Biomolecular Modelling: Goals, Problems, Perspectives 1. Goal simulate/predict processes such as 1.polypeptide foldingthermodynamic 2.biomolecular associationequilibria governed 3.partitioning between solventsby weak (nonbonded) 4.membrane/micelle formationforces common characteristics: -degrees of freedom: atomic (solute + solvent)hamiltonian or -equations of motion:classical dynamicsforce field -governing theory:statistical mechanicsentropy

Processes: Thermodynamic Equilibria Folding Micelle Formation Complexation Partitioning folded/nativedenaturedmicellemixture boundunbound in membrane in waterin mixtures

Methods to Compute Free Energy Classical Statistical Mechanics: Free Energy: Free Energy Differences: - between two systems:and - depending on a parameter: - along a (phase space) coordinate:

Methods to Compute Free Energy Counting of Configurations: one simulation, but sufficient events sampled? Thermodynamic Integration many simulations:ensemble average for each value then numerical integration Perturbation Formula one simulation, sufficient overlap?

Free Energy Difference via Thermodynamic Integration - Accurate: sufficient sampling sufficient  -points i - many (10 – 100) separate simulations - for each new pair of states A and B a new set of simlulations is required - for each the state is unphysical Very time consuming FF state A state B

Free Energy Calculations One-step perturbation technique and efficient sampling of relevant configurations Thermodynamic Integration N inhibitors: unboundbound 2 M N simulations

Free Energy Calculations One-step Perturbation 2 simulations of an unphysical state which is chosen to optimise sampling for entire set of N inhibitors Idea:use soft-core atoms for each site where the inhibitors possess different (or no) atoms The reference state simulation (R) should produce an ensemble that contains low-energy configurations for all of the Hamiltonians (inhibitors) H 1, H 2, …,H N

conformational space A A A B B B R B’ C D E

H2OH2OProtein A B C Unphysical Reference Ligand R  G A bind  G B bind  G AR H2O  G BR H2O  G AB =  G B bind –  G A bind =  G AR H2O –  G AR protein –  G BR H2O +  G BR protein Unphysical Reference Ligand R A B C ……

Y1 (C) U1 (A) U8 Y2 (T)Y3Y4Y5Y6Y7Y8Y9Y10 U2 (G)U3 U4U5U6U7 U9U10U11U12U13 Free energies of base insertion, stacking, pairing in DNA

(CGCGAXYTCGCG) 2.0 ns3.4 ns 2.0 ns 2.0 ns Five MD simulations to obtain free energies of base insertion, stacking, pairing

Double helix d(CGCGAXYTCGCG) 2 in water

Free energy of insertion and stacking for particular pairs of central bases

ABC A, G, C, TA, U 13, C, TA, G, Y 9, T Stacking of adjacent central bases

10x13x10x13 – 1 = values (in fact we did 1024) U1-2 – Y1-2 and U1-13 – Y1-10, and vice versa (520 free energies) Decompose the double free energies of pairing into single free energies of pairing AG C T kJ/mol purine pyrimidine Free energies of double base pairing in (CGCGAXYTCGCG) 2

U5Y10 U10 Y9Y7U2 (G) U4Y4 14 kJ/mol 99 kJ/mol 105 kJ/mol 65 kJ/mol

(CGCGAXYTCGCG) 2.0 ns3.4 ns 2.0 ns 2.0 ns Five MD simulations to obtain free energies of base insertion, stacking, pairing

Free Enthalpy of Solvation by Thermodynamic Integration Make Hamiltonian (Interaction) dependent on a coupling parameter solute-solute assume = 0 (for simplicity) solute-solvent small solvent-solvent very large =0nosolute-solvent interaction (solute in gas phase) =1fullsolute-solvent interaction (solute in solution)

(Free) Enthalpy and Entropy of Solvation difficult to calculate due to U vv same term assumed: only solute-solvent interaction U uv ( ) depends on solvent-solvent term U vv does not

(Free) Enthalpy and Entropy of Solvation U vv terms are absent  computable Calculate instead of  H S and T  S S : both computable yield insight into enthalpic and entropic effects

(Free) Enthalpy and Entropy of Solvation Nico van der Vegt reference: J. Phys. Chem. B. (2004) mole fraction

Solvation of Methane in Na + Cl - Solutions methane solvation in salt  U * uv triangles T  S * uv squares relative to neat water concentration Na + Cl - free enthalpy energy (enthalpy) entropy Entropy disfavours solvation increasingly with salt concentration (non-linear).

Solvation of Methane in Acetone Solution methane solvation in acetone  U * uv triangles T  S * uv squares relative to neat water: SPC water SPC/E water free enthalpy entropy energy (enthalpy) Entropy favours solvation. mole fraction

Solvation of Methane in Dimethylsulfoxide (DMSO) Solutions free enthalpy entropy energy (enthalpy) Energy favours solvation (non-linearly). mole fraction reference: J. Chem. Phys. B. (2004)

 G S  U uv T  S uv Relative to Solvation in Pure Water enthalpy relative and absolute contributions do vary entropy dominantcounteracts enthalpyenthalpy and entropy co-act counteract changes sign mole fraction  different models relative values of  U uv, T  S uv change,  G s not so much

Computer-aided Chemistry: ETH Zuerich Molecular Simulation Package GROMOS = Groningen Molecular Simulation + GROMOS Force Field Generally available: Research Topics searching conformational space force field development –atomic –polarization –long range Coulomb techniques to compute free energy 3D structure determination –NMR data –X-ray data quantum MD: reactions solvent mixtures, partitioning interpretation exp. data applications –proteins, sugar, DNA, RNA, lipids, membranes, polymers –protein folding, stability –ligand binding –enzyme reactions

Computer-aided Chemistry: ETH Zuerich Group members Dirk Bakowies Indira Chandrasekhar David Kony Merijn Schenk (Alex de Vries) (Thereza Soares) (Nico van der Vegt) (Christine Peter) Alice Glaettli Yu Haibo Chris Oostenbrink Peter Gee Markus Christen Riccardo Baron Daniel Trzesniak Daan Geerke Bojan Zagrovic