SAXS Investigation of Hierarchical Structures in Biological Materials

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Analysis of hierarchical, biological materials using small-angle X-ray scattering.

Abstract

Biological materials are distinguished by their hierarchical structures in which the organization of the basic building blocks is precisely controlled on many discrete length scales. This biophysical organization, i.e., the structure, along with the biochemical attributes, dictates their properties and function. This article is a review, and also a tutorial, that describes the use of small-angle X-ray scattering (SAXS) for determining the structures at the nanometer and sub-micron length scales in three distinct classes of scattering patterns that arise from fibrous structures, lamellae, and solutions. Fibrous structures are discussed using results from collagen, bone, hair, feathers, and silk. The use of SAXS to study the lamellar structures is illustrated using the results from myelin and membranes. SAXS in solutions is discussed by highlighting the results from multidomain proteins such as monoclonal antibodies and facile structures in intrinsically disordered proteins and protein condensates. The goal is to describe the different methods for analyzing the distinct classes of scattering patterns arising from 1- and 2-D ordered structures and from 3D structures in solutions and to illustrate how the structure imparts unique functions and properties to the biological materials. An understanding of the hierarchical structures in biology is expected to be useful in medical diagnosis and serve as a guide for fabricating functional biomaterials by mimicking these structures.

1. Introduction

Many biological materials are hierarchically structured by the organization of molecular units that are controlled with high precision on many length scales. Larger structures are formed by the self-assembly of smaller molecular units into more intricate structures through a step-by-step process at progressively higher levels of the organization [1]. This way, biological materials can perform increasingly complex functions using a few basic rules and principles of structural organization. Hierarchical structures have also been observed in more facile supramolecular assemblies, where non-covalent interactions assemble the building blocks [2]. Characterization of the different types of order at various length scales in both rigid and facile structures will enhance our understanding of the influence of structure on the function of these assemblies. Since function derives from the structure, knowledge of the structure will help understand the disease process resulting from cellular disruption. This also helps us to mimic these structures and functionalities in non-native environments.

This review focuses on the characteristics of hierarchical structures and how these structural features give rise to their distinctive small-angle X-ray scattering (SAXS) patterns. The article is also a tutorial for the analyses of SAXS data to determine these structures at 1 to 100 nm length scales [3,4,5]. Complex cellular processes occur at these mesoscales, i.e., at length scales between the individual molecules and the cell’s overall size. Structures at these length scales can also be imaged by microscopy, such as atomic force microscopy (AFM) and transmission electron microscopy (TEM) [6,7,8,9]. X-ray scattering has the advantage of minimal sample preparation, and samples can be studied in any desired environment [10]. Microdiffraction patterns obtained with submicron-sized beams are used to characterize the spatial variation in structures, such as skin-core morphologies [11,12,13,14,15]. The basic information that will be useful for the beginner to benefit from our paper is succinctly presented in the guide published by Anton Paar [16].

SAXS provides information about the structure of nanoscale structures in the form of size, shape, aggregation states, voids, and fractal dimensions [17,18,19,20]. TEM and AFM give similar structural information. The advantage of these techniques over SAXS is that they provide the structural information directly, while in SAXS, the data appears in reciprocal space, and need to be interpreted either by modeling, Fourier transformation, or in terms of structural parameters, and therefore can sometimes be ambiguous. The advantage of SAXS is that unlike TEM and SAXS, the information is not limited to one spatiotemporal region but is a statistical average over a large volume and duration [17,21,22,23]. Microdiffraction techniques with beam sizes of <1 μm are used to map the spatial variations in structure, thus approaching the capabilities of TEM and AFM. Additionally, sample preparation is much easier, and the sample environment in SAXS is flexible and easily altered, and not necessarily so in TEM and AFM. SAXS can be coupled with other techniques, such as thermal analysis (e.g., differential scanning calorimetry, DSC), spectroscopy (e.g., Fourier Transform Infrared (FTIR) and Raman), optical microscopy, and, in the case of solution, size-exclusion chromatography (SEC). The utility of small-angle scattering is enhanced by carrying out small-angle neutron scattering (SANS) if samples can be deuterated [19,24,25].

There are a large number of review articles on SAXS [26,27,28,29,30]. However, most of these cover solution scattering, and few discuss the use of SAXS for analyzing fibrous structures that occur in biological systems. This is because protein solution scattering yields results that can be directly used to understand their functional therapeutic molecules. The response of fibers and lamellae, the basic organizational features of which were investigated decades ago [31,32,33,34,35], to the changes in the environments, e.g., disease, are subtle and difficult to extract from the features of the underlying structure. Recent advances in X-ray sources, detectors, and automated analysis make this now possible. This review emphasizes the analysis of fibrous structures while also discussing lamellar and solution scattering to the extent necessary to highlight the differences in the analysis of these three classes of scattering patterns. This paper is less a review of the vast SAXS literature but a discussion of the different methods of analysis to determine the wide range of structures that are present in biological systems. The goal is to illustrate, through examples, practical applications for SAXS to assess strength and degradation in tissue scaffolds (tendons, meniscus, and blood vessels) and monitor diseases (cancer, degenerative diseases) by examining the structure.

Three classes of materials that encompass the most common hierarchical structures in biology, each of which give rise to their own distinctive SAXS patterns, are discussed in this article. These are fibrous materials (keratinous structures in hair and feathers, silk, collagenous structures), lamellar structures (membrane and multilamellar structures), and solutions (proteins) [33,36,37,38]. SAXS data from these materials are used to illustrate the different methods of analyses in the three distinct classes of SAXS patterns.

2. Methods

In a SAXS experiment, the sample is exposed to an X-ray beam with energy ranging from 8 keV (typical of Cu K_α from in-house laboratory sources) to 20 keV (used in synchrotron sources to minimize radiation damage). Data can be obtained in fractions of a second by recording the scattering pattern on electronic detectors, making it possible to routinely follow the effect of temperatures, solvent conditions, and deformation [39].

The scattered intensity is measured as a function of the scattering angle, 2θ, but is usually expressed as a function of the scattering vector, q, also referred to as the momentum transfer vector. The relations between 2θ (degrees or radians), q (Å⁻¹), and the widely used Bragg spacing between diffraction planes, d (Å) and λ (Å), the X-ray wavelength, are given in Equations (1) and (2).

$$q={\displaystyle \frac{4\pi }{\lambda }}\sin \theta$$

(1)

$$d={\displaystyle \frac{2\pi }{q}}$$

(2)

Peaks or reflections in the scattering pattern at certain q-values correspond to specific characteristic distances in the material. For SAXS, q is typically less than ~0.25 Å⁻¹ (d > 25 Å), whilst for WAXS (wide-angle X-ray scattering), q > ~0.25 Å⁻¹ (d < 25 Å).

The coherence size of the scattering domains, L, also known as the crystallite size (Å), is calculated from the width of the peak using the Scherrer equation (Equation (3)),

$$L={\displaystyle \frac{2\pi K}{∆q}}$$

(3)

where Δq is the peak’s full width at half maximum, FWHM, and K is a constant, typically 0.9 [40,41]. L represents the length over which the structure is ordered. The three distinct classes of materials, fibers, lamellae, and solutions, are discussed in the following sections along with examples.

2.1. Analysis Procedures

The SAXS pattern is strongly dependent on the class of the sample being analyzed. Correspondingly, the methods of analysis of the SAXS data from organized systems, such as fibers and lamellar structures (Figure 1b,c), are quite different from those for disordered systems, such as solutions (Figure 1d). In the case of fibers and lamellar structures, where there are planes of atoms that diffract coherently, Bragg’s law can be used to determine the spacing between these planes. The patterns are analyzed by slicing along and perpendicular to the orientation axes to obtain meridional and equatorial scans, respectively. As described in Section 3 and Section 4, the full data can also be analyzed by using models derived by iterations starting with an approximate prior knowledge of the structure.

The scattering in solutions is governed by the Debye function [42]. The 2D pattern is typically converted to a 1D pattern by azimuthal averaging. Since there are no crystalline regions and periodicity, methods based on Bragg’s law are not meaningful. Instead, the diffuse scattering near q = 0 is analyzed to understand the uncorrelated structures in protein solutions and particulates embedded in a matrix and voids. These scans are analyzed by methods described in Section 5. Briefly, the small-angle region of the SAXS data is analyzed using the Guinier equation [40]. The asymptotic behavior is interpreted using Porod’s law to understand the contribution of the interfaces. These are discussed in detail in Section 5.

2.2. Robustness of the Results

Unlike wide-angle crystallographic data, SAXS data are not amenable to direct structure determination. Hence, the structures are usually derived from SAXS data using one of the many modeling methods. While a structure gives a unique SAXS pattern, a SAXS pattern can be fitted to many structures. This ambiguity is usually resolved by using results from alternative sources, such as direct imaging by TEM or AFM, molecular simulations, or by comparison with similar structures that have been validated. In many instances, rather than insisting on a definitive model, it is sufficient to parameterize the scattering pattern so that the changes due to the environment, processing, and disease states can be monitored. This can be carried out quite reliably by using parameters that are derived from Bragg, Guinier, and Porod and Scherrer equations, or variations of these [41,42]. The uncertainty in the parameters derived from these is typically <1% and is sufficient to discern the effect of environmental conditions such as temperature, humidity, and stress, and, more importantly, disease conditions. These will be illustrated in Section 3, Section 4 and Section 5.

2.3. Sample Limitations

SAXS requires homogeneous or monodisperse samples to obtain unambiguous interpretation. In analyzing solution scattering, improper solvents or background subtractions can introduce artifacts that can affect the derived parameters [43,44]. At high concentrations, intermolecular interactions make data interpretation more difficult [45]. Dilute series are required to extrapolate to infinite dilution [46,47]. X-ray-induced degradation can damage biological samples [48,49]. Cryo-cooling the sample during testing can reduce degradation but can prevent the examination of the conformations at elevated temperatures [49,50,51]. Reducing the exposure time can also reduce the amount of damage but results in poor-quality data. Oscillating the sample through the beam and rastering the beam across the sample reduces radiation damage. In translating the laboratory results into clinical applications, difficulty in accessing a large number of patient tissues will affect the reproducibility of these diseased tissues.

3. Fibrous Structures

Fibers are the most ubiquitous biological structures in the solid phase. There is a long history of SAXS studies with these structures, starting with collagen [31] and extending to silk and hair. Collagen is the primary extracellular matrix (ECM) molecule that supports cell growth and is responsible for the mechanical strength in connective tissues. Bone is an example of a composite consisting of two major components, collagen microfibrils and crystallites of hydroxyapatite [52]. Many other biomineralized materials, including bone, exhibit a highly organized hierarchical structure at several length scales [53,54]. Hair is one of the many fibrous keratinous structures that have been well studied because of the importance of wool in the textile industry [55,56]. Hair is a model structure for other structures that contain intermediate filaments that interact with the nuclear and cytoplasmic membranes in tissues to provide mechanical support [57]. Keratinous materials other than wool include quill and nail. Like wool, silk is another important textile fiber because its specific strength can be engineered to be stronger than steel [38,58]. SAXS data are used to determine axial periodicity, fibril diameter, orientation, and structure along the fiber axis and in the equatorial plane. These fibrous structures will be discussed in this section.

3.1. Collagen

Collagen is a structural protein in extracellular matrices and connective tissue. The basic unit of collagen is made of three polypeptide chains (α-chains) with repeating glycine-proline-lysine (Gly-X-Y) arranged in a hierarchical structure (Figure 2a). Three α-chains self-assemble to form tropocollagen, a ~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 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 (~1010 nm). The breadth of the reflections in the lateral direction provides information about the fibril diameter (~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 ~11 Å (dry state) to ~15 Å, resulting from water uptake separating the molecules [31,61].

Burger 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, ρ(x), was calculated from the integrated intensities using the following equation:

$$\rho \left(x\right)=L^{-1}\sum _n\left|F_n\right|\exp \left(i{\phi }_n\right)\exp \left(-{\displaystyle \frac{2\pi inx}{L}}\right)\exp (-B{n}^2)$$

(4)

where x is the position along the fibril axis, L is the period of the collagen fibril, |F_n| is the structure factor (=$\sqrt{I\left(n\right)}$), φ_n is the phase angle of the nth 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 ≈ 66.6 nm [61]. The lateral integral width of the meridional reflections is represented by b₁₂, which can be used to determine the fibril diameter. b₁₂ increases with the diffraction order, n, due to an increase in tilt and disorder; therefore, a zero-order extrapolation approach was used using the equation below:

$$b_{12,n}=b_{12,0}+Cn$$

(5)

where b_12,0 is the width without any broadening effect, b_12,n is the width of a reflection at a given n, and C is a proportional constant. The average fibril diameter is estimated from the reciprocal of b_12,0. An average fibril diameter of 85 nm was calculated, which was corroborated by the TEM result.

The width of meridional reflections along the axial direction corresponds to the coherence length along the fibril. The total peak broadening (b_3,n) is the combination of the finite fibril size effect (b_size), instrumental broadening (b_inst), and lattice disorder (b_dis) as given below:

$$b_{3,n}=\sqrt{b_{size}^2+b_{inst}^2+b_{dist}^2}$$

(6)

Broadening from disorder effects was quantified using the equation:

$$b_{3,n}=b_{3,0}+B^{\prime }n+B^{″}{n}^2$$

(7)

where b_3,0 is the broadening without the disordering effects, and B′ and B″ are fitting parameters for strain and paracrystalline disorders, respectively. The longitudinal coherence length is the reciprocal of b_3,0, which is ~1010 nm.

The azimuthal intensity distribution in the diffraction patterns is related to the orientation distribution of the molecules. This is usually quantified by Hermann’s orientation function [3]. Alternatively, the narrowness of the azimuthal distribution can be empirically expressed by an orientation index (OI) calculated using the equation below:

$$OI={\displaystyle \frac{90°-OA}{90°}}$$

(8)

where OA is the azimuthal angle range containing 50% of the fibrils. If OI = 1, the fibril is perfectly aligned; if OI = 0, the fibrils are entirely disordered. Higher OI was observed in stretched-tanned leather, meaning that fibrils are more aligned, making the leather stiffer. The lower OI in the control leather indicates a random fibril orientation, indicating greater flexibility [63].

Fibril diameter distribution was analyzed by Goh et al. using the equatorial reflections. The intensity was modeled based on the form factor, F(q) of the cylindrical fibrils, as shown below:

$$F\left(q\right)=\rho A{\displaystyle \frac{2J_1\left(qr\right)}{qr}}$$

(9)

where J₁ is the first-order Bessel function of the first kind, ρ is the number of excess electrons per unit area and unit length, and A is the cross-sectional area of the fibril. The form factor describes how the size and shape of fibrils influence the SAXS intensity. The relationship between I(q) and F(q)² for polydisperse systems was derived as:

$$I\left(q\right)=b+c\sum _{i=1}^N{r}_i^4{w}_i{[{\displaystyle \frac{2J_1\left(qr\right)}{qr}}]}^2$$

(10)

where variables b and c describe the baseline corrections and scaling factor, respectively, N is the total number of bins, and w_i is the frequency of a fibril in the system. This model considers various fibril sizes and accounts for the frequency of different diameters (Figure 2c) [62]. The fibril size distribution dictates the overall shape of the scattering [64]. This approach provides statistical analysis of a larger scale compared to the small sample size of TEM [62].

A practical application of SAXS analysis of collagen is in the processing of leather. Figure 3 shows a diffraction pattern and structure of leather before and after tanning. The diffraction pattern was used to determine the D-period, fibril diameter and orientation, and intermolecular spacing [63]. Figure 3a shows that the diffraction pattern of normal leather is nearly isotropic from the random orientation of collagen fibrils compared to the diffraction pattern of tanned leather in Figure 3b, which has an elliptical shape, indicating anisotropy. The diffraction rings in these figures arise from the axial periodicity (D-period) The sixth-order diffraction peak in the integrated intensity profile was fitted with Gaussian curves to determine the D-period. The fibrillar structure’s lateral intermolecular spacing, the spacing between collagen molecules, was determined using Equation (2). The diffuse areas of the pattern transform into well-defined arcs in the stretched-tanned leather where collagen molecules are aligned, indicating fibrils have a smaller diameter and are more densely packed. Figure 3c and Figure 3d show the collagen alignment in normal and stretched-tanned leather, respectively. Figure 3c shows the collagen’s random alignment, which reflects the isotropy shown in Figure 3a. In Figure 3d, the collagen fibrils are aligned, reflecting the anisotropy in Figure 3b. The water molecules added to the leather act as a lubricant and decrease the stiffness. However, the stretched-tanned leather is more densely packed, inhibiting the ability of water to decrease the stiffness. Stretched-tanned leather has a larger D-spacing because of the residual strain from tanning under tension. Stretch-tanned leather has a smaller lateral spacing, indicating denser collagen packing results in higher stiffness. This orientation increased the stiffness of the leather. The fibrils were found to swell when exposed to moisture, resulting in larger D-spacing. L, the lateral interplanar spacing, increased with moisture content [63].

Meridional periodicity in collagen is an extremely useful proxy to monitor the structural change in collagen. An example is the study in which the reason collagen materials are resistant to gross structural change in the early stages of dehydration was explored. In one experiment, the effect of dehydration was simulated by exposure to various concentrations of isopropanol [65]. Dehydration was found to be a two-stage process. During initial dehydration (<90% isopropanol), a decrease in the overlap length and an increase in the gap length keep the D-period unchanged. At the second stage of dehydration, the D-period decreases and the overlap length decreases, leading to a significant decrease in the collagen fibril length. The changes in the D-period were also used to compare in situ the strength of stabilization against the heat of the intermolecular structure of collagen in skins due to natural and synthetic crosslinks [66]. The D-period changes when collagen denatures. It was found that although natural crosslinks occur during self-assembly, the number of natural crosslinks decreases, and their contribution to heat stability is overwhelmed by a synthetic crosslink, chromium sulfate.

Another example of the use of meridional periodicity can be found in the work of Zhang et al., who carried out in situ SAXS measurements using synchrotron radiation to analyze changes in the collagen structure during leather processing with different tanning agents [67]. Unlike electron microscopy, SAXS can reliably examine large areas. SAXS was used to analyze collagen in a diseased state [62]. Health professionals can better understand and remedy diseases by comparing SAXS-derived structural differences between collagen in healthy tissues and those in the diseased state [68]. For example, Moger et al. used SAXS to examine variations in collagen orientations within articular cartilage and subchondral bone in horse joints to understand osteoporosis better. The collagen orientation could be determined by using a color map of the SAXS pattern [69].

The last example for the use of SAXS data from collagen to understand the structural alteration due to disease and surgical intervention and hence the biomechanical properties can be illustrated by using results from collagen in the corneal stroma, the thickest (~90%) layer of the cornea. Collagen in this layer is arranged to make the layer transparent and tough [70]. Unlike in ligaments and bone, collagen in stroma has a layered structure, such that each lamella runs parallel to the surface. Fibrils (see Figure 1a) within a given lamella are parallel to each other but at a greater angle to those in the adjacent lamellae. SAXS/WAXS measurements were used to study the structural responses of corneal collagens under tensile load to understand the effect of deformations on the hierarchical structure of collagen (Figure 1a) [71]. At the molecular and fibril levels, under physiological strains, the changes in fibril elongation (D-period, Figure 1), fibril diameter, and tilt (see Figure 3c,d) were found to be indicative of a sub-fibrillar “spring-like” deformation mechanism. The variation in the thickness and orientation of healthy collagen within the corneal stroma provides a structural basis for the differences in biomechanical strength and optical transparency. A good structural and mechanical model of the cornea derived from these results will be useful for making artificial corneas by using tissue engineering methods to reproduce the complex arrangement of collagen that will result in a tough and transparent lens.

3.2. Bone

The collagen microfibrils discussed in the previous section form fibers and are modified by specific tissues with either proteoglycans or mineral depositions to form bone [9,59]. Bone is a biological, hierarchical composite of hydroxyapatite and collagen [72]. Figure 4a shows the hierarchical structure of a human cortical bone, starting with collagen molecules and the collagen fibrils preferentially aligned along the longitudinal direction, thus contributing to bone stiffness as the bone carries the load in the longitudinal direction [73]. Hydroxyapatite crystals nucleate on the c-axis parallel to the fibrils in the gaps between the ends of the tropocollagen molecules, thus mimicking the staggered arrangement of fibrils [72]. Mineralized collagen fibrils arrange themselves alternately to create a lamellae structure. These lamellae surround a central canal, forming osteons that create the cortical bone together [72,74].

Figure 4b (i) shows the 2D SAXS pattern of a trabecular bone [75]. The diffuse scattering at the center of the pattern originates from the nanocrystalline hydroxyapatite platelets. A radially integrated scan (Figure 4b (ii)) of the 2D pattern is used to analyze the structure. The first- and third-order peaks seen in this radial scan come from the meridional D-periodicity of collagen. Because the first-order peak has large diffuse scattering, the third-order peak, shown in blue, as seen in Figure 4b (i), is used for analysis [75]. The degree of alignment of the platelets is determined from the azimuthal intensity distribution of the third-order diffraction from collagen. The thickness of the mineral platelets was calculated from the surface-to-volume ratio calculated from the invariant, $Q=\int q^2·I\left(q\right),$ and the Porod parameter, the y-intercept in the Porod-Debye plot, I(q). q⁴ vs. q⁴ [77].

Figure 4c shows the SAXS pattern of a wild-type femur of a mouse. Figure 4c (i) is from a healthy bone, and Figure 4c (ii) is from a femur afflicted with a form of arthritis. The diffraction pattern of the healthy bone shows oriented features in the third-order collagen reflection as well as in the scattering due to platelets. In the afflicted bone, the collagen becomes unoriented, and the diffuse scattering shows lower platelet ordering. These broader azimuthal intensity distributions show the random orientation of the fibrils in the afflicted bone compared to the healthy bone. This decrease in ordering resulted in poorer mechanical properties of the afflicted bone [76]. Changes in the collagen structure can also be used to evaluate samples for osteoporosis by monitoring the changes in the position of the collagen peak, which can be related to the strain in the bone and can be evaluated [78].

3.3. 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 α-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].

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 Å [57]. Lipid bilayers in keratinous tissues or plasma membranes also contribute to the second peak at 45 Å. 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 Å peak shifts to lower q-values and changes the shape, indicating that IFs are disrupted. The 67 Å peak loses intensity, resulting from the denaturing or unfolding of filaments. The 45 Å peak shifts to 57 Å 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]. Rafik 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:

$$I\left(q\right)=L\times P\left(q\right)\times {\left|F\left(q\right)\right|}^2\times Z(q)$$

(11)

where L is the Lorentz factor, ${\left(\cos 2\theta \right)}^3,$ 2θ 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_j) and the position vector (r_j) for the jth atom (see Figure 6b,c).

$$F\left(q\right)={\sum }_j{f}_j\left(q\right)\times e^{-2i\pi qrj}$$

(12)

Another constraint must be applied to generate the models. Equation (13), given below, implements the condition of uniform density, indicating a repeating, regular motif.

$$F\left(q\right)=2\pi r_2^2\rho J_1{\displaystyle \frac{{(u}_2)}{u_2}}-2\pi r_1^2\rho J_1{\displaystyle \frac{\left(u_1\right)}{u_1}}$$

(13)

This equation represents the structure factor as a hollow cylinder with an outer radius of r₂, an inner radius of r₁, and a density ρ where u_i = 2πr_iq and J₁ is the first-order Bessel function [57]. By comparing simulated and observed patterns, as depicted in Figure 6a, a model for the intermediate filament with eight coiled-coil α-helices in a herringbone arrangement was proposed. This model accounts for the diffraction peaks at 90 Å and 45 Å and includes the possibility of significant disorder in the positions and orientations of the tetramers [57].

More recently, based on small-angle neutron scattering (SANS) studies of hair in dry and hydrated states, Murthy et al. suggested a new model (Figure 6b–e) [38]. This model arranges two twisted a-helices in a nested dumbbell (‘T’ shape) to form a tetrameric unit. Eight tetrameric units are assembled to give either a ring-core structure (Figure 6b) or a ring structure (Figure 6c), both observed in electron micrographs. This arrangement of the helices allows the relative orientations of the tetrameric units, protofilaments, to change with little cost in terms of energy. It thus accounts for the prevalence of the ring-core and ring structures in hair. This helps to understand the diffusion behavior of water in hair. The two proposed models, the ring-core and ring models, give insight into the variability of the intermediate filament arrangement when exposed to water, oil, or heat, while the earlier models do not [38].

3.4. Feather

Feather is another keratinous material like hair. The hierarchical structure that gives the feather its strength and flexibility is shown in Figure 7a [89]. In the barb of feathers, the β-keratins are arranged into a quasi-ordered, sponge-like structure. The barb’s microstructure is composed of either sphere- or channel-type nanostructures. In contrast to hair, where pigments produce their color, light coherently scattered from feathers’ nanostructure at specific wavelengths gives them their characteristic color [89,90]. These optical properties can be explained in terms of the nanostructures in the feather that can be determined from its SAXS patterns [36,40,90,91,92,93]. Knowledge of network connectivity, periodicity, size distribution, and order can lead to a better understanding of the structural origins of the optical properties for engineering this class of nanostructures.

Figure 7b shows three types of SAXS diffraction patterns from (i) rudimentary structure, (ii) channel-type, and (iii) sphere-type structures. In these patterns, the ring-like diffraction pattern indicates an isotropic, short-range order in the quasi-ordered nanostructure [40,91]. The central diffuse scattering in Figure 7b (i) indicates the typical absence of the organization of β-keratin molecules for rudimentary structures. The sharper rings in 7b (ii) and 7b (iii) reflect higher-order β-keratin in channel-type and sphere-type structures, respectively. The interconnected channels caused the broader rings observed in channel-type structures [91].

The 2D diffraction patterns in Figure 7b were analyzed by plotting the azimuthally integrated intensities in the form of a 1D Kratky plot (Iq² vs. q) [90]. The peak position in this plot (Figure 7c) refers to the periodicity in the structure. The full-width at half maximum (FWHM) of the peak represents the nanostructure size distribution. The breadth of the peak is a good indicator of the size distribution; a narrow peak demonstrates a small deviation in size, whilst a broader peak demonstrates the broad size distribution. The third parameter, the peak height, represents the populations of ordered structures; a higher maximum peak height represents a more ordered arrangement [90]. SAXS has been used to analyze various bird species in a single taxom. By comparing and analyzing the diffraction patterns alongside a phylogenetic tree, evolutionary changes in the gyroid’s structure are tracked, moving from quasi-ordered to ordered, as indicated by sharper higher-order peaks [36].

3.5. Silk

Silk is another important fibrous structure of biological origin, the SAXS patterns of which have been investigated in detail. The hierarchical structure of a silk fiber from a silkworm (B. mori) is shown in Figure 8a. The basic building block of this structure is a polypeptide chain. As the silkworm extrudes the silk solution from its salivary glands through its spinneret, the shear flow causes the hydrophilic regions to form random coils and α-helices, while the hydrophobic regions form β-sheets. These alternating amorphous (random coils and α-helices) and crystalline (β-sheets) phases form ~3 nm nanofibrils. Hydrogen bonds between β-sheets and random coils result in a more ordered structure; in earlier models, β-sheets were thought of as cross-linkers connecting neighboring nanofibrils. As multiple nanofibrils combine, they form a 20–100 nm silk fibril, which then bundles into a ~10-μm diameter fibroin filament. Within the molecular chain, nanofibrils, and fibrils, there are globular protrusions that increase the silk’s mechanical strength [94].

One example of the SAXS diffraction pattern from dry spider silk is shown in Figure 8b (i). The strong equatorial streak in the pattern (perpendicular to the fiber) arises from fiber surfaces and oriented voids that are invariably present in fibers. However, there is structural information in the equatorial streak related to the lateral organization and orientation of the fibrils [35,96]. The weak, meridional reflections with a spacing of 8.34 nm correspond to the disordered regions (alternating crystalline β-polyalanine and amorphous regions) of the fiber. When hydrated, as the fiber shrinks and the lamellar spacing decreases to 5.86 nm (compression of the crystalline and amorphous regions), a new reflection on the equator appears, corresponding to a new periodic structure (Figure 8b (ii)). The central streak becomes diffuse due to the increased disorder. After drying, as shown in Figure 8b (iii), the strong, equatorial reflection disappears, confirming that water is responsible for the new equatorial reflection. In contrast, when the fiber is hydrated while held stretched, the lamellar peak becomes more prominent without shifting (Figure 8b (iv)). The equatorial streak becomes intense and extends to higher scattering angles as a result of increased tension. Unlike in Figure 8b (ii), the new equatorial reflection is less prominent, suggesting a different structure is present under tension. When a fiber relaxes while wet, Figure 8b (v) shows the equatorial streak broadening, indicating disordering. The lamellar reflections remain unchanged [35].

A second example of silk’s SAXS pattern shown in Figure 8c (i) is from bagworm silk, Eumeta variegate. A Kratky plot of the data, as shown in Figure 8c (ii), better delineates the peaks in the diffraction scan. The meridional peaks are associated with alternating sections of crystalline and amorphous regions with a periodicity of ~40 nm (red, meridional plot). On the other hand, the diffuse scattering in the equatorial direction signifies a disordered structure, as seen in the featureless Kratky plot (green, equatorial plot). By highlighting the electron density differences, the meridional Kratky plot shows the periodic nature of the alternating amorphous and crystalline phases. This was seen earlier in various diffraction patterns [95,97,98,99]. This difference in the diffraction pattern along two different directions indicates structural anisotropy within the fiber [95]. Similarly, Yoshioka et al. analyzed the silk from another species, A. assama, and observed a 7.6 nm peak at the meridional and a 4.6 nm peak at the equatorial, indicating the presence of two phases [97]. This anisotropy explains the fiber’s mechanical performance. Understanding structural anisotropy and the relationship between hierarchy and mechanical properties is important for replicating these materials.

4. Lamellar Structures

In addition to fibrous materials, lamellar structures are common in biological organisms. For instance, these thin, plate-like layers occur naturally in cell membranes. Other lamellar structures include bones, nacre, feathers, and wood [7,100,101,102]. The hierarchical architecture of lamellae gives them distinct properties [103,104]. Lipid bilayers are the most ubiquitous example of lamellar structures arising from a primary phospholipid (primary structure) containing a hydrophobic tail and hydrophilic head. These phospholipids form monolayers (secondary structure) that stack into bilayers and organize into domains (ternary structure), which exhibit coordinated behavior. The quaternary structure emerges from the interactions of nearby domains [105]. The lamellar organization of lipids in biological materials is often found in cell membranes. Cell membranes play vital roles in biological processes such as molecular transport, signaling, and structure maintenance. The lamellar structure provides a matrix for various proteins and controls the permeability of organic molecules, water, and ions. Lamellae also prevent nonspecific protein-protein aggregation, whereas they allow protein diffusion and conformational changes in the membrane [106]. An example of a non-lipid lamellar structure is collagen fibrils organized into stacks in the cornea [107]. Electron density differences between the layers give rise to distinct SAXS patterns that can be analyzed to determine the detailed organization and spacing between these layers. The observed Bragg peaks are used to calculate the electron density profiles across the layers from which the protein distribution and phase behavior can be determined.

Cell membranes and single-walled liposomes contain just one lamellar bilayer. But multilamellar structures with stacks of lipids have various advantages over unilamellar membranes; e.g., they spatially organize processes and compartmentalize molecules. They are also quite common biologically. Multilayer liposomal membranes, also called multilamellar vesicles (MLVs), with multiple concentric lipid bilayers, are widely used in drug delivery. Kiselev et al. used SAXS to show polymorphic transformation in bilayer thickness, inter-bilayer spacing, and lipid packing as temperature, humidity, and chemical conditions varied [108]. An analysis of the lamellar structure can be illustrated by using data from myelin, which has long been used as a model system for studying the structure of membranes. Myelin consists of a stack of lipid bilayers made by oligodendrocytes that form an insulating sheath around nerves, including those in the brain and spinal cord, to facilitate the transmission of electrical impulses. In the sciatic nerve, for example, a dense lipid bilayer from the cytoplasmic surfaces repeatedly alternates with a layer derived from compact outer membranes to form the myelin sheath [109]. Figure 9a shows the myelin structure characterized by its multilayered, compact nature [110].

An example of SAXS pattern for the multilayer structure of myelin can be seen in the SAXS from a dried frog sciatic nerve (Figure 9b). Reflections 146, 73, and 49Å are the first, second, and third reflections of the double layer; also present is the 61Å, the first-order reflection of the single bilayer [34]. Most interesting in this diffraction pattern is splitting the 44 Å reflections into six diffraction spots. This is the evidence for the formation of the near-hjexagonal arrangment or ripple phase in lamellar structures.

While dried nerve patterns are useful for understanding the lamellar structures, analysis of hydrated structures is clinically more relevant. Figure 9c shows the SAXS profiles of aqueous suspensions of phosphatidylglycerol (PG, scan a) and a PG-21% myelin basic protein (MBP) complex (scan b). These features are typical of membrane monolayers. Multilayers are formed at protein concentrations > 35%, as shown in Figure 9d. The data in Figure 9c were used to obtain the electron density profile ρ(x) across the thickness of the membrane using Equation (14).

$$\rho \left(x\right)={\int }_{q_{min}}^{q_{max}}\left(\pm \right)q\sqrt{I\left(q\right)}\cos \left(qx\right)dq$$

(14)

where the $\sqrt{I\left(q\right)}$ is the scattering amplitude of the vesicle, which is a randomly oriented bilayer, and hence is multiplied by h. The electron density profiles shown in the inset show that the MBP does not significantly affect the electron density in the interior of the layer (away from the polar groups). However, the lipid bilayer’s hydrophilic polar groups interact with MBP, increasing the electron density in that region, as indicated by large increases in intensity near h = 0 in the diffraction scan (Figure 9d inset). This, and the increases in the width of the bilayer determined by the position (q) of the peak in the scattering pattern, support the idea that the protein alters the structure of the lipid membrane [33].

Hamley presented a detailed analysis of the lamellar diffraction patterns by focusing on the diffuse scattering that is present in the diffraction patterns using thermal disorder and a paracrystalline model of lamellar structures [112]. In this analysis, the structure factor, S(q), for perfect and evenly spaced layers in the lamellar structure, is written as:

$$S\left(q\right)=N+2\sum _{k=1}^{N-1}(N-k)\cos (kqd)$$

(15)

where N is the total number of layers inside the scattering domain, d is the layer spacing, q is the scattering vector, and k represents each layer. S(q) essentially accounts for how the X-rays scattered by one layer interfere with those from other layers. This equation needs to be modified to take into account the various types of defects and imperfections. This thermal disorder model uses the Debye–Waller factors to account for random positional fluctuations. Another main model is the paracrystalline model, in which small fluctuations of layer spacing are considered. In the third model, the Caillé model, the fluctuations are quantified in terms of the flexibility of the membranes:

$$S\left(q\right)=N+2\sum _{k=1}^{N-1}(N-k)\cos (kqd)\times \exp [-{\left({\displaystyle \frac{qd}{2\pi }}\right)}^2\eta \gamma ]{(\pi k)}^{-{({\displaystyle \frac{qd}{2\pi }})}^2\eta }$$

(16)

where γ is Euler’s constant and η is the Caillé parameter. η depends on the lamellae’s bulk compression modulus, B, and the bending rigidity, K. This model considers fluctuations from elastic membrane deformations. Figure 9e shows an example of the experimental data fitted with this model.

$$\eta ={\displaystyle \frac{\pi k_{B}T}{2d^2{\left(BK\right)}^{1/2}}}$$

(17)

The paracrystalline and the Caillé models were used to model the bilayer form factor between the lipid and peptide. To fit the model, four parameters were used: z_H, σ_H (center and width of the headgroup Gaussian, respectively), σ_c, (width of the hydrocarbon chain Gaussian), and ρ_R (ratio of the methyl-terminus electron density amplitude to the headgroup amplitude) [111,112]. The data were fitted using the lamellar bilayer models (Figure 9e).

SAXS studies of the structural features of myelin under different neurological diseases are useful in diagnosis and treatment. De Felici et al. investigated the lamellar structure of a human cerebral myelin sheath, revealing a periodicity of 15–20 nm using a randomly oriented white matter sample, which can reveal changes during demyelination [113]. SAXS-TT (tensor tomography) was used to map the myelin density with axon orientation in 3D to differentiate between the central and peripheral nervous systems’ myelin periodicities. They found that the diseased mice lacked myelin in the white matter portion [114].

The structure of the mutated and wild-type cystic fibrosis transmembrane regulator (CFTR) was studied to understand diseases such as cystic fibrosis. The electron density profiles were determined from SAXS data of the mutated and wild-type. The difference between them indicated that there was an altered protein structure at the membrane [115]. Similarly, analyzing the SAXS data for amyloid β-peptides and its membrane was used to determine its role in Alzheimer’s disease. The change in membrane thickness affects the amyloid β-peptide behavior that may signal the early stages of Alzheimer’s disease [116].

5. Solution Scattering

While the examples presented thus far are in the solid phase, in this last section, we discuss hierarchical structures that are also present in the solution. Proteins are the fundamental building blocks in biology, and their structure is important to understand biological functions and is implicated in many diseases. Proteins are polypeptide chains, the folding pattern of which is determined by their amino acid sequence (primary structure). These polypeptide chains are folded into α-helices or β-sheets (secondary structure), the basic building blocks of the more complex structure. The α-helices and β-sheets are arranged into the tertiary structure. Multiple ternary structures form the quaternary structure. Each tertiary structure can be considered a domain, a distinct functional, structural protein unit with a unique independent three-dimensional shape [117,118]. Scattering from protein solutions has been discussed extensively in the literature [26,27,29]. Therefore, we illustrate the analysis of solution SAXS data using more recent examples drawn from intrinsically disordered proteins (IDPs), multidomain proteins, and protein condensates as examples in this class of materials [28,30,119]. SAXS data from solutions are analyzed using methods that are quite different from those discussed thus far for solids, fibers, and lamellar structures. Solution scattering data provide information about the size and shape of the scattering particles, proteins, or the assembly of proteins, as well as their internal structure at nm resolutions.

5.1. Intrinsically Disordered Proteins

One of the common mechanisms that drives cellular organization is the formation of biomolecular condensates via liquid-liquid phase separation (LLPS). By using SAXS to determine the structural features of proteins that encode LLPS, one can obtain critical insights into many biological processes. The driving force behind the biomolecular phase separation is thought to be IDPs. IDPs lack a fixed three-dimensional structure under physiological conditions, existing instead as dynamic ensembles of conformations. IDPs can interact with multiple other proteins and adapt to changing conditions crucial for signaling, recognition, and regulation. They fold when binding to specific sites, bridging their dynamic nature with essential biological functions [120].

Figure 10a shows an example of raw SAXS data. These featureless data are typical of solution scattering. The size of the scattering particle, IDP, is most easily obtained in the form of the radius of gyration R_g (Å) using the Guinier approximation.

$$I\left(q\right)\approx I\left(0\right)\exp (-{\displaystyle \frac{q^2{R}_g^2}{3}})$$

(18)

where I(q) is the scattering intensity at the scattering vector q and I(0) is the scattering intensity at the zero angle (q = 0). R_g can be obtained from the slope, –(R_g)²/3 of the linear region of the curve in the ln(I(q)) vs. q² plot, Guinier plot (Figure 10b). The intercept yields I(0), which is determined by the concentration and is proportional to the molecular weight and electron density contrast of the particle. Thus, particles with larger molecular mass and larger contrast scatter more X-rays, resulting in a higher I(0). The equation is valid at low q, qR_g < 1.3 for globular particles or qR_g < 0.8 for elongated particles [27]. R_g measures the size of the scattering particle, representing the root-mean-square distance of all scattering points from their center of mass. The molecular mass equation is shown in Equation (19), where MM is the molecular mass (g/mol) and c is the sample concentration (mol/L). The subscript st corresponds to a standard.

$${MM}_p={\displaystyle \frac{I\left(0\right)}{c}}*{\displaystyle \frac{{MM}_{st}}{\raisebox{1ex}{${I\left(0\right)}_{st}$}\!\left/ \!\raisebox{-1ex}{$c_{st}$}\right.}}$$

(19)

The volume of the particle can be obtained from the Porod’s Invariant Q (Equation (20)). Porod’s invariant quantifies the total scattering power of the system; it is independent of the particles’ size, shape, or arrangement, and depends on the electron density contrast and the volume of the particle. Information about the surface area and the volume of the particle is obtained by calculating Equation (21).

$$Q={\int }_0^{\infty }{q}^{2}I\left(q\right)dq$$

(20)

$$Q=2{\pi }^2{\left(\mathsf{\Delta }\rho \right)}^{2}V$$

(21)

For a system with sharp boundaries, Δρ is the electron density difference between the sample and solvent. Information about the internal order, flexibility, and the shape of the particle are obtained from a Kratky plot, a plot of I(q) q² vs. q (Figure 10c) [121]. Well-folded, globular, monodomain proteins have bell-shaped Kratky plots, whereas disordered proteins show continuous increase and unfolded proteins appear hyperbolic [122].

Figure 10. (a) Raw data for solution scattering. (b) Guinier transformation. (c) Kratky plot; reprinted from Martin et al. (2021) [123] with permission from Elsevier. (d) Image of an antibody. (e) Scattering intensity of various immunoglobins. (f) Pair distribution functions of the transformed scattering intensity; reprinted from Tian et al. (2015) [124] with permission from International Union of Crystallography.

Figure 10. (a) Raw data for solution scattering. (b) Guinier transformation. (c) Kratky plot; reprinted from Martin et al. (2021) [123] with permission from Elsevier. (d) Image of an antibody. (e) Scattering intensity of various immunoglobins. (f) Pair distribution functions of the transformed scattering intensity; reprinted from Tian et al. (2015) [124] with permission from International Union of Crystallography.

Another parameter useful in characterizing particles in solution is the fractal dimension, D_s, which is associated with the interfaces or boundaries between particles and the surrounding medium. D_s can be determined from the scattered intensity at the tail end, i.e., the high q-region, of the SAXS curve using the Equation (22)

$$I\left(q\right)\alpha q^{{-(6-D}_s)}$$

(22)

In the limit of when the surface of the particle has sharp and smooth interface, then D_s = 2, and the Equation (2) reduces to Porod’s law states that I(q) ∝ q⁻⁴. The generalized form of Porod’s law is used to charcereize surface roughness in terms of fractals of dimension D_s, with (2 ≤ D_s <3) for surface fractals, and I(q) ∝ q^−(6−D_m⁾, with (1 ≤ D_m < 3) for mass fractals. If the particle is a sphere with well-defined boundaries, d_f = 4, where intensity decays q⁻⁴, characteristic of smooth, solid surfaces. For fractal surfaces, d_f < 4 represents rough, irregular surfaces. For 2D (sheet) or 1D (rod-like) particles, d_f = 2 or 1, respectively.

Fractal dimensions also provide insights into the structural organization and solvation of intrinsically disordered proteins (IDPs). The fractal dimension (D_m) reflects how mass scales within the protein, with values around 1.7 indicating a well-solvated, expanded random coil. For the bacteriophage λN protein, a fractal dimension of 1.76 confirmed its extensively unfolded and well-solvated nature, even under varying urea concentrations or ionic strength conditions. This highlights the protein’s flexibility and suitability for dynamic interactions, making D_m a valuable metric for studying protein behavior and solvation in diverse environments [125].

The parameters, such as the R_g related to size, the Porod exponent, and the shape of the Kratky plot related to internal order, are implicit in the pair distribution function, P(r) (PDF) (Equation (23)), which can be obtained from the scattered intensity, I(q), by carrying out indirect Fourier transform (IFT, Equation (16)), where D_max is the maximum dimension of the particle.

$$P\left(r\right)={\displaystyle \frac{1}{2{\pi }^2}}{\int }_0^{\infty }{q}^{2}I(q){\displaystyle \frac{\sin \left(qr\right)}{qr}}dq$$

(23)

$$I\left(q\right)=4\pi {\int }_0^{D_{max}}P\left(r\right){\displaystyle \frac{\sin \left(qr\right)}{qr}}dr$$

(24)

P(r) describes the frequency of occurrence of the distance, r, between pairs of scattering centers within the particle. Peaks in P(r) correspond to interatomic or intermolecular distances. At r > D_max, P(r) goes to 0. When P(r) is graphed against r, the shape of the plot gives insight into the particle’s geometry. A symmetrical P(r) function indicates a spherical particle. An asymmetric P(r) indicates an elongated or irregular shape. The P(r) function is valuable for unraveling the details of multidomain proteins. This information is used for shape determination for well-folded systems [126]. However, since the use of IFT requires the P(r) function to converge to zero at a fixed maximum dimension (D_max), its use is difficult with the data from highly flexible, disordered systems, such as IDPs. It is often recommended that IFT be used to obtain a more reliable value of radii of gyration compared to Guinier analysis because it uses scattering data from the full q range. Although the problems resulting from the use of P(r) calculated from IFT outweigh the benefits for IDPs, these functions are useful with multidomain proteins as will be discussed in the next section.

5.2. Multidomain Proteins

Single-domain proteins such as myoglobin, albumin, and lysozyme consist of just one polypeptide chain. A multidomain protein is made of multiple independent structural units. Hemoglobin, which transports oxygen in the blood, is an example of a multidomain protein containing four globin units. Other examples include laminin, essential for the formation and functioning of muscle, nerve, kidney, lung, skin, vasculature, and other tissues, pyruvate kinase, RNA-binding proteins, and proteins, such as fibronectin, involved in blood clotting and cell adhesion. We use a multidomain protein, monoclonal antibody (mAb), to illustrate the application of SAXS for this class of proteins. mAbs bind specifically to antigens on the surface of target cells, enabling precise immune responses or direct cell destruction. A detailed study is useful in engineering molecules to mimic or enhance the immune system’s ability to target and destroy harmful cells, such as cancer cells. Small-angle neutron scattering (SANS) using contrast-matching techniques with deuterated solvents has been used to study the subdomains, fragment antigen binding (Fab), and crystallizable fragment (F_c) (Figure 10d). These curves are used to generate PDF plots (Figure 10e). The figure shows the SAXS data from three of the four human IgG subclasses, namely IgG1, IgG2, IgG3, and IgG4 [124]. The R_g for the three are 49.4 (IG1) and 47.6 Å (IG2 and IG4). The differences in R_g are reflected in the PDFs. Unlike in IDPS, the PDFs are most useful in understanding the small differences in the raw SAXS data.

SAXS is also used to investigate protein-protein interactions in concentrated mAb solutions that result in self-association, reversible dimer formation, aggregation and appearnce of larger clusters that are correlated with viscosity [127,128,129]. Insights into colloidal stability obtained by integrating light scattering and coarse-grained modeling with SAXS to study aggregation can lead to formulation of higher-concentration and thermally stable mAb solutions [130]. Larpent et al. analyzed the crystalline mAb suspensions under various conditions to optimize batch processes for vaccination [131]. In contrast, SAXS gave structural features (R_g and aggregation). SAXS gave insight into colloidal stability and aggregation, whilst SANS gave insight into molecular and domain-level dynamics [132].

6. Conclusions

SAXS is a powerful tool for investigating hierarchical materials. The non-destructive nature of SAXS, and the ease with which SAXS can be coupled with many other characterization techniques under varying environmental conditions, makes it a versatile tool for understanding the dynamic nature of these structures. The analysis of the large variety of scattering patterns is presented by dividing the biological materials into three classes: 1D fibrous structures in collagen, hair, feather, and silk; 2D lamellar structures in membranes; and 3D disordered structures as in protein solutions. The analysis of the distinctive patterns that are characteristic of each of these classes of materials is discussed. It is hoped that such a detailed analysis of these structures will prove to be useful in biomedical applications such as biomimetic designs of new materials and disease diagnoses.