Skip to main content

Stochastic Simulation of the Kinetics of Multiple Interacting Nucleic Acid Strands

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9211)

Abstract

DNA nanotechnology is an emerging field which utilizes the unique structural properties of nucleic acids in order to build nanoscale devices, such as logic gates, motors, walkers, and algorithmic structures. Predicting the structure and interactions of a DNA device requires effective modeling of both the thermodynamics and the kinetics of the DNA strands within the system. The kinetics of a set of DNA strands can be modeled as a continuous time Markov process through the state space of all secondary structures. The primary means of exploring the kinetics of a DNA system is by simulating trajectories through the state space and aggregating data over many such trajectories. We expand on previous work by extending the thermodynamics and kinetics models to handle multiple strands in a fixed volume, in a way that is consistent with previous models. We developed data structures and algorithms that allow us to take advantage of local properties of secondary structure, improving the efficiency of the simulator so that we can handle reasonably large systems. Finally, we illustrate the simulator’s analysis methods on a simple case study.

Keywords

  • Single Base Pair
  • Continuous Time Markov Process
  • Gillespie Algorithm
  • Loop Graph
  • Pseudoknotted Structure

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-21999-8_13
  • Chapter length: 18 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   54.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-21999-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   69.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.

Notes

  1. 1.

    We calculate \(V_0\) as the volume in which we would have exactly one molecule at a standard concentration of 1 mol/L: \(V_0 = 1 / (N_a * 1\,\text{ mol/L })\), where \(N_a\) is Avogadro’s number, and thus \(V_0\) is in liters. Similarly, we may wish to calculate V based on the concentration u in mol/L of a single strand such that the volume V is chosen such that exactly one molecule is present in that volume. In this case we have \(V = \frac{1}{u * N_a}\) and the relative number of states in the box is then \(\frac{V}{V_0} = \frac{N_a}{u * N_a} = \frac{1}{u}\).

References

  1. Crothers, D.M., Bloomfield, V.A., Tinoco Jr., I.: Nucleic Acids: Structures, Properties, and Functions. University Science Books, Sausalito (2000)

    Google Scholar 

  2. 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)

    CrossRef  Google Scholar 

  3. Chitsaz, H., Salari, R., Sahinalp, S.C., Backofen, R.: A partition function algorithm for interacting nucleic acid strands. Bioinformatics 25(12), i365–i373 (2009)

    CrossRef  Google Scholar 

  4. Dirks, R.M., Bois, J.S., Schaeffer, J.M., Winfree, E., Pierce, N.A.: Thermodynamic analysis of interacting nucleic acid strands. SIAM Rev. 49(1), 65–88 (2007)

    MathSciNet  CrossRef  Google Scholar 

  5. Dirks, R.M., Lin, M., Winfree, E., Pierce, N.A.: Paradigms for computational nucleic acid design. Nucleic Acids Res. 32(4), 1392–1403 (2004)

    CrossRef  Google Scholar 

  6. Dirks, R.M., Pierce, N.A.: Triggered amplification by hybridization chain reaction. Proc. Natl. Acad. Sci. U. S. A. 101(43), 15275–15278 (2004)

    CrossRef  Google Scholar 

  7. Flamm, C., Fontana, W., Hofacker, I.L., Schuster, P.: RNA folding at elementary step resolution. RNA 6, 325–338 (2000)

    CrossRef  Google Scholar 

  8. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)

    CrossRef  Google Scholar 

  9. Hongzhou, G., Chao, J., Xiao, S.-J., Seeman, N.C.: A proximity-based programmable DNA nanoscale assembly line. Nature 465(7295), 202–205 (2010)

    CrossRef  Google Scholar 

  10. Kawasaki, K.: Diffusion constants near the critical point for time-dependent Ising models. Phys. Rev. 145, 224–230 (1966)

    MathSciNet  CrossRef  Google Scholar 

  11. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953)

    CrossRef  Google Scholar 

  12. Muscat, R.A., Bath, J., Turberfield, A.J.: A programmable molecular robot. Nano Lett. 11(3), 982–987 (2011)

    CrossRef  Google Scholar 

  13. Omabegho, T., Sha, R., Seeman, N.C.: A bipedal DNA Brownian motor with coordinated legs. Science 324(5923), 67–71 (2009)

    CrossRef  Google Scholar 

  14. Qian, L., Winfree, E.: Scaling up digital circuit computation with DNA strand displacement cascades. Science 332(6034), 1196–1201 (2011)

    CrossRef  Google Scholar 

  15. SantaLucia, J.: A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics. Proc. Natl. Acad. Sci. 95(4), 1460–1465 (1998)

    CrossRef  Google Scholar 

  16. SantaLucia, J., Allawi, H.T., Seneviratne, P.A.: Improved nearest-neighbor parameters for predicting DNA duplex stability. Biochemistry 35(11), 3555–3562 (1996)

    CrossRef  Google Scholar 

  17. SantaLucia, J., Hicks, D.: The thermodynamics of DNA structural motifs. Ann. Rev. Biophys. Biomol. Struct. 33(1), 415–440 (2004)

    CrossRef  Google Scholar 

  18. Schaeffer, J.M.: Stochastic simulation of the kinetics of multiple interacting nucleic acid strands. PhD thesis, California Institute of Technology (2013)

    Google Scholar 

  19. Seelig, G., Soloveichik, D., Zhang, D.Y., Winfree, E.: Enzyme-free nucleic acid logic circuits. Science 314(5805), 1585–1588 (2006)

    CrossRef  Google Scholar 

  20. Shin, J.-S., Pierce, N.A.: A synthetic DNA walker for molecular transport. J. Am. Chem. Soc. 126(35), 10834–10835 (2004)

    CrossRef  Google Scholar 

  21. Turberfield, A.J., Mitchell, J.C., Yurke, B., Mills Jr., A.P., Blakey, M.I., Simmel, F.C.: DNA fuel for free-running nanomachines. Phys. Rev. Lett. 90(11), 118102 (2003)

    CrossRef  Google Scholar 

  22. Venkataraman, S., Dirks, R.M., Rothemund, P.W., Winfree, E., Pierce, N.A.: An autonomous polymerization motor powered by DNA hybridization. Nat. Nanotechnol. 2(8), 490–494 (2007)

    CrossRef  Google Scholar 

  23. Wetmur, J.G.: Hybridization and renaturation kinetics of nucleic acids. Ann. Rev. Biophys. Bioeng. 5(1), 337–361 (1976)

    CrossRef  Google Scholar 

  24. Wilkinson, D.J.: Stochastic dynamical systems. In: Stumpf, M.P., Balding, D.J., Girolami, M. (eds.) Handbook of Statistical Systems Biology, pp. 359–375. Wiley, New York (2011)

    CrossRef  Google Scholar 

  25. Xayaphoummine, A., Bucher, T., Isambert, H.: Kinefold web server for RNA/DNA folding path and structure prediction including pseudoknots and knots. Nucleic Acids Res. 33(suppl 2), W605–W610 (2005)

    CrossRef  Google Scholar 

  26. Yin, P., Choi, H.M., Calvert, C.R., Pierce, N.A.: Programming biomolecular self-assembly pathways. Nature 451(7176), 318–322 (2008)

    CrossRef  Google Scholar 

  27. Yurke, B., Turberfield, A.J., Mills, A.P., Simmel, F.C., Neumann, J.L.: A DNA-fuelled molecular machine made of DNA. Nature 406(6796), 605–608 (2000)

    CrossRef  Google Scholar 

  28. Zadeh, J.N., Steenberg, C.D., Bois, J.S., Wolfe, B.R., Pierce, M.B., Khan, A.R., Dirks, R.M., Pierce, N.A.: NUPACK: analysis and design of nucleic acid systems. J. Comput. Chem. 32(1), 170–173 (2011)

    CrossRef  Google Scholar 

  29. Zhang, D.Y., Seelig, G.: Dynamic DNA nanotechnology using strand-displacement reactions. Nature Chem. 3(2), 103–113 (2011)

    CrossRef  Google Scholar 

  30. Zhang, D.Y., Turberfield, A.J., Yurke, B., Winfree, E.: Engineering entropy-driven reactions and networks catalyzed by DNA. Science 318(5853), 1121–1125 (2007)

    CrossRef  Google Scholar 

  31. Zhang, W., Chen, S.-J.: RNA hairpin-folding kinetics. Proc. Natl. Acad. Sci. 99(4), 1931–1936 (2002)

    CrossRef  Google Scholar 

Download references

Acknowledgements

We are greatly indebted to years of insights, suggestions, and feedback from Niles Pierce, Robert Dirks, Justin Bois, and Victor Beck, especially their contributions to the formulation of the energy model and the first step simulation mode.This work has been funded by National Science Foundation grants DMS-0506468, CCF-0832824, CCF-1213127, CCF-1317694, and the Gordon and Betty Moore Foundation through the Caltech Programmable Molecular Technology Initiative.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Erik Winfree .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Schaeffer, J.M., Thachuk, C., Winfree, E. (2015). Stochastic Simulation of the Kinetics of Multiple Interacting Nucleic Acid Strands. In: Phillips, A., Yin, P. (eds) DNA Computing and Molecular Programming. DNA 2015. Lecture Notes in Computer Science(), vol 9211. Springer, Cham. https://doi.org/10.1007/978-3-319-21999-8_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-21999-8_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21998-1

  • Online ISBN: 978-3-319-21999-8

  • eBook Packages: Computer ScienceComputer Science (R0)