Читать книгу Diatom Morphogenesis - Группа авторов - Страница 41

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 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.