||Study on biological transport network utilizing plasmodium of Physarum polycephalum
Vein networks found in various organisms such as leaf veins, bronchial tree, and blood ves-sels have a crucial role to transport substances fast and efficiently, so that it is expected to provide physical signi cance on morphology by revealing relationship between geometrical structure of transport network and transport ability. However, most biological transport networks have difficulties on culture methods in experimental environment, discouraging repetitive experiment. Moreover, design of transport network is a multi-objective prob-lem such as total volume cost of network and coverage of region, which make it harder to evaluate trasport ability. Meanwhile, a true slime mold, Physarum polycephalum, attracts attention recently as a model organism of biological transport network. In a stage of its life-cycle called plas-modium, it forms a giant multi-nuclear single amoeboid cell, creating tubular structure for ow of prosoplasms. This tubular network also has an important role of locomotion and distribution of nutrients, and therefore, if plasmodium touches some food sources, vein network connecting them emerges overnight. Some researches exploit this behavior to solve geometrical problems such as maze and design of railway network and analyzed mainly the connectivity of network, whilst the view point of transport has been rarely discussed ever. For the purpose of evaluation of transport ability, we developed an experiment uti-lizing Physarum polycephalum and analyzed the results. In the experiment, a uniform plasmodium without vein structure was put on a con ned space created on agar, which had one narrow exit to a food source. Thus, as the plasmodium moves through the exit, a vein network to transport body mass emerges and develops with the evacuation from the arena. This experiment enables us to observe emergence of transport network and draining process from a two-dimensional area to one point by the network. As an analysis of transportation by the emerged network, we estimated plasmodial thickness and its decreasing speed at each point of the con ned space from the time-lapse images. The result showed that the plasmodium was drained with the same time course despite the distance from the exit, except for the points of main veins. Moreover, this property was con rmed for different shapes of arena, which implied shape-adaptive ability of plasmodium to construct transport networks causing homogeneous draining. To investigate what factors originate in this homogeneous draining, we analyzed cover-age rate of network to arena, suggesting to use Hausdorff distance as an index of coverage. Hausdorff distance between a network and an arena means, in brief, the least distance by which any point in the arena can reach a point of the network. Therefore, lower Hausdorff distance implies higher coverage, and vice versa. The calculation for plasmodial network showed Hausdorff distance of 10% to 20% of diameter of the arena. This higher coverage of networks may be a factor of the homogeneous draining. Furthermore, we tried uid dynamical approach under the assumption of Hagen-Poiseuille ow, according to which ow rate and applied pressure of a tube consisted proportional relationship of which coefficient, called conductivity, was determined by the length and the radius. Under this assumption, we calculated hydrodynamical conductivity from each point of the network to the exit. Then the contour line of conductivity expanded similarly to the shape of arena, that is, the hydrodynamic accesibility is not developed with eucridian distance, which supports the mass draining independent of distance. In addition to these analysis, we validated a well-known rule of transport network, Murray's law, which was stated by Murray in 1926. According to the law, when a artery with radius r0 rami es into arteries with radii r1 and r2, they have relationship r30 = r31+r32. Because Murray derived this relationship by considering minimization of frictional loss of ow and volume cost, establishment of the law implies optimization of them. Therefore, we estimated the exponent of the relationship for plasmodial network in the experiment, and 95% con dence interval [2:53; 3:29] was obtained. This result supports Murray's law on vein network of Physarum polycephalum and implies optimization of trade-off between frictional energy loss and total volume cost. To investigate the mathematical background of these network formation, we applied a model called current-reinforcement to simulate the experimental environment. In the model, slime mold is expressed as a random mesh network, and quantity D of each edge, which corresponds to diameter of edge, develops according to a formula dD=dt=QD, where Q and are ow rate and a positive parameter, respectively. In other words, a vein with larger ow is thickened, and with lesser ow is thinned. Here, because it needs information of sources and sinks to nd ow rate of each edge, we set every node except for the exit as a source of the same quantity and the exit as the only one sink. With these settings, the simulation converged to a steady state, in which tree structure without loop is obtained for > 1, and relatively uniform vein structure with loops for 1. In the similar manner to the analysis of experimental data, the Hausdorff distance was calculated for the simulated networks. The result shows that networks with > 1 gave similar values to the experimental ones, while 1 had worse coverage. Especially, networks of = 4=3 are found to be optimized according to evaluation of friction energy and total volume. In fact, the parameter = 4=3 is closely bounded with Murray's law which was derived from optimization of friction energy and total volume cost. Assuming a steady state of current-reinforcement, we can derive proportional relationship between Q and r4= , so that = 4=3 corresponds to Murray's law. Hence, the experimental validation of Murray's law means suitability of = 4=3 as a parameter of current-reinforcement. This apparent parameter correspondence has not been discussed on studies of current-reinforcement model and is, therefore, new discovery on plasmodial network. Moreover, the revealed relationship between models of blood vessels and slime mold is important result showing usefulness of slime mold as a model organism of morphological study.
Hokkaido University（北海道大学）. 博士(生命科学)