Distributed self-regulation of living tissue. Effects of nonideality

arXiv:0812.0157v1 [q-bio.TO] 30 Nov 2008 Distributed self-regulation of living tissue. Effects of nonideality Wassily Lubashevsky,1, ∗Ihor Lubashevsky,2, 1, † and Reinhard Mahnke3, ‡ 1Moscow Technical University of Radioengineering, Electronics, and Automation, Vernadsky 78, 119454, Moscow Russia 2A.M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilov Str. 38, 119991 Moscow, Russia 3Universit¨at Rostock, Institut f¨ur Physik, 18051 Rostock, Germany (Dated: November 9, 2018) Self-regulation of living tissue as an example of self-organization phenomena in active fractal systems of biological, ecological, and social nature is under consideration. The characteristic feature of these systems is the absence of any governing center and, thereby, their self-regulation is based on a cooperative interaction of all the elements. The paper develops a mathematical theory of a vascular network response to local effects on scales of individual units of peripheral circulation. First, it formulates a model for the self-processing of information about the cellular tissue state and cooperative interaction of blood vessels governing redistribution of blood flow over the vascular network. Mass conservation (conservation of blood flow as well as transported biochemical com- pounds) plays the key role in implementing these processes. The vascular network is considered to be of the tree form and the blood vessels are assumed to respond individually to an activator in blood flowing though them. Second, the constructed governing equations are analyzed numerically. It is shown that at the first approximation the blood perfusion rate depends locally on the activator concentration in the cellular tissue, which is due to the hierarchical structure of the vascular network. Then the distinction between the reaction threshold of individual vessels and that of the vascular network as a whole is demonstrated. In addition, the nonlocal component of the dependence of the blood perfusion rate on the activator concentration is found to change its form as the activator concentration increases. I. DISTRIBUTED SELF-REGULATION IN ACTIVE HIERARCHICAL SYSTEMS For a wide class of biological, ecological and social sys- tems, including economic ones a generalized notion of “homeostasis” can be introduced. By this term we under- stand the internal conditions required for such a system to survive as well as its ability to maintain them. Living multicellular organisms can exist if only the tempera- ture, oxygen concentration, etc. are in certain intervals, giving rise to a large number of mechanisms controlling these conditions (for introduction see, e.g., [1]). Eco- logical communities comprising many species continue to exist under environmental perturbations due to complex “predator-prey” relationships maintaining their struc- ture. In particular, in microbial communities predator microorganisms act as homeostatic regulators to correct microbial imbalances and restor the balanced environ- ment [2], density-dependent migration sustains the pop- ulations of birds, fishes, and insects [3], complex species interaction was experimentally revealed in the dynam- ics of ecosystems of several deserts in Arizona [4]. There are a variety of social and economic systems such as large firms and enterprizes, product and service markets whose dynamics is governed by self-regulation. In these systems the behavioral standards, social norms, can be regarded also as the homeostasis characteristics (see, e.g., [5, 6, 7]). ∗Electronic address: kloom@mail.ru †Electronic address: ialub@fpl.gpi.ru ‡Electronic address: reinhard.mahnke@uni-rostock.de When one of the homeostasis parameters deviates from its normal value and comes close to the destruction threshold the system has to respond to this event in or- der to prevent a further variation of this parameter. For a system with many elements its response to such per- turbations is usually implemented via reaction of a spe- cial subsystem governing the functioning of the system as a whole. In living tissue blood flow through the ves- sel network transporting various biochemical compounds governs the homeostasis of the cellular tissue. In a rela- tively large product market agents of various levels form a trade network and perturbations in the supply-demand equilibrium are damped, in principle, via the reaction of this trade network as a whole. We will call these subsys- tems life-support networks. A life-support network is to be of hierarchical struc- ture, because it not only controls the system homeostasis but also supplies the elements with the required “nutri- ents” and withdraws their “life activity” products. Typ- ically “nutrients” come into a system centrally from the external environment, so, before getting every element their flow branches many times. In the ideal case a life-support network has to operate in such a manner that every element be supplied with the amount of “nutrients” required for its cur

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