Beyond Steady-State Adaptation: Evaluating the Dynamic Performance of Biological Feedback Control

Abstract Organisms rely on feedback control mechanisms to maintain key biological variables within functional ranges despite persistent perturbations. Understanding how effectively these mechanisms maintain homeostasis is fundamental, yet quantitative evaluation of feedback performance remains challenging in nonlinear biological systems. Our previously developed framework, Control Ratio ( CoRa ), addresses this challenge by isolating the contribution of a feedback interaction through controlled comparison with an otherwise identical system in which that interaction has been removed. However, because CoRa evaluates adaptation solely through steady-state responses, it cannot distinguish controllers that ultimately recover to the same state but follow markedly different transient trajectories, despite the potentially profound physiological consequences of those dynamics. Here, we introduce CoRaDyn , a framework for evaluating adaptation as a dynamic process rather than solely as a steady-state outcome. Building on the comparative strategy introduced in CoRa, CoRaDyn quantifies the cumulative effect of feedback throughout the post-perturbation response, generating a time-dependent characterization of feedback performance. This approach reveals how the contribution of feedback depends not only on system parameters but also on the time horizon over which adaptation is evaluated, allowing distinct physiological objectives—such as rapid recovery or the generation of transient pulses—to be systematically compared. We demonstrate CoRaDyn using a gene regulatory circuit implementing proportional-integral-derivative (PID) control. Whereas CoRa predicts identical perfect adaptation for all controllers containing integral feedback, CoRaDyn discriminates their transient performance, quantifies the dynamic contributions of proportional and derivative control, and reveals trade-offs that are invisible to steady-state analyses. By extending feedback evaluation from endpoints to the full adaptation process, CoRaDyn broadens the scope of biological questions that can be addressed using the CoRa framework and provides a general approach for comparing feedback architectures when transient dynamics are central to biological function. Author summary Living cells continuously experience changes in their internal and external environments. Feedback control helps them cope with these disturbances by regulating biological processes to maintain proper function. Two concepts are commonly used to describe these behaviors: homeostasis , which emphasizes the ability of a system to recover after a disturbance, and biochemical adaptation , which focuses on the transient response that allows cells to detect and respond to environmental changes. Although these phenomena are often studied separately, they can emerge from the same feedback system, simply reflecting different aspects of how feedback control shapes its response to perturbation. Most existing methods for evaluating biological feedback focus on the final outcome of the response, asking whether a system eventually returns to its original state. However, the path taken to reach that state can be equally important, as different transient responses may have profoundly different physiological consequences. In this work, we present CoRaDyn , a computational framework that quantifies the contribution of feedback as the response to a perturbation unfolds over time, rather than only at steady state. Using a gene regulatory circuit with proportional-integral-derivative (PID) control, we show that CoRaDyn distinguishes feedback strategies that appear identical under steady-state analysis while revealing differences in their dynamic performance. By evaluating adaptation as a process instead of only as an endpoint, CoRaDyn provides a general framework for studying how feedback shapes biological behavior across diverse physiological contexts.