χ-value analysis for a voltage-dependent system comprising two particles. (A) The particle diagram for a system comprising a voltage sensor (which can exist in two states, R and A) and a pore (which exists in two states, C and O). K₁ and K₂ represent the intrinsic equilibrium constants of the voltage sensor and the pore in the absence of any other interactions. There are four state-dependent interaction terms between the two particles (θ_RC, θ_AC, θ_RO, and θ_AO; as in Fig. 1 A). The energetic parameters of this system can be normalized through the relations shown. The voltage dependencies of K_V and K_P can be expressed as: $Ki=Ki0exp(ziFV/RT)$ (i = V, P), where $Ki0$ is the voltage-independent part of the equilibrium constant and z_i is its voltage dependence; F, R, and T represent the Faraday constant, the universal gas constant, and the temperature, respectively. Because the four coupling constants are voltage independent, K_V has the same voltage dependence as K₁, and K_P has the same voltage dependence as K₂. The coupling parameter in the normalized version, θ, is the ratio of the like state interactions (θ_RCθ_AO) and the unlike state interactions (θ_ACθ_RO). (B) Using arbitrary values of the energetic parameters, the ln(P₀/1 − P₀) vs. voltage plot for the pore was simulated for the allosteric model in A (closed symbols). The broken red lines are the extrapolations of the linear regimes (obtained at high and low voltages). The slope of the red lines is governed by z_P. The difference of the intercepts created by the linear extrapolations on the V = 0 axis is the χ^diff for pore, which is linearly proportional to the difference in the like state interaction energies (−RTln(θ_AOθ_RC)) and the unlike state interaction energies (−RTln(θ_ACθ_RO)). (C) The state diagram for an allosteric model comprising two particles, using the normalized parameters: there are four possible states of the system, depending on the conformations of each of the two particles. When K_P becomes very low, the allosteric model reduces to a linear sequential scheme representing an obligatorily coupled system. In terms of the un-normalized parameters, this could occur when K₂ << 1, θ_RO << 1, and/or θ_RC >> 1. The ln(P₀/1 − P₀) vs. V plot for the pore, for the obligatorily coupled model, shown in B (open symbols), keeps on decreasing steeply at hyperpolarizing voltages.