Читать книгу Diatom Morphogenesis - Группа авторов - Страница 38

Stability analysis was performed using symmetries calculated from the valve formation simulation of cumulative concentric rings from annulus to valve margin for eight taxa. Lyapunov exponents were calculated to determine the behavior of symmetry changes. Initial conditions for valve formation are v_{Valve formation} (0) = v_annulus. Each valve formation simulation produced a series of Lyapunov exponents comprising a Lyapunov spectrum [2.166]. The largest real Lyapunov exponent was positive and characterized the overriding behavior of symmetry changes during valve formation. For all taxa, sum of the positive Lyapunov exponents signified chaotic asymmetric behavior characterizing instability (Table 2.2). The remainder of the real Lyapunov exponents in descending order were very small (large negative values), summed, and signified stability (Table 2.2). For Lyapunov exponents calculated from KS entropy, some random instability was present during valve simulation as well (Table 2.2).

Figure 2.16 Cyclotella meneghiniana symmetry for normal and abnormal valves.

Figure 2.17 Cyclotella meneghiniana symmetry values for normal forming and external vegetative and initial valves.

The change in symmetry was compared with the change in chaotic instability for taxa used in the valve formation simulation. Symmetry increased monotonically in the order of least symmetric to most symmetric: Actinoptychus splendens, Cyclotella meneghiniana, Arachnoidiscus ornatus, Aulacodiscus oregonus, Coscinodiscus sp., Arachnoidiscus ehrenbergii (1), Arachnoidiscus ehrenbergii forming valve, Asterolampra marylandica, Arachnoidiscus ehrenbergii (2), and Actinoptychus senarius (Table 2.2). Rescaling symmetry and chaotic instability values, a comparative bar graph showed that chaotic instability was more evident in less symmetric taxa Actinoptychus splendens, Cyclotella meneghiniana, Arachnoidiscus ornatus, and Aulacodiscus oregonus than higher symmetric taxa with the exception of Arachnoidiscus ehrenbergii (2) (Figure 2.23). Rescaled random instability was more evident in less symmetric taxa Actinoptychus splendens, Cyclotella meneghiniana, and Aulacodiscus oregonus as well, in contrast to higher symmetric taxa, with the exception of the Arachnoidiscus ehrenbergii forming valve (Figure 2.24). Overall, less symmetry is associated with higher chaotic and random instability (Figures 2.23 and 2.24).

Figure 2.18 Asterolampra marylandica (SanDBay02_03dx1100) as a sequence of 24 cumulative concentric rings simulating valve formation from annulus to valve margin.