Marco CecchiniLaboratoire d’Ingénierie des Fonctions Moléculaires !

ISIS (UMR7006), Faculté de Chimie!Université de Strasbourg

Computational Approaches to Protein-Ligand Binding

Lab d’Ingénierie des Fonctions Moléculaires

Joel Montalvo-Acosta

Outline

• Protein-ligand binding!

• Computational approaches to ligand-binding affinity!

• Provide a classification of methods based on stat mech!

• Results on a host-guest system

Protein-Ligand Binding

Buch, Giorgino & De Fabritiis, PNAS (2011)

Ligand Binding Affinity

P + L ↵ PL

10

�510

�210

110

4

0

0.2

0.4

0.6

0.8

1

A

B

Initial concentration

MolarFraction

µM < nM < pM

ligand potency (Kd)

[L] = Kdwhen

[PL][PL] + [P ]

=12

Kd = 1/Keq =[P ][L][PL]

Isothermal Titration Calorimetry

Canonical Approach

Keq � �µ�b � QL, QP , QPL

exp

✓��µ

�b

kT

◆=

CPL (C�)

CP CL= KeqC

�

Chemical thermodynamics

F (N,V, T ) = �kT lnQ

µ(V, T ) =✓

@F

@N

◆

V,T

= �kT✓

@ lnQ@N

◆

V,T

Classical Statistical Mechanics

Chemical Potentials

µi,v(V, T ) = �kT ln��

2�mkTh2

� 32 V

N

�+

�kT ln��

�

�

�8�2kT

h2

� 32 �

IXIY IZ

�+

�kT3n�6�

j=1

ln��

kT

h�j

��+

�De

In the limit of RRHO/BO & in vacuum

µi(V, T ) = µi,v(V, T ) + Wbulk(X0)

McQuarrie, Statistical Mechanics, New York: Harper & Row, 1976

1. Absolute Chemical Potentials

P + L ↵ PL

µ�i (T ) = µ�i,v(T ) + Wbulk(X0)

Grand Canonical Approach

µ�i (T ) = �kT ln✓Qi(p, T )

V

◆

exp

✓��µ

�b

kT

◆=

(QPL/VPL)(QP /VP ) (QL/VL)

= Keq

Qi(p, T ) ⇡QN,1QN,0

@ infinite dilution

Hill, Cooperativity Theory in Biochemistry, Springer Verlag 1985

Keq =QN,LP /QN,P

(QN,L/QN,0)/VL

& small ligand

2. Ligand Partitioning

Lfree � Lbound

Keq =QL(P )

(QL/VL)

Rigorous Approach

Keq =

Rsite dL

RdX exp (��U)R

bulk dL � (rL � r⇤)R

dX exp (��U)

Keq =QL(P )

(QL/VL)

µi(V, T ) = µi,v(V, T ) + Wbulk(X0)1. Absolute Chemical Potentials

2. Ligand Partitioning

Double Decoupling

Jorgensen et al, J Chem Phys (1988) Gilson et al, Biophys J (1997)

Keq =

Rsite dL

RdX exp (��U1)R

bulk dL � (rL � r⇤)R

dX exp (��U0)⇥

Rbulk dL � (rL � r

⇤)

RdX exp (��U0)R

bulk dL � (rL � r⇤)R

dX exp (��U1)

Keq =QL(P )

(QL/VL)

Solution

Gas phase

Alchemical RouteKeq = QL(P )(QL/VL)

Solution

Gas phase

�F = �kT ln�e��(U1�U0)

�

U0FEP

…with Restraints

Gumbart, Chipot & Roux, J Chem Theory Comput (2013)

Keq =�

site dL�

dX exp (��U1)�site dL

�dX exp [�� (U1 + uc)]

��

site dL�

dX exp [�� (U1 + uc)]�site dL

�dX exp [�� (U1 + uc + uo)]

��

site dL�

dX exp [�� (U1 + uc + uo)]�site dL

�dX exp [�� (U1 + uc + uo + up)]

��

site dL�

dX exp [�� (U1 + uc + uo + up)]�bulk dL � (rL � r�)

�dX exp [�� (U0 + uc + uo + up)]

��

bulk dL � (rL � r�)

�dX exp [�� (U0 + uc + uo + up)]�

bulk dL � (rL � r�)�

dX exp [�� (U0 + uc + uo)]�

�bulk dL � (rL � r

�)�

dX exp [�� (U0 + uc + uo)]]�bulk dL � (rL � r�)

�dX exp [�� (U0 + uc)]

��

bulk dL � (rL � r�)

�dX exp [�� (U0 + uc)]]�

bulk dL � (rL � r�)�

dX exp [�� (U1 + uc)]�

�bulk dL � (rL � r

�)�

dX exp [�� (U1 + uc)]]�bulk dL � (rL � r�)

�dX exp [�� (U1)]

Uc, Uo, Up!conformational, orientational &

positional restraints

2 alchemical transformations

Rigorous Methods

End-points Approach

MM/PBSA

Kollman et al, Acc Chem Res (2000)

Solution

Gas phase

µi(V, T ) = µi,v(V, T ) + Wbulk(X0)

MM/PBSAµi(V, T ) = µi,v(V, T ) + Wbulk(X0)

MM/PBSA assumptions

By separating out enthalpy vs entropy contributionsµi(V, T ) = (3n� 3) kT �De,v � TSi(V ) + Wsolv

Wsolv � GPBSA1.�De,v = hUi �

X 12kT2.

µi(V, T ) =32nikT + hUi + hGPBSAi � TSi(V )

implicit solvent

Force Field

MM/PBSA (HOWTO)

µi

(V, T ) = 32nikT + Ebond + Eelec + EvdW + Gpol + Gnp � TSi(V )

•Run explicit-water MD for P, L & PL!•Extract sampling & compute ensemble

averages of both the internal energy in a vacuum & the solvation free energy!

•Evaluate the configurational entropy at standard state (1M) by statistical mechanics formulas & NMA

Linear Interaction Energy (LIE)

Åqvist, Medina & Samuelsson, Protein Eng (1994)

Solution

Gas phaseKeq =

QL(P )(QL/VL)

LIEKeq =QL(P )

(QL/VL)

�µ�b = �kT log ⇣ + Wsite �Wbulk

q = qtr

qrot

qvib

qelec

In the limit of RRHO & rigid ligands:

Keq =qCM

qrock

e��Wsite

(qtr

/V ) qrot

e��Wbulk

�µ�b =12

hhUelecl/s isite � hUelecl/s ibulk

i+ ↵

hhUvdwl/s isite � hUvdwl/s ibulk

i+ �

Wnpi ⇡ ↵DUvdWl/s

EW pol

i

=12

DUelec

l/s

ELIE’s assumptions

1. 2.

LIE (HOWTO)

• Run two explicit-water MD for L & PL!• Compute ensemble averages of the

electrostatic & the Van Der Waals interaction energy of the ligand with the surroundings!

• Assign appropriate β for your ligand(s)!• Determine α,γ by fitting on experiments

�µ�b = �hhUelecl/s isite � hUelecl/s ibulk

i+ ↵

hhUvdwl/s isite � hUvdwl/s ibulk

i+ �

Guitiérrez-de-Terán & Åqvist, Computational Drug Discovery and Design (2012)

End-points Approach

Empirical Approach

virtual High-Throughput Screening (vHTS)

Molecular DockingBUT to be fast…!•Bound State!•Rigid Receptor!•Flexible Ligand!•No/Implicit solvent!•Ligand Fitness by a

Scoring Function

Force-Field Scoring

�µ�b

=protX

i

ligX

j

A

ij

r12ij

� Bijr6ij

+ 332.0qi

qj

✏ (rij

) rij

!DOCK score:

�µ�b

=h⇣

Uelecl/s

⌘

site

� 0i

+h⇣

Uvdwl/s

⌘

site

� 0i

=protX

i

ligX

j

A

ij

r12ij

� Bijr6ij

+qi

qj

✏ (rij

) rij

!

FF’s assumptions h. . . i = (. . . )⇣U

elec

l/s

⌘

bulk=

⇣U

vdw

l/s

⌘

bulk= 0

NO sampling

NO desolvation

Empirical Scoring�µ�b =X

i

Wi�µi

�µ�b = �G0 + �Ghb�

hbonds

f (�R,��) +

�Gio�

io int.

f (�R,��) +

�Glipo�

lipo cont.

Alipo +

�Garo�

aro int.

f (�R) + �Grot �Nrot

Böhm’s score:

FRESNO’s score:�µ�b = K + ↵ (HB) + � (LIPO) + � (ROT ) + � (BP ) + ✏ (DESOLV )

Montalvo-Acosta & Cecchini, Mol Inform (2016)

Summary

Host-Guest SystemCucurbit-[7]-uril (CB7)

−14.1

−11.8

−13.4

Moghaddam et al, J Am Chem Soc (2011)

AUTODOCK

LIEMM/PBSA

FEP

Benchmark

Keq =�

site dL�

dX exp (��U1)�site dL

�dX exp [�� (U1 + uc)]

��

site dL�

dX exp [�� (U1 + uc)]�site dL

�dX exp [�� (U1 + uc + uo)]

��

site dL�

dX exp [�� (U1 + uc + uo)]�site dL

�dX exp [�� (U1 + uc + uo + up)]

��

site dL�

dX exp [�� (U1 + uc + uo + up)]�bulk dL � (rL � r�)

�dX exp [�� (U0 + uc + uo + up)]

��

bulk dL � (rL � r�)

�dX exp [�� (U0 + uc + uo + up)]�

bulk dL � (rL � r�)�

dX exp [�� (U0 + uc + uo)]�

�bulk dL � (rL � r

�)�

dX exp [�� (U0 + uc + uo)]]�bulk dL � (rL � r�)

�dX exp [�� (U0 + uc)]

��

bulk dL � (rL � r�)

�dX exp [�� (U0 + uc)]]�

bulk dL � (rL � r�)�

dX exp [�� (U1 + uc)]�

�bulk dL � (rL � r

�)�

dX exp [�� (U1 + uc)]]�bulk dL � (rL � r�)

�dX exp [�� (U1)]

�µ�b =12

��Uelecl/s �site � �Uelecl/s �bulk

�+

���Uvdwl/s �site � �Uvdwl/s �bulk

�

�µ�b = Wvdwprot�

i

lig�

j

�Aijr12ij

� Bijr6ij

�+

Wele

prot�

i

lig�

j

�qiqj

� (rij) rij

�+

Whbond

prot�

i

lig�

j

�E (t)

Cijr12ij

� Dijr10ij

�+

Wsol

prot�

i

lig�

j

(SiVj + SjVi) exp

��r2ij2�2

�

µi(V, T ) =32nikT + hUi + hGPBSAi � TSi(V )

Binding Free Energy

-16

-14

-12

-10

-8

-6

-4

-16 -14 -12 -10 -8 -6 -4

Exp

erim

enta

l [kc

al/m

ol]

Calculated [kcal/mol]

FEPMM/PBSA

LIEAUTODOCK

Accuracy/Efficiency y = 0.22 3p

x

Take Home MessageOur statistical mechanics interpretation of protein-ligand binding:!• provides a useful classification of existing

methods to the binding constant!• highlights their inherent approximations and

provides guidelines for future development!• has allowed to quantify their performances

(accuracy/efficiency) on a model host-guest system

Montalvo-Acosta & Cecchini, Mol Inform (2016)

Acknowledgments

Joel Montalvo-Acosta!!

Nicolas Muzet (Sanofi)!Michael Schaefer (Novartis)

• Explicit-water MD simulations for three host-guest systems!

• Initial coordinates for the complexes obtained from the Cambridge Structural Database!

•Dodecahedron box with a layer of >14Å around the solute !

•Force-Field parameters for the guests by CGenFF!

•NPT @ 298K & 1atm!•Production runs of 20 ns with

GROMACS 5.1 compiled with PLUMED plugin

5073 atoms Simulation Setup