David Soloveichik, Georg Seelig and Erik Winfree- DNA as a universal substrate for chemical kinetics

8/3/2019 David Soloveichik, Georg Seelig and Erik Winfree- DNA as a universal substrate for chemical kinetics

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DNA as a universal substrate for chemical kineticsDavid Soloveichika,1, Georg Seeliga,b,1, and Erik Winfreec,1

aDepartment of Computer Science and Engineering, University of Washington, Seattle, WA 98195; bDepartment of Electrical Engineering, University ofWashington, Seattle, WA 98195; and cDepartments of Computer Science, Computation and Neural Systems, and Bioengineering, California Institute ofTechnology, Pasadena, CA 91125

Edited by Jos N. Onuchic, University of California San Diego, La Jolla, CA, and approved January 29, 2010 (received for review August 18, 2009)

Molecular programming aims to systematically engineer molecular

and chemical systems of autonomous function and ever-increasing

complexity. A key goal is to develop embedded control circuitrywithin a chemical system to direct molecular events. Here we show

that systems of DNA molecules can be constructed that closely ap-

proximate the dynamic behavior of arbitrary systems of coupled

chemical reactions. By using strand displacement reactions as a

primitive, we construct reaction cascades with effectively unimole-

cular and bimolecular kinetics. Our construction allows individual

reactions to be coupled in arbitrary ways such that reactants can

participate in multiple reactions simultaneously, reproducing the

desired dynamical properties. Thus arbitrary systems of chemical

equations can be compiled into real chemical systems. We illustrate

our method on the LotkaVolterra oscillator, a limit-cycle oscillator,

a chaotic system, and systems implementing feedback digital logicand algorithmic behavior.

molecular programming mass-action kinetics strand displacementcascades chemical reaction networks nonlinear chemical dynamics

Chemical reaction equations and mass-action kinetics providea powerful mathematical language to describe and analyzechemical systems. For well over a century, mass-action kineticshas been used to model chemical experiments and to predictand explain their dynamical properties. Both biological and non-biological chemical systems can exhibit complex behaviors such asoscillations, memory, logic and feedback control, chaos, and pat-tern formationall of which can be explained by the correspond-

ing systems of coupled chemical reactions (1

4). Whereas the useof mass-action kinetics to describe existing chemical systems iswell established, the inverse problem of experimentally imple-menting a given set of chemical reactions has not been consideredin full generality. Here, we ask: Given a set of formal chemicalreaction equations, involving formal species X1; X2;; Xn, can

we find a set of actual molecules M1;M2;;Mm that interactin an appropriate buffer to approximate the formal systemsmass-action kinetics? If this were possible, the formalism ofchemical reaction networks (CRNs) could be treated as an effec-tive programming language for the design of complex networkbehavior (59).

Unfortunately, a formally expressed system of coupled chemi-cal equations may not have an obvious realization in knownchemistry. In a formal system of chemical reactions, a species

may participate in multiple reactions, both as a reactant and/oras a product, and these reactions progress at relative rates deter-mined by the corresponding rate constants, all of which imposesformidable constraints on the chemical properties of the speciesparticipating in the reactions. For example, it is likely hard to finda physical implementation of arbitrary chemical reaction equa-tions using small molecules, because small molecules have a lim-ited set of reactivities.

Thus, formal CRNs may appear to be an unforgiving target forgeneral implementation strategies. Indeed, most experimental

work in chemical and biological engineering has started withparticular molecular systemsgenetic regulatory networks (10),RNA folding and processing (11), metabolic pathways (12), signaltransduction pathways (13), cell-free enzyme systems (14, 15),and small molecules (16, 17)and found ways to modify or re-

wire the components to achieve particular functions. Attempts tosystematically understand what functional behaviors can be ob-tained by using such components have targeted connections toanalog and digital electronic circuits (10, 18, 19), neural networks(2022), and computing machines (15, 20, 23, 24); in each case,complex systems are theoretically constructed by composingmodular chemical subsystems that carry out key functions, suchas boolean logic gates, binary memories, or neural computingunits. Despite its apparent difficulty, we directly targeted CRNsfor three reasons. First, shoehorning the design of syntheticchemical circuits into familiar but possibly inappropriate comput-ing models may not capture the natural potential and limitationsof the chemical substrate. Second, there is a vast literature on

the theory of CRNs (25, 26) and even on general methods to im-plement arbitrary polynomial ordinary differential equations asCRNs (27, 28). Third, as a fundamental model that capturesthe essential formal structure of chemistry, implementation ofCRNs could provide a useful programming paradigm for mole-cular systems.

Here we propose a method for compiling an arbitrary CRNinto nucleic-acid-based chemistry. Given a formal specificationof coupled chemical kinetics, we systematically design DNA mo-lecules implementing an approximation of the system scaled to anappropriate temporal and concentration regime. Formal speciesare identified with certain DNA strands, whose interactions aremediated by a set of auxiliary DNA complexes. NonconservingCRNs can be implemented because the auxiliary species implic-

itly supply energy and mass.Conveniently, the base sequence of nucleic acids can deter-mine reactivity not only through direct hybridization of single-stranded species (29) but also through branch migration andstrand displacement reaction pathways (3032). These powerfulreaction primitives have been used previously for designingnucleic-acid-based molecular machines with complex behaviors,such as motors, logic gates, and amplifiers (3337). Here we usethese reaction mechanisms as the basis for the implementation ofarbitrary CRNs. Our work advances a systematic approach thataims to provide a general mechanism for implementing a well-specified class of behaviors.

Molecular Primitive: Strand Displacement CascadesBecause simple hybridization reactions cannot be cascaded, weuse the more flexible strand displacement reaction as a molec-ular primitive. [We use strand displacement as a shorthand fortoehold-mediated branch migration and strand displacement

A preliminary version of this work appeared as ref. 52.

Author contributions: D.S., G.S., and E.W. designed research, performed research,and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Freely available online through the PNAS open access option.

1To whom correspondence may be addressed. E-mail: [email protected], [email protected], or [email protected].

This article contains supporting information online at www.pnas.org/cgi/content/full/0909380107/DCSupplemental.

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reactions (33, 38), combined with the principles of toeholdsequestering (34) and toehold exchange (35).]

Intuitively, in a strand displacement reaction (Fig. 1) a strand(X) displaces another strand (Y) from a complex (G). In our dia-grams, each subsequence that acts as a single functional unit(a domain) is labeled with a unique number (e.g., the two func-tionally distinct domains of strand X are 1 and 2). We assumestrong sequence design such that domain x is complementaryto domain x and interacts with no other, allowing us to analyzeDNA systems entirely at the domain level. Short (dashed) do-mains are called toeholds. A strand displacement reaction starts

when two single-stranded toehold domains (e.g., 1 and 1) bindeach other. Then, a random walk process (branch migration) fol-lows, where two domains with identical sequences compete forthe same complementary domain. Domain 2 of strand 23 com-petes with and is partially displaced by domain 2 of strand 12.Finally, the initially bound strand 23 is released when toeholds 3and 3 separate. To obtain a fast and reliable reaction, the toeholddomains must be short (e.g., 6 nt) whereas the branch migrationdomains must be long (e.g., 20 nt). Despite the complex mecha-nism, a strand displacement reaction is well modeled by a singlereversible reaction (38, 39) under a large range of experimentalconditions where toehold binding is rate limiting (Fig. 1, Inset).Removing toehold 3 slows the reverse reaction enough that wecan consider the forward reaction effectively irreversible (38, 40).

Strand displacement reactions are programmable by the designof sequences. A single base mismatch significantly impedesbranch migration (32, 41). By varying the binding strength ofthe initiating toeholds, the reaction rate constant can be con-trolled over 6 orders of magnitude (38, 39). In turn, the lengthand sequence composition of the toeholds controls their bindingstrength. The forward and reverse rate constants q and r increase

with the hybridization energy of their respective toehold domains(1 to 1 and 3 to 3). The same strand (X) can react at differentrates to different complexes, e.g., G and G0. To attain smallerq0 < q for G0, we can use a toehold domain 1q0 that is not a fullcomplement of 1. For example, 1q0 can be a truncation of1

(39)(Fig. S1).

In the following, we construct systems of molecules whose in-teractions are mediated by strand displacement reactions. To en-

sure desired reactions and exclude undesired reactions, we usemodular components that incorporate three key design princi-ples. First, we design the system so that all long domains arealways double-stranded, thus ensuring that toeholds and theircomplements are the only single-stranded domains capable of hy-bridizing together. Second, we require that toeholds are short en-ough that this interaction is fleeting unless it triggers stranddisplacement. Thus we need to consider only strand displace-ment, and not hybridization, of long domains. Third, only the de-sired target must have the correct combination of toehold anddisplacement regions, preventing any undesired strand displace-ment reactions. Satisfying these three design principles, whichcan be verified by inspection of the modules, guarantees that ar-bitrarily complex systems will function as intended for idealstrand displacement reactions.

Implementing NetworksWe implement CRNs in DNA by using three types of modules,

which specify the DNA molecules to create for each target reac-tion (either unimolecular or bimolecular) and for each target spe-cies. Initially, we suppose that our target CRN uses rate constantsand concentrations that are realistic for aqueous-phase nucleic-acid strand displacement reactions. Additionally, the signal spe-cies must always be at much lower concentrations than theauxiliary species. Thus, the feasibility of a CRN depends upon

its behavior; for example, we can implement explosive reactionslike X1 X1 X1 for only a limited time. If the target CRN op-erates outside of a physically realistic regime, we will need to re-lax the goal of implementing the exact target system and insteadallow a uniform scaling down of concentrations and/or slowingdown of the kinetics. We will show that such scaling can also ar-bitrarily increase the accuracy of our construction.

Unimolecular Reactions

DNA Implementation. Standardizing the molecular equivalents ofthe formal species facilitates their assignment to roles as reactantsor products in multiple reactions. We implement a formal speciesas a single-stranded DNA molecule (signal strand) consisting ofan inert history domain and a unique species identifier (a longdomain flanked by two toehold domains; Fig. 2, orange boxes)

that regulates its interactions with other molecules. The historydomain depends on the formal reaction producing the species: IfXj is a product of multiple reactions, signal strands with differinghistory domains but the same species identifier would representthe same formal species Xj. In our scheme, the signal strand isactive when fully single-stranded and inactive when bound to an-other DNA molecule. Signal strands react exclusively by stranddisplacement reactions initiated at the left toehold of their spe-cies identifier; thus, sequestering the left toehold into a doublehelix is sufficient to inactivate them. The sum of the concentra-tions of the active signal strands for Xj gives Xj, and the changesin Xj are effected either by the inactivation of a signal strand

when it binds to a complementary DNA molecule or by the re-lease of a previously bound strand.

Being composed of entirely distinct domains, DNA signalstrands do not interact with each other directly, but rather aset of auxiliary DNA complexes present at large concentrationsmediates exactly the desired reactions. Suppose the ith formalreaction is unimolecular, as in Fig. 2. We implement reactionequation 1 as inactivation of a signal strand with species identifierX1 coupled with activation of strands with identifiers X2 and X3.These chemical steps are done through two strand displacementreactions involving two auxiliary complexes Gi and Ti (for gateand translater, respectively) as shown in Fig. 2. A two-step cas-cade ensures that arbitrary species identifiers can be connected inthis reaction because there is no sequence dependence betweenX1,X2, andX3. To implement other effective net reactionsevencatalytic or autocatalytic reactions such as X1 X1 X2 orX1 X1 X1only the auxiliary complexes need to be modi-

fied. A system of multiple coupled unimolecular reactions isimplemented simply by starting with the auxiliary complexesfor all the desired reactions, along with appropriate initial con-centrations of the signal strands. Implementing unimolecular re-actions with differing numbers of products requires extending orshrinking the intermediate output Oi. Removing auxiliary com-plex Ti altogether results in unimolecular decay reactions likeX1 , where indicates the absence of products.

Kinetic Analysis. To approximate ideal unimolecular mass-actionkinetics, we require the dynamics of the target reaction networkto be slow relative to the fastest strand displacement steps and theconcentration of auxiliary DNA species to be much larger thanthe signal concentrations. Here and in the next section, wepresent an informal argument that the desired kinetics will be

1* 2* 3*

2 3

1* 2* 3*

2 3

2 323

1 2

1

2

1 2

1 * 2 * 3*

1 2

1* 2* 3*

(3)(2)(1)

single-reaction model

Fig. 1. Strand displacement molecular primitive. Domains are labeled bynumbers, with * denoting WatsonCrick complementarity. Multiple elemen-tary steps are indicated: (1) binding of toeholds 1 and 1; (2) branch migra-tion, a random walk process where domain 2 of strand 23 is partially

displaced by domain 2 of strand 12; (3) the separation of toeholds 3 and3. (Inset) The single-reaction model of strand displacement.

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attained under these conditions. In SI Text we prove the conver-gence to the desired kinetics in the limit of high concentration ofappropriate auxiliary species relative to the concentrations of thesignal species.

For simplicity, we assume all full toehold domains have equal

binding strength yielding a maximum strand displacement rateconstant qmax. Let Cmax be the starting concentration of auxiliarycomplexes Gi and Ti. The strand displacement cascade of Fig. 2follows reaction equations 2 and 3, where qi qmax is a partial-toehold strand displacement rate constant controlled by the bind-ing energy of domains 1qi and 1, and 1

qi

is chosen to obtain thedesired rate constant for formal reaction i. Letting be the ex-periments duration, we assume a regime where ki maxX1 Cmax, and ki qmaxCmax, and set the partial-toehold strand dis-placement rate constant qi kiCmax. Then over a sufficientlyshort duration , Gi and Ti remain effectively constant at theirinitial concentration Cmax, and the cascade becomes equivalent toa pair of unimolecular reactions (Fig. 2, Eqs. 4 and 5, whereqiCmax ki). Further, reaction 4 is the rate-limiting step, andwe obtain the desired unimolecular kinetics for formal reaction 1:

The instantaneous rate of the consumption ofX1 and the produc-tion ofX2 and X3 in this module is kiX1.

Adding Bimolecular Reactions

DNA Implementation. We now extend the construction to realizebimolecular in addition to unimolecular reactions. Suppose theith formal reaction is bimolecular, as in Fig. 3. Implementing re-action equation 6, X1 must not be consumed in the absence ofX2and vice versa, which seems difficult because X1 and X2 cannotreact directly, and therefore one of them (say, X1) must react withan auxiliary complex first without knowing whether the other ispresent. The challenge here is to minimize the X2-independentdrain on X1; later we will show how we can exactly compensatefor it by rescaling rate constants.

Our construction (Fig. 3) solves this problem by introducing areversible first step. Strand X1 reversibly displaces Bi (the back-ward strand) from auxiliary complexLi, producing activated in-termediate Hi. If no X2 is present, Bi can reverse the reaction andrelease X1 back into solution. But ifX2 is present, it can react

with intermediate Hi to release intermediate output Oi. As inthe unimolecular module, this in turn releases the final outputX3 and makes the overall reaction irreversible. The species iden-tifiers of the reactants and all products can be arbitrarily specifiedas in the unimolecular modules. Not all Bi are necessarily uniqueDNA species: If reactions i and i0 have the same reactants, thenBi

would have the same sequence as Bi0 .

Kinetic Analysis. The kinetics of the module of Fig. 3 is modeledusing reaction equations 79. Set the initial concentration ofLi,

Bi, and Ti to Cmax and partial-toehold rate constant qi ki.By assuming ki maxX1 maxX2 Cmax and maxX1 Cmax, the concentrations Li, Bi, and Ti remain effectively con-stant throughout the duration of the experiment, and reaction 8 isthe rate-limiting step of the pair 89. This system of reactions can

then be approximated by reaction equations10

and11

. Further,(informally) if in the target reaction network X1 varies on aslower time scale than that of the equilibration of reaction 10,

we can assume that X1 and Hi attain instant equilibrium throughreaction 10 with X1Hi qmaxki. Then the instantaneous rateof reaction 11 is qmaxX2Hi kiX1X2. At first glance this ap-pears to give the correct kinetics for the production ofX3; how-ever, X3 is buffered: Some fraction of it will quickly equilibrate

with Hi0 of other reactions in which X3 is the first reactant, result-ing in a lower effective rate of production ofX3. X1 and X2 arelikewise buffered. Let frX1 be the set of bimolecular reactionsin which X1 is the first reactant. Reactions 10 in the different re-action modules create instant equilibrium between all species inthe equilibrium set ofX1, esX1, consisting ofX1 and everyHi,i frX1. Producing or consuming a unit of any species in the

equilibrium set of X1 results in the fraction 1 X1esX1of a unit change in X1. Thus the effective instantaneous ratesof the consumption of X1, consumption of X2, and produc-tion of X3 because of reaction 11 in the above module are1kiX1X2, 2kiX1X2, and 3kiX1X2, respectively.

Making the additional assumptionki qmax shifts equilibria ofreactions 10 to the left such that j approaches 1. Then the in-stantaneous rates of variation in X1, X2, and X3 because of thisreaction module approach the desired bimolecular kinetics valueofkiX1X2.

Canceling out the Buffering Effect. We can exactly cancel out thebuffering effect, thereby easing the requirement ki qmax for bi-molecular reactions and allowing the implementation of much

larger rate constants. Intuitively, because the buffering effect sys-tematically counteracts desired concentration changes ofXj whenj < 1, we should be able to compensate by increasing the rateconstants of all reactions involving Xj. Because a reaction be-tween species with different j will behave as if it has incorrectstoichiometry, we want all j to be equal. However, even withequal j, we cannot simply multiply all ki by

1j because that

would in turn change j.Let j ifrXjki be the sum of formal rate constants of

bimolecular reactions with Xj as the first reactant, and let maxjfjg. We ensure that the fraction j is equal for all Xjby using a new buffering module (Fig. 4) for each Xj for whichj < . Each buffering module is modeled as reaction equation12. InitiallyLSj BSj Cmax, and as for the bimolecular mod-ule we can reduce 12 to 13. The equilibrium set of Xj now

A

B

? 1 2 3

98

7

7*

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11*

65

410

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3 410 11 7

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2 3

2

104

117

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?1 2 3

7*11*4*10*3*

23 410 11 7

speciesidentifier

10 4 5 6

speciesidentifier

11 7 8 9

speciesidentifier

3 410 11 72qi

qiX1Gi

Oi

OiTi

X2 X3

reaction

implement

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waste

waste

1*qi

[1]

[2]

[3]

[4]

[5]

Fig. 2. Unimolecular module: DNA implementation of the formal unimolecular reactionX1 X2 X3 with reaction index i. Orange boxes highlight the DNAspecies that correspond to the formal species X1 (species identifier 1-2-3), X2 (4-5-6), and X3 (7-8-9). Domains identical or complementary to species identifiersfor X1, X2, and X3 are colored red, green, and blue, respectively. Black domains (10 and 11) are unique to this formal reaction. To reduce rate constant qi,toehold domain 1qi may not be a full complement of domain 1. (A) Complex Gi undergoes a strand displacement reaction with strand X1, with X1 displacingstrand Oi. (B) Oi displacesX2 and X3 from complex Ti. Without buffer cancellation, qi kiC

1max; with buffer cancellation,qi 1kiC1max. Reaction equations of

type 23 are used in simulations (Figs. 5 and 6); simplified reaction equations 45 are useful for analysis.

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includes HSj. It is not hard to show that with the buffering-scalingfactor 1 qmaxqmax

1, settingqsj 1 j, and repla-

cing ki byk0i

1ki for the formal rate constants when setting qiin reactions 2 and 7, we obtain a common buffer fraction j ,

which is exactly compensated for by the 1

scaling of formal rateconstants. For initial concentrations cj of formal species Xj, westart with initial concentrations 1cj of DNA strands Xj. Thenall species in esXj quickly equilibrate yielding Xj cj.

See SI Text for a summary of our algorithm for compiling anarbitrary CRN into DNA-based chemistry.

Rescaling for Feasibility and AccuracyThe above procedure will not yield a functional DNA system ifthe original formal CRN has infeasibly high reaction rates or con-centrations. In that case, we scale the system to use lower rateconstants and concentrations while maintaining the same, albeitscaled, behavior. If Xjt are solutions to differential equationsarising from a set of unimolecular and bimolecular reactions,then Xjt are solutions to the same set of reactions but

in which we multiply all unimolecular rate constants by 1and all bimolecular rate constants by1 . We introduce amixed concentration-time-scaling parameter with and 1, which scales down the concentration and slows downthe dynamics by a factor of without increasing the largest rateconstant.

We justified the accuracy of our construction by assuming thetarget system operates in a regime with concentrations suffi-ciently smaller than Cmax, a physically determined parameter.This may not hold without rescaling, but thankfully, arbitrarilyhigh accuracy for arbitrarily large duration of interest can stillbe attained in a regime of smaller concentrations of formal spe-cies and slowed down dynamics. We prove the convergence of theDNA-based kinetics with buffer cancellation to the target CRN in

the limit Cmax

by using singular perturbation theory (27, 42)(see SI Text). Whereas taking the limit Cmax is more math-ematically convenient, increasing Cmax by a factor of is equiva-lent, up to scaling in concentration of the experimentalimplementation, to decreasing all Xj by a factor of, decreasing

all formal unimolecular rate constants by a factor of , and in-creasing by a factor of to simulate the same behavior.

Examples

We illustrate our method on the Lotka

Volterra chemical oscil-lator shown in Fig. 5A. The concentration oscillations are in therange of about 02. Under typical nucleic-acid experimental con-ditions, maximal second-order rate constants for strand displace-ment reactions are about 106Ms, and maximum concentrationsare on the order of105 M. To fit into this regime with reasonablesimulation accuracy and time span, we scale the original system bya time-scaling factor of 300, and a concentration-scaling fac-tor 108 employing units of seconds and molar, and useCmax 10

5 M and qmax 106Ms. The DNA species for

our implementation, including buffering modules, are shown inFig. S2 and the possible strand displacement reactions inFig. S3. The equations governing our DNA implementation asderived by using the transforms described in Figs. 24 are shownin Fig. 5B. Simulations shown in Fig. 5C confirm that the DNA

implementation nicely approximates the ideal formal chemicalsystem. Because Cmax < , deviations between the DNA imple-mentation and the target system gradually develop, as the deple-tion of complexes L1, G1, and G2 and the buildup of strands B1alter the effective rate constants.

We next apply our construction to more complex systems(Fig. 6). For fastest behavior, all systems are scaled so that thelargest qi or qsj is qmax 10

6Ms, and Cmax 105 M, leaving

only one free scaling parameter , which determines both imple-mentation accuracy and the speed of the dynamics. The Orego-nator limit-cycle oscillator (Fig. 6A) is a simplified model of theBelousovZhabotinsky reaction (43). The DNA implementationof this system is relatively slow because of the wide range of rateconstants. The chaotic system due to Willamowski and Rssler

(44) exhibits complex concentration fluctuations and is particu-larly sensitive to perturbations at long time scales (Fig. 6B).The DNA implementation follows the ideal system relatively wellfor a few revolutions around the strange attractor. Fig. 6Cdemonstrates the efficacy of our construction for implementing

waste

waste

+

+

+

+

+

+

??

1 2 3

speciesidentifier

? 4 5 6

2 4 2 435 6

6* 6* 12*7*12* 7*

speciesidentifier

12 7 8 912 7 126 78

9 speciesidentifier

A

B

C

5

5

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3*

1

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127

2 5 6

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?1 2 54 6

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3

3* 5*4*2* 6*3*

127

?

5 6 12 7

qi

X1 Li Hi

HiOi

Oi Ti

Bi

X2

X3

reaction

1*qi1*qi

1*qi 1*qi

implement

simplify

[6]

[7]

[8]

[9]

[10]

[11]

Fig. 3. Bimolecular module: DNA implementation of the formal bimolecular reaction X1 X2 X3 with reaction index i. The black domain (12) is unique tothis formal reaction. (A) X1 reversibly displaces Bi from complex Li producing complex Hi. (B) X2 displaces Oi from complex Hi. Occurrence of reaction B pre-cludes the backward reaction of A. (C) Oi displaces X3 from complex Ti. Without buffer cancellation, qi ki; with buffer cancellation, qi

1ki. Reactionequations of type 79 are used in simulations (Figs. 5 and 6); simplified reaction equations 1011 are useful for analysis.

?? qsj4 5 6

speciesidentifier

5 13 5 136

13*5*

6

6*

4 5

13*5*

6

6*

qs2

X2 LS2 HS2BS2

4*qs24*qs2

simplify[12] [13]

Fig. 4. Bufferingmodule:DNA implementation of thebuffering moduleusedto cancelout thebuffering effect. A bufferingmodule is neededfor each formalspecies Xj for which j < . The buffering module for speciesX2 is shown. X2 reversibly displaces BS2 from complex LS2 to produce complex HS2 similarly to thefirst reaction of the bimolecular module. The black domain (13) is unique to this buffering module for species X2. Set qsj 1 j. Reaction equations 12are used in simulations (Figs. 5 and 6); simplified reaction equations 13 are useful for analysis.

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a chemical logic circuit responding to an external input signal(black trace). The 2-bit counter is a classic example of a digitalcircuit with feedback (45); the high or low values of the red andgreen output species give the binary count of the number of inputpulses, 03. This feedback circuit contrasts with the use-once cir-cuits of ref. 34. Fig. 6D shows a different style of algorithmic be-havior: a state machine (5, 6). This state machine increments thenumber of green spikes between consecutive red spikes by 1every time.

Experimental ConsiderationsThe correctness of our systematic construction was predicated onseveral idealizations of DNA behavior, and it is worth consideringthe deviations that we would expect in practice. A good approxi-mation to strong sequence design (domain x binds exclusivelyto x) should be possible for several thousand long domains byusing existing techniques developed for strand displacement sys-tems (46). There are a limited number of short toehold domainsequences available, but it is straightforward to modify our con-struction to reuse toehold sequences without introducing errors.

This limit also constrains choices for reaction rate constants butcan be countered by adjustment of auxiliary complex concentra-tions. More serious issues are presented by leak reactions in

which an output is produced even if no input is present. Althoughexperimentally characterized strand displacement systems exhibitleak rate constants up to a million times slower than the fastestdesired reactions (35, 39), a leak could pose a problem for sometarget CRNs. One way to ameliorate a leak, while also allowingfor unbounded running times, would be to provide auxiliary com-plexes at low concentrations in a continuous-flow stirred-tank re-actor (1). These issues are discussed further in SI Text.

ConclusionsWith a rich history and extensive theoretical and software tools,formal CRNs are a powerful descriptive language for modelingchemical reaction kinetics. By providing a systematic methodfor compiling formal CRNs into DNA molecules, our worksuggests that CRNs can also be regarded as an effective program-ming language and used prescriptively for the synthesis of uniquemolecular systems. This view is bolstered by the fact that CRNs

1:

12:

3:

2

3

buffering module

time (hrs)

concentration(nM)

unscaled1.5

1

1

5.105 /M/s

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1/300 /s

scaled

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

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20A B CIdeal chemical reactions DNA reaction modules Simulation of ideal and DNA reactions

ideal DNA

Fig. 5. LotkaVolterra chemical oscillator example. (A) The formal chemical reaction system to be implemented with original (unscaled) and scaled rate con-stants. Desired initial concentrations of X1 and X2 are 2 and 1 unscaled and 20 and 10 nM scaled. (B) Reactions modeling our DNA implementation. Each formalreaction corresponds to a set of DNA reactions as indicated. Species X2 requires a buffering module because 2 < ( 1 k1 and 2 0). Maximum stranddisplacement rate constant qmax 106 M1 s1 and initial concentration of auxiliary species Gi, Ti, Li, Bi, LSj, and BSj is Cmax 10 M. Buffering-scaling factor1 qmaxqmax

1 2. The initial concentrations of strandsX1 and X2 introduced into the system is 120 nM 40 nM and 110 nM 20 nM. (C) Plot ofthe concentrations of X1 (Red curve) and X2 (Green curve) for the ideal system (Dashed line) and the corresponding DNA species (Solid line).

00

20 40 60 80 100 hr

nM

nMOregonator (limit cycle oscillator) Rssler (chaotic)

Incrementer state machine (algorithmic)2-bit pulse counter(digital circuit)

nM

nM

hr

30

60

0

30

60

x

x(0)+y(1) onw

onw+w(0) onw+w(1)

y logic

where

where thresholding catalyzed by clk1

anytime

z

w(0)+w(0) offz

w(1)+w(1) onz

onz+z(0) onz+z(1)

offz+z(1) offz+z(0)

x(1)+w(1) x(1)+w(0)

y(0)+w(1) y(0)+w(0)

hrhr

10

20

0 20 40 60 80 100

10

20

0

0

0 50 100 150 200 250

1

2

3

4

5

dual railrepresentation:

species valuex(0) x(1)

high low 0

low high 1

x

v1 v2 + v3

c1 c2 + c3

v2+ c2

w2 w1

r1 r2+ r3

r2 + v2 d + v2

d + v1

c2 + r2 c2+ i + w1

i + w1 i + v1+ w2

v2

c2

i

v3 v1

c3 c1

r3 r1

v > 0?

v:=v-1

yesno

w:=w+1v:=w

v:=0w:=0

catalyzed by clk2 catalyzed by clk3

clk1

clock madefrom chemical

oscillator:

clk2clk3

A B

C D

0 10 20 30 40

1

2

3

4

5

Fig. 6. Examples showing more complex behavior. In all the maximum strand displacement rate constant qmax 106 M1 s1 and the initial concentration ofauxiliary species Cmax 10 M. See Figs. S4 and S5 and SI Textfor the rate constants used in AD, as well as the details of Cand D. Plots show the ideal CRN(Dashed lines) and the DNA reactions (Solid lines).

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can implement arbitrary ordinary differential equations, analogand digital circuits, Turing-universal behavior, andin the limitof small signal species concentrations and finite reaction volumestochastic CRNs (47, 48). It is perhaps surprising that a simpleprimitive such as sequence-specific strand displacement provessufficient for the implementation of arbitrarily complex chemicalreaction kinetics, somewhat akin to how a few basic electronicelements such as transistors and wires are sufficient for the con-struction of arbitrarily complex logic circuits.

Our construction has several notable features for future mo-lecular programming efforts. First, the signal species, entirely un-reactive by themselves, react only when immersed in a complexbuffer of auxiliary complexes. Thus, when implementing a system,one focusses on designing the buffer rather than the signalspecies, which stay the same regardless of the system. Our con-struction also highlights the rich dynamical behavior possible inmultistranded nucleic-acid systems even without enzymes andcovalent bond modification. Furthermore, established techniquesfor interfacing DNA to other chemistries (e.g., ref. 49) can enableprogrammable DNA reaction networks to respond to or controlnon-nucleic-acid systems, such as complex chemical synthesispathways, biomedical diagnostics in vitro, or cellular functionsin vivo. Finally, our construction and variants thereof (48) relyexclusively on a very simple molecular primitive, sequence-

specific strand displacement reactions, suggesting that it couldbe adapted for other molecular substrates, such as RNA (11)or even peptides (17).

Many important chemical behaviors, such as pattern formationand self-assembly, involve geometric aspects that are beyond thescope of well-stirred reactions considered here. However, with aslight extension of our approachcontrolling the relative diffu-sion rates of signal speciesit would be possible to implementarbitrary reaction-diffusion systems (1). Though more challen-

ging, regulating molecular assembly and disassembly steps withDNA signal species would allow for qualitatively more powerfulcomputational systems (50) and systems capable of movementand construction of molecular-scale objects. We hope that themethods presented here provide a framework for successfullygeneralizing and unifying the existing experimental achievementsin this direction (36, 37, 51).

ACKNOWLEDGMENTS. We thank L. Cardelli, D. Dotty, L. Qian, D. Zhang,J. Schaeffer, and M. Magnasco for useful discussions. Thiswork was supportedby National Science Foundation Grants EMT-0728703 and CCF-0832824 andHuman Frontier Science Program Award RGY0074/2006-C.D.S. D.S. was sup-ported by the CIFellows project. G.S. was supported by the Swiss NationalScience Foundation and a Burroughs Wellcome Fund CASI award.

1. Epstein IR, Pojman JA (1998) An Introduction to Nonlinear Chemical Dynamics: Oscilla-tions, Waves, Patterns, and Chaos (Oxford Univ Press, London).

2. Arkin A, Ross J (1994) Computational functions in biochemical reaction networks. Bio-phys J 67:560578.

3. Tyson J, Chen K, Novak B (2003) Sniffers, buzzers, toggles and blinkers: Dynamics ofregulatory and signaling pathways in the cell. Curr Opin Cell Biol 15:221231.

4. Wolf D, Arkin A (2003) Motifs, modules and games in bacteria. Curr Opin Microbiol6:125134.

5. Liekens AML, Fernando CT (2007) Turing Complete Catalytic Particle Computers,Lecture Notes in Computer Science (Springer, Berlin), Vol 4648, pp 12021211.

6. Soloveichik D, Cook M, Winfree E, Bruck J (2008) Computation with finite stochasticchemical reaction networks. Nat Comp 7:615633.

7. Cook M, Soloveichik D, Winfree E, Bruck J (2009) Algorithmic Bioprocesses (Springer,Berlin), pp 543584.

8. Fett B, Bruck J, Riedel M (2007) Synthesizing Stochasticity in Biochemical Systems (As-sociation for Computing Machinery, New York), pp 640645.

9. Cardelli L (2008) From processes to ODEs by chemistry. TCS 2008, IFIP International

Federation for Information Processing, (Springer, Boston), Vol 273, pp 261

281.10. Andrianantoandro E, Basu S, Karig DK, Weiss R (2006) Synthetic biology: New engi-neering rules for an emerging discipline. Mol Syst Biol 2:2006.0028.

11. Win M, Liang J, Smolke C (2009) Frameworks for programming biological functionthrough RNA parts and devices. Chem Biol 16:298310.

12. Keasling J (2008) Synthetic biology for synthetic chemistry. ACS Chem Biol 3:6476.13. Levskaya A, et al. (2005) Synthetic biology: Engineering Escherichia coli to see light.

Nature 438:441442.14. Noireaux V,Bar-ZivR, LibchaberA (2003) Principles of cell-freegenetic circuitassembly.

Proc Natl Acad Sci USA 100:1267212677.15. Ran T, Kaplan S, Shapiro E (2009) Molecular implementation of simple logic programs.

Nat Nanotechnol 4:642648.16. Benner S, Sismour A (2005) Synthetic biology. Nat Rev Genet 6:533543.17. AshkenasyG, Ghadiri M (2004) Boolean logic functionsof a syntheticpeptide network.

J Am Chem Soc126:1114011141.18. Magnasco MO (1997) Chemical kinetics is Turing universal.Phys RevLett78:11901193.19. Simpson Z, Tsai T, Nguyen N, Chen X, Ellington A (2009) Modelling amorphous com-

putations with transcription networks. J R Soc Interface 6:S523S533.20. Hjelmfelt A, Weinberger ED, Ross J (1991) Chemical implementation of neural net-

works and Turing machines. Proc Natl Acad Sci USA 88:1098310987.21. Buchler NE, Gerland U, Hwa T (2003) On schemes of combinatorial transcription logic.

Proc Natl Acad Sci USA 100:51365141.22. Kim J, Hopfield JJ, Winfree E (2004) Neural Network Computation by in Vitro Tran-

scriptional Circuits.NeuralInformationProcessing Systems, edSaul LK(MIT Press,Cam-bridge, MA), Vol 17, pp 681688.

23. Okamoto M, Sakai T, Hayashi K (1987)Switching mechanismof a cyclicenzyme system:Role as a chemical diode. Biosystems 21:111.

24. Hjelmfelt A, Weinberger ED, Ross J (1992) Chemical implementation of finite-statemachines. Proc Natl Acad Sci USA 89:383387.

25. Horn F, JacksonR (1972) General mass action kinetics.ArchRation Mech An 47:81116.26. rdi P, Tth J (1989) Mathematical Models of Chemical Reactions: Theory and Applica-

tions of Deterministic and Stochastic Models (Manchester Univ Press, Manchester).27. Klonowski W (1983) Simplifying principles for chemical and enzyme reaction kinetics.

Biophys Chem 18:7387.

28. Korzuhin M (1967) Oscillatory Processes in Biological and Chemical Systems (Nauka,Moscow), (in Russian), pp 231251.

29. Morrison L, Stols L (1993) Sensitive fluorescence-based thermodynamic and kineticmeasurements of DNA hybridization in solution. Biochemistry32:30953104.

30. Green C, Tibbetts C (1981) Reassociation rate limited displacement of DNA strands bybranch migration. Nucleic Acids Res 9:19051918.

31. Weinstock PH, Wetmur JG (1990) Branch capture reactions: Effect of recipient struc-ture. Nucleic Acids Res 18:42074213.

32. Panyutin I, Hsieh P (1993) Formation of a single base mismatch impedes spontaneousDNA branch migration. J Mol Biol 230:413424.

33. Yurke B, Turberfield AJ, Mills AP, Jr, Simmel FC, Neumann J (2000) A DNA-fuelled mo-lecular machine made of DNA. Nature 406:605608.

34. Seelig G, Soloveichik D, Zhang DY, Winfree E (2006) Enzyme-free nucleic acid logiccircuits. Science 314:15851588.

35. Zhang DY, Turberfield AJ, Yurke B, Winfree E (2007) Engineering entropy-driven re-actions and networks catalyzed by DNA. Science 318:11211125.

36. YinP, ChoiH, Calvert C, PierceN (2008) Programmingbiomolecular self-assembly path-

ways. Nature 451:318322.37. Bath J, Turberfield A (2007) DNA nanomachines. Nat Nanotechnol 2:275284.38. Yurke B, Mills AP (2003) Using DNA to power nanostructures. Genet Program Evol M

4:111122.39. Zhang DY, Winfree E (2009) Control of DNA strand displacement kinetics usingtoehold

exchange. J Am Chem Soc 131:1730317314.40. Reynaldo LP, Vologodskii AV, Neri BP, Lyamichev VI (2000) The kinetics of oli gonucleo-

tide replacements. J Mol Biol 297:511520.41. Panyutin I, Hsieh P (1994) The kinetics of spontaneous DNA branch migration. Proc

Natl Acad Sci USA 91:20212025.42. Khalil HK (1996) Nonlinear Systems (Prentice Hall, Englewood Cliffs, NJ).43. Field RJ, Noyes RM (1974) Oscillations in chemical systems. IV. Limit cycle behavior in a

model of a real chemical reaction. J Chem Phys 60:18771884.44. WillamowskiKD, Rssler OE (1980) Irregular oscillationsin a realistic abstractquadratic

mass action system. Z Naturforsch A 35:317318.45. Marcovitz A (2004) Introduction to Logic Design (McGrawHill, New York), 2nd Ed, pp

369441.46. Qian L, Winfree E (2009) A simple DNA gate motif for synthesizing large-scale circuits.

DNA Computing 14, LectureNotes in ComputerScience (Springer, Berlin), Vol 5347, pp7089.

47. Gillespie D, et al. (1977) Exact stochastic simulation of coupled chemical reactions.J Phys Chem 81:23402361.

48. Cardelli L (2009) Strand algebras for DNA computing. DNA Computing 15, LectureNotes in Computer Science (Springer, Berlin), Vol 5877, pp 1224.

49. Gartner Z, et al. (2004) DNA-templated organic synthesis and selection of a library ofmacrocycles. Science 305:16011605.

50. Bennett C (1982) The thermodynamics of computationA review. Int J Theor Phys21:905940.

51. Rothemund P, Papadakis N, Winfree E (2004) Algorithmic self-assembly of DNAsierpinski triangles. PLoS Biol 2:e424.

52. Soloveichik D, Seelig G, Winfree E (2009) DNA as a Universal Substrate for ChemicalKinetics (Extended Abstract). DNA Computing 14, Lecture Notes in Computer Science(Springer, Berlin), Vol 5347, pp 5769.


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David Soloveichik, Georg Seelig and Erik Winfree- DNA as a universal substrate for chemical kinetics