Although, fewer taxa were analyzed with respect to forming valves, a similar result emerged when compared to external valves. Four-part valve or greater featured or shaped taxa were the most symmetric (Figure 2.14). Taxa with higher forming than external valve symmetry were Triceratium favus and Triceratium pentacrinus fo. quadrata (Figure 2.14). Different species of Asteromphalus forming valves were found to be similar in symmetry to those of Asterolampra marylandica, contrary to the situation with external valves.
Figure 2.23 Valve formation simulation comparison of symmetry and chaotic instability. See Table 2.2 for numerical values.
Figure 2.24 Valve formation simulation comparison of symmetry and random instability. See Table 2.2 for numerical values.
A comparison of symmetric and asymmetric external valve surfaces resulted in higher symmetry for Arachnoidiscus and Aulacodiscus over Asteromphalus (Figure 2.15). Steeper slopes for the symmetric genera of 1.276 and 2.2925 also indicate more symmetry change among species than Asteromphalus species with a slope of 1.1878. Generally, rotationally asymmetric diatom surfaces exhibit less symmetry than morphologically similar taxa such as Asterolampra marylandica (Figure 2.13). However, asymmetric surfaced Spatangidium arachne has higher symmetry than Asteromphalus species (Figure 2.13). The five rays of Spatangidium arachne are thin, straight and regularly spaced and sized, which may be recovered as more dihedrally symmetric than Asteromphalus species which are more variable in these surface features [2.146].
Normal and abnormal valves for Cyclotella meneghiniana were compared in terms of symmetry (Figure 2.16). Normal valves had a steeper slope than abnormal valves, indicating a greater change in symmetry. That is, abnormal valves have less symmetry or greater asymmetry than normal valves of Cyclotella meneghiniana.
Initial valves were compared to normal vegetative valves of Cyclotella meneghiniana. The average symmetry of the initial valves was much higher than average symmetry for external and forming vegetative valves (Figure 2.17). The difference in symmetry of the central area was the key to the difference between initial and vegetative valves. The undulated central area of the vegetative valve has less symmetry than the symmetry of the more uniform convex shape in initial valves.
In the valve formation simulation, least symmetric taxa ranged from Actinoptychus splendens to the highest symmetric taxon, Actinoptychus senarius (Figures 2.23 and 2.24). The more distinct and regular the valve surfaces features were present, the more symmetric was the taxon. Less symmetric taxa had the highest chaotic and random instability (Figures 2.23 and 2.24). This relation between asymmetry and instability is supported for other organisms [2.54]. The main difference in our analysis is that the sources of instability are different for different degrees of symmetry.
2.4.1 Symmetry and Scale in Diatoms
Symmetry in diatoms is a characteristic of morphology that is evident at various scales. Discrete or continuous symmetry [2.167] as well as local or global symmetry are issues of scale. In our study, we used discrete measures at a local scale. Symmetry may be scale dependent over the 3D morphology of the organism. For different structural parts, e.g., honeycomb, repeating hexagonal pores in layers [2.138], of the diatom valve, there may be a hierarchical cumulative symmetry of morphological characters at the nano to micro to macro scale, across the valve face as a result of epigenesis with respect to morphogenesis, differentiation, and growth [2.32]. Hierarchical cumulative symmetry may also change from organelle to the whole diatom cell as a result of cytokinesis and mitosis with respect to morphogenesis. At the 3D surface and boundary shape of the cell, diatoms have structural symmetry changes at each developmental stage as each subsequent whole daughter cell undergoes size diminution, auxosporulation, and production of an initial cell. Our focus on centric diatom vegetative cells and valve formation as well as vegetative and initial cell symmetry was used in assessing symmetry during diatom morphogenesis.
Nano-, micro-, or macro-scale dependent symmetry may be evident as individual or groups of structural features that may not have the same symmetry at the same time (Figure 2.3). Concomitantly, these different kinds of symmetry are measurable implicitly at multiple scales. By isolating sections of the valve face in terms of micro or nanostructure, each kind of symmetry can be measured. Scale dependency occurs if a valve formation sequence is not monotonic or linear. For example, centric diatom symmetry in Cyclotella meneghiniana may be an example of scale dependency (Figure 2.19). In some cases, scale symmetry as scale invariance may be present (Figure 2.3k). Pattern repetition at multiple scales and fractal scaling may be determined by testing for the Hausdorff dimension [2.29].
2.4.2 Valve Formation and Stability
We modeled our diatom morphogenetic valve formation system as dynamical and in equilibrium in order to test for stability behavior as it is associated to symmetry. Uncanny symmetry as a measure was useful in determining the degree of stability in valve formation. We determined that valve formation simulation proceeded primarily as deterministic chaotic instability mixed with lesser periods of stability. While instability may look like randomness, we found that for our simulation of valve morphogenesis, accretionary behavior is essentially chaotic. Valve formation PDF depicted a gamma distribution with a stretched exponential tail (Figure 2.20), potentially indicating intermittency in this dynamical system [2.115], provided that as time approaches infinity, the Lyapunov exponent is equal to zero [2.74].
Stability analysis in the form of Lyapunov systems provided a way to assess the sources of instability with regard to valve formation in diatom morphogenesis. Finding a deterministic chaotic component is noteworthy because the assumption is that only random behavior dictates instability in developmental systems [2.54], or that if chaos is present, it cannot be quantified separately from randomness [2.53]. Our results indicate that the behavior of instability varies chaotically and randomly throughout the valve formation process; however, less symmetric forms have the highest instabilities. In spite of the presence of chaotic and random instabilities, valve formation overall is a regular, stable process when considering the end product. The chaotic component of instability may be indicative of multi-scalability of symmetry during valve formation. Fluctuations of chaotic and random instability may be embedded in the valve formation system so that at times, scale symmetry as well as scale-dependent symmetries may be present.
Diatom valve formation in morphogenesis may be viewed as exhibiting different dynamical system qualities. Some elements of valve formation may be dissipative at equilibrium. Other elements may exhibit a reactive-diffusive, non-equilibrium state. Self-assembly at the molecular level may induce periodic micro- and nanoscale pore assemblies that can be modified via nanoparticles in a prescribed way in diatoms [2.165]. Silica deposition may exhibit phase separation at the chemical level of valve formation [2.138] leading to pore formation [2.154]. Elastic instability may induce buckling [2.95] as a phenomenon in diatoms during morphogenesis [2.52] in contrast to Turing instabilities via reaction-diffusion [2.91, 2.99]. Buckling may be characterized by Lyapunov exponents to determine degree of chaotically unstable behavior [2.141].
Valve formation as a dynamical system has aspects of regularity as well. Cross-costae formation exhibits a regularly spaced homogeneous growth pattern, with initiation of the valve formation process starting with a central organizing structure [2.154]. The dynamic growth of a diatom valve
