Water Determines the Structure and Dynamics of Proteins

2.2. Water in the Interior of Globular Proteins

So far, we have focused on the dynamics of the majority of water molecules interfacing the external protein surface, but the protein interior can be hydrated as well.⁴⁷ The exchange dynamics of internal water generally occurs on the nanosecond time scale and longer. This exchange dynamics was first investigated by NMR experiments and correlated to slow rearrangement of the protein matrix.⁴⁷ The first detailed investigation at atomistic resolution have been carried out by computer simulations but limited to the nanosecond time scale.^48–50 Nowadays, thanks to technological development, it is possible to characterize this exchange process (e.g., individuate the path for the exchange and correlated protein motions) via computer simulation at much longer time scales and with an unprecedented resolution and richness of details, as shown in Figure 2.^51,52 Moreover, from the experimental perspective, the possibility to link proteins in a matrix of gel, hence reducing rotational tumbling, allowed NMR to dissect this in/out water exchange at a much finer temporal resolution.⁵³ Interior water molecules can be characterized as localized and fluctuating water molecules. Localized water molecules are identified by high resolution X-ray crystallography and by water ¹⁷O and ²H nuclear magnetic relaxation dispersion (NMRD) measurements⁴⁷ and can have residence times on the millisecond time scale. Fluctuating water molecules cannot be easily detected as single water molecules but can be characterized by density or their relaxation properties. A computational study of a peptide-MHC complex show that fluctuating water molecules at the complex interface can have higher entropy than in bulk and can enhance the peptide binding affinity.⁵⁴ The nonpolar cavity in the protein interleulcin-l beta (IL-1β) has been reported to be filled by water on the basis of some experiments and simulations^55,56 and to be empty on the basis of others.⁵⁷ Yin et al.⁵⁸ studied the thermodynamics of filling the central nonpolar cavity of interleukin-1 beta by molecular dynamics simulation and found that water in the central nonpolar cavity is thermodynamically unstable. The favorable contribution of internal water to protein thermal stability has been recently pointed out^59,60 and complements previous investigations on the role of internal hydration to pressure induced unfolding discussed later in the review.

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Figure 2

Fluctuations in the interloop region during 1 ms. (A) Volume of the convex hull spanning the four internal hydration sites. (B) Backbone RMSD, relative to crystal structure 5pti, for the residues defining the hull. (C) Number of water molecules inside the hull. (D) Dihedral angle χ1(C14), with disulfide isomeric states color-coded: M1 (blue), M2 (magenta), M3 (yellow), mC14 (red), mC38 (green), and other (gray). The time series values plotted here are either averaged over consecutive 25 ns windows (A–C) or sampled with 25 ns resolution (D). The full-trajectory distributions are projected on the right-hand axis. This figure is reprinted from ref ⁵¹. Copyright 2013 American Chemical Society.

2.3. Coupling of Protein/Water Dynamics

How fast dynamics at the protein surface couples to the motion of the external exposed to solvent amino acids as well as to the protein interior and the escape dynamics of internal water relates to protein soft modes has been the object of intense research. Neutron scattering techniques, which are very sensitive to protons, are particularly suitable for these studies.⁶¹ However, dielectric spectroscopy,⁶² nuclear magnetic resonance (NMR),⁶³ and molecular dynamics simulation (MD) have led to significant results.^64–67

In seminal studies of myoglobin over a wide temperature range by Doster et al.,⁶⁸ it was shown, by using Inelastic Neutron Scattering (INS), that at low temperatures (below 80 K) the protein behaves like a harmonic system and displays a linear increase in mean square displacement (MSD) with temperature. However, above 180 K, there is a dramatic increase in the MSD (Figure 3).⁶⁸ This increase in MSD was attributed to a dynamic transition [later labeled a protein dynamic transition (PTD)] arising from the excitation of nonvibrational motion, first attributed to torsional jumps between states and later attributed to methyl group rotations.⁶⁹ Earlier Mossbauer spectroscopy studies have shown similar behavior, but Mossbauer probes the protein (the heme iron) dynamics on a much slower time scale.^70,71 This sudden change of slope in the temperature dependence of mean-square displacement of hydrated proteins around 220 K has been extensively studied.⁷² This effect is present in other proteins, although at different temperatures, and MD simulations reproduce the MSD temperature dependence.⁷³

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Figure 3

Mean square displacements in dry and wet myoglobin.⁶⁸ The sharp increase in mean square displacements at around 200 K for the hydrated sample is absent in the dehydrated sample. Reproduced with permission from ref ⁶⁸. Copyright 1989 Macmillan Publishers Ltd., Nature.

The interest in the protein’s “dynamical transition” is due to the fact that this transition is supposed to be intimately connected to protein function and that this connection can be made for a wide variety of systems from small soluble globular proteins to membrane proteins. In the early steps of the transition, the role of the solvent surrounding the proteins has been recognized, since in the absence of hydration the 220 K dynamical transition vanishes. Moreover the transition temperature is controlled by the viscosity of the solvent, and the transition temperature is raised in the presence of cosolvents like sugars.^74,75 A clear connection among the role of solvent in controlling such thermal activation of protein vibrations has become more explicit by comparing the paralleling of the thermal dependence of protein and water motions. A strong parallel evolution at 150 and 220 K between the mean-square displacements related to interfacial water rotational dynamics and to protons dynamics of a hydrated lysozyme protein⁷⁶ has been obtained (see Figure 4). This connection is made at the local scale (a few Angstroms) and on the time scale of nanoseconds. These observations provided evidence that interfacial water rotational dynamics are the real source of entropy driving protein dynamics. In the context of the previous findings, it has been possible to reach a final view of the protein-hydration water interaction⁷⁷ and how this interaction can drive the protein function: the protein external side-chains⁷⁸ short time motions, induced by fast water reorientational motion, propagate in a hierarchical way, along the protein structure from the residue side chains down to the protein core to induce the longer time scale motion of protein backbone necessary for its function.⁷⁹ The dynamical crossover experienced by water at 150 and 220 K are also detected on the protein dynamics, even though the time scales of the crossover can be different (longer times for protein than interfacial water). This was also observed by taking advantage of a deuterated protein, the maltose binding protein (MBP), and combining elastic incoherent neutron scattering with MD simulations.^73,80,81 The temperature-dependence of experimental and simulated dynamics of MBP and its hydration water, probed individually on the same sample, affirm the existence of a dynamical coupling between protein and solvent motion on the ps–ns time scale. These observations have been confirmed further using a different deuterated protein which allowed one to distinguish between protein dynamics and hydration water dynamics.⁸² QENS and IR experiments on lysozyme powders suggest that the protein dynamical transition may have its origin in the liquid–liquid critical point in the protein hydration water.^83,84

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Figure 4

Temperature dependence if the hydrogen atom mean-square displacement of lysozyme hydrated with a monolayer of D2O. Zanotti et al.⁸⁵ observed a strong correlation between the local reorientational transition observed in water at 220 K, <u² _{rot water}>, and the onset of the long time (1 ns) large amplitude overdamped motions responsible for the <u² _protein> to increase above 220 K. The correlation between protein dynamical crossovers at 150 and 220 K and interfacial water rotational dynamics (no correlation with translational dynamics): it was suggested that the sole hydration water rotational dynamics trigger the protein dynamics. This figure is reproduced with permission from ref ⁸⁵. Copyright 2007 Springer.

Nevertheless, the interpretation of the origin of the dynamic transition has been controversial.^86–91 Young et al.^87,88 have argued that the dynamical transition in Mossbauer experiments is caused by the incorrect separation of the spectrum into sharp and broad components. They argue that, given that proteins are not harmonic, the entire spectrum is inhomogeneous composed of sharp lines, with each sharp line comin,g from conformational substates in a complex, hierarchically ordered, free energy landscape. Frauenfelder and collaborators proposed a unified model of protein dynamics (UMPD) and showed that the Mossbauer and INS spectrum of myoglobin (Mb) can be explained above 180 K, without any fitting parameters, by the dielectric fluctuations in the hydration shell.⁹⁰ The UMPD can also explain DSC and INS in dehydrated samples without invoking dynamical transitions.⁹¹ In this model, the striking change in dynamics near 180 K is a kinetic behavior that depends on the time scale probed by the experiment and not an equilibrium behavior. Fenimore et al. proposed that motions in proteins can be characterized into two classes: slaved and nonslaved.⁸⁹ Slaved processes are tightly coupled to the solvent, with the solvent being responsible for the activation enthalpy, and the protein and its hydration shell control the activation entropy. The dominant conformational motions are slaved by the hydration shell and the bulk solvent. The protein fluctuations are called α and β _h, following studies of the fluctuations of glass-forming liquids.⁹² The alpha fluctuations are structural, and their rate constant is inversely proportional to the viscosity of the medium. The β _h fluctuations come from the hydration shell and influence the protein motions.^87,93 The β _h fluctuations proposed by Fenimore et al. are different than the typical β fluctuations in supercooled liquids and glasses, which is a bulk solvent property. The microscopic origin of the β _h is not known but may be related to the behavior of interfacial fluids on the nanometer scale of proteins.⁹⁴ Molecular dynamics simulations and QENS on lysozyme-hydrated powders exhibit a logarithmic decay in the 10 ps to 1 ns time range, which is in the β-relaxation range of the protein.^95,96 Readers interested in the specific arguments related to this controversy are encouraged to read recent articles by Doster,⁸⁶ Young and Fenimore,⁹¹ and Frauenfelder et al.⁹⁰

4.2. Proton Wires

In 1978, Nagle and Morowitz¹⁷¹ introduced the notion of hydrogen-bonded chains that would form through a series of sequential hydrogen bonds involving the hydroxyl side chain group of several serine amino acids. These hydrogen-bonded wires could serve as conduits along which proton transfer occurred. Numerous experimental and theoretical studies later on indeed confirmed that fairly long-lived water-wires formed in a plethora of biological systems particularly membrane proteins and allowed for correlated or concerted protonhopping events in which several protons hop over many water molecules simultaneously leading to enhanced mobility.^172–175

Water and proton wires may not be a unique feature of water in confined environments. Taking the view of water as a percolating 3D hydrogen bond network yields the presence of closed rings or loops, the segments along which lead to wirelike structures. In the presence of topological defects such as the proton or hydroxide ion, concerted proton hopping can also occur.¹⁷⁶ Rather than being able to form more linearshaped and long-lived topological structures like in proteins, the proton wires in water will be a lot more coiled up and instead much shorter lived. Understanding both the structural and dynamical properties of these networks will likely play an important role in attracting and funneling protons to the active site such as in proteins like the green fluorescence protein.^177,178

Besides their role in conducting protons, the fluctuations and evolution of hydrogen bond networks and wires has been shown to play an important role in the binding of protein interfaces. Extensive molecular dynamics simulations of the interaction of two hydrophilic proteins have shown that the interfacial water between the proteins acts as an adhesive-directed network, resulting in a dielectric shielding which results in highly directed electrostatic interactions between the hydrophilic groups of the protein.¹⁶⁸ The role of collective hydrogen bond networks has also been observed recently in the folding of antifreeze proteins. In particular, this protein is an alanine rich polypeptide forming helical bundles that are stabilized by clathrate-like pentagonal hydrogen bond net-works.^162,179

One of the most fundamental processes in protein biophysics is the exchange of the amide backbone hydrogen atoms with the surrounding solvent. The time scales of amide exchange is a very useful probe into the flexibility of the protein. Despite its ubiquitous occurrence, the molecular mechanism of amide backbone hydrogen exchange is still not completely understood, although it is believed that the process is catalyzed by the close proximity of a hydroxide ion near the amide backbone. Using long classical molecular dynamics simulations, Persson et al.¹⁸⁰ have proposed a mechanism whereby a hydroxide ion approaches the N–H bond through a water wire. They suggest, based on the interpretation of their classical simulations, that the number of water molecules and orientation of waters within the wire plays an important role in ensuring successful amide proton exchange.

5.1. Pressure-Assisted Cold Denaturation

The study of proteins at low or subzero temperatures is based on the unusual behavior of water under pressure, which allows for the study of protein denaturation without ice crystal formation. An extensive study of pressure-assisted cold denaturation on β-lactoglobulin has been done by combining fourth derivative UV spectroscopy under pressure (300 MPa) with gel permeation chromatography and DSC measurements.²²⁶ It has been shown that at a given pH, low temperature-pressurization is very useful to minimize structural changes and therefore losses in protein functionality.

Thermodynamically, the pressure denaturation and pressure-assisted cold denaturation observed by Bridgman can be described by a simple parabolic P-T stability diagram, first described by Hawley. Figure 8 shows a P–T stability diagram obtained from molecular dynamics simulations on a small protein.²²⁸ Notice that at low temperature and high pressure the protein can unfold upon cooling. This diagram shows isentropic and isochoric lines on which the entropy and volume changes of the transition are zero. For P–T states below the ΔV = 0 curve, the volume of the unfolded state is higher than the volume of the folded state, while above this curve the volume of the folded state is higher. We must emphasize that here volumes and entropy changes refer to the overall changes on the system, not just the protein. This suggests that for the region above, ΔV = 0 (i.e., ΔV = V _unfolded − V _folded < 0), as required by Le Chattelier’s principle, the unfolded state should be preferred at higher pressures. Equally interesting are the P–T regions separated by the ΔS = 0 curve. Below this curve, the protein solution has larger entropy in the unfolded state over the folded state, as is typically described in textbooks. However, above the ΔS = 0 curve, the folded state will have higher entropy than the unfolded state, and, as the system is driven toward the more ordered state as temperature is lowered, the protein will adopt the (cold denatured) unfolded state. This behavior has been observed experimentally for model proteins and has been described by molecular dynamics simulations.^228–232

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Figure 8

Calculated P–T stability diagram for the Trp-cage miniprotein shows an elliptical shape that predicts pressure and cold denaturation at high pressures. The ΔG = 0 curve is the innermost contour. Other contours show ΔG = −15, −10, and 15 kJ mol^-1. The dotted lines show the thermodynamic states sampled in two independent REMD simulations. The black line shows the ΔV = 0 isochore; the solid red line shows the ΔS = 0 isentrope. This figure is reproduced with permission from ref ²²⁸. Copyright 2010 Wiley-Liss, Inc.

The P–T diagram of the Trp-cage for an alpha helical peptide (AK16) and for a β-hairpin forming peptide known as the GB1 peptide calculated by Hatch et al.²³² are shown in Figure 9. The diagram for the Trp-cage shows pressure denaturation at low pressures as well as high pressures. However, the GB1 β-hairpin only shows high-pressure denaturation. Hatch et al. suggest that the behavior of the Trp-cage and GB1 β-hairpin under negative pressure is related to the position of the center of the elliptical stability diagram relative to the spinoidal for the bulk liquid described by Tip3P water.

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Figure 9

Calculated P–T diagram for a small protein and peptides. The colored lines are the loci of the 0.5 fraction folded for the Trp-cage (magenta), 0.5 fraction folded for the GB1 β-hairpin (blue), and 0.25 fraction folded for the AK16 (green) α helix. Dashed lines are in the vicinity of the simulation data. The back line is a numerical estimate of the spinoidal of the bulk solution. The figure and caption are reprinted from ref ²³². Copyright 2014 American Chemical Society.

5.2. Cavities in Folded Proteins

The structure of the temperature-unfolded state of a protein is modeled as a random coil with little ordered content of secondary structure. However, the pressure-denatured state (at room temperature) of a protein is much more compact than the random coil and preserves much of the protein’s secondary structure.^214,233 The compactness of the pressure-unfolded state of a protein can be explained in terms of the interaction of hydrophobic groups in water as a function of pressure. A computational study of the potential of mean force of two hard spheres in water as a function of pressure by Hummer et al. showed that as pressure is increased the solvent separated minimum becomes stabilized relative to the contact minimum.²¹² This led the authors to suggest a model in which the pressure denatured states of a protein consists of an ensemble in which the hydrophobic core of the protein is disrupted by water molecules that are stabilized between hydrophobic groups in the protein. To test the hypothesis that water will penetrate the hydrophobic core of a protein at high pressures, Collins et al. studied a mutant of T4 lysozyme (L99A), which opens a 160 A³ cavity in the protein by X-ray crystallography at various pressures.^234,235 It was found that at 0.1 MPa (1 atm), the cavity was empty, but at pressures of 100 MPa, they observed water density corresponding to a population of half a water molecule in the cavity. At higher pressures, 200 MPa, they found that four molecules occupy the cavity. Therefore, these experiments demonstrated that modest pressure can change the hydration of protein cavities, and this change occurs within a narrow range of pressures, suggesting a cooperative effect. At 200 MPa, T4 lysozyme remains folded.

The susceptibility of T4 lysozyme L99A mutant to pressure is suppressed when the protein hydrophobic cavity is occupied by a benzene molecule.²³⁶ Merski and co-workers determined the structure of eight alkyl benzenes bound to the L99A mutant of T4 lysozyme and pointed out that low-energy excited states of the protein can be able to accommodate increasingly larger ligands.²³⁷ Lerch and co-workers found that under high pressure, there is a small fraction of the population of the protein in which protein side chains fill the hydrophobic cavity.²³⁸ The balance between volume changes and water penetration is fragile. For example, the effect of pressure on T4 lysozyme L99A mutant is suppressed when the protein is encapsulated in reverse miscelles. Nucci and collaborators found that under these confinement conditions, the hydrophobic cavity is preferentially filled with water (as found in crystal structures) rather than driving the system to the unfolded state, which is unfavorable under the nanoscale volume of the reverse micelle.²³⁶ Maeno et al. combined high-pressure NMR, MD simulations, and 3D RISM calculations to conclude that the water containing cavity in the L99A mutant of T4 lysozyme are the source of conformational fluctuations.²³⁹

Roche et al. have shown that protein cavities are the main source of volume changes upon pressure denaturation of proteins.^{211,215–217,240} Solvation affects are also present but are not the dominant contributor to volume changes. MD dynamics simulations on a small protein have been used to characterize the changes in solvation between folded and unfolded states.²⁴¹ It has been shown that most of the solvation changes occur around hydrophobic groups, while polar groups do not exhibit significant changes in water density with pressure. The small changes in water density around polar and charged groups occur since the density of water around these groups is already higher at low pressures due to electrostriction.

Molecular dynamics studies of the pressure denaturation of α-helices showed that pressure does not destabilize α-helices,^232,242 yet experimental studies have shown that pressure stabilizes α-helices since they form compact, low-volume conformations.^243–245 This is consistent with the observation of high α-helical content in the unfolded state of proteins at high pressures and room temperature.^214,246

The kinetics of proteins is also affected by pressure. Brun et al. studied the folding kinetics of mutants of SNase by fluorescence spectroscopy and determined the structural and physical character of the transition state ensemble of the protein.²¹⁹ They found a large activation volume for folding, showing that the major effect of pressure is to decrease the folding rate. When the protein occupies the transition state ensemble (TSE), the core of the protein is dehydrated. However, mutants that introduced ionizable groups into the protein core show an increase in the unfolding rate due to a large negative activation volume for unfolding. This suggests that the effect of pressure on proteins depends on the protein, since a single mutation can result in large variations in the activation volume of proteins.

The role of hydration on protein folding has been studied by molecular dynamics simulation. Umbrella sampling,^247–249 REMD,²⁵⁰ coarse-grained models,²⁵¹ and knowledge-based (structure prediction) models²⁵² have been used to study the hydration of the protein along the folding progress, as represented by structure-based order parameters. These calculations showed that, in addition to inducing the hydrophobic collapse of the chain, water guides the folding process by forming folding intermediates with hydrated cores. The hydrophobic cores are then dehydrated in the final steps of folding.²⁵¹ Predictions obtained for SH3 using a modified minimalist model²⁵¹ that used solvent-mediated hydrophobic interactions²⁵³ were confirmed experimentally.²⁵⁴ The presence of a hydrated hydrophobic core in unfolded states is consistent with pressure having an effect on the folding rate of proteins.⁴³

5.3. Characterization of the Unfolded State of Proteins

The influence of the secondary structure on thermal and chemical denatured states has been investigated for different proteins in aqueous solutions: phosphoglycerate kinase (PGK),²⁵⁵ β-casein,²⁵⁶ neocarzinostatine (NCS),²⁵⁷ bovine pancreatic trypsin inhibitor (BPTI),²⁵⁸ apo-calmodulin,²⁵⁹ and β-lactoglobulin (βLG).²²⁵ Recently, the heat-denatured states of apo-calmodulin that is a small, mainly a, soluble protein have been studied. Conformational differences between native and thermal denatured states have been detected by using SANS and other biophysical techniques. The results, shown in Figure 10, show that secondary and tertiary structures of apo-calmodulin evolve in a synchronous way, indicating the absence in the unfolding pathway of molten-globule state sufficiently stable to affect transition curves. From SANS experiments, at 85 °C, apo-calmodulin adopts a polymer chain conformation with some residual local structures. After cooling down, apo-calmodulin recovers a compact state, with a secondary structure close to the native one but with a higher radius of gyration and a different tyrosine environment. In fact, on a time scale of a few minutes, heat denaturation of apo-calmodulin is partially reversible, but on a time scale of hours (for SANS experiments), the long exposure to heat lead to nonreversibility due to some chemical perturbation of the protein. Mass spectrometry measurements have permitted evidence of dehydration and deamidation of heated apo-calmodulin.²⁵⁹

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Figure 10

(A) Guinier representation [ln I = f(Q2)] of SANS spectra of apo-calmodulin at 5 g/L for several temperatures, between 20 °C (in blue) and 85 °C (in red). Straight full lines are fits using Guinier laws leading to radius of gyration. The red curved full line is a fit of SANS spectra of apo-calmodulin at 85 °C, using the Debye law. (B) Radius of gyration, R _g, of apo-calmodulin in D₂O at 5 g/L as a function of temperature. In black is the fit with a three-state transition model, with the same thermodynamic parameters as for circular dichroism. (C) Adapted Kratky [Q2.3.I(Q) = f(Q)] representation of SANS spectra at 18 g/L of apo-calmodulin at several temperatures between 20 °C (in blue) and 77 °C (in red). (D) Adapted Kratky [Q2.3. I(Q) = f(Q)] representation of SANS spectra at 5 g/L of apo calmodulin at 20 °C (in blue) and 85 °C (in red). The samples are solutions of apo-calmodulin in a 50 mM D2O Tris buffer, pD = 7.6 with 80 mM KCl and 500 μM EDTA. The figure and caption are reproduced with permission from ref ²⁵⁹. Copyright 2012 Elsevier B.V.

p_f is Linked to the Water Diffusion Coefficient within the Pore

p _f is defined by the volume of water that a narrow channel conducts per unit of time in response to an osmotic pressure difference:

where Φ_w and Δc _s are the osmotic water flux (in mole per second) and the difference in impermeant solute concentration in the two solutions at both channel ends (in mole per unit volume). With the assumption that the water densities inside and outside of the channel are equal to each other, Φ_w can also be expressed as the product of the average linear drift velocity, υ, and water concentration, c _w:²⁸⁶

where A, F, and γ are the channel cross section, the force acting on an individual water molecule, and the molecular friction coefficient, respectively. Eq 2 can be rewritten since F = A Π = AΔc_sN_Ak_bT, where A = v _w/z:

where D _w, N _A, and z and are the diffusion constant of the water molecules within the channel, Avogadro’s number, and the distance between water molecules, respectively. Eqs 1 and 3 combine to give²⁸⁹

Eq 4 corresponds well with p_f = υ _w k ₀ ^290,291 or p_f = 1/2Φ₀ υ_w,²⁹² where k ₀ and Φ₀ are defined as the rate by which all single-file water molecules shift by the separation distance between two neighboring water molecules or the intrinsic flux, respectively.

7.1. Mobility of Single-File Water Molecules as Judged from p_f Values

To visualize single-file water diffusion, molecular dynamics simulations have been employed. Bulk water models^293,294 are commonly used since the calculated p _f values do not deviate much from experimental ones; however, it is not really clear to which extent these models reflect the physical and chemical properties of confined water.²⁹⁵ For example, p _f of the human red cell water channel protein, aquaporin 1 (AQP1), varied in silico between 0.7 and 1 × 10^-13 cm³ s^-1.^293,294 Part of these data are in reasonable agreement with recent experiments in which p _f was found to be 5 × 10^-13 cm³ s^-1.²⁸⁷ Thus, they confirm the observation that confinement of water to subnanometer films does not grossly alter its fluidity.²⁹⁶

In accordance with eq 4, the p _f of 5 × 10^-13 cm³ s^-1 indicates that water molecules retain bulk water mobility. Water rushes through the slightly wider bacterial glycerol facilitator GlpF even faster as indicated by its p _f of 19 × 10^-13 cm³ s^-1.²⁸⁷ That is, within the shorter single-file region of GlpF, water molecules attain a mobility that is slightly above that of bulk water molecules.

7.2. Energetics of Single-File Water Transport

When water enters the pore, it is partially dehydrated because every molecule may retain no more than two of the four hydrogen bonds that it forms with neighboring water molecules. Usually only a fraction of the lost energy can be recovered through interactions with pore-lining residues. In the extreme case of water conduction through the hydrophobic channel of a carbon nanotube, there is no replacement for the lost hydrogen bonds. The carbon atoms of the nanotube only allow for van der Waals interactions, which recover no more than 4 kcal mol^-1 of the 10 kcal mol^-1 entrance penalty in silico.^262,297

The energetics of water passage through gramicidin-like peptidic pores is different. Hydrogen bonds may form between the single-file waters and the backbone of amino acids lining the pore.²⁹⁸ Although the backbone carbonyls may act as surrogates for the waters of hydration, the rate at which water can transform its hydration environment from that of the channel exterior to that of the interior was identified as the major source of the resistance to water permeation.²⁹⁸ Molecular dynamics simulations of water transport through the pore of a d,l-polyalanine helix revealed an access barrier of 3.6 k _B T (or 2.1 kcal mol^-1). Since the calculated energy for movements between binding sites within the pore was on the order of k _B T,²⁸⁹ the access barrier appeared to dominate the whole transport process. It has however remained unclear why the smaller transport barrier did not translate into a higher p _f of the peptidic channels: p _f was more than an order of magnitude smaller than that calculated for carbon nanotubes.^289,297 Systematically varying the polarity of the d,l-polyalanine helix with the gramicidin A fold did not solve the conundrum: the access penalty was always larger than the barrier for permeating the pore.²⁹⁹ Transient channel occlusions by lipid head-groups³⁰⁰ explained part of the phenomenon: gramicidin acetylation increased p _f by sterically preventing pore occlusion.³⁰¹

Yet, even without pore occlusion, water permeation through the peptidic pores remained slower than through carbon nanotubes. The experimentally found dependence of p _f on channel length³⁰² provided the first clue for a better understanding of the permeation process. p _f rapidly decreases with increasing channel length because the number of backbone carbonyls which may form hydrogen bonds with the single-file waters increases. Similarly, an increase in the total number of hydrogen bonds (N _h) that any of the single-file water molecules may form with pore-lining residues while traversing aquaporins decreases their p _f.²⁸⁷ In order to obtain that result, the accuracy of p _f measurements was improved by (i) minimizing unstirred layer effects due protein reconstitution into vesicles that were not larger than 120 nm, (ii) precisely counting the number of reconstituted proteins per proteoliposome using both fluorescence correlation spectroscopy and high-speed atomic force microscopy, and (iii) by a new adaptation of the Rayleigh-Gans-Debye equation to follow the kinetics of vesicle shrinkage from the intensity of scattered light. The bacterial potassium channel KcsA, which may also conduct water^290,303 falls into the same logarithmic dependency between N _h and p _f (Figure 13).

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Figure 13

Total number N _H of hydrogen bonds that single-file water molecules may form with pore-lining residues while traversing the channel governs water mobility. The latter is expressed in terms of the water diffusion coefficient D _w in the pore. The data for the peptidic channels gramicidin, midigramicidin, and minigramicidin are from refs ³⁰² and ³⁰⁵ for the potassium channel KcsA from Hoomann et al., 2013 (with corrections as outlined in ref ²⁸⁷), for the aquaporins AQP1, AQPZ, and GlpF from ref ²⁸⁷ and for nanotubes from ref ²⁹⁷. The figure is reproduced with permission from ref ²⁸⁷. Copyright 2015 American Association for the Advancement of Science.

It is not entirely clear as to why the dependency should be logarithmic. Assuming that the breaking and forming of a hydrogen bond occurs at a constant rate would rather lead to a linear dependence. Strictly speaking, this is only true if there were a lone water molecule in the pore that may have formed just one bond at a time. If, however, a whole column of water molecules was moving in a concerted fashion, all of the singlefile waters would have had to break their hydrogen bonds with pore-lining residues at the same time. Even if such an event was to happen, neighboring water binding sites may be situated very close to each other, so that there is only slight progress (if any) after bond reformation. For example, yeast aquaporin-1 offers four water positions in the selectivity filter that are too closely spaced to be simultaneously occupied.³⁰⁴ Thus, it seems to be the joint multiplicity of all the binding options of all singlewater molecules taken together that result in the logarithmic dependence of p _f on N _H, since every one of them is capable of simultaneously forming four hydrogen bonds.

7.3. Channel Geometry

Apart from the gating peculiarities of individual water channels, the opening angle of the pore is also predicted to affect p _f. The calculated p _f was lowest for channels with cylinder geometry and highest for channels with an opening angle that matched that of the hourglass shape of aquaporins.²⁸⁸ The single-file movement inside the central part of the AQP was not captured in these calculations at all, since a continuum description was used. That is, the single-file movement was assumed to occur free of any resistance.

Experimental support for this oversimplified view has yet to be obtained. Figure 13 compares channels with cylindrical geometry (peptidic pores: gramicidin, minigramicidin, and midigramicidin), hourglass symmetry (aquaporin-1, GlpF, and AqpZ), and channels (KcsA) with a cylindrical entrance on one side and a vestibule on the other. They all fit into the logarithmic dependence of p _f on N _H, suggesting that the main resistance has to be attributed to the narrowest region (constriction site) and not to the much wider entrance regions. Nevertheless, the shorter cylindrical peptidic pores (minigramicidin) are likely to have a smaller hydrophobic thickness than the surrounding lipid so that an hourglass-shaped lipidic entrance may have formed, which could be responsible for the increment in p _f above the regression line (Figure 13). That is, pore geometry may well act to modulate the p _f value which is chiefly determined by N _H.

This work was supported by the National Science Foundation in the USA (MCB-1050966 to AEG) and the Austrian Science Fund (FWF, Grant P23679 to P.P.). F.S. acknowledges funding from the European Research Council (ERC) under the European Community’s Seventh Framework Programme (FP7/2007–2013), Grant 258748. A.E.G. acknowledges support from Los Alamos National Laboratory LDRD funds. This review was initiated during the NORDITA (Nordic Institute for Theoretical Physics) scientific program “Water - the Most Anomalous Liquid”. Additional financial support for this program was provided by the Royal Swedish Academy of Sciences through its Nobel Institutes for Physics and Chemistry, by the Swedish Research Council, and by the Department of Physics at Stockholm University.