Oscillation

Overview

Oscillation can be found in the nature easily. It is a phenomenon describing objects changing periodically with time. Seasons change, sunrise and sunset, heartbeat, biological clock, they are all a form of oscillation. And now, we want to apply oscillation to E.coli, making it express different genes periodically, in order to realize the purpose of changing the code book. So, how can we achieve this goal?

Solution

Through the preliminary experiment, we can see that the bacteria concentration of fluorescent protein expression of AHL and the strength of bacteria is positively correlated. Therefore, we through modeling of the external AHL concentration, the results can be used to characterize the fluorescence intensity.

After being catalyzed, AHL, which is generated by bimolecular components of luxR and luxl will combine together, and this compound will promote the expression of pluxR. pluxR controls the expression of aiiA, encoding the enzyme which catalyze the degradation of AHL. And all of these form an AHL single-cycled oscillation.

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 function

describes 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.

In the equation（1）（2）,P(α，τ) represents the Hill function

describes 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.

Results

By consulting relevant literatures and measuring in our experiment, we get most of the coefficients in the differential equation above and solve the equations in Matlab. The followed picture shows the result solved by Matlab.
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Zhejiang University, YuHangTang Road NO.866

Hangzhou, China

**iGEM ZJU-China 2016 Team**

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