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. More detailed work is necessary to determine the contributions of stability and instability as well as symmetry at each morphogenetic stage. A more exacting representation is needed of diatom valve formation in morphogenesis concerning irreversibility of the process and indistinguishability of symmetry states and their associated valve formation stages.

If successive states of instability via symmetry states are found during diatom valve formation or morphogenesis more broadly, how does this impact diatom morphological complexity [2.107] over time? Additionally, what happens to morphological complexity over long time periods at stationarity? The expectation is that dynamical complexity is related to chaotic instability [2.22] and that increasing complexity occurs over time. Algorithmic information theory may be used to tie symmetry to complexity and to determine the role of instability over time, and in turn, gain an understanding of another facet of diatom morphogenesis.

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