Fractional Modelling of H₂O₂-Assisted Oxidation by Spanish broom peroxidase

Abstract

The $H_2{O}_2$-assisted oxidation by a peroxidase enzyme takes place to help plants maintain the concentrations of organic compounds at physiological levels. Cells regulate the oxidation rate by inhibiting the action of this enzyme. The cells use two inhibitory processes to regulate the enzyme: a noncompetitive substrate inhibitory process and a competitive substrate inhibitory process. Numerous applications of peroxidase have been developed in clinical biochemistry, enzyme immunoassays, the treatment of waste water containing phenolic compounds, the synthesis of various aromatic chemicals, and the removal of peroxide from industrial wastes. The kinetic mechanism of the Spanish broom peroxidase enzyme is a Ping Pong Bi Bi mechanism with the presence of competitive inhibition by substrates. A mathematical model may help in identifying the key mechanism from amongst a set of competing mechanisms. In this study, we developed a fractional mathematical model to describe the $H_2{O}_2$-supported oxidation by the enzyme Spanish broom peroxidase. Numerical simulations of the model produced results that are consistent with the known behaviour of Spanish broom peroxidase. Finally, some future investigations of the study are briefly indicated as well.

1. Introduction

Enzymes, natural proteins found in living organisms, catalyse biochemical reactions essential for cell metabolism by reducing the activation energy. They are not consumed during reactions and can be biologically degradable, suggesting an important role in environmental protection [1,2,3,4,5]. Cells regulate metabolite concentrations through various mechanisms, including enzymatic inhibition processes such as competitive, noncompetitive, and uncompetitive inhibition. Competitive inhibition involves substrate and inhibitor molecules competing for enzyme binding sites, hindering catalytic activity. The competitive substrate inhibition of an enzyme is a competitive inhibition process in which the substrate plays the role of inhibitor, and enzymes often are bi-substrate enzymes. Some enzymes, for example, Spanish broom peroxidase, can be inhibited by their substrates [6,7].

Enzymes offer numerous benefits due to the features of enzymes. Nowadays, scientific and technological advances facilitate studies on enzymes and their applications [6,7,8]. Novel enzymes are increasingly being extracted and investigated. A variety of applications of enzymes have been developed for biotechnology, industry, and medicine. Some common applications of enzymes take place in pharmaceuticals, food processing, biofuels, and so on [8,9,10,11]. Understanding enzyme mechanisms is crucial for application development, utilising both experimental and mathematical modelling approaches [12,13,14,15,16,17].

Plants use oxygen as a terminal electron acceptor. Class III plant peroxidases (EC 1.11.1.7; donor: hydrogen peroxide oxidoreductases) help maintain low levels of hydrogen peroxide [18]. Peroxidases are enzymes that catalyse the oxidation of numerous substrates such as halide, aromatic amines, phenols, and thiosanisoles through $H_2{O}_2$ reduction. The reducing substrate is dependent on the type of peroxidase enzyme [19]. Enzymatic biocatalysis plays a crucial role in the development of many chemical industries. Due to the catalytic features of peroxidases, they become attractive enzymes for biotechnological processes. Investigating substrate specificity and the effectors affecting peroxidase enzyme activities may assist in developing the use of the catalytic potential of peroxidases [20]. Peroxidases also play important roles in clinical biochemistry and enzyme immunoassays [21]. Recently, peroxidases were used in the treatment of waste water containing phenolic compounds, the synthesis of various aromatic chemicals, and the removal of peroxide from industrial wastes [22].

In recent years, experimental evidence has increasingly pointed to the relevance of fractional calculus analysis in understanding dynamic phenomena in nature. This burgeoning field of research has seen rapid growth, driven by its wide-ranging applications across diverse areas of engineering and science. From chemical models to physics and signal and image processing to quantum mechanics and control theory to nonlinear dynamics, fractional calculus finds utility in a multitude of disciplines. Furthermore, its application extends to biological population models, optimisation theory, and beyond [23,24,25,26,27,28].

Alicea [29] and Alawneh [30] independently tackled the integer order-model for the system of the following chemical reaction:

$$S+E\rightleftharpoons ES\to E+P$$

(1)

using different mathematical methods. Alicea employed the method of multiple time scales to obtain asymptotic solutions, while Alawneh utilised a generalised differential transform method with multisteps and discussed estimation analysis using fractional order derivatives. However, despite their efforts, the solutions obtained mathematically for the concentrations of E, S, $ES$, and P at any time t often did not align with those of the experimental results. To address this discrepancy, alternative approaches have been explored, including the use of fractional derivatives instead of integer-order derivatives. Fractional derivatives offer greater flexibility and accuracy compared to classical derivatives, making them particularly appealing for modelling complex systems. Interested readers are encouraged to explore the works referenced in [26,31] for further insights into the advantages of fractional derivatives in mathematical modelling.

In this study, we developed a fractional mathematical model describing hydrogen peroxide-assisted oxidation by Spanish broom peroxidase enzyme. The model consists of the Spanish broom peroxidase enzyme, the substrates $H_2{O}_2$ and $A{H}_2$, and their complexes. Numerical simulations provided insights into the model behaviour, using Python software for numerical integration. The notations employed in subsequent sections are elucidated in Figure 1.

The rest of this paper is organised as follows. In Section 2, we describe the formulation of the mathematical model and methods used in the current study. Numerical solutions for the model are presented in Section 3. Finally, we discuss some concluding remarks in Section 4.

2. Materials and Methods

2.1. Mathematical Model

In this section, we present a minimal fractional-order model that describes the $H_2{O}_2$-assisted oxidation by Spanish broom peroxidase enzyme, in which the substrates competitively inhibit the enzyme by binding to the enzyme molecule at the binding site for the other substrate. The fractional derivative operator, which is used in this study, is the Caputo one. The fractional derivative of order $\alpha$ in the Caputo sense is defined as the operator $D_t^{\alpha }f\left(t\right)$ such that

$$D_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(m-\alpha )}}{\int }_0^t{\displaystyle \frac{f^{\left(m\right)}\left(t\right)}{{(t-s)}^{\alpha +1-m}}}ds,m-1<\alpha <m,$$

where $\mathsf{\Gamma }$ is the Gamma function defined as follows:

$$\mathsf{\Gamma }\left(\alpha \right)={\int }_0^{\infty }{u}^{\alpha -1}{e}^{-u}du,\alpha >0.$$

Here, we list several common formulae for the Gamma function as follows:

$$\begin{array}{cc}\hfill & \mathsf{\Gamma }(\alpha +1)=\alpha \mathsf{\Gamma }\left(\alpha \right),\hfill \\ \hfill & \mathsf{\Gamma }(\alpha +n)=\alpha (\alpha +1)\cdots (\alpha +n-1)\mathsf{\Gamma }\left(\alpha \right),\hfill \\ \hfill & \mathsf{\Gamma }(n+1)=(n+1)!,n=0,1,2,\cdots ,\hfill \\ \hfill & \mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }(1-\alpha )={\displaystyle \frac{\pi }{sin\left(\pi \alpha \right)}}.\hfill \end{array}$$

For more details, the readers can find them in [32]. In the current study, we used $\alpha \in (0,1)$. Next, we provide a brief description of the kinetic mechanism that the model was based on.

2.1.1. The Kinetic Mechanism

It is known that Spanish broom peroxidase catalyses the $H_2{O}_2$-mediated oxidation in a Ping Pong mechanism with the presence of competitive inhibition by the substrates [6]. The kinetic mechanism for the $H_2{O}_2$-mediated oxidation by Spanish broom peroxidase is summarised in Figure 2.

For convenience, the mechanism in Figure 2 can be illustrated using separated stages as in Figure 3.

In this mechanism, an enzyme molecule has at least two binding sites for its substrates and produces two types of product molecules. The Ping Pong mechanism includes two stages: Stage 1 and Stage 2. In Stage 1, an enzyme molecule E absorbs a molecule of substrate $H_2{O}_2$ to form a substrate-enzyme complex $E·H_2{O}_2$. The bound enzyme molecule catalyses the complex to form a product-enzyme complex $E^{\prime }·H_{2}O$. It should be noticed that these reactions are reversible reactions. Then, the enzyme releases the product molecule $H_{2}O$ and an intermediate enzyme molecule $E^{\prime }$ into the medium. This is an irreversible reaction.

When intermediate enzyme molecules $E^{\prime }$ occur at the end of Stage 1, each of them may be able to initiate Stage 2 by absorbing a substrate molecule $A{H}_2$ to form a substrate–enzyme complex $E^{\prime }·A{H}_2$. The bound enzyme molecule catalyses the complex $E^{\prime }·A{H}_2$ to form a product–enzyme complex $E·A{H}^{•}$. These reactions are reversible also. The product molecule $A{H}^{•}$ and original enzyme molecule E are then irreversibly released into the medium.

The competitive inhibition by the substrates includes the inhibition by $H_2{O}_2$ and the inhibition by $A{H}_2$. First, we explain the competitive inhibition by substrate $H_2{O}_2$. The inhibition is interpreted as follows: a substrate molecule $H_2{O}_2$ that is able to bind to an enzyme molecule $E^{\prime }$ to form a substrate–enzyme complex $E^{\prime }·H_2{O}_2$. The binding of a molecule $H_2{O}_2$ to an enzyme molecule $E^{\prime }$ prevents the $A{H}_2$ molecules from binding to the enzyme molecule $E^{\prime }$ [6,33,34].

The competitive inhibition by the substrate $A{H}_2$ is represented as follows: a substrate molecule $A{H}_2$ that is capable of binding to an enzyme molecule E to form a substrate–enzyme complex $E·A{H}_2$. The binding of a molecule $A{H}_2$ to an enzyme molecule E prevents $H_2{O}_2$ molecules from binding to the enzyme molecule E [6,33,34].

To simplify the modelling, we reduced the above kinetic mechanism using a minimal set of chemical reactions as follows (Figure 4).

Assumptions are needed for developing a mathematical model. In the next section, we list some assumptions for the model.

2.1.2. Modelling Assumptions

The model developed here was based on the law of mass action. The system is quite complex due to the number of binding sites of an enzyme molecule. Here are the necessary assumptions for the model.

• The mixture of peroxidase enzyme, $H_2{O}_2$, and $A{H}_2$ (such as Ferulic acid, Guaiacol, Catechol, etc.) is well stirred throughout. This implies that diffusive effects in the process can be omitted and that the concentrations of the various species in the mixture can be described by functions of time only. This further implies that the evolution of the system can be modelled using a coupled system of nonlinear fractional differential equations and that a partial differential equation model is not required [15,35].
• We assume that mass action kinetics occur throughout; this implies that the rate of a reaction is taken to be proportional to the product of the concentrations of the reactants. We emphasise here that more complex formulas, such as the Michaelis–Menten formula for the rate of product formation in an enzyme-catalysed reaction, are derivable from more fundamental mass action considerations under simplifying assumptions [15,35].

2.1.3. Construction of the Governing Fractional Differential Equations

For the convenience of modelling, we can rewrite the chemical reactions in Figure 4 as follows:

$$\begin{array}{cc}\hfill & E+H_2{O}_2\underset{k_{-1}}{\overset{k_1}{\rightleftharpoons }}E·H_2{O}_2,\hfill \\ \hfill & E·H_2{O}_2\stackrel{k_2}{\to }{E}^{\prime }+H_{2}O,\hfill \\ \hfill & E^{\prime }+A{H}_2\underset{k_{-3}}{\overset{k_3}{\rightleftharpoons }}{E}^{\prime }·A{H}_2,\hfill \\ \hfill & E^{\prime }·A{H}_2\stackrel{k_4}{\to }E+A{H}^{•},\hfill \\ \hfill & E+A{H}_2\underset{k_{-5}}{\overset{k_5}{\rightleftharpoons }}E·A{H}_2,\hfill \\ \hfill & E^{\prime }+H_2{O}_2\underset{k_{-6}}{\overset{k_6}{\rightleftharpoons }}{E}^{\prime }·H_2{O}_2.\hfill \end{array}$$

Under the above assumptions and using the law of mass action, the model equations that describe the concentrations of the species in the mixture are given by

$$\begin{array}{cc}\hfill & D_t^{\alpha }\left[E\right]=-k_1\left[E\right]\left[H_2{O}_2\right]-k_5\left[E\right]\left[A{H}_2\right]+k_{-1}[E·H_2{O}_2]+k_4[E^{\prime }·A{H}_2]+k_{-5}[E·A{H}_2],\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & D_t^{\alpha }\left[H_2{O}_2\right]=-k_1\left[E\right]\left[H_2{O}_2\right]-k_6\left[E^{\prime }\right]\left[H_2{O}_2\right]+k_{-1}[E·H_2{O}_2]+k_{-6}[E^{\prime }·H_2{O}_2],\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & D_t^{\alpha }[E·H_2{O}_2]=-(k_{-1}+k_2)[E·H_2{O}_2]+k_1\left[E\right]\left[H_2{O}_2\right],\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & D_t^{\alpha }\left[E^{\prime }\right]=-k_3\left[E^{\prime }\right]\left[A{H}_2\right]-k_6\left[E^{\prime }\right]\left[H_2{O}_2\right]+k_2[E·H_2{O}_2]\hfill \\ \hfill & +k_{-3}[E^{\prime }·A{H}_2]+k_{-6}[E^{\prime }·H_2{O}_2],\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & D_t^{\alpha }\left[H_{2}O\right]=k_2[E·H_2{O}_2],\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & D_t^{\alpha }\left[A{H}_2\right]=-k_3\left[E^{\prime }\right]\left[A{H}_2\right]-k_5\left[E\right]\left[A{H}_2\right]+k_{-3}[E^{\prime }·A{H}_2]+k_{-5}[E·A{H}_2],\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & D_t^{\alpha }[E^{\prime }·A{H}_2]=-(k_{-3}+k_4)[E^{\prime }·A{H}_2]+k_3\left[E^{\prime }\right]\left[A{H}_2\right],\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & D_t^{\alpha }\left[A{H}^{•}\right]=k_4[E^{\prime }·A{H}_2],\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & D_t^{\alpha }[E·A{H}_2]=-k_{-5}[E·A{H}_2]+k_5\left[E\right]\left[A{H}_2\right],\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & D_t^{\alpha }[E^{\prime }·H_2{O}_2]=-k_{-6}[E^{\prime }·H_2{O}_2]+k_6\left[E^{\prime }\right]\left[H_2{O}_2\right],\hfill \end{array}$$

(11)

where $\left[X\right]=\left[X\right]\left(t\right)$ denotes the concentration of species X at time t.

It is not necessary to provide discussions of these equations here. However, we do briefly discuss two of them to illustrate how the model equations are constructed. The chemical reactions for the model are displayed in Figure 4. We begin by considering the Equation (2) for E given by

$$D_t^{\alpha }\left[E\right]=\underset{1}{\underbrace{-k_1\left[E\right]\left[H_2{O}_2\right]}}\underset{2}{\underbrace{-k_5\left[E\right]\left[A{H}_2\right]}}\underset{3}{\underbrace{+k_{-1}[E·H_2{O}_2]}}\underset{4}{\underbrace{+k_4[E^{\prime }·A{H}_2]}}\underset{5}{\underbrace{+k_{-5}[E·A{H}_2]}},$$

where the numbered parts are described as follows:

1

This term accounts for the reduction in the concentration of E due to substrate $H_2{O}_2$ binding.

2

The reduction in the concentration of E due to substrate $A{H}_2$ binding.

3

The increase in the concentration of E due to the substrate unbinding from the complex $E·H_2{O}_2$.

4

The increase in the concentration of E due to the enzyme catalysing the complex $E^{\prime }·A{H}_2$ and releasing the product and the original enzyme molecule.

5

The increase in the concentration of E due to the substrate unbinding from the complex $E·A{H}_2$.

Now, we turn our attention to Equation (4) for complex $E·H_2{O}_2$:

$$D_t^{\alpha }[E·H_2{O}_2]=\underset{a}{\underbrace{-(k_{-1}+k_2)[E·H_2{O}_2]}}\underset{b}{\underbrace{+k_1\left[E\right]\left[H_2{O}_2\right]}},$$

where the parts are as follows:

a

This accounts for the reduction in the concentration of $E·H_2{O}_2$ due to the substrate unbinding from $E·H_2{O}_2$ and the enzyme catalysing $E·H_2{O}_2$ to form product $H_{2}O$.

b

The increase in the concentration of $E·H_2{O}_2$ due to the enzyme binding to the substrate $H_2{O}_2$.

The remaining equations, Equations (3) and (5)–(11), are interpreted similarly.

2.1.4. Initial Conditions

The equations described in the previous subsection are solved under the initial conditions:

$$\begin{array}{ll}\left[E\right](t=0)=e_0\mathrm{mM},\hfill & \left[E^{\prime }\right](t=0)=0.0\mathrm{mM},\hfill \\ \left[H_2{O}_2\right](t=0)=a_0\mathrm{mM},& \left[A{H}_2\right](t=0)=b_0\mathrm{mM},\\ [E·H_2{O}_2](t=0)=0.0\mathrm{mM},& [E^{\prime }·A{H}_2](t=0)=0.0\mathrm{mM},\\ \left[H_{2}O\right](t=0)=0.0\mathrm{mM},& \left[A{H}^{•}\right](t=0)=0.0\mathrm{mM},\\ [E·{AH}_2](t=0)=0.0\mathrm{mM},& [E^{\prime }·H_2{O}_2](t=0)=0\mathrm{mM},\end{array}$$

where, e₀, a₀, and b₀ give the initial constant concentrations of the enzyme, H₂O₂, and AH₂, respectively. The initial concentrations for all of the enzyme complexes were taken to be zero. Finally, the initial concentrations of the rest of the species were set to be zero also.

2.1.5. Conservation Laws

Computing the sum of Equations (2) + (4) + (5) + (8) + (10) + (11), (3) + (4) + (6) + (11) and (7) + (8) + (9) + (10) and integrating both sides yields

$$\begin{array}{cc}\hfill & \left[E\right]+[E·H_2{O}_2]+\left[E^{\prime }\right]+[E^{\prime }·A{H}_2]+[E·A{H}_2]+[E^{\prime }·H_2{O}_2]=e_0,\hfill \end{array}$$

(12a)

$$\begin{array}{cc}\hfill & \left[H_2{O}_2\right]+[E·H_2{O}_2]+\left[H_{2}O\right]+[E^{\prime }·H_2{O}_2]=a_0,\hfill \end{array}$$

(12b)

$$\begin{array}{cc}\hfill & \left[A{H}_2\right]+[E^{\prime }·A{H}_2]+\left[A{H}^{•}\right]+[E·A{H}_2]=b_0,\hfill \end{array}$$

(12c)

which are the expressions of the conservation of enzyme E and substrates $H_2{O}_2$ and $A{H}_2$, respectively.

2.2. Computational Methods

In this section, we describe the computational tools used to analyse the model equations. The software developed for this paper was coded using the Python programming language [36].

2.2.1. Numerical Method for Solving the Fractional Differential Equations

The numerical integration of fractional-order ordinary differential equations was performed using the fodeint solver, alongside the SciPy and Numpy libraries [37,38,39]. SciPy [39] is an open-source Python [36] library that provides numerical routines for scientific and engineering applications. The fodeint solver, a Python package, numerically integrates fractional ordinary differential equations using an explicit one-step Adams–Bashforth (Euler) method [40,41]. However, the convergence and accuracy of this method have not yet been evaluated. Recently, novel numerical methods have been developed to solve fractional-order ordinary differential equations efficiently, and their analyses have yielded several interesting results [42,43]. The employment of these methods in creating a new Python solver for fractional-order ordinary differential equations could lead to intriguing research opportunities. In the present study, our primary aim involved investigating a specific phenomenon utilising well-established methodologies, including theoretical frameworks and computational libraries. Our focus did not extend to creating a computational library specifically for solving non-integer differential equations. Nevertheless, we maintained an open-minded approach and may consider exploring such endeavors opportunistically in the days ahead.

2.2.2. Model Parameter Values

Table 1. Some model parameter values and their literature sources.

Parameter	Value	Unit	Ref.
$k_1$	$9.1$	mM−1s−1
$k_{-1}$	$3.2$	s−1
$k_2$	$14.5$	s−1
$k_3$	$9.8$	mM−1s−1
$k_{-3}$	$2.4$	s−1
$k_4$	$12.6$	s−1	[6]
$k_5$	$8.0$	mM−1s−1
$k_{-5}$	$2.5$	s−1
$k_6$	$7.0$	mM−1s−1
$k_{-6}$	$1.5$	s−1

shows some of the model parameter values, together with their literature sources. Typically, the parameter values are rare in the literature, except a few of them.

3. Results and Discussion

3.1. Numerical Results

The Section 2 introduced the mathematical model. It also described the computational methods used to integrate the model equations, and included some discussion of the numerical method, the choice of parameter values, and the initial conditions. In the current section, we describe some of the numerical results obtained some initial conditions. The initial conditions used here correspond to $e_0=5.0$ mM, $a_0=9.0$ mM, and $b_0=15.0$ mM; see Section 2.1.4 for details.

The principal purpose of the numerical solutions displayed here was to gain insight into the $H_2{O}_2$-assisted oxidation by Spanish broom peroxidase. To focus attention on the oxidation process itself, we made no attempt to model the evolution of the physiological levels of the species in the mixture. Instead, we simply assumed the constant initial concentrations of the species and then tracked its subsequent conversion via the enzymatic reactions. It should be noted that the derivative order used here was $\alpha =0.8$. The nearer to zero that the $\alpha$ is, the longer the time consumption for the computation.

In Figure 5, we plotted the numerical solutions of the model corresponding to the initial conditions and the parameter values aforementioned. Each line corresponds to the concentration of one species in the mixture with respect to time t. Some points of discussion on these numerical results are as follows.

• The line represents the concentration of the enzyme during the process. In the first stage, the concentration of enzyme drops rapidly due to the binding of substrates to form substrate–enzyme complexes $E·H_2{O}_2$ and $E·A{H}_2$. As time goes on, the enzyme converts the substrates to products. This reduces the concentrations of the substrates and makes the increases in the concentrations of products continuous. The concentration of the enzyme goes up and reaches a steady state at the end of the process. It should be noted that the steady concentration of the enzyme is lower than its initial concentration.
• The line displays the concentration of the substrate $H_2{O}_2$. The concentration decreases rapidly and reaches a steady state at the end of the process. This occurs because the intermediate enzyme $E^{\prime }$ cannot convert the product $H_{2}O$ to the substrate $H_2{O}_2$.
• The line describes the concentration of the substrate–enzyme complex $E·H_2{O}_2$. It can be seen that the concentration rises up rapidly in the early stage because of the binding of the substrate to the enzyme. Then, the concentration drops down quickly since the enzyme catalyses the complex to form product $H_{2}O$, and the substrate unbinds from the complex. In the end, the concentration of the complex is completely catalysed to form the product $H_{2}O$ and tends to zero then. This is because the reaction is irreversible.
• The line corresponds to the concentration of the intermediate enzyme $E^{\prime }$. The concentration increases gradually and tends to zero at the end of the process. This agrees with the nature of the process. That is, the intermediate enzyme is incapable of converting the product $H_{2}O$ to the substrate $H_2{O}_2$ and the original enzyme E, and the substrate $A{H}_2$ binds to the enzyme $E^{\prime }$ to form complexes $E^{\prime }·A{H}_2$.
• The line shows the concentration of the product $H_{2}O$. The concentration increases rapidly and reaches a steady state. It is clear that the concentration approaches the initial concentration of the substrate $H_2{O}_2$. In the end, the substrate $H_2{O}_2$ is completely converted to the product $H_{2}O$. This is in line with the nature of the process since the enzyme is not able to convert the product $H_{2}O$ to the substrate $H_2{O}_2$.
• The line represents the concentration of the substrate $A{H}_2$. The rapid decrease in the concentration of $A{H}_2$ is due to the binding of the substrate $A{H}_2$ to the intermediate enzyme $E^{\prime }$ to form the substrate–enzyme complex $E^{\prime }·A{H}_2$. The concentration tends to the concentration of $H_{2}O$ at the end of the process since this process will not take place once the substrate $H_2{O}_2$ is completely consumed. This agrees with the fact that the enzyme is not able to convert the product $A{H}^{•}$ to the substrate $A{H}_2$ at all.
• The line shows the concentration of the substrate–enzyme complex $E^{\prime }·A{H}_2$. At the early stage, the rapid increase in the concentration is due to the binding of the substrate $A{H}_2$ to the intermediate enzyme $E^{\prime }$ to form the complex $E^{\prime }·A{H}_2$. The concentration approaches zero at the end of the process since the complex is totally catalysed to form the product $A{H}^{•}$, and the substrate $A{H}_2$ is completely converted to the product $A{H}^{•}$. This is in line with the fact that the conversion of the complex to the product $A{H}^{•}$ is an irreversible reaction.
• The line displays the concentration of the product $A{H}^{•}$ with respect to time t. The concentration increases quickly at the early stage since the concentration of the complex $E^{\prime }·A{H}_2$ increase quickly and the enzyme rapidly catalyses the complex and releases the product then. The concentration tends to the initial concentration of the substrate $A{H}_2$ or that of $H_2{O}_2$. The reason is that the reactions that form the product $A{H}_2$ will be terminated at once if the substrate $H_2{O}_2$ or $A{H}_2$ is exhausted.
• The line shows the concentration of the substrate–enzyme complex $E·A{H}_2$. At the early stage, the rapid increase in the concentration is due to the binding of the substrate $A{H}_2$ to the enzyme E to form the complex $E·A{H}_2$. It should be noted that this reaction is reversible. The concentration approaches a steady state at the end of the process since the substrate $H_2{O}_2$ is completely converted to the product $H_{2}O$. This means that the enzyme $E^{\prime }$ is exhausted.
• The line displays the concentration of the substrate–enzyme complex $E^{\prime }·H_2{O}_2$. At the early stage, the increase in the concentration is due to the binding of the substrate $H_2{O}_2$ to the enzyme $E^{\prime }$ to form the complex $E^{\prime }·H_2{O}_2$. It should be noted that this reaction is reversible. The concentration tends to zero at the end of the process since the substrate $H_2{O}_2$ is completely converted to the product $H_{2}O$.

3.2. Further Numerical Results

We now display some further numerical solutions inspired by the fact that the fractional order $\alpha$ is in $(0,1)$. That is, we conducted some numerical experiments for the different values of $\alpha$ to obtain insights into the behaviours of the solutions of the model. These values of $\alpha$ were taken to be $\alpha =0.7$, $\alpha =0.8$, and $\alpha =0.9$ to reduce the computation time. In these calculations, the default values used for the parameters are shown in Table 1, and the initial concentrations for the enzyme, $H_2{O}_2$, and $A{H}_2$ are those used in Section 3.1. We used the solutions of the model for the product $H_{2}O$ and the substrate $A{H}_2$ for this investigation.

In Figure 6, the solutions of the model for the product $H_{2}O$ were used to study $\alpha =0.7$, $\alpha =0.8$, and $\alpha =0.9$. We illustrate how the new numerical results shown in Figure 6 were generated by considering a particular example. Here, we consider the curves displayed in Figure 6. The solid blue curve was generated using $\alpha =0.7$. The solid orange curve was generated using $\alpha =0.7$, and the dash–dot green curve was generated using $\alpha =0.9$. It can be clearly seen that the higher the value of $\alpha$, the faster the increase in the concentration of $H_{2}O$.

Figure 7 shows the plots of the concentrations of $A{H}_2$ with respect to time t for different values $\alpha =0.7$, $\alpha =0.8$, and $\alpha =0.9$. Clearly, the smaller the value of $\alpha$, the slower the decrease in the concentration of $A{H}_2$.

4. Conclusions

Peroxidase enzymes facilitate oxidation with the assistance of $H_2{O}_2$ to assist plants in maintaining optimal concentrations of organic compounds essential for physiological functions. The cellular regulation of the oxidation rates involves inhibiting the enzyme activity. Cells employ two inhibitory mechanisms to modulate enzyme activity: a noncompetitive substrate inhibition process and a competitive substrate inhibition process. This paper presents a fractional mathematical model elucidating the $H_2{O}_2$-mediated oxidation catalysed by Spanish broom peroxidase. Our mathematical model serves to dissect the regulatory mechanisms governing the behaviour of Spanish broom peroxidase. The biological evidence utilised in the modeling process has been previously established. However, this study marks the first instance where these concepts have been synthesised into a fractional mathematical model. The model incorporates numerous bound states for the enzyme, along with their corresponding activation statuses. The model was numerically integrated using the fodeint, from the SciPy Python library, and the solutions obtained were found to align with the established behaviour of Spanish broom peroxidase. Furthermore, the model output demonstrated sensitivity to the fractional order.

While the model developed in this study was thoroughly investigated, there exists potential for improvement and future research. For instance, one avenue for further exploration is the estimation of model parameters using experimental data. Additionally, within the confines of this study, we did not specifically examine the sensitivity of the model output to small variations in parameter values [44,45]. Consequently, employing the homotopy analysis method to derive analytic solutions for the model [46,47] may enhance the sensitivity analysis, as these analytic solutions could potentially reduce computational expenses. Furthermore, it may be of interest to conduct a research study aimed at developing a computational tool for numerically solving the model, based on the methods proposed in the referenced papers [42,43]. In addition, addressing the positivity of the solutions of the model is an important consideration [48]. Finally, a notable limitation of the study lies in the absence of validation against experimental data. While the theoretical and computational aspects were explored, the lack of empirical validation leaves room for further investigation and refinement.