dna data-structure

A DNA Data Structure Ben Shirt-Ediss & Natalio Krasnogor

The Dr. Jekyll Side of DNA

Source: http://compbio.pbworks.com

Mr. Hide: DNA as a Dynamic Polymer1.4.1 DNA Hybridization Reactions

A well-known example of DNA hybridization reactions is the Watson-Crick DNA hybridization between twocomplimentary ssDNA strands as discussed in Section 1.2.2. Two ssDNA strands can attach to each other.However, they can also detach from one another, if the temperature is greater than the melting temperatureof the strands (Figure 6). The melting temperature of a dsDNA is defined as the temperature at which 50%of the dsDNA has converted to single stranded form.

Figure 6: DNA Denaturation Renaturation [60]

Toehold Mediated Strand Displacement Yurke et al. reported an interesting DNA hybridization re-action through their DNA tweezers system [124]. As illustrated in Figure 7a), two ssDNA strands (B andCB are bound to one another, with one strand, called the incumbent strand (strand B) completely bound,while the other CB has a few unbound bases. These bases (C) can together be called a sticky end, overhang,or a toehold. A third ssDNA, called the incoming strand (strand BC), complementary to the ssDNA withthe toehold (CB), can hybridize to the toehold region (C) and displace the incumbent strand (B). Thisprocess is termed toehold-mediated strand displacement. Typical toehold lengths used for toehold-mediatedstrand displacement hybridization reactions range from 3 to 7 nucleotides. The rate constant of the toehold-mediated strand displacement ranges from 1 M�1s�1 to 6⇥ 106 M�1s�1.

Toehold Exchange Toehold exchange is an extension to toehold mediated strand displacement, but itis extremely powerful. Zhang and Winfree [129] made the reaction in the previous section reversible, byintroducing a small exit toehold. As seen in Figure 7b), toehold exchange proceeds in the same manner

Hybridisation Toehold-Mediated Strand Displacement

Toehold Exchange

and therefore can be considered effectively unreactive. This isbecause the complex L does not possess a single-strandedcomplement to strand Y’s R domain, so Y cannot be easilycolocalized to complex L. Thus, the presence and properties ofthe toehold domain are instrumental to the kinetic control ofDNA strand displacement reactions.5,53

Because of the toehold’s role in initiating strand displacementreactions, strands can be rendered effectively unreactive if thetoehold domain is made inaccessible by toehold sequestering.Toehold sequestering can be achieved in a number of ways, thetwo most common of which are hybridization of the toehold to acomplementary domain9,15,16 and isolation of the toehold in ashort hairpin structure where helix formation is difficult6,52,55

(see Figure 1C). Programmed sequestering and subsequentexposure of toehold domains allows precise control of order andtiming over the reactions and has been used in conjunction withtoehold-mediated strand displacement to construct molecularmotors,5,8,9 polymerization reactions,6,9 catalytic reactions,9,51,52 andlogic gates.15,48,49

Recently, the toehold exchange mechanism was introducedas a method for improved control of strand displacementkinetics.16 Toehold exchange is similar to toehold-mediatedstrand displacement in that an invading strand (X) binds by atoehold to initiate branch migration but differs from the latterin that the incumbent strand (Y) possesses a unique toeholdthat must spontaneously dissociate for the reaction to complete.Expanding on the example strand displacement reaction inFigure 1B, the toehold exchange reaction and mechanism thatwe study experimentally is illustrated in Figure 1DE for aninvading toehold of length n and an incumbent toehold of lengthm (m, n > 0): Strand X(m, n) binds to complex S via invadingtoehold γn and displaces strand Y’s "m domain by branchmigration. Strand Y’s incumbent toehold "m then spontaneouslydissociates, yielding free strand Y and complex L(m, n). Theend result of the toehold exchange reaction is that the originallyactive toehold γn is sequestered while the formerly sequestered(55) Green, S. J.; Lubrich, D.; Turberfield, A. J. Biophys. J. 2006, 91, 2966.

Figure 1. (A) DNA abstraction. A DNA complex (top) is typically abstracted as several directional lines, one for each strand, with bases identities shown. Here,we abstract DNA strands and complexes one step further by grouping contiguous nucleotides into domains, functional regions of DNA that act as units in binding.Because the principles and mechanisms studied in this paper are expected to be generalizable to most DNA sequences, we typically do not show the sequences ofDNA strands in figures. For sequences, refer to Table 1. (B) A toehold-mediated strand displacement reaction. The displacement of strand Y by strand X is facilitatedby strand X’s toehold domain γ. (C) Two examples of toehold sequestration. A strand of DNA can be rendered unreactive by inactivating its toehold domains. Inthe figure, toehold γ is sequestered through isolation in a hairpin (middle) and through hybridization to a complementary domain (bottom). (D) A toehold exchangereaction and its mechanism. Invading strand X(m, n) binds to substrate complex S by toehold γn (known as the invading toehold) to form intermediate I(m, n).Intermediate I(m, n) represents all branch migration states in which Y is bound to more bases of "m than X(m, n). Intermediate I(m, n) rearranges to form intermediateJ(m, n), which analogously represents all states in which X(m, n) binds more bases of "m than Y. Domain "m (known as the incumbent toehold) spontaneouslydissociates, releasing products Y and L(m, n). The toehold exchange reaction is reversible, because strand Y can subsequently bind to complex L(m, n) via strandY’s toehold "m. (E) Comparison of various invading strands X(m, n). Strand X(m, n) is the concatenation of domains "m and γn and consequently has length(b + n - m) nt, where b is the length of the full " domain. In a toehold exchange reaction using X(m, n), the invading toehold has length n and the incumbenttoehold has length m. For our experiments, we used three sets of invading toeholds, γn, γsn, and γwn. The sequence composition of the latter two are purely A/T’sand purely G/C’s, respectively, to characterize the kinetics of toehold exchange given weak and strong toeholds, respectively. Substrates using γs are labeled Ss, andinputs using γs are labeled Xs(m, n), and similarly for γ and γw. (F) Schematic of the experimental system used for rate measurements. Reporter complex R reactsstoichiometrically with product Y to yield increased fluorescence.

Table 1. Domain Sequences

domain sequence length (nt)

R 5′- CCACATACATCATATT -3′ 16" 5′- CCCTCATTCAATACCCTACG -3′ b ≡ 20γs 5′- CCCGCCGCCG -3′ 10γ 5′- TCTCCATGTCACTTC -3′ 15γw 5′- ATTTATTATA -3′ 10

J. AM. CHEM. SOC. 9 VOL. 131, NO. 47, 2009 17305

Control of DNA Strand Displacement Kinetics A R T I C L E S

104 NATURE CHEMISTRY | VOL 3 | FEBRUARY 2011 | www.nature.com/naturechemistry

Several subsequent works used Yurke’s basic reaction sequence (a hybridization step followed by strand displacement to reverse the effect of the initial hybridization) for controlling complex nanos-cale structures. Simmel and Yurke34 demonstrated a nanoactuator related to Yurke’s original tweezer design. Addition of a first input strand pushed the two arms of the nanoactuator apart; addition of a second input strand set them free. In further work they built a device that could be switched between three distinct states using two pairs of fuel strands35. Tian and Mao36 built a device consisting of two DNA complexes reminiscent of interlocking gears that could be repeatedly cycled through three different states.

Reconfiguring self-assembled structures. Strand displacement can be combined with structural self-assembly to enable dynamic reconfiguration of larger DNA nanostructures post-assembly, and

can be used to induce changes at macroscopic scales. A first exam-ple of this was described by Yan and co-workers37 who used the toehold-mediated cycling technique of Yurke et al. to construct a rotary DNA device. Their device could be switched between two states corresponding to different DNA tile motifs, called PX and JX2 (Fig. 1b). They also assembled multiple devices into a linear structure large enough to be visualized with an atomic force micro-scope and demonstrated switching of a DNA multi-stranded struc-tural motif relative to the main axis of a larger structure (Fig. 1c). Their device was based on an earlier example of a switchable DNA nanomachine30 that responded to ambient salt concentration rather than DNA inputs.

Chakraborty et al.38 later extended this basic design to a system that could be switched between three different states, and Zhong and Seeman39 demonstrated that switching could be indirectly controlled

DNA is represented as directional lines, with the hook denoting the 3 end (panel a). For many strand-displacement-based designs, it is convenient to abstract contiguous DNA bases into functional DNA domains that act as a unit in hybridization, branch migra-tion or dissociation. Domains are represented here by numbers; a starred domain denotes a domain complementary in sequence to the domain without a star (for example, domain 2* is comple-mentary to domain 2). The sequences of the nucleotide bases are not typically shown because it is expected that DNA devices based on strand displacement will work for many if not most choices of domain sequences.

The key reaction that has allowed the construction of the dynamic assemblies shown in this review is DNA strand displace-ment. Panel b shows one example of this reaction. Single-stranded DNA molecule A reacts with multi-stranded DNA complex X to release strand B and complex Y. Throughout the text we will refer to single-stranded reactants (such as A) that initiate a reaction as ‘inputs’ and to single-stranded reactants that are released from a complex (such as B) as ‘outputs’. The strand-displacement reaction is facilitated by the ‘toehold’ domains 3 and 3*: the hybridization of these single-stranded toeholds co-localizes A and X, and allows the 2 domain to ‘branch migrate’. Branch migration is the random

walk process in which one domain displaces another of identical sequence through a series of reversible single nucleotide dissocia-tion and hybridization steps24. At the completion of branch migra-tion, complex Y is formed and strand B is released. The concept of toeholds was introduced to DNA nanotechnology by Yurke et al.29, and studied in detail by Yurke and Mills17, Li et al.18 and Zhang and Winfree19.

Panel c shows that the kinetics of strand displacement can be accu-rately modelled and predicted from the length and sequence of the toehold domain19 (nt = nucleotide). The rate constant of the strand-displacement reaction varies over a factor of 106, from 1 M–1 s–1 to 6 × 106 M–1 s–1. The green trace shows the kinetics of using a strong toehold composed of only G/C nucleotides, the red trace shows the kinetics of using a toehold composed only of A/T nucleotides, and the black trace shows the kinetics of a toehold composed of roughly equal numbers of all four nucleotides. The grey region spanned by the green and red traces roughly shows the range of potential kinet-ics based on toehold length. The progress of strand-displacement reactions is typically assayed using fluorescence, either by means of reporter complexes that stoichiometrically react with the output, or by using dual-labelled probes as output strands. Part c reproduced with permission from ref. 19, © 2009 ACS.

Box 1 | Overview of DNA strand displacement

2* 3*3

1 222

2

31

5'

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c

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32Input A

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3. Strand displacement completes

1. Toehold domains initiate binding

Complex Y

Output B

(toehold)

(toehold)

0 10 155

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8

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Maximum

GGGAGTAAGTTATGGGATGC 3*2*

2

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21

Toehold length (nt)

log 1

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-1 s

-1)

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2

CCACATACATCATATT

1

––– –––

REVIEW ARTICLE NATURE CHEMISTRY DOI: 10.1038/NCHEM.957

nchem_.957_FEB11.indd 104 11/1/11 11:08:45

© 2011 Macmillan Publishers Limited. All rights reserved

Source: Zhang & Winfree. 2011. Nat Chem.

Source: Zhang & Winfree. 2009

DNA: Dynamic and also Well Understood

Predictable tertiary structure

Can use methods in Mol. Bio to quantify experimental success

Enzymatic reactions cutting/joining DNA well characterised

Hybridisation, branch migration and strand displacement reactions well understood

PAGE, Fret, PCR, AFM…

234901-3 Snodin et al. J. Chem. Phys. 142, 234901 (2015)

FIG. 1. Schematics contrasting the original oxDNA model (left), with equalgroove widths, with oxDNA2 (right), which has di↵erentiated major andminor grooves. (a) A cross section of a duplex with one base pair displayed.The large dashed circle shows the helix radius, and each nucleotide is repre-sented by three circles joined by a line; the large solid circles represent thebackbone sites, while the small solid circles represent the stacking (closer tothe backbone) and hydrogen-bonding (at the end of the nucleotide) sites. ForoxDNA2, a value of 20� was chosen for the angle �. (b) A representation ofa DNA duplex for each model.

were incorporated into the backbone site’s excluded volume,an approximation which can be justified by the very short De-bye screening length at that relatively high ion concentration.

Second, the original oxDNA model30 represented eachnucleotide as a linear rigid body (Fig. 1). The optimal configu-ration for base-pairing occurs when the two nucleotides pointdirectly at each other. As a consequence, the DNA double helixwas symmetric, with the two grooves having equal widths.

Third, in the original oxDNA model introduced in Ref. 30,all four types of base were treated equally except that only A-Tand G-C base pairs could be formed. Later, Šulc et al.60 intro-duced sequence-dependent thermodynamics into the model bymaking the strengths of the hydrogen-bonding and stackingterms depend on the identities of the interacting nucleotides.The nearest-neighbour DNA model of SantaLucia,49 to whichoxDNA was parameterised, does not resolve the di↵erencebetween AA and TT stacking as it works on the base-pair steplevel—therefore AA and TT stacking strengths were set to bethe same in Ref. 60.

In this work, we mostly work from the original, sequence-averaged parameterisation of the model rather than thesequence-dependent one, as it is more e�cient to fit the ther-modynamic parameters to sequence-averaged duplex melt-ing temperatures as given by the SantaLucia model. Theexception is the parameterisation of the AA and TT stackingstrengths, which did use the sequence-dependent parametersfrom Ref. 60 as a starting point, to allow the best possiblecomparison between the model and the experimental resultsthat were used for the fitting. After the parameters for oxDNA2had been obtained, including new values for the sequence-averaged hydrogen bonding and stacking strengths, we then

rescaled the sequence-dependent interaction strengths fromRef. 60 accordingly for use with the new model.

III. INTRODUCING DIFFERENT WIDTHS FOR MAJORAND MINOR DNA GROOVES

B-DNA in the original oxDNA model has equal groovewidths, while in reality DNA has a larger major groove anda smaller minor groove. Having realistic widths for the majorand minor grooves is equivalent to having appropriately posi-tioned backbone sites in the model, an important feature forthe physical properties of many DNA motifs. For example,in DNA origami, antiparallel double helices are joined bycrossovers, for which the position of the backbone has beenshown to be crucial for origami structure.2,63 Another exampleis anisotropic duplex bending: the duplex can be expected tobend more easily into the major groove than into the minor, ifthe groove widths are unequal.

The oxDNA nucleotide is composed of three interactionsites: the hydrogen-bonding, stacking, and backbone sites. Weintroduce di↵erent groove widths by changing the position ofthe backbone site while keeping the duplex radius unchanged(Fig. 1), such that, rather than lying on a straight line, thethree interaction sites lie in a plane. The new nucleotide shapeintroduces an additional parameter into the model, the angle� between the line from the duplex centre to the backbonesite and the line from the duplex centre to the stacking site(Fig. 1(a)). Given the coarse graining of the 18 atoms of thesugar-phosphate DNA backbone into a single interaction site,there is no definitive choice for the precise position of thebackbone site and thus the value of the model parameter �(Fig. 1(a)). We set � = 20�, a value which maps onto a full-atom representation of a DNA duplex well by visual inspec-tion, although values of � between 15� and 25� would give anequally satisfying visual match.

The backbone site is moved such that the duplex radiusis unchanged, and we note that the modification has a negli-gible added computational cost when simulating the model.However, the thermodynamic and mechanical properties areslightly a↵ected. For the thermodynamics, we found a changeof 1-2 K in the duplex melting temperatures, and we modifiedthe hydrogen-bonding and stacking strengths using the histo-gram reweighting method described in Sec. II C 2 of Ref. 64,so that the agreement with experimental melting temperatureswas as good as for the original model. The mechanical prop-erties of DNA are less well constrained experimentally and sowere not refitted. The mechanical properties for the new modelcan be found in Sec. VIII.

One illustration of the importance of the groove widths inoxDNA for the structural properties of DNA assemblies is pro-vided by systems of 3-arm star tiles that are designed to formtriangular prismatic polyhedra. We find that modifying thegroove widths qualitatively changes the structure of trimers ofthese tiles (Fig. 2). Specifically, the body of the trimer definesa plane with two distinct faces. Zhang et al.65 found that one oftwo possible isomers of the polyhedron preferentially formed,implying that the free arms of the trimer systematically pointedin the direction of one of these two faces. We find a consistentresult when the groove widths specified by oxDNA2 are used.

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oxDNA

DNA Computing

eral minutes to obtain near-field fluores-cence spectra with good signal-to-noise ra-tios. Furthermore, the recent work of Xieand Dunn (33) and by Ambrose et al. (34)showed that the metal-coated probe tip cansignificantly perturb the electronic proper-ties of the molecule being detected. In con-trast, the far-field confocal fluorescence ap-proach provides unlimited laser throughputand a three-dimensional sectioning capabil-ity and is truly noninvasive, although itsresolution is diffraction limited. These fea-tures are expected to allow important appli-cations such as enhanced Raman spectrosco-py at the single-molecule level and on-linefluorescence identification and sorting of in-dividual molecules and quantum-confinednanostructures. The extraordinary sensitivityachieved in this work allows the direct, real-time study of the dynamics of a single mol-ecule and the chemical and biochemical re-actions that such a molecule may undergo insolution.

REFERENCES AND NOTES

1. W. E. Moerner, Science 265, 46 (1994), and refer-ences therein.

2. M. Orrit, J. Bernard, R. I. Personov, J. Phys. Chem.97,10256(1993).

3. F. GOttler, T. Irngartinger, T. Plakhotnik, A. Renn, U.P. Wild, Chem. Phys. Lett. 217, 393 (1994).

4. M. Ishikawa, K. Hirano, T. Hayakawa, S. Hosoi, S.Brenner, Jpn. J. Appl. Phys. 33,1571 (1994).

5. E. Betzig and R. J. Chichester, Science 262, 1422(1993).

6. J. K. Trautman, J. J. Macklin, L. E. Brus, E. Betzig,Nature 369, 40 (1994).

7. D. C. Nguyen, R. A. Keller, J. H. Jett, J. C. Martin,Anal. Chem. 59, 2158 (1987).

8. K. Peck, L. Stryer, A. N. Glazer, R. A. Mathies, Proc.Natl. Acad. Sci. U.S.A. 86, 4087 (1989).

9. W. B. Whitten, L. M. Ramsey, S. A. Arnold, B. V.Bronk, Anal. Chem. 63,1027 (1991); K. C. Ng, W. B.Whitten, S. A. Arnold, L. M. Ramsey, ibid. 64, 2914(1992).

10. M. Eigen and R. Rigler, Proc. Natl. Acad. Sci. U.S.A.91, 5740 (1994).

11. In fluorescence correlation spectroscopy, the inten-sity recorded at time t is multiplied by that recordedat t + At and the product is integrated over a finiteperiod of time; see D. E. Koppel, Phys. Rev. A 10,1938 (1974).

12. T. T. Perkins, D. E. Smith, S. Chu, Science 264, 819(1994); T. T. Perkins, S. R. Quake, D. E. Smith, S.Chu, ibid., p. 822.

13. S. B. Smith, L. Finzi, C. Bustamante, ibid. 258,1122(1992); C. Bustamante, Annu. Rev. Biophys. Bio-phys. Chem. 20, 415 (1991).

14. M. Washizu and 0. Kurosawa, IEEE Trans. Ind. Appl.26,1165 (1990).

15. S. B. Smith, P. K. Aldridge, J. B. Callis, Science 243,203 (1989).

16. N. J. Rampino and A. Chrambach, Anal. Biochem.194, 278 (1991).

17. K. Morikawa and M. Yanagida, J. Biochem. 89, 693(1981).

18. I. Auzanneau, C. Barreau, L. Salome, C. R. Acad.Sci. Paris 316, 459 (1993).

19. H. Kabata et al., Science 262, 1561 (1993).20. D. A. Schafer, J. Gelles, M. P. Sheetz, R. Landick,

Nature 352, 444 (1991).21. Laser excitation at 488.0 and 514.5 nm was provided

by an argon ion laser (Lexel Lasers, Fremont, CA). Thelaser beam entered the microscope through a backport and was directed to an oil-immersion objective(x100, NA = 1.3, Nikon Instrument Group, Melville,NY) by a dichroic beamsplitter (505DRLP02 or

540DRLP02, Omega Optical Inc., Brattleboro, VT).The laser beam was focused to a diffraction-limitedspot by the high NA objective in our study, which wasverified qualitatively by comparing the laser focal sizeand 1 -pm polystyrene microspheres (Duke Scientific,Palo Alto, CA). Fluorescence was collected by thesame objective, passed the same dichroic beamsplit-ter, and was then directed to a side port by a reflectivemirror. Efficient rejection of out-of-focus signals wasachieved by placing a pinhole (50 to 100 pum diame-ter, Newport Corp., Irvine, CA) in the primary imageplane. A single interference bandpass filter (OmegaOptical Inc., Brattleboro, VT) was used to reject thelaser light and the Rayleigh and Raman scatteredphotons. The fluorescence signal was then focusedon a photon-counting Si avalanche photodiode(quantum efficiency, 55% at 630 nm, and dark noise,7 counts per second) (Model SPCM-200, EG&G Can-ada, Vaudreuil, Quebec). Time-dependent data wereacquired by using a multichannel scalar (EG&G OR-TEC, Oak Ridge, TN) run on a personal computer(IBM PC-AT). Fluorescent dyes and other materialswere purchased from Molecular Probes, Inc. (Eugene,OR), Eastman Chemicals (Kingsport, TN), and SigmaChemical Corp. (St. Louis, MO).

22. M. B. Schneider and W. W. Webb, Appl. Opt. 20,1382 (1981).

23. R. Rigler, U. Mets, J. Widengren, P. Kask, Eur. Bio-phys. J. 22,169 (1993).

24. W. Feller, An Introduction to Probability Theory andits Applications (Wiley, New York, ed. 3, 1968).

25. S. A. Soper, H. L. Nutter, R. A. Keller, L. M. Davis, E.B. Shera, Photochem. Photobiol. 57, 972 (1993).

26. A complicating factor is photobleaching, which con-verts the molecule being detected into a nonfluores-cent state and prevents its further detection. The mul-tiple detection and similar fluorescence intensity ob-served for molecules of greatly different photode-struction efficiencies (that is, R6G and fluorescein)indicate however that photobleaching is not signifi-cant in this study.

27. This calculation is based on the diffusion equation TD= W2/2D, where TD is the diffusion time, w is thediffusion distance in one dimension, and D is thediffusion coefficient (2.8 x 10-6 cm2 s-1 for rhoda-mine 6G in water/ethanol).

28. D. Magde, E. L. Elson, W. W. Webb, Biopolymers13, 1 (1974); ibid., p. 29.

29. D. N. Dempster, T. Morrow, M. F. Quinn, J. Photo-chem. 2, 343 (1973).

30. M. M. Asimov, V. N. Gavrilenko, A. N. Rubinov, J.Lumin. 46, 243 (1990).

31. H. Qian and E. L. Elson, Appl. Opt. 30,1185 (1991).32. E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner,

R. L. Kostelak, Science 251, 1468 (1991); E. Betzigand J. K. Trautman, ibid. 257, 189 (1992).

33. X. S. Xie and R. C. Dunn, ibid. 265, 361 (1994).34. W. P. Ambrose, P. M. Goodwin, J. C. Martin, R. A.

Keller, ibid., p. 364.35. S.N. acknowledges the Whitaker Foundation for a

young investigator award. D.T.C. is a Beckman CellScience Scholar of Stanford University. This workwas supported by Beckman Instruments, Inc.

18 July 1994; accepted 19 September 1994

Molecular Computation of Solutions toCombinatorial Problems

Leonard M. Adleman

The tools of molecular biology were used to solve an instance of the directed Hamiltonianpath problem. A small graph was encoded in molecules of DNA, and the "operations" ofthe computation were performed with standard protocols and enzymes. This experimentdemonstrates the feasibility of carrying out computations at the molecular level.

In 1959, Richard Feynman gave a visionarytalk describing the possibility of buildingcomputers that were "sub-microscopic" (1).Despite remarkable progress in computerminiaturization, this goal has yet to beachieved. Here, the possibility of comput-ing directly with molecules is explored.A directed graph G with designated ver-

tices vn and vout is said to have a Hamilto-nian path (2) if and only if there exists asequence of compatible "one-way" edges el,e2, ... ., e, (that is, a path) that begins at in,ends at v., and enters every other vertexexactly once. Figure 1 shows a graph thatfor vn = 0 and v01u = 6 has a Hamiltonianpath, given by the edges 0-*1, 1->2, 2->3,3---4, 4->5, 5->6. If the edge 2->3 wereremoved from the graph, then the result-ing graph with the same designated verti-ces would not have a Hamiltonian path.Similarly, if the designated vertices werechanged to vin = 3 and vout = 5 there

Department of Computer Science and Institute for Molec-ular Medicine and Technology, University of Southern Cal-ifornia, 941 West 37th Place, Los Angeles, CA 90089,USA.

SCIENCE * VOL. 266 * 11 NOVEMBER 1994

would be no Hamiltonian path (because,for example, there are no edges enteringvertex 0).

There are well-known algorithms for de-ciding whether an arbitrary directed graphwith designated vertices has a Hamiltonianpath or not. However, all known algorithmsfor this problem have exponential worst-casecomplexity, and hence there are instances ofmodest size for which these algorithms re-quire an impractical amount of computertime to render a decision. Because the direct-ed Hamiltonian path problem has beenproven to be NP-complete, it seems likelythat no efficient (that is, polynomial time)algorithm exists for solving it (2, 3).

The following (nondeterministic) algo-rithm solves the directed Hamiltonian pathproblem:

Step 1: Generate random paths through thegraph.

Step 2: Keep only those paths that begin with vinand end with v,,f.

Step 3: If the graph has n vertices, then keeponly those paths that enter exactly n vertices.

Step 4: Keep only those paths that enter all of

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Science 1994

the vertices of the graph at least once.Step 5: If any paths remain, say "Yes"; otherwise,

say "No."

The graph shown in Fig. 1 with designatedvertices Vin = 0 and vtut = 6 was solved withthe algorithm above implemented at the mo-lecular level. Note that the labeling of thevertices in such a way that the (unique)Hamiltonian path enters the vertices in se-quential order is only for convenience in thisexposition and provides no advantage in thecomputation. The graph is small enough thatthe Hamiltonian path can be found by visualinspection; however, it is large enough todemonstrate the feasibility of this approach.It seems clear that the methods describedhere could be scaled-up to accommodatemuch larger graphs.

To implement Step 1 of the algorithm,each vertex i in the graph was associatedwith a random 20-mer sequence of DNAdenoted 0j. For each edge i->j in the graph,an oligonucleotide 0i i was created that wasthe 3' 10-mer of Oi (unless i = 0, in whichcase it was all of 0) followed by the 5'10-mer of Oj (unless j = 6, in which case itwas all of 0). Notice that this constructionpreserves edge orientation. For example,02-3 will not be the same as O3 2. The20-mer oligonucleotide with the sequencethat is Watson-Crick complementary to (9was denoted 0, (Fig. 2).

For each vertex i in the graph (except i -0 and i = 6) and for each edge i->j in thegraph, 50 pmol of 0 and 50 pmol of Oij,respectively, were mixed together in a singleligation reaction (4). The 0 oligonucleo-tides served as splints to bring oligonucleo-tides associated with compatible edges to-gether for ligation (Fig. 2). Hence the liga-tion reaction resulted in the formation ofDNA molecules encoding random pathsthrough the graph.

The scale of this ligation reaction farexceeded what was necessary for the graphunder consideration. For each edge in thegraph, approximately 3 X 1013 copies of theassociated oligonucleotide were added to theligation reaction. Hence it is likely thatmany DNA molecules encoding the Hamil-tonian path were created. In theory, thecreation of a single such molecule would besufficient. As a result, for this graph quanti-ties of oligonucleotides less than an attomolewould probably have been sufficient. Alter-natively, a much larger graph could havebeen processed with the picomole quantitiesused here.

To implement Step 2 of the algorithm,the product of Step 1 was amplified bypolymerase chain reaction (PCR) usingprimers O0 and °6 (5). Thus, only thosemolecules encoding paths that begin withvertex 0 and end with vertex 6 were ampli-fied. To implement Step 3 of the algorithm,the product of Step 2 was run on an agarose

1022

02 TATCGG&TCGGTATATCCGA

0 GCTATTCGAQCTTAAAGCTA

04 GGCTAGGTACCAGCATGT

02-3 GTATATCCQGACTATTCQG

03-4 CTTA&AGCTAQGCTAQGTAC

03 CG&TA&GCTCGA.&TTTCG&T

0,),

Fig. 1. Directed graph. When vi, = 0 and v,,t = 6,a unique Hamiltonian path exists: 0-s1, 1 -s2,2-s3, 3->4, 4-s5, 5-s6.

gel, and the 140-base pair (bp) band (cor-responding to double-stranded DNA en-

coding paths entering exactly seven verti-ces) was excised and soaked in doubly dis-tilled H20 (ddH2O) to extract DNA (6).This product was PCR-amplified and gel-purified several times to enhance its purity.

To implement Step 4 of the algorithm,the product of Step 3 was affinity-purifiedwith a biotin-avidin magnetic beads system.This was accomplished by first generatingsingle-stranded DNA from the double-stranded DNA product of Step 3 and thenincubating the single-stranded DNA with°1 conjugated to magnetic beads (7). Onlythose single-stranded DNA molecules thatcontained the sequence 01 (and hence en-

coded paths that entered vertex 1 at leastonce) annealed to the bound O1 and were

retained. This process was repeated succes-

sively with 02, 03, 04, and 05. To imple-ment Step 5, the product of Step 4 was

amplified by PCR and run on a gel.Figure 3 shows the results of these pro-

cedures. In Fig. 3A, lane 1 is the result ofthe ligation reaction in Step 1. The smear

with striations is consistent with the con-

struction of molecules encoding randompaths through the graph (8). Lanes 2through 5 show the results of the PCR reac-

tion in Step 2. The dominant bands corre-

spond to the amplification of molecules en-

coding paths that begin at vertex 0 and endat vertex 6.

Figure 3B shows the results of a "gradu-ated PCR" performed on the single-strandedDNA molecules generated from the bandexcised in Step 3. Graduated PCR is a meth-od for "printing" results and is performed byrunning six different PCR reactions with theuse of 00 as the right primer and Ol as theleft primer in the ith tube. For example, on

the molecules encoding the Hamiltonianpath 0-1, 1->2, 2-s3, 3->4, 4--5, 5->6,graduated PCR will produce bands of 40, 60,80, 100, 120, and 140 bp in successive lanes.On the molecules encoding the path 0-si,1-s3, 3-s4, 4--5, 5->6, graduated PCR willproduce bands of 40, x, 60, 80, 100, and 120bp in successive lanes, where x denotes the


03,4

j3

Fig. 2. Encoding a graph in DNA. For each vertex iin the graph, a random 20-mer oligonucleotide Oi isgenerated (shown are 02, 03, and 04, for vertices2, 3, and 4, respectively). For edge ij in the graph,an oligonucleotide Oj is derived from the 3' 10-mer of 0, and from the 5' 1 0-mer of Oj (shown are02 3 for edge 2--3 and 034 for edge 3-s4). Foreach vertex i in the graph, 0i is the Watson-Crickcomplement of OC (shown is 03, the complement of03). 03 serves as a splint to bind 02 3 and 034 inpreparation for ligation. All oligonucleotides arewritten 5' to 3', except 03.

absence of a band in lane 2 (correspondingto the omission of vertex 2 along this path).On molecules encoding the path 0->3, 3->2,2->3, 3->4, 4->5, 5-s6, graduated PCR willproduce bands of x, 60, 80-40, 100, 120, and140 bp in successive lanes, where 80-40 de-notes that both a 40-bp and an 80-bp bandwill be produced in lane 3 (corresponding tothe double passage of vertex 3 along thispath). The most prominent bands in Fig. 3Bappear to be those that would arise from thesuperimposition of the bands predicted forthe three paths described above. The bandscorresponding to path 0-sI, 1->3, 3--4,4-s5, 5->6 were not expected and suggestthat the band excised in Step 3 containedcontamination from 1 20-bp molecules.However, such low weight contamination isnot a problem because it does not persistthrough Step 4. Figure 3C shows the resultsof graduated PCR applied to the moleculesin the final product of Step 4. These bandsdemonstrate that these molecules encodethe Hamiltonian path 0-sI, 1 ->2, 2->3,3->4, 4-s5, 5-s6 (9).

This computation required approximately7 days of lab work. Step 4 (magnetic beadseparation) was the most labor-intensive, re-quiring a full day at the bench. In general,with use of the algorithm above the numberof procedures required should grow linearlywith the number of vertices in the graph.The labor required for large graphs might bereduced with use of alternative procedures,automation, or less labor-intensive molecu-lar algorithms.

The number of different oligonucleotidesrequired should grow linearly with the num-ber of edges. The quantity of each oligonu-

start stop

A unique hamiltonian path exists: 0>1,1>2,2>3,3>4,4>5,5>6

C-150-100-50

1 2 3 4 5 6

-150-100-50

1 2 3 4 5 6 7

-150-100-50

1 2 3 4 5 6 7

Fig. 3. Agarose gel electrophoresis of various products of the experiment. (A) Product of the ligationreaction (lane 1), PCR amplification of the product of the ligation reaction (lanes 2 through 5), andmolecular weight marker in base pairs (lane 6). (B) Graduated PCR of the product from Step 3 (lanes 1through 6); the molecular weight marker is in lane 7. (C) Graduated PCR of the final product of theexperiment, revealing the Hamiltonian path (lanes 1 through 6); the molecular weight marker is in lane 7.

cleotide needed is a rather subtle graph the-oretic question (8). Roughly, the quantityused should be just sufficient to insure thatduring the ligation step (Step 1) a moleculeencoding a Hamiltonian path will be formedwith high probability if such a path exists inthe graph. This quantity should grow expo-nentially with the number of vertices in thegraph. The molecular algorithm used herewas rather naive and inefficient, and as withclassical computation, finding improved al-gorithms will extend the applicability of themethod.

As the computation is scaled up, the pos-sibility of errors will need to be looked atcarefully. During Step 1, the occasional liga-tion of incompatible edge oligonucleotidesmay result in the formation of moleculesencoding "pseudopaths" that do not actuallyoccur in the graph. Although such moleculesmay be amplified during Step 2 and persistthrough Step 3, they seem unlikely to sur-vive the separation in Step 4. Nonetheless,at the completion of a computation, it wouldbe prudent to confirm that a putative Ham-iltonian path actually occurs in the graph.During the separation step, molecules encod-ing Hamiltonian paths may fail to bind ad-equately and be lost, whereas molecules en-coding non-Hamiltonian paths may bindnonspecifically and be retained. The latterproblem might be mitigated by more strin-gent or repeated separation procedures. Onemight deal with the former problem by pe-riodically applying PCR with primers de-signed to amplify Hamiltonian paths (in theexample above, primers 00 and U6). Thebalanced use of these techniques may beadequate to control such errors.

The choice of random 20-mer oligonucle-otides for encoding the graph was based onthe following rationale. First, because 42020-mer oligonucleotides exist, choosing ran-domly made it unlikely that oligonucleotidesassociated with different vertices would sharelong common subsequences that might resultin "unintended" binding during the ligationstep (Step 1). Second, it was guessed thatwith high probability potentially deleterious(and presumably rare) features such as severehairpin loops would not be likely to arise.Finally, choosing 20-mers assured that bind-

ing between "splint" and "edge" oligonucle-otides would involve 10 nucleotide pairs andwould consequently be stable at room tem-perature. This approach was successful forthe small graph considered above; however,how to best proceed for larger graphs mayrequire additional research.

What is the power of this method ofcomputation? It is premature to give defini-tive answers; however, some remarks seem inorder. A typical desktop computer can exe-cute approximately 106 operations per sec-ond. The fastest supercomputers currentlyavailable can execute approximately 1012operations per second. If the ligation (con-catenation) of two DNA molecules is con-sidered as a single operation and if it isassumed that about half of the approximately4 X 1014 edge oligonucleotides in Step 1were ligated, then during Step 1 approxi-mately 1014 operations were executed.Clearly, this step could be scaled-up consid-erably, and 1020 or more operations seemsentirely plausible (for example, by using mi-cromole rather than picomole quantities).At this scale, the number of operations persecond during the ligation step would exceedthat of current supercomputers by more thana thousandfold. Furthermore, hydrolysis of asingle molecule of adenosine triphosphate toadenosine monophosphate plus pyrophos-phate provides the Gibbs free energy (AG =-8 kcal mol-1) for one ligation operation(10, 11 ); hence in principle 1 J is sufficientfor approximately 2 X 1019 such operations.

This is remarkable energy efficiency, con-sidering that the second law of thermody-namics dictates a theoretical maximum of 34x 1019 (irreversible) operations per joule (at300 K) (12, 13). Existing supercomputers arefar less energy-efficient, executing at most109 operations per joule. The energy con-sumed during other parts of the molecularcomputation, such as oligonucleotide syn-thesis and PCR, should also be small incomparison to that consumed by current su-percomputers. Finally, storing information inmolecules of DNA allows for an informationdensity of approximately 1 bit per cubicnanometer, a dramatic improvement overexisting storage media such as videotapes,which store information at a density of ap-


proximately 1 bit per 1012 nm3.Thus, the potential of molecular compu-

tation is impressive. What is not clear iswhether such massive numbers of inexpen-sive operations can be productively used tosolve real computational problems. One ma-jor advantage of electronic computers is thevariety of operations they provide and theflexibility with which these operations canbe applied. Whereas two 100-digit integerscan be multiplied quite efficiently on anelectronic computer, it would be a dauntingtask to do such a calculation on a molecularcomputer using currently available protocolsand enzymes (14).

Nonetheless, for certain intrinsicallycomplex problems, such as the directedHamiltonian path problem where existingelectronic computers are very inefficient andwhere massively parallel searches can be or-ganized to take advantage of the operationsthat molecular biology currently provides, itis conceivable that molecular computationmight compete with electronic computationin the near term. It is a research problem ofconsiderable interest to elucidate the kindsof algorithms that are possible with the use ofmolecular methods and the kinds of prob-lems that these algorithms can efficientlysolve (12, 15, 16).

For the long term, one can only speculateabout the prospects for molecular computa-tion. It seems likely that a single molecule ofDNA can be used to encode the "instanta-neous description" of a Turing machine (17)and that currently available protocols andenzymes could (at least under idealizedconditions) be used to induce successivesequence modifications, which would cor-respond to the execution of the machine.In the future, research in molecular biol-ogy may provide improved techniques formanipulating macromolecules. Researchin chemistry may allow for the develop-ment of synthetic designer enzymes. Onecan imagine the eventual emergence of ageneral purpose computer consisting ofnothing more than a single macromole-cule conjugated to a ribosomelike collec-tion of enzymes that act on it.


1. R. P. Feynman, in Minaturization, D. H. Gilbert, Ed.(Reinhold, New York, 1961), pp. 282-296.

2. M. R. Garey and D. S. Johnson, Computers andIntractability (Freeman, San Francisco, CA, 1979).

3. R. M. Karp, in Complexity of Computer Computa-tions, R. E. Miller and J. W. Thatcher, Eds. (Plenum,New York, 1972), pp. 85-103.

4. Each oligonucleotide (50 pmol) with 5'-terminal phos-phate residue, 5 units of T4 DNA ligase (Boehringer-Mannheim, Germany), ligase buffer, and ddH20 to atotal volume of 100 [.1 was incubated for 4 hours atroom temperature.

5. All PCR amplifications were performed on a Perkin-Elmer (Norwalk, CT) 9600 thermal cycler. For amplifi-cation in Step 2,50 pmol of each primer and 5 units ofTaq DNA polymerase (Gibco-BRL, Grand Island, NY)in PCR buffer to a total volume of 50 [LI were pro-cessed for 35 cycles at 940C for 15 s and at 300C for

1023

illonis ----

Formation of DNA molecules encode possible paths

PCR & cut gel to select paths starting at 0, ending at 6

DNA paths entering all vertices found by 6 washings with magnetic beads

7 days of lab work

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competing with silicon

embedding control in molecular systems and cells

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AND/OR logic gates Source:

Qian & Winfree 2011

represented by a wire. Each side of the node canbe connected to any number of wires. Each wireconnects two different sides of two nodes. Eachred number indicates one DNA species with itsinitial relative concentration: Each number on awire corresponds to a free signal strand; eachnumber within a node at the end of a wire cor-responds to a bound signal strand (positive num-ber) or a threshold that absorbs a signal when itarrives at the gate (negative number). A reporterthat transforms a DNA signal into a fluorescencesignal is represented by half a node with a zigzagarrow (Fig. 1B), with its initial relative concen-tration written similar to a threshold.

Each signal is a single-stranded DNA mole-cule that has two recognition domains identify-ing the two gates it connects, one on either sideof a central toehold domain. Each gate is asso-ciated with a gate base strand that has (the com-plement of ) one recognition domain flanked bytwo toehold domains. When a signal strand isbound to a gate, it forms a gate:signal complexwith the gate’s base strand. At any given mo-ment (not counting the transient states duringreactions shown in fig. S1C), a gate base strandalways has a signal strand bound to one side,leaving the toehold on the other side uncovered.

There are three basic reactions involved ina seesaw network (Fig. 1C and fig. S1C). Thefirst one is seesawing: A free signal on one sideof a gate can release a signal bound on the otherside of the gate by toehold-mediated strand dis-placement. The process starts with the free signalstrand (e.g., w2,5) hybridizing to the gate:signalcomplex (e.g., G5:5,6) at the uncovered toeholddomain (e.g., T*) and then undergoing branch mi-gration through the recognition domain (e.g., S5).The previously bound signal will fall off when itis attached to the gate base strand only by theshort toehold. The resulting gate:signal complex(e.g., G2,5:5) will have an uncovered toehold onthe other side, and therefore the now-free signal(e.g.,w5,6) can reverse the process symmetrically.The second reaction is thresholding: A thresh-old species associated with a gate and an imping-ing signal can react with the signal by meansof a longer toehold (e.g., s2*T*), producingonly inert waste species that have no exposedtoehold. Thresholding is much faster than see-sawing because the toehold-mediated strand dis-placement rate grows exponentially with toeholdlength for short toeholds (7, 8). As a result, see-sawing effectively only happens when the inputsignal exceeds the threshold. The third reactionis reporting: A reporter species similar to a thresh-old, but modified with a fluorophore and quench-er pair, can absorb an impinging signal whilegenerating a fluorescence signal. Unlike thresh-olding, reporting does not compete with seesaw-ing, and it therefore does not require a longertoehold.

DNA signals can play different roles such asinput (signals that arrive at a gate), output (signalsthat are produced by a gate), and fuel (signals thathelp to catalytically produce the output). One see-

saw gate with a few wires can create a catalyticcycle in which input transforms free fuel into freeoutput without being consumed in the process(Fig. 1D and fig. S1, B and C). Initially, the outputsignal is bound to the right side of the gate; theinput and fuel signals are free (in our analogy, theoutput is riding on the right side of the seesawboard; the input and fuel are wandering around).The input signal first releases the output signal andbinds to the gate instead (the input jumps onto theleft side of the board and makes the output jumpoff). The fuel signal then displaces the input signalby binding to the gate in the same way (the fuelpushes off the input). A catalytic cycle has beencompleted. In general, a free signal on one side ofa seesaw gate can catalyze the exchange of signalson the other side, and this exchange will not hap-pen without the catalyst. These reactions are driv-en forward by the entropy of equilibration for theseesawing reactions. A small amount of free inputcan catalyze the release of a large amount of freeoutput (fig. S2).

Thresholding can be directly combinedwith aseesaw catalyst to support a digital abstraction—which is the basic principle underlying digital

logic in electronics—by pushing the intrinsicallyanalog signal toward either the ideal ON or OFFvalue. Fluorescence kinetics experiments (Fig. 1E)demonstrated the circuit in Fig. 1A connected tothe reporter in Fig. 1B. The input-versus-outputrelationship (plotted in Fig. 1F) reveals a sharpthreshold, ideal for signal restoration.

A cascade of two seesaw gates can computethe logic function OR or AND. To explain this,we introduce two composable seesaw compo-nents for digital circuits. We first define thegross production of signal X as the total amounteventually released from the gate:

⟨X ⟩ ¼ ∫þ∞

0X prodðtÞdt ð1Þ

Motivated by sequence design constraints (figs. S3and S4), we then define two types of feedforwardseesaw gates, each assuming an irreversible down-stream drain. The first type is called an amplifyinggate. It has a threshold and fuel. If the grossproduction of its input is greater than the initialamount of threshold, the output will keep beingreleased catalytically until it reaches the max-imum, which is the initial amount of bound

Fig. 2. Digital logic gates implemented with the seesaw DNA motif. (A) Abstract diagram of aseesaw circuit that computes either OR or AND, depending on the initial concentration of thethreshold. Input signals x1 (w1,2) and x2 (w3,2) are summed together at gate 2 and, if they exceedthe threshold, are amplified by gate 5 to generate output signal y (w5,6), which is reported by theROX fluorophore in reporter 6. (B) Domain-level DNA implementation of the two-input AND or ORgate. (C) Kinetics experiments. Input strands were at 0.1× (0, logic OFF) or 0.9× (1, logic ON),where 1× = 100 nM. Sequences of strands are listed in tables S2 and S3, circuit 3. Experimentswere performed at 20°C.

3 JUNE 2011 VOL 332 SCIENCE www.sciencemag.org1198

REPORTS

represented by a wire. Each side of the node canbe connected to any number of wires. Each wireconnects two different sides of two nodes. Eachred number indicates one DNA species with itsinitial relative concentration: Each number on awire corresponds to a free signal strand; eachnumber within a node at the end of a wire cor-responds to a bound signal strand (positive num-ber) or a threshold that absorbs a signal when itarrives at the gate (negative number). A reporterthat transforms a DNA signal into a fluorescencesignal is represented by half a node with a zigzagarrow (Fig. 1B), with its initial relative concen-tration written similar to a threshold.

Each signal is a single-stranded DNA mole-cule that has two recognition domains identify-ing the two gates it connects, one on either sideof a central toehold domain. Each gate is asso-ciated with a gate base strand that has (the com-plement of ) one recognition domain flanked bytwo toehold domains. When a signal strand isbound to a gate, it forms a gate:signal complexwith the gate’s base strand. At any given mo-ment (not counting the transient states duringreactions shown in fig. S1C), a gate base strandalways has a signal strand bound to one side,leaving the toehold on the other side uncovered.

There are three basic reactions involved ina seesaw network (Fig. 1C and fig. S1C). Thefirst one is seesawing: A free signal on one sideof a gate can release a signal bound on the otherside of the gate by toehold-mediated strand dis-placement. The process starts with the free signalstrand (e.g., w2,5) hybridizing to the gate:signalcomplex (e.g., G5:5,6) at the uncovered toeholddomain (e.g., T*) and then undergoing branch mi-gration through the recognition domain (e.g., S5).The previously bound signal will fall off when itis attached to the gate base strand only by theshort toehold. The resulting gate:signal complex(e.g., G2,5:5) will have an uncovered toehold onthe other side, and therefore the now-free signal(e.g.,w5,6) can reverse the process symmetrically.The second reaction is thresholding: A thresh-old species associated with a gate and an imping-ing signal can react with the signal by meansof a longer toehold (e.g., s2*T*), producingonly inert waste species that have no exposedtoehold. Thresholding is much faster than see-sawing because the toehold-mediated strand dis-placement rate grows exponentially with toeholdlength for short toeholds (7, 8). As a result, see-sawing effectively only happens when the inputsignal exceeds the threshold. The third reactionis reporting: A reporter species similar to a thresh-old, but modified with a fluorophore and quench-er pair, can absorb an impinging signal whilegenerating a fluorescence signal. Unlike thresh-olding, reporting does not compete with seesaw-ing, and it therefore does not require a longertoehold.

DNA signals can play different roles such asinput (signals that arrive at a gate), output (signalsthat are produced by a gate), and fuel (signals thathelp to catalytically produce the output). One see-

saw gate with a few wires can create a catalyticcycle in which input transforms free fuel into freeoutput without being consumed in the process(Fig. 1D and fig. S1, B and C). Initially, the outputsignal is bound to the right side of the gate; theinput and fuel signals are free (in our analogy, theoutput is riding on the right side of the seesawboard; the input and fuel are wandering around).The input signal first releases the output signal andbinds to the gate instead (the input jumps onto theleft side of the board and makes the output jumpoff). The fuel signal then displaces the input signalby binding to the gate in the same way (the fuelpushes off the input). A catalytic cycle has beencompleted. In general, a free signal on one side ofa seesaw gate can catalyze the exchange of signalson the other side, and this exchange will not hap-pen without the catalyst. These reactions are driv-en forward by the entropy of equilibration for theseesawing reactions. A small amount of free inputcan catalyze the release of a large amount of freeoutput (fig. S2).

Thresholding can be directly combinedwith aseesaw catalyst to support a digital abstraction—which is the basic principle underlying digital

logic in electronics—by pushing the intrinsicallyanalog signal toward either the ideal ON or OFFvalue. Fluorescence kinetics experiments (Fig. 1E)demonstrated the circuit in Fig. 1A connected tothe reporter in Fig. 1B. The input-versus-outputrelationship (plotted in Fig. 1F) reveals a sharpthreshold, ideal for signal restoration.

A cascade of two seesaw gates can computethe logic function OR or AND. To explain this,we introduce two composable seesaw compo-nents for digital circuits. We first define thegross production of signal X as the total amounteventually released from the gate:

⟨X ⟩ ¼ ∫þ∞

0X prodðtÞdt ð1Þ

Motivated by sequence design constraints (figs. S3and S4), we then define two types of feedforwardseesaw gates, each assuming an irreversible down-stream drain. The first type is called an amplifyinggate. It has a threshold and fuel. If the grossproduction of its input is greater than the initialamount of threshold, the output will keep beingreleased catalytically until it reaches the max-imum, which is the initial amount of bound

Fig. 2. Digital logic gates implemented with the seesaw DNA motif. (A) Abstract diagram of aseesaw circuit that computes either OR or AND, depending on the initial concentration of thethreshold. Input signals x1 (w1,2) and x2 (w3,2) are summed together at gate 2 and, if they exceedthe threshold, are amplified by gate 5 to generate output signal y (w5,6), which is reported by theROX fluorophore in reporter 6. (B) Domain-level DNA implementation of the two-input AND or ORgate. (C) Kinetics experiments. Input strands were at 0.1× (0, logic OFF) or 0.9× (1, logic ON),where 1× = 100 nM. Sequences of strands are listed in tables S2 and S3, circuit 3. Experimentswere performed at 20°C.

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NATURE CHEMISTRY | VOL 3 | FEBRUARY 2011 | www.nature.com/naturechemistry 111

The focus on a single material and the systematic application of a few basic design principles means that different devices are compat-ible with each other and can be modularly integrated into increas-ingly complex assemblies.

Functional nucleic acids, such as ribozymes and aptamers, can be used to broaden the set of chemistries that can be controlled by strand-displacement circuits62,66,74. Systems combining protein func-tion with strand displacement further increase the variety of behav-iours that can be programmed52,57.

Devices with practical applications may benefit from inte-grating the programmability of nucleic acids with physical and/or chemical properties of other materials8. Liu and co-workers, for example, used the specific hybridization of nucleic acids to

speed up the reactions of organic molecules functionalized to the complementary strands, in a process known as DNA-templated synthesis108. Mirkin and co-workers directed the aggregation of gold nanoparticles using DNA hybridization109. Le et al. showed improved control over gold nanoparticle positioning by using DNA self assembly110. Maune et al. assembled crossbar connec-tions between carbon nanotubes by using DNA origami as a tem-plate, and demonstrated an ensemble with field-effect transistor properties111. Inclusion of strand displacement in the design of these DNA scaffolds could potentially allow precise modulation of metamaterial behaviour46.

Control of gene expression is a primary goal of synthetic biol-ogy; dynamic DNA nanotechnology provides a practical approach

Figure 6 | An autonomous, processive and directional DNA walker based on strand displacement95. a, Fuel hairpins H1 and H2 are present in solution, and react with the track to push the walker forward. In the absence of the walker, however, the fuels do not react with the track. The track behind the walker is different from that ahead of the walker. b, Schematic of the walker taking one step. Hairpin H1 displaces the hind leg of the walker through a series of strand-displacement reactions. The lengths of the walker legs constrain the walker such that the freed hind leg cannot bind any track molecule other than that directly in front of it. The hind leg is now the leading leg, and H2 can initiate a similar reaction to drive the new hind leg forward.

a

b

11 *5*

6*4

3*2*

1*76*3*

41 1

521

11*5*2*

1*

Track base

Track base Track base Track base

Track baseTrack base

1*1

7 2*3*

1*1

6711*

5*15

6*3411*

2*12

3*6711*

5*15

6*7*1*

4*1* 1*

7*

17*5

6

Walker Hairpin H1 Hairpin H2

1*2* 5*

1*125

114

3* 6* 7 2*3* 3*

2*76*3*

152

1 1*5*2*

1* 14

12

33411*

2* 5*1*1

2 51

3* 6* 7 2*3*

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3*2*76*3*

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1 1*5*2*

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1*1

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Directionof motion

32 4*

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32 4*

REVIEW ARTICLENATURE CHEMISTRY DOI: 10.1038/NCHEM.957

nchem_.957_FEB11.indd 111 11/1/11 11:08:51


Source: Omabegho, T., Sha, R. & Seeman, N. C. 2009Molecular

walkers

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Universal substrate for reaction kinetics

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 José 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 molecularand chemical systems of autonomous function and ever-increasingcomplexity. A key goal is to develop embedded control circuitrywithin a chemical system to direct molecular events. Here we showthat systems of DNA molecules can be constructed that closely ap-proximate the dynamic behavior of arbitrary systems of coupledchemical reactions. By using strand displacement reactions as aprimitive, we construct reaction cascades with effectively unimole-cular and bimolecular kinetics. Our construction allows individualreactions to be coupled in arbitrary ways such that reactants canparticipate in multiple reactions simultaneously, reproducing thedesired dynamical properties. Thus arbitrary systems of chemicalequations can be compiled into real chemical systems. We illustrateour method on the Lotka–Volterra 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 analyze

chemical 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 formation—all 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, canwe find a set of actual molecules M1;M2;…;Mm that interactin an appropriate buffer to approximate the formal system’smass-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 (5–9).

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 speciesmay 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 experimentalwork in chemical and biological engineering has started withparticular molecular systems—genetic 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(20–22), 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 onthe 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 (30–32). These powerfulreaction primitives have been used previously for designingnucleic-acid-based molecular machines with complex behaviors,such as motors, logic gates, and amplifiers (33–37). 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.

www.pnas.org/cgi/doi/10.1073/pnas.0909380107 PNAS ∣ March 23, 2010 ∣ vol. 107 ∣ no. 12 ∣ 5393–5398

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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, 0–3. 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 inwhich 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

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

ideal DNA

Fig. 5. Lotka–Volterra 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 σ2 < σ (σ ¼ σ1 ¼ k1 and σ2 ¼ 0). Maximum stranddisplacement rate constant qmax ¼ 106 M−1 s−1 and initial concentration of auxiliary species Gi , Ti , Li , Bi , LSj , and BSj is Cmax ¼ 10 μM. Buffering-scaling factorγ−1 ¼ qmaxðqmax − σÞ−1 ¼ 2. The initial concentrations of strands X1 and X2 introduced into the system is γ−120 nM ¼ 40 nM and γ−110 nM ¼ 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

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Fig. 6. Examples showing more complex behavior. In all the maximum strand displacement rate constant qmax ¼ 106 M−1 s−1 and the initial concentration ofauxiliary species Cmax ¼ 10 μM. See Figs. S4 and S5 and SI Text for the rate constants used in A–D, as well as the details of C and D. Plots show the ideal CRN(Dashed lines) and the DNA reactions (Solid lines).

Soloveichik et al. PNAS ∣ March 23, 2010 ∣ vol. 107 ∣ no. 12 ∣ 5397

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DNA Computing

NAM data storage









































Leonard M. Adleman













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366 NATURE MATERIALS | VOL 15 | APRIL 2016 | www.nature.com/naturematerials

commentary

Nucleic acid memoryVictor Zhirnov, Reza M. Zadegan, Gurtej S. Sandhu, George M. Church and William L. Hughes

Nucleic acid memory has a retention time far exceeding electronic memory. As an alternative storage media, DNA surpasses the information density and energy of operation offered by flash memory.

Information and communication technologies generate vast amounts of data that will far eclipse today’s data flows

(Fig. 1). Memory materials must therefore be suitable for high-volume manufacturing. At the same time, they must have elevated information stability and limit the energy consumption and trailing environmental impacts that such flows will demand. Analysts estimate that global memory demand — at 3 × 1024 bits — will exceed projected silicon supply in 2040 (Fig. 1b and Supplementary Information sections 1 and 2). To meet such requirements, flash-memory manufacturers would need ~109 kg of silicon wafers even though the total projected wafer supply is ~107–108 kg (Supplementary Figs 1 and 2). Such forecasts motivate an exploration of unconventional materials with cost-competitive performance attributes. With information retention times that range from thousands to millions of years, volumetric density 103 times greater than flash memory and energy of operation 108 times less, we believe that DNA used as a memory-storage material in nucleic acid memory (NAM) products promises a viable and compelling alternative to electronic memory.

In this Commentary, we discuss the information retention, density and energetics of NAM — specifically related to DNA — for non-biological and non-volatile memory applications, ranging from letters to libraries. The potential of NAM has often been dismissed, as nucleic acids are believed by some to be fragile and therefore unreliable. This is not the case. For example, the room-temperature half-life of ancient DNA exceeds 100 years1,2. Indeed, the complete genomes of an ~50,000-year-old Neanderthal3 recovered from Siberia and an ~700,000-year-old horse4 recovered from the Arctic permafrost (approximate average temperature –4 °C) have been sequenced. Still, the long-term stability of DNA and its decay kinetics are poorly understood at a per-bit (that is, base) level. As an energy-barrier model shows (Methods), DNA has a retention time far exceeding electronic memory, and it can store information reliably over time. Through first-principle calculations, DNA has been validated as a model material for future NAM products (Supplementary Information section 8). Therefore, we call for increased cooperation between the biotechnology and semiconductor sectors to pair previously

unfathomable technological advances — such as those from the Human Genome Project — with the scaling expertise of the semiconductor industry.

Nucleic acid memory as a materialAs a material, nucleic acids are negatively charged polyelectrolytes with four monomers (the nucleotides A, T or U, C and G). Monomers are covalently bonded to form polymer chains. Once polymerized, an individual chain can hydrogen-bond with itself or with other chains that satisfy base complementarity. These attributes endow nucleic acids with the power of molecular self-assembly, which is made possible by the thermal fluctuations between complementary hydrogen bonds during Watson–Crick hybridization. During DNA hybridization, adenine (A) forms a base-pair with thymine (T), and guanine (G) pairs with cytosine (C). In RNA, thymine is substituted by uracil (U). By encoding sequence complementarity, molecular self-assembly can be exploited to pull nucleic acids like a rope5, weave them like a fabric6,7, decorate them like a scaffold8,9 and recycle10 them like a thermoplastic. Beyond their recyclability, nucleic acids and potential

iii iii

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Zhirnov et al. 2016

10cm

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Abstract Data Types

Container

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Record occurence of events in a cell

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b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

Push binds by toehold


stack empty

Recording signals

DNA Molecular Stack Recorderc a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

Branch migration


stack empty

Recording signals


a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

X


stack empty

X

Recording signals


a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

XHybridisation


1 signal on stack

XRecording signals


a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

Push

X


1 signal on stack

XRecording signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

XPush binds by toehold


1 signal on stack

XRecording signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

XBranch migration


1 signal on stack

XRecording signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

X

YSignal Y


1 signal on stack

X

Y

Recording signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

X YHybridisation


2 signals on stack

XYRecording signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

Push

X Y


2 signals on stack

XYRecording signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

X YPush binds by toehold


2 signals on stack

XYRecording signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

Branch migrationX Y


2 signals on stack

XYRecording signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

X Y

XSignal X


2 signals on stack

XY

X

Recording signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

X Y XHybridisation


3 signals on stack

XYXRecording signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

X Y XHybridisation


3 signals on stack

XYXPopping signals


c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Y XX

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Read


3 signals on stack

XYXPopping signals


c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Y XX

Read binds by toehold,Branch migration follows


3 signals on stack

XYXPopping signals

DNA Molecular Stack Recorderc a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Y

X

X Strand displacement

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Signal X released


2 signals on stack

XY

X

Popping signals


c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Y

X

X

Pop

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*


2 signals on stack

XYPopping signals


c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Y

X

X

Pop binds by toehold,Branch migration follows


2 signals on stack

XYPopping signals


c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Y

X

X Hairpinreforms

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Pop-Pushdouble strand


2 signals on stack

XYPopping signals


c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Y

X

X

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Read


2 signals on stack

XY

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Popping signals


c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Y

X

X



2 signals on stack

XY

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Popping signals


c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Y

X

X Strand displacement

Signal Y released


1 signal on stack

X

Y

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Popping signals


c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

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c a b a*

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c* a* b* a d* f* g* f e*

d* e* c*

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c* a* b* a d* f* g* f e*

YX

X



1 signal on stack

Xc a b a*

c* a* b* a d*f* f

g*

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d e c a b a*

c* a* b* a d*f* f

g*

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c a b a*

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d* e* c*

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c a b a*

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ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Popping signals


c* a* b* a d*f* f

g*

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d e c a b a*

c* a* b* a d*f* f

g*

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d e c a b a*

c* a* b* a d*f* f

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c a b a*

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c a b a*

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c a b a*

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c a b a*

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c a

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c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Strand displacement Signal X released


stack empty

X

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

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c a b a*

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c* a* b* a d* f* g* f e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

Popping signals

Naive Chemical View

S + P + Xb'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Real Well-Mixed Chemistry

S + P + Xb'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

— partly formed complexes— partially bound complexes

— unintended side reactions

— DNA complexes have finite diffusion and reaction rates

+ P b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

+ Xb'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Sb'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

"1-6�>-.71�2:<�=-8;71�� /"1-6

$5C1*.;+

�:9/�*;3�D7+

�:7-<5>B*;8:7�7+

!.=1<@->5:9=

�<1- �753910��53<->5:9%581�*=+

"1-6�1534>

"1-6(50>4

��:2%:>-7

%581/:<<1/>10�-<1-

� � �� =E4@!0@94@

��

� �� )>>4@

!0@94@ ��

#135:9�>-.71�2:<�=-8;71�� /�<:8*.;+

%:�*.;+ �:<<��<1-

��:2%:>-7

�@1<-31�$5C1*.;+

$5C1�05=><5.?>5:9�59�'�*�+

�:9/�*;3�D7+

�:7-<5>B*;8:7�7+

�:7:<

� ��

"2

!@1<-77�#1=?7>=�2:<�=-8;71�� /

�$��!�� "��$�� "��

��!!��!��

��F>4@B��'�� I��=>G@867B�� 68:4<B�(427<=:=684A��<2� $@8<B43� ��

��1A;1<>,�534�$19=5>5@5>B�� ==-B,��,��,�� A-0 $064 =5� ��

�@40B43�!=385843�

�� 0B0�$0B7�

�867�'4<A8B8D8BG��"��AA0G��-��67�'4<A8B8D8BG��"��AA0G/��/��/��F03

�AA0G��:0AA�

�71/><:;41<:3<-8�$?88-<B��:9>59?10��Real Well-Mixed Chemistry

Required: Rule-based stochastic model to rigorously capture the reactions happening in-vitro

To know if stack chemistry is operating correctly:

core reactions

Which Rules? — Reaction Space

all reactionswith full domains

bound

all possible reactions,including partially bound

complexes

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

core reactions

all reactionswith full domains

bound

all possible reactions,including partially bound

complexes

Microsoft DSD 2.0


Multi-strandthermodynamic

prediction software

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

core reactions


pop

push

read

X…stack

stack…push

stack…X…stack

stack…Xpush

stack…push

stack…push

Core Reactions: 1-Step Rulesstack…

…stack

start push

X

readRate parameters

pop

start

…stack

push

X…stack

+

+

+

(1)

(2)

(3)

(4) +…stack + X

…stack +…stack + poppush

Recording

Popping

kf(h)⌦kr

1

kf(h+b)⌦kr

1

kf(h+b)⌦kr

1

!

!

kf(h) = 106M�1s�1

kf(h+b) = 105M�1s�1

core reactions

(5)kf(s1) = 105M�1s�1

kf(s2) = 104M�1s�1

kf(s2)

kf(s1)

kr = 0 (No ring complexes allowed)

Simulation ResultsRecording signals

t = 5 min t = 1 hour

S200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mP200nM

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

spsp

p


S200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mP200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

X200nM

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

spx

spxpx

spx



S200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mP200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

X200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mP200nM

spx spxpxp


spxp

psp

spxp

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b


S200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mP200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

X200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mP200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

X200nM


spx

spxpx

spxpxpx

spxpxpxpx

spxpx

spxpxpx

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b


S200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mP200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

X200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mP200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

X200nM

P200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

pspxp

spxpx

spxpxp

spxpxpx

spxpxpxp

spxpxpxpxp

spxpxp

spxpxpxp


S200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mP200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

X200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mP200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

X200nM

P200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mX200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

spxpxpx

spxpx spxpxpxpxspxpxpxpxpx

spxpxpxpxpxpx

spxpxpx

spxpxpxpxspxpx

Simulation Results

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Popping signals


[xr]

spxpspxpx

spxpxp spxpxpspxpxpx

spxpxpxp

spxpxpxpxspxpxpxpxp

[xr]

spxp spxpxpxp

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

Simulation Results

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Q200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Popping signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

q

[xr]

spx

spxpspxpx

spxpxp

spxpxpxspxpxpxp

spxpxpxpxspxpxpxpxp

[xr][pq]

[pq]

spxpx

spxpxpspxpxpx

Simulation Results

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Q200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Popping signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

r qsp

[xr][pq]

spxp

spxpx

spxpxp

spxpxpxp

spxpxpxpxpsp

[xr][pq]

spxp

spxpxp

Simulation Results

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Q200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Q200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Popping signals


[xr][pq]

[xr][pq]

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

spx

spxspxp

spxpx

spxpxpspxpxpx

spxpxpxpspxpxpxpxp spxp

spxpx

Simulation Results

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Q200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Q200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Popping signals


c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

[xr][pq]

sp

[xr][pq]

spxp spxpxpspxpxpxp

sp spxp

Simulation Results

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Q200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Q200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

R200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Q200nM

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

Popping signals


c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

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d e c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

d* e* c*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*e*d*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

d* e* c*

ca a*

b

df f*

g

e

c a b a*

c* a* b* a d*f* f

g*

e*

e f*g

fd a*b

ac

c a

a*b

c a b a* d f g f* e

c* a* b* a d* f* g* f e*

sqsp

[xr][pq][xr]

[pq]

sqsp spxp spxpxp

spxpxpxp

Experimental Results

●But results are hard to interpret because of the superposition of stacks.●Concentration of read and pop needs to be controlled.

1 signal 2 signals 3 signals 1 signal2 signals

SPXSPXP

SPXPXSPXPXP

SPXPXPX

P RX PQ

Reading and Popping


Recording Popping

Selected bioanalyzer results qualitatively show same pattern as simulations

PAGE gel results still pose some questions…!

P P P P P

X XX X

XP

S S

P P P

S S S

S dimer

S dimer

S

SP SP SP SP SP

SPP?

SSP?

S

S dimer

P

SP

SSP?

X

SPP?

XP

P+X S+P+X+P S+PS+X+P+P

Simulation ResultsBrick order matters!

t = 1 hour t = 1 hour

S

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

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read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

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x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

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x'm

P

b'

aa'd'c'

g'

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a'b

c a

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d ehh'

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start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

X

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'mP

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

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e

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x'm

X

spxpx

spxpxpx

S

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

X

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

X

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

x'm

P

b'

aa'd'c'

g'

ff'e'c a

a'b

c a

a'b

d ehh'

lk

x

d' c'e'

start push write-X

read pop

b

a'a

dc

g

f'f

e

report

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P

In order Out of order

spxpxspxpxpx

spx

s

spxpxpxpxspxpxpxpxpx

spxpxpxpxpxpxspxpxpxpxpxpxpx

Challenges Establishing In-Silico/In-Vitro Equivalence

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EXPERIMENT

“a qualitative agreement exists”

“pure” output:all species and

concentrations known

“proxy” output:chemistry state encoded by indirect variables

y scale: nt y scale: migration time


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EXPERIMENT

y scale: nt y scale: migration time


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EXPERIMENT

gel band positions

polymer gel

gel band intensitiesWhat is the mapping from brick

structure to migration time through the gel?

What is the mapping from mass conc of a brick to the FU of a band?

GG

GA

GA

GC

CA

UC

UC

CC

GC

AC

AC

AC

UU

CG

GG

AG

AC

CA

AAU

UA

GU

AG

GUAG

ACAAA

AA

AAGACCG

CUAAA

CU C U A A U

CAC A

CC

UA

CU

AA

UA

CA

CC

AC

UU

CG

GG

AG

AG

CC

AU

CU

CC

CG

CA

CA

CA

CU

UC

GG

GA

GA

CC

AAAUUAGUAGGUA

GAC

AAAAAAA

GACCG

CU A A A C U

CUAAU CAC

A C C U A C U A A UA

CA

CC

AC

UU

CG

GG

AG

AG

CC

AU

CU

CC

CG

CA

CA

CA

CU

UCU U

GAAGUGUGUGCGGGAGAUGGCUCUCCCGAAGUGGUGU C

CGCCG

GG C A

GCGGCGGU

UG

GU

CU

CC

CG

AA

GU

GU

GU

GC

GG

GA

GA

UG

GC

UC

UC

CC

GA

AG

UG

GU

GU CCGCCG

G G CA

GCGGCGGU

UG

GU

CU

CC

CG

AA

GU

GU

GU

GC

GG

GA

GA

UG

GC

UC

UC

CC

GA

AG

UG

GU

GU C C G C C G G G

CAG

CGGCGGU U G G U C U C C C

G G G A G A GC

CAUCUCCC

GC

ACACA

CU

U C GGGAGACCAAAUUAGUAGGU

AGA C A A A

AAAAG

ACCGCUAA

AC

UCUA

A UCA

CA

CC

UA

CU

AA

U

AC

AC

CA

CU

UC

spxpxpx

y scale: nt

(ladder is for dsDNA only)

y scale: migration time

Electrophoresis

Washing Steps

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a

a*b

c*a* a

b*

d*f* f

g*

e*

ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a

a*b

c*a* a

b*

d*f* f

g*

e*

c a b a*

c* a* b* a d*f* f

g*

e*

d e ca a*

b

c*a* a

b*

d*f* f

g*

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c a b a*

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g*

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d e c a b a*

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g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

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g*

e*

d e c a

a*b

c*a* a

b*

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g*

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c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

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g*

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d e ca a*

b

c*a* a

b*

d*f* f

g*

e*

c a b a*

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g*

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d e c a b a*

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g*

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d e c a b a*

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g*

e*

d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

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d e c a b a*

c* a* b* a d*f* f

g*

e*

d e c a b a*

c* a* b* a d*f* f

g*

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d e c a

a*b

c

a

a*b

c*a*

ab*

d*f*f

g*e*

caa*

b

c*a*

ab*

d*f*f

g*e*

c

a

b

a*

c*a*

b*a

d*f*f

g*e*

d

e

c

a

a*b

c

a

b

a*

c*a*

b*a

d*f*f

g*e*

d

e

c

a

a*b

c*a*

ab*

d*f*f

g*e*

c

a

b

a*

c*a*

b*a

d*f*f

g*e*

d

e

caa*

b

c*a*

ab*

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g*e*

c

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b

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g*e*

d

e

c

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b

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d

e

c

a

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c

a

b

a*

c*a*

b*a

d*f*f

g*e*

d

e

c

a

b

a*

c*a*

b*a

d*f*f

g*e*

d

e

c

a

a*b

c*a*

ab*

d*f*f

g*e*

c

a

b

a*

c*a*

b*a

d*f*f

g*e*

d

e

c

a

b

a*

c*a*

b*a

d*f*f

g*e*

d

e

caa*

b

c*a*

ab*

d*f*f

g*e*

c

a

b

a*

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b*a

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g*e*

d

e

c

a

b

a*

c*a*

b*a

d*f*f

g*e*

d

e

c

a

b

a*

c*a*

b*a

d*f*f

g*e*

d

e

c

a

a*b

c

a

b

a*

c*a*

b*a

d*f*f

g*e*

d

e

c

a

b

a*

c*a*

b*a

d*f*f

g*e*

d

e

c

a

b

a*

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b*a

d*f*f

g*e*

d

e

c

a

a*b

ca

a* b

c*a* a

b*

d*f*f

g*

e*

caa*

b

c*a* a

b*

d*f*f

g*

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ca

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c*a*

b*a

d*f*f

g*

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d

ec

a

a* b

ca

ba*

c*a*

b*a

d*f*f

g*

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d

ec

a

a* b

c*a* a

b*

d*f*f

g*

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ca

ba*

c*a*

b*a

d*f*f

g*

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d

eca

a*

b

c*a* a

b*

d*f*f

g*

e*

ca

ba*

c*a*

b*a

d*f*f

g*

e*

d

ec

ab

a*

c*a*

b*a

d*f*f

g*

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d

ec

a

a* b

ca

ba*

c*a*

b*a

d*f*f

g*

e*

d

ec

ab

a*

c*a*

b*a

d*f*f

g*

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d

ec

a

a* b

c*a* a

b*

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g*

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ca

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c*a*

b*a

d*f*f

g*

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d

ec

ab

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c*a*

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d*f*f

g*

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d

eca

a*

b

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b*

d*f*f

g*

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ca

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d*f*f

g*

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d

ec

ab

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c*a*

b*a

d*f*f

g*

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d

ec

ab

a*

c*a*

b*a

d*f*f

g*

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d

ec

a

a* b

ca

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c*a*

b*a

d*f*f

g*

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d

ec

ab

a*

c*a*

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d*f*f

g*

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d

ec

ab

a*

c*a*

b*a

d*f*f

g*

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d

ec

a

a* b

c a

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b*

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g*

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ca a*

b

c*a* a

b*

d*f* f

g*

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c a b a*

c* a* b* a d*f* f

g*

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d e c a

a*b

c a b a*

c* a* b* a d*f* f

g*

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d e c a

a*b

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b*

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Modelling Metaphor

Diego Velázquez, 1656Las Meninas Las Meninas

Pablo Picasso, 1957

Thank you

Diego Velázquez, 1656Las Meninas Las Meninas

Pablo Picasso, 1957

In vitro implementation of a stack data structure based on DNA strand displacement

Unconventional Computation and Natural ComputationManchester, UK, July 11-15 2016

Harold Fellermann, Annunziata Lopiccolo,Jerzy Kozyra, Ben Shirt -Ediss and Natalio Krasnogor

Interdisciplinary Computing and Complex Biosystems Research GroupSchool of Computing, Newcastle University

Newcastle-upon-Tyne UK

Jerzy Kozyra Ben Shirt-Ediss Natalio KrasnogorHarold FellermannNunzia Lopiccolo

UCNC2016, Annunziata Lopiccolo – Newcastle University 60

Acknowledgments

Jerzy Kozyra Ben Shirt-Ediss Natalio KrasnogorHarold Fellermann

In collaboration with: