John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA.
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, Hong Kong.
Proc Biol Sci. 2023 Sep 27;290(2007):20230420. doi: 10.1098/rspb.2023.0420.
Adaptive avian radiations associated with the diversification of bird beaks into a multitude of forms enabling different functions are exemplified by Darwin's finches and Hawaiian honeycreepers. To elucidate the nature of these radiations, we quantified beak shape and skull shape using a variety of geometric measures that allowed us to collapse the variability of beak shape into a minimal set of geometric parameters. Furthermore, we find that just two measures of beak shape-the ratio of the width to length and the normalized sharpening rate (increase in the transverse beak curvature near the tip relative to that at the base of the beak)-are strongly correlated with diet. Finally, by considering how transverse sections to the beak centreline evolve with distance from the tip, we show that a simple geometry-driven growth law termed 'modified mean curvature flow' captures the beak shapes of Darwin's finches and Hawaiian honeycreepers. A surprising consequence of the simple growth law is that beak shapes that are not allowed based on the developmental programme of the beak are also not observed in nature, suggesting a link between evolutionary morphology and development in terms of growth-driven developmental constraints.
与鸟类喙部多样化相关的适应性辐射，形成了多种形式，从而实现了不同的功能，以达尔文雀和夏威夷蜜旋木雀为例。为了阐明这些辐射的本质，我们使用多种几何度量来量化喙的形状和颅骨的形状，这些度量使我们能够将喙形状的可变性简化为一组最小的几何参数。此外，我们发现喙形状的两个度量——宽度与长度的比值以及标准化的锐化率（喙尖端附近的横向喙曲率相对于喙基部的增加）——与饮食密切相关。最后，通过考虑喙中心线的横向截面如何随距离尖端的变化而变化，我们表明，一种称为“修正平均曲率流”的简单几何驱动生长规律可以捕捉到达尔文雀和夏威夷蜜旋木雀的喙形状。简单生长规律的一个令人惊讶的结果是，根据喙的发育程序不允许的喙形状在自然界中也没有观察到，这表明在生长驱动的发育约束方面，进化形态学和发育之间存在联系。
