Team:NCTU Formosa/modeling

Measurement

This new measurement method can calculate the protein expression rate in the different E. coli population density for the protein expression device (=Promoter_RBS device).

In previous studies, the transcriptional strengths of promoter and transcriptional strengths of RBSs was defined as a constant values. But in biological concepts, we know the expression rates of most proteins decreased dramatically while the bacteria at the stationary phase. To overcome this problem, our team developed a new measurement method can calculate the protein expression activity of promoter_RBS device changes with cell density in the culture tube directly. We provide a simple polynomial equation which can describe linear relationship between the protein expression activity p(s) of promoter_RBS device and cell desity s (the value of OD600).

p(s) = p₀+p₁s

where p0 denotes zero-order coefficient, p1 denotes first-order coefficient. This dynamic model captured the experimentally observed differences of green fluorescence intensity for each protein expression circuit. As we expect, promoter-RBS activity decrease linearly related with cell population density. Therefore, if the P0 and P1 parameters of a promoter_RBS device are characterized, we can calculate the protein expression activity of a promoter_RBS device by OD600 value. Furthermore, this model enable us to rational connect a promote_RBS device with different strength to obtain a target protein expression level in a synthetic genetic circuit.

Motivation
In previous studies to model a protein expression, the transcription rates of promoters and translation rates of ribosome binding sites (RBSs) are defined as a constant value. However, the values are not constant during cell growth. For example, the transcription rates of promoters and the translation rates of RBSs are lager in log phase than bacteria growth in stationary phase. To overcome this problem, we selected four promoters and three RBSs with different regulation strength and constructed 12 protein expression devices which combine promoter, RBS and green fluorescent protein (GFP) in Escherichia coli. The GFP expression levels with time were measured using a flow cytometry, and the experimental data can used to characterize a protein expression rate of a protein expression device which contains a promoters and a RBS. A dynamic model that captured the experimentally observed differences for each protein expression device was developed in this study. Using this method, we can measurement the protein expression rate in the different E. coli population density for the protein expression device.

Methods
Two strategies will be applied in this method:
1.Construction of a Promoter-RBS Library and Assay of GFP Expression
We selected four promoters and three RBSs with different regulation strength and constructed 12 protein expression devices which combine promoter, RBS and green fluorescent protein (GFP) in E. coli (Fig.1 &table.1). The GFP expression level with time was measured using a flow cytometry (Fig.3 &Fig.4)

2.Build a Mathematical Model to Characterize Protein Expression Ability of Promoter-RBS Devices.
The logistic growth model is commonly used to describe the bacterium growth curve under the nutrient limited condition as follow [1-2]:

Where s denotes the cell density parameter (optical density at 600 nm (O.D. 600)), smax denotes the maximum value of cell density, and ks denotes growth rate constant. The cell density time course data of the bacteria with different GFP expression devices were used to solve Eq. (1) (Fig. 4). The growth rate constant ks was calculated and listed in Table 2.

The Ptet promoter (R0040) combined with RBS (B0034) had higher GFP protein expression level and had the smaller ks i.e., slower growth rate. The results indicated that protein expression level of a genetic circuit is negatively correlated to the growth rate of a host bacterium. Bacteria with a low metabolic load have greater growth rate. The protein expression ability of promoter-RBS devices is indirectly measured by fluorescence intensity of GFP expression devices (Fig. 1 and 3). Since the protein concentration per cell is diluted due to the cell growth and protein degradation, a dynamic model to character the protein expression ability of promoter-RBS devices can be built as follows:

Where denotes the protein expression level of a GFP expression device (Fig. 1). p(s) denotes the protein expression ability of a promoter-RBS device, which is the function of cell density parameter s (optical density at 600 nm (O.D. 600)), and rGFP denotes the protein degradation rate of GFP. g(s) denotes the cell growth rate and can be determined from Eq. (1) as:

The cell growth rate g(s) can be calculated directly by experimental data of O.D. 600 (Fig.4). The negative correlation between cell growth rate g(s) and O.D. 600 indicated that bacteria loss their growth potential gradually when grow in the nutrient limited condition (Fig.5).

To calculate the protein expression ability of a promoter-RBS devices p(s) , we assume p(s) is related to cell density parameter s (optical density at 600 nm (O.D. 600)), and a polynomial Eq. (4) is used to trace p(s) as fellow:
p(s) = p₀+p₁s             (4)
Where p₀ denotes zero-order coefficient, p1 denotes first-order coefficient. The experimental data and trace results were showed in Fig.6. Parameters of our model were determined by the nonlinear regression based on the minimum sum of squired residuals. This dynamic model captured the experimentally observed differences for each genetic circuit (Fig.6). The fitting results indicate this model is reasonable. A universal mathematical model was found to explain the protein expression activity of Promoter_RBS devices change affected by the cell density (OD600). As we expect, promoter-RBS activity decrease linearly during cell growth (Fig. 7). Similarly, promoter-RBS activity also decreases with time (Fig. 8).

In previous test, we also assumed p(s) is a polynomial as Eq. (5),

but the analysis of extra sum of squares was used to compare this alternative. The Eq. (5) did not fit the data significant better than Eq. (4). We concluded that Eq. (5) was overparametrized, and protein expression activity of promoter_RBS can be described simply as Eq. (4).