Team:Kyoto/Modeling

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===Equations in model1===
===Equations in model1===
===Equaitons in model2===
===Equaitons in model2===
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LacI binds to lactose promoter and repress it
 +
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 not binding LacI and [XD] means the concentration of the lactose promoter DNA sequence binding 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.
 +
        kon
 +
     
 +
        koff
 +
                  (1)
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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.
 +
 
 +
                          (2)
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If this reaction is in stable statement, d[XD]/dt=0 and equation2 become as below.
 +
 
 +
                          (3)
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  When [DT] is the concentration of total DNA where LacI binds, equation4 is established.
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 +
                          (4)
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In addition, let Kd=kof/kon, then equation 3 can be modified as follows from equation4.
 +
 
 +
                  (5)
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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.
 +
 
 +
                                  (6)
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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.
 +
          kon’
 +
 
 +
          koff’
 +
                      (7)
 +
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.
 +
 +
                                                            (8)
 +
If the reaction is in stable statement, then d[SXnX]/dt=0 and equatin8 become as follows.
 +
=0
 +
                            (9)
 +
Here, let [XT] as the concentration of total LacI, and equation10 is applied.
 +
[X] + [SXnX] = [XT]
 +
                    (10)
 +
Let KXn = koff’/kon’ and equation11 is established from equation9 and
 +
 +
(11)
 +
 +
Lactose and IPTG are inducer of lactose promoter
 +
From equation6 and 11, the lactose promoter activity is described as follows.
 +
 +
 +
(12)

Revision as of 12:45, 9 October 2010

Contents

Indroduction

Our purpose is establish the models which adjust the real relationship between expression level of 'Killer gene' and that of 'Anti-killer gene' in 'Lysis Box' when cell lysis occurs. We made these models from experiments we did practically.

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'.

Model2. The cell lysis and the ratio of expression level of 'Killer gene' and that of 'Anti-killer gene'

About equations

Equations in model1

Equaitons in model2

LacI binds to lactose promoter and repress it 

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 not binding LacI and [XD] means the concentration of the lactose promoter DNA sequence binding 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.
       kon
     
       koff 
                 (1)
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.
 
                         (2)
If this reaction is in stable statement, d[XD]/dt=0 and equation2 become as below.
 
                         (3)
 When [DT] is the concentration of total DNA where LacI binds, equation4 is established.
 
                         (4)
In addition, let Kd=kof/kon, then equation 3 can be modified as follows from equation4.
 
                  (5)
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.
 
                                 (6)

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.
          kon’
 
          koff’
                      (7)
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.

                                                           (8)
If the reaction is in stable statement, then d[SXnX]/dt=0 and equatin8 become as follows.
=0
                           (9)
Here, let [XT] as the concentration of total LacI, and equation10 is applied.

[X] + [SXnX] = [XT] 

                    (10)
Let KXn = koff’/kon’ and equation11 is established from equation9 and

(11)

Lactose and IPTG are inducer of lactose promoter 

From equation6 and 11, the lactose promoter activity is described as follows.


(12)

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