1. Background

With the rapid advancement of biotechnology, the development and utilization of marine polysaccharide resources have garnered increasing attention. Agar, as a crucial seaweed polysaccharide, finds extensive applications in the food, pharmaceutical, and other industries. However, traditional agar processing techniques suffer from low efficiency and high energy consumption, necessitating the development of novel biotransformation technologies.

Traditional agar treatment methods primarily rely on physicochemical approaches, including high-temperature and high-pressure hydrolysis, as well as acid-alkali treatment. Although these methods can achieve agar degradation, they exhibit drawbacks such as harsh reaction conditions and uneven product distribution. In contrast, enzymatic degradation has attracted significant interest due to its high specificity and mild reaction conditions. Among various enzymes, α-agarase demonstrates unique advantages in degrading agarose. This enzyme specifically hydrolyzes α-1,3 glycosidic bonds, converting agarose into agar oligosaccharides (AOS) with diverse biological activities. These oligosaccharides hold substantial application prospects in the fields of functional foods and pharmaceuticals.

Metal ions play a critical regulatory role in the catalytic process of α-agarase. Studies have shown that monovalent metal ions such as sodium ions (Na⁺) influence enzyme activity primarily through the following mechanisms: (1) Electrostatic stabilization: Maintaining the active conformation of the enzyme protein by neutralizing the negative charges on the enzyme molecule surface; (2) Substrate binding regulation: Altering the solubility and spatial conformation of agarose molecules to facilitate the formation of enzyme-substrate complexes; (3) Catalytic center modulation: Participating in charge balance at the active center and stabilizing transition-state intermediates.

To realize the industrial application of efficient agarose conversion, precise control of enzyme reaction conditions is essential. Based on the principles of enzyme reaction kinetics, establishing a quantitative relationship model between sodium ion concentration and enzyme activity will provide a theoretical basis for process optimization. This model enables precise regulation of reaction conditions, enhances conversion efficiency, reduces production costs, and promotes the high-value utilization of agarose resources. Future research may further explore the synergistic effects of different metal ions, offering new insights for the development of efficient and stable α-agarase preparations.

2. Experimental Principle and Design

In this experiment, the DNS method (3,5-dinitrosalicylic acid method) was employed to determine the activity of α-agarase. The DNS method is a classic technique based on reducing sugar determination, which quantifies enzyme activity by detecting the reducing sugars produced during the enzymatic hydrolysis reaction. During the experiment, α-agarase and agarose substrate first react under appropriate temperature and pH conditions, allowing the enzyme to specifically hydrolyze α-1,3 glycosidic bonds and generate reducing oligosaccharides. After the reaction, DNS reagent is added to terminate the reaction, followed by heating in a boiling water bath for color development. In an alkaline environment, 3,5-dinitrosalicylic acid in the DNS reagent undergoes a redox reaction with the aldehyde groups of reducing sugars, producing reddish-brown 3-amino-5-nitrosalicylic acid. The intensity of the color is proportional to the content of reducing sugars. After removing precipitates by centrifugation, the absorbance value is measured at a wavelength of 540 nm using spectrophotometry, and the amount of reducing sugars produced is calculated based on a standard curve.

The enzyme activity unit is defined as the amount of enzyme required to catalyze the production of 1 μmol of reducing sugar per minute under specific conditions (e.g., 35°C, pH 7.0). The DNS method offers advantages such as simple operation, stable color development, and high sensitivity, making it particularly suitable for determining the activity of polysaccharide-degrading enzymes (e.g., α-agarase) and accurately reflecting the catalytic efficiency of the enzyme. This method is not only applicable to laboratory research but also can be used for activity detection and quality control of enzyme preparations in industrial production.

3. Initial Experimental Data and Analysis

3.1 Preparation of D-Fructose Standard Curve

3.1.1 Preparation of Standard Solutions

Since D-fructose is a reducing sugar, it was selected as the standard for product identification in this experiment. Standard D-fructose solutions with concentrations of 0, 0.2, 0.4, 0.6, 0.8, and 1 mg/mL were prepared respectively.

3.1.2 Detection

200 μL of each standard solution was taken in sequence, and an equal volume of DNS solution was added for mixing. The mixed samples were heated in a boiling water bath for 5 minutes, then cooled in an ice water bath to terminate the reaction, and 1 mL of deionized water was added to each sample for dilution. The wavelength of the microplate reader was adjusted to 540 nm, and 200 μL of D-fructose solutions with different concentrations were sequentially added to a 96-well plate. The plate was placed in the microplate reader for absorbance measurement, and a standard curve was plotted based on the results.

3.1.3 Initial Data

According to the experimental procedure, preliminary data on the concentration and absorbance of D-fructose in the solution after the start of the enzymatic hydrolysis reaction were obtained. Three sets of different data were collected, the average value was calculated, and the difference between the absorbance of D-fructose at different concentrations and the control group (background value) was determined based on the average value. The results are shown in Tab. 1.

Tab 1.Data table of D-fructose concentration and absorbance

Furthermore, a standard curve of D-fructose (Fig. 1) was plotted, which shows the corresponding absorbance values at different D-fructose concentrations.

The scatter plot in Figure 1 indicates an extremely strong linear relationship between D-fructose concentration and absorbance change. This means that in subsequent experiments, it is feasible and reasonable to determine the concentration of products generated during the enzyme reaction by monitoring changes in absorbance.

Enzymatic Reaction Kinetics Experiment

3.2 Determination of Enzymatic Reaction Kinetic Parameters of AgaE

3.2.1 Substrate Preparation

Agarose substrates with concentrations of 0.5%, 1%, 2%, 2.5%, 3%, and 4% were prepared as follows: 0.05 g, 0.1 g, 0.2 g, 0.25 g, 0.3 g, and 0.4 g of agarose were accurately weighed respectively and placed in 15 mL centrifuge tubes. 10 mL of PB buffer (pH 7.0) was added to each tube, and the mixtures were heated in a boiling water bath with constant shaking until the agarose was completely dissolved (the solution became colorless and transparent). The solutions were then placed in a 60°C oven for later use.

3.2.2 Determination of AgaE Activity

Based on the optimal enzymatic reaction time and kinetic parameters obtained earlier, an experimental protocol was designed to investigate the effect of sodium ion concentration on AgaE activity. With the AgaE concentration fixed, 50 μL of substrate solutions with different concentrations (0.5%-4%) were sequentially added to 2 mL EP tubes, followed by the addition of 50 μL of AgaE solution. According to the above reaction system, seven groups of 50 μL NaCl solutions with different concentrations (0-100 mM gradient) were added respectively. The reaction was carried out at 35°C for 3 minutes.

After the reaction was terminated, the samples were treated using the same DNS method: inactivation in a boiling water bath for 5 minutes, centrifugation at 12000 rpm for 10 minutes, 200 μL of the supernatant was mixed with an equal volume of DNS reagent, heated in a boiling water bath for 5 minutes for color development, then cooled in an ice bath, and 1 mL of deionized water was added for dilution. The absorbance was measured at a wavelength of 540 nm using a microplate reader, and the amount of reducing sugars produced was calculated based on the standard curve.

By comparing the changes in reaction rate under different Na⁺ concentrations, the effects of sodium ions on enzyme kinetic parameters (Km, Vmax) were analyzed, and the regulatory mechanism was revealed. This protocol kept other conditions consistent and only changed the Na⁺ concentration as the variable, ensuring the comparability of the experimental data.

3.2.3 Initial Data

Tab. 2 provides detailed information on the differences between the original OD data of the enzyme hydrolysis products and the control group, showing how the original OD data of the enzyme hydrolysis products changed over time. The data were recorded in a standardized manner with an appropriate sample size, truly reflecting the absorbance information at each condition. During the experiment, the data exhibited good continuity and representativeness, with no significant outliers or abnormal values, thus no data cleaning or correction was required.

Tab. 2 Changes in absorbance at different substrate concentrations and different concentrations of Na⁺

4. Modeling and solving

4.1 Linear Modeling of Absorbance and D-Fructose Concentration

Table 1 and Figure 1 present the standard curve of D-fructose, showing a clear linear relationship between absorbance change and D-fructose concentration. Therefore, a linear model was established as follows:

Where:

• y represents the absorbance change, which is a function of D-fructose concentration (expressed as y=y(c));
• c represents the D-fructose concentration, which varies with the yield under enzymatic reaction;
• a (a>0) represents the ratio coefficient between the growth rate of D-fructose concentration and absorbance;
• b is the background absorbance value (b≥0);
• ε represents the random disturbance term, which is assumed to follow a normal distribution.

The least squares estimation method was used to regress Equation (1). The least squares method involves searching for the value of a that minimizes the sum of the squares of residuals. Geometrically, this means finding the best-fitting regression line that minimizes the sum of the squares of the distances from the observed values to the line, which satisfies the following equations:

Thus, the values of a and b can be calculated, where c and ӯ represent the sample means. Therefore, a univariate regression result describing the change in absorbance with D-fructose concentration was obtained through regression, as shown in Fig. 2.

The R² statistic of the regression model was 0.9969, and the estimated values of the F-statistic, error variance, and other parameters indicated that the regression model was statistically significant and well explained the linear relationship between D-fructose concentration and absorbance. Since absorbance and D-fructose concentration exhibit a linear relationship, this model provides a theoretical basis for the indirect determination of reducing sugar concentration. The corresponding reducing sugar concentration can be calculated by measuring absorbance data.

The regression equation can be written as: y=2.79c+0.09. Based on the univariate linear regression model derived from the absorbance property of reducing sugars, the corresponding reducing sugar concentration data under different absorbance levels were calculated, as shown in Tab. 3.

Tab. 3 Yields of reducing sugars under Different conditions

4.2 Enzymatic Reaction Rate

To ensure the accuracy of the enzymatic reaction rate, the ideal reaction time should be selected during the linear phase where the reaction rate is proportional to the substrate concentration. In this phase, the consumption of the substrate is minimal, and the product generation is proportional to time. During this period, the enzyme activity remains stable, and the reaction rate reflects the true catalytic efficiency of the enzyme. In previous experiments, it was confirmed that the optimal enzymatic reaction time is 3 minutes. Subsequent verification and analysis of enzyme reaction data at other substrate concentrations proved that 3 minutes can be selected as the optimal enzymatic reaction time for all cases.

Therefore, a dataset representing the relationship between substrate concentration and initial reaction rate can be plotted, based on which further calculations and analyses can be conducted. The data in Table 4 show the relationship between substrate concentration and initial reaction rate calculated in our experiment.

Tab.4 The Relationship Between v₀ and [S]

4.3 Regression and Results

Enzymatic reactions are usually described by the classic Michaelis-Menten Kinetics model. This model is based on the interaction between enzymes and substrates and describes the relationship between reaction rate and substrate concentration, especially the enzymatic reaction rate at different substrate concentrations. Based on the data in the above table, we can calculate 1/S and 1/V₀, as shown in Tab. 5:

Tab. 5The Value of V₀ & 1/[S] for AgaE

This is a typical Michaelis-Menten kinetic analysis based on the given enzymatic reaction data (1/S and 1/V). We can conduct the analysis by plotting a graph. Typically, for Michaelis-Menten kinetics, the model follows the Michaelis-Menten equation:

Where, Vmax represents the maximum reaction rate of the enzyme, and Km is the Michaelis constant, which reflects the substrate concentration when the reaction rate reaches half of the maximum rate. The equation can be linearized by plotting the Lineweaver-Burk double reciprocal plot:

Non-linear regression (such as fitting techniques based on the least squares method) was used to fit the data respectively, and the fitting results are shown in Fig. 3 below:

The results obtained from the linear regression analysis are shown in the table below:

These results conform to the typical Michaelis-Menten kinetic reaction pattern and reflect the kinetic characteristics of the AgaE enzyme in its reaction with agarose.

4.4 Activation-Inhibition Kinetic Model

To quantitatively analyze the biphasic regulatory mechanism of Na⁺ on enzyme activity (i.e., activation at low concentrations and inhibition at high concentrations), this study employed a classic enzyme kinetic regulation model for fitting analysis. The model is based on the following assumption: Na⁺ can act as an activator to bind to one site of the enzyme (with an activation constant Ka) and also act as an inhibitor to bind to another site of the enzyme (with an inhibition constant Ki), and there is a competitive balance between the two processes. Its mathematical expression is as follows:

Where:

• B = Basal catalytic efficiency without Na⁺ (min⁻¹)
• A = Maximum activatable catalytic efficiency range induced by Na⁺ (min⁻¹)
• Kₐ = Activation constant (mM)
• Kᵢ = Inhibition constant (mM)

Using Origin software, non-linear least squares method was used to fit the model equation, with Na⁺ concentration as the independent variable and VmaxKmas the dependent variable, to solve the key parameters in the model.

The values of each key parameter and relevant statistical information obtained through fitting are as follows:

• B = 0.0089 min⁻¹
• A = 0.0098 min⁻¹
• Kₐ = 32.7 mM
• Kᵢ = 58.4 mM

Meanwhile, by taking the derivative of the fitting curve and finding the peak point, the optimal concentration was determined to be approximately:

Ka = 32.7 mM, which reflects the characteristics of the activation effect on the enzyme. When the Na⁺ concentration is lower than Ka, Na⁺ mainly exerts an activation effect, causing (Vmax/Km) to increase with the increase in concentration. This may be because it promotes the binding of the enzyme to the substrate or enhances the function of the enzyme's catalytic active center.

Kᵢ = 58.4 mM, which reflects the characteristics of the inhibitory effect on the enzyme. When the Na⁺ concentration is higher than Kᵢ , the inhibitory effect of Na⁺ dominates, leading to a significant decrease in Vmax/Km with the increase in concentration. It is speculated that high concentrations may destroy the spatial structure of the enzyme or compete with the substrate for the active site of the enzyme.

From the experimental data and fitting curve, it can be seen that the effect of Na⁺ concentration on VmaxKm exhibits a dual effect of "activation at low concentrations and inhibition at high concentrations". The optimal Na⁺ concentration is approximately 43.7 mM. This result provides an important reference for practical applications (such as industrial enzyme-catalyzed reactions and the construction of biochemical experimental systems), that is, in relevant reaction systems, controlling the Na⁺ concentration at around 43.7 mM can maintain the enzyme at a high catalytic efficiency.

4.5 Model Verification

4.5.1 Model Prediction

Based on the following model, we can predict the catalytic efficiency at a Na⁺ concentration of 43.7 mM:

Substituting each parameter, the predicted value of Vmax/Km is calculated as follows:

4.5.2 Experimental Verification

The kinetic parameters at 43.7 mM were determined using the same experimental method for enzymatic reaction kinetics as described above, and the difference between the original OD data of the enzyme hydrolysis product and the control group was obtained, as shown in Tab. 6.

Tab. 6 Changes in absorbance at 43.7 mM Na⁺

Based on the univariate linear regression model derived from the absorbance property of reducing sugars, the corresponding reducing sugar concentration data under different absorbance levels were calculated, as shown in Tab. 7.

Tab.7. Yields of Reducing Sugars under Different conditions

Based on the data in the above table, we can calculate 1/S and 1/V₀, as shown in Tab. 8：

Tab. 8 The Value of 1/V₀ & 1/[S] for AgaE

Non-linear regression (such as fitting techniques based on the least squares method) was used to fit the data respectively, and the fitting results are shown in the following figure:

The results obtained from the linear regression analysis are as follows:

• Maximum reaction rate Vmax = 0.0327 mg/mL/min
• Michaelis constant Km = 2.6725 mg/mL
• Regression correlation coefficient R² = 0.9945, indicating an excellent linear relationship between the data and the fitting line.

After calculation:

4.5.3 Error Calculation

From the above calculation results, it can be seen that the relative error is approximately 7.4%, indicating that the prediction results of the above model are relatively accurate.

5. Discussion

This study confirmed through experimental verification that the mixed activation-inhibition kinetic model can accurately describe the regulatory effect of Na⁺ on enzyme activity, and determined that 43.7 mM is the optimal Na⁺ concentration for the enzyme. Based on the predictive ability of this model, it is recommended to control the Na⁺ concentration within the range of 40-45 mM in experimental research, and adopt 43.7 mM Na⁺ in industrial applications to achieve the maximum catalytic efficiency.

Future work should improve the accuracy by increasing data points around 43.7 mM, further study the interaction between factors such as temperature and pH with Na⁺, and explore the applicability of this model in other similar enzyme systems, so as to provide a theoretical basis for the optimization of enzyme activity conditions.