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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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