# Team:Kyoto/Modeling

## Modeling

### Models

#### Model1. The characterization of R0011, a latosepromoter

Before making the model for lysis cassette, we must characterize R0011, a lactose promtoer, because we use it to change the expression level of lysis cassette. We made some mathmatical model for the lcatose promoter activity.

### Model1. The characterization of R0011, a latosepromoter

#### Indroduction

It is too laborious to measure the activity of R0011 with all IPTG concentrations, such as 0.01mM, 0.02mM, , , , , 10mM. Measurement with some appropriate IPTG concentrations and making a mathematical model from the experimental data is beneficial to estimate the promoter activity with the other IPTG concentration and save the trouble.

#### LacI binds to lactose promoter and represses it

```figure
```

LacI, the repressor of lactose promoter, binds to DNA sequence of lactose promoter and represses it. R0011, a lactose repressor, has two operator regions where LacI binds .　The equilibrium reaction of binding and dissociation of LacI and the lactose promoter can be described as equation1.

```equation1.
```

Here, [X] means the concentration of LacI, [D] means the concentration of the lactose promoter not binding LacI, [DX] means the concentration of the lactose promoter binding a lacI [DX2] means the concentration of the lactose promoter binding two LacI.

```K1 means the equilibrium constant for the reaction between [D] and [X],
K2 means the equilibrium constant for the reaction between [DX] and [X].
The equilibrium constants K1 and K2 are described as equation2 and equation3 respectively.
```
```equation2
```
```equation3
```

From equation2 and equation3, equation4 is established.

```equation4
```

Here, α is the promoter activity when all the lactose promoter bind two LacI respectively, β is the promoter activity when all the lactose promoter bind a LacI respectively,

```is the ratio of [D] to the total concentration of lactose promoter,
is the ratio of [DX] to the total concentration of lactose promoter.
```

In this model, we hypothesized that the promoter activity is 0 when all the lactose promoter bind two LacI respectively.

#### LacI binds to lactose promoter and represses it

```figure
```

The inducer of lactose promoter, Lactose and IPTG, binds to LacI, and changes LacI conformation so that LacI dissociates from lactose promoter. The equilibrium reaction of the inducer and LacI is described as follows.

```equation5
```

Here, [SX] is the inducer concentration, [X] is the concentration of LacI not binding with the inducer, n is the number of the inducer molecule binds to one LacI molecule. The equbilium constant, KXn, is described as equation6.

```equation6
```

We assume that the total concentration of LacI in a cell does not change in the log phase growth in which the cell growth does not change because the dilution effect for LacI due to the cell growth does not change and the expression and the degradation of LacI is constitutive. Therefore, let XT as the concentration of total LacI, and equation7 is applied.

```equation7
```

From equation6 and equation7, equation8 is established .

```equation8
```

#### Lactose and IPTG are inducer of lactose promoter

```figure
```

From equation4 and equation8, the relationship between IPTG and lactose promoter activity is described as follows.

```equation9
```

#### Simulation

We apply equation9 to the result of characterization of R0011 and parameters are decided by using MATLAB 7.10.0 (MathWorks).

```graph
```

The orange markers and error bars are experimental data and blue line is the expectation from this model. The data of 0.5mM IPTG does not be used in fitting the model.

Parameters are shown in table below.

```table
```

#### Reference

 Uri Alon (2006) “An Introduction toSystems Biology” Chapman & Hall