Molecular Mechanics Force Fields

Basic Premise

If we want to study a protein, piece of DNA, biological membranes, polysaccharide, crystal lace, nanomaterials, diffusion in liquids,… the number of electrons (i.e. the number of energy calculaons) make quantum mechanical calculaons impossible even with present‐day computers.

Instead, we replace the nuclei and electrons, and their interacons, by new potenal funcons: ”Classical” atoms.

Based on simple physical concepts

Enables the systems under study to be VERY large (100,000 atoms). Molecular mechanics force fields

The molecular interacons, also known as the potenals, together form a force field,

A force field is a mathemacal descripon of the classical forces or energies between parcles (atoms). Energy = funcon of atomic posions (x,y,z)

The force field equaon consists of several funcons that describe molecular properes both within and between molecules

The force field also contains parameters (numbers) in the potenal funcons that are tuned to each type of molecule (protein, nucleic acid, carbohydrates)

There are many different force field equaons and parameter sets

A force field must be simple enough that it can be evaluated quickly, but sufficiently detailed that it reproduces the key features of the physical system being modelled. Force field classificaon

In general, force fields can be classified as either:

Specific (many parameters, limited applicability, high accuracy) Oen developed in academic labs for study of specific molecular classes or Generic (fewer parameters, more generalizaons, wide applicability, poor accuracy) Easiest to use in point‐and click soware

Force Field Parameters can come from:

Experimental sources (mainly from x‐ray diffracon) or Theorecal calculaons (mainly from QM)

Many force fields employ similar mathemacal equaons but differ in the parameters used in the equaons. It is therefore extremely dangerous mix to parameters between force fields. Different Force Fields:

AMBER ( Assisted Model Building with Energy Refinement).

CHARMM (Chemistry at HARvard using Molecular Mechanics).

GROMOS (GROenigen Molecular Simulaon)

OPLS (Opmized Parameters for Large‐scale Simulaons)

MMFF (the Merck Molecular Force Field)

DREIDING Generic force field due to Mayo et al. (1990)

UNIVERSAL (UFF)Generic force field due to Rappeet al. (1992)

CVFF/PCFF Force fields for fluorinated hydrocarbons

MM2, MM3, MM4 Developed by Allinger et al. for calculaons on small molecules

COMPASS Commercial force field marketed by Accelrys Inc. Different Force Fields:

AMBER ( Assisted Model Building with Energy Refinement).

CHARMM (Chemistry at HARvard using Molecular Mechanics).

GROMOS (GROenigen Molecular Simulaon)

OPLS (Opmized Parameters for Large‐scale Simulaons)

MMFF (the Merck Molecular Force Field)

DREIDING Generic force field due to Mayo et al. (1990)

UNIVERSAL (UFF)Generic force field due to Rappeet al. (1992)

CVFF/PCFF Force fields for fluorinated hydrocarbons

MM2, MM3, MM4 Developed by Allinger et al. for calculaons on small molecules

COMPASS Commercial force field marketed by Accelrys Inc. Force Field Potenal Funcons

The potenal funcons may be divided into bonded terms, which give the energy contained in the internal degrees of freedom, and non‐bonded terms, which describe interacons between molecules. E V V V V V pot = ∑ r + ∑ θ + ∑ τ + ∑ vanderWaals + ∑ electrostatics bonds angles torsions atoms atoms

Potenals between bonded atoms Potenals between non‐bonded atoms

Total potenal Energy, Epot or Vtot Force Field Potenal Energy Funcons

12 6 σ  σ   V = 4ε  ij  − ij   vanderWaals     (John Lennard‐Jones – 1931)  Rij Rij  Rij      i j qiq j VElectrostatic = (Charles Augusn de Coulomb ‐1785) 4πεRij

r ij 1 2 i j ij 0 Vbonds = kr (rij − rij ) 2

(Robert Hooke ‐ 1660) k j

τijkl j i 1 ijk 0 2 Vangles = kθ (θij −θij ) 2 l i θijk k 1 ijkl Vtorsions = kn (1− cos(nτ )) 2 ∑n (Jean Bapste Joseph Fourier – 1822)

Alternavely, a power‐series expansion of the Morse potenal can be used

Graphical comparison of Morse and power law potenals

Problem with harmonic approximaon:

Bonds cannot break (essence of Molecular Mechanics; no bonds are broken or formed, cannot be used for chemical reacons).

Torsion Angle or Dihedral Angle Energy

The torsional energy is defined between every four bonded atoms (1‐4 interacons), and depends on the torsion (aka dihedral) angle ϕ made by the two planes incorporang the first and last three atoms involved in the torsion

Torsion terms account for any interacons between 1‐4 atom pairs that are not already accounted for by non‐bonded interacons between these atoms

For example: they could be used to describe barriers to bond rotaon from electron delocalizaon (double bonds or paral double bonds), or stereo‐electronic effects Torsion Example – The Single Bond

Using the standard cos3φ potenal, there are three equilibrium posions: ϕ = 180° (trans state) and ± 60° (gauche states).

In pracce, the energies of the gauche states are slightly different than that of the trans state, depending on the atoms involved in the torsion.

To introduce a difference between the stabilies of the gauche and trans conformaons, the torsion funcon can be expanded with addional terms, each with it’s unique contribuon to the rotaonal energy: Electrostacs

Difference in electronegavity between atoms generates unequal charge distribuon in a molecule

Oen electronegavity differences are represented as fraconal point charges (q) within the molecule (normally centered at the nuclei (paral atomic charges)

Electrostac interacon energy is calculated as a sum of interacons between paral atomic charges, using Coulombs law

Naturally, this equaon is also used for modeling interacons between integral charges, such as between ions

qiq j VElectrostatic = 4πεRij

The problem with this approach is that there is no such thing as a fraconal electron, therefore there is no perfect method to derive the paral atomic charges Van der Waals Interacons

Non‐bonded interacon that are not electrostac (e.g. between atoms in noble gas) are labeled van der Waals interacons

Contains dispersion and short‐range components

Dispersion interacons always aracve. Arise from instantaneous dipoles that occur during fluctuaons within the molecular electron cloud

Short‐range interacons are always unfavorable. Also labeled exchange, or overlap, forces. They occur between electrons with the same spin so they do not occupy same region in space (Pauli exclusion principle)

SUMMARY – Force Field Terms Electrostac energy is represented using a set of paral atomic charges van der Waals energy has both weakly aracve and strongly repulsive components and arises from represents electron correlaon

The dispersion term is always negave whereas short‐range energy is always repulsive.

Torsion terms describe bond rotaonal properes that arise from non‐classical effects, such as electron delocalizaon

The remaining bond and angle terms describe covalent bonding

Once we have our force field, what can we do with it?

–Energy minimisaon – Molecular Dynamics –Conformaonal analysis

The accuracy of the output from all these techniques will obviously be sensive to a greater or lesser extent on the parameterizaon of the force field