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2010 | Buch

Biosensor Action Kapitel lesen Erstes Kapitel lesen

Mathematical Modeling of Biosensors

An Introduction for Chemists and Mathematicians

verfasst von: Romas Baronas, Feliksas Ivanauskas, Juozas Kulys

Verlag: Springer Netherlands

Buchreihe : Springer Series on Chemical Sensors and Biosensors

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Biosensors are analytical devices in which speci?c recognition of the chemical substances is performed by biological material. The biological material that serves as recognition element is used in combination with a transducer. The transducer transforms concentration of substrate or product to electrical signal that is amp- ?ed and further processed. The biosensors may utilize enzymes, antibodies, nucleic acids, organelles, plant and animal tissue, whole organism or organs. Biosensors containing biological catalysts (enzymes) are called catalytical biosensors. These type of biosensors are the most abundant, and they found the largest application in medicine, ecology, and environmental monitoring. The action of catalytical biosensors is associated with substrate diffusion into biocatalytical membrane and it conversion to a product. The modeling of bios- sors involves solving the diffusion equations for substrate and product with a term containing a rate of biocatalytical transformation of substrate. The complications of modeling arise due to solving of partially differential equations with non-linear biocatalytical term and with complex boundary and initial conditions. The book starts with the modeling biosensors by analytical solution of partial differential equations. Historically this method was used to describe fundamental features of biosensors action though it is limited by substrate concentration, and is applicable for simple biocatalytical processes. Using this method the action of biosensors was analyzed at critical concentrations of substrate and enzyme activity.

Inhaltsverzeichnis

Frontmatter

Analytical Modeling of Biosensors

Frontmatter
Biosensor Action
Abstract
The biosensors contain immobilized enzymes or other biological catalysts [128, 131, 258]. The biocatalyst catalyzes the conversion of the substrate to the product. Biological catalysts (enzymes) show high activity and specificity. The activity of enzymes may exceed the rate of chemically catalyzed reaction by a factor 4. 6 ×105 − 1. 4 ×1017 [58]. The enzymatic activity of the enzymes depends on many factors, i.e. the free energy of reaction, the substrate docking in the active center of enzyme, the proton tunneling and other factors [88, 124, 132, 143].
The general principles of catalytic activity of enzymes are known, but particular factors that determine high enzyme activity are often not established [178]. The specificity of enzymes depends on the enzyme type [65, 81]. There are enzymes which catalyze the conversion of just one substrate. Other enzymes show broad substrates specificity. Oxidoreductases, i.e. enzymes that catalyze electron transfer, may catalyze, for example, the oxidation or reduction of many substrates. To characterize the substrates with diverse activity a slang expression “good substrate” and “bad substrate” is used.
Romas Baronas, Ivanauskas Feliksas, Juozas Kulys
Modeling Biosensors at Steady State and Internal Diffusion Limitations
Abstract
The most popular glucose biosensor is based on glucose oxidase (GO) that catalyzes β-D-glucose oxidation with oxygen [261, 273],
$$\beta \text{ -D-glucose} +{ \mathrm{O}}_{2}{ \text{ D-glucose oxidase} \atop \rightarrow } \text{ D-glucono-}\delta \text{ -lactone} +{ \mathrm{H}}_{2}{\text{ O}}_{2}$$
(1)
The hydrogen peroxide produced is oxidized on platinum electrode acting as a transducer. One of the first tasks of modeling of this type of the biosensors was devoted to evaluate the dependence of biosensors response on enzymatic parameters [128]. The action of the biosensors was analyzed at the internal diffusion limiting conditions and at the steady state conditions.
The biosensor response (the current density) was calculated as
$$i(t) = {n}_{e}{D}_{e}F\,\frac{\partial P} {\partial x}{ \bigg\vert}_{x=0},$$
(2)
where n e – the number of electrons (for hydrogen peroxide n = 2), F – the Faraday number, D e – the diffusion coefficient of the substrate and the product in the biocatalytical membrane.
Romas Baronas, Ivanauskas Feliksas, Juozas Kulys
Modeling Biosensors at Steady State and External Diffusion Limitations
Abstract
The modeling of the biosensors action at an external diffusion limitation is much easer due to the linear gradient of the substrates concentration in a stagnant layer. The analysis of such systems, however, did not receive a lot of attention since the internal diffusion problems are intrinsic for the catalytical biosensors. For a biosensor acting at the external diffusion limitation and at the steady state conditions the flux of the substrate through a stagnant layer is equal to the enzyme reaction rate on the surface of the transducer:
$${D}_{0}\frac{{S}_{0} - {S}_{s}} {\delta } = \frac{{V }_{max,s}{S}_{s}} {{K}_{M} + {S}_{s}}\,,$$
(1)
where V max, s corresponds to the maximal enzyme rate on the surface expressed as mol∕cm2s.
Romas Baronas, Feliksas Ivanauskas, Juozas Kulys
Modeling Biosensors Utilizing Microbial Cells
Abstract
The biocatalytical system of microbial cells can be used as biocatalyzers for the biosensor preparation [127, 130]. They can show very high specificity for some substrates. For example, yeast cells Hansenula anomala grown in lactate reach breeding media induce cytochrome b 2, and shows high specificity to L-lactate [125]. The scheme of substrates distribution in a microbial biosensor is depicted in Fig. 1.
The peculiarity of the modeling of the microbial biosensors is a slow substrate and product transport through the microbial cell wall. If the substrate transport is slower than the diffusion through the bulk solution and the semipermeable membrane the substrate and the product concentration change in the cell can be written:
$$\begin{array}{rcl} & & \frac{\mathrm{d}{S}_{c}} {\mathrm{d}t} = k({S}_{0} - {S}_{c}) - {V }_{c}\,, \\ & & \frac{\mathrm{d}{P}_{c}} {\mathrm{d}t} = {V }_{c} - k'{P}_{c}\,, \end{array}$$
(1)
where S c and P c are the concentrations of the substrate and the product in the cell, respectively, k and k′ are constants of substrate transport into the cell and the product transport from the cell, respectively, and V c is the rate of the enzymatic process in the cell. The constants k and k′ are related to permeability (h) of the cell wall, that can be expressed as
$$\begin{array}{rcl} & & k = {h}_{s}/l\,, \\ & & k' = {h}_{p}{\mathit{sur}}_{c}/{\mathit{vol}}_{c}\end{array}$$
(2)
where sur c and vol c are the surface and the volume of the microbial cell, respectively.
Romas Baronas, Ivanauskas Feliksas, Juozas Kulys
Modeling Nonstationary State of Biosensors
Abstract
The biosensors response at a transition state can be modeled solving partial differential equations (PDE) of the substrates diffusion and the biocatalytical conversion with the initial and the boundary conditions. The analytical solutions, however, exist at very limited cases. The Laplace transformation that is typically used for solving the diffusion equations is no longer applicable for the solution of such problems. Therefore, for the modeling of the diffusion and enzymatic reactions the other methods of PDE solving are used.
Carr [61] used the Fourier method to solve (Chapter 1, eq. 7) at S ≪ K M and S ≫ K M with the initial and the boundary conditions: S = P = 0 at 0 < x < d and t = 0; ∂S ∕ ∂x = ∂P ∕ ∂x = 0 at x = 0; S = S 0, P = 0 at x = d.
Romas Baronas, Ivanauskas Feliksas, Juozas Kulys

Numerical Modeling of Biosensors

Frontmatter
Mono-Layer Mono-Enzyme Models of Biosensors
Abstract
A membrane biosensor may be considered as an electrode, having a layer of an enzyme applied onto the electrode surface [69]. Consider a scheme where the substrate (S) combines reversibly with the enzyme (E) to form a complex (ES). The complex then dissociates into the reaction product (P) and the enzyme is released [65, 78, 229, 258],
$$\text{ S} + \text{ E} \rightleftarrows \text{ ES} \rightarrow \text{ E} + \text{ P}$$
(1)
Romas Baronas, Ivanauskas Feliksas, Juozas Kulys
One-Layer Multi-Enzyme Models of Biosensors
Abstract
The amperometric biosensors have proved to be reliable and low-cost in various analytical systems with applications in biotechnology, medicine and environmental monitoring [106, 218, 229, 246, 275]. However, amperometric biosensors possess a number of serious drawbacks. One of the main reasons restricting wider use of the biosensors is a relatively short linear range of the calibration curve. Increasing the concentration range of detectable analyte, the sensitivity and specificity of the detection event improves the prospects for commercialising biosensors [176, 196, 217, 228, 246].
One way of overcoming those problems is to couple different enzymes either in sequence, in competition or in recycle pathways. Due to the appropriate combination of enzymes, the range of analyte species accessible to measurement, the selectivity and the sensitivity of the biosensor may be enhanced [63, 94, 137, 150, 274].
Romas Baronas, Ivanauskas Feliksas, Juozas Kulys
Multi-Layer Models of Biosensors
Abstract
There are various reasons for applying a multi-layer approach to the modeling of biosensors. Multi-layer models are usually used in the following cases [35, 119, 235, 236]:
  • The bulk solution is assumed to be slightly-stirred or non-stirred. This assumption leads to two-compartment models [51, 60, 102, 120, 286].
  • The enzyme layer is covered with an inert outer membrane [234]. The membrane stabilizes the enzyme layer and creates a diffusion limitation to the substrate, i.e. lowers the substrate concentration in the enzymatic layer and thereby prolongs the calibration curve of the biosensor [152, 166, 229, 238, 258].
  • The electrode is covered with a selective membrane [236]. Selective membranes are usually impermeable to certain molecules and permeable to a desired substance. This arrangement can notably increase the biosensor selectivity. The selective layer can also protect the metal interface of the electrode [16, 47, 91, 157].
  • In multienzyme systems, enzymes are often immobilized separately in different active layers packed in a sandwich-like multi-layer arrangement [13, 14, 15, 113, 184, 242]. This approach seems to be a rather fast and cheap method to design biosensors for different purposes.
Romas Baronas, Ivanauskas Feliksas, Juozas Kulys
Modeling Biosensors of Complex Geometry
Abstract
Usually, when modeling a biosensor as a flat electrode having one or several layers sandwich-likely applied onto the electrode surface, a mathematical model of the biosensor is formulated in a one-dimensional-in-space domain. This chapter deals with the modeling of biosensors for which two-dimensional-in-space domains are used when describing mathematically the biosensor action.
Romas Baronas, Ivanauskas Feliksas, Juozas Kulys

Numerical Methods for Reaction-Diffusion Equations

Frontmatter
The Difference Schemes for the Diffusion Equation
Abstract
Contemporary numerical methods for solving problems of the mathematical chemistry are gaining increasing popularity. The aim of this chapter is to introduce the reader with the relevant facts about the basic concepts of the theory of the difference schemes for the linear diffusion equations. The linear diffusion equations play an important and crucial role in most models of a biosensor theory. The selection of the difference methods for the solution of these equations is motivated by the following two arguments:
  • The geometry (size) of a biosensor really does not change during the measurements;
  • The simplicity and efficiency of the difference method.
The most popular simple and together effective difference schemes are presented here. These difference schemes are extensively applied to the solution of a biosensor problems in the next chapter. This method is being frequently used in solving applied problems not only by professional mathematicians, but also by laymen. The concepts presented below are of a primary nature and are sufficient for the solution of the problems of the biosensor. In this book the notations of [222] are mainly applied. The many aspects of the numerical methods for the solution of the partial differential equations are presented in [5, 12, 187, 222].
Romas Baronas, Ivanauskas Feliksas, Juozas Kulys
The Difference Schemes for the Reaction–Diffusion Equations
Abstract
This chapter is devoted to various difference approximations of the reaction–diffusion equations. The difference technique, developed in a previous chapter, is employed for the construction of the difference scheme for the system of the reaction–diffusion equations. The main subject of investigation is the system of two nonlinear reaction–diffusion equations in one and two dimensional in space cases.
Romas Baronas, Ivanauskas Feliksas, Juozas Kulys
Backmatter
Metadaten
Titel
Mathematical Modeling of Biosensors
verfasst von
Romas Baronas
Feliksas Ivanauskas
Juozas Kulys
Copyright-Jahr
2010
Verlag
Springer Netherlands
Electronic ISBN
978-90-481-3243-0
Print ISBN
978-90-481-3242-3
DOI
https://doi.org/10.1007/978-90-481-3243-0

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