NONLINEAR MATHEMATICAL MODELS OF HORMONAL SYSTEMS

Abstract

The paper considers various approaches to mathematical modelling of endocrine systems. The functional and operational complexity of hormonal activities turns out to be the result of the cooperation of three factors: global feedback structures on the level of glands, subtle feedback and regulatory mechanisms on the level of single cells and molecules (including messengers, receptors and functional proteins like G-proteins) and finally, coupling to other organs (predominantly to the brain, e.g. via hypothalamus). To date, it is practically impossible to construct a mathematical model comprising together all these aspects. The paper aims at providing some major building bricks to such an endeavor. In the first part we summarize some of our recent models on the gobal structure of hormonal systems, in the form of nonlinear differential equations containing delay terms; oscillatory input from the brain is taken into account. Solutions of the equations display nearly all kinds of dynamical behaviour as stable limit cycles, phase locking, quasi-periodic and chaotic motions. Special emphasis is put on developing a mathematical model for the fine-tuned sequence of hormone-induced transmembrane signalling, where agonist couples to some cellular effector via transfer-proteins — this principle is widely spread among the hormone-targeted cells and crucially involved in regulating cells' behaviour towards external stimuli, e.g. their ability to desensitize as a reaction to sustained hormonal input.