It is well known that lipid bilayers have different thicknesses and
different areas per lipid molecule for different lipids (Rand and Parsegian,
1989; Thurmond et al, 1991; McIntosh and Simon, 1986). However,
the actual values of these quantities are rather poorly determined (Tristram-Nagle
et al, 1993; Nagle, 1993) , so that the differences between different
lipid bilayer systems are comparable to the experimental uncertainty, especially
for the biologically relevant fluid 
phase.
One source of uncertainty in structural determinations of lipid bilayers is endogenous, namely, these are not crystalline systems, so the methods of crystallography can not necessarily be expected to apply. Nor can a realistic goal be to obtain atomic structure at Angstrom resolution, since the systems are disordered and fluctuating. The fluctuations are described by various kinds of correlation functions, some of which are more fluid-like than crystal-like (Wiener and White, 1991). Indeed, the most popular sample preparation for structure determination consists of multilamellar vesicles (MLVs) that are best characterized as liquid crystals.
Even though the liquid crystal nature of MLV samples of lipid bilayers is well known, the occurrence of very sharp, well separated, small angle scattering peaks means that there are well defined D-spacings in MLVs. This, in turn, has motivated the determination of low resolution structure along the bilayer normal by measuring the intensities of the scattering peaks and applying the usual Lorentz correction to obtain the square of the form factors (Torbet and Wilkins, 1976; Worthington and Khare, 1978; Franks and Lieb, 1979; McIntosh and Simon, 1986; Kim et al, 1987; Wiener et al, 1989). After applying various methods to obtain the phases, electron density profiles have then been obtained.
The standard procedure in the preceding paragraph does not take into account the liquid crystalline nature of MLVs. The analysis assumes that each scattering peak is a Bragg peak, with perhaps some broadening due to finite size of the scattering domains. However, there are two theories that both show that disorder removes scattering intensity from the central peaks and pushes it into the troughs between the peaks where it merges with the background and cannot be accurately measured. Most importantly, this effect becomes progressively larger as the order h of the scattering peaks increases. This is a major factor accounting for the absence of higher order peaks. Even for those peaks that one can observe, measuring only the intensities under the central peaks systematically underestimates the higher order form factors, thereby degrading the electron density profile.
One theory that allows the above artifact to be corrected is paracrystalline theory (PT) (Hosemann and Bagchi, 1962; Guinier, 1963); this theory has been applied to multilamellar arrays of retinal rod membranes (Schwartz et al., 1975; Worthington, 1981) and nerve myelin (Blaurock and Nelander, 1976). Another theory is the Caillé theory (CT) (Caillé, 1972), recently modified (MCT) (Zhang et al., 1994); this theory has been applied to multilamellar arrays of lipid bilayers and to various liquid crystal systems (Roux and Safinya, 1988; Zhang et al., 1995). While these two theories are both based on the general notion of disorder, the details of the theories are quite different and, most importantly, the predicted corrections are different, as we show in the theory section of this paper. The primary goals of this paper, then, are to determine whether either of these two theories describes scattering from lipid bilayers and, if so, which one is better. If this is successful, appropriate corrections to the form factors can be made in subsequent work to obtain better electron density profiles.
This kind of work requires that the experimental shapes of the peaks
and their tails be well resolved. Fortunately, even though the peaks are
very sharp, it is possible to resolve their shapes (not just their separations,
which is easy) using high instrumental resolution diffraction (HWHM in
of 0.0001
Å
). Since very
few photons are scattered by lipid bilayers in such small angular ranges
(
0.001
),
we also use a synchrotron source. This combination enables us to achieve
our primary goals.