Introduction

Here we take biosensor HbpR as an example to demonstrate how our fine-tuning improves the performance of our biosensors. We constructed Ordinary Differential Equations (ODEs) based on single molecule kinetics and simulate the performance of our biosensors under different Pc and RBS strength with the steady state solution of the ODEs.

Figure1.Genetic regulation circuit of the biosensor HbpR.

Construction of ODEs

The genetic regulation circuit is shown in figure 1. HbpR is constitutively expressed under the constitutive promoter(Pc). When the cell is exposed to its inducer X, HbpR can bind to X and form a complex HbpRX. Considering cooperation may exists in this binding reaction, the steady state concentration of HbpRX can be written as

Where K_H is a constant and n_H is the Hill coefficient of this reaction.

HbpRX is an active state, which can activate its promoter PHbpR. We assume that there are a small proportion of HbpR can change into an active state without binding to its inducer X. Therefore, the concentraion of HbpR in active state(HbpRA) can be writen as

Where α is the proportion of HbpR in active state without binding to X.

HbpRA can bind to its promoter PHbpR and initiate the transcription. Concerning the number of HbpRA is much larger than the number of PHbpR the concentration of HbpRAPHbpR complex satisfies

Where k_P1 and k_P2 are the reaction rate constants of forward and reverse reactions.

The concentration of mRNA satisfies

Where A_m and D_m are constants.

The concentraion of mRNA ribosome complex(mRNArib) satisfies

Where k_RBS1 is the reaction rate constant of the forward reaction which is influened by the RBS strength and k_R2 is the reaction rate constant of the reverse reaction.

The concentration of sfGFP satisfies

Where A_G and D_G are constants.

The fluorescence can be written as

Where A_F is a constant.

We deduced the steady state solution of the ODEs above

Where