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Free energy simulations: theory and applications
O. Michielin(1,2,4)
(1) Ludwig Institute for Cancer Research Epalinges, Switzerland
(2) Swiss Institute for BioinformaticsEpalinges, Switzerland
(4) Centre Pluridiscipinaire d'oncologieCHUV, Lausanne, Switzerland
Dynamical aspects of molecular recognition
Free energy: classical definition
+
Enthalpic Entropic
! Hydrogen bonds! Polar interactions! Van der Waals interactions! ...
! Loss of degrees of freedom! Gain of vibrational modes! Loss of solvent/protein structure! ...
Theoretical Predictions: ! Approximate: empirical formula for all contributions
! Exact: using statistical physics definition of G
G = -KBT ln(Z)
�
!G = !H "T!S
The free energy is the energy left for once you paid the tax to entropy:
Examples of factors determining the binding free energy
Electrostatic interactions
- Strength depends on microscopic environment (!)
- Case of hydrogen bonds
-2.4 to -4.8 kcal/molCharge assisted :
-1.2 ± 0.6 kcal/molNeutral :
OH
O
H
H
O
H
H
O
H
H
O
H
H
O
H
H
OHO
H
S
solvent complex
EH-bond (solv.) - EH-bond (comp.)
determines if H-bondscontributes to affinity or not
Unpaired polar groups upon binding are detrimental
Strong directional nature
Specificity of molecular recognition
Free energy: statistical mechanics definition
G = !kBT ln(Z ) Z = e
!"Ei
i#where
is the partition function
Free energy differences between 2 states (bound/unbound, …)
are, therefore, ratios of partition functions
!G = GA"G
B= "k
BT ln
ZA
ZB
#
$%&
'(
Free energy simulations aim at computing this ratio using various
techniques.
Relation with chemical equilibrium
A + B "# A’B’
A + B"#A’B’
Kb : binding constant, Kd : dissociation constant, Ki : inhibition constant
KD (mol/l)
"Gbinding (kcal/mol) -2 -4 -6 -8 -10 -12 -14 -16
10 -1210
-910 -610
-3
Weak asso. Strong asso.
KD= K
i=
A[ ] B[ ]A'B'[ ]
KA
= Kb=
A'B'[ ]A[ ] B[ ]
!Gbinding = "RTlnKA = RTlnKD = !H " T!S
Connection micro/macroscopic: thermodynamics and kinetics
Free Energy Association Constant
e - RT!G = KA
Microscopic Structure Biological function
Relative bindingfree energies: !!G
" KA’ / KA
Absolute binding free energies: !G
" KA
Binding free energyprofiles: !G(#)" KA, Kon, Koff
The free energy is the main function behind all process
A) Chemical equilibrium
B) Conformational changes
C) Ligand binding
D) …
+!Gbinding = RTlnKA
!Gconf
= kBTlnPConf 1
PConf 2
!Gbinding = kBTlnPUnbound
PBound
KA=
AB[ ]A[ ] B[ ]
A B ABKD= 1 / K
A
R = kBN
A
Free energy: computational approaches
!G = GA"G
B= "k
BT ln
ZA
ZB
#
$%&
'(
Free energy simulations techniques aim at computing ratios of
partition functions using various techniques.
Z = e!"Ei
i#
Sampling of important
microstates of the system
(MD, MC, GA, …)
Computation of energy
of each microstate
(force fields, QM, CP)
Connection micro/macroscopic: intuitive view
E1, P1 ~ e-$E1
E2, P2 ~ e-$E2
E3, P3 ~ e-$E3
E4, P4 ~ e-$E4
E5, P5 ~ e-$E5
Where
is the partition function
Expectation value
�
O =1
ZOie!"Ei
i
#
�
Z = e!"Ei
i#
Central Role of the Partition Function
G = -kBT ln(Z)
. . .
Expectation Value
Internal Energy Pressure Gibbs free energy
Z = e!"Ei
i#
O =1
ZOie!"Ei
i#
E =!
!"ln(Z ) =U p = kBT
! ln(Z )!V
"#$
%&'N ,T
Molecular Modeling Principles
1) Modeling of molecular interactions
2) Simulation of time evolution (Newton)
3) Computation of average values
O = < O >Ensemble = < O >Temps (Ergodicity)
Macroscopic value Average simulation value
Connectionmicroscopic/macroscopic
Free energy landscape
Electrostatics Van der Waals
Covalentbonds Solvent
The CHARMM Force Field
�
V = Kb
Bonds
! (b " b0)2
+ K#
Angles
! (# "#0)2
�
+ K!
Impropers
" (! #!0)2
�
+ K!
Dihedrals
" 1# cos(n!! #$! )[ ]
�
+qiq j
4!"i> j
#1
ri, j
�
+ 4!ij (" ij /rij )12# (" ij /rij )
6[ ]i> j
$
#$
E
3N Spatial coordinates
Ergodic Hypothesis
NVT simulation
?
MD Trajectory
NVE simulation
Dialanine Protein
OEnsemble
=1
ZO(!," )e#$E (! ," )% d!d" =
1
&O(t)dt
0
&
% = OTime
Free energy calculation: Main approaches
Free Energy Perturbation (FEP) Thermodynamical Integration (TI)
CP
U T
ime
Linear Interaction Energy (LIE) Molecular Mechanics/Poisson-Boltzmann/Surface area (MM-PBSA)
Quantitative Structure Activity Relationship (QSAR)
Non Equilibrium Statistical Mechanics (Jarzynski)
Sam
plin
g, E
xact
Sam
plin
g, A
ppro
x.
Appro
x.
G k 0 k i X i (X is a descriptor)
�
!G = F(X)
Free energy calculation: Main approaches
Free Energy Perturbation (FEP) Thermodynamical Integration (TI)
CP
U T
ime
Linear Interaction Energy (LIE) Molecular Mechanics/Poisson-Boltzmann/Surface area (MM-PBSA)
Quantitative Structure Activity Relationship (QSAR)
Non Equilibrium Statistical Mechanics (Jarzynski)
Sam
plin
g, E
xact
Sam
plin
g, A
ppro
x.
Appro
x.
G k 0 k i X i (X is a descriptor)
�
!G = F(X)
Theoretical approaches for the estimation of binding affinities
Without 3D structure of the complex- 2D - QSAR
With 3D structure of the complex
- Binding free energy decomposition (MM-PBSA, MM-GBSA)
- Knowledge based approaches • regression based methods • potential of mean force
- Free energy simulation
- Linear interaction energy
- 3D - QSAR
Quantitative Structure Activity Relationships (QSAR)
Advantage
No need for structural information about the target
Assumption
Affinity is a function of the ligand physico-chemical properties
Requirements (drawbacks)Known affinities for a series of ligands
(Drug design)
Structurally related ligands or similar binding modes
Chemical similarity of ligands Similarity of biological response
2D - QSAR
N
N
O
R1
R2
R3
O
R4
n
.
.
.
3
2
1
"Gbind
Xm
.
.
.
X2
X1
!+="ii0bindXkkG
n structurally related molecules
Quantitative description Measured activities
( )ibind
Xnetwork neuralG =!
Descriptors (Xi):- Substituents: surface, volume, electrostatics (%), hydrophobicity (&), partial charges, etc...- Molecule : volume, MR, HOMO, dipole moment, etc...
C. Hansch and T. Fujita, JACS, 1964, 86, 1616
S.S. So and M. Karplus, J. Med. Chem., 1996, 39, 1521
S.S. So and M. Karplus, J. Med. Chem., 1996, 39, 5246
2D - QSARLimitations
Needs the experimental activity of a series ligands
Not for ab initio studies
Use limited to the descriptor’s range in the training set :
N
N
O
R1
R2
R3
O
R4
If only hydrophobic groups at R1 in the training set
Influence of a hydrophilic group at R1 ?
If only methyl, ethyl, propyl, butyl at R1 in the training set
Contribution of pentyl, hexyl, etc... ?
Limited to structurally related molecules
Overfitting- Method for selecting the descriptor (genetic algorithm)- Estimation of the predictive ability (test set, randomization test, Leave-One-Out method, ...)
3D - QSARExample : Comparative molecular field analysis (CoMFA)
R.D. Cramer et al., JACS, 1988, 110, 5959
n
.
.
.
3
2
1
"Gbind
.
.
.
E2
E1
.
.
.
S2
S1
! !++="ii0bindESkGii
#$
Steric field Electrostatic field
(x1,y1,z1) (x2,y2,z2) (x1,y1,z1) (x2,y2,z2)Ligands mutually alignedCommon 3D lattice
PLS
(xi,yi,zi)
(xj,yj,zj)
3D - QSARLimitations
Needs the experimental activity of a series of ligands
Not limited to structurally related molecules
Alignment of the molecules in their bioactive conformation. Use of:
- known structure of a complex (QSAR?)
- conformationally rigid example in the dataset
- functional groups in agreement with a pharmacophore hypothesis
Others : CoMSIA, HASL, Compass, APEX-3D, YAK, ...
Knowledge based approaches. Regression based methodsExample : LUDI score
H.J. Bohm, J. Comput.-Aided Mol. Des., 1994, 8, 623H.J. Bohm, J. Comput.-Aided Mol. Des., 1998, 12, 309
Trained using a 82 protein-ligand complexes dataset with known experimental "Gbind
Polar interactions
"Ghb = -0.81, "Gion = -1.41 and "Gesrep = +0.10 kcal/mol
Apolar interactions
"Glipo = -0.81 and "Garo = -0.62 kcal/mol
Desolvation effect
Active site filled with water molecules
MD Unbound water molecules
"Glipo water = -0.33 kcal/mol
Ligand flexibility
"Grot = +0.26 kcal/mol
Nrot : number of rotable bonds (acyclic sp3-sp3, sp3-sp2)
!Gbind = !G0 + !Gpolar + !Gapolar + !Gsolv + !Gflexi
!Gpolar = !Ghb f !R,!"( ) # f Nneighb( ) # fpcshb
$
+!Gion f !R,!"( ) # f Nneighb( ) # fpcsion
$
+!GesrepNesrep
!Gapolar = !GlipoAlipo + !Garo f (R)aro
"
!Gsolv = !Glipo wat unbound water"
!Gflex
= !GrotNrot
Knowledge based approaches. Regression based methodsExample : LUDI score
SD ~2 kcal/mol
Advantages : Drawbacks :
- Rapid estimation of the affinities
- Structurally different ligands
- Allows identification of high affinity ligands
- “Universal” (different proteins)
- Method biased:
- Somewhat large errors
• certain type of proteins• only good complementarity protein/ligand
- Some interactions ignored (cation – &)
82 complexes of the training set
Others : ChemScore, VALIDATE
Knowledge based approaches. Potential of mean forceExample : PMF score
I. Muegge et al., J. Med. Chem., 1999, 42, 791I. Muegge et al., Persp. In Drug Disc. And Des., 2000, 20, 99
Trained using 697 complexes from the PDB.No need for experimental "Gbind
atom type i for protein and j for ligand
%ij(r) : number density of atom pairs of type ij at distance r
%ijbulk(r) : number density of atom pairs of type ij in a sphere with radius R
f jvolv_corr : ligand volume correction
16 protein atom types, 34 ligand atom types
PM
F (
kc
al/
mo
l)
Atom pair distance (Å) Atom pair distance (Å) Atom pair distance (Å)
NC positively charged nitrogen
ND nitrogen as hydrogen bond donor
OC negatively charged oxygen
OD oxygen as hydrogen bond donnor
OA oxygen as hydrogen bond acceptor
Aij r( ) = !kbTln fvol_corrj (r)
" ij(r)"bulkij
#
$%
&
'(
PMF score = Aij (r)kl of type ij
r<rcutoff
!ij
!
Knowledge based approaches. Potential of mean forceExample : PMF score
PM
F s
co
re
log Ki (experimental)
77 complexes, 5 different proteinsSD ~2 kcal/mol
Advantages : Drawbacks :
- Rapid estimation of the affinities
- Structurally different ligands
- Allows identification of high affinity ligands
- “Universal” (different ligand and protein types)
- Somewhat large errors
- No measure of directionality of H-bonds
- No fitting parameters to measured "Gbind
Others : SMoG-Score, BLEEP, DrugScore
Free energy calculation: Main approaches
Free Energy Perturbation (FEP) Thermodynamical Integration (TI)
CP
U T
ime
Linear Interaction Energy (LIE) Molecular Mechanics/Poisson-Boltzmann/Surface area (MM-PBSA)
Quantitative Structure Activity Relationship (QSAR)
Non Equilibrium Statistical Mechanics (Jarzynski)
Sam
plin
g, E
xact
Sam
plin
g, A
ppro
x.
Appro
x.
G k 0 k i X i (X is a descriptor)
�
!G = F(X)
Linear interaction energy (LIE)J. Åqvist, J. Phys. Chem., 1994, 98, 8253
Free state
“solvent” = water
Bound state
“solvent” = water + protein
Two MD runs : free state and bound state
J. Åqvist, J. Phys. Chem., 1994, 98, 8253
'=0.165 and (=0.5
T. Hansson et al., J. Comp.-Aided Molec. Design, 1998, 12, 27
'=0.181 and (=0.5, 0.43, 0.37, 0.33
W. Wang, Proteins, 1999, 34, 395
' function of binding site hydrophobicity
!Gbind
= " El# svdw
bound# E
l# svdw
free( ) + $ E
l# selec
bound# E
l# selec
free( )
Linear interaction energy (LIE)
Advantages : Drawbacks :
- Faster than free energy simulation
- More structurally different ligands than for free energy simulation. But generally restricted to rather similar ligands.
- Slower than scores based on a single conformation (LUDI, PMF, ...)
- Not really universal (' and ( system dependent)
Modifications :
- Additional term proportional to buried surface upon complexation
- Use of continuum solvent model instead of explicit solventR. Zhou and W.L. Jorgensen et al., J. Phys. Chem., 2001, 105, 10388
D.K. Jones-Hertzog and W.L. Jorgensen, J. Med. Chem., 1997, 40, 1539
- Need experimental binding affinities of known complexes
elec
ligprotE!
desolvG!
buriedSASA
Electrostatic interactions between protein and ligand
Electrostatic contribution to desolvation energy (continuum model)
Solvent accessible surface of protein and ligand buried upon complexation
New LIE-like methodV. Zoete, O. Michielin and M. Karplus, J. Comput.-Aided Molec. Design, 2004, in press
MD simulation of the complex without explicit solvent
(, ) and % fitted using a training set
!Gbind = " Eprot# ligelec
+ $ !Gdesolv +% SASAburied
New LIE-like method
Training set :
Trained with a set of 16 known inhibitors of the HIV-1 protease belonging to different chemical families
Used to rank new HIV-1 protease ligands
New LIE-like method
SD ~ 0.85 kcal/mol
Training set New ligands
Advantages : Drawbacks :
- Faster than free energy simulation LIE
- Ligands with very different chemical structures
- Slower than scores based on a single conformation- Not universal
- Need experimental binding affinities of known complexes
- Restricted to ligands uncharged when bound
-17
-15
-13
-11
-9
-17 -15 -13 -11 -9
Exp. !Gbind (kcal/mol)
Calc
. !
Gb
ind
(kcal/m
ol)
-14
-13
-12
-11
-10
-9
4 5 6 7 8 9
Exp. pIC50
Calc
. !
Gb
ind
(kcal/m
ol)
Free energy calculation: Main approaches
Free Energy Perturbation (FEP) Thermodynamical Integration (TI)
CP
U T
ime
Linear Interaction Energy (LIE) Molecular Mechanics/Poisson-Boltzmann/Surface area (MM-PBSA)
Quantitative Structure Activity Relationship (QSAR)
Non Equilibrium Statistical Mechanics (Jarzynski)
Sam
plin
g, E
xact
Sam
plin
g, A
ppro
x.
Appro
x.
G k 0 k i X i (X is a descriptor)
�
!G = F(X)
Binding free energy decomposition: MM-PBSA, MM-GBSA
Lig + Prot Lig:Prot
Lig:Prot!Gbind
Lig + Prot
Gaz
Sol
Averaged over an MD simulation trajectoryof the complex (and isolated parts)
B. Tidor and M. Karplus, J. Mol. Biol., 1994, 238, 405
Molecular mechanics – Poisson-Boltzmann Surface Area (MM- PBSA)
Molecular mechanics – Generalized Born Surface Area (MM- GBSA)
J. Srinivasan, P.A. Kollmann et al., J. Am. Chem. Soc., 1998, 120, 9401
H. Gohlke, C. Kiel and D.A. Case, J. Mol. Biol., 2003, 330, 891
Depending on the way !Gsolv,elec is calculated:
�
!Gbind = !Egaz + !Gdesolv "T !S
�
Egaz = Eelec + Evdw + !Eint ra
�
!Gdesolv = !Gsolv
comp" !Gsolv
lig+ !Gsolv
prot( )
�
!T"S = !T(Scomp
! (Sprot
+ Slig))
�
!Gsolv
lig
�
!Gsolv
prot
�
!Gsolv
comp
�
S = Strans
+ Srot
+ Svib
�
!Gsolv = !Gsolv,elec + !Gsolv,np
�
!Gdesolv = !Gsolv,elec
comp" !Gsolv,elec
lig+ !Gsolv,elec
prot( ) + # SASAcomp
" SASAlig
+ SASAprot( )( )
�
!Egaz
Exp
érim
enta
lT
héo
riq
ue
MM-GBSA Method: application to TCR-p-MHC
�
!Gbind = !Egaz + !Gdesolv "T !S
MM-GBSA Method: application to TCR-p-MHC
�
!Gbind = !Egaz + !Gdesolv "T !S
Examples of TCR optimization: 2C TCR
Improvement offavorable TCR
residues
Replacement ofunfavorable TCR
residues
Examples of TCR optimization: 2C TCR
Binding free energy decomposition
Advantages : Drawbacks :
- Used for ligand:protein and protein:protein complexes
- Could be applied to structurally different ligands (but in fact applied to similar ones)
- “Universal” (no parameter to be fitted)
- Rather time consuming
- In some cases, found unable to rank ligands
-T"S is necessary to find the order of magnitudeof the absolute binding free energies but, in some
cases, it is not necessary to estimate relative bindingfree energies
- MM-GBSA allows a per-atom decomposition of "Gbind (e.g. contribution of side chains)
MM- PBSA, MM-GBSA
H. Gohlke, C. Kiel and D.A. Case, J. Mol. Biol., 2003, 330, 891
W. Wang and P.A. Kollman, J. Mol. Biol., 2000, 303, 567
Free energy calculation: Main approaches
Free Energy Perturbation (FEP) Thermodynamical Integration (TI)
CP
U T
ime
Linear Interaction Energy (LIE) Molecular Mechanics/Poisson-Boltzmann/Surface area (MM-PBSA)
Quantitative Structure Activity Relationship (QSAR)
Non Equilibrium Statistical Mechanics (Jarzynski)
Sam
plin
g, E
xact
Sam
plin
g, A
ppro
x.
Appro
x.
G k 0 k i X i (X is a descriptor)
�
!G = F(X)
Free energy calculation: Main approaches
Free Energy Perturbation (FEP) Thermodynamical Integration (TI)
CP
U T
ime
Linear Interaction Energy (LIE) Molecular Mechanics/Poisson-Boltzmann/Surface area (MM-PBSA)
Quantitative Structure Activity Relationship (QSAR)
Non Equilibrium Statistical Mechanics (Jarzynski)
Sam
plin
g, E
xact
Sam
plin
g, A
ppro
x.
Appro
x.
G k 0 k i X i (X is a descriptor)
�
!G = F(X)
Use of thermodynamical cycles
L1 + Prot
L2 + Prot
L1:Prot
L2:Prot
"G1
"G2
"G3 "G4 ""Gbind="G2-"G1="G4-"G3
Thermodynamic cycle perturbation approach:
HN
N
O
CH2O
OH
HN
N
O
CH2O
Cl
HN
N
O
CH2O
OH Cl
( )( )i
n
i
iii
!!!"
#
=+
###=$$1
0
bind RTHHexp ln RTG
*=0 * *=1
Coupling parameter *
“Alchemical” reaction. MD or MC at different *.
Free energy perturbation (FEP) Thermodynamic integration (TI)
!=
= "
"=##
1
0bind
HG
$
$
$ $$$d
"G4-"G3 is computationally accessible
H!= H
0+ ! H
L1
+ 1" !( ) HL2
Alchemical free energy formalism
“Alchemical” Free Energy Calculations
Hybrid Side Chain for P "A Mutation
Results of the Free Energy Simulations
Total (Path Independent)
Experimental: 2.9 (0.2) kcal/molTheoretical: 2.9 (1.1) kcal/mol
Components (Path Dependent)
Experimental: -Theoretical:
TCR 25%Solvent 20%
HLA A2 40%Peptide 15%
}
}
45%
55%
Free energy formalism
From the statistical definition of the free energy,
Free energy derivative
100% Pro 100% Ala
Free energy derivative components
100% Pro 100% Ala
Solvent Contribution
Solvent contribution to ""A:
- vdW: +1.4 kcal/mol - Elec: -0.7 kcal/mol
- Total: +0.7 kcal/mol
The solvent favors association with Tax(P6)-A2 compared tothe A6 mutant that is more so-luble.
p-MHC conformational change contribution
p-MHC conformational change observed upon TCR binding is indirectly computed along the horizontal paths:
- the free energy cost of the P+A mutation is higher in the boundsimulation because of the numerous favorable interaction of P6 with HLA A2. This brings a net addition of +1.2 kcal/mol to !!A.
Combined approach for peptide design
p-MHC affinity
TCR-(p-MHC) affinity
Check on repertoire selection
Select modifications that increase MHC affinity
Concluding Remarks
1) Free energy simulations can reproduce accurately experimental changesin association constant between to closely related protein systems ifdetailed structural knowledge is available (X-ray, NMR or model)
2) The formalism is exact from a statistical physics stand point and accuratetreatment of entropic terms, solvent effect or conformational changes can be obtained
3) Convergence of the free energy derivative is still problematic. The situationshould improve with new methodological enhancements as well as longersimulation time
4) Absolute free energies can also be computed but the convergence is evenmore difficult
5) Much details about the specificity of the association can be gained usingcomponent analysis, opening the door to rational peptide or protein design
Free energy calculation: Main approaches
Free Energy Perturbation (FEP) Thermodynamical Integration (TI)
CP
U T
ime
Linear Interaction Energy (LIE) Molecular Mechanics/Poisson-Boltzmann/Surface area (MM-PBSA)
Quantitative Structure Activity Relationship (QSAR)
Non Equilibrium Statistical Mechanics (Jarzynski)
Sam
plin
g, E
xact
Sam
plin
g, A
ppro
x.
Appro
x.
G k 0 k i X i (X is a descriptor)
�
!G = F(X)
time
react
ion
Independent starting pts
(canonical ensemble)
reference trajectory
Wadia = "G KA (Infinitely slow)
W = Wadia + Wdiss (Finite rate)
Let G be the free energy and W the work,
WC
VPulld
Pull
Computation of absolute TCR binding free energy
(Jarzynsky)
�
e!"W
= e!"#G
“Proof” of the Jarzynski equation: Park & al. 2004
Suppose a system S in equilibrium with a thermal bath B at temperature T, and! composite system SB is thermally isolated! interaction energy of SB is negligible:
where & and ' denotes the phases (p,q) of S and B, resp.then
�
H!
SB(",#) = H
!
S(") + H
B(#)
�
Z!SB
= d"d#exp $%H!SB(",#)[ ]& = d"exp $%H!
S(")[ ]& d#exp $%HB
(#)[ ]& = Z!SZB
Let’s now consider a process for which the (time dependent) ( parameter goesfrom 0 to , and drives the system from (&0,'0) to (&),')). The work done during the process is
�
W = H!
SB("
#,$
#) %H
0
SB("
0,$
0)
Since SB is an isolated system, it follows a micorcanonical distribution.However, for large system, one can use the thermodynamical limit and usea canonical distribution:
�
1
Z0SBexp !"H0
SB(#0,$0)[ ]
“Proof” of the Jarzynski equation: Park & al. 2004
The expectation value for the exponential work is
�
W = H!
SB("
#,$
#) %H
0
SB("
0,$
0)
�
e!"W
= d#0$ d%0
1
Z0SBexp !"H0
SB(#0,%0)[ ] x exp !" H&
SB(#' ,%' ) !H0
SB(#0,%0)[ ]{ }
�
= d!0" d#0
1
Z0SBexp $%H&
SB(!' ,#' )[ ]
The last step consists in transforming the integration variable from (&0,'0) to (&),')). According to Liouville’s theorem (cave!) the Jacobian of thetransformation is unity and d&0d'0 = d&)d')
�
e!"W
= d#$% d&$
1
Z0SBexp !"H'
SB(#$ ,&$ )[ ] =
Z'SB
Z0SB
Using the negligible interaction between S and B (see above)
�
e!"W
=Z#SB
Z0SB
=Z#SZB
Z0SZB
=Z#S
Z0S
= exp(!"$GS)
�
e!"W
= exp(!"#GS)i.e
Simulation setup
- Gromos96 Force Field
- Gromacs Engine
- Particle Mesh Ewald (PME)
- Periodic boundary conditions
- Box: 80x80x150 A
- 26000 Water molecules
- 85000 Atoms
- Hydrogen shaken
- 2 fs timestep
- 0.5 ns / 24h on 4 alpha CPU
Absolute free energy results
Pull
Binding free energy profile:
�
e!"W
= e!#G
8.313
10.611
1924
Exp.Jarzynski
Agreement with experimental values
Free energy simulation: conclusion
Advantages : Drawbacks :
- Rigorous
- Estimates influence of small modifications
- Partitioning of the free energy (TI)
- Restricted to small mutations of ligand or protein
- Time consuming
S. Boresch et al., Proteins, 1994, 20, 25S. Boresch and M. Karplus, J. Mol. Biol., 1995, 254, 801
... )(H)(H
G1
0
angles1
0
bondsbind +
!
!+
!
!="" ##
=
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... GGG anglesbondbind +!!+!!=!!
- No parameter to be fitted- Most often: relative "Gbind