Molecular Mechanics Yohann Moreau yohann.moreau@ujf-grenoble.fr November 26, 2015 Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 1 / 29
Introduction A so-called Force-Field is based on an empirical approach No explicit description of electrons (no QM) - Electrons are implicitely included in atomic parameters Bonds are not a result of the calculation (as in QM) but are parameters - Balls & Springs model - Energy is a function of geometry AND topology! Different types of atoms that interact with defined potentials - Parameters depend on the nature of interacting atoms A force field is made of a huge number of parameters A force field must be limited to a defined kind of system - DNA, Solids, fatty acids, proteins, etc... Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 2 / 29
Generality The total energy of a system is the sum of different terms They account for interactions between all the atoms of the system: Interactions between bound atoms - Bonding term between 2 atoms a & b: E r ab =f(r ab) - Angle (3 atoms): E=f(θ abc ) - Torsional (4 atoms): E fct of the dihedral between 4 atoms - Improper Torsional (4 atoms): for planar env. (e.g. sp2 carbon) Atoms separated by 4 bonds or more are unbound - Their interaction is made of Van der Waals and electrostatic forces (charge/charge) Some other specific terms can also be added (after all it s empirical) The total energy can thus be written: E tot = E bond + E angle + E dihedral + E VdW + E Elst Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 3 / 29
Illustration of the different terms E tot = E bond + E angle + E dihedrah + E VdW + E Elst Termes liés E a =f(θ) E l =f(r) E d =f(φ) E d =f(χ) Termes non liés E VdW =f(r) δ+ δ- E Elec =f(r) Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 4 / 29
Determination of the energy of a system To know the energy as a function of the geometry, one has to: know the topology of the molecule - how atoms are bound together? know the type of each atom and their position Count all the interaction terms associated Once all of this is gathered, energy can be computed If one parameter is missing, energy can t be determined Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 5 / 29
Atom types The atomic type depends on - its nature - its hybridization - its neighbours Hence a FF is generally restrained to a given type of system such as proteins Parameters are considered transferable (similar system) See charmm22.prm e.g. from Introduction to Computational Chemistry, p23, F. Jensen Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 6 / 29
Exercise: propanal in interaction with H 2 O Using the following representation of the system: - determine the number of degrees of freedom - identify all the terms needed to calculate the total potential energy of the system Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 7 / 29
Application: propanal in interaction with H 2 O Count of the different terms - 8 bonds: 3 C-H, 1 C-C, 1 C-O, 1 C-H ; 2 O-H - 10 angles : 3 H-C-H, 3 H-C-C, 1 C-C-H, 1 C-C-O, 1 H-O-H - 6 dihedral: 3 H-C-C-H, 3 H-C-C-O - 1 improper dihedral: C-C-O-H - 3 7 = 21 VdW interactions and 21 electrostatic interactions - 1 possible H-bond (not in all force fields) Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 8 / 29
Potentials of interaction Once terms identified, energy associated to each can be computed Each potential accounts for the energy modification in fonction of the modification of the degree of freedom associated This must allow to reproduce the PES around the equilibrium geometry (region for wich E is 40 kj/mol or less above minimum) Potentials have a simple analytic expression so as to be easily derived for dynamics and optimisation Remarks: - The equilibrium point of an interaction is a force-field parameter! - A given degree of freedom will remain close to this value (vide infra) - Exemple: r CC 0 varies close to 1,523Å in MM2 Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 9 / 29
math point: the Taylor development near a given point x 0, a function can be expressed as a sum of the derivatives (polynomial approx.): f (x) = f (k) (x 0 ) (x x 0 ) k + O(k + 1) n! k=0,n At the second order, one gets a parabola: f (x) = f (x 0 ) + (x x 0 ).f (x 0 ) + 1/2.(x x 0 ) 2.f (x 0 ) f (x 0 ): value of the function at x 0 - if x 0 is the equilibrium parameter, this is the lowest point (equilibrium energy) f (x 0 ) = 0 if x 0 is a minimum: the curve is flat f (x 0 ) = force constant af an harmonic potential The goal is to mimic the unknown fonction f(x) around x 0, the equilibrium position by a parabola Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 10 / 29
Application: Bonding potential between two atoms (stretching) The energy variation in function of the interatomic distance is complex At the vicinity of the equilibrium distance, the curve is flat (f (x)=0) At the actual equilibrium distance r 0, energy is zero relative to the equilibrium energy The stretch energy between atoms i and j is expressed by a harmonic potential E lij = 1 2 k ij(r ij rij 0)2 Where: - rij 0 is the equilibrium distance between i & j - k ij is the force constant of the spring, usually in kcal/mol/å2 Stretching terms can also be more complex (including exponentials, e.g.) but the harmonic form is the most common Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 11 / 29
Beyond the harmonic approximation Anharmonic potential -Taylor development can be made at a greater order (4, e.g.) E l,anh (r r 0 ) = k ij 2 (r r 0) 2 + k ij 3 (r r 0) 3 + k ij 4 (r r 0) 4 More parameters to define and higher computational cost Morse potential: - E l,morse (r r 0 ) = D(1 exp [ α(r r 0)] 2 ), α = k 2D - D: dissociation energy, α = f (k) - good reproduciton of the dissociation curve but less usable Common force fields use quadratic potentials Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 12 / 29
Illustration: Different C-H stretching potentials from: Introduction to Computational Chemistry, p28, F. Jensen Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 13 / 29
How is the parameter k ij determined? The fonction to fit has an unknown analytical expression The value of k depends on the type of atoms bound Different ways to get the force constant and equilibrium distance: - From vibrational spectra (IR) - From a dissociation profile obtained by quantum chemistry the creation of a force field is a huge work of parametrization Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 14 / 29
Bending terms Increase or decrease the angle makes the energy increase One uses a harmonic potential in function of the angle θ E a = 1 2 k ijk.(θ θ 0 ) 2 k ijk is expressed in kcal/mol/degrees 2 and depends on the nature of atoms i, j and k The harmonic approximation is good enough at ±30 around θ 0 Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 15 / 29
Torsion: I- out of plane bending This term accounts for the cost of puting an atom out of plane - Example: pyramidalization of an sp 2 atom (planar environment at equilibrium) E oop (φ) = k.φ 2 ; at equilibrium: φ 0 = 0 Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 16 / 29
Torsion: II- dihedral For 4 bound atoms, the term represents the angle between tyhe two external bonds - Torsions have a periodic character: E(360 ) = E(0 ) - several minima and maxima (weak rotational barriers) - A quadratic potential (harmonic) is not suitable cosine-type function is often used: x E diedre (φ) = V n cos(nφ) n=1 - x: nb. of minima (max: 5), V n fixes the barrier height φ [0 360 ] Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 17 / 29
Example: torsional potential with 3 minima A term (1+) is added to shift position of zero, as well as a phase factor τ E = 1 2 [0.5(1+cos(1φ τ) 0.2(1 cos(2φ τ))+0.5(1+cos(3φ τ))] Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 18 / 29
Non bonded terms: electrostatic interactions This strong interactions takes place between charges atoms - Charge accounts for the difference between the amount of negative (e ) and positive (protons) charges carried by an atom. - It depends on the nature of each type of atom (electronegativity and neighbours) The pair interaction writes: E elec,ij = 1 4πɛ 0 q i.q j r ij The electrostatic energy is thus: - positive (repulsive) for two atoms with charge of the same sign - negative (attractive) between a negative and a positive atom - proportionnal to the product of the two charges - inverse to the interatomic distance This interaction is counted only for atoms with more than 3 bonds between Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 19 / 29
Non bonded terms: electrostatic interactions Determination of charges Generally, each type of atom has a specific charge Charges are determined by mean of quantum calculation - always the same level of theory (MP2/6-31G* e.g.) for consistency Charges are fitted so as to reproduce the QM electrostatic potential - In protein force fields, amino-acids are split into groups: side chain and backbone - Each group has a whole charge (zero for backbone) See charmm topology file eg Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 20 / 29
Higher order electrostatic terms The electronic structure around an atom can be developped as a series of multipoles of different orders - Charge is a multipole of zeroth order - An atomic dipole can be added, rarely higher order terms - It can account for the local dissymmetry of the electronic structure This adds supplementary terms: charge/dipole et dipole/dipole General force fields don t make use of such terms There are also polarizable force fields in which environment modifies the atomic multipolar structure - totally beyond our scope Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 21 / 29
Non bonded terms: Van der Waals interactions The Van der Waals interaction is the sum of non-polar interactions between atoms These interactions originate in quantum phenomena - Pauli repulsion: at short distance, two electronic clouds can t interpenetrate - Dispersion interactions (London, Keesom, Debye forces): slightly attractive ( behave as 1/r 6 ) A Van der Waals potential is written as the sum of the two phenomena and vanishes quickly: E VdW (r ij ) = E repulsion (r ij ) C ij r 6 ij The expression of E repulsion can be written in different ways Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 22 / 29
Van der Waals potentials Lennard-Jones-type potential is the most common - repulsion varies as r 12 : E LJ (r ij ) = 4ε ij [( σ ij r ij ) 12 ( σ ij r ij ) 6 ] - σ ij, VdW radius, distance for which E=0: σ ij = σ i + σ j - ε is the well-depth (energy) ε ij = ε i ε j There are other expressions for the repuslive part: - Example: Buckingham/Hill potential: E Hill = A exp BR C R 6 Other specific terms, such as H-bonds: - Very rare: E LH = ε ij [5( σ ij r ij ) 12 6( σ ij r ij ) 10 ] At long distance, such interactions can be neglected beyond a cut-off value (often around 12 to 14 Å) Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 23 / 29
Lennard-Jones potential between two non polar hydrogens Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 24 / 29
Common ForceFields A force field is the whole set of parameters and equations that allows the energetic description of a given system force fields are specific to defined kind of systems: - CHARMM, Gromos & Amber for proteins - MM2/3/4: general application ( - UFF (Universal Force Field): parameters are calcultaed on-the-fly Whatever the force field, the molecular topology as well as all parameters MUST be known and remain unchanged - Applied to a given geometry, one can compute the associated energy Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 25 / 29
Example: the CHARMM Force Field Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 26 / 29
A specific force field for water: TIP3P Rather old: 1983 for the first version but still in use -only O has L-J params: σ O = 1, 7683 Å, ε O = 0, 1520 kcal.mol 1 - geometry is fixed: 104,52 ; r OH = 0,9572 Å - charges: q H = +0,417 et q O = -0,834 It was built to reproduce properties of liquid water: -density, strucutre, diffusion Exercise: calculate the dipole moment of TIP3P water with: 1D = 2,54 u.a. ; µ exp = 1, 86D ; discuss your result Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 27 / 29
Application force fields Strength: very fast (energy and derivatives) - allows the study of very large systems and/or carry out a large number of calculation Drawbacks: very specific to the kind of system studied - impossible to modify the topology and atomic types Common applications: - Molecular dynamics: evolution along time of a system - Statistical sampling in general - Conformational analysis (see labs) - Structure refinement for X-ray or NMR structure determination - Molecular docking between a protein and a ligand - Qantitative Structure-Activity Relation Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 28 / 29
Other force field-like methods Force fields have been defined at the atomic level but other cases exist: Coarse grained FFs: -A particle represents several atoms such as the side chain of an amino-acid Allows to study way larger systems Allows to study slow phenomena such as protein folding/assembly Reactive force fields: parameters vary in function of the geometry - Narrow domain of application, quantum-based methods are more suited Yohann Moreau (UJF) Molecular Mechanics, Label RFCT 2015 November 26, 2015 29 / 29