Promoter activity

The problem of gene regulation could be represented mathematically as the probability of n molecules — RNAP, activators, repressors and inducers — are bound to a target regions.[4][2]

To compute the probability of bound, it is needed to sum the Boltzmann weights over all possible states of ${\displaystyle P}$ polymerase molecules

on DNA.[8] Here in this deduction ${\displaystyle P}$ is the effective number of RNAP molecules available for binding to the promoter.

This approach is based in statistical thermodynamics of two possible microscopic outcomes:[4]

1. one state where all P polymerases molecules are distributed among all the non-specific sites (sites not participating in gene expression)
2. a promoter occupied and the remaining P-1 polymerases distributed among the non-specific sites.

The statistical weight of promoter unoccupied Z(P) is defined:

${\displaystyle Z(P)=~{\frac {N_{NS}!}{P!*(N_{NS}-P)!}}*{e}^{-~{\frac {E_{NS}}{KbT}}}}$

Where the first term is the combinatorial result of taken ${\displaystyle P}$ polymerase of ${\displaystyle N_{NS}}$ non-specific sites available, and the second term are the Boltzmann weights, where ${\displaystyle E_{NS}}$ is the energy that represents the average binding energy of RNA polymerase to the genomic background (non-specific sites).

Then, the total statistical weight ${\displaystyle Z(Ptotal)}$, can be written as the sum of the ${\displaystyle Z(P)}$ state and the RNA polymerase on promoter state:

${\displaystyle Z(Ptotal)=Z(P)+Z(P-1)*{e}^{-~{\frac {E_{S}}{KbT}}}}$

Where ${\displaystyle E_{S}}$ in the ${\displaystyle Z(P-1)}$ state is the binding energy for RNA polymerase on the promoter (where the s stands for specific site).

Finally, to find the probability of a RNA polymerase to binding (${\displaystyle Prob_{bound}}$ ) to a specific promoter, we divide ${\displaystyle Z(P)}$ by ${\displaystyle Z(Ptotal)}$ which produces:

${\displaystyle Prob_{bound}={\frac {1}{1+{\frac {N_{N}S}{P}}*{e}^{-{\frac {\Delta E}{KbT}}}}}}$

Where, ${\displaystyle \Delta E=E_{S}-E_{NS}}$

An important result of this model is that any transcription factor, regulator or perturbation could be introduced as a term multiplying ${\displaystyle P}$ in the probability of binding equation. This term for any transcriptional factor (here called factor regulators) modify the probability of binding to:

${\displaystyle Prob_{bound}={\frac {1}{1+{\frac {N_{N}S}{P*F_{R}}}*{e}^{-{\frac {\Delta E}{KbT}}}}}}$

Where ${\displaystyle F_{R}}$ is the term for transcriptional factors, and it has the value of ${\displaystyle F_{R}>1}$ for increase of ${\displaystyle F_{R}<1}$ for decrease of the number of RNA polymerase available to bind.

This result has an important significance to represent mathematically all the possible configurations of transcriptional factor by derive different models to estimate ${\displaystyle F_{R}}$ (for further developments, see also [4]).

Core Promoter elements.

The process of activation and binding in eukaryotes is different from bacteria in the way that specific DNA elements bind the factors for a functional pre-initiation complex. In bacteria there is a single polymerase, that contain catalytic subunits and a single regulatory subunits known as sigma, which transcribe for different type of genes.[9]

In eukaryotes, the transcription is performed by three different RNA polymerase, RNA pol I for ribosomal RNAs (rRNAs), RNA polymerase II for messenger RNAs (mRNAs) and some small regulatory RNAs, and the RNA polymerase III for small RNAs such as transfer RNAs (tRNAs). The process of positioning of the RNA polymerase II and the transcriptional machinery require the recognition of a region known as "core promoter".[9] The elements that could be found in the core promoter include the TATA element, the TFIIB recognition element (BRE), the initiator (Inr), and the downstream core promoter element (DPE).[10] Promoters in eukaryotes contain one or more of these core promotes elements (but any of them are absolutely essential for promoter function),[9] these elements are binding sites for subunits of the transcriptional machinery and are involve in the initiation of the transcription, but also they have some specific enhancer functions.[10] In addition, the promoter activity in eukaryotes include some complexities in the way of how they integrate signals from distal factors with the core promoter.[11]