Coherent states of Gompertzian growth.
The origin of the Gompertz function G(t)=G(0)e(b/a(1-e(-at))) widely applied to fit the biological and medical data, particularly growth of organisms, organs, and tumors is analyzed. It is shown that this function is a solution of a time-dependent counterpart of the Schrödinger equation for the Morse oscillator with anharmonicity constant equal to 1. The coherent states of the Gompertzian systems, which minimize the time-energy uncertainty relation, have been found. These are eigenstates of the annihilation operator identified with the operator of growth, whereas eigenstates of the creation operator represent the Gompertzian states of regression. The coherent formation of the specific growth patterns in the Gompertzian systems appears as a result of the nonlocal long-range cooperation between the microlevel (the individual cell) and the macrolevel (the system as a whole).
Molski
Marcin
M
Department of Theoretical Chemistry, Faculty of Chemistry, A. Mickiewicz University of Poznań, ulnica Grunwladzka 6, PL60-780 Poznań, Poland. marcin@rovib.amu.edu.pl
Konarski
Jerzy
J
D000818:Animals; D055592:Biophysical Phenomena; D001703:Biophysics; D002455:Cell Division; D006801:Humans; D015233:Models, Statistical; D008962:Models, Theoretical; D009369:Neoplasms / Q000473:pathology; D013997:Time Factors