Wave-like oscillations of clamped microtubules driven by collective dynein transport

Abstract

Microtubules (MTs) are observed to move and buckle driven by ATP-dependent molecular motors in both mitotic and interphasic eukaryotic cells as well as in specialized structures such as flagella and cilia with a stereotypical geometry. In previous work, clamped MTs driven by a few kinesin motors were seen to buckle and occasionally flap in what was referred to as flagella-like motion. Theoretical models of active-filament dynamics and a following force have predicted that, with sufficient force and binding-unbinding, such clamped filaments should spontaneously undergo periodic buckling oscillations. However, a systematic experimental test of the theory and reconciliation to a model was lacking. Here, we have engineered a minimal system of MTs clamped at their plus ends and transported by a sheet of dynein motors that demonstrate the emergence of spontaneous traveling-wave oscillations along single filaments. The frequencies of tip oscillations are in the millihertz range and are statistically indistinguishable in the onset and recovery phases. We develop a 2D computational model of clamped MTs binding and unbinding stochastically to motors in a “gliding-assay” geometry. The simulated MTs oscillate with a frequency comparable to experiment. The model predicts the effect of MT length and motor density on qualitative transitions between distinct phases of flapping, regular oscillations, and looping. We develop an effective “order parameter” based on the relative deflection along the filament and orthogonal to it. The transitions predicted in simulations are validated by experimental data. These results demonstrate a role for geometry, MT buckling, and collective molecular motor activity in the emergence of oscillatory dynamics.

Significance

Previous work has demonstrated buckling and flapping oscillations of end-clamped microtubules (MTs) driven by collective force generation of plus-end-directed molecular motors, kinesins. Models of such buckling instabilities have predicted general properties based on following forces, which remain to be tested. In this study, we have systematically immobilized MTs by their plus ends and allowed them to interact with a sheet of minus-end-directed motors, dyneins, in a “gliding-assay” setup. In the presence of ATP, the free MT end spontaneously oscillates with a frequency in the millihertz range. A computational model and a geometric order parameter predict the generality of such self-organized motor-driven MT oscillations. The phase space of patterns predicted in simulations is matched by experiments.

Introduction

Collective force generation by molecular motors and microtubules (MTs) is responsible for multiple vital cellular processes, including spindle assembly and segregation, intracellular cargo transport, and the dynamics of flagella and cilia. Collective mechanical effects arising from the activity of multi-motors has typically involved measurement of vesicle transport by both plus- and minus-end-directed motors (i.e., kinesins and dyneins). The presence of both antagonistic motors results in a tug of war that is resolved by multiple mechanisms, as reviewed elsewhere (1). Typically these studies used motor-decorated vesicles or beads with diameters ranging between ${10}^{-2}$ and a few micrometers, with only a small fraction of motors encountering the MT track, due to the spherical geometry that limits the contact area (2). This limits the range of collective motor activity that can be examined. In contrast, in MT “gliding assays,” typically one type of motor—kinesin or dynein—is immobilized on a surface with MTs transported as a linear cargo. The number of motors encountered is proportional to the length of the MT. Such assays have been used to show kinesin-dependent transport is independent of density (3), whereas varied linker lengths of kinesins provided evidence for “loose coupling” in driven MT transport (4). Dynein collective transport velocity did not change with increasing numbers of dyneins, but the study was limited to $<10$ motors due to the low rigidity the synthetic DNA-origami-based cargo used (5). Gliding assays of dyneins with up to $\sim$100 motors transporting a single MT demonstrated transport velocity and directionality increasing with number of motors, potentially as a result of the cooperativity of dynein (6,7). Thus, gliding assays can examine the effect of collective motor transport over a large range, not limited by cargo geometry.

Such large numbers of dynein motors transporting MTs in a physiological context can be seen in axonemal mechanics where dynein stepping on adjacent MTs and their coordination result in bending mechanics (8). Mathematical models of such sliding forces combined with elastic coupling of adjacent filaments demonstrate the emergence of wave-like oscillations (9). The spatial regulation of dynein activity is required to generate the coordinated asymmetric forces as described in the “switch inhibition” mechanism (10,11). Theoretical models have suggested aspects of such oscillations can be self-organized (12,13,14). However, the connection to detailed mechanics of collective motor transport still remains unclear. A model of collective transport by multiple ($\sim {10}^4$) molecular motors walking on filaments was shown reproduce oscillations that were quantitatively distinct from thermal fluctuations (15). Experimental observations of dynamics of spirals and curvature propagation in single filaments in actin gliding assays where motility was driven by a sheet of immobile myosin motors were reproduced by theoretical models of motor-driven filament-bending dynamics (16). Similar models have more recently predicted flagella-like motion of a chain of active Brownian particles anchored at one end with the pinned length, filament length, and noise levels determining the dynamics (17). The closest experimental test has been the observation of transient waves of bending observed when free MTs were bound to randomly distributed clusters of the plus-end-directed motor kinesin, nonspecifically bound to the surface (18). Although such experiments demonstrate the possibility of spontaneous onset of oscillations, they arise from defects. A systematic experimental test of this theory has not been attempted so far.

Here, we have reconstituted a minimal system of MT filaments clamped by their plus ends with the free ends interacting with immobilized dynein motors in a gliding-assay format, resulting in spontaneous, slow (millihertz) oscillations with wave-like deformations from base to tip. In contrast to previous work summarized above, where the kinesin motors were used, we demonstrate the emergence of such regular waves using a cytoplasmic yeast dynein as the force generator. We observe a qualitative change in spatio-temporal patterns with motor density and MT length in our minimal system. Compared to filaments in the absence of motors, we demonstrated a specific, dominant, millihertz frequency of oscillations. A computational model using discrete stochastic motors interacting with semi-flexible rods, MTs, predicts qualitatively comparable patterns. Using a deflection-based “order parameter,” we quantitatively classify the patterns and compare the model predictions to experiment. Our results could help in an improved understanding of the emergence of periodic oscillations in the context of collective motor-filament interactions.

Materials and methods

Protein purification and labeling

Tubulin was purified from goat brain using the high-molarity PIPES-based activity cycling protocol described for porcine brain tubulin (19) and used previously for goat brain tubulin (7).

Tubulin was labeled with either 5(6)-carboxytetramethylrhodamine N-succinimidyl ester or Alexa Fluor 488 NHS ester (Molecular Probes, Eugene, OR, USA) by incubating it with polymerized tubulin followed by a depolymerization-polymerization cycle, similar to the activity-based cycling protocol (19). This ensures that the labeled tubulin is active. Biotin-labeled tubulin was prepared by the same method by incubating (+)-biotin N-hydroxysuccinimide ester with polymerized goat brain tubulin. The molar ratios of label to tubulin monomer obtained for rhodamine, Alexa 488, and biotin were 3.66, 0.275, and 2.88, respectively. Dye concentrations were measured using absorbance measurements with minimal volumes (uCuvette G1, BioSpectrophotometer, Eppendorf, Germany).

Saccharomyces cerevisiae (yeast) strain VY208 was used to express a genomic copy of a modified cytoplasmic dynein fusion construct zz-GFP-GST-Dyn1331 under the control of a galactose inducible promoter with the N-terminal ZZ-tag for purification, GFP for visualization, and GST for dimerization as described by Reck-Peterson et al. (20). Cells were grown in batch cultures and induced with 2% (w/v) galactose (HiMedia, India) followed by affinity purification using the ZZ-tag using immunoglobulin G beads (GE Healthcare, Sweden). The protein was cleaved from the beads using a 6×-His tagged TEV which was expressed in E. coli BL21 DE3 and purified using Co-NTA resin (Thermo Scientific, IL, USA) followed by elution using imidazole (HiMedia, India).

To specifically immobilize dynein on glass surface, anti-GFP nanobody was used. The pGEX6P1-GFP-nanobody construct was a gift from Kazuhisa Nakayama (Addgene plasmid 61838, http://n2t.net/addgene:61838. RRID: Addgene_61838). The nanobody was transformed and expressed in E. coli BL21-DE3 cells that were induced with 1 mM isopropyl β-D-1-thiogalactopyranoside, IPTG (SRL, India) for 4.5 h at ${30}^{\circ }$C. The protein was purified from the clarified lysate using glutathione sepharose (GE Healthcare, Sweden) and eluted using reduced glutathione (21).

MT beating assay

MT filament assembly

MTs were prepared by polymerizing 20 μM unlabeled tubulin with 5 μM rhodamine-labeled tubulin in BRB80 and 10% glycerol at ${37}^{\circ }$C. The MTs were stabilized by addition of 20 μM Taxol (paclitaxel, Cytoskeleton, CO, USA). Free monomers were removed by pelleting MTs at 150,000 × g in TLA-100.3 rotor (Beckman Coulter, CA, USA). For plus-end biotinylation, the MT pellet was resuspended in BRB-80 and 10% glycerol followed by addition of either 20 μM biotinylated tubulin and 5 μM rhodamine-labeled (single-color filaments) or 10 μM biotinylated tubulin and 10 μM Alexa Fluor 488-labeled tubulin (dual-labeled filaments). The filaments were allowed to polymerize by incubating the mixture at ${37}^{\circ }$C for 30 min. Free monomers were once more removed by centrifuging at 150,000 × g in TLA-100.3 rotor (Beckman Coulter, CA, USA) and MT pellet resuspended in BRB-80 and 20 μM Taxol and used immediately.

Dynein motors and flow-chamber assembly for motility

The beating assays were performed in a flow-chambers constructed using 22 × 22-mm coverslip (no. 1.5, VWR) adhered to a slide (22 × 60 mm, HiMedia, India) by two parallel strips of double-backed Kapton polyimide tape (Ted Pella, USA) spaced 7 mm apart, creating a chamber of area 22 × 7 mm. The flow chamber was coated with a 1:1 molar ratio mixture of nanobody and streptavidin (each 20 μM concentration). After blocking the surface with casein, one chamber volume of dynein of a fixed concentration was flowed in. To test the effect of dynein density on patterns, purified dynein ranging in mass between 1.05 μg (4.53 pmol) and 2.46 μg (8.46 pmol) were introduved into the chamber and incubated for 8 min. Dilutions were used, resulting in effectively 1.07 μg (3.23 pmol) and 1.18 μg (3.56 pmol) dynein expected to occupy the chamber. Using a GFP-intensity calibration method (described later), this resulted in densities of 27 and 46 motors/${\mu \mathrm{m}}^2$ respectively. The dynein solution was incubated for 10 min and unbound dynein removed by washing with lysis buffer (30 mM HEPES (HiMedia, India), 2 mM Mg-Acetate (Amresco, OH, USA), 50 mM K-Acetate (Fisher Scientific, India), 1 mM EGTA, 10% glycerol). The density was estimated at the imaging surface, since the washes were optimized to result in no further outflow of dynein. We have assumed all the motors bound to the surface are active. This was followed by flowing in plus-end biotinylated MTs and incubating for 10 min. The unbound MTs were washed using wash buffer (lysis buffer + 20 μM Taxol). The beating was observed after addition of motility buffer (lysis buffer + 4 mM ATP + Antifade mix (0.005 mg glucose oxidase, 0.0015 mg catalase, 7.2 mM glucose in 100 μL 10× PBS (SRL Chemicals, Mumbai, India))). All assays were performed at 37°C. All reagents, unless otherwise stated, were from Sigma-Aldrich, MO, USA.

Microscopy and image analysis

The filaments were imaged using a 60× oil immersion lens (NA = 1.45) mounted on a Nikon Ti-E inverted microscope (Nikon, Tokyo, Japan) using a Tetramethylrhodamine Isothiocyanate (TRITC) filter (and fluorescein isothiocyanate in case of dual-color images) with temperature control system (Okolab, Pozzuoli, Italy). The filaments were imaged every 10 s for 10–20 min using an Andor Clara2 CCD camera (Andor Technology, Belfast, UK). The images were denoised by applying a median filter followed by background subtraction using FIJI (22). Filaments were either interactively traced using the NeuronJ plugin (23) in FIJI and when feasible using active contours and skeletonization using the MATLAB Image Processing Toolbox (Mathworks, Natick, MA, USA). Time-projected images with encoding of time was based on the FIJI plugin ZstackDepthColorCode (https://github.com/ekatrukha/ZstackDepthColorCode). The xy coordinates of the tips were generated by tracking the tip using plugin MTrackJ 1.5.1 also in FIJI. Drift observed in some videos was corrected by subtracting the tracked coordinates of the minus end of the MT by the position coordinates of the immobilized plus tip. All the analyses and plotting were performed in MATLAB (R2021b, Mathworks, MA, USA).

Estimating motor surface density

The surface density of dynein in motors per unit area (${\mu \mathrm{m}}^2$) was estimated based on the principle that the dynein is surface immobilized via the anti-GFP nanobody and an image in one plane of the epifluorescence microscope can provide an estimate of the number of molecules after calibration of the point spread function (PSF) and intensity to molecules conversion with a GFP standard, as previously described (7). Such an approach is based on comparable approaches used to estimate the surface density of myosin (24,25). To this end, a range of concentrations of bacterially expressed and affinity-purified 6xHis-eGFP molecules were introduced into the flow chamber and imaged. The number of molecules in the field of view was estimated as the product of the image volume and independently measured protein concentration. The image volume (cuboid) is the product of the PSF (height) and image area (in 2D). The PSF was estimated by measuring the fluorescence intensity ($I_z$) along the z axis (z) of 0.1 μm diameter TetraSpeck beads (Thermo Fisher Scientific, IL, USA). The data from multiple beads were fitted to the equation $I_z=a+\left(b-a\right)·e^{-\frac{{\left(z-c\right)}^2}{2·d^2}})$ (Fig. S4 A–C) where a, b, and c are scaling parameters and d is the SD, using the function CurveFitter in FIJI (26). The PSF is 1.45 μm, based on the relation $PSF=2.35·d$ (27). The fluorescence intensity of the eGFP dilution series on coverslips was used to plot a calibration curve of intensity of the image (y axis) as a function of protein surface density (x axis) and fitted to a straight line (Fig. S4 D). The image intensity of dynein-GFP from the coverslip surface was used to estimate monomer density resulting in a factor of 0.5 for the dynein dimer density. The resulting motor densities obtained from different dilutions of the stock protein ranged from 5.25 to 62.86 motors/${\mu \mathrm{m}}^2$, with control experiments performed without motors.

Filament mobility and contour analysis

The 2D filament deflections in x and y axes were quantified in terms of the free-tip (minus-end) mobility by plotting the deflection in x and y, angle subtended at the immobile tip with the positive x axis, the tip angle φ, and a combined “order parameter” that we refer to as deflection or sweep range, S. The fixed end (plus end) is considered the origin and the clamped segment, approximated as a straight rod, determines the x axis. The tip position at every time, t, is calculated as Euclidean distance between the clamped filament and the free end. The tip angle φ is the minimal angle subtended at the fixed end by two lines, the ray connecting the free (minus) end with the immobilized end of the filament (plus) and the line connecting the rigid segment (Fig. 1 G). Due to the difficulty in detecting the exact length of the clamped segment in experiments, the calculation of the tip angle starts from the immobile end of the filament. This position of the vertex of the angle does not alter the frequency of oscillations. The time series of φ for multiple filaments were smoothed using a moving average with a window size of 0.1 (fraction of total length) to denoise the signal (Fig. 2 A–C (insets)). The frequency of the smoothed oscillation was estimated using the fast Fourier transform (FFT), using the function fft in MATLAB (Mathworks, Natick, MA, USA) (Fig. 2 D). The difference in tip angles at the consecutive time points is defined as the angular velocity, $\omega =\left(\phi \left(t+\Delta t\right)-\phi \left(t\right)\right)/\Delta t$, where $\Delta t$ is the time interval between frames. The shape of filament is quantified as the local tangent angle (ψ) calculated at every discrete point along the filament contour and is given by:

$$\psi \left(s,t\right)={\tan }^{-1}\left(\frac{y_{n+1}-y_n}{x_{n+1}-x_n}\right)\text{,}$$ (1)

where ($x_n$, $y_n$) are the coordinates of the filament segment ($s_n$) at time, t. The beat frequency, amplitude, and the time period are inferred from time series of the angle φ(s, t), which is denoised using the function smoothdata and Fourier analysis performed to extract the frequency. Filament length estimates from our image analysis pipeline are subject to orientation-dependent discretization errors; to avoid such variation in length that may not have anything to do with the experimental data, a spline interpolation of the filament and running average MT length contour is plotted for the various kymographs of tangent angle. All the analyses and plotting were performed in MATLAB (R2021b, Mathworks, MA, USA).

Figure 1.

In vitro reconstitution of dynein-driven flagella-like MT oscillations. (A) The schematic represents the in vitro reconstitution setup with a single MT with a biotinylated plus-end (magenta) pinned to the surface by streptavidin, whereas the remainder of the filament is free (cyan) and interacts with surface-immobilized dynein, which can propel the MT. (B) The montage of a representative microscopy time series (Video S1) shows MT plus ends labeled with biotin- and Alexa 488-tubulin (magenta) bound to streptavidin on glass, together with free minus ends labeled with rhodamine-tubulin (cyan), transported by dynein undergoing bending dynamics. Here, $L_{MT}$ = 6.7 μm and ${\rho }_m$ = 47/${\mu \mathrm{m}}^2$. Time, mm:ss. Scale bar, 4 μm. (C) Filament contours from a time series of rhodamine-tubulin-labeled MT of length $L_{MT}$ = 4.7 μm similarly immobilized by a biotinylated plus end (Video S2) were tracked and contours projected in time. Dynein density, 27/${\mu \mathrm{m}}^2$. Color bar, time (s). (D and E) The dynamics of the free filament tip (minus end) are plotted in terms of (D) x (blue) and y (red) coordinates and (E) the tip angle φ in radians as a function of time in seconds. (F) the tangent angle (ψ) is plotted (colors) over time (y axis) and MT length (x axis) as a kymograph at 10-s intervals. Color bar, angle in radians. (G) The schematic represents the measures used to quantify bending (dashed lines): tip angle φ, span in x ($S_x$) and y ($S_y$), and local tangent angle $\psi \left(s,t\right)$. To see this figure in color, go online.

Figure 2.

Frequency analysis of MT tip angle oscillation. (A–C) The tip angle, φ, in radians is plotted with time in seconds for three representative filaments (colored lines) and overlaid with a smoothed curve based on an average filter (dashed lines). (Insets) A representative frame of the time series with segmented MTs (white) are overlaid with the linear fit to pinned segment of the filament (blue line) and the line connecting the two ends of the filament (red line). φ is the minimal angle between the blue and red lines. (D) The amplitude spectrum, $\left|P1\left(f\right)\right|$ (y axis) of the frequencies, f (x axis) from FFT analysis of the smoothed time series of φ is plotted (dashed lines). Colored lines correspond to the filaments in (A)–(C), gray lines are additional data. (Inset) The dominant frequencies of multiple (n = 25) filament time series analyzed with the mean frequency $⟨f⟩$ = 10 mHz. Dominant frequencies of individual trajectories are (A) 5.6, (B) 5.9 and (C) 1.2 mHz. To see this figure in color, go online.

Classification of MT motility patterns

We quantify the pattern of the free end of the filament, based on the deflection in x and y, as $S_x$ and $S_y$, where $S_y=y_{\mathit{max}}-y_{\mathit{min}}$ and $S_x=x_{\mathit{max}}-x_{\mathit{min}}$ (Fig. 1 G). We combine the deflections $S_x$ and $S_y$ into a variable we refer to as S metric, a measure of sweep, which is the log base 2 of the ratio of y and x extent represented by Eq. 2.

$$S={\log }_2\left(S_y/S_x\right)$$ (2)

The ratio was transformed by ${\log }_2$ to obtain a range where small changes were magnified and large changes compressed to infer qualitative patterns. We use this score to quantify the spatial patterns seen in simulations and examine any systematic effect of parameter changes on the patterns, as described in the “results” sections.

Computational model of MT-motor mechanics

The 2D stochastic computational model consists of the mechanics of semi-flexible rods representing MTs, stochastic spring-like motors with the emergence of collective effects arising from motors being bound to the same MT. All movement is subject to both thermal noise (random) and viscous drag. The simulator solves the Langevin dynamics of deterministic and stochastic components using an agent-based approach in C++ simulation code Cytosim (28). Collective mechanical effects then emerge from this, as previously described for gliding assays (7) and aster transport in confinement (29). The dynamical behavior of the individual components is described in the following section.

MTs

MTs are modeled as semi-flexible filaments. The lengths of all filaments ($L_{MT}$) are constant during a simulation run. The model points along the filament are spaced regularly at intervals of 0.05 μm. The bending elastic modulus of the filament (κ), thermal energy, drag experienced by the filament, and the motor forces determine the filament dynamics, with most values taken from literature (Table 1).

Table 1.

MT and Motor Parameters Used in Simulations

Symbol	Description	Value	Reference
MTs
Κ	MT bending rigidity	20 pN ${\mu \mathrm{m}}^2$	Gittes et al. (34)
$L_{MT}$	MT length	5-25 μm	this study
Dynein
${\mathrm{v}}_0$	single-motor velocity	0.10 μm/s	Reck-Peterson et al. (20)
$r_a$	attachment rate	5 ${\mathrm{s}}^{-1}$	Leduc et al. (52)
$d_a$	attachment range	0.02 μm	Estimated from kinesin-binding range (53) and molecular size of dynein
$r_0$	load-free detachment rate	0.04 ${\mathrm{s}}^{-1}$	Reck-Peterson et al. and Rao et al. (20,54)
$F_d$	detachment force	3 pN	Nicholas et al. (38)
$F_s$	stall force	5 pN	Gennerich et al. (55)
$k_m$	linker stiffness	100 pN/μm	Lindemann and Hunt; Burgess et al.; Kamiya et al. (56,57,58)

The values for mechanochemical properties of the MTs, motors, and the simulation system are taken from literature where available.

Motors

Motors are modeled as Hookean springs, represented as a bead with a spring (Fig. 4 A). The bead represents the motor head and the base of the spring depicts the tail, which is immobilized on the surface. The motor head can bind to the MT with a probabilistic attachment rate of $r_a$ if it is within a certain distance to the filament, the binding range $d_a$. The motors are modeled as continuous steppers with a fixed load-free velocity ($v_0$) resulting in an instantaneous velocity v at each time step dt, following a linear force-velocity relation:

$$v=v_0·\left(1-f_{\Vert }/F_s\right)$$ (3)

Figure 4.

Simulating motor-driven flagellar beating of a pinned MT. (A) The schematic of the model of MTs clamped at one end from their plus ends with stiff attachment (magenta) over the length of ($L_p$), whereas motors (gray circles) stochastically attach to an MT (gray circle), step a distance $d_v$ (violet circle), and detach at a force-dependent rate $r_d$. Motors are modeled as springs (black) with a constant $k_m$ resulting in a force $F_m$ when extended. $d_a$, attachment distance (dashed line); $r_a$, attachment rate. (B) Representative image time series of the 2D simulation of filament beating. Gray circles, motors; red line, clamped segment; gray line, free segment. Time in seconds. Scale bar, 2 μm. (C) Filament contours projected in time at 15-s intervals over 1200-s simulation. (D and E) The dynamics of the free plus end are plotted in terms of the time dependence of (D) the tip angle φ (radians) and (E) the x (blue) and y (red) positions. (F) The local tangent angle along the MT filament ψ (radians) is represented as a kymograph over time (y axis) and along the MT length (x axis). Here, filament length ${\mathrm{L}}_{MT}$ = 15 μm, motor density ${\rho }_m$ = 10 dyneins/${\mu \mathrm{m}}^2$, pinned MT length L_p = 2 μm. Color bar, angle in radians. To see this figure in color, go online.

Here, the forces $f_{\Vert }$ and $F_s$ are respectively the component parallel the MT and the stall force. The motor is modeled to step forward and backward depending on the ratio of the parallel to stall force since the step size is given by $\delta r=\delta v\times \delta t$ and $\delta v$ can be positive or negative. The force experienced by the motors is calculated as $F_m=-k_m·\Delta L_{MT}$, where $\Delta L_{MT}$ is the extension along the motor and the $k_m$ the stiffness constant of the spring. The motor can detach probabilistically with a detachment rate ($r_d$), dependent on the magnitude of the force experienced by the motor under the stretch ($F_m$) and the basal load-free detachment rate ($r_0$). The detachment force ($F_d$) is the threshold force determining single-motor force-dependent detachment kinetics governed by the Kramers law (30):

$$r_d=r_0·e^{\left|F_m\right|/F_d}$$ (4)

wherein $r_d$ is the instantaneous and $r_0$ is the basal load-free detachment rate of a single motor. Motor parameters are also taken largely from experiment (Table 1).

MT pinning sites

MTs are pinned to anchors, which are similar to motors but immobile (i.e., they have no velocity). They are modeled as already bound to filaments, with extremely high detachment forces, which results in no detachment during the simulation. These pinning sites are also very stiff with a high elasticity constant $k_a$. The spacing between the anchors determines the length of MT pinning ($L_p$).

Simulations

Simulations were run in a 2D space with a periodic boundary condition using Cytosim version f7921c4 from the gitlab (https://gitlab.com/f.nedelec/cytosim). The effects of thermal energy and viscosity on the MT-motor interactions are also modeled, with k_BT = 4.3 pN nm, viscosity η of 0.826 cP based on the viscosity of the buffer that contains 10% glycerol (31). Simulations were integrated with a time step of 0.01 s. The motors are initialized randomly in the space with MT positioned at the center aligned along the x dimension. The simulation parameter file relating to the scenario described can be downloaded from https://github.com/CyCelsLab/MT-beating and run in Cytosim.

Results

Dynein gliding assay with plus-end-clamped MTs produces traveling-wave oscillations

We have developed a minimal experimental reconstitution by modifying a typical 2D gliding assay by clamping one end of the MT. The system has only two key components: 1) cytoplasmic dynein motors grafted to a glass coverslip surface, and 2) MTs whose plus ends are immobilized via a biotin-streptavidin linkage with the rest of the filament free to bind to motors (Fig. 1 A). The dynein has a GFP-tag in the tail domain, which is used to orient the motor with the motor head pointing away from the glass surface by immobilizing an anti-GFP nanobody on the coverglass surface. MT tip-immobilization is achieved by labeling the plus ends with biotinylated tubulin monomers combined with streptavidin layered on the glass surface. We use the yeast cytoplasmic dynein due to the high processivity, well-established methods for purification and activity (20), and unidirectional transport of filaments in gliding assays (7). The intrinsic variability of MT lengths combined with density variation allowed us to test the effect of increasing motor numbers with motor density, ${\rho }_m$, ranging from 5.25 motors/${\mu \mathrm{m}}^2$ to 62.86 motors/${\mu \mathrm{m}}^2$, measured using calibrated microscopy (as described in the “materials and methods” section).

The collective force generation on MTs clamped at one end and driven by motors is akin to a cantilever beam with a uniformly distributed force along the entire length. To achieve this, the clamp involves labeling between 2% and 20% of the total length of the filament with biotinylated tubulin (described in the “materials and methods” section). Pinning at a point with rotational degrees of freedom corresponds to the simplest case and results in filament swiveling, as previously shown in MT gliding assays when one single molecule attaches to the tip of a filament (7,32). When the clamp is sufficiently long, the MT plus end remains straight (magenta), whereas the free minus end (cyan) bends and buckles in response to increasing elastic force (Fig. 1 A). Once the filament bending exceeds the energy of the MT-motor attachment, the motors detach resulting in a recovery stroke, as seen in time series of both dual-labeled (plus end, magenta; minus end, cyan) filaments (Video S1) and single-dye-labeled filaments (Video S2). Typical tip trajectories complete one cycle in $\sim$200–400 s. Time-projected filament contours demonstrate the oscillatory nature of the movement (Fig. 1 C). The x and y coordinates of the free end of the MT describe oscillatory movement of filament tip deflections over three to five cycles (Fig. 1 D). The amplitude of the deflections in y are higher than in x because the clamped part of the filament defines the x axis, and, as expected from bending and buckling, deflections in y (orthogonal to the length) dominate. The tip angle φ as a function of time combines the x and y deflections and shows periodic, and somewhat regular, oscillations between −1 and 1 radians (i.e., −57° to 57°, a range of 114°) (Fig. 1 E). In ciliary motion, tip oscillations are observed to happen at 1 Hz and faster, whereas these oscillations are $\sim {10}^3$ slower. To quantify the apparent traveling-wave nature of the oscillations, we also measured the filament tangent angle ψ (radians), pixel-wise, along the contour and over time, resulting in a kymograph (a space-time plot) of filament dynamics (Fig. 1 F). Such kymographs have been used to demonstrate the oscillatory and wave-like nature of minimal reconstituted filaments driven by motors, both MT-kinesin systems (18) and more recently actin-myosin systems (33). The kymograph of the filament curvature ψ of the representative filament suggests a propagation of curvature over $\sim$300 s, after which the filament snaps back and repeats the cycle, suggesting a slow oscillation with a frequency in millihertz. The tangent angles (ψ) at the same position along the MT length appear to show some variability or “noise,” suggesting that, as the filament bends, local effects of binding-unbinding and thermal effects influence the propagation of the buckling along the filament length. Such noise is also seen in the tip position (Fig. 1 D) and angle dynamics (Fig. 1 E).

Video S1. Dynamics of dual-color-labeled MT flagella-like MT-beating assay

A representative time series of an MT in dual-channel microscopy with an Alexa 488-labeled plus end (magenta) and the minus end labeled with rhodamine-labeled tubulin (cyan) is seen (left) oscillating in time (right) overlaid with the tracked contour of the MT. Assay conditions are described in detail in the “materials and methods” section. $L_{MT}$ = 6.73 μm, ${\rho }_m$ = 46/${\mu \mathrm{m}}^2$, pinned MT = 11.5$\%$. Time, mm:ss (minutes:seconds); frame rate, 5 fps. Scale bar, 2 μm.

Video S2. Dynein-driven beating of individual MT in beating assay

A representative time series of rhodamine-labeled MTs (gray) undergoing dynein-driven oscillations (left). The plus end was co-polymerized with biotinylated tubulin and the coverslip was coated with a mixture of streptavidin and anti-GFP nanobody (1:1 molar ratio), passivated with casein followed by flowing in of a minimal GFP-dynein (GFP-Dyn) construct showing (left) oscillatory wave-like patterns (right) overlaid with the filament contour after tracking. Assay conditions are described in detail in the “materials and methods” section. $L_{MT}$ = 4.8 μm, ${\rho }_m$ = 27/${\mu \mathrm{m}}^2$, pinned MT = 11$\%$. Time, mm:ss; frame rate, 5 fps. Scale bar, 2 μm.

The measures used to quantify MT oscillation dynamics (schematically represented in Fig. 1 G) are 1) tip angle φ, angle subtended at the plus end by the straight line of the pinned segment (cyan) and the free minus-end tip (cyan); 2) tangent angle ψ, the angle made by the tangent to each point along the contour with the projection of the clamped segment (cyan), treated as the positive x axis; 3) kymographs (space-time plot) of filament bending are based on the tangent angle ψ along the length of the MT with time; 4) span $S_x$ and $S_y$, the maximal deflection of the free end of the MT in x and y axes, respectively. These measures are used throughout our study to further quantify the dynamics to explore the principles driving the observed phenomenon.

Active, slow, symmetric oscillations of the free end of MTs in experiment

The tip angle φ was analyzed from multiple, representative filament time series showing periodic dynamics with noise and some instability (Fig. 2 A–C). We smoothed the data and used them to examine the oscillation frequency and amplitude spectrum using a FFT (as described in the section “materials and methods”). The power spectra of time-dependent φ dynamics from multiple pinned filaments (n = 25) showed frequencies ranging between 0 and 50 mHz (Fig. 2 D). The frequency with the maximal amplitude, i.e., dominant frequency, was found to be 10 mHz after averaging (n = 25) for multiple filaments (Fig. 2 D (inset)). This assay is robust since the filaments are seen to be either transported in a processive manner similar to a 2D gliding assay or oscillating (Video S3), with comparable statistics of 77% gliding as compared to 23% oscillating between replicates and densities (Fig. S1). The higher proportion of gliding filaments could be either because not all filaments were biotinylated equally or not all biotinylated filament ends are captured by the streptavidin on the surface, related to the streptavidin density. Our choice of the proportion of streptavidin to anti-GFP nanobody to immobilize motors was optimized to increase the proportion of oscillating filaments, since there appears to be a trade-off between plus-end pinning and motor accessibility to the minus ends. At the same time the presence of such gliding MTs confirms that oscillations arise due to motor activity, serving as an internal control for the active nature of the process. In absence of motors, filaments undergo passive oscillations undergoing rapid and random thermal fluctuations of very small amplitude, less than $\sim 1\mu$m (Video S4), with filaments visualized using rhodamine-labeled tubulin. Filaments are clamped in a manner similar to the oscillating filaments, but their motion arising from thermal energy shows comparatively small amplitude bending and lacks any buckling or traveling waves.

Video S3. Dynein based flagella-like beating assay

The video represents an entire field of view of experimental time series of rhodamine-labeled MTs in a dynein-driven flagellar-beating assay. The MTs are pinned to the surface by streptavidin coating with biotinylated tubulin labeling the plus ends. Green boxes: filaments that appear to undergo flagella-like beating. Time, seconds. Scale bar, 10 μm.

Video S4. Swiveling and fluctuation of MTs in absence of motors

Time-lapse videos of representative MTs clamped at the plus end in the absence of motors (${\rho }_m$ = 0/${\mu \mathrm{m}}^2$) and arranged in order of increasing lengths (left to right) L = 2.2, 8.6, 14.2, and 18.4 μm, corresponding to the time projections in Fig. 8. Time, mm:ss. Scale bar, 4 μm.

We examined the symmetry of the oscillations in time by examining the filament time series (Fig. 3 A and B). The angle made by the free end with the clamped segment, φ, showed oscillations as before (Fig. 3 C). The oscillatory data were denoised and peaks and troughs detected. The instantaneous angular velocity (dφ/dt), the discrete derivative of the angle between subsequent time-steps, was classified as either belonging to onset, the stage between one trough and a succeeding peak, or recovery, from a peak to the succeeding trough (Fig. 3 C (right y axis)). The goal was to use the angle to address the question of whether there was any quantifiable difference in the free end, as it oscillated from a minimum (trough) to the maximum (peak), as compared to maximum (peak) to trough, representative of the entire cycle through onset and recovery. The values of the angular velocity distribution (ignoring the sign) of onset and recovery strokes were found to be statistically indistinguishable (Fig. 3 D).

Figure 3.

Analysis of the symmetry of free-end oscillations of MTs. (A) The representative fluorescence microscopy time series of a single plus-end-pinned MT (gray) in presence of surface-immobilized dynein. Time, mm:ss. Scale, 2 μm. Color scale, start (blue) to end (yellow) of filament contour. (B) Detected filament contours are overlaid with an interval of 10 s over a total period of 940 s. Color bar, contour length of the MT from plus-tip (blue) to minus end (yellow). (C) Tip angle is plotted for the time series in (A). The raw data (black dashed lines) is smoothed using a Savitzky-Golay filter (blue lines). Peak finding to determine the local peaks of the smoothed trajectory was used to plot the absolute instantaneous angular velocity (|dφ/dt|), right y axis) for the onset (green) and recovery (red) strokes of the free end of the MT. (D) Frequency distributions of the onset (green) and recovery (red) phases of the absolute value of tip angular velocities are plotted and compared using Kolmogorov-Smirnov statistics showing them to be statistically similar. Test value = 0.24, p value = 0.12. To see this figure in color, go online.

Previous reports have included comparable cytoskeletal filament oscillations driven by other motors such as MTs with kinesin (18) as well as actin-myosin (16). This suggests such oscillations could constitute a general class of pattern-forming systems. Identifying the theoretical minimum of mechanical components and interactions could help better understand both the generality of such patterns and the limitations within which our experimental setup may function. We therefore proceeded to develop a computational model to examine the principles governing the patterns and explore the sensitivity of the system to motor density and MT length.

2D agent-based model of clamped MTs driven by collective motor activity results in regular wave-like oscillations

The computational model consists of two components, MTs and motors, with MT plus ends pinned. MTs are modeled as semi-flexible polymers with plus ends clamped to the surface and the remaining filament free to bind molecular motors, which are minus-end directed with probabilistic binding and unbinding and linear force-velocity relations (Fig. 4 A). The model description lists the equations governing the MT-motor model (“materials and methods” section), whereas model parameters are taken from the literature wherever possible (Table 1). MT pinning is modeled by introducing multiple rigid attachments that clamp a segment of the MT plus end to the surface. The attachment is through binding elements that have a high spring constant, $k_a$, of ${10}^3$ pN/μm and do not detach from the MTs. In the 2D simulation box, MTs are initialized to be straight, with the free segment rapidly binding to a sheet of motors along its length, similar to previous models of gliding assays. Once motors have bound to the MT, their stepping collectively produces forces, which, combined with the clamped plus end, spontaneously result in bending and buckling if the force exceeds the critical force and motors detach, resulting in a recovery stroke (Fig.s 4A and Video S5). The collective behavior at the spatial scale of micrometers emerges over time, based on the interaction rules of each component (model) and their single-parameter values at nanometer range, and is not explicitly encoded in the model formalism. The patterns of bending appear symmetric between the onset and recovery strokes (Fig. 4 B), also seen in time-projected filament plots (Fig. 4 C). The periodic movement of the free tips can be observed in the time dependence of the tip angle φ (Fig. 4 D) as well as the x and y positions of the free end (Fig. 4 D). We also observe some stochasticity in the position-time and angle-time plots, which originates from stochastic binding-unbinding. One cycle completed in $\sim$300 s, corresponding to a frequency of $\sim$3 mHz and corresponding closely to experimental observations. The curvature propagates in a base-to-tip manner, as seen from the kymograph of the tangent along the contour (ψ, described in detail in the “materials and methods” section and Eq. 1) with time (Fig. 4 E). Interestingly our simulations are both qualitatively comparable to experiments in terms of base-to-tip bending, symmetry of onset and recovery strokes, as well as quantitatively in terms of amplitude and timescale of free-end oscillation angles. This is despite the lack of optimization of parameters that we believe arises from the choice of mechanical and dynamical parameters of the underlying components, taken from experimental literature. The dynamics in that sense are then emergent.

Video S5. Simulating anchored MT beating driven by surface-immobilized motors

A simulation video over the 1200 s of a 15-μm MT filament anchored at the plus end at two sites (red circles) 2 μm apart. The motor density corresponds to 10 motors/${\mu \mathrm{m}}^2$.

To further explore the limits of the theory, we explore parameter sensitivity of our model and compare the predictions to experiment.

Simulations predict qualitative transitions in oscillations depending on motor density and filament length

Based on a previously developed model of cytoskeletal filament bending by motors on a surface where the radius of curvature of spirals (R) formed by pinned filaments scaled as $R\sim {\left(k_{B}T{\xi }_p/f\right)}^{1/3}$, where ${\xi }_p$ is the persistence length and f is the linear force density, i.e., motors acting per unit filament length (16). Therefore MT filament lengths ($L_{MT}$) were varied over a range of 5–25 μm, based on typical experimental observations, and dynein motor density (${\rho }_m$) was varied over three orders of magnitude, based on previous experimental reports (6,7). The time-projected images of filament contours demonstrate the formation of multiple patterns, qualitatively classified as fluctuations (irregular and uncorrelated bending), oscillations (prominent curvature with changing direction), and looping (multiple segments of the filament take on different curvatures) (Fig. 5 A). Fluctuations are observed in the complete absence of motors, similar to those reported for cantilevers in a thermal bath. As the motor density increases, the patterns appear to depend on the MT length. We observe short filaments of length 5 μm subjected to increasing motor density appear to undergo bending to one side, do not recover, but higher length or density both demonstrate oscillations (Fig. 5 A). In representative kymographs, we examine the effect of constant length and constant motor density on the local tangent angle with time and length, indicative of the traveling-wave oscillations. The length dependence is seen when a constant motor density of ${\rho }_m={10}^1$ motors/${\mu \mathrm{m}}^2$ shows onset of oscillations when MT length is greater than 15 μm (Fig. 5 B). On the other hand, a constant MT length of 15 μm in presence of low motor densities (${\rho }_m=$ ${10}^0$ motors/${\mu \mathrm{m}}^2$) results in only fluctuations with a density of ${\rho }_m=$ ${10}^1$ motors/${\mu \mathrm{m}}^2$ or greater required to result in regular filament buckling and oscillation (Fig. 5 C Video S6). Looping also emerge at lower motor densities for longer filaments (10–25 μm). In the combined matrix of ${\rho }_m$ and $L_{MT}$ variation, along a diagonal we observe a narrow region of patterns that resemble regular oscillations (Fig. 5 A).

Figure 5.

Simulating the effect of MT length and motor density on flagellar beating. (A) The dynamics of filaments are simulated with MT lengths ranging from 5 to 25 μm (rows) and motor densities ranging from 1 to ${10}^3$ motors/${\mu \mathrm{m}}^2$ (columns). The projected filament contours are plotted at 3-s intervals over a total time of 1200 s. (B and C) Representative kymographs of the local tangent angles, ψ (radians), plotted as a function of position along the filament (x axis) and time (y axis). Dashed lines indicate either (B) increasing motor density with MTs lengths constant, here 15 μm, or (C) increasing MT lengths with motor density constant, here ${\rho }_m$ = 10 motors/${\mu \mathrm{m}}^2$. To see this figure in color, go online.

Video S6. Simulating the effect of motor density on MT beating

Simulation videos in presence of increasing motor density, ${\rho }_m$ increases (from left to right) as 0, ${10}^0$, ${10}^1$, ${10}^2$, and ${10}^3$ motors/${\mu \mathrm{m}}^2$ with an MT of length 15 μm with 2 μm pinned to the surface at the plus end. MTs are depicted as black lines, whereas motors bound to the MT are displayed as blue circles. Total time of simulation: 20 min.

Although the motor density and length suggest that there is a clear motor number dependence on patterns, we find that the number of motors bound to a filament fluctuate in time for all lengths, as observed for representative traces from simulations of filaments of increasing lengths from 5 to 25 μm interacting with motors of density ${\rho }_{2D}=$ 10 motors/${\mu \mathrm{m}}^2$ (Fig. S2 A). Fluctuations themselves were expected from the stochastic binding-unbinding kinetics, forces acting on individual motors, and thermal noise, and the number of bound motors is lower than the expected value $N_{expected}$ (Fig. S2 B). Here, the expected number serves as an upper limit, obtained from a square lattice approximation of the distribution of motors, i.e., $N_{expected}=L_{MT}\times \sqrt{{\rho }_{2D}}$. Interestingly, although tip angles oscillate in a regular manner for $L_{MT}$ of 15 μm and longer, this correlates with more than 10 motors being bound to the MT on average. The time-dependent number fluctuations, however, do not correlate with the tip oscillations (Fig. S2 C). This could result from the fact that, although a minimal number of motors is required to generate the bending and buckling forces, filament mechanics also plays a role in the oscillatory patterns of MTs.

These results suggest the onset of regular oscillations requires a minimal MT length and motor density (motor number), with both factors determining frequency of oscillations. We test whether a specific frequency dependence can be seen in simulations by analyzing simulated filament oscillations in presence of a specific motor density (Fig. 6 A) for the free-end angle φ dynamics (Fig. 6 B) by examining the Fourier spectrum and identifying the highest-amplitude oscillations, $f_{\mathit{max}}$ (Fig. 6 C). A systematic analysis of the effect of MT length and motor density suggests the oscillations of the free end show a distinct coordinated increase with both variables in amplitude (Fig. 6 D) and a decrease in corresponding frequency to $\sim 10$ mHz (Fig. 6 E). Data generated from multiple runs demonstrate a high degree of variability in individual runs in terms of both amplitude (Fig. 6 F) and frequency (Fig. 6 G). This further emphasizes the stochastic nature of the oscillations.

Figure 6.

Frequency analysis of filament oscillations from simulations. (A–C) Representative simulations of motor-driven filament beating were used to plot (A) time-projected contours (color bar, time) of MTs of length 15 μm in presence of a motor density ${\rho }_m$ of 10 motors/${\mu \mathrm{m}}^2$, (B) tip angles (φ) with time, and (C) the amplitude of oscillation frequency from FFT analysis. $f_{\mathit{max}}$ is the frequency with highest amplitude. (D) The amplitude and (E) frequency of oscillations from simulated filaments as a function of varying lengths ($L_{MT}$ = 5–25 μm) and motor densities (${\rho }_m$ = 0 (no motors) to ${10}^3$ motors/${\mu \mathrm{m}}^2$) are shown as mode of multiple simulation runs (n = 15). (F and G) The relationship of the (F) dominant frequency and (G) amplitude as a function of filament length (x axis) are plotted at different motor densities (color). Circles, mean $\pm$ SD. To see this figure in color, go online.

To compare experiments to theory we proceed to examine the qualitative nature of the transition in observed patterns of MT oscillations.

Geometric measure to classify spatial patterns of clamped filaments driven by motor-sheet

Our experiments and simulations demonstrate the systematic emergence of periodic beating patterns arising from a simple 2D system that are specific to the filament length and motor density. Simulations were run for a range of filament lengths and densities and the outputs evaluated in terms of the time-projected filament profiles (Fig. 7). These demonstrate that short filaments ($\sim$5 μm) will only undergo (I) fluctuations even when ${\rho }_m$ is varied from 0 to ${10}^2$ motors/${\mu \mathrm{m}}^2$, whereas long filaments ($\sim$20 μm) will exhibit (II) regular oscillatory motion, even at lower dynein densities of ${10}^{-1}$ motors/${\mu \mathrm{m}}^2$. Filaments undergo (III) looping when the MTs are long ($\ge 10\mu$m) and motor density sufficiently high ($\ge {10}^2$ motors/${\mu \mathrm{m}}^2$).

Figure 7.

Quantifying MT patterns in terms of space explored by the free tip in experiment and simulation. The qualitative phases are distinguished based on the sweep value $S={\log }_2\left(S_y/S_x\right)$, where $S_x$ and $S_y$ are the extent of tip movement along x and y axes as described in Fig. 1G. The phases are as (I) fluctuations, (II) regular oscillations, (III) looping, and (IV) swiveling based on the time-projected patterns of representative simulations and experiments. In experiment, the patterns observed are fluctuations, S = 3.8 ($L_{MT}=$ 9.7 μm, no motors); flagella-like, S = 0.41 ($L_{MT}=$ 4.7 μm, ${\rho }_m=$ 27 motors/${\mu \mathrm{m}}^2$); looping, S = −1 ($L_{MT}=$ 24 μm, ${\rho }_m=$ 27 motors/${\mu \mathrm{m}}^2$); and swiveling, S = 0 ($L_{MT}=$ 2.2μm, no motors). Scale bar, 2 μm. Color bar, minutes. ${\rho }_m$, motor density. To see this figure in color, go online.

The state (IV), swiveling, refers to movement that corresponds to very short pinned regions akin to a filament being bound at the very end and encountering a very low density of motors, or their absence. We find that no single quantitative measure of filament dynamics used to describe the dynamics in simulations can help distinguish between the visibly different scenarios, i.e., the maximal range of values explored in x and y ($S_x$, $S_y$), oscillation frequency f, maximal tip angle ${\phi }_{mx}$, angular velocity ω, and number of motors bound (Fig. S3). To address this, we came up with a simple metric that serves as an order parameter for this system, which we refer to as the sweep range, the S metric (Eq. 2; “materials and methods” section). This metric, S, is the binary logarithm (${\log }_2$) of the ratio of the deflection along the respective y and x axes, $S_y$ and $S_x$ (schematic in Fig. 1 E). We arrived at this measure after testing multiple alternative order parameter measures to quantitatively identify distinct states of the filament dynamics. This captures the relative proportion of space explored by the tip along the y axis relative to the x axis (the span) and the binary logarithm scales the range. The specific values of the S metric corresponded to distinct patterns of MT dynamics (Fig. 7). As a result, the following thresholds were used to classify patterns:

$$\begin{array}{cc}S\ge 3& \left(\mathrm{I}\right)\text{Fluctuations,}\\ 3>S>0& \left(\text{II}\right)\text{Regular}\text{oscillations,}\\ 0>S& \left(\text{III}\right)\text{Looping}\end{array}$$ (5)

S = 0 corresponds to the special case of swiveling that arises when the point of attachment of a filament is one molecule with universal degree of rotation. The value of $S=0$ corresponds to the situation when the range of values covered in x and y are equal, i.e., $S_y=S_x$.

To test of the utility of our order parameter, we asked if it also matches the patterns observed in experiments. We examined the tracked coordinates of multiple experimentally measured time series of clamped MTs of different lengths showing diverse patterns, with multiple motor densities and MT lengths. (I) Fluctuations were observed in the absence of motors, whereas increasing MT lengths and motor density gave rise to (II) oscillations and (III) looping. We also observe (IV) swiveling when a very short segment of the MT end is anchored with the rest of the filament free to move. We find not just that experimentally observed filament pattern dynamics match the qualitative behavior of simulation outputs but even the quantitative measure of the sweep variable (S) thresholds match (Fig. 7).

We proceed therefore to test if this novel order parameter, the filament sweep value S, could be used to quantitatively classify the systematic effect of MT and motor properties on the emergent dynamics of the system.

Comparing simulation predictions to experiment: Effect of MT length and motor density

Using the combined x- and y-deflection measure, we classify simulation outputs into the categories 1) fluctuations, 2) regular oscillations, and 3) looping, and we find that a combined variation of MT length and motor density corresponds to a transition between these patterns in an objective manner (Fig. 8 A). Based on the thresholds that correspond to qualitatively distinct patterns (Fig. 7), we mapped interpolated boundaries of $⟨S⟩$ = 0 and 3 through the MT-length density space (Fig. 8 A). We proceeded to test the validity of the predicted phase-diagram boundaries by plotting experimentally analyzed $⟨S⟩$ values (colors) from multiple filament time series (n = 65) for motor densities: 0 (no motors), 5.25, 16.04, 31.06, and 62.86 dyneins per ${\mu \mathrm{m}}^2$ and the MT lengths from 5 to 25 μm, as calculated in simulations (Fig. 8 B). Each point corresponds to one filament time series with the color based on the value of the S metric. Motor densities were estimated in three biological replicates using a GFP calibration approach as described in the “materials and methods” section and Fig. S4, with biological replicates showing small differences in the exact density obtained. We find that filaments in the absence of motors (density = 0 motors/${\mu \mathrm{m}}^2$) indeed correspond to the $S>3$ threshold predicted from simulations, independent of MT length. The high value of the span measure can be understood to arise from the fact that the pinned segment of filaments corresponds to the x axis in both experiments and simulations and shows very little movement ($S_x$ in Eq. 2) compared to other patterns, whereas the y axis value emerges from the interplay of the thermal and filament bending energy scales (the numerator in in Eq. 2). Looping predicted in simulations to correspond to $S<0$ is observed experimentally at high motor density and long filaments (upper right quadrant of Fig. 8 B), arising from the movement of the tip closer to the base of the MT along the x axis and therefore having a high value of $S_x$ (Eq. 2). This is expected to arise due to the high motor density and resultant force. Regular wave-like oscillations were predicted in simulations to correspond to $0<S<3$ (between green and blue lines). Although experimental values of the S metric of short filaments ($<10$ μm) do correspond to the predicted boundaries and show oscillations, at high motor densities, the S metric also indicates looping (Fig. 8 B). Thus, although experimental data in the absence of motors and at high motor densities and long MTs are consistent with simulations, some differences are observed between simulation predictions and experimental observations in cases of short filaments at high motor density. The S metric value predicted by simulation should indicate fluctuations ($S\ge 3$) but, in experiment, it appears to correspond to wave-like oscillations ($0<S<3$). This could arise either from filament bending rigidity differences in stabilized MTs (34,35) as compared to those used in simulations or the deviation from a linear force-velocity and detachment rate dynamics of the yeast dynein (36,37,38). A qualitative match in terms of the effect of increased motor density and lengths resulted in transitions from 1) fluctuations through 2) wave-like oscillations to 3) looping seen in experimental filament time series (Fig. 8 C and Video S7). The quantitative deviations from experiment suggest the need for further improved experimental control over MT lengths, geometry, and motor immobilization in future.

Figure 8.

Motor density and MT-length-dependent transitions in experiment and simulation. (A and B) The parameter space for varying motor densities (${\rho }_m$) and MT lengths ($L_{MT}$) from (A) simulations and (B) experiments are divided into three regions based on their S values of $⟨S⟩$ = 0 (blue) and $⟨S⟩$ = 3 (green), which distinguish qualitative filament tip dynamics such that (I) $⟨S⟩$ > 3 corresponds to fluctuations, (II) 3> $⟨S⟩$ > 0 to regular oscillations, and (III) $⟨S⟩$ < 0 to looping. (A) Simulated values are averaged over 10 runs for each parameter, whereas (B) experimental data are plotted for multiple filament time series analyzed (circles) where the motor density was varied from 0, 5.25, 16.04, 31.06, and 63.02 motors/${\mu \mathrm{m}}^2$ from three biological replicates (n = 65). (C) Representative time-projected regions of interest of filaments taken from (B) demonstrate the qualitative change in MT dynamics with MT length and motor density. The corresponding time series sampled from multiple density and length values are seen in Video S4 (no motors) and SV7 (increasing density and length). To see this figure in color, go online.

Video S7. Oscillating filament time series with multiple motor densities and varying MT lengths in experiment

The time series of representative rhodamine-labeled MTs (gray) for increasing MT length in the presence of a calibrated motor density (${\rho }_m$) of (top to bottom) 5.25, 16.04, 31.06, and 62.86 motors/${\mu \mathrm{m}}^2$. MT lengths, L, vary between 2.57 and 24.6 μm. These filaments correspond to the time projections seen in Fig. 8C. Each region of interest extends over slightly different times (mm:ss). Scale bar, 5 μm.

In summary, our experimental reconstitution of a two-component system of clamped MTs and a 2D sheet of dynein motors drives low-frequency (millihertz) wave-like oscillations that are qualitatively consistent with a theoretical model with some quantitative deviations of experimental data from in predicted phase boundaries. The model predicts the self-organized emergence of regular oscillatory movement of filaments. These patterns emerge at an intermediate motor density and length of MTs in simulation. We find our in vitro reconstitution experiments validate the predicted phase space providing a quantitative match between theory and experiment. The abrupt transition between fluctuating and regular wave-like oscillatory states merits further exploration of potential phase-transition-like behavior.

Discussion

We have described a minimal experimental system and the constraints that can spontaneously result in wave-like oscillatory patterns. The experiment involves a modified dynein gliding assay where MT plus ends are biotinylated and bind to surface-bound streptavidin. In the presence of ATP, this system results in regular wave-like oscillations that propagate from the clamp to the tip of the MT. We find these oscillations are low frequency (in the millihertz range), are symmetric between peaks and troughs, and are driven by active motor activity. We combine this observation with a computational model of semi-flexible polymers driven by point models of surface-bound motors that stochastically bind and unbind from MTs and, when bound, generate forces resulting in bending and buckling due to the clamping of one end. Using a simple geometric measure of deflections, simulations show a motor density and MT length dependence of transitions between distinct patterns that are qualitatively comparable to experiments.

Comparable in vitro approaches have indeed been taken in the past with a minus-end-clamped MT showing kinesin-driven bending in a gliding-assay geometry (39). However, they did not report oscillations, possibly due to the low density of motors chosen. Actin filaments gliding on a sheet of myosins were reported to occasionally get pinned by their ends and move in spirals, and longer filaments were observed to oscillate (16). This is qualitatively comparable to our findings, with the notable difference that the stiffer filaments such as MTs will be expected to quantitatively differ in the forces required and filament lengths that drive the behavior. Kinesin itself was shown to result in flagella-like beating patterns when one section of MT filaments was passively immobilized and the free end was stochastically bound to kinesin complexes (18). However, that experiment lacked the control of our experiments, where we were able to control the length of the MT clamp and average surface density of the motors. Additionally, unlike previous studies, we used dynein, thus demonstrating the generality of the process. MT bundles driven by kinesin multimers were also demonstrated to self-organize into ciliary and flagellar beating waves (40). However, the study lacked information about or control over motor numbers, which we have addressed in our work. Indeed, in future, combinations of our approach and the bundles could provide a deeper insight into mechanisms. A theoretical model of a single semi-flexible chain clamped at one end by a section, driven by forces, was predicted to produce wave-like oscillations sensitive to boundary conditions, filament length, and total force (17). Although the relation between frequency and motor forces cannot be directly tested in our experiments, we do for the first time validate their predictions with the geometry of clamped filaments and forces acting along the MT lengths. Thus, our study demonstrates the self-organized emergence of smooth wave-like oscillations from a single semi-flexible filament driven by dynein motors binding and pushing it along the length, with patterns observed depending on length and motor density.

Although our observations are based on an artificial setup, comparable processes of point-clamped MTs and load exerted either along the length or at the free end of a filament have been invoked to explain in vivo observations of highly curved MTs that also show buckling (41). The absence of wave-like oscillations inside cells could arise from multiple causes, such as motor force, detachment, speed, density, and MT geometry. On the other hand, when oscillatory beating patterns have been observed previously in vitro, based on pinned MTs and motors, they were based on nonspecific pinning of kinesin complexes and accidental clamping of MTs (18), suggesting a distinct lack of control. In addition, in case of beating patterns of MT bundles driven by kinesin complexes, both the number and lengths of MTs were unclear (40). In addition, our experiments directly address model predictions, including the length of the filament (17), that have not been reported before. Therefore, our approach demonstrates both a simpler set of components and constraints and a quantitative experimental approach that validate model predication of oscillatory wave-like patterns in single MTs driven by motor forces.

The single-filament oscillatory patterns observed in both our experiments (Fig. 1) and simulations (Fig. 4) propagate from base to tip, appearing to be flagella-like (42). However, for the moment, this comparison is purely qualitative. In previous comparable work with clamped MTs, kinesin-driven buckling was observed (39), but not periodic oscillations. Other work has also shown waving and rotational motion (16,43) and loop formation at high motor density (44). The generality of such structures was illustrated by spirals and flagellum-like bending of actin formed when filament tips were pinned to surface defects and myosin motors generated forces (16). To test the effect of motor-specific properties on our simulations, we varied the single-molecule velocity of the motors by a factor of 20 and found the dominant frequency of oscillations increases over the range by 10-fold in a nonlinear manner (Fig. S5). This predicts that faster motors could indeed increase oscillation frequency. The yeast cytoplasmic dynein that we used in our experiments has a reported in vitro velocity of 0.1 μm/s (7,20). In contrast, conventional kinesin-1 has an unloaded velocity of 0.8 μm/s (45), whereas the outer-arm dynein from Tetrahymena transports MTs in a gliding assay at a velocity of 5 μm/s (46). The simulations over one order of magnitude demonstrate an almost identical increase in frequency. However, single-motor velocities required to approach the frequency observed in more complex oscillating structures, such as flagella, require single-molecule velocities exceeding known natural MT-dependent motor velocities. It is no surprise that a single filament structure and random encounters with a sheet of motors cannot therefore achieve the high-speed oscillations, and further additions to the model, such as filament bundling, geometry mimicking the axoneme, and motor arrangement, bring us closer to understanding the principles driving in vivo MT-motor wave-like oscillations seen in flagella.

In our experiments, during wave-like oscillatory motion of filaments, we occasionally observe segments moving in and out of plane (Fig. S6). This effect is likely to result from torsional movement of the filament, consistent with the reported ability of cytoplasmic dynein to generate torque based on its motion in a right-handed helical path (47). Such helical movement has also been reported in cases of mammalian cytoplasmic dynein (48) and Tetrahymena axonemal dynein (49). Reports of such MT rotation appear to also extend from mammalian dynein to kinesin-1 and kinesin-8 (50). At high motor density, the torque produced by motors has even been shown to result in supercoiling of filaments, as seen in the case of pinned actin filaments transported by myosin (51). In our study, we do not observe such dramatic supercoiling as often, potentially due to the low motor density used. Additionally, our simulations are, for the sake of simplicity, conducted in 2D. Extensions to our work to use both higher density of motors as well as more quantitative estimation of 3D displacement could demonstrate consistency with other work in terms of torsional motion, and improved simulations could be used to test the generality of the process.

Conclusions

The self-organized nature of the observed patterns suggests that a minimal system consisting of mechanical components of filaments and motors can drive qualitatively comparable oscillations to the more complex flagella. At the same time, the absence of the geometric complexity, regulation, and coordination seen in bona fide flagella suggests that future work designing a bottom-up minimal synthetic flagellum could benefit from some of the design principles defined in this study, along with highlighting the fundamental properties of collective mechanics that arise from combinations of components and geometric constraints.

Author contributions

S.A.Y. performed the experiments and analyzed the data. N.K. and A.S. performed simulations and wrote the analysis codes. S.A.Y., N.K., D.K., and A.S. prepared the figures. D.K. analyzed the image data. N.K., S.A.Y., D.K., and C.A.A. wrote the manuscript. C.A.A. conceptualized and supervised the work and obtained the funding.

Editor: Kristen Verhey.

Supporting material

Figure S1. Frequency of MT patterns observed

The proportion of filaments seen to undergo wave-like oscillations as compared to gliding transport in replicates (n = 4) was interactively classified in time-lapse image series for an assay with motor density of 46 motors/${\mu \mathrm{m}}^2$. Details of how the motor density was estimated are described in the “materials and methods” section. On an average, 23% were observed to have wave-like motion, whereas 77% appeared to undergo gliding transport. Transients stops were ignored in this analysis.

Figure S2. Number of bound motors driving filament dynamics in simulations

(A) The number of motors bound to the MTs at different MT lengths from 5 to 25 μm (colors) are plotted as a function of time (seconds). (B) The number of motors interacting with MTs that are expected to interact $N_{expect}$ (black circles) are compared to the average bound motors $⟨N_{bound}⟩$ from simulations (blue circles). $N_{\mathit{exp}ect}=L_{MT}\times \sqrt{{\rho }_{2D}}$, where ${\rho }_{1D}$ is the linear motor density and $L_{MT}$ MT length. Here ${\rho }_{2D}=10$ motors/${\mu \mathrm{m}}^2$. (C) The corresponding MT tip angles are plotted as a function of time to indicate oscillation onset.

Figure S3. Dependence of multiple metrics of filament dynamic patterns with MT length and motor density

The effect of varying motor density (${\rho }_m$) and MT lengths ($L_{MT}$) was quantified from multiple runs ($N_{runs}$ = 10) from 600 s of simulations, when the system is at steady state. Mean values of the following measures are displayed: the span on the x and y axis (maximal values, μm), frequency (1/s) at the mid-point of the filament, the maximal tip angle φ, maximal angular velocity of the tip ω, and maximum number of motors bound to a filament. Color bars indicate the variable plotted.

Figure S4. Dynein density estimation using EGFP standard calibration curve

(A–C) Representative plots of z profile of fluorescence intensity of three 0.1-μm beads (black circles) fitted to a Gaussian profile (blue) using the equation displayed on the graph (a). This plot was used to estimate the PSF as described in the “materials and methods” section. (D) The mean image fluorescence intensity in gray values (blue circles) with error bars representing SD (y axis) is plotted as a function of GFP equivalents of an EGFP concentration series (x axis). The linear fit (red line with equation) was used to calculate the number of dynein molecules (red circle) in the system. $R^2$ indicates the regression coefficient.

Figure S5. Effect of motor velocity on oscillation frequency

The dominant frequency (y axis) from the Fourier analysis of oscillations of filament tips is plotted as a function of increasing motor velocity (x axis) for two different densities: 50 and 100 motors/${\mu \mathrm{m}}^2$. Each point in the graph is the mean $\pm$ SD (error bar) of n = 3 simulation runs. $L_{MT}$ = 15 μm.

Figure S6. Out-of-plane movement of filament sections during oscillations

Pinned MTs on a surface of dynein displaying oscillatory wave-like beating in presence of dynein density of 27 motors/${\mu \mathrm{m}}^2$. Segments of a representative filament (yellow arrows) show reduced fluorescence intensity and a return to brightness comparable to the rest suggesting out of plane movement. Time, mm:ss. Scale bar, 2 μm.