DNA Sequence and Structure under the Prism of Group Theory and Algebraic Surfaces

Page 1

International Journal of Molecular Sciences Article

DNASequenceandStructureunderthePrismofGroupTheory andAlgebraicSurfaces

MichelPlanat 1,*,† ,MarceloM.Amaral 2,† ,FangFang 2 ,DavidChester 2 ,RaymondAschheim 2 andKleeIrwin 2

1 InstitutFEMTO-STCNRSUMR6174,UniversitédeBourgogne-Franche-Comté,F-25044Besançon,France

2 QuantumGravityResearch,LosAngeles,CA90290,USA

* Correspondence:michel.planat@femto-st.fr

†Theseauthorscontributedequallytothiswork.

Citation: Planat,M.;Amaral,M.M.; Fang,F.;Chester,D.;Aschheim,R.; Irwin,K.DNASequenceand StructureunderthePrismofGroup TheoryandAlgebraicSurfaces. Int.J. Mol.Sci. 2022, 23,13290. https://doi.org/ 10.3390/ijms232113290

AcademicEditors:GiuseppeZanotti andZhongzhouChen

Received:16September2022 Accepted:26October2022 Published:31October2022

Publisher’sNote: MDPIstaysneutral withregardtojurisdictionalclaimsin publishedmapsandinstitutionalaffiliations.

Abstract: TakingaDNAsequence,awordwithletters/basesA,T,GandC,astherelationbetween thegeneratorsofaninfinitegroup π,onecandiscriminatebetweentwoimportantfamilies:(i) thecardinalitystructureforconjugacyclassesofsubgroupsof π isthatofafreegroupononeto fourbases,andtheDNAword,viewedasasubstitutionsequence,isaperiodic;(ii)thecardinality structureforconjugacyclassesofsubgroupsof π isnotthatofafreegroup,thesequenceisgenerally notaperiodicandtopologicalpropertiesof π havetobedetermineddifferently.Thetwocasesrelyon DNAconformationssuchasA-DNA,B-DNA,Z-DNA,G-quadruplexes,etc.Wefoundafewsalient results:Z-DNA,wheninvolvedintranscription,replicationandregulationinahealthysituation, implies(i).Thesequenceoftelomericrepeatscomprisingthreedistinctbasesmostofthetimesatisfies (i).Fortwo-basesequencesinthefreecase(i)ornon-freecase(ii),thetopologyof π maybefoundin termsofthe SL(2, C) charactervarietyof π andtheattachedalgebraicsurfaces.Thelinkingoftwo unknottedcurves—theHopflink—mayoccurinthetopologyof π incasesofbiologicalimportance, intelomeres,G-quadruplexes,hairpinsandjunctions,afeaturethatwealreadyfoundinthecontext ofmodelsoftopologicalquantumcomputing.Forthree-andfour-basesequences,otherknotting configurationsarenoticedandabuildingblockofthetopologyisthefour-puncturedsphere.Our methodshavethepotentialtodiscriminatebetweenpotentialdiseasesassociatedtothesequences.

Keywords: DNAconformations;transcriptionfactors;telomeres;infinitegroups;freegroups; algebraicsurfaces;aperiodicity;charactervarieties

1.Introduction

Copyright: ©2022bytheauthors. LicenseeMDPI,Basel,Switzerland. Thisarticleisanopenaccessarticle distributedunderthetermsand conditionsoftheCreativeCommons Attribution(CCBY)license(https:// creativecommons.org/licenses/by/ 4.0/).

Grouptheoryandalgebraicgeometryservethedeciphermentof‘thebookoflife’[1], abookmadeofalanguageemployingfourletters/nucleotides:A(adenine),T(thymine), G(guanine)andC(cytosine),asdescribedinthiswork.Therearefinitegroups,groups madeofafinitenumberofgeneratorsandafinitenumberofelementsthatmaybeusedto mapthecodonstoaminoacids,ascarriedoutinourpapers[2,3].Suchanapproachtoward thegeneticcodeismadepossiblebyidentifyingtheirreduciblecharactersofthegroupto theaminoacids.Themultipletsofcodonsattachedtoaselectedaminoacidcorrespond totheirreduciblecharactershavingthecorrespondingdimensionoftherepresentation (Table3in[2],Table4in[3]).Avirtueoftheapproachisthattheusedirreduciblecharacters arealsoseenasquantumstatescarryingcompletequantuminformation.

FormodelingDNAinitsvariousconformationstakenintranscriptionfactors,telomeresandotherbuildingblocksofmolecularbiology,weneedinfinitegroupsdefinedfrom amotif.AsequenceoftheDNAnucleotidesservesasthegeneratorofthegroup[4].Inthis context,ithasbeenfoundthatagroupthatisnotfreeisoftenthewitnessofapotential disease.Wecoinedtheterm‘syntacticalfreedom’forrecognizingthisproperty,withinspirationfromanearlierwork[5].Wealsoshowedthatsuchfreegroupshavethedistinctive

Int.J.Mol.Sci. 2022, 23,13290. https://doi.org/10.3390/ijms232113290 https://www.mdpi.com/journal/ijms

propertyofgeneratinganaperiodicsubstitutionrule,providingaconnectionbetween (group)syntacticalfreedomandirrationalnumbers(Section4in[4]).Foraninfinitegroup, therepresentationcannotbebasedoncharactersbutontheso-calledcharactervariety. ThistopicleadstoarelationshipbetweenDNA,algebraictopologyandalgebraicgeometry. Toolsalreadyproposedfortopologicalquantumcomputing[6]arealsousedinthecontext ofDNAconformations.

InSection 2,webrieflyaccountforthemanytypesoftopologiesthatDNAcanshow, intermsofdoublestrandsormorestrands.Then,werecallthemathematicalconcepts employedinourpaperwithsomeredundancywithearlierwork[4,6].

InSection 3,weexplaintheconceptofan SL(2, C) charactervarietyassociatedto aninfinitegroupwithtwoorthreegenerators.TheformercasecorrespondstoDNA motifshavingonlytwodistinctnucleotides.Insuchacase,thevarietyoftencontains theCayleycubicassociatedtotheHopflink,thenondisjointunionoftwocirclesinthe three-dimensionalspace.Inthelatercase,thevarietycontainstheFricke–Kleinseventh variablepolynomialthatischaracteristicofthetopologyofthethree-dimensionalsphere withfourpointsremoved.

InSection 4,weapplythesemathematicalmethodstotranscriptionfactors,telomeric sequencesandaspecificDNAdecamersequence,wherealmostallofitsconformations havebeencrystallized.

2.MaterialsandMethods

MathematicalcalculationsperformedinthispaperareonthesoftwareMagma[7](for groups)oronSagesoftware[8](forcharactervarieties).

2.1.DNAConformations

DNAisalongpolymermadefromachainofthenucleotidesA,T,CorG.DNAexists inmanypossibleconformations,whichincludeadouble-strandedhelixofA-DNA,B-DNA andZ-DNA,althoughonlyB-DNAandZ-DNAhavebeendirectlyobservedinfunctional organisms[9,10].TheB-DNAformismostcommonundertheconditionsfoundincells, butZ-DNAisoftenpreferredwhenDNAbindstoaprotein.Aviewofadoublehelix intheA-,B-andZ-DNAformsisgiveninFigure 1 OtherDNAconformationsalsoexist, suchasasingle-strandedhairpinusedmostlyinmacromolecularsynthesisandrepair, atriple-strandedH-DNAfoundinpeptides,aG-quadruplexstructurefoundintelomeres andaHollidayjunction.

Figure1. Fromlefttoright,thestructuresofA-,B-andZ-DNA.Theviewofthedoublehelixfrom above(orbelow)showsdistinctsymmetries, 11-foldfortheA-DNA, 10-foldfortheB-DNAand 6-foldfortheZ-DNA[9,10].

Int.J.Mol.Sci. 2022, 23,13290 2of16

2.2.FinitelyGeneratedGroups,FreeGroupsandTheirConjugacyClasses,andAperiodicity ofSequences

Thefreegroup Fr on r generators(ofrank r)consistsofalldistinctwordsthatcanbe builtfrom r letterswheretwowordsaredifferentunlesstheirequalityfollowsfromthe groupaxioms.Thenumberofconjugacyclassesof Fr ofagivenindex d isknownandisa goodsignatureoftheisomorphism,orthecloseness,ofagroup π to Fr.Inthefollowing, thecardinalitystructureofconjugacyclassesofindex d in Fr iscalledthecardinality sequence(cardseq)of Fr,andweneedthecasesfrom r = 1 to 3 tocorrespondtothenumber ofdistinctbasesinaDNAsequence.Thecardseqof Fr isinTable 1 forthethreesequences ofinterestinthecontextofDNA[11].

Table1. Numberofconjugacyclassesofsubgroupsofindex d infreegroupofrank r = 1 to 3 [11]. Thelastcolumnistheindexofthesequenceintheon-lineencyclopediaofintegersequences[12].

rCardSeqSequenceCode

1 [1,1,1,1,1,1,1,1,1, ] A000012

2 [1,3,7,26,97,624,4163,34470,314493, ] A057005 3 [1,7,41,604,13753,504243,24824785,1598346352, ··· ] A057006

Next,givenafinitelygeneratedgroup fp witharelation(rel)givenbythesequence motif,weareinterestedinthecardseqofitsconjugacyclasses.Often,theDNAmotifin thesequenceunderinvestigationisclosetothatofafreegroup Fr,with r + 1 beingthe numberofdistinctbasesinvolvedinthemotif.However,thefinitelygeneratedgroup f p = x1, x2|rel(x1, x2) ,or f p = x1, x2, x3|rel(x1, x2, x3) or f p = x1, x2, x3, x4|rel(x1, x2, x3, x4) (wherethe xi aretakeninthefourbasesA,T,GandC,andrelisthemotif),maynotbethe freegroup F1 = x1, x2|x1 x2 ,or F2 = x1, x2, x3|x1 x2 x3 or F3 = x1, x2, x3, x4|x1 x2 x3 x4 Theclosenessof f p to Fr canbecheckedbyitssignatureinthefiniterangeofindicesofthe cardseq.

2.2.1.Groups f p ClosetoFreeGroupsandAperiodicityofSequences

Accordingtoreference[5],aperiodicitycorrelatestothesyntacticalfreedomofordering rules.Thisstatementwascheckedintherealmoftranscriptionfactors(Section4in[4]). Letusintroducetheconceptofageneralsubstitutionruleinthecontextoffreegroups. Ageneralsubstitutionrule ρ onafinitealphabet Ar on r lettersisanendomorphismofthe correspondingfreegroup Fr (Definition4.1in[13]).Theendomorphismpropertymeans thetworelations ρ(uv)= ρ(u)ρ(v) and ρ(u 1)= ρ 1(u),forany u, v ∈ Fr.

Aspecialroleisplayedbythesubgroup Aut(Fr ) ofautomorphismsof Fr.Weintroduce themap α : Fr → Zr from Fr totheAbeliangroup Zr inordertoinvestigatethesubstitution rule ρ withthetoolsofmatrixalgebra.

Themap α inducesahomomorphism M :End(Fr ) → Mat(r, Z).Under M, Aut(Fr ) mapstothegenerallineargroupofmatriceswithintegerentries GL(r, Z).Given ρ,thereis auniquemapping M(ρ) thatmakesthemapdiagramcommutative[13](p.68).Thesubstitutionmatrix M(ρ) of ρ maybespecifiedbyitselementsatrow i andcolumn j asfollows: (M(ρ))i,j = card(ρai (aj ))

Thisapproachwasappliedtobindingmotifsoftranscriptionfactors[4].Thebindingmotifrelinthefinitelypresentedgroup fp = A, T, G, C|rel(A,T,G,C) issplitinto appropriatesegmentssothat rel = relArelT relGrelC withthesubstitutionrules A → relA, T → relT , G → relG, C → relC Weareinterestedinthesequenceoffinitelygeneratedgroups f (l) p = A, T, G, C|rel(rel(rel (A, T, G, C))) (withrelapplied l times)

Int.J.Mol.Sci. 2022, 23,13290 3of16

whosecardseqisthesameateachstep l andequaltothecardseqofthefreegroup Fr (in thefiniterangeofindicesthatitispossibletocheckwiththecomputer).

Undertheseconditions,(group)syntacticalfreedomcorrelatestotheaperiodicity ofsequences.

2.2.2.AperiodicityofSubstitutions

Thereisnodefinitiveclassificationofaperiodicorder,theintermediatebetween crystallineorderandstrongdisorder,butinthecontextofsubstitutionrules,somecriteria canbefound.First,weneedafewdefinitions.

Anon-negativematrix M ∈ Mat(d, R) isonewhoseentriesarenon-negativenumbers. Apositivematrix M (denoted M > 0)hasatleastonepositiveentry.Astrictlypositive matrix(denoted M >> 0)hasallpositiveentries.Anirreduciblematrix M =(Mij )1≤i,j≤d isoneforwhichthereexistsanon-negativeinteger k with (Mk )ij > 0 foreachpair (i, j) Aprimitivematrix M isonesuchthat Mk isastrictlypositivematrixforsome k

APerron–Frobenius(PFforshort)eigenvector v ofanirreduciblenon-negativematrix istheonlyonewhoseentriesarepositive: v > 0.Thecorrespondingeigenvalueiscalled thePFeigenvalue.

Wewillusethefollowingcriterion(Corollary4.3in[13]).Aprimitivesubstitution rule ρ ofsubstitutionmatrix M(ρ) withanirrationalPF-eigenvalueisaperiodic.

Awell-studiedprimitivesubstitutionruleistheFibonaccirule ρ = ρF : a → ab, b → a ofsubstitutionmatrix MF = 11 10 andPF-eigenvalueequaltothegoldenratio

λPF = φ =(√5 + 1)/2 (Example4.6in[13]).Asexpected,theirrationalityof λ correspondstotheaperiodicityoftheFibonaccisequence.

ThesequenceofFibonacciwordsisasfollows:

a, b, ab, aba, abaab, abaababa, abaababaabaab, ThewordshavelengthsequaltotheFibonaccinumbers1,1,2,3,5,8,13,21, Allfinitelygeneratedgroups f (l) p whoserelations rel(a, b)= ab, aba, abaab, abaababa, haveacardseqwhoseelementsare 1s,asforthecardseqofthefreegroup F1.TheFibonacci sequenceisourfirstexamplewheregroupsyntacticalfreedomcorrelatestoaperiodicity.

2.2.3.AFour-LetterSequencefortheTranscriptionFactoroftheFosGene

Letusnowapplythemethodtoatranscriptionfactorofimportance.Thetranscription factorofgeneFoshasselectedmotif rel = TGAGTCA [14].Forthiscase,thefour-letter generatedgrouphasacardseqsimilartothefreegroup F3 giveninTable 1 Wesplitrelintofoursegmentssothat rel = relArelT relGrelC withthesubstitution maps A

Int.J.Mol.Sci. 2022, 23,13290 4of16
= T
T →
A, T, G, C, ATGC, TGAGTCA, GAGTCTAGTCGAT ··· Thesubstitutionmatrixforthissequenceis M =     0011 1010 0110 0010     .Itisaprimitivematrix(M4 >> 0)whoseeigenvaluesfollowfromthevanishingofthepolynomial λ4 λ3 λ2 λ 1.Therearetworealeigenvalues λ1 ≈ 1.92756 and λ2 ≈−
λ3,4 ≈−
±
is λPF = λ1
relA
,
relT = G, G → relG = AGTC, C → relC = A toproducethe substitutionsequence
0.77480,as wellastwocomplexconjugateeigenvalues
0.07637
0.81470i.ThePF-eigenvalue
,withaneigenvectorof(positive)entries
(1,0.37298,0.40211,0.20861)T .It followsthattheselectedsequencefortheFosgeneisaperiodic. Allofthefinitelygeneratedgroups f (l) p whoserelationsare rel(A, C, G, T)= ATGC, TGAGTCA, GAGTCTAGTCGAT, ,

haveacardseqwhoseelementsare

1,7,41,604,13753,504243, , whichisthecardseqofthefreegroup F3.FortheFostranscriptionfactor,groupsyntactical freedomcorrelatestoaperiodicityasexpected.

FurtherexamplesareobtainedinthecontextofDNAsequencesfortranscription factors(Section4in[4])andbelow,inrelationtoDNAconformationsandtelomeres.

3.Discussion

Inthefollowing,wemakeuseof SL(2, C) representationsoftheinfinitegroups π arisingfromspecificDNAsequences.Thecharactervarietyhasmanyinterpretations inmathematicsandphysics.Forinstance,inmathematics,thevarietyisthespaceof representationsofhyperbolicstructuresofthree-manifolds M withfundamentalgroup π(M),andthevarietyofthecharactersof SL(2, C) representationsof π(M) isreflected inthealgebraicgeometryofthecharactervariety[15,16].Inphysics,thegroup SL(2, C) expressesthesymmetriesoffundamentalphysicallaws.ItisalsoknownastheLorentz group;moreprecisely,thedoublecoveroftherestrictedLorenzgroupis SL(2, C),whichis thespingroup.

Figure2. (Left):theHopflink.(Right):thelink L = A ∪ B isattachedtotheplane R2 inthe half-space R3 + .Itisnotsplittable.Thiscanbeprovedbycheckingthatthefundamentalgroup π = π2(L) isnotfree[17]andp.90in[18].Onegets π2 = x, y, z|(x, (y, z))= z ,where(.,.) meansthegrouptheoreticalcommutator.Thecardinalitysequenceofccofsubgroupsof π2 is [1,3,10,51,164,1365,9422,81594,721305, ] (Figure3in[4]).

3.1.SL(2, C) CharacterVarietiesandAlgebraicSurfaces

Recently,wefoundthattherepresentationtheoryoffinitegroupswiththeircharacter tableallowsustoderiveanapproachofthegeneticcode[3].

Forinfinitegroups π suchasthosedefinedbyDNAsequences,itisusefultodescribe therepresentationsof π intheLorentzgroup SL(2, C),thegroupof (2 × 2) matriceswith complexentriesanddeterminant 1.Suchagroupexpressesthefundamentalsymmetryof allknownphysicallaws,apartfromgravitation.

Representationsof π in SL(2, C) arehomomorphisms ρ : π → SL(2, C) withcharacter κρ (g)= tr(ρ(g)), g ∈ π.Thesetofcharactersallowsustodefineanalgebraicsetbytaking thequotientofthesetofrepresentations ρ bythegroup SL2(C),whichactsbyconjugation onrepresentations[15,19].

Fortwo-generatorgroups,thecharactervarietymaybedecomposedintotheproduct ofsurfaces,whichrevealsthetopologyof M.Werecentlyfoundaconnectionbetweensome groupswhosetopologyisbasedontheHopflinkandamodeloftopologicalquantum computing[6].TheHopflinkunderliesmanyDNAsequenceswhosegroupstructureis (orisnot)thatofthefreegroup F1.Theclassificationoftheinvolvedalgebraicsurfacesin

Int.J.Mol.Sci. 2022, 23,13290 5of16

thevarietyisperformedusingspecifictoolsavailableinMagma[7];see(Section2.1in[6]) fordetails.

Forthree-generatorgroups,wefindthattheFricke–Kleinquarticispartofthecharactervariety.

3.2.TheHopflink

TakingthelinkingoftwounknottedcurvesasinFigure 2 (Left),theobtainedlinkis calledtheHopflink H =L2a1,whoseknotgroupisdefinedasthefundamentalgroupof theknotcomplementinthethree-sphere S3

Π1(S3 \ L2a1)= a, b|[a, b] = Z2,(1)

where [a, b]= abAB (with A = a 1 , B = b 1)isthegrouptheoreticalcommutator. Thereareinterestingpropertiesoftheknotgroup Π1 oftheHopflink.

Figure3. Left:athree-dimensionalpictureofthe SL2(C) charactervariety ΣH fortheHopflink complement H.Right:amodifiedcharactervarietyofdefiningequation f H (x, y, z) withsimilarsingularities.

First,thenumberofcoveringsofdegree d of Π1 (whichisalsothenumberofconjugacy classesofindex d)ispreciselythesumofdivisorfunction σ(d) [20].

Second,aninvarianceof Π1 underarepetitiveactionofthegoldenratiosubstitution (theFibonaccimap) ρ : a → ab, b → a orunderthesilverratiosubstitution ρ : a → aba, b → a exists.ThetermsgoldenandsilverrefertothePerron–Frobeniuseigenvalueofthe substitutionmatrix(Examples4.5and4.6in[13]).SuchanobservationlinkstheHopflink, thegroup Π1 ofthe2-torusandaperiodicsubstitutions.

UsingSagesoftware[8]developedfromRef.[19],the SL2(C) charactervarietyisthe polynomialcorrespondingtotheso-calledCayleycubic f H (x, y, z)= xyz x2 y2 z2 + 4.(2)

Asexpected,thethree-dimensionalsurface Σ : f H (x, y, z)= 0 isthetraceofthe commutatorandisknowntocorrespondtothereduciblerepresentations(Theorem3.4.1 in[21]).ApictureisgiveninFigure 3 (left).

Intheperspectiveofalgebraicgeometry,weclassifythehomogenizationofequation f H asarationalsurfaceofdegree3delPezzotype.Itdisplaysfoursimplesingularities.

3.3.BeyondtheHopfLink

Asshownin[6],theHopflinkistheirreduciblecomponentofmanycharactervarietiesrelevanttoamodeloftopologicalquantumcomputing.InthecontextofDNA groupsinvestigatedinthenextsection,wealsofindanothersurfacewithsimilarsimple singularitiesasshowninFigure 3 (right).Thedefiningpolynomialis

f H (x, y, z)= z4 2xyz (+z3)+ 2x2 + 2y2 3z2( 4z) 4.(3)

Int.J.Mol.Sci. 2022, 23,13290 6of16

Thehomogenizationofequation f H (x, y, z) allowsustoclassifyitasaconicbundlein thefamilyof K3 surfaces.

FortheDNAsequence,whosegroup πH containsthecomponent f H (x, y, z),werefer tothethirdsubsectionoftheResultssectionbelowandthefirsttableinthissubsection. Therelevanttripletnnn=CGGofthedodecamericsequence d(CCnnnN6 N7 N8GG) leadsto aDNAconformationwiththelabel1ZEYinthePDBbank.

TheDNAdodecamersequenced(CCCCCGCGGGGG)isalsofoundinthePDBbank withlabel 2D47,correspondingtoacompleteturnofA-DNA.Thecharactervarietyforthe groupdefinedbythissequencecontainsthepolynomial f H andapolynomialsimilarto f H withoutthethird-orderterm z3 andthefirst-orderterm 4z,butinthesamefamily.

3.4.TheFricke–KleinSeventhVariablePolynomial

TheCayleycubicisasubsetofthecharactervarietyforthefour-puncturedthreedimensionalsphere S2 4 (thesphereminusfourpoints).Itsfundamentalgroup Π1 isisomorphictothefreegroup F3 ofrank 3, Π1(S2 4)= α, β, γ, δ|αβγδ ,wherethefourhomotopy classes α, β, γ, δ correspondtoloopsaroundthepunctures. The SL(2, C) charactervarietyfor Π1(S2 4) satisfiesaquarticequationintermsofthe Fricke–Kleinseventhvariablepolynomial[21](p.65)and[22]: f (x, θ)= xyz + x2 + y2 + z2 θ1 x θ2 x θ3z + θ4,(4) with θ1 = uv + wk, θ2 = uw + vk, θ3 = uk + vw, θk = uvwk + u2 + v2 + w2 4.

4.Results

Inthissection,weapplythe SL(2, C) representationtheorytospecificnon-canonical DNAsequenceshavingregulatoryfunctionsingeneexpression(thetranscriptionfactors), replication(thetelomeres)andDNAconformations.

4.1.GroupStructureandTopologyofTranscriptionFactors

Inatranscriptionfactor,amotif-specificDNAbindingfactorcontrolstherateofthe transcriptionofagenefromDNAtomessengerRNAbybindingaproteintotheDNA motif.Inreference[4],wefoundacorrelationbetweenmotifswhosesubgroupstructure isthatofafreegroupandthelackofapotentialdiseasewhilethegeneisactivatedin transcription,thepropertyof‘syntacticalfreedom’.

InTable 2,thisideaisillustratedbyrestrictingtoafewtranscriptionfactorswhose motifcomprisestwobases.Thecardseqofthemotifiseitherthefreegroup F1,closeto F1 orawayfromafreegroupwhenthecardseqisthatofthemodulargroup H3,ofthe Baumslag–Solitargroup BS( 1,1) orthatofgroups π1 and π1.Comparedtotheresults providedin[4],thereistheadditionalfourthcolumnthatsignalswhentheGroebnerbase fortheidealringofthe SL(2, C) charactervarietycontainstheCayleycubic,theunique componentinthecaseoftheHopflink[6],adegree 3 delPezzosurface(denotedHL),or not.Anadditionalfifthcolumnisfilledtocheckthepresenceofasurfaceoftype K3.Only thelastrowofthetableforthetranscriptionfactorofgeneEHFdoesnotshowthisproperty.

Int.J.Mol.Sci. 2022, 23,13290 7of16

Table2. Groupstructureofmotifsforafewtwo-lettertranscriptionfactors.Thecardseqforthemodulargroup H3 is [1,1,2,3,2,8,7,10,18,28, ].TheBaumslag–Solitargroup BS( 1,1) isthefundamentalgroupoftheKleinbottle.Thecardseqfor BS( 1,1) is [1,3,2,5,2,7,2,8,3,8,2,13,2,9,4, ] Thecardseqfor π1 is [1,4,1,2,4,2,1,7,2,2,4,2,2,8,1,2,7,2,3, ];for π1,itis [1,1,1,2,1,3,3,1,2,2,1,1,9,2,14,2,1, ··· ].ThesymbolHLmeansthattheCayleycubicis partoftheGroebnerbasefortheidealringofthecorresponding SL(2, C) charactervariety. Forthree-lettertranscriptionfactors,theidealringofthecorresponding SL(2, C) charactervariety containstheFricke–Kleinseventhvariablepolynomial 4,whichisafeatureofthefour-punctured spheretopology.Thegroupstructureofthree-lettertranscriptionfactorsnotleadingtofreegroupsis shownin(Table5in[4]).

GeneMotifCardSeqLinkTypeLiterature

DBXTTTATTA F1 HL K3 [23],MA0174.1

SPT15TATATATAT....,MA0386.1 PHOX2ATAATTTAATTA ≈F1 ...,MA0713.1

FOXATGTTTGTTT F1 ..[24,25]

FOXGTTTGTTTTT...[24]

NKX6-2TAATTAA H3 no K3 [23],[MA0675.1,MA0675.2]

FOXGTGTTTG BS( 1,1) no K3 [23,26],MA1865.1 HoxA1,HoxA2TAATTA π1 no K3 [23],[MA1495.1,MA0900.1] POU6F1,Vax.,[MAO628.1,MA0722.1] RUNX1TGTGGT.no..,MA0511.1 RUNX1TGTGGTT π1 no K3 [23],MA0002.2 EHFCCTTCCTC.HL.,MA0598.1

TheCharacterVarietyfortheTranscriptionFactoroftheDBXGene

Weexplicitlyshowthe SL(2, C) charactervarietyforthetranscriptionfactorofthe DBXgene.

fDBX (x, y, z)= f H (x, y, z)(yz2 y2 xz y + 2)(xy2 z3 yz x + 3z) (5) (y3 z2 3y + 2)(y2z xy yz + x z)(z4 x2y + xz 4z2 + y + 2)

Thefactorsin(5)arethreedegree 3 delPezzosurfaces(includingtheCayleycubic f H ), tworationalruledsurfacesanda K3 surfacebirationallyequivalenttotheprojectiveplane, respectively.Thelatterfactoralsobelongstothecharactervarietyofgroup Π1(S4 \ E6), where S4 isthefour-sphereand E6 isthesingularfiber IV∗ inKodaira’sclassificationof minimalellipticsurfaces(Figure4bin[6]).

Itisimportanttomentionthat,forthree-lettertranscriptionfactors,theidealringof thecorresponding SL(2, C) charactervarietycontainstheFricke–Kleinseventhvariable polynomial(4),whichisafeatureofthefour-puncturedspheretopology.

Table 3 providesashortaccountofthefunctionorpotentialdysfunctionofthegenes underconsideration.Asmentionedbefore,mostofthetime,suchadysfunctioniscorrelatedtoacardseqawayfromthatofthefreegroup F1

Inviewofourresults,itisinterestingtocorrelatethepresenceoftheHopflinkHLin thecharactervarietywiththepossibleremodelingofB-DNAintoZ-DNAoranotherDNA conformation.Toourknowledge,generalinformationaboutthissubjectisstilllacking. Fromabiologicalpointofview,itisknownthatsomeoftheZ-DNA-formingconditionsthat arerelevantinvivoarethepresenceofDNAsupercoiling,Z-DNA-bindingproteins[27] andbasemodifications.Whentranscriptionoccurs,themovementofRNApolymeraseII alongtheDNAstrandgeneratespositivesupercoilinginfrontof,andnegativesupercoiling behind,thepolymerase[28].

Int.J.Mol.Sci. 2022, 23,13290 8of16

Table3. Ashortaccountofthefunctionordysfunction(throughmutationsorisoforms)ofgenes associatedwithtranscriptionfactorsandsectionsinTable 2

GeneTypeFunctionDysfunction

DBXdrosophilasegmentation

SPT15TATA-boxgeneexpression,regulation bindingproteininSaccharomycescerevisiae

PHOX2Ahomeodomaindifferentiation,maintenancefibrosis ofnoradrenergicphenotypeofextraocularmuscles FOXproteinsforkheadboxgrowth,differentiation,

FOXA2.insulinsecretiondiabete longevity

NKX6-2homeoboxcentralnervoussystem,pancreasspasticataxia

FOXGforkheadboxnotochord(neuraltube)chordoma

HoxA1homeoboxembryonicdevtoffaceandhearautism HoxA2..cleftpalate

Pou6F1.neuroendocrinesystemclearcelladenocarcinoma

Vax.forebraindevelopmentcraniofacialmalform.

RunX1Runt-relatedcelldifferentiation,painneuronsmyeloidleukemia

EHFhomeoboxepithelialexpressioncarcinogenesis,asthma

PerhapsthelackofHLinthecharactervarietyfortranscriptionfactorsofgenesin Table 2 meansthattheZ-DNA-formingconditionisnotrealized.

4.2.GroupStructureandTopologyofDNATelomericSequences

TerminalstructuresofchromosomesaremadeofshorthighlyrepetitiveG-richsequenceswithproteinsknownastelomeres.Theyhaveaprotectiveroleagainsttheshorteningofchromosomesthroughsuccessivedivisions.Mostorganismsuseatelomere-specific DNApolymerasecalledtelomerasethatextendsthe3’endoftheG-richstrandofthe telomere[29].Telomereshorteningisassociatedwithaging,mortalityandaging-related diseasessuchascancer.

AlistofresultsobtainedbyusingourgrouptheoreticalapproachisinTable 4.For two-letter telomeresequences,the SL(2, C) charactervarietycontainstheCayleycubic, thecharacteristicoftheHopflinkHL,onlyinthefirstrow.InadditiontotheCayley cubic,onefindssurfacesofageneraltype.Inthenexttworows,theCayleycubicisnot found.Therearedegree 3 delPezzosurfacesinthefactorsofthecharactervarietybutnot generalsurfaces.

AsfortheHopflink,thesequenceisfoundtobeaperiodicwiththePerron–Frobenius eigenvalue λPF equaltothegoldenratio.Forthree-lettertelomeresequences,thecardseq isthatofthefreegroupof F2,exceptforthelastrow,wheretheidentifiedgroupis π2;see Figure 2 (right)forthedefinitionofsuchagroup.Intheformersevencases,theDNA topologyisknowntobeaG-quadruplexstructure[30–36].Wecouldidentifyanaperiodic structureofthetelomeresequencewiththePerron–Frobeniuseigenvalue λPF asshownin column5.Inthelattercase,thetopologyisofthebaskettype[36]andnoaperiodicityof thetelomeresequencecouldbefound.

Figure 4,takenfromtheproteindatabank(PDB2HY9),illustratestheG-quadruplex structureofthetelomeresequenceinvertebrates.

Int.J.Mol.Sci. 2022, 23,13290 9of16

Table4. Groupanalysisofthetelomeresequencefoundinsomeeukaryotes.Thefirstcolumnis forthetelomererepeat,thesecondcolumnistheorganismunderinvestigation,thethirdcolumnis forthePDBcode,thefourthcolumnisforthecardseqofthegroup π orthatofthecorresponding groupthatisidentified,thefifthcolumnisforthePerron–Frobeniuseigenvaluewhenthesequence isfoundtobeaperiodic,thesixthcolumnidentifiesthepresenceoftheHopflink(intwo-base sequences)ortheDNAconformation(inthree-basesequences)andtheseventhcolumnisarelevant reference.ThenotationG-quadrisfortheG-quadruplex;seeFigure 4.Thecardseqfor π1 is [1,3,2,16,16,69,118,719,1877,8949 ].TheHeckegroup H4 isdefinedin(Table2in[4]).

SeqOrganismPDBCardSeq λPF Link/DNAConfRef

G4T4G4Oxytricha1D59 π1 (√5 + 1)/2HL[37] TG4Tuniversal244D_1 H4 .no[38]

T2G4Tetrahymena230D H4 .no[29]

T2AG3Vertebrates2HY9 F2 2.5468G-quadr.[30] TAG3Giardia2KOW F2 2.2055G-quadr[31]

T2AG2Bombysmoriunknown F2 noG-quadr[32] T4AG3Greenalgaeunknown F2 3.07959unknown[33] G2T2AGHumanunknown F2 2.5468G-quadr[34] TAG3T2AG3Human2HRI F2 3.3923G-quadr[35] G3T2AG3T2AG3THumanunknown F2 4.3186G-quadr[36] (GGGTTA)3G3THumanunknown π2 nobasket[36]

Figure4. HumantelomereDNAquadruplexstructureinK+solutionhybrid-1form,PDB2HY9[30].

4.3.GroupStructureandTopologyoftheDNADecamerSequenced(CCnnnN6 N7 N8GG) [10]

AchallengingquestionofstructuralbiologyistodetermineifandhowaDNA(or RNA)sequencedefinesthethree-dimensionalconformation,aswellasthesecondaryand tertiarystructureofproteins.Intheprevioustwosubsections,wetackledtheproblem withregardtotranscriptionfactorsandtelomericsequences,respectively.Intheformer case,werestrictedtotheDNApartofthetranscriptionsincetheDNAmotifisalmost exactlyknownfromX-raytechniqueswhilethesecondarystructureofthebindingprotein stronglydependsonthemodelemployedandthechoicemadetorecognizethesections ofthesecondarystructures(e.g.,alphahelices,betasheetsandcoils)[39].Inthelatter case,inmanyorganisms,natureinventedtelomerasefortakingcareofthereplication withoutdamagingthesequencesatthe 3-endstoomuch,whilekeepingthecatalyzing actionofDNApolymerase.Again,thereisaloopcomplexintelomerasecomprising telomere-bindingproteins,withsecondarystructuresnotbeinganalyzedsofarwithour grouptheoreticalapproach.

Int.J.Mol.Sci. 2022, 23,13290 10of16

Inthissection,wealsostudyDNAconformationsandtheirrelationshiptoalgebraic topologyinaspecificDNAdecamersequenceinvestigatedinreference[10]byastandard crystallizationtechniquefollowedbyX-raydiffractiondiscrimination.Inthesequence d(CCnnnN6 N7 N8GG),thefactors N6, N7 and N8 aretakeninthetwonucleotidesGand C,andnnnisspecifiedinordertomaintaintheself-complementarityofthesequence. Thisinverserepeatedmotifistheminimummotifusedtodistinguishbetweenthedoublestrandformsof B-and A-DNA,whileexcludingthe Z-DNAforms.Athirdconformation isallowedandcalledthefour-strandedHollidayjunction J.Wereferto(Table1in[10])for themainresults.

Onourside,thecardseqofeachsequencewasdeterminedandthe SL(2, C) character varietywasobtained.OurresultsaresummarizedinthefourTables 5–8

Table5. Groupanalysisofthesequence d(CCnnnN6 N7 N8 GG),where N6, N7 and N8 aretakenin thetwonucleotides G and C and nnn isspecifiedinordertomaintaintheself-complementarity ofthesequence[10].Thefirstcolumnisfortheselectedtriplet N6 N7 N8,thesecondcolumnisfor thecodeintheproteindatabank,thethirdcolumnisfortheDNAconformationwhenknown(see Table1in[10]),thefourthcolumnisforthecardinalitystructureofsubgroupsof π andthefifth columncheckstheoccurrenceofasurfacecorrespondingtotheHopflinkinthefactorizationofthe SL(2, C) of π.ThesymbolsA,BandJareforA-DNA,B-DNAandafour-strandedHollidayjunction; lowercaseisusedwhentheconformationisnotconfirmedin[10].

TripletPDBConformationCardSeq(π)Knot CCC1ZF1A [1,1,1,1,7,1,1,2,9,6, ] HL CCC1ZF2JidemHL CCG1ZEXAidemHL CGG1ZEYA [1,1,1,2,6,3,1,4,2,6, ] HLlike CGCnoneunknown [1,1,2,1,6,3,2,1,3,6, ] no GGG1ZF9A [1,1,1,1,10,25,25,9,2,1798, ] no GCCnoneb/J [1,1,1,1,6,1,2,1,1,6, ] HL GCGnoneunknown [1,1,2,2,7,5,1,4,5,9, ] no GGCnoneB/a [1,1,1,1,6,11,9,5,2,208, ] no (cardseqofHeckegroup H5)

Table6. Groupanalysisofthesequence d(CCnnnN6 N7 N8 GG),where N6, N7 and N8 aretaken inthetwonucleotidesA,T[10].Groups π3 and π3 areasin(Table5in[4]).Thecardseqfor π3 is [1,7,50,867,15906,570528, ];for π3 ,itis [1,7,50,739,15234,548439, ];for π(4) 3 ,itis [1,7,59,1258,24787, ].Groups π3 and π3 maybesimplifiedtoagroupwhosecardseqisthatof π2,thefundamentalgroupofthelink L = A ∪ B describedinFigure 3 (right).

5, N6, N7 and N8 aretakeninthetwonucleotides G and C,formingeight tripletsandtheassociatedtwo-letterdecamersequences.Notethatthetriplet CCC producestwodistinctDNAconformations A and J.ThecharactervarietyoftheHopflinkHL (theCayleycubic)ispresentinthefactorsoftheidealringofthe SL(2, C) charactervariety infivecasesovertheninepossibilities,whereonecase(withtriplet CGG andcode1ZEYin theproteindatabank)showsanalgebraicsurfacesimilartotheCayleycubic(withfour

Int.J.Mol.Sci. 2022, 23,13290 11of16
TripletPDBConformation π TTA1ZFHB π3 → π2 TAAnoneB π3 → π2 AATnoneb π3 → π2 ATTnoneunknown π3 → π2 AAAnoneb π3 → π2 TTTnoneunknown π3 → π2 ATAnoneunknown π(4) 3 TATnoneunknown π(4) 3
InTable

simplesingularities)asdefinedinEquation(3)andasshowninFigure 3 (right).Wedonot observeaclearcorrelationbetweenthetypeofDNAconformationandtheunderlyingHL topology,butthepresenceofHLinthevarietyseemstoexcludetheB-DNAconformation. Inaddition,thecharactervarietyalwayscontainsasurfaceoftype K3 initsfactors.

Table7. Groupanalysisofthesequence d(CCnnnN6 N7 N8 GG) [10],where N6, N7 and N8 aretaken inthetwonucleotidesA,G(left)andA,C(right).Groups π3 and π3 areasin(Table5in[4]).Thecard seqfor π(3) 3 is [1,7,41,668,14969, ]and,for π(5) 3 ,itis [1,7,41,604,28153, ].

TripletPDBConformation π

TripletPDBConformation π

AGA1ZEWB F3 ACAnoneunknown π(3) 3 → π2

AGGnoneunknown π(5) 3 ACCnoneJ F3 GGA1ZFAA F3 CCAnoneunknown F3 AAGnoneunknown F3 AAC1ZF0B π3 ” → π2

TGTnoneunknown F3 TCTnoneb π(3) 3 → π2 TGG1ZF6A F3 TCCnoneunknown F3 GGT1ZF8A F3 CCTnoneb F3 TTGnoneunknown π3 TTCnoneB π3 ” → π2

Table8. Groupanalysisofthesequence d(CCnnnN6 N7 N8 GG) [10],where N6, N7 and N8 are takeninthethreenucleotidesA,G,C(left)andA,T,C(right).Thecardseqfor π(6) 3 is [1,7,59,874,20371,748320 ··· ]

TripletPDBConformation π TripletPDBConformation π

AGC1ZFMB F3 ATC1ZFC/1ZF3B/J π(6) 3

ACGnoneunknown F3 ACTnoneB π(3) 3 → π2 GCA1ZFEB F3 TCAnoneunknown π(3) 3 → π2 GAC1ZF7B F3 TACnoneunknown π(6) 3 CAGnoneunknown F3 CATnoneunknown F3 CGAnoneunknown F3 CTAnoneunknown F3

InTable 6, N6, N7 and N8 aretakeninthetwonucleotides A and T,formingeight triplets.TheDNAconformation(whenknown)isoftype B.Thecardseqthatweobtain isgroups π3, π3 or π(4) 3 asdescribedinthecaptionofTable 5.Insixcasesovertheeight possibilities,thegroupsencapsulatethetopologyoftherank 2 group π2,whoseassociated linkisshowninFigure 2 (right).Asalreadymentioned,the SL(2, C) charactervariety containstheFricke–Kleinseventhvariablepolynomial 4 InTable 7, N6, N7 and N8 aretakeninthetwonucleotides A, G (leftpart)and A, C (rightpart).Thistime,eitherthecardseqofthegroup π isthatofthefreegroup F3, ofrank 3 (10casesoverthe 16 possibilities),ornot.Inthelattercase,thegroupencapsulates thetopologyof π2 onlyattherightsideofthetable.SimilarconclusionsholdinTable 8 when N6, N7 and N8 aretakeninthetwonucleotides A, G, C (leftpart)and A, T, C (right part).

Tosummarizethissection,noclearcorrelationisobservedbetweentheDNAconformationsoftheconsidereddecamerandourgroupanalysis.Longersequencesmaybe neededtoobtainsuchacorrelation.Forinstance,thetwo-letterDNAdodecamersequence d(CCCCCGCGGGGG) (PDB2D47)correspondingtoacompleteturnofA-DNA—see Figure 5 (right)—featuresthepolynomial f H (forHL)andafourth-orderpolynomialsimilarto f H withfoursimplesingularities,asannouncedattheendofSection 3

Int.J.Mol.Sci. 2022, 23,13290 12of16

Figure5. (Left)Thefour-strandHollidayjunction J:PDB 1ZF2,(Right)AcompleteturnofADNA:PDB 2D47.ItisassociatedtoDNAdodecamersequence d(CCCCCGCGGGGG) with SL(2, C) containingthefactor f H = xyz x2 y2 z2 + 4 (theCayleycubic)andthefactor f H = z4 2xyz + 2x2 + 2y2 3z2 4.

5.Conclusions

Inthepresentpaper,followingearlierworkaboutthegeneticcode[2,3]andabout theroleoftranscriptionfactorsingenetics[4],wemadeuseofgrouptheoryappliedto appropriateDNAmotifsandwecomputedthecorrespondingvarietyof SL(2, C) representations.TheDNAmotifsunderconsiderationmaybecanonicalstructures,suchas (double-stranded)B-DNA,ornon-canonicalDNAstructures[40],suchas(single-stranded) telomeres,(double-stranded)A-DNA,Z-DNAorcruciforms,(triple-stranded)H-DNA, (four-stranded)i-motifsorG-quadruplexes,etc.Oneobjectiveoftheapproachistoestablish acorrespondencebetweenthealgebraicgeometryandtopologyofthe SL(2, C) character varietyandthetypesofcanonicalornon-canonicalDNA-forms.Forexample,fortwo-letter transcriptionfactors,onecancorrelatethepresenceoftheCayleycubicand/ora K3 surface inthevarietywithDNAsupercoilingintheremodelingofB-DNAtoZ-DNA.Forthree-or four-letterDNAstructures,ourworkneedstobedevelopedinordertoputthefeatures ofthevarietyandpotentialdiseasesincorrespondence.Inaseparateworkdevotedto topologicalquantumcomputing,thetopologyofthefour-puncturedsphereandtherelatedFrickesurfaces(generalizingtheCayleycubic)arerelevant[41].Inaddition,Fricke surfacesmaybeputinparallelwithdifferentialequationsofthePainlevéVItype.Itwill beimportanttocomparethesemodelswithotherqualitativemodelsbasedonnon-linear differentialequations[42,43].Inthenearfuture,weintendtoapplythesemathematical toolstothecontextofDNAstructures.

AuthorContributions: Conceptualization,M.P.,F.F.andK.I.;methodology,M.P.,D.C.andR.A.;software,M.P.;validation,R.A.,F.F.,D.C.andM.M.A.;formalanalysis,M.P.andM.M.A.;investigation, M.P.,D.C.,F.F.andM.M.A.;writing—originaldraftpreparation,M.P.;writing—reviewandediting, M.P.;visualization,F.F.andR.A.;supervision,M.P.andK.I.;projectadministration,K.I.;funding acquisition,K.I.Allauthorshavereadandagreedtothepublishedversionofthemanuscript.

Funding: FundingwasobtainedfromQuantumGravityResearchinLosAngeles,CA,USA.

InstitutionalReviewBoardStatement: Notapplicable.

InformedConsentStatement: Notapplicable.

DataAvailabilityStatement: Thedatasetsusedand/oranalyzedduringthecurrentstudyare availablefromthecorrespondingauthoronreasonablerequest.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

Int.J.Mol.Sci. 2022, 23,13290 13of16

References

1. Nerlich,B.;Dingwall,R.;Clarke,D.D.Thebookoflife:HowthecompletionoftheHumanGenomeProjectwasrevealedtothe public. HealthInterdiscip.J.Soc.StudyHealthIlln.Med. 2002, 6,445–469.

2. Planat,M.;Aschheim,R.;Amaral,M.M.;Fang,F.;Irwin,K.CompletequantuminformationintheDNAgeneticcode. Symmetry 2020, 12,1993.

3. Planat,M.;Chester,D.;Aschheim,R.;Amaral,M.M.;Fang,F.;Irwin,K.FinitegroupsfortheKummersurface:Thegeneticcode andquantumgravity. QuantumRep. 2021, 3,68–79.

4. Planat,M.;Amaral,M.M.;FangF.;Chester,D.;Aschheim,R.;Irwin,K.GrouptheoryofsyntacticalfreedominDNAtranscription andgenomedecoding. Curr.IssuesMol.Biol. 2022, 44,1417–1433.

5. Irwin,K.Thecode-theoreticaxiom;thethirdontology,Rep.Adv.Phys.Sci. 2019, 3,1950002.

6. Planat,M.;Amaral,M.M.;FangF.;Chester,D.;Aschheim,R.;IrwinK.Charactervarietiesandalgebraicsurfacesforthetopology ofquantumcomputing. Symmetry 2022, 14,915.

7. Bosma,W.;Cannon,J.J.;Fieker,C.;Steel,A.(Eds). HandbookofMagmaFunctions,Australia,Edition2.23;UniversityofSydney: Sydney,Australia,2017;5914p.

8. PythonCodetoComputeCharacterVarieties.Availableonline: http://math.gmu.edu/~slawton3/Main.sagews (accessedon 1May2021).

9. DNA.Availableonline: https://en.wikipedia.org/wiki/DNA (accessedon1January2022).

10. Hays,F.A.;Teegarden,A.;JonesZ.J.R.;Harms,M.;Raup,D.;Watson,J.;Cavaliere,E.,Ho,P.S.Howsequencedefinesstructure: AcrystallographicmapofDNAstructureandconformation. Proc.Natl.Acad.Sci.USA 2005, 20,7157–7162.

11. Kwak,J.H.;Nedela,R.Graphsandtheircoverings. Lect.NotesSer. 2007, 17,118.

12. TheOn-LineEncyclopediaofIntegerSequences.Availableonline: https://oeis.org/book.html (accessedon1June2022).

13. BaakeM.,GrimmU. AperiodicOrder,Vol.I:AMathematicalInvitation;CambridgeUniversityPress:Cambridge,UK,2013.

14. Glover,J.N.;Harrison,S.C.CrystalstructureoftheheterodimericbZIPtranscriptionfactorc-Fos-c-JunboundtoDNA. Nature 1995, 373,257–261.

15. Culler,M.;ShalenP.B.Varietiesofgrouprepresentationsandsplittingof3-manifolds. Ann.Math. 1983, 117,109–146.

16. Cooper,D.;Culler,M.;Gillet,H.;Long,D.D.;Shalen,P.B.Planecurvesassociatedtocharactervarietiesof 3-manifolds. Invent.Math. 1994, 118,47–84.

17. ZeemanE.C.Linkingspheres. Abh.Math.Sem.Univ.Hambg. 1960, 24,149–153.

18. RolfsenD. KnotsandLinks;AMSChelseaPublishing:Providence,RI,USA,2000.

19. Ashley,C.;BurelleJ.P.;Lawton,S.Rank1charactervarietiesoffinitelypresentedgroups. Geom.Dedicata 2018, 192,1–19.

20. Liskovets,V.;Mednykh,A.Onthenumberofconnectedanddisconnectedcoveringsoveramanifold. ArsMath.Contemp. 2009, 2, 181–189.

21. Goldman,W.M.TracecoordinatesonFrickespacesofsomesimplehyperbolicsurfaces.In HandbookofTeichmüllerTheory; EuropeanMathematicalSociety:Zurich,Switzerland,2009;Volume13,pp.611–684.

22. Cantat,S.;Loray,F.Holomorphicdynamics,PainlevéVIequationandcharactervarieties. arXiv 2007,arXiv:0711.1579v2.

23. Sandelin,A.;Alkema,W.;Engström,P.;Wasserman,W.W.;Lenhard,B.JASPAR:Anopen-accessdatabaseforeukaryotic transcriptionfactorbindingprofiles. NucleicAcidsRes. 2004, 32,D91–D94.Availableonline: https://jaspar.genereg.net/ (accessed on1January2022).

24. LambertS.A.;JolmaA.;CampitelliL.F.;DasP.K.;YinY.;AlbuM.;ChenX.;TalpaleJ.;HughesT.R.;WeirauchM.T.Thehuman transcriptionfactors. Cell 2018, 172,650–665.Availableonline: http://www.edgar-wingender.de/huTF$_$classification.html (accessedon1January2022).

25. Lantz,K.A.;Vatamaniuk,M.Z.;Brestelli,J.E.;Friedman,J.R.;Matschinsky,F.M.;Kaestner,K.H.Foxa2regulatesmultiplepathways ofinsulinsecretion. J.Clin.Investig. 2004, 114,512–520.

26. José-Edwards,D.S;Oda-Ishii,I.;KuglerJ.E.,Passamaneck,Y.J.;Katikala,L.;Nibu,Y.;DiGregorio,A.Brachyury,Foxa2andthe cis-RegulatoryOriginsoftheNotochord. PLoSGenet. 2015, 11,e1005730.https://doi.org/10.1371/journal.pgen.1005730.

27. Ray,B.K.;Dhar,S.;Shakya,Ray,A.Z-DNA-formingsilencerinthefirstexonregulateshumanADAM-12geneexpression. Proc.Natl.Acad.Sci.USA 2011, 108,103–106.

28. Ravichandran,S.;Subramani,V.K.;Kil,K.K.Z-DNAinthegenome:Fromstructuretodisease. Biophys.Rev. 2019, 11,383–387.

29. Nugent,C.I;Lundblad,V.Thetelomerasereversetranscriptase:Componentsandregulation. GeneDev. 1998, 12,1073–1085.

30. Dai,J.;Punchihewa,C.;Ambrus,A.;ChenD.;JonesR.A.;Yang,D.StructureoftheintramolecularhumantelomericG-quadruplex inpotassiumsolution:Anoveladeninetripleformation. Nucl.AcidsRes. 2007, 35,2440–2450.

31. Hu,L.;Lim,K.W.;Bouazuz,S.;Phan,A.T.GiardiaTelomericSequenced(TAGGG)4FormsTwoIntramolecularG-Quadruplexes inK+Solution:EffectofLoopLengthandSequenceontheFoldingTopology. J.Am.Chem.Soc. 2009, 131,16824–16831.

32. Kettani,A.;Bouazziz,S.;Wang,W.;Jones,R.A.;Patel,D.J.BombyxmorisinglerepeattelomericDNAsequenceformsa G-quadruplexcappedbybasetriads. Nat.Struc.Biol. 1997, 4,383–389.

33. Fulne˘cková,J.;Hasíková,T.;FajkusJ.;Luke˘sová,A.;Eliá˘s,M.Sýkorová,E.DynamicEvolutionofTelomericSequencesinthe GreenAlgalOrderChlamydomonadales. GenomeBiol.Evol. 2012, 4,248–264.

34. Gavory,G.;Farrow,M.;Balasubramanian,S.MinimumlengthrequirementofthealignmentdomainofhumantelomeraseRNA tosustaincatalyticactivityinvitro. Nucl.AcidsRes. 2002, 30,4470–4480.

Int.J.Mol.Sci. 2022, 23,13290 14of16

Int.J.Mol.Sci. 2022, 23,13290

35. Parkinson,G.N.;Ghosh,R.;Neidle,S.Structuralbasisforbindingofporphyrintohumantelomeres. Biochemistry 2007, 46, 2390–2397.

36. Zhang,N.;Phan,A.T.;Patel,D.J.(3+1)assemblyofthreehumantelomericrepeatsintoanasymmetricdimericG-Quadruplex. J.Am.Chem.Soc. 2005, 127,17277–17285.

37. Kang,C.;Zhang,X.;Ratliff,R.;Moyzis,R.;Rich,A.Crystalstructureoffour-strandedOxytrichatelomericDNA. Nature 1992, 356, 126–131.

38. Laughlan,G.;Murchie,A.I.H.;NormanD.H.;Moore,M.H.;Moody,P.C.E.;LilleyD.M.J.;Luisi,B.TheHigh-ResolutionCrystal StructureofaParallel-StrandedGuanineTetraplex. Science 1994, 265,520–524.

39. M.Planat,R.Aschheim,M.M.Amaral,F.FangandK.Irwin.Quantuminformationintheproteincodes, 3-manifoldsandthe Kummersurface. Symmetry 2021, 13,39.

40. Bansal,A.;Kaushik,S.;Kukreti,S.Non-canonicalDNAstructures:Diversityanddiseaseassociation. Front.Genet. 2022, 13, 959258.https://doi.org/10.3389/fgene.2022.959258.

41. Planat,M.;Chester,D.;Amaral,M.;IrwinK.Fricketopologicalqubits. Preprints 2022 .https://doi.org/10.20944/preprints202210.0125.v1.

42. Matsutani,S.;Previato,E.Analgebro-geometricmodelforthshapeofsupercoliedDNA. Phys.DNonlinearPhenom. 2022, 430,133073.

43. Rand,D.A.,Raju,A.;Saez,M.;Siggia,E.D.Geometryofgeneregulatorydynamics. Proc.Natl.Acad.Sci.USA 2021, 118, e2109729118.https://doi.org/10.1073/pnas.2109729118.

15of16

Turn static files into dynamic content formats.

Create a flipbook
Issuu converts static files into: digital portfolios, online yearbooks, online catalogs, digital photo albums and more. Sign up and create your flipbook.
DNA Sequence and Structure under the Prism of Group Theory and Algebraic Surfaces by Klee Irwin - Issuu