Equivalent circuit and kinetic model. (A) Patch-clamp elements. V_c, command potential; V_os, voltage offset; I, membrane current; I_os, current offset; R_s, series resistance; V_m, membrane potential; V_rev, reversal potential (assigned zero for symmetric solutions); C_m, membrane capacitance; R_m, leak resistance. The governing equation is I′ = (V′ − V_m)/R_s = C_m(dV_m/dt) + G(V_m − V_rev), solved numerically for V_m with initial conditions V_m(0) = V′(0)/(1 + R_s/R_m). The membrane conductance G is given by G_min + P_o(G_max – G_min), where P_o is open probability, G_min = 1/R_m, and G_max – G_min = Ng_L (product of channel number and unit conductance). (B) Kinetic scheme for the n = 1 allosteric model of Kir. The four states are labeled according to their contribution to the partition function. Expressions for rate constants are shown next to transition arrows. Ratios of forward to backward rate constants simplify to one of these transition equilibrium constants: L, J, LD, or JD, which also equal to the ratio of product to reactant terms in the partition function. Prefactors ν_L and ν_J are transition frequencies for the L and J particles, respectively. The linear free energy variables x_L and x_J range from 0 to 1. State probabilities were calculated by numerically integrating the rate differential equations governing the model using V_m as input, with initial conditions determined by the starting equilibrium distribution. (C) Newly formed inside-out patch without leak, capacity, or series resistance compensation (experiment R1). A fit (black line) of the ionic current (blue) to the linear elements of the equivalent circuit yielded R_s = 10.9 Mohm and C_m = 7.33 pF. The “leak” current contains contributions from both R_m and channel openings. The inset shows an expanded view of the inward capacity transient. The voltage protocol is shown at the bottom (red, V_c; black, V_m). V_os = −0.04 mV; I_os = 0.0019 pA. (D) Same patch as in C, but after capacity and 70% series resistance compensation. Equivalent circuit parameters were changed to R_s = 3.23 Mohm, C_m = 1.0 pF. Same patch subject to double ramp protocol. Experimental recordings of membrane current I and command voltage V_c are in blue and red, respectively. Dashed horizontal lines indicate zero values for current and voltage tracings. The inset is an enlargement of the current at very positive voltages. The black lines represent calculations of I and V_m using simultaneously equivalent circuit and gating models in response to V_c with adjustable variables derived from the Kir kinetic model and simultaneously fitted to I and the Hill plot. I_os = −0.00074 pA; V_os = 0 mV. R_s and C_m, same as in D. (F) Hill plots derived from Pos (blue) and Neg (orange) ramp currents, showing mild hysteresis. The black lines are the dynamic fits. The center green line is the calculated equilibrium Hill energy. Model variables: Ng_L = 30 nS; Δq_L = 0.17 e_o; V_L = −44 mV; ν_L = 2.0 kHz; x_L = 0.2; Δq_J = 1.69 e_o; V_J = 55 mV; ν_J = 0.15 kHz; x_J = 0.65; W_D = −90 meV (−2.1 kcal/mol).