Team:NCTU Formosa/modeling

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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).

Summary

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)

Figure 1: Combinatorial promoter-RBS architecture reveals constitutive expression.
The reporter gene, green fluorescence protein (GFP) locates downstream of promoter-RBS devices as output. The output expression is controlled by differently combinatorial promoter-RBS devices. Each promoter-RBS device is expected to show the significant expression diversity. The combinatorial library contains 12 sets of promoter-RBS devices. Each set consists of different promoters and RBSs.

Table 1: The name of 12 green fluorescence protein devices and their composed biobricks
A (*) refers to construct in two different backbone.

Figure 2: The green fliorescence intensity changed with time provided the useful hints of the interactiofluorescence intensity per cell was measured using a flow cytometer.
The O.D.ratio and the fluorescence data n function between promoters and RBSs.

Figure 3: The time-course expression results were measured with time for 12 sets of GFP expression devices.
(a) promoter-RBS devices which contained the same promoter J23105 combined with three different RBSs (B0030, B0032, and B0034) were selected for measuring. (b) promoter-RBS devices which contained the same promoter J23106 combined with three different RBSs (B0030, B0032, and B0034) were selected for measuring. (c) promoter-RBS devices which contain the same promoter J23114 combined with three different RBSs (B0030, B0032, and B0034) were selected for measuring. (d) promoter-RBS devices which contained the same repressible promoter R0040 combined with three different RBSs (B0030, B0032, and B0034) were selected for measuring. Each measuring even was detected every 15 minutes. And all of the data represented the average of three independent measurements. Error bars indicated standard deviations. X-axis indicated the time units, and Y-axis indicated the fluorescence units with different scales. Furthermore, the fluorescent signals changed with time per cell were measured by using a flow cytometer.

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]:

(1)

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.

Figure 4:The bacterial population simulated results of 12 sets of promoter-RBS expression devices. Blue dotted lines were the value of OD600 of individual expression devices. Red curves were the simulated result based on the least squares to estimate parameters.

Table 2: The growth constant ks for GFP expression devices
The defined promoter-RBS devices carried out different protein level and comprised different growth constant with their corresponding cell growth rate. We used equation (2) to identify the growth constant of devices via experimental data fitting (figure 4). Each promoter-RBS device corresponded to their growth constant.

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:

(2)

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:

(3)

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).

Figure 5: The growth rate changed with cell density (O.D. 600) in GFP expression devices. Blue dotted lines were the growth rate of individual expression devices with different promoters and RBSs. Red curves were the simulated result based on the least squares estimated parameters in table 2. Figure labels corresponded to table 1. The growth rate showed a significant decreasing current with cells growing. In the saturated stage, cell density reached to the maxima concentration and the cell growth rate almost decreased to 0. All of the data represented the average of three independent measurements. Error bars indicated standard deviations. X-axis indicated the cell density units, and Y-axis indicated the growth rate with the different scales.

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).

Figure 6: The simulated result of 12 sets of promoter-RBS expression devices. Blue dotted lines were the fluorescence intensities of individual expression devices with different promoters and RBSs. Red curves were the simulated result based on the least squares to estimate parameters in Table 2. Figure labels corresponded to table 1. And all of the data represented the average of three independent measurements. Error bars indicated standard deviations. X-axis indicated the time units, and Y-axis indicated the fluorescence units with different scales.

Figure 7: The activity of different promoter-RBS devices changed with cell density (O.D. 600). We proposed that the activity of promoter-RBS devices will decrease and change with cell growing. Figure labels corresponded to table 1. And all of the data represented the average of three independent measurements. X-axis indicated the cell density units, and Y-axis indicated the device expressed units with different scales.

Figure 8: The activity of different promoter-RBS devices changed with time. We proposed that the activity of promoter-RBS devices will decrease and change with cell growing. Figure labels corresponded to table 1. And all of the data represented the average of three independent measurements. X-axis indicated the cell density units, and Y-axis indicated the device expressed units with different scales.

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

(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).

Reference
1. You, L., et al., Programmed population control by cell-cell communication and regulated killing. Nature, 2004.
428(6985): p. 868-71.
2. RICHARDS, F.J., A Flexible Growth Function for Empirical Use. Journal of Experimental Botany, 1959. 10(2): p. 290-301.