Functional Group Distributions, Partition Coefficients, and Resistance Factors in Lipid Bilayers Using Site-Identification by Ligand Competitive Saturation (SILCS)

Abstract

Small molecules such as metabolites and drugs must pass through the membrane of the cell, a barrier primarily comprised of phospholipid bilayers and embedded proteins. To better understand the process of passive diffusion, knowledge of the ability of various functional groups to partition across bilayers and the associated energetics would be of utility. In the present study, the SILCS (Site Identification by Ligand Competitive Saturation) methodology has been applied to sample the distributions of a diverse set of chemical solutes representing the functional groups of small molecules across phospholipid bilayers composed of 0.9:0.1 POPC/Cholesterol, and a mixture of 0.52:0.18:0.3 DOPS:DOPC:cholesterol used in parallel artificial membrane permeability assay (PAMPA) experiments. A combination of oscillating chemical potential Grand Canonical Monte Carlo and Molecular Dynamics in the SILCS simulations was applied to achieve solute sampling through the bilayers and surrounding aqueous environment from which the distribution of solutes and the functional groups they represent were obtained. Results show differential distribution of aliphatic versus aromatic groups with the former having increased sampling in the center of the bilayers versus in the region of the glycerol linker for the latter. Variations in the distribution of different polar groups are evident, with large differences between negative acetate and positive methylammonium with accumulation of the polar-neutral and acetate solutes above the bilayer head groups. Conversion of the distributions to absolute free energies allows for a detailed understanding of energetics of functional groups in different regions of the bilayers and for calculation of absolute free energy profiles of multifunctional drug-like molecules across the bilayers from which partition coefficients and resistance factors suitable for insertion into the homogenous solubility diffusion equation for calculation of permeability were obtained. Comparisons of the calculated bilayer/solution partition coefficients with 1-octanol/water experimental data for both drug-like molecules and the solutes show overall good agreement, validating the calculated distributions and associated absolute free energy profiles.

Graphical Abstract

Introduction

Membranes are an essential component of cells, defining their spatial extent as well as creating compartments in eukaryotic systems. Inherent in their function as cellular boundaries is their ability to selectively allow the passage of molecules, including via passive diffusion. The amphipathic character of phospholipids and cholesterol drives the spontaneous formation of bilayers in aqueous solution with the hydrophobic tails facing the interior and the hydrophilic head groups exposed to the aqueous medium. This physical feature yields the crucial ~4-5 nm barrier that supports the compartments in cells that essential solutes must pass through and is required for cellular function. While many molecules, including some drugs, pass through membranes using an active transport mechanism, a large number instead rely on passive diffusion. As defined in the solubility-diffusion model, a molecule driven by the concentration gradient penetrates the membrane by partitioning through the polar head group region and into the hydrophobic region, diffuses across the bilayer, exits the membrane, and dissolves in the intracellular aqueous phase.^{1, 2} The solubility-diffusion model provides the essential framework for estimating the permeation of a molecule across a membrane bilayer. Specifically, it relates the rate of permeation (P) of a molecule to the resistance (R) that the molecule experiences in the membrane according to the permeation resistance equation. ³

The ability of molecules to distribute into and to diffuse across membranes is strongly correlated to the physiochemical properties of the molecule in combination with those of the bilayer itself. Charged molecules, such as ions and highly polar molecules, tend to stay in the aqueous phase with a negligible concentration inside the bilayer while hydrophobic, non-polar, molecules partition preferentially to the hydrophobic core of the membrane bilayer.² A number of theoretical studies have investigated the distribution of various small molecules in lipid bilayers (see references ^4,5,6,7 for recent reviews). These include efforts based on atomistic and coarse-grained molecular dynamics (MD) simulations ^{8,9,10,11,12,13,14,15} as well as approaches using implicit solvent models. ^{16,17,18,19,20} While these investigations have yielded insights into the probability distribution and free energy profiles of individual molecules in an aqueous-bilayer system, information on the probability distributions and associated absolute free energy profiles of different classes of functional groups across lipid bilayers are not available. Such information would allow for an improved detailed molecular understanding of the forces impacting passive diffusion and how different membrane compositions impact the passive diffusion of molecules.

This study presents a detailed molecular picture of the distributions and free energies of various functional groups across two lipid bilayers; 0.9:0.1 POPC/Cholesterol (henceforth POPC/Chol), and 0.52:0.18:0.3 DOPC:DOPS:Cholesterol. The latter system models the bilayer composition used in a parallel artificial membrane permeability assay (PAMPA) experimental study²¹ and thus will be referred to as PAMPA throughout the remainder of the manuscript. We note that there are variations in the content of the lipids in different PAMPA experiments, including the inclusion of n-dodecane. ^22–24 The functional group distributions were calculated using the site identification by ligand competitive saturation (SILCS) ^{25, 26} method that involves exhaustive sampling of small solutes representative of different chemical functional groups throughout a system. Comprehensive sampling in complex heterogeneous systems is achieved through a combination of oscillating excess chemical potential, μ_ex, Grand Canonical Monte Carlo (GCMC) simulations to facilitate solute and water sampling through the system and MD simulations to account for conformational sampling of the lipids in the bilayer, as well as additional sampling of the solute and water distributions. From these calculations the distributions of the solutes throughout the simulation systems were obtained along with probability distributions and absolute free energies of the studied functional groups across the bilayers.

Implications of the functional group absolute free energy profiles for passive diffusion are discussed and the spatial distributions of the functional group free energies, termed grid free energies (GFE), are used to calculate the free energy profiles of “drug-like molecules,” including butanol, testosterone, caffeine, and the neutral and charged forms of ibuprofen, alprenolol, and pindolol, across the bilayers. SILCS GFE distributions, termed GFE FragMaps, have been used in a number of studies to estimate the free energies of the binding of ligands to proteins. ^27,28,26 This is based on the assignment of ligand atoms to different FragMap atom types. Based on the overlap of each ligand atom with its corresponding FragMap, as defined by the atom type, the individual atomic GFE values are obtained. The ligand affinity is then calculated as the sum of all GFE values from all classified ligand atoms to yield a total ligand GFE (LGFE). ²⁸ This approach allows for the overall affinity to be estimated, as well as yielding information on the contribution of the different functional groups to the affinity. The same approach is utilized here to obtain the free energy profiles for selected drug-like molecules across the lipid bilayers. Notably, comparison of calculated partition coefficients of both the solutes and drug-like molecules show good agreement with experimental water/octanol data, validating the method.

Methods

Set up of the lipid bilayers

Two bilayer systems were built using the membrane builder available in the CHARMM-GUI.^{29, 30} The systems consisted of (i) 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) and cholesterol (90:10 ratio) and (ii) a multi-lipid bilayer system consisting of 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), 1,2-dioleoyl-sn-glycero-3-phospho-L-serine (DOPS) and cholesterol (52:18:30 ratio) with 154 and 150 components in each leaflet for systems i and ii, respectively. As system ii is comprised of two lipids and cholesterol, we performed initial molecular dynamics (MD) simulation to equilibrate the system. The MD simulation was performed for 500 ns using the standard settings generated by CHARMM-GUI for GROMACS³¹ simulations using the CHARMM36 lipid force field parameters.³² 0.15 M sodium chloride was included in this equilibration simulation. The system was initially subjected to a steepest decent (SD) minimization for 5000 steps with a convergence criterion of 10 kJ/mol/nm. This was followed by 250 ps of MD simulation using the md (leap-frog algorithm) integrator and the Berendsen thermostat using an integration step size of 1 fs and a subsequent 1.125 ns simulation with Berendsen pressure coupling at 1 bar. The initial 125 ps of that simulation used an integration step of 1 fs, after which it was increased to 2 fs. Production runs were performed with the md integrator with a time step of 2 fs. The Nose-Hoover method^{33, 34} was used to maintain a temperature at 303.15 K and the pressure was maintained at 1 bar using the Parrinello-Rahman barostat. ³⁵ The time constants used for temperature and pressure coupling were set to 1 and 5 ps, respectively. The particle mesh Ewald (PME) method³⁶ was used to treat long range electrostatic interactions with a 12 Å cutoff; van der Waals interactions were switched off in the range of 10-12 Å.

SILCS simulations and generation of FragMaps

Both systems were prepared for SILCS simulations using the SILCS-Membrane protocol initially developed for membrane-bound proteins^{26, 27} using the SilcsBio suite of programs (SilcsBio LLC). The lipid bilayers were placed in a rectangular water box containing approximately 0.25 M of each small solute; acetaldehyde, acetate, benzene, formamide, imidazole, methanol, methylammonium and propane. To prevent ion pairing/aggregation of charged and hydrophobic solutes a repulsive energy term was introduced only between the followed solute pairs: benzene:benzene, benzene:propane, propane:propane, acetate:acetate, acetate:methylammonium, and methylammonium:methylammonium, as previously described. ²⁷ Parameters for the solutes were taken from the CHARMM General Force Field (CGenFF)³⁷ and water was treated with the CHARMM-modified TIP3P model.³⁸ Initial equilibration of each SILCS system involved 5000 steps of energy minimization using the SD method. The minimized system was then subjected to 100 ps MD at 300 K using the velocity rescaling thermostat with randomized initial velocities, and the Berendsen barostat³⁹ to allow for initial relaxation of the system volume. This was followed by 0.5 ns MD using the leap-frog integrator. The equilibrated systems were then subjected to 25 cycles of oscillating excess chemical potential (μ_ex) Grand-Canonical Monte Carlo (GCMC) comprised of 200,000 MC steps to redistribute water and the SILCS solutes. Here, the first step involved removal of all non-lipid molecules from a sub-volume of the entire simulation box with dimensions that are 30 Å smaller than the full simulation system (eg. X-30 Å, Y-30 Å, Z-30 Å), denoted the active GCMC box. This volume was then re-solvated with water and solute molecules over the 25 cycles of GCMC. The final coordinates from this procedure served as the initial coordinates for the production combined GCMC/MD simulation. Each system was subjected to 10 individual simulations, starting from the 25 cycle GCMC redistribution phase so the initial water and solute distributions differed for each simulation. The MD simulations were performed using GROMACS (version 2019.4) and the GCMC calculations were conducted using in-house code. ⁴⁰ The simulation temperature was set to 303 K.

Data were collected over 200 cycles of GCMC/MD in the production run. Each cycle consists of 200,000 GCMC steps, followed by 5000 steps of SD minimization and 100 ps MD equilibration followed by one ns of production MD for a total of 200 ns of MD. During production simulations a time step of 2 fs was used with neighbor list being updated every 20 ps. For long-range electrostatic interactions the PME method²² were used with a direct cut-off set to 8 Å, with the same cutoff distance used for the Lennard-Jones interactions in conjunction with a isotropic long-range dispersion corrections, as previously described.²⁶ Water geometries and covalent bonds involving a hydrogen atom were treated with the LINCS algorithm. ⁴¹ Simulation temperature and pressure were maintained using the Nose-Hoover thermostat^{33, 34} and the Parrinello-Rahman barostat.³⁵ Periodic boundary conditions were used in all directions. Atom coordinates were saved every 20 ps.

The final coordinates from each MD simulation were used as initial coordinates for the GCMC portion of the subsequent cycle. In the oscillating μ_ex GCMC method the excess μ_ex can be tuned, allowing solutes to be either added or removed from the system to achieve a target concentration.⁴⁰ GCMC including both insertions and deletions of the solutes and water, as well as translations, rotations and dihedral rotations. Tuning of μ_ex was performed every three cycles in response to the change in solute concentration.

SILCS FragMaps were calculated from the production MD trajectories across the 10 × 200 ns systems, yielding an accumulative MD time of 2 μs with snapshots saved every 20 ps. The FragMaps are based on probability distributions of selected atoms for each solute or water by partitioning the system into 1 ų volume elements and calculating each selected atoms’ local occupancy throughout the entire trajectory. For example, the FragMaps for benzene and propane utilize all carbon atoms, as they are indistinguishable, thus all atoms contribute to the FragMap, whereas for methanol and acetaldehyde, each molecules’ respective oxygen was selected and by itself represent the MEOO and AALD oxygen FragMaps. For a complete atom selection scheme, we refer to table 1 in Ustach et al.²⁶ The calculated solute probability distribution was then normalized with respect to the concentration of each solute in the simulation box volume and converted into free energies based on a Boltzmann transformation yielding grid free energy (GFE) FragMaps.²⁶ The GFE values, due to the normalization with respect to the concentration of each solute in the entire simulation box, correspond to absolute free energies, such that the average GFE over the entire simulation system for each solute is 0 kcal/mol. Convergence of the SILCS simulations was based on the overlap coefficients of the GFE FragMaps as described in the supporting information (Table S1). During the SILCS simulations, the target concentration for the different solutes is 0.25 M based on the total volume of the simulation system. However, variations occur dependent on the oscillating excess chemical potential. ⁴⁰ The calculated average SILCS solute concentrations calculated over the 2 μs of MD simulations are shown in Table 1.

Table 1.

Average concentration over the entire SILCS simulation for POPC/Chol and the PAMPA system calculated relative the entire simulation box volume.

	Benzene	Propane	Acetaldehyde	Methanol	Formamide	Imidazole	Acetate	Methylammonium
POPC	0.165	0.172	0.197	0.198	0.172	0.174	0.171	0.176
PAMPA	0.170	0.172	0.196	0.196	0.179	0.177	0.178	0.205

Calculation of absolute free energy profiles

Absolute free energy profiles were calculated for the solutes and for drug-like molecules across the bilayers. For the functional groups associated with the solutes, the free energy profiles were calculated by obtaining the average GFE across all the voxels at a given value of Z. In the case of formamide and imidazole the profiles for the individual donor and acceptor groups were similar and, therefore, averaged.

For the drug-like molecules the free energy profiles were obtained by calculating the LGFE of the molecules along the normal to the bilayer using the SILCS-MC approach.²⁸ In SILCS-MC the ligand is docked in the targeted position with MC sampling of rotational, translational and dihedral degrees of freedom based on intramolecular energies and the LGFE score. The LGFE score is determined from the overlap of selected ligand atoms with the SILCS GFE FragMaps. This allows for assigning a GFE score to each classified atom in the ligand with the summation over all the classified atoms yielding the LGFE. A complete description of how ligand atoms are assigned to SILCS FragMaps is detailed in Ustach et al. ²⁶ To calculate the absolute free energy profiles the center of mass of the ligand is assigned an initial position along the normal (Z-axis) of the bilayer. Free energy profiles were calculated from Z = −35 to 35 Å with 1 Å increments yielding a total of 71 points across the bilayer. Each ligand was subjected to an initial intramolecular energy minimization for 10,000 steps using the BFGS method with a gradient tolerance of 3×10⁻⁸ kcal/mol/Å based on the intramolecular energies defined by the CGenFF energy function that includes a 4r dielectric constant for the electrostatic term. At each Z value the ligand was given a random initial conformation based on its center-of-mass (COM) with the COM position allowed to vary by 1.0 Å from the assigned Z value during MC sampling. This was followed by 10,000 steps of MC sampling in the field of the FragMaps and ligand intramolecular energy with molecular translation, molecular rotation and dihedral rotation sampled within a range of [0:1] Å, [0:180]° and [0:180]°, respectively. Next the ligand was subjected to simulated annealing (SA) from 300 to 0 K over 40,000 MC steps, where the ligand is allowed to sample within a range of [0:0.2] Å, [0:9]° and [0:9]° for translations, rotations and dihedral rotations, respectively. This describes one round of the SILCS-MC docking and is repeated up to 250 rounds or until 3 ligand poses with the most favorable LGFE scores fall within 0.5 kcal/mol, then the docking was determined to have converged and was stopped. A minimum of 50 rounds was always performed to ensure the ligand have sampled a sufficient amount in the given search volume. To ensure proper statistics, free energy profiles were generated at 9 different (X, Y) positions along the plane of the bilayer, with the final profiles being the average and standard deviations over those 9 free energy profiles.

Calculation of partition and resistance coefficients

The partition coefficient K is defined as the equilibrium ratio of a permeants’ concentration c_m and c_w for membrane and water, respectively. Thus, the partition coefficient was evaluated from free energy profiles as described by Ghysels et al:⁴²

$$K\left(h\right)=\frac{c_m\left(h\right)}{c_w}=\frac{1}{h}\underset{-h/2}{\overset{h/2}{{\displaystyle \int }}}{e}^{\beta \left(G\left(z\right)-G_{ref}\right)}dz$$ (1)

where G(z) is the position dependent free energy at a given z coordinate, β = 1/k_BT, k_B and T are the Boltzmann constant and the absolute temperature, respectively, and h is the bilayer thickness. G_ref is the reference free energy corresponding to the water phase and was set to the average value of G(z) between 25-35 Å from the bilayer midplane (z = 0). As a check, the partition coefficient was calculated by directly counting those solutes located in the membrane and water. As is clear from Eq. (1), the calculated partition coefficient depends on the choice of membrane thickness. This was assumed here to be the distance between the phosphate planes: 38.40 Å for POPC/Chol and 40.52 Å for PAMPA. Plots of log $K\left(h\right)$ were used to examine details of the partitioning, including its dependence on the assumed bilayer thickness.

Estimates of the resistance R are based on the inhomogeneous solubility diffusion equation,⁴³ where the permeability P is written as:

$$\frac{1}{P}=e^{-\beta G_{ref}}\underset{-h/2}{\overset{h/2}{{\displaystyle \int }}}\frac{e^{\beta G\left(z\right)}}{D_{\perp }(z)}dz$$ (2)

Where $D_{\perp }(z)$ is the position dependent diffusion coefficient. The diffusion constant is not calculated here so will be assumed to be the constant $〈D〉$. Eq. (2) is then

$$\frac{1}{P}={\langle D\rangle }^{-1}{e}^{-\beta G_{ref}}\underset{-h/2}{\overset{h/2}{{\displaystyle \int }}}{e}^{\beta G\left(z\right)}dz=\frac{R}{\langle D\rangle }$$ (3)

where R can be interpreted as the resistance to permeability associated with the free energy that molecules experience as they cross the membrane. Because the free energy is exponentiated in Eq (3), variations associated with the diffusion constant are not a significant consideration for interpreting the permeability. The resistance associated with the hydrocarbon portion of the bilayer, $R_{CC}$ is defined as:

$$R_{CC}=e^{-\beta G_{ref}}\underset{-h_{C2}/2}{\overset{h_{C2}/2}{{\displaystyle \int }}}{e}^{\beta G\left(z\right)}dz$$ (4)

where h_c2 is the distance between the C2 planes of the acyl chain carbons on opposite leaflets. The specific values of h_c2 are 27.72 Å and 29.86 Å for POPC/Chol and PAMPA, respectively.

Results

SILCS solute distributions

An example of the information content accessible from the SILCS simulations of selected functional groups associated with the different solutes in the POPC/Chol bilayer is shown in Figure 1. Figure 1A shows the apolar aromatic and aliphatic group carbon GFE FragMaps at two isocontour levels. At a favorable GFE value of −0.6 kcal/mol the solutes are located in the hydrophobic core of the bilayer, as expected. However, it is evident that there are differences in the distributions, with the aliphatic groups being populated at this favorable energy level more towards the center of the bilayer with the aromatic groups more populated closer to the glycerol linker region. At an unfavorable free energy of 0.6 kcal/mol both types populate the polar head group region with subtle differences in the locations still occurring. Figure 1B shows FragMaps of the hydrogen bond donor formamide nitrogen and acceptor acetaldehyde oxygen at GFE values of −0.3 and 0.3 kcal/mol. At −0.3 both functional groups are favorably solvated in the aqueous solution away from the bilayer. Interestingly, at 0.3 kcal/mol the region in the head groups is populated with the acceptor in the vicinity of the POPC choline moieties and a region adjacent to the glycerol linker region at the edge of the hydrophobic core while the donor groups are occupying the region of the glycerol groups at this energy. With the charged groups in Figure 1C, their favorable solvation in aqueous solution is evident at −0.3 kcal/mol. In addition, the region above the positively charged choline moieties is occupied by the negative groups associated with acetate while the positive groups associated with methylammonium are occupying the region populated by the phosphates and into the glycerol linker region at this energy. At an unfavorable GFE of 0.3 kcal/mol the negative group FragMaps occur in the region just above the phosphates while the positive group FragMaps are observed in the aliphatic region just below the glycerol linker regions. Further increasing the GFE value to 2.0 kcal/mol, the positively charged FragMaps occur in the portion of the aliphatic region adjacent to the glycerol linkers with the negative group FragMaps located between the phosphate head and the carbonyl groups of the lipid. These results indicate an increased, though still highly unfavorable ability of the positive groups to penetrate into the upper region of the hydrophobic core. This qualitative analysis shows the utility of the SILCS approach for mapping the free energy profiles of the functional groups across the bilayers and the resolution of the method for obtaining subtle details of those profiles.

Figure 1.

Functional group distributions in the 0.9:0.1 POPC/Cholesterol (POPC/Chol) bilayer at different Grid Free Energy (GFE) cutoffs. (A) Aromatic (benzene carbons in purple) and aliphatic (propane carbons in green) at GFE values of −0.6 and +0.6 kcal/mol. (B) Hydrogen bond donor (formamide nitrogen in blue) and acceptor (acetaldehyde oxygen in red) at GFE values of −0.3 and +0.3 kcal/mol. (C) Negatively charged (acetate oxygen in orange) and positively charged (methylammonium nitrogen in cyan) at GFE values of −0.3, 0.0, 0.3, and 2.0 kcal/mol. Coloring as is as follows: POPC chains (sticks with nonpolar hydrogens not shown); phosphorous atoms (orange spheres); cholesterol (yellow sticks); water molecules (red and white spheres for oxygen and hydrogens, respectively).

Distribution of lipid components

Lipid probability distributions of different moieties were calculated along the bilayer normal (Z-axis) in 1 Å bins. The lipid probability distributions were calculated using the phosphorus atom to determine the location of the phosphates and the nitrogen was used to define the location of the choline for POPC and DOPC. The nitrogen and the carboxylate oxygens were used to define the location of the serine for DOPS. For all lipids, the terminal methyl group carbon was used to define the aliphatic tail group and the carbonyl oxygens were used for the glycerol linker region (Figure S1). Additionally, probability densities were calculated for the hydroxyl oxygen and the terminal methyl group carbon of cholesterol. This allows for evaluation of the overall directionality of the cholesterol molecules. By separating the lipid and cholesterol molecules into different moieties we can survey the distributions according to the four regions defined by Marrink and Berendsen.^44–46 These regions are defined proceeding from the aqueous phase to the center of the membrane bilayer; 1) low headgroup density, 2) high headgroup density, 3) low tail density, and 4) high tail density. They will subsequently be referred to as M-B1 through M-B4. The probability distributions were calculated over the SILCS 200 ns MD trajectories and were then averaged over the 10 replicate SILCS simulations from which standard deviations were obtained.

Figure 2 shows the distributions of the lipid components of the POPC/Chol and PAMPA bilayers, with their individual components included in each panel. Distributions of the components in the individual bilayers are shown in Figure S3. The distributions of the different groups in the POPC/Chol bilayers are consistent with those previously reported.⁴² Similarly, the average peak-to-peak phosphate distance across the entire bilayer is around 38 Å, comparable to the experimental range,⁴⁷ as well as that seen in computational studies using the CHARMM36 lipid force field.⁴⁸ Hence the presence of the solutes in the SILCS simulations do not significantly perturb the bilayer structures.

Figure 2.

Lipid functional group distribution for the studied bilayers. Relative number density for (A) phosphate groups (orange) and aliphatic tail methyl groups (black), (B) carbonyl groups (C) choline and DOPS serine nitrogen (blue) and serine carboxylate oxygens (red), and (D) cholesterol hydroxyl (red) and methyl tail group (black). Densities are relative to the phosphorous densities that were set to 1 at the maxima based on the POPC and DOPC lipid distributions for the POPC/Chol and PAMPA systems, respectively. The four M-B regions, which are delineated by vertical lines at Z = [6, 13, 20, 27] and correspond to the distance (Å) from the center of the bilayer Z = 0, are noted at the top of each region.

The distributions of the multi-lipid system consisting of DOPC, DOPS, and cholesterol are also presented in Figure 2. The lipid types and ratio of the bilayer was chosen to represent PAMPA membranes that have previously been used experimentally to determine permeability of small molecules.²¹ The PAMPA system follows the general trend of distributions as seen for POPC/Chol. However, the head group distribution of the bilayer is slightly shifted to larger distances consistent with the longer aliphatic tails with possible contributions due to the increased cholesterol concentration. The presence of cholesterol is known to contribute to the straightening of the hydrophobic tails and thus an increase of bilayer thickness.^{49, 50} There is little positional difference between the phosphate groups for the two different phospholipids, DOPC and DOPS, despite the different head groups.

The cholesterol hydroxyl group distributions (Figure 2D) have maxima in the vicinity of 15 Å with the peak shifted outward in the PAMPA system. The density for the cholesterol hydroxyl group aligns very well with the carbonyl region of both POPC and PAMPA, consistent with the shift in the location of the carbonyl distributions in Figure 2B. The cholesterol molecules are strongly aligned along the bilayer normal, as the peaks for hydroxyl and the cholesterol tail are separated by ~15 Å, only slightly shorter than the distance of approximately 17.5 Å in a fully extended cholesterol molecule. The conformational sampling of the cholesterol can be quantified by calculating the average angle of cholesterol with respect to the membrane normal.^51–54 This was performed by calculating the distance between the hydroxyl group and the C20 atom along the membrane normal based on the density peak of respective atoms (along the Z-axis) and measuring the same distance of the extended molecule. This generated an average tilt angle of 29.6° in the POPC system. The sterol tilt angle has been reported around 10-40 degrees. ^{53, 55–58} The maximum of the hydroxyl distribution in the PAMPA system indicates that the cholesterol molecules are more aligned along the bilayer normal, with an average tilt angle of ~ 23°. Although this comparison is between different lipid systems, it agrees with previous data suggesting that the cholesterol tilt angle decreases with an increase in cholesterol concentration.^{57, 58} This is consistent with the lipid tails tending to be more ordered and less flexible going to an increased cholesterol concentration.

Distributions and free energies of the SILCS solutes in phospholipid bilayers

Figure 3 shows the average SILCS solute density distributions for the POPC/Chol and PAMPA bilayers along the lipid bilayer normal along with the headgroup distributions. The specific composition of different lipids used here offers a unique perspective into the distribution of small solutes into lipid bilayers with complex compositions. As expected, the lipophilic solutes, benzene and propane, populate the interior of the bilayers among the aliphatic tails. In both systems the propane densities rapidly decrease moving away from maxima at the center of the bilayers with an inflection point at Z ~ 8 Å. In this region the propane and benzene densities cross with the benzene densities reaching maxima at 11-12 Å, adjacent to the glycerol carbonyl oxygen distribution. The presence of benzene solutes in close proximity to the polar region of the lipid may be explained by the greater polarity of the aromatic benzene as compared to propane (Figure 3 upper panels). There is a noticeable local accumulation of benzene and propane at the center of the bilayer (M-B region 4). This appears to be correlated to the presence of cholesterol in the POPC and PAMPA systems, as a similar enhancement of molecular oxygen was observed in simulations of a cholesterol-rich liquid ordered phase.⁴² There is a greater density of propane in the center of the PAMPA bilayer with the shoulder of the distribution at ~10 Å being smaller. A previous MD study of different bilayers reported that the partitioning of both polar and nonpolar solutes into the hydrophobic tail region of lipid bilayers from ~5 to 15 Å increases in the presence of cholesterol, but there is no change in the center of the bilayer.⁵⁹ The present results indicate that the phospholipid type and the presence of cholesterol do impact the distribution of the nonpolar solutes in the bilayer interior, though it is not possible to separate the contributions from the individual lipid types and from those of the presence of cholesterol based on the available data.

Figure 3.

SILCS solute distribution along the membrane normal. (A) POPC/Chol and (B) PAMPA bilayers. The solute distributions are overlaid with the functional groups of the different phospholipids. Lipid head groups phosphorus, choline nitrogen/DOPS serine oxygen and carbonyl are displayed as solid transparent filled regions in yellow, blue and red, respectively. Standard deviations for SILCS solutes are shown as light grey filling. Vertical lines delineate the 4 M-B regions. The distributions are for hydrophobic, polar and charged fragments in top, center and bottom panels, respectively. All densities have been normalized based on the POPC and DOPC phosphorous densities for visualization.

The polar-neutral solutes (Figure 3 middle panel), acetaldehyde, formamide, imidazole and methanol, show low probabilities inside the aliphatic region of the bilayer, as expected. However, some sampling is evident, consistent with the ability of polar-neutral molecules to diffuse through lipid bilayers. There is a gradual increase of density for the polar solutes moving away from the center of bilayer with larger increases occurring in the region occupied by the lipid carbonyl groups. The shapes of the methanol and acetaldehyde densities are similar moving from the aliphatic region into the region occupied by the carbonyl moieties, with both solutes showing distinct peaks followed by minima in the region where the lipid phosphate densities are at a maximum. The acetaldehyde distribution shows a maximum at ~14 Å with the maxima for methanol occurring at slightly larger distances; it is significantly larger for acetaldehyde than methanol in both bilayers. Local minima for both solutes occur at Z = 20 – 22 Å corresponding to the maxima of the phosphate distributions, and then gradually increase upon going into aqueous solution.

The methanol and acetaldehyde distributions align well with the distribution for cholesterol hydroxyl (Figure 2 and Figure 3). This occurs at Z values between M-B regions 2 and 3. This is especially true with the methanol O distributions indicating the presence of favorable interactions of the hydroxyls on either methanol or cholesterol with the phosphates.

The formamide and imidazole solutes, which both contain hydrogen-bond donor and acceptor groups, show a moderately linear increase in density that starts within the region of high tail density. The formamide and imidazole profiles are similar out to Z ~ 15 Å in both bilayers. Upon going further from the aliphatic core to the region dominated by the phosphate groups the formamide density increases more rapidly with that trend being the largest in the PAMPA system. This suggests more favorable interactions of the primary amide donor with the phosphate versus the N-H donor of imidazole. Upon moving beyond the phosphates at Z ~ 25 Å the two densities become more similar with maxima occurring in aqueous solution. There is a small maximum just beyond the head group (choline or serine distributions) indicating a tendency for the polar-neutral solutes to accumulate just above the bilayer to a greater extent than in solution.

Analysis for the charged solutes, methylammonium and acetate (Figure 3 bottom panels) reveals the expected rapid increase of methylammonium around the phosphate moieties. This increase is more prominent in the PAMPA system where a large maximum occurs at Z ~ 24 Å. This is the maxima of the serine moiety density of DOPS and indicates the impact of the acid group on the positively charged methylammonium distribution. With acetate, the density is close to zero out until approximately Z = 15 Å, followed by a large increase in the region of the choline/serine moieties seen in both bilayer systems. A maximum with acetate occurs just beyond the region occupied by the choline/serine moieties indicating that this group, along with the polar-neutral groups, accumulates just above the bilayer surfaces.

The inclusion of GCMC along with MD simulations in the SILCS methodology allows the solutes, as well as water, to partition throughout the entire simulation systems. From the MD trajectories the populations of each solute, in 1 ų voxels, are calculated throughout the entire simulation system. Notably, these populations are normalized based on the overall concentration of each solute in the entire simulation system. Subsequent Boltzmann transformations of these normalized populations yield absolute free energies for each of the solutes, as well as water, throughout the simulation system. Accordingly, the average GFE over the entire simulation system for each functional group is 0 kcal/mol with the GFE values for each individual voxel in the system representing the absolute free energy in that local volume element.

Presented in Figure 4 are the averaged free energies along the membrane normal Z, referred to as GFE profiles. We note that these free energy surfaces differ from those traditionally extracted from potential of mean force (PMF) calculations as it is not necessary to offset the energies to a selected point in the simulation system, although the shapes of the free energy surfaces from both methods will be, in practice, similar.

Figure 4.

Absolute free energy profiles of the different solute functional groups along the normal to the lipid bilayers. The free energies were calculated by averaging over the GFE values in the XY-plane over all voxels at each Z value across the bilayer for the (A) 0.9:0.1 POPC/Chol and (B) PAMPA systems. Solutes are shown as benzene (purple), propane (green), methanol O (red solid), acetaldehyde O (red dashed), formamide N/O (blue), imidazole N/NH (purple) methylammonium N (cyan) and acetate O (orange).

Analysis of the absolute GFE profiles shows the expected trends (Figure 4). The nonpolar solutes benzene and propane have favorable free energies in the bilayers and unfavorable free energies in aqueous solution at Z > 30 Å. In all three systems, propane is more favorable at the very core of the bilayer and more unfavorable in aqueous solution as compared to benzene. This is consistent with the hydration free energy of propane being more unfavorable than that of benzene (1.96 vs. −0.83 kcal/mol; hydration free energies as reported in Lakkaraju et al. ⁴⁰ except where noted). With the polar solutes, the free energies are unfavorable in the bilayer and favorable in aqueous solution. Methanol is significantly more unfavorable in the bilayer interior than acetaldehyde consistent with methanol having an experimental hydration free energy of −5.1 kcal/mol versus −3.5 kcal/mol for acetaldehyde.⁶⁰ The formamide and imidazole free energy profiles are very similar, consistent with the presence of individual donors and acceptors on both molecules. However, the hydration of free energy of formamide and imidazole are −14 and −9.63 kcal/mol,⁶¹ respectively, suggesting that larger differences should be present between the free energy profiles. Furthermore, for the charged species the expected trend is observed where the free energies are unfavorable inside the bilayers and favorable in aqueous solution. The overall difference between the bilayer interior and the aqueous solution free energies are slightly larger for acetate versus methylammonium consistent with its more favorable hydration free energy of −79.9 kcal/mol versus −71.3 kcal/mol for methylammonium. ⁶² More quantitative comparison of the solute bilayer versus solution partitioning with respect to water/octanol partition coefficients is presented below.

Figure 5 presents the densities and absolute free energies of water in the two bilayer systems. The overall patterns are consistent with previous simulation studies of water penetration into bilayers^{9, 44} using potential of mean force calculations that obtained a free energy barrier of ~6 kcal/mol in DPPC bilayers. Notably, the water free energies are < 0 kcal/mol in aqueous solution. This is expected, as the free energies correspond to the entire, but finite simulation system, such that the low density of water in the membrane leads to the free energy of water being < 0 in aqueous solution when the volume of the entire system is considered. ⁵²

Figure 5.

Water GFE and average number densities along the lipid bilayers. (Left scale) Calculated GFE of water for POPC/Chol (green line) and PAMPA (red line). (Right scale) Atom number density for water oxygen POPC/Chol (green dotted line) and PAMPA (red dotted line) calculated to ~30 mol /nm³ showing bulk properties corresponding to a 55 M standard concentration. Vertical lines indicate the Marrink-Berendsen regions.

Partition and resistance coefficients of the SILCS solutes in the bilayers

Table 2 lists the membrane/water partition coefficients evaluated from (Eq. 1) and from counting for the SILCS solutes and compares them to experimentally determined octanol/water partition coefficients. The calculated values based on (Eq. 1) and counting are quite similar as expected; differences arise when the free energy in the water layer is not constant (see Figure 4) Agreement of calculations and experiments is qualitatively very good, and differences with experiment can reasonably be attributed to the significant difference between molecular details of the membranes and octanol/water. Concerning acetate and methylammonium in the SILCS simulations, the compounds are in their ionized, charged states while in the experimental studies the compounds are likely in their neutral forms in the non-polar phase and ionized in the aqueous phase, further making direct comparison of the results difficult.

Table 2.

Log of membrane/water partition coefficients K(h) of solutes evaluated from Eq. (1) and from solute counting with h (membrane thickness) equal to the distance between phosphate planes of each bilayer, and octanol/water from experiment.

Compound	POPC/Chol		PAMPA		Expta (octanol/water)
	Eq 1	Counting	Eq 1	Counting
benzene	1.09	1.08	1.22	1.28	2.13
propane	1.42	1.34	1.58	1.61	2.36
methanol	−0.63	−0.67	−0.51	−0.65	−0.77
acetaldehyde	−0.46	−0.50	−0.42	−0.44	−0.34
formamide	−0.78	−0.82	−0.61	−0.82	−1.51
imidazole	−0.88	−0.91	−0.72	−0.87	−0.08
acetate	−1.63	−2.02	−1.45	−1.94	−0.29
methylammonium	−0.66	−0.72	−0.30	−0.71	−0.7

^aExperimental data from https://pubchem.ncbi.nlm.nih.gov under LogP for the respective compounds.

As noted in the Methods, the partition coefficient $K\left(h\right)$ is a function of membrane thickness. The differences in $\log K\left(h\right)$ for the two bilayers for acetate shown in Figure 6 are relatively small, and consistent with the partition coefficients listed in Table 2. However, the variation in the headgroup region highlights the uncertainty in defining the membrane boundary: $\log K\left(h\right)$ for PAMPA ranges from −2.47 to −0.80 for h/2 = 15 and 25 Å, respectively. $\mathrm{Log}K\left(h\right)$ for all of the model compounds in PAMPA are plotted in Figure 6b. The variation in this set is largest for acetate and gives a measure of the uncertainty in the estimate based on the membrane boundary. Lastly, the shape of $K\left(h\right)$ is similar for related compounds, and the ratio of their partition coefficients will be somewhat insensitive to the boundary definition.

Figure 6.

Partition coefficient (log K(h)) across the bilayers for A) acetate in POPC/Chol and PAMPA and B) for all the SILCS solutes in the PAMPA bilayer.

Table 3 lists the resistance coefficients R (Eq. 3) and R_CC (Eq. 4), and their ratios. As found for the partition coefficients there are mostly small differences in the results for POPC/Chol and PAMPA. Also, as anticipated, R is lowest for the nonpolar molecules benzene and propane and increases with polarity. From (Eq. 3) a rough estimate of the permeability of these compounds may be obtained by assuming an average diffusion constant across the bilayer and modeling the membrane thickness, h/2, as approximately 30 Å. The ratio R/R_CC provides insight as to the location of the barrier to permeation. If it is close to unity, the barrier is in the acyl chain region of the bilayer; the ratio is far from 1.0 for benzene and propane, indicating that the primary resistance to permeation arises in the headgroup region. This is consistent with the free energy profiles in Figure 4. In addition, the larger R/R_CC of propane is consistent with the higher calculated barrier than that of benzene.

Table 3.

Resistances of total bilayer R and acyl chains R_CC (in units of 10⁻⁶ cm) and their ratio for POPC/Chol and PAMPA bilayers for the solutes.

Compound	POPC/Chol			PAMPA
	R	RCC	R/RCC	R	RCC	R/RCC
benzene	0.323	0.0220	14.7	0.419	0.0171	24.50
propane	0.314	0.0108	29.0	0.471	0.00904	52.10
methanol	18.6	13.5	1.37	31.6	24.4	1.30
acetaldehyde	3.18	2.61	1.22	4.61	4.07	1.13
formamide	32.9	32.3	1.02	49.6	49.1	1.01
imidazole	33.6	32.9	1.02	55.4	54.9	1.01
acetate	165	163	1.01	220	219	1.01
methylammonium	35.1	34.6	1.01	48.3	47.9	1.01

Free energy profiles of drug-like molecules in the bilayers

Free energy profiles of the drug-like molecules were calculated for both bilayers based on the SILCS LGFE scores. Absolute free energies were calculated by restraining the center of mass of each ligand to a value of Z and then performing a Monte-Carlo optimization of the ligand rotational, translation and dihedral degrees of freedom in the field of the SILCS FragMaps along with intramolecular energy contributions, termed SILCS-MC. Performing this across the lipid bilayers allows for the absolute free energy profiles to be obtained, where the free energy is based on the LGFE value alone. This was applied to a number of drug-like molecules with varying size and chemical functional group complexity (Figure S4).

Figure 7 shows the LGFE profiles of the studied molecules along the normal of the lipid bilayers. In addition, the plots include the sum of the different types of FragMap GFE contributions from which information on the contributions of the different classes of functional groups to the free energy profiles are obtained. The free energy profiles calculated here may be compared to the PMF profiles reported in earlier studies. ^{63, 64} The simplest molecule, 1-butanol, has a free energy minimum around 14 Å from the bilayer center and a maximum when crossing through the polar lipid headgroups. The shape of the overall profile is similar to the previous study,⁶³ with a maximum upon moving into the bilayer, a minimum moving into the aliphatic tail region of the bilayer and a second maximum at the center of the bilayer. However, in that study, an approximate 3 kcal/mol difference was calculated between the global minimum and maximum at the bilayer center with the difference between bulk solution and the center of the bilayer being ~1 kcal/mol. In the present study, the difference between the global minimum and the center of the bilayer is 1.5 kcal/mol while the energy difference between bulk solution and the center of the bilayers is actually favorable at −1 kcal/mol. Thus, while the overall free energy profiles are similar quantitative differences are evident, with the SILCS method indicating that 1-butanol is more favorable in the bilayer versus in bulk solution, even in the center of the bilayer. The functional group GFE plots clearly show that the aliphatic carbons, represented by propane SILCS atom types, are dictating the overall LGFE profile. The alcohol group counterbalances the aliphatic contribution, dominating the free energy at the center of the bilayer. The alcohol GFE plot for butanol shows that the free energy becoming unfavorable corresponds to the transition between the M-B regions 2 and 1. Analysis of the contributions of the functional-group GFEs emphasizes the amphiphilic characteristics of the short-chained alcohol.

Figure 7.

Neutral drug-like molecule LGFE and atom type GFE profiles in the PAMPA system. Top panel shows the LGFE profiles for (A) 1-butanol, (B) Testosterone and (C) Caffeine as black dashed lines. Lower panels show the functional group contributions to the total LGFE based on the atom type GFE terms for propane (green), generic hydrogen-bond acceptor (solid red), methanol oxygen (dashed red) and generic heterocycle carbons (dashed purple).

The same analysis was subsequently applied to more complex molecules. The free energy profile for the steroid testosterone shows large variations along the bilayer normal. Steroids are nearly insoluble in water, which is reflected by the large unfavorable LGFE value in the aqueous phase, Z > 20. Testosterone has three different atom classifications that contribute to the LGFE, that is, carbons are represented by aliphatic propane carbons (PRPC), the cyclohexanone oxygen by generic acceptor (GENA) and the hydroxyl group being classified as methanol oxygen (MEOO) (Figure S4). As expected, the free energy surface is dominated by the GFE contributions of the aliphatic carbons of testosterone. The polar hydrogen bond acceptor (GENA) and alcohol (MEOO) make relatively small contributions to the free energy profile, with the biggest impact being the free energy being less favorable at the center of the bilayer. In previous studies based on PMFs the free energy profiles for testosterone show that the steroid is largely favored inside the bilayer, with the energy minimum ~8.5 kcal/mol.^{17, 64} While the overall shapes of the free energy profiles are similar inside the bilayer the SILCS free energies show a difference of ~10.5 kcal/mol between bulk solution and the lipid interior with the free energy at the center of the bilayer still favorable while in the previous study by Essex and coworkers⁶⁴ the energy difference is zero. The study by Brocke et al¹⁷ based on Generalized Born continuum models show the minimum energy in the bilayer being 2 to 4 kcal/mol below that bulk solution phase based on the force field with the free energy at the center of the bilayer actually less favorable then in bulk solution for one force field.

Caffeine, which is a highly polar but neutral molecule, has a free energy profile that shows a rather large unfavorable energy at the center of the bilayer (Figure 7). This is largely contributed by the GEHC FragMaps, associated with the carbons of the heterocycle imidazole. Figures 3 and 4 show the low density and unfavorable free energy, respectively, for imidazole compared to the other polar solutes, with a GFE at the bilayer core that is ~3 kcal/mol. As the purine-based caffeine is predominantly classified as GEHC (Figure S4) the GEHC GFEs dominate the absolute free energy profile. Notably, the free profile shows a very steep decrease in free energy from the center of the bilayer out to Z~12 Å. The profiles presented here suggest that caffeine is more unfavorable deep in the bilayer than reported in previous studies. ^{17, 65}

Additional free energy profiles were calculated for two beta blocker molecules, Pindolol and Alprenolol, as well as the NSAID Ibuprofen. These compounds contain a wider range of chemical group types along with significant conformational flexibility and ionizable groups. Accordingly, the molecules were treated as both neutral and in their ionized states. Both free energy profiles for each molecule are shown in Figure 8.

Figure 8.

Charged drug-like molecule GFE and atom GFE profiles for the PAMPA system. Top panel shows the LGFE profiles for (A) Ibuprofen, (B) Pindolol and (C) Alprenolol. LGFE profiles for the neutral and charged molecules are shown as black dashed and solid lines, respectively. The middle and bottom panels show the individual functional group contributions to the total LGFE based on the atom type GFE terms for propane (green), generic hydrogen-bond donor (blue), generic hydrogen-bond acceptor (solid red), methanol oxygen (dashed red), generic heterocycle carbons (dashed purple), positive charged groups (solid cyan) and negative charged groups (solid orange).

With Ibuprofen, there is a significant difference between the charged and the neutral free energy profiles. As expected, the negatively charged species is much less favorable than the neutral species in the aliphatic region of the lipid bilayer, while the free energies in the bulk solution are both favorable. Notably, the free energy of charged Ibuprofen in still somewhat favorable in the interior of the bilayer. This is due to the significant decrease in the free energy contributions going from bulk solution to the lipid interior for both the propane and benzene carbons. In the charged species the unfavorable GFE contribution of the acid group is evident, but it is not sufficient to make the overall free energy unfavorable in the bilayer, though is likely associated with limitations in the GFE estimates of the charged solutes in the center of the bilayer, as discussed below. Another interesting feature is the similarity of the free energies in aqueous solution for both species. This is due to the absolute free energies being similar for the neutral and charged solutes as shown in Figure 5. Contributing to this similarity is the presence of the aliphatic methyl group on the charged groups which favor their partitioning into the bilayer versus in vacuum that lead to large favorable free energies of hydration. In combination, the Ibuprofen free energy profiles show how the contributions of the different moieties in a ligand can lead to the free energy in some regions of the lipid interior not being overall unfavorable despite the presence of an ionized carboxylate group.

While Alprenolol and Pindolol are to a large degree composed of similar functional groups, their free energy profiles show significant differences (Figure 8 B, C). In addition, the neutral and charged free energy profiles are similar. Pindolol shows a significantly more unfavorable free energy at the bilayer center while Alprenolol is much more unfavorable in bulk solution. These differences can be explained based on the GFE contributions. Firstly, in the neutral state, the beta blockers have the amino group represented by the generic donor classification (GEND). This leads to more unfavorable energies in the center of the bilayer in Pindolol as compared to Alprenolol due to the presence of 2 versus 1 NH donor groups, respectively. Additional contributions leading to Pindolol being more unfavorable in the bilayer come from the generic heteroatom type (GEHC) associated with the 5-membered ring in the indole moiety. With Alprenolol the additional alkyl sidechain, which is classified as a propane type (PRPC), leads to the free energy being more favorable inside the bilayer. The contribution of the aromatic carbons (BENC) is similar for the two molecules. These results are consistent with previous calculations⁶⁴ showing that Pindolol is more favorable in solution than Alprenolol.

Comparison of the Alprenolol and Pindolol free energy neutral and charged profiles show them to be rather similar in the regions from the lipid interior out to bulk solution (Figure 8). The largest difference occurs at the center of the bilayer where both the charged states of both molecules exhibit ~2 kcal/mol more unfavorable free energy compared to the neutral species. This behavior may be explained by analyzing the methylammonium free energy profile in Figure 4. Upon going from bulk solution to the bilayer center, the free energy stays favorable down to almost 15 Å due to the favorable interactions with the phosphates, as shown in Figure 3. In contrast, the negative charge on Ibuprofen makes significantly more unfavorable interactions from 15 Å to near the center of the bilayer where both positive and negative charge contributions approach 3 – 4 kcal/mol. Analysis of the methylammonium free energy profile in Figure 4 shows the energy to become sharply unfavorable at Z ~ 15 Å, consistent with the phosphate moiety in the head groups. Combined with the Ibuprofen charged free energy profile which does not become unfavorable until Z ~15 Å, these results indicate that the ability of the molecule to reorient in the bilayer allows the nonpolar ring and aliphatic moieties to insert further into the bilayer thereby minimizing the unfavorable contribution from the acidic group. In the experimental regimen it may be anticipated that the ionization state of the acid, as well as the amine group, will shift favoring the neutral state depending on position in the bilayer. This issue will be addressed in a future study.

Potential of mean force (PMF) calculations on multiple drug-like molecules have been reported on Ibuprofen as well as Salbutamol/Atenolol and Theophylline, which are similar to Alprenolol and Caffeine, respectively. ⁶⁶ Overall, the reported PMFs from Carpenter et al. are similar to the absolute free energy profiles in the present study. Both Caffeine and Theophylline show minima in the vicinity of the Z=10-20 Å with unfavorable energies at the center of the bilayer. Interestingly, in both studies upon moving from 20 Å towards the center of the membrane there is a small barrier followed by a local minimum before the large barrier at Z=0 Å. With Ibuprofen, minima are observed in both studies at Z~10 Å with unfavorable energies at the center of the bilayer for the charged species and favorable with the neutral ones. In the case of Alprenolol versus Salbutamol/Atenolol the overall profiles are again similar with minima in the vicinity of Z=10-20 Å while the free energies of both the charged and neutral species are unfavorable at Z=0 Å in all cases. Concerning the magnitudes of the free energies, with Ibuprofen the global minima were more favorable values in the present study, while the opposite was observed with both Theophylline vs. Caffeine and Salbutamol/Atenolol vs. Alprenolol. While differences between the free energy profiles are to be expected due to the use of different types of bilayers, different free energy calculation methods and different force fields, the overall similarity of the free energy profiles further indicates the potential utility of the SILCS-based approach for the calculation of free energy profiles of drug-like molecules through bilayers.

Partition and resistance coefficients of drug-like molecules in the bilayers

Table 4 lists R, and R_CC, and $\log K$ for the set of neutral drug-like molecules in the PAMPA bilayer; Figure 9 plots log K(h) for both charged and neutral species. Consistent with the differences in the chemical structures the variation in partition and resistance coefficients is significantly larger than for the SILCS solutes (Tables 2 and 3). Testosterone is an extreme example, where the high barrier to escape in the headgroup region (Fig. 7B) dominates R. Caffeine is opposite, where the barrier is in the center (Fig. 7C). Nevertheless, the thickness dependence of $\log K\left(h\right)$ is low between 15 to 25 Å from the bilayer center, allowing a precise estimate of the partition coefficient within the context of the method. Comparison of the $\log K$ values with experimental water/octanol partition coefficients show the agreement to be quite good with the respect to both the magnitudes and the rank ordering. The biggest difference occurs with testosterone, where the calculated value is significantly larger than the experimental values. However, such a difference is likely associated with the differences between the properties of a water/octanol emulsion versus a membrane as the overall structure of the bilayer, including long and ordered fatty acid chains is well-tuned to accommodate cholesterol, likely making it a good solvent for testosterone leading to increased partitioning into the bilayer. Such differences in partition coefficients based on membranes versus octanol/water partition coefficients have been previously presented.²⁴ These results indicate the utility of applying functional group free energies based on the small SILCS solutes to larger, more complex drug-like molecules.

Table 4.

Log partition coefficients K, resistances R (cm) and R_CC (cm), and their ratios for the set of neutral drug-like molecules in PAMPA bilayers for the drug-like molecules.

Compound	log K		R	RCC	R/RCC
	Calculated	Experimental
Butanol	1.15	0.88	4.29×10−7	3.96×10−8	10.8
Testosterone	7.10	3.32	4.71×10−7	1.10×10−13	4.27×106
Caffeine	−0.80	−0.07	3.67	3.67	1.00
Ibuprofen	3.31	3.97	3.25×10−7	2.99×10−10	1.09×103
Alprenolol	4.11	3.1	3.45×10−7	5.62×10−8	6.14
Pindolol	1.65	1.75	6.78×10−5	6.74×10−5	1.01

a) Calculated values based on equation 1.

b) Experimental data from https://pubchem.ncbi.nlm.nih.gov under LogP for the respective compounds.

Fig 9.

Partition coefficient (log K(h)) along the bilayer normal for the drug-like molecules in PAMPA.

Discussion and conclusions

In the present study, we have utilized the SILCS methodology and examined the distribution and energetics of small solute molecules, representative of different types of chemical functional groups, in two lipid bilayer membrane systems. The SILCS simulations involved 2 microseconds of MD simulations in combination with oscillating chemical potential GCMC simulations to enhance the sampling of the solutes and water. Thus, each solute type and water samples the entire lipid bilayer and the surrounding bulk solution based on the CHARMM36 energy function, from which probability distributions were obtained. Normalization of the probability distributions was performed based on the concentrations in the full simulation system followed by Boltzmann transformation yielding the absolute free energies of each solute type and water throughout the simulation systems mapped onto a 1 Å cubic grid termed grid free energies (GFE). This information has been used to determine the distribution of functional probes with respect to the 4 regions originally defined by Marrink and Berendsen.^44–46

The distributions and absolute free energy profiles of the solutes and water throughout the studied bilayers were largely as expected given the physiochemical properties of the solutes and their associated functional groups. Yet, subtle aspects of the profiles did produce some novel insights. For example, analysis of the hydrophobic solutes, benzene and propane, shows a clear separation of the two apolar solutes within bilayer interiors. The aliphatic propane solutes are energetically favored over benzene at the center of the bilayer in M-B4. With benzene there is a minimum in the free energy profile in the slightly polar region in M-B3 that overlaps with the glycerol linker region, indicating the more polar nature of aromatic versus aliphatic groups.

With polar solutes differential distributions are observed. More favorable free energy minima are observed with acetaldehyde and methanol in the glycerol linker region versus imidazole and formamide. However, with all four polar-neutral solutes there is some sampling throughout the interior of the bilayer and the free energies in aqueous solution are similar. The positively charged solute methylammonium shows minima in the head group region not present with the anionic functionality of acetate. This emphasizes how the positively charged group can penetrate further into layer due to favorable interactions the lipid phosphate group. In addition, with both polar-neutral solutes and acetate there is an accumulation of the solutes in the region just above the bilayer head groups, with the trend most notable with the PAMPA system.

A limitation with the charged SILCS solutes should be noted. While both methylammonium and acetate are unfavorable in the central region of the bilayers, as expected, the GFE profiles plateau from Z~10 (acetate, ~3.6 kcal/mol) or 5 (methylammonium, ~3.1 kcal/mol) to 0 Å, whereas previous studies suggest that the free energies of charged solutes should be more unfavorable.^{67, 68} This is associated with the calculation of the free energies directly from the sampled populations throughout the simulation system. For example, as the SILCS GFE values are directly obtained via Boltzmann transformation of the normalized probabilities with respect to the average sampling in the entire simulation system, the maximum unfavorable GFE energy is limited by the total amount sampling. As GFE = −RTln(N/N_total), where R is the Boltzmann constant, T is the temperature, N is the number of counts of a given atom type in a voxel and N_total is that averaged over the entire simulation system, then the value of N_total dictates the smallest defined ratio. For example, if N_total = 100, then the smallest defined ratio in a voxel is 0.01 and GFE = 2.76 kcal/mol while if N_total = 10000 then the smallest defined ratio is 0.0001 and GFE = 5.53 kcal/mol. In practice a factor of 0.1 is added to N and N_total to avoid calculation of the natural log of zero. Alternatively, the potential for over sampling of the charged solutes in the center of the bilayer associated with the GCMC insertions into that region is a possibility, potentially making the free energies in this region less unfavorable. These limitations will also impact the free energies of the charged states of the drug-like molecules in the central regions of the bilayers. Nevertheless, sampling in the headgroup regions is reliable and valuable, and the profiles are therefore presented.

Analysis of the two bilayer systems showed both similarities and differences. The distributions of the moieties in the lipid groups and the methyl tail of the aliphatic chains were generally similar with shifts in the location of the maxima associated with changes on chain length and the presence of cholesterol evident. The presence of increased cholesterol potentially leads to increases in the probability of both polar-neutral and nonpolar solutes in the hydrophobic tail region in the PAMPA system, but not in the center of the bilayers for the polar neutral solutes, consistent with a previous report.⁵⁹ Some of the most notable differences as a function of bilayer was increased sampling in the center of the bilayer in the PAMPA system by benzene and propane, again likely due to the presence of 30% cholesterol in that system. In addition, a significant increase in the probability of methylammonium in the PAMPA head group region is evident, consistent with the presence of the phosphorserine groups. This may contribute to the larger amounts of acetate accumulating in the region above the headgroups. Importantly, the presence of the eight solutes in the simulation systems did not substantially alter the structure of the bilayers based on comparison with previously reported simulations.

The availability of the absolute free energy profiles of the solutes allowed for calculation of bilayer/water partition and resistance coefficients; partition coefficients were also calculated based on direct counting. While experimental partition coefficients for bilayer systems are not directly available, comparison of the calculated values with experimental 1-octanol/water partition coefficients shows the agreement to be quite good with respect to both the values for the individual solutes and their rank ordering (Table 2). Notably, this level of agreement was also obtained with the drug-like molecules (Table 4). This level of agreement is notable despite the differences between the partially ordered bilayer and an octanol/water emulsion, validating the ability of the SILCS to quantitatively model the free energies of both small solutes and larger, drug-like molecules in biological membranes.

Calculation of resistances in the total bilayers and in the acyl chains showed the resistance to be larger in the polar and charged solutes, as expected. The locations of the barriers to passive diffusion with the polar species is in the acyl regions (R/R_CC ~1, Table 3) consistent with the free energy profiles (Figure 4). Similar results were obtained with the drug-like molecules (Table 4) again pointing towards the utility of the SILCS GFE FragMaps in modeling the behavior of larger, multifunctional molecules in bilayers.

A particular advantage of the SILCS approach is the availability of the individual functional group GFE FragMaps allowing for absolute free energy profiles across the lipid bilayers for larger, drug-like molecules with varying complexity and size to be rapidly calculated using the SILCS-MC procedure. For example, once the SILCS FragMaps have been calculated, which may take one to two weeks on 10 NVIDA 2080TI GPUs on 8 core Ryzen CPUs, a single free energy profile for one ligand may be calculated in approximately 30 minutes on a single AMD EPYC 7702P processor core. The advantage of this is demonstrated in Figures 7 and 8 where the free energy profiles are presented for both the neutral polar and charged species, respectively. Again, the overall free energy profiles are as expected, showing a wide range of behavior with the largest barrier with testosterone occurring in solution followed by caffeine with the barrier occurring in the center of the bilayer. With ionizable compounds, the computational efficiency of the SILCS-MC method readily allows calculation of free energy profiles of both the neutral and charged species (Figure 8).

Beyond the overall absolute free energy profiles the ability to obtain the free energy contributions of the different classes of functional group in each molecule leads to interesting observations. For example, Pindolol and Alprenolol, to a large extent, share similar functional groups, but present distinctively different free energy profiles. This difference can be traced to the indole moiety of Pindolol being significantly more unfavorable inside the bilayer compared to the phenylpropene in Alprenolol. Overall, analysis of the different functional group contributions to the free energy profiles reveals how their different individual contributions combine to yield the total absolute free energies. For example, with all the drug-like molecules it is seen that the unfavorable contributions of polar and charged groups at the center of the bilayer are countered by the favorable free energies of the apolar benzene and propane classified atoms. This sort of analysis offers the potential to facilitate ligand design with respect to passive diffusion during late-stage drug discovery projects.