Lepidoptera demonstrate the relevance of Murray’s Law to circulatory systems with tidal flow

Background Murray’s Law, which describes the branching architecture of bifurcating tubes, predicts the morphology of vessels in many amniotes and plants. Here, we use insects to explore the universality of Murray’s Law and to evaluate its predictive power for the wing venation of Lepidoptera, one of the most diverse insect orders. Lepidoptera are particularly relevant to the universality of Murray’s Law because their wing veins have tidal, or oscillatory, flow of air and hemolymph. We examined over one thousand wings representing 667 species of Lepidoptera. Results We found that veins with a diameter above approximately 50 microns conform to Murray’s Law, with veins below 50 microns in diameter becoming less and less likely to conform to Murray’s Law as they narrow. The minute veins that are most likely to deviate from Murray’s Law are also the most likely to have atrophied, which prevents efficient fluid transport regardless of branching architecture. However, the veins of many taxa continue to branch distally to the areas where they atrophied, and these too conform to Murray’s Law at larger diameters (e.g., Sesiidae). Conclusions This finding suggests that conformity to Murray’s Law in larger taxa may reflect requirements for structural support as much as fluid transport, or may indicate that selective pressures for fluid transport are stronger during the pupal stage—during wing development prior to vein atrophy—than the adult stage. Our results increase the taxonomic scope of Murray’s Law and provide greater clarity about the relevance of body size. Supplementary Information The online version contains supplementary material available at (10.1186/s12915-021-01130-0).


Continuous structural stiffness law for load carrying veins
In addition to transporting fluid, insect veins also have a structural function. The veins have to support the aerodynamic and inertial forces generated during flight. The corrugated profile of insect wings is a key feature that increases their load carrying ability [6,7]. Although insect wings must carry load, insects cannot afford their wings to break when overloaded due to impact with the natural environment or turbulent gusts [8]. To avoid structural failure, in particular during hard impact, insect wings have resilin integrated in a number of vein joints [9,10] that enable them to effectively buckle instead of break when a critical load is reached [11]. Considering this morphological specialization, we assume that vein bifurcations are not optimized for continuous strength, because they fail safely beyond a critical load. Instead we identify the critical role of providing continuous stiffness to bear the (below critical) aerodynamic and inertial loads without deforming too much. This enables insect wings to attain and maintain an effective aerodynamic shape. The vein's cross-sectional shape critically contributes to the bending stiffness of the wing (as well as torsional stiffness through differential bending). Here we simply consider the vein's contribution to stiffness via the second moment of area in two cases, (1) negligible corrugation and (2) stiffness dominating corrugation. For these two limit cases we determine the relationship between the vein radii before and after the junction for a continuous stiffness contribution along the bifurcating vein.
1. Continuous stiffness along a bifurcating vein in a wing with negligible corrugation. In this case the requirement for a continuous vein stiffness is I 0 = I 1 + I 2 , in which I is the second moment of area of the vein's cross-sectional shape resisting bending. Given that the second moment of area of a thin walled vein is I = πr 3 t [12], we find πr 3 0 t 0 = πr 3 1 t 1 + πr 3 2 t 2 . So assuming the vein thickness remains approximately constant across the bifurcation, t 0 = t 1 = t 2 , we find the radii across the bifurcation must relate as in Murray's Law: r 3 0 = r 3 1 + r 3 2 .
2. Continuous stiffness along a bifurcating vein in a wing with dominant corrugation. This is the case when h c /r v 1 ; the stiffness contribution of the corrugation height, h c , with respect to the neutral line, dominates the contribution from the vein's radius, r v . In this case the requirement for a continuous vein stiffness is again I 0 = I 1 + I 2 , in which I is the second moment of area contribution of vein's cross-sectional area located at the extremities of the corrugated wing. The extremities enable the veins to contribute maximal geometric bending stiffness. Given that the second moment of area of a thin walled vein located at the extremity of the corrugation is I = h 2 c · 2πrt [12], we find h 2 c0 2πr 0 t 0 = h 2 c1 2πr 1 t 1 +h 2 c2 2πr 2 t 2 . Under the reasonable assumptions that the corrugation height and vein thickness remain approximately constant over the bifurcation we find r 0 = r 1 + r 2 .
To determine which of the two cases best approximates the vein stiffness relationship across the bifurcation, we consider the full second moment of area expression without assuming either one dominates: The ratio ξ = I v /I c = r 2 /2h 2 c determines which case applies. For ξ 1 the wing's stiffness is corrugation dominated, for ξ 1 the second moment of the vein's circular cross section area dominates the wing's stiffness. Given the wing corrugation height is typically several vein radii in highly corrugated paleopteran insect wings [6,7], it is most parsimonious to assume case 2 represents the ancestral pterygote state for continuous vein stiffness contribution across a bifurcation: r 0 = r 1 + r 2 . For effectively flat wings (compared to the vein radius) the Murray-like r 3 0 = r 3 1 + r 3 2 law applies for continuous stiffness.
In addition to the radius power law, the orientation of the mother vein along the spanwise direction and in particular the bifurcation angle of the daughter vein will modify the continuous stiffness evaluation. The second moment of area of the associated effectively elliptical thin-walled cross sections can be found in [12].
Here we chose to not include these angles in our second moment of area calculation in order to keep our analysis as simple as possible and because there is no obvious correlation between the radius and bifurcation angle ( Figure ??h-n).
The wings of Neoptera are less corrugated than those of Paleoptera [6,13], and the extent of corrugation can vary within an order [14]. In the few species of Lepidoptera whose wings have been examined in this context, corrugation is minimal-especially among branches of the R vein [15].