Self-regulation in tip-growth: the role of cell wall ageing

We develop a model to describe the effect of cell wall ageing on the local expansion rate of tip-growing cells. Starting from an exact equation for the stationary age-distribution of the wall material, we propose a generic measure for the local expansion propensity of the wall if the ageing process is described by a constant rate Poissonian decay process. This ageing process may be either interpreted as biochemical in nature describing the finite lifetime of regulatory proteins, or as mechanical in nature describing the gradual "hardening" of the wall through cross-linking or gelation of the wall polymers. In this way we can construct models for tip-growth in which material deposition, evolving wall properties and surface expansion are self-consistently intertwined. As a proof of principle, we implement our ageing approach in two different idealised models of tip-growth, obtaining the stationary tip shapes as a function of the ageing parameter. In the first, the spatial distribution of delivery of growth material is determined by the local curvature of the cell and the growth mode is orthogonal. In the second, the growth material originates from a Vesicle Supply Center, a point-like representation of the Spitzenkörper as found in fungal hyphae, and the growth mode is isometric.