Journal of Solid State Electrochemistry https://doi.org/10.1007/s10008-019-04320-7 ORIGINAL PAPER Square-wave protein-film voltammetry: new insights in the enzymatic electrode processes coupled with chemical reactions Rubin Gulaboski1 & Valentin Mirceski2,3 & Milivoj Lovric4 Received: 4 April 2019 /Revised: 9 June 2019 /Accepted: 9 June 2019 # Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract Redox mechanisms in which the redox transformation is coupled to other chemical reactions are of significant interest since they are regarded as relevant models for many physiological systems. Protein-film voltammetry, based on surface confined electro- chemical processes, is a methodology of exceptional importance, which is designed to provide information on enzyme redox chemistry. In this work, we address some theoretical aspects of surface confined electrode mechanisms under conditions of square-wave voltammetry (SWV). Attention is paid to a collection of specific voltammetric features of a surface electrode reaction coupled with a follow-up (ECrev), preceding (CrevE) and regenerative (EC’) chemical reaction. While presenting a collection of numerically calculated square-wave voltammograms, several intriguing and simple features enabling kinetic char- acterization of studied mechanisms in time-independent experiments (i.e., voltammetric experiments at a constant scan rate) are addressed. The aim of the work is to help in designing a suitable experimental set-up for studying surface electrode processes, as well as to provide a means for determination of kinetic and/or thermodynamic parameters of both electrode and chemical steps. Keywords Kinetics of electrode reactions and chemical reactions . Surface EC′ catalytic mechanism . Surface ECrev mechanism . Surface CrevE mechanism . Square-wave voltammetry Introduction modulation applied, however, SWV is seldom explored as a technique for mechanistic evaluations. In spite of the fact that In the past three decades, square-wave voltammetry (SWV) numerous theoretical models for various electrode mechanisms emerged as a leading technique in the family of voltammetric arepresentedsofar,awideapplication of SWV for kinetic and methods. Its unique features stemfromthespecificshapeofthe mechanistic studies is still a challenge. This holds true even for applied exciting signal and the current sampling procedure that some electrode mechanisms considered to be Bsimple^.For efficiently discriminates faradaic against charging current instance, the so-called simple surface confined electrode reac- [1–5]. Hence, SWV is a highly sensitive voltammetric tech- tion under conditions of square-wave voltammetry is attributed nique capable to detect miscellaneous analytes at nanomolar with rather complex voltammetric pattern [1, 6–10]. concentration level [6, 7]. Due to the complexity of the potential Enzymatic electrode reactions in the so-called protein film voltammetry [11–17] are probably the most important exam- ples in the family of surface confined electrode processes. * Rubin Gulaboski Indeed, electrode mechanisms of uniformly adsorbed redox [email protected] active biological molecules coupled with follow-up or preced- ing chemical reactions attract significant attention as they are 1 Faculty of Medical Sciences, Goce Delcev University, considered as adequate models for some physiological sys- Stip, Macedonia tems [7, 13, 16, 17]. We have presented recently several the- 2 Institute of Chemistry, Faculty of Natural Sciences and Mathematics, oretical models of surface processes coupled to chemical re- POB 162, 1000 Skopje, Macedonia actions [18–28], revealing a set of intriguing, unique 3 Department of Inorganic and Analytical Chemistry, University of voltammetric features under conditions of SWV.In the present Lodz, Faculty of Chemistry, Tamka 12,, 91-403 Lodz, Poland work, a further attempt is made to throw a light to the specific 4 Divkovićeva 13, 10090 Zagreb, Croatia features of surface electrode processes coupled with a follow- J Solid State Electrochem k up, preceding or regenerative chemical reaction. The nomen- ÀÀÀÀf → Y þ ZadsðÞ OxðÞ ads clature of these systems is surface ECrev, CrevE and EC′ (or ←ÀÀÀÀ kb ð4Þ surface catalytic mechanism), respectively. The aim of this k θ ÀÀÀÀs → cumulative work is to help experimentalists to characterize OxðÞþ ads ne− ←ÀÀÀÀ RedðÞ ads qualitatively particular electrode mechanism and to design a suitable experimental approach for estimation of kinetic and In all mathematical models, we assume that all species are thermodynamic parameters governing surface electrode firmly immobilized on the electrode surface and the mass processes. transfer is neglected. In addition, it is considered that the immobilized species form a uniform monolayer free of lateral – B ^ Mathematical models interactions. In the schemes (2 4), the symbol Y is assigned to an electrochemically inactive compound that is present in a large excess. Consequently, the concentration of Y is virtually We consider theoretically four electrode mechanisms ade- constant at the electrode surface in the course of voltammetry, quate to the redox chemistry of firmly adsorbed enzymes in and all chemical steps in reaction mechanisms (2–4) are con- protein-film voltammetry: sidered to be of pseudo-first order. The solution of simple surface electrode mechanism (1) under SW voltammetric con- 1. Simple surface confined electrode reaction (E) ditions can be found elsewhere [1, 7, 8, 17], while the corre- k θ sponding solutions of the surface EC′, EC and surface C E ÀÀÀÀs → rev rev OxðÞþ ads ne− ←ÀÀÀÀ RedðÞ ads ð1Þ mechanisms are reported in [1, 17–19, 26], respectively. The recurrent formulas for calculating dimensionless currents Ψ of 2. Surface confined catalytic (regenerative) electrode mech- theoretical SW voltammograms as a function of applied po- anism (surface EC′) tential are given with Eqs. (5–8): ! Φ k θ m m−1 s −α⋅Φ 1 þ e ðÞþ− ÀÀÀÀ→ ðÞ ⋅ m ⋅ − ⋅∑ Ψ Ox ads ne ←ÀÀÀÀ Red ads ð2Þ K e 1 j 50 j−1 kc RedðÞþ ads Y ÀÀÀÀ→ OxðÞ ads Ψ m≔ ð5Þ −α⋅Φm ÀÁ K⋅e Φ 1 þ ⋅ 1 þ e m 50 3. Surface confined electrode reaction coupled with a follow-up reversible chemical reaction (surface ECrev) (5) is the recurrent formula for mechanism (1) i.e., simple surface redox reaction. k θ ÀÀÀÀs → ! OxðÞþ ads ne− ←ÀÀÀÀ RedðÞ ads Φ 1 þ e m m−1 k f ð3Þ −α⋅Φm ÀÀÀÀ→ K⋅e ⋅ 1− ⋅∑ Ψ j⋅Mm− jþ1 RedðÞþ ads Y ←ÀÀÀÀ ZadsðÞ Kchem j−1 kb Ψ m≔ ð6Þ −α⋅Φ Φ M1 1 þ K⋅e m ⋅ðÞ1 þ e m ⋅ Kchem 4. Surface confined electrode reaction coupled with a pre- ′ ceding reversible chemical reaction (surface CrevE) (6) is the recurrent formula for mechanism (2) (surface EC ). − − − K −α⋅Φ m 1 K⋅Keq − Φ ⋅ðÞ−α m 1 Kchem ðÞ−α ⋅Φ m 1 −α⋅Φm m 1 m 1 1 m K⋅e − ⋅e ⋅∑ Ψ j− ðÞKchem ⋅e ⋅∑ ðÞΨ i⋅Mi − ⋅e ⋅∑ðÞΨ i⋅Mi 50 i¼1 1 þ Keq i¼1 1 þ Keq i¼1 Ψ m≔ ð7Þ K −α⋅Φ K⋅Keq − Φ ⋅ðÞ−α Kchem ðÞ−α ⋅Φ 1 þ ⋅e m þ ðÞKchem 1⋅e m 1 ⋅M þ ⋅e 1 m ⋅M 50 1 þ Keq 1 1 þ Keq 1 (7) is the recurrent formula for mechanism (3) (surface ECrev). ! −α⋅Φ − − − K⋅e m ⋅Keq 1 m 1 − 1 m 1 ÀÁK m 1 1 −α⋅Φm Φm⋅ðÞ1−α ⋅ 1 þ ⋅∑ Ψ j þ ðÞKchem ⋅K⋅ ⋅e ⋅∑ Ψ i⋅Sm− jþ1 − ⋅e ⋅∑ Ψ j 1 þ Keq 50 ¼ 1 þ Keq ¼ 50 ¼ Ψ ≔ j 1 j 1 j 1 m −α⋅Φ (8) ⋅ m ⋅ K e Keq 1 −1 1 −α⋅Φ K Φ ⋅ðÞ1−α ⋅ þ 1 −ðÞKchem ⋅K⋅ ⋅M ⋅e m þ ⋅e m 1 þ Keq 50 1 þ Keq 1 50 J Solid State Electrochem (8) is the recurrent formula for mechanism (4) (surface CrevE). In the recurrent formulas (I–IV), the reduction current is considered to be positive and the dimensionless current is defined as Ψ = I/[(nFSfΓ*)]. In the last equation, I is electric current, n is stoichiometric number of electrons, S is active electrode surface area, f is SW frequency (f =1/2tp, where tp is the duration of a single potential pulse in SWV), and Γ*is the total surface concentration (i.e., the initial surface concen- Φ tration ofÀÁ Ox species). refers to the dimensionless potential Φ ¼ nF − ϕ0 ϕ’ α ( RT E E ,whereE is the formal redox potential, is the electron transfer coefficient, while other symbols have their common meaning. o The dimensionless electrode kinetic parameter K = ks /f re- Scheme 1 Schematic representation of a potential cycle in SWV o consisting of two adjacent oppositely oriented potential pulses. The lates the formal rate constant ks to the SWV frequency. For ε current is measured at the end of potential pulses (in the sectors the surface ECrev and CrevE mechanisms, Kchem = /f is the B ^ B ^ ε assigned with If and Ib ). In the so-called dead time, the current is dimensionless chemical kinetic parameter, where =(kf + kb) not measured, but the electrode reactions go on also in this time segment is the sum of the first-order rate constant kf and kb of the forward and backward chemical reactions, respectively. In ad- current measuring in SWV takes place in a narrow time seg- dition, the equilibrium constant is defined as Keq = kf/kb.For ment at the end of each potential pulse (I and I in the Scheme ′ f b the surface EC catalytic mechanism, the dimensionless chem- 1). In this way, one discriminates significantly faradaic from ical kinetic parameter is defined as Kchem = kc/f,wherekc is the the charging current. Of course, the surface concentration var- first-order rate constant of the regenerative (catalytic) chemi- iation of redox species prior to the current sampling in the so- cal reaction. called dead time (see scheme 1) plays an important role in In addition, M is numerical integration factor defined as overall features of SW voltammograms [25, 26].