Application of the fractional-stable distributions for approximation of the gene expression profiles

Nowadays, there are reliable scientific data highlighting that the probability density function of the gene expression demonstrates a number of universal features commonly observed in the microarray experiments. First of all, these distributions demonstrate the power-law asymptotics and, secondly, the shape of these distributions is inherent for all organisms and tissues. This fact leads to appearance of a number of works where authors investigate various probability distributions for an approximation of the empirical distributions of the gene expression. Nevertheless, all these distributions are not a limit distribution and are not a solution of any equation. These facts, in our opinion, are essential shortcoming of these probability laws. Besides, the expression of the individual gene is not an accidental phenomenon and it depends on the expression of the other genes. This suggests an existence of the genic regulatory net in a cell. The present work describes the class of the fractional-stable distributions. This class of the distributions is a limit distribution of the sums of independent identically distributed random variables. Due to the power-law asymptotics, these distributions are applicable for the approximation of the experimental densities of the gene expression for microarray experiments. The parameters of the fractional stable distributions are statistically estimated by experimental data and the functions of the empirical and theoretical densities are compared. Here we describe algorithms for simulation of the fractional-stable variables and estimation of the parameters of the the fractional stable densities. The results of such a comparison allow to conclude that the empirical densities of the gene expression can be approximated by the fractional-stable distributions.