Journal of Mathematical Chemistry

, Volume 52, Issue 5, pp 1460–1476 | Cite as

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

  • Maria Emelianenko
  • Diego Torrejon
  • Matthew A. DeNardo
  • Annika K. Socolofsky
  • Alexander D. Ryabov
  • Terrence J. Collins
Original Paper


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


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



The authors are grateful to Dan Anderson, Pak-Wing Fok, Angela Dapolite and Joshua Patent for fruitful discussions and contributions at the early stages of the work. In addition, support is acknowledged of the Heinz Endowments (T.J.C.) and the Institute for Green Science (T.J.C). NMR instrumentation at CMU was partially supported by National Science Foundation (CHE-0130903 and CHE-1039870). M.E. acknowledges National Science Foundation support under DMS-202340.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Maria Emelianenko
    • 1
  • Diego Torrejon
    • 1
  • Matthew A. DeNardo
    • 2
  • Annika K. Socolofsky
    • 2
  • Alexander D. Ryabov
    • 2
  • Terrence J. Collins
    • 2
  1. 1.Department of Mathematical SciencesGeorge Mason UniversityFairfaxUSA
  2. 2.Department of ChemistryCarnegie Mellon UniversityPittsburghUSA

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