The MCT fits to our x-ray scattering data taken at high resolution are
quite good, as seen visually by the fits in Figs. 5-7.
Although the reduced 
values in Table I are systematically greater than one, this is to be expected
because the only errors used in computing the 
were the counting statistics; accounting for other errors, such as small
random fluctuations in the two-theta angles, would reduce the 
values. Therefore, the modified Caillé theory appears to be appropriate
for lipid bilayers. By comparison, the PT fits are clearly worse than MCT
fits, with much larger 
values and with systematic residuals in the tails and the central peaks
that are not present in the MCT fits.
Theoretically, it is not surprising that MCT is more appropriate than
PT for L
phase
multilamellar vesicles. Whereas PT attempts to guess the nature of the
fluctuations by imposing a stochastic assumption on the correlations between
bilayers, MCT utilizes a reasonable free energy to describe the fluctuations.
Furthermore, MCT allows for both undulation fluctuations as well as compressional
fluctuations; PT only considers the latter.
PT fits consistently underestimate the central peak and the tails and overestimate the shoulders. PT tries to compensate for the sharp central peak by predicting a domain size larger than that given by MCT and by underestimating the size of the fluctuations of the distance between nearest neighboring membranes (see Table I and Fig. 3). As will be shown in subsequent work, the larger root mean square fluctuation in nearest neighbor separations required by MCT is still safely less than the mean water spacing, which is of order 20 Å when D=67.2 Å.
Most importantly, PT predicts that the scattering in the tails of the peaks should fall off in a Lorentzian fashion as in Eq. 7. This contrasts strongly with the slower power law decay of MCT (Eq. 8) which gives rise to a significant fraction of the scattering occurring in the tails. PT cannot account for this scattering in the tails and it therefore underestimates the form factors for the higher orders, as can be seen in the last row of Table I and in part C of the Theory Section.
It may also be emphasized that simple integration of the experimental scattering data underestimates the tail scattering by about the same amount as PT. While the signal is small in these tails, the range of q values is five times the range shown in Fig. 3, so that the lost intensity is significant. Even if one were able to measure the intensity throughout this larger range of q, where the number of counts is small and the background is larger than the signal, the form factors F(q) are continuously varying, so that straightforward integration would yield a distorted result (Zhang et al, 1994). However, with a good theory such as MCT, one can reliably extrapolate the tails into this region from the measurable data shown in Figs. 5-7 for the region near the peaks where the form factor is nearly constant.
The present paper provides the basis for obtaining more reliable form
factors for disordered and fluctuating L
phases. In a subsequent paper, these form factors will be used to determine
better electron density profiles and to address various structural issues
in lipid bilayers.
Acknowledgments: This research was supported by NIH Grant No. GM44976. CHESS is supported by NSF Grant DMR-9311772 and MacCHESS is supported by NIH Grant RR01646.