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Boltzmann distribution:
–How the probability of a given microstate of a system depends on its energy

Define microstate:
A microstate is one particular realization of the microscopic arrangement of the constituents for the problem of interest

Assuming L ligands in a solution which is modeled as Ω boxes, The number of microstates or arrangements corresponding to an open receptor:

Boltzmann distribution, the probability of different microstates as determined by their energy: (The summation of the probabilities included)

Boltzmann partition function:

Ion channel: probability of microstates determined by the Boltzmann distribution: (State, Energy, Weight, Probability)

Average Energy equation using boltzmann distribution:

Mathematical trick for computing averages from Boltzmann distribution

States and weights for ligandreceptor binding (L ligands and 1 receptor): (State, Energy, Multiplicity, Weight)

The probability when the receptor is occupied:

Partition function for ligand receptor bindings:

Most RNA polymerase molecules are bound to DNA:
•We assume that all RNA polymerase molecules are bound to DNA (either the promoter of interest or nonspecifically)
•We are to find the probability of RNA polymerase binding to targeted sites in the presence of the competition of nonspecific binding sites

Different states with different energies: specific binding vs. nonspecific binding
P: number of RNA polymerase molecule
NNS: nonspecific binding sites
(State, Energy, Multiplicity, Weight)

The probability that the RNA polymerase (RNAP) is bound to promoter of interest:

