I'm reading a paper where the authors constructed a toggle switch that uses bidirectional $P_R/P_{RM}$ promoter found in the $\lambda$ phage. There are 3 binding sites - $O_{R1}$, $O_{R2}$, $O_{R3}$ - located between the two promoters. CI is expressed by $P_{RM}$ and CI dimers bind to these binding sites:
What I don't get is that CI seems to function as both an activator and a repressor of $P_{RM}$.
First, my understanding was that $P_{RM}$ is an inducible promoter activated by CI which also represses a constitutive promoter $P_R$:
CI is a transcriptional activator of $P_{RM}$, thus producing positive feedback.
CI preferentially binds to $O_{R1}$ and $O_{R2}$, repressing the constitutive promoter, $P_R$.
However, it is also stated:
In high concentrations, CI binds to $O_{R3}$ and acts as a repressor to $P_{RM}$.
This seems to suggest that the CI can function either as an activator or as a repressor depending on the binding site bound; specifically, CI dimer binding to $O_{R1}$ and $O_{R2}$ will repress $P_R$ and at the same time activate $P_{RM}$, while CI dimer binding to $O_{R3}$ will repress $P_{RM}$. Is this correct?
I (a computer scientist by training) was always given the impression that a transcription factor is either inherently a repressor or an activator of a promoter, but this doesn't seem to be the case here. I would expect that e.g. CI would act as a repressor to $P_R$ and an activator to $P_{RM}$ regardless of the binding site. So why does CI binding to $O_{R2}$ (and $O_{R1}$ as well?) activate $P_{RM}$, while binding to $O_{R3}$ represses it - why the inconsistency? Note that I'm primarily interested in this from the modeling perspective.
What I also don't understand is whether or not $P_{RM}$ promoter is inducible or constitutive (e.g. in the absence of bound transcription factors, how leaky is it)?
Answer
There's no rule that says a transcription factor must be either a repressor or an activator. The lambda repressor (CI) is in fact a repressor and activator of transcription, depending on where it is bound and to what promoter you are referring to. I know your question isn't directly about lambda phage, but I think this mechanism may be best explained in the context of where it was first invented: nature. So, I will try to briefly explain how the lambda repressor's dual role allows regulation of its own expression and controls lysogenic induction.
The cI gene is under the control of the PRM promoter. The PRM promoter is weak because the sigma factor of RNA polymerase (RNAP) holoenzyme has low affinity for it. Binding of RNAP to PRM, and thus transcription from there, at a reasonable level requires an activator. There is another protein involved in this process as well, called Cro, that can also bind the OR operators. The cro gene is under the control of the PR promoter. PRM is not a constitutive promoter because it requires an activator. I have heard PR called constitutive because it is a strong promoter and can bind RNAP without an activator. But, at the same time, it is under the control of the lambda repressor. I suppose it depends on which definition of 'constitutive promoter' you use.
The OR operators are not identical and thus CI and Cro bind them with different affinities: CI has the highest affinity for OR1, followed distantly by OR2 and OR3, while Cro binds OR3 with highest affinity. Binding (of any two molecules) is dependent on both their affinity to each other as well as their effective concentration. At lower concentrations, CI can only bind OR1 and OR2 cooperatively. That is, each lambda repressor dimer binds to another, forming a tetramer, with one half binding OR1 and the other binding OR2. Binding both operators as tetramer allows for a stronger interaction. At higher concentrations, CI can bind OR3. Furthermore, OR1 and OR2 are located in the PR promoter, while OR3 is located in the PRM promoter.*
Most lambda genes are under the control, directly or indirectly, of the PR and PL promoters (PL is similar to PR, but the promoter itself is not directly involved in cI regulation). When lambda exists as a prophage, its DNA is integrated into the host genome and most of its gene expression is repressed. This requires CI repression of PL and PR. When the viral DNA is first integrated, there is no CI in the cell (because it is a phage gene). Expression of cI is initially dependent on a different promoter (PRE), but I won't get into this here. Once CI concentration is high enough, it can bind to OR1 and OR2 as a tetramer. Binding here physically blocks the PR promoter and prevents expression of cro (CI acts as a repressor of PR here). It also interacts with RNAP (actually the sigma factor) and facilitates is binding to the PRM promoter, which leads to further cI expression. This process is commonly called recruitment: since CI is bound tightly to the DNA and has some affinity for RNAP, it can create a stronger interaction between the DNA and the RNAP-CI complex. Simply put, it induces the PRM promoter and is thus functioning as an activator of cI transcription. As CI concentration increases, it will bind OR3 more (which is located in the PRM promoter) and repress the PRM promoter by physically blocking RNAP from binding. In these ways, cI can autoregulate its expression level, both positively and negatively.
When the lysogen is in some kind of stressful condition, normal host processes will result in proteolysis of CI and its DNA/sigma binding domain is cleaved from the dimer- and tetramerization domains. This means that CI can no longer oligomerize and is no longer able to bind cooperatively to OR1 and OR2. The result is loss of PRM activation (leading to vastly reduced cI expression) and also a loss of PR repression (leading to cro expression). Cro can then bind OR3 and represses PRM. Lack of PR repression also allows expression of early phage genes (which are under the control of PR).
Note that a similar situation occurs at the PL promoter region, and binding of CI to both OR and OL as well as CI octomerization is important in prophage regulation, but that seems even further out of the scope of your question. Hopefully you can see how he lambda "repressor" actually functions as both a repressor and activator of transcription. If this kind of stuff interests you, I highly recommend a textbook called Molecular Biology of the Gene by Watson et al.
*See comments. Upon further research, the PRM and PR promoters both overlap OR2 (Autoregulation and Function of a Repressor in Bacteriophage Lambda, Ptashne et al.), as the diagram in the OP shows. It appears that the -35 region of both PRM and PR overlap OR2 by 2 and 3 base pairs, respectively. That led me to wonder that if CI binds to OR2, which is in PRM, wouldn't it physically inhibit RNAP binding and thus repress transcription? Well it appears that CI interacts with the same region of sigma that is responsible for binding the -35 region of PRM (Amino acid substitutions in the -35 recognition motif of sigma 70 that result in defects in phage lambda repressor-stimulated transcription, Kuldell and Hochschild). So instead of blocking sigma binding here, it enhances it.
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