Dynamics of Proteostasis and Cellular Energy Balance: Stability, Bifurcations, and Time-Scale Separation in a Two-Dimensional ODE Model

Cellular function depends on the balance between energy metabolism and protein homeostasis (proteostasis), yet the dynamical relationship between these systems remains poorly understood. While energy imbalance is often viewed as the primary driver of cellular dysfunction, proteostasis collapse may arise independently and even precede energetic failure. To investigate these interactions, we developed a two-dimensional system of ordinary differential equations describing protein folding and unfolding dynamics within the proteostasis network. Through bifurcation analysis, phase plane dynamics, and time-scale separation, we identified conditions associated with recovery, resilience, and collapse. These results provide a foundation for extending the model to include coupled energy homeostasis dynamics, allowing us to examine how energetic stress influences proteostasis stability and how disruptions in proteostasis may affect energy regulation. Future analysis will employ dimensionality reduction techniques, including Principal Component Analysis (PCA), together with stability tools such as the Routh-Hurwitz criteria to identify conditions under which failure in one system propagates to the other.