Measuring Cohesion between Macromolecular Filaments One Pair at a Time:Depletion-Induced Microtubule Bundling

Feodor Hilitski,1 Andrew R. Ward,1 Luis Cajamarca,2 Michael F. Hagan,1 Gregory M. Grason,3 and Zvonimir Dogic1,*1Department of Physics, Brandeis University, Waltham, Massachusetts 02454, USA

2Department of Physics, University of Massachusetts at Amherst, Amherst, Massachusetts 01003, USA3Department of Polymer Science and Engineering, University of Massachusetts at Amherst, Amherst, Massachusetts 01003, USA

(Received 18 August 2014; published 2 April 2015)

In the presence of nonadsorbing polymers, colloidal particles experience ubiquitous attractiveinteractions induced by depletion forces. Here, we measure the depletion interaction between a pair ofmicrotubule filaments using a method that combines single filament imaging with optical trapping. Byquantifying the dependence of filament cohesion on both polymer concentration and solution ionicstrength, we demonstrate that the minimal model of depletion, based on the Asakura-Oosawa theory, failsto quantitatively describe the experimental data. By measuring the cohesion strength in two- and three-filament bundles, we verify pairwise additivity of depletion interactions for the specific experimentalconditions. The described experimental technique can be used to measure pairwise interactions betweenvarious biological or synthetic filaments and complements information extracted from bulk osmotic stressexperiments.

DOI: 10.1103/PhysRevLett.114.138102 PACS numbers: 87.16.Ka, 82.70.-y, 87.80.Cc, 87.80.Nj

Ranging from elastic nanopillar arrays [1] to ropes ofcarbon nanotubes [2] to dense chromatin structures [3],numerous materials of synthetic or biological origin areassembled from filamentous building blocks. The macro-scopic properties of such filamentous materials are gov-erned not only by the mechanical properties of theconstituents, but also by the interactions between them.These interactions are traditionally measured using bulkosmotic stress experiments in which one applies an externalpressure of known magnitude while simultaneously meas-uring the filament spacing using x-ray scattering [4–6].Here, we describe a complementary single-filament tech-nique that directly measures cohesive interactions betweena pair of filamentous macromolecules. This approachallows us to assemble bundles in a controlled fashion,with predetermined filament number and binding geometry,yielding information that cannot be accessed by bulkmethods. It extends microscopy-based methods developedfor measurement of interactions between isotropic, colloi-dal particles [7,8] to the case of extreme particle anisotropy(e.g., macromolecular filaments).We study cohesive interactions between microtubules

(MTs), cytoskeletal filaments that are assembled fromtubulin heterodimers to form rigid tubular structures withan outer diameter of 25 nm, a contour length that can reachtens of microns, and a persistence length of a few milli-meters [9]. MTs carry significant negative charge atphysiological pH [10]. To assemble bundles, MT filamentswere placed in a suspension of nonadsorbing polymers,which induce attractive interactions by the depletionmechanism. We used either poly(ethylene glycol) (PEG,MW ¼ 20 kDa) or dextran (MW ¼ 500 kDa) as a

depletant. The radius of gyration (Rg) of 20 k PEG is∼7 nm [11], dextran 500 k has Rg ≈ 18 nm [12]. Thedepleting polymers were in the dilute regime for allmeasurements. As two MTs form a bundle, an additionalfree volume becomes available to polymer coils, thusincreasing the overall system entropy and resulting in aneffective attraction between the rods [Fig. 1(a)]. Its strengthand range can be tuned by changing the polymer concen-tration and size, respectively [13]. Bundle formation

(a) (b)

FIG. 1 (color online). Formation of a microtubule (MT) bundlein quasi-2D chamber. (a) Schematic depiction of the PEG-induced MT bundle, showing depleting polymer, excludedvolume shells inaccessible to depleting coils, and disorderedpoly-aminoacid c-terminus tails of tubulin monomers. Inset:Relevant length scales of the bristle-stabilized model: Rh,effective radius of the polymer coils; h, bristle height; D,surface-to-surface MT separation. (b) Sequence of darkfieldmicroscopy images showing evolution of a bundle (yellowarrows) formed by two filaments (MT1 and MT2). Short(∼1 μm) fragment bound to MT1 exhibits 1D diffusion alongthe filament.

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requires high ionic strength in order to screen the repulsiveelectrostatic interactions between the filaments [14].We directly visualized multistep formation of bundled

MTs in the presence of the depleting agent [Fig. 1(b)]. First,two diffusing MTs encountered each other and formed abundle with a random overlap. Subsequently, the bundlemaximized the MT overlap. To understand such a process,consider the free energy of a bundle given by G ¼ −λL,where λ is the cohesive free energy per unit length and L isthe filament overlap length. The magnitude of the retractionforce is given by f ¼ −∂G=∂L ¼ λ. Therefore, measuringthe retraction force directly quantifies the depletion inter-action as characterized by the cohesion energy per unitlength, λ.The experimental system used to directly measure G and

determine λ is illustrated in Fig. 2. Briefly, we preparedsegmented MTs in which only a short portion (seed) islabeled with biotin. The biotinylated seeds and the biotin-free elongated segments of MTs were labeled with differentfluorescent dyes [15]. The entire MTs were stabilizedwith GMPCPP, a nonhydrolysable analog of GTP, thusensuring that there is no dynamic instability. Opticallytrapped Neutravidin coated silica beads were manuallyattached to biotin labeled MT seeds [Figs. 2(a) and 2(b)].

Subsequently, the two MTs, each attached to a single bead,were brought into close proximity to facilitate bundling[27]. Once the MT bundle formed [Fig. 2(c)], we verifiedthat the MTs overlapped only along the biotin-free elon-gated segments. This ensured that Neutravidin, whichdesorbs from beads, did not cross-link biotinylated partsof both filaments. Our experimental setup allowed forcontrol of the MT overlap length with nanometer accuracy[Fig. 2(d)].Before bundle formation, each bead with an attached MT

fluctuated around the minimum of the harmonic potentialimposed by the optical traps. Upon bundle formation, theaverage bead separation, R, decreased due to the retractionforce [Fig. 3(a)]. While measurement of the reduction inaverage spacing for known trap stiffness directly yields themagnitude of the retraction force f ¼ λ, we improvedstatistics by incorporating the fluctuations around theequilibrium position. Specifically, we experimentallyimplemented a form of umbrella sampling to measurethe free energy, G, of the bundle as a function of R [28,29].As beads fluctuated in the optical traps [30], we measuredtheir center-to-center separation R and constructed normal-ized probability distributions: PbundleðRÞ and PcalibrationðRÞfor the bundled and unbundled configurations, respectively[Fig. 3(b)]. Both distributions had Gaussian shapes of equalwidth, which was determined by the stiffness of the opticaltraps alone. Presence of the bundle reduced the mean beadseparation. Since the separation R and the overlap length L

FIG. 2 (color online). Experimental setup used to measure MTcohesion strength. (a) Left: two unbundled optically trappedbead-MT complexes. Right: upon bundle formation the depletionforce pulls beads closer together, away from the trap centers.(b) Composite fluorescence image of the MT filaments showingbiotin-free elongated segments (red) and biotin labeled seeds(green). Beads attached to seeds are not visible. Optical traps areindicated by the dashed lines. (c) Composite fluorescence imageshowing MTs from panel (b) forming a bundle, visible as thebright red region. Bundle overlap region and trap locations areindicated by the yellow and blue dashed lines, respectively.(d) Bundle overlap length (yellow arrows) is controlled by theindependent movement of two optical traps (blue dashed lines).

FIG. 3 (color online). Analysis of the experimental data.(a) Fluctuations in the separation of two beads (R) with andwithout bundled MTs (blue line). MT bundle formation isaccompanied by a rapid reduction of R (red line). (b) Probabilitydistributions PbundleðRÞ and PcalibrationðRÞ are obtained by sam-pling statistically independent configurations of the two beads.(c) The bundle free energy, G, scales linearly with the beadseparation. Slope of the weighted linear fit gives the value of thecohesion strength, λ. (d) λ is independent of the bundle overlap,L. Dashed lines correspond to the mean cohesion strength. Errorbars for the individual measurements represent 95% confidenceintervals.

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are simply related [31], the bundle free energy as a functionof the overlap length, L, is given by:

GðLÞ ¼ −kBT log

�PbundleðLÞ

PcalibrationðLÞ�þ const: ð1Þ

We extracted GðLÞ by using the measured distributions inEq. (1). The cohesion energy per unit length is given byλ ¼ −∂G=∂L, which was calculated from the slope of theweighted linear fit to the experimental points [Fig. 3(c)].For example, the probability distributions depicted inFig. 3(b) yielded λ ¼ 25� 1 kBT=μm, which is equal tothe retraction force of ∼0.1 pN. A single experimentsamples only a small region of the bundle overlap(≈100–150 nm) that is accessible by thermal fluctuations.To extend this range, we manually changed the overlaplength and repeated the measurement. We found λ to beindependent of the MT bundle overlap, L, in agreementwith our initial assumptions [Fig. 3(d)].We measured how the cohesion strength depends on both

the suspension ionic strength and depleting polymer con-centration. For a fixed ionic strength, λ increased linearlywith increasing polymer concentration. For a fixed polymerconcentration, λ decreased with decreasing ionic strength(Fig. 4). These results can be qualitatively understood byconsidering that the cohesive potential between alignedMTs is governed by a combination of the attractive depletioninteraction and electrostatic repulsion [Fig. 5(a)]. In theAsakura-Oosawa (AO) model, the depletion strengthdepends linearly on the depletant concentration, in agree-ment with the experimental findings. Upon decreasing ionicstrength, the range of the electrostatic repulsion increases,thus leading to decreased cohesion strength as observed inthe experiments. When the ionic strength is sufficiently low,the depletion attraction is unable to overcome repulsionbetween the negatively charged filaments, thus suppressingbundling.

To quantitatively assess the relationship between mea-sured MT cohesion and physical and structural features ofMTs, we developed two theoretical models. In the simplestprimitive model, MTs were treated as uniformly charged,hollow cylinders of radius a ¼ 12.5 nm with surfacecharge density σs ¼ −0.23 e=nm2 (−23 e per tubulindimer) [10]. Electrostatic repulsion was computed viathe linearized Poisson-Boltzmann theory, and thedepletion-induced attraction was computed by modelingpolymer coils as effective hard spheres with radius Rh ¼2Rg=

ffiffiffiπ

p[32]. The binding free energy predicted by the

minimal-energy separation of the primitive model over-estimates the measured cohesion strength by a factor of∼5–7, implying an overestimation of depletion attractionand/or an underestimation of repulsions. To explore thepossibility that repulsive forces generated by negativelycharged, flexible “bristlelike” c-terminal tails of tubulin[26,33] contribute to this discrepancy, we developed abristle-stabilized MT model, which includes additionalsalt-dependent electrostatic forces between c-terminal bris-tles anchored to MT surfaces [Fig. 1 (a, inset)]. Salt-dependent longer-range repulsive forces generated by thec-terminus bristles have been implicated in previousexperimental measurements of inter-MT spacing and tubu-lin interactions [34,35]. Here, we adopted the electrostaticbrush model of Pincus et al. [36] to calculate the depend-ence of bristle height, h, on the ionic strength. Treating thebristles as 13-segment chains [37] (segment size 0.5 nm)with chargeQ ¼ −8 e per bristle as (an approximately) flatbrush of areal density of one bristle per tubulin monomerσB ¼ 1=42 nm−2, we predicted bristle heights to vary from

FIG. 4 (color online). Dependence of λ on the counterionconcentration [Kþ] for several PEG concentrations. Inset:Dependence of λ on [Kþ] for 2.0% dextran and 1.5% PEG.Each point is an average of N ¼ 5–60 independent measure-ments. Error bars represent standard deviation.

FIG. 5 (color online). Comparisons of theoretical models of thecohesion strength. (a) Effective interactions between two parallelMTs (solid black curve) as a function of filament surfaceseparation, D. The effective potential is composed of thedepletion attraction (dash-dotted blue curve) and repulsivepotential due to charged hollow cylinders (red dashed curve)and charged bristles (dotted light brown curve). Inset: Thepredicted bristle height h vs salt concentration. (b) Theoreticalpredictions of the cohesion strength for the primitive model(dashed curve), the bristle-stabilized model (solid curve), andexperimental results (squares) for PEG 1.0%.

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2.6 to 4.8 nm over the range of salt concentrationsmeasured, scaling roughly as h ∼ ½Kþ�−1=3 in the high-saltlimit [Fig. 5(a, inset)]. We assumed uniformly stretchedbristles with the bristle charge density at radial distancer from the MT center given by ρBðrÞ ¼ QaσB=rh.Electrostatic repulsion in the bristle-stabilized model wascomputed from the linearized-PB interaction betweensuperposed bare surface charge of MTs and bristle chargedensity, ρfðrÞ ¼ σsδðr − aÞ þ ρBðrÞ, while depletion wastreated as before, such that polymers were assumed topenetrate the bristles freely. These interactions increasefilament separation by roughly twice the bristle height atthe expense of decreasing depletion attractions. Whilecalculated binding energies were closer to the experimentalvalues, they still overestimated the measured λ by a factorof ∼2–3 [Fig. 5(b)]. Comparing the primitive and bristle-stabilizedmodel predictions suggests that the larger surfaceseparation due to charged c-terminus bristles is fundamen-tal to MT-MT interactions. However, in light of apparentoverestimation of the polymer-induced depletion by the AOtheory, a more detailed representation of polymer inter-actions with the MT geometry is needed to quantitativelyrecapitulate measured cohesion strengths.Depletion interactions between spherical colloids and

polymers are pairwise additive as long as the depletingpolymer is significantly smaller than the spherical colloid[38,39]. To investigate the additivity of the depletionbetween rodlike particles, we characterized properties ofmultifilament bundles. We induced the formation of athree-MT bundle by bringing a third filament near anexisting two-MT bundle. Once adsorbed, the third MTalways migrated to the existing two-filament overlapregion, since such configurations minimize the depletantexcluded volume [Figs. 6(a) and 6(b)]. The cohesionstrength, λ, for the 3-MT bundles was consistently twicethe value measured for the 2-MT bundles [Fig. 6(c)]. Theseexperiments indicate that depletion interactions with a∼7 nm range are still pairwise additive for the filamentswith hardcore diameters of ∼25 nm. The electrostaticinteractions decrease the range of the depletion attractionand thus likely further reduce the importance of many-bodyoverlap configurations.Polymer-induced depletion interactions with nanometer

range have been extensively studied between micron-sizedcolloidal particles [8,40,41]. Here, we have measured thedepletion interactions in the protein limit [42,43], in whichthe size of the nonadsorbing polymer (Rg) approaches thesize of the colloidal object (MT filament diameter). Wefound significant discrepancies between our measurementsand predictions of a simplified AO model. These were onlypartially reconciled by a more refined microscopic descrip-tion of MT structure. To ultimately discern the source ofthis discrepancy, several other effects, beyond the scopeof this manuscript, need to be investigated. For example,the AO model is only accurate when polymer coils are

significantly smaller than colloidal particles [42]. Thoughthe accuracy of the AO model has been studied forspherically symmetric particles, there are few equivalentstudies for rodlike colloids [44]. Furthermore, bendingdegrees of freedom are severely constrained in a bundle,corresponding to an additional entropic cost of bundling forfinite-persistence length filaments. This entropy loss wasimportant for understanding depletion-induced bundling ofactin filaments [45]; since the bending entropy loss scalesinversely with the fourth root of the bending stiffness, itcould yield significant corrections even for stiff MTs.Finally, we have treated the depletant as ideal nonadsorbingpolymers; excluded volume interactions could affect themagnitude of the depletion attraction [46].In general, depletion-induced filamentous bundles can

exist in two distinct states—dynamical, where the slidingfriction of constituent filaments is governed by hydro-dynamic interactions, and static, where solidlike frictiondominates [47]. Our technique measures cohesion strengthfor dynamical bundles, whose filaments are freely slidingwith respect to each other and thus minimize the freeenergy on experimental time scales. For MT filaments,transition between the two regimes is governed by theconcentrations of both depletant and counterions as well asby some structural modifications of tubulin (for example,removal of the c-terminus brush) [47].In conclusion, we have developed a technique to

measure attractive depletion interactions between filamen-tous structures and applied it to reveal intriguing propertiesof the depletion-driven MT bundles. Most materials obeyHooke’s law, so that the deformation energy scales withsquare of the applied strain. In comparison, the free energyof MT bundles scales linearly with the applied strain,indicating that such assemblages behave as a constant forcetransducer with effectively zero stiffness. Our results arerelevant for biological or synthetic systems where depletion

(a) (b)

(c)

FIG. 6 (color online). Pairwise additivity of the depletioninteraction. (a) Schematic representation and (b) fluorescenceimage of a three-filament bundle. The third MT is not attached tothe beads. Scale bar, 5 μm. (c) For a wide range of experimentalconditions, λ for three-MT bundles is double the value for thetwo-MT bundles. Error bars represent standard deviation.

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forces due to globular polymers or other nonadsorbingparticles play an important role [48,49]. They are alsoimportant in materials science, where the depletion effect isused to engineer interactions between colloids and to driveself-assembly [44,50–56]. The methods described here canbe used in studies of MT-associated motor enzymes andcross-linking proteins (MAPs) [57–59] and extended toother filamentous systems and alternative interactionssuch as counterion-induced bundling of polyelectrolytefilaments [60–64].

F. H. and Z. D. were supported by Department of Energy,Office of Basic Energy Sciences under Award No. DE-SC0010432TDD. A.W. acquired preliminary data withsupport of Grant No. NSF-MRI-0923057. M. F. H. devel-oped the theoretical model with support of GrantsNo. NSF-CMMI-1068566 and No. NSF-MCB-1329623.Theoretical work of L. C. and G. G. was supported byGrant No. NSF-CMMI-1068852. We also acknowledge useof MRSEC optical microscopy facility which is supportedby Grant No. NSF-MRSEC-1420382.

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