The distribution function f has the form of:

Since the aforementioned distribution of free energy is the Gaussian in this study, one can easily obtain the distribution function of the equilibrium constant K as log-normal.

So, the distribution function f of the equilibrium constants K or the affinity can be shown to be: which is a log-normal distribution near the mean above TC While which shows a power law decay near the low K value tail of the distribution (near or below TC).

We have confirmed the analytical form of the distribution functions with 720 diversified small molecules binding with a specific receptor: Cox-2 protein enzyme.

The probability and physical relevant quantity range is found to fit adequately the distribution functions from the analytical model, and the two kinds of curves are fitted well to the data in the center and near the tail.

Radial distribution functions (S4 Fig) showed an increased degree of solvation in the sugar group area of the membrane, such that on average 20 water molecules were present in this region after 2 pi-croseconds, compared to ~12 molecules prior to PMB1 binding.

Radial distribution functions along the z—axis showing solvent penetration relative to the membrane center of mass (top row A—C).

Radial distribution function (all axis) showing solvent proximity to the DAB amine (bottom row F—I)

However it can be normalized, by defining: which is a proper density distribution function .

Further, a proper cumulative distribution function (GDP) for the 19175 (the relative haplotype frequencies) can be derived by normalizing VH(x) where Pr(pj g x) = 1 — vH (x) / H , since vH (x) / H represents the complementary CDF (i.e.1-Pr(pj§x)).

pH (x) Density of p; values—the number of haplotypes in a region = the number of haplotypes pH(X) = VHlfIX) Density distribution function for the probability of obsen/ing a clone of frequency x.