Hill coefficient for estimating the magnitude of cooperativity in gating transitions of voltage-dependent ion channels

Abstract

A frequently used measure for the extent of cooperativity in ligand binding by an allosteric protein is the Hill coefficient, obtained by fitting data of initial reaction velocity (or fractional binding saturation) as a function of substrate concentration to the Hill equation. Here, it is demonstrated that the simple two-state Boltzmann equation that is widely used to fit voltage-activation data of voltage-dependent ion channels is analogous to the Hill equation. A general empiric definition for a Hill coefficient (n(H)) for channel gating transitions that is analogous to the logarithmic potential sensitivity function of Almers is derived. This definition provides a novel framework for interpreting the meaning of the Hill coefficient. In considering three particular and simple gating schemes for a voltage-activated cation channel, the relation of the Hill coefficient to the magnitude and nature of cooperative interactions along the reaction coordinate of channel gating is demonstrated. A possible functional explanation for the low value of the Hill coefficient for gating transitions of the Shaker voltage-activated K(+) channel is suggested. The analogy between the Hill coefficients for ligand binding and for channel gating transitions further points to a unified conceptual framework in analyzing enzymes and channels behavior.

FIGURE 1

The KNF allosteric model applied to voltage-dependent gating. (A) Scheme of the different states considered by the KNF model. A homodimeric channel undergoes two sequential voltage-dependent subunit transitions from the closed (square) to open (circle) states. K_A(V) and K_B(V) are the equilibrium constants for the first and second transitions, respectively, and display voltage dependence of the form K_i(V) = K_i exp(Z_iFV/RT), where K_i is the chemical equilibrium constant for the corresponding transitions at 0 mV (i = A, B), Z_i (+++) is the gating charge associated with subunit transition from the closed to open states (it is assumed to be equal for both transitions, Z_A = Z_B = Z), F is Faraday constant, and all other constants have their usual thermodynamic meaning. For this gating scheme, the open probability function is given by (B) Double-mutant cycle representation of the KNF model, above (A). The “mutation” in this cycle is a transition of a subunit from the closed to the open conformation. Cooperativity arises when the free energy associated with this “mutation” depends on the conformational state of the adjacent subunit. Such cooperativity is a function of the magnitude of intersubunit interactions between the monomers in the different conformations and is manifested by K_B/K_A.

FIGURE 2

Three-dimensional surfaces of n_H (V = V_1/2) and V_1/2 as a function of both K_A and K_B using identical scales. The graph for n_H was plotted based on Eq. 9 and that of V_1/2 according to the equation derived using the condition of half-activation (P_O = 1/2). Unitary gating charge of 3 was assumed for Z. The main diagonal trajectory where K_A = K_B separates positive ((K_B/K_A) > 1) and negative ((K_B/K_A) < 1) coupling domains to the right and left of this diagonal, respectively. Three extreme scenarios may be envisaged: 1), along the main diagonal trajectory; increasing K_A and K_B upon mutations, for example, would shift the midpoint activation voltage to the left to more negative voltages; however, n_H would not change. 2), The trajectory in bold corresponds to a case in which increasing K_B only would result in a shift of the midpoint activation voltage to the left in parallel with steeper n_H slopes. 3), Opposite dependence is observed between n_H and V_1/2 for a case in which only K_A is changed upon mutations.

FIGURE 3

MWC-type allosteric model applied to voltage-dependent gating. (A) Scheme for the different channel states considered by the MWC model. According to this model, voltage-induced charge movement transitions between closed and between open states (horizontal transitions, K_C(V) and K_O(V) transitions, respectively) are separated by concerted voltage-independent subunit conformational changes (vertical transitions, L (= [C₁]/[O₁]) transition). K_C(V) and K_O(V) display Boltzmann-type voltage dependence as specified in Fig. 1 legend. For this MWC gating scheme the open probability function is given by It is assumed that identical gating charge Z moves across the membrane electric field in transitions between closed and between open states. (B) Dependence of n_H on voltage at different values of c (= K_C/K_O). Graphs were generated according to Eq. 11 with different c-values ranging from 0.0001 to 10,000 in 10-fold increments and assuming a unitary Z of 3.

FIGURE 4

Dependence of the maximal Hill coefficient () and the activation voltage at this slope (V_max) on c. The graph for was plotted based on Eq. 12 (shaded, left y-axis) and that of V_max (black, right y-axis) according to the equation V_max = (RT/2ZF)ln(1/(K_OK_C)), derived by solving the equation ∂n_H(V)/∂V = 0, which is a normalized second derivative of the MWC open probability expression. In this plot, for convenience, K_C was assigned a value of 1, whereas K_O values range from 10⁻⁵ to 10⁵. See text for further discussion.