Membrane Structure Characterization Using Variable-Period X-Ray Standing Waves  Ruitian Zhang, Rosangela Itri, Martin Caffrey  Biophysical Journal  Volume 74, Issue 4, Pages 1924-1936 (April 1998) DOI: 10.1016/S0006-3495(98)77901-9 Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 1 Reflection and refraction (transmission) of x-rays at an interface between vacuum (air) and a mirror. k1 and k1R are, respectively, the incident and reflected wave vectors of the x-rays in vacuum (air), and k2 is the refracted wave vector in the mirror. θ1≡θ is the incident angle, and θ2 is the angle of refraction. E1, E1R, and E2 are the electric fields of the incident, reflected, and refracted (transmitted) x-rays, respectively, all of which are perpendicular to the incident (x-z) plane. The refractive index, n, of the two media is also indicated in the figure. qz is the z component of the momentum transfer in vacuum (air). Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 2 Ray trace diagram of x-rays for three adjacent layers (m−1, m, and m+1) in a multilayer system. Am−1,mi represents the complex amplitude of the incident x-rays striking interface (m−1, m) from layer (m−1); Am−1,mR,(0) represents the complex amplitude of x-rays reflected directly from interface (m−1, m) back into layer (m−1) without penetrating layer m; Am−1,mR,(1) represents the complex amplitude of x-rays transmitted back into layer (m−1) from interface (m−1, m) after traveling once into and then back out of layer m. Am,m+1i and Am,m+1R,(0) refer to layers m and m+1 in the same way that Am−1,mi and Am−1,mR,(0) refer to layers (m−1) and m. The origin of the z coordinate is at the interface (m, m+1). The thickness of layer m is dm. The amplitude of the electric field in layer m is Em(0) just above the (m, m+1) interface and Em(dm) just below the (m−1, m) interface. The electric fields in layers (m−1) and (m+1) are Em−1 and Em+1, respectively. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 3 The effect of an organic adlayer on a mirror surface on x-ray reflectivity and standing-wave electric field intensity. Reflectivity, R(θ) (solid line), and standing-wave electric field intensity, I(θ, z) (solid circles) at z=50Å above the gold mirror surface, were calculated using Eqs. 5–7 for a flat, two-layer (air/gold on bulk silicon) system. Similar reflectivity, R1,2(θ) (dashed line), and intensity, I2(θ, z) (open circles) at z=50Å calculations using Eqs. Eqs. 11–12 and 18–19 were performed for a three-layer system consisting of air/lipid/gold on bulk silicon. The values used in the calculations: organic adlayer thickness, dL=100Å; gold mirror thickness, dgold=1000Å; refractive indices for gold, δ=2.99×10−5, β=2.20×10−6; for the organic adlayer, δ=2.50×10−6, β=2.24×10−9; and for bulk silicon, δ=3.80×10−6, β=4.00×10−7 (Itri et al., 1997). The critical angle of the gold mirror is θc=7.73mrad (indicated by the arrow), and that of the lipid adlayer is θ2c=2.24mrad. The x-ray wavelength is λ=1.265Å. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 4 Schematic representation of a typical lipid sample used in x-ray standing-wave measurements and how interfacial roughness, σr, convolutes with intrinsic marker atom spread, σin, in the marker atom distribution that is actually measured, σ. Mirror roughness is illustrated in A as a series of vertically displaced facets whose surfaces lie parallel to the x-y plane. The z direction is normal to this surface. The average mirror surface height is assigned a z value of zero. z′ refers to the distance from this average height to the height of a given facet at the mirror surface. zin is the vertical distance separating a marker atom from the surface of a facet in the mirror directly below it. Marker atoms, lipid molecules, and octadecanethiol (ODT) are represented by solid circles, lollipop figures, and vertical lines, respectively. A and B represent the sample before and after mirror roughness has been deconvoluted. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 5 Experimental (circles) and theoretical (lines) x-ray reflectivity profiles for a multilayer sample in air consisting of an inverted bilayer of zinc arachidate on an octadecanethiol monolayer self-assembled on a 1000-Å-thick gold mirror (Itri et al., 1997). The data were fit in the 3–9-mrad angular range using the Nevot-Croce, rNC (solid line), and Debye-Waller, rDW (dashed line), factors to account for mirror surface roughness, using Eqs. 25 and 22, respectively. The critical angle of the gold mirror, θc, is indicated. The following conditions for the experimental and theoretical aspects of the work apply: x-ray wavelength, 1.265Å; refractive indices for gold, lipid, and silicon are as reported in the legend to Fig. 3. Data were collected at beam line X15A of the National Synchrotron Light Source (NSLS) of Brookhaven National Laboratory, using a highly collimated and monochromatic x-ray beam and a pin diode detector. See Itri et al. (1997) for complete details. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 6 The effect of gold mirror thickness, dgold, on reflectivity from a flat, three-layer system consisting of air, gold, and bulk silicon and calculated by using Eqs. 11 and 12. θc is the critical angle of the gold layer. Refractive indices for gold and silicon are reported in the legend to Fig. 3. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 7 The effect of using truncated Gaussian distribution functions on calculated fluorescence yield profile. The profiles were generated based on typical x-ray standing-wave measurements using Eqs. 18–20 and 25, and the following conditions: gold mirror thickness, dgold=280Å; organic adlayer thickness, dL=80Å; interfacial roughness, σr=5.6Å. The Gaussian distribution functions in real space are illustrated in the inset. The corresponding 〈z〉 and σ values are 16.7Å and 12.4Å for distribution 1 (solid line, inset), and 16.4Å and 10.9Å for distribution 2 (dashed line, inset). Sample composition and refractive indices for gold, lipid adlayer, and silicon are as reported in the legend to Fig. 3. The critical angle of the gold mirror, θc, is 7.73mrad. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 8 The effect of organic adlayer thickness, dL, on calculated reflectivity profile. The profiles were generated based on typical x-ray reflectivity measurements, using Eqs. 9–12 and 25 and the following conditions: gold mirror thickness, dgold=280Å; mirror roughness, σr=5.6Å. Sample composition and refractive indices used for gold, lipid adlayer, and silicon are as reported in the legend to Fig. 3. The critical angle of the gold mirror, θc, is indicated. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 9 The effect of mirror roughness, σr, on calculated reflectivity profile. The profiles were generated based on typical x-ray reflectivity measurements, using Eqs. 9–12 and 25 and the following conditions: gold mirror thickness, dgold=280Å; organic adlayer thickness, dL=80Å. Sample composition and refractive indices used for gold, lipid adlayer, and silicon are as reported in the legend to Fig. 3. The critical angle of the gold mirror, θc, is indicated. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 10 The effect of organic adlayer thickness, dL, on calculated fluorescent yield profile. The profiles were generated based on typical x-ray standing wave measurements, using Eqs. 18–20 and 25 and the following conditions: gold mirror thickness, dgold=280Å; mirror roughness, σr=5.6Å; marker atom position, 〈z〉=50Å; distribution half-width, σ=7Å. Sample composition and refractive indices used for gold, lipid adlayer, and silicon are as reported in the legend to Fig. 3. The critical angle of the gold mirror, θc, is indicated. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 11 The effect of mirror roughness, σr, on calculated fluorescent yield profile. The profiles were generated based on typical x-ray standing-wave measurements, using Eqs. 18–20 and 25 and the following conditions: gold mirror thickness, dgold=280Å; organic adlayer thickness, dL=80Å; marker atom position, 〈z〉=50Å; distribution half-width, σ=7Å. Sample composition and refractive indices used for gold, lipid adlayer, and silicon are as reported in the legend to Fig. 3. The critical angle of the gold mirror, θc, is indicated. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 12 The effect of marker atom position, 〈z〉, in the lipid adlayer above the reflecting mirror surface on calculated fluorescent yield profile. The profiles were generated based on typical x-ray standing-wave measurements, using Eqs. 18–20 and 25 and the following conditions: gold mirror thickness, dgold=280Å; organic adlayer thickness, dL=100Å; mirror roughness, σr=5.6Å; distribution half-width, σ=7Å. Sample composition and refractive indices used for gold, lipid adlayer, and silicon are as reported in the legend to Fig. 3. The critical angle of the gold mirror, θc, is indicated. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions

Figure 13 The effect of marker atom distribution width, σ, and of mirror roughness, σr, on calculated fluorescence yield profile. The profiles were generated based on typical x-ray standing-wave measurements, using Eqs. 18–20 and 25 and the following conditions: gold mirror thickness, dgold=280Å; organic adlayer thickness, dL=100Å; marker atom position, 〈z〉=50Å. Sample composition and refractive indices used for gold, lipid adlayer, and silicon are as reported in the legend to Fig. 3. The critical angle of the gold mirror, θc, is indicated. Biophysical Journal 1998 74, 1924-1936DOI: (10.1016/S0006-3495(98)77901-9) Copyright © 1998 The Biophysical Society Terms and Conditions