Modeling

Theoretically, smtB-OP device may also be activated by other metal ions (Zn²⁺ or Co²⁺). As Cd²⁺ was the most sensitive metal ion to our device at low concentration, which was previously proved in our experiments, we had a conclusion that our reconstructive smtB-OP device would report the presence of Cd²⁺ by inducing the expression the reporter gene. In this part, we deeply discussed the data mentioned in wet lab-work by mathematical modeling.

1. Equation constructed

We divided the process of our reporter gene's expressing induced by Cd²⁺ into three steps. First, the entrance of cadmium ions into the bacteria. Second, the combination of cadmium ions and SmtB protein (exposure of the smtO-P). Third, the expression of the reporter gene.

For these three steps, we designed a set of equations. They are listed as following:

According to the three equations above; we can get the following equation:

As the p will be getting close to P in a short time, we can simplify the equation (4) and get the new equation:

In those equations:

• Y-----The expression level of reporter gene.
• y₀-----The maximum amount of reporter gene expression at per unit time.
• P-----Concentration of Cd²⁺ in environment.
• p-----Concentration of Cd²⁺ in cell.
• r₁-----The rate of Cd²⁺ move into the cell.

Therefore, we construct a simple relationship between concentration of Cd²⁺ (P)、expression of reporter gene(Y) and experiment time(t). Combining with the experimental data, we use the fitting method to work out the parameter of the equation.

2. Experimental data

Basing on the fig.1, we can find that Rosetta pLysS (1*106 cells/ ml) carrying the smtB-OP-amilCP element were grown with Cd²⁺ (1μM) supplement for 1-2h immediately before assay and the expression was monitored by measuring the OD₆₀₀ value. Clearly, they have a linear relationship. It means that the data of amilCP expression supplement for 1-2h is typical.

For this reason, we can use the experiment data of different amilCP expression level with different Cd²⁺ concentration supplement for 1h to fit the equation.

Fig.2. Experiment data of reporter gene (amilCP) expression level with different concentrations of cadmium.

Clearly, the relationship between P and Y is not a linear correlation. When there is a low concentration of Cd²⁺ in environment (namely the value of P is low), the Y increases sensitively. While the concentrations of Cd²⁺ in environment (P) is reaching to a higher level, the Y will tend to saturation.

According to fig.2, we estimated that the tend or our data should fit in an exponential equation or a polynomial.

3.Fitting

The following figure is the first time we fit the experiment data. Due to the Y increases quickly in the beginning and tends to saturation as the value of P is higher and higher, we choose quadratic polynomial as the first basic equation to fit our data.

So, we get this equation:

Y=-0.0003145P²+0.0448P+1.021 fit(1)

In the fig.3, we can find the curve is not fitting well with our data (RMSE=0.3657).

Besides, the decrease of Y after P > 70μM contradicts the math model we constructed before [equation (5)]. Therefore, the monotone increasing equation may fit better with our experiment data.

The exponential equation is our second choice. The result is shown as following:

As the fig.4 shows, if we consider the relationship between P and Y is exponential, we can get the following equation:

Y=3.063-2.324P^-0.2802 fit(2)

The fitting result is pretty good (RMSE=0.0589).

In fact, differential coefficient of equation fit (2) is equal to or greater than zero and the second derivative of it is less than zero. It means that Y grows faster with low concentrations of cadmium, and then grows gradually slower as the concentration of cadmium increases.

In summary, we got a suitable equation fit (2) to describe the relationship between the concentration of Cd²⁺ in environment(P) and the expression level of reporter gene(Y),and our designed genetic construct is available to detect Cd²⁺ at a certain degree.