Force fields and molecular modelling Carine Clavagu´era Laboratoire de Chimie Physique, CNRS & Universit´eParis Sud, Universit´eParis Saclay [email protected] Carine Clavagu´era Force fields and molecular modelling 1 / 76 Outline Class I and class II force fields Class III force fields Solvent modelling Applications Reactive force fields Carine Clavagu´era Force fields and molecular modelling 2 / 76 Multiscale modelling Carine Clavagu´era Force fields and molecular modelling 3 / 76 History 1930 : Origin of molecular mecanics : Andrews, spectroscopist 1960 : Pioneers of force fields : Wiberg, Hendrickson, Westheimer 1970 : Proof of concept : Allinger 1980 : Basis of force fields for classical molecular dynamics : Scheraga, Karplus, Kollman, Allinger 1990 : Commercial distribution of program packages Since 2000 I More and more publications about classical molecular dynamics simulations I Wide availability of codes I New generation force fields Carine Clavagu´era Force fields and molecular modelling 4 / 76 Potential energy r N : 3N coordinatesfx,y,zg for N particules (atoms) Force field = mathematical equations to compute the potential energy + parameters N U (r ) = Us + Ub + Uφ + Uvdw + Uelec + Upol | {z } | {z } intramolecular interactions intermolecular interactions 1-4 long-range interaction terms = interactions between 2 rigid spheres Carine Clavagu´era Force fields and molecular modelling 5 / 76 Class I force fields CHARMM, AMBER, OPLS, GROMOS. I Intramolecular interactions : harmonic terms only I Intermolecular interactions : Lennard-Jones 6-12 and Coulomb term based on atomic charges X 2 X 2 X V (r) = ks (r − ro ) + kb (θ − θo ) + Vn [1 + cos((nφ) − δ)] s b torsion 12 6 X qi qj r0 r0 + + 4" − "rij r r i<j Carine Clavagu´era Force fields and molecular modelling 6 / 76 Bond stretching / bond angle Harmonic approximation Taylor series expansion around Bond energy versus bond length bo P 2 Ubond = b Kb (b − b0) Chemical Kbond b0 type (kcal/mol/A˚2)(A)˚ C−C 100 1.5 C−C 200 1.3 C−C 400 1.2 Taylor series expansion around θo X 2 Uangle = Ka (θ − θo ) a Carine Clavagu´era Force fields and molecular modelling 7 / 76 Torsion May be treated by direct 1-4 interaction terms Much more efficient to use a periodic form X Utorsion = Vn [1 + cos((nφ) − δ)] torsion Carine Clavagu´era Force fields and molecular modelling 8 / 76 Improper torsion To describe out-of-plane movements Mandatory to control planar structures Example : peptidic bonds, benzene P 2 Uimproper = φ Kφ(φ − φo ) Carine Clavagu´era Force fields and molecular modelling 9 / 76 van der Waals interactions London dispersion forces in 1=R6 and repulsive wall at short distances Lennard-Jones 6-12 potential " # R 12 R 6 E (R) = 4" o − o LJ R R Diatomic parameters 1 RAB = (RA + RB ) o 2 o o p "AB = "A + "B Carine Clavagu´era Force fields and molecular modelling 10 / 76 Electrostatic interaction X qi qj Interactions between punctual atomic charges Eelec = "rij i<j Quantum chemical calculation of atomic charges 1- Quantum chemical calculations of molecular models. 2- Electrostatic potential on a grid of points surrounding a molecule. 3- Atomic charges derived from the electrostatic potential. Carine Clavagu´era Force fields and molecular modelling 11 / 76 Electrostatic potential fit Electrostatic potential (ESP) from the wave function (QM calculations) nuc Z ∗ 0 0 X ZA Ψ (r )Ψ(r ) 0 φesp (r)) = − 0 dr jRA − rj jr − rj A Basic idea : fit this quantity with point charges Minimize error function Npoints Natoms ! X X QA(Ra ) ErrF (Q) = φesp (r) − jRa − rj r a An example : Restrained Electrostatic Potential (RESP) charges ! potential is fitted just outside of the vdW radius Carine Clavagu´era Force fields and molecular modelling 12 / 76 Class I force fields CHARMM, AMBER, OPLS, GROMOS... X 2 X 2 X V (r) = ks (r − ro ) + kb (θ − θo ) + Vn [1 + cos((nφ) − δ)] s b torsion 12 6 X qi qj r0 r0 + + 4" − "rij r r i<j Force field bond and angle vdw Elec Cross Molecules AMBER P2 12-6 and 12-10 charge none proteins, nucleic acids CHARMM P2 12-6 charge none proteins OPLS P2 12-6 charge none proteins, nucleic acids GROMOS P2 12-6 charge none proteins, nucleic acids Carine Clavagu´era Force fields and molecular modelling 13 / 76 Class I force fields : differences AMBER (Assisted Model Building with Energy Refinement) I Few atoms available, some calculations are impossible I Experimental data + quantum chemical calculations I Possible explicit terms for hydrogen bonds and treatment of lone pairs CHARMM (Chemistry at HARvard Molecular Mechanics) I Slightly better for simulations in solution I Charges parameterized from soluted-solvent interaction energies I No lone pair, no hydrogen bond term OPLS (Optimized Potentials for Liquid Simulations) I Derived from AMBER (intramolecular) I Non-bonded terms optimized for small molecule solvation I No lone pair, no hydrogen bond term Carine Clavagu´era Force fields and molecular modelling 14 / 76 How to obtain intramolecular parameters ? Quantum chemistry and experimental data ! Force constants in bonded terms : vibrational frequencies, conformational energies ! Geometries : computations or experiments (ex. X-ray) Example : potential energy surfaces of molecules with several rotations Carine Clavagu´era Force fields and molecular modelling 15 / 76 Class II force fields MMFF94, MM3, UFF I Anharmonic terms (Morse, higher orders,...) I Coupling terms (bonds/angles, . .) I Alternative forms for non-bonded interactions MM2/MM3 I Successful for organic molecules and hydrocarbons MMFF (Merck Molecular Force Field) I Based on MM3, parameters extracted on ab initio data only I Good structures for organic molecules I Problem : condensed phase properties Improvements I Conformational energies I Vibrational spectra Carine Clavagu´era Force fields and molecular modelling 16 / 76 Bond stretching Morse potential ! Taylor series expansion around ro Beyond harmonic approximation : X 2 3 4 Ubond = K2(b − b0) + K3(b − b0) + K4(b − b0) + ::: Carine Clavagu´era Force fields and molecular modelling 17 / 76 Bond angle Harmonic approximation : X 2 Uangle = Ka (θ − θo ) a Taylor series expansion around θo MM3 : beyond a quadratic expression θ3 term mandatory for deformation larger than 10-15◦. X 2 −5 2 Uangle = Ka (θ − θo ) [1 − 0:014(θ − θo ) + 5:6:10 (θ − θo ) a −7 3 4 − 7:0:10 (θ − θo ) + 2:2:10−8(θ − θo ) ] Carine Clavagu´era Force fields and molecular modelling 18 / 76 Couplings Coupling terms between at least 2 springs Stretch-bend coupling X 0 Esb = Klθ (l − l0) + l − l0 (θ − θ0) l;l0 Other couplings : stretch-torsion, angle-torsion, ... Carine Clavagu´era Force fields and molecular modelling 19 / 76 Interactions de van de Waals Alternative to Lennard-Jones potential : Buckingham potential 2 3 aRo 6 − R E (R) = " e R − o vdw 4 R 5 Used in the MM2/MM3 force fields However, LJ potential is usually preferred in biological force fields ! Computational cost of the exponential form in comparison with R−12 Force field bond and angle vdw Elec Cross Molecules MM2 P3 Exp-6 dipole sb general MM3 P4 Exp-6 dipole sb, bb, st general MMFF P4 14-7 charge sb general UFF P2 or Morse 12-6 charge none all elements Carine Clavagu´era Force fields and molecular modelling 20 / 76 Comparison of conformational energies for organic molecules Carine Clavagu´era Force fields and molecular modelling 21 / 76 Comparison of conformational energies for organic molecules Carine Clavagu´era Force fields and molecular modelling 22 / 76 Comparison of conformational energies for organic molecules Carine Clavagu´era Force fields and molecular modelling 23 / 76 Biological force fields M.R. Shirts, J.W. Pitera, W.C. Swope, V.S. Pande, J. Chem. Phys. 119, 5740 (2003) Carine Clavagu´era Force fields and molecular modelling 24 / 76 Limitations "Additive force fields" I No full physical meaning of the individual terms of the potential energy I No inclusion of the electric field modulations I No change of the charge distribution induced by the environment ! No many-body effects Transferability I Better accuracy for the same class of compounds used for parameterization I No transferability between force fields Properties from electronic structure unavailable (electric conductivity, optical and magnetic properties) ! Impossible to have breaking or formation of chemical bonds Carine Clavagu´era Force fields and molecular modelling 25 / 76 Class III force field class Next Generation Force Fields ! To overcome the limitation of additive empirical force fields I Polarizable force fields : 3 models based on F Induced dipoles F Drude model F Fluctuating charges X-Pol, SIBFA, AMOEBA, NEMO, evolution of AMBER, CHARMM I Able to reproduce physical terms derived from QM : total electrostatic energy, charge transfer, ... I Optimized for hybrid methods (QM/MM) I To account for the electronegativity and for hyperconjugation I Reactive force fields Carine Clavagu´era Force fields and molecular modelling 26 / 76 Accurate description of electrostatic effects H2OH2O + q Cisneros et al. Chem. Rev. 2014 Errors (V) for electrostatic potential on a surface around N-methyl propanamide Multipole expansion 3 Dipole-Dipole interaction : (µ1 µ2)/R 4 Dipole-Quadrupole interaction : (µ1 Q2)/R Quadrupole-Quadrupole interaction : 5 (Q1 Q2)/R Point charges vs. multipole expansion Stone Science, 2008 Carine Clavagu´era Force fields and molecular modelling 27 / 76 Multipolar distributions Distributed multipole analysis (DMA) : multipole moments extracted from the quantum wave function Multipoles are fitted to reproduce the electrostatic potential A. J. Stone, Chem. Phys. Lett. 1981, 83, 233 Acrolein : molecular electrostatic potential (a) QM (reference) (b) DMA (c) Fitted multipoles Difference between ab initio ESP and (d) DMA (e) Fitted multipoles C. Kramer, P. Gedeck, M. Meuwly, J. Comput. Chem. 2012, 33, 1673 Carine Clavagu´era Force fields and molecular modelling 28 / 76 Solvent induced dipoles • The matter is polarized proportionally to the strength of an applied external electric field.