Chemical Master Equation Closure for Computer-Aided Synthetic Biology

  • Patrick Smadbeck
  • Yiannis N. KaznessisEmail author
Part of the Methods in Molecular Biology book series (MIMB, volume 1244)


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.

Key words

Synthetic biology Computer-aided design Multiscale models Chemical master equation Closure schemes 



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.


  1. 1.
    Elowitz MB, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403:335–338PubMedCrossRefGoogle Scholar
  2. 2.
    Gardner TS, Cantor CR, Collins JJ (2000) Construction of a genetic toggle switch in Escherichia coli. Nature 403:339–342PubMedCrossRefGoogle Scholar
  3. 3.
    Andrianantoandro E, Basu S, Karig DK, Weiss R (2006) Synthetic biology: new engineering rules for an emerging discipline. Mol Syst Biol 2:2006.0028PubMedCentralPubMedCrossRefGoogle Scholar
  4. 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: 23808914PubMedCentralPubMedCrossRefGoogle Scholar
  5. 5.
    Ramalingam K, Maynard J, Kaznessis YN (2009) Forward engineering of synthetic bio-logical AND gates. Biochem Eng J 47:38–47CrossRefGoogle Scholar
  6. 6.
    Alon U (2003) Biological networks: the tinkerer as an engineer. Science 301:1866–1867PubMedCrossRefGoogle Scholar
  7. 7.
    Endy D (2005) Foundations for engineering biology. Nature 438:449–453PubMedCrossRefGoogle Scholar
  8. 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):1107–1116PubMedCentralPubMedCrossRefGoogle Scholar
  9. 9.
    Kaern M, Blake WJ, Collins JJ (2003) The engineering of gene regulatory networks. Annu Rev Biomed Eng 5:179–206PubMedCrossRefGoogle Scholar
  10. 10.
    Keasling J (2005) The promise of synthetic biology. Bridge Natl Acad Eng 35:18–21Google Scholar
  11. 11.
    Salis H, Kaznessis YK (2005) Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. J Chem Phys 122:1–13CrossRefGoogle Scholar
  12. 12.
    Haseltine EL, Rawlings JB (2002) Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J Chem Phys 117:6959–6969CrossRefGoogle Scholar
  13. 13.
    Salis H, Kaznessis YN (2005) Numerical simulation of stochastic gene circuits. Comp Chem Eng 29:577–588CrossRefGoogle Scholar
  14. 14.
    Cao Y, Li H, Petzold L (2004) Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. J Chem Phys 121:4059–4067PubMedCrossRefGoogle Scholar
  15. 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 21:2136–2137PubMedCrossRefGoogle Scholar
  16. 16.
    Tian T, Burrage K (2004) Binomial leap methods for simulating stochastic chemical kinetics. J Chem Phys 121:10356–10364PubMedCrossRefGoogle Scholar
  17. 17.
    W E, Liu D, Vanden-Eijnden E (2005) Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J Chem Phys 123:194107CrossRefGoogle Scholar
  18. 18.
    Munsky B, Khammash M (2006) The finite state projection algorithm for the solution of the chemical master equation. J Chem Phys 124:044104PubMedCrossRefGoogle Scholar
  19. 19.
    McQuarrie DA (1967) Stochastic approach to chemical kinetics. J Appl Prob 4:413–478CrossRefGoogle Scholar
  20. 20.
    Moyal JE (1949) Stochastic processes and statistical physics. J Roy Stat Soc Ser B 11:150–210Google Scholar
  21. 21.
    Oppenheim I, Shuler KE (1965) Master equations and Markov processes. Phys Rev 138:1007–1011CrossRefGoogle Scholar
  22. 22.
    Oppenheim I, Shuler KE, Weiss GH (1967) Stochastic theory of multistate relaxation processes. Adv Mol Relax Process 1:13–68CrossRefGoogle Scholar
  23. 23.
    Oppenheim I, Shuler KE, Weiss GH (1977) Stochastic processes in chemical physics: the master equation. The MIT Press, Cambridge, MAGoogle Scholar
  24. 24.
    Gillespie DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22:403–434CrossRefGoogle Scholar
  25. 25.
    Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361CrossRefGoogle Scholar
  26. 26.
    Gibson MA, Bruck J (2000) Efficient exact stochastic simulation of chemical systems with many species and many channels. J Phys Chem 104:1876–1889CrossRefGoogle Scholar
  27. 27.
    Cao Y, Gillespie DT, Petzold LR (2005) Avoiding negative populations in explicit Poisson tau-leaping. J Chem Phys 123:054104PubMedCrossRefGoogle Scholar
  28. 28.
    Chatterjee A, Vlachos DG (2006) Temporal acceleration of spatially distributed kinetic monte Carlo simulations. J Comput Phys 211:596–615CrossRefGoogle Scholar
  29. 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 123:214106PubMedCrossRefGoogle Scholar
  30. 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 128:014103PubMedCrossRefGoogle Scholar
  31. 31.
    Kaznessis Y (2006) Multi-scale models for gene network engineering. Chem Eng Sci 61:940–953CrossRefGoogle Scholar
  32. 32.
    Kaznessis Y (2007) Models for synthetic biology. BMC Syst Biol 1:47PubMedCentralPubMedCrossRefGoogle Scholar
  33. 33.
    Harris LA, Clancy PA (2006) A “partitioned leaping” approach for multiscale modeling of chemical reaction dynamics. J Chem Phys 125:144107PubMedCrossRefGoogle Scholar
  34. 34.
    Tuttle L, Salis H, Tomshine J, Kaznessis YN (2005) Model-driven design principles of gene networks: the oscillator. Biophys J 89:3873–3883PubMedCentralPubMedCrossRefGoogle Scholar
  35. 35.
    Tomshine J, Kaznessis YN (2006) Optimization of a stochastically simulated gene network model via simulated annealing. Biophys J 91:3196–3205PubMedCentralPubMedCrossRefGoogle Scholar
  36. 36.
    Gillespie CS (2009) Moment closure approximations for mass-action models. IET Syst Biol 3:52–58PubMedCrossRefGoogle Scholar
  37. 37.
    Sotiropoulos V, Kaznessis YN (2011) Analytical derivation of moment equations in stochastic chemical kinetics. Chem Eng Sci 66:268–277PubMedCentralPubMedCrossRefGoogle Scholar
  38. 38.
    Smadbeck P, Kaznessis YN (2012) Efficient moment matrix generation for arbitrary chemical networks. Chem Eng Sci 84:612–618PubMedCentralPubMedCrossRefGoogle Scholar
  39. 39.
    Smadbeck P, Kaznessis YN (2013) A closure scheme for chemical master equations. Proc Natl Acad Sci U S A 110(35):14261–14265PubMedCentralPubMedCrossRefGoogle Scholar
  40. 40.
    Schlögl F (1972) Chemical reaction models for non-equilibrium phase transition. Z Phys 253:147–161CrossRefGoogle Scholar
  41. 41.
    Salis H, Sotiropoulos V, Kaznessis YN (2006) Multiscale Hy3S: hybrid stochastic simulations for supercomputers. BMC Bioinform 7(93):2006Google Scholar
  42. 42.
    Hill A, Tomshine J, Wedding E, Sotiropoulos V, Kaznessis YK (2008) SynBioSS: the synthetic biology modeling suite. Bioinformatics 24:2551–2553PubMedCrossRefGoogle Scholar
  43. 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):394–402PubMedCentralPubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Chemical Engineering and Materials ScienceUniversity of MinnesotaMinneapolisUSA

Personalised recommendations