Lulu Qian and Erik Winfree- A simple DNA gate motif for synthesizing large-scale circuits

8/3/2019 Lulu Qian and Erik Winfree- A simple DNA gate motif for synthesizing large-scale circuits

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A simple DNA gate motif forsynthesizing large-scale circuits

Lulu Qian1 and Erik Winfree1,2,3,*

1Bioengineering, 2Computer Science, and 3Computation and Neural Systems,California Institute of Technology, Pasadena, CA 91125, USA

The prospects of programming molecular systems to perform complex autonomous tasks havemotivated research into the design of synthetic biochemical circuits. Of particular interest tous are cell-free nucleic acid systems that exploit non-covalent hybridization and strand displa-cement reactions to create cascades that implement digital and analogue circuits. To date,circuits involving at most tens of gates have been demonstrated experimentally. Here, wepropose a simple DNA gate architecture that appears suitable for practical synthesis oflarge-scale circuits involving possibly thousands of gates.

Keywords: nucleic acids; synthetic biology; strand displacement circuits;molecular programming

1. INTRODUCTION

RNA- and DNA-based catalysts [13] and logic gates[4 6] have been proposed as general-purpose com-ponents for synthesizing chemical circuits [6 8] withapplications in medical therapeutics [9], nanotechnol-ogy [10] and embedded control of chemical reactions[11]. Progress in this direction will depend uponadvances in three areas: (i) developing input/output

interfaces between the DNA circuits and biomedicallyrelevant molecules [12], DNA nanomachines [13,14],and general chemistries [15,16]; (ii) developing DNAcircuit construction techniques that scale up so thatlarge and interesting circuits can be systematically cre-ated [6,8,17,18]; and (iii) extending the DNAprogramming methodology beyond well-mixed sol-utions to include spatial structures at the molecular[19 21] and macroscopic scales [22 24].

In this paper, we focus on the second challenge, usingDNA strand displacement cascades [3,6,17,18]. Weintroduce a DNA gate motif suitable for scaling up tolarge circuits, along with an abstract circuit formalism

that aids the design and understanding of circuit behav-iour. To illustrate the rich potential of this approach, weshow how to implement arbitrary feedforward digitallogic circuits, arbitrary relay circuits and analogue cir-cuits exhibiting a variety of temporal dynamics. Largecircuits can be made fast and reliable because first,the gates can act catalytically to amplify small signalspropagating through the circuit, and second, the gatescan incorporate a threshold to clean up noise and erro-neous signals. Further, thanks to the modular design ofthe gate motif, systematic construction can be auto-mated by a straightforward compiler that converts ahigh-level description of a circuit function into a mol-

ecular implementation at the level of DNA sequences.Finally, we argue that synthesis and preparation of

molecular components can be parallel and scalable.Our estimates, based on the available sequence designspace for typical DNA lengths, current synthesis tech-nologies using parallel DNA microarrays, andplausible mass-action rates and concentrations, suggestthat circuits involving thousands of distinct gates maybe designed, synthesized and executed.

2. A SIMPLE DNA GATE MOTIF

We begin by describing the elementary building blockfor our circuits. Figure 1a shows the abstract diagramof a single gate, configured so that an input catalyti-cally converts fuel to output when the inputconcentration exceeds a threshold level. Signals on thewires, e.g. input, fuel and output, will correspondto single-stranded DNA molecules we call signalstrands, while the gate node itself will correspondto partially double-stranded DNA molecules we callgate:signal complexes and threshold complexes. Thecatalytic cycle is achieved by a series of interactions

between signal strands and gate complexes, based onthe underlying mechanism of toehold-mediated DNAstrand displacement [25] in which a single-strandedDNA molecule displaces another from a double-stranded complex with the help of a short toeholddomain.

Signal strands have a uniform format consisting ofa left recognition domain, a central toehold, and aright recognition domain ( figure 1b). In our example,the input is S1TS2, the output is S2TS3 and the fuelis S2TS4. Recognition domain S2 is the active domainparticipating in the catalytic reactions associated withthis gate, while S1, S3 and S4 can similarly react with

other gates. The recognition domains are relativelylong (e.g. 15 nt) to ensure stable hybridization,while the toehold is relatively short (e.g. 5 nt) toensure fast release of strand displacement products,*Author for correspondence ([email protected]).

J. R. Soc. Interface

doi:10.1098/rsif.2010.0729

Published online

Received20 December 2010Accepted14 January 2011 1 This journal is q 2011 The Royal Society

on February 7, 2011rsif.royalsocietypublishing.orgDownloaded from
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but still long enough to effectively initiate strand dis-placement. While many distinct recognition domainsmay be used, all molecules use the same universaltoehold sequence; identity is conferred by the recog-nition domain sequences. The structure andmechanism of the gate are intrinsically symmetric,so that labels input, output and fuel referonly to roles played by the signal strands in thisasymmetrically configured example.

A gate is represented abstractly as a two-sided node

with one or more wires connected to each side. A signalstrand may either be free in solution with its toeholdexposed, indicating activity on a wire, or else boundto a gate base strand (e.g. the bottom strand

T0S02T0 of the gate:output complex) with its toehold

sequestered and thus inactive. Because of their DNAimplementation, gates have an intrinsic polarity thatdistinguishes the two sides, which we will refer to asthe left side and the right side. Wires must connectthe right side of one gate to the left side of another.(For example, the output strand S2TS3 uses subse-quence S2T to react with one gate from the right sideand subsequence TS3 to react with another gate fromthe left side.) Gate:signal complexes also have a uniform

structure. The gate base strand consists of a recognitiondomain identifying the gate, flanked by two toeholddomains, and is always complexed with a signalstrand facing either left or right; only one toehold is

input (1x)

gate:output (10x)

fuel (10x)

threshold (0.5x)

1

output

input 10

10

fuel

gate

0.5

wastewaste

S2

TS1

S2

threshold

S2

input

T S2

S1

inputgate:output

fueloutput

gate:fuel

gate:input

S4

TS2

S1

T S2

TS2

S3 S1

T S2

TS2

S4

S3

TS2 T S2

S1

S3

TS2

S4

TS2

(a) (b)

(c)

(d)

S2

S3

TS2

S4

TS2

T S2

S1

T S2

S1

T S2

T T T T

T S2

S2

S2 S

2

S2

S2 S

2

S2T

s1

s1

s1

s1 T

TT

TT S2

TT

TT S2

TT

Figure 1. The DNA motif for seesaw gates. (a) Abstract gate diagram. Red numbers indicate initial concentrations. (b) TheDNA gate motif and reaction mechanism. S1, S2, S3 and S4 are the recognition domains; T is the toehold domain; T

0 is theWatsonCrick complement of T, etc. Arrowheads mark the 30 ends of strands. Signal strands are named by their domainsfrom 30 to 50, i.e. from left to right, so the input is S1TS2; gate base strands and threshold bottom strands are named by theirdomain from 50 to 30. All reactions are reversible and unbiased; solid lines indicate the dominant flows for the initial concen-trations shown in (a), while the reverse reactions are dotted. (c) The threshold motif and reaction mechanism. The toehold isextended by a few bases (s01, the complement of the first few 5

0 bases of S1), providing an increased rate constant relative tothe gate itself. Branch migration intermediate states are omitted from the diagram. ( d) Example sequences. Gate complexesand signal molecules are shown at the domain level (second column) and at the sequence level (third column). Here, recognitiondomain sequences are 15 nt, the toehold domain sequence is 5 nt, and the toehold is extended by 3 nt for the threshold. Otherlengths are possible, so long as they ensure that recognition domains will not spontaneously dissociate, toehold exchange is fast,and thresholding is sufficiently faster.

2 A simple DNA gate motif L. Qian and E. Winfree


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exposed at any given time. In our example, the gate:out-put complex comprises gate base strand T0S02T

0 boundto signal strand S2TS3 facing right (figure 1b). In thefollowing, signal strand will mean free signal strand,unless otherwise specified.

A threshold may also be associated with a gate,absorbing a signal strand at a faster rate compared to

the gate. Threshold complexes are similar to gate com-plexes, but with an extended toehold on one side and notoehold on the other sideconsequently, the top strandhas no toehold and is therefore inert when released. Thetoehold is extended by a few nucleotides (e.g. 3 nt) toensure faster reaction rates [26]. In our example, thethreshold complex that absorbs input S1TS2 comprisesbottom strand s01T

0S02 and top strand S2 (figure 1c).Each position within the abstract diagram

(figure 1a) corresponds to a specific molecule, andeach red number in that position indicates the (relative)initial concentration for that molecule: positions oneach wire correspond to free signal strands, while pos-

itions within the node at the end of each wirecorrespond to gate:signal complexes (positive numbers)or threshold complexes (negative numbers).

Gates support three elementary behaviours. Stoi-chiometric triggering occurs when a free signal strand(e.g. input) binds to the exposed toehold of a gate com-plex (e.g. gate:output), initiating branch migration andthe subsequent release of the signal strand previouslysequestered in the gate complex (e.g. output;figure 1b, left pathway). Thus, toehold exchange [27]has been effected; the resulting gate complex (e.g.gate:input) now has a toehold exposed on the oppositeside. Consequently, the overall reaction is reversible:

the released signal strand can bind to the gate again,triggering the release of the original signal strand. Over-all, stoichiometric triggering allows the exchange ofequal amounts of activity from a wire on one side of agate to a wire on the other side. This back-and-forthmotion is the inspiration for our name for this motif:seesaw gates.

If fuel is present, then a catalytic cycle may occurwherein after stoichiometric triggering by the input toproduce the output, fuel may bind to the gate and dis-place the input, freeing it to trigger more release of theoutput (figure 1b, both pathways). This cycle, a simpli-fication of Zhang et al. [3] that was demonstrated in

Zhang & Winfree [27], allows an arbitrarily smallamount of input, over time, to catalyse the release ofan arbitrarily large amount of output. Because thereis also a reverse action whereby the output (ratherthan fuel) displaces the gate-bound input, as well asan action whereby the input displaces gate-bound fuel(rather than gate-bound output), we expect a single iso-lated gate to establish an equilibrium rather going tocompletion. Critically, if no input is present, there isno fast pathway for the fuel to directly displace theoutput; strand displacement without a toehold ismany orders of magnitude slower [26 28], and weneglect it here.

Finally, thresholding occurs whenever a thresholdcomplex is present (figure 1c). Due to the thresholdcomplexs extended toehold, and the exponentialdependence of reaction rate on toehold length [26,27],

input strands will react with threshold complexesfaster than they react with gate complexesweassume a 20-fold speed-up in this paper. The two result-ing waste complexes are inert, as they have no exposedtoehold domains. Consequently, only if the input con-centration exceeds the threshold concentration, canthe excess input partake significantly in stoichiometric

triggering or catalysis. With subthreshold input levels,there will still be a small leak due to inputs thatreact with gates before encountering a threshold. Fortu-nately, even leaky thresholding behaviour can enablereliable circuit function by cleaning up after upstreamleaky reactions.

It is worth considering the driving forces for thesethree behaviours. Previous strand displacement circuitsprimarily made use of additional base pairing [6] orexploited the release of additional molecules [3] todrive reactions forward, but neither mechanism is essen-tial in seesaw gate circuits. Stoichiometric triggeringinvolves a reversible reaction with the same number of

molecules and the same number of base pairs in thereactants as in the products, so the standard freeenergy difference is approximately zero. Thus, thedriving force comes from the entropic free energy ofconcentration imbalances, which encourages equaliza-tion. The catalytic cycle is similarly driven exclusivelyby entropic free energy, but introduces an auxiliaryspecies, the fuel, whose initial concentration canserve as an energy source to drive the reaction in adesired direction. On the other hand, thresholdinginvolves an essentially irreversible reaction with a netgain of base pairs (8 bp in our example) and thus asignificant standard free energy change. A down-

stream threshold can therefore act as a drain on anupstream entropy-driven reaction, further pulling thereaction forward.

The single-gate circuit shown in figure 1a is config-ured for catalysis with thresholding. The diagramspecifies the gate:output complex, fuel signal strand,input signal strand and threshold complex with respect-ive concentrations 10x, 10x, 1x and 0.5x, where 1x is astandard concentration, perhaps 50 nM. With theseinitial concentrations, the input strand will first over-come the threshold and then act catalytically tofacilitate the equilibration of the output strand andthe fuel strand to approximately 5x each. This level

can be estimated by noting that by symmetry the twowires will have similar activity at equilibrium, whilethe total concentration on each wire and the total con-centration within the gate remain constant at 10xeach.Thus, a single seesaw gate can robustly amplify aninput signal that exceeds a threshold. Other behaviours,including stoichiometric triggering, are possible withdifferent configurations of wires and concentrations.When connected into circuits involving many interact-ing seesaw gates, complex functional behaviour can beobtained.

3. ABSTRACT CIRCUIT FORMALISM

The abstract network representation introduced infigure 1a facilitates concise reasoning about circuits

A simple DNA gate motif L. Qian and E. Winfree 3


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involving many interacting gates. In general, a circuitconsists of a number of gate nodes and a number ofwires between gate nodes. Each gate node consists ofa left side and a right side, and it may connect to anynumber of wires on each side ( figure 2a). Each wireconnects exactly two gates (figure 2b), from the leftside of one to the right side of the other. Some wires,such as input, output and fuel, may connect to virtualgates whose total base strand concentration is zero.Virtual gates provide a consistent naming scheme for

wires that appear connected on just one side, makingit easy to extend a circuit or compose circuits together.For clarity, gates with non-zero base strand concen-tration can be referred to as realized gates. A seesawcircuit then consists of a number of gate nodes con-nected by wires between their left and right sides(figure 2c). The diagram can be annotated to indicatethe initial state of the circuit ( figure 2a).

The implementation of any such circuit diagramusing DNA molecules is straightforward. A circuitwith N (realized or virtual) gates requires a single uni-versal toehold sequence T and a set of N sufficientlydistinct m-mer sequences S1, S2, . . ., SN, one for each

gate. The wire from the right side of gate i to the leftside of gate j is called wi,j and is implemented as afree signal strand with sequence SiTSj. The gate i basestrand, T0S0iT

0, is always part of a gate complex: gi:i,jrefers to the gate complex in which the base strand isbound to the left side of signal strand SiTSj, and gk,i:irefers to the gate complex in which the base strand isbound to the right side of signal strand SkTSi. Notethat gi:i,i and gi,i:i are both complexes of signal strandSiTSi and gate i base strand T

0S0iT0, but the former

has the signal strand bound on its left side, leavingthe gates left toehold exposed, while in the latter casethe signal strand is bound on its right side, leaving

the gates right toehold exposed. The threshold complexfor wire wi,j within gate node j is called thi,j:j and isimplemented as a complex of top strand Sj andbottom strand si

0T0Sj0; on the other side of the same

wire, thi:i,j within gate node i also absorbs wire wi,j,and is implemented as Si and Si

0T0sj0.

Because all components are in a standard form,the set of chemical reactions modelling a seesawcircuit can be written concisely. The reversibletoehold exchange steps, where a free signal strand dis-places a bound signal strand from a gate complex, aremodelled by

wj;i gi:i;kOks

ks

gj;i:i wi;k; 3:1

where ks is a bimolecular rate constant and i,j,k[f1, 2, . . ., Ng. Similarly, the irreversible thresholdingsteps, where a signal strand is absorbed by a thresholdcomplex in the gate node on either end of its wire, aremodelled by

wi;j thi;j:j !kf

waste kf

thi:i;j wi;j; 3:2

where kf is a bimolecular rate constant much faster thanks. Using standard mass action chemical kinetics, thisgives rise to a system of ordinary differential equations(ODEs) for the dynamics:

d

dtwi;j ks

XNn1

wn;i gi:i;j wj;n gi;j:j

ksXNn1

wi;j gn;i:i wi;j gj:j;n

kfwi;j thi:i;j wi;j thi;j:j;

3:3

d

dtgi:i;j ks

PNn1

wi;j gn;i:i wn;i gi:i;j

ddt

gi;j:j ksPNn1

wi;j gj:j;n wj;n gi;j:j

9

>>>=>>>;3:4

(a) (b) (c)

j

ii

1,iw

2,iw

N,iw i,Nw

i,2w

i,1w

i,jw1, :i ig

2, :i ig

N, :i i i ig

: ,g

1

2...

N

1

2...

.

.. ......

...

N

1

2 7

8

9

106

4

5

3

N

i i: ,g

2

i i: ,g

1

Figure 2. Abstract diagrams for seesaw gate circuits. (a) The general form of a gate node. Each gate imay be connected to manywires on each side, potentially all Nnodes in the network, including itself. For each wire from the right side of gate ito the left sideof gate j, the initial concentration of the free signal wi,jmay be written above the wire, and the initial concentrations of gate com-plex gj,i:i (wj,i bound to gate i) and gi:i,j (wi,j bound to gate i) may be written within the node at the ends of the correspondingwires. Gate concentrations are simply omitted if they are zero. Initial concentrations of thj,i:i (the threshold for wj,i arriving atgate i) and thi:i,j (the threshold for wi,j arriving at gate i) may be written in the same locations as gj,i:i and gi:i,j, respectively,but as negative numbersor omitted if they are zero. (b) The general form of a wire. Each wire is specifically connected onits left end to the right side of a gate node, and connected on its right end to the left side of a gate node. ( c) An example circuitwith five realized gates (numbered circles), five virtual gates (numbers at ends of wires), and 11 wires. Each wire is identified bythe two gates it connects; thus the virtual gates serve to provide full names (and sequences) to their incident wires. Note thatcircuit diagrams may be drawn without providing gate numbers, as they are not relevant to circuit function.



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and

d

dtthi:i;j kf wi;j thi:i;j

d

dtthi;j:j kf wi;j thi;j:j

9>=>;: 3:5

These dynamics have conserved quantities for each

gate node and for each wire. First, the total amountof the gate i base strand T0S0iT

0, whether its left toe-hold is covered ([g1,i:i] . . . [gN,i:i]) or its right toehold iscovered ([gi:i,1] . . . [gi:i,N]), remains constant for alltimes t:

ci def

XNn1

gn;i:it gi:i;nt: 3:6

Second, the total amount of each signal strand SiTSj,whether free in solution ([wi,j]), bound in a gate complex([gi:i,j] and [gi,j:j]), or absorbed by a threshold([thi,j:j]02 [thi,j:j]t and [thi:i,j]02 [thi:i,j]t), remains con-

stant, so

ci;j def

gi:i;jt wi;jt gi;j:jt thi:i;jt thi;j:jt: 3:7

Note that we omitted the constant values [thi:i,j]0and [thi,j:j]0 from this sum; ci,j represents the amountby which the total signal strand concentration (in anyform) exceeds the total threshold complex concen-tration. If there is more threshold than signal strand,then ci,j could be negative; in that case, any activityon the wire will be immediately suppressed.

Additional constraints come from equilibrium, if andwhen it is obtained. Because threshold complexes inhi-

bit activity on their wires until the threshold isovercome, and because equilibration of wires on oneside of a gate node can depend upon activity in a wireon the other side, a seesaw circuit can settle downinto a state with a complex pattern of active andinactive wires, with only certain gates and wires partici-pating in the equilibrium. Consider, then, a networkthat has settled down in such a configuration, andtreat concentrations arbitrarily near zero as exactlyzero, for convenience. Consider further a gate node ithat has at least one active wire on each side,undergoing active exchange with gate complexes.Equilibrium enforces a simple relationship between

the free and bound forms of the signal strands, namelythat their ratio with respect to a particular gate must beidentical for all active wires connected to that gate. Thisfollows immediately from equation (3.1) and the detailedbalance equation ks. [wj,i] . [gi:i,k] ks. [gj,i:i]. [wi,k]. Towit, for each gate i, at equilibrium all active wires achievethe ratio

ri def wj;i1

gj;i:i1

wi;k1gi:i;k1

; 3:8

where j and k refer to any active wires on the left andright sides of the gate node.

As an example, we can calculate the equilibrium con-centrations for a single realized gate, i, connected to anumber of virtual gates, as in figure 1a. Without lossof generality, we assume that all wires have initial

concentrations larger than their respective thresholds,i.e. all ci,j are positive, and all thresholds will eventuallybe consumed. We can derive all equilibrium wire andgate concentrations from our knowledge of the con-stants ci and ci,j, which can be calculated directlyfrom the initial conditions. The equilibrium wireconcentration [wi,j]1 depends upon the gate ratio ri

and the initial concentrations on that wire, as can bederived from equations (3.7) and (3.8) and the defi-nition of a virtual gate (i.e. [gi,j:j]t [thi,j:j]t 0 forvirtual gate j):

wi;j1 wi;j1

gi:i;j1 wi;j1gi:i;j1 wi;j1

ri

1 rigi:i;j0 wi;j0 thi:i;j0

ri

1 rici;j:

3:9

Symmetrically, where gate k is also a virtual gate:

wk;i1 ri

1 rick;i: 3:10

To solve for the equilibrium gate ratio ri, we canrewrite equation (3.6) using equation (3.8), to obtain:

ci 1

ri

XNn1

wn;i1 wi;n1; 3:11

whereupon substituting in equation (3.9) immediatelygives us

ri

1 ri 1

ciPn cn;i ci;n

: 3:12

It is easy to work out the equilibrium wire con-centrations directly from a diagram, such asfigure 1a. We first calculate ri=1 ri 110=10 10 1 0:5 % 0:51, where the numerator isthe sum of all non-negative red numbers within thegate node, and the denominator is the sum of all rednumbers in the entire diagram. Then, the input[w1,2]1 % 0.5x 0.51 % 0.26x, where 0.5x is the excessof input over the threshold. Similarly, the output[w2,3]1 % 10x 0.51 5.1x, where 10x is the sum ofred numbers on the output wire, and likewise the fuel

[w2,4]1 % 10x 0.51

5.1x.In conclusion, the abstract seesaw circuit formalism

provides a precise executable interpretation for arbi-trary seesaw circuit diagrams, in terms of formalchemical reaction networks and their associated massaction ODEs. The uniformity of components in seesawgate circuits also facilitates their analysis and simu-lation. We have written routines for conciselyrepresenting, constructing and simulating models ofseesaw gate circuits in Mathematica. For efficient simu-lation of large circuits, we output to SBML [29] andimported into COPASI [30].

The merit of seesaw circuits is that it is straight-

forward to compile them into systems of DNAmolecules, with the implicit claim that the experimen-tally synthesized systems will closely approximate themodel ODEs. This claim needs to be tempered by



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acknowledging a number of additional reactions, notincluded in our model, that will certainly occur inexperimental systems.

First, even without a toehold, a single-stranded DNAdomain can displace an identical domain from a doublehelix, albeit at a rate five or six orders of magnitudeslower than kf, the fastest rate for toehold-mediated

strand displacement [26 28]. Examples include (i)fuel-gate leak where, with reference to figure 1b, fuelreacts directly with gate:output, yielding output andgate:fuel even in the absence of input, and (ii) gategate leak where the single-stranded S3 domain ofgate:output directly interacts with a downstream gatewith base strand T0S3T

0, causing it to release its topstrand. So long as leak is not too large, it can be cleanedup with threshold gates.

Second, because we use a universal toehold sequenceT, anysignal strand can bind to the toehold ofanygatecomplex. However, the uniqueness of the recognitiondomains Si ensures that no such binding will lead to

branch migration, and the invading signal strand willquickly fall off. While this spurious toehold bindingwill not result in an incorrect strand displacementreaction, it will change the effective reaction ratesby temporarily disabling some fraction of gate andthreshold complexes and reducing the signal strand con-centrations. If all reactions are slowed down to the samedegree, then the behaviour changes are inconsequential.Furthermore, the disruption is less at higher tempera-tures and lower concentrations, and therefore can beavoided at the cost of overall speed.

Third, there is one case where unproductive spurioustoehold binding has the potential to cause significantly

more disruption. If a signal strand shares only theupstream domain with a threshold complex, it couldspuriously bind the entirety of the threshold complexsextended toehold. For example, if the signal strandS1TS4 co-existed in a system with the threshold complexof figure 1c, it could bind to the 8 nt toehold s01T

0 butcould not undergo strand displacement. Such inter-actions will take significantly longer to resolve,because nucleic acid dissociation rates decrease expo-nentially with the number (more accurately, freeenergy) of base pairs that must be broken [31]. As a con-sequence, these threshold reactions will be slowed downmore than gate reactions, decreasing their effectiveness

as thresholds. We call this threshold inhibition. It can beminimized by optimizing toehold lengths, temperatureand concentrations.

Fourth, in gates with more than one input, there willbe some crosstalk if one or both of the inputs have athreshold. The threshold for the first input can absorbthe second input signal strand and visa versa, becauseboth thresholds have the same initial toehold sequenceT0 even if the extra bases are different. Consequently,after one input exceeds its own threshold, havingreacted at a fast rate, it will continue to be absorbedby the other thresholdat roughly the same slowerrate with which it reacts with the gate:output complex.

We call this threshold crosstalk. Unlike the previousthree problems, threshold crosstalk is intrinsic to thedesign and thus cannot be reduced by optimizingexperimental conditions.

Rather than explicitly add these effects into ourmodel, here we prefer to keep the model as simple aspossible, while noting in which circuits we expect thatside reactions would have a significant effect. Ourmain concerns are threshold inhibition and thresholdcrosstalk. Since threshold inhibition cannot occur in cir-cuits in which every gate immediately upstream of a

threshold has exactly one output and no fuel, andthreshold crosstalk cannot occur in circuits in whichonly gates with a single input are allowed to have athreshold, seesaw circuits satisfying both these con-ditions are called clean circuits and we expect themto be well modelled by our simple equations. In prin-ciple, any unclean circuit may be converted into aclean one by inserting a new gate node in the middleof offending wires, moving offending thresholds to thenew nodes as necessary. Although the temporaldynamics and equilibrium will not be identical, inmany cases the resulting clean circuit will closelyapproximate the original circuits behaviour. How-

ever, as we will see in the following sections, often thenon-idealities of unclean circuits do not significantlydisrupt correct behaviour, and conversely it is oftenpossible to find efficient clean circuit implementationsdirectly.

4. FEEDFORWARD DIGITAL CIRCUITS

Digital logic has two compelling features for circuit con-struction: first, it has proved to be very expressive forthe synthesis of a wide range of desired behaviours;and second, it is intrinsically robust to a variety of man-

ufacturing and operational defects. The basic principleunderlying digital logic is that an intrinsically analoguesignal carrier may be considered simply to be either ONor OFF if at each stage of computation, signals areeither pushed toward the ideal ON value or pushedtoward the ideal OFF value. This is called signal restor-ation, because if noise or device imperfections slightlycorrupt a signal, that deviation from ideal behaviouris cleaned up (perhaps not completely) by subsequentprocessing without altering the interpretation of thesignal as ON or OFF. For example, a digital abstractionmight consider signal levels between 0x and 0.2x asOFF, while signal levels between 0.8x and 1x are

considered ON. Intermediate signal levelsin thetransition regionmust be transient or else they areconsidered a fault, because proper behaviour of logicgates is no longer guaranteed for input levels in theintermediate range. Devices that support wider digitalabstract ranges are therefore more robust to noise andother imperfections.

In principle, digital logic behaviour can be achievedin analogue mass-action chemistry by exploiting cataly-sis and non-linearity for signal restoration [32]; catalysisis also essential for ensuring that signal propagation inmulti-layer circuits is fast [33]. Seesaw gates easily pro-vide both catalysis and thresholding to implement

signal restoration and fast signal propagation. However,once a seesaw gate has been activated and reaches equi-librium, it cannot be re-used, putting a limit on whatclass of circuits can be implemented.



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Feedforward digital circuits are an important class ofcombinational circuits, i.e. memoryless circuits in whichthe inputs uniquely determine the outputs withoutthe need to re-use any parts of the circuit. It is wellknown that any boolean function can be computed(often quite efficiently) by a well-designed feedforwardcircuit built from AND, OR and NOT gates. Interest-

ingly, with a small cost in circuit size and a minorchange in representation, called dual-rail logic, ANDand OR by themselves suffice. We will therefore provideconstructions for these basic operations, incorporatingsignal fan-in, fan-out, routing, and thresholding asmay be required to paste the computing gates togetherinto large circuits. We present two schemes for imple-menting digital logic with seesaw gates. The firstscheme, called the 1-4 scheme because each OR gaterequires one seesaw gate and each AND gate requiresfour seesaw gates, exhibits ideal digital behaviourwith large digital abstraction ranges. However, theseesaw circuits are not clean, and therefore one

would expect degraded performance in experimentalimplementations due to threshold crosstalk andthreshold inhibition. The second scheme, called the2-2 scheme, also exhibits excellent digital behaviour,although with smaller margins. Moreover, the 2-2scheme circuits are clean, alleviating experimentalimplementation concerns.

In the 1-4 scheme, the OR gate ( figure 3a) is built asa simple extension of the basic catalyst ( figure 1a) inwhich there are two input wires, w1,3 and w2,3, thateach have a threshold. If either exceeds its threshold,it can serve as catalyst for the release onto outputw3,4, driven by exchange with the fuel wire w3,5.

Rather than analyse this subcircuit in isolation, wetreat the case where a downstream circuit provides aload on the output, in the sense that free outputsignal strands are irreversibly absorbed, for exampleby a threshold in the downstream circuit. We plot thetotal amount of output that is absorbed by the down-stream process. With this approach, the behaviour ofcircuits can be understood without having to calculateexact equilibria: once an input wire exceeds itsthreshold, catalysis will proceed at some rate until allthe output is released. Under the assumption that thethreshold reaction rate constant, kf, is 20 times fasterthan the gate reaction rate constant, ks, the OR gate

exhibits ideal digital behaviour in simulations. A digitalabstraction with 00.4x being OFF and 0.61x beingON is possible. For smaller ratios kf/ks or shortertimes, the sharpness of the threshold would decreasecorrespondingly.

The AND gate for the 1-4 scheme is a little trickier(figure 3b). The idea here is to put two seesaw gatesin series (gates 2 and 4), with one gates fuel being pro-vided by another gates outputthus gate 2 becomescatalytically active only when both inputs are ON.Gate 8 is a routing device that effectively allows awire to go from the right side of one gate to the rightside of another gate, and gate 5 provides signal restor-

ation to clean up stoichiometric triggering of gate 2when only its input is ON. Let us verify the four casesfor a digital abstraction with 00.4x being OFF and0.61x being ON. First, if w1,2 and w3,4 are both OFF,

they will be absorbed by their respective thresholds,and nothing more will happen, so the output w5,6 willremain OFF. Second, if only w3,4 is ON, then it willdrive wire w4,8 high, which will release the routing cat-alyst w8,10 from the routing gate, allowing activity toflow to gate 2s fuel wire, w2,8. However, with inputw1,2 being OFF, nothing more will happen, and the

output will remain OFF. In the third case, bothinputs are ON; as before, gate 2s fuel wire is drivenhigh, but now input w1,2 exceeds its threshold and cat-alyses activity to flow to wire w2,5, which in turnexceeds its threshold and catalyses the output to turnON. In the final case, input w1,2 is ON but w3,4 isOFF. In this case, gate 2s fuel w2,8 remains inactive,but as much as 0.5xcould be pushed onto w2,5 from stoi-chiometric triggering by input w1,2. However, this doesnot exceed gate 5s threshold, so the output remainsOFF. For this qualitative argument to go throughquantitatively, the initial gate and threshold concen-trations need to be adjusted carefully; a working

choice is shown in figure 3b, for which simulationsshow ideal digital behaviour.Having established that seesaw circuits can

implement digital logic if threshold crosstalk andthreshold inhibition are neglected, we set out to find aclean implementation where neither issue can arise inan experimental system. This is accomplished by the2-2 scheme, which can support a digital abstract with020.3x being OFF and 0.721x being ON. In the ORgate (figure 3c), the first seesaw gate has two inputsbut no threshold and no fuel, thus producing stoichio-metric activity on the intermediate wire w3,4 capableof summing the two inputs. The threshold on the

second gate is set at 0.66x to be above the maximumsum of two OFF inputs (0.3 0.3 0.6) and belowthe minimum sum of an ON input plus an OFF input(0.0 0.7 0.7). Thus, catalysis can drive the outputwire w4,5 ON only if either or both inputs are ON.Simulations show a sharp threshold along the line[w1,3]0 [w2,3]0 0.66x.

The AND gate (figure 3d) has the same wiring dia-gram as the OR gate, but uses a suitably increasedthreshold. If neither or one input signal is ON, atmost 1.3x can be pushed onto wire w3,4, which will beabsorbed by the 1.33x threshold of gate 4. Only whenboth input signals are ON, can a signal equal to or

greater than 1.4x be pushed onto wire w3,4 and exceedthe threshold. Simulations show a sharp thresholdalong the line [w1,3]0 [w2,3]0 1.33x.

Fan-in and fan-out are handled easily in bothschemes. More than two inputs (additional fan-in) toan AND/OR gate can be implemented by a binarytree of two-input AND/OR gates. More than oneoutput (fan-out) from an AND/OR gate simply entailsconnecting more output wires and increasing theconcentration of the fuel.

NOT gates appear to be difficult to implementdirectly. The problem is that a NOT gate must dis-tinguish between a low input signal computed by an

upstream gate, in which case it should release itsoutput strand, and an input signal that is low simplybecause it has not yet been computed, in which casethe NOT gate should not do anything yet. If the



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2

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w4,5

w1,3

w2,3

(a) input 1: w1,3 input 2: w2,3 output: w3,4=w1,3 OR w2,3

(b)

(c)

(d)

input 1: w1,3 input 2: w2,3 output: w4,5=w1,3 OR w2,3

input 1: w1,2 input 2: w3,4 output: w5,6=w1,2 AND w3,4

input 1: w1,3 input 2: w2,3 output: w4,5=w1,3 AND w2,3

Figure 3. Circuit diagrams and input/output behaviour of boolean logic gates. Output wires with arrowheads indicate that a down-stream load is assumed, which consumes signal strands as they are released. ( a2b) A two-input OR gate and a two-input AND gateusing, respectively, 1 and 4 seesaw gates, the 1-4 scheme. Circuits constructed using the 1-4 scheme are not clean, and thus wouldperform worse if threshold crosstalk and threshold inhibition were modelled. (cd) A two-input OR gate and a two-input AND gateusing two seesaw gates each, the 2-2 scheme. Circuits constructed using the 2-2 scheme are clean. All simulations were performedwith the reference concentration 1x 50 nM, and stopped at t 10 h. Here and in all other simulations, kf 2 10

6 M21 s21 andks 10

5 M21 s21.



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NOT gate releases its output too soon, and later theinput goes high, the damage is done and cannot beundone: downstream gates may have already acted onthe NOT gates output, and those gates cannot turnOFF after they have (prematurely) turned ON.

To avoid this problem, we use the dual-rail conven-tion [34,35]. We convert a circuit of AND, OR, NOT,NAND, NOR and XOR gates into an equivalent dual-rail circuit that uses only AND and OR, as illustratedin figure 4a,b for a circuit of NAND gates. In the new cir-

cuit, which will contain roughly twice as many gates,each wire z is replaced by two new wires, z0 and z1. Ifneither new wire is ON, this indicates that the logicalvalue ofz has not been computed yet; if only z0 is ON,

this indicates that the logical value of z must be OFF;while if only z1 is ON, this indicates that the logicalvalue ofzmust be ON. (If both z0 and z1 are ON, thenthe circuit is experiencing a fault.) With this represen-tation, each original AND, OR, NAND, or NOR gatecan be implemented using one AND gate and one ORgate; XOR requires four OR gates and two AND gates;and a NOT gate simply requires rerouting wires andswapping their labels. For example, the original gatez x NAND y becomes the two gates, z1 x0 OR y0

and z0 x

1

AND y1

, which can be verified by inspection.Furthermore, dual-rail logic effectively solves the problemof NOT gates because no computation will take placebefore the input signals arrive.

1(a)

(c)

(e) (f)

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t(h)

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3 4 5

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Figure 4. Compiling boolean logic circuits. (a) A sample circuit with six gates. (b) Translation into an equivalent dual-rail circuitwith 12 gates. (c,d) Translation into an equivalent seesaw gate circuit with 32 gates (1-4 scheme) and 26 gates (2-2 scheme). (e,f )Simulation results for all 32 possible input vectors in the 1-4 scheme and in the 2-2 scheme. The concentrations of all four dual-railoutput species are shown as a function of time. Delays vary with the input, depending the shortest decision path through the net-work. Simulations were run using the concentration 1x 50 nM, with ON inputs at 0.9x and OFF inputs at 0.1x. For the 1-4scheme, the simulated reaction equations were augmented to also model threshold crosstalk, which degrades the performance ofOR gatesbut the system still works.



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To demonstrate the process of making feedforwarddigital logic circuits out of seesaw gates, we translateda six NAND gate circuit ( figure 4a) to its equivalentdual-rail circuit (figure 4b), and then to its equivalentseesaw circuits using both the 1-4 scheme ( figure 4c)and the 2-2 scheme (figure 4d). Extra fan-out gateswere introduced for inputs that were used more thanonce, and a downstream load was included to pull theoutput high. The corresponding system of ODEsdescribing the networks mass action kinetics was thensimulated for all 32 possible input combinations. As

shown in figure 4e,f, in every case both outputs reachedeither a clear OFF or ON concentration level, which wasverified to be correct even with imperfect input concen-trations. Even for the unclean circuit (figure 4e)simulated with threshold crosstalk equations included,the imperfect behaviour of each logic operation wasgreatly improved by the signal restoration built intothe downstream operations, and correct digital functionwas observed. This provides concrete evidence that thedigital logic circuits compose well.

With realistic rate constants, our simulations suggestthat the timescales for seesaw circuits are the order ofan hour per layer of digital logic. This being a

bit slow, it is worth noting that there are interestingcomputations that require not too many layers. Forexample, a standard 74L85 4-bit magnitude compara-tor, with roughly 30 logic gates, requires only four

layers. We compiled this circuit to seesaw gates usingthe 2-2 scheme, and ODE simulations demonstratedcorrect behaviour ( figure 5).

5. RELAY CONTACT CIRCUITS

In his seminal Masters thesis [36], Claude Shannonestablished a systematic symbolic approach to theanalysis and design of digital circuits. A prevalent tech-nology at the time was relay contact circuits, in which

input switching signals (A, B, C, etc.) opened orclosed electrical contacts in a network, either allowingcurrent to flow through the network, or not. Like circuitsmade of AND, OR, and NOT gates, relay contact circuitscan concisely implement arbitrary boolean functions. Toillustrate the flexibility of the seesaw gate motif, we pro-vide a general method to compile relay contact circuitsdown to equivalent seesaw gate circuits. (Here, we con-sider only circuits where relays are directly controlledby external input signals. Using the output currentsignal of a relay circuit as the input switching signal toanother relay circuit is left as an exercise to the reader.)

The basic primitives for constructing relay contact

circuits are simple. The function of a relay contactswitch (figure 6a, left) is similar to that of a singleseesaw gate configured as a catalyst: current signalflows only if the switching signal is ON. However, for

(a)

A3

B3

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A[3:0]=[1,1,1,0] B[3:0]=[0,1,1,1]

[AB, A=B]=[1,0,0]

A[3:0]=[1,0,1,1] B[3:0]=[1,1,0,1]

[AB, A=B]=[0,1,0]

A[3:0]=[1,1,0,1] B[3:0]=[1,1,0,1]

[AB, A=B]=[0,0,1]

Figure 5. A 74L85 standard 4-bit magnitude comparator (four layers deep) and its seesaw circuit simulation, with 1 x 50 nM.(a) The digital logic circuit diagram. The corresponding seesaw circuit has roughly 100 seesaw gates. ( b) Seesaw circuit simulationwith selected input vector of A greater than B. (c) Seesaw circuit simulation with selected input vector of A smaller than B.

(d) Seesaw circuit simulation with selected input vector of A equal to B.



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a modular implementation that maintains the orien-tation and intensity of the current flow, we use threeseesaw gates to implement each relay contact switch(figure 6a, middle). The input current signal arriveson the wire at the left, the switching signal provides cat-alytic input to the leftmost seesaw gate, and after achain of events, the output current signal goes high ifand only if the switching signal A is ON. The thresholdshelp to clean up any leak in the current signal or switch-ing signal. (For example, when the switching signal isON but the current input is OFF, a small amount ofsignal will still be produced on the wire that connectsthe leftmost and the middle seesaw gates, due to stoi-

chiometric triggering.) The middle of the three seesawgates is a routing gate added to accommodate theintrinsic polarity of seesaw gates, so that the outputcurrent signal of one relay can be directly used as the

input current signal to another (as can be easily verifiedusing the shading of gate nodes in figure 6a). The right-most seesaw gate acts as a signal restorer that pushesthe current signal to the standard ON/OFF levels, com-pensating for input current signal decay that is aconsequence of its role as a fuel. (Consider a catalyticgate without downstream load, as in figure 1a: at equi-librium, the output signal level is strictly less than theinitial fuel level.) The simulation of this circuit showsthat only when the current signal is ON and the switch-ing signal A is ON, can the output signal reach an ONstate (figure 6a, right).

More complex regulatory logic can be implemented

by composing relay contact subcircuits. Two switches(or subcircuits) in series perform an AND operation(figure 6b). Two switches (or subcircuits) in parallelperform an OR operation (figure 6c). The only

(a)

(b) (c)

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currentoutput

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currentoutput

Figure 6. Implementation of relay circuits. (a) A simple circuit with current source (battery) and controlled device (denoted by aresistor), the corresponding seesaw gate circuit, and its simulation using 1 x 50 nM. Shaded and unshaded sides of seesaw gatesassist checking that a wire always connects different sides of two seesaw gates as required by node polarity, i.e. each wire connectsthe shaded side of one seesaw gate to the unshaded side of another. Switching signal A is provided at 1xif ON, or else 0.1xif OFF.Input current signal was provided at 10x; to verify that no output signal is produced when the current input is OFF, a 1 xsignalwas provided. (b) AND logic. (c) OR logic. (d) A more complex circuit. Overlapping trajectories (orange and light blue) wereshifted to the left by 100 s to make them visible. ( e) Switching signal fan-out, current signal fan-out and current signal fan-in.



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innovation here is that two distinct current input andoutput wires are used. Fan-out of the current input(figure 6e, middle) can provide such a signal, and

fan-in of the current output (figure 6e, right) canconsolidate the signal into a single wire, if desired.

In general, any relay contact circuit can be compileddown to seesaw gates, with each relay contactimplemented using the three gate scheme. Since thecurrent flow direction on a wire in the relay contact cir-cuit may be unknown (e.g. the wire connected by switchC in figure 6d, left), but the seesaw implementation isdirectional, in such cases two triples of seesaw gatesare needed for each relay contact. However, after thepower supply is added, only some of the directionswill be active. A more complex circuit ( figure 6d) wasautomatically compiled to seesaw gates in this

manner. Besides current signal fan-out, current signalfan-in, we also need switching signal fan-out(figure 6e, left) to produce different switching signalstrands for different seesaw gates representing the

same logical input (such as the multiple instances ofA and B in figure 6d).

6. ANALOGUE TIME-DOMAIN CIRCUITS

The behaviour of seesaw gate circuits is intrinsicallyanalogue. Following the approach of Zhang et al. [3],we construct amplifier cascades with initial quadraticgrowth and with initial exponential growth. More com-plex temporal dynamics can also be synthesized, such asa pulse generator.

The amplifier shown in figure 7a is a two-stage feed-forward cascade. Input signal w1,2 catalytically pushesstrands onto wire w2,4, which exhibits initially lineargrowth with time. Signal w2,4 also serves as a catalyst

for the release of output strand w4,5, which thereforeinitially grows quadratically with time. While theoutput remains below 0.5x, a good approximation is[w4,5] % a [w1,2]0.t

2 with a %125 h22.

(a)

(c)

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6100

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21 3

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input: 0.0001x0.1x

input: w1,2 output: w4,5

w4,5w1,2

w1,2

w1,2

w2,3

w2,3



[w4,5] a[w1,2]0t2 initially

[w2,3] [w1,2]0ebt initially

input: 0.1x1x

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00 21

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Figure 7. Analogue time-domain circuits. (a) A catalytic cascade that exhibits initially quadratic growth, with a % 125 h22. Tem-poral trajectories are shown for a series of exponentially decreasing initial input concentrations. (b) A positive feedback circuitthat exhibits initially exponential growth, with b% 17 h21. The same series of exponentially decreasing input concentrationsnow yields a series of trajectories with linearly increasing half-completion times. (c) A pulse-generating circuit. Pulse amplitudedepends on the input concentration. Here, we use a linear series of input concentrations between 0 xand 1x. All simulations use1x 50 nM.



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The amplifier shown in figure 7b is a one-stage feed-back cascade. Initially, input signal w1,2 catalyticallyreleases strands onto both the feedback wire w2,2 andthe output wire w2,3. However, signal strand w2,2 con-tains the recognition domain of gate 2 on both its leftside and its right side, so therefore once it has beenreleased, it can play the role of the input signal in cat-

alysing output of more w2,2 and w2,3 from the gate.Because strand w2,2 is catalytically active in releasingadditional copies of itself, its concentration increasesexponentially (until gate concentrations drop to lowlevels). The output strand w2,3, being released atthe same rate as the feedback strand, also grows rou-ghly exponentially. We found that for small inputs,[w2,3] % [w1,2]0.e

bt with b% 17 h21.The pulse generator shown in figure 7c illustrates a

non-amplifying temporal dynamic. The basic idea isthat the input strand w1,2 initially releases a significantamount of output w2,3, but later this strand is pulledback into gate 2 as gate 4 becomes active. Why does

this happen? Free w2,3 can stoichiometrically triggerrelease of w4,2, which is now entropically sucked intogate 4 as the catalyst w6,4 is activated and equilibriumis established. The wire/gate ratio r4 cannot exceed1/99 for wire w4,5; this same ratio must apply to wirew4,2, with the result that almost all the free signalstrand will be absorbed by gate 4. Thus, at equilibrium[g4:4,2] % 0.99x% [g4,2:2] and [w4,2] % 0.01x. So thewire/gate ratio r2 must also be low, which in turnpulls w2,3 back into gate 2, similarly pulling theoutput low. (Effectively, gate 4 behaves like a thresholdcomplex, and could be replaced by one if desired.)

These three simple examples make it clear that

seesaw circuits can support interesting dynamical beha-viours, even without using explicit threshold complexes.Further examples could be given, for example, addingthresholds to the circuit in figure 7a and extending itin series would yield a delay line, from which fan-outwires could provide signals that turn ON at specifiedtimes after the input arrives. Characterizing the fullrange of analogue behaviours that can be achieved,with and without threshold complexes, is an importantopen question.

7. DISCUSSION

This project was inspired by the remarkable successof scaffolded DNA origami [20,37,38] for program-ming the self-assembly of hundreds of DNA strandsinto a single target structure. The self-assembly ofDNA origami is extraordinarily reliable despite thatDNA sequences cannot be optimized to avoid unde-sired binding interactions and that unpurified DNAstrands are used, implying that the system is surpris-ingly robust to spurious binding, to impreciselyknown strand concentrations, and to subpopulationsof incomplete or damaged molecules. We were there-fore looking for an analogous design for DNA strand

displacement-based circuitsone that would requireminimal sequence design effort and work well evenwith unpurified strands and unreliable concentrations.Does our proposed seesaw gate motif live up to our

hopes and expectations as a DNA circuit componentsuitable for scaling up to large and complex circuits?We see some encouraging features, some concerns andsome clear challenges.

First, design of large feedforward digital circuitslooks promising. At the highest level, abstract specifica-tions for circuit function can be expressed concisely

using existing hardware description languages such asVerilog [39,40] and VHDL [41], then compiled downto a gate level netlist specifying elementary gates(AND, OR, NOT, NOR, NAND, XOR) and their con-nectivity. Thus, the sheer complexity of large-scalecircuit design can be managed by off-the-shelf tools.The next step is compiling the digital logic netlistdown to the seesaw gate circuit abstraction, using theconstructions described above for dual-rail logic. Thisis straightforward if no circuit size optimizations areattempted. To achieve the final step of designingmolecules, we must assign sequences to each gate basestrand. For this purpose, a single large set of sufficiently

distinct domain sequences would enable constructionof any circuit containing up to the given number ofseesaw gates.

Seesaw circuits make use of two kinds of sequencedomains: long recognition domains and short toeholds.In principle, multiple distinct toehold sequences couldbe used, which would reduce the spurious toehold bind-ing problem. However, the toehold length is severelyconstrained by physical factors, and for a fixed lengththere are a limited number of sufficiently distinctsequences. With the goal of a scalable architecture inmind, we thought it was simplest to confront this limit-ation early on, and we adopted the universal toehold

scheme, understanding that to mitigate the effects ofunproductive spurious binding, the 1xstandard concen-tration may have to be lowered as circuits get larger.Thus, the burden of making reactions highly specificfalls to the recognition domains.

Can we design sufficiently many recognition domainsequences to scale up to circuits with many gates? Weconsider two design criteria: (i) signal strands shouldnot have strong secondary structure and should notinteract with each other, and (ii) strand displacementshould be unable to proceed if the invading sequenceis not the desired signal strand. The first criterion canbe satisfied by standard DNA sequence design methods

[42,43]; here we take the easy approach by using a threeletter code (A, C and T) for the signal strands, thusensuring that problematic secondary structures andinteractions are unlikely [44 46]. The gate base strandwill therefore consist of (A, G, and T). Because of thecomplete independence of domains within the seesawgate motif, no system-level conflicts arise when strandsequences are generated by concatenation. The secondcriterion may be formalized as combinatorial sequenceconstraints. For example, we could require that atleast 30 per cent of bases are different for any two dis-tinct recognition domains; as each mismatch impedesbranch migration speed by a factor of roughly 10

[47,48], even five mismatches will dramatically reducecrosstalk. Additionally, we require that mismatchesare spread out, so that when the wrong signal strandinteracts with a gate, it will quickly encounter



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difficulties and dissociate; specifically, the longest run ofmatches must be less than 35 per cent of the domainlength. Finally, to reduce synthesis errors and ensurecomparable melting temperatures, we require thatthere are no more than four As or Ts in a row, nomore than three Cs or Gs in a row, and that sequenceshave between 30 and 70 per cent GC-content (cf. con-

straints 1,7,8 of [49]). Using a sphere-packingtechnique [50], we have found sets of sizes 67, 399,3097 and 8343 for recognition domains of lengths 10,15, 20 and 25, respectively, confirming the theoreticallyexpected exponential growth in available sequencespace (empirically, N% 1.5 2L/2). This is enough toconstruct some interesting circuits. The caveat is thatwhile reasonable, the criteria used here are ad hoc; weare not certain that molecules designed using these cri-teria will work consistently in the laboratory, andadditional or alternative design criteria may need tobe articulated.

Can so many distinct sequences be synthesized and

handled in the laboratory? A substantial difficultywith our previous work [6,3,18] was that each gate mol-ecule was a complex of multiple strands that had to beseparately annealed together, and each complex had tobe purified to remove excess single-stranded species andmalformed gate substrates. In the case of seesaw gates,an additional difficulty is presented by self-loop gatessuch as g2,2:2 of figure 7b, for which straightforwardannealing would yield an equal amount of the undesir-able g2:2,2; instead, a multi-step purification procedurewould be necessary. Thankfully, the simplicity of theseesaw gate motif makes laboratory procedures forsynthesizing gates and circuits plausible to carry out

on a large scale in a manner that avoids this problem.Here, we aim to simultaneously prepare all gate com-plexes together in a single test tube; to do so, we mustensure that different gate species do not interact, andthat the strands needed to form a given gate complexfind each other efficiently. For our solution, we drawinspiration from the observation [51,52] that mixturesof hairpin molecules, when annealed, are likely toform non-interacting intramolecular hairpins even if atroom temperature there exist lower free energy statesinvolving intermolecular complexes. This occursbecause the intramolecular hairpins are typicallystable at some moderately high temperature, above

the melting temperature of the intermolecular com-plexesthus, during annealing, the hairpins form firstand become kinetically trapped. The implication forgates is that if each gate species can be synthesizedinitially as a hairpin precursor, annealing all such gateprecursors in a single reaction will result in a highyield of properly formed non-interacting molecules.Figure 8 shows our realization of this scheme. Afterannealing, incubation with appropriate restrictionenzymes removes the now-undesired linker subsequence,resulting in a well-formed complex of two strands. Theentire solution could be purified by gel, since all gatesare the same size; all threshold gates could be purified

similarly. One way or another, making circuit functionrobust to sloppy parallel gate preparation methodswill be crucial to scaling up existing DNA circuits tohundreds or thousands of gates.

The hairpin gate precursor architecture also facili-tates the synthesis of the DNA strands themselves.We envision all hairpin strands being synthesized inparallel on a DNA microarray, such as those offeredby Agilent and NimbleGen that are currently capableof synthesizing approximately 106 distinct sequenceswith lengths approximately 50 nt in quantities of

approximately 10

6

molecules per spot [53]. With 5 nttoeholds and 15 nt recognition domains, our hairpinswould be up to 91 nt, while with 25 nt recognitiondomains our hairpins would be up to 121 nt, whichappears within reach using new micro array technol-ogies [54]. To estimate the circuit complexity thatcould be synthesized, consider the 2-2 scheme for digitallogic circuits. If each spot provides a 0.33x concen-tration in our chosen reaction volume (this is theminimal increment we use in the 2-2 scheme), thensynthesizing the material for an AND or OR gate(which on average require 6x) would require the use of18 spots. This corresponds to approximately 50 000

gates or 5000 gates if we want 10 times more moleculesof each species. After synthesis, linkers attachingstrands to the slide can be cleaved, and a mixture ofall strands can be collected in a single tube, annealedto form hairpins, digested with restriction enzymes toproduce gates, and then gel-purified to eliminate non-functional molecules (cf. [55]). Thus, all molecules forthe entire circuit are synthesized and processed in par-allel in a single tube.

Once designed and synthesized, will the DNAcircuits work? The first question is speed. If the maxi-mum total concentration for reliable DNA gateoperation is 100 mM (perhaps optimistic), then 1x

would be 3 nM for a 5000 logic gate circuit, and theslowest reaction half-times (the effective gate delay)would be roughly one day. If this is considered tooslow, either one must resign oneself to smaller circuits,for which the concentrations can be higher, or onemust find a way to speed up hybridization reactionsin dilute complex mixtures. For example, the phenolemulsion reassociation technique (PERT) has beenreported to speed-up hybridization dynamics by fourorders of magnitude [56,57]. If the exponential depen-dence on toehold lengths is preserved under theseconditions, this would reduce the gate delay to roughly10 seconds.

The second question is whether the computation willbe correct. For feedforward digital circuits, thresholdingand signal restoration (the digital abstraction) isexpected to provide some robustness to variations inconcentrations, to leak, and to minor crosstalk. How-ever, experimental exploration of seesaw gate circuitswill be needed to evaluate the potential for producingreliable function in practice. We anticipate that therewill be many unforeseen difficulties. In summary, wehave proposed a new catalytic DNA gate that appearsto be suitable for scaling up to analogue and digital cir-cuits incorporating thousands of gates with areasonable expectation that adequate speed and

reliability could be achieved.However, a limitation of this work is that the circuit

constructions we described all function by completelydepleting key gate and fuel species, hence each circuit



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preparation can be used only once. This prevents usfrom implementing sequential circuits containing buf-fers, flip-flops, resets and clocks that orchestrate the

re-use of circuit elements and can process time-varyinginput signals. We do not know at this point whetherthis limitation is essential to the seesaw gate motif.Because threshold complexes function by being comple-tely consumed, it appears that this question is relatedto characterizing the behaviours possible in seesaw cir-cuits without thresholds. Especially interesting is howthe equilibrium concentrations within such circuitsrespond to changes in total signal placed on input/output wires. Similarly, we do not yet have a character-ization of the class of analogue dynamics that can beachieved in the use-once setting, although it appearsto be a rich space of behaviours. In considering what

can be achieved with the seesaw motif, one might betempted to add additional molecular mechanisms inorder to obtain new behaviours; however, one mustkeep in mind that fully general digital, analogue, andalgorithmic behaviours have already been shown to bepossible using more complex multi-stranded gate com-plexes [18,58]. Unfortunately, such gates are moredifficult to prepare experimentally; the value of theseesaw motif lies in its simplicity and experimentaltractability.

It is not yet clear what applications seesaw circuitswill find in practice. However, other DNA and RNAreaction cascades based on toehold-mediated strand dis-

placement have already been demonstrated successfullyas programmable cancer drugs that work in vivo [59],and as programmable amplification mechanisms forimproving in situ fluorescence imaging of mRNA [60].

While the cellular environment may interfere withproper seesaw circuit function, it is not implausiblethat seesaw circuits could be adapted or modified to

work in that context. Already, the entropy-driventoehold-exchange catalytic mechanism used in seesawcircuits has been investigated as an amplifier for invitro biomedical diagnostics [61]. Additional input andoutput interfaces between nucleic acid circuits andother chemical species would allow nucleic acid circuitsto serve as embedded controllers for a wide range ofmolecular events. Finally, the simplicity and power ofthe basic seesaw architecture makes one wonderwhether circuits based on RNA toehold exchange pro-cesses could exist in biological systems. While it maybe a superficial similarity between seesaw gates andmiRNAs and siRNAs [62,63], all of which are short

duplex nucleic acids with single-stranded overhangsthat arise by cleavage of hairpin precursors, it isharder to imagine biological systems avoiding thanexploiting seesaw-like mechanisms.

The authors thank Dave Zhang for discussion of the catalyticmechanism, Marc Riedel for providing example netlists fromlogic synthesis benchmarks, Virgil Griffith for suggestinguseful techniques for DNA sequence design, Ho-Lin Chenand Shuki Bruck for suggesting the connection to relaycircuits, David Soloveichik for Mathematica code simulatingmass-action chemical reaction networks, and Georg Seelig,Bernard Yurke, and everyone else for discussions andsupport. This work has been supported by NSF grant nos.

0728703, and 0832824 (The Molecular ProgrammingProject) and HFSP award no. RGY0074/2006-C. Apreliminary version of this paper appeared as Qian &Winfree [64].

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Lulu Qian and Erik Winfree- A simple DNA gate motif for synthesizing large-scale circuits

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Transcript of Lulu Qian and Erik Winfree- A simple DNA gate motif for synthesizing large-scale circuits