School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA.
Proc Natl Acad Sci U S A. 2010 Feb 23;107(8):3356-60. doi: 10.1073/pnas.0911575107. Epub 2010 Feb 16.
Evolution by natural selection has resulted in a remarkable diversity of organism morphologies that has long fascinated scientists and served to establish the first relations among species. Despite the essential role of morphology as a phenotype of species, there is not yet a formal, mathematical scheme to quantify morphological phenotype and relate it to both the genotype and the underlying developmental genetics. Herein we demonstrate that the morphological diversity in the beaks of Darwin's Finches is quantitatively accounted for by the mathematical group of affine transformations. Specifically, we show that all beak shapes of Ground Finches (genus Geospiza) are related by scaling transformations (a subgroup of the affine group), and the same relationship holds true for all the beak shapes of Tree, Cocos, and Warbler Finches (three distinct genera). This analysis shows that the beak shapes within each of these groups differ only by their scales, such as length and depth, which are genetically controlled by Bmp4 and Calmodulin. By measuring Bmp4 expression in the beak primordia of the species in the genus Geospiza, we provide a quantitative map between beak morphology and the expression levels of Bmp4. The complete morphological variation within the beaks of Darwin's finches can be explained by extending the scaling transformations to the entire affine group, by including shear transformations. Altogether our results suggest that the mathematical theory of groups can help decode morphological variation, and points to a potentially hierarchical structure of morphological diversity and the underlying developmental processes.
自然选择导致了生物体形态的显著多样性，这长期以来一直令科学家着迷，并为建立物种之间的最初关系提供了依据。尽管形态作为物种的表型具有重要作用，但目前还没有一种正式的、数学的方案来量化形态表型，并将其与基因型和潜在的发育遗传学联系起来。本文中，我们证明了达尔文雀类的喙部形态多样性可以用仿射变换的数学群来定量解释。具体来说，我们表明，地雀属（Geospiza）的所有鸟喙形状都是由缩放变换（仿射群的一个子群）相关联的，而树雀、寇克斯雀和莺雀（三个不同的属）的所有鸟喙形状也存在同样的关系。这种分析表明，这些组内的喙形状仅在其尺度上有所不同，例如长度和深度，这些尺度由 Bmp4 和钙调蛋白基因控制。通过测量地雀属物种喙原基中的 Bmp4 表达，我们提供了喙形态和 Bmp4 表达水平之间的定量关系图。通过将缩放变换扩展到整个仿射群，并包括剪切变换，可以解释达尔文雀类喙内的所有形态变化。总之，我们的研究结果表明，群论的数学理论可以帮助解码形态变化，并指出了形态多样性和潜在发育过程的潜在层次结构。
