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

Original Paper

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DOI: 10.1007/s10910-014-0322-4

- Cite this article as:
- Emelianenko, M., Torrejon, D., DeNardo, M.A. et al. J Math Chem (2014) 52: 1460. doi:10.1007/s10910-014-0322-4

## 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## Copyright information

© Springer International Publishing Switzerland 2014