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* Laboratory of Physics and Helsinki Institute of Physics, Biophysics and Statistical Mechanics Group, Laboratory of Computational Engineering, Helsinki University of Technology, Helsinki, Finland and Wihuri Research Institute, Helsinki, Finland
Correspondence: Address reprint requests to Emma Falck, Laboratory of Physics, Helsinki University of Technology, PO Box 1100, 02015 HUT, Finland. Tel.: +358-9-451-5804; E-mail: emma.falck{at}hut.fi.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMODEL AND SIMULATION DETAILSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMODEL AND SIMULATION DETAILSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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; Ohvo-Rekilä et al., 2002, and references therein). It is abundant in the plasma membranes of higher organisms: depending on the exact lipid composition, the plasma membrane may contain the order of 20–50% cholesterol (Alberts et al., 1994).
Eukaryotic cells do not seem to be able to grow and differentiate properly without cholesterol. It has been firmly established that cholesterol modulates the physical properties of the plasma membrane (McMullen and McElhaney, 1996). A finite cholesterol content has been said to improve the characteristics of a simple phospholipid bilayer and allow for wider variations in the lipid composition of the membrane (Vist and Davis, 1990). Perhaps not surprisingly, cholesterol is one of the primary molecules in lipid rafts (Edidin, 2003; Silvius, 2003; Simons and Ikonen, 1997, and references therein), i.e., microdomains rich in cholesterol, sphingomyelin, and saturated phospholipids. Rafts have been thought to confine proteins involved in, e.g., signal transduction events, and hence act as platforms for adhesion and signaling. Consequently, one could well imagine that as cholesterol alters the properties of the bilayer, it might affect the functioning of the embedded proteins (Cantor, 1999; Yeagle, 1991).
The effects of cholesterol on the properties of phospholipid bilayers are diverse. In the physiologically relevant fluid phase, adding cholesterol to the bilayer leads to increased orientational order in the phospholipid tails (Chiu et al., 2002; Hofsäß et al., 2003; McMullen and McElhaney, 1996; Sankaram and Thompson, 1990b) and smaller average areas per molecule (Petrache et al., 1999). In other words, cholesterol modifies the packing of molecules in bilayers. Other important effects are changes in passive permeability of small solutes (Jedlovszky and Mezei, 2003; Xiang, 1999, and references therein) and suppressed lateral diffusion of phospholipids in bilayers with cholesterol (Almeida et al., 1992; Galla et al., 1979; Hofsäß et al., 2003; Polson et al., 2001; Vattulainen and Mouritsen, 2003). Both permeability and lateral diffusion, in turn, are strongly affected by the amount and distribution of free volume or area in a membrane, i.e., space not occupied by phospholipids, cholesterols, or water. Cholesterol thus seems to simultaneously influence packing, free area, diffusion, and permeability in lipid bilayers, and it is reasonable to expect that the changes in these properties are somehow coupled.
Although there is a wealth of information on the effects of cholesterol on lipid bilayers, the interplay of packing, free area, diffusion, and permeability has not yet been studied systematically. Experimental electron density profiles (McIntosh, 1978) and deuterium nuclear magnetic resonance (NMR) data (Sankaram and Thompson, 1990b) suggest that cholesterol should influence the packing inside membranes. Fluorescence recovery after photobleaching (FRAP) experiments, in turn, have been used to study the dependence of lateral diffusion coefficients on free area (Almeida et al., 1992). More information at the atomic level, however, is essential for gaining a detailed understanding of the effect of cholesterol on lipid bilayers. Such atomic-level information can be obtained from computer simulations. Molecular dynamics in particular provides a unique tool to investigate both the structure and dynamics of lipid membranes with a level of detail missing in any experimental technique. Until recently, however, systematic simulation studies have been limited by the extensive computational requirements.
In the present study, we investigate the cholesterol-induced changes in packing, free area, ordering, and lateral diffusion in phospholipid bilayers. Specifically, we study the presumptive interplay between these changes. To this end, we employ 100-ns molecular dynamics simulations on dipalmitoylphosphatidylcholine (DPPC)/cholesterol bilayers, with cholesterol concentrations ranging from 0 to 50 mol %. Although detailed multi-nanosecond simulation studies on the atomic level have emerged only very recently (Hofsäß et al., 2003; Scott, 2002; Tieleman et al., 1997), there exist large amounts of experimental studies for DPPC/cholesterol bilayers (McMullen and McElhaney, 1996; Sankaram and Thompson, 1990a,b; Vist and Davis, 1990, and references therein). These previous studies and the experimental results in particular offer us an excellent platform for comparison.
To further enhance the understanding of the effect of cholesterol on bilayers, we introduce a novel method for investigating the packing and free area in bilayers. The scope of this technique is very wide. It allows us to estimate how much space DPPC, cholesterol, and water molecules on average occupy in different regions of the bilayer. Consequently, it yields information on the amount and location of free space in the bilayer. As discussed below, this is related to various structural aspects such as the ordering of lipids in a membrane. Our method also provides valuable insight into dynamic properties. For example, our approach allows us to determine the area compressibility modulus across a membrane, and hence yields information on rate-limiting regions for lateral diffusion. In addition, as the method enables us to examine changes in free area with an increasing cholesterol content, we may estimate diffusion coefficients in terms of free area theories for lateral diffusion. The present approach can be applied to a wide range of different kinds of membrane systems, including one- and multicomponent bilayers, and bilayers with embedded solutes, probes, and proteins.
We find that cholesterol strongly affects the amount of space occupied by molecules in different parts of a phospholipid bilayer. The close-packed areas occupied by the tails of DPPC molecules can be explained by the ordering of the tails, and a simple relation (Petrache et al., 1999) can be used for quantifying the dependence of close-packed area on ordering. The amount and location of free space is significantly reduced by an increasing cholesterol content, and clearly reflect the total space occupied by DPPC and cholesterol molecules. The lateral diffusion coefficients, too, show a substantial decrease with an increasing cholesterol concentration. We find that so-called free area theories (Almeida et al., 1992; Cohen and Turnbull, 1959; Galla et al., 1979), which are essentially two-dimensional mean-field models, correctly predict this reduction, but are not applicable to quantitatively describing lateral diffusion in lipid bilayers.
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MODEL AND SIMULATION DETAILS |
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TOPABSTRACTINTRODUCTIONMODEL AND SIMULATION DETAILSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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= 0%, 4.7%, 12.5%, 20.3%, 29.7%, and 50.0%.
The starting point was a united atom model for a fully hydrated pure DPPC bilayer that has been validated previously (Tieleman and Berendsen, 1996; Patra et al., 2003). The parameters for bonded and nonbonded interactions for DPPC molecules were taken from a study of a pure DPPC bilayer (Berger et al., 1997) available at http://moose.bio.ucalgary.ca/Downloads/lipid.itp. The partial charges are from the underlying model description (Tieleman and Berendsen, 1996) and can be found at http://moose.bio.ucalgary.ca/Downloads/dppc.itp. For water, the SPC model (Berendsen et al., 1981) was used. As our initial configuration for the pure DPPC bilayer we used the final structure of run E discussed in Tieleman and Berendsen (1996) and available at http://moose.bio.ucalgary.ca/Downloads/dppc128.pdb. The bilayer is aligned such that it lies in the x,y plane, i.e., the bilayer normal is parallel to the z axis.
The cholesterol force field and the initial shape of an individual cholesterol molecule were taken from http://www.gromacs.org/topologies/uploaded_molecules/cholesterol.tgz (Höltje et al., 2001). Cholesterols were introduced to the bilayer by choosing DPPC molecules from the pure phospholipid bilayer at random and replacing them by cholesterols. The same number of DPPC molecules was replaced in each of the two monolayers. In practice, the center of mass (CM) of a cholesterol molecule was moved to the CM position of the removed DPPC molecule. The main axis of inertia of each inserted cholesterol was parallel to the z axis, and each molecule was rotated by a random angle around the z axis.
The molecular dynamics (MD) simulations were performed at a temperature T = 323 K using the GROMACS (Lindahl et al., 2001) molecular simulation package. The time step for the simulations was chosen to be 2.0 fs. The lengths of all bonds were kept constant with the LINCS algorithm (Hess et al., 1997). Lennard-Jones interactions were cut off at 1.0 nm without shift or switch functions. Long-range electrostatic interactions were handled using the particle-mesh Ewald (Essman et al., 1995) method, which has been shown to be a reliable method to account for long-range interactions in lipid bilayer systems (Patra et al., 2003). The details of the implementation of particle-mesh Ewald have been discussed elsewhere (Patra et al., 2004).
After an initial energy minimization, we needed to equilibrate the system to fill the small voids left by replacing DPPC molecules by somewhat smaller cholesterol molecules. The equilibration was commenced by 50 ps of NVT molecular dynamics with a Langevin thermostat using a coupling time of 0.1 ps, i.e., every 0.1 ps the velocities of all particles were completely randomized from a Maxwell distribution corresponding to the target temperature. This complete loss of memory after 0.1 ps reduces the amount of ballistic motion of atoms inside a void. The equilibration was continued by 500 ps of NpT molecular dynamics at a pressure of 1 bar with a Langevin thermostat and a Berendsen barostat (Berendsen et al., 1984). The time constant for the latter was set to 1 ps, and the height of the simulation box was allowed to vary separately from the cross-sectional area of the box.
Finally, for every cholesterol concentration, we performed 100 ns of MD in the NpT ensemble with a Berendsen thermostat and barostat (Berendsen et al., 1984). The barostat was the same as the one described above, and the thermostat was set to separately couple the DPPC, cholesterol, and water molecules to a heat bath with a coupling time of 0.1 ps. With such a setup, in the case of pure DPPC, the average dimensions of the simulation box are 6.5 nm x 6.5 nm x 6.5 nm. For 29.7% cholesterol the dimensions are 5.2 nm x 5.2 nm x 9.0 nm.
The six simulations took a total of 
60,000 h of CPU time. For all systems up to and including the cholesterol molar fraction of 29.7%, a simulation time of 100 ns guarantees a good sampling of the phase space. The results for 50% cholesterol should be regarded with some caution, as the diffusion of the DPPC and cholesterol molecules is already quite slow; see Lateral Diffusion and Free Area, below. As mixing of DPPC and cholesterol molecules in this case is quite limited, the system probably bears traces of its initial configuration. This applies to all state-of-the-art simulation studies of phospholipid/cholesterol systems, and has been mentioned by other authors (Smondyrev and Berkowitz, 1999).
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMODEL AND SIMULATION DETAILSRESULTS AND DISCUSSIONSUMMARY AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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; Hofsäß et al., 2003). As for experimental results, we are only aware of an accurate measurement for the average area per molecule in the case of a pure DPPC bilayer (Nagle and Tristram-Nagle, 2000). In this case the average area per molecule was determined to be 0.64 nm2 at T = 323 K, in good agreement with (0.655 ± 0.005) nm2 obtained here. Measurements of the average area per molecule in DPPC/cholesterol monolayers (McConnell and Radhakrishnan, 2003) show trends similar to ours. The exact correspondence between average areas per molecule measured for bilayers and monolayers, however, is not evident (Nagle and Tristram-Nagle, 2000).
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), which are a measure of the orientational order of the phospholipid tails. Order parameters can be obtained from deuterium NMR experiments (Seelig and Seelig, 1974) or computer simulations (Tieleman et al., 1997). In united atom simulations such as ours, the orientational order can be characterized using tensors with elements S
ß such that
 | (1) |
is the angle between the molecular -axis and the bilayer normal (Tieleman et al., 1997). The molecular axes must be defined separately for each segment of an acyl chain: usually for the nth methylene group denoted as Cn, the z axis points in the Cn–1–Cn+1 direction, and Cn–1, Cn, and Cn+1 span the y,z plane. If the motion of the segments is assumed to be symmetric at the bilayer normal, the experimental deuterium order parameter SCD can be easily acquired, as
 | (2) |
), it is useful to compute the order parameters separately for both chains.
The order parameter profiles for the sn-1 and sn-2 chains are depicted in Fig. 3. The ordering effect of cholesterol is clearly visible: the order parameters grow significantly with an increasing cholesterol content. For pure DPPC and low cholesterol concentrations, the order parameter profiles show a plateau for small and intermediate values of n and decay near the center of the bilayer. When the cholesterol content increases, the plateau disappears, and there is a clear maximum at intermediate n. The ordering effect of cholesterol is most pronounced for n 6–10 and quite modest for segments near the phospholipid headgroups and bilayer center. This is due to the position of the cholesterol ring system in the bilayer along the bilayer normal (Smondyrev and Berkowitz, 1999): the largest ordering occurs for segments at roughly the same depth as the ring system. For instance, with 29.7% cholesterol, the order parameters for n 6–10 are increased roughly by a factor of 2.
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; Chiu et al., 2002; Hofsäß et al., 2003). However, as most force fields yield qualitatively similar results, and various technical details may influence the detailed form of the order parameter profile (Patra et al., 2003), it is more interesting to make comparisons to experimental findings.
The results for the pure DPPC system are in good agreement with experiments (Brown et al., 1979; Douliez et al., 1995; Petrache et al., 2000). As for mixtures of DPPC and cholesterol, Sankaram and Thompson found that when 50% of the DPPC molecules were substituted by cholesterols in a pure DPPC bilayer at T = 325 K, the order parameter for intermediate n was increased by a factor of 2.65 (Sankaram and Thompson, 1990b). Similarly, when 30% of the dimyristoylphosphatidylcholines (DMPCs) were replaced by cholesterols in a pure DMPC bilayer at T = 308 K, the order parameter increased by a factor of 2. Vist and Davis, in turn, observed an increase by a factor of 2 when replacing 24% of the DPPC molecules by cholesterol at T = 323 K (Vist and Davis, 1990). Similar agreement is found when our results are compared to other experiments (Douliez et al., 1996; Kintanar et al., 1986). In all, our simulations agree well with experimental findings. The only detail which our, or any other, united-atom MD simulations cannot reproduce is the behavior of the experimental deuterium order parameter for sn-2 at n = 2 (Sankaram and Thompson, 1990b; Seelig and Seelig, 1975).
Electron density profiles
Additional information about the structure of the bilayer along the normal or z direction can be obtained by computing density profiles for the whole system, different molecular species, or certain atomic groups of interest. In simulations it is possible to calculate atom density, mass density, and electron density profiles. These give information on the distribution of atoms in the normal direction. Related information can be acquired from x-ray and neutron diffraction studies. Due to fluctuations, x-ray diffraction studies on fully hydrated bilayers in a fluid phase only yield total electron density profiles, whose maxima are associated with the electron dense phosphate groups (Nagle and Tristram-Nagle, 2000). The distance between the maxima allows one to estimate the distance between the headgroups in the opposite leaflets, but does not yield accurate predictions for the hydrocarbon thickness or the true phosphate-phosphate distance (Nagle et al., 1996). Additional information, most importantly about the average location of various atomic groups, can be gained from neutron diffraction studies either with selective deuteration or in combination with x-ray diffraction (Nagle and Tristram-Nagle, 2000).
Fig. 4 shows the total electron densities calculated for the different cholesterol concentrations. The density profiles have a characteristic shape reminiscent of x-ray diffraction studies, with maxima approximately corresponding to the location of the phosphate groups, and a minimum, a so-called methyl trough, in the bilayer center, where the terminal methyl groups reside. For pure DPPC and low cholesterol concentrations, the densities decrease monotonically from the maxima to the minimum in the bilayer center. This medium density region corresponds to the methylene groups in the DPPC tails. When more cholesterol is present, the headgroup-headgroup distance increases, i.e., the bilayer gets thicker, and the electron density in the bilayer center decreases slightly. In addition, the density in the tail region increases, and the density profile between the center and the headgroups is no longer monotonically decreasing. The elevation is due to the fact that the cholesterol ring structure, which resides in the phospholipid tail region, has a higher electron density than do phospholipid tails.
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2 nm. We can conclude that cholesterol is located in the hydrophobic interior of the bilayer. The penetration of water into the bilayer becomes more difficult with increasing amounts of cholesterol: this reflects both the thickening of the bilayer and the increasing densities in the headgroup region. The lipid/water interface also seems to become steeper. The electron density of DPPC in the hydrophobic tail region decreases with the cholesterol content, which is compensated by an increasing cholesterol electron density. By comparing the electron densities for cholesterol and cholesterol ring systems, we can conclude that only the short acyl chain of cholesterol can approach the bilayer center.
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; Smondyrev and Berkowitz, 1999; Tu et al., 1998). In all simulation studies, the peaks that indicate the location of headgroups for pure DPPC are located approximately at the same distance from the bilayer center. With 10–12.5% cholesterol, only minor changes in the total densities can be observed. Except in the case of Tu et al., increasing amounts of cholesterol lead to a larger bilayer thickness and a slightly decreased total density in the bilayer center. All studies show an increased density in the phospholipid tail region. By investigating the DPPC or water densities, one may also note that all studies clearly indicate that the lipid/water interface becomes more abrupt. Our findings for the distribution of phosphorus atoms agree well with those of Smondyrev and others: when the cholesterol content increases, the peaks are narrowed and shifted toward the water phase. In all, density profiles computed using slightly different force fields are, for the most part, consistent with each other.
Our results are also consistent with diffraction experiments on DPPC and DMPC bilayers. Nagle et al. (1996) have determined the structure of a fully hydrated pure DPPC bilayer in the liquid-disordered phase using x-ray diffraction. The form of the density profile from our simulations of pure DPPC closely resembles Nagle's electron density profile for pure DPPC at T = 323 K. The head-head distance obtained from Nagle's experiment and that determined from our density profiles also are in good agreement. As for the influence of cholesterol, McIntosh (1978) has published x-ray diffraction experiments on model membranes containing cholesterol and phospholipids with saturated tails containing 12–18 carbons. His DLPC/cholesterol systems in the fluid phase behave in a qualitatively similar way as do our DPPC/cholesterol bilayers. By comparing the electron densities from systems with different phospholipids and cholesterol to the densities from pure phospholipid bilayers, McIntosh also establishes the location of the cholesterol ring structure in the bilayer. Our studies support his view. In addition, there are more recent neutron diffraction studies of DMPC/cholesterol bilayers. The studies by Douliez et al. (1996) and Léonard et al. (2001) clearly show that substituting 30% of the phospholipids by cholesterol in a pure DMPC bilayer in the liquid-disordered phase increases the bilayer thickness. Léonard and co-workers have also investigated the location of cholesterol in the bilayer, and concluded that cholesterol is located well within the hydrophobic core. Although DPPC has longer hydrocarbon tails than DMPC, the cholesterol ring structure should be located in the same region of the bilayer (McIntosh, 1978). Our simulations indicate that cholesterol is indeed situated in the nonpolar region, as is the case in Douliez's and McIntosh's experiments.
Radial distribution functions
Together, the above results ascertain that our model correctly describes the behavior of the dimensions of the bilayer and the ordering of the nonpolar phospholipid tails as functions of the cholesterol content. Further, the structure of our DPPC/cholesterol bilayer in the normal direction is consistent with results from previous computations and experiments. This is very satisfactory, but in addition, we need to ensure that our bilayers truly are in the fluid state, i.e., that there is no translational long-range order. This can be ascertained by examining the radial distribution functions for, e.g., phosphorus and nitrogen atoms in the DPPC headgroups. For instance, the N–N radial distribution functions calculated in two dimensions for various cholesterol concentrations have large nearest-neighbor peaks at r 0.82 nm and show essentially no structure beyond r = 1.5 nm (data not shown). Additional calculations for other pairs of atoms and for the CM positions of the DPPC and cholesterol molecules lead to a similar conclusion, i.e., that there is no lateral long-range structure. Hence, we can be confident that our bilayers are either in the liquid-disordered or liquid-ordered phase, as they should. With this, we consider our model to be valid.
Estimating average areas per molecule in multicomponent bilayers
The average area per molecule, which is obtained by dividing the total area of the bilayer by the total number of molecules, is a well-defined concept in one-component lipid bilayers. It includes both area actually occupied by a lipid, the so-called close-packed area, and some free area. A similar quantity can be defined for multicomponent bilayers. It is a useful quantity when simulation results are compared to experiments. Its interpretation, however, is less clear: different lipids and sterols could occupy significantly different amounts of area. Hence, it would be desirable to be able to estimate the average area occupied by each molecular species present in the bilayer.
The average area per molecule 
A
as a function of cholesterol concentration is portrayed in Fig. 6. As mentioned in Equilibration, above, it is evident that A decreases with the cholesterol content, and that the results agree well with previous simulation studies (Chiu et al., 2002; Hofsäß et al., 2003).
We would not, however, like Chiu et al. (2002), conclude that A decreases linearly with and use this assumption to compute the average areas per phospholipid and cholesterol. It is not obvious, in the first place, that the average area per cholesterol or DPPC is independent of cholesterol content, as these authors seem to imply.
Another way to divide the total area between DPPC and cholesterol molecules has also been suggested (Hofsäß et al., 2003). By computing the total area and volume of the simulation box as functions of the cholesterol content and making a number of assumptions, one can arrive at estimates for the average areas occupied by DPPC and cholesterol molecules. In this case, an important assumption is that the average volume of a cholesterol molecule can be, for all concentrations, taken to be the volume occupied by a cholesterol molecule in a cholesterol crystal. Further, it is assumed that all space is occupied by DPPC, cholesterol, or water, i.e., that there is no free volume or area. The average areas per DPPC and cholesterol, 
and 
obtained along these lines from our data, are shown in the inset of Fig. 6. These closely resemble the corresponding results by Hofsäß et al.
A yet further method of distributing the area among the molecular species in a bilayer is to apply Voronoi analysis in two dimensions (Jedlovszky et al., 2004; Patra et al., 2003; Shinoda and Okazaki, 1998). In Voronoi tessellation for a bilayer, the center of mass (CM) coordinates of the molecules comprising the bilayer are projected onto the x,y plane. An arbitrary point in this plane is considered to belong to a particular Voronoi cell, if it is closer to the CM position associated with that cell than to any other one. In this way one can calculate the total area associated with the CM positions of, e.g., the DPPC molecules and then scale this quantity by the number of DPPC molecules in a monolayer. The resulting average areas per DPPC and cholesterol, 
and 
as functions of the cholesterol content, are depicted in Fig. 7.
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; Venable et al., 2000). The differences are in part due to the fact that the assumptions inherent to the respective methods lead to different ways of distributing the free area in the bilayer. This, however, does not fully explain the large differences and the peculiar values obtained using the Voronoi analysis at high cholesterol concentrations. A more important reason is that basic Voronoi analysis does not, in any way, allow one to take into account the close-packed sizes of the molecules. It may well be that the area (which from the point of view of the Voronoi analysis belongs to cholesterol), as a matter of fact would be covered by projected coordinates of atoms from a DPPC molecule. Concluding, although Voronoi analysis may be a useful tool for studying, e.g., phase separation or local effects, the values for molecular areas from such analysis have no quantitative meaning.
Slicing membranes
We are now confronted by fundamental questions relevant to both one- and multicomponent bilayer systems. How can we find estimates for the average close-packed cross-sectional areas for the molecular species present in a one-component or composite bilayer? Further, how can we estimate the average amount of free area in a membrane?
Our approach to answer these questions bears a certain resemblance to tomography. Related grid approaches have been used in other applications (see, e.g., Kandt et al., 2004, and references therein). We map each configuration on a number of rectangular three-dimensional grids as follows. If a grid point lies within the van der Waals radius of an atom belonging to a DPPC molecule, this point is considered occupied, and otherwise empty, on a grid keeping account of DPPC molecules. Grid points within van der Waals radiae of atoms belonging to cholesterol, in turn, will be occupied on a grid characterizing the cholesterol molecules. Finally, a grid for water molecules is constructed analogously. In the x,y plane the grids have 100 x 100 elements. Because the system size fluctuates weakly, the size of an element will vary slightly from configuration to configuration. In the z direction, on the other hand, the size of the elements has been fixed to 0.1 nm, and we only consider grid points within 3 nm from the bilayer center.
The grids can be used to view given slices of the bilayers: they show cross sections of DPPC, cholesterol, and water molecules, as well as patches of free area. Pictures of slices can be illustrative as such, and Fig. 8 contains a selection of such slices for the case of 20.3% cholesterol. From Figs. 5 and 8 a we can conclude that there are quite large amounts of free area in the bilayer center, and that cholesterol tails from a given monolayer extend to the opposite monolayer. Fig. 8 b portrays the region where DPPC tails and cholesterol ring structures should, according to Fig. 5, dominate. DPPC tails can be recognized as circular red structures, and the green formations are cross sections of cholesterol ring structures. Fig. 8 c is a cross section of the bilayer at a distance z 1.7 nm from the bilayer center. Some cholesterol is still present in this slice, and there are also small amounts of water. The amount of free area is significantly smaller than in the bilayer center. Fig. 8 d finally shows a cross section at z 2 nm: there are DPPC headgroups, substantial amounts of water, and very little cholesterol.
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to characterize a pure DPPC bilayer. Fig. 9 exemplifies the computation of the various area profiles for a bilayer with 20.3% cholesterol.
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ADPPC(z) and Achol(z), for the different cholesterol concentrations are computed in the manner described above.
To find the average numbers of DPPC and cholesterol molecules in each slice, we locate the maximum and minimum z coordinates of each molecule with respect to the bilayer center, taking into account the finite size of the constituent atoms. The molecule is considered to be present in all the slices between these points. By averaging over all molecules of a certain species and over all configurations, we arrive at the average numbers of DPPC molecules and cholesterols as functions of the distance from the bilayer center, denoted by NDPPC(z) and NChol(z), shown in Fig. 10. Perhaps the most notable feature in Fig. 10 is that all curves peak in the bilayer center. This is due to so-called interdigitation: a substantial part of both DPPC and cholesterol molecules extend to the opposite monolayer. On both sides of the peak, there are broad plateaus, which reflect the amount of molecules of a certain species in a monolayer. Eventually, at 3 nm from the bilayer center for DPPC and 2 nm for cholesterol, the curves decay to zero.
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). Further, the tilt of the DPPC molecules with respect to the bilayer normal decreases (Róg and Pasenkiewicz-Gierula, 2001). Cholesterol, on the other hand, with its rigid ring structure and short tail, does not undergo such significant extension.
The second effect partially responsible for the thickening is that with a larger cholesterol content , a smaller amount of DPPC and cholesterol molecules extend to the opposite monolayer. Further, the ones that protrude do not penetrate quite as deep into the opposite leaflet as they do at low cholesterol concentrations. In a pure DPPC bilayer, 53% of the DPPC molecules protrude to the opposite monolayer, whereas at 29.7% cholesterol the corresponding figure is 40%. The effect is stronger for cholesterol: at 4.7% and 29.7% concentrations, respectively, 41% and 17% of the molecules extend to the opposite bilayer. As the cholesterol hydroxyl is thought to be anchored to the DPPC headgroup via direct hydrogen bonding or through water bridges (Chiu et al., 2002; Pasenkiewicz-Gierula et al., 2000), this effect may be coupled to the elongation of the DPPC molecules.
Equipped with the total areas occupied by the molecular species together with the average numbers of these molecules as functions of distance from the bilayer center, we can now compute the average cross-sectional areas for DPPC and cholesterol across a membrane, aDPPC(z) 
ADPPC(z)/NDPPC(z) and achol(z) Achol(z)/Nchol(z), shown in Fig. 11.
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1 nm from the bilayer center for a pure DPPC bilayer becomes at intermediate cholesterol concentrations a plateau at 0.5–1.5 nm from the center, and finally with 29.7% cholesterol in the bilayer develops into two small maxima at 0.5 nm and 2 nm with a shallow minimum in between. These changes in aDPPC(z) are in part due to the behavior of the phospholipid tails: significant changes occur in regions where the tail densities are high and where there are few or no headgroups. This can be deduced by comparing the electron densities for DPPC molecules and DPPC tails in Fig. 5, a and d. This allows us to partially interpret the behavior of aDPPC(z) from the point of view of ordering. The most substantial ordering effect with large amounts of cholesterol present in the bilayer occurs for carbons in the middle of the tail; see Fig. 3. Close to the headgroups and in the bilayer center the ordering effects of cholesterol are more modest. As increased order correlates with a decreasing area occupied by the tails, one expects that with an increased cholesterol content the cross-sectional area per DPPC approximately at a distance 1 nm from the bilayer should decrease. Our findings are consistent with this picture.
This is not to say, however, that there would exist a simple way of mapping aDPPC(z) with order parameter profiles (see also Close-Packed Areas from Ordering of Acyl Chains, below). The maximum that develops at z 2 nm, e.g., is a result of contributions from glycerol, phosphate, and choline groups. From separate cross-sectional area profiles for the two tails on one hand and the glycerol, phosphate, and choline groups on the other hand (data not shown), we found that when the cholesterol concentration increases, the cross-sectional area occupied by the tail portion of a DPPC molecule decreases as a consequence of ordering, whereas the area occupied by the glycerol, phosphate, and choline groups seems to be increasing (data not shown). The increase is probably related to changes in the orientation of these groups. Concluding, the maximum at z 2 nm at intermediate and high cholesterol concentrations is related to the interplay of the decreasing tail contribution with a plateau centered at z 1 nm and the increasing head contribution that peaks at z 2 nm.
In the case of cholesterol the cross-sectional close-packed area of a molecule is changed only weakly when the cholesterol concentration is increased. The slight decrease with an increasing can be explained by the tilt of the cholesterol molecules. At high concentrations, almost all cholesterols are oriented nearly parallel to the bilayer normal (data not shown). At low concentrations, on the other hand, the distribution of the angle between the bilayer normal and the ring structure becomes more broad and flat, i.e., the molecules are more tilted with respect to the bilayer normal. Hence the cross sections appear larger at low concentrations.
The general form of achol(z) is compatible with our idea of the structure of the cholesterol molecule: narrow in the bilayer center where the small cholesterol tails reside and broad where the ring structure is located. It also reflects the thickening of the bilayer, as the maxima associated with the ring structures are pushed toward the water phase when more cholesterol is present. This picture, overall, supports the common belief that the average area per cholesterol in a phospholipid bilayer is largely unaltered by the amount of cholesterol in the bilayer.
Our results for achol(z) can be compared to the outcome of an old experiment (Rothman and Engelman, 1972), where a model of cholesterol made of plastic was immersed in a tube filled with water. This experiment resulted in a steric profile for cholesterol, i.e., a profile of the cross-sectional area occupied by cholesterol. This steric profile and our achol(z), especially at high cholesterol concentrations, bear a surprisingly good resemblance to each other. The steric profile measured by Rothman and Engelman displays a plateau where the cholesterol rings are located, with cross-sectional areas of the order of 0.25 nm2. In the region where the cholesterol tail is located, they report a small maximum: here the cross-sectional areas are of the order of 0.15 nm2.
It is clearly difficult to describe the close-packed area of a DPPC or cholesterol molecule by a single number. Of course, we could attempt to define the close-packed area of, e.g., a DPPC molecule in a given DPPC/cholesterol bilayer as the maximum of the relevant aDPPC(z) profile, but this would not give accurate information about the packing of DPPC and cholesterol molecules in a composite bilayer. Despite this, we may note that the maximum values are useful at least when assessing the plausibility of the close-packed area profiles for DPPC and cholesterol molecules.
In the case of DPPC the maxima assume values between 0.36 nm2 and 0.42 nm2. These values can be compared to the average area per molecule in a pure DPPC bilayer in the gel state, where the contribution of the free area to the total area assigned to a phospholipid molecule is expected to be rather minor. Experiments have yielded an area per molecule of 0.48 nm2 (Nagle and Tristram-Nagle, 2000), and MD simulations suggest that A = 0.46 nm2 (Venable et al., 2000). An exact comparison is not meaningful, since DPPC/cholesterol mixtures, especially with high cholesterol concentrations, have structures quite different from a pure DPPC bilayer in the gel state. However, the comparison shows that the magnitude of the close-packed areas for DPPC molecules is rational.
In a similar fashion, the maxima of the achol(z) profiles can be compared to values extracted from experiments on cholesterol crystals. The maxima found in this study decrease monotonically from 0.33 nm2 to 0.29 nm2 when the cholesterol concentration changes from 4.7% to 29.7%. In a cholesterol crystal, the area per cholesterol, which in this case contains both occupied and free area, has been reported to be 0.38 nm2 (Craven, 1979; Chiu et al., 2002; Hofsäß et al., 2003; and references therein).
Free area
We now turn our attention to the behavior of free area profiles for bilayers with different amounts of cholesterol. In Fig. 12, we show the average amount of free area per molecule, i.e., afree Afree/N, where N is the total number of molecules—both phospholipids and cholesterol—in a monolayer. The figure clearly shows that the amount of free area per molecule decreases in all regions of the bilayer, i.e., for all values of z, with an increasing cholesterol content. Compared to the case of pure DPPC, 4.7% cholesterol in the bilayer leads to a free area per molecule reduced by 7% in all regions of the bilayer. With 12.5%, 20.3%, and 29.7% cholesterol in the bilayer, the free area per molecule is decreased by 20%, 35%, and 45%. One may note that the behavior of the total area of the bilayer cannot be explained by the reduced free area only. The occupied area, i.e., the area taken up by DPPC, cholesterol, or water molecules, also decreases with more cholesterol. For instance, when 29.7% of the DPPC molecules are substituted by cholesterol, the amount of occupied area decreases by 30%. The behavior of the total free and occupied volumes in a bilayer with an increasing cholesterol concentration will be discussed in detail elsewhere.
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1.7 nm. For large cholesterol concentrations the minimum is still present, but due to the thickening of the bilayer it is pushed toward larger z: e.g., for 29.7% it can be found at z 2 nm. This minimum can, for all cholesterol concentrations, be associated with a peak in the density profile of DPPC molecules located slightly behind the headgroups in a region where tails, glycerol, phosphate, and choline groups are present. The density of water in this region is already substantial, whereas there is very little cholesterol. When the cholesterol concentration is increased, also another flat, plateau-like minimum starts to develop between the bilayer center and the minimum associated with the maximum in the DPPC density, i.e., at z 1–2 nm. The plateau is almost constant through the tail and headgroup regions. It has counterparts in the area profiles of DPPC and cholesterol: the cross-sectional DPPC area displays here a flat minimum and the cholesterol area a broad maximum (see Fig. 11). We can thus conclude that the changes in the form of the free area profile are intimately related to modifications in the packing of the molecules in the bilayer.
It is evident that the free area profiles are related to the relocation and diffusion of solutes inside membranes. MD simulations suggest that solutes such as ubiquinone (Söderhäll and Laaksonen, 2001) and benzene (Bassolino-Klimas et al., 1993) are preferentially located in the hydrophobic core region of a membrane. Also, it is known that certain nonpolar probe molecules, e.g., diphenylhexatriene, prefer the bilayer center to the lipid/water interface (Lentz, 1993). These observations are in accord with our suggestion that the free area is largest in the bilayer center.
There are two other simulation studies where quantities similar in nature to our free area profile have been calculated for DPPC or DPPC/cholesterol bilayers. Marrink et al. (1996) have calculated a so-called empty free volume profile for pure DPPC. This should give essentially the same information about the amount of average free area in a given cross section of the bilayer as does our free area profile for pure DPPC. Our profile does indeed show the same general features as Marrink's: a maximum in the bilayer center and minima near the headgroup region. Tu et al. (1998) have also looked at the influence of 12.5% cholesterol on a so-called empty free volume fraction, which is equivalent to our total free area scaled by the total area of the bilayer. If we compare such scaled free areas (data not shown) to Tu's data, we see that the scaled profiles have many features in common. One difference is that the bilayer thickening is not visible in Tu's results, whereas it can be clearly distinguished from ours. The thickening has also been verified experimentally. Further, there are some differences in the detailed form of the profiles in the bilayer interior, i.e., the location of the minima are slightly different. As Tu et al. point out, the differences are probably due to different computational models.
Lateral diffusion and free area
We have seen that an increasing cholesterol concentration reduces the amount of free area per molecule in the bilayer and simultaneously alters the packing of the molecules. On the other hand, it is well known from experiments that lateral diffusion of both DPPC and cholesterol molecules is affected by changes in the cholesterol content (Almeida et al., 1992; Filippov et al., 2003a; König et al., 1992). It is reasonable to expect that these properties of the bilayer and the observed modifications in them with the cholesterol concentration are related. Free volume theory is a simple but appealing model for explaining such dependencies.
Free volume theory was originally developed for describing the transport properties of glass-forming fluids (Cohen and Turnbull, 1959; Macedo and Litovitz, 1965; Turnbull and Cohen, 1961, 1970). It was subsequently adapted to modeling two-dimensional diffusion (Galla et al., 1979; MacCarthy and Kozak, 1982; Vaz et al., 1985; Almeida et al., 1992) and is usually in this context dubbed free area theory. Free area theory, a two-dimensional mean-field model for diffusion, can be used to at least qualitatively describe lateral self-diffusion in lipid bilayers (Almeida et al., 1992). According to free area theory, lateral diffusion of a lipid or sterol in a bilayer is restricted by the occurrence of a free area greater than some critical area adjacent to the diffusing molecule. A diffusing molecule spends a comparatively long time—of the order of tens of nanoseconds (Tieleman et al., 1997; Vattulainen and Mouritsen, 2003)—in a cage formed by its neighbors, and then, given a large enough activation energy and an adjacent free area, jumps.
More specifically, free area theory predicts that the lateral diffusion coefficient of a lipid or sterol diffusing in a bilayer depends on the free area and the packing properties as follows (Almeida et al., 1992):
 | (3) |
To examine the validity of Eq. 3 we compute the lateral diffusion coefficients for DPPC and cholesterol molecules at different cholesterol concentrations. The lateral tracer diffusion coefficients can be computed using the Einstein relation
 | (4) |
is the CM position of molecule i at time t and the sum is over all molecules of a given species. The lateral diffusion coefficients have been calculated by following the position of each molecule in the upper (lower) monolayer with respect to the CM position of the corresponding upper (lower) monolayer. Thus, should there be any drift, the motion of the CM of each monolayer has been taken into account.
Results for lateral diffusion coefficients are shown in Fig. 13. The lateral diffusion coefficients for both DPPC and cholesterol decrease monotonically with an increasing cholesterol content. This reduction is qualitatively consistent with experiments (Almeida et al., 1992; Filippov et al., 2003a; König et al., 1992). Quantitative comparisons should preferably be made to experimental techniques that probe lateral diffusion of individual molecules at timescales comparable to those reached in MD simulations. Fluorescence correlation spectroscopy measurements should hence give us a good reference. In fluorescence correlation spectroscopy measurements for DLPC/cholesterol systems, Korlach et al. (1999) found that when the cholesterol concentration was increased from 0% to 60%, DT for DLPC was reduced by a factor of 10. Even though the acyl chains of DLPC molecules are shorter than those of DPPC molecules, our findings are in reasonable accord with Korlach's experiments.
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In our opinion, one should at least not expect free area theory to yield quantitative results. It might, however, give qualitative predictions about trends in cases where, e.g., the cholesterol content in a bilayer is increased. With this in mind, let us assume that the cholesterol concentration in a DPPC/cholesterol bilayer rises from 4.7% to 29.7%. If we now use the largest possible values for the fractions a0/af, free area theory will give us upper bounds for the reduction of the lateral diffusion coefficients. The lateral diffusion coefficient for DPPC should, according to free area theory, now be reduced by a factor of 3 at most, and DT for cholesterol should decrease by a factor of 2. As a matter of fact, the lateral diffusion coefficients for both DPPC and cholesterol computed from the simulation data are reduced much more strongly; see Fig. 13. We can conclude that Eq. 3 tends to underestimate the changes in the values of the lateral diffusion coefficients.
Even though the discrepancies in the predictions of Eq. 3 and the computed lateral diffusion coefficients do exist, we cannot immediately declare free area theory incomplete. There is a detail that has been overlooked in our discussion so far, and the significance of this detail will now be considered. To jump to an adjacent empty site, a diffusing molecule needs energy to overcome an activation barrier. In free area theory this is accounted for by letting the lateral diffusion coefficient be proportional to a Boltzmann factor exp(–Ea/kBT), where Ea is the activation barrier. As a growing cholesterol concentration increases the ordering of the DPPC tails and therefore reduces the area per molecule, it seems reasonable to expect that Ea should increase with the cholesterol content. Experimental results (Almeida et al., 1992) do support this idea but are partly contradictory. This is, however, probably due to the fitting procedure used (Almeida et al., 1992). In a more recent study, Filippov et al. (2003b) used NMR to study the lateral diffusion in palmitoyloleoylphosphocholine/cholesterol and dioleoylphosphocholine/cholesterol bilayers over a cholesterol concentration range of 0–45 mol %. At small , they found the apparent (Arrhenius) diffusion barrier to be approximately constant, whereas for large the diffusion barrier increased markedly. Hence, the neglect of the energy term might in our case lead to slight underestimates for the reduction of the lateral diffusion coefficients.
Summarizing, we have found that free area theory correctly predicts the reduction of the lateral diffusion coefficients with an increasing cholesterol concentration. At the same time it seems unnecessary to aim for a quantitative description with such a simple framework. Instead of being based on mean-field arguments, a full theoretical description of lateral diffusion should account for local free volume fluctuations in the vicinity of diffusing molecules. Atomic-scale MD studies in this direction should be feasible in the near future.
Area compressibility modulus
Lateral diffusion is clearly influenced by the average amount of free area in the bilayer. However, not only the average free area, but also fluctuations in the amount of free area should play a role here. Recall that free area theory states that a diffusion jump is not possible unless there is a large enough free area next to the diffusant (Almeida et al., 1992). Large enough free areas are a result of fluctuations, and hence we would expect diffusion to depend on the magnitude of the fluctuations: decreasing fluctuations and slowed lateral diffusion should be coupled. For similar reasons, it is likely that permeation of molecules across membranes can at least partially be explained by area fluctuations in membranes.
We may quantify area fluctuations in different regions of the membrane as follows. The starting point is the average occupied area Aocc(z), i.e., the area which is not free but occupied by DPPC, cholesterol, or water molecules. The occupied area obviously varies with the distance from the bilayer center z. Based on the definition of compressibility modulus given in Feller and Pastor (1999) and Hofsäß et al. (2003), we now define an area compressibility modulus for the occupied area as
 | (5) |
), i.e., 
), 20.3% (
), 29.7% (
), and 50.0% (
), and the index n increases toward the center of the bilayer.
computed using Voronoi tessellation. The errors for DPPC are smaller than the markers.
