Non-Equilibrium Spatial Encoding of Nanoscale Mechanical Relaxation in Growing Plant Epithelial cells

A central problem in soft and biological physics is how molecular-scale activity and remodelling coarse-grain into emergent mechanical laws at larger scales. In growing cell walls (polymeric composite materials that surround 90% of living organisms' cells) irreversible deformation is not controlled by elastic stress alone. Instead, growth depends on the interplay between energy storage, dissipation, and the local timing of viscoelastic relaxation. Although dynamic atomic force microscopy (AFM) resolves storage and loss moduli (E', E'') of living walls at nanometre resolution, these observables have remained phenomenological and disconnected from constitutive field variables. Here we introduce a physics-based inversion framework that converts AFM measurements of epidermal cells of living Arabidopsis plants into spatially resolved fields of stiffness k, viscosity η, and relaxation time τ. By analysing the spatial gradients of E' and E'', we uncover organized mechanical heterogeneities governed by cellular confinement and stress focusing. We demonstrate that the local relaxation time is encoded directly in the coupling between storage and dissipation, yielding the pointwise relation τ = (1/ω)∂ E'/∂ E'' is the indentation frequency. This relation enables model-independent extraction of mechanical timescales and establishes a general route from nanoscale non-equilibrium rheology to continuum descriptions of growth in living and active soft materials. ### Competing Interest Statement The authors have declared no competing interest. BBSRC, BB/P01979X/1