Bulletin of Mathematical Biology

, Volume 81, Issue 5, pp 1303–1336 | Cite as

Quasi-Steady-State Approximations Derived from the Stochastic Model of Enzyme Kinetics

  • Hye-Won Kang
  • Wasiur R. KhudaBukhsh
  • Heinz Koeppl
  • Grzegorz A. RempałaEmail author


The paper outlines a general approach to deriving quasi-steady-state approximations (QSSAs) of the stochastic reaction networks describing the Michaelis–Menten enzyme kinetics. In particular, it explains how different sets of assumptions about chemical species abundance and reaction rates lead to the standard QSSA, the total QSSA, and the reverse QSSA. These three QSSAs have been widely studied in the literature in deterministic ordinary differential equation settings, and several sets of conditions for their validity have been proposed. With the help of the multiscaling techniques introduced in Ball et al. (Ann Appl Probab 16(4):1925–1961, 2006), Kang and Kurtz (Ann Appl Probab 23(2):529–583, 2013), it is seen that the conditions for deterministic QSSAs largely agree (with some exceptions) with the ones for stochastic QSSAs in the large-volume limits. The paper also illustrates how the stochastic QSSA approach may be extended to more complex stochastic kinetic networks like, for instance, the enzyme–substrate–inhibitor system.


Michaelis–Menten kinetics Stochastic reaction network Multiscale approximation QSSA 

Mathematics Subject Classification

60J27 60J28 34E15 92C42 92B25 92C45 



This work has been co-funded by the German Research Foundation (DFG) as part of project C3 within the Collaborative Research Center (CRC) 1053—MAKI (WKB) and the National Science Foundation under the Grants RAPID DMS-1513489 (GR) and DMS-1620403 (HWK). This research has also been supported in part by the University of Maryland Baltimore County under Grant UMBC KAN3STRT (HWK). This work was initiated when HWK and WKB were visiting the Mathematical Biosciences Institute (MBI) at the Ohio State University in Winter 2016–2017. MBI is receiving major funding from the National Science Foundation under the Grant DMS-1440386. HWK and WKB acknowledge the hospitality of MBI during their visits to the institute.


  1. Anderson DF, Kurtz TG (2011) Continuous time markov chain models for chemical reaction networks. In: Koeppl H, Setti G, di Bernardo M, Densmore D (eds) Design and analysis of biomolecular circuits: engineering approaches to systems and synthetic biology. Springer New York, New york, NY, pp 3–42.
  2. Anderson DF, Cappelletti D, Koyama M, Kurtz TG (2017) Non-explosivity of stochastically modeled reaction networks that are complex balanced. ArXiv e-prints arXiv:1708.09356
  3. Assaf M, Meerson B (2017) WKB theory of large deviations in stochastic populations. J Phys A 50(26):263001MathSciNetzbMATHCrossRefGoogle Scholar
  4. Ball K, Kurtz TG, Popovic L, Rempala GA (2006) Asymptotic analysis of multiscale approximations to reaction networks. Ann Appl Probab 16(4):1925–1961MathSciNetzbMATHCrossRefGoogle Scholar
  5. Barik D, Paul MR, Baumann WT, Cao Y, Tyson JJ (2008) Stochastic simulation of enzyme-catalyzed reactions with disparate timescales. Biophys J 95(8):3563–3574CrossRefGoogle Scholar
  6. Bersani AM, Dell’Acqua G (2011) Asymptotic expansions in enzyme reactions with high enzyme concentrations. Math Methods Appl Sci 34(16):1954–1960MathSciNetzbMATHCrossRefGoogle Scholar
  7. Bersani AM, Pedersen MG, Bersani E, Barcellona F (2005) A mathematical approach to the study of signal transduction pathways in MAPK cascade. Ser Adv Math Appl Sci 69:124MathSciNetzbMATHGoogle Scholar
  8. Biancalani T, Assaf M (2015) Genetic toggle switch in the absence of cooperative binding: exact results. Phys Rev Lett 115:208101CrossRefGoogle Scholar
  9. Borghans JAM, De Boer RJ, Segel LA (1996) Extending the quasi-steady state approximation by changing variables. Bull Math Biol 58(1):43–63zbMATHCrossRefGoogle Scholar
  10. Bressloff PC (2017) Stochastic switching in biology: from genotype to phenotype. J Phys A 50(13):133001MathSciNetzbMATHCrossRefGoogle Scholar
  11. Bressloff PC, Newby JM (2013) Metastability in a stochastic neural network modeled as a velocity jump markov process. SIAM J Appl Dyn Syst 12(3):1394–1435MathSciNetzbMATHCrossRefGoogle Scholar
  12. Briggs GE, Haldane JBS (1925) A note on the kinetics of enzyme action. Biochem J 19(2):338CrossRefGoogle Scholar
  13. Choi BS, Rempala GA, Kim J (2017) Beyond the Michaelis–Menten equation: accurate and efficient estimation of enzyme kinetic parameters. Sci Rep 7:17018CrossRefGoogle Scholar
  14. Cornish-Bowden A (2004) Fundamentals of enzyme kinetics. Portland Press, LondonGoogle Scholar
  15. Darden TA (1979) A pseudo-steady-state approximation for stochastic chemical kinetics. Rocky Mt J Math 9(1):51–71MathSciNetzbMATHCrossRefGoogle Scholar
  16. Darden TA (1982) Enzyme kinetics: stochastic vs. deterministic models. In: Reichl LE, Schieve WC (eds) Instabilities, bifurcations, and fluctuations in chemical systems. University of Texas Press, Austin, pp 248–272Google Scholar
  17. Dell’Acqua G, Bersani AM (2011) Quasi-steady state approximations and multistability in the double phosphorylationdephosphorylation cycle. In: International joint conference on biomedical engineering systems and technologies, pp 155–172Google Scholar
  18. Dell’Acqua G, Bersani AM (2012) A perturbation solution of Michaelis–Menten kinetics in a “total” framework. J Math Chem 50(5):1136–1148MathSciNetzbMATHCrossRefGoogle Scholar
  19. Dingee JW, Anton AB (2008) A new perturbation solution to the Michaelis–Menten problem. AlChE J 54(5):1344–1357CrossRefGoogle Scholar
  20. Ethier SN, Kurtz TG (1986) Markov processes: characterization and convergence, vol 282. Wiley, LondonzbMATHCrossRefGoogle Scholar
  21. Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361CrossRefGoogle Scholar
  22. Goeke A, Walcher S (2014) A constructive approach to quasi-steady state reductions. J Math Chem 52(10):2596–2626MathSciNetzbMATHCrossRefGoogle Scholar
  23. Gómez-Uribe CA, Verghese GC, Mirny LA (2007) Operating regimes of signaling cycles: statics, dynamics, and noise filtering. PLoS Comput Biol 3(12):e246MathSciNetCrossRefGoogle Scholar
  24. Grima R, Schmidt DR, Newman TJ (2012) Steady-state fluctuations of a genetic feedback loop: an exact solution. J Chem Phys 137(3):035104. CrossRefGoogle Scholar
  25. Hammes G (2012) Enzyme Catalysis and Regulation. Elsevier, New YorkGoogle Scholar
  26. Kang H-W, Kurtz TG (2013) Separation of time-scales and model reduction for stochastic reaction networks. Ann Appl Probab 23(2):529–583MathSciNetzbMATHCrossRefGoogle Scholar
  27. Kim H, Gelenbe E (2012) Stochastic gene expression modeling with hill function for switch-like gene responses. IEEE/ACM Trans Comput Biol Bioinform 9(4):973–979CrossRefGoogle Scholar
  28. Kim JK, Josić K, Bennett MR (2014) The validity of quasi-steady-state approximations in discrete stochastic simulations. Biophys J 107(3):783–793CrossRefGoogle Scholar
  29. Kim JK, Josić K, Bennett MR (2015) The relationship between stochastic and deterministic quasi-steady state approximations. BMC Syst Biol 9(1):87CrossRefGoogle Scholar
  30. Kim JK, Rempala GA, Kang H-W (2017) Reduction for stochastic biochemical reaction networks with multiscale conservations. arXiv preprint arXiv:1704.05628
  31. Kurtz TG (1972) The relationship between stochastic and deterministic models for chemical reactions. J Chem Phys 57(7):2976–2978CrossRefGoogle Scholar
  32. Kurtz TG (1992) Averaging for martingale problems and stochastic approximation. Appl Stoch Anal 177:186–209MathSciNetCrossRefGoogle Scholar
  33. Laidler KJ (1955) Theory of the transient phase in kinetics, with special reference to enzyme systems. Can J Chem 33(10):1614–1624CrossRefGoogle Scholar
  34. Lin CC, Segel LA (1988) Mathematics Applied to Deterministic Problems in the Natural Sciences. SIAM, BangkokzbMATHCrossRefGoogle Scholar
  35. McQuarrie DA (1967) Stochastic approach to chemical kinetics. J Appl Probab 4(3):413–478MathSciNetzbMATHCrossRefGoogle Scholar
  36. Michaelis L, Menten ML (1913) Die kinetik der invertinwirkung. Biochem Z 49(333–369):352Google Scholar
  37. Newby JM (2012) Isolating intrinsic noise sources in a stochastic genetic switch. Phys Biol 9(2):026002CrossRefGoogle Scholar
  38. Newby JM (2015) Bistable switching asymptotics for the self regulating gene. J Phys A 48(18):185001MathSciNetzbMATHCrossRefGoogle Scholar
  39. Pedersena MG, Bersanib AM, Bersanic E (2006) The total quasi-steady-state approximation for fully competitive enzyme reactions. Bull Math Biol 69(1):433CrossRefGoogle Scholar
  40. Perez-Carrasco R, Guerrero P, Briscoe J, Page KM (2016) Intrinsic noise profoundly alters the dynamics and steady state of morphogen-controlled bistable genetic switches. PLoS Comput Biol 12(10):1–23CrossRefGoogle Scholar
  41. Rao CV, Arkin AP (2003) Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm. J Chem Phys 118(11):4999–5010CrossRefGoogle Scholar
  42. Sanft KR, Gillespie DT, Petzold LR (2011) Legitimacy of the stochastic Michaelis–Menten approximation. IET Syst Biol 5(1):58–69CrossRefGoogle Scholar
  43. Sauro HM, Kholodenko BN (2004) Quantitative analysis of signaling networks. Prog Biophys Mol Biol 86(1):5–43CrossRefGoogle Scholar
  44. Schneider KR, Wilhelm T (2000) Model reduction by extended quasi-steady-state approximation. J Math Biol 40(5):443–450MathSciNetzbMATHCrossRefGoogle Scholar
  45. Schnell S, Maini PK (2000) Enzyme kinetics at high enzyme concentration. Bull Math Biol 62(3):483–499zbMATHCrossRefGoogle Scholar
  46. Schnell S, Mendoza C (1997) Closed form solution for time-dependent enzyme kinetics. J Theor Biol 187(2):207–212CrossRefGoogle Scholar
  47. Segel IH (1975) Enzyme Kinetics, vol 360. Wiley, New YorkGoogle Scholar
  48. Segel LA (1988) On the validity of the steady state assumption of enzyme kinetics. Bull Math Biol 50(6):579–593MathSciNetzbMATHCrossRefGoogle Scholar
  49. Segel LA, Slemrod M (1989) The quasi-steady-state assumption: a case study in perturbation. SIAM Rev 31(3):446–477MathSciNetzbMATHCrossRefGoogle Scholar
  50. Smith S, Cianci C, Grima R (2016) Analytical approximations for spatial stochastic gene expression in single cells and tissues. J R Soc Interface 13(118):20151051CrossRefGoogle Scholar
  51. Stiefenhofer M (1998) Quasi-steady-state approximation for chemical reaction networks. J Math Biol 36(6):593–609MathSciNetzbMATHCrossRefGoogle Scholar
  52. Thomas P, Straube AV, Grima R (2011) Communication: limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks. J Chem Phys 135(18):181103CrossRefGoogle Scholar
  53. Tian T, Burrage K (2006) Stochastic models for regulatory networks of the genetic toggle switch. Proc Natl Acad Sci USA 103(22):8372–8377CrossRefGoogle Scholar
  54. Tzafriri AR (2003) Michaelis–Menten kinetics at high enzyme concentrations. Bull Math Biol 65(6):1111–1129zbMATHCrossRefGoogle Scholar
  55. Van Slyke DD, Cullen GE (1914) The mode of action of urease and of enzymes in general. J Biol Chem 19(2):141–180Google Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Maryland Baltimore County BaltimoreUSA
  2. 2.Department of Electrical Engineering and Information TechnologyTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Division of Biostatistics and Mathematical Biosciences InstituteThe Ohio State UniversityColumbusUSA

Personalised recommendations