Eur. Phys. J. Special Topics 191, 159-172 (2010)
https://doi.org/10.1140/epjst/e2010-01348-2

Regular article

A model for oscillations and pattern formation in protoplasmic droplets of Physarum polycephalum

¹ Physikalisch-Technische Bundesanstalt, Department for Mathematical Modelling and Data Analysis, Abbestr. 2-12, 10587 Berlin, Germany
² Technische Universität Berlin, Institute for Theoretical Physics, Hardenbergstr. 36, 10623 Berlin, Germany

^a e-mail: markus.radszuweit@ptb.de
^b e-mail: h.engel@physik.tu-berlin.de
^c e-mail: markus.baer@ptb.de

Received: 22 November 2010
Revised: 23 December 2010
Published online: 18 February 2011

Abstract

A mechano-chemical model for the spatiotemporal dynamics of free calcium and the thickness in protoplasmic droplets of the true slime mold Physarum polycephalum is derived starting from a physiologically detailed description of intracellular calcium oscillations proposed by Smith and Saldana (Biopys. J. 61, 368 (1992)). First, we have modified the Smith-Saldana model for the temporal calcium dynamics in order to reproduce the experimentally observed phase relation between calcium and mechanical tension oscillations. Then, we formulate a model for spatiotemporal dynamics by adding spatial coupling in the form of calcium diffusion and advection due to calcium-dependent mechanical contraction. In another step, the resulting reaction-diffusion model with mechanical coupling is simplified to a reaction-diffusion model with global coupling that approximates the mechanical part. We perform a bifurcation analysis of the local dynamics and observe a Hopf bifurcation upon increase of a biochemical activity parameter. The corresponding reaction-diffusion model with global coupling shows regular and chaotic spatiotemporal behaviour for parameters with oscillatory dynamics. In addition, we show that the global coupling leads to a long-wavelength instability even for parameters where the local dynamics possesses a stable spatially homogeneous steady state. This instability causes standing waves with a wavelength of twice the system size in one dimension. Simulations of the model in two dimensions are found to exhibit defect-mediated turbulence as well as various types of spiral wave patterns in qualitative agreement with earlier experimental observation by Takagi and Ueda (Physica D, 237, 420 (2008)).