Introduction

Under certain conditions, a fluid flowing through a linear channel can form secondary currents transversal to the primary flow1. Such helical or swirling flows are widespread at different length scales in nature and have a significant impact on physical, chemical, and biological processes. Helical streams are the major factor of cut banks erosion and cliffs formation in river bends1,2. Vortices formation accompanies the transition into a superfluidic state in liquid 2He and 3He, being considered a signature of superfluidity3. The blood helical flows in the human aorta enhance the oxygen flux to the arterial wall thus protecting the aorta from atherosclerosis4. At the microscale, swirling flows in microfluidic channels can be induced by the channel geometry5,6 or by patterning the surface charges7 or the wettability8,9 of the walls. Such channels are used in microfluidic systems and lab-on-a-chip devices for promoting the solutions mixing5,6 or implementing unidirectional flows in microfluidic valves and diodes10,11.

Although water flows in nanochannels such as aquaporins, carbon or peptide nanotubes (NTs) have been actively studied experimentally12,13,14,15,16,17, theoretically18,19,20, and by molecular dynamic simulations20,21,22,23,24,25,26, it is generally assumed that flows in nanometer-sized channels are predominantly laminar and uniaxial due to the low Reynolds numbers5,27. There are a few studies reporting the modelling of transverse flows in the nanochannels with patterned surfaces21,22,23, but the experimental evidence of such flows is still missing.

Herein, on the example of archetypical self-assembling diphenylalanine (H–Phe–Phe–OH, FF) peptide NTs28, we demonstrate the existence of unusual helical water flows in nanotubular channels of a sub-nanometer diameter. FF is one of the most studied dipeptides demonstrating the self-assembly into micro- and nanotubes with spectacular physical properties, such as efficient water diffusion16, remarkable piezoelectric29, pyroelectric30, electronic28,31,32, and optical33 properties. Moreover, peptide NTs are generally considered as models of transmembrane channels34,35,36. Therefore, the effects observed in peptide NTs may find an analogy in biological systems.

Results and discussion

Three types of water molecules in diphenylalanine nanotubes

FF NTs filled with H2O (compound 1) and D2O (compound 2) are formed via a self-assembly process following the standard assembling scheme29 (see the “Methods” section). The individual open-ended helical NTs with an inner diameter of 0.92 nm37 and typical helix step c = 5.46 Å38 assemble in hexagonal microbundles belonging to the P61 space group (Fig. S1). The bundles can possess one or several microscopic holes or no holes at all (Fig. S2). The inner hydrophilic surface of the NTs consists of positively charged amino and negatively charged carboxyl groups, whereas aromatic phenyl groups form the hydrophobic outer surface (Fig. S1b). During the self-assembly, water molecules from the solution are captured inside the nanochannels16, where they form layers of hydrogen-bounded and dynamically disordered mobile water along the NT axis (Fig. 1a). The structure of these layers dramatically changes the physical properties of FF NTs31,32,39,40,41, and has been a matter of detailed research in numerous studies16,37,40,41.

Fig. 1: The structure of FF NTs and characteristics of three types of water therein.
figure 1

a Molecular structure of FF nanochannel and preferable locations of water molecules therein. Bound water molecules are derived from single crystal X-ray diffraction data and combined with DFT calculations32. Red spheres denote preferable positions of oxygen atoms pertaining to mobile water molecules. Cyan lines depict hydrogen bonds between bound water and FF molecules. b Typical 2H MAS NMR spectrum of FF MNTs filled with D2O (compound 2) and c deconvolution of the central lines and one of the side bands. d–f Three components extracted from the deconvolution of 2H MAS NMR spectrum (intensities are normalized by the maximum for better visibility) and related to D2O molecules bound to NH3+ (d) and COO– groups (e), and dynamically disordered mobile D2O molecules located in the cavities (f). g Determination of the transverse relaxation time T2 for three types of D2O molecules captured in FF NTs. Inset shows the fitting equation. The obtained values of T2 are presented in Table 1.

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The FF NTs filled with D2O (compound 2) is a convenient object to be studied by 2H solid-state NMR, which is a powerful method for investigating the dynamics of water confined in various micro- and mesoporous materials42. Indeed, the 2H MAS NMR spectrum of FF NTs shown in Fig. 1b depicts the typical signal features of a rigid water environment: a pronounced central line flanked by a set of side bands, whose envelope is modulated by the 2H quadrupole coupling interaction. The deconvolution of the 2H NMR spectrum reveals three components with central lines (isotropic chemical shifts) centred at 0.55, 0.51, and 0.21 kHz (Fig. 1c). Two of these components participate in quadrupole interaction with the quadrupole coupling constants (CQ) listed in Table 1. The first spectral component with the isotropic chemical shift centred at 0.55 kHz (8.99 ppm) displays a Pake doublet pattern with CQ = 197.61 kHz and the asymmetry parameter ηQ = 0.2 (Fig. 1d) typical of a “rigid” water with a strong quadrupole coupling (CQ varies from 160 to 340 kHz43). A second, bell-shaped component with the isotropic chemical shift centred at 0.51 kHz (8.25 ppm), exhibits a CQ = 110.19 kHz and ηQ = 0.65 demonstrating a weaker quadrupole coupling (CQ < 150 kHz) compared to the first component (cf. Fig. 1d, e). The third spectral component consists of a single sharp peak centred at 0.21 kHz (3.41 ppm), suggesting that water molecules associated with this resonance undergo isotropic motion (Fig. 1f).

Table 1 NMR parameters and water diffusion coefficients retrieved from 2H NMR spectra
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The presence of several components in the 2H NMR spectra points towards the existence of at least three types of water molecules confined in FF nanochannels. This is in accordance with the layered water structure revealed from the X-ray diffraction32,37, thermal analysis, and water vapour sorption measurements16. Detailed study of the dielectric relaxation times32 also revealed three types of water with different dielectric behaviour: (1) water hydrogen-bounded to NH3+ groups of FF molecules, (2) water hydrogen-bounded to COO– groups, and (3) water confined in cages and not interacting with FF molecules. These types of water molecules are well correlated with the quadrupolar parameters and chemical shifts extracted from the experimental spectral components (Table 1). In particular, CQ is extremely sensitive to the rigidity of water molecules from which the strength of hydrogen bonds involving those water molecules can be estimated. In other words, a larger CQ corresponds to a more rigid water environment. The strongest hydrogen bonding is established for Type 1 water presenting the highest CQ (197.61 kHz), while water of Type 2 exhibits the smallest CQ value (110.19 kHz). It is also worth mentioning that water molecules engaged in stronger hydrogen bonding led to 1H resonances appearing at higher chemical shifts44. A similar trend is observed in our data with the 2H resonance frequency values, which increase for water molecules engaged in stronger hydrogen bonds (Table 1). Finally, noninteracting Type 3 water molecules (CQ = 0 kHz) possess much higher mobility than the other two water species. This component disappears from the NMR spectrum upon the NTs dehydration for 3 h at 80 °C in open air (Fig. S3a).

Density Functional Theory (DFT) calculations were performed to verify the experimental 2H central lines assignments using a molecular structure of the NT determined from X-ray diffraction38. The oxygen atoms corresponding to bound water molecules were complemented by deuterium atoms (D1D2O and D3D4O for Types 1 and 2, respectively, Fig. S3b) as their positions cannot be obtained by X-ray diffraction. Full structure geometry optimization under periodic boundary conditions has been implemented. The theoretically obtained 2H frequencies (Table S1), corresponding to deuterons D1 (from Type 1 water molecules, 0.55 kHz) and D3 (from Type 2 water molecules, 0.46 kHz) involved in the formation of rigid hydrogen bonds with the peptide molecules are in good agreement with the experimental spectrum (0.55 and 0.51 kHz, respectively, Fig. S3b). The two other deuterons, D2 and D4, do not show corresponding lines in the NMR spectrum, supposedly because they are involved in the formation of disordered/mobile hydrogen bond networks with other water molecules.

2H NMR spectroscopy can also be used to estimate the diffusion coefficient (D) of various molecules confined in porous materials42. The method is based on the general Stokes–Einstein equation for diffusivity: D = 〈l2〉/6τ, where 〈l2〉 is the mean square displacement of the molecule in the pores, and τ is the diffusion time. If the geometry of the pore in the material and the expected diffusional motion of the molecule are known, 〈l2〉 can be substituted with the square of the mean distance between adsorption sites, whereas the diffusion time τ can be associated with the reorientational correlation time τc derived from 2H NMR measurements42. The correlation time, in turn, is τc = s–1ωQ–2T2–1, where T2 stands for the spin–spin relaxation time, ωQ = 0.24 MHz is the 2H quadrupole frequency, and s is a numerical coefficient depending on the rotational motion45. For a two-fold rotation of the D2O molecule with the rotation angle α = 104.45°, s = ½ (3cos2α–1) = 0.4146.

The relaxation times T2 for three types of D2O molecules were determined using the Quadrupolar Carr–Purcell–Meiboom–Gill (QCPMG) sequence47. The central lines shown in the 2H NMR spectra have been deconvoluted, and the peak areas (A) were determined and plotted against the delay time (τd) between pulses (Fig. 1g). An exponential decay function A = a exp(–τd/T2) was used for the curve fitting, allowing to determine the transverse relaxation times T2 and the correlation times τc (Table 1). It is important to note the absence of exchange between bound water molecules (Types 1 and 2) and mobile water (Type 3) because their correlation times, associated with the water mean residence time at one adsorption site, differ by three orders of magnitude (Table 1).

The distance between adjacent adsorption sites of Types 1 and 2 water molecules can be found from the crystallographic data complemented with DFT calculations32, while the longitudinal jump-like diffusion of mobile water should occur along the nanochannel axis to the same position in the next equivalent FF ring. In this case, 〈l2〉 for the mobile water (Type 3) should be equal to the square of the helix step c = 5.46 Å of the helical NT37. Due to their similar correlation time, water molecules of Types 1 and 2 can exchange, and, therefore, the nearest adsorption sites for bound water molecules are located at an average distance of about 3.31 Å (Fig. S4), noticeably closer than that of the mobile water molecules.

The calculated diffusion coefficients D are given in Table 1. The value of D = 1.09 × 10−10 m2 s−1 obtained for the mobile water is in good agreement with D = 1.30 × 10−10 m2 s−1 determined earlier from direct measurements by dynamic vapour sorption16 and is comparable with the values measured in a variety of other nanostructured materials (Table S2). The diffusion coefficients of Type 1 and 2 water are similar, although still marginally higher for the Type 2 molecules bounded to the COO– anion due to weaker hydrogen bonds (Table 1). Naturally, the diffusion coefficient of the Type 3 mobile water is about three orders of magnitude higher than that of the two types of bound water (Table 1).

The obtained results allow us to draw an important conclusion. Since water molecules of Types 1 and 2 tend to interact strongly with the peptide shell and do not exchange with the mobile water, their trajectories, once the flow is initiated, should reproduce the helical structure of the NT. The difference in the diffusion coefficients makes the slow helical flow of bound water molecules experimentally distinguishable from the fast axial and translational flow of the mobile water.

Controllable axial and helical flows

To initiate the diffusion-governed water flows in the FF NTs and determine their main characteristics we have performed room temperature H2O and D2O dynamic water vapour sorption (DVS) measurements for compounds 1 and 2, respectively, similar to those made in ref. 16. Briefly, the NTs were first dried at 65 °C for 2 h and then refilled with H2O or D2O vapour under different partial pressures (p/p0). For both compounds, the sorption isotherms are of type IV typical for mesoporous materials16. The D2O uptake is somewhat higher than that of H2O (Fig. 2a), which is in good quantitative agreement with the higher mass of D2O (molar mass ratio \({M}_{{{\rm {D}}}_{2}{\rm {O}}}/{M}_{{{\rm {H}}}_{2}{\rm {O}}}=1.112\)). The adsorption part of the isotherms allowed controlled filling of the NTs and getting the maximum number of adsorbed water molecules for both compounds of around 11.7 per unit cell (see the Supplementary Notes). However, stable water flows can be obtained at the desorption stage of the experiment because the water motion during desorption is free of kinetic limitations related to the potential barrier at the NT entrance16.

Fig. 2: Experimental observation of axial and helical water flows in FF nanochannels and their dynamic characteristics.
figure 2

a H2O and D2O adsorption−desorption isotherms measured by dynamic vapour sorption technique. b Temporal variations of the mass loss rate during the water desorption at different partial pressures and their fitting by Eq. (1). c Diffusion coefficients, and d fluxes of H2O and D2O molecules belonging to axial and helical flows appeared during the desorption experiments at various partial pressures (note the inverse p/p0 scale representing desorption).

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The time-dependence of the adsorbed water mass changes for a given partial pressure step (examples of which are shown in Fig. 2b) can be analysed using the diffusion equation already adopted in our previous work16 and modified herein for two independent flows in the nanochannel:

$$\frac{\partial M}{\partial t}(t)= \mathop{\sum}_{i=1,\,2}\frac{\partial {M}_{i}}{\partial t}=\mathop{\sum}_{i=1,\,2}\left\{4{C}_{0}\frac{{D}_{i}}{{x}_{0}}\right\}\left[\exp \left(-\frac{{D}_{i}}{{x}_{0}^{2}}\frac{{\pi }^{2}}{4}t\right) \right. \\ \left.+\exp \left(-\frac{{D}_{i}}{{x}_{0}^{2}}\frac{9{\pi }^{2}}{4}t\right)+\exp \left(-\frac{{D}_{i}}{{x}_{0}^{2}}\frac{25{\pi }^{2}}{4}t\right)\right]$$
(1)

where M is the total mass of the sample, Mi and Di are the masses and diffusion coefficients of bound (helical flux) and mobile (axial flux) water (H2O or D2O), C0 is the water concentration outside the NTs, and x0 corresponds to a half of the average length of the NTs (see the Supplementary Notes and Fig. S5). Fitting of the transient data to Eq. (1) is straightforward (Fig. 2b) and yields estimates of the helical and axial D values and the corresponding fluxes (J) as summarized in Table 2 (see the Supplementary Notes for details).

Table 2 Main characteristics of axial and helical flows in FF NTs filled with H2O and D2O at partial pressure 0.5
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For both compounds 1 and 2, two distinct values of D were found at each water vapour partial pressure (Fig. 2c). One of the flows is characterized by D values of about 10−10 m2 s−1, which is comparable with the diffusion coefficient of mobile water (attributed to the axial flow) derived from 2H NMR spectra. Another type of flow exhibits a much lower D of about 10−12 m2 s−1 (Fig. 2c), and can thus be attributed to the diffusivity of the bound water molecules associated with the proposed helical flow. For each flow type, D values for D2O and H2O are very close and gradually decrease with lowering partial pressure. The sudden drop in D values occurred at relative pressures around 0.15–0.20 (Fig. 2c) is likely related to the decomposition of water clusters16.

The flux for the axial flow exceeds that of the helical flow for most partial pressures, whereas at p/p0 below 0.1, they become equal (Fig. 2d). Notice that in the pressure range from 0.28 to 0.38, the flux of the helical flow is close to zero, and thus only an axial flow occurs in the NTs (Fig. 2d). A subsequent reduction of the helical flow occurs at around 0.15. These features can also be related to the reconstruction of water clusters in the NTs accompanied by the change of the flow regime.

Molecular dynamic simulation of the water flows

Further mechanistic insights on the trajectories of the axial and helical water flows were obtained by molecular dynamics (MD) simulations. The study was done on a NT consisting of 50 consequent helical steps (Fig. S6) completely saturated with 1200 water molecules, which correspond to 24 molecules per FF ring. The flows were induced by the application of a constant axial force of 0.63 kJ mol−1 Å−1 to each water molecule (about 13 MPa).

The coordinates of water molecules were recorded after each 1k simulation steps (1 ps), and the positions of their oxygen atoms were projected on the NT cross-sectional and longitudinal planes (Fig. 3). The cross-sectional density map consists of a series of dark spots indicating a higher concentration of water molecules in two preferential regions: near the peptide walls and around its longitudinal axis, corresponding to the bound and mobile water molecules, respectively. The bound water forms a hexagonally shaped solvation shell with a higher density (darker spots) in the vicinity of COO− groups (Fig. 3a), thus suggesting that water tends to interact more with the carboxyl groups. This is additionally confirmed by the distribution of the water hydrogen atoms (Fig. S7), in agreement with previous findings40. An interesting feature of these spots is the presence of long diffuse tails oriented along the peptide shell (Fig. 3a). These tails denote regions occupied by the water molecules during much shorter times and can be interpreted as the result of a jump-like motion (diffusion) of the molecules between equivalent adsorption sites of the subsequent helical steps. Moreover, the orientation of these tails indicates that the bound water molecules diffuse along the NT following a clockwise direction, which is also apparent in the periodical helical arrangement along the NT shown in the longitudinal density map (Fig. 3b). In fact, the helical flow twisting direction should depend on the chirality of the NTs depending, in turn, on the chirality of their FF molecules. Thus, in right-handed FF NTs assembled of l,l-enantiomers as used in this work, the diffusion of bound water should indeed lead to a clockwise flow, whereas a counter-clockwise flow is expected for a left-handed helix of NTs made of d,d-enantiomers48.

Fig. 3: Molecular dynamics simulation of water flows in FF NT under an external pressure of 13 MPa.
figure 3

a Projection of the coordinates of the water oxygen atoms on the NT cross-section plane during the flow simulation highlighting the hexagonal solvation shell formed by the bound water and the diffuse circular region around the NT longitudinal axis formed by the mobile water. b The projection of bound water oxygen atoms on the longitudinal section plane, where the solid lines show the water helical arrangement (pressure is applied from left to right, and oxygen atoms related to the mobile water are removed for clearness).

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The cross-sectional density map for mobile water represents a diffuse circular region around the NT longitudinal axis (Fig. 3a). Contrary to the bound water, mobile water molecules do not present any significant ordering due to much weaker interaction with the wall ionic groups. Nevertheless, the hydrogen distribution map reveals the formation of relatively stable hydrogen bonds with oxygen atoms of Type 1 bound water (Fig. S7). Though this interaction may endow mobile water with a certain ordering, however, this effect is blurred by thermal effects and is difficult to be observed. Thus, the resulting flow of the bulk molecules should not obey a special structural order, confirming the proposed laminar nature.

The D values estimated from MD simulations for waters are presented in Table 2, where they are compared with the DVS and NMR results. For mobile water, the MD value is comparable with that estimated by DVS. As expected, the analysis of the axial velocity of the bound water yields lower D values than for the mobile water. However, they are higher by one or two orders of magnitude than the DVS and NMR data. This is probably due to the fact that MD simulations did not take into account bound water dissociation and, thus, additional electrostatic interactions with the peptide shell. Indeed, to maintain overall electroneutrality, the charged amino and carboxyl functional groups in the peptide shell should be screened by OH− and H3O+ ions, respectively, eventually leading to a more complicated pattern of water motion. The transition of water molecules from bound to mobile states and vice-versa, occasionally observed in the simulations, can also be attributed to this effect. If OH− and H3O+ ions participate in the helical flow, the exchange between bound and mobile states would be strongly reduced. These effects will be considered in future work.

Future perspectives

The helical water flows studied in this work may have a potentially high impact on various fields of science and applications. In biophysics, self-assembling peptide nanochannels and helical proteins are recognized models of transmembrane ion channels34,35,36, where the diffusion of the ions, such as potassium, sodium, or chlorine, is largely determined by the dynamics of the surrounding water molecules. Currently, this effect is still poorly studied, and water is often represented as a structureless dielectric medium49. However, helical flows in such protein-based ion channels as voltage-gated human CaV3.3 channel50 or OmpF porin51 may influence the ion's osmotic transportation through the cell's membranes, and thus affect various biochemical processes, including cellular metabolism.

In nanochemistry, the helical flows amenable to the control by the external water vapour pressure may find application in controlling chemical reactions in nanoconfinement52,53, in water harvesting by MOFs54, or in osmotic power generation55, where the mixing of picoliter volumes of salt solutions are required. These effects can be further exploited in various nanofluidic, lab-on-a-chip, and organ-on-a-chip devices.

We hypothesize that similar helical flows of water or other molecules may occur in other helical peptide and protein-based NTs, some kinds of zeolites, covalent organic and metal-organic frameworks (COFs and MOFs, respectively), porous organic polymers (POPs), and other types of nanochannels with helicoidally distributed ionic groups, thus further expanding the range of applications, where the helical flows can be important.

To conclude, a combination of quadrupolar solid-state NMR spectroscopy, DFT calculations, molecular dynamics simulation, and dynamic water vapour sorption measurements was used to analyse water diffusion inside the self-assembling FF nanochannels with a diameter below 1 nm. The obtained results indicate a helical water flow coexisting with an axial laminar flow. The helical trajectory of the flow originates from the screw-like distribution of ionic groups in the channel walls, while its flux can be controlled by external water vapour pressure. These two flows are independent of each other, immiscible, and are associated with two distinct diffusion coefficients that differ by several orders of magnitude. Helical flows of water or other molecules may occur in other types of nanochannels with helicoidally distributed ionic groups such as zeolites, COFs, MOFs, POPs, etc., and be exploited for controlling chemical reactions, water harvesting, osmotic power generation, advanced lab-on-a-chip and organ-on-a-chip devices, and many other applications.

Methods

Samples preparation

A 100 mg/mL stock solution was prepared by dissolving the lyophilized powder of l,l-diphenylalanine (H-l-Phe-l-Phe-OH, FF, Bachem, Switzerland) in 1,1,1,3,3,3-hexafluoro-2-propanol (HFIP, Merk, Germany). The nanotube growth was initiated by mixing 100 mL of the stock solution and 900 mL of deionized water. The self-assembly started immediately and after 24 h, the obtained solution with FF NTs was evaporated at room temperature to get NTs. For 2H NMR measurements, the deionized water was replaced by D2O (Sigma-Aldrich, USA).

2H solid-state NMR measurements

The 2H solid-state NMR measurements were performed using a Bruker Avance III WB 400 MHz spectrometer (9.4 T) using a magic angle spinning (MAS) frequency of 10 kHz. A 4 mm double-resonance MAS probe (Bruker) was used, and the chemical shifts were calibrated in a static mode using liquid D2O as a secondary standard centred at 4.8 ppm. A π/2–τ–π/2 rotor-synchronized solid echo pulse sequence was employed to acquire the 2H spectra. The pulse parameters are as following: π/2 pulse width is 2.5 μs, solid-echo delay τ = 97.5 μs, π/2 pulse strength of 100 kHz, 1360 scans, recycle delay 6.5 s. The fitting of 2H MAS NMR spectra was done in the Dmfit software.

The true relaxation times T2 for D2O molecules of all three types were determined using the Quadrupolar Carr–Purcell–Meiboom–Gill (QCPMG) sequence47. The same Bruker Avance III WB 400 MHz spectrometer (9.4 T) with an MAS frequency of 10 kHz was used. The pulse parameters are as following: π/2 pulse width 2.5 μs, solid-echo delay time τ varied from 0.5 μs to 8 ms, π/2 pulse strength of 100 kHz, 64 scans, recycle delay 6.5 s. The area (A) under each central line in the obtained 2H MAS NMR spectra has been determined and plotted against the delay time (τd) between pulses (Fig. 1g). Fitting of this dependence by the exponential decay function A = a exp(–τd/T2) allowed determining the true rotational relaxation times T2 for all types of water molecules.

Water sorption measurements

H2O and D2O vapour sorption isotherms were obtained at 25.0 °C using a dynamic vapour sorption (DVS) device from Surface Measurement Systems with dry nitrogen (<3 ppm H2O) as a carrier gas with a total flow 200 sccm for both pretreatment and measurements. About 20 mg of the sample was loaded in a steel pan and suspended in the measuring chamber. A 120 min pretreatment at 65 °C in a dry nitrogen atmosphere and subsequent 60 min stabilization at 30 °C were performed to dry out the samples. Adsorption data were obtained under variable H2O or D2O vapour partial pressure (p/p0) steps from 0 to 0.5. Each pressure step was maintained until the rate of the mass change over time was lower than 0.002% for at least 10 min. In a few cases, where the stability criterion was not attained, the maximum stage time at each pressure step was limited to 360 min. Desorption curves were recorded after each adsorption isotherm by decreasing the p/p0 in the same steps and following the same procedure. The mass sensitivity of the equipment is 0.1 μg, vapour pressure accuracy 1%, and the temperature accuracy is 0.1 °C.

DFT calculations

Periodic DFT calculations were carried out with CASTEP version 19.1156. Atomic positions were converged with a fixed unit cell, using ultrasoft pseudopotentials57,58, the Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional59, a plane wave cutoff energy of 750 eV, and a k-point spacing of 0.05 × 2π Å−1. The Tkatchenko–Scheffler scheme was used to account for van der Waals interactions60. The convergence criteria were set to 1 × 10−7 eV per atom for the total energy, a maximum atomic force of 5 × 10−3 eV Å−1, and a maximum atomic displacement of 5 × 10−4 Å. NMR calculations were carried out using the gauge including projector augmented wave (GIPAW) method61, with the same parameters used in the geometry optimization step. 2H chemical shifts of D2O molecules were determined by referencing them to the calculated 1H chemical shielding of phenyl and CH groups of FF molecules, considering the experimental values of 7.26 and 4.6 ppm, respectively.

Molecular dynamics simulation

The molecular dynamics simulation has been done in LAMMPS package62. The peptide shell was simulated within CHARMM Generic Force Field (CGenFF)63, and a four-point TIP4P-Ew rigid water model64 was used. The energy of water-filled NT was minimized using molecular mechanics relaxation for 100 ps in the NVT ensemble with the Nosé–Hoover thermostat65 maintaining the constant bath temperature at 298 K and a time coupling of 0.1 ps. Time integration was conducted using a velocity-Verlet algorithm66 with a timestep of 1 fs. A cutoff distance of 1.2 nm was applied for both Lennard–Jones and electrostatic interactions, with the particle–particle particle-mesh (P3M) algorithm67 for electrostatic interactions. To induce water flows, axial external forces of 0.63, 1.05, 1.47, and 2.09 kJ mol−1 Å−1 were applied to the oxygen atoms of each water molecule that corresponds to the effective axial pressure of 13, 22, 31, and 44 MPa. The water behaviour was simulated for 7 ns (a longer simulation, for 12 ns, has yielded comparable results), and the positions of water molecules were saved every 1 ps. The obtained data were processed and analysed using Python scripting and MDAnalysis package68. The mean-squared-displacement (MSD) was computed, and the diffusion coefficient, D, was determined using the Stokes–Einstein equation. For more details, see the Supplementary Methods.