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A study of complex systems : from magnetic to biological
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Lovelady, Douglas Carroll
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Cancer
Fractional Brownian Motion
Magnetic Anisotropy
Magnetic Multilayers
Random Walk
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ABSTRACT: This work is a study of complex many-body systems with non-trivial interactions. Many such systems can be described with models that are much simpler than the real thing but which can still give good insight into the behavior of realistic systems. We take a look at two such systems. The first part looks at a model that elucidates the variety of magnetic phases observed in rare-earth heterostructures at low temperatures: the six-state clock model. We use an ANNNI-like model Hamiltonian that has a three-dimensional parameter space and yields two-dimensional multiphase regions in this space. A low-temperature expansion of the free energy reveals an example of Villain's `order from disorder' 81, 60 when an infinitesimal temperature breaks the ground-state degeneracy. The next part of our work describes biological systems. Using ECIS (Electric Cell-Substrate Impedance Sensing), we are able to extract complex impedance time series from a confluent layer of live cells. We use simple statistics to characterize the behavior of cells in these experiments. We compare experiment with models of fractional Brownian motion and random walks with persistence. We next detect differences in the behavior of single cell types in a toxic environment. Finally, we develop a very simple model of micromotion that helps explain the types of interactions responsible for the long-term and short-term correlations seen in the power spectra and autocorrelation curves extracted from the times series produced from the experiments.
Thesis:
Disseration (Ph.D.)--University of South Florida, 2011.
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by Douglas Carroll Lovelady.
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AStudyofComplexSystems:fromMagnetictoBiological by DouglasC.Lovelady Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofPhysics CollegeofArtsandSciences UniversityofSouthFlorida Co-MajorProfessor:DavidA.Rabson,Ph.D. Co-MajorProfessor:Chun-MinLo,Ph.D. GarrettMatthews,Ph.D. LiliaWoods,Ph.D. SagarPandit,Ph.D. DateofApproval: March29,2011 Keywords:magneticmultilayers,magneticanisotropy,can cer,randomwalk,fractional Brownianmotion Copyright c r 2011,DouglasC.Lovelady

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DEDICATION Tomywife

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ACKNOWLEDGMENTS Iwouldliketoacknowledgeallthepeoplewhohaveassistedm eonvariouspartsof thiswork,manyofwhomwereundergraduatesatthetimeofthe irparticipation.They includeRoxaneRokicki,HeatherM.Harper,I.E.Brodsky,Hi epQ.Le,TysonC.Richmond,A.N.Maggi,J.Friedman,andS.Patel.Iwouldalsolike tothankDr.Samuel MokfromtheHarvardMedicalSchoolforprovidinguswiththe celllinesusedinour studyofcancer.Financialsupportwasprovidedatdierents tagesbyResearchCorporation(D.A.Rabson),theFloridaSpaceResearchInstitute (C.-M.Lo)andbyagrant fromNIH/NCI1R03CA123621-01A1(C.-MLo).Iwouldalsolike tothankmyadvisors Drs.DavidA.RabsonandChun-MinLofortheirinvaluableand enthusiasticadviceand participationinthisproject.Finally,IthankDr.Gangara mLaddeforagreeingtobethe chair.

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TABLEOFCONTENTS LISTOFTABLES iii LISTOFFIGURES iv ABSTRACT vi CHAPTER1INTRODUCTION 1 CHAPTER2EXTENDEDSIX-STATECLOCKMODEL4 2.1Introduction 4 2.2TheModelanditsGroundStates 6 2.3Low-TemperatureExpansion 10 2.3.1FirstOrder 10 2.3.2ExpansiontoHigherOrders13 2.4Implications 15 2.5Appendix:Transfer-MatrixTechniques17 CHAPTER3DISTINGUISHINGCANCEROUSFROMNONCANCEROUS CELLS 25 3.1Introduction 25 3.2ExperimentalMethods 25 3.3StatisticalMeasuresofNoise 28 3.4TwoSimpleModels 35 3.5Applications 37 3.6AppendixA:ComparingDiscriminants383.7AppendixB:KurtosisandCorrelationLength40 CHAPTER4DETECTINGEFFECTSOFLOWLEVELSOFCYTOCHALASINBIN3T3FIBROBLASTCULTURES45 4.1Introduction 45 4.2ExperimentalMethods 46 4.3StatisticalMeasuresofNoise 50 4.3.1Long-termcorrelations 50 4.3.2Short-TermCorrelations 53 4.4MeasureSpace 55 4.5Discussion 58 i

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CHAPTER5STATISTICAL-MECHANICALMODELOFMICROMOTION60 5.1Introduction 60 5.2Hamiltonian 62 5.3SimulatingCellMovementwithKineticMonteCarlo63 5.3.1HistoryofKMC 64 5.3.2Thekinetic-Monte-CarloAlgorithm64 5.4Applicationtooursystem 68 5.5Results 69 5.6Appendix:DerivationofthePoissondistributionforou rsystem75 REFERENCES 79 ABOUTTHEAUTHOR EndPage ii

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LISTOFTABLES Table2.1Boltzmannfactorcontributionstothefreeenergy forparticularmagneticexcitations. 11 Table3.1Power-spectralmeasuresofHOSE(non-cancerous) andSKOV(cancerous)resistiveandcapacitivenoiseseries.30 Table3.2Additionalmeasuresoflong-timecorrelationint henoisetimeseries, Hurstanddetrended-ructuationexponents.30 Table3.3Measuresofshort-timecorrelationinthenoiseti meseries.31 Table3.4Percentagesofcorrectidentications.34Table3.5Kurtosisaveragedoverallruns,standarddeviati onofkurtoses,and standarddeviationofthemeans. 34 Table4.1Long-termcorrelation.Power-spectral,Hurst,a ndDFAmeasuresof conruentlayersof3T3broblastresistivenoiseseriesave ragedover allrunsatdierentconcentrationsofthetoxincytochalasi nB.51 Table4.2Short-termcorrelation.Firstzero( Z )andrst1/ecrossing( E )of theautocorrelationfunction. 53 Table4.3Theresultsfromcalculationsdoneon concentration spheresinour constructed measure space. 58 iii

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LISTOFFIGURES Figure2.1Theground-statephasediagramfor(partof)the y = Z planefor L =4. 9 Figure2.2Adiagramofahierarchyofphasesemanationfroma multiphasepoint atzerotemperatureuptoaCurietemperature.16 Figure2.3SpinEnvironments 24 Figure3.1Schemeofdataextractionfromnoise.27Figure3.2Projectionalongthersttwoprincipalcomponen ts.42 Figure3.3Theincrementsofaniterandomwalkwithpersist ence.43 Figure3.4Fractionaldiscrepancies. 44 Figure4.1Resistanceovertimefordierentexposerstothet oxincytochalasin B. 47 Figure4.2AverageresistancesofECISruns. 48 Figure4.3Power-spectral-densityplots. 49 Figure4.4Powerslopeversusconcentrationofthetoxincyt ochalasinB.54 Figure4.5First 1 e crossingoftheautocorrelationfunctionplottedagainstc oncentrationofthetoxincytochalasinB.56 Figure4.6Therstandsecondprincipalcomponents57Figure5.1Thebarrierenergybetweentwopotentialwells66Figure5.2Thebarrierenergybetweentwopotentialwells66Figure5.3Timeseriesforarunwith1024MCtimesteps70Figure5.4log-logplotofthederivativeofthetimeseries7 1 Figure5.5Changingslopeofthelog-logplotofthepowerspe ctrumforthederivativeofthetimeseriesasthevaluesoftheinteractionenerg ies J and S alsochangebut J + S =5 72 iv

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Figure5.6Cell-cellsignalingstrengthversusthepowersl ope74 v

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ABSTRACT Thisworkisastudyofcomplexmany-bodysystemswithnon-tr ivialinteractions.Many suchsystemscanbedescribedwithmodelsthataremuchsimpl erthanthe real thingbut whichcanstillgivegoodinsightintothebehaviorofrealis ticsystems.Wetakealookat twosuchsystems.Therstpartlooksatamodelthatelucidat esthevarietyofmagnetic phasesobservedinrare-earthheterostructuresatlowtemp eratures:thesix-stateclock model.WeuseanANNNI-likemodelHamiltonianthathasathre edimensionalparameter spaceandyieldstwo-dimensionalmultiphaseregionsinthi sspace.Alow-temperature expansionofthefreeenergyrevealsanexampleofVillain's `orderfromdisorder'[81,60] whenaninnitesimaltemperaturebreakstheground-stated egeneracy.Thenextpartof ourworkdescribesbiologicalsystems.UsingECIS(Electri cCell-SubstrateImpedance Sensing),weareabletoextractcompleximpedanceseriesfr omaconruentlayeroflive cells.Weusesimplestatisticstocharacterizethebehavio rofcellsintheseexperiments. WecompareexperimentwithmodelsoffractionalBrownianmo tionandrandomwalks withpersistence.Wenextdetectdierencesinthebehavioro fsinglecelltypesinatoxic environment.Finallywedevelopaverysimplemodelofmicro motionthathelpsexplain thetypesofinteractionsresponsibleforthelong-termand short-termcorrelationsseen inthepowerspectraandautocorrelationcurvesextractedf romthetimesseriesproduced fromtheexperiments. vi

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CHAPTER1 INTRODUCTION Thisdissertationisastudyoftwocomplexsystems,onemagn eticandtheotherbiological.Bothprojectshaveenoughinteractingdegreesof freedomthatIamobligedto usecomputerstostudythem,althoughIhavealsobeenableto applysymboliccomputer algebra(inthemagneticproject)andanalyticaltechnique s.TheknowledgeIgainedfrom thisrstprojecthelpedmetounderstandhowtomodelcomple xsystemsandwasofgreat benetwhenIbegantothinkonhowImightcreateasimplemode lofasignalgenerated byacellcultureplacedintoanelectriccell-substrateimp edancesensing(ECIS)device. Inchaptertwoweshalluseaspinsystemtomodelamagneticsy stemandinthenal chapterweuseaspinsystemtomodelabiologicalsystem.Inb othwehaveapplied statistical-mechanicaltechniquesinourstudyoftheseco mplexsystems. Thesecondchapterdescribeshowanextendedsix-statecloc kmodelisusedtostudya multiphaseregionofhelimagneticsuperlatticesatlowtem perature.Weusethemodelto studytherichvarietyofmagneticphasesthatareobservedi nrare-earthheterostructures atlowtemperatures[42].Section2.1motivatesthesix-sta teclockmodelandintroduces notationthatisusedthroughoutthechapter.Themodelcons idersonlynearestandnextnearestneighborinteractiontodescribethemagneticphas esseeninmaterialssuchas holmium.Inthenextsectionweextendthemodeltoincludetw omoreenergyterms thatmodeltheinteractionofspinsoneithersideanonmagne ticspacerthatliesbetween magneticlayers.Indoingsowegofromaone-dimensionalpar ameterspacetoathreedimensionalparameterspace.Wendinthisspacemulti-pha seregionsasopposedtothe multiphasepointfoundwiththesix-stateclockmodel.Tomy knowledgewearetherst tondsuchregions.Alow-temperatureexpansiontorstord erisachievedinsection3, 1

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wherewendmanyofthemultiphaseregionsarechangedintos eparateareaseachhaving justonephaseseparatedbyboundaries.However,wediscove ratrianglewhereaninnite numberofphasesstillpersist.Thisbringsupaninterestin gquestion.Howisitpossible byraisingthetemperaturetogofromacompletelydisordere dstatetoanorderedstate? ThisisanexampleofVillain's`orderfromdisorder'[81].I nsection4weexpandthefree energytostillhigherorderandinthelastsectiondiscusst heimplications. InChapter3wediscusselectriccell-substrateimpedances ensing(ECIS).Thisdevice isusedtomonitorcellbehaviorandissensitivetocellmorp hologicalchangesandcell motility.InourexperimentsweuseECIStoproducestimeser iesdatafromaconruent layeroflivecells.Theructuationsinthetimeseriesarepr oducedbythemovementsof thecellsastheydisturbacurrentofabout1 A.Wedevelopatechniquethat,usingonly thenoiseinthetimeseries,distinguishescancerousfromn oncancerouscultures.Section1 motivatesourstudywhileSection2introducestheexperime ntalmethodsweusetoobtain ourtimeseries.Section3describesourmeasuresofnoise,o uranalysis,andourresults. Section4wecompareourresultswithtwosimplemodels:frac tionalBrownianmotionand therandomwalk.Thenalsectiondiscussesapplications. Wenextturnourattentiontotestingthesensitivityofthet echniquebytryingto detecttheeectsoflowlevelsofcytochalasinBin3T3brobl astcultures.Thisisthe subjectofChapter4.Onceagainweworkwithconruentlayers ofcells,butthistime weintroducethetoxincytochalasinBintothecultures.Thi stoxinbreaksdownthe cytoskeletonstructureofthecells.Usingouranalysis,we areabletoshowthatthenoise spectrumdistinguishesdierentconcentrationsmoreeecti velythanaverageresistance. Asbefore,section1motivatesourstudyandintroducesthek eyideas.Section2describes ourexperimentalmethods.Insections3wepresentourresul ts.Insection5wediscussthe meaningofourresultsandthepossibilitythatthecorrelat ionsthatweseeinoursystems areinsomesenseameasureofthecommunicationbetweencell sinourcultures. Wedevelopinthelastchapterasimplemodelofmicromotion. Weuseaversionofthe q -statePottsmodeldevelopedbyGranerandGlazier[37]tomo deltheenergyinteractions 2

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onasquarelatticeofspins.Eachofthe q dierentcolorsofthePottsmodelrepresents adierentcell,andsoonecanrepresentabiologicalsystemo nalatticeofspins.We developamodelofcell-cellcommunicationbasedonParBak' ssand-pilemodel[4]and modelthedynamicsofthesystemusingkineticMonteCarlo.W eextractfromourmodel atimeseriesthatcanbecomparedwiththetimeseriesproduc edfromatypicalECIS experiment.Asusual,section1ofthechaptermotivatesand introducesthekeyconcepts. Section2introducesandderivestheenergyfunctionforour system.Adescriptionof kineticMonteCarloisrealizedinsection3.Section4tiesa llthetoolstogetherandshows howweuseitforoursimplemodelofECIS,andthelastsection presentsourpreliminary resultsandpossibledirectionsforthefuture. 3

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CHAPTER2 EXTENDEDSIX-STATECLOCKMODEL 1 2.1Introduction Layeredplanesofrare-earthmetalsexhibitawealthofmagn etically-orderedphasesat lowtemperature.Inhelimagneticphases,spins(treatedcl assically)alignferromagneticallywithineachplane,withanaxialRKKYinteractionresp onsibleforaprogressionof spinanglesthroughsuccessiveplanes[8,43].Strongeasyaxisanisotropymayfrustrate thenaturalRKKYpitchangle,leadingtoamultitudeofpossi blephasescharacterized bythenumberoflayersseparatingskips,or\walls,"inthep atternofpitchangles.In theaxial-next-nearest-neighborIsing(ANNNI)[27,28,78 ,29]andrelatedclockmodels [88,86,68,70,61,55],asingleparametercontrolstherela tivestrengthsofcompeting interactions,andatasinglevalueofthisparameter,inni telymanyphasescoexist;this iscalledamultiphasepoint.Sincethesephasescoverallal lowedspacingsbetweenwalls, suchphasesareindistinguishablefromrandomsequences.T husthezero-temperature stateisdisordered.Thisdisorderisbrokenatinnitesima ltemperatureinanexampleof \orderfromdisorder"[81,60].Wenowaskwhathappensinamo delofhelimagneticheterostructureswithathree-dimensionalparameterspace:w eidentifyfullytwo-dimensional multiphaseregionsandinvestigatethetopologyofthelowtemperaturephasediagram. Withthegiantmagnetoresistiveeect[3]inferromagnetic/ nonmagneticsuperlattices havingspawnedimportanttechnologicalapplicationsthat reachedthemarketaround1997 [21,2],itseemspractical,aswellastheoreticallyintere sting,toexaminethepossiblephases ofhelimagnetic/nonmagneticsuperlattices.Suchsuperla tticeshavebeendepositedusing 1 Thischapterhasbeenpublished,inslightlydierentform, byD.C.Lovelady,H.M.Harper,I.E. Brodsky,andD.A.RabsoninJ.Phys.A:Math.Gen. 39 5681(2006). 4

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molecular-beamepitaxy,alternatingdysprosium[24],erb ium[11],orholmium[42]with non-magneticyttriumspacerlayersaswellasholmiumwithl utetium[77].Surprisingly, neutron-scatteringexperimentsshowthatthehelicityoft hespinsintherare-earthlayers ispreservedacrossthespacers,withthemagneticmomentsf orminglong-period\spin-slip" phases[42].RKKY-likepolarization[15]ofconductionele ctronsinthenon-magneticlayersisagainimplicated[18,19];inanycase,wecanmodelthe indirectexchangeacross non-magneticspacersinparallelwiththatbetweensuccess ivemagneticplanes.Iftheexchangeparameterscanbecontrolledwithpressure,externa lelds,orspacer-layerthickness, suchsystemscouldpossiblybeusefulasmagneticsensorsor indata-storageapplications. Axiallymodulated,high-order,commensuratephasesareno tlimitedtorare-earthheterostructures:SzpilkaandFisher[78]citehalfadozenoth ersystemsinwhichsuchphases havebeenobserved,rangingfromCeSb[67]toferroelectric thiourea[54,22]. Seno etal. [70]appliedtheANNNIideastoacaseofinnitehexagonalan isotropy,the six-stateclockmodel,relevant,forexample,tobulkholmi um. 2 Aspin inthe j th plane pointsinadirectionthatisanintegralmultiple, n j ,of2 = 6.Atzerotemperature,allthe spinsinaplanepointinthesamedirection( n j ),andthemodeliscontrolledbyasingle parameter,theratio x ofthestrengthofthenext-nearest-axial-neighborantife rromagnetic ( J 2 )tonearest-axial-neighborferromagnetic( J 1 )interaction,withtheaxialtermsinthe Hamiltoniansumming J 1 cos(2 ( n j +1 ; n j; ) = 6)and+ J 2 cos(2 ( n j +2 ; n j; ) = 6).For 0 1ahelimagnetinterruptedbywallseverysecondlayer.Atth e singlepoint x =1intheone-dimensionalphasediagram,innitelymanypha sescoexist inthegroundstate.Werepresentthehelimagneticphase(1 = 3 1arerepresented by ::: 00330033 ::: and ::: 01 j 34 j 01 j 34 ::: :thislastisthoughtofasamodicationofthe helicalphasebytheinsertionofskips,orwalls(denoted\ j "),everysecondlayer.Thewalls 2 Anextensionofthisworkpresentedasmall-inverse-anisot ropyexpansionabouttheclockmodeland againfoundahierarchyofphasesemanatingfromthemultiph asepointatinniteanisotropy[71]. 5

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areanalogoustodomainwallsintheANNNImodel.Atthemulti phasepoint, x =1,in additionto ::: 00330033 :::; ahelicalphasewithwallsplacedanywhereatleasttwolayer s apartisagroundstateofthesystem, e.g. ::: 01 j 345 j 12 j 450 ::: .Aconvenientnotationin ANNNI-typemodelslabelsaperiodicphasebythespacingsbe tweensuccessivewalls:thus, thislastexampleis h 23 i ,thephasewithwallseverysecondlayer h 2 i ,andthebarehelical phasewithoutwalls h1i .Inalow-temperatureexpansion,Seno etal. followedahierarchy ofphases(similartowhatwedescribebelow)andshowedthat eachphasebetween h 23 i and h1i acquiresaregionofstabilityatinnitesimaltemperature Theforgoingmodelsimpliestheactualmagneticstructure ofbulkholmium.Neutron scatteringgivestheturnangleperatomiclayeras30 ratherthan60 ,withmoments bunchedinpairsaroundthesixeasyaxes[9,42],andwhileth eaverageturnangleincreases inlms,theeectisthoughttobeduetointerspersalofsingl etsamongthepairs;thus the h 3 i phaseinthesimpliedmodelmightactuallyrepresentmomen ts ::: 00122344 ::: wherepairsofrepeatedspinslieafewdegreesbeforeandaft ertheeasy-axisdirection(see Fig.14ofReference[42]).Themodel,oritspresentextensi ontosuperlattices,wasmeant nottoreproducerealisticdetailsofaparticularrare-ear thhelimagnetbutrathertoreduce asystemwithcompetingcrystal-eldandexchangeinteract ionstothesimplestform,in whichexactresultsarepossible,soastoinvestigateunive rsalpropertiesoftheresulting hierarchyofcommensurate,longitudinally-modulatedspi n-slipphases. 2.2TheModelanditsGroundStates Weconsiderasuperlatticeinwhichblocksof L magneticlayersareseparatedbynonmagneticspacerscharacterizedbyeectivecouplings J 0 1 and J 0 2 ;thissimpleextensionof thebulkmodelof[70]givesthefullHamiltonian H = 1 2 J 0 X i;; ( ) cos 2 6 ( n i n i ) 6

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J 1 X i; cos 2 6 ( n i n i +1 ; ) + J 2 X i; cos 2 6 ( n i n i +2 ; ) J 0 1 X i; 0 cos 2 6 ( n i n i +1 ; ) + J 0 2 X i; 0 cos 2 6 ( n i n i +2 ; ) ,(2.1) where i labelslayers, aspinwithina(simple-hexagonal)layer,and ( )itsnearest neighbors.Theunprimedsumsinthesecondlinearetakenonl yoverbondsthatdonot straddleanon-magneticspacer,whiletheprimedsumsinthe thirdlinearetakenonlyover bondsthatdo.Forpurposesofthelow-temperatureexpansio n,thein-planeferromagnetic couplingconstant J 0 istakentobepositiveandmuchstrongerthananyoftheaxial couplings[70].Sincewearelookingforhelicalphases,wet akealloftheremainingfour couplingsalsotobepositive.(Certainnegativecouplings areinfactrelatedtothepositive sectorbysymmetriesof H .)Themodelreducestothatof[70]when J 0 1 = J 1 and J 0 2 = J 2 or,equivalently,when L =1.Thethree-dimensionalcouplingspaceisgivenby x = J 2 =J 1 y = J 0 1 =J 1 ,and z = J 0 2 =J 1 ;itisconvenienttoset J 1 =1. Wegeneralizethepreviousnotationtoaccommodatestateso fasuperstructureinwhich blocksof L magneticlayersareseparatedbynon-magneticspacers,den otedby jj ,with thearrangementrepeatedperiodically.(Thesymbol jj maydenoteanynumberofatomic layersofthenon-magneticmetal.)Sincethedirectinterac tionsin(2.1)extendamaximum oftwolayersintheaxialdirection,wallsareclassiedint hreecategories.Awallatleast twolayersfromaspacerhasthesameenergycostasinthebulk modelandistermeda type-1wall,forexample( L =5) ::: jj 0123 j 50 jj 12 ::: .(2.2) Insertionofawallonelayerfromanon-magneticspacerhasa dierentenergycost,since a J 0 2 bondisbroken.Thisistermedatype-2wall: ::: jj 01234 j 0 jj 12 ::: .(2.3) 7

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Atype-3wallcoincideswithanon-magneticspacer: ::: jj 012345 jjj 12 ::: .(2.4) Helicalcongurations,including h1i itself,thatdierfrom h1i onlybytheinsertion ofwallsarecalledwallstates.Thesestatespreservethese nseofhelicity(positiveor negative).Weconsider L 3,as L =1isthesameasbulk,while L =2omitsthe J 2 ( x ) parameterandsohasonlyatwo-dimensionalparameterspace .Itisalsolesslikelytobe ofexperimentalinterest. Astraightforwardcalculationyieldsthetotalenergyofaw allstateasafunctionofthe densities W i ofwallsofthethethreetypes: E wall = 1 2 L (1+ x )( L 2)+1+ y +2 z +(1 x ) W 1 + 1 x + z 2 W 2 +( y z ) W 3 .(2.5) Asintheoriginalmodel,successivewallsareenergeticall yforbidden.Weseekregionsof thethree-dimensionalparameterspaceinwhichtheinserti onofawallofsometypecosts noenergy:thisoccurswhenthecoecientofoneormoreofthe densities W i vanishes. Thustheplanes x =1,( x + z ) = 2=1,and y = z allpotentiallyconstitutemultiphase regions;however,itisalsonecessarytoconsidercompetin gnon-wallstates,whichmay havelowerenergies.Forpresentpurposes,weshallconcent rateonthe y = z plane,for whichtype-3wallscostnoenergy.Sinceanegativeenergyfo rtype-2wallswouldshut type-3wallsout,weexaminethepartofthe y = z planetotheleftofthe x + z =2line. Byconsideringpointstotheleftoftheline x =1,weexcludetype-1wallsaswell.For L =4,directcalculationgivesthephasediagramofFigure2.1 .Anexhaustivecomputer search(ofphasesoflength3 L =12withtwistedperiodicboundaryconditions)veried thatthewall-stateenergy(2.5)islowerthanthatofanycom petingphaseinsideatriangle inthe y = z plane,whichconstitutesamultiphaseregion.Comparablew all-stateregions werecalculatednumericallyfor L intherange3{11. 8

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Figure2.1.Ground-statephasediagramfor(partof)the y = z plane, L =4.The horizontalaxisgivesthenormalizedbulksecond-neighbor coupling,theverticalthecouplingsacrossnon-magneticspacers.Outsidethetriangled elimitedbydot-dashedlines,the groundstatesareasindicated.Insidethetriangle,wallst atesarethegroundstates.(An exhaustivesearchfoundnolower-energystatesoflengthup to3 L .)Similarground-state phasediagramswerecalculatedforothervaluesof L .Therst-orderlow-temperatureexpansiongives h1i totheleftofthedottedlinewithinthetriangleandthe h 1 i phasetothe right;onthelineitself,thesephasesandtheirprogenycoe xist,requiringahigher-order low-temperatureexpansiontodistinguish.Onroughlytheu pperhalfoftheleftlegofthe dottedline,from z =11 = 9to z =13 = 9,webelieveinnitelymanyphasesoftheform h 1 k 2 i coexisttoallorders. 9

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Sinceweareconcentratingonaregioninwhichtype-1andtyp e-2wallsareexcluded, whiletype-3wallscostnoenergy,weadaptthenotationof[7 0]tocountmagneticblocks, ratherthanmagneticlayers,betweenwalls.Thus,forexamp le,with L =4, h 1 i hasawall coincidingwitheachspacer,whilethe h 2 i phasehasawallateveryotherspacer.Sinceno restrictionpreventsadjacentwallsofthistype,thecount ofpossiblephasesissimply2to thepowerofthenumberofmagneticblocks;thisrepresentsa simplicationrelativetothe ANNNIandotherrelatedmodels[65].2.3Low-TemperatureExpansion2.3.1FirstOrder Thenovelfeaturepresentedbythecurrentproblemisthemul tiphasetriangle(for L =4 orasimilarpolygonforother L )throughoutwhichinnitelymanyphasescoexistatzero temperature.Aninterestingtheoreticalquestionishowth ermaldisordercandistinguish thefreeenergiesofallthesephasesinthegivenregion. AlthoughtheHamiltonian(2.1)containsonlyrst-andseco nd-neighboraxialterms,a non-zerotemperatureintroduceseectivelong-rangeinter actionsthroughanaxialchainof thermally-excitedspins,eachpointinginadirectionatva riancewithitsin-planeneighbors [27,28,70].ByanalogytotheANNNImodel,wecallsuchexcit ations\spinrips."Since thenumberofwaysanexcitationofaparticularenergymayoc curdependsonthestate, rippedspinsprovideanentropicmechanismfordistinguish ingthefreeenergiesofwall statesatinnitesimaltemperature.Ifthe i th excitation,whichmayinvolveseveralspins, hasanenergy E i relativetotheground-stateenergyperspin E 0 andcanbeplacedonthe latticeof N spins g i dierentways,thefreeenergyperspinisgivenbythelinkedcluster theorem[84]: f = E 0 k B T X i r i e E i ,(2.6) 10

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where r i =lim N 0 g i =N istheintensivepartof g i =N .(Thelimitdiscardsthoseterms in g i thatgoashigherpowersof N ;suchtermscomefromindependentclustersofspin excitations.) Weapplythemethodrsttoanisolatedspinrip,whichmayocc urinalayeradjacent tooronelayerseparatedfromaspacer,oritmay( L> 4)occurinbulk.Anisolated spinripinbulkgivesthesamecontributionto f regardlessofphase,sowecalculatethe energiesandcounts r i justforthersttwocases,leadingtotheweightsinTable2. 1.The case L =3requiresspecialtreatmentbecausethecostofanexcitat ioninthelayerinthe middleofablockdependsonthepresenceorabsenceofwallso nbothsides. Table2.1.Contributionsto(2.6)areformedbyacount(pers pin)ofthenumberofways offormingtheexcitationtimesaBoltzmannfactor.Theleft columngivesanexampleof theexcitationunderconsideration,wherethecaret( ^ )markstheplaneinwhichasingle spinisrotated(\ripped")plus60 orminus60 fromtheangleofitsneighborsinthe plane.ThesecondcolumngivestheBoltzmannfactor,andthe remainingcolumnsgivethe intensivecounts r i weightingtheBoltzmannfactorforthecases h 1 i h 2 i ,and h1i L is thenumberofmagneticlayersinablock.Thelastthreerowsa pplyonlyto L =3.Here, istheinversetemperature, q =exp( J 0 = 2), t [=6]thenumberofin-planenearest neighbors,and r =exp( J 1 = 2). intensivecount excitationBoltzmannfactor h 1 ih 2 ih1i 1.45 ^01 jj 23 q t r 1 x +2 z + r 1+2 x z 01 =L 2 =L 2.450 ^1 jj 23 q t r 2 x y +2 z + r 1+2 x +2 y z 01 =L 2 =L 3.45 ^01 jjj 34 q t r 1 x + z + r 1+2 x + z 2 =L 1 =L 0 4.450 ^1 jjj 34 q t r 2 x 2 y + z + r 1+2 x + y + z 2 =L 1 =L 0 5.0 jj 1 ^23 jj 42 q t r 1+ z 001 =L 6.0 jjj 2 ^34 jj 5 q t r + r 1+3 z 01 =L 0 7.0 jjj 2 ^34 jjj 02 q t r 1+2 z 1 =L 00 Weconsider L 4rst.Iftherearenotype-3walls,theonlysingle-spinexc itations (otherthanbulk)willbeofoneofthetypesinthersttworow sofTable2.1.Thisdescribes the h1i phase.Ifaphasehasthemaximumdensityoftype-3walls,the excitationswillbe 11

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ofthetypesinthesecondtworows.Thisisthe h 1 i phase.Tothisrstorderinthelowtemperatureexpansion,anyotherwallphase( e.g. h 2 i )willhaveafreeenergyintermediate betweenthesetwocases.Thuswelookrstforthecoexistenc eof h 1 i and h1i .Subtracting rows1and2fromthesumofrows3and4givesthefree-energydi erence f = f h 1 i f h1i = 2 L k B Tq t r 1 x + z + r 1+2 x + z + r 2 x 2 y + z + r 1+2 x + y + z r 1 x +2 z r 1+2 x z r 2 x y +2 z r 1+2 x +2 y z ,(2.7) where =1 = ( k B T )istheinversetemperature, t thenumberofin-planenearestneighbors, q =exp( J 0 = 2),and r =exp( J 1 = 2).Setting f =0and y = z yieldstheexpression r 3 x = r z + r 1 r 1 2 z 1+ r z 2 r 2 z r 2 .(2.8) Inthezero-temperaturelimit, r 0,sothepowerof r withthesmallestexponentdominates.Thisallowustosolveforthecoexistenceline, x = 8>>>>>><>>>>>>: 2 3 for0
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interpolatebetweenthecountsof h 1 i and h1i :thatis,asingle-spinexcitationinthe middleplaneofamagneticblockdistinguishesnotonly h 1 i from h1i butalsoeachfrom h 2 i .Thus,therst-orderexpansionmustpotentiallyconsider threecoexistencelines.In theevent,thethreecollapsetoone.For z> 0,allwallphasescoexistontheline z = 3 2 (1 x )( L =3).(2.10) For z> (3 = 2)(1 x ),the h 1 i phasehasthelowestfreeenergy,whileforsmaller z ,the h1i phasehasthelowestfreeenergy.2.3.2ExpansiontoHigherOrders Thehierarchyofpotentialphasesinthelow-temperatureex pansionhasbeendescribed wellelsewhere[27,28,86,87]andsowillonlybesummarized .Atanyorderoftheexpansion,acoexistenceregionhasbeenestablishedbetweentwo \parent"phasesandinnitely manyotherwallstates.(InFigure2.1for L =4,thisregionisthezig-zagline,onwhich, torstorder,parents h 1 i and h1i coexistwithallotherwallstates.)Spinexcitationsto thisorderoftheexpansioncannotdistinguishtheparentsf romtheotherwallstates,but byaddingsomenumberofadditionalspinexcitations,linke dtothoseofthegivenorder, wecandistinguishthetwoparentsfroma\child"phasemadeb yconcatenatingoneperiod ofeachparent.Asexamples,thechildof h 1 i and h1i is h 2 i ,whilethatof h 1 i and h 2 i is h 12 i .Aconnectedchainofspinexcitationscan\see"thepresenc eorabsenceofwallsover itslength;viewedanotherway,thisleadstoaneectivelong -rangeinteractionbetween walls. Whileinprincipleonecouldcontinuetheenumerationofcon nectedexcitationsoftwo, three,andmorespinsalongthelinesofTable2.1,atransfer -matrixtechnique[88,70]iswell suitedtocomputersymbolicalgebra.Wedeferimplementati ondetailstotheappendix. Thematricesaremoreinvolvedthanthosein[70],sotheresu ltsareforspeciccases,from whichweconjecturegeneralizations. 13

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Inrstorder,wehavealreadyseenthetwo-dimensionalmult iphaseregionshrinktoone dimension(Figure2.1).Wewishtondoutwhetherthelinesh rinksfurthertoapointor setofpoints,orwhethertheline,oraportionoftheline,be haveslikeamultiphasepoint, withtheadditionaldegreeoffreedomessentiallyirreleva nt.Itisalsoofinterestwhether allwallstatesdescendingfrom h 1 i and h1i attainstabilityoronlyasubset. Wecarriedoutthelow-temperatureexpansionformagneticb locksoflength L between 3and17;exceptfortheinterestingcaseof L =4,thehierarchyterminatesafterjusta fewphases.Asidefrom h1i ,theonlystablephasesfoundfor L =4wereoftheform h 1 k 2 i ,0 k 27(thehighestcalculated)and k = 1 ( i.e. h 1 i ).Thisresemblesthe ANNNImodel[27,28]morethansomeclockmodelsinthatthere donotexisttwophases 4 all ofwhoseprogenyattainstability.VillainandGordon[82]( seealso[78])distinguish aDevil'sstaircase[6]froma\harmless"one.Inboth,amult iphasepointgivesrisetoa largenumberofphasesthatapproachesinnityat T 0.However,inthelattercase,at any nite T> 0,itisarguedthatonlynitelymanyphasesarestable.Sinc eourmodel failstondaninnitehierarchyof\mixedphases"[86],wec onjecturethatourstaircase maysimilarlybeharmless. Thewaythe h 1 i h1i coexistencelinebreaksupfor L =4isalsoofinterest.Itintersects themultiphasetriangle(Figure2.1)for1 = 3 z 13 = 9;outsidethisregion,itceasesto describecoexistenceof ground states.Thesymbolictransfer-matrixcalculationndstha t h 2 i isstableonthelineonlyfor3 = 4 z 13 = 9.Below3 = 4,thereisarst-orderphase transitionbetween h 1 i and h1i .Thephase h 12 i isstableat z =3 = 4andthenagain for11 = 9 z 13 = 9.Allsubsequentphases h 1 k 2 i forwhichwewereabletoextract symbolicresults( k 5)arestablefor11 = 9
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For L =3,thecoexistenceline(2.10)intersectswiththeregioni nwhichwallstates havethelowestenergyfor0 z 1.Thestates h 1 i h1i h 2 i h 12 i ,and h 3 i arestableon thislinesegment,butnootherphases. For L =5,thecoexistencelineisagain(2.9),whichpassesthroug hthewall-stateregion for0 z 13 = 9.Thesamephasesarestableasfor L =3: h 2 i for0 6:thecoexistenceline(2.9)intersects withthewall-stateregionfor0 z 13 = 9.Evenvaluesfor L (wecomputed8,10,12,14, and16)givearst-ordertransitionbetween h 1 i and h1i allalongthecoexistenceline.No otherphasesarestable.Forodd L (7,9,11,13,15,and17),thephases h 2 i h 12 i ,and h 3 i arealsostablefor0
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Figure2.2.Diagramofahierarchyofphasesemanatingfroma multiphasepointatzero temperatureuptoaCurietemperature.Thehorizontalaxisr epresentsaratio x ofcoupling strengths,whichat1leadstozero-temperaturedisorder.R aisingthetemperaturefrom thevicinityof x =1givesasuccessionofstablephases.AdaptedfromFigure1 of[70], whereitshowsanumericalmean-eldcalculationonthebulk six-stateclockmodel.Inthe presentcontext,itcanbethoughtofasschematicforthe T> 0behaviorofthesystemof Figure2.1atsomepointalongthezig-zagline,where x representsatransversedimension. thetemperatureincreases,thevolumesofstabilitydoaswe ll,sothatnophaseslienearby, thusstabilizingasinglephase. Interestingly,thecoherencelengths ofthebasal-planeholmiummomentsinHo/Er superlatticeshavebeenfoundto increase withtemperature T between8Kto100K[74]. SinceEracquiresamomentbelow100K,theexperimentalsyst emisconsiderablymore complexthanoursimplemodel;moreover,toofewtemperatur esweremeasuredtopermit acomparisontotheplateauxonewouldexpectin ( T )fromFigure2.2.Asimilareect isobservedinEr/Lu[73,18]. Thequestionofcommensurate versus incommensuratemagneticmodulationalsoawaits experimentalresolution.Inthelow-temperatureexpansio n,incommensuratephasesare onlyapproached,asthelimitofahierarchyofcommensurate phases,whilethebulkmean16

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eldcalculation(Figure2.2)suggeststhattheselimiting phaseswilloccupyavolumeof measurezerointhephasediagram.Whileseveralrare-earth systemsunambiguouslyshow commensuratephases[9,11,73],othersuperlatticesappea rtoshowacontinuousincrease withtemperatureintheaverageturnangleperatomiclayer, suggestingthatincommensuratephasesaregeneric[42,20].Wecannotruleoutanaverag ingeectbeingresponsible, butthiswouldappearinconsistentwiththeabsenceofplate auxandtheexpectationof vanishingmeasureforhigh-orderphases.Itwillbeparticu larlyinterestingtoinvestigate whetherastatistical-mechanicalmodelnotmuchmorecompl exthanthatconsideredhere canincorporatemoreofthequalitativebehaviorseeninrar e-earthsuperlattices. Wehaveshownthatasuperlatticeofhelimagneticandnon-ma gneticlayersexhibits behaviordierentfromthatofthebulksix-stateclockmodel [70].Therearemultiphase regions,ratherthanasinglemultiphasepoint.Whenprecis elyfourmagneticlayerslie betweennon-magneticspacers,alinesegmentinthemultiph asetriangleappearstosupport asetofphasesmorelikethatintheANNNImodel[27,28]thanl ikethebulksix-state clockmodel.Forothervaluesof L ,thelow-temperatureexpansionndsonlyafewstable phases.Thisraisestheinterestingexperimentalquestion ofwhetherrichmagneticphase diagramsinarticialsuperlatticescouldappearforcerta inmagicspacingswhilebeing absentforothers.Ifthephasediagramweretodependassens itivelyon L asinourmodel, itmightbediculttogrowlmssucientlyuniformtotestth ehypothesis;however,if theextentofthemagiccouplingwerebroader(say,L=4{6),t heeectcouldbeobservable. Further,amultiphaseregionofcouplingspacemightbemore amenabletoexperiment thanamultiphasepointthatrequiresexacttuning;suchare gion,however,wouldneedto havethefulldimensionalityofthecouplingspace,somethi ngwehavenotyetconstructed. 2.5Appendix:Transfer-MatrixTechniques Inordertocalculatethefree-energydierenceofachildfro mitsparents,weadapt thetransfer-matrixtechnique[88,29,70]totheregionwit honlytype-3walls.Webegin inaregionoverwhich,totheorderalreadycalculatedinthe low-temperatureexpansion, 17

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parentphases h a i = h a 1 a 2 ::: i ofperiod p a = P i a i andfreeenergyperspin f a and h b i = h b 1 b 2 ::: i ofperiod p b = P i b i andfreeenergyperspin f b coexistandhavelower freeenergiesthantheirparentphases. 5 Wethenseekthedoublefree-energydierence a h ab i = f h ab i p a p a + p b f h a i p b p a + p b f h b i (2.11) toleadingorder.If a h ab i < 0,thechildphase h ab i acquiresaregionofstability.Isolated spinrotations(asinTable2.1)cannotdeterminethesignof (2.11),sincethethreephases, h a i h b i ,and h ab i ,havethesamefreeenergiestorstorder.Wemustconsiderc onnected spinexcitations:ingeneral,theBoltzmannweightoftwo(o rmore)spinrotationsthat shareanaxialbondwilldierfromtheweightofthesamerotat ionssituatedintheir respectiveplanessuchthattheydonotshareabond.Sinceth e J 0 (in-plane)bondis assumedthemostexpensivetobreak,theshortestexcitatio nthatdistinguishes h ab i from itsparentsprovidestheleadingterminthelow-temperatur eexpansion.Thisrequiresthat theconnectedexcitationshouldspan( p a + p b 1)blocksoflength L ,inthesensethat bondsoneachendextendthroughtheterminatingspacerlaye rsandsosensewhether thesespacerscoincidewithwalls.Thetransfer-matrixtec hniquekeepstrackofallthe combinationsofconnectedanddisconnectedexcitationsof thislength. Asin[70],twocasesarise.Whentheproduct( p a + p b 1) L isodd,anexcitationof connectedspinseverysecondlayerdistinguishesthechild fromthetwoparents,and2 2 matricessuce.Whentheproductiseven,weshallneed4 4matrices. Theprinciplesarebestillustratedbyanexample.Consider distinguishing h 2 i fromits parents h1i and h 1 i when L =5.Inthefollowingdiagramshowingjustoveroneperiod ofthe h 2 i phase, S representamagneticlayer,while ^ S representsamagneticlayerwitha rippedspin: S jjj S ^ SS ^ SS jj SSSSS jjj S .(2.12) 5 Theperiodof h1i ,forthepurposeof(2.11),is1. 18

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Inthe h 2 i phase,thetwoextremalspacers( jj )coincidewithwalls.Inthe h 1 i -phaseparent, allthreespacerscoincidewithwalls,whileinthe h1i -phaseparent,therearenowalls. Thepicturedconnectedspinexcitations,spanning p h 1 i + p h1i 1=1block,istheshortest thatispossiblefor h 2 i butimpossibleforeitherparent. Theenergydierence(2.11)subtractsfromthefreeenergyof diagram(2.12)theparentdiagramfreeenergies.Weaccomplishthiswithaproductofv ectors(lowercaseGreek letters)andmatrices.Forthisexample,weget a h 2 i / ( y y ) A ( ),(2.13) where representsadiagram S ^ SS jj S thediagram S ^ SS jjj S ,and A thediagram S ^ SS ^ SS Thedualityoperator,denedforvectorsby v y =( Qv ) T ,with Q having 1allalongthe antidiagonal,describesthereverseddiagram, e.g. y = S jj S ^ SS ,withtheclockdirections alsoreversed.Sincethespininanexcitationcanberotated 60 counterclockwise(+)or clockwise( ),bothconditionsmustbeaccountedfor.Thefourentriesof amatrixstand forthefourwaysthetwoconnectedspinsinamatrixdiagramc anberipped: 0B@ + ++ + 1CA .(2.14) Theentriesofarowvectorare(+ ),thoseofacolumnvector + ,sothateachcontractioninamatrixproductsumsoverthepossibilitiesforasin glespin.Each2 2matrix entrygivesBoltzmannweightsforconnectedanddisconnect edcombinationsofthetwo constituentspins,asillustratedinFigure2.3a. Inaddition,eachmatrixentryisadierencebetweentheconn ectedBoltzmannfactor andthedisconnectedfactor,asspeciedbythelinked-clus tertheorem,(2.6).Vectors terminatetheproduct(Figure2.3b).Thefollowing2 2matricesarerequired;common factorsof q t areomitted,sinceonlythesignsofthematrixproductsinth ezero-temperature limitmatter. 19

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S ^ SS ^ SS A = r 0B@ 1 r x r 3 x r 4 x 1 r 2 x 1 r x 1CA (2.15) S jj ^ SS ^ SS B = 0B@ r z r x + z r z ( r 3 x r 4 x ) r 3 2 z 2 (1 r 2 x ) r 3 2 z 2 (1 r x ) 1CA (2.16) S ^ S jj S ^ SS C = r z 0B@ r 3 2 3 z 2 (1 r z ) r 3 2 + 3 z 2 (1 r z ) 1 r 2 z 1 r z 1CA (2.17) S jjj ^ SS ^ SS D = r z 2 0B@ 1 r x r 3 x r 4 x r 3 2 (1 z ) (1 r 2 x ) r 3 2 (1 z ) (1 r x ) 1CA (2.18) S ^ S jjj S ^ SS E = r z 2 0B@ r 3 2 (1 z ) (1 r 2 z ) r 3 2 (1 z ) ( r 3 z r 2 z ) r 3 z r 2 z 1 r 2 z 1CA (2.19) S ^ SS jj S = r 1 2 0B@ r 2 z r z 1CA (2.20) S ^ SS jjj S = r 1 2 0B@ r z r z 1CA (2.21) Thefollowingenvironmentsoccuronlywhen L =3: 20

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S jjj ^ SS ^ S jjj S F = 0B@ r 1 2 z 2 (1 r x ) r 3 x + z 1 (1 r x ) r 2 2 z (1 r 2 x ) r 1 2 z 2 (1 r x ) 1CA (2.22) S jjj ^ SS ^ S jj S G = 0B@ r 1 2 (1 r x ) r 1+3 x + 3 z 2 (1 r x ) r 2 3 z 2 (1 r 2 x ) r 1 2 (1 r x ) 1CA (2.23) S jj ^ SS ^ S jj S H = 0B@ r 1 2 + z 2 (1 r x ) r 3 x +2 z 1 (1 r x ) r 2 z (1 r 2 x ) r 1 2 + z 2 (1 r x ) 1CA (2.24) When( p a + p b 1) L iseven,thereisnouniqueshortestleading-orderdiagramo n themodelof(2.12).Rather,afamilyofsuchdiagramswithri ppedspinseverysecond layerexceptforonepairofaxiallyadjacentrippedspinsal lspantherequisitedistance. Toaccountforasingleadjacentpairanywherealongtheleng thofanexcitation,Seno et al. [70]introduced4 4transfermatricesoftheform 0BBBBBBB@ +0+0+0 0+00++00 0+0 0 0 00+ 00 0++00+ 00+0+0+0 0 +00 00 0+0 0 1CCCCCCCA (2.25) eachentryofwhichconsidersfouradjacentplanesinwhicha spinhasrotatedinthe positive(+)ornegative( )clockdirection,ornotrotatedatall(0).SeeFigure2.3c. Thefourentriesoftheupper-rightquadrantcontainnoconn ectedspinexcitationsandso vanish.End-capvectors(Figure2.3d)accountforthenalp airofplanes,oneofwhichwill containaspinrip.Thefollowingmatricesandend-capvecto rsresult(again,thecommon factorof q t isomitted): 21

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c SS c SSA = 0BBBBBBB@ r (1 r x ) r 1 2 ( r 3 x r 4 x )00 r 5 2 (1 r 2 x ) r (1 r x )00 r 1 2 + x (1 r ) r 4 x ( r 2 1) r (1 r x ) r 1 2 ( r 3 x r 4 x ) r 2 x ( r 2 r 3 ) r 1 2 + x (1 r ) r 5 2 (1 r 2 x ) r (1 r x ) 1CCCCCCCA (2.26) c SS d S jj SB = 0BBBBBBB@ r 3 2 z 2 (1 r x ) r z 3 2 ( r 3 x r 4 x )00 r 3 z 2 (1 r 2 x ) r z (1 r x )00 r 3 z 2 x ( r r 2 ) r 2 x +3 z ( r r 1 ) r z (1 r z ) r 5 z 2 r 7 z 2 r 3 z 2 x ( r 5 2 r 7 2 ) r 2 x ( r 1 2 r 1 2 ) r 3 2 + z (1 r 2 z ) r 3 2 z 2 (1 r z ) 1CCCCCCCA (2.27) c SS jj c SSC = 0BBBBBBB@ r 2 z 1 (1 r z ) r 1 2 ( r 2 z r 3 z )00 r 1 2 ( r 2 z 1) r 2 ( r z 1)00 r 1 2 + z (1 r z ) r 2+2 z ( r 2 z 1) r 2 z (1 r z ) r 1 2 +2 z (1 r z ) r z 1 (1 r z ) r 1 2 + z (1 r z ) r 1 2 +2 z (1 r 2 z ) r 2 z 1 (1 r z ) 1CCCCCCCA (2.28) c SS \ S jjj SD = 0BBBBBBB@ r 3 2 z (1 r x ) r z 2 3 2 ( r 3 x r 4 x )00 r 3 z (1 r 2 x ) r z 2 (1 r x )00 r x ( r r 2 ) r 3 z 2 +2 x ( r r 1 ) r z 2 (1 r 2 z ) r z ( r z 1) r x ( r 5 2 r 7 2 ) r 2 x + 3 z 2 ( r 1 2 r 1 2 ) r 3 2 + 5 z 2 ( r z 1) r 3 2 z (1 r 2 z ) 1CCCCCCCA (2.29) c SS jjj c SSE = 0BBBBBBB@ r z 1 (1 r 2 z ) r 1 2 ( r z 1)00 r 1 2 +3 z ( r z 1) r 2 2 z (1 r 2 z )00 r 1 2 +2 z (1 r z ) r 2 z (1 r z ) r 2 ( r 2 z 1) r 1 2 ( r z 1) r 2 z 1 (1 r 2 z ) r 1 2 +2 z (1 r z ) r 1 2 +3 z ( r z 1) r z 1 (1 r 2 z ) 1CCCCCCCA (2.30) 22

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c SS jj SSa = 0BBBBBBB@ r 2 z 1 2 r 1 z r 1+ z r z 1 2 1CCCCCCCA (2.31) c SS jjj SSb = 0BBBBBBB@ r z 1 2 r 1+ z r 1 z r 2 z 1 2 1CCCCCCCA (2.32) Astraightforwardcomputeralgorithmgeneratesthereleva ntsequenceofmatrices;as oneexample,for L =4, a h 23 i =( a y + b y ) ACAEACA ( a b ).(2.33) Theprogramexpandsandsymbolicallydeterminestheleadin gbehaviorof a inthezerotemperaturelimit;if a isnegative,thechildattainsaregionofstabilitywithres pecttoits parents.Forsucientlylongchainsofmatrices,itwasimpr acticaltoexpandthematrix products,andanumericalapproachwassubstituted. 23

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(a) sssss ^^ (c) ss ss (b) ssss ^ s (d) ss ss Figure2.3.Amatrixelementrepresentstworippedspins,av ectorelementone.Boldface bondsarecountedatfullstrengthintheBoltzmannweights, whileeachoftheotherbonds iscountedintwodierentdiagramsandsocomesinathalfstre ngth.(a)A2 2matrix representsrippedspins( ^ S )inthesecondandfourthplanes.(b)A(column)2-vector contractswitha2 2matrixtoitsleft.(c)A4 4matrixrepresentsarippedspininone (andonlyone)ofthersttwolayersandinone(andonlyone)o fthesecondtwo.(d)A (column)4-vectorcontractswitha4 4matrixtoitsleft.(AdaptedfromReference[70].) 24

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CHAPTER3 DISTINGUISHINGCANCEROUSFROMNONCANCEROUSCELLS 1 3.1Introduction Electricalcell-substrateimpedancesensing(ECIS)hasbe eninusesince1984[33]to monitorchangesincellculturesduetospreadingorinrespo nsetochemicalstimuli,infection,orrow.Applicationsincludestudiesofcellmigratio n,barrierfunction,toxicology, angiogenesis,andapoptosis.Severalpapershavenotedtha timpedanceructuationsare associatedwithcellularmicromotion[49].However,weare notawareofanypreviouswork applyingstatisticaltechniquestotheseructuationsinor dertodistinguishtwodierent celltypes.Here,wedemonstratethatmeasuresoftheelectr icalnoisefromculturesof cancerousandnon-canceroushumanovariansurfaceepithel ialcellsdistinguishthem.We ndthatthenoiseinbothcancerousandnon-cancerouscultu resshowscorrelationson manytimescales,butbyallmeasures,thesecorrelationsar eweakerorofshorterduration inthecancerouscultures.3.2ExperimentalMethods WeusedtheECISsystemtocollectmicromotiontime-seriesd ata,theructuationsin whicharecausedbythemovementsinaconruentlayeroflivec ells.Thesystemcan bemodeledasanRCcircuit[34,35,46,47].Thecellsarecult uredonasmallgold electrode(5 10 4 cm 2 ),whichisconnectedinseriestoa1-Megaohmresister,anAC signalgeneratoroperatingat1voltand4000Hz,andnallyt oalargegoldcounterelectrode(0 : 15cm 2 ).Thisnetworkisconnectedinparalleltoalock-inamplie r,andthe 1 Thischapterhasbeenpublished,indierentform,byD.C.Lo velady,T.C.Richmond,A.N.Maggi, C.-M.Lo,andD.A.RabsoninPhys.Rev.E 76 041908(2007). 25

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in-phaseandout-of-phasevoltagesarecollectedonceasec ond,fromwhichweextract timeseriesofresistanceandcapacitivereactance(Figure 3.1a).InECISexperiments, theructuationsincompleximpedancecomeprimarilyfromch angesinintercellulargaps andinthenarrowspacesbetweenthecellsandthesmallgolde lectrode[35,47,46].A currentofaboutonemicroampisdriventhroughthesample,a ndtheresultingvoltagedrop ofafewmillivoltsacrossthecelllayerhasnophysiologica leect:thisisanoninvasive, in-vitro technique.Anovariancancerline(SKOV3,forSloan-Ketter ingOvarian)anda normalhumanovariansurfaceepithelial(HOSE)cellline(H OSE15)wereprovidedbyDr. SamuelMokatHarvardMedicalSchool.Thesecellsweregrown inM199andMCDB 105(1:1)(Sigma,St.Louis,MO)supplementedwith10%fetal calfserum(Sigma),2mM L-glutamine,100units/mlpenicillin,and100microgram/m lstreptomycinunder5%CO 2 anda37 C,high-humidityatmosphere.ForECISmicromotionmeasure ments,cellswere takenfromslightlysub-conruentcultures48hoursafterpa ssage,andamono-disperse cellsuspensionwaspreparedusingstandardtissue-cultur etechniqueswithtrypsin/EDTA. Thesesuspensionswereequilibratedatincubatorconditio nsbeforeadditiontotheECIS electrodewells.Conruentlayerswereformed24hoursafter inoculation,resultingina densityof10 5 cell = cm 2 Figure3.1ashowsarepresentative4096-secondrun(justov eronehour)measuringthe realpartofimpedanceasafunctionoftime;theexampleshow saHOSEculture,butto theeye,SKOVculturesdonotappearverydierent.Whilethee xampleshowsincreasing resistancewithtime,othersshowadecrease;atthistimesc ale,thereisnoevidencefor anoveralltrend.Wecollected,undersimilarconditions,1 8timeseriesforHOSEcultures, ofwhich16wentfor8192secondsandtwofor4096seconds.Eac h8192-secondrunwas splitintwohalves,sothateectivelywehadthirty-four409 6-secondruns;however,where appropriateintheanalysisbelow,wediscardthesecondhal vesofthelongerrunsinorder toavoidinadvertentlyintroducingcorrelations.Similar ly,forSKOVcultureswetookdata ineight8192-secondrunsandten4096-secondruns,yieldin geectivelytwenty-six4096secondruns.Wenumericallydierentiatedtheresistancean dcapacitancetimeseriesto 26

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Figure3.1.Schemeofdataextractionfromnoise.(a)Timese riesofresistanceforone oftheexperimentalruns.Takingthediscretetimederivati veandnormalizingtozero meanandunitvariancegivesthenoise,(b).Thepowerspectr umofnoiseisshownin (c),usinghalf-overlappingwindowsof512pointsinordert oreducescatter.Fitstothe rsthundredandlasthundredfrequenciesestimatelow-and high-frequencypower-laws, f .Whitenoisewouldhaveappearedfrequency-independent( =0).TheFourier transformofthepowerspectrumgivestheautocorrelation, (d),whichwettoashifted power-lawdecayandextractthemeasure 0 .Asexplainedinthetext,subtledierences intheunivariatenoisedistribution(e)(smoothed)discri minatebetweencancerousand non-cancerousmicromotion. 27

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obtainnoisetimeseriesforeach,whichwenormalizedtozer omeanandunitvariance (Figure3.1b).3.3StatisticalMeasuresofNoise Weseekinformationfromthenormalizednoiseseries.Ther stquestiontoposeis whetherthenoisecandistinguishcancerousfromnon-cance rouscultures,butmoregenerallythemeasuresweextractmaybeusedtotestmodelsofce llmicromotion.Broadly, suchmodelsmaybecharacterizedbyshort-termandlong-ter mcorrelation,sowelookat severalmeasuresforeach. First,thepowerspectraldensity(Figure3.1c)looksverym uchmorelike\pinknoise" than\whitenoise;"thatis,itshowssignsoflong-timecorr elations.Alog-logplotofspectraldensityagainstfrequency, f ,suggestsanintensitygoingas f inthelow-frequency limit.(Wediscussbelowtheextenttowhichatruewhite-noi seprocessmaymimicpink noiseduetothenitetimeofarun.)Foreachrun,wesplitthe 4096noiseamplitudes intohalf-overlappingwindowsof512seconds,multipliedb yaHannwindow,Fouriertransformed,andsquared,averagingtheresultingspectrainord ertoreducescatter[62]. Asintheexampleofthegure,somerunsshowacrossoverbetw eenlow-andhighfrequencyvaluesfor ,whichweestimatedwithleast-squaresstraight-linetso fpower attherst100(excludingzerofrequencyandtheverylowest frequency)andlast100 frequencies(outof256non-zerofrequencies).Inmanyruns ,low-andhigh-frequency alphaestimateswereequal,withinttingerrors.Table3.1 summarizestheresults,giving inthecolumnslabeled\ave"themeansoverallHOSErunsoral lSKOVrunsforthegiven measures;thecolumnslabeled\ "givethestandard-deviationestimatorforthepopulation ofalllikeruns.ThedierencesbetweenalphasforHOSEandSK OV,bothlow-andhighfrequency,exceedseveralstandarderrors(orstandarddev iationsofthemean, = p N where isthestandarddeviationand N isthenumberofruns).Moreover,theStudentt -testandKolmogorov-SmirnovtestshowthattheHOSEandSKO Vpopulationsdier 2 2 Typically,Kolmogorov-Smirnovistakentorejectthe(null )hypothesisthattwopopulationsweredrawn fromthesamedistributionifityieldsaprobabilitylessth an5%.Threeofthefouralphameasuresmeet 28

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Thelow-frequencyexponentsaremoresignicant.Thefactt hatthesemeasuresarelarger forHOSEthanforSKOVsuggestsadierenceinlong-timecorre lationsinmicromotionand isconsistentwiththehypothesisthatnon-cancerousHOSEc ellsmoveinamoreorderly mannerthancancerousSKOV. Anon-zero low isindicativeoflong-time,\fractal"[7],correlation,bu tasRangarajan andDing[64]pointout,relyingonpower-lawbehavioralone canleadtoincorrectidenticationofsuchcorrelationswhennoneexist.Tworelatedmea suresaretheHurstexponent andtheexponentofdetrendedructuationanalysis[51,7,25 ,58,59,53];bothmethods splitthetimeseriesofnoiseintobinsofduration T ,thendeterminehowameasurescales with T .ForHurst,onesubtractsthemeanfromallthedatainabinan dcharacterizes thatbinbyitsstandarddeviation, S .Theseriesisintegrated,andtheminimumvalue subtractedfromthemaximum,yieldingtherange, R .Foreachbin,onerecordstheratio R=S andaveragesoverbinsofthesamesize.Theprocedureisrepe atedforsuccessively largerbins( T ).Astraight-linettoalog-logplotof R=S againstbinsize T revealsa powerlaw, R=S T H ,where H istheHurstexponent.Detrendedructuationanalysis runsalongsimilarlines,butwithineachbinonesubtractsa best-tline,thusdetrending thedata.Thedatainthebinarethencharacterizedbystanda rddeviation S T D ,where D istheDFAexponent.Table3.2showstheresults;again,with highcondence(based particularlyonStudent's t -test)wecanconcludethatHOSEandSKOVnoisecomefrom dierentdistributions.However,sincethemeansaresepara tedbylessthanapopulation standarddeviation,manyruns(of4096seconds)wouldbenec essarytodeterminethe provenanceof one particularculture. While low H ,and D weredesignedtoestimatecorrelationsatdivergingtimesc ales, short-timecorrelationisconvenientlydeterminedfromau tocorrelation,Figure3.1d,normalizedtounityatzerolag.Thelagofrstzerocrossingpro videsonenaturalmeasure ofwhencorrelationislost,butsinceautocorrelationcurv esmaysometimesreachvery small,yetpositive,plateausbeforecrossingzero,wealso measuredthelagatwhichthe thiscriterion.Asacontroltest,halfofHOSErunswerechec kedagainsttheotherhalfandSKOVagainst SKOV,andineverycasethealphameasurementswerecompatib lewiththenullhypothesis,asexpected. 29

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Table3.1.Power-spectralmeasuresofHOSE(non-cancerous )andSKOV(cancerous)resistiveandcapacitivenoiseseries.Shownareestimatesfo r1 =f behaviorathighand lowfrequencies.Themeansofthealphasdierbymanystandar derrors( = p N ,where isthestandarddeviation),allowingustodistinguishthep opulationscomposedof N runs,althoughnotbyenoughtodistinguishreliablya single HOSErunfroma single SKOV.The F -testand t -testgivetheprobabilitiesthatthevariancesandmeansof the distributionsofvaluesof woulddierbyasmuchasormorethantheydoifthetwopopulationshadcomefromthesameGaussiandistribution.KSgi vestheprobabilityunderthe Kolmogorov-Smirnovtestthatthetwopopulations'cumulat ivedistributionscoulddier asmuchastheydo.Smallprobabilitiesindicatethatthepop ulationsdier;aprobability of0 : means < 10 6 N =34forHOSE, N =26forSKOV.Inallcases,weapplythe approximate t -testfordistributionswithunequalvariances[62]. HOSESKOVprob.fromsamedistribution measure ave = p N ave = p N F -test t -testKS-test resistance low 0 : 9910 : 1320 : 020 : 8000 : 1480 : 030 : 544 : 10 6 4 : 2 10 4 high 1 : 580 : 5580 : 101 : 090 : 6480 : 130 : 424 : 10 3 0 : 024 capacitance low 0 : 9090 : 09880 : 020 : 7340 : 1310 : 030 : 130 : 9 : 10 6 high 1 : 1330 : 4460 : 080 : 9800 : 3570 : 070 : 250 : 150 : 37 Table3.2.Additionalmeasuresoflong-timecorrelationin thenoisetimeseries,Hurstand detrended-ructuationexponents.SeeTable-3.1captionfo rcolumndescriptions. HOSESKOVprob.fromsamedistribution measureave = p N ave = p NF -test t -testKS-test resistance Hurst H 0 : 7700 : 04420 : 0080 : 7440 : 08760 : 0173 : 10 4 0 : 170 : 099 DFA D 0 : 8540 : 04730 : 0080 : 8060 : 07930 : 0160 : 0069 : 8 10 3 0 : 057 capacitance Hurst H 0 : 7920 : 04740 : 0080 : 7310 : 08860 : 0179 : 10 4 3 : 1 10 3 0 : 012 DFA D 0 : 8430 : 04790 : 0080 : 7880 : 07480 : 0150 : 0172 : 5 10 3 3 : 4 10 3 autocorrelationrstcrosses1 =e .Inamodelwithonlyshort-timecorrelation,the1 =e timeestimatestheexponentialdecaytime.However,aswedi scussbelow,weobserved signicantdeviationsfromexponentialdecay,ndingbett ertstoashiftedpower-law decay, autocorrelation= t + t 1 t 1 0 .(3.1) Wetautocorrelation,forlagsintheheuristicinterval t =1to t =20seconds,using Levenberg-Marquardtleast-squaresminimizationtothisf ormtond 0 .Table3.3summarizesresultsforthetwocrossingsand 0 ;thelastdistinguishesthepopulationsofHOSE andSKOVrunsonlyinthat(cancerous)SKOVshowsmuchgreate rscatterin 0 ,asmea30

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Table3.3.Measuresofshort-timecorrelationinthenoiset imeseries:thelagatwhich normalizedautocorrelation(seeFigure3.1d)fallsto1 =e ,therstzero-crossingofautocorrelation,andtheexponent 0 fromttingtherstfewlagswithashiftedpowerlaw. SeetheTable-3.1captionforthestatisticallabels.Ofthe semeasures,the1 =e crossing(in resistance)andthezerocrossing(incapacitance)havethe greatestsignicanceindistinguishingthepopulations; 0 issignicantonlyinthesensethatthe scatter isverymuch greaterforcancerousSKOVthanfornon-cancerousHOSE. HOSESKOVprob.fromsamedistribution measureave = p N ave = p NF -test t -testKS-test resistance 1 =e 6 : 351 : 760 : 304 : 912 : 620 : 510 : 0320 : 0209 : 1 10 3 zero132 : 88 : 015111 : 115 : 23 : 0 : 140 : 440 : 068 0 1 : 180 : 5650 : 105 : 1112 : 82 : 500 : 0 : 130 : 48 capacitance 1 =e 5 : 771 : 400 : 244 : 403 : 960 : 780 : 0 : 106 : 0 10 5 zero194 : 136 : 23 : 97 : 5111 : 22 : 0 : 293 : 7 10 3 8 : 6 10 3 0 1 : 171 : 160 : 201 : 933 : 350 : 660 : 0 : 280 : 71 suredbythe F -test.Bothcrossingsvarygreatlyfromruntorun,butthe1 =e crossing inresistanceandzerocrossingincapacitancedistinguish thepopulationsofHOSEand SKOVexperimentsatbetterthanthe95%condencelevelasme asuredbyStudent's t testandtheKolmogorov-Smirnovtest.Inparticular,theav eragedmeasuresshowshorter crossingtimesandsteeperdescents( 0 )forSKOVthanforHOSE,againconsistentwith thehypothesisthatthemicromotionofcancerousculturesi slesscorrelatedthanthatof non-cancerouscultures. WiththefourteenmeasuressummarizedinTables3.1{3.3,ea chrunof4096secondscan bethoughtofasapointinafourteen-dimensionalspace.Ins uchproblems,thepopulations mightseparateintotwodistinct,compactclusters[45, x 4.2];whiletheidenticationof clustersinhigh-dimensionalspacesremainsanopenproble minstatisticalresearch,it iscommontousethevariance-maximizingprincipal-compon entanalysisintroducedby Hotellingtoprojectontooptimalsubspaces,usuallytaken tobetwo-dimensional[40]. Figure3.2plotsthersttwoprincipalcomponents.Whileth eplotshowsacleardierence betweenthetwopopulationsconsistingofallrunsofHOSEan dallrunsofSKOV,overlap betweenthetwoclustersmakesitdiculttoapplythetechni quediagnostically.Wefound thisproblemtobegeneric:anexhaustiveexaminationofpai rsofprincipalcomponents (beyondthersttwo)producedsimilarplots,withthetwopo pulationsusuallylessdistinct 31

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inhigher-ordercomponents,whileaddingorsubtractingse veralmeasurestothelistof fourteenmeasuresdidnotimproveclustering. Thusfar,thenoisemeasuresconsideredhaveshownthatelec tricalnoisefromHOSE andSKOVexperimentshave,onaverage,dierentcorrelation s,buttheydonotprovidea reliablewaytodeterminewhetherthecellsinasinglerunof 4096secondsareHOSEor SKOV.However,fromthenormalized(zero-mean,unit-varia nce)noisetimeseriesofFigure 3.1b,wecanextractaprobabilitydistributionofnoiseamp litudes,asinFigure3.1e.Not surprisingly,thedistributionisapproximatelyGaussian ;however,subtledeviationsfrom normalformdodistinguishHOSEfromSKOV,eveninasingleru n,ifweapplythe Kolmogorov-Smirnovtestdirectlytothenoise.Thistestlo oksonlyatdistributionsof noiseamplitudes,ratherthancorrelations. Tothisend,weconcatenatetherstnine4096-secondHOSEre sistanceruns(discarding,forthispurpose,thesecondhalvesofthe8192-sec ondruns)tocreateaHOSE resistancereferencedistribution.Similarly,wecreatea SKOVresistancereferencebyconcatenatingtherstnine4096-secondSKOVruns.Eachofther emainingrunsistested againstthetworesistancereferencesets.Thesameprocedu reisappliedwithcapacitance data.Inmanycases,Kolmogorov-Smirnovdoesnotshowamatc hwitheitherdistribution withhighprobability,butwecancomparethetwoprobabilit ies:onetypicalHOSErun matchestheHOSEreferencewithprobability0.02andSKOVwi thprobability4 : 7 10 8 sowe(correctly)identifythisrunasHOSEbasedontheratio ofprobabilities.Of56tested datasets(noneofwhichwentintotheconstructionoftheref erencesets),42(75%)matched thecorrectreferencesetbythiscriterion,anoutcomethat wouldhappenbychancewith probabilityapproximately1 : 2 10 4 .Werepeatedtheprocedureusingasecondcollection offourreferencesets(HOSE/SKOV,resistance/capacitanc e)eachconstructedfromnine runsnotusedinmakingtherstreferencesets.Of64trials( noneusedinthenewreferencesets),53(83%)wereidentiedcorrectly,withcorresp ondingprobability5 10 8 .The resultsfromthetwosetsoftrialsareaddedandsummarizedi nTable3.4.Wecanreduce percentagesofincorrectidenticationsbyinsistingonag reementbetweenresistanceand 32

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capacitancetimeseries;thislowerstheoverallincorrect identicationrateto11.7%,with acorrectrateof70.0%anda\not-sure"rateof18.3%. Viewedbyeye,thekernel-smoothedprobabilitydistributi onfunctionofanoisetime serieslooksapproximatelynormal(Gaussian),althoughof tenwithoutliers;seeFigure3.1e. Deviationsfromnormalityarecharacterizedinpartbykurt osis 3 ,whichislargerforHOSE thanforSKOV:seeTable3.5.Apossibleexplanationisthatk urtosishereisaproxyfor correlation.Underthishypothesis,twoeectscouldbeatwo rk.First,whileallofour runshavethesamenumberoftimesteps,Table3.3showsthatS KOVcorrelationtimes areshorterthanHOSEcorrelationtimes;thus,aSKOVruncou ldbesaidtohavemore independent timestepsthanaHOSErunofthesamelength.Thestandarddev iationof theestimatorofkurtosisscalesas1 = p N ,with N thenumberofindependentsamples[3]. Theratiosof18inTable3.5betweenstandarddeviationsofk urtosesfromthepopulations ofallHOSEandallSKOVimplymuchtoolargearatioofcorrela tiontimes(18 2 ),but qualitativelytheysupporttheideaofmoreindependentsam plesinSKOV.However,this rsteectwouldnotresultintheobservedstatisticallysig nicantdierencesinmean kurtoses.Asecondpossiblemanifestationofcorrelationg etstotheheartofwhythenoise distributionsappearapproximatelyGaussian:alargenumb erofcellscontributetothe overallmeasurementofresistanceorcapacitance.Wewould expectanormaldistribution inthelimitofinnitelymanycells;however,convergenceu nderthecentral-limittheoremis non-uniform,withadistributionapproachingaGaussiansl owlyinthetailsasthenumber ofindependentcellularmotionsincreases.If,aswebeliev e,HOSEmotionismorecorrelated thanSKOV,itwouldcomprise fewer independentcellularmotionsandsohavealarger kurtosis. Temporalcorrelationcannotexplainthewholeeect:aswear gueinAppendixAofthis chapter,bothkurtosisandKolmogorov-Smirnovappeartobe betterdiscriminantsthan adirectmeasure,the1 =e crossing.Thissuggeststhattheunivariatenoisedistribu tion 3 Conventionsforkurtosisabound.Specically,wemeantheu nbiasedestimator g 2 = k 4 =k 2 2 ,where k i aretheFisherstatistics:seeE.Keeping, IntroductiontoStatisticalInference (vanNostrand,Princeton, 1962,republishedDover,1995).Sincethequantityestimat edby g 2 iszeroforanormaldistribution,isis sometimesreferredtoas\kurtosisexcess." 33

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Table3.4.Percentagesofcorrectidentications.TheKolm ogorov-Smirnovtestisapplied todistributionsofnoiseamplitudesagainstHOSEandSKOVr eferencesets.Twononoverlappingchoicesofreferencesetsareused;inneitherc asedidanytrialrungureina referencesetagainstwhichitwastested.The\average"col umngivespercentagesweighted bynumbersoftrials(16HOSEand12SKOVintherstset,18HOS Eand14SKOVin thesecond). rstsetsecondsetaverage HOSEcapacitance62 : 5%72 : 2%67 : 6% HOSEresistance87 : 566 : 776 : 5 SKOVcapacitance83 : 3100 : 92 : 3 SKOVresistance66 : 7100 : 84 : 6 allresistance78 : 681 : 380 : 0 allcapacitance71 : 484 : 478 : 3 allHOSE75 : 069 : 472 : 1 allSKOV75 : 0100 : 88 : 5 all75 : 082 : 879 : 2 Table3.5.Kurtosisaveragedoverallruns,standarddeviat ionofkurtoses,andstandard deviationofthemeans. F -testprobabilitiesforHOSEandSKOVtocomefromthesame distributionwereboth < 10 6 ; t -testswere0 : 022forresistanceand0 : 005forcapacitance. TheKolmogorov-Smirnovtestgaveprobability < 10 6 forresistanceand6 10 5 for capacitance. average = p N HOSEresistance74 : 4173 : 129 : 7 SKOVresistance3 : 009 : 571 : 9 HOSEcapacitance17 : 632 : 25 : 5 SKOVcapacitance0 : 941 : 800 : 35 ismorethanjustaproxyforcorrelationtime.InAppendixB, weconsiderwhetherthe observedkurtosiscouldresultfromspatialcorrelations 4 proportionaltothemeasured temporalcorrelationsandarguethatthekurtosiseectisto ostrongandthecoupling betweenkurtosisandtemporalcorrelationtooweaktosuppo rtthishypothesis. 4 Wewillreportelsewhereondirectmeasuresofspatialcorre lationinmicromotion. 34

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3.4TwoSimpleModels Havingmotivatedandinterpretedourmeasuresofnoiseinte rmsofshort-andlongtimecorrelations,wenowcompareourdatatothesimplestpo ssiblediscrete-timemodels, thebinaryrandomwalkwithpersistence[32],displayingon lyshort-timecorrelation,and adiscretefractionalBrownianmotion[50,64],whichhasco rrelationsonalltimescales. Forpresentpurposes,itsucestoconsideronlytheincreme ntsratherthanthewalks themselves;thatis,wecomparetoFigure3.1b,notFigure3. 1a. First,considertheincrementsofadiscreterandomwalkwit hpersistence.Letthe incrementattime j t ,where t isthetimestep,be x j ,drawnfrom f +1 ; 1 g .Then x j +1 = x j withprobability a and x j +1 = x j withprobability1 a ;onerecoverstheusual discretebinaryrandomwalkfor a =1 = 2.Sincewethinkofthisprocessasapproximating acontinuousone,andthereisnonaturalwaytotakethelimit t 0foranticorrelated increments,werestrict1 = 2 a 1.Forconvenience,weset t =1.Asimpleinductive argumentshowsthat h x 0 x n i =(2 a 1) n =exp( n= ),(3.2) wherethecorrelationtime = 1 = ln(2 a 1).Fortimesmuchlargerthan ,thisMarkov processlookslikeanordinarybinaryrandomwalkwitharesc aledtime,andbytheusual arguments[66],thepowerspectrumapproacheswhitenoise, i.e. ,itbecomesindependent offrequencyinthelow-frequencylimit.However,foranit erun,thepowerspectrummay mimiccorrelated(pink)noiseeven,surprisingly,fora asshortas4inarunaslongas 4096,asinFigure3.3a.However,therandomnoiselevelson oticeablyatlowfrequencies, whiletheexperimentaldata(Figure3.3b)appeartofollowa 1 =f powerlawtothelowest frequencies 5 .Thissupportsthepresenceofcorrelationsatalltimescal es.Theshortness ofthelow-frequencyplateauinFigure3.3aismisleading.T oseemoreoftheratpartof thespectrum,nerfrequencyresolutionisnecessary.Taki nglargerwindows,wecan(at leastforarunlongerthan4096)extendthegraphmanydecade stotheleftandverify 5 Indeed,ourFigure3.3aresemblesFigure6bofReference[64 ].Thatprocessalsohasnotruelong-time correlations. 35

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thatthespectrumremainsrat(white),butatthecostofgrea terscatter.Parts(c)and (d)ofthegureshowautocorrelationforrandomnoiseandex perimentaldatawithts toexponentialdecay(dotted)andtheshiftedpowerlaw(3.1 )(solid).Thetwotsfall ontopofoneanotherfortheprocesssatisfying(3.2).Thate xponentialdecaydoesnot approximatetheexperimentaldataaswellasthepowerlawco rroboratesthehypothesis oflonger-than-short-timecorrelations. MandelbrotandvanNess[50]introducethenotionoffractio nalBrownianmotionwith correlationsbetweenincrementsseparatedbyarbitraryti medierencesandwitha1 =f powerspectrum.RangarajanandDing[64]describeaparticu larlysimplewayofgeneratingatimeseriesofincrementswithsuchproperties:star twithaGaussian-distributed uncorrelatedtimeseries f x j g ,Fourier-transform,multiplyby f = 2 ,andFourier-transform back.TheresultingprocesshasaHurstexponentgivenby H =(1+ ) = 2.(3.3) Determinationofexponents and H issubjecttotheusualnumericalvicissitudes,but RangarajanandDingarguethattruelong-rangedprocessess houldsatisfy(3.3)atleast approximately. Figure3.4plotsfractionaldiscrepanciesbetween(3.3)an dmeasuredHurstexponents asfunctionsofmeasuredspectralexponents .Atthebottomareplottedarticiallygeneratedlong-time-correlateddatafollowingtheprescr iptionofRangarajanandDing (plottingsymbols+);themeasuredexponents arealwaysclosetotheknownvalues,so themeasurementerrorsoccurinestimating H .Wenoteasystematictrendtowardlarger errorsawayfrom 0 : 5,butgenerallytheerrorsstaysmall.Atthetopofthegraph (plottingsymbols )arearticially-generatedrandomwalkincrementswithpe rsistence timesrangingfrom2attheleftto7attheright.Measuredval uesof followthesame prescriptionasusedabove,althoughasnotedearlier(Figu re3.3),thetsfailforlow frequencies;indeed,every shouldbezero.Hurstestimatesrangefrom0.45to0.67; 36

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thetruevalueineverycaseshouldbe1 = 2.AsdiscussedbyRangarajanandDing,the discrepanciesbetweenmeasuredHurstandHurstestimatedf rommeasured arelarge. Inthemiddleandatthebottomareplottedourexperimentald ata(HOSE ,SKOV ). Agreementbetweentheexponents H and isgenerallynotasgoodasforthelong-rangecorrelatedprocessesbutnotsopoorasfortheshort-time-c orrelatedrandomwalk.On average,theexperimentalpointslieclosertotheformerth antothelatter.Weinterpret thisresultassupportingtheexistenceofcorrelationson, attheveryleast,manydierent timescales.Amodelofcellmotionwillneedtoexplainbotht heshort-timeandlong-time correlationswehaveobserved.3.5Applications Wehavedemonstratedthatelectrical-noisemeasurementso nhumanovariansurface epithelialcellscandistinguishcancerousandnon-cancer ouscultures.Thisisnotintended asadiagnostictool;foronething,itiseasiertodistingui shthemunderamicroscope.We nditisalsopossibletodistinguishHOSEfromSKOVbasedpu relyon average electrical resistanceorcapacitance.Ourmainfocushasratherbeenon developingstatisticaltools withwhichtotestmoresophisticatedstatistical-mechani calmodelsandindevelopinga databaseofcharacteristicsofmanydierentcelltypes,for whichasinglemeasurement ( e.g. ,averageelectricalresistance)willsurelybeinadequate .TheapplicationoftheECIS methodologytoinvestigatecellmotilityincultureunderd ierentenvironmentalconditions mayprovideausefultoolinthiseort. Motilityofcellsintissueculturehasbeenwidelyobserved andisthoughttobean expressionofabasiccellularmechanisminvolvedinnumero usphysiologicalandpathologicalprocesses,suchasmorphogenesis,woundhealing,andt umormetastasis.Inaddition tolocomotion,motilitymaytaketheformofmembraneruing andundulationsandthe extensionofregionsofthecytoplasmintheformofblebsand lamellipodia.Whilenormalcellsexhibitsteadycontroloftheirgrowthrateandmot ilebehaviorinresponseto cell-substrateandcell-cellinteractions,thelackofsuc hcontactinhibitionincancercellsis 37

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directlyresponsiblefortheirinvasivebehavior[38].Ina nECISmeasurement,asthecells attachandspreadontheelectrodesurface,theelectriccur rentmustrowinthespaces underandbetweenthecells,asthecellmembranesareessent iallyinsulators.Inreducing theareaavailableforcurrentrow,thisinitialmotioncaus esalargeincreaseinimpedance. Thisgenerallypeaksafewhoursintotheexperiment;ourdat aweretakenwellafterthe peak.Subsequentsmallerchangesinthecell-substrateand cell-cellinteractionsdueto cellmotionscausetheimpedancetoructuatewithtime.Then umericalmethodsusedin thispaperopenthewaytoanalyzingimpedanceructuationsm easuredbyECISandmay provideinformationaboutcellulardynamicssuchasdieren tbehaviorbetweencancerous andhealthycells. Ourobservationofshortercorrelationtimesincancerousc ulturesisconsistentwiththe pictureofthesecellsmovinginalessregulatedmanner.Now thatithasbeenestablished thatdierentcelltypesgeneratedistinguishablenoisepat terns,futureresearchinthis areawillfocusonthedevelopmentofrealisticmodelsofcel lularmotilityforhealthyand malignantcells.3.6AppendixA:ComparingDiscriminants Weclaimnooriginalitytothefollowingelementaryapplica tionofstatisticsbutcould notndatextbookdiscussionofquitethispoint.Giventwod istributions, A and B (for instance,thekurtosesofHOSEdatasetsandthoseofSKOV),a ssumedtobeGaussian andcharacterizedbymeans A < B andstandarddeviations A B ,thereareseveral choicesofwheretoplaceadividingpoint x 0 soastoidentifyall xx 0 topopulation B .Onenaturalchoiceistopick x 0 sothatthe expectedratesofcorrectidenticationofthetwopopulati onswillbethesame, i.e. ,that x 0 A shouldbethesamemultipleof A as B x 0 isof B ,or x 0 = A B + B A A + B .(3.4) 38

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Anyotherchoicewilldecreasetheexpectedrateofincorrec tidenticationofonepopulation atthecostofincreasingtheother.Asecondplausiblechoic eistoseektomaximizethe sumoftheexpectedcorrectidenticationrates, C A = 1 2 + 1 2 erf x 0 A A p 2 C B = 1 2 + 1 2 erf B x 0 B p 2 ; (3.5) itiseasytoshowthattheseparatrix x 0 isthen x 0 = B 2 A B A B A p ( A B ) 2 +2( 2 A 2 B )ln( A = B ) 2 A 2 B .(3.6) (OnerootmaximizesC A +C B .Notethat(3.6)reducesto( A + B ) = 2when A = B .) Athirdnaturalchoice,maximizingtheproductC A C B ,requiresnumericalsolution.Of course,amorecomplicatedriskfunctioncouldapply,forin stanceinmedicaldiagnosis, whereafalsenegativeismuchworsethanafalsepositive. Tocomparethepredictivevaluesofthreeofthestatistical measuresdevelopedinthe text,1 =e crossingfromTable3.3,kurtosisfromTable3.5,andtheKol mogorov-Smirnovtest ofTable3.4,weapplythesimplestseparatrix,(3.4)tothem eansandstandarddeviations estimatedforthersttwo.(Thischoiceismotivatedbythes imilarcorrect-identication percentagesforHOSEandSKOVinTable3.4,butasanalternat iveto(3.5),usingthe actualdatasetsgivescomparableanswers).Thentheexpect edcorrect-identicationrate (3.5)for1 =e asadiscriminantis62%andthatforkurtosis67%.Theserate sarebothlower thanthe79%(Table3.4)fortheKolmogorov-Smirnovtestapp liedtothenoisedistribution, underminingtheideathatthedeviationofthisdistributio nfromnormalformisstrictlya proxyforcorrelationtime. 39

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3.7AppendixB:KurtosisandCorrelationLength Inthepreviousappendix,wehavearguedfromthedatathatbe cause1 =e decaytime asameasureofcorrelationtimedoesnotdiscriminateHOSEc ulturesfromSKOVaswell astheKolmogorov-Smirnovtestonthenoisedistributions, thelattermustbemorethan aproxyfortemporalcorrelation.Wenowconsiderwhethersp atialcorrelation,whichthis experimentdoesnotmeasuredirectly[4],mightenhancekur tosisbyreducingthenumber ofindependentmotionsresponsibleforthemeasuredtimese ries,asdiscussedinSection 3.3. Theresistanceorcapacitancemeasuredatagiventimeisthe resultofmotioninvolving manycells.Ifweviewthetotalsignalasthesumofmanycompo nents,andifeachofthese hasanon-divergentvariance,thecentral-limittheoremho lds,andweexpectapproximately aGaussiandistribution,whichweobserve(Figure3.1e).As iswellknown,convergence underthecentral-limittheoremasthenumberofcomponents n !1 isnon-uniform,and fornite n ,outlierscanaectthekurtosis.Thebinomialdistribution (fordeniteness,with equalprobabilitiesforindividualevents 1)providesafamiliarexample.Here,kurtosis r 2 = 2 =n [1,(26.1.20)],butmoregenerallywewouldexpect r 2 n 1 forthewholeclass ofrelatedmodels.Ifaspatialcorrelationlengthissuppos edproportionaltothecorrelation time, ,thenwewouldexpect n 2 ,sincethecultureistwo-dimensional,sothat r 2 2 .(3.7) Acomparisonof average 1 =e timesfromTable3.3,estimating ,toaverageestimated kurtosesfromTable3.5showsamonotonicincreaseofkurtos iswith ,aspredicted.However,theincreaseisverymuchmorerapidthan 2 ,roughly 12 ,accordingtothesefour datapoints.TheverylargeratiosofkurtosisbetweenHOSEa ndSKOVsamples(factor of25forresistance,19forcapacitance)formodestincreas esin1 =e times(29%and31%) suggeststhatspatialcorrelationsarestrongerthanthete mporalones.Ontheotherhand, scatterplots(forresistanceandforcapacitancedata)ofk urtosis versus 1 =e timeforthe60 40

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runsshowtremendousvariationandnoevidenttrend;onlyon averagedoweseemonotonic behavior.Thissuggeststhattemporalcorrelationandkurt osis,whilebothdiscriminants betweenHOSEandSKOV,maynotbestronglycoupled. 41

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Figure3.2.(Coloron-line.)Projectionalongthersttwop rincipalcomponentsofthe fourteen-dimensionalspacedeterminedbyTables3.1{3.3. Blueopensymbolsmarkthe 34HOSEruns,redcrossesthe26SKOVexperiments.Aspopulat ions,thesetwosetsare distinct,buttheoverlapofclustersmakesitdiculttodis tinguishindividualrunsinthis typeofprojection. 42

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Figure3.3.Theincrementsofaniterandomwalkwithpersis tence(left)maymimic certainaspectsoftheexperimentaldata(right),butwithn otabledierences.Therandom processhas a =0 : 8894,soanexponentialdecaytime =4 : 00.Theexperimentisatypical capacitancenoisetimeseriesofHOSE,withameasured1 =e crossingof5 : 7.(a)and(b) showthebest-tlinestotherst100points(excludingzero andthelowestfrequency)of thepowerspectrum;bothgiveslopes 1 : 0,buttherandomdatalevelonoticeably atlowfrequencies,aswouldbeexpectedofwhitenoise.Auto correlationcurves(c)and (d)showtstoexponential(dottedline)andshiftedpowerlaw(3.1)(solid)decays.For therandomnoise,thetwotsfallontopofoneanother,butfo rtheexperimentaldata,a powerlawtsbetterthanexponentialdecay. 43

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Figure3.4.(Coloron-line.)Fractionaldiscrepanciesbet ween H pred givenby(3.3)and measuredHurstexponentasfunctionsofmeasuredspectrale xponent low .Nearthe bottom,plottedwithlarge+symbols,arearticially-gene rateddatawithknownlongtimecorrelations.Attoparegenerateddata(large )fromrandom-walkincrementswith persistencetimesrangingfrom2(smallervaluesof )to7(largervalues).Inthemiddle areexperimentalresultsforHOSE(blue )andSKOV(red ).Mostoftheexperimental datalookmorelikethecorrelateddatathantheuncorrelate d,butafewoverlapwith uncorrelatednoise;alloftheseareSKOV. 44

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CHAPTER4 DETECTINGEFFECTSOFLOWLEVELSOFCYTOCHALASINBIN 3T3FIBROBLASTCULTURES 1 4.1Introduction ThetoxincytochalasinBinterfereswithcytoskeletonfunc tionbyinhibitingactinpolymerization[72,79,10,14].Atsucientlyhighconcentrati on,cytochalasinpoisoningof 3T3broblastsleadstoanumberofmorphologicaleects[72, 79,31,57].Theelectrical resistanceofacellculturejumpsrapidlywiththeaddition ofalowconcentrationofthe toxintothemediuminarowcell;thisjumpisreversedjustas rapidlywhenthetoxin isrushed[13].Althoughthejumpsinresistanceprovideanu nmistakablesignaturefor theadditionandsubsequenteliminationofcytochalasinB, the absolute resistancegives afarlesssensitivesignal:givenanaverageresistanceove rhalfanhour,onecanperhaps distinguisha2 : 5 Mconcentration 2 inthemediumfromnotoxin,butthecorrelationbetweenabsoluteresistanceandconcentrationistooweakfor nercomparisonswhenthe experimentdoesnotpermitdynamiccontroloverthelevelso ftoxininthemedium.The techniqueofelectriccell-substrateimpedancesensing(E CIS)hasbeenintroducedtomonitorelectricalimpedanceofcellcultures[33];however,i nmostpublishedECISwork( e.g. [33,46,39,16,69,23]),onlyatime-averagedsignal,orase culartrendinresistanceover manyhours,isused.Wewillarguethatastatisticalanalysi sofimpedancenoisedoes abetterjobofdistinguishinglowlevelsofcytochalasinBi nculturethandoesaverage impedance. 1 Thischapterhasbeenpublished,indierentform,byD.C.Lo velady,J.Friedman,S.Patel,D.A. Rabson,andC.-MLoinBiosen.&Bioele. 24 2250(2009). 2 M=micromole/liter 45

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AtthebottomofFig.4.1weshowmicrographsofconruentlaye rsof3T3broblasts underdierentamountsofthetoxin.Itisdiculttodistingu ishamongcultures(b),(c), and(d):exposedtotoxinlevelsof0,0.1,and1 M,theyremainconruent.However, culturesexposedtohigherconcentrations(notshown)deve lopeasilyrecognizedholesand othergrossfeatures.Theseresultsareevenclearerintheg raph,(a),showingtheresistance versustimeofconruentlayersof3T3cellsexposedtosixcon centrationsofthetoxin.While thethreehigherconcentrationsareeasilydistinguishedf romthethreelowest,onecannot tellthedierencesamongthethreelowestconcentrations.T hisisconrmedinFig.4.2, showingthatevenaveragingontheorderoften2048-secondr uns(5{6hoursofdata) cannotdistinguishthethreelowesttoxinlevels. Inreference[48],weintroducedastatisticaltechniquefo ranalyzingtherapidand apparently\random"noiseructuationsseeninECISexperim entsanddemonstratedthat suchanalysiscandistinguishcancerousfromnon-cancerou sculturesofhumanovarian surfaceepithelialcells.Wenowapplytheseideastocultur esof3T3broblastswithlevels ofcytochalasinBinthemediumrangingfromzeroto10 Mandshowthatthenoise spectrumdistinguishesdierentconcentrationsmoreeecti velythanaverageresistance. Aswiththepreviouswork,thestatisticalmeasuresinclude thepowerspectrumofthe noise,Hurstexponents,detrendedructuationanalysis,an dstatisticaltestsofpopulation dierences.4.2ExperimentalMethods WeusedtheECISsystemtocollectmicromotiontime-seriesd ata,theructuationsin whicharecausedbythemovementsinaconruentlayeroflivec ells.Thesystemcanbe modeledasanRCcircuit[34,35,46,47].Thecellsarecultur edonasmallgoldelectrode (5 10 4 cm 2 ),whichisconnectedinseriestoa1-Megaohmresister,anAC signalgenerator operatingat1voltand4000Hz,andnallytoalargegoldcoun ter-electrode(0 : 15cm 2 ). Thisnetworkisconnectedinparalleltoalock-inamplier, andthein-phaseandout-ofphasevoltagesarecollectedonceasecond,fromwhichweext racttimeseriesofresistance 46

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(a) (b) (c) (d) Figure4.1.Graph(a)displaysresistanceasafunctionofti meinconruentculturesof3T3 broblastsexposedtomediumcontainingthetoxincytochal asinBinconcentrationsof0, 0 : 1,1 : 0,2 : 5,5 : 0,and10 M.Theopticalmicrographs(b){(d)showthestatesofcultur es 20hoursafterexposureforthethreelowestconcentrations .Thecircularobject(250 diameter)ineachisthegoldelectrode.Theimagesweretake nwithacooledCCDcamera(PrincetonInstrumentsMicroMax:512BFT)throughaZei ssAxiovert200microscope equippedwitha40xNA0.65Achromatphaseobjectivelensand stageincubatorand wereprocesseddigitallytoenhancecontrast(ImageMagick -equalize option).Ifone ignorestheoverallbrightnessasanartifactofthephase-c ontrastmicroscopy,itisdicult todistinguishimages(b){(d). 47

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Figure4.2.AverageresistancesofECISruns(asinFig.4.1a )donotdistinguishlowconcentrationsofcytochalasinB.Symbols( )plottheaveragesofmultiplerunsasfunctions ofconcentration;controlruns(0 M)areplottedat0 : 01 M.Foreachconcentration,the outererrorbargivesthepopulationstandarddeviation,th einnererrorbarthestandard errorofthemean.Thestandarderrorsofthemeanofthelowes tthreeconcentrationsall overlap,consistentwiththepicturepresentedinFig.4.1a .Weaveragednine2048-second runsat0 M,twelveat0 : 1 M,twelveat1 : 0 M,tenat2 : 5 M,elevenat5 : 0 M,andten at10 M. 48

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(a) (b) (c) Figure4.3.Power-spectral-densityplotsforrepresentat iverunsattoxinconcentrationsof (a)0 M,(b)0 : 1 M,and(c)1 : 0 M.Thepowersareforthenumericalderivativesof individual2048-stimeserieswithhalf-overlapping,256secondHannwindows;theseries arenormalizedtounitvariancebeforethepowerspectraare estimated,sotheunitsare arbitrary.Low-frequencyexponents characterizing1 =f noiseareestimatedbystraightlinetstotherst100frequencies(excludingzeroandthen extlowest);fortheexamples shown,theseare0.83,0.75,and0.60.Estimatesfromapopul ationofrunsateachconcentrationareaveragedtoproducethedatainthethirdthrough fthcolumnsofTable4.1 andinFigure4.4.andcapacitivereactance.InECISexperiments,theructuat ionsincompleximpedance comeprimarilyfromchangesinintercellulargapsandinthe narrowspacesbetweenthecells andthesmallgoldelectrode[35,47,46].Acurrentofabouto nemicroampisdriventhrough thesample,andtheresultingvoltagedropofafewmillivolt sacrossthecelllayerhasno physiologicaleect:thisisanoninvasive,invitro-techni que.The3T3broblasts,obtained fromtheAmericanTypeCultureCollection(Manassas,VA),w eregrowninDMEM(4.5g/L D-glucose)(Mediatech,Manassas,VA)supplementedwith10 %FBS(Mediatech),50g/mL streptomycin,50units/mLpenicillin,and250ng/mLamphot ericinBunder5%CO 2 anda37 C,high-humidityatmosphere.ForECISmicromotionmeasure ments,cellswere harvestedandgrowntoconruence24hoursbeforeadditionof cytochalasinBintothe electrodewells,resultinginacelldensitythatwascontro lledat10 5 cell = cm 2 .Cytochalasin B(Sigma,ST.Louis,MO)wasdilutedinDMSOasa10mMsolution beforeuse. Toexpose3T3celllayerstocytochalasinB,0.4mLofcomplet eculturemediumwas usedineachwellbeforeaddingthecytochalasin-Bsolution .Serialdilutionswereprepared inculturemedium,and0.1mLoftoxinsolutionwascarefully addedtoeachwelltoachieve 49

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thenaldesiredconcentration.Incontrolexperiments,ea chwellreceivedthesameamount ofculturemediumwithoutcytochalasinB.NotethatintheEC ISapparatus,thewellsare completelyindependent.Weemphasizethatweare not dynamicallychangingtoxinlevels asinrow-cellexperiments[13];toseetheeectsofthetoxin willrequirestatisticalanalysis. From64separatecultures,wecollectedtimeseriesofwhich ninewereatzeroconcentrationofthetoxincytochalasinB.Wetooktwelverunsat0 : 1 M,twelveat1 : 0 M,ten at2 : 5 M,elevenat5 : 0 M,andtenat10 M.Each2048-srun(justover1/2hour)was taken24hoursaftertheintroductionofthetoxintoensuret hatthecellsreceivedthefull eectateachconcentration.Wenumericallydierentiatedth eresistanceandcapacitance timeseriestoobtainnoisetimeseriesforeach,whichwenor malizedtozeromeanandunit variance.4.3StatisticalMeasuresofNoise Whilethepowerspectrum,Hurstexponent,anddetrendedruc tuationanalysisgive uswaysofquantifyinglong-termcorrelationsinthenoise, theFouriertransformofthe powerspectrumyieldsautocorrelation,whichenablesusto quantifyandstudyshort-term correlations.Inwhatfollowsweexaminetherealpartofimp edancefromtheexperiments. At4000Hz,theimaginaryorcapacitivepartwasnotasuseful atdistinguishingthedierent toxinslevels;wewouldexpectcapacitancetobecomemorese nsitivetocellmotionsat higherfrequencies.4.3.1Long-termcorrelations Werstexaminelong-termcorrelations.Inthelow-frequen cylimit,thepowerspectrum atthelowerconcentrationsshowssignsoflong-timecorrel ations,withthecorrelations gettingweakerastheconcentrationisincreased.Alog-log plotofspectraldensityagainst frequency f suggestsanintensityvaryingas f ,withthistrendbecominglessclearat higherconcentrations.Weestimated withleast-squaresstraight-linetsofpowerfrom onlythelowest100frequencies(excludingzerofrequencya ndthenextlowestfrequency). 50

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Table4.1.Long-termcorrelation.Power-spectral,Hurst, andDFAmeasuresofconruent layersof3T3broblastresistivenoiseseriesaveragedove rallrunsatdierentconcentrationsofthetoxincytochalasinB.Therstandsecondcol umnsgivetheconcentration in Mandthenumber( N )ofindependentexperiments.Shownareestimatesfor1 =f behavior( ),Hurst( H ),anddetrendedructuationanalysis( D )exponentsatlowfrequencies.Themeansofthe valuesdierbymanystandarderrorsofthemean( p N ,where isthestandarddeviation),allowingustodistinguishthep opulationscomposedof N runs justfromthepowerspectrum.Noticethatforthepowerspect rum,concentrationsofzero, 2.5,and5 : 0 Mareseparatedby ,thestandarddeviation. conc. N p N H H H p N D D D p N 0 : 090 : 8540 : 1020 : 0340 : 7640 : 0620 : 0200 : 8320 : 0360 : 012 0 : 1120 : 7800 : 0970 : 0280 : 7680 : 0380 : 0110 : 8370 : 0450 : 013 1 : 0120 : 6640 : 1310 : 0380 : 7460 : 0450 : 0130 : 7870 : 0350 : 010 2 : 5100 : 4000 : 1040 : 0330 : 6900 : 0320 : 0100 : 7060 : 0320 : 010 5 : 011 0 : 2830 : 3610 : 1090 : 6930 : 0740 : 0220 : 5580 : 0520 : 016 10 : 010 0 : 8430 : 4630 : 1470 : 6050 : 1180 : 0370 : 4030 : 1070 : 034 Foreachrun,wesplitthe2048noiseamplitudesintohalf-ov erlappingwindowsof256s, multipliedbyaHannwindow,Fouriertransformed,andsquar ed,averagingtheresulting spectrainordertoreducescatter[62].WeshowinFigure4.3 sometypicalscatterplotsat thethreelowestconcentrations,alongwiththeirts. Havingobtainedan foreveryexperiment,wethenaveragedtheseseparatelyfor eachconcentration.OurresultsareshowninTable4.1,wher e istheaverage, isthe standarddeviation,and p N ,thestandarderrorofthemean,is dividedbythesquare rootof N ,thenumberofexperiments.Therstthingwenoticeisthatt hemeansof the sareseparatedbyseveralstandarderrors( p N ).Thisimpliesthatalreadywiththe powerspectrumalone,givenenoughexperiments,onecouldi nprincipledistinguishthe concentrations.Moresignicantly,themeansatconcentra tionsofzero,2.5,and5 : 0 Mare separatedfromtheirneighboringmeansbyatleastastandar ddeviation ;thusthepower spectrummightbethestrongestindicatorfordistinguishi ngdierentlevelsofthistoxinin 3T3broblasts.TheStudentt andKolmogorov-Smirnovtestsconrmedtheseparations ofpopulationsby exceptforthetwolowestconcentrations(zeroand0 : 1 M). Thesepower-slopeaveragesalsoshowacleartend,asFig.4. 4bringsout.Asconcentrationincreases,thepowerslopedecreases.Thus,thelon g-termcorrelations,whichare strongestatzeroconcentration,aredisruptedbyaddition ofthetoxin. 51

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Thereisadangerinlookingonlyatpower-lawbehaviortodet erminetheexistence oflong-termcorrelationsintime-seriesdata[63,17].Ino rdertoavoidthisproblem,we lookattwootherindicatorsoflong-termcorrelation,thee xponentsprovidedbyrescaled rangeanalysis(Hurstanalysis)anddetrendedructuationa nalysis[7,51,25,58,59].Both methodsarebinningtechniques.Atimeseriesissplitintob insofduration ,anditisthen determinedhowameasure S ( )scaleswith .ForHurst,onesubtractsthemeanfrom allthedatainabinandcharacterizesthatbinbyitsstandar ddeviation, .Theseries isintegrated,andtheminimumvaluesubtractedfromthemax imum,yieldingtherange, R .Foreachbin,onerecordstheratio R= andaveragesoverbinsofthesamesize.The procedureisrepeatedforsuccessivelylargerbins( ).Astraight-linettoalog-logplot of R= againstbinsize revealsapowerlaw, R= H ,where H istheHurstexponent. Detrendedructuationanalysisrunsalongsimilarlines,bu twithineachbinonesubtracts abest-tline,thusdetrendingthedata.Thedatainthebina rethencharacterizedby standarddeviation D ,were D istheDFAexponent. Table4.1showstheresults. H ( D )istheHurst(DFA)exponentaveragedoverall experimentsforeachconcentration. i isthestandarddeviationand i p N thestandard errorformeasure i = D;H NeitherHurstnorDFAisasclear-cutasthepower-spectruma nalysis.ForHurst,concentrationsthatareclosetoeachotheroverlapevenwhenco nsideringjustthestandard error.DFAisalittlebetter,butusingthestandarderrorca nnotdistinguishzeroconcentrationfromthenextlowest.BothHurstandDFAshowadeclin eintheexponentsfrom thelowestconcentrationtothehighest.Thattheoverallse parationseemstobebetter forDFAthanforHurstmightbeduetonite-sizeeectsinthet imeseries.Aspointed outbyCoronadoandCarpena,DFAdoesabetterjobwithnitet imeseriesthandoes Hurst[17].TheStudentt testseemstosupportthis.ForDFAitisabletondthatall concentrationsdierexceptforzeroandthenextlowest,whi leforHurstitcannottellthe dierencebetweenthethreelowestconcentrations.Forneit herDFAnorHurstcanthe 52

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Table4.2.Short-termcorrelation.Firstzero( Z )andrst1/ecrossing( E )oftheautocorrelationfunctionusedasmeasuresforconruentlayerso f3T3broblastresistivenoise seriesaveragedoverallrunsatdierentconcentrationsoft hetoxincytochalasinB. concentration( M) Z Z Z p N E E E p N 0 : 075 : 177108 : 56836 : 1894 : 7372 : 0160 : 672 0 : 149 : 70032 : 2079 : 2973 : 5851 : 5220 : 439 1 : 047 : 36938 : 36811 : 0762 : 8621 : 3550 : 391 2 : 527 : 3319 : 8483 : 1142 : 2010 : 2860 : 091 5 : 010 : 14718 : 3415 : 5301 : 5780 : 0660 : 020 10 : 01 : 7880 : 06950 : 02201 : 4980 : 0440 : 014 Kolmogorov-Smirnovtestdistinguishthethreelowestconc entrations,butforDFAitcan distinguishalltheothers. Bothshowthesametrendofstrongcorrelationatzeroconcen trationwiththatcorrelationgettingweakerastheconcentrationisincreased.With allthreemeasuresoflong-term correlationshowingthistrend,webelievethatitisactual lythere.Thistrendisalsowhat wewouldintuitivelyexpectconsideringtheeectofcytocha lasinBonthecytoskeleton. HurstandDFAcanbeusedtovalidatepower-spectral( )results[63].Werepeatedthe techniqueweintroducedinRef.[48,Fig.4],computingdisc repanciesbetweenthevalues forbothHurstandDFApredictedfromthenumericalestimate sfor andthosemeasured inthetimeseries.Asinthepreviouswork,theexperimental discrepanciesweremore consistentwiththesmalldiscrepanciesmeasuredinartic ialtimeseriesconstructedto havelong-timecorrelationsthanwiththelargerdiscrepan ciesofarticialwhitenoise. 4.3.2Short-TermCorrelations Inordertoinvestigatetheshort-termcorrelationsinthes esystems,welookatthe rstzeroandrst1 =e crossingsoftheautocorrelationfunction.Ourresultsare shownin Table4.2.The1 =e crossingcandistinguishzeroconcentrationfrom2 : 5 Mand2 : 5 M from5 : 0 Musingthestandarddeviation,whiletherstzerocrossing candosowith thestandarderrorofthemean.Student's t testforthe1 =e crossingdistinguishesall concentrationsbutzerofromthenextlowestand1 : 0 Mfrom0 : 1 Mand2 : 5 M.Thezero crossingdoesnotdonearlyaswell;itcan'tdistinguishany ofthelowestconcentrationsfrom 53

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Figure4.4.Theplotshowstheconcentrationofthetoxincyt ochalasinBonthehorizontal axiswithzeroconcentrationplottedat10 2 M(semilogplotforclarity)versustheslope ofthelog-logplotofpowerversusfrequencyaveragedovera llexperimentalrunsforeach concentration.Thelargererrorbarsinthegraphshowthest andarddeviationasgiven inTable4.1,whilethesmallerbarsgivethestandarderror. Noticethecleartrendof decreasingpowerslopeatincreasingconcentration,indic atingtheweakeningoflong-term correlation. 54

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eachother.TheKolmogorov-Smirnovtestgenerallyfailsto distinguishthezero-crossing distributionsatthetestedconcentrations. Thetrendforeachalsosupportsourhypothesisthatthesyst emisbecominglesscorrelatedwithhigherconcentrationofthetoxin.Figure4.5dis playsthistrendbyplottingthe concentrationonthehorizontalaxisandthe1 =e crossingontheverticalaxisalongwith thestandarderrorateachconcentration.Astheconcentrat ionisincreased,therst1 =e crossingisreduced,implyingthatthesystemislesscorrel atedforthehigherconcentrations. 4.4MeasureSpace Usingourmeasuresoflong-andshort-termcorrelationsint henoise,wecanconstruct amultidimensionalspaceeachaxisofwhichrepresentsoneo fourmeasures,andeach experimentisthenapointinthisspace.Weconstructsuchas pacewiththefourdimensions D H ,and1 =e crossing,normalizingtounitvariance. Wendthevectorinthisspacethatrepresentstheaveragepo sitionforeachconcentrationandthenconstructasphereabouteachaverageposition withradiusgivenbytheroot meansquareofthestandarderrorsalongthefouraxes.Divid ingthedistancebetween theaveragepositionofthepopulationsattwoconcentratio nsbythesumoftheirradii measuresroughlytheirseparation,witharatiosignicant lylargerthanunityindicating goodseparation.Table4.3comparesthesixspheres.Eventh oseclustersthatoverlap underthiscriterion,suchasthezero-concentrationand0. 1Mspheres,haveseparation parametersclosertounitythantozero,suggestingthatlon gerdatasetsmightseparate themeasurementsmoreclearly. Inordertogetsomepictureofwhatishappeninginthisfourdimensionalspace,we projectontoaplanewhosetwoaxesmaximizethevariance, i.e. ,thersttwoprincipalcomponents[40].Fig.4.6showsclusteringofthedieren tconcentrations.Forthis principal-componentanalysis,weomittedtherunsat10 M.Startingfromthehighest concentration,weseethattheclustersaredistinctbuttha tastheconcentrationisreduced,theclustersgetclosertogetherandbegintooverlap 55

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Figure4.5.Theplotshowstheconcentrationofthetoxincyt ochalasinBonthehorizontal axiswithzeroconcentrationplottedat10 2 M(semilogplotforclarity)versustherst 1 =e crossingaveragedoverallexperimentalrunsforeachconce ntration.Theerrorbarsin thegraphshowthestandarderrorofthemeanasgiveninTable 4.2.Noticethecleartrend ofdecreasingrst1 =e crossingwithincreasingconcentration,indicatingthewe akeningof short-termcorrelation. 56

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Figure4.6.Therstprincipalcomponentonthehorizontala xisisplottedagainstthesecond principalcomponentonthevertical.Oneseesseparationof dierentconcentrationsofthe toxin. 57

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Table4.3.Theresultsfromcalculationsdoneon concentration spheresinourconstructed measure space.Therowsandcolumnsbothrepresentthedierentconce ntrationswehave usedinourexperiments.Anentryinthetablethentellswhic htwospheresarebeing compared.Wedividedthecenter-to-centerdistancebetwee nanytwospheresbythesum oftheradiiofthosesametwospheres.Unitymeansthesphere sjusttouch. 0 : 0 M0 : 1 M1 : 0 M2 : 5 M5 : 0 M 0 : 1 M0 : 8 1 : 0 M1 : 51 : 1 2 : 5 M3 : 33 : 72 : 5 5 : 0 M4 : 14 : 73 : 93 : 0 10 : 0 M4 : 14 : 64 : 32 : 90 : 7 4.5Discussion Wepreviouslydemonstrateduseofthesestatisticaltoolso nelectrical-noisemeasurementsfromECIStodistinguishcancerousandnoncanceroush umanovariansurfaceepithelialcellsinculture[48].Wehavenowusedthesesameto olstodierentiatetoxinlevels inculturesof3T3broblasts.Wehaveobservedthat,asthet oxinlevelisincreased,the long-termcorrelations,asmeasuredbythepowerspectrum, Hurstexponent,anddetrended ructuationanalysis,decrease.Inaddition,theshort-ter mcorrelations,asmeasuredbythe rstzeroand1 =e crossingoftheautocorrelationfunction,decreaseasthet oxinlevelis increased.Ifweinterpretthesecorrelationsasadescript ionofthelevelofcommunication andcooperationbetweencells,thenthesemeasuresaredesc ribingasystemthatisinsome sensecoordinated,thatcoordinationbeingdisruptedbyad ditionofthetoxincytochalasin B.Eventually,asthetoxinreachesathreshold,thesystemi sunabletoworktogether,and themeasuresapproachvaluestypicalofrandomsystems.For example,theHurstexponent dropsfrom0 : 764inthecontrolruns,indicatingcorrelation,towardthe valueof1 = 2expectedforwhitenoise.Atthehighestconcentrations, appearstogonegative;however, thelog-logpower-spectralplotsalsobecomehardertointe rpret,soweexpectthatlonger runswouldgive =0(whitenoise).Thelossoftemporalcorrelationwithincr easingtoxin concentrationcanbeexplainedbytheeectsofcytochalasin Bonthecytoskeleton[72,79]. 58

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AsthecytochalasinBinterfereswithcellfunction,theabi lityofthecellstomaintaintight celljunctionsisdisrupted. NoiseanalysisofECISdatacanbeusedtoteststatistical-m echanicalmodelsofmicromotion.Lookingfurther,weenvisionadatabaseofelectric al(ECIS)characteristics,with thecollectionofnoisemeasuresforeachcelltypeandenvir onment( e.g. ,toxin)constitutingakindofngerprint:thiscouldopenthedoortofurthera pplications,includingdrug screeningandenvironmentalsensing. 59

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CHAPTER5 STATISTICAL-MECHANICALMODELOFMICROMOTION 1 5.1Introduction Wenowintroduceoneofthesimplestwayspossibletomodelmi cromotion.Thetime seriesproducedbyECISmotivatesthemodel,andwewishtoex tractfromourmodelatime seriesthatcanbecomparedwiththeECISexperiment.Wecons iderlocalinteractionsin cellculturesandmodelthecell-cellandcell-mediuminter actions.Theadhesionofnormal cellstoothercellsandtheextracellularmatrixiscausedb yreceptorswhichareexpressedon thesurfaceofthecells[75,80].Thecell-celladhesionofa mutatedcellchangesdepending onthestageofmetastasisandisacomplicatedprocess[41,8 0].Itistoomuchtoexpect thatasimplemodelwillcaptureallaspectsofcellinteract ionssuchastheelasticityofthe cellmembrane,contactinhibition,andadhesion.Themanyf actorsthatareinvolvedin cell-cellinteractioncannotbemappeddirectlyontooneor twoenergyparametersinaspin model.However,suchsimpliedparameterscan,atleastinp rinciple,modelafewaspects oftheexperimentalsystem,andeventhoughtherearemanyco mplicatedprocessesgoing on,wefeelthatourmodelwillbeabletosaysomethingaboutt hesimplestwaythese cellsbehave.Wethinkofasimpleinteractionenergyascoup lingspecicallytothecontact inhibitionofthecell.Withthisinmindwebuildamodelofce llinteractionsbyconsidering thecellsurfaceenergyasitreactstothepresenceofotherc ellsandthemedium.One wayofdoingthisisbyusingamodied q -statePottsmodeldevelopedbyGranerand Glazier[37].Intheirmodel,every for 2f 1 ; 2 ; 3 ; ;q g representsadierentcell. Theysimulatedthesortingoftwodierentcelltypeswithdie rentialadhesivity.They 1 ThischapterisinpreparationforsubmissionbyD.C.Lovela dyandD.A.Rabson. 60

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wereabletoshowlong-distancecellmovementleadingtosor tedclustersintwophases. AlthoughwecannotjustifytheiruseofMetropolis\dynamic s"(andneithercanthey[36]), wewillusetheirHamiltonianasourenergyfunctiontodescr ibetheinteractionenergyof singlecelltypes.Thesecelltypescanbeeectivelychanged bychangingtheinteraction energiesofthecellswitheachotherandthemedium.Sincewe areinterestedinthe collectivebehaviorofbiologicalsystemsthatarebiggert hanasinglecell,butaresmaller thananorgan(smallclustersofcellsthatmightbedenedas asortofmesoscalefor biophysics)andsincethetimeseriesofthesesystemsshowc orrelationsonmanytime scales(thisshowsupinalog-logplotofthepowerspectruma mongothermeasures),we takeinspirationfromPerBak'ssand-pilemodel[4,5].PerB aketalia,inaneortto explainthepower-lawbehavior 1 f thatisseeninmanysystemsinnature,developeda modeltoshowthatdynamicalsystemswithmanyspatialdegre esoffreedomwillevolve intowhattheycallaself-organizedcriticalpoint.Theyar guethatspatialscalingfora systemnearthiscriticalpointwillleadtonoisethatpropa gatesthroughoutthesystemand thatthesystemwillthusshowtemporalructuationsonallti mescales.Theysuggested thatundercertainconditionsasandpilewouldobeythisbeh avior.Sandisrandomly sprinkledontoanexistingsandpilecomposedofmanypeakso fvaryingheights.Whenthe averageslopeofaparticularpeakexceedssomecriticalval ue,theheightofthatpeakis reducedbytransferringsandtoitsnearestneighbors.This cancauseanavalanchethat hasthepotentialofcreatingmultipleavalanchesbydistur bingitsneighbors,andthus adisturbancecantravelacrosstheentiresandpile.Theycl aimthatforabigenough system,onecanseeavalanchesonallspatialandtemporalsc ales.Weadaptthistechnique tocreateamodelofcell-cellcommunicationinoursystem.I nourmodel,cellsproduce andstoreasignalingchemicalwhich,afteracertainlimiti sreached,isdumpedintothere surroundingenvironment.Someofthechemicalistransferr edtoneighboringcells,butif acellbordersagapbetweencells,someofthechemicalwillb esweptawayanddegraded. Thisisnecessarytopreventunlimitedbuild-upofthechemi calandtakestheplaceofthe openboundaryconditionsinthesand-pilemodel.Thusthece llsinoursystemwillquickly 61

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reachthisbarelycriticalpointbyachemicalbuild-upinea chcell.Asthecellsreach theirtolerancelevel,theydumpthechemicalintothesurro undingenvironmentincluding theirneighboringcells.Thiscancauseanavalancheofchem icaldumping.Thechemical concentrationineachcellcouplestoitsareaandsoalsotot heareaofthegapsbetween thecellswhichiswhatismeasuredinoursimulations.Thesp atialrelationshipsthat resultcarryoverintotemporalrelationshipsthatshowupi nthepowerspectrumofthe system.Thepowerspectrumcapturesthedynamicsofoursyst emasitevolvesfrompoint topointinphasespace.Inaneorttohaveourtoymodel(atlea stinprinciple)mimicthe dynamicsofECIS,weusekineticMonteCarlotoproducetheti meevolutionofoursystem. Webelievethistobeanimprovementuponpreviousworkthath asusedMetropolis. 5.2Hamiltonian The q -statePottsmodelhasbeeninvokedsuccessfullytoexplain collectivebehaviorin manydierenteldsofscienceincludingphysics,economics ,andbiology.Wewishtouse itinmodiedformtoelucidatetimeseriesproducedfromECI Sexperiments.The q -state PottsHamiltonianis H = J X i j (5.1) wherethesumrunsovernearestneighborsand i representsthestateofsite i [85].Graner andGlaziermodiedthistodescribeacollectionof N cellsona2 D latticeofspinswhere ij isthespinvalueonsite ij .Thus ij =1 ; 2 ;:::;N .If ij = i 0 j 0 ,thelatticesite belongstothesamecellandtodierentcellsiftheyarenoteq ual.Let beanindex thatidentiescellsofdierenttypes.Anindividualcellca nthenbedescribedbytwo indices, and ,where isthecelland isitstype.Inthepresentwork takesonlytwo values,zeroforagapbetweencellsandoneforanycell. J ( ij ) ( i 0 j 0 ) willthenrepresent thecell-cellinteractionwhichdescribesthesurfaceener gybetweenneighboringcellsorin theabsenceofaneighboringcelltheextracellularmatrix. Withthesemodicationsthe 62

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systemisdescribedbytheHamiltonian H = X ij X i 0 j 0 J ( ij ) ( i 0 j 0 ) 1 ij i 0 j 0 : (5.2) Noticethatif ij = i 0 j 0 ,theKroneckerdeltagoestounity,andthetermdropsoutof thesum.Thisisbecausethesites ij and i 0 j 0 belongtothesamecellandthusdonot contributetothesurfaceenergybetweentwodierentcells. Theenergyrequiredforthecelltobeawayfromitstargetare a V T is ( V T ) 2 (5.3) where isitsareaatsometimestepand isaconstant.Withthisdescriptionofacell's internalenergy,theHamiltonianbecomes H = X ij X i 0 j 0 J ( ij ) ( i 0 j 0 ) 1 ij i 0 j 0 + X ( V T ) 2 : (5.4) ThisHamiltonian,asitstandsorwithslightmodications, hasbeenusedtomodelcell sorting,tumorgrowthinhealthycells,andtumormigration inthepresenceofanutrient gradient[37,80,76].Weshalltakethisasthedescriptiono ftheenergyforourcells. Strictlyspeaking,energyisnotconservedforoursystem,a ndsowecallitanenergy functionasopposedtothetermHamiltonianwhichmustberes ervedforsystemsthatdo conserveenergy.5.3SimulatingCellMovementwithKineticMonteCarlo Cellmovementandtheassociatedtimeevolutionofthismove mentaresimulatedby useofatechniquecalledbyphysicistsKineticMonteCarlo( KMC)andchemistsdynamic MonteCarlo.AsmostreaderswillbeunfamiliarwithKMC,ath oroughexplanation follows.Ishallstartwithaverybriefhistoryofthedevelo pmentofKMC. 63

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5.3.1HistoryofKMC SystemsinequilibriumcanbestudiedusingtheMetropolism ethodofMonteCarlo developedin1953byateamatLosAlamos[52].Foragivenseto finitialconditions,one mayrunaMonte-Carlosimulationandonceequilibriumisrea chedcalculatethedesired properties.However,thismethodofMonteCarlocantellusn othingaboutnonequilibrium systems,andthereisnothingmeaningfulinthewaythesyste mreachesequilibrium.In otherwords,thetimeevolutionofthesystemunderstudyhas beenlost.Inessence,time hasbeenintegratedout,andthereisnotruedynamicsthatca nbestudiedbyMetropolis. AccordingtoKaiNordlund[56],therststepinthedirectio nofkineticswasmadeby FlinnandMcManus[30],whoconsideredtransitionprobabil itiesandratesinvacancy motion.However,theydidnotderiveanyexplicittimescale .Thiswasrstdescribed byYoungandElcock[89],whoprobablyarethersttooutline thebasicsoftheKMC method.Inparticular,theyarethersttouseacumulativep robabilityfunctiontoselect aneventandatimescalecalculationoftheformusedinKMC.I n1975,Bortz,Kalos, andLebowitzdevelopedaKMCalgorithmforsimulatinganIsi ngSpinsystemthatthey calledthe\ n -foldway"[12].Itwascreatedinordertospeedupcomputert ime,butitalso simulatesthedynamicsofthesystemunderstudy.Theymaken omentionoftheworkby YoungandElcock,andperhapstheydidnotknowofit.Mostphy sicistsseemtocredit Bortzetaliawiththediscoveryofwhatisnowknownasthekin etic-Monte-Carlomethod. Thereareseveralpapersandtutorialsthatgiveamoremoder nperspectiveofthebasic theorybehindKMCanditsuses[26,83,56].Ishallnowtrytog iveanexplanationofthe particularmethodthatweuseinourwork.5.3.2Thekinetic-Monte-CarloAlgorithm Typically,kineticMonteCarloisusedtodescribethedynam icsofsystemsthathave clearlydenedstatetransitions.Thesestatesmaybethoug htofaspotentialwellsthatare separatedbybarriersthatservetotrapthesystem.Inorder forthesystemtoleavethe well,itwillhavetojumpoverabarrier.Thesystemthenrema insinanewwellforsome 64

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characteristictimebeforemakingajumptoanotheravailab lewell.Statedmoreprecisely, themicroscopicstatechangesfrequently,butallthemicro scopicstatesexploredoversome substantiallengthoftimeremaininthesamepotentialwell andsointhesamecoarsegrainedstate.Thusthesystemmayexperienceructuationsw hileinacoarse-grainedstate (apotentialwell),buttheseareusuallynotenergeticenou ghtokickthesystemoutofthis coarse-grainedstateandintoanotherexceptonrareoccasi ons.Forapossibletransition betweenaninitialstatewithenergy E 1 andanalstatewithenergy E 2 ,thereisabarrier energydescribedby E b =max( E 1 ;E 2 )+ E 1 ; (5.5) where E b isthebarrierenergyandisthedierencebetweenthetopoft heenergyrise betweenthetwoenergystatesandthehigherofthetwostates initialandnal.Figure5.1 illustratesthecasefor E 1 E 2 .Thentheprobability thatthesystemleavesitscurrentstatecanfollowaBoltzma nndistribution, P / exp( E b k b T )(5.6) where k b isBoltzmann'sconstant,and T isthetemperature.Wecanusethistocreate aratebyconstructingtheprobabilityperunittimethatthe systemleavesthecurrent state.Let o beanattemptfrequency,thatis,howoftenittriestoexceed thebarrieror, equivalently,howoftenittriestoleavethecurrentstate. Thenwecanwrite rate= = 0 exp( E b k b T ) : (5.7) Ifthisrateisindependentofprevioushistoryanddoesnotc hangeintime,thenthe transitionprobabilitywillbeauniformfunctionoftimean disaPoissonprocess[56,26]. Infact,inoursystems,ourenergiesdochangeintime,buton atimescalelongerthan thatconsideredinthekineticMonteCarlo.Thiswillallowu stodevelopanexpression forthetimedependenceofthesystem.Intheappendixtothis chapter,wegiveasimple 65

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Figure5.1.Thegureshowstheenergybarrierforaparticle thatistrappedinapotential wellthatleadstoahigherenergystate.Noticethatthat E 1 E 2 andso E b =. 66

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derivationofthePoissondistributionforoursystemfromw hichourtimedependenceis explained.Fromtheappendix,onewillseethattheprobabil itydensityorprobabilityper unittimecanbewrittenas P d (1; t )= re rt (5.8) Thisistheprobabilitydistributionoftherstescapetime Wecannowgetatimeforaneventbyconsideringthepreviouse quation.Firstnote thatifwehaveasystemthatismadeupofalargenumberofPois sonprocesses,thenit willbehavelikeonebigPoissonprocess,andonecanwritefo rtherates R = X i r i : (5.9) Theprobabilitydensityfortheentiresystemnowbecomes P system (1; t )= Re Rt (5.10) Thisequationgivestheprobabilitythataneventoccursina time t fortheentiresystem. Ifwenowchoosearandomnumber u between0and1excludingzero,thenweseethat thetimeforaneventtooccuris t = ln( u ) R : (5.11) Thisishowtimeisincrementedinthekinetic-Monte-Carlom ethod. Inowpresentthekinetic-Monte-Carloalgorithm.1.Setthetime t =0. 2.Formalistofalltherates r i ofall N possibletransitionsinthesystem. 3.Calculatethecumulativeprobabilityfunction R j = P ji r i .Notethat R = R N 4.Produceauniformrandomnumber u 1 2 [0 ; 1]. 5.Findtheeventtocarryoutbyndingthe i forwhich R i 1
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6.Carryoutevent i 7.Findallpossibleeventsthatmayhappen,andrecalculate all r i that mayhavechangedduetocarryingoutevent i 8.Getanewuniformrandomnumber u 2 2 [0 ; 1]. 9.Updatethetimewith t = t + t where t = ln( u ) R : (5.13) 10.Returntostep2. 5.4Applicationtooursystem Inowsummarizehowthisisappliedtooursystem.Oursystemc onsistsofasquare latticeofspinswherethespinshavebeendividedamongthec ellsandthespacebetween thecells.Ourcellsareputintosomeinitialconguration, andtimeissettozero.Cell movementissimulatedbysinglespinrips.Whenaspinrips,i tnolongerbelongsto thecellthatclaimeditbutnowbelongstoanothercell,orpe rhapsitbecomespartof thespacebetweenthecells.Thespinscanriponlytoneighbo ringvalues,i.e.,only atboundariesbetweendierentcellsorbetweenacellandaga pbetweencells.Every possibletransition(spinrip)isfound,andtheprobabilit yforthattransitioncalculated. Thecumulativeprobabilityfunction R j = P ji r i isthencalculated,and R = R N isfound fromthisfunction,where N isthetotalnumberoftransitions.Thespintoberippedis thenfoundandtheeventcarriedout(step6oftheoutlinedpr ocedure).Whenaspinrips, thiscorrespondstoacelladvancingorretreating.Thenumb eroftransitionsandtheir ratesarethenupdated.Anotherrandomnumberisfound,andu singitwendthetime ittookforthespintoripsincethelastspinrippedusing t = ln( u ) R .Thenthetimeis updatedwith t new = t old + t WiththeHamiltonianfromGranerandGlazier[37]asourener gyfunctiontocalculate therates,thesand-pilemodelfromParBaketalia[4,5]toex plorecell-cellcommunica68

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tion,andthekinetic-Monte-Carlomethodtosimulatecelld ynamics,weusethismodel toinvestigatetimeseriesimitatingECISexperiments.Wes etupagridwithperiodic boundaryconditions.Weallowedtheprogramtorununtilita chievedaconruentlayerof cellsformingacompactsystemwiththemediumshowingthrou ghintheinterstices.At everyMCtimestep,werecordedthenumber n ofgridpointsnotoccupiedbythecells butbythesubstrate.Thenumber n isproportionaltotheconductancethesystemoers tothesmallcurrentgoingthroughthecellculture.Thisiso urtimeseries.Iturntoour preliminaryresults.5.5Results Wepresenttheresultsof30computerrunsona30-by-30squar espinlatticewith30 cells.Theareainteractionterm wassettotwelve.Theinitialareaofthecellswasset to30,andwassetto0.01.Thetemperatureissettounity.We ran J from0to5 with J + S =5.Foreachpairof S and J valuesweranveseparaterandomseedsin ordertosimulateveseparatedynamicalconditionsandthu svedierentexperiments. Eachrunwentfor50,000timesteps.Theoutputoftheprogram isatimeseries.Atthe beginningoftheseriesthereisatransientthatoccursasar esultofthecellsmovingfrom energeticallyunfavorableinitialconditionstoamoresta bleconguration.Wesuppress thispartofthetimeseries.Itisalwayslessthanathirdoft heseries.Alphaisrelatively insensitivetowherethetransientiscuto.Figure5.3shows anexcerptfromonesuch run.NoticehowsimilarthisistotheECIStimeserieswehave seeninearlierchapters. AswithourECISexperiments,wetakethederivativeoftheti meseries,subtractoutthe average,andnormalizetheresulttounitvarianceinordert oextractthenoise.(However, unlikeECISdata,theoutputofourprogramisnotinniceunif ormtimesteps.Beforewe doanythingwithourtimeseries,welinearlyinterpolatein ordertocreateatimeseries ofequaltimesteps.)Wethentakeapowerspectrum.Figure5. 4showsalog-logplotof thepowerspectrumforthetimeseriesapieceofwhichwassho wningure5.3.Thiswas takenwithaHannwindowandaveragedoverawindowsizeof512 .Theoneshownis 69

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Figure5.3.Atimeseriesfortheinteractionenergies J =3and S =2.Thisisfor30cells ona30-by-30squarespinlattice.Itlooksremarkablylikea nECIStimeseries. typicalofthepowerspectrumseeninallofourtimeseries.T hepowerspectrumhasfour distinctregions.Readingthegraphfromrighttoleft,theh igh-frequencyregionisdue tothedynamicaleectsofthealgorithmthatweareusing.Eve nifweset S = J =0, wendthatwehavesomeslopeinthisregion.Webelievethisi sbecauseofabsorbing boundaryconditionsthatareplacedonthecellsduetoourpr ocedure.Inparticular,the spinscanriponlyatboundaries,andisolatedspinsseparat edfromthemainbodyofa cellarelikelytovanish.Oncetheyvanish,theycannotspon taneouslyreappear(except asnewbudsattheboundaryofthemaincellbody).Thisdynami calconstrainttendsto preservecellsassimply-connectedobjectsevenintheabse nceofsurfaceenergies S and J Theseconstraintsintroducelocaldynamicalcorrelations ,whichweobserveaspinknoise atthehighestfrequencies. Nextcomestheplateauregion,whichlooksmostlylikewhite noise.Thecauseofthis isuncleartousatthistime. 70

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Figure5.4.Alog-logplotofthepowerspectrumforthederiv ativeofthetimeseries.The lineisalineofbestt.Thisisfromthetimeseriesthatwase xcerptedingure5.3.The slopeofthelineis-0.20. Thencomesarapidriseatlowerfrequencies.Webelievethis toduetothechemical scalingcausedbyuseofthesand-pilemodel.Oncecansimply noticebyeyethattheslope ofthisregionissteeperthanthelineofbesttfortheentir espectrum. Manyofthespectrahaveattheverylowestfrequenciesadrop o.Thisisduetothe nitesizeofoursystemandperiodicboundaryconditions. Figure5.5showsaplotof J S ontheabscissaand ,theslopeofthebesttlineto thelog-logplotofthepowerspectrumaveragedovereachoft heveruns,foreachvalue of J and S whileholding J + S =5. J S representstheexcessofcontactinhibitionover surfacetension.As J increases(whichcorrespondstoasimultaneousdecreasein S ), ,the averageslopeofthebesttlinetoalog-logplotofthepower spectrumofthederivative ofthetimeseries,increases.Apositive J isarepulsiveforceterm.Ifwethinkofthisasa modelofcontactinhibitionforcells,thenas J increases,contactinhibitionalsoincreases. Figure5.5demonstratesthat increaseswithincreasedcontactinhibition,whichiswhat 71

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Figure5.5.Aplotshowingthechangingvaluesof ,theslopeofthebesttlinetothe log-logtofthepowerspectrumforthederivativeofthetim eseries,withchangingvalues ofthesurfaceinteractionenergies J and S whileholding J + S constantandequalto5. J isthecell-cellinteractiontermandisrepulsiveifpositi ve. S isthecell-mediuminteraction termandistherepulsivesurfaceenergybetweenacellandag apbetweencells.Notethat J getsbiggerand S getssmallerwhile trendsupwards.If J and S aremodelingthe contactinhibitionofcancerthenthisimpliesthatasconta ctinhibitiondecreases, trends inthedirectionwewouldexpectformutatedcells.wewouldexpect.Normalcellshaveahighercontactinhibiti onandahigher thando cancercells. Anotherwaytomodelthedierencebetweennormalcellsandmu tatedcellspresents itselfifonelooksathowthecellsarecommunicating.Weass umethatnormalcellshave morecommunicationthanmutatedcells.Inourmodel,wecanr eplicatethisbycontrolling howmuchthechemicalisabletoaectthetargetareaofacell( seeequation5.3).Inour program,targetareasarecontrolledbythefunction V target = A tanh( B (chemical))+ C (5.14) 72

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where A istheparameterthatcontrolsthestrengthofthechemicalc ommunication,and C isaconstantaroundwhichthetargetareawillructuate. B isjustascalingfactor. Bysetting A toahighvalue,thechemicalbuild-upineachcelldramatica llycontrolsthe targetareaofthecell.If A issettoalowvalue,thenthechemicaldoesnotcontrolthe targetareaasmuch,thereforethereislessstrengthinthec ell-cellcommunications.We imaginethatcancerbehavesinthiswaydependingonthestag eofmetastasis.Normal cellsbehaveintheoppositeway. Withthisinmindwemade40separatecomputerrunswith A changinginvaluefrom 5inincrementsof5andthen A =38forthelastvalue.Wewouldliketoholdtheaverage simulatedECISsignal,thatis,thenumberofspinstakingth evalue0,correspondingto gapsbetweencells,constantsothat,invarying A ,wekeeptherunsasmuchalikein short-time(highfrequency)dynamicsaspossiblewithinth emodelwhilechangingtheir long-time(low-frequency)correlations.Wehavederiveda semi-empiricalformulatoaid usinoureorts.Ifwepretendthatthechemicalconcentratio n,averagedoverallcellsand time,isuniformlydistributedbetween0and r ,thechemicalcuto,andifwecanequate theaverageareaofacelltotheaveragetargetarea,wehave V = A B r Z B r 0 tanh( x ) dx + C (5.15) where x isthechemicalconcentration.Fortherunsinthiswork, B =2and r =1which yieldsthefollowing: V =0 : 663 A + C: (5.16) Here V istheaverageareaofacell,averagedovertimeandoverallc ellsinthesimulation. Thisdenesarelationshipbetween A and C .Toseeifweneededanumericalcorrection, wetto V 0 : 663 A C = A + : (5.17) 73

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ForseveraldierentrunsofAandCwefoundthat =0butthat =0 : 03.Putting thesevaluesinfor and andsolvingfor C wegetthecorrectedformula C = V 0 : 69 A (5.18) Proceeding,wethenfoundtheslopeofthebesttlinetothel og-logplotofthepower spectrumforeachrunandplottedeachpointingure5.6.The abscissais A ,thestrength ofcell-cellcommunication,andtheordinateis ,theslopeofthebesttline.Fromthe Figure5.6.Theordinateis ,theslopeofthebesttlinetoalog-logplotofthepower spectrum.Ittrendsupwardasthecell-cellsignalingstren gthincreases,shownonthe abscissa.graph,oneseesthatasthecell-cellsignalingstrengthisi ncreased, alsoincreases.As A isincreasedfromalowvalue,theeectismoredramaticandth enseemstorattenout. Ifwewentanyhigherwith A ,then C wouldhavetobenegativetokeep V thesame.For anynitesystemthecell-cellsignalingstrengthwillbeco mesaturatedatsomepoint.The trendisexactlywhatonewouldexpect.Ascellsignalingstr engthisincreased,thesystem becomesmorecorrelatedandjusttheoppositeforadecrease insignalingstrength.These 74

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resultsshowthatourmodelmightbeusefulintryingtounder standtheeectofcell-cell signalingstrengthonthebehaviorofdierentcelltypesint ermsofpowerspectra. Thismodel,assimpleasitis,seemstobetoocomplicated.We wouldliketond thesimplestmodelpossiblethatcandemonstrate 1 f behavior.However,anotherasyet untriedideawiththepresentmodelistoreplacethechemica lcutoconcentrationat whicheachcelldumpsitschemicalintotheenvironmentwith acutochemicalgradient measuringthedierencebetweenthechemicalbuildupinacel landitsnearestneighbors. Thiswouldseemtobemoreinlinewiththeideasbehindthesan d-pilemodelandmay helptosimplifythefourregimesweseeinourspectra.5.6Appendix:DerivationofthePoissondistributionforoursystem Considerasystemthatistrappedinapotentialwell.Wewish tondtheprobabilitythatitwillescapeandultimatelythetimeittakesforit toescape.KineticMonte Carlomakesthefollowingassumptionsthatareconsistentw iththederivationofaPoisson distribution.Foralessstraightforwardwayofderivingth ePoissondistributionseeVan Kampen[44]. 1.Theprobabilityofthesystemleavingthewellinatime t when t issmallis proportionalto t P (leaving; t )= r t (5.19) where r isaconstanttobedetermined. 2.Theprobabilitythatthesystemleavesmorethanonepoten tialwellin t isnegligible when t isverysmall.Therefore,wecanwrite P (not-leaving; t )+ P (leaving; t )=1 : (5.20) 3.Thenumberofwellsthatthesystemleavesinonetimeinter valisindependentof thenumberofwellsitleavesinanyothernon-overlappingin terval. 75

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Nowndtheprobabilitythatthesystemdoesnotleavethepot entialwellinatime interval t .Theprobabilitythatoursystemdoesnotleavethewellin t isequaltothe probabilitythatthesystemdoesnotleavein t t andtheprobabilitythatthesystem doesnotleavein t .Theintervalsdonotoverlapandareindependentofeachoth er,so wecanwrite P (not-leaving; t )= P (not-leaving; t t ) P (not-leaving; t ) : (5.21) Wenowuseassumptions(1)and(2)toget P (nl; t ) P (nl; t t ) t = aP (nl; t t )(5.22) wherenlmeansnot-leavingthewell.Inthelimitas t goestozero,wehavethefollowing dierentialequation: d dt P (nl; t )= rP (nl; t ) : (5.23) Thesolutiontothisequationis P (nl; t )= C exp( rt ) ; (5.24) andwhenweconsidertheboundaryconditionwhen t 0weknowthat P ( nl ;0)=1so that C =1andweget P (nl; t )=exp( rt ) : (5.25) Thisistheprobabilitythatthesystemhasnotleftthepoten tialwellinatime t Inowconsidertheprobabilityofthesystemleaving k numberofwellsinaninterval t + t .Withthepreviousassumptions,thiscanonlyhappeninoneo ftwoways. 1. k wellsareexitedintime t and0areexitedin t or 76

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2. k 1wellsareexitedintime t and1isleftintime t .Either1or2canhappen butnotboth.Theyaremutuallyexclusive.Therefore, P ( k ; t + t )= P ( k ; t ) P (0; t )+ P ( k 1; t ) P (1 ; t )(5.26) Since P (0; t )=1 P (1; t )=1 r t and P (1; t )= r t wecanwrite P ( k ; t + t )= P ( k ; t )(1 r t )+ P ( k 1; t ) r t: (5.27) Thiswillleadto P ( k ; t + t ) P ( k ; t ) t + rP ( k ; t )= rP ( k 1; t )(5.28) andinthelimitwehavethefollowingdierentialequation d dt P ( k ; t )+ rP ( k ; t )= rP ( k 1; t ) : (5.29) Wemaymultiplythroughby e rt toget e rt d dt P ( k ; t )+ re rt P ( k ; t )= re rt P ( k 1; t ) : (5.30) Ifweexpressthetermontheleftasatotalderivativewemayw rite d dt ( e rt P ( k ; t ))= re rt P ( k 1; t )(5.31) andnowintegratewithrespecttottoget e rt P ( k ; t )= Z t 0 re rt 0 P ( k 1; t 0 ) dt 0 + C: (5.32) Since P ( k ;0)=0, C =0andwehave P ( k ; t )= re rt Z t o e rt 0 P ( k 1; t 0 ) dt 0 : (5.33) 77

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Nowapplyrecursionstartingwith k =1toobtain P (1; t )whichistheprobabilitythatthe systemhasleftonewellintime t .Afterintegrationweobtain P (1 ;t )= rte rt (5.34) Theprobabilitydensityorprobabilityperunittimecanbew rittenas P d (1; t )= re rt (5.35) where r cannowbethoughtofasaprobabilityrateatwhichthesystem leavesonepotential wellorcoursed-grainedstate.Weshallrefertothisasthe rstescapetime.Thisisallwe needforthesystemswearestudyingbutonecouldcontinueto get p ( k ; t )= ( at ) k e at k : (5.36) 78

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ABOUTTHEAUTHOR DouglasCarrollLoveladywasborninDallasTexasonMay20th ,1969.In1998,he receivedhisB.A.ineconomics,andhisB.A.inphysicsin200 1,bothfromtheUniversity ofSouthFlorida.HewentontoreceivehisM.S.inphysicsin2 003,alsofromUSF.His hobbiesincludeplayingwithhiscat,chess,thebassguitar ,reading,skeetshooting,and mostrecentlyscubadiving.


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ABSTRACT: This work is a study of complex many-body systems with non-trivial interactions. Many such systems can be described with models that are much simpler than the real thing but which can still give good insight into the behavior of realistic systems. We take a look at two such systems. The first part looks at a model that elucidates the variety of magnetic phases observed in rare-earth heterostructures at low temperatures: the six-state clock model. We use an ANNNI-like model Hamiltonian that has a three-dimensional parameter space and yields two-dimensional multiphase regions in this space. A low-temperature expansion of the free energy reveals an example of Villain's `order from disorder' [81, 60] when an infinitesimal temperature breaks the ground-state degeneracy. The next part of our work describes biological systems. Using ECIS (Electric Cell-Substrate Impedance Sensing), we are able to extract complex impedance time series from a confluent layer of live cells. We use simple statistics to characterize the behavior of cells in these experiments. We compare experiment with models of fractional Brownian motion and random walks with persistence. We next detect differences in the behavior of single cell types in a toxic environment. Finally, we develop a very simple model of micromotion that helps explain the types of interactions responsible for the long-term and short-term correlations seen in the power spectra and autocorrelation curves extracted from the times series produced from the experiments.
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