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

doi:10.1073/pnas.0603931103

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

*Cell Biology and Biophysics Unit, European Molecular Biology Laboratory, Meyerhofstrasse 1, D-69117 Heidelberg, Germany;
^‡Department of Medical Biochemistry, Biology, and Physics, Innovative Technologies for Signal Detection and Processing Center and Instituto Nazionale Fisica Nucleare, Università di Bari, Piazza Giulio Cesare 11, 70124 Bari, Italy;
^§Center for Nonlinear Dynamics, University of Texas, Austin, TX 78712; and
^¶Arnold Sommerfeld Center for Theoretical Physics and Center for Nano Science, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 Munich, Germany

Communicated by Robert H. Austin, Princeton University, Princeton, NJ, May 13, 2006
↵ ^†F.P. and G.L. contributed equally to this work. (received for review June 25, 2005)

Abstract

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 ℓ_p. 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 ℓ_p 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 ℓ_p for microtubules of different contour length L. Upon varying L between 2.6 and 47.5 μm, we find a systematic increase of ℓ_p 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 subnanometer 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.

Footnotes

^‖To whom correspondence may be addressed. E-mail: frey{at}lmu.de or florin{at}chaos.utexas.edu
Author contributions: F.P., G.L., E.F., and E.-L.F. designed research; F.P. and G.L. performed research; A.J. contributed new reagents/analytic tools; F.P. and G.L. analyzed data; and F.P., G.L., T.S., E.F., and E.-L.F. wrote the paper.
Conflict of interest statement: No conflicts declared.
↵ ** This experiment is also relevant to confirming the reliability of our approach: Thermal fluctuations of the MT's tip are basically related to the first bending mode of the MT, which is very sensitive to flows of the liquid in the sample (3–5) and induces a large experimental error. This problem is less critical when measuring equilibrium fluctuations, as in our case, but still requires a very good control over the experimental conditions. By measuring the fluctuations of two beads at two different positions of the same MT, we obtain two independent and consistent measurements of ℓ_p, thus confirming the reliability of our assay.
↵ †† According to ref. 22, the relaxation time for the first bending mode of a semiflexible polymer is τ = (ζ/ℓ_p k _BT)(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 τ = 76.6 ms for a MT of 10 μm: This estimate has been compared with relaxation times extracted from our data by analyzing the power spectra of the thermal fluctuations and, as an independent consistency check, by calculating the histograms of the tip displacements at varying delay times. Interestingly, we found that the experimental relaxation times were systematically longer than expected from theoretical estimates. Unexpectedly long relaxation time for MTs were already reported in the literature, and they may be interpreted as the result of an internal friction effect within the MT (4). As far as our measure of P(y) is concerned, the set charge-coupled device camera integration time is short enough compared with the relaxation time of the MT thermal fluctuations, and the width of the experimental P(y) is not artificially narrowed. However, the total observation time largely exceeds the MT relaxation time in the range of contour lengths examined, so that the width of P(y) is always measured at its equilibrium value (see ref. 30).
↵ ‡‡ Symmetry considerations impose that E ₂/ν₂₁ = E ₁/ν₁₂, so that there are four independent elastic parameters.
↵ §§ The total number of independent elastic parameters is thus 5.
Abbreviations:

Abbreviations:

AFM,

atomic force microscopy;

MT,

microtubule;

PF,

protofilament