e-oscillators

Two genetic oscillators, dual-feedback oscillator and quorum-sensing oscillator, are built in our wetlab, so we first analyze the important biological processes and construct ODE sets to simulate these two oscillators separately. However, due to the time interval between the expression of regulating protein and their binding to promoters, we improve our models more precise by DDEs. Once the better genetic oscillator is selected through experiments, we will adopt the corresponding one as the virtual part.

Quorum-sensing oscillator

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ODEs

As for quorum-sensing oscillator, luxI proteins could generate AHL from some substrates consisting of acyl-ACPs and Sadenosylmethionine (SAM). We assume that the amount of substrates is sufficient.

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AHL(Acyl homoserine lactones) is a kind of auto-inducer, which could combine with luxR protein and activate the promoter. In addition, the lactonic ring of the AHL will be hydrolyzed once in presence of AiiA protein.

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For transcription activation, we use Hill function to describe the rate of production. is the maximal transcription rate. When the promoter is bound by the transcription factor, gene will be transcribed at the maximal transcription rate. So the Hill function is

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In addition, many genes have a non-zero minimal expression level, namely basal expression level. It can be described by adding a term.

The translation and degradation processes are governed by the following set of reactions:

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Y refers to luxI and AiiA. is the translation rate. The degradation of mRNA is a much swifter process compared to protein, so we use parameter representing the degradation rate. LuxI and AiiA protein added with LVA-tag can be degraded by the same enzymatic reaction, so they can be modeled to follow Michaelis-Menten enzyme kinetics. Different values of and represent different preferential binding dynamics of LuxI and AiiA to degrading enzyme.

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According to the above processes，we can describe the genetic circuit by the following equations:

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refers to mRNA while is protein. The internal and external AHL are described by AHL and Ae. LuxR is constitutively produced at a constant level and the protein is degraded at at a proportion , since it is not tagged for fast degradation.

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