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Optimization of Isopolar Microtubule Arrays Rodney R. Agayan,† Robert Tucker,‡,# Takahiro Nitta,†,§ Felix Ruhnow,∥ Wilhelm J. Walter,∥ Stefan Diez,∥,⊥ and Henry Hess*,†,‡ †

Department of Biomedical Engineering, Columbia University, 351L Engineering Terrace, MC 8904, New York, New York 10027, United States ‡ Department of Materials Science and Engineering, University of Florida, Gainesville, Florida 32611, United States § Department of Mathematical and Design Engineering, Gifu University, Gifu 501-1193, Japan ∥ B CUBE − Center for Molecular Bioengineering, Technische Universität Dresden, 01307 Dresden, Germany ⊥ Max-Planck-Institute of Molecular Cell Biology and Genetics, 01307 Dresden, Germany S Supporting Information *

ABSTRACT: Isopolar arrays of aligned cytoskeletal filaments are components in a number of designs of hybrid nanodevices incorporating biomolecular motors. For example, a combination of filament arrays and motor arrays can form an actuator or a molecular engine resembling an artificial muscle. Here, isopolar arrays of microtubules are fabricated by flow alignment, and their quality is characterized by their degree of alignment. We find, in agreement with our analytical models, that the degree of alignment is ultimately limited by thermal forces, while the kinetics of the alignment process are influenced by the flow strength, the microtubule stiffness, the gliding velocity, and the tip length. Strong flows remove microtubules from the surface and reduce the filament density, suggesting that there is an optimal flow strength for the fabrication of ordered arrays.

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INTRODUCTION Hybrid nanodevices integrating biomolecular motors and their associated cytoskeletal filaments have been developed for a number of applications, such as biosensing, surface imaging, and force measurements.1−8 Biomolecular motors, such as myosin and kinesin,9 achieve a level of performance in terms of speed, lifetime, and energy efficiency that is unmatched by current synthetic molecular motors10,11 and can be close to the theoretical maximum.12,13 As a result, they enable the exploration of application concepts enabled by nanoscale force generation as well as the discovery of general engineering principles at the nanoscale.14−16 Central components of several device designs are arrays of microtubules that support the directed transport of structures covered with kinesin motors.17−23 In particular, isopolar arrays of cytoskeletal filaments combined with arrays of biomolecular motors could potentially generate macroscopic forces.24 In muscle tissue, the interaction of aligned and dense arrays of actin filaments with arrays of myosin motor proteins can produce optimal force outputs25 with high energy efficiency.26,27 Similarly, the performance of hybrid nanoactuators will depend on the availability of dense and ordered arrays of cytoskeletal filaments. Isopolar arrays of cytoskeletal filaments can be produced by exposing filaments gliding on motor-coated surfaces to a uniform force field (Figure 1). The force field will cause filaments moving against the force to reorient until their direction of motion is aligned with the force. Forces can been exerted on filaments by electric fields,28−31 magnetic fields,32,33 and fluid flow fields.20,22,34 © 2013 American Chemical Society

After the alignment process has been completed, the oriented microtubules can be permanently attached to the surface using chemical cross-linkers, such as glutaraldehyde.35 This approach complements the creation of isopolar filament arrays by taking advantage of the differential growth velocities at the ends of microtubules,36 which can generate either distinct regions of unipolar microtubules,21 or a strongly biased distribution in the polarity everywhere.37 Other studies described the creation of isopolar arrays by capturing specific filament ends and subsequent combing,18 or the creation of disordered arrays.38,39 The assembly of isopolar arrays from individual, gliding filaments is also mechanistically distinct from the formation of assemblies of interacting filaments, such as coils and bundles.40−48 Here, we investigate theoretically what degree of alignment can be obtained and how fast the system will reach that state, and we experimentally test the predictions using the alignment of microtubules gliding on a kinesin-coated surface by fluid flow.

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THEORY The movement of a microtubule as it glides on a surface coated with kinesins is controlled by the “tip” of the microtubule, whose attachment to processive motors defines a path for the “tail” to follow. In the absence of external forces, the Brownian Received: September 20, 2012 Revised: January 14, 2013 Published: January 18, 2013 2265

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Figure 1. (a) Microtubules (red) with a diameter of 25 nm and a length of several micrometers glide on kinesin motor proteins adhered to the surface. In the presence of fluid flow (or other external fields exerting force), drag forces align most microtubules, while some microtubules become immobilized or are removed from the surface. The orientational distribution of moving microtubules converges toward a Gaussian distribution with a width determined by the thermal energy and the drag force exerted by the fluid. (b) Fluorescence microscopy images of microtubules before and after exposure to fluid flow (using setup 2 described below) show the alignment but also the removal of microtubules. The height of the microtubules above the surface is a critical parameter for the calculation of the drag force.

where U(ϕ) is the potential energy of the segment in the flow field. The potential energy of a microtubule tip of length L deflected by an angle ϕ is given by

motion of the tip will lead to a trajectory with the character of a wormlike chain.49 Fluid flow exerts a drag force on the tip that biases it in the direction of the flow; however, thermal fluctuations let the microtubule deviate from perfect alignment. The rate of change in the angle ϕ between a microtubule tip and the direction of fluid flow is given by50,51 dϕ L2ωv = sin ϕ dt 3kBTLp

U (ϕ) =

(1)

p(ϕ) =

σϕ 2 = σ 2(t = ∞) = 2kBT /ωL2

(5)

(6)

Equation 6 indicates that the alignment improves as the drag force per length exerted on the filament increases. However, the increasing force will lead to unbinding of the microtubule from the surface, so that the resulting loss in microtubule density eventually exceeds the benefit derived from improved alignment. The unbinding rate koff of the microtubules is difficult to predict because it is not known how the kinesin motors holding the microtubule share the applied load and in what sequence the motors unbind.52 Moreover, the rate may change as the microtubule changes its orientation relative to the flow. A number of different scenarios are discussed in the Supporting Information.53 The simplest scenario is that the microtubule is held by N motors that must release the microtubule simultaneously. The Bell equation54 can then be written as

(2)

The alignment by the fluid flow is opposed by random thermal forces, so that a microtubule trajectory fluctuates around the flow lines. At steady state, the probability that a segment of the trajectory is oriented with an angle ϕ to the direction of fluid flow is proportional to the Boltzmann factor: ⎛ U (ϕ) ⎞ p(ϕ) ∝ exp⎜ − ⎟ ⎝ kBT ⎠

⎛ ϕ2 ⎞ ⎟ exp⎜⎜ − 2⎟ ⎝ 2σϕ ⎠ 2πσϕ 2 1

with

3kBTLp L2ωv

(4)

where we use the beginning and the end of the microtubule to define the orientation ϕ relative to the flow field. If the standard deviation of the angular distribution is defined as σϕ = ⟨(ϕ − 1 ⟨ϕ⟩)2⟩ /2, then the probability distribution of the orientation angles at equilibrium becomes a Gaussian function:

where L is the length of the tip, ω is the applied force per length when the microtubule is perpendicular to the fluid flow, v is the velocity of the gliding microtubule, kB is the Boltzmann constant, T is the temperature, and Lp is the persistence length of the tip (equal to the trajectory persistence length). In the small angle approximation sin ϕ ≈ ϕ, the solution of this differential equation is an exponential decay. Simulations of the alignment process using the method of Nitta et al.50 show that the presence of large angles and the distribution of angles around the average roughly double the average time constant τ relative to that for the small angle approximation but still lead to an approximately exponential decay in the standard deviation of the microtubule orientation angles, so that τ=2×

ωL2 ωL2 2 (1 − cos ϕ) ≈ ϕ 2 4

(3) 2266

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Langmuir koff ≈

⎛ ⎞ ⎛ E ⎞ f 1 exp⎜(1 − N ) b ⎟exp⎜⎜ ⎟⎟ Ntoff,1 ⎝ kBT ⎠ ⎝ fβ ⎠

Article

Once the Nescofilm melted (∼10 s), the assembly was removed from heat and placed on a surface at room temperature to cool. The assembly was placed in a custom-built mount for observation on a microscope stage. The mount coupled the ends of Tygon tubing (0.25 mm inner diameter) directly in contact with the outer coverslip surface via set screws aligned with the laser-drilled holes. This ensured that the fluid could be delivered to a single channel without leaking. Protein-containing or alignment solutions were fed via the inlet tube while the outlet tube lead to a computer-controlled, stepper-motor-driven syringe pump (GeSIM, Grosserkmannsdorf, Germany) that drew fluid into a 250 μL Hamilton syringe glass barrel. The syringe pump included a switchable valve allowing the syringe to be quickly (1 s) emptied to waste or to replenish the alignment solution once full. This enabled semicontinuous flow rates up to 12.0 μL/s with 0.1 μL/s resolution. To align microtubule arrays within the channel, the inner surfaces were first coated with casein by flowing in 20 μL of 0.5 mg/mL casein in BRB80 and waiting 5 min. A 20 μL amount of kinesin solution diluted (0.3−1 μg/mL) in BRB80 supplemented with 0.2 mg/mL casein, 1 mM magnesium adenosine 5′-triphosphate (MgATP), and 10 mM dithiothreitol (DTT) was flowed in and left undisturbed for 5 min. A 20 μL amount of motility solution consisting of microtubules (total tubulin: ∼40 nM for taxol-stabilized microtubules, ∼100 nM for doublestabilized microtubules), antifade (40 mM D-glucose, 110 μg/mL glucose oxidase, 22 μg/mL catalase, 10 mM DTT), and 1 mM MgATP in BRB80T was flowed into the channel. Before alignment, microtubules bound to kinesin were allowed to glide in the absence of flow for 3 min in order to randomize their orientations. A 2 mL amount of alignment solution (motility solution with microtubules replaced by BRB80T) was flowed through the channel and allowed to recirculate after each 250 μL pass of the syringe pump. This allowed the microtubules to be imaged during alignment at a fixed flow rate. Room temperature was 22 °C. Setup 2. For alignment at increased flow strengths, a flow cell was made by creating a sandwich of double-sided tape between two microscope coverslips (35 × 75 mm2 and 22 × 22 mm2, thickness no. 0, FisherFinest, Fisher Scientific) with double-sided tape as spacers resulting in a channel approximately 22 mm × 5 mm × 74 μm. The inner surfaces of the flow cell were coated with casein by flowing in 0.5 mg/mL casein in BRB80 buffer for 5 min, followed by ∼10 nM kinesin solution containing 0.2 mg/mL casein and 1 mM ATP in BRB80 for 5 min. Microtubules (128 nM total tubulin), antifade (20 mM D-glucose, 20 μg/mL glucose oxidase, 8 μg/mL catalase, 10 mM DTT), and adenosine 5′-(β,γimido)triphosphate (AMP-PNP, a nonhydrolyzable ATP analogue) in BRB80 buffer were perfused into the flow cell and allowed time (∼10 min) to bind to the kinesin-coated surface. Alignment solution (buffer, antifade, and 1 mM ATP, no microtubules) was perfused into the flow cell to initiate motility. To achieve high flow rates for alignment, the alignment solution was added continuously to one end of the flow cell while being removed from the other end of the flow cell with a P200 pipet tip (Neptune esp, VWR, West Chester, PA) connected to a vacuum line (Fisherbrand, 1/4 in. inner diameter, 3/32 in. wall width) attached to a vacuum pump (0.02 HP, Barnant Company, Barrington, IL). Each alignment used approximately 700 μL of alignment solution, and the alignment time (between 75 and 200 s) was recorded. After the solution was used, a new solution containing buffer, antifade, and 1 mM AMPPNP was perfused into the flow cell, and the resulting microtubule array was imaged. Room temperature was 20 °C. Imaging and Measurement. Setup 1. Microtubules were imaged using a Zeiss Axio Observer.Z1 inverted microscope (Zeiss, Jena, Germany) equipped with an arc lamp (HXP 120, Zeiss), a rhodamine filter cube (filter set 43HE, Zeiss), a 40× oil objective (NA 1.3, Zeiss), and a CCD camera (Zeiss AxioCam MRm). Images were acquired at a frame rate of 1 Hz and an exposure time of 300 ms. Each acquisition was started after microtubules were allowed to randomize their orientations. Flow was applied several seconds after acquisition began. Frames 1, 25, 50, 75, and 100 were analyzed using ImageJ (v.1.47a, National Institutes of Health, Bethesda, MD). Microtubules in each image satisfying a “guard box region” criteria56 were manually traced, and the outlines were spline-fitted, filled-in to 1 pixel spacing, and smoothed. End point coordinates for each microtubule were estimated as the points along the outline with the highest curvature. Microtubule

(7)

or in terms of the filament length LM and the motor spacing Δsm: koff ≈

⎛ ⎞ ⎛⎛ L ⎞ E ⎞ Δsm ωL exp⎜⎜⎜1 − m ⎟ b ⎟⎟exp⎜⎜ m ⎟⎟ Lmtoff,1 ⎝⎝ Δsm ⎠ kBT ⎠ ⎝ fβ ⎠

(8)

where toff,1 is the average attachment time of a kinesin motor to a microtubule in the absence of load, Eb is the bond energy of the kinesin−microtubule bond, and fβ = kBT/xβ is a thermal force scale where xβ is the distance between the equilibrium bond length and the transition state to unbinding. Equation 8 indicates that shorter microtubules experience larger dissociation rates than longer ones. In summary, our considerations predict that (1) the angle between microtubules and flow direction decays exponentially after flow is initiated, (2) the orientation distribution after flow alignment will be approximately Gaussian in shape and can be characterized by the standard deviation, and (3) shorter microtubules are removed first from the surface by the flow due to the smaller number of kinesin motors holding the microtubule.

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METHODS

Kinesin and Microtubules. Full length, wild-type kinesin-1 from Drosophila melanogaster expressed with a C-terminal histidine-tag in Escherichia coli was purified using a Ni−NTA column as described previously.55 Kinesin densities below 0.3 μg/mL did not support microtubule gliding. Taxol-stabilized microtubules were polymerized as follows: Polymerization buffer was prepared by supplementing BRB80 (80 mM piperazine-N,N′-bis(2-ethanesulfonic acid) (PIPES), pH 6.9, with KOH, 1 mM ethylene glycol tetraacetic acid (EGTA), 1 mM MgCl2) with 24% dimethyl sulfoxide, 20 mM MgCl2, and 5 mM guanosine 5′-triphosphate (GTP). A 1.25 μL amount of this polymerization buffer was added to 5 μL of 4 mg/mL tubulin (purified from pig brains, 75% unlabeled and 25% labeled by rhodamine) in BRB80 resulting in a final tubulin concentration of 30 μM. The solution was incubated at 37 °C for 30 min and then diluted ∼80-fold in BRB80 supplemented by 10 μM taxol (BRB80T) for stabilization and stored at room temperature. Double-stabilized microtubules were polymerized by supplementing 80 μL of BRB80 with 2.5 μM tubulin (purified from pig brains, 75% unlabeled and 25% labeled by rhodamine), 1.25 mM guanosine 5′[α,β-methyleno]triphosphate (GMPCPP), and 1.25 mM MgCl2. This solution was incubated first on ice for 5 min and then at 37 °C for 2 h. The microtubule solution was centrifuged in a Beckman Airfuge at 25 psi for 5 min. The supernatant was discarded, and the pellet was resuspended in 200 μL of BRB80T and stored at room temperature. Microtubule Array Preparation. Setup 1. Microtubules were aligned with an isopolar orientation using fluid flow. A flow cell was constructed such that solutions could be delivered at controlled flow densities. The flow cell consisted of two 22 × 22 mm2 coverslips (Corning, no. 1.5) separated by Nescofilm (100 μm thickness) laser cut (Speedy 100, Trotec, Wels, Austria) to have three separate flow channels. Coverslips were first sonicated for 15 min in mild detergent, rinsed with deionized water for 1 min, bathed in ethanol for 10 min, rinsed with Nanopure water for 1 min, and dried with nitrogen gas. By using the laser cutter (100 Hz pulse, 15 passes/layer, power setting 30, speed setting 5), six 0.4 mm diameter holes were drilled into the top coverslip to serve as the inlets and outlets for the three channels. The three layer sandwich of Nescofilm between top and bottom coverslips was placed on a hot plate at ∼150 °C while applying gentle pressure from the blunt end of plastic tweezers until the Nescofilm adequately melted creating sealed channels with final dimensions 3 mm × 17 mm × 100 μm. 2267

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length was estimated as LM = P(1 + (1 − (16A/P2)) /2/4 where P and A are the outline perimeter and area, respectively. This assumes that, for a microtubule outline with width w, P = 2(LM + w) and A = LMw. This approximation breaks down when P2/16 < A, for example, when the outline is the trace of a round region such as one in which LM ≈ w. In these cases, the length is set to the end-to-end distance. Microtubule gliding direction was determined “manually” using MATLAB (The Mathworks, Natick, MA) by overlaying three consecutive frames of the recorded fluorescence imaging videos with the first frame false-colored red, the second green, and the third blue. The resulting composite showed images of a slightly extended microtubule with the front end in blue, the trailing end in red, and the region in between either grayscale or intermediary colors of the RGB palette depending on the gliding speed. In this way, the forward direction of the microtubule trajectory could easily be discerned by the user as moving from the red end toward the blue end. Microtubules that stalled during the three frames showed no color at the ends in the composite image and were thus easily distinguished from moving microtubules. Microtubule orientation was calculated as the angle of the vector pointing from the trailing end to the leading end of the microtubule with 0° along the direction of flow. Setup 2. The flow cells were imaged using a Nikon Eclipse TE2000U microscope (Nikon, Melville, NY), an X-Cite Hg-arc lamp (EXFO, Ontario, Canada), a rhodamine filter cube (no. 48002, Chroma Technologies, Rockingham, VT), a 40× (Nikon S Fluor, NA 1.3) or 100× (Nikon Plan Apo, NA 1.45) oil objective, and an iXon EMCCD camera (Andor DV885-JCS-VP, South Windsor, CT). Images were taken before and after flow alignment; thus, no directional information is provided. For this reason, microtubule orientation was calculated using the end-to-end vector such that the angle fell within the range from −90° to 90°. Gliding Velocity and Trajectory Persistence Length. Microtubule gliding velocity and trajectory persistence lengths were determined by first manually tracking the coordinates of the front end of a microtubule in each frame. To calculate mean velocity, instantaneous velocity (calculated using central differences) was averaged first over all frames and then over microtubules. Trajectory persistence length was determined by calculating ⟨cosΔθi⟩ = ⟨ν⃗0·Δν⃗i/|ν⃗0||ν⃗i|⟩ where Δθi is the relative tangent angle at the ith point, ν⃗i is the microtubule velocity vector at the ith point, v0 is the initial velocity, and ⟨ ⟩ represent averaging over an ensemble of microtubules. The persistence length Lp is then calculated by performing a weighted fit of cos Δθi to an exponential of the form exp(−Lt/2Lp) where Lt is the contour length along the trajectory. Calculation of the Force Density. The velocity profile of pressuredriven fluid flow in a rectangular channel with height H and width B in flat channels (H/B < 2/3) is given by57

v vmax

1.54 ⎞ ⎛ ⎛ 2 2 2⎞ ⎟ · ⎜1 − x y ⎟ = ⎜1 − B H ⎠ ⎝ ⎠⎝

by Gast et al.61 allowed drag force densities at the surfaces up to 0.6 pN/μm. Setup 2 allowed higher force densities of 1−2 pN/ μm but without close control of the shear rate. Using setup 1, we observed the alignment process over time for three different force densities (0, 0.24, 0.61 pN/μm), for two different microtubule stiffnesses (taxol-stabilized, and double-stabilized with taxol and GMPCPP, a nonhydrolyzable analogue of GTP62), and for two different kinesin densities (nominally 0.6−1.0 μg/mL and 0.3−0.5 μg/mL). Microtubules were introduced into the flow cell, were bound to the kinesin motors, and were allowed to glide for several minutes in order to randomize their initial orientation. Fluid flow was then applied, and the microtubules gradually changed their orientations as they glided on the surface. Figure 2a shows a sequence of fluorescence microscopy images of the rhodaminelabeled microtubules as they aligned under the influence of fluid flow. Moving and nonmoving microtubules were traced and distinguished using imaging analysis software. Only moving microtubules were analyzed in detail. The end-to-end vector of each microtubule (trailing to leading) defined the orientation angle, resulting in values of ϕ that ranged from −180° to 180° with respect to the flow direction. The standard deviation of the orientation angle, σϕ, was calculated for moving microtubules in individual images and used to quantify the evolution of alignment. For measurements without flow or at early times, random distributions that showed no sign of a clear peak were expected. For these cases, σϕ was calculated as the standard deviation of the raw data, and error bars represent the expected value of the standard deviation of σϕ. For measurements with nonzero flow and orientation distributions that showed a discernible peak (Figure 2b), the distribution was fit to a Gaussian profile with a width σϕ as given by eq 5. In the absence of flow, the standard deviation of the orientation distribution matched that of a random distribution (σ0 = 104°) for all microtubule types and kinesin densities. When shear flow was applied, σϕ decreased with time until a steady state was reached within about 100 s (Figure 2c). Microtubules aligned faster with increased flow density, as expected, but also with reduced kinesin density. The faster average gliding velocity observed by us and others34 for microtubules at lower kinesin densities (⟨vlow⟩ = 650 ± 20 nm/s; ⟨vhigh⟩ = 560 ± 20 nm/s) also contributes to faster alignment, but it is not sufficient to explain the effect. The temporal evolution of the standard deviation of the orientation angle (Figure 2c) was fit by an exponential decay with an offset:

(9)

.

The volumetric flow rate Q is obtained by integrating over the channel cross section, which allows the determination of the maximal flow velocity vmax given the volumetric flow rate used in the experiments and gives the following expression for the velocity at height y at the center of the channel (x = 0):

v = 2.47

Q ⎛ 2 2⎞ y ⎟ ⎜1 − BH ⎝ H ⎠

⎡ t − t0 ⎤ σϕ(t ) = (σ0 − σϕ)exp⎢ − ⎥ + σϕ ⎣ τ (ω , L , L P ) ⎦

(11)

where t0 is a temporal offset used to normalize each time trace of σϕ(t) to account for initial angular spreads that are not perfectly random. σϕ and τ describe the equilibrium and kinetic behavior, respectively. For taxol-stabilized microtubules at high kinesin density and high flow (ω = 0.61 pN/μm), the fits yielded σϕ = 16° ± 10° and τ = 42 ± 19 s. For taxol-stabilized microtubules at high kinesin density and low flow (ω = 0.24 pN/μm), σϕ and τ cannot be simultaneously fit, since equilibrium is not reached on the time scale of the experiment. If the fits are performed such that σϕ·√ω remains constant (σϕ = 27° ± 2°), then the best fit gives τ = 155 ± 44 s.

(10)

The drag force per length is given by ω = C⊥ηv where C⊥ = 6.5 is the perpendicular drag coefficient58 on a cylinder, η is the dynamic viscosity of water at temperature T,59 and v is determined from eq 9 using 30 nm as the height of the center of the microtubule above the surface60 (y = H/2 − 30 nm).

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RESULTS AND DISCUSSION Microtubules gliding on kinesin-coated surfaces were flowaligned in two different experimental setups. Setup 1 described 2268

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Figure 2. Evolution of microtubule alignment. (a) Rhodamine-labeled microtubules gliding on kinesin were tracked during exposure to fluid flow (flow direction is toward the right). (b) In the absence of flow (top), taxol-stabilized microtubules are uniformly oriented. With fluid flow, microtubules that have not fully equilibrated (middle) can be aligned (bottom), and their orientation distribution can be approximated by a Gaussian. Error bars indicate the standard deviation of the Poisson-distributed number of microtubules in each bin. (c) Decay of the standard deviation of the orientation distribution during alignment. Increasing the force density from 0.24 pN/μm (triangles, number of moving microtubules M = 159−179) to 0.61 pN/μm (crosses, M = 165−175) produces faster and better alignment of taxol-stabilized microtubules at a high kinesin density. Taxol-stabilized microtubules exposed to 0.61 pN/μm align even faster at a low kinesin density (closed circles, M = 138−156) but reach the same plateau. Increasing the microtubule rigidity has little effect at low kinesin densities, as seen for the alignment of taxol/GMPCPP-stabilized microtubules exposed to 0.61 pN/μm (open circles, M = 60−70). Each data set is overlaid by an individually fit curve determined from eq 11. Error bars indicate the standard deviation of the σϕ of the Gaussians fitted to the orientation histograms as shown in (b). The middle and bottom histograms in (b) correspond to the first and last points, respectively, of the taxol-stabilized, high force density, low kinesin density data set (closed circles). Additional data sets are shown in the Supporting Information, Figure S1.

The relationships defined by eq 2 are well reflected in the fits. Given a persistence length of taxol-stabilized microtubule trajectories of 0.1 mm,49 an observed gliding velocity of ⟨vhigh⟩ = 560 nm/s, a force density of ω = 0.61 pN/μm, and a tip length of 0.4 μm, eq 2 predicts a time constant of 40 s. For a reduction in the force density from ω = 0.61 pN/μm to ω = 0.24 pN/μm, eq 2 predicts an increase in the time constant to 100 s. Therefore, eqs 2 and 6 provide a good overview of the alignment process at high kinesin densities. At low kinesin densities, the fits yielded σϕ = 17° ± 2° and τ = 16 ± 4 s for taxol-stabilized microtubules and σϕ = 19° ± 2° and τ = 15 ± 7 s for double-stabilized microtubules. Equation 6 predicts that the steady-state orientation distribution is independent of the kinesin density as well as the microtubule stiffness, in agreement with these data. In contradiction to eq 2, however, the data at low kinesin density show an alignment

A steady-state standard deviation of 16° for ω = 0.61 pN/μm implies according to eq 6 a tip length of 0.4 ± 0.04 μm independent of kinesin density. While we initially interpreted the tip length to be the distance between the tip of the microtubule and the first attached kinesin,63 the independence of the steady-state standard deviation from the kinesin density leads us to interpret the tip length differently (Figure 3). In this view, the “tip” is defined as the section of the microtubule that can fluctuate as it is being constrained by the limited extensibility of the kinesins attached to it. The kinesins, whose tail can be modeled as a freely jointed chain due to hinge regions connecting coiled coil domains,64 exert only very weak entropic forces onto the microtubule until it is fully stretched, which is thought to constrain but not dampen the microtubule fluctuations. Clearly, this interpretation will have to be supported or refuted by a future, more detailed analysis. 2269

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that are too high can pull off microtubules from the surface. The flow-induced removal of microtubules lowers the microtubule number density thereby reducing the overall array quality. Using setup 2, we aligned taxol-stabilized microtubules (Figure 1, lower panel). If we assume the orientation distribution of moving microtubules is Gaussian while nonmoving microtubules are uniformly distributed, then the standard deviation of the orientation distribution after alignment was 14° (σϕ = 0.25), which implies ω ≈ 0.8 pN/μm when the standard deviation is compared with the standard deviations obtained with setup 1 and the force density is scaled according to eq 6. At this high force density, the removal of the microtubules showed a pronounced length dependence (Figure 5).

Figure 3. Tip length interpretation. Kinesin motors attach in a variety of geometries. The flexibility of the tail does not create a defined attachment point; instead, the observed tip length can incorporate the leading segment of the microtubule that can perform a swiveling motion despite attachment to multiple motors.

process that is faster than the alignment at high densities and an alignment process that is not affected by the 3-fold higher persistence length (Lp = 0.22 mm) of the double-stabilized microtubules (Figure 4). This deviation from the predicted

Figure 5. Survival of microtubules at the surface after exposure to very high force densities (total number of microtubules: Mbefore = 716; Mafter = 483). Error bars represent the standard deviation of the survival fraction.

The histogram in Figure 5 can be fit with eq 8 using the parameters toff,1 = 1 s, Eb = 0.07kT, fβ = 3.1 pN,66 and Δsm = 0.2 μm, which suggests that, at high flow conditions, the kinesins release the microtubule simultaneously from a very weakly bound state, which may be reached when the two heads of a kinesin motor simultaneously detach due to their finite processivity. In our opinion, a more detailed theoretical description of the unbinding process including, for example, the changes of the force vectors as the microtubule aligns, has to await the measurement of angledependent detachment rates and the direct observation of the unbinding process before it could deliver significant additional insights.

Figure 4. Measurements of the trajectory persistence lengths of taxolstabilized (O) and taxol/GMPCPP-stabilized (X) microtubules. Fits of exponential decays to the plots of angular correlation as a function of the contour length give persistence lengths of 60 and 220 μm for taxoland taxol/GMPCPP-stabilized microtubules, respectively. Average microtubule lengths were 4.9 and 4.0 μm for taxol- and taxol/ GMPCPP-stabilized microtubules, respectively (microtubule length distributions are shown in the Supporting Information, Figure S2).

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CONCLUSIONS The present study uncovers several insights for the assembly of polar arrays of microtubules as they are transported on a surface by kinesin motors. First, thermal fluctuations limit the degree of alignment. In the steady state, the alignment is determined by the effective tip length and the force exerted per unit length of the filament. Second, high force densities lead to the removal of microtubules. Due to the multivalency of the attachment of microtubules to the surface via multiple motors, short microtubules are removed first. It is therefore advisable to engineer long microtubules for these arrays. Third, to minimize microtubule loss, the time of alignment should be minimized. More flexible filaments and higher gliding velocities permit this. Low motor densities can accelerate alignment; however, since microtubules are attached to fewer motors, they also facilitate the removal of microtubules from the surface. Together, these insights add detail to the studies by Bö hm et al.,20 Yokokawa et al.,22 and Kim et al.34 describing the flow-induced alignment of microtubules gliding on kinesin motors.

behavior can be explained by the fact that, at low kinesin densities, short microtubules have the opportunity to change their orientation by rotating around their center without bending.65 A shortcoming of the double-stabilized microtubules is that the percentage of nonmoving microtubules was 50% on average compared to 15% for taxol-stabilized microtubules. The number of both moving and nonmoving microtubules remained steady during data acquisition suggesting that most of the stalling occurred during the first few unrecorded minutes of randomization. For double-stabilized microtubules, stalling percentages at the high force density (70%) were larger than at both the lower (30%) and zero force densities (40%) in contrast to taxolstabilized microtubules that had a stalling percentage of 15% for all force densities. Increased flow strengths provided faster alignment and narrower orientation distributions (Figure 2c), but flow strengths 2270

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ASSOCIATED CONTENT

S Supporting Information *

Models of microtubule removal; derivation of statistical error; and plots of the decay of the standard deviation of the orientation distribution during alignment and length distributions. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address #

Dr. Robert Tucker is now affiliated with Hansen Medical Inc., Mountain View, California 94043, United States. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Volkswagen Foundation for their support of this project as part of the program “Integration of molecular components into functional macroscopic systems”. The authors also thank the Diez group and A. Agarwal for experimental support and helpful discussions. T.N. was supported by the Marubun Research Promotion Foundation.

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