A thermodynamic cycle for a channel dually modulated by voltage and ligand. The reference state of the system is that in which all gating charges are in their resting position and all ligand binding sites are unoccupied (at hyperpolarizing voltages and zero ligand concentration). The final state of the system is where all gating charges have moved and all ligand binding sites are occupied by the ligand (depolarizing voltages and saturating ligand concentrations). The free-energy difference between these two states can be calculated by two different ways. In Path I, Q-V is measured at zero ligand concentration followed by the ligand binding curve at highly depolarized voltages. In Path II, the ligand binding curve is measured at hyperpolarizing voltages, whereas the Q-V curve is measured at saturating ligand concentration. The free energy expressions for each of the steps are shown alongside.