Abstract
DNA based nanostructures and devices are becoming ubiquitous in nanotechnology with rapid advancements in theory and experiments in DNA self-assembly which have led to a myriad of DNA nanodevices. However, the modeling methods used by researchers in the field for design and analysis of DNA nanostructures and nanodevices have not progressed at the same rate. Specifically, there does not exist a formal system that can capture the spectrum of the most frequently intended chemical reactions on DNA nanostructures and nanodevices which have branched and pseudo-knotted structures. In this paper we introduce a graph rewriting system for modeling DNA nanodevices. We define pseudo-DNA nanostructures (\(\mathbf {PDN}\)s), which describe the sequence information and secondary structure of DNA nanostructures, but exclude modeling of tertiary structures. We define a class of labeled graphs called DNA graphs, that provide a graph theoretic representation of PDNs. We introduce a set of graph rewrite rules that operate on DNA graphs. Our DNA graphs and graph rewrite rules provide a powerful and expressive way to model DNA nanostructures and their reactions. These rewrite rules model most conventional reactions on DNA nanostructures, which include hybridization, dehybridization, base-stacking, and a large family of enzymatic reactions. A subset of these rewrite rules would likely be used for a basic graph rewrite system modeling most DNA devices, which use just DNA hybridization reactions, whereas other of our rewrite rules could be incorporated as needed for DNA devices for example enzymic reactions. To ensure consistency of our systems, we define a subset of DNA graphs which we call well-formed DNA graphs, whose strands have consistent \(5^\prime \) to \(3^\prime \) polarity. We show that if we start with an input set of well-formed DNA graphs, our rewrite rules produce only well-formed DNA graphs. We give four detailed example applications of our graph rewriting system on (1) Yurke et al. [82] DNA tweezer system, (2) Yurke et al. [77] catalytic hairpin-based triggered branched junctions, (3) Dirks and Pierce [17] HCR, and (4) Qian and Winfree [59] scalable circuit of seesaw gates. Finally, we have a working software prototype (DAGRS) that we have used to generate automatically well-formed DNA graphs using a basic rewriting rule set for some of the examples mentioned.
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References
Andersen, J.L., Flamm, C., Merkle, D.: Inferring chemical reaction patterns using rule composition in graph grammars. J. Syst. Chem. 4(1), 4 (2013)
Bath, J., Green, S., Turberfield, A.: A free-running DNA motor powered by a nicking enzyme. Angew. Chem. Int. Edit. 44(28), 4358–4361 (2005)
Birac, J.J., Sherman, W.B., Kopatsch, J., Constantinou, P.E., Seeman, N.C.: Architecture with GIDEON, a program for design in structural DNA nanotechnology. J. Mol. Gr. Model. 25(4), 470–480 (2006)
Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)
Cardelli, L.: Strand algebras for DNA computing. Nat. Comput. 10(1), 407–428 (2011)
Chen, Y., Wang, M., Mao, C.: An autonomous DNA nanomotor powered by a DNA enzyme. Angew. Chem. Int. Edit. 43(27), 3554–3557 (2004)
Chen, Y.-J., Dalchau, N., Srinivas, N., Phillips, A., Cardelli, L., Soloveichik, D., Seelig, G.: Programmable chemical controllers made from DNA. Nat. Nanotechnol. 8(10), 755–762 (2013)
Chhabra, R., Sharma, J., Liu, Y., Yan, H.: Addressable molecular tweezers for DNA-templated coupling reactions. Nano Lett. 6(5), 978–983 (2006)
Chomsky, N.: Three models for the description of language. IRE Trans. Inf. Theory 2(3), 113–124 (1956)
Chomsky, N.: Syntactic Structures, The Hague (1971)
Claus, V., Ehrig, H., Rozenberg, G. (eds.): Graph-Grammars and Their Application to Computer Science and Biology. Lecture Notes in Computer Science, vol. 73. Springer, Berlin (1979)
Courcelle, B.: Graph rewriting: an algebraic and logic approach. Handbook of Theoretical Computer Science, pp. 194–242. Elsevier, Amsterdamm (1990)
Danos, V., Feret, J., Fontana, W., Harmer, R.: Graphs, rewriting and causality in rule-based models (2012)
Danos, V., Harmer, R., Honorato-Zimmer, R.: Thermodynamic Graph-Rewriting. Springer, Berlin (2013)
Danos, V., Laneve, C.: Graphs for Core Molecular Biology. Springer, Berlin (2003)
Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. Handbook of Theoretical Computer Science, vol. B. North-Holland, Amsterdam (1991)
Dirks, R., Pierce, N.: Triggered amplification by hybridization chain reaction. Proc. Natl. Acad. Sci. USA 101(43), 15275–15278 (2004)
Doye, J.P.K., Ouldridge, T.E., Louis, A.A., Romano, F., Šulc, P., Matek, C., Snodin, B.E.K., Rovigatti, L., Schreck, J.S., Harrison, R.M., Smith, W.P.J.: Coarse-graining DNA for simulations of DNA nanotechnology. Phys. Chem. Chem. Phys. 15(47), 20395–20414 (2013)
Ehrig, H.: Introduction to the algebraic theory of graph grammars (a survey). Proceedings of the International Workshop on Graph-Grammars and Their Application to Computer Science and Biology, pp. 1–69. Springer, London (1979)
Ehrig, H., Pfender, M., Schneider, H.J.: Graph-grammars: an algebraic approach. In: Automata Theory, pp. 167–180
Feret, J., Krivine, J.: Kasim: a simulator for kappa (2008–2013)
Flamm, C., Andersen, J.L., Merkle, D., Stadler, P.F.: Inferring chemical reaction patterns using rule composition in graph grammars. arXiv.org (2012)
Genot, A., Zhang, D., Bath, J., Turberfield, A.: Remote toehold: a mechanism for flexible control of DNA hybridization kinetics. J. Am. Chem. Soc. 133(7), 2177–2182 (2011)
Ghrist, R., Lipsky, D.: Grammatical self assembly for planar tiles. In: 2004 International Conference on MEMS, NANO and Smart Systems (ICMENS’04), pp. 205–211. IEEE (2004)
Green, S., Bath, J., Turberfield, A.: Coordinated chemomechanical cycles: a mechanism for autonomous molecular motion. Phys. Rev. Lett. 101, 238101 (2008)
Grun, C., Sarma, K., Wolfe, B., Shin, S.W., Winfree, E.: A domain-level DNA strand displacement reaction enumerator allowing arbitrary non-pseudoknotted secondary structures. http://dna.caltech.edu/Papers/Peppercorn2014-VEMDP.pdf (2014). Accessed 4 Nov 2014
Gu, H., Chao, J., Xiao, S.-J., Seeman, N.: A proximity-based programmable DNA nanoscale assembly line. Nature 465(7295), 202–205 (2010)
He, Y., Liu, D.: Autonomous multistep organic synthesis in a single isothermal solution mediated by a DNA walker. Nat. Nanotechnol. 5(11), 778–782 (2010)
Hopcroft, J.E., Tarjan, R.E.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)
Ibuki, K., Fumiaki, T., Masami, H.: MPS. Abstraction of DNA graph structures for efficient enumeration and simulation. 2011(12), 1–6 (2011)
Jonoska, N., Karl, S.A., Saito, M.: Graph structures in DNA computing. Computing with Bio-Molecules, Theory and Experiments, pp. 93–110. Springer, Berlin (1998)
Kawamata, I., Aubert, N., Hamano, M., Hagiya, M.: Abstraction of graph-based models of bio-molecular reaction systems for efficient simulation. Computational Methods in Systems Biology, pp. 187–206. Springer, Berlin (2012)
Klavins, E.: Universal self-replication using graph grammars. In: 2004 International Conference on MEMS, NANO and Smart Systems (ICMENS’04), pp. 198–204. IEEE (2004)
Klavins, E.: Programmable self-assembly. IEEE Control Syst. Mag. 27(4), 43–56 (2007)
Klavins, E., Ghrist, R., Lipsky, D.: Graph grammars for self assembling robotic systems. In: Proceedings. ICRA’04. 2004 IEEE International Conference on Robotics and Automation, 2004, vol. 5, pp. 5293–5300 (2004)
Klavins, E., Ghrist, R., Lipsky, D.: A grammatical approach to self-organizing robotic systems. IEEE Trans. Autom. Control 51(6), 949–962 (2006)
Krause, C., Giese, H.: Probabilistic Graph Transformation Systems. New Trends in Image Analysis and Processing—ICIAP 2013, pp. 311–325. Springer, Berlin (2012)
Krishnan, Y., Simmel, F.C.: Nucleic acid based molecular devices. Angew. Chem. Int. Edit. 50(14), 3124–3156 (2011)
Kumara, M.T., Nykypanchuk, D., Sherman, W.B.: Assembly pathway analysis of DNA nanostructures and the construction of parallel motifs. Nano Lett. 8(7), 1971–1977 (2008)
Lakin, M.R., Cardelli, L., Youssef, S., Phillips, A.: Abstractions for DNA circuit design. J. R. Soc. Interface 9(68), 470–486 (2012)
Lakin, M.R., Parker, D., Cardelli, L., Kwiatkowska, M., Phillips, A.: Design and analysis of DNA strand displacement devices using probabilistic model checking. J. R. Soc. Interface R. Soc. 9(72), 1470–1485 (2012)
Lakin, M.R., Youssef, S., Polo, F., Emmott, S., Phillips, A.: Visual DSD: a design and analysis tool for DNA strand displacement systems. Bioinform. (Oxf. Engl.) 27(22), 3211–3213 (2011)
Lilley, D.M.J.: Structures of helical junctions in nucleic acids. Q. Rev. Biophys. 33(02), 109–159 (2000)
Liu, D., Balasubramanian, S.: A proton-fuelled DNA nanomachine. Angew. Chem. Int. Edit. 42(46), 5734–5736 (2003)
Lund, K., Manzo, A.J., Dabby, N., Michelotti, N., Johnson-Buck, A., Nangreave, J., Taylor, S., Pei, R., Stojanovic, M.N., Walter, N.G., Winfree, E., Yan, H.: Molecular robots guided by prescriptive landscapes. Nature 465(7295), 206–210 (2010)
Machinek, R.R.F., Ouldridge, T.E., Haley, N.E.C., Bath, J., Turberfield, A.J.: Programmable energy landscapes for kinetic control of DNA strand displacement. Nat. Communi. 5, 5324 (2014)
Mann, M., Ekker, H., Flamm, C.: The graph grammar library-a generic framework for chemical graph rewrite systems. arXiv.org (2013)
Mao, C., Sun, W., Shen, Z., Seeman, N.: A Nanomechanical device based on the B-Z transition of DNA. Nature 397, 144–146 (1999)
McCaskill, J.S., Niemann, U.: Graph replacement chemistry for DNA processing. DNA Comput. 2054, 103–116 (2001) (Chapter 8)
Modi, S., Krishnan, Y.: A method to map spatiotemporal pH changes inside living cells using a pH-triggered DNA nanoswitch, pp. 61–77 (2011)
Nupponen, K.: The design and implementation of a graph rewrite engine for model transformations. Master’s thesis (2005)
Ouldridge, T.E., Louis, A.A., Šulc, P., Romano, F., Doye, J.P.K.: DNA hybridization kinetics: zippering, internal displacement and sequence dependence. Nucleic Acids Res. 41, 8886–8895 (2013)
Peixoto, T.P.: Graph-tool: efficient network analysis (Version 2.2.31) [Software]. http://graph-tool.skewed.de/ (2014). Accessed 23 June 2014
Phillips, A., Cardelli, L.: A programming language for composable DNA circuits. J. R. Soc. Interface 6(11), 419–436 (2009)
Pinaud, B., Melançon, G., Dubois, J.: PORGY: a visual graph rewriting environment for complex systems. Comput. Gr. Forum 31(3), 1265–1274 (2012)
Potoyan, D.A., Savelyev, A., Papoian, G.A.: Recent successes in coarse-grained modeling of DNA. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 3(1), 69–83 (2012)
Python Software Foundation: Python™. https://www.python.org/download/releases/2.7/ (2001–2014)
Qian, L., Winfree, E.: A simple DNA gate motif for synthesizing large-scale circuits. DNA Computing, pp. 70–89. Springer, Berlin (2009)
Qian, L., Winfree, E.: Scaling up digital circuit computation with DNA strand displacement cascades. Science 332(6034), 1196–1201 (2011)
Reif, J.: Parallel biomolecular computation: models and simulations. Algorithmica 25(2–3), 142–175 (1999)
Reif, J.: The design of autonomous DNA nano-mechanical devices: walking and rolling DNA. DNA Computing, pp. 439–461. Springer, Berlin (2003)
Reif, J., Chandran, H., Gopalkrishnan, N., LaBean, T.: Self-assembled DNA nanostructures and DNA devices, pp. 299–328. Nanofabrication Handbook. CRC Press, Taylor and Francis Group, New York (2012)
Riverbank Computing Limited: PyQt5 (version 5.3.2) [Software]. http://www.riverbankcomputing.com/software/pyqt/download5 (2014)
Rozenberg, G.: Handbook of Graph Grammars and Computing by Graph Transformation: Volume I. Foundations. World Scientific, Singapore (1997)
Rozenberg, G., Ehrig, H.: Handbook of Graph Grammars and Computing by Graph Transformation, vol. 1. World Scientific, Singapore (1999)
Sekiguchi, H., Komiya, K., Kiga, D., Yamamura, M.: A design and feasibility study of reactions comprising DNA molecular machine that walks autonomously by using a restriction enzyme. Nat. Comput. 7(3), 303–315 (2008)
Sherman, W., Seeman, N.: A precisely controlled DNA biped walking device. Nano Lett. 4, 1203–1207 (2004)
Shin, J.-S., Pierce, N.: A synthetic DNA walker for molecular transport. J. Am. Chem. Soc. 126(35), 10834–10835 (2004)
Sinden, R.R.: DNA Structure and Function. Gulf Professional Publishing (1994)
Tian, Y., He, Y., Chen, Y., Yin, P., Mao, C.: A DNAzyme that walks processively and autonomously along a one-dimensional track. Angew. Chem. Int. Edit. 44(28), 4355–4358 (2005)
Tian, Y., Mao, C.: Molecular gears: a pair of DNA circles continuously rolls against each other. J. Am. Chem. Soc. 126(37), 11410–11411 (2004)
Torrini, P., Heckel, R., Ráth, I.: Stochastic simulation of graph transformation systems. Fundamental Approaches to Software Engineering, pp. 154–157. Springer, Berlin (2010)
Ullmann, J.R.: An algorithm for subgraph isomorphism. J. ACM 23(1), 31–42 (1976)
Wang, Z.-G., Elbaz, J., Willner, I.: DNA machines: bipedal walker and stepper. Nano Lett. 11(1), 304–309 (2011)
Wei-Kuan, S., Wen-Lian, H.: A new planarity test. Theor. Comput. Sci. 223(1–2), 179–191 (1999)
Woo, S., Rothemund, P.W.K.: Programmable molecular recognition based on the geometry of DNA nanostructures. Nat. Chem. 3(8), 620–627 (2011)
Yin, P., Choi, H., Calvert, C., Pierce, N.: Programming biomolecular self-assembly pathways. Nature 451(7176), 318–322 (2008)
Yin, P., Turberfield, A., Sahu, S., Reif, J.: Designs for autonomous unidirectional walking DNA devices. DNA Comput. pp. 410–425. Springer, Berlin (2004)
Yin, P., Yan, H., Daniell, X., Turberfield, A., Reif, J.: A unidirectional DNA walker moving autonomously along a linear track. Angew. Chem. Int. Edit. 116(37), 5014–5019 (2004b)
Yordanov, B., Kim, J., Petersen, R.L., Shudy, A., Kulkarni, V.V., Phillips, A.: Computational design of nucleic acid feedback control circuits. ACS Synth. Biol. 3(8), 600–616 (2014)
Yordanov, B., Wintersteiger, C.M., Hamadi, Y., Phillips, A., Kugler, H.: Functional Analysis of Large-Scale DNA Strand Displacement Circuits. Springer International Publishing, Cham (2013)
Yurke, B., Turberfield, A., Mills, A., Simmel, F., Neumann, J.: A DNA-fuelled molecular machine made of DNA. Nature 406(6796), 605–608 (2000)
Zhang, D.Y., Winfree, E.: Control of DNA strand displacement kinetics using toehold exchange. J. Am. Chem. Soc. 131(48), 17303–17314 (2009)
Acknowledgments
This work was supported by the National Science Foundation under NSF CCF 1217457 and NSF CCF 1320360.
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Mokhtar, R., Garg, S., Chandran, H., Bui, H., Song, T., Reif, J. (2017). Modeling DNA Nanodevices Using Graph Rewrite Systems. In: Adamatzky, A. (eds) Advances in Unconventional Computing. Emergence, Complexity and Computation, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-33921-4_15
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