Modeling

1. Background of project

Agarose is a commonly used biocompatible polysaccharide extracted from seaweed. It is abundant in nature, inexpensive, and can self-assemble into a fibrous network in aqueous solutions without the need for crosslinking agents, making it an ideal material for forming hydrogels[1]. Additionally, seaweed polysaccharides possess numerous beneficial biological effects, and hydrogels made from these polysaccharides have found various applications in drug delivery systems[2,3]. For instance, agarose nanoparticles have been developed as injectable carriers for protein and peptide delivery[4]. Based on this, using agarose as a raw material to create hydrogels for skin wound applications is a promising option[5].

In our contribution experiment, we developed an agarose-based hydrogel incorporating AgaA, and experimental results demonstrated that this hydrogel exhibits enhanced antioxidant properties compared to traditional sodium alginate hydrogels. This improvement can help eliminate excess ROS generated in skin wounds, thereby promoting wound healing. However, as a protein-derived material, in addition to biocompatibility and functionality, the stability of this hydrogel is a critical consideration.

Therefore, we designed an experiment where AgaA was degraded by proteinase-K at a certain concentration, allowing us to obtain data on the concentration of free amino acids over time, reflecting the degradation effects. Additionally, under the same experimental conditions, we compared the resistance of AgaA to degradation with that of two other proteins, skim milk powder, and bovine serum albumin (BSA).

Specifically, we analyzed the principles of biodegradation, established a mathematical model to simulate the degradation process of AgaA, and performed theoretical derivations and numerical analyses to address the problem. Furthermore, we inferred fundamental patterns based on the experimental data we obtained and combined the theoretical analysis with experimental data to construct a model that explains the resistance of AgaA to proteinase-K.

2. Experiment preparation

2.1 Experiment principle

In our experiments, we added AgaA to the hydrogel, and then used proteinase-K to simulate the human environment to degrade AgaA into amino acids, and then judged the degree of degradation/hydrolysis of AgaA according to the content of amino acids. As direct measurement of amino acid concentration in solution is difficult and costly, the OPA method was chosen for the determination of amino acid content. OPA, also known as o-phenylenedialdehyde, is a white, needle-like crystalline substance that is commonly used as a reagent for the determination of amino acid amines and bases.

2.2 Experimental design

In this mainly test the stability of AgaA under the degradation of proteinase-K ,which simulate environment of human compare the trend of amino acid concentration over time under the degradation effect between AgaA, BSA and skimmed milk powder. And finally build a mathematical model to deduce the degradation law of AgaA. According to our experimental designed a two-stage experiment.

Stage 1: Standard curve drawing by OPA method. This stage of the experiment mainly obtains different amino acid concentrations and the corresponding absorbance data, to obtain the linear relationship between amino acid concentration and absorbance, to lay the foundation for the later experiment to calculate the size of the protein enzymatically dissolved into amino acid concentration by measuring absorbance. Here the amino acid concentration is by configuring the standard solution, so it is easy to obtain different concentrations, and then use the OPA method to measure the corresponding absorbance under different amino acid concentrations.

Stage 2: Test the data of different proteins in the degradation process and the change of their absorbance with time. We set up three groups of different proteins (AgaA, skimmed cow's milk powder, and BSA) to show the trend of their absorbance changes over time when they are broken down into amino acids under the action of proteinase-K, respectively. The relationship between absorbance and amino acid concentration was used to solve the reaction equation of amino acid concentration over time, which was used to deduce the enzymatic decomposition pattern of the biological protein AgaA.

3. Experimental produce

3.1 Standard curve experiment of OPA method

3.1.1 Preparation of standard solution

Accurately weigh 0.0131 g of L-leucine in a centrifuge tube with an electronic balance to 10 mL (10 mM) and then dilute to 0, 0.4, 0.6, 0.8 and 1 mM in turn.

3.1.2 Configuration of OPA solution

Take 80 mg of O-phthalaldehyde (OPA) with 2 mL of anhydrous ethanol in the dark place away from light to dissolve, and then take 1.9068 g of sodium tetraborate, 0.1 g of SDS (sodium dodecyl sulphate) with water to completely dissolve, and then add 88 mg of DTT (dithiothreitol), after dissolution, all transferred to a brown volumetric flask with water to 100 mL.

3.1.3 Determination of absorbance of standard solution

The specific method of OPA method is as follows: firstly, adjust the wavelength of the enzyme counter to 340 nm, and then suck 5 µl of water, 20 µl of different concentrations of L-Leucine and 150 µl of OPA solution into 96-well plate, and at the same time, use a stopwatch to start the timing, and then, after 2 min, put it into the enzyme counter to determine the readings, and then plot the standard curve.

3.1.4 Preliminary results

According to the experimental steps, we obtained preliminary data on the concentration and absorbance of leucine in solution after the enzymatic reaction started, we obtained three different sets of data and calculated the mean values, and calculated the difference between the absorbance value of leucine of different concentrations and the control group (Background value) according to the mean values, the results are displayed in Tab. 1.

Tab. 1: Data table of L-leucine concentration and absorbance

Concentration (mM)	0.000	0.400	0.600	0.800	1.000	Background value
OD value of the first set of data	0.250	0.339	0.326	0.414	0.463	0.210
OD value of the second set of data	0.227	0.284	0.357	0.352	0.452	0.212
OD value of the third set of data	0.234	0.329	0.374	0.450	0.448	0.208
The mean of the three sets of OD value	0.237	0.317	0.352	0.405	0.454	0.210
OD difference	0.027	0.107	0.142	0.195	0.244	-

Further, we plotted the standard curve of the OPA method as shown in Fig. 1, i.e., the absorbance difference between the control group and the control group at different amino acid concentrations.

Fig.1: Standard curve of OPA method

The scatter plot in Fig. 1 shows that the concentration of leucine and absorbance changes show an extremely strong linear relationship, which implies that it is feasible and reasonable to solve the concentration of amino acids during the enzymatic reaction by observing the absorbance changes in our later experiments.

3.2 Experiment to determine the absorbance

3.2.1 Preparation of different protein solutions

1g bovine serum protein (BSA) and skimmed milk powder were respectively weighed and dissolved with 20 mM Tris-HCl (pH 7.0, 3 mM Ca2+) buffer, then transferred to a 100 ml bottle with a fixed volume of buffer to prepare 1% bovine serum protein solution and 1% defatted milk powder solution, and then diluted as the appropriate ratio of mother liquor. Dilute to approximate the protein concentration to be measured (0.05 mg/ml).

3.2.2 Determination of absorbance

Reaction groups were arranged according to the following system, with three parallel samples in each group:

1. 400 µL BSA+80 µL proteinase-K
2. 400 µL Milk+80 µL proteinase-K
3. 400 µL AgaA+80 µL proteinase-K

In the control group, an equal amount of buffer was added to each group to replace proteinase-K.

Incubation was carried out at room temperature. 25 µL reaction solution was taken out successively at 0, 5, 10, 15, 20, 25, 30, 45 and 60 min, and added into 96-well plates respectively. Then 150µl OPA solution was added into each sample hole, and the solution was placed into the enzymoleter for 2 min. Read the absorption value at wavelength 340 nm.

3.2.3 Preliminary data on absorbance

We have detailed in Table 2 the difference between the raw OD data of AgaA, skimmed milk powder, and BSA under degradation by proteinase-K, respectively, over time. Our data were recorded in a standardized manner with a moderate sample size that truly reflected the absorbance information at each time and location. As the experiment progressed, the data showed good continuity and representativeness, with no obvious outliers or outliers, and no need for data cleaning and correction.

Tab. 2: absorbance changes data table at each time and place

Time(min)	proteinase-K（AgaA）	proteinase-K（Milk）	proteinase-K（BSA）
0	0.051	0.039	0.025
5	0.070	0.071	0.035
10	0.100	0.127	0.056
15	0.129	0.172	0.079
20	0.153	0.189	0.127
25	0.193	0.200	0.168
30	0.215	0.214	0.185
45	0.210	0.224	0.184
60	0.219	0.227	0.188

Note: It should be noted that to avoid the appearance of excessive errors, each set of observations was obtained by taking the mean of three observations.

To observe the relationship more intuitively between amino acid concentration and absorbance changes in solution after the enzymatic reaction, and for the modeling of the relationship between the two later. We drew a scatter plot between leucine concentration and absorbance, as shown in Fig. 2.

Fig.2: The absorbance of each experiment varies with time.

The scatterplot in Fig. 1 depicts the trend of absorbance over time for three different proteins (containing AgaA, which is our focus) in the presence of proteinase-K, respectively. We observed the following results:

(1) In the presence of protease K, the absorbance changes of BSA and Milk is always higher than that of AgaA, indicating that in this case, BSA and milk are degraded to contain more amino acids.

(2) The absorbance values of the three different proteins showed a stable trend after a certain period after the enzymatic reaction, i.e., the presence of convergence provided data support for the mathematical modeling later.

4. Modeling and solving

In the previous phase, we obtained some raw data, such as the absorbance data under different standard leucine concentrations, and the absorbance data of AgaA, BSA and skimmed milk powder under the action of proteinase-K, and their product amino acids were measured in the OPA method. Here we need to solve two problems, one is to establish a linear equation between absorbance changes and amino acid concentration, through which we can solve the amino acid concentration data in the second phase of the experiment; the second is to establish a model of the amino acid concentration data over time based on the absorbance solved, and to study the degradation pattern of AgaA in the action of proteinase-K.

4.1 Model assumption

Before our model is constructed, it is also necessary to make assumptions about our model based on some basic principles of enzymatic reactions.

(1) The basic structure of proteins is amino acids, which are small polar molecules, so they have a certain solubility in water, which is greater in dilute acids or dilute bases. Therefore, proteins are slowly degraded to amino acids in solution because the reaction process is so slow that it can be considered as a constant rate. Therefore, we propose hypothesis 1.

Hypothesis 1: In the absence of enzymes involved in biological reactions, the rate of natural degradation of proteins to amino acids is slow and steady.

(2) According to the principle of chemical reactions, the higher the concentration of the product, the more the reaction equilibrium will move in the opposite direction. Especially when the amount of enzyme is no longer increased, the reaction inhibition effect is more obvious. Accordingly, we propose hypothesis 2.

Hypothesis 2: The growth rate of free amino acid concentration in the experiment is inversely proportional to the existing concentration of amino acids, i.e., the higher the concentration of amino acids, the slower the rate of formation of amino acids by the continuous degradation reaction.

(3) The hydrolysis of proteins is very slow and negligible compared to the rate of protease reaction. The half-life of proteins such as actin and myosin has been reported to be more than one month. Therefore, we propose hypothesis 3.

Hypothesis 3: During enzymatic degradation, the autohydrolysis reaction of proteins had a negligible effect on the experiment.

(4) The authenticity and accuracy of the data is not only the key to the experimental results, but also the basic prerequisite for solving the model parameters. The experimental principle of this project is clear, the steps are reasonably designed, the preparation of the operation process is standardized, the original records are complete, and the data obtained from several experiments can to a certain extent overcome the contingency of biological experimental results and reduce the occurrence of experimental errors. Accordingly, we propose hypothesis 4.

Hypothesis 4: The data collected in the experiment are accurate and can truly and objectively reflect the experimental results.

4.2 Linear modeling of absorbance and amino acid concentration

Both Table 1 and Figure 1 show the standard curve of the OPA method, which shows a clear linear relationship between absorbance changes and amino acid concentration. Accordingly, we established a linear model as shown below:

$$y = ac + b + \varepsilon$$

y represents the absorbance changes as a function of amino acid concentration and time and can be written as y=y(c,t); c represents the amino acid concentration, which varies with time under the enzymatic reaction, i.e. c=c(t); a(a>0)represents the ratio coefficient between the growth rate of amino acid concentration and amino acid concentration is -a. b is the natural rate of protein degradation to amino acids, b>=0. ɛ denotes the random disturbance term, which is assumed to conform to a normal distribution.

We use least squares estimation to regress equation (1). Least squares are the search for a that minimizes the sum of squares of the residuals, which geometrically means finding the best-fitting regression line that minimizes the sum of squares of the distances from the observations to that regression line. That is, the following.

$$\left\{ \begin{aligned} &\sum_{i=1}^N (y_i - ac_i - b) &= 0 \\ &\sum_{i=1}^N c_i(y_i - ac_i - b) &= 0 \end{aligned} \right.$$

equation is satisfied:

Therefore, the values of a and b can be calculated.

$$\left\{ \begin{aligned} a &= \sum_{i=1}^N \frac{(c_i - \overline{c})(y_i - \overline{y})}{(c_i - \overline{c})^2}\\ b &=(\overline{y} - a\overline{y}) \end{aligned} \right.$$

$\overline{c}$ and $\overline{y}$ denotes the sample mean.

This is known as the least squares estimate. Linear regression function $y(t)= 0.2159c(t)+0.0224$, denoted as

$$y = 0.2159c + 0.0224$$

The R' statistic value of the regression model was 0.9945, and the F statistic, error variance and other estimated values showed that the regression model was statistically significant, which well explained the linear relationship between amino acid concentration and absorbance changes. Because of the linear relationship between absorbance and amino acid concentration, this model provides a theoretical basis for indirect measurement of amino acid concentration. The corresponding amino acid concentration can be calculated by measuring absorbance data:

$$c = \frac{y-0.0224}{0.2159}$$

Fig.3: The standard curve of L-leucine and absorbance

The regression equation can be written as: $y= 0.2159c+0.0224$. Based on the univariate linear regression model obtained by the OPA method, we can solve the corresponding amino acid concentration data at different absorbance levels, as shown in Tab. 3.

Tab. 3 Absorbance and amino acid concentration

Time (min)	AgaA		Milk		BSA
	OD difference	concentration	OD difference	concentration	OD difference	concentration
0	0.0513	0.1340	0.0393	0.0784	0.0250	0.0120
5	0.0697	0.2189	0.0710	0.2251	0.0347	0.0568
10	0.1000	0.3594	0.1267	0.4829	0.0560	0.1556
15	0.1287	0.4922	0.1717	0.6914	0.0787	0.2606
20	0.1530	0.6049	0.1890	0.7717	0.1267	0.4829
25	0.1927	0.7886	0.2000	0.8226	0.1677	0.6728
30	0.2147	0.8905	0.2137	0.8859	0.1847	0.7516
45	0.2103	0.8705	0.2240	0.9338	0.1840	0.7485
60	0.2187	0.9091	0.2270	0.9477	0.1877	0.7655

4.3 Modeling of amino acid concentrations over time

Here we will study the degradation pattern of AgaA in the presence of proteinase-K. Therefore, modeling is needed to solve the trend of amino acid concentration over time.

4.3.1 Theoretical derivation

In Section 4.1, we made some basic assumptions about the model, such as the increasing concentration of amino acids in the reactants with time as mentioned in Assumption 2, which is accompanied by a decreasing rate of new degradation and formation of amino acids, and which we initially notate in the following form:

$$\frac{dc}{dt} = -ac (a>0)$$

Integrating both sides to obtain $c(t)=ke^{-at}$, where the k is a constant.

The exponential model described above reveals that the amino acid concentration decreases with time and tends infinitely to 0. This is not consistent with the principle of protein enzymatic degradation. Recalling the model assumptions of 4.1, we add the process of protein autohydrolysis and assume that the growth rate of protein degradation into amino acids is constant under natural conditions and that the growth rate of amino acid concentration is inversely proportional to the concentration. Under experimental conditions, the rate of change of amino acid concentration is an affine function of the current amino acid concentration. Therefore, there is,

$$\frac{dc}{dt} = -ac + b$$

Similarly, we can obtain a theoretical model of amino acid concentration over time:

$$c(t) = -\frac{b}{a} - ke^{-at}$$

k is the undetermined coefficient, which is usually determined by the initial conditions, we use the nonlinear least square method to fit the solution.

4.3.2 Nonlinear least squares method

Next, the experimental data are used to fit the coefficients a and b in the model expression. The basic method of solution is still least squares estimation. We take a numerical analysis approach to its solution.

The programming steps are as follows(The specific programming program commands we put after the references):

(1) Data preparation: the absorbance data at each time was converted into amino acid concentration data.

(2) Set the fitted function: confirm the expression of the exponential function with parameters to be fitted.

(3) Nonlinear fitting function call.

(4) Validity checks of calculation results.

4.3.3 Results analysis

After performing some simulations, we drew fitted plots of amino acid concentration over time with 90% confidence intervals and compared the trends of AgaA with Milk and BSA, and the results are displayed in Figures 4 to 6.

$$c(t) = \frac{b}{a} - ke^{-at}$$

Fig.4: Changes in amino acid concentration with time (AgaA)

Fig.5: Changes in amino acid concentration with time (Milk)

Fig.6 Changes in amino acid concentration with time (BSA)

Fig. 4, 5 and 6 reflect the trend of amino acid concentration of different proteins over time during enzymatic digestion, respectively.

Milk is rapidly hydrolyzed under the action of protease K, and the final amino acid release concentration is the highest. AgaA protein has the slowest hydrolysis rate, relatively high stability and low amino acid release concentration during the reaction, which may indicate its strong anti-hydrolysis ability to protease K.

These results showed that Milk protein was the most easily hydrolyzed under the action of protease K, followed by BSA, while AgaA protein showed stronger resistance to hydrolysis and relatively high stability.

4.4 Discussion

In this project, we utilized proteinase-K to simulate an in vitro environment to assess the tolerance of AgaA. We expected AgaA to degrade at a low and stable rate and conducted a comparative experiment with two other proteins. To achieve this, we designed a two-stage experimental model: the first stage analyzed the linear relationship between absorbance and amino acid concentration; the second stage examined the trend of amino acid concentration over time (considering that the newly produced amino acids would be influenced by the pre-existing amino acid concentration in the solution and other factors). The modeling results showed that the stability of AgaA was significantly higher than that of BSA and Milk.

It is worth noting that this project only focused on the degradation pattern and tolerance of AgaA under the action of proteinase-K. In practical applications, AgaA will be exposed to more complex environments, and its tolerance needs further exploration. Additionally, the frequency of data collection during the experiment was not sufficient both for the absorbance and amino acid concentration data as well as the time intervals for sampling free amino acids. We plan to improve data collection to obtain more representative and comprehensive experimental results.

Finally, the model used in this project has certain applicable scopes. Firstly, it can help evaluate the stability of different biological materials in various environments. Secondly, it provides quantified performance indicators for improving the application of AgaA in agarose hydrogels.

References

[1] Kumar, A., & Han, S. S. (2021). Agarose-based biomaterials for tissue engineering applications: A review. Journal of Polymer Research, 28(6), 1-15.

[2] Li, L., et al. (2020). Seaweed polysaccharides for soft tissue regeneration. International Journal of Biological Macromolecules, 150, 456-465.

[3] Wang, S., et al. (2019). Advances in agarose-based biomaterials for drug delivery. Journal of Controlled Release, 313, 1-10.

[4] Gupta, R. K., & Anjum, R. (2020). Development of agarose nanoparticles for drug and protein delivery. Materials Science & Engineering C, 112, 110901.

[5] Wu, Z., et al. (2021). Agarose hydrogels for skin wound healing: Recent advances and future prospects. Carbohydrate Polymers, 267, 118168.