Modeling

In this part, we combined experimental data and numerical model results to study the variation of fluorescence signal GFP with time and arsenite concentration. Based on the results, the variation characteristics of GFP were analyzed to provide reference for the inversion of arsenite concentration. We did four sets of experiments, and the average GFP values are shown in the following tables.

Table 1.The average GFP values of ArSA, ArSD and ArSR

ArSA-GFP	0h	1h	2h	3h
0(μg/L)	2172756	2156646	2303444	2200693.8
20(μg/L)	2168052	2082483	2498665	2345364.8
50(μg/L)	2137249	2101361	2520527	2388312.8
100(μg/L)	2130052	2151532	2490965	2364019
150(μg/L)	2092658	2088392	2437316	2325883.5

ArSD-GFP	0h	1h	2h	3h
0(μg/L)	2114005	2021859	2322535	2178671
20(μg/L)	2159168	2031231	2444486	2288187
50(μg/L)	2171890	2048819	2485226	2361719
100(μg/L)	2102311	2011604	2482411	2380690
150(μg/L)	2154590	2078689	2338642	2328193

ArSR-GFP	0h	1h	2h	3h
0(μg/L)	2007906	1917922	2208283	2135826
20(μg/L)	2035052	2004930	2381238	2319252
50(μg/L)	2050514	1928481	2331246	2273673
100(μg/L)	1984137	1955064	2335800	2329591
150(μg/L)	2054256	1926563	2349759	2318759

According to the experimental data, the two-dimensional interpolation method was used to establish the fitting model.

The Matlab code is shown below:

clear;clc
time=[0 1 2 3];
con=[0 20 50 100 150]; %
%
ArSA1=[2172756.25   2156646.25  2303444 2200693.75;...
2168052.25  2082482.75  2498664.5   2345364.75;...
2137248.75  2101361.25  2520527 2388312.75;...
2130052.25  2151532.25  2490964.75  2364019;...
2092658 2088392.25  2437316.25  2325883.5];
%
ArSD2=[2114004.75   2021858.5   2322534.75  2178671.25;...
2159167.75  2031231 2444486.25  2288186.75;...
2171889.5   2048819 2485226 2361718.5;...
2102310.5   2011604.25  2482411 2380690.25;...
2154589.5   2078688.5   2338642 2328193];
%
ArSR3=[2007906  1917921.5   2208283.25  2135825.75;...
2035052.25  2004929.5   2381238.25  2319251.5;...
2050513.75  1928480.75  2331246 2273673;...
1984137.25  1955064 2335799.5   2329591;...
2054255.75  1926562.5   2349758.5   2318759.25];
%
t=0:0.1:3;  %
C=0:5:150;    %
z1=interp2(time,con,ArSA1,t,C','spline');%
z2=interp2(time,con,ArSD2,t,C','spline');%
z3=interp2(time,con,ArSR3,t,C','spline');%
figure,surf(t,C,z1),title('ArSA-GFP'),xlabel('time (h)'),ylabel('C (ug/L)'),zlabel('GFP')
figure,surf(t,C,z2),title('ArSD-GFP'),xlabel('time (h)'),ylabel('C (ug/L)'),zlabel('GFP')
figure,surf(t,C,z3),title('ArSR-GFP'),xlabel('time (h)'),ylabel('C (ug/L)'),zlabel('GFP')
%
z1max=max(max(z1))
z1min=min(min(z1))
z2max=max(max(z2))
z2min=min(min(z2))
z3max=max(max(z3))
z3min=min(min(z3))
[i1,j1]=find(z1==max(max(z1)));
t1_max=t(j1)
C1_max=C(i1)
[ii1,jj1]=find(z1==min(min(z1)));
t1_min=t(jj1)
C1_min=C(ii1)
[i2,j2]=find(z2==max(max(z2)));
t2_max=t(j2)
C2_max=C(i2)
[ii2,jj2]=find(z2==min(min(z2)));
t2_min=t(jj2)
C2_min=C(ii2)
[i3,j3]=find(z3==max(max(z3)));
t3_max=t(j3)
C3_max=C(i3)
[ii3,jj3]=find(z3==min(min(z3)));
t3_min=t(jj3)
C3_min=C(ii3)

Model Results：

The interpolation curved surfaces of ArSA-GFP, ArSD-GFP, and ArSR-GFP are obtained as shown in Figure 1.

Figure 1. Fluorescence signal GFP against arsenite concentration and time.

According to the model results, we can obtain the maximum and minimum values of GFP within the range of experimental conditions (0<t<3h and 0<c<150ug/L):

For ArSA, the maximum value of GFP is 2.6060e+06, when t=2.4h and c=35ug/L; the minimum value of GFP is 1.9814e+06, when t=0.5h and c=30ug/L.

For ArSD, the maximum value of GFP is 2.5811e+06, when t=2.4h and c=85ug/L; the minimum value of GFP is 1.9134e+06, when t=0.5h and c=110ug/L.

For ArSR, the maximum value of GFP is 2.4677e+06, when t=2.5h and c=25ug/L; the minimum value of GFP is 1.8446e+06, when t=0.6h and c=65ug/L.

Analysis:

By analyzing the above model results, we found that when the arsenite concentration was fixed, the change of ArSA-GFP and ArSD-GFP signal with time was consistent with the Fourier function (Figure 2 shows the fitting results of ArSA-GFP). The Fourier function is expressed as follows:

here we set n=1, and the Fourier function can be simplified as:

Where w represents frequency.

Figure 2. ArSA-GFP signal changes with time under different arsenite concentrations (c=0, 20, 50, 100, and 150ug/L).

We found that the frequency of ArSA- and ArSD-GFP signal decreased gradually with the increase of arsenic concentration (as shown in the following figures and tables).

Table 2. The frequency of ArSA-GFP signal

C (ug/L)	0	20	50	100	150
w (h-1)	1.905	1.808	1.732	1.69	1.701

Figure 3. The model of ArSA-GFP signal frequency

Table 3. The frequency of ArSD-GFP signal

C (ug/L)	0	20	50	100	150
w (h-1)	1.899	1.858	1.805	1.736	1.701

Figure 4. The model of ArSD-GFP signal frequency

Conclusion

The model can simulate the GFP fluorescence signals of ArSA, ArSD and ArSR at different arsenite concentrations and time. Analyzing the model results, we found that the arsenite concentration is correlated with the frequency of GFP fluorescence signal. For ArSA-GFP, the relationship is w=-0.0012C+1.847, and for ArSD-GFP, the relationship is w=-0.0013C+1.8846. Therefore, we can deduce the arsenite concentration according to the above relationship. It should be noted that the above relationships are obtained from our four sets of experimental data. The practical application still needs to be verified by a large number of experiments.