A closed-loop mathematical structure of mechanics-turnover coupling for mechanical adaptation in living systems

Mechanical adaptation underlies mechanical homeostasis by allowing living systems to restore characteristic mechanical variables under sustained perturbations. Across biological scales, turnover-mediated remodeling enables mechanical adaptation by continuously renewing internal structures under load. Despite extensive progress in this field, it remains to be established what closed-loop mathematical structure of mechanics-turnover coupling is sufficient to guarantee homeostasis and how the characteristic adaptation timescale emerges from this coupling. Here, we identify the minimal mathematical structure of closed-loop mechanics-turnover coupling, providing a unifying description of mechanically adaptive remodeling across scales. We derive an analytical expression for the adaptation timescale as a function of the coupling between internal mechanical parameters and turnover kinetics, enabling direct cross-system comparison. To isolate this structure, we formulate a dynamical model linking mechanics and turnover, and establish conditions under which the closed-loop dynamics exhibit integral action. Specifically, our model describes how deviations in the mechanical state modulate the turnover of an internal structural state, and the renewed structure feeds back onto mechanics in a negative-feedback direction, driving recovery toward a reference state. We define systems satisfying this structure as Feedback Adaptive Turnover-mediated Environment-Dependent (FATED) systems. As an experimental example, we formulate mechanical adaptation in terms of mechanically regulated actin turnover. With the generalization of this architecture, we evaluate cross-system consistency by comparing reported adaptation and turnover timescales across representative remodeling systems.