# Probing Globular Protein Self-Assembling Dynamics by Heterodyne Transient Grating Experiments
## 3 Results and Discussion
In fact, the parameters that described the propagation of the acoustic wave into the materials were \(\Gamma_{A}\) = 13.2 MHz, \(C_{s}\) = 1.52 km/s for the concentrated solution and \(\Gamma_{A}\) = 14.3 MHz, \(C_{s}\) = 1.53 km/s for the gel system. Figure 3 plots the variation of the speed of sound in the studied samples. In all conditions, the speed of sound curves had a similar trend as a function of temperature. This was related to the sound velocity temperature behavior of deuterium oxide (D\({}_{2}\)O), in fact, the solvent was the most abundant  
Figure 1: Heterodyne transient grating (HD-TG) decay signal (gray dots) and its relative fit (red line) of 120 mg/mL lysozyme aqueous solution at T = 23 \({}^{\circ}\)C and wave-vector \(q\) = 2.1 \(\upmu\)m\({}^{-1}\). Figure 2: HD-TG decay signals for the wave vector \(q\) = 2.1 \(\upmu\)m\({}^{-1}\) of: Hen egg white lysozyme (HEWL) 120 mg/mL solution (purple line), 240 mg/mL solution (orange line) and gel (blue line) at 23 \({}^{\circ}\)C (left panel) and 60 \({}^{\circ}\)C (right panel). component in the samples. Figure 3a shows the temperature dependence of the speed of sound of the diluted HEWL solution, which was very close to that of the pure solvent. In fact, \(C_{s}\) increased rapidly with temperature, until 50 \({}^{\circ}\)C, and less rapidly between 50-75 \({}^{\circ}\)C and then decreased after 75 \({}^{\circ}\)C, as experimentally observed in D\({}_{2}\)O [42] and H\({}_{2}\)O [29]. The \(C_{s}\) data of the concentrated (240 mg/mL) HEWL solution, reported in Figure 3b, showed an overall increase of the velocities with a monotonic temperature dependence similar to the diluted solution dependence up to 55 \({}^{\circ}\)C. At this temperature value, there was a small but clear deviation. The gel material also showed the solvent trend, but again a dip in the sound velocity showed up at a lower temperature of about 45 \({}^{\circ}\)C.  
We must remember that the present HD-TG experiment measured the propagation of longitudinal acoustic bulk waves, whereas the gel/aggregation formations were expected to modulate mainly the transverse components of the sound velocities. Nevertheless, the self-assembling phenomena modulated the sound velocities, changing its overall value and producing the observed dips at about 55 \({}^{\circ}\)C (Figure 3b), due to the formation of a high content of aggregate species. With regard to the gel matrix, the deviation of the sound velocity data trend occurred at 45 \({}^{\circ}\)C due to the transition from gel to liquid state (Figure 3c). This was also in agreement with the damping rate trends that are considered in the following. Figure 4 shows the temperature dependence of the acoustic damping rate of the studied samples. For the 120 mg/mL HEWL solution (Figure 4a), starting from room temperature to \(\sim\)45 \({}^{\circ}\)C, a decrease from 11.1 MHz to 7.3 MHz of the damping rate occurred, similar to what was observed for a pure liquid [30]. In the range 45-55 \({}^{\circ}\)C, the damping rate remained constant, this behavior could be connected to the transition of proteins from a folded to unfolded state. The literature reports a temperature of about 75 \({}^{\circ}\)C [4; 43] for the folding-unfolding transition of the lysozyme in pure aqueous solution. Nevertheless, in our sample, the acidification of the solution destabilized the protein's  
Figure 3: Temperature dependence of the sound velocity on: (**a**) The diluted (120 mg/mL) lysozyme solution, (**b**) the concentrated (240 mg/mL) lysozyme solution and (**c**) the gel system. The vertical lines point out the temperatures where the speed sound variations take place. tertiary and secondary structure, so that the unfolding process occurred at lower temperature [44], in agreement with the damping rate observed variation. During the folding-unfolding transition, the damping rate did not change, which could be due to a stabilization of the solution viscosity, induced by the increasing concentration of denatured proteins.