Scale-invariant pattern in the alga Micrasterias

Abstract

Spatial structures arise in a variety of different physical, chemical and biological systems. A striking example is found during morphogenesis in the single-celled alga Micrasterias, where cell extensions called lobes branch repeatedly to produce a highly regular, apparently self-similar pattern. Lobe outgrowth in Micrasterias is thought to be controlled by the local accumulation of growth determinants at the lobe tips. These tip-growth sites undergo successive spatial bifurcations, leading to the recursively branched, final cell form. I have tested for scale invariance of this form, by measuring the distribution of tips as a function of position along the cell perimeter in mature Micrasterias cells of four different species. This tip distribution should reflect the steady-state distribution of growth determinants at the end of the spatial bifurcation process. For each cell measured, the distribution of tips resembled a Cantor set with three levels of constant, nested scaling. Significantly, roughly the same scale factor (∼3.0) was found at each scaling level in individual cells, and among different cells in each of the four species measured. These data suggest that scaling by this factor is intrinsic to the pattern formation process in Micrasterias.