## 3 Fibrous Structures
### Hair  
Hair is one of nature's most complex biological composites [79]. Keratin, the basic building block of hair, is a polypeptide made of amino acids produced in specific epithelial cells and assembled into right-handed \(\alpha\)-helices (Figure 5a). A total of 32 helices assemble into intermediate filaments (IFs), which form a macrofibril that makes up the hair fiber [80; 81; 82]. The intermediate filaments (IFs) are embedded within an elastomeric protein matrix, as in a fiber-reinforced matrix. The intermediate filaments are organized differently in the three distinct regions of hair: the medulla, the cortex, and the cuticle [32; 83]. The medulla, the innermost core, is spatially and orientationally disordered and provides mechanical integrity [80; 84; 85]. The cortex surrounding the medulla comprises densely packed IFs and pigments. The cuticle is the outermost layer, which provides environmental protection for the medulla and cortex [86]. Figure 5: (**a**) Hierarchical structure of hair starting with the \(\alpha\)-helix chains to the hair fiber; reprinted from [PERSON] et al. (2017) [82] with permission from Elsevier. (**b**) SAXS diffraction pattern of hair with a \(q\) range of 0.01–2.0 nm\({}^{-1}\); reprinted from [PERSON] et al. (2013) [86] with permission from Wiley. A typical SAXS diffraction pattern of hair fiber is shown in Figure 5b. The outermost ring on the diffraction pattern arises from the lipid structure inside the cell membrane complex (CMC). The broad equatorial interference reflections at -9 nm correspond to the lateral packing of the keratin IFs. This is accompanied by two other higher-order equatorial peaks, not visible in the photograph, at -45 and -28 A [57]. Lipid bilayers in keratinous tissues or plasma membranes also contribute to the second peak at 45 A. The weak meridional arc reflection at 6.7 nm corresponds to the staggered arrangement of the keratin IFs. The diffuse scatter around the beam stop corresponds to amorphous/disordered structures within the hair [86]. The SAXS patterns of hair were taken when exposed to acid-straightening and bleaching to mimic common hair alterations. When the hair is acid-straightened (low pH and high heat), the 89 A peak shifts to lower \(q\)-values and changes the shape, indicating that IFs are disrupted. The 67 A peak loses intensity, resulting from the denaturing or unfolding of filaments. The 45 A peak shifts to 57 A as water is absorbed into the lipid layers and increases the spacing. When bleached, the keratin oxidizes and removes lipids, but the water intake is lower compared to acid straightening. When bleached and acid straightened, the hair is denatured and more porous and takes up more water [87; 88]. The structure from these fiber patterns is usually determined by trial and error. The hair's structure was determined by comparing the observed pattern with a simulated pattern generated from a model [38; 57; 86]. [PERSON] et al. simulated diffraction patterns using a number of different models and Equations (11)-(13) to replicate the scattered intensity [57]. Equation (11) represents the intensity, \(I\), as a function of \(q\), as shown below:  
\[I(q)=L\times P(q)\times\left\lvert\,F(q)\right\rvert^{2}\times Z(q) \tag{11}\]  
where \(L\) is the Lorentz factor, \(\left(\cos\,\,2\theta\right)^{3}\), \(2\theta\) is the diffraction angle), \(P(q)\) is polarization correction, \(F(q)\) is the structure factor of an IF, and \(Z(q)\) is the interference term from the relative lateral positions of an IF. \(L\) and \(P\) take into account the details of the interactions of the X-rays with the electrons of the atoms. \(F\) captures the shape of the individual scattering entity, in this case IFs. \(Z(q)\) takes into account the lattice in which the IFs are arranged (Figure 6d,e). The structure factor in Equation (12) is calculated from the atomic scattering factor (f\({}_{\text{j}}\)) and the position vector (r\({}_{\text{j}}\)) for the \(j\)th atom (see Figure 6b,c).