Walled cells have the ability to remodel their shape while
sustaining an internal turgor pressure that can reach values
up to 10 atmospheres [1–7]. Although it is undisputed that
this requires a tight and simultaneous regulation of cell
wall assembly and mechanics, previous theoretical studies
on tip growth focused either on the mechanical behavior of
the cell wall or on its assembly [8–14]. To study the interplay
between growth and mechanics in shaping a walled cell, we
examine the particularly simple geometry of tip-growing
cells [1, 3, 15, 16], which elongate via the assembly and
expansion of cell wall in the apical region of the cell. We
describe the observed irreversible expansion of the cell
wall during growth as the extension of an inhomogeneous
viscous fluid shell under the action of turgor pressure, fed
by a material source in the neighborhood of the growing
tip. This allows us to determine theoretically the radius of
the cell and its growth velocity in terms of the turgor pressure and the secretion rate and rheology of the cell wall
material. We derive simple scaling laws for the geometry of
the cell and find that a single dimensionless parameter,
which characterizes the relative roles of cell wall assembly
and expansion, is sufficient to explain the observed variability in shapes of tip-growing cells. More generally, our
description provides a framework to understand cell growth
and remodeling in plants (pollen tubes [17], root hairs, etc.
[18]), fungi (hyphal growth [19, 20] and fission and budding
yeast [3]), and some bacteria [21], in the context of both tip
growth and diffuse growth.