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The potential of mean force of nitrous oxide in a 1,2-dimyristoylphosphatidylcholine lipid bilayer
Chemical Physics Letters 489 (2010) 96–98
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
The potential of mean force of nitrous oxide in a 1,2-dimyristoylphosphatidylcholine lipid bilayer Eric R. Pinnick a, Shyamsunder Erramilli a,b, Feng Wang c,* a
Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, United States Department of Biomedical Engineering and the Photonics Center, Boston University, 8 Saint Mary’s Street, Boston, MA 02215, United States c Department of Chemistry, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, United States b
a r t i c l e
i n f o
Article history: Received 7 January 2010 In final form 16 February 2010 Available online 19 February 2010
a b s t r a c t The free-energy profile for N2O in a hydrated 1,2-dimyristoylphosphatidylcholine (DMPC) phospholipid bilayer is calculated. The N2O molecule is preferentially concentrated in the tail region of the DMPC lipid and there is no free-energy barrier for the diffusion of N2O from an aqueous environment into the lipid. Although the dipole moment of N2O is rather small, our calculation indicates that the N2O partition coefficient is strongly influenced by electrostatic interactions. On the other hand, the van der Waals interaction overwhelms the steric effect and causes N2O concentration to be greatest at a location close to the bcarbon of the 1-myristoyl group. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Nitrous oxide, also called the laughing gas, is perhaps the oldest drug in continuous use in modern pharmacology. It has generally been regarded as a safe and effective anesthetic for more than one and a half centuries. However, the mechanism of the anesthetic action for this linear triatomic molecule is far from being understood. Mechanisms have been proposed that suggest N2O acts primarily either by interacting with the lipid membrane or by binding to specific proteins, especially channel proteins [1–5]. Previous studies suggest that mutating a key membrane integrated protein, syntaxin, confers resistance to volatile general anesthetics. These studies seem to indicate a protein-based mechanism [6]. However, it is possible N2O interacts with trans-membrane proteins indirectly by modulating physical properties of the membrane [1,7]. Regardless of the actual mechanism, understanding the free-energy profile for N2O penetration and partitioning between lipid and water is an important step toward elucidating the function of N2O as a general anesthetic. After all, N2O has to penetrate through membranes before it reaches specific binding sites. The penetration barrier correlates strongly with the action rate and the partition coefficient determines the equilibrium concentration of N2O in lipid. In addition to its role as an anesthetic, N2O serves as a promising probe molecule for biological systems. N2O is small in size and possesses a small dipole moment (0.166 D [8]). This allows N2O to access both polar and nonpolar solvent environments while causing a relatively small perturbation to the molecule being probed. The asymmetric stretch (v 3 ) of N2O at roughly 2215—2230 cm1
* Corresponding author. E-mail address: [email protected] (F. Wang). 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.02.047
has a large extinction coefficient (1:5 103 M1 cm1 ); most biomolecules and water on the other hand do not have strong adsorption in this spectral region. The vibrational lifetime of the asymmetric stretching mode, which can be readily probed by ultrafast infrared spectroscopy, is very sensitive to the solvation environment. In a recent work, the utility of N2O as a reporter molecule for the solvation dynamics of DOPC bilayers is demonstrated [9]. However, further interpretation of these results will benefit from an understanding of how N2O partitions and diffuses from aqueous to lipid regions. In order to address these concerns, we investigate the free-energy profile of N2O in hydrated 1,2-dimyristoylphosphatidylcholine (DMPC) lipids through molecular dynamics simulations. The freeenergy profile is reported as the potential of mean force (PMF) calculated using umbrella sampling [10] and the weighted histogram analysis method [11]. Such a free-energy profile gives information about specific binding sites, partition coefficients of N2O in DMPC and water, and the activation barrier for N2O penetration both into and across the lipid. This information provides physical insight into the equilibrium distribution of N2O in lipids and the readiness of N2O penetration through lipids to action sites, and is thus valuable for understanding the anesthetic action of N2O and for interpreting experimental studies on N2O as a probe molecule. We have also tried to elucidate the fundamental interaction that gives rise to such a free-energy profile for N2O. 2. Methods The PMF was calculated from a set of 40 independent molecular dynamics simulations using umbrella sampling and the weighted histogram analysis method. Simulations and analysis were
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3. The PMF of nitrous oxide The PMF of N2O is reported in Fig. 1. It is clear from this figure that N2O prefers to stay in the lipid tail region. The transport of N2O Table 1 Summary of N2O model.
a
0
−6
Density (arbitrary units)
PMF Gibbs Surface Carboxyl Density Phosphate Density β-Carbon Density
−3
PMF (kJ/mol)
performed with the GROMACS package [12,13]. The lipid–water bilayer was modeled in the La phase using the united-atom lipid force field and the previously-equilibrated bilayer system of Tieleman and collaborators [14] with SPC water [15]. Each simulation system consisted of 128 DMPC molecules fully hydrated by 3655 water molecules, along with a single N2O molecule. The water to lipid number ratio is 28.5, which is above the excess hydration point at the simulation temperature. The equilibrium box size is 6.15 6.15 6.64 nm, giving an average head group area of 59.1 Å2 with an equilibrium normal pressure of 1 bar. N2O geometry and charge values were fitted using the ab initio QCISD method [16] with the aug-cc-pVTZ basis set [17], and summarized in Table 1. The QCISD calculation predicted a N2O dipole moment of 0.166 D, which is identical to the experimental value. The atomic charges were determined to best reproduce the electrostatic potential of N2O while constraining the dipole moment. The Lennard-Jones parameters for N2O were taken from the modified GROMOS87 forcefield in Gromacs. The force constant and spacing of the umbrella potentials were optimized by trial simulations on a scaled-down bilayer containing 24 lipids, with the same average head group area and water to lipid number ratio as the full-sized system. A series of umbrella potential force constants ranging from 40 to 200 kJ=ðmol nm2 Þ and a series of window spacings from 1 to 2.5 Å were experimented to achieve a population overlap of at least 50% between adjacent windows. Based on these trial simulations, we selected an umbrella potential constant of 80 kJ=ðmol nm2 Þ and a spacing between umbrella potentials of 1.66 Å. This spacing leads to a total of 40 windows for the PMF study. One N2O molecule was inserted into each window. The xy position of the N2O was randomly chosen. N2O insertion was accomplished following the protocol of Goodfellow and collaborators [18] with a total equilibration time of 200 ps. Typically, there are two approaches to get a well-converged PMF. One approach is to perform long simulations with a relatively small number of windows. The second approach is to perform relatively short simulations but with a large number of windows. We took the second approach because a larger scale of parallelism can be achieved trivially with a large number of windows. We picked a total of 40 windows and simulated under the canonical ensemble for 4 ns with a time step of 1 fs in each window. Although 4 ns do not seem long, the standard error bars of our simulation have an average value of 0.4 kJ/mol and a maximum of 0.6 kJ/mol. This small error is achieved since the overlap between adjacent windows in our simulation ranges from 40% to 90%, with an average overlap of 75%. The simulations from adjacent windows thus help the sampling of the configuration space. During PMF simulations, a simulation temperature of 310 K was maintained by a Nosé–Hoover thermostat [19,20] with a thermostat time constant of 0.5 ps. Periodic boundary conditions were employed in all directions. Lennard-Jones interactions were calculated using a 14 Å cutoff. Electrostatic interactions were treated exactly using the particle mesh Ewald method [21].
−9 −12 −15 −18
−30
−20
−10
0
10
20
30
˚ Z-Axis Position (A) Fig. 1. PMF of N2O (red). The Gibbs surface (dot-dashed blue) and density profiles of the phosphate P (dashed black), carboxyl group C (dashed magenta), and b-carbon of the 1-myristoyl group (dashed green) are provided for reference. The curves have been averaged to reflect the symmetry of the box. The raw data look very close to the averaged curves. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
from water into the lipid is a barrier-less process. Once inside the lipid, the N2 O will be most stable at about 10 Å from the lipid central plane, which is on the tail side of the carboxyl group carbon. The position of this minimum coincides with the average location of the b-carbon of the 1-myristoyl tail. This is the same as the most favorable location for halothane in a dipalmitoylphosphatidylcholine bilayer [22] and that of benzocaine in a mixed phospholipid bilayer [23]. Furthermore, a ‘shoulder’ feature exists at around 18 Å from the lipid central plane, coinciding with the phosphorus density maximum. A shallow free-energy barrier resides at the lipid central plane. The barrier for N2 O to penetrate the tail region of DMPC is about 2 kJ/mol, which is comparable to kT at room temperature. This is much smaller than the free energy difference between the bulk water phase and the free-energy minima in the lipid, 16.4 ± 0.4 kJ/mol. Detailed partition information of N2O in hydrated lipid can be obtained from the PMF. Fig. 1 marks the Gibbs surface of interlammellar water, which almost coincides with the phosphorus density maximum [24]. We will separate the head and tail regions of the lipid by the carboxylate carbon density maxima. For N2O inside the Gibbs surface of water, 92% of the N2O is in the tail group region, while only 8% resides in the head group region. The Molar partition coefficient K for N2O partitioning between DMPC and water is calculated as the thermodynamic equilibrium constant for the reaction
N2 Oaq N2 Olipid :
ð1Þ
Obviously, the K depends on the overall hydration level of the lipid and how the water–lipid interface is defined. Using the Gibbs surface of water to separate the water phase and the lipid phase, the partition coefficient K for our excess hydration DMPC is 2.5 using molarity as units of concentration. We are not aware of an experimental partition coefficient for N2O between DMPC and water. The octanol–water experimental K value of 2.3 [25] and the olive oil–water K value of 2.97 [26] provide useful guidelines [4]. 4. The origins of the PMF features
Atom
q/e
Pos.a (Å)
r
(kJ/mol)
(Å)
N N0 O
0.239 0.523 0.284
1.12 0.000 1.19
0.8767 0.8767 0.8490
2.976 2.976 2.955
Position taken relative to central N atom, N0 .
N2O is a very labile molecule with good solubility in both water and phospholipids. It is worth mentioning that some other molecules such as halothane and benzocaine mentioned previously that have different relative solubility actually have the same relative site of enrichment. In order to identify the fundamental interaction
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E.R. Pinnick et al. / Chemical Physics Letters 489 (2010) 96–98
20
PMF (kJ/mol)
10 0 −10
N2O LJ-N2O
−20
HS-N2O N2O Min.
−30 −30
−15
0
15
30
˚ Z-Axis Position (A) Fig. 2. PMF of N2O (red), LJ-N2 O (blue), and HS-N2 O (green) in hydrated DMPC. Vertical lines correspond to the free-energy minima of N2O. The curves have been averaged to reflect the symmetry of the box. The raw data look very close to the averaged curves. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
that gives rise to these partition and penetration characteristics, umbrella sampling calculations were performed with pseudoN2O molecules. In order to understand the importance of the electrostatic interactions, we investigated pseudo-N2O with all the partial charges set to zero. Such N2O molecules still retain the Lennard-Jones interactions with the lipid and water. We will refer these molecules as LJ-N2 O. In order to understand the importance of the van der Waals interaction, we created pseudo-N2O with only the repulsive part of the Lennard-Jones interaction. Thus, both the van der Waals and electrostatic interaction between N2O and the rest of the system are eliminated. We will refer these molecules as hard-sphere ðHSÞ-N2 O. Fig. 2 reports the PMF for LJ-N2 O and HS-N2 O. For LJ-N2 O, the PMF inside the tail region of the lipid looks almost identical to that of the real N2O, with a shallow barrier at the center of the lipid. The height of the barrier for LJ-N2 O is also about 2 kJ/mol, identical to that of the real N2O. However, the relative stability of N2O in the hydrophobic lipid tail region and in water is changed dramatically. The free-energy difference between the hydrophobic lipid tail region and water is nearly twice as high for LJ-N2 O. The electrostatic interaction evidently stabilizes the N2O in water relative to a hydrophobic environment. This is quite distinct from other small anesthetics such as Xe. LJ-N2 O has a partition coefficient of 70, which is much higher than N2O. Once absorbed in a lipid, a LJ-N2 O is thus more likely to be trapped in the lipid instead of diffusing back to an aqueous environment. This may explain the clinical observation of Xe accumulating much more slowly in the gastrointestinal tract than N2O when administrated as a general anesthetic [27]. Assuming Xe is very similar to LJ-N2 O, while it dissolves readily in lipid, it will take longer for Xe to diffuse through living tissues into the digestive system. It is worth noting the shoulder in the N2O PMF disappears in the LJ-N2 O PMF. The N2O PMF gradually reduces as it enters into the lipid due to dehydration. The shoulder, which indicates a slow down of the dehydration, is probably caused by phosphate being highly hydrophilic. Removing the electrostatic interaction between N2O and the environment visibly eliminates the stabilization due to hydrophilic phosphate groups. The HS-N2 O has a completely different PMF, with the minimum coinciding with the lipid central plane. HS-N2 O is a probe for vacancies and will be most abundant at places where there are
many local vacancies. The HS-N2 O being most stable at the lipid central plane indicates many vacancies are created at the region where the tails of the DMPC monolayers meet. The maximum of the HS-N2 O PMF reflects the most spatially crowded region, which coincides with phosphorus density maximum. We conclude from these observations that steric hindrance does not dominate the interactions between N2O and the phospholipid. The enrichment of N2O close to the b-carbon of the 1-myristoyl is mostly due to the van der Waals force. For larger molecules, however, it is conceivable that steric hindrance can be more important. This may explain an observed shallow minimum in the PMF of benzocaine around the central plane of the lipid [23]. Although the dipole moment for N2O is very small, the electrostatic interaction does strongly influence the overall partition coefficient of N2O between DMPC and water, making N2O much more hydrophilic. The significant contribution of electrostatics indicates that it may be possible to adjust the partition coefficient by changing the salinity of lipid hydration water, thus providing leverage in using N2O as a probe for solvation dynamics [9]. Acknowledgments This work is supported by a startup grant from Boston University and by the National Science Foundation under DGE-0221680 and DMR0821450. The computational resource for this work is provided by the generous computer time allocation from the National Center for Supercomputing Applications under MRAC TGCHE070060 and by the Boston University Center for Scientific Computing and Visualization. References [1] S.M. Gruner, E. Shyamsunder, Ann. N.Y. Acad. Sci. 625 (1991) 685. [2] N.P. Franks, W.R. Lieb, Nature 36 (1994) 607. [3] V. Jevtovic-Todorovic, S.M. Todorovc, S. Mennerick, S. Powell, K. Dikranian, et al., Nat. Med. 4 (1998) 460. [4] B.W. Urban, M. Bleckwenn, M. Barann, Pharmacol. Ther. 111 (2006) 729. [5] R. Eckenhoff, W. Zheng, M. Kelz, Clin. Pharmacol. Ther. 84 (2008) 144. [6] B. van Swinderen, O. Saifee, L. Shebester, R. Roberson, M.L. Nonet, C.C. Michael, Proc. Natl. Acad. Sci. USA 96 (1999) 2479. [7] R.S. Cantor, Biochemistry 36 (1997) 2339. [8] R.G. Shulman, B.P. Dailey, C.H. Townes, Phys. Rev. 78 (1950) 145. [9] L.R. Chieffo, J.T. Shattuck, E. Pinnick, J.J. Amsden, M.K. Hong, S. Erramilli, L. Ziegler, J. Phys. Chem. B 112 (2008) 12776. [10] G.M. Torrie, J.P. Valleau, J. Comput. Phys. 23 (1977) 187. [11] S. Kumar, J.M. Rosenberg, D. Bouzida, R.H. Swendsen, P.A. Kollman, J. Comput. Chem. 13 (1992) 1011. [12] H.J.C. Berendsen, D. van der Spoel, R. van Drunen, Comput. Phys. Comm. 91 (1995) 43. [13] B. Hess, C. Kutzner, D. van der Spoel, E. Lindahl, J. Chem. Theory Comput. 4 (2008) 435. [14] B.L. de Groot, D.P. Tieleman, P. Pohl, H. Grübmuller, Biophys. J. 82 (2002) 2934. [15] H.J.C. Berendsen, J.P.M. Postma, W.F. van Gunsteren, J. Hermans, in: B. Pullman (Ed.), Intermolecular Forces, Reidel, Dordrecht, 1981, p. 331. [16] J.A. Pople, M. Head-Gordon, K. Raghavachari, J. Chem. Phys. 87 (1987) 5968. [17] R.A. Kendall, T.H. Dunning Jr., R.J.J. Harrison, J. Chem. Phys. 96 (1992) 6796. [18] J.M. Goodfellow, M. Knaggs, M.A. Williams, J.M. Thornton, Faraday Discuss. 103 (1996) 339. [19] S. Nosé, Mol. Phys. 51 (1984) 255. [20] W.G. Hoover, Phys. Rev. A 31 (1985) 1695. [21] U. Essmann, L. Perera, M.L. Berkowitz, T. Darden, H. Lee, L.G. Pedersen, J. Chem. Phys. 103 (1995) 8577. [22] L. Koubi, M. Tarek, M.L. Klein, D. Scharf, Biophys. J. 78 (2000) 800. [23] R.D. Porasso, W.F. Drew Bennett, S.D. Oliveira-Costa, J.J. López Cascales, J. Phys. Chem. B 113 (2009) 9988. [24] P.C. Hiemenz, R. Rajagopalan, Principles of Colloid and Surface Chemistry, third ed., Marcel Dekker, New York, 1997. [25] M.A. Hernandez, A. Rathinavelu, Basic Pharmacology: Understanding Drug Actions and Reactions, CRC, Boca Raton, 2006. [26] A. Steward, P.R. Allott, A.L. Cowles, W.W. Mapleson, Br. J. Anaesth. 45 (1973) 282. [27] H. Reinelt, U. Schirmer, T. Marx, P. Topalidis, M. Schmidt, Anesthesiology 94 (2001) 475.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
The potential of mean force of nitrous oxide in a 1,2-dimyristoylphosphatidylcholine lipid bilayer Eric R. Pinnick a, Shyamsunder Erramilli a,b, Feng Wang c,* a
Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, United States Department of Biomedical Engineering and the Photonics Center, Boston University, 8 Saint Mary’s Street, Boston, MA 02215, United States c Department of Chemistry, Boston University, 590 Commonwealth Avenue, Boston, MA 02215, United States b
a r t i c l e
i n f o
Article history: Received 7 January 2010 In final form 16 February 2010 Available online 19 February 2010
a b s t r a c t The free-energy profile for N2O in a hydrated 1,2-dimyristoylphosphatidylcholine (DMPC) phospholipid bilayer is calculated. The N2O molecule is preferentially concentrated in the tail region of the DMPC lipid and there is no free-energy barrier for the diffusion of N2O from an aqueous environment into the lipid. Although the dipole moment of N2O is rather small, our calculation indicates that the N2O partition coefficient is strongly influenced by electrostatic interactions. On the other hand, the van der Waals interaction overwhelms the steric effect and causes N2O concentration to be greatest at a location close to the bcarbon of the 1-myristoyl group. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Nitrous oxide, also called the laughing gas, is perhaps the oldest drug in continuous use in modern pharmacology. It has generally been regarded as a safe and effective anesthetic for more than one and a half centuries. However, the mechanism of the anesthetic action for this linear triatomic molecule is far from being understood. Mechanisms have been proposed that suggest N2O acts primarily either by interacting with the lipid membrane or by binding to specific proteins, especially channel proteins [1–5]. Previous studies suggest that mutating a key membrane integrated protein, syntaxin, confers resistance to volatile general anesthetics. These studies seem to indicate a protein-based mechanism [6]. However, it is possible N2O interacts with trans-membrane proteins indirectly by modulating physical properties of the membrane [1,7]. Regardless of the actual mechanism, understanding the free-energy profile for N2O penetration and partitioning between lipid and water is an important step toward elucidating the function of N2O as a general anesthetic. After all, N2O has to penetrate through membranes before it reaches specific binding sites. The penetration barrier correlates strongly with the action rate and the partition coefficient determines the equilibrium concentration of N2O in lipid. In addition to its role as an anesthetic, N2O serves as a promising probe molecule for biological systems. N2O is small in size and possesses a small dipole moment (0.166 D [8]). This allows N2O to access both polar and nonpolar solvent environments while causing a relatively small perturbation to the molecule being probed. The asymmetric stretch (v 3 ) of N2O at roughly 2215—2230 cm1
* Corresponding author. E-mail address: [email protected] (F. Wang). 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.02.047
has a large extinction coefficient (1:5 103 M1 cm1 ); most biomolecules and water on the other hand do not have strong adsorption in this spectral region. The vibrational lifetime of the asymmetric stretching mode, which can be readily probed by ultrafast infrared spectroscopy, is very sensitive to the solvation environment. In a recent work, the utility of N2O as a reporter molecule for the solvation dynamics of DOPC bilayers is demonstrated [9]. However, further interpretation of these results will benefit from an understanding of how N2O partitions and diffuses from aqueous to lipid regions. In order to address these concerns, we investigate the free-energy profile of N2O in hydrated 1,2-dimyristoylphosphatidylcholine (DMPC) lipids through molecular dynamics simulations. The freeenergy profile is reported as the potential of mean force (PMF) calculated using umbrella sampling [10] and the weighted histogram analysis method [11]. Such a free-energy profile gives information about specific binding sites, partition coefficients of N2O in DMPC and water, and the activation barrier for N2O penetration both into and across the lipid. This information provides physical insight into the equilibrium distribution of N2O in lipids and the readiness of N2O penetration through lipids to action sites, and is thus valuable for understanding the anesthetic action of N2O and for interpreting experimental studies on N2O as a probe molecule. We have also tried to elucidate the fundamental interaction that gives rise to such a free-energy profile for N2O. 2. Methods The PMF was calculated from a set of 40 independent molecular dynamics simulations using umbrella sampling and the weighted histogram analysis method. Simulations and analysis were
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3. The PMF of nitrous oxide The PMF of N2O is reported in Fig. 1. It is clear from this figure that N2O prefers to stay in the lipid tail region. The transport of N2O Table 1 Summary of N2O model.
a
0
−6
Density (arbitrary units)
PMF Gibbs Surface Carboxyl Density Phosphate Density β-Carbon Density
−3
PMF (kJ/mol)
performed with the GROMACS package [12,13]. The lipid–water bilayer was modeled in the La phase using the united-atom lipid force field and the previously-equilibrated bilayer system of Tieleman and collaborators [14] with SPC water [15]. Each simulation system consisted of 128 DMPC molecules fully hydrated by 3655 water molecules, along with a single N2O molecule. The water to lipid number ratio is 28.5, which is above the excess hydration point at the simulation temperature. The equilibrium box size is 6.15 6.15 6.64 nm, giving an average head group area of 59.1 Å2 with an equilibrium normal pressure of 1 bar. N2O geometry and charge values were fitted using the ab initio QCISD method [16] with the aug-cc-pVTZ basis set [17], and summarized in Table 1. The QCISD calculation predicted a N2O dipole moment of 0.166 D, which is identical to the experimental value. The atomic charges were determined to best reproduce the electrostatic potential of N2O while constraining the dipole moment. The Lennard-Jones parameters for N2O were taken from the modified GROMOS87 forcefield in Gromacs. The force constant and spacing of the umbrella potentials were optimized by trial simulations on a scaled-down bilayer containing 24 lipids, with the same average head group area and water to lipid number ratio as the full-sized system. A series of umbrella potential force constants ranging from 40 to 200 kJ=ðmol nm2 Þ and a series of window spacings from 1 to 2.5 Å were experimented to achieve a population overlap of at least 50% between adjacent windows. Based on these trial simulations, we selected an umbrella potential constant of 80 kJ=ðmol nm2 Þ and a spacing between umbrella potentials of 1.66 Å. This spacing leads to a total of 40 windows for the PMF study. One N2O molecule was inserted into each window. The xy position of the N2O was randomly chosen. N2O insertion was accomplished following the protocol of Goodfellow and collaborators [18] with a total equilibration time of 200 ps. Typically, there are two approaches to get a well-converged PMF. One approach is to perform long simulations with a relatively small number of windows. The second approach is to perform relatively short simulations but with a large number of windows. We took the second approach because a larger scale of parallelism can be achieved trivially with a large number of windows. We picked a total of 40 windows and simulated under the canonical ensemble for 4 ns with a time step of 1 fs in each window. Although 4 ns do not seem long, the standard error bars of our simulation have an average value of 0.4 kJ/mol and a maximum of 0.6 kJ/mol. This small error is achieved since the overlap between adjacent windows in our simulation ranges from 40% to 90%, with an average overlap of 75%. The simulations from adjacent windows thus help the sampling of the configuration space. During PMF simulations, a simulation temperature of 310 K was maintained by a Nosé–Hoover thermostat [19,20] with a thermostat time constant of 0.5 ps. Periodic boundary conditions were employed in all directions. Lennard-Jones interactions were calculated using a 14 Å cutoff. Electrostatic interactions were treated exactly using the particle mesh Ewald method [21].
−9 −12 −15 −18
−30
−20
−10
0
10
20
30
˚ Z-Axis Position (A) Fig. 1. PMF of N2O (red). The Gibbs surface (dot-dashed blue) and density profiles of the phosphate P (dashed black), carboxyl group C (dashed magenta), and b-carbon of the 1-myristoyl group (dashed green) are provided for reference. The curves have been averaged to reflect the symmetry of the box. The raw data look very close to the averaged curves. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
from water into the lipid is a barrier-less process. Once inside the lipid, the N2 O will be most stable at about 10 Å from the lipid central plane, which is on the tail side of the carboxyl group carbon. The position of this minimum coincides with the average location of the b-carbon of the 1-myristoyl tail. This is the same as the most favorable location for halothane in a dipalmitoylphosphatidylcholine bilayer [22] and that of benzocaine in a mixed phospholipid bilayer [23]. Furthermore, a ‘shoulder’ feature exists at around 18 Å from the lipid central plane, coinciding with the phosphorus density maximum. A shallow free-energy barrier resides at the lipid central plane. The barrier for N2 O to penetrate the tail region of DMPC is about 2 kJ/mol, which is comparable to kT at room temperature. This is much smaller than the free energy difference between the bulk water phase and the free-energy minima in the lipid, 16.4 ± 0.4 kJ/mol. Detailed partition information of N2O in hydrated lipid can be obtained from the PMF. Fig. 1 marks the Gibbs surface of interlammellar water, which almost coincides with the phosphorus density maximum [24]. We will separate the head and tail regions of the lipid by the carboxylate carbon density maxima. For N2O inside the Gibbs surface of water, 92% of the N2O is in the tail group region, while only 8% resides in the head group region. The Molar partition coefficient K for N2O partitioning between DMPC and water is calculated as the thermodynamic equilibrium constant for the reaction
N2 Oaq N2 Olipid :
ð1Þ
Obviously, the K depends on the overall hydration level of the lipid and how the water–lipid interface is defined. Using the Gibbs surface of water to separate the water phase and the lipid phase, the partition coefficient K for our excess hydration DMPC is 2.5 using molarity as units of concentration. We are not aware of an experimental partition coefficient for N2O between DMPC and water. The octanol–water experimental K value of 2.3 [25] and the olive oil–water K value of 2.97 [26] provide useful guidelines [4]. 4. The origins of the PMF features
Atom
q/e
Pos.a (Å)
r
(kJ/mol)
(Å)
N N0 O
0.239 0.523 0.284
1.12 0.000 1.19
0.8767 0.8767 0.8490
2.976 2.976 2.955
Position taken relative to central N atom, N0 .
N2O is a very labile molecule with good solubility in both water and phospholipids. It is worth mentioning that some other molecules such as halothane and benzocaine mentioned previously that have different relative solubility actually have the same relative site of enrichment. In order to identify the fundamental interaction
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E.R. Pinnick et al. / Chemical Physics Letters 489 (2010) 96–98
20
PMF (kJ/mol)
10 0 −10
N2O LJ-N2O
−20
HS-N2O N2O Min.
−30 −30
−15
0
15
30
˚ Z-Axis Position (A) Fig. 2. PMF of N2O (red), LJ-N2 O (blue), and HS-N2 O (green) in hydrated DMPC. Vertical lines correspond to the free-energy minima of N2O. The curves have been averaged to reflect the symmetry of the box. The raw data look very close to the averaged curves. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
that gives rise to these partition and penetration characteristics, umbrella sampling calculations were performed with pseudoN2O molecules. In order to understand the importance of the electrostatic interactions, we investigated pseudo-N2O with all the partial charges set to zero. Such N2O molecules still retain the Lennard-Jones interactions with the lipid and water. We will refer these molecules as LJ-N2 O. In order to understand the importance of the van der Waals interaction, we created pseudo-N2O with only the repulsive part of the Lennard-Jones interaction. Thus, both the van der Waals and electrostatic interaction between N2O and the rest of the system are eliminated. We will refer these molecules as hard-sphere ðHSÞ-N2 O. Fig. 2 reports the PMF for LJ-N2 O and HS-N2 O. For LJ-N2 O, the PMF inside the tail region of the lipid looks almost identical to that of the real N2O, with a shallow barrier at the center of the lipid. The height of the barrier for LJ-N2 O is also about 2 kJ/mol, identical to that of the real N2O. However, the relative stability of N2O in the hydrophobic lipid tail region and in water is changed dramatically. The free-energy difference between the hydrophobic lipid tail region and water is nearly twice as high for LJ-N2 O. The electrostatic interaction evidently stabilizes the N2O in water relative to a hydrophobic environment. This is quite distinct from other small anesthetics such as Xe. LJ-N2 O has a partition coefficient of 70, which is much higher than N2O. Once absorbed in a lipid, a LJ-N2 O is thus more likely to be trapped in the lipid instead of diffusing back to an aqueous environment. This may explain the clinical observation of Xe accumulating much more slowly in the gastrointestinal tract than N2O when administrated as a general anesthetic [27]. Assuming Xe is very similar to LJ-N2 O, while it dissolves readily in lipid, it will take longer for Xe to diffuse through living tissues into the digestive system. It is worth noting the shoulder in the N2O PMF disappears in the LJ-N2 O PMF. The N2O PMF gradually reduces as it enters into the lipid due to dehydration. The shoulder, which indicates a slow down of the dehydration, is probably caused by phosphate being highly hydrophilic. Removing the electrostatic interaction between N2O and the environment visibly eliminates the stabilization due to hydrophilic phosphate groups. The HS-N2 O has a completely different PMF, with the minimum coinciding with the lipid central plane. HS-N2 O is a probe for vacancies and will be most abundant at places where there are
many local vacancies. The HS-N2 O being most stable at the lipid central plane indicates many vacancies are created at the region where the tails of the DMPC monolayers meet. The maximum of the HS-N2 O PMF reflects the most spatially crowded region, which coincides with phosphorus density maximum. We conclude from these observations that steric hindrance does not dominate the interactions between N2O and the phospholipid. The enrichment of N2O close to the b-carbon of the 1-myristoyl is mostly due to the van der Waals force. For larger molecules, however, it is conceivable that steric hindrance can be more important. This may explain an observed shallow minimum in the PMF of benzocaine around the central plane of the lipid [23]. Although the dipole moment for N2O is very small, the electrostatic interaction does strongly influence the overall partition coefficient of N2O between DMPC and water, making N2O much more hydrophilic. The significant contribution of electrostatics indicates that it may be possible to adjust the partition coefficient by changing the salinity of lipid hydration water, thus providing leverage in using N2O as a probe for solvation dynamics [9]. Acknowledgments This work is supported by a startup grant from Boston University and by the National Science Foundation under DGE-0221680 and DMR0821450. The computational resource for this work is provided by the generous computer time allocation from the National Center for Supercomputing Applications under MRAC TGCHE070060 and by the Boston University Center for Scientific Computing and Visualization. References [1] S.M. Gruner, E. Shyamsunder, Ann. N.Y. Acad. Sci. 625 (1991) 685. [2] N.P. Franks, W.R. Lieb, Nature 36 (1994) 607. [3] V. Jevtovic-Todorovic, S.M. Todorovc, S. Mennerick, S. Powell, K. Dikranian, et al., Nat. Med. 4 (1998) 460. [4] B.W. Urban, M. Bleckwenn, M. Barann, Pharmacol. Ther. 111 (2006) 729. [5] R. Eckenhoff, W. Zheng, M. Kelz, Clin. Pharmacol. Ther. 84 (2008) 144. [6] B. van Swinderen, O. Saifee, L. Shebester, R. Roberson, M.L. Nonet, C.C. Michael, Proc. Natl. Acad. Sci. USA 96 (1999) 2479. [7] R.S. Cantor, Biochemistry 36 (1997) 2339. [8] R.G. Shulman, B.P. Dailey, C.H. Townes, Phys. Rev. 78 (1950) 145. [9] L.R. Chieffo, J.T. Shattuck, E. Pinnick, J.J. Amsden, M.K. Hong, S. Erramilli, L. Ziegler, J. Phys. Chem. B 112 (2008) 12776. [10] G.M. Torrie, J.P. Valleau, J. Comput. Phys. 23 (1977) 187. [11] S. Kumar, J.M. Rosenberg, D. Bouzida, R.H. Swendsen, P.A. Kollman, J. Comput. Chem. 13 (1992) 1011. [12] H.J.C. Berendsen, D. van der Spoel, R. van Drunen, Comput. Phys. Comm. 91 (1995) 43. [13] B. Hess, C. Kutzner, D. van der Spoel, E. Lindahl, J. Chem. Theory Comput. 4 (2008) 435. [14] B.L. de Groot, D.P. Tieleman, P. Pohl, H. Grübmuller, Biophys. J. 82 (2002) 2934. [15] H.J.C. Berendsen, J.P.M. Postma, W.F. van Gunsteren, J. Hermans, in: B. Pullman (Ed.), Intermolecular Forces, Reidel, Dordrecht, 1981, p. 331. [16] J.A. Pople, M. Head-Gordon, K. Raghavachari, J. Chem. Phys. 87 (1987) 5968. [17] R.A. Kendall, T.H. Dunning Jr., R.J.J. Harrison, J. Chem. Phys. 96 (1992) 6796. [18] J.M. Goodfellow, M. Knaggs, M.A. Williams, J.M. Thornton, Faraday Discuss. 103 (1996) 339. [19] S. Nosé, Mol. Phys. 51 (1984) 255. [20] W.G. Hoover, Phys. Rev. A 31 (1985) 1695. [21] U. Essmann, L. Perera, M.L. Berkowitz, T. Darden, H. Lee, L.G. Pedersen, J. Chem. Phys. 103 (1995) 8577. [22] L. Koubi, M. Tarek, M.L. Klein, D. Scharf, Biophys. J. 78 (2000) 800. [23] R.D. Porasso, W.F. Drew Bennett, S.D. Oliveira-Costa, J.J. López Cascales, J. Phys. Chem. B 113 (2009) 9988. [24] P.C. Hiemenz, R. Rajagopalan, Principles of Colloid and Surface Chemistry, third ed., Marcel Dekker, New York, 1997. [25] M.A. Hernandez, A. Rathinavelu, Basic Pharmacology: Understanding Drug Actions and Reactions, CRC, Boca Raton, 2006. [26] A. Steward, P.R. Allott, A.L. Cowles, W.W. Mapleson, Br. J. Anaesth. 45 (1973) 282. [27] H. Reinelt, U. Schirmer, T. Marx, P. Topalidis, M. Schmidt, Anesthesiology 94 (2001) 475.