Electrostatic shape control of a charged molecular membrane from ribbon to scroll

Significance

Controlling the shape and internal architecture of assemblies of amphiphiles is critical for many technologies. The structure, and thus the function, of these assemblies reconfigures in response to stimuli, via mechanisms that are often elusive. Here, we observe and explain how molecular reordering driven by variations in electrostatic screening length induce micrometer-scale structural changes in crystalline membranes of charged, chiral molecules: The transformation of high aspect ratio, planar bilayers into scroll-like cochleates by increasing the solution salt content is described and explained. Our study suggests that this transformation should be general to charged bilayers possessing a spontaneous curvature.

Abstract

Bilayers of amphiphiles can organize into spherical vesicles, nanotubes, planar, undulating, and helical nanoribbons, and scroll-like cochleates. These bilayer-related architectures interconvert under suitable conditions. Here, a charged, chiral amphiphile (palmitoyl-lysine, C₁₆-K₁) is used to elucidate the pathway for planar nanoribbon to cochleate transition induced by salt (NaCl) concentration. In situ small- and wide-angle X-ray scattering (SAXS/WAXS), atomic force and cryogenic transmission electron microscopies (AFM and cryo-TEM) tracked these transformations over angstrom to micrometer length scales. AFM reveals that the large length (L) to width (W) ratio nanoribbons (L/W > 10) convert to sheets (L/W → 1) before rolling into cochleates. A theoretical model based on electrostatic and surface energies shows that the nanoribbons convert to sheets via a first-order transition, at a critical Debye length, with 2 shallow minima of the order of thermal energy at L/W >> 1 and at L/W = 1. SAXS shows that interbilayer spacing (D) in the cochleates scales linearly with the Debye length, and ranges from 13 to 35 nm for NaCl concentrations from 100 to 5 mM. Theoretical arguments that include electrostatic and elastic energies explain the membrane rolling and the bilayer separation–Debye length relationship. These models suggest that the salt-induced ribbon to cochleate transition should be common to all charged bilayers possessing an intrinsic curvature, which in the present case originates from molecular chirality. Our studies show how electrostatic interactions can be tuned to attain and control cochleate structures, which have potential for encapsulating, and releasing macromolecules in a size-selective manner.

Amphiphilic molecules can self-assemble into a variety of 3D, 2D, and 1D nano- and mesoscale structures. These structures serve as simplified models for understanding biological assemblies and their functions and have applications in drug delivery (1–5), regenerative medicine (6, 7), biosensing (8), hydrogen production (9, 10), and clean water technologies (11). An interesting assembly structure is the nanoribbon, which is a high aspect ratio (10:1 or greater) bilayer. Nanoribbons are a gateway to a number of other morphologies with distinct functionalities. For example, nanoribbons of a charged chromophore amphiphile can transform to a scroll-like (cochleate) morphology when the solution ionic strength is increased (9). These cochleates serve as efficient charge-transfer agents for photocatalysts in hydrogen production. Cochleate formation from liposomes of negatively charged phospholipids in the presence of multivalent cations also involves a nanoribbon intermediate (3, 12, 13). Biocompatible phospholipid cochleates are being explored as drug-delivery agents because they can trap macromolecules, such as proteins, and DNA, and provide protection against degradation due to their multilayer geometry. Nanoribbons have also been observed in peptide amphiphiles (PAs), which consist of a sequence of amino acids covalently linked to an alkyl tail (14, 15). For example, a peptide amphiphile that stimulates collagen production has been found to self-assemble into nanotapes with an internal bilayer structure (16). In a PA with alternating charged and neutral amino acids, nanoribbons were found to transform into helical ribbons as the PA concentration was reduced (17) and into helical and twisted nanoribbons when the amino acid sequence was permuted (18). Helical assemblies have been previously used to template semiconductor nanohelices (19). Despite the progress, the correlation between experimental conditions such as molecular design, ionic strength, pH, amphiphile concentration, and the attained nanoribbon-related morphology are not fully established. Therefore, precise control of nanoribbon-related architecture requires further understanding of the delicate interplay between intermolecular interactions and elastic and interfacial energies.

A recent theoretical study showed that for charged molecules, tuning the range of electrostatic interactions could induce transitions between different nanoribbon-related morphologies (20). Specifically, a phase diagram was deduced for a 2D lattice of charged points, which interacted via long-range repulsive electrostatic interactions and short-range attractive interactions. Planar nanoribbon to wavy ribbon with periodic undulations to helical ribbon transitions were predicted as the range of the electrostatic interactions is increased. This study suggests a facile method for accessing distinct nanoribbon architectures by varying the ionic strength (µ) of the solution because the range of electrostatic interactions as parametrized by Debye length (λ_d) scales as µ^−1/2. Recent experiments also attest that tuning the ionic strength leads to predictable changes in the nanoribbon-related assembly morphology. For example, the period of the twists in amyloid fibril aggregates monotonically decreases with decreasing ionic strength (21).

In this study, we analyze morphological changes in charged planar nanoribbons as a function of increasing ionic strength. In this regime, nanoribbon to cochleate transformations have been observed in phospholipids (12) and chromophore amphiphiles (9, 10, 22, 23). However, the generality and the mechanistic details of this transition are still unknown. In particular, the correlation between the ionic strength induced changes in the molecular packing and the mesoscopic morphology transformations are elusive. The principal aim of this study is to start with a nanoribbon structure and experimentally trace the micrometer to angstrom length-scale transformations in the membrane structure as a function of ionic strength by using a combination of cryotransmission electron microscopy (cryo-TEM), liquid-atomic force microscopy (liquid-AFM), and in situ small- and wide-angle X-ray scattering (SAXS/WAXS). The experiments are coupled with theoretical models that qualitatively explain the observed morphological transitions.

We chose a simple peptide amphiphile (PA), C₁₆K₁, with a single ionizable amino acid lysine (K) covalently linked to a palmitoyl (C₁₆) alkyl tail (Fig. 1A). This PA was chosen because our recent study (24) on C₁₆K₂ found spherical micelle to cylindrical micelle to a mixture of cylindrical micelle and nanoribbon transformations as the molecular charge was reduced by increasing the solution pH. Therefore, we hypothesized that removing one of the charged lysines from the headgroup could yield a macroscopic state consisting purely of nanoribbons. Second, the choice of this PA ensures that the interheadgroup interactions are Coulombic, unlike the case of PAs with multiple amino acids, where the assembly is strongly modulated by intermolecular hydrogen bonding.

Fig. 1.

(A) Molecular structure of +1 charged C₁₆K₁ with estimates for hydrophobic tail and hydrophilic headgroup lengths. The molecular conformation was derived from an MD simulation for a single C₁₆K₁ in water using the universal force field (25). (B) AFM image from a silica/water interface showing high aspect ratio C₁₆K₁ nanoribbons. (C) The height profile across a C₁₆K₁ ribbon (green line in B).

We first describe the C₁₆K₁ assembly in the absence of added salt. For this, L-C₁₆K₁ was dispersed in pure water at 4 mM concentration. Unless otherwise stated, the enantiomeric form of the amino acid [left-handed (L)] and the PA concentration (4 mM) are the same in all of the samples. The pH of this C₁₆K₁ dispersion was ∼4.6, which is much lower than the dissociation constant pK ∼ 7.4 for C₁₆K₁ in their aggregates (SI Appendix, Fig. S1). Therefore, under the experimental conditions, nearly all of the C₁₆K₁ are expected to be in their +1 ionized state.

The AFM image of C₁₆K₁ assemblies at a silica (SiO_x)/water interface (Fig. 1B) and other AFM images collected at different spots on the substrate reveal that in the absence of added salt, C₁₆K₁ assembles into flat ribbons, with widths (W) in the range of a few hundred nm, lengths (L) ranging from 2 to 20 µm, and aspect ratio (L/W) as high as 30. All of the ribbons exhibit the same thickness of ∼4.0 nm, as shown by a representative AFM height scan (Fig. 1C). This thickness is less than twice the length (5.4 nm) of fully extended C₁₆K₁ molecules (Fig. 1A), suggesting that the C₁₆K₁ ribbons are bilayers with the alkyl tails of the 2 leaflets interdigitated. The interdigitated bilayer configuration, which has also been observed in C₁₆K₂ (24) and other PAs (17), is expected for molecules with headgroup cross-sectional areas much larger than that for the alkyl tails.

Screening effects are analyzed in C₁₆K₁ dispersions that contain NaCl at concentrations (c) ranging from 0 to 100 mM. Fig. 2 A–D show AFM images of C₁₆K₁ assemblies at SiO_x/NaCl solution interfaces for c = 0, 1, 3, and 5 mM. Peak-force error images are shown as they deliver better 3D representation of the morphologies (26, 27). With increasing NaCl concentration the ribbon aspect ratio decreases, and at c = 3 mM nearly isotropic sheets of 1 to 3-µm diameter are observed (Fig. 2C). At or above this threshold concentration (c_th), the sheets roll into cochleates (Fig. 2D). AFM images depicting semirolled membranes and detailed screw-like features of the cochleates are shown in SI Appendix, Figs. S2 and S3. The multilayered nature of the scrolls is also observed in cryo-TEM images (Fig. 2 E–H and SI Appendix, Fig. S4). Fig. 2 F–H further reveal that the interbilayer spacing (D) within the cochleates monotonically decreases with increasing c. Overall, AFM and cryo-TEM show that increasing the ionic strength even over a narrow range first induces the ribbon to sheet to cochleate transitions, and thereafter reduces the interlamellar spacing within the cochleates.

Fig. 2.

(A–D) AFM peakforce error images of drop-cast C₁₆K₁ membranes at SiO_x/NaCl solution interface. As NaCl concentration increases, structural transformations are observed from nanoribbon to isotropic sheet and to rolled-up cochleates, which exhibit a screw-like pitch. (E–H) Cryogenic TEM images of cochleates exhibiting scroll morphology and the internal multilayer features. The interbilayer spacing D within the cochleate structure decreases as NaCl concentration increases.

We obtain ensemble-averaged, quantitative details of the mesoscopic morphology by SAXS and the molecular packing by WAXS in the ribbons and the cochleates. Fig. 3 A and B show the background-subtracted SAXS and WAXS data for C₁₆K₁ ribbons in pure water, as a function of scattering vector magnitude q = 4πsin(θ)/λ. Here, λ = 0.827 Å is the X-ray wavelength, and 2θ is the scattering angle. For q < 0.4 nm⁻¹ (Fig. 3A), the intensity I ∝ q⁻² is indicative of structures with extended sizes in 2D, which is consistent with the AFM observation of flat ribbons with length and width both greater than 2π/q_min ∼ 300 nm, where q_min is the minimum accessible q in the measurements. Fig. 3B shows strong Bragg reflections in the range 14 < q < 16 nm⁻¹ corresponding to 0.45 > 2π/q > 0.4 nm distances, which are close to the diameter of the alkyl tails. Therefore, within the ribbons, the alkyl tails pack on a crystalline lattice such that the nearest-neighbor distances are commensurate with the tail diameter.

Fig. 3.

(A) Background-subtracted in situ SAXS intensity profile for C₁₆K₁ nanoribbon. The solid black curve is the best fit over the range of 0.1 < q < 6 nm⁻¹ based on a symmetric bilayer model. (B) Background subtracted in situ WAXS intensity profile of C₁₆K₁ nanoribbon shows diffraction peaks over the range of 10 < q < 30 nm⁻¹. The solid black curve is a simulation based on a parallelepiped model for alkyl tails and the unit cell in D. (C) The interdigitated C₁₆K₁ bilayer structure and electron density profile determined from analysis of SAXS data. (D) Two-dimensional unit cell and lattice parameters for alkyl tail packing, derived from WAXS.

A symmetric bilayer model with a hydrophobic tail region sandwiched between 2 hydrophilic head regions is used for fitting the SAXS data (Fig. 3A). The tail region electron density (ED) was fixed at ρ_t = 320 e⁻/nm³, the value for crystalline, densely packed alkyl tails (28). Setting t_h ≤ 0.85 nm, based on a molecular dynamics (MD) simulation for a single C₁₆K₁ molecule in water (Fig. 1A), the best fit for the SAXS data (black curve, Fig. 3A) was obtained with thicknesses $t_t={2.3}_{-0.0}^{+0.5}$ nm and $t_h={0.85}_{-0.45}^{+0.0}$ nm for the tail and the head region, respectively, and ${\rho }_h={386}_{-0}^{+61}$ e⁻/nm³ for the head region ED. The most robust parameter, the bilayer thickness $2t_h+t_t={4.0}_{-0.4}^{+0.0}$ nm, is consistent with the AFM measurements (Fig. 1C). Consistent with the interdigitation hypothesis, SAXS reveals a much lower thickness of the hydrophobic region (2.3 nm) as compared to the length of 2 fully extended C₁₆ tails (2 × 1.9 nm). The SAXS observations are summarized in Fig. 3C.

The WAXS data for ribbons (Fig. 3B) show 3 strong Bragg peaks at q = 13.9, 15.0, and 16.2 nm⁻¹; consistent with crystalline packing of the alkyl tails into an oblique unit cell with lattice constants a_t = 0.49 nm, b_t = 0.53 nm, and γ_t = 127°. To distinguish between the tails from the 2 leaflets, the unit cell is recast with a = 0.49 nm, b = 0.85 nm, and γ = 100° (Fig. 3D). In this unit cell, the tails at the vertices and the center belong, respectively, to the 2 opposing leaflets. By using these 2D unit cell parameters and an untilted parallelepiped to model each tail (SI Appendix, section 2), we are able to simulate the positions and the intensities of the Bragg reflections for q > 10 nm⁻¹ (Fig. 3B, black curve). Complementary grazing incidence X-ray scattering (GIXS) measurement on C₁₆K₁ ribbons drop-casted onto a Si substrate revealed diffraction peaks at q_xy positions, which were identical to the q positions of the diffraction peaks observed in solution WAXS (SI Appendix, Fig. S5). Here, q_xy is the component of the scattering vector in the bilayer plane. The above observation validates the WAXS-derived oblique lattice (Fig. 3D), and the near-zero tilt of the tails with respect to the bilayer normal. Specifically, GIXS yields the maximum tilt of the alkyl tails ∼6° (SI Appendix, Fig. S5). Line-shape analysis of WAXS peaks (Fig. 3B) reveals an average 2D crystal domain size of ∼15 nm, which is significantly smaller than the ribbon size. This implies that the ribbons are polycrystalline.

The SAXS/WAXS data from the ribbons also exhibit weak, but sharp Bragg reflections for q ≤ 10 nm⁻¹. The most prominent of these reflections is at q = 2.5 nm⁻¹ (Fig. 3A). A much weaker reflection is also observed at q = 7.5 nm⁻¹ (Fig. 3B). In fact, GIXS measurements of ribbons revealed q_z-extended intensity rods at q_xy = 2.5, 5.0, 7.5, and 10.0 nm⁻¹ (SI Appendix, Fig. S6). Here, q_z is the scattering vector component along the bilayer normal. Thus GIXS shows that the low-q Bragg reflections originate from an in-plane ordering within the bilayer, which we attribute to a preferred orientational ordering of C₁₆K₁ headgroups. This headgroup ordering is commensurate with the alkyl tail lattice, and can be defined by a unit cell that is a 1 × 3 supercell of the alkyl tail unit cell in Fig. 3D because the position of the first Bragg reflection (q₁ = 2.5 nm⁻¹) equals one-third of the magnitude of the (0 1) reciprocal lattice vector for the alkyl tail lattice [i.e., q₁ = b*/3 = 2π /(3b sinγ)]. Perhaps headgroups of neighboring C₁₆K₁ along the b axis in each leaflet are rotated 120° clockwise (or anticlockwise) relative to each other about the bilayer normal. However, proving this hypothesis is beyond the scope of the current work. Overall, the SAXS/WAXS analysis of C₁₆K₁ ribbons in pure water proves that ribbons are bilayers with interdigitated leaflets and that the packing of alkyl tails and headgroups exhibits crystalline ordering.

We traced the ribbon to cochleate transformation and the changes in cochleates as a function of NaCl concentration via SAXS/WAXS. Fig. 4 A and B show X-ray scattering from 4 mM C₁₆K₁ at low (c = 1–5 mM) and high (c = 5–50 mM) NaCl concentrations, respectively. Fig. 4A shows that for c ≥ 2 mM and for q < 0.1 nm⁻¹_, the monotonic fall in intensity (Figs. 3A and 4A, bottom curve) is replaced by multiple intensity modulations due to cochleates. The SAXS-deduced NaCl concentration of 2 mM for ribbon to cochleate transition is only slightly lower than the threshold c_th = 3 mM in AFM experiments. More importantly, c_th is independent of the C₁₆K₁ concentration, as indicated by SAXS experiments on 10 mM C₁₆K₁ solutions (SI Appendix, Fig. S7). Therefore, rolling of the membrane into cochleates is driven by the solution ionic strength; i.e., the range of intermolecular electrostatic interactions controls the ribbon to cochleate transition.

Fig. 4.

(A) Background-subtracted in situ SAXS/WAXS data for 4 mM C₁₆K₁ as the NaCl concentration is increased from 1 to 5 mM. The datasets are offset vertically for clarity. With increasing NaCl concentration, the appearance of multiple intensity modulations for q < 0.1 nm⁻¹ and the smearing of the sharp WAXS diffraction peaks are connected with the ribbon to cochleate transition. (B) Background-subtracted in situ SAXS data for 4 mM C₁₆K₁ as the NaCl concentration is increased from 5 to 50 mM. The datasets are offset vertically for clarity. The position of first-order small-angle diffraction peaks (0.1 < q < 1 nm⁻¹) is used to determine the interbilayer spacing D inside the cochleates. (C) SAXS-derived interbilayer spacing in the cochleates varies linearly as a function of c^−1/2, where c is NaCl molar concentration. The solid black line is the best fit with equation: D (nm) = 6.40 + 2.05 × c^−1/2.

Fig. 4A shows intensity modulations across the entire q range of 0.02–30 nm⁻¹, which are divided into 4 groups, each yielding information at a different length scale: 1) the 0.02 < q < 0.06 nm⁻¹ modulation (Fig. 4A, left red box) arises from the cross-section of the cochleates. If we assume the overall shape of a cochleate is a cylinder of radius R, then the scattering amplitude $F_{cyl}\left(q\right)\propto J_1\left(qR\right)/\left(qR\right)$ where J₁ is the first-order Bessel function of the first kind. The first zero of $J_1\left(qR\right)/\left(qR\right)$ occurs at qR = 3.8. Therefore, the minimum at q ∼ 0.022 nm⁻¹ yields an average cochleate radius R = 3.8/0.022 ∼173 nm, which is consistent with the TEM image of the cochleates in 5 mM NaCl (Fig. 2D). 2) For 0.1 < q < 1 nm⁻¹, the intensity maxima positions follow the sequence q_max: 2 × q_max: 3 × q_max. Therefore, these modulations are Bragg reflections due to periodic lamella, with a spacing D = 2π/q_max within the cochleates. Based on q_max (Fig. 4B) for 0.005 ≤ c ≤ 0.1 M, D (nm) = 6.40 + 2.05 × c^−1/2 (Fig. 4C). That is, for the range of NaCl concentration used, the interlamellar spacing could be continuously tuned from 13 to 35 nm, and the interlamellar spacing varies linearly with the electrostatic screening length (λ_d ∝ c^−1/2). Thus, the range of electrostatic interactions also controls the interbilayer spacing. 3) As noted in the SAXS analysis of ribbons, the broad intensity modulation for 0.8 < q < 3 nm⁻¹ is due to an individual C₁₆K₁ bilayer. Fig. 4A shows that the minimum position at q ∼0.9 nm⁻¹ shifts to a lower q when the ribbons are transformed into cochleates. This shift is observed up to c = 4 mM (Fig. 4B), and is consistent with an increase in the bilayer thickness from 4.0 nm (ribbon) to 4.3 nm (cochleate) due to an increase in the thickness of the tail region (SI Appendix, Fig. S8). Thus, a curvature-induced strain in the cochleates reduces the extent of interdigitation between the bilayer leaflets. 4) The curvature also reduces the degree of order in the packing of C₁₆K₁ tails and headgroups. For the case of 4 mM C₁₆K₁, the sharp diffraction peaks corresponding to the crystalline ordering smear out for c ≥ 3 mM. A similar behavior is observed for 10 mM C₁₆K₁ solutions (SI Appendix, Fig. S7). Overall, X-ray scattering analysis reveals that above a threshold NaCl concentration of 2–3 mM, C₁₆K₁ assembles into cochleates. The interbilayer spacing within these cochleates depends linearly on the electrostatic screening length, and the curved morphology of the cochleates induces a reduction in the interdigitation between the bilayer leaflets and the crystallinity in the molecular packing.

We developed simple theoretical models to rationalize the observed structural changes. We focused on 3 aspects: 1) The decrease in the ribbon aspect ratio with increasing salt concentration, 2) The rolling of the membranes, and 3) the linear relationship between the interbilayer separation within the cochleates and the electrostatic screening length.

We first discuss the ribbon to sheet transformation. For this, we model the membrane as a thin parallelepiped (Fig. 5 A, Inset) of length L, width W, thickness δ, and uniform charge density$\rho =Q/LW$ for the top and bottom surfaces. The membrane energy is formulated as the sum of electrostatic interactions $H_S$, interfacial energy $H_I$, and the edge energy $H_L$ that accounts for the exposure of hydrophobic tails to water on the edge surfaces of the membrane, $H_T=H_S+H_I+H_L$:

$$H_T={\displaystyle \underset{0}{\overset{Q}{\int }}{\phi }_s\left(q´\right)\text{d}q´}+\sigma {\displaystyle \int \text{dA}}+\gamma \delta {\displaystyle \int dl.}$$ [1]

Here,${\phi }_s$ is the screened electrostatic potential evaluated on the membrane surface, $\sigma$ is the interfacial tension, and $\gamma$ is the energy density for the membrane edge surfaces. For details, see SI Appendix, section 3. Short-range interactions, such as the intermolecular van der Waals are neglected. Furthermore, the second term in Eq. 1 can be ignored because AFM images (Fig. 2) show 1–5-µm² membranes, independent of the electrolyte concentration. We evaluated Eq. 1 numerically for rectangular membranes of a fixed area A that are constituted by a fixed number of charges interacting through the Debye–Hückel potential (Fig. 5A). The numerical values of parameters ρ, γ, δ, A are listed in SI Appendix, Table S1. Briefly, at very high salt concentrations (λ_d → 0), the electrostatic interactions are weak, short-ranged, and thus insensitive to the membrane shape. Here, the interfacial energy dominates and leads to square sheets (Fig. 5A, ${\lambda }_d\le 2.96\hspace{0.17em}\text{nm}$), a configuration that minimizes the exposed edge surfaces or the ratio of perimeter to area for the top and bottom membrane surfaces. For the same reason, in experiments quasicircular sheets are observed (Fig. 2C). In the very low salt condition, the electrostatic interactions are strong and long-range and H_S becomes dominant. This leads to high aspect ratio ribbons (Fig. 5A, ${\lambda }_d\ge 2.97\hspace{0.17em}\text{nm}$). Fig. 5B shows the optimal membrane aspect ratio for different Debye lengths (salt concentrations) revealing a first-order transition from narrow ribbons to square sheets. The energy difference between the two phases is much lower than $k_{B}T$ (Fig. 5 B, Inset). Therefore, the two phases can coexist near the transition, which is consistent with experimental observation of membranes of varied aspect ratios near the transition (Fig. 2B). We note that ${\lambda }_d=2.97\hspace{0.17em}\text{nm}$ corresponds to ∼10 mM NaCl, which is larger than the experimental 3 mM NaCl for the ribbon to sheet transition. This minor difference may be due to the approximate parameters used for the surface tension and membrane charge density.

Fig. 5.

(A) The membrane energy per molecule (Eq. 1) as a function of the inverse of membrane aspect ratio W/L. At low salt concentrations (large Debye lengths), an elongated ribbon structure is the equilibrium morphology. By contrast, at very high salt concentrations (small Debye lengths), the membrane energy is minimized when L/W = 1. (Inset) Schematic representation of a C₁₆K₁ nanoribbon for numerical calculation. (B) The optimal inverse aspect ratio (minimum energy) for different Debye lengths. The first-order transition happens at ${\lambda }_d=2.96\hspace{0.17em}\text{nm}$. (Inset) The energy difference between the optimal aspect ratio and the sheet (aspect ratio = 1). The energy difference near the transition point is much smaller than $k_{B}T$.

Next, we consider the cochleate formation. Previous theoretical studies have shown that membranes of chiral molecules will experience an out-of-plane bending force (29–31) if the molecules are tilted with respect to the bilayer normal. Briefly, because of chirality, the molecules do not pack parallel to each other, but exhibit a twist with respect to their neighbors. This relative orientation constraint and the constraint of a preferred tilt angle with respect to the bilayer normal can be simultaneously satisfied by shapes exhibiting cylinder-like curvature, such as closed tubes. In qualitative agreement with these theories, SAXS/WAXS measurements show that at high salt concentrations, cochleates are formed for (left-handed) L- and (right-handed) D-C₁₆K₁. Under identical conditions, planar bilayers are observed for a racemic mixture (1:1 mixture of L- and D-C₁₆K₁) (SI Appendix, Fig. S9). Furthermore, molecular chirality induces chirality in the assemblies at all length scales: At the nanoscale, the 2D lattice for tail packing is oblique (Fig. 3D), and at the mesoscale the cochleates have a screw-like handedness (Fig. 2D and SI Appendix, Fig. S3). For these reasons, we use the Helfrich–Prost model (29) to show that the combined effects of molecular chirality and tilt not only lead to helical ribbon and cylinder (29–31) morphologies, but can also stabilize the spiral-helicoidal shape of cochleates. This model also yields insights into the relationship between interbilayer separation in cochleates and salt concentration.

The scroll morphology (SI Appendix, Fig. S3) resembles a spiral-helicoidal surface (Fig. 6A) that can be parameterized as:$\mathit{X}\left(\theta ,z\right)=\left(D\theta \sin \theta ,D\theta \hspace{0.17em}\text{cos}\theta ,p\theta +z\right)$. Here, D is the sheet separation in the cochleate, and p is the pitch of the helical windings along the cochleate long axis. The relevant interactions for such a membrane are the elastic energy, the long-range electrostatic interaction, the short-range attractive van der Waals interaction, and the short-range hydration repulsion (32). Based on SAXS/WAXS measurements, the aqueous layer thickness (D–δ) varies between ∼31 and 9 nm when the salt concentration is varied between 5 and 100 mM. This thickness is much larger than the hydration decay length (32). Therefore, the hydration energy term can be neglected. Besides, in a mean-field description, short-range attractive forces can be neglected (29, 33). Theoretical arguments and experimental observations above suggest that lipid tilt and chirality are relevant in the membrane description. Thus, the energy for a cochleate can be written as:$H_T=H_F+H_S+H_B$. Here, $H_F$ is the Frank interaction describing the increment in the energy due to the molecular reordering and distortions from their uniformly aligned configuration. H_S and H_B are the electrostatic and the bending energies, respectively. The electrostatic interactions $H_s$ renormalize the physical properties of the membrane. In particular, the membrane bending rigidity changes as: $\kappa ={\kappa }_0+{\kappa }_{el}\left({\lambda }_d\right)$ (34). Here, ${\kappa }_0$ is the intrinsic membrane bending rigidity and ${\kappa }_{el}$ is an electric contribution that depends on the membrane geometry and the Debye length, ${\lambda }_d$. This electric contribution to bending has been experimentally verified for some lipid membranes (35). Thus, the combined effect of bending and electrostatic energies can be written as $H_s+H_B\approx \kappa {\displaystyle \int dA{K}^2}$. Here $K=\frac{2+{\theta }^2}{D{\left(1+{\theta }^2\right)}^{3/2}}$ is twice the mean curvature of the cochleate, and $dA=D\sqrt{1+{\theta }^2}$ is the area element. If we assume that the molecules orient uniformly such that the tilt projection $\mathit{m}$ in the local tangent plane forms an angle ${\phi }_0$ with the azimuthal direction, then $\mathit{m}=\cos {\phi }_0\widehat{\mathit{\theta }}+\sin {\phi }_0\widehat{\mathit{z}}$, where $\widehat{\mathit{\theta }},\widehat{\mathit{z}}$ are the unit vectors in the azimuthal and axial directions (Fig. 6A). Therefore, the Frank energy takes the simple form

$$\frac{H_F}{W}=\frac{\kappa \prime }{2}{K}^2{\cos }^2²{\phi }_0-{\lambda }_{HP}\hspace{0.17em}K\sin {\phi }_0\cos {\phi }_0.$$ [2]

Here,$\kappa \prime$ is the difference between elastic constants for bending the membrane in the parallel and perpendicular directions to the tilt vector, and ${\lambda }_{HP}$ measures the strength of the intermolecular chiral interactions (29, 30). The minimization of the total energy H_T with respect to ${\phi }_0$ predicts a critical tilt angle $\sec 2{\phi }_0=-\left(2\kappa +\kappa \prime \right)/\kappa \prime$. If the energetic costs of bending the membrane parallel or perpendicular to the tilt direction are almost the same, (i.e., $\kappa \prime \approx 0$), then ${\phi }_0\approx 45°$. This is roughly equal to the angle of the helical windings with the cochleate’s principal axis (SI Appendix, Fig. S3) suggesting that molecular tilt direction coincides with the membrane folding direction. This hypothesis needs further exploration. While the molecular tilt orientation is related only to the membrane elastic parameters, the interbilayer separation D depends on κ/λ_HP, i.e., the ratio of the membrane bending rigidity and the molecular chiral interaction parameter (SI Appendix, Eq. S16). In particular, D decreases nearly linearly with the Debye length for $R\ll k/{\lambda }_{HP}$, where $R$ is the cochleate external radius. Furthermore, a theoretical curve that approximately reproduces the D vs. λ_d experimental data (Fig. 6B) yields κ₀/λ_HP ∼ 200 m. (See SI Appendix, section 3 for details.) The slight quantitative deviation between the experiment and theory (Fig. 6B) is likely due to neglecting the stretching degrees of freedom and the thickness of the membrane. Nevertheless, the accuracy of the qualitative predictions of the model clearly highlights the collective effect of molecular tilt and chirality in inducing the spontaneous membrane curvature. This combined with the electrostatic effects, which rigidifies the membrane, enable us to deduce qualitatively the key structural features of the cochleates. In particular, this simplified theoretical model suggests that the linear relationship between the interbilayer separation and the electrostatic screening length is not a result of system-specific design, but of the interplay between electrostatic energy and the membrane internal degrees of freedom. Therefore, it is not surprising that similar linear relationships have been observed in other charged layered systems, such as clay mineral montmorillonite (36, 37). We note that the linear relationship is not valid in the presence multivalent ions. For example, negatively charged phospholipid cochleates show little or no dependence of interbilayer spacing on the CaCl₂ concentration (38). It is possible that the multivalent cations are tightly bound to the molecules resulting in interbilayer electrostatic interactions that cannot be parameterized by the screening length λ_d alone. By contrast, the use of monovalent salts to induce the C₁₆K₁ cochleate structure leads to tunable interbilayer spacing over ∼10–40 nm. This structural feature may have application for controlled encapsulation and release of drug particles within a specific size range.

Fig. 6.

(A) The geometry of the cochleates. D is the interbilayer separation. Arrows represent the projection of the tilt vector in the local tangent plane. (B) Theoretical prediction showing a roughly linear relationship between interbilayer spacing D and c^−1/2, where c is NaCl molar concentration.

Finally, we note that while the experimental results for cochleates are in qualitative agreement with the predictions of theoretical models for assembly of chiral molecules, there are still unresolved questions. First, these continuum models (29–31) are strictly applicable to fluid-like membranes or membranes with hexatic order. That is for cases where there are no long-range intermolecular positional correlations. However, the WAXS data from L- and D-C₁₆K₁ ribbons clearly show sharp diffraction peaks indicating crystalline bilayers (Fig. 3C and SI Appendix, Figs. S5 and S9). Second, membrane curvature is expected for cases where the molecules are tilted with respect to the bilayer normal. Our WAXS and GIXS data are currently inconclusive in this regard: In the planar ribbon phase, the molecules may have a slight (<6°) tilt. Whether, the molecules in bilayer have a small tilt or the molecules undergo a tilting transition just prior to the sheet to cochleate transformation will be a subject of future studies.

Conclusions

We designed a peptide amphiphile C₁₆K₁ to investigate the electrolyte-induced transformation of planar bilayers to scroll-like cochleates. We show that with the addition of NaCl, the high aspect ratio C₁₆K₁ ribbons formed in zero salt conditions transform to isotropic sheets, prior to rolling up to form cochleates. This ribbon to cochleate transformation also induces a reduction in the crystallinity in the molecular packing. Further addition of salt reduces, within the cochleates, the interbilayer separation, which scales linearly with the Debye length. A simplified model demonstrates that the ribbon to sheet transformation is a first-order transition induced by the reduction in the range of electrostatic interactions. Theoretical models show that rolling of membranes into cochleates is the combined effect of molecular chirality and tilt. The linear relationship between the interbilayer separation and the screening length in cochleates is qualitatively explained by the competition between electrostatic and the effective elastic interactions that include the internal degrees of freedom of tilt and chirality. These results suggest that the salt-induced structural transitions in the C₁₆K₁ system should be observed in other charged bilayer membranes. Our combined experimental and theoretical study yields insight into attaining the cochleate structures and controlling their internal architecture. These results should be useful for optimizing the structure and function of cochleates in many applications, including drug delivery and photocatalytic production of hydrogen.

Materials and Methods

Peptides synthesis and SAXS/WAXS, Cryo-TEM, and liquid-AFM measurements are described in SI Appendix.

Data Availability

All the X-ray data shown in the manuscript and the SI Appendix, as well as the code for simulating the WAXS intensity along with the data from theory calculations, have been deposited at Bitbucket (39). The files are in folders labeled with corresponding figure numbers.