# Estimation of rate constants in nonlinear reactions involving chemical inactivation of oxidation catalysts

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## Abstract

Over the last decades, copious work has been devoted to the development of small molecule replicas of the peroxidase enzymes that activate hydrogen peroxide in metabolic and detoxifying processes. TAML activators that are the subject of this study are the first full functional, small molecule peroxidase mimics. As an important feature of the catalytic cycle, TAML reactive intermediates (active catalysts, Ac) undergo suicidal inactivation, compromising the functional catalysis. Herein the relationship between suicidal inactivation and productive catalysis is rigorously addressed mathematically and chemically. We focus on a generalized catalytic cycle in which the TAML inactivation step is delineated by its rate constant \(k_{\mathrm{i}}\) where the revealing data is collected in the regime of incomplete conversion of substrate (S) artificially imposed by the use of very low catalyst concentrations. The system exhibits a nonlinear conservation law and is modeled via a singular perturbation approach, which is used to obtain closed form relationships between system parameters. A new method is derived that allows to compute all the rate constants in the catalytic cycle, \(k_{\mathrm{I}},k_{\mathrm{II}}\), and \(k_{\mathrm{i}}\), with as little as two linear least squares fits, for the minimal data set collected under any conditions providing that the oxidation of S is incomplete. This method facilitates determination of \(k_{\mathrm{i}}\), a critical rate constant that describes the operational lifetime of the catalyst, and greatly reduces the experimental work required to obtain the important rate constants.The approach was applied to the behavior of a new TAML activator, the synthesis and characterization of which are also described.

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \hbox {Resting catalyst (Rc)} + \hbox {Oxidant} \rightarrow \hbox {Ac} &{} (k_{\mathrm{I}})\\ \hbox {Ac + Substrate (S)}\rightarrow \hbox {Rc}+\hbox {Product} &{} (k_{\mathrm{II}})\\ \hbox {Ac} \rightarrow \hbox {Inactive catalyst} &{} (k_{\mathrm{i}}) \end{array} \right. \end{aligned}$$

### Keywords

Iron TAMLs Hydrogen peroxide Catalyst inactivation Ordinary differential equations Mathematical modeling Perturbation methods### References

- 1.H.B. Dunford,
*Heme Peroxidases*(Wiley-VCH, New York, 1999)Google Scholar - 2.T.J. Collins, S.W. Gordon-Wylie et al., in P.T. Anastas, T.C Williamson, ed. by
*Green Chemistry*(Oxford University Press, Oxford, 1998), pp. 46–71Google Scholar - 3.T.J. Collins, TAML oxidant activators: a new approach to the activation of hydrogen peroxide for environmentally significant problems. Acc. Chem. Res.
**35**, 782–790 (2002)Google Scholar - 4.T.J. Collins, Designing ligands for oxidizing complexes. Acc. Chem. Res.
**27**, 279–285 (1994)Google Scholar - 5.T. Turanyi, A.S. Tomlin, M.J. Pilling, On the error of the quasi-steady-state approximation. J. Phys. Chem.
**97**, 163–172 (1993)CrossRefGoogle Scholar - 6.W. Richardson, L. Volk, K.H. Lau, S.H. Lin, H. Eyring, Application of the singular perturbation method to reaction kinetics. Proc. Natl. Acad. Sci. USA
**70**, 1588–1592 (1973)Google Scholar - 7.M. Lazman, G. Yablonsky, in
*Advances in Chemical Engineering: Mathematics and Chemical Engineering and Kinetics*. Vol. 34 (Academic Press, 2008)Google Scholar - 8.N.S. Shuman, T.M. Miller, A.A. Viggiano, J. Troe, Electron attachment to CF3 and CF3Br at temperatures up to 890 K: experimental test of the kinetic modeling approach. J. Chem. Phys.
**138**, 204316 (2013)Google Scholar - 9.C. Jiménez-Borja, B. Delgado, F. Dorado, J.L. Valverde, Experimental data and kinetic modeling of the catalytic and electrochemically promoted \(\text{ CH }_{4}\) oxidation over Pd catalyst-electrodes. Chem. Eng. J.
**225**, 315–322 (2013)Google Scholar - 10.A.D. Ryabov, T.J. Collins, Mechanistic considerations on the reactivity of green FeIII-TAML activators of peroxides. Adv. Inorg. Chem.
**61**, 471–521 (2009)CrossRefGoogle Scholar - 11.A. Ghosh, A.D. Ryabov, S.M. Mayer, D.C. Horner, D.E. Prasuhn Jr., S. Sen Gupta, L. Vuocolo, C. Culver, M.P. Hendrich, C.E.F. Rickard et al., Understanding the mechanism of \(\text{ H }^{+}\)-induced demetalation as a design strategy for robust iron(III) peroxide-activating catalysts. J. Am. Chem. Soc.
**125**, 12378–12379 (2003)Google Scholar - 12.V. Polshin, D.-L. Popescu, A. Fischer, A. Chanda, D.C. Horner, E.S. Beach, J. Henry, Y.-L. Qian, C.P. Horwitz, G. Lente et al., Attaining control by design over the hydrolytic stability of Fe-TAML oxidation catalysts. J. Am. Chem. Soc.
**130**, 4497–4506 (2008)CrossRefGoogle Scholar - 13.A. Chanda, A.D. Ryabov, S. Mondal, L. Alexandrova, A. Ghosh, Y. Hangun-Balkir, C.P. Horwitz, T.J. Collins, The activity-stability parameterization of homogeneous green oxidation catalysts. Chem. Eur. J.
**12**, 9336–9345 (2006)CrossRefGoogle Scholar - 14.N. Chahbane, D.-L. Popescu, D.A. Mitchell, A. Chanda, D. Lenoir, A.D. Ryabov, K.-W. Schramm, T.J. Collins, \(\text{ Fe }^{\rm III}\)-TAML-catalyzed green oxidative degradation of the azo dye orange II by \(\text{ H }_{2}\text{ O }_{2}\) and organic peroxides: products, toxicity, kinetics, and mechanisms. Green Chem.
**9**, 49–57 (2007)Google Scholar - 15.D.-L. Popescu, M. Vrabel, A. Brausam, P. Madsen, G. Lente, I. Fabian, A.D. Ryabov, R. van Eldik, T.J. Collins, Thermodynamic, electrochemical, high-pressure kinetic, and mechanistic studies of the formation of Oxo FeIV-TAML species in water. Inorg. Chem.
**49**, 11439–11448 (2010)CrossRefGoogle Scholar - 16.M.H. Holmes,
*Introduction to Perturbation Methods*(Springer, New York, 1995)CrossRefGoogle Scholar - 17.Holmes, M.H.
*Introduction to the Foundations of Applied Mathematics*(Springer Texts in Applied Mathematics, 2009)Google Scholar - 18.L.F. Shampine, M.W. Reichelt, The MATLAB ODE suite. SIAM J. Sci. Comput.
**18**, 1–22 (1997)CrossRefGoogle Scholar - 19.A. Ghosh, D.A. Mitchell, A. Chanda, A.D. Ryabov, D.L. Popescu, E. Upham, G.J. Collins, T.J. Collins, Catalase-peroxidase activity of iron(III)-TAML activators of hydrogen peroxide. J. Am. Chem. Soc.
**130**, 15116–15126 (2008)CrossRefGoogle Scholar - 20.W.C. Ellis, C.T. Tran, R. Roy, M. Rusten, A. Fischer, A.D. Ryabov, B. Blumberg, T.J. Collins, Designing green oxidation catalysts for purifying environmental waters. J. Am. Chem. Soc.
**132**, 9774–9781 (2010)CrossRefGoogle Scholar - 21.P. George, The chemical nature of the second hydrogen peroxide compound formed by cytochrome c peroxidase and horseradish peroxidase. Biochem. J.
**54**, 267–262 (1953)Google Scholar

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