文档介绍：该【calculation of π and classification of self-avoiding lattices via dna configuration anshula tandon外文参考 】是由【宝钗文档】上传分享，文档一共【11】页，该文档可以免费在线阅读，需要了解更多关于【calculation of π and classification of self-avoiding lattices via dna configuration anshula tandon外文参考 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..entificreportsopeNCalculationofπandClassificationofself-avoidingLatticesviaDNAConfigurationReceived:6July2018Anshulatandon1,seungjaeKim1,Yongwoosong1,HyunjaeCho1,saimaBashar1,Accepted:8January2019Jihoonshin2,taiHwanHa2,3&sungHapark1Published:xxxxxxxxNumericalsimulation(),studiescalculatingnumericalconstantsinmathematics,andestimationofgrowthbehaviorviaanon-conventionalself-assemblyinconnectionwithDNAnanotechnology,,amethodtocalculatethenumericalvalueofπ,andwaytoevaluatepossiblepathsofself-avoidingwalkwiththeaidofMonteCarlosimulation,,experimentallyobtainedvariationoftheπasfunctionsofDNAconcentrationandthetotalnumberoftrials,andthebehaviourofself-avoidingrandomDNAlatticegrowthevaluatedthroughnumberofgrowthsteps,(πexp)obtainedbydoublecrossoverDNAlatticesandDNArings,-avoidingrandomlatticesgrownbythethree-pointstarDNAmotifs,-(MC),generateusefulmathematicalfunctions,plicatedalgo-,plexbiologicalprocessmechanismssuchasthebiologicalself-assemblybehaviour,biomoleculedynamics,andtheinteractionbetweenbiomoleculesandnanomaterials2–,therearetwointerestingones;calculatingπ(oneofmostimpor-tantmathematicalconstantsdefinedastheratioofacircle’scircumferencetoitsdiameter),andinterpretingaself-avoidingwalk(anabstractmodeldescribingthebehaviourofchainlikeentitieswherenotwopointscanoccupythesameplace),amongwhichthefamouslyusedoneisBuffon’-avoidingwalk,-turesfindvariousapplicationsasphysical,chemical,orbiomedicaldevicesandsensors26–32,calculatingmathe-(SAint),SungkyunkwanUniversity,Suwon,16419,,KoreaResearchInstituteofBioscienceandBiotechnology(KRIBB),Daejeon,34141,,KRIBBSchoolofBiotechnology,KoreaUniversityofScienceandTechnology(UST),Daejeon,34113,.(email:******@).(email:******@)ScientificRepoRts|(2019)9:2252|https:///-019-38699-01:..entificreports/.(a)(=πest,whereeststandsforestimation)isdefinedas(ND-in/ND)×4,whereND-inandNDstandforthenumberofdotsinsideaquadrantofacircle(witharadiusofR)andthetotalnumberofdotsinasquare(withalengthofR).Bydefinition,πestwithfourdifferentND,,50,100,(=6/10×4),,,urateknownvalueofπ(πknown≈).(b)AflowchartdepictingalgorithmicstepstoobtainπestwithvariousNDandtotalnumberoftrials(nT).(c)πestasafunctionofnTatagivenND(,50,100,or1000).Ingeneral,πestapproachestoπknownwiththeincreasingnTatrelativelylargerNDvalues,asexpected.(d)πestasafunctionofNDatafixednT(,5,10,50,or100markedasadottedlinein(c).Fromobservation,πestapproachestoπknownwiththeincreasingNDatrelativelysmallernTbutπestisroughlyindependentwithNDatrelativelylargernT.(e),,wedevelopwaystocalculateπandevaluatetheapplicablenumberofself-,weexperimentallydemonstratethecalculationofπandeval-uateapplicableself-avoidingwalkpathswithtwodifferentDNAnanostructures(doublecrossoverDNAlatticesandDNArings)andself-avoidingrandomDNAlattices(constructedbyathree-pointstarDNAmotifhavingablunt-end),,weanalyzethetrendofnumericalπvariationscontrolledbyDNAconcentration,thetotalnumberoftrials,andthecharacteristicgrowthbehaviourofself-avoidingrandomDNAlatticesevalu-atedthroughthetotalnumberofgrowthstepsfortheself-.?(=πest,whereeststandsforestimation),arandomeventneedstobeconsideredwhichcanbedefinedasdrawinguni-formlydistributeddots(likethrowingdartsrandomlyataboard)overasquareboundingboxwithintheregionScientificRepoRts|(2019)9:2252|https:///-019-38699-02:..entificreports/,theratioofthequadrantareatothesquareareaisapproximatelyequaltotheratioofthetotalnumberofdotsfallinginsidethequadrant(ND-in,markedasblue)tothetotalnumberofdotsinsidethesquare(ND),πestcanbedefinedas(ND-in/ND)×,representativeπestwithfourdifferentND(,50,100,and1000)(=6/10×4),,(πknown≈).Consequently,themagnitude(=|–|,,,)ofthedeviationofπestfromπknown(?πexp=|πest?πknown|)willbesmallerasND(10,50,100,and1000)increasesatagivenoptimumND-in(,39,79,and785,paredtoπknownatagivenND).Figure?1bshowsaflowchartrepresentingalgorithmicstepsinordertoobtainπestasafunctionofeitherND,orthetotalnumberoftrials(nT).ByassigninganinitialinputofNDwiththeunit-stepincrementofj,,ND-inarecounteduntiljreachestoNDfollowedbyevaluationofπestcalcu-latedas(ND-in/ND)×,whentheunit-stepincrementofireachesaninitialinputofnT,,numericalvaluesoftheπestasfunctionsofNDandnTcanbeobtainedandana-(,50,100,and1000)areobtained,whichapproachesπknownwiththeincreasingnTatanygivenNDvalues,asexpected(Fig.?1c).TheπestwithvaryingNDatafixednT(,5,10,50,or100markedasadottedlineinFig.?1c)areextractedinordertoevaluatethetrendofπestasafunctionofNDwhichshowsthatπestheavilyreliesonNDatrelativelysmallernTbutitisroughlyindependentofNDatlargernT(Fig.?1d).ArepresentativegraphofπestasafunctionofNDisshowninFig.?,-entrangesofND(~20and980~1000),whichclearlyshowsthatfluctuationofπestfromπknowntendstodecreasewiththeincreaseinND,,thedifferentiationofπestperunitnumberofdots(=Δπest/ΔND)asafunctionofNDisshowninSupplementaryFig.?(πexp)isdemonstratedbyconstructingtwotypesofDNAnanostructures,(DX)DNAlattices33,34andDNArings35–37(Fig.?2).TwosetsofDXDNAmotifs()aredesignedforconstructionofDXDNAlat-,PstandsforPi(π)andR/Sindicateoppositehelicaldirectionalitiesoftheduplexeswithinthemotifs(SeeSupplementaryFig.?2,SupplementaryTables?1and2).EachsethastwoDXmotifs,withoutandwithhairpinsmarkedasPR(S)0andPR(S)1,respectively(Fig.?2a).ADXmotifhavinghairpins~().DXandDXHmotifs,havingidenticalsetsofstickyendsineachsetwiththeequalprobabilityofbinding(twoexemplifiedbindingsitesareindicatedbyquestionmarksinFig.?2b),plementarycolour-codedandshape-,prisedofTmotifs(non-crossoverbasedDNAmotifshavingthreedouble-.?2c,SupplementaryFig.?3,andSupplementaryTables?3and4)-pairsofthestickyendsinTmotifs(Fig.?2d).RepresentativestructuralconfigurationsofDXDNAlatticesandDNAringsareshowninFig.?2e,h,respec-(AFM)imagesofDXlatticeswithdifferentconcentrationsofDXH(0,25,50,100,150,and200nMsymbolizedasDXH0,,,,,,respectively)(showninblue)(,,,,,,)throughimagesinFig.?2eareobtainedby(NH-in/NH)×4,whereNH-,([DXH])-dependentNHvalues(roughlylinearlydependent)obtainedbytheoreticalcalculationandanalyzedbyAFMimagesaredisplayedinFig.?,AFMimagesofDNAringswithdifferentconcentrationsofaTmotif(1,2,5,8,10,and20nMindicatedasR1,R2,R5,R8,R10,andR20,respectively)wereannealedthroughamica-assistedgrowthmethod38–40(Fig.?2h).Arcsaredrawninthirdquadrantsandcorrespondingπexp((NR-in/NR)×4,whereNR-inandNRindicatenumberofringsinsideaquadrantandtotalnumberofringsinanimage),aplotofNRasafunctionof[T](roughlysigmoidal)analyzedbyAFMimagesisshowninFig.?[DX]andnTarecon-ductedandresultsaredisplayedinFig.?.?3ashowsanaverageofπexp(?πexp?obtainedfrommorethanfourdatasetsatagiven[DXH])asafunctionof[DXH](theconcentrationsumofDXHsineachsetofmotif,[DXPR1]+[DXPS1]),50,75,100,150,and200nMatafixed[DXPR]and[DXPS]-ple,150nMof[DXH]indicates150nMof[DXPR1]+[DXPS1]with50nMof[DXPR0]+[DXPS0].Althoughthestandarddeviationofanerrorbargenerallydecreasesas[DXH]increases,themagnitudeofthedeviationofπexpfromπknown(?πexp≡|πexp-πknown|)isalmostconstantabove50nMof[DXH].Aplotofπexpasafunctionof[DX](=[DXPR(S)]with[DXPR(S)0]=[DXPR(S)1])isshowninFig.?,100nMof[DX]u-rateπexp(?πexpof~)than50(~)or200nM(~)of[DX].Figure?3cdisplays?πexp(arrangedinadescendingorder)and?πexp?(definedas∑nTπ/)nasafunctionofnTatafixed[DX]of100nMwhichpro-n=1expn,videthegeneralbehaviourofπexpapproachingπknownwithincreasingnT,,πexpand?πexpasfunctionsof[T].?3d,thestandarddeviationofanerrorbarroughlydecreasesas[T]increasesand?πexpisapproximatelyindependentwith[T],whichmightbeduetotheuniformdistributionoftheDNAringsonaScientificRepoRts|(2019)9:2252|https:///-019-38699-03:..entificreports/.(a,b)Cartoonrepresentationsoftwosets?PRandPS?ofDNAdouble-crossover(DX)motifsandcorrespondingDXlatticeformedbycomplementarycolour-,withoutandwithhairpinsmarkedasPR(S)0andPR(S)1,