## 3 Fibrous Structures
### Collagen  
Collagen is a structural protein in extracellular matrices and connective tissue. The basic unit of collagen is made of three polypeptide chains (\(\alpha\)-chains) with repeating glycine-proline-lysine (Gly-X-Y) arranged in a hierarchical structure (Figure 2a). Three \(\alpha\)-chains self-assemble to form tropocollagen, a \(\sim\)300 nm long right-handed triple helix [9; 59]. tropocollagen molecules arrange themselves into a staggered structure, a microfibril with a band pattern observed in electron micrographs, known as the D-band. The periodicity of these D-bands, the D-period, is 67 nm, with a 40 nm space between the ends of the tropocollagen and a 27 nm overlap area. These tropocollagen molecules assemble into microfibrils in a quasi-hexagonal lattice stabilized by hydrophobic interactions and cross-linking [9; 59]. SAXS has been instrumental in analyzing the details of this structure, as discussed below. Figure 2: (**a**) Hierarchical structure of collagen; reprinted from [PERSON] et al. (2023) [60] with permission from Springer Nature [[[https://doi.org/10.1186/s42825-023-00127-5](https://doi.org/10.1186/s42825-023-00127-5)]([https://doi.org/10.1186/s42825-023-00127-5](https://doi.org/10.1186/s42825-023-00127-5))]. (**b**) SAXS diffraction pattern of collagen. The white circle in the middle of the pattern is from the beam stop to prevent the X-rays from damaging the detector; reprinted from [PERSON] et al. (2008) [61] with permission from Applied Crystallography. (**c**) A bar chart showing the fibril diameter distribution calculated from SAXS data. Each bar represents a 4 nm range of diameters; reprinted from [PERSON] et al. (2005) [62] with permission from Elsevier. (**d**) Electron density distribution in collagen along the fiber axis; reprinted from Ref. [61] with permission from Applied Crystallography. Figure 2b shows a SAXS diffraction pattern of the collagen from an unmineralized intramuscular herringbone [61]. The first SAXS pattern of collagen was taken in 1944 [31]. The meridional peaks (shown along the horizontal axis) come from the structures aligned with the fiber axis, which has a periodicity of 64 nm (D-period) corresponding to the staggered arrangement of the collagen molecules. When hydrated, the periodicity increases to 68 nm due to swelling. This swelling brings about a systematic change in the intensities of the meridional peaks: odd-order reflections are more intense than the even-order reflections in hydrated collagen, whilst even orders are stronger in dry collagen. The sharpness of the meridional reflection relates to the longitudinal coherence length (\(\sim\)1010 nm). The breadth of the reflections in the lateral direction provides information about the fibril diameter (\(\sim\)85 nm). The equatorial peaks arise from intermolecular spacing perpendicular to the fiber axis, representing the fibril's lateral molecular packing (spacing between collagen molecules). When hydrated, the spacing increases from \(\sim\)11 A (dry state) to \(\sim\)15 A, resulting from water uptake separating the molecules [31; 61]. [PERSON] et al. recently analyzed the meridional reflections to determine the periodicity (\(L\)) of the fibrils, fibril diameter, and longitudinal coherence length (\(l_{c}\)) by taking into account preferred orientation and fibril size [61]. The electron density projected onto the fiber axis, \(\rho(x)\), was calculated from the integrated intensities using the following equation:  
\[\rho(x)=L^{-1}\!\!\sum_{n}|F_{n}|\exp(i\varphi_{n})\exp\!\left(-\frac{2\pi inx }{L}\right)\exp(-Bn^{2}) \tag{4}\]  
where \(x\) is the position along the fibril axis, L is the period of the collagen fibril, \(\mid\!F_{n}\!\mid\) is the structure factor (=\(\sqrt{I(n)}\)), \(\varphi_{n}\) is the phase angle of the \(n\)th order, and \(B\) is the disorder parameter. The electron density shown in Figure 2c fluctuates from high values where collagen molecules overlap to low values in gap zones. The distance between peaks is the axial periodicity, \(L\approx 66.6\) nm [61].