We use single-particle tracking to study the elastic properties of single
microtubules grafted to a substrate. Thermal fluctuations of the free
microtubule's end are recorded, in order to measure position distribution
functions from which we calculate the persistence length of microtubules with
contour lengths between 2.6 and 48 micrometers. We find the persistence length
to vary by more than a factor of 20 over the total range of contour lengths.
Our results support the hypothesis that shearing between protofilaments
contributes significantly to the mechanics of microtubules.Comment: 9 pages, 3 figure

A. Jonas

Chretien

Chr  tien

de Pablo

E. Frey

E.-L. Florin

F. Pampaloni

Felgner

G. Lattanzi

Gittes

Humphrey

Hunyadi

Janson

J  nosi

Kasas

Keskin

Kroy

Kurachi

Le Goff

Mickey

Needleman

Nogales

N  d  lec

Schaap

Sept

Speidel

T. Surrey

VanBuren

Venier

Wilhelm

Wong

English

arXiv

Crossref

Thermal fluctuations of grafted microtubules provide evidence of a length-dependent persistence length

We use single-particle tracking to study the elastic properties of single microtubules grafted to a substrate. Thermal fluctuations of the free microtubule's end are recorded, in order to measure position distribution functions from which we calculate the persistence length of microtubules with contour lengths between 2.6 and 48 micrometers. We find the persistence length to vary by more than a factor of 20 over the total range of contour lengths. Our results support the hypothesis that shearing between protofilaments contributes significantly to the mechanics of microtubule

Pampaloni, Francesco

Lattanzi, Gianluca

Jonás, Alexandr

Surrey, Thomas

Frey, Erwin

Florin, Ernst-Ludwig

Open Access LMU ( Ludwig-Maximilians-Univ. München)

LMU-ASC 26/05Elastic properties of grafted microtubulesFrancesco Pampaloni1,5, Gianluca Lattanzi2,3,4, Alexandr Jona´sˇ5,Thomas Surrey1, Erwin Frey6 and Ernst-Ludwig Florin5∗1EMBL, Cell Biology and Biophysics Programme,Meyerhofstr. 1, D-69117 Heidelberg, Germany2Department of Medical Biochemistry, Biology and Physics, Bari, Italy3TIRES-Center of Innovative Technologies for SignalDetection and Processing, Universita` di Bari, Bari, Italy4Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Bari, Italy5Center for Nonlinear Dynamics, University of Texas at Austin, Austin, Texas 78712 (USA)6Arnold Sommerfeld Center and CeNS, Department of Physics,Ludwig-Maximilians-Universita¨t Mu¨nchen,Theresienstrasse 37, D-80333 Mu¨nchen, Germany(Dated: February 9, 2008)AbstractWe use single-particle tracking to study the elastic properties of single microtubules grafted toa substrate. Thermal fluctuations of the free microtubule’s end are recorded, in order to measureposition distribution functions from which we calculate the persistence length of microtubuleswith contour lengths between 2.6 and 48 µm. We find the persistence length to vary by morethan a factor of 20 over the total range of contour lengths. Our results support the hypothe-sis that shearing between protofilaments contributes significantly to the mechanics of microtubules.PACS numbers: 87.15.Ya, 87.15.La, 87.16.Ka, 36.20.Ey∗ Corresponding author. FP and GL have equally contributed to this work.1The mechanics of living cells is determined by the cytoskeleton, a self-organizing andhighly dynamic network of filamentous proteins of different lengths and stiffnesses [1]. Un-derstanding the elastic response of purified cytoskeletal filaments is fundamental for theelucidation of the rheological behaviour of the cytoskeleton. Microtubules (MTs) are hollowcylindrical cytoskeletal filaments formed by in average thirteen tubulin protofilaments as-sembled in parallel. The outer and inner diameters are about 25 and 15 nm, respectively. Incells, MTs are generally 1− 10 µm long while in axons their length can be 50− 100 µm [2].The tubular structure of MTs implies a minimal cross–sectional area, hence a high strengthand stiffness combined with low density. Although usually modelled as isotropic cylinders,MTs have anisotropic elastic properties determined by their discrete protofilament structure.Their response to longitudinal tensile/compressive stress is determined by the strength of thehead-tail αβ-αβ tubulin bonds along the protofilament, whereas the response to shear stressby the weaker interprotofilament bonds [3, 4, 5, 6] which allow adjacent protofilaments toskew past each other (Fig 1b). In the language of continuum mechanics and linear elasticitytheory, it is thus expected that the shear modulus G of MTs differs significantly from theYoung’s modulus E [7, 8]. For instance, mechanically probing MTs with AFM yields a shearmodulus that is 2-3 orders of magnitude smaller than the Young’s modulus [7, 8]. Moreover,a transversal buckling stress of Pr = 600 Pa has been recently measured on MTs [9]. This isfour orders of magnitude smaller than the buckling stress calculated by modelling MTs asisotropic solids with E ∼ 1.6 GPa. The main parameter for the description of the mechanicsof polymer filaments is the ratio ℓp/L between the persistence length ℓp and the contourlength L, which quantitatively describes the stiffness of a filament. ℓp is expected to dependon both G and E, but a G ≪ E implies also a dependence of ℓp on L, as suggested by therecent AFM experiments in [8]. The most sensitive and elegant way to determine ℓp/L ismeasuring the probability distribution function of the end-to-end distance of the filament,P (R) [10]. P (R) and ℓp/L have been measured with great accuracy for actin [11]. However,a detailed study of the mechanics of MTs based on probability distributions is complicatedby their high stiffness, requiring an assay with very high spatial resolution. In this Letter,we describe a method based on single-particle tracking (SPT) for measuring the probabilitydistribution function P (x, y) of grafted MTs. From the measured probability distributionswe obtain the MTs persistence length and investigate its dependence on L over a broadrange of contour lengths (2.6 − 48µm). In the assay, single fluorescent colloidal beads are2attached at the tip of grafted MTs and the thermal fluctuations of the MT-bead systemare recorded. Indeed, thermal fluctuations are the most natural way to mechanically probebiological nanometer-sized specimens non-destructively and over extended times. The exper-imental situation is depicted in Fig 1a. Biotinylated, rhodamine-labelled MTs, polymerizedaccording to a standard protocol [12] and stabilized with taxol are covalently attached to amicrostructured gold substrate made of parallel bars (bar width 10 µm, distance betweenthe bars 50 µm) (Fig 1a). The gold surface is chemically modified to ensure well-definedand reproducible conditions for the attachment of MTs [27]. A flow cell is assembled asshown in Fig 1a, setting a distance of 20− 30 µm between the gold substrate and the upperand lower boundaries of the cell, so that any influence of the chamber’s walls is negligi-ble. A MTs suspension is then flowed into the cell perpendicularly to the bars, inducingMTs to attach preferentially along this direction, with just a segment of MT grafted to thegrid, while the free portion fluctuates unconstrained in the medium (Fig 1a). A dilutedsuspension of yellow-green fluorescent, avidin-coated colloidal beads (diameter 200 or 500nm, Molecular Probes F-8774 and Bang Laboratoires CP01F, respectively) is then flowedinto the chamber. The beads bind strongly to the biotinylated MTs. Finally, the chamberis sealed with silicon grease, to avoid evaporation of the medium over time. In living cells,MTs usually irradiate from a microtubule organizing center toward the cell cortex [1], hencethe boundary conditions adopted in our assay are close to MT configurations in cells. Singlegrafted MTs with one bead attached proximally to the tip are selected for the measurement.Alternatively, optical tweezers are employed to attach the bead to the MT’s tip. Since thebeads and the MTs emit fluorescence at two different wavelengths, they can be observedindependently by exchanging the filter set in the microscope. Fig. 2a shows an exampleof the distribution resulting from the projection of the three-dimensional fluctuations ofthe MT’s tip on the xy plane. The distribution is obtained by measuring the position ofthe fluorescent bead attached at the MT’s end in at least 104 frames (average frame rate16 frames/s). The bead’s position is automatically tracked frame-by-frame with a customparticle-tracking routine. The coordinates of the bead’s centroid can be determined withan accuracy of 1 nm and better under optimal conditions [13]. Here, the accuracy wasslightly reduced to ∼ 10 nm. A stack containing the fluorescent MTs is recorded separatelyfor the determination of L. The recorded particle position is used to obtain the probabilitydistribution function P (x, y) of the free end in the xy plane and the corresponding reduced3FIG. 1: Representation of the experimental situation. (a) The MT is covalently grafted on a goldsubstrate. A fluorescent colloidal bead (not to scale) is attached at the MT’s tip, which freelyfluctuates in the medium (δ ≡ deflection). Both δ and the MT bending profile are exaggeratedfor illustration purposes. The distance of the MT from both sides of the chamber is at least 20µm. (b) Bending generates a tensile/compressive stress P = E1ǫ and a shear stress s = G12γbetween adjacent protofilaments (E1, G12 longitudinal Young’s and shear modulus, respectively; ǫtensile strain, γ shear strain) . Given the anisotropy in the tubulin interaction across and alongprotofilaments, E1 and G12 may differ by several orders of magnitude (x, y, z geometrical axes;1, 2, 3 material axes).distribution functions P (x) and P (y) (Fig. 2b-c), obtained by integrating P (x, y) over thecoordinates y and x respectively [28]. These can be directly compared with theoretical pre-dictions based on the wormlike chain model, as reported in [14]. Given the high persistencelength of a microtubule (ℓp/L can be as high as several hundreds), the longitudinal distribu-tion function P (x) is peaked towards full stretching, as in Fig. 2c, and varies over a typicallength scale L‖ = L2/ℓp. A non-Gaussian and asymmetric shape of P (x), is clearly visible inFig. 2b. In contrast, P (y) is a Gaussian whose width is given by the transverse length scale4FIG. 2: (a) Bead’s position tracked over 104 frames. The bead has a diameter of 200 nm and isattached at the tip of a MT with a contour length of 7.2 µm. (b) Reduced distribution functionalong the x-axis (see Fig. 1a). (c) Reduced distribution function along the y-axis.L⊥ =√(2L3/3ℓp), as represented in Fig. 2c [15, 16]. By measuring the contour length L, ℓpcan be extracted from both P (x) or P (y). It is important to notice that the measured P (x)and P (y) correspond to convolution products between the theoretical functions Pth(x) andPth(y) predicted by the wormlike chain model [15, 16] and the function that represents theexperimental precision in the measurement of the distance between the tip of the filamentand the attachment point on the substrate. Since longitudinal and transverse fluctuationsscale differently, the convolution has a negligible effect on Pth(y), in contrast to Pth(x) whoseshape is much more sensitive to small variations of the experimental precision. For this rea-son, we focused on P (y) for the determination of ℓp in this work. The persistence lengthsobtained from P (y) for 48 grafted microtubules of different contour lengths are shown inFig. 3. We observe a clear dependence of ℓp on L. This result is a direct consequence of theanysotropic tubulin interactions along and across protofilaments. In facts, as a consequenceof their anisotropic protofilament structure, MTs are expected to have five independentelastic moduli across the longitudinal (xy) and transversal (yz) planes (i.e. E1, E2 = E3,G12 = G13, G23, and the Poisson’s ratios ν13 = ν12; 1, 2, 3 are the material axes, Fig. 1a),similarly to fiber-reinforced materials [19]. In the following discussion we will model MTs asanisotropic hollow cylindrical beams and assume that the small fluctuations in the tip’s posi-tion can be interpreted as deflections due to thermal forces. Given the boundary conditionsand the geometry of the system, only the elastic moduli E1 and G12 are involved in bendingon the xy-plane. Microscopically, E1 is expected to arise from the strong αβ-αβ tubulin5bonds along single protofilaments, whereas the origin of G12 is expected to be in the muchweaker interaction between tubulin dimers on adjacent protofilaments (see Fig. 1b), whichare allowed to skew past each other in order to accommodate the shear strain associatedwith MTs bending. Thus, the total deflection of MTs (δ) is the sum of both bending (δB)and shear deformations (δS) [8, 17]:δ = δB + δS =PL3αE1I2+PLG12kA=PL3αℓpkBT, (1)where P is the lateral load at the MT’s tip, I2 = (1/4)π(R4e −R4i ) is the second moment ofthe tube’s cross-sectional area A, k a geometrical correction factor whose value is 0.72 for ahollow cylindrical beam [20], and α = 3 for a grafted beam. The last equality, with kBT thethermal energy, represents the linear response of a semiflexible polymer [21]. Through therelation E1I2 = ℓpkBT , Eq. 1 can be used to obtain the length dependence of the persistencelength:ℓp = ℓ∞p(1 +3E1I2G12kAL2)−1, (2)where ℓ∞p is the persistence length for long microtubules (L≫√3E1I2/G12kA ≃ 21µm).Eq. 2 provides a good fit to the experimental points, giving a value of ℓ∞p = 6.3 ± 0.8mm in agreeement with previous measurements on long MTs [22]. This value correspondsto E1 = 1.51± 0.19 GPa and a ratio δS/δB = (21/L)2 when L is expressed in micrometers,corresponding to a ratio between Young’s and shear modulus of ∼ 106, three orders ofmagnitude higher than previously reported [7, 8]. This can be partially explained by thefact that in ref. [7, 8] MTs were stabilized with glutharaldeyde, a cross-linker used forchemical fixation of proteins, whereas we use the drug taxol, which insert in the MTs latticeand may relieve the internal tension inducing a further decrease of the shear modulus [23].In addition, experiments reported in ref. [7, 8] suggest a strong dependence of the shearmodulus on temperature. Whereas E ∼ G in conventional materials, a shear modulussignificantly smaller than the Young’s modulus is typical of biological materials (from woodto bones) and engineered composites. Down to nanometer scale, a ratio E \ G ∼ 102has also been measured for single-walled carbon nanotubes [24]. As shown in Fig. 3, thevalues of ℓp measured on MTs spread with increasing L. This spreading is partially dueto differences in the structures of single MTs. In facts, in MTs polymerized in vitro the6FIG. 3: The persistence length of MTs as a function of their contour lengths. The contour lengthranges from 2.6 µm to 48 µm. The fit with Eq. 2 is superimposed on the experimental points.number of protofilaments may vary from one MT to another and even along a given MT,and several lattice types are possible [25]. The probability of defects or inhomogeneities inthe MT lattice may also increase with the contour length. Since there are also evidences thatthe stiffness of MTs can vary with the polymerization velocity [26], we carefully monitoredthe MT polymerization temperature. A further potential experimental source of error mightbe a loose binding of the bead to the MT, which could lead to an underestimation of ℓp.This has been tested by pulling the bead with optical tweezers at high laser power, ensuringthat it attaches firmly to the MT and does not roll or twist on it. In addition, by stronglybending the MT with optical tweezers we ensured that the segment of the MT anchored tothe substrate does not change its orientation during thermal fluctuations, a further source ofuncertainty in the measurement of L. Theoretical calculations show also that the fluctuationsof the angle of torsion along the whole microtubule are below 7 degrees and can be safelyneglected. In summary, we developed an assay based on single-particle tracking to measurewith high spatial resolution the reduced distribution functions P (x) and P (y) for the tip’sposition of grafted MTs. Through a systematic measure of the persistence length of MTswith contour lengths between 2.6 and 48 µm, we observe a dependence of the MTs stiffnesson L. For short MTs (L = 2−3µm) ℓp is at least one order of magnitude smaller than for longones (L ≥ 30−40µm). This property can be explained by the anisotropic structure of MTs,and described by linear elasticity theory. Our results together with those in [7, 8, 9] suggestthat the interprotofilament bonds are much weaker than the αβ-αβ tubulin bonds along theprotofilaments, leading to a very low shear modulus in MTs. The dependence of ℓp on L in7MTs might also contribute to explain the broad range of measured MTs persistence lengthsfound in literature (0.5−5 mm) [22, 23, 26]. Noticeably, the maximum variability of ℓp occursat contour lengths close to the typical length scale of a cell (L ≃ 12 µm). It is interestingto speculate about the biological significance of this property. MTs are able to accomplishvarious tasks in differently sized cells and during different stages of a cell’s life cycle. Onone hand, flexibility of MT’s tip allows MTs to search space laterally for binding partnersor to bend and continue growing in a modified direction when encountering obstacles, forexample when growing MTs hit the side of a small fission yeast cell. On the other hand,sufficient rigidity is important in situations in which MTs need to resist to pushing forcessuch as in large vertebrate cells during anaphase of mitosis, when the spindle elongates. Thelength dependence of MTs rigidity might have allowed cells during evolution to use MTsin an optimal way both for lateral searching and for exerting pushing forces in differentcontexts and especially on different length scales. We are grateful to A. Rohrbach, E. H.K. Stelzer and J. Swoger for their substantial support and many interesting discussions.GL’s research has been partially supported by a Marie Curie Fellowship under contractno. HPMF-CT-2001-01432. FP has been supported by the German Research Foundation,grant no. FL351/2-1.[1] F. Ne´de´lec et al., Curr. Op. Cell Biol. 15, 118 (2003).[2] D. Bray, Cell Movements - from molecules to motility (Garland Publishing, New York, 2001).[3] D. Van Buren et al., Proc. Natl. Acad. Sci. USA 99, 6035 (2002).[4] P. J. de Pablo et al., Phys. Rev. Lett. 91, 098101 (2003).[5] I. A. T. Schaap et al., Eur. Biophys. J. 33, 462 (2004).[6] D. Sept et al., Protein Sci. 12, 2257 (2003).[7] S. Kasas et al., ChemPhysChem 5, 252 (2004).[8] A. Kis et al., Phys. Rev. Lett. 89, 1216 (2002).[9] D. J. Needleman et al., Phys. Rev. Lett. 93, 198104 (2004).[10] J. Wilhelm and E. Frey, Phys. Rev. Lett. 77, 1216 (1996).[11] L. LeGoff et al., Physical Review Letters 89, 258101 (2002).[12] K. L. Carraway and C. A. C. Carraway, The Cytoskeleton – a practical approach (Oxford8University Press, New York, 1992).[13] M. Speidel et al., Opt. Lett. 28, 69 (2003).[14] G. Lattanzi et al., Phys. Rev. E 69, 021801 (2004).[15] P. Benetatos and E. Frey, Phys. Rev. E 67, 051108 (2003).[16] J. Wilhelm and E. Frey, unpublished.[17] S. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970).[18] L. D. Landau and E. M. Lifshits, Course of Theoretical Physics, Vol. 7 (Pergamon Press,London, 1959).[19] L. P. Kolla´r and G. S. Springer, Mechanics of Composite Structures (Cambridge UniversityPress, 2003).[20] A. Kis, Doctoral Thesis (EPFL, Lausanne http://library.epfl.ch/theses, 2003) p. 118.[21] K. Kroy and E. Frey, Phys. Rev. Lett. 77, 306 (1996).[22] F. Gittes et al., J. Cell. Biol. 120, 923 (1993).[23] H. Felgner et al., J. Cell. Sci. 109, 509 (1996).[24] J. P. Salvetat et al., Phys. Rev. Lett. 82, 944 (1999).[25] D. Chre´tien et al., J. Cell. Biol. 117, 1031 (1992).[26] M. E. Janson, and M. Dogterom, Bioph. J. 87, 2723 (2004).[27] The grid surface is cleaned with a solution H2O2/NH3/H2O 1:1:5 and incubated overnight withmercaptoundecanoic acid (5 mM in ethanol), producing a self-assembled monolayer with ex-posed carboxylic groups. These are activated by an aqueous solution of N-hydroxysuccinimideand 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide both 100 mM to ensure the covalent at-tachment of MTs.[28] According to [11] the longest relaxation time for a semiflexible polymer is τC =(ζ/ℓpkBT )(L/A)4, where ζ = 2.47×10−3 Pa s is an effective friction coefficient and A = 1.875for a polymer with one clamped and one free end. We find that τc = 76.6 ms for a MT of 10µm: end-to-end fluctuations saturate quite fast to the equilibrium value within the time ofthe observation9

Elastic properties of grafted microtubules

Microtubules are hollow cylindrical structures that constitute one of the three major classes of cytoskeletal filaments. On the mesoscopic length scale of a cell, their material properties are characterized by a single stiffness parameter, the persistence length Lp. Its value, in general, depends on the microscopic interactions between the constituent tubulin dimers and the architecture of the microtubule. Here, we use single-particle tracking methods combined with a fluctuation analysis to systematically study the dependence of Lp on the total filament length L. Microtubules are grafted to a substrate with one end free to fluctuate in three dimensions. A fluorescent bead is attached proximally to the free tip and is used to record the thermal fluctuations of the microtubule’s end. The position distribution functions obtained with this assay allow the precise measurement of Lp for microtubules of different contour length L. Upon varying L between 2.6 and 47.5 μm, we find a systematic increase of Lp from 110 to 5,035 μm. At the same time we verify that, for a given filament length, the persistence length is constant over the filament within the experimental accuracy. We interpret this length dependence as a consequence of a nonnegligible shear deflection determined by sub- nanometer relative displacement of adjacent protofilaments. Our results may shine new light on the function of microtubules as sophisticated nanometer-sized molecular machines and give a unified explanation of seemingly uncorrelated spreading of microtubules’ stiffness previously reported in literature

PAMPALONI F

JONAS A

SURREY T

FREY E

AND FLORIN E. L.

LATTANZI, GIANLUCA

Archivio istituzionale della ricerca - Università di Bari

Universität München: Elektronischen Publikationen

LMU-ASC 26/05
Elastic properties of grafted microtubules
Francesco Pampaloni1,5, Gianluca Lattanzi2,3,4, Alexandr Jonáš5,
Thomas Surrey1, Erwin Frey6 and Ernst-Ludwig Florin5∗
1EMBL, Cell Biology and Biophysics Programme,
Meyerhofstr. 1, D-69117 Heidelberg, Germany
2Department of Medical Biochemistry, Biology and Physics, Bari, Italy
3TIRES-Center of Innovative Technologies for Signal
Detection and Processing, Università di Bari, Bari, Italy
4Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Bari, Italy
5Center for Nonlinear Dynamics, University of Texas at Austin, Austin, Texas 78712 (USA)
6Arnold Sommerfeld Center and CeNS, Department of Physics,
Ludwig-Maximilians-Universität München,
Theresienstrasse 37, D-80333 München, Germany
(Dated: February 9, 2008)
Abstract
We use single-particle tracking to study the elastic properties of single microtubules grafted to
a substrate. Thermal fluctuations of the free microtubule’s end are recorded, in order to measure
position distribution functions from which we calculate the persistence length of microtubules
with contour lengths between 2.6 and 48 µm. We find the persistence length to vary by more
than a factor of 20 over the total range of contour lengths. Our results support the hypothe-
sis that shearing between protofilaments contributes significantly to the mechanics of microtubules.
PACS numbers: 87.15.Ya, 87.15.La, 87.16.Ka, 36.20.Ey
∗ Corresponding author. FP and GL have equally contributed to this work.
1
The mechanics of living cells is determined by the cytoskeleton, a self-organizing and
highly dynamic network of filamentous proteins of different lengths and stiffnesses [1]. Un-
derstanding the elastic response of purified cytoskeletal filaments is fundamental for the
elucidation of the rheological behaviour of the cytoskeleton. Microtubules (MTs) are hollow
cylindrical cytoskeletal filaments formed by in average thirteen tubulin protofilaments as-
sembled in parallel. The outer and inner diameters are about 25 and 15 nm, respectively. In
cells, MTs are generally 1− 10 µm long while in axons their length can be 50− 100 µm [2].
The tubular structure of MTs implies a minimal cross–sectional area, hence a high strength
and stiffness combined with low density. Although usually modelled as isotropic cylinders,
MTs have anisotropic elastic properties determined by their discrete protofilament structure.
Their response to longitudinal tensile/compressive stress is determined by the strength of the
head-tail αβ-αβ tubulin bonds along the protofilament, whereas the response to shear stress
by the weaker interprotofilament bonds [3, 4, 5, 6] which allow adjacent protofilaments to
skew past each other (Fig 1b). In the language of continuum mechanics and linear elasticity
theory, it is thus expected that the shear modulus G of MTs differs significantly from the
Young’s modulus E [7, 8]. For instance, mechanically probing MTs with AFM yields a shear
modulus that is 2-3 orders of magnitude smaller than the Young’s modulus [7, 8]. Moreover,
a transversal buckling stress of Pr = 600 Pa has been recently measured on MTs [9]. This is
four orders of magnitude smaller than the buckling stress calculated by modelling MTs as
isotropic solids with E ∼ 1.6 GPa. The main parameter for the description of the mechanics
of polymer filaments is the ratio ℓp/L between the persistence length ℓp and the contour
length L, which quantitatively describes the stiffness of a filament. ℓp is expected to depend
on both G and E, but a G ≪ E implies also a dependence of ℓp on L, as suggested by the
recent AFM experiments in [8]. The most sensitive and elegant way to determine ℓp/L is
measuring the probability distribution function of the end-to-end distance of the filament,
P (R) [10]. P (R) and ℓp/L have been measured with great accuracy for actin [11]. However,
a detailed study of the mechanics of MTs based on probability distributions is complicated
by their high stiffness, requiring an assay with very high spatial resolution. In this Letter,
we describe a method based on single-particle tracking (SPT) for measuring the probability
distribution function P (x, y) of grafted MTs. From the measured probability distributions
we obtain the MTs persistence length and investigate its dependence on L over a broad
range of contour lengths (2.6 − 48µm). In the assay, single fluorescent colloidal beads are
2
attached at the tip of grafted MTs and the thermal fluctuations of the MT-bead system
are recorded. Indeed, thermal fluctuations are the most natural way to mechanically probe
biological nanometer-sized specimens non-destructively and over extended times. The exper-
imental situation is depicted in Fig 1a. Biotinylated, rhodamine-labelled MTs, polymerized
according to a standard protocol [12] and stabilized with taxol are covalently attached to a
microstructured gold substrate made of parallel bars (bar width 10 µm, distance between
the bars 50 µm) (Fig 1a). The gold surface is chemically modified to ensure well-defined
and reproducible conditions for the attachment of MTs [27]. A flow cell is assembled as
shown in Fig 1a, setting a distance of 20− 30 µm between the gold substrate and the upper
and lower boundaries of the cell, so that any influence of the chamber’s walls is negligi-
ble. A MTs suspension is then flowed into the cell perpendicularly to the bars, inducing
MTs to attach preferentially along this direction, with just a segment of MT grafted to the
grid, while the free portion fluctuates unconstrained in the medium (Fig 1a). A diluted
suspension of yellow-green fluorescent, avidin-coated colloidal beads (diameter 200 or 500
nm, Molecular Probes F-8774 and Bang Laboratoires CP01F, respectively) is then flowed
into the chamber. The beads bind strongly to the biotinylated MTs. Finally, the chamber
is sealed with silicon grease, to avoid evaporation of the medium over time. In living cells,
MTs usually irradiate from a microtubule organizing center toward the cell cortex [1], hence
the boundary conditions adopted in our assay are close to MT configurations in cells. Single
grafted MTs with one bead attached proximally to the tip are selected for the measurement.
Alternatively, optical tweezers are employed to attach the bead to the MT’s tip. Since the
beads and the MTs emit fluorescence at two different wavelengths, they can be observed
independently by exchanging the filter set in the microscope. Fig. 2a shows an example
of the distribution resulting from the projection of the three-dimensional fluctuations of
the MT’s tip on the xy plane. The distribution is obtained by measuring the position of
the fluorescent bead attached at the MT’s end in at least 104 frames (average frame rate
16 frames/s). The bead’s position is automatically tracked frame-by-frame with a custom
particle-tracking routine. The coordinates of the bead’s centroid can be determined with
an accuracy of 1 nm and better under optimal conditions [13]. Here, the accuracy was
slightly reduced to ∼ 10 nm. A stack containing the fluorescent MTs is recorded separately
for the determination of L. The recorded particle position is used to obtain the probability
distribution function P (x, y) of the free end in the xy plane and the corresponding reduced
3
FIG. 1: Representation of the experimental situation. (a) The MT is covalently grafted on a gold
substrate. A fluorescent colloidal bead (not to scale) is attached at the MT’s tip, which freely
fluctuates in the medium (δ ≡ deflection). Both δ and the MT bending profile are exaggerated
for illustration purposes. The distance of the MT from both sides of the chamber is at least 20
µm. (b) Bending generates a tensile/compressive stress P = E1ǫ and a shear stress s = G12γ
between adjacent protofilaments (E1, G12 longitudinal Young’s and shear modulus, respectively; ǫ
tensile strain, γ shear strain) . Given the anisotropy in the tubulin interaction across and along
protofilaments, E1 and G12 may differ by several orders of magnitude (x, y, z geometrical axes;
1, 2, 3 material axes).
distribution functions P (x) and P (y) (Fig. 2b-c), obtained by integrating P (x, y) over the
coordinates y and x respectively [28]. These can be directly compared with theoretical pre-
dictions based on the wormlike chain model, as reported in [14]. Given the high persistence
length of a microtubule (ℓp/L can be as high as several hundreds), the longitudinal distribu-
tion function P (x) is peaked towards full stretching, as in Fig. 2c, and varies over a typical
length scale L‖ = L
2/ℓp. A non-Gaussian and asymmetric shape of P (x), is clearly visible in
Fig. 2b. In contrast, P (y) is a Gaussian whose width is given by the transverse length scale
4
FIG. 2: (a) Bead’s position tracked over 104 frames. The bead has a diameter of 200 nm and is
attached at the tip of a MT with a contour length of 7.2 µm. (b) Reduced distribution function
along the x-axis (see Fig. 1a). (c) Reduced distribution function along the y-axis.
L⊥ =
√
(2L3/3ℓp), as represented in Fig. 2c [15, 16]. By measuring the contour length L, ℓp
can be extracted from both P (x) or P (y). It is important to notice that the measured P (x)
and P (y) correspond to convolution products between the theoretical functions Pth(x) and
Pth(y) predicted by the wormlike chain model [15, 16] and the function that represents the
experimental precision in the measurement of the distance between the tip of the filament
and the attachment point on the substrate. Since longitudinal and transverse fluctuations
scale differently, the convolution has a negligible effect on Pth(y), in contrast to Pth(x) whose
shape is much more sensitive to small variations of the experimental precision. For this rea-
son, we focused on P (y) for the determination of ℓp in this work. The persistence lengths
obtained from P (y) for 48 grafted microtubules of different contour lengths are shown in
Fig. 3. We observe a clear dependence of ℓp on L. This result is a direct consequence of the
anysotropic tubulin interactions along and across protofilaments. In facts, as a consequence
of their anisotropic protofilament structure, MTs are expected to have five independent
elastic moduli across the longitudinal (xy) and transversal (yz) planes (i.e. E1, E2 = E3,
G12 = G13, G23, and the Poisson’s ratios ν13 = ν12; 1, 2, 3 are the material axes, Fig. 1a),
similarly to fiber-reinforced materials [19]. In the following discussion we will model MTs as
anisotropic hollow cylindrical beams and assume that the small fluctuations in the tip’s posi-
tion can be interpreted as deflections due to thermal forces. Given the boundary conditions
and the geometry of the system, only the elastic moduli E1 and G12 are involved in bending
on the xy-plane. Microscopically, E1 is expected to arise from the strong αβ-αβ tubulin
5
bonds along single protofilaments, whereas the origin of G12 is expected to be in the much
weaker interaction between tubulin dimers on adjacent protofilaments (see Fig. 1b), which
are allowed to skew past each other in order to accommodate the shear strain associated
with MTs bending. Thus, the total deflection of MTs (δ) is the sum of both bending (δB)
and shear deformations (δS) [8, 17]:
δ = δB + δS =
PL3
αE1I2
+
PL
G12kA
=
PL3
αℓpkBT
, (1)
where P is the lateral load at the MT’s tip, I2 = (1/4)π(R
4
e − R
4
i ) is the second moment of
the tube’s cross-sectional area A, k a geometrical correction factor whose value is 0.72 for a
hollow cylindrical beam [20], and α = 3 for a grafted beam. The last equality, with kBT the
thermal energy, represents the linear response of a semiflexible polymer [21]. Through the
relation E1I2 = ℓpkBT , Eq. 1 can be used to obtain the length dependence of the persistence
length:
ℓp = ℓ
∞
p
(
1 +
3E1I2
G12kAL2
)−1
, (2)
where ℓ∞p is the persistence length for long microtubules (L ≫
√
3E1I2/G12kA ≃ 21µm).
Eq. 2 provides a good fit to the experimental points, giving a value of ℓ∞p = 6.3 ± 0.8
mm in agreeement with previous measurements on long MTs [22]. This value corresponds
to E1 = 1.51 ± 0.19 GPa and a ratio δS/δB = (21/L)
2 when L is expressed in micrometers,
corresponding to a ratio between Young’s and shear modulus of ∼ 106, three orders of
magnitude higher than previously reported [7, 8]. This can be partially explained by the
fact that in ref. [7, 8] MTs were stabilized with glutharaldeyde, a cross-linker used for
chemical fixation of proteins, whereas we use the drug taxol, which insert in the MTs lattice
and may relieve the internal tension inducing a further decrease of the shear modulus [23].
In addition, experiments reported in ref. [7, 8] suggest a strong dependence of the shear
modulus on temperature. Whereas E ∼ G in conventional materials, a shear modulus
significantly smaller than the Young’s modulus is typical of biological materials (from wood
to bones) and engineered composites. Down to nanometer scale, a ratio E \ G ∼ 102
has also been measured for single-walled carbon nanotubes [24]. As shown in Fig. 3, the
values of ℓp measured on MTs spread with increasing L. This spreading is partially due
to differences in the structures of single MTs. In facts, in MTs polymerized in vitro the
6
FIG. 3: The persistence length of MTs as a function of their contour lengths. The contour length
ranges from 2.6 µm to 48 µm. The fit with Eq. 2 is superimposed on the experimental points.
number of protofilaments may vary from one MT to another and even along a given MT,
and several lattice types are possible [25]. The probability of defects or inhomogeneities in
the MT lattice may also increase with the contour length. Since there are also evidences that
the stiffness of MTs can vary with the polymerization velocity [26], we carefully monitored
the MT polymerization temperature. A further potential experimental source of error might
be a loose binding of the bead to the MT, which could lead to an underestimation of ℓp.
This has been tested by pulling the bead with optical tweezers at high laser power, ensuring
that it attaches firmly to the MT and does not roll or twist on it. In addition, by strongly
bending the MT with optical tweezers we ensured that the segment of the MT anchored to
the substrate does not change its orientation during thermal fluctuations, a further source of
uncertainty in the measurement of L. Theoretical calculations show also that the fluctuations
of the angle of torsion along the whole microtubule are below 7 degrees and can be safely
neglected. In summary, we developed an assay based on single-particle tracking to measure
with high spatial resolution the reduced distribution functions P (x) and P (y) for the tip’s
position of grafted MTs. Through a systematic measure of the persistence length of MTs
with contour lengths between 2.6 and 48 µm, we observe a dependence of the MTs stiffness
on L. For short MTs (L = 2−3µm) ℓp is at least one order of magnitude smaller than for long
ones (L ≥ 30−40µm). This property can be explained by the anisotropic structure of MTs,
and described by linear elasticity theory. Our results together with those in [7, 8, 9] suggest
that the interprotofilament bonds are much weaker than the αβ-αβ tubulin bonds along the
protofilaments, leading to a very low shear modulus in MTs. The dependence of ℓp on L in
7
MTs might also contribute to explain the broad range of measured MTs persistence lengths
found in literature (0.5−5 mm) [22, 23, 26]. Noticeably, the maximum variability of ℓp occurs
at contour lengths close to the typical length scale of a cell (L ≃ 12 µm). It is interesting
to speculate about the biological significance of this property. MTs are able to accomplish
various tasks in differently sized cells and during different stages of a cell’s life cycle. On
one hand, flexibility of MT’s tip allows MTs to search space laterally for binding partners
or to bend and continue growing in a modified direction when encountering obstacles, for
example when growing MTs hit the side of a small fission yeast cell. On the other hand,
sufficient rigidity is important in situations in which MTs need to resist to pushing forces
such as in large vertebrate cells during anaphase of mitosis, when the spindle elongates. The
length dependence of MTs rigidity might have allowed cells during evolution to use MTs
in an optimal way both for lateral searching and for exerting pushing forces in different
contexts and especially on different length scales. We are grateful to A. Rohrbach, E. H.
K. Stelzer and J. Swoger for their substantial support and many interesting discussions.
GL’s research has been partially supported by a Marie Curie Fellowship under contract
no. HPMF-CT-2001-01432. FP has been supported by the German Research Foundation,
grant no. FL351/2-1.
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mercaptoundecanoic acid (5 mM in ethanol), producing a self-assembled monolayer with ex-
posed carboxylic groups. These are activated by an aqueous solution of N-hydroxysuccinimide
and 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide both 100 mM to ensure the covalent at-
tachment of MTs.
[28] According to [11] the longest relaxation time for a semiflexible polymer is τC =
(ζ/ℓpkBT )(L/A)
4, where ζ = 2.47×10−3 Pa s is an effective friction coefficient and A = 1.875
for a polymer with one clamped and one free end. We find that τc = 76.6 ms for a MT of 10
µm: end-to-end fluctuations saturate quite fast to the equilibrium value within the time of
the observation
9


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Elastic properties of grafted microtubules

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