Thermodynamical Stability of Biomolecular Systems:

Insights from Molecular Dynamics Free Energy Simulations and Continuum Electrostatics

Georgios ArchontisDepartment of Physics, University of Cyprus

Overview

• Relative Binding Strength to Glycogen Phosphorylase of Catalytic-site inhibitors,by MDFE calculations.

• Dinucleotide inhibitor binding to Ribonuclease A by MD/PB calculations.

• A continuum electrostatics/linear response method to calculate pKa shifts in proteins.

Relative Strength of Binding of Ligands in the Catalytic Site of

Glycogen Phosphorylase,by MDFE calculations

P

P

Gpa, active Gpb, inactive

Gpb, activeGpa, inactive

Kinase

Phosphatase

Glucose Glucose-6-PAMP

Regulation of Glycogen Phosphorylase (simplified)

Interactions of Glucose with Catalytic Site

Interactions of Hydantoin with Catalytic Site

The Simulation System

R

R = H HYDANR = CH3 MHYDANR = NH2 NHYDAN

Ligand Structure

Technical Details

• Dual topology.• Linear Dependence of Hamiltonian on λ.• Fitting of van der Waals Free Energy

Derivatives to a power law ~λ-3/4 at end points.

Summary of Free Energy Runs

Run # DG vw elecProteinHM 1 10.5 8.7 1.8

2 8.0 6.5 1.53 7.4 5.7 1.7

Average 8.6 ± 2.0

SolutionHM 1 4.9 3.8 1.1

2 4.8 3.7 1.1Average 4.85 ± 0.05ΔΔG 3.75 Exp: 3.6

Summary of Free Energy Runs

Run # DG vw elecProteinHN 1 1.5 4.6 -3.1

2 1.6 3.2 -1.6HH’N 1.75 1.1 0.55Average 1.65 ± 0.95

SolutionHN 1 0.5 2.0 -1.5

2 0.8 2.3 -1.5Average 0.65 ± 0.15ΔΔG 1.0 Exp: 2.3

Left: Interaction of the HYDAN ring with the catalytic site residues, observed in the simulations.

D283N284

E88G134

W200W176

N1

H377

H377

Right: Typical MD snapshot of the M-HYDAN: GP complex. W176 ismost of the time displaced by ~1 A and interacts with E88.

Interactions of GP with HYDAN/MHYDAN

H377

G134 E88

W176

W200

D283N284

N1

M-HYDAN

D283W176

E88

G134

N284

H377

N-HYDANN1

Right: Typical MD snapshot of the N-HYDAN:GP complex. As in M-Hydan, W176 is most of the time displaced by ~1 A and interacts with E88.

Interactions of GP with N-HYDAN

Van der Waals Free Energy derivatives

H M

H M

λ=0.06

Log(dF/dλ)

a) H M b) H N Components: vw elec vw elec Runs: 1 2 3 1 2 3 1 2 1 2------------------------------------------------------------------------------------------------D283 8.0 8.0 6.6 2.6 2.5 2.5 6.0 4.7 0.2 0.4W176 4.3 3.2 2.8 0.3 0.1 0.1 2.2 2.2 0.8 0.4W200 0.6 1.1 0.7 -0.1 -0.1 0.0 0.3 0.7 -0.7 -0.5N284 -0.4 -0.8 -0.4 0.3 0.1 0.1 -0.7 -0.5 0.1 0.2 G135 -0.4 -0.6 -0.7 -0.1 0.1 -0.1 -0.8 -0.6 0.0 0.1----------------------------------------------------------------------------------------------------Sum: 8.7 5.8 5.7 1.8 1.5 1.7 4.6 3.2 -3.1 -1.6a) H M: Residue D283 opposes the creation of the M methyl group, due to steric interference and loss of electrostatic interactions, as shown by the large vw and elec components. Water W176 also opposes the transformation, mainly due to steric repulsion. Other, more distant residues favor the M state mainly due to improveddispersion interactions. b) H N: Replacement of the HYDAN proton by the NH2 group introduces steric interference mainly with D283 and W176. Electrostatic interactions are improved, but mainly with distant residues.

Residue Free Energy Decomposition of G ( in protein )

Superposition of MD structures of the various ligands in the GP Catalytic site

Free Energy Profile for W176 (Asp283 Glu88)

D283 E88

HYDAN

MHYDAN

HYDAN MHYDAN

Conclusions

• The experimental relative binding order of hydantoin inhibitors is reproduced by MDFE calculations.

• Interactions between HYDAN and GP or water residues in the catalytic site are optimal.

• Introduction of CH3 or NH2 causes steric interference , that mostly accounts for the reduced binding strength of M- and NHYDAN relative to the best inhibitor (HYDAN).

Free Energies of Binding between Ribonuclease A and dinucleotide inhibitors, evaluated by MD/PB

schemes

Recognition Subsites in the RNAse A Active Site

Putative catalysis mechanism by RNAseARaines & co/rs, Biochem. 40:4949 (2001)

RNAse inhibitors considered in this work

dUppA (deoxyuridine-3-pyropho-sphate (P->5) adenosine)

pdUppAp (5΄-phospho-3΄-deoxyuridine 3-pyrophosphate (P->5) adenosine 3-phosphate)

PD

PB PAPG

PB PA

Typical interactions between pdUppAp and surrounding protein residues, observed in the MD simulations

PD

PBPA PGHIS12

HIS119

GLN11 LYS7

Typical interactions between dUppA and surrounding protein residues, observed in the MD simulations

PB PA

HIS12

HIS119

LYS7GLN11

RNAse complex with pdUppAp

In a standard PB binding calculation, the protein and ligand are assumed to have the same conformation in the biomolecul-ar complex (left) and in the infinitely separated, solvated states (right). The binding free energy is obtained by

G*bind = G*

pl – G*p – G*

l

+

G*p + G*

lG*pl

Standard scheme to calculate Binding Free Energies by PB

P* L* P* L*

Binding Free Energies evaluated by the standard scheme(a, b).

L i g a n dεprot pdUppAp(c) dUppA (c) Diff (d) ----------------------------------------------------------------

1.0 -39.6 -23.3 +16.3 2.0 -28.1 -17.5 +10.6 4.0 -22.1 -14.3 +7.8

8.0 -18.7 -12.3 +6.4 16.0 -16.6 -10.8 +5.820.0 -16.0 -10.4 +5.6

----------------------------------------------------------------(a) Experimental Km values are 27 nM (dpUppAp) and 11.3 μM (dUppA),corresponding to a ΔΔG = 3.6 Kcal/mol. (b) All energies in Kcal/mol;(c) Binding energies averaged over 200 structures taken from a 4-ns simulation. (d) Binding free energy relative to the RNAse:pdUppAp complex.

At values of εprot used typically in PB binding calculations (2.0 - 4.0), dUppA is predicted to bind more weakly by 10.6 - 4.9 Kcal/mol.

Expanded scheme to calculate Binding Free Energies by PB

Upon dissociation, thesystem is allowed to relaxto equilibrium structuresof the separated states.(step 2).

+P L

Gbind = G*bind +G2

+1

2

P* L* P* L*

G*bind

G2

Thermodynamic cycle used to calculate PB binding energies in the

expanded scheme

Infinite HomogeneousMedium (ε solv=εprot=ε)

ΔGbind = 1 + 5 = 1 + 2 + 3 + 4

P*:L1* P* + L1* P + L1

P* + L1* P + L1

1

2

3

4

5

Standard scheme

Solution(εsolv=80;εprot=ε).

Binding Free Energies evaluated by the expanded scheme(a).

L i g a n dεprot pdUppAp (b) dUppA (b) Diff (c)

----------------------------------------------------------------- 1.0 -3456.2 -3458.2 -2.0 2.0 -1716.7 -1715.2 +1.5 4.0 -847.8 -844.2 +3.6 8.0 -414.0 -409.6 +4.4

16.0 -198.2 -193.2 +5.0 20.0 -155.7 -150.4 +5.3 -----------------------------------------------------------------

(a) Experimental Km values are 27 nM (pdUppAp) and 11.3 μM (dUppA),corresponding to a ΔΔG = 3.6 Kcal/mol. (b) Binding energies averaged over 200 structures taken from a 4-ns simulation; (c) Binding free energy relative to the RNAse:pdUppAp complex.At values of εprot used typically in PB binding calculations (2.0-4.0), dUppA is predicted to bind more weakly by 1.5 – 3.6 Kcal/mol.

Conclusions

• PB/MD calculations, averaged over multi-ns trajectories can predict correctly the order of binding of dinucleotide inhibitors to RNAse.

• Accuracy is improved when the relaxation of protein and ligands after dissociation is included.

A method to calculate pKa shifts in proteins.

Calculations with Continuum Electrostatics and Linear Response

Thermodynamic Cycle for pKa shift

P-RH P-R- + H+

R- + H+RH

(1)

(2)

ΔΔG = ΔG1 – ΔG2

pK a, prot – pK a, model = 1/(2.303 kBT)*ΔΔG

Calculation of Ionization Free Energies

Two-step Procedure to Calculate Charge Insertion Free Energies (Marcus, 1956)

I : Insert q with environment fixed: ΔGx, stat = q Vx q

II: Allow environment to relax: ΔGx, stat = ½ q Vx, q q

q q

I II

“Reactant” State “Product” State

Gstatic Grelax

Static Free Energy

ΔGx, stat = q Vx q

(Average Interaction Energy between q and restof the system, at equilibrium state x)

Relaxation Free Energy

ΔGx, stat = ½ q Vx, q q

(Born self-energy of inserted charge)

q q

I II

Greac, static Greac, relax

ΔG = ΔGreac, stat+ ΔGreac, relax (1)

q

II I

ΔG’ = -ΔG = ΔGprod, stat+ ΔGprod, relax (2)

Gprod, relax Gprod, static

2ΔG = (ΔGreac, stat- ΔGprod, stat) + (ΔGreac, relax - ΔGprod, relax)

= q(Vreac, stat - Vprod, stat)

ΔG = ΔGreac, stat+ ΔGreac, relax (1)

-ΔG = ΔGprod, stat+ ΔGprod, relax (2)

(Linear Response)= 0

ΔG = q/2 (Vreac, stat - Vprod, stat)=1/2 (ΔGreac, stat- ΔGprod, stat)

Average Electrostatic Energy over reactant and product

Application

• Thioredoxin Asp26 (pKa = 7.5; ΔΔGexp = 4.8)

• Thioredoxin Asp20 (pKa = 4.0; ΔΔGexp = 0)

• RNAse A Asp14 (pKa = 2.0; ΔΔGexp = -2.7)

A. Αsp26 (Thioredoxin) # of waters ΔΔG 0 1 MDFE (CHARMM)½(ASPH+ASP) 4.7 5.1 10.9¼ (ASPH+2*Mid+ASP) 10.2 4.8 Exp: 4.8 B. Asp14 (Rnase A)½(ASPH+ASP) -3.9 -- 1.1¼ (ASPH+2*Mid+ASP) -3.8 Exp: -2.7 • Averages over 100-200 structures, spanning ~ 4-6 ns• Protein dielectric constant ε = 1

Static Step Free Energies A. Αsp26 (Thioredoxin)

# of explicit waters State 0 1 ASPH -28.7 -6.9ASP -95.5 -116.6Midpoint -50.5 -61.8½(ASPH+ASP) -62.1 -61.7¼ (ASPH+2*Mid+ASP) -56.3 -61.8

B. Model CompoundASPH 6.0ASP(-) -139.6Midpoint -66.4½(ASPH+ASP) -66.8¼ (ASPH+2*Mid+ASP) -66.5

C. Αsp20 (Thioredoxin) ΔΔG MDFE (CHARMM)

½ ( reac + prod ) 5.5 0.9

Midpoint 5.9

¼ (reac + 2*midpoint + prod) 5.7 Exp: 0.0

•Continuum Assumption.•Equilibrium structures consistent with MD force-field.

Conclusions

• PKa shifts can be calculated with a continuum electrostatics/LR method.

• Averaging over end-states accounts for structural reorganization upon titration.

• Optimal Protein dielectric constant for static calculation ε ~1.

• Relaxation Free Energies require a different dielectric constant (optimal values, ε ~2-6).

Acknowledgements

• Glycogen Phosphorylase– Dr. Qian Xie (MDFE, University of Cyprus). – Dr. Nikos Oikonomakos (x-ray, NHRF, Greece).

• Ribonuclease A– Dr. Demetres Leonidas (x-ray, NHRF, Greece).

• pKa work– Thomas Simonson (MDFE/CHARMM, Ecole

Polytechnique).• University of Cyprus ($).