This page presents a general thermodynamic framework for describing the energetics of protein folding in membranes by means of a four-step thermodynamic model. The energetic components of the partitioning of peptides and proteins into bilayers are described and equations for calculating free energies of transfer from partitioning measurements are presented.
If a membrane protein (MP) is at thermodynamic equilibrium, one can think of its folding and stability in terms of thermodynamic models that need not mirror the biological assembly process. Such models are nevertheless important for the design of biological experiments because they describe the thermodynamic context within which biological processes must proceed. These processes have evolved to take useful advantage of thermodynamic equilibrium states by either regulating the heights of barriers separating such states or by using metabolic energy to work against them.
Jacobs and White (1) proposed a three-step thermodynamic model for membrane protein folding (interfacial partitioning, interfacial folding, and insertion) based upon structural and thermodynamic measurements of the partitioning of small hydrophobic peptides and the so-called helical hairpin insertion model (2). An essential feature of their model, subsequently supported by several theoretical studies (3, 4), was that the bilayer interface provided a free energy well for initial binding and folding of hydrophobic peptides. At about the same time, Popot and Engelman (5, 6) proposed a two-stage model for the assembly of alpha-helical proteins in which the helices are first "established" across the membrane and then assemble into functional structures. The idea for this model came from a series of experiments which demonstrated that isolated fragments of bR in lipid bilayers can reassemble spontaneously into a fully functional form (5), consistent with the native protein residing in a free energy minimum. Combined, these two lines of thought represent a four-step thermodynamic cycle (8) (Figure 1, below). The four steps, partitioning, folding, insertion, and association, can proceed along an interfacial path, a water path, or a combination of the two. Determination of the free energies (ΔG) for each step along a path allow thermodynamic stabilities to be computed.
Figure 1. A four-step thermodynamic cycle for describing the energetics of the partitioning, folding, insertion, and association of an alpha-helix into a fluid lipid bilayer. The process can follow an interfacial path, a water path, or a combination of the two. Studies of folding along the interfacial path are experimentally more tractable (7, 8). The Δ>G symbols indicate standard transfer free energies. The subscript terminology indicates a specific step in the cycle. The subscript letters are defined as follows: i = interface, h = hydrocarbon core, u = unfolded, f = folded, and a = association. With these definitions, for example, the standard free energy of transfer from water to interface of a folded peptide would by ΔG_{wif}.
The four-step model is useful for both constitutive and non-constitutive MPs. For non-constitutive MPs, the steps progressing from left to right describe the energetics of the natural folding process, while for constitutive MPs, the steps progressing from right to left describe "unfolding". Besides providing a useful thermodynamic scheme, the four-step model summarizes the types of experiments on MP folding that are being pursued in several laboratories.
One way to obtain the free energy changes of the four-step is by measurements of the partitioning of peptides and proteins between the aqueous and membrane phases. The primary causes of favorable partitioning of a peptide or protein from water into a membrane are non-polar (np) interactions, due to expulsion of non-polar compounds from water (hydrophobic effect), and electrostatic (qE) attraction between basic amino acid residues and anionic lipids. Upon binding to the membrane, the peptide can change its conformation (con) and its motional degrees of freedom due to immobilization (imm) in the membrane. In addition, there can be electrostatic (elc) effects arising from differences in the dielectric constants of the water and membrane related to the cost of partitioning H-bonded peptide bonds (4). Finally, the partitioning of the peptide can perturb the lipid (lip). The standard transfer free energy, ΔG^{0}, associated with partitioning can be decomposed into a sum of contributions from the various effects (1, 4, 9, 10):
ΔG^{0} = ΔG^{0}_{np} + ΔG^{0}_{elc} + ΔG^{0}_{qE} + ΔG^{0}_{con} + ΔG^{0}_{imm} + ΔG^{0}_{lip} (1a)
Because the first two terms are related to changes in the solvation of the protein upon partitioning, Ben-Tal et al (4) suggest defining a solvation free energy ΔG^{0}_{solv} = ΔG^{0}_{np} + ΔG^{0}_{elc} so that
ΔG^{0} = ΔG^{0}_{solv} + ΔG^{0}_{qE} + ΔG^{0}_{con} + ΔG^{0}_{imm} + ΔG^{0}_{lip} (1b)
Equation 1 is useful for two reasons. First, it allows the possibility of computing ΔG^{0} from first principles if the contributions from each term can be calculated individually. The computation of ΔG^{0}_{qE} has been discussed extensively by several authors (11, 12, 13) and approaches for calculating ΔG^{0}_{elc} , ΔG^{0}_{con} , ΔG^{0}_{imm} , and ΔG^{0}_{lip} have been discussed in detail by Honig and his colleagues (4, 10) and, earlier, by Jähnig (9). Second, and more pragmatically, when partitioning is driven primarily by the hydrophobic effect, Equation 1 allows membrane partitioning to be considered in the context of bulk-phase partitioning. For simple non-polar solutes, the desolvation free energy arising from the hydrophobic effect (14, 15, 16) is given by
ΔG^{0}_{np} = σ · A (2)
where A is the solute's accessible surface area in water and σ the solvation parameter, estimated to be about −20 to −25 cal mol^{−1} Å^{−2} for water-to-hydrocarbon transfer (15, 17, 18, 19; but see below). For uncharged molecules (ΔG^{0}_{qE} = 0), Equation 1b may be written as
ΔG^{0} = ΔG^{0}_{solv} + ΔG^{0}_{bilayer} (3)
where ΔG^{0}_{bilayer} = ΔG^{0}_{con} + ΔG^{0}_{imm} + ΔG^{0}_{lip} represents the contribution of bilayer effects (20) to the partitioning process. The conformational and immobilization components are included because they arise specifically from the association of the peptide with the bilayer. These effects, sometimes called the non-classical hydrophobic effect (21), are significant for even simple non-polar molecules. For the transfer of n-hexane from water to an alkane phase at 25°C, for example, ΔG^{0} = −7.74 kcal mol^{−1}, whereas for partitioning into dioleoylphosphocholine bilayers ΔG^{0} = −5.77 kcal mol^{−1} on a per acyl-chain basis (22). Thus, ΔG^{0}_{bilayer} = 1.97 kcal mol^{−1}, which is a very significant effect. This simple case demonstrates that membrane partitioning cannot be reliably predicted on the basis of bulk-phase partitioning.
Difficulties can arise when transfer free energy data from different laboratories are compared because different standard states for the transfers are used. Although the preferential association of a solute with the lipid bilayer is frequently treated as macromolecule binding-site problem, the simplest and most rigorous approach is to treat the association as simple partitioning, and use mole-fraction partition coefficients in the computation of standard transfer free energies. Figure 2 summarizes various methods of calculating these free energies based upon rigorous discussions of several authors (4, 10, 23, 24). Experimental methods for determining partition coefficients have been reviewed in detail (25).
There is considerable controversy regarding the preferred system of units for calculating free energies, concerned primarily with possible corrections to account for size differences between solutes and solvents, the so-called the Flory-Huggins correction [reviewed in (26)]. Because water is so small compared to lipid molecules, the size correction can be substantial in bilayer partitioning (27). One important consequence of the size correction is that the non-polar solvation parameter (Equation 2) is about −45 kcal mol^{−1} rather than −25 kcal mol^{−1} (28, 29). The controversy revolves around the computation of the so-called cratic entropy. Chan and Dill (26) have given a lucid and detailed account of the issues. The best course is to use mole fraction partition coefficients for computing free energies (Figure 2, below).
K_{x} = x_{bil} / x_{w}, ΔG^{0}_{x} = −RTlnK_{x}
K_{C} = C_{bil} / C_{w} , ΔG^{0}_{C} = −RTlnK_{C}
K_{C} = K_{x}(v_{water} / v_{lipid})
ΔG^{0}_{x} = ΔG^{0}_{C} + RTln(v_{water} / v_{lipid}) = ΔG^{0}_{C} − 2.2 kcal mol^{−1}
for L + P ↔ LP : K_{assoc} = [PL] / [P][L], ΔG_{assoc} = −RTlnK_{assoc}
K_{assoc} = K_{x} / [W]
ΔG^{0}_{x} = ΔG_{assoc} − RTln([W]) = ΔG_{assoc} − 2.38 kcal mol^{−1}
Figure 2. (8) Various systems used for partition coefficients, association constants, and transfer free energies. Transfer free energy data from different laboratories must be compared with caution because (i) free energies are often calculated (inappropriately) from association constants (units of M^{−1}) based on binding site models and (ii) the standard states for the transfers are not always defined clearly. The simplest and most rigorous approach is to treat the association of peptides with membranes as a partitioning rather than a binding-site problem and to use mole-fraction partition coefficients for calculating standard state transfer free energies, ΔG^{0}. This figure summarizes different systems encountered frequently in the literature and relates them to the mole-fraction system. Conversions of molar and association free energies to mole-fraction standard transfer free energies involve only additive terms. Therefore, differential free energy terms, ΔΔG_{assoc} and ΔΔG^{0}_{C} , will be identical to ΔΔG^{0}_{x} . All equations assume partitioning from the water (w) phase to the bilayer (bil) phase. Abbreviations: [L], molar concentration of Lipid; [P] molar concentration of Peptide; [PL], molar concentration of peptide bound to lipid; [W], molar concentration of water (55.3). The molar volumes of lipid and water are v_{lipid} and v_{water} , respectively. For the typical phospholipid, v_{lipid} » 1300 Å^{3} and v_{water} = 30 Å^{3}.