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The Total Quasi-Steady-State Approximation for Fully Competitive Enzyme Reactions

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Abstract

The validity of the Michaelis–Menten–Briggs–Haldane approximation for single enzyme reactions has recently been improved by the formalism of the total quasi-steady-state approximation. This approach is here extended to fully competitive systems, and a criterion for its validity is provided. We show that it extends the Michaelis–Menten–Briggs–Haldane approximation for such systems for a wide range of parameters very convincingly, and investigate special cases. It is demonstrated that our method is at least roughly valid in the case of identical affinities. The results presented should be useful for numerical simulations of many in vivo reactions.

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References

  • Albe, K.R., Butler, M.H., Wright, B.E., 1990. Cellular concentrations of enzymes and their substrates. J. Theor. Biol. 143, 163–195.

    Article  Google Scholar 

  • Atkinson, D.E., 1977. Cellular Energy Metabolism and its Regulation. Academic Press, New York.

    Google Scholar 

  • Baker, G.A., Jr., 1975. Essentials of Padé approximants. Academic Press, London.

    MATH  Google Scholar 

  • Bhalla, U.S., Iyengar, R., 1999. Emergent properties of networks of biological signaling pathways. Science 283, 381–387.

    Article  Google Scholar 

  • Bisswanger, H., 2002. Enzyme Kinetics. Principles and Methods. Wiley-VCH.

  • Borghans, J., de Boer, R., Segel, L., 1996. Extending the quasi-steady state approximation by changing variables. Bull. Math. Biol. 58, 43–63.

    Article  MATH  Google Scholar 

  • Briggs, G.E., Haldane, J.B.S., 1925. A note on the kinetics of enzyme action. J. Biochem. 19, 338–339.

    Google Scholar 

  • Burack, W.R., Sturgill, T.W., 1997. The activating dual phosphorylation of MAPK by MEK is nonprocessive. Biochem. 36, 5929–5933.

    Google Scholar 

  • Ferrell, J.E., Bhatt, R.R., 1997. Mechanistic studies of the dual phosphorylation of mitogen-activated protein kinase. J. Biol. Chem. 272, 19008–19016.

    Article  Google Scholar 

  • Goldbeter, A., Koshland, D.E., Jr., 1981. An amplified sensitivity arising from covalent modification in biological systems. Proc. Natl. Acad. Sci. 78, 6840–6844.

    Google Scholar 

  • Henri, V., 1901a. Recherches sur la loi de l’action de la sucrase. C. R. Hebd. Acad. Sci. 133, 891–899.

    Google Scholar 

  • Henri, V., 1901b. Über das gesetz der wirkung des invertins. Z. Phys. Chem. 39, 194–216.

    Google Scholar 

  • Henri, V., 1902. Théorie générale de l’action de quelques diastases. C. R. Hebd. Acad. Sci. 135, 916–919.

    Google Scholar 

  • Huang, C.-Y.F., Ferrell, J.E., 1996. Ultrasensitivity in the mitogen-activated protein kinase cascade. Proc. Natl. Acad. Sci. 93, 10078–10083.

    Google Scholar 

  • Kholodenko, B.N., 2000. Negative feedback and ultrasensitivity can bring about oscillations in the mitogen-activated protein kinase cascades. Eur. J. Biochem. 267, 1583–1588.

    Article  Google Scholar 

  • Kishore, N., Sommers, C., Mathialagan, S., Guzova, J., Yao, M., Hauser, S., Huynh, K., Bonar, S., Mielke, C., Albee, L., Weier, R., Graneto, M., Hanau, C., Perry, T., Tripp, C.S., 2003. A selective IKK-2 inhibitor blocks NF-κ B-dependent gene expression in interleukin-1β-stimulated synovial fibroblasts. J. Biol. Chem. 278, 32861–32871.

    Article  Google Scholar 

  • Kv{å}lseth, T.O., 1985. Cautionary note about r 2 . The American Statistician 39, 279–285.

    Article  Google Scholar 

  • Markevich, N.I., Hoek, J.B., Kholodenko, B.N., 2004. Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades. J. Cell Biol. 164, 353–359.

    Article  Google Scholar 

  • Michaelis, L., Menten, M.L., 1913. Die kinetik der invertinwirkung. Biochem. Z. 49, 333–369.

    Google Scholar 

  • Pedersen, M.G., Bersani, A.M., Bersani, E., 2006. Quasi steady-state approximations in intracellular signal transduction—a word of caution. Preprint Me. Mo. Mat. no. 3/2006, Department of Mathematical Methods and Models, “La Sapienza” University, Rome, Italy.

  • Pi, N., Leary, J.A., 2004. Determination of enzyme/substrate specificity constants using a multiple substrate ESI-MS assay. J. Am. Soc. Mass Spectrom. 15, 233–243.

    Article  Google Scholar 

  • Rubinow, S., Lebowitz, J., 1970. Time-dependent Michaelis–Menten kinetics for an enzyme–substrate–inhibitor system. J. Am. Chem. Soc. 92, 3888–3893.

    Article  Google Scholar 

  • Schnell, S., Maini, P., 2000. Enzyme kinetics at high enzyme concentrations. Bull. Math. Biol. 62, 483–499.

    Article  Google Scholar 

  • Schnell, S., Maini, P., 2003. A century of enzyme kinetics: Reliability of the k m and v max estimates. Comm. Theor. Biol. 8, 169–187.

  • Schnell, S., Mendoza, C., 1997. Closed-form solution for time-dependent enzyme kinetics. J. Theor. Biol. 187, 207–212.

    Article  Google Scholar 

  • Schnell, S., Mendoza, C., 1997a. Enzymological considerations for a theoretical description of the quantitative competitive polymerase chain reaction (QC-PCR). J. Theor. Biol. 184, 433–440.

    Article  Google Scholar 

  • Schnell, S., Mendoza, C., 1997b. Theoretical description of the polymerase chain reaction. J. Theor. Biol. 188, 313–318.

    Article  Google Scholar 

  • Schnell, S., Mendoza, C., 2000. Time-dependent closed-form solutions for fully competitive enzyme reactions. Bull. Math. Biol. 62, 321–336.

    Article  Google Scholar 

  • Segel, L., 1988. On the validity of the steady state assumption of enzyme kinetics. Bull. Math. Biol. 50, 579–593.

    MATH  MathSciNet  Google Scholar 

  • Segel, L.A., Slemrod, M., 1989. The quasi steady-state assumption: A case study in pertubation. SIAM Rev. 31, 446–477.

    Article  MATH  MathSciNet  Google Scholar 

  • Sols, A., Marco, R., 1970. Concentration of metabolites and binding sites. Implications in metabolic regulation. In: Current Topics in Cellular Regulation, vol. 2. Academic Press, New York.

  • Stayton, M.M., Fromm, H.J., 1979. A computer analysis of the validity of the integrated Michaelis–Menten equation. J. Theor. Biol. 78, 309–323.

    Article  Google Scholar 

  • Straus, O.H., Goldstein, A., 1943. Zone behavior of enzymes. J. Gen. Physiol. 26, 559–585.

    Article  Google Scholar 

  • Turner, T.E., Schnell, S., Burrage, K., 2004. Stochastic approaches for modelling in vivo reactions. Comp. Biol. Chem. 28, 165–178.

    Article  MATH  Google Scholar 

  • Tzafriri, A.R., 2003. Michaelis–Menten kinetics at high enzyme concentrations. Bull. Math. Biol. 65, 1111–1129.

    Article  Google Scholar 

  • Tzafriri, A.R., Edelman, E.R., 2004. The total quasi-steady-state approximation is valid for reversible enzyme kinetics. J. Theor. Biol. 226, 303–313.

    Article  MathSciNet  Google Scholar 

  • Tzafriri, A.R., Edelman, E.R., 2005. On the validity of the quasi-steady state approximation of bimolecular reactions in solution. J. Theor. Biol. 233, 343–350.

    Article  Google Scholar 

  • Zhao, Y., Zhang, Z.-Y., 2001. The mechanism of dephosphorylation of extracellu-lar signal-regulated kinase 2 by mitogen-activated protein kinase phosphatase 3. J. Biol. Chem. 276, 32382–32391.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Technical University of Denmark, Matematiktorvet, Building 303, DK-2800, Kgs. Lyngby, Denmark

    Morten Gram Pedersena

  2. Department of Mathematical Methods and Models, “La Sapienza” University, Rome, Italy

    Alberto M. Bersanib

  3. ISMAC Genova, Genova, Italy

    Enrico Bersanic

Authors
  1. Morten Gram Pedersena
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  2. Alberto M. Bersanib
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  3. Enrico Bersanic
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Correspondence to Morten Gram Pedersena.

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Pedersena, M.G., Bersanib, A.M. & Bersanic, E. The Total Quasi-Steady-State Approximation for Fully Competitive Enzyme Reactions. Bull. Math. Biol. 69, 433–457 (2007). https://doi.org/10.1007/s11538-006-9136-2

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  • DOI: https://doi.org/10.1007/s11538-006-9136-2

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