# Introduction

We aim to convert the antibiotic signal into an AHL molecule signal by using a specific promoter in combination with LUX. And set up a positive feedback system based on the population induction system of Vibrio califlora. The input AHL molecular signal is amplified by a positive feedback system, then outputs fluorescent signal.The previous detection system is mostly between "0" and"1", only detect the presence of the measured object while cannot measured on the quantitative. The fluorescence signal reaches the threshold time is different in contrast to inputting different concentrations of AHL signal molecular. Based on this we can build a relationship between the threshold time and the input signal like the qPCR, achieving quantitative effect.

We build a forecasting model and optimize it on the base of that principle. Compared with most biological systems, our system has an effect of local expression. So we made a modeling prediction of the impact of background expression and the stability of the system, proving the feasibility of our system.

Compare to the previous AHL-t map, we can see that the time to reach threshold is increased significantly after we consider the growth and extend our time of detection then, which is what we do not expect. So it is more advantages that we initially chose to open the system in a smooth period.

We build a forecasting model and optimize it on the base of that principle. Compared with most biological systems, our system has an effect of local expression. So we made a modeling prediction of the impact of background expression and the stability of the system, proving the feasibility of our system.

# Forecast Model

1. Assumption

1) The prediction model is an experimental analysis which based on the experimental principle and the Hill function by drawing up the relevant parameters.

2) It is assumed that there is less attenuation of the AHL when it is in low concentration.

3) The molecular weight of AHL-LuxR does not vary with time and remains stable.

4) The fixed parameters used in the model are based on the experimental principle and related literature hypothesis, for there may be about the predicted trend of curve and the problems which may arise in the process of experiment.

5) The model does not consider the impact of environmental factors on the change of natural causes.

6) The threshold can be chosen according to the experimental phenomena, and the threshold in ours prediction model is chosen as [LR]/2.

7) The model does not consider the impact of background expression on GFP accumulation.

2) It is assumed that there is less attenuation of the AHL when it is in low concentration.

3) The molecular weight of AHL-LuxR does not vary with time and remains stable.

4) The fixed parameters used in the model are based on the experimental principle and related literature hypothesis, for there may be about the predicted trend of curve and the problems which may arise in the process of experiment.

5) The model does not consider the impact of environmental factors on the change of natural causes.

6) The threshold can be chosen according to the experimental phenomena, and the threshold in ours prediction model is chosen as [LR]/2.

7) The model does not consider the impact of background expression on GFP accumulation.

2. Theoretical basis

- Character definition
- Hill function
- Derivation process of discrete forecasting model

3. Model building

- The effect of different initial AHL concentration：
- The effect of background expression on AHL accumulation：
- The relationship between time and concentration when the AHL-LuxR is reaching the threshold([LR]=[LR]/2):
- The change of fluorescence with time under different initial AHL concentration：

4. Feasibility analysis

Based on the discrete forecasting model, we can see that the positive feedback system is very suitable for the detection of trace. If the background expression accumulation can be controlled at a low level, then the relationship between fluorescence and time will be more obvious at the same group of initial AHL concentration. The minimum detection limit of the system is that the AHL expressed in the background is completely degraded by the AiiA hydrolase.

# Model optimization

1. The impact of background expression

In the search for the relationship between the initial concentration of AHL and the time at which the threshold is reached, we found that when we changed the background expression only, we can see that when the background is expressed as a certain value, the concentration of added AHL can be linearly related to the time at which the threshold is reached. By analyzing the data, we determined the optimal background expression in a highe range, where the concentration of AHL can be linearly related to the reached threshold time, that is, when the background expression of the system is expressed within this range, it is considered that the concentration of added AHL (concentration range (0,1000)) is linearly related to the time , through the linear treatment, we can calculate the actual production of the initial concentration of AHL better. At the same time, it also lay the theoretical basis for elimination of background expression through AiiA hydrolase later and optimization system.Figure 1

2. Effects of Bacterial Growth on Threshold Time

We expect to build a system that can work after Stable Period. But in the experiment we found that when the bacteria grow to a stable period, the number of intracellular LuxR protein is not enough to make the system work. We introduce a growth curve based on the logistic equation to construct a model that takes into account the growth of bacteria in order to analysis the current stage of the experiment, although we will solve this problem in the future.Compare to the previous AHL-t map, we can see that the time to reach threshold is increased significantly after we consider the growth and extend our time of detection then, which is what we do not expect. So it is more advantages that we initially chose to open the system in a smooth period.

Figure 2

Figure 3

According to the following comparison chart, we can see that when the sensor is introduced, the linear range of the concentration of AHL and time reached threshold is greatly reduced, and cause 2 problems: (1) When the concentration of addde AHL is low, the time to reach the threshold will be greatly extended; (2) When the concentration is haigh, the time to reach the threshold will be no significant gap. This is why we get this result: no fluorescence or fluorescence is too strong after we add AHL. But it is certain that the introduction of the sensor can reduce the time to reach the threshold. Thus we add AiiA to offset this part and the background expression produced by the AHL later. # Stability verification of the system

The transfer of metabolic molecules

[2]Wang H O, Abed E H. Bifurcation control of a chaotic system ☆[J]. Automatica, 1995, 31(9):1213-1226.

[3]PEI YU, GUANRONG CHEN. HOPF BIFURCATION CONTROL USING NONLINEAR FEEDBACK WITH POLYNOMIAL FUNCTIONS[J]. International Journal of Bifurcation & Chaos, 2004, 14(05):1683-1704.

[4]Le H N, Hong K S. Hopf bifurcation control via a dynamic state-feedback control[J]. Physics Letters A, 2012, 376(4):442-446.

^{[1]}as fig1.Specified P is AHL molecule, Pn is LuxR-AHL polymer, GR inhibitory promoter, GA activated promoter, M is LuxI.First LuxR and AHL combine to form a complex,which dimerize into a transcrip -tional activator, LuxR-AHL.According to the theory of system biology^{[2-4]},We can get the formula: The above process is described by ordinary differential equations: When the system is balanced: The number of solutions is related to the parameters.n> 1, there are 1-3 solutions;Whenδ_{2}<δ_{ 3}. δ=δ_{2}orδ_{3},the system has two equilibrium solutions.when δ< δ_{2}or δ >_{3}，.The system has only one equilibrium solution.Whenδ_{2}< δ < δ_{3}, the system has three equilibrium solutions.When the eigenvalue satisfies Re <0, the equilibrium point is stable, but only if the Re satisfies <0, the positive feedback system is stable and the system does not need to convert high and low steady state. The parameters in the program are Re <0, so we can think that the expression of our system is stable.References:

[1]Haseltine E L, Arnold F H. Implications of Rewiring Bacterial Quorum Sensing[J]. Applied & Environmental Microbiology, 2008, 74(2):437.[2]Wang H O, Abed E H. Bifurcation control of a chaotic system ☆[J]. Automatica, 1995, 31(9):1213-1226.

[3]PEI YU, GUANRONG CHEN. HOPF BIFURCATION CONTROL USING NONLINEAR FEEDBACK WITH POLYNOMIAL FUNCTIONS[J]. International Journal of Bifurcation & Chaos, 2004, 14(05):1683-1704.

[4]Le H N, Hong K S. Hopf bifurcation control via a dynamic state-feedback control[J]. Physics Letters A, 2012, 376(4):442-446.

# Program

clear

i=1;

yc1=0.0000001; % The rate of LuxI molecules produced by background expression

n=2; % Hill factor

v1=0.001; % The rate at which a AHL-LuxR molecule produces LuxI molecules per unit of time

Kd=100; % Dissociation constant

R=100; % The content of LuxR in a cell

v2=0.001; % LuxI catalyzes the generation of AHL

N=1.73772; % Total number of cells

for L=0.001:0.001:1 % Initial AHL molecule

t=1;

L(1)=L;

I(1)=0; % Initial LuxI value

LR=0;

t=2;

while L<3

if t>7200

% M=I1(t-7200);% Green fluorescent protein (LuxI) attenuation

M=0;

else

M=0;

end

% Y=N/(1+exp((-1.29336)*((0.001*t)-4.35987)));% The cell growth curve corresponds to the number of cells

Y=N;

I(t)=R*v1*(L(t-1).^n/(Kd+L(t-1).^n))+I(t-1)+yc1-M;% The amount of LuxI molecule in unit cells +L(1)/v2

L(t)=I(t)*v2*(Y/N)+L(t-1); % The amount of AHL molecule in unit cells

LR=R*(L(t).^n/(Kd+L(t).^n));% The amount of LuxR-AHL molecule in unit cells

t=t+1;

end

tt(i)=t-1;

i=i+1;

end

figure

plot(tt)

xlabel('Initial AHL concentration')

ylabel('t')

grid on;

i=1;

yc1=0.0000001; % The rate of LuxI molecules produced by background expression

n=2; % Hill factor

v1=0.001; % The rate at which a AHL-LuxR molecule produces LuxI molecules per unit of time

Kd=100; % Dissociation constant

R=100; % The content of LuxR in a cell

v2=0.001; % LuxI catalyzes the generation of AHL

N=1.73772; % Total number of cells

for L=0.001:0.001:1 % Initial AHL molecule

t=1;

L(1)=L;

I(1)=0; % Initial LuxI value

LR=0;

t=2;

while L<3

if t>7200

% M=I1(t-7200);% Green fluorescent protein (LuxI) attenuation

M=0;

else

M=0;

end

% Y=N/(1+exp((-1.29336)*((0.001*t)-4.35987)));% The cell growth curve corresponds to the number of cells

Y=N;

I(t)=R*v1*(L(t-1).^n/(Kd+L(t-1).^n))+I(t-1)+yc1-M;% The amount of LuxI molecule in unit cells +L(1)/v2

L(t)=I(t)*v2*(Y/N)+L(t-1); % The amount of AHL molecule in unit cells

LR=R*(L(t).^n/(Kd+L(t).^n));% The amount of LuxR-AHL molecule in unit cells

t=t+1;

end

tt(i)=t-1;

i=i+1;

end

figure

plot(tt)

xlabel('Initial AHL concentration')

ylabel('t')

grid on;

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