free energy simulations: theory and...

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)

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


- 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



CP

U T

ime




Sam

plin

g, E

xact

Sam

plin

g, A

ppro

x.

Appro

x.


�

!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)


- 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


- 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)



CP

U T

ime




Sam

plin

g, E

xact

Sam

plin

g, A

ppro

x.

Appro

x.


�

!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

�


MM-GBSA Method: application to TCR-p-MHC

�


Examples of TCR optimization: 2C TCR

Improvement offavorable TCR

residues

Replacement ofunfavorable TCR

residues

Examples of TCR optimization: 2C TCR

Binding free energy decomposition


- 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



CP

U T

ime




Sam

plin

g, E

xact

Sam

plin

g, A

ppro

x.

Appro

x.


�

!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



CP

U T

ime




Sam

plin

g, E

xact

Sam

plin

g, A

ppro

x.

Appro

x.


�

!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


- 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 +

!

!+

!

!="" ##

=

=

=

=

$$$

$$

$$

$ $

$

$

$dd

... GGG anglesbondbind +!!+!!=!!

- No parameter to be fitted- Most often: relative "Gbind

free energy simulations: theory and...

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