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Complex Dynamics of Water in Protein Confinement Daniel R Martin, James E Forsmo, and Dmitry V Matyushov J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b10448 • Publication Date (Web): 05 Dec 2017 Downloaded from http://pubs.acs.org on December 8, 2017

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The Journal of Physical Chemistry

Complex Dynamics of Water in Protein Confinement Daniel R. Martin,∗,† James E. Forsmo,‡ and Dmitry V. Matyushov†

∗,¶

†Department of Physics, Arizona State University, PO Box 871504, Tempe, Arizona 85287 ‡Georgia Institute of Technology ¶School of Molecular Sciences, Arizona State University, PO Box 871504, Tempe, AZ 85287-1504 E-mail: [email protected]; [email protected]

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Abstract This paper studies single-molecule and collective dynamics of water confined in protein powders by means of molecular dynamics simulations. The single particle dynamics show a modest retardation compared to the bulk, but become highly stretched in the powder, with the stretching exponent of ≃ 0.2. The collective dynamics of the total water dipole are affected by intermolecular correlations inside water and by cross-correlations between the water and the protein. The dielectric spectrum of water in the powder gets two nearly equal-amplitude peaks: a Debye peak with ≃ 16 ps relaxation time and a highly stretched peak with the relaxation time of ≃ 13 ns and a stretching exponent of ≃ 0.12. The slower relaxation component is not seen in the single-molecule correlation functions and can be assigned to elastic protein motions displacing water in the powder. The loss spectrum of the intermediate scattering function reported by neutron scattering experiments is also highly stretched, with the highfrequency wing scaling according to a power law. Translational dynamics can become much slower in the powder than in the bulk, but are overshadowed by the rotational loss in the overall loss spectrum of neutron scattering.

Introduction

charged residues. This picture resembles the phenomenology established for relaxor ferroelectrics, 11 which also form nano-scale domains of polarized material. The distinction of the protein-water interface is that polarized domains are formed by and pinned to the surface charges, in contrast to the spontaneous polarization of domains in the bulk of a relaxor ferroelectric. In other words, whatever happens to waters in the hydration shell is fundamentally an interfacial problem. The positional and orientational structure of water in the interface are determined by the strength of intermolecular interactions, which enter Boltzmann weights for specific configurations. In contrast, the dynamics are driven by forces entering the equations of motion. One then anticipates that the short-ranged interactions produce larger forces as derivatives of the corresponding interaction potentials and, in turn, a stronger effect on the dynamics than the long-ranged interactions, even if the latter are stronger in magnitude and significantly affect the structure. An extreme example of this scenario would be the mean-field uniform potential of the van der Walls theory, which produces no forces, but strongly affects the liquid thermodynamics. From this general perspective of the anticipated effect of the interaction range vs merely its strength, changes in the interfacial dynamics do not necessarily mirror changes in the interfacial structure. Neverthe-

Proteins fold into compact structures with high densities of ionized residues at their surfaces. 1 The specific folded state is stabilized by the free energy of surface solvation, which also enhances solubility of globular proteins in solution. A high density of surface charged and polar groups also affects the structure of water in the interface, which gains properties distinct from the bulk. 2,3 The density of water in protein’s first hydration shell is higher than in the bulk. 4,5 Increased packing of the water molecules breaks the network of hydrogen bonds established in the bulk, 6 making hydration water more structurally disordered. 7,8 From the viewpoint of its density, the properties of hydration water can be compared to those of high-density amorphous ice 9 (no Bragg peaks characteristic of hexagonal ice appear in neutron diffraction 2,10 ). This comparison, usually referring to the density of hydration water, does not, however, extend to the orientational structure, which is substantially driven by the charge distribution at the protein’s surface. Water molecules tend to form domains around ionized residues with preferential orientations driven by the local fields. The overall statistics of orientations thus reflects both the collective character of these domains and their mutual frustration caused by the requirement to satisfy electric fields from nearby oppositely


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Collective dynamics involve both singleparticle rotations and dynamic correlations of orientations of different molecules of the medium. 19 Collective rotational dynamics are different from single-particle rotations even in bulk polar liquids. 20 Figure 1 presents loss spectra of TIP3P water calculated from singleparticle rotations (blue shaded area) and from the time correlation function of the entire dipole moment of the sample (dielectric loss, blue dashed line). The collective dynamics of the dipole moment, involving cross correlations of different dipoles, are slower as expected from general theories (the retardation ratio is of the order of the Kirkwood correlation factor 19 ). Single-particle rotations of water confined in the protein powder relax on the time-scale close to the collective dynamics in the bulk (green shaded area with the peak at the relaxation time τ ≃ 6 ps). The collective dynamics of confined water (dashed green line) are even slower, developing a very slow component with the relaxation time τ ≃ 13 ns (Figure 1). Collective dynamics in protein-water systems are typically reported by time-resolved Stokes shift of optical dyes 21–23 and by dielectric spectroscopy. 24–26 In both cases, the water dynamics are not resolved separately and in fact even water’s dynamics are affected by the motions of the protein with corresponding slower timescales. 27–32 In contrast to these techniques, absorption of radiation in the THz window of frequencies has an advantage of effectively freezing the solute motions and probing the response of solely the water component of the solution. 33–35 However, this frequency-domain technique does not provide information about relaxation times of hydration water. While dielectric spectroscopy reports the dynamics of the macroscopic dipole of the sample, timeresolved Stokes shift is a more local probe sensitive to the motions of molecular charges within a few solvation layers from the optical dye. Despite these differences, both techniques report significantly slower relaxation times than those produced by the NMR. The Stokes-shift dynamics involve both fast motions, on the timescale of single-molecule rotations and translations, and much slower collective tails, which

less, the dynamics of hydration water, like its structure, are significantly perturbed compared to the bulk. The extent of this perturbation is affected by a number of factors. Among them, the perturbation of the hydration shell reported by experiment can significantly depend on the experimental technique. Two widely used techniques, NMR 12 and neutron scattering, 13,14 probe single-particle dynamics. The analysis of NMR data, reporting single-particle rotational dynamics in terms of the second-order Legendre polynomial of the OH unit vector, requires assigning a specific hydration sub-ensemble distinct from the bulk. The first hydration layer of the solute is usually chosen, and the result is the retardation factor of 2–4 for single-particle rotations. 12 This result, and the assignment of the slower dynamics to the first layer, 15 are in general accord with computer simulations. 16,17 A somewhat related technique is the Raman probe of the OH stretch resolving dangling OH bonds in the water interface 18 and allowing an estimate of the number of molecules perturbed by the solute.

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Figure 1: Loss functions ǫ′′ (ν) ∝ χ′′ (ν) describing single-particle rotations (shaded curves) and collective rotational dynamics (dashed lines) of TIP3P water (simulation results obtained here). Shown are TIP3P water in the bulk (blue) and in confinement in powders of lysozyme proteins (green). The susceptibilities χ′′ (ν) are produced from normalized correlation functions to fit to the plot and therefore are not scaled with their corresponding static susceptibilities. This representation is used to stress on the positions of the characteristic loss peaks and not to specify the loss intensities.


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typically contribute 10–20% of the total signal. 36 Nevertheless, it seems to be generally confirmed that collective dynamics of hydration water affecting optical dyes show more significant retardation than single-particle rotations probed by the NMR. Qualitatively, this observation is consistent with the simulation results shown in Figure 1 and discussed in more detail below. Both Stokes-shift dynamics and dielectric response can be measured for protein solution. In contrast, neutron scattering, 13,14 reporting single-particle dynamics, and some dielectric measurements 37,38 are performed with protein powders, where water is strongly confined by the protein molecules. The dynamics of hydrogen displacements reported by neutron scattering 39–41 and of the dipole moment from dielectric spectroscopy 37,38,42,43 are complex. This implies that the corresponding linear response functions cannot be represented by a single Debye process. 44 The complexity of the dynamics is usually gauged by fitting the time correlation functions to a stretched exponential functionality ∝ exp[−(t/τ )α ], where α = 1 represents the Debye limit. Alternatively, one considers the frequency-resolved loss functions, which typically show power laws in their high-frequency wings, χ′′ (ω) ∝ ω −γ , with γ = 1 corresponding to Debye relaxation. A low value of stretching exponent, γ ≃ 0.2, is commonly observed for protein powders. Such low exponents are associated with so-called nearly constant loss introduced by Ngai and co-workers and believed to be a general property of confined water. 40,41,45 A large body of experimental work performed on protein hydration water has not offered complete answers to some of the most fundamental questions posed by these observations. One first wants to know what are the reasons for different results reported by measurements sensitive to single-particle and collective dynamics. Those differences, still existing for bulk liquids 19 (Figure 1), seem to be significantly amplified in interfaces. The answer to this question might well be related to the challenge posed by low magnitudes of stretching exponents. 37,39 Two reasons are typically considered to cause complex dynamics. Even though they are often

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viewed as alternatives, they do not necessarily oppose each other. The first view is that the dynamics are intrinsically complex, that is describing the dynamics of a single molecule in the bulk or in the interface requires equations of motions that capture intermolecular structural correlations and memory effects in the dynamics. Molecular hydrodynamics 44,46 offers a general umbrella for describing such process in terms of susceptibilities depending on both the wavevector (spatial correlations) and frequency (memory effects). One then understands the complex dynamics as generally a dependence of the relaxation time τ (k) on the wavevector k. The complexity of the dynamical response arises from integrating the corresponding susceptibilities, each characterized by a specific τ (k), over the wavevectors. 47 The problem becomes even more entangled when the hierarchy of memory functions is introduced, but the basic result that the dynamics of any single molecule picked from the solution are complex remains intact. The alternative view of the complex dynamics is in terms of spatial heterogeneity. 48 In this view, the dynamics of any given molecule is essentially Debye and the only reason for the complex dynamics is that the Debye relaxation times are distributed in space. The stretched exponential dynamics are then obtainedR as an ensemble average of Debye decays, g(τ ) exp[−t/τ ]dτ , with a proper chosen distribution g(τ ) of relaxation times τ . The two paradigms are difficult to separate since they produce essentially equivalent results in the linear response. Nonlinear spectroscopic techniques are required to interrogate the reasons for complex dynamics. 49,50 The resolution of the problem of complex dynamics might well hold the promise of answering the conundrum of the distinction between single-particle and collective dynamics of hydration water. Bagchi and co-workers have recently noted 3,51,52 that even though the average relaxation time of single-particle rotations of hydration water is indeed not much slower than in the bulk, the distribution of the single-particle relaxation times is dramatically broader for hydration shells compared to


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(water/protein mass ratio) is strongly confined. The simulation cell includes four protein molecules and 1272 water molecules (Figure 2). We produce the frequency-dependent dielectric susceptibility function of the simulation cell from its overall dipole moment and analyze the protein and water components of the dielectric loss. A highly stretched water peak with the relaxation time of ∼ 13 ns and the stretching exponent of γ ≃ 0.12 is found (Figure 3). The analysis of single-particle rotations of hydration water indicates highly stretched dynamics for individual water molecules, with a distribution of relaxation times and stretching exponents. We find stretched dynamics as well from the calculations of the self intermediate scattering function of water in the powder, commonly related to quasielastic neutron scattering. 14 The relaxation times, and the stretching exponents, extracted from these data are consistent with the results for single-molecule rotations. However, the overall loss of the dynamical self structure factor is a complex convolution of the rotational and translational contributions. While translational times can reach ∼ 100 ps at k ≃ 1 ˚ A−1 , they are hidden in the total loss by dominant rotations at the frequencies below the loss peak. Above the loss peak, both rotational are translational dynamics are highly stretched, with the power-law tail ∝ ω −0.35 .

bulk water. Similar observations were previously reported for hydrated DNAs. 53,54 These observations offer an appealing picture that the collective dynamics, as recorded by dielectric and time-resolved Stokes-shift spectroscopies, weigh differently, compared to singleparticle measurements, the long-time tail in the wide distribution of relaxation times, thus resulting in generally much slower relaxation times. From this perspective, the dynamic heterogeneity (spread of the relaxation times), the dynamic complexity (stretched dynamics), and the distinction between the collective and single-particle dynamics are all connected issues. Nevertheless, while offering an explanation of many observations, this view neglects the role of cross molecular correlation in the collective dynamics. 20 Such cross correlations are always present and are mostly responsible for intermediate, between water and protein, relaxation components (δ-relaxation) in dielectric spectroscopy of protein solutions. 25,26,55 Therefore, if cross-correlations is a significant part of the picture for collective dynamics, one cannot deduce collective dynamics by re-weighting the single-particle dynamics.

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Figure 2: A cartoon of the simulation cell with four lysozyme proteins (colored differently) and TIP3P water molecules shown explicitly.

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Figure 3: Dielectric loss spectrum ǫ′′ (ν) = 4πχ′′ (ν) of the protein powder. The total spectrum (black) is separated into the protein (green) and water (blue) contributions. The two components do not add to the total (black) because of significant cross-correlations between protein and water dipole moments.

In this article, we are addressing these questions by performing long (3 µs) simulations of lysozyme powders at 300 K (see Supporting Information (SI) for the simulation protocol). Water in the powder with h ≃ 0.4


5


Results

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Dielectric Response. The dielectric loss spectrum of the protein powder is shown in Figure 3. Two relaxation peaks of the composite system is the stretched-exponential relaxation of the protein dipole with the time-scale of ∼ 215 ns (Table 1) and the Debye peak of water close to its relaxation peak in the bulk. The new component present here, as well as in other systems involving confined water, 40,41 is slow and highly stretched relaxation of water with the characteristic time-scale of ∼ 13 ns (Table 1). As we show in more detail in Figure 4, the slow and fast Debye components are nearly equally divided in the entire dielectric loss spectrum of water in the powder. The dielectric response of the protein-water system was determined from the dynamics of the P overall dipole moment of the sample M(t) = a Ma (t), where the index a = p, w is used for the protein (p) and water (w) components, respectively. The normalized time self-correlation function can be defined for each component separately −1 φa (t) = h(δMa )2 i hδMa (t) · δMa (0)i (1)

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Figure 4: The dielectric loss spectrum ǫ′′w (ν) of water in the protein powder. The Debye peak is shown in darker blue and the stretched exponential relaxation in lighter blue. The black line refers to the total dielectric loss (not equal to the sum of the components due to crosscorrelations). 4πχ′′a (ω) yields the dielectric loss spectrum of the components a = p, w shown in Figures 3 and 4. As is seen from Table 1, the Debye component of water relaxation is about 16 ps, which is about a factor of two slower than τwD ≃ 7 ps of bulk TIP3P water. 56 This slowing factor is consistent with what is typically expected for a geometrically confined water. 57 The second relaxation time, ∼ 13 ns, is obviously much longer. Not unlike the δ-peaks in dielectric spectra of protein solutions, 25,55 this relaxation time falls between the Debye relaxation time of water and the peak of the protein. The latter most likely reflects sub-microsecond tumbling of the protein dipole, which is much slower in the powder compared to ∼ 10 − 20 ns time-scale of tumbling in solution. 26,27,55 Obviously, protein dipole is incompletely frozen in the powder, which needs to be taken into account when interpreting dielectric spectroscopy of solid protein samples. 37,38 Most importantly, the slow relaxation component of water does not appear in single-molecule rotational dynamics which we consider next. This carries a significant message that nanosecond dynamics of water in confinement can be observed in collective dynamics only. It is likely related to elastic motions of the protein displacing large parts of the hydration shell. 28–32 Such motions

(2)

The results of fitting are listed in Table 1. The dielectric susceptibility functions of the components follow from φã (ω) hδMa2 i 1 + iω φã (ω) 3kB T Ω

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χ ã (ω) =

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(3)

Here, Ω is the volume of the simulation cell in MD simulations. The function χ ã (ω) is complex-valued and its imaginary part, ǫ′′a (ω) =


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Table 1: Fits of the normalized time correlation functions for the fluctuations of the dipole moment components of the powder (eq 1) to the analytical form given by eq 2. Component Water Protein Total

Aa τaD (ps) 0.45 16 0.05 1 0.38 17

are not expected to produce new dynamics in single-particle rotations, as indeed confirmed by our results. Before we turn to the singleparticle dynamics, we emphasize that both the time-resolved Stokes-shift experiments 21–23 and simulations 28,30–32 and dielectric spectroscopy, which all have reported nanosecond dynamics of hydration water, measure collective response, in agreement with our interpretation. Single-Molecule Rotational Dynamics. In an attempt to understand the origin of the stretched dynamics of the water component, we have looked at the single-particle dynamics to complement the dynamics of the total water dipole shown in Figure 4. The single-particle dynamics average out over all protein sites at which individual water molecules can reside and thus pick the spatial distribution of potentials and forces. The anticipated consequence is the spatial heterogeneity of the relaxation times, which is not influenced by the cross-correlations of different dipoles. We provide the results of calculating the first-order Legendre polynomial of the unit vector µ(t) ˆ of the water dipole most relevant for comparing with the dielectric response C1 (t) = hµ(t) ˆ · µ(0)i ˆ (4)

τa (ns) 13 215 325

βa 0.12 0.36 0.29

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Figure 5: The distribution of the stretching relaxation times τw (A) and of the stretching exponents βw (B) for the time self-correlation function of single water rotations in the protein powder (eq 4). Panel C shows the distributions of the Debye relaxation times τwD . Panel D shows the distribution of the Debye amplitudes Aw and of the stretched-exponential amplitudes 1 − Aw . ters shown in Figure 5 (see SI for the examples of the fits). Our procedure, therefore, separates the time average usually performed in computer simulations from an ensemble average common to statistical theories of time correlation functions. In the case of an infinite observation trajectory a single particle would be able to visit all sites within the system, converging to a single time correlation function. The heterogeneity of the relaxation parameters shown in Figure 5 thus reflects an inequality between time and ensemble averages (nonergodic sampling) at a given observation window specified by the length of the trajectory. Such dependence on resolution, with qualitatively similar consequences, is expected for any experimental situation probing the system within a specific observation window. In our case, the distribution of the stretching exponent β has a single

Similarly to the relation of the total dipole moment of water, this time correlation function is fitted to a sum of a Debye and stretchedexponential terms (eq 2). Fitting with two exponential decays, in addition to the stretched exponential term, was also attempted and the results are given in the SI. Both fitting procedures produce consistent results. The correlation function C1 (t) was calculated for each out of 1272 water molecules in the powder on the time-scale of the simulation trajectory, ∼ 3 µs. The results are used to produce the distributions of the fitting parame-


7


peak around 0.2 spanning the range from ≃ 0.1 to ≃ 0.4 (Figure 5B). As mentioned above, this low stretching exponent is consistent with nearly constant loss universally assigned to confined water. 41,45 Our results clearly show that complexity of the water dynamics is revealed already at the single-molecule level and does not require crosscorrelations between the molecules. In fact, the single-particle correlation function C1 (t) is dominated by the stretched-exponential decay (≃ 0.8 amplitude, Figure 5D). The stretching exponents found for water (βw ≃ 0.12 from collective dipole and ≃ 0.2 from single-molecule relaxation) are also consistently lower than the corresponding stretching exponents for the protein (βp ≃ 0.36, Table 1). This results suggests a higher extent of heterogeneity, or potentially complexity, of the water dynamics compared to the protein dynamics. Intermediate Scattering Function. Incoherent scattering of neutrons provides a probe of single-particle dynamics in terms of the self intermediate scattering function (ISF) 14,44 NH

ik·[ri (t)−ri (0)] 1 X Fs (k, t) = e NH i=1

and NH

ik·[d (t)−d (0)] 1 X i i e i FR (k, t) = NH i=1

(5)

Figure 6: Fitting to eq 10 of the translational ISF for water in the lysozyme powder across all k-values sampled. A) The stretching exponent βT , B) the average translational relaxation times (eq 11), C) the relaxation times (j) τj = (DT k 2 )−1 , and D) the diffusion constants (j) DT . In C and D the blue points correspond to the first term in eq 10 (j = 1) while the green points correspond to the second term (j = 2). The connecting lines in C and D are to guide the eye. The power-law fit hτ i ∝ k −γ in B changes from γ = 1.23 at low k to γ = 1.89 at higher k values.

(6)

with FT (k, t) and FR (k, t) being the translational and rotational ISF, respectively. They are given by the following equations NH

ik·[r (t)−r (0)] 1 X 0i e 0i FT (k, t) = NH i=1

(8)

In Eqs. (7) and (8) r0i are the center of mass positions of the water molecules and di are the position of the hydrogen atoms relative to the center of mass. Since TIP3P water molecules are rigid, the hydrogen atoms are at a fixed distance di = 0.957 ˚ A from the center of mass. Therefore, only rigid-body rotations contribute to fluctuations of the phase of the plane wave in FR (k, t).

where NH is the number of hydrogen atoms, ri are their positions, and k is the wavevector of the momentum transfer of the incident neutron. Here, we consider the ISF of water hydrogens only. To a good approximation, in particular for low values of k, the ISF can be split into the contribution from the center of mass motions and rotations about the center of mass 58–62 Fs (k, t) = FT (k, t)FR (k, t)

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The Gaussian approximation for the ensemble average in eq 7 yields 44,46 FT (k, t) = e−k

2D

Tt

(9)

(7) where DT is the translational diffusion coefficient. For heterogeneous environments, one has to introduce heterogeneity in the diffusion coefficients accounted for by two terms in the trans-


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lational ISF FT (k, t) = Ae−k

2 D (1) t T

+(1−A)e−(k

2 D (2) t)βT T

(10)

The average translational relaxation time follows from eq 10 as (11) hτ i = Aτ1 + (1 − A)τ2 Γ 1 + βT−1 (j)

where τj = (DT k 2 )−1 . The fitting parameters for the translational ISF are shown in Figure 6. We find a stretching exponent βT ≃ 0.4−0.5 essentially independent of k and the k-dependent relaxation times. The exponent in the power law of the average relaxation time hτ i ∝ k −γ changes from 1.2 at low k values to 1.9 at high k values. A very similar crossover from γ ≃ 1.2 to γ ≃ 2 was previously reported from molecular dynamics simulations of supercooled SPC/E water in ref 63. The rotational ISF was produced by applying the Rayleigh expansion 64,65 in terms of spherical harmonics 66,67 to the plane wave in eq 8. This expansion, which leads to a series of decaying exponents, can be modified to incorporate the dynamical heterogeneity. 68 The result is 2

FR (k, t) = [j0 (ka0 )] +

∞ X

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Figure 7: Translational (green), rotational (blue) and total (black, obtained from eq 13) loss functions kB T χ′′T,R (ν) of the self intermediate scattering function for water in the lysozyme powder. The spectrum is highly stretched, with the high-frequency tail approximated by a power law, χ′′ (ν) ∝ ν −a , a = 0.35 (black dashed line). The violet points show the fit of experimental χ′′ (ν) for neutron scattering (NS) from lysozyme only in the protein-D2 O contrast. 69 The calculations are performed at k = 0.7 ˚ A−1 and the orange dots refer to the fit of neutron scattering data at k = 0.7 ˚ A−1 . 70 The experimental curves are arbitrary shifted vertically to fit the scale of the plot.

(2ℓ + 1) [jℓ (ka0 )]2

× exp −(ℓ(ℓ + 1)DR t)βR

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(12)

where jℓ (x) is a spherical Bessel function. 64 This equation assumes axial symmetry of the rotor, which is not the case for the water molecule. The parameter a0 is therefore an effective length of the dumbbell rotor representing water’s rotations. It can be found by taking the t → ∞ limit in eq 12, which yields a0 = 0.911 ˚ A across all k-vectors, with only slight variation from low to high k values. This effective length is in fact close to the O-H distance of 0.957 ˚ A in TIP3P water. Fitting the simulation data to eq 12 has produced the stretching exponent βR ≃ 0.48 and the rotational relaxation time τR = (2DR )−1 ≃ 6 ps both essentially independent of k (Figure S10 in the SI), in accord with the expectations.


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Table 2: Relaxation parameters for the single-particle dynamics of water represented by C1 (t) (eq 4) and FR (k, t) (eq 12). Function Aw τwD (ps) τw (ps) βw C1 (t) 0.22a 5a 18 0.25 FR (k, t) 6b 0.48 a relaxation times at the peaks of the corresponding distributions shown in Figure 6. b first-rank (ℓ = 1) rotational relaxation time τR = (2DR )−1 . The value of the rotational relaxation time τR agrees with the average Debye relaxation time, hτD i ≃ 6.7 ps, obtained from the single-particle rotational dynamics of the water dipole moment (eq 4 and Figure 5). The results from the singleparticle rotational dynamics extracted from the dipole time-correlation function and from the rotational part of the ISF are summarized in Table 2. We have applied the results of fitting to produce the loss functions 44 for translational and rotational dynamics χ′′T,R (ω, k) ∝ ωST,R(ω, k), where ST,R (ω, k) are the self dynamic structure factors for the translational and rotational motions of water. These are shown in Figure 7 for water translations (green) and rotations (blue). The total loss function, corresponding to Fs (k, t) in eq 6, was produced from the frequency convolution of the translational and rotational components according to the relation Z 2ω ∞ ′′ ′ ′′ kB T χ (ω, k) = χ (ω , k) π −∞ T × χ′′R (ω − ω ′ , k)dω ′/[ω ′(ω − ω ′ )] (13)

high-frequency scaling is shared between the protein and water components of the powder. The maximum of water’s loss spectrum in the powder is consistent with the measurements in bulk water 70 (orange dots in Figure 7). This comparison again underscores a week effect of the protein confinement on the single-molecule water dynamics. Nevertheless, the translational dynamics occurs at the time-scale of ≃ 100 ps at k≃1˚ A−1 (Figure 6B). The loss spectrum from neutron scattering, however, does not pick this effect since it is dominated by water rotations at the frequencies below the peak.

Discussion The single-particle rotational dynamics in water are essentially exponential, with the relaxation time of 2-3 ps, as follows experimentally from NMR and closely matched (with some variations between the force-fields) by simulations. On the contrary, the single-particle dynamics of water confined in the protein powder are highly stretched with the stretching exponent of 0.2. The stretched dynamics follow from both direct calculations of the single-particle time correlation functions and from fitting the self intermediate scattering functions recorded by neutron scattering experiments. The distinction between the collective dynamics and the single-particle dynamics comes from the cross molecular correlations, in this case between the dipole momemts of different water molecules. Such cross correlations produce the retardation factor amounting approximately to the Kirkwood correlation factor in the bulk liquids, 19 gK ≃ 2.7 for water. 20 For water confined in the protein powder, two collective relaxation components appear, with approximately equal

The result of this convolution is shown by the black solid line in Figure 7. As is seen, the total loss function is dominated by water rotations at frequencies below the peak and by translations at frequencies above the peak. The loss spectrum of water in the powder is highly stretched. The high-frequency wing is well approximated by the power-law decay χ′′ (ω, k) ∝ ω −0.35 (dashed black line in Figure 7). A very similar scaling ∝ ω −0.3 was reported for the loss spectrum from neutron scattering of lysozyme only (with D2 O contrast). 69 While the peak of the lysozyme spectrum is shifted to lower frequencies (violet dots in Figure 7), the


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Corresponding author: ∗ [email protected] Notes: The authors declare no competing financial interest.

weights. The Debye relaxation component (≃ 16 ps) is slower than the bulk Debye relaxation (≃ 7 ps for TIP3P water 56 ) by a factor ≃ 2 often quoted for liquids in confinement. 57 However, the longer component (≃ 13 ns) is significantly slower. Since this component does not appear in the single-molecule time-correlation function, it seems reasonable to assign it to displacements of confined water produced by elastic fluctuations of the proteins in the powder sample. 28–32 The single-particle translational dynamics of water in the powder can potentially be significantly slower: hτ i ≃ 100 ps (eq 11) is found here at k ≃ 1 ˚ A−1 typically reported by neutron scattering. 69,70 The frequency spectrum is, however, a convolution of the rotational and translational spectra with the result that the slow translational dynamics are hidden in the total loss by the dominance of rotations at the low-frequency wing (Figure 7). The overall spectrum of the intermediate scattering function of water in the powder is highly stretched, with the power-law decay ∝ ω −0.35 at the highfrequency wing. The main conclusion of our study is that water in the protein confinement is strongly modified compared to the bulk and is characterized by complex and highly stretched dynamics both in its translational and rotational degrees of freedom. The collective dynamics of water are affected by the elastic protein fluctuations, which are distinct from the dipole tumbling and do not show up as a separate relaxation process in the dielectric spectrum. The self intermediate scattering function is a complex convolution of the rotational and translational contributions, with rotations and translations dominating, correspondingly, the low- and high-frequency wings of the loss spectrum.

Acknowledgement This research was supported by the National Science Foundation (CHE1464810). CPU time was provided by the National Science Foundation through XSEDE resources (TG-MCB080071).

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Supporting Information Available: Sim-

(7) Lerbret, A.; Hédoux, A.; Annighöfer, B.; Bellissent-Funel, M.-C. Influence of pressure on the low-frequency vibrational modes of lysozyme and water: A complementary inelastic neutron scattering and molecular dynamics simulation study.

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