Modeling::Simplified DDE Model

Contents 1 Simplification 2 Parameters 3 Results 3.1 General results 3.2 Stability analysis 3.3 Proportion of cell volume 3.4 Period adjustment 3.5 Phase adjustment

Simplification

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Although ODEs provide a thorough, precise description of the whole system, they contain too many equations and parameters which would act as a barrier for simulation and further analysis. A simplification of complicated ODEs is necessary. We simplify every single ODE according to certain appropriate assumptions. Finally, we came up with a set of DDE equations.

Assumptions we have made to deduct the original equations involve:

●Relatively faster reactions such as transcription reactions and binding reactions will reach to Quasi-equilibrium.

●Basal expression of protein is so meager that they can be ignored in modeling.

●Two series of Hill Kinetics equations can be estimated through a single Hill Kinetics equation.

●Protein which is translated from mRNA is proportional to corresponding mRNA in a previous time.

Consider gene lasR and luxR which are all constant genes, since AHL has a relatively low concentration compared to other substances, we assume that protein concentrations of PlasR and PluxR are irrelevant to AHLs. Therefore, concentrations of these proteins will reach to constant quantities. Moreover, certain mRNAs will also be constant. That is:

We have:

Then focus on compound LA1 and LA2. The binding process is far quicker than other reactions such as translation reactions, thus we assume that the compounds are Quasi-equilibrium:

We have:

Take the expressions to equation (6) (15), the functions are transformed to:

The transformed functions have nothing to do with feedback factors compared to original ones. Knowing that such feedback is essential for our system, we add additional feedback factors k_a,k_b manipulatively.

Then we want to come up a clear expression of A2_c1,A1_c2.They are generated by protein PluxI and PlasI, whose translation rate is much slower than the transcription rate. So we assume that certain mRNA is Quasi-equilibrium:

We have:

By ignoring the basal expression:

Knowing that protein PtetR is controlled by LA2 through Hill Function, we assume that concentration of mRNA is directly related to LA2:

Parameter KM4 is relevant to both KM1 and KM3, however accurate function depicting the relationship is unknown. We estimate the quantity of KM4 by multiplying KM1 and KM3. Noting LA1 and LA2 have already been expressed, we have:

Then we have trouble in expressing protein PluxI and PlasI. Since concentration of protein is the integrals of its mRNA, we assume that it is proportional to concentration of mRNA in a previous time. Thus we have a DDE function:

Equations concerning environmental AHLs remain unchanged.

The original ODEs can be transformed to a much simple DDEs by above deductions.

Parameters

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Some of the parameters are derived from original ones, some of them are created to describe the new equations, and others are set for further testing.

Table 2 Parameters of DDEs

Results

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General results

Stability analysis

Proportion of cell volume

Period adjustment

Phase adjustment