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Bending Dynamics of Fluctuating Biopolymers Probed by Automated

High-Resolution Filament Tracking

Clifford P. Brangwynne,* Gijsje H. Koenderink,* Ed Barry,yZvonimir Dogic,yFrederick C. MacKintosh,z

and David A. Weitz*§

*Harvard School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts;yRowland Institute at Harvard,

Cambridge, Massachusetts;zDepartment of Physics and Astronomy, Vrije Universiteit, Amsterdam, The Netherlands; and§Department

of Physics, Harvard University, Cambridge, Massachusetts

ABSTRACT

and dynamics. However, successful extraction of this information requires precise localization of filament position and shape

from thousands of noisy images. Here, we present careful measurements of the bending dynamics of filamentous (F-)actin and

microtubules at thermal equilibrium with high spatial and temporal resolution using a new, simple but robust, automated image

analysis algorithm with subpixel accuracy. We find that slender actin filaments have a persistence length of ;17 mm, and

display a q?4-dependent relaxation spectrum, as expected from viscous drag. Microtubules have a persistence length of several

millimeters; interestingly, there is a small correlation between total microtubule length and rigidity, with shorter filaments

appearing softer. However, we show that this correlation can arise, in principle, from intrinsic measurement noise that must be

carefully considered. The dynamic behavior of the bending of microtubules also appears more complex than that of F-actin,

reflecting their higher-order structure. These results emphasize both the power and limitations of light microscopy techniques for

studying the mechanics and dynamics of biopolymers.

Microscope images of fluctuating biopolymers contain a wealth of information about their underlying mechanics

INTRODUCTION

Actin filaments and microtubules are semiflexible biopoly-

mers that form the elastic cytoskeletal network within cells

that controls cell migration, division, cargo transport, and

mechanosensing (1). Understanding the mechanical behav-

ior of these filaments is thus of central importance for

establishing how they function as a dynamic mechanical

scaffold within living cells. Indeed, mechanical models of

the cell require accurate measurements of the stiffness and

dynamical behavior of the component filaments (2–4).

However, the semiflexible macromolecular backbone of

microtubules and actin filaments results in physical behavior

that remains incompletely understood.

Using light microscope images to directly measure the

shape fluctuations of individual biopolymers is a powerful

technique for studying their dynamics and mechanical

behavior (5–9). By extracting a set of filament coordinates

from each image, the variance of the curvature fluctuations

induced by thermal motion can be used to obtain the bending

rigidity. The rigidity is typically expressed in terms of the

length scale beyond which the filament shows significant

curvature due to thermal forces, known as the persistence

length, lp: Changes in thermally-induced curvature occur on

a timescale that is set by viscous drag from the surrounding

fluid. This time can be obtained from the relaxation time of a

shape autocorrelation function, such as the mean-squared

difference in curvature calculated for increasing lag times.

Utilizing variations of this technique, actin filaments have

been shown to have a persistence length of ;17 mm (5,8,10),

although they were initially suggested to have a length scale-

dependent rigidity (11). Microtubules have been shown to be

orders of magnitude more stiff, with a persistence length on

the order of millimeters (8). However, recent studies have

suggested that microtubule rigidity may depend on their

growth velocity (7) and other factors related to their mac-

romolecular structure (12–14). Indeed, it was recently

suggested that inter-protofilament shearing leads to a soft-

ening of shorter microtubules (15). Other studies have

suggested that internal dissipation mechanisms may domi-

nate over hydrodynamic drag to increase the relaxation times

of microtubules on short length scales (7,16).

Successfully extracting such mechanical information from

microscope images is limited by position uncertainties re-

sulting from image noise, requiring precise filament tracking.

Despite the need for accurate filament localization, reports

utilizing this technique often use semi-manual filament

tracking methods and do not report a noise floor. Automated

video tracking of objects with spherical symmetry, such as

colloidal particles and fluorescent point-sources, is a well-

developed technique for quantifying their dynamic behavior.

Tracking of large numbers of spherical probe particles is

central to microrheological measurements of the mechanical

behavior of soft materials such as biopolymer gels and even

living cells (17–19). Particle tracking algorithms typically

employ an initial particle localization, such as intensity

Submitted September 7, 2006, and accepted for publication February 15,

2007.

Address reprint requests to David A. Weitz, Gordon Mckay Professor of

Applied Physics and Professor of Physics, Harvard University, Pierce Hall,

Rm. 321, 29 Oxford St., Cambridge, MA 02138. Tel.: 617-496-2842; Fax:

617-495-2875; E-mail: weitz@seas.harvard.edu.

Editor: Marileen Dogterom.

? 2007 by the Biophysical Society

0006-3495/07/07/346/14 $2.00

doi: 10.1529/biophysj.106.096966

346Biophysical JournalVolume 93July 2007 346–359

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maxima, followed by particle position refinement using an

intensityweighted center of mass (20) or fitting to a Gaussian

or parabolic function. These algorithms can be used to obtain

particle positions with subpixel accuracy. Positions are then

linked in time to establish the trajectories of the particles.

Precise tracking of objects with nonspherical symmetry,

such as biopolymer filaments, presents more of a challenge.

Unlike colloidal particles, the shape of the filament is not

known a priori and thus conventional automated fitting

techniques are no longer applicable. There is an extensive

computer vision literature that describes identification of lin-

ear structures, such as roads or capillary tubes, from complex

landscapes (21). A related technique for automated tracking

of differential interference contrast (DIC) images of linear

structures such as microtubules has been described (22). A

recent study describes automated tracking of uniformly-sized

fluorescent colloidal rods (23). There are also a number of

papers describing filament-centroid tracking routines for in

vitro motility assays of fluorescent actin filaments gliding

over myosin-coated surfaces (24,25). However, to our know-

ledge there have been no reports describing automated

contour tracking of fluctuating fluorescent filaments with

sub-pixel accuracy.

In this paper, we develop and utilize a robust automated

image analysis algorithm for tracking fluorescently-labeled

biopolymer filaments with subpixel accuracy. We first dem-

onstrate the accuracy of this technique by tracking im-

mobilized fluorescent biopolymers and show that we obtain a

root mean-square precision of ;0.15 pixel (;20 nm), even

as the filaments begin to be photobleached after hundreds of

exposures. We then use this technique to study the shape

fluctuations of microtubules and actin filaments at thermal

equilibrium down to short length and time scales. This

allows us to address recent conflicting reports of the

mechanics and dynamics of biopolymer bending fluctu-

ations, particularly those of microtubules. The persis-

tence length of actin filaments obtained from our

measurements agrees well with previous experiments

(5,8,10), lp? 17 mm. We obtain microtubule persistence

lengths on the order of a few mm, also in agreement with

previous measurements (7,8,13,14). Our data show a

slight correlation between filament length and stiffness,

consistent with a recent study (15). However, we dem-

onstrate that this correlation can in principle arise from

inherent noise limitations that occur even with high precision

measurements. Thus, over the range we study, there does not

appear to be a systematic dependence of the bending rigidity

on length or wavevector, in agreement with other studies

(7,8,13,14). However, after a careful analysis of the relax-

ation times of the fluctuating normal modes, we find

evidence that microtubules do display anomalous bending

dynamics that reflects their more complex molecular struc-

ture (26) compared to actin filaments. Although the relax-

ation times of fluctuating actin filaments are consistent with

simple viscous drag, for microtubules they are much longer

than expected from hydrodynamic drag on short length

scales. This effect may be due to a surprisingly large internal

dissipation that dominates the relaxation times of microtu-

bules on biologically relevant length and time scales. Our

findings emphasize both the power and limitations of light

microscopy techniques for probing the complex mechanical

behavior of semiflexible polymers.

Filament tracking algorithm

The analysis algorithm used to track the shape of fluo-

rescently labeled biopolymers in successive images consists

of the steps detailed below. Briefly, an initial rough estimate

of the position of a filament is first determined by thresh-

olding the image. Pixels above threshold are then skeleton-

ized to obtain a 1-pixel-wide representation of the filament.

A polynomial fit to this skeleton is then used to walk along

the contour of the filament, refining the position estimate by

finding the intensity maximum along perpendicular cuts

across the filament.

Background noise reduction

We first performed an image noise reduction by convolving

each pixel of the image with a Gaussian kernel over a local

region of size w:

AGaussðx;yÞ ¼

1

Bðx;yÞ

+

w

i;j¼?w

Aðx1i;y1jÞexp ?i21j2

4

??

;

(1)

where Bðx;yÞ is an appropriate normalization. The long

wavelength background intensity variation is also

accounted for by averaging each pixel of the unfiltered

image over a local region of size w, to obtain Abackgroundðx;yÞ:

The final filtered image is then obtained from: Afilterðx;yÞ ¼

abs AGaussðx;yÞ ? Abackgroundðx;yÞ

localization, the size w was set to be around three times

larger than the width of the filament, ;5 pixels in our

system. For refinement of the initial filament localization, we

limit the convolution of local position information with

neighboring position information by setting the filter size to

approximately the width of the filament. An example of the

result of this procedure for a typical image of a fluorescently

labeled microtubule (Fig. 1 A) is shown in Fig. 1 B. Filters

designed specifically for highlighting linear structures are

another alternative (27).

??(20). For initial filament

Thresholding

Thresholding was done to separate the filament from the

remaining background intensity fluctuations. Pixels with a

value greater than the threshold value are assigned a value of

1, whereas those with a value less than the threshold value

are assigned a value of 0. The threshold value is given by

Biopolymer Bending Dynamics 347

Biophysical Journal 93(1) 346–359

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f ¼ ÆIijæ1msij; where ÆIijæ is the average intensity value of

all pixels ij, and sij is the standard deviation of the pixel

intensity, both of which are dominated by the nonfilament

background pixels. For high-frequency image acquisition,

the signal/noise ratio (s/n) is usually small; thus, it is critical

to determine an appropriate threshold value by determining

the number of standard deviations away from the average, m.

If m is too small, some background may be included, whereas

if it is large, the entire filament may not be included. We

typically start with a small value of m(;3) and successively

increase its value in small increments (0.2) until the backbone

of the thresholded region is well fit by a polynomial. An

example of thresholding is shown in Fig. 1 C.

Skeletonization

For good s/n and an appropriate threshold value, pixels

above threshold will cover the fluorescent filament in a

cluster several pixels wide. This cluster can be thinned to a

1-pixel-wide line using a skeletonization routine that erodes

pixels while maintaining cluster connectedness (28). Skel-

etonization has the advantage that small clusters correspond-

ing to nonfilament background fluctuations above threshold

are typically eliminated by the erosion process. Persistent

fluorescent specks that are still not eliminated are ignored

using a simple routine to exclude objects above threshold

from specific regions of the image. If necessary, more

complex pixel clustering algorithms may be used to identify

pixels associated with the desired filament (28). An example

of skeletonization of a thresholded filament is shown in Fig.

1 D. This line, corresponding to a set of position coordinates

x ¼ i, y ¼ j such that Iij¼ 1; provides a first, imprecise,

estimate of the filament location.

Polynomial fitting and filament rotation

We refine the initial position estimate by analyzing the inten-

sity distribution of sections taken perpendicular to the local

slope of the filament (Fig. 2 A). To obtain the local slope,

we fit the xy coordinates of the skeleton with a polynomial

of order p. The polynomial degree required depends on the

amount of curvature in the filaments under study; we

typically use p ¼ 3–5 for microtubules and actin filaments.

We thus obtain a curve fðxÞ ¼ +p

ferentiated to obtain an estimate of the local slope of the

filament at position x0: uðx0Þ ¼ tan?1df

i¼0cixithat can be dif-

dxjx¼x0: To obtain the

FIGURE 1

localization algorithm for a typical fluorescence micrograph of an Alexa-488

dye-labeled microtubule constrained to move in a quasi-two-dimensional

chamber. (A) Unprocessed image, and image after bandpassing, as in Eq.

1 (B), followed by thresholding (C), and skeletonization (D). Note that the

spurious speck in the thresholded image is eliminated upon skeletonization.

Example of the initial image processing stages of the filament

FIGURE 2

intensity profiles along cuts perpendicular to the filament contour. (A)

Skeleton (see Fig. 1 D) in the xy coordinate frame with polynomial fit and

schematic perpendicular cuts. (B) The intensity profile along a perpendicular

cut of an unprocessed image is shown (solid squares). After bandpassing the

image, the intensity distribution (open squares) is fit to a Gaussian (solid

line). The filament location is taken to be the maximum of the Gaussian.

Example of filament position refinement by fitting Gaussian

348Brangwynne et al.

Biophysical Journal 93(1) 346–359

Page 4

intensity profile of the filament perpendicular to this local

slope, we rotate the original image by an angle uðx0Þ around

the position, ? xrot¼ ðxo;fðxoÞÞ; to orient the filament hor-

izontally. Successful image rotation requires the use of a

good interpolation algorithm that maps the intensity of all

pixels onto the rotated image without introducing artifacts.

Image rotation routines implementing good interpolation

algorithms accompany software packages such as IDL and

Matlab. The process is computationally intensive, and thus

we restrict image rotation to a local region of interest around

the filament with coordinates ½ðx0?bÞ:ðx01bÞ;ðfðx0Þ?bÞ:

ðfðx0Þ1bÞ?; where b is larger than the extent of the Gaussian

kernel, w.

Because the skeletonization procedure erodes pixels at the

ends of the thresholded filament, we sample perpendicular

sections along the polynomial backbone from, typically, 4

pixels before to 4 pixels after the end points of the skeleton.

We calculate the integrated signal along the perpendicular

column of pixels k, M ¼ +kIk; and set a threshold value

(‘‘masscut’’) for identified filament positions. The first and

last filament positions with a signal mass above threshold

define the two ends of the filament. Because fluorescently

labeled biopolymers frequently display nonuniform labeling

along their length, masscut thresholding often leaves holes in

the position identification of the filament, particularly in

low-s/n images. We fill in these gaps by linear interpolation

between the nearest successfully sampled points.

Position refinement by Gaussian fitting

We proceed to analyze the intensity distribution of the

vertical column of pixels in each rotated region. An example

of such an intensity distribution is shown in Fig. 2 B (solid

squares). By applying the Gaussian convolution kernel and

background subtraction to this rotated local region, we

improve the s/n ratio by eliminating both high- and low-

frequency noise (Fig. 2 B, open squares). The center of the

filament can be identified from this intensity distribution

using either functional fitting or an intensity-weighted center

of mass. It is important to note that the Gaussian filter leads

to averaging of pixel intensities over a local region defined

by the size of the Gaussian kernel (;5–10 pixels); averaging

the filament centroid obtained from adjacent perpendicular

cuts of unbandpassed images could provide an alternative s/n

improvement, without undesirable averaging in the perpen-

dicular direction. We choose to fit the intensity distribution

to a Gaussian intensity profile (Fig. 2 B, solid line), which

yields the highest precision for tracking of spherical objects

(29). We find that the primary advantage of Gaussian fitting

over an intensity-weighted center-of-mass approach is its

relative insensitivity to any residual background signal.

The position of the center of the Gaussian, y9g; identifies

the position of the filament in the rotated frame. This center

position is then mapped back onto the unrotated full image

coordinate system to obtain the real-space position of the

mthsampled point ? xm; using ym¼fðx0Þ1f½y9g?fðx0Þ?3

cosðuðx0ÞÞg and xm¼x0?f½y9g?fðx0Þ?3sinðuðx0ÞÞg: This

procedure is repeated along the polynomial at the desired

sampling frequency, typically in steps of size Ds¼2 pixels,

until a full set of refined filament coordinates is obtained.

MATERIALS AND METHODS

Monomeric (G) actin was purified from rabbit skeletal muscle (30), with an

additional gel-column chromatography step (Sephacryl S-200) to remove

residual cross-linking and capping proteins. Actin filaments stabilized with

phalloidin were prepared by polymerizing G-actin in AB-buffer (25 mM

imidazole,50mMKCl,2mMMgCl2,1mMEGTA,and1mMdithiothreitol,

pH 7.4). Filaments were fluorescently labeled by using Alexa-488-modified

G-actin,orbystabilizingunlabeledactinfilamentswithAlexa-488phalloidin

(Molecular Probes). We analyzed actin filaments with contour lengths

between 8 and 15 mm. Tubulin was purified from bovine brain according to

standard procedures and fluorescently labeled with Alexa-488. Microtubules

were polymerized in K-Pipes buffer (80 mM K-Pipes, 1 mM EGTA, and

1 mM MgCl2, pH 6.8) at 37?C and stabilized with 10 mM taxol.

Microtubules were imaged in a quasi-two-dimensional sample chamber

that was made by placing a small volume (typically 0.3 mL) of a dilute

solution of stabilized filaments augmented with a standard antioxidant

mixture (glucose oxidase, catalase, glucose, 2-mercaptoethanol) (31) to slow

photobleaching between a microscope slide and coverslip. The edges of

the coverslip were sealed with mineral oil to prevent fluid flow due to

evaporation. Adsorption of filaments onto the glass surfaces was avoided by

passivating the glass with adsorbed bovine serum albumin or covalently

attached poly-(ethylene glycol) chains (MPEG-silane-5000, Nektar, San

Carlos, CA). All experiments were performed at room temperature.

Fluorescent images of microtubules were acquired on a Leica DM-IRB

inverted microscope equipped with either a Hamamatsu intensified charge-

coupled device (CCD) camera C7190-21 (exposure time 33 ms, 0.136 mm/

pixel) or a Hamamatsu ORCA CCD camera C4742-95 (exposure time 59

ms, 0.128 mm/pixel) (Hamamatsu City, Japan). Images of actin filaments

and some microtubules were obtained using a Nikon inverted microscope

with Coolsnap HQ camera (exposure time of 5–100 ms, 0.129–0.194 mm/

pixel). Microtubules were also imaged with a 10-ms exposure using a high-

speed camera (Phantom V7, Vision Research, Wayne, NJ) equipped with an

image intensifier (Model VS4-1845HS, Video Scope International, Dulles,

VA). Some actin filament bending data (see Fig. 7) was obtained using a

secondmethod to slow down the filament dynamicsby adding nonadsorbing

polymer (2–3%, Dextran 500,000 g/mol). The polymer induces an attractive

depletion interaction between the coverglass and the filaments, leading to

confinement of the filaments close to the bottom surface. Using these

samples, we were able to acquire images with either epi-illumination or total

internal reflection fluorescence (TIRF) illumination. Due to the low levels of

background fluorescence,we were ableto decrease exposure time with TIRF

illumination to a few milliseconds.

The data in Fig. 11 B were fit to a line to obtain a slope of ?0.050 6

0.066, indicating no statistically significant correlation between persistence

length, lp; and wavevector, q. In addition, the correlation coefficient was

calculated usingr ¼ slpq=slpsq;whereslpqisthecovariancebetweenlpand

q, and slpand sqis the standard deviation of lpand q, respectively. The

values of r vary from 1 (strongly correlated) to ?1 (strongly anticorrelated).

We obtained a value of ?0.096, which is a statistically insignificant

correlation (p . 0.005).

RESULTS

Evaluation of tracking precision

As a first test to evaluate the performance of our algorithm

we determined the uncertainty in filament localization

Biopolymer Bending Dynamics 349

Biophysical Journal 93(1) 346–359

Page 5

arising from camera shot noise and tracking limitations. This

was accomplished by analyzing the shape fluctuations of

microtubules that were immobilized by physically adhering

them to poly-L-lysine coated surfaces. These microtubules

were locked tightly to the glass coverslip surface, so any

residual filament shape fluctuations that we measured would

be due to uncertainty in filament localization inherent in our

technique. To visualize the typical noise level, we overlay

the extracted shapes of an immobilized filament for 10

consecutive frames of a movie in Fig. 3. The insets show

higher-magnification views of two representative positions

along the filament against a pixel-sized grid. This explicitly

demonstrates that we obtained subpixel tracking accuracy

with a noise level of ;0.1–0.2 pixels (;20 nm). This degree

of accuracy is representative of the filaments we track in typ-

ical conditions, although, as we quantify below, the tracking

accuracy decreased with decreasing s/n. Since spherically

symmetric colloidal particles are much more straightforward

to track, and typically result in ;0.1-pixel accuracy, our fil-

ament tracking algorithm performed well, with a comparable

root mean-squared (RMS) noise level.

Bending rigidity of microtubules and actin

filaments from mode analysis

Wetrackmicrotubulesandactinfilamentsfreelyfluctuatingin

quasi-two-dimensional chambers. We obtain the filament

bending rigidities by analyzing the shape fluctuations using a

Fourier decomposition technique developed in previous stud-

ies (8,9). From the pixel coordinates ðxm;ymÞ of the filaments

we first calculate the tangent angle uðsÞ as a function of

arclength s, using: uðsmÞ ¼ tan?1ðym11? ymÞ= ðxm11? xmÞ;

where

sm¼

+

j¼0

m?1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffi

L

ðxj11? xjÞ21ðyj11? yjÞ2

q

r

q

()

11=2

ðxm11? xmÞ21ðym11? ymÞ2

:

The tangent angle is then decomposed into a sum of cosines,

uðsÞ ¼

2

+

n¼0

N

aqcosðqsÞ;

(2)

where the wave vector q is defined as q ¼ np=L; with n the

mode number and L the filament contour length (8). A cosine

expansion of the tangent angle using different normalization

prefactors has been used in other studies (9), but we find the

normalization used by Gittes et al. most natural since it re-

sults in a length-independent variance of the mode ampli-

tude. Use of this pure cosine mode decomposition assumes a

zero-curvature boundary condition at the free filament ends,

appropriate for freely fluctuating filaments.

The amplitudes, aq; of the first 24 bending modes of a

freely fluctuating microtubule are plotted as a function of

time in each of the subpanels of Fig. 4. The microtubule has a

nonzero intrinsic curvature, u0ðsÞ ¼

resulting in a nonzero mean amplitude. The variance of the

lower modes contains information about the flexural rigidity,

whereas the amplitude of the higher modes is dominated by

experimental noise. The amplitudes of these higher modes

widen with time (Fig. 4, inset) due to photobleaching and

concomitant reduction of the s/n.

A convenient feature of the cosine expansion in Eq. 2 is

that random noise due to errors in filament localization obeys

the following simple relation (8):

ffiffiffiffiffiffiffiffi

2=L

p

+N

n¼0a0

qcosðqsÞ;

FIGURE 3

charged, poly-L-lysine-coated glass surface. (Insets) Higher-magnification

views of 10consecutivefilamentcontoursfrom a videoacquisition;the pixel

size is represented by the gray grid, demonstrating that the filament tracking

algorithm performs with subpixel accuracy.

Filament contours for a microtubule fixed to an oppositely

FIGURE 4

in Fig. 1, plotted as a function of time in each of the subpanels. The micro-

tubule has intrinsic curvature, resulting in a nonzero mean amplitude. The

variance of the lower modes contains information about the flexural rigidity,

whereas the amplitude of the higher modes is dominated by experimental

noise. The amplitudes widen with time (see inset) due to photobleaching and

concomitant reduction of the signal/noise ratio.

Theamplitudeof24bendingmodesforthemicrotubuleshown

350Brangwynne et al.

Biophysical Journal 93(1) 346–359