Fig 1.
Characterisation of the wing vein network through topological transformation.
(a) Schematic of the wing venation in D. melanogaster. (b) Topologically transformed vein network. The black or coloured lines denote the wing veins, and circles indicate the bifurcations of the veins. The veins with numbers starting with “VE” and “VB” are the edge veins and base veins, respectively. “VC” represents the connecting veins. The light blue and dark blue veins are separated into edge- and base-side segments that are respectively subscripted with “E” and “B”. The red vein that is captioned “PCV” is the posterior cross vein (PCV). The numbers starting with “M” in brackets represent the membrane cells. The grey-coloured cells with subscripted “E” and “B” indicate the edge- and base-side portions of the membrane cell divided by the PCV. The circles at both ends of the network captioned “IN” and “OUT” denote the inlet and outlet of haemolymph flow.
Table 1.
Geometries of the wing veins of D. melanogaster.
Table 2.
Surface and contact areas for D. melanogaster wing membrane cells.
Fig 2.
Illustration of the numerical modelling of a fluidic circuit with five resistors and three closed loops.
The white circles denote nodes where fluid mass is conserved. The white rectangles denote the hydraulic resistors. The straight arrows indicate the assumed flow directions. The looped arrows denote the closed loops of the circuit whose sums of pressure losses are zero. Red physical quantities are unknown values in our calculation to calculate in the wing vein network, whereas black ones are known.
Fig 3.
Schematic modelling of the wing membrane cells, M6, divided by the posterior cross vein (PCV).
(a) Actual membrane cells. (b) Schematic model of the membrane cells. The membrane cell containing the PCV is surrounded by the edge vein, VE6; base vein, VB6; and anterior and posterior connecting veins, VC5 and VC6, respectively. The base vein and segments of connecting veins are represented by straight-line segments connecting their ends. The red circles labelled “Node A” and “Node P” indicate the anterior and posterior connections of the PCV, respectively, separating these connecting veins. The lengths of their base-side segments, VC5B and VC6B, are respectively designated as the variables lA and lP.
Fig 4.
Distribution of AS / AC values of the wing membrane cells.
The membrane cells are coloured based on the values of AS / AC. High values are represented by dark colours, and low values by light colours.
Fig 5.
Influence of the posterior cross vein (PCV) position on AS / AC.
Variation in AS / AC values of the separated membrane cells on the edge side. The light blue vertical and dark blue horizontal axes indicate the positions of the PCV connections on the anterior and posterior connecting veins, Nodes A and P, respectively. High values are represented by dark colours, and low values by light colours. The red circle captioned “Actual position” shows the actual positions of the connections in the simplified model and the number below is the AS / AC value at the actual positions. The maximum is obtained at the position of the black dot overlying the red circle. The minimum value is obtained at the position of the white dot in the upper-right corner of the colourmap.
Fig 6.
Changes in the flow rate and flow directions owing to the presence of the PCV.
The colours of the wing veins are based on their values of Δq /qo, where Δq = q − qo, and q and qo are the flow rates in the presence and absence of the PCV, respectively. Orange indicates an increase, and grey a decrease in the values from those in the absence of the PCV. The white triangles on the veins denote the directions of the flow, which are analytically obtained, and the red triangles denote the changed flow direction owing to the PCV’s presence. The colour bar is logarithmic.
Fig 7.
Overall pressure loss in the wing with cross-veins.
“Membrane cell with CV” indicates the membrane cell where the cross vein is located. “Topology” and “Vein map” show the locations of the corresponding cross veins in the topological venation model and the actual venation as the red veins. “Decrease rate in pressure-loss” represents the decrease in the overall pressure loss due to the presence of each cross vein. “CV” in this figure denotes “cross vein”.
Fig 8.
Normalised combined resistance of a pair of edge and base veins with or without a cross vein in each membrane cell.
The vertical axis shows the combined resistance of a group of veins that is normalised by that of the sixth pair of edge and base veins without the PCV. Grey and red bars respectively denote normalised combined resistances of vein pairs without and with the alternative cross veins in corresponding membrane cells. “CV” in this figure denotes “cross vein”. For the sixth membrane cell, M6, “CV” corresponds to the PCV.