An evolutionary multiobjective method for low-rank and sparse matrix decomposition

Abbreviations:

$[E_T]$: Total enzyme concentration, units: $mM$;

$[E_{OX}]$: Enzyme concentration of the oxygen, units: $mM$;

$[ES]$: Enzyme concentration of the substrate, units: $mM$;

$[E_{red}]$: Reduced enzyme concentration, units: $mM$;

$[Med_{OX}]$: Concentration of oxidized mediator at any position in the enzyme layer, units: $mM$;

$[Med_{red}]$: Concentration of reduced mediator at any position in the enzyme layer, units: $Mol cm^{-3}$;

$D_M$: Diffusion coefficient of oxidized mediator, units: $cm^2 s^{-1}$;

$D_S$: Diffusion coefficient of substrate, units: $cm^2 s^{-1}$;

$d$: Thickness of the planar matrix, units: $cm$;

$[Med_{OX}]_b$: Oxidized mediator concentration at the enzyme layer electrode boundary, units: $mM$;

$[Med_{OX}]_\infty$: Oxidized mediator concentration in bulk solution, units: $mM$;

$[S]$: Concentration of substrate at any position in the enzyme layer, units: $mM$;

$[S]_b$: Concentration of substrate at any position in the enzyme layer electrode boundary, units: $mM$;

$[S]_\infty$: Substrate concentration in bulk solution, units: $mM$;

$k_1, k_3, k_4$: Rate constants, units: $M^{-1} s^{-1}$;

$k_{-1}, k_2$: Rate constants, units: $s^{-1}$;

$\phi_{O}^{2}$: Thiele modulus for the oxidized mediator, units: None

1.   Introduction

The development of biosensing in recent decades has affected several fields, including environmental and biomedical monitoring. In modern biosensors, miniaturization, mass production, and simple transport processes are possible. Biological fluids constantly change, making biosensors an excellent real-time tool for monitoring these changes. Polymer membranes have been widely used as carriers for immobilizing enzymes in recent years ^[1]. Biocatalysts immobilized on membranes perform optimally. Catalytic reactions can be more selective toward desired products when substrate partitioning occurs at the membrane/fluid interphase (see ^[2]). A new approach to enzyme immobilization ^[3] based on molecular recognition has recently been used successfully for building chemically active membranes ^[4] as well as for building enzymatic biosensors. A great deal of effort has gone into developing biosensors with biologically sensitive components and transformers in the past decades, devices that have many possible applications ^[5]. Robeson demonstrated certain changes in membrane chemistry ^[6]. By enhancing the membrane area per unit volume, separation can be expedited by altering the membrane geometry. Recent research has identified increased surface area as a research priority for membranes ^[7].

Experiments were performed using a two-substrate model for enzyme electrodes incorporating nonlinear enzyme reactions ^[8]. Models of glucose oxidation electrodes have been developed using this approach ^[9]. When the mediators and natural co-substrates are both present in the assay solution, it has been found that the mediators cannot solely replace the co-substrate, so a three-substrate model is required. The calibration curve of the enzyme electrode is complex in these cases ^[10]. Although biosensors have been extensively tested experimentally, very few studies have focused on modeling or theoretical design. Efficient and productive biosensor design can be enhanced using a digital model ^[11,12,13]. The semi-analytical properties of biosensors have been optimized using mathematical models (see ^{[14,15,16,17]}). Biosensor models have been studied under steady-state ^[18,19] and transient conditions ^[20,21] using synergistic substrate conversion schemes.

A theoretical model for an enzyme-membrane amperometric oxidase electrode was recently presented by Loghambal et al. ^[22]. Novel enzyme electrodes were numerically analyzed in ^[23,24]. Semi-analytical expressions of the substrate concentration for planar, cylindrical, and spherical particles under steady-state conditions were derived ^[25]. Lyons et al. ^[26] examined the problem of describing the transport and kinetics of catalytic reactions in a bounded region such as a conductive polymer-modified electrode.

Biological sensors have been assessed using electrochemical impedance spectroscopy, which is both nondestructive and sensitive to electrochemical properties ^[27,28,29]. An analysis of the influence of complex homogeneous and heterogeneous reactions on the sensor response was performed using mathematical models. A one-dimensional model for amperometric sensors was developed by Bartlett et al. and coworkers ^[30,31,32] based on the Michaelis-Menten approximation. Michaelis-Menten kinetics assumes a very large substrate concentration and that complex-forming reactions are equilibrated. A porous rotating disk electrode was recently analyzed by Visuvasam et al. ^[33]. The mathematical solution is semi-analytical, and more often, it is numerical. This method can be applied to various systems, and has been described in several publications ^{[34,35,36,37]}. This model can be understood better by reading ^{[38,39,40,41,42]} and its references.

This paper presents a semi-analytical solution for the model. By solving the system of nonlinear reaction-diffusion equations using Akbari-Ganji's (AGM) method (see ^[43,44]), we can derive semi-analytical expressions for the substrate and oxidized and reduced mediator concentrations. These models provide information about the enzyme electrode mechanisms and their kinetics. These modeling results can be helpful for sensor design and optimization, and for determining how the electrode will react.

2.   Mathematical formulation of the problem

The dimensionless nonlinear mass transport equation for this model was derived by Gooding and Hall ^[23]. A solid substrate encases the biological layer, and an outer permeable electrode is in contact with the sample in the enzyme-membrane geometry. This model employs a permeable electrode to facilitate the penetration of the substrate and co-substrate into the enzyme layer, where it reduces to the form of a co-substrate that diffuses oxygen back to the electrode. In the presence of two oxidants, immobilized oxidase can be described by the following general reaction scheme ^[23]:

with respect to the $m^{th}$ reaction, ${k_m}$ and ${k_{-m}}$ represent the forward and backward rate constants, where $m = 1, 2, 3, \ldots,$ respectively. If $[E_{T}]$ is the total enzyme concentration in the matrix, then at all times

where $[E_{OX}]$, $[ES]$, and $[E_{red}]$ are the oxidized mediator, enzyme-substrate complex, and reduced mediator enzyme concentrations, respectively. When a substrate diffuses into an enzyme layer at steady-state, its diffusion rate is equal to its reaction rate within the matrix. We consider a planar matrix of thickness $y = d$, where diffusion is only considered in the y-direction (see Figure 1).

On the enzyme electrode, the governing equations for the planar diffusion and reaction are as follows ^[23]:

and from the above equations, we have

where $D_{M}$ and $D_{S}$ are the diffusion coefficient of the mediator and substrate within the enzyme layer. $\left[\operatorname{Med}_{\mathrm{OX}}\right], \left[\mathrm{Med}_{\mathrm{red}}\right]$, and $[S]$ are the concentration of oxidized, reduced mediators and the substrate within the enzyme layer.

and

are the rate constants in dimensionless form.

The corresponding boundary conditions are the following: at the far wall, $y = 0$,

At the electrode, $y = d$,

where $\left[\operatorname{Med}_{\mathrm{OX}}\right]_{b}$ and $[S]_{b}$ are the bulk concentration of the oxidized mediator and substrate at the enzyme layer electrode boundary. They are the bulk solution concentrations and are the equilibrium partition coefficients for the oxidized mediator and the substrate, respectively.

By defining the following dimensionless variables, we can reduce the nonlinear differential Eqs (2.5)–(2.7) to dimensionless form

where $F_{O}$, $F_{S}$, and $F_{R}$ are the normalized concentration of the oxidized mediator, substrate, and reduced mediator, respectively, and $\chi$ is the normalized distance. $B_{\mathrm{O}}$ and $B_{\mathrm{S}}$ are the normalized surface concentration of the oxidized mediator and substrate. $\phi_{O}$ is the Thiele modulus for the oxidized mediator. Non-dimensionalized expressions for the oxidized mediator, substrate, and reduced mediator are as follows:

From the above equations, we obtain the following relations:

Corresponding boundary conditions are given by:

From Eq (2.14), we get

and

Solving Eqs (2.18) and (2.19) we get

The following expression gives the normalized current response:

3.   Approximate analytical expressions for the oxidized mediator, substrate, and reduced mediator using AGM

AGM ^{[45,46,47,48,49]} was used to solve the boundary value problem and its associated boundary conditions represented by Eqs (2.12)–(2.17), which has a minimum number of unknowns. This is an appropriate and simple method for nonlinear differential equations ^[50]. This is a particular case of the exponential function method proposed by He et al. ^[51]. Using this method, a general semi-analytical expression for the normalized concentrations can be obtained as follows:

where

From these relations, the following current response formula is derived:

4.   Previous analytical results for concentration for the oxidized mediator, substrate, and reduced mediator ^[52]

Loghambal et al. ^[52] derived the approximate semi-analytical expressions for the concentration for the oxidized mediator, substrate, and reduced mediator using the adomian decomposition method

The normalized current becomes

where

5.   Numerical simulation

Numerical methods are used to solve the nonlinear differential Eqs (2.12)–(2.14). A numerical solution of the nonlinear differential Eqs (2.12)–(2.14) has been performed via MATLAB. Comparing our numerical solutions with our analytical results is shown in Figures 2–4 regarding species concentrations. The numerical solution, which is shown in Figures 2–4, yields a satisfactory result when compared to the AGM.

6.   Discussion

Equations (3.1)–(3.3) provide semi-analytical expressions of the concentrations of the substrate, oxygen, and reduced mediators that are obtained by using AGM with the normalized current given in Eq (3.5). Figure 2a–c depicts the normalized steady-state concentrations of species obtained using Eq (2.12)–(2.14) for various values of $\phi_{0}$ and for some fixed values of $\mu_{\mathrm{S}}, B_{\mathrm{O}}$, and $B_{\mathrm{S}}$. The concentration is uniform when $\phi_{0} \le 1$ for all species. Figure 2a, b shows that for some fixed values of other parameters $\mu_{\mathrm{S}}, B_{\mathrm{O}}$, and $B_{\mathrm{S}}$, the normalized concentrations of the oxidized mediator and substrate decrease with the increasing Thiele modulus. In contrast, this modulus has an opposite effect on the normalized concentrations of the reduced mediator for some fixed values of the $\mu_{\mathrm{S}}, B_{\mathrm{O}}$ and $B_{\mathrm{S}}$ as shown in Figure 2c.

Figure 3a–c illustrates the normalized concentrations of substrates, the oxidized mediator and reduced mediator, versus the dimensionless distance, respectively. Figure 3a, b shows that, for some fixed values of the parameters $\mu_{\mathrm{S}}$, $\phi_{0}$, and $B_{\mathrm{S}}$ concentrations of substrates and the oxidized mediator decrease as $B_{\mathrm{O}}$ increases. Figure 3c illustrates that for some constant measurements of the parameters $\mu_{\mathrm{S}}$, $\phi_{0}$, and $B_{\mathrm{S}}$, the normalized surface concentrations of the reduced mediator increase as $B_{\mathrm{O}}$ increases.

According to Eqs (2.12)–(2.14), the normalized steady-state concentrations of substrate and various mediator species can be determined for various values of $B_{\mathrm{S}}$ as shown in Figure 4a–c. We conclude that when the dimensionless parameter $B_{\mathrm{S}}$ increases, the concentrations of substrate and oxidized mediator decrease for some fixed values of the parameters $\mu_{\mathrm{S}}$, $\phi_{0}$, and $B_{\mathrm{O}}$ in Figure 4a, b. In Figure 4c, it can be seen that the influence of the increasing dimensionless parameter $B_{\mathrm{S}}$ can result in normalized concentrations of the mediator increasing, even when the other parameter is at fixed values.

Figure 5a–c illustrates the normalized concentrations of substrates, oxidized mediator, and reduced mediator versus the dimensionless distance $\chi$, respectively. Figure 5a, b shows that, for some fixed values of the parameters $\phi_{0}$, $B_{\mathrm{S}}$, and $B_{\mathrm{O}}$, concentrations of substrates and reduced mediator decrease as $\mu_{\mathrm{S}}$ increases. Figure 5c illustrates that for some constant measurements of the parameters $\phi_{0}$, $B_{\mathrm{S}}$, and $B_{\mathrm{O}}$, the normalized surface concentrations of the oxidized mediator increase as $\mu_{\mathrm{S}}$ increases.

The effect of the various parameters on the current in the three-dimensional is displayed in Figure 6a–c. The dependencies of the steady-state current $I$ on the concentration of the substrate, oxygen, and reduced mediators versus the Thiele modulus are displayed in Figure 6. It is noticed from these figures that the steady-state current increases as the values of the parameters $\mu_{\mathrm{S}}$, $\phi_{0}$, $B_{\mathrm{O}}$, and $B_{\mathrm{S}}$ increase. The proposed empirical concentration models are compared with the corresponding numerical data in Tables 1 and 2.

7.   Conclusions

An amperometric enzyme-based biosensor combines chemistry, biology, electrochemistry, materials science, polymer synthesis, enzymology, and electrochemistry to provide a powerful analytical tool.

Our main goal was to develop a biosensor that responds to oxidase linked tests independently of biorecognition matrix thickness for oxidase linked tests. The mathematical models (2.12)–(2.14) of the biosensor utilizing the synergistic scheme of substrates conversion can be successfully used to investigate the peculiarities of the biosensor response and sensitivity at steady as well as at transition state. In the amperometric biosensor system, a nonlinear differential equation was used to determine the semi-analytical solution of the species concentration. Using the AGM, an approximate general semi-analytical expression for the concentration of substrate, mediator and current of amperometric biosensor at the enzyme-membrane electrode geometry is derived for all values of parameters $\phi_{O}^{2}$, $B_O$, and $B_S$. The effects of these parameters on the concentration and effectiveness were also explored. The results were satisfactory compared to those of the numerical simulation. It is possible to determine the qualitative behavior of biosensors by using this hypothetical model. Furthermore, the results of this study provide an option for extending this method to measure substrate concentrations and diffusion currents. As the reciprocal of the squares of the Thiele modulus decreases, the current density increases. The results of this study can be used to optimize and design biosensors.

Acknowledgments

The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-92).

Conflict of interest

All authors declare no conflicts of interest in this paper.