Dynamics of protein – water mixtures: insight from combined 1H spin-lattice and spin-spin relaxation studies for myoglobin

Dynamics of protein–water mixtures: insight from combined 1 H spin-lattice and spin-spin relaxation studies of myoglobin Kahinga Kamau a , Elzbieta Masiewicz a , Pedro Jos´ e Sebasti˜ ao b,* , Danuta Kruk a,* a Department of Physics and Biophysics, University of Warmia and Mazury in Olsztyn, Oczapowskiego 4, 10-719 Olsztyn, Poland b CeFEMA −Center of Physics and Engineering of Advanced Materials and Department of Physics, Instituto Superior T´ ecnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal ARTICLE INFO Keywords: Proteins Dynamics NMR Relaxation Myoglobin ABSTRACT Aiming to reveal the dynamical properties of highly concentrated protein–water systems and validating models of motion, 1 H spin-lattice and spin-spin relaxation experiments were performed for myoglobin – H 2 O (40 % wt. of myoglobin) mixture versus temperature, from 268 K to 310 K. The spin-lattice relaxation studies were conducted using Fast Field Cycling (FFC) NMR relaxometry, in the frequency range from 10 kHz to 20 MHz. The comprehensive set of relaxation data was interpreted in terms of a superposition of relaxation contributions expressed in terms of Lorentzian spectral densities and corresponding to the time scales of the order of 10 −6 s, 10 −8 –10 −7 s and 10 −9 –10 −8 s. The model was validated against the 1 H spin-spin relaxation data obtained from Time Domain (TD) NMR experiments at 18.5 MHz. The studies were complemented by 1 H spin-lattice (FFC NMR and TD NMR) and spin-spin (TD NMR) relaxation experiments for myoglobin – D 2 O (40 % wt. of myoglobin) mixture. Bi-exponential spin-spin relaxation processes were observed for both myoglobin – H 2 O and myoglobin – D 2 O mixtures. A thorough comparison of the spin-lattice relaxation and spin-spin relaxation rates (both components) for the H 2 O and D 2 O containing mixtures led to the conclusion that the fast component of the spin-spin relaxation process originates from the pool of 1 H nuclei of myoglobin and is associated with slow dynamics of the protein, while the slow component of the spin-spin relaxation is a counterpart of the spin-lattice relaxation process (observed by FFC NMR and TD NMR) and they both reflect the dynamics of water molecules strongly bound to myoglobin (for myoglobin – H 2 O) and a similar dynamics of myoglobin (for myoglobin – D 2 O). 1. Introduction Understanding the dynamic properties of biomolecular systems is one of the cornerstones of molecular science. These properties, which encompass time scales and mechanisms of molecular motion, are fundamental to our comprehension of biological processes. In this context, Nuclear Magnetic Resonance (NMR) relaxation studies at high resonance frequencies give insight into the fast molecular motion [1,2]. As far as slower dynamical processes, occurring on the timescale from milliseconds to nanoseconds, are concerned, Fast Field Cycling (FFC) NMR relaxometry has become a highly appreciated method in molecular science because of its unique advantages [3–8]. Using FFC NMR relaxometry one can perform 1 H spin-lattice experiments covering at least three orders of magnitude in resonance frequency (from 10 kHz to 10 MHz or even higher) which makes it possible to probe dynamical processes on different time scales in a single experiment, In addition to this ability, frequency dependent relaxation studies have the potential to reveal the mechanism of the molecular motion – one can distinguish, for instance, between three – dimensional (3D) isotropic translation diffusion [9–11], anisotropic (two – dimensional, 2D) diffusion [5,12–19] or polymer specific motion [7]. Relaxation rates are given as linear combinations of spectral density functions, being Fourier transforms of the corresponding time correlation functions [20–25]. The mathematical forms of the correlation functions (and, consequently, the spectral densities) depend on the mechanism of the motion [5,16–19,26–31]. This implies that the shape of the frequency dependencies of the spinlattice relaxation rates reflect the mechanism of the motion. To illustrate this statement one should point out that a linear dependence of spin-lattice relaxation rates on square root of the resonance frequency is observed at low frequencies for isotropic translation diffusion * Corresponding authors. E-mail addresses: [email protected] (K. Kamau), [email protected] (E. Masiewicz), [email protected] (P.J. Sebasti˜ ao), [email protected] (D. Kruk). Contents lists available at ScienceDirect Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr https://doi.org/10.1016/j.jmr.2025.107993 Received 24 August 2025; Received in revised form 12 November 2025; Accepted 12 November 2025 Journal of Magnetic Resonance 382 (2026) 107993 Available online 17 November 2025 1090-7807/© 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ). [19,26,32,33], a linear dependence of spin-lattice relaxation rates on logarithm of the resonance frequency is characteristic of 2D (surface) diffusion [5,17–19], while power law dependencies ( ω − α , where ω denotes the resonance frequency) have been attributed to polymer dynamics, with the factor α characteristic of the specific regimes of polymer motion [7]. NMR relaxometry has been exploited to address the subject of dynamical properties of proteins. The studies are, however, centred on two limits. The first one is the case of protein solutions in water with low concentrations of the proteins [6,21,34–39]. On the opposite limit are the studies of solid proteins [40–42], although they are rare. Even rarer are cases of highly concentrated protein – water mixtures; here one can mention only a few examples [43–46]. In all cases the theoretical analysis of the relaxation data poses a considerable challenge. This has given rise to relaxation models relaxation contributions associated with 3D and 2D water diffusion (for hydrated proteins) [40–44] or models based on power laws attributed to vibrational dynamics involving protein backbones [47–49]. One of the proposed theoretical descriptions of relaxation data for protein systems is referred to as a “model free” approach [36,50–52]. The model represents the 1 H spin-lattice relaxation rates as a sum of relaxation contributions expressed in terms of Lorentzian spectral densities. Often one cannot reproduce relaxation data for protein systems using models of motion associated with the well-defined dynamical scenario (this is not surprising taking into account the complexity of the systems including several pools of hydrogen atoms), while the “model free” approach makes it possible to reach a reasonable agreement with the experimental data, for the price of including many parameters, especially when the relaxation data are reproduced in terms of three relaxation contributions [40,50–52]. One can find a justification of this approach, attributing the relaxation contributions to dynamics of water molecules bound to the protein at specific binding sites and reflecting the protein motion (in this way one can indirectly probe the protein dynamics). Nevertheless, one can still wonder to what extent the description really represents the time scales of the main dynamical processes and argue that this is rather a parametrization that does not provide deeper insight into the dynamics. This question is of high importance, considering that this theoretical approach is widely used for investigating slow dynamics of biomolecular systems. With the aim of shedding light on this subject, we performed 1 H spinlattice NMR relaxation experiments in the frequency range from 10 kHz to 20 MHz, for myoglobin – H 2 O and myoglobin – D 2 O mixtures including 40 % wt. of myoglobin (highly concentrated systems). Myoglobin, a heme-containing globular protein, is renowned for its role in oxygen storage and release, processes that are critical for muscle metabolism and function. Structurally, myoglobin consists of 153 amino acids and a heme prosthetic group, which is integral to its oxygenbinding capacity [53,54]. After establishing that the relaxation data could not be reproduced in terms of a motion specific model, we have used the “model free” approach. The 1 H spin-lattice relaxation data for the myoglobin – D 2 O system directly reflect the protein motion. Therefore, the comparison between the results obtained for this system and the myoglobin – H 2 O mixture helps to identify the origin of the relaxation contributions. The 1 H spin-lattice relaxation data obtained from FFC NMR relaxometry experiments have been complemented by 1 H spin-lattice and spin-spin relaxation studies performed at 18.5 MHz in time domain. The terminology can be confusing, because all relaxation experiments are, in fact, performed in the time domain – one extracts the relaxation rates from the time evolution of the magnetization. The name “Time – Domain (TD) NMR” stems from the use of the so-called Time Domain NMR equipment produced by Resonance Systems (Germany). The advantage of the TD NMR experiments is the possibility to detect multi-exponential relaxation processes. Although FFC NMR relaxometry can also reveal multi-exponential relaxation, the relative resolution of this method is lower. Moreover, the TD NMR equipment allows for measuring shorter relaxation times; in case of FFC NMR this is the price to be paid for varying the magnetic field. The possibility to detect multi-exponential relaxation processes with a very good resolution applies also to the spin-spin relaxation experiments. To our knowledge this is the first example of combining FFC NMR and TD NMR relaxation studies for protein systems. The consistency of the theoretical approach requires that one can reproduce the 1 H spin-lattice and spin-spin relaxation data in terms of the same parameters. This brings us to the two goals of this work: to validate the “model free” approach, which is frequently used to gain information about slow dynamical processes in biomolecular systems, and gain deeper insight into dynamical properties of highly concentrated protein–water mixtures by probing the relaxation processes associated with the water as well as the protein fractions. 2. Theory 1 H relaxation processes in protein systems are predominantly caused by 1 H – 1 H magnetic dipole-dipole interactions. For systems including equivalent nuclei, the spin-lattice relaxation rate, R1( ω )( ω denotes the resonance frequency in angular frequency units), is given as [20–25]: R1( ω ) = CDD[J( ω ) + 4J(2 ω ) ] (1) where the J( ω )quantities are referred to as spectral density functions (they are Fourier transforms of the corresponding time correlation functions characterizing the dynamical processes causing stochastic fluctuations of the dipole-dipole interactions), while CDD denotes the dipolar relaxation constant reflecting the amplitude of the dipole-dipole interactions. The corresponding spin-spin relaxation rate, R2( ω ), is given as [20–25]: R2( ω ) = CDD 2[3J(0) + 5J( ω ) + 2J(2 ω ) ] (2) For complex systems one can expect several relaxation contributions. According to the spin relaxation theory [20–25] the relaxation process being most efficient at a given resonance frequency is the one associated with dynamics occurring on the time scale matching the reciprocal value of the resonance frequency. One can attempt to interpret the spin-lattice relaxation data in terms of three relaxation contributions associated with dynamical processes referred to as slow, intermediate and fast ones [36,40,41]: R1( ω ) = Cs[ τ s 1+ ( ωτ s)2+4 τ s 1+ (2 ωτ s)2]+Ci[ τ i 1+ ( ωτ i)2+4 τ i 1+ (2 ωτ i)2] +Cf[ τ f 1+( ωτ f)2+4 τ f 1+(2 ωτ f)2]+A1 (3) where τ c denotes a characteristic time constant describing the time scale of the motion – it is referred to as a correlation time. Following this line, the pairs of the parameters: (Cs, τ s), (Ci, τ i)and (Cf, τ f)denote the correlation times for the slow, intermediate and fast dynamics ( τ s, τ i and τ f, respectively), while Cs, Ci and Cf are the corresponding dipolar relaxation constant. The parameter A1 denotes a frequency independent term that can be associated with dynamical processes occurring on a time scale of about few ns or shorter, so the corresponding relaxation contribution is frequency independent. The corresponding relaxation expression for the spin-spin relaxation rate takes the form: R2( ω )=Cs 2[3 τ s+5 τ s 1+( ωτ s)2+2 τ s 1+(2 ωτ s)2]+Ci 2[3 τ i+5 τ i 1+( ωτ i)2 +2 τ i 1+(2 ωτ i)2]+Cf 2[3 τ f+5 τ f 1+( ωτ f)2+2 τ f 1+(2 ωτ f)2]+A2 (4) K. Kamau et al. Journal of Magnetic Resonance 382 (2026) 107993 2 where A2 denotes the spin-spin relaxation counterpart of A1. 3. Experimental details 1 H spin-lattice relaxation experiments have been performed for myoglobin – H 2 O and myoglobin – D 2 O mixtures (including 40 % wt. of myoglobin). Two kinds of spin-lattice relaxation data have been collected. The first set includes 1 H spin-lattice relaxation rates in the frequency range from about 10 kHz to about 20 MHz at 310 K, 298 K, 288 K and 278 K (with the accuracy of 1 K) using Fast Field Cycling (FFC) NMR relaxometer (STELAR s.r.l. Mede, Italy). Pre-polarization was applied for frequencies below 3.5 MHz. The magnetization curves (magnetization versus time) include 32 logarithmically spaced points, and the relaxation rates were obtained from the expression: M(t) = Aexp( − Rit) + A0(5a) where i=1,2 stands for spin-lattice and spin-spin relaxation, respectively, and (A+A0) = 1. In cases where, this approach was not sufficient, the data were analysed in terms of a bi-exponential function: M(t) = Aslowexp(−Ri,slowt)+Afastexp(−Ri,fastt)+A0(5b) The relaxation rates Ri,slow and Ri,fast refer to the lower and higher relaxation rates, respectively, while Aslow and Afast denote the corresponding amplitudes of these relaxation components (Aslow +Afast + A0=1). The FFC NMR studies were complemented with spin-lattice and spinspin relaxation experiments performed at 18.5 MHz using the equipment produced by Resonance Systems GmbH (Kirchheim unter Teck, Germany). For the spin-lattice relaxation measurements the saturation recovery sequence [55] was used. The spin-spin relaxation experiments were performed applying the Carr-Purcell-Meiboom-Gill (CPMG) sequence [56] with 417 echoes and 32 scans. The temperature accuracy was also 1 K. 4. Results and analysis 4.1. 1 H spin-lattice relaxation by FFC NMR for myoglobin – H 2 O Fig. 1a show 1 H spin-lattice relaxation data for the mixture containing 40 % wt. of solid myoglobin and 60 % wt. of H 2 O collected in the frequency range from about 10 kHz to about 20 MHz using the FFC NMR method. The spin-lattice relaxation process was found to be singleexponential at all temperatures in the whole frequency range – examples of the magnetization curves were shown in Supplementary Material (Fig. S1 a-d). The data were analysed in terms of. Eq. 3. It has turned out that obtaining satisfactory fits requires that both the correlation times and the corresponding dipolar relaxation constants vary with temperature. This was to be expected considering the non-monotonic changes with temperature of the relaxation rates in the low frequency limit. The outcome of the analysis for the data collected at 278 K is shown in Fig. 1b, while the corresponding results for 288 K, 298 K and 310 K are shown in Supplementary Material (Fig. S2 a-c). The obtained parameters are included into Table 1. It is worth noting the temperature independent value of the correlation time τ s. The model contains seven parameters, which raises the question about the ambiguity of the analysis. It is important to point out at this stage that attempts to reproduce the data in terms of relaxation models representing anisotropic translation diffusion [10,11,19,26,32,44], surface diffusion [5,12,16–18] in combination with other contributions (such as power laws [47–49]) failed. 4.2. 1 H spin-lattice and spin-spin relaxation by TD NMR for myoglobin – H 2 O The results of the 1 H spin-lattice and spin-spin relaxation TD NMR experiments for the myoglobin – H 2 O mixture are shown in Fig. 2a and Fig. 2b, respectively. The 1 H spin-lattice relaxation process was found to be single-exponential (as expected from the FFC NMR data). Examples of the single exponential fits are shown in Fig. 2a. Table 2 includes the outcome of the single-exponential fits. In contrast to the spin-lattice relaxation, the spin-spin relaxation process was found to be bi-exponential. Table 2 includes the result of the bi-exponential analysis in terms of Eq. 5. Fig. S3 a-d show the outcomes of single-exponential and bi-exponential fits of the spin-spin relaxation data. The bi-exponentiality of the relaxation process indicates the presence of molecular fractions engaged in different kinds of dynamics, with exchange effects being too slow to lead to an averaging. 4.3. Comparison of 1 H relaxation data by FFC NMR and TD NMR for myoglobin – H 2 O The relaxation rates obtained by means of FFC NMR and TD NMR at 18.5 MHz are compared in Fig. 3a. The first observation is the agreement between the spin-lattice relaxation rates obtained by means of both methods – the agreement is satisfactory, although somewhat worse at lower temperatures. Fig. 4a includes also the slow component of the Fig. 1. a) 1 H spin-lattice relaxation data for the myoglobin – H 2 O mixture; b) fit of the data at 278 K in terms of Eq. 3(solid line) decomposed into the individual relaxation contributions: R1,s (dashed line), R1,i (dashed-dotted line), R1,f (dashed-dotted-dotted line) and A1(dotted line). K. Kamau et al. Journal of Magnetic Resonance 382 (2026) 107993 3 spin-spin relaxation process, R2,slow. The spin-spin relaxation rates are larger than the spin-spin ones, as expected. However, one can see that the temperature dependence of R2,slow does not follow the well–known predictions of Eq. 2(spin-spin relaxation rates decrease with increasing temperature). One can explain this effect treating the R2,slow relaxation process as a counterpart of the spin-lattice relaxation, R1. Using the parameters obtained from the analysis of the FFC NMR data (Table 1) we have calculated the expected values of the spin-spin relaxation rates (Eq. 4; it has been set: A2=A1), achieving a good agreement with the experimental data as shown in Fig. 3a. The temperature dependence of the spin-spin relaxation rates R2,fast (Fig. 3b) can be reproduced in terms of Eq. 6, including the Arrhenius expression for the correlation time [57,58]: R2( ω ) = C 2[3 τ c+5 τ c 1+ ( ωτ c)2+2 τ c 1+ (2 ωτ c)2]; τ c= τ 0exp(EA RT)(6) where EA denotes an activation energy, while R is the gas constant. The parameters yield: C= (8.9±0.4)×10 9 Hz 2 , τ 0= (2.3 ±0.05) ×10 −13 s and EA= (27.1 ±0.2)kJ/mol. 4.4. 1 H relaxation for myoglobin – D 2 O With this relaxation picture in mind, we turned attention to 1 H spinlattice relaxation studies of the myoglobin (40 % wt.) - D 2 O system by means of FFC NMR. The experiment was very demanding and, because of the limitations of FFC NMR, it was performed only at 288 K, 298 K and 310 K. Even at these temperatures, significant scattering of the data occurred below 100 kHz. The data are presented in Fig. 4a. The results for 298 K and 288 K do not show significant differences. The data at 310 K can be interpreted as the sum of two relaxation contributions expressed in terms of Lorentzian spectral densities, with two pairs of parameters: (C1, τ 1)and (C2, τ 2)complemented with a frequency independent term, A1 (a simplification of Eq. 3): R1( ω )=R1,1( ω )+R1,2( ω ) =C1[ τ 1 1+( ωτ 1)2+4 τ 1 1+(2 ωτ 1)2]+C2[ τ 2 1+( ωτ 2)2+4 τ 2 1+(2 ωτ 2)2]+A1 (7) The obtained parameters are collected in Table 3. Fig. 4b show the fit decomposed into the individual contributions. The data for 298 K and 288 K can be reproduced using a single relaxation contribution (parameters C and τ ) and the A1 term (Table 3). The data shown in Fig. 4a have been complemented by TD NMR results obtained for the myoglobin (40 % wt.) - D 2 O mixture; the outcome of the TD NMR experiments is shown in Fig. 5. For the myoglobin – D 2 O mixture the spin-lattice relaxation process can be treated as single – exponential; examples of the singleexponential fits of the magnetisation are shown in Fig. 6a. The obtained spin-lattice relaxation rates are collected in Table 4. The spin-spin relaxation process for the myoglobin – D 2 O mixture is bi-exponential. Examples of bi-exponential fits of the magnetisation Table 1 Parameters obtained from the analysis of the 1 H spin–lattice relaxation data for the myoglobin – H 2 O mixture in terms of Eq. 3. T [K] Cs [10 5 Hz 2 ] τ s [s] Ci [10 7 Hz 2 ] τ i [s] Cf [10 8 Hz 2 ] τ f [s] A1[s −1 ] 310 2.0 ±0.3 (2.9 ±0.4) ×10 −6 1.8 ±0.7 (5.6 ±1.2) ×10 −8 3.8 ±0.5 (8.8 ±1.1) ×10 −9 21 ±1 298 1.9 ±0.3 (2.9 ±0.5) ×10 −6 1.12 ±0.01 (1.15 ±0.08) ×10 −7 2.35 ±0.04 (9.0 ±0.3) ×10 −9 13 ±0.4 288 2.1 ±0.1 (2.9 ±0.5) ×10 −6 1.39 ±0.06 (1.55 ±0.05) ×10 −7 1.6 ±0.2 (1.03 ±0.07) ×10 −8 9.7 ±0.3 278 2.4 ±0.5 (2.9 ±0.6) ×10 −6 1.59 ±0.07 (2.00 ±0.09) ×10 −7 1.2 ±0.1 (1.5 ±0.1) ×10 −8 9.5 ±0.3 Fig. 2. Normalized 1 H magnetization curves obtained in TD NMR spin-lattice a) and spin-spin b) relaxation experiments for the myoglobin – H 2 O system. Solid lines in a) show single exponential fits. Table 2 1 H spin-lattice and spin-spin relaxation rates obtained for the myoglobin – H 2 O mixture. T [K] R1 [s −1 ]R2,slow [s −1 ]R2,fast [s −1 ]Aslow/Afast 268 11.76 ±0.07 23.73 ±0.02 629 ±18 6.9 ±0.1 273 11.75 ±0.05 20.97 ±0.02 555 ±14 5.9 ±0.1 278 12.05 ±0.10 19.29 ±0.02 414 ±8 5.4 ±0.07 283 12.16 ±0.09 18.32 ±0.02 363 ±7 5.1 ±0.06 288 13.13 ±0.10 17.98 ±0.02 308 ±6 4.9 ±0.05 293 14.36 ±0.08 18.84 ±0.02 287 ±5 4.8 ±0.04 298 16.26 ±0.09 20.94 ±0.03 252 ±4 4.7 ±0.04 303 19.03 ±0.13 23.54 ±0.03 243 ±4 4.7 ±0.04 310 25.40 ±0.10 30.32 ±0.04 222 ±3 4.7 ±0.04 K. Kamau et al. Journal of Magnetic Resonance 382 (2026) 107993 4 curves obtained in the spin-spin relaxation experiments are presented in Fig. 6b including a decomposition into the individual relaxation components. Further fits are shown in Supplementary Material (Fig. S4). The obtained relaxation parameters are also included into Table 4. 4.5. Comparison of 1 H relaxation for myoglobin – H 2 O and myoglobin – D 2 O at 18.5 MHz Fig. 7a show a comparison between the spin-lattice relaxation rates and the slow spin-spin relaxation rates for the myoglobin – H 2 O and myoglobin – D 2 O mixtures. One can observe that the relaxation rates for the myoglobin – D 2 O mixture are somewhat higher than the corresponding relaxation rates for the myoglobin – H 2 O mixture and the shapes of the temperature dependencies show some differences. Fig. 7b show the R2,fast rates for the myoglobin – D 2 O mixture accompanied by the fit using of Eq. 6for C= (1.3 ±0.4) ×10 10 Hz 2 , τ 0=(2.1 ±0.03) ×10 −13 s, EA =27.6 ±0.1 kJ/ mol. It is worth noting that the parameters obtained from the analysis of the FFC NMR data for the myoglobin – D 2 O mixture (Table 3) give the following values of the corresponding spin – spin relaxation rates: 178 s −1 (310K), 254 s −1 (298 K) and 249 s −1 (288 K). Fig. 3. a) Comparison of 1 H spin lattice relaxation rates obtained by means of FFC NMR and TD NMR and spin-spin relaxation rates, R2,slow, for the myoglobin – H 2 O mixture. Theoretical values of spin-spin relaxation rates obtained in terms of Eq. 4using the parameters collected in Table 1 are shown (stars). b) Spin-spin relaxation rates, R2,fast; solid lines – fit in terms of Eq. 6including the Arrhenius dependence of the correlation time. Fig. 4. a) 1 H spin-lattice relaxation data for the myoglobin – D 2 O mixture; b) fit of the data at 310 K (solid line) decomposed into the individual relaxation contributions: R1,1 (dashed-dotted line), R1,2 (dashed-dotted-dotted line) and A1(dotted line). Table 3 Parameters derived from analyzing the 1 H spin-lattice relaxation data for the myoglobin – D 2 O mixture, based on Eq. 7. T [K] C1[10 8 Hz 2 ] τ 1[s] C2[10 8 Hz 2 ] τ 2[s] A1[s −1 ] 310 2.2 ±0.1 (2.1 ±0.5) ×10 −7 1.20 ±0.09 (4.4 ±0.8) ×10 −8 23.4 ± 0.7 C[10 9 Hz 2 ] τ [s] 298 1.55 ±0.02 (1.02 ±0.03) ×10 −7 19.2 ± 0.6 288 1.52 ±0.04 (1.02 ±0.03) ×10 −7 18.3 ± 0.6 K. Kamau et al. Journal of Magnetic Resonance 382 (2026) 107993 5 5. Discussion The analysis of the 1 H spin-lattice relaxation data for the myoglobin – H 2 O mixture collected in the frequency range from 10 kHz to 20 MHz at 310 K, 298 K, 288 K and 278 K has led to a series of dipolar relaxation constants: Cs, Ci and Cf. They can be associated with fractions of water molecules engaged in the dynamical processes described by the corresponding correlation times ( τ s, τ i and τ f). However, one should treat this interpretation with caution, as the dipolar relaxation constants are also affected by exchange processes. The correlation time τ s turned out to be temperature independent, yielding the value of about 3 ×10 −6 s. The intermediate correlation time, τ i, varies between about 6 ×10 −8 s and 2.0 ×10 −7 s, while τ f varies between about 9 ×10 −9 s and 1.5 ×10 −8 s. The temperature dependence of the dipolar relaxation constants is necessary to explain the temperature changes of the spin-lattice relaxation rates. This statement is supported by the temperature dependence of the spin-spin relaxation rates, R2,slow, for the myoglobin – H 2 O mixture at 18.5 MHz (Fig. 3a). For temperature independent pre-factors (dipolar relaxation constants), spin-spin relaxation rates increase with decreasing temperature (for slower dynamics), while the R2,slow values exhibit a minimum. We have used the parameters obtained from the analysis of the FFC NMR spin-lattice relaxation data, including the frequency independent term (Table 1) to calculate corresponding spin-spin relaxation rates using Eq. 4. The agreement between the experimental values of R2,slow and the calculated values is very good. This finding indicates that the slow spin-spin relaxation component is the counterpart of the spin-lattice relaxation process. Moreover, this agreement serves, to some extent, as validation of the model used for the analysis of the spin-lattice relaxation data (FFC-NMR) and a verification of the plausibility of the obtained parameters (the dipolar relaxation constants and Fig. 5. Normalized 1 H magnetization curves obtained in the spin-lattice (a) spin-spin and (b) relaxation experiments at 18.5 MHz for the myoglobin – D 2 O mixture. Fig. 6. 1 H magnetisation versus time for the myoglobin – D 2 O mixture at 278 K and 310 K obtained in spin-lattice (a) and spin-spin relaxation experiments at 18.5 MHz. Solid lines in (a) – single-exponential fits; solid lines in (b) - bi-exponential fits decomposed into the relaxation contribution associated with R2,slow and R2,fast (dashed and dotted lines, respectively). Table 4 1 H spin-lattice and spin-spin relaxation rates obtained for the myoglobin – D 2 O mixture. T [K] R1 [s −1 ]R2,slow [s −1 ]R2,fast [s −1 ]Aslow/Afast 273 15.9 ±0.3 20.4 ±0.1 920 ±18 0.89 ±0.01 278 16.6 ±0.3 19.7 ±0.3 735 ±32 0.82 ±0.03 283 16.2 ±0.3 18.9 ±0.2 646 ±21 0.75 ±0.02 288 16.6 ±0.3 19.9 ±0.3 572 ±16 0.64 ±0.01 293 17.6 ±0.2 20.6 ±0.3 488 ±13 0.62 ±0.01 298 18.9 ±0.3 23.5 ±0.3 423 ±10 0.60 ±0.01 303 20.3 ±0.2 28.2 ±0.4 402 ±9 0.61 ±0.01 310 22.3 ±0.4 38.9 ±0.5 366. ±9 0.67 ±0.01 K. Kamau et al. Journal of Magnetic Resonance 382 (2026) 107993 6 the correlation times). One should, however, be aware that the dominating contribution to the relaxation rate, R2,slow, stems from the dynamical process denoted as “fast”, and the frequency independent term, A1. The spin-spin relaxation rates, R2,fast, increase with decreasing temperature and can be reproduced using the Arrhenius dependence of the correlation time (Eq. 6) and a single relaxation term. As far as 1 H spin-lattice relaxation rates for the myoglobin – D 2 O mixture obtained by means of FFC NMR are concerned, the data for 298 K and 288 K are similar and can be reproduced in terms of a single relaxation contribution (and a frequency independent term) over the whole frequency range. The correlation time of about 1 ×10 −7 s (that is of the order of the τ i value obtained for the relaxation data for the myoglobin – H 2 O mixture), while the dipolar relaxation constant is by two orders of magnitude higher than Ci. At 310 K the analysis requires using two relaxation contributions (complemented with a frequency independent term). The correlation time τ 1 ranges between τ S and τ i, while the correlation time τ 2 is close to τ i. The dipolar relaxation constants are about an order of magnitude higher (compared to Ci and Cf). Dipolar relaxation constants depend on r−6, where r denotes an effective 1 H – 1 H distance). This implies that decreasing the value of r by a factor of 2, increases the dipolar relaxation constant by two orders of magnitude. This effect can be attributed to changes in protein flexibility (reduced flexibility), altering folding properties and increasing protein stability caused by D 2 O (compared to H 2 O used as a solvent) [59–62]. One should realize that the more significant (compared to the myoglobin – H 2 O mixture scatter of the relaxation rates makes the fits less sensitive to the details of the theoretical model. The outcome of the TD-NMR relaxation experiments for the myoglobin – H2O and myoglobin – D 2 O mixtures have been compared in Fig. 7. The spin-spin relaxation process for the myoglobin – D 2 O mixture is also biexponential. The fast components of the spin-spin relaxation process for the myoglobin – D 2 O and myoglobin – H 2 O mixtures match (in a good approximation) after multiplying the last one by a factor of about 1.7 (Fig. 7b). Consequently, they can be reproduced in terms of similar τ 0 and EA (values) Following this line, as for the myoglobin – D 2 O mixture the primary source of 1 H relaxation is 1 H nuclei of myoglobin (combined with a contribution from 1 H nuclei originally belonging to the protein, but appearing in the hydration shell as a result of exchange processes) one can conclude that the relaxation component R2,slow stems from (very) slow dynamics of myoglobin for both D 2 O and H 2 O mixtures. The spinlattice relaxation rates and the R2,slow values for the myoglobin – D 2 O mixture (Fig. 7a) can be attributed to the protein motion occurring on a shorter time scale. One should point out that the motion is heterogenous – the analysis of the FFC-NMR spin-lattice relaxation data for the D 2 O containing mixture indicate the range of the correlation time of the order of 10 −7 s-10 −9 s (or even shorter). Following this line, the R1 and R2,slow relaxation rates for the myoglobin – H 2 O mixture as similar to those for the D 2 O system (Fig. 7a). This comparison supports the concept that strongly bound water molecules follow the dynamics of the protein and, consequently, in this indirect way the 1 H spin-lattice relaxation data for the myoglobin – H 2 O mixture reflects the protein motion. This concept is further supported by the comparison of the Aslow Afast ratios for the H 2 O and D 2 O containing mixtures. The ratio is much higher for the first case (of the order of 5–7 compared to below 1). This can be explained by the contribution of the bound H 2 O molecules to the amplitude of the slow component of the spin-spin relaxation process. As already pointed out (referring to the FFC NMR data for the myoglobin – D 2 O mixture), the generally faster dynamics associated with R1 and R2,slow (compared to that associated with R2,fast) is heterogeneous an occurs on the time scale from 10 −6 s to 10 −9 s (or even below), as indicated in Table 1. The motion on the time scale of 10 −6 s (represented by the correlation time τ s) might be associated with the formation of the hydrogen network in the presence of H 2 O. Eventually, it is worth mentioning that the differences in the R1 and R2,slow relaxation rates for the H 2 O and D 2 O containing mixtures can be attributed to the fact that not all H 2 O molecules are strongly bound to the protein (and, hence, follow the protein motion); the water dynamics likely includes also other mechanisms. Finishing this section, it is worth to point out that in parallel to contributing to the validation of the “model free” approach, the differences in the dipolar relaxation constants obtained for the myoglobin – H 2 O and myoglobin – D 2 O mixtures likely reflecting the changes in myoglobin flexibility and folding properties, can also be considered as indicators of the stiffness of the hydration shell (D₂O forms slightly stronger and more tetrahedral hydrogen bonds than H₂O, which alters hydration and the hydrophobic effect [59]) with consequences for enzymatic regulation under crowding and stress [63,64]. 6. Conclusion 1 H spin-lattice relaxation data for a highly concentrated myoglobin – H 2 O mixture (including 40 % wt. of myoglobin) in the frequency range studied can be reproduced as a sum of relaxation contributions expressed in terms of Lorentzian spectral densities with temperature Fig. 7. a) Comparison of 1 H spin-lattice relaxation rates and spin-spin relaxation rates, R2,slow, at 18.5 MHz (TD NMR) for the myoglobin – H 2 O and myoglobin – D 2 O mixtures (the relaxation rates for the myoglobin – H 2 O mixtures were already shown in Fig. 3a). b) Spin-spin relaxation rates, R2,fast for the myoglobin – D 2 O mixture; solid lines – fit in terms of Eq. 6including the Arrhenius dependence of the correlation time. R2,fast (multiplied by 1.7) for the myoglobin – H 2 O mixture is added for comparison. K. Kamau et al. Journal of Magnetic Resonance 382 (2026) 107993 7 dependent dipolar relaxation constants. The spin-spin relaxation process is bi-exponential (at 18.5 MHz), and the slow relaxation component can be treated as the counterpart of the spin-lattice relaxation process. The shape of the temperature dependence of the relaxation rates of the slow spin-spin relaxation component confirms the need for temperature dependent dipolar relaxation constants and validates the model. A comparison of the spin-lattice and spin-spin relaxation processes for myoglobin – H 2 O and myoglobin – D 2 O mixtures indicates that the fast component of the spin-spin relaxation stems from the slow dynamics of myoglobin, Moreover, the comparison show that strongly bound H 2 O molecules follow the dynamics of the protein and, by contributing to the spin-lattice relaxation and the slow component of the spin-spin relaxation process carry information about the protein motion. The demonstrated consistency of the data analysis serves as a validation of the model-free approach. It has been shown that using the parameters obtained from the analysis of the spin-lattice relaxation data for the myoglobin-H 2 O mixture (covering a broad frequency data) in terms of the model free approach one can obtain a very good agreement with the experimental spin-spin relaxation rates. Following the line, the consistency was further strengthen by the analysis of the 1 H spin-lattice and spin-spin relaxation data for the myoglobin-D 2 O mixture, in terms of the model-free approach. This sequence of analyses and comparisons confirms that the model-free approach provides a robust and physically meaningful description of molecular dynamics rather than serving merely as a parametrization. CRediT authorship contribution statement Kahinga Kamau: Writing – review & editing, Writing – original draft, Methodology, Investigation, Conceptualization. Elzbieta Masiewicz: Investigation. Pedro Jos´ e Sebasti˜ ao: Writing – review & editing, Methodology, Conceptualization. Danuta Kruk: Writing – review & editing, Writing – original draft, Supervision, Methodology, Funding acquisition, Conceptualization. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This work has been supported by the European Commission through HORIZON MSCA-DN project FC-RELAX (grant agreement no. 101072758). Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.jmr.2025.107993. Data availability The supporting data are available at: https://doi. org/10.5281/zenodo.13928220 References [1] F.-A. Chao, R.A. Byrd, Protein dynamics revealed by NMR relaxation methods, Emerg. Top Life Sci. 2 (2018) 93–105, https://doi.org/10.1042/ETLS20170139. [2] V.-V. Telkki, V.V. Zhivonitko, Ultrafast NMR diffusion and relaxation studies, Annu. Rep. 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