# 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.

### Models

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

Before making the model for cell lysis, we must characterize R0011, a lactose promtoer, because we use it to change the ratio between expression of 'Killer gene' and that of 'Anti-killer gene'.

LacI, the repressor of lactose promoter, binds to DNA sequence of lactose promoter and repress it. When [X] means the concentration of LacI, [D] means the concentration of the lactose promoter DNA sequence which is not bound to LacI and [XD] means the concentration of the lactose promoter DNA bound to LacI, the equilibrium reaction, binding and dissociation of LacI and lactose promoter can be described as equation1. In this equation, kon is the rate constant of the reaction LacI binds to lactose promoter and koff is the rate constant of the reaction LacI dissociates from lactose promoter.

The synthesis rate of [XD] is described as kon[X][D], and the dissociation rate of [XD] is described as koff[XD]. Therefore, equation2 is established.

If this reaction is in stable statement, d[XD]/dt=0 and equation2 become as below.

When [DT] is the concentration of total DNA where LacI binds, equation4 is established.

In addition, let Kd=kof/kon, then equation 3 can be modified as follows from equation4.

Here, [D]/[DT] means the proportion of non-repressed DNA, that is ,when [D]/[DT]=50%, then, promoter activity is the half of its max level. Let βas maximum of lactose promoter and promoter activity is described as follows.

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

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, [SXnX] is the concentration of LacI binding with the inducer, kon’ is the rate constant of reaction that inducers binds to LacI and koff’ is the rate constant of reaction that inducers dissociates from LacI. The synthesis rate of [SXnX] is described as kon’[S]n[X] and the dissociation rate of [SXnX] is described as koff’ [SXnX], accordingly, equation 8 is established.

If the reaction is in stable statement, then d[SXnX]/dt=0 and equatin8 become as follows.

Here, let [XT] as the concentration of total LacI, and equation10 is applied.

Let KXn = koff’/kon’ and equation11 is established from equation9 and

Lactose and IPTG are inducer of lactose promoter From equation6 and 11, the lactose promoter activity is described as follows.