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 ().
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
concentration in molar.
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
graph below.
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
abandoned in favor of a more complex model taking in to account CRO and RNAP
concentration inside the cell as well as repressor concentration.
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
for Gene Regulation published in 1983. This model had 40 configurations of operator
region in it while the previous paper only had 8, which makes for a more accurate
representation of real life. Once again it relied on a statistical thermodynamic model of
the form:
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
the OR region respectively and have values from 0 to 3.
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
concentrations.
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
protein in nM. This model shows the deactivating nature of CI repressor at PRM and the
activating qualities of CRO. At very low concentrations of CI repressor the promoter site
is relatively transcriptionally active, while at higher concentrations the RNAP is unable to
transcribe protein.
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
operator region. The negative effects of CRO on the OR3 operating region can be seen
as the concentration of CRO increases the transcriptional probability goes down as CRO
binds to OR3 blocking RNAP. The positive effects of CI can also be seen as the increase
in CI concentration prevents CRO from being formed allowing RNAP to bind to OR3 and
proceed with transcription.
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
of transcription and kinetic constants.
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.
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
consists of CRO and C1 repressor.
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
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.
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