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Mean Lifetime and First-Passage Time of the Enzyme Species Involved in an Enzyme Reaction. Application to Unstable Enzyme Systems

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Abstract

Taking as starting point the complete analysis of mean residence times in linear compartmental systems performed by Garcia-Meseguer et al. (Bull. Math. Biol. 65:279–308, 2003) as well as the fact that enzyme systems, in which the interconversions between the different enzyme species involved are of first or pseudofirst order, act as linear compartmental systems, we hereby carry out a complete analysis of the mean lifetime that the enzyme molecules spend as part of the enzyme species, forms, or groups involved in an enzyme reaction mechanism. The formulas to evaluate these times are given as a function of the individual rate constants and the initial concentrations of the involved species at the onset of the reaction. We apply the results to unstable enzyme systems and support the results by using a concrete example of such systems. The practicality of obtaining the mean times and their possible application in a kinetic data analysis is discussed.

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References

  • Ahlers, J., Arnold, A., Van Döhren, F.R., Peter, H.W., 1982. Enzymkinetik, Stuttgart

  • Anderson, D.H., 1983. Compartmental Modelling and Tracer Kinetics. Springer, Berlin.

    Google Scholar 

  • Arias, E., Picazo, M., Garcia-Sevilla, F., Alberto, A., Arribas, E., Varon, R., 2007. Matlab-based software to obtain the time course equations of dynamic linear systems. Appl. Math. Sci. 1, 551–70.

    MATH  Google Scholar 

  • Aumente, D., Buelga, D.S., Lukas, J.C., Gomez, P., Torres, A., Garcia, M.J., 2006. Population pharmacokinetics of high-dose methotrexate in children with acute lymphoblastic leukaemia. Clin. Pharmacokinet. 45, 1227–238.

    Article  Google Scholar 

  • Bronstein, I.N., Semendjajew, K.A., Musiol, G., Mühlig, H., 2005. Taschenbuch der Mathematik. Verlag Harri Deutsch, Frankfurt and Main.

    Google Scholar 

  • Cutler, D.J., 1987. Definition of mean residence times in pharmacokinetics. Biopharm. Drug Dispos. 8, 87–7.

    Article  Google Scholar 

  • Darvey, I.G., 1977. Transient phase kinetics of enzymes reactions where more than one species of enzyme is present at the start of the reaction. J. Theor. Biol. 65, 465–78.

    Article  Google Scholar 

  • Duggleby, R.G., 1986. Progress curves of reactions catalyzed by unstable enzymes. A theoretical approach. J. Theor. Biol. 123, 67–0.

    Article  Google Scholar 

  • Empey, P.E., McNamara, P.J., Young, B., Rosbolt, M.B., Hatton, J., 2006. Cyclosporin A disposition following acute traumatic brain injury. J. Neurotrauma 23, 109–16.

    Article  Google Scholar 

  • Fenoll, L.G., Rodriguez-Lopez, J.N., Garcia-Sevilla, F., Tudela, J., Garcia-Ruiz, P.A., Varon, R., Garcia-Canovas, F., 2000. Oxidation by mushroom tyrosinase of monophenols generating slightly unstable o-quinones. Eur. J. Biochem. 267, 5865–878.

    Article  Google Scholar 

  • Fenoll, L.G., Rodriguez-Lopez, J.N., Garcia-Sevilla, F., Garcia-Ruiz, P.A., Varon, R., Garcia-Canovas, F., Tudela, J., 2001. Analysis and interpretation of the action mechanism of mushroom tyrosinase on monophenols and diphenols generating highly unstable o-quinones. Biochim. Biophys. Acta 1548, 1–2.

    Google Scholar 

  • Frank, S.A., Nowak, M.A., 2003. Cell biology: Developmental predisposition to cancer. Nature 422, 494.

    Article  Google Scholar 

  • Ganguli, A., Persson, L., Palmer, I.R., Evans, I., Yang, L., Smallwood, R., Black, R., Qwarnstrom, E.E., 2005. Distinct NF-kappaB regulation by shear stress through Ras-dependent IkappaBalpha oscillations: real-time analysis of flow-mediated activation in live cells. Circ. Res. 96, 626–34.

    Article  Google Scholar 

  • Garcia-Meseguer, M.J., Vidal de Labra, J.A., Garcia-Canovas, F., Havsteen, B.H., Garcia-Moreno, M., Varon, R., 2001. Time course equations of the amount of substance in a linear compartmental system and their computerized derivation. Biosystems 59, 197–20.

    Article  Google Scholar 

  • Garcia-Meseguer, M.J., Vidal de Labra, J.A., Garcia-Moreno, M., Garcia-Canovas, F., Havsteen, B.H., Varon, R., 2003. Mean residence times in linear compartmental systems. Symbolic formulae for their direct evaluation. Bull. Math. Biol. 65, 279–08.

    Article  Google Scholar 

  • Garcia-Sevilla, F., Garrido del Solo, C., Duggleby, R.G., Garcia-Canovas, F., Peyro, R., Varon, R., 2000. Use of a windows program for simulation of the progress curves of reactants and intermediates involved in enzyme-catalyzed reactions. Biosystems 54, 151–64.

    Article  Google Scholar 

  • Garrido del Solo, C., Varon, R., 1995. Kinetic behaviour of an enzymatic system with unstable product: conditions of limiting substrate concentration. An. Quim. 91, 13–8.

    Google Scholar 

  • Garrido del Solo, C., Garcia-Canovas, F., Havsteen, B.H., Varon, R., 1993. Kinetic analysis of a Michaelis–Menten mechanism in which the enzyme is unstable. Biochem. J. 294, 459–64.

    Google Scholar 

  • Garrido del Solo, C., Garcia-Canovas, F., Havsteen, B.H., Valero, E., Varon, R., 1994a. Kinetics of an enzyme reaction in which both the enzyme-substrate complex and the product are unstable or only the product is unstable. Biochem. J. 303(Pt 2), 435–40.

    Google Scholar 

  • Garrido del Solo, C., Varon, R., Garcia-Canovas, F., Havsteen, B.H., 1994b. Kinetic analysis of the Michaelis–Menten mechanism in which the substrate and the product are unstable. Int. J. Biochem. 26, 645–51.

    Article  Google Scholar 

  • Garrido del Solo, C., Garcia-Canovas, F., Havsteen, B.H., Varon, R., 1995. The influence of product instability on Michaelis–Menten kinetics under steady-state and rapid equilibrium assumptions. Int. J. Biochem. Cell Biol. 27, 475–79.

    Article  Google Scholar 

  • Garrido del Solo, C., Garcia-Canovas, F., Havsteen, B.H., Varon, R., 1996a. Analysis of progress curves of enzymatic reaction with unstable species. Int. J. Biochem. Cell Biol. 28, 1371–379.

    Article  Google Scholar 

  • Garrido del Solo, C., Havsteen, B.H., Varon, R., 1996b. An analysis of the kinetics of enzymatic systems with unstable species. Biosystems 38, 75–6.

    Article  Google Scholar 

  • Gilabert, M.A., Hiner, A.N., Garcia-Ruiz, P.A., Tudela, J., Garcia-Molina, F., Acosta, M., Garcia-Canovas, F., Rodriguez-Lopez, J.N., 2004a. Differential substrate behaviour of phenol and aniline derivatives during oxidation by horseradish peroxidase: kinetic evidence for a two-step mechanism. Biochim. Biophys. Acta 1699, 235–43.

    Google Scholar 

  • Gilabert, M.A., Fenoll, L.G., Garcia-Molina, F., Garcia-Ruiz, P.A., Tudela, J., Garcia-Canovas, F., Rodriguez-Lopez, J.N., 2004b. Stereospecificity of horseradish peroxidase. Biol. Chem. 385, 1177–184.

    Article  Google Scholar 

  • Green, M.H., 1992. Introduction to modeling. J. Nutr. 122, 690–94.

    Google Scholar 

  • Hearon, J.Z., 1963. Theorems on linear systems. Ann. N.Y. Acad. Sci. 108, 36–8.

    Article  MATH  MathSciNet  Google Scholar 

  • Hiner, A.N., Hernandez-Ruiz, J., Rodriguez-Lopez, J.N., Arnao, M.B., Varon, R., Garcia-Canovas, F., Acosta, M., 2001. The inactivation of horseradish peroxidase isoenzyme A2 by hydrogen peroxide: an example of partial resistance due to the formation of a stable enzyme intermediate. J. Biol. Inorg. Chem. 6, 504–16.

    Article  Google Scholar 

  • Hiner, A.N., Hernandez-Ruiz, J., Arnao, M.B., Rodriguez-Lopez, J.N., Garcia-Canovas, F., Acosta, M., 2002. Complexes between m-chloroperoxybenzoic acid and horseradish peroxidase compounds I and II: implications for the kinetics of enzyme inactivation. J. Enzym. Inhib. Med. Chem. 17, 287–91.

    Article  Google Scholar 

  • Isoherranen, N., Lavy, E., Soback, S., 2000. Pharmacokinetics of gentamicin C(1), C(1a), and C(2) in beagles after a single intravenous dose. Antimicrob. Agents Chemother. 44, 1443–447.

    Article  Google Scholar 

  • Jacquez, J.A., 1985. Compartmental Analysis in Biology and Medicine. Ann Arbor, Michigan.

    Google Scholar 

  • Kong, A.N., Jusko, W.J., 1988. Definitions and applications of mean transit and residence times in reference to the two-compartment mammillary plasma clearance model. J. Pharm. Sci. 77, 157–65.

    Article  Google Scholar 

  • Koukouraki, S., Strauss, L.G., Georgoulias, V., Schuhmacher, J., Haberkorn, U., Karkavitsas, N., Dimitrakopoulou-Strauss, A., 2006. Evaluation of the pharmacokinetics of 68Ga-DOTATOC in patients with metastatic neuroendocrine tumours scheduled for 90Y-DOTATOC therapy. Eur. J. Nucl. Med. Mol. Imaging 33, 460–66.

    Article  Google Scholar 

  • Luebeck, E.G., Moolgavkar, S.H., 1996. Biologically based cancer modeling. Drug Chem. Toxicol. 19, 221–43.

    Article  Google Scholar 

  • Mayhew, T.M., Wadrop, E., Simpson, R.A., 1994. Proliferative versus hypertrophic growth in tissue subcompartments of human placental villi during gestation. J. Anat. 184(Pt 3), 535–43.

    Google Scholar 

  • Moolgavkar, S.H., 2004. Commentary: Fifty years of the multistage model: remarks on a landmark paper. Int. J. Epidemiol. 33, 1182–183.

    Article  Google Scholar 

  • Moolgavkar, S.H., Luebeck, G., 1990. Two-event model for carcinogenesis: biological, mathematical, and statistical considerations. Risk Anal. 10, 323–41.

    Article  Google Scholar 

  • Paccaly, A., Frick, A., Rohatagi, S., Liu, J., Shukla, U., Rosenburg, R., Hinder, M., Jensen, B.K., 2006. Pharmacokinetics of otamixaban, a direct factor Xa inhibitor, in healthy male subjects: pharmacokinetic model development for phase 2/3 simulation of exposure. J. Clin. Pharmacol. 46, 37–4.

    Article  Google Scholar 

  • Resat, H., Ewald, J.A., Dixon, D.A., Wiley, H.S., 2003. An integrated model of epidermal growth factor receptor trafficking and signal transduction. Biophys. J. 85, 730–43.

    Article  Google Scholar 

  • Rescigno, A., 1999. Compartmental analysis revisited. Pharmacol. Res. 39, 471–78.

    Article  Google Scholar 

  • Rescigno, A., Gurpide, E., 1973. Estimation of average times of residence, recycle and interconversion of blood-borne compounds using tracer methods. J. Clin. Endocrinol. Metab. 36, 263–76.

    Article  Google Scholar 

  • Rodriguez-Lopez, J.N., Lowe, D.J., Hernandez-Ruiz, J., Hiner, A.N., Garcia-Canovas, F., Thorneley, R.N., 2001. Mechanism of reaction of hydrogen peroxide with horseradish peroxidase: identification of intermediates in the catalytic cycle. J. Am. Chem. Soc. 123, 11838–1847.

    Article  Google Scholar 

  • Schuster, S., Heinrich, R., 1987. Time hierarchy in enzymatic reaction chains resulting from optimality principles. J. Theor. Biol. 129, 189–09.

    Article  MathSciNet  Google Scholar 

  • Sines, J.J., Hackney, D.D., 1987. A residence-time analysis of enzyme kinetics. Biochem. J. 243, 159–64.

    Google Scholar 

  • Tozer, E.C., Carew, T.E., 1997. Residence time of low-density lipoprotein in the normal and atherosclerotic rabbit aorta. Circ. Res. 80, 208–18.

    Google Scholar 

  • Varon, R., 1979. Estudios de sistemas de compartimentos y su aplicación a la fase de transición de ecuaciones cinéticas. Tesis Doctoral, Universidad de Murcia, Murcia

  • Varon, R., Valero, E., Havsteen, B.H., Garrido del Solo, C., Rodriguez-Lopez, J.N., Garcia-Canovas, F., 1992. Comments on the kinetic analysis of enzyme reactions involving an unstable irreversible modifier. Biochem. J. 287(Pt 1), 333–34.

    Google Scholar 

  • Varon, R., Havsteen, B.H., Valero, E., Garrido del Solo, C., Rodriguez-Lopez, J.N., Garcia-Canovas, F., 1993a. The kinetics of an enzyme catalyzed reaction in the presence of an unstable irreversible modifier. Int. J. Biochem. 25, 1889–895.

    Article  Google Scholar 

  • Varon, R., Valero, E., Garrido del Solo, C., Garcia-Canovas, F., Havsteen, B.H., 1993b. Kinetic analysis of a Michaelis–Menten mechanism with an unstable substrate. J. Mol. Catal. 83, 273–85.

    Article  Google Scholar 

  • Varon, R., Garrido del Solo, C., Garcia-Moreno, M., Sanchez-Gracia, A., Garcia-Canovas, F., 1994. Final phase of enzyme reactions following a Michaelis–Menten mechanisms in which the free enzyme and/or the enzyme-substrate complex are unstable. Biol. Chem. Hoppe-Seyler 375, 35–2.

    Google Scholar 

  • Varon, R., Garcia-Meseguer, M.J., Garcia-Canovas, F., Havsteen, B.H., 1995a. General linear compartment model with zero input: I. Kinetic equations. Biosystems 36, 121–33.

    Article  Google Scholar 

  • Varon, R., Garcia-Meseguer, M.J., Havsteen, B.H., 1995b. General linear compartment model with zero input: II. The computerized derivation of the kinetic equations. Biosystems 36, 135–44.

    Article  Google Scholar 

  • Varon, R., Garcia-Meseguer, M.J., Valero, E., Garcia-Moreno, M., Garcia-Canovas, F., 1995c. General linear compartment model with zero input: III. First passage residence time of enzyme systems. Biosystems 36, 145–56.

    Article  Google Scholar 

  • Varon, R., Garrido del Solo, C., Garcia-Moreno, M., Garcia-Canovas, F., Moya-Garcia, G., Vidal de Labra, J.A., Havsteen, B.H., 1998. Kinetics of enzyme systems with unstable suicide substrates. Biosystems 47, 177–92.

    Article  Google Scholar 

  • Varon, R., Ruiz-Galea, M.M., Garrido del Solo, C., Garcia-Sevilla, F., Garcia-Moreno, M., Garcia-Canovas, F., Havsteen, B.H., 1999. Transient phase of enzyme reactions. Time course equations of the strict and the rapid equilibrium conditions and their computerized derivation. Biosystems 50, 99–26.

    Article  Google Scholar 

  • Varon, R., Picazo, M., Alberto, A., Arribas, E., Masia-Perez, J., Arias, E., 2007. General solution of the set of differential equations describing the time invariant linear dynamics systems. Application to enzyme systems. Appl. Math. Sci. 1, 281–00.

    MATH  MathSciNet  Google Scholar 

  • Veng-Pedersen, P., 1989. Mean time parameters in pharmacokinetics. Definition, computation and clinical implications (Part II). Clin. Pharmacokinet. 17, 424–40.

    Article  Google Scholar 

  • Venkatasubramanian, R., Henson, M.A., Forbes, N.S., 2006. Incorporating energy metabolism into a growth model of multicellular tumor spheroids. J. Theor. Biol. 242, 440–53.

    Article  MathSciNet  Google Scholar 

  • Wastney, M.E., House, W.A., Barnes, R.M., Subramanian, K.N., 2000. Kinetics of zinc metabolism: variation with diet, genetics and disease. J. Nutr. 130, 1355S–359S.

    Google Scholar 

  • Watts, G.F., Chan, D.C., Ooi, E.M., Nestel, P.J., Beilin, L.J., Barrett, P.H., 2006. Fish oils, phytosterols and weight loss in the regulation of lipoprotein transport in the metabolic syndrome: lessons from stable isotope tracer studies. Clin. Exp. Pharmacol. Physiol. 33, 877–82.

    Article  Google Scholar 

  • Weiss, E., 1967. Transaminase activity and other enzymatic reactions involving pyruvate and glutamate in Chlamydia (psittacosis-trachoma group). J. Bacteriol. 93, 177–84.

    Google Scholar 

  • Wilson, P.D., Dainty, J.R., 1999. Modelling in nutrition: an introduction. Proc. Nutr. Soc. 58, 133–38.

    Article  Google Scholar 

  • Yamaoka, K., Nakagawa, T., Uno, T., 1978. Statistical moments in pharmacokinetics. J. Pharmacokinet. Biopharm. 6, 547–58.

    Article  Google Scholar 

  • Zimmermann, T., Laufen, H., Yeates, R., Scharpf, F., Riedel, K.D., Schumacher, T., 1999. The pharmacokinetics of extended-release formulations of calcium antagonists and of amlodipine in subjects with different gastrointestinal transit times. J. Clin. Pharmacol. 39, 1021–031.

    Article  Google Scholar 

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Authors and Affiliations

  1. Applied Physics Department, University of Castilla-La Mancha, Albacete, Spain

    E. Arribas

  2. Departamento de Química-Física, University of Castilla-La Mancha, Albacete, Spain

    A. Muñoz-Lopez, M. J. Garcia-Meseguer, L. Avalos, M. Garcia-Moreno & R. Varon

  3. Medical Science Department, University of Castilla-La Mancha, Albacete, Spain

    A. Lopez-Najera

  4. Departamento Bioquímica y Biología Molecular A, Universidad de Murcia, Murcia, Spain

    F. Garcia-Molina

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  1. E. Arribas
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  2. A. Muñoz-Lopez
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  3. M. J. Garcia-Meseguer
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Arribas, E., Muñoz-Lopez, A., Garcia-Meseguer, M.J. et al. Mean Lifetime and First-Passage Time of the Enzyme Species Involved in an Enzyme Reaction. Application to Unstable Enzyme Systems. Bull. Math. Biol. 70, 1425–1449 (2008). https://doi.org/10.1007/s11538-008-9307-4

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