Design methods for 3D wireframe DNA nanostructuresOrponen – Design methods for 3D wireframe DNA nanostructures 2 by an open mesh of double-helix edges. This family of designs can

Orponen–Designmethodsfor3DwireframeDNAnanostructures

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Designmethodsfor3DwireframeDNAnanostructures

PekkaOrponen†

AbstractThefieldofstructuralDNAnanotechnologyaimsat the systematic development of self-assemblingnanostructures using DNA as the construction material.Research in this area is progressing rapidly, and thecontrolled,computer-aideddesignofincreasinglycomplexstructures is becoming feasible. One thread of thisendeavour is the design and characterisation of self-assembling 3D nanostructures based on wireframepolyhedralmodels.Thisarticleaims to illustrate someofthekeydevelopmentsinthisdirection,insufficientdetailsothatthereadercanachieveageneralunderstandingofthemainconceptsandapproaches.Theemphasisisonthedesign principles rather than experimentalmethodology,andtheroleofcomputerscienceandcomputationaltoolsissetforth.

Keywords DNA nanotechnology · DNA origami · Self-assembly·3Dnanostructures·Wireframemodels

1Introductionandbriefhistory

The powerful idea of using synthetic DNA to creategeometricallywell-characterised nanostructureswas firstintroduced by Nadrian Seeman in the early 1980’s(Seeman 1981, 1982). Since then this conception hasspawned the flourishing field of structural DNAnanotechnology(Pinheiroetal.2011;Seeman2015),whichnow has an increasing number of applications inbiochemicalandbiomedicalresearch,aswellasinseveralareas of nanoscience and –technology (Seeman 2010;Krishnan and Simmel 2011; Zhang et al. 2014;Chandrasekaranetal.2016a;Linkoetal.2016).Seeman’soriginal main motivation was in creating regular three-dimensionalDNAlatticesthatcouldbeusedasframeworksto position macromolecules in a spatially periodicarrangement for further uses. Crystallographic studies ofproteins were suggested in (Seeman 1982), but laterproposedapplicationsincludee.g.artificialcatalysts(Zhaoetal.2016)andplasmonicdevices(Kuzyketal.2012;Tianetal.2015;Güretal.2016).Inadditionto3Dlattices,alsolattices in two dimensions and finite polyhedra weresuggestedin(Seeman1982).

The immediate follow-up to Seeman’s early worksconsideredthedetaileddesignandcharacterisationoftheimmobile Holliday junctions that were proposed in

† Department of Computer Science, Aalto University, FI-00076 Aalto,Finland.E-mail:[email protected]=single-strandedDNA,dsDNA=double-strandedDNA.

(Seeman1981,1982)asthefundamentalbuildingblocksoftheenvisionedDNAlattices(SeemanandKallenbach1983;Kallenbachetal.1983;CooperandHagerman1987).Thenin 1991, Chen and Seeman presented the first detaileddesignofaDNApolyhedron,acubecomposedbyligatingandhybridisingtogethertensyntheticssDNAstrands(Chenand Seeman 1991). This work was also the first to beaccompaniedbya (partial) experimental characterisationof the designed structure. A design and partialcharacterisationofatruncatedDNAoctahedron,basedonasomewhatdifferentapproach,waspresentedthreeyearslater in (Zhang and Seeman 1994), and a design (notsynthesised)foraregularoctahedronwasproposedin(Lietal.1996).

A new approach towards the development of 3D DNApolyhedra was introduced by Shih, Quispe and Joyce in2004, when they presented a design for a regularoctahedron that was based on folding a single syntheticlongssDNA1stranduponitself(Shihetal.2004),usingasvertexmotifsthefour-wayjunctionsfrom(Seeman1981,1982)andasedgemotifstheDXandPXcrossoverschemesfrom(FuandSeeman1993;Seeman2001).Theresultingmolecular complex was experimentally characterised bymeans of both gel-shift analysis and cryo-electronmicroscopy.

Two years later, Paul Rothemund simplified andgeneralisedthesingle-strandfoldingapproachremarkablyby his idea of DNA origami (Rothemund 2006a). Thispowerful technique is based on folding a generic longscaffoldstrand,commonlythe7,249nt2ssDNAgenomeofbacteriophageM13mp18,tothedesiredtargetshapewiththehelpofshort,20-50nt,shape-specificstaplestrands.Rothemund’s seminal publication demonstrated theversatilityofthistechniquebyanumberofimpressive2Ddesigns,andhassincethenledtoalargenumberoffurtherdevelopmentsandapplications.Inparticular,anumberof3Dextensionsof theapproachhavebeendemonstrated,someofwhichwillbereviewedhereinmoredetail.

Several of the earliest 3D origami designs, e.g. (Ke et al.2009;Dietz et al. 2009;Hanet al. 2011),werebasedonvolume-or face-fillingpackingsofdoublehelices,3but inthe present overview we shall focus exclusively onwireframe-typedesigns,wherethedesiredshapeisformed

2nt=nucleotide.3Aninterestingnon-origamiapproachtovolume-filling3Ddesigns,usingDNA“bricks”,ispresentedin(Keetal.2012).


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by an open mesh of double-helix edges. This family ofdesigns can be further conceptually partitioned into“braided” (non-origami) designs, where the mesh iscomposedbyentwiningtogetheranumberofelementarymotifs(ChenandSeeman1991;ZhangandSeeman1994;Goodman et al. 2004; Shih et al. 2004; Goodman et al.2005;Heetal.2008;Zhangetal.2008),“modular”origamidesigns, where the mesh is built up from distinctelementaryDNAorigamicomponents(Rothemund2006b;Douglas et al. 2009; Iinuma et al. 2014), and “global”origamidesigns,wheretheorigamiscaffoldstrandisfirstrouted according to the graph-theoretic structure of themeshandthenstapledtogethertoachievethedesired3Dconformation(Hanetal.2013;Bensonetal.2015;Zhangetal.2015;Venezianoetal.2016).

Inthefollowing,weshalldiscusssomesampledesignsfromeachofthesecategoriesinroughlyhistoricalorder,startingfromthebraideddesignsof(ChenandSeeman1991;Shihet al. 2004;Heet al. 2008) in Section2, followedby themodularorigamidesignsof(Rothemund2006b;Douglasetal.2009; Iinumaetal.2014) inSection3,andconcludingwiththeglobalorigamidesignsof(Hanetal.2013;Bensonetal.2015;Venezianoetal.2016)inSection4.InSection5we discuss briefly the role of computer science andcomputationinthisresearcharea,andconcludewithsomesummaryremarksinSection6.

2Braideddesigns

2.1ChenandSeeman,1991

Thefirstdetaileddesignfora3DDNApolyhedronwastheDNA cube presented by Chen and Seeman in (Nature,1991). Theirdesign comprises ten carefully crafted80ntssDNAstrands(containinge.g.norepeated6ntsegments)whichareligatedandhybridisedtogetherassummarisedinFig.1,eventually forming12dsDNAedges thatboundthesixfacesofthecube.

4Infact,acubeisnotanidealteststructureforDNApolyhedraldesigns,because unless the corner joints are stiff, a wireframe cube is notstructurallyrigid,i.e.inarod-hingemodelitflexes.Atetrahedronorany

Fig. 1 Design of a DNA cube by braiding together ten ssDNAstrands. Reprinted with permission from (Chen and Seeman1991)

In the firststep (Fig.1, rows1-2), linearssDNAstrands1and6 are (a) cyclisedby ligation and (b) hybridisedwithlinearstrands2-5and7-10,respectively,toformtheLeft(L) and Right (R) faces of the cube,with complementarystickyendsA,B,C’,D’forLandA’,B’,C,DforR.

Inthesecondstep(Fig.1,rows2-3),thesticky-endpairs(C,C’)and(D,D’)arecombinedbyligation,joiningthefacesLandRandconcomitantlyalsocreatingthefront(F)faceofthe cube. In two further steps (not shown in Fig. 1), theresulting L-F-R complexes are purified and rehybridised,and the nicks between strand pairs (4, 9) and (2, 7) aresealed.Inafinalstep(Fig.1,row3),sticky-endpairs(A,A’)and (B, B’) are ligated together, closing the design andcreatingawireframeDNAstructurewiththeconnectivityofacube.

Thearticle(ChenandSeeman1991)presentsagelanalysisofeachstageofthissynthesisprocess,validatingthatthecorrect intermediateand final compoundsareproduced.However, since there is no characterisation of thegeometryof the finalmolecularcomplex, theauthorsdonotclaimthat itactuallyhastheoverallshapeofacube,onlythatithastherightconnectivity.4

Thesynthesisprotocolisrelativelycomplicated,becauseittakestheapproachofcatenatingtogetherdistinctcircularssDNAstrandsthatconstitutethefacesofthecube,byasophisticated sequence of hybridisations and ligations.Latertechniquesthatuseasinglebackboneor“scaffold”ssDNA strand that runs through the whole structure, orfocus on the vertex rather than the face motifs, do notcreate catenanes and hence achieve similar results with

otherconvexpolyhedronwithonlytriangularfaceswouldnothavethisproblem(Cromwell1999,Chapter6).


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much simpler protocols, ideally even in a single-potreaction.

A furtherproblem identified in thisearlyapproach is theneedfordetailedstoichiometryofthecomponentstrands.If the proportions of the reactants are not exactly right,thenincompletestructuresform,andpurificationmaybedifficult because the sizes of the correct and incorrectstructuresareclosetoeachother(Rothemund2006b).

2.2Shihetal.,2004

An innovative alternative approach to the creation ofpolyhedralDNAstructureswaspioneeredbyShih,Quispeand Joyce in (Shihetal.2004). Insteadofcomposing thetarget structure from shorter strands in a multi-stageprocess,theseauthorsproposefoldingitfromasinglelongstrand, joined by a small number of helper strands, in aone-pot synthesis. Some of the advantages of thisapproacharethat(a)thesynthesiscanbecompletedinasinglemixingandcoolingprocess,(b)sincethebackboneofthestructureismadeupofasinglestrand,stoichiometryis not a concern, and (c) since the strand folds (in asuccessful design) from its denatured state into theeventualstructurewithoutanytopologicalorkinetictraps,the structures could in principle be mass-produced bycloningtherequisiteDNAsequences.

Fig. 2 Folding of a DNA octahedron from a single long ssDNAstrand.Reprintedwithpermissionfrom(Shihetal.2004)

Asademonstrationofthisidea,thearticle(Shihetal.2004)presents the folding of a regular octahedron from asynthetic1,669ntssDNAbackbonestrand,augmentedbyfive40nt“helper”strands(Fig.2).Thedesignbuildsinan

innovativewayonanumberofmotifsintroducedearlier.The six vertices of the octahedron follow the four-wayjunctiondesignfrom(Seeman1981,1982),andofthe12edges, five are constituted according to the two-helixdouble-crossover(DX)schemefrom(FuandSeeman1993)and seven according to the paranemic crossover (PX)scheme from (Seeman2001). Eachedgeof the structurethuscomprisestwointerleaveddouble-helicaldomainsofDNA. (In Fig. 2(a) and Fig.2(b), the DX-based edgestructuresor“struts”areindicatedincyanblueandthePX-based struts in rainbow colours. Fig. 2(c) presents thecrossoverschematicofaPXstrut.)Thefivehelperstrandspartake in the forming of the DX-based struts. Thenucleotide sequences for the backbone and the helperstrands are carefully designed to form the correctsecondary structures and to minimise the possibility ofspurioushybridisations.

Thefoldingofthestructureproceedsintwophases.Inthefirst phase, the five helper strands hybridise with theirrespective complementary domains in the backbone toformthefiveDX-basedstruts,andtherestofthebackbonefolds upon itself to form fourteen 76 nt loops, eachcorresponding to one half of an eventual PX-based strut(Fig. 2(b)). In the second phase, the backbone loopshybridiseinpairstoformthesevenPXstruts,resultinginthe structure of Fig. 2(a). In Fig. 2(b) the PX pairings areindicatedbymatchingcoloursofthestrandloops.

Fig.3VisualisationoftheDNAoctahedronbasedoncryo-electronmicroscopy. Scale bar 20 nm. Reprinted with permission from(Shihetal.2004)

Fromgelmobility-shiftdata, theyieldof correctly foldedstructureswasestimatedatapproximately50%(Shihetal.2004), and the geometry of the resulting molecularcomplexeswascharacterisedbycryo-electronmicroscopy.Fig.3presentsasummaryofthischaracterisation:(a)rawmicrograph,withimagesofindividualstructuresboxed,(b)single-particlereconstruction,basedona largesampleofindividualimagesandimposedsymmetryassumptions,(c)correspondencesofsixrawimageswiththereconstructedstructure.


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As a design method, this single-strand folding approachwould generalise also to other polyhedra, contingent ontheavailabilityofappropriatedegree-𝑘vertexmotifs,andlimitedbytheconstraintsondesigningappropriatelengthversions of the DX and PX motifs. These concernsnotwithstanding, the general approach – formulated ingraph-theoretic terms –would be to consider the graph𝐺 = 𝑉, 𝐸 representingthevertices𝑉andedges𝐸ofthepolyhedronofinterest,andselectsomespanningtree𝑇 ⊆𝐸 for it.5 Thentheedges𝑒 ∈ 𝑇comprisingthespanningtreewouldberepresentedasDXmotifs,andtheedges𝑒 ∈𝐸 ∖ 𝑇 complementing the spanning tree into the fullpolyhedronasPXmotifs.

To what extent this general procedure for creating 3Dwireframestructureswouldworkexperimentallyremainslargelyunexplored,becausethemethodrequiresdetailedcontrolofthenucleotidesequenceofthebackbonestrand,and synthesising designer strands on the order ofthousandsofnucleotideshasatleastsofarbeenexpensiveand error-prone. Hence the technique was largelysuperseded by the introduction of Rothemund’s DNAorigamimethod(Rothemund2006a),whichonlyrequiresthescaffoldstrandtobe(sufficientlycloseto)random,andmoreover enables the use of a single generic scaffoldstrand for a great variety of designs. However, recentprogressinthesynthesisofgenome-lengthstrands(Gibsonet al. 2009, 2010; Hughes and Ellington 2017) mayeventuallyprovideanopportunitytorevisitalsothesingle-strandfoldingframework.6

2.3Heetal.,2008

Aratherdifferentapproachtothesynthesisofwireframestructures was reported in the article (He et al. 2008).Basedonexperiencefromearlierworkwith2Dstructures(Heetal.2005),theauthorssurmisedthatbyadjustingthestructural flexibility of elementary motifs and theirconcentration in a one-pot assembly process, one couldpromote the spontaneous formation of higher-orderstructuresofadesiredtype.Intheworkreportedin(Heetal.2008),basicthree-point-star(tri-star)motifswithsticky-endterminalsweremadetoself-assembleintotetrahedra,dodecahedra and truncated icosahedra (buckyballs),dependingonthenumberofunpairedbasesatthecentreofthetri-starmotifandtheDNAconcentrationinsolution.

5AspanningtreeofagraphGisacycle-freesetofedgesthatconnectsallverticesofG (Diestel2017,Section1.5).Efficientalgorithms for findingspanningtreesexistandareinwideuse(Cormenetal.2009,Chapter23).6 It isnoted inpassingat theendof (Shihetal.2004)thattheirdesigncouldbesimplifiedfurther,withsomelossofrigidity,byreplacingtheDXstrutsbysimpleduplexstruts,andthePXstrutsbyhairpinloopsthatarefirstcleavedopenbyrestrictionenzymesandthenjoinedby ligationto

Fig.4Schemataforself-assemblingtri-starmotifsintopolyhedralstructures.Reprintedwithpermissionfrom(Heetal.2008)

The basic tri-starmotif used in (He et al. 2005, 2008) ispresented in Fig. 4. It can be seen as a three-armedgeneralisationoftheDXdouble-crossoverschemefrom(FuandSeeman1993)andconsistsofalongrepetitivecentralstrand𝐿(blue+red),threeidenticalmedium-lengthstrands𝑀(green)thatpassthroughthemotifbycrossingfromonearmtoanother,andthreeidenticalshortstrands𝑆(black)thatcomplementthe𝑀strandsattheterminalendsofthearms. The central strand 𝐿 contains three short single-stranded segmentsor “loops” (red)whose lengthhasanimportant effect on the outcome of the self-assemblyprocess. Adding bases to the central loops increases theflexibilityofthemotifanditsamenabilitytosharperspatialangles.

Thefoldingofatargetpolyhedronproceedsintwostages:firsttheindividual𝐿,𝑀and𝑆strandscoalesceintotri-starmotifs, and in the second stage these hybridise at theirterminal sticky ends to form the polyhedral complex.Accordingtotheexperimentaldatapresentedin(Heetal.2008), theoutcomeof the second stagedependson thelength of the central loops in the 𝐿 strands and theconcentrationofDNA in thesolution:shortcentral loopsimposeobtuseangles,andhighDNAconcentrationfavourslarge assemblies. At central loop lengths of 5 nt each,tetrahedral complexeswere observed to formwhen theDNA strand concentration was < 100 nM, whereas at

theirpartners.OneofthereviewersofthepresentsurveyalsoaskedifthedesigncouldbebasedonDX-typestrutsalone.Fromthestrandroutingpointofviewthiswouldseemtobepossible,howeverwiththebackbonestrandcrossingitselfatsomepoint(s),sothatonewouldneedtotakecarealsothattheplannedroutingdoesnotformaknot(cf.Section4.2).Thefolding of this type of design hasmost likely not been experimentallytested.


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centrallooplengthsof3nt,theoutcomewasdodecahedraatlowDNAconcentration(~50nM)andbuckyballsathighDNA concentration (~ 500 nM). From gel-shift data, theyield for tetrahedra was estimated at 90%, fordodecahedraat76%andforbuckyballsat69%.

Fig.5Visualisationofself-assembledtetrahedralstructuresbasedoncryo-electronmicroscopy.Reprintedwithpermissionfrom(Heetal.2008)

Fig.5presentsacryo-electronmicroscopycharacterisationof the geometry of the tetrahedral complexes (imagingdata for the other structures is available in the originalpublication): (a) raw micrograph, (b) single-particlereconstruction, (c) correspondences of five raw imageswiththereconstructedstructure.

Along the same research line, article (Zhang et al. 2008)reports on applying a five-point-star motif to synthesiseDNA icosahedra and large nanocages. The advantage ofthis methodology is that it achieves one-pot, high-yieldself-assembly of well-characterised structures from verysimple designs and minimal control mechanisms. Itslimitation, on the other hand, is that because of thesymmetric nature of the synthesis process, it is onlyapplicabletohighlyregularstructuresanddoesnotseemtoprovideapathwaytoirregularones.

3Modularorigamidesigns

ThelandscapeofDNAnanostructuredesignchangedwiththe introductionof Rothemund’sDNAorigami technique(Rothemund 2006a), which made possible the general-purpose, computer-aided design of structures of almostarbitrary complexity, limitedmainly by the length of theavailable scaffold strand. Because of the simplicity,robustness and generality of the procedure, mostsubsequent research on DNA nanostructures has beendrawntoapplicationsandextensionsofthismethod,withalternative approaches to some extent having beenneglected.

WeshallnotreviewthebasicDNAorigamitechniquehere,because ithasbeenamplycovered inother sources,but

7 A polyhedron is simplicial if it is homeomorphic (i.e. inflatable to) asphere.Itistrivalentifallverticeshaveexactlythreeneighbours.

shallmovedirectlyto itsapplications inthesettingof3Dwireframe structures. In this section, we discussapproaches where the origami scaffolding is used toconnect together elementary motifs, and in the nextsection we cover methods where the target structureemergesfromaglobalroutingofthescaffoldstrandalongtheedgesofthewireframemesh.

3.1Rothemund,2006

InaveryinterestingFestschriftpaper(Rothemund2006b),contemporaneoustotheseminalDNAorigamipublication,RothemunddiscussespolygonalnetworkschemesderivingfromSeeman’swork,andproposesanewmodulardesignapproachbasedon connecting tri-star (andby extensionalso 𝑘-star) motifs with an origami-type scaffold strand(Fig.6).Theviabilityofthisschemehasapparentlyneverbeentestedinthelaboratory,althoughsomeofthemorerecent approaches (Zhang et al. 2015; Veneziano et al.2016) come close to the same result from a differentdirection.

Fig.6 (a)Tri-starmotifswith0-3openends. (b)Schematics forscaffold and helper joins. (c) Join details. Reprinted withpermissionfrom(Rothemund2006b)

The basic building blocks of the proposed approach arefour tri-star motifs with 0, 1, 2 or 3 “open” (unpaired)terminalends,presentedinFig.6(a).Atri-starwith𝑖openterminalsissaidtobeof“Type𝑖“.ToachievetheDNAself-assemblydesignofagiventrivalentsimplicialpolyhedron𝑃,7 the polyhedron is first represented in terms of itsSchlegeldiagram𝐺2,which is aplanargraphdelineatingthe vertex-edge relationships of 𝑃. (Intuitively, one canobtain𝐺2by“pulling𝑃flat”fromoneofitsfaces,sothatthechosenfacebecomestheouterboundaryof𝐺2.)Many


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interesting polyhedra, e.g. all Platonic solids except theoctahedron and the icosahedron, are trivalent, and thisproperty is of course reflected also in their Schlegeldiagrams.Fig.7depictsthetrivalentSchlegeldiagramsofthedodecahedron(top)andthebuckyball(bottom).

Fig. 7 Designs for a dodecahedron and a buckyball based oncombining tri-star elements by scaffold and helper joins.Reprintedwithpermissionfrom(Rothemund2006b)

Thedesignthenproceedsbycoveringtheverticesof𝐺2bytri-starelementsonebyone,inthefollowingsteps:

1. Atri-starofType0 isplacedonsomearbitrarilychosen initialvertex.Theouterboundaryof thistri-starconstitutestheinitialscaffoldstrandloop.

2. Agivenpartialcoveringisextendedby:

a. choosinganuncoveredneighbourvertexofsomealreadyplacedtri-star𝑡,

b. placingatri-star𝑡′ofType0onthenewvertex,c. “opening up” the terminal ends of 𝑡 and 𝑡5

abutting eachother and connecting the two tri-starsbya“scaffoldjoin”pairing(Fig.6(b)-(c),left).Thus,if𝑡waspreviouslyofType𝑖and𝑡5ofType0,now𝑡isofType𝑖 + 1and𝑡5isofType1.

Step2isrepeateduntilalltheverticeshavebeencovered.

8Thisspecificconnection isrecognisedandsummarised in(Rothemund2006b)withelegantconciseness:“WilliamShihhasobservedthatsingle-strandedorigamimaybeusedtocreatearbitrarypolygonalnetworks.To

Notethatthroughoutthisextensionprocess,(i)thetri-starcoveredpartof𝐺2staysconnected,(ii)alltheterminalsofthetri-starcoverare“closed”andonlyopenedonebyonewhenanewtri-starisadded,(iii)thescaffoldstrandloop,whichisinitialisedinStep1andextendedineachrepeatofStep 2, stays connected. The eventual outcome of thisprocessisthusaroutingofthescaffoldstrandaroundtheverticesof𝐺2,with“gaps”betweenneighbouring,butnotscaffold-paired tri-star terminals (cf. Fig. 7). To completethe design, the gaps are finally closed by “helper join”pairingsusingthenon-scaffoldstrandsinthetri-stararms(Fig.6(b)-(c),right).

Example designs for the dodecahedron and buckyballgraphsfollowingthisschemearepresentedinFig.7,wherethepartofthescaffoldstrandtraversing“inwards”intheSchlegel diagram is indicated with black and the parttraversing“outwards”isindicatedwithred.(Thescaffold-connectedtri-stararmsareextendedtovaryinglengthsintheseschematicstomakeplanarrepresentationpossible.In the actual 3D folding the tri-star motifs would besymmetric. Note also that the directions “inwards” and“outwards” in theplanarSchlegeldiagramcorrespondto“awayfromtheviewer”and“towardstheviewer”in3D.)Also many other routings of the structure are possible,dependingontheorderinwhichthetri-starelementsareplacedontheverticesofthediagraminStep2ofthedesignprocess.

Acomplementarypointofviewtothisdesignscheme,notexplicitly mentioned in (Rothemund 2006b), is that thescaffold-connectededgesinfactconstituteaspanningtree𝑇 of𝐺2, and the scaffold strand traverses “twicearoundthe tree” similarly as in e.g. the well-known TSPapproximationheuristicincomputerscience(Cormenetal.2009, Section 35.2). One could thus alternativelysummarise the approach by saying that one chooses anarbitrary spanning tree 𝑇 of 𝐺2 = 𝑉, 𝐸 ,routes thescaffoldstrandtwicearound𝑇,andthencreatesthenon-spanningtreeedges𝑒 ∈ 𝐸 ∖ 𝑇byhelperjoins.

ThisgeneralpointofviewalsorelatesRothemund’sdesignapproach to that of (Shih et al. 2004), which is similarlyimplicitlybasedonbuildingaspanningtreeofapolyhedralgraph using one type of connectors (DX struts), andcreating the non-spanning tree edges with another type(PXpairings).8

Rothemundfinallypointsoutanddiscussesthepossibilityofextendingthedesignmethodfromtrivalenttoarbitrarysimplicialpolyhedra,contingentontheavailabilityofwell-

seethis,replacehelperjoinswithparanemiccohesionmotifsandscaffoldjoins with Shih’s double-crossover struts in all the diagrams of thissection.”


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behaving𝑘-starmotifs. From the spanning-tree point ofviewitiseasytoseethatthemethodinfactgeneralisestoallconnectedspatialgraphs–notjustpolyhedra–withaminimumdegreeofat least3,butwhether suchdesignswould actually fold in the laboratory remains open toexperiment.

3.2Douglasetal.,2009

Possiblythefirstlaboratory-testedapproachtocreating3Dwireframenanostructuresbasedontheorigamitechniquewas presented in (Douglas et al. 2009). The article isprimarily concernedwith 3D structures constructed by ahelix-packingmethod,butitoutlinesalsothesynthesisofa wireframe icosahedron composed by hybridisingtogetherthree2Dorigami-basedflexibledouble-triangles(Fig.8).

Fig. 8 Design and characterisation of a modular DNA origamiicosahedron.Scalebars100nm.Reprintedwithpermissionfrom(Douglasetal.2009)

Thesynthesisoftheicosahedronisatwo-stageprocess.Inthe first stage, each of the three component double-triangles(Fig.8,topleft)isfoldedfroman8,100ntscaffoldstrand, a handcrafted variant of the commonly usedM13mp18 strand. To form these double-triangularstructures, the scaffold strand is initially stapled into abranchingtreeoffour5-starmotifvertices,connectedbythreescaffoldjoinedges(eachofwhichisinfactasix-helixbundle). Sticky-end strands at the ends of the half-strutlongarms(whichareagainsix-helixbundles)ofthevertexmotifsaredesignedsothattwopairsofthearmshybridise,in order create the remaining two edges in the double-trianglestructure.Fig.8(topmiddle)presentsaTEMimageofonesuchcomplex.

Eventhoughthedouble-trianglesaregeometricallysimilarand based on the same scaffold strand, they are madechemicallydifferentbyusingdifferentcyclicpermutationsofthescaffoldineachdouble-triangle.Inthesecondstage

oftheprocess,thethreecomponentdouble-trianglesaremixedtogether,andbyadesignedinter-componentsticky-endhybridisationoftheremainingtenunpairedhalf-strutarms in eachof the structures, the target icosahedron iscreated (Fig. 8, top right). The two lower rows of Fig. 8presentorthographicprojectionmodelsandTEMimagesoffouricosahedralcomplexessynthesisedbythismethod.

3.3Iinumaetal.,2014

Theworkpresentedin(Iinumaetal.2014)addressesthetask of creating large (edge-length ~100 nm) regularwireframe structures using the DNA origami technique.Thebasicdesignmotifisa“DNAtripod”(Fig.9(b)):avertexelement consisting of three 50 nm long arms, eachconstitutedasastiff16-helixbundle,andatriangulartwo-helixstrutthatcanbeusedtocontroltheinternalanglesbetweenthearms.DNAtripodarmscanbeconnectedintopolyhedral edges by sticky-end “connector” elements,severalofwhichweretriedoutin(Iinumaetal.2014),andby this approach all trivalent convex polyhedra withisometricedgesareinprincipleconstructible.

Fig.9TheDNAtripodmotifandpolyhedraldesignsbasedonit.Reprintedwithpermissionfrom(Iinumaetal.2014)

The authors of (Iinuma et al. 2014) demonstrate thisapproach by synthesising the structures displayed in Fig.9(a), which require adjusting the inter-edge anglesbetween 60° and 120°. The resulting complexes wereimagedinsolution,usingtheauthors’newly-developed3Dsuper-resolutionfluorescentmicroscopytechnique“DNA-PAINT”.

4Globalorigamidesigns

Themethodspresented inthissectiontakeamore“top-down”viewofthedesigntaskthanthepreviousones.Thisis reflected both in the design methodology and in thecomplexityofthestructuressynthesised.


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4.1Hanetal.,2013

The article (Han et al. 2013) may be the first one toformulatethe3DwireframeDNAstructuredesigntaskintermsofaglobalroutingofthescaffoldstrandthroughameshframework, insuchawaythattheself-crossingsofthescaffoldformtheverticesofthemesh.Inthiswork,themeshgraphisnotyetcompletelygeneral,butiscomposedof square motifs (Fig. 10(a)), whose two sides areconstitutedbyantiparallelsegmentsofthescaffoldstrand(red),connectedbystaplestrandsthatspantheperimetersofthesquares(grey).Thehelixarrangementaroundeachsquareiscarefullybalancedsothatthehelicalstrutsmeeteachotherata90°angle,ratherthaninanative60°angleconformation that each junctionmotifwould take on itsownbecauseofintrinsichelixgeometry(Fig.10(b)-(e)).Atypicalscaffoldroutingpathfora2DDNAgridironpatternispresentedinFig.10(f),(g).

Fig. 10 The DNA gridiron design principles. Reprinted withpermissionfrom(Hanetal.2013)

9 Although the design methodologies in (Shih et al. 2004; Rothemund2006b)arehighlygeneraltoo,thisisnotquiteexplicitinthearticlesandnotreflectedintheexperimentalwork.

Fig. 11 Designs and AFM images of 2D and 3D structuressynthesised as DNA gridiron meshes. Scale bars 200 nm.Reprintedwithpermissionfrom(Hanetal.2013)

Thearticle(Hanetal.2013)focusesmostlyonthedesignand synthesis of various 2Dmeshes, including oneswithglobal curvature (Fig.11(a)),butanumberof3Ddesignsarepresentedtoo(Fig.11(b),(c)).Asregards3Dstructures,theworkexceedsingeneralitypreviousapproaches,sincebyusingdifferentlengthDNAsegmentsindifferentpartsofthedesign,anyconvex3Dwireframemodelcomposedoftrapezoidalfacescouldinprincipleberendered,possiblysomenonconvexmodelstoo.9Apossibleconcernisthatameshcomposedoftrapezoidalelementsisnotnecessarilystructurallyrigid(cf.Footnote4);howeverthisissuedoesnot seem tobe affecting the complexesdisplayed in theexperimentaldatain(Hanetal.2013),possiblybecauseofthetightintertwiningofthecrossingscaffoldsegmentsatthe vertices of the mesh, which makes the vertices nothinge-like.

4.2Bensonetal.,2015

The article (Benson et al. 2015) presents a general top-down design methodology for rendering simplicialpolyhedra as wireframe DNA origami structures. Thestarting point is the planar Schlegel diagram of a givenpolyhedron, and the goal is to route the scaffold strandalong theedgesof thediagramand staple it togetheratlocations corresponding to the vertices, so that in self-assemblyamolecularcomplexcorrespondingtothetargetpolyhedroniscreated.


9

Amaindifferencetopreviousapproaches is thatthere isno notion of composing the structure out of elementaryvertexoredgemotifs.Anotherdifferenceincomparisontothetechniquein(Rothemund2006b),whichcouldalsobeseenasmotif-less,isthatalledgesarescaffold-connected,i.e. there are no cross-scaffold helper joins, and to thetechniquein(Hanetal.2013)thatthevertexdegreeofthepolyhedron isnot fixed,andmoreover themeshverticesarenotcreatedbycrossingsbutratherbyco-incidentturnsofthescaffoldstrand.

Fig. 12 A scaffold-routing based design process for the DNArendering of a subdivided icosahedron. Reprinted withpermissionfrom(Bensonetal.2015)

Fig. 12 presents an outline of the design process for asphere-like polyhedron 𝑃, technically a subdividedicosahedronor“dualbuckyball”(Fig.12(a)).ApartoftheSchlegeldiagram𝐺2of𝑃isshowninFig.12(b);ideallythecircular scaffold strandwouldbe routed so that it traceseach of the edges in 𝐺2 exactly once, thus forming anEulerian circuit in the graph (Diestel 2017, Section 1.8).However,byawell-knowntheoremofEuler(1741;Diestel2017, Section 1.8), this is possible if and only if all theverticesin𝐺2areofevendegree.Hencealltheodd-degreevertices – of which there are always an even number(Diestel 2017, Section 1.2) – are matched pairwise byaugmenting paths of auxiliary edges (Fig. 12(c)). Thenumber of added edges is minimised by using theminimum-cost graph matching algorithm of Edmonds(1965).

Infact,theroutingisaimingforaspecialtypeofEuleriancircuit which (i) does not cross itself at vertices, and (ii)evenmorestrictlyturnsback,uponenteringavertex,alongone of the two faces incident to the entering edge(left/right). An example of such an A-trail (Fleischner1990)10for𝐺2isdisplayedinFig.12(d).Onereasonforthisconstraintisthatonewishestostaplethesegmentsofthe10ThenameA-trailpresumablyderivesfromthesharp“A-like”turnsthepathmakesateachvertex.

scaffoldtogetherateachvertexasshowninFig.12(e),bystaplestrandsthatjointwoadjacent,butnotconsecutivesegments of the scaffold. Another, and morefundamentally important reason is thatanA-trail routingon a sphere is guaranteed to be unknotted, i.e.continuouslydeformableintoacircle(andviceversa),andsocreatesnotopologicaltrapsforthefoldingofthecircularscaffoldstrand.

AfteranappropriateA-trailroutingforthescaffoldstrandhasbeenfound,thedetailedpositions,lengthsandphasesoftheDNAhelicesareoptimisedinordertominimisethestrain created by the interhelix gaps at the vertex sites,where the scaffold strand needs to transit from oneincidenthelixtothenext.Theeventualresultisastrand-levelDNAmodelofthetargetpolyhedron(Fig.12(f)),fromwhich the requisite staple strand sequences can becomputed,giventhescaffoldstrandsequence.Thedesignprocess,includingstaplestrandsequencegeneration,hasbeen automated in a tool called vHelix,11 a DNAnanostructure design plugin for the Autodesk Maya 3Dcomputergraphicsandmodellingplatform.

Fig. 13Designs and imagesof simplicial polyhedra renderedaswireframeDNAorigamimeshes.Scalebars50nm.Reprintedwithpermissionfrom(Bensonetal.2015)

Fig.13presentsanumberofDNApolyhedradesignedandsynthesisedusingthisapproach.Thefirstrowshowsthe3Dmesh designs, the second row the strand-level DNAdesigns,andthethreefurtherrowselectronmicrographsoftheresultingcomplexes.(Theonefurthesttotherightisa variant of the “Stanford bunny”,12 a widely-used 3D

11http://www.vhelix.net/12https://en.wikipedia.org/wiki/Stanford_bunny


10

computer graphics test model.) All imaging is fromtransmission electronmicroscopy, except for the sphereimageatbottomleftandthebunnyimageatbottomright,whichareobtainedbycryo-electronmicroscopy.

Thetargetpolyhedraareinthisworkgenerallyassumedtobetriangulated(haveonlytriangularfaces),whichentailsthat their respective Schlegel diagrams are fullytriangulatedplanargraphs.Thisisnotstrictlynecessaryforthe method, but simplifies some considerations andguarantees that thewireframemodel is structurally rigid(Cromwell1999,Chapter6).

ItshouldalsobenotedthatwhilefindingordinaryEuleriancircuits in graphs is a standard task in computer sciencewithasimpleandefficientsolutionmethod(Cormenetal.2009, Section 22), finding A-trails is an NP-completeproblem.13 In (Benson et al. 2015), a branch-and-boundsearch space exploration algorithm is developed for thistask,andforthestrainrelaxationofthehelix-leveldesign,aphysicalsprings-and-rodsmodelisused.

According to (Benson et al. 2015), the DNA meshespresented inFig.13 ingeneral foldwell,withanaverageyield of 43%. The highest reported yield of 92% wasobtainedforthesubdividedicosahedron(𝑛 = 42vertices,𝑚 = 120edges) and the lowest yield of 4.5% for thesuperficiallysimple“stickman”figure(𝑛 = 92,𝑚 = 282).The visually complex “bunny” design (𝑛 = 70,𝑚 = 204)achieved an intermediate yield of 22%. An additionalbenefit isthatthefoldingseemstobestablealsoin low-salt physiological buffers. A possible concern at least forsome applications may be the floppiness of the single-duplexstruts,althoughnosystematiccomparisonstoe.g.DX-basedstrutshavesofarbeenmade.

4.3Venezianoetal.,2016

Therecentarticle(Venezianoetal.2016)presentsanotherhighly systematic and automated top-down approach totheDNArenderingofwireframepolyhedralstructures.Itismore closely connected to previously established designprinciples than theone in (Bensonet al. 2015), and is insome ways more general, in others slightly moreconstrainedthanit.Itwillbeinterestingtoseehowthetwoapproaches compare experimentally, and what newdevelopmentswillemergefromthem.

Thestartingpointofthedesignmethodin(Venezianoetal.2016) isagaintheSchlegeldiagram𝐺2=(V,E)ofagiventargetpolyhedron𝑃(Fig.14),forwhichaspanningtree𝑇

13Moreprecisely, theproblem isknowntobeNP-complete inEulerianpolyhedral graphs (Døvling Andersen and Fleischner 1995), but not intriangulated Eulerian polyhedral graphs. In fact, Fleischner (1990)

is firstdetermined(Fig.14(i)).Thenthescaffoldstrand isrouted twice around the spanning tree𝑇 similarly as in(Rothemund2006b;Fig.14(ii)),andstapledtogether intodouble-duplex strutswithDX-type connectormotifs (Fig.14(iii),2).Inadditiontothespanning-treeedges𝑒 ∈ 𝑇,thestrutsalsocovertwoopposinghalvesofeachnon-spanningtreeedge𝑒′ ∈ 𝐸 ∖ 𝑇, and theseare connectedby sticky-endhelperjoinmotifstoconstitutetheseedges(Fig.14(iii),3).Thevertices,whichinthisapproachcaninprinciplebeofarbitrarydegree,arereinforcedbyvertexstaples,eachofwhich spans either threeor four adjacent strut stems(Fig.14(iii),1).

Fig. 14 A spanning-tree routing based design process forwireframe DNA polyhedra. Reprinted with permission from(Venezianoetal.2016)

Many of these design principles derive from a previousarticlebypartlythesameauthors(Zhangetal.2015),but(Venezianoetal.2016)seemstobethefirstpublicationon3DwireframeDNA structures that expresses the idea ofspanningtreeroutings,augmentedbyhelperjoinpairingstocreatethenon-spanningtreeedges,reallyexplicitlyandmakes it the basis of a systematic, automated andempiricallyvalidateddesignmethod.Theoutcomeisquiteclose to thedesignsproposed in (Rothemund2006b), orindeed (Shih et al. 2004), but neither of these articlesformulates their approachexplicitly in termsof spanningtreeroutings.Theideaisalsopresentedbrieflyin(Bensonetal.2015,ExtendedDataFig.1),butonlytocomparethe

conjecturedthateverytriangulatedEulerianpolyhedralgraphhasanA-trail, but even under this assumption, no polynomial time algorithm isknownforfindingsuch.


11

A-trailroutingmethodtopreviousdesignstrategies.Withrespect to the apparently un-implemented proposal in(Rothemund 2006b), one should also note that althoughthe results of the high-level designs in (Veneziano et al.2016)arequitesimilar,thedetailededgeandvertexmotifssuggestedinthetwoarticlesareratherdifferent.

Thedesignprocessdescribedin(Venezianoetal.2016)hasbeenautomatedinthesoftwaretoolDAEDALUS,14andthearticledisplaysexamplesofa largenumberofsymmetricDNA polyhedra designed using this tool. Experimentalcharacterisationsarepresentedforsixofthepolyhedrainthe main article (three shown in Fig. 15), and in thesupplementarymaterialforaseventh.Thereportedyieldsforthedesignsareveryhigh,withanaverageof82%andarange between 60% (cube, 𝑛 = 8, 𝑚 = 12) and 93%(tetrahedron,𝑛 = 4,𝑚 = 6).Similarlyasin(Bensonetal.2015),thestructuresfoldwellandarestablealsoinlow-saltphysiologicalbuffers.

Fig. 15 Designs and images of spanning-tree routedwireframeDNAicosahedra,tetrahedraandoctahedra.Scalebars10nmforatomic models, 20 nm for AFM and cryo-EM. Reprinted withpermissionfrom(Venezianoetal.2016)

Theadvantagesofthespanning-treebaseddesignmethodin(Venezianoetal.2016),ascomparedtotheA-trailbasedmethodin(Bensonetal.2015),arethatitistheoreticallysimplerandcomputationallymoreefficient(althoughthedifference is probably not noticeable in practice), and it

14http://daedalus-dna-origami.org/

applies as such to arbitrary polyhedra, whereasgeneralising the A-trail based method to nonsimplicialpolyhedra leads to quite nontrivial theoreticalconsiderations. (However an extension to triangulatedpolyhedra on surfaces of any fixed genus was recentlypresented in (Mohammed and Hajij 2017)). A partialdrawbackofthespanning-treemethodisthatbecauseofthedouble-routingofmeshedges,itusestwicetheamountofscaffoldDNAoftheA-trailmethod,sothatsomeofthecomplexessynthesisedin(Bensonetal.2015,cf.alsoFig.13) would be unreachable with this approach using thestandard M13mp18 scaffold strand. On the other hand,double-routing visibly increases the rigidityof theedges,anddevelopments for longer scaffold strands areon theway: e.g. (Marchi et al. 2014) presents origami designsbasedon a biologically derived51,466nt scaffold strandand (Chandrasekaran et al. 2016b) surveys recentdevelopmentsinthisdirectionmorebroadly.

Initspresentform,theapproachof(Venezianoetal.2016)imposes some regularity on the achievable structuresbecauseof the internaldesignof theDNA struts (e.g. alledgelengthsarerequiredtobemultiplesofafullturnofDNA). But this is more of a compromise between thereliability and generality of themethod, not an inherentconstraint.Apossibleconcernrelatedtothescalabilityofthemethodistherelativelylargenumberofcross-scaffoldhelper joins (𝐹 − 1 for a simplicial polyhedron with𝐹faces), which could in complex designs lead tononspecific interactions across the termini of unpairedhalf-edge struts. Considering the high yields of thesynthesesreportedin(Venezianoetal.2016),thisdoesnotseemtobeaproblematleastforthestructuresshowninFig. 15. These are relatively simple, however, so furtherexperimentationwithanddevelopmentof themethod isneededtoexposemoreofitscharacteristics.

5Theroleofcomputerscienceandcomputation

ComputerscienceandcomputationalmethodsimpactthedevelopmentofDNAnanotechnology,inthepresentcaseresearchon3DwireframeDNAnanostructures,inatleastfourways:

1. Computational tools are crucial for the design ofincreasingly complex structures and to implementhigh-level,genericdesignapproaches.Thisaspecthasbeen well recognised in the DNA nanotechnologycommunityfromthebeginning,andparticularlysincethe introduction of Rothemund’s DNA origamitechnique.Inthepresentcontext,inparticulartheA-trailbaseddesigns(Bensonetal.2015)couldnothavebeenachievedwithoutcomputersupport,andalsoin


12

the other approaches at least the detailed strandsequencelistingsaretoolaborioustobeproducedbyhand. For non-origami schemes, an important andhighlynontrivialcomputationaltaskisalsothedesignandoptimisationofthenucleotidesequencesothatitindeed folds into the desired conformation(Brenneman and Condon 2002; Dirks et al. 2004;FeldkampandNiemeyer2006).

2. Well packaged and easily accessible software isnecessaryformakingthetechnologywidelyavailableoutsideofthecoreresearchcommunity.Thisissuehasalso beenwell recognised, and in the domain of 3Dwireframe structures, packages such as vHelix andDAEDALUS are aiming formaximally automated anduser-friendlyinterfacesforthenon-specialistusersofthedesigntools.Inotherdirections,e.g.theCadnano15softwarepackageiswidelyusedforthedesignofhelix-packed3DDNAnanostructures,andtheMfold16andNUPACK17 packages for sequence design andsecondarystructurepredictiontasks.

3. A computer science perspective brings clarity andgeneralitytotheplanningofnewdesignapproaches.Whilenanostructuredesignisfundamentallyrootedinthe characteristics of thematerial, a high-level viewcan provide valuable complementary insights. In thepresent setting, this is illustrated for instanceby theprevalenceoftheconceptofspanningtreeroutingsinseveral of the design approaches (Shih et al. 2004;Rothemund2006b;Zhangetal.2015;Venezianoetal.2016), even though the concept was not explicitlyrecognised and fully utilised until (Veneziano et al.2016). As another example, the A-trail based designtechnique(Bensonetal.2015)isdeeplyembeddedingraph-theoretic concepts and algorithms. Furtherapplicationsof graph-theoretic and topological ideasto the design of wireframe nanostructures arepresentedin(KlavžarandRus2013;Ellis-Monaghanetal.2015,2017;MohammedandHajij2017;Morseetal.2017).

4. Computationalmodellingandsimulationtoolswillbeincreasinglyusefultospeedupthedesignprocessandreducelaboratorytimeandcosts.Asthecomplexityofthe targeted structures increases, it becomesprogressivelymoreimportanttobeabletopre-screenthem computationally, before committing expensivelaboratory resources to their experimentalcharacterisation. In the context of 3D wireframedesigns,importanttoolsincludee.g.theCanDofinite-element modelling framework for DNAnanostructures,18which is being integratedwith theDAEDALUSdesignpackage,andtheoxDNAmolecular

15http://cadnano.org/16http://unafold.rna.albany.edu/17http://www.nupack.org/18http://cando-dna-origami.org/

dynamics modelling framework,19 which has beenusedtosimulateandvisualisemany interestingDNAstructures and processes, including e.g. the A-trailbased sphere20 and bunny21molecular complexes. Acloser integrationof thedesign and simulation toolscan be expected in the future, so that eventuallyautomated, model-based pre-assessment of thestructures can also be used to guide the designprocess.

6Summary

Three-dimensional DNA nanostructure designs based onwireframe polyhedral models have evolved into aninteresting alternative to the more established helix-packing3Ddesigns:severaldistinctapproachesexist,andfunctionalisations are beginning to emerge. Wireframedesigns are appealing both because they make moreefficient use of the DNA scaffold than helix-packingapproaches, and because they seem to fold with higheryield and remain more stable in low-salt, physiologicalbufferconditions.

Because of the inherent combinatorial complexity ofwireframedesigns,automationofthedesignprocess isacentral task already for exploratory research, and evenmoresowhenaimingtomakethemethodologyrobustandgenerally available. Several computerised tools alreadyexistforassistinginthehigh-leveldesignstage,butthesestill need to be complemented by and integrated tonumericalmodellingsoftwaretohelpinvalidatingandfine-tuningthedesignsbeforemovingtothesynthesisstage.

AcknowledgmentsPreparingthissurveywasmadepossiblebyasabbaticalleavefromAaltoUniversity,andmostofitwaswrittenduring stays at ETH Zürich (host Prof. RogerWattenhofer), theUniversity of Electro-Communications in Tokyo (host Prof.ShinnosukeSeki)andtheTokyoInstituteofTechnology(hostProf.OsamuWatanabe).Iamverygratefulforthisopportunity.Ialsothank the two anonymous reviewers for their constructive andinsightfulcomments.

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