Understanding such key biological processes as molecular recognition, protein insertion, and molecular self-assembly in living biological membranes remains a challenge. Membrane materials mimicking biological ones (biomimetic membranes) provide model systems to address this issue, and, because of its special sensitivity to hydrogen, neutron reflectivity (N R) offers a unique way to reveal thin film structures in biomimetic membranes.
The feasibility of using phase-inversion techniques in N R to reveal such structural details has been demonstrated (Refer to reference 1). It is possible to directly measure the real part, r sub 1 of (Q), of the complex reflection coefficient r of (Q), as a function of wave vector Q and to mathematically invert it to obtain the scattering length density (S L D) depth profile rho of (x), of the film. There remains much to do, however, in defining the reliability of the results, especially in accessing the unavoidable reduction of spatial resolution induced by data truncated at a maximum wave vector, Q sub max. Wavelet analysis provides a systematic and useful approach to the problem (Refer to reference 2).
In using Wavelet Multi Resolution Analysis (W M R A), S L D profiles are characterized by a finest length scale, ell, sub zero, approximately equals one Angstrom, and coarser scales, ell sub j, equals two to the minus j power, times ell sub zero, with j negative. rho of (x), is viewed as a coarse “trend”, rho sub J, of (x), at resolution level J, with added “detail”, delta sub capital J, rho of (x), giving the trend at the next finer scale, rho of (x), sub capital J, plus 1 = rho of (x), sub capital J, plus delta sub capital J, rho of (x). Relative to a base scale of resolution ell sub capital J, the S L D profile thus can be represented by the trend plus all remaining detail, rho of (x), equals rho sub capital J of (x), plus sum over capital J less than or equal to lower case j, of delta sub capital J, rho of (x). The experimental rho of (x), associated with a given Q sub max determined by the instrument, is a blurred representation of the veridical S L D. It can be thought of as a coarse image bracketed by neighboring trends for this Q sub max.
W M R A provides spatially localized orthonormal bases for this description. A family of wavelets called Daubechies - 8 seems well suited to N R analysis. For illustration, we use a realistic S L D profile obtained by molecular modeling to represent a hybrid lipid membrane on a thin gold film (diagram on top of Figure 1), a biomimetic system typical of those being studied in many laboratories. The model rho of (x), seen in the back panes of Figures 1 and 2, consists of the gold layer, a hydrogenated alkanethiol layer, and a deuterated lipid monolayer.
Figure 1. Diagram of model (top) for which the computed scattering length density rho of (x), is the thick grey curve. Trends converge: rho of x, sub minus 4, rho of x, sub minus 3, etcetera, converge to rho of (x).
Figure 2. Trend plus detail converges: rho sub minus 4 of (x), plus delta sub minus 4 rho of (x), plus delta sub minus 3 rho of (x), etcetera, converges to rho of (x).
Figures 1 and 2 depict the convergence of the W M R A descriptions of rho of (x), as trend and detail. The “overall” shape of rho of (x), effectively is determined by the trend rho sub minus 4 of (x), and detail delta sub minus 4, rho of (x), i.e., by trend, rho sub minus 3 of (x). However, emergence of the prominent double peaked structure of the lipid head group near x over ell sub zero = 100, needs detail delta sub minus 3, rho of (x). Subsequent detail mainly acts to sharpen the edges between the film’s components.
Figure 3 shows the effective contributions of the trend and the successive spatial detail to the reflection coefficient r of (Q), each calculated exactly. The edge-sharpening detail seen in Figure 3 is not revealed in the reflection spectrum below Q ell sub 0, approximately equal to 0.6. Figure 4 shows the “smeared” rho of (x), obtained by direct inversion of the reflection from rho of (x), using data truncated at Q, sub max approximately equal to 0.2 inverse angstroms. This is seen to fall “between” the trends rho sub minus 4 of (x), and rho sub minus 3 of (x), stemming from low-pass filters with roll-offs at Q approximately equal to 0.2 inverse angstroms and Q approximately equal to 0.4 inverse angstroms, respectively. This is expected from the fact that the Fourier transforms of trends and detail overlap to a degree. Thus, a “pure” trend cannot be observed in the truncated data.
Wavelet analysis thus provides a systematic method for assessing the correctness of density profiles measured by N R. It promises to add reliability in unraveling structures of importance in biomimetic membranes.
Figure 3. Effective contributions to Q squared, r sub 1 of (Q), generated by the trend rho, sub minus 4 of (x), (in red), and the details delta sub lowercase j, rho of (x), for j equals minus 4, minus 3, minus 2, minus 1, shown in Figure 2. The Q squared, r sub 1 of (Q) contributions are labeled by the S L D’s that produced.
Figure 4. Inverted r sub 1 of (Q), (black) using only data truncated at Q, times ell sub zero, equals 0.2. Trends rho of (x), sub minus 4, (red), and rho of (x), sub minus 3, (blue), of the actual rho of (x).
 C. F. Majkrzak, N. F. Berk, S. Krueger, J. Dura, M. Tarek, D. Tobias, V. Silin, C. W. Meuse, J. Woodward, and A. L. Plant, Biophysical J., 79, 3330 (2000).
 Wavelets were first applied to x-ray reflectivity with different focus and using different techniques by I. R. Prudnikov, R. D. Deslattes, and R. D. Matyi, J. Appl. Phys., 90, 3338 (2001).
N. F. Berk and C. F. Majkrzak
NIST Center for Neutron Research
National Institute of Standards and Technology
Gaithersburg, MD 20899-8562
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