Skip to main content
SpringerLink
Log in

Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors

  • Chapter
  • First Online:
Mathematical Modeling of Biosensors

Part of the book series: Springer Series on Chemical Sensors and Biosensors ((SSSENSORS,volume 9))

Abstract

This chapter deals with applying a multi-layer approach to the modeling of biosensors. The multi-layer mathematical models of amperometric biosensors are considered at stationary and transient conditions. First, a multi-layer approach is demonstrated for a biosensor having several mono-enzyme layers sandwich-likely applied onto the electrode surface. Then, a two-compartment model involving an enzyme layer, where the enzyme reaction as well as the mass transport by diffusion take place, and a diffusion limiting region, where only the mass transport by diffusion takes place, is analysed. The dependencies of the internal and external diffusion limitations on the response and sensitivity of amperometric biosensors are investigated, and the conditions when the mass transport outside the enzyme membrane may be neglected to simulate the biosensor response accurately in a well-stirred solution are considered. The mathematical models are based on the reaction–diffusion equations containing a nonlinear term related to Michaelis–Menten kinetics. The computer simulation was carried out using the finite difference technique.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Achi F, Bourouina-Bacha S, Bourouina M, Amine A (2015) Mathematical model and numerical simulation of inhibition based biosensor for the detection of Hg(II). Sens Actuator B Chem 207(A):413–423

    Article  Google Scholar 

  2. Amatore C, Szunerits S, Thouin L, Warkocz JS (2001) The real meaning of Nernst’s steady diffusion layer concept under non-forced hydrodynamic conditions. A simple model based on Levich’s seminal view of convection. J Electroanal Chem 500(1–2):62–70

    Article  Google Scholar 

  3. Ames W (1977) Numerical methods for partial differential equations, 2 edn. Academic, New-York

    MATH  Google Scholar 

  4. Aris R (1975) The mathematical theory of diffusion and reaction in permeable catalysts: The Theory of the Steady State. Oxford Studies in Physics, vol 1. Oxford University Press, Oxford

    Google Scholar 

  5. Aris R (1999) Mathematical modeling: a chemical engineer’s perspective. Academic, London

    MATH  Google Scholar 

  6. Ašeris V, Gaidamauskaitė E, Kulys J, Baronas R (2014) Modelling the biosensor utilising parallel substrates conversion. Electrochim Acta 146:752–758

    Article  Google Scholar 

  7. Ašeris V, Baronas R, Petrauskas K (2016) Computational modelling of three-layered biosensor based on chemically modified electrode. Comp Appl Math 35(2):405–421

    Article  MathSciNet  MATH  Google Scholar 

  8. Bacha S, Bergel A, Comtat M (1995) Transient response of multilayer electroenzymic biosensors. Anal Chem 67(10):1669–1678

    Article  Google Scholar 

  9. Bacha S, Montagné M, Bergel A (1996) Modeling mass transfer with enzymatic reaction in electrochemical multilayer microreactors. AICHE J 42:2967–2976

    Article  Google Scholar 

  10. Bacon NC, Hall E (1999) A sandwich enzyme electrode giving electrochemical scavenging of interferents. Electroanal 11(10–11):749–755

    Article  Google Scholar 

  11. Baeumner A, Jones C, Wong C, Price A (2004) A generic sandwich-type biosensor with nanomolar detection limits. Anal Bioanal Chem 378(6):1587–1593

    Article  Google Scholar 

  12. Banica FG (2012) Chemical sensors and biosensors: fundamentals and applications. Wiley, Chichester

    Book  Google Scholar 

  13. Baronas R (2017) Nonlinear effects of diffusion limitations on the response and sensitivity of amperometric biosensors. Electrochim Acta 240:399–407

    Article  Google Scholar 

  14. Baronas R, Ivanauskas F, Kulys J (1999) Modeling a biosensor based on the heterogeneous microreactor. J Math Chem 25(3–4):245–252

    Article  MATH  Google Scholar 

  15. Baronas R, Ivanauskas F, Kulys J (2003) Computer simulation of the response of amperometric biosensors in stirred and non stirred solution. Nonlinear Anal Model Control 8(1):3–18

    Article  MATH  Google Scholar 

  16. Baronas R, Ivanauskas F, Kulys J (2003) The influence of the enzyme membrane thickness on the response of amperometric biosensors. Sensors 3(7):248–262

    Article  MATH  Google Scholar 

  17. Baronas R, Ivanauskas F, Kulys J (2004) The effect of diffusion limitations on the response of amperometric biosensors with substrate cyclic conversion. J Math Chem35(3):199–213

    Google Scholar 

  18. Baronas R, Kulys J, Ivanauskas F (2006) Computational modeling of biosensors with perforated and selective membranes. J Math Chem 39(2):345–362

    Article  MathSciNet  MATH  Google Scholar 

  19. Baronas R, Ivanauskas F, Kaunietis I, Laurinavicius V (2006) Mathematical modeling of plate-gap biosensors with an outer porous membrane. Sensors 6(7):727–745

    Article  Google Scholar 

  20. Baronas R, Kulys J, Petkevičius L (2019) Computational modeling of batch stirred tank reactor based on spherical catalyst particles. J Math Chem 57(1):327–342

    Article  MathSciNet  MATH  Google Scholar 

  21. Bartlett PN (2008) Bioelectrochemistry: fundamentals, experimental techniques and applications. Wiley, Chichester

    Book  Google Scholar 

  22. Bartlett P, Whitaker R (1987) Electrochemical immobilisation of enzymes: Part 1. Theory. J Electroanal Chem 224:27–35

    Article  Google Scholar 

  23. Bergel A, Comtat M (1984) Theoretical evaluation of transient responses of an amperometric enzyme electrode. Anal Chem 56(14):2904–2909

    Article  Google Scholar 

  24. Bieniasz L (2017) A specialised cyclic reduction algorithm for linear algebraic equation systems with quasi-tridiagonal matrices. J Math Chem 55(9):1793–1807

    Article  MathSciNet  MATH  Google Scholar 

  25. Blaedel W, Kissel T, Boguslaski R (1972) Kinetic behavior of enzymes immobilized in artificial membranes. Anal Chem 44(12):2030–2037

    Article  Google Scholar 

  26. Britz D, Strutwolf J (2015) Digital simulation of chronoamperometry at a disk electrode under a flat polymer film containing an enzyme. Electrochim Acta 152:302–307

    Article  Google Scholar 

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

    Google Scholar 

  28. Britz D, Baronas R, Gaidamauskaitė E, Ivanauskas F (2009) Further comparisons of finite difference schemes for computational modelling of biosensors. Nonlinear Anal Model Control 14(4):419–433

    Article  MathSciNet  MATH  Google Scholar 

  29. Buerk D (1995) Biosensors: theory and applications. CRC Press, Lancaster

    Google Scholar 

  30. Cambiaso A, Delfino L, Grattarola M, Verreschi G, Ashworth D, Maines A, Vadgama P (1996) Modeling and simulation of a diffusion limited glucose biosensor. Sens Actuator B Chem 33(1–3):203–207

    Article  Google Scholar 

  31. Chen L, Tseng K, Ho K (2006) General kinetic model for amperometric sensors based on Prussian blue mediator and its analogs: application to cysteine detection. Electroanal 18(13–14):1313–1321

    Article  Google Scholar 

  32. Ciliberto A, Capuani F, Tyson JJ (2007) Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation. PLoS Comput Biol 3(3):e45

    Article  MathSciNet  Google Scholar 

  33. Crank J (1975) The mathematics of diffusion. Oxford University Press, London

    MATH  Google Scholar 

  34. Frew JE, Hill HAO (1987) Electrochemical biosensors. Anal Chem 59(15):933A–944A

    Article  Google Scholar 

  35. Goeke A, Schilli C, Walcher S, Zerz E (2012) Computing quasi-steady state reductions. J Math Chem 50(6):1495–1513

    Article  MathSciNet  MATH  Google Scholar 

  36. Goldbeter A (2013) Oscillatory enzyme reactions and Michaelis–Menten kinetics. FEBS Lett 587(17):2778–2784

    Article  Google Scholar 

  37. Gooding JJ, Hall EAH (1996) Parameters in the design of oxygen detecting oxidase enzyme electrodes. Electroanalysis 8(5):407–413

    Article  Google Scholar 

  38. Grieshaber D, MacKenzie R, Vörös J, Reimhult E (2008) Electrochemical biosensors - sensor principles and architectures. Sensors 8(3):1400–1458

    Article  Google Scholar 

  39. Gutfreund H (1995) Kinetics for the life sciences. Cambridge University Press, Cambridge

    Book  Google Scholar 

  40. Ha J, Engler CR, Lee SJ (2008) Determination of diffusion coefficients and diffusion characteristics for chlorferon and diethylthiophosphate in Ca-alginate gel beads. Biotechnol Bioeng 100(4):698–706

    Article  Google Scholar 

  41. Hameka H, Rechnitz G (1983) Theory of the biocatalytic membrane electrode. J Phys Chem 87(7):1235–1241

    Article  Google Scholar 

  42. Hassan M, Atiqullah M, Beg S, Chowdhury M (1995) Analysis of non-isothermal tubular reactor packed with immobilized enzyme systems. Chem Eng J Biochem Eng J 58(3):275–283

    Article  Google Scholar 

  43. Hickson RI, Barry SI, Mercer GN, Sidhu HS (2011) Finite difference schemes for multilayer diffusion. Math Comput Model 54(1–2):210–220

    Article  MathSciNet  MATH  Google Scholar 

  44. Iliev I, Atanasov P, Gamburzev S, Kaisheva A, Tonchev V (1992) Transient response of electrochemical biosensors with asymmetrical sandwich membranes. Sens Actuator B Chem 8(1):65–72

    Article  Google Scholar 

  45. Ivanauskas F, Baronas R (2008) Modeling an amperometric biosensor acting in a flowing liquid. Int J Numer Meth Fluids 56(8):1313–1319

    Article  MATH  Google Scholar 

  46. Ivanauskas F, Baronas R (2008) Numerical simulation of a plate-gap biosensor with an outer porous membrane. Simul Model Pract Th 16(8):962–970

    Article  Google Scholar 

  47. Ivanauskas F, Katauskis P, Laurinavičius V (2014) Mathematical modeling of biosensor action in the region between diffusion and kinetic modes. J Math Chem 52(2):689–702

    Article  MathSciNet  MATH  Google Scholar 

  48. Jobst G, Moser I, Urban G (1996) Numerical simulation of multi-layered enzymatic sensors. Biosens Bioelectron 11(1–2):111–117

    Article  Google Scholar 

  49. Jochum P, Kowalski BR (1982) A coupled two-compartment model for immobilized enzyme electrodes. Anal Chim Acta 144:25–38

    Article  Google Scholar 

  50. Kulys J, Razumas V (1986) Bioamperometry. Mokslas, Vilnius

    Google Scholar 

  51. Laurinavicius V, Kulys J, Gureviciene V, Simonavicius K (1989) Flow through and catheter biosensors with an extended concentration range. Biomed Biochem Acta 48(11–12):905–909

    Google Scholar 

  52. Lemke K (1988) Mathematical simulation of an amperometric enzyme-substrate electrode with a pO2 basic sensor. Part 2. Mathematical simulation of the glucose oxidase glucose electrode. Med Biol Eng Comput 26(5):533–540

    Article  Google Scholar 

  53. Levich V (1962) Physicochemical hydrodynamics. Prentice-Hall, London

    Google Scholar 

  54. Li B, Shen Y, Li B (2008) Quasi-steady-state laws in enzyme kinetics. J Phys Chem A 112(11):2311–2321

    Article  Google Scholar 

  55. Lyons MEG, Greer JC, Fitzgerald CA, Bannon T, Barlett PN (1996) Reaction/diffusion with Michaelis-Menten kinetics in electroactive polymer films. Part 1. The steady state amperometric response. Analyst 121(6):715–731

    Article  Google Scholar 

  56. Lyons M, Bannon T, Hinds G, Rebouillat S (1998) Reaction/diffusion with Michaelis-Menten kinetics in electroactive polymer films. Part 2. The transient amperometric response. Analyst 123(10):1947–1959

    Article  Google Scholar 

  57. Lyons M, Murphy J, Rebouillat S (2000) Theoretical analysis of time dependent diffusion, reaction and electromigration in membranes. J Solid State Electrochem 4(8):458–472

    Article  Google Scholar 

  58. Meskauskas T, Ivanauskas F, Laurinavicius V (2013) Degradation of substrate and/or product: mathematical modeling of biosensor action. J Math Chem 51(9):2491–2502

    Article  MathSciNet  MATH  Google Scholar 

  59. Meskauskas T, Ivanauskas F, Laurinavicius V (2013) Numerical modeling of multilayer biosensor with degrading substrate and product. In: AlBegain K, AlDabass D, Orsoni A, Cant R, Zobel R (eds) 8th EUROSIM congress on modelling and simulation (EUROSIM). Cardiff, Wales, pp 24–29

    Google Scholar 

  60. Meyerhoff M, Duan C, Meusel M (1995) Novel nonseparation sandwich-type electrochemical enzyme immunoassay system for detecting marker proteins in undiluted blood. Clin Chem 41(9):1378–1384

    Article  Google Scholar 

  61. Mullen W, Keedy F, Churchouse S, Vadgama P (1986) Glucose enzyme electrode with extended linearity: application to undiluted blood measurements. Anal Chim Acta 183:59–66

    Article  Google Scholar 

  62. Nernst W (1904) Theorie der Reaktionsgeschwindigkeit in heterogenen systemen. Z Phys Chem 47(1):52–55

    Google Scholar 

  63. Özisik MN (1980) Heat conduction. Wiley, New York

    Google Scholar 

  64. Pfeiffer D, Scheller F, Setz K, Schubert F (1993) Amperometric enzyme electrodes for lactate and glucose determinations in highly diluted and undiluted media. Anal Chim Acta 281(3):489–502

    Article  Google Scholar 

  65. Romero MR, Baruzzi AM, Garay F (2012) Mathematical modeling and experimental results of a sandwich-type amperometric biosensor. Sens Actuator B Chem 162(1):284–291

    Article  Google Scholar 

  66. Rong Z, Cheema U, Vadgama P (2006) Needle enzyme electrode based glucose diffusive transport measurement in a collagen gel and validation of a simulation model. Analyst 131(7):816–821

    Article  Google Scholar 

  67. Sadana A, Sadana N (2011) Handbook of biosensors and biosensor kinetics. Elsevier, Amsterdam

    MATH  Google Scholar 

  68. Samarskii A (2001) The theory of difference schemes. Marcel Dekker, New York-Basel

    Book  MATH  Google Scholar 

  69. Scheller FW, Schubert F (1992) Biosensors. Elsevier Science, Amsterdam

    Google Scholar 

  70. Schulmeister T (1987) Mathematical treatment of concentration profiles and anodic current of amperometric enzyme electrodes with chemically amplified response. Anal Chim Acta 201:305–310

    Article  Google Scholar 

  71. Schulmeister T (1990) Mathematical modelling of the dynamic behaviour of amperometric enzyme electrodes. Sel Electrode Rev 12(2):203–260

    Google Scholar 

  72. Schulmeister T, Pfeiffer D (1193) Mathematical modelling of amperometric enzyme electrodes with perforated membranes. Biosens Bioelectron 8(2):75–79

    Google Scholar 

  73. Segel LA, Slemrod M (1989) The quasi-steady-state assumption: a case study in perturbation. SIAM Rev 31(3):446–477

    Article  MathSciNet  MATH  Google Scholar 

  74. Senda M, Ikeda T, Miki K, Hiasa H (1986) Amperometric biosensors based on a biocatalyst electrode with entrapped mediator. Anal Sci 2(6):501–506

    Article  Google Scholar 

  75. Somasundrum M, Aoki K (2002) The steady-state current at microcylinder electrodes modified by enzymes immobilized in conducting or non-conducting material. J Electroanal Chem 530(1–2):40–46

    Article  Google Scholar 

  76. Sorochinskii V, Kurganov B (1996) Amperometric biosensors with a laminated distribution of enzymes in their coating. Steady state kinetics. Biosens Bioelectron 11(1–2):45–51

    Article  Google Scholar 

  77. Tang L, Koochaki Z, Vadgama P (1990) Composite liquid membrane for enzyme electrode construction. Anal Chim Acta 232:357–365

    Article  Google Scholar 

  78. Turner APF, Karube I, Wilson GS (eds) (1990) Biosensors: fundamentals and applications. Oxford University Press, Oxford

    Google Scholar 

  79. Velkovsky M, Snider R, Cliffel DE, Wikswo JP (2011) Modeling the measurements of cellular fluxes in microbioreactor devices using thin enzyme electrodes. J Math Chem 49(1):251–275

    Article  MathSciNet  MATH  Google Scholar 

  80. Wang J (2000) Analytical electrochemistry, 2 edn. Wiley, New-York

    Book  Google Scholar 

  81. Yang H (2000) Mathematical model for liquid-liquid phase-transfer catalysis. Chem Eng Comm 179(1):117–132

    Article  Google Scholar 

  82. Ylilammi M, Lehtinen L (1988) Numerical analysis of a theoretical one-dimensional amperometric enzyme sensor. Med Biol Eng Comput 26(1):81–87

    Article  Google Scholar 

  83. Zhao W, Xu J, Chen H (2006) Electrochemical biosensors based on layer-by-layer assemblies. Electroanal 18(18):1737–1748

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics & Informatics, Institute of Computer Science, Vilnius University, Vilnius, Lithuania

    Romas Baronas & Feliksas Ivanauskas

  2. Life Sciences Center, Vilnius University, Vilnius, Lithuania

    Juozas Kulys

Authors
  1. Romas Baronas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Feliksas Ivanauskas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Juozas Kulys
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Baronas, R., Ivanauskas, F., Kulys, J. (2021). Effects of Diffusion Limitations on the Response and Sensitivity of Biosensors. In: Mathematical Modeling of Biosensors. Springer Series on Chemical Sensors and Biosensors, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-030-65505-1_2

Download citation

Publish with us

Policies and ethics

Search

Navigation