peng yin, harry m. t. choi, colby r. calvert and niles a. pierce- programming biomolecular...

8/3/2019 Peng Yin, Harry M. T. Choi, Colby R. Calvert and Niles A. Pierce- Programming biomolecular self-assembly pathways

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LETTERS

Programming biomolecular self-assembly pathwaysPeng Yin1,2, Harry M. T. Choi1, Colby R. Calvert1 & Niles A. Pierce1,3

In nature, self-assembling and disassembling complexes of pro-teins and nucleic acids bound to a variety of ligands performintricate and diverse dynamic functions. In contrast, attempts torationally encode structure and function into synthetic amino acidand nucleic acid sequences have largely focused on engineeringmolecules that self-assemble into prescribed target structures,rather than on engineering transient system dynamics1,2. Todesign systems that perform dynamic functions without humanintervention, it is necessary to encode within the biopolymersequences the reaction pathways by which self-assembly occurs.Nucleic acids show promise as a design medium for engineeringdynamic functions, including catalytic hybridization36, triggeredself-assembly7 and molecular computation8,9. Here, we programdiverse molecular self-assembly and disassembly pathways using areaction graph abstraction to specify complementarity relation-ships between modular domains in a versatile DNA hairpin motif.Molecular programs are executed for a variety of dynamic func-tions: catalytic formation of branched junctions, autocatalyticduplex formation by a cross-catalytic circuit, nucleated dendriticgrowth of a binary molecular tree, and autonomous locomotionof a bipedal walker.

The hairpin motif (A in Fig. 1a) comprises three concatenateddomains, a, b and c. Each domain contains a special nucleation sitecalled a toehold10, denoted at, bt and ct. Two basic reactions can be

programmed using this motif, as illustrated for the example of cata-lytic duplex formation in Fig. 1b. First, an assembly reaction (1)occurs when a single-stranded initiator I, containing an exposedtoehold at*, nucleates at the exposed toehold at of hairpin A, initiat-ing a branch migration that opens the hairpin. Hairpin domains band c, with newly exposed toeholds bt and ct, can then serve asassembly initiators for other suitably defined hairpins, permittingcascading (for example, in reaction (2), domain b of hairpin A assem-bles with domain b* of hairpin B, opening the hairpin). Second, adisassembly reaction (3) occurs when a single-stranded domain (a*of B) initiates a branch migration that displaces the initiator I from A.In this example, I catalyses the formation of duplex ANB through aprescribed reaction pathway.

To assist in programming more complex reaction pathways, we

abstract the motif of Fig. 1a as a node with three ports (Fig. 1c): atriangular input port and two circular output ports. The state of eachport is either accessible (open triangle/circle) or inaccessible (solidtriangle/circle), depending on whether the toehold of the corres-ponding motif domain is exposed or sequestered. Functional rela-tionships between ports within a node are implicit in the definitionof the nodal abstraction corresponding to a particular motif (forexample, for the node of Fig. 1c, the output ports flip to accessiblestates if the input port is flipped to an inaccessible state through aninteraction with a complementary upstream output port). By depict-ing assembly reactions by solid arrows and disassembly reactionsby dashed arrows (each directed from an output port to a comple-mentary input port of a different node), reaction pathways can be

specified abstractly in the form of a reaction graph, representing aprogram to be executed by nucleic acid molecules.

The reactions depicted in the secondary structure mechanism ofFig. 1b are specified using a reaction graph in Fig. 1d. The initialconditions for this program are described via the state of each portin the reaction graph. Figure 1e depicts the execution of this reactiongraph through cascaded assembly and disassembly reactions. Anassembly reaction is executed when ports connected by a solid arroware simultaneously accessible. For the initial conditions depicted inFig. 1d, the program must start with the execution of reaction (1).

Reaction 1 (assembly): in an assembly reaction (executed here bythe accessible output port of I and the complementary accessibleinput port of A), a bond is made between the ports and they areflipped to inaccessible states; the two output ports of A are flipped

1Department of Bioengineering, 2Department of Computer Science, 3Department of Applied & Computational Mathematics, California Institute of Technology, Pasadena, California91125, USA.

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Figure 1 | Programming biomolecular self-assembly pathways.a, Secondary structure of the hairpin motif. Coloured lines represent stranddomains; short black lines representbase pairs;arrowheads indicate 39 ends.Domain c is optional. b, Secondary structure mechanism illustratingassembly and disassembly reactions during catalytic duplex formation.Asterisks denote complementarity. c, Abstraction of the motif A as a node

with three ports (colour use is consistent with a). d, A reaction graphrepresenting a molecular program executed schematically in b ande. e, Execution of the reaction graph of d. f, Hierarchical design process.

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to accessible states (based on the internal logic of node A). Reaction 2(assembly): a bond is made between the newly accessible blue outputport of A and the complementary accessible input port of B and bothports are flipped to inaccessible states; the output port of B is flippedto the accessible state (based on the internal logic of node B).Reaction 3 (disassembly): in a disassembly reaction (executed hereby the newly accessible output port of B, the inaccessible input port ofA, and the inaccessible output port of I), the bond between the outputport of I and the input port of A is displaced by a bond between the

output port of B and the input port of A; the states of the two outputports are flipped (see Supplementary Information 2 for additionaldetails).

The reaction graph provides a simple representation of assembly(and disassembly) pathways that can be translated directly intomolecular executables: nodes represent motifs, ports representdomains, states describe accessibility, arrows represent assemblyand disassembly reactions between complementary ports. Startingfrom a conceptual dynamic function, a molecular implementationis realized in three steps (Fig. 1f): (1) pathway specification via areaction graph; (2) translation into secondary structure motifs; (3)computational design of motif primary sequences (see Methods fordetails). We demonstrate the utility of this hierarchical design pro-cess by experimentally executing molecular programs encoding four

distinct dynamic functions.Program 1: Catalytic geometry. Current protocols for self-assembling synthetic DNA nanostructures often rely on annealingprocedures to bring interacting DNA strands to equilibrium on thefree-energy landscape1113. By contrast, self-assembly in biologyproceeds isothermally and assembly kinetics are often controlled bycatalysts. Until now, synthetic DNA catalysts36 have been used tocontrol the kinetics of the formation of DNA duplex structures.The next challenge is to catalyse the formation of branchedDNA structures, the basic building blocks for DNA structuralnanotechnology14,15.

First, we demonstrate the catalytic formation of a three-arm DNAjunction. The assembly and disassembly pathways specified in thereaction graph of Fig. 2a are translated into the motif-based mole-cular implementation of Fig. 2b (see Supplementary Information 3.1for details). The complementarity relationships between the seg-ments of hairpins A, B, and C are specified (Fig. 2b, top) so that inthe absence of initiator strand I, the hairpins are kinetically impededfrom forming the three-arm junction that is predicted to dominate atequilibrium. In the reaction graph, this property is programmed bythe absence of a starting point if node I is removed from the graph(that is, no pair of accessible ports connected by an assembly arrow).The introduction of I into the system (Fig. 2b, bottom) activates acascade of assembly steps with A, B and C, followed by a disassembly

step in which C displaces I from the complex, freeing I to catalyse theself-assembly of additional branched junctions.

Gel electrophoresis confirms that the hairpins assemble slowly inthe absence of initiator and that assembly is markedly accelerated bythe addition of initiator (Fig. 2c). Disassembly of the initiatorleads tocatalytic turnover, as indicated by the nearly complete consumptionof hairpins even at substoichiometric initiator concentrationsInterestingly, only minimal assembly is achieved by annealing thehairpin mixture, illustrating the utility of pathway programming

for traversing free-energy landscapes with kinetic traps that cannobe overcome by traditional annealing approaches.

Direct imaging of the catalysed self-assembly product ANBNCby atomic force microscopy (AFM) reveals the expected three-armjunction morphology (Fig. 2d). In principle, the reaction pathwaycan be extended to the catalytic self-assembly of k-arm junctions(Supplementary Information 3.5). We illustrate k5 4 with the reac-tion graph and AFM image of Fig. 2e and f.

Program 2: Catalytic circuitry. By programming cross-catalyticself-assembly pathways in the reaction graph of Fig. 3a, we obtainan autocatalytic system with exponential kinetics. In the corresponding molecular implementation, four hairpin species, A, B, C and Dcoexist metastably in the absence of initiator I (Fig. 3b, top). Theinitiator catalyses the assembly of hairpins A and B to form duplex

ANB (steps 12, Fig. 3b, bottom), bringing the system to an exponen-tial amplification stage powered by a cross-catalytic circuit: theduplex ANB has a single-stranded region that catalyses the assemblyof C and D to form C ND (steps 34); duplex CND in turn has a single-stranded region that is identical to I and can thus catalyse A and B toform ANB (steps 56). Hence, ANB and CND form an autocatalytic secapable of catalysing its own production. Disassembly (steps 2b, 4band 6b) is fundamental to the implementation of autocatalysis andsterically uninhibited exponential growth.

Each step in the reaction is examined using native polyacrylamidegel electrophoresis (Supplementary Fig. 12), showing the expectedassembly and disassembly behaviour. System kinetics are examinedin a fluorescence quenching experiment (Fig. 3c). Spontaneousinitiation in the absence of initiator reflects the finite timescale assoc

iated with the metastability of the hairpins and yields a sigmoidatime course characteristic of an autocatalytic system16. As expectedthe curve shifts to the left as the concentration of initiator isincreased. A plot of 10% completion time against the logarithm ofthe concentration shows a linear regime, consistent with exponentiakinetics and analytical modelling (Fig. 3c, inset). The minimaleakage of a system containing only A and B (labelled A1B inFig. 3c) emphasizes that the sigmoidal kinetics of spontaneous initiation for the full system (A1B1C1D) are due to cross-catalysis

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Figure 2 | Programming catalytic geometry: catalytic self-assembly ofthree-arm and four-arm branched junctions. See SupplementaryInformation 3 for details. a, Reaction graph for three-arm junctions.b, Secondary structure mechanism. Each letter-labelled segment is sixnucleotides in length. The initially accessible(a* forstep1) or newly exposed(b* for Step 2, c* for step 3) toeholds that mediate assembly reactions arelabelled with purple letters. c, Agarose gel electrophoresis demonstrating

catalytic self-assembly for the three-arm system with 750-nM hairpins.Nearly complete conversion of hairpins to reaction products usingstoichiometric or substoichiometric initiator I (lanes 14). Minimalconversion in the absence of initiator (lane 5), even with annealing (lane 6).d, AFM image of a three-arm junction. Scale bar: 10 nm. e, Reaction graphand f, AFM image for a four-arm junction. Scale bar: 10 nm.

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This system demonstrates synthetic biomolecular autocatalysis1720

driven by the free energy of base-pair formation. Autocatalysis andexponential system kinetics can also be achieved through entropy-driven hybridization mechanisms21. Forsensing applications, the trig-gered exponential growth of these systems suggest the possibility ofengineering enzyme-free isothermal detection methods.

Program 3: Nucleated dendritic growth. The molecular programin Fig. 4a depicts the triggered self-assembly of a binary moleculartree of a prescribed size. The reaction starts with the assembly of an

initiator node I with a root node A1. Each assembled node subse-quently assembles with two child nodes during the next generation ofgrowth, requiring two new node species per generation. In theabsence of steric effects, a G-generation dendrimer requires 2G 1node species and yields a binary tree containing 2G1 monomers,that is, a linear increase in the number of node species yields an

exponential increase in the size of the dendrimer product. Figure 4bdepicts the motif based implementation of the program depictedin Fig. 4a: hairpins are metastable in the absence of initiator; the

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Figure 3 | Programming catalytic circuitry: autocatalytic duplex formationby a cross-catalytic circuit with exponential kinetics. See SupplementaryInformation 4 for details. a, Reaction graph. Multiple assembly arrowsentering the same input port depict parallel processes on separate copies ofthe nodal species. b, Secondary structure mechanism. c, System kineticsexamined by fluorescence quenching.Formationof ANB is monitored by theincrease in fluorescence resulting from increased spatial separation betweenthe fluorophore (green star in b) and the quencher (black dot in b) at either

end of A. Raw data for two independent reactions are displayed for eachinitiator concentration (20-nM hairpins). Single traces are shown for thecontrols containing only A and B or only A. Inset: linear fit of the 10%completion time against the logarithm of the relative concentration of I(0.0033# [I]# 0.053). High-concentration end points ([I]$ 0.13) areexcluded based on theoretical analysis; low-concentration end points([I]# 0.0013) are excluded because of signal poisoning by leakage. SeeSupplementary Information 4.4 for a detailed treatment.

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Figure 4 | Programming nucleated dendritic growth: triggered assembly ofquantized binary molecular trees. See Supplementary Information 5 fordetails. a, Reaction graph.Multipleassemblyarrows entering thesame inputport depict parallel processes on separate copies of the nodal species.b, Secondary structure mechanism. c, Agarose gel electrophoresisdemonstrating triggered self-assembly. Lanes 16: the dominant reaction

band shifts with the addition of each generation of hairpins. Subdominant

bands are presumed to represent imperfect dendrimers. Lane 7: minimalconversion to reaction products in the absence of initiator. Hairpins A1, A2,B2 at 62.5 nM; the concentration doubles for each subsequent generation ofhairpins. Initiator I at 50 nM. d, Linear relationship between amplificationsignal (putative G5 reaction product) and initiator for three independentexperiments (cross, diamond, circle). See Supplementary Fig. 17 for details.

e, AFM images of G3, G4 and G5 dendrimers. Scale bars: 30 nm.

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initiator I triggers the growth of a dendrimer with five generations ofbranching (G5).

We constructed trees with G5 1, 2, 3, 4 and 5. The nucleatedgrowth of the tree is examined using native agarose gel electro-phoresis. Band shifting demonstrates increasing dendrimer size witheach generation of growth (Fig. 4c). Figure 4d demonstrates that theconcentration of dendrimer depends linearly on the concentrationof the initiator in the system. Finally, AFM imaging of dendrimersfor G5 3, 4 and 5 reveals the expected morphologies (Fig. 4e).

Measurements of the dendrimer segment lengths agree well withthe design (Supplementary Information 5.4).

In contrast to previous work in which DNA dendrimer targetstructures were synthesized by sequential ligation of structural sub-units22, here we program self-assembly pathways so that DNA mono-mers form dendrimers only on detection of a target nucleationmolecule. By growing to a prescribed size, these dendrimers providequantitative signal amplification with strength exponential in thenumber of constituent species.

Program 4: Autonomous locomotion. The challenge of engineer-ing molecular machines capable of nanoscale autonomous loco-motion has attracted much interest in recent years2327. Inspired by

the bipedal motor protein, kinesin, which hauls intracellular cargo bystriding along microtubules28, we have developed an autonomousenzyme-free bipedal DNA walker capable of stochastic locomotionalong a DNA track.

Joined by a duplex torso, each of two identical walker legs, I, iscapable of catalysing the formation of waste duplex A NB from meta-stable fuel hairpins A and B through a reaction pathway in which Iassembles with A, which assembles with B, which subsequently dis-assembles I fromthe complex (see Fig. 5a and b for the reaction graph

and corresponding molecular implementation). The track consists ofive A hairpins arranged linearly at regular intervals along a nickedDNA duplex. In the presence of hairpin B, a subpopulation of walkersis expected to move unidirectionally along the track by sequentiallycatalysing the formation of ANB. Because of the one-dimensionaarrangement of anchor sites, this processive motion occurs only forthose walkers that use a foot-over-foot gait by stochastically liftingthe back foot at each step.

We investigate walker locomotion using a bulk fluorescence assaythat tests whether there is a subpopulation of walkers that movesprocessively through positions 3, 4 and 5, starting from an initialcondition with legs anchored at positions 1 and 2. Quenchers areattached to the walkers legs and spectrally distinct fluorophores arepositioned proximal to anchorages 3, 4 and 5. Consistent with pro-

cessivity, the anticipated sequential transient quenching of the fluor-ophores at positions 3, 4 and 5 is observed (Fig. 5c). To rule out thepossibility that this signal arises from non-processive walker dif-fusion through the bulk solution from one position to the next, werepeated the experiments using monopedal walkers that lack a mechanism for achieving processivity. In this case, the sequential transientquenching no longer matches the ordering of the fluorophores alongthe track (Fig. 5d) and the timescale for visiting any one of the threeanchorages is longer than the timescale to visit all three anchoragefor the bipedal system (Fig. 5e). Additional control experiments(Supplementary Information 6.9) show that this difference in time-scales cannot be explained by the relative rates with which freelydiffusing bipedal and monopedal walkers land on the track. As afurther test of processivity for the bipedal walker, reordering the

fluorophores along the track leads to the expected change in theordering of the transient quenching (Fig. 5f).

The experimental execution of these four molecular programsdemonstrates that the hairpin motif functions as a modular pro-grammable kinetic trap, and that rewiring the connections betweennodes in the reaction graph corresponds to rewiring the connectionsbetween kinetic traps in the underlying free-energy landscape. In thephysical systems, metastable hairpins are initially caught in engi-neered kinetic traps; the introduction of initiator molecules begina chain reaction of kinetic escapes in which the hairpin species inter-act through programmed assembly and disassembly steps to imple-ment dynamic functions. It is important that the timescale ometastability for kinetically trapped molecules is longer than thetimescale relevant for the execution of the program. We found it

helpful to incorporate clamping segments at the ends of helices todiscourage the initiation of non-toehold-mediated branch migra-tions (see Supplementary Information 3.1). We also found thatimpure strand syntheses artificially reduce the strength of metastabletraps and increase leakage rates. System fidelity was improved byligating hairpins out of two shorter segments to increase strand pur-ity (Supplementary Information 7.1).

Reaction graphs can be extended beyond the present versatilemotif by defining new nodal species that abstract the functionarelationships between domains in other motifs. The present hie-rarchical approach to encoding dynamic function in nucleic acidsequences represents a promising step towards the goal of construct-ing a compiler for biomolecular functionan automated designprocess that requires as input a modular conceptual system design

and provides as output a set of biopolymer sequencesthat encode the

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Figure 5 | Programming autonomous locomotion: stochastic movement ofa bipedal walker. See Supplementary Information 6 for details. a, Reactiongraph.Bondsbetweenoutput portson I andinput portson A representinitialconditions. Static structural elements are depicted by grey line segments.b, Secondary structure mechanism depicting processive locomotion. SeeSupplementary Information 6.1 and 6.3 for non-processive trajectories.cf, Fluorescence quenching experiments measuring the proximity of thequenchers (black dots) on the walker feetto the fluorophores (coloured stars)decorating the track. Fitted curves (solid) are used to determine the time at

which the minimum fluorescence (maximum quenching) was observed(dashed vertical line) for each fluorophore. c, Bipedal walker with tracklabelled by fluorophores JOE (green star)RTAMRA (red)RFAM(blue) asin b. For each pair of consecutive minima (JOER TAMRA and TAMRARFAM), we testthe nullhypothesisthat the mediantime difference betweentheminima is zero against the alternative hypothesis that the time difference ispositive. Based on a statistical analysis of six independent experiments (seeSupplementary Information 6.6, 6.7), the null hypothesis can be rejected forboth time differences with the same P-value of 0.0156, supporting theinterpretation that the observed minima are sampled from a distribution in

which the ordering of the minima matches the physical ordering of thefluorophores along the track. Similar interpretations apply to the ordering ofminima for d and f. d, Monopedal walkers on the same track (JOE (orangestar)R TAMRA (pale green)R FAM (pale blue)). e, Comparison of timescales for bipedal and monopedal walkers (eighteen traces per walker type:three fluorophores, six experiments). f, Bipedal walker with track labelledTAMRA (red star)R JOE (green)R FAM (blue).

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desired dynamic system behaviour (Supplementary Information7.2).

METHODS SUMMARY

Starting from a conceptual dynamic function, a molecular implementation isrealized in three steps summarized in Fig. 1f.See Supplementary Information3.1for an example illustrating the design of the catalytic three-arm junction system.Step (1): pathway specification. We specify thepathway that implements a targetdynamic function using a reaction graph. Step (2): translation to motifs. Thereaction graphis directly translatedto motifsecondarystructures. First,the basic

complementarity requirements are defined and then clamping/padding seg-ments are added (as in Supplementary Information 3.1). Initial dimensioningof the number of nucleotides in each segment is performed using the NUPACKserver (www.nupack.org), which models the behaviour of strand species in thecontext of a dilute solution (including unintended species of complexes)29. Step(3): sequence design. Sequences are designed by considering a suite of structuresthat punctuate the intended reaction pathway or that explicitly precludeundesired off-pathway interactions (for example, structures specifying theabsence of an interaction between two strands that should not pair). Thesequencesare optimizedcomputationally (J.N. Zadeh andR. M.Dirks, personalcommunication) to maximize affinity and specificity for this suite of structuresby minimizing the average number of incorrectly paired bases at equilibrium30.We then synthesize and verify the system using gel electrophoresis, bulk fluore-scence quenching, or single-molecule AFM.

Full Methods and any associated references are available in the online version of

the paper at www.nature.com/nature.

Received 20 July; accepted 31 October 2007.

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13. Rothemund, P. W. K. Folding DNA to create nanoscale shapes and patterns.

Nature 440, 297302 (2006).

14. Seeman, N. C. Nucleic acid junctions and lattices. J. Theor. Biol. 99, 237247

(1982).

15. Feldkamp, U. & Niemeyer, C. M. Rational design of DNA nanoarchitectures.

Angew. Chem. Int. Edn Engl. 45, 18561876 (2006).

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Soc. Rev. 29, 141152 (2000).

17. von Kiedrowski, G. A self-replicating hexadeoxynucleotide. Angew. Chem. Int. Edn

Engl. 25, 932935 (1986).

18. Paul, N. & Joyce, G. F. A self-replicating ligase ribozyme. Proc. Natl Acad. Sci. USA

99, 12733

12740 (2002).19. Levy, M. & Ellington, A. D. Exponential growth by cross-catalytic cleavage of

deoxyribozymogens. Proc. Natl Acad. Sci. USA 100, 64166421 (2003).

20. Lee, D. H., Granja, J. R., Martinez, J. A., Severin, K. & Ghadiri, M. R. A self-

replicating peptide. Nature 382, 525528 (1996).

21. Zhang,D. Y.,Turberfield, A. J.,Yurke, B. & Winfree, E. Engineering entropy-driven

reactions and networks catalyzed by DNA. Science 318, 11211125 (2007).

22. Li, Y. et al. Controlled assembly of dendrimer-like DNA. Nature Mater. 3, 3842

(2004).

23. Yin, P., Yan, H., Daniell, X. G., Turberfield, A. J. & Reif, J. H. A unidirectional DNA

walker that moves autonomously along a track. Angew. Chem. Int. Edn Engl. 43,

49064911 (2004).

24. Tian, Y., He, Y., Chen, Y., Yin, P. & Mao, C. A. DNAzyme that walks processively

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43554358 (2005).

25. Bath, J., Green, S. J. & Turberfield, A. J. A free-running DNA motor powered by a

nicking enzyme. Angew. Chem. Int. Edn Engl. 44, 43584361 (2005).

26. Pei, R. et al. Behavior of polycatalytic assemblies in a substrate-displaying matrix.J. Am. Chem. Soc. 128, 1269312699 (2006).

27. Venkataraman, S.,Dirks,R. M.,Rothemund,P. W.K., Winfree,E. & Pierce, N.A. An

autonomous polymerization motor powered by DNA hybridization. Nature

Nanotechnol. 2, 490494 (2007).

28. Asbury, C. L. Kinesin: worlds tiniest biped. Curr. Opin. Cell Biol. 17, 8997 (2005).

29. Dirks,R. M.,Bois, J.S., Schaeffer, J.M., Winfree,E. & Pierce, N.A. Thermodynamic

analysis of interacting nucleic acid strands. SIAM Rev. 49, 6588 (2007).

30. Dirks, R. M., Lin, M., Winfree, E. & Pierce, N. A. Paradigms for computational

nucleic acid design. Nucleic Acids Res. 32, 13921403 (2004).

Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements We thank the following for discussions: J. S. Bois, R. M. Dirks,

M. Grazier GSell, R. F. Hariadi, J. A. Othmer, J. E. Padilla, P. W. K. Rothemund,T. Schneider, R. Schulman, M. Schwarzkopf, G. Seelig, D. Sprinzak,

S. Venkataraman, E. Winfree, J. N. Zadeh and D. Y. Zhang. We also thank

J. N. Zadeh,R. M. Dirksand J. M. Schaefferfor theuse of unpublishedsoftware, andR. F. Hariadi and S. H. Park for advice on AFM imaging. This work is funded by theNIH, the NSF, the Caltech Center for Biological Circuit Design, the Beckman

Institute at Caltech, and the Gates Grubstake Fund at Caltech.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare competing financial interests:details accompany the paper on Natures website (http://www.nature.com/

nature). Correspondence and requests for materials should be addressed to N.A.P.([email protected]).

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METHODSSystem design. A molecular implementation is realized in three steps summar-ized in Fig. 1f and illustrated in Supplementary Information 3.1. Step (1): path-way specification. Step (2): translation to motifs. Following initial dimensioningusing the NUPACK server, the segment dimensions are sometimes further opti-mizedbased on subsequent experimentaltesting.Step (3):sequencedesign. Aftercomputational optimization, occasional further manual optimization was per-formed using the same design metric on a subset of crucial target structures. Wethen further analysed the thermodynamic behaviour of the sequences using theNUPACK server. For some systems, stochastic kinetic simulations31 (J. M.

Schaeffer, personal communication) were carried out to confirm the absenceof significant kinetic traps along the target reaction pathways. The sequences areshown in Supplementary Information 8.System synthesis. DNA was synthesized and purified by Integrated DNATechnologies. The purified DNA strands were reconstituted in ultrapure water(resistance of 18 MV cm). We determined the concentrations of the DNA solu-tions by measuring ultraviolet light absorption at 260 nm.

Hairpins were synthesized as two pieces which were then ligated to producethe full hairpin (see Supplementary Information 7.1 for details). We performedthe ligation using T4 DNA ligase (New England Biolabs) at either room tem-peratureor 16 uC fora minimum of2 h. Wefurther purified ligated strands usingdenaturing polyacrylamide gel electrophoresis. The bands corresponding to theDNA strands of expected sizes were visualized by ultraviolet shadowing andexcised from thegel.The DNAstrandswerethenelutedandrecoveredby ethanolprecipitation.

For monomer preparation, we diluted the concentrated DNA strands to reac-tion conditions:50 mMNa2HPO4,0.5MNaCl,pH5 6.8 for speciesin Fig. 2 andSupplementary Fig. 4; and 20 mM Tris, pH5 7.6, 2 mM EDTA, 12.5 mM Mg21

(13TAE/Mg21 buffer) for species in Fig. 3, Supplementary Fig. 12, and Fig. 4.Wethen annealed thehairpinsby heating for5 minat 90 uC, andthenturning offthe heating block to allow the system to cool to room temperature (requiring atleast 2 h). For walker system assembly, see Supplementary Information 6.4.Gel electrophoresis. For the gel in Fig. 2c, 12 ml of each 3-mM hairpin specieswere mixed by pipetting. Portions of this master mix were aliquoted into fiveseparate tubes (6ml per tube). To these tubes we added 2ml of either 3mM I (lane1), 1.5mM I (lane 2), 0.75mM I (lane 3), 0.3mM I (lane 4), or 13 reaction buffer(50mM Na2HPO4, 0.5 M NaCl, pH5 6.8) (lane 5) to reach a total reactionvolume of 8ml. The samples were then mixed by pipetting and allowed to reactfor 2.5 h at room temperature. The annealed reaction (lane 6), prepared 0.5 h inadvance, was made by mixing 2 ml of each hairpin with 2 ml of the 13 reactionbuffer, and then annealing as described in monomer preparation. A 2% native

agarose gel was prepared for use in 13 LB buffer (Faster Better Media, LLC). Wethen mixed 1ml of each sample with 1ml of 53 SYBR Gold loading buffer: 50%glycerol/50% H2O/SYBR Gold (Invitrogen) and loaded this into the gel. The gelwasrun at 350V for10 minat room temperature andimaged usingan FLA-5100imaging system (Fuji Photo Film).

Forthe gelin Fig. 4c,we annealed thehairpinsat thefollowing concentrations:A1, A2, B2, A3 and B3 at 1 mM; A4 and B4 at 2 mM; A5 and B5 at 4 mM. Theinitiator I was prepared at 800nM. The following sample mixtures wereprepared: lane 1, A1; lane 2, I1A1; lane 3, I1A11A21B2; lane 4,I1A11A21B21A31B3; lane 5, I1A11A21B21A31B31A41B4;lane 6, I1A11A21B21A31B31A41B41A51B5; lane 7, A11A21B21A31B31A41B41A51B5. Here, I, A1, A2 and B2 were added at 1 ml;A3, B3, A4, B4, A5 and B5 at 2 ml. We added 13 reaction buffer (20 mM Tris,pH5 7.6, 2 mM EDTA, 12.5mM Mg21) to bring the total volume of eachsample to 16ml. We mixed the samples by pipetting and allowed them to reactfor 2 h at room temperature. A 1% native agarose gel was prepared in 13 LB

buffer. We added 8ml of each sample to 2 ml 53 SYBR Gold loading buffer andloaded 8ml of this sample/loading-buffer mix into the gel. The gel was run at350 V for 10 min at room temperature and then imaged using an FLA-5100imaging system. For the reactions in Fig. 4d, the hairpins were mixed to reachthe followingfinal concentration: A1-Cy5 (see Supplementary Information 8.4),

A2, B2, 100nM; A3, B3, 200 nM; A4, B4, 400 nM; A5, B5, 800 nM. We then

aliquoted portions of this mix into 10 separate tubes (9 ml per tube). To thesetubes we added either 13TAE/Mg21 reaction buffer or the initiator I to give the

indicated final concentration of I and a final volume of 11 ml. The samples weremixed by pipetting and allowed to react for 1 h at room temperature. We then

mixed the sample with 53 LB loading buffer(Faster BetterMedia, LLC) to reach13 loading buffer concentration (8 ml sample, 2 ml loading buffer). We loaded

the sample/loading buffer mix into a 1% native agarose gel prepared in 13 LB

buffer. Thegel wasrun at350 V for10 minat room temperature andthenimaged

and quantified using an FLA-5100 imaging system. The experiments were per-

formed with 10mM inert 25-nt poly-T carrier strands21

in the reaction solutionAFM imaging. We obtained AFM images using a multimode scanning probemicroscope (Veeco Instruments), equipped with a Q-Control module for ana-

logue AFM systems (Atomic Force F&E). The images were obtained in liquid

phase under tapping mode using DNP-S oxide sharpened silicon nitride canti-

levers (Veeco). We first diluted samples in 13 TAE/Mg21 buffer to achieve thedesired imaging density. We applied a 20 ml drop of 13 TAE/Mg21 and a 5mdrop of sample to the surface of freshly cleaved mica and allowed them to bindfor approximately 2 min. We added supplemental Ni21 (1530 mM) to increase

the strength of DNAmica binding32. Before placing the fluid cell on top of the

mica puck,we added anadditional1520mlo f 13TAE/Mg21bufferto thecavity

between the fluid cell and the AFM cantilever chip to avoid bubbles.

Fluorescence experiments. For catalytic circuitry experiments, we obtained

fluorescence data using a QM-6/2005 steady state spectrofluorometer (Photon

Technology International), equipped with a Turret 400TM four-position cuvetteholder (Quantum Northwest) and 3.5-ml QS quartz cuvettes (Hellma). The

temperature was set to 25 uC. We set the excitation and emission wavelengths

to 520 nm (2-nm bandwith) and 540 nm (4-nm bandwidth), respectively. Fothe experiments in Fig. 3c, we prepared hairpin monomers, A, B, C and D, and

initiator, I, separately as described above. We added 40 ml 1-mM A to 1.8 ml 13TAE/Mg21 buffer and mixed it by rapid pipetting eight times using a 1-ml tip

We recorded the baseline signal for,16 min. Then we added 40ml of 1-mM B, Cand D and the appropriate concentration of I (or 13 TAE/Mg21 buffer in th

case of 03 I) to the cuvette (to reach the target concentrations described inFig. 3c) and mixed by rapid pipetting eight times using a 1-ml tip. The contro

with 20-nM A alone was monitored continuously. The final volume was 2 m

for all experiments. We carried out the experiments with 10-mM inert 25-n

poly-T carrier strand21 in the individual hairpin and initiator stock solutionand,1-mM inert 25-nt poly-T carrier strands in the final reaction solution.

For autonomous locomotion experiments, we used the same spectrofluoro-

meter as above with the temperature controller set to 21uC. We used two 3.5-m

QS quartz cuvettes (Hellma) in each set of experiments. Excitation and emissionwavelengths were set to 492nm and 517 nm (for FAM), 527 nm and 551 nm (fo

JOE), and 558 nm and 578 nm (for TAMRA), respectively, with 4-nm bandwidths. The assembly of the walker system is described in Supplementary

Information 6.4. We snap-cooled hairpin B in the reaction buffer (4mM

MgCl2, 15 mM KCl and 10 mM Tris-HCl, pH5 8.0): heating at 95 uC for 90 s

rapid cooling at room temperature, sitting at room temperature for 30 minbefore use. The system was assembled using 4 nM track and 3.5 nM bipeda

walker. We used a substochiometric amount of walker to ensure that no free-floating walker would bind to hairpin A on the track. For the same reason, we

used substoichiometric monopedal walker (7 nM) in the diffusion experiments

The final concentration of hairpin B was 20 nM, which was equimolar with th

five A hairpins on the track (5 3 4 nM5 20 nM). The assembled track was firsintroduced to record the fluorescence baselines for FAM, JOE and TAMRA. We

then introduced hairpin B andmixed100 timesby rapid pipettingto start walke

locomotion.31. Flamm, C., Fontana, W., Hofacker, I. L. & Schuster, P. RNA folding at elementar

step resolution. RNA 6, 325338 (2000).

32. Hansma, H.G. & Laney,D. E. DNAbindingto mica correlates with cationic radius

assay by atomic force microscopy. Biophys. J. 70, 19331939 (1996).

doi:10.1038/nature06451

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Supplementary Information

Programming Biomolecular Self-Assembly Pathways

Peng Yin1,2, Harry M.T. Choi1, Colby R. Calvert1 & Niles A. Pierce1,3,

1

Department of Bioengineering,2

Department of Computer Science,3

Department of Applied & Computational MathematicsCalifornia Institute of Technology, Pasadena, CA 91125, USAEmail: [email protected]

Contents

S1 Summary figure 1

S2 Reaction graph conventions 2

S3 Catalytic geometry 3

S3.1 System design for catalytic formation of a 3-arm junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

S3.2 E xecution of the reaction graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

S3.3 C atalytic formation of a 4-arm junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5S3.4 AFM image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

S3.5 Design for the catalytic formation of a k-arm junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

S4 Catalytic circuitry 10

S4.1 E xecution of the reaction graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

S4.2 D etailed secondary structure mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

S4.3 Stepping gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

S4.4 System kinetics analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

S5 Nucleated dendritic growth 16



S5.3 Q uantitative amplification gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19S5.4 AFM image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

S6 Autonomous locomotion 23


S6.2 S econdary structure of the walker system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24


S6.4 A ssembly of the walker system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

S6.5 C haracterization of the fuel system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

S6.6 Raw data for the fluorescence quenching experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

S6.7 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

S6.8 C omparison of walker time scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

S6.9 C ontrol for walker landing effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

S7 Discussion 37

S7.1 Leakage and ligation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

S7.2 Molecular compiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

S8 DNA sequences 39

S8.1 C atalytic 3-arm junction system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

S8.2 C atalytic 4-arm junction system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

S8.3 Autocatalytic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

S8.4 N ucleated dendritic growth system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

S8.5 Fuel for the walker system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

S8.6 Walker system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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S1 Summary figure

Motif

Abstraction

Catalytic geometry

Catalytic circuitry

Autonomous locomotion

Nucleated dendritic growth

Molecular programs Molecular implementation Molecular execution

30 nm

10 nm

Time (hr)

Signal

Time (hr)

Exponential

kinetics

50

Signal

Sequential

quenching

10

Figure S1. Summary. Diverse biomolecular self-assembly (and disassembly) pathways are programmed using an abstraction of a versatile DNA

hairpin motif.

1



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S2 Reaction graph conventions

Using the definitions introduced in the main text, the reaction graph abstraction obeys the following conventions.

Initial conditions

The initial condition of the system is defined by the state of each port and the initial bonds between the ports. An initial bond between an output port and an input port implies that an assembly reaction has already occurred prior to the

execution of the reaction graph (see, e.g., the bond between the output port of I and the input port of A in Fig. 5a).Static structural elements

Static structural elements are depicted by gray line segments (e.g. the substrate of Fig. 5a) and are inert during execution ofthe reaction graph.

These elements can be used to impose geometric constraints on the execution of the reaction graph (e.g. the rigid substrateand inextensible torso of the walker system, Sect. S6.1).

Execution starting points

Execution begins with any solid arrow (assembly reaction) connecting two accessible ports. In a system lacking two accessible ports connected by a solid arrow, execution cannot begin (e.g., the removal of node I

prevents execution of all programs described in the present work).

Assembly reaction

An assembly reaction is depicted by a solid arrow that points from an input port to a complementary output port of a differentnode.

An assembly reaction is executed when these two ports are simultaneously accessible. In the execution of an assembly reaction, a bond is formed between the two ports, they are flipped to their inaccessible states,

and the internal logic of the node with the affected input port is applied to its output ports (e.g., for the present motif, the

output ports are flipped to their accessible states).

Multiple solid arrows entering the same input port depict parallel processes on separate copies of the nodal species (e.g., theinput port of node A in Fig. 3a and the input ports of nodes A2-A5 and B2-B5 in Fig. 4a).

Disassembly reaction

A disassembly reaction is depicted by a dashed arrow that points from an input port to a complementary output port of adifferent node.

Using nodal abstractions of the present hairpin motif, a disassembly arrow must complete a disassembly cycle. For a cycleinvolving k nodes: input port 1 blue output port 2 input port 3 blue output port 4 . . . blue output port 2k input port1, where denotes an assembly reaction, denotes a disassembly reaction, and denotes the internal logical connectionbetween two ports on the same node. For example, Fig. 1d contains a disassembly cycle for k = 2: input port of A blueoutput of A input port of B blue output port of B input port of A. Fig. 2a contains a disassembly cycle for k = 3:input port of A blue output port of A input port of B blue output port of B input port of C blue output port of C input port ofA. In physical terms, this corresponds to requiring that the displacing strand and the strand to be displacedemanate as adjacent branches for a k-arm junction, allowing nucleation of the displacement branch migration (e.g., Figs 2b,

S4b, and S7b). The special case of k = 2 corresponds to standard toehold-mediated strand displacement1 (e.g., Fig. 1b,

where the whole of domain b of hairpin A serves as the toehold).

A disassembly reaction is executed when the participating output port is accessible and the participating input port is inacces-sible (using nodal abstractions of the present motif, the requirement that a disassembly arrow must complete a disassembly

cycle implies that the participating output port can only become accessible after the participating input port becomes inacces-

sible).

In the execution of a disassembly reaction (e.g., Fig. 1e), the existing bond from an (inaccessible) output port to an (inac-cessible) input port is replaced by a new bond to the displacing (accessible) output port; the states of both output ports are

flipped.

Multiple dashed arrows entering the same input port depict parallel disassembly cycles involving separate copies of the nodalspecies (no such example is presented in this paper).

2



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S3 Catalytic geometry

S3.1 System design for catalytic formation of a 3-arm junction

a

a b

b*c*

A B C

c

c*a*

b a

b*a*

c

I

b*a*

(2.1) Complementarity relationships

A B C

a x z*b

b* y* c*

y

x*

y c z

c* z* a*

x*b

y*

z a x y*

b*a*x*

c

z*

I

x* b* y*a*

b

c

(2.2) Clamping/padding

(1) Pathway specifiation

Catalytic formation of a 3-arm DNA junctionTarget dynamic function:

(2.3) Dimensioning

|a| = |b| = |c| = |x| = |y| = |z| = 6 nt

(3) Sequence design

d

e

f

DNA sequences

A B CI

ACGT

A

BC

I

Figure S2. Procedure for designing the catalytic 3-arm junction system.

Here we describe the design procedure for the catalytic 3-arm junction system presented in Fig. 2.

Step (1) Pathway specification. The desired dynamic behavior (Fig. S2a) is specified using a reaction graph (Fig. S2b).

Step (2) Translation into secondary structure motifs. The reaction graph can be translated directly into secondary struc-

ture motifs.

Step (2.1) Basic complementarity relationships. In the reaction graph, two ports connected by an arrow are complemen-

tary to each other. These portal complementarity relationships specify the complementarity relationships between the motif

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domains modeled by the ports, and thus enable a direct translation of the reaction graph to the secondary structure motifs

(Fig. S2c). For example, the assembly arrow connecting the brown output port of node I and the orange input port of node A

(Fig. S2b) indicates these two ports are complementary, and hence the initiator and the orange domain of hairpin A in Fig. S2c are

complementary. Similarly, the disassembly arrow connecting the blue output port of node C and the orange input port of node A

(Fig. S2b) indicates these two ports are complementary, and hence the blue domain of hairpin C is complementary to the orange

domain of hairpin A in Fig. S2c.

Step (2.2) Clamping/padding. The basic implementation (Fig. S2c) is modified by adding clamping/padding segments (x,

y, z, x*, y*, and z* in Fig. S2d). These segments serve two purposes. First, they serve as padding segments to modulatethe lengths of a hairpins sticky-end, stem, and loop regions, permitting more flexible dimensioning in Step (2.3). Second, the

segments serve as clamps to decrease spurious leakage reactions in the absence of the initiators. Consider un-clampedhairpin

A and hairpin B in Fig. S2c. When the left-end of the stem of hairpin A breathes, the 3 end of segment b* will be transientlyexposed, revealing a partial toehold that is complementary to the toehold b of hairpin B. This transient toehold exposure would

permit hairpin A and hairpin B to react spuriously and form AB (which would then react with C to form A BC). By contrast,the breathing of the left end of the clampedhairpin A stem in Fig. S2d exposes x* instead of b*. Thus, b* remains sequestered,

discouraging spurious nucleation between A and B at b*.

Step (2.3) Segment dimensioning. The purpose of segment dimensioning is to assign the length of each segment (number

of nucleotides) such that under specified conditions, spurious reactions are suppressed and the desired reaction proceeds smoothly.

The NUPACK server (www.nupack.org) is used for dimensioning. For the catalytic 3-arm junction system described here, the

thermodynamic analysis of the interacting DNA strands suggests that assigning 6-nt to each segment (Fig. S2e) stabilizes critical

structures in the reaction pathway in the context of a dilute solution of interacting nucleic acid strands.

Step (3) Sequence design. See Methods in main text.

S3.2 Execution of the reaction graphs

(1) (2) (5)(3)b

(4)

a

A B

C

I

D

IA B

CD

(4)

D

A B

C

I

C

A B

I

B

A

II

A

(1)

(4)

(5)

A B

C

I(2)

(3)

D

I A

BC

A

BC

I(3)

C

A

B

I(2)

B

AII(1)

A

I(1)

(3)

(2)(4)

A

BC

Figure S3. Execution of reaction graphs for catalytic 3-arm/4-arm junction systems . a, Execution of the reaction graph of Fig. 2a. Reaction

1 (assembly): A bond is made between the accessible output port of I and the accessible input port of A and both ports are flipped to inaccessible

states; the output port of A is flipped to the accessible state (based on the internal logic of node A). Reaction 2 (assembly): A bond is made

between the newly accessible output port of A and the accessible input port of B and both ports are flipped to inaccessible states; the output

port of B is flipped to the accessible state (based on the internal logic of node B). Reaction 3 (assembly): A bond is made between the newly

accessible output port of B and the input port of C and both ports are flipped to inaccessible states; the output port of C is flipped to the accessible

state (based on the internal logic of node C). Reaction 4 (disassembly): The bond between the inaccessible output port of I and the inaccessible

input port of A is displaced by a bond between the newly accessible blue output port of C and the input port of A; the states of the two output

ports are flipped. b, Execution of the reaction graph of Fig. 2e.

Fig. S3 depicts the step-by-step execution of the reaction graphs in Figs 2a and e. Note that the reaction graph in Fig. S3a

contains a k = 3 disassembly cycle: input port of A blue output port of A input port of B blue output port of B input portof C blue output port of C input port of A; the reaction graph in Fig. S3b contains a k = 4 disassembly cycle: input port ofA blue output port of A input port of B blue output port of B input port of C blue output port of C input port of D blue output port of D input port of A.

4



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S3.3 Catalytic formation of a 4-arm junction

b Metastable monomers

I

w*b*x*a*

Initiator

(1) (2) (4a) (4b)

Cataytic formation of

a 4-arm junction

c

1 2 3 4 6A

I.A

A.B.C.D

5 87

A+I

A I+A+B

I+A+B

+C

0.5I+

A+B

+C+D

0.25

I+

A+B

+C+D

0.1I+

A+B

+C+D

A+B

+C+D

I.A.B

Step 1Step 2

Step 3

A+B

+C+D

.annealed

9

LeakageI+

A+B

+C+D

10

I.A.B.C

Step 4

Aew q

a w y*b

b*x* c*

A

x

w*

q*

x c y

c* y* d*

z*

B

b

x*

r*

y d z w*

a*d*z*

c

y*

s*

z a w x*

b*a*w*

D

d

z*

t*

z* d*y* c*

z d y

y

c

x

b

x*

cz*

z*a*w*

d*

z

a

w

d

y*

r*

s*

t*

A.B.C.D

Be

Cey s

Dez t

(3)

I.A.B.CI.A.B.CI.A.B

w b xay*c*x*b*w*

q*

I.A

a* w* b* x*

I

w*b*x*a*

I

w*b*x*a*

d

Extend armsfor AFM imaging

A.B.C.D.Ae.Be.Ce.De

x r

AFM imaging

a

w b xa

y*c*x*

b*w*

q*

a* w* b* x*

z* d*y* c*

z d y

y

c

x

b

x*

c

z*a*w*

d*

y*

r*

s*

w b xay*c*x*

b*w*

q*

a* w* b* x*

z* d*y* c*y

c

x

b

x* r*w b xay*c*x*

b*w*

q*

a* w* b* x*

z* d*y* c*

z d y

y

c

x

b

x*

c

z*a*w*

d*

z

a

w

d

y*

r*

s*

w b xa

y*c*x*

b*w*

q*

a* w*b*x*

z*

t*

a*

w*

b*

x*

z* d*y* c*

z d y

y

c

x

b

x*

cz*

z*a*w*

d*

z

a

w

d

y*

r*

s*

t*

w b xa

y*c*x*

b*

w*

q*

a* w* b* x*

w

q

x r

y

s

zt

C(1)

(4a)

(4b)

A B

C

I(2)

(3)

D

Figure S4. Catalytic formation of a 4-arm DNA junction. a, Reaction graph. Note that since the green output ports do not serve as initiators

for any downstream reaction, they are omitted here for simplicity. See Sect. S3.2 for step-by-step execution of the graph. b, Secondary structure

schematic of the reaction. The lengths of segments q, q*, r, r*, s, s*, t, and t* are 18 nt; the lengths of the other segments are 6 nt. Hairpins A,

B, C, and D are metastable in the absence of the initiator I. The initiator I catalyzes monomers A, B, C, and D to form a 4-arm DNA junction, as

follows: (1) segment a* of I nucleates at the toehold a of hairpin A and initiates a strand displacement that results in the opening of hairpin A;

(2) newly exposed b* of A nucleates at toehold b of B and results in the opening of B; (3) newly exposed c* of B nucleates at toehold c of C and

results in the opening of C; (4a) newly exposed d* of C nucleates at d of hairpin D and results in the opening of D; (4b) D displaces I from A. c,Agarose gel electrophoresis demonstrates the catalytic formation of the 4-arm junction. Lanes 1-5: A gel shifting assay validates each reaction

step depicted in panel (b). Lanes 5-9: Effects of different concentrations of I (1, 0.5, 0.25, 0.1, and 0) on the formation of ABCD.

600 nM reactants were incubated at room temperature for 2 hours. Lane 10: ABCD annealed over 2.5 hours (600 nM hairpin species heated at

95 C for 5 minutes and cooled to room temperature over 2.5 hrs). The 2% agarose gel was prepared in 1 LB buffer (Faster Better Media, LLC)with 0.5 g/ml ethidium bromide. The gels were run at 150 V for 30 min at room temperature and then visualized using UV transillumination.

The hairpins used for these reactions did not contain the 3 tails (q*, r*, s*, and t*). d, AFM images of two 4-arm junctions. To assist in AFMimaging of the 4-arm junction, four strands (Ae, Be, Ce, and De) were incubated with the catalytically formed 4-arm junction ABCD. Note

that the duplex portion of the arms of the final structure ABCDAeBeCeDe are twice as long as the duplex portion of the arms of A BCD.

Two AFM images of ABCDAeBeCeDe are presented. Scale bar, 10 nm. See Sect. S3.4 for length measurements.

Fig. S4a and b depict the reaction graph and reaction schematic for the catalytic formation of a 4-arm junction. In the absence

of initiator I, hairpins A, B, C, and D are kinetically impeded from forming the 4-arm junction that is predicted to dominate

at equilibrium. Introduction of I into the system (Fig. S4b, bottom) activates a cascade of assembly steps with A, B, C, and D

followed by a disassembly step in which D displaces I from the complex, freeing I to catalyze the self-assembly of additional

branched junctions.

Native agarose gel electrophoresis (Fig. S4c) confirms that the hairpins assemble slowly in the absence of the initiator (Lane 9)

and that assembly is dramatically accelerated by the addition of initiator (Lane 5). Disassembly of the initiator enables catalytic

turnover as indicated by the nearly complete consumption of hairpins even at sub-stiochiometric initiator concentrations (Lanes

6-8). As in the 3-arm junction case, only minimal assembly is achieved by annealing the hairpin mixture (Lane 10).

AFM imaging of the catalyzed self-assembly product (augmented with strands that extend the duplex portion of each arm as

described in the caption) reveals the expected 4-arm junction morphology (Fig. S4d).

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S3.4 AFM image analysis

7.7 nm8.1 nm7.7 nm

10nm

7.1 nm 7.7 nm7.7 nm

10nm

6.6 nm 7.1 nm 7.0 nm

7.6 nm 14.5 nm 14.2 nm

10nm

7.0 nm7.5 nm

12.5 nm 14.3 nm

7.1 nm

8.2 nm

10nm

b

a

~7.8 nm

~7.8 nm ~7.8 nm

Figure S5. AFM image analysis of 3-arm/4-arm junctions. AFM measurements of the 3-arm (panel a) and 4-arm (panel b) junctions described

in Fig. 2 and Fig. S4. The small images are screenshots of the measurement section files. The distance between the two arrows is listed above

the image.

Using a B-DNA model where one helical turn contains 10.5 base pairs and measures 3.4 nm, we calculate the expected arm

length for the 3-arm junction as follows: (24 / 10.5) 3.4 nm = 7.8 nm. Similarly, the arm length for the 4-arm junctionis calculated to be 7.8 + 7.8 = 15.6 nm. The measured lengths of the arms are roughly consistent with the calculated lengths

(Fig. S5). Fig. S6 shows AFM images with a larger field of view for 3-arm (panel a) and 4-arm (panel b) junctions.

6



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100 nm

100 nm

b

a

Figure S6. Large-field-of-view AFM images of the 3-arm (a) and 4-arm (b) junction systems.

7



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S3.5 Design for the catalytic formation of a k-arm junction

. . .

I

H1.H2....Hk

x* 1 a* 2 x* 2x* k a* 1

x* 3

a*

4

x*

4

x*

2

a*

3

x1 a2 x2a1

x*2

a*3

x*3

x2

a3

x3

a2

x*1

a*2

xk-2ak-1xk-1 ak-2

x*k-1

a*k

x*k

xk-1

ak

xk

ak-1 x*

k-2

a*k-1

x*

k

a*

1

x*

1

xk

a1

x1

ak

x*

k-1

a*

k

x* 1 a* 2 x* 2a* 1

x* k-2 a* k-1 x* k-1

xk-2 ak-1 xk-1ak-2

a* k

x* kHk-2

x* k-1 a* k x* k

xk-1 ak xkak-1

a* 1

x* 1Hk-1

x* k a* 1 x* 1

xk a1 x1ak

a* 2

x* 2Hk

x* 1 a* 2 x* 2

x1 a2 x2a1

a* 3

x* 3H1

x* 2 a* 3 x* 3

x2 a3 x3a2

a* 4

x* 4H2

H3

Hk-3

. . . . . .

Ix* 1 a* 2 x* 2a* 1

Metastable monomers

InitiatorCatalytic formation of a k-arm junction

I

a

H1

H2 H3

Hk Hk-1

b

......

Figure S7. Catalytic formation of a k-arm junction. a, Reaction graph. b, Reaction schematics. Hairpins H1, H2, . . . , Hk are metastable in the

absence of the initiator I. The initiator I catalyzes monomers H1, H2, . . . , Hk to form a k-arm DNA junction.

The catalytic system described in Fig. 2 and Fig. S4 can, in principle, be generalized to a system capable of the catalytic forma-

tion of a k-arm junction. Fig. S7 describes the reaction graph and the secondary structure schematic for the catalytic formation of

a k-arm junction. Fig. S8 gives an example when k = 6.

8



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17/56

S4 Catalytic circuitry

S4.1 Execution of the reaction graph

A B

D C

(1) (2b)

(3)

(4a)

(4b)(5)

(6b)

(2a)(6a)

I

(5) (6a) (6b)

CD A

CD

A

B

CD

A A B

CD

(3) (4a) (4b)A B

C

A B

C

A B

CD

A B

CD

D

(1) (2a) (2b)I

B

I

A B

I

A B

A

I

A

B

Figure S9. Execution of the reaction graph for the autocatalytic system of Fig. 3.

Fig. S9 describes the step-by-step execution of the reaction in Fig. 3a. The reaction starts at solid arrow (1) that connects the

accessible output port of I and the accessible input port of A. Note that by convention, the two arrows entering the same input port

of A depict parallel processes on separate copies of the nodal species.

Reaction 1 (assembly): A bond is made between the accessible output port of I and the accessible input port of A and bothports are flipped to inaccessible states; the output port of A is flipped to the accessible state (based on the internal logic of

node A).

Reaction 2a (assembly): A bond is made between the newly accessible output port of A and the accessible input port of B andboth ports are flipped to inaccessible states; the two output ports of B are flipped to accessible states (based on the internal

logic of node B).

Reaction 2b (disassembly): The bond between the inaccessible output port of I and the inaccessible input port of A is displaced

by a bond between the newly accessible blue output port of B and the input port of A; the states of the two output ports are

flipped.

Reaction 3 (assembly): A bond is made between the newly accessible green output port of B and the accessible input port ofC and both ports are flipped to inaccessible states; the output port of C is flipped to the accessible state (based on the internal

logic of node C).

Reaction 4a (assembly): A bond is made between the newly accessible output port of C and the accessible input port of Dand both ports are flipped to inaccessible states; the output ports of D are flipped to accessible states (based on the internal

logic of node D).

Reaction 4b (disassembly): The bond between the inaccessible green output port of B and the inaccessible input port of C isdisplaced by a bond between the newly accessible blue output port of D and the input port of C; the states of the two output

ports are flipped.

Reaction 5 (assembly): A bond is made between the newly accessible green output port of D and the accessible input port ofA and both ports are flipped to inaccessible states; the output port of A is flipped to the accessible state (based on the internal

logic of node A).

Reaction 6a (assembly): A bond is made between the newly accessible output port of A and the accessible input port of Band both ports are flipped to inaccessible states; the output ports of B are flipped to accessible states (based on the internal

logic of node B).

Reaction 6b (disassembly): The bond between the inaccessible green output port of D and the inaccessible input port of A isdisplaced by a bond between the newly accessible blue output port of B and the input port of A; the states of the two output

ports are flipped.

10



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S4.2 Detailed secondary structure mechanism

a x y x* a* c* y* b*b x*v u* v*

(3)

(4)(5)

(6)

Ia* x* b* y*v*

A

(1)

(2)

a* x* b* y*v*

C c dy v

d*y* v*

c*

a*

u*

u*

u

x*

D dd*

a cx u

a* c*x* u*

v*

b*y*

v

v*y*

I

a* x* b* y*v*

Metastable monomers

Initiator

Self-replication

bb*

c ay x

c* a*y* x*

y*

d*u*

u

v*

B

v*

A a a*b

c*

x y

b*x* y*

x*v

v*u*

a a*b

c*

x y

b*x* y*

x*v

v*u*

bb*

c ay x

c* a*y* x*

y*

d*u*

u

v*

B

v*a* x* y* x a c y bb*v* d* y* c*


v* uu*

bb*

c ay x

c* a*y* x*

y*

d*u*

u

v*

B

v*

Cc dy v

d*y* v*

c*

a*

u*

u*

u

x*

Dd

d*a cx u

a* c*x* u*

v*

b*y*

v

v*y*

A

a a*b

c*

x y

b*x* y*

x*v

v*u*

ud y cv

x*

a*c*

y*d*

u*

v*

u*

a* x* y* x a c y bb*v* d* y* c*


v* uu*

v* d* y* c*x a ucd a* b*x* y*

ud y cx* a* vc*y* d*u* v* u*

v u* v*

x*

a*

c*

y*

b*

x*

u*

v*

a x ybv



v u* v*

A.B.C

C.D

C.D.A

A.B

I.A

Figure S10. Detailed reaction schematic for the autocatalytic system of Fig. 3 . The length of each segment is 6 nt. Green star, fluorophore;

black dot, quencher.

Fig. S10 describes the detailed reaction flow of the autocatalytic system described in Fig. 3. Fig. S11 describes additionalintermediate steps. Steps 1-2 are the initiation stage; steps 3-6 are the exponential amplification stage.

Step 1: the toehold a* of I nucleates at the toehold a of A, resulting in the opening of the hairpin and the formation of theproduct IA.

Step 2: IA, with b* newly exposed, opens hairpin B (step 2a); B subsequently displaces I from A (step 2b), producingAB and bringing the system to the exponential amplification stage. The single-stranded tail (v*-d*-y*-u*-c*) of AB nextcatalyzes C and D to form CD (in steps 3 and 4).

Step 3: AB, with c* newly exposed, opens hairpin C. Step 4: ABC, with d* newly exposed, opens hairpin D (step 4a); D subsequently displaces C from B, separating A B and

C

D (step 4b). The single-stranded tail (a*-x*-v*-b*-y*) of C

D is identical to I and next catalyzes A and B to form A

B (in

steps 5 and 6).

Step 5: CD, with a* newly exposed, opens hairpin A. Step 6: CDA, with b* newly exposed, opens B (step 6a); B subsequently displaces A from D, separating C D and AB (step

6b).

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D

d

d*

a cx u

a* c*x* u*

v*

b*

y*

v

v*y*

+

+

x*

a*

c*

y*

b*

x*

u*

v*

a x ybv

+

x a ucd


vv*

d*y*

c*

a*

b*

x*

y*

u*

v*


a x b x* a* c* y* by x*v u* v*

v* uu*

x a c y b


uv* d* y* c*x a ucd a* b*x* y*


v u* v*

a*

x*

y*

b*

v*

d*

y*

c*

v*

u*

C

c dy v

d*y* v*

c*

a*

u*

u*

u

x*

+

ud y cv

x*a*c*

y*d*

u*

v*

u*

(3)

(4a)



v u* v*

+(4b)

(5)

(6a)

(6b)

Aa a*b

c*

x y

b*x* y*

x*v

v*

I a* x* b* y*v*

+


a* x* b* y*v*

(1)

u*

+

B

bb*

c ay x

c* a*y* x*

y*

d*

u*

u

v*



v* uu*

I a* x* b* y*v*

+

x a c y bu

a*

x*

y*

b*

v*

d*

y*

c*

v*

u*

(2a) (2b)

v*

Step (1)

Step (2)

Step (3)

Step (4)

Step (5)

Step (6)


a* x* b*y*v*


a* x* b* y*v*



v* uu*



v* uu*

ud y cv

x*a*c*

y*

d*

u*

v*

u*



v* uu*



v* uu*



v u* v*

Aa a*b

c*

x y

b*x* y*

x*v

v*u*



v u* v*

B

bb*

c ay x

c* a*y* x*

y*

d*

u*

u

v*

v*

x*

a*

c*

y*

b*

x*

u*

v*

a x ybv



v u* v*



v u* v*

+a* x* y* x a c y bb*v* d* y* c*


v* uu*

I.A

I.A

A.B.C

C.D.A

A.B.C.D

C.D.A.B

A.B.C

C.D

A.B

I.A.B

A.B

A.B

C.D

C.D.A

C.D

A.B

Figure S11. Step-by-step reaction schematic for the autocatalytic system of Fig. 3 .

12



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S4.3 Stepping gel

1 2 3 4 5 6 7 8 9 10 11 12 13 14

A+I

A (AI)

(AI)+

B

(AB)

(AB)

+C

(ABC

)

(ABC

)+D

(CD)

(CD)

+A

(CDA

)

(CDA)+B

A

A.I A.I

A.B A.B

A.B.C A.B.C

A.B,

C.D

C.D

C.D.A C.D.A

C.D, A.B

Step 1 Step 2 Step 3 Step 4 Step 5 Step 6

A.I

C.D

C

D

A.B

A.B.C

(3)

(4)

C.D.A

(5)

(6)

I

B

(1)

(2)

B

Exponential

amplification

Initiation

A

Self-replication

A

a

b

Figure S12. Stepping gel for the autocatalytic system. a, Reaction schematic. b, Native polyacrylamide gel electrophoresis demonstrates the

step-by-step reaction depicted in Fig. 3b. The symbol () indicates annealing; + indicates 15 minute reaction at room temperature. The hairpins

used for these reactions were synthesized and purified by IDT DNA and used without further purification. The annealed samples were annealedat 2 M reactant concentrations: heating at 95 C for 5 minutes followed by cooling to room temperature over approximately 2.5 hours. Theroom temperature reactions were conducted with each reactant species at 1 M concentration. Consider the sample, (AI) + B, in Lane 5. The

sample was prepared by first annealing a mix containing 2 M A and 2 M I to produce (AI). Then 2 L of (AI), at 2 M concentration, was

mixed with 2 L of B at 2 M concentration and allowed to react at room temperature for 15 minutes. Lanes 1 and 14 are 20-1000 bp DNA

ladders (Bio-Rad). The 5% native polyacrylamide gel was prepared in 1 TAE/Mg++ buffer (20 mM Tris, pH = 7.6, 2 mM EDTA, 12.5 mM

Mg++). The samples were loaded with 10% glycerol. The gel was run at 100 V for 90 minutes at room temperature, post-stained with 0.5

g/mL ethidium bromide, and visualized by UV transillumination. The blue line delineates the boundary between two gels.

The autocatalytic system was validated on a step-by-step basis using native polyacrylamide gel electrophoresis (PAGE)

(Fig. S12):

Step 1. Hairpin A reacts with initiator I and produces a band that corresponds to product A

I (Lane 3), which migrates at

about the same speed as the annealed product AI (Lane 4), as expected. Step 2. Annealed sample AI reacts with hairpin B and produces a band that corresponds to product A B (Lane 5), which

migrates at about the same speed as the annealed product AB (Lane 6), as expected. Step 3. Annealed sample AB reacts with hairpin C and produces a band that corresponds to product ABC (Lane 7), which

migrates at about the same speed as the annealed product ABC (Lane 8), as expected. Step 4. Annealed sample ABC reacts with hairpin D and produces a band that corresponds to product AB and CD (Lane

9), which migrates at about the same speed as the annealed product AB (Lane 6) and the annealed product CD (Lane 10),as expected.

Step 5. Annealed sample CD reacts with hairpin A and produces a band that corresponds to product CDA (Lane 11), whichmigrates at about the same speed as the annealed product CDA (Lane 12), as expected.

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Step 6. Annealed sample CDA reacts with hairpin B and produces a band that corresponds to product CD and AB (Lane13), which migrates at about the same speed as the annealed product CD (Lane 10) and the annealed product AB (Lane 6),as expected.

S4.4 System kinetics analysis

Analytical modeling

The autocatalytic system is modeled using the following reactions:

I+ Ak1 IA

IA + B k2 I+ AB

AB + C k3 ABC

ABC+ D k4 AB + CD

CD + A k5 CDA

C

D

A + B

k6

C

D + A

B.

To make the system tractable for analytical treatment, we make the following simplifying assumptions:

Assumption 1. The forward reaction rates are all the same: ki = k, for i = 1, . . . , 6. This is based on the fact that all thereactions are strand-displacement reactions mediated by 6-nt toe-holds. Under the experimental conditions, the rate limiting

step of the toe-hold mediated reactions is the nucleation step,2 the rate of which is determined primarily by the toe-hold

length.

Assumption 2. The reactions are irreversible. This approximation is justified by the fact that at 25 C, the equilibriumconstants for these reactions (e.g., K1 QIAQIQA for the first equation here, Q denotes the partition function for a givencomplex species) are all calculated to be greater than 109 using multi-stranded partition function analysis3 with NUPACK(www.nupack.org).

Assumption 3. CD and I are treated as identical species at the level of mass action kinetics modeled here. This is based onthe fact that the a

peng yin, harry m. t. choi, colby r. calvert and niles a. pierce- programming biomolecular...

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