1. Cell Biology
  2. Structural Biology and Molecular Biophysics
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Microtubules soften due to cross-sectional flattening

  1. Edvin Memet
  2. Feodor Hilitski
  3. Margaret A Morris
  4. Walter J Schwenger
  5. Zvonimir Dogic
  6. L Mahadevan  Is a corresponding author
  1. Harvard University, United States
  2. Brandeis University, United States
  3. University of California, Santa Barbara, United States
Research Article
Cite this article as: eLife 2018;7:e34695 doi: 10.7554/eLife.34695
8 figures, 3 videos, 2 tables and 4 additional files

Figures

Probing the bending and stretching of microtubules using an optical trap.

(A) Schematic of the experimental setup for microtubule compression and stretching (note: not to-scale). Beads of radius 0.5 μm manipulated through optical traps are attached to the filament via biotin-streptavidin bonds. The left trap remains fixed, while the right one is incrementally displaced towards or away. The bead separation db is different from the trap separation dt due to the displacement between the centers of each bead and the corresponding trap. The force exerted by the traps on the beads is proportional to this displacement, with the constant of proportionality being the optical trap stiffness. The (effective) MT length L is the length between the attachment points of the two beads (but note that multiple bonds may form). The arclength coordinate s is measured from the left attachment point and takes value between 0 and L: 0<s<L. A cross-sectional profile is shown to illustrate our definition of ‘eccentricity’ or ‘flatness’ e=(Ra)/R (Equation 4) where 2a is the length of the small axis and R is the radius of the undeformed MT cross-section, taken to be R=12 nm in our simulations. (B) Series of fluorescence images from microtubule compression experiment. Trap positions represented through red circles of arbitrary size are drawn for illustration purposes and may not precisely represent actual trap positions, since neither beads nor traps are visible in fluorescence imaging. The mobile trap is moved progressively closer to the detection trap in the first four snapshots, resulting in increasingly larger MT buckling amplitudes. The mobile trap then reverses direction, decreasing the buckling amplitude as seen in the last two snapshots. The scale bar represents 5 μm, the end-to-end length of the MT is approximately 18 μm, and the length of the MT segment between the optical beads is 7.7 μm.

https://doi.org/10.7554/eLife.34695.002
Discrete mechanical network model of microtubule.

(A) Schematic of spring network model of microtubule. Red particles discretize the filament while yellow particles represent the optical beads. Top inset: closeup of a region of the microtubule model, showing the 3D arrangement of the particles. Right inset: further closeup of the region, showing the defined interactions between particles. In the circumferential direction, ten particles are arranged in a circle of radius R=12 nm, (only three particles are shown in the inset) with neighboring pairs connected by springs of stiffness kc and rest length p (light green). Neighboring triplets in the circumferential direction form an angle ϕc and are characterized by a bending energy interaction with parameter κc (dark green). In the axial direction, cross-sections are connected by springs of stiffness ka (light blue) running between pairs of corresponding particles. Bending in the axial direction is characterized by a parameter κa and the angle ϕa formed by neighboring triplets (dark blue). All other triplets of connected particles that do not run exclusively in the axial or circumferential direction encode a shearing interaction with energy given by Vbend(κs,ϕs,π/2) (Table 1) with parameters κs and ϕs (yellow), where the stress-free shear angle is π/2. (B) Force-strain curve for a microtubule of length 7.1 μm from experiment (blue) and simulation using a 1D model (green, dashed) or a 3D model (red). Arrows on the red curve indicate forward or backward directions in the compressive regime. The bending rigidity for the 1D model fit is B=12pNμm2. The fit parameters for the 3D model are Ea=0.6 GPa, Ec=3 MPa, and G=1.5 GPa. Errorbars are obtained by binning data spatially as well asaveraging over multiple runs. Inset: Raw data from individual ‘forward’ (green) and ‘backward’ (yellow) runs.

https://doi.org/10.7554/eLife.34695.006
Discrete mechanical model of bacterial flagellum.

(A) Schematic of spring network model for bacterial flagellum. Green particles discretize the flagellum while blue particles represent the optical beads. Right inset: closeup of a region of the filament, showing that neighboring masses are connected by linear springs of stiffness k which give rise to a stretching energy Vstretch(k,l,l0) according to Equation 2, where l0 is the stress-free length of the springs. In addition, each triplet of neighbors determines an angle ϕ which dictates the bending energy of the unit: Vbend(κ,ϕ,π) (Equation 3). Left inset: closeup of the region where the bead connects to the flagellum. The bead particle is connected to a single particle of the flagellum by a very stiff spring kb of stress-free length b corresponding to radius of the optical bead. In addition, there is a bending interaction Vbend(κb,ϕb,π/2), where ϕb is the angle determined by the bead, its connecting particle on the flagellum, and the latter’s left neighbor. (B) Force-strain curve from experiment (blue) and simulation (yellow) for a flagellum of length L= 4.1 μm. Errorbars are obtained by binning data spatially as well as averaging over multiple runs. The bending rigidity obtained from the fit is B=4pN×μm2. The classical buckling force for a rod of the same length and bending rigidity with pinned boundary conditions is Fc2.3pN.

https://doi.org/10.7554/eLife.34695.007
Plots of force F as a function of strain ϵ (Equation 1) for microtubules ranging in length from 3.6 to 11.1μm (blue).

Best-fit curves from simulation of the 3D model (red) and fit parameters (top) are shown for each plot. Arrows on the red curves indicate forward or backward directions in the compressive regime.

https://doi.org/10.7554/eLife.34695.009

(A) Force-strain curve for a microtubule of length 8.3 μm from experiment (blue) and simulation using a 1D model (green, dashed) or a 3D model (red). Arrows on the red curve indicate forward or backward directions in the compressive regime. Errorbars are obtained by binning data spatially as well as averaging over multiple runs. The fit parameters for the 3D model are Ea=0.9 GPa, Ec=4 MPa, and G=4 GPa. The bending rigidity for the 1D model fit is B=23pNμm2. Labels a, b, and c indicate specific points in the 3D simulation model. (inset) Snapshot from simulation corresponding to the point labeled c. Smaller insets show the shape of the cross-section at indicated points. (B) Profiles of flatness e (Equation 4) along the dimensionless arclength coordinate s^=s/L (see Figure 1A) at points identified in panel A as a (green), b (blue), and c (red). Three insets show the shape of the cross-section at indicated points for the curve labeled c (red). A fourth inset shows the curvature profiles corresponding, by color, to the flatness profiles in the main figure.

https://doi.org/10.7554/eLife.34695.011
(bottom) Expected (Equation 6, with a numerical factor of unity) versus observed critical bending moments MB of simulated microtubules of various lengths, elastic moduli, and optical bead sizes.

Data points that correspond to actual experimental parameters are marked with stars, while the rest are marked with black circles. The shaded region is zoomed-in on (top), to show that most experimental points fall in the range about 1000–2000 pN×nm.

https://doi.org/10.7554/eLife.34695.012
Author response image 1
(A) Force-strain curve for a microtubule of length 7.7 μm from simulation using 10 protofilaments (red), 10 protofilaments with thermal noise (yellow), and 14 protofilaments (green).

The fit parameters are E a = 0.6 GPa, E c = 4 MPa, and G = 6 GPa. B) Profiles of flatness e (Eq. 4, manuscript) along the dimensionless arclength coordinate ŝ = s/L (see Fig. 1A, manuscript) at pointsidentified in panel A with dots.

https://doi.org/10.7554/eLife.34695.018
Author response image 2
Comparison of shape between experiment (filament: (green); approx-imate trap positions: (blue)) and simulation (filament: (red); optical beads:(gray)) for a microtubule of length 8.3 μm (same as in figure 5, manuscript) at astrain ε ≈ −0.21 by A) aligning filament profiles and B) aligning bead positions(estimated).
https://doi.org/10.7554/eLife.34695.019

Videos

Video 1
Microtubule buckling experiment, showing two ‘back-and-forth’ cycles.

The microtubule is labeled with fluorescent dye Alexa-647 and is actively kept in the focal plane. The scale bar represents 5 μm. Approximate positions of the optical traps are shown by red circles of arbitrary size.

https://doi.org/10.7554/eLife.34695.003
Video 2
Flagellum buckling experiment.

(left) Simulation of microtubule buckling experiment shown in Figure 4, first panel. The length of the microtubule between the optical bead attachment points is 3.6 μm while the size of the optical beads is 0.5 mum. The shape of the cross-section located halfway along the microtubule is also shown. (right) Force-strain curves from experiment (blue) and simulation (red). The simulation curve is synchronized with the visualization on the left.

https://doi.org/10.7554/eLife.34695.004
Video 3
Flagellum buckling experiment, showing three ‘back-and-forth’ cycles.

The scale bar represents 5 μm. Approximate positions of the optical traps are shown by green circles of arbitrary size. Measured force vector and magnitude are also indicated.

https://doi.org/10.7554/eLife.34695.005

Tables

Table 1
List of microscopic parameters for the 3D model, their physical significance, the energy functional associated with them (Equation2 or Equation 3), and their connection to the relevant macroscopic elastic parameter.

Parameters, angles, and lengths are highlighted in Figure 2A. where h2.7 nm and h01.6 nm are the microtubule’s thickness and effective thickness (see Materials and methods), respectively and factors involving the Poisson ratios νa and νc are neglected.

https://doi.org/10.7554/eLife.34695.008
MicroscopicPhysicalAssociatedRelation to
ParameterSignificanceEnergyMacroscopic moduli

ka

Axial stretching

12ka(lad)2

Eahpd

κa

Axial bending

12κa(ϕaπ)2

16Eah03pd

kc

Circumferential stretching

12kc(lcp)2

Echdp

κc

Circumferential bending

12κc(ϕc144π/180)2

16Ec h03d/p

κs

Shearing

12κs(ϕsπ/2)2

2Ghpd

Table 2
Range of macroscopic elastic parameters that fit experimental data for microtubules and flagella.
https://doi.org/10.7554/eLife.34695.010
SystemFit parametersRange that fits data

Ea

0.6–1.1 GPa
Microtubule

Ec

3–10 MPa

G

1.5–6 GPa
Flagellum

B

3–5 pNμm2

Additional files

Source Code 1

Source code files and source data for Figure 6 and source code files along with instructions for generating the data in Figure 2.

https://doi.org/10.7554/eLife.34695.013
Supplementary file 1

List of parameters used in the paper.

https://doi.org/10.7554/eLife.34695.014
Supplementary file 2

(left) Summary of previous experiments examining microtubule stiffness, adapted from (Hawkins et al., 2010).

(right) Box plot comparing bending stiffness values obtained via thermal fluctuations and mechanicalbending for microtubules stabilized with GDP Tubulin + Taxol (orange), GMPCPP Tubulin (green), and GDP Tubulin (blue)

https://doi.org/10.7554/eLife.34695.015
Transparent reporting form
https://doi.org/10.7554/eLife.34695.016

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