Predicting Partition Coefficients Of Neutral And Charged Solutes In The Mixed SLES-fatty acids Micellar System

Abstract

Sodium laureth sulfate (SLES) and fatty acids are common ingredients in many cosmetic products. Understanding how neutral and charged fatty acid compounds partition between micellar and water phases is crucial to achieve the optimal design of the product formulation. In this paper, we first study the formation of mixed SLES and fatty acids micelles using molecular dynamics (MD) simulations. Micelle/water partition coefficients of neutral and charged fatty acids are then calculated using COSMOmic as well as a MD approach based on Potential of Mean Force (PMF) calculations performed using Umbrella Sampling (US). The combined US/PMF approach was performed with both the additive, non-polarizable CHARMM General Force Field (CGenFF) and the classical Drude polarizable force field. The partition coefficients for the neutral solutes are shown to be accurately calculated with the COSMOmic and additive CGenFF US/PMF approaches while only the US/PMF approach with the Drude polarizable force field accurately calculated the experimental partition coefficient of the charged solute. These results indicate the utility of the Drude polarizable force field as a tool for the rational development of mixed micelles.

Graphical Abstract

1. Introduction

Pharmaceutical and cosmetic products often have complex formulations involving cationic or anionic surfactants which form complex microstructures such as mixed micelles. Understanding how active solutes partition in such microstructures helps in achieving their optimal delivery and maximum efficacy of functional benefits such as health, nutrition, hygiene and wellbeing. The solute partition coefficient in multiphase materials is a thermodynamic property that is related to the free energy change associated with the transfer of a solute molecule from one phase to another. It is defined as the ratio of the solute concentrations between the two phases at equilibrium. Experimentally, partition coefficients can be measured in different ways. Direct measurements are usually performed by using High-Pressure Liquid Chromatography^1,2 (HPLC), Micellar Electrokinetic Chromatography^3,4 (MEKC) and microemulsion electrokinetic chromatography⁵ (MEEKC). While solute partitioning in complex formulations can be measured by experimental methods, these are often quite expensive and time consuming. For this reason, development of in-silico methodologies for accurate prediction of partition coefficients would make product development more cost effective.

Early in-silico models have been reported for predicting solutes partitioning in model biphasic systems⁶. Group contributions (GC) and Quantitative structure-activity relationship (QSAR) methods have been extensively used and can often predict partition coefficients with good accuracy⁶. These methods require an extensive amount of experimental data for the estimation of model parameters and therefore their applicability is limited to molecules groups for which a large number of experimental data are available. An alternative method for the prediction of thermodynamic properties of mixed solvents is the “conductor-like screening model for real solvents”, the so-called COSMO-RS^7,8 theory that is implemented in the COSMOtherm software⁶. The COSMO-RS theory predicts partition coefficients using quantum mechanical (QM) calculations as a basis and its applicability is much wider than that of GC and QSAR methods. COSMOmic⁹ method is an extension of the COSMO-RS theory for inhomogeneous systems such as micelles and other molecular assemblies. In the COSMOmic approach, the molecular structure of assemblies is explicitly considered at the atomistic scale where the resolution of the molecular assembly is usually achieved by performing molecular dynamics (MD) simulations. This is also referred as the MD/COSMOmic approach. Several publications showed that the MD/COSMOmic approach accurately predicts partition coefficients for neutral solutes not only in homogeneous fluids such as octanol-water¹⁰ but also in structured fluids such as micelles containing anionic, cationic, zwitterionic surfactants mixtures^11–16 and microemulsions¹⁷. However it fails in accurately predicting the partition coefficients of charged solutes in micellar systems¹⁵.

Alternatively, a molecular mechanics¹⁸ (MM) modelling approach, that employs MD simulations and Umbrella Sampling^19,20 (US), from which potentials of mean force (PMF) are obtained, can be employed for the in-silico prediction of partition coefficients. This approach is hereafter referred to as the US/PMF approach. The US/PMF approach was used by Yordanova et al.¹⁵ for predicting partition coefficients of neutral and charged solutes in micellar systems. The US/PMF approach showed good accuracy in predicting partition coefficients of neutral solutes but severe inaccuracy in predicting partition coefficients for charged solutes. Yordanova et al.¹⁵ suggested that the inaccuracy of the US/PMF approach in predicting partition coefficients for charged solutes was due to the employment of a non-polarizable force field that cannot accurately model the electrostatic interactions.

In this paper we apply both the combined MD/COSMOmic and the US/PMF approaches to predict the partition coefficient of neutral capric acid and charged capric acid anion (caprate) in the mixed micelle of sodium laureth sulfate and capric acid (mixed SLES/CA micelle). Calculations were also performed for the partition coefficient of neutral palmitic acid in the mixed micelle of SLES and palmitic acid (mixed SLES/PA micelle). The combination of SLES and fatty acids (capric and palmitic acid) is ubiquitous in hair shampoos and liquid soap formulations^21,22. The interactions between fatty acids and SLES anionic surfactants have been extensively studied by Tzocheva et al.²³ by experimentally measuring the free energies of transfer of fatty acid molecules from water to the mixed SLES/fatty acid (SLES/FA) micelles. In the current work, MD simulations are initially performed to predict the self-assembly of mixed SLES/capric acid (SLES/CA) and SLES/palmitic acid (SLES/PA), at the experimental concentration of fatty acids in SLES²³, using the CGenFF²⁴ (CHARMM General Force Field) non-polarizable force field. COSMOmic is subsequently used for predicting the partition coefficients based on the structures from the MD simulations. The comparison of the predicted partition coefficients with the experimental data²³ shows that the combined MD/COSMOmic approach is accurate for predicting the micelle/water partition coefficients of the mixed SLES/PA micelles for the neutral solutes (capric and palmitic acid) whereas it lacks accuracy for the charged solute (caprate). Therefore, the non-polarizable force field (CGenFF) and the CHARMM classical Drude polarizable force field are used for performing the US/PMF calculations to predict solute partitioning in the mixed SLES/FA micellar systems. To the best of our knowledge, this is the first time that a polarizable force field, that can accurately model the effect of the polarizability of anionic surfactant molecules near the water/micelle interface, is employed for predicting the micelle/water partition coefficients of a charged solute. The comparison of the predicted values with the experimental data shows that the use of the polarizable force field in the US/PMF approach is accurate and robust for predicting the partition coefficient of both neutral and charged solutes in the SLES/FA micellar solution.

2. Experimental dataset

The reported experimental value of critical micelle concentration of the SLES surfactant, in pure water at 25 °C, is 0.003 M²⁵. Values for the solubility limits of capric acid and palmitic acid in the respective mixed SLES/CA and SLES/PA micellar solutions were from Tzocheva et al.²³ who reported saturation molar fractions of 0.301 and 0.0909 for capric acid and palmitic acid, respectively, using light absorbance measurements. The free energies of transfer of capric acid and caprate ion from water to the mixed SLES/CA micelle phase and that of palmitic acid from water to the mixed SLES/PA micelle were also collected from Tzocheva et al.²³ These values along with the relative values of the micelle/water partition coefficients are listed in Table 1. Capric acid ion values were derived from monomer concentration in water and in the micelle phase (calculation of K _{mic, A} in supplementary information). In order to derive the free energies of transfer of capric acid, caprate and palmitic acid from water to the respective mixed micellar phase, the authors used a semi-empirical approach^26,27, based on the measured solubilities of fatty acids in water and in the SLES micelle.

Table 1:

Micelle/water partition coefficients, K _{mic, A}, and free energy of transfer, ΔG_transfer, for fatty acids at 25 °C in the mixed SLES/FA micelles.

Compound	log Kmic,A(mol/mol)*	ΔGtransf(kJ/mol)*
Capric acid	3.37	−19.24
Palmitic acid	6.36	−36.20
Capric acid anion	1.03	−5.88

^*Data collected from Tzocheva et al.²³

3. Methods

3.1. CHARMM classical Drude polarizable force field parametrization

The fixed partial atomic charges used in the non-polarizable force field do not take into account the induced polarization arising from the perturbation of the electronic structure of molecules in response to the external electric field²⁸. On the contrary, the CHARMM classical Drude polarizable force field explicitly models the effect of polarization by attaching a charged particle (Drude oscillator) to each polarizable atom through a harmonic spring. As a consequence, a finite induced dipole is created, and the atomic dipole varies by changing the spatial relationship between the atomic nucleus and the Drude particles. The CHARMM Drude force field was first proposed by Lamoureux et al.^29,30 for water and it was further developed for a range of small molecules and biomolecules³¹. Li et al. ³², Chowdhary et al. ³³ and Harder et al.⁶⁵ showed how the use of the Drude force field in simulating lipid membranes leads to significantly different profiles for the electrostatic potential compared to the additive CHARMM36 force field. The authors reported that the most significant difference between the polarizable and the non-polarizable force fields appears in the lipid/water interface region. In that region the effect of the induced polarization between water and lipids headgroups is a particularly important feature that cannot be captured by an additive force field. Similarly, in the case of the mixed SLES/CA micellar systems, the interaction of the headgroups of the SLES and CA with water leads to significant degree of induced polarization. The effect of the polarizability of anionic surfactant molecules near the water/micelle interface is particularly important when predicting the micelle/water partition coefficients of the charged solutes, whereas it does not affect predictions for neutral solutes, as shown by Yordanova et al.¹⁵ Accordingly, the polarizable Drude force field was extended in this study to SLES, capric acid and caprate. Details of the parameter optimization procedure are presented in the parametrization section of supplementary information.

3.2. MD simulations of SLES/FA mixed micelles

For simulations using the non-polarizable force field all MD simulations were performed with GROMACS 5.5.1³⁴. Non-polarizable force field parameters for SLES and fatty acids were obtained from the CGenFF²⁴ program and the TIP3P^35,36 force field was used for water molecules. The Verlet cut-off scheme was employed and both the short-range electrostatic cut-off and the short-range van der Waals (vdW) cut-off were set at 1.2 nm. The vdW interactions were smoothed over 1.0 to 1.2 nm using the forced switch method³⁷ while Particle Mesh Ewald (PME) method was used for long-range electrostatic interactions³⁸. The Nosé-Hoover³⁹ thermostat with a coupling constant of τ_t = 1 ps was used for maintaining a constant temperature at T = 298.15 K and the Parrinello-Rahman⁴⁰ barostat with a coupling constant of τ_p = 1 ps was used for maintaining the pressure at a constant value of P = 1 bar.

All simulations performed with the polarizable force field were carried out using the OpenMM software⁴¹ (http://openmm.org/). The Drude polarizable force field was employed for the SLES and fatty acid molecules along with the SWM4-NDP⁴² model for the water molecules. The PME method³⁸ was used to calculate electrostatic interactions with a real-space cutoff of 1.2 nm. The van der Waals potential was smoothed to zero from 1.0 to 1.2 nm using a potential switch function. Covalent bonds to hydrogen atoms were constrained and the Drude particle to atom nucleus separation were limited to 0.2 Å by using the hard-wall constraint. The thermostat was set to a reference temperature of 298.15 K and maintained with a friction coefficient of 5 ps⁻¹. The Drude oscillator thermostat was set to 1 K with a friction coefficient of 20 ps⁻¹. The pressure was maintained at 1 bar using the Monte-Carlo barostat in OpenMM.

3.2.1. Micelles self-assembly

MD simulations were performed to simulate the self-assembly of the mixed SLES/CA and the SLES/PA micelles under the experimental conditions reported by Tzocheva et al. ²³, using the CGenFF non-polarizable force field. For the mixed SLES/CA system, 216 SLES molecules were randomly placed in a cubic simulation box of 8 nm × 8 nm × 8 nm, by means of the insert-molecules command in GROMACS. Subsequently, 95 capric acid molecules were added in order to match the experimental value of the saturation molar fraction of 0.301²³. The system was then solvated with 22,216 water molecules of which 216 water molecules were replaced by 216 sodium ions to achieve electroneutrality. The resulting SLES concentration in water was ${\widehat{\mathrm{c}}}_{\mathrm{SLES}}$ = 0.47 M. For the mixed SLES/PA system, the procedure for generating the MD simulations system is identical to that for the SLES/CA, except that 20 palmitic acid molecules, rather than 95 of capric acid, were added to match the experimental value for the saturation molar fraction of 0.0909. The energy of the system was minimized by means of the steepest descent algorithm. After the energy minimization, a short MD equilibration simulation of 600 ps was carried out in the isothermal-isochoric ensemble (NVT) with a timestep of 2 fs. The production simulation was run in the isothermal-isobaric ensemble (NPT) for 45 ns with a timestep of 2 fs. Snapshots were saved at intervals of 100 ps. By the end of the MD simulation, several micelles of different sizes had formed. The criterion to identify the micelle to which each SLES and fatty acid molecule belongs followed the method originally proposed by Sammalkorpi⁴³ for pure SDS micelles and further developed by Koneva et al.⁴⁴ for mixed micelles. For the SLES/CA system three sets of distances between the selected atoms are computed for all pairs of SLES and capric acid molecules (Figure 1 and Table 2). Similarly for the SLES/PA system three sets of distances are computed for all pairs of SLES and palmitic acid molecules (Figure 1 and Table 2). SLES and fatty acid molecules are considered to be in the same micelle if at least one of the computed distances in either set 1, set 2 or set 3 is shorter than r1_cutoff = 0.55, or if two distances from two different sets are shorter than r2_cutoff = 0.68 or if all the three distances from the three different sets are shorter than r3_cutoff = 0.70. Values of the cut-offs were chosen in accordance with the work conducted by Storm et al.⁴⁵ on similar mixed surfactant systems.

Figure 1:

Reference atoms for SLES, capric acid and palmitic acid.

Table 2:

Distances between carbon atoms and cutoffs used for the definition of micelles.

Molecules	Distances between atoms			r1	r2	r3
SLES and capric acid	C3_SLES/ C3_SLES or C3_SLES/C1_Capric or C1_Capric/C1_Capric	C7_SLES/ C7_SLES or C7_SLES/C5_Capric or C5_Capric/C5_Capric	C10_SLES/ C10_SLES or C10_SLES/C10_Capric or C10_Capric/C10_Capric	0.55	0.68	0.70
SLES and palmitic acid	C3_SLES/ C3_SLES or C3_SLES/C1_Palmitic or C1_Palmitic/C1_Palmitic	C8_SLES/ C8_SLES or C8_SLES/C6_Palmitic or C6_Palmitic/C6_Palmitic	C16_SLES/ C16_SLES or C16_SLES/C16_Palmitic or C16_Palmitic/C16_Palmitic	0.55	0.68	0.70

3.2.2. Partition coefficient predictions from the US/PMF approach

The largest mixed SLES/CA and SLES/PA micelles were extracted from the final configuration of the MD simulations and each of the micelles was transferred to a water box of 8 nm × 8 nm × 8 nm for setting up the steered molecular dynamic (SMD) simulations in preparation for the US/PMF calculations (Figure 2). 3 US/PMF simulations were performed using the non-polarizable force field that included the capric acid and caprate solutes in the mixed SLES/CA micelle and palmitic acid in the mixed SLES/PA micelle. 2 US/PMF simulations were performed using the polarizable force field that included the capric acid and caprate solutes in the mixed SLES/CA micelle. In each of the SMD simulations, the solute molecule was placed at a distance of 3.5 nm from the micelle centre of mass (COM). For the non-polarizable force field simulation, the system was relaxed in order to minimize the energy with the steepest descent algorithm for 1000 steps and equilibrated in the NVT ensemble for 600 ps. For the polarizable force field simulation, the energy was minimized by allowing the Drude particles to move while the positions of all the atoms were kept fixed. Subsequently all atoms and Drude particles were relaxed with an energy minimization of 1000 steps of steepest descent followed 1000 steps of the adopted basis Newton-Raphson algorithm. Minimization was followed by 50 ns of MD equilibration in the NPT ensemble with a timestep of 1 fs. At the end of the equilibration, the COM of each micelle was constrained by means of a harmonic potential of 3000 kJ/mol/nm². PLUMED⁴⁶ was used to apply the COM distance restraints in the simulations performed with the polarizable force field. US/PMF calculations were performed from the starting COM of each solute at 3.5 nm over 36 US windows to the COM of the micelle. The initial configuration for each US window was obtained through the SMD simulations by pulling each solute molecule towards the respective micelle COM by applying a velocity of 10 nm/ns when the non-polarizable force field was employed (Figure 3). For polarizable systems this velocity was reduced to 2 nm/ns. For each of the US/PMF simulations, the respective US simulations were run for each configuration for 10 ns in the NPT ensemble. The force constant for the umbrella potential was set to 3000 kJ/mol/nm². Subsequently, calculation of the PMF to account for the US biasing harmonic potential was performed using the weighted histogram analysis method (WHAM)⁴⁷ in the WHAM software⁴⁸, version 2.0.9 (http://membrane.urmc.rochester.edu/page_id=94) in which the first 5 ns of sampling in each 10 ns window was considered as equilibration and discarded with the PMFs calculated over the final 5 ns. This yielded the unbiased free energy profile (i.e. PMF) for the transfer of each solute from the water to the micellar phase. The convergence of the PMFs was tested by running WHAM for different time portions of the 36 windows (i.e. 0–2.5 ns, 0–5ns, 0–7.5ns, 0–10ns). Plots of PMF showing the achieved convergence for the caprate solute in both the polarizable and the non-polarizable systems are shown in Figure S1 of the supplementary information. Error estimates for the PMFs were calculated as the averaged absolute differences between the free energies calculated over the intervals 5–6 ns and 5–8 ns and that computed over the interval 5–10 ns. As the US from the COM of the micelle to the water phase involves sampling in discretized shells of increasing volumes, the Jacobian correction^49,50 was implemented in the free energy calculations in order to take into account the effect of transforming the Cartesian coordinates into the distance reaction coordinate, using the following equation proposed by Ciccotti et al.⁵¹

$$\mathrm{\Delta }\mathrm{G}({\mathrm{r}}_{\mathrm{i}})={\mathrm{\Delta }\mathrm{G}}^{\mathrm{WHAM}}({\mathrm{r}}_{\mathrm{i}})+2{\mathrm{k}}_{\mathrm{B}}\mathrm{Tln}\frac{{\mathrm{r}}_{\mathrm{i}+1}}{{\mathrm{r}}_{\mathrm{i}}}$$ (1)

where i is the index that runs over the bins of the discretised shell. ΔG^WHAM(r_i) is the unbiased free energy computed by WHAM and $2{\mathrm{k}}_{\mathrm{B}}\mathrm{T}\mathrm{l}\mathrm{n}\frac{{\mathrm{r}}_{\mathrm{i}+1}}{{\mathrm{r}}_{\mathrm{i}}}$ is the applied Jacobian correction. Subsequently, the partition coefficient of each solute is calculated from the corrected free energy profiles as follows:

$$\mathrm{Log}{\mathrm{K}}_{\mathrm{mic}/\mathrm{w}}=\frac{{\mathrm{\Delta }\mathrm{G}}_{\mathrm{transf}}}{-2.303\mathrm{RT}}$$ (2)

where ΔG_transf is the difference between the free energy in the water phase (0) and the minimum free energy value in the PMF profile that corresponds to the most populated state of the solute in the micelle.

Figure 2:

Extracted micelle configurations used for the steered molecular dynamic simulation for capric acid (a), palmitic acid (b) in the respective SLES/FA mixed micelles. SLES molecules in blue, capric acid in red, palmitic acid in orange; water molecules are excluded. Solute molecules are highlighted by representing oxygen atoms in red, carbon atoms in green and hydrogen atoms in white as spheres.

Figure 3:

Self assembled mixed micelles of SLES/FA predicted by MD simulation: (a) SLES/CA and (b) SLES/PA surrounded by a water shell, used as input for COSMOmic calculations. Water molecules are colored in cyan, SLES molecules in blue, capric acid in red, and palmitic acid in orange.

3.2.3. Partition coefficient predictions from the MD/COSMOmic approach

The largest micelle from each of the two simulated SLES/FA micellar systems, i.e. SLES/CA and SLES/PA, with an adjacent shell of water molecules, as shown in Figure 3, was extracted and fed into COSMOmic for predicting the micelle/water partition coefficient of capric acid and caprate in the SLES/CA micelle and that of palmitic acid in the SLES/PA micelle. COSMOmic⁹ is an extension of the COSMO-RS theory for inhomogeneous systems in which the chemical potential of a solute within its surrounding solvent is computed from the screening charge density (σ) on the surface of molecules^7,8,52. Screening charge densities are calculated by quantum mechanics (QM). The σ values for capric acid, palmitic acid, caprate and SLES were computed using density functional theory (DFT) with the Becke-Perdew^53,54 (BP) functional, the triple-zeta valence polarization^55,56 (TZVP) basis set and the resolution of identity⁵⁷ (RI) approximation. QM calculations were carried out using Turbomole 7.3⁵⁸ package (http://www.turbomole.com). Details about parameters needed for performing DFT QM calculations can be found in the Turbomole user manual (http://www.cosmologic.de/files/downloads/manuals/TURBOMOLE-Users-Manual_70.pdf). COSMOmic 19.0 (http://www.cosmologic.de) is used in this work (http://www.cosmologic.de/files/downloads/manuals/COSMOmic_Manual.pdf), along with the COSMO-RS parameter file BP_TZVP_19.ctd. In COSMOmic, each of the two micelles was discretized into 30 layers along the radius with the last layer exclusively consisting of water molecules. The micelle/water partition coefficients were computed by the default equations implemented in COSMOmic 19.0.

4. Results and discussion

4.1. Micelle structure

4.1.1. Self-Assembly of the mixed SLES/FA micelles

The evolutions of the maximum and mean aggregation numbers along with the numbers of micelles over the course of the simulated self-assembly are plotted in Figures 4a and 4b for the SLES/CA and the SLES/PA systems, respectively. The numbers of molecules in the largest micelles reached the maximum values of 120 for SLES/CA and 74 for SLES/PA by 5ns and remained constant afterwards. The number of micelles decreases as coalescence leads to the formation of bigger micelles characterized by a higher value of the average aggregation number. At the end of the simulation, the SLES/CA and the SLES/PA systems reached average aggregation numbers of 63 and 48, respectively. Five micelles were distinguishable for both the SLES/CA and the SLES/PA systems.

Figure 4:

MD simulation of self-assembly of the mixed SLES/CA micelles (a) and the mixed SLES/PA micelles (b), as characterized by the aggregation number (left axis) and number of formed micelles (right axis).

From the MD simulation, the probability distributions of aggregation numbers are obtained for the mixed SLES/CA and the mixed SLES/PA systems. The aggregation numbers that occurred more often are 21, 24, 50, 91 and 121 for SLES/CA and 19, 34, 45, 69 and 74 for SLES/PA, as shown in Figure 5. The aggregation numbers for pure SLES micelles were experimentally measured to be 43 by Aoudia et al.⁵⁹ and between 67–79 by Anachkov et al⁶⁰. Observed aggregation numbers for both the SLES/CA and the SLES/PA systems are in these ranges. The aggregation numbers for the mixed SLES/PA system are closer to the experimental values of SLES pure micelles compared to the SLES/CA system; this is due to the lower solubility of palmitic acid compared to capric acid in SLES micelles.

Figure 5:

Probability distribution for aggregation numbers for the SLES/CA system (a) and the SLES/PA system (b).

4.1.2. Density profile and probability distribution of terminal atoms

The free energy profiles of fatty acids in the mixed SLES/FA micelles depend on how fatty acid and SLES molecules assemble. In order to have a better understanding of the predicted free energy, we calculated the density profiles of SLES, fatty acids, and water, as well as the probability distribution of hydrophobic and hydrophilic groups of SLES and fatty acids within the selected micelles. The density profiles of capric acid, SLES and water in the largest SLES/CA micelle are shown in Figure 6a and those of palmitic acid, SLES and water in the largest SLES/PA micelle are shown in Figure 6b. In the mixed SLES/CA micelle, the water profile first intersects SLES at r = 2.17 nm and then with capric acid at r _micelle = 2.31 nm. The latter distance is therefore chosen as the radius of the micelle. For the SLES/CA micelle the density of capric acid is higher than that of SLES at the micelle/water interface. The water profile intersects first with palmitic acid at r = 1.74 nm and then with SLES at r _micelle = 2.07 nm. This is due to the concentration of palmitic acid in the micelle being much lower than that of capric acid. The density of palmitic acid is higher at the centre of the micelle, whereas for capric acid the maximum value occurs at 1.5 nm, due to palmitic acid having a longer carbon chain than capric acid. Palmitic acid and SLES have the same number of carbon atoms (C16), therefore the carbon chains of palmitic acid molecules reach the centre of mass of the micelle. The radius of gyration for the two extracted micelles was computed using the following equation suggested by Bogusz et al.⁶¹:

$${\mathrm{R}}_{\mathrm{SLES}/\mathrm{FA}}=\sqrt{5/3}{\mathrm{R}}_{\mathrm{gyration}}$$ (3)

The calculated mean radii for the mixed SLES/CA and the SLES/PA micelles were R _SLES/CA = 2.21 nm and R _SLES/PA = 1.97 nm, respectively, which are in good agreement with the radii of the micelles selected from the density profiles, i.e. 2.31 nm for SLES/CA and 2.07 nm for SLES/PA.

Figure 6:

Density profiles for SLES, capric acid and water (a) and SLES, palmitic acid and water (b) in the MD simulated mixed SLES/FA micelle assemblies.

Figure 7 shows the probability distributions of the hydrophobic groups of the terminal carbon atoms of SLES and fatty acids, and hydrophilic groups of the sulfate group of the SLES molecule and the hydroxyl group of fatty acids in the two self-assembled micelles. For the mixed SLES/PA micelle, the probability distributions of SLES_C16 and PALMITIC_C16 overlapped due to SLES and palmitic acid having the same carbon chain lengths. This is not the case for the SLES/CA micelle where the mismatch in the molecular carbon chains (10 carbons for capric acid and 16 for SLES) resulted in the shifting of the CAPRIC_C10 distribution towards the micelle/water interface. The probability distribution of the hydrophobic groups provides a preliminary estimation of where the fatty acid solute will preferentially distribute in the micelle. Capric acid atoms preferentially distribute between the peak of the distribution for the CAPRIC _C10 atoms at 1.58 nm and the peak of the distribution for the CAPRIC _OH atoms at 2.08 nm from the COM of the micelle. In the case of palmitic acid, the hydrophobic atoms distribute between the peak of the distribution for the PALMITIC _C16 atoms at 1.15 nm and the peak of the distribution for the PALMITIC _OH atom and 1.83 nm.

Figure 7:

Probability distributions of hydrophobic and hydrophilic reference groups for (a) the SLES/CA mixed micelle and (b) the SLES/PA mixed micelle.

4.2. Free energy profiles and micelle/water partition coefficients

4.2.1. Neutral solutes

Free energy profiles for the transfer of capric acid and palmitic acid solutes from water to the respective mixed SLES/PA micelle are presented in Figure 8a (capric acid) and 8b (palmitic acid). The free energy profile for the capric acid solute in the mixed SLES/CA micelle was calculated by means of COSMOmic and US/PMF; for the latter both the non-polarizable (US_CGenFF) and the polarizable (Drude) force fields were used. When using US/PMF, free energy profiles were calculated for the intervals of 5–6 ns, 5–8 ns and 5–10 ns in order to further estimate the extent of convergence. In Figures 8a and 8b, free energy profiles obtained with US/PMF using the non-polarizable and the polarizable force fields for the interval 5–10 ns are shown as solid lines whereas error bars represent the differences of the free energies between the values computed in the intervals 5–6 ns and 5–8 ns and the one computed in the interval 5–10 ns. The energy profile predicted by the US/PMF with the polarizable force field (US/PMF-P) is about ∼20 kJ/mole higher at the centre of the micelle than those predicted by the US/PMF with the non-polarizable force field (US/PMF-N) and COSMOmic. The free energy predicted by the US/PMF-N and COSMOmic showed a rapid decrease in the first 0.5 nm and 0.7 nm, respectively. The free energy then remains constant in the plateau region where only the SLES carbon chain is present; this is also evident from the probability distribution of terminal carbon atoms (Figure 7a). The end of the plateau corresponds to the beginning of the capric acid molecules carbon chain and therefore a subsequent drop in the free energy occurs. The plateau region is not observed in the US/PMF-P as the free energy continuously decreased to ∼15 kJ/mol from 0.5 nm to 1 nm along the reaction coordinate. This steady decrease indicates the higher electrostatic potential of the induced dipole. This dipole effect cannot be captured by the non-polarizable force field ^32,62,63. From 1 to 3.5 nm there are no major differences between the free energy profiles of US/PMF-P and that of US/PMF-N. The minima of all three profiles occurred between 1.5 nm and 2 nm from the centre of the micelle, in agreement with the results from the probability distributions. The minima of the US/PMF-N and COSMOmic profiles are close to 2 nm, whereas the minimum of the polarizable force field is close to 1.5 nm. With regards to the mixed SLES/PA micelle, the minima of US/PMF-N and COSMOmic are located just after 1.5 nm from the centre of the micelle, in agreement with the results from the probability distributions. The minima for palmitic acid are closer to the centre of the micelle, in agreement with the results from density profiles which showed a maximum for the density of palmitic acid at the centre of the micelle.

Figure 8:

Free energy profiles for the transfer of fatty acids from water to the mixed SLES/FA micelle, (a, left panel) for capric acid and (b, right panel) for palmitic acid. Error bars in the US profiles represent the differences of the free energies between the values computed in the intervals 5–6 ns and 5–8 ns and the one computed in the interval 5–10 ns.

The free energies of transfer of fatty acids were used for calculating the micelle/water partition coefficients as given in Table 3. The predicted values for the free energies of transfer of capric acid and palmitic acid are very close to the experimental values for both the non-polarizable and polarizable force field US/PMF simulations. From Table 3 we can see that COSMOmic also performed well in predicting the partition coefficients of the neutral fatty acids, capric and palmitic acids, in both the mixed SLES/CA and SLES/PA micelles. The partition coefficient for palmitic acid is slightly underpredicted by COSMOmic since COSMOmic tends to underpredict the partition coefficients of highly hydrophobic compounds⁶⁴. The prediction errors of US/PMF using the non-polarizable force field are comparable with COSMOmic, although the predicted value for palmitic acid is slightly closer to the experimental data than COSMOmic. The US/PMF-P showed a good prediction accuracy for the partition coefficient of capric acid, in line with values predicted by COSMOmic and the non-polarizable force field.

Table 3:

Predicted free energies of transfer and micelle/water partition coefficients for capric acid and palmitic acid, by US/PMF-N, US/PMF-P and COSMOmic methods. The averaged absolute differences, calculated as the average between the values of the intervals 5–10 ns and 5–8 ns and those of the intervals 5–10 ns and 5–6 ns, are reported in parenthesis next to reported values computed at 10 ns.

Capric acid
ΔGtransf				ΔGtransf Difference vs. Exp.
Non-polarizable force field	Polarizable force field		exp	US_ CGenFF	Polarizable force field
−19.91 (0.47)	−18.89 (0.57)		−19.24	0.67	−0.35
Log Kmic/w				Log Kmic/w Difference vs. Exp.
Non-polarizable force field	Polarizable force field	COSMOmic	exp	Non-polarizable force field	Polarizable force field	COSMOmic
3.49	3.31	3.52	3.37	0.12	−0.06	0.15
Palmitic Acid
ΔGtransf				ΔGtransf Difference vs. Exp.
Non-polarizable force field	Polarizable force field		exp	Non-polarizable force field	Polarizable force field
−35.57 (0.35)	N.E.		−36.2	−0.63	N.E.
Log Kmic/w				Log Kmic/w Difference vs. Exp.
Non-polarizable force field	Polarizable force field	COSMOmic	exp	Non-polarizable force field	Polarizable force field	COSMOmic
6.23	N.E.	5.99	6.36	−0.13	N.E.	−0.37

^*Data collected from Tzocheva et al.²³ N.E. indicates not evaluated.

4.2.2. Caprate anion

The free energy profiles for the transfer of the charged fatty acid caprate from water to the mixed SLES/CA micelle are shown in Figure 9. As with the neutral solutes, the predicted free energy profiles of the US/PMF with both force fields in the interval 5–10 ns are shown as solid lines with the error bars representing the differences of the free energies computed in the intervals 5–6 ns and 5–8 ns and the one computed in the interval 5–10 ns.

Figure 9:

Free energy profiles for the transfer of caprate anion from water to the mixed SLES/CA micelle. Error bars represent the differences of the free energies computed in the intervals 5–6 ns and 5–8 ns and the one computed in the interval 5–10 ns.

The free energy profiles for the transfer of caprate predicted by the US/PMF-P and the US/PMF-N are similar in shape. The free energy of caprate at the centre of the micelle predicted by the US/PMF-P is higher than the one predicted by the US/PMF-N but the difference is relatively small. Towards the micelle/water interface the energy profile of caprate from the US/PMF-P shows a less favourable minimum compared to that of the non-polarizable US/PMF. In the interfacial region, where the anionic oxygen of caprate interacts with the sulfate anion of SLES, the induced polarization effect is an important feature that can be captured by the polarizable force field but not by the non-polarizable force field. Values for the free energies of transfer for caprate, predicted by the US/PMF with both force fields are compared with the experimental data in Table 4. The predicted value obtained from the US/PMF-P is much closer to the experimental value as compared to US/PMF-N. The minimum in the free energy profile as predicted by US/PMF-P results in a higher value of the free energy of transfer and consequently a considerably improved prediction of the logarithm of the micelle-water partition coefficient compared to US/PMF-N. The COSMOmic profile shows a minimum in correspondence with the water phase and the free energy profile does not assume negative values along the micelle radius. This results in a positive value for the free energy of transfer and, consequently, in a negative value for the logarithm of the micelle-water partition coefficient (Log K_mic/w).

Table 4:

Free energies of transfer and micelle/water partition coefficients for caprate solute in the mixed SLES/CA micelle, predicted by US/PMF-N and US/PMF-P and COSMOmic. The averaged absolute errors are calculated between the values of the intervals 5–10 ns and 5–8 ns and those between 5–10 ns and 5–6 ns are reported in parenthesis next to reported values of the interval 5–10 ns.

ΔGtransf				ΔGtransf Difference vs Exp.
Non-polarizable force field	Polarizable force field		exp	Non-polarizable force field	Polarizable force field
−13.14 (0.64)	−8.41 (0.57)		−5.88	−7.26	−2.53
Log Kmic/w				Log Kmic/w Difference vs. Exp.
Non-polarizable force field	Polarizable force field	COSMOmic	exp	Non-polarizable force field	Polarizable force field	COSMOmic
2.34	1.46	−0.8	1.03	1.31	0.43	−1.83

^*Data collected from Tzocheva et al²³.

Based on the PMFs both COSMOmic and US/PMF-N showed substantial disagreements for the micelle-water partition coefficient (Table 4). US/PMF-P improves considerably the prediction of the caprate partition coefficient, yielding an absolute prediction difference of 0.43 that is almost more than one logarithmic unit smaller than the ones of COSMOmic (−1.83) and US/PMF-N (1.31). Thus, the use of a polarizable force field is necessary for improving the prediction accuracy of the micelle-water partition coefficients of charged solutes in anionic micelles.

Figure 10 shows example conformations of the caprate solute at the minimum of the profiles for the two force fields. With the polarizable force field, the hydrophilic group is more embedded in the water phase (Figure 10b). In contrast, the ionic group is more shifted towards the micellar phase in the case of the non-polarizable force field (Figure 10a). This difference is consistent with the higher value of the free energy of transfer predicted by the polarizable force field, which indicates a higher affinity of the solute for the water phase. Such high affinity is consistent with the presence of the electronic polarization, leading to the more favourable interactions of the charged head group with the aqueous environment.

Figure 10:

Capric acid anion (caprate), at the minima of the free energy, interacting with capric acid and SLES molecules predicted by the non-polarizable force field (a) and polarizable force field (b). Capric acid and SLES atoms are represented as pipes (hydrogen atoms are excluded for clarity), water atoms as lines and caprate atoms as spheres. Carbon atoms are colored in cyan, hydrogen atoms in white, oxygen atoms in red, sulfur atoms in yellow; and water molecules are colored in red.

Finally, the behaviour of sodium ions when the ionic caprate solute is absorbed in the micelle is analyzed. In Figure 11 the probability distributions of sodium ions, carboxyl oxygen atoms of capric acid and sulphur atoms of SLES in the mixed SLES/capric acid micelle are shown for both the non-polarizable and the polarizable systems. As is evident, both the peaks of probability distributions for carboxyl oxygens (2.30 and 2.31 nm for the polarizable and the non-polarizable systems, respectively) and sulphurs (2.51 and 2.55 nm, respectively) are close to each other for the polarizable and the non-polarizable systems. On the contrary, the peaks of the probability distributions of sodium ions are at 2.75 and 2.97 nm from the micelle center of mass for the polarizable and the non-polarizable systems, respectively. In both models the sodium ions condensed in the vicinity of the negatively charged micelle, as expected based on simple electrostatic consideration. However, with the polarizable force field the sodium ions cluster at a shorter distance from the micelle center of mass, suggesting that attractive interactions between the negatively charged micelle and sodium cations, as modelled by the polarizable force field, are stronger as compared to the additive force field.

Figure 11:

Probability distributions of sodium ions around the mixed SLES/capric acid micelle for the non-polarizable and the polarizable systems when the caprate anion is at its minimum location in the PMFs.

5. Conclusions

In this paper, the performance of the MD/COSMOmic and the US/PMF approaches for predicting solute partition coefficients of fatty acids in the mixed SLES/FA micelles have been evaluated. MD simulations have been performed to obtain structures from the self-assembly of the SLES/CA and the SLES/PA mixed micelles. A number of micelles were formed for both simulated systems and the largest micelle of each system was selected to predict solute partition coefficients of capric acid, palmitic acid and caprate anion in the mixed micelles. The selected micelles of the two systems were first studied using COSMOmic. The predicted micelle/water partition coefficients showed good agreement with the experimental data for the neutral solutes of capric acid and palmitic acid, but a severe underprediction occurred for the micelle-water partition coefficient of the capric acid anion (caprate). Subsequently, the US/PMF approach has been explored for predicting partition coefficients using both polarizable and non- polarizable force fields. The results of the non-polarizable force field produced similar results to COSMOmic for neutral solutes but not for the charged solute though the agreement with experiment is still poor, showing that the use of a non-polarizable force field is limited to good prediction accuracy for neutral solutes. Good prediction accuracy for the micelle-water partition coefficient of the charged caprate solute was achieved by using the Drude polarizable force field. Thus, for simulating charged solutes in anionic surfactant micelles, the use of an accurate polarizable force field is crucial to model the electrostatic interactions. This appears to be due the polarizable force field’s improved ability to accurately model the dipole potential as previously reported for the lipid bilayer⁶⁵. The present results indicate that with the availability of polarizable force fields, such as the CHARMM Drude polarizable force field, in-silico prediction of partition coefficients for charged solutes will be a more robust low-cost alternative to laboratory experiments.