By what mechanism do changes in bilayer composition modulate conformational equilibria of intrinsic membrane proteins?

Introduction

The function of an intrinsic membrane protein often requires a change in shape that implies a transition among conformational states. Typically, the conformational equilibrium strongly favors an "inactive" state until an external stimulus causes a shift in the equilibrium significantly toward an "active" state. The stimulus may be, for example, a change in electrical potential across the membrane or binding of ligands to sites on the protein, as is common for ion channel proteins, or the absorption of light, as for rhodopsin. The activity of the protein, i.e., the distribution of its conformational states (both in the absence and the presence of the stimulus) is often strongly affected by the molecular composition of the bilayer in which it is anchored. Variation in lipid characteristics, such as head group type and the unsaturation and length of the acyl chains, or addition of cholesterol or small solutes such as alcohol and anesthetics can all contribute to modulation of protein function.

The molecular mechanisms of such lipid-protein coupling remain poorly understood. The underlying interactions between bilayer components (lipids or other membrane-soluble molecules) and proteins can be distinguished on the basis of "specificity": the former can influence the latter either through direct binding to localized protein sites, or indirectly, by altering the structural, thermodynamic or dynamic properties of the bilayer, which in turn modulates protein behavior. Certainly, there are many systems, particularly involving peripheral membrane proteins, in which the presence of a specific lipid submolecular fragment is crucial to protein function, strongly suggesting local recognition and binding; such interactions are not considered here. Rather, we examine the consequences of a putative nonspecific (i.e., indirect) mechanism of lipid modulation of protein conformational equilibria and peptide aggregation.

At present, the physical underpinnings of the nonspecific mechanisms by which membrane components influence embedded proteins are largely unknown. Proposed mechanisms have involved correlations of variations in bilayer composition with altered structural properties of the membrane such as thickness, coupled to the free energy through hydrophobic mismatch or directly to thermodynamic properties such as microphase separation that can effect the free energy change of protein conformational equilibria. Curvature elastic stress is likely to play an important role in view of the considerable experimental evidence that proper membrane function requires incorporation of "nonlamellar" lipids (which, when the only lipid present, form inverse hexagonal phases) and that membrane homeostasis may involve proximity to the lamellar/hexagonal transition. Of particular importance is the coupling between hydrophobic mismatch and curvature stress.

Whereas small variations in lipid composition can influence protein activity significantly, the accompanying changes in membrane structural properties such as thickness are typically small, although it has been argued that relatively small changes in thickness can have a significant effect on protein equilibria. However, such structural changes can often be achieved by small variations in temperature or other external variables that have relatively little effect on proteins. In addition, the composition of a membrane can be varied in such a way as to maintain, for example, constant membrane thickness, but which nonetheless has a significant effect on protein function, suggesting that other, non-structural properties may play a major role in lipid modulation of protein activity.

So: the question arises: Is there another membrane property that is highly sensitive to small changes in membrane composition, and has a clear mechanistic link to protein function? It is suggested that the lateral pressure profile within the membrane may serve as such a property for those intrinsic proteins whose function involves a conformational change accompanied by a depth-dependent variation in the cross-sectional area of the protein in the transmembrane domain.

First, the distribution of lateral pressures in lipid bilayers is discussed. Then, a simplified description of the equilibrium between conformational states of proteins is developed. Finally, a simple thermodynamic argument is applied that provides a mechanistic link between changes in the lateral pressure profile (the depth-dependent distribution of lateral stresses within the membrane) and protein conformational (or aggregation) equilibria.

Lateral pressure profile in lipid bilayers

In a "self-assembled" membrane, i.e., in the absence of any lateral constraints, the bilayer is free to adjust its molecular area (expand or contract laterally) so as to minimize its free energy. In other words, once equilibrium is reached, the sum of the forces acting in the plane of the bilayer (lateral pressures) is essentially zero. Figure 1. However, since the bilayer is of finite thickness, the various contributions to the total lateral pressure will in general act at different depths; positive lateral pressures occuring at some depth must therefore be balanced by negative pressures (tensions) elsewhere. To be more explicit, imagine dividing up the bilayer into thin planar slices. Within a slice centered at a depth z in the bilayer, a nonzero local lateral pressure π(z) may exist, constrained only insofar as the sum of the pressures over the thickness of the entire bilayer gives the total lateral pressure, which must be zero: ∑_z π(z) = 0. While there is as yet no direct measurement of these localized lateral pressures, there is both experimental and theoretical evidence of their magnitude and distribution with respect to depth in the bilayer. For example, the curvature elastic properties (spontaneous curvature and curvature elastic moduli) of bilayers are integral moments of the pressure "profile" (i.e., its depth-dependence) and its curvature derivatives, as discussed below. In addition, predictions of the pressure profile have been obtained using analytical theory and Monte Carlo simulations for various lipid systems.

The nonzero lateral pressure profile π(z) arises in large part from the competition between contributions of opposite sign: a tension (negative pressure) largely localized near the interfaces, and more broadly distributed positive pressures arising predominantly from chain conformational entropy, as well as from head-group repulsions. The interfacial tension derives from the large free energy cost of contact between hydrocarbon and water at each of the two hydrophilic/hydrophobic interfaces. This contribution to the free energy is approximately proportional to the area of interfacial contact, the constant of proportionality being roughly .05 J per m² of interface (equivalently, a constant interfacial tension of g = .05 N/m = 50 dyn/cm), using a typical value for fluid hydrocarbon/water interfaces. Acting alone, this contribution would induce the bilayer to minimize the area per molecule, e.g., for saturated chains, to align the chains in their all-trans configuration. However, the chain conformational entropy, which reflects the degree of chain conformational disorder, also makes a large contribution to the free energy of the bilayer. In contrast to the interfacial free energy, the chain conformational contribution to the pressure depends sensitively on molecular area. This pressure is very large at small molecular areas (when the acyl chains are necessarily very orientationally ordered, so even a small increase in molecular area allows for a large increase in conformational freedom), but at larger areas per molecule, at which the chains are already quite conformationally disordered, the change in entropy upon lateral expansion is much smaller. (The entropy eventually goes through a maximum with increasing area, beyond which the conformational freedom of the chains is reduced.) It is important to note that upon lateral expansion, the volume occupied by the lipids changes very little, since the energetic cost of creating free volume (the increase in van der Waals energy among the hydrocarbon chains) would be enormous; rather, the bilayer thins as it expands laterally, the chains becoming increasingly bent and intertwined, thus able to sample more of the enormous number of their configurational states, while filling up all the space in the bilayer interior at roughly constant bulk density.

The free energy minimum that defines the bilayer equilibrium derives from the compromise between these opposing forces, at which the pressures arising from interfacial tension and chain conformational (and head group) interactions are just balanced. In Figure 2, the dependence on molecular area of these two main contributions to the free energy is simplistically portrayed using a square lattice model of short (4-segment) flexible chains in a two-dimensional bilayer. While the picture does not accurately describe the details of chain rotational isomeric states, it correctly illustrates the balance between the opposing effects of varying molecular area on the interfacial and chain conformational (and head group) contributions to the free energy that occurs at constant bulk density in the hydrocarbon domain. Although this picture is very simple, it nonetheless clearly indicates that it can be misleading to describe lipid molecules as having a characteristic "shape", e.g., conical or cylindrical. The average shape of a lipid can only indicate the depth-dependence of the the volume it occupies on average, which is overwhelmingly determined by the geometric constraints of the aggregate. Furthermore, there is no obvious relation, for example, between the degree of chain orientational order (a well-defined thermodynamic equilibrium property, either as a function of bilayer depth, or of position along the acyl chains) and less well-defined concepts such as free volume or the existence of packing defects. In any case, there is no simple relation between molecular details (chain unsaturation, etc.) and the distribution of lateral pressures in the bilayer, as has been suggested.

A rough estimate of the magnitude of the lateral pressures in the bilayer interior is readily obtained by noting that the sum of the lateral stresses distributed over the hydrophobic interior of the fluid bilayer (ignoring head group contributions) must balance the pair of interfacial tensions at the interfaces. Using g = .05 N/m as the tension of each interface, the chains must generate an opposing lateral pressure of equal magnitude distributed over the hydrophobic interior of the bilayer, of thickness (2h) somewhere in the range of 25Å to 30Å. The lateral pressure density (force per unit area), i.e., the lateral pressure (force per unit length) per unit thickness of the bilayer is thus on average roughly 2g/2h ≈ 350 atm. While other contributions to the lateral pressure will alter this number somewhat, it nonetheless provides a measure of the magnitude of the lateral pressure densities acting upon an inclusion such as a protein or peptide aggregate that passes through the bilayer interior. The actual pressure profile will be nonuniform, in a manner that depends sensitively upon the molecular composition of the bilayer.

Protein conformational equilibrium

The trans-membrane domain of each of the conformational states (s = r, t, ...) of a membrane protein is characterized by a cross-sectional area A_r, A_t, ..., that depends (in general) on the depth z within the bilayer. Figure 3. In a conformational change r —> t, the cross-sectional area of the protein at depth z thus changes by an amount ∆A(z) = A_t(z) - A_r(z). As with the bilayer, it is convenient to imagine dividing the transmembrane region of the protein into thin slices of thickness ∂z. Let 2h be the thickness of the bilayer (assumed for simplicity to be comprised of identical monolayers), and let z = 0 be the location of the bilayer midplane. For a slice centered at depth z, the mechanical work accompanying the protein conformational change depends both on the volume change of the protein ∆V(z) = ∂z ∆A(z), and the lateral pressure density of the bilayer p(z) = π(z)/∂z against which it either expands [∆V(z) > 0] or contracts [∆V(z) < 0] laterally. The total work is obtained by summing over the entire bilayer (-h < z < h):

W = -∑_z p(z) ∆V(z) = -∑_z π(z) ∆A(z) = - p(z) ∆A(z) dz

Define p₀(z) to be the pressure profile of a bilayer having some standard lipid composition (i.e., the "standard state" of the bilayer), in which the protein conformational equilibrium is given by K₀ = [t]₀/[r]₀. A change in bilayer composition will result in a different pressure profile p(z), and conformational equilibrium K = [t]/[r]. The relation between K and K₀ is readily obtained by a simple thermodynamic argument. Equating the chemical potentials of the two conformational states, µ_r = µ_t, both in the standard and in the altered bilayer environments, and assuming that A(z) is independent of p(z) gives K = K₀e^-ß, where

ß  = (k_BT)^-1 ∆p(z) ∆A(z) dz

and p(z) = p(z) - p₀(z). A positive value of ß corresponds to protein inhibition (K < K₀), while a negative value corresonds to activation (K > K₀). The shift in conformational equilibrium required to have a significant inhibitory or excitatory effect on protein function (for example, as a shift in a dose-response curve) will depend on the protein, and how its activity is measured. As an order-of-magnitude estimate, we might expect that a difference between K and K₀ by a factor of at least two, i.e., |ß| > ln(2), will significantly alter protein activity. Note that since the relative change in protein conformational equilibrium K/K₀ depends exponentially on ß, it is thus very sensitive both to the redistribution of lateral pressures resulting from altered bilayer composition and on the change in the cross-sectional area profile of the protein.