# Team:Tsinghua/Modeling

### From 2013.igem.org

# Modeling

The **regulatory pathway** is modeled as systems of ordinary equations dependent on **time**. After setting **initial concentration** of all species, a time series of the concentration of each species can be generated by simulation. **Regression analysis** relates concentration of the reporter gene product (ADE2) to yeast color. Simulation of the model predicts how yeast color changes with time. Sensitivity analysis predicts the relationship between **input (AHL concentration)** and **output (yeast color)**. A **dose-response curve** (yeast color to AHL concentration) can be obtained by simulation of the model with different initial AHL concentration. Then the AHL concentration in the environment can be estimated from yeast color using the dose-response curve. As the AHL concentration is proportional to the population bacteria, the population of bacteria can also be estimated. By fitting the model to experiment data, we can detect the concentration of specific bacteria from yeast color.

## Introduction

After mating, the fused yeast cell gains both the **sensor** and **reporter** system. Then the yeast cell is capable of detection AHL in the environment and reports them.

There are three stages in the detection of AHL from bacteria. First, AHL in the environment **diffuses** across the cell membrane of the yeast. Second, AHL binds to modified **LuxR transactivator** and forms a complex, which enters the **nucleus** and binds to the **LuxR promoter**. Upon binding, the **AHL-LuxR complex** activates the expression of the transcription factor tTA^{1}. **tTA** enters the nucleus and binds to **Tet** operator, activating the reporter gene **ADE2**. Expression of ADE2 **changes the color** of the yeast from red to white. An overview of the biochemical process is shown in Figure 1. The figure is drawn with CellDesigner^{2} 4.3.

Figure 1. Overview of the biochemical process

## Assumptions

AHL is secreted by bacteria and **diffuses** across the cell membrane of the yeast. It is assumed that the diffusion process reaches **equilibrium** within a **short time** so the concentration of AHL inside and outside the yeast cell membrane is the same.

After AHL binds to **modified LuxR protein** to form an **AHL-LuxR complex**, the complex must be transported into the cell nucleus. The nuclear localization sequence (NLS) on the LuxR protein is recognized by importin and then imported into the cell nucleus. To model the cell more accurately, the **rate of transportation** must be considered. However, without sufficient experiment data, it is difficult to estimate the **kinetic parameters**. In a **simplified model**, the concentrations of transcription factor inside and outside cell nucleus are assumed to be equal.

Three steps are required to **activate expression** of a protein: **transcription factor binding**, **transcription** and **translation**. If transportation of proteins and mRNAs are considered, there will be more steps. To simplify the model, we assume that the concentrations of transcription factors and mRNAs inside and outside the cell nucleus are equal. Transcription and translation can be modeled as a **single process** as they are tightly coupled.

Activation of transcription is modeled as a **stochastic process**. A promoter is either bound or unbound by one transcription factor molecule at a moment. Binding of transcription factor increases **transcription rate** of the target gene. The probability of transcription factor binding is determined by the **concentration of transcription factor**, **gene copy number** and **binding affinity** (or disassociation rate).

## Model

The biochemical process is modeled as ordinary differential equations. The variables and equations are list as follows.

### Species

- AHL (concentration remains constant)
- LuxR – LuxR in cytoplasm
- LuxRC – LuxR-AHL complex (dimer)
- tTA
- ADE2

### Kinetic parameters

Name | Description |
---|---|

k_{1} | basal expression rate under constitutive promoter |

k_{2} | dimerization rate of AHL and LuxR |

k_{3} | degradation rate of LuxR |

k_{4} | degradation rate of LuxRC |

k_{5} | expression rate of tTA |

k_{6} | activation coefficient of LuxRC |

k_{7} | degradation rate of tTA |

k_{8} | basal expression rate of tTA |

k_{9} | expression rate of ADE2 |

k_{10} | activation coefficient of tTA |

k_{11} | degradation rate of ADE2 |

k_{12} | basal expression rate of ADE2 |

### Equations

LuxR protein is synthesize at a constant rate k1.
AHL binds to LuxR to form a complex.
Then AHL-LuxR complex dimerizes to form a transcription factor^{3}.

Activation of tTA expression is modeled using Hill function.
Hill functions is commonly used to model the interactions between transcription factors and promoters^{4}.
The transcription factor cooperativity is 1 (single binding site).
k_{5} is the expression rate of tTA if the promoter is fully activated.

Activation of ADE2 expression is also modeled in Hill function.

## Sensitivity Analysis

Among all species considered in the model, initial AHL concentration is the main factor that determines the output of the system. The main output of the system is the color of the yeast which is correlated with the concentration of ADE2. The relationship between the concentration of ADE2 and the initial concentration of AHL will be analyzed.

Time series of the concentration of each species can be generated by simulation of the model with **initial conditions**. When AHL is added to the system, the concentration of ADE2 will increase in the initial phase. Finally the concentration of ADE2 will reach its **maximum** and keep **steady** for a time. The color of the yeast will also turn from red to white. The time it takes for the concentration of ADE2 to reach its **maximum** is defined as **response time** and the **maximum** concentration of ADE2 is defined as **response value**.

To analyze the sensitivity of the system to AHL concentration, we set different **initial AHL concentrations**. A **dose-response curve** can be drawn from a series of AHL concentrations and response values. The parameters of the dose-response curve can be estimated from experiment data. Then the curve can be used to estimate the AHL concentration in the environment from yeast color.

However, we didn’t collect sufficient data to estimate the model parameters. We **set parameters adapted from literature**^{3}. A time series data is plotted as shown in Figure 2 by setting initial AHL concentration to 1 μM. A dose-response curve is shown in Figure 3. However, the parameters were estimated from experiments with ** E. coli**, and might not be applied to

**yeast**system. We can only know the shape of the curves. We will try to collect more data points to increase the predictive power of our model.

Figure2. Time series of the concentration of each species

Figure3. Dose-response curve

## References

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*Nature***434**, 1130–1134 (2005). - Goutelle, S. et al.
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