**1. Introduction**

Wide-line 1H NMR is an accepted method to delineate the structures of hydrogen-containing molecules determined primarily by X-ray and, to a lesser extent, by neutron-scattering. This way, it can provide information on the location and structural environment of hydrogen atoms in proteins. It has a unique capability, on the other hand, in the direct observation of translational and rotational movements of molecules in the condensed phase.

NMR characteristics of aqueous solutions rapidly frozen and then slowly thawed through equilibrium thermal states provide direct information on the immobile and partially or fully mobile parts of the molecules. We have previously reviewed relevant features of this approach in our works "Hydrogen skeleton, mobility and protein architecture" [1] and "Studying molecular motions in solid states by NMR" [2].

Based on these studies, we state that molecular motions in the sample result in narrowing of the wide-line NMR spectrum. This phenomenon is known as motional narrowing in the literature [3]. Our goal is to advance from this observation to arrive at the thermodynamic characterization of protein systems.

In Figure 1a, we show the typical 1H NMR free-induction decay (FID) signal of a set of spins containing proton–proton pairs of different mobilities. In Figure 1b, we present the deduced NMR spectrum. Similar FID signals and NMR spectra are observed at certain temperatures when studying the aqueous solution of a protein that contains hydrogen pairs.

**Figure 1.** Free induction decay (FID, panel (**a**)) and spectrum (panel (**b**)) of a motionally two-state spin system. (We focus on the slow component of the FID, the initial part of which is lost in the dead time of the spectrometer, marked by dashed line, and can be disregarded).

Time domain (Figure 1a) and frequency or energy domain (Figure 1b) representation of the spectra are linked through Fourier transformation, yet it may be useful to consider both, as they provide information on different practical utilities. The amplitude of the FID signal (even considering its slow component) extrapolated to time zero gives the number of relevant nuclei (spins) through nuclear magnetization. The amplitude of response to the 90◦ radiofrequency pulse is proportional to the relevant *x*-*y* component of nuclear magnetization that is further proportional to *M*<sup>0</sup> ≈ (*nB*0)/*T*, in which *B*<sup>0</sup> is the constant magnetic induction, *T* is the absolute temperature, and *n* is the number of resonant nuclei (in our case, it equals the number of protons in water). On the other hand, the width of the spectrum gives direct information on the motional characteristics of proton spin pairs. In a system of two components (e.g., one that contains both mobile and immobile spin-pairs), it is important to have direct information on both parameters.

It is questionable whether such a simple approach can give significant novel information on the dynamics of a complex system, such as a protein and its environment in an aqueous solution. The independent measurement over a broad temperature range of the two parameters of the slow FID component (FID amplitude extrapolated to *t* = 0 and the spectral width) is debatable.

Therefore, here we address the behavior of the slow-FID, and the narrow-spectral component. Our working hypothesis (that we already partially proved) is that the narrow spectral component comes from water molecules bound to the protein, termed bound water-molecules [4]. One may ask a range of relevant questions about their number, their strength of binding to the protein vs. the neighboring water molecules, and about their potential field following molecular changes of the protein, etc. Similar questions can also be asked for the broad-spectrum component, which we have already addressed before [1,2].

In earlier studies [5–8], we have addressed in detail the behavior of globular and intrinsically disordered proteins (IDPs) in aqueous solutions and provided an initial and partial interpretation of experimental observations. As relevant examples, we refer to results with proteins, such as ubiquitin, bovine serum albumin, α-synuclein (and its point mutants), calpastatin, ERD10 (early response to dehydration 10), and lysozyme. Here, we demonstrate our point by focusing on two proteins, ubiquitin (Ubq) and ERD10, as they have been thoroughly studied earlier; one (Ubq) is a globular/structured protein and the other (ERD10) is intrinsically disordered, i.e., they are representatives of these distinct structural classes. We show the temperature dependence of the slowly-decaying component of the FID extrapolated to *t* = 0 (which gives directly the ratio of relevant mobile water protons). We show the observed behavior in the form of a melting diagram (*MD*). In Figure 2, we show the *MD* of three

studied systems (bulk water and the aqueous solution of two proteins, ubiquitin and ERD10) in the usual ◦C scale.

The melting process of inhomogeneous systems (such as the protein solutions we study), basically differs from the first-order phase transition of homogeneous, single component material, such as the melting of ice at a given transition temperature.

**Figure 2.** "Old fashioned" melting diagrams, i.e., the total number of mobile water molecules (through protons) normalized to the total number of water molecules, as a function of temperature (blue squares: bulk water, green circles: ubiquitin, red stars: ERD10 proteins in aqueous solutions). The data are given for 50 mg/mL protein concentration.

We consider melting as the process of the beginning of movement of a component of the mixture (such as a bound water-molecule, or a fragment of the protein of high symmetry, e.g., a methyl group or other terminal moiety), in which either translation or rotation begins. In our case, these (individual) events of initial movements show a temperature distribution characteristic of the given molecule, and the derived *MD*s link the well-defined, directly measurable NMR characteristics with atomic/molecular motions.

These characteristics can thus also give direct information on molecular interactions. The water molecules associated with the protein molecule constitute an integral part of the system. Thus, their nuclei, rather than large energy particles applied in scattering techniques (such as X-ray crystallography), monitor the potential energy surface of the protein as built-in probes. In our previous works [5–8], however, we only drew qualitative conclusions from the *MD*s.

These were as follows. In aqueous solutions, melting (that is, beginning of molecular motions) of protein-bound water molecules begins at a much lower temperature than the melting of bulk ice. Each protein has a unique *MD* (individual profile or fingerprint) that results from its individual thermodynamic characteristics. The *MD* of globular and ID proteins vastly differ. They can be characterized by temperature-independent FID amplitudes—a plateau (globular protein)—or they can lack a plateau (IDP) or can have a plateau of small temperature extension (partly IDP).

#### *1.1. Energetic Interpretation of Melting Diagrams*

We have made significant advances in several respects of interpreting our results [9,10] since we last addressed these questions [5–8]. Key steps are detailed in chapters 4–6 of ref [9]; here, we add a new element and summarize these steps in more detail, following the logical order of the application.

As a reminder, we are following the beginning of the movement—probably the rotation—of water molecules bound to the surface of the protein, by observing motional narrowing in wide-line 1H NMR spectroscopy. For the first time in the field—following the seminal work of Kittel and Kroemer [11]—we introduced the concept of fundamental temperature, *T*f, and also introduced here the idea suggested by Waugh and Fedin [12] for connecting the thermal excitation energy, *V*0, in which molecular motions begin with the temperature, *T*, as *V*<sup>0</sup> = constant × *T*.

In some detail, the key steps taken are as follows.
