Modeling
In this part, we use numerical models to simulate the relationship between the fermentation time of recombinant yeast in synthetic medium L2FL and the yield of 2'-FL, as well as the relationship between the fermentation time in sweet potato residue medium L2FL and the yield of 2'-FL. Then, according to the simulation results, we can be predicted the ability of fermented sweet potato residue to produce 2'-FL, and the high-value utilization of sweet potato residue. Table 1 shows the experimental data of fermentation time and 2'-FL yield of recombinant yeast in synthetic medium L2FL. Table 2 represents the experimental data of fermentation time and 2'-FL production of recombinant yeast in sweet potato residue medium L2FL.
Table 1. The experimental data of fermentation time and 2'-FL yield in synthetic medium L2FL
Fermentation Time (h) 2'-FL（g/L）
0 0
4 0
8 0
12 0
16 0.09
20 0.185
24 0.26
28 0.325
32 0.485
36 0.505
48 0.74
Table 2. The experimental data of fermentation time and 2'-FL production in sweet potato residue medium L2FL
Fermentation Time (h) 2'-FL（g/L）
0 0
8 0
16 0.045
24 0.2
32 0.405
48 0.615
Models apply common models of biological growth
Logistic Equation（1）:
dt = RN(K-N)/K(1)
Where N is the total number of individuals, t is the time, R is the population growth potential index, and K is the maximum environmental capacity.
The analytical solution (2):
N = 1+ae-Rt(2)
Note: the dependent variable (N) represents the 2'-FL yield in this model.
Coding
    
      After verification, the two sets of experimental data are in line with the characteristics of the model, showing an s-shaped growth trend:
      clear;clc;
      t1=[0 4 8 12 16 20 24 28 32 36 48];
      t2=[0 8 16 24 32 48];
      L2FL_1=[0 0 0 0 0.09 0.185 0.26 0.325 0.485 0.505 0.74];
      L2FL_2=[0 0 0.045 0.2 0.405 0.615];
      Glucokinase=@(a,t)a(1)./(1+a(2)*exp(a(3)*t));%Glucokinase
      a0=[0.1 0.1 -1];
      b0=[0.1 0.1 -1];%
      a=nlinfit(t1,L2FL_1,Glucokinase,a0)%
      b=nlinfit(t2,L2FL_2,Glucokinase,b0)%
      t=0:0.1:50;  
      Glucose_activity_1=Glucokinase(a,t);%
      Glucose_activity_2=Glucokinase(b,t);%
      plot(t1,L2FL_1,'b*',t,Glucose_activity_1,'r','linewidth',2');
      hold on
      plot(t2,L2FL_2,'k+',t,Glucose_activity_2,'g','linewidth',2');
      hold off
Model Results：
1. Model diagram of time and 2 ' -FL in synthetic medium L2FL
Figure 1. The model diagram of time and 2 ' -FL in synthetic medium L2FL
General model:
y = 1+be-ct
Coefficients (with 98% confidence bounds):
a=0.7714
b=78.0795
c=0.1472
Note: when the anonymous function is defined in the program, the parameter c is not negative, so the program calculates the negative value.
2.Model diagram of time and 2' -FL in sweet potato residue medium L2FL
Figure 2. The model diagram of time and 2 ' -FL in sweet potato residue medium L2FL
General model:
y = 1+be-ct
Coefficients (with 98% confidence bounds):
a=0.6294
b=207.0859
c=0.1866
Note: when the anonymous function is defined in the program, the parameter c is not negative, so the program calculates the negative value.
Conclusion
The increase first and then stabilized yield of 2' -FL of synthetic medium L2FL was attributed to an increase in the fermentation time. The yield of 2' -FL in sweet potato residue medium L2FL increased first, and then stabilized with the increase of fermentation time. The model can predict the ability of fermented sweet potato residue to produce 2'-FL, which is of great significance for the high-value utilization of sweet potato residue and the production of 2'-FL by the biological method.