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 is spatially controlled, and sequential deposition contributing to the height of silica on the valve surface contributes to species-specific pattern formation [2.154]. Given constraints in the process, at the valve micro-scale surface, chaotic or random instability may be present, while at the nano-scale level, hexagonal or round pores with varying patterns of cribra within areolae pores may be in a regular periodic stabilizing pattern. Morphometric noise of multiple traits has weak control over pore shape and size. Development of a morphological character such as a pore is affected by inhibitors in contrast to the spatial arrangement of pores [2.154]. With further study, parsing the relation between unstable and stable states at multiple scales may be useful in understanding symmetry breaking [2.41, 2.102] during valve morphogenesis.
2.4.3 Symmetry, Stability and Diatom Morphogenesis
Diatoms and valve morphogenesis are a combination of static structure formation at specific time junctures and a dynamic progression to the formation of the vegetative cell. As static structures, diatoms have high symmetry, termed uncanny symmetry. As selforganizing siliceous structures, they are some of the most stable static structures. Entropy as an ensemble measure [2.84] is instrumental in measuring of silica-deposited dynamics during valve formation that lead to the various regularly created geometric patterns defining species-specific valve surfaces.
The dynamical process from a more to less chaotic unstable state may be a characterization of self-organization. In the valve formation process, self-organization is the recovery from a chaotic unstable state, even though the chaotic state could be construed to be “normal.” Symmetry is a reflection of these states. In contrast, abnormally developed valves could be construed to be in a higher “abnormal” chaotic state, and stress during valve formation is evidenced by the resultant vegetative cell and its increasingly asymmetry state. Self-organization as the appearance of new structures may be a transition from a more chaotic, unordered to less chaotic, more ordered symmetry state, or vice versa in the case of a final abnormal symmetry state. In this way, equilibrium states during non-linear dissipative fluctuations in a dynamical system may end up being the most chaotic states [2.71].
The picture that symmetry and its relation to stability exhibits during morphogenesis is not straight-forward. In fact, no quantitative generalized overall framework for evolutionary dynamics, including stability of structural inheritance or structural integrity during reproduction for changing environmental conditions, exists with respect to diatom morphogenesis [2.131]. From our study, we give a speculative understanding and interpretation of our results on symmetry and stability as they pertain to the three major steps of centric diatom morphogenesis [2.126].
Figure 2.25 Average sequence of 24 simulated valve formation steps of symmetry for eight centric diatom taxa. Symmetry increases exponentially from the annulus to the completed valve.
Overall, symmetry increased over the valve formation simulation with highest symmetry at the finished vegetative valve (Figure 2.19). The basal layer that forms from silica deposition horizontally during the first major step of diatom valve morphogenesis, starting with the annulus, is roughly covered by the first 8 of 24 “stages” of simulation (Figure 2.18). Symmetry with the lowest values occurring at approximately the same value (Figure 2.25). For roughly “stages” 9–16 representing the second major step, vertical silica deposition is represented by the changing luminosity of pixels in the central area of the images as the indication of the formation of rays and areolae emerges (Figure 2.18). Symmetry changes become more evident, going from low to higher symmetry, yet still within a small range (Figure 2.25). For the final major step, horizontal silica deposition is represented by even more changes in pixel luminosity over the valve face in “stages” 17 to 24 where indications of the formation of cribra going toward the valve margin is present (Figure 2.18). Symmetry changes are highest in these “stages”, representing completion of silica deposition of the valve (Figure 2.25). The valve forms present during the three major valve formation steps may be viewed as symmetries functioning as silica depositional ensembles changing in direction and magnitude (horizontally to vertically to horizontally again) on the valve face. This dynamical system operates at equilibrium with exponentially changing symmetry occurring via chaotic instability during valve formation.
2.4.4 Future Research—Symmetry, Stability and Directionality in Diatom Morphogenesis
We measured and characterized instability, but what are the characterizations of stability—i.e., is stability periodic, monotonic, or something else? Characterization of stability may be informative concerning the end products of epigenetic, morphogenetic, or other processes, or at times during the morphogenetic process when apparent stasis occurs as no discernable change in morphology, symmetry, or other structural state. Characterization of the behavior of stability during morphogenesis may enable inferences about directionality and irreversibility at an evolutionary level. Research along this line has implications for studies of diatom valve formation and morphological evolution.
Improvements to assessing such systems for the factors enabling directionality with respect to stability or instability are in the offing by using Lyapunov co-vectors [2.39]. These co-vectors are associated with stable and unstable manifolds via Lyapunov exponents along trajectories in a dynamical system [2.39, 2.101], and they are also associated with entropies with respect to Lyapunov exponents [2.39, 2.110]. There are positive Lyapunov exponents associated with forward Lyapunov co-vectors, and negative Lyapunov exponents associated with backward Lyapunov co-vectors [2.77]. These Lyapunov co-vectors do not directly indicate time; however, they can indicate directionality, and inferences about directionality with respect to time may be determined. Directionality with respect to valve formation changes at species-level or higher taxonomic category may be informative about diatom evolution.
To determine directionality, irreversibility and degree of indistinguishability among symmetry states need to be assessed within a dynamical morphogenetic system. Changes in instability behavior, at which points, stability occurs, are potentially a multidimensional and multiscale problem. Chaotic instability may be a part of directionality for shorter time spans in contrast to long term random instability potentially for all dimensions and scales during the morphogenetic process.