ch3-分子动力学模拟基础

主讲教师：卢本卓计算数学与科学工程计算研究所第三章分子动力学模拟基础生物分子模拟与计算时间平均与系综平均•ValuessuchaspressureorheatcapacitygenerallydependuponthepositionsandmomentaoftheNparticlesthatcomprisethesystem.TheinstantaneousvalueofthepropertyAcanthusbewrittenas:))(),((trtpANN•AveragevalueofthepropertyAcanbecalculatedusingintegralapproach:0))(),((1limtNNavedttrtpAA一，遍历假设（Ergodichypothesis）Formacroscopicnumbersofatomsormolecules(oftheorderof1023,Avogadro'snumberis6.02214199×1023)itisnotfeasibletodetermineaninitialconfigurationofthesystem,andtolaterintegrateequationofthemotionwhichdescribeitstemporalevolution.BoltzmannandGibbsdevelopedstatisticalmechanics。遍历性假设：一个分子系统的足够长时间的演化（保持能量守恒）可遍历（或无限接近）任何微观状态。Asystemofmoleculeswillassume,inthelongrun,allconceivablemicro-statesthatarecompatiblewiththeconservationofenergy.),(),(NNNNNNrprpAdrdpA时间平均与系综平均Asinglesystemevolvingintimeisreplacedbyalargenumberofreplicationsofthesamesystemthatareconsideredsimultaneously.Anaveragequantitymeasuredoverlongenoughtimewillapproachtheensembleaverage：时间平均与系综平均AAMdttAtAMitttt11)(1lim00这一是遍历性假设的另一种表达。IstheErgodicHypothesisTrueinGeneral?•No.–Becausesamplingtimeistooshort.•Theergodichypothesisistrueformanyspecialcases.–Wherebarriersbetweenstatesaresmall(ontheorderofRT),wecanmodelthethermodynamicbehaviorofthetwostatesinrelationtoeachother.二，分子动力学模拟•SchrödingerequationiximFdtxdi22U(r)mHΨ(r),HtΨ(r)iiii222ˆˆ•NewtonianmotionTooexpensiveforbiosystems!运动方程•MonteCarlomethodGenerateanensemblewithp.d.f:TkqpHBeqp/)),(),(Moleculardynamicssimulations(Xray/NMR),r(t=0)ΔtmoveatomsΔtTimehistoryThermodynamicandkineticpropertiesmillionsofmoves)(rUrmiiithepotentialenergyU(r1;r2;:::;rN)isasumof~O(N)potentialsforbondedforces,~O(N2)potentialsfornonbondedforces.AboutMolecularDynamicsAboutMolecularDynamics•AtomicNewtonianmotion-approachbiochemistryaccuracy•Allowsfor“bottom-up”approachtopredictingsystemphenomenaatthenanoscale•ComputingAgeImplementation•Computationallyintensive•Advantage:simpletheoreticalbackground•Disadvantage:limitedsystemsizeSimulationmethodologiesUncontrolledapproximations---modeling•predeterminedmodels,•computedmodels,e.g.,topologyfile+diffusioncoeff.Controlledapproximations---algorithms•samplingprotocols,•integrators/propagators,•minimizers,•fastforceevaluationmethods.GoalandconcernsofMDsimulations•Goal:Thermodynamics,kinetics,potencyofchemicalreactionevent.•Concerns:smalltimescale-insufficientsamplingKineticsTheequationsofmotionarechaotic.Tomakesenseoftrajectories,incorporateuncertaintystochasticallyandaskonlyforaverages:TocomputeexpectationofA(Γ(t))whereΓ(t)={r(t),p(t)},UsewheretheΓ(0)arerandomwithp.d.f.ρ(Γ).trialsNiitrialstAN))((1ThermodynamicsThermodynamicscalculations(structureandenergetics)canbeexpressedGoalandconcernsofMDsimulations•Goal:Thermodynamics,kinetics,potencyofchemicalreactionevent.•Concerns:smalltimescale-insufficientsamplingNIHResourceforBiomolecularModelingandBioinformatics•Coarse-grainedparticleMD•Browniandynamics:•Continuumdiffusion•MD/MCwithbiasedpotential/temperatureorotherconditions…iiiiiRrrVrm)(三，力场jiijijijijjiijmjidihedimpaabbrBrArqqnKKKllKrU,612,0.2.020204cos(1lψθ-+anglebondimproperdihedralselectrostaticsvdWω)(rUrmForcefield(力场)referstothefunctionalformandparametersetsusedtodescribethepotentialenergyofasystemofparticles(typicallybutnotnecessarilyatoms).AtypicalAMBERforcefield(力场):•Simple,fixedalgebraicformforeverytypeofinteraction.•Formstemsfromcompromisebetweenaccuracyandspeed.•Variableparametersdependontypesofatomsinvolved.Potentialfunctionalform•Simple,fixedalgebraicformforeverytypeofinteraction.•Formstemsfromcompromisebetweenaccuracyandspeed.•Variableparametersdependontypesofatomsinvolved.Forcefieldfunctionsandparametersetsarederivedfrombothexperimentalworkandhigh-levelquantummechanicalcalculations.All-atomforcefieldsprovideparametersforeveryatominasystem,includinghydrogen,whileunited-atomforcefieldstreatthehydrogenandcarbonatomsinmethylandmethylenegroupsasasingleinteractioncenter.Coarse-grainedforcefields,whicharefrequentlyusedinlong-timesimulationsofproteins,provideevenmoreabstractedrepresentationsforincreasedcomputationalefficiency.各个作用项分解•练习：每一项对应的作用力？ThevanderWaalsPotential•Atomswithnonetelectrostaticchargewillstilltendtoattracteachotheratshortdistances,aslongastheydon’tgettooclose.•Oncetheatomsarecloseenoughtohaveoverlappingelectronclouds,theywillrepeleachotherwithastoundingforceThevanderWaalsPotentialTheConstantsεandRmindependontheatomtypes,andarederivedfromexperimentaldata.pairsnonbondedijijijijijrRrR6min,12min,)(2)(TheElectrostaticPotential•OppositeChargesAttract•LikeChargesRepel•TheforceoftheattractionisinverselyproportionaltothesquareofthedistancepairsnonbondedikkiticelectrostarqqEBondedAtomsTherearethreetypesofinteractionbetweenbondedatoms:•Stretchingalongthebond•Bendingbetweenbonds•Rotatingaroundbondsbondalongrot