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 , and relaxation time{tau} . 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{tau} = (1/{omega}) {partial}E/{partial}E, where{omega} 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. SignificanceHow molecular-scale activity gives rise to tissue-scale form is a central challenge in biological physics. Although growth is fundamentally a non-equilibrium mechanical process, experimental measurements at the nanoscale have not been directly connected to the constitutive parameters that govern morphogenesis. We introduce a framework that converts dynamic atomic force microscopy maps of storage and loss moduli into spatially resolved fields of stiffness, viscosity, and relaxation time in living cell walls. By revealing that mechanical relaxation is encoded in the local coupling between elastic storage and viscous dissipation, our work provides a route from nanoscale rheology to growth-relevant mechanical timing. This establishes a quantitative bridge between molecular remodeling and continuum mechanics, enabling direct experimental constraints on multiscale theories of morphogenesis.