Free energy simulations: theory and applications O. Michielin (1,2,4) - PowerPoint PPT Presentation

Free energy simulations: theory and applications O. Michielin (1,2,4) (1) Ludwig Institute for Cancer Research Epalinges, Switzerland (2) Swiss Institute for Bioinformatics Epalinges, Switzerland (4) Centre Pluridiscipinaire d'oncologie CHUV, Lausanne, Switzerland

Dynamical aspects of molecular recognition

Free energy: classical definition + The free energy is the energy left for once you paid the tax to entropy: � G = � H � T � S Enthalpic Entropic � Hydrogen bonds � Loss of degrees of freedom � Polar interactions � Gain of vibrational modes � Van der Waals interactions � Loss of solvent/protein structure � ... � ... Theoretical Predictions: � Approximate: empirical formula for all contributions � Exact: using statistical physics definition of G G = -K B T ln(Z)

Examples of factors determining the binding free energy Electrostatic interactions - Strength depends on microscopic environment ( � ) - Case of hydrogen bonds Neutral : -1.2 ± 0.6 kcal/mol Charge assisted : -2.4 to -4.8 kcal/mol E H-bond (solv.) - E H-bond (comp.) determines if H-bonds H O H contributes to affinity or not H O H H H O H O O H O Unpaired polar groups upon binding H are detrimental H H O S O H H Strong directional nature solvent complex Specificity of molecular recognition

Free energy: statistical mechanics definition � where G = � k B T ln( Z ) e � � Ei Z = i is the partition function Free energy differences between 2 states (bound/unbound, …) are, therefore, ratios of partition functions � � � G = G A � G B = � k B T ln Z A � � � � Z B Free energy simulations aim at computing this ratio using various techniques.

Relation with chemical equilibrium [ ] A'B' � A + B � K A = K b = A’B’ [ ] B [ ] A [ ] B [ ] K D = K i = A � A’B’ A + B � [ ] A'B' K b : binding constant, K d : dissociation constant, K i : inhibition constant � G binding = � RTlnK A = RTlnK D = � H � T � S � G binding (kcal/mol) -2 -4 -6 -8 -10 -12 -14 -16 Strong asso. Weak asso. K D (mol/l) 10 -3 10 -6 10 -9 10 -12

Connection micro/macroscopic: thermodynamics and kinetics Absolute binding free energies: � G � K A e - � G/RT = K A Relative binding free energies: �� G Free Energy Association Constant � K A’ / K A Binding free energy profiles: � G( � ) � K A , K on , K off Microscopic Structure Biological function

The free energy is the main function behind all process [ ] AB K A = [ ] B [ ] A) Chemical equilibrium A + � G binding = � RTlnK A K D = 1/ K A A B AB B) Conformational changes � G conf = � k B Tln P Conf 1 R = k B N A P Conf 2 C) Ligand binding � G binding = � k B Tln P Bound P Unbound D) …

Free energy: computational approaches � � � G = G A � G B = � k B T ln Z A � � � � Z B Free energy simulations techniques aim at computing ratios of partition functions using various techniques. � e � � Ei Z = i Sampling of important Computation of energy microstates of the system of each microstate (MD, MC, GA, …) (force fields, QM, CP, …)

Connection micro/macroscopic: intuitive view E 1 , P 1 ~ e - � E1 E 2 , P 2 ~ e - � E2 Expectation value O = 1 � E 3 , P 3 ~ e - � E3 O i e � � Ei Z i � e � � Ei Z = Where i E 4 , P 4 ~ e - � E4 is the partition function E 5 , P 5 ~ e - � E5

Central Role of the Partition Function � Z = e � � Ei i O = 1 � O i e � � Ei . . . Z i Expectation Value E = � � ln( Z ) � � ln( Z ) = U p = k B T G = -k B T ln (Z) � � � � � V � � N , T Internal Energy Pressure Gibbs free energy

Molecular Modeling Principles 1) Modeling of molecular interactions Electrostatics Van der Waals Covalent Solvent bonds Free energy landscape 2) Simulation of time evolution (Newton) 3) Computation of average values Connection microscopic/ O = < O > Ensemble = < O > Temps (Ergodicity) macroscopic Macroscopic value Average simulation value

The CHARMM Force Field ( b � b 0 ) 2 + � � V = ( � � � 0 ) 2 K b K � Bonds Angles � + ( � � � 0 ) 2 K � Impropers � [ ] + 1 � cos( n � � � � � ) K � Dihedrals q i q j � 1 + 4 �� r i > j i , j ij ) 12 � ( � ij / r [ ] � + 4 � ij ( � ij / r ij ) 6 i > j

Ergodic Hypothesis MD Trajectory NVT simulation E NVE simulation � � 3 N Spatial coordinates “Alanine” Protein � ? O Ensemble = 1 d � d � = 1 � � O ( � , � ) e � � E ( � , � ) = O Time O ( t ) dt � Z 0

Free energy calculation: Main approaches Sampling, Exact Free Energy Perturbation (FEP) Thermodynamical Integration (TI) Non Equilibrium Statistical Mechanics (Jarzynski) CPU Time Sampling, Approx. Linear Interaction Energy (LIE) Molecular Mechanics/Poisson- Boltzmann/Surface area (MM-PBSA) Approx. Quantitative Structure Activity Relationship (QSAR) � G = F ( X ) G k 0 k i X i ( X is a descriptor)

Free energy calculation: Main approaches Sampling, Exact Free Energy Perturbation (FEP) Thermodynamical Integration (TI) Non Equilibrium Statistical Mechanics (Jarzynski) CPU Time Sampling, Approx. Linear Interaction Energy (LIE) Molecular Mechanics/Poisson- Boltzmann/Surface area (MM-PBSA) Approx. Quantitative Structure Activity Relationship (QSAR) � G = F ( X ) G k 0 k i X i ( X is a descriptor)

Theoretical approaches for the estimation of binding affinities Without 3D structure of the complex - 2D - QSAR - 3D - QSAR With 3D structure of the complex - Knowledge based approaches • regression based methods • potential of mean force - Free energy simulation - Linear interaction energy - Binding free energy decomposition (MM-PBSA, MM-GBSA)

Quantitative Structure Activity Relationships (QSAR) Assumption Chemical similarity of ligands Similarity of biological response Affinity is a function of the ligand physico-chemical properties Advantage No need for structural information about the target Requirements (drawbacks) Known affinities for a series of ligands Structurally related ligands or similar binding modes

2D - QSAR n structurally related molecules O R 4 R 1 N O N R 2 R 3 Measured activities Quantitative description Descriptors (X i ): - Substituents: surface, volume, electrostatics ( � ), hydrophobicity ( � ), partial charges, etc... - Molecule : volume, MR, HOMO, dipole moment, etc... � � = + C. Hansch and T. Fujita, JACS , 1964 , 86 , 1616 G k k X bind 0 i i ( ) � = S.S. So and M. Karplus, J. Med. Chem ., 1996 , 39 , 1521 G neural network X bind i S.S. So and M. Karplus, J. Med. Chem ., 1996 , 39 , 5246

2D - QSAR: limitations Limited to structurally related molecules Needs the experimental activity of a series ligands Not for ab initio studies Overfitting - Method for selecting the descriptor (genetic algorithm) - Estimation of the predictive ability (test set, randomization test, Leave-One-Out method, ...) Use limited to the descriptor’s range in the training set : If only hydrophobic groups at R 1 in the training set O Influence of a hydrophilic group at R 1 ? R 4 R 1 N If only methyl, ethyl, propyl, butyl at R 1 in the training set O N R 2 R 3 Contribution of pentyl, hexyl, etc... ?

3D - QSAR Example : Comparative molecular field analysis (CoMFA) R.D. Cramer et al ., JACS , 1988 , 110 , 5959 Ligands mutually aligned (x 1 ,y 1 ,z 1 ) (x 2 ,y 2 ,z 2 ) (x 1 ,y 1 ,z 1 ) (x 2 ,y 2 ,z 2 ) Common 3D lattice (x i ,y i ,z i ) (x j ,y j ,z j ) Steric field Electrostatic field PLS � � � = + � + � G k S E bind 0 i i i i

3D - QSAR: limitations 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 methods Example : LUDI score H.J. Bohm, J. Comput.-Aided Mol. Des., 1994 , 8 , 623 H.J. Bohm, J. Comput.-Aided Mol. Des., 1998 , 12 , 309 � G bind = � G 0 + � G polar + � G apolar + � G solv + � G flexi Trained using a 82 protein-ligand complexes dataset with known experimental � G bind Polar interactions Desolvation effect ( ) � fpcs � ( ) � f N neighb MD � G polar = � G hb f � R, � � Active site filled with Unbound water water molecules molecules hb ( ) � fpcs � ( ) � f N neighb + � G ion f � R, � � � � G solv = � G lipo wat unbound water ion + � G esrep N esrep � G lipo water = -0.33 kcal/mol � G hb = -0.81, � G ion = -1.41 and � G esrep = +0.10 kcal/mol Ligand flexibility Apolar interactions � G flex = � G rot N rot � � G apolar = � G lipo A lipo + � G aro f (R) � G rot = +0.26 kcal/mol aro � G lipo = -0.81 and � G aro = -0.62 kcal/mol Nrot : number of rotable bonds (acyclic sp 3 -sp 3 , sp 3 -sp 2 )