Team:Imperial College London/Drylab/M1/Protein production/Analysis/Detailed

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==In the absence of IPTG==
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=Genetic circuit=
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[[Image:m1gci.jpg | 700px]]
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=Equations=
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<font face='Calibri' size='3'><b>In the absence of IPTG</b></font><br><br>
<b>Equation 1:</b> Equation describing the rate of transcription of LacI MRNA (M<sub>LacI</sub>): <br>
<b>Equation 1:</b> Equation describing the rate of transcription of LacI MRNA (M<sub>LacI</sub>): <br>
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[[Image:II09_MOD1_M1.jpg]]<br><br>
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[[Image:II09_MOD1_M1.jpg|centre]]<br><br>
k<sub>mlacI</sub> is the transcription rate of M<sub>lacI</sub> (a measure of promoter strength) and d<sub>mlacI</sub> is the degradation rate. At steady state:
k<sub>mlacI</sub> is the transcription rate of M<sub>lacI</sub> (a measure of promoter strength) and d<sub>mlacI</sub> is the degradation rate. At steady state:
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[[Image:II09_M1_SS1.jpg|100px]]
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[[Image:II09_M1_SS1.jpg|200px|centre]]
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<br><br>
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<b>Equation 2:</b> Equation describing the rate of translation of LacI protein (P<sub>lacI</sub>)
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[[Image:II09_MOD1_M1_E2.jpg|centre]]<br>
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At steady state:<br>
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[[Image:II09_M1_SS2.jpg|150px|centre]]
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where k<sub>lacI</sub> is the translation rate of lacI protein and d<sub>placI</sub> is the degradation rate of P<sub>lacI</sub>.Equations 3 and 4 describe the transcription and translation of the protein of interest P<sub>out</sub>.
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<b>Equation 3:</b> Transcription of P<sub>out</sub><br><br>
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Unlike in the previous case, the output promoter is inducible. In the absence of further information, we model the effect of LacI on transcription/ POPS activity with a Hill function, which represses when amounts are above the threshold K, and activates when P<sub>lacI</sub> amounts fall below threshold. Such assumption can be revised in the light of contradicting experimental data.
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[[Image:II09_MOD1_M1_E3.jpg|centre]]
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At steady state:
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[[Image:II09_M1_SS3.jpg|500px|centre]]
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<b>Equation 4:</b> Equation describing the rate of translation of protein of interest  P<sub>out</sub>:
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[[Image:II09_MOD1_M1_E4.jpg|centre]]
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At steady state:
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[[Image:II09_M1_SS4.jpg|500px|centre]]
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<font face='Calibri' size='3'><b>When IPTG is introduced</b></font><br><br>
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When IPTG is added into the system, LacI can bind to it, forming an intermediate complex [IPTG-LacI]:
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[[Image:II09_M1_reaction.jpg|centre]]
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k1 and k2 are the dissociation constants of the forward and reverse reactions. Therefore, we can modify Equation 2 to include the effects of IPTG on the LacI system as follows:<br><br>
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<b>Equation 2 modified:</b>
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[[Image:II09_M1_EQ2A.jpg|650px|centre]]
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Note that now, the effects of IPTG have been included. The –k2[Plac][IPTG] term contributes negatively, as it removes PLac from the system. The k1[IPTG-Plac] term contributes positively, as it re-stores the levels of PLac in the system.
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The equations that describe the evolution of IPTG and IPTG-LacI are-<br>
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<b>Equation 5:</b> Rate of change of IPTG in the system:
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[[Image:II09_M1_EQ5.jpg|centre]]
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==When IPTG is introduced==
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<b>Equation 6:</b> Rate of change of intermediate [IPTG-Plac] complex in the system.
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[[Image:II09_M1_EQ6.jpg|centre]]
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==Parameters and values==
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==Parameters==
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[[Image:II09_M1_PARAMETERS.jpg|400px]]
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Latest revision as of 14:50, 14 October 2009



Genetic circuit

M1gci.jpg

Equations

In the absence of IPTG

Equation 1: Equation describing the rate of transcription of LacI MRNA (MLacI):

II09 MOD1 M1.jpg


kmlacI is the transcription rate of MlacI (a measure of promoter strength) and dmlacI is the degradation rate. At steady state:

II09 M1 SS1.jpg



Equation 2: Equation describing the rate of translation of LacI protein (PlacI)

II09 MOD1 M1 E2.jpg

At steady state:

II09 M1 SS2.jpg

where klacI is the translation rate of lacI protein and dplacI is the degradation rate of PlacI.Equations 3 and 4 describe the transcription and translation of the protein of interest Pout.


Equation 3: Transcription of Pout

Unlike in the previous case, the output promoter is inducible. In the absence of further information, we model the effect of LacI on transcription/ POPS activity with a Hill function, which represses when amounts are above the threshold K, and activates when PlacI amounts fall below threshold. Such assumption can be revised in the light of contradicting experimental data.

II09 MOD1 M1 E3.jpg

At steady state:

II09 M1 SS3.jpg

Equation 4: Equation describing the rate of translation of protein of interest Pout:

II09 MOD1 M1 E4.jpg

At steady state:

II09 M1 SS4.jpg

When IPTG is introduced

When IPTG is added into the system, LacI can bind to it, forming an intermediate complex [IPTG-LacI]:

II09 M1 reaction.jpg

k1 and k2 are the dissociation constants of the forward and reverse reactions. Therefore, we can modify Equation 2 to include the effects of IPTG on the LacI system as follows:

Equation 2 modified:

II09 M1 EQ2A.jpg

Note that now, the effects of IPTG have been included. The –k2[Plac][IPTG] term contributes negatively, as it removes PLac from the system. The k1[IPTG-Plac] term contributes positively, as it re-stores the levels of PLac in the system. The equations that describe the evolution of IPTG and IPTG-LacI are-

Equation 5: Rate of change of IPTG in the system:

II09 M1 EQ5.jpg

Equation 6: Rate of change of intermediate [IPTG-Plac] complex in the system.

II09 M1 EQ6.jpg

Parameters

II09 M1 PARAMETERS.jpg

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