Water Dynamics in Highly Concentrated Protein Systems—Insight from Nuclear Magnetic Resonance Relaxometry

Citation: Kruk, D.; Kasparek, A.; Masiewicz, E.; Kolodziejski, K.; Cybulski, R.; Nowak, B. Water Dynamics in Highly Concentrated Protein Systems—Insight from Nuclear Magnetic Resonance Relaxometry. Int. J. Mol. Sci. 2023,24, 4093. https://doi.org/10.3390/ ijms24044093 Academic Editor: Todd M. Alam Received: 31 December 2022 Revised: 7 February 2023 Accepted: 10 February 2023 Published: 17 February 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). International Journal of Molecular Sciences Article Water Dynamics in Highly Concentrated Protein Systems— Insight from Nuclear Magnetic Resonance Relaxometry Danuta Kruk 1,* , Adam Kasparek 1, Elzbieta Masiewicz 1, Karol Kolodziejski 1, Radoslaw Cybulski 2 and Bartosz Nowak 2 1Department of Physics and Biophysics, University of Warmia & Mazury in Olsztyn, Oczapowskiego 4, 10-719 Olsztyn, Poland 2Department of Mathematical Methods of Informatics, University of Warmia & Mazury in Olsztyn, Sloneczna 54 Street, 10-710 Olsztyn, Poland *Correspondence: danuta.kr[email protected] Abstract: 1 H spin-lattice relaxation experiments have been performed for water–Bovine Serum Albumin (BSA) mixtures, including 20%wt and 40%wt of BSA. The experiments have been carried out in a frequency range encompassing three orders of magnitude, from 10 kHz to 10 MHz, versus temperature. The relaxation data have been thoroughly analyzed in terms of several relaxation models with the purpose of revealing the mechanisms of water motion. For this purpose, four relaxation models have been used: the data have been decomposed into relaxation contributions expressed in terms of Lorentzian spectral densities, then three-dimensional translation diffusion has been assumed, next two-dimensional surface diffusion has been considered, and eventually, a model of surface diffusion mediated by acts of adsorption to the surface has been employed. In this way, it has been demonstrated that the last concept is the most plausible. Parameters describing the dynamics in a quantitative manner have been determined and discussed. Keywords: dynamics; relaxation; proteins 1. Introduction The dynamical properties of molecular systems are one of the most fundamental questions of molecular science. The question encompasses not only the time scale of the motion but also its mechanism—in other words: one is not satisfied with determining the time scale of a specific dynamical process (by providing, for instance, diffusion coefficients), one wishes to get insight into the geometry of the motion (for instance the dimensionality of the translation displacements). The experimental means allowing to enquire into the characteristic features of molecular motion are very limited. Nuclear Magnetic Resonance (NMR) methods are broadly appreciated as a source of information about molecular structure and dynamics. As far as dynamics are concerned, NMR relaxation studies are of primary importance. However, “classical” NMR relaxation experiments are commonly performed at a single, high magnetic field (resonance frequency). According to spin relaxation theories [ 1 – 3 ], the relaxation process is most efficient when the time scale of the fluctuations of the spin interactions causing the relaxation is of the order of the reciprocal resonance frequency. This implies that at high frequencies, one mostly probes fast dynamics. Consequently, to probe dynamical processes occurring over a broad time scale, one has to vary the magnetic field (resonance frequency). This kind of study is referred to as NMR relaxometry. In the present studies, the resonance frequency is varied from about 10 kHz to 10 MHz ( 1 H resonance frequency), which gives three orders of magnitude. Consequently, one can probe molecular motion on the time scale from about 10 −4 s to about 10 −8 s in a single experiment. This potential of NMR relaxometry has been widely exploited for molecular and ionic systems of varying complexity—from liquids [ 4 – 6 ] via polymers and proteins [ 7 – 20 ] to tissues [ 21 , 22 ] and Int. J. Mol. Sci. 2023,24, 4093. https://doi.org/10.3390/ijms24044093 https://www.mdpi.com/journal/ijms Int. J. Mol. Sci. 2023,24, 4093 2 of 21 liquid and solid electrolytes [ 23 – 30 ]. The great advantage of NMR relaxometry is the ability to give insight into the mechanism of motion. Relaxation rates (reciprocal relaxation times) are given as linear combinations of so-called spectral density functions. A spectral density function is defined as a Fourier transform of a corresponding time correlation function characterizing the stochastic fluctuations (caused by the molecular motion) of the spin interactions. The mathematical form of the correlation function (and, hence, the spectral density) depends on the mechanism of the motion (for instance, such as isotropic and anisotropic rotational motion, free (three-dimensional) translation diffusion, or restricted (two-dimensional or one-dimensional) translation motion). Via the form of the spectral density function, the shape of the frequency dependence of the relaxation rates is a fingerprint of the mechanism of motion. At this stage, one should point out that single-frequency relaxation studies hardly contain information about the characteristic features (mechanisms) of the dynamical process leading to the relaxation at that frequency. The advantages of NMR relaxometry (the ability to probe molecular motion over a broad time scale and the ability to reveal the mechanism of motion) interfere with each other. The reason for that is several relaxation contributions present over such a broad frequency range and constitute the overall relaxation rate. The relaxation contributions stem from different relaxation pathways. For instance, magnetic dipole-dipole interactions (being the dominating origin of 1 H relaxation) can be of intra-molecular or inter-molecular origin. The first ones fluctuate in time as a result of rotational and internal dynamics, while the second ones are mostly modulated by translation diffusion. For simple systems, the two contributions can be unambiguously identified and disentangled [ 5 , 6 ], profiting from the time scale separation of translational and rotational dynamics. In such a case, one can fully profit from the unique advantages of NMR relaxometry (investigating rotational and translational dynamics in a single experiment and identifying the mechanism of the observed dynamical processes) in a relatively straightforward way. The task becomes much more cumbersome for multi-component systems due to several relaxation contributions and not clear time scale separation of the dynamical processes associated with the relaxation contribution. Examples of such systems are highly concentrated protein—water mixtures. The systems include a macromolecular fraction (proteins) forming a matrix entrapping water molecules. Consequently, one can expect pools of water molecules to perform different kinds of complex motions. The purpose of this work is twofold. The first one is to enquire into the mechanism of water motion in the presence of a substantial fraction of proteins (in contrast to highly diluted protein solutions [ 18 ]), profiting from the unique potential of NMR relaxometry. For this purpose, Bovine Serum Albumin (BSA) has been chosen as an example. In this context, one should mention NMR relaxometry studies for sedimented proteins showing much different dynamics than proteins in solution [ 19 ] and studies addressing the subject of water diffusion on protein surfaces in the presence of ions [ 20 ]. The second goal has methodological aspects. We present a thorough analysis of 1 H spin-lattice relaxation data for BSA–water mixtures using different forms of spectral density functions. In this way, we demonstrate the challenges of revealing the mechanisms of molecular motion for complex systems. At the same time, the work presents an overview of theoretical models that can potentially be exploited to reproduce NMR relaxometry data for systems including water and a macromolecular fraction and illustrates by examples their verification. Theory 1 H NMR spin-lattice relaxation processes are predominantly caused by magnetic dipole-dipole interactions. According to the spin relaxation theory [ 1 – 3 ], the spin-lattice relaxation rate, R 1 , originating from 1 H1 H dipole-dipole interactions, is given as the following combination of spectral density functions: R1(ω)=CDD[J(ω)+4J(2ω)], (1) Int. J. Mol. Sci. 2023,24, 4093 3 of 21 where ω denotes the resonance frequency in angular frequency units, while CDD is referred to as a dipolar relaxation constant reflecting the amplitude of the magnetic dipole-dipole interactions causing the relaxation process. The form of the spectral density function, J(ω) (Fourier transform of the corresponding time correlation function), depends on the mechanism of the motion responsible for stochastic time fluctuations of the dipole-dipole interactions. For exponential correlation functions, the Fourier transform (and, hence, the spectral density) takes a Lorentzian form. Consequently, the relaxation rate is given as [1–3]: R1(ω)=CDD"τc 1+(ωτc)2+4τc 1+(2ωτc)2#, (2) where τc denotes a time constant characterizing the time scale of the motion, referred to as a correlation time. As already pointed out in the Introduction, the broad frequency range covered in NMR relaxometry experiments implies that several dynamical processes can be probed in a single experiment. The simplest way to get some insight into the molecular motion is to attempt to decompose the overall relaxation process into contributions associated with dynamics occurring on different timescales. In such a case, the relaxation rate can be expressed as [12,16,17]: R1(ω)=CDD sτs 1+(ωτs)2+4τs 1+(2ωτs)2+CDD iτi 1+(ωτi)2+4τi 1+(2ωτi)2+ CDD fτf 1+(ωτf)2+4τf 1+(2ωτf)2+A, (3) where τs , τi , and τf denote correlation times characterizing slow, intermediate, and fast dynamics (in a relative scale), respectively, while CDD s , CDD i , and CDD f are the corresponding dipolar relaxation constants. The frequency-independent factor, A , accounts for a relaxation contribution associated with a very fast motion for which the condition: ωτc 1 is fulfilled in the whole frequency range. An example of such dynamics can be the movement of water molecules in bulk. The decomposition assumes that the contributing dynamical processes can be characterized by exponential correlation functions. One can go beyond the simple description (parametrization) and attempt to get insight into the mechanism of the molecular motion. In water—protein mixtures, it is expected that water molecules perform translation diffusion that is considerably affected by the presence of the macromolecules. Discussing translation diffusion, one should consider the dimensionality of this process—the translation motion can be isotropic (three-dimensional) or anisotropic (two-dimensional in this case). The two-dimensional translation diffusion one envisages a motion occurring near the surface of the macromolecules (surface diffusion). The spectral density function for three-dimensional diffusion, J3D(ω) , takes the form [31–33]: J3D(ω)=72 5Z∞ 0 u4 81 +9u2−2u4+u6 τtrans 1+(ωτtrans)2du, (4) Consequently, the corresponding expression for the spin-lattice relaxation rate, R1(ω) , can be expressed as a sum of a relaxation contribution associated with three-dimensional translation diffusion and Lorentzian terms. Limiting ourselves to a single Lorentzian term, one obtains: R1(ω)=3 2µ0 4πγ2 H}21 d3NHR∞ 0u4 81+9u2−2u4+u6τtrans u4+(ωτtrans)2+4τtrans u4+(2ωτtrans)2du+ CDDτc 1+(ωτc)2+4τc 1+(2ωτc)2+A, (5) where γH is 1 H gyromagnetic factor, µ0 is the vacuum permeability, } is reduced Planck constant, NH denotes the number of hydrogen atoms per unit volume (referring to the Int. J. Mol. Sci. 2023,24, 4093 4 of 21 fraction of water molecules undergoing the translation diffusion), while d denotes a distance of the closest approach [ 31 , 32 ]. The model is called force free hard sphere model—it assumes that molecules have a form of hard spheres with 1 H nuclei placed in their centers. In this approximation, the distance of the closest approach is given as a sum of the radii of the interacting molecules—in case of identical molecules, this gives the molecular diameter. The correlation time τtrans is given as τtrans =d2 2Dtrans , where Dtrans denotes the translation diffusion coefficient. In the low-frequency range, when ωτtrans < 1, the spectral density for three-dimensional translation diffusion (Equation (4)) shows a linear dependence on √ω [ 31 – 33 ]. Consequently, when the dominating relaxation contribution at low frequencies stems from intermolecular dipole-dipole interactions modulated by three-dimensional translation diffusion, the relaxation rates, R1(ω) , show a linear dependence on √ω in this range. In case the diffusion process is restricted to two dimensions—in other words, it occurs in the vicinity of a surface, the corresponding spectral density, J2D(ω) , takes the form [4,15,19,34,35]: J2D(ω)=τtransln   1+(ωτtrans)2 τtrans τres 2+(ωτtrans)2  , (6) where τres denotes a residence lifetime of water molecules on the surface of the macromolecules. For a long residence lifetime, when τtrans τres ωτtrans, Equation (6) converges to: J2D(ω)=τtranslnh1+(ωτtrans)−2i, (7) This implies that at low frequencies, when ωτtrans < 1, the spectral density shows a linear dependence on lnω [ 35 ]. Therefore, in analogy to the case of three-dimensional diffusion, when the relaxation contribution associated with translation dynamics dominates in the low-frequency range, the relaxation rate shows a linear dependence on lnω . For two-dimensional translation diffusion, the counterpart of Equation (5) takes the form: R1(ω)=Ctransτtransln1+(ωτtrans)2 (τtrans τres )2+(ωτtrans)2+4ln1+(2ωτtrans)2 (τtrans τres )2+(2ωτtrans)2+ CDDτc 1+(ωτc)2+4τc 1+(2ωτc)2+A, (8) where Ctrans denotes a dipolar relaxation constant. When neglecting the effect of the residence lifetime, Equation (8) converges to: R1(ω)=Ctransτtranshln1+(ωτtrans)−2+4ln1+(2ωτtrans)−2i+ CDDτc 1+(ωτc)2+4τc 1+(2ωτc)2+A,(9) In biomolecular systems one can also expect a relaxation contribution originating from 1 H14 N dipole-dipole interactions. 14 N nuclei possess quadrupole moments. This implies that in case of slow molecular dynamics, the energy level structure of 14 N nuclei stems from a superposition of their Zeeman and quadrupole interactions. As the quadrupole coupling is independent of the magnetic field, there are magnetic fields at which the 1 H resonance frequency matches the transition frequencies of the 14 N nucleus between its energy levels. When the 1 H and 14 N transition frequencies match, the 1 H magnetization can be transferred to (taken over by) the 14 N nucleus [ 16 , 17 , 36 – 43 ]. This manifests itself as a faster decay of the 1 H magnetization (a higher relaxation rate) at specific frequencies. The faster decay leads to a frequency-specific enhancement of the spin-lattice relaxation Int. J. Mol. Sci. 2023,24, 4093 5 of 21 rate, referred to as Quadrupole Relaxation Enhancement (QRE). The 1 H14 N relaxation contribution, RH−N 1(ω)can be expressed as [43]: RH−N 1(ω) =CHN DD ×         1 3+sin2θcos2φτQ 1+(ω−ω−)2τ2 Q +τQ 1+(ω+ω−)2τ2 Q+ 1 3+sin2θsin2φτQ 1+(ω−ω+)2τ2 Q +τQ 1+(ω+ω+)2τ2 Q+ 1 3+cos2θτQ 1+(ω−ω0)2τ2 Q +τQ 1+(ω+ω0)2τ2 Q         (10) where the frequencies ω− , ω+ and ω0 are defined as: ω− 2π=aQ1−η 3 , ω+ 2π=aQ1+η 3 and ω0=ω+−ω− , aQ denotes the quadrupole coupling constant, while η is the asymmetry parameter. The angles θ and φ describe the orientation of the principal axis system of the electric field gradient tensor with respect to the 1 H14 N dipole-dipole axis, while the correlation time τQ characterizes time fluctuations of the 1 H14 N dipole-dipole coupling. The dipolar relaxation constant, CHN DD , is defined as: CHN DD =2 3µ0 4π γHγN} r3 HN 2 , where rHN denotes the 1H-14N inter-spin distance, while γNdenotes 14N gyromagnetic factor. 2. Results 1 H spin-lattice relaxation data for BSA–water mixtures, 20%wt and 40%wt of BSA, versus temperature, are shown in Figure 1a,b. Int. J. Mol. Sci. 2023, 24, x FOR PEER REVIEW 5 of 24 The faster decay leads to a frequency-specific enhancement of the spin-lattice relaxation rate, referred to as Quadrupole Relaxation Enhancement (QRE). The 1H-14N relaxation contribution, 𝑅1 𝐻−𝑁(𝜔) can be expressed as [43]: 𝑅1 𝐻−𝑁(𝜔) =𝐶𝐷𝐷 𝐻𝑁× [ (1 3+𝑠𝑖𝑛2𝜃𝑐𝑜𝑠2𝜙)( 𝜏𝑄 1+(𝜔−𝜔−)2𝜏𝑄 2+𝜏𝑄 1+(𝜔+𝜔−)2𝜏𝑄 2)+ (1 3+𝑠𝑖𝑛2𝜃𝑠𝑖𝑛2𝜙)( 𝜏𝑄 1+(𝜔−𝜔+)2𝜏𝑄 2+𝜏𝑄 1+(𝜔+𝜔+)2𝜏𝑄 2)+ (1 3+𝑐𝑜𝑠2𝜃)( 𝜏𝑄 1+(𝜔−𝜔0)2𝜏𝑄 2+𝜏𝑄 1+(𝜔+𝜔0)2𝜏𝑄 2)] (10) where the frequencies 𝜔−,𝜔+ and 𝜔0are defined as: 𝜔− 2𝜋 =𝑎𝑄(1−𝜂 3), 𝜔+ 2𝜋 =𝑎𝑄(1+𝜂 3) and 𝜔0=𝜔+−𝜔−, 𝑎𝑄 denotes the quadrupole coupling constant, while 𝜂 is the asymmetry parameter. The angles 𝜃 and 𝜙 describe the orientation of the principal axis system of the electric field gradient tensor with respect to the 1H-14N dipole-dipole axis, while the correlation time 𝜏𝑄 characterizes time fluctuations of the 1H-14N dipole-dipole coupling. The dipolar relaxation constant, 𝐶𝐷𝐷 𝐻𝑁, is defined as: 𝐶𝐷𝐷 𝐻𝑁 =2 3(𝜇0 4𝜋𝛾𝐻𝛾𝑁ℏ 𝑟𝐻𝑁 3)2, where 𝑟𝐻𝑁 denotes the 1H-14N inter-spin distance, while 𝛾𝑁 denotes 14N gyromagnetic factor. 2. Results 1H spin-lattice relaxation data for BSA–water mixtures, 20%wt and 40%wt of BSA, versus temperature, are shown in Figure 1a,b. Figure 1. 1H spin-lattice relaxation rates for BSA–water mixtures versus temperature, (a) 20%wt of BSA, (b) 40%wt of BSA. Stars show the changes in the relaxation rates upon cooling down. Looking at Figure 1a, one sees that between 268 K and 263 K, the dynamics of the system changed due to the freezing of the water fraction. Actually, the freezing process has been captured—stars in Figure 1a. The temperature was set to 263 K, and, after 60 min, the experiment began. The relaxation rates at the highest frequency correspond to those at 268 K, then in the course of time with progressing freezing, the relaxation rates reach the values of the relaxation data represented by blue squares that have been obtained at 263 K after waiting the next 60 min. The data for 263 K and below show QuadFigure 1. 1 H spin-lattice relaxation rates for BSA–water mixtures versus temperature, ( a ) 20%wt of BSA, (b) 40%wt of BSA. Stars show the changes in the relaxation rates upon cooling down. Looking at Figure 1a, one sees that between 268 K and 263 K, the dynamics of the system changed due to the freezing of the water fraction. Actually, the freezing process has been captured—stars in Figure 1a. The temperature was set to 263 K, and, after 60 min, the experiment began. The relaxation rates at the highest frequency correspond to those at 268 K, then in the course of time with progressing freezing, the relaxation rates reach the values of the relaxation data represented by blue squares that have been obtained at 263 K after waiting the next 60 min. The data for 263 K and below show Quadrupole Relaxation Enhancement (QRE) effects (quadrupole peaks). For the mixture including 40%wt of BSA (Figure 1b) , the freezing temperature has been carefully investigated—it has turned out that at 266 K, the system remains liquid, while it freezes at 265 K. Here, one also sees QRE effects. Int. J. Mol. Sci. 2023,24, 4093 6 of 21 Before proceeding with a quantitative analysis of the relaxation data, it is worth noting some effects (Figure 2a). The ratio between the relaxation rates for the mixture containing 40%wt of BSA and 20%wt at 268 K and 273 K has a characteristic shape that, in fact, repeats itself at 278 K (after multiplying the ratio by 0.87). At low temperatures, the ratio reaches a factor close to one in the whole frequency range—that means that the relaxation data tend to overlap. The overlapping is seen in Figure 2b, which also shows, for comparison, relaxation data for solid BSA at 293 K taken from Ref. [16]. Int. J. Mol. Sci. 2023, 24, x FOR PEER REVIEW 6 of 24 rupole Relaxation Enhancement (QRE) effects (quadrupole peaks). For the mixture including 40%wt of BSA (Figure 1b), the freezing temperature has been carefully investigated— it has turned out that at 266 K, the system remains liquid, while it freezes at 265 K. Here, one also sees QRE effects. Before proceeding with a quantitative analysis of the relaxation data, it is worth noting some effects (Figure 2a). The ratio between the relaxation rates for the mixture containing 40%wt of BSA and 20%wt at 268 K and 273 K has a characteristic shape that, in fact, repeats itself at 278 K (after multiplying the ratio by 0.87). At low temperatures, the ratio reaches a factor close to one in the whole frequency range—that means that the relaxation data tend to overlap. The overlapping is seen in Figure 2b, which also shows, for comparison, relaxation data for solid BSA at 293 K taken from Ref. [16]. Figure 2. (a) Ratio between spin-lattice relaxation rates for BSA–water mixtures (40%wt of BSA and 20%wt of BSA); (b) rescaled spin-lattice relaxation data for BSA–water mixtures compared with data for solid BSA. We begin the analysis of the relaxation data with the mixture including 20%wt of BSA and the simplest concept of decomposing the relaxation data into contributions expressed in terms of Lorentzian spectral densities and attributed to dynamical processes referred to as slow, intermediate, and fast ones, according to Equation (2). The outcome of the analysis is shown in Figure 3, while the obtained parameters are collected in Table 1. Figure 2. ( a ) Ratio between spin-lattice relaxation rates for BSA–water mixtures (40%wt of BSA and 20%wt of BSA); ( b ) rescaled spin-lattice relaxation data for BSA–water mixtures compared with data for solid BSA. We begin the analysis of the relaxation data with the mixture including 20%wt of BSA and the simplest concept of decomposing the relaxation data into contributions expressed in terms of Lorentzian spectral densities and attributed to dynamical processes referred to as slow, intermediate, and fast ones, according to Equation (2). The outcome of the analysis is shown in Figure 3, while the obtained parameters are collected in Table 1. Table 1. Parameters obtained from the analysis of 1 H spin-lattice relaxation data for BSA–water mixtures at higher temperatures (268 K and above) in terms of Equation (1). The dipolar relaxation constants, CDD i and CDD f , for 20%wt concentration of BSA yields: CDD i = 7.84 × 10 6 Hz 2 , CDD f = 2.16 × 10 7 Hz 2 . The dipolar relaxation constants for 40%wt concentration of BSA are: CDD s= 5.61 ×106Hz2,CDD i= 1.45 ×107Hz2and CDD f= 9.89 ×107Hz2. 20%wt of BSA Temp. [K] τi[s] τf[s] A[s−1] 268 1.97 ×10−72.28 ×10−82.52 273 1.65 ×10−71.91 ×10−82.25 278 1.27 ×10−71.39 ×10−81.62 298 8.33 ×10−89.62 ×10−90.98 Int. J. Mol. Sci. 2023,24, 4093 7 of 21 Table 1. Cont. 40%wt of BSA Temp. [K] τs[s] τi[s] τf[s] A[s−1] 266 1.16 ×10−62.66 ×10−72.24 ×10−84.68 268 1.10 ×10−62.41 ×10−72.09 ×10−84.50 273 9.67 ×10−71.95 ×10−71.84 ×10−83.91 278 8.53 ×10−71.56 ×10−71.59 ×10−83.40 Int. J. Mol. Sci. 2023, 24, x FOR PEER REVIEW 7 of 24 Figure 3. 1H spin-lattice relaxation data for BSA–water mixture (20%wt of BSA) reproduced in terms of Equation (3); black solid line—overall fit decomposed into a contribution associated with intermediate dynamics (dashed-dotted black line) and fast dynamics (dashed black line), there is no relaxation contribution associated with slow dynamics. Comparison fits obtained in terms of Equation (5) are shown as the corresponding color lines (the lines are hardly visible as they almost overlap with the black ones), they are decomposed into Lorentzian term (dashed-dotted line), a term associated with three-dimensional translation diffusion (dashed line) and frequency independent term (dotted line). Table 1. Parameters obtained from the analysis of 1H spin-lattice relaxation data for BSA–water mixtures at higher temperatures (268 K and above) in terms of Equation (1). The dipolar relaxation constants, 𝐶𝑖𝐷𝐷 and 𝐶𝑓𝐷𝐷, for 20%wt concentration of BSA yields: 𝐶𝑖𝐷𝐷 = 7.84 × 106 Hz2, 𝐶𝑓𝐷𝐷 = 2.16 × 107 Hz2. The dipolar relaxation constants for 40%wt concentration of BSA are: 𝐶𝑠𝐷𝐷 = 5.61 × 106 Hz2, 𝐶𝑖𝐷𝐷 = 1.45 × 107 Hz2 and 𝐶𝑓𝐷𝐷 =9.89 × 107 Hz2. 20%wt of BSA Temp. [K] 𝜏𝑖 [s] 𝜏𝑓 [s] 𝐴 [s−1] 268 1.97 × 10−7 2.28 × 10−8 2.52 273 1.65 × 10−7 1.91 × 10−8 2.25 278 1.27 × 10−7 1.39 × 10−8 1.62 298 8.33 × 10−8 9.62 × 10−9 0.98 40%wt of BSA Figure 3. 1 H spin-lattice relaxation data for BSA–water mixture (20%wt of BSA) reproduced in terms of Equation (3); black solid line—overall fit decomposed into a contribution associated with intermediate dynamics (dashed-dotted black line) and fast dynamics (dashed black line), there is no relaxation contribution associated with slow dynamics. Comparison fits obtained in terms of Equation (5) are shown as the corresponding color lines (the lines are hardly visible as they almost overlap with the black ones), they are decomposed into Lorentzian term (dashed-dotted line), a term associated with three-dimensional translation diffusion (dashed line) and frequency independent term (dotted line). For the mixture including 20%wt of BSA, the relaxation data can be reproduced using only two Lorentzian terms (plus the frequency independent term). The obtained parameters have been associated with intermediate and fast dynamics. The association has been made on the basis of the comparison with the parameters obtained for the mixture including 40%wt of BSA. In that case, all three relaxation contributions are needed to reproduce the data, as shown in Figure 4. The order of the values of the longer correlation times obtained for 20%wt of BSA matches that for the correlation times characterizing Int. J. Mol. Sci. 2023,24, 4093 8 of 21 intermediate dynamics for 40%wt BSA (Table 1). The analysis of the relaxation data for 40%wt of BSA is shown in Figure 4. Int. J. Mol. Sci. 2023, 24, x FOR PEER REVIEW 8 of 24 Temp. [K] 𝜏𝑠 [s] 𝜏𝑖 [s] 𝜏𝑓 [s] 𝐴 [s−1] 266 1.16 × 10−6 2.66 × 10−7 2.24 × 10−8 4.68 268 1.10 × 10−6 2.41 × 10−7 2.09 × 10−8 4.50 273 9.67 × 10−7 1.95 × 10−7 1.84 × 10−8 3.91 278 8.53 × 10−7 1.56 × 10−7 1.59 × 10−8 3.40 For the mixture including 20%wt of BSA, the relaxation data can be reproduced using only two Lorentzian terms (plus the frequency independent term). The obtained parameters have been associated with intermediate and fast dynamics. The association has been made on the basis of the comparison with the parameters obtained for the mixture including 40%wt of BSA. In that case, all three relaxation contributions are needed to reproduce the data, as shown in Figure 4. The order of the values of the longer correlation times obtained for 20%wt of BSA matches that for the correlation times characterizing intermediate dynamics for 40%wt BSA (Table 1). The analysis of the relaxation data for 40%wt of BSA is shown in Figure 4. Figure 4. 1H spin-lattice relaxation data for BSA–water mixture (40%wt of BSA) reproduced in terms of Equation (3); black solid line—overall fit decomposed into a contribution associated with slow dynamics (dashed-dotted black line), intermediate dynamics (dashed grey line), and fast dynamics (dashed black line). Comparison fits obtained in terms of Equation (5) are shown as the corresponding color lines, they are decomposed into a Lorentzian term (dashed-dotted line), a term associated with three-dimensional translation diffusion (dashed line) and a frequency independent term (dotted line). Figure 4. 1 H spin-lattice relaxation data for BSA–water mixture (40%wt of BSA) reproduced in terms of Equation (3); black solid line—overall fit decomposed into a contribution associated with slow dynamics (dashed-dotted black line), intermediate dynamics (dashed grey line), and fast dynamics (dashed black line). Comparison fits obtained in terms of Equation (5) are shown as the corresponding color lines, they are decomposed into a Lorentzian term (dashed-dotted line), a term associated with three-dimensional translation diffusion (dashed line) and a frequency independent term (dotted line). The obtained parameters give insight into the time scale of the molecular motion, however, we aim at revealing not only the time scale but also the mechanism of the movement of water molecules. Therefore, in the second step we have reproduced the data in terms of Equation (5) as a sum of a relaxation contribution associated with threedimensional translation diffusion and a Lorentzian term. The fits have been performed with the following adjustable parameters: CDD , τc , Dtrans , NH and A ; the distance of the closest approach has been set to the diameter of a water molecule: d = 2.7Å. The parameters obtained for the case of 20%wt of BSA are collated in Table 2. The dipolar relaxation constant, CDD = 7.29 × 10 6 Hz 2 is very close to that obtained for the intermediate dynamics (Equation (3)), CDD i = 7.84 × 10 6 Hz 2 ; the values of the correlation time, τc , are also similar to that for τi . The fits are shown in Figure 3for comparison. The same approach has been applied to the relaxation data for the BSA–water (40%wt) mixture. The values of the obtained parameters are discussed in the next section. At this stage one should notice that this approach has led to a reduction in the number of the adjustable parameters—instead of the two pairs of parameters: CDD i , τi and CDD f , τf , characterizing the intermediate and Int. J. Mol. Sci. 2023,24, 4093 9 of 21 slow dynamics, the model involves only the translation diffusion coefficient, Dtrans , and the NHnumber. The translation diffusion coefficients are rather small. Table 2. Parameters obtained from the analysis of 1 H spin-lattice relaxation data for BSA–water mixtures in terms of Equation (5). The dipolar relaxation constant, CDD , for 20%wt concentration of BSA yields: CDD = 7.29 × 10 6 Hz 2 , NH = 1.52 × 10 27 /m 3 ; the corresponding values for 40%wt concentration yield: CDD = 7.69 × 10 6 Hz 2 , NH = 2.84 × 10 27 /m 3 ; the distance of closest approach has been set in all cases to d = 2.7Å. The correlation time τtrans has been obtained from the relationship: τtrans =d2 2Dtrans . Temp. [K] τc[s] Dtrans [m2/s] A[s−1]τtrans [s] 20%wt of BSA 268 1.85 ×10−72.08 ×10−12 1.98 3.50 ×10−8 273 1.54 ×10−72.44 ×10−12 1.79 2.98 ×10−8 278 1.18 ×10−73.28 ×10−12 1.25 2.22 ×10−8 298 7.60 ×10−84.56 ×10−12 0.69 1.60 ×10−8 40%wt of BSA 266 9.08 ×10−75.56 ×10−13 5.00 1.31 ×10−7 268 8.47 ×10−76.10 ×10−13 4.84 1.20 ×10−7 273 7.32 ×10−77.23 ×10−13 4.31 1.01 ×10−7 278 6.35 ×10−78.68 ×10−13 3.82 8.40 ×10−8 In the pursuit of the mechanism of water diffusion, we have attempted to exploit the model of two-dimensional translation diffusion (surface diffusion) represented by Equation (7) . The model of two-dimensional translation diffusion combined with a Lorentzian relaxation contribution (Equation (9)) has led to the fits shown in Figure 5for 20%wt of BSA and in Figure 6for 40%wt of BSA. The obtained parameters are collated in Table 3. Table 3. Parameters obtained from the analysis of the 1 H spin-lattice relaxation data for BSA – water mixtures in terms of Equation (9). The dipolar relaxation constant, CDD , for 20% and 40%wt concentration of BSA yield: CDD = 9.81 × 10 6 Hz 2 and CDD = 8.28 × 10 6 Hz 2 , respectively, the relaxation constant associated with two dimensional translation diffusion is Ctrans = 7.09 × 10 7 Hz 2 for both concentrations of BSA. The translation diffusion coefficient has been obtained from the relationship: Dtrans =d2 2τtrans . Temp. [K] τc[s] τtrans [s] A[s−1]Dtrans [m2/s] 20%wt of BSA 268 1.37 ×10−77.26 ×10−10 1.70 5.02 ×10−11 273 1.14 ×10−76.15 ×10−10 1.55 5.93 ×10−11 278 9.05 ×10−84.05 ×10−10 1.19 9.00 ×10−11 298 5.99 ×10−82.50 ×10−10 0.77 1.46 ×10−10 40%wt of BSA 266 6.92 ×10−77.14 ×10−94.32 5.11 ×10−12 268 6.39 ×10−76.64 ×10−94.01 5.49 ×10−12 273 5.42 ×10−75.67 ×10−93.50 6.43 ×10−12 278 4.63 ×10−74.79 ×10−93.03 7.61 ×10−12 Int. J. Mol. Sci. 2023,24, 4093 16 of 21 ation contribution expressed in terms of Lorentzian spectral densities in the low frequency range, the mathematical properties of the corresponding spectral density functions could not be used as a discriminating factor. These examples demonstrate that unambiguous analysis of NMR relaxometry data for complex molecular systems requires situations in which the relaxation data follow the mathematical form of a specific spectral density over a broad frequency range and this effect is not masked by other relaxation contributions. This has been achieved for the model of two-dimensional translation diffusion modulated by acts of adsorption to the surface with a residence lifetime not being much longer (orders of magnitude) than the correlation time of the translation motion, rendering the conclusion that the diffusion process is of two-dimensional character. The presented strategy of the data analysis demonstrates the need for thorough evaluation of the applied models to profit from the potential of NMR relaxometry. Author Contributions: Conceptualization, D.K.; methodology, D.K. and E.M.; software, R.C. and B.N.; investigation, A.K., E.M. and K.K.; writing—original draft preparation, D.K.; writing—review and editing, D.K. and E.M.; funding acquisition, D.K. All authors have read and agreed to the published version of the manuscript. Funding: This research has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 899683 (project “HIRES-MULTIDYN”). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: https://doi.org/10.5281/zenodo.7595111. Conflicts of Interest: The authors declare no conflict of interest. Appendix A Selected 1 H magnetization curves ( 1 H magnetization versus time) for BSA–water mixtures. Int. J. Mol. Sci. 2023, 24, x FOR PEER REVIEW 19 of 24 0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.2 1.6 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.2 0.0 0.2 0.4 0.6 0.8 1.0 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz 298K 278K Time [s] 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz Normalized magnetization [a.u.] 273K 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz 268K Figure A1. Normalized 1H magnetization curves for BSA–water mixture (20%wt BSA) at selected resonance frequencies in the temperature range from 298 K to 268 K. Solid lines—single exponential fits. Figure A1. Normalized 1 H magnetization curves for BSA–water mixture (20%wt BSA) at selected resonance frequencies in the temperature range from 298 K to 268 K. Solid lines—single exponential fits. Int. J. Mol. Sci. 2023,24, 4093 17 of 21 Int. J. Mol. Sci. 2023, 24, x FOR PEER REVIEW 20 of 24 0.0 0.2 0.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.8 1.0 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz 278K 273K Time [s] 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz Normalized magnetization [a.u.] 268K 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz 266K Figure A2. Normalized 1H magnetization curves for BSA–water mixture (40%wt BSA) at selected resonance frequencies in the temperature range from 278 K to 266 K. Solid lines—single exponential fits. Figure A2. Normalized 1 H magnetization curves for BSA–water mixture (40%wt BSA) at selected resonance frequencies in the temperature range from 278 K to 266 K. Solid lines—single exponential fits. Int. J. Mol. Sci. 2023,24, 4093 18 of 21 Int. J. Mol. Sci. 2023, 24, x FOR PEER REVIEW 21 of 24 0.00 0.05 0.10 0.15 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.0 0.2 0.4 0.6 0.8 1.0 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz 263K 263K Time [s] 4.26 MHz 3.30 MHz 2.55 MHz 1.97 MHz 1.52 MHz 701 kHz 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz Normalized magnetization [a.u.] 258K 4.26 MHz 3.30 MHz 2.55 MHz 1.97 MHz 1.52 MHz 701 kHz 258K 4.26 MHz 3.30 MHz 2.55 MHz 1.97 MHz 1.52 MHz 701 kHz 258K 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz 253K Figure A3. Normalized 1H magnetization curves for BSA–water mixture (20%wt BSA) at selected resonance frequencies in the temperature range from 263 K to 253 K. Solid lines—single exponential fits. On the left frequencies are selected from the entire range measured and, on the right, only from areas with quadrupole peaks. Figure A3. Normalized 1 H magnetization curves for BSA–water mixture (20%wt BSA) at selected resonance frequencies in the temperature range from 263 K to 253 K. Solid lines—single exponential fits. On the left frequencies are selected from the entire range measured and, on the right, only from areas with quadrupole peaks. Int. J. Mol. Sci. 2023,24, 4093 19 of 21 Int. J. Mol. Sci. 2023, 24, x FOR PEER REVIEW 22 of 24 0.00 0.04 0.08 0.12 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.04 0.08 0.12 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.04 0.08 0.12 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.04 0.08 0.12 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.2 0.4 0.6 0.8 1.0 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz 265K 265K 4.26 MHz 3.30 MHz 2.55 MHz 1.97 MHz 1.52 MHz 701 kHz -0.004 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz Normalized magnetization [a.u.] 263K 4.26 MHz 3.30 MHz 2.55 MHz 1.97 MHz 1.52 MHz 701 kHz 263K 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz Time [s] 253K 4.26 MHz 3.30 MHz 2.55 MHz 1.97 MHz 1.52 MHz 701 kHz 258K 3.74 MHz 1.03 MHz 285 kHz 78.7 kHz 21.7 kHz 258K 4.26 MHz 3.30 MHz 2.55 MHz 1.97 MHz 1.52 MHz 701 kHz 253K Figure A4. Normalized 1H magnetization curves for BSA–water mixture (40%wt BSA) at selected resonance frequencies in the temperature range from 263 K to 253 K. Solid lines—single exponential fits. The frequencies are selected from the entire range measured. Figure A4. Normalized 1 H magnetization curves for BSA–water mixture (40%wt BSA) at selected resonance frequencies in the temperature range from 263 K to 253 K. Solid lines—single exponential fits. The frequencies are selected from the entire range measured. References 1. Slichter, C.P. Principles of Magnetic Resonance, 3rd ed.; Springer: Berlin, Germany, 1990. 2. Kruk, D. Understanding Spin Dynamics; CRC Press–Pan Stanford Publishing: Boca Raton, FL, USA, 2015. 3. Kowalewski, J.; Maler, L. Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications, 2nd ed.; CRC Press–Taylor & Francis Group: Boca Raton, FL, USA, 2019. Int. J. Mol. Sci. 2023,24, 4093 20 of 21 4. Korb, J. 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