C. Ensembles

A molecular dynamics simulation consists of numerically solving the equations of motion of a set of particles (atoms), given the forces on the particles. Classical MD simulations that solve Newton’s equations of motion generate trajectories belonging to the microcanonical [constant number of particles, volume, and energy (NVE)] statistical mechanical ensemble. It is generally desirable to perform simulations in other ensembles such as the isobaricisothermal ensemble (constant NPT). In addition to being the natural choice for correspondence with typical experimental conditions, the NPT ensemble is useful for validating force fields by checking their ability to reproduce important structural parameters known from experimental measurements, such as the surface area per lipid, the interlamellar spacing, and the membrane thickness, and for predicting these quantities when they are not known (e.g., in membrane–protein systems). Constant pressure and temperature are enforced in simulations by controlling the fluctuations of the particle kinetic energy and system volume, respectively, and there are various ways to do this. The best algorithms, in terms of their ability to rigorously generate the NPT ensemble, are based on the ‘‘extended system’’ approach [46], in which additional dynamic variables are introduced, for example a time-dependent friction coefficient (‘‘thermostat’’) to control the temperature and a piston to control the pressure. The equations of motion and a conserved energy are consistently formulated so that the microcanonical distribution function for the extended phase space gives the isobaric-isothermal distribution function for the particles after integration over the additional dynamic variables [47].

It is now widely accepted that it is best to simulate membranes at constant pressure, but there is some disagreement concerning the assumed form of the pressure tensor, i.e., whether the pressure should be the same in all directions (isotropic) or whether the pressure in the plane of the membrane should be different from the pressure perpendicular to the membrane [11]. The latter is appropriate for a true interfacial system, such as the air/ water and air/hydrocarbon interfaces, with a significant surface tension. Because membrane/water interfaces have either a vanishing or very small surface tension [48], in principle membranes should be simulated with an isotropic pressure tensor. However, some simulators have argued that to correct for finite size effects in simulations of small membrane patches, a modest surface tension should be imposed [49]. As larger and larger bilayer patches are being simulated ( 1000 lipids), it is becoming evident that there are finite-size effects in systems containing 100 lipids. For example, in a recent comparison of bilayers simulated at constant isotropic pressure, the area per lipid in a system of 100 lipids was about 3% less than in a system of 1000 lipids [75]. However, the explanation of the origin of this contraction and the best remedy for avoiding it in small systems (1000 lipids is still out of reach for most simulators) are debatable. On the one hand, one could argue that a surface tension should be applied to stretch a small membrane patch to counteract the shrinkage due to finite-size effects. However, an imposed surface tension can be expected to change the spectrum of fluctuations of the interface (in effect, stiffening the interface). On the other hand, one could impose constant isotropic pressure, keeping in mind the systematic error that results from finite-size effects. The results presented in this chapter were obtained by constant isotropic pressure calculations on bilayers containing 64 lipids [50,51], and hence the areas per lipid reported here are likely a couple of percent below the large system limit for the potential function employed.

Figure 4 Comparison of average distances from the bilayer center along the bilayer normal for deuterated methyl and methylene groups distributed throughout the DPPC molecule computed from constant-pressure MD calculations and neutron diffraction measurements on gel and liquid crystalline phase DPPC bilayers.

The first atomic scale picture of the average structure of the interface was produced by measurement of the density distributions of different types of atoms along the bilayer normal by a combined neutron and X-ray diffraction study of a phosphatidylcholine bilayer at low hydration [6]. The corresponding picture for fully hydrated bilayers has been provided by MD simulations and is exemplified by our results for the electron density profiles of the liquid crystalline phase of DPPC shown in Figure 5. Defining the bilayer thickness

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as the distance between the peaks in the total electron density, we obtain 37.2 A, which is in reasonable agreement with the values determined by X-ray diffraction analysis [5]. In Figure 5b we show the decomposition of the electron density profile into contributions from the lipid polar groups. The chemical heterogeneity of the membrane/water interface is clearly evident in the overlapping distributions of the lipid polar groups and water molecules in the broad interfacial region connecting the bulk water in the middle of the interlamellar space to the hydrocarbon in the middle of the bilayer. In Figure 5a the total contributions from the water and the lipid show that the water density decays smoothly

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from the bulk value roughly 30 A from the bilayer center and penetrates deeply into the bilayer. This is in contrast to the water density oscillations observed next to flat hydrophobic surfaces [53] and the relatively narrow, noninterpenetrating air/water and hydrocarbon/water interfaces [54]. Defining the interface as the range over which the water density goes from 90% to 10% of its bulk value, we find that the two interfaces

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occupy 23 A, which is more than half of the total bilayer thickness. Thus, the membrane should be thought of as a broad hydrophilic interface, with only a thin slab of pure fluid hydrocarbon in the middle.

Figure 5 Electron density distributions along the bilayer normal from an MD simulation of a fully hydrated liquid crystalline phase DPPC bilayer. (a) Total, lipid, and water contributions; (b) contributions of lipid components in the interfacial region.

C. Solvation of the Lipid Polar Groups

To discuss the interactions between water molecules and specific lipid polar groups, we consider radial distribution functions for the water oxygen atoms surrounding the phosphate P, choline N, and carbonyl C atoms (Fig. 6). The PEO g(r ) has a sharp first peak

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at 3.8 A, indicating tight solvation of the negatively charged phosphate by an average of four water molecules [obtained by integrating g(r ) to the first minimum]. Inspection of

Figure 6 Radial distributions of water oxygen atoms around sites in the polar groups in a DPPC bilayer.

the individual phosphate O–water O g(r ) functions (not shown; see Ref. 55) reveals that the water molecules are almost exclusively associated with the unesterified phosphate

E ˚

oxygens. The N O g(r ) displays a broader first peak at 4.8 A corresponding to 15 water molecules in the first solvation shell of the positively charged choline ammonium group. The relatively inaccessible carbonyl groups at the glycerol–fatty acyl ester linkages are solvated by an average of only 1.5 water molecules each.

The radial distributions of water around the lipid polar groups all contain welldefined first peaks signaling the existence of well-defined solvation shells. In subsequent analyses, we distinguish the water molecules that are closely associated with lipids by defining ‘‘bound’’ waters as those that fall within the first peaks of the water–phosphate, water–choline, or water–carbonyl g(r ) functions [55]. Thus, on average, roughly four waters are bound to the phosphate (‘‘P-bound’’), 15 to the choline (‘‘N-bound’’), and 1.5 to each carbonyl group (‘‘CO-bound’’). Some of these bound waters are shared between the polar groups, and overall roughly half of the 28 waters per lipid in the fully hydrated fluid phase bilayer are considered bound to lipids by our definition. We distinguish these bound waters from the ‘‘bulk’’ waters, a slab of water molecules roughly two to three water molecules thick in the middle of the interlamellar space, which we define as being

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more than 30 A from the bilayer center. Of course, our definitions of bound and bulk water molecules are somewhat arbitrary, as are alternative definitions based on interaction energies, residence times, etc. However, we see in the next section that these definitions are useful for showing that the dynamics of water molecules near membrane surfaces depend on the strength of their interactions with specific lipid polar groups.

D.Water Orientational Polarization and the Membrane Dipole Potential

In membranes containing phospholipids such as DPPC, the negatively charged phosphate groups exert a strong influence on the structure of the water molecules. As the unesterified

PEO bonds are preferentially oriented, on average pointing away from the membrane and into the water, the P-bound water molecules tend to orient their dipoles with their positive ends pointing toward the negative phosphates, resulting in a net orientational ‘‘polarization’’ [55]. The orienting power of the phosphate becomes clear when one observes that the orientational polarization is much less pronounced at water interfaces with lipid components that do not contain phosphate groups, e.g., glycerol, decane, and decyl- β-glycoside [54].

These orientationally polarized water molecules appear to have a profound influence on the electrostatic properties of membranes. Experimental measurements on lipid monolayers and bilayers have demonstrated that there is an electric potential difference across lipid/water interfaces, typically a few hundred millielectronvolts, negative on the water side relative to the hydrocarbon [56,57]. Thus, for the purpose of describing the electrostatics, the membrane can be thought of as a planar array of dipoles whose negative ends point toward the water [58]. The molecular origins of the resulting ‘‘dipole potential’’ are of interest because the electrical properties of the bilayer surface influence the binding and passive transport of charged species. Experiments on phospholipid bilayers suggest that the primary negative contributions to the dipole potential arise from oriented water molecules [57] and, to a lesser extent, from the carbonyl groups in the acyl ester linkages [58].

The dipole potential can be easily calculated from a simulation as a double integral of the average charge density (for example, see Ref. 59). The total dipole potential profile from our DPPC bilayer simulation plotted in Figure 7 monotonically decreases in the membrane/water interface to a value in the bulk water of about 500 mV. The lipid contribution is nonzero only in the interfacial region. The negative lipid contribution on the hydrocarbon side of the interface is canceled by a positive contribution on the water side, while the water contribution is monotonically decreasing throughout the interface. Thus, we conclude that the net dipole potential is due primarily to an excess of water

Figure 7 The electric potential relative to the hydrocarbon (‘‘dipole’’ potential) as a function of distance from the center of a fully hydrated DPPC bilayer.