Hierarchical thermodynamic linkage analysis is used to study ion channel gating. (A) A unified thermodynamic linkage cycle for assessing the magnitude and sign of interaction between any X and Y protein (P) components. The cycle comprises four protein states, one in which no changes occurred in components X and Y (PXY), two states in which one of the components has undergone a change (PXY and PXY), and a fourth state in which changes occurred in both components (PXY). ΔG1 and ΔG2 correspond to the free-energy changes for a change in component X to occur, in the absence and presence of a change in the second Y component, respectively. The nonadditive component of the two changes, as given by ΔG1 − ΔG2 (=Δ2GXY), determines the magnitude and sign of the interaction between the X and Y components. (B) An obligatory (subunit) channel gating scheme characteristic of the Shaker Kv channel. In this scheme, only after all four voltage sensors undergo two transitions (K1(V) and K2(V)) does late concerted pore opening (L) occur. C and O denote the closed and open states, respectively. (C) Combination of sequential mutant interaction cycles to evaluate changes in the interaction free energy between the XY residue pair along the gating pathway described in B. In each cycle, the different wild-type (wt) and mutant (m) chemical equilibrium constants are indicated above the appropriate channel protein. In the left cycle, Δ2GXY(C1→C2) denotes the interaction energy between the XY residue pair in the C2 channel state relative to the C1 state and is calculated according to −RTln((K1wt K1m1m2)/(K1m1 K1m2)). The other residue interaction energies are calculated in a similar manner. In all schemes, arrows represent equilibrium between the states connected.
