Molecular Simulation of the Transport of Drugs across Model

Apr 24, 2014 - This approach can be utilized to probe the diffusion and aggregation of the drug in phospholipid bilayers, which has proved successful ...
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Molecular Simulation of the Transport of Drugs across Model Membranes Sharon M. Loverde* Department of Chemistry, City University of New York, College of Staten Island, 2800 Victory Boulevard, Staten Island, New York 10314, United States ABSTRACT: This Perspective describes recent progress in the area of the molecular simulation of the interactions of hydrophobic and hydrophilic solutes with membranes. The ability to predict drug solubility prior to synthesis is an extremely desirable goal for pharmaceutical research. A major advantage of molecular dynamics is the ability to computationally probe both the drug solubility as well as the pathway for the transport of drugs across membranes. Computational methods to predict the interaction free energy of solutes with membranes have advanced significantly in recent years and can characterize the intra- and intermolecular state of the drug as well as perturbations of the drug to the membrane environment itself. In addition to a brief review on computational methods to characterize the transport of drugs across membranes, we will highlight recent discoveries and discuss the challenges and opportunities in the field.

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he transport of hydrophilic and hydrophobic compounds (drugs) across model membranes (phospholipid or polymeric) is of interest to physical chemistry from both a fundamental and applied perspective.1−4 Mimicking the cell membrane in order to characterize small molecule interactions is an experimental challenge. In particular, bicelles have emerged as candidates to mimic the cell membrane for NMR spectroscopy.5,6 Molecular dynamics simulations have advanced significantly in the past decade, particularly in regards to membrane simulations,7 with a multitude of simulation methodologies, such as coarse-grained molecular simulations,8 emerging to capture the long-time and large-scale behavior of membranes. These simulation methodologies can aid in the design and optimization of drug formulations and liposomal formulations for drug delivery as well as add additional insight into the interaction of solutes (drugs) with polymeric selfassemblies such as micelles.9 Furthermore, modeling the solubility and phase behavior of drugs in lipid bilayers can provide valuable information about the liposomal stability and permeability, rationalizing the design of liposomal-based drug delivery systems.10 Moreover, investigation of the transport of drugs across model cellular membranes and even more complex interfacial (or cellular) environments can allow for the regulation of the solubility as well as the release behavior of solutes (drugs).Investigation of the transport of drugs across model cellular membranes and even more complex interfacial (or cellular) environments can allow for the regulation of the solubility as well as the release behavior of solutes (drugs). At the simplest level, a membrane can be considered a homogeneous slab, as shown in Figure 1. The solute concentrations (per unit volume), CmD and CmA, immediately inside of the membrane, with a smooth concentration gradient between them, are related to CD and CA, the concentrations outside of the membrane, by the partition coefficient K = (CmD/CD) = (CmA/CA). From Ficks’s law, we can define the © 2014 American Chemical Society

Investigation of the transport of drugs across model cellular membranes and even more complex interfacial (or cellular) environments can allow for the regulation of the solubility as well as the release behavior of solutes (drugs). solute flux per unit area, J = D((CmD − CmA)/2L). Additionally, the membrane permeability, P, can be expressed in terms of the bulk properties such that P = (KD/2L), where D is the diffusion coefficient of the solute in the membrane and 2L is the thickness of the membrane.11,12 As the real bilayer system, however, contains heterogeneities in both molecular components and density, this model can be expanded upon in terms of an inhomogeneous solubility-diffusion model,13,14 such that P = [1/(∫ z2 z1(exp(ΔG(z)/kBT))/(Dz(z)))], where ΔG(z) is the free energy along the membrane normal and Dz(z) is the z component of the diffusion coefficient of the solute along the membrane normal. The binding free energy of the solute (drug) with the membrane as well as solute diffusion in the membrane can be experimentally obtained, but the intrinsic variation of the free energy along the membrane normal, as well as the inhomogeneous diffusion of the solute (drug), can be obtained from molecular dynamics techniques. Furthermore, while the membrane system is heterogeneous, likewise the solute (drug) can possess multiple degrees of freedom, Received: February 13, 2014 Accepted: April 24, 2014 Published: April 24, 2014 1659

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anticancer drug, binding with a dioleoylphosphatidylcholine (DOPC) lipid membrane shown in Figure 2, as compared with

Figure 1. Schematic illustration of diffusion across a lipid bilayer including the quantities that enter the solubility-diffusion model: the solute concentrations outside (CD and CA) and immediately inside (CmD and CmA) of the membrane, with a smooth concentration gradient between them, the solute flux (J), and the thickness of the membrane (2L).11 Reprinted from ref 11. Figure 2. (Top) The fractional thickness changes of DOPC bilayers containing curcumin as a function of the curcumin/DOPC molar ratio in comparison with experimental observations by Hung et al.20 (Bottom) The calculated value of XS (the mole fraction of drug in the S phase per lipid) and XI (the mole fraction drug in the I phase per lipid) from the solution of the two-phase model.21 Reprinted with permission from ref 21. Copyright 2009 Elsevier.

orientational and conformational, which can influence the interaction free energy with the membrane. The effect of the drug conformation on the free-energy profile across the bilayer interphase can be explored with advanced sampling techniques in molecular dynamics, as highlighted later in this Perspective.15 Correspondingly, the solubility-diffusion model can be expanded in terms of the translational and rotational degrees of freedom of the solute.11 It is found that for fast solute reorientation, the classical solubility-diffusion model is recovered. The transport of solutes across biological membranes can take place via multiple paths, dependent on both the size and nature of the solute. Hydrophobic or lipophilic solutes can easily partition into the cellular membrane, while small hydrophilic solutes are thought to move between cells.16 The transport of the solute itself across the membrane structure can take either a passive1 or else a more cooperative mechanism whereby the solute perturbs the membrane structure and leads to transient pore formation.2 In a passive mechanism, the solute diffuses across the membrane structure, from water to the membrane and then water again via the solubility-diffusion model. In a more direct mechanism, due to cooperative effects, the solute can initiate fluctuations in the hydrophobic density such that water can locally diffuse in and create a transient pore, bending the membrane.17,18 For example, this mechanism has been observed with molecular dynamics simulations by Garcia et al.;2 charged peptides act cooperatively to facilitate the bending of the phospholipid membrane and subsequent pore formation. Additionally, antimicrobial peptides can insert in the membrane and oligimerize to create pores, either via the barrel stave model or else the toroidal pore model, which can be observed by experimental methods such as solid-state NMR19 as well as molecular dynamics. A unique property of pore-forming compounds, including some drugs, is that they exhibit a threshold concentration for pore formation.21 This behavior can be described with a simple two-state model for binding,22 one mode for binding at the membrane interface and one mode for binding in the membrane core. This two-phase model is analogous to micellar solution theory. For example, we see the fit of the two-state model describing membrane thickness for curcumin, an

experimental results from Hung et al.20 The model suggests that the initial binding is in the S phase (mole fraction XS), where the drug binds to the bilayer interphase and thins the membrane. The fraction of drug in the I phase, XI, gradually increases, following an increase in the mole fraction of bound drugs (NB/L). This corresponds to a fraction of drugs that partition more deeply into the membrane. Lipid membrane ordering and thinning with incorporated curcumin in the bilayers has been experimentally observed by Barry et al.23 through solid-state NMR of curcumin interacting with mixed DHPC (1,2-dihexanoyl-sn-glycero-3-phosphocholine)/DMPC (1,2-dimyristoyl-sn-glycero-3-phosphocholine) bicelles. As shown in Figure 3, curcumin induces segmental ordering in the membrane. The concentration-dependent effect on the order parameter (SCH) is highest for the carbons in the glycerol backbone, although an increase in ordering is seen for the hydrophilic head group, glycerol backbone, and hydrophobic tails. It is suggested that at low concentrations (0.25−0.5%), curcumin remains in its monomeric state, while at increased concentration (>1%), curcumin oligomerizes, increasing its interaction and thinning the membrane,20 reducing the order induced with lower concentrations of curcumin. These observations are consistent with the previously described twophase model for membrane binding. We hypothesize that the two-phase model may describe the interaction behavior of additional hydrophobic anticancer drugs such as paclitaxel. Paclitaxel is one of the first anticancer drugs discovered and acts by binding and stabilizing microtubules to frustrate cell division,24 and for similar reasons, it is also used in numerous polymer coatings on drug-eluting stents to help prevent restenosis.25 The main group of paclitaxel, the taxane portion of the molecule (one eight- and two six-membered rings) is rigid but slightly more hydrophilic than the three phenyl groups as side chains. The packing and interactions 1660

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Figure 3. Order parameter, SCH, for curcumin determined by solid-state NMR. The effect on ordering is greatest in (B), the order parameter of the carbons from the glycerol backbone, as compared with (A), the order parameter of the hydrophilic head group, and (C), the order parameter of the hydrophobic tails. (D) Schematic of the effect of curcumin concentration on the order of the tails. At higher concentrations, it is proposed that curcumin oligomerizes within the hydrophobic tails, decreasing the lipid order.23 Reprinted from ref 23.

membranes suggest that the behavior might be well-described by the two-phase model. Following, we review the history of molecular simulation models to characterize small molecule and/or drug interactions with membranes, as highlighted in several references.1,3,32 Some of the first simulations of solute (benzene) in a DMPC bilayer confirmed the nonheterogeneous nature of the membrane environment.33,34 The authors observed in simulations on the single nanosecond time scale that benzene diffuses through a “hopping” mechanism, jumping from void to void during conformational rotations of the phospholipid tail. The calculated diffusion coefficient (on the order of 10−6 cm2/s) was comparable to experimental measurements of small molecule diffusion.35 Following, Marrink and Berendson14 calculated the water permeation rate through a phospholipid bilayer, utilizing molecular dynamics indirectly via Widom insertion and thermodynamic integration techniques from the free-energy and diffusion rate profiles of a water molecule across the bilayer. Furthermore, they introduced the fourregion model of the lipid bilayer, consisting of low head group density, high head group density, high tail density, and low tail density. Shinoda et al. examined the permeability of small solutes in branched and unbranched lipid bilayers, utilizing a particle insertion technique as well as thermodynamic integration.36 They found that decreased permeability of small solutes in branched lipid bilayers is mainly correlated to the higher degree of branched structure and lower water mobility in the branched bilayers. This is complementary to the results of Sugii et al,37 who found that the length of the hydrocarbon tails of the phospholipids in the membrane is correlated with a decreased water permeability in the

between paclitaxel are both solvent-and concentration-dependent.26 The crystal structure in water is an alternating head-totail packing as described by Mastropaolo et al.27 However, the structure in hydrophobic environments such as chloroform is proposed such that paclitaxel stacks in a head-to-head manner.26 Additionally, it exhibits a fluidizing effect on phospholipid membranes, both dipalmitoylphosphatidylcholine (DPPC) liposomes and biological membranes, which have been characterized utilizing fluorescence polarization.28 The order parameter (S) for the packing of the hydrophobic chains of gel-phase DPPC decreases approximately 17% upon incorporation of 1 mol % of paclitaxel.28 The calculated lipid order parameter is defined as the inverse of fluidity.29 A similar effect is seen with curcumin in a mixed DMPC/DHPC bicelle membrane at concentrations greater than 1 mol %, as shown in Figure 3. Similar effects are observed with differential scanning calorimetry (DSC) on the main DPPC gel transition. At concentrations less than 2 mol % of the drug, the transition broadens due to the paclitaxel location in the outer hydrophobic domain. At higher concentrations, the transition temperature begins to drop due to the drug partitioning deeper into the hydrophobic region of the membrane. At much higher concentrations, it is observed that the drug aggregates and phase separates within the liposome.30 We propose that similar to curcumin, paclitaxel oligimerizes within the hydrophobic tails. However, in contrast to curcumin, recent experimental and computational investigations have suggested that paclitaxel can contribute to pore formation in phosphatidylcholine and phosphatidylserine membranes.31 These observations of the concentration-dependent nature of paclitaxel partitioning in 1661

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membrane. Bemporad et al.38 calculated the permeability coefficients for several larger organic molecules such as acetamide, acetic acid, benzene, ethane, methanol, methyl acetate, methylamine, and water. They found that the calculated permeability coefficients are generally an order of magnitude larger than experimental data, but the relative permeability coefficients are reproduced. The permeability of small solutes has been relatively well explored with molecular simulations. Next, we highlight molecular simulation models to characterize the interactions of larger solutes, or drugs, with lipid membranes. The interaction and diffusion of nifedipide, a blood pressure medication, with a DMPC lipid bilayer were characterized with molecular dynamics by Stouch et al.39 As compared with the hopping mechanism, as previously observed for smaller solutes, such as benzene, the diffusion rate did not change based on the z position. Without voids in the bilayer of a comparable size to that of the drug, the nifedipine is found to translate to a much higher degree in the xy plane versus that in the perpendicular direction. The movement in the z direction is described by a “bobbing” motion. This motion in consistent with the directional diffusion of halothane, an anesthetic, at higher concentrations.40 The interaction of anesthetics with lipid bilayers has been extensively explored with molecular dynamics simulations.40−42 In particular, the concentrationdependent effects are explored with a coarse-grained model of the drug and lipid.40 It is found that the maximum in drug population, for the case of halothane, remains in the same z position close to the lipid head groups. However, for higher concentrations, the maxima increase in population, but a fraction of the drug populates the center of the bilayer. This drug at high concentrations is found to diffuse from maxima to maxima with a slower rate than that at lower concentrations due to the increased partitioning in the center of the bilayer, consistent with a bobbing motion. Asymmetry of diffusion in the parallel and perpendicular directions was also observed by Orsi et al.43 for the motion of beta blockers in the DMPC lipid bilayer. The model consists of a dual resolution approach, a coarse-grained representation of the bilayer in combination with an all-atomistic representation of the drug to increase sampling in time and length scales. A coarse-grained model for an anticancer drug, paclitaxel, has also been utilized by Loverde et al.44 to probe the free-energy profile and diffusion of the drug in diblock copolymer micelles. This approach can be utilized to probe the diffusion and aggregation of the drug in phospholipid bilayers, which has proved successful to characterize the diffusion and aggregation of buckyballs in lipid bilayers as in Jusufi et al.45 Spontaneous permeation rates of solutes across membranes are relatively slow. Given the slow diffusion of larger solutes in the membrane (∼10−8 cm2/s), advanced sampling techniques in molecular simulation can be further utilized to assess the free-energy profile across the membrane. Here, we highlight several different advanced sampling techniques in molecular dynamics as methods to accurately obtain quantitative thermodynamic information about the interaction free energy of drugs in a membrane environment coupled with the conformation or charge on the drug. A multitude of freeenergy techniques, such as thermodynamic integration (zconstraint) or umbrella sampling,46−48 metadynamics,15 and adaptive biasing force (ABF)49,50 calculations have been practically applied to assess the potential of mean force between compounds and membranes. For example, Jambeck et al.15 utilized well-tempered metadynamics to probe the free-

energy surface of aspirin, diclofenac, and ibuprofen interacting with bilayer membranes. It was found that the cis and trans conformations of ibuprofen resulted in different barriers to cross the membrane, as shown in Figure 4. It is predicted that the drug will transition from its cis to trans conformation in order to cross the membrane with a lower free-energy barrier.

Figure 4. Free-energy surface of ibuprofen within a lipid bilayer sampled with well-tempered metadynamics (A) and the projection of the free-energy profile for the cis and trans conformations of ibuprofen to cross the membrane (B).15 Reprinted from ref 15.

Advanced sampling techniques offer an ideal approach to assess the free-energy profile of larger solutes (drugs) across the membrane. However, obtaining an accurate and converged free-energy profile may be difficult when considering the hidden sampling barriers, such as solvent and membrane reorganization times. 48 With more advanced sampling techniques, we can aim to probe (i) the conformational state of the solute, (ii) the heterogeneous nature of the membrane 1662

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With more advanced sampling techniques, we can aim to probe (i) the conformational state of the solute, (ii) the heterogeneous nature of the membrane environment, (iii) the concentration of solute, and, quite possibly, (iv) the curvature of the membrane. Furthermore, coarse-grained molecular dynamics, in complement to advanced sampling techniques, can explore relevant time and length scales for solute−solute and solute-membrane interactions.

Figure 5. Solute distribution from the outer to inner bilayer as a function of membrane curvature calculated with mean field lattice theory. The dashed line represents a flat bilayer, while the solid line represents a positively curved vesicle with equal surface densities for the inner and outer bilayers. The dotted line represents the same vesicle with a more realistic lower outer membrane surface density than inner membrane surface density. The solute partitioning toward the inner bilayer increases with increased membrane curvature, with an overall decrease in net partitioning.51 Reprinted with permission from ref 51. Copyright 1986 AIP Publishing LLC.

environment, (iii) the concentration of solute, and, quite possibly, (iv) the curvature of the membrane. Furthermore, coarse-grained molecular dynamics, in complement to advanced sampling techniques, can explore relevant time and length scales for solute−solute and solute-membrane interactions. For example, the concentration of ibuprofen in multilamellar phospholipid bilayers has been studied utilizing diffraction studies, as well as molecular simulation.46 Changes in the bilayer structure as studied by molecular dynamics are consistent with the experimental results. Increasing the concentration of drug acts to thin the lipid membrane; however, there is no significant effect of the concentration on the density distribution of ibuprofen positioning. It is suggested that these concentrations are below the point at which phase segregation occurs. Additionally, the effect of interfacial curvature on solute partitioning has been investigated theoretically by several studies51−53 but remains to be investigated with molecular simulations. For example, Dill et al.51 used mean field theory to describe the solute distribution from the outer to inner bilayer as a function of membrane curvature, as shown in Figure 5. In the figure, the dashed line represents a flat bilayer, and the solid line represents a positively curved vesicle with equal surface densities for the inner and outer bilayer. The dotted line represents the same vesicle with a more realistic lower outer membrane surface density than inner membrane surface density. The solute partitioning toward the inner bilayer increases with increased membrane curvature, with an overall decrease in net partitioning. However, the overall partitioning for a membrane of typical elastic modulus (100 dyn/cm) predicts a decrease of only 10% solubility for a vesicle with a radius of 10 nm as compared with that of a flat bilayer.51 This is consistent with experimental observations of benzene partitioning into vesicles as opposed to flat bilayers.54 On the contrary, Gruen et al.52 found the solubility of alkane in curved multilamellar vesicles to be 17−65% of the solubility in a comparably flat bilayer. While the degree of correlation between curvature and solute partitioning may vary, it is indeed agreed that curvature effects should perturb the overall solution partitioning. Several other effects need to be accounted for, such as the conformational state of the solute, the concentration of the solute, as well as the heterogeneous

nature of the membrane itself. The predicted curvature effects for bilayer membranes are also prevalent in nonlamellar structure such as micelles, composed of surfactants or polymers.53 As the Flory−Huggins interaction parameter, χ, becomes increasing attractive (more negative), the favorable micellar state changes from a spherical to cylindrical to wormlike micelle. The favorable effect of curvature on the polymeric micelle loading capacity has been found experimentally55 and computationally44 yet remains to be investigated with model phospholipid membranes. The challenges and opportunities of the field involve development of systematic and transferrable methods to coarse-grained drug molecules, characterization of their interactions with various solvents, such as phospholipid membranes, liposomes, as well as polymeric biomaterials such as micelles or hydrogels, as well as prediction a priori of the solubility of the drug on the specific membrane environment.56 For example, coarse-grained models or multiple-resolution molecular models of drug molecules such as paclitaxel44 and beta blockers43 have shown the potential to capture the transfer free energy and diffusion in a membrane environment. Moreover, an additional challenge would be to predict the pathway of membrane permeation, either passive, active, or via the assistance of additional components or a transport mechanism. There exists a unique opportunity in the field to develop dynamic interactions between computational chemists and experimentalists working in the field, with an active feedback mechanism. Knowledge and techniques developed would aid in the development of fundamentally new methodologies for molecular simulation; in addition, they would be extremely beneficial for the pharmaceutical industry. 1663

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 718-982-4075. Notes

The authors declare no competing financial interest. Biography Sharon Loverde received her Ph.D. in Materials Science from Northwestern University (2007). For her postdoc, she worked with Michael L. Klein (Temple University) and Dennis E. Discher (UPenn). She is an assistant professor at the College of Staten Island and the Graduate Center of the City University of New York (CUNY). https://sites.google.com/site/loverdelaboratory/

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ACKNOWLEDGMENTS The author acknowledges start-up funding awarded by the College of Staten Island and City University of New York (CUNY).

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