Hydration shells of proteins probed by depolarized light scattering and dielectric spectroscopy: Orientational structure is significant, positional structure is not Daniel R. Martin and Dmitry V. Matyushov Citation: The Journal of Chemical Physics 141, 22D501 (2014); doi: 10.1063/1.4895544 View online: http://dx.doi.org/10.1063/1.4895544 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/22?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The remarkable hydration of the antifreeze protein Maxi: A computational study J. Chem. Phys. 141, 22D510 (2014); 10.1063/1.4896693 In silico studies of the properties of water hydrating a small protein J. Chem. Phys. 141, 22D502 (2014); 10.1063/1.4895533 Unusual structural properties of water within the hydration shell of hyperactive antifreeze protein J. Chem. Phys. 141, 055103 (2014); 10.1063/1.4891810 Depolarized light scattering and dielectric response of a peptide dissolved in water J. Chem. Phys. 140, 035101 (2014); 10.1063/1.4861965 Depth dependent dynamics in the hydration shell of a protein J. Chem. Phys. 133, 085101 (2010); 10.1063/1.3481089
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THE JOURNAL OF CHEMICAL PHYSICS 141, 22D501 (2014)
Hydration shells of proteins probed by depolarized light scattering and dielectric spectroscopy: Orientational structure is significant, positional structure is not Daniel R. Martin1 and Dmitry V. Matyushov2,a) 1
Department of Physics, Arizona State University, PO Box 871504, Tempe, Arizona 85287-1504, USA Department of Physics and Department of Chemistry and Biochemistry, Arizona State University, PO Box 871504, Tempe, Arizona 85287-1504, USA 2
(Received 9 June 2014; accepted 12 August 2014; published online 12 September 2014) Water interfacing hydrated proteins carry properties distinct from those of the bulk and is often described as a separate entity, a “biological water.” We address here the question of which dynamical and structural properties of hydration water deserve this distinction. The study focuses on different aspects of the density and orientational fluctuations of hydration water and the ability to separate them experimentally by combining depolarized light scattering with dielectric spectroscopy. We show that the dynamics of the density fluctuations of the hydration shells reflect the coupled dynamics of the solute and solvent and do not require a special distinction as “biological water.” The orientations of shell water molecules carry dramatically different physics and do require a separation into a sub-ensemble. Depending on the property considered, the perturbation of water’s orientational structure induced by the protein propagates 3–5 hydration shells into the bulk at normal temperature. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895544] I. INTRODUCTION
This paper interrogates the structure and dynamics of hydration shells of proteins by means of molecular dynamics (MD) simulations. We are asking here the question of whether hydration shells can show the dynamics (time-dependent properties) and/or statistics (time-independent properties) significantly different from bulk water. The focus here is on comparing the structure and dynamics of the hydration shell as determined based on either its density or orientational profile. Our main conclusion is that orientations of shell waters are significantly different from the bulk and need to be characterized as a special sub-ensemble, not reducible dynamically or statistically to the properties of either the protein or water. On the contrary, the density perturbation of water by the protein is short-ranged and involves fewer water molecules. Correspondingly, the dynamics of the density fluctuations of the hydration shell reflect the coupling between the slow dynamics of the protein and the fast dynamics of water and do not require special shell dynamics for its description. The density fluctuations of bulk water can be studied experimentally by depolarized light scattering (DLS)1, 2 or optical Kerr effect (OKE).3 The polarizability of water is highly isotropic. As a consequence, the polarizability anisotropy relaxation, reported as the DLS signal, is mostly caused by dipole-induced-dipole (DID) interactions produced by molecular translations.1, 4 On the contrary, the dipolar response of water is dominated by rotational diffusion of its dipolar polarization density. Therefore, translational and rotational dynamics of bulk water can be separately studied by, correspondingly, the DLS/OKE and dielectric spectroscopies.5 a) Electronic mail: [email protected]
0021-9606/2014/141(22)/22D501/8/$30.00
A clear separation of different nuclear modes is much harder to achieve for solutions and interfaces. The problem is commonly addressed by the analysis of time correlation functions and spectral intensities reported by linear-response techniques.6–11 The general phenomenology reported for loss functions of solutions is the appearance of absorption bands intermediate in frequency to those representing the solute and solvent.12, 13 These new bands have been broadly assigned to the specific dynamics of the interface, even though concerns have been risen that bands of intermediate frequency may be caused by cross solute-solvent correlations.14 The problem of separating the dynamics of the interface from the dynamics arising from interactions between the solute and solvent is notoriously hard due to the lack of direct experimental means to distinguish between the two. The difficulty is not much helped by numerical simulations either because of practical complications of calculating dynamical cross correlations from simulations. We do not resolve this general difficulty in this study. However, we provide a number of careful tests to show that both cross correlations and interfacial dynamics can contribute to the observed signals. In particular, we do not find any specific interfacial modes in the DLS response. A separate relaxation component, falling between solute and solvent relaxation peaks, and having a negative amplitude, is indeed found, but is assigned to the protein-water dynamic cross correlations. The assignment is consistent with a phenomenological model allowing a bi-linear coupling between slow and fast Brownian oscillators. This analysis fails, however, for the dipolar response of the protein hydration shells. A significantly higher in amplitude, compared to the DLS response, contribution to the dynamics is found. It cannot be assigned to either the solute, solvent, or to their dynamical
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cross-correlations. We, therefore, assign the orientational manifold of hydration shell waters to a separate subensemble, which can be called the “biological water.” Its dynamics cannot be explained solely by mixing the dynamics of the solute and the bulk and its structural properties are clearly distinct from those of bulk water. II. GENERAL CONSIDERATIONS
The general problem we are facing here is that of measuring the time correlation function of a composite system including a solute (symbol “0”) with the Hamiltonian H0 and a solvent (symbol “s”) with the Hamiltonian Hs . Their interaction is described by the Hamiltonian H0s such that the overall system Hamiltonian is H = H0 + H0s + Hs . One observes the time autocorrelation function C(t) = δX(t)δX(0), which can be separated into the solute, C0 (t), solvent, Cs (t), and cross-correlation, C0s (t), components. The question this separation poses is whether the cross-correlation will show the dynamics distinct from the dynamics of the solute and solvent separately. This question connects to a number of recent time-resolved experiments where separate peaks in the loss spectrum χ (ω) ∝ ωC(ω) were observed (C(ω) is the time Fourier transform of C(t)). The fitting of the loss spectra, typically achieved by adding separate Debye relaxation processes, suggests the appearance of specific interfacial dynamics. The possibility of new dynamics distinct from that of either the solute or solvent and caused by cross-correlations is rarely considered,14, 15 but should not be excluded. In order to first approach the problem from a general perspective, one can start with the Liouville representation of the time cross-correlation function16, 17 C0s (t) = δX0 (t)δXs (0) = δX0 (0)δXs (−t) = δX0 eiLt δXs .
(1)
Here, iLX = (H, X) is the Liouville operator and round brackets denote the Poisson bracket. Since Xs depends on coordinates and momenta of the solvent only, exp (iLt)δXs = exp (i(Ls + L0s )t)δXs . On the other hand, the Liouville operator is hermitian and can act on δX0 to the left. δX0 , however, depends only on the variables of the solute and, therefore, Ls disappears from the series of Poisson brackets. Finally, one arrives at C0s (t) = δX0 e
iL0s t
δXs .
(2)
The dynamics of the cross-correlation function is fully determined by the solute-solvent Hamiltonian. If H0s produces forces significantly different from those produced by either Hs or H0 , one should expect specific dynamics originating from the solute-solvent cross correlations. On the other hand, the spatial extent of static cross correlations is determined not by forces, but by the range of solute-solvent interactions and the number of solvent molecules involved. The static (time-independent) correlations are, therefore, expected to be influenced primarily by long-range interactions, producing weak forces. On the contrary, dynamic signatures of cross correlations arise from short-range specific interactions, producing strong forces, such as hydrogen bonds.11 The dynamic
and static cross-correlations are, therefore, affected by different types of solute-solvent interactions. In order to provide a physically transparent perspective on possible observable consequences of dynamic crosscorrelations, we consider here a simple model of two overdamped Ornstein-Uhlenbeck stochastic processes18 coupled by a bilinear term in the Hamiltonian. The system Hamiltonian thus has the form 1 1 H = k0 (δX0 )2 + ks (δXs )2 + γ δX0 δXs , (3) 2 2 where k0 and ks are the force constants of harmonic displacements and γ is the coupling constant. Each of the stochastic processes is described by the corresponding Langevin equation, which are coupled through the coupling term in the Hamiltonian. These equations are solved in the supplementary material.19 The solution provides the changes of the ˜ Fourier-Laplace transforms C(ω) of the time correlation functions C(t) caused by the solute-solvent coupling. The main result is given by the simple equation τ C˜ C˜ s = 0 (0)0 . (0) τs C˜ 0 C˜ s
(4)
Here, τ 0 and τ s are the (Debye) relaxation times of the two (0) modes without coupling and C˜ 0,s are the corresponding correlation functions when the coupling is switched off. Further, (0) are the changes in the correlation funcC˜ 0,s = C˜ 0,s − C˜ 0,s tions when the coupling is introduced. For typical conditions of solutes exceeding in size the solvent one has τ 0 /τ s 1. Equation (4) then indicates that the perturbation introduced by the solute-solvent coupling to the time correlation function affects more the solvent than the solute. To produce a specific example of such change, we consider the alteration of the loss spectrum of the solvent when the relaxation of the solute is described by a Debye process. In order to model the change of the solute relaxation caused by the solute-solvent coupling, we will assume that the relaxation time of the solute τ0sol in the solution is different from the relaxation time τ 0 in an imaginary configuration with no solute-solvent coupling. Note that only the coupling between the modes X0 and Xs is considered here, and it can be switched off by changing the solvent. In this model, Eq. (4) becomes C˜ s = C˜ s(0)
−1 τ0sol (1 − ξ ) 1 − iωτ0sol , τs
(5)
where ξ = τ0 /τ0sol is the relative change in the solute relaxation time. We will use this equation below in the analysis of the DLS response of the lysozyme solution to show that the simulation results are in a general agreement with this model. To simplify the problem even further, we next consider the case of a Debye relaxation of the solvent in C˜ s(0) , which is consistent with the Ornstein-Uhlenbeck stochastic process. The change in the loss spectrum of the solvent component,16 χs (ω) = βωRe[C˜ s ], when the coupling between X0 and Xs is switched on follows from the fluctuation-dissipation theorem16 ω¯ , (6) ks χs (ω) ¯ = κ (1 − ξ ) Re (1 − i ω)(1 ¯ − iκ ω) ¯
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D. R. Martin and D. V. Matyushov
ωτ FIG. 1. Change in the loss spectrum of the solvent χs (ω) caused by the solute-solvent coupling. Calculations are done according to Eq. (6). The two curves are obtained with κ = τ0sol /τs = 100 and ξ = τ0 /τ0sol = 0.6 (solid line) and 0.8 (dashed line).
where ω¯ = ωτs and κ = τ0sol /τs . It is easy to see that χs (ω) alters its sign (Fig. 1) at the frequency (7) ω−1 = τ0sol τs determined by the geometric mean of the solute and solvent relaxation times and thus positioned between their corresponding Debye peaks. If τ0sol > τ0 , i.e., the solute-solvent coupling leads to slowing down of the solute, χs (ω) is negative at the position of the solvent relaxation peak ωτ s = 1 (Fig. 1). The solutesolvent coupling then results in the loss of the solvent intensity, as is usually observed in solution experiments7, 8 and discussed in more detail below. At the same time, a positive peak appears near the solute characteristic relaxation frequency, thus increasing the spectral loss in that region. We also note that, depending on the solute, the position of the solvent relaxation band can either be unaffected by the solute or shift to usually a lower frequency.20 To account for the latter effect, the solvent relaxation time τ s needs adjusting within the present model or, alternatively, the force constant ks in Eq. (3) needs to become a function of the solute concentration. The main result of this simple model is the appearance of a relaxation peak characteristic of the solute in the loss spectrum of the solvent. If the solute relaxation is characterized by a sum of several relaxation processes, they all will contribute, with different weights, to the solvent loss spectrum. It is also possible that some of these processes will produce negative contributions to the loss, while others will increase the loss intensity. These relaxation peaks might be incorrectly assigned to new, slower dynamics of the hydration layer by experiments probing exclusively the relaxation of the solvent (such as NMR6 or time-resolved infra-red spectroscopy8 ). We now turn to the polarizability anisotropy (DLS) response of the hydrated lysozyme, followed by the response of the solution dipole moment.
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both the single particle polarizabilities of all molecules in the solutions and their DID polarizabilities. The polarizability of water is assigned to its oxygen with the principal component values α xx = 1.408, α yy = 1.497, and α zz = 1.417 Å3 with the water molecule assumed to be in the yz-plane.21 The polarizability of lysozyme is calculated from its atomic polarizabilities according to the Thole formalism.22–24 It involves the dipole-dipole interactions between the atomic induced dipoles such that the resulting polarizability of the solute is calculated by inverting the matrix of binary interactions of the induced dipoles at each instantaneous configuration of the system (see the supplementary material19 ). The total polarizability of the solution αβ was used to calculate the time autocorrelation function in the depolarizedscattering geometry, which includes only off-diagonal components 1 δ αβ (t)δ αβ (0), (8) C (t) = 6 α=β where α = x, y, z are the Cartesian components of the 2-rank Cartesian tensor. The corresponding normalized time correlation function becomes φ (t) = C (t)/C (0).
(9)
The time correlation function was used to define the OKE response function25 χ (t) = −β C˙ (t), where β = 1/(kB T) is the inverse temperature. The response function can be FourierLaplace transformed to connect to the loss function via the classical limit of the fluctuation-dissipation relation17 χ (ω) = βC (0)Re[ωφ˜ (ω)]. ˜
(10)
Here, φ (ω) is the Fourier-Laplace transform of φ (t) in Eq. (9). A. TIP3P water
The DLS response of bulk water was calculated from separate MD simulations of TIP3P water at 300 K.19 The data were analyzed by assigning anisotropic polarizability21 to water, with the principal-axes components listed above and in the caption to Fig. 2. Three peaks of the experimental DLS spectrum of water7 are also seen in simulations (Fig. 2), although they are shifted to different extent depending on the force field used. The 0.4 0.3
χs''(ν)
Δχ ″(ω)
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Experiment TIP3P SPC/E
0.2 0.1
III. POLARIZABILITY RESPONSE
The simulation protocol and calculation of the solvent and solution polarizabilities are described in the supplementary material.19 Here, we focus on the data analysis and the results. The second-rank tensor of the polarizability (t) of the solution containing TIP3P water and lysozyme includes
0.0 0.001
0.01
0.1
1
ν (THz)
10
FIG. 2. The DLS loss spectrum of water: experiment7 (points), MD simulations of TIP3P (blue line), and SPC/E29 (orange line) force field water. The principle-axes polarizability of water is: α xx = 1.408, α yy = 1.497, and α zz = 1.417 Å3 with the water molecule placed in the yz-plane.21
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highest-frequency peak represents hindered rotations (librations) of the water molecules.26, 27 The two lower-frequency peaks are assigned the H-bond bending (lower frequency) and H-bond stretching (higher frequency).26, 27 Given that some of these motions carry some ballistic character, damped harmonic components have been used in fitting of the DLS spectra.28 We, therefore, fit the normalized time correlation function φs (t) = Cs (t)/Cs (0) to a linear combination of a Gaussian decay, a damped harmonic oscillator, and two exponential Debye terms. Two restrictions, φs (0) = 1 and φ˙ s (0) = 0, are additionally imposed, resulting in the following fitting function:29 φs (t) = e−ωg t
2 2
/2
+
2
Bi gi (t),
(11)
i=1
where Bi are the component amplitudes and gi (t) = e−αi t + (αi τh − 1)e−ωg t
2 2
/2
− αi τh e−t/τh cos ωh t. (12)
Seven free parameters obtained by fitting Eqs. (11) and (12) to MD data are listed in Table S1 in the supplementary material.19 We also show in Fig. 2 the DLS loss spectrum of SPC/E water reported previously by us.29 The TIP3P force field overestimates the intensity of the lower-frequency DLS peak compared to the SPC/E model, but gives a fair account of the experimental low-frequency tail of the loss spectrum. The oscillatory feature in the time correlation function φ (t), well resolved for SPC/E water, is nearly absent for TIP3P water (Fig. S1 in the supplementary material19 ). The result is the disappearance of the highest frequency peak in the DLS spectrum of TIP3P water in Fig. 2. B. Lysozyme solution
In order to quantify separate component contributions to the DLS response of the lysozyme solution, we first calculated the anisotropy polarizability relaxation of the lysozyme solute. The corresponding autocorrelation function 1 δ 0,αβ (t)δ 0,αβ (0) (13) C0 (t) = 6 α=β describes the relaxation of the single solute polarizability 0 (see the supplementary material19 for more details). The main contribution to this time correlation function comes from a single-exponential decay with the relaxation time τ0 = 1.6 ns (Table S2 in the supplementary material19 ) corresponding to rotational diffusion of lysozyme in solution. This relaxation time is obtained from the dynamics of the secondrank tensor. If the same rotational diffusion (given by the longest relaxation in a multi-exponential fit) is responsible for the relaxation of first- and second-rank observables, the relaxation time of the second-rank observable should be three times shorter.1 One can, therefore, compare τ0 to τ0M extracted from the time autocorrelation function of the first-rank dipole moment M0 (t) of lysozyme C0M (t) = δM0 (t) · δM0 (0).
(14)
The main component of this correlation function decays exponentially with the relaxation time τ0M = 9.1 ns (Table S2 in the supplementary material19 ), which is longer than 3τ0 . The anticipated scaling applies to rotation of objects with axial symmetry1 and can only approximately hold for solutes of irregular shape, such as lysozyme. The normalized function φ0 (t) = C0 (t)/C0 (0) is used to represent the dynamics of the solute polarizability in the correlation function of the solution φ (t) [Eqs. (8) and (9)]. The other two components are the relaxation of water [Eq. (11)] and all possible relaxation process that cannot be attributed to either the solute or water alone. They are lumped (t) in the equation into φcross φ (t) = B1 φ0 (t) + B2 φs (t) + φcross (t),
(15)
where Bi are the component amplitudes considered as fitting parameters. The usual constraints,17 φ (0) = 1 and φ˙ (0) = 0, are imposed on φ (t) when fitting the MD data. The subscript “cross” in the last term in Eq. (15) reflects the expectation that at least cross-correlations between the bulk and the solute will contribute to this term. An independent dynamic process, such as new dynamics of hydration shells, can be a part of (t) as well. This term was modeled by two decaying exφcross ponents (Ne = 2) when fitting the MD data (Table S2 in the supplementary material19 ) (t) φcross
=
Ne
Ai e−t/τi .
(16)
i=1
The fitting of Eqs. (15) and (16) to MD data yields only (t) (yellow in Fig. 3). The ∼6% of the intensity from φcross DLS response of the solution is, therefore, mostly a linear combination of the water and protein components. The cross component has the relaxation time of 0.6 ns (Table S2 in the supplementary material19 ) and a negative intensity. This latter result suggests solute-solvent cross correlations as the origin of this component in a general agreement with our model considerations above. Looking at the DLS response originating from the water component of the solution helps to further identify the source of this intermediate relaxation component.
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χ''(ν)
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Bulk water Total Solute Water Cross
0.2
0.0 0.00001
0.001
0.1
ν (THz)
10
FIG. 3. The DLS loss spectrum of lysozyme in TIP3P water. The overall spectrum (black line) is decomposed into the solute (green), water (blue), and cross (yellow) terms according to Eq. (15). The solute peak represents rotational diffusion of the solute. The dashed line refers to TIP3P water; solution and water loss spectra are equally normalized to produce unity for the integral of χ (ν)/ν.
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sol
χs ''(ν)
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Bulk water Total Solute Solvent Cross
0.3 0.2 0.1 0.0 0.00001
0.001
0.1
ν (THz)
10
FIG. 4. The DLS loss spectrum of TIP3P water in the solution of lysozyme. The overall spectrum (black line) is decomposed into the solute (green), water (blue), and cross (yellow) terms according to Eq. (15). The dashed line refers to TIP3P water; solution and water loss spectra are equally normalized to produce unity for the integral of χs (ν)/ν.
Figure 4 shows the DLS loss spectrum (χssol ) (ω) of the water component of the lysozyme solution. As in the case of the loss peak of the overall solution polarizability in Fig. 3, there is a drop in the water peak caused by the solute and the appearance of a peak identified with rotational diffusion of lysozyme. This relaxation component makes 13% of the normalized autocorrelation function (B1 , Table S2 in the supplementary material19 ). The splitting of the water spectrum into the bulk and solute components is achieved by applying Eq. (15). Similar to the analysis of φ (t) in Eq. (15), the cross term in the correlation function amounts to ∼5% of the intensity and is characterized by the relaxation time of 0.5 ns, also similar to the analysis of φ (t). The water, solute, and cross terms in the loss spectrum are color coded in Fig. 4. Even more revealing is the difference loss spectrum χs (ω) = (χssol ) (ω) − χs (ω) of water between the solution and the bulk. It is compared to Eq. (5) in Fig. 5. The agreement between the simple bilinear coupling model and the MD results is quite satisfactory given that water dynamics are more complex than assumed in the Ornstein-Uhlenbeck model. To produce a single relaxation time for water in Eq. (5), τ s was calculated from the equation ∞ φs (t)dt. (17) τs = 0
One gets τ s = 0.33 ps when this equation is applied to bulk TIP3P water (Eq. (11), Table S1 in the supplementary material19 ). This relaxation time is typically assigned to hindered rotations (librations) of water dipoles.30 The average
relaxation time, however, grows significantly to τssol = 210 ps when applied to water in the lysozyme solution due to slow water molecules from the hydration shells. Note that the timescales of 100–300 ps have been reported for the lifetimes of hydrogen-bonds of water bound to the protein.31, 32 In contrast, we are dealing here with the average relaxation time of all water molecules in the simulation box, and the results obtained for waters bound to the protein do not apply here. Formally, the appearance of the solute relaxation peak in the water spectrum (χssol ) (ω) is responsible for a significant increase of τ s . This phenomenon, therefore, will be reported by any spectroscopic technique targeting solvent in the solution. The present model, however, clearly shows that the slowing down cannot be directly interpreted as the appearance of a specific sub-ensemble of the solvation shell. It should instead be related to the coupling between the slow and fast subsystems in the system Hamiltonian. The slowing down does accompany the appearance of a sub-ensemble of hydration water, as we show below for the orientational structure of water interfacing lysozyme, but its existence cannot be established solely from the slower dynamics of water in the solution. Figure 5 also shows the application of Eq. (5) to produce χs (ω) assuming that the relaxation time of the solute is increased by 15% due to the lysozyme-water coupling (ξ = 0.85 in Eq. (5)). One gets a positive peak at the solute rotational frequency and a negative peak at the water relaxation frequency, consistent with the MD results. In addition, the difference spectrum χs (ω) crosses zero at 7 GHz, while Eq. (7) predicts the crossing point at 9 GHz. Given the simplicity of the model, it clearly captures the main physics of the alteration of the water spectrum introduced by the solute. IV. DIPOLAR RESPONSE A. Dipolar response of the lysozyme solution
The loss spectrum of the dipole moment of the lysozyme solution, which is experimentally recorded by dielectric spectroscopy,12, 13 is shown in Fig. 6. It is separated into the rotational diffusion of the solute’s dipole moment (φ0M (t) = C0M (t)/C0M (0), Eq. (14)), dipolar relaxation of bulk TIP3P M (t) water (φsM (t)), and an extra term given as φshell M (t), φ M (t) = B1 φ0M (t) + B2 φsM (t) + φshell
(18)
where, as above, Bi are the amplitudes of the solute and water components in the time correlation function. Model Simulations
-0.05 0.00001
Pure TIP3P Lys - Tot Lys - Solute Lys - Water Lys - Debye 1 Lys - Debye 2
0.4
0.00
χ''(ν)
Δχs''(ν)
0.05
0.001
0.1
ν (THz)
0.2
10
FIG. 5. The difference loss spectrum χs (ν) of the DLS response of water in solution and in the bulk. The loss spectra are from MD simulations (dotted line) and from Eq. (5) (solid line). The retardation factor of the solute rotational diffusion due to the lysozyme-water coupling ξ = τ0 /τ0sol = 0.85 was used in the calculations, the water relaxation time of τw = 0.33 ps was calculated from Eqs. (11) and (17).
0.0 0.00001
0.001
0.1
10
ν (THz) FIG. 6. Loss spectrum of the solution dipole moment. The spectrum is separated into the solute (lysozyme), water, and shell contributions according to Eq. (18).
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Pure TIP3P Total Solute Water Debye 1 Debye 2
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0.2 0.0 0.00001
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ν (THz) FIG. 7. Loss spectrum of the dipole moment of TIP3P water in the lysozyme solution. The spectrum is separated into the solute (lysozyme), bulk TIP3P water, and shell contributions according to Eq. (18).
Both correlation functions φsM (t) and φ0M (t) are fitted by two decaying exponents before applying Eq. (18). The solvent fit produces the relaxation times of 4.5 ps (74%) and 15 ps (26%) and the solute fit yields 0.7 ns (12%) and 9.1 ns (88%) (Tables S1 and S2 in the supplementary material19 ). The solute relaxation contributes only 7% to the overall φ M (t), while the contribution of bulk water is 28%. The contribuM (t) is, therefore, 65%, with the main relaxation tion of φshell time of 6 ps. This extra component significantly exceeds 6% contribution of the cross-correlation term to the polarizability correlation function in Eq. (15). The arguments presented below indicate that, in contrast to the DLS response, the orientational manifold of waters in the hydration shell presents a separate sub-ensemble, with its properties distinct from those of either the solute or solvent and not reducible to their cross correlations. The subscript “shell” in the last summand in Eq. (18) is used to stress this distinction. The dipolar loss spectrum of water in the solution is shown in Fig. 7. Here, in contrast to the DLS response, where the solute component is well distinguished in the overall response (Fig. 5), its contribution to the correlation function is only 0.5%. In contrast, 62% of the correlation function is given by the shell component dominated by an exponential decay with the relaxation time of 6 ps. A small relaxation component with a ∼100 ps relaxation time is consistent with dielectric δ-relaxation.12 The dominant 6 ps component is the same relaxation process, and roughly the same ampliM (t) of φ M (t) in tude, as is seen for the shell component φshell Eq. (18). These results are inconsistent with the bilinear coupling model that has described the cross correlations in the DLS response so well. They clearly suggest that orientations of water dipoles in the hydration shell of the protein must be attributed to a separate sub-ensemble including many water molecules and not reducible dynamically to a coupling between fast and slowly relaxing subsystems. Consistent with this view, the difference spectrum χs , obtained by taking the difference between the dipolar loss spectra of water in solution and of bulk water (Fig. S5 in the supplementary material19 ), is very different from what is shown in Fig. 5 for the DLS response. We now take a closer look at the dynamics of water molecules in the hydration shell of lysozyme.
vide the dipolar dynamics of hydration shells of increasing thickness around lysozyme. For that purpose the correlation function of the dipole moment Ms (a, t) of the water shell of thickness a from the protein’s van der Waals surface is correlated with the dipole moment Ms (0) of the entire water subsystem. Such correlation functions appear in linear response calculations17 of the dipole moment of the shell induced by a long-wavelength external field interacting with the dipoles of water only.33 In dielectric experiment, the latter condition requires measurements at frequencies exceeding the frequency of the solute rotational diffusion (τ0M )−1 to dynamically freeze the solute response.29 The correlation function we calculate here becomes CsM (a, t) = δMs (a, t) · δMs (0).
(19)
The normalized function φsM (a, t) = CsM (a, t)/CsM (a, 0) is fitted, as above, to a weighted sum of the solute and solvent correlation functions and an additional term represented by a sum of two decaying exponents (Table S3 in the supplementary material19 ) φsM (a, t) = B1 (a)φ0M (t) + B2 (a)φsM (t) +
2
Ai (a)e−t/τi (a) .
i=1
(20) Above a = 12 Å, the decay of the last term in Eq. (19) becomes essentially single-exponential and only one exponent, with the relaxation time of ∼6 ps, is used. For the first hydration layer, about 2% of shell waters are strongly bound to the protein and follow its rotational dynamics with the relaxation time of ∼9 ns. The bulk dynamics is absent, and a part of the shell not rotating with the protein (∼12%) shows an additional slow component with the relaxation time of ∼1.2 ns. This component, in turn, disappears beyond the shell thickness of ∼9 Å and is replaced with a single relaxation with the 6 ps decay. The average relaxation time ∞ φsM (a, t)dt, (21) τ (a) = 0
which accommodates a weighted average of all these processes, is shown by circles in Fig. 8. It decays with increasing shell thickness to the limit of ∼6 ps within the range of a 20 Å. A relatively weak decay of the average relaxation time with increasing the shell size is a signature of a deep propagation of the orientational structural perturbation produced in 300
(ps)
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200 100 0 0
10
20
30
40
a (Å) B. Dipolar response of hydration shells
To additionally highlight the distinctions between the dynamics of the polarizability and the dipole moment, we pro-
FIG. 8. The average (Eq. (21)) dipolar relaxation time obtained from the dynamics of the dipole moment of the water shells of thickness a measured from the van der Waals surface of lysozyme [Eq. (19)].
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χ0s
0.05 0.00 -0.05 -0.10 4
8
12
16
20
a (Å) FIG. 9. Susceptibility χ 0s (a) [Eq. (22)]. The dotted line marks the zero value of χ 0s (a), which is reached at a 7.5 Å.
the water layers by the protein.15, 29 The structural effect of the protein on water is reflected by static dipolar correlations. To illustrate the spatial range of these correlations, Fig. 9 shows the cross-correlation of the lysozyme dipole moment with the dipole moment of the shell of thickness a. Specifically, we calculate the static susceptibility χ0s (a) =
β δM0 · δMs (a), 3vs Ns (a)
(22)
where Ns (a) is the number of water molecules, with the molecular volume vs , within the shell. The cross correlation extends to 12 Å into the bulk. Moreover, the correlation is negative for the closest shells, but alters its sign with the shell growth. Note that χ 0s is expected to be negative in dielectric theories for macroscopically large shells (a → ∞).15 Breaking this prediction is just another manifestation of specific orientational structure of water around proteins. On a more qualitative level, water molecules in the first hydration shell face a mosaic of positive and negative surface residues often establishing strong H-bonds with the nearest water molecules. The H-bond structure of bulk water is broken in the first hydration layer, and the positions and orientations of the surface waters are driven by the local interactions with the surface residues. There is, therefore, little correlation of the surface waters with the overall multipole of the solute (dipole in Eq. (22)). This correlation builds up only slowly in the outer hydration layers, and this slow build-up of the dielectric response is responsible for the long range character of χ 0s (a). The results for the relaxation times of hydration layers reported in Fig. 8 are significantly different from a similar analysis done for the solution of a N-Acetyl-leucine-methylamide dipeptide (NALMA) in SPC/E water.29 Just a small increase, ∼10%, of the dipolar relaxation time in the first solvation shell compared to the bulk was reported for the longest exponential decay. Further, τ (a) was found to slightly decrease compared to the bulk due to a compensation between slowing down of the first layer by a slight speed-up of the second layer. No long relaxation times in the nanoseconds range could be observed for NALMA, in stark contrast to lysozyme. There are obviously qualitative differences between the dynamics of hydration shells of a relatively small and weakly polar NALMA dipeptide and lysozyme. The difference can be attributed both to the difference in polarity of the surface groups
and to significantly different sizes of their hydration shells: Ns (4 Å) = 48 for NALMA and 624 for lysozyme (Ns (3 Å) = 423). It is only in the case of the protein that there is a sufficient number of shell molecules to form a sub-ensemble with new dynamical and statistical properties. The hydration shell sub-ensemble will not be detected by experiments probing the statistics and dynamics of collective density fluctuations (translational motions). In addition, only a weak dynamic signature of bound waters (∼1% of the amplitude), with the relaxation time of ∼100 ps characteristic of the dielectric δ-relaxation,12 appears in the dynamics of the overall dipole moment of solution recorded by dielectric spectroscopy (Fig. 6). The rest of the signal is the solute tumbling and shell relaxation that is very close in the relaxation time to bulk water (4.5 ps in the bulk vs. 6 ps in the solution). In experimental fitting, it can potentially be assigned to a shifted “solvent” peak. Observables that probe the collective orientational dynamics of water separately have a better chance of sensing the shell waters and their specific dynamics. Along these lines, Stokes shift dynamics of optical dyes attached to the protein do demonstrate relaxation times in the range of nanoseconds.34–36
V. CONCLUSIONS
Water interfacing proteins needs to be distinguished as a separate sub-ensemble displaying properties distinct from the bulk. Hydration shells can be considered as “biological water” potentially affecting biological function in ways not deducible from either bulk water or from solvation properties of small model solutes.37 What we stress here is that one has to be extremely careful not to generalize this statement to all properties of the protein hydration shell and specify which parameters and corresponding observables require the notion of “biological water.” In particular, we show that the dynamics of the collective density fluctuations of the hydration shells as probed by depolarized light scattering reflect the coupled dynamics of the slow solute and fast solvent and do not require a separate sub-ensemble. The density perturbations produced by solutes in the solvent in general and by proteins in particular are local and do not propagate far from the solute surface into the bulk.38, 39 The orientational structure and dynamics of the shell carry dramatically different physics when probed with collective variables affected by many-body correlations of the shell dipoles. Proteins significantly distort the orientational structure of bulk water due to a high density of oppositely charged surface residues and surface sites providing H-bonds. Restoring the orientational bulk structure takes a relatively long distance from the protein surface into the bulk. The layer in which the bulk structure is restored involves a significant number of water molecules and needs separation into a subensemble of “biological water.” This sub-ensemble carries dynamical and statistical signatures not reducible to either the solute or the bulk. Depending on the property considered, the perturbed layer can propagate from two to five hydration shells into the bulk at the normal temperature. The new orientational structure of the shell dipoles thus emerges from a
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fairly large sub-ensemble of correlated dipoles characterized by a broken network of bulk-like hydrogen bonds. ACKNOWLEDGMENTS
This research was supported by the National Science Foundation (NSF) (CHE-1213288). CPU time was provided by the National Science Foundation through XSEDE resources (TG-MCB080116N). 1 B.
J. Berne and R. Pecora, Dynamic Light Scattering (Dover Publications, Inc., Mineola, NY, 2000). 2 D. Fioretto, L. Gomez, M. E. Gallina, A. Morrezi, L. Palmieri, M. Paolantoni, P. Sassi, and F. Scarponi, Chem. Phys. Lett. 441, 232 (2007). 3 K. Mazur, I. A. Heisler, and S. R. Meech, J. Phys. Chem. A 116, 2678 (2012). 4 D. A. Turton, J. Hunger, G. Hefter, R. Buchner, and K. Wynne, J. Chem. Phys. 128, 161102 (2008). 5 T. Fukasawa, T. Sato, J. Watanabe, Y. Hama, W. Kunz, and R. Buchner, Phys. Rev. Lett. 95, 197802 (2005). 6 J. Qvist, E. Persson, C. Mattea, and B. Halle, Faraday Discuss. 141, 131 (2009). 7 S. Perticaroli, L. Comez, M. Paolantoni, P. Sassi, A. Morresi, and D. Fioretto, J. Am. Chem. Soc. 133, 12063 (2011). 8 K.-J. Tielrooij, J. Hunger, R. Buchner, M. Bonn, and H. J. Bakker, J. Am. Chem. Soc. 132, 15671 (2010). 9 F. Sterpone, G. Stirnemann, and D. Laage, J. Am. Chem. Soc. 134, 4116 (2012). 10 S. T. van der Post, K.-J. Tielrooij, J. Hunger, E. H. G. Backus, and H. J. Bakker, Faraday Discuss. 160, 171 (2013). 11 S. Perticaroli, M. Nakanishi, E. Pashkovski, and A. P. Sokolov, J. Phys. Chem. B 117, 7729 (2013). 12 A. Oleinikova, P. Sasisanker, and H. Weingärtner, J. Phys. Chem. B 108, 8467 (2004). 13 C. Cametti, S. Marchetti, C. M. C. Gambi, and G. Onori, J. Phys. Chem. B 115, 7144 (2011). 14 T. Rudas, C. Schröder, S. Boresch, and O. Steinhauser, J. Chem. Phys. 124, 234908 (2006).
J. Chem. Phys. 141, 22D501 (2014) 15 D.
V. Matyushov, J. Chem. Phys. 136, 085102 (2012). Kubo, Rep. Prog. Phys. 29, 255 (1966). 17 J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, Amsterdam, 2003). 18 C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1997). 19 See supplementary material at http://dx.doi.org/10.1063/1.4895544 for details of the simulation protocol. 20 M. Sajadi, F. Berndt, C. Richter, M. Gerecke, R. Mahrwald, and N. P. Ernsting, J. Phys. Chem. Lett. 5, 1845 (2014). 21 M. Adrian-Scotto, G. Mallet, and D. Vasilescua, J. Mol. Struct (THEOCHEM) 728, 231 (2005). 22 B. T. Thole, Chem. Phys. 59, 341 (1981). 23 P. T. van Duijnen and M. Swart, J. Phys. Chem. A 102, 2399 (1998). 24 M. D. Elola and B. M. Ladanyi, J. Chem. Phys. 126, 084504 (2007). 25 L. C. Geiger and B. M. Ladanyi, Chem. Phys. Lett. 159, 413 (1989). 26 E. W. Castner, Jr., Y. J. Chang, Y. C. Chu, and G. E. Walrafen, J. Chem. Phys. 102, 653 (1995). 27 M. Paolantoni, P. Sassi, A. Morresi, and S. Santini, J. Chem. Phys. 127, 024504 (2007). 28 S. Perticaroli, L. Comez, M. Paolantoni, P. Sassi, L. Lupi, D. Fioretto, A. Paciaroni, and A. Morresi, J. Phys. Chem. B 114, 8262 (2010). 29 D. R. Martin, D. Fioretto, and D. V. Matyushov, J. Chem. Phys. 140, 035101 (2014). 30 D. Laage, G. Stirnemann, F. Sterpone, R. Rey, and J. T. Hynes, Annu. Rev. Phys. Chem. 62, 395 (2011). 31 K. Bagchi and S. Roy, J. Phys. Chem. B 118, 3805 (2014). 32 J. Yoon, J.-C. Lin, C. Hyeon, and D. Thirumalai, J. Phys. Chem. B 118, 7910 (2014). 33 A. D. Friesen and D. V. Matyushov, Chem. Phys. Lett. 511, 256 (2011). 34 S. Lampa-Pastirk and W. F. Beck, J. Phys. Chem. B 108, 16288 (2004). 35 K. Sahu, S. K. Mondal, S. Ghosh, D. Roy, and K. Bhattacharyya, J. Chem. Phys. 124, 124909 (2006). 36 D. Toptygin, A. M. Gronenborn, and L. Brand, J. Phys. Chem. B 110, 26292 (2006). 37 P. Ball, Chem. Rev. 108, 74 (2008). 38 D. I. Svergun, S. Richard, M. H. J. Koch, Z. Sayers, S. Kuprin, and G. Zaccai, Proc. Natl. Acad. Sci. U.S.A. 95, 2267 (1998). 39 J. C. Smith, F. Merzel, A. N. Bondar, A. Tournier, and S. Fischer, Philos. Trans. R. Soc. B 359, 1181 (2004). 16 R.
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THE JOURNAL OF CHEMICAL PHYSICS 141, 22D501 (2014)
Hydration shells of proteins probed by depolarized light scattering and dielectric spectroscopy: Orientational structure is significant, positional structure is not Daniel R. Martin1 and Dmitry V. Matyushov2,a) 1
Department of Physics, Arizona State University, PO Box 871504, Tempe, Arizona 85287-1504, USA Department of Physics and Department of Chemistry and Biochemistry, Arizona State University, PO Box 871504, Tempe, Arizona 85287-1504, USA 2
(Received 9 June 2014; accepted 12 August 2014; published online 12 September 2014) Water interfacing hydrated proteins carry properties distinct from those of the bulk and is often described as a separate entity, a “biological water.” We address here the question of which dynamical and structural properties of hydration water deserve this distinction. The study focuses on different aspects of the density and orientational fluctuations of hydration water and the ability to separate them experimentally by combining depolarized light scattering with dielectric spectroscopy. We show that the dynamics of the density fluctuations of the hydration shells reflect the coupled dynamics of the solute and solvent and do not require a special distinction as “biological water.” The orientations of shell water molecules carry dramatically different physics and do require a separation into a sub-ensemble. Depending on the property considered, the perturbation of water’s orientational structure induced by the protein propagates 3–5 hydration shells into the bulk at normal temperature. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4895544] I. INTRODUCTION
This paper interrogates the structure and dynamics of hydration shells of proteins by means of molecular dynamics (MD) simulations. We are asking here the question of whether hydration shells can show the dynamics (time-dependent properties) and/or statistics (time-independent properties) significantly different from bulk water. The focus here is on comparing the structure and dynamics of the hydration shell as determined based on either its density or orientational profile. Our main conclusion is that orientations of shell waters are significantly different from the bulk and need to be characterized as a special sub-ensemble, not reducible dynamically or statistically to the properties of either the protein or water. On the contrary, the density perturbation of water by the protein is short-ranged and involves fewer water molecules. Correspondingly, the dynamics of the density fluctuations of the hydration shell reflect the coupling between the slow dynamics of the protein and the fast dynamics of water and do not require special shell dynamics for its description. The density fluctuations of bulk water can be studied experimentally by depolarized light scattering (DLS)1, 2 or optical Kerr effect (OKE).3 The polarizability of water is highly isotropic. As a consequence, the polarizability anisotropy relaxation, reported as the DLS signal, is mostly caused by dipole-induced-dipole (DID) interactions produced by molecular translations.1, 4 On the contrary, the dipolar response of water is dominated by rotational diffusion of its dipolar polarization density. Therefore, translational and rotational dynamics of bulk water can be separately studied by, correspondingly, the DLS/OKE and dielectric spectroscopies.5 a) Electronic mail: [email protected]
0021-9606/2014/141(22)/22D501/8/$30.00
A clear separation of different nuclear modes is much harder to achieve for solutions and interfaces. The problem is commonly addressed by the analysis of time correlation functions and spectral intensities reported by linear-response techniques.6–11 The general phenomenology reported for loss functions of solutions is the appearance of absorption bands intermediate in frequency to those representing the solute and solvent.12, 13 These new bands have been broadly assigned to the specific dynamics of the interface, even though concerns have been risen that bands of intermediate frequency may be caused by cross solute-solvent correlations.14 The problem of separating the dynamics of the interface from the dynamics arising from interactions between the solute and solvent is notoriously hard due to the lack of direct experimental means to distinguish between the two. The difficulty is not much helped by numerical simulations either because of practical complications of calculating dynamical cross correlations from simulations. We do not resolve this general difficulty in this study. However, we provide a number of careful tests to show that both cross correlations and interfacial dynamics can contribute to the observed signals. In particular, we do not find any specific interfacial modes in the DLS response. A separate relaxation component, falling between solute and solvent relaxation peaks, and having a negative amplitude, is indeed found, but is assigned to the protein-water dynamic cross correlations. The assignment is consistent with a phenomenological model allowing a bi-linear coupling between slow and fast Brownian oscillators. This analysis fails, however, for the dipolar response of the protein hydration shells. A significantly higher in amplitude, compared to the DLS response, contribution to the dynamics is found. It cannot be assigned to either the solute, solvent, or to their dynamical
141, 22D501-1
© 2014 AIP Publishing LLC
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cross-correlations. We, therefore, assign the orientational manifold of hydration shell waters to a separate subensemble, which can be called the “biological water.” Its dynamics cannot be explained solely by mixing the dynamics of the solute and the bulk and its structural properties are clearly distinct from those of bulk water. II. GENERAL CONSIDERATIONS
The general problem we are facing here is that of measuring the time correlation function of a composite system including a solute (symbol “0”) with the Hamiltonian H0 and a solvent (symbol “s”) with the Hamiltonian Hs . Their interaction is described by the Hamiltonian H0s such that the overall system Hamiltonian is H = H0 + H0s + Hs . One observes the time autocorrelation function C(t) = δX(t)δX(0), which can be separated into the solute, C0 (t), solvent, Cs (t), and cross-correlation, C0s (t), components. The question this separation poses is whether the cross-correlation will show the dynamics distinct from the dynamics of the solute and solvent separately. This question connects to a number of recent time-resolved experiments where separate peaks in the loss spectrum χ (ω) ∝ ωC(ω) were observed (C(ω) is the time Fourier transform of C(t)). The fitting of the loss spectra, typically achieved by adding separate Debye relaxation processes, suggests the appearance of specific interfacial dynamics. The possibility of new dynamics distinct from that of either the solute or solvent and caused by cross-correlations is rarely considered,14, 15 but should not be excluded. In order to first approach the problem from a general perspective, one can start with the Liouville representation of the time cross-correlation function16, 17 C0s (t) = δX0 (t)δXs (0) = δX0 (0)δXs (−t) = δX0 eiLt δXs .
(1)
Here, iLX = (H, X) is the Liouville operator and round brackets denote the Poisson bracket. Since Xs depends on coordinates and momenta of the solvent only, exp (iLt)δXs = exp (i(Ls + L0s )t)δXs . On the other hand, the Liouville operator is hermitian and can act on δX0 to the left. δX0 , however, depends only on the variables of the solute and, therefore, Ls disappears from the series of Poisson brackets. Finally, one arrives at C0s (t) = δX0 e
iL0s t
δXs .
(2)
The dynamics of the cross-correlation function is fully determined by the solute-solvent Hamiltonian. If H0s produces forces significantly different from those produced by either Hs or H0 , one should expect specific dynamics originating from the solute-solvent cross correlations. On the other hand, the spatial extent of static cross correlations is determined not by forces, but by the range of solute-solvent interactions and the number of solvent molecules involved. The static (time-independent) correlations are, therefore, expected to be influenced primarily by long-range interactions, producing weak forces. On the contrary, dynamic signatures of cross correlations arise from short-range specific interactions, producing strong forces, such as hydrogen bonds.11 The dynamic
and static cross-correlations are, therefore, affected by different types of solute-solvent interactions. In order to provide a physically transparent perspective on possible observable consequences of dynamic crosscorrelations, we consider here a simple model of two overdamped Ornstein-Uhlenbeck stochastic processes18 coupled by a bilinear term in the Hamiltonian. The system Hamiltonian thus has the form 1 1 H = k0 (δX0 )2 + ks (δXs )2 + γ δX0 δXs , (3) 2 2 where k0 and ks are the force constants of harmonic displacements and γ is the coupling constant. Each of the stochastic processes is described by the corresponding Langevin equation, which are coupled through the coupling term in the Hamiltonian. These equations are solved in the supplementary material.19 The solution provides the changes of the ˜ Fourier-Laplace transforms C(ω) of the time correlation functions C(t) caused by the solute-solvent coupling. The main result is given by the simple equation τ C˜ C˜ s = 0 (0)0 . (0) τs C˜ 0 C˜ s
(4)
Here, τ 0 and τ s are the (Debye) relaxation times of the two (0) modes without coupling and C˜ 0,s are the corresponding correlation functions when the coupling is switched off. Further, (0) are the changes in the correlation funcC˜ 0,s = C˜ 0,s − C˜ 0,s tions when the coupling is introduced. For typical conditions of solutes exceeding in size the solvent one has τ 0 /τ s 1. Equation (4) then indicates that the perturbation introduced by the solute-solvent coupling to the time correlation function affects more the solvent than the solute. To produce a specific example of such change, we consider the alteration of the loss spectrum of the solvent when the relaxation of the solute is described by a Debye process. In order to model the change of the solute relaxation caused by the solute-solvent coupling, we will assume that the relaxation time of the solute τ0sol in the solution is different from the relaxation time τ 0 in an imaginary configuration with no solute-solvent coupling. Note that only the coupling between the modes X0 and Xs is considered here, and it can be switched off by changing the solvent. In this model, Eq. (4) becomes C˜ s = C˜ s(0)
−1 τ0sol (1 − ξ ) 1 − iωτ0sol , τs
(5)
where ξ = τ0 /τ0sol is the relative change in the solute relaxation time. We will use this equation below in the analysis of the DLS response of the lysozyme solution to show that the simulation results are in a general agreement with this model. To simplify the problem even further, we next consider the case of a Debye relaxation of the solvent in C˜ s(0) , which is consistent with the Ornstein-Uhlenbeck stochastic process. The change in the loss spectrum of the solvent component,16 χs (ω) = βωRe[C˜ s ], when the coupling between X0 and Xs is switched on follows from the fluctuation-dissipation theorem16 ω¯ , (6) ks χs (ω) ¯ = κ (1 − ξ ) Re (1 − i ω)(1 ¯ − iκ ω) ¯
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ωτ FIG. 1. Change in the loss spectrum of the solvent χs (ω) caused by the solute-solvent coupling. Calculations are done according to Eq. (6). The two curves are obtained with κ = τ0sol /τs = 100 and ξ = τ0 /τ0sol = 0.6 (solid line) and 0.8 (dashed line).
where ω¯ = ωτs and κ = τ0sol /τs . It is easy to see that χs (ω) alters its sign (Fig. 1) at the frequency (7) ω−1 = τ0sol τs determined by the geometric mean of the solute and solvent relaxation times and thus positioned between their corresponding Debye peaks. If τ0sol > τ0 , i.e., the solute-solvent coupling leads to slowing down of the solute, χs (ω) is negative at the position of the solvent relaxation peak ωτ s = 1 (Fig. 1). The solutesolvent coupling then results in the loss of the solvent intensity, as is usually observed in solution experiments7, 8 and discussed in more detail below. At the same time, a positive peak appears near the solute characteristic relaxation frequency, thus increasing the spectral loss in that region. We also note that, depending on the solute, the position of the solvent relaxation band can either be unaffected by the solute or shift to usually a lower frequency.20 To account for the latter effect, the solvent relaxation time τ s needs adjusting within the present model or, alternatively, the force constant ks in Eq. (3) needs to become a function of the solute concentration. The main result of this simple model is the appearance of a relaxation peak characteristic of the solute in the loss spectrum of the solvent. If the solute relaxation is characterized by a sum of several relaxation processes, they all will contribute, with different weights, to the solvent loss spectrum. It is also possible that some of these processes will produce negative contributions to the loss, while others will increase the loss intensity. These relaxation peaks might be incorrectly assigned to new, slower dynamics of the hydration layer by experiments probing exclusively the relaxation of the solvent (such as NMR6 or time-resolved infra-red spectroscopy8 ). We now turn to the polarizability anisotropy (DLS) response of the hydrated lysozyme, followed by the response of the solution dipole moment.
J. Chem. Phys. 141, 22D501 (2014)
both the single particle polarizabilities of all molecules in the solutions and their DID polarizabilities. The polarizability of water is assigned to its oxygen with the principal component values α xx = 1.408, α yy = 1.497, and α zz = 1.417 Å3 with the water molecule assumed to be in the yz-plane.21 The polarizability of lysozyme is calculated from its atomic polarizabilities according to the Thole formalism.22–24 It involves the dipole-dipole interactions between the atomic induced dipoles such that the resulting polarizability of the solute is calculated by inverting the matrix of binary interactions of the induced dipoles at each instantaneous configuration of the system (see the supplementary material19 ). The total polarizability of the solution αβ was used to calculate the time autocorrelation function in the depolarizedscattering geometry, which includes only off-diagonal components 1 δ αβ (t)δ αβ (0), (8) C (t) = 6 α=β where α = x, y, z are the Cartesian components of the 2-rank Cartesian tensor. The corresponding normalized time correlation function becomes φ (t) = C (t)/C (0).
(9)
The time correlation function was used to define the OKE response function25 χ (t) = −β C˙ (t), where β = 1/(kB T) is the inverse temperature. The response function can be FourierLaplace transformed to connect to the loss function via the classical limit of the fluctuation-dissipation relation17 χ (ω) = βC (0)Re[ωφ˜ (ω)]. ˜
(10)
Here, φ (ω) is the Fourier-Laplace transform of φ (t) in Eq. (9). A. TIP3P water
The DLS response of bulk water was calculated from separate MD simulations of TIP3P water at 300 K.19 The data were analyzed by assigning anisotropic polarizability21 to water, with the principal-axes components listed above and in the caption to Fig. 2. Three peaks of the experimental DLS spectrum of water7 are also seen in simulations (Fig. 2), although they are shifted to different extent depending on the force field used. The 0.4 0.3
χs''(ν)
Δχ ″(ω)
22D501-3
Experiment TIP3P SPC/E
0.2 0.1
III. POLARIZABILITY RESPONSE
The simulation protocol and calculation of the solvent and solution polarizabilities are described in the supplementary material.19 Here, we focus on the data analysis and the results. The second-rank tensor of the polarizability (t) of the solution containing TIP3P water and lysozyme includes
0.0 0.001
0.01
0.1
1
ν (THz)
10
FIG. 2. The DLS loss spectrum of water: experiment7 (points), MD simulations of TIP3P (blue line), and SPC/E29 (orange line) force field water. The principle-axes polarizability of water is: α xx = 1.408, α yy = 1.497, and α zz = 1.417 Å3 with the water molecule placed in the yz-plane.21
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J. Chem. Phys. 141, 22D501 (2014)
highest-frequency peak represents hindered rotations (librations) of the water molecules.26, 27 The two lower-frequency peaks are assigned the H-bond bending (lower frequency) and H-bond stretching (higher frequency).26, 27 Given that some of these motions carry some ballistic character, damped harmonic components have been used in fitting of the DLS spectra.28 We, therefore, fit the normalized time correlation function φs (t) = Cs (t)/Cs (0) to a linear combination of a Gaussian decay, a damped harmonic oscillator, and two exponential Debye terms. Two restrictions, φs (0) = 1 and φ˙ s (0) = 0, are additionally imposed, resulting in the following fitting function:29 φs (t) = e−ωg t
2 2
/2
+
2
Bi gi (t),
(11)
i=1
where Bi are the component amplitudes and gi (t) = e−αi t + (αi τh − 1)e−ωg t
2 2
/2
− αi τh e−t/τh cos ωh t. (12)
Seven free parameters obtained by fitting Eqs. (11) and (12) to MD data are listed in Table S1 in the supplementary material.19 We also show in Fig. 2 the DLS loss spectrum of SPC/E water reported previously by us.29 The TIP3P force field overestimates the intensity of the lower-frequency DLS peak compared to the SPC/E model, but gives a fair account of the experimental low-frequency tail of the loss spectrum. The oscillatory feature in the time correlation function φ (t), well resolved for SPC/E water, is nearly absent for TIP3P water (Fig. S1 in the supplementary material19 ). The result is the disappearance of the highest frequency peak in the DLS spectrum of TIP3P water in Fig. 2. B. Lysozyme solution
In order to quantify separate component contributions to the DLS response of the lysozyme solution, we first calculated the anisotropy polarizability relaxation of the lysozyme solute. The corresponding autocorrelation function 1 δ 0,αβ (t)δ 0,αβ (0) (13) C0 (t) = 6 α=β describes the relaxation of the single solute polarizability 0 (see the supplementary material19 for more details). The main contribution to this time correlation function comes from a single-exponential decay with the relaxation time τ0 = 1.6 ns (Table S2 in the supplementary material19 ) corresponding to rotational diffusion of lysozyme in solution. This relaxation time is obtained from the dynamics of the secondrank tensor. If the same rotational diffusion (given by the longest relaxation in a multi-exponential fit) is responsible for the relaxation of first- and second-rank observables, the relaxation time of the second-rank observable should be three times shorter.1 One can, therefore, compare τ0 to τ0M extracted from the time autocorrelation function of the first-rank dipole moment M0 (t) of lysozyme C0M (t) = δM0 (t) · δM0 (0).
(14)
The main component of this correlation function decays exponentially with the relaxation time τ0M = 9.1 ns (Table S2 in the supplementary material19 ), which is longer than 3τ0 . The anticipated scaling applies to rotation of objects with axial symmetry1 and can only approximately hold for solutes of irregular shape, such as lysozyme. The normalized function φ0 (t) = C0 (t)/C0 (0) is used to represent the dynamics of the solute polarizability in the correlation function of the solution φ (t) [Eqs. (8) and (9)]. The other two components are the relaxation of water [Eq. (11)] and all possible relaxation process that cannot be attributed to either the solute or water alone. They are lumped (t) in the equation into φcross φ (t) = B1 φ0 (t) + B2 φs (t) + φcross (t),
(15)
where Bi are the component amplitudes considered as fitting parameters. The usual constraints,17 φ (0) = 1 and φ˙ (0) = 0, are imposed on φ (t) when fitting the MD data. The subscript “cross” in the last term in Eq. (15) reflects the expectation that at least cross-correlations between the bulk and the solute will contribute to this term. An independent dynamic process, such as new dynamics of hydration shells, can be a part of (t) as well. This term was modeled by two decaying exφcross ponents (Ne = 2) when fitting the MD data (Table S2 in the supplementary material19 ) (t) φcross
=
Ne
Ai e−t/τi .
(16)
i=1
The fitting of Eqs. (15) and (16) to MD data yields only (t) (yellow in Fig. 3). The ∼6% of the intensity from φcross DLS response of the solution is, therefore, mostly a linear combination of the water and protein components. The cross component has the relaxation time of 0.6 ns (Table S2 in the supplementary material19 ) and a negative intensity. This latter result suggests solute-solvent cross correlations as the origin of this component in a general agreement with our model considerations above. Looking at the DLS response originating from the water component of the solution helps to further identify the source of this intermediate relaxation component.
0.4
χ''(ν)
22D501-4
Bulk water Total Solute Water Cross
0.2
0.0 0.00001
0.001
0.1
ν (THz)
10
FIG. 3. The DLS loss spectrum of lysozyme in TIP3P water. The overall spectrum (black line) is decomposed into the solute (green), water (blue), and cross (yellow) terms according to Eq. (15). The solute peak represents rotational diffusion of the solute. The dashed line refers to TIP3P water; solution and water loss spectra are equally normalized to produce unity for the integral of χ (ν)/ν.
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sol
χs ''(ν)
0.4
J. Chem. Phys. 141, 22D501 (2014)
Bulk water Total Solute Solvent Cross
0.3 0.2 0.1 0.0 0.00001
0.001
0.1
ν (THz)
10
FIG. 4. The DLS loss spectrum of TIP3P water in the solution of lysozyme. The overall spectrum (black line) is decomposed into the solute (green), water (blue), and cross (yellow) terms according to Eq. (15). The dashed line refers to TIP3P water; solution and water loss spectra are equally normalized to produce unity for the integral of χs (ν)/ν.
Figure 4 shows the DLS loss spectrum (χssol ) (ω) of the water component of the lysozyme solution. As in the case of the loss peak of the overall solution polarizability in Fig. 3, there is a drop in the water peak caused by the solute and the appearance of a peak identified with rotational diffusion of lysozyme. This relaxation component makes 13% of the normalized autocorrelation function (B1 , Table S2 in the supplementary material19 ). The splitting of the water spectrum into the bulk and solute components is achieved by applying Eq. (15). Similar to the analysis of φ (t) in Eq. (15), the cross term in the correlation function amounts to ∼5% of the intensity and is characterized by the relaxation time of 0.5 ns, also similar to the analysis of φ (t). The water, solute, and cross terms in the loss spectrum are color coded in Fig. 4. Even more revealing is the difference loss spectrum χs (ω) = (χssol ) (ω) − χs (ω) of water between the solution and the bulk. It is compared to Eq. (5) in Fig. 5. The agreement between the simple bilinear coupling model and the MD results is quite satisfactory given that water dynamics are more complex than assumed in the Ornstein-Uhlenbeck model. To produce a single relaxation time for water in Eq. (5), τ s was calculated from the equation ∞ φs (t)dt. (17) τs = 0
One gets τ s = 0.33 ps when this equation is applied to bulk TIP3P water (Eq. (11), Table S1 in the supplementary material19 ). This relaxation time is typically assigned to hindered rotations (librations) of water dipoles.30 The average
relaxation time, however, grows significantly to τssol = 210 ps when applied to water in the lysozyme solution due to slow water molecules from the hydration shells. Note that the timescales of 100–300 ps have been reported for the lifetimes of hydrogen-bonds of water bound to the protein.31, 32 In contrast, we are dealing here with the average relaxation time of all water molecules in the simulation box, and the results obtained for waters bound to the protein do not apply here. Formally, the appearance of the solute relaxation peak in the water spectrum (χssol ) (ω) is responsible for a significant increase of τ s . This phenomenon, therefore, will be reported by any spectroscopic technique targeting solvent in the solution. The present model, however, clearly shows that the slowing down cannot be directly interpreted as the appearance of a specific sub-ensemble of the solvation shell. It should instead be related to the coupling between the slow and fast subsystems in the system Hamiltonian. The slowing down does accompany the appearance of a sub-ensemble of hydration water, as we show below for the orientational structure of water interfacing lysozyme, but its existence cannot be established solely from the slower dynamics of water in the solution. Figure 5 also shows the application of Eq. (5) to produce χs (ω) assuming that the relaxation time of the solute is increased by 15% due to the lysozyme-water coupling (ξ = 0.85 in Eq. (5)). One gets a positive peak at the solute rotational frequency and a negative peak at the water relaxation frequency, consistent with the MD results. In addition, the difference spectrum χs (ω) crosses zero at 7 GHz, while Eq. (7) predicts the crossing point at 9 GHz. Given the simplicity of the model, it clearly captures the main physics of the alteration of the water spectrum introduced by the solute. IV. DIPOLAR RESPONSE A. Dipolar response of the lysozyme solution
The loss spectrum of the dipole moment of the lysozyme solution, which is experimentally recorded by dielectric spectroscopy,12, 13 is shown in Fig. 6. It is separated into the rotational diffusion of the solute’s dipole moment (φ0M (t) = C0M (t)/C0M (0), Eq. (14)), dipolar relaxation of bulk TIP3P M (t) water (φsM (t)), and an extra term given as φshell M (t), φ M (t) = B1 φ0M (t) + B2 φsM (t) + φshell
(18)
where, as above, Bi are the amplitudes of the solute and water components in the time correlation function. Model Simulations
-0.05 0.00001
Pure TIP3P Lys - Tot Lys - Solute Lys - Water Lys - Debye 1 Lys - Debye 2
0.4
0.00
χ''(ν)
Δχs''(ν)
0.05
0.001
0.1
ν (THz)
0.2
10
FIG. 5. The difference loss spectrum χs (ν) of the DLS response of water in solution and in the bulk. The loss spectra are from MD simulations (dotted line) and from Eq. (5) (solid line). The retardation factor of the solute rotational diffusion due to the lysozyme-water coupling ξ = τ0 /τ0sol = 0.85 was used in the calculations, the water relaxation time of τw = 0.33 ps was calculated from Eqs. (11) and (17).
0.0 0.00001
0.001
0.1
10
ν (THz) FIG. 6. Loss spectrum of the solution dipole moment. The spectrum is separated into the solute (lysozyme), water, and shell contributions according to Eq. (18).
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D. R. Martin and D. V. Matyushov
Pure TIP3P Total Solute Water Debye 1 Debye 2
0.4
χ''(ν) s
J. Chem. Phys. 141, 22D501 (2014)
0.2 0.0 0.00001
0.001
0.1
10
ν (THz) FIG. 7. Loss spectrum of the dipole moment of TIP3P water in the lysozyme solution. The spectrum is separated into the solute (lysozyme), bulk TIP3P water, and shell contributions according to Eq. (18).
Both correlation functions φsM (t) and φ0M (t) are fitted by two decaying exponents before applying Eq. (18). The solvent fit produces the relaxation times of 4.5 ps (74%) and 15 ps (26%) and the solute fit yields 0.7 ns (12%) and 9.1 ns (88%) (Tables S1 and S2 in the supplementary material19 ). The solute relaxation contributes only 7% to the overall φ M (t), while the contribution of bulk water is 28%. The contribuM (t) is, therefore, 65%, with the main relaxation tion of φshell time of 6 ps. This extra component significantly exceeds 6% contribution of the cross-correlation term to the polarizability correlation function in Eq. (15). The arguments presented below indicate that, in contrast to the DLS response, the orientational manifold of waters in the hydration shell presents a separate sub-ensemble, with its properties distinct from those of either the solute or solvent and not reducible to their cross correlations. The subscript “shell” in the last summand in Eq. (18) is used to stress this distinction. The dipolar loss spectrum of water in the solution is shown in Fig. 7. Here, in contrast to the DLS response, where the solute component is well distinguished in the overall response (Fig. 5), its contribution to the correlation function is only 0.5%. In contrast, 62% of the correlation function is given by the shell component dominated by an exponential decay with the relaxation time of 6 ps. A small relaxation component with a ∼100 ps relaxation time is consistent with dielectric δ-relaxation.12 The dominant 6 ps component is the same relaxation process, and roughly the same ampliM (t) of φ M (t) in tude, as is seen for the shell component φshell Eq. (18). These results are inconsistent with the bilinear coupling model that has described the cross correlations in the DLS response so well. They clearly suggest that orientations of water dipoles in the hydration shell of the protein must be attributed to a separate sub-ensemble including many water molecules and not reducible dynamically to a coupling between fast and slowly relaxing subsystems. Consistent with this view, the difference spectrum χs , obtained by taking the difference between the dipolar loss spectra of water in solution and of bulk water (Fig. S5 in the supplementary material19 ), is very different from what is shown in Fig. 5 for the DLS response. We now take a closer look at the dynamics of water molecules in the hydration shell of lysozyme.
vide the dipolar dynamics of hydration shells of increasing thickness around lysozyme. For that purpose the correlation function of the dipole moment Ms (a, t) of the water shell of thickness a from the protein’s van der Waals surface is correlated with the dipole moment Ms (0) of the entire water subsystem. Such correlation functions appear in linear response calculations17 of the dipole moment of the shell induced by a long-wavelength external field interacting with the dipoles of water only.33 In dielectric experiment, the latter condition requires measurements at frequencies exceeding the frequency of the solute rotational diffusion (τ0M )−1 to dynamically freeze the solute response.29 The correlation function we calculate here becomes CsM (a, t) = δMs (a, t) · δMs (0).
(19)
The normalized function φsM (a, t) = CsM (a, t)/CsM (a, 0) is fitted, as above, to a weighted sum of the solute and solvent correlation functions and an additional term represented by a sum of two decaying exponents (Table S3 in the supplementary material19 ) φsM (a, t) = B1 (a)φ0M (t) + B2 (a)φsM (t) +
2
Ai (a)e−t/τi (a) .
i=1
(20) Above a = 12 Å, the decay of the last term in Eq. (19) becomes essentially single-exponential and only one exponent, with the relaxation time of ∼6 ps, is used. For the first hydration layer, about 2% of shell waters are strongly bound to the protein and follow its rotational dynamics with the relaxation time of ∼9 ns. The bulk dynamics is absent, and a part of the shell not rotating with the protein (∼12%) shows an additional slow component with the relaxation time of ∼1.2 ns. This component, in turn, disappears beyond the shell thickness of ∼9 Å and is replaced with a single relaxation with the 6 ps decay. The average relaxation time ∞ φsM (a, t)dt, (21) τ (a) = 0
which accommodates a weighted average of all these processes, is shown by circles in Fig. 8. It decays with increasing shell thickness to the limit of ∼6 ps within the range of a 20 Å. A relatively weak decay of the average relaxation time with increasing the shell size is a signature of a deep propagation of the orientational structural perturbation produced in 300
(ps)
22D501-6
200 100 0 0
10
20
30
40
a (Å) B. Dipolar response of hydration shells
To additionally highlight the distinctions between the dynamics of the polarizability and the dipole moment, we pro-
FIG. 8. The average (Eq. (21)) dipolar relaxation time obtained from the dynamics of the dipole moment of the water shells of thickness a measured from the van der Waals surface of lysozyme [Eq. (19)].
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J. Chem. Phys. 141, 22D501 (2014)
χ0s
0.05 0.00 -0.05 -0.10 4
8
12
16
20
a (Å) FIG. 9. Susceptibility χ 0s (a) [Eq. (22)]. The dotted line marks the zero value of χ 0s (a), which is reached at a 7.5 Å.
the water layers by the protein.15, 29 The structural effect of the protein on water is reflected by static dipolar correlations. To illustrate the spatial range of these correlations, Fig. 9 shows the cross-correlation of the lysozyme dipole moment with the dipole moment of the shell of thickness a. Specifically, we calculate the static susceptibility χ0s (a) =
β δM0 · δMs (a), 3vs Ns (a)
(22)
where Ns (a) is the number of water molecules, with the molecular volume vs , within the shell. The cross correlation extends to 12 Å into the bulk. Moreover, the correlation is negative for the closest shells, but alters its sign with the shell growth. Note that χ 0s is expected to be negative in dielectric theories for macroscopically large shells (a → ∞).15 Breaking this prediction is just another manifestation of specific orientational structure of water around proteins. On a more qualitative level, water molecules in the first hydration shell face a mosaic of positive and negative surface residues often establishing strong H-bonds with the nearest water molecules. The H-bond structure of bulk water is broken in the first hydration layer, and the positions and orientations of the surface waters are driven by the local interactions with the surface residues. There is, therefore, little correlation of the surface waters with the overall multipole of the solute (dipole in Eq. (22)). This correlation builds up only slowly in the outer hydration layers, and this slow build-up of the dielectric response is responsible for the long range character of χ 0s (a). The results for the relaxation times of hydration layers reported in Fig. 8 are significantly different from a similar analysis done for the solution of a N-Acetyl-leucine-methylamide dipeptide (NALMA) in SPC/E water.29 Just a small increase, ∼10%, of the dipolar relaxation time in the first solvation shell compared to the bulk was reported for the longest exponential decay. Further, τ (a) was found to slightly decrease compared to the bulk due to a compensation between slowing down of the first layer by a slight speed-up of the second layer. No long relaxation times in the nanoseconds range could be observed for NALMA, in stark contrast to lysozyme. There are obviously qualitative differences between the dynamics of hydration shells of a relatively small and weakly polar NALMA dipeptide and lysozyme. The difference can be attributed both to the difference in polarity of the surface groups
and to significantly different sizes of their hydration shells: Ns (4 Å) = 48 for NALMA and 624 for lysozyme (Ns (3 Å) = 423). It is only in the case of the protein that there is a sufficient number of shell molecules to form a sub-ensemble with new dynamical and statistical properties. The hydration shell sub-ensemble will not be detected by experiments probing the statistics and dynamics of collective density fluctuations (translational motions). In addition, only a weak dynamic signature of bound waters (∼1% of the amplitude), with the relaxation time of ∼100 ps characteristic of the dielectric δ-relaxation,12 appears in the dynamics of the overall dipole moment of solution recorded by dielectric spectroscopy (Fig. 6). The rest of the signal is the solute tumbling and shell relaxation that is very close in the relaxation time to bulk water (4.5 ps in the bulk vs. 6 ps in the solution). In experimental fitting, it can potentially be assigned to a shifted “solvent” peak. Observables that probe the collective orientational dynamics of water separately have a better chance of sensing the shell waters and their specific dynamics. Along these lines, Stokes shift dynamics of optical dyes attached to the protein do demonstrate relaxation times in the range of nanoseconds.34–36
V. CONCLUSIONS
Water interfacing proteins needs to be distinguished as a separate sub-ensemble displaying properties distinct from the bulk. Hydration shells can be considered as “biological water” potentially affecting biological function in ways not deducible from either bulk water or from solvation properties of small model solutes.37 What we stress here is that one has to be extremely careful not to generalize this statement to all properties of the protein hydration shell and specify which parameters and corresponding observables require the notion of “biological water.” In particular, we show that the dynamics of the collective density fluctuations of the hydration shells as probed by depolarized light scattering reflect the coupled dynamics of the slow solute and fast solvent and do not require a separate sub-ensemble. The density perturbations produced by solutes in the solvent in general and by proteins in particular are local and do not propagate far from the solute surface into the bulk.38, 39 The orientational structure and dynamics of the shell carry dramatically different physics when probed with collective variables affected by many-body correlations of the shell dipoles. Proteins significantly distort the orientational structure of bulk water due to a high density of oppositely charged surface residues and surface sites providing H-bonds. Restoring the orientational bulk structure takes a relatively long distance from the protein surface into the bulk. The layer in which the bulk structure is restored involves a significant number of water molecules and needs separation into a subensemble of “biological water.” This sub-ensemble carries dynamical and statistical signatures not reducible to either the solute or the bulk. Depending on the property considered, the perturbed layer can propagate from two to five hydration shells into the bulk at the normal temperature. The new orientational structure of the shell dipoles thus emerges from a
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fairly large sub-ensemble of correlated dipoles characterized by a broken network of bulk-like hydrogen bonds. ACKNOWLEDGMENTS
This research was supported by the National Science Foundation (NSF) (CHE-1213288). CPU time was provided by the National Science Foundation through XSEDE resources (TG-MCB080116N). 1 B.
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