Fig 1.
Introduction to pattern types and spiral wave formation.
A: (Left to right) spatial and spatiotemporal pattern examples. Spot and stripe Turing patterns both in coupled Schnakenberg elements; target wave formation from a central pacemaker and established spiral wave, both in coupled FitzHugh-Nagumo oscillators. B: Snapshots of spiral wave patterns from diverse biological systems: (left to right) cAMP signaling in a Dictyostelium discoideum colony, local contraction in neonatal rat cardiac monolayer cultures, MinD protein density in a lipid bilayer and simulated cytokine levels in a two-dimensional grid of cells. See Acknowledgments for image sources. C (upper row): The update rules of the minimal three-state cellular automaton model lead to spiral wave formation, when applied to an open wave front (consisting of a layer of excited cells, depicted in black, and an adjacent layer of refractory cells, depicted in gray). Lower row: a similar numerical experiment for the model from [6,7].
Table 1.
Sources of variability affecting details of macroscale patterns in Dictyostelium and their representations in computer-based mathematical models.
Fig 2.
Event identification in simulated and real spiral wave patterns.
Left column: snapshots of the lattices; middle column: corresponding space-time event plots; right column: top-down views of the event plots. Target wave origins are red asterisks, left- and right-handed spiral waves are blue and green diamonds, respectively. A: In the "Levine" model, spiral waves evolve early on due to colliding wavefronts, without a sustained target-wave phase. B: Development of spiral waves from the interaction of target waves in the "Goldbeter" model with a developmental path. C: Development of spiral waves from target waves in experimental Dictyostelium discoideum data. All simulated lattices are 100x100 and experimental data is rescaled to the same size.
Fig 3.
Effect of "pacemaker" location in the "Levine" model from [6,7].
Pacemaker locations are shown as grey or black crosses and spiral tip occupancy in the red scale. A: A conserved pacemaker pattern leads to different spiral wave patterns of excitability (upper row). Spiral tip occupancy over 1,000 runs favors locations free of pacemakers. The probability of observing a spiral wave at a particular spatial site is determined by the spatial distribution of cell properties. A simple pacemaker grouping gives rise to a less coherent, more variable tip occupancy in the Levine model compared to the clear geometric shaping in the "Goldbeter" model (C), based on 250 runs. D: Quantification of the relationships between pattern types in the Goldbeter model: over 100 runs, the correlation coefficient of (left to right) target wave origins to time offsets on the developmental path, spiral tips to target origins and spiral tips to time offsets; against the radius of the Gaussian filter.