MolecularMechanicsForceFields

BasicPremiseIfwewanttostudyaprotein,pieceofDNA,biologicalmembranes,polysaccharide,crystalla;ce,nanomaterials,diffusioninliquids,…thenumberofelectrons(i.e.thenumberofenergycalculaAons)makequantummechanicalcalcula-onsimpossibleevenwithpresent‐daycomputers.Instead,wereplacethenucleiandelectrons,andtheirinteracAons,bynewpotenAalfuncAons:”Classical”atoms.BasedonsimplephysicalconceptsEnablesthesystemsunderstudytobeVERYlarge(100,000atoms).

ThemolecularinteracAons,alsoknownasthepotenAals,togetherformaforcefield,Aforcefieldisamathema8caldescrip8onoftheclassicalforcesorenergiesbetweenpar8cles(atoms).Energy=func%onofatomicposi%ons(x,y,z)Theforcefieldequa%onconsistsofseveralfunc%onsthatdescribemolecularproper%esbothwithinandbetweenmoleculesTheforcefieldalsocontainsparameters(numbers)inthepotenAalfuncAonsthataretunedtoeachtypeofmolecule(protein,nucleicacid,carbohydrates)TherearemanydifferentforcefieldequaAonsandparametersetsAforcefieldmustbesimpleenoughthatitcanbeevaluatedquickly,butsufficientlydetailedthatitreproducesthekeyfeaturesofthephysicalsystembeingmodelled.

Molecularmechanicsforcefields

Ingeneral,forcefieldscanbeclassifiedaseither:Specific(manyparameters,limitedapplicability,highaccuracy)O>endevelopedinacademiclabsforstudyofspecificmolecularclassesorGeneric(fewerparameters,moregeneraliza-ons,wideapplicability,pooraccuracy)Easiesttouseinpoint‐andclickso>wareForceFieldParameterscancomefrom:Experimentalsources(mainlyfromx‐raydiffrac-on)orTheore-calcalcula-ons(mainlyfromQM)ManyforcefieldsemploysimilarmathemaAcalequaAonsbutdifferintheparametersusedintheequaAons.Itisthereforeextremelydangerousmixtoparametersbetweenforcefields.

Forcefieldclassifica8on

AMBER(AssistedModelBuildingwithEnergyRefinement).CHARMM(ChemistryatHARvardusingMolecularMechanics).GROMOS(GROenigenMolecularSimulaAon)OPLS(OpAmizedParametersforLarge‐scaleSimulaAons)

DifferentForceFields:

MMFF(theMerckMolecularForceField)DREIDINGGenericforcefieldduetoMayoetal.(1990)UNIVERSAL(UFF)GenericforcefieldduetoRappeetal.(1992)CVFF/PCFFForcefieldsforfluorinatedhydrocarbonsMM2,MM3,MM4DevelopedbyAllingeretal.forcalculaAonsonsmallmoleculesCOMPASSCommercialforcefieldmarketedbyAccelrysInc.

AMBER(AssistedModelBuildingwithEnergyRefinement).CHARMM(ChemistryatHARvardusingMolecularMechanics).GROMOS(GROenigenMolecularSimulaAon)OPLS(OpAmizedParametersforLarge‐scaleSimulaAons)

DifferentForceFields:

MMFF(theMerckMolecularForceField)DREIDINGGenericforcefieldduetoMayoetal.(1990)UNIVERSAL(UFF)GenericforcefieldduetoRappeetal.(1992)CVFF/PCFFForcefieldsforfluorinatedhydrocarbonsMM2,MM3,MM4DevelopedbyAllingeretal.forcalculaAonsonsmallmoleculesCOMPASSCommercialforcefieldmarketedbyAccelrysInc.

ThepotenAalfuncAonsmaybedividedintobondedterms,whichgivetheenergycontainedintheinternaldegreesoffreedom,andnon‐bondedterms,whichdescribeinterac8onsbetweenmolecules.

ForceFieldPoten8alFunc8ons

∑∑∑∑∑ ++++=atoms

ticselectrostaatoms

svanderWaaltorsionsanglesbonds

rpot VVVVVE τθ

Poten8alsbetweenbondedatoms Poten8alsbetweennon‐bondedatoms

Totalpoten8alEnergy,EpotorVtot

jiRij

jirij

ij

kθijk

jk

l i

τijkl

−

=

612

4ij

ij

ij

ijsvanderWaal RR

Vσσ

ε

( )20

21

ijijijk

angles kV θθθ −=

ij

jiticElectrosta R

qqV

πε4=

( )20

21

ijijijrbonds rrkV −=

( ))cos(121

τnkVn

ijklntorsions −= ∑

(JohnLennard‐Jones–1931)

(CharlesAugus-ndeCoulomb‐1785)

(RobertHooke‐1660)

(JeanBap-steJosephFourier–1822)

ForceFieldPoten8alEnergyFunc8ons

AlternaAvely,apower‐seriesexpansionoftheMorsepotenAalcanbeused

GraphicalcomparisonofMorseandpowerlawpotenAals

ProblemwithharmonicapproximaAon:

Bondscannotbreak(essenceofMolecularMechanics;nobondsarebrokenorformed,cannotbeusedforchemicalreacAons).

Thetorsionalenergyisdefinedbetweeneveryfourbondedatoms(1‐4interacAons),anddependsonthetorsion(akadihedral)angleϕmadebythetwoplanesincorporaAngthefirstandlastthreeatomsinvolvedinthetorsion

TorsionAngleorDihedralAngleEnergy

TorsiontermsaccountforanyinteracAonsbetween1‐4atompairsthatarenotalreadyaccountedforbynon‐bondedinteracAonsbetweentheseatomsForexample:theycouldbeusedtodescribebarrierstobondrotaAonfromelectrondelocalizaAon(doublebondsorparAaldoublebonds),orstereo‐electroniceffects

Usingthestandardcos3φpotenAal,therearethreeequilibriumposiAons:ϕ=180°(transstate)and±60°(gauchestates).InpracAce,theenergiesofthegauchestatesareslightlydifferentthanthatofthetransstate,dependingontheatomsinvolvedinthetorsion.

TorsionExample–TheSingleBond

TointroduceadifferencebetweenthestabiliAesofthegaucheandtransconformaAons,thetorsionfuncAoncanbeexpandedwithaddiAonalterms,eachwithit’suniquecontribuAontotherotaAonalenergy:

DifferenceinelectronegaAvitybetweenatomsgeneratesunequalchargedistribuAoninamoleculeOmenelectronegaAvitydifferencesarerepresentedasfracAonalpointcharges(q)withinthemolecule(normallycenteredatthenuclei(parAalatomiccharges)ElectrostaAcinteracAonenergyiscalculatedasasumofinteracAonsbetweenparAalatomiccharges,usingCoulombslawNaturally,thisequaAonisalsousedformodelinginteracAonsbetweenintegralcharges,suchasbetweenions

Electrosta8cs

ij

jiticElectrosta R

qqV

πε4=

TheproblemwiththisapproachisthatthereisnosuchthingasafracAonalelectron,thereforethereisnoperfectmethodtoderivetheparAalatomiccharges

Non‐bondedinteracAonthatarenotelectrostaAc(e.g.betweenatomsinnoblegas)arelabeledvanderWaalsinteracAonsContainsdispersionandshort‐rangecomponentsDispersioninteracAonsalwaysaoracAve.ArisefrominstantaneousdipolesthatoccurduringfluctuaAonswithinthemolecularelectroncloudShort‐rangeinteracAonsarealwaysunfavorable.Alsolabeledexchange,oroverlap,forces.Theyoccurbetweenelectronswiththesamespinsotheydonotoccupysameregioninspace(Pauliexclusionprinciple)

VanderWaalsInterac8ons

ElectrostaAcenergyisrepresentedusingasetofparAalatomicchargesvanderWaalsenergyhasbothweaklyaoracAveandstronglyrepulsivecomponentsandarisesfromrepresentselectroncorrelaAonThedispersiontermisalwaysnegaAvewhereasshort‐rangeenergyisalwaysrepulsive.TorsiontermsdescribebondrotaAonalproperAesthatarisefromnon‐classicaleffects,suchaselectrondelocalizaAonTheremainingbondandangletermsdescribecovalentbonding

SUMMARY–ForceFieldTerms

Oncewehaveourforcefield,whatcanwedowithit?–EnergyminimisaAon–MolecularDynamics–ConformaAonalanalysisTheaccuracyoftheoutputfromallthesetechniqueswillobviouslybesensi8vetoagreaterorlesserextentontheparameteriza8onoftheforcefield