Important coordinates

Eff

ecti

ve p

oten

tial

Effective Potentials for Protein Folding and BindingWith Thermodynamic Constraints

• The AGBNP effective solvation potential

•Optimization for structure prediction

•Free energy surfaces for -hairpin and -helical peptide folding

•Dynamics and kinetics

Roadmap to GB/NP Effective Potential Models for Solvation

•Electrostatic Component—Dielectric Continuum approximation. Generalized Born models—Parameterization (atomic radii) against FEP explicit solvent calculations with OPLS-AA force field

•Non-Polar Component—Novel non-polar estimator from FEP explicit solvent studies—Parameterization against experimental gas solubilities of small molecules—Parameterization for macromolecules: binding, folding

R.M. Levy, L. Y. Zhang, E. Gallicchio, and A.K. Felts, JACS, 125, 9523 (2003)

E. Gallicchio, L. Y. Zhang, and R.M. Levy, JCC, 23, 517, (2002)

E. Gallicchio, M. Kubo, and R.M. Levy, JPC, 104, 6271 (2000)

The AGBNP Implicit Solvent Model

AGBNP: Analytical Generalized Born + Non-Polar

Requirements:• Applicable to small ligands and large biomolecules,

many different functional groups• Applicable to study small and large conformational

changes: sensitive to molecular geometry.• Analytical with analytical gradients: MD sampling

E. Gallicchio, R. Levy, J. Comp. Chem., 25, 479-499 (2004)

AGBNP

Gsolv Gelec Gnp

Gcav GvdW

• Novel pairwise descreening Generalized Born model.• Separate models for cavity free energy and solute-solvent van der Waals interaction energy.• Fully analytical.• Sensitive to conformational change.• Equally applicable to small molecules and macromolecules.

Generalized BornSurface area model Born radius-based estimator

Generalized Born Model

Charging Free Energy in linear dielectric medium:

Gelec 12

1 in

1w

qiq j

fij (rij )ij

fij rij2 BiB j exp

rij2

4Bi Bj

1/ 2

Bi is the Born radius of atom i defined by:

Gsinglei 1

21 in

1w

qi2

Bi 1

81 in

1w

qi2

r ri4

V d3r

AGBNP: Pairwise Descreening Scheme

i

Born radii: rescaled pairwise descreening approximation:

1

Bi

1

Ri 1

4 s jQijj

Rescale according to self-volume of j:

s j Vj (self)

Vj

Vj (self) Vj 1

2 Vjkk 1

3 Vjklkl

Self-volume of j (Poincarè formula, ca. 1880):

E. Gallicchio, R. Levy, J. Comp. Chem. (2004)Hawkins, Cramer, and Truhlar, JPC 1996Schaefer and Karplus, JPC 1996Qiu, Shenkin, Hollinger, and Still, JPC 1997

j

Accuracy of Born Radii:Ligand Binding (free - bound)

Bi 1

(AGBNP)[Å-1]

Bi 1

(Numerical)[Å-1]

Non-Polar Hydration Free Energy

Gnp i Ai iW(Bi ) i

Non-polar hydration free energy estimator:

Gnp Gcav GvdW

Wi w-4i i

6

| r ri |6slv. 16wi i

6

3Ci3

Ci 3

41

| r - ri |6slv.

1/ 3

Bi

: Surface area of atom i

: Estimator based on Born radius

: Surface tension and van der Waals adjustable parameters

Ai

W(Bi )

i ,i

R.M. Levy, L. Y. Zhang, E. Gallicchio, and A.K. Felts, JACS, 125, 9523 (2003)

E. Gallicchio, M. Kubo, and R.M. Levy, JPC, 104, 6271 (2000)

Enthalpy-Entropy and Cavity Decomposition of Alkane Hydration Free Energies: Numerical Results and

Implications for Theories of Hydrophobic SolvationEmilio Gallicchio, Masahito Kubo, Ronald Levy, J. Phys. Chem., 104, 6271 (2000)

Solute-Solvent van der Waals Energy of Proteins: Comparison of Surface Area and Continuum Solvent Models

SASA (A2) SASA (A2)

Uvd

W (

kca

l/m

ol)

Figure: Continuum solvent solute-solvent van der Waals interaction energies of various peptides and proteins conformations plotted against their accessible surface area. (A) Data with accessible surface area between 3000 and 12000 A2. Filled circles denote 98 native peptide and protein conformations, open triangles denote 12 extended protein conformations, and filled triangles denote 273 decoy conformations of 4 native proteins. (B) Data with accessible surface area between 6000 and 10000 A2. Filled triangles denote decoy conformations of of protein lz1 (the native conformation of lz1 is circled).The lines are the linear least square fit to all native and extended protein conformations examined, respectively.

(A) (B)

Optimization of the AGBNP Effective Potential for Structure Prediction with thermodynamic constraints

Geff = Uint + GAGB + Gnp

Gnp = i k Ai + k(16ii6 / 3Bi

3)

where k indicates atomtype of atom i

Z-score: Zn = ave(Gi Gn)/d

Maximize: Zn 2

Summary of Fitting Results (in kcal/mol)Molecular class Number of Fit Prediction molecules

Compounds containing only C and Halkanes 19 0.21 0.24alkenes and dienes 11 0.16 0.22alkynes 6 0.20 0.33arenes 21 0.38 0.44Subtotal 57 0.26 0.32

Compounds containing only C, H and Oalcohols 27 0.25 0.28ethers 17 0.62 0.66ketones and aldehydes 22 0.23 0.26carboxylic acids 3 0.23 0.44esters 15 0.23 0.25Subtotal 103 0.29 0.33

Compounds containing only C, H, O and Namines 23 0.27 0.33amides 5 0.19 0.40nitriles 5 0.42 1.15nitro compounds 6 0.78 2.57nitrogen heterocyclic 13 0.33 1.22Subtotal 52 0.35 0.89

Compounds containing C, H, O, N and Sthiols 3 0.72 1.24sulfides 3 0.43 0.74Subtotal 6 0.57 0.99

Total 199 0.32 0.50

Protein Loops Modeling

7RSA (13-24)

Prediction of native loop conformation using the OPLS/AGBNP effective energy function

AGBNP: Applications

• Protein Folding- Peptides.- Protein Decoys.

• Ligand binding- Binding Mode Prediction.- Binding Free Energy Prediction.

• Structure Prediction- Protein Loop Modeling.

The -Hairpin of B1 Domain of Protein G

The hydrophobic sidechains are in green.

Pande, PNAS, 1999Nussinov, JMB, 2000Garcia et al., Proteins, 2001Zhou & Berne, PNAS, 2002

Dinner, Lazaridis, Karplus, PNAS, 1999Pande et al., JMB, 2001Zhou & Berne, PNAS, 2002

Replica Exchange Sampling for -hairpin FoldingReplica Exchange Sampling for -hairpin Folding

• Replica exchange sampling* is a method to effectively sample rough energy landscapes which have high dimensionality - the hairpin has 768 degrees of freedom •~20 MD simulations of the -hairpin run in parallel over the temperature range 270 K -690 K.

• Every 50 MD steps MC replica exchange moves are attempted

• Total sampling time: 20 processors x 4 x 106 step/processor = 80 x 106 steps

Time series of the temperature for one replica Time series of the replicas for one Temp., T = 442 K

0

5

10

15

20

0 1 105

2 105

3 105

4 105

5 105

Th

e n

um

ber

of

pro

cess

or

Step number (fsec)

250

300

350

400

450

500

550

600

650

0 1 105

2 105

3 105

4 105

5 105

Tem

pe

ratu

re

Step number (fsec)

* Y. Sugita, and Y. Okamoto, Chem. Phys. Let., 314, 141 (1999)

The -Hairpin of B1 Domain of Protein G

The potential of mean force of the capped peptide.

Simple nonpolar model. AGB-NP with Scharged=0.5

A Felts, Y. Harano, E. Gallicchio, and R. Levy, Proteins, 56, 000 (2004)

The -Hairpin of B1 Domain of Protein G

The potential of mean force of the capped peptide.

Simple nonpolar model. AGB-NP with Scharged=0.5

A Felts, Y. Harano, E. Gallicchio, and R. Levy, Proteins, 56, 000 (2004)

Estimated -Hairpin and -Helical Populations(native peptide from protein G, T=298K)

No WHAM T-WHAM

Gmax=5 kcal/mol Gmax=10 kcal/mol

-hairpin > 90%-helix < 10%G ~ 2 kcal/mol

T-WHAM: PMF contains information from high temperature walkers

Solve for (E) and insert into

expression for P(E;Ti).

T-WHAM

• A way to combine data from simulations at various temperatures to obtain properties at one given temperature.

Energy distribution:

P(E;Ti ) Z(T0 )

Z(Ti )e (i 0 )E P(E;T0 )

P(E;T0 )(E) e 0E

Z(T0 )

- Given P(Ej;T0) can predict histogram of energies n(Ej;Ti) at any temperature. - Select P(Ej;T0) that best reproduces observed histograms (maximum likelihood solution assuming multinomial-distributed counts).

P(E j ;T0 ) n(E j ;Ti )i

Ni e fi e (i 0 )E j

ie fi

Z(Ti )

Z(T0 ) P(E j ;T0 )e

(i 0 )E jj

WHAM equations: {Same derivation for joint probability P(x,E;T).

Alternative Coordinates for the -HairpinProjections onto the first four principal components

Alternative Coordinates for the -HairpinTemperature dependence

T = 298 K T = 400 K

T = 328 K T = 488 K

Free Energy Surface of the Protein G -Hairpin With Respect to the (1,4) Principle Components

T-WHAM

In Silico Mutation Study of the protein G -Hairpin Sequence

Sequence coilnative GEWTYDDATKTFTVTE 88% 8% 4%

W43S mutant GESTYDDATKTFTVTE 42% 40% 18%Y45S mutant GEWTSDDATKTFTVTE 23% 71% 6%W43S, Y45S GESTSDDATKTFTVTE 0.1% 83% 17%

44% homol* GEQVAREALKHFAETE 0% 95% 5%

random #1 VTGADFTKYTTEDWTE 35% 4% 61% random #2 VYEWDGTTKTEFADTT 31% 13% 56%

*C-terminal -helix of 1b6g: 44% BLAST homology with sequence from protein G

Free Energy Surfaces Generated with REM and OPLS-AA/AGBNP

-Hairpin of C-terminusof B1 domain of protein G

-Helix of C-peptideof ribonuclease A

GEWTYDDATKTFTVTE KETAAAKFERQHM

Important coordinates

Eff

ecti

ve p

oten

tial

Effective Potentials for Protein Folding and BindingWith Thermodynamic Constraints

• The AGBNP effective solvation potential

Emilio Gallicchio, Tony Felts

• Optimization for structure prediction

Emilio Gallicchio, Tony Felts

• Free energy surfaces for -hairpin and -helical peptide folding

Yuichi Harano, Tony Felts, Emilio Gallicchio, M. Andrec

• Dynamics and kinetics

Dimitriy Chekmarev, Tateki Ishida, Michael Andrec

Important coordinatesE

ffec

tive

pot

enti

al

Effective Potentials for Protein Folding and BindingWith Thermodynamic Constraints