Modeling carbohydrates oxidation by oxygen catalyzed by bienzyme glucose dehydrogenase/laccase system immobilized into microreactor with carbon nanotubes


This paper presents a mathematical model of a batch stirred tank reactor based on an array of identical spherical porous microbioreactors loaded with non specific glucose dehydrogenase and oxygen reducing enzyme, i.e. laccase. The model was validated by experimental data. The microreactors (MR) are mathematically modeled by a two-compartment model, based on reaction–diffusion equations containing nonlinear terms related to the Michaelis–Menten kinetics of two enzymatic reactions with addition of the mass transport. The dynamics of oxygen concentration change is analysed numerically using the finite difference technique. The transient effectiveness factor and the process duration are investigated at different initial concentration of carbohydate (lactose) as well as at internal and external diffusion resistances. The simulation results show a non-monotonic effect of the initial concentration of lactose and nonlinear effects of the internal and external diffusion limitations on the transient effectiveness.

Graphic Abstract

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    A.E. Al-Muftah, I.M. Abu-Reesh, Effects of simultaneous internal and external mass transfer and product inhibition on immobilized enzyme-catalyzed reactor. Biochem. Eng. J. 27, 167–178 (2005)

    CAS  Google Scholar 

  2. 2.

    M. Al-Shannag, Z. Al-Qodah, J. Herrero, J.A.C. Humphrey, F. Giralt, Using a wall-driven flow to reduce the external mass-transfer resistance of a bio-reaction system. Biochem. Eng. J. 38, 554–565 (2008)

    Google Scholar 

  3. 3.

    R. Aris, Mathematical Modeling: A Chemical Engineer’s Perspective (Academic Press, London, 1999)

    Google Scholar 

  4. 4.

    R. Baronas, F. Ivanauskas, J. Kulys, Modelling a biosensor based on the heterogeneous microreactor. J. Math. Chem. 25, 245–252 (1999)

    CAS  Google Scholar 

  5. 5.

    R. Baronas, J. Kulys, L. Petkevičius, Modelling the enzyme catalysed substrate conversion in a microbioreactor acting in continuous flow mode. Nonlinear Anal. Model. Control 23, 437–456 (2018)

    Google Scholar 

  6. 6.

    R. Baronas, J. Kulys, L. Petkevičius, Computational modeling of batch stirred tank reactor based on spherical catalyst particles. J. Math. Chem. 57, 327–342 (2019)

    CAS  Google Scholar 

  7. 7.

    P.N. Bartlett, Bioelectrochemistry: Fundamentals, Experimental Techniques and Applications (Wiley, Chichester, 2008)

    Google Scholar 

  8. 8.

    L.A. Belfiore, Transport Phenomena for Chemical Reactor Design (Wiley, Hoboken, 2003)

    Google Scholar 

  9. 9.

    C.M. Bidabehere, J.R. García, U. Sedran, Use of stirred batch reactors for the assessment of adsorption constants in porous solid catalysts with simultaneous diffusion and reaction. Theoretical analysis. Chem. Eng. Sci. 61, 2048–2055 (2006)

    CAS  Google Scholar 

  10. 10.

    C.M. Bidabehere, J.R. García, U. Sedran, Transient effectiveness factor in porous catalyst particles. Application to kinetic studies with batch reactors. Chem. Eng. Res. Des. 118, 41–50 (2017)

    CAS  Google Scholar 

  11. 11.

    C.M. Bidabehere, J.R. García, U. Sedran, Transient effectiveness factor. simultaneous determination of kinetic, diffusion and adsorption equilibrium parameters in porous catalyst particles under diffusion control conditions. Chem. Eng. J. 345, 196–208 (2018)

    CAS  Google Scholar 

  12. 12.

    C.M. Bidabehere, U. Sedran, Transient effectiveness factors in the dynamic analysis of heterogeneous reactors with porous catalyst particles. Chem. Eng. Sci. 137, 293–300 (2015)

    CAS  Google Scholar 

  13. 13.

    R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nd edn. (Wiley, New York, 2006)

    Google Scholar 

  14. 14.

    K. Bizon, B. Tabiś, Dynamics of an isothermal catalyst pellet with simultaneous chemical reaction and adsorption. Chem. Eng. Res. Des. 115, 221–229 (2016)

    CAS  Google Scholar 

  15. 15.

    D. Britz, R. Baronas, E. Gaidamauskaitė, F. Ivanauskas, Further comparisons of finite difference schemes for computational modelling of biosensors. Nonlinear Anal. Model. Control 14, 419–433 (2009)

    CAS  Google Scholar 

  16. 16.

    D. Britz, J. Strutwolf, Digital Simulation in Electrochemistry, 4th edn., Monographs in Electrochemistry (Springer, Cham, 2016)

    Google Scholar 

  17. 17.

    H.-C. Chang, C.-C. Wu, S.-J. Ding, I.-S. Lin, I.-W. Sun, Measurement of diffusion and partition coefficients of ferrocyanide in protein-immobilized membranes. Anal. Chim. Acta. 532, 209–214 (2005)

    CAS  Google Scholar 

  18. 18.

    D.S. Clark, H.W. Blanch, Biochemical Engineering, 2nd edn. (Marcel Dekker, New York, 1997)

    Google Scholar 

  19. 19.

    M.E. Davis, R.J. Davis, Fundamentals of Chemical Reaction Engineering (McGraw-Hill, New York, 2003)

    Google Scholar 

  20. 20.

    P.M. Doran, Bioprocess Engineering Principles, 2nd edn. (Academic Press, Waltham, MA, 2013)

    Google Scholar 

  21. 21.

    R.F. Fonseca, C.C.B. Melo, B.C.P. Beatriz, V. Sanches, C.S. Bertucci-Neto, W.H.K. Farinas, Modelling of solid-state fermentation over wide operational range for application in process optimization. Can. J. Chem. Eng. 96, 1723–1734 (2018)

    CAS  Google Scholar 

  22. 22.

    J. Iqbal, S. Iqbala, C.E. Müller, Advances in immobilized enzyme microbioreactors in capillary electrophoresis. Analyst 138, 3104–3116 (2013)

    CAS  PubMed  Google Scholar 

  23. 23.

    B. Kaoui, M. Lauricella, G. Pontrelli, Mechanistic modelling of drug release from multi-layer capsules. Comput. Biol. Med. 93, 149–157 (2018)

    CAS  PubMed  Google Scholar 

  24. 24.

    A.Y. Khan, S.B. Noronha, R. Bandyopadhyaya, Glucose oxidase enzyme immobilized porous silica for improved performance of a glucose biosensor. Biochem. Eng. J. 91, 78–85 (2014)

    CAS  Google Scholar 

  25. 25.

    J. Kulys, The development of new analytical systems based on biocatalysts. Anal. Lett. 14, 377–397 (1981)

    CAS  Google Scholar 

  26. 26.

    J. Leszczynski, Handbook of Computational Chemistry (Springer, Dordrecht, 2012)

    Google Scholar 

  27. 27.

    M.F. Luna, E.C. Martínez, Optimal design of dynamic experiments in the development of cybernetic models for bioreactors. Chem. Eng. Res. Des. 136, 334–346 (2018)

    CAS  Google Scholar 

  28. 28.

    G. Maria, Enzymatic reactor selection and derivation of the optimal operation policy, by using a model-based modular simulation platform. Comput. Chem. Eng. 36, 325–341 (2012)

    CAS  Google Scholar 

  29. 29.

    E. Nagy, Survey on biocatalytic membrane reactor and membrane aerated biofilm reactor. Curr. Org. Chem. 21, 1713–1724 (2017)

    CAS  Google Scholar 

  30. 30.

    E. Papadakis, S. Pedersen, A.K. Tula, M. Fedorova, J.M. Woodley, R. Gani, Model-based design and analysis of glucose isomerization process operation. Comput. Chem. Eng. 98, 128–142 (2017)

    CAS  Google Scholar 

  31. 31.

    S. Petronis, M. Stangegaard, C.B.V. Christensen, M. Dufva, Transparent polymeric cell culture chip with integrated temperature control and uniform media perfusion. Biotechniques 40, 368–376 (2006)

    CAS  PubMed  Google Scholar 

  32. 32.

    W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007)

    Google Scholar 

  33. 33.

    D. Ratautas, L. Marcinkevičienė, R. Meškys, J. Kulys, Mediatorless electron transfer in glucose dehydrogenase/laccase system adsorbed on carbon nanotubes. Electrochim. Acta 174, 940–944 (2015)

    CAS  Google Scholar 

  34. 34.

    D. Ratautas, E. Ramonas, L. Marcinkevičienė, R. Meškys, J. Kulys, Wiring gold nanoparticles and redox enzymes: a self-sufficient nanocatalyst for the direct oxidation of carbohydrates with molecular oxygen. ChemCatChem 10, 971–974 (2018)

    CAS  Google Scholar 

  35. 35.

    U. Rinas, H. El-Enshasy, M. Emmler, A. Hille, D.C. Hempel, H. Horn, Model-based prediction of substrate conversion and protein synthesis and excretion in recombinant Aspergillus niger biopellets. Chem. Eng. Sci. 60, 2729–2739 (2005)

    CAS  Google Scholar 

  36. 36.

    A. Sagiv, Exact solution of mass diffusion into a finite volume. J. Membr. Sci. 186, 231–237 (2001)

    CAS  Google Scholar 

  37. 37.

    D. Schäpper, M.N.H.Z. Alam, N. Szita, A.E. Lantz, K.V. Gernaey, Application of microbioreactors in fermentation process development: a review. Anal. Bioanal. Chem. 395, 679–695 (2009)

    PubMed  Google Scholar 

  38. 38.

    S. Skoneczny, M. Cioch-Skoneczny, Mathematical modelling and approximate solutions for microbiological processes in biofilm through homotopy-based methods. Chem. Eng. Res. Des. 139, 309–320 (2018)

    CAS  Google Scholar 

  39. 39.

    W. Tischer, F. Wedekind, Immobilized enzymes: methods and applications, in Biocatalysis-From Discovery to Application, vol. 200, Topics in Current Chemistry. Topics in Current Chemistry, ed. by W.D. Fessner, et al. (Springer, Berlin, 1999), pp. 95–126

    Google Scholar 

  40. 40.

    M. Velkovsky, R. Snider, D.E. Cliffel, J.P. Wikswo, Modeling the measurements of cellular fluxes in microbioreactor devices using thin enzyme electrodes. J. Math. Chem. 49, 251–275 (2011)

    CAS  PubMed  Google Scholar 

  41. 41.

    G. Vidriales-Escobar, R. Rentería-Tamayo, F. Alatriste-Mondragón, O. González-Ortega, Mathematical modeling of a composting process in a small-scale tubular bioreactor. Chem. Eng. Res. Des. 120, 360–371 (2017)

    CAS  Google Scholar 

  42. 42.

    J. Villadsen, J. Nielsen, G. Lidén, Bioreaction Engineering Principles, 3rd edn., Monographs in Electrochemistry (Springer, New York, 2011)

    Google Scholar 

  43. 43.

    H.J. Vos, P.J. Heederik, J.J.M. Potters, K.C.A.M. Luyben, Effectiveness factor for spherical biofilm catalysts. Bioprocess Eng. 5, 63–72 (1990)

    CAS  Google Scholar 

  44. 44.

    S. Whitaker, The Method of Volume Averaging, vol. 13 (Springer, Berlin, 2013)

    Google Scholar 

  45. 45.

    L.-T. Zhu, W.-Y. Ma, Z.-H. Luo, Influence of distributed pore size and porosity on MTO catalyst particle performance: modeling and simulation. Chem. Eng. Res. Des. 137, 141–153 (2018)

    CAS  Google Scholar 

Download references


The work of R. Baronas and L. Petkevičius was supported by the Research Council of Lithuania under Grant No. S-MIP-17-98. J. Kulys thanks to Eimantas Ramonas for the CPG modification and the performance of the oxygen consumptions experiments.

Author information



Corresponding author

Correspondence to Linas Petkevičius.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 802 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Baronas, R., Kulys, J. & Petkevičius, L. Modeling carbohydrates oxidation by oxygen catalyzed by bienzyme glucose dehydrogenase/laccase system immobilized into microreactor with carbon nanotubes. J Math Chem 59, 168–185 (2021).

Download citation


  • Batch reactor
  • Modeling
  • Bienzyme
  • Enzyme wiring
  • Effectiveness factor