The structure and properties of optimal networks depend on the cost functional being minimized and on constraints to which the minimization is subject. We show here two different formulations that lead to identical results: minimizing the dissipation rate of an electrical network under a global constraint is equivalent to the minimization of a power-law cost function introduced by Banavar et al. [Phys. Rev. Lett. 84, 4745 (2000)10.1103/PhysRevLett.84.4745]. An explicit scaling relation between the currents and the corresponding conductances is derived, proving the potential flow nature of the latter. Varying a unique parameter, the topology of the optimized networks shows a transition from a tree topology to a very redundant structure with loops; the transition corresponds to a discontinuity in the slope of the power dissipation.
Crack patterns, as they can be observed in the glaze of ceramics or in desiccated mud layers, are formed by successive fractures and divide the two-dimensional plane into distinct domains. On the basis of experimental observation, we develop a description of the geometrical structure of these hierarchical networks. In particular, we show that the essential feature of such a structure can be represented by a genealogical tree of successive domain divisions. This approach allows for a detailed discussion of the relationship between the formation process and the geometric result. We show that--with some restraints--it is possible to reconstruct the history of the system from the geometry of the final pattern.
The forms resulting from growth processes are highly sensitive to the nature of the driving impetus, and to the local properties of the medium, in particular, its isotropy or anisotropy. In turn, these local properties can be organized by growth. Here, we consider a growing plant tissue, the shoot apical meristem of Arabidopsis thaliana. In plants, the resistance of the cell wall to the growing internal turgor pressure is the main factor shaping the cells and the tissues. It is well established that the physical properties of the walls depend on the oriented deposition of the cellulose microfibrils in the extracellular matrix or cell wall; this order is correlated to the highly oriented cortical array of microtubules attached to the inner side of the plasma membrane. We used oryzalin to depolymerize microtubules and analyzed its influence on the growing meristem. This had no short-term effect, but it had a profound impact on the cell anisotropy and the resulting tissue growth. The geometry of the cells became similar to that of bubbles in a soap froth. At a multicellular scale, this switch to a local isotropy induced growth into spherical structures. A theoretical model is presented in which a cellular structure grows through the plastic yielding of its walls under turgor pressure. The simulations reproduce the geometrical properties of a normal tissue if cell division is included. If not, a "cell froth" very similar to that observed experimentally is obtained. Our results suggest strong physical constraints on the mechanisms of growth regulation.shoot apical meristem | turgor regulation | microtubules | development | modeling
The leaf venation of dicotyledons forms complex patterns. In spite of their large variety of morphologies these patterns have common features. They are formed of a hierarchy of structures, which are connected to form a reticulum. Excellent images of these patterns can be obtained from leaves from which the soft tissues have been removed. A numerical image processing has been developed, specially designed for a quantitative analysis of this type of network. It provides a precise characterization of its geometry. The resulting data reveals a surprising property of reticula's nodes: the angles between vein segments are very well defined and it is shown that they are directly related by the radii of the segments. The relation between radii and angles can be expressed very simply using a phenomenological analogy to mechanics. This local organization principle is universal; all leaf venation patterns studied show the same behavior. The results are compared with physical networks such as fracture arrays or soap froth in terms of hierarchy and reorganization.
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