Revival of the Intermolecular Nuclear Overhauser Effect for Mapping

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Revival of the Intermolecular Nuclear Overhauser Effect for Mapping Local Protein Hydration Dynamics Daniel Braun, Michael Schmollngruber, and Othmar Steinhauser J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b01013 • Publication Date (Web): 07 Jul 2017 Downloaded from http://pubs.acs.org on July 11, 2017

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The Journal of Physical Chemistry Letters

Revival of the Intermolecular Nuclear Overhauser Effect for Mapping Local Protein Hydration Dynamics Daniel Braun, Michael Schmollngruber, and Othmar Steinhauser∗ Department of Computational Biological Chemistry, University of Vienna, Währinger Straße 17, 1090 Vienna, Austria E-mail: [email protected]

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Almost three decades ago, the intermolecular nuclear Overhauser effect (NOE) between protein and water protons was found to be a suitable tool to study the local hydration dynamics at the protein-water interface. 1 Besides improvements to make the measurements more sensitive, 2,3 the method was soon advanced with the aid of analytical models which suggested that mutual translational diffusion of protein and water (rather than, e.g., rotation of water molecules) plays a dominant role in the NOE relaxation process. 4 This mutual diffusion is naturally connected to the time that a water molecule spends at a specific site of the protein surface. 4–10 The present theoretical work based on molecular dynamics (MD) simulation, despite considering characteristic features of the intermolecular NOE, surprisingly, leads to a similar interpretation of the experiment as these early studies on the topic. In particular, we report on a remarkable correlation between the protein-water NOE and local water residence times, challenging the more recent but widespread opinion that the intermolecular NOE is unable to measure local hydration dynamics 11–16 due to its long-range nature. The quantity of prime interest 4,6,7 is the ratio σL /σR between NOE cross-relaxation rates in the laboratory frame σL = 0.6J(2ν0 ) − 0.1J(0)

(1)

σR = 0.3J(ν0 ) + 0.2J(0)

(2)

and rotating frame

with ν0 , the spectrometer frequency in MHz. The spectral density function (SDF) J(ν) is the Fourier cosine transform of a time correlation function (TCF) G(t)

J(ν) = 2 · K ·

Z

∞

dt cos(2πνt) G(t)

(3)

0

with the dipolar coupling constant K in Å6 s−2 . Here, G(t) is the sum of all TCFs of a specific P spin I of the protein with one of the spins S of all water molecules in the system S GIS (t) 3



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In practice, σL /σR can take on values from 1, representing fast dynamics, to -0.5, representing slow dynamics. We have calculated σL /σR between 629 protons of the protein ubiquitin (UBQ) and water protons from MD simulations with a total length of 1 µs (for computational details on simulation and analysis please refer to the Supporting information). Fig. 1a shows σL /σR (at a spectrometer frequency ν0 of 600 MHz) mapped on the surface of UBQ (for a 360o view, cf. Fig. S1). A large part of the protein surface exhibits positive σL /σR values, mostly between 0.4 and 0.8, suggesting overall fast hydration dynamics. Only few sites show negative σL /σR , indicating slow to very slow hydration dynamics. In order to test the hypothesis in question whether the intermolecular NOE is able to represent local hydration dynamics, we compare σL /σR to locally resolved residence times of water at the protein surface

τRES =

Z

∞

dt 0

P D w

nw (0) · nw (t) D E 2 nw (0)

E

(6)

with the sum over all water molecules w and the binary function

nw (t) =

   1, if rIw (t) ≤ 4Å

(7)

  0, if rIw (t) > 4Å, where rIw (t) is the distance between a specific protein proton (analogous to spin I) to the center-of-mass of a water molecule (analogous to spin S). Fig. 1b shows τRES mapped on the protein surface and is, indeed, congruent with σL /σR (also compare Fig. S1 and Fig. S2). This means that, from a theoretical standpoint, σL /σR is very well able to represent local hydration dynamics, i.e. local residence times τRES .

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Having established the correlation between the experimental observable σL /σR and local residence times, we provide an appropriate model function

model τRES =

hν −1 1 i−1 1.5 0 − 1 − 105 σL /σR + 0.5 τP

(8)

model with τRES and τP in ps and ν0 in MHz. It is derived in a consistent way from the logistic

function (for details please refer to the Supporting information) and also includes a correction for the overall tumbling of the protein, modulating NOE relaxation via the angular part of eq. (4) in the form of the second-rank Legendre polynomial relaxation time τP (3000 ps in this work). τP can be determined via 15 N NMR relaxation or employing hydrodynamic models. 17,18 From eq. (8) we also see that residence times longer than τP cannot be determined, 10 which is relevant in cases of very small proteins or very long residence times. Scatter plots of σL /σR vs. τRES (cf. Fig. S3) at different spectrometer frequencies ν0 from 400 to 1000 MHz confirm the suitability of our model function eq. (8). The Pearson correlation coefficient of model log(τRES ) vs. log(τRES ) for surface protons (cf. Fig. S5) is 0.95.

Looking at the distribution of τRES at the UBQ surface (ranging from 20 ps to > 3 ns; cf. Fig. 1, Fig. S4, Tab. S1), we observe a high heterogeneity of hydration dynamics. More precisely, most hydration sites exhibit fast τRES (below 100 ps), while a small fraction is retarded by up to two orders of magnitude or more. This is in accordance with previous computational studies 19–21 describing the hydration dynamics of ubiquitin via single-particle rotation of water molecules. In particular, the analysis in Ref. 21 is based on the very same simulation system and reports on the nuclear quadrupole relaxation of water around UBQ. Observations in the computational studies, including this one, are perfectly consistent with previous experimental results. 14,22 Besides the well-known 22,23 position of a water molecule found in the crystal structure of UBQ (site A in Fig. 1), we find a second highly retarded site (site B in Fig. 1) involving the side-chains of Asp21, Asp25 and Lys29. In a previous MD study using a completely

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different force field, 24 the authors reported a similar site involving Asp21, but neither Asp25 nor Lys29. We observe that the interaction with the side chain of Lys29 is crucial for a maintained residence of the water molecule. Furthermore, a third retardation site with considerably shorter τRES is marked as site C in Fig. 1. In general, we confirm that the topology of the protein surface is a crucial factor for τRES (cf. Ref. 21 and therein). The fact that the characteristic distribution of residence times is represented by σL /σR across the whole protein surface is remarkable considering the widespread opinion that σL /σR is unable to map local hydration dynamics due to the influence of long-range interactions. 11–16 This view originates from the fact that the intermolecular NOE measures cross-relaxation rates between spin I and many spins S and, moreover, that both the number of spins S and the NOE relaxation time increase with r2 . 11–13,25,26 Hence, the intermolecular NOE characteristically differs 27,28 from the intramolecular NOE, particularly with respect to its much slower convergence over r. Indeed, for a long time, this has not been considered in the interpretation of experiments. 1,3,5,6,9 On the other hand, up until now, analysis of the proteinwater NOE has been restricted to analytical models, 4,11,13,28,29 which might lack the necessary level of detail and accuracy and be prone to overinterpretation. 9,28,29 Therefore, the need for a corresponding analysis on the basis of MD simulation has long been recognized. 9,28,29 In order to assess the influence of long-range dipolar interactions on the intermolecular NOE and why it does not obscure the representation of local hydration dynamics by σL /σR , a distance resolution of the calculated NOE quantities is needed. Therefore, first we resolve the NOE TCF into contributions from spherical shells at different distances r from the protein reference spin I as G(r, t) =

X S

δ r − rIS (0) GIS (t).

7


(9)


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As explained above, the amplitude A of the total TCF cancles out in the ratio σL /σR . However, the spatially resolved amplitude

A(r) =

  P   S δ(r − rIS )   

1 , r6

if r ≤ 30Å

4πρ r2 r16 ,

if r > 30Å

(10)

is crucial for the relative weighting of contributions at different r. Furthermore, for each spherical shell we can calculate a normalized TCF g(r, t) = G(r, t)/A(r) , and with that the spatially resolved NOE relaxation time

τNOE (r) =

 R    ∞ dt g(r, t) , 0

  

if r ≤ 30Å

(11)

r2 /(6DT ), if r > 30Å.

While contributions to the intermolecular NOE up to 30 Å are calculated directly from the MD trajectory, its asymptotic behavior is expected to be described by established analytical formulae. 28 Accordingly, above r = 30 Å we calculate A(r) from the average number density of water protons ρ (66.5·10−3 Å

−3

in this work) and τNOE (r) from the bulk water translational 2

diffusion coefficient DT (2.7 · 10−5 Å s−1 in this work). In Fig. 2, we see A(r) and τNOE (r) resolved in 1 Å bins, below and above 30 Å (indicated by a vertical dashed line). At small r (close to the protein), there exists a large spread of both A(r) and τNOE (r) among different spins I, indicating highly site-specific structure and dynamics. On the other hand, this spread decreases fast with increasing distance. Starting from r ∼ 15 Å, all protein spins I experience the same interactions with water spins S, regarding both amplitude and dynamics of the NOE time correlation function. Moreover, in this region, the NOE data calculated from MD simulation already behaves according to the analytical formulae used for the calculation of the asymptotic region, resulting in a smooth transition between numerical and analytical data. This allows us to calculate contributions to the intermolecular NOE at a high level of detail and accuracy in the close-range region 8


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Figure 2: Left: A(r), eq. (10), where black and grey circles represent surface and non-surface protons, respectively. Right: τNOE (r), eq. (11) for surface protons; the color axis shows the correlation between τNOE and τRES . Both A(r) and τNOE (r) are depicted in bins of ∆r = 1 Å. The vertical dashed line at 30 Å indicates the transition between data calculated from the MD trajectory and the asymptotic description. and at the same time up to any desired distance. On the basis of eqs. (9) to (11), radial contributions to the spectral density function (omitting prefactors) are given by

J(r, ν) = A(r)

 R    ∞ dt cos(2πνt) g(r, t), if r ≤ 30Å 0   

τNOE (r) , 1+(2π ν τNOE (r))2

(12)

if r > 30Å,

where in the asymptotic region r > 30Å we approximate the SDF with the commonly used Lorentzian function. Fig. 3a shows the SDF of an example protein spin Q31-HA (cf. Tab. S2). Cumulative contributions to the SDF imply a dominance of long-range interactions to the protein-water NOE (similar as Fig. 2 in Ref. 11). Close-range interactions up to 6 Å internuclear distance only account for ∼20% of the total J(0) and even spins beyond 33 Å make a significant contribution. This shows, similar to previous analytical models, 11–13 that the influence of long-range dipolar interactions with many water protons cannot be neglected in the interpretation of the intermolecular NOE. 9



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Figure 3: Spectral density functions J(ν) at different internuclear distances (eq. (12)) a) cumulative J(ν) (in analogy to Fig. 2 in Ref. 11) of an example protein spin Q31-HA (cf. Tab. S2) normalized by its total J(0); b) mean and standard deviation of J(ν) (normalized by the same value) of all protein protons. Please note the large spread at small internuclear distances vs. the marginal spread at larger distances. However, this scenario is not representative of the whole set of protein spins I. In Fig. 3b, we see that long-range contributions remain more or less uniform for all protein spins, while in the close-range part, we observe enormous heterogeneity and much larger contributions than in Fig. 3a. This is grounded in both close-range structure and dynamics. Regarding the structural component, the underlying quantity of interest is the pair correlation function 26 between protein spin I and all water spins S. A typical property of this pair correlation function is its high initial peak. Although this is extremely important for the estimation of close-range contributions to the NOE, except in Ref. 29, it was not considered in previous analytical models analyzing the protein-water NOE. 11–13,28 Furthermore, we find that the location of spin I on the protein and its accessibility by water determine the onset of the pair correlation function, which also affects the intermolecular NOE via A(r). This can be seen in Fig. 2 upon comparison of surface and non-surface protons represented by green and open yellow circles, respectively. For small r, A(r) of surface protons is tendentially larger than that of non-surface protons. Since A(r) determines the relative contribution of 10


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J(r, ν) to the complete SDF (cf. eq. (12)), non-surface protons carry larger contributions from longer distance r, as compared to surface protons. Since in the asymptotic region the relaxation time of the internuclear vector is a function of r2 (cf. eq. (11) and Fig. 2), this ultimately results in tendentially lower values of σL /σR . 11,13,28 On the other hand, since the aim is to interpret σL /σR in terms of residence times, it is important that the concept of water residence appears already misguided in places where the protein is little or not at all in direct contact with water molecules. This is best shown by explicit calculation of residence times for non-surface protons, which gives values that tend to be unreasonably small or even zero (cf. Fig. S4). The reason for this is that contacts (defined via eq. (7) in this work), if ever achieved, remain brief. As a consequence, we argue that interpretation of σL /σR should in general 30 be limited to surface protons (cf. Fig. S5).

Figure 4: Distance resolved σL /σR of surface protons at a binwidth of 1 Å: a) σL /σR at distance r according to eq. (13); the vertical dashed line at 30 Å indicates the transition between data calculated from the MD trajectory and the asymptotic description (cf. eq. (12)). b) Cumulative value of σL /σR over r according to eq. (14). Finally, σL /σR is resolved into contributions from spherical shells via

σL /σR (r) =

0.6J(r, 2ν0 ) − 0.1J(r, 0) . 0.3J(r, ν0 ) + 0.2J(r, 0)

11


(13)


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In Fig. 4a, we see σL /σR (r) for surface protons up to 100 Å from the protein. Similar to A(r) and τNOE (r) (cf. Fig. 2), the initial spread of σL /σR (r) decreases fast over distance r, again allowing the transition between numerical and analytical description. The spread of σL /σR (r) at small r (close to the protein) in Fig. 4a correlates well with the corresponding residence time τRES at the reference spin I (similar to τNOE (r) in Fig. 2), as can be seen by the color axis. In order to see how this correlation is altered by more distant contributions, we calculate the ratio of the running integrals of σL (r) and σR (r) over r up to a distance R as RR

dr 0.6J(r, 2ν0 ) − 0.1J(r, 0) σL /σR (R) = R0 R , dr 0.3J(r, ν ) + 0.2J(r, 0) 0 0

(14)

where the integral over r, in fact, is the discrete sum in steps of ∆r = 1Å, which is shown in Fig. 4b. We observe that the systematic correlation between σL /σR and τRES , which is established in the vicinity of the protein, persists over the entire observed range up to 100 Å, where the curves are fully converged. In fact, already after ∼20 Å, no significant change of σL /σR can be seen. In contrast, for non-surface protons (cf. Fig. S6) a considerable change of σL /σR might be observed even after ∼40 Å from the reference spin I, depending on its accessibility by water. The low accessibility of non-surface protons leads to a small A(r) at close distance r (cf. Fig. 2), resulting in a larger relative weigthing of long-range contributions. In conclusion, the experimental observable σL /σR is very well able to represent local residence times of hydration water at the protein surface. The influence of long-range interactions does not obscure the representation of characteristic local hydration dynamics. This can be explained by several properties of the protein-water NOE and especially the underlying molecular system that were not accounted for in previous analytical model theories. Restricting the discussion to surface protons, as explained above, the close-range structural distribution of protein and water protons favors local contributions of water dynamics in the

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signal. Still, long-range contributions have a considerable influence on the signal. However, this influence only changes the absolute values of σL /σR but not their order and thus not their systematic relation to τRES , which is established in the vicinity of the protein (cf. Fig. 4b). Therefore, we developed a simple model function (eq. (8)), which allows experimenters to calculate local water residence times on the protein surface from the experimentally accessible observable σL /σR .

Acknowledgement This work was supported by the FWF Austrian Science Fund (project No. P23494) as well as the Faculty of Chemistry, University of Vienna.

Supporting Information Available The Supporting Information is available free of charge on the ACS Publications website at DOI: - Methods (simulation and analysis) - Derivation of the model function eq. (8) - Supplementary figures and tables

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(3) Otting, G.; Liepinsh, E.; Farmer, B. T.; Wüthrich, K. Protein hydration studied with homonuclear 3D1H NMR experiments. J. Biomol. NMR 1991, 1, 209–215. (4) Otting, G.; Liepinsh, E.; Wüthrich, K. Protein hydration in aqueous solution. Science 1991, 254, 974–980. (5) Clore, G. M.; Bax, A.; Wingfield, P. T.; Gronenborg, A. M. Identification and Localization of Bound Internal Water in the Solution Structure of Interleukin 1, 6 by Heteronuclear Three-Dimensional’H Rotating-Frame Overhauser 15N-’H Multiple Quantum Coherence NMR Spectroscopy. Biochemistry 1990, 29, 5671–5676. (6) Wüthrich, K.; Otting, G.; Liepinsh, E. Protein hydration in aqueous solution. Faraday discuss. 1992, 93, 35–45. (7) Otting, G.; Liepinsh, E.; Wuethrich, K. Polypeptide hydration in mixed solvents at low temperatures. J. Am. Chem. Soc. 1992, 114, 7093–7095. (8) Brunne, R.; Liepinsh, E.; Otting, G.; Wüthrich, K.; Van Gunsteren, W. Hydration of proteins: a comparison of experimental residence times of water molecules solvating the bovine pancreatic trypsin inhibitor with theoretical model calculations. J. Mol. Biol. 1993, 231, 1040–1048. (9) Otting, G.; Liepinsh, E. Protein hydration viewed by high-resolution NMR spectroscopy: implications for magnetic resonance image contrast. Acc. Chem. Res. 1995, 28, 171–177. (10) Otting, G. NMR studies of water bound to biological molecules. Prog. Nucl. Magn. Reson. Spectrosc. 1997, 31, 259–285. (11) Halle, B. Cross-relaxation between macromolecular and solvent spins: The role of longrange dipole couplings. J. Chem. Phys. 2003, 119, 12372–12385.

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(12) Halle, B. Protein hydration dynamics in solution: a critical survey. Philos. Trans. R. Soc., B 2004, 359, 1207. (13) Modig, K.; Liepinsh, E.; Otting, G.; Halle, B. Dynamics of protein and peptide hydration. J. Am. Chem. Soc. 2004, 126, 102–114. (14) Mattea, C.; Qvist, J.; Halle, B. Dynamics at the protein-water interface from 17 O spin relaxation in deeply supercooled solutions. Biophys. J. 2008, 95, 2951–2963. (15) Nucci, N. V.; Pometun, M. S.; Wand, A. J. Mapping the hydration dynamics of ubiquitin. J. Am. Chem. Soc. 2011, 133, 12326–12329. (16) Nucci, N. V.; Pometun, M. S.; Wand, A. J. Site-resolved measurement of water-protein interactions by solution NMR. Nat. Struct. Mol. Biol. 2011, 18, 245–249. (17) Schneider, D. M.; Dellwo, M. J.; Wand, A. J. Fast internal main-chain dynamics of human ubiquitin. Biochemistry 1992, 31, 3645–3652. (18) Blake-Hall, J.; Walker, O.; Fushman, D. Characterization of the overall rotational diffusion of a protein from 15 N relaxation measurements and hydrodynamic calculations. Protein NMR Techniques 2004, 139–159. (19) Fogarty, A. C.; Laage, D. Water dynamics in protein hydration shells: the molecular origins of the dynamical perturbation. J. Phys. Chem. B 2014, 118, 7715–7729. (20) Duboué-Dijon, E.; Laage, D. Comparative study of hydration shell dynamics around a hyperactive antifreeze protein and around ubiquitin. J. Chem. Phys. 2014, 141, 22D529. (21) Braun, D.; Schmollngruber, M.; Steinhauser, O. Rotational dynamics of water molecules near biological surfaces with implications for nuclear quadrupole relaxation. Phys. Chem. Chem. Phys. 2016, 18, 24620–24630.

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(22) Denisov, V. P.; Halle, B. Protein hydration dynamics in aqueous solution: a comparison of bovine pancreatic trypsin inhibitor and ubiquitin by oxygen-17 spin relaxation dispersion. J. Mol. Biol. 1995, 245, 682–697. (23) Vijay-Kumar, S.; Bugg, C. E.; Cook, W. J. Structure of ubiquitin refined at 1.8 A resolution. J. Mol. Biol. 1987, 194, 531. (24) Tian, J.; García, A. E. Simulations of the confinement of ubiquitin in self-assembled reverse micelles. J. Chem. Phys. 2011, 134, 06B603. (25) Gabl, S.; Steinhauser, O.; Weingärtner, H. From Short-Range to Long-Range Intermolecular NOEs in Ionic Liquids: Frequency Does Matter. Angew. Chem. 2013, 125, 9412–9416. (26) Braun, D.; Steinhauser, O. The intermolecular NOE is strongly influenced by dynamics. Phys. Chem. Chem. Phys. 2015, 17, 8509–8517. (27) Abragam, A.; Carr, H. The principles of nuclear magnetism. 1961. (28) Brüschweiler, R.; Wright, P. E. Water self-diffusion model for protein-water NMR cross relaxation. Chem. Phys. Lett. 1994, 229, 75–81. (29) Frezzato, D.; Rastrelli, F.; Bagno, A. Nuclear Spin Relaxation Driven by Intermolecular Dipolar Interactions: The Role of Solute- Solvent Pair Correlations in the Modeling of Spectral Density Functions. J. Phys. Chem. B 2006, 110, 5676–5689. (30) It should be mentioned that protein protons near water molecules buried inside the protein (not topic of this work) are also expected to have meaningful σL /σR .

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Revival of the Intermolecular Nuclear Overhauser Effect for Mapping

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