Team:NAU-CHINA/Model/Conditions Optimization Model
Conditions Optimization Model
1 Abstract
2 Brief Introduction
3 Experimental scheme design
4 Model establishment and solving
1 Abstract
In order to optimize the yield of SeNPs by engineered bacteria and provide theoretical guidance for later engineered production, response surface method is used to study the yield of SeNPs under different conditions.
In Response Surface Method, according to Box-Behnken Design(BBD)’s central combination test principle, we take temperature, selenite concentration and pH as test factors, take yield of SeNPs as response value and carry out experimental design of 3 factors and 3 levels. We obtain regression equation and make variance analysis and response surface analysis by establishing the model with Design-Export 8.0.1. Combined with the intuitive contour map and three-dimensional graph, the interactive influence of different factors on the experimental results is obtained.
Finally, the optimal experimental conditions are as follows: the temperature is 30.26 ℃, the pH is 6.75, and the selenite concentration is 14.33mM.
Figure 1.1 Flow chart
2 Brief Introduction
Response Surface Method
Box-Behnken Design (BBD) is a commonly used method in response surface design. For example, we take three factors and design them with three levels of each and code them as (-1, 0, 1). Set 0 as the center point, +1 and -1 as the high and low values corresponding to the cubic point respectively. The distribution of experimental points designed by BBD is shown in the Figure 2.1, and the points on the figure represent the test carried out. There are 17 test groups in total:
Figure 2.1 Distribution test points of BBD
Through quadratic regression fitting, the quadratic equation is obtained, including primary term, quadratic term and interaction term, the main effect and interaction effect are analyzed and the optimal value is finally obtained.
The multivariate quadratic regression equation is described as:
where $Y$ is the response value, $x_{i}$ and $x_{j}$ are the coding variables, $\beta_{0}$ are the intercept, and $\beta_{i}$, $\beta_{j}$, $\beta_{ij}$ are the coefficients of the primary term, quadratic term and interaction term respectively.
3 Experimental scheme design
3.1 Determination of factors
There are many factors affecting the yield of SeNPs, and each factor has many levels to screen.
After reviewing relevant literature and communicating with the wet lab, we finally determine three factors on the response surface: temperature, selenite concentration and pH value, which are the main influencing factors on SeNPs yield.
3.2 Experiment process of producing SeNPs
The effects of temperature, concentration of sodium selenite concentration and pH value on the yield of SeNPs are investigated. The specific lab steps and details in this section can be seen in part Engineering.
3.3 Single factor experiment
3.3.1 Effect of temperature
When the pH value is 7 and the selenite concentration is 12mM, the relationship between the yield of SeNPs and the change of temperature is shown in Figure 3.1.
Figure 3.1 The relationship between the yield of SeNPs and the change of temperature
As can be seen from the Figure 3.1, the yield of SeNPs increases with the temperature increasing. But when temperature reaches 30℃, if we keep increasing the temperature, the yield of SeNPs begins to decrease, which may be due to the reason that within a certain range, the yield of SeNPs obtained by reduction is proportional to the concentration of selenite, and the more raw materials, the more production can be get, but if selenite concentration exceeds a certain range, it will be toxic to E.coli, inhibit its normal physiological activity and growth and even cause death. Thus 30 ℃ is more suitable.
3.3.2 Effect of selenite concentration
When the pH value is 7 and the temperature is 30 ℃, the relationship between the yield of SeNPs and the change of selenite concentration is shown in Figure 3.2.
Figure 3.2 The relationship between the yield of SeNPs and the change of selenite concentration
As can be seen from the Figure 3.2, the yield of SeNPs increases with selenite concentration increasing. But when selenite concentration reaches 12mM, if we keep increasing selenite concentration, the yield of SeNPs begins to decrease, which may be due to the reason that within a certain range, the yield of SeNPs obtained by reduction is proportional to the concentration of selenite, and the more raw materials, the more production can be get, but if selenite concentration exceeds a certain range, it will be toxic to E.coli, inhibit its normal physiological activity, growth and even cause death. Thus 12mM is more suitable.
3.3.3 Effect of pH
When the temperature is 30 ℃ and the selenite concentration is 12mM, the relationship between the yield of SeNPs and the change of pH value is shown in Figure 3.3.
Figure 3.3 The relationship between the yield of SeNPs and the change of pH value
As can be seen from the Figure 3.3, the yield of SeNPs increases with the increase of pH. But when pH reaches 8, if we keep increasing pH, the yield of SeNPs begins to decrease, which may be due to the reason that necessary enzymes for bacterial metabolism need to be kept at a certain range of pH to work effectively, if the pH is too low or too high, the protein conformation of the enzyme may change, resulting in decreased enzyme activity, or even inactivation. Thus pH 7-8 is more suitable.
Finally, combined with the actual biological situation and the results of previous experiments, we determine that the three levels of temperature, concentration of selenite and pH are (16 ℃, 30 ℃, 37 ℃), (12 mM, 15mM, 18mM) and (6, 7, 8).
4 Model establishment and solving
4.1 Establishment of model
Based on single factor experiment, yield of SeNPs is taken as the response value. We take three factors, temperature (A 16℃,30℃,37℃), selenite concentration (B 12mM,15mM,18mM) and pH (C 6,7,8) as the experimental factors, which significantly affected the yield of SeNPs. The factors and levels of Box-Behnken Design are shown in Table 1, and the test results are shown in Table 2.
| Factors | Level | ||
|---|---|---|---|
| -1 | 0 | 1 | |
| $A$.Temperature/℃ | 16 | 30 | 37 |
| $B$.Selenite concentration | 12 | 15 | 18 |
| $C$.pH | 6 | 7 | 8 |
| Number | Actual levels of variables | Code levels of variables | The yield of SeNPs by engineered bacteria | ||||
|---|---|---|---|---|---|---|---|
| 1 | 30 | 15 | 7 | 0 | 0 | 0 | 0.031 |
| 2 | 30 | 15 | 7 | 0 | 0 | 0 | 0.0288 |
| 3 | 30 | 15 | 7 | 0 | 0 | 0 | 0.0293 |
| 4 | 30 | 15 | 7 | 0 | 0 | 0 | 0.03 |
| 5 | 16 | 15 | 6 | -1 | 0 | -1 | 0.0001 |
| 6 | 37 | 15 | 6 | 1 | 0 | -1 | 0.0204 |
| 7 | 16 | 15 | 8 | -1 | 0 | 1 | 0.0001 |
| 8 | 37 | 15 | 8 | 1 | 0 | 1 | 0.0001 |
| 9 | 16 | 12 | 7 | -1 | -1 | 0 | 0.0001 |
| 10 | 30 | 12 | 8 | 0 | -1 | 1 | 0.0202 |
| 11 | 30 | 12 | 6 | 0 | -1 | -1 | 0.0222 |
| 12 | 30 | 12 | 7 | 0 | -1 | 0 | 0.0268 |
| 13 | 30 | 12 | 8 | 0 | -1 | 1 | 0.0235 |
| 14 | 37 | 18 | 7 | 1 | 1 | 0 | 0.0229 |
| 15 | 30 | 18 | 6 | 0 | 1 | -1 | 0.0216 |
| 16 | 16 | 18 | 7 | -1 | 1 | 0 | 0.0001 |
| 17 | 30 | 18 | 8 | 0 | 1 | 1 | 0.0001 |
Fitting the results in Table 2 with Design expert V8.0.1, we can get regression equation as follows:
ySeNPs=0.028+0.00827A-0.00266B-0.0042C-0.00093AB-0.00478AC-0.00582BC-0.013A2-0.0035B2-0.0093C24.2 Description of regression equation
4.2.1 Normal plot of Residuals
Figure 4.1 Normal plot of Residuals
It can be seen from Figure 4.1 that the points are uniformly distributed near the straight line, so we think that the distribution of residuals accords with the normal distribution, and the equation has high credibility.
4.2.2 Residual vs. Predict
Figure 4.2 Plot of Residual vs. Predict
It can be seen from Figure 4.2, there is no regularity between the residual distribution of the predicted value and the residual value. In line with the correct distribution law of the residual of the predicted value and residual value, it can be seen that the regression equation of the model has high credibility.
4.2.3 Predict vs.Actual
Figure 4.3 Plot of Predict vs.Actual
It can be seen from the Figure 4.3 that the predicted value and the actual value are evenly distributed in a straight line. And by comparing the correct distribution law between the predicted value and the actual value, it can be seen that the regression equation of the model has high credibility.
4.2.4 Residual vs. Factors
Figure 4.4 Plot of Residual vs. Temperature
Figure 4.5 Plot of Residual vs. Selenite concentration
Figure 4.6 Plot of Residual vs. pH
Through the analysis of Figure 4.4, Figure 4.5 and Figure 4.6, we can see that the absolute values of our residuals are all less than the maximum threshold.
To sum up, the regression equation calculated by the model can precisely reflect our experiment, and the regression equation obtained by the model can accurately obtain the best experimental conditions.
4.3 Results analysis of Response Surface Method
| Source | SS | df | Mean Square | F Value | Prob > F | |
|---|---|---|---|---|---|---|
| Model | 0.002441 | 9 | 0.000271 | 11.66635 | 0.0019 | significant |
| $A$ | 0.000417 | 1 | 0.000417 | 17.92302 | 0.0039 | |
| $B$ | 0.0000565 | 1 | 0.0000565 | 2.430858 | 0.1629 | |
| $C$ | 0.000143 | 1 | 0.000143 | 6.149162 | 0.0422 | |
| $AB$ | 0.00000268 | 1 | 0.00000268 | 0.115278 | 0.7442 | |
| $AC$ | 0.0000097 | 1 | 0.000097 | 4.171979 | 0.0804 | |
| $BC$ | 0.000158 | 1 | 0.000158 | 6.812719 | 0.0349 | |
| $A^2$ | 0.000416 | 1 | 0.000416 | 17.87148 | 0.0039 | |
| $B^2$ | 0.0000476 | 1 | 0.0000476 | 2.048208 | 0.1955 | |
| $C^2$ | 0.00034 | 1 | 0.00034 | 14.63052 | 0.0065 | |
| Residual | 0.000163 | 7 | 0.0000233 | |||
| Lack of Fit | 0.000155 | 3 | 0.0000515 | 25.22224 | 0.0046 | significant |
| Pure Error | 0.0000082 | 4 | 0.00000204 | |||
| Cor Total | 0.002604 | 16 |
Furthermore, the error statistical analysis of the fitting regression equation is carried out, and the precision, multivariate correlation coefficient, credibility and accuracy are calculated by Design-Expert software, which are shown in table 4.
| Std. Dev. | 0.004822 | R-Squared | 0.937498 |
|---|---|---|---|
| Mean | 0.016312 | Adj R-Squared | 0.857139 |
| C.V. % | 29.56214 | Pred R-Squared | -0.14278 |
| PRESS | 0.002976 | Adeq Precision | 8.896265 |
From this table, $R_{adj}^2-R_{pred}^2<0.2$, these two values are high and close, indicating that our regression equation can fully explain the technological process. Adeq Precision is the ratio of effective signal to noise, and is considered reasonable (more than 4).
The response surface method also overcomes the defect that the orthogonal test can not give intuitive graphs. According to the quadratic equation model, the three-dimensional response surface and contour map of the interaction between the experimental factors are made respectively to investigate the effect of the interaction of the other two factors on the yield of SeNPs when a factor is fixed at the center value.
The response surface is a three-dimensional surface diagram formed by response values on various factors. The greater the influence of factors on response values it has, the steeper the surface. The shape of contour line can reflect the significance of interaction between two factors. Ellipse represents significant interaction between two factors, while circle represents insignificant interaction between two factors.
As for the response surface curves of the interaction of temperature, selenite concentration, and pH on the yield of SeNPs, please look at Figure 4.7, Figure 4.8, Figure 4.9.
Figure 4.7 Interactive effect between temperature and selenite concentration on the yield of SeNPs
Figure 4.8 Interactive effect between temperature and pH on the yield of SeNPs
Figure 4.9 Interactive effect between selenite concentration and pH on the yield of SeNPs
From the contour lines in Figure 4.7, Figure 4.8, Figure 4.9, we can directly see that the interaction between selenite concentration and pH is significant, while the interaction between selenite concentration and temperature, pH and temperature is not significant.
Through the model analysis, the optimum production conditions are as follows: the temperature is 30.26 ℃, the pH is 6.75, and the selenite concentration is 14.33mM. Under this condition,the predicted yield of SeNPs is 0.0315562g. When it is applied in large-scale production, we'll see striking improvement.
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