The Mathematics Behind Surface CirrE1CrevE2C’ Mechanism in Protein-Film Square-Wave Voltammetry

Protein-film square-wave voltammetry provides a sensitive approach for investigating complex electron-transfer and coupled chemical reactions of surface-confined redox proteins. In this work, the mathematical framework of the surface CirrE₁CrevE₂C′ mechanism is developed and analyzed. The mechanism combines a preceding irreversible chemical step (Cirr), two successive electron-transfer reactions (E₁ and E₂), an intermediate reversible chemical transformation (Crev), and a regenerative catalytic reaction (C′). All participating species are considered strongly adsorbed at the working electrode, allowing the system to be described exclusively through the time-dependent surface concentrations. Butler–Volmer kinetics are applied to both electron-transfer steps, while the accompanying chemical reactions are represented by their corresponding kinetic rate expressions. The resulting set of differential equations links the individual kinetic and thermodynamic parameters with the forward, backward, and net components of the square-wave voltammetric response. The mathematical model provides a basis for systematically exploring how coupled chemical and electrochemical steps shape the voltammetric patterns of surface-confined redox proteins. A Mathcad simulation protocol in which the model is implemented enables straightforward variation of the relevant parameters and simulation of different kinetic scenarios. The proposed framework can therefore serve as a practical theoretical tool for mechanistic discrimination and kinetic analysis of complex processes in protein-film square-wave voltammetry.