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

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

Abstract

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.

Keywords

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)

F

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

Notes

References

  1. 1.
    A.T. Yahiro, S.M. Lee, D.O. Kimble, Bioelectrochemistry. I. Enzyme utilizing bio-fuel cell studies. Biochim. Biophys. Acta 88, 375–383 (1964)Google Scholar
  2. 2.
    M.J. Cooney, V. Svoboda, C. Lau, G. Martin, S.D. Minteer, Enzyme catalysed biofuel cells. Energy Environ. Sci. 1, 320–337 (2008)CrossRefGoogle Scholar
  3. 3.
    J. Kim, H.F. Walsh, P. Wang, Challenges in biocatalysis for enzyme-based biofuel cells. Biotechnol. Adv. 24, 296–308 (2006)CrossRefGoogle Scholar
  4. 4.
    S.C. Barton, J. Gallaway, P. Atanassov, Enzymatic biofuel cells for implantable and microscale devices. Chem. Rev. 104, 4867–4886 (2004)CrossRefGoogle Scholar
  5. 5.
    A. Heller, Miniature biofuel cells. Phys. Chem. Chem. Phys. 6, 209–216 (2004)CrossRefGoogle Scholar
  6. 6.
    R.A. Butten, T.C. Arnot, J.B. Lakeman, F.C. Walsh, Biofuel cells and their development. Biosens. Bioelectron. 21, 2015–2045 (2006)CrossRefGoogle Scholar
  7. 7.
    H.Y. Eileen, S. Keith, Enzymatic biofuel cells-fabrication of enzyme electrodes. Energies 3, 23–42 (2010)CrossRefGoogle Scholar
  8. 8.
    I. Ivan, V.K. Tanja, S. Kai, Recent advances in enzymatic fuel cells: experiments and modeling. Energies 3, 803–846 (2010)CrossRefGoogle Scholar
  9. 9.
    M.H. Osman, A.A. Shah, R.G.A. Wills, F.C. Walsh, Mathematical modelling of an enzymatic fuel cell with air-breathing cathode. Electrochim. Acta 112, 386–393 (2013)CrossRefGoogle Scholar
  10. 10.
    R. Baronas, F. Ivanauskas, J. Kulys, Modelling a biosensor based on the heterogeneous microreactor. J. Math. Chem. 25(2–3), 245–252 (1999)CrossRefGoogle Scholar
  11. 11.
    W.F. Ames, Numerical methods for partial differential equations, 2nd edn. (Academic Press, New York, 1977)Google Scholar
  12. 12.
    L. Rajendran, G. Rahamathunissa, The application of He’s variational iteration method to nonlinear boundary value problems in enzyme-substrate reaction diffusion processes: part 1. The steady-state amperometric response. J. Math. Chem. 44, 849–861 (2008)CrossRefGoogle Scholar
  13. 13.
    A. Meena, A. Eswari, L. Rajendran, Mathematical modelling of enzyme kinetics reaction mechanisms and analytical solutions of non-linear reaction equations. J. Math. Chem. 48, 179–186 (2010)CrossRefGoogle Scholar
  14. 14.
    R. Senthamarai, L. Rajendran, Traveling wave solution of non-linear coupled reaction diffusion equation arising in mathematical chemistry. J. Math. Chem. 46, 550–561 (2009)CrossRefGoogle Scholar
  15. 15.
    V.M. PonRani, L. Rajendran, Mathematical modelling of steady-state concentration in immobilized glucose isomerase of packed-bed reactors. J. Math. Chem. 50, 1333–1346 (2012)CrossRefGoogle Scholar
  16. 16.
    C. Hettige, D. Minteer, S.C. Barton, Simulation of multi-step enzyme electrodes. ECS Trans. Electrochem. Soc. 13, 99–109 (2008)CrossRefGoogle Scholar
  17. 17.
    P. Kar, H. Wen, H. Li, S.D. Minteer, S.C. Bartona, Simulation of multistep enzyme-catalyzed methanol oxidation in biofuel cells. J. Electrochem. Soc. 158, 580–586 (2011)CrossRefGoogle Scholar
  18. 18.
    J.-H. He, Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons Fractals 26(3), 695–700 (2005)CrossRefGoogle Scholar
  19. 19.
    Q.K. Ghori, M. Ahmed, A.M. Siddiqui, Application of homotopy perturbation method to squeezing flow of a Newtonian fluid. Int. J. Nonlinear Sci. Numer. Simul. 8(2), 179–184 (2007)CrossRefGoogle Scholar
  20. 20.
    T. Özis, A. Yildirim, A comparative study of He’s homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities. Int. J. Nonlinear Sci. Numer. Simul. 8(2), 243–248 (2007)CrossRefGoogle Scholar
  21. 21.
    S.-J. Li, Y.-X. Liu, An improved approach to nonlinear dynamical system identification using PID neural networks. Int. J. Nonlinear Sci. Numer. Simul. 7(2), 177–182 (2006)CrossRefGoogle Scholar
  22. 22.
    M.M. Mousa, S.F. Ragab, Application of the homotopy perturbation method to linear and nonlinear Schrödinger equations. Zeitschrift fur Naturforschung—J. Phys. Sci. 63(3-4), 140–144 (2008)Google Scholar
  23. 23.
    J.H. He, Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999)CrossRefGoogle Scholar
  24. 24.
    L. Rajendran, S. Anitha, Reply to ‘Comments on analytical solution of amperometric enzymatic reactions based on HPM’. Electrochim. Acta 102, 474–476 (2013)CrossRefGoogle Scholar
  25. 25.
    J.H. He, Homotopy perturbation method with an auxiliary term. Abstr. Appl. Anal. 2012, 857612 (2012)Google Scholar
  26. 26.
    J.H. He, Homotopy perturbation method with two expanding parameters. Indian J. Phys. 88, 193–196 (2014)CrossRefGoogle Scholar
  27. 27.
    Yusry El-dib, Multiple scales homotopy perturbation method for nonlinear oscillators. Nonlinear Sci. Lett. A 8, 352–364 (2017)Google Scholar
  28. 28.
    Z.J. Liu, M.Y. Adamu, E. Suleiman, J.H. He, Hybridization of homotopy perturbation method and Laplace transformation for the partial differential equations. Therm. Sci. 21, 1843–1846 (2017)CrossRefGoogle Scholar
  29. 29.
    A.A. Hemeda, Homotopy perturbation method for solving systems of nonlinear coupled equations. Appl. Math. Sci. 6, 4787–4800 (2012)Google Scholar
  30. 30.
    J.H. He, Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178, 257–262 (1999)CrossRefGoogle Scholar
  31. 31.
    J.H. He, Application of homotopy perturbation method to nonlinear wave equations. Chaos, Solitons Fractals 26, 695–700 (2005)CrossRefGoogle Scholar

Copyright information

© 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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