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A theoretical model for self-assembly of flexible tiles

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A theoretical model for self-assembly of flexible tiles
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Staninska, Ana
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DNA
Spectrum
Pot graph
Probabilistic model
Complete complexes
Cyclic molecules
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ABSTRACT: We analyze a self-assembly model of flexible DNA tiles and develop a theoretical description of possible assembly products. The model is based on flexible branched DNA junction molecules, which are designed in laboratories and could serve for performing computation. They are also building blocks for make of even more complex molecules or structures. The branched junction molecules are flexible with sticky ends on their arms. They are modeled with "tiles", which are star like graphs, and "tile types", which are functions that give information about the number of sticky ends. A complex is a structure that is obtained by gluing several tiles via their sticky ends. A complex without free sticky ends is called "complete complex". Complete complexes are our main interest. In most experiments, besides the desired end product, a lot of unwanted material also appears in the test tube (or pot). The idea is to use the proper proportions of tiles of different types.The set of vectors that represent these proper proportions is called the "spectrum" of the pot. We classify the types of pots according to the complexes they acan admit, and we can identify the class of each pot from the spectrum and affine spaces. We show that the spectrum is a convex polytope and give an algorithm (and a MAPLE code), which calculates it, and classify the pots in PTIME. In the second part of the dissertation, we approach molecular self-assembly from a graph theoretical point of view. We assign a star-like graph to each tile in a pot, which induces a "pot-graph". A pot-graph is a labeled multigraph corresponding to a given pot type, whose vertices represent tile types. The complexes can be represented by "complex-graphs", and each such graph is mapped homomorphically into a pot-graph. Therefore, the pot-graph can be used to distinguish between pot types according to the structure of the complexes that can be assembled.We begin the third part of the dissertation with a pot containing uniformly distributed DNA junction molecules capable of forming a cyclic graph structure, in which all possible Watson-Crick connections have already been established, and compute the expectation and the variance of the number of self-assembled cycles of any size. We also tested our theoretical results in wet lab experiments performed at Prof. Nadrian C. Seeman's laboratory at New York University. Our main concern was the probability of obtaining cyclic structures. We present the obtained results, which also helped in defining an important parameter for the theoretical model.
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Dissertation (Ph.D.)--University of South Florida, 2007.
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ATheoreticalModelforSelf-AssemblyofFlexibleTiles by AnaStaninska Adissertationsubmittedinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy DepartmentofMathematics CollegeofArtsandSciences UniversityofSouthFlorida Co-MajorProfessor:NatasaJonoska,Ph.D. Co-MajorProfessor:GregoryL.McColm,Ph.D. ArunavaMukherjea,Ph.D. MasahikoSaito,Ph.D. StephenSuen,Ph.D. DateofApproval: May1,2007 Keywords:DNA,spectrum,potgraph,probabilisticmodel,c ompletecomplexes, cyclicmolecules c r Copyright2007,AnaStaninska

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Dedication TomysisterTanja.

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Acknowledgements AlthoughIamthesoleauthorofthisdissertation,manypeop lehavehelpedme reachthisgoal.Firstandforemost,Iwouldliketoexpressm ysincereanddeepestgratitudetomyresearchsupervisors,dissertationadv isors,andmentors,Natasa JonoskaandGregoryMcColm.Theirguidanceandsupporthelp edmeandinspired meinmyresearch,mystudiesandeverydaylife.Besidesover seeingmyresearch, theytaughtmeveryvaluablelessonsinandoutsidetheUnive rsity.Ihavelearnedso muchfromthem,andcanjusthopetobenearlyasgoodanadviso roneday.Nothing inthisdissertationwouldhavebeenpossiblewithoutthem. Iconsidermyselfthe luckieststudenttohavethebestmentorspossible. Iextendmythankstoallthecommitteemembers,ArunavaMukh erjea,Masahiko Saito,andStephenSuen,forcreatinganexcellentresearch atmosphereandproviding mewithvaluablecommentsandideas.ThetimeIspentinNadri anSeeman'slaboratoryatNYUwasanunforgettableandveryvaluableexperie nce.Iamgratefulto himforgivingmethisopportunitytoworkonwetlabexperime nts,andtoBanani Chakrabortyforintroducingmetotheworldofchemistry.Iw ouldalsoliketothank theotherstudentsinthelaboratoryforbeinghelpfulandpa tientwithamathematician.InthisoccasionIwouldliketothankmyundergraduate thesisadvisorDonco Dimoskiforsettingmeontherightwayonmymathematicaledu cation.Ialsowishto thankMohamedEddaoudi,chairmanofmycommittee,fortakin gontheobligation withoutaqualm. Iwouldnothavecomethisfarifitwasnotformyimmediatefam ily.Theyhave beenmyrock,mysupporters,andtheygavemeencouragementw henitwas

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mostneeded.TherearenowordswithwhichIcanexpressmygra titudetowardsmy parentsKrumandIlinka,mysisterTanja,herhusbandLjupco ,andthecenterofmy world,theirdaughters,BojanaandKalina. ItisbeyondanyformalacknowledgmentwithwhichIcanexpre ssmythanksto Ferenc,forhisfriendship,help,love,support,andunders tanding.Mostofallfor bearingtheseyearswithmeandbeingalwaysthereforme. Iamveryfortunatetohaveabigandsupportivefamily.Theyh avebeengiving meloveandsupportfromafar,followingmyprogressandmaki ngmysummersin Macedoniaverypleasantandeventful.Iamexpressingmylov eandgratitudemy auntSnezana,uncleGligor,auntZorica,uncleLambe,auntB uba,uncleTome,and mycousins:Saso,Deni,Keti,Goko,Irina,Klime,Vlatko,D ime,VeselaandVesna, withherfamily,Kire,GoceandIlina. DuringthesixyearsIhavespentinTampa,Icouldalwaysrely onmyfriends.I metverydearandgoodfriendswhosefriendshipwilllastali fetime.Iamexpressing mythankstoallofthemandtomylongtimefriends. Finally,Iexpressmyappreciationtoallmyprofessorsfrom theDepartmentof MathematicsatUSF,tothechairMarcusMcWaters,associate chairScottRimbey, toBeverly,Maryann,Aya,Nancy,BarbaraandSarinaforthei rgeneroushelp.The musicofToseProeskihadbeenkeepingmecompanyallthesey ears. TheresearchforthisdissertationwassupportedbytheNati onalScienceFoundationgrantsCCF#0432009andCCF#0523928.

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TableofContents ListofFigures................................iiiAbstract....................................viii1Introduction................................12IntroductiontotheModel........................6 2.1ThestructureofDNA.....................62.2DenitionoftheModel.....................82.3PotTypeClassication.....................16 3SpectrumofaPot............................23 3.1Denitions............................233.2GeometricRepresentationoftheSpectrum..........253.3AlgebraicRepresentationoftheSpectrum..........363.4MapleProgram.........................38 4GraphofaPotwithDNAcomplexes..................40 4.1Introduction...........................404.2DenitionofaPotGraph...................414.3Homomorphisms........................46 MinimalandMaximalCompleteComplexGraphs.....66 5ProbabilisticAnalysis...........................70 5.1Introduction...........................70 OneTypePot.........................72MultiTypePot........................77 5.2AnotherMethodforObtainingtheExpectedNumberofCycl es77 Expectations.........................78 i

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Variance............................81 5.3TheoreticalBasefortheExperimentalResults........ 84 6DesignoftheExperiment........................91 6.1TheProtocol..........................986.2Results..............................99 Firstset............................99Secondset...........................104Thirdset............................108 6.3Conclusion............................112 7Conclusion.................................113References...................................117Appendices..................................122 AppendixA-MapleProgram....................122AppendixB-ExistenceoftheProbabilitySpace.........12 6 AbouttheAuthor..............................EndPage ii

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ListofFigures 2.1a)Schematicrepresentationofonenucleotide.b)Schem aticrepresentationoftwosinglestrandedDNAmoleculesconnectedth rough Watson-Crickcomplementarity...................7 2.2Above:Watson-CrickbondingoftwoDNAjunctionmolecul es.Below:JunctiongraphthatrepresentsbondingofthetwoDNAju nction moleculesdepictedontheleft....................9 2.3a)1-branchedjunctionmolecule,i.e.,hairpinb)2-bra nchedjunction molecule................................10 2.4a)Tiles t 1 t 2 t 3 b)Possiblecomplexesthatcanbeobtainedbygluing thetilesina)............................10 2.5Thecomplex C = h T;S;J i .....................17 2.6Thecomplexes C 1 and C 2 .....................18 2.7a)Thecomplex C C ........................19 2.8a)Satisablepottypethatisnotstronglysatisable( t 3 and t 4 cannot beapartofacompletecomplex)b)weaklysatisablepottype that isnotsatisable(thestickyend c cannotbeapartofacomplete complex)...............................21 2.9a) j P j = j H + j b) j P j < j H + j c) j P j > j H + j : .........21 3.1a)Theminimalcompletecomplexesforthepottypegiveni nFigure2.9 cb)Acompletecomplexthatisminimal,butnotextremalcomp lete complex................................32 iii

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3.2a)Stronglysatisablepottypewithspectrum f ( 1 4 ; 1 2 ; 1 4 ) g ,b)Strongly satisablepottypewithspectrum S ( P )= f ( u; 4 u 1 ; 2 5 u ): 1 4 u 2 5 g .c)Stronglysatisablepottypewithspectrum f (1 u v;u;v ): 0 u 1 ; 0 v 1 ;u + v 1 g ..................35 3.3TheclosureofthespectrumofthepottypegiveninExampl e3.2.20b) isthelinesegment;theclosureofthespectrumofthepottyp egiven inExample3.2.20c)isthetriangleboundedbythedottedlin esalong withitsinterior............................35 4.1Pottype P anditspotgraph...................42 4.2a)Apottype P = f t 1 ; t 2 ; t 3 g .b)Thepotgraphcorrespondingtothe pottype P givenina).......................43 4.3Labeledgraphthatisnotapotgraph...............434.4Elementsfromthesetofcompletecomplexgraphsforthep ottype giveninExample4.1........................45 4.5Acompletecomplexgraphandapotgraph...........484.6 t = t 0 viaanisomorphismthatpreservesstickyends. t = t 00 viaan isomorphismthatdoesnotpreservesstickyends.......... 50 4.7Thepottypes P 1 and P 2 areequivalent.............52 4.8Twopottypes P 1 and P 2 withequivalentsetsofcompletecomplex graphs................................53 4.9Thesetsofcompletecomplexgraphsforthepottypesgive ninFigure 4.8..................................54 4.10Thesetsofcomplexgraphs G ( P 1 )and G ( P 2 )areequivalent,but P 1 P 2 ..................................57 4.11 G P 1 = G P 2 ,but P 1 P 2 ......................59 4.12Apottype P anditscorrespondingpottype e P thatdon'tcontaintile typeswiththesamestickyends...................65 4.13Elementsfromthesetsofcompletecomplexgraphsforth epottypes P and e P giveninFigure4.12....................65 iv

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4.14a)Twocompletecomplexesthatshareastickyendincomm onb)A completecomplexbuildfromthecomplexesina)........68 4.15pottypesthatarenotsimilar,butwiththesamespectru m S = f ( 1 2 u; 1 2 u;u;u ):0 u 1 4 g andsamesupport Supp = f (1 ; 1 ; 1 ; 1) g 69 5.1(a)Three2-armedtilesformatrianglewhichrepresents a K 3 complex. (b)Threetilegraphsusedinapottoassemble K 3 .(c)Complete complexesforthispotwillbecyclesoflengthdivisibleby3 .Cycles K 3 and C 6 aredepicted.......................85 6.1Thedesiredproduct........................916.2Thisisa12%nativegeltochecktheformationofMolecule 1.A10 nucleotidemarkerisintherstlane.InthesecondlaneisMo lecule 1,inthethirdlaneisacomplexconsistingofthestrandsAS1 1and AS12,inthefourthlaneisacomplexconsistingofstrandsAS 11and AS13,inthefthlaneisacomplexconsistingofthestrandsA S12and AS13,inthesixthlaneisthestrandAS11,intheseventhlane isthe strandAS12,andintheeightlaneisthestrandAS13......94 6.3Thisisa12%nativegeltochecktheformationofMolecule 2.A10 nucleotidemarkerisintherstlane.InthesecondlaneisMo lecule 2,inthethirdlaneisacomplexconsistingofthestrandsAS2 1and AS22,inthefourthlaneisacomplexconsistingofstrandsAS 21and AS23,inthefthlaneisacomplexconsistingofthestrandsA S22and AS23,inthesixthlaneisthestrandAS21,intheseventhlane isthe strandAS22,andintheeightlaneisthestrandAS23......95 v

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6.4Thisisa12%nativegeltochecktheformationofMolecule 3.A10 nucleotidemarkerisintherstlane.InthesecondlaneisMo lecule 3,inthethirdlaneisacomplexconsistingofthestrandsAS3 1and AS32,inthefourthlaneisacomplexconsistingofstrandsAS 31and AS33,inthefthlaneisacomplexconsistingofthestrandsA S32and AS33,inthesixthlaneisthestrandAS31,intheseventhlane isthe strandAS32,andintheeightlaneisthestrandAS33......96 6.5(a)10nucleotidemarker,(b)ligatedmoleculesfromthe testtubeT5 of1 M concentration,(c)exotreatmentoftheligatedproductfro m testtubeT5of1 M concentration,(d)ligatedmoleculesfromtest tubeT4of1 : 5 M ,(e)exotreatmentoftheligatedproductfromtest tubeT4of1 : 5 M ,(f)ligatedmoleculesfromtesttubeT3of2 M (g)exotreatmentoftheligatedproductfromtesttubeT3of2 M ,(h) ligatedmoleculesfromtesttubeT2of2 : 5 M .(i)exotreatmentofthe ligatedproductfromtesttubeT2of2 : 5 M ...........101 6.6(a)10nucleotidemarker(b)ligatedmoleculesfromthet esttubeT10 of0.1 M concentration,(c)exotreatmentoftheligatedproductfro m testtubeT10of0.1 M concentration,(d)ligatedmoleculesfromtest tubeT9of0 : 2 M ,(e)exotreatmentoftheligatedproductfromtest tubeT9of0 : 2 M ,(f)ligatedmoleculesfromtesttubeT8of0 : 3 M (g)exotreatmentoftheligatedproductfromtesttubeT8of0 : 3 M (h)ligatedmoleculesfromtesttubeT7of0 : 4 M ,(i)exotreatmentof theligatedproductfromtesttubeT7of0 : 4 M ,(j)moleculesfrom testtubeT6of0 : 5 M ,(k)exotreatmentoftheligatedproductfrom thetesttubeT6of0 : 5 M ......................102 6.7(a)10nucleotidemarker,(b)ligatedmoleculesfromthe testtubeT1 of1 M concentration,(c)exotreatmentoftheligatedproductfro m testtubeT1of1 M concentration.................106 vi

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6.8(a)10nucleotidemarker,(b)ligatedmoleculesfromthe testtubeT2 of0 : 75 M concentration,(c)exotreatmentoftheligatedproduct of0 : 75 M concentration,(d)ligatedmoleculesfromtesttubeT3of 0 : 5 M concentration,(e)exotreatmentoftheligatedproductfro m testtubeT3of0 : 5 M concentration................107 6.9(a)10nucleotidemarker,(b)ligatedmoleculesfromthe testtubeof 3 M concentration,(c)exotreatmentoftheligatedproductof3 M concentration,(d)ligatedmoleculesfromthetesttubeof1 : 5 M concentration,(e)exotreatmentoftheligatedproductof1 : 5 M concentration,(f)ligatedmoleculesfromthetesttubeof1 M concentration, (g)exotreatmentoftheligatedproductof1 M concentration.109 6.10(a)10nucleotidemarker,(b)ligatedmoleculesfromth etesttube of0 : 5 M concentration,(c)exotreatmentoftheligatedproductof 0 : 5 M concentration,(d)ligatedmoleculesfromthetesttubeof0 : 1 M concentration.............................110 6.11(a)10nucleotidemarker,(b)ligatedmoleculesfromth etesttubeof 1 M concentration,(c)exotreatmentoftheligatedproductof1 M concentration,(d)ligatedmoleculesfromthetesttubeof0 : 5 M concentration,(e)exotreatmentoftheligatedproductof0 : 5 M concentration,(f)ligatedmoleculesfromthetesttubeof0 : 1 M concentration,(g)exotreatmentoftheligatedproductof0 : 1 M concentration..................................111 vii

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MathematicalModelsforMolecularSelf-assembly AnaStaninska Abstract Weanalyzeaself-assemblymodelofrexibleDNAtilesanddev elopatheoretical descriptionofpossibleassemblyproducts.Themodelisbas edonrexiblebranched DNAjunctionmolecules,whicharedesignedinlaboratories andcouldserveforperformingcomputation.Theyarealsobuildingblocksformake ofevenmorecomplex moleculesorstructures. Thebranchedjunctionmoleculesarerexiblewithstickyend sontheirarms.They aremodeledwith\tiles",whicharestarlikegraphs,and\ti letypes",whichare functionsthatgiveinformationaboutthenumberofstickye nds.Acomplexisa structurethatisobtainedbygluingseveraltilesviatheir stickyends.Acomplex withoutfreestickyendsiscalled\completecomplex".Comp letecomplexesareour maininterest. Inmostexperiments,besidesthedesiredendproduct,aloto funwantedmaterial alsoappearsinthetesttube(orpot).Theideaistousethepr operproportionsoftiles ofdierenttypes.Thesetofvectorsthatrepresentthesepr operproportionsiscalled the\spectrum"ofthepot.Weclassifythetypesofpotsaccor dingtothecomplexes theyacanadmit,andwecanidentifytheclassofeachpotfrom thespectrumand anespaces.Weshowthatthespectrumisaconvexpolytopean dgiveanalgorithm (andaMAPLEcode),whichcalculatesit,andclassifythepot sinPTIME. Inthesecondpartofthedissertation,weapproachmolecula rself-assemblyfroma graphtheoreticalpointofview.Weassignastar-likegraph toeachtileinapot,which viii

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inducesa\pot-graph".Apot-graphisalabeledmultigraphc orrespondingtoagiven pottype,whoseverticesrepresenttiletypes.Thecomplexe scanberepresentedby \complex-graphs",andeachsuchgraphismappedhomomorphi callyintoapot-graph. Therefore,thepot-graphcanbeusedtodistinguishbetween pottypesaccordingto thestructureofthecomplexesthatcanbeassembled. Webeginthethirdpartofthedissertationwithapotcontain inguniformlydistributedDNAjunctionmoleculescapableofformingacyclic graphstructure,inwhich allpossibleWatson-Crickconnectionshavealreadybeenes tablished,andcomputethe expectationandthevarianceofthenumberofself-assemble dcyclesofanysize. Wealsotestedourtheoreticalresultsinwetlabexperiment sperformedatProf. NadrianC.SeemanslaboratoryatNewYorkUniversity.Ourma inconcernwasthe probabilityofobtainingcyclicstructures.Wepresentthe obtainedresults,whichalso helpedindeninganimportantparameterforthetheoretica lmodel. ix

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1Introduction Self-assemblycanbenaturalorsynthesizedprocessthatis understoodasaspontaneousorganizationofsimplerstructuresintomorecompl exones.Molecularselfassemblyisoneofthemostimportantaspectsofnanotechnol ogythatmayleadtoward understandingmanyprocessesinnatureandhasapotentialf ormanyapplications. Inthisdissertation,weconsiderseveralproblemstoadvan ceourunderstandingof DNAself-assembly. Inrecentyears,understandingself-assemblyasaprocessa nddiscoveringnewways tousethemoleculehasledtomanyscienticadvances,bothe xperimentallyand theoretically.Manynanostructures,nanomaterials,nano devicesandcomputational modelshavebeendevelopedbasedontheprinciplesofself-a ssembly[10,40,60,61]. In1987,TomHeadinhispaper[15]giveanideaforusingDNAfo rcomputational purposes.In1994,LenAdlemansolvedsmallinstanceoftheH amiltonianPathProblem[1]usingDNAmolecules.Sincethenseveraldierentmod elsforbiomolecular computationhavebeendevelopedandmanyinstancesofNPcom pleteproblemshave beenaddressedusingDNAcomputing[23,30,41].Inaddition ,molecularsimulations ofnitestateautomata,cellularautomataandTuringmachi neshavebeendesigned mainlythroughself-assemblyandenzymerestrictions[6,4 2]. Whiletherehasbeensignicantexperimentalprogressinse lf-assembly,thetheoreticalunderstandingisstilllaggingbehind.Severalth eoreticalmodelsforDNA self-assemblyhaveappeared,mostlyusingrigidsquaretil es[2,4,26,38,39].They modelWangtilesandthereforesimulateauniversalTuringM achine.Althoughthere areresultsinthetheoreticalself-assembly,thereisstil laneedforunderstandingthe 1

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limitationsandthecomplexityoftheprocess.Partialresu lts,mainlyconcerningrigid tilemodelshavealreadybeenobserved;theminimalnumbero ftiletypesneededto buildan N N squaresisknowntobe O (log N= loglog N )[3,39];theminimaltime neededtobuildthesquaresis O ( N )[3];computingthesmallesttilesetneededfor uniqueself-asseblyinagivenshapeisNPhard[2];andanarb itraryshapecanbeassembledwithO(Kolmogorovcomplexity)tileswithscaling[ 48].Akinetictilemodel wasdevelopedbyWinfree,thatisoftenusedinsilicosimula tions[54]. Alsoquestionsaboutthedesignanderrorcorrectionofther igidtilemodelhave beenaddressedandansweredusingdierentapproachesas:p roofreadingtiles[56], snakedtiles[9],andself-healingtiles[55]. Apartfromtheothertheoreticalmodelsonself-assembly,t hemodelthatisdescribedinthisdissertationusesrexibletilesasitsmainb uildingblock.Itwasrst reportedin[20],andfurtheronelaboratedin[23,24].Prob lemssolvablebytherexibletilemodelofDNAassemblyarepreciselytheNPtimeprobl ems[23].Therexible tilesmodelbranchedjunctionmoleculeswithfreestickyen ds(singlestrandedsequences)ontheirbranches.DuetothenaturalWatson-Crick complementarity,complementarystickyendsoftwomoleculescangluetogetheran dformmorecomplex structures. BesidesencodingNPcompleteproblems,rexibletilesareus edforbuildingnanotstructureandnanomaterials.Forexamplein[14]atetrah edronwasbuiltfrom rexibletiles,in[31]Borromeanringswereconstructed.Al sotheyhavebeenused forconstructionoftwodimensionalarrays[32],andsugges tedforgrowingaDNA fractal-likemolecule[8].Severalexperimentshavebeend oneusingtheself-assembly processofrexibletilesforDNAcomputingpurposes[19,20, 35,34].Inthisdissertationweusetherexibletilemodelfortheoreticalanalysiso ftheself-assemblyprocess anditsproducts.Thisistherstattempttohaveasystemati ctheoreticalstudyof thismodel. Duetothecomplexityoftheproblem,weconcentrateonthest aticmodelanddo notconsideranythermodynamicpropertiesofthesolution. Thestaticmodeldeals 2

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onlywiththeinputandtheoutputofanexperiment.Wehopeto extendthismodel toadynamicversion. TheWatson-Crickcomplementarityisrepresentedwithanin volutionfunction. Eachbranchedjunctionmoleculeisrepresentedasatileofc ertaintype,wherea tiletype isafunctionthatgivesinformationaboutthenumberofstic kyendsonthe molecule.A pot isacollectionoftilesofvarioustypesanda pottype isthesetof dierenttiletypes. Acomputationalproblemcanbeencodedwithrexibletiletyp esinsuchaway thatasolutiontotheproblemexistsifandonlyifacomplete complex(acomplex withnostickyends)ofcertainsizeisformedinthesucient lylargepotsofthatpot type.Thatisoneofthereasonswhyweconcentrateoncomplet ecomplexes. AlthoughthemaininspirationforthemodelcamefromDNAcom puting,with thismodelweaddressissuesrelatedtotheself-assemblypr ocess,suchas:predicting thepossibleoutcomes,anddeterminingtheperfectmixofti lesforexperiments. Inthisdissertationthreedierentproblemsareinvestiga ted.Theyare: anecessaryconditionforobtainingonlycompletecomplexes,ades criptionofpottypesand complexeswithgraphs,andtheprobabilityoftheformation ofcyclicstructures. ANecessaryconditionforobtainingonlycompletecomplexe s Givenasetofrexiblebranchedjunctionmoleculeswithstic kyendsweconsider thequestionofdeterminingtheproperstoichiometrysucht hatallstickyendscould endupconnected.Thenecessaryconditionforobtainingonl ycompletecomplexesat theendofanexperimentistousetheproperproportionofeac htypeofmolecules, whichingeneralisnotuniform.Thesetofvectorsforthepro perproportionsiscalled the\spectrum". Thepottypesareclassiedinfourclassesaccordingtoposs iblecomponentsthat assembleincompletecomplexes:unsatisable,weaklysati sable,satisable,and stronglysatisable.Throughinvestigatingsubsetsofan espaces,weprovidean algorithmtoidentifytheclassofagivenpottype.Thepotcl assicationisPTIME computableandcanbedeterminedfromthespectrumandthesu pportofapottype, 3

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wherethesupportisabinomialvectorofdimensionequaltot henumberofdierent stickyendtypes.WealsogiveaMapleprogramthatperformst histask.Thisstudy isincludedinChapter3. DescriptionofPotTypesandComplexeswithGraphs.Sinceeverycomplexhasanaturallyarisinggraphstructure ,weturntograph theorytobetterdescribetheproductsoftheself-assembly process.Thegraphmodel isusedtodeterminewhatcompletecomplexescanbeassemble dfromagivenpottype aswellascompareandclassifythepottypesthemselves.Itr epresentsanapplication ofgraphhomomorphismtheory. Therexibletilesandthewaytheyconnectlendthemselvesna turallytograph representation.Eachpottypecanberepresentedasalabele dmultigraph,calleda pot-graph .Eachvertexrepresentsatiletypewhileeachedgerepresen tsaconnection betweentwotiletypesthathavecomplementarystickyends. Iftwotiletypescan connectviacomplementarystickyends,forexample h and b h ,thenthetiletypesare representedastwoverticesinthegraphconnectedwithaned gelabeled h Similarly,eachcompletecomplexisrepresentedbyagraph. Thisgraphiscalled c ompletecomplex-graph.Everycompletecomplex-graphisan isomorphicpre-image ofasubgraphofthepot-graph.Usingthepot-graphweareabl etodeterminethe typesofcompletecomplexesonepottypeadmits.Wealsochar acterizethepots accordingtothepot-graphs. Twopottypesare equivalent iftheyhaveequalnumberoftiletypesandequal numberofstickyendtypeswithaproperbijectionbetweenth em.Sincetheproperties ofthepottypescanbedeterminedbylookingatthecorrespon dingpotgraphs,it canbeshownthatequivalentpottypeshaveisomorphicgraph s.Pottypesthathave isomorphiccompletecomplex-graphsaresaidtobe similar .Itcanbeshownthat equivalentpottypesarealsosimilar,butnotviceversa.Th isstudyispresentedin Chapter4. TheProbabilityoftheFormationofCyclicStructures.Ifonedesignsapotwithtilesthatcouldbuildaparticularg raphstructure(com4

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pletecomplex),besidesthedesiredcompletecomplex.othe rcompletecomplexesmay appearaswell.Inthischapterweconsiderasetoftilesthat canbuildcycliccomplete complexesandperformastudyoftheexpectednumbersofcycl iccompletecomplexes ofanysize. Theprocessofself-assemblyisstochasticandasarstappr oximationwepresenta newrandomgraphmodeloftheproductsofself-assemblyproc esses.Ourmodeldiers fromotherexistingrandomgraphmodelsbecauseitusesnonuniformprobabilities fortheedgeappearance. Weshowthatthesmallestcyclesaremorelikelytoappeartha nlargerones.The expectednumberofthedierentcyclesisinverselyproport ionaltotheirlengthand thestandarddeviationoftheexpectednumberofcycliccomp lexesisverysmall. Tocheckthetheorydeveloped,weconductedawetlabexperim ent.Theexperimentconsistedofthreedierent2-branchedjunctionm olecules,uniformlydistributed,capableofformingacycliccompletecomplexofle ngth3.Wewantedto ndthelowestconcentrationforwhichonlycyclicmolecule swillbeformed.However, eveninverydilutedsolution,appearanceofdoublecycles( dimers)wereobserved. TheseresultsaregiveninChapter6.Wealsousedtheobtaine dresultstodenean importantparameterfortherandomgraphthemodel. Therandomgraphmodelisusedtopredictthestabilityofthe experimentand calculatetheexpectednumberofcyclicmoleculesofcertai nsizewhichappearatthe endofanexperiment.However,itcannotbeusedtocalculate theexpectednumber ofcomplexesthatarenotcyclic;furtheradjustmentswould berequiredforthat. Someoftheshortandlongtermsgoalsconcerningthemodelan ddirectionsfor futureresearcharegiveninChapter7. 5

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2IntroductiontotheModel Themaininspirationforthemodelpresentedinthisdissert ationistheself-assembly ofbranchedjunctionDNAmolecules.Althoughmostofthewor kisdedicatedto DNAself-assembly,themethodsandtheideaspresentedcanb eappliedtoother self-assemblyprocessesaswell. Westartthischapterwithashortintroductiononthestruct ureoftheDNA molecule,totheextentneededforthisdissertation. 2.1ThestructureofDNA DNAmoleculesareverywidespreadandcanbefoundineveryli vingorganism.They haveveryimportantrolesinthelivingcell,carryingthege neticinformationfromone generationintothenextoneandplayingacrucialroleinthe synthesisandregulation ofproteins. DNAisanabbreviationforDeoxyribonucleicAcid;itisapol ymerconsisting ofsequencesofmonomers,calleddeoxyribonucleotidesors hortly,nucleotides.Each nucleotideconsistsofthreecomponents:asugar,aphospha tegroupandanitrogenous base(SeeFig2.1). Thesugarcomponent,called Deoxyribose ,consistsofvecarbonatomsnumbered 1'through5'.Thephosphategroupisattachedtothe5'carbo n,whilethebaseis attachedtothe1'carbon.Thereisalsoahydroxylgroup(OH) attachedtothe3' carbonofthesugarmolecule.Whenthe5'phosphategroupofo nenucleotidejoins withthe3'hydroxylgroupofanothernucleotide,theyforma strongcovalentbond alsoknownasphosphodiesterbond,whichishardtobreak.Th isconnectiongives 6

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orientationtothemolecule,becauseonelinearstrandwill haveafreephosphategroup hangingononesideandafreehydroxylgroupontheotherside .Usuallywhenwe depictastrandofDNA,wedrawanarrowfromthe5'tothe3'end ofthemolecule, sincethemoleculecanbeextendedbyaddingnucleotidesoni ts3'end. Nucleotidesdierbytheirbases.Therearefourbasisandth eyaredividedinto twogroups:PurinesandPyrimidines.Adenineandguanine,o rAandGforshort, arepurins,andcytosineandthymine,orCandTforshort,are pyrimidines. Thebaseofonenucleotidecanjoinwiththebaseofanothernu cleotideina certainway,formingaweakhydrogenbond .Adeninecanbondwiththymine,and cytosinewithguanine;nootherbaseconnectionsarepossib le(A-Tpairinginvolves theformationoftwohydrogenbonds,whiletheC-Gpairingin volvestheformation ofthreehydrogenbondsbetweenthetwonucleotides.SotheC -Gbondisstronger thanA-T;(however,thisfactdoesnotplayaroleinourinves tigation).Wesaythat AiscomplementarytoTandCiscomplementarytoG.Thisprinc ipleofpairingis calledWatson-Crickcomplementarity(namedafterJamesD. WatsonandFrancisH. C.Crick). Strong covalent bond 5' 3' OH A P 3' 5' 3' 5' 3' 5' 3' 5' 3' 5' 3' 5' 3' 5' 3' 5' P OH OH AT C G C A G T a) b) P Weak hydrogen bond Figure2.1:a)Schematicrepresentationofonenucleotide. b)Schematicrepresentationof twosinglestrandedDNAmoleculesconnectedthroughWatson -Crickcomplementarity. TwosinglestrandedDNAmoleculeswithcomplementaryseque ncesontheirbases 7

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connectthroughWatson-Crickconnectioninananti-parall elfashion.Thismeansthat thefree5'endofonestrandisonthesamesideasthe3'endoft heotherstrand andthetwostrandsbondtoeachotherthroughtheirbases,fo rmingadoublehelix molecule.Thedoublehelixisorganizedsothatthesugar-ph osphatebondisonthe outerside,whilethebasesareontheinnerside.Inthispres entation,Watson-Crick complementaritywillbethemainbuildingtoolforassembly AlthoughtheDNAmoleculeismostlyknownasalinearhelixst ructure,itcanbe constructedintomorecomplexform.Examplesincludebranc hedjunctionmolecules, double[29]andtriplecrossovermolecules[28](calledDXa ndTXmolecules),etc. 2.2DenitionoftheModel Thisdissertationexploresthetheoreticalaspectsconcer ningDNAself-assembly.Itis basedonatheoreticalmodelmotivatedbytheweakhydrogenb ondingoftheDNA molecules.Themodelcanbeadjustedforinvestigatingthes elf-assemblyofDNA tilesandself-assemblyofotherstructures.Thethermodyn amicpropertiesofthe moleculesinthetesttube(pot)arenotincludedinthedescr iptionoftheassembly process.Also,arelativelyuniformmeltingtemperaturefo rthestickyendsisassumed. Themainbuildingblocksforthemodelareinspiredbythebra nchedjunctionDNA molecules.Thesearesynthesizedstarlikemolecules[44], thathaverexiblearmswith stickyends(seeFigure2.2(a)totheleft).Eacharmhastwop arts:abodyand astickyendextendingfromthebody.Thebodypartisadouble strandedDNA molecule,whilethestickyendpartisasinglestrandedDNAm olecule.Whensingle strandedpartsoftwoarmswithcomplementarystickyendshy bridize,theygluethe moleculesbythestickyends,formingamorecomplexstructu re. Forsimplicityweignoresomeofthetechnicalities(liketh esequencesofthe molecules,modelthesequenceofstickyendswithasymbol,r epresentingthecomplexesasadoublestrandedmolecules)oftheself-assemble dcomplexesandrepresent themaslabeledgraphs.Forexample,Figure2.2(a)represen tsthethree-andthe four-branchedjunctionmoleculegluedtogether,depicted inawayclosetoreality, 8

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a) b) d b e 2 a t 1 t a d b c t 1 c t 2 e Figure2.2:Above:Watson-CrickbondingoftwoDNAjunction molecules.Below:Junction graphthatrepresentsbondingofthetwoDNAjunctionmolecu lesdepictedontheleft. whileFigure2.2(b)isagraphrepresentationofthreeandfo urbranchedjunction moleculesbeforeandaftergluingoccurred.InFigure2.2(b )thegraphontheleft representsafour-branchedjunctionmoleculewithfoursti ckyendslabeled a;b;c ,and d ,whilethegraphinthemiddlerepresentsathreebranchedju nctionmoleculewith stickyends b a b b ,and e .Thestickyends b a and a arecomplementarytoeachother, andthestickyends b b and b arecomplementarytoeachother.Inthegluingprocess, thecomplementarystickyendsconnect,hencethecomplexth atisaproductofthe gluingofthosetwographs(representedintheFigure2.2ont herighthasonlysticky ends c d ,and e A1-branchedjunctionmoleculeisahairpinstructurewitho nlyonestickyend,a 2-branchedjunctionmoleculeisadoublehelixwithtwostic kyends,oneateachend (seeFigure2.3).Ingeneral n -branchedjunctionmoleculewillhave n stickyends. 9

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b) a) Figure2.3:a)1-branchedjunctionmolecule,i.e.,hairpin b)2-branchedjunctionmolecule. Inorderforallconnectionstobepossible,thebranchesoft hejunctionmolecules needtoberexible.Therexibilityisobtainedbyaddingbulg edT'sonthejunction sequences,likein[21,22]and[35]. Twojunctionmoleculescanconnectinmanydierentways.Fo rexamplethe threetilesgiveninFigure2.4(a)canconnectinfourdiere ntways,asdepictedin Figure2.4(b). t t 3 t 3 t 3 1 t 1 t 1 t e c a a c 2 a) b) a c c a a b c d a c 1 3 t t e c d a b 2 t e 2 t t e 2 t 3 e 2 t t 1 Figure2.4:a)Tiles t 1 t 2 t 3 b)Possiblecomplexesthatcanbeobtainedbygluingthetile s ina). 10

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Inourmodel,weconsideratesttube,orsocalled(whichweca ncalla pot ,with DNAbranchedjunctionmolecules( tiles )init.Usingappropriatechemicalprotocols, thecomplementarypartsoftheDNAmoleculesinthetesttube hybridizeandform morecomplexstructures.Wewanttoperformastudyontheass emblyprocessandon thepossibleoutcomes.Forthat,aformalmathematicalden itionofthecomponents andoftheprocessisneeded. Westartwiththestickyends.Iftwostickyendshavethesame sequencesof nucleotides,wesaythattheyareofthesame stickyendtype andwedenoteby H the setofstickyendtypes.Eachstickyendinthetesttubeisofa certainstickyendtype andsincethereareonlynitelymanystickyendtypes, H isnite.A stickyend of type h isacopyofthestickyendtype h andhasthesamesequenceonnucleotides asthestickyendtype h TheWatson-Crickcomplementarityismodeledwithafunctio n : H H whichisaninvolution,i.e., ( ( h ))= h forall h 2 H .Wecall ( h ) 2 H the complementarystickyendtype to h suchthatstickyendsoftypes h andtypes ( h ) bond.If ( h )= h 0 thenthestickyends h and h 0 canconnect.Foreach h 2 H weassumethat ( h ) 6 = h = ( ( h )).Thus H canbepartitionedintotwosets, H + and H suchthatif h isanelementof H + then ( h )isanelementof H .Toease thenotationwewrite( H; )forthesetofstickyends H forwhich representsthe complementaryfunction. Weoftensimplifythenotationbywriting b h for ( h )andwex H .Weusenotation [ n ]= f 1 ; 2 ; 3 :::n g and N = f 0 ; 1 ; 2 ;::: g inwhatfollows. Denition2.2.1. Atiletype over ( H; ) isafunction t : H N .Atileoftype t has t ( h ) stickyendsoftype h .The degree ofatiletype t is d = d ( t )= X h 2 H t ( h ) Weposearestrictiononthetiletypes,sothatitisnotpossi bleforatiletypeto havestickyendsoftype h and b h forsome h 2 H + atthesametime,i.e.,itisnot possibleforagiventiletype t t ( h ) > 0and t ( b h ) > 0. Informally,a tile representsatypeofbranchedjunctionmolecules.Formally a tileisastar-likegraphwith(seeFigure2.8)withonecentr alvertexofdegree d ( t ), 11

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and d ( t )verticesofdegreeonelabeledwithstickyends.Ifatile t isoftype t then foreachstickyend h oftype h 2 H t ( h )= t ( h )= t ( h ),meaningthatthetile t has exactly t ( h )stickyendsoftype h .Ifthestickyend h isonthetile t ,wewillwrite t ( h )=1,otherwise t ( h )=0.Astickyend h canonlybeononetile,soif t 1 ( h )=1 and t 2 ( h )=1,then t 1 = t 2 .InapotwithDNAmoleculestherearemanycopiesof agiventypeofjunctionmolecules,andhencewecanassumepo tentiallyaninnite supplyoftilesofeachtype. Forexample,Figure2.4(a)showsexamplesofthreetileswit hdegrees3,5,and3, respectively.Thecentralvertexisrepresentedwithablac kcircleandthestickyends areindicatedwithdierentcolorsandshapes.Thosetilety pesareformallydened asfollows: t 1 ( a )=1 t 2 ( b a )=1 t 2 ( b d )=1 t 3 ( a )=1 t 1 ( b )=1 t 2 ( b b )=1 t 2 ( b e )=1 t 3 ( c )=1 t 1 ( c )=1 t 2 ( b c )=1 t 3 ( d )=1 Denition2.2.2. Apottype over ( H; ) isaset P oftiletypesover ( H; ) suchthat forany h 2 H and t 2 P ,if t ( h ) > 0 thenthereexists t 0 2 P suchthat t 0 ( b h ) > 0 Wewrite P ( H; ) forapottypeover ( H; ) (ForexamplethepottypeofFigure2.4is P = f t 1 ; t 2 ; t 3 g .)A pot P over P is acollectionoftilesfromtypesin P .Wemodelatesttubeanditscontentwitha pot P .Henceweworkwithapot P oftype P ,where P containsinnitesupplyof distincttilesofeachtiletype.Denition2.2.3. Let T beasetoftilesand S asetofstickyends.Thefunction type : T [ S P [ H iscalledatypefunction andisdenedas: -forevery h 2 S type ( h )= h ,ifthestickyend h isoftype h -forevery t 2 T type ( t )= t ,ifthetile t isoftype t Nextwegiveaformalmathematicaldenitionofthegluingpr ocessandformation ofcomplexes. 12

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Denition2.2.4. Acomplex overapottype P isatriple C = h T;S;J i ,where T isasetoftileswithtiletypesin P S isasetofstickyendswithstickyendtypes in H ,and J isasetofunorderedpairs c = f ( t;h ) ; ( t 0 ;h 0 ) g satisfyingthefollowing properties:a) foreach c = f ( t;h ) ; ( t 0 ;h 0 ) g2 J;t;t 0 2 T;h;h 0 2 S ,suchthat type ( h )= h ;type ( h 0 )= b h ,and t ( h ) ;t 0 ( b h ) > 0 ; (cindicatestheconnectionbetweentwo complementarystickyends)and b) foreach h 2 H ,either X t 2 T t ( h )= X h 2 S type ( h )= h jf c :( t;h ) 2 c gj or X t 2 T t ( b h )= X h 0 2 S type ( h 0 )= b h jf c :( t;h 0 ) 2 c gj ,(thispreventsacomplexfromhavingcomplementary stickyends), c) thesumofthecardinalities X h 2 S type ( h )= h jf c :( t;h ) 2 c gj t ( h ) foreach t 2 T and h 2 H (thispreventsthetilefrommakingmoreconnectionsthanit hassticky ends), d) jf h 2 S : type ( h )= h gj = X t 2 T t ( h ) (thetotalnumberofstickyendsoftype h inS equalsthetotalnumberofstickyendsoftype h onthetilesin T ). Astickyend h canbeonlyononetile,soif t 1 ( h )=1and t 2 ( h )=1,then t 1 = t 2 Acomplex C = h T;S;J i schematicallycanbepresentedasagraph G ( C )=( V;E ) denedinthefollowingway: V = V T [ V S E = E T [ E H ,where V T = f t : t 2 T g V S = f h 2 S : h 2 S; for t 2 T;t ( h )=1 ; ( t;h ) = 2 c forany c 2 J g ; deg( t )= d ( t ) anddeg( h )=1 E T = ff t;t 0 g :thereexists c 2 J;c = f ( t;h ) ; ( t 0 ;h 0 ) g forsome h;h 0 2 S g E H = ff t;h g : t ( h )=1 ; ( t;h ) = 2 c forevery c 2 J g 13

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Fromthedenitionofthegraphforagivencomplexitfollows that j V j = j T j + j S j 2 j J j and j E j = j J j +( j S j 2 j J j )= j S jj J j Ifacomplex C = h T;S;J i hasastickyend h 2 S suchthat,forthetile t 2 T satisfying t ( h )=1,( t;h ) = 2 c ,forevery c 2 J ,thenthatstickyendiscalleda freestickyend Weassumethattheassemblyprocessoccursinanextremelydi lutedsolution,so thatwhentwocomplexesmeet,alloftheircomplementaryfre estickyendsjoinup andthattherearenocomplementaryfreestickyendsleft.Th isisthepartwherethe rexibilityisneeded.Iftheywererigid,thisassumptionwo uldhavenotbeenpossible. However,DNAjunctionmoleculeswithanadditionofbulgedT 'sontheirjunction sequencescanbemadetoberexible.Thus,acomplex C = h T;S;J i representsone oftheoutcomesofgluingallpossiblestickyendsonthetile sin T Denition2.2.5. Thetype ofacomplex C = h T;S;J i isthefunctiontype ( C ): H N denedby type ( C )( h )= X t 2 T t ( h ) X h 2 S type ( h )= h jf c :( t; h ) 2 c gj : Informally,acomplextyperecordsthenumberandthetypeso fthestickyends thatarefree. Note:Tileisalsoacomplex t = hf t g ;S; ;i andatiletypeisalsoacomplextype. Boththetiletypeandthecomplextypekeeptheinformationa boutthestickyends andnotabouttheunderlyinggraphstructure.Therefore,we denea structuretype Structuretypesareequivalenceclasses:twocomplexesare ofthesamestructuretype ifthereisagraphisomorphismfromonetotheotherthatpres ervesthetiletypes, stickyendtypesandedges. Forexample,thecomplexesinFigure2.4(b)areallofthesam etype( C ( a )=1, C ( c )=1, C ( b e )=1,and C ( k )=0,for k 2 H = f a;c; b e g ),butallofthemareof dierentstructuretypes. Moregenerally,twocomplexes C 1 = h T 1 ;S 1 ;J 1 i and C 2 = h T 2 ;S 2 ;J 2 i canbeglued bytheircomplementarystickyendstoformabiggercomplex C = h T;S;J i .Asfor 14

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thetiles,complexesmayglueinseveraldierentways,buta llstickyendsthatcan connectmustbeindeedconnected. Formallywedenethegluingprocessinthefollowingway. Denition2.2.6. Wesaythat C = h T;S;J i isobtainedbygluingcomplexes C 1 = h T 1 ;S 1 ;J 1 i and C 2 = h T 2 ;S 2 ;J 2 i if T = T 1 [ T 2 ;S = S 1 [ S 2 ; and J = J 1 [ J 2 [4 J; where 4 J isasetofunorderedpairs c = f ( t 1 ;h 1 ) ; ( t 2 ;h 2 ) g satisfyingthefollowing properties: a) foreach c = f ( t 1 ;h 1 ) ; ( t 2 ;h 2 ) g24 J t 1 2 T 1 t 2 2 T 2 h 1 2 S 1 and h 2 2 S 2 arefreestickyendssuchthat ( type ( h 1 ))= type ( h 2 ) ( c indicatestheconnection betweentwotilesfrombothcomplexes). b) foreach h 2 H ,type ( C )( h )= max f type ( C 1 )( h )+ type ( C 2 )( h ) type ( C 1 )( b h ) type ( C 2 )( b h ) ; 0 g (foreach h 2 H ,asmanyfreestickyendsoftype h aspossible arejoined). Denition2.2.7. Acomplex C iscalledcomplete ifithasnofreestickyends,i.e., forallstickyends h ,type ( C )( h )=0 : Ifinapotsomecomplexeshavefreestickyends,thatmeansth eyarestillfree togluewithothercomplexes.Soifwewanttodesignacertain graphstructure withtiles,wewouldlikethenaloutcometobeacompletecom plex.Also,aDNA computingproblemcanbeencodedinthetiles,andasolution totheproblemwill beacompletecomplexofcertainsize.Hence,ourmainintere stare,inparticular, completecomplexes. Forapottype P wedenoteby C ( P ) thesetofallcompletecomplexes thatcan beobtainedbytilesoftiletypesin P 15

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2.3PotTypeClassication Weclassifythepotsinfourclasses:\unsatisable",\weak lysatisable",\satisable" and\stronglysatisable"accordingtopossibleassemblie sofcompletecomplexes. Denition2.3.1. Acomplex C = h T;S;J i (representedbyagraph G ( C )=( V;E )= ( V T [ V S ;E T [ E S ) )isembedded inacomplex C 0 = h T 0 ;S 0 ;J 0 i (representedbyagraph G 0 =( V 0 ;E 0 )=( V 0 T [ V 0 S ;E 0 T [ E 0 S ) )if T T 0 S S 0 J J 0 ,andthereexistsa function : V V 0 .Thefunction isdenedasfollows:for t 2 V T ( t )= t ;for h 2 V S ( h )= 8<: h for t 2 T suchthat t ( h )=1 ; ( t;h ) = 2 c; forany c 2 J 0 t 0 for t 2 T suchthat t ( h )=1 ; thereexists c 2 J 0 ; ( t;h ) 2 c: Naturally,atile t is embedded inacomplex C = h T;S;J i if t 2 T Nowweclassifythepottypes. Denition2.3.2. Apottype P isweaklysatisable ifitadmitsacompletecomplex, i.e, C ( P ) 6 = ; .Otherwiseitisunsatisable Apottype P issatisable if,foreach h 2 H ,thereisacompletecomplex C 2C ( P ) ofthepotcontainingatleastonestickyendoftype h Apottype P isstronglysatisable ifeverycomplexthatcanbegeneratedby P can beembeddedintoacompletecomplexof P Sinceweareonlyinterestedincompletecomplexes,wewould liketoobtainonly completecomplexesastheproductsoftheassemblyprocess. Therefore,strongsatisabilityisthenotionofmostimmediateinterestinourst udy. Denition2.3.3. Acomplex C = h T;S;J i iscalled k -tilecomplex if j T j = k Lemma2.3.4. Apottype P isstronglysatisableifandonlyifforeverytiletype t 2 P ,thereexistsacompletecomplex C = h T;S;J i2C ( P ) with t 2 T suchthat type ( t )= t 16

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Proof. Oneimplicationofthelemmaistrivial;if P isstronglysatisable,sinceevery tileisacomplex,itcanbeembeddedintoacompletecomplex. Theconverseisobtainedbymathematicalinductiononthenu mberoftilesin acomplex.Ifacomplexconsistsofonlyonetile,i.e.,if C = hf t g ;S; ;i ,thenthe complex C itselfisthetile t .Bytheassumptionthatanytilecanbeembeddedina completecomplex,itfollowsthat t canbeembeddedinacompletecomplex. Assumethestatementholdsfor k -tilecomplexes;weclaimitholdsfor( k +1) tile Let C = h T;S;J i bea( k +1)-tilecomplexwith T = f t 0 ;t 1 ;:::;t k g .If C is completethenthetheoremisproved.Assumeitisnot,i.e.,f orsome h 2 H type( C )( h )= k X i =0 t i ( h ) X h 2 S type ( h )= h jf c :( t i ;h ) 2 c gj > 0 : C' 0 t Figure2.5:Thecomplex C = h T;S;J i Considerthecomplex C 0 = h T 0 ;S 0 ;J 0 i with T 0 = f t 1 ;t 2 ;:::;t k g and J 0 = J f c 2 J :( t 0 ;h ) 2 c g ,andthetile t 0 = hf t 0 ;S 0 ; ;gi .Thecomplexobtainedbygluingthe k -tilecomplex C 0 andthetile t 0 is,infact, C CASE1:Thecomplex C 0 isnotcomplete. Bytheinductionalhypothesis, C 0 and t 0 canjoinwithcomplexes cC 0 = h b T 0 ; b S 0 ; b J 0 i ( b T 0 6 = ; since C 0 isnotcomplete)and c C t 0 = h b T 0 ; b S 0 ; b J 0 i ( b T 0 6 = ; ),respectively toformcompletecomplexes cC 0 and c C t 0 havestickyendsthatarecomplementary to C 0 and t 0 respectively,i.e., b S 0 = f b h 0 : ( type ( b h 0 ))= type ( h 0 )for h 0 2 S 0 g and 17

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b S 0 = f b h 0 : ( type ( b h 0 ))= type ( h 0 )for h 0 2 S 0 g Thecomplexesformedbygluing C 0 and cC 0 arecomplete,andthecomplexes obtainedbygluing c C t 0 and t 0 arealsocomplete(aswesawbeforethegluingprocessis notunique,sowecangluetwocomplexesinmanydierentways ,eachwayobtaining anewcomplex).Letchooseonecomplexfromthesetofcomplex esobtainedby gluing C 0 and cC 0 ,say C 1 = h T 1 ;S 1 ;J 1 i .Also,letchooseanothercomplexfromthe setofcomplexesobtainedbygluing c C t 0 and t 0 C 2 = h T 2 ;S 2 ;J 2 i .Thecomplexes C 1 = h T 1 ;S 1 ;J 1 i and C 2 = h T 2 ;S 2 ;J 2 i arecompletecomplexes,i.e.,type( C 1 )( h )=0 andtype( C 2 )( h )=0forevery h 2 H Notethatforevery h 2 H ,since C 1 isacompletecomplex,in S 1 thereareequal numberofstickyendsoftype h and b h .Samethingholdfor S 2 C' C' t 0 C t 0 Figure2.6:Thecomplexes C 1 and C 2 Considerthecomplex,obtainedbygluing cC 0 and c C t 0 ,saythecomplex b C = h b T; b S; b J i ,with b T = b T 0 [ b T 0 and b J = b J 0 [ b J 0 [ b J ,where b J isthesetofunordered pairs f ( p;h p ) ; ( q;h q ) g p 2 b T 0 and q 2 b T 0 type ( h p )= h forsome h 2 H ,and type ( h q )= b h Weclaimthateverycomplexformedbygluing C and b C isacompletecomplex. Consideracomplex C C = h T C ;S C ;J C i formedbygluing C and b C .Bythedenition ofgluingwehave T C = T [ b T S C = S [ b S ,and J C = J [ b J [4 J ,where 4 J isasetof unorderedpairs f ( r;h 1 ) ; ( s;h 2 ) g suchthatafreestickyendof h 1 ,with type ( h 1 )= h 1 forsome h 1 2 H ,fromtile r 2 T connectstoafreestickyend h 2 oftype b h 1 fromtile s 2 b T Fromthepreviousobservation, T C = T [ b T = T [ b T 0 [ b T 0 =( f t 1 ;:::;t k g[ b T 0 ) [ 18

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( f t 0 g[ b T 0 )= T 1 [ T 2 .Also, S C = S [ b S = S 0 [ S 0 [ b S 0 [ b S 0 = S 1 [ S 2 Fromthedenitionofthecomplexitfollowsthatforevery h 2 H jf h 2 S C : type ( h )= h gj = X t 2 T C t ( h )and jf b h 2 S C : type ( b h )= b h gj = X t 2 T C t ( b h ).Sincein S C ,forevery h 2 H ,thereareasmanystickyendsoftype h asoftype b h itfollows that jf h 2 S C : type ( h )= h gj = jf b h 2 S C : type ( b h )= b h gj = X h 2 S c type ( h )= h jf c 2 J c : c = f ( t;h ) ; ( t 0 ;h 0 ) g ;type ( h )= h ;type ( h 0 )= b h ;t;t 0 2 T C ;h;h 0 2 S C gj .Hence, type( C C )( h )= X t 2 T C t ( h ) X h 2 S C type ( h )= h jf c 2 J C :( t;h ) 2 c gj = X t 2 T C t ( h ) jf h 2 S C : type ( h )= h gj = X t 2 T C t ( h ) X t 2 T C t ( h )=0 ; i.e., C C isacompletecomplex. 0 C C' t 0 C' t 0 C' C' t 0 C t Figure2.7:a)Thecomplex C C STEP2:Thecomplex C 0 iscomplete. Since P isstronglysatisable,thereexistsacomplex c C t 0 = h b T 0 ; b S 0 ; b J 0 i suchthat acomplexobtainedbygluing t 0 and c C t 0 iscomplete.Consideracomplexobtainedby gluingthecomplexes C and c C t 0 ,sayacomplex C = h T ;S ;J i .Fromthedenition forgluingoftwocomplexes, T = T [ b T 0 S = S [ c S t 0 ,and J = J [ b J 0 [4 J ,where 4 J = f c = f ( t 0 ;h ) ; ( t;h 0 ) g : h 2 S 0 S;t 2 b T 0 ;h 0 2 b S 0 ; ( type ( h 0 ))= type ( h ) g 19

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Fromthedenitionofthecomplexitfollowsthatforevery h 2 H jf h 2 S : type ( h )= h gj = X t 2 T t ( h )and jf b h 2 S : type ( b h )= b h gj = X t 2 T t ( b h ).Sincein S ,forevery h 2 H ,thereareasmanystickyendsoftype h asoftype b h itfollows that jf h 2 S : type ( h )= h gj = jf b h 2 S : type ( b h )= b h gj = X h 2 S type ( h )= h jf c 2 J c : c = f ( t;h ) ; ( t 0 ;h 0 ) g ;type ( h )= h ;type ( h 0 )= b h ;t;t 0 2 T ;h;h 0 2 S gj Thecomplex C iscompletesinceforevery h 2 H : type( C )( h )= X t 2 T t ( h ) X h 2 S type ( h )= h jf c 2 J :( t;h ) 2 c gj = X t 2 T t ( h ) jf h 2 S : type ( h )= h gj = X t 2 T t ( h ) X t 2 T t ( h )=0 ; Itisstraightforwardtoseethatallstronglysatisablepo ttypesarealsosatisable andthatallsatisablepottypesareweaklysatisable.But theconverseisnot necessarytrue.Figure2.8a)showsapottypethatissatisa ble,butnotstrongly satisable,sincetilesofthetiletypes t 3 and t 4 canneverbeembeddedintoacomplete complex.ThepottypeofFigure2.8b)isanexampleofapottyp ethatisweakly satisable,butnotsatisablesincethestickyendtype c canneverbeapartofany nitecompletecomplex. 20

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a 2 t 1 a b c t 3 t 4 t 1 t 2 b a c t 3 t 4 b ) a ) b a c a a b b a b a c t Figure2.8:a)Satisablepottypethatisnotstronglysatis able( t 3 and t 4 cannotbeapart ofacompletecomplex)b)weaklysatisablepottypethatisn otsatisable(thestickyend c cannotbeapartofacompletecomplex). Notethatthenumberofstickyendtypesdoesnotdependonthe numberoftile types.ThepottypesinallthreeexamplesinFig2.9arestron glysatisable;inthe rstexamplethenumberoftiletypesandstickyendtypesare equal;inthesecond onethenumberoftiletypesarelessthenfreestickyendtype s;andinthethird examplethenumberoftiletypesisgreaterthenstickyendty pes. b 2 t 1 a c b t 3 t 2 t 1 a a t 1 t 2 b a a a ) b ) b a c c ) a a a a t Figure2.9:a) j P j = j H + j b) j P j < j H + j c) j P j > j H + j : Fortherestofthedissertationwereserve m = j P j ( P = f t 1 ; t 2 :::; t m g ),and 2 n = j H j ( H = f h 1 ;:::; h n ; c h 1 ;:::; c h n g ). Toeachcomplextype C weassociateavector z C =( z 1 ;z 2 ;:::;z n )from Z n such that 21

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z i : H + Z z i =type( C )( h i ) type( C )( b h i ) : Weassumethatthepotisdilutedandthethermodynamiccondi tionsaresuchthat allstickyendsthatcanconnectwouldbeableto.Inthissens e z C givesinformation abouttheremainingfreestickyendtypesonthecomplex C Therefore,eithertype( C )( h i ) > 0ortype( C )( b h i ) > 0,butnotboth.Iftype( C )( h i ) > 0,then z i > 0,andiftype( C )( b h i ) > 0,then z i < 0. Sinceatileisacomplex,notilehascomplementarystickyen ds.So,forevery t 2 P andevery h 2 H ,if t ( h ) > 0then t ( b h )=0.Inthiscase,foreverytile t ,we associateavector z t =( z t ( h 1 ) ;z t ( h 2 ) ;:::;z t ( h n ))from Z n suchthat z t ( h i )= 8>>><>>>: t ( h i )if t ( h i ) > 0 t ( h i )if t ( b h i ) > 0 0otherwise : NOTE:Themethodandthetheorydescribedhereiscorrecteve niftileswith complementarystickyendsareaccepted. 22

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3SpectrumofaPot 3.1Denitions FromreportsonDNAassembliesweknowthatwhenonerunsanex periment,the desiredcomplexesarenottheonlythingsthatshowsupinthe pot;theremaybealot ofincompletecomplexes.Theyincreasetheerrorrateandth ecostoftheexperiment. Ifthestoichiometryinthetesttubeisbad,i.e.,animprope rratioofeachofthe moleculesisused,thenunderanyconditionsincompletecom plexeswillbepresent.In thissectionweproposeamethodwhichtheoretically(ignor ingalldynamicconsiderationssuchasthosein[27])eliminatesthepresenceofincom pletecomplexesassuming thatassemblyoccursinidealconditionsinawellmixeddilu tedpot.Thesetsof vectorsoftheratioofthemoleculesiscalledthe spectrumofthepot ,andwegive analgorithmforcalculatingitusingtheJordan-Eliminati onmethod.Theclosureof aspectrumofapotinEuclidianspaceisasimplex,whosevert icescorrespondto connectedcompletecomplexes.Also,inthissectionthroug hinvestigatingsubsetsof anespaceswegiveamethodforidentifyingtheclassofagiv enpottype. Denition3.1.1. The spectrum of P istheset S ofallvectors r =( r t : t 2 P ) suchthat: 1)Foreach t r t 0 and X t 2 P r t =1 ; (3.1.1) 2)foreach h X t 2 P r t t ( h )= X t 2 P r t t ( b h ) ; (3.1.2) 23

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i.e,foreach h 2 H thereareasmanystickyendsoftype b h asthereareoftype h Usingthevector z t ( h )= t ( h ) t ( b h ),associatedtotile t whenitisconsideredas acomplex,thesecondpartofthedenitioncanberewritteni nthefollowingform 2 ) X t 2 P r t ( t ( h ) t ( b h ))=0 ; X t 2 P r t z t ( h )=0 : Anobviousobservationfromthedenitionofthespectrum,i sthatthespectrum canberepresentedasanintersectionofthehyperplane H 1 = f x 2 R m : m X i =1 x i =1 g withthekernelofthelineartransformations.Also,noteth atthespectrum S ( P ) [0 ; 1] j P j Ifapothasamixtureoftileswhoseproportionscorrespondt oavectorinthe spectrum,theninperfectconditionsonlycompletecomplex esneedbeexpected.If theusedproportionofthemoleculesisnotinthespectrum,t herearenoconditions underwhichattheendoftheexperimentonlycompletecomple xeswillbepresent inthetesttube. Note thatapottypeadmitsacompletecomplexifandonlyifits spectrumisnonempty. Foravector r =( r 1 ;r 2 ;:::;r m ) 2S ( P ), r i 2 [0 ; 1]for i 2 [ m ]and m X i =1 r i =1. Therefore,thevector r canbeconsideredasavectorofprobabilitiesfortilestobe onacompletecomplex,i.e., r j canbeconsideredastheprobabilitythatarandomly selectedtileisoftype t j Example3.1.2. ThespectrumofthepottypegiveninFigures2.8(a),(b)each containingthreetiletypes,isthesolutionofthefollowin gsystemsofequationsfor 24

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r 1 ;r 2 ;r 3 0. (a) r 1 + r 2 + r 3 + r 4 =1(b) r 1 + r 2 + r 3 + r 4 =1 r 1 r 2 +2 r 3 =0 r 1 r 2 + r 3 + r 4 =0 r 1 r 2 +2 r 4 =0 r 1 r 2 =0 r 1 r 2 =0 r 3 r 4 =0 : Bothsystemshavethesamesolution,i.e.,thespectrumofbo thpottypesis S = f ( 1 2 ; 1 2 ; 0 ; 0) g ,buttherstpottypeissatisable,whilethesecondisonly weakly satisable.Thesetwoexamplesshowthatspactracannotbeu sedtodistinquish betweenaweaklysatisablepottypeandstronglysatisabl epottype. Intheaboveexamplefornostickyendstoremainfree,i.e.,o nlycompletecomplexestobeassembled,thespectrumpointsoutthatoneneed stouseequalnumber ofmoleculesofthersttwotypes,andnouseofanymolecules fromtheothertwo types. Useofproportionoftiletypesfromthespectrumisnecessar yforeliminatingthe incompletecomplexesattheendofanexperiment,butitdoes notgiveinformation aboutthetypeofcompletecomplexes.Wecanonlyassumethat inaverydiluted solution,thesmallestcomplexeswillbemostfavorable. Therearenitenumberoftilesinagivenpottype,sotheprop ortionofeachtile isarationalnumber.Forthepracticalpurposesweconsider S ( P ) Q m Q being thesetofrationalnumbers. 3.2GeometricRepresentationoftheSpectrum First,wegivesomedenitionsfromLinearProgrammingthat willbeused.[11,33] givesagoodintroductiontothesubject.Denition3.2.1. Apolyhedron in R n istheset f x 2 R n : A x b g ,where A 2 R m n isamatrixand b 2 R m isavector.Aboundedpolyhedroniscalledapolytope 25

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Denition3.2.2. Let a 1 ;a 2 ;:::;a n bescalarsand b beapointin R .Ahyperplane isthesetofallpoints x =( x 1 ;x 2 ;:::;x n ) 2 R n satisfying n X i =1 a i x i = b Denition3.2.3. Let P = f x 2 R n : A x b g beanonemptypolyhedron.If c isa nonemptyvectorforwhich =max f cx : x 2 P g isnite,then f x : cx = g iscalled asupportinghyperplane of P .Aface of P is P itselfortheintersectionof P witha supportinghyperplaneof P .Apoint v forwhich f v g isafaceiscalledavertex of P Denition3.2.4. Aconvexcombination ofnitenumberofpoints a 1 ;a 2 ;:::;a n is n X i =1 i a i ; where n X i =1 i =1 and i 0 for i =1 ; 2 ;:::;n: Denition3.2.5. Aconvexhull ofasetofpoints S isthesetofallconvexcombinationsofthepointsfrom S .Anextremepoint of S isapointthatcannotbewritten asaconvexcombinationoftwootherpointsfrom S Denition3.2.6. Considerthesystemofequations A x = b ; x 0 ,where x isan n -vector, b 2 R m A is m n matrix.Afeasiblesolution tothesystemisavector x =( x 1 ;x 2 ;:::;x n ) with x i 0 ,for i 2 [ n ] ,and A x = b Denition3.2.7. Let P = f x 2 R n ; A x = b ; x 0 g ,where A is m n matrix, x =( x 1 ;x 2 ;:::;x n ) .Let B beasubsetof [ n ] ,with j B j m suchthat A B ,the matrixconsistingofcolumnsof A thatcorrespondtotheindicesofB,isinvertible. Abasicfeasiblesolution isafeasiblesolution x with x j =0 ,for j= 2 B; A B x B = b ,or x B = A 1 B b ,where x B isthevectorconsistingofelementsof x restrictedtotheindicesofB. Itisknownthattheconvexhullofaset S isthesmallestconvexsetcontaining S [7].Also,everypolytopeistheconvexhullofitsextremepo intsandtheextreme pointsaretheverticesofthepolytope[7]. 26

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Also,fromLinearprogramming[7,33],itisknownthatavect or r isanextreme pointoftheconvexpolyhedron S = f r : A r = b ; r 0 g ( A is n m ,rank( A )= n< m b 2 R n ,and r 2 R m )ifandonlyif r isabasicfeasiblesolutionto A r = b : Denoteby H m theintersectionofthesubspaceof Q m+ = f ( r 1 ;r 2 ;r 3 :::r m ): r i 2 Q ;r i 0for i 2 [ m ] g ,andthehyperplane r 1 + r 2 + ::: + r m =1.Denition3.1.1 showsthatthespectrumofapotwith m tiletypesisasubsetoftheset H m andthe n hyperplanes(foreach h X t 2 P r t z t ( h )=0). Theintersectionofthehyperplanesand R m+ isapolytope,i.e.,itisaconvexhull ofitsvertices.Thereforethespectrumofanygivenpotisde nseinthecorresponding convexhullandcontainsalltheverticesofthehull.Proposition3.2.8. Thespectrum S ( P ) ofapottype P with j P j = m andcorrespondingsetofstickyends H with j H + j = n isanintersectionof n hyperplanesand theset H m .MoreovertheclosureofaspectruminEuclidianspaceisaco nvexhull whoseverticesarerationalpoints.Proposition3.2.9.a) Apottypeisweaklysatisableifandonlyifitadmitsa nonemptyspectrum. b) TheclosureinEuclidianspaceofthespectrum S ( P ) ofapottype P isaconvex hull:if u ; v 2S ( P ) ,andif z 2 [0 ; 1] ,then z u +(1 z ) v isintheclosureof S ( P ) Proof.a) Let P beaweaklysatisablepottypeand C = h T;S;J i beacompletecomplex ( T 6 = ; )assembledfromtilesoftypesin P .Denoteby k i thenumberoftilesin T oftype t i andlet r i = k i j T j (Note j T j > 0and r i 0).Obviously m X i =1 r i =1.Since C isacompletecomplex,forevery h 2 H thereareasmanystickyendsoftype h amongthetilesin T asoftype b h .Consequently m X i =1 k i t i ( h )= m X i =1 k i t i ( b h ),orby dividingbothsidesoftheequalityby j T j ,itfollowsthat m X i =1 r i t i ( h )= m X i =1 r i t i ( b h ). 27

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Fromthedenitionofthespectrumitfollowsthatthevector r =( r 1 ;r 2 ;:::;r m ) isanelementofthespectrum.Conversely,if S ( P ) 6 = ; ,thereexistsanonzerovector r =( r i : t i 2 P )of rationalnumbersin S ( P ).Eachcoordinateof r canbewrittenas r i = q i d i for q i 0and d i > 0bothintegers.Denoteby d =lcm( d 1 ;d 2 ;:::d m )and p i d i = d foraninteger p i .Thus d r =( dr 1 ;dr 2 ;:::;dr m )=( p 1 q 1 ;p 2 q 2 ;:::;p m q m ).A(not necesseralyconnected)complexthathas p i q i tilesoftype t i ,for i 2 [ m ]isa completecomplex. b) Followsimmediatelyfromthefactthatthespectrumisaconv exhull. Proposition3.2.10. Thespectrum S ( P ) ofagivenpot P iseitherempty,asingleton oraninniteset.Proof. Thespectrumofanunsatisablepotisempty(Proposition3. 2.9).Sincethe spectrumisdenseinaconvexhullandincludestheverticeso fthatconvexhull,ifit containstwopointsthenitcontainsatleasttwovertices,a ndhenceeveryrational pointbetweenthosevertices,sothespectrumisinnite. Proposition3.2.11. Let P = f t 1 ; t 2 ;:::; t m g beapottypeand S ( P ) itsspectrum. Foreveryextremepoint, s = k 1 d 1 ; k 2 d 2 ;:::; k m d m of S ( P ) ,thereexistsacompleteconnectedcomplex C = h T;S;J i2C ( P ) with d k i d i tilesoftype t i ,for i 2 [ m ] where d = lcm ( d 1 ;d 2 ;:::;d m ) and gcd( k j ;d j )=1 ,for j 2 [ m ] Proof. Theratioofthetilesofthecomplex C = h T;S;J i consistingof d k i d i tilesof type t i ,for i 2 [ m ]is s = k 1 d 1 ; k 2 d 2 ;:::; k m d m (Notethat, m X i =1 k i d i =1).Consequently thenumberoftilesinthecomplexis j T j = m X i =1 d k i d i = d m X i =1 d i k i = d: 28

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Since s 2S (andfromthedenitionofthespectrum), C isacompletecomplex.Next wehavetoshowthat C isaconnectedcompletecomplex. Assumethat C isnotaconnectedcompletecomplex.Withoutlossofgeneral ity, wecanassumethat C consistsoftwononemptycompletecomplexes C 1 = h T 1 ;S 1 ;J 1 i and C 2 = h T 2 ;S 2 ;J 2 i with p i and q i thenumbersoftilesoftype t i ,respectively.The spectrumpointscorrespondingto C 1 and C 2 are s 1 = p 1 j T 1 j ; p 2 j T 1 j ;:::; p m j T 1 j and s 2 = q 1 j T 2 j ; q 2 j T 2 j ;:::; q m j T 2 j .Since C consistsof C 1 and C 2 ,then j T j = j T 1 j + j T 2 j ,and sinceboth C 1 and C 2 arenonempty, j T 1 j 1.In therstcase,whengcd( p i ; j T 1 j )=1, p i = k i and d i = j T 1 j ,forevery i 2 [ m ].So, d 1 = d 2 = ::: = d m = d ,i.e., j T 1 j = d ,whichiscontradictswith j T 1 j
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nectionsbetweenthetiles.Similarly,theremightbemoret hanonecompletecomplex associatedwithangivenpointsfromthespectrumofagivenp ottype. Denition3.2.13. Let P = f t 1 ; t 2 ;:::; t m g beapottypewithitsspectrum S Let s i = k i 1 d i1 ; k i 2 d i2 ;:::; k i m d im for i 2 [ l ] betheextremepointsof S andlet d i = lcm ( d i1 ;d i2 ;:::;d im ) Theset S i ofcompletecomplexes C consistingof d j k j i d ji tilesoftypes t l iscalledthe setof extremecompletecomplexescorrespondingto s i andthecomplexesarecalled extremalcomplexes Denition3.2.14. Acomplex C = h T;J i iscalleda minimalcompletecomplex if theredoesnotexistsacompletecomplex C 0 = h T 0 ;J 0 i with T 0 T and T 6 = T 0 FromProposition3.2.11followsthateveryextremecomplex isalsoaminimal completecomplex.Buttheconverseisnotnecessarytrue,as wewillseeinExample 3.2.16.Proposition3.2.15. Let P = f t 1 ; t 2 ;:::; t m g beapottype, S ( P ) thespectrumof P with l extremepoints,and C 1 ;C 2 ;:::;C l beextremecompletecomplexesforthepot typecorrespondingtotheextremepoints.Thevectorofthen umberoftilesofevery completecomplexin C ( P ) isalinearcombinationofthevectorsofthenumberoftiles oftheextremalcompletecomplexes.Proof. Let C = h T;S;J i beacompletecomplexbuiltfromtilesoftypesin P ,let r =( r 1 ;r 2 ;:::;r m )bethevectoroftheratiosofthetiletypesin C andletthe correspondingvectorofthenumbersoftilesbe c =( c 1 ;c 2 ;:::;c m )= j T j r If r isanextremepointof S ( P ),thenthenumberoftilesof C isamultipleofthe numberoftilesoftheextremalcompletecomplexescorrespo ndingtothatextreme point. Let r beanon-extremepointof S ( P ),then r canbewrittenasaconvexcombinationoftheextremepointsin S ( P ),say r = l X i =1 i r i ,for r i rangingovertheextreme pointsof S ( P )and l X i =1 i =1. 30

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For i 2 [ l ], r i =( r i 1 ;r i 2 ;:::;r i m ),where r i j 0for j 2 [ m ]and m X j =1 r i j =1.Let r i j = x ij d ij and d i =lcm( d i1 ;d i2 ;:::;d im ),then d i r i =( d i r i 1 ;d i r i 2 ;:::;d i r i m )=( c i1 ;c i2 ;:::;c im ) and m X j =1 d i r i j = d i ,where( c i1 ;c i2 ;:::;c im )isthevectorcorrespondingtotheextreme point r i .Withotherwords,theminimalcompletecomplexhas d i tiles. Denote D =lcm( d 1 ;d 2 ;:::;d l ).Foreach i 2 [ l ],wecanwrite D = d i m i ,for an m i 2 N .Thespectrum S ( P )consistsofpointswithrationalcoordinates,say i = p i q i i 2 [ l ].Denoteby q thelcm( q 1 ;q 2 ;:::;q k ),i.e., q = s i q i ,forappropriate s i 2 N i 2 [ l ].Then, c = j T j r = l X i =1 j T j i ( r i 1 ;r i 2 ;:::;r i m ) = l X i =1 j T j Dq Dq p i q i ( r i 1 ;r i 2 ;:::;r i m ) = l X i =1 j T j Dq d i m i s i q i p i q i ( r i 1 ;r i 2 ;:::;r i m ) = l X i =1 j T j Dq p i s i m i ( r i 1 d i ;r i 2 d i ;:::;r i m d i ) = l X i =1 j T j p i s i m i Dq ( c i1 ;c i2 ;:::;c im ) : Thevectorofthenumberoftilesin C canbewrittenasalinearcombinationof thevectorsofthenumberoftilesfortheextremalcompletec omplexes. Thepreviouspropositionstatesthatthevectorofthenumbe roftilesofevery completecomplexcouldbewrittenasalinearcombinationof thevectorsofthe 31

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numberoftilesoftheextremecomplexes.Thislinearcombin ationisnotnecessarily anintegercombination.Foragivenpottype P ,thesetofcompletecomplexes, C ( P )canbecharacterizedthroughtheextremecompletecomplex es.Notallminimal complexesareextremecomplexesaswecanseeinFigure3.1.Example3.2.16. ConsiderthePottypefromFigure2.9c.Itspectrumistheset of solutionstothefollowingsystemofequations. r 1 + r 2 + r 3 =1 3 r 1 2 r 2 r 3 =0 : Hence,thespectrumis S ( P )= f ( u; 4 u 1 ; 2 5 u ): 1 4 u 2 5 g .Tondthe extremepoints,weneedtondthebasicfeasiblesolutionfo rthesystemgivenabove, i.e.,tondsolutionstothefollowingsystems.24 113 2 35 24 r 1 r 2 35 = 24 10 35 ; 24 113 1 35 24 r 1 r 3 35 = 24 10 35 ; 24 11 2 1 35 24 r 2 r 3 35 = 24 10 35 for r 1 0 ;r 2 0 ;r 3 0. Theextremepointsforthespectrumare s 1 = 1 4 ; 0 ; 3 4 and s 2 = 2 5 ; 3 5 ; 0 .The minimalcompletecomplexesforthispottypearegiveninthe gurebelow. a) t 1 t 3 t 2 t 3 t 3 t 1 t 1 t 2 t 2 t 2 t 3 t 1 b) Figure3.1:a)Theminimalcompletecomplexesforthepottyp egiveninFigure2.9cb)A completecomplexthatisminimal,butnotextremalcomplete complex. 32

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Thespectrumpointcorrespondingto C 1 is 1 4 ; 0 ; 3 4 .Thespectrumpointcorrespondingto C 2 is 2 5 ; 3 5 ; 0 .Thespectrumpointcorrespondingto C 3 is 1 3 ; 1 3 ; 1 3 Althoughallthreecomplexesareminimalone, C 1 and C 2 areextremecomplexes, while C 3 isnot. Corollary3.2.17. Ifthespectrumconsistsofonlyonepoint, S ( P )= f ( r t 1 ;r t 2 ;:::;r t m ) g andif r t k > 0 forsome k 2 [ m ] ,theneverycompletecomplexin C ( P ) containsatile oftype t k .Moreover,if r t k > 0 forall k 2 [ m ] P isstronglysatisable. Proof. Ifthespectrumconsistsofonlyonepoint,thenthatpointmu stbeanextremepoint.FromProposition3.2.11itfollowsthataminim alcompletecomplex correspondingtotheextremepointisconnectedandcontain stilesoftypes t k for which r t k > 0.Consequently,everyothercompletecomplexisalinearco mbination ofthesecomplexes,i.e.,thenumberoftilesoftype t k ,for k 2 [ m ],ofanyother competecomplexisamultipleofthenumberoftilesoftype t k onthecomplexes correspondingtotheextremepoint. If r t k > 0forall k 2 [ m ],thenthecompletecomplexcorrespondingtotheextreme pointfromthespectrumcontainstilesofeachtype,so P isstronglysatisable. FromCorollary3.2.17wecanconcludethatifthespectrum S ( P ),ofagiven pottype P ,consistsofonlyonepointwhosecoordinatesarepositive, thenevery completecomplexin C ( P )containstilesofeachtype.Nowlet'sconsiderexamples forthespectraofstronglysatisablepottypes.Example3.2.18. ConsiderthepottypesdepictedinFigure2.9.Theirspectra can becomputedsimilarlyasinExample3.1.2.Thespectrumofth epottypeinFigure 2.9(a)is S = f ( 1 2 ; 1 2 ) g ,thespectrumofthepottypeinFigure2.9(b)is S = f ( 1 2 ; 1 2 ) g whiletheoneforFigure2.9(c)is S = f ( u; 4 u 1 ; 2 5 u ): u 2 R ; 1 4 u 2 5 g Intwo-dimensionalspace(correspondingtoapottypewithe xactlytwotiletypes) thespectrumisapartofthelinesegment( r 1 + r 2 =1 ; 0 r 1 1 ; 0 r 2 1) 33

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connectingthepoints(0 ; 1)and(1 ; 0).Sothespectrumiseitherapointofthatline segment,oritistheentirelinesegment,oritistheemptyse t.Thespectrumis theentirelinesegmentifandonlyifcomplementarystickye ndsofsametypeona singletileisallowed.Because,inthiscasethesystemofeq uationsandinequalities becomes r 1 + r 2 =1 ; 0 r 1 1 ; 0 r 2 1,i.e.,whenatileofthersttypeformsa completecomplex,andatileofthesecondtypeformsacomple tecomplex.Sincewe donotallowthat,thespectrumintwo-dimensionalcasewill alwaysconsistofonly onepoint.Proposition3.2.19. Thespectrum, S ( P ) ,ofthepottype P = f t 1 ; t 2 ;:::; t m g is aconvexhullwithvertices f (1 ; 0 ; 0 ;:::; 0) ; (0 ; 1 ; 0 ;:::; 0) ;:::; (0 ; 0 ;:::; 0 ; 1) g ifand onlyifeachtilefromatypein P iaacompletecomplexes. Proof. Assumethespectrum S ( P ),ofthepottype P = f t 1 ; t 2 ;:::; t m g isaconvexhullwithvertices f (1 ; 0 ; 0 ;:::; 0) ; (0 ; 1 ;:::; 0) ;:::; (0 ; 0 ;:::; 1) g : FromProposition3.2.15followsthatforeach i 2 [ m ]thecomplex C i consistingofatileoftype t i isconnectedcompletecomplex.Becauseofthateachtilefro matypein P canform acompletecomplex. Assumethateachtilefromatypein P isacompletecomplex.Then,foreach h 2 Hz t i ( h )=0,andhencewedonotgetanyequationfromthesecondcondi tioninthe denitionofthespectrum.So,thespectrumof P is: S ( P )= f ( r 1 ;r 2 ;:::;r m ): r i 0 ; for i 2 [ m ]and P mi =1 r i =1 g ,whichisapolytopewithvertices f (1 ; 0 ; 0 ;:::; 0) ; (0 ; 1 ;:::; 0) ;:::; (0 ; 0 ;:::; 1) g : Example3.2.20. Allthreeexampleshaveathreetiletypepottype P = f t 1 ; t 2 ; t 3 g andthesetofstickyendtypes H = f a ; b ; b a ; b b g a ) a b b b t 3 t 1 t 2 a b a b z t 1 =(2 ; 1), z t 2 =( 1 ; 1), z t 3 =(0 ; 3), 34

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a 1 a a t 2 t 3 a a b ) a t z t 1 =(3), z t 2 =( 2), z t 3 =( 1), c ) 1 a t 3 b c t 2 a b c t z t 1 =(0 ; 0), z t 2 =(0 ; 0), z t 3 =(0 ; 0) Figure3.2:a)Stronglysatisablepottypewithspectrum f ( 1 4 ; 1 2 ; 1 4 ) g ,b)Stronglysatisable pottypewithspectrum S ( P )= f ( u; 4 u 1 ; 2 5 u ): 1 4 u 2 5 g .c)Stronglysatisable pottypewithspectrum f (1 u v;u;v ):0 u 1 ; 0 v 1 ;u + v 1 g Notethespectrumcontainsvectorswithrationalentries,i tsclosureinEuclidian spaceisboundedandthereforecompactsubsetof R m : 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 Figure3.3:Theclosureofthespectrumofthepottypegiveni nExample3.2.20b)isthe linesegment;theclosureofthespectrumofthepottypegive ninExample3.2.20c)isthe triangleboundedbythedottedlinesalongwithitsinterior 35

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3.3AlgebraicRepresentationoftheSpectrum Thespectrumistheintersectionof n hyperplanes(foreach h X t 2 P r t z t ( h )=0)and H m .Henceitisthesolutionof n homogeneousand1non-homogeneousequations with m variablesover Q + : r 1 + r 2 + + r m =1 z t 1 ( h 1 ) r 1 + z t 2 ( h 1 ) r 2 + + z t m ( h 1 ) r m =0 z t 1 ( h 2 ) r 1 + z t 2 ( h 2 ) r 2 + + z t m ( h 2 ) r m =0 ... ... ... ... z t 1 ( h n ) r 1 + z t 2 ( h n ) r 2 + + z t m ( h n ) r m =0 : (3.3.3) AnecientwaytosolvethissystemisbytheGauss-Jordaneli mination,which transformstheaugmentedmatrixofsystem(3.3.3)intother ow-echelonform.The computationalcomplexityofsolvingthissystemwiththeai dofGauss-Jordaneliminationis O ( m 2 n ). FromExample3.1.2andExample3.2.18itcanbeseenthatfors atisableand weaklysatisablepots(butnotnecessarilystronglysatis ablepots)vectorsofthe spectrummayhavezerocoordinates.Ifthespectrumofstron glysatisablepottypes isasingleton,thenallofitscoordinatesarepositivenumb ers(Proposition3.3.2). Denition3.3.1. Let A beasetof n dimensionalvectors.The support of A istheset supp ( A )= f i 2 [ n ]: thereexistsavector u =( u 1 ;u 2 ;:::;u n ) 2 A suchthat u i 6 = 0 g .Inotherwords,if i= 2 supp ( A ) ,thenthe i th coordinateofeverypointin A is0. Proposition3.3.2. Suppose S ( P ) isthespectrumofagivenpottype P with j P j = m a) supp ( S ( P ))=[ m ] ifandonlyif P isstronglysatisable. b) ;6 = supp ( S ( P )) ( [ m ] ifandonlyif P isweaklysatisablebutnotstrongly satisable. Proof. a) supp ( S ( P ))=[ m ]ifandonlyifeverytilehasapositiveprobabilityof beingonacompletecomplex,i.e.,everytiletypeoccureson acompletecomplex, 36

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whichmeansthatthepottypeisstronglysatisable. b) supp ( S ( P )) ( [ m ]ifandonlyifthereisacoordinatethatiszeroineveryvect or ofthespectrum,i.e.,atleastonetiletypecannotbeembedd edintoacomplete complex,sothepotisnotstronglysatisable,butitisweak lysatisablesince ithasanonemptyspectrum. Denition3.3.3. Let P ( H; )= f t 1 ;:::; t m g beapottype.The m dimensional vectors l h =( l 1 ( h ) ;l 2 ( h ) ;:::;l m ( h )) suchthatsuchthat l t ( h )= 8<: 1 if t ( h ) 1 or t ( b h ) 1 0 otherwise arecalledstickyendsvectors Wedenoteby l h [ i ]the i th coordinateofthevector l h Proposition3.3.4. Classicationofpottypesintoweaksatisability,satis ability andstrongsatisabilityisinPTIME.Proof. Let P beapottypewith m tiletypesand n stickyendtypes.Inorder toobtainthespectrumforthegivenpotweneedtosolvesyste m(3.3.3)of n +1 equationswith m variables.Ifthereisasolutionwithallpositivecoordina tes,then thespectrumisnonemptyandfromProposition3.3.2itfollo wsthatthepotisstrongly satisable.Ifthereisasolutiontothesystem(3.3.3)and supp ( S ( P )) ( [ m ],then thespectrumisnonemptyandfromPropositions3.2.9and3.3 .2followsthatthepot isweaklysatisablebutnotstronglysatisable.Therefor eweaksatisabilityand strongsatisabilityareinPTIME. Nowsupposethatthepotisweaklysatisablebutnotstrongl ysatisable,i.e., supp ( S ( P )) ( [ m ]. Considerthestickyendvectors l h =( l t ( h ): t 2 P )for P .Ifthereexistsasticky end h 2 H suchthat l h [ i ]=0forall i 2 supp ( S ( P )),then h couldnotbeembedded 37

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intoanycompletecomplex,hencethepotisweaklysatisabl e,butnotsatisable (otherwisethepotissatisable).Consequently,if supp ( S ( P )) ( [ m ]andthereexists an h 2 H forwhich m X i =1 l t i ( h ) s i =0(where S ( P )= f ( s 1 ;s 2 ;:::;s m ) g ),thenthepot typeisweaklysatisable.Tocheckthatoneneedstoformthe dotproductsbetween the l vectorsandthespectrum.Ifoneofthedotproductis0,thent hepottypeis notsatisable. Hence,thecomputationalcomplexityforclassifyingpotty pesaccordingtotheir typeifsatisabilityis O ( m 2 n )+ O ( mn )= O ( m 2 n ),i.eitisinPTIME. Corollary3.3.5. Apottype P ( H; ) issatisableifandonlyifforevery h 2 H l h [ i ]=1 forevery i 2 supp ( S ( P )) Example3.3.6. ThepottypesgiveninFigure2.8havesamespectrum S ( P )= 1 2 ; 1 2 ; 0 ; 0 (therefore supp ( P )= f 1 ; 2 g ),althoughoneofthem(Figure2.8a))is satisable,whiletheother(Figure2.8b))isonlyweaklysa tisable.Proposition 3.3.4helpstoclassifythem. ForthepottypegiveninFigure2.8a)the l vectorsare: l a =(1 ; 1 ; 1 ; 0), l b = (1 ; 1 ; 0 ; 1).Thedotproductsbetweenthespectrumandthe l vectorsare: S ( P ) l a =1, and S ( P ) l b =1.Therefore,thispottypeissatisable. ForthepottypegiveninFigure2.8b)the l vectorsare: l a =(1 ; 1 ; 1 ; 1), l b = (1 ; 1 ; 0 ; 0),and l c =(0 ; 0 ; 1 ; 1).Thedotproductsbetweenthespectrumandthe l vectorsare: S ( P ) l a =1, S ( P ) l b =1,but S ( P ) l c =0.Therefore,thispottypeis notsatisable. 3.4MapleProgram WewroteaMapleProgramthatcomputesthespectrumofagiven pottype,the support,andthatclassifythepottypeinoneofthefourclas ses.Theprogramis givenintheAppendixA.Todescribetheprogram,wewillgive anexampletheway programworksfortheFigure3.2a).For m weinputthenumberoftiletypes,and 38

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for n weinputthenumberofstickyendtypes.Firstweformamatrix a =[ a s;t ]isa matrixofsize( n +1) ( m +1),where a s;t = z t s mbh t 1 )for s =2 ;:::;n +1, t 2 [ m ], a 1 ;i =1for i 2 [ m +1], a i;m +1 =0for i =2 ;:::;n +1.Wealsoneedamatrix L =[ l s;t ] ofsize n m forthestickyendsvectors,i.e., l s;m =1if t m ( h s ) > 0or t m ( b h s ) > 0, otherwise l s;m =0.FortheFigure3.2,thatmatricesare: a = 26664 11112 100 11 30 37775 ; L = 24 110111 35 : TheMapleprogramcalculatesthesupportofthepot,calcula testhespectrumof thepotandclassifythepottypes. 39

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4GraphofaPotwithDNAcomplexes 4.1Introduction Inthischapterweaddressseveralstructuralquestionsabo utassembledcomplexes. Someinclude:Whatkindofcomplexescanresultfromagivenp ottype?Couldtwo dierentpottypeshavethesamesetofcompletecomplexes?W hatkindofrelations couldbedenedonthesetofpottypes?Tohelpanswertheseas wellasother questions,weapproachself-assemblyfromgraphtheoretic alpointofview. ToeverytiletypefromapotofDNAmoleculesweassignalabel edmultigraph, andweassignalabeledmultigraphtothepotofDNAmolecules andtoeverycomplex whichcanbeproducedfromself-assembly.Themainideaisto classifythetypeof complexesthatcanappearinagivenpot. Wecomparetwopotstypesaccordingtotheirtiletypesandac cordingtothe completecomplexesthatcanbeassembled.Firstwegiveden itionsthatwewilluse throughthechapter.Denition4.1.1. Alabeledmultigraph G isaquadruple G =( V;E;l;L ) ,where V isthesetofvertices, E isthesetofedges, L isthesetoflabels,and l isthelabeling function l : E L thatassignslabeltoeveryedge.Toeveryedge e 2 E ,weassign thevertexsetof e vs ( e ) ,denedasthesetoftwoverticesincidenttotheedgeand thelabel l ( e ) Denition4.1.2. Foragivenmultigraph G =( V;E;l;L ) ,thedegreeofavertex w 2 V withrespecttothelabel a ,isdenedas deg( w;a )= jf e 2 E : l ( e )= a and w 2 vs ( e ) gj : 40

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Denition4.1.3. Thelabeledmultigraph G 1 =( V 1 ;E 1 ;l 1 ;L 1 ) ishomomorphic to thelabeledmultigraph G 2 =( V 2 ;E 2 ;l 2 ;L 2 ) ifthereexistsahomomorphism h : V 1 [ E 1 [ L 1 V 2 [ E 2 [ L 2 suchthat h j V 1 = h V : V 1 V 2 h j E 1 = h E : E 1 E 2 and h j L 1 = h L : L 1 L 2 issuchthat vs ( h E ( e ))= h V ( vs ( e )) forevery e 2 E 1 ,i.e.,for u;w 2 V 1 and e 2 E 1 f h V ( u ) ;h V ( w ) g = vs ( h E ( e )) whenever f u;w g = vs ( e ) ,and l ( h E ( e ))= h L ( l ( e )) Toeasethenotation,forthehomomorphismsbetweenthegrap hs G 1 and G 2 we willusethenotation h : G 1 G 2 4.2DenitionofaPotGraph Denition4.2.1. Let P ( H; ) beapottype.Denethepotgraph G P of P ( H; ) as alabeledmultigraph G P =( V;E;l;H + ) asfollows.Let V = f v t : t 2 P g bethesetof vertices, E bethesetofedges,and l : E H + bethesetoflabelswiththefollowing proviso:foreachstickyendtype h 2 H + andeachpairoftiletypes s ; t 2 P ( H; ) s ( h ) > 0 and t ( b h ) > 0 ,thereexistsanedge e 2 E with vs ( e )= f v s ;v t g and l ( e )= h Remark4.2.2. Foreach s ; t 2 P ( H; ) andeach h 2 H + ,inthepotgraph G P there isatmostoneedgeconnecting v s v t withlabel h Notethatwhenthepottypeis P = f t 1 ; t 2 ;:::; t m g ,thecorrespondingpotgraph, thesetofverticesisdenedas V = f v i : t i 2 P g .Andsinceitknownthatthesetof labelsis H + ,wewillomititfromthedenitionofthepotgraph,i.e.,fro mnowon thepotgraphwillbedenotedas G P =( V;E;l ). Example4.2.3. Figure4.1isanexampleofapottypeanditspotgraph. 41

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3 a b c a c b a b c 2 4 v v v 1 2 a a a a b b b b c c c c 1 3 t t tt v P ac b 4 Figure4.1:Pottype P anditspotgraph Itfollowsfromtheabovedenitionsthatforagiventile t 2 P ,if t ( h ) > 0for h 2 H + thenitscorrespondingvertex v t willbeadjacenttoallverticesofdierent tiletypes v s forwhich s ( b h ) > 0,orif t ( b h ) > 0itscorrespondingvertex v t willbe adjacenttoallverticesofdierenttiletypes v s forwhich s ( h ) > 0.Thereforefor each h 2 H + deg( v t ; h )= X s 2 P I s ( b h ) I t ( h )+ X s 2 P I s ( h ) I t ( b h ) ; where I t ( h )= 8<: 1 t ( h ) > 0 0otherwise : Fromtheassumptionthatthereisnotilewithcomplementary stickyendsoftypes h and b h ,forevery h 2 H ,intheabovedenition,either I t ( h )=0or I t ( b h )=0. Foragivenpottype P anditscorrespondingpotgraph G P ,foreach t 2 P deg( v t ; h )doesnotdependonthenumberofstickyendtypesoftype h on t ,i.e., doesnotdependon j z t ( h ) j .ThepottypeinFigure4.2isstronglysatisable,and j z t 1 ( h ) j < deg( v t 1 ; h ), j z t 2 ( h ) j =deg( v t 2 ; h ),and j z t 3 ( h ) j > deg( v t 3 ; h ). 42

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t 1 v 2 v 3 h h 2 1 tt b) a) h h h h 3 v Figure4.2:a)Apottype P = f t 1 ; t 2 ; t 3 g .b)Thepotgraphcorrespondingtothepottype P givenina) Remark4.2.4. Noteverylabeledmultigraphisapotgraph.Thefollowingex ample conrmsthat. 1 v 4 v 3 v 2 v h h h Figure4.3:Labeledgraphthatisnotapotgraph. Withoutlossofgeneralityletusassumethat t 1 ( h ) > 0 ,fromthegivenpottype. Sincethereisanedgewithvertexset f v 1 ;v 2 g labeled h ,andthereisanedgewith vertexset f v 1 ;v 3 g labeled h ,andsincethereisnoedgewithvertexset f v 1 ;v 4 g labeled h ,wecanconcludethat t 2 ( b h ) > 0 t 3 ( b h ) > 0 ,and t 4 ( h ) > 0 .Thatmeansthat thereshouldexistsanedgewithvertexset f v 2 ;v 4 g labeled h andanedgewithvertex set f v 3 ;v 4 g labeled h .Sincethereisnoedgewithvertexset f v 2 ;v 4 g labeled h ,this multigraphcannotbeapotgraph. WiththefollowingPropositionweclassifythepotgraphs,w egiveanecessary andsucientconditionforagraphtobeapotgraph.Proposition4.2.5. Alabeledmultigraph G =( V;E;l ) ( l : E H + )isapotgraph 43

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ifandonlyifforevery h 2 H + thesubgraphspannedbytheedgeswithlabel h isa completebipartitegraph.Proof. Let G P =( V;E;l )beapotgraph.Forevery h 2 H + constructtwosubsetsof V V h = f v t : t 2 P ; t ( h ) > 0 g and b V h = f v t : t 2 P ; t ( b h ) > 0 g .Bytheassumption thatnotilehasstickyendsoftype h and b h atthesametime, V h and b V h aredisjoint. In G P everyvertexof V h isincidenttoeveryvertexof b V h byanedgelabeled h and therearenootheredgeslabeled h in G P .Notwoverticesof V h (or b V h )areadjacent inthesubgraphspannedbytheedgeswithlabel h .Therefore,ifwedenotewith E h = f e : e 2 E and l ( e )= h g ,thenthegraph G =( V h [ b V h ;E h )isacomplete bipartitegraph. Conversely,supposethat G =( V;E;l )isalabeledgraph(withnitesetofvertices andedges)andforevery h 2 H + thesubgraphspannedbytheedgeswithlabel h isacompletebipartitegraph G h =( V h [ b V h ;E h ),where V h \ b V h = ; .Wedenea pottype P thathasasmanytiletypesasverticesin V .Sincethesetofvertices isnite,say j V j = m ,wecannumbertheverticesof V = f v i : i 2 [ m ] g .Toevery vertex v i 2 V weassignatiletype t i denedinthefollowingway:forevery h 2 H + if v i 2 V h then t i ( h )=deg( v i ; h ),andif v i 2 b V h then t i ( b h )=deg( v i ; h ),otherwise t i ( h )= t i ( b h )=0.Considerthepotgraph G P =( V P ;E P ;l P )of P .Thesetofvertices forthepotgraphis V P = f v i : t i 2 P ;i 2 [ m ] g ,therefore V P = V .Forevery h 2 H + if t i ( h ) 1and t j ( b h ) 1,therewillbeanedge e 2 E P with vs ( e )= f v i ;v j g and l ( e )= h .Fromthedenitionofthesets V h and b V h itfollowsthatwhere v i 2 V h and v j 2 b V h ,i.e., e 2 E h .Hence, E P E h Supposeastickyend h 2 H + isgiven,andanedge e 2 E h with vs ( e )= f v i ;v j g Since G h isacompletebipartitegraphitwillbethecasethat v i 2 V h and v j 2 b V h or v i 2 b V h and v j 2 V h .Fromthedenitionofthepotitfollowsthat t i ( h ) 1and t j ( b h ) 1,or t i ( b h ) 1and t j ( h ) 1,i.e.,thereexistsanedgein G P labeled h incidentto v i to v j ,i.e, e 2 E P .Hence E h E P ,fromwhereitfollowsthat G = G P 44

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Supposewearegivenacomplex C = h T;S;J i ,where T isthesetoftiles, S isthesetofstickyends,and J isthesetofconnections.Wedeneafunction type : T [ S P [ H suchthat type ( h )= h ifthestickyend h 2 S isoftype h 2 H and type ( t )= t isthetile t isoftype t 2 P .Besidesassigningpotgraphstoagiven pottype,wealsoassigngraphstocomplexes.Denition4.2.6. Let C = h T;S;J i beacompletecomplexof C ( P ) .Denethe completecomplexgraph G C of C asamultilabeledmultigraph G C =( V C ;E C ;l C ) ,as follows. V C = f v t : t 2 T g isthesetofvertices, l C : E C H + isthesetoflabels. Thesetofedges, E C ,isdenedsuchthatforeveryconnection c = f ( t;h ) ; ( t 0 ; b h ) g2 J for t;t 0 2 T ,type ( h )= h 2 H + ,type ( b h )= b h 2 H ,thereexistsanedge e 2 E C with vs ( e )= f v t ;v t 0 g and l ( e )= h Fromthedenitionofacompletecomplexgraphitfollowstha tforevery t 2 T deg( v t ; h )= j z type( t ) ( h ) j Example4.2.7. Severalelementsfromthesetofcompletecomplexgraphsari sen fromthepottypegiveninExample4.1aregiveninFigure1.4. v v v v 1 2 3 4 b a b a c c v v v v 1 2 3 4 a b c a b c c v v v v 1 2 3 4 c a b a b v v v v 1 2 3 4 b b a c a c v v v v 1 2 3 4 a a b c b c v v v v 1 2 3 4 b a b a c c v v v v 1 2 3 4 a a c c b b Figure4.4:Elementsfromthesetofcompletecomplexgraphs forthepottypegivenin Example4.1Denition4.2.8. Let C = h T;S;J i beacomplexover P ( H; ) .Denethecomplex graph G C of C asamultilabeledmultigraph G C =( V C ;E C ;l C ) withnitesetofvertices andedges.Thesetofvertices V C ispartitionedintoadisjointunion V C = V T [ 45

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V H ,where V T = f v t : t 2 T g and V H = f v h : 9 t 2 T forwhich t ( h ) > 0 ;h 2 S isoftype h 2 H and ( t;h ) = 2 c forany c 2 J g .Thesetofedges E C ,whichisalso partitionedintoadisjointunion E C = E T [ E T;H ,togetherwiththelabelingfunction l C : E C H + aredenedsuchthat -foreveryconnection c = f ( t;h ) ; ( t 0 ; b h ) g2 J for t;t 0 2 T ,type ( h )= h 2 H + and type ( b h )= b h 2 H ,thereexistsanedge e 2 E T with vs ( e )= f v t ;v t 0 g ,and l ( e )= h -forevery t 2 T suchthat t ( h ) > 0 and ( t;h ) = 2 c forsomestickyend h 2 S oftype h 2 H and c 2 J ,thereexistsanedge e 2 E T;H with vs ( e )= f v t ;v h g ,and l ( e )= h if h 2 H + or l ( e )= b h if h 2 H Remark4.2.9. Everycompletecomplexgraphisacomplexgraphforwhichthe set ofedges V H isempty. Remark4.2.10. Thetilegraph, G t ,correspondingtoatile t ,isacomplexgraphwith V T = f v t g and V H = f v h : type ( h )= h ;t ( h ) > 0 g E T = ; andforevery h 2 H + Aswementionedpreviously,themainmotiveformodelingpot typeswithpot graphswastostudytheoutcomesoftheprocessofself-assem blywithtoolsthatwe arefamiliarwithandtoolsthatcanhelpusinunderstanding oftheprocess.The nextsubsection,weshowthatthedenitionsusedforpotgra phsandcomplexgraphs areverynatural,andthatcanbeveryeasilyestablishedhom omorphismbetweena complexgraphandpotgraphofasamepottype. 4.3Homomorphisms Inthissection,weshowthatthegraphofeverycompletecomp lexfromagivenpot type P ( H; )ishomomorphictothepotgraphof P ( H; ).Thedenitionofequivalent andsimilarpottypesaregiveninthissection,andweshowth atequivalentpottypes haveisomorphicpotgraphs,andequivalentsetsofcomplexg raphs.Attheendwe showthatforeverypottype P thereexistsasimilarpottype e P ( P and e P have isomorphiccomplexgraphs)whosetiletypeshavestickyend sofdistincttypes. 46

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Lemma4.3.1. Let G P =( V P ;E P ;l P ) bethepotgraphofthepot P ( H; ) andlet G C = ( V C ;E C ;l C ) bethecompletecomplexgraphofacompletecomplex C = h T;S;J i2 C ( P ) .Thereexistsahomomorphism : G C !G P denedinthefollowingway: V ( v t )= v type ( t ) ,for v t 2 V C l ( E ( e ))= l ( e ) ,for e 2 E C Proof. iswelldened: If v t = v t 0 ,for v t ;v t 0 2 V C then t and t 0 mustbeofthesametype,therefore V ( v t )= V ( v t 0 ). If e = e 0 ,for e;e 0 2 E C ,then vs ( e )= vs ( e 0 )and l ( e )= l ( e 0 ).Since isa homomorphismand V iswelldeneditfollowsthat V ( vs ( e ))= V ( vs ( e 0 ))and l ( E ( e ))= l ( e )= l ( e 0 )= l ( E ( e 0 ))i.e., vs ( E ( e ))= vs ( E ( e 0 ))and l ( E ( e ))= l ( E ( e 0 )).ByRemark4.2.2,therearenotwoedgesin G P withthesamevertex setandsamelabel.Therefore, E ( e )= E ( e 0 ). ishomomorphism:Let e 2 E C with vs ( e )= f v t ;v t 0 g and l ( e )= h .There existsaconnection c = f ( t;h ) ; ( t 0 ; b h ) g2 J ,where h isastickyendoftype h 2 H + b h isastickyendoftype b h 2 H t isatileoftype t 2 P and t 0 is atileoftype t 0 2 P .Withoutlossofgeneralitywemayassumethat t ( h ) > 0 and t 0 ( b h ) > 0,fromwhereitfollowsthat t ( h ) > 0and t 0 ( b h ) > 0i.e.,by thedenitionofthepotgraphitfollowsthatthereisanedge e P 2 E P with vs ( e P )= f v t ;v t 0 g = f V ( v t ) ;' V ( v t 0 ) g and l ( E ( e ))= l ( e P )= h = l ( e ). Example4.3.2. ConsiderthepotgraphgiveninFigure4.1andonecompleteco mplexgraphfromtheFigure4.4,saytheonegivenintheFigure 4.5. 47

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3 v v v v 1 2 3 4 b a b a c c j 1 a b b c b c v v 2 c a a a b c v v 4 Figure4.5:Acompletecomplexgraphandapotgraph Thehomomorphism : G C !G P ,denedas: ( v 1 )= v 1 ( v 2 )= v 2 ( v 3 )= v 4 ( v 4 )= v 3 ,and ( e )= e for e 2f a;b;c g ,mapsthecompletecomplexgraphfrom Figure4.5inthepotgraphinFigure4.1. Thefollowingpropositionisacharacterizationofcomplet ecomplexes,i.e.,wegive necessaryandsucientconditionforagraphtobeacomplete complexgraph. Proposition4.3.3. Supposeapottype P ( H; ) (withitspotgraph G P =( V P ;E P ;l P ) ) andalabeledmultigraph G =( V;E;l ) ( l : E H + )aregiven.Ifthereexists ahomomorphism : G !G P satisfying: V ( v )= v t ifandonlyif deg( v; h )= j z t ( L ( h )) j ,then G isacompetecomplexgraphforthatpot. Proof. Supposeapottype P = f t 1 ; t 2 ;:::; t m g andalabeledmultigraph G =( V;E;l ) aregiven.Let V : V V P beahomomorphismsuchthat, V ( v )= v t ifandonly ifdeg( v; h )= j z t ( L ( h )) j forevery v 2 V Forevery v t j 2 V ( V ), j 2 [ m ],thereexistsaset f w t j 1 ;w t j 2 ;:::;w t j k g2 V such that V ( w t j i )= v t j for i 2 [ k ].Let T j = f t jw 1 ;t jw 2 ;:::;t jw k g besetsoftiles,suchthat alltilesin T j areoftype t j ,for v t j 2 V ( V )and j 2 [ m ]. Next,weconstructacomplex C = h T;S;J i ,wherethesetoftilesis T = [ v t j 2 V ( V ) T j whilethesetofconnections, J ,isdenedinthefollowingmanner.To everyedge e 2 E with vs ( e )= f w t i r ;w t j s g and l ( e )= type ( h )weassociateasticky end h e 2 S suchthatwhere type ( h e )= type ( h ).Toeveryedge e 2 E with vs ( e )= f w t i r ;w t j s g and l ( e )= type ( h )weassociateaconnection c = f ( t iw r ;h ) ; ( t jw s ;h 0 ) g2 J where type ( h 0 )= ( type ( h )).Fromthedenitionofthehomomorphismitfollows 48

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thatthereareasmanyconnectionsbetween t iw r and t jw s viastickyendtype h asthe edgesbetween w t i r and w t j s labeled 1 L ( h )(notethat L isabijectionthatmaps H + into H + ).Becauseofthehomomorphism,if e 2 E with vs ( e )= f w t i r ;w t j s g thereis anedge e P 2 E P with vs ( e P )= f v t i ;v t j g labeled l ( E ( e ))inthepotgraph G P .From thedenitionofthepotgraphitfollowsthat t i ( l ( E ( e ))) > 0and t j ( ( l ( E ( e )) > 0 (or t i ( ( l' E ( e ))) > 0and t j ( l ( E ( e )) > 0),i.e., t iw r ( l ( e )) > 0and t jw s ( ( l ( e )) > 0(or t iw r ( ( l ( e ))) > 0and t jw s ( l ( e )) > 0). If C isnotacompletecomplex,thenthereexistsatile t kw l 2 T andastickyend h oftype h 2 H suchthat t k ( h ) > 0,but( t kw l ;h ) = 2 c forany c 2 J ,i.e.,forthat h w t k l doesnothaveincidentedge.Thereforedeg( w t k l ; h ) < j z t k ( h ) j ,whichisnotpossible. So C isacompletecomplex. Nextwewillshowthat G isisomorphictothecompletecomplexgraphof C Suppose G C =( V C ;E C ;l C )isthecompletecomplexgraphofC.Then V C = f v t jw i : t jw i 2 T g and e C 2 E C ,with vs ( e C )= f v t iw r ;v t jw s g and l ( e C )= type ( h ),ifandonlyif f ( t iw r ;h ) ; ( t jw s ;h 0 ) g2 J Wedeneafunction : V V C by ( w t j i )= v t jw i .Bytheconstructionofthe complex C itfollowsthat V C = V ,so isabijection.Itishomomorphismbecause: e 2 E C with vs ( e )= f v t iw r ;v t jw s g and l ( e )= h ifandonlyif f ( t iw r ;h ) ; ( t jw s ;h 0 ) g2 J fortype( h )= h ifandonlyifthereexistsanedge e 0 2 E with vs ( e 0 )= f w t i r ;w t j s g and l ( e 0 )= 1 L ( h ). BythehomomorphismdenedinLemma4.3.1, G ishomomorphicto G P Thenweproceedtodenewhentwopottypesareisomorphic.In ordertodo thatrstwedeneisomorphismbetweentwotiletypes(onefr omeachpottype)and stickyendtypes,andthendenewhentwopotsareequivalent .Twopottypesare equivalentifthereisabijection betweenthesetsofstickyendtypesandabijection betweenthepottypes,sothatifatiletype t ismappedtoatiletype t 0 ,thenthere isabijectionfromthestickyendsof t tothoseof t 0 suchthatastickyendoftype, say h ,isassignedtooneoftype ( h ). 49

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Denition4.3.4. Let t 1 beatiletypeofthepottype P 1 ( H 1 ; 1 ) and t 2 beatiletype ofthepottype P 2 ( H 2 ; 2 ) .Thetiles t 1 and t 2 areisomorphic (notation: t 1 = t 2 )if thereexistsagraphisomorphism fromthetilegraph G t 1 tothetilegraph G t 2 Whentwotiles, t 1 and t 2 areisomorphic,wesaythattheisomorphism between G t 1 =( V H 1 [f v t 1 g ;E 1 ;l 1 )and G t 2 =( V H 2 [f v t 2 g ;E 2 ;l 2 )preservesstickyends iffor every h 1 and h 2 ofthesamestickyendtype h 2 H 1 and ( v h 1 )= v 0 h 01 ( v h 2 )= v 0 h 02 v h 1 ;v h 2 2 V H 1 v 0 h 01 ;v 0 h 02 2 V H 2 h 01 and h 02 areofthesamestickyendtype h 0 2 H 2 ,i.e., deg( v t 1 ; h )=deg( v t 2 ; h 0 ). Example4.3.5. ConsidertheFigure4.6.Theisomorphismbetween t and t 0 preservesstickyends,whiletheisomorphismbetween t and t 00 doesnotpreservesticky ends. a c 1 t b x y z t' x x z t'' Figure4.6: t = t 0 viaanisomorphismthatpreservesstickyends. t = t 00 viaanisomorphism thatdoesnotpreservesstickyends.Denition4.3.6. Thepairs ( H 1 ; 1 ) and ( H 2 ; 2 ) areisomorphic (notation: ( H 1 ; 1 ) = ( H 2 ; 2 ) )ifthereisbijection f : H 1 H 2 suchthatthefollowingdiagramcommutes. H 1 H 2 H 1 H 2 f ? 1 ? 2 f f 1 = 2 f Werequirethediagramtocommute,becausewewantallWatson -Crickconnectionstobepreserved.Fortherestofthedissertationwex f tobetheisomorphism denedabove. 50

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Proposition4.3.7. Theisomorphismbetweenthestickyendsisanequivalencere lation.Proof. = isrerexive. Given( H 1 ; 1 ).Considertheidentitymap f : H 1 H 1 ,i.e., f ( h )= h forevery h 2 H 1 .Thenforany h 2 H 1 f ( 1 ( h ))= 1 ( h )= 1 ( f ( h )),i.e., f 1 = 1 f = issymmetric. Assume( H 1 ; 1 ) = ( H 2 ; 2 ).Thereexistsabijectivemap f : H 1 H 2 such that f 1 = 2 f .Since f isabijection,itfollowsthat f 1 : H 2 H 1 isa bijection,and 1 f 1 = f 1 1 ,i.e.,( H 2 ; 2 ) = ( H 1 ; 1 )viathemap f 1 = istransitive. Let( H 1 ; 1 ) = ( H 2 ; 2 )and( H 2 ; 2 ) = ( H 3 ; 3 ).Thereexistsabijection f : H 1 H 2 suchthat f 1 = 2 f ,andabijection g : H 2 H 3 suchthat g 2 = 3 g .Thebijection h : H 1 H 3 denedas h = g f satises h 1 = g f 1 = g 2 f = 3 g f = 3 h .Therefore( H 1 ; 1 ) = ( H 2 ; 3 ). Denoteby G ( P )= fG C : C 2C ( P ) g thesetofallcompletecomplexgraphsof P ( H; ) : DenotebyG( P )= fG C : C isacomplexover P g thesetofallcomplexgraphsof P ( H; ). Finallywedenewhentwopottypesareequivalent.Theideab ehindpottype equivalenceistotransferallthepropertiesofonepottype totheoneequivalentto it,nomatterhowdierenttiletypesorstickyendtypesthey mighthave. Denition4.3.8. Thepottypes P 1 ( H 1 ; 1 ) and P 2 ( H 2 ; 2 ) areequivalent (notation P 1 = P 2 )if ( H 1 ; 1 ) = ( H 2 ; 2 ) viatheisomorphism f ,andthereisabijection : P 1 P 2 suchthatforevery t 2 P 1 G t = G ( t ) viaauniqueisomorphism that preservesstickyendtypes,i.e.,forevery h 2 H 1 L ( h )= f ( h ) 51

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Fortheabovedenition,theisomorphismtopreservesticky endtypesmeans:if P 1 ( H 1 ; 1 ) = P 2 ( H 2 ; 2 )viabijection ,andfor G t =( v t [ V H 1 ;E t ;H 1 ;l t ), G t = G ( t ) viaisomorphism ,thenforevery v h 2 V H 1 ( v h )= v 0 h 0 2 V H 2 suchthat type ( h 0 )= f ( h ) 2 H 2 whenever type ( h )= h 2 H 1 ,and ( v t )= v 0 t 0 ,where type ( t 0 )= ( t )and G ( t ) =( v 0 t 0 [ V H 2 ;E ( t ) ;H 2 ;l ( t ) ). Example4.3.9. LetusshowthatthepottypesgiveninFigure4.7areequivale nt. Theyaredenedasfollows: P 1 = f t 1 ; t 2 g H 1 = f a;b; 1 ( a ) ; 1 ( b ) g ; P 2 = f t 01 ; t 02 g H 2 = f x;y; 2 ( x ) ; 2 ( y ) g .Thecorrespondingtilegraphsaredenedasusual. 2 a 1 1 t a b t q ( ) b 2 P 1 1 x y q ( ) q ( ) y x 1 t' t' 2 P 2 2 q ( ) Figure4.7:Thepottypes P 1 and P 2 areequivalent First,( H 1 ; 1 ) = ( H 2 ; )viatheisomorphism f : H 1 H 2 ,denedas f ( a )= x f ( b )= 2 ( y ), f ( 1 ( a ))= 2 ( x ),and f ( 1 ( b ))= y .Obviously, f 1 = 2 f ,since f ( 1 ( a ))= 2 ( x )= 2 ( f ( a ))and f ( 1 ( b ))= y = 2 ( 2 ( y ))= 2 ( f ( b )). Second,thereexistsabijection : P 1 P 2 denedas ( t 1 )= t 01 and ( t 2 )= t 02 Third, G t 1 = G t 01 and G t 2 = G t 02 viatheisomorphism denedas ( v 1 )= v 0 1 ( v 2 )= v 0 2 ( a )= x ( 1 ( a ))= 2 ( x ), ( b )= 2 ( y ),and ( 1 ( b ))= y Denition4.3.10. Forthepottypes P 1 ( H 1 ; 1 ) and P 2 ( H 2 ; 2 ) ,thesetsofcomplete complexgraphsareequivalent G ( P 1 ) = G ( P 2 ) ,ifthereexisttwofunctions 1 : G ( P 1 ) G ( P 2 ) and 2 : G ( P 2 ) G ( P 1 ) suchthatforevery G C 1 2 G ( P 1 ) 1 ( G C 1 ) = G C 1 ,andforevery G C 2 2 G ( P 2 ) 2 ( G C 2 ) = G C 2 Hereweneedtopointoutthatintheabovedenitionthecompl etecomplexgraphs arenotnecessarilyisomorphicwithauniqueisomorphism.F orexampleif G C 1 and 52

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G C 2 aretwocompletecompletegraphsfrom G ( P 1 ),thenwecouldhave 1 ( G C 1 ) = G C 1 viaanisomorphism 1 and 1 ( G C 2 ) = G C 2 viaanotherisomorphism 2 Example4.3.11. ConsiderthepottypesgiveninFigure4.8. y t 3 t 4 t 5 t a b x a b y 3 t' 4 t' x 5 t' 6 t' a b t 2 P 1 1 t' t' 2 P 2 x a x a b 1 Figure4.8:Twopottypes P 1 and P 2 withequivalentsetsofcompletecomplexgraphs Theircorrespondingsetsofcompletecomplexgraphs G ( P 1 )= f G ( C 1 ) ;G ( C 2 ) g and G ( P 2 )= f G ( C 0 1 ) ;G ( C 0 2 ) g aregiveninFigure4.9.Thegraphs G ( C 1 )and G ( C 0 1 ) areisomorphicviatheisomorphism 1 : G ( C 1 ) G ( C 0 1 )denedas 1 ( v 1 )= v 0 1 1 ( v 3 )= v 0 3 ,and 2 ( v 4 )= v 0 4 .Thegraphs G ( C 2 )and G ( C 0 2 )areisomorphicvia theisomorphism 2 : G ( C 2 ) G ( C 0 2 )denedas 1 ( v 2 )= v 0 2 1 ( v 3 )= v 0 5 ,and 2 ( v 5 )= v 0 6 .Obviously,theisomorphisms 1 and 2 aredierent,butthesetsof completecomplexgraphsareequivalent. 53

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1 v 3 v 4 v a b 3 v 2 v 5 v a x G (C ): 1 G (C ): 2 2 v' 5 v' 6 v' 1 v' 3 v' 4 v' x y a b G (C' ): 2 G (C' ): 1 Figure4.9:Thesetsofcompletecomplexgraphsforthepotty pesgiveninFigure4.8 Althoughthepottypes P 1 and P 2 haveequivalentsetsofcompletecomplex graphs,theyarenotequivalent. Similarlywecandenewhenthesetsoftwocomplexgraphsare isomorphic. Denition4.3.12. Forthepottypes P 1 ( H 1 ; 1 ) and P 2 ( H 2 ; 2 ) ,thesetsofcomplex graphsareequivalent G ( P 1 ) = G ( P 2 ) ,ifthereexisttwofunctions 1 : G ( P 1 ) G ( P 2 ) and 2 : G ( P 2 ) G ( P 1 ) suchthatforevery G C 1 2 G ( P 1 ) 1 ( G C 1 ) = G C 1 andforevery G C 2 2 G ( P 2 ) 2 ( G C 2 ) = G C 2 Also,inthisdenition,itisnotnecessarilytruethattwoc omplexgraphsare isomorphicviaasingleisomorphism,butratherseveraldi erentones. Proposition4.3.13. Ifthepottypes P 1 ( H 1 ; 1 ) and P 2 ( H 2 ; 2 )areequivalent,then thecorrespondingsetsofcomplexgraphsareequivalent,i. e. P 1 = P 2 ) G ( P 1 ) = G ( P 2 ) : Proof. Let P 1 = P 2 ,so( H 1 ; 1 ) = ( H 2 ; 2 )(i.e,thereexistsabijection f : H 1 H 2 suchthat f 1 = 2 f )andthereexistsanbijection : P 1 P 2 s.t.forevery t 2 P 1 G t = G ( t ) viaanisomorphism Supposethetiletypes s and t from P 1 havecomplementarystickyendoftype h 2 H 1 suchthat t ( h ) > 0and s ( 1 ( h )) > 0.Considerthetilegraphs G t and G s from 54

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G ( P 1 )andtheirequivalenttilegraphs G ( t ) and G ( s ) .Thereexistsanedge e t 2G t with vs ( e t )= f v t ;v h g type ( h )= h ,andanedge e s 2G s with vs ( e s )= f v s ;v b h g ,where type ( b h )= 1 ( h ).Because G t = G ( t ) and G s = G ( s ) ,thereexistsanedge e 0t 2G ( t ) with vs ( e 0t )= f v 0 t 0 ;v 0 h 0 g type ( t 0 )= ( t ), type ( h 0 )= f ( h ),andanedge e 0s 2G ( s ) with vs ( e 0s )= f v 0 s 0 ;v 0 b h 0 g type ( s 0 )= ( s ), type ( b h 0 )= 2 ( f ( h )).Fromtheconstructionof thetilegraphsfollowsthat ( t )( f ( h )) > 0and ( s )( 2 ( f ( h ))) > 0,i.e.,thetiletypes ( t )and ( s )from P 2 canconnectviathestickyend f ( h ).Withotherwordsiftiles oftypes t and s canconnectviaastickyendtype h ,thentilesofthetypes ( t )and ( s )canconnectviathestickyendoftype f ( h ). Nextwewillshowthattoeverycomplexgraphfrom G ( P 1 ),thereexistsaunique complexgraphin G ( P 2 )isomorphictoit. Let G C =( V T [ V H ;E T [ E T;H ;l ) 2 G ( P 1 ),thenthereexistsacomplex C = h T;S;J i of P 1 s.t. G C isacomplexgraphof C .Wecanconstructacomplex C 0 = h T 0 ;S 0 ;J 0 i of P 2 thatisisomorphicto C C 0 hastilesoftypesthatareisomorphic tothetiletypesin P 1 suchthatthenumberoftilesin T oftype t isequaltothe numberoftilesin T 0 oftype ( t ) 2 P 2 .Hence,wecandeneabijection g : T T 0 g ( t )= t 0 if ( t )= t 0 ,where type ( t )= t and type ( t 0 )= t 0 .Thecomplex C 0 ,foreach h 2 H ,thenumberofstickyendsoftype h onthetilein T isequaltothenumberof stickyendsoftype f ( h )onthetilesin T 0 .Wecandeneabijection k : S S 0 such that k ( h )= h 0 if type ( h )= h type ( h 0 )= f ( h ),andif t ( h )=1,then g ( t )( k ( h ))=1. Thesetofconnectionsfor C 0 isdenedas J 0 = f ( t 0 ;h 0 ) ; ( s 0 ; b h 0 ) g : type ( h 0 )= h 0 ;type ( b h 0 )= 2 ( h 0 ) ; f ( g 1 ( t 0 ) ;k 1 ( h 0 )) ; ( g 1 ( s 0 ) ;k 1 ( b h 0 )) g2 J; where type ( k 1 ( h 0 ))= f 1 ( h 0 )and type ( k 1 ( b h 0 ))= f 1 ( 2 ( h )) g .(Theabovediscussionfollowsthedenition ofthesetofconnectionsfor C 0 .) Thereforethecomplexgraphfor C 0 G C 0 =( V T 0 [ V H 0 ;E T 0 [ E T 0 ;H 0 ;l C 0 ),isdenedasthefollowing: V T 0 = f v 0 g ( t ) : v t 2 V T g V H 0 = f v 0 k ( h ) : v h 2 V H and v h 2 vs ( e )for e 2 E T;H g .Thesetofedgesandthesetoflabelsaredenedas e 0 2 E T 0 with vs ( e 0 )= f v 0 s ;v 0 t g andlabel l C 0 ( e )ifandonlyifthereexistsandedge e 2 E T with 55

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vs ( e )= f v g 1 ( s ) ;v g 1 ( t ) g and f ( l C ( e ))= l C 0 ( e 0 ).Anedge e 0t with vs ( e 0t )= f v 0 t 0 ;v 0 h 0 g (type( h )= h )andlabel l C 0 ( e 0t )isin E T 0 ;H 0 ifandonlyifthereexistsanedge e t 2 E T;H with vs ( e t )= f v g 1 ( t 0 ) ;v k 1 ( h 0 ) g (type( h 0 )= f 1 ( h ))and f ( l C ( e t ))= l C 0 ( e 0t ).Inother words,thereexistsabijectionbetweenG( P 1 )andG( P 2 ),i.e.,toeverycomplexgraph G C 2 G( P 1 )wecanbijectivelycorrespondacomplexgraph G C 0 2 G( P 2 ).Nextwe willshowthatthesetwographsareisomorphic, G C = G C 0 Deneamap : V C V C 0 suchthat ( v t )= v 0 g ( t ) and ( v h )= v 0 k ( h ) .Fromthe constructionofthecomplexgraphitisclearthat ishomomorphism. isone-to-one:Fromtheconstruction,if ( v s )= ( v t )followsthat v 0 g ( s ) = v 0 g ( t ) ,so g ( s )= g ( t )andsince g isbijection, s = t ,i.e. v s = v t .If ( v h 1 )= ( v h 2 ),then v 0 k ( h 1 ) = v 0 k ( h 2 ) ,itfollowsthat k ( h 1 )= k ( h 2 )fromwherefollowsthat h 1 = h 2 i.e. v h 2 = v h 2 isonto:If v 0 t 2 V T 0 thenthereexists v g 1 ( t ) 2 V T s.t. ( v g 1 ( t ) )= v 0 t .If v 0 h 0 2 V H 0 thenthereexists v k 1 ( h 0 ) 2 V H s.t. ( v k 1 ( h 0 ) )= v 0 ( h 0 ) : isahomomorphism:Followsfromthedenitionof G C 0 : Weshowedthatforevery G C 2 G ( P 1 )thereexists G C 0 2 G ( P 2 )satisfying G C = G C 0 .Consequentlythereexistsamap 1 : G ( P 1 ) G ( P 2 ),denedas 1 ( G C )= G C if G C = G C 0 ,i.e., 1 ( G C ) = G C for G C 2 G ( P 1 ). Nextwewillshowthatthereexistsamap 2 : G ( P 2 ) G P 1 ,suchthat 2 ( G C 0 ) = G C 0 for G C 0 2 G ( P 2 ).Inaverysimilarmanneraspreviouslyinthisproof,wesho w thatforeverycomplexgraph G C 0 =( V T 0 [ V H 0 ;E T 0 ;E T 0 ;H 0 ;l C 0 ) 2 G ( P 2 )withcomplex C 0 from P 2 ,wecanbuildacomplex C in P 1 whosegraph, G C ,isisomorphicto G C 0 Hence 2 isdenedsuchthat 2 ( G C 0 )= G C ,i.e., 2 ( G C 0 ) = G C 0 Forthesereasons,wecanconcludethat G ( P 1 ) = G ( P 2 ). Remark4.3.14. Theconverseisnottrueingeneral.Themainreasonwhythe conversefailsisthemultipleisomorphismsallowedbetwee nthesetsofcompletecom56

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plexgraphs.ConsiderthepottypesdepictedonFigure4.10. Thepottypesarenot equivalent,whiletheirsetsofcomplexgraphsare. v 1 G P 2 G v 2 v 1 v x x v 1 v 2 v 1 v a v 2 v b v c v 3 P 1 x x 1 2 t t P 2 a b c 3 2 1 t tt P Figure4.10:Thesetsofcomplexgraphs G ( P 1 ) and G ( P 2 ) areequivalent,but P 1 P 2 Denition4.3.15. Thepottypes P 1 ( H 1 ; 1 ) and P 2 ( H 2 ; 2 )aresimilar (notation P 1 P 2 )iftheircorrespondingsetsofcompletecomplexgraphsare equivalent G ( P 1 ) = G ( P 2 ) Corollary4.3.16. Ifthestronglysatisablepottypes P 1 ( H 1 ; 1 ) and P 2 ( H 2 ; 2 )are equivalent,thentheyaresimilar. Theproofimmediatelyfollowsfromthepreviousone. Proposition4.3.17. Ifthetypes P 1 ( H 1 ; 1 ) and P 2 ( H 2 ; 2 )areequivalent,thenthe correspondingpotgraphs, G P 1 and G P 2 ,areisomorphic. Proof. Since P 1 = P 2 ,thereexistsabijection : P 1 P 2 suchthatforevery t 2 P 1 G t = G ( t ) andtheisomorphismpreservesstickyendtypes.Also,( H 1 ; 1 ) = ( H 2 ; 2 ), i.e.,thereexistsabijectivefunction f : H 1 H 2 suchthat f 1 = 2 f Letusethefollowingnotation: G P 1 =( V P 1 ;E P 1 ;l P 1 )and G P 2 =( V P 2 ;E P 2 ;l P 2 ), where V P 1 = f v t : t 2 P 1 g and V P 2 = f v 0 t : t 2 P 2 g .Since j P 1 j = j P 2 j itisclearthat j V P 1 j = j V P 2 j Deneamap r : G P 1 !G P 2 ,s.t r ( v t )= v 0 ( t ) 57

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r iswelldened:Fromthedenitionofthepotgraph,if v t = v t 0 ,for v t ;v t 0 2 V P 1 then t = t 0 i.e ( t )= ( t 0 ).Hence v 0 ( t ) = v 0 ( t 0 ) i.e. r ( v t )= r v t 0 r isone-to-one:Assume r ( v t )= r ( v t 0 ),bythedenitionof r itfollowsthat v 0 ( t ) = v 0 ( t 0 ) .Therefore ( t )= ( t 0 ).Since isabijection, t = t 0 i.e., v t = v t 0 : r isonto:Forevery t 0 2 P 2 ,thereexists t 2 P 1 s.t. ( t )= t 0 .Hence r ( v t )= v 0 ( t ) = v 0 t 0 r isahomomorphism:Let v t and v t 0 areadjacentviaanedgelabeled h ( v t and v t 0 areverticesof G P 1 ).Bythedenitionof V P 1 ,itfollowsthateither t ( h ) > 0 and t 0 ( 1 ( h )) > 0or t ( 1 ( h )) > 0and t 0 ( h ) > 0.Because preservessticky endtypes,thesecasescorrespondto ( t )( f ( h )) > 0and ( t 0 )( 2 ( f ( h ))) > 0, or ( t )( 2 ( f ( h ))) > 0and ( t 0 )( f ( h )) > 0,respectively,i.e., v 0 ( t ) = r ( v t )and v 0 ( t 0 ) = r ( v t 0 )areadjacentviaanedgelabeled f ( h ). Remark4.3.18. Theconverseingeneralisnottruei.e.,from G P 1 = G P 2 itdoesn't followthat P 1 = P 2 .ThepottypespresentedinFigure4.11a)haveequivalentpo t graphs,presentedinFigure4.11b),butthepottypesarenot equivalent.Thereason forthatisthatthesetofcompletecomplexesforthepottype P 1 isinnite,whilethe setofcompletecomplexesforthepottype P 2 issingleton,i.e., P 1 and P 2 arenot similar,thereforetheyarenotequivalent. 58

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b) a vv 12 a a a t' 1 2 t' P 1 P 2 P 1 G P 2 G = ~ a a t 1 2 t a a) Figure4.11: G P 1 = G P 2 ,but P 1 P 2 InorderfortheconverseofProposition4.3.17tobetrue,we needtostrengthen thehypothesis.Proposition4.3.19. Let t ( h ) 1 8 h 2 H and 8 t 2 P 1 [ P 2 .If G P 1 = G P 2 via isomorphism ,then P 1 ( H 1 ; 1 ) = P 2 ( H 2 ; 2 ) viaisomorphism Proof. Let G P 1 = G P 2 viaanisomorphism .Deneamap : P 1 P 2 such that ( t )= s ifandonlyif ( v t )= v 0 s .Itisstraightforwardtoshowthat is abijection.Weshouldjustshowthat isahomomorphismandpreservessticky ends.Let v t from G P 1 isadjacentto v t 1 ;v t 2 :::v t k viaedgeslabeled h .Thatmeans thateither t ( h )=1or t ( 1 ( h ))=1.Withoutlossofgeneralityletstake t ( h )=1, then t 1 ( 1 ( h ))=1 ; t 2 ( 1 ( h ))=1 ;:::; t k ( 1 ( h ))=1.Alsobecauseofthepot graphisomorphismfollowsthat ( v t )isadjacentto ( v t 1 ) ; ( v t 2 ) ;:::; ( v t k )via edgeslabeled ( h ).Thatmeans v 0 ( t ) isadjacentto v 0 ( t 1 ) ;v 0 ( t 2 ) ;:::;v 0 ( t k ) viaedges labeled ( h ),i.e. ( t )( ( h ))=1(or ( t )( 2 ( ( h )))=1)and ( t i )( 2 ( f ( h )))=1(or ( t i )( ( h ))=1)for i 2 [ m ].Thereforetheisomorphismpreservesstickyendsi.e., deg( v t ; h )=deg( v 0 ( ( t )) ; ( h ))=1forall h 2 H and t 2 P 1 .Hence P 1 = P 2 Remark4.3.20. Ifthepottypeswereonlysimilar,butnotequivalentthenth ecorrespondingpotgraphsarenotnecessarilyequivalent.Cons iderthefollowingexample 59

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depictedintheguresgivenbelow.Thesetsofcompletecomp lexgraphsareequivalent sincethereexistsabijection : G ( P 1 ) G ( P 2 ) denedas:for i 2 [8] ( G C i )= G C 0 i suchthat G C i = G C 0 i forviadierentisomorphisms i : G C i !G C 0 i .Howeverthepot typesarenotisomorphicsincetheirpotgraphsarenotisomo rphic. t x y f y e d h y a b c x d x h y g f e a b c c g d h g b a y c d h g f e e f b a P 1 1 x 2 x 3456 1314 1112 910 y y 7 8 1516 P 2 12 x 34 x t 56 t 7 8 t 1516 h t t 1314 t 1112 y 910 y f aa bb c e e g g d h h d c b a ef gh b a c d g f e f c ttttttt d t t t t t t t t t tttttttt t Thesetsofcompletecomplexgraphs, G ( P 1 ) ; G ( P 2 ) for P 1 and P 2 areobviously equivalent,hence P 1 P 2 60

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e y y v 6 v 3 v 4 v 5 v 6 v 7 v 8 v 11 v 12 v 7 v 8 v 13 v 14 y y e e y e e y e e v 7 v 15 v 8 v 16 v 9 v 10 v 11 v 12 v 9 v 10 v 14 v 9 v 15 v 16 v 10 C 2 C 3 C 4 C 8 C 7 C 6 5 C C' 3 C' 4 C' 5 C' 6 7 C' C' 8 C 1 C' 1 v 1 v 5 v 2 v 13 v 1 v 2 v 11 v 13 v 9 v 10 v 3 v 5 v 1 v 6 v 4 v 6 v 2 v 7 v 8 v 3 v 7 v 8 v 3 v 9 v 10 v 13 v 14 v 12 y v 14 y v 15 v 16 v 11 v 12 v 15 v 16 v 5 P 1 x x f f ac a c b b c c d d dd gg ddhhff g g h h )) G( G( C' 2 c a x x h y x y ff f P 2 x d b d d b d a a c c b b c a e g g g g hh f h e e Butaswecanseethepotgraphsarenotequivalent. v 1 v 6 v 7 v 8 v 9 v 10 v 3 v 2 v 4 v 11 v 12 v 13 14 v 15 v 16 v v 5 x x a b c x d e y y f y y g h x G P 2 v 1 v 6 v 7 v 8 v 9 v 10 v 3 v 2 v 4 v 5 v 11 v 12 v 13 14 v 15 v 16 v a b c d e f g h x x y y y y y y G P 1 Alsowewouldliketopointoutthatthisexamplecanalsoserv easacounter examplefortheconverseofProposition4.3.13,if G( P 1 ) = G( P 2 ) itdoesnotfollow that P 1 = P 2 61

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Nextwewillshowthatforeverypottype P ,thereexistsapottype e P similarto P suchthateverytiletypein e P hasstickyendsofdistincttype.Hence,wecould onlyconsiderpottypeswithtiletypesthathasdistinctsti ckyendtypes.Therefore, fromProposition4.3.19andProposition4.3.17willfollow thatequivalenttwopot typesareequivalentifandonlyiftheirpotgraphsareequiv alent. Proposition4.3.21. Foreverypottype P ( H; ) thereexistsapottype e P ( e H; e ) such thateverytilein e P ( e H; e ) hasstickyendsofdistincttypeand P ( H; ) e P ( e H; e ) Proof. Suppose j P j = m ,say P = f t i : i 2 [ m ] g j H + j = n ,say H + = f h i : i 2 [ n ] g andmax t 2 P j z t ( h i ) j = k i for i 2 [ n ]. Constructapottype e P ( e H ; e )denedinthefollowingway: e H = e H + [ e H ,where e H + isadisjointunionofthesets e H + i = f h ji : j 2 [ k i ] g and e H isadisjointunionofthesets e H i = f e ( h ji ): j 2 [ k i ] g e : e H e H isaninvolutionandfor h ji 2 e H + ( e ( h ji ) 2 e H ), e ( h ji )and h ji can connect. Foreach t j 2 P weconstructaset e P j oftiletypesin e P denedinthefollowing way.If t j ( h i )= l i for i 2 [ n ]and h i 2 H + ,eachtiletypefrom e P j willhave l i stickyendsfromtheset e H + i ,elseif h i 2 H willhave l i stickyendsfrom e H i Weassignonenewtiletypetoeach n Y i =1 k i l i combinationpossible.Withother wordseachtiletype t 0 2 e P j willsatisfy t 0 ( h 1i )+ t 0 ( h 2i )+ + t 0 ( h k i i )= t j ( h i ) for h 1i ;:::; h k i i 2 e H +i if h i 2 H + ,or h 1i ;:::; h k i i 2 e H i if h i 2 H .Everytile typein e P hasdierentstickyendsandsincewedeneallthepossiblet iletypes in P j suchthateachtiletypehas l i stickyendsfrom e H + orfrom e H ,thereare n Y i =1 k i l i tiletypesin e P j e P = [ t j 2 P e P j CLAIM: P ( H; ) e P ( e H; e ) 62

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Supposeacompletecomplex C = h T;S;J i in C ( P )isgiven,andsupposethere are r i tilesoftype t i in T ,for i 2 [ m ],i.e., T = f t 11 ;:::;t r 1 1 ;t 12 ;:::;t r 2 2 ;:::;t 1m ;:::;t r m m g andtype( t ki )= t i ,for i 2 [ m ]and k 2 [ r i ].Let G C =( V C ;E C ;l )bethecomplete complexgraphof C ,where V C = f v t : t 2 T g Weconstructacomplex e C = h e T; e S; e J i2C ( e P ),suchthat j e T j = j T j andthereare r i tilesoftypesin e P i in e T ,for i 2 [ m ].Thecomplexgraphis e G =( e V; e E; e l ),forwhich e V = f v 0 e t : e t 2 e T g .Since j e T j = j T j ,fromthedenitionofthecomplexgraphsitfollows that j e V j = j V j Consideranytwotilesfrom T ,say t i (oftiletype t i ), t j (oftiletype t j ),such thattheyconnectwith k l connectionsviastickyends h l and ( h l ),for h l 2 H +l and l 2 [ n ]. Forthetilesin T givenabovethereexisttwotilesin e T e t i and e t j suchthat type ( e t i ) 2 e P i and type ( e t j ) 2 e P j andtheyhave k l complementarystickyendsfrom e H + i and e H i respectively,say e t i ( h il )= e t j ( e ( h il ))=1for i 2 [ k l ].Basedonthe originaltileconnections,theotherconnectionsareestab lishedinasimilarmanner, i.e.thereexistsconnections f ( e t i ;h il ) ; ( e t j ;f i l ) g2 e J ,for i 2 [ k l ],and type ( h il )= h il type ( f i l )= e ( h il ).Thisconstructionalwaysworks,because e P i hastileswithallthe possiblecombinationsof l i stickyendsfrom e H + i or f H i dependingwhether t j ( h i )= l i or t j ( ( h i ))= l i ./ Thenumberofconnectionsbetween t i and t j isthesameasthenumberofconnectionsbetween e t i and e t j .Notethatthenumberofstickyendsonthetilesin T isthe sameasthenumberofstickyendsonthetilesin e T (thiscomesfromthedenitionof thetilesin e P ).Sinceforeveryconnectionin C ,thereexistsexactlyoneconnection in e C ,wecanconcludethat e C isacompletecomplex. Wedeneamap : V e V suchthat ( v t i )= v 0 e t i and ( v t j )= v 0 e t j ifthe aboveissatised.Therefore,thenumberofedgesbetween v t i and v t j isthesameas thenumberofedgesbetween v 0 e t i and v 0 e t j .Since C and e C havethesamenumberof connections,fromthedenitionofcompletecomplexgraphs itfollowsthat j E j = j e E j Fromherewecanconcludethatthereare x numberofedgesbetween v t i and v t j if 63

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andonlyifthereare x numberofedgesbetween v 0 e t i and v 0 e t j ,i.e., isanisomorphism. CONVERSE:Supposethat e C = h e T; e S; e J i isacompletecomplexin C ( e P )for whichthereare r i tiles(in e T )from e P i for i 2 [ n ]andsome r i .Supposethatthe correspondingcompletecomplexgraphis e G C =( e V; e E; e l ). Weconstructacomplex C = h T;S;J i in P suchthat T has r i tilesoftype t i Consideranytwotiles e t i (oftiletypein e P i )and e t j (oftiletypein e P j )from e T ,such thattheyconnectwith w l connectionsviastickyendsfrom e H + l and e H l respectively, for l 2 [ n ]. Forthetiletypes t i and t j in P givenaboveweassigntwotilesin T t i t j such that type ( t i )= t i and type ( t j )= t j Fromtheconstructionofthepottype e P ,i.e., e P i and e P j ,itfollowsthat t i ( h l ) w l and t j ( ( h l )) w l ,for l 2 [ n ].Tothegiventiles t i and t j weassign w l connections viathestickyends h l and ( h l )for l 2 [ n ]. Therefore,thenumberofconnectionsbetween e t i and e t j issameasthenumberof connectionsbetween t i and t j .Notethatthenumberofstickyendsonthetilesin T isthesameasthenumberofstickyendsonthetilesin e T (thiscomesfromthe denitionofthetilesin e P ).Sinceforeveryconnectionin e C ,thereisoneconnection in C ,wecanconcludethat C isacompletecomplex. Wedeneamap : e V V suchthat ( v 0 e t i )= v t i and ( v 0 e t j )= v t j iftheabove issatised.Therefore,thenumberofedgesbetween v 0 e t i and v 0 e t j isthesameasthe numberofedgesbetween v t i and v t j .Since C and e C havesamenumberofconnections, fromthedenitionofcompletecomplexgraphsitfollowstha t j e E j = j E j .Fromhere wecanconcludethatthereare x numberofedgesbetween v 0 t i and v 0 t j ifandonlyif thereare x numberofedgesbetween v t i and v t j ,i.e., isanisomorphism. Example4.3.22. Thisisanexampleofapottype P anditscorrespondingpottype e P thatdon'tcontaintiletypeswiththesamestickyends. 64

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cc b aab bc a bc a bc a bc a 1 2 11 12 111 112 211 22 1 P ~ aab bc a cc b P t tt tt t 123 2 2 3 2 1 t 4 2 t 2 1 1 1 3 t Figure4.12:Apottype P anditscorrespondingpottype e P thatdon'tcontaintiletypes withthesamestickyends. Thecorrespondingsetsofcompletecomplexgraphs. t 2 v 2 t' v t 3 v t 1 v t 1 v t 2 v 2 t' v t 3 v ( P ) G ( P ) G a 1 b 1 a 2 c 1 b 1 c 2 t 11 v t 3 1 v t 2 1 v t 2 4 v t 11 v a 1 b 1 b 1 c 1 c 2 a 2 t 3 1 v t 2 2 v t 2 3 v c 1 a 1 a 2 b 1 b 1 c 2 t 11 v t 3 1 v t 2 1 v t 2 4 v b 1 c 1 a 2 b 1 c 2 a 1 t 11 v t 2 2 v t 2 3 v t 3 1 v a a bbc c c c a abb ~ Figure4.13:Elementsfromthesetsofcompletecomplexgrap hsforthepottypes P and e P giveninFigure4.12. 65

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MinimalandMaximalCompleteComplexGraphsDenition4.3.23. Let G C 0 and G C becomplexgraphsofthesamepot.Thecomplex graph G C 0 isacoveringcomplexgraph of G C ifthereexistsasurjectivehomomorphism : V C 0 V C whichpreservestheneighborhoodofeveryvertex,i.e.,for everyvertex v 2 V C 0 andevery h 2 H deg( v; h )=deg( ( v ) ;f ( h )) ,where f isthemapping denedinDenition4.3.6.Denition4.3.24. Thecomplex C 0 isacoveringcomplex of C ifthecorresponding complexgraphof C 0 G C 0 ,isacoveringcomplexgraphofthecorrespondingcomplex graphof C G C Denition4.3.25. Let P beapot.Acompletecomplex C = h T;J i2C ( P ) isa minimalcompletecomplex ifeveryothercompletecomplex C 0 = h T 0 ;J 0 i2C ( P ) such that T and T 0 havethesamesetoftiletypes,isacoveringcomplexof C Thesetofminimalcompletecomplexesof C ( P ) isdenotedby MC ( P ) Denition4.3.26. Let P beapot.Acompletecomplex C = h T;J i2C ( P ) isa maximalcompletecomplex ifitisnotacoveringcomplexofanyothercomplexfrom P ,andeveryothercompletecomplex C 0 = h T 0 ;J 0 i2C ( P ) suchthat T T 0 and J J 0 isacoveringcomplexofC.(viz,therearenobiggercomplexe swiththesame tiles,exceptthecoveringcomplexes.)Proposition4.3.27. Thepotgraph G P ofastronglysatisablepot P withasingle pointspectrumisconnected.Proof. InChapter3wediscussedthatifthespectrumofastronglysa tisablepot P isasinglepoint,theneverycompletecomplexin C ( P )containatilefromeverytile type.Considertwotiles t 1 and t 2 withcorrespondingvertices v 1 and v 2 in G P .If thereisnopathbetween v 1 and v 2 ,then t 1 and t 2 mustbeontwodierentcomplete complexes,whichisnotpossible.Therefore,thepotgrapho fastronglysatisable potwithasinglepointspectrumisconnected. 66

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Denition4.3.28. Aseparatingset ofacomplex C = h T;J i isasetofconnections S J suchthat h T;J S i isnotaconnectedcomplex.Theconnectivityofacomplex C k ( C ) ,istheminimumsizeofaseparatingset S s.t. J S dissolves C intotwo complexes.Acomplexis k -connected ifitsconnectivityisatleast k .Apot P is k -connected ifallofitscompletecomplexesare k -connected. Remark4.3.29. If P isastronglysatisablepotsuchthateveryincompletecomp lex hasatleasttwofreestickyends,theneverycompletecomple xof P is2-connected. Proposition4.3.30. Let P bea2-connectedstronglysatisablepottype.Thepot graphof P isconnectedifandonlyifthereisaconnectedcompletecomp lex C 2C ( P ) thatcontainsallthetiletypes.Proof. Supposethatthepotgraph, G P ,of P isconnectedandthereisnocomplete complexin C ( P )thatcontainsallthetiletypes.Weclaimthatthereexistm aximal connectedcompletecomplexes C i = h T i ;J i i for i 2 [ n ]suchthatforeverytiletype, thereisatilethatisembeddedinoneofthemaximalcomplete complexesandno twoofthemaximalcomplexescanhaveatileofthesametype,m oreovernotwoof themaximalcomplexescanhaveastickyendofthesametypein common. Let C i k = h T i k ;S i k ;J i k i and C i m = h T i m ;S i m ;J i m i bemaximalcompletecomplexes thathaveastickyendoftype h 2 H incommon.Thereexisttiles t 1 ;t 01 2 T i k and t 2 ;t 02 2 T i m suchthat t 1 ( h ) > 0, t 01 ( b h ) > 0, t 2 ( h ) > 0,and t 02 ( b h ) > 0.Since P isa 2-connectedstronglysatisablepottype,thetiles t 1 and t 02 canconnect,andthetiles t 2 and t 01 canconnectformingacomplex C = h T i k [ T i m ;S i k [ S i m ;J i .Thatisnot possible,becausethecomplexes C i k and C i m aremaximalcompletecomplexes.We canconcludethatnotwomaximalcompletecomplexesshareas tickyendofthesame typeincommon,thereforetheycannotshareatileofthesame typeincommon. 67

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a) 1 t' 1 t 2 t' 2 C 1 C 2 t 1 t' 1 t 2 t' 2 C 1 C 2 b) t Figure4.14:a)Twocompletecomplexesthatshareastickyen dincommonb)Acomplete complexbuildfromthecomplexesina) Let G C i for i 2 [ n ]bethecorrespondingcompletecomplexgraphsforthemaxim al completecomplexes,and bethehomomorphismfromcompletecomplexgraphsand potgraphs.Then ( G C i )for i 2 [ n ]arepairwisedisjoint,sincetheydon'thavetiles norfreestickyendofthesametypes.Alsothereisnovertexo ranedgethatisin G P thatitisnotinsomeofthe ( G C i ).Thisisnotpossiblesince G P isconnected. Hencethereisacompletecomplexin C ( P )thatcontainstilesofeverytiletype. Supposethatthereexistsaconnectedcompletecomplex C = h T;S;J i2C ( P )that containstilesofeverytiletype.Thecompletecomplexgrap hof C G C ,ismapped ontowithrespecttotheverticesto G P andsinceitisconnected,forevery t;t 0 2 T thereexistsapathconnectingthem,i.e.forevery v t ;v t 0 2 V thereexistsapath connectingthem(becausewecorrespondandedgetoeverycon nection).Therefore G P isconnected. Theinducedsubgraphforacoveringofaminimalcompletecom plexwillbethe 68

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sameastheinducedsubgraphfortheminimalcompletecomple x.Therefore,from aninducedsubgraphwecanobtaininformationonlyaboutthe \minimal"complete complexes. Fromthespectrumitselfandfromthesupportofagivenpot P wearenotableto ndalltheinformationabouttheproductsofself-assembly .Thestudyofpotgraphs expandsourview,andinconjunctionwithotherstudiescang ivemorecomplete pictureofthepossiblecomplexesassemblingfromapotofti les.Thefollowingis anexampleoftwopottypeswithsamespectrumandsamesuppor t,butdierent completecomplexes.Wecanconcludethattheyhavedierent completecomplexes onlybecausethecorrespondingpotgraphsarenotequivalen t. a a j 1 j 2 j 3 j 1 j 2 j 3 a b P 1 P 1 G ( ) j 1 c c v 1 v 2 v 3 v 4 a a a a c b G ( ) P 2 v 1 v 2 v 3 v 4 a b P 2 a b c b a j 4 c dd b a c d Figure4.15:pottypesthatarenotsimilar,butwiththesame spectrum S = f ( 1 2 u; 1 2 u;u;u ):0 u 1 4 g andsamesupport Supp = f (1 ; 1 ; 1 ; 1) g Inthischapterwejustscratchedthesurfaceofpotgraphs.W eplantoextend thethisstudyandtoconnectitwiththestudyofthespectrum andprobability. 69

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5ProbabilisticAnalysis 5.1Introduction Forbetterunderstandingoftheself-assemblyprocessaswe llaspossiblyto predict whatweshouldexpecttoobtainwithinapot withDNAmoleculesoneneedstostudy thedistributionofthepossibleoutcomesoftheprocess. Attheendoftheexperimenttheremaybesomecompletecomple xesofthedesired type,someothercompletecomplexesandsomeincompletecom plexes.Sinceour majorconcernistheconstructionofcompletecomplexes(of certainsizes),wewant toexploretheproportionofeachcompletecomplex,forthep urposeofevaluatingthe resultsoftheexperiment.Weapproachthisproblembyconsi deringaspecialcase, graphassemblyofuniformlydistributedtiles,thatcanass embleintocycliccomplexes. Webelievethatmorecomplicatedstructureshaveadditiona lgeometricandother intrinsicconstraintsthatwouldmakeourassumptionsunre alisticandsuperruous. Goodconditionsforstudyingsuchcomplicatedstructuresr emaintobediscovered. Inthischapterweraisequestionsrelatingtheprobability ofobtainingacyclic completecomplexandapproachthemusingtwodierentmetho ds.Althoughthisisa rsttheoreticalstudyoftheprobabilityofself-assemble dcomplexes,itisnottherst studyoftheprobabilityofobtainingcyclicmolecules[13, 16,17,46,47,51,52,58,59]. Thedistributionoflinearandcyclicmoleculesprovidesab etterunderstandingofthe rexibilityandintrinsicpropertiesoftheDNAmolecules.A lsothecorrelationbetween theconcentrationofthemoleculesandtheprobabilityoftw ostickyendsconnecting canbebetterrealizedbythesamedistribution. TheDNAmoleculeinlivingorganismsusuallyappearsasacyc licmoleculeof 70

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dierentsizes.Thecongurationofthosemoleculesandthe probabilityoftheir formationsisofinteresttomanybiologists,chemists,mat hematicians,andphysicists. Therstpaperconcerningtheprobabilityofringformation ofDNAmolecules appearedin1950byH.JacobsonandW.H.Stockmayer[16,17]. Theyintroduced thenotionofringclosureprobabilityofDNAmoleculeswhic hisknownasthe j factor.Itisdenedastheratiooftwoequilibriumconstant s K c and K a ,where K c isthecyclizationconstantand K a isthebimolecularequilibriumconstantforjoining twomolecules. Fewyearslater,J.C.WangandN.Davidson[51,52]usedtherm odynamicand kineticpropertiesofDNAmoleculesinthestudyofringclos ureprobabilityorcyclization.TheymeasuredtheentropyandenthalpychangeofDNAmo leculesduringthe interconversionfromlinearintocircularmolecules.Thei rexperimentsweresimilar toH.JacobsonandW.H.Stockmayer's.Theirworkinvolvedth reedierenttypesof DNAmolecules;onetypehadtwostickyendsandthosemolecul eswereabletoform cycles,whiletheothertwotypeshadonestickyendeach(for exampleoneofthem hadthestickyendontheleftside,andtheotheronehaditont herightside)and onebluntend,andthosetypesofmoleculeswerenotabletoco ngureintocycles, butrathertoself-assembleintolinearstructureswithtwo bluntends. D.ShoreandR.L.Baldwin[46,47]achivedastrikingresults howingthatthe j -factormainlydependsonthefractionaltwistoftheDNAmol ecule:thedierence betweenthetotalhelicaltwistandthenearestinteger.The ywerealsoabletodistinguishdierenttopoisomersobtainedattheendoftheexperi ment.Intheirresearch, theystudiedseriesof12linearDNAmoleculesthathadcompl ementarystickyends oneachside,andtheDNAmoleculesdieredonlyintheirleng th. A.Dugaiczyk,H.W.Boyer,andH.M.Goodman[13]studiedthep robabilityof aringclosureofDNAmolecules.Theyconsideredonlyonetyp eofDNAmolecule intheirexperiments,anditwasalinearmoleculewithcompl ementarystickyends onthesides.Theirresultsshowedthatthe j -factordependsonthecontourlengthof themolecule,therandomcolisegmentandthemolarconcentr ationofthesolution. 71

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Theringclosureprobabilityhasbeencomputedasafunction ofmolecularlengthofa DNAmoleculebyYamakawaandStockmayer[58,59].Theyconsi deredthemolecule asawormlikemodelanddenedthe j -factorasaGreen'sfunctionofthemolecular length. ThecyclizationoftheDNAmoleculehasbeenthetopicofscie nticresearchfor manyyears.Unfortunatelywestilldon'thaveaprecisedesc riptionoftheprocess andoftheconformationtheDNAmoleculetakes.Inthenextfe wsectionsweoutline thecalculationoftheprobabilityofself-assemblingcycl icmoleculesofdierentsizes. Forsimplicityweavoidthermodynamicpropertiesandconsi deronlytheselfassemblyprocessforwhichallWatson-Crickconnectionsar eequallylikelyandno freestickyendsremainafterthecompletionoftheexperime nt.Thatmeansthat afterself-assemblyhasoccurredonlycompletecomplexesa represent.Weproposea staticmodel(similartomodelsstudiedindiscreteprobabi litytheory)withuniformly distributedtiles,andWatson-Crickcomplementarypairs. Eachstickyendhasequal probabilitytoconnect. Theevolutionoftheself-assemblyprocessisnotexamined, onlydistributionof obtainedcompletecomplexesattheendoftheprocessiscons idered.Wearemainly interestedintheinputandtheoutputoftheprocess.Weprov idesomeinsighttothe question:Ifwehaveacertainamountoftilemolecules,howm uchofeachcomplex typesarethereintheoutcome? Inthischapterweonlyprovidetheoreticalresults,whilee xperimentalresultsare giveninthenextchapter.Theexistenceoftheprobabilitys paceisdemonstratedin AppendixB.Inthenextsectionweaddressthequestionsrais edintheintroduction bytwodierentmethods.OneTypePotToexplorethestaticmodel,webeginwithabasicset-up,pot type P = f t g ofonly onetiletypeand H = f h ; b h g thesetofstickyends( b h denotestheW-Ccomplement of h ),s.t.forthegiventiletype t ( h )= t ( b h )=1.Assumethatwehave m tilesin 72

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thegivensolution. Let t beatilefromthetiletype t ,set P ( t isonacycleoflength k )= r m;k : ExpectationDenoteby:A m;t;k :theeventthatthetile t isona k -cycle, I m;t;k :theassociatedindicatorrandomvariable, I m;k = P t I m;t;k :thenumberoftilesin k -cycles, X m;k = I m;k k :thenumberof k -cycles. Then E ( I m;t;k )= P ( I m;t;k =1)= r m;k ; and E ( I m;k )= X t E ( I m;t;k )= mr m;k ; hence E ( X m;k )= 1 k E ( I m;k )= m k r m;k istheexpectednumberof k cycles : Proposition5.1.1. Let P beapot(oftype P = f t g )whichcontains m 2-branched tilesoftype t and H = f h ; b h g bethesetofstickyendtypesforthepot P ,suchthat t ( h )= t ( b h )= 1 .Let X m;k denotethenumberofcyclesoflength k in P and r m;k the probabilitythatagiventilefromthepot P isonacycleoflength k .Then E ( X m;k )= m k r m;k ;k 2 [ m ] : Var ( X m;k )= m k r m;k (1 r m;k ) : NOTE1 Weassumethatduringtheself-assemblyprocessallWatsonCrickconnectionsare establishedandthatonlycompletecomplexesareobtained. Thatmeansafterthe 73

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annealingprocesseverytileisonacompletecycle,i.e., m X k =1 r m;k =1.Moreoversince wehaveatotalof m tilesand mr m;k k k -cyclesinthepot(ina k -cycle, k tilesare engaged) m X k =1 k mr m;k k = m; shouldhold,whichisanywaysatised,since m X k =1 k mr m;k k = m m X k =1 r m;k = m: NOTE2 Basedontheoutcomesofsomeexperiments,wecanassumethat probabilityofobtainingasmallercycleisgreaterthantheprobabilityofob tainingabiggercycle. Thereforeprobabilityofonetilebeingonasmallercycleis greaterthantheprobabilityofbeingonalargercyclei.e., r m; 1 r m; 2 r m;m Proof. Variance Fromthedenitionoftheevents,since X m;k = I m;k k ; Var( X m;k )= 1 k 2 Var( I m;k )= 1 k 2 Var( X t I m;t;k ) : Fromthepropertiesofthevariance,itfollowsthat Var( X m;k )= 1 k 2 X t 1 ;t 2 Cov( I m;t 1 ;k ;I m;t 2 ;k ) : When t 1 = t 2 ; Cov( I m;t 1 ;k ;I m;t 1 ;k )=Var( I m;t 1 ;k )= E ( I 2 m;t 1 ;k ) ( E ( I m;t 1 ;k )) 2 = r m;k (1 r m;k ) ; 74

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whilewhen t 1 6 = t 2 ; Cov( I m;t 1 ;k ;I m;t 2 ;k )= E ( I m;t 1 ;k I m;t 2 ;k ) E ( I m;t 1 ;k ) E ( I m;t 2 ;k ) : Bythedenitionoftheexpectation, E ( I m;t 1 ;k I m;t 2 ;k )=1 P ( I m;t 1 ;k I m;t 2 ;k =1)+0 P ( I m;t 1 ;k I m;t 2 ;k =0) = P ( I m;t 1 ;k I m;t 2 ;k =1)= P ( I m;t 1 ;k =1) P ( I m;t 2 ;k =1 j I m;t 1 ;k =1) : Giventhatthetile t 1 isona k -cycle,theprobabilitythat t 2 isalsoona k -cycleis equaltotheprobabilitythat t 2 isonthesamecycleas t 1 (whichisoneoftheother k 1tilesonthecycle)oritisoneoftheremaining m k tilesthatarejoinedin other k -cycle(eachonewithprobability r m k;k beingonacycleoflength k ). Therefore, P ( I m;t 2 ;k =1 j I m;t 1 ;k =1)= k 1 m 1 + m k m 1 r m k;k Hencefor k m 2 (inthiscasetwotilescanbeeitheronasamecycleorontwo dierentones)weobtain Cov( I m;t 1 ;k ;I m;t 2 ;k )= r m;k k 1 m 1 + m k m 1 r m k;k r 2 m;k = r m;k k 1 m 1 + m k m 1 r m k;k r m;k whilefor k> m 2 (inthiscasetwotilescanbeonlyonasamecycle), Cov( I m;t 1 ;k ;I m;t 2 ;k )= r m;k k 1 m 1 r 2 m;k = r m;k k 1 m 1 r m;k : 75

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Now, Var( X m;k )= 1 k 2 X t 1 ;t 2 Cov( I m;t 1 ;k ;I m;t 2 ;k ) = 1 k 2 h X t 1 = t 2 Cov( I m;t 1 ;k ;I m;t 2 ;k )+ X t 1 6 = t 2 Cov( I m;t 1 ;k ;I m;t 2 ;k ) i = 1 k 2 h mr m;k (1 r m;k )+ m ( m 1) r m;k k 1 m 1 + m k m 1 r m k;k r m;k i NOTE3 Usuallyinapotconsistingofalargenumberoftilemolecule s(say10 15 )weexpectthat ifa k -cycleisadmitted,thenalmostsurelywewillobservemanyo ther k -cyclesi.e., k m 2 .Moreoverifa k -cycleisobtainedintheoutcomeofthepot,thenwithgreat probabilitycyclesofsmallerlengththan k arealsoobtained.Hencewecanmakeour assumptionevenstrongerandassumethatonlycyclesofleng thmuchsmallerthan m areassembledinthepoti.e., k m .Thiswillleadtowardsapproximationofthe probabilities r m;k by lim m !1 r m k;k r m;k =1 ; Cov( I m;t 1 ;k ;I m;t 2 ;k )= r m;k k 1 m 1 + m k m 1 r m;k r m;k for k m: Usingthefactthatfor k m Var( X m;k )= m k 2 r m;k (1 r m;k + k 1+( m k ) r m;k ( m 1) r m;k ) = m k r m;k (1 r m;k ) : 76

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MultiTypePotLet P = f t 1 ; t 2 ;::: t n g beapottype,and P apotthatcontains m dierent2armedtilesfromtypesin P fromwhichan n -cyclecanbeconstructed.Let H = f h 1 ; h 2 ;::: h n ; b h 1 ; b h 2 ::: b h n g bethesetofstickyendtypessuchthat t i ( h i )= t i ( b h i + 1 )= 1for i 2 [ n 1] ; and t n ( h n )= t n ( b h 1 )=1.Weassumetohaveuniformlydistributed tiletypesi.e, n tilesfromeachtype,and P ( t isonacycleoflength k )= r k : Fromtheselectionofthetiles,itisclearthatonlycyclesw hoselengthsaremultiples of m areassembled,i.e, r l =0if m l: DenotebyA t;k :theeventthatthetile t isona k -cycle, I t;k :theassociatedindicatorrandomvariable, I k = P t I t;k :thenumberoftilesina k -cycle, X k = I k k :thenumberof k -cycles. Wecancalculate E ( I t;kn )= P ( I t;kn =1)= r kn E ( I kn )= X t E ( I t;kn )= mnr kn E ( X kn )= E I kn kn = mn kn r kn = m k r kn : 5.2AnotherMethodforObtainingtheExpectedNumberofCycl es Toexplorethestaticmodel,webeginwithabasicset-up,pot type P = f t g ofonly onetiletypeand H = f h ; b h g ( b h denotestheW-Ccomplementof h ),s.t.forthe giventiletype t ( h )= t ( b h )=1.Assumethatwehave m tilesinthegivensolution. 77

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Let t beatilefromthetiletype t ,set P ( t isonacycleoflength k )= r k S k :asetof k tilesfromtypein P A S k :theeventthatthetilesfrom S k formacycleoflength k X S k :theassociatedrandomvariablefor A S k Expectations CASE1 : S 1 = f t 1 g Since X S 1 istheindicatorrandomvariablefortheevent A S 1 X 1 = P S 1 X S 1 willdenotethenumberofcyclesoflength1inthepoti.e,mon omersand E ( X S 1 )= P ( X S 1 =1)= r 1 : Wehave m sets S 1 andbythelinearityoftheexpectationtheexpectednumber ofmonomersinthepotis E ( X 1 )= X S 1 EX S 1 = mr 1 : CASE2 : S 2 = f t 1 ;t 2 g Since X S 2 istheindicatorrandomvariablefortheevent A S 2 X 2 = P S 2 X S 2 willdenotethenumberofcyclesoflength2inthepoti.e,dim ersand E ( X S 2 )= P ( X S 2 =1) Note:Giventhat t 1 isonadimer,weknowthatitislinkedtoanothertile,so astheprobabilitythatislinkedtothetile t 2 ,foranyparticulartile t 2 6 = t 1 ,is 1 m 1 ,so 78

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E ( X S 2 )= P (( t 1 and t 2 arelinked) ^ ( t 1 isonadimer)) = P ( t 1 isonadimer) P ( t 1 and t 2 arelinked j t 1 isonadimer) = r 2 1 m 1 = r 2 m 1 : Wehave m 2 sets S 2 andbythelinearityoftheexpectationtheexpectednumber ofdimersinthepotis E ( X 2 )= X S 2 EX S 2 = m 2 r 2 m 1 = m 2 r 2 : CASE3 : S 3 = f t 1 ;t 2 ;t 3 g E ( X S 3 )= P ( X S 3 =1) Note:Giventhat t 1 isonatrimer,weknowthatitislinkedtotwoothertile molecules,theprobabilitythatislinkedviathestickyend oftype b h tothetile t 2 ,forsome t 2 6 = t 1 is 1 m 1 .Giventhat t 1 isonatrimerandthatitislinkedto t 2 ,weknowthatitmustbealsolinkedtooneothertileviathest ickyendof type b h ,theprobabilitythatthisothermoleculeis t 3 is 1 m 2 ,forgiven t 3 .There aretwopossibletrimersobtainedfromthesetiles( t 1 t 2 t 3 and t 1 t 3 t 2 ).Therefore E ( X S 3 )=2 P (( t 1 and t 2 arelinked) ^ ( t 1 and t 3 arelinked) ^ ( t 1 isonatrimer)) =2 P ( t 1 isonatrimer) P (( t 1 and t 2 arelinked) j ( t 1 isonatrimer)) 2 P (( t 1 and t 3 arelinked) j ( t 1 isonatrimer) ^ ( t 1 and t 2 arelinked)) =2 r 3 1 m 1 1 m 2 = 2 r 3 ( m 1)( m 2) : Wehave m 3 sets S 3 andbythelinearityoftheexpectationtheexpectednumber 79

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oftrimersinthepotis E ( X 3 )= X S 3 EX S 3 = m 3 2 r 3 ( m 1)( m 2) = m 3 r 3 : GENERALCASEFOR k : S k = f t 1 ;t 2 ;:::;t k g Calculatingtheexpectationforthiscasewillgothesamewa yasforallprevious casesuptopermutation.Thistilemoleculescanform( k 1)!\dierent" k cycles.Therefore E ( X S k )=( k 1)! r k ( m 1)( m 2) ::: ( m k +1) =( k 1) 1 ( k 1)! r k m 1 k 1 = m k r k m k : Wehave m k sets S k andbythelinearityoftheexpectationtheexpectednumber of k -cyclesinthepotis E ( X k )= X S k EX S k = m k m k r k m k = m k r k : Proposition5.2.1. Let P = f t g beapotwhichcontains m 2-branchedtilesoftype t and H = f h ; b h g bethesetofstickyendtypesforthepot P ,suchthat t ( h )= t ( b h )= 1 Let X k denotethenumberofcyclesoflength k inthe P and r k theprobabilitythata giventileisonacycleoflength k .Then E ( X k )= m k r k ;k 2 [ m ] : 80

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OBSERVATION Weassumedthatduringtheself-assemblyprocessallWatson -Crickconnectionsare establishedandthatonlycompletecomplexesareobtained. Thatmeansthatafter theannealingprocesseverytileisonacompletecyclei.e., m X k =1 r k =1.Moreover, sincewehaveatotalof m tilesand mr k k k cyclesinthepot(ina k -cycle k molecules areengaged)wearenotsurprisedthat m X k =1 k mr k k = m m X k =1 r k = m: Variance GENERALCASEFORVAR( X k )( 2 k m ) : { S k \ T k = ; Wehave m k m k k choicesforthiskindofsets,andinthiscase E ( X S k X T k )= P ( X S k =1 ;X T k =1) = P ( X S k =1 j X T k =1) P ( X T k =1) = ( m k ) r k k m k k mr k k m k EX S k EX T k = m 2 k 2 m k 2 r 2 k ; hence Cov( X S k ;X T k )= ( m k ) r k k m k k mr k k m k m 2 k 2 m k 2 r 2 k ; andhence 81

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X S k \ T k = ; Cov( X S k ;X T k )= m k m k k ( m k ) r k k m k k mr k k m k m k m k k m 2 k 2 m k 2 r 2 k = m ( m k ) k 2 r 2 k m k k m 2 4 k 2 m k r 2 k : { 0 < j S k \ T k j
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E ( X S k X T k )= P ( X S k =1 ;X T k =1) = P ( X S k =1 j X T k =1) P ( X T k =1) = m k m k r k ; and EX S k EX T k = m 2 k 2 m k 2 r 2 k ; hence Cov( X S k ;X T k )= m k m k r k 1 m k m k r k ; andhence X S k = T k Cov( X S k ;X T k )= m k m k m k r k 1 m k m k r k = m k r k 1 m k m k r k = m k r k m 2 k 2 m k r 2 k : CONCLUSION Var( X k )= X S k ;T k Cov( X S k ;X T k ) = m ( m k ) k 2 r 2 k m k k m 2 k 2 m k r 2 k m 2 k 2 r 2 k + m k k m 2 k 2 m k r 2 k + m 2 k 2 m k r 2 k + m k r k m 2 k 2 m k r 2 k = m ( m k ) ( k ) 2 r 2 k m 2 k 2 r 2 k + m k r k = m 2 k 2 r 2 k m k r 2 k m 2 k 2 r 2 k + m k r k = m k r k (1 r k ) : 83

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Var( X k )= m k r k (1 r k ) GENERALCASEFORVAR( X k )( 2 k>m ) : Inthiscasetheonlypossibleoutcomeisonecycleoflength m .Itisnotpossible tohavetwodisjointsetsof k elements,justonewith m elements S m = T m .Fromthe previousobservation,weobtain Var( X m )= X S m = T m Cov( X S m ;X T m ) = m m r m m 2 m 2 mm r 2 m = r m (1 r m ) : Ourassumptionisthat r m =0,sowecanconcludethatVar( X m )=0,and E ( X m )= m m r m mm =0. 5.3TheoreticalBasefortheExperimentalResults Inthissectionweestablishthetheoreticalbasefortheexp erimentalresultsgiven inthenextchapter.Webeginwithapotcontaininguniformly distributedDNA moleculescompatibleofformingacyclicgraphstructure.A ftertheannealingprocessitisassumedthatnofreestickyendsremain.Weproveth atundercertain probabilityconditionsalmostallstructuresrepresentth eoriginallyencodedgraph i.e.,theappearanceofdimer(doublecover)ortrimer(trip lecover)moleculesiswith smallprobability. Themainassumptionofthemodelisthattheprobability r thatthelastofthe possibleconnectionswithinacomplexappearsaftertheoth erconnectionshavebeen establishedisveryhigh.Undertheseconditionsweshowtha tprobabilityofappear84

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anceofdimersandtrimersinapotdesignedtoformmonomercy clesapproaches 0. Westartwithaspecialcaseofobtainingcyclicmoleculeswi ththree2-armed tiles.Thiscorrespondstobuildingatriangle.Forthispur pose,weconsiderthree dierenttypesof2-armedtiletypes P = f t 1 ; t 2 ; t 3 g whichcontain3dierenttypesof complementaryfreestickyends H = f h 1 ; h 2 ; h 3 ; b h 1 ; b h 2 ; b h 3 g Thesetilesareuniformly distributedinapotandarecapableofadmittingacomplete K 3 complex,meaning thatwehaveequalamountoftilesfromeachtiletype.Weconv enientlyrepresent thisamountwithaninteger m .Thestickyendtypesareadequatelyarranged(see Fig.5.1). 3 (a) j 1 j j 2 1 2 hh 3 h 3 2 1 1 2 3 1 h (b) 3 2 j j j j j j hh j 3 1 j j 2 j 2 j j 1 3 C C 3 6 Figure5.1:(a)Three2-armedtilesformatrianglewhichrep resentsa K 3 complex.(b) Threetilegraphsusedinapottoassemble K 3 .(c)Completecomplexesforthispotwill becyclesoflengthdivisibleby3.Cycles K 3 and C 6 aredepicted. t 1 ( h 1 )= t 1 ( b h 2 )=1 ; t 2 ( h 2 )= t 2 ( b h 3 )=1 ; t 3 ( h 3 )= t 3 ( b h 1 )=1 : Withthiskindofselectionforthetilemolecules,complete complexesthatare obtainedwouldbecyclicandwouldinvolve3 k tilesforsome k (1 k m ).Tiles fromtypesin P arealwaysassemblingaccordingtoaspecicpattern, t 1 t 2 t 3 repeatedly or t 1 t 3 t 2 repeatedly,dependingontheorientation.Wesaythatacycl eisof length 3 k ifithas k tilesfromeachtypeadequatelyarranged,forexample t 1 t 2 t 3 repeatedly 85

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k times,suchthatthelast t 3 tileisgluedtotherst t 1 tile.Wewillusethenotation C 3 k forcyclesoflength3 k k 1,(Note:Foracycleoflength3,wewillusethe notation K 3 .) Weemploytheprobabilisticmethod,oftenusedinrandomgra phtheory[18,49]to obtaintheresults.Westartouranalysisbycomputingthepr obabilityofappearance ofatleastone K 3 complexinthepotdescribedabove. Proposition5.3.1. Let P = f t 1 ; t 2 ; t 3 g beapottypeand P apotwhichcontains uniformlydistributed2-armedtilesoftypein P capableofadmittinga K 3 complex. Let X denotethenumberofcomplete K 3 complexesin P ,and r theprobabilitythat threeconnectedtilesbytwostickyendswillcloseinacompl ete K 3 complex.Under assumptionthattheconditionalprobabilityofthesecondc onnectionisthesameas theprobabilityoftherst,theexpectednumberof K 3 completecomplexesinthepot is E ( X )= mr ; moreover,when r =1 X = m almostsurely,where m denotestheamountoftilesin P ofeachtype. Proof. Fortheproofweusethefollowingnotation: S :asetof3tilesfrom t 3 ,oneofeachtype, A S :theeventthatthetilesfromSformacomplete K 3 X S :theassociatedindicatorrandomvariablefor A S B i :theeventthat h i and b h i willconnect,for i =1 ; 2 ; 3and i :theassociatedindicatorrandomvariablefor B i Forthesetofstickyendtypes H = f h 1 ; h 2 ; h 3 ; b h 1 ; b h 2 ; b h 3 g in S ,theprobability thatthetilesfrom S willformacomplete K 3 isequaltotheprobabilitythatalltiles 86

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of S wouldconnect.Thatis, P ( 1 =1 ; 2 =1 ; 3 =1)= P ( 1 =1) P ( 2 =1 j 1 =1) P ( 3 =1 j 1 =1 ; 2 =1) = ppr = p 2 r: (Noticeweassumetheconditionalprobabilityofthesecond connectionisthesame astheprobabilityoftherst). Since X S istheindicatorrandomvariablefortheevent A S X = P X S willdenote thenumberofcomplete K 3 'sinthepot,and E ( X S )= P ( A S )= p 2 r: Wehave m 3 sets S andbythelinearityofexpectationtheexpectednumberofco mplete K 3 'sinthepotis E ( X )= m 3 p 2 r: Ignoringthethermodynamicpropertiesofthesolution,the probabilityofonesticky endconnectingwithitscomplementaryis p = 1 m ,fromwhichitfollowsthat E ( X )= mr Tocalculatethevarianceforthenumberofcomplete K 3 'sinthepot Var( X )= X S;T Cov( X S ;X T ) weneedtocalculatethecovariancesrst: Cov( X S ;X T )= E ( X S X T ) E ( X S ) E ( X T ) : Inordertodothatweneedtolookattwosets S and T ,eachoneconsistingofthe threedierenttilesfromthepot,onefromeachtype.Again, fortheanalysisofthe covarianceweconsiderthecasewhen p = 1 m Case1: S \ T = ; 87

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Wehave m 3 ( m 1) 3 choicesforthiskindofpairofsets,andinthiscase E ( X S X T )= P ( X S =1 ;X T =1) = P ( X S =1 j X T =1) P ( X T =1) = r ( m 1) 2 r m 2 = r 2 m 2 ( m 1) 2 ; and E ( X S ) E ( X T )= r m 2 r m 2 = r 2 m 4 ; henceCov( X S ;X T )= r 2 m 2 ( m 1) 2 r 2 m 4 = r 2 (2 m 1) m 4 ( m 1) 2 ; X S \ T = ; Cov( X S ;X T )= m 3 ( m 1) 3 r 2 (2 m 1) m 4 ( m 1) 2 = r 2 (2 m 1)( m 1) m : Case2: j S \ T j =1 Wehave m 3 31 ( m 1) 2 =3 m 3 ( m 1) 2 choicesforthiskindofset,andweget: E ( X S X T )= P ( X S =1 ;X T =1) = P ( X S =1 j X T =1) P ( X T =1) =0 ; as P ( X S =1 j X T =1)=0,andas E ( X S ) E ( X T )= r m 2 r m 2 = r 2 m 4 ; X j S \ T j =1 Cov( X S ;X T )= 3 m 3 ( m 1) 2 r 2 m 4 = 3 ( m 1) 2 r 2 m : Case3: j S \ T j =2 88

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Wehave m 3 32 ( m 1)=3 m 3 ( m 1)choicesforthiskindofpairofsets,so E ( X S X T )= P ( X S =1 ;X T =1) = P ( X S =1 j X T =1) P ( X T =1)=0 ; and E ( X S ) E ( X T )= r m 2 r m 2 = r 2 m 4 ; andhence X j S \ T j =2 Cov( X S ;X T )= 3 m 3 ( m 1) r 2 m 4 = 3 ( m 1) r 2 m : Case4: j S \ T j =3 ; i.e. S = T Wehave m 3 choicesforthosekindofsets,so E ( X S ;X T )= P ( X S =1 ;X T =1) = P ( X S =1 j X T =1) P ( X T =1)= r m 2 ; and EX S EX T = r m 2 r m 2 = r 2 m 4 ; andhenceCov( X S ;X T )= r m 2 r 2 m 4 = r m 2 (1 r m 2 ) ; so X j S \ T j =3 Cov( X S ;X T )= m 3 r ( m 2 r ) m 4 = r ( m 2 r ) m : Fromtheobtainedinformationabove,addingsumstogether, weobtain Var( X )= X S;T Cov( X S ;X T )= mr (1 r ) : (NotethatVar( X ) 0if m 0and r 1.) When r =1,Var X = E ( X E ( X )) 2 =0,andsince X isanonegativerandom 89

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variableitfollowsthatalmostsurely X = E ( X )= m .Thatmeansalmostsurely only K 3 complexesareobtainedin P When r 1,forevery > 0,lim r 1 P ( j X E ( X ) j ) lim r 1 Var( X ) 2 ,i.e., lim r 1 P ( j X E ( X ) j )=0.Therefore,lim r 1 P ( j X E ( X ) j < )=1. Thecasewhen p< 1 m wouldresultwithincompletecomplexes,andwedonot considerthis,butcertainlywebelievethatsuchanalysism ayprovidevaluableinformationforunderstandingtheself-assemblyprocess. Torecapitulate,given m ,dependingontheamountofsolution,and r ,depending onthemoleculardynamics,theexpectednumberoftilesin K 3 cyclesis mr ,with standarddeviation p mr (1 r ),thelaterbeingunobservableundercontemporary laboratoryconditions. Wecangeneralizetheresult(obtainedforcomplete K 3 )forcircularcomplexesof anylength.Considerapotthatcontains n 2-brancheddierenttiletypesuniformly distributed,capableofformingacycleoflength n 90

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6DesignoftheExperiment Inthepreviouschapter,wedevelopedatheoryforcalculati ngtheexpectednumber ofcycliccomplexes.Inordertocheckthevalidityofthethe ory,wedidanexperiment correspondingtothetheoryandinthischapterwepresentou rndings. Figure6.1:Thedesiredproduct Thegoaloftheexperimentistocalculatetheprobabilityof obtainingcyclicproductswhenthree2-armjunctionmoleculesaregiven.Eachmol eculeiscomposedof threestrands.Onestrandis48nucleotideslong,anotherst randis19nucleotides longandthethirdoneis23nucleotideslong.(Seethepictur eabove.) Forrexibilityofthemolecules,bulges(6Tnucleotides)we readdedatthejunc91

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tionsandalsoanick(gapinthestrand,i.e.,alackofaphosp hodiesterbondbetween twoconsecutivenucleotides)attheothersideofthejuncti on. Eachofthejunctionmoleculesisformedfromthreeoligonuc leotides,singlestranded DNAmolecules.UsingthecomputerprogramSEQUINwedesigne dthestrandsfor themoleculesinsuchawaythattherewereno5bpmismatchesa ndthenumberof 4bpmismatcheswasminimal. Thedesignedsequenceswerethefollowing:MOLECULE1-M1 STRAND#1(AS11)CONSISTSOF:CGTAGTCACTGTGCGTCGCTGGTTTTTTTTGTCGTTGATGCTGATACASTRAND#2(AS12)CONSISTSOF:ACCAGCGACGCACAGTGACSTRAND#3(AS13)CONSISTSOF:CTGGTGTATCAGCATCAACGACAMOLECULE2-M2 STRAND#1(AS21)CONSISTSOF:CCAGCGATGTCGTCACTGTAGTATTTTTTATGGTAGCACACGCATCAGSTRAND#2(AS22)CONSISTSOF:TACTACAGTGACGACATCGSTRAND#3(AS23)CONSISTSOF:TCAACTGATGCGTGTGCTACCATMOLECULE3-M3 STRAND#1(AS31)CONSISTSOF:TTGACTACAACATCGCAGCATCATTTTTTGACCAGCGTGTGCTACTGTSTRAND#2(AS32)CONSISTSOF:TGATGCTGCGATGTTGTAG 92

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STRAND#3(AS33)CONSISTSOF:TACGACAGTAGCACACGCTGGTCThreeoligonucleotides(strands#3fromeachmolecule)wer epurchasedfrom IntegratedDNATechnologyandthenpuriedwiththestandar dpuricationprocess inthelaboratory.Theremainingsixoligonucleotideswere designedtohaveanextra Phosphorattachedtotheir5'end,whichisneededforthelig ationprocess.Those strandsweremadeinDr.Seeman'sLaboratorybyRujieSha. Themoleculeswereformedbyannealing.Annealingisaproce ssbywhichtwo complementarysinglestrandedDNAmoleculesbondthrought heWatson-Crickcomplementarity.Weformedthemoleculesbythefastannealing protocol(seeSection 6.1).Allthreemoleculesformedwell,ascanbeseenfromthe guresgivenbelow. Sincetheprojectitselfrequiresconsiderableprecision, inordertocalculatethe percentageofmoleculesinacomplex,oneneedstoreadthere sultsusingaverysensitivemethod.Onesuchmethodmeasuresthepercentagesbas edontheradioactive counts(seeSection6.1).Onasmallportionofthestrands#1 weattachedaradioactivephosphate(P 32 ),inordertorecord(usingPhosphorImager)everycomplex formed.Hence,onestrandofeachjunctionmoleculeisradio activelylabeled,and thepercentageofmoleculesthatformacomplexcanbeeasily determinedforeach complex.Whenweconstructedeachjunctionmoleculeweadde d10%oftheradioactivelylabeledstrand#1and90%ofthestrand#1thathadregu larPhosphateon its5'end. Thecomplexeswereformedbyslowannealingofthejunctionm olecules.We designedseveraltesttubeswithdierentconcentrationso fthemolecules.Theslow annealingprocesslastedforseveralhoursandafterthatth ecomplexeswereligated. (LigationisaprocessbywhichabackboneofaDNAstrandisre covered,i.e.,ifthere isanickinaDNAmoleculesuchthatthereisaPhosphoronthe5 'endatthenick ofthemolecule,thenaligationenzymesealsthenickandrec oversthebackboneof themolecule.) 93

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Figure6.2:Thisisa12%nativegeltochecktheformationofM olecule1.A10nucleotide markerisintherstlane.InthesecondlaneisMolecule1,in thethirdlaneisacomplex consistingofthestrandsAS11andAS12,inthefourthlaneis acomplexconsistingof strandsAS11andAS13,inthefthlaneisacomplexconsistin gofthestrandsAS12and AS13,inthesixthlaneisthestrandAS11,intheseventhlane isthestrandAS12,andin theeightlaneisthestrandAS13. 94

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Figure6.3:Thisisa12%nativegeltochecktheformationofM olecule2.A10nucleotide markerisintherstlane.InthesecondlaneisMolecule2,in thethirdlaneisacomplex consistingofthestrandsAS21andAS22,inthefourthlaneis acomplexconsistingof strandsAS21andAS23,inthefthlaneisacomplexconsistin gofthestrandsAS22and AS23,inthesixthlaneisthestrandAS21,intheseventhlane isthestrandAS22,andin theeightlaneisthestrandAS23. 95

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Figure6.4:Thisisa12%nativegeltochecktheformationofM olecule3.A10nucleotide markerisintherstlane.InthesecondlaneisMolecule3,in thethirdlaneisacomplex consistingofthestrandsAS31andAS32,inthefourthlaneis acomplexconsistingof strandsAS31andAS33,inthefthlaneisacomplexconsistin gofthestrandsAS32and AS33,inthesixthlaneisthestrandAS31,intheseventhlane isthestrandAS32,andin theeightlaneisthestrandAS33. 96

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Afterthecomplexeswereligated,eachtesttubecontainedc yclicmoleculesaswell aslinearones.Inordertodistinguishthelinearonesfromt hecyclic,wesplitthe solutionofeachtesttubeintotwoequalonesandaddedanExo IandExoIIIenzyme inoneofthetwohalves.Exoenzymesworkinthefollowingway .Iftheyrecognize anickinamolecule,theychopupthemolecule,nucleotideby nucleotide,likethe gamepackman.IfanExoenzymeisaddedintoatesttubethatco ntainsbothlinear andcircularmolecules,theenzymewilldestroyalllinearm olecules,leavingonlythe cycliconesinthesolution. Theresultsareobtainedbymeasuringthesizesoftheobtain edcomplexes.A standardmethodformeasuringthelengthofaDNAmoleculeis by gelelectrophoresis TheelectrophoresistechniqueisbasedonthefactthatDNAm oleculesarenegatively chargedandiftheyareplacedinanelectriceld,theywillm ove(migrate)towards thepositiveelectrode.Thegelisprepared,anditisinsert edbetweentwo20cm.long squareglassplates.Beforethegelthickens,acombisinser tedbetweentheplates,to formwells(andlateronlanes),anditisremovedafterthege lthickens.TheDNA solutionisputinthewells,andthegelisconnectedonanele ctriceld. InonewellweaddonehalfofaDNAsolutionthatdoesnotconta inExoenzymes, andinthelanenexttoitweaddtheotherhalfofthesolution, theonethatcontains ExoIandExoIIIenzymes.Thiswayitiseasytomakethecompar isonwhichband oftherstlanecorrespondstoacycliccomplex. Aftertheelectricsystemisturnedo,theresultsareobtai nedbyexposingthegel toaPhosphorImagersystem.APhosphorImagersystemisaqua ntitativeimaging devicethatusesstoragephosphortechnologyinlifescienc eimagingapplications.It lookslikeacassettewithawhiteboardinit,andthegelisex posedonthewhite board.Theboardcountstheradioactivityemittedfromtheg el,andtheresultsare obtainedbyscanningtheboardwithaspecialscanner.Since thePhosporImager systemcountsonlyradioactivity,thegelresultswilldepe ndonthestrand#1(48 nucleotideslong)fromeachmolecule(sincethatstrandwas radioactivelylabeled). Hence,alinearmonomerwillbe48nt.,alineardimerwillhav ealengthof96nt., 97

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alineartrimerandcyclicmonomerwillhavealengthof144nt .,etc.Usuallycyclic moleculestravelslowerthenthelinearonessothesmallone stravelfaster,etc.,and wedetectedthatcyclic144nt.runssimilarlyaslinear240n t. Inthenextsection,adetaileddescriptionoftheprotocols usedisgiven.InSection 6.2wegivetheresults. 6.1TheProtocol Hereisamoreprecisedescriptionoftheprotocolsusedinth eexperiment. STEP0:KINATION(Radioactivelabeling)bythePhosphoryla tion Wemixtogether1 L DNA(1pmole),1 L kinationbuer,1 L labeledATP, 6 L ddH 2 O,and1 L kinase.Thereactionproceedsat37degreeforabout1 hour.Thenweinactivatethekinase,byleavingthesolution on90degreesfor5min. Afterwards,welterthesolutionthroughtheG-25microspi ncolumn(Pharmacia) toremoveunincorporatedg-32P-ATP.Thenwedophenolextra ctionandethanol precipitation.Attheend,thehotDNAispuriedbya10to15% denaturinggel. STEP1:ANNEALINGTHEJUNCTIONMOLECULESForeachmolecule,wecombinethethreestrandstogethertha tthemoleculeconsistsof(in1XTAEMg ++ Buer(12.5 M ,where M standsformicromolar)) andthenfastannealthem(5minon90 C ,15minon65 C ,20minon45 C ,20min on37 C and20minonroomtemperature). STEP2:ANNEALINGTHECOMPLEXESAfterthejunctionmoleculeswereformed,theywerecombine dtogethertoanneal inasolutioncontaining1XTAEMg ++ ,1XLigaseBuerandddH 2 Odependingon thedesiredconcentration.Theywereannealingslowlyinap eriodofmorethan24 hoursfrom45 C toRoomTemperature.(Thetemperatureuniformlydeclined. ) Intherstsetofexperiments,thevolumewasconstant(30 L ),thereforethe countsineachconcentrationweredierent.Inthesecondse tofexperiments,we usedthesamecounts,thereforeweuseddierentvolumefore achconcentration. STEP3:LIGATINGTHECOMPLEXES 98

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Afterthecomplexeswereannealed,weputthetesttubesinth eincubator(temperature16 C )for10minutes,thenweadded2 L ofLigaseenzymeandleftthem intheincubatoragainforatleast16hours. STEP4:EXOTREATMENTAfterligation,wedividedthesolutionofeachtesttubeint otwoequalparts.In oneofthemweadded1 L ofExoIand1 L ofExoIIIandputthattesttubefor1 houron37 C .Thesecondonestayedthesame. STEP5:GELWesawtheresultsonan8%denaturinggel,on20cmlongsquare glassplates, whichranforatleastanhour. STEP5:SCANNINGTHERESULTSWescannedtheresultsusingPhosphorimagerandtheaccompa nyingsoftware MolecularDynamics. 6.2Results FirstsetFirstwedidasetof10experimentswithconcentrationsrang ingfrom10 M to 0 : 083 M .WesetupthetesttubesinthewaygiveninFigure6.2.Forexa mple,the junctionmoleculeM1isconstructedbycombiningthestrand sAS11,AS12,AS13, andAS11*,whichistheradioactivelylabeledstrand,plusa buer.Inparticular,in atesttubewecombined11 L of30 M molecule AS 11,12 L of30 M molecule AS12,12 L of30 M moleculeAS13,and1 L ofthemoleculeAS11*and4 L of TAEMg ++ (abueralwaysconstitutes10%ofthetotalvolumeanditisn ecessary fortheannealingprocess).Inthetesttubethereis12 30=360pmolsofeach strand.Since,1pmolofAS11+1pmolofAS12+1pmolofAS13=1p molMin thetesttubethereare360pmolsofthemoleculeM1,ina40 L volume,i.e.,the concentrationofthemoleculeM1inthetesttubeis 360 40 =9 M Todescribethewaycomplexeswereformedweuseasanexample thetesttube 99

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5(T5*)whoseconcentrationis1 M .InthetesttubeT5*,3 L ofeachofthe threejunctionmoleculeswasadded,andalso1 : 8 L TAEMg ++ buerwasadded (notethattherewasalready0 : 3 L ofTAEMg ++ inthesolutionforeachjunction molecule),3 L ofLigasebuerand9 L ddH 2 0todilutethesolutionforthedesired concentration.ThetotalvolumeinthetesttubeT5*is31.8 L andthereare3 9= 27pmolsofeachcomplex,sotheconcentrationonthetesttub eis 27 31 : 8 =0 : 85 M Inthisset,wekeptthevolumeofthesolutioninthetesttube sthesame(30 L), andthereforethenumberofradioactivecountswasdierent 100

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Figure6.5:(a)10nucleotidemarker,(b)ligatedmolecules fromthetesttubeT5of1 M concentration,(c)exotreatmentoftheligatedproductfro mtesttubeT5of1 M concentration,(d)ligatedmoleculesfromtesttubeT4of1 : 5 M ,(e)exotreatmentofthe ligatedproductfromtesttubeT4of1 : 5 M ,(f)ligatedmoleculesfromtesttubeT3of2 M (g)exotreatmentoftheligatedproductfromtesttubeT3of2 M ,(h)ligatedmolecules fromtesttubeT2of2 : 5 M .(i)exotreatmentoftheligatedproductfromtesttubeT2of 2 : 5 M 101

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Figure6.6:(a)10nucleotidemarker(b)ligatedmoleculesf romthetesttubeT10of0.1 M concentration,(c)exotreatmentoftheligatedproductfro mtesttubeT10of0.1 M concentration,(d)ligatedmoleculesfromtesttubeT9of0 : 2 M ,(e)exotreatmentofthe ligatedproductfromtesttubeT9of0 : 2 M ,(f)ligatedmoleculesfromtesttubeT8of 0 : 3 M ,(g)exotreatmentoftheligatedproductfromtesttubeT8of 0 : 3 M ,(h)ligated moleculesfromtesttubeT7of0 : 4 M ,(i)exotreatmentoftheligatedproductfromtest tubeT7of0 : 4 M ,(j)moleculesfromtesttubeT6of0 : 5 M ,(k)exotreatmentofthe ligatedproductfromthetesttubeT6of0 : 5 M 102

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CalculationonthepercentageswasdoneusingtheSoftware\ MolecularDynamics".Thissoftwaregivesanestimateonthepercentagesbas edontheintensityofthe bands.Herearetheresults. Ligationisaprocesswithlowyield.Unfortunately,thatis ouronlywaytocheck whetheracomplexiscompleteornot.Ifagivencomplexforms acyclethenitis complete,butsometimesacomplexcanformacycle,butnotal lofthenicksatthe stickyendsareligated.Inthatcasethecomplexappearsasa linearone,althoughit isactuallycomplete.Whenthreejunctionmoleculesconnec t,theyconnectbytheir stickyends.Eachstickyendconnectionleavesonenickatth eedgeofthetriangle. Inorderforthosethreemoleculestoformacyclewithnonick s,threenicksshould beligatedandtheprobabilityofthreeligationsisnothigh .Henceinanalyzingthe resultsweassumethatthecircularmoleculesoflength144n t.andlinearmoleculesof length144nt.bothformedacompletecomplex,justthersto nesligatedthreenicks, whileinthesecondonlytwonickswereligated.Henceifwein terpretatetheresults 103

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inthefollowingway.Fortheconcentration2 : 5 M 44.32%ofcyclicmonomers,for 2 M 46.12%cyclicmonomers,for1 : 5 M 53.58%cyclicmonomersandfor1 M 57.42%.Fromthisitisclearthatbydecreasingtheconcentr ation,thenumberof cyclicmonomersincreases. For0 : 5 M ,46.01%,for0 : 4 M ,50.43%,for0 : 2 M 51.72%.Itisimportantto mentionthatthebestwaytocomparethepercentageofthecyc licmonomersisby consideringthesolutionsthatrunonasamegel.Sinceasoft wareisusedtoobtain thepercentages,theaccuracyoftheresultsdependsonthea ccuracyofthesoftware. Inthetesttubeof0 : 3 M forcalculatingthepercentagesofthecyclicmolecules, themoleculesoflength48nt.werenottakenintoconsiderat ion,hencewearenot consideringtheresultsforthattesttube.SecondsetIntherstsetofexperiments,wekeptthevolumeofeachsolu tionthesame,thusthe amountofradioactivecountsineachtesttubewasdierent. Theresultsshowedno correlationbetweentheamountofcyclictrianglesandconc entrationofthesolution, sowechangedtheset-upofthetesttubesforthenextsetofex periments.For thesecondsetofexperimentswedecidedtokeeptheamountof radioactivecounts thesame,andchangethevolumeaccordingly.Thiswayfromon elookofthegel, onecanseewhetherthenumberofcyclesoflength3increases aswedecreasedthe concentration,whichiswhatweassumed. Thelowerconcentrationtesttubehadhighervolume,conseq uentlythehigher concentrationtesttubehadlowervolume.Eachbuerconsti tutes10%ofthetotal volume,thusinalowconcentrationsolutiontheamountofio ns(fromthebuers)was highandthatpreventstheDNAmoleculesfromfreemovementi nagel.Although mostofthegelsweredestroyedwewereabletodrawaconclusi onfromtheresults. Accordingtothetheoreticalanalysis,weassumedthatinav erylowconcentration,almostsurelyallofthejunctionmoleculesshouldbec omeapartofatriangle. However,thatwasnotthecaseintheprevioussetofexperime nts,andneitherisit 104

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withthisone. Wesetupthetesttubesinthefollowingway: Theresultsarethefollowing 105

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Figure6.7:(a)10nucleotidemarker,(b)ligatedmolecules fromthetesttubeT1of1 M concentration,(c)exotreatmentoftheligatedproductfro mtesttubeT1of1 M concentration. 106

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Figure6.8:(a)10nucleotidemarker,(b)ligatedmolecules fromthetesttubeT2of0 : 75 M concentration,(c)exotreatmentoftheligatedproductof0 : 75 M concentration,(d)ligated moleculesfromtesttubeT3of0 : 5 M concentration,(e)exotreatmentoftheligated productfromtesttubeT3of0 : 5 M concentration. 107

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Likewementionedbefore,thebestwaytocomparethepercent agesisbyconsideringonegelatatime.Inthatcaseitisclearfromthissetof experimentsthatasthe concentrationdecreases,theproportionofmoleculesthat areintrianglesincreases.It isalsoobviousthatbesidesthemonomers(triangles),dime rsalsoappearregardless oftheconcentration,i.e.,wecannotconcludethatinavery dilutedsolution,the probabilityofthelaststickyendtoclose, r ,approaches1. ThirdsetWerealizedthatthelengthonthegelplateswasshort,i.e.f orbetterdierentiation ofthelengthoftheDNAmolecules,werepeatedtheprocesson longergelplates(40 cm.). 108

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Figure6.9:(a)10nucleotidemarker,(b)ligatedmolecules fromthetesttubeof3 M concentration,(c)exotreatmentoftheligatedproductof3 M concentration,(d)ligated moleculesfromthetesttubeof1 : 5 M concentration,(e)exotreatmentoftheligatedproductof1 : 5 M concentration,(f)ligatedmoleculesfromthetesttubeof1 M concentration, (g)exotreatmentoftheligatedproductof1 M concentration. 109

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Figure6.10:(a)10nucleotidemarker,(b)ligatedmolecule sfromthetesttubeof0 : 5 M concentration,(c)exotreatmentoftheligatedproductof0 : 5 M concentration,(d)ligated moleculesfromthetesttubeof0 : 1 M concentration. 110

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Figure6.11:(a)10nucleotidemarker,(b)ligatedmolecule sfromthetesttubeof1 M concentration,(c)exotreatmentoftheligatedproductof1 M concentration,(d)ligated moleculesfromthetesttubeof0 : 5 M concentration,(e)exotreatmentoftheligatedproductof0 : 5 M concentration,(f)ligatedmoleculesfromthetesttubeof0 : 1 M concentration, (g)exotreatmentoftheligatedproductof0 : 1 M concentration. 111

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Theresultswerethefollowing: 6.3Conclusion Fromtheseveralsetsofexperimentswedid,wecanconcludet hatwiththedecrease intheconcentrationtheamountofcyclicmonomersincrease s.Also,wecanconclude thatthenumberofcyclicdimersdecreaseswiththeincrease intheconcentration, buttheydonotdisappear,asweassumedinthetheoreticalse ttings.Appearanceof cyclictrimerswasnoticedinthelastsetofexperiments(si ncethelengthofthegel plateswasdoubled),butitwasnotconrmed.Wehavetoconr mtheappearanceof cyclictrimersaswellastocheckiftherearecyclicmolecul esoflargersize.Besides, examiningthesizeofallthecyclicmolecules,weshouldals oexaminethestructure. Inparticular,whethertheyrepresentonebigcyclicmolecu le,ortheyaretwosmaller cyclicmoleculeslinkedtogether,oronetwistedmolecule, etc.? Theresultsthatweobtainedfromtheexperimentsweplantou sethemforstudyingthethermodynamicsandkineticpropertiesofthemolecu le.Inthenextsection, werepresentsomeoftheideasthatwewanttopursueforfutur eresearch.Oneidea istouserateequations,andforthatweneedtoknowallthepo ssiblecomplexes, bothlinearandcyclic.Wehopetoextendthisstudyfromcycl icmolecules,toany typeofcomplexes,regardlessoftheirstructure. 112

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7Conclusion Giventhepossibilitiesofestablishingdierentconnecti ons,DNAmoleculescanconnectinmanydierentwaysbuildingdierentnewstructures .Althoughmostofthe researchmethodsinthisdissertationtargetDNAself-asse mbly,theyareverygeneral inasensethattheycanbeextendedandappliedtoanyotherty peofself-assembly. Thiscouldbedonesothattheelementsthatself-assemblewi llcorrespondtotiles, andthepartsofthem(orconnectionrules)bywhichtheeleme ntsconnectwillcorrespondtostickyends.Thentheprocessofgluingandformin gcomplexesshouldbe denedsimilarlyaswedidinChapter2,andtheotherdeniti onsandpropositions shouldbechangedaccordingly. Inthisdissertationweinvestigatefundamentalquestions relatedtoDNAselfassemblyandhavedevelopedtheappropriatetoolsaccordin gly.Thequestionswe haveaddressedare:Whatarethepossiblecomplexesobtaine dattheendofthe self-assemblyprocess?Whichofthosecanbeactuallyexpec tedandwithwhatprobability?Whatarethenecessaryconditionstoeliminatethe byproducts?Howto pre-designatesttubetominimizethoseby-products?Whatr elationscanbedened ontesttubes? Knowingthatmanytimesinscienticlaboratories,besides asolutiontoagiven problem,alotofextraneousmaterial(non-completecomple xes)alsoappears,we addressedtheproblemofeliminating(oratleastminimizin g)thesludgeinChapter 3.Weprovedthatanecessaryconditionforobtainingonlyco mpletecomplexesat theendofanexperimentistousetheproperproportionofeac htypeofmolecules present.Thesetofvectorsfortheproperproportionsiscal led\spectrum",and 113

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withtheMAPLEprogramgiveninAppendixAitcanbeveryquick lyandeasily determined. Thepottypesareclassiedinthreeclassesaccordingtopos siblecomponents thatassembleincompletecomplexes,andbasedonthespectr umtheclassicationis PTIMEcomputable.Thequestionthataroserecentlyiswheth erthroughthespectrumwecanclassifythepossiblecompletecomplexes.Using computationalgeometry, weshowedthatbasedontheextremalpointsofthespectrum,w ecandescribeaportionofthecompletecomplexes,butnotallofthem.Nextstep topursueistond waystodescribethoserandomobjectsthroughalgebraicrep resentation,orthrough algebraicandgraphicalrepresentationcombined.Another questionthatweshould addressisifitisdecidablewhetherornotapottypecontain sacompetecomplex thathastilesofeachtype.Ifweareabletoclassifyallthec ompletecomplexeswith algebraicrepresentaton,thenthisquestionshouldbeansw ered. Thegraphicalrepresentation,giveninChapter4,wasdevel opedtostudythe productsofself-assemblywithfamiliartoolsthathelpinu nderstandingtheprocess. Thegraphmodelisusedtodeterminewhatcompletecomplexes canbeassembled fromagivenpottypeaswellascompareandclassifythepotty pesthemselves.It representsanapplicationofgraphhomomorphismtheory.We onlyscratchedthe surfaceofthetopic,thereismuchmoretobedoneinthatdire ction.Immediate questionsthatarisefromthissectionare:Isthereaconnec tionbetweenthepot classicationandgraphrepresentation,i.e.,isthereawa ytoclassifythepottypes usingtheirpotgraphs?(Weknow,thatweshouldnotlookatgr aphconnectivity, becauseitispossibleforastronglysatisablepottypetoh avedisconnectedgraph, whileanunsatisablepottypetohaveaconnectedgraph.)Ar eminimalcomplex graphs,primegraphs?Canwededucesomekindofconclusionf orthemaximal completecomplexgraphs,basedonthetheoryofprimegraphs ? Soastodeterminethepossiblecomplexesattheendofthesel f-assemblyprocess andwithwhatprobabilitytheyoccur,inChapters5and6weco ncentratedonthe formationofcycliccomplexes.Inaquesttocalculatetheex pectednumberofcyclic 114

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moleculesthatshowattheendofanexperimentandtheprobab ilityoftheirformation,webuiltanewrandomgraphmodelthatconsidersself-a ssembly.Naturally, thenextsteptopursueistoadjusttherandomgraphmodelsoa stoincludeother moleculesbesidescyclicones.Inordertodothat,rstones houldunderstandthe numberofdierentprobabilitiesofthesystemandbuildthe probabilityspaceaccordingly.Although,theintermediatestepsinthestudyof thecyclicmoleculeswere known,weshouldperformastudyforthemodelwitharbitrary (notonlycyclic) complexes.Thatwillgivebetterinsightintothepossiblep robabilitiesandthenal distributionoftherandomstructures.Thestudyoftheinte rmediatestepsshould bedonewithbranchingprocesses,tounderstandhowonecomp lexmayevolveover time,andgiveacompletepictureoftheprocessandthenalo utcomes. Thermodynamicshasn'tbeenincorporatedintherandommode ls.Wearecurrentlyconsideringwayshowtoincludeit.Therstapproach thatwetookiswitha systemofrateequations. Hereisanideahowtoincludethesystemofrateequations.As sumewehavea pottypewithonlyonetiletype,suchthattilescancloseina completecomplex,and thereareexactly m tilesinthepot.Denotewith C n thenumberofcycliccomplexes consistingof n tilesinstep t L n ( t )thenumberoflinearcomplexesconsistingof n tilesinstep t k i;j therateatwhichalinearcomplexoflength i connectstoalinear complexoflength j ,and r i therateatwhichacycleoflength i closes.Thefollowing systemofequationsmodeltheevolutionoftheselfassembly process. dL n ( t ) dt = 1 2 n 1 X i =1 k ( n i;i ) L n i L i r n L n dC n ( t ) d t = r n L n m X i =1 i ( L i + C i )= m ,and L 1 (0)= m C 1 (0)=0, L (laststep)=0, P mi =1 iC i (laststep)= m Thesystemitselfisdiculttosolve,andweplantostudythi ssystemfurther. 115

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Inthisdissertationweapproachedtheself-assemblyproce ssthroughdiscrete mathematicaltheory.Weaddressedsomeofthefundamentalq uestionsrelatedto theprocess,buttherearemoretobeconsidered.Wehopeyoue njoyedthejourney, aswedid,throughthisdissertation. 116

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[52]J.C.Wang,N.Davidson. OntheProbabilityofRingClosureofLambdaDNA ,Journalof MolecularBiology19(1966),469-482. [53]Y.Wang,J.E.Mueller,B.Kemper,N.C.Seeman. Theassemblyandcharacterizationof5 armand6armDNAjunctions ,Biochemistry30(1991),5667-5674. [54]E.Winfree. AlgorithmicSelf-AssemblyofDNA:TheoreticalMotivation sand2DAssemblyExperiments ,JournalofBiomolecularStructureandDynamics,11(S2)(2 000),263-270. [55]E.Winfree. Self-HealingTileSets ,Nanotechnology:ScienceandComputation(edt.J.Chen, N.Jonoska,G.Rozenberg),(2006),55-78. [56]E.Winfree,R.Bekbolatov. ProofreadingTileSets:ErrorCorrectionforAlgorithmicS elfAssembly ,LNCS2943(2004),126-144. [57]E.Winfree,F.Liu,L.A.Wenzler,N.C.Seeman. Designandself-assemblyoftwo-dimentsional DNAcrystals ,Nature394(1998),539-544. [58]H.Yamakawa,W.H.Stockmayer. StatisticalMechanicsofWormlikeChains.I.Asymptotic Behavior ,TheJournalofChemicalPhysics,Volume57,Issue7,(1972) ,2839-2843. [59]H.Yamakawa,W.H.Stockmayer. StatisticalMechanicsofWormlikeChains.II.Excluded VolumeEect ,TheJournalofChemicalPhysics,Volume57,Issue7,(1972) ,2843-2854. [60]H.Yan,X.Zhang,Z.Shen,N.C.Seeman, ArobustDNAmechanicaldevicecontrolledby hybridizationtopology ,Nature415(2002),62-65. [61]B.Yurke,A.J.Turbereld,A.P.Mills,F.C.Simmel, ADNAfueledmolecularmachinemade ofDNA ,Nature406(2000),605-608. [62]Y.Zhang,N.C.Seeman. TheconstructionofaDNAtruncatedoctahedron ,JournalofAmerican ChemicalSociety116(5)(1994),1661-1669. 121

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Appendices AppendixA-MapleProgram Inthissectionwegivethemapleprogramforcalculatingthe spectrumandapotclass ofagivenpot.Whatfollowsisasolutionforthespectrumoft hepottypeexplained inFig3.2.Thisisaprogramthatcancalculatethespectrumofagivenpo t.Assumethata pot P with m junctiontypesand n stickyendtypesisgiven. > restart: > with(LinearAlgebra): Inputthenumberofjunctiontypes > m:=3: Inputthenumberofstickyendtypes > n:=2: > c:=0: Enterthecorrespondingz(h)vectors.Thecoordinatefora[ s,t]=z f j f s gg (h f t-1 g ) > a:=Matrix([[1,1,1,1],[2,-1,0,0],[1,1,-3,0]]); a := 26664 11112 100 11 30 37775 > L:=Matrix([[1,1,0],[1,1,1]]); L := 24 110111 35 122

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> A:=ReducedRowEchelonForm(a):evalm(A): > B:=Matrix(n+1,m): > forifrom1ton+1do > forjfrom1tomdo > B[i,j]:=A[i,j]:enddo:enddo;B: > b:=Vector(n+1): > forifrom1ton+1do > b[i]:=A[i,m+1]:enddo:b; 26666664 1 41 21 4 37777775 > if(evalb(Rank(A)=Rank(B)))=false > thenc:=-1:endif: > r:=LinearSolve(B,b,free='t'); r := 26666664 1 41 21 4 37777775 > forifrom1tomdo > eq[i]:=r[i]>=0; > enddo: > Sys:=seq(eq[i],i=1..m); > d:=seq(t[i],i=1..m); > s:=solve( f Sys g f d g ); Sys :=0 1 4 ; 0 1 2 ; 0 1 4 d := t 1 ;t 2 ;t 3 s := f t 1 = t 1 ;t 2 = t 2 ;t 3 = t 3 g 123

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> forifrom1tomdo > forjfrom1tomdo > ifevalb(s[i]=(t[j]=0))=truethenr:=subs( f t[j]=0 g ,eval(r)) > endif: > enddo:enddo: > Supp:=[]:forifrom1tomdo > ifevalb(r[i]>0)=falsethenSupp:=[op(Supp),0]; > elseSupp:=[op(Supp),1];endif:enddo:'Supp'=Supp; Supp =[1 ; 1 ; 1] > ifevalb(c=0)=truethen > forifrom1tondo > forjfrom1tomdo > ifevalb(Row(L,i)[j]=1andSupp[j]=1)=truethenc:=0: > break;elsec:=1:endif;enddo; > enddo;endif;c; 0 > ifevalb(c=0)=truethen > forifrom1tomdo > ifevalb(r[i]>0)=falsethenc:=2:break; > endif;enddo;endif; > ifevalb(c=0)=truethenprint("Thegivenpotisstronglysa tisfiable > anditsspectrumis");'r'=r;s;endif; > ifevalb(c=1)=truethenprint("Thegivenpotisweaklysati sfiable and> itsspectrumis");'r'=r;s;endif; > ifevalb(c=2)=truethenprint("Thegivenpotissatisfiabl eandits > spectrumis");'r'=r;s;endif; > ifevalb(c=-1)=truethenprint("Thegivenpotisnotweakly > satisfiable");endif; \Thegivenpotisstronglysatisableanditsspectrumis" r = 26666664 1 41 21 4 37777775 124

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f t 1 = t 1 ;t 2 = t 2 ;t 3 = t 3 g > forifrom1tomdo > e[i]:=solve(r[i]=0, f d g ); > subs(e[i],eval(r)); > enddo: > sys1:=seq(e[i],i=1..m): 125

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AppendixB-ExistenceoftheProbabilitySpace Inthissectionofthedissertationwepresentamathematica llysupportforChapter5, i.e.,aformalprooffortheexistenceoftheprobabilitymea sureandthesamplespace, andtogiveanideahowthesamplespaceshouldlooklike.Asab ovewestartwith observationofthesamplespace(n m )consistingoftheoutcomesintheself-assembly processforaonetypepotwith m tiles, P m = f t g .Weassumethatduringtheprocess allWatson-Crickconnectionsareestablishedandthatonly completecomplexesare obtained.Thatmeansaftertheannealingprocesseverytile isonacompletecycle. AlsoweassumethatallWatson-Crickconnectionsareequall ylikely.Specically, anoutcomeisacomplexconsistingofacomplexgraphof m vertices,asa2-regular graph.Ourassumptionsassurethattwoisomorphicoutcomes areequallylikely. Denotewith r m;k theprobabilityinn m ,thatagiventileisonacycleoflength k Theprobabilitymeasureforthissamplespacewewilldenei trecursively. Considertwodierenttiles,say t 1 and t 2 ,fromtype t 2 P m .AssumingallW-C connectionsareequallylikely,foraxednumber k m P ( t 1 isonacycleoflength k )= P ( t 2 isonacycleoflength k )= r m;k : Thenwecandene r m;k recursively. Fix t 1 t 2 For1 k
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P ( t 1 isonacycleoflength k ) = P ( t 1 isonacycleoflength k j t 1 and t 2 arenotonthesamecycle ) P ( t 1 and t 2 arenotonthesamecycle ) + P ( t 1 isonacycleoflength k j t 1 and t 2 areonthesamecycle ) P ( t 1 and t 2 areonthesamecycle ) When t 1 isonacycleoflength k ,and t 1 and t 2 arenotonthesamecycle,then t 2 willbeonanyothercycleoflength n ,forsome1 n m k .Inthatcase P ( t 1 isonacycleoflength k j t 1 and t 2 arenotonthesamecycle ) P ( t 1 and t 2 arenotonthesamecycle ) = m k Xn =1 P ( t 1 isonacycleoflength k j t 1 and t 2 arenotonthesamecycle ; t 2 isonacycleoflength n ) P ( t 1 and t 2 arenotonthesamecycle ;t 2 isonacycleoflength n ) = m k Xn =1 P ( t 1 isonacycleoflength k j t 1 and t 2 arenotonthesamecycle ; t 2 isonacycleoflength n ) P ( t 1 and t 2 arenotonthesamecycle j t 2 isonacycleoflength n ) P ( t 2 isonacycleoflength n ) = m k Xn =1 r m n;k m n m 1 r m;n : When t 1 and t 2 areonthesamecycleandisgiventhat t 2 isonacycleoflength n = k ,then t 1 needstobeoneoftheremaining k 1tilesonthatcycle.Therefore theprobability t 1 isonacycleoflength k underthegivenconditionswillbe k 1 m 1 i.e., 127

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P ( t 1 isonacycleoflength k j t 1 and t 2 areonthesamecycle ) P ( t 1 and t 2 areonthesamecycleoflength k ) = k 1 m 1 r m;k : Mergingbothequationstogetherweobtain P ( t 1 isonacycleoflength k )= r m;k = m k Xn =1 r m n;k m n m 1 r m;n + k 1 m 1 r m;k Hence,for1 k
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Theprobabilitiesforthesamplespacen m canbecalculatedfromthefollowing systemofequations: 8>>>><>>>>: m X k =1 r m;k =1 r m;k = 1 m k m k X n =1 ( m n ) r m n;k r m;n ; for1 kn 129

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Letusexpandtheaboveequations. r m; 1 = 1 m 1 h ( m 1) r ( m 1) ; 1 r m; 1 +( m 2) r ( m 2) ; 1 r m; 2 +( m 3) r ( m 3) ; 1 r m; 3 + ::: +3 r 3 ; 1 r m; ( m 3) +2 r 2 ; 1 r m; ( m 2) + r 1 ; 1 r m; ( m 1) i r m; 2 = 1 m 2 h ( m 1) r ( m 1) ; 2 r m; 1 +( m 2) r ( m 2) ; 2 r m; 2 +( m 3) r ( m 3) ; 2 r m; 3 + ::: +3 r 3 ; 2 r m; ( m 3) +2 r 2 ; 2 r m; ( m 2) i r m; 3 = 1 m 3 h ( m 1) r ( m 1) ; 3 r m; 1 +( m 2) r ( m 2) ; 3 r m; 2 +( m 3) r ( m 3) ; 3 r m; 3 + ::: +3 r 3 ; 3 r m; ( m 3) i ...r m; ( m 3) = 1 3 h ( m 1) r ( m 1) ; ( m 3) r m; 1 +( m 2) r ( m 2) ; ( m 3) r m; 2 +( m 3) r ( m 3) ; ( m 3) r m; ( m 3) i r m; ( m 2) = 1 2 h ( m 1) r ( m 1) ; ( m 2) r m; 1 +( m 2) r ( m 2) ; ( m 2) r m; 2 i r m; ( m 1) =( m 1) r ( m 1) ; ( m 1) r m; 1 (7.0.1) Wewilltransformthissystemintoahomogeneousone. ( r ( m 1) ; 1 1) r m; 1 + m 2 m 1 r ( m 2) ; 1 r m; 2 + m 3 m 1 r ( m 3) ; 1 r m; 3 + ::: + 3 m 1 r 3 ; 1 r m; ( m 3) + 2 m 1 r 2 ; 1 r m; ( m 2) + 1 m 1 r 1 ; 1 r m;m 1 =0 m 1 m 2 r ( m 1) ; 2 r m; 1 + r ( m 2) ; 2 r m; 2 + m 3 m 2 r ( m 3) ; 2 r m; 3 + ::: + 3 m 2 r 3 ; 2 r m; ( m 3) + 2 m 2 r 2 ; 2 r m; ( m 2) =0 m 1 m 3 r ( m 1) ; 3 r m; 1 + m 2 m 3 r ( m 2) ; 3 r m; 2 + r ( m 3) ; 3 r m; 3 + ::: + 3 m 3 r 3 ; 3 r m; ( m 3) +0+0=0 ... m 1 3 r ( m 1) ; ( m 3) r m; 1 + m 2 3 r ( m 2) ; ( m 3) r m; 2 + m 3 3 r ( m 3) ; ( m 3) r m; ( m 3) +0+ ::: r m;m 3 +0+0=0 m 1 2 r ( m 1) ; ( m 2) r m; 1 + m 2 2 r ( m 2) ; ( m 2) r m; 2 +0+0+ ::: +0 r m; ( m 2) +0=0 ( m 1) r ( m 1) ; ( m 1) r m; 1 +0+0+0+ ::: +0+0 r m; ( m 1) =0 (7.0.2) Multiplyingbythereciprocalofthecoecientinfrontofth erstterm,weobtainthefollowing130

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( r ( m 1) ; 1 1) r m; 1 + m 2 m 1 r ( m 2) ; 1 r m; 2 + m 3 m 1 r ( m 3) ; 1 r m; 3 + ::: + 3 m 1 r 3 ; 1 r m; ( m 3) + 2 m 1 r 2 ; 1 r m; ( m 2) + 1 m 1 r 1 ; 1 r m; ( m 1) =0 r ( m 1) ; 2 r m; 1 + m 2 m 1 r ( m 2) ; 2 r m; 2 + m 3 m 1 r ( m 3) ; 2 p 3 ;m + ::: + 3 m 1 r 3 ; 2 r m;m 3 + 2 m 1 r 2 ; 2 r m; ( m 2) +0=0 r ( m 1) ; 3 r m; 1 + m 2 m 1 r ( m 2) ; 3 r m; 2 + m 3 m 1 r ( m 3) ; 3 r m; 3 + ::: + 3 m 1 p 3 ; 3 r m; ( m 3) +0+0=0 ...r ( m 1) ; ( m 3) r m; 1 + m 2 m 1 r ( m 2) ; ( m 3) r m; 2 + m 3 m 1 r ( m 3) ; ( m 3) r m; ( m 3) +0+ ::: r m; ( m 3) +0+0=0 r ( m 1) ; ( m 2) r m; 1 + m 2 m 1 r ( m 2) ; ( m 2) r m; 2 +0+0+ ::: +0 r m; ( m 2) +0=0 r ( m 1) ; ( m 1) r m; 1 +0+0+0+ ::: +0+0 r m; ( m 1) =0 (7.0.3) ( r ( m 1) ; 1 + r ( m 1) ; 2 + ::: + r ( m 1) ; ( m 2) + r ( m 1) ; ( m 1) 1) r m; 1 + m 2 m 1 ( r ( m 2) ; 1 + r ( m 2) ; 2 + r ( m 2) ; 3 + ::: + r ( m 2) ; ( m 3) + r ( m 2) ; ( m 2) 1) r m; 2 + m 3 m 1 ( r ( m 3) ; 1 + r ( m 3) ; 2 + r ( m 3) ; 3 + ::: + r ( m 3) ; ( m 4) + r ( m 2) ; ( m 3) 1) r m; 3 + + ::: + 2 m 1 ( r 2 ; 1 + r 2 ; 2 1) r m; ( m 2) + 1 m 1 ( r 1 ; 1 1) r m; ( m 1) =0 (7.0.4) i.e.,addingtheequationsweobtain 0 r m; 1 + m 2 m 1 0 r m; 2 + ::: + 2 m 1 0 r m; ( m 2) + 1 m 1 0 r m; ( m 1) =0 i.e.,wecanchoose r m; 1 ;r m; 2 :::r m;m independetntly.131

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Wehave m +1 2 equationsand m +1 2 unknowns,soweshouldbeabletodeneall theprobabilities.Sincesomeoftheseequationsaredepend ent,wehavedegreesof freedomandwehaverexibilityforchoosingsomeofthosepro babilitiesbasedonthe componentinthesolutionspace. Thesamplespaceforapotwithtiles,isthesetofallpossibl eoutcomesfromthe self-assemblyprocess.Soif S m isasamplespaceforapotwith m tiles, E k r; isthe setofalloutcomeswhoseproportionoftilesin k -cyclesisbetween r and r + weget measure( E k r; )= P ( E k r; ) : Considerthepottype P = f t g ,with t ( h )= t ( b h )=1.Denoteby P i = f t 1 ;t 2 ;t 3 ;:::;t i g thepotwith i tilesfromthetypein P ,andby S i thesamplespacefor P i Givenasamplespace S k ,wecandenecertainprobabilitiestoeachoftheelements inthespace,andthiswayfordierentprobabilitieswegetd ierentfamilyofprobabilities.Onesuchfamilyisuniformdistributionofeache lementsintheprobability space. Sincewecandeneoneprobabilityspace,theexistenceofth eprobabilityspace isproved. 132

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AbouttheAuthor AnaStaninskawasborninSkopje,Macedonia.Shereceivedhe rsB.S.degreein MathematicsattheUniversity\Sv.KiriliMetodij",Skopje ,Macedoniain2000.She enteredtheMasterprograminMathematicsattheUniversity ofSouthFloridain2001 andreceivedherM.ADegreein2003.SheenteredtheP.h.D.pr ograminMathematics attheUniversityofSouthFlorida.Sheworksonamathematic almodelofDNAselfassembly.HerPh.D.advisorsareDr.NatasaJonoskaandDr. Gregory.L.McColm. AtUSF,AnataughtseveralundergraduatecoursesasaTeachi ngAssistant.She spentthreemonthsworkingonaprojectinDr.NedSeeman'sLa boratoryatNYU. Analikesallkindsofsports,especiallybiking,runningan daerobics.


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A theoretical model for self-assembly of flexible tiles
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ABSTRACT: We analyze a self-assembly model of flexible DNA tiles and develop a theoretical description of possible assembly products. The model is based on flexible branched DNA junction molecules, which are designed in laboratories and could serve for performing computation. They are also building blocks for make of even more complex molecules or structures. The branched junction molecules are flexible with sticky ends on their arms. They are modeled with "tiles", which are star like graphs, and "tile types", which are functions that give information about the number of sticky ends. A complex is a structure that is obtained by gluing several tiles via their sticky ends. A complex without free sticky ends is called "complete complex". Complete complexes are our main interest. In most experiments, besides the desired end product, a lot of unwanted material also appears in the test tube (or pot). The idea is to use the proper proportions of tiles of different types.The set of vectors that represent these proper proportions is called the "spectrum" of the pot. We classify the types of pots according to the complexes they acan admit, and we can identify the class of each pot from the spectrum and affine spaces. We show that the spectrum is a convex polytope and give an algorithm (and a MAPLE code), which calculates it, and classify the pots in PTIME. In the second part of the dissertation, we approach molecular self-assembly from a graph theoretical point of view. We assign a star-like graph to each tile in a pot, which induces a "pot-graph". A pot-graph is a labeled multigraph corresponding to a given pot type, whose vertices represent tile types. The complexes can be represented by "complex-graphs", and each such graph is mapped homomorphically into a pot-graph. Therefore, the pot-graph can be used to distinguish between pot types according to the structure of the complexes that can be assembled.We begin the third part of the dissertation with a pot containing uniformly distributed DNA junction molecules capable of forming a cyclic graph structure, in which all possible Watson-Crick connections have already been established, and compute the expectation and the variance of the number of self-assembled cycles of any size. We also tested our theoretical results in wet lab experiments performed at Prof. Nadrian C. Seeman's laboratory at New York University. Our main concern was the probability of obtaining cyclic structures. We present the obtained results, which also helped in defining an important parameter for the theoretical model.
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Dissertation (Ph.D.)--University of South Florida, 2007.
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Advisor: Nataa Jonoska, Ph.D.
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DNA.
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Probabilistic model.
Complete complexes.
Cyclic molecules.
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