Faculty of Life Sciences, University of Manchester, Michael Smith Building, Oxford Road, Manchester M13 9PT, UK.
BMC Evol Biol. 2011 Sep 29;11:280. doi: 10.1186/1471-2148-11-280.
Studies of symmetric structures have made important contributions to evolutionary biology, for example, by using fluctuating asymmetry as a measure of developmental instability or for investigating the mechanisms of morphological integration. Most analyses of symmetry and asymmetry have focused on organisms or parts with bilateral symmetry. This is not the only type of symmetry in biological shapes, however, because a multitude of other types of symmetry exists in plants and animals. For instance, some organisms have two axes of reflection symmetry (biradial symmetry; e.g. many algae, corals and flowers) or rotational symmetry (e.g. sea urchins and many flowers). So far, there is no general method for the shape analysis of these types of symmetry.
We generalize the morphometric methods currently used for the shape analysis of bilaterally symmetric objects so that they can be used for analyzing any type of symmetry. Our framework uses a mathematical definition of symmetry based on the theory of symmetry groups. This approach can be used to divide shape variation into a component of symmetric variation among individuals and one or more components of asymmetry. We illustrate this approach with data from a colonial coral that has ambiguous symmetry and thus can be analyzed in multiple ways. Our results demonstrate that asymmetric variation predominates in this dataset and that its amount depends on the type of symmetry considered in the analysis.
The framework for analyzing symmetry and asymmetry is suitable for studying structures with any type of symmetry in two or three dimensions. Studies of complex symmetries are promising for many contexts in evolutionary biology, such as fluctuating asymmetry, because these structures can potentially provide more information than structures with bilateral symmetry.
对称结构的研究为进化生物学做出了重要贡献，例如，通过使用波动不对称作为发育不稳定性的度量，或研究形态整合的机制。大多数关于对称和不对称的分析都集中在具有双侧对称性的生物体或部分上。然而，这不是生物形状中唯一类型的对称，因为在植物和动物中还存在许多其他类型的对称。例如，一些生物体具有两条反射对称轴（二辐射对称；例如，许多藻类、珊瑚和花）或旋转对称轴（例如，海胆和许多花）。到目前为止，还没有一种用于分析这些类型对称的通用形状分析方法。
我们将目前用于双侧对称物体形状分析的形态计量方法推广，以便可以用于分析任何类型的对称。我们的框架使用基于对称群理论的对称数学定义。这种方法可用于将形状变化分为个体之间对称变化的分量和一个或多个不对称分量。我们使用具有模糊对称性的群体珊瑚的数据来说明这种方法，因此可以以多种方式对其进行分析。我们的结果表明，在这个数据集不对称变化占主导地位，其数量取决于分析中考虑的对称类型。
分析对称和不对称的框架适用于研究二维或三维中具有任何类型对称的结构。复杂对称的研究在进化生物学的许多背景下都很有前途，例如波动不对称，因为这些结构可能比具有双侧对称性的结构提供更多的信息。
