Programming biomolecular self- assembly pathwaysreif/courses/molcomplectures/... · 2016-03-08 · Visual Communication/Models LETTERS Programming biomolecular self-assembly pathways

Programming biomolecular self-assembly pathways

Yin, P., Choi, H. M. T., Calvert, C. R., & Pierce, N. A. (2008). Programming biomolecular self-assembly pathways. Nature, 451(7176), 318–322. doi:10.1038/nature06451

Slides by Reem Mokhtar

DNA Modeling

l  Visual communication or modeling paradigms/concepts (cartoons, states, etc)

l  Graphical and Data representation (software, data structures, etc)

l  Grammatical/Language representation of DNA and reactions.

Visual Communication/Models

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 hybridization3–6, triggeredself-assembly7 and molecular computation8,9. Here, we programdiverse molecular self-assembly and disassembly pathways using a‘reaction 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 beprogrammed 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, weabstract 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.

Input port a(accessible state)

Output port c(inaccessible)

Output port b(inaccessible)

a

c d

(1) (2) (3)

f Pathway programming

b

Reaction graph

Translateto motifs

Motif Assembly and disassembly reactions

Nodal abstraction

Catalytic formation of a DNA duplex

Dynamicfunction

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sequences

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a*a*

at*

Specifypathways

Designsequences

Figure 1 | Programming biomolecular self-assembly pathways.a, Secondary structure of the hairpin motif. Coloured lines represent stranddomains; short black lines represent base 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 nodewith 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.

Vol 451 | 17 January 2008 | doi:10.1038/nature06451

318Nature Publishing Group©2008

l Purpose: ease of visual communication or modeling of a specific paradigm/concept

l May be conceptual abstractions or realistic representations of:

- Structure - Reactions

l Notation used in literature

l Representation depends on needs of communication & conceptualization:

- Tiling - Hybridization reactions - Animation of movement - Seesaw gates

616 ROWEIS ET AL.

have been proposed which suggest that DNA based computers may be flexible enough to tackle a wide rangeof problems (Adleman, 1994, 1996; Amos et al, 1999; Lipton, 1995; Boneh et al, 1996; Beaver, 1995;Rothemund, 1996), although fundamental issues such as the volumetric scale of materials and fidelity ofvarious laboratory procedures remain largely unanswered.In this paper we introduce a new model ofmolecular computation that we call the sticker model. Like many

previous proposals, it makes use of DNA strands as the physical substrate in which information is representedand of separation by hybridization as a central mechanism. However, unlike previous models, the stickersmodel has a random access memory that requires no strand extension, uses no enzymes, and (at least in theory)its materials are reusable.The paper begins by introducing a new way of representing information in DNA, followed by an ab-

stract description of the basic operations possible under this representation. Possible means for physicallyimplementing each operation are discussed. Finally, we go on to propose a specific machine architecture forimplementing the stickers model as a microprocessor-controlled parallel robotic workstation, employing onlytechnologies which exist today.

2. THE STICKERS MODEL

2.1. Representation of informationThe stickers model employs two basic groups of single stranded DNA molecules in its representation of

a bit string. Consider a memory strand N bases in length subdivided into K nonoverlapping regions eachM bases long (thus, N >MK). Each region is identified with exactly one bit position (or equivalently oneboolean variable) during the course of the computation. We also design K different sticker strands or simplystickers. Each sticker is M bases long and is complementary to one and only one of the K memory regions.If a sticker is annealed to its matching region on a given memory strand then the bit corresponding to thatparticular region is on for that strand. If no sticker is annealed to a region, then that region's bit is off. Figure 1illustrates this representation scheme.Each memory strand along with its annealed stickers (if any) represents one bit string. Such partial duplexes

are called memory complexes. A large set of bit strings is represented by a large number of identical memorystrands each of which has stickers annealed only at the required bit positions. We call such a collection ofmemory complexes a tube. This differs from previous representations of information using DNA in whichthe presence or absence of a particular subsequence in a strand corresponded to a particular bit being on or

off (e.g., see Adleman, 1994; Lipton, 1995). In this new model, each possible bit string is represented by a

unique association of memory strands and stickers whereas previously each bit string was represented by a

unique molecule.To give a feel for the numbers involved, a reasonable size problem (for example, breaking DES as discussed

in Adleman et al, 1999), might use memory strands of roughly 12,000 bases (N), which represent 580 binaryvariables (K) using 20 base regions (M).The information density in this storage scheme is (l/M) bits/base, directly comparable to the density of

previous schemes (Adleman, 1994; Boneh et al, 1996; Lipton, 1995). We remark that while information stor-age in DNA has a theoretical maximum value of 2 bits/base, exploiting such high values in a separation-basedmolecular computer would require the ability to reliably separate strands using only single base mismatches.

bit... bit I bit ¡+2 bit... (up to bit K)

|T C A T A

0FIG. 1. A memory strand and associated stickers (together called a memory complex) represent a bit string. The topcomplex on the left has all three bits off; the bottom complex has two annealed stickers and thus two bits on.

Nodal Abstraction 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)

Sign

al

Time (hr)

Exponentialkinetics

50

Sign

al

Sequentialquenching

10

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

1

Aim: diagramming reaction execution process as a “program”

Rules IC:

1) defined by state of each port + initial bonds

2) bond between output port/input port à reaction already took place.

Elements:

1) Static à gray line segments. inert during execution.

2) can be used to impose geometric constraints

Starting point:

1) beginning of execution à solid arrow connecting two 'accessible' ports.

2) system without 2 accessible ports + solid arrow --> no execution begins.

Rules Assembly reaction:

1) input port à output port: solid arrow depicts assembly reaction

2) occurs when 2 ports are accessible

3) when executed, ports flip to inaccessible state à internal logic applied to output ports (example: accessible state)

4) 2+ solid arrows entering input port à parallel processes on copies of node species

Disassembly:

1) input port à output port: dashed arrow

2) occurs when input port inaccessible, output port accessible

3) Bond from output port to input port (both inaccessible) is replaced by displacing (inaccessible) output port. Both output port states are flipped.

Nodal Abstraction

Secondary structure of hairpin motif.


LETTERS












a

c d

(1) (2) (3)


b

Reaction graph

Translateto motifs


Nodal abstraction


Dynamicfunction

I A

B


IA

B

A.B

Nucleic acidprimary

sequences

I

A B

c b

at

ct bt

a

A

eExecution of

reaction graph

IA

B

(1)

IA

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(3)

IA

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IA

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(2)A

NodeI

A

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(1)

(2)(3)

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at

ct bt

at ctbt

at*

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B

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IA

at*

c

ba

cba

a*

b*a*

bt*

a*

B

b*

a*

bt*

I

(1) Assembly

(2) Assembly

(3) Disassembly

a*a*

at*

Specifypathways

Designsequences




Nodal Abstraction

Secondary structure mechanism to illustrate assembly/

disassembly reactions during catalytic duplex formation


LETTERS












a

c d

(1) (2) (3)


b

Reaction graph

Translateto motifs


Nodal abstraction


Dynamicfunction

I A

B


IA

B

A.B

Nucleic acidprimary

sequences

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A B

c b

at

ct bt

a

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eExecution of

reaction graph

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NodeI

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ct bt

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at*

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cba

a*

b*a*

bt*

a*

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b*

a*

bt*

I

(1) Assembly

(2) Assembly

(3) Disassembly

a*a*

at*

Specifypathways

Designsequences




Nodal Abstraction

Node with 3 ports. 2 types of ports: 1.  Input (circle) 2.  Output (square) State of toehold:

1.  Exposed or accessible (open triangle/circle)

2.  Sequestered or Inaccessible (closed triangle/circle)


Nodal Abstraction

l  Reaction graph l  Molecular program

executed schematically in b,e

LETTERS












a

c d

(1) (2) (3)


b

Reaction graph

Translateto motifs


Nodal abstraction


Dynamicfunction

I A

B


IA

B

A.B

Nucleic acidprimary

sequences

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c b

at

ct bt

a

A

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reaction graph

IA

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ct bt

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a*

b*a*

bt*

a*

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b*

a*

bt*

I

(1) Assembly

(2) Assembly

(3) Disassembly

a*a*

at*

Specifypathways

Designsequences




Nodal Abstraction

Execution of reaction graph


LETTERS












a

c d

(1) (2) (3)


b

Reaction graph

Translateto motifs


Nodal abstraction


Dynamicfunction

I A

B


IA

B

A.B

Nucleic acidprimary

sequences

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ct bt

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reaction graph

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ba

cba

a*

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bt*

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B

b*

a*

bt*

I

(1) Assembly

(2) Assembly

(3) Disassembly

a*a*

at*

Specifypathways

Designsequences




Nodal Abstraction

LETTERS












a

c d

(1) (2) (3)


b

Reaction graph

Translateto motifs


Nodal abstraction


Dynamicfunction

I A

B


IA

B

A.B

Nucleic acidprimary

sequences

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A B

c b

at

ct bt

a

A

eExecution of

reaction graph

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(1)

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at ctbt

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bt*

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b*

a*

bt*

I

(1) Assembly

(2) Assembly

(3) Disassembly

a*a*

at*

Specifypathways

Designsequences




Hierarchical design process


Nodal Abstraction

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 theoutput 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 fourdistinct 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 landscape11–13. By contrast, self-assembly in biologyproceeds isothermally and assembly kinetics are often controlled bycatalysts. Until now, synthetic DNA catalysts3–6 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 initiator leads tocatalytic turnover, as indicated by the nearly complete consumptionof hairpins even at substoichiometric initiator concentrations.Interestingly, only minimal assembly is achieved by annealing thehairpin mixture, illustrating the utility of pathway programmingfor traversing free-energy landscapes with kinetic traps that cannotbe 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 k 5 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 correspond-ing molecular implementation, four hairpin species, A, B, C and D,coexist metastably in the absence of initiator I (Fig. 3b, top). Theinitiator catalyses the assembly of hairpins A and B to form duplexANB (steps 1–2, 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 CND (steps 3–4); duplex CND in turn has a single-stranded region that is identical to I and can thus catalyse A and B toform ANB (steps 5–6). Hence, ANB and CND form an autocatalytic setcapable 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 sigmoidaltime course characteristic of an autocatalytic system16. As expected,the 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 exponentialkinetics and analytical modelling (Fig. 3c, inset). The minimalleakage of a system containing only A and B (labelled A 1 B inFig. 3c) emphasizes that the sigmoidal kinetics of spontaneous ini-tiation for the full system (A 1 B 1 C 1 D) are due to cross-catalysis.

d

1 2 3 4 65

1× in

itiat

or5.0× 52.0×

1.0×

delaennA

e f

Leakage

x* b* y*a*

A.B.C

x b ya

z*

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cz

b*x

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x

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x* b*y*a*

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x b ya

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Catalytic formation of three-arm junction

0×

(1)

(3)(2)(4) z a x y*

b*a* x*C

c

z*

a x

x*

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b* y* c*A

yB

a*

x*y c z

c* z*

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Metastable monomers

Ix* b* y*a*Initiator

A

BC

Ia

(1)

(4)(5)

A B

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I(2)

(3)

D

b c

0×

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* for step 1) 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 1–4). 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.

NATURE | Vol 451 | 17 January 2008 LETTERS


Reaction graph:

1. Only indicates sequence of reactions – not actual execution

2. Only focuses on toeholds of the species (everything else is ignored/discarded)


Nodal Abstraction

1.  Initiator(I)+HairpinsA

2.  I�A+B

3.  I�A�B+C

4.  A�B�C3-armjunction+I



Nodal Abstraction 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 ab*

a*

c

I

b*a*

(2.1) Complementarity relationships

A B C

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y

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(2.2) Clamping/padding

(1) Pathway specifiation

Catalytic formation of a 3-arm DNA junction Target 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

3

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 arecomplementary. 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 orangedomain 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 hairpin’s sticky-end, stem, and loop regions, permitting more flexible dimensioning in Step (2.3). Second, thesegments serve as ‘clamps’ to decrease spurious ‘leakage’ reactions in the absence of the initiators. Consider un-clamped hairpinA 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 wouldpermit hairpin A and hairpin B to react spuriously and form A·B (which would then react with C to form A·B·C). By contrast,the ‘breathing’ of the left end of the clamped hairpin 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 (numberof 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, thethermodynamic analysis of the interacting DNA strands suggests that assigning 6-nt to each segment (Fig. S2e) stabilizes criticalstructures 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)

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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. Reaction1 (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 inaccessiblestates; 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 madebetween the newly accessible output port of A and the accessible input port of B and both ports are flipped to inaccessible states; the outputport 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 newlyaccessible 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 accessiblestate (based on the internal logic of node C). Reaction 4 (disassembly): The bond between the inaccessible output port of I and the inaccessibleinput 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 outputports 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. S3acontains 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

Nodal Abstraction

Programming catalytic circuitry: autocatalytic duplex formation by a cross-catalytic circuit with exponential kinetics.


Nodal Abstraction

This system demonstrates synthetic biomolecular autocatalysis17–20

driven by the free energy of base-pair formation. Autocatalysis andexponential system kinetics can also be achieved through entropy-driven hybridization mechanisms21. For sensing 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 2G–1 monomers,that is, a linear increase in the number of node species yields anexponential 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

C.D

C

D

A.B

A.B.C

(3)

(4)C.D.A

(5)

(6)

I

(1) (2a)

Exponentialamplification

Initiation

I.A

A

Autocatalysis

B

A

0.001×

0× (No initiator) 0.0005× (10 pM initiator)

0.003×

0.01×0.005×1× 0.3× 0.1× 0.05× 0.03× 0.02×

A

a a*b

c*

x y

b*x* y*

x*v

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*

v

v*y*

Ia* x* b* y*v*

Metastablemonomers Initiator

bb*

c ay x

c* a*y* x*

y*

d*u*

u

*vB

u*

ba

10% completion

v* y*

B

A B

D C

(1) (2b)

(3)

(4a)

(4b)(5)

(6b)

(2a) (6a)

I

2

3

4

0 1 2 3 4 5

Tim

e (h

)

1.5

c

Time (h)

Em

issi

ons

(cou

nts

s–1)

×105

log10 [I]–1.6–2–2.4 –1.2

1.2

0.8

0.4A+B+C+D

A+B A

I.A.B(2b)

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. Formation of 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.

a

G4

(4)(3)(2)(1)I

G2

A2 B2

A1I

G3

A2 B2

A1

A3 B3

A3B3

I A2 B2

A1

A3 B3

A3B3

B4 A4 I B4 A4

A4 B4 A4 B4

G1

A1I

(5)

G5

A2 B2

A1

A3 B3

A3B3

B4 A4I

B4 A4

A4 B4 A4 B4

A5

B5

A5

B5

B5

A5

B5

A5

A5B5 A5

B5

A5B5 A5

B5

1 2 3 4 5 6 7

G1 G2 G3 G4 G5A1


G4G5

b B3

A3

B2

A2

B4

A4

B5

A5A1Metastablemonomers

IInitiator

G5w/o I

Leakage

0 10 20 30I (nM)

40Sig

nal (

arbi

trar

y un

its)

50 60 700

1c d e

G3

A3 B3

A2 B2

A1

I

A4 B4

A5 B5

(4)

(1)

(3)

(2)

(5)

Figure 4 | Programming nucleated dendritic growth: triggered assembly ofquantized binary molecular trees. See Supplementary Information 5 fordetails. a, Reaction graph. Multiple assembly arrows entering the same inputport depict parallel processes on separate copies of the nodal species.b, Secondary structure mechanism. c, Agarose gel electrophoresisdemonstrating triggered self-assembly. Lanes 1–6: the dominant reactionband 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.

LETTERS NATURE | Vol 451 | 17 January 2008


Programming catalytic circuitry: autocatalytic duplex formation by a cross-catalytic circuit with exponential kinetics.

1. Fluorophore (green star)

2. Quencher (black dot)


Nodal Abstraction





C.D

C

D

A.B

A.B.C

(3)

(4)C.D.A

(5)

(6)

I

(1) (2a)


Initiation

I.A

A

Autocatalysis

B

A

0.001×


0.003×

0.01×0.005×1× 0.3× 0.1× 0.05× 0.03× 0.02×

A

a a*b

c*

x y

b*x* y*

x*v

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*

v

v*y*

Ia* x* b* y*v*


bb*

c ay x

c* a*y* x*

y*

d*u*

u

*vB

u*

ba

10% completion

v* y*

B

A B

D C

(1) (2b)

(3)

(4a)

(4b)(5)

(6b)

(2a) (6a)

I

2

3

4

0 1 2 3 4 5

Tim

e (h

)

1.5

c

Time (h)

Emis

sion

s (c

ount

s s–1

)

×105

log10 [I]–1.6–2–2.4 –1.2

1.2

0.8

0.4A+B+C+D

A+B A

I.A.B(2b)



a

G4

(4)(3)(2)(1)I

G2

A2 B2

A1I

G3

A2 B2

A1

A3 B3

A3B3

I A2 B2

A1

A3 B3

A3B3

B4 A4 I B4 A4

A4 B4 A4 B4

G1

A1I

(5)

G5

A2 B2

A1

A3 B3

A3B3

B4 A4I

B4 A4

A4 B4 A4 B4

A5

B5

A5

B5

B5

A5

B5

A5

A5B5 A5

B5

A5B5 A5

B5

1 2 3 4 5 6 7

G1 G2 G3 G4 G5A1


G4G5

b B3

A3

B2

A2

B4

A4

B5


IInitiator

G5w/o I

Leakage

0 10 20 30I (nM)

40Sig

nal (

arbi

trar

y un

its)

50 60 700

1c d e

G3

A3 B3

A2 B2

A1

I

A4 B4

A5 B5

(4)

(1)

(3)

(2)

(5)









C.D

C

D

A.B

A.B.C

(3)

(4)C.D.A

(5)

(6)

I

(1) (2a)


Initiation

I.A

A

Autocatalysis

B

A

0.001×


0.003×

0.01×0.005×1× 0.3× 0.1× 0.05× 0.03× 0.02×

A

a a*b

c*

x y

b*x* y*

x*v

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*

v

v*y*

Ia* x* b* y*v*


bb*

c ay x

c* a*y* x*

y*

d*u*

u

*vB

u*

ba

10% completion

v* y*

B

A B

D C

(1) (2b)

(3)

(4a)

(4b)(5)

(6b)

(2a) (6a)

I

2

3

4

0 1 2 3 4 5

Tim

e (h

)

1.5

c

Time (h)

Emis

sion

s (c

ount

s s–1

)

×105

log10 [I]–1.6–2–2.4 –1.2

1.2

0.8

0.4A+B+C+D

A+B A

I.A.B(2b)



a

G4

(4)(3)(2)(1)I

G2

A2 B2

A1I

G3

A2 B2

A1

A3 B3

A3B3

I A2 B2

A1

A3 B3

A3B3

B4 A4 I B4 A4

A4 B4 A4 B4

G1

A1I

(5)

G5

A2 B2

A1

A3 B3

A3B3

B4 A4I

B4 A4

A4 B4 A4 B4

A5

B5

A5

B5

B5

A5

B5

A5

A5B5 A5

B5

A5B5 A5

B5

1 2 3 4 5 6 7

G1 G2 G3 G4 G5A1


G4G5

b B3

A3

B2

A2

B4

A4

B5


IInitiator

G5w/o I

Leakage

0 10 20 30I (nM)

40Sig

nal (

arbi

trar

y un

its)

50 60 700

1c d e

G3

A3 B3

A2 B2

A1

I

A4 B4

A5 B5

(4)

(1)

(3)

(2)

(5)






Programming nucleated dendritic growth: triggered assembly of quantized binary molecular trees

Nodal Abstraction


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*

2a

*3

x*

3x

2a

3x

3

a2

x*

1a

*2

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

x*

k-1

a*

k

x*

kx

k-1

ak

xk

ak-1 x

*k-2

a*

k-1

x* k

a* 1

x* 1

x ka

1x 1

ak

x* k-

1a

* 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

. . . . . .

I

x*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 theabsence 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 ofa k-arm junction. Fig. S8 gives an example when k = 6.

8

The DNA motif for ‘seesaw’ gates.

Qian L , Winfree E J. R. Soc. Interface doi:10.1098/rsif.2010.0729

©2011 by The Royal Society

Seesaw Gates

Implementation of relay circuits.

Qian L , Winfree E J. R. Soc. Interface doi:10.1098/rsif.2010.0729

©2011 by The Royal Society

Seesaw Gates

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