Analytical expressions of the substrate and mediator of multi-step enzyme electrodes

  • K. M. Dharmalingam
  • M. VeeramuniEmail author
  • T. Praveen
Original Paper


Mathematical models of multi-step enzyme systems immobilized on porous electrodes are discussed. The model contains a non-linear term related to Monod kinetics. This work presents a better approximate analytical solution for non-linear differential equations of concentrations profile is derived using new approach of Homotopy perturbation method. Moreover, the derived analytical expressions are compared with numerical simulations. In addition, analytical expression for current density is derived and also it is compared with experimental data of already published work of Hettige et al [16]. The analytical expressions derived here are good enough to predict the behavior of the system. The influence of dimensionless governing parameters are discussed and presented graphically. The presented solutions are more reliable and easy to predict the dynamic behavior of the system by varying the parameters.


Mathematical modeling Non-linear equations Biofuel cells Enzymes Homotopy perturbation method 

List of symbols

\( a \)

Electrode area/volume (cm−1)

\( b \)

Tafel slope (V)

\( C_{Cx} \)

Initial concentration of substrate (mol cm−3)

\( Dai_{j}\)

Damkohler number

\( D_{C} \)

Diffusivity of species C (m2 s−1)

\( D_{M} \)

Diffusivity of species M (m2 s−1)


Faraday constant (s A/mol)

\( k \)

Electron transfer rate constant at the electrode (mol s−1 cm−3)

\( K_{cat} \)

Turnover rate (s−1)

\( K_{M} \)

Michaelis constant for mediator (mM)

\( K_{S} \)

Michaelis constant for substrate (mM)

\( l \)

Length of the electrode (cm)

\( M_{Mx} \)

Initial concentration of mediator (mol cm−3)

\( Pe_{i} \)

Peclet numbers

\( r_{j} \)

Rate of reaction j

\( r_{j}^{\prime} \)

Dimensionless rate of reaction j

\( s_{ij} \)

Stoichiometric coefficient of species i

\( v \)

Flow velocity (m2 s−1)

\( \eta \)

Electrode overpotential (V)

\( x \)

Distances (cm)

\( x' \)

Dimensionless distances



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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe Madura CollegeMaduraiIndia
  2. 2.Department of Mathematics, School of Advanced SciencesVITVelloreIndia

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