Journal of Biological Physics 17: 235-243, 1990. 0 1990 Kluwer Academic Publishers. Printed in the Netherlands. 235 Scale-Invariant Pattern in the Alga Micrasterias JAMES G. MCNALLY Institute for Biomedical Computing, and Department of Biology? Washington University 700 S. Euclid Ave., St. Louis, MO 63110, U.S.A. (Received: 14 December 1989; revised: 25 July 1990) 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 exten- sions 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. Key words. Spatial bifurcations, pattern formation, Cantor set, Micrasterias, scale invariance 1. Introduction Natural examples of spatial pattern formation abound. When one fluid displaces another of lower viscosity, fractal viscous fingers form [ 11. During Rayleigh-Binard convection, when a fluid is subjected to gravity and a temperature gradient, rolls and traveling waves arise [2]. Dendritic growth of crystals generates snowflake patterns [3]. The Belousov-Zhabotinsky chemical reaction produces target and spiral traveling waves in two dimensions [4], and intricate scroll waves in three dimensions [S]. During the development of multicellular organisms, cells differen- tiate according to well defined patterns that ultimately determine the characteristic distribution of cell types and shape of the adult [6]. A variety of theoretical approaches have been employed to model the formation of spatial structures. Nonlinear partial differential equations, such as the Navier- Stokes equations for fluid flows or reaction-diffusion equations for chemical and biological patterns [7,8], have been analysed extensively. In an effort to reduce complexity, simpler model systems such as diffusion-limited aggregation [9], cellu- lar automata [lo] and coupled-map lattices [ll] have also been investigated. Even these simpler systems can exhibit remarkably intricate spatiotemporal behavior that sometimes mirrors natural phenomena.