Interpretation of small angle X-ray measurements guided by molecular dynamics simulations of lipid bilayers
Introduction
X-ray diffraction studies of lipid bilayers have been specifically geared towards elucidation of the electron density profile, from which structural parameters such as the bilayer thickness, DB, and the area per lipid, A, can be calculated (Nagle and Tristram-Nagle, 2000). Obtaining accurate estimates of these quantities is important for two main reasons. First, the close matching between the bilayer thickness and the hydrophobic part of membrane proteins appear to control protein function Huang, 1986, Bloom et al., 1991, Harroun et al., 1999. It has been hypothesized that structural matching is biologically regulated via selection of lipids with proper chain length and saturation (Bloom et al., 1999). Depending on the length and saturation, bilayer thicknesses can differ by 10–15 Å Rand and Parsegian, 1989, Nagle and Tristram-Nagle, 2000. Yet, significantly more subtle variations are known to influence ion channel lifetimes (Elliott et al., 1983) and conformations (Greathouse et al., 1994), as well as the orientations of hydrophobic helical peptides Killian and Heijne, 2000, Petrache et al., 2000, Petrache et al., 2002. Similarly, structural properties associated with the lipid cross-sectional area and lateral stress Gruner, 1989, Brown, 1994, Botelho et al., 2002 are known to modulate protein function, as well as membrane permeability.
Second, accurate measurements of structural parameters are needed for quantification of interbilayer interactions McIntosh and Simon, 1986a, Rand and Parsegian, 1989, Israelachvili, 1992, Leikin et al., 1993, Zimmerberg and Chernomordik, 1999. At small distances, interbilayer forces depend exponentially on the interbilayer separation, changing by 20% with a 1 Å change in separation McIntosh and Simon, 1986a, McIntosh and Simon, 1993, Marsh, 1989, Rand and Parsegian, 1989, Petrache et al., 1998a. Here the relevant structural parameter is the water spacing DW, which is calculated from the lamellar repeat D, as DW=D−DB. The accuracy of DW then depends on the determination of DB.
Measuring structural parameters for soft, highly fluctuating materials such as the lipid membrane, is a difficult task. Structural descriptions are more readily obtained for the lower temperature states: gel Nagle and Wiener, 1989, Sun et al., 1996a, Tristram-Nagle et al., 2002, subgel Tristram-Nagle et al., 1994, Katsaras, 1995 and, obviously, crystal (Small, 1986), where lipid hydrocarbon chains are ordered. Both normal (thickness) and lateral (cross-sectional area) parameters can be determined directly from the low and the wide angle X-ray scattering, respectively. For the fluid (melted chain) phase, the apparent structural resolution is 5–10 Å, corresponding to the spatial extent of the molecular distributions, as presented by the electron density profiles Worthington, 1969, Blaurock et al., 1971, Wiener and White, 1992. However, by parsing the lipid bilayer into molecular components, such as the lipid headgroup and acyl chains, average structural parameters are commonly reported with a precision of 1 Å or less Rand and Parsegian, 1989, Nagle and Tristram-Nagle, 2000, Rawicz et al., 2000.
The most readily available parameter from the electron density profile is the spacing DHH between the electron rich headgroup peaks. By itself, DHH is not sufficient for a complete description of structure, because its relationship with lateral parameters (area per lipid, A) is complicated by the broad lipid–water interface. One needs to relate DHH to better defined, and thermodynamically relevant parameters such as the hydrocarbon thickness DC and the total bilayer thickness DB, which are related to A through molecular volumes. Therefore, it is necessary to combine electron density and volumetric analyses in order to describe fluid phase bilayer structure McIntosh and Simon, 1986b, Nagle et al., 1996, Petrache et al., 1997, Armen et al., 1998, Tristram-Nagle et al., 1998, Nagle and Tristram-Nagle, 2000.
Given the experimental scattering data, i.e. the form factor ratios rh=Fh/F1, there are two main methods to construct the electron density profile, ρ*(z). The first method is model-free and consists of direct Fourier reconstruction (Worthington, 1969), where ρW* is the bulk water electron density, D is the lamellar repeat spacing, αh=±1 are form factor phases, and hmax is the number of observed diffraction orders. There are two quantities in Eq. (1) that are usually not available from X-ray. These are F1, which in this formalism sets ρ*(z) on an absolute scale, and F(0) which gives the total bilayer contrast (offset) relative to the water electron density ρ*W, in which A, VL, and nL* denote the area, volume, and number of electrons per lipid molecule, respectively. Handling the scale has been an issue for the fluid phase, as this requires additional information or certain assumptions with regard to the shape of the electron density profile (Petrache et al., 1998b). The shape, however, is strongly influenced by the number of diffraction orders available, hmax, which truncate the sum in Eq. (1). Fourier truncates complicates comparison between lipid bilayers.
The second method for constructing electron density profiles is functional modeling. This is done by assuming a particular functional form for the bilayer profile, with a number of free parameters to be determined by fitting to scattering form factors. Several approaches have been undertaken, from simple step-function models (Worthington, 1969), to more realistic Gaussian models Nagle and Wiener, 1989, Wiener et al., 1989, Wiener and White, 1992 and to more detailed component models Wiener and White, 1992, Schalke and Losche, 2000. Each of these models attempt to breakdown the electron density into component distributions by integrating knowledge from other measurements, such as specific volume, to reduce the number of fitting parameters.
Atomic-level computer simulations provide a new perspective on bilayer structure. There are numerous valuable contributions to the field addressing the underlying molecular disorder and heterogeneity Chiu et al., 1995, Berger et al., 1997, Tieleman et al., 1997, Feller et al., 1997, Tobias et al., 1997, Smondyrev and Berkowitz, 1999, Huber et al., 2002. In the hierarchy of models just discussed, these in fact constitute the most elaborate. One consequence of such detail, however, is that molecular dynamics simulations cannot be cast as fitting procedures as with the models above, as tuning the force-field parameters to a particular scattering dataset is unfeasible. Of course, the aims of simulations are more ambitious than just modeling of electron density profiles, but in this work we will focus just on this aspect. Because simulated electron density profiles result from all-atom representations which implicitly obey volume conservation, they can be used to evaluate assumptions employed in structure determination from scattering data.
A “bootstrap” method has been used in the literature to obtain structural parameters for a new bilayer by comparison with a reference structure McIntosh and Simon, 1986b, Nagle and Tristram-Nagle, 2000. It is assumed that the difference in the hydrocarbon thickness ΔDC can be estimated from the shift of the headgroup peak, ΔDHH/2, if two lipids have the same headgroup. (A factor of 1/2 is needed because DC is conventionally defined as half-thickness.) Having estimated DC from the DHH shift, the area per lipid is then obtained as the ratio between hydrocarbon volume and thickness. DHH does not depend on F(0) or the density scale, but needs to be corrected for Fourier truncation effects Blaurock et al., 1971, Lesslauer et al., 1972. To minimize these effects, DHH values have customarily been compared at similar resolution, D/hmax McIntosh and Simon, 1986a, McIntosh and Simon, 1986b, McIntosh and Simon, 1993, Rawicz et al., 2000, and correction terms were estimated using functional modeling (Sun et al., 1996a). With the availability of detailed atomic simulations, this aspect calls for renewed attention.
Here we consider previously reported MD simulations of fluid phase DMPC and DPPC bilayers as the basis for our discussion of structure determination from X-ray. We have focused on the Fourier reconstruction method and the calculation of the area per lipid from DHH and density measurements. The main goal is a comprehensive exercise with electron density profiles, and not necessarily simulation refinement. We have used simulated bilayers with structural parameters within the experimental uncertainty (1.5% or better), as a scaffolding for construction of self-consistent methodologies for structure determination from experiment. Because the bilayer form factors are the primary X-ray data, we have compared simulation and experiment in the Fourier space. We show an overall agreement, especially in the low q range, which seems to have the major influence on the main structural parameters. We process the simulated continuous transform by artificial sampling and cutoff at high q-values to mimic the measurement of head–head spacing DHH. We then compare DHH differences between DMPC and DPPC with differences in the hydrocarbon thicknesses DC to test the bootstrapping assumption. By analysis of Fourier truncation effects, we identify conditions for ΔDC≈ΔDHH/2 and estimate a possible deviation range.
By comparison of continuous transforms, we identify and fix a scale discrepancy between simulation and experiment. We propose an improved method for setting experimental electron density profiles on an absolute scale by using structural and volumetric parameters obtainable from unscaled profiles. While not directly relevant for measurements of DHH and subsequent determination of bilayer thicknesses, a correct absolute scale is critical for X-ray contrast (substitution) experiments (Franks et al., 1978).
Section snippets
Simulation setup
Simulations were performed using CHARMM software (Brooks et al., 1983) version 26 and were previously reported (Petrache et al., 2002). Periodic boundary conditions were used with constant number of atoms (N), temperature (T), lateral area (A), and normal pressure (PN) to generate NAPNT ensembles. Two lipids were considered: dimyristoylphosphatidylcholine (DMPC), and dipalmitoylphosphatidylcholine (DPPC). Simulation temperatures were C for DMPC and C for DPPC. For both lipids, bilayers
Fourier truncation and head–head spacing DHH
Simulated electron density profiles for DMPC at C and DPPC at C are shown in Fig. 1. Electrons were counted in bins of (in the z-direction normal to the bilayer) and averaged over the simulation. The most prominent features of the profiles are the headgroup peaks at the lipid–water interface and the methyl troughs at the bilayer center. The two profiles shown are similar in shape, with DPPC being broader by about 2–3 Å due to its longer hydrocarbon chains. Relative to the water
Discussion
Guided by molecular dynamics simulations of DMPC and DPPC, we have taken a critical look at the Fourier reconstruction method used to obtain structural parameters from X-ray. We have addressed two main aspects. First, we have estimated the effect of Fourier truncation on the headgroup peak location, and the uncertainty in measuring ΔDC using the headgroups peaks. We have used smoothed (4-order) electron density profiles corresponding to the typical resolution achievable by experiments. Between
Acknowledgements
We thank Dr. John F. Nagle for valuable discussions and comments on the manuscript, and Dr. Thomas Huber for discussions of continuous transforms and of his simulation results. J.N.S. thanks the Whitaker Foundation for Biomedical Engineering for graduate fellowship support.
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