Free energy calculations and the potential of mean force Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science 100 Washington Square East New York University, New York, NY 10003 IMA Workshop on Classical and Quantum Approaches in Molecular Modeling
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Free energy calculations and the potential of mean force · Free energy calculations and the potential of mean force Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical
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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)