Refined definition of the critical micelle concentration and application to alkyl maltosides used in membrane protein research

The critical micelle concentration (CMC) of nonionic detergents is defined as the breaking point in the monomer concentration as a function of the total detergent concentration, identified by setting the third derivate of this function to zero. Combined with a mass action model for micelle formation, this definition yields analytic formulae for the concentration ratio of monomers to total detergent at the CMC and the relationship between the CMC and the free energy of micellization gmic. The theoretical breaking point is shown to coincide with the breaking point of the experimental titration curve, if the fluorescence enhancement of 8-anilino-1-naphthalene-sulfonic acid (ANS) or a similar probe dye is used to monitor micelle formation. Application to a series of n-alkyl-β-d-maltosides with the number of carbon atoms in the alkyl chain ranging from 8 to 12 demonstrates the good performance of a molecular thermodynamic model, in which the free energy of micellization is given by gmic = σΦ + gpack + gst. In this model, σ is a fit parameter with the dimension of surface tension, Φ represents the change in area of hydrophobic molecular surfaces in contact with the aqueous phase, and gpack and gst are contributions, respectively, from alkyl chain packing in the micelle interior and steric repulsion of detergent head groups. The analysis of experimental data from different sources shows that varying experimental conditions such as co-solutes in the aqueous phase can be accounted for by adapting only σ, if the co-solutes do not bind to the detergent to an appreciable extent. The model is considered a good compromise between theory and practicability to be applied in the context of in vitro investigations of membrane proteins.

b Root mean square deviation in mM.
c Model 7a has the same ݃ ୫୧ୡ as model 7, but employs eq. (21) instead of eq. (14).   5 ESI Figures   Fig. 7 Determination of the surface area of alkyl tails from alkyl maltosides. The solvent-excluded surface of molecular models of maltose, alkanes and alkyl maltosides was measured using the MSMS script. 11 Then, using the linear relations in eq. (39) of the main text, the surface of the alkyl tail was calculated by subtracting the maltose portion of the surface from the surface area of the whole detergent molecule. The molecular models in this figure do not represent the solvent-excluded surface. Instead, a ball and stick model is used with coloring according to the CPK convention.

Fig. 8
When experimental values for the aggregation number ݉ of alkyl maltosides are plotted against the alkyl chain length ݊, an exponential dependence is obtained. The data was compiled from several sources (see Table 4).  ESI Text S1: Modelling the steric free energy ‫ܜܛ‬ The steric free energy accounts for the repulsion between the polar head groups of adjacent detergent molecules within the micelle and is generally considered to be the component that limits the micelle growth to a certain aggregation number. The mathematical expression for ݃ ୱ୲ in eq. (32) of the main text was suggested based on the van-der-Waals equation of state: 12,13 where ܲ is the pressure, ܴ the gas constant, ܶ the absolute temperature, ‫ݒ‬ the molar volume, ܾ the volume excluded by a mole of molecules, and ܽ the parameter representing attractive interactions. Since we are interested only in excluded volume effects, we neglect the attractive interactions. Then, with ܽ = 0, we can formally compute the work ܹ required to "blow up" the molecules from point particles to real molecules excluding the volume ܾ: For one molecule, this can be translated into a standard chemical potential difference between a real molecule with excluded volume ܸ and actually occupied volume ܸ ୬ and a point particle occupying the volume ܸ ୬ : where ݇ is Boltzmann´s constant.
In order to arrive at eq. (32), we consider the motion of detergent head groups to be restricted to the surface area of the micellar core. The steric repulsion then is a function of the ratio of the excluded area (cross-sectional area) of one head group ‫ܣ‬ ୮ and the total surface area of the micellar core per detergent molecule ‫:ܣ‬ Dividing by ݇ ܶ, we obtain ݃ ௦௧ .

S2: Alternative derivation of eq. (20)
Phillips investigated the formation of micelles for the case of ionic detergents, where the micelles bear an effective charge ‫‬ and are interacting with counter ions. 14 We repeat his derivation for nonionic detergents with ‫‬ = 0 and no counter ions. With the assumption of a fixed aggregation number ݉ (mass action model), we can write the equilibrium constant ‫ܭ‬ for micelle formation as where ‫ݖ‬ ݉ ⁄ is the mole fraction of micelles, ‫ݕ‬ the mole fraction of detergent monomers, and ݃ ୫୧ୡ the micellization free energy per detergent molecule in the micelle divided by the thermal energy ݇ ܶ. Then, ‫ݖ‬ = ‫ݕܭ݉‬ (S6) Phillips assumes ߶ to be an ideal property, i. e., a measurable quantity that is proportional to the mole fraction (or concentration) of detergent monomers or micelles, so that the breaking point in the experimental titration curve coincides with the breaking point in the dependence of ‫ݕ‬ or ‫ݖ‬ ݉ ⁄ on the total detergent concentration ‫.ݔ‬ Corresponding to his equations (5) and (6), we have in our symbolism (with ‫ܣ‬ ଵ and ‫ܣ‬ ଶ being proportionality constants) as well as ‫ݔ‬ = ‫ݕ‬ + ‫ݖ‬ = ‫ݕ‬ + ‫ݕ݉ܭ‬ (S8) Differentiating both equations (S7) and (S8) with respect to ‫,ݕ‬ we find Then, and Plugging eq. (S16) into (S15) and rearranging yields eq. (20) of the main text.

S3: Draft of a non-ideal solution model
In the draft model, the Gibbs free energy of the solution is given by ‫ܩ‬ = ‫ܩ‬ + ‫ܩ‬ ୫୧୶ + ‫ܩ‬ ୧୬୲ (S17) where ‫ܩ‬ is given by eq. (3) of the main text. The difference to the ideal solution model is in the free energy of mixing, ‫ܩ‬ ୫୧୶ , which we want to improve by considering excluded volume effects, and in the free energy of interaction between solutes, ‫ܩ‬ ୧୬୲ , which we shall describe in the framework of a mean field approach.
The root idea of considering excluded volume effects in the entropy of mixing as formulated by Hildebrand 15 is based on an equation of state of the van-der-Waals type, where as in the modelling of the steric free energy ݃ ௦௧ (see Text S1 above), ܽ = 0. Then, the entropy change of expanding the gas isothermally from a volume ܸ ଵ to a volume ܸ ଶ is given by Thus, the entropy change is determined by the change in free volume ܸ − ܾ.
In the following we work again with particle numbers and the Boltzmann constant instead of mole numbers and the gas constant. To obtain the entropy of mixing for a binary liquid mixture, we consider the transfer from ܰ ଵ molecules of type 1 and ܰ ଶ molecules of type 2 with excluded volumes ܾ ଵ and ܾ ଶ , respectively, from the pure state, where they occupy the respective volumes v ଵ and v ଶ , to a solution with total particle number ܰ ଵ + ܰ ଶ . Since the considered molecules are polyatomic, they have internal degrees of freedom (DOFs) due to vibrations, rotations, and librations. We follow Hildebrand 15 and assume that the energy in these internal DOFs is not significantly different in the pure state and in the solution. Then, the entropy of mixing is obtained by considering for each component the expansion from its free volume ܰ (v − ܾ ) in the pure state to its free volume ܸ − ܰ ଵ ܾ ଵ − ܰ ଶ ܾ ଶ in the mixture, where ܸ is the actual volume of the solution: A simplifying assumption is that the solution is additive, 15 i. e., there is no volume change due to mixing: ܸ = ܰ ଵ v ଵ + ܰ ଶ v ଶ (S20) Then, eq. (S19) becomes With these settings, we now consider an aqueous solution of a detergent with -for the sake of simplicity -only one type of co-solute. We shall use the index "w" for water, "c" for the cosolute, and "ν" for a detergent aggregate with ν detergent molecules. We introduce the abbreviations so that the free energy of mixing becomes To model ‫ܩ‬ ୧୬୲ , we employ the Bragg-Williams approximation: Here, we only consider interactions between solutes, and ‫ܬ‬ is the exchange parameter of the respective interaction. 16 Note that the total particle number is different from ܰ ୲୭୲ defined in the main text, as ܰ ఔ counts the micelles rather than the detergent molecules in micelles.
Eq. (9) requires the computation of the partial derivative We have and These equations are still quite general. To make progress in understanding, we simplify further by making two approximations: 1) We adopt a mass action model, i. e. we assume that only one type of detergent micelle with aggregation number ݉ exists. This is the same assumption that we made in the main text within the ideal solution model. Thus, we have only ߥ = 1 and ߥ = ݉.
2) We neglect interactions between micelles, between detergent monomers as well as between detergent monomers and micelles. Note that ‫ܩ‬ ୧୬୲ does not account for micelle formation, but only for the interaction between separated solutes. Thus, we consider only such interactions between the co-solute and detergent monomers as well as micelles and between co-solute molecules.
With these approximations, we obtain from eq. (9) together with eqs. (S27) to (S30): Eq. (S31) can be rearranged to give This equation can be further simplified by assuming that detergent and co-solute are diluted enough so that terms in the order of ܰ ଶ ܰ ഥ ଶ ⁄ can be neglected. Considering the definition in eq. (12), we thus obtain: The first three terms on the right-hand side of eq. (S33) originate from the consideration of excluded volume effects in the entropy of mixing, while the last term is due to interactions between detergent and co-solute. Note that the latter interaction does not include an attachment of co-solute molecules to either detergent monomers or micelles. Further, eq. (S33) is not a model for ݃ ୫୧ୡ , but rather describes the relation of ݃ ୫୧ୡ to the various particle numbers as does eq. (13) of the main text in the ideal solution model.
To find the connection to the mole fractions ܺ defined in the main text, we make another simplifying assumption discussed by Hildebrand: 15 We assume that the free volumes ߱ are proportional to the molar volumes ‫ݒ‬ with a species-independent proportionality constant. (Hildebrand actually discussed the molal volumes. 15 ) This approximation implies that the volume fraction ܻ of species ݅ (with mole number ݊ and molarity ܿ ) is given by Recall that for micelles, we defined ܺ ఔ as the mole fraction of detergent in micelles rather than as the mole fraction of micelles. Thus, eq. (6) of the main text implies, that in the particular case ݅ = ݉, we have Here, ܻ and ‫ݒ‬ are the volume fraction and molar volume, respectively, of micelles with aggregation number ݉. Eq. (S34) also implies that where ܰ is Avogadro´s number, so that Here, we neglected the contribution of detergent to ܰ ഥ in the last step, so that ܰ ഥ ≈ ܰ ୲୭୲ and ܿ ୲୭୲ ≈ ܰ ഥ (ܰ ܸ) ⁄ . Note that this approximation was already applied in Section 2.6, where the contribution of detergent to ܿ ୲୭୲ ≈ ܰ ୲୭୲ (ܰ ܸ) ⁄ was neglected. Then, To have a short notation, we introduced the quantity which represents the volume difference between ݉ detergent monomers in solution and a micelle with aggregation number ݉ per detergent molecule in the micelle. It may be argued that this quantity is practically zero, but ߦ = 0 would be an approximation. Further, with the neglect of detergent contributions to ܰ ഥ , we obtain for the co-solute ܰ ୡ ܰ ഥ ⁄ ≈ ܺ ୡ . Similar to eq. (S39), we introduce the quantity representing the difference in interaction of the co-solute with a micelle with aggregation number ݉ and with ݉ detergent monomers per detergent molecule in the micelle. Then, eq. (S33) becomes: We now see clearly that with respect to the ideal solution model represented by eq. (13), we have three additional terms. While (1 − ݉) is known, the other two of these additional terms remain to be quantified.
If ߦ ≈ 0, we may use ‫ݒ‬ ≈ ‫ݒ݉‬ ଵ and compute ‫ݒ‬ according to where v is now approximated by the volume of a micelle with the parameters ܽ and ܾ describing the spheroidal core and ݀ being the thickness of the head group layer (for the volume of an oblate spheroid, see eq. (16)  are compiled in Table 6.