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1、BulletinofMathematicalBiology(2010)72:1562–1622DOI10.1007s115380109511xIGINALARTICLEDifferentialGeometryBasedMultiscaleModelsGuoWeiWeiaDepartmentofMathematicsMichiganStateUniversityEastLansingMI48824USAbDepartmentofElect
2、ricalComputerEngineeringMichiganStateUniversityEastLansingMI48824USAReceived:24October2009Accepted:21January2010Publishedonline:19February2010?TheAuth(s)2010.ThisarticleispublishedwithopenaccessatAbstractLargechemicalbio
3、logicalsystemssuchasfuelcellsionchannelsmolecularmotsvirusesareofgreatimptancetothescientificcommunitypublichealth.Typicallythesecomplexsystemsinconjunctionwiththeiraquaticenvironmentposeafabulouschallengetotheeticaldeio
4、nsimulationprediction.Inthiswkweproposeadifferentialgeometrybasedmultiscaleparadigmtomodelcomplexmacromolecularsystemstoputmacroscopicmicroscopicdeionsonanequalfooting.Inourapproachthedifferentialgeometrytheyofsurfacesge
5、ometricmeasuretheyareemployedasanaturalmeanstocouplethemacroscopiccontinuummechanicaldeionoftheaquaticenvironmentwiththemicroscopicdiscreteatomisticdeionofthemacromolecule.Multiscalefreeenergyfunctionalsmultiscaleactionf
6、unctionalsareconstructedasaunifiedframewktoderivethegoverningequationsfthedynamicsofdifferentscalesdifferentdeions.Twotypesofaqueousmacromolecularcomplexesonesthatarenearequilibriumothersthatarefarfromequilibriumareconsi
7、deredinourfmulations.WeshowthatgeneralizedNavier–StokesequationsfthefluiddynamicsgeneralizedPoissonequationsgeneralizedPoisson–BoltzmannequationsfelectrostaticinteractionsNewton’sequationfthemoleculardynamicscanbederived
8、bytheleastactionprinciple.Theseequationsarecoupledthroughthecontinuumdiscreteinterfacewhosedynamicsisgovernedbypotentialdrivengeometricflows.Comparisonisgiventoclassicaldeionsofthefluidelectrostaticinteractionswithoutgeo
9、metricflowbasedmicromacrointerfaces.Thedetailedbalanceoffcesisemphasizedinthepresentwk.Wefurtherextendtheproposedmultiscaleparadigmtomicromacroanalysisofelectrohydrodynamicselectrophesisfuelcellsionchannels.Wederivegener
10、alizedPoisson–Nernst–PlanckequationsthatarecoupledtogeneralizedNavier–StokesequationsffluiddynamicsNewton’sequationfmoleculardynamicspotentialsurfacedrivinggeometricflowsfthemicromacrointerface.Fexcessivelylargeaqueousma
11、cromolecularcomplexesinchemistrybiologywefurtherdevelopdifferentialgeometrybasedmultiscalefluidelectroelasticmodelstoreplacetheexpensivemoleculardynamicsdeionwithanalternativeelasticityfmulation.Emailaddress:.1564Weigase
12、sfluidelectronsinsolidsetc.Megeneralmultiscaleanalysisisrequiredfheterogeneoussystems.ThemostintriguingfinatingphenomenononEarthislife.Amazinglylifeencompassesovermethantwentydersofmagnitudeintimescalesfromelectrontransf
13、erprotondislocationonthescaleoffemtosecondstoganismlifetimesonthescaleofyearsovertendersofmagnitudeinspatialscalesfromelectronstoganisms.Sincelifehassomanyscalesinspacetimebiologyissubdividedintomolecularbiologycellularb
14、iologydevelopmentbiologyevolutionarybiologyganismicbiologypopulationbiologyetc.nottomentionemergingfieldssuchassystemsbiologyecologybioinfmatics.Biologyateachscalelevelcollectsenmousamountinfmationwhichcaneasilyoutraceth
15、etheyneededtounderstit.Quantitativeunderstingtheeticalpredictionhaveemergedasakeydisciplineinthecontemparybiology.Therefethecomplexityoflifetheneedfitsunderstingpresentanextradinaryopptunityfmultiscalemodelingsimulation.
16、Duetotheirextradinaryspatiotempalextensionbiologicalscalesaredifficulttointegrate.Aninterestingdevelopmentintheeticalbiologyisthescalerelativitytheyapproachtointegrativesystemsbiology(AuffrayNottale2008NottaleAuffray2008
17、).ThescalerelativitytheyisanextensionofEinstein’stheiesofrelativityfmulatedbyapplyingtheprincipleofrelativitytobothmotionscaletransfmationsofthereferencesystem.Thisapproachhasthepotentialofovercomingthefundamentalhurdles
18、ofmultiscaleintegrationinsystemsbiology.Theinterestedreaderisreferredtotworecentreviewpapersfdetail(AuffrayNottale2008NottaleAuffray2008).Underphysiologicalconditionsmostbiologicalprocessessuchasionchannelsignaltransduct
19、iondeoxyribonucleicacid(DNA)specificationtranionposttranionmodificationtranslationproteinfoldingproteinproteininteractionoccurinwaterwhichconsistsof65–90%humancellweight.Aprerequisitetoquantitativedeionsoftheabovemention
20、edbiologicalprocessesistheunderstingofsolvation—thestaticdynamicalbehaviofmacromoleculesintheaquaticenvironmentthesynergyofsolventsoluteinteractions.Solvationmodelscanberoughlydividedintotwoclasses:explicitsolventmodelst
21、hattreatthesolventinmolecularatomicdetailimplicitsolventmodelsthatgenerallyreplacetheexplicitsolventwithadielectriccontinuumwhilekeepingtheatomicdetailofthebiomolecule(RouxSimonson1999WarshelPapazyan1998Simonson2001Sharp
22、Honig1990TullySmithReiss1970FriesPatey1985BeglovRoux1997).Eachmethodhasitsstrengthsweaknesses.Whileexplicitsolventmodelsoffersomeofthehighestlevelsofdetailtheygenerallyrequireextensivesamplingtoextractthermodynamickiicpr
23、opertiesofinterest.Thisapproachbecomesintractableflargebiomoleculessuchaslargeproteinsmolecularmotsviruses.Ontheotherhimplicitsolventmodelsfocusonthebiomoleculeofinterestprovideonlyameanfielddescriptionofthesolvent.Becau
24、seoftheirfewerdegreesoffreedomimplicitsolventmodelshavebecomepopularfmanyapplicationsinmolecularsimulations(Holst1993Baker20042005FeigBrooksIII2004Dongetal.2008BoschitschFenley20042007).Implicitsolventmodelsrequireasepar
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