Flagella-like Beating of a Single Microtubule

Kinesin motors can induce a buckling instability in a microtubule with a fixed minus end. Here we show that by modifying the surface with a protein-repellent functionalization and using clusters of kinesin motors, the microtubule can exhibit persistent oscillatory motion resembling the beating of sperm flagella. The observed period is of the order of 1 min. From the experimental images we theoretically determine a distribution of motor forces that explains the observed shapes using a maximum likelihood approach. A good agreement is achieved with a small number of motor clusters acting simultaneously on a microtubule. The tangential forces exerted by a cluster are mostly in the range 0–8 pN toward the microtubule minus end, indicating the action of 1 or 2 kinesin motors. The lateral forces are distributed symmetrically and mainly below 10 pN, while the lateral velocity has a strong peak around zero. Unlike well-known models for flapping filaments, kinesins are found to have a strong “pinning” effect on the beating filaments. Our results suggest new strategies to utilize molecular motors in dynamic roles that depend sensitively on the stress built-up in the system.

T he propulsion of motile cells such as sperms relies on undulating bending of flagella, appendages of the cellular body able to perform periodic oscillations. This beating is driven by the interaction between microtubules and motor proteins, and the mechanism that regulates these interactions as well as how they can orchestrate complex vital processes are not well understood yet. The functional reactions of living systems operating out of equilibrium are the result of component rearrangements due to the intrinsic stochasticity in the system itself. Building a synthetic molecular system in which components continuously rearrange and reorganize in the presence of energy sources would enable a deep understanding of these molecular interactions.
Here we present experimental and theoretical results on a minimal system made of a single microtubule with a fixed end and a small number of kinesin-1 motor proteins (kinesin hereafter) that in the presence of ATP perform a continuous motion resembling flagellar beating.
In the past decades these components were used in in vitro gliding assays 1,2 where the motor proteins were attached on the surface of a glass coverslip at high density. Directed sliding motion of filaments was enabled when they came into contact with the surface bound motors. Previous studies showed microtubule buckling in a low density gliding assay due to a single kinesin motor 3 and waving and rotating motion of biofilaments in a gliding assay 4,5 as well as spiral formation of microtubules caused by pinning at the leading end 4 and by the interactions with neighboring microtubules. 6 In this study we quantitatively analyze the buckling instabilities of single microtubules clamped at one end and subjected to the forces exerted by motor proteins. We found that by using a specific surface treatment for the substrate and organizing the motors in randomly distributed clusters we obtain filaments that perform continuous beating. We can describe the system as follows ( Figure 1): (i) the kinesin motors in the cluster simultaneously bind either to the microtubules or unspecifically to the surface; (ii) the motors exert forces on the microtubule which are directed along the tangential direction. Above a threshold the straight configuration of the filament becomes unstable inducing the buckling of the filament; (iii) the force that bends the filament has a longitudinal component as well as a perpendicular component that allows the microtubule to snap back to its straight configuration. This can be explained by either the filament detaching from the motors or by the motors breaking the bond with the protein-repellent surface and being pulled away while remaining attached to the filament. Both lead to a subsequent buckling when the filament reaches the new position. The behavior of the system is self-organized by the elasticity of the filament, the active forces exerted by motor proteins, and by using attachment and detachment between the building blocks of this minimal system.
Results and Discussion. Here, we report the quantitative analysis of buckling instabilities of clamped microtubules caused by compressive forces exerted by kinesin motors. Diluted polymerized microtubules of approximately 8 μm length were decorated with motor protein clusters by mixing them with biotinylated kinesin clustered by using streptavidin. The surface was functionalized with PLL-g-PEG (see Materials and Methods section). This nonionic polymer has been proven to limit biological interactions, 7 and in this setup it prevents irreversible protein adsorption on the surface. A sample of the microtubule-kinesin mixture was poured on the PLL-g-PEG modified coverslip, and the experimental chamber was sealed in order to avoid any artifacts caused by fluid streaming. In our experiments the chosen functionalization of the glass surface almost entirely prevents the microtubules adsorption. However, we could occasionally observe microtubules with one tip clamped (typically 2 μm) to the surface (1−2 samples in each experiment), likely due to electrostatic interaction with the polylysine backbone of the PLL-g-PEG functionalization. The anchor point had no rotational or translational degree of freedom. The free part of the microtubule was observed to perform oscillatory motion as shown in Figure 2. This motion is attributed to the kinesin motors which could bind with their heads attached to the microtubule and unspecifically bound to the surface simultaneously, due to their organization in clusters. When the kinesin-clusters were bound in this manner the motors transiently pushed the filament forward resulting in cyclic conformational changes for time intervals up to 5 min.
The wave propagation from the bound toward the free end of the filament can be understood as follows. After the first buckling instability, the resulting bend is subject to continued forcing from the direction of the free end, which makes it bulge out further. As more filament is entering the bend, its position on the arclength automatically moves toward the free end. At the same time, each bend leads to a counter-bend with opposite curvature at the clamped end, which eventually increases in size and moves toward the free end, like the original bend did. The result is a periodic undulatory motion.
The oscillations of the microtubule can be divided into cycles, defined as the time that the filament needs to move from one portion of the space to the adjacent one and to come back ( Figure 2

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Letter without a PLL-g-PEG surface functionalization showed no buckling filaments, proving that the unspecific binding of both motor proteins and microtubules does not allow any filament to move freely.
The motor concentration needed to buckle the filament can be estimated from the following simple argument. A filament of length L is attached to motor clusters with an average spacing λ, which we treat as pinning points. If we assume that the filament is torque-free at the motor positions (which is a simplification), the force needed to buckle the segment is The compressive force, exerted by the ensemble of motors, is highest in the segment next to the clamped end and corresponds to F M L/λ, where F M is the stall force of a motor cluster. Both become equal at a critical length which is inversely proportional to the motor spacing or proportional to their density. With the values F M = 8 pN, EI = 0.4 × 10 −23 Nm 2 , and λ = 1 μm, we obtain L C = 5 μm. A filament of this length will generally buckle if the number of motors acting on it is in the range 1−5.
We verified the importance of the motor density by repeating the experiments with a different motor protein concentration. The experiments showed a reduced capacity to buckle continuously. When using a 100-fold higher concentration, no continuous oscillations were observed. The buckling events were less frequent and limited in time compared to the analyzed case (movie S2). By reducing 2fold the concentration again the buckling events were less frequent and limited in time as the sliding of filaments was followed by the filaments leaving the focal plane due to the missing grasp (movie S3). We analyze the movement pattern of our filaments. Their beating was observed to be quasi 2D, and this allowed us to track the filament shape. We characterized the tracked filament shape by a tangent angle ψ(s,t), which describes the angle between the straight line representing the initial position of the filament and the local tangent of the arclength s along the filament at the time t with 0 ≤ s ≤ L, where L is the length of the free moving filament portion (Figure 2-D). We tracked the shape changes for n frames with n up to 300. We obtained a matrix that represents the kymograph of the filament movement (Figure 2-E). We can observe a pattern indicating a wavy motion propagating from one end of the filament along its entire length and reflecting the periodicity of the filament beating, i.e. the four cycles which its movement can be divided into, as defined above and depicted in Figure 2-A. Thus, the beating profile shown in Figure 2-E is similar to the beating of eukaryotic flagella. One has to bear in mind, however, that there are also essential differences between the two systems. Filaments in our experiments interact with motors attached to the surface, and

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Letter the motor forces therefore have the nature of force monopoles. Dyneins in cilia and flagella act between axonemal microtubule doublets and therefore appear as force dipoles.
Filaments subject to tangential compressive forces and their analogy with flagellar beating have already been discussed in several theoretical studies. The basic problem consists of a semiflexible polymer with one clamped end, subject to local drag and an active tangential force. Sekimoto et al. 5 showed the existence of a dynamical instability, above which the filament undergoes flapping motion, whereas Bourdieu et al. 4 briefly presented a waving motion in actin and microtubules. The full phase diagram of a tangentially propelled particle chain was discussed in ref 8. A related problem with a single follower force at the free filament tip was solved in ref 9. If the filament is free to move in 3 dimensions, there is an additional spinning state. 10 If the clamped end is replaced by a cargo, resembling a sperm head, the numerically obtained solutions show rotation or sperm-like beating. 11 Despite the similarities, there are also essential differences between our system and these models. The small number of motors involved makes the dynamics less deterministic, while the ability of kinesins to sustain off-axis loads largely reduces the lateral motion of the filament, except during periods of fast relaxation, presumably caused by the detachment of motors from the distal region.
The motor concentrations in our experiment are significantly lower than in conventional motility assays, 4 such that a small number of multiheaded motors can lead to sustained oscillations. We have the conjecture that the continuous buckling dynamics might be also due to the reversible attachment of the motors to the surface, which was tailored to allow a binding that is sufficiently stable for activating the 'walk' on the microtubule but so weak to dissociate under mechanical stimulus. In this way the system self-organized by using the force of motor proteins that continuously rearranged by breaking and reforming bonds to the surface. The weak bond to the surface was tested by applying a fluid flow at a shear rate estimated as 4 × 10 4 s −1 that was able to wash out the motors unspecifically bound to the surface. Other studies show that surface modifications are able to create reversible motor binding that leads to self-organization of the microtubule dynamics due to a continuous rearrangement of motors. 12 We believe that the force exerted by the motors in one portion of the filament inducing its buckling reached a threshold above which it was able to pull the other motors from the surface. With the motors detached, the microtubule relaxed (fully or partially) to the straight shape due to its elastic properties, thus transporting the motors to new positions. After this rapid position change the reattachment of the motors occurred again, leading to new sliding of the filament.
We used a maximum likelihood method to reconstruct a distribution of motor forces that gives the observed filament shape. Examples of filaments with corresponding forces are shown in Figure 3 and Movie S1. The tangential components of the forces (Figure 3-B) show a strong accumulation in the range between −8 pN and 0 pN (a negative sign denotes forces pushing the filament toward the clamped end, which is the MT minus end), consistent with the forces produced by single kinesin motors. Larger values, up to −20 pN, could result from the action of multiple motors, either within a cluster or sufficiently close together that they become unresolvable. Normal forces are mostly in the range −10 to 10 pN, also consistent with 1−2 kinesins. The tangential velocity of the filament shows a highly skewed distribution, mainly in the range −400 to 0 nm/s (negative sign shows the filament moving toward the clamped end, consistent with the action of kinesins). However, both velocities sometimes exceed 2 μm/s, namely as the free filament end snaps to a relaxed position.
In conclusion, we report an example of an autonomous molecular system that dynamically self-organizes through its elasticity and the interaction with the environment represented by the active forces exerted by motor proteins. Assembling such minimal systems that can mimic the behavior of much more complex biological structures might help to unveil the basic mechanism underlying the beating of real cilia and flagella.
Materials and Methods. Nonadsorbing Surface Coatings and Experimental Chamber Assembly. Glass coverslips (64 × 22 mm 2 , VWR) were cleaned by washing with 100% ethanol and rinsing in deionized water. They were further sonicated in acetone for 30 min and incubated in ethanol for 10 min at room temperature. This was followed by incubation in a 2% Hellmanex III solution (Hellma Analytics) for 2 h, extensive washing in deionized water, and drying with a filtered airflow. The cleaned coverslips are immediately activated in oxygen plasma (FEMTO, Diener Electronics, Germany) for 30 s at 0.5 mbar and subsequently incubated in 0.1 mg/mL poly(L-lysine)-graf t-poly(ethylene glycol) (PLL-g-PEG) (SuSoS AG, Switzerland) in 10 mM HEPES, pH = 7.4, at room temperature for 1 h on parafilm (Pechiney, U.S.A.). Finally, the coverslips were lifted off slowly, and the remaining PLL-g-PEG solution was removed for a complete surface dewetting. The experimental chamber was obtained by cutting a window (8 mm × 8 mm) on double-sided tape of thickness 10 μm (No. 5601, Nitto Denk Corporation, Japan) and sandwiched between two functionalized coverslips.
Imaging and Tracking. Image acquisition was performed using an inverted fluorescence microscope Olympus IX81 (Olympus, Japan) with a 63× oil-immersion objective (Olympus, Japan). For excitation, a Lumen 200 metal arc

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Letter lamp (Prior Scientific Instruments, U.S.A.) was applied. The data was recorded with a Photometrics Cascade II EMCCD camera (512 × 512 px). The images were acquired every 1 s with an exposure time of 100 ms. The microtubule trajectory was manually tracked by tracing the filament contour using JFilament, an ImageJ plugin for segmentation and tracking. 16 Reconstruction of Kinesin Forces. To determine a minimal configuration of forces that can lead to the observed shapes, we maximized the quantity Here Δ denotes the shortest distance between a point on the experimental and the shape, calculated from the forces. The integral runs over the part of the filament that is free to move, as obtained from an analysis of lateral fluctuations. The second and the third terms were introduced to avoid overfitting: they penalize solutions with excessive forces or with too many parameters. For a collection of forces, f 1 , ..., f N f , located at positions s 1 , ..., s N f , the expected filament shape is calculated from the equations Ä = s x t d d (6) with the boundary condition M(L) = 0. We assume a planar filament shape, such that f i , x, and t lie in the horizontal plane, and M is perpendicular to it. We solve the differential eqs 4−6 numerically and evaluate the deviation of the solution from the experimental shape Δ(s). Examples of reconstructed force distributions for one experiment are shown in Figure 3