accurate Atomic Radii:Basis of Quantitative StudiesBasis of Quantitative Studies

Different cavity radii Different cavity radii for PB solvents will for PB solvents will result in different result in different agreements with agreements with explicit solvent. explicit solvent.

Atomic cavity radii are responsible for the Atomic cavity radii are responsible for the desolvation penalties of amino acids and desolvation penalties of amino acids and nucleotides.nucleotides.

18

Quality of radius refitQuality of radius refit

-100 -80 -60 -40 -20 0

-100

-80

-60

-40

-20

0

Gelec

by PB (kcal/mol)

ΔGelec

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

19

Conformation dependenceConformation dependencePeptide reaction field Peptide reaction field

energiesenergies• Three helical conformationsThree helical conformations

3310 10 helix helix αα helix π helix helix π helix

• Ten beta-strand conformationsTen beta-strand conformations

5 Parallel 5 Anti-parallel5 Parallel 5 Anti-parallel• Peptides with salt bridgePeptides with salt bridge

HD3 HD4 HD5 HD3 HD4 HD5

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

20

Peptide reaction field Peptide reaction field energies energies

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

-100

-80

-60

-40

-20

Gelec

by PB (kcal/mol)

ΔGelec

by TI (kcal/mol)

Correlation Coefficient: Correlation Coefficient:

0.9970.997

RMSD: RMSD:

2.90 kcal/mol 2.90 kcal/mol

TI statistical uncertainties less than 0.6 kcal/mol.

21

Size dependenceSize dependenceSalt-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.

22

Salt-bridge charging free Salt-bridge charging free energiesenergies

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

-100

-80

-60

-40

-20

G_ele exp

( / )kcal mol

Gele_imp

(kcal/mol)

PEPp53

PEPenh

ENHP53

Tan and Luo, In Prep

23

Electrostatic solvationElectrostatic solvation

• Conformation dependent energetics is Conformation dependent energetics is consistent between PB and TI.consistent between PB and TI.

• PB correlate very well with TI from short PB correlate very well with TI from short peptides up to proteins of typical sizes.peptides up to proteins of typical sizes.

24

Nonpolar SolvationNonpolar Solvation

25

• 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)

λλλ

λ

dd

)dH(G

1

0∫=Δ

Tan et al, JPC-B, 111, In Press, 2007

26

Nonpolar repulsive free Nonpolar repulsive free energiesenergies

(A) SESCC: 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

G_rep exp

( / )kcal mol

Grep_imp (kcal/mol)0 5 10 15 20 25

Tan et al, JPC-B, 111, In Press, 2007

27

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

G_att exp

( / )kcal mol

Gatt_imp

(kcal/mol)

Error bars too small to be seenTan et al, JPC-B, 111, In Press, 2007

28

Total nonpolar free Total nonpolar free energiesenergies

(A) SESCC: 0.981RMSD: 0.33kcal/mol

(B) SEVCC: 0.891RMSD: 0.67kcal/mol

(C) SASCC: 0.984RMSD: 0.31kcal/mol

(D) SAVCC: 0.986RMSD: 0.28kcal/mol

Tan et al, JPC-B, 111,

In Press, 2007

29

Conformation and size Conformation and size dependencedependence

Nonpolar free energies of TYRNonpolar 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.

30

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

G_att exp

( / )kcal mol

Gatt_imp

(kcal/mol)

PEPα

PEPβ

PGBP53

Error bars too small to be seen

Tan and Luo, In Prep.

31

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

PEPα

PEPβ

PGB P53

G_rep exp

( / )kcal mol

Grep_imp

(kcal/mol)

-4

0

4

8

12

16

B

A

Tan and Luo, In Prep.

32

Behavior of Two Estimators for Behavior 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

γ ( / /kcal mol

Å2)

Conformations

PEPα

PEPβ

PGB P53 AVG_HLX

0 2 4 6 8 10 120.02

0.03

0.04

0.05

0.06

p (kcal/mol/

Å3)

Conformations

PEPα

PEPβ

PGB P53 AVG_HLX

Tan and Luo, In Prep.

33

Nonpolar solvationNonpolar solvation

• Both attractive and repulsive nonpolar component works Both attractive and repulsive nonpolar component works well from tested peptides to proteins of different scales if well from tested peptides to proteins of different scales if the volume estimator is used.the volume estimator is used.

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

34

AcknowledgementsAcknowledgements

Jun Wang, Siang YipJun Wang, Siang Yip

Chuck Tan, Yuhong TanChuck Tan, Yuhong Tan

Qiang Lu, MJ HsiehQiang Lu, MJ Hsieh

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

Morris Chen, Emmanuel ChancoMorris Chen, Emmanuel Chanco

NIH/GMSNIH/GMS

35

How does implicit How does implicit solventssolvents

perform in perform in dynamics?dynamics?

36

PMEMD simulations at PMEMD simulations at 450K450K

• 10 independent trajectories for at least 8ns in NVT at 10 independent trajectories for at least 8ns in NVT at 450K450K

• Pre-equilibrated for 2ns in NPT at 300KPre-equilibrated for 2ns in NPT at 300K• Simulation parameters:Simulation parameters:

• TIP3P waterTIP3P water• Truncated octahedron box with a buffer of 11ÅTruncated octahedron box with a buffer of 11Å• Real space cutoff 9ÅReal space cutoff 9Å• Continuum van der Waals energy correction beyond cutoffContinuum van der Waals energy correction beyond cutoff• Berendsen heat bathBerendsen heat bath

37

PBMD simulations at 450KPBMD simulations at 450K

• P3M treatment of electrostaticsP3M treatment of electrostatics• Modified VDW surface for dielectricsModified VDW surface for dielectrics• Nonpolar contributions reweighted after MD simulationsNonpolar contributions reweighted after MD simulations

(C, Tan et al. (C, Tan et al. JPCJPC, 2007 ), 2007 )

• Simulation parameters:Simulation parameters:• Solvent probe radius: 0.6Å Solvent probe radius: 0.6Å (C, Tan et al. (C, Tan et al. JPCJPC, 2006 ), 2006 )• Finite difference solver grid spacing: 0.5ÅFinite difference solver grid spacing: 0.5Å• Finite difference solver tolerance: 0.0001Finite difference solver tolerance: 0.0001• Cutoff for PM electrostatic interaction: 7.0ÅCutoff for PM electrostatic interaction: 7.0Å• No VDW cutoffNo VDW cutoff• Langevin heat bathLangevin heat bath

Lu and Luo, JCP, 119, 11035-11047, 2003

38

-150

-100

-50

0

50

100

150

-150 -100 -50 0 50 100 150

-150

-100

-50

0

50

100

150Ψ [ ]Degree

-150 -100 -50 0 50 100 150

Φ [Degree]

0

1.000

2.000

3.000

4.000

5.000

6.000

7.000

((ΦΦ,,ΨΨ) Free Energy ) Free Energy LandscapeLandscape

ARG

GLU

PME PB

39

((ΦΦ,,ΨΨ) Free Energy ) Free Energy LandscapeLandscape

-150

-100

-50

0

50

100

150 0

1.000

2.000

3.000

4.000

5.000

6.000

7.000

-150 -100 -50 0 50 100 150

-150

-100

-50

0

50

100

150Ψ [ ]Degree

-150 -100 -50 0 50 100 150

Φ [Degree]

PME PB

LEU

GLN

40

Alpha contentAlpha content

PB vs PMEPB vs PME PBNP vs PMEPBNP vs PME

RMSDRMSD 0.140.14 0.090.09

UAVGUAVG 0.110.11 0.070.07

-180 -90 0 90 180-180

-90

0

90

180

Ψ [ ]Degree

Φ [Degree]

ALA ARG ASN ASP CYS GLN GLU HID HIE HIP ILE LEU LYS MET PHE SER THR TRP TYR VAL0.0

0.2

0.4

0.6

0.8

1.0

-log(SSC)

PME PB PBNP

(Hu et al. PROTEINS, 2003)

41

PBMDPBMD

• No systematic bias in secondary structure propensities in No systematic bias in secondary structure propensities in PBMD for the tested dipeptidesPBMD for the tested dipeptides

• More challenging tests: More challenging tests: • HD4 Alpha Helix : AAAAAHAAADAAAAAAHD4 Alpha Helix : AAAAAHAAADAAAAAA

• HPN Beta hairpin: GEWTYNDATKTFTVKQHPN Beta hairpin: GEWTYNDATKTFTVKQ

• Observables: Observables: secondary structures, secondary structures, salt bridges, salt bridges, Hydrophobic contacts, and free energy landscapesHydrophobic contacts, and free energy landscapes

42

Beta contentBeta content

PB vs PMEPB vs PME PBNP vs PMEPBNP vs PME

RMSDRMSD 0.100.10 0.070.07

UAVGUAVG 0.070.07 0.060.06

-180 -90 0 90 180-180

-90

0

90

180

Ψ [ ]Degree

Φ [Degree]

ALA ARG ASN ASP CYS GLN GLU HID HIE HIP ILE LEU LYS MET PHE SER THR TRP TYR VAL0.0

0.2

0.4

0.6

0.8

1.0

1.2

-log(SSC)

PME PB PBNP

(Hu et al. PROTEINS, 2003)