Modeling
In this part, we used numerical models to simulate the influence of temperature and pH on enzymes (including W1-lipase and SP-lipase) activity. Then, the optimal temperature and pH corresponding to the peak activity of the enzymes were predicted according to the numerical results. Table 1 presents the experimental data of the effect of temperature on the activities of W1-lipase and SP-lipase. Table 2 shows the experimental results of the influence of pH on the activities of W1-lipase and SP-lipase.
Table 1. Activities of W1-lipase and SP-lipase under different temperature.
Temperature (ºC) W1-lipase SP-lipase
25 17.94339623 19.45283019
30 24.07295597 25.89308176
35 32.96226415 33.69811321
40 36.75471698 27.83018868
45 29.27232704 24.31320755
50 19.80188679 20.63836478
55 15.53584906 19.20754717
60 12.5408805 15.45283019
65 6.628930818 8.001257862
70 7.305031447 6.139622642
Table 2. Activities of W1-lipase and SP-lipase under different pH.
PH W1-lipase SP-lipase
3 19.23018868 17.49056604
5 23.94103774 22.34528302
7 29.80943396 26.62641509
9 37.95125786 34.60377358
11 22.13899371 28.93207547
12 15.00990566 21.37735849
Here, we used Model (1) to simulate the effect of temperature and pH on the activities of W1-lipase and SP-lipase.
f(x) = (p1·x2 + p2·x + p3)/(x2 + q1·x + q2) (1)
Where p1，p2，p3，q1 and q2 are the parameters need to be determined. Model (2) was applied to simulate the relationship between pH and W1-lipase activities.
g(x) = (p'1·x2 + p'2·x + p'3)/(x3 + q'1·x2 + q'2x + q'3) (2)
Where p'1，p'2，p'3，q'1，q'2 and q'3 are the parameters need to be determined.
Model (3) was applied to simulate the relationship between pH and SP-lipase activities.
h(x) = p*1·x4 + p*2·x3 + p*3·x2 + p*4·x + p*5 (3)
Where p*1，p*2，p*3，p*4 and p*5 are the parameters need to be fitted.
Coding
    
      clear;clc;
      % exp. data temperature
      Data_T=importdata('data_t.txt');
      tem=Data_T(:,1); 
      W1=Data_T(:,2); 
      SP=Data_T(:,3);
      % exp. data pH
      Data_pH=importdata('data_ph.txt');
      pH=Data_pH(:,1); 
      W1_pH=Data_pH(:,2); 
      SP_pH=Data_pH(:,3);
      % model simulation
      p1=[3.549 -44.49];
      p2=[-296.7 4442];
      p3=[1.116e+04 -6.977e+04];
      q1=[-78.57 -27.61];
      q2=[1680 770.9];
      % temperature
      x_t=25:0.1:70;
      % W1-lipase
      y_t1= (p1(1)*x_t.^2 + p2(1)*x_t + p3(1))./(x_t.^2 + q1(1)*x_t + q2(1));
      % SP-lipase
      y_t2= (p1(2)*x_t.^2 + p2(2)*x_t + p3(2))./(x_t.^2 + q1(2)*x_t + q2(2));
      figure,plot(tem,W1,'*')
      hold on, plot(x_t,y_t1)
      figure,plot(tem,SP,'o')
      hold on, plot(x_t,y_t2)
      %% pH
      x_ph=3:0.1:12;
      % W1-lipase
      p1_w=1.983e+04;
      p2_w=-5.256e+05;
      p3_w=3.874e+06;
      q1_w=2360;
      q2_w=-4.683e+04;
      q3_w=2.496e+05;
      y_p3=(p1_w*x_ph.^2 + p2_w*x_ph + p3_w)./(x_ph.^3 + q1_w*x_ph.^2 + q2_w*x_ph + q3_w);
      % SP-lipase
      p1_s=-0.01441;
      p2_s=0.2927;
      p3_s=-1.933;
      p4_s=7.124;
      p5_s=6.89;
      y_p4=p1_s*x_ph.^4 + p2_s*x_ph.^3 + p3_s*x_ph.^2 + p4_s*x_ph + p5_s;
      figure,plot(pH,W1_pH,'s')
      hold on, plot(x_ph,y_p3)
      figure,plot(pH,SP_pH,'+')
      hold on, plot(x_ph,y_p4)
Model Results：
1. Temperature versus W1-lipase activity.
Figure 1. Comparison of numerical and experimental results for the influence of
temperature on W1-lipase activity.
General model:
f(x) = (p1*x^2 + p2*x + p3) / (x^2 + q1*x + q2)
Coefficients (with 95% confidence bounds):
p1 = 3.549 (-2.742, 9.84)
p2 = -296.7 (-903.2, 309.8)
p3 = 1.116e+04 (-1275, 2.36e + 04)
q1 = -78.57 (-84.96, -72.18)
q2 = 1680 (1453, 1906)
Goodness of fit:
SSE: 9.239
R-square: 0.9905
Adjusted R-square: 0.9829
RMSE: 1.359
2. Temperature versus SP-lipase activity.
Figure 2. Comparison of numerical and experimental results for the influence of
temperature on SP-lipase activity.
General model:
f(x) = (p1*x^2 + p2*x + p3) / (x^2 + q1*x + q2)
Coefficients (with 95% confidence bounds):
p1 = -44.49 (-276.9, 188)
p2 = 4442 (-1.837e + 04, 2.725e + 04)
p3 = -6.977e+04 (-4.647e + 05, 3.252e + 05)
q1 = -27.61 (-264.3, 209)
q2 = 770.9 (-3747, 5288)
Goodness of fit:
SSE: 29.58
R-square: 0.9552
Adjusted R-square: 0.9194
RMSE: 2.432
3. pH versus W1-lipase activity.
Figure 3. Comparison of numerical and experimental results for the effect of pH on
W1-lipase activity.
General model:
f(x) = (p1*x^2 + p2*x + p3) / (x^3 + q1*x^2 + q2*x + q3)
Coefficients:
p1 = 1.983e + 04
p2 = -5.256e + 05
p3 = 3.874e + 06
q1 = 2360
q2 = -4.683e + 04
q3 = 2.496e + 05
Goodness of fit:
SSE: 2.848
R-square: 0.9914
Adjusted R-square: NaN
RMSE: NaN
4. pH versus SP-lipase activity.
Figure 4. Comparison of numerical and experimental results for the effect of pH on
SP-lipase activity.
Linear model:
f(x) = p1*x^4 + p2*x^3 + p3*x^2 + p4*x + p5
Coefficients (with 95% confidence bounds):
p1 = -0.01441 (-0.3712, 0.3424)
p2 = 0.2927 (-10.44, 11.03)
p3 = -1.933 (-116, 112.1)
p4 = 7.124 (-492.7, 506.9)
p5 = 6.89 (-740.9, 754.7)
Goodness of fit:
SSE: 7.153
R-square: 0.9617
Adjusted R-square: 0.8083
RMSE: 2.675
Conclusion
From the simulation results, our models can accurately simulate the experimental data (R-square>0.95). Based on the above numerical results, we can predict the optimal temperature and pH for the activities of W1-lipase and SP-lipase. As shown in Figure 1~ Figure 4, the optimal temperature for W1-lipase and SP-lipase activities were 39.0 ºC and 35.3 ºC, respectively. The best pHs for W1-lipase and SP-lipase activities were 8.9 and 9.6, respectively.