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

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.

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.

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

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:

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 parameterrepresenting 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 ofandrepresent different preferential binding dynamics of LuxI and AiiA to degrading enzyme.

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

refers to mRNA whileis 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.

DDEs

Due to the time interval between expression of regulating protein and transcriptional activation by the luxR and AHL complex which depends on the past concentration of internal AHL, we decide to use DDEs model. Transcription, translation, and maturation rate of proteins are combined into a single time-delay parameter. We modeled based on the paper “Rapid and tunable post-translational coupling of genetic circuits”.

The simulation is carried out using custom written software in MATLAB. We get the simulation result of oscillator, as shown in Figure 1.

Figure 1. Simulation result of quorum-sensing oscillator.

Dual-feedback oscillator

ODEs

In the genetic circuit of Dual-feedback oscillator, the three promoters are the same hybrid promoter composed of an activation site and a repression site. They can be activated by the araC protein in the presence of arabinose and repressed by the LacI protein in the absence of IPTG. The mechanism of dual-feedback makes oscillatory behaviour stable and adjustable.

There are four states of promoter: original promoter without inducer, bound with araC dimers (), bound with lacI tetramers (), bound with araC and lacI tetramers together.stands for the proportion of the four states of promoters.

when reached the quasi steady state, we can get

Because of, we get

We assume that araC dimers and lacI tetramers combine with promoter separately without interference. So it means that

AraC dimers and lacI tetramers are generated through their monomeric versions.

When reached the quasi steady state, we can get

We define that，，and plug them into D~D3, then we have

In addition, from the consequence of Jesse Stricker(2008), we know that the binding rates of the activator dimer and repressor tetramer to the operator sites are dependent upon the concentrations of arabinose and IPTG.

are the maximum and minimum affinities of AraC dimers to the binding sites, [ara] is the concentration of arabinose (in % w/v), [IPTG] is the IPTG concentration (in mM),,,,.

are the maximum and minimum affinities of LacI tetramers to the binding site,,.

We put the relationships betweenandinto the above equations.

The transcription, translation and degradation processes are listed by the following reactions:

Based on the above analysis,We could construct a set of ODEs to describe the dual-feedback oscillator:

stand for the number of plasmid copies. The degradation rate of protein,，,X is the total number of tags in the system.

DDEs

In consideration of the delay of transcription, translation and maturation, DDEs may be more capable. Based on the project of HUST-China 2013, we simulate the genetic circuit by DDE sets

Due to the delay, we denoteas the time interval parameters of araC dimers and lacI tetramers. And attaching delay to three parameters:

Then we code in MATLAB, and the simulation result is shown in Figure 2.

Figure 2. Simulation result of dual-feedback oscillator.

Parameters

Parameters play an important role in simulating, which would determine the behaviors of the system. In our modeling part, parameters in quorum-sensing e-oscillator are from parameter part of Original Full Model by Tsinghua-A iGEM 2011 and the paper “Rapid and tunable post-translational coupling of genetic circuits” by Arthur Prindle et al. And parameters used in dual-feedback e-oscillator are based on the paper “A fast, robust and tunable synthetic gene oscillator” and parameter table in HUST-China iGEM 2013. Although these previous work give us some experience, we still adjust some parameters more reasonable in terms of our system and some experimental tests.

Reference

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