Molecular Flexibility Esther Kellenberger Faculté de Pharmacie

1. 1/27 Molecular Flexibility Esther Kellenberger Faculté de Pharmacie UMR 7200, Illkirch Tel: 03 68 85 42 21 e-mail: ekellen@unistra.fr

2. introduction Force field Geometry-based sampling Energy-based sampling conclusion 2/27 Molecules have geometries… … « good » geometries in bioactive conformations Methotrexate, used in treatment of cancer, autoimmune diseases methotrexate bound to therapeutcal targets (dihydrofolate reductase and thymidilate synthase)

3. introduction Force field Geometry-based sampling Energy-based sampling conclusion 3/27 Molecules have geometries… … and there are imposible conformations  unusualbond length, steric collisions, distordedring, …

4. introduction Force field Geometry-based sampling Energy-based sampling conclusion 4/27 The number of molecular conformations … depends on the molecular degrees of freedom = Number of rotatable bonds (NROT) Appr. number of simple bonds betweentwo non-hydrogenatoms. For methotrexate, NROT= 10 Considering 3 possible angular values for each NROT yields 310 = 59 049 different conformations

5. introduction Force field Geometry-based sampling Energy-based sampling conclusion 5/27 How to evaluate the conformations? potentialenergy In physics, potential energy exists when a force acts upon an object that tends to restore it to a lower energy configuration. Potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration (SI unit= Joules, common unit = kcal/mol, 1 cal = 4.1868 J) A force field is a vector field that describes a non-contact force acting on a particle at various positions in space. stable (good) conformation lowenergy unstable(bad) conformation highenergy

6. introduction Force field Geometry-based sampling Energy-based sampling conclusion 6/27 Experimentalproperties of a molecularis an mean of properties of populatedconformers Boltzmann’sprobability distribution Boltzmann’sprobability distribution P (conformer of energy E) ~ exp ( -E / kb T) Boltzmann averaging for the observedproperty Property (molecule) = Σ P(conformer) X property(conformer)

7. 7/27 • Chapter1: • Evaluation • of the potentialenergy • of conformers

8. introduction Force field Geometry-based sampling Energy-based sampling conclusion 8/27 Molecularmechanics • Molecular systems are modeled using Newton’s laws: • each atom is simulated as a single particle • each particle is assigned a radius (van der Waals), polarizability, and a constant net charge • bonded interactions are treated as "springs" with an equilibrium distance equal to the bond length • Molecular system's potential energy (E) in a given conformation as a sum of individual energy terms: • E = E covalent + E non covalent

9. introduction Force field Geometry-based sampling Energy-based sampling conclusion 9/27 Covalent contributions to E Bond stretching Angle stretching Torsion correction term Ex. of « standard » values: θ0= 109.5° for Csp3 θ0= 120° for Csp2 θ0= 180° for Csp Ex.of values: for Csp3‐Csp3 n= 3, γ= 0 Etors = 0 at 60°, 180° & -60° Ex.of « standard » values: r0=1.53Å for Csp3‐Csp3 r0=1.09Å for C‐H

10. introduction Force field Geometry-based sampling Energy-based sampling conclusion 10/27 Non covalent contributions to E • Van derWaals term • Lennard Jones potential (6-12) • EVdW = A / rij12 – B/rij6 • where A = 4 εσ12 B = 4 εσ6 • ε = depth of the well • σ ~ distance with minimum EVdW • Electrostatic term • Coulomb’s law • Ecoulomb = δ+ δ-/ 4πε0rij • where δ = charge • ε0= solventdielectric constant Desolvation and hydrophobic term

11. introduction Force field Geometry-based sampling Energy-based sampling conclusion 11/27 Key points on the energy surface highbarrier Energy lowbarrier « ugly » geometries Local minimum Local minimum Global minimum « good » geometries Conformational state

12. introduction Force field Geometry-based sampling Energy-based sampling conclusion 12/27 Energyminimization Given a startinggeometry, deterministicalgorithmsallow the discovery of the adjacent local minimum. Energy starting final starting final Conformational state

13. introduction Force field Geometry-based sampling Energy-based sampling conclusion 13/27 The limits of conformational exploration by moleculardynamics Molecular dynamics trajectory may be seen as an exchange of potential and kinetic energy, with total energy being conserved. The dynamic system consists of moving particles (i.e. molecular atoms with coordinates and velocities). Particle position as a function of time is obtained by solving equation from the Newton’s laws. samplingdepends on the number of frames (time) Energy Amplitude of motion controled by heat starting heating minimisation Conformational state

14. 14/27 • Chapter2: • exploration of the molecularenergylandscape

15. introduction Force field Geometry-based sampling Energy-based sampling conclusion 15/27 Torsions : the gateway to conformationalsampling Energy surface with respect to two torsions

16. introduction Force field Geometry-based sampling Energy-based sampling conclusion 16/27 SystematicSearch and randomsearch angularincremental or random change of selectedrotatablebonds Solutions sorted by Energy (relative)

17. introduction Force field Geometry-based sampling Energy-based sampling conclusion 17/27 Generation of haloperidol 3D conformers by omega http://www.eyesopen.com/products/applications/omega.html 1. Enumerating ring conformations and invertible nitrogen atoms (fragment library) 2. Torsion alteration 3. Reassembly 4. Evaluation MMFF force field Knowledgebased Tables pairwisermsd>2.5Å, Energy threshold  28 conformers

18. introduction Force field Geometry-based sampling Energy-based sampling conclusion 18/27 Increasingcomplexity of energyhypersurface … • Geometry-basedsamplingmethods: • a systematicsearchis possible if NROT < 4-5 • Enumerationrestricted to a fixednumber of conformers for flexible compounds (Ex: 200 in omega) • Energy-basedsamplingmethods: • (moleculardynamics ) • stochasticsampling: Monte-Carlo and Geneticalgorithm

19. introduction Force field Geometry-based sampling Energy-based sampling conclusion Energi Energy 19/27 Monte Carlo random modification of conformations combinedwithacceptation criteria  motion towardenergeticallyfavoredregions Conformational state

20. introduction Force field Geometry-based sampling Energy-based sampling conclusion 20/27 • Montecarloalgorithm Initial state X steps Χ11 Χ12 …Χ1n Perform move Evaluate E(x) Χ21 Χ22 …Χ2n Randomly chosen torsionalaxis Random rotation around that axis yes Better energy no no, restore previous state yes, acceptance test replace state

21. Geometry-based sampling introduction Force field Energy-based sampling conclusion 21/27 Acceptation criteriaThe Boltzmann statistics: P isalsocalled the Bolzmann factor • Test • if Ef < Einew pose isaccepted • if Ef> Eicalculateprobability P of acceptance • Compare P with random number h • if h < P new pose accepted • if h > P restart based on last accepted pose Large energy differences and low temperature lower the Boltzmann factor P  acceptance range goes down k: boltzman constant T: temperature Ef -Ei æ æ P e exp = = ç ç ÷ ÷ - - æ ç ç ÷ ÷ ç kT è è ç ç è è

22. introduction Force field Geometry-based sampling Energy-based sampling conclusion 22/27 Geneticalgorithm Genetic in the real world Genotype : ensemble of genescontained in chromosomes. Diploidorganism : 2 copies of eachgene. Phenotype : ensemble of individualfeatures, resultingfromgene expression. Evolution environmentselection pressure  survival if adaptedphenotype parent 1 parent 2 Chromosomes generation 1 gene 2 copies + Reproduction child 3 child 1 child 2 genera- tion 2 evolution dominant genesadaptedphenotype recessivegenesinadaptedphenotype & & generation 3

23. introduction Force field Geometry-based sampling Energy-based sampling conclusion 23/27 Geneticin the real world (continued) increaseddiversityafter: Cross-over mutation * parent 1 parent 2 generation 1 * + Reproduction generation 2 child 1 child 2

24. introduction Force field Geometry-based sampling Energy-based sampling conclusion 24/27 « virtualgenetic » « chromosome »:fingerprintwhichcodes ligand conformation (e.g., Torsions: binarycoding of the angle value) • parent 1 1101100100110110 • parent 2 1100111000011110 • « crossover » : mixing2 chromosomes (random position) • parent 1 11011 | 00100110110 • parent 2 11001 | 11000011110 • child 1 11011 | 11000011110 • child 2 11001| 00100110110 • « mutation » : randommodification of one (or more) string • parent 1 1101111000011110 • parent 2 1100100100110110 • child 1 1101011000011110 • child 2 1101101100110110 • « selection»: energybelow a selectionthreshold(fitness)

25. introduction Force field Geometry-based sampling Energy-based sampling conclusion 25/27 initial population Size (4) individualssorted by energy (color: high fitness low fitness) Intermediate population Final population Χ11 Χ12 …Χ1n Χ21 Χ22 …Χ2n Χ31 Χ32 …Χ3n Χ41 Χ42 …Χ4n mutation rate crossover rate Geneticoperators max number of generations Χ11 Χ12 …Χ1n Χ51 Χ52 …Χ5n Convergence: evolution of the average/best fitness Χ21 Χ22 …Χ2n Χ61 Χ62 …Χ6n Χ31 Χ32 …Χ3n Χ71 Χ72 …Χ7n Χ41 Χ42 …Χ4n Χ81 Χ82 …Χ8n random Selectionfitness score (green), Survival rate (4) Χ11 Χ12 …Χ1n Χ21 Χ22 …Χ2n Χ31 Χ32 …Χ3n Χ41 Χ42 …Χ4n

26. introduction Force field Geometry-based sampling Energy-based sampling conclusion 26/27 Genetic algorithm is an optimization method: How to preserve the diversity? • Selection pressure: child chromosome replace the worst members of the population / bias in the selection of parent chromosomes (towards high fitness or favoring torsion values seen in in previous populations) • Multiple islands model: population split into sub-populations, with parallel simulations and occasionally swapping solutions (migration) • Discard of redundant chromosomes (requires a metric to evaluate the similarity of individuals) the niche model: a niche is a ensemble of similar individuals in a population (as estimated by RMSD). If there a more than niche size individuals in the niche, then the new individual is replaces the worst individual of the niche rather than the worse individual of the population, in order to preserve diversity within the population.

27. introduction Force field Geometry-based sampling Energy-based sampling conclusion 27/27 CONCLUSION • Conformational Sampling is the key element for understanding of molecular behavior • It may range from very simple to extremely difficult, to impossible • If you don’t do it well, better don’t do it at all: empirical methods based on molecular topology only may be more accurate than 3D models based on wrong – or too few – conformations • Two main sources of errors: A.) wrong calculated energy- geometry landscape (poor Force Field parameterization) and B.) – insufficient sampling! • Thanks to DragosHowarth!