Energetic analysis of Na⁺/K⁺-ATPase using bond graphs

The sodium-potassium ATPase (NKA) accounts for 19-28% of the total ATP consumption in mammals and is critical for maintaining ion homeostasis. Understanding its energetic efficiency is essential for comprehending cellular physiology and pathophysiology. We develop bond graph models of the NKA that ensure thermodynamic consistency by enforcing conservation of mass, charge, and energy. A simplified 6-state model captures biophysics comparable to a 15-state model while remaining computationally tractable. Through detailed energetic analysis, we demonstrate that under physiological conditions for the mammalian cardiac cell, approximately 66% of the energy from ATP hydrolysis is stored as chemical energy in ion gradients, 12% as electrical energy in the membrane potential, and 22% is dissipated as heat, yielding an overall efficiency of ∼78%. We investigate how the free energy of ATP hydrolysis (ΔG_MgATP), intracellular Na⁺, and extracellular K⁺ affect NKA efficiency and activity. A critical threshold exists at ΔG_MgATP ≈ - 48 kJ/mol below which chemoelectrical transduction drops dramatically, consistent with NKA inhibition under ischemic conditions. The bond graph framework enables quantitative comparison of different NKA models and provides a systematic approach for analyzing ion pumps.