# Team:NCTU Formosa/Modeling

### From 2009.igem.org

## The dynamics model constructed for ‘Bacterial referee’

**Motivation**

Our team constructed a controllable synthetic genetic circuit in E. coli which has timer and counter functions. The ability to design a DNA sequence that reliably implements a desired cellular function with quantitative precision is still lack. Model construction and simulation become more important as the size of genetic circuit grows. Therefore, computational models are used in our project for genetic circuit design and analysis.

**Model equations**

Because the memory modules (D and E strands) do not affect the system operation, we reduced our mathematical model as Fig. 1. It consists of six genes: LacI, LuxR, TetR, GFP-LVA, LuxI, and mRFP. Lactose and AHL are the input to the system. The fluorescence of the system, due to the protein GFP and mRFP, are the measured output. This system can be modeled by differential equations as fellows:

*Alpha-a, b, c and f* are production rates of the corresponding promoter, which are assumed to be given constants. [Lact] is the concentration of added lactose and Alpha-out is the production rate of luxI protein that produced by the external bacteria which invading the system. The *k1* is the degradation rate of lactose that consumed by bacteria. The *k2* is the production rate of AHL that synthesized by the LuxI protein. *Gamma-GFP, gamma-tetR, gamma-LuxI and gamma-RFP *are decay rates of the corresponding proteins. Since the proteins are relatively stable, we neglected protein degradation rate and assumed that the degradation rate parameters were simply equal to the dilution rate corresponding to the observed division time of the cell (about 45 min) (Batt, 2007). The Hill function can be derived from considering the equilibrium binding of the transcription factor to its site on the promoter region. For a repressor of LacI and tetR protein, Hill function is an S-shaped curve which can be described in the form **1 / [ 1 + ( x/Kr)^n ]**.

*Kr*is the repression coefficient. The Hill coefficient

*n*governs the steepness of the input function. For an activator AHL/LuxR complex, Hill function can be described in the form

**.**

*x*^n / (*K*^*n*+*x*^*n*)*K*is the activation coefficient. n determines the steepness of the input function (Alon, 2007). From the gene network design procedure, we give the initial values of proteins and the parameter of production rates

*Alpha*and decay rates

*Gamma*as the EXCEL file. That is, given the initial condition and numerical configurations, several rounds of simulations were lead to exactly the results (Fig.2 and Fig.3).

**A. Simulation results of timer function**

The timer function was simulated in Fig. 2. The simulation data indicated the concentration of lactose determines timer’s working length. Low concentration of lactose decreases the intesity and duration of green fluorescent. After added lactose is consumed by *E. coli*, the RFP is translated to remind us that time’s up. Moreover, the counting-time can be controlled from 8 hours to 20 hours using different concentration of lactose.

**B. Simulation results of counter function**

The counter function of different outer bacteria invasion intensity was simulated in Fig. 3. The simulation data indicated the concentration of AHL that synthesized by Lux I protein determines the counter’s response time. When external bacteria invade the system, they increase the production rate of LuxI protein (*i.e. Alpha-out *). Thus, the concentration of AHL increases and and bind to LuxR protein to form AHL/LuxR complex. The complex binds the Plux/cIIp22 promoter to activate transcription of the downstream RFP. Therefore, RFP is produced quicker with higher concentration of invading bacteria. In our simulation results, the counter system can detect the invading bacteria over two decades. Outer bacteria invasion intensity affects the production rate of RFP, but not affects the production of GFP.

According to the simulation results, the model equations tally with our expectations of timer and counter functions. The model equations present interesting mathematical properties that can be used to explore how qualitative features of the genetic circuit depend on reaction parameters. Results of the analysis were used to guide the choice of genetic ’parts’ (genes, promoters and ribosome-binding sites), devices and growth conditions to favor a successful implementation of designed circuit function.

**Important concept**

The engineering process of this project: ‘Bacterial Referee’ involved multiple cycles of design, modeling, optimization, implementation, and revision. The implementation methods are listed in the following steps (Fig. 4).

- Building a mathematical model that can capture the dynamics of ‘Bacterial Referee’.
- Explore dynamics
*in silico*to capture the qualitative behavior of the engineered*E. coli.* - Based on our design, the necessary bio-bricks are selected to construct the plasmids. We implement our circuit on a plasmid and need to decide on copy number, what promoters, ribosome binding sites, transcription terminators, and perhaps degrade tags to use.
- The biological circuits will be transformed into the host cells to test whether it work as our design.
- If the biological circuits can not work as predicted, we retest, revise or redesign the circuit to address critical parameter changes, and perhaps ‘fine tune’ the promoter activity by the degenerated PCR.

**Expected achievements of the project**

At present, we have satisfactory in silico results. However, in order to upgrade our project, we need to implement the ‘Bacterial Referee’, to insert these genetic circuits into the host bacteria and to develop the control schemes. These need the cooperation of our research team. The expected achievements of the project are given as follows

- Improve the experimental results.
- Finish the implementation in the host cell and develop control schemes.
- Find the application of ‘Bacterial Referee’.

**References**

Alon, U. (2007) *An Introduction to Systems Biology: Design Principles of Biological Circuits. *Chapman & Hall/CRC.
Batt, G., Yordanov, B., Weiss, R. and Belta, C. (2007) Robustness analysis and tuning of synthetic gene networks, *Bioinformatics*, 23, 2415.