Relative Binding Free Energies for Protein-Inhibitor Complexes

Relative Binding Free Energies for Protein-Inhibitor Complexes

CATFEE: Critical Assessment of Techniques for Free Energy Evaluation

Blind test of 10 compounds. Determine relative affinities to Factor Xa, Thrombin and Trypsin.

Our Interest: Test protocols and convergence

Journal of Computer-Aided Molecular Design 17: 673–686, 2003.

Alessandra Villa, Ronen Zangi, Gilles Pieffet

1 3 4 5 6

Inhibitor 2

7 8

9 10

Inhibitors

Trypsin Template

PDB entry 1QB1Acta Cryst. D (1999) D55 1395-1404

2,6-diphenoxypyridineligand

Factor Xa Template

Removed from simulations

2,6-diphenoxypyridineligand

PDB entry 1FJS Biochemistry (2000) 39, 12534-42

Integration Formula(work along reversible path)

Methods to Compute Free Energies

dH

FB

A

AB

Simulate at fixed and integrate numerically

1 432

H

Requirements (at each )•Equilibrium.•Ensemble average must converge.•F() must be a smooth function.

Soft core potential: Addition of distance to shift the potential


In principle each point is independent

I3 to I6

I3 to I4


NH

H

H

H

N

N OO

O

F F

OH

R2

+

R1

OO

OH

-

Cl

OO

OO

OCH3CH3O

-CH3CH2

3OO

CH3CH3

2

4

6

1

CH3CH2

2

N

N

CH37

H

H

N H

HN

H

N+

N

H9

N

H10

8

N

N

CH3

N

N

H

N

HN

NH

Cl

Cycle Closure: water (kJ/mol)

-277.89.7

52.6

268.4

-321.5

83.0

238.8-141.8

-97.9

-28.6

2.8

61.0 -65.9

cycles

-0.5

-3.4

-0.9

3.4

-2.1


NH

H

H

H

N

N OO

O

F F

OH

R2

+

R1

OO

OH

-

Cl

OO

OO

OCH3CH3O

-CH3CH2

3OO

CH3CH3

2

4

6

1

CH3CH2

2

N

N

CH37

H

H

N H

HN

H

N+

N

H9

N

H10

8

N

N

CH3

N

N

H

N

HN

NH

Cl

Cycle Closure: Factor Xa (kJ/mol)

-281.917.4

57.9

252.4

-304.3

74.2

245.7-127.2

-97.9

-36.5

2.1

72.3-71.2

cycles

6.0

12.1

5.3

-5.5

3.2

6, 6*

5, 5*

4, 4*

1, 1*

Asymmetric Ligands

hindered rotation in protein

Treat rotational isomers as independent compounds

+

-

+

+

3

6

4

1

1*

2 7

3*

6*

5 5*

9

8

10

74.2

Asymmetric Ligands: Factor Xa

-56.9

-307.3

-304.357.9

252.4

269.5

?

Preferredconformers in

factor Xa


G = Gfactor Xa - Gwater

Inhibitor 1 2 3 4 5 6 7 8 9 10G kJmol-1 0.0 8.8 15.1 7.4 ~ 9.8 25.9 11.2 18.0 18.7

Ranking 1 >> 4 ~ 2 > 6 ~ 8 > 15 > 9 > 10 >> 7

Experimental data on the compounds investigated not yet released!.

Comparison to Experiment

Summary

•Converged and reproducible results for mutations > 10 atoms (soft core potentials). •Large numbers of mutations possible (cluster computing).•Initial screening? (extrapolation approaches)

Estimating changes in the monomer-dimer equilibrium of

SUC1 upon mutationDissociation free-energy

calculated using thermodynamic integration

Gilles Pieffet

SUC1: Structure

• Protein of 113 residues

andfold– 4 stranded sheets– 3 short helices

• Domain swapping of the dimer:

• C-terminal strand

Crystallographic structure of the monomer Crystallographic structure of the dimer

Gilles Pieffet

Experiment

D M + MKd

Gdiss = - RT ln KdKd

GDM = - RT ln ( )

Kd

mutant

WT

=Gdiss (M) - Gdiss (WT)

Look at difference in disassociation constant upon mutation(experimental data not without question)

Gilles Pieffet

Thermodynamic cycle

D (WT) M (WT) + M (WT)

D (M) M (M) + M (M)

Gdiss (WT)

Gdiss (M)

GWT M (mono)GWT M (dimer)

GDM = Gdiss (M) - Gdiss (WT)

= - GWT M (dimer) + 2 GWT M (mono)

Gilles Pieffet

Simulation parameters

• Gromos96 forcefield• time step of 2 fs• T = 323 K• T and P coupling• Twin range cut-off of 0.9 and 1.4 nm• Reaction-Field 78)• Equilibration: 100 ps with position restraint

10 ns without position restraint• Free-energy calculation:

– relaxation: 100 ps for each lambda point – data collection:

• 400 ps for the monomer• 200 ps for the dimer

18 points are used for the integration Gilles Pieffet

Mutations studied

Mutation GDM

(kJ /mol)Mutation GDM

(kJ /mol)Mutation GDM

(kJ /mol)LA10 -5.2 LA95 0.4 KA49 -3.5

LA18 -2.4 LA96 2.7 KA98 3.6

LA43 1.4 VA41 -2.7 EA86 -7.0

LA48 2.9 VA87 -0.2

LA63 -0.2 VA89 -2.3

LA74 -4.4 YA38 5.3

All mutations correspond to the transformation of a residue into an alanine.

GDM < 0 Dimer of WT is more stable than dimer in mutant

GDM > 0 Dimer of WT is less stable than dimer in mutant Gilles Pieffet

Monomer: G = 11.6 kJ/mol Sim: G = -2.2 kJ/mol

dimer: G = 25.4 kJ/mol Exp: G = -2.4 kJ/mol Gilles Pieffet

Monomer: G = -5.2 kJ/mol Sim: G = -27.5 kJ/mol


Results

Mutation LA10 LA18 LA43 LA48 LA63 LA74 LA95 LA96 VA41 VA87 VA89 YA38 KA49 KA98 EA86

Mono 2.9 11.6 9.4 5.7 14.6 11.8 4.6 13.9 1.5 -1.4 -5.2 79.5 193.1 216.3 276.6

Dimer 21.6 25.4 28.7 26.8 21.1 13.9 28.1 12.6 8.1 -7.7 17.1 153.3 381.3 450.6 586.9

Simul. -15.8 -2.2 -9.9 15.4 8.1 9.7 -18.9 15.2 -5.1 4.9 -27.5 5.7 4.9 -18 -33.7

Exper. -5.2 -2.4 1.4 2.9 -0.2 -4.4 0.4 2.7 -2.7 -0.2 -2.3 5.3 -3.5 3.6 -7.0

Gilles Pieffet

Relative stability of the wild type dimer with respect to some mutants (kJ/mol).

Case of the LA95 mutation

Monomer

Gilles Pieffet

Case of the LA95 mutation

Dimer

Results

Mutation Monomer Dimer Simulation experiment

LA95 4.6 28.1 -18.9 0.4

LA95-rev -0.4 3.4 -4.2 0.4

LA95-rand -3.5 24.2 -31.2 0.4Gilles Pieffet


dimer: G = 3.4 kJ/mol Exp: G = 0.4 kJ/mol Gilles Pieffet

Monomer: G = -3.5 kJ/mol Sim: G = -31.2 kJ/mol

dimer: G = 24.2 kJ/mol Exp: G = 0.4 kJ/mol Gilles Pieffet

Divergence for specific values:

= 0.40, 0.45, 0.50, 0.55 for the monomer

= 0.30, 0.40, 0.45, 0.50 for the dimer Gilles Pieffet

Gilles Pieffet

Results

Mutation Monomer 6ns per value

Dimer Simulation experiment

LA95 3.7 (4.6) 27.6(28.1) -20.2 (-18.9) 0.4

LA95-rev 1.0 (-0.4) 3.0 (3.4) -1.0 (-4.2) 0.4

LA95-rand 0.0 (-3.5) 24.1(24.2) -24.1 (-31.2) 0.4

Gilles Pieffet

Conclusions

• no simple mutations when it comes to proteins

•sampling on a multi ns timescale needed to get convergence due to protein fluctuations.

•Not possible to tell if sampling/force field/structural problems.

Gilles Pieffet

Incorporating the effect of ionic strength in free-energy calculations

using explicit ionsSerena Donnini and Alessandra Villa

Different protocols: ignore ions (ions independent, water high dielectric)neutralize the system (neutral system is more natural)add lots of ions

Also: physiological ionic strength 0.1-0.2 molar.different claims concerning the creation of a net charge(create counter charge?)

Generally ambiguous.

Why worry?

Incorporation of explicit ions in free energy calculations

Serena Donnini and Alessandra Villa

Cancellation of effects within a thermodynamic cycle?

Protein charge = +2Ligand charge = -2

Should one incorporate ions in the unlighted state?

Will effects cancel?

Consider a very simple mutation:

1. Only a change in dipole.

2. No change in number or atoms or net charge

3. Atoms partly buried.



Consider the same mutation in:

1. a charged molecule.2. a neutral molecule

Ionic environment

1. no ions2. just enough to neutralize charged system3. 0.04M ionic strength4. 0.1M ionic strength5. 0.2 M ionic strength

Thermodynamic integration18 values

simulate at fixed integrate numerically

B

A

BA dV

G


Mutation of 2-phosphoglycolic acid (PGA) to 3-phosphonopropanoic acid (3PP)(triosephosphate isomerase inhibitors)

pH ~7.0

pH ~2.0

Bonded parameters

Mutation

2-phosphoglycolic acid (PGA)

to

3-phosphonopropanoic acid (3PP)


charged form 2-

some difference to neutral

neutral form not much effect

What did I expect?

+ 2 Na+ similar to neutral form more ions less effect

+

+

Internal terms irrelevant

Reaction field of the solvent

Free energies are in kJ mol-1. Ionic strengths are in M.

Uncharged Species Charged SpeciesIonic

Strength



Slight trend

Charged Species

Uncharged Species


200 ps sampling at each value

200 ps sampling at each value



No. of ions > 1 in close proximity negligible effect.


Average lifetime each ion ~ 7ns

need 100’s ns at each lambda value


no ions in close proximity

minimum distanceclosest ion

Conclusions

• Experimentally the effect of the ionic strength on free energy differences is not

expected to be very large.

•Close proximity of ions has major effect even for mutations that do not involve

change in net charge.

•Inclusion of explicit ions can lead to severe sampling problems.

Options:1. No inclusion of ions and accept errors associated with an overall charged system.2. Perform simulations at high ionic strength to ensure sampling of ionic distribution.

Calculating free energies from non-equilibrium work?

Free Energy Calculations for Dummies

Two methods to estimate the difference in free energy between two states of a system A and B.

free energy = work to go from A to B via a reversible path

Method 1.

Method 2.

free energy = -kT ln [equilibrium probability (partition) function]

Method 1. Reversible work

work = force x distance

force = derivative of a potential w.r.t. coordinate

V

forceaverage force w.r.t. arbitrary coordinate

B

A

BA dV

G

thermodynamic integration

equilibrium ensemble

Method 2. Probability function

+

free

boundK

Relative probability of finding the system in 1 of 2 states.

dekTFB

A

kTHHAB /)()(lnPerturbation Formula

for an equilibrium ensemble

KkTF ln

Non-Equilibrium Simulation

If not in equilibrium do work against the environment -> overestimate free energy

In a dissipative system wA->B FA->B

kTwBA

BAekTG /ln But now I am told

No need to worry about intermediate statesno need to worry about equilibrium

Does not fit to either of the two general methods

Has been described as “remarkable”, “amazing”, “unexpected”.

Can I think of a trivial test case to convince myself it must be correct?

Grow Particle

Work to slowly grow particle

System in always in equilibrium

B

A

BA dV

G

Particle Insertion

kTVexcess ekT /ln

Energy of adding a particle weighted by probability of finding an appropriate

location.

(in an equilibrium ensemble)

works well in low density systems but.....

... also works in principle for a ship in the ocean

Particle deletion

Can not have sampled statesappropriate to the

ensemble were there isno particle

If particle insertion is good for low density systemswhen does particle deletion work?

NEVER

for systems with excluded volume

... also fails boat test.

water and boat cannot occupy the same space at the same time.

A BGA->B = 0

Move particle from location A to location B in box of water.

Very simple test case for non-equilibrium pullingin a dissipative environment.

A B

GA->B = 0

Move particle from A to B

Method 1. Reversible work

a. Very slowly pull particle from A to B (system always in equilibrium).

b. Determine average force.

x

B

ABA dV

G

x

0force average

V

B

A

BA dV

G

0

trivial

A B

GA->B = 0


Method 2. Probability function

a. Particle insertion at A and particle insertion at Bb. Probability of finding particle at A or B

x

0 ABBAG

trivial

kTVexcess ekT /ln

A B

GA->B = 0


Non-equilibrium pulling

x

0ln /

kTwBA

BAekTG

1/ kTw BAemany paths w > 0some paths w = 0must find paths for which w < 0

longer the pathfaster I pull

more likely w > 0

must find path with w <<0

A B

GA->B = 0



x

0ln /

kTwBA

BAekTG

What about instantaneous move (passing through intermediates)

? 0ln /

kTVBA

BAekTG

Doesn't work

particle deletion at Aparticle insertion at B

average never converges

A B

GA->B = 0



x

0ln /

kTwBA

BAekTG

Do any paths give w = 0

Yes

but rare

A B

GA->B = 0



x

0ln /

kTwBA

BAekTG

Which if any paths give w < 0

Particle must be pushed from A to

B by the environment

The boat test

If I pull my boat from A to B often enough one day the ocean will feel sorry for me and push me for free.

I gain energy and disprove the 2nd law of thermodynamics.

What to do

In a dissipative system wA->B FA->B

kTwBA

BAekTF /ln is heavily weighted towards the lowest value of wA->B

However the best estimate of FA->B (excluding statistical fluctuations) is simply the lowest value rather than the exponential average.

Round Table 2: When are free energy calculations useful?

1. For academia

2. Industry (drug design, prediction of properties)

I am only ever interested in things that don’t work

Computational methods only have to be fast when they do not really work

Fast free energy methods? Extrapolation approaches?

Force fieldGromos 43a2

Villa et al. J.Comp.Chem. (2002) 23 p.548

For other force fields see: J.Comp.Chem. (2003) 24 p 1930 J.Chem. Phys. (2003) 119 p. 5740

Parameterization of force fields

Explaining things in detail: Binding Benzamidine inhibitors to trypsin

NH2

NH2

R

R=Me, Et, n-Pr, i-Pr, n-Bu, t-Bu, n-Pent, n-Hex

Cl-

Explaining things in detail: Binding Benzamidine inhibitors to trypsin

Actually we do well

1. Vijay is doing free energy calculations as I would like to do. (except for using perturbation approaches)

2. Conformational preferences of peptides.

3. Prediction of binding conformations of ligands. (first step toward predicting energies)

Recurring trends

Relative Binding Free Energies for Protein-Inhibitor Complexes

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