Mechanical tweezer action by self-tightening knots in surfactant nanotubes

Abstract

Entanglements and trefoil knots on surfactant nanotubes in the liquid phase were produced by a combination of network self-organization and micromanipulation. The resulting knots are self-tightening, and the tightening is driven by minimization of surface free energy of the surfactant membrane material. The formation of the knot and the steady-state knot at quasi-equilibrium can be directly followed and localized by using fluorescence microscopy. Knots on nanotubes can be used as nanoscale mechanical tweezers for trapping and manipulation of single nano- and micrometer-sized high-aspect ratio objects. Furthermore, we demonstrate that by controlling the surface tension, objects captured by a knot can be transported along given trajectories defined by the nanotube axes.

Knots and entanglements occur naturally on microscopic scales. They play an important role in most macromolecular systems, such as gels and rubbers. DNA in our cells contains complex knots that are controlled by topoisomerase enzymes. It has been demonstrated also that tight knots exist deep inside the folded structure of proteins (1), and trefoil knots in synthetic RNA have revealed properties of enzymes (2). Catenanes and knots have been synthesized in small molecules with the aim of constructing, for e.g., molecular machines and motors (3). Intricately linked and knotted structures have been made from single-stranded DNA, and such systems have been proposed as building blocks for computational biochips and nanorobotics applications (4, 5). With the advancement of clever micromanipulation techniques, and sensitive imaging techniques, knots have been recently tied (6) in single-actin filaments and DNA strands by using optical tweezers in combination with beads attached to the macropolymers as handles. It has been suggested (6) that molecular-scale knots might function as micromanipulation tools in biological systems.

Here, we demonstrate that it is possible to tie knots in nanoscale surfactant systems in the fluid state. The technology and procedures involved are based on methods described (7–9) for building nanofluidic networks of surface-immobilized vesicles connected by suspended nanotubes (50–300 nm diameter) made from continuous sheets of lipid-bilayer membranes. Lipidbilayer membranes are 2D fluids above the gel–fluid transition and possess properties both of a liquid (fluidity) and a solid (bending rigidity and resistance to area dilation). Nanofluidic networks can be prepared with controlled connectivity, vesicle size, geometry, and topology. Vesicles residing in a single plane can be connected by nanotubes that cross each other in a controllable fashion at different planes to create multilayer systems of nanotubes (8, 9). The diameter of the surface-immobilized vesicles is ≈10–50 μm, the length of the suspended nanotubes is between a few tens and hundreds of micrometers, and the diameter of the nanotubes is ≈200 nm (7). Here, we show that it is possible to design such networks to evolve toward knotted nanotube formations by using a combination of self-organization and micromanipulation, and that the resulting knot can be used as a mechanical torus tweezer for manipulation of singular objects in a liquid environment.

Materials and Methods

Vesicle Preparation. Vesicles were prepared from soybean lecithin dissolved in chloroform (100 mg/ml) as a stock solution. The dehydration–rehydration method described by Criado and Keller (10), with modifications (11), was used to prepare unilamellar liposomes. In short, a droplet of 5 μl of lipid dispersion (1 mg/ml) was placed on the coverslip and dehydrated in a vacuum desiccator for 25 min. When the lipid film was dry, it was rehydrated with buffer solution (5 mM Trizma base/30 mM K₃PO₄/30 mM KH₂PO₄/1 mM MgSO₄/0.5 mM EDTA, pH 7.8) containing the fluorescent dye FM 1–43 {N-(3-triethylammoniumpropyl)-4-[4-(dibutylamino)styryl]pyridinium dibromide} at 1 μg/ml. After a few minutes, giant unilamellar liposomes were formed. A small sample of lipid suspension was carried over into a droplet of buffer solution positioned on SU8-coated borosilicate coverslips.

Preparation of SU8-Coated Coverslips. Circular, 28-mm-diameter coverslips (Merck) with a thickness of 130 μm were spin-coated with SU8 2002 (MicroChem, Newton, MA) at 4,000 rpm for 30 s, resulting in a film thickness of ≈1.1 μm. The substrates were baked at 95°C on a hotplate for 60 s, exposed on a mask aligner for a dose of 100 mJ·cm^–2, and baked after exposure at 95°C on a hotplate for 60 s.

Microscopy and Fluorescence. The coverslips were placed directly on the stage of a DM IRB inverted microscope (Leica, Wetzlar, Germany). For epifluorescence illumination, the 488-nm line of an Ar⁺ laser (2025-05; Spectra-Physics) was used. To break the coherence and scatter the laser light, a transparent spinning disk was placed in the beam path. The light was sent through a polychroic mirror (Leica) and an objective to excite the fluorophores. The fluorescence was collected by a three-chip color CCD camera (Hamamatsu, Kista, Sweden) and recorded by using a DSR-11 digital video camera (Sony, Tokyo). Digital images were edited by using premiere and photoshop graphic software (Adobe Systems, Mountain View, CA).

Formation of Networks. A carbon fiber microelectrode (5 μm diameter; Dagan Instruments, Minneapolis) and a tapered micropipette, controlled by high-graduation micromanipulators [MWH-3 and MC-35A (coarse manipulator); Narishige, Tokyo], were used to create the unilamellar nanotube-interconnected vesicles by using a microelectroinjection technique (7). In brief, by applying a combination of mechanical and electrical forces, the membrane of a surface-immobilized unilamellar vesicle was penetrated. By pulling the micropipette away from the vesicle, a nanotube was created. Subsequently, buffer solution was injected through the micropipette into the nanotube orifice, resulting in the formation of a new vesicle at the tip of the pipette. After the vesicle had reached the desired size, it was positioned and immobilized on the surface. Tapered micropipettes were made from GC100TF-10 borosilicate capillaries (Clark Electromedical Instruments, Reading, UK), pulled on a P-2000 CO₂ laser-puller instrument (Sutter Instruments, Novato, CA). For electroinjections, a microinjection system (Eppendorf femtojet) and a pulse generator (DS9A digitimer stimulator; Welwyn, Garden City, UK) were used.

Chemicals and Materials. Trizma base, glycerol, and potassium phosphate were obtained from Sigma–Aldrich. Soybean lecithin (a polar lipid extract) was obtained from Avanti Polar Lipids. Chloroform, EDTA (titriplex III), magnesium sulfate, potassium dihydrogen phosphate, potassium chloride, sodium chloride, and magnesium chloride were obtained from Merck. Deionized water (Millipore) was used to prepare the buffer. FM 1–43 {N - (3 - triethylammoniumpropyl)-4-[4-(dibutylamino)s-tyryl]pyridinium dibromide} was obtained from Molecular Probes. The polar lipid extract consisted of a mixture of phosphatidylcholine (45.7%), phosphatidylethanolamine (22.1%), phosphatidylinositol (18.4%), phosphatidic acid (6.9%), and other lipids (6.9%).

Results and Discussion

Network Self-Organization. To understand how the nanotubeconjugated vesicle networks evolve into knotted and entangled geometries, let us first consider a simple system consisting of three surface-immobilized vesicles connected by two suspended nanotubes, initially far apart (Fig. 1a). This initial geometry represents a stable, kinetically arrested state of the network. To trigger a change in network geometry, we force the two nanotubes to come close together at the surface of the liposome where they are connected by using mechanical and electrical forces. When the separation distance between the two nanotubes is sufficiently small, they merge (Fig. 1b). Merging nanotubes form a three-way junction or bifurcation point (Fig. 1b) that propagates spontaneously toward the configuration with the shortest total tube length. By minimizing the length of the nanotubes, the network reaches its minimal surface free-energy state that represents the global minimum for this three-vesicle system (Fig. 1c) (8). By using simple trigonometry, the energy minimum is found when the angle between nanotubes is 120° (12). Excess of membrane material during network reorganization is reabsorbed by the immobilized vesicles. This propensity of the fluid-state membrane material to self-organize into lower surface free-energy states by propagation of bifurcation points was used as the basis for creating complex networks that could be triggered to evolve into entangled and knotted topologies.

Fig. 1.

Fluorescence micrographs showing formation and evolution of a fluid-state bifurcating surfactant network. (a) Initially (time, 0 s), the nanotube–vesicle network, immobilized on a SU8-coated substrate, has a V-shaped geometry consisting of three vesicles connected by two nanotubes. By applying a focused electrical field (40–60 V/cm) through a micromanipulator-controlled glass micropipette (tip diameter, ≈2 μm) (6) locally over the two nanotubes emanating from the central vesicle in the network (arrowheads), the two tubes are forced to coalesce. (b) As the nanotubes coalesce, a three-way junction is formed that propagates, typically with an average speed of 20 μm/s, toward the state representing the shortest total tube length. This process is driven by minimizing the surface free energy (6). (c) After 2.3 s, the system reaches the state with minimal total-tubes length, at which the angle between nanotubes at the junction point is 120°. This geometry represents the minimum energy configuration and, thus, the final geometry of the network. Images were edited digitally to improve quality. (Scale bar, 15 μm.)

Entanglements in Surfactant Nanotubes Are Reversible. We used the capacity for self-organization in 2D fluids in combination with micromanipulation tools (7–9) to design starting network geometries that would evolve toward knotted and entangled nanotube geometries by propagation of bifurcation points. For example, if we build networks with several vesicles, 5, 6, 7, etc., we can correspondingly produce 3, 4, 5, etc. propagating nanotube junctions. Provided that we have the right connectivity arrangement of nanotubes (i.e., that they cross each other in the correct planes), we may create entangled networks as well as trefoil, figure-8, and higher-order crossing knots.

First, we demonstrate that we can create an entangled network in which a propagating nanotube junction is used to fork around another nanotube (Fig. 2 a–d). To perform this experiment, we created a system consisting of two independent networks with four surface-immobilized vesicles and one additional vesicle immobilized on a pipette tip that is freely movable. One of the vesicles was doubly conjugated with nanotubes as shown in Fig. 2a. After triggering the nanotubes to coalescence, the network forms a traveling junction point that forks around the nanotube connecting vesicles I and II (Fig. 2b). In the fluorescence micrograph shown in Fig. 2c, the entanglement appears at the point at which the three nanotubes meet. We demonstrate that the entanglement shown in Fig. 2c, which has a large bending deformation (as shown schematically in the 3D graphic Inset), is fully reversible as the pipette-coupled vesicle is moved down towards liposome IV and then back to its original position (Fig. 2d). Thus, nanotubes can be tightly wrapped around each other without causing topological changes (such as fusion or hemifusion) or mechanical collapse (i.e., the nanotubes stay intact when challenged by bending deformations with loop radii approaching the diameter of the nanotubes).

Fig. 2.

Fluorescence micrographs and 3D graphical illustrations showing entanglement of surfactant nanotubes. (a) The experimental system consists of two separate nanotube–vesicle networks, one linear two-container network supporting one nanotube (2, red) and a V-shaped three-container network supporting two nanotubes (1 and 3, green). Nanotube 2 is positioned above nanotube 1 and under nanotube 3. Vesicles I, II, IV, and V are immobilized on the substrate surface, whereas vesicle III is attached to a micromanipulator-controlled micropipette and can, thus, be translated in any direction. (b) Liposome III is moved slightly upward by translation of the micropipette, bringing nanotube 3 sufficiently close to nanotube 1 to facilitate tube coalescence on the surface of liposome IV (arrows). When tubes 1 and 3 merge, a propagating bifurcation point is formed that drives the system toward the state of minimal surface energy, as described in Fig. 1. Inset shows a 3D graphic representation of the spatial arrangement of nanotubes as the bifurcation point propagates. (c) As the junction point moves, it promotes bending of nanotube 2. The 3D graphic Inset shows that the junction region can be viewed as an entanglement formed by a Y-shaped connection composed of nanotube 1, nanotube 3, and the bent nanotube 2 passing through it. (d) By moving the micropipette with the liposome III attached to it toward liposome IV and than upward again, we untwine the entanglement and bring the network to its initial state. Experimental conditions were otherwise as described for Fig. 1. The vesicles and nanotubes were all extracted from the same giant vesicle. Therefore, only one fluorescent dye could be used. To accentuate the spatial separation of the nanotubes at the entanglement region, images were enhanced digitally and false-colored. The 3D graphic representation is used as a qualitative tool for understanding how the nanotubes are related to each other in 3D space. It is not meant to represent the true surface geometry of the system. (Scale bar, 15 μm.)

Formation of Trefoil Knots on Surfactant Nanotubes. To demonstrate the formation of a trefoil knot, we created a system consisting of five surface-immobilized vesicles with a particular 3D arrangement of nanotubes between them, as shown in Fig. 3 a and d. Three of the vesicles were doubly nanotube-conjugated. After triggering the nanotubes to coalescence, the network forms three traveling junction points, each of which represents a loop of a trefoil knot (Fig. 3 b and e). In the fluorescence micrograph shown in Fig. 3f, the knot appears at the point at which the five nanotubes meet, and each of the foils of the knot is suspended to the three vesicles that were initially tube-conjugated.

Fig. 3.

Fluorescence micrographs and 3D graphic displays showing formation and tightening of a trefoil knot. The 3D graphics show the following in sequence. (a) Initial geometry of the network with five nanotube-conjugated surface-immobilized vesicles. The network is prepared in such a way that the nanotubes reside in different planes for trefoil-knot formation. (b) The nanotubes are triggered to coalesce and to form three propagating junction points. As the junction points advance and membrane material is reabsorbed by the stationary vesicles, the knot is tightened. (c) Graphical illustration of the tightened knot. Inset shows the geometry of the suspended trefoil knot. The corresponding fluorescence images to the above sequence are shown in d–f.(d) Arrows indicate the regions where the nanotubes were coalesced by using the same method as described in Fig. 1. (e) The network 4 s after nanotube coalescence. As the nanotubes merge, they form three junction points that propagate and tighten the knot. (f) The ready knot together with the five nanotubes connected to it. The nanotube arrangement at the knot remains the same as in Fig. 2 b and e. The size of the knot is comparable with the diameter of the nanotubes. Experimental conditions were otherwise as described for Fig. 1. The 3D graphic representation is used as a qualitative tool for understanding how the nanotubes are related to each other in 3D space. It is not meant to represent the true surface geometry of the system. Images were edited digitally to improve image quality. (Scale bar, 15 μm.)

As the total tube length gets progressively shorter because of minimization of surface free energy, it effectively leads to the tightening of the knot (Fig. 3 c and f). We can follow this process by measuring the radius of the knot as a function of time. It takes ≈4–5 s to tighten the knot when the distance between the doubly nanotube-conjugated vesicles is ≈40 μm.

To estimate the size of the knotted region, we assume that the tube behaves effectively as a string with bending resistance and tension. Generally, two regimes are possible for such a string: tight and open knot (6). For low tension (open knot) the knotted region is much larger than the diameter of the string and the knotted part of the string is approximately a circle with a radius R. If the tension is sufficiently high, the knotted region becomes comparable with the diameter of the string (tight knot). The radius of a straight membrane tube is a function of the bending modulus κ and membrane tension σ (13), such that

To get a semiquantitative understanding of knot size, we replaced the knotted part of the tube by a torus of radius R, an approach that has been used (14) to determine the tightness of knots in polymers. The energy of the torus is the sum of the bending and surface-tension energy:

This energy is minimal for R = 1.3a, for any combination of bending modulus and tension. These calculations suggest that the knots in tubes should always be tight, i.e., have a size comparable with the tube radius.

Of course, as soon as the knot becomes tightened, other effects come into play. For example, the tube changes shape and becomes thinner inside the knot. Knotted ropes usually break at the point of the knot when the tension is increased (15). The same phenomenon is also observed at molecular scales: the critical tensile force for breaking actin is two orders of magnitude lower if it contains a knot, whereas DNA appears stronger (6). Molecular dynamics simulations show that polyethylene strands are weakened significantly by knots (16). We found that surfactant membrane knots represents a quite stable configuration and can be observed for tens of minutes (no tests on long-term stability were conducted). When systems were challenged by high force loads, nanotube rupture was typically achieved at the vesicle-nanotube junctions and not in the knot region (data not shown).

Trefoil Knots on Surfactant Nanotubes as Mechanical Tweezers. To demonstrate that a trefoil nanotube knot can be used to capture and transport small high-aspect ratio objects, we tightened a knot around another nanotube and displaced it from its original location (Figs. 4 a–h). The initial geometry of the membrane network was designed in such a way that the translated nanotube will pass through the knot as it gets tightened (Fig. 4 a and e). In Fig. 4 b and f, a tightened knot has evolved with the nanotube passing through it, and Fig. 4 c and g display the same knot after we have cut the suspension tubes by microelectrofission (rupture of nanotubes by focused electric fields applied by micropipettes). The nanotube containing the knot is connected between two vesicles in an approximate cross configuration, and to displace the knot in the x–y plane, we deform one of the vesicles with a local force application by using the tip of a micropipette (Fig. 4 c and d). The deformation of the vesicle causes an increase in surface tension, and transport is governed by a moving boundary (membrane walls) directed toward the region of highest surface tension (17). As the knot moves toward the deformed vesicle, the captured nanotube is translated as shown in Fig. 4 d and h. The total length of the translated nanotube is increased. This observation implies that the force exerted by the knot must exceed f = 2sin(θ)f₀, where θ is the angle by which the tube is tilted and f₀ is the minimal force required to pull out membrane tubes from vesicles, such that

(13). By inserting typical values of bending rigidity κ = 10^–19J, and tube radius, a = 100 nm, f₀ corresponds to 6 pN. We were able to tilt captured tubes by angles of the order θ ≈ 15⁰ (data not shown), with a force f = f₀/2. Thus, these forces required to translocate objects with surfactant knots are in the range of where optical tweezers operate (18).

Fig. 4.

Fluorescence micrographs and 3D graphic displays showing formation and tightening of a trefoil knot around a suspended nanotube. The 3D graphics shows the following in sequence. (a) Five-vesicles network connected by nanotubes (green) together with a separate network consisting of two vesicles connected by a nanotube (red); the two networks are arranged in 3D space in such a way that the tubes in the 5-vesicle networks will form a knot around the nanotube in the two-vesicle network. (b) The tightened knot together with the nanotube that goes through it. Inset shows an expanded representation of the suspended trefoil knot. (c) After the three suspension tubes have been cut, the trefoil knot resides on one single suspended nanotube. Inset shows an expanded representation of the trefoil knot capturing the nanotube after the suspension tubes have been removed. (d) Deformation of one of the vesicles connected to the knot by mechanical action from a micropipette results in translation of the knot toward the region of highest surface tension. The nanotube captured inside the knot is thereby transported. The corresponding fluorescence images are shown in e–h.(e) Arrows indicate the area where nanotubes are triggered to coalesce by the experimental means described in Fig. 1. The knot is formed together with the nanotube passing through it. (f) The suspension nanotubes were cut by applying one or several short electric pulses (60–100 V/cm) over the micropipette when positioned close to the nanotubes (≈5 μm). (g) Image showing the nanotube containing the knot (green) and the nanotube spanned by the knot (red). In this instance, the nanotube spanned by the knot is straight. (h) Displacement of the knot is induced by changing the surface-to-volume ratio of a vesicle by using mechanical force from a micropipette. As a result of this deformation, the position of the knot is changed toward deformed vesicle (translation) and the nanotube passing through the knot is bent. Experimental conditions were otherwise as described for Fig. 1. All vesicles and nanotubes were extracted from the same lipid source, allowing the use of only one fluorescent dye. To aid visualization of the process of knot formation, images were enhanced digitally and false-colored. The 3D graphic representation is used as a qualitative tool for understanding how the nanotubes are related to each other in 3D space. It is not meant to represent the true surface geometry of the system. (Scale bar, 15 μm.)

Because the direction of transport is controlled by a gradient in surface tension and the trajectory of transport by network geometry (orientation of nanotube axis), both of which can be well controlled, these systems can form the foundation for networks with the capability to capture, manipulate, and transport elongated objects in aqueous solution. For example, it should be feasible to tie knots around high-aspect ratio rigid rods, such as carbon fibers or microtubules, and it should be possible also to grip and manipulate single large DNA and RNA strands in extended forms as well as single chromosomes and to move these captured objects along a given trajectory defined by the linear extension of the nanotube. It should be feasible also to tie several knots from independent tubes around the same object. This capability would then allow extensive and detailed control over the position and orientation of trapped rod-shaped objects.

This trapping and transport capability of small high-aspect ratio objects in water-based media complements optical trapping (18, 19) and magnetic trapping (20) but, in contrast to these methods, does not require that the objects have particular dielectric or magnetic properties. Both optical and magnetic tweezers are “noncontact methods,” whereas direct contact is achieved between the captured object and the knot (mechanical torus tweezer). Furthermore, because the surface properties of the knot can be tailored quite easily with a wide range of different functionalities, binding interactions, or particular mechanical interactions between the knot and the captured object can thus additionally be studied.

Conclusions and Outlook

To the best of our knowledge, this is the first demonstration of creating knots and entanglements in a fluid matrix. The knot is self-tightening, remains tight after it has been formed, and is stable for prolonged periods of time. The estimated size of the trefoil knot is comparable with the radius of the nanotube. It is also, to the best of our knowledge, the first demonstration of the use of self-organization to form knotted structures. We also show that this knot can be used as a mechanical tweezer for a submicrometer object and that is possible to transport it along the nanotube axis. As such, self-organizing nanotube networks in combination with transport control might lead to development of nanoscale manipulation tools. The results described here could also have implications for biological systems. For example, entanglements might affect the properties (tension, geometry, and topology) of cell-membrane networks as well as the transport of material through these networks (21). In addition, these structures can also serve as topological templates for production of knotted gels and polymers because such materials can be included into these structures and thereafter become solidified by appropriate means (12).