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Before experiment, a mathematical model is developed to describe our system and make a prediction about our following experiment. Our model is based on the following assumption.

(1) Transcription follows Hill equation, other reactions follows mass action principle and saturation kinetics.

(2) Parameter values are partly found on current publications[1,2] and partly guess in a reasonable way.

(3) The amount of protein binding to promoter is negligible because the amount of protein binding to promoter is trace compared to the total amount of the protein

Model of AND gate

As described before, HrpS binds to HrpR forming a complex HrpRS. HrpRS then binds to promoter hrpL triggering transcription of the downstream gene. We use two inducible promoters--the isopropylthiogalactoside (IPTG)-inducible Plac and the arabinose-inducible PBAD as the AND gate inputs. A quantitative dynamic model based on protein and protein-promoter interactions for the hrp gene regulatory is described below:

1 Transcription of hrpR and hrpS mRNA

A transcription function named Hill function[3] was used to explain how IPTG, Arabinose and HrpRS influence the promoters. The Hill function and its curve are

θj is the threshold for the regulatory influence of the xj on a target gene, m is the steepness parameter. h+ explain the positive regulation on the target gene, while h- is the negative regulation function.

IPTG binds to promoter Plac and trigger the transcription of hrpR_mRNA at a maximum rate constant vmr,

while hrpR_mRNA degrade at a rate constant Drnar, so the dynamic change for hrpR_mRNA is

Similarly, we get the reaction for PBAD, they are

2 Translation of HrpR and HrpS protein

To simplify our model, we made some assumption

(1) the level of translation product is assumed to be proportional to mRNA levels

(2) There is no delay in synthesis of either component or delay because of protein transportation

HrpR and HrpS are synthesized at the rate constants Kr and Ks respectively, and degrade at the rate constant Dr and Ds.

The net forming rate of HrpR and HrpS are

3 Interaction between protein HrpRS and promoter hrpL

HrpR and HrpS binding rate constant is define as Krs, and their dissociation rate constant is K_rs. Complex HrpRS degrade at a rate constant Drs

The net forming rate of HrpRS is

Activating for the promoter hrpL is like Plac and PBAD

Promoter hrpL trigger the transcription of our target protein. Ko and Do denote the synthetic and degradation rate constant for output protein.

The whole model for the hrp AND gate are summarize below