Statistical Mechanics for Proteins

Transcription

1 The Partition Function From Q all relevant thermodynamic properties can be obtained by differentiation of the free energy F: = kt q p E q pd d h T V Q ), ( exp 1! 1 ),, ( ),, ( ln ),, ( T V Q kt T V F = T V F p, = V T F S, = T V F, = μ Statistical Mechanics for Proteins

2 Exact evaluation of Q only possible for idealized systems (free particles, harmonic oscillator) From this: 2 / 3 3 free 2! 1 ),, ( h mkt T V Q = π Statistical Mechanics for Proteins Λ + = 3 ln 1 ),, ( V kt T V F V kt p = Λ + = 3 ln 2 5 V k S Λ = 3 kt ln V μ kt TS F U 2 3 = + =

3 Statistical Mechanics for Proteins Since force field is not entirely harmonic, Q for a protein can not be evaluated in closed form: Q(, V, T ) = Q = free (, V, T )* Q 1 2πmkT 3! h (, V, T ) The second term is called configurational integral and describes the contribution of the interaction part to Q. Free energies can therefore be decomposed into 3 / 2 inter * ( q) V dq exp kt ( ln Q ln Q ) F = kt ln Q = kt + free inter

4 Statistical Mechanics for Proteins Two primary strategies to evaluate the excess (nonideal, interaction-related) contribution to Q: Molecular Dynamics Simulations Sampling of the phase space (p,q) Monte Carlo Sampling of the integral

5 Free energy: classical definition + The free energy is the energy left for once you paid the tax to entropy:!g =!H"T!S Enthalpic! Hydrogen bonds! Polar interactions! Van der Waals interactions!... Entropic! Loss of degrees of freedom! Gain of vibrational modes! 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)

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

7 Free energy: statistical mechanics definition G =!k B Tln(Z) where Z = # e!"ei i is the partition function Free energy differences between 2 states (bound/unbound, É) are, therefore, ratios of partition functions #!G =G A "G B = "k B Tln Z A $ % Z B & ' ( Free energy simulations aim at computing this ratio using various techniques.

8 Relation with chemical equilibrium A + B " # AÕBÕ K A = K b = [ A'B' ] A [ ] [ B] AÕBÕ "# A + B [ ] [ B] [ ] K D = K i = A 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) Weak asso. K D (mol/l) Strong asso.

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

10 The free energy is the main function behind all process A) Chemical equilibrium!g binding =RTlnK A + K A = [ AB] A [ ] [ B] A B K D =1/K A AB B) Conformational changes!g conf =k B Tln P Conf1 P Conf2 R=k B A C) Ligand binding D) É!G binding =k B Tln P Unbound P Bound

11 Free energy: computational approaches #!G =G A "G B = "k B Tln Z A $ % Z B & ' ( Free energy simulations techniques aim at computing ratios of partition functions using various techniques. Z = # i e!"ei Sampling of important microstates of the system (MD, MC, GA, É) Computation of energy of each microstate (force fields, QM, CP)

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

13 Central Role of the Partition Function Z = # i e!"ei O = 1 Z # i O i e!"ei... Expectation Value E =!!" ln(z)=u " p =k B T!ln(Z) % # $!V & ',T G = -k B T ln(z) Internal Energy Pressure Gibbs free energy

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

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17 Ergodic Hypothesis MD Trajectory E VT simulation $ # VE simulation 3 Spatial coordinates Dialanine Protein O Ensemble = 1 % Z O(!,")e#$E(!,") d!d" = 1 % O(t)dt = O & Time? & 0

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

19 Linear interaction energy (LIE) J. qvist, J. Phys. Chem., 1994, 98, 8253 Two MD runs : free state and bound state Free state ÒsolventÓ = water Bound state ÒsolventÓ = water + protein vdw!g bind =" E l#s ( ) + $ ( elec E ) l#s # E vdw bound l#s free # E elec bound l#s free J. qvist, J. Phys. Chem., 1994, 98, 8253 '=0.165 and (=0.5 T. Hansson et al., J. Comp.-Aided Molec. Design, 1998, 12, 27 '=0.181 and (=0.5, 0.43, 0.37, 0.33 W. Wang, Proteins, 1999, 34, 395

20 Linear interaction energy (LIE) Advantages : Drawbacks : - Faster than free energy simulation - More structurally different ligands than for free energy simulation. But generally restricted to rather similar ligands. - Slower than scores based on a single conformation (LUDI, PMF,...) - ot really universal (' and ( system dependent) - eed experimental binding affinities of known complexes Modifications : - Additional term proportional to buried surface upon complexation D.K. Jones-Hertzog and W.L. Jorgensen, J. Med. Chem., 1997, 40, Use of continuum solvent model instead of explicit solvent R. Zhou and W.L. Jorgensen et al., J. Phys. Chem., 2001, 105, 10388

21 Binding free energy decomposition: MM-PBSA, MM-GBSA Gaz lig!g solv Lig + Prot prot!g solv!e gaz comp!g solv Lig:Prot Averaged over an MD simulation trajectory of the complex (and isolated parts)!g bind =!E gaz +!G desolv "T!S Sol Lig + Prot!G bind Lig:Prot E gaz =E elec +E vdw +!E intra!g desolv =!G comp solv "!G solv lig prot ( +!G solv )!T"S =!T(S comp!(s prot +S lig )) S =S trans +S rot +S vib B. Tidor and M. Karplus, J. Mol. Biol., 1994, 238, 405!G solv =!G solv,elec +!G solv,np comp!g desolv =!G solv,elec Depending on the way!g solv,elec is calculated: prot ( +!G solv,elec ) +# SASA comp " SASA lig +SASA prot lig "!G solv,elec Molecular mechanics Ð Poisson-Boltzmann Surface Area (MM- PBSA) J. Srinivasan, P.A. Kollmann et al., J. Am. Chem. Soc., 1998, 120, 9401 Molecular mechanics Ð Generalized Born Surface Area (MM- GBSA) H. Gohlke, C. Kiel and D.A. Case, J. Mol. Biol., 2003, 330, 891 ( ) ( )

22 Binding free energy decomposition MM- PBSA, MM-GBSA Advantages : Drawbacks : - Used for ligand:protein and protein:protein complexes - Could be applied to structurally different ligands (but in fact applied to similar ones) - ÒUniversalÓ (no parameter to be fitted) - MM-GBSA allows a per-atom decomposition of "G bind (e.g. contribution of side chains) - Rather time consuming - In some cases, found unable to rank ligands -T"S is necessary to find the order of magnitude of the absolute binding free energies but, in some cases, it is not necessary to estimate relative binding free energies W. Wang and P.A. Kollman, J. Mol. Biol., 2000, 303, 567 H. Gohlke, C. Kiel and D.A. Case, J. Mol. Biol., 2003, 330, 891

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24 Use of thermodynamical cycles L1 + Prot "G 1 L1:Prot Thermodynamic cycle perturbation approach: "G 3 "G 4 ""G bind ="G 2 -"G 1 ="G 4 -"G 3 L2 + Prot "G 2 L2:Prot "G 4 -"G 3 is computationally accessible ÒAlchemicalÓ reaction. MD or MC at different *. Coupling parameter * H! = H 0 +! H L1 + ( 1"! ) H L2 O O H H O CH 2 O OH CH 2 OH Cl O H O CH 2 Cl *=0 * *=1 Free energy perturbation (FEP) Thermodynamic integration (TI) $$ G bind =# RT n " # 1 i= 0 ln exp (# ( H # H ) RT)! i! i+ i! i ## G bind = $! = 1 " H $ = 0 " $ $ d$ $

25 Alchemical free energy formalism

26 ÒAlchemicalÓ Free Energy Calculations

27 Hybrid Side Chain for P "A Mutation

28 Results of the Free Energy Simulations Total (Path Independent) Experimental: Theoretical: 2.9 (0.2) kcal/mol 2.9 (1.1) kcal/mol Components (Path Dependent) Experimental: - Theoretical: TCR 25% Solvent 20% HLA A2 40% Peptide 15% } } 45% 55%

29 Free energy formalism From the statistical definition of the free energy,

30 Concluding Remarks 1) Free energy simulations can reproduce accurately experimental changes in association constant between to closely related protein systems if detailed structural knowledge is available (X-ray, MR or model) 2) The formalism is exact from a statistical physics stand point and accurate treatment of entropic terms, solvent effect or conformational changes can be obtained 3) Convergence of the free energy derivative is still problematic. The situation should improve with new methodological enhancements as well as longer simulation time 4) Absolute free energies can also be computed but the convergence is even more difficult 5) Much details about the specificity of the association can be gained using component analysis, opening the door to rational peptide or protein design