Emergence of a Low-Dimensional Energetic Landscape Governing Calcium Binding in Troponin C

Calcium binding regulates muscle contraction through coordinated structural and dynamical mechanisms, yet a unified description of how mutations perturb binding energetics remains incomplete. Mutational effects on protein are often described as high-dimensional and difficult to interpret mechanistically. Here we show that calcium-binding in troponin C can be represented within a reduced low-dimensional framework. We constructed physics-informed descriptors capturing geometric proximity to the coordination environment, directional properties of the local electric field, and dynamical communication derived from elastic network models. Structural, electrostatic, and dynamical domains independently achieve Pearson correlations of R _p =0.84, 0.70, and 0.81, respectively, with experimentally measured free energies. Integration of these domains yields a composite reaction coordinate that predicts mutational energetics with strong accuracy (R _p =0.92 under leave-one-out validation). Projection onto this coordinate reveals a hierarchical organization in which structural proximity defines the dominant energetic axis, while electrostatic alignment and dynamical communication introduce orthogonal modulation corresponding to distinct energetic regimes. These results suggest that complex mutational responses can be interpreted through low-dimensional coordinates that unify multiple interacting mechanisms, providing a physics-informed framework for understanding mutation-induced changes in regulatory proteins.