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Molecular computing for Markov chains

  • Chuan ZhangEmail author
  • Ziyuan Shen
  • Wei Wei
  • Jing Zhao
  • Zaichen Zhang
  • Xiaohu You
Article
  • 11 Downloads

Abstract

In this paper, it is presented a methodology for implementing arbitrarily constructed time-homogenous Markov chains with biochemical systems. Not only discrete but also continuous-time Markov chains are allowed to be computed. By employing chemical reaction networks as a programmable language, molecular concentrations serve to denote both input and output values. One reaction network is elaborately designed for each chain. The evolution of species’ concentrations over time well matches the transient solutions of the target continuous-time Markov chain, while equilibrium concentrations can indicate the steady state probabilities. Additionally, second-order Markov chains are considered for implementation, with bimolecular reactions rather than unary ones. An original scheme is put forward to compile unimolecular systems to DNA strand displacement reactions for the sake of future physical implementations. Deterministic, stochastic and DNA simulations are provided to enhance correctness, validity and feasibility.

Keywords

Molecular computing DNA strand displacement Markov chain Mass action kinetics Gillespie algorithm 

Notes

Acknowledgement

This work is supported in part by NSFC under grants 61871115 and 61501116, Jiangsu Provincial NSF for Excellent Young Scholars under grant BK20180059, the Six Talent Peak Program of Jiangsu Province under grant 2018-DZXX-001, the Distinguished Perfection Professorship of Southeast University, the Fundamental Research Funds for the Central Universities, the SRTP of Southeast University, and the Project Sponsored by the SRF for the Returned Overseas Chinese Scholars of MoE.

References

  1. Adleman LM (1994) Molecular computation of solutions to combinatorial problems. Science 266(5187):1021CrossRefGoogle Scholar
  2. Anderson DF, Kurtz TG (2011) Continuous time Markov Chain models for chemical reaction networks. Springer, New York, pp 3–42Google Scholar
  3. Bennett CH (1982) The thermodynamics of computation—a review. Int J Theor Phys 21(12):905CrossRefGoogle Scholar
  4. Berry G, Boudol G (1992) The chemical abstract machine. Theor Comput Sci 96(1):217MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bolch G, Greiner S, de Meer H, Trivedi KS (2006) Queueing networks and Markov chains: modeling and performance evaluation with computer science applications. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  6. Cardelli L (2013) Two-domain DNA strand displacement. Math Struct Comput Sci 23(2):247MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cardona M, Colomer M, Conde J, Miret J, Miró J, Zaragoza A (2005) Markov chains: computing limit existence and approximations with DNA. Biosystems 81(3):261CrossRefGoogle Scholar
  8. Chen HL, Doty D, Soloveichik D (2014) Deterministic function computation with chemical reaction networks. Nat Comput 13(4):517MathSciNetCrossRefzbMATHGoogle Scholar
  9. Ching WK, Huang X, Ng MK, Siu TK (2013) Markov chains. Springer, Berlin, pp 141–176Google Scholar
  10. DeGroot MH, Schervish MJ (2012) Probability and statistics. Addison-Wesley, BostonGoogle Scholar
  11. Érdi P, Tóth J (1989) Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models. Manchester University Press, ManchesterzbMATHGoogle Scholar
  12. Gillespie DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22(4):403MathSciNetCrossRefGoogle Scholar
  13. Hjelmfelt A, Weinberger ED, Ross J (1991) Chemical implementation of neural networks and turing machines. Proc Natl Acad Sci 88(24):10983CrossRefzbMATHGoogle Scholar
  14. Horn F, Jackson R (1972) General mass action kinetics. Arch Ration Mech Anal 47(2):81MathSciNetCrossRefGoogle Scholar
  15. Jiang H, Salehi SA, Riedel MD, Parhi KK (2013a) Discrete-time signal processing with DNA. ACS Synth Biol 2(5):245CrossRefGoogle Scholar
  16. Jiang H, Riedel MD, Parhi KK (2013b) Proceedings of the IEEE/ACM international conference on computer-aided design (ICCAD), pp 721–727Google Scholar
  17. Jiang H, Riedel M, Parhi KK (2011) Proceedings of the design automation conference, pp 836–841Google Scholar
  18. Kannan KS, Vallinayagam V, Venkatesan P (2007) Markov chain Monte Carlo methods in molecular computing. In: IJISEGoogle Scholar
  19. Kharam AP, Jiang H, Riedel MD, Parhi KK (2011) Proceedings of the Pacific symposium on biocomputing, pp 302–313Google Scholar
  20. Kurtz TG (1972) The relationship between stochastic and deterministic models for chemical reactions. J Chem Phys 57(7):2976CrossRefGoogle Scholar
  21. Liekens A, Fernando C (2007) Turing complete catalytic particle computers. In: Advances in artificial life, pp 1202–1211Google Scholar
  22. Lund K, Manzo AJ, Dabby N, Michelotti N, Johnson-Buck A, Nangreave J, Taylor S, Pei R, Stojanovic MN, Walter NG, Winfree E, Yan H (2010) Molecular robots guided by prescriptive landscapes. Nature 465(7295):206CrossRefGoogle Scholar
  23. Magnasco MO (1997) Chemical kinetics is turing universal. Phys Rev Lett 78(6):1190CrossRefGoogle Scholar
  24. McQuarrie DA (1967) Stochastic approach to chemical kinetics. J Appl Probab 4(03):413MathSciNetCrossRefzbMATHGoogle Scholar
  25. Ouyang Q, Kaplan PD, Liu S, Libchaber A (1997) DNA solution of the maximal clique problem. Science 278(5337):446CrossRefGoogle Scholar
  26. Păun G, Rozenberg G (2002) A guide to membrane computing. Theor Comput Sci 287(1):73MathSciNetCrossRefzbMATHGoogle Scholar
  27. Qian L, Winfree E (2011) Scaling up digital circuit computation with DNA strand displacement cascades. Science 332(6034):1196CrossRefGoogle Scholar
  28. Rothemund PWK (1995) A DNA and restriction enzyme implementation of turing machines. DNA Based Comput 27:75MathSciNetCrossRefGoogle Scholar
  29. Salehi SA, Riedel MD, Parhi KK (2014) Proceedings of the IEEE Asilomar conference on signals, systems and computers, pp 1767–1772Google Scholar
  30. Salehi SA, Riedel MD, Parhi KK (2015a) Proceedings of the IEEE international conference on digital signal processing (DSP), pp 689–693Google Scholar
  31. Salehi SA, Jiang H, Riedel MD, Parhi KK (2015b) Molecular sensing and computing systems. IEEE Trans Mol Biol Multi Scale Commun 1(3):249CrossRefGoogle Scholar
  32. Salehi SA, Parhi KK, Riedel MD (2016) Chemical reaction networks for computing polynomials. ACS Synth Biol 6(1):76CrossRefGoogle Scholar
  33. Shen Z, Zhang C, Ge L, Zhuang Y, Yuan B, You X (2016) Proceedings of the IEEE international workshop on signal processing systems (SiPS) IEEE, pp 27–32Google Scholar
  34. Soloveichik D (2009) CRNSimulator Mathematica Package. http://users.ece.utexas.edu/~soloveichik/crnsimulator.html. Accessed 20 Dec 2018
  35. Soloveichik D, Cook M, Winfree E, Bruck J (2008) Computation with finite stochastic chemical reaction networks. Nat Comput 7(4):615MathSciNetCrossRefzbMATHGoogle Scholar
  36. Soloveichik D, Seelig G, Winfree E (2010) DNA as a universal substrate for chemical kinetics. Proc Natl Acad Sci 107(12):5393CrossRefGoogle Scholar
  37. Stemmer WP (1995) The evolution of molecular computation. Science 270(5241):1510CrossRefGoogle Scholar
  38. Van Kampen NG (1995) Stochastic processes in physics and chemistry. Elsevier, LondonzbMATHGoogle Scholar
  39. Yurke B, Mills AP (2003) Using DNA to power nanostructures. Genet Program Evolv Mach 4(2):111CrossRefGoogle Scholar
  40. Zhang DY, Winfree E (2009) Control of DNA strand displacement kinetics using toehold exchange. J Am Chem Soc 131(47):17303CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Lab of Efficient Architectures for Digital-Communication and Signal-Processing (LEADS), Quantum Information Center of Southeast University, National Mobile Communications Research LaboratorySoutheast UniversityNanjingChina
  2. 2.State Key Laboratory of Coordination Chemistry, School of Chemistry and Chemical Engineering, State Key Laboratory of Pharmaceutical Biotechnology, School of Life SciencesNanjing UniversityNanjingChina

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