INTERCONNECTION OF LOCAL AND GLOBAL OPTIMIZATION IN TRANSPORTATIONAL NETWORKS
Conducting systems for long-distance delivering of liquids in plant and animal tissues possess optimal properties. The total energy expenses W for the liquid delivering and the conducting system construction and nutrition are minimal at given total volume V of the system. Solution of the optimization problem for the conducting element as long thin rigid tube gives relation between the volumetric rate Q and diameter d of the tube. In the optimal tube the shear stress at the wall is constant. When volumetric rate in a bifurcation of three optimal tubes remains constant, one can obtain the relation between the diameters of the vessels in the optimal bifurcation. Optimal geometrical relations have been found in numerous measurements in arterial systems of mammals and in venations of plant leaves. Possible mechanism of formation of optimal vessels during growth and development of the organism is connected with maintaining the constant shear stress at the vessels’ wall by mechanoreceptors of the wall. It is the mechanism of local optimality. Reasonable explanations of global optimality of arterial systems are absent nowadays. The model of the plant conducting system as a network of thin rigid tubes with permeable walls has been proposed in our previous papers. Solution of the optimization problem for both the separate vessel and network of vessels gives the same relation between the diameters of the vessels in a bifurcation that had been observed in arterial beds. As flow characteristics can not be traced in the plant vessels because they are presented by cellular walls without live cell contents, the mechanisms of interconnection of local optimality at the level of separate vessels and global optimality of the whole transportation network are connected with processes in the surrounding tissues. In the present paper the optimization problem (Wmin, V=const) as applied to both a single tube and a bifurcating system with a given geometry is proposed. The cases of permeable and impermeable vessel wall are considered. The relations between the geometrical parameters when conditions of local and global optimality coincide are obtained. It is shown that formation of optimal transportation networks in a living body (global optimality of the system) can be connected with balance between input of the liquid into the transportation system and its consumption by the tissues. The obtained relation corresponds to the form and structure interconnection that is widely discussed in biological literature.
Coresponding author e-mail: nnk_[at]bk[dot]ru
Session: Process Optimisation