text: 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.
  To compute the probability of bound, it is needed to sum the Boltzmann weights over all possible states of  polymerase molecules
  on DNA.  Here in this deduction    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:  
  # one state where all P polymerases molecules are distributed among all the non-specific sites (sites not participating in gene expression)  
  # 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:
  Where the first term is the combinatorial result of taken   polymerase of   non-specific sites available, and the second term are the Boltzmann weights, where  is the energy that represents the average binding energy of RNA polymerase to the genomic background (non-specific sites).
  Then, the total statistical weight , can be written as the sum of the  state and the RNA polymerase on promoter state:
  Where  in the  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 ( )  to a specific promoter, we divide  by  which produces:
  Where,  
  An important result of this model is that any transcription factor, regulator or perturbation could be introduced as a term multiplying  in the probability of binding equation. This term for any transcriptional factor (here called factor regulators) modify the probability of binding to:
  Where  is the term for transcriptional factors, and it has the value of  for increase of  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  (for further developments, see also ).