Linear rate-equilibrium relations arising from ion channel-bilayer energetic coupling

Abstract

Linear rate-equilibrium (RE) relations, also known as linear free energy relations, are widely observed in chemical reactions, including protein folding, enzymatic catalysis, and channel gating. Despite the widespread occurrence of linear RE relations, the principles underlying the linear relation between changes in activation and equilibrium energy in macromolecular reactions remain enigmatic. When examining amphiphile regulation of gramicidin channel gating in lipid bilayers, we noted that the gating process could be described by a linear RE relation with a simple geometric interpretation. This description is possible because the gating process provides a well-understood reaction, in which structural changes in a bilayer-embedded model protein can be studied at the singlemolecule level. It is thus possible to obtain quantitative information about the energetics of the reaction transition state and its position on a spatial coordinate. It turns out that the linear RE relation for the gramicidin monomer-dimer reaction can be understood, and the quantitative relation between changes in activation energy and equilibrium energy can be interpreted, by considering the effects of amphiphiles on the changes in bilayer elastic energy associated with channel gating. We are not aware that a similar simple mechanistic explanation of a linear RE relation has been provided for a chemical reaction in a macromolecule. RE relations generally should be useful for examining how amphiphile-induced changes in bilayer properties modulate membrane protein folding and function, and for distinguishing between direct (e.g., due to binding) and indirect (bilayer-mediated) effects.Brønsted | linear free energy relationships | phi value | bilayer elasticity I t is a common observation that manipulations that alter the energetics of a reaction:alter the rate constants (k 1 and k −1 ) and equilibrium constant (K eq ¼ k 1 ∕k −1 ), such that lnfk 1 g is a linear function of lnfK eq g. This linearity is interpreted to mean that, for a series of manipulations of the reaction, the activation free energy (ΔG ‡ ) varies as a linear function of the equilibrium free energy (ΔG 0 ):where a is a slope factor and b a constant (1-3). Such linear relations, known as linear rate-equilibrium (RE) relations (4), linear free energy relations (5), or linear rate-equilibrium free energy relations (6), have been observed and studied since the early 1900s. First observed in acid-based reactions (7-9), they have been identified in a wide range of reactions (1-3). Leffler proposed, based on studies of covalent reactions, that the slope factor a (Eq. 2) identifies the position of the transition state on a reaction coordinate, where S and P are positioned at 0 and 1, respectively (5). Marcus and coworkers deduced the principles underlying linear RE relations in electron transfer processes using a model in which the transfer is described as a one-step transition between two harmonic energy wells (10, 11). Fersht and coworkers used linear RE relations associated with p...

Connection between Oligomeric State and Gating Characteristics of Mechanosensitive Ion Channels

“…We predict that the oligomeric state and hydrophobic shape of a membrane protein are reflected in the energetic cost of the lipid bilayer deformations necessary to accommodate the protein within the membrane. Thus, our results suggest that, in addition to the hydrophobic mismatch between membrane proteins and the surrounding lipid bilayer [11]–[17], the symmetry and shape of the hydrophobic cross section of membrane proteins, and resulting structure of elastic membrane deformations, play an important role in the regulation of protein function by bilayer membranes.…”

“…In particular, the bilayer hydrophobic core couples to the hydrophobic regions of membrane proteins [7]–[10]. The resulting deformations in the lipid bilayer membrane from its unperturbed state can be described quantitatively [11]–[17] using the continuum elasticity theory of membranes [18]–[20]. The energetic cost of protein-induced membrane deformations depends on the protein conformational state as well as on the bilayer material properties, which allows [11]–[17] the lipid bilayer to act as a regulator of protein function.…”

“…The resulting deformations in the lipid bilayer membrane from its unperturbed state can be described quantitatively [11]–[17] using the continuum elasticity theory of membranes [18]–[20]. The energetic cost of protein-induced membrane deformations depends on the protein conformational state as well as on the bilayer material properties, which allows [11]–[17] the lipid bilayer to act as a regulator of protein function. A widely-studied model system for the coupling between membrane protein function and the elastic deformation of lipid bilayers is provided by mechanosensitive ion channels.…”

A distinct mechanism for activating uncoupled nicotinic acetylcholine receptors

“…Although further work is required to quantify how lipids influence both nAChR conformational equilibria and the rates of nAChR conformational change, the markedly different effects of anionic lipids and membrane hydrophobic thickness on these parameters contrast the typical linear effects of hydrophobic thickness that are observed on equilibrium-rate relations 40 . Furthermore, both the unique effects of hydrophobic thickness on nAChR conformational transitions and the high activation energy barrier (or barriers) between uncoupled and coupled conformations suggest that the structural differences between uncoupled and coupled states are more substantial than the typical rotations or tilting movements of transmembrane α-helices that underlie membrane protein conformational change, including nAChR channel gating 41,42 .…”

Interactions of drugs and amphiphiles with membranes: modulation of lipid bilayer elastic properties by changes in acyl chain unsaturation and protonation

“…The amphiphile-induced changes in H B can be estimated as [12, 14] $$H_{\mathrm{normalB}}^{\text{amph}}-H_{\mathrm{normalB}}^{\text{cntrl}}=\frac{k_{\mathrm{normalB}}T}{2·\delta ·(l_{13}-l_{15})}\text{ln}true\{\frac{{\mathrm{normal\tau }}_{15}^{\text{cntrl}}\cdot {\mathrm{normal\tau }}_{13}^{\text{amph}}}{{\mathrm{normal\tau }}_{15}^{\text{amph}}\cdot {\mathrm{normal\tau }}_{13}^{\text{cntrl}}}true\},$$ where the subscripts 13 and 15 denote the two channel types. δ ≈ 0.16 nm and l 15 − l 13 ≈ 0.32 nm [26, 62]. Using Eq.…”