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The Uncanny Symmetry of Some Diatoms and Not of Others: A Multi-Scale Morphological Characteristic and a Puzzle for Morphogenesis^*

       Janice L. Pappas1+, Mary Ann Tiffany2 and Richard Gordon3,4

       1Department of Mathematics, Michigan Math and Science Scholars, University of Michigan, Ann Arbor, MI, United States

       2Dept. of Biology, San Diego State University, San Diego, CA, United States

       3Gulf Specimen Marine Laboratory & Aquarium, Panacea, FL, United States

       4C.S. Mott Center for Human Growth & Development, Department of Obstetrics & Gynecology, Wayne State University, Detroit, MI, United States

       Abstract

      Organism symmetry via body plan is the balanced arrangement of structural features (internally or externally) that is definable in terms of dimension (2D or 3D) and geometry (contours, surfaces, boundaries). Symmetry is evident at the endpoint of morphogenesis and at each scale, from genes to cells to organism to population or higher taxonomic level. Symmetry may change from juvenile to adult stages as well as between sexual and asexual reproduction. The arrangement of morphological features reiterates symmetry antecedents so that at any given time, degrees of symmetry may be directed, random or chaotic outcomes signifying degrees of instability throughout morphogenesis.

      With their geometric surface ornamentation, diatoms exhibit a high degree of rotational and/or reflective symmetry depending on the ontogenetic path, either commencing with an annulus or a sternum for centrics and pennates, respectively. Sometimes this symmetry is remarkable, whence we call it “uncanny symmetry”. Using centric diatom examples, surface features are used in uncanny rotational symmetry assessment via digital image processing techniques and include analysis of vegetative valve formation stage, initial valves, and abnormally developed cells. Quantitatively, entropy as digital image information content is related to symmetry states and is used to determine random and chaotic instability in morphogenesis.

      Keywords: Rotational symmetry, reflective symmetry, centric diatoms, diatom morphogenesis, fluctuating asymmetry, uncanny symmetry, entropy, developmental instability

2.1 Introduction

      Morphology is a conduit to assessing evolutionary relationships as well as ecological interactions among organisms. Different aspects of morphology provide information on size, shape, color, pattern of the external form of an organism, as well as internal anatomical features. Morphology is informative regarding commonalities among organisms at the species level in population studies and among larger taxonomic groupings from generic to phylum level in macroevolutionary studies. At the phylum level, body plan (bauplan) assessment enables morphological analysis across taxonomic levels, and characteristics of phylogenetic importance can be gleaned as comparative markers of continuity in arrangement of structural form.

      A body plan is the group of structural characteristics of a phylum that is shared via development [2.41]. Genetic and embryogenetic processes manifest themselves throughout development, culminating in the adult form as the end member of morphogenesis. The connection among morphogenetic forms throughout development is exemplified via symmetry and symmetry breaking [2.3, 2.19, 2.24]. It is symmetry that enables assessment of body plans not only among developmental stages within a given phylum but also among phyla exhibiting commonality in some aspect of structural form.

      Symmetry as an external morphological characteristic is determined via perception or observation of the “balance” of an organism. That is, we perceive or observe whether the organism exhibits some evenly distributed “likeness” of form, albeit as a reflective [2.159] or rotational [2.159] impression. To determine the “balance” or “likeness” of a given form, we necessarily perceive or observe the geometry of the organism explicitly or implicitly as an impression. To arrive at the point of recognition of the form as an identifiable entity, we perceive such characteristics as degree of curvedness (roundness) or angularity (sharpness) as well as distinct surface peaks, valleys, or saddles (local maxima or minima as extrema), if present. Surface geometry plays a key role in our perception of the quality of form [2.104–2.108]. Contours and edges are the pieces of information we gather and connect to the extrema or points (dots) that enable us to concoct a contiguous picture or image about the form we perceive or observe. This provides the initial point of determination in the assessment of symmetry of a given organism. The boundary shape of an organism as a composite of contours is instrumental in symmetry determination.

The product of morphogenesis culminates in a fixed or adult shape in which the organism will exhibit specific external and internal symmetries, usually having symmetries common to a higher taxonomic group. Structural aspects of morphology such as symmetry are evident from the morphogenetic process of each succeeding step in development. Arrangement of morphological features reiterate internal and external symmetry antecedents so that to some extent, symmetry is a directed process with regard to morphogenesis [2.54]. Symmetry is not only evident at the endpoint of morphogenesis but also at each scale of the materials involved, from