at its perimeter [2.50], as a tensegrity structure that may explain the shape of polygonal centric diatoms [2.48]. All of these models presume that the SDV membrane constrains the silica precipitation, the latter at best somehow enlarging the SDV, by unstated mechanisms. However, the opposite effect, of the precipitated silica altering the shape of the SDV, not just its size, can also be conceived.
Pattern formation diatom studies on cellular organelles during mitotic stages of ontogeny preceding valve formation indicate changes in organelle location and symmetry (cell organization) at the subcellular level as well [2.114, 2.123, 2.155]. For example, during interphase the arrangement of vacuoles on either side of the central area with respect to microtubules and the microtubule organizing center [2.113, 2.122] and spindle formation via the position of the microtubules are indicative of reflective symmetry. As cytokinesis proceeds, at metaphase chromosomes assemble around the spindle during metaphase, while at the end of anaphase complete cleavage of the spindle occurs. The position and cleavage of the spindle exhibit reflective symmetry just prior to the formation of new daughter cells [2.28] (cf. [2.1]).
Each of the modeling systems proposed for diatom valve formation and morphogenesis are incomplete in that they either have not been found to be present during all parts of the process or indirectly attempt to account for morphogenetic processes from a chemical/molecular or cytogenetic viewpoint. As has been emphasized, each “new model of diatom morphogenesis”, which, although highly attractive, is a model, awaiting to be substantiated by TEM-micrographs of sufficiently prepared cells, and experiments undertaken with the living cell” [2.124]. This remains the case. Experiments such as manipulation of salinity have morphogenetic effects, which can be interpreted at the level of silica precipitation [2.157], and the presence of (probably) sodium chloride crystals inside the silicalemma [2.46] is consistent with easy access of small ions into the SDV. The same goes for pH [2.58]. Gene regulation of silica transporters may occur at multiple levels [2.120], perhaps corresponding to multiscalar effects on morphogenesis.
In any case, no model so far accurately simulates the mature silica structure at all scales. This is not surprising, since multiscalar diatom silica structures span an extraordinary 8 orders of magnitude [2.37], and it would seem unlikely that the same mechanism prevails at all size scales (although nested hexagons made of silica spheres of decreasing sizes have been postulated, but not computer simulated, to account for three levels of structure in Coscinodiscus [2.138, 2.139]. The formation of these nanospheres has also been simulated [2.81]. A seven-state cellular automaton model for pennate diatom morphogenesis sets the striae and pores at specific spacings, by unspecified mechanisms [2.12]. At a lower scale, organic microrings of diameter 6 nm may correspond to fultoportulae [2.75]. By looking at each of the models as potential partial mechanisms for producing different aspects of symmetry during different morphogenetic stages and scales, the models may be implicitly useful in addressing how symmetry becomes evident during morphogenesis and how symmetry may be instrumental in understanding stability changes during morphogenesis. However, we presently have no global models that work at all scales. “Better models are needed, therefore, but for this we need a better formulation of the problems we are trying to solve” [2.93].
None of these models for diatom pattern formation involve the linked chemical reactions of Turing reaction-diffusion processes [2.148] which may lead to instabilities that have been invoked for multicellular morphogenesis [2.91, 2.92, 2.99]. However, Turing models have recently been simulated for multilayer situations [2.153], producing patterns somewhat similar to arrays of diatom pores so that such models are still under consideration [2.43].
In a dynamical system, changes in structural stability may be related to changes in symmetry via pattern formation changes not only over time but also at hierarchical spatial scales [2.166]. Pattern formation as an indicator of symmetry states may be evident as self-similarity as well [2.96]. The role of symmetry in developmental processes is discernable through surface pattern and shape changes over time, and these changes may be used in a morphogenetic dynamical system to explicitly assess the contribution of symmetry to structural stability.
In our work on morphogenesis with the late Antone G. Jacobson, we conceived of and repeatedly traversed an intellectual triangular loop of 1) experiments and observations, 2) computer simulation, and 3) formal mathematics, to converge to an understanding of how nature did it [2.49, 2.61–2.64]. This loop has not been traversed or ever closed even once for diatoms. Curiously, in the research with Jacobson, we invoked the bilateral symmetry of the organism we were simulating, which both reduced computer time and made the simulation match reality much better. A similar two axis symmetry was placed on a pennate diatom simulation [2.12]. However, in a real organism there is no global “force” creating its symmetry. Symmetry is a consequence of morphogenesis, not a cause, and deviations from symmetry or its perfection are hints to the underlying processes.
2.1.4 Diatoms and Uncanny Symmetry
Microorganisms are generally not well studied in terms of quantified symmetry changes over time. Diatoms have distinct amorphous silica frustules that exhibit a variety of geometric shapes and surfaces that lend themselves to analyses of symmetries. Diatoms are pigmented protists that are considered to be a monophyletic phylum. Morphogenesis is a topic of great interest not only to phycologists but also to nanotechnologists [2.42, 2.44, 2.51, 2.85]. In girdle view, diatoms are asymmetrical because of the parent-daughter cell division that occurs within the previously formed cell. In valve view, shape and surface are both open to symmetry considerations.
We have proposed that diatoms possess “uncanny symmetry” [2.134]. By uncanny symmetry is meant that (some) diatoms in valve view (may) exhibit near perfect symmetry. It was shown that by subtracting the rotated image of a given diatom from its original image, an almost completely blacked out image would result, indicating near perfect matching of image pixels, suggesting near perfect symmetry [2.134]. Such results were obtained for Aulacodiscus oregonus and Triceratium formosum var. quinquelobata (Plate 7, Figures 1–4 in [2.134]). “The apparently high degree of perfection of this noncrystalline precipitate deserves quantification, which may prove comparable with the (small) degree of imperfection of crystalline snowflakes [2.83]” [2.134].
Centric diatoms exhibit rotational symmetry in shape and surface, but other symmetries such as dihedral symmetry are present as well. A case in point is Auliscus (Plate 5, Figure 1 in [2.134]). From dihedral symmetry, reflective symmetry of this diatom may be characterized as a 180° rotational symmetry, thereby identifying rotational symmetry as a starting point in the measurement of uncanny symmetry. That multiple symmetries occur simultaneously in centric diatom may be recovered by an uncanny symmetry measurement.
As suggested in [2.134], we propose to quantify centric diatom symmetry generally and uncanny symmetry specifically using concepts from information theory and image processing methods. Because symmetry involves the acquisition of information regarding the “balance” of an organism such as a diatom, information theory is used to develop a measure of centric diatom valve uncanny symmetry.
The incongruity of diatom morphogenesis and uncanny symmetry can be summarized in the question, why should diatoms possess uncanny symmetry? Symmetry is about pattern, and morphogenesis is about process. Why should this specific process produce a near perfect valve pattern and shape in diatoms? We propose to address this question by quantifying uncanny symmetry, quantifying the relation of uncanny symmetry to stability, and analyzing these results to determine the implications that symmetry via stability has on the diatom morphogenetic process.
2.1.5 Purpose of This Study
This study was conducted to measure rotational symmetry in centric diatoms and relate changes in symmetry to instability in a diatom morphogenetic dynamical system. We will demonstrate that symmetry is measurable explicitly as a deterministic quantity, and instability may be parsed to quantify deterministic and non-deterministic behavior in a morphogenetic dynamical system. Information contained in valve morphology
