How to probe water dynamics in hydration shells?
Several experimental techniques can be used to probe water reorientational dynamics in biomolecular hydration shells. If all of them measure a slowdown in hydration shell water dynamics, diﬀerent techniques lead to strikingly diﬀerent pictures of the hydration shell, at least regarding two aspects. The first one is the extent of the hydration shell, that is the distance over which the behavior of water molecules is aﬀected by the presence of the biomolecule, and the second much debated aspect is the amplitude of the slowdown. There-fore, it seems necessary to briefly review the most frequently used experimental techniques, insisting on the diﬀerences in the approaches they adopt and on their respective limitations, while technical details can be found elsewhere [1–3, 20, 21].
One can divide experimental techniques to probe water orientational dynamics into two groups. A first group of techniques probe the reorientation of individual water molecules. Among them, ultrafast infrared (IR) pump-probe spectroscopy is a very rich source of infor-mation because it makes it possible to follow water reorientation with a femtosecond (fs) time resolution through the anisotropy decay of the excitation of the OH/OD stretching mode, measured by combining the signals collected with parallel and perpendicular polarizations of the pump and probe pulses . This was, for example, used to measure water reorientation in the hydration shell of small hydrophobic solutes, where the presence of nearly immobilized water molecules was suggested . However, this technique is limited to short-time decays (< 10 ps) because the anisotropy cannot be reliably measured for delays much longer than the vibrational lifetime.
Nuclear Magnetic Resonance (NMR), and especially one of its recent flavors Magnetic Relaxation Dispersion (MRD), probes water individual reorientation by measuring the fre-quency dependence of the longitudinal relaxation rate R1(!). This technique provides a measure of the average water reorientation in the sample. The dynamics in the hydration shell is singled out using a simple two-state model: assuming that the kind of individual reorientational motions probed by NMR is only aﬀected in the first hydration layer of the biomolecule (typically 3.5 – 4 Å), a two-state model with a bulk-like population and a dy-namically perturbed hydration shell can well describe the averaged reorientation, and makes it possible to single out the hydration shell contribution. MRD has been used to study a wide range of solutes, from small hydrophobic solutes , to proteins, including e.g. globular  and anti-freeze proteins . In all cases, the retardation of water dynamics in the hydration shell was found to be very moderate at room temperature, from a 1.5-fold retardation next to hydrophobic solutes to a 3 to 6-fold average slow down next to proteins. Hence, in con-trast with the fs-IR experiments mentioned above, MRD did not find any immobilized water molecules in hydrophobic hydration shells and instead pictures a very mobile hydration layer, only moderately retarded with respect to bulk water molecules, with a perturbation limited to the first shell.
A second group of techniques do not probe single molecule motions, but are instead sensitive to collective motions. This is the case of THz spectroscopy, which probes lower-frequency, more collective motions involving many water molecules and possibly also the biomolecule. THz experiments inferred a thickness of the hydration layer with perturbed water dynamics as large as 18 Å . In addition to the fact that the collective motions probed by THz spectroscopy diﬀer from the individual ones probed by NMR, which may explain the discrepancy between the results obtained with the two techniques, the models used in THz spectroscopy to single out hydration shell dynamics have recently been seriously questioned . Dielectric relaxation spectroscopy is also sensitive to collective motions, and measures the relaxation of the total dipole moment of the sample. The spectrum obtained in the frequency domain reveals the characteristic time scales of the processes involved in the relaxation of the dipole. The presence of a nanosecond timescale was traditionally explained by the very strong retardation of half the hydration shell water molecules, but more recent work  suggested that it originates from slow protein motions to which water dynamics would respond. Eventually, Time-Dependent Stokes-Shift (TDSS) experiments  follow solvent reorganization after excitation of a chromophore through the time-evolution of the Stokes shift (diﬀerence between the absorption and emission wavelengths). The long timescales (20-200 ps) found in the TDSS decay  were interpreted as originating from slow water dynamics, possibly coupled with slow protein dynamics.
The diﬀerent experiments available thus provide contrasted pictures of water dynamics in the hydration shell of biomolecules. Even putting aside experiments probing collective motions, which cannot be easily related to individual dynamics, a huge discrepancy remains between the diﬀerent types of measurements: from a very labile, only slightly retarded hydra-tion shell (NMR) to a very viscous, nearly frozen one (fs-IR). Moreover, all these experiments measure the average reorientation dynamics over the entire hydration shell and cannot read-ily provide a spatially resolved picture. Only few attempts have been made so far in this direction. A first one used TDSS with chromophores placed in diﬀerent locations  but the signal does not only reflect local motions and can be sensitive to long-range rearrangements. A second one  used Nuclear Overhauser Eﬀect (NOE) NMR to map the hydration dy-namics of a small globular protein, ubiquitin, encapsulated in a reverse micelle, where it is not clear how the encapsulation may aﬀect water dynamics.
Molecular dynamics simulations
In this context of conflicting experiments with a need for a spatially resolved picture of the hydration shell, classical molecular dynamics (MD) simulations provide a way to probe the dynamics of water in biomolecular hydration shells directly at the molecular level.
In classical molecular dynamics simulations, the system of interest is defined by an en-semble of atoms (typically a few tens of thousand for the systems studied in this thesis), interacting through an empirical potential, optimized to reproduce experimental properties, such as solvation enthalpies, dipole moments or results of electronic structure calculations. For a system composed of N atoms interacting through the potential U(rN ), propagating molecular dynamics trajectories consists in numerically solving Newton’s equations of mo-tion:
MD simulations thus yield the trajectory of each atom in the system, with a typical length of a few tens up to a few hundreds of nanoseconds in the present work, and thus provide highly valuable dynamical and structural information at the molecular scale.
Early simulations of proteins with explicit solvent pictured a reduced mobility in the hydration shell with lower diﬀusion constants than in bulk water , and computed water residence times in the hydration shell ranging from a few ps next to apolar groups to over 50 ps next to charged groups . Several recent MD studies confirmed that water in the hydration shell is on average moderately retarded with respect to bulk water (3-7 times) [34, 35], in agreement with NMR simulations.
Defining the hydration shell of a biomolecule is much easier in MD simulations than in experiments: a layer thickness can be defined, for example based on the probability to find a water molecule at a given distance of the protein surface. Water molecules lying within this cutoﬀ of the protein are then considered to belong to the hydration shell. Such a geometric definition can be refined to account for the fact that the hydration shell is more contracted around polar and charged groups by defining site specific geometric criteria that involve site-dependent cutoﬀ distances and angles , as will be done in the following work. Moreover, such a site-specific geometric definition makes it possible to map water dynamical properties on the biomolecule surface , thus providing a highly valuable spatially resolved information which is so far impossible to obtain experimentally.
The hydration shell being defined, water reorientation is followed through the second order time correlation function (TCF) of the molecular orientation C2(t):
where P2 is the 2nd order Legendre polynomial . For a given vector attached to the molecule, such as the water OH bond, this function measures how fast the water molecule looses the memory of its initial orientation. The integrated reorientation time 2reor is then simply defined as the time integral of the reorientation TCF:
Restricted to water molecules initially lying in the hydration shell [resp. next to a protein site], it provides a good estimate of the characteristic reorientation time in the hydration shell [resp. next to a protein site]. We must note that this definition is not perfect because at long delays the water molecule may escape the hydration shell or move next to another protein site, which reduces the spatial resolution of the analysis. The slow down in water dynamics in the hydration shell of a solute with respect to the bulk is quantified through the retardation factor reor, defined as the ratio between the reorientation time in the shell and that in the bulk:
The reorientational TCF of bulk water, as presented in Fig. 1.1, clearly reveals the char-acteristic time scales of water reorientation: a very fast (< 200 fs) decay, followed by a longer time decay, with a picosecond time-scale. However, the underlying mechanism for water re-orientation was determined only about 10 years ago, and I will now summarize the key points of this mechanism.
Water reorientation mechanism: Extended Jump Model
Bulk water reorientation mechanism
The very fast decay (< 200 fs) of the reorientation TCF in bulk water displayed Fig. 1.1 is due to librations, that is to say fast fluctuations of each OH bond around its optimal linear hydrogen bond (H-bond) geometry with the H-bond acceptor. The longer time decay (> ps) of the TCF in bulk water had been described for a long time by the Debye rotational diﬀusion picture , in which water reorientation occurs through very small angular steps. However Laage and Hynes suggested in 2006 a completely new molecular picture of water reorienta-tion , based on molecular simulations and analytic modeling. This new mechanism was subsequently evidenced for water in aqueous salt solutions by 2D-IR experiments [40, 41] and has now become the new standard description for water reorientation. In this new picture, water reorientation proceeds through large amplitude angular jumps, very diﬀerent from the small angular steps of the Debye rotational diﬀusion picture, which occur when a water molecule exchanges H-bond acceptors (Fig. 1.2). The jump can be modeled as a chemical reaction: in the reactant state, the water is initially H-bonded to its initial oxygen acceptor Oa. For this OH bond to reorient, a new H-bond acceptor oxygen Ob has to penetrate the first hydration shell, and when the initial and final oxygen H-bond acceptors are equidistant from the rotating water oxygen O∗, the water OH bond suddenly executes a large-amplitude jump from one acceptor to the other, forming at the transition state a bifurcated H-bond. The H-bond with the new partner eventually stabilizes, while the initial partner leaves.
where nX = 1 if the OH bond is in its stable state X (i.e. forming a stable H-bond with its initial or final acceptor respectively), and 0 otherwise. This TCF is computed with absorbing boundary conditions in the final state in order to exclusively consider the first jump out of the initial state . The jump correlation function (Fig. 1.3) is purely monoexponential in bulk water at room temperature, and no longer exhibits the very fast initial decay that was present in the reorientation TCF and was due to librations, which are on purpose not captured in the stable state picture used for the jump TCF calculation. The jump time is obtained by a numerical integration of the 1 Cjump(t) function. One advantage of jump times over reorientation times for the analysis of water dynamics next to biomolecules is that the definition of jump times is more local: a water molecule initially lying next to a given protein site has to jump to end up next to another protein site. Hence, using jump times to study water dynamics provides a highly spatially-resolved determination of water dynamics.
Hydrophobic solutes: transition-state excluded volume (TSEV) factor
This simple model works very well in bulk water but needs to be extended to account for the presence of a solute. Before studying complex biologically relevant solutes as proteins, with a strong topographical and chemical surface heterogeneity, even the influence of a small hy-drophobic solute on water dynamics has been the subject of a long ongoing debate. Following the initial picture of a « microscopic iceberg » forming around a hydrophobic solute suggested by Franck and Evans  on the basis of thermodynamic considerations, fs-IR experiments inferred the presence of immobilized water molecules next to a paradigm hydrophobic so-lute . In contrast, NMR experiments found a very modest slowdown (2-3-fold) of water dynamics in the hydration shell of several hydrophobic solutes . These seemingly contra-dictory results can be reconciled within the EJM framework in terms of a transition-state excluded-volume (TSEV) eﬀect for the H-bond exchange, without any water immobilization: the approach of a new H-bond acceptor is hindered by the presence of the hydrophobic solute, inducing an excluded volume. This eﬀect can be rationalized by considering the fraction f of the ring representing the possible TS locations that overlap with the solute excluded volume (Fig. 1.4). This leads to a TSEV slowdown factor V :
This TSEV description correctly reproduces the moderate ( V ’ 1:4) retardation in water dynamics next to small convex hydrophobic solutes, as obtained both from MD simulations and NMR experiments. The apparent contradiction between NMR and fs-IR simulations was shown to be due to the high solute concentration used in fs-IR experiments, that led to very high fractions of excluded volume and much slower dynamics in the hydration shell .
Hydrophilic solutes: transition-state hydrogen-bond (TSHB) factor
In the case of hydrophilic solutes, the TSEV factor alone cannot explain the eﬀect of the solute on water dynamics. For example, while the TSEV model can only explain a slowdown in water dynamics, some ions accelerate water dynamics, whereas others slow it down .
Hydrophilic solutes can be divided in two groups: those which can donate a H-bond to water molecules and those accepting a H-bond from water molecules. For H-bond donor solutes, the reorientation dynamics of a neighboring water OH bond stems solely from an excluded-volume eﬀect, as already observed for hydrophobic solutes, and water reorientation is only moderately slowed down. Next to a solute that can accept H-bonds from water, the initial H-bond is formed with the solute itself, rather than with another water molecule (Fig. 1.5). One thus has to take into account the free energy cost Gz to elongate the H-bond between water and the H-bond acceptor from its equilibrium length, Req, to the TS one, Rz: Gz = G(Rz) G(Req), as compared to the free energy cost to elongate a water-water frac-tion. Considering a stable H-bond within the hydration shell (red dashed line), possible TS locations for the new H-bond acceptor lie on a ring defined by the distance Rz to the reorienting water and the jump angle . The fraction f of this ring overlapping with the solutes is represented in pink, and the complementary accessible fraction is shown in green. Reproduced from Ref. .
which leads to a slowdown in water dynamics if the initial H-bond is stronger than a water-water H-bond, and an acceleration if it is weaker .
which was shown to provide a quasi quantitative description of water slowdown in the hy-dration shell of a protein . If ρHB is low enough (if the H-bond with the solute acceptor is weak enough), it can overcompensate ρV to lead to an acceleration of the dynamics next to the solute.
Biomolecular hydration shells: spatial heterogeneity
In bulk water or in dilute aqueous solutions of small solutes, the average reorientation TCF is usually mono exponential after the fast initial sub-picosecond decay (Fig. 1.1). In contrast, the average reorientation TCF for water molecules in the hydration shell of a protein is highly non monoexponential , as illustrated in Fig. 1.6. This non monoexponential decay was shown to originate from an underlying distribution of reorientation times (Fig. 1.6) due to the chemical and topological heterogeneity of the surface : water molecules lying next to diﬀerent groups in diﬀerent environments exhibit distinct reorientation behaviors. This was rationalized within the framework of the jump model previously developed and in all cases, the average retardation was found to be moderate (around 2-4), in very good agreement with NMR experiments .
All the results and models described so far pertained to the room temperature situation, where the models developed provide a quantitative description of water reorientation dynamics. However, recent NMR experiments measured a non-monotonic temperature dependence of the retardation next to hydrophobic solutes  and proteins . This could not be explained by any of the previous models, putting into question their relevance and renewing the debate about the molecular origin of the perturbation, as will be detailed in Chapter 2. Water dynamics in protein hydration shells at supercooled temperature has a limited biological relevance, but it is a stringent test of the accuracy of the theoretical models developed to describe the perturbation of water dynamics next to biomolecules.
Hence, we describe in Chapter 2 the development of a new theoretical model to account for the temperature dependence of water dynamics next to a paradigm hydrophobic solute.
In the following two chapters, we then focus on biologically relevant systems, where hydration shell water dynamics is debated. We start in Chapter 3 with the detailed study of hydration structure and dynamics of an anti-freeze protein over a broad range of temperatures. The conclusions of the model developed in Chapter 2 are used to rationalize the behavior of this complex system, and a similar study is performed for comparison purposes on a paradigm globular protein, ubiquitin, in order to clearly point out the specificity of the anti-freeze sys-tem. We then conclude this part on hydration dynamics in Chapter 4, with a comprehensive description of water dynamics in DNA hydration shell. We particularly focus on the complex behavior of water in the minor groove and identify the diﬀerent sources of heterogeneity in such an environment.
The behavior of water molecules next to hydrophobic groups governs a wide range of fun-damental biochemical processes, including e.g. protein folding, protein-ligand binding, and the assembling of lipid membranes . Yet, despite numerous experimental and theoretical studies, the dynamical properties of these interfacial water molecules are still debated. At room temperature, all experimental measurements and numerical simulations show a slow-down in water reorientation dynamics in hydrophobic hydration shells relative to the bulk, but no consensus has been found regarding the magnitude of this retardation factor. Esti-mates range from a moderate approximately two-fold slowdown in NMR  and simulation studies [44, 48, 49] to an immobilization of some hydration shell water molecules inferred from ultrafast infrared spectroscopy [23, 50]. Two opposing molecular interpretations of this slowdown have been suggested. The first explanation involves an excluded volume eﬀect , as described in Chapter 1, which leads to a moderate slowdown in dilute solutions but a pronounced water retardation in concentrated solutions. In another picture, the slowdown is large and concentration independent, and arises from a collective frustration of the hydration shell structural dynamics [23, 50].
However, none of these descriptions can explain recent NMR results  on a series of small and mostly hydrophobic solutes where the retardation factor in water reorientation dynamics was found to change in a non-monotonic fashion with temperature  (Fig. 2.1). Intriguingly, extrapolating these results at very low temperature suggests that the dynamical perturbation induced by hydrophobic solutes would even disappear and the rotational dynam-ics of water within those hydration shells would become as fast as in the bulk (Fig. 2.1). None of the existing descriptions of hydrophobic hydration dynamics can describe this behavior. For example, the excluded volume eﬀect developed for solutions at room temperature  has an entropic origin and thus cannot explain the observed changes with temperature. Within the collective frustration picture, the activation energy for water reorientation was suggested to be higher in the shell than in the bulk , which would lead to a monotonic increase in the retardation factor upon cooling, in clear contradiction with the NMR results.
A first qualitative interpretation of the NMR results invoked  the limited changes induced by temperature in the local arrangements of water molecules within the shell relative to those in the bulk, due to the constraints imposed by the solute interface. However, it is not clear how this can explain the non-monotonic behavior of the perturbation. Furthermore, several important questions raised by these experiments still need to be solved, including whether the molecular origin of the perturbation and its entropic or enthalpic character change with temperature.
Here, we employ molecular dynamics (MD) simulations and analytic modeling to design a new model capable of properly describing the evolution of hydrophobic hydration dynamics over a broad temperature range spanning the supercooled and liquid domains. This model is then used to clearly identify the molecular factors at play (e.g. enthalpic vs. entropic, local structure vs. excluded volume) and their respective importance. As detailed below, this model builds upon the entropic excluded-volume factor initially developed at room tem-perature  and combines it with an additional factor arising from local structural fluctua-tions. The paradigm hydrophobic solute we have selected is trimethylamine-N-oxide (TMAO) whose hydration dynamics has been extensively studied at room temperature with a range of techniques including NMR , ultrafast infrared [23, 50] and optical Kerr-eﬀect  spectro-scopies together with MD simulations [44, 53, 54], and for which the temperature-dependent retardation factor was recently determined by NMR .
Classical molecular dynamics simulations are performed for a series of dilute TMAO aqueous solutions at diﬀerent temperatures. We use the TIP4P/2005 water model  which provides one of the best available descriptions of the water phase diagram and which was verified to properly describe water reorientation dynamics over the supercooled and liquid temperature range (see Ref.  and ). To describe TMAO, we employ a specific rigid force field  which has been shown to provide a good description of water dynamics within its hydration shell . Each simulation box contains a single TMAO solute with 490 water molecules, corresponding to a molality of approximately 0.1 mol/kg. At each temperature, the system is first equilibrated for at least 500 ps in the NPT ensemble using a Langevin thermostat and a Langevin barostat. It is then equilibrated for 500 ps in the NVT ensemble with a Langevin thermostat before a final equilibration in the NVE ensemble for at least 500 ps and the production run with a 1 fs timestep. The lengths of the production runs are respectively 70 ns at 218 K, 50 ns at 231 K, 40 ns at 244 K, 50 ns at 249 K, and 3 ns at 264 K, 278 K, 285 K, 300 K and 350 K. The simulations are performed with NAMD , with periodic boundary conditions and a Particle Mesh Ewald treatment of long-range electrostatic interactions. An 11 Å cutoﬀ was applied to non-bonded interactions with a switching function between 9 and 11 Å. While neat liquid water is not stable below approximately 231 K , we simulated the solution at 218 K because the presence of the solute is known to decrease the homogeneous nucleation temperature  and because the TIP4P/2005 water model was shown to yield a shifted phase diagram, resulting in a stable supercooled liquid at this temperature . The error bars indicated in the following plots correspond to the standard deviations calculated by dividing each trajectory in 5 independent blocks (except at 218 K where 4 blocks are used).
H-bond exchanges in water occur through jumps (see Chapter 1) which can be treated as chemical reactions whose rate constant is determined by the free energy barrier Gz to reach the transition state. The barrier essentially results from the free energy costs for the elongation of the H-bond with the initial acceptor and the approach of the final acceptor to the transition state configuration [42, 56].
For water molecules within the solute hydration shell, the rdf must be corrected for the fraction of the space which is not accessible to water molecules because of the presence of the solute. We therefore define an excluded-volume corrected rdf , where x(r) is the fraction of the spherical shell, of radius r and centered on a water oxygen site within the solute hydration layer, which is not blocked by the solute sites. The solute sites are defined as hard spheres centered on each TMAO heavy atom and their radius is the minimum approach distance between the site and a water oxygen atom. x(r) is evaluated numerically from the simulation by discretizing each spherical shell of radius r centered on each water molecule within the TMAO hydration layer into 150 uniformly distributed points on the surface of the spherical shell and determining whether each point overlaps with a solute site.
Table of contents :
1 Introduction to Part I: Biomolecular hydration shell dynamics
1.1 How to probe water dynamics in hydration shells?
1.1.1 Experimental techniques
1.1.2 Molecular dynamics simulations
1.2 Water reorientation mechanism: Extended Jump Model
1.2.1 Bulk water reorientation mechanism
1.2.2 Hydrophobic solutes: transition-state excluded volume (TSEV) factor
1.2.3 Hydrophilic solutes: transition-state hydrogen-bond (TSHB) factor
1.2.4 Biomolecular hydration shells: spatial heterogeneity
2 Temperature dependence of hydrophobic hydration dynamics
2.3.1 Temperature dependence of perturbation from simulations
2.3.2 Excluded volume with local structural fluctuations
2.3.3 Interpretation of the temperature dependence of the perturbation factor
2.3.4 Validity of the excluded-volume picture at room temperature
2.4 Concluding remarks
3 Hydration shell dynamics around a hyperactive antifreeze protein and ubiquitin
3.3 Hydration dynamics of ubiquitin
3.3.1 Average water reorientation dynamics in the hydration shell
3.3.2 Hydration dynamics distributions and mapping
3.3.3 Temperature dependence of the dynamical perturbation
3.4 Hydration Dynamics of CfAFP
3.4.1 Average retardation factor in the Cf AFP hydration shell
3.4.2 Site-resolved analysis of Cf AFP hydration dynamics
3.4.3 Local structural order
3.4.4 Short-ranged structural perturbation
3.4.5 Water–Cf AFP H-bond strength
3.5 Concluding remarks
4 Heterogeneous water dynamics in DNA hydration shell
4.2.1 System preparation
4.2.2 Water dynamics analysis
4.2.3 Groove width analysis
4.3 Orientational dynamics in the hydration shell: spatial heterogeneity
4.3.1 Heterogeneity in hydration dynamics
4.3.2 Spatial heterogeneity
4.3.3 Mechanism for reorientation
4.4 Dynamical heterogeneity in the minor groove
4.4.1 Temporal heterogeneity
4.4.2 Hydration structure and kinetics in the A-tract
4.4.3 Minor groove fluctuations: dynamical heterogeneity
4.4.4 Dynamical heterogeneity model
4.5 Concluding remarks
5 Introduction to part II: Environmental effects in enzyme catalysis
5.1 Context of the study
5.1.1 Enzyme catalysis: controversies and challenges
5.1.2 Transition-State Theory
5.2 Computational approach
6 Role of active site residues in DHFR catalysis
6.1.1 DHFR: a paradigm system
6.1.2 Recent experiments and open questions
6.2.1 System Preparation
6.2.2 EVB simulations
6.2.3 pKa shifts calculations
6.3.1 Hydride transfer step
6.3.2 Protonation step
6.3.3 Overall picture
6.4 Concluding remarks
7 Enzyme catalysis in organic solvents: role of water
7.1.1 Non-aqueous enzymology
7.1.2 Subtilisin Carlsberg: a model system
7.2.1 System preparation
7.2.2 EVB description
7.2.3 Equilibration procedure
7.2.4 Hydration levels
7.3 How does water affect the chemical rate constant?
7.3.1 Hydration level and activation free-energy Gz
7.3.2 Hydration level and transmission coefficient
7.3.3 Existence of a stable non-productive bound state at low hydration
7.4 Concluding remarks
8 Résumé de la thèse
8.2 Dynamique de l’eau dans la couche d’hydratation de biomolécules
8.3 Effets de l’environnement sur la catalyse enzymatique