A mechanical cusp catastrophe imposes a universal developmental constraint on the shapes of tip-growing cells

Understanding the mechanistic basis for cell morphology is a central problem in biology. Evolution has converged on tip growth many times, yielding filamentous cells, yet tip-growing cells display a range of apical morphologies. To understand this variability, we measured the spatial profiles of cell-wall expansion for three species that spanned the phylogeny and morphology of tip-growth. Profiles were consistent with a mechanical model whereby the wall was stratified and stretched by turgor pressure during cell growth. We calculated the spatial profiles of wall mechanical properties, which could be accurately fit with an empirical two-parameter function. Combined with the mechanical model, this function yielded a "morphospace" that accounted for the shapes of diverse tip-growing species. However, natural shapes were bounded by a cusp bifurcation in the morphospace that separated thin, fast-growing cells from (nonexistent) wide, slow-growing cells. This constraint has important implications for our understanding of the evolution of tip-growing cells.