The protein–water nuclear Overhauser effect (NOE) as an indirect microscope for molecular surface mapping of interaction patterns

Abstract

In this computational study, the intermolecular solute–solvent Nuclear Overhauser Effect (NOE) of the model protein ubiquitin in different chemical environments (free, bound to a partner protein and encapsulated) is investigated. Short-ranged NOE observables such as the NOE/ROE ratio reveal hydration phenomena on absolute timescales such as fast hydration sites and slow water clefts. We demonstrate the ability of solute–solvent NOE differences measured of the same protein in different chemical environments to reveal hydration changes on the relative timescale. The resulting NOE/ROE-surface maps are shown to be a central key for analyzing biologically relevant chemical influences such as complexation and confinement: the presence of a complexing macromolecule or a confining surface wall modulates the water mobility in the vicinity of the probe protein, hence revealing which residues of said protein are proximate to the foreign interface and which are chemically unaffected. This way, hydration phenomena can serve to indirectly map the precise influence (position) of other molecules or interfaces onto the protein surface. This proposed one-protein many-solvents approach may offer experimental benefits over classical one-protein other-protein pseudo-intermolecular transient NOEs. Furthermore, combined influences such as complexation and confinement may exert non-additive influences on the protein compared to a reference state. We offer a mathematical method to disentangle the influence of these two different chemical environments.

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1 Introduction

In order to understand the actual mechanisms underlying the function of biomolecules, we need to study the natural environment they operate in, i.e. aqueous solution^1,2 and confinement in a biological cell, surrounded by a high concentration of other solutes.^3,4 Many contemporary hydration dynamics measurement methods such as time-resolved fluorescence spectroscopy⁵ or ODNP have been used to study local hydration dynamics near globular,⁶ fibrous⁷ and membrane-bound⁸ proteins but require the insertion of an artificial probe (chromophore or stable radical) into the native protein type. In contrast, intermolecular protein–water NOEs provide an intrinsic probe to measure site-resolved hydration dynamics of proteins.^9,10 Another advantage of the protein NOE is that hydrogen atoms cover the entire surface of the protein, allowing for a comprehensive mapping of hydration dynamics. Taking the example of the popular model protein UBQ,^9–11 one can probe up to 629 hydrogen atoms (about 315 surface protons¹²) but only one intrinsic UV/Vis chromophore, TYR-59,^13–15 which is furthermore buried in the protein scaffold^16,17 and subject to intramolecular quenching mechanisms.^18,19 A list of the abbreviations used in this article is given in Table 1.

Table 1 List of abbreviations used in the article

DMAG	1-Decanoyl-rac-glycerol
LDAO	Lauryldimethylamine-N-oxide
NOE	Nuclear Overhauser effect
NOESY	Nuclear Overhauser effect correlation spectroscopy
NQR	Nuclear quadrupole resonance
ODNP	Overhauser dynamic nuclear polarization
RDF	Radial distribution function
RM	Reverse micelle
ROE	Rotating-frame nuclear Overhauser effect
ROESY	Rotating-frame nuclear Overhauser effect correlation spectroscopy
SDF	Spectral density function
TCF	Time-correlation function
UBQ	Ubiquitin

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In essence, the TCF relevant to dipole–dipole magnetization transfer NOE captures the average dynamics of the vectors r_IS joining a protein atom (typically hydrogen) with spin I and the water protons with spin S:

where r_IS is the distance between the two spins and θ_IS is the angle swept by the connecting vector during time t. The Fourier transform of G_IS(t) yields the SDF J(ω)whose linear combinations can be measured experimentally as the self- and cross-relaxation rates ρ and σ:²⁰with

σ_NOE(ω) = 0.6J(2ω) − 0.1J(0) (5)

σ_ROE(ω) = 0.3J(ω) + 0.2J(0) (6)

ρ(ω) = 0.6J(2ω) + 0.3J(ω) + 0.1J(0) (7)

where ω is the Larmor frequency in the homo-nuclear case. The intramolecular NOESY signal is exploited regularly in organic-chemical and biochemical structure elucidation due to its strict 1/r⁶ distance dependence. In the past, this inspired experimentalists to tacitly assume the 1/r⁶ short-range dependency for intermolecular NOEs as well.^21–25 However, in this case, one reference spin I interacts with many solvent spins S. For instance, using above σ_NOE relation and assuming exponential correlation decaya measured protein–water σ_NOE rate of 0.01 m s⁻¹ between an UBQ surface proton and water would yield a spin–spin vector relaxation time τ close to the P2 tumbling time of the protein itself and would thus refer to the extreme case of a water molecule sticking strongly to the surface.¹² It was shown that intermolecular NOEs lead to distance dependencies 1/rⁿ with n ranging from 1 to 3.^26,27 Using these long-ranged distance dependencies in above example, we obtain more realistic water residence times of tens of picoseconds.²⁸

In light of this difference, the explanatory qualities of intermolecular NOE studies have been called into question.^29–32 Halle first described the long-ranged nature of the intermolecular SDF in his groundbreaking work using simple step functions to describe the RDFs of interacting nuclei.²⁹ Halle's finding holds even when accounting for local structure by using more sophisticated model functions,³³ RDFs from MD simulations^26,34 or calculating the SDF directly from MD simulations.^27,34 While carrying chemically specific structural information, the actual NOESY signal is contaminated by many unspecific long-ranged spin interactions^29,33 with the range depending on the frequency of the spectrometer²⁶ which casts doubt on the locality of the intermolecular NOE.

However, more recent computational studies by our group have shown that while absolute NOE entities (σ_NOE, σ_ROE, ρ,…) are in fact long-ranged, relative observables like the ratio σ_NOE/σ_ROE correlate well with the actual site-specific hydration dynamics¹² due to relative error compensation eliminating chemically unspecific long-ranged contributions.³⁵ While the first hydration shell shows a mere overall retardation of a factor of about 5 around proteins^28,30,36,37 and 3 around interface walls,^38,39 NMR possesses the unique ability to separate contributions by chemical identity and reveal a surprisingly large range of dynamics. We reported comprehensive hydration data^12,35 mapped on each protein hydrogen and were thus able to point out three water pockets holding slow structural water in the protein UBQ as opposed to the two previously known ones.^37,40

Additionally, comparative NOE/ROE ratio calculations Δ_confined(σ_NOE/σ_ROE) of UBQ dissolved in diluted bulk water and UBQ in a living-matter mimic (macromolecular crowding by other proteins, encapsulation in a RM or both, hitherto referred to as confined system)

showed that the NOE/ROE ratio picks up the slow dynamics of water molecules captured interstitially between the measured protein and a foreign interface and is thus able to map the inherent heterogeneity of confined systems by measuring the same protein–water cross-peak of a protein hydrogen in a confined environment and in diluted bulk solution both with NOESY and ROESY, form the ratio and subtract them.³⁵ This encapsulation process corresponds to paths B → E symbolized by green arrows in Fig. 1.

Fig. 1 Overview of the six systems necessary to analyze the combined effect of complex formation (paths 1 → 3 and 2 → 3, purple arrows) and encapsulation (paths (B → E)1, (B → E)2, (B → E)3, green arrows). The combined effect is described either by pathway B(1, 2 → 3); (B → E)3 (complex formation prior to encapsulation) or, alternatively, by path (B → E)1, 2; E(1, 2 → 3) (encapsulation before complex formation).

Note that proteins in bulk solution are prone to chemical proton exchange and exchange-relayed NOE, making experimental interpretation difficult and dependent on assumptions when separating magnetization transfers into the actual dipole–dipole coupling signal and the exchange contributions on the time scale. Quite generally, protein hydration dynamics in bulk solutions are fast⁴¹ and may be difficult to measure, requiring e.g. hyperpolarized solvent NMR spectroscopy.⁴² Water dynamics in RMs, however, are slow in the high-field NMR⁴³ and hydrogen exchange is slowed down drastically.⁹

Typically, a σ_NOE/σ_ROE term varies from −0.5 to 1.¹² Hence, our ratio difference varies from −1.5 = −0.5 − (+1) (hydration dynamics at this site is much slower in confinement than in bulk) to 1.5 = +1 −(−0.5) (much faster in confinement than in bulk). In practice, the difference is limited up to ≈0 since solute and solvent molecular motion are retarded in confined systems compared to bulk solutions.^44,45

Instead of the ratio of two observables σ_NOE and σ_ROE, one may also measure just one observable per sample, the NOE enhancement, and form the difference.³⁵ This may be experimentally advantageous for various reasons, e.g. σ_ROE is often merely estimated,⁴⁶ though interpretation may be more complicated due to spin diffusion and hydrogen exchanges. See the ESI† document for the equivalence of using either the NOE to ROE ratio or the NOE enhancement.

Since the intermolecular solute–solvent NOE is able to map anisotropic hydration dynamics brought about by confinement effects, we raise the question whether it is also able to map hydration dynamics changes brought about by complex formations, e.g. between two proteins. These specific protein–protein interactions are important in many biological processes involving molecular recognition,⁴⁷ for instance enzymatic catalysis⁴⁸ and docking.⁴⁹

Thus we performed molecular dynamics simulations, allowing us to use explicit atomistic time-resolved data. Despite the high computational effort, we avoided the far more common analytical models and their intrinsic assumptions like strictly exponentially TCFs, which were experimentally found to be non-applicable to some molecular liquids.^51,52 We selected UBQ, a protein involved in many protein–protein complex formations in protein-degradation pathways.⁵³ UBQ is a small protein (76 amino acids) of near-spherical geometry and a stable conformation. Due to its robustness and availability, it is widely used in theoretical and experimental proof-of-concept studies, e.g.ref. 9 and 10. We selected the ubiquitin protein ligase Cbl-b⁵⁴ as its partner. They form a complex as depicted in the bottom row of Fig. 1.

2 Methods

This section details the computational experiments carried out. Fig. 2 summarizes the work-flows employed.

Fig. 2 Schematic representation of the MD/computational spectroscopy work-flows performed in the frame of this study.

2.1 Molecules and force field

The geometries of the three proteins studied were taken from the RCSB database:⁵⁵

• ligase: entry 2OOA⁵⁴

• ubiquitin: entry 1UBQ⁴⁰

• ligase–ubiquitin complex: entry 2OOB⁵⁴

The proteins were parametrized using appropriate CHARMM36 atom types.^56–59

We picked Berendsen's SPC/E water model⁶⁰ to represent the explicit solvent. This water model has been shown to reproduce experimental observables like dielectric relaxation,⁶¹ terahertz spectra,⁶² neutron scattering⁶³ and, most importantly, NMR data⁶⁴ better than the TIP family of water models. It was demonstrated to combine well with the CHARMM force field.⁶⁵

In the RM systems, we used isooctane as the immersion medium surrounding the RM. While this phase fills most of the simulation box in our systems, it is not relevant to the post-production analysis per se. Hence, both isooctane and the aliphatic tails of the surfactant were modelled as coarse-grained to save computational resources with one particle representing a CH, CH₂ or CH₃ group. For this aliphatic phase, we set the negligible atomic charges strictly to zero. All other parameters were selected from van Gunsterens GROMOS 45A3 force field which was designed specifically for aliphatic aggregates like lipids and micelles.⁶⁶ In practice, it has been shown to combine excellently with the polar phase described above, yielding stable RMs which accurately reproduce dielectric absorptions,⁶⁷ terahertz bands⁶⁸ and NQR relaxation times.⁶⁹

We picked a novel surfactant consisting of a 1 : 2 mixture of the zwitterionic LDAO and the monoglyceride DMAG. This mixture was demonstrated by Wand et al. to safely encapsulate a variety of proteins and even t-RNA, conserving the macromolecular structures faithfully.^70,71 The aliphatic tails were modelled analogously to isooctane described above, the polar heads of the LDAO and DMAG surfactants were modelled atomistically with appropriate CHARMM36 atom types^56–59 except for the charge, which was derived from quantum-chemical calculations. For details regarding the surfactant force field creation, we refer the reader to ref. 72.

Recent works showed that current biomolecular force fields are optimized to model simply one protein in aqueous solution, yielding considerable artifacts when using several proteins capable of interacting with one another.^73,74 These simulations lead to erroneous solvation energies⁷³ and exaggerated protein aggregation behaviour.⁷⁴ By extension, this defect is also true for RMs where the studied protein may interact with the surfactant wall. Best and Mittal pointed out that these errors can be counteracted fairly well when scaling the Lennard-Jones interaction of protein–water atom pairs by a factor λ:⁷³

In essence, one makes the protein surface stickier for water molecules, efficiently increasing hydration. Setting λ = 1.1 already improves the simulation accuracy considerably.⁷³ Thus, in order to avoid conclusions based on simulational artifacts, we created RM simulations using different levels of Mittals scaling λ = 1.1:

1. The RM systems with standard force field parameters (unscaled system)

2. RM systems with the protein–water interactions scaled up as proposed by Best and Mittal (half-scaled system)

3. As above, but with the water–surfactant interactions scaled up as well (fully scaled system)

Overall, we find that our results are consistent across all three levels of λ-scaling, thus we do not discuss this aspect further.

2.2 Simulation setup and trajectory production

Table 2 summarizes the systems used in the frame of this work. For each system, we created five independent replica using numerical seeds to ensure statistical stability in the following analysis. Furthermore, note that for the confined systems, we employed three different levels of λ-scaling as described above. This ensures independence from force field artifacts known to occur in simulations with multiple biological interfaces.^73,74

Table 2 Overview of the systems simulated. Both the ligase and the ubiquitin–ligase complex simulations were produced for the purposes of this study. The ubiquitin bulk system was created for this project as well, but the ubiquitin micelle simulations were reused from ref. 50

System	# H2O	# 2OOA	# 1UBQ	# 2OOB	# Na^+	# LDAO	# DMAG	# isooctane
Ligase bulk	6500	1	—	—	3	—	—	—
Ligase confined	1500	1	—	—	3	50	100	9000
Ubiquitin bulk	6500	—	1	—	—	—	—	—
Ubiquitin confined	1500	—	1	—	—	50	100	9000
Complex bulk	6500	—	—	1	3	—	—	—
Complex confined	1500	—	—	1	3	50	100	9000

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The starting geometries of the simulation boxes were constructed using the program PACKMOL.⁷⁵ In the simple case of the bulk systems, the protein was put into the box center and surrounded by the specified amount of water molecules and counter ions. In the case of an RM, the protein was set into the box center, then surrounded by a sphere of water molecules and counter ions. This sphere was covered by a layer of properly oriented surfactant molecules. The cubic simulation box containing the proto-micelle in its center was then filled with the proper amount of isooctane molecules.

The following molecular dynamics simulation steps were performed using DOMDEC CHARMM.^76,77 The initial PACKMOL geometries were energetically minimized by 2000 steps of steepest descent. The resulting simulation box was then equilibrated as a toroidal isobaric–isothermal (NpT) ensemble at T = 300 K for 4 ns. The converged cubic box length was then used to start the actual production of the trajectory as a toroidal isochoric–isothermal (NVT) ensemble. The temperature was kept constant at T = 300 K using the Nosè-Hoover thermostat.^78,79 The equations of motion were integrated using a leap-frog scheme.⁸⁰ High-frequency vibrations of covalent bonds involving hydrogen atoms were constrained using the SHAKE algorithm⁸¹ which allowed for a generous time step of 2 fs, a write frequency of 1 ps and a total trajectory length of 450 ns. Non-bonded interactions were smoothly switched off between 10 and 12 Å. The long-ranged latter interactions were calculated using the Particle Mesh Ewald method^82,83 with cubic splines of order 6 on a 128 × 128 × 128 grid with κ = 0.41 Å⁻¹ (tinfoil boundary conditions).

For the subsequent computational NOE analysis, we used 60 independent trajectories, a total of 21 000 ns of simulational data.

2.3 Post-production analysis

The resulting CHARMM trajectories were read into a Python3 program with the MDAnalysis module.^84,85 The dipole–dipole TCF was evaluated using accelerated Cython functions. All pairs formed by a protein hydrogen atom (397 in the ligase, 629 in ubiquitin and 1026 in the ubiquitin–ligase complex) and a water hydrogen atom (13 000 in the bulk systems and 3000 in the RM systems) were analyzed, totaling between 1 191 000 and 13 338 000 interacting pairs per trajectory depending on the system. In total, we have thus recorded 225 720 000 protein–water correlation functions. Furthermore, we calculated the correlation functions in a radially resolved manner to examine long-range contributions, using an increment of 0.1 nm up to 3.0 nm.

In essence, the same TCF construction method presented in ref. 12 was employed in this study. In all systems, we have chosen a trajectory length of 350 ns and a TCF length of 50 ns with 500 equally spaced starting points and a time step of 10 ps. We discarded the initial 100 ns of each trajectory to avoid slow equilibration artifacts, in particular systematic protein motion within the RMs.⁶⁷ At each starting point, the protein was centered in the simulation box, ensuring minimum distance of the protein and the water molecules. For every starting point, the trajectory was unfolded allowing a natural diffusive motion of water, correcting toroidal coordinate jumps caused by periodic boundary conditions. This is essential for correctness because the collision-induced dipole–dipole TCF tracks the diffusive motion of the spin pair, hence spatial context matters.

The resulting TCFs were averaged over the five independent replica to minimize the influence of random simulation events, then real-part Fourier transformed using the numpy.fft module to obtain the spectral density function, which in turn served as a base to calculate the frequency-dependent self- (ρ(ω)) and cross-correlation rates (σ_NOE(ω), σ_ROE(ω)). The rates used in the main article were taken at ν = 600 MHz, a common operating frequency in ¹H-NMR spectrometers.

3 Results and discussion

3.1 Ubiquitin–ligase complex in liquid-bulk solution

Protein–protein interactions are profoundly important in biological processes⁴⁷ such as molecular docking⁴⁹ and enzymatic catalysis,⁴⁸ hence inspiring interest in protein–protein geometry in solution. The difference in hydration dynamics was calculated as a difference of NOESY and ROESY signals from the same protein–water cross-peakfor each protein hydrogen. This corresponds to the complex formation paths 1 → 3 and 2 → 3 in Fig. 1 (purple arrows). Note that a difference of NOEs refers to forming the ratio of the hydration dynamics, since the context between translational diffusion and σ_NOE/σ_ROE is logarithmic.¹² The resulting maps thus represent the relative timescale.

The result of this procedure is shown in Fig. 3. Using the example of UBQ, the ligase selectively coordinates the β-sheets over the opposite alpha-helix, as is typical for UBQ complexes.^54,86–88 Local hydration anisotropies like water pockets do not appear in this difference if their hydration dynamics do not change systematically upon complexation. Thus, we see that Δ_complex(σ_NOE/σ_ROE) is able to map spatial anisotropy brought about by complex formation as well.

Fig. 3 Δ _complex(σ_NOE/σ_ROE) calculated for the UBQ–ligase complex. The top figure shows this difference for each protein hydrogen with colors indicating neutral (red) to negative (blue) values. This corresponds to unaffected hydration dynamics versus strongly retarded ones. To present this entity for each partner in the complex, the two sub-panels resolve Δ_complex(σ_NOE/σ_ROE) in detail while representing the partner as a green ribbon cartoon shadow. Note that in precisely those regions where the two partners cover each other, the hydration water limited to interstitial molecules exhibits strongly retarded dynamics, thus reflecting complex formation in an indirect, but spatially resolved way.

Note that position of the complex partner is revealed by an indirect measure, i.e. the hydration dynamics, standing for the spatially resolved protein–water interaction. This is in sharp contrast to classical NOE studies of complex formation via direct intermolecular NOEs between the complex partners. This indirect method offers potential experimental advantages:

• Instead of relying on an isolated spin pair I and S residing on different complex partners, the interaction with the singular S spin is replaced by a set of interactions with many water spins S.

• As a further consequence, the indirect way of probing interactions within the complex can be applied to short-lived complexes as well. The direct measurement of the transient NOE between complex partners is of quasi-intramolecular character and thus limited to complexes surviving on the NMR time scale. The protein–water NOE, on the other hand, always represents an average hydration dynamics not requiring stable complexes.

3.2 Encapsulated proteins

At this point, it is important to remember that the biomolecules forming the machinery of life exist and function in the context of a biological cell, crowded by other cell contents. This environment significantly reduces the volume available to biomolecules and thus alters their behaviour.^3,4 Aside from unspecific excluded volume effects, directed interaction with other biological interfaces^44,89,90 impacts protein enthalpy terms⁹¹ and dynamics^45,92 which in turn alters the equilibria and reaction rates molecules partake in compared to usually studied diluted buffer solutions.^4,93 These confinement conditions influence protein stability, aggregation behaviour, folding rates^3,4,94–98 and hydrogen exchange rates,^41,70,71 while the solvating water exhibits greater viscosity,^43,99–108 a smaller dielectric constant^67,109–116 and normal^50,117 to elevated⁷² collective dynamics. Depending on the system studied, protein secondary structures are reinforced^118–121 or destabilized.^121,122

Thusly motivated, we simulated the three protein systems encapsulated in RMs as shown in the top row of Fig. 1. Of course, these RMs do not resemble actual biological cells in size, wall curvature or wall composition, but they do model some aspects of cellular environments, for instance the presence of a volume limiting wall. In previous work, we have shown that Δ_confined(σ_NOE/σ_ROE) is capable of spatially revealing the specific influence of the cell wall on one encapsulated protein.³⁵ We confirm these previous findings in Fig. 4 where Δ_confined(σ_NOE/σ_ROE) specifically resolves the sites in proximity to the cell wall. For instance, UBQ preferentially interacts with the micelle wall via its β-sheets. This is the same position containing the hydrophobic patch (Leu-8, Ile-44, Val-70), the preferential region for UBQ complex formation^53,123 as can be seen in Fig. 3. The secondary structure stays intact during this interaction as was found empirically.¹¹ Experimentally, protein–surfactant interactions have been known to occur¹²⁴ and have been described both by the groups of Wand^125,126 and Flynn.^127–129 In particular, Flynn studied encapsulated UBQ using HSQC, NMR chemical shift perturbation and Lipari–Szabo order parameters and reported interactions between the β-sheet and the surfactant wall.^127–129 While the group of Flynn used another surfactant in their experiments, their findings regarding UBQ qualitatively agree with ours.

Fig. 4 The difference Δ_confined(σ_NOE/σ_ROE) between the unconfined and the confined protein (ligase, top; UBQ, middle; complex, bottom) can be used to detect contact with the cellular wall. Available experimental NOE data for UBQ encapsulated in AOT confirm slow water dynamics at the β-sheets and fast water dynamics at the α-helix.^9,10 Note that in the complex (bottom figure), the NOE/ROE difference caused by complex formation as discussed above does not show up. This is because we form the differences corresponding to paths (B → E)1, (B → E)2 and (B → E)3 in Fig. 1 corresponding to the green arrows. In these paths, the complex binding state (unbound vs. bound) does not change for the sample proteins, thus the influences of complex formation are effectively ruled out.

Surfactant–wall interactions can profoundly impact enzymatic activity. For instance, Cabral et al. found that encapsulation of cutinase inhibits its catalytic activity.^130–132 Furthermore, note how the protein sites facing the RM boundary and thus seeing extremely retarded interstitial water show very low NOE/ROE ratios. In fact, the existence of such slowed-down mono-layers of water has been connected recently to the cryopreservation of biomolecules.^133,134

3.3 Distinction of different chemical influences

In the encapsulated complex system E3, mimicking protein–protein interaction in a biological cell, we now face retarding influences brought about by both the complex partner and the cellular wall as expressed by eqn (9) and (11) (position in the cell vs. complex binding state). Fig. 5 displays ΔΔ_{confined,complex}(σ_NOE/σ_ROE) for the system studied here.

Fig. 5 Expression tree showing the calculation of ΔΔ_{confined,complex} (eqn (12)) in a pictorial way. The top row presents the protons of the studied proteins, coloured by their σ_NOE/σ_ROE from high (red, fast dynamics) to low (blue, slow dynamics) values. In case of the unbound proteins, the few slow hydration sites correspond to water pockets. In the next row, we form the encapsulation difference of the protein–ligase complex (left-hand side) as well as the complex formation difference of the complex in bulk dilution. Red indicates faster hydration dynamics while blue marks retarded sites. In case of the encapsulation difference, this retardation occurs at the sites facing the RM wall, while for the complex formation, water caught between the two proteins causes the strong difference. Forming the difference of differences, we obtain the proton map at the bottom. Note how it clearly distinguishes the protein regions (a) facing the complex partner, (b) facing the cell wall or (c) facing neither. Values below zero are marked in red and are indicative of the first subtraction term (in this case encapsulation of the complex) while positive values are marked in green and point to the second subtraction term being dominant (complex formation in free bulk solution). Neutral (white) values indicate equal influence of both leading to cancellation.

Forming the difference between the encapsulated complex E3 and the diluted unbound proteins B1 and B2 would yield an unspecific retardation map. In the specific example of UBQ, we cannot tell whether the complex partner or the cell wall coordinates the β-sheets, as both foreign interfaces have shown a tendency to do so above. Thus, in order to disentangle the two different influences, we need to consider paths in Fig. 1. In formal language, this corresponds to taking the double difference

This corresponds to forming the complex, then encapsulating it (pathway B(1, 2 → 3), (B → E)3). Alternatively, one may first encapsulate the unbound protein, followed by complexation within the capsule, though we discourage forming ΔΔ_{confined,complex} this way. This double difference is not a state function and calculating another path will result in a different surface pattern. We discuss the path dependence in the ESI† in detail.

In total, this calculation requires three different samples of the protein to be measured in NOESY and ROESY experiments (or in NOE enhancement experiments): In term (1), the protein (e.g. UBQ) is in complex with its partner (e.g. ligase) and confined in a capsule (system E3); in term (2) the protein complex is dissolved in a diluted buffer (system B3); term (3) requires the protein to be without binding partner, dissolved in solution (system B1 or B2).

From this calculation procedure, which can be done both for computational and empirical data, it now becomes clear which part of UBQ interacts with the ligase and which interacts with the micelle wall. Thus we are now in a position to discriminate whether a surface proton is influenced by the proximity of the complex partner or the cellular wall.

Eqn (12) may also be generalized and applied to discern any two different influences, for instance two different possible binding partners or the influence of a specific binding partner vs. the influence of an unspecific crowder.

4 Conclusions

Intermolecular solute–solvent NOE in principle contains hydration structure and dynamics information. While the raw NOESY observable is contaminated with a multitude of chemically unspecific long-ranged interactions, short ranged observables such as the ratio NOE/ROE reveal local hydration phenomena on the absolute timescale such as fast surface hydration and slow water trapped in pockets. In this study, we focus on differences of NOE signals from the same protein in different environments, thus accessing relative timescales instead: the absolute hydration dynamics does not matter, but the extent of change does. For example, on a relative timescale, a very slow hydration site and a very fast hydration site slowed by the same factor appear as equal. Experiments measuring the relative time scale require both a test system and a reference system.

Generally, retardations are caused by the proximity of foreign interfaces, which trap water interstitially between their surface and the probe protein, causing it to slow down. Thus, the surface map of the relative timescale (i.e. differences of NOE/ROE) is able to spatially resolve the presence of this interface. We show this for a protein complex of UBQ with a ligase. The coordination of a protein site by a different biomacromolecule leads to hydration retardation, which is measurable via indirect protein–water NOEs. This approach may in future offer experimental advantages over traditional transient direct NOEs between complex partners. As a second test case, nanoconfined UBQ feels the surfactant wall proximity due to strongly retarded water specifically at the β-sheets.

Both informations can be combined to detect, e.g., which site of a protein preferentially interacts with the cell wall and which site interacts with the binding partner. Thus we propose the intermolecular solute–solvent NOE, represented either by the NOE/ROE ratio or by the NOE enhancement, as an experimental means to actually map the influences of the cell and the other cell contents on the hydrated surface of the protein and thus elucidate the intracellular processes directly.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We thank Dr Dennis Kurzbach for providing an exemplary experimental protein–water σ_NOE relaxation rate discussed in the introductory section and Fiona Kearns for proofreading the manuscript.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9cp04752b

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