The steady state current at a microdisk biosensor

Mathematical Analysis of Reaction-Diffusion Equations Modeling the Michaelis-Menten Kinetics in a Micro-Disk Biosensor

In this study, we have investigated the mathematical model of an immobilized enzyme system that follows the Michaelis-Menten (MM) kinetics for a micro-disk biosensor. The film reaction model under steady state conditions is transformed into a couple differential equations which are based on dimensionless concentration of hydrogen peroxide with enzyme reaction (H) and substrate (S) within the biosensor. The model is based on a reaction-diffusion equation which contains highly non-linear terms related to MM kinetics of the enzymatic reaction. Further, to calculate the effect of variations in parameters on the dimensionless concentration of substrate and hydrogen peroxide, we have strengthened the computational ability of neural network (NN) architecture by using a backpropagated Levenberg-Marquardt training (LMT) algorithm. NNs-LMT algorithm is a supervised machine learning for which the initial data set is generated by using MATLAB built in function known as "pdex4". Furthermore, the data set is validated by the processing of the NNs-LMT algorithm to find the approximate solutions for different scenarios and cases of mathematical model of micro-disk biosensors. Absolute errors, curve fitting, error histograms, regression and complexity analysis further validate the accuracy and robustness of the technique.

A Detailed Model of Electroenzymatic Glutamate Biosensors To Aid in Sensor Optimization and in Applications in Vivo

Simulations conducted with a detailed model of glutamate biosensor performance describe the observed sensor performance well, illustrate the limits of sensor performance, and suggest a path toward sensor optimization. Glutamate is the most important excitatory neurotransmitter in the brain, and electroenzymatic sensors have emerged as a useful tool for the monitoring of glutamate signaling in vivo. However, the utility of these sensors currently is limited by their sensitivity and response time. A mathematical model of a typical glutamate biosensor consisting of a Pt electrode coated with a permselective polymer film and a top layer of cross-linked glutamate oxidase has been constructed in terms of differential material balances on glutamate, H₂O₂, and O₂ in one spatial dimension. Simulations suggest that reducing thicknesses of the permselective polymer and enzyme layers can increase sensitivity ∼6-fold and reduce response time ∼7-fold, and thereby improve resolution of transient glutamate signals. At currently employed enzyme layer thicknesses, both intrinsic enzyme kinetics and enzyme deactivation likely are masked by mass transfer. However, O₂-dependence studies show essentially no reduction in signal at the lowest anticipated O₂ concentrations for expected glutamate concentrations in the brain and that O₂ transport limitations in vitro are anticipated only at glutamate concentrations in the mM range. Finally, the limitations of current biosensors in monitoring glutamate transients is simulated and used to illustrate the need for optimized biosensors to report glutamate signaling accurately on a subsecond time scale. This work demonstrates how a detailed model can be used to guide optimization of electroenzymatic sensors similar to that for glutamate and to ensure appropriate interpretation of data gathered using such biosensors.

Digital simulation of chronoamperometry at an electrode within a hemispherical polymer drop containing an enzyme: comparison of a hemispherical with a flat disk electrode

Current-time and steady state current behaviour was simulated for the cases of a hemispherical and flat inlaid disk electrodes located under a hemispherical polymer drop containing an enzyme which converts a substrate diffusing into the drop into a product that is electroactive at the electrode. As well, a cylindrical electrode with length much greater than its diameter and coated with a layer of polymer/enzyme was treated. The ratio of steady state currents at the hemispherical to the disk electrode is not, as has sometimes been assumed, always equal to π/2; indeed this is only approached for polymer drops with large spillover ratio, that is, having a radius much larger than that of the electrodes. Steady state currents for all electrode geometries (including the cylinder) go through a maximum for some spillover ratio and then approach a constant value for larger spillover ratios. This constant value is the same as that for the diffusion limited current in a semi-infinite medium. For a cylindrical electrode, the steady state current tends towards zero for large spillover ratios.

Modeling the measurements of cellular fluxes in microbioreactor devices using thin enzyme electrodes

An analytic approach to the modeling of stop-flow amperometric measurements of cellular metabolism with thin glucose oxidase and lactate oxidase electrodes would provide a mechanistic understanding of the various factors that affect the measured signals. We divide the problem into two parts: (1) analytic formulas that provide the boundary conditions for the substrate and the hydrogen peroxide at the outer surface of the enzyme electrode layers and the electrode current expressed through these boundary conditions, and (2) a simple diffusion problem in the liquid compartment with the provided boundary conditions, which can be solved analytically or numerically, depending on the geometry of the compartment. The current in an amperometric stop-flow measurement of cellular glucose or lactate consumption/excretion is obtained analytically for two geometries, corresponding to devices developed at the Vanderbilt Institute for Integrative Biosystems Research and Education: a multianalyte nanophysiometer with effective one-dimensional diffusion and a multianalyte microphysiometer, for which plentiful data for metabolic changes in cells are available. The data are calibrated and fitted with the obtained time dependences to extract several cellular fluxes. We conclude that the analytical approach is applicable to a wide variety of measurement geometries and flow protocols.

Theory of Polymer Entrapped Enzyme Ultramicroelectrodes: Fundamentals

We have developed a theoretical description of the amperometric response of ultramicroelectrode (UME) biosensors formed via enzyme entrapment. Our model allows for multiple enzymes and co-substrates, and results in a closed-form analytical expression for the steady-state current response of the disk ultramicroelectrode. It captures the effects of enzyme-entrapment domain size, species transport properties (which can be different in the polymer and surrounding electrolyte), enzyme kinetics, and axisymmetric diffusion. Assumptions inherent in the derivation are carefully explained, as are the resulting limits on the applicability of the results. The ability to theoretically predict the response of enzyme entrapped UMEs should enable improved design, operation, and data interpretation for this important class of biosensors.

Control of the oxygen dependence of an implantable polymer/enzyme composite biosensor for glutamate

Biosensors for glutamate (Glu) were fabricated from Teflon-coated Pt wire (cylinders and disks), modified with the enzyme glutamate oxidase (GluOx) and electrosynthesized polymer PPD, poly(o-phenylenediamine). The polymer/enzyme layer was deposited in two configurations: enzyme before polymer (GluOx/PPD) and enzyme after polymer (PPD/GluOx). These four biosensor designs were characterized in terms of response time, limit of detection, Michaelis-Menten parameters for Glu (J max and K(M)(Glu)), sensitivity to Glu in the linear response region, and dependence on oxygen concentration, K(M)(O2). Analysis showed that the two polymer/enzyme configurations behaved similarly on both cylinders and disks. Although the two geometries showed different behaviors, these differences could be explained in terms of higher enzyme loading density on the disks; in many analyses, the four designs behaved like a single population with a range of GluOx loading. Enzyme loading was the key to controlling the K(M)(O2) values of these first generation biosensors. The counterintuitive, and beneficial, behavior that biosensors with higher GluOx loading displayed a lower oxygen dependence was explained in terms of the effects of enzyme loading on the affinity of GluOx for its anionic substrate. Some differences between the properties of surface immobilized GluOx and glucose oxidase are highlighted.

Oxygen tolerance of an implantable polymer/enzyme composite glutamate biosensor displaying polycation-enhanced substrate sensitivity

Biosensors were fabricated at neutral pH by sequentially depositing the polycation polyethyleneimine (PEI), the stereoselective enzyme L-glutamate oxidase (GluOx) and the permselective barrier poly-ortho-phenylenediamine (PPD) onto 125-microm diameter Pt wire electrodes (Pt/PEI/GluOx/PPD). These devices were calibrated amperometrically at 0.7 V versus SCE to determine the Michaelis-Menten parameters for enzyme substrate, l-glutamate (Glu) and co-substrate, dioxygen. The presence of PEI produced a 10-fold enhancement in the detection limit for Glu (approximately 20 nM) compared with the corresponding PEI-free configurations (Pt/GluOx/PPD), without undermining their fast response time (approximately 2 s). Most remarkable was the finding that, although some designs of PEI-containing biosensors showed a 10-fold increase in linear region sensitivity to Glu, their oxygen dependence remained low.

The efficiency of immobilised glutamate oxidase decreases with surface enzyme loading: an electrostatic effect, and reversal by a polycation significantly enhances biosensor sensitivity

The apparent Michaelis constant, K(M), for glutamate oxidase (GluOx) immobilised on Pt electrodes increased systematically with enzyme loading. The effect was due, at least in part, to electrostatic repulsion between neighbouring oxidase molecules and the anionic substrate, glutamate (Glu). This understanding has allowed us to increase the Glu sensitivity of GluOx-based amperometric biosensors in the linear response region (100+/-11 nA cm(-2)microM(-1) at pH 7.4; SD, n=23) by incorporating a polycation (polyethyleneimine, PEI) to counterbalance the polyanionic protein. Differences in the behaviour of glucose biosensors of a similar configuration highlight a limitation of using glucose oxidase as a model enzyme in biosensor design.

Needle enzyme electrode based glucose diffusive transport measurement in a collagen gel and validation of a simulation model

Rapid response needle enzyme electrodes were fabricated to measure the glucose concentration at the centre of a cylindrical spiralled collagen gel, which is a relevant constituent for tissue engineering scaffolds. The experimental data were based on a low consumption glucose sensor which minimised the distorting effect of enzymatic degradation. As the measurement was carried out within a collagen gel the stirring independence was compulsory for the biosensor. Glucose concentration changes were derived from a model based on the solution to Fick's Second Law. This had two different expressions for different dimensionless time (T) domains. The expression for large T and a first order approximation for small T were known. The expression for high order approximation for small T was then derived. An analytical expression consisting of fast convergent parts of these two expressions is proposed, which operates for the entire time region. A computational model for glucose concentration evolution where an electrode is located is proposed to operate for extended time periods. The model was confirmed by agreement between the simulated and observed data. An experimental technique is developed here to determine glucose diffusion coefficient by fitting the simulated concentration profile to the observed one. The glucose diffusion coefficient within the collagen gel was estimated to be 1.3 x 10(-6) cm(2) s(-1); higher accuracy is achieved here because errors due to noise, baseline and zero time determination are minimised with best fit.