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.

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. Theoretical basis

Character definition

Hill function

Derivation process of discrete forecasting model

3. Model building

The effect of different initial AHL concentration：

Fig. 1 represents the variation of AHL-LuxR corresponding to time at different initial AHL concentrations.And it can be seen that the distance between curves from left to right is getting larger and larger, which indicates that the lower the initial AHL concentration, the higher the accuracy of the measurement data.

The effect of background expression on AHL accumulation：

Fig.2 As shown, when the other parameters are all the same values, we only transform the value of the background expression, but show completely different results. Due to the background expresses the LuxI rather than the AHL, which can produce AHL constantly. So in the case of not adding the AHL, the background expression on AHL accumulation may also have a greater impact on the system after the switch is switched on with IPTG. It also suggests that we are likely to add AiiA hydrolase to inhibit the effect of background expression. To avoid the effect of initial AHL addition, the expression intensity of AiiA hydrolase is also worthy of attention, and it can not express too strongly.

The relationship between time and concentration when the AHL-LuxR is reaching the threshold([LR]=[LR]/2):

Fig.3 is the main basis for the feasibility, which reflects the different initial concentration of AHL corresponding to reach the threshold of time, so as to reflect the initial concentration through time, used to trace detection. According to the forecast curve, it shows that the slope of the curve with the increasing of the concentration gradually decreased. When the initial concentration of AHL molecules is at a low level, the same changes of AHL molecules, and the corresponding time difference vary greatly, which indicates that the positive feedback system is more suitable for the accurate detection of low concentration substances.

The change of fluorescence with time under different initial AHL concentration：

Fig.4 Combining with Fig.1 and Fig.3, it’s easy to find out that in Fig.4, also exist such a relationship. This shows that when the initial AHL at low concentrations, by measuring the changes of the fluorescence intensity in real time, finding out a suitable threshold, the relationship can also be found between the initial AHL and time.When the initial concentration of AHL molecules is at a low level, the same changes of AHL molecules, and the corresponding time difference vary greatly. It is also shown that the lower the initial AHL concentration, the higher the accuracy of the measurement results under low background expression.

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 ^[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;Whenδ₂<δ₃. δ=δ₂orδ₃,the system has two equilibrium solutions.when δ< δ₂ or δ > ₃，.The system has only one equilibrium solution.Whenδ₂ < δ < δ₃, 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;

Team:SiCAU-China/Model

Introduction

Forecast ModelModeling

Model optimization

Stability verification
of the system

Program

Introduction

Forecast Model

Model optimization

Stability verification of the system

Program