Model
1. Background of project
Epidermal growth factor (EGF), also known as human oligopeptide 1, is an active substance in the body that can promote the growth of epidermal cells and accelerate wound healing. However, the water-soluble nature of EGF makes it easy for it to leak from the tissue fluid into blood vessels after contact with a wound. This may lead to some unknown physical diseases on the one hand, and on the other hand, it may cause material waste and frequent dressing changes. Considering that spider silk proteins are highly adhesive, we planned to express EGF-R formed by fusion of spider silk proteins and human epidermal growth factor EGF and test its stability.
Therefore, we designed experiments in which proteinase-K degraded EGF-R at certain concentrations and obtained data on the concentration of free amino acids that could reflect the degradation effect over time. In addition, we compared the tolerance of EGF-R with two different proteins, skimmed milk powder and BSA(bovine serum protein), under the same experimental conditions.
Specifically, we analyzed the principle of biodegradation to establish a mathematical model for simulating the degradation process of EGF-R, and performed theoretical derivation and numerical analysis to solve the problem. In addition, we speculated the basic laws through the actual data obtained from the experiments, and combined the theoretical analysis and experimental data to build a model to explain the law of tolerance of EGF- R to proteinase-K.
2. Experiments preparation
2.1 Experimental principle
Deducing (quantifying) the enzymatic/hydrolytic processes of different proteins by measuring the changes in the concentration of free amino acids produced during enzymatic/hydrolytic processes (over time) is the basic principle of the experiments in this project. However, direct measurement of amino acid concentration in solution is costly and difficult to operate, so we used the OPA method to detect the degradation of proteins. OPA, known as o-phthalaldehyde, is a white needle-like crystalline material, which is commonly used as a reagent for the determination of amino acid amines and alkaloids. The principle of the OPA method is that the aldehyde group of o-phthalaldehyde can combine with free amine groups of amino acids to form a yellow complex, which can be measured by a spectrophotometer at 340 nm and then analyzed by a spectrophotometer at 340 nm. The absorbance can be measured at 340 nm with a spectrophotometer. The linear relationship between absorbance and amino acid concentration is then used to solve for the concentration data of different proteins degraded to amino acids at different times, and a reaction model (i.e., the trend of amino acid concentration over time) is established based on the solved amino acid concentration data.
2.2 Experimental design
In this project, we mainly test how EGF-R is tolerated under the degradation of proteinase K, i.e., what is the trend of amino acid concentration over time under the degradation effect, and compare the tolerance of EGF-R with that of BSA and skimmed milk powder, and finally build a mathematical model to deduce the degradation law of EGF-R. According to our experimental principle, we 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 (EFG-R, 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 EFG-R.
3. Experimental procedure and preliminary results
3.1 Standard curve experiment of OPA method
3.1.1 Preparation of standard solution
0.32795 g of L-leucine was accurately weighed in a 1000 mL volumetric flask with an electronic balance and fixed, then 0, 0.2, 0.4, 0.6, 0.8, 1.0 mL of this solution were taken to six 25 mL colorimetric tubes,0.4, 0.6, 0.8, 1.0 mL of the solution to six 25 mL colorimetric tubes to 25 mL, and according to the concentration of the respective numbers0, 1, 2, 3, 4 and 5 respectively.
3.1.2 Configuration of OPA solution
Take 80 mg of OPA and dissolve it with 2 mL of anhydrous ethanol in the dark place away from light, then take 1.9068 g of sodium tetra borate, 0.1 g of SDS (sodium dodecyl sulfate), 88 mg DTT (dithiothreitol) were dissolved with water. Then all transferred to 100mL brown volumetric flask with water.
3.1.3 Determination of absorbance of the standard solution
OPA method to measure the absorbance of the specific method is as follows: first, the wavelength of the enzyme standard instrument is adjusted to 340nm, 5Ml of water, 20µl of water to be transferred to the test solution, 150 µl of OPA. The method of OPA method is as follows: firstly, set the wavelength of the enzyme counter to 340nm, transfer 5Ml water, 20µl of the solution to be measured and 150 µl of OPA solution into 96-well plate, and at the same time, use the stopwatch to start timing. After 2 min, put into the enzyme calibrator to determine the reading and 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, the results are displayed in Table 1.
Table 1 Data table of L-leucine concentration and absorbance
Further, we plotted the standard curve of the OPA method as shown in Fig. 1, i.e., the corresponding absorbance 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 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 in our later experiments.
3.2 Experiments to determine the absorbance
3.2.1 Preparation of different protein solutions
Weigh 1g of BSA (bovine serum protein) and skimmed milk powder, dissolve them with water, transfer them to a 100 ml bottle and add water to form a solution of 1% bovine serum protein.1% BSA solution and 1% skimmed milk powder solution, and then as the mother solution diluted by appropriate times, respectively, diluted to a similar concentration with the egg to be measured. The concentration of white was diluted to approximate that of the egg white to be measured.
3.2.2 Determination of absorbance
The configured BSA、skimmed milk powder and EGF-R were added into 96-well plates, and the wavelength was adjusted to 340 nm, and the absorbance was measured at this time as a control.
Take 150 of OPA solution and add it three times into three wells of the 96-well plate, add 5 µl of 2 mg/ml proteinase K and 20 to two kinds of solutions (control: equal volume of PBS) and quickly put them into a micro plate reader for reading. The power curve parameter was set to a total time of 30 min, the first 10 min, every 1 min readings, and the last 20 min, every 5 min readings into the Enzyme Labeling Instrument for reading.
3.2.3 Preliminary data on absorbance
We have detailed in Table 2 the raw OD data of EGF- R, skimmed milk powder, and BSA under degradation by trypsin and 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.
Table 2 Absorbance data table of each time and place
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 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 Figure 2.
Fig.2 The absorbance of each experiment varies with time.
The scatterplot in Figure 1 depicts the trend of absorbance over time for three different proteins (containing EGF-R, which is our focus) in the presence of proteinase K, respectively. We observed the following results:
(1) In the presence of proteinase K, the absorbance of BSA was consistently higher than that of EGF- R and skimmed milk powder, indicating that in this case, BSA was degraded to contain more leucine.
(2) The trend of absorbance of EGF- R and skimmed milk powder was close to each other, i.e., there was not much difference in the change of amino acid concentration between them after being degraded.
(3) 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 EGF- R, 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 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 EGF-R 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 and amino acid concentration. Accordingly, we established a linear model as shown below:
y represents the absorbance 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 is b, b>=0. 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.
equation is satisfied:
Therefore, the values of a and b can be calculated.
denotes the sample mean. Accordingly, we regressed to obtain the univariate regression results for the variation of absorbance with amino acid concentration as shown in Figure 3.
Fig.3 One-way linear regression of absorbance and amino acid concentration
The regression equation can be written as: y=0.4424c+0.1343. 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 Table 3.
Table 3 Absorbance and amino acid concentration
4.3 Modeling of amino acid concentrations over time
Here we will study the degradation pattern of EGF-R 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, implying that the rate equation for the change in amino acid concentration over time is an inverse function of the amino acid concentration and time, and which we initially notate in the following form:
Integrating both sides to obtain 
, 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,
Similarly, we can obtain a theoretical model of amino acid concentration over time:
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 EGF- R with skim milk powder and BSA, and the results are displayed in Figures 4 to 6.
Fig.4 Changes in amino acid concentration with time (EGF-R)
Fig.5 Changes in amino acid concentration with time (skimmed milk powder)
Fig.6 Changes in amino acid concentration with time (BSA)
Figures 4, 5 and 6 reflect the trend of amino acid concentration of different proteins over time during enzymatic digestion, respectively. In the previous section, we explored the initial concentration (expressed as absorbance) of the three, and found that the highest initial concentration of amino acids was produced by the enzymatic digestion of BSA, while the EGF- R, which we were concerned about, was much lower than that of BSA, and at the same time, the confidence intervals of the EGF- R showed that the decomposition of EGFR into amino acids was more stable under the action of proteinase K, which was close to the results we wanted to verify.
4.4 Discussion
In this project, we mainly borrowed proteinase k to mimic the environment outside the human body and observe the tolerance of EGF-R, which we expect to be broken down at a low and stable rate in a low and stable interval. For a simple comparison, we also observed the degradation process of two other different proteins in our experiments. For this purpose, we performed two stages of experiments with different models, one for the linear relationship between absorbance and amino acid concentration, and the other for the trend of the amino acid concentration over time (here we take into account the fact that the newly produced amino acids will also be affected by the concentration of amino acids in the solution itself, among other factors). The modeling results show that the stability of EGF- R is much higher than that of BSA, but closer to that of skim milk powder.
It should be noted that this project only considered the degradation pattern and tolerance of EGF- R under the action of proteinase K. In the real-life application, we have only considered the degradation pattern and tolerance of EGF- R under the action of proteinase K. In real daily life applications, our product (considering the EGF-R application scenario) will be in a more complex environment, and its tolerance should be further explored. In addition, our experimental data collection frequency is not high enough, whether it is the data corresponding to absorbance and amino acid concentration, or the collection time of free amino acids, we will continue to follow up to obtain more representative and complete experimental data.
Finally, the model of this project has a certain range of applicability. One is that it can help assess the stability of different biological materials in different environments. Secondly, it can provide quantitative performance indicators for us to improve the performance of arachnid protein biomaterials.
Reference
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