Team:Penn State/Modelling
From 2011.igem.org
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of the OR operator region in configuration s (fs) to the Gibbs free energy of binding of | of the OR operator region in configuration s (fs) to the Gibbs free energy of binding of | ||
repressor to that site and the concentration of unbound repressor dimer (). | repressor to that site and the concentration of unbound repressor dimer (). | ||
+ | </br> | ||
+ | <center><img src="https://static.igem.org/mediawiki/2011/3/36/PSUequ1.JPG"></center> | ||
+ | </br> | ||
The following graph was produced showing the probability distributions of all eight | The following graph was produced showing the probability distributions of all eight | ||
possible configurations of the model as a function of the unbound repressor dimer | possible configurations of the model as a function of the unbound repressor dimer | ||
concentration in molar. | concentration in molar. | ||
- | + | </br> | |
+ | <center><img src="https://static.igem.org/mediawiki/2011/9/98/Psumodelgraph1.png"></center> | ||
+ | </br> | ||
By summing the probability distributions of specific configurations an overall fraction of | By summing the probability distributions of specific configurations an overall fraction of | ||
repressed operator as a function of unbound dimer can be determined as is shown in the | repressed operator as a function of unbound dimer can be determined as is shown in the | ||
graph below. | graph below. | ||
- | + | </br> | |
- | + | <center><img src="https://static.igem.org/mediawiki/2011/7/79/PSUmodelgraph2.png"></center> | |
+ | </br> | ||
This model however was too simplistic in that it only took in to account repressor | This model however was too simplistic in that it only took in to account repressor | ||
concentration and not CRO concentration. Therefore this model was ultimately | concentration and not CRO concentration. Therefore this model was ultimately | ||
abandoned in favor of a more complex model taking in to account CRO and RNAP | abandoned in favor of a more complex model taking in to account CRO and RNAP | ||
concentration inside the cell as well as repressor concentration. | concentration inside the cell as well as repressor concentration. | ||
- | + | </br> | |
- | + | </br> | |
This new model, also developed by Madeline Shea and Gary Ackers appeared in | This new model, also developed by Madeline Shea and Gary Ackers appeared in | ||
the paper The OR Control System of Bacteriophage Lambda A Physical-Chemical Model | the paper The OR Control System of Bacteriophage Lambda A Physical-Chemical Model | ||
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representation of real life. Once again it relied on a statistical thermodynamic model of | representation of real life. Once again it relied on a statistical thermodynamic model of | ||
the form: | the form: | ||
- | + | </br> | |
- | + | <center><img src="https://static.igem.org/mediawiki/2011/c/c2/PSUEqu2.png"></center> | |
+ | </br> | ||
where Cs is the probability of the operating region being in the specified form, and the | where Cs is the probability of the operating region being in the specified form, and the | ||
values i, j, and k refer to the number of repressor, CRO, and RNAP molecules bound to | values i, j, and k refer to the number of repressor, CRO, and RNAP molecules bound to | ||
the OR region respectively and have values from 0 to 3. | the OR region respectively and have values from 0 to 3. | ||
- | + | </br> | |
- | + | </br> | |
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In this model when specific configurations were summed the probability of RNAP | In this model when specific configurations were summed the probability of RNAP | ||
expressing protein could be determined as a function of the repressor and CRO protein | expressing protein could be determined as a function of the repressor and CRO protein | ||
concentrations. | concentrations. | ||
- | + | </br> | |
+ | </br> | ||
The following contour plot shows the probability of RNAP being bound to OR3 | The following contour plot shows the probability of RNAP being bound to OR3 | ||
and repressor at OR2 as a function of the concentrations of CRO protein and repressor | and repressor at OR2 as a function of the concentrations of CRO protein and repressor | ||
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is relatively transcriptionally active, while at higher concentrations the RNAP is unable to | is relatively transcriptionally active, while at higher concentrations the RNAP is unable to | ||
transcribe protein. | transcribe protein. | ||
+ | </br> | ||
+ | <center><img src="https://static.igem.org/mediawiki/2011/f/fc/PSUmodelgraph3.png"></center> | ||
+ | </br> | ||
This particular graph was equal to the sum of configurations 24, 37, and 40. The graph | This particular graph was equal to the sum of configurations 24, 37, and 40. The graph | ||
below represents the sum of configurations 9, 16, 23, and 26 and corresponds to the PR | below represents the sum of configurations 9, 16, 23, and 26 and corresponds to the PR | ||
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in CI concentration prevents CRO from being formed allowing RNAP to bind to OR3 and | in CI concentration prevents CRO from being formed allowing RNAP to bind to OR3 and | ||
proceed with transcription. | proceed with transcription. | ||
- | + | </br> | |
+ | <center><img src="https://static.igem.org/mediawiki/2011/8/8f/PSUmodelgraph4.png"></center> | ||
+ | </br> | ||
Using this distribution and sum of probabilities the following two differential equations | Using this distribution and sum of probabilities the following two differential equations | ||
could be produced relating the rate of change of protein concentration to the probabilities | could be produced relating the rate of change of protein concentration to the probabilities | ||
of transcription and kinetic constants. | of transcription and kinetic constants. | ||
- | + | </br> | |
+ | <center><img src="https://static.igem.org/mediawiki/2011/7/7e/PSUequ3.png"></center> | ||
+ | </br> | ||
The above equation corresponds to the rate of repressor production R that occurs at PRM, | The above equation corresponds to the rate of repressor production R that occurs at PRM, | ||
while the following equation corresponds to the rate of CRO production at PR. | while the following equation corresponds to the rate of CRO production at PR. | ||
+ | </br> | ||
+ | <center><img src="https://static.igem.org/mediawiki/2011/d/db/PSUequ4.png"></center> | ||
+ | </br> | ||
By solving this simultaneous system of nonlinear differential equations, the production | By solving this simultaneous system of nonlinear differential equations, the production | ||
rate of protein at PRM and PR can be determined which in the case of the lambda phage | rate of protein at PRM and PR can be determined which in the case of the lambda phage | ||
consists of CRO and C1 repressor. | consists of CRO and C1 repressor. | ||
- | + | </br> | |
+ | <center><img src="https://static.igem.org/mediawiki/2011/e/e9/PSUmodelgraph5.png"></center> | ||
+ | </br> | ||
The above graph shows the concentration of CRO protein and C1 repressor as a function | The above graph shows the concentration of CRO protein and C1 repressor as a function | ||
of time. As can be expected initially both repressor and CRO are produced but as the | of time. As can be expected initially both repressor and CRO are produced but as the | ||
repressor protein degrades or is cleaved by recA protease, the rate of production of C1 | repressor protein degrades or is cleaved by recA protease, the rate of production of C1 | ||
repressor approaches zero due to the inhibitory effects of the CRO protein on OR3. | repressor approaches zero due to the inhibitory effects of the CRO protein on OR3. | ||
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</div> | </div> | ||
</html> | </html> |
Revision as of 04:05, 26 September 2011
Initially, to model the OR operating region a statistical thermodynamic model developed by Gary Ackers, Alexander Johnosn, and Madeline Shea was used. This model appeared in a 1981 paper entitled Quantitative model for gene regulation by lambda phage repressor. The model consisted of the following equation, which related the probability of the OR operator region in configuration s (fs) to the Gibbs free energy of binding of repressor to that site and the concentration of unbound repressor dimer ().