design of autonomous dna cellular automata · 2015-10-28 · tion, dna robotics, and dna computing...

Design of Autonomous DNA Cellular Automata�

Peng Yin�

Sudheer Sahu�

Andrew Turberfield�

John H. Reif�

Abstract

Recent experimental progress in DNA lattice construc-tion, DNA robotics, and DNA computing provide thebasis for designing DNA cellular computing devices,i.e. autonomous nano-mechanical DNA computing de-vices embedded in DNA lattices. Once assembled, DNAcellular computing devices can serve as reusable, com-pact computing devices that perform (universal) compu-tation, and programmable robotics devices that demon-strate complex motion. As a prototype of such devices,we recently reported the design of an Autonomous DNATuring Machine, which is capable of universal sequentialcomputation, and universal translational motion, i.e. themotion of the head of a single tape universal mechanicalTuring machine.

In this paper, we describe the design of an Au-tonomous DNA Cellular Automaton (ADCA), which canperform parallel universal computation by mimicking aone-dimensional (1D) universal cellular automaton. Inthe computation process, this device, embedded in a 1DDNA lattice, also demonstrates well coordinated parallelmotion. The key technical innovation here is a molec-ular mechanism that synchronizes pipelined “molecularreaction waves” along a 1D track, and in doing so, re-alizes parallel computation. We first describe the designof ADCA on an abstract level, and then present detailedDNA sequence level implementation using commerciallyavailable protein enzymes. We also discuss how to ex-tend the 1D design to 2D.

1 IntroductionDNA has recently been used, with great success,to fabricate nanoscale lattices [5, 9, 10, 13, 15,�The work is supported by NSF ITR Grants EIA-0086015

and CCR-0326157, NSF QuBIC Grants EIA-0218376 and EIA-0218359, and DARPA/AFSOR Contract F30602-01-2-0561.�

Department of Computer Science, Duke Uni-versity, Durham, NC, USA. � py, sudheer,reif � @cs.duke.edu�

University of Oxford, Department of Physics,Clarendon Laboratory, Parks Road, Oxford OX 1 3PU,[email protected]

20, 22, 23], to construct nanomechanical devices[2, 6, 7, 16, 17, 18, 24, 25, 27, 28], and to per-form computation [1, 4, 3, 11, 12, 14, 19]. Theprogress in these three fields together provides thebasis for the next step forward: designing and con-structing autonomous DNA computing devices em-bedded in well defined DNA lattices, which are ca-pable of (universal) computation. We call such de-vices DNA cellular computing devices.

Once assembled, DNA cellular computing de-vices can serve as reusable, compact computing de-vices that perform (universal) computation, and pro-grammable robotics devices that demonstrate com-plex motion. First, DNA cellular computing de-vices represent an exciting step beyond the priormolecular computing schemes, such as algorithmicself-assembly of DNA tiles, which performs a one-time computation during the assembly. In contrast,DNA cellular computing devices, once assembled,can perform multi-round computations. The out-put, embedded in the DNA lattices, is preserved andcan serve as input for future computations. In ad-dition, DNA cellular computing devices are morecompact than the tiling scheme. A 1D AutonomousDNA Turing machine holds equivalent computa-tional power as a 2D tiling lattice. Second, DNAcellular computing devices can demonstrate sophis-ticated programmable motion, for example, univer-sal translational motion, which we define as the mo-tion of the head of a single tape universal mechan-ical Turing machine. As such, DNA cellular com-puting devices may promise interesting applicationsin nanorobotics and nano-computation, as well as,nano-fabrication, nano-sensing, and nano-actuatedelectronics.

As one prototype of the DNA cellular comput-ing devices, we previously reported the design of anautonomous unidirectional DNA mechanical devicecapable of universal sequential computation, termed

as Autonomous DNA Turing Machine [26]. Here,we extend our previous work and obtain the de-sign of an Autonomous DNA Cellular Automaton(ADCA). By mimicking a 1D universal cellular au-tomaton, the ADCA can perform parallel universalcomputation, and in the process, demonstrate wellcoordinated parallel motion. The parallel computa-tion and motion is realized using a novel molecularmechanism that synchronizes pipelined “molecularreaction waves” along a 1D track.

A cellular automaton is a set of “colored” cellson a grid of specified shape that evolves throughdiscrete time steps according to a set of transitionrules based on the colors of neighboring cells [21].If the lattice is a one (resp. two) dimensional lat-tice, the cellular automaton is called a one (resp.two) dimensional cellular automaton. Figure 1 (a)shows the cells of an example one-dimensional cel-lular automaton. Each cell of this automaton canhave one of two states, or equivalently two colors,��

and �� . The evolving process of a cel-lular automaton is specified by the transition rules.Figure 1 (b) illustrates one example rule set for thecellular automaton shown in Figure 1 (a). The ruleset consists of 8 transition rules (numbered (1) -(8) in the figure). For example, according to rule(1), if the current cell and both of its neighborshave color �� , at next time step, the middle cellwill change to color

�� . This rule is denoted

as �� . Applying therules in Figure 1 (b) to the initial configuration inFigure 1 (a), we obtain the evolving table depicted inFigure 1 (c). Cellular automaton can hold universalcomputing power. The cellular automaton depictedin Figure 1 is one such universal cellular automa-ton, known as rule 110, as described in [21]. Thiscellular automaton can be simulated by the ADCAdescribed in this paper.

The rest of the paper is organized as follows. InSect. 2, we describe the structural design and opera-tional process of our ADCA. In Sect. 3, we describea detailed molecular implementation of the designpresented in Sect. 2. We then briefly describe howto extend the design of ADCA to two-dimensions inSect. 4 and close in Sect. 5.

(2)(1) (4)(3) (6)(5) (8)(7)

Figure 1: A universal cellular automaton with two colors:Rule 110. The figure is adapted from Wolfram [21]

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Figure 2: Top panel: the initial configuration of anabstract cellular automaton. Bottom Panel: schematicdrawing of the structure of an ADCA corresponding tothe abstract cellular automaton in the top panel. Thebackbones of DNA strands are depicted as directed linesegments. The short bars represent base pairing betweenDNA strands

Figure 3: Three endonucleases used in the molecular im-plementation of the ADCA. The recognition site of an en-zyme is bounded by a box and the cleavage site indicatedwith a pair of bold arrows. The symbol “ a ” indicatesthe position of a base that does not affect endonucleaserecognition

2 Design2.1 StructureThe ADCA operates in a solution system. Figure 2illustrates an example abstract cellular automaton inthe top panel, and the structure of the corresponding

ADCA in the bottom panel. The ADCA is com-posed of four parts: a rigid symbol track, a linear ar-ray of dangling DNA molecules tethered to the sym-bol track, a set of floating DNA molecules, and agroup of floating protein enzymes.b Symbol track. The symbol track pro-

vides a rigid structural platform on which thedangling-molecules are tethered. It can be im-plemented, for example, as a rigid addressableDNA lattice, such as the barcode DNA latticereported in [22].b Dangling DNA molecules. The array

of dangling-molecules, also called symbol-molecules, tethered to the symbol track rep-resent the array of cells (symbols) in the cel-lular automaton (and hence the name symbol-molecule). A dangling-molecule is a duplexDNA fragment, with one end tethered to thesymbol track via a flexible single strand DNAfragment and the other end possessing a singlestrand DNA extension (the sticky end). Due tothe flexibility of the single strand DNA link-age, a dangling-molecule moves rather freelyaround its joint on the symbol track. We re-quire that the only possible interactions be-tween two dangling-molecules are those be-tween two immediate neighbors. This require-ment can be ensured by properly spacing thedangling-molecules along the rigid track.b Floating DNA molecules. In addition to the

array of dangling-molecules, the system con-tains floating-molecules. A floating-moleculeis a free floating (unattached to the symboltrack) duplex DNA segment with a singlestrand overhang at one end (sticky end). Afloating-molecule floats freely in the solutionand thus can interact with another floating-molecule or a dangling-molecule providedthat they possess complementary sticky ends.There are two kinds of floating-molecules:the rule-molecules and the assisting-molecules.The rule-molecules collectively specify thecomputational rules and are the programmablepart of the ADCA, while the assisting-molecules assist in the carrying out the opera-tions of the ADCA, which we describe in detailin Sect. 2.2.

b Protein enzymes. The system also containsfloating DNA ligase and three types of DNAendonucleases. The enzymes perform ligationsand cleavages on the DNA molecules to effectthe designed structural changes and hence theinformation processing. The cleavage patternsof the endonucleases are described in Figure 3.

2.2 Operation2.2.1 Structural ChangesFigure 4 illustrates the structural changes during theoperation of ADCA.

Initial configuration. Figure 4 (a) depicts an ex-ample abstract cellular automaton in its top panel,and a corresponding ADCA in its bottom panel. Forsimplicity and clarity, the floating enzymes and thefloating DNA molecules in the ADCA are omittedfrom the figure; the symbol track, as well as the du-plex and sticky end portions of a dangling-molecule,is depicted as a thick line segment; the flexible hingeof a dangling-molecule as a thin curve. The leftmostsymbol-molecule is a special initiator dangling-molecule, c , representing the cell d in the abstractcellular automaton (see Figure 4). To the right of c ,three types of dangling-molecules, e , f , and g , arepositioned evenly along the track in a periodic ordersuch that cells h�ikjml , h�ikjon , and h�ikjoh , wherei is a non-negative integer, in the abstract cellularautomaton are represented in the ADCA by symbol-molecules e , f , and g , respectively. The symbol-molecules differ in their default sticky ends, i.e. thesticky ends they possess in their respective initialconfigurations before the reaction starts. As we shallsee later, this is essential for the synchronization ofthe operation of the ADCA. The color of each cell inthe abstract cellular automaton is encoded in a cor-responding symbol-molecule in the ADCA.

Pipelined reaction waves. Figure 4 (b) illus-trates structural changes. During the operation ofthe ADCA, the initiator molecule c sends out a “re-action wave” that travels down the track from left toright. A critical novel property of the ADCA is thatmultiple reaction waves can travel down the track ina “pipelined” fashion. However, we have carefullyengineered a “synchronization” mechanism so thata reaction wave that starts at a later stage can never

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overtake one that starts earlier. This ensures the syn-chronization of the state changes of the ADCA, andhence its correct operation.

Figure 4 (b) illustrates two consecutive reac-tion waves, respectively indicated with red/dark andgreen/grey boxes and arrows. Next, we focus onthe first reaction wave (indicated by dark boxesand arrows), and describe the structural changes ofADCA.

In Stage 0, the reaction wave starts at the initia-tor c at position d , then travels sequentially to e inStage 1, f in Stage 2, and g in Stage 3. The reac-tion wave finishes one full cycle in Stages 1, 2, and3, and thus goes on inductionally down the track.

In Stage i , where i��d��Yl��`n , and h , three types ofreactions occur, namely reactions i`�Bd , iU�Tl . and iU�Rn .b In Stage 0, c has a complementary sticky end

to its right neighbor e and is thus ligated toe , and the ligation product is subsequentlycleaved by an endonuclease (Reaction d��Tl ).Next, c is “modified” by an assisting-molecule,depicted as a pink (grey) line segment, andrestored to its default configuration (Reactiond��Rn ). The “modification” will be implementedas ligation and cleavage events and will be de-scribed in detail in Sect. 3. In a parallel reaction

d��Rh , e is also modified by another assisting-molecule such that e will possess a comple-mentary sticky end to f , and thus the reactionwave is ready to enter Stage 1 (Reaction d��Rh ).b In Stage 1, similar structural changes occur as

in Stage 0. However, after reaction l��Tl , e willpossess a sticky end that encodes the state, i.e.color, information of itself, its left neighbor c ,and its right neighbor f . In the ensuing re-action l��Rn , a rule-molecule corresponding to atransition rule in Figure 1 recognizes e ’s stickyend and effects a state transition of moleculee . e will then be modified by an assisting-molecule and restored to its default configura-tion, encoding its new state. In the exampleshown in Figure 4 (b), a rule-molecule corre-sponding to rule (7) in Figure 1 changes thecolor encoded in e from

�� to �� . In

the parallel reaction l��Rh , f will be modified toposses a complementary sticky end to g .

b In Stages 2 and 3, reactions of the same natureas in Stage 1 will occur. Details are omitted forbrevity.

Figure 7: Information flow

Figure 5: Encoding state information in DNA molecules.The backbones of DNA strands are depicted as directedline segments. � and � �� represent the duplex portion andthe sticky end of the DNA molecule, respectively. Thebases shaded in blue are used to encode state information,� or � . “ � bp” indicates � DNA base pairs, where � is anon negative integer. In Figure (b), the number � is usedto encode state information � and is thus shaded in blue.The red (dark) box indicates the recognition site for anendonuclease, in this case, EcoPl5 I.

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Figure 6: Initial configuration

2.2.2 Information Flow

Notation. We next describe the information flowduring the operation of the ADCA. For ease of ex-position, we first introduce some notation. An in-formation encoding DNA molecule is denoted as�� D�

, where�

is its duplex portion,� ��

is itssticky end portion, and � and � respectively rep-resent the state information encoded in

�and� ��

.This is illustrated in Figure 5. As shown in the fig-ure, there are two ways to encode information � inthe duplex

�. In Figure 5 (a), � is encoded as a

unique DNA sequence ��e ; in Figure 5 (b), � isencoded as the number of base pairs ( � bp in theFigure) between an endonuclease recognition siteand the sticky end of DNA molecule. The sequenceof the sticky end

� ��, in this case g��g , encodes the

state information � . Furthermore, we use��

to de-note the complementary sticky end of

� ��. Finally,

we describe how to represent ligation and cleavageevents. The ligation of two molecules

�� D�and��D `¡�¢

is described by the equation

� � � �� j �� ¡ ¢ � ��£ �

Suppose��£

incorporates an endonuclease recog-nition site and is cut into

�¤�U¥P� ¦��D�§¥and��¦�� ¥8¡ ¢ ¥

. Thisis represented as

��£ � � �U¥ � ¦�� §¥ j �§�¦�� ¥ ¡ ¢ ¥ �We can combine the above two equations and ob-tain,

� � � �� j �� ¡ ¢ � � �U¥ � ¦¨� �§¥ j ��¦¨� ¥ ¡ ¢ ¥ �Initial configuration. Figure 6 shows the ADCA

in its default configuration before the reaction starts.As mentioned in Sect. 2.2.1, molecules e , f , and gpossess different default sticky ends:

��¦¨�,�T�© � , and�ª�« � . Note that the state information � , � , and ¬

are encoded in the duplex portions of e , f , andg , not their sticky ends. This is essential to en-sure that repeated reactions between neighboringsymbol-molecules can occur for multiple rounds, asdescribed below.

Information flow. Figure 7 illustrates the infor-mation flow. We follow the framework of four-stagestructural changes presented in Sect. 2.2.1 and enu-merate the involved reactions below.

1. Reaction 0.1. Initiator molecule c � ¦�� and itsimmediate right neighbor

��¦¨� e � share comple-mentary sticky ends

¦and�§�¦��

, and result in re-action,

c � ¦¨� j �§�¦�� e � �®c � ¯°� � j � �¯±� � e��Note that the sticky end

� �¯±�of product e en-

codes both the state information i from reactantc and the state information � from reactant e .

2. Reaction 0.2. The rule-molecule� �¯±� �ª² re-

stores c � ¯°� � to its default configuration in re-action,

c � ¯±� � j � �¯±� � ² �®c � ¦¨� j ��¦¨�D² �

3. Reaction 0.3. Molecule� �¯±� � e is modified by

assisting-molecule �� ¯±� � in reaction

� � � ¯±� � j � �¯±� � e³�´� �T�© � j � © � e � �Now e is transformed to

� © � e � . This es-sentially shifts or transduces the state infor-mation i°� initially encoded in the sticky end

of e to its duplex portion. Hence we termthe assisting-molecule �µ �� ¯°� � as a transducer-molecule. The above reaction also modifies e ’ssticky end to

� © � , which is complementary tothe default sticky end of e ’s immediate rightneighbor f . This effectively makes e ready tointeract with f .

4. Reaction 1.1. Molecule eµ �� © � interacts withits right neighbor

�T�© � f � in reaction,

e � � © � j �T�© � f � �¶e � ¯±� �`� j � �¯±� �`� f·�Now the sticky end of the product e encodesstate information i°�� , i.e. the current state ofe ’s left neighbor, the current state of e , andthe current state of e ’s right neighbor. Thissuffices to specify a transition rule shown inFigure 1 and results in Reaction 1.2 below.

5. Reaction 1.2. Reaction 1.2 has two steps. Instep 1.2.1, e � ¯±� �`� interact with a rule-molecule� �¯P� �`��²¸�U¥ in reaction,

e � ¯±� �¹� j � �¯±� �`� ² � ¥ �®e �\º»�D¼ � ¥ j �T�ºY�D¼ � ¥ ² �This essentially effects a state transition ofmolecule e , as specified by the rule i°��µ�½��¾ .However, for e to repeatedly perform compu-tation, we need to restore e to its default con-figuration, i.e. a configuration with a defaultsticky end

��¦��and encoding its new state �¿¾

in its duplex portion. This task is carried outby another kind of assisting-molecule calledextension-molecule À . However, as a floatingmolecule, À not only needs to recognize e ’scurrent state but also distinguish e from theother two types of symbol-molecules, f andg . As such, we require the sticky end of e en-codes not only its state information but also itstype information Á , where Á�ÂÄÃªÁ�ÅÆ�UÁ�ÇÈ�UÁªÉËÊ .Hence, in the above equation, the product epossesses a sticky end

�\º»�encoding both its

type information Á and its new state �¿¾ ( ��¾ notshown in Figure 7 (b)). This molecule e �\º»� ¼ � ¥is then modified by extension-molecule À �)�ºÌ� ¼ � ¥in reaction 1.2.2,

e �\º»� ¼ � ¥ j �T�ºY� ¼ � ¥ ÀÍ�¶e � ¥ ��¦¨� j � ¦�� ÀÎ�

Figure 8: Example molecular implementation of reaction0.1. The red (dark) box and red (dark) arrows respec-tively indicate the recognition and cleavage sites for en-donuclease Bsl I. The encoded state information is indi-cated with blue (grey) region. Base pair Ï»Ð�ÑÏ is a pair ofunnatural bases

6. Reaction 1.3. Molecule� �¯°� �¹� f is modified by

transducer-molecule � �`�Ì� ¯°� �¹� in reaction,

� �`� � ¯±� �¹� j � �¯±� �¹� fÒ�´� �ª�« � j � « � f �`� �Note that now f encodes state �� in its du-plex portion (state i is not kept since it is notrequired for effecting f ’s transition), and pos-sesses sticky end

� « � , which is complementaryto the default sticky end of g

7. Other reactions. Similar to reactions 1.1, 1.2,and 1.3.

3 Molecular Implementation3.1 Step-by-step ImplementationTo demonstrate the practicality of our design, wenext give a detailed description of the molecularimplementation of the ADCA. The complete DNAmolecule set will be described in Sect. 3.2.

1. Reaction 0.1. Figure 8 depicts an examplemolecular implementation of reaction 0.1,

c � ¦¨� j �§�¦�� e � �®c � ¯°� � j � �¯±� � e��For simplicity, only the end sequences ofdangling-molecule e are depicted; for full se-quences, see Figure 15 (a) in Appendix. Pan-els (a) and (b) respectively illustrate caseswhen �Ó� ��

and �Ó� �� . Inmolecule

��¦�� e � , the state information �ÔÂ

Figure 9: Example molecular implementation of reaction0.2.

Ã �� Ê , is encoded by the presence orabsence of a DNA base pair between the stickyend��¦��

(sequence � ) and the half recognition

site for endonuclease Bsl I (sequence Õ�Õ¿Ö�� )in the duplex portion. This is further indicatedin Figure 8 by the shaded blue (grey) region.In the case �Í� �� , the cleavage of lig-ation product c�e by Bsl I produces a stickyend sequence ÕÕ for molecule c , and �� formolecule e . Both these unique sticky end se-quences encode state information i°� . When�×� �� , a different pair of sticky ends� ¯±� Ö � �¯±� are generated ( ��ÕÕ�Ö�Õ�� ).Molecule e contains a pair of unnatural bases,i.e. synthetic bases other than the natural bases� , � , Õ , and

. They are required because the

four-letter ��Õ natural vocabulary do not pro-vide sufficient encoding space for our construc-tion. For a survey on experimental synthesis ofunnatural bases, see [8].


c � ¯±� � j � �¯±� � ² �®c � ¦�� j ��¦��D² �The endonuclease involved is EcoPl5 I. Thisrestores c to its default configuration withsticky end sequence

� ¦¨� �� .3. Reaction 0.3. Figure 10 depicts an example

molecular implementation of reaction 0.3,

� � � ¯±� � j � �¯±� � eØ�Ù� �T�© � j � © � e � �Here, we only illustrate the case �Ú� �� ,and omit the similar case �Û� �� for

Figure 10: Example molecular implementation of reac-tion 0.3.

brevity. Note that the state information i°� ini-tially encoded in the sticky end

� �¯±�(sequence

�� ) of e is now encoded in the blue (grey) re-gion in its duplex portion (sequence ��Õ�¿Ö�Õ� ).


e � � © � j �)�© � f � �®e � ¯°� �¹� j � �¯±� �¹� f·�Again, we only illustrate the case �Ü� �� and �Ý� �� for brevity. This reaction issimilar to reaction 0.1. Both the sticky ends

� ¯±�and� �¯°�

encode state information i°�� .Two technical points warrant explanation inthis reaction. First, the bases labeled with red(dark) circles contain phosphorothioate bondand are hence resistant to enzyme cleavage.This modification of the bases is required toprevent the unwanted cleavage of the DNA du-plex by Mwo I, whose recognition sites are in-dicated with blue (grey) boxes in the figure.This trick will be used again in reaction 1.2.2.Second, we assume here that the cleavage byendonuclease Bsl I will occur, but the cleav-age by EcoP15 I will not occur (note that bothmolecule e and molecule f contain EcoP15 Irecognition site, i.e. ��Õ��Õ�Ö�� Õ� Õ ). Thisassumption is based on the fact that Bsl I, aType II endonuclease, can act far more effi-ciently than EcoP15 I, a Type III endonuclease.

5. Reaction 1.2.1. Figure 12 (a) depicts an ex-ample molecular implementation of reaction1.2.1,

e ¼ � ¯°� �`� j � �¯°� �`� ² � ¥ �®e �\º»� ¼ � ¥ j �T�ºY� ¼ � ¥ ² �

Figure 11: Example molecular implementation of reac-tion 1.1. The red (dark) box and red (dark) arrows re-spectively indicate the recognition and cleavage sites forendonuclease Bsl I. The bases labeled with red (dark) cir-cles contain phosphorothioate bond and are hence resis-tant to enzyme cleavage. The light blue (grey) box indi-cate the Mwo I recognition site

Recall that this reaction effects a state transi-tion for molecule e , as specified by the rulei°��Þ�®� ¾ . Here, the rule molecule incorporatesa spacer region, the length of which ( � bp) en-codes the new state information � ¾ . In particu-lar, when �ß�àn , ��¾á�â�� ; when �ß�àã ,��¾¨� �� .When ��¾Ý� �� (the EcoP15 I cleavagestep not shown in Figure 12 (a)), molecule eis restored to the desired target configuration��¦�� e � ¥ , with the default sticky end

�§�¦�� ä� .In this case, the next step, reaction 1.2.2, is notrequired. The reaction can thus be rewritten as,

e ¼�� ¯±� �`� j � �¯±� �`� ² �U¥ �®e �U¥ �§�¦�� j � ¦��D² �However, when �¾¤� �� (the EcoP15 Icleavage step shown in the figure), molecule eis modified into e �\º»� ¼ � ¥ , with a unique stickyend�\º»� �æå�Õ that encodes both e ’s type infor-

mation Á¸�ØÁYÅ and e ’s new state �¾ (Recall thatÁçÂÝÃªÁ Å �UÁ Ç �UÁ É Ê encodes type information, inthe case illustrated here, ÁÜ�èÁ�Å . This infor-mation is initially encoded in the blue (grey)duplex portion of e , in the form of sequenceå�Ö�éå ). Then the reaction proceeds to the nextstep, reaction 1.2.2, which will finish the statetransition for e .

In our molecular implementation, in a transi-tion ê ��ë � � ¾ , the values of

�,ë, and

ë ¾ coop-eratively determines the spacer length � , which

Figure 12: Example molecular implementation of reac-tion 1.2. Panel (a): reaction 1.2.1. Panels (b): reaction1.2.2.

in turn decides the result of the transition. Fordetail, see Figure 15 (c) in Appendix.

6. Reaction 1.2.2. Figures 12 (b) depicts the case�� U��¾k�ì�� , which follows fromthe case illustrated in Figures 12 (a).

e �\ºÌ� ¼ � ¥ j �)�ºÌ� ¼ � ¥ Àí�®e � ¥ ��¦�� j � ¦¨� Àî�Here, extension-molecule À �T�ºY� ¼ � ¥ restores e toits default configuration with sticky end

��¦¨� �� . However, now e encodes new state ��¾ inits duplex portion.

7. Other reactions. Similar to the above reactions,and hence omitted for brevity.

3.2 Complete Molecule SetThe complete set of DNA molecules constitutingADCA are described graphically in Figure 15 in theAppendix. The dangling-molecules (Figure 15 (a))and the floating rule-molecules (Figure 15 (c)) arethe programmable parts of the ADCA: the selec-tions of dangling-molecules and rule-molecules re-spectively determine the initial configuration and the

Figure 13: An example futile reaction.

transition rules of the ADCA. Note that all the stickyends of rule-molecules are unique. In contrast,the floating assisting-molecules, i.e. transducer-molecules (Figure 15 (d)) and extension-molecules(Figure 15 (b)), only assist in the proper operationof the ADCA and are non-programmable.

3.3 Futile ReactionsBesides the main reactions described above, thereexist reversible futile reactions in the system. Thesefutile reactions are carefully engineered such thatthey will not block the main reactions. Futile re-actions, however, can also be used to maintain a dy-namic balance among the floating molecules, and,in doing so, ensure the proper operation of the de-vices. Figure 13 shows an example where futile re-actions are used to maintain a balance between

²and � molecules that have complementary stickyends. For a more detailed discussion of futile re-actions, see [26].

3.4 Computer SimulationFully debugging the above molecular implementa-tion requires meticulous inspection of every step ofthe ADCA operation, which can become exceed-ingly tedious. We thus developed a computer simu-lator and used it to test and debug the ADCA. Thesimulator takes as input the DNA sequences whichspecifies an ADCA instance, simulates the opera-tions of ADCA, and gives graphical output. For de-tail, see http://pengyin.org/paper/dnaCA/.

4 Two-Dimensional ADCATo illustrate the operational principle of 2-D ADCA,we first present an abstract view of the 1-D ADCA inFigure 14 (a) and (b). Figure 14 (a) illustrates a reac-tion wave of the 1-D ADCA: the reaction wave startsat initiator c and travels sequentially down the one-dimensional track. Figure 14 (b) examines one indi-

vidual dangling-molecule�

, where� �me��Uf·�¹g .

Assume w.l.o.g.,� �àf . As shown in Figure 14

(b), f in a 1-D ADCA undergoes the following fourphases in one full reaction cycle.

1. Phase 1. f has a sticky end that is complemen-tary to its left neighbor e (indicated by a solidsquare to the left of f in Figure 14 (b)). Thisis before reaction 1.1 as depicted in Figure 7.In this phase, f encodes in its duplex portionits own state information denoted by � ( � forcenter).

2. Phase 2. In reaction 1.1, f interacts withits left (i.e. west) neighbor, and enters phase2. Now f encodes in its sticky end both thestate information of itself, � , and the state in-formation of its west neighbor,

�. This sticky

end is complementary to a floating transducer-molecule, indicated by a circle around f .

3. Phase 3. After reaction 1.3, f now encodesstate information � � in its duplex portion, andpossesses a sticky end complementary to itsright (i.e. east) neighbor (indicated by a solidsquare to the right of f ).

4. Phase 4. In reaction 2.1, f interacts with itseast neighbor, and enters phase 4. Now f en-codes in its sticky end the state information ofitself � , its west neighbor

�, and its east neigh-

bor�

. This sticky end thus encodes all the stateinformation required to effect a state transitionfor f , and is recognized by a floating rule-molecule (indicated by a thick circle). And thisstate transition restores f to its default config-uration, finishing a full cycle.

With the above understanding of the 1-D ADCA,we can extend it to 2-D as follows. First, we takecare of reaction waves by positioning two arrays ofinitiators as shown in Figure 14 (c). Each initiatorcan send out a reaction wave that travels either hor-izontally or vertically. Next, we take care of the in-formation flow and synchronization, by again exam-ining one single molecule

�. As shown in Figure 14

(d), we engineer the system such that molecule�

undergoes 8 phases. During these 8 phases,�

se-quentially interacts with its west (

�), north ( ï ), east

(�

), and south ( ð ) neighbors to garner the state infor-mation from each of them. As such, upon entering

Figure 14: (a) (b) Operational overview of 1-D ADCA.(c) (d) Operational overview of two-dimensional ADCA.In panels (b) and (d), black numbers indicate the phasesof � ; blue (grey) numbers indicate reactions correspond-ing to reactions depicted in Figure 7; red (dark) lettersindicate the state information carried by �

phase 8,�

carries in its sticky end the state infor-mation � � ï � ð . This state information is sufficient toeffect a state transition for

�. As in the 1-D case,

�will undergo a state transition and re-enters phase 1,completing a full circle.

5 ConclusionIn this paper, we present the theoretical design andmolecular implementation of 1-D ADCA and de-scribe how to extend it to 2-D.

Open question: can we simplify the current com-plex design?

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Figure 15: (a)Dangling-molecules. Left and right panels respectively depict the cases when the encoded infor-mation � Ðª�`Ð��! #"%$'& and (!)'*!+', . (b) Extension-molecules. Left and right panels respectively depict the caseswhen the transition is -/.0�! #"%$1&2.43657(8)'*!+9, and -/.0(8)'*!+',:.;3<5=�! #">$1& , where -?.;3A@CB�(8)9*!+',8.D�1 #"%$1&8E . (c) Rule-molecules. The eight columns (1-8) correspond to the eight possible configurations of -��!3 in the rule -��!3�5®�#F ,where -?.°�#.;32.°� F @�BG(8)'*1+',8.D�! H"%$1&8E . The symbol ('(1� stands for the configuration -��!3��IB�(!)'*!+',8.0(8)'*!+',:.0�! #"%$1&8E .The value of � is determined cooperatively by � , 3 , and 3 F . (d) Transducer-molecules. The bases labeled with red(dark) circles contain phosphorothioate bond and are hence resistant to enzyme cleavage (see reaction 1.1, Figure 11)

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