On time-space of nonlinear phenomena with Gompertzian dynamics

Neuronal Fractal Dynamics

Synapse formation is a unique biological phenomenon. The molecular biological perspective of this phenomenon is different from the fractal geometrical one. However, these perspectives are not mutually exclusive and supplement each other. The cornerstone of the first one is a chain of biochemical reactions with the Markov property, that is, a deterministic, conditional, memoryless process ordered in time and in space, in which the consecutive stages are determined by the expression of some regulatory proteins. The coordination of molecular and cellular events leading to synapse formation occurs in fractal time space, that is, the space that is not only the arena of events but also actively influences those events. This time space emerges owing to coupling of time and space through nonlinear dynamics. The process of synapse formation possesses fractal dynamics with non-Gaussian distribution of probability and a reduced number of molecular Markov chains ready for transfer of biologically relevant information.

[Fractal geometry in the objective grading of prostate carcinoma]

BACKGROUND

A possible approach to objectively classify complex patterns in tumor tissue is a mathematical and statistical investigation of the distribution of cell nuclei as a geometric representation of cancer cells by fractal dimensions. Both the existence and changes in the fractal structure of tumor tissue have important consequences for the objective system of tumor grading. In addition, the complexity of growth in different carcinomas or their intercellular interactions can be compared to each other.

RESULTS

We present a theoretical introduction into fractal geometry as well as in the computer algorithms based upon the Rényi family of fractal dimensions. Finally, a geometric model of prostate cancer is introduced and the relationship between geometric patterns of prostate tumor and the fractal dimensions of the Rényi family are explained.

Neuronal differentiation and synapse formation in the space-time with temporal fractal dimension

An improvement of the Waliszewski and Konarski approach ([2002] Synapse 43:252-258) to determine the temporal fractal dimension b(t) and scaling factor a(t) for the process of neuronal differentiation and synapse formation in the fractal space-time is presented. In particular the analytical formulae describing the time-dependence of b(t)(t) and a(t)(t), which satisfy the appropriate boundary conditions for t-->0 and t-->infinity, are derived. They have been used to determine the temporal fractal dimension and scaling factor from the two-parametric Gompertz function fitted to experimental data obtained by Jones-Villeneuve et al. ([1982] J Cell Biol 94:253-262) for embryonal carcinoma P19 cells treated by retinoic acid. The results of the calculations differ from those obtained previously by making use of the three- and four-parametric Gompertz function as well as other S-shape functions (Chapman, Hill, Logistic, Sigmoid) evaluated by the fitting of the experimental curve. The temporal fractal dimension can be used as a numerical measure of the neuronal complexity emerging in the process of differentiation, which can be related to the morphofunctional cell organization. A hypothesis is formulated that neuronal differentiation and synapse formation have a lot in common with the process of tumorigenesis. They are qualitatively described by the same Gompertz function of growth and take place in the fractal space-time whose mean temporal fractal dimension is lost during progression.

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.