A principle of fractal-stochastic dualism and Gompertzian dynamics of growth and self-organization.

The emergence of Gompertzian dynamics at the macroscopic, tissue level during growth and self-organization is determined by the existence of fractal-stochastic dualism at the microscopic level of supramolecular, cellular system. On one hand, Gompertzian dynamics results from the complex coupling of at least two antagonistic, stochastic processes at the molecular cellular level. It is shown that the Gompertz function is a probability function, its derivative is a probability density function, and the Gompertzian distribution of probability is of non-Gaussian type. On the other hand, the Gompertz function is a contraction mapping and defines fractal dynamics in time-space; a prerequisite condition for the coupling of processes. Furthermore, the Gompertz function is a solution of the operator differential equation with the Morse-like anharmonic potential. This relationship indicates that distribution of intrasystemic forces is both non-linear and asymmetric. The anharmonic potential is a measure of the intrasystemic interactions. It attains a point of the minimum (U(0), t(0)) along with a change of both complexity and connectivity during growth and self-organization. It can also be modified by certain factors, such as retinoids.