Simple Predictive Models of Passive Membrane Permeability

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Simple Predictive Models of Passive Membrane Permeability Incorporating Size-Dependent Membrane-Water Partition Siegfried S. F. Leung,†,§ Daniel Sindhikara,‡ and Matthew P. Jacobson*,† †

Department of Pharmaceutical Chemistry, University of California, San Francisco, California 94158, United States Schrödinger, Inc., 120 West 45th Street, 17th Floor, New York, New York 10036, United States

‡

S Supporting Information *

ABSTRACT: We investigate the relationship between passive permeability and molecular size, in the context of solubility-diffusion theory, using a diverse compound set with molecular weights ranging from 151 to 828, which have all been characterized in a consistent manner using the RRCK cell monolayer assay. Computationally, each compound was subjected to extensive conformational search and physics-based permeability prediction, and multiple linear regression analyses were subsequently performed to determine, empirically, the relative contributions of hydrophobicity and molecular size to passive permeation in the RRCK assay. Additional analyses of Log D and PAMPA data suggest that these measurements are not size selective, a possible reason for their sometimes weak correlation with cell-based permeability.

environment also imposes a decrease in entropy on the permeant, which loses conformational freedom in the compact environment and may change conformation in response to the lipophilic surroundings. Analogously, protein−ligand binding also presents a similar scenario in which entropic costs are incurred by both ligand and protein upon binding. Solubility-diffusion theory provides a framework for computing key physical parameters underlying permeation. In this formalism, transmembrane permeability (Pm) depends on the membrane/water partition of the permeant (Km/w), the permeant’s diffusion rate across the membrane (Dm), and the membrane’s width (dm).6 By assuming that 1) the membrane can be approximated as a homogeneous rate-limiting barrier and 2) both membrane partition and diffusion are independent of the permeant’s position in the membrane, the permeability rate can be expressed as the following

Optimizing membrane permeability is a critical component in the development of small molecule drugs.1 Many relevant biological processes, such as intestinal absorption, skin penetration, or blood-brain barrier permeation, involve permeation of molecules across biological membrane via passive transmembrane diffusion and/or active transport mechanisms.2−4 The present study focuses solely on passive membrane permeability. Hydrophobicity has been considered as the key physicochemical property that contributes to the passive membrane permeation of a molecule by driving the membrane partition process. The partition or distribution coefficient measured between water and an organic solvent (ex. octanol) of a molecule is often used as a de facto estimate for permeability and other related properties, like absorption. However, the biophysics of membrane permeability is more complex than simple partitioning between two different solvent phases. The membrane lipid bilayer is anisotropic, thus its physical characteristics cannot be fully captured by an isotropic bulk organic solvent. For example, the packed membrane environment gives rise to molecular size and shape selectivity of the rate of membrane permeation. Size-related terms, most commonly the molecular weight, are included in rules of thumb for drug-likeness described by Lipinski’s rule-of-five and QSPR models for permeability.5 The size/shape selectivity of passive permeation is driven largely by entropy, especially as manifested in the hydrophobic effect in water and diffusion across the packed membrane structure. The hydrophobic effect in water is primarily entropic, as the desolvation of a permeant results in a favorable entropy gain by water molecules losing the solvation shell structure, which is a key size-dependent driving force for partition into the membrane. When the permeant enters the membrane, the free energy cost of opening a cavity of the correct size and shape for the ligand is largely entropic as well. The membrane © XXXX American Chemical Society

Pm = K m/wDm /dm

(1)

Size-dependence can be incorporated in the calculations of both the partition and diffusion terms. For computing sizedependent membrane/water partition, we have used the barrier domain model, proposed by Xiang and Anderson.7,8 As shown in eq 2, the membrane/water partition, Km/w, is estimated as a partition coefficient between water and an organic solvent, Korg/w, in conjunction with a size-based scaling factor, ξv K m/w = ξvKorg/w

(2)

The size selectivity factor is formulated as the reverse work done for creating a cavity in membrane to accommodate the permeant, which depends on the membrane’s interior pressure exerted on the permeant and the permeant’s size Received: January 6, 2016

A

DOI: 10.1021/acs.jcim.6b00005 J. Chem. Inf. Model. XXXX, XXX, XXX−XXX

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Journal of Chemical Information and Modeling ξv = exp( −ϱVp/kBT )

Table 1. RRCK Data Sets

(3)

where kB is the Boltzmann constant, T is the temperature, Vp is the permeant’s volume, and ϱ is a pressure descriptor term based on lateral and normal pressures in the membrane. The resulting partition coefficient decreases exponentially with increasing volume of the permeant. Below, we will use the logarithmic version of eq 2 Log K m/w = Log Korg/w − βVp

1 2 3 4 5 6 7 8

(4)

where β = ϱ/kBT. We and others have developed predictive computational models based on this formalism, using molecular mechanics methods to perform conformational sampling, for flexible molecules, and to estimate the various terms. Unlike QSPRtype statistical modeling that requires training on experimental data, the approach aims to quantitate the key physical parameters underlying passive membrane diffusion, allowing us to predict relative passive permeability ab initio. In brief, the model assumes that the permeant adopts a neutral membranephilic conformation when diffusing across the membrane. As described in our prior works,7,9−15 the partitioning term is estimated using implicit solvent models to represent water and the membrane as dielectric continuums.16−18 The generality of this approach has been demonstrated in various studies by predicting relative permeability, without training on experimental data, of diverse chemical classes, ranging from drug-like small molecules to macrocyclic peptides, as measured in different in vitro permeability assays, including PAMPA and different cell-based assays.7,9−15 However, we have previously been unable to predict absolute permeability, especially for compounds differing significantly in size. In this work, we explore the idea of reweighting the key terms in the model based on experimental data. This approach is not unlike the training of QSPR models, but our goal remains to capture the key physics of passive membrane permeation in the trained model; and the resulting models have only a few adjustable parameters. Specifically, the size/shape selectivity is further investigated here by studying the relationship between membrane permeability and the permeant’s size/shape in different in vitro assays. For our present purpose, it is important to use permeability data that has been measured, under consistent conditions, for highly diverse classes of ligands with a wide range of molecular sizes. For this reason, we use primarily data generated by Pfizer, Inc. from in vitro RRCK (Ralph Russ Canine Kidney) cellbased permeability assay, which uses an MDCK cell line with a low expression of P-glycoprotein and has exhibited lower active efflux than both the MDCK-WT and Caco-2 cell lines.19 When compounds with active influx (via transporters) are excluded, the RRCK measurements primarily reflect passive permeation. The RRCK data sets surveyed in this study are summarized in Table 1. These sets include small molecule drugs, congeneric drug-like compounds, and macrocyclic peptides, with a wide range of molecular weight (MW: 151−828) that extends beyond the typical drug-like chemical space described by Lipinski’s rule-of-five.5 The data spans more than 3 orders of magnitude in rates of permeation, from 10−7 to 10−4 cm/s. Figure 1a shows poor correlation between the RRCK data and predicted permeability using our previously described method, which showed good ability to predict relative permeability among compounds within several of the different

chemical type

N

congeneric

cyclic peptides drug-like drugs cyclic peptides drug-like drug-like drug-like drug-like

8 21 104 16 9 23 34 22

yes yes no yes yes yes yes yes

other data Log Log Log Log Log

D D,1,22 PAMPA1 D D D

Log D

ref 12 20 21 14 23 24 25 26

Figure 1. Physics-based in silico models for predicting RRCK permeability. (a) The untrained model based on solubility-diffusion theory. (b) The MLR model based on calculated partition free energies and the permeant’s volume (Model 4). Data points are sized based on volume. Compounds excluded in the MLR analysis due to known active transport mechanisms or issues in assay (ex. poor recovery) are labeled as “Excluded”.

data sets. In brief, the conformational prediction employs a semiexhaustive torsional sampling algorithm to identify steriB

DOI: 10.1021/acs.jcim.6b00005 J. Chem. Inf. Model. XXXX, XXX, XXX−XXX

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Journal of Chemical Information and Modeling cally favorable conformations using PLOP.29 The macrocycle sampling protocol in MacroModel was utilized for macrocycles with nonpeptide backbone. 30 Energy evaluations were performed with PLOP using the OPLS force field in conjunction with implicit solvent models.16−18,29,31,32 The high dielectric aqueous environment is represented by a SGB model of water. The low dielectric membrane environment is represented by using a Generalized Born model of chloroform, which only accounts for the electrostatic interactions. The nonpolar term for cavitation is not included, because, opposite to the size selectivity effect of membrane partitioning, the term becomes more favorable with increasing solute size. The membranephilic conformation of the permeant is modeled by the low dielectric conformation (LDC), which is the conformation predicted to be energetically most favorable in the low dielectric continuum, and all property predictions, including the size-dependent term in eq 3, were performed on the predicted LDCs. The free energy cost of retaining the neutral membranephilic state, ΔGstate, was also calculated to account for neutralization of ionizable functional groups and tautomerization. Using EPIK,33 this “state penalty” was computed at pH 7.4. Compounds with well-known active influx transport mechanisms, such as L-DOPA, methotrexate, and tetracycline,27,28 as anticipated, were predicted to be less permeable than their experimentally measured permeability. However, the correlation between prediction and experiment remains poor even when excluding these compounds. We hypothesized that the poor correlation across this highly diverse set of compounds was due, in part, to underweighting of the size-dependence of permeation. In general, larger compounds like the cyclic peptides were predicted to be too permeable, relative to small molecules; while size-dependence was incorporated in the calculations as described above, it made little quantitative difference relative to the desolvation term. Motivated by the functional form of eq 4, we introduced empirical weighting constants for each free energy term Log Papp(RRCK)‐calc = aΔGpart + bV + c

the relative weights of these two terms dramatically improved correlation with experimental permeability. Multiple linear regression (MLR) analyses were performed using CANVAS.35 Compounds in the RRCK data set with known active influx transport mechanisms or low recovery in the assays were identified and excluded, yielding a subset of 201 compounds for subsequent analysis. A training set of 151 compounds was randomly selected for the MLR analysis, and the remaining 50 compounds constituted a test set for validation. The performance of different combinations of partition and size terms was examined (Supporting Information), and selected results are listed in Table 2. As previously Table 2. MLR Models for Predicting RRCK Permeability Based on Calculated Free Energy Terms and Size Descriptorsa MLR models of RRCK permeability (LogPapp(RRCK)) 1. 2. 3. 4. 5. 6. 7. a

−3.03e-2 ΔGdesolv − 4.88 −4.36e-2 (ΔGdesolv + ΔGstate) − 4.68 −4.53e-2 ΔGdesolv − 0.133 ΔGstate − 4.56 −4.95e-2 ΔGdesolv − 0.168 ΔGstate − 7.39e-4 V − 3.51 −5.38e-2 ΔGdesolv − 0.150 ΔGstate − 1.00e-2 CSA − 3.82 −4.63e-2 ΔGdesolv − 0.166 ΔGstate − 1.77e-3 SASA − 3.28 −4.90e-2 ΔGdesolv − 0.172 ΔGstate − 1.96e-3 MW − 3.63

training set: r2

test set: r2, q2

0.10 0.18 0.26 0.51

0.04, 0.12, 0.26, 0.52,

0.04 0.11 0.20 0.50

0.42

0.49, 0.45

0.49

0.48, 0.46

0.49

0.50, 0.48

Training set: N = 151; test set: N = 50.

shown,7 including ΔGstate in the MLR models, which accounts for neutralization of ionizable groups, is critical (Models 2 and 3). Fitting ΔGdesolv and ΔGstate separately (Model 3) yields better correlation than the sum of both terms (Model 2). This may be due to the difference in the dielectric environments between the actual membrane and the reference low dielectric model. Hence, decoupling the free energy terms allowed better calibration with the experimental data. Overall, models augmented with a size descriptor (Models 4−7) perform notably better than those with only partition terms (Models 1− 3). The signs of coefficients in these models are as expected. The negative coefficients of free energy terms indicate that a favorable desolvation cost would result in favorable membrane partitioning, i.e. smaller free energy values predict higher membrane partitioning. The negative coefficients of size descriptors reflect the anticipated size dependence, i.e. larger permeant size results in lower permeability. Relative to the untrained model (r2 = 0.11, Figure 1a), these simple MLR models of size-dependent partition with 2 free energy terms and a size descriptor (Models 4−7) perform markedly better. The size descriptors V, SASA, and MW produced a better correlation than CSA, which is related to both molecular size and shape. Model 4, which is comprised of desolvation free energy, state penalty, and molecular volume, displays the best agreement with RRCK measurements (r2 = 0.51, q2 = 0.50, Figure 1b). The linear dependence on molecular volume also has some theoretical justification, i.e. based on eq 4. This sizedependent partition approach (Model 4) has been implemented as part of the physics-based permeability prediction module in Schrödinger’s Small-Molecule Drug Discovery Suite (Supporting Information). Predictions by this model also produced reasonable agreement with Caco-2 cell-based

(5)

where ΔGpart is the computed partitioning free energy (between water and either vacuum or a low dielectric solvent, i.e., chloroform), and V is the molecular volume. We also generalized this expression further in two ways, both of which should be viewed as purely empirical rather than theoretically justified. First, in addition to volume, which was calculated as solvent-accessible volume, we also used crosssectional area (CSA), solvent-accessible surface area (SASA), and molecular weight (MW) as measures of molecular size. All 3D size descriptors, i.e. V, CSA and SASA, are computed based on the predicted LDC. Volume, SASA, and MW were computed using QikProp,34 and both volume and SASA were calculated using a probe with a radius of 1.4 Å. CSA was calculated with in-house software using the OPLS force field parameters.7 Although CSA is conventionally used as a shape descriptor, the CSA values in this study reflect both size and shape given the diverse molecular sizes represented in the data sets. The comparison of these size descriptors is included in the Supporting Information. Second, ΔGpart is itself composed of two terms, (1) ΔGdesolv, the free energy of desolvating the electrically neutral form of the molecule, computed using implicit solvent model; and (2) ΔGstate, the free energy of neutralizing the molecule, computed from estimated pKa values using EPIK. We found empirically that independently adjusting C

DOI: 10.1021/acs.jcim.6b00005 J. Chem. Inf. Model. XXXX, XXX, XXX−XXX

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Journal of Chemical Information and Modeling permeability measurements (r2 = 0.4−0.6) from various studies that include small molecule drugs and peptidomimetics. To further examine the effect of including the size descriptor, we compared predictions by Model 3 (free-energy-based partition model) and Model 4 (size-dependent partition model) for each compound series (Supporting Information). In general, Model 4 performed best for compound series with significant size variability and a dynamic range of permeability measurements of at least 1 Log unit. The addition of a size descriptor, however, did not yield improvement when modeling similar-sized congeneric compounds. For almost all congeneric series in this data set, estimating partition free energy from desolvation and neutralization (Model 3) yielded stronger correlation than the size-dependent partition approach (Model 4). Next, we explored the use of experimental partition measurements rather than computed solvation energies. The MLR analyses were performed on a subset of 137 compounds (MW = 151−826) from the RRCK data set that have Log D values. As described above, compounds with known active transport mechanisms and recovery issues in assays were not included in the MLR analysis. The resulting MLR models for predicting RRCK permeability from Log D values and size/ shape descriptors are listed in Table 3. Table 3. MLR Models for Predicting RRCK Permeability Based on Log D and Size Descriptors (N = 137) MLR models of RRCK permeability (Log Papp(RRCK))

r2

0.112 Log D − 5.40 −7.77e-4 V − 4.19 −8.49e-3 CSA − 4.64 −1.97e-3 SASA − 3.82 −2.02e-3 MW − 4.32 0.230 Log D − 1.06e-3 0.201 Log D − 1.28e-2 0.233 Log D − 2.68e-3 0.259 Log D − 3.09e-3

0.07 0.29 0.12 0.30 0.25 0.53 0.31 0.55 0.54

8. 9. 10. 11. 12. 13. 14. 15. 16.

V − 4.26 CSA − 4.75 SASA − 3.78 MW − 4.36

Figure 2. (a) Scatter plot of RRCK permeability versus Log D values (N = 137). (b) The MLR model for predicting RRCK permeability based on Log D and permeant’s volume (Model 13). Data points were sized based on volume. Compounds with known active transport mechanism or issues in assay that were excluded in the MLR analysis were labeled as “Excluded”.

Overall, the results are similar to those using computed values. Neither Log D (Models 8, Figure 2a) nor the size descriptor alone (Models 9−12) correlate strongly with the RRCK data. Correcting the Log D values with molecular size (Models 13, 15, 16) results in much improved agreement (r2 > 0.5, Figure 2b), slightly better than the corresponding models using only computed values (Models 4−7). The signs of coefficients are consistent with the expected size-selective permeation process across cellular membrane, i.e. positive for Log D and negative for size descriptors. The significant improvements brought by the size descriptors support the idea that the size selectivity observed in membrane permeation is not adequately captured by bulk organic phase partitioning measurements. Finally, we repeated our analysis for PAMPA measurements to compare cellular and artificial membranes. From the RRCK data set, a subset of 72 drug molecules with available PAMPA measurements, and no known active influx, was selected for this analysis. Members of this subset remain diverse in size (MW: 151−781). Similar to the analysis of RRCK permeability, MLR models were derived for PAMPA permeability (Log Pm(PAMPA)) using the same physical terms describing desolvation and molecular size (Table 4).

Table 4. MLR Models for Predicting PAMPA Permeability Based on Calculated Free Energy Terms and Size Descriptors (N = 72) MLR models of PAMPA permeability (Log Pm(PAMPA)) 17. 18. 19. 20. 21. 22. 23.

−0.126 −0.136 −0.131 −0.125 −0.125 −0.123 −0.131

ΔGdesolv − 3.06 (ΔGdesolv + ΔGstate) − 2.68 ΔGdesolv − 0.237 ΔGstate − 2.52 ΔGdesolv − 0.227 ΔGstate + 1.52e-3 ΔGdesolv − 0.245 ΔGstate + 3.93e-2 ΔGdesolv − 0.228 ΔGstate + 3.78e-3 ΔGdesolv − 0.233 ΔGstate + 3.14e-3

V − 4.23 CSA − 4.72 SASA − 4.83 MW − 3.62

r2 0.27 0.32 0.34 0.39 0.44 0.39 0.38

Unlike the results for cell-based permeability, the results for PAMPA permeability suggest that the size dependence is not as prominent. That is, including volume, SASA, MW, or CSA terms only improved correlations very slightly. Moreover, contrary to models of RRCK permeability, the coefficients of D

DOI: 10.1021/acs.jcim.6b00005 J. Chem. Inf. Model. XXXX, XXX, XXX−XXX

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Journal of Chemical Information and Modeling

adjustable parameters to describe membrane partition and permeant’s size were derived from MLR analysis. Predictions by these simple models (Models 4 and 13) yielded reasonable correlation with RRCK permeability measurements, demonstrating the importance of such size dependency in cellular membrane permeation and the feasibility of the approach. The MLR analyses on Log D and PAMPA data suggest that these measurements lack the size selectivity as observed in the cellbased assays. This finding might help to explain the inconsistency often observed between different in vitro measurements (Figures 2a and 3a). As illustrated by our approach in supplementing calculated partition free energies or experimental Log D values with a size descriptor (Figures 1b and 2b), such a size-dependent partition model might be deployed to obtain a more accurate permeability prediction than based on partition/distribution coefficients alone.

the size descriptors in all these models are positive, indicating that larger molecules have higher PAMPA permeability (to the extent that size contributes to the models at all). The modeling results here suggest that the PAMPA membrane might behave more like bulk organic solvent than a cellular membrane. In support of this hypothesis, PAMPA data for this set of compounds correlates poorly (r2 = 0.15) with the corresponding RRCK measurements (Figure 3a), while showing much stronger correlation with the Log D values (r2 = 0.66, Figure 3b).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jcim.6b00005. RRCK data sets and predictions, plot of RRCK permeability against molecular weight, comparison of size-descriptors, results of MLR analysis between RRCK permeability and various calculated and experimental descriptors, computational RRCK permeability model implemented by Schrödinger and its predictive performance (PDF) The data set in 2D SDF format (TXT)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address §

Circle Pharma, Inc. 280 Utah Ave, Suite 100, South San Francisco, CA 94080-6883. Notes

The authors declare the following competing financial interest(s): M.P.J. is a consultant to Schrodinger LLC, which distributes some of the software used in this work; was formerly a consultant to Pfizer, the source of the experimental data; and is a co-founder of Circle Pharma.

■

ACKNOWLEDGMENTS Gratitude is expressed to the National Institutes of Health (GM086602) for financial support and to Morena Spreafico, Alan Mathiowetz, Markus Boehm, and Kenneth Borrelli for helpful discussions.

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Figure 3. (a) Scatter plot of RRCK permeability versus PAMPA measurements. Compounds with known active transport mechanisms are labeled as “Active transport”. (b) Scatter plot of PAMPA measurements versus Log D measurements. Data points were sized based on volume.

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In summary, with a focus on understanding size selectivity, we have applied a size-dependent membrane partitioning approach to model RRCK permeability data for a large and diverse compound set. Theoretically, the size-dependent partition model accounts not only for the membrane/water partition driven by permeant’s hydrophobicity but also the entropic effects involving both membrane and permeant in the permeation process. Simple models that contained only 2 to 4 E

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DOI: 10.1021/acs.jcim.6b00005 J. Chem. Inf. Model. XXXX, XXX, XXX−XXX