We analyzed the experimental data through modeling.

We made four assumptions in order to simplify some of the aspects of the model. Many similar assumptions have been made in the literature.

The degradation of and other substances are linear.

Each cell has the same state.

The reaction in the process satisfies the first-order kinetic reaction.

Using the assumption about the first-order kinetic reaction,

Give：

Ordinary differential equations are too difficult to be solved, in order to get the solution of the equations, quasi-steady-state approximation is necessary.

We use the approximate conditions as little as possible and try to make the quasi-steady-state of matter consistent with the actual situation.

According to the formula’s deformation and derivation,

it gives：

Under incomplete repression, the bacteria express red fluorescent proteins without the presence of nickel ions.

The incomplete repression of promoters is a major issue in our fluorescence system. This issue was particularly observed in following diagram.

The common formulation of the Hill equation is as follows:

Consequently, we modeled incomplete repression by using the ratio of occupied promoter concentration to total promoter concentration.

In view of the fact that is difficult to be measured and have same trend to , we used the concentration of nickel ions to approximate the equation.

So, the fluorescence intensity can be obtained without considering the cytotoxicity of Nickel Ions:

The para meters for the production function are:

A-basal expression level of promoter

B-maximal expression level of promoter

K_d-half maximal effective concentration of N-Ni

n-Hill coefficient for induction.

Added Variable, The differential equation of was updated by:

The solution of this differential equation is as follows：

We designed a kit, including 10ml centrifuge tubes, filter columns (containing 0.22 M filter membrane), the national standard concentration of nickel ion solution 50ml(3μg/L), pure water 20ml, engineering bacteria 5ml.

Instructions：

1. Add 3ml chloramphenicol containing LB medium and 30μL of original bacteria to the test tube. Incubate at 37°C 200rpm for overnight, and dilute the cultures to a target OD600 of 0.8;

2.Take out 10ml centrifuge tubes for each sample besides one for blank one for stdrand. Add 2.7ml chloramphenicol containing LB medium and 30μL overnignt bacteria into each tube;

3.Add 2.7ml chloramphenicol containing LB medium and 30μL of the overnight culture solution into each tube.

4.Place the filter columns into each centrifuge tube. Add 300μL pure water into one column，add 300μL nickel ion standard solution into another column, and add 300μL sample solution into the other columns.

5. Detect the fluorescence after incubating at 37°C, 200rpm for 4hr.