Supplementary materials for the publication:
Modeling the Synchronization of the Movement of Bacillaria paxillifer by a Kuramoto Model with Time Delay
Author: Thomas Harbich
Chapter 8 in the book:
The Mathematical Biology of Diatoms [DMTH, Volume in the series: Diatoms: Biology & Applications, series editors: Richard Gordon & Joseph Seckbach]. J.L. Pappas and R. Gordon, (eds.) Wiley-Scrivener, Beverly, MA, USA: in preparation.
The supplementary material consists of videos recorded in cultures of Bacillaria paxillifer (O. F. Müller) Hendey and computer simulations.
Abstract:
Bacillaria paxillifer (also known as Bacillaria paradoxa) forms band-shaped colonies in which the diatoms can actively move relative to each other. Often a synchronization of sections or the whole colony is observed. According to the tracking data of adjacent diatoms, an easily calculable function is proposed for its relative positions as a function of time. Observations made on small colonies and individual diatoms indicate that the propulsion systems of neighboring diatoms interact in such a way that these systems can be regarded as one oscillator. The interaction includes the control of the own movement as well as the recognition of the position of the adjacent diatom. The correlated motion of the colony is assumed to be due to the interaction of neighboring oscillators, which presupposes communication between them. The hypothesis is made that the coupling of the oscillators according to the Kuramoto model is suitable for describing the movement of Bacillaria colonies. The phase shifts always observable in colonies require the assumption of a time delay in the coupling. This reproduces the typical observed dynamic movement patterns such as linear, bowed and S-shaped colonies. The Kuramoto model predicts that for oscillator frequencies which are near to each other, the movement from resting states in extreme positions starts from the ends of the colony, as is often observed. An analysis of the equations and numerical simulations allow statements to be made about the critical coupling above which the chain of oscillators oscillates synchronously and about the phenomenon of partial synchronization. An extended Kuramoto model, which additionally considers a periodic illumination whose frequency is close to that of the oscillators, suggests that the diatoms synchronize their motion with it. This could be confirmed experimentally.