# Interplay of Hydropathy and Heterogeneous Diffusion in the Molecular Transport within Lamellar Lipid Mesophases

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model

^{2}/s. For instance, at 25 °C, one has ${D}_{wat}=0.7-0.9\times {10}^{-9}$ m

^{2}/s for amino acids [49,50,51,52,53], and ${D}_{wat}=0.5,0.7,0.8\times {10}^{-9}$ m

^{2}/s for ibuprofen [54], aspirin [55], and paracetamol [56], respectively. The value of ${D}_{wat}$ is expected to be dependent on temperature, T. When small temperature differences are considered (such as estimation of ${D}_{wat}$ at physiological temperature starting from room-temperature measurements), a simple yet effective approach to estimate the effect of T is to assume a Stokes–Einstein relation ${D}_{wat}={k}_{B}T/\left(6\pi \eta \left(T\right)R\right)$, where ${k}_{B}$ is Boltzmann’s constant, R is the size of the particle, and $\eta \left(T\right)$ is the temperature-dependent viscosity of water. This approach has enabled accurate predictions of transport of glucose molecules in monolinolein-based cubic phases [57]. Unless stated otherwise, in our simulations, we consider ${D}_{wat}=0.7\times {10}^{-9}$ m

^{2}/s.

^{2}/s [26], which gives values in the range $1.7\u20132\times {10}^{-11}$ m

^{2}/s for the lateral diffusion coefficient when accounting for the geometric constraint imposed by the minimal surface at the mid-plane of the lipid bilayer [58]. Amino acids and drugs such as the ones mentioned above are smaller than lipid molecules, so that ${D}_{lip}$ is expected to be somewhat larger for them. Here, we fix ${D}_{lip}=0.09{D}_{wat}$, based on molecular dynamics simulations of paracetamol in DPPC [47].

#### 2.2. Brownian Dynamics

^{2}/s = 0.7 nm

^{2}/ns, one has $\tau \simeq 1.4$ ns. The value of $\tau $ is important in the determination of the integration of the timestep $dt$. Our rationale in its choice was to consider the largest possible value of $dt$ which correctly recovers the equilibrium distribution of a collection of particles. In this regard, for the system corresponding to $U\left(z\right)$ and $D\left(z\right)$ as in Figure 1, we performed simulations of ${10}^{4}$ particles initially located at $z=0$ for a total time $2\times {10}^{4}\tau $, by considering several values of $dt$; the optimal choice turned out to be $dt=3\times {10}^{-4}\tau $, which at the end of the simulation led to the correct sampling of the implemented $U\left(z\right)$ (orange points Figure 1b).

^{2}. We found that this threshold value was sufficient to ensure a good estimation of the long-time diffusion coefficient.

#### 2.3. Relationship between $logP$ and $\u2329D\u232a$

#### 2.4. Amino Acids Simulations

^{2}/ns (obtained for Trp) and 1.3 nm

^{2}/ns (obtained for Cys), with an average equal to 1.04 nm

^{2}/ns. As in the toy model, the value of ${D}_{lip}$ was set to ${D}_{lip}=0.09{D}_{wat}$ for each residue.

## 3. Results

#### 3.1. Assessing the Importance of Each Physical Ingredient

#### 3.1.1. Impact of $\Delta {U}_{b}$

^{2}/ns, so as to isolate the effect of $\Delta {U}_{b}$ alone. Examples of potentials $U\left(z\right)$ are reported in the insets of Figure 2. In the figure, the main plots show the mean-square displacement (MSD) for selected values of $\Delta {U}_{b}$, in order to highlight the role played by the magnitude and sign of this parameter. Particularly, in Figure 2a, we show the MSD for $\Delta {U}_{b}=-4{k}_{B}T$ (green squares) and $\Delta {U}_{b}=4{k}_{B}T$ (orange circles). The empty symbols correspond to the MSD computed along planes parallel to the lipid/water interface, MSD

_{‖}. Since in the considered systems the diffusion coefficient is constant throughout space, lateral diffusion is unaffected by the value of $U\left(z\right)$. Therefore, MSD

_{‖}shows a standard diffusive behavior characterized by a diffusion coefficient ${D}_{\Vert}={D}_{wat}$, i.e., MSD

_{‖}$=4{D}_{wat}t$ (grey dashed line in Figure 2a). In a log-log plot such as the ones considered in the figure, this corresponds to a linear function with slope one shifted according to the value of ${D}_{\Vert}$.

#### 3.1.2. Impact of $\Delta U$

#### 3.1.3. Impact of ${D}_{lip}/{D}_{wat}$

#### 3.2. Putting the Physical Ingredients Together

#### 3.3. Application: Large-Scale Transport of Amino Acids

^{2}/ns), which indicates that ${D}_{\Vert}\gg {D}_{\perp}$ for large enough barriers, hence suggesting parallel diffusion to be dominant in such scenario. Similar to ${D}_{\perp}$, also in this case, we find a clustering of the points according to the physico-chemical properties of the residues. Particularly, apolar residues (green empty squares) have a stronger affinity for the lipid phase, as denoted by the larger values of $logP$ (taken with sign). This implies that a larger fraction of time is spent in the slowly-diffusing region corresponding to the lipids, thus yielding low values of ${D}_{\Vert}$. Similarly, polar residues (red empty circles) have lower affinity for lipids, thus resulting in faster parallel diffusion. Unintuitive results are obtained instead for charged residues (golden empty stars), for which one would imagine a strong depletion from the lipid phase. While Glu (E) and Asp (D) abide by the expected behavior (low affinity for lipids, a large value of ${D}_{\Vert}$), quite surprisingly, Lys (K) and Arg (R) show instead the opposite behavior. An inspection of the corresponding $U\left(z\right)$ profiles for these residues (Figure 7a) reveals the presence of a deep well ($\simeq -10{k}_{B}T$) in correspondence with the lipid heads. Thus, despite being strongly depleted from the lipid tails, Lys and Arg spend most of the time sitting at the lipid/water interface, which is characterized by slow diffusion. In terms of the toy model, one can locate these residues in Figure 6a in correspondence with $\Delta U>0$ and $\Delta {U}_{b}<0$, both with large magnitudes. However, it is important to stress that the quantitative value of ${D}_{\Vert}$ for Lys and Arg (and more in general for all the amino acids with positive values of $logP$) strongly depends on the profile chosen for $D\left(z\right)$ (Figure 1c) and on the value of ${D}_{lip}$, so that a more precise quantitative estimation requires a proper determination of $D\left(z\right)$ from ad hoc atomistic simulations. The cases of Lys and Arg are also an instructive example of how the size of barriers (${U}_{max}-{U}_{min}$) and the hydropathy of the molecule ($logP$) are not necessarily correlated with each other.

## 4. Discussion

- The lower boundary of ${D}_{\Vert}$ is equal to ${D}_{lip}$, obtained for highly-hydrophobic guest molecules. Together with point 1, this indicates that, for large enough barriers, parallel diffusion is dominant.

_{a}values close to the pH of the solution. Indeed, in this case, the possibility of dynamic protonation has to be taken into account. For instance, for the charged amino acids in Figure 7a, we considered for $U\left(z\right)$ the value corresponding to the charged or neutral state according to their relative thermodynamic stability, which changes with z [63]. The best practice is to run constant-pH atomistic simulations [81], which yield directly the correct free-energy profile [64]. However, we believe that, in practice, this will result in small changes in the predicted transport values, as a molecule with preference for charged states (in pure water) tends to be depleted by the hydropobic center of the lipid bilayer, thus resulting in a high free-energy barrier for bilayer permeation [82]. In turn, this yields negligible values for ${D}_{\perp}$ (Figure 7b), making parallel diffusion the dominant transport mechanism. While charge neutralization lowers the barrier at the center of the bilayer, it is expected that such barrier is still present and large in magnitude [82]; hence, while promoting perpendicular diffusion, the value of ${D}_{\perp}$ is still expected to be too small to significantly affect overall diffusion.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Briggs, J.; Chung, H.; Caffrey, M. The temperature-composition phase diagram and mesophase structure characterization of the monoolein/water system. J. Phys. II
**1996**, 6, 723–751. [Google Scholar] - Templer, R.; Seddon, J.; Warrender, N.; Syrykh, A.; Huang, Z.; Winter, R.; Erbes, J. Inverse bicontinuous cubic phases in 2: 1 fatty acid/phosphatidylcholine mixtures. The effects of chain length, hydration, and temperature. J. Phys. Chem. B
**1998**, 102, 7251–7261. [Google Scholar] - Tyler, A.I.; Barriga, H.M.; Parsons, E.S.; McCarthy, N.L.; Ces, O.; Law, R.V.; Seddon, J.M.; Brooks, N.J. Electrostatic swelling of bicontinuous cubic lipid phases. Soft Matter
**2015**, 11, 3279–3286. [Google Scholar] [PubMed] - Mezzenga, R.; Meyer, C.; Servais, C.; Romoscanu, A.I.; Sagalowicz, L.; Hayward, R.C. Shear rheology of lyotropic liquid crystals: A case study. Langmuir
**2005**, 21, 3322–3333. [Google Scholar] - Negrini, R.; Mezzenga, R. Diffusion, molecular separation, and drug delivery from lipid mesophases with tunable water channels. Langmuir
**2012**, 28, 16455–16462. [Google Scholar] [PubMed] - Barauskas, J.; Johnsson, M.; Tiberg, F. Self-assembled lipid superstructures: Beyond vesicles and liposomes. Nano Lett.
**2005**, 5, 1615–1619. [Google Scholar] - Turner, D.R.; Wang, Z.G.; Gruner, S.; Mannock, D.; Mcelhaney, R. Structural study of the inverted cubic phases of di-dodecyl alkyl-β-D-glucopyranosyl-rac-glycerol. J. Phys. II
**1992**, 2, 2039–2063. [Google Scholar] [CrossRef] - Salvati Manni, L.; Assenza, S.; Duss, M.; Vallooran, J.J.; Juranyi, F.; Jurt, S.; Zerbe, O.; Landau, E.M.; Mezzenga, R. Soft biomimetic nanoconfinement promotes amorphous water over ice. Nat. Nanotechnol.
**2019**, 14, 609–615. [Google Scholar] - Landau, E.M.; Rosenbusch, J.P. Lipidic cubic phases: A novel concept for the crystallization of membrane proteins. Proc. Natl. Acad. Sci. USA
**1996**, 93, 14532–14535. [Google Scholar] - Mulet, X.; Boyd, B.J.; Drummond, C.J. Advances in drug delivery and medical imaging using colloidal lyotropic liquid crystalline dispersions. J. Colloid Interface Sci.
**2013**, 393, 1–20. [Google Scholar] - Mezzenga, R.; Schurtenberger, P.; Burbidge, A.; Michel, M. Understanding foods as soft materials. Nat. Mater.
**2005**, 4, 729–740. [Google Scholar] [CrossRef] [PubMed] - Fong, C.; Le, T.; Drummond, C.J. Lyotropic liquid crystal engineering–ordered nanostructured small molecule amphiphile self-assembly materials by design. Chem. Soc. Rev.
**2012**, 41, 1297–1322. [Google Scholar] [PubMed] - Speziale, C.; Salvati Manni, L.; Manatschal, C.; Landau, E.M.; Mezzenga, R. A macroscopic H+ and Cl− ions pump via reconstitution of EcClC membrane proteins in lipidic cubic mesophases. Proc. Natl. Acad. Sci. USA
**2016**, 113, 7491–7496. [Google Scholar] [PubMed] - Assenza, S.; Mezzenga, R. Soft condensed matter physics of foods and macronutrients. Nat. Rev. Phys.
**2019**, 1, 551–566. [Google Scholar] [CrossRef] - Yao, Y.; Zhou, T.; Färber, R.; Grossner, U.; Floudas, G.; Mezzenga, R. Designing cryo-enzymatic reactions in subzero liquid water by lipidic mesophase nanoconfinement. Nat. Nanotechnol.
**2021**, 16, 802–810. [Google Scholar] - Boyd, B.J.; Clulow, A.J. The influence of lipid digestion on the fate of orally administered drug delivery vehicles. Biochem. Soc. Trans.
**2021**, 49, 1749–1761. [Google Scholar] - Negrini, R.; Mezzenga, R. pH-responsive lyotropic liquid crystals for controlled drug delivery. Langmuir
**2011**, 27, 5296–5303. [Google Scholar] [CrossRef] - Martiel, I.; Baumann, N.; Vallooran, J.J.; Bergfreund, J.; Sagalowicz, L.; Mezzenga, R. Oil and drug control the release rate from lyotropic liquid crystals. J. Control. Release
**2015**, 204, 78–84. [Google Scholar] - Zabara, A.; Mezzenga, R. Controlling molecular transport and sustained drug release in lipid-based liquid crystalline mesophases. J. Control. Release
**2014**, 188, 31–43. [Google Scholar] - Negrini, R.; Fong, W.K.; Boyd, B.J.; Mezzenga, R. pH-responsive lyotropic liquid crystals and their potential therapeutic role in cancer treatment. Chem. Commun.
**2015**, 51, 6671–6674. [Google Scholar] [CrossRef] - Nazaruk, E.; Miszta, P.; Filipek, S.; Gorecka, E.; Landau, E.M.; Bilewicz, R. Lyotropic cubic phases for drug delivery: Diffusion and sustained release from the mesophase evaluated by electrochemical methods. Langmuir
**2015**, 31, 12753–12761. [Google Scholar] [CrossRef] [PubMed] - Clogston, J.; Craciun, G.; Hart, D.; Caffrey, M. Controlling release from the lipidic cubic phase by selective alkylation. J. Control. Release
**2005**, 102, 441–461. [Google Scholar] [CrossRef] [PubMed] - Lee, K.W.; Nguyen, T.H.; Hanley, T.; Boyd, B.J. Nanostructure of liquid crystalline matrix determines in vitro sustained release and in vivo oral absorption kinetics for hydrophilic model drugs. Int. J. Pharm.
**2009**, 365, 190–199. [Google Scholar] [PubMed] - Phan, S.; Fong, W.K.; Kirby, N.; Hanley, T.; Boyd, B.J. Evaluating the link between self-assembled mesophase structure and drug release. Int. J. Pharm.
**2011**, 421, 176–182. [Google Scholar] [CrossRef] [PubMed] - Zabara, A.; Negrini, R.; Onaca-Fischer, O.; Mezzenga, R. Perforated Bicontinuous Cubic Phases with pH-Responsive Topological Channel Interconnectivity. Small
**2013**, 9, 3602–3609. [Google Scholar] [CrossRef] - Meikle, T.G.; Yao, S.; Zabara, A.; Conn, C.E.; Drummond, C.J.; Separovic, F. Predicting the release profile of small molecules from within the ordered nanostructured lipidic bicontinuous cubic phase using translational diffusion coefficients determined by PFG-NMR. Nanoscale
**2017**, 9, 2471–2478. [Google Scholar] - Ghanbari, R.; Assenza, S.; Saha, A.; Mezzenga, R. Diffusion of polymers through periodic networks of lipid-based nanochannels. Langmuir
**2017**, 33, 3491–3498. [Google Scholar] - Assenza, S.; Mezzenga, R. Curvature and bottlenecks control molecular transport in inverse bicontinuous cubic phases. J. Chem. Phys.
**2018**, 148, 054902. [Google Scholar] [CrossRef] - Ghanbari, R.; Assenza, S.; Mezzenga, R. The interplay of channel geometry and molecular features determines diffusion in lipidic cubic phases. J. Chem. Phys.
**2019**, 150, 094901. [Google Scholar] - Ghanbari, R.; Assenza, S.; Zueblin, P.; Mezzenga, R. Impact of Molecular Partitioning and Partial Equilibration on the Estimation of Diffusion Coefficients from Release Experiments. Langmuir
**2019**, 35, 5663–5671. [Google Scholar] [CrossRef] - Salvati Manni, L.; Duss, M.; Assenza, S.; Boyd, B.J.; Landau, E.M.; Fong, W.K. Enzymatic hydrolysis of monoacylglycerols and their cyclopropanated derivatives: Molecular structure and nanostructure determine the rate of digestion. J. Colloid Interface Sci.
**2021**, 588, 767–775. [Google Scholar] [PubMed] - Chang, C.; Meikle, T.G.; Drummond, C.J.; Yang, Y.; Conn, C.E. Comparison of cubosomes and liposomes for the encapsulation and delivery of curcumin. Soft Matter
**2021**, 17, 3306–3313. [Google Scholar] [CrossRef] [PubMed] - Rajesh, S.; Zhai, J.; Drummond, C.J.; Tran, N. Novel pH-Responsive Cubosome and Hexosome Lipid Nanocarriers of SN-38 Are Prospective for Cancer Therapy. Pharmaceutics
**2022**, 14, 2175. [Google Scholar] [CrossRef] [PubMed] - Dubbeldam, D.; Snurr, R. Recent developments in the molecular modeling of diffusion in nanoporous materials. Mol. Simul.
**2007**, 33, 305–325. [Google Scholar] [CrossRef] - Krishna, R. Diffusion in porous crystalline materials. Chem. Soc. Rev.
**2012**, 41, 3099–3118. [Google Scholar] [PubMed] - Bagchi, B. Water dynamics in the hydration layer around proteins and micelles. Chem. Rev.
**2005**, 105, 3197–3219. [Google Scholar] [PubMed] - Laage, D.; Elsaesser, T.; Hynes, J.T. Water dynamics in the hydration shells of biomolecules. Chem. Rev.
**2017**, 117, 10694–10725. [Google Scholar] - Bourg, I.C.; Steefel, C.I. Molecular dynamics simulations of water structure and diffusion in silica nanopores. J. Phys. Chem. C
**2012**, 116, 11556–11564. [Google Scholar] - Hande, V.R.; Chakrabarty, S. How Far Is “Bulk Water” from Interfaces? Depends on the Nature of the Surface and What We Measure. J. Phys. Chem. B
**2022**, 126, 1125–1135. [Google Scholar] - Vallooran, J.J.; Assenza, S.; Mezzenga, R. Spatiotemporal control of enzyme-induced crystallization under lyotropic liquid crystal nanoconfinement. Angew. Chem.
**2019**, 131, 7367–7371. [Google Scholar] - Kasim, N.A.; Whitehouse, M.; Ramachandran, C.; Bermejo, M.; Lennernäs, H.; Hussain, A.S.; Junginger, H.E.; Stavchansky, S.A.; Midha, K.K.; Shah, V.P.; et al. Molecular properties of WHO essential drugs and provisional biopharmaceutical classification. Mol. Pharm.
**2004**, 1, 85–96. [Google Scholar] [PubMed] - Takagi, T.; Ramachandran, C.; Bermejo, M.; Yamashita, S.; Yu, L.X.; Amidon, G.L. A provisional biopharmaceutical classification of the top 200 oral drug products in the United States, Great Britain, Spain, and Japan. Mol. Pharm.
**2006**, 3, 631–643. [Google Scholar] [CrossRef] [PubMed] - Eason, T.; Ramirez, G.; Clulow, A.J.; Salim, M.; Boyd, B.J. Revisiting the Dissolution of Praziquantel in Biorelevant Media and the Impact of Digestion of Milk on Drug Dissolution. Pharmaceutics
**2022**, 14, 2228. [Google Scholar] [PubMed] - Kim, J.; Lu, W.; Qiu, W.; Wang, L.; Caffrey, M.; Zhong, D. Ultrafast hydration dynamics in the lipidic cubic phase: Discrete water structures in nanochannels. J. Phys. Chem. B
**2006**, 110, 21994–22000. [Google Scholar] [CrossRef] - Helfrich, W. Effect of thermal undulations on the rigidity of fluid membranes and interfaces. J. Phys.
**1985**, 46, 1263–1268. [Google Scholar] [CrossRef] - Angelov, B.; Ollivon, M.; Angelova, A. X-ray diffraction study of the effect of the detergent octyl glucoside on the structure of lamellar and nonlamellar lipid/water phases of use for membrane protein reconstitution. Langmuir
**1999**, 15, 8225–8234. [Google Scholar] - Nademi, Y.; Amjad Iranagh, S.; Yousefpour, A.; Mousavi, S.Z.; Modarress, H. Molecular dynamics simulations and free energy profile of Paracetamol in DPPC and DMPC lipid bilayers. J. Chem. Sci.
**2014**, 126, 637–647. [Google Scholar] - Zhou, T.; Vallooran, J.J.; Assenza, S.; Szekrenyi, A.; Clapés, P.; Mezzenga, R. Efficient asymmetric synthesis of carbohydrates by aldolase nano-confined in lipidic cubic mesophases. ACS Catal.
**2018**, 8, 5810–5815. [Google Scholar] - Ma, Y.; Zhu, C.; Ma, P.; Yu, K. Studies on the diffusion coefficients of amino acids in aqueous solutions. J. Chem. Eng. Data
**2005**, 50, 1192–1196. [Google Scholar] - Longsworth, L. Diffusion measurements, at 25, of aqueous solutions of amino acids, peptides and sugars. J. Am. Chem. Soc.
**1953**, 75, 5705–5709. [Google Scholar] - Ribeiro, A.C.; Barros, M.C.; Verissimo, L.M.; Lobo, V.M.; Valente, A.J. Binary diffusion coefficients for aqueous solutions of l-aspartic acid and its respective monosodium salt. J. Solut. Chem.
**2014**, 43, 83–92. [Google Scholar] [CrossRef] - Ribeiro, A.C.; Rodrigo, M.; Barros, M.C.; Verissimo, L.M.; Romero, C.; Valente, A.J.; Esteso, M.A. Mutual diffusion coefficients of L-glutamic acid and monosodium L-glutamate in aqueous solutions at T = 298.15 K. J. Chem. Thermodyn.
**2014**, 74, 133–137. [Google Scholar] - Umecky, T.; Ehara, K.; Omori, S.; Kuga, T.; Yui, K.; Funazukuri, T. Binary diffusion coefficients of aqueous phenylalanine, tyrosine isomers, and aminobutyric acids at infinitesimal concentration and temperatures from (293.2 to 333.2) K. J. Chem. Eng. Data
**2013**, 58, 1909–1917. [Google Scholar] - Mendes, F.S.; Cruz, C.E.; Martins, R.N.; Ramalho, J.P.P.; Martins, L.F. On the diffusion of ketoprofen and ibuprofen in water: An experimental and theoretical approach. J. Chem. Thermodyn.
**2022**, 2022, 106955. [Google Scholar] [CrossRef] - Edwards, L. The dissolution and diffusion of aspirin in aqueous media. Trans. Faraday Soc.
**1951**, 47, 1191–1210. [Google Scholar] [CrossRef] - Ribeiro, A.C.; Barros, M.C.; Veríssimo, L.M.; Santos, C.I.; Cabral, A.M.; Gaspar, G.D.; Esteso, M.A. Diffusion coefficients of paracetamol in aqueous solutions. J. Chem. Thermodyn.
**2012**, 54, 97–99. [Google Scholar] [CrossRef] - Antognini, L.M.; Assenza, S.; Speziale, C.; Mezzenga, R. Quantifying the transport properties of lipid mesophases by theoretical modelling of diffusion experiments. J. Chem. Phys.
**2016**, 145, 084903. [Google Scholar] - Anderson, D.M.; Wennerstroem, H. Self-diffusion in bicontinuous cubic phases, L3 phases, and microemulsions. J. Phys. Chem.
**1990**, 94, 8683–8694. [Google Scholar] [CrossRef] - Doi, M.; Edwards, S.F.; Edwards, S.F. The Theory of Polymer Dynamics; Oxford University Press: Oxford, UK, 1988; Volume 73. [Google Scholar]
- Van Kampen, N.G. Stochastic Processes in Physics and Chemistry; Elsevier: Amsterdam, The Netherlands, 1992; Volume 1. [Google Scholar]
- Marsaglia, G.; Tsang, W.W. The ziggurat method for generating random variables. J. Stat. Softw.
**2000**, 5, 1–7. [Google Scholar] [CrossRef] - Martin, Y.C. Quantitative Drug Design: A Critical Introduction; CRC Press: Boca Raton, FL, USA, 2010. [Google Scholar]
- MacCallum, J.L.; Bennett, W.D.; Tieleman, D.P. Distribution of amino acids in a lipid bilayer from computer simulations. Biophys. J.
**2008**, 94, 3393–3404. [Google Scholar] [CrossRef] - Yue, Z.; Li, C.; Voth, G.A.; Swanson, J.M. Dynamic protonation dramatically affects the membrane permeability of drug-like molecules. J. Am. Chem. Soc.
**2019**, 141, 13421–13433. [Google Scholar] [PubMed] - Koirala, R.P.; Bhusal, H.P.; Khanal, S.P.; Adhikari, N.P. Effect of temperature on transport properties of cysteine in water. AIP Adv.
**2020**, 10, 025122. [Google Scholar] - Rodrigo, M.M.; Valente, A.J.; Barros, M.C.; Verissimo, L.M.; Romero, C.; Esteso, M.A.; Ribeiro, A.C. Mutual diffusion coefficients of L-lysine in aqueous solutions. J. Chem. Thermodyn.
**2014**, 74, 227–230. [Google Scholar] [CrossRef] - Heller, W.T. Small-Angle Neutron Scattering for Studying Lipid Bilayer Membranes. Biomolecules
**2022**, 12, 1591. [Google Scholar] [PubMed] - Johansson, A.C.; Lindahl, E. Position-resolved free energy of solvation for amino acids in lipid membranes from molecular dynamics simulations. Proteins Struct. Funct. Bioinform.
**2008**, 70, 1332–1344. [Google Scholar] [CrossRef] [PubMed] - Zhu, F.; Hummer, G. Theory and simulation of ion conduction in the pentameric GLIC channel. J. Chem. Theory Comput.
**2012**, 8, 3759–3768. [Google Scholar] [CrossRef] [PubMed] - Vallooran, J.J.; Bolisetty, S.; Mezzenga, R. Macroscopic alignment of lyotropic liquid crystals using magnetic nanoparticles. Adv. Mater.
**2011**, 23, 3932–3937. [Google Scholar] [PubMed] - Boggara, M.B.; Krishnamoorti, R. Partitioning of nonsteroidal antiinflammatory drugs in lipid membranes: A molecular dynamics simulation study. Biophys. J.
**2010**, 98, 586–595. [Google Scholar] [PubMed] - Keyvanfard, M.; Hatami, M.; Gupta, V.K.; Agarwal, S.; Sadeghifar, H.; Khalilzadeh, M.A. Liquid phase analysis of methyldopa in the presence of tyrosine using electrocatalytic effect of a catechol derivative at a surface of NiO nanoparticle modified carbon paste electrode. J. Mol. Liq.
**2017**, 230, 290–294. [Google Scholar] [CrossRef] - Fagerholm, U.; Lennernäs, H. Experimental estimation of the effective unstirred water layer thickness in the human jejunum, and its importance in oral drug absorption. Eur. J. Pharm. Sci.
**1995**, 3, 247–253. [Google Scholar] [CrossRef] - Lu, Y.; Li, M. Simultaneous rapid determination of the solubility and diffusion coefficients of a poorly water-soluble drug based on a novel UV imaging system. J. Pharm. Sci.
**2016**, 105, 131–138. [Google Scholar] [PubMed] - Vilt, M.E.; Ho, W.W. Supported liquid membranes with strip dispersion for the recovery of Cephalexin. J. Membr. Sci.
**2009**, 342, 80–87. [Google Scholar] [CrossRef] - Tajik, S.; Beitollahi, H.; Aflatoonian, M.R.; Mohtat, B.; Aflatoonian, B.; Shoaie, I.S.; Khalilzadeh, M.A.; Ziasistani, M.; Zhang, K.; Jang, H.W.; et al. Fabrication of magnetic iron oxide-supported copper oxide nanoparticles (Fe
_{3}O_{4}/CuO): Modified screen-printed electrode for electrochemical studies and detection of desipramine. RSC Adv.**2020**, 10, 15171–15178. [Google Scholar] [CrossRef] - Beitollahi, H.; Hamzavi, M.; Torkzadeh-Mahani, M. Electrochemical determination of hydrochlorothiazide and folic acid in real samples using a modified graphene oxide sheet paste electrode. Mater. Sci. Eng. C
**2015**, 52, 297–305. [Google Scholar] [CrossRef] - Pyka, A. Lipophilicity investigations of ibuprofen. J. Liquid Chromatogr. Related Technol.
^{®}**2009**, 32, 723–731. [Google Scholar] [CrossRef] - Barros, M.C.; Ribeiro, A.C.; Esteso, M.A.; Lobo, V.M.; Leaist, D.G. Diffusion of levodopa in aqueous solutions of hydrochloric acid at 25 °C. J. Chem. Thermodyn.
**2014**, 72, 44–47. [Google Scholar] [CrossRef] - Ye, F.; Yaghmur, A.; Jensen, H.; Larsen, S.W.; Larsen, C.; Østergaard, J. Real-time UV imaging of drug diffusion and release from Pluronic F127 hydrogels. Eur. J. Pharm. Sci.
**2011**, 43, 236–243. [Google Scholar] - Radak, B.K.; Chipot, C.; Suh, D.; Jo, S.; Jiang, W.; Phillips, J.C.; Schulten, K.; Roux, B. Constant-pH molecular dynamics simulations for large biomolecular systems. J. Chem. Theory Comput.
**2017**, 13, 5933–5944. [Google Scholar] [CrossRef] - Johansson, A.C.; Lindahl, E. Titratable amino acid solvation in lipid membranes as a function of protonation state. J. Phys. Chem. B
**2009**, 113, 245–253. [Google Scholar] - Enkavi, G.; Javanainen, M.; Kulig, W.; Róg, T.; Vattulainen, I. Multiscale simulations of biological membranes: The challenge to understand biological phenomena in a living substance. Chem. Rev.
**2019**, 119, 5607–5774. [Google Scholar] - Prajapati, J.D.; Kleinekathöfer, U.; Winterhalter, M. How to enter a bacterium: Bacterial porins and the permeation of antibiotics. Chem. Rev.
**2021**, 121, 5158–5192. [Google Scholar] [PubMed]

**Figure 1.**General features of the theoretical model. (

**a**) sketch of a repeating unit in a lamellar mesophase and definition of the main parameters; molecules are not at scale; (

**b**) representative periodic potential of mean force $U\left(z\right)$ corresponding to the system in (

**a**). In this case, $U\left(z\right)$ corresponds to a hydrophilic molecule ($\Delta U>0$) with low affinity for the lipid heads ($\Delta {U}_{b}>0$). The orange circles correspond to the free energy extracted from a control simulation to determine the optimal timestep, as described in the Methods; (

**c**) representative periodic, position-dependent diffusion coefficient $D\left(z\right)$ corresponding to the system in (

**a**), smoothly changing from ${D}_{lip}$ within the lipid phase to ${D}_{wat}$ in the water phase far from the lipid/water interface.

**Figure 2.**Representative evolution of mean-square displacement as a function of time for $\left(\right)open="|"\; close="|">\Delta {U}_{b}$ (

**a**) and $\left(\right)open="|"\; close="|">\Delta {U}_{b}$ (

**b**), while $\Delta U=0$ and ${D}_{lip}={D}_{wat}$. In both panels, filled and empty symbols correspond to $MS{D}_{\perp}$ and $MS{D}_{\Vert}$, respectively, as reported in the legends. The dashed lines correspond to the formula $4{D}_{wat}t$. In the insets, the potential of mean force for each case is reported.

**Figure 3.**(

**a**) Dependence of long-time diffusion coefficients on the size of the barriers $\left(\right)$, for $\Delta U=0$ and ${D}_{lip}={D}_{wat}$. The horizontal dashed line corresponds to ${D}_{wat}=0.7$ nm

^{2}/ns. The red continuous line is obtained by fitting the values for $\left(\right)open="|"\; close="|">\Delta {U}_{b}$ via an exponential function. In the inset, we report a representative potential of mean force $U\left(z\right)$ for this study, corresponding to $\Delta {U}_{b}=4{k}_{B}T$; (

**b**) same as (

**a**), but focusing on varying $\left(\right)$ in systems with $\Delta {U}_{b}=0$ and ${D}_{lip}={D}_{wat}$. In the inset, we report $U\left(z\right)$ for $\Delta U=4{k}_{B}T$.

**Figure 4.**Effective diffusion coefficients in a system with position-dependent $D\left(z\right)$ and no external potential. The red continuous line is the prediction from Equation (12). The blue dashed line corresponds to a fit of the simulation data by a power law.

**Figure 5.**Dependence of the perpendicular long-time diffusion coefficient ${D}_{\perp}$ on hydropathy ($\Delta U$) and affinity for lipid heads ($\Delta {U}_{b}$), implemented according to the corresponding potential of mean force $U\left(z\right)$ (Figure 1a); (

**a**) is obtained by assuming ${D}_{lip}={D}_{wat}$, while (

**b**) considers a more realistic diffusion profile with ${D}_{lip}=0.09{D}_{wat}$ (Figure 1c). The contours correspond to the indicated values of ${D}_{\perp}/{D}_{wat}$.

**Figure 6.**(

**a**) Dependence of the parallel long-time diffusion coefficient ${D}_{\Vert}$ on hydropathy ($\Delta U$) and affinity for lipid heads ($\Delta {U}_{b}$), implemented according to the corresponding potential of mean force $U\left(z\right)$ (Figure 1a). The diffusion profile corresponds to Figure 1c with ${D}_{lip}=0.09{D}_{wat}$; (

**b**) dependence of ${D}_{\Vert}/{D}_{wat}$ on $logP$ for the simulation data (golden stars) and for an extended set of systems (purple triangles), for which the lateral diffusion coefficient was computed according to Equation (13). The extended systems with $logP<0$ were obtained by considering $\Delta U=\Delta {U}_{b}=20{k}_{B}T,200{k}_{B}T,2000{k}_{B}T$; the extended systems with $logP>0$ were obtained by setting $\Delta U=\Delta {U}_{b}=-10{k}_{B}T,-9{k}_{B}T,-8{k}_{B}T,-7{k}_{B}T,-6{k}_{B}T$. The red continuous line corresponds to Equation (14).

**Figure 7.**(

**a**) Potential of mean force for 16 amino acids as computed in Ref. [63]. The plots were colour-coded according to the physico-chemical properties of the side chain: gold ↔ charged, red ↔ polar and green ↔ apolar; (

**b**) perpendicular diffusion coefficient ${D}_{\perp}$ for charged (filled gold stars), polar (filled red circles) and apolar (filled green squares) residues as a function of the difference ${U}_{max}-{U}_{min}$ between maximum and minimum height of the corresponding potential of mean force. In the inset, the full range of ${U}_{max}-{U}_{min}$ is considered to include the case of Arg, for which ${U}_{max}-{U}_{min}\simeq 32{k}_{B}T$. The dashed grey line is a best fit via an exponential decay; (

**c**) parallel diffusion coefficient ${D}_{\Vert}$ for charged (empty gold stars), polar (empty red circles) and apolar (empty green squares) residues as a function of the logarithm of the partition coefficient $logP$. The dashed grey line corresponds to the theoretical prediction according to Equation (14).

**Figure 8.**Approximation error on ${D}_{\mathrm{eff}}$ when employing Equation (15) for the toy model (

**a**) and the amino-acids simulations (

**b**). The error is computed as $100\xb7\left(\right)open="|"\; close="|">1{D}_{\mathrm{eff},\mathrm{pred}}/{D}_{\mathrm{eff},\mathrm{sim}}$, where ${D}_{\mathrm{eff},\mathrm{pred}}$ is the predicted value according to Equation (15), and ${D}_{\mathrm{eff},\mathrm{sim}}$ is computed directly from the simulations.

**Table 1.**Estimated values of effective diffusion coefficient for release of various drugs from a lamellar mesophase at 43 °C with geometric parameters chosen from the literature. The values of ${D}_{wat}$ were obtained by renormalizing experimental values obtained at different temperatures via the Stokes–Einstein equation, as discussed in the Methods [57]. Experimental values of ${D}_{wat}$ and $logP$ were taken from Refs. [42,54,72,73,74,75,76,77,78,79,80].

Name | $log\mathit{P}$ | ${\mathit{D}}_{\mathit{wat}}$ (nm^{2}/ns) | ${\mathit{D}}_{\mathbf{eff}}$ (nm^{2}/ns) |
---|---|---|---|

Cephalexin | −0.67 | 0.70 | 0.27 |

Hydrochlorothiazide | −0.15 | 1.69 | 0.43 |

Levodopa | 0.00 | 0.95 | 0.21 |

Piroxicam | 0.29 | 0.85 | 0.14 |

Methyldopa | 0.39 | 1.14 | 0.18 |

Paracetamol | 0.46 | 1.06 | 0.15 |

Antipyrine | 1.01 | 1.04 | 0.11 |

Carbamazepine | 2.93 | 1.13 | 0.10 |

Ketoprofen | 3.31 | 0.67 | 0.06 |

Desipramine | 3.94 | 0.46 | 0.04 |

Ibuprofen | 3.99 | 0.77 | 0.07 |

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**MDPI and ACS Style**

Bosch, A.M.; Assenza, S.
Interplay of Hydropathy and Heterogeneous Diffusion in the Molecular Transport within Lamellar Lipid Mesophases. *Pharmaceutics* **2023**, *15*, 573.
https://doi.org/10.3390/pharmaceutics15020573

**AMA Style**

Bosch AM, Assenza S.
Interplay of Hydropathy and Heterogeneous Diffusion in the Molecular Transport within Lamellar Lipid Mesophases. *Pharmaceutics*. 2023; 15(2):573.
https://doi.org/10.3390/pharmaceutics15020573

**Chicago/Turabian Style**

Bosch, Antonio M., and Salvatore Assenza.
2023. "Interplay of Hydropathy and Heterogeneous Diffusion in the Molecular Transport within Lamellar Lipid Mesophases" *Pharmaceutics* 15, no. 2: 573.
https://doi.org/10.3390/pharmaceutics15020573