Therefore,basing on the biological reaction process above, we derived the following set of delay-differential equation model for intracellular concentrations of LuxI (I),AiiA (A), internal AHL (Hi), and external AHL (He)In the equation（1）（2）,P(α，τ) represents the Hill functiondescribes the delayed production of corresponding proteins, it depends on the past concentration of the internal AHL, and τ is the time delay. These delayed reactions mimic the complex cascades of processes (transcription, translation, maturation, etc.) leading to formation of functional proteins. The pre-factor [1 − (d/d0)4] describes slowing down of protein synthesis at high cell density d due to lower nutrient supply and high waste concentration. In the equation (3), AHL synthetase is encoded by luxI and aiiA with LAA degradation tag is controlled by pluxR promoter and aiiA encodes the enzyme catalyzing AHL to degradation. We use the Michaelis-Menten kinetics to describe these two processes. Also, the internal AHL will diffuse into the extracellular space because of the difference of the concentration of AHL inside and outside with the diffusion coefficient D0. The equation (4) describes the external AHL concentration outside the cells. The factor d/(1 − d) comes from the total mass conservation of AHL inside and outside the cells. The micro fluid in the main channel of the microfluidic chip will take the external AHL off, which we have built in the penetration model. The last term in equation for He describes the diffusion of external AHL. In our model, the trap is considered a unit, so it is meaningless for us to calculate this term and we set D1=0.
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