# Probing Globular Protein Self-Assembling Dynamics by Heterodyne Transient Grating Experiments
## 1 Introduction  
In order to fulfill protein functions, the polypeptide chains must be organized in their native conformation. In fact, after protein biosynthesis the folding process occurs, giving the protein a three-dimensional structure that makes it biologically active. Sometimes, erroneous protein structuring (misfolding) may occur, resulting in the onset of many diseases [1]. In general, the development of protein structural organization is given by the tendency to minimize interactions between the hydrophobic residues and the polar solvent. Depending on solvating conditions, folding or misfolding and aggregation can be favored. It is possible to modulate the protein's unfolding and aggregation processes by changing the environmental conditions, such as pH, ionic strength, solvent composition and temperature. These self-assembling phenomena are characterized by different steps, in which the protein undergoes conformational rearrangements and intermolecular associations to form stable structures of increasing complexity [2, 3], such as amyloid fibrils and hydrogel networks [4, 5]. Protein-based hydrogels have recently received great attention since they can be the key to develop novel biomaterials. In fact, hydrogels have the capability to retain inside their matrix a great amount of water (up to 97%) [6], which make these materials similar to biological tissues [7, 8]. Protein hydrogels are therefore of great interest in many fields spanning from the medical to the nanotechnological [9, 10, 11]. Moreover, hydrogels hold the promise to help in the understanding of tissue formation and transformation. For example, the \(\beta\)-sheet intermolecular motifs, that are responsible ofseveral diseases, form hydrogel structures, which self-organize into three-dimensional networks. The hydrogel matrices have been utilized as scaffolds for stem cells [12], highlighting how the mechanical properties of the gel can modulate cellular differentiation processes. In the class of globular proteins, an excellent study case is hen egg white lysozyme (HEWL). In fact, HEWL is a small globular protein easy to find, that is capable of producing a variety of aggregated structures, including the intermolecular \(\beta\)-sheet. The self-assembly of HEWL--and more in general of globular proteins--give rise to much interest in biomaterial science since they can originate biocompatible hydrogels whose properties can be modulated by changing the aggregation condition. Lysozyme hydrogels are promising candidates to produce successful cell scaffolds, thanks to their high cytocompatibility [13]. With the aim of studying the protein aggregation processes and following the evolution of the system gelation, we choose to work with an acid (pH = 1.4) and highly concentrated (240 mg/mL) HEWL solution [14]. In particular, the low pH value favors the unfolding process and the high protein concentration facilitates the aggregation/gelation of the system [15]. The investigation of protein solutions and hydrogels by ultrasonic propagation has been performed by a series of research studies [16; 17; 18; 19; 20; 21]. In fact, the sound speed and the acoustic wave damping can be used to provide information about protein hydration in an aqueous environment and their conformational changes or phase transformations induced by the external stimuli [22]. In particular, [PERSON] and [PERSON] [16] study the behavior of the acoustic wave through a solution of whey proteins, finding that the acoustic attenuation in protein solutions is due both to scattering contributions and protein relaxation phenomena. The latter could be related to hydration, proton transfer [23; 24; 25] and conformational changes [26; 27]. Despite several studies on this argument, one of the major challenges of research in the field of soft matter remains that of relating intra- and intermolecular interactions that govern the structure of individual proteins and their aggregates with the elastic macroscopic characteristics of the whole system.