Modeling Properties of Diatoms with Fibonacci Growth Using Lindenmayer Systems

In most cases, the sizes of the daughter cells of diatoms follow the MacDonald–Pfitzer rule, whereby in many species all diatoms divide once in each generation. In contrast, there are division schemes in which the smaller or larger daughter cell is delayed in its division by one generation and therefore leads to Fibonacci growth. Several properties of diatoms, especially in chain-like colonies, that exhibit such delayed division can be modeled by Lindenmayer systems. These include, above all, the size and orientation of the diatoms. Certain sequences of properties, such as the differences in size indices of neighboring diatoms, are aperiodic and represent self-similar fractal structures. For the division schemes studied, explicit solutions can be found for the number of diatoms of a certain size in each generation. For the experimental differentiation of the division schemes in a diatom chain, in addition to the observation of the division processes over several generations, methods are available that only require the analysis of the structure of a sufficiently large sample. This includes the investigation of the differences in the sizes of neighboring diatoms, the orientations of the diatoms and the frequencies of size indices in a culture. These methods provide a toolbox for investigating diatom properties, applicable to the division models described. Additionally, a mathematical framework is presented that has the potential to be transferable to other properties and other division schemes.