Skip to main content
SpringerLink
Log in
Book cover

Computational Methods in Synthetic Biology pp 179–191Cite as

Chemical Master Equation Closure for Computer-Aided Synthetic Biology

  • Protocol
  • First Online:

Part of the book series: Methods in Molecular Biology ((MIMB,volume 1244))

Abstract

With inexpensive DNA synthesis technologies, we can now construct biological systems by quickly piecing together DNA sequences. Synthetic biology is the promising discipline that focuses on the construction of these new biological systems. Synthetic biology is an engineering discipline, and as such, it can benefit from mathematical modeling. This chapter focuses on mathematical models of biological systems. These models take the form of chemical reaction networks. The importance of stochasticity is discussed and methods to simulate stochastic reaction networks are reviewed. A closure scheme solution is also presented for the master equation of chemical reaction networks. The master equation is a complete model of randomly evolving molecular populations. Because of its ambitious character, the master equation remained unsolved for all but the simplest of molecular interaction networks for over 70 years. With the first complete solution of chemical master equations, a wide range of experimental observations of biomolecular interactions may be mathematically conceptualized. We anticipate that models based on the closure scheme described herein may assist in rationally designing synthetic biological systems.

This is a preview of subscription content, log in via an institution.

eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Springer Nature is developing a new tool to find and evaluate Protocols. Learn more

References

  1. Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403:335–338

    Article  CAS  PubMed  Google Scholar 

  2. Gardner TS, Cantor CR, Collins JJ (2000) Construction of a genetic toggle switch in Escherichia coli. Nature 403:339–342

    Article  CAS  PubMed  Google Scholar 

  3. Andrianantoandro E, Basu S, Karig DK, Weiss R (2006) Synthetic biology: new engineering rules for an emerging discipline. Mol Syst Biol 2:2006.0028

    Article  PubMed Central  PubMed  Google Scholar 

  4. Volzing K, Borrero J, Sadowsky MJ, Kaznessis YN (2013) Antimicrobial peptides targeting gram-negative pathogens, produced and delivered by lactic acid bacteria. ACS Synth Biol 2(11):643–650, PubMed PMID: 23808914

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  5. Ramalingam K, Maynard J, Kaznessis YN (2009) Forward engineering of synthetic bio-logical AND gates. Biochem Eng J 47:38–47

    Article  Google Scholar 

  6. Alon U (2003) Biological networks: the tinkerer as an engineer. Science 301:1866–1867

    Article  CAS  PubMed  Google Scholar 

  7. Endy D (2005) Foundations for engineering biology. Nature 438:449–453

    Article  CAS  PubMed  Google Scholar 

  8. Volzing K, Biliouris K, Kaznessis YN (2011) proTeOn and proTeOff, new protein devices that inducibly activate bacterial gene expression. ACS Chem Biol 6(10):1107–1116

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  9. Kaern M, Blake WJ, Collins JJ (2003) The engineering of gene regulatory networks. Annu Rev Biomed Eng 5:179–206

    Article  CAS  PubMed  Google Scholar 

  10. Keasling J (2005) The promise of synthetic biology. Bridge Natl Acad Eng 35:18–21

    Google Scholar 

  11. Salis H, Kaznessis YK (2005) Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. J Chem Phys 122:1–13

    Article  Google Scholar 

  12. Haseltine EL, Rawlings JB (2002) Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J Chem Phys 117:6959–6969

    Article  CAS  Google Scholar 

  13. Salis H, Kaznessis YN (2005) Numerical simulation of stochastic gene circuits. Comp Chem Eng 29:577–588

    Article  CAS  Google Scholar 

  14. Cao Y, Li H, Petzold L (2004) Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. J Chem Phys 121:4059–4067

    Article  CAS  PubMed  Google Scholar 

  15. Chatterjee A, Mayawala K, Edwards JS, Vlachos DG (2005) Time accelerated monte carlo simulations of biological networks using the binomial {tau}-leap method. Bioinformatics 21:2136–2137

    Article  CAS  PubMed  Google Scholar 

  16. Tian T, Burrage K (2004) Binomial leap methods for simulating stochastic chemical kinetics. J Chem Phys 121:10356–10364

    Article  CAS  PubMed  Google Scholar 

  17. W E, Liu D, Vanden-Eijnden E (2005) Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J Chem Phys 123:194107

    Article  Google Scholar 

  18. Munsky B, Khammash M (2006) The finite state projection algorithm for the solution of the chemical master equation. J Chem Phys 124:044104

    Article  PubMed  Google Scholar 

  19. McQuarrie DA (1967) Stochastic approach to chemical kinetics. J Appl Prob 4:413–478

    Article  Google Scholar 

  20. Moyal JE (1949) Stochastic processes and statistical physics. J Roy Stat Soc Ser B 11:150–210

    Google Scholar 

  21. Oppenheim I, Shuler KE (1965) Master equations and Markov processes. Phys Rev 138:1007–1011

    Article  Google Scholar 

  22. Oppenheim I, Shuler KE, Weiss GH (1967) Stochastic theory of multistate relaxation processes. Adv Mol Relax Process 1:13–68

    Article  CAS  Google Scholar 

  23. Oppenheim I, Shuler KE, Weiss GH (1977) Stochastic processes in chemical physics: the master equation. The MIT Press, Cambridge, MA

    Google Scholar 

  24. Gillespie DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22:403–434

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  26. Gibson MA, Bruck J (2000) Efficient exact stochastic simulation of chemical systems with many species and many channels. J Phys Chem 104:1876–1889

    Article  CAS  Google Scholar 

  27. Cao Y, Gillespie DT, Petzold LR (2005) Avoiding negative populations in explicit Poisson tau-leaping. J Chem Phys 123:054104

    Article  PubMed  Google Scholar 

  28. Chatterjee A, Vlachos DG (2006) Temporal acceleration of spatially distributed kinetic monte Carlo simulations. J Comput Phys 211:596–615

    Article  Google Scholar 

  29. Salis H, Kaznessis YN (2005) An equation-free probabilistic steady state approximation: dynamic application to the stochastic simulation of biochemical reaction networks. J Chem Phys 123:214106

    Article  PubMed  Google Scholar 

  30. Sotiropoulos V, Kaznessis YN (2008) An adaptive time step scheme for a system of SDEs with multiple multiplicative noise. Chemical Langevin equation, a proof of concept. J Chem Phys 128:014103

    Article  PubMed  Google Scholar 

  31. Kaznessis Y (2006) Multi-scale models for gene network engineering. Chem Eng Sci 61:940–953

    Article  CAS  Google Scholar 

  32. Kaznessis Y (2007) Models for synthetic biology. BMC Syst Biol 1:47

    Article  PubMed Central  PubMed  Google Scholar 

  33. Harris LA, Clancy PA (2006) A “partitioned leaping” approach for multiscale modeling of chemical reaction dynamics. J Chem Phys 125:144107

    Article  PubMed  Google Scholar 

  34. Tuttle L, Salis H, Tomshine J, Kaznessis YN (2005) Model-driven design principles of gene networks: the oscillator. Biophys J 89:3873–3883

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  35. Tomshine J, Kaznessis YN (2006) Optimization of a stochastically simulated gene network model via simulated annealing. Biophys J 91:3196–3205

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  36. Gillespie CS (2009) Moment closure approximations for mass-action models. IET Syst Biol 3:52–58

    Article  CAS  PubMed  Google Scholar 

  37. Sotiropoulos V, Kaznessis YN (2011) Analytical derivation of moment equations in stochastic chemical kinetics. Chem Eng Sci 66:268–277

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  38. Smadbeck P, Kaznessis YN (2012) Efficient moment matrix generation for arbitrary chemical networks. Chem Eng Sci 84:612–618

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  39. Smadbeck P, Kaznessis YN (2013) A closure scheme for chemical master equations. Proc Natl Acad Sci U S A 110(35):14261–14265

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  40. Schlögl F (1972) Chemical reaction models for non-equilibrium phase transition. Z Phys 253:147–161

    Article  Google Scholar 

  41. Salis H, Sotiropoulos V, Kaznessis YN (2006) Multiscale Hy3S: hybrid stochastic simulations for supercomputers. BMC Bioinform 7(93):2006

    Google Scholar 

  42. Hill A, Tomshine J, Wedding E, Sotiropoulos V, Kaznessis YK (2008) SynBioSS: the synthetic biology modeling suite. Bioinformatics 24:2551–2553

    Article  CAS  PubMed  Google Scholar 

  43. Weeding E, Houle J, Kaznessis YN (2010) SynBioSS designer: a web-based tool for the automated generation of kinetic models for synthetic biological constructs. Brief Bioinform 11(4):394–402

    Article  CAS  PubMed Central  PubMed  Google Scholar 

Download references

Acknowledgements

This work was supported by a grant from the National Institutes of Health (American Recovery and Reinvestment Act grant GM086865) and a grant from the National Science Foundation (CBET-0644792) with computational support from the Minnesota Supercomputing Institute (MSI). Support from the University of Minnesota Digital Technology Center and the University of Minnesota Biotechnology Institute is also acknowledged.

Author information

Authors and Affiliations

  1. Department of Chemical Engineering and Materials Science, University of Minnesota, 253 Amundson Hall, 421 Washington Ave SE, Minneapolis, MN, 55455-0531A, USA

    Patrick Smadbeck & Yiannis N. Kaznessis

Authors
  1. Patrick Smadbeck
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yiannis N. Kaznessis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yiannis N. Kaznessis .

Editor information

Editors and Affiliations

  1. School of Life Science and Technology, Harbin Institute of Technology, Harbin, China

    Mario Andrea Marchisio

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media New York

About this protocol

Cite this protocol

Smadbeck, P., Kaznessis, Y.N. (2015). Chemical Master Equation Closure for Computer-Aided Synthetic Biology. In: Marchisio, M. (eds) Computational Methods in Synthetic Biology. Methods in Molecular Biology, vol 1244. Humana Press, New York, NY. https://doi.org/10.1007/978-1-4939-1878-2_9

Download citation

  • DOI: https://doi.org/10.1007/978-1-4939-1878-2_9

  • Published:

  • Publisher Name: Humana Press, New York, NY

  • Print ISBN: 978-1-4939-1877-5

  • Online ISBN: 978-1-4939-1878-2

  • eBook Packages: Springer Protocols

Publish with us

Policies and ethics

Search

Navigation