Water Partition Coefficients

Article pubs.acs.org/Langmuir

Prediction of Micelle/Water and Liposome/Water Partition Coefficients Based on Molecular Dynamics Simulations, COSMO-RS, and COSMOmic Thomas Ingram,*,† Sandra Storm,*,† Linda Kloss,† Tanja Mehling,† Sven Jakobtorweihen,‡ and Irina Smirnova† †

Institute of Thermal Separation Processes, ‡Institute of Chemical Reaction Engineering, Hamburg University of Technology, Eissendorfer Straße 38, D-21073 Hamburg, Germany S Supporting Information *

ABSTRACT: Liposomes and micelles find various applications as potential solubilizers in extraction processes or in drug delivery systems. Thermodynamic and transport processes governing the interactions of different kinds of solutes in liposomes or micelles can be analyzed regarding the free energy profiles of the solutes in the system. However, free energy profiles in heterogeneous systems such as micelles are experimentally almost not accessible. Therefore, the development of predictive methods is desirable. Molecular dynamics (MD) simulations reliably simulate the structure and dynamics of lipid membranes and micelles, whereas COSMO-RS accurately reproduces solvation free energies in different solvents. For the first time, free energy profiles in micellar systems, as well as mixed lipid bilayers, are investigated, taking advantage of both methods: MD simulations and COSMO-RS, referred to as COSMOmic (Klamt, A.; Huniar, U.; Spycher, S.; Keldenich, J. COSMOmic: A Mechanistic Approach to the Calculation of Membrane−Water Partition Coefficients and Internal Distributions within Membranes and Micelles. J. Phys. Chem. B 2008, 112, 12148−12157). All-atom molecular dynamics simulations of the system SDS/water and CTAB/water have been applied in order to retrieve representative micelle structures for further analysis with COSMOmic. For the system CTAB/water, different surfactant concentrations were considered, which results in different micelle sizes. Free energy profiles of more than 200 solutes were predicted and validated by means of experimental partition coefficients. To our knowledge, these are the first quantitative predictions of micelle/water partition coefficients, which are based on whole free energy profiles from molecular methods. Further, the partitioning in lipid bilayer systems containing different hydrophobic tail groups (DOPC (1,2-dioleoyl-sn-glycero-3-phosphocholine), SOPC (stearoyl-oleoylphosphatidylcholine), DMPC (1,2-dimyristoyl-sn-glycero-3-phosphocholine), and POPC (1-palmitoyl-2-oleoylsn-glycero-3-phosphocholine)) as well as mixed bilayers was calculated. Experimental partition coefficients (log P) were reproduced with a root-mean-square error (RMSE) of 0.62. To determine the influence of cholesterol as an important component of cellular membranes, free energy profiles in the presence of cholesterol were calculated and shown to be in good agreement with experimental data.

1. INTRODUCTION

direction, with large gradients in density and polarity on a nanometer length scale. According to Marrink and Berendsen,10 a lipid bilayer can be divided into four regions based on their chemical properties. Region I almost exclusively contains lipid tails. Region II contains a diverse mixture of functional groups such as the lipid tail, the carbonyl group from the fatty acid backbone of the lipid, as well as a portion of the headgroups, Region III contains a diverse mixture of functional groups, including a small portion of carbonyl groups, the headgroup, and water. Region IV can be considered as bulk water. Due to the absence of the glycerol backbone for surfactants, micelles can be divided into three regions: the

Liposomes and micelles can be regarded as potential drug delivery systems in research and industry.2 Consequently, numerous studies investigate novel micelle forming polymers as well as targeted uptake and release kinetics from nanostructured materials such as liposomes or micelles.3−9 One of the crucial factors, representing the capacity of potential drug delivery systems, is the partition coefficient of a potential drug candidate between the aggregate and the surrounding water. Depending on the hydrophobicity of the solute, it partitions into different regions of the aggregate. However, opposed to common two phase systems such as the system octanol/water, where two clearly defined phases can be observed, the interactions in micelles and liposomes are governed by interfacial phenomena. Hence, liposomes and micelles must be considered as inhomogeneous in the normal or radial © XXXX American Chemical Society

Received: May 14, 2012 Revised: February 8, 2013

A

dx.doi.org/10.1021/la305035b | Langmuir XXXX, XXX, XXX−XXX

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micelles or liposomes are calculated using molecular dynamics simulations and COSMO-RS. Structural information such as the atomic distribution in heterogeneous systems as well as different conformers of the amphiphiles is derived from MD simulations.26 Subsequently, these data are used to calculate free energy profiles with COSMOmic. 2.1.1. Micelles. Starting from a nonaggregated configuration, the micelle self-assembly of the ionic surfactants CTAB and SDS was studied. The numbers of surfactant and water molecules were chosen according to the desired surfactant concentration as shown in Table 1.

hydrophobic core, the headgroup region in contact with water, and an aqueous region outside the micelle. The preference of a solute for a certain region can be quantified in form of a free energy profile,11 whereas the minimum of the free energy indicates the most favorable position of the solute. Further, free energy profiles reveal various thermodynamic and transport processes, governing membrane binding and transport in the aggregate. Therefore, the knowledge of free energy profiles can be directly applied to the design and optimization of potential drug delivery systems.12 However, at the molecular level properties such as free energy profiles in heterogeneous systems are difficult to explore in laboratory experiments. Therefore, significant efforts have been made to develop and validate methods to reliably simulate the structure and dynamics of lipid membranes and micelles.13,14 Latest developments have shown that molecular dynamics (MD) simulations can be applied to directly simulate free energy profiles of small molecules in lipid membranes.12,15−18 Still these simulations are computationally demanding. Moreover, although much effort was spent on developing new force field parameters recently, there is still a lack of universal force field parameters.19−21 Therefore, calculated free energies have rarely been evaluated by means of experimental membrane/water partition coefficients. Besides almost no studies of solubilization phenomena in micellar systems starting from intermolecular interactions have been performed so far.22 Alternatively, thermodynamic models such as UNIFAC and COSMO-RS have been used to predict partition coefficients between micelles and water.23,24 These methods consider the micelle as a macroscopic phase in equilibrium with the aqueous surrounding (pseudophase approach). Mokrushina et al.23 predicted micelle/water partition coefficients, which showed reasonable quantitative agreement with experimental data. Nevertheless, potential transport resistances due to the inhomogeneity in micelles and liposomes cannot be determined. To overcome this limitation, Klamt et al.1 proposed a new method which combines the structural information of lipid bilayers and COSMO-RS, referred to as COSMOmic. For a large set of experimental liposome water partition coefficients, Endo et al.25 demonstrated the advantages of COSMOmic compared to other predictive methods based only on the molecular structure. In their study, the model system DMPC/ water was used for all calculations. So far, the influence of different lipid structures from MD simulations as well as the selection of representative conformers for the COSMOmic calculation has not been investigated. In this work, the combination of molecular dynamics simulations and COSMO-RS for micellar systems is investigated. Micellar structures of the systems SDS/water as well as CTAB/water were obtained by all-atom molecular dynamics simulations. The corresponding structural information from MD simulations is used to predict free energy profiles along the radius of the spherical micelles using COSMOmic. The results are validated based on a large set of experimental micelle/water partition coefficients. Further, different lipid systems including mixed bilayers are simulated and the structural information is used to investigate the influence of additives such as cholesterol on the partitioning of alcohols in mixed lipid systems.

Table 1. Configurations of the Micellar Systems for the Molecular Dynamics Simulations surfactant

csurfactant [M]

surfactant

ion

water

simulation time [ns]

SDS CTAB CTAB

1.00 0.73 0.10

216 216 216

216 216 216

8535 12 000 120 000

400 100 100

Counterions of the ionic surfactant molecules were taken explicitly into account. After energy minimization of the initial system, a 600 ps equilibration simulation in the NVT ensemble was carried out. Thereafter, the self-assembly was simulated in the NPT ensemble. Criteria to identify micelles were defined according to Sammalkorpi et al.27 In brief, for each surfactant, three to four different cut-offs (rc1− rc4) were defined, depending on the molecule length (SDS, three cutoffs; CTAB, four cut-offs). As described by Sammalkorpi et al.,27 the distance between the center of mass of two surfactants is always the first distance and in addition two or three other distances between carbon atoms have been selected. The last distance is always between the terminal carbon atoms of the hydrophobic tails. Micelles are identified if at least one of the distances is smaller than the first and smallest cutoff rc1 or if two distances are smaller than the second cutoff rc2 and so on, while rc1 < rc2 < rc3. The surfactant self-assembly is faster for high surfactant concentrations compared to lower concentrations. Micelle structures of the system CTAB/water have been simulated at two different concentrations: c1,CTAB = 0.73 M and c2,CTAB = 0.10 M. The first relatively high concentration has been chosen in order to reduce the simulation time until stable micelle sizes are reached, neglecting the poor solubility of CTAB in water. The second concentration (c2,CTAB = 0.10 M) has been selected to analyze the self-assembly process taking into account the poor solubility of CTAB in water. Nevertheless, in the time scale of nanoseconds, it is not likely to observe any phase separation due to the solubility limits of the surfactants. All micelles chosen for further investigations with COSMOmic have been compared to experimental data in respect of size. The micelle structures were analyzed according to their probability of occurrence. Only the most probable micelle sizes were selected for further calculations. For each size, three different micelles, each at a different time, were selected randomly to ensure uncorrelated structures. Surfactant conformers having a solvent accessible surface (SAS) close to the average SAS were chosen according to the conformerselection procedure developed for lipid bilayers by Jakobtorweihen et al.26 In brief, a surfactant molecule having a total SAS close to the average total SAS is chosen, where the average is calculated over all lipids in the MD trajectory. The SAS was calculated with the g_sas program of the Gromacs software package. 2.1.2. Liposomes. Starting structures of the lipid bilayers were taken from the CHARMM-GUI Web site or created with the CHARMMGUI Membrane Builder.28−30 The systems contained 30−33 water molecules per lipid. Hence, the lipids were fully hydrated. All bilayers were simulated for at least 40 ns. The mixed bilayer (30% cholesterol and 70% DPPC) was simulated for 200 ns. Depending on the equilibration of the area per lipid, 10−20 ns of these simulations were assigned as equilibration periods and not further analyzed. The remaining production phase was long enough for the analysis done in

2. COMPUTATIONAL DETAILS 2.1. Molecular Dynamics Simulations of Micelles and Bilayers. Free energy profiles in nanostructured materials such as B


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this work.31 Recently, we have developed a procedure to obtain system structures for COSMOmic from MD simulations.26 Rather than taking one system snapshot as input for the atomic distribution over the bilayer axis,1 it was shown that it is more physically sound and that the results are more reliable when the atomic distributions are calculated as time averages. The lipid conformer needed for the COSMOmic calculations was chosen from the MD trajectory as explained in ref 26. 2.1.3. Simulation Details. All MD simulations were performed with the Gromacs 4.5.3 simulation package.32,33 The CHARMM36 all-atom force field was used for all lipids31 and the CHARMM TIP3P model for water.34 The Lennard−Jones interactions were smoothly switched to zero between 0.8 and 1.2 nm. As the Coulomb interactions were truncated at 1.0 nm, the long-range part of these interactions was taken into account via the particle-mesh Ewald method.35 For the integration of the equations of motion, a time step of 2 fs were used. If not otherwise mentioned, all simulations were carried out in the NPT ensemble. The temperature was coupled to a Nosé-Hoover thermostat36 with a coupling constant of τT = 1 ps. The pressure was kept at 1 bar with the Parrinello-Rahman barostat.37,38 For micelle simulations, an isotropic coupling with τp = 1 ps was used. Whereas for bilayer simulations a semi isotropic coupling with τp = 5 ps was employed. For further analysis, the system configurations were stored every 10 ps. 2.2. COSMO-RS. COSMO-RS is an effective model to calculate thermodynamic properties such as the chemical potential of a compound in a homogeneous liquid system. A detailed description of the model can be found elsewhere.39−41 Briefly, COSMO-RS describes the interactions of a compound in any solvent mixture based on pair wise interactions of surface segments. Each surface segment is characterized by its COSMO polarization charge density σ.42 In this work, the COSMO polarization charge density on the molecular surface of the solutes were calculated with a full DFT geometry optimization, while the charge density of the surfactants and lipids were calculate with a single point calculation. The structures of the amphiphiles were taken from MD simulations (for details, see Jakobtorweihen et al.26) All quantum chemical calculations were performed with Turbomole 5.10 using density functional theory with the BP functional, the TZVP basis set, and the RI approximation.43−48 The polarization charge densities of all surface segments, the corresponding atoms, and structural information of the molecules are saved in a so-called cosmo-file. 2.2.1. Extension of COSMO-RS: COSMOmic. To account for the anisotropy in micelles and liposomes with COSMO-RS, COSMOmic was introduced by Klamt et al.1 Structural information can be obtained with all-atom molecular dynamics simulations. Starting from the center of mass of all surfactant or lipid molecules, the aggregate structure is divided in n arbitrary layers along the normal (bilayer) or radial direction (micelle). Each layer is considered as a separate homogeneous phase with a specific molecular composition as shown in Figure 1.

In this work, these atomistic compositions were calculated from MD simulations. The surface segments representing a single layer can be derived from the system configuration and the corresponding cosmofiles of the molecules. Therefore, the chemical potential of a solute segment μi,S can be calculated as function of the distance to the center of mass. The chemical potential of a solute i is evaluated in each layer for m different orientations d. In the case of perfectly symmetrical molecules such as methane, the orientations are all equal. As a result, surface segments of a single molecule might be located in different layers. Thus, the chemical potential of a single solute depends on the position and the orientation of the molecule. μi (r , d) =

∑ aeff μi ,S(r , d) + μi ,comb (r , d) S∈i

(1)

Similar to COSMO-RS the chemical potential of a solute with COSMOmic includes a combinatorial contribution, for details see.1 aeff denotes the effective contact area of a surface segment. The partition function Zi and the probability pi of the solute at any position related to bulk water can be calculated as m

Zi(rj) =

∑ k

⎧ − μ (rj , dk) ⎫ i ⎬ exp⎨ kT ⎭ ⎩ ⎪

⎪

⎪

⎪

(2)

and

Zi(rj)

pi (rj) =

Zi(rn)

(3)

where rn denotes the outermost layer consisting exclusively of bulk water. Then the difference in free energy ΔGi of a solute between any position in the aggregate and bulk water can be determined:

ΔGi(rj) = − RT ln pi (rj)

(4)

Assuming that the solute partitions only to the position with the minimum of free energy ΔGi,min, the partition coefficient can be estimated from

⎛ −ΔGi ,min ⎞ Pi ,min = exp⎜ ⎟ ⎝ RT ⎠

(5)

Pi,min gives a good approximation of the partition coefficient. However, if the free energy profile as function of the distance to the center of the aggregate is known, the probability at each position should be taken into account. All free energy and probability calculations are based on the solubilization of a single solute in the system. No interactions between the solute molecules with each other are taken into account. Besides, the free energy is equal to the free enthalpy for the properties studied within this paper, since they do not strongly depend on the pressure. In this work partition coefficients Pi were calculate by means of free energy profiles as follows. Taking the probability distribution pi(rj) into account, a system size dependent partition coefficient Ki can be derived: n

Ki =

n w (rj)

(

∑ j V (rj) pi (rj) − V (rn) pi (rn) n V (rn) n

ngw w (rn)

pi (rn)

w (rn)

) (6)

where V(rj) and nw(rj) denote the volume of a layer and the number of water molecules in the layer, respectively, and nwg is the total number of water molecules in the system. Ki represents the total probability related to the number of amphiphiles in the system divided by the probability related to all water molecules in the system. This value can be converted into a system size independent partition coefficient Pi [Lw/kgamph] as follows:

Figure 1. Principle of COSMOmic presented for a lipid bilayer and an example solute. The bilayer shows lipid headgroups (red), lipid tailgroups (yellow), and water (blue).

Pi = f ·K i C

(7) dx.doi.org/10.1021/la305035b | Langmuir XXXX, XXX, XXX−XXX

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Figure 2. Self-assembly of SDS at cSDS= 1.00 M for the first 100 ns. Clustering of the micelles is presented for the cut-offs rc1 = 0.52, rc2 = 0.57 ,and rc3 = 0.67 [nm] (red line, maximum aggregation number; green line, average aggregation number; blue line, number of aggregates). f=

NAV (rn)

ngw

l ∑t namphMamph

n w (rn)

micelles sizes would approximate a Gaussian shape. However, our procedure to obtain suitable structures for the prediction of the partition behavior of solutes with COSMOmic, is justified by the good results of the predictions compared to experimental results. The aggregation number of SDS has been estimated experimentally to be in the range between 50 and 80 surfactant molecules per micelle for a 0.1 M SDS solution at 20 °C.49 Hence, our simulations are reasonable in comparison with experimental data. This finding is in line with previous simulation work of the system SDS/water at the same conditions and surfactant concentration.27 Nevertheless, it has to be noted that the self-assembly process is not likely to have reached equilibrium conditions even for the long simulation time of over 200 ns as already mentioned by Sammalkorpi et al.27 Although the system might have not reached equilibrium, the micelle structures have reached an appropriate state such that they can be used for COSMOmic calculations. Hence, the simulation time is sufficient for the aim of this study. For the system CTAB/water, it could be observed from MD simulations that the average micelle size increases significantly with increasing surfactant concentration. At a CTAB concentration c2CTAB = 0.10 M, the most probable aggregation number is 16 surfactant molecules per micelle. The largest aggregation number found was 23 due to the slow aggregation process at this low CTAB concentration In comparison, experimental aggregation numbers measured at surfactant concentrations in the range of 0.2−100 mM scatter in the range of 80−15050,51 for the system CTAB/water. In the present study, we aim to determine solvation free energies along the radius of the micelles, and hence, a realistic representation of the micelle structures is desirable. Therefore, we conducted further simulations at a higher CTAB concentration in order to reach more realistic micelle sizes in a reasonable time scale. The most probable micelle size found during our simulation of 100 ns is 119 surfactant molecules per micelle, which is in the range of experimental data and has been selected for further analysis with COSMOmic. Representative micelle structures for each size are shown in Figure 3. The MD simulations reveal that different aggregation numbers may occur during single simulation run depending on the system conditions and concentrations. For this reason, different structures have been selected for calculating free energy profiles with COSMOmic along the radius of SDS micelles. The aggregation number in the system CTAB/water

(8)

where NA is the Avogadro constant, V(rn) is the volume of the outermost layer, namph,s is the number of amphiphiles of the type t in the system, t is the number of different amphiphiles, and Mamph,s is the molecular mass of an amphiphile. 2.2.2. Pseudophase Approach. Using the pseudophase approach, partition coefficients were predicted assuming that micelles and lipid bilayers can be considered as homogeneous pure surfactant and lipid phases, respectively.23 Activity coefficients γi in the pseudophase (pp) and in pure water (w) were calculated with COSMO-RS. The partition coefficient Pi[Lw/ kgamph] is calculated as follows: Pi =

γ wvw

i pp

γi Mamph

(9)

where νW denotes the molar volume of water.

3. RESULTS AND DISCUSSION 3.1. Partition Coefficients in Micellar Systems. One of the relevant properties representing the capacity of micellar systems as potential drug carrier is the partition coefficient. In this work, structural information of the systems SDS/water and CTAB/water were derived from MD simulations. By means of free energy profiles along the radius of the micelles, we predicted the partitioning of solutes with different hydrophobicities and compared our data to experimental micelle/ water partition coefficients from literature. 3.1.1. Micellar Structures. Starting from a nonaggregated configuration, micelle self-assembly of the ionic surfactants CTAB and SDS was simulated by molecular dynamics simulations. The first 100 ns of the self-assembly of the SDS surfactants at cSDS= 1.00 M is presented in Figure 2. Figure 2 illustrates that in the first 20 ns the micelle growth is most evident, whereas afterward the micelles are staying rather stable. In the system SDS/water, the micelle size distribution from MD simulations was analyzed over 400 ns. As a result, three micelle sizes with an aggregation number of 36, 57, and 71 occurred most often and were most stable. Nevertheless, we also performed multiple simulations from different random configurations and observed always different probabilities of micelle sizes. It seems to be most likely that if an infinite number of different simulations (or a very long simulation in the range of μs) would be performed, the probability of the D


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in the core is in accordance. However, the minimum free energy for methane was reported to be around 0 kcal/mol, whereas for benzene the minimum free energy was reported to be ca. −5.5 kcal/mol and for ethylbenzene ca. −8 kcal/mol. These absolute values differ from our calculations with COSMOmic, which might be due to the different models and methods. The contributions of each layer to the partition coefficient calculated with COSMOmic are shown in Figure 5. For

Figure 3. Randomly selected micelle structures. The number behind the surfactant type indicates the size of the micelle.

depends on the initial surfactant concentration. At c2CTAB = 0.10 M, an average aggregation number of 16 was found. Increasing the CTAB concentration leads to more realistic CTAB micelle structures with 119 surfactant molecules per micelle, which is in the range of experimental data. 3.1.2. Free Energy Profiles in Micellar Systems. Structural information from self-assembled micelle structures, as described in the previous section, was combined with COSMOmic to predict free energy profiles along the radius of the micelle. Figure 4 shows the free energy profile of methane, benzene, and Figure 5. Probability distribution of ethylbenzene, benzene, and methane in a SDS micelle consisting of 71 surfactant molecules, T = 25 °C.

benzene and ethylbenzene, the partition coefficient is dominated by the transfer from the aqueous bulk phase to the hydrophobic core region. In the case of methane, the aqueous bulk phase has a significant contribution to the overall probability (compare Figure 5). Therefore, the contribution of the water molecules in the headgroup region must be accounted for calculating the partition coefficient. Using eq 6, free energies as well as the distribution of water molecules along the radius of the micelles are considered. According to eq 6, the partition coefficients derived from the whole free energy profiles are 1.34 (0.67),53 2.12 (1.94),54 and 3.02 (2.77)54 for methane, benzene, and ethylbenzene, respectively. The experimental partition coefficients are given in brackets. To our knowledge, these are the first quantitative predictions of micelle/water partition coefficients, which have been derived from whole free energy profiles, based on molecular methods. The partition coefficients have been determined not only taking into account the minimum of the free energy profile, as done usually, but rather taking the whole profile for the determination. 3.1.3. Partition Coefficients in Micellar Systems. System SDS/Water. In order to calculate free energies along the radius of the micelle, we have studied three randomly selected micellar configurations for each micelle size (compare Figure 3). Partition coefficients for more than 200 solutes were predicted for all nine system configurations. The results are summarized in Table 2. The root-mean-square error (RMSE) deviates between 0.44 and 0.57 for the nine configurations. In comparison, partition coefficients predicted with the pseudophase approach and the nine representative conformers from molecular dynamics simulations deviate significantly (compare Table 2). Still, the pseudophase approach is a considerably

Figure 4. Free energy profile of methane, benzene, and ethylbenzene along the radius of a SDS micelle consisting of 71 surfactant molecules, T = 25 °C.

ethylbenzene in a SDS micelle consisting of 71 surfactant molecules. The three solutes were selected because they were used as model systems in a previous simulation work.52 It is evident that all three solutes methane, benzene, and ethylbenzene have the lowest free energy of transfer in the micellar core region. An increase in free energy relative to bulk water can be observed in the transition region from the polar core to the headgroups. The transition region from the headgroups to bulk water is rich in charged species; hence, a small but significant maximum in relative free energy can be observed. Free energy profiles in SDS micelles can be calculated qualitatively with COSMOmic. In agreement with previous simulation work,52 the three solutes partition uniformly in the core region. In comparison with the publication of Matubayasi et al.,52 the order of the solutes with regard to their free energy E


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Table 2. Comparison of Different Micellar Structures and Conformers for the Prediction of Partition Coefficients in the System SDS/Watera micelle

SDS 36 (I)

SDS 36 (II)

SDS 36 (III)

SDS 57 (I)

SDS 57 (II)

SDS 57 (III)

SDS 71 (I)

SDS 71 (II)

SDS 71 (III)

COSMOmic pseudophase approach

0.48 0.68

0.48 0.81

0.45 0.72

0.50 0.73

0.57 0.76

0.46 0.69

0.44 0.70

0.44 0.68

0.48 0.69

a The number listed with the surfactant type indicates the size of the micelle (36, 57, 71) and different independent simulation snapshots (I−III). Values present the RMSE for more than 200 solutes found in the literature (for details, see the Supporting Information).

simple and useful tool to predict partition coefficients, at least qualitatively, in micellar systems. According to Mokrushina et al.,23 predictions using the pseudophase approach can be further improved by selecting appropriate conformer mixtures. The difference between COSMOmic, the pseudophase approach, and experimental data is visualized in Figure 6.

Figure 7. Free energy profile of benzene in two different CTAB micelles T = 25 °C: large micelle (119 surfactant molecules), equilibrium structure at cCTAB = 0.73 M; small CTAB micelle (16 surfactant molecules), equilibrium structure at cCTAB= 0.10 M.

micelles, the hydrophobic core region becomes very small. Hence, the partitioning is dominated by the interfacial region between the hydrophobic core and the headgroups. The free energy of benzene at the center of both micelles is equal. However, the contribution of the core region to the partition coefficient deviates due to the different volumes of the center regions; therefore, the partition coefficient of a hydrophobic solute such as benzene increases with increasing micelle size. In total, more than 80 experimental partition coefficients of neutral solutes were found in literature for the system CTAB/ water. In Figure 8, the predicted partition coefficients are shown against experimental data. The RMSE for both, the small and the large micelle, are 0.53 and 0.51, respectively. Again larger deviations are found for PAHs. Partition coefficients were also predicted using the pseudo phase approach (compare Figure 8). The RMSE (0.85 and 0.86) increases significantly compared to the calculations with COSMOmic. Hence, for a realistic calculation of the free energy of transfer from water into micelles, the structure of the micelles must be taken into account. The larger RMSE for the system CTAB/water compared to the system SDS/water can be predominately attributed to amine compounds such as caffeine, dopamine, lidocaine, and butylurea. The model seems to underestimate the repulsive interactions between the amine groups of the solute and the headgroups of CTAB. Still, considering the known limitation of COSMO-RS for ternary and quaternary amines,56 a RMSE of 0.51 can be considered as very satisfying, and it illustrates the robustness of COSMOmic. 3.2. Liposome/Water Partition Coefficients. Numerous authors investigated the partitioning of small organic solutes in lipid membranes.25,57 Generally, theoretical studies focus on

Figure 6. Comparison of experimental (for details, see the Supporting Information) and predicted SDS-micelle/water partition coefficients at T = 25 °C. Micelle structure consists of 71 surfactant molecules (RMSE = 0.44); the diagonal indicates a perfect agreement between predicted and experimental values; and the two dotted lines (above and below the diagonal) represent the average experimental error of 0.3 in the log P scale.23

The result demonstrates that the combination of molecular dynamics simulations and COSMOmic can reproduce the partitioning in micellar systems for a wide range of different solutes with different functional groups and different hydrophobicities. Larger deviations were found for hydrophobic solutes such as Benzo(a)pyren (log Pexp = 6.49). Although experimental micelle water partition coefficients of hydrophobic compounds must be considered with caution,24 our results show that COSMO-RS underestimates the hydrophobicity off polycyclic aromatic hydrocarbons (PAHs). Neglecting the four PAHs, the RMSE further decreases from 0.44 to a value of 0.39 log units. Considering the present accuracy of experimental partition coefficients available in the literature, further improvements can be considered to be rather unlikely. System CTAB/Water. Experimental studies as well as molecular dynamics simulation show, that the micelle size depends on the surfactant concentration.55 For the system CTAB/water, we studied the influence of different surfactant concentrations on the micelle structure and, thus, on the free energy profile of a solute in the micelle. The influence of the aggregate structure on the free energy profile of the model solute benzene is shown in Figure 7. In the case of small F


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same system (log P = 13),58 our result is in excellent agreement with experimental partition coefficients (log P = 3.87).59 The core region represents a significant energy barrier of 3.2 kcal/ mol transferring ibuprofen from the minimum to the center position. The average atomic distribution of the DOPC bilayer as well as the representative conformers were sampled from molecular dynamics simulations according to the method described by Jakobtorweihen et al.26 It is known from literature that free energy barriers in the membrane depend on the composition of the headgroups,60 as well as on the chain length of the hydrophobic tails.61 Therefore, we calculated free energy profiles of more than 200 compounds in various pure phosphocholine (PC) lipid bilayers with different tails, including DOPC (1,2-dioleoyl-snglycero-3-phosphocholine), SOPC (stearoyl-oleoylphosphatidylcholine), DMPC (1,2-dimyristoyl-sn-glycero-3-phosphocholine), and POPC (1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine) (for details, see the Supporting Information). Figure 10 shows the predicted partition coefficients against experFigure 8. Comparison of experimental (for details, see the Supporting Information) and predicted CTAB/water partition coefficients at T = 25 °C. Micelle structure consists of 119 surfactant molecules (RMSE= 0.51); the diagonal indicates a perfect agreement between predicted and experimental values; and the two dotted lines (above and below the diagonal) represent the average experimental error of 0.3 in the log P scale.23

solutes such as amino acid residues11 or drug molecules.17 At present, molecular dynamics simulation methods to calculate free energy profiles in biomembranes such as umbrella sampling are computationally demanding and restricted to existing force field parameters. Figure 9 shows the free energy profile of ibuprofen in a DOPC lipid bilayer predicted with COSMOmic. Ibuprofen partitions preferably into the interfacial region between the hydrophobic core and the more polar headgroup region. The free energy minimum is equal to ΔGmin = 6 kcal/mol, resulting in a partition coefficient of log P = 4.01. In contrast to the results from recent MD simulations of the Figure 10. Comparison of experimental (for details, see the Supporting Information) and predicted liposome (SOPC, DOPC, POPC, DMPC)/water partition coefficients, T = 22−37 °C. Average atomic distributions are used in order to represent the structure of the lipid bilayers (RMSE= 0.62); the diagonal indicates a perfect agreement between predicted and experimental values; and the two dotted lines (above and below the diagonal) represent the average experimental error of 0.3 in the log P scale.23

imental data. Predicted partition coefficients using the pseudophase approach scatter widely (RMSE > 1.1) around the diagonal. The predictions can be significantly improved incorporating the molecular structure from molecular dynamics simulations with COSMOmic (RMSE = 0.62). These results are in line with previous work by Endo et al.25 and Klamt et al.1 Again, larger deviations are found for hydrophobic compounds. Beside experimental difficulties determining partition coefficients of extremely hydrophobic compounds, COSMO-RS systematically underestimates the hydrophobicity of PAHs. Accurate molecular dynamics simulations combined with COSMOmic can predict the partition in inhomogeneous systems such as lipid bilayers. Free energy profiles in different systems such as DOPC, SOPC, DMPC, and POPC can be reproduced. Hence, the influence of the tail length (for details,

Figure 9. Free energy profile of ibuprofen in a DOPC bilayer at T = 25 °C. The average atomic distributions (snapshot represents one potential configuration) of a DOPC bilayer are used in order to represent the membrane structure. G


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Figure 11. Number density profiles in 30% cholesterol DPPC bilayers (left) and 0% cholesterol DPPC bilayer (middle). Representations include water (black), DPPC tails (blue), DPPC phosphocholine (green), cholesterol (red); probability distribution of SAS of DPPC conformers in 30% cholesterol (solid) and 0% cholesterol (dashed) DPPC bilayers (right).

see the Supporting Information) on potential energy barriers in biomembranes can be accounted for. 3.2.1. Liposome/Water Partition Coefficients for Mixed Bilayers. Experimental as well as theoretical studies generally investigate the partitioning of solutes in pure lipid bilayers such as DMPC16,62 or DOPC.57,58 However, it is well-known that the local composition and organization in biomembranes varies significantly and controls specific biophysical properties.63 Therefore, we studied the influence of cholesterol (approximately 25−40 mol % in human plasma membranes)63 on the partition coefficient of alcohols in lipid bilayers. Figure 11 shows the molecular distributions of a DPPC and a 30% cholesterol DPPC (30 mol % of the lipid molecules in the system are cholesterol molecules) bilayer, respectively. In the headgroup, region almost no cholesterol can be determined. As a consequent no changes of the free energy profile of any solute in the headgroup region can be expected with COSMOmic. For the determination of free energy profiles with COSMOmic the selection of a representative conformer of the lipid mixture is required.26 It is known from literature that cholesterol influences the flexibility of the DPPC molecules;63 therefore, we analyzed the SAS area of the DPPC conformers in presence of cholesterol (compare Figure 11). The SAS distribution of the DPPC conformers becomes more narrow compared to a pure DPPC bilayer, and consequently, different conformers were selected for the prediction of free energies in DPPC and DPPC/cholesterol bilayers. Figure 12 shows the free energy profile of heptanol in a DPPC-cholesterol bilayer. Consistent with previous simulation work and experimental data,63 the thickness of the layer increases with increasing cholesterol concentrations. Consequently, the thickness of the core region increases and the minimum of free energy moves farther from the center of the membrane. Further, the minimum value increases which leads to a decreasing partition coefficient at higher cholesterol concentrations. Accordingly, Yacoub et al.64 and Bennett et al.63 reported a similar shift of the minimum free energy of Doxorubicin and DPPC with increasing cholesterol concentrations. For a series of alcohols ranging from hexanol to nonanol, Rowe et al.65 found a moderate decrease of the partition coefficient in the range of 0.55−0.15 in the log P scale using cholesterol concentrations up to 30 mol % in their experiments. We predict a decrease of 0.28−0.13 in log P units which is very close to the experimental data. Recently, Wennberg et al.66 reported that cholesterol decreases solute partitioning into lipid bilayers in MD simulations much more strongly (factor 10 2 −10 7) than expected from experi-

Figure 12. Free energy profile of heptanol in 0% and 30% cholesterol DPPC bilayers at T = 45 °C. Average atomic distributions (snapshot represents one potential configuration) are used to represent the structure of the lipid bilayers.

ments.65,67−69 The author attributed this large discrepancy to inhomogeneous cholesterol concentrations in the membrane. These limitations are unlikely to effect the simulations of this work as the atomic distributions from MD simulations are averaged over sufficient long production runs (see section 2.1.2). Hence, free energy profiles in complex mixed aggregate systems can be predicted. However, structural contributions, which are governed by local phenomena such as water pores, defects in the aggregate structure,11 as well as membrane deformation64 due to the volume of the solute are presently not explicitly considered with COSMOmic. Therefore, it can be concluded that the partitioning of small organic compounds is dominated by electrostatic and van der Waals interactions, which can be very well represented with COSMO-RS. In summary, MD simulations reliably simulate the structure and dynamics of lipid membranes and micelles, whereas COSMO-RS accurately reproduces solvation free energies in different solvents. In this work, the combination of both methods was used and validated by means of experimental partition coefficients found in the literature. To our knowledge, these are the first quantitative predictions of micelle/water partition coefficients, which are based on whole free energy profiles from molecular methods. Free energy profiles in pure H


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ACKNOWLEDGMENTS The authors appreciate the financial support of the DFG (Project SM 82/4-2). Computational resources have been provided by The North-German Supercomputing Alliance (HLRN).

and mixed lipid systems were calculated and validated by means of experimental partition coefficients. In concordance with experimental studies, a decreasing partition coefficient for alcohols in DPPC/cholesterol lipid bilayers was found. In this work, we determined free energy profiles of neutral organic solutes in heterogeneous systems. Alternatively, free energy profiles in heterogeneous systems can be determined using methods such as umbrella sampling;70 a comparison of different methods can help to determine potential challenges for future developments such as force field parameters. At present, the dominant interactions which govern the partition of ionic species into micelles as well as lipid bilayers are still not fully understood.60,71,72 An efficient method, which is able to handle a large variety of different structures in a short period of time, can help to answer questions such as the transport of ionic species in lipid membranes. Previous work has shown that COSMO-RS is principally able to calculate thermodynamic properties in systems containing charge species.73,74 Therefore, the combination of MD simulations and COSMO-RS can be directly applied to study free energy profiles of charged solutes.

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ASSOCIATED CONTENT

S Supporting Information *

List of all experimentally determined partition coefficients of the surfactant and lipid systems either by our group or by others. This material is available free of charge via Internet at http://pubs.acs.org.

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REFERENCES

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4. CONCLUSIONS Free energy profiles in potential drug delivery systems, such as micelles and liposomes, provide valuable information concerning thermodynamic and kinetic processes governing membrane binding and transport in the aggregate. In this work, free energy profiles in micellar systems were first predicted, combining molecular dynamics simulations with the thermodynamic model COSMO-RS. Partition coefficients derived from predicted free energy profiles along the radius of the corresponding micelle were validated by means of more than 200 experimental micelle/water partition coefficients from literature. For the systems CTAB/water and SDS/water, a RMSE of 0.44 and 0.51 was found, respectively. Considering the present accuracy of experimental micelle/water partition coefficients available in literature, further improvements can be considered to be rather unlikely. In addition, average structural properties derived from MD simulation can be used to predict partition coefficients in various lipid bilayer/water systems with an average RMSE of 0.62. The influence of the lipid tail length as well as the influence of an important cell component such as cholesterol on the membrane/water partition coefficient can be investigated with COSMOmic. Hence, reliable predictions of the partitioning in even more complex systems such as plasma membranes or mitochondrial membranes seem to be realistic in the future.

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AUTHOR INFORMATION

Corresponding Author

*Telephone: +49 (40) 42878 2988. Fax: +49 (40) 42878 4072. E-mail: [email protected] (T.I.); [email protected] (S.S.). Notes

The authors declare no competing financial interest. I


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