Modeling
In this part, we used numerical models to simulate the following two problems: (1) the relationship between the drug
resistance of tumor cells and the concentration of TMZ; (2) how the cell activities of Lv-sh1 and Lv-sh2 change with
time after knocking down the mRNA level of PDRG1 in tumor cells (T98G and U118). Tables 1~4 are the corresponding
experimental data.
Here, we used Model (1) to simulate the drug resistance of tumor cells change with TMZ concentration.
Here, we used Model (1) to simulate the drug resistance of tumor cells change with TMZ concentration.
f(x)=(p1∙x+p2 )/(x+q1) (1)
Where p1, p2 and q1 are the parameters need to be determined.
Model (2) was applied to simulate the relationship between cell activity and time.
Model (2) was applied to simulate the relationship between cell activity and time.
g(x)=p'1∙x2+p'2∙x
+p'3 (2)
Where p'1, p'2 and
p'3
are the parameters need
to be fitted.
Table 1. Experimental results of drug resistance of tumor cells (T98G
and
T98G-R)
| TMZ | T98G(%) | T98G-R(%) |
|---|---|---|
| 0 | 100.0000033 | 100 |
| 50 | 90.86329 | 95.398697 |
| 100 | 57.39058667 | 87.50356 |
| 200 | 48.1125 | 77.973933 |
| 400 | 26.16935 | 79.046223 |
| 800 | 10.25597833 | 67.699043 |
Table 2. Experimental results of drug resistance of tumor cells (U118
and
U118-R) change with TMZ concentration..
| TMZ | U118(%) | U118-R(%) |
|---|---|---|
| 0 | 100.00001 | 99.999973 |
| 200 | 83.5312 | 92.48128 |
| 400 | 71.27007333 | 86.48513 |
| 800 | 45.79195667 | 79.899577 |
| 1600 | 16.81491333 | 70.371573 |
| 3200 | 0.770985333 | 58.765933 |
Table 3. After knocking down the mRNA level of PDRG1 in tumor cells
(T98G),
the activity of three kinds of cells (Lv-shNC, Lv-sh1, and Lv-sh2) changed with time.
| Time(h) | Lv-shNC | Lv-sh1 | Lv-sh2 |
|---|---|---|---|
| 0 | 0.48386 | 0.47354 | 0.47144 |
| 24 | 0.80324 | 0.7465 | 0.8041 |
| 48 | 1.21532 | 0.93478 | 1.06982 |
| 72 | 1.880275 | 1.23156 | 1.32092 |
Table 4. After knocking down the mRNA level of PDRG1 in
tumor cells (U118),
the activity of three kinds of cells (Lv-shNC, Lv-sh1, and Lv-sh2) changed with time.
| Time(h) | Lv-shNC | Lv-sh1 | Lv-sh2 |
|---|---|---|---|
| 0 | 0.1616 | 0.1626 | 0.1768 |
| 24 | 0.2638 | 0.2506 | 0.2432 |
| 48 | 0.6214 | 0.4212 | 0.4334 |
| 72 | 1.1634 | 0.6442 | 0.737 |
Coding
clear;clc;
%% exp.data--TMZ T98
Data_1=importdata('TMZ_T98.txt');
TMZ1=Data_1(:,1);
T98=Data_1(:,2);
T98R=Data_1(:,3);
figure,plot(TMZ1,T98,'*',TMZ1,T98R,'o')
hold on
% model simulation
% f(x) = (p1*x + p2) / (x + q1)
p1=[-16.23 60.62];
p2=[2.3e+04 2.216e+04];
q1=[223.7 220.2];
% TMZ
x_1=0:2:1500;
x_1=x_1';
% T98
y_t98= (p1(1)*x_1 + p2(1))./(x_1 + q1(1));
% T98R
y_t98R= (p1(2)*x_1 + p2(2))./(x_1 + q1(2));
plot(x_1,y_t98,'-',x_1,y_t98R,'--')
%
syms x
ans1=solve((-16.23*x+2.3e+04)/(x+223.7)==0,x)
ans2=solve((60.62*x+2.216e+04)/(x+220.2)==0,x)
%% exp.data--TMZ U118
Data_2=importdata('TMZ_U118.txt');
TMZ2=Data_2(:,1);
U118=Data_2(:,2);
U118_R=Data_2(:,3);
figure,plot(TMZ2,U118,'*',TMZ2,U118_R,'o')
hold on
% model simulation
% f(x) = (p1*x + p2) / (x + q1)
p1_u=[-43.94 38.56];
p2_u=[1.326e+05 1.641e+05];
q1_u=[1301 1651];
% TMZ
x_2=0:5:3500;
x_2=x_2';
% T98
y_u118= (p1_u(1)*x_2 + p2_u(1))./(x_2 + q1_u(1));
% T98R
y_u118R= (p1_u(2)*x_2 + p2_u(2))./(x_2 + q1_u(2));
plot(x_2,y_u118,'-',x_2,y_u118R,'--')
%
syms x
ans3=solve((-43.94*x+1.326e+05)/(x+1301)==0,x)
ans4=solve((38.56*x+1.641e+05)/(x+1651)==0,x)
%% exp.data--T98 sh
Data_3=importdata('T98G.txt');
Time1=Data_3(:,1);
shNC=Data_3(:,2);
sh1=Data_3(:,3);
sh2=Data_3(:,4);
figure,plot(Time1,shNC,'*',Time1,sh1,'o',Time1,sh2,'s')
hold on
% model simulation
% f(x) = p1*x^2 + p2*x + p3
p1_T=[0.00015 1.034e-05 -3.54e-05];
p2_T=[0.008373 0.009515 0.01427];
p3_T=[0.4919 0.4832 0.4741];
time1=0:1:80;
time1=time1';
shNC_T=p1_T(1)*time1.^2 + p2_T(1)*time1 + p3_T(1);
sh1_T=p1_T(2)*time1.^2 + p2_T(2)*time1 + p3_T(2);
sh2_T=p1_T(3)*time1.^2 + p2_T(3)*time1 + p3_T(3);
plot(time1,shNC_T,'-',time1,sh1_T,'--',time1,sh2_T,'.-')
%% exp.data--U118 sh
Data_4=importdata('U118.txt');
Time2=Data_4(:,1);
shNC_u=Data_4(:,2);
sh1_u=Data_4(:,3);
sh2_u=Data_4(:,4);
figure,plot(Time2,shNC_u,'*',Time2,sh1_u,'o',Time2,sh2_u,'s')
hold on
% model simulation
% f(x) = p1*x^2 + p2*x + p3
p1_U=[0.0001909 5.859e-05 0.000103];
p2_U=[0.0002687 0.002512 0.0003825];
p3_U=[0.1581 0.1611 0.1763];
time2=0:1:80;
time2=time2';
shNC_U118=p1_U(1)*time2.^2 + p2_U(1)*time2 + p3_U(1);
sh1_U118=p1_U(2)*time2.^2 + p2_U(2)*time2 + p3_U(2);
sh2_U118=p1_U(3)*time2.^2 + p2_U(3)*time2 + p3_U(3);
plot(time2,shNC_U118,'-',time2,sh1_U118,'--',time2,sh2_U118,'.-')
Model Results：
1. T98G and T98G-R change with TMZ.
Figure 1 The spatial structure of BHET protein
T98G model results:
Model:
f(x) = (p1*x + p2) / (x + q1)
Coefficients (with 95% confidence bounds):
p1 = -16.23 (-71.01, 38.54)
p2 = 2.3e+04 (-6041, 5.204e+04)
q1 = 223.7 (-86.67, 534.1)
Goodness of fit:
SSE: 181.2
R-square: 0.9708
Adjusted R-square: 0.9513
RMSE: 7.772
T98G-R model results:
model:
f(x) = (p1*x + p2) / (x + q1)
Coefficients (with 95% confidence bounds):
p1 = 60.62 (35.6, 85.63)
p2 = 2.216e+04 (-1.846e+04, 6.277e+04)
q1 = 220.2 (-197.3, 637.7)
Goodness of fit:
SSE: 38.81
R-square: 0.9463
Adjusted R-square: 0.9105
RMSE: 3.597
2. U118 and U118-R change with TMZ.
Figure 2. Comparison of numerical and experimental results for U118 and U118-R .
U118 model results:
Model:
f(x) = (p1*x + p2) / (x + q1)
Coefficients (with 95% confidence bounds):
p1 = -43.94 (-83.05, -4.833)
p2 = 1.326e+05 (4.781e+04, 2.174e+05)
q1 = 1301 (386.2, 2216)
Goodness of fit:
SSE: 46.65
R-square: 0.9938
Adjusted R-square: 0.9897
RMSE: 3.944
U118-R model results:
Model:
f(x) = (p1*x + p2) / (x + q1)
Coefficients (with 95% confidence bounds):
p1 = 38.56 (26.28, 50.85)
p2 = 1.641e+05 (8.768e+04, 2.405e+05)
q1 = 1651 (854.6, 2447)
Goodness of fit:
SSE: 2.795
R-square: 0.9975
Adjusted R-square: 0.9959
RMSE: 0.9652
3. The change of cell activity with time (T98G knock-down).
Figure 3. Comparison of numerical and experimental results for T98G knock-down.
Lv-shNC model results：
Linear model:
f(x) = p1*x^2 + p2*x + p3
Coefficients (with 95% confidence bounds):
p1 = 0.00015 (-0.0002451, 0.000545)
p2 = 0.008373 (-0.02131, 0.03805)
p3 = 0.4919 (0.0483, 0.9354)
Goodness of fit:
SSE: 0.001283
R-square: 0.9988
Adjusted R-square: 0.9965
RMSE: 0.03582
Lv-sh1 model results：
Linear model:
f(x) = p1*x^2 + p2*x + p3
Coefficients (with 95% confidence bounds):
p1 = 1.034e-05 (-0.0004661, 0.0004868)
p2 = 0.009515 (-0.02628, 0.04531)
p3 = 0.4832 (-0.05177, 1.018)
Goodness of fit:
SSE: 0.001866
R-square: 0.9939
Adjusted R-square: 0.9817
RMSE: 0.0432
Lv-sh2 model results：
Linear model:
f(x) = p1*x^2 + p2*x + p3
Coefficients (with 95% confidence bounds):
p1 = -3.54e-05 (-0.0001644, 9.364e-05)
p2 = 0.01427 (0.00458, 0.02397)
p3 = 0.4741 (0.3292, 0.6189)
Goodness of fit:
SSE: 0.0001369
R-square: 0.9997
Adjusted R-square: 0.999
RMSE: 0.0117
4. The change of cell activity with time (U118 knock-down).
Figure 4. Comparison of numerical and experimental results for U118 knock-down .
Lv-shNC model results：
Linear model:
f(x) = p1*x^2 + p2*x + p3
Coefficients (with 95% confidence bounds):
p1 = 0.0001909 (1.578e-05, 0.000366)
p2 = 0.0002687 (-0.01289, 0.01342)
p3 = 0.1581 (-0.03857, 0.3547)
Goodness of fit:
SSE: 0.000252
R-square: 0.9996
Adjusted R-square: 0.9988
RMSE: 0.01588
Lv-sh1 model results：
Linear model:
f(x) = p1*x^2 + p2*x + p3
Coefficients (with 95% confidence bounds):
p1 = 5.859e-05 (-1.589e-05, 0.0001331)
p2 = 0.002512 (-0.003084, 0.008108)
p3 = 0.1611 (0.07746, 0.2447)
Goodness of fit:
SSE: 4.56e-05
R-square: 0.9997
Adjusted R-square: 0.999
RMSE: 0.006753
Lv-sh2 model results：
Linear model:
f(x) = p1*x^2 + p2*x + p3
Coefficients (with 95% confidence bounds):
p1 = 0.000103 (7.73e-05, 0.0001286)
p2 = 0.0003825 (-0.001545, 0.00231)
p3 = 0.1763 (0.1475, 0.2051)
Goodness of fit:
SSE: 5.408e-06
R-square: 1
Adjusted R-square: 0.9999
RMSE: 0.002326
Conclusion
From the simulation results, our models can accurately simulate the experimental data with high R-square. The
experimental data and simulation results showed that the drug resistance of tumor cells decreased with the increase
of TMZ concentration.
By solving the model (1), we can know that with the increase of TMZ concentration, the tumor cell T98G-R will not be completely killed. No matter how high the concentration of TMZ is, at least 60% of tumor cells T98G-R survive. Similarly, according to the results of model (1), tumor cells U118 will not be completely killed, even if the concentration of TMZ is very high, 35% of tumor cells U118murr still survive.
After knocking down the mRNA level of PDRG1 in tumor cells (T98G and U118), the increasing trend of cell viability of Lv-sh1 and Lv-sh2 was significantly lower than that of Lv-shNC group. It is suggested that knocking down the mRNA level of tumor cell PDRG1 will reduce its resistance to TMZ.
By solving the model (1), we can know that with the increase of TMZ concentration, the tumor cell T98G-R will not be completely killed. No matter how high the concentration of TMZ is, at least 60% of tumor cells T98G-R survive. Similarly, according to the results of model (1), tumor cells U118 will not be completely killed, even if the concentration of TMZ is very high, 35% of tumor cells U118murr still survive.
After knocking down the mRNA level of PDRG1 in tumor cells (T98G and U118), the increasing trend of cell viability of Lv-sh1 and Lv-sh2 was significantly lower than that of Lv-shNC group. It is suggested that knocking down the mRNA level of tumor cell PDRG1 will reduce its resistance to TMZ.