Resolving anisotropic distributions of correlated vibrational motion in protein hydration water Matthias Heyden Citation: The Journal of Chemical Physics 141, 22D509 (2014); doi: 10.1063/1.4896073 View online: http://dx.doi.org/10.1063/1.4896073 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 influence of water on protein properties J. Chem. Phys. 141, 165104 (2014); 10.1063/1.4900500 Hofmeister anionic effects on hydration electric fields around water and peptide J. Chem. Phys. 136, 124501 (2012); 10.1063/1.3694036 Far-infrared spectroscopy on free-standing protein films under defined temperature and hydration control J. Chem. Phys. 136, 075102 (2012); 10.1063/1.3686886 Vibrational spectroscopy of water in hydrated lipid multi-bilayers. II. Two-dimensional infrared and peak shift observables within different theoretical approximations J. Chem. Phys. 135, 164506 (2011); 10.1063/1.3655671 Thermal breaking of spanning water networks in the hydration shell of proteins J. Chem. Phys. 123, 224905 (2005); 10.1063/1.2121708
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THE JOURNAL OF CHEMICAL PHYSICS 141, 22D509 (2014)
Resolving anisotropic distributions of correlated vibrational motion in protein hydration water Matthias Heyden Max-Planck-Institut für Kohlenforschung, Theoretical Chemistry, 45470 Mülheim an der Ruhr, Germany
(Received 11 July 2014; accepted 8 September 2014; published online 3 October 2014) In this study, we analyze correlations of vibrational motion on the surface of a small globular protein and in its hydration shell. In contrast to single particle hydration water dynamics, which are perturbed by interactions with the protein solute only in the first few hydration layers, we find that correlated, collective motions extend into the surrounding solvent on a 10 Å length scale, specifically at farinfrared frequencies below 100 cm−1 . As a function of frequency, we analyze the distribution of correlated longitudinal motions in the three-dimensional environment of the protein solute, as well as in the vicinity of different protein-water interfaces. An anisotropic distribution of these correlations is observed, which is related to specific protein-water vibrations and interactions at the interfaces, as well as flexibilities of solvent exposed sites. Our results show that coupling of protein and water dynamics leaves a three-dimensional imprint in the collective dynamics of its hydration shell, and we discuss potential implications for biomolecular function, e.g., molecular recognition and binding, and the dynamical coupling of proteins to their native solvation environment. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4896073] I. INTRODUCTION
Water is an essential component of living organisms and the natural solvent for manifold biochemical processes taking place in them.1 It is found as a crucial part of biomolecular structures,2, 3 to participate directly in enzymatically catalyzed chemical reactions,4 and its unique solvent properties stabilize native protein structures and lubricate their dynamics.5 A detailed description of solute-water interactions is needed to understand not only static or thermodynamic properties of proteins and other biomolecules, e.g., the stability of their native conformation, specificity of target molecule binding and aggregation,6, 7 but also their dynamics. Proteinwater interactions do not provide a constant background for the potential energy surface of a solvated protein, but instead fluctuate themselves on timescales characteristic of dynamical processes in bulk and hydration water. This results in coupled protein and solvent dynamics, which, for example, give rise to coinciding dynamical transition temperatures in solvated proteins and their hydration water, observed in neutron scattering and molecular dynamics (MD) simulations.8, 9 The observation of a correlated temperature dependence of dynamical processes in proteins and their solvent on a multitude of timescales has coined the term of solvent slaving,10 which suggests that many biologically relevant dynamical processes in proteins are dominated by the dynamics of the surrounding solvent. Arguably, protein and solvent dynamics are subject to mutual influences, however, the observations indicate an intimate coupling of protein and water dynamics. We aim to understand the role of this dynamical coupling for the biological function of proteins and enzymes in their respective native environment, as well as in non-native environments in which enzymes are often employed to catalyze a desired reaction.11 Nuclear magnetic resonance experiments and molecular dy-
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namics simulations12 recently led to the conclusion that fast atomic fluctuations on pico- to nanosecond timescales, which we note are likely to be directly influenced by solvent dynamics occurring on similar timescales, facilitate much slower micro- to millisecond dynamics linked to biological function. Dynamical processes, which depend on each other over a hierarchy of timescales,12 also observed in the context of solvent slaving,10 are key in understanding the dynamical role of the solvent for biochemical processes, which have evolved for a particular solvation environment. At room temperature, timescales of relevant dynamical processes in bulk water range from 10’s of femtoseconds (fs) for fast librational motions of water molecules, 100’s of fs for hydrogen bond vibrations,13 1–10 picoseconds (ps) corresponding to rearrangement dynamics in the water hydrogen bond network,14 and approximately 10 ps for the collective total dipole relaxation of water.15 These dynamic processes are perturbed in protein hydration water relative to bulk. The fast, sub-picosecond vibrational motions of the hydrogen bond network can increase in frequency due to strong protein-water interactions,16, 17 while hydrogen bond rearrangements and water molecule rotational relaxation are slowed down in the first hydration layer.17–19 In addition, dielectric spectroscopy observes relaxation modes of the collective total dipole moment in protein solutions, which are attributed to protein hydration water with an increased relaxation time relative to bulk.20 Studies of the dynamical perturbation of protein hydration water provide a direct probe for coupled proteinwater dynamics under native conditions, e.g., without the need to perturb protein or solvent dynamics externally. Therefore, numerous experiments have been employed to study this perturbation, ranging from nuclear magnetic resonance and relaxation18, 21, 22 to femtosecond resolved
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fluorescence Stokes shifts,23–25 terahertz,26–30 and dielectric spectroscopy,20, 31 in combination with constantly growing numbers of molecular dynamics simulations.14, 16, 17, 32–38 It is worth noting, that the influence of biomolecular solutes on hydration water dynamics is likely to be dependent on the underlying dynamic process being probed. Recently, we have pointed out differences in particular between single particle and collective dynamics.35 Single particle dynamics, e.g., water molecule rotation probed by spin relaxation18 or extracted from simulations,17 hydrogen bond dynamics and water diffusion,14, 32, 33 as well as vibrational spectra obtained from velocity auto correlations (vibrational density of states, VDOS),16, 17, 35 exhibit non-bulk behavior only in the first 1–2 hydration layers. However, experimental probes that report on collective properties, e.g., fluctuations of total dipole moments probed in terahertz and dielectric spectroscopy, and collective vibrational modes observed in incoherent scattering, may observe more long-ranged soluteinduced perturbations,26, 35, 36 depending on the delocalization and the collective character of the probed motions. In our previous study, we describe an analysis of time cross correlations of atomic velocities from atomistic molecular dynamics simulations to study specifically the collective character of coupled protein-water vibrations in the hydration water of a small globular protein at far-infrared and terahertz frequencies.35 Particularly at frequencies below 100 cm−1 , we observed correlated vibrational motion of atoms on the protein surface and hydration water molecules with separation distances of 8– 10 Å between them. The correlations could be traced back to propagating, collective modes originating from vibrations of water molecules in the first hydration shell against the protein surface, e.g., protein-water hydrogen bonds in the presence of hydrogen bond donor and acceptor sites of solvent exposed protein residues.35 We showed that the properties of these collective modes, e.g., intensity and propagation velocity, are influenced by protein-water interactions at solvent-exposed hydrophobic and hydrophilic interfaces, resulting in an inhomogeneous, long-ranged influence of the protein solute on vibrational dynamics in its hydration water.35 Here, we extend this analysis to allow for a three-dimensional, site-specific resolution of correlated vibrations between protein surface atoms and hydration water. In doing so, we observe the heterogeneity and distribution of protein-water vibrations at the solute-solvent interface as a function of frequency, as well as the spatial extent of long-ranged correlations of the resulting collective motions that propagate in the hydration water hydrogen bond network. II. METHODS
The study presented here is based on MD simulations of the solvated λ∗ 6-85 -repressor DNA-binding domain, an engineered model protein studied previously in terahertz spectroscopy experiments26 and simulations.35, 36 See the supplementary material for details on the protein and the simulation protocol.39 To analyze the coupling of vibrational motion between the protein and its hydration water, we utilized a generalization of the standard expression of the vibrational density
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of states, which describes the vibrational spectrum of a system via the Fourier transform of mass-weighted time autocorrelation functions of atomic velocities, v(t)dt, (1) I (ω) = eiωt mv(0)v(t)dt = eiωt ˜v(0) √ with the weighted atomic velocity v˜ = mv and brackets describing ensemble averaging. As introduced in Ref. 35, we extend this description by including cross correlations of the atomic velocities, resolved in time and space, (2) I (ω, ri , rj ) = eiωt ˜vi (0, ri )˜vj (t, rj )dt, with the position vectors of atom i described by ri . Collective vibrations and long-ranged correlations between the motions of atoms are predominantly found in the far-infrared spectral range below 400 cm−1 ,40–42 where contributions from heavy atoms are dominating the vibrational spectrum. Therefore, we focus our analysis on velocities of water oxygens, v˜ O , and heavy (non-hydrogen) protein atoms, v˜ P , to analyze the correlations of vibrational motion in the protein and its hydration water. The practical use of Eq. (2) requires a proper localization of the atoms whose velocities are being sampled and correlated. Otherwise, the distance vector r between them is not well defined during the time window needed to sample the correlation function, which is at least the inverse of the lowest frequency to be observed. This requirement is largely fulfilled for the protein atoms. Translational and rotational motions of the protein can be removed from coordinates and velocities prior to the analysis via RMSD (root mean squared deviations) fitting of the protein backbone to a common reference structure. However, diffusive motion of water results in insufficient localization of hydration water oxygens to sample correlated vibrational motion in the far-infrared with spatial resolution, except for very few tightly bound water molecules with residence times beyond 10 ps. Following our previous work,35, 43, 44 we introduce a localized, smooth density of water oxygen atomic velocities, ρ v˜ (t, r) =
n
δ[r − ri (t)]˜vi (t, ri (t))
i
≈
n i
1 |r − ri (t)|2 v˜ i (t, ri (t)), (3) exp − (2π σ 2 )3/2 2σ 2
summing over all n water oxygens in the system. In this expression, we approximate the exact velocity density, characterized by infinitely sharp delta functions, δ[r − ri (t)], by smooth three-dimensional Gaussians. These Gaussian functions blur the position of the individual atoms, depending on the parameter σ ; however, they simplify sampling this velocity density and its fluctuations at any given point in the system. Setting σ to 0.4 Å allows proper sampling of fluctuations of ρ v˜ at any position r in the protein hydration shell, while still ensuring dominating contributions from single particles at any given time. During the observation time window, water molecules will diffuse in and out of the spatial region surrounding a given position r at which the smooth density ρ v˜ is sampled. The time-dependent fluctuations of ρ v˜ at this position will therefore not always be caused by the
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same molecules. However, with the chosen value of σ the contributions to the smooth density of atomic velocities ρ v˜ will be dominated by single molecules at a given time. Hence, the fluctuations of the local velocity density, ρ v˜ , still resemble single particle vibrations.35 Therefore, for practical considerations, we replace v˜ j (t, rj ) in Eq. (2) by the density of weighted water oxygen velocities ρ v˜ ,OW (t, r) sampled at position r, and v˜ i (0, ri ) by the weighted atomic velocity of the closest heavy atom of the protein surface v˜ P (0, rP ) at its averaged position rP . In this study, we focus particularly on the components of velocities/velocity densities parallel to the normal vector of the protein-water interface, which are obtained via projection on the vector r = r − rP . The resulting longitudinal correlated vibrations were previously shown to be particularly sensitive to the chemical properties of the protein surface.35 r r ρ v˜ ,OW (t, r)· dt. I (ω, r) = eiωt v˜ P (0, rP )· |r| |r| (4) The brackets denote an ensemble average over time origins and the analyzed trajectories. The densities ρ v˜ ,OW (t, r) were evaluated from 100 microcanonical trajectories on a 32 × 32 × 32 cubic grid with a grid constant of 1.5 Å, defined in a reference coordinate system, which follows the translational and rotational motion of the protein. The spectrum of the cross correlation functions between water oxygen velocity densities and velocities of the closest protein atom, i.e., their components parallel to the surface normal, were computed using a relation based on the convolution theorem,45 iωt iωt −iωt e A(t)dt e B(t)dt , e A(0)B(t)dt = (5) where we imply a non-complex time dependent property B(t). Under equilibrium conditions, ensemble averaged time cross correlation functions are symmetric, i.e., ˜vP (0, rP ) ρ v˜ ,OW (t, r) = ˜vP (0, rP )ρ v˜ ,OW (−t, r). While numerically, this criterion is not exactly fulfilled by cross correlation functions obtained from simulations, this behavior can be imposed by symmetrization resulting in a non-complex Fourier transform of the correlation function. Thus we can distinguish negative and positive correlations of the fluctuations of the atomic velocities in the cross correlation spectra. In particular for the case of longitudinal velocity components exam-
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ined here, negative correlations indicate a stretch-like vibration along the vector r, while positive correlations describe a concerted motion of protein atoms and water oxygens along the protein surface normal.35 In summary, the described approach allows us to analyze the spectra of correlated vibrations of protein surface atoms and hydration water oxygens along the protein surface normal with full spatial resolution in the solvation environment of the protein solute. To analyze spectral intensities at fixed distances to various parts of the protein surface, linear interpolation between nearest grid points was used to determine spectral intensities at 10 000 sampling points, equally distributed on the respective surface for each specific distance, which ranged from 2.0 Å to 10.0 Å in steps of 0.5 Å. Surfaces employed for this analysis included the full protein, i.e., the surface formed by all its heavy atoms, surfaces formed by heavy atoms of hydrophobic residues excluding the backbone, and hydrophilic residues including the backbone. From interpolated intensities at the sampling points for a specific distance |r| to a respective surface, average intensities (shown in Fig. 1) as well as their probability distributions on the surface (shown for the full protein surface in Fig. 2) were computed. The correlations resolved in the three-dimensional environment of the protein are shown in Fig. 3. For DNA and protein binding sites, inferred from RMSD alignment of the simulated protein to the crystal structure of the complex of the λ-repressor DNA-binding domain dimer and its target DNA sequence (PDBID: 1LMB),46 from 10 000 sampling points covering the entire surface of the protein at each distance, those were selected which overlap (distance < 1.5 Å) with heavy atoms of the bound DNA or the second λ-repressor monomer of the superimposed complex structure, respectively. Hydrogen bonds between individual protein donor and acceptor atoms and water molecules were analyzed via hyˆ h(t)/ ˆ ˆ h(0), ˆ h(0) with drogen bond correlation functions,13 h(0) ˆ evaluating to 1 if a hydrogen bond is intact the operator h(t) and zero otherwise. These functions were computed from the set of microcanonical trajectories for correlation times of up to 20 ps. Average hydrogen bond lifetimes are computed as the integral over these functions, resulting in a lower bound in cases where the correlation function does not fully decay to zero in this interval. Hydrogen bonds were considered intact for donor-acceptor distances 150◦ .13
III. RESULTS A. Correlated vibrations versus separation distance
The conformational stability of the protein allowed us to analyze correlated vibrational motion between atoms on the protein surface and its hydration water relative to an average structure without having to distinguish between conformational substates. In Figure 1, we present the results obtained from Eq. (4) for averaged spectra of cross correlation functions of protein surface heavy atom and hydration water oxygen velocity components parallel to the protein surface normal, resolved by the separation distance, i.e., the distance to the respective protein-water interface. This result corresponds closely to our previous study on this system,35 apart from the increased signal to noise ratio due to the analysis of a larger number of microcanonical trajectories and increased sampling time in each, particularly noticeable for the correlations in the vicinity of the hydrophobic surface, and the slightly deviating approach of sampling the correlations on a cubic grid first. We observe long-ranged correlations between vibrations of protein surface atoms and hydration water, extending 10 Å from the protein surface at frequencies below 100 cm−1 . We have shown previously, that the continuous areas of negative and positive correlations in Fig. 1 can be interpreted as propagating collective motions, which were analyzed by their dispersive behavior in reciprocal space, i.e., a linear relation between the frequency (at maximum intensity) and the inverse distance.35 The collective motions, which can be interpreted as local sound waves, propagate from the protein surface into surrounding hydration water (and vice versa). They provide crucial insights into the long-ranged character of the mechanical coupling of protein and solvent dynamics. Differences in intensity and propagation velocity of these collective modes in the hydration water of hydrophobic and hydrophilic parts of the protein indicate sensitivity of those long-ranged collective motions to chemical properties of the protein–solvent interface.35 The results obtained from our analysis can be compared in spirit to longitudinal current spectra CL (k,ω) = (ω2 /k2 )S(k,ω), obtained from the dynamic structure factor S(k,ω) observed in neutron and x-ray scattering experiments42, 47 as well as simulations,37, 38, 48 which describes collective motion via the spatial correlations of density fluctuations. However, our analysis based on correlations of atomic velocities in real space allows us to select specific motions extending from the protein-water interface into the surrounding hydration shell. The negative correlations in the first hydration layer (∼3 Å) are representative of vibrations of water molecules against the protein surface, e.g., of protein-water hydrogen bonds at frequencies of approximately 240 cm−1 at hydrophilic parts of the protein surface. Notably, the peak for negative intensities is shifted to below 200 cm−1 and increased distances of 3.5 Å for hydrophobic surfaces, where no direct protein-water hydrogen bonds are formed. These neg-
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ative correlations extend from the protein surface far into the hydration water, where with increasing distance the frequencies decrease proportional to the inverse distance in accord with their dispersive behavior.35 At low frequencies, positive correlations are observed, indicating a concerted motion of protein and water atoms in the same direction. In the first hydration shell, these correlations exhibit a maximum intensity at finite frequencies, approximately 30–40 cm−1 . The positive correlations also extend to large distances, where they become diffusive, i.e., exhibit a maximum at 0 cm−1 . We interpret these positive correlation intensities as vibrations of the protein and its first hydration layer against the surrounding water, while the long ranged correlations can be understood as a manifestation of hydrodynamic interactions.
B. Distribution of cross correlation intensities
While Figure 1 depicts averaged correlation intensities versus frequency and separation distance from the protein surface, we analyze the probability distributions underlying these averages in Figure 2. The distributions are primarily unimodal with a well-defined single maximum. Only for frequencies at which the transition between primarily positive and negative correlations occurs for a given distance to the protein, a distortion of the unimodal behavior is observed (e.g., for 80 cm−1 at 3 Å in Fig. 2(b)). At distances of 3 Å, i.e., in the first hydration layer, positive correlations at 40 cm−1 follow a wide symmetric distribution. With increasing distance to the protein surface this distribution becomes increasingly narrow, while approaching decreasing average intensities in accord with the results in Fig. 1. The negative intensities at higher frequencies follow a narrow distribution. At distances of 3 Å and 4 Å, the distributions at frequencies between 200 cm−1 and 300 cm−1 exhibit a pronounced tail toward more intense negative correlations, as shown in the upper two panels of Figs. 2(a) and 2(b). This indicates the existence of a distinguishable fraction of hydration water molecules in the first hydration layers, which exhibits more strongly correlated stretch vibrations (negative correlations) with bonding partners at the protein surface at these frequencies. At distances of 6 Å and 9 Å, negative correlations are observed for frequencies between ≈50 cm−1 and 120 cm−1 , while positive correlations remain below 50 cm−1 . With increasing distance to the protein surface the distributions of correlations become increasingly narrow and symmetric.
C. Collective vibrations resolved in 3D
After analyzing the statistical distributions of correlated motions, we proceed to analyze how the intensities of the correlated vibrations are distributed spatially in the threedimensional environment of the protein hydration water at selected frequencies in Figure 3. The volume enclosed by isosurfaces, indicating areas of increased positive and negative intensities of correlated vibrations, shrink and approach the protein surface with increasing frequencies, indicating their
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long-ranged character at low frequencies and localized motion at frequencies >100 cm−1 . At the lowest frequency depicted, 40 cm−1 , the whole protein is surrounded by positive correlations of atomic velocities in the hydration water and the protein surface, describing concerted motions of both, e.g., due to hydrodynamics. However, at 80 cm−1 , the frequency used in THz absorption experiments which detected a protein-induced change of the absorption coefficient for a >10 Å dynamical hydration shell,26 heterogeneous patterns of positive and negative correlations are observed. Positive correlations are restricted to the protein-water interface at this frequency, consistent with the averaged results in Fig. 1 and the distributions in Fig. 2. However, significant negative correlations extend 6–9 Å from the protein surface at 80 cm−1 and 4–7 Å at 120 cm−1 . Full three-dimensional spatial resolution of these long-ranged negative correlations is limited by the signal-tonoise ratio; however, its heterogeneous spatial distribution is evident. At frequencies >120 cm−1 , the positive correlations disappear, while negative correlations approach the protein surface, now corresponding to the correlated motion that results from directly bound water molecules vibrating against the protein surface. The frequency at which strongly negative correlations are found in this regime is therefore indicative of the strength of protein-water interactions at a specific site. This is furthermore convoluted by the localization of the involved atoms of the protein surface and bound surface waters. A functional group in a flexible sidechain, which is able to visit various conformations, will result in a reduced local correlation of atomic velocities. The correlations due to vibrations of non-covalent interactions between this particular functional group and solvating water molecules will be distributed over a larger volume. Therefore, we use the ratio of the hydrogen bond strength, characterized by the average lifetime of hydrogen bonds between protein atoms and water molecules, and the root mean squared fluctuations (RMSF) of the protein atoms to encode the color of the protein surface in Fig. 3. Qualitatively, a correlation between the presence of rigid hydrogen bond donors and acceptors on the protein surface and negative correlations is observed at frequencies ≥200 cm−1 , with increasing frequency indicating increased interaction strength. In addition, we observe a marked similarity in the distribution of positive correlations on the protein surface at low frequencies, e.g., 80 cm−1 , and negative correlations at high frequencies, e.g., 240 cm−1 , in the first hydration layer. This observation suggests that both types of correlations, positive intensities indicating a concerted motion in the same direction, as well as negative intensities describing a stretch vibration, depend on the direct protein-water interactions at short range. In general, we find that protein-water interactions lead to specific correlations in hydration water and protein atom motions in the studied frequency range, which are long-ranged below 100 cm−1 and heterogeneous and anisotropic in the protein environment. The data shown in Fig. 3 can therefore be interpreted as a frequency dependent three-dimensional fingerprint of the protein’s influence on hydration water collective dynamics.
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FIG. 4. DNA binding complex of the λ-repressor DNA binding domain dimer. (a) Simulated λ∗ 6-85 -repressor DNA binding domain in cartoon representation, and color gradient from blue to white to red indicating the sequence, after backbone Cα RMSD alignment with the complex crystal structure (chain 3 in PDBID: 1LMB); the perspective relative to the simulated λ∗ 6-85 -repressor DNA binding domain is equivalent to Fig. S1(b) in the supplementary material39 and Fig. 3; the second λ-repressor monomer is shown in cyan and the DNA single strands in yellow and magenta; (b) isosurface indicating strongly negative correlation intensities (160 cm−1 in the first hydration layers (3 Å and 4 Å) of hydrophobic surface residues, compensated by an increased probability to observe correlation intensities around zero. Instead the probability of negative correlations is increased between 100 cm−1 and 150 cm−1 , particularly at 3.5 Å (data not shown) and 4 Å. This is readily explained by the absence of protein-water hydrogen bonds at this surface and the presence of lower frequency vibrations due to weaker interactions between hydration water molecules and hydrophobic sidechains. Positive correlation intensities at frequencies below 100 cm−1 are redistributed from high intensities (3 arb.u.) toward lower intensities (1 arb.u.) around hydrophobic surfaces at 3 Å, which is attributed to a slightly increased preferential distance between hydrating water molecules in the first hydration shell and hydrophobic sidechains. At larger distances, e.g., at 6 Å, this shift leads to increased probabilities of high positive correlations (≈3 arb.u.) and low positive intensities (1 arb.u.) in the low-frequency regime below 50 cm−1 due to the propagating nature of the underlying motions. When we analyze the distribution of correlation intensities in the vicinity of the DNA and protein binding sites, we observe specific shifts that characterize correlated vibrational motion of hydration water and the protein surface at
these sites. In particular, a reduced intensity of negative correlations at typical protein-water hydrogen bond vibrational frequencies (≥200 cm−1 ) in the first hydration shell (3 Å) at these binding sites is observed, in accord to our discussion of Fig. 4. For the protein binding site, missing negative correlations are replaced by increased probabilities around zero intensity, i.e., the high frequency protein-water vibrations are absent, similar to the results for hydrophobic surfaces. Also here, an increase in the probability of negative correlations is observed at 100 cm−1 , indicative of weaker interactions instead. At the DNA-binding surface, a slightly different picture is found. The probability of strongly negative correlations (−1 to −3 arb u.) is decreased, while the probability of intermediate negative correlations is increased. This increase is observed over the entire frequency range between 100 cm−1 and 300 cm−1 at 3 Å and 4 Å distances. It is accompanied by a reduced probability of uncorrelated vibrations (zero intensity) in the first hydration shell in the frequency windows of 100 cm−1 to 150 cm−1 and 200 cm−1 to 300 cm−1 , characteristic for weak bonding and strong hydrogen bonds, respectively. This indicates that the overall number of interactions is increased at these frequencies. Due to the fact that we are sampling a spatial density of correlated vibrations, the increased
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FIG. 6. Quantitative analysis of the collective modes probed by the cross-correlations of protein and water velocities. (a) Linear behavior of the frequency at the maximum (negative) intensity of the cross-correlations shown in Fig. 1 for hydrophilic (blue) and hydrophobic (red) protein surfaces as a function of the inverse distance k = 2π /r. (b) Behavior of the mode intensity at the maximum (relative to the total maximum at 3.0 Å and 3.5 Å for the hydrophilic and hydrophobic case, respectively) versus separation distance between protein and water oxygen atoms. The grey areas indicate regions used for linear fits.
probabilities for intermediate correlation intensities suggest that at the DNA binding surface, water molecules vibrating against the protein are less localized, i.e., their correlated motion is distributed over a larger volume. Changes in the distribution of vibrational correlations are not restricted to the first hydration shell, but propagate to higher hydration shells. Accordingly, we observe changes in the distribution of correlations with increasing distance at lower frequency.35 However, these changes resemble in shape our observations directly at the surface. It is tempting to speculate that our observations resemble a specific property of these binding surfaces, which may play a role in molecular recognition. However, we cannot draw this conclusion from studying a single protein and without deeper mechanistic insights. On the other hand, our results do show that protein and hydration water motions are mechanically coupled, even on the 10 Å length scale, and that this mechanical coupling influences hydration water collective motion in a non-isotropic way. IV. DISCUSSION
Our study reports on a specific aspect of coupled proteinwater dynamics, collective vibrations, which we analyze via frequency dependent correlations of protein surface atom and hydration water oxygen velocities. Our results show, that collective motions extend at least 10 Å from the protein. The frequency at which these correlations are observed decreases with the inverse separation distance from the protein surface, which indicates linear dispersion of a propagating mode.35, 38 In Figure 6(a), we display the frequency of the maximum (negative) intensity of the velocity correlations shown in Fig. 1 as a function of the inverse distance (k = 2π /r) from the protein for the hydrophilic and hydrophobic surfaces. The linear dispersion, e.g., the k-dependence of the frequency of the underlying collective modes, is apparent in both cases and from the slope of linear fits between 0.8 Å−1 and 1.6 Å−1 , we obtain propagation velocities c of 3600 m/s and 2100 m/s from the dispersion relation c = dω/dk for the collective modes extending from hydrophilic and hydrophobic protein
surfaces into the hydration water, respectively. We note, that due to the increased signal-to-noise ratio and the focus on the linear dispersive region associated to “fast” sound,38, 48, 49 we obtain excellent agreement with the previously reported “fast” sound propagation velocity of 3800 m/s in protein hydration water at 300 K38 for collective longitudinal motion extending from hydrophilic protein surfaces. Additionally, the propagation velocities obtained from our simulations are in close agreement with recently reported neutron scattering experiments by Russo et al.50 on solutions of hydrophilic and hydrophobic peptides, which observed collective density fluctuations propagating with 3600 m/s and 2500 m/s, respectively. The significantly reduced propagation velocity obtained for collective longitudinal motions extending from solvent exposed hydrophobic protein surfaces into the hydration water, as previously reported,35 indicates differences in the underlying molecular motions, which may be further investigated. In Figure 6(b), we analyze the evolution of the maximum (negative) intensity of the modes analyzed in Fig. 6(a) with increasing distance to the protein. For the mode extending from the hydrophilic surface, we observe a strict linear decrease of the correlation intensity beyond a separation distance of 5 Å. At distances
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THE JOURNAL OF CHEMICAL PHYSICS 141, 22D509 (2014)
Resolving anisotropic distributions of correlated vibrational motion in protein hydration water Matthias Heyden Max-Planck-Institut für Kohlenforschung, Theoretical Chemistry, 45470 Mülheim an der Ruhr, Germany
(Received 11 July 2014; accepted 8 September 2014; published online 3 October 2014) In this study, we analyze correlations of vibrational motion on the surface of a small globular protein and in its hydration shell. In contrast to single particle hydration water dynamics, which are perturbed by interactions with the protein solute only in the first few hydration layers, we find that correlated, collective motions extend into the surrounding solvent on a 10 Å length scale, specifically at farinfrared frequencies below 100 cm−1 . As a function of frequency, we analyze the distribution of correlated longitudinal motions in the three-dimensional environment of the protein solute, as well as in the vicinity of different protein-water interfaces. An anisotropic distribution of these correlations is observed, which is related to specific protein-water vibrations and interactions at the interfaces, as well as flexibilities of solvent exposed sites. Our results show that coupling of protein and water dynamics leaves a three-dimensional imprint in the collective dynamics of its hydration shell, and we discuss potential implications for biomolecular function, e.g., molecular recognition and binding, and the dynamical coupling of proteins to their native solvation environment. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4896073] I. INTRODUCTION
Water is an essential component of living organisms and the natural solvent for manifold biochemical processes taking place in them.1 It is found as a crucial part of biomolecular structures,2, 3 to participate directly in enzymatically catalyzed chemical reactions,4 and its unique solvent properties stabilize native protein structures and lubricate their dynamics.5 A detailed description of solute-water interactions is needed to understand not only static or thermodynamic properties of proteins and other biomolecules, e.g., the stability of their native conformation, specificity of target molecule binding and aggregation,6, 7 but also their dynamics. Proteinwater interactions do not provide a constant background for the potential energy surface of a solvated protein, but instead fluctuate themselves on timescales characteristic of dynamical processes in bulk and hydration water. This results in coupled protein and solvent dynamics, which, for example, give rise to coinciding dynamical transition temperatures in solvated proteins and their hydration water, observed in neutron scattering and molecular dynamics (MD) simulations.8, 9 The observation of a correlated temperature dependence of dynamical processes in proteins and their solvent on a multitude of timescales has coined the term of solvent slaving,10 which suggests that many biologically relevant dynamical processes in proteins are dominated by the dynamics of the surrounding solvent. Arguably, protein and solvent dynamics are subject to mutual influences, however, the observations indicate an intimate coupling of protein and water dynamics. We aim to understand the role of this dynamical coupling for the biological function of proteins and enzymes in their respective native environment, as well as in non-native environments in which enzymes are often employed to catalyze a desired reaction.11 Nuclear magnetic resonance experiments and molecular dy-
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namics simulations12 recently led to the conclusion that fast atomic fluctuations on pico- to nanosecond timescales, which we note are likely to be directly influenced by solvent dynamics occurring on similar timescales, facilitate much slower micro- to millisecond dynamics linked to biological function. Dynamical processes, which depend on each other over a hierarchy of timescales,12 also observed in the context of solvent slaving,10 are key in understanding the dynamical role of the solvent for biochemical processes, which have evolved for a particular solvation environment. At room temperature, timescales of relevant dynamical processes in bulk water range from 10’s of femtoseconds (fs) for fast librational motions of water molecules, 100’s of fs for hydrogen bond vibrations,13 1–10 picoseconds (ps) corresponding to rearrangement dynamics in the water hydrogen bond network,14 and approximately 10 ps for the collective total dipole relaxation of water.15 These dynamic processes are perturbed in protein hydration water relative to bulk. The fast, sub-picosecond vibrational motions of the hydrogen bond network can increase in frequency due to strong protein-water interactions,16, 17 while hydrogen bond rearrangements and water molecule rotational relaxation are slowed down in the first hydration layer.17–19 In addition, dielectric spectroscopy observes relaxation modes of the collective total dipole moment in protein solutions, which are attributed to protein hydration water with an increased relaxation time relative to bulk.20 Studies of the dynamical perturbation of protein hydration water provide a direct probe for coupled proteinwater dynamics under native conditions, e.g., without the need to perturb protein or solvent dynamics externally. Therefore, numerous experiments have been employed to study this perturbation, ranging from nuclear magnetic resonance and relaxation18, 21, 22 to femtosecond resolved
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fluorescence Stokes shifts,23–25 terahertz,26–30 and dielectric spectroscopy,20, 31 in combination with constantly growing numbers of molecular dynamics simulations.14, 16, 17, 32–38 It is worth noting, that the influence of biomolecular solutes on hydration water dynamics is likely to be dependent on the underlying dynamic process being probed. Recently, we have pointed out differences in particular between single particle and collective dynamics.35 Single particle dynamics, e.g., water molecule rotation probed by spin relaxation18 or extracted from simulations,17 hydrogen bond dynamics and water diffusion,14, 32, 33 as well as vibrational spectra obtained from velocity auto correlations (vibrational density of states, VDOS),16, 17, 35 exhibit non-bulk behavior only in the first 1–2 hydration layers. However, experimental probes that report on collective properties, e.g., fluctuations of total dipole moments probed in terahertz and dielectric spectroscopy, and collective vibrational modes observed in incoherent scattering, may observe more long-ranged soluteinduced perturbations,26, 35, 36 depending on the delocalization and the collective character of the probed motions. In our previous study, we describe an analysis of time cross correlations of atomic velocities from atomistic molecular dynamics simulations to study specifically the collective character of coupled protein-water vibrations in the hydration water of a small globular protein at far-infrared and terahertz frequencies.35 Particularly at frequencies below 100 cm−1 , we observed correlated vibrational motion of atoms on the protein surface and hydration water molecules with separation distances of 8– 10 Å between them. The correlations could be traced back to propagating, collective modes originating from vibrations of water molecules in the first hydration shell against the protein surface, e.g., protein-water hydrogen bonds in the presence of hydrogen bond donor and acceptor sites of solvent exposed protein residues.35 We showed that the properties of these collective modes, e.g., intensity and propagation velocity, are influenced by protein-water interactions at solvent-exposed hydrophobic and hydrophilic interfaces, resulting in an inhomogeneous, long-ranged influence of the protein solute on vibrational dynamics in its hydration water.35 Here, we extend this analysis to allow for a three-dimensional, site-specific resolution of correlated vibrations between protein surface atoms and hydration water. In doing so, we observe the heterogeneity and distribution of protein-water vibrations at the solute-solvent interface as a function of frequency, as well as the spatial extent of long-ranged correlations of the resulting collective motions that propagate in the hydration water hydrogen bond network. II. METHODS
The study presented here is based on MD simulations of the solvated λ∗ 6-85 -repressor DNA-binding domain, an engineered model protein studied previously in terahertz spectroscopy experiments26 and simulations.35, 36 See the supplementary material for details on the protein and the simulation protocol.39 To analyze the coupling of vibrational motion between the protein and its hydration water, we utilized a generalization of the standard expression of the vibrational density
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of states, which describes the vibrational spectrum of a system via the Fourier transform of mass-weighted time autocorrelation functions of atomic velocities, v(t)dt, (1) I (ω) = eiωt mv(0)v(t)dt = eiωt ˜v(0) √ with the weighted atomic velocity v˜ = mv and brackets describing ensemble averaging. As introduced in Ref. 35, we extend this description by including cross correlations of the atomic velocities, resolved in time and space, (2) I (ω, ri , rj ) = eiωt ˜vi (0, ri )˜vj (t, rj )dt, with the position vectors of atom i described by ri . Collective vibrations and long-ranged correlations between the motions of atoms are predominantly found in the far-infrared spectral range below 400 cm−1 ,40–42 where contributions from heavy atoms are dominating the vibrational spectrum. Therefore, we focus our analysis on velocities of water oxygens, v˜ O , and heavy (non-hydrogen) protein atoms, v˜ P , to analyze the correlations of vibrational motion in the protein and its hydration water. The practical use of Eq. (2) requires a proper localization of the atoms whose velocities are being sampled and correlated. Otherwise, the distance vector r between them is not well defined during the time window needed to sample the correlation function, which is at least the inverse of the lowest frequency to be observed. This requirement is largely fulfilled for the protein atoms. Translational and rotational motions of the protein can be removed from coordinates and velocities prior to the analysis via RMSD (root mean squared deviations) fitting of the protein backbone to a common reference structure. However, diffusive motion of water results in insufficient localization of hydration water oxygens to sample correlated vibrational motion in the far-infrared with spatial resolution, except for very few tightly bound water molecules with residence times beyond 10 ps. Following our previous work,35, 43, 44 we introduce a localized, smooth density of water oxygen atomic velocities, ρ v˜ (t, r) =
n
δ[r − ri (t)]˜vi (t, ri (t))
i
≈
n i
1 |r − ri (t)|2 v˜ i (t, ri (t)), (3) exp − (2π σ 2 )3/2 2σ 2
summing over all n water oxygens in the system. In this expression, we approximate the exact velocity density, characterized by infinitely sharp delta functions, δ[r − ri (t)], by smooth three-dimensional Gaussians. These Gaussian functions blur the position of the individual atoms, depending on the parameter σ ; however, they simplify sampling this velocity density and its fluctuations at any given point in the system. Setting σ to 0.4 Å allows proper sampling of fluctuations of ρ v˜ at any position r in the protein hydration shell, while still ensuring dominating contributions from single particles at any given time. During the observation time window, water molecules will diffuse in and out of the spatial region surrounding a given position r at which the smooth density ρ v˜ is sampled. The time-dependent fluctuations of ρ v˜ at this position will therefore not always be caused by the
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same molecules. However, with the chosen value of σ the contributions to the smooth density of atomic velocities ρ v˜ will be dominated by single molecules at a given time. Hence, the fluctuations of the local velocity density, ρ v˜ , still resemble single particle vibrations.35 Therefore, for practical considerations, we replace v˜ j (t, rj ) in Eq. (2) by the density of weighted water oxygen velocities ρ v˜ ,OW (t, r) sampled at position r, and v˜ i (0, ri ) by the weighted atomic velocity of the closest heavy atom of the protein surface v˜ P (0, rP ) at its averaged position rP . In this study, we focus particularly on the components of velocities/velocity densities parallel to the normal vector of the protein-water interface, which are obtained via projection on the vector r = r − rP . The resulting longitudinal correlated vibrations were previously shown to be particularly sensitive to the chemical properties of the protein surface.35 r r ρ v˜ ,OW (t, r)· dt. I (ω, r) = eiωt v˜ P (0, rP )· |r| |r| (4) The brackets denote an ensemble average over time origins and the analyzed trajectories. The densities ρ v˜ ,OW (t, r) were evaluated from 100 microcanonical trajectories on a 32 × 32 × 32 cubic grid with a grid constant of 1.5 Å, defined in a reference coordinate system, which follows the translational and rotational motion of the protein. The spectrum of the cross correlation functions between water oxygen velocity densities and velocities of the closest protein atom, i.e., their components parallel to the surface normal, were computed using a relation based on the convolution theorem,45 iωt iωt −iωt e A(t)dt e B(t)dt , e A(0)B(t)dt = (5) where we imply a non-complex time dependent property B(t). Under equilibrium conditions, ensemble averaged time cross correlation functions are symmetric, i.e., ˜vP (0, rP ) ρ v˜ ,OW (t, r) = ˜vP (0, rP )ρ v˜ ,OW (−t, r). While numerically, this criterion is not exactly fulfilled by cross correlation functions obtained from simulations, this behavior can be imposed by symmetrization resulting in a non-complex Fourier transform of the correlation function. Thus we can distinguish negative and positive correlations of the fluctuations of the atomic velocities in the cross correlation spectra. In particular for the case of longitudinal velocity components exam-
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ined here, negative correlations indicate a stretch-like vibration along the vector r, while positive correlations describe a concerted motion of protein atoms and water oxygens along the protein surface normal.35 In summary, the described approach allows us to analyze the spectra of correlated vibrations of protein surface atoms and hydration water oxygens along the protein surface normal with full spatial resolution in the solvation environment of the protein solute. To analyze spectral intensities at fixed distances to various parts of the protein surface, linear interpolation between nearest grid points was used to determine spectral intensities at 10 000 sampling points, equally distributed on the respective surface for each specific distance, which ranged from 2.0 Å to 10.0 Å in steps of 0.5 Å. Surfaces employed for this analysis included the full protein, i.e., the surface formed by all its heavy atoms, surfaces formed by heavy atoms of hydrophobic residues excluding the backbone, and hydrophilic residues including the backbone. From interpolated intensities at the sampling points for a specific distance |r| to a respective surface, average intensities (shown in Fig. 1) as well as their probability distributions on the surface (shown for the full protein surface in Fig. 2) were computed. The correlations resolved in the three-dimensional environment of the protein are shown in Fig. 3. For DNA and protein binding sites, inferred from RMSD alignment of the simulated protein to the crystal structure of the complex of the λ-repressor DNA-binding domain dimer and its target DNA sequence (PDBID: 1LMB),46 from 10 000 sampling points covering the entire surface of the protein at each distance, those were selected which overlap (distance < 1.5 Å) with heavy atoms of the bound DNA or the second λ-repressor monomer of the superimposed complex structure, respectively. Hydrogen bonds between individual protein donor and acceptor atoms and water molecules were analyzed via hyˆ h(t)/ ˆ ˆ h(0), ˆ h(0) with drogen bond correlation functions,13 h(0) ˆ evaluating to 1 if a hydrogen bond is intact the operator h(t) and zero otherwise. These functions were computed from the set of microcanonical trajectories for correlation times of up to 20 ps. Average hydrogen bond lifetimes are computed as the integral over these functions, resulting in a lower bound in cases where the correlation function does not fully decay to zero in this interval. Hydrogen bonds were considered intact for donor-acceptor distances 150◦ .13
III. RESULTS A. Correlated vibrations versus separation distance
The conformational stability of the protein allowed us to analyze correlated vibrational motion between atoms on the protein surface and its hydration water relative to an average structure without having to distinguish between conformational substates. In Figure 1, we present the results obtained from Eq. (4) for averaged spectra of cross correlation functions of protein surface heavy atom and hydration water oxygen velocity components parallel to the protein surface normal, resolved by the separation distance, i.e., the distance to the respective protein-water interface. This result corresponds closely to our previous study on this system,35 apart from the increased signal to noise ratio due to the analysis of a larger number of microcanonical trajectories and increased sampling time in each, particularly noticeable for the correlations in the vicinity of the hydrophobic surface, and the slightly deviating approach of sampling the correlations on a cubic grid first. We observe long-ranged correlations between vibrations of protein surface atoms and hydration water, extending 10 Å from the protein surface at frequencies below 100 cm−1 . We have shown previously, that the continuous areas of negative and positive correlations in Fig. 1 can be interpreted as propagating collective motions, which were analyzed by their dispersive behavior in reciprocal space, i.e., a linear relation between the frequency (at maximum intensity) and the inverse distance.35 The collective motions, which can be interpreted as local sound waves, propagate from the protein surface into surrounding hydration water (and vice versa). They provide crucial insights into the long-ranged character of the mechanical coupling of protein and solvent dynamics. Differences in intensity and propagation velocity of these collective modes in the hydration water of hydrophobic and hydrophilic parts of the protein indicate sensitivity of those long-ranged collective motions to chemical properties of the protein–solvent interface.35 The results obtained from our analysis can be compared in spirit to longitudinal current spectra CL (k,ω) = (ω2 /k2 )S(k,ω), obtained from the dynamic structure factor S(k,ω) observed in neutron and x-ray scattering experiments42, 47 as well as simulations,37, 38, 48 which describes collective motion via the spatial correlations of density fluctuations. However, our analysis based on correlations of atomic velocities in real space allows us to select specific motions extending from the protein-water interface into the surrounding hydration shell. The negative correlations in the first hydration layer (∼3 Å) are representative of vibrations of water molecules against the protein surface, e.g., of protein-water hydrogen bonds at frequencies of approximately 240 cm−1 at hydrophilic parts of the protein surface. Notably, the peak for negative intensities is shifted to below 200 cm−1 and increased distances of 3.5 Å for hydrophobic surfaces, where no direct protein-water hydrogen bonds are formed. These neg-
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ative correlations extend from the protein surface far into the hydration water, where with increasing distance the frequencies decrease proportional to the inverse distance in accord with their dispersive behavior.35 At low frequencies, positive correlations are observed, indicating a concerted motion of protein and water atoms in the same direction. In the first hydration shell, these correlations exhibit a maximum intensity at finite frequencies, approximately 30–40 cm−1 . The positive correlations also extend to large distances, where they become diffusive, i.e., exhibit a maximum at 0 cm−1 . We interpret these positive correlation intensities as vibrations of the protein and its first hydration layer against the surrounding water, while the long ranged correlations can be understood as a manifestation of hydrodynamic interactions.
B. Distribution of cross correlation intensities
While Figure 1 depicts averaged correlation intensities versus frequency and separation distance from the protein surface, we analyze the probability distributions underlying these averages in Figure 2. The distributions are primarily unimodal with a well-defined single maximum. Only for frequencies at which the transition between primarily positive and negative correlations occurs for a given distance to the protein, a distortion of the unimodal behavior is observed (e.g., for 80 cm−1 at 3 Å in Fig. 2(b)). At distances of 3 Å, i.e., in the first hydration layer, positive correlations at 40 cm−1 follow a wide symmetric distribution. With increasing distance to the protein surface this distribution becomes increasingly narrow, while approaching decreasing average intensities in accord with the results in Fig. 1. The negative intensities at higher frequencies follow a narrow distribution. At distances of 3 Å and 4 Å, the distributions at frequencies between 200 cm−1 and 300 cm−1 exhibit a pronounced tail toward more intense negative correlations, as shown in the upper two panels of Figs. 2(a) and 2(b). This indicates the existence of a distinguishable fraction of hydration water molecules in the first hydration layers, which exhibits more strongly correlated stretch vibrations (negative correlations) with bonding partners at the protein surface at these frequencies. At distances of 6 Å and 9 Å, negative correlations are observed for frequencies between ≈50 cm−1 and 120 cm−1 , while positive correlations remain below 50 cm−1 . With increasing distance to the protein surface the distributions of correlations become increasingly narrow and symmetric.
C. Collective vibrations resolved in 3D
After analyzing the statistical distributions of correlated motions, we proceed to analyze how the intensities of the correlated vibrations are distributed spatially in the threedimensional environment of the protein hydration water at selected frequencies in Figure 3. The volume enclosed by isosurfaces, indicating areas of increased positive and negative intensities of correlated vibrations, shrink and approach the protein surface with increasing frequencies, indicating their
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long-ranged character at low frequencies and localized motion at frequencies >100 cm−1 . At the lowest frequency depicted, 40 cm−1 , the whole protein is surrounded by positive correlations of atomic velocities in the hydration water and the protein surface, describing concerted motions of both, e.g., due to hydrodynamics. However, at 80 cm−1 , the frequency used in THz absorption experiments which detected a protein-induced change of the absorption coefficient for a >10 Å dynamical hydration shell,26 heterogeneous patterns of positive and negative correlations are observed. Positive correlations are restricted to the protein-water interface at this frequency, consistent with the averaged results in Fig. 1 and the distributions in Fig. 2. However, significant negative correlations extend 6–9 Å from the protein surface at 80 cm−1 and 4–7 Å at 120 cm−1 . Full three-dimensional spatial resolution of these long-ranged negative correlations is limited by the signal-tonoise ratio; however, its heterogeneous spatial distribution is evident. At frequencies >120 cm−1 , the positive correlations disappear, while negative correlations approach the protein surface, now corresponding to the correlated motion that results from directly bound water molecules vibrating against the protein surface. The frequency at which strongly negative correlations are found in this regime is therefore indicative of the strength of protein-water interactions at a specific site. This is furthermore convoluted by the localization of the involved atoms of the protein surface and bound surface waters. A functional group in a flexible sidechain, which is able to visit various conformations, will result in a reduced local correlation of atomic velocities. The correlations due to vibrations of non-covalent interactions between this particular functional group and solvating water molecules will be distributed over a larger volume. Therefore, we use the ratio of the hydrogen bond strength, characterized by the average lifetime of hydrogen bonds between protein atoms and water molecules, and the root mean squared fluctuations (RMSF) of the protein atoms to encode the color of the protein surface in Fig. 3. Qualitatively, a correlation between the presence of rigid hydrogen bond donors and acceptors on the protein surface and negative correlations is observed at frequencies ≥200 cm−1 , with increasing frequency indicating increased interaction strength. In addition, we observe a marked similarity in the distribution of positive correlations on the protein surface at low frequencies, e.g., 80 cm−1 , and negative correlations at high frequencies, e.g., 240 cm−1 , in the first hydration layer. This observation suggests that both types of correlations, positive intensities indicating a concerted motion in the same direction, as well as negative intensities describing a stretch vibration, depend on the direct protein-water interactions at short range. In general, we find that protein-water interactions lead to specific correlations in hydration water and protein atom motions in the studied frequency range, which are long-ranged below 100 cm−1 and heterogeneous and anisotropic in the protein environment. The data shown in Fig. 3 can therefore be interpreted as a frequency dependent three-dimensional fingerprint of the protein’s influence on hydration water collective dynamics.
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FIG. 4. DNA binding complex of the λ-repressor DNA binding domain dimer. (a) Simulated λ∗ 6-85 -repressor DNA binding domain in cartoon representation, and color gradient from blue to white to red indicating the sequence, after backbone Cα RMSD alignment with the complex crystal structure (chain 3 in PDBID: 1LMB); the perspective relative to the simulated λ∗ 6-85 -repressor DNA binding domain is equivalent to Fig. S1(b) in the supplementary material39 and Fig. 3; the second λ-repressor monomer is shown in cyan and the DNA single strands in yellow and magenta; (b) isosurface indicating strongly negative correlation intensities (160 cm−1 in the first hydration layers (3 Å and 4 Å) of hydrophobic surface residues, compensated by an increased probability to observe correlation intensities around zero. Instead the probability of negative correlations is increased between 100 cm−1 and 150 cm−1 , particularly at 3.5 Å (data not shown) and 4 Å. This is readily explained by the absence of protein-water hydrogen bonds at this surface and the presence of lower frequency vibrations due to weaker interactions between hydration water molecules and hydrophobic sidechains. Positive correlation intensities at frequencies below 100 cm−1 are redistributed from high intensities (3 arb.u.) toward lower intensities (1 arb.u.) around hydrophobic surfaces at 3 Å, which is attributed to a slightly increased preferential distance between hydrating water molecules in the first hydration shell and hydrophobic sidechains. At larger distances, e.g., at 6 Å, this shift leads to increased probabilities of high positive correlations (≈3 arb.u.) and low positive intensities (1 arb.u.) in the low-frequency regime below 50 cm−1 due to the propagating nature of the underlying motions. When we analyze the distribution of correlation intensities in the vicinity of the DNA and protein binding sites, we observe specific shifts that characterize correlated vibrational motion of hydration water and the protein surface at
these sites. In particular, a reduced intensity of negative correlations at typical protein-water hydrogen bond vibrational frequencies (≥200 cm−1 ) in the first hydration shell (3 Å) at these binding sites is observed, in accord to our discussion of Fig. 4. For the protein binding site, missing negative correlations are replaced by increased probabilities around zero intensity, i.e., the high frequency protein-water vibrations are absent, similar to the results for hydrophobic surfaces. Also here, an increase in the probability of negative correlations is observed at 100 cm−1 , indicative of weaker interactions instead. At the DNA-binding surface, a slightly different picture is found. The probability of strongly negative correlations (−1 to −3 arb u.) is decreased, while the probability of intermediate negative correlations is increased. This increase is observed over the entire frequency range between 100 cm−1 and 300 cm−1 at 3 Å and 4 Å distances. It is accompanied by a reduced probability of uncorrelated vibrations (zero intensity) in the first hydration shell in the frequency windows of 100 cm−1 to 150 cm−1 and 200 cm−1 to 300 cm−1 , characteristic for weak bonding and strong hydrogen bonds, respectively. This indicates that the overall number of interactions is increased at these frequencies. Due to the fact that we are sampling a spatial density of correlated vibrations, the increased
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FIG. 6. Quantitative analysis of the collective modes probed by the cross-correlations of protein and water velocities. (a) Linear behavior of the frequency at the maximum (negative) intensity of the cross-correlations shown in Fig. 1 for hydrophilic (blue) and hydrophobic (red) protein surfaces as a function of the inverse distance k = 2π /r. (b) Behavior of the mode intensity at the maximum (relative to the total maximum at 3.0 Å and 3.5 Å for the hydrophilic and hydrophobic case, respectively) versus separation distance between protein and water oxygen atoms. The grey areas indicate regions used for linear fits.
probabilities for intermediate correlation intensities suggest that at the DNA binding surface, water molecules vibrating against the protein are less localized, i.e., their correlated motion is distributed over a larger volume. Changes in the distribution of vibrational correlations are not restricted to the first hydration shell, but propagate to higher hydration shells. Accordingly, we observe changes in the distribution of correlations with increasing distance at lower frequency.35 However, these changes resemble in shape our observations directly at the surface. It is tempting to speculate that our observations resemble a specific property of these binding surfaces, which may play a role in molecular recognition. However, we cannot draw this conclusion from studying a single protein and without deeper mechanistic insights. On the other hand, our results do show that protein and hydration water motions are mechanically coupled, even on the 10 Å length scale, and that this mechanical coupling influences hydration water collective motion in a non-isotropic way. IV. DISCUSSION
Our study reports on a specific aspect of coupled proteinwater dynamics, collective vibrations, which we analyze via frequency dependent correlations of protein surface atom and hydration water oxygen velocities. Our results show, that collective motions extend at least 10 Å from the protein. The frequency at which these correlations are observed decreases with the inverse separation distance from the protein surface, which indicates linear dispersion of a propagating mode.35, 38 In Figure 6(a), we display the frequency of the maximum (negative) intensity of the velocity correlations shown in Fig. 1 as a function of the inverse distance (k = 2π /r) from the protein for the hydrophilic and hydrophobic surfaces. The linear dispersion, e.g., the k-dependence of the frequency of the underlying collective modes, is apparent in both cases and from the slope of linear fits between 0.8 Å−1 and 1.6 Å−1 , we obtain propagation velocities c of 3600 m/s and 2100 m/s from the dispersion relation c = dω/dk for the collective modes extending from hydrophilic and hydrophobic protein
surfaces into the hydration water, respectively. We note, that due to the increased signal-to-noise ratio and the focus on the linear dispersive region associated to “fast” sound,38, 48, 49 we obtain excellent agreement with the previously reported “fast” sound propagation velocity of 3800 m/s in protein hydration water at 300 K38 for collective longitudinal motion extending from hydrophilic protein surfaces. Additionally, the propagation velocities obtained from our simulations are in close agreement with recently reported neutron scattering experiments by Russo et al.50 on solutions of hydrophilic and hydrophobic peptides, which observed collective density fluctuations propagating with 3600 m/s and 2500 m/s, respectively. The significantly reduced propagation velocity obtained for collective longitudinal motions extending from solvent exposed hydrophobic protein surfaces into the hydration water, as previously reported,35 indicates differences in the underlying molecular motions, which may be further investigated. In Figure 6(b), we analyze the evolution of the maximum (negative) intensity of the modes analyzed in Fig. 6(a) with increasing distance to the protein. For the mode extending from the hydrophilic surface, we observe a strict linear decrease of the correlation intensity beyond a separation distance of 5 Å. At distances
