Binding Free Energy Calculation Using Molecular Simulations

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Binding Free Energy Calculation Using Molecular
Simulations
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Overview

• Description of the free energy problem
• Methods to Compute Free Energies
• Coupling parameter approach
• Free energy components method
• Configurational entropy from MD trajectories
• Application in the analysis of Protein-Ligand and
  Protein-DNA Binding Free Energies

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Classical Statistical Mechanics
Hamiltonian
Probability
Partition function
Free Energy
Probability of sampling states contributing
little to F ? High! Probability of sampling
states contributing a lot to F ? Low!
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Free Energy Measurements

• Experimentally only the Relative Free Energies
  are measured

Molecular Mechanics ? Free Energy
Simulation Coupling Parameter Approach and
Thermodynamic Cycle
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Coupling Parameter Approach
The coupling parameter, l, may be considered a
generalized extent parameter, defining the
progress of a system along a path, A to B.
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Coupling Parameter Approach
Free energy being a state function, the path
between the two states A and B could correspond
to an alchemical (nonphysical) transformation.
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• Single Topology

Common coordinate set for the initial and final
states
Potential Energy Function
Parameter Mixing
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• Dual Topology

Different coordinate set for the initial and
final states
Potential Mixing
Potential Energy Function
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Free Energy Perturbation
Convergence depends on overlap of low energy
regions of configurational space sampled at l
and lDl.
Simulations at fixed l
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Thermodynamic Integration

• Simulate at fixed l and integrate numerically
  (multiconfigurational TI)
• Make l function of time and slowly change during
  simulation (slow growth)

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Jarzynski Procedure
For a reversible process
Slow growth simulation
Non-equilibrium work relation
Fast Growth Method
In the limit t ? 0,
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Thermodynamic Cycle

• DG for Any Closed Path 0

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Thermodynamic Cycle Binding Free Energy

• Double Annihilation cycle

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Molecular Dynamics Procedure

• NAMD Not Another Molecular Dynamics
• Scalable Molecular Dynamics Program
• Data and Message Driven Parallelism
• Object Oriented Design and C programming
  language
• Particle Mesh Ewald
• Constant Temperature and Pressure Control Schemes
• Multiple Time Stepping
• File Compatibility with CHARMm and AMBER
• Easy interoperatability with VMD (Visual
  Molecular Dynamics)
• Production quality and developing
• Free download with source code
• NAMD was developed by the Theoretical Biophysics
  Group at the University of Illinois,
  Urbana-Champaign.

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Benchmark Results on CHARMm and NAMD
-Real time for 100 steps of MD of the
Biotin-Streptavidin system (27702 atoms) with
NAMD and CHARMm -Speed-up ration (CPU time for 1
processor job/CPU time for n Processor job) with
CHARMm and NAMD for Carboxy-myoglobin in water
(14026 atoms) -Speed-up ratio with CHARMm and
NAMD for Biotin-Streptavidin system in water
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Biotin-Streptavidin Complex
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Simulation Protocol

• 27702 atoms, periodic boundary conditions
• Nosé Anderson pressure control scheme
• Particle Mesh Ewald
• 1 fs time step
• 25 ps equilibration 25 ps production per window
• 10 windows, linear change in l
• Electrostatic decoupling
• 1 ns simulation in each direction gt 15 days on
  16 R12000 processors
• Forward and reverse simulation for error
  estimation

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The estimated binding free energy of
Biotin-streptavidin complex using the FEP and TI
methodologies.
Free Energy Perturbation (FEP)
Thermodynamic Integration (TI)
The reported errors are based on forward and
reverse simulations. Experimental values From
Weber et al. (1992) J. Am. Chem. Soc. 114, 3197.
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Effect of long range electrostatics treatment ?
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Distance root mean square deviation (RMSD) of the
atomic positions of biotin over a 25 ps
time-scale In aqueous solution and in the active
site. The mobility of biotin is strongly
restricted due to packing and hydrogen bonding
interactions.
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Potential of Mean Force Umbrella Sampling
Bias or focus sampling in important regions of
phase space.
where
Generating the Umbrella Distributions carries the
sampling to a Non-Boltzmann regime.
The distribution can be unbiased to generate the
unbiased probability Distribution. (Torrie and
Valleau)
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Molecular Dynamics Procedure

• NAMD Not Another Molecular Dynamics
• Scalable Molecular Dynamics Program
• Data and Message Driven Parallelism
• Object Oriented Design and C programming
  language
• Particle Mesh Ewald
• Constant Temperature and Pressure Control Schemes
• Multiple Time Stepping
• File Compatibility with CHARMm and AMBER
• Easy interoperatability with VMD (Visual
  Molecular Dynamics)
• Production quality and developing
• Free download with source code
• NAMD is developed by the Theoretical Biophysics
  Group at the University of Illinois,
  Urbana-Champaign.
• The free energy simulation procedure in NAMD
  was implemented at LCTN by S. B. Dixit.

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Biotin-Streptavidin Complex Effect of long
range electrostatics treatment ?
(Binding energies in kcal/mol)
Dixit SB Chipot C. J Phys Chem A, 2001, 107,
9795.
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Binding of lowaffinity, nonpeptide
inhibitorsto the SH2 domain of the src protein.
(Binding energies in kcal/mol)
Chipot C, Xavier R Dixit SB J Comput Aided Mol
Des 2005, 19, 765.
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Modes of Protein-DNA Binding

• Repressor-OL1 Operator
• (Beamer et al. 1988)

CAP-DNA Complex (Schultz et al. 1991)
TBP-DNA Complex (Kim et al. 1993)
IHF-DNA Complex (Rice et al. 1996)
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Free Energy Analysis of Protein-DNA Binding
Jen-Jacobson (1997) Biopolymers 44, 153-180.
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Thermodynamics of Protein-DNA Binding
Free Energy Component Analysis to Predict Binding
Free Energy MM GB/PB SA
?G ? ( best affordable theoretical estimate of
each component )
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Interaction Energy
Empirical Energy Functions
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Solvation Energy
Electrostatic
Non-Electrostatic
A Solvent Accessible Surface Area
where
Generalized Born Equation
g 7.2 cal/mol/Å2 gcav 47.0 cal/mol/Å2 gvdW
-39.8 cal/mol/Å2
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Physical Chemistry

• Translational and Rotational Entropies
• Configurational Entropy
• Condensed Counter Ions
• Free Counter Ions

Janin J (1995) Biochimie 77, 497.
-TDS0 2 kcal/mol per ion
DG0 -98.3 kcal/mol per unit charge
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Free Energy Analysis of Protein-DNA Binding

• What is the contribution of
• Electrostatics?
• Van der Waals effects?
• Cavitation effects?
• Internal entropy effects?
• Structural adaptations?
• Ion release?

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Application EcoRI Endonuclease - DNA Complex
Mc Clarin JA et al. (1986) Science 234, 1526.
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Calculated Values of Various Component
Contributions to the Free Energy of Binding
Jayaram B, Dixit SB et al., J Comput Phys, 1999,
151, 333.
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EcoRI Endonuclease Complex
Jayaram B, Dixit SB et al., J Comput Phys, 1999,
151, 333.
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EcoRI Endonuclease-DNA Energy Map vdW and
Cavitation Components
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EcoRI Endonuclease DNA Complex Interaction
Energy Protein Monomer 1 Perspective
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Consensus View of 40 Protein-DNA Complex Crystal
Structure Binding Energies
Jayaram B, Dixit SB et al., J Comput Chem, 2002,
23, 1.
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The Dynamics in Thermodynamics of Binding
Enthalpy-Entropy Contributions to Binding Free
Energy
Strategies for Site-Specific Protein-DNA Binding.
Jen-Jacobson et. al. (2000) Structure 8,
1015-1023.
Structural Deformation ? Entropy Driven Regular
structure ? Enthalpy Driven
J.D. Forman-Kay (1999), Nature Struct Biol 6,
1086-1087
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Molecules in motion!
HIV protease inhibitor Ritonavir binding to the
protease http//www.umass.edu/microbio/chime/expl
orer
Information flow through membrane
receptors http//bio.winona.msus.edu/berg/ANIMTNS/
Recep.htm
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Time Scales of Biological Motions
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Molecular Dynamics Simulation
Discrete simulation of conformational change in
timesteps of 10-15 second
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Bending In CAP DNA Structures During MD
Simulations
Starting Structure
Bent DNA in CAP-DNA Complex
Straight Canonical B-form DNA
Simulation Time ?
Red MD Structures of DNA in Complex Blue MD
Structures of Unbound DNA
Dixit SB Beveridge DL Biophys J 2005, 88, L04.
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Molecular Dynamics Simulation of Protein-DNA
Complexes
Catabolite Activator Protein-DNA Complex

• Repressor-Operator Complex

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Entropy Change During Protein-DNA complexation

Covariance matrix
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Configurational Entropy in Protein-DNA Complexes
Dixit SB, Andrews DQ, Beveridge DL Biophys J 2005
88, 3147 .
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Structure ? Dynamics ? Thermodynamics
X-Ray
Keq
Simulations
Thermodynamics
Structure
NMR
Cp
Simulations
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Conclusions

• Availability of faster computational resources
  makes free energy simulation techniques a
  tractable procedure
• Traditional free energy simulation can provide
  highly accurate results but is still not a back
  box procedure
• The free energy component analysis method is an
  useful technique to obtain qualitative estimate
  of binding free energy
• Challenges to free energy simulation methods
  entropic effects in solute and solvent, capture
  allosteric changes on slower time scales,
  polarization effects, high throughput while
  maintaining high accuracy
• Free energy simulations are useful tools in the
  later stages of drug discovery research and
  provide valuable methods for the prediction of
  relative binding strength of molecules