Free energy calculations and the potential of mean force

Mark TuckermanDept. of Chemistry

and Courant Institute of Mathematical Science100 Washington Square East

New York University, New York, NY 10003

IMA Workshop on Classical and Quantum Approaches in Molecular Modeling

Free Energy

( , ) ( , , )3 ( )

1( , , ) = !

N N H A N V TN D V

Q N V T d d e eN h

β β− −= ∫ ∫ p rp r

Canonical Ensemble (Helmholtz free energy):

( ( , ) ) ( , , )3 0 ( )

0

1( , , ) = !

N N H PV G N P TN D V

N P T dV d d e eV N h

β β∞ − + −∆ = ∫ ∫ ∫ p rp r

Isothermal-Isobaric Ensemble (Gibbs free energy):

( , , ) ln ( , , )A N V T kT Q N V T= −

( , , ) ln ( , , )G N P T kT N P T= − ∆

Free Energy (cont’d)

P

VT

1

2

State function:

21 2 1A A A∆ = −

Free energy and work

• If an amount of work W is required to change the thermodynamic state of the system from 1 to 2, then

• Equality holds when the work is performed infinitely slowly or reversibly.

• Jarzynski’s equality [PRL, 78 2690, (1997)] shows how to relate irreversible work to the free energy difference. Let W21(x) be a microscopic function whose ensemble average is the thermodynamic work W21.

21 21W A≥ ∆

21 21

1

Ae eβ β− − ∆=W

1 2q d d= −

d1 d2

Free energy profiles

Ak e βκ −= ‡

A‡

Protein Folding EnergeticsFrom G. Bussi, et al. JACS 128, 13435 (2006)

1

2

( , ) ( ) ( )( , ) ( ) ( ) ( , )

E I E E I I

E I E E I I EI E I

U U UU U U U

= += + +

r r r rr r r r r r

1 2( , , ) ( ) ( , ) ( ) ( , )E I E I E IU f U g Uλ λ λ= +r r r r r r(0) 1 (1) 0(0) 0 (1) 1

f fg g

= == =

[ ][ ][ ]

bindGi

E IK e

EIβ− ∆= =

1

bind 0

UG dλ

λλ

∂∆ =

∂∫

Binding Free Energies

Inhibition constant:

Thermodynamic state potentials:

Meta-potential:

Thermodynamic integration (Kirkwood, 1935)

Binding free energies: Thermodynamic perturbation

( , )3 3( )

( ) 2

( )

1 ( , , )( , , ) ! !

( , , ) / 2

N N HN ND V

N U

D V

Z N V TQ N V T d d eN h N

Z N V T d e h m

β

β

λ

λ β π

−

−

= =

= =

∫ ∫

∫

p r

r

p r

r

2 221

1 1

ln lnQ ZA kT kTQ Z

∆ = − = −

2 1 2 1

2 1

( ) ( ) ( ( ) ( ))2

1 1 1

( )

1

1 1

U U U U

U U

Z d e d e eZ Z Z

e

β β β

β

− − − −

− −

= =

=

∫ ∫r r r rr r

Free energy difference related to partition function ratio:

Perturbation formula:

Need sufficient overlap between two ensembles

λ dynamics methods

Use molecular dynamics to sample λ via a Hamiltonian:2 2

1

2 2

1 1 2 1

( ,..., , )2 2

( ) ( ,..., ) ( ) ( ,..., )2 2

iN

i i

iN N

i i

pH Um m

p f U g Um m

λλ

λ

λ

λ

λ

λ λ

= + +

= + + +

∑

∑

p r r

p r r r r

Free energy from probability distribution of λ:

( , )( ) UP d e β λλ −= ∫ rr

21

( ) ln ( )(1) (0)

A kT PA A Aλ λ= −

∆ = −

Need to have best sampling at the endpoints of the λ-path, which arenormally the most difficult to sample.

λ dynamics methods

( )A λ

0λ = 1λ =

Aim for a profile with a barrier:

In order to generate such a profile, we need:

1. A high temperature Tλ >> T to ensure barrier crossing2. An adiabatic decoupling between λ and other degrees of freedom3. Choose mλ >> mi.

λ dynamics methodsUnder adiabatic conditions, we generate a free energy profile at Tλ

( ; ) ( ; ) ( , ) A A Ue e d eλλ

λ

βββ λ β β λ β β λ ββ− − − = = ∫ rr

Free energy profile at temperature Tfrom probability distribution generated under adiabatic conditions:

adb( ; ) ln ( ; , )A kT Pλ λλ β λ β β= −

Chemical Potential of Lennard-Jones Argon

( ) 24 ]1[ −= λλf 24 ]1)1[()( −−= λλg

2000 200m m T Tλ λ= =

TI

[ ]bins

exact1bins

1( ) ( ; ) ( )N

i ii

t P x t P xN

ς=

= −∑

HO

CH

H

0.145

0.06

0.06-0.683

0.418

Backbone

HO

C

H

HH

0.085

0.06

0.06-0.683

0.418

(Serine) (Methanol)

0.06

H

CH

H

-0.27

0.09

0.090.09

Backbone

H

C

H

HH

-0.36

0.09

0.090.09

(Alanine) (Methane)

0.09

Solvation free energies of amino acid side-chain analogs

1 Solute (CHARMm22 Parameters)

• 256 TIP3P Water molecules• Cubic Simulation Box (L = 19.066 A)• Periodic Boundary Conditions• Ewald Summation Technique for charges• System Temperature: 298 K• NVT via GGMT Thermostats (Liu,MET 2000)

λ-AFED Parameters:• kTλ = 12,000 K = 50 kT• mλ = 16,000 g.mol-1

a. Wolfenden, et al. Biochem. 20, 849 (1981)b. Shirts, et al. JCP 119, 5740 (2003); JCP 122, 134508 (2005)c. Yin and Mackerell J. Comp. Chem. 19, 334 (1998)

The Free Energy Profile

( , )1

1( ) ( ( ,..., ) ) N N HNP q d d e q q

Qβ δ−= −∫ p rp r r r

Probability distribution function:

Free energy profile in q:

( ) ln ( )F q kT P q= −

Bluemoon Ensemble Approach

1( ,..., )Nq q=r r

1 1( ,..., , ,..., ) 0iN N i

i ii i i

q qqm

∂ ∂= = =

∂ ∂∑ ∑ pr r p p rr ri i

Impose a constraint of the form:

However, constraints also require:

But in constrained MD, what we are actually computing is:

( , )1 1 1

1( ) ( ( ,..., ) ) ( ( ,..., , ,..., ))N N HN N NP q d d e q q q

Qβ δ δ−= −∫ p rp r r r r r p p

E. A. Carter, et al.Chem. Phys. Lett. 156. 472 (1989); M. Sprik and G. Ciccotti, J. Chem. Phys. 109, 7737 (1998).

Bluemoon Ensemble

[ ]1/ 211

1/ 211

Z kTdFdq Z

λ−

−

− + ℑ=

**When using SHAKE/RATTLE, λ must be the SHAKE multiplier!! M. Sprik and G. Ciccotti, J. Chem. Phys. 109, 7737 (1998).

Using the Lagrange multiplier to compute free energy:

2

2,

1 1 1 i i ji i i i j i i j j

q q q q qzm z m m

∂ ∂ ∂ ∂ ∂= ℑ =

∂ ∂ ∂ ∂ ∂ ∂∑ ∑r r r r r ri i i

[ ]1/ 2

constr1/ 2

constr

z kTdFdq z

λ−

−

− + ℑ=

1 2d dδ = −

d1 d2

Free energy profiles

‡

A‡

From D. Marx and MET, PRL 86, 4946 (2001).

Variable transformations and statistical mechanics

Adiabatic dynamics and free energy profiles

Hamiltonian from transformation:

2 23

1 31 1

( ,..., )2 2

n Ni i

Ni i ni i

p pH U q qm m= = +

= + +∑ ∑

Adiabatic conditions:

k k

q

m mT T

L. Rosso, P. Minary, Z. Zhu and MET, J. Chem. Phys. 116, 4389 (2000); Maragliano, Vanden Eijnden CPL 426, 168 (2006)

Free energy surface:

1 1( ,..., ) ln ( ,..., )n q nA q q kT P q q= −

Conformational sampling of the solvated alanine dipeptide

ψφ

AFED Tφ,ψ = 5T, Mφ,ψ = 50MC 4.7 nsUmbrella Sampling 50 ns

CHARMm22αR

β

[L Rosso, J. B. Abrams and MET J. Phys. Chem. B 109, 2099 (2005)]

C7ax αL

7.64.50.00.2

4.43.91.410.0

7.44.80.01.0Meta1

US2

AFED

Time FF β αR C7ax αL

5ns

5ns

400ns

CHm27

CHm22

CHm22

1. Ensing, et al. ACR 39, 73 (2005)2. Smith, JCP 111, 5568 (1999)

NH

NH O

CH3

N

O

H CH3

φ

ψ

Why NATMA?• Small compared to actual proteins and can be easily studied

• Tryptophan side-chain gives rise to a free energy landscape endemic of actual proteins

• Experimental1 and DFT1 Minimization data available

1. Dian, B.C., et al. Science, 296, 2369 (2002); J. Chem. Phys. 117, 10688. (2002)

NATMA (gas phase)(N-acetyl-Tryptophan-methyl-amide)

C7eq

C5(AP)

C5(AΦ)

kT(φ,ψ)=20kT, m(φ,ψ)=600mC t=2.5 ns

AFED Minima Predictions:

Conf. Energy Location (φ,ψ)C5(AP) 0.0 kcal/mol (-160, 160)C5(AΦ) +1.03 kcal/mol (-140, 140)C7eq +2.18 kcal/mol (-100, 100)

Ab initio energies1

Conf. Rel. EnergyC5 (AP) 0.0 kcal/molC5(AΦ) +0.65 kcal/molC7eq +2.28 kcal/mol

φ ψ

F(φ,ψ)

Metadynamics

Add bias potential to Hamiltonian:

A. Laio and M. Parrinello, PNAS 99, 12562 (2002); A. Laio, et al. JPCB 109, 6714 (2005)

2

1 1( ,..., ) ( ,..., , )2

iN G N

i i

H U U tm

= + +∑ p r r r r

( )2

1 2,2 ,... 1

( ) ( ( )( ,..., , ) exp

2G G

nk k G

G Nt k

q q tU t W

τ τ= =

−= −

∆ ∑ ∑

r rr r

Free energy is negative of bias potential:

( )2

1 2,2 ,... 1

( ( )( ,..., ) exp

2G G

nk k G

nt k

q q tA q q W

τ τ= =

−= − −

∆ ∑ ∑

r

REPSWA (Reference Potential Spatial Warping Algorithm)

No Transformation Transformation

5kT

10kT

10kT

V‡

How it works

V‡=kT

V‡=5kT

V‡=10kT

Forces:

( )

ref

ref

( )( )

( ) ( )

V V xF ux u

xF x F xu

∂ − ∂= −

∂ ∂∂

= −∂

/x u∂ ∂ becomes large in the barrierregion!

Barrier Crossing Transformations (cont’d)

‘

‘

‘

‘

ref ( )V φ

ref tors 1 inter 4 5 2 4 5( ,{ }) ( ) ( ) (| ( ,{ }) |) (| ( ,{ }) |)V V S V Sφ φ φ α φ φ= + − −r r r r r r r

P. Minary, G. J. Martyna and MET SIAM J. Sci. Comp. (accepted)

Comparison for 50-mer using TraPPE with all interactionsPT replicas = 10; PT exchange prob. = 5%, REPSWA α = 0.8; Every 10th dihedral not transformed

Comparison to parallel tempering and CBMCSiepmann and Frenkel, Mol. Phys. 75, 59 (1992)

End-to-end distance fluctuations

Comparison for 50-mer using CHARMM22 all interactions

Comparison of 50-mer using CHARMM22 all interactions

Honeycutt and Thirumalai

Model sheet protein βNo TransformationParallel TemperingSDC-REPSWA

PT replicas = 16; PT exchange prob. = 5%