Relationships between structural dynamics and functional kinetics in oligomeric membrane receptors

Abstract

Recent efforts to broaden understanding of the molecular mechanisms of membrane receptors in signal transduction make use of rate-equilibrium free-energy relationships (REFERs), previously applied to chemical reactions, enzyme kinetics, and protein folding. For oligomeric membrane receptors, we distinguish between a), the Leffler parameter alpha(L), to characterize the global transition state for the interconversion between conformations; and b), the Fersht parameter, varphi(F), to assign the degree of progression of individual residue positions at the transition state. For both alpha(L) and varphi(F), insights are achieved by using harmonic energy profiles to reflect the dynamic nature of proteins, as illustrated with single-channel results reported for normal and mutant nicotinic receptors. We also describe new applications of alpha(L) based on published results. For large-conductance calcium-activated potassium channels, data are satisfactorily fit with an alpha(L) value of 0.65, in accord with REFERs. In contrast, results reported for the flip conformational state of glycine and nicotinic receptors are in disaccord with REFERs, since they yield alpha(L) values outside the usual limits of 0-1. Concerning published varphi(F) values underlying the conformational wave hypothesis for nicotinic receptors, we note that interpretations may be complicated by variations in the width of harmonic energy profiles.

Figure 1

Static and dynamic energy diagrams for conformational isomerizations of normal (A–C) and mutant (D–F) nAChR with 0, 1, or 2 bound ligands. (A) Static energy diagram for normal nAChR based on published data analyzed previously, with α_L = 0.8 and L = 10⁹ (14). (In the original publication, the TS parameter was ^BAp, where ^BAp = 1 − α_L). (B) Dynamic representation of the diagram in A as profiles of free energy versus r_‡, with harmonic wells that broaden with ligand binding and the TS at intersection points (small open circles). (C) Dynamic representation as in B, but with profile widths that remain unchanged upon ligand binding; the double vertical arrows labeled a and b indicate the gap to the TS energy levels in B for unliganded and diliganded states, respectively. (D) Static energy diagram for nAChR myasthenic variant ɛT246P (49), as analyzed previously with α_L = 0.55 and L = 100 (14). (E) Dynamic representation of the diagram in D as energy profiles with harmonic wells, which broaden upon ligand binding. (F) Dynamic representation as in E, but with energy profiles as in B and with the open state shifted to lower energies corresponding to L = 10². The energies indicated by the double arrows labeled c and d indicate the gaps to the TS energy levels in E for unliganded and monoliganded states, respectively. Free-energy versus r_‡ profiles were computed from a general equation for harmonic wells: r_‡= k + ((x − h)²/(4p)), where h is the horizontal placement along r_‡ (h = 0 or 1); k is the vertical placement; and p is a flatness parameter. For profiles with minima at 0 and 1, the intersection value, r_‡^X is given by $r_{‡}^x=\left\{-p_0+\sqrt{p_0{p}_1\left[1-4\left(k_0+k_1\right)\left(p_0-p_1\right)\right]}\right\}/\left(p_1-p_0\right)$ Intersection points of B, C, E, and F occur at progressively lower values of r_‡ as the number of bound ligands increases (for example, in B, r_‡ = 0.85 for the unliganded state and 0.76 for the diliganded states, based on a value of α_L = 0.8 for the monoliganded state; in E, r_‡ = 0.51, 0.55, and 0.59 for the diliganded, monoliganded, and unliganded states, respectively). This progression to lower r_‡ values is a consequence of the geometry of the parabolic profiles.

Figure 2

Progression of opening and closing rates of the large-conductance calcium-activated potassium channels according to the number of calcium ions bound (19). Each point represents the average of the three values for independent patches (presented in Table 3 of Cox et al. (19)), with blue diamonds for opening rates and green squares for closing rates. The straight lines are calculated using α_L = 0.65 from least-squares fitting of successive rates with Eq. 1, with ^BAc = 0.11, the average for successive L values obtained from the ratio of interconversion rates for each degree of ligand binding.

Figure 3

Application of REFERs to reactions of GlyR. (A) Reaction cycles with two or three molecules of taurine (Tau) bound. Rate constants and dissociation constants (K_B and K_F) are from Lape et al. (20). Binding of the third molecule of taurine stabilizes F relative to B by a factor of K_B/K_F = 1.5 and A relative to F by a factor of K_F/K_A = 42.2; their reciprocals give ^BFc and ^FAc, respectively, used in the calculations in B. A value for K_A was not presented by Lape et al., but was calculated from linkage relations. (B) Energy diagram for the reactions in A comparing predicted and observed energy barriers for the diliganded states (red, observed; green, predicted), based on the values reported for the triliganded states (blue). The calculated values are obtained from the data in A using Eq. 1. TS heights (kcal/mole) were calculated as described (4). Green areas correspond to the limits for δ₂ (A, left), with the lower TS barrier for x′ = x with α_L = 0 and with the upper barrier for y′ = y with α_L = 1 and for β₂ (A, right), with the lower TS barrier for y′ = y with a_L = 0 and with the upper barrier for z′ = z with a_L = 1. Error bars from estimates in Lape et al. are presented for the diliganded state only, since the errors are essentially within the thickness of the line for the triliganded states. Barrier heights may vary depending on specific assumptions of the TS theory (85), but the energy differences x = x′, y = y′, and z = z′ would remain unchanged.