Modeling
In this part, the relationship among time, tryptophan concentration and the fluorescent protein GFP is simulated by numerical fitting method. Below is our experimental data, reflecting the effect of time and tryptophan concentration on the GFP value of fluorescent protein.
C(mmol/L)Time (h) 0.75 1 1.5 2 2.5 3
1 8441.333 -84008 72548 27250.33 1559 38315.67
2 300625 264969.7 289920.3 236277.7 242421.3 239773.7
3 229463 202330 252103.7 206195 255064 263058.3
4 455967 440415.7 496753 511516 540326.7 563790.3
5 660409 670000.7 729505.3 716617.3 785243 816712.7
6 969054.7 1043204 1110555 1110348 1196770 1233489
7 1246483 1293688 1356562 1338142 1363047 1400867
According to the experimental data, we choose to adopt the two-dimensional interpolation method to establish the fitting model.
It should be noted that one data point in the table is an outlier (the red number), so we use the following model to find its surrogate value.
f(x)=p1·x2+p2·x+p3(1)
Where p1, p2 and p3 are the parameters need to be determined.
Matlab Code:
    
      clear;clc;
      p1=3.761e+04;
      p2=-1.129e+05;
      p3=2.891e+05;
      x=1;
      y_renew=p1*x^2+p2*x+p3;
      %
      t0=1:7;
      c0=[0.75 1 1.5 2 2.5 3];
      data=[8441.333333   y_renew 72548   27250.33333 1559    38315.66667;...
          300625  264969.6667 289920.3333 236277.6667 242421.3333 239773.6667;...
          229463  202330  252103.6667 206195  255064  263058.3333;...
          455967  440415.6667 496753  511516  540326.6667 563790.3333;...
          660409  670000.6667 729505.3333 716617.3333 785243  816712.6667;...
          969054.6667 1043204 1110555 1110348.333 1196770.333 1233489.333;...
          1246482.667 1293688 1356562 1338141.667 1363047.333 1400867];
      %
      t=0:0.1:7; 
      c=0:0.05:3;
      CZ=interp2(c0,t0,data,t,c','spline'); 
      surf(t,c,CZ);
      CZmax=max(max(CZ))
      [ii,jj]=find(CZ==max(CZmax));
      t1=t(jj)
      c1=c(ii)
Model Results：
1. Use the Model (1) to find the surrogate value for the outlier in the table. As shown in Figure 1 we replace the outlier (-84008) with 213810.
Figure 1. When the tryptophan concentration is 1 mmol/L, the relationship between GFP value and time is fitted by Model (1).
Conclusion
With the increase of xylan concentration, the model showed a trend of increasing first and then stabilizing. We can use this model to predict the maximum absorption rate (Vmax) and Michaelis constant (Km) of the enzyme solution CCxynA, thereby predicting the reaction rate and maximum activity of the enzyme.
Coefficients (with 95% confidence bounds):
p1 = 3.761e+04 (-3251, 7.847e+04)
p2 = -1.129e+05 (-4.854e+05, 2.597e+05)
p3 = 2.891e+05 (-4.751e+05, 1.053e+06)
Goodness of fit:
SSE: 1.846e+10
R-square: 0.9808
Adjusted R-square: 0.968
RMSE: 7.845e+04
2. A three-dimensional surface model is established to evaluate the relationship among time, tryptophan concentration and GFP value.
Figure 2. The changing trend of fluorescent protein GFP value with time and tryptophan concentration.
Conclusion
According to the data, under the same tryptophan concentration, the fluorescent protein GFP value increased with time. The simulation results of this study can be used to predict the tryptophan concentration through the reaction time and GFP signal, and then judge whether the patient has autism.