Page 1

Bipedal Locomotion in Crawling Cells

Erin L. Barnhart,†‡Greg M. Allen,†‡Frank Ju ¨licher,§and Julie A. Theriot†‡{*

†Department of Biochemistry, Stanford University School of Medicine, Stanford, California;‡Howard Hughes Medical Institute, Stanford

University School of Medicine, Stanford, California;§Max Planck Institute for the Physics of Complex Systems, Dresden, Germany;

and{Department of Microbiology and Immunology, Stanford University School of Medicine, Stanford, California

ABSTRACT

force. Force-generating systems that act on elastic cytoskeletal elements are prone to oscillating instabilities. In this work, we

have measured spontaneous shape and movement oscillations in motile fish epithelial keratocytes. In persistently polarized,

fan-shaped cells, retraction of the trailing edge on one side of the cell body is out of phase with retraction on the other side, re-

sulting in periodic lateral oscillation of the cell body. We present a physical description of keratocyte oscillation in which periodic

retraction of the trailing edge is the result of elastic coupling with the leading edge. Consistent with the predictions of this model,

the observed frequency of oscillation correlates with cell speed. In addition, decreasing the strength of adhesion to the substrate

reduces the elastic force required for retraction, causing cells to oscillate with higher frequency at relatively lower speeds. These

results demonstrate that simple elastic coupling between movement at the front of the cell and movement at the rear can

generate large-scale mechanical integration of cell behavior.

Many complex cellular processes from mitosis to cell motility depend on the ability of the cytoskeleton to generate

INTRODUCTION

Cell migration requires temporal and spatial integration of

multiple force-generating systems (1–3). At the front of the

cell, actin polymerization drives protrusion of the leading

edge (4–6), and at the rear, actin depolymerization and

myosin contraction facilitate retraction of the trailing edge

and translocation of the cell body (7). Contractile forces

generated by myosin II activity and by turnover of the elastic

actin network are balanced by adhesions between the cell

and the underlying substrate, enabling generation of traction

force and net forward movement (3,8–11). Each of these

processes—polymerization and depolymerization of the

actin meshwork, myosin contraction, and adhesion—are

complex, highly-regulated processes that have been well

characterized individually, but the molecular and mechanical

mechanisms that couple protrusion of the leading edge with

retraction of the trailing edge are not well understood.

Fish epithelial keratocytes are notoriously well coordi-

nated cells; in many keratocytes, protrusion of the leading

edge is so tightly coupled with retraction of the trailing

edge that migrating cells appear to glide across the substrate

while maintaining a constant shape and speed (12). Recently,

however, careful quantification of cell shape has shown that

keratocytes from primary fish skin cultures are heteroge-

neous (13–15). Stereotypical, ‘‘coherent’’ keratocytes are

fast-moving and fan-shaped, with smooth leading edges,

whereas ‘‘decoherent’’ cells, in which protrusion and retrac-

tion are more loosely coupled, are rounder, slower-moving,

and have a rough leading-edge morphology (13,14). More-

over, coherent keratocytes are directionally persistent,

moving in one direction over many cell lengths of move-

ment, whereas decoherent keratocytes tend to move in

curved trajectories (13), suggesting that the protrusive,

contractile, and adhesive forces required for migration are

more tightly balanced in coherent keratocytes than in deco-

herent keratocytes.

The dynamic organization and mechanics of the kerato-

cyte cytoskeleton have been extensively characterized,

particularly in coherent keratocytes (2). Keratocytes have

a broad, flat lamellipodium that consists of a densely

branched actin meshwork (16). In coherent keratocytes, the

anticapping protein Ena/VASP and filamentous actin are

both enriched in the front center of the leading edge

(13,14), and AFM measurements indicate that the elastic la-

mellipodium is stiffest near the front (17). The actin mesh-

work is organized with barbed ends primarily oriented

toward the leading edge (16) and polymerization of the actin

meshwork is tightly coupled to protrusion of the leading

edge; photoactivation experiments and quantitative fluores-

cent speckle microscopy have demonstrated that the actin

network is nearly stationary with respect to the underlying

substrate (4,18). Adhesion proteins such as integrin and talin

localize to the leading edge in fan-shaped keratocytes (19),

and local disruption of adhesions with forces too small to

stall actin polymerization nonetheless stall protrusion of

the leading edge (20). In the rear of the cell, myosin contrac-

tion exerts force on the substrate perpendicular to the direc-

tion of cell movement (10,11,21), and these contractile

forces are balanced by large adhesions on either side of the

cell body (19,22). In decoherent cells, the cytoskeleton is

less well organized, with no enrichment of Ena/VASP or fila-

mentous actin in the front center of the cell (13,14).

The tight coupling of protrusion and retraction in coherent

cells makes keratocytes an ideal model system for eluci-

dating the manner in which events at the front of the cell

Submitted August 25, 2009, and accepted for publication October 30, 2009.

*Correspondence: theriot@stanford.edu

Editor: Douglas Nyle Robinson.

? 2010 by the Biophysical Society

0006-3495/10/03/0933/10 $2.00

doi: 10.1016/j.bpj.2009.10.058

Biophysical Journal Volume 98 March 2010 933–942 933

Page 2

are coupled with events at the rear. In this work, we have

observed keratocytes that, rather than gliding across the

substrate, take small steps forward. In these cells, retraction

of the trailing edge on one side of the cell body is out of

phase with retraction on the other side, resulting in periodic

lateral oscillation of the cell body. These oscillations are

more prevalent in coherent keratocytes than in decoherent

cells, suggesting that they may be the result of efficient inte-

gration of protrusive, contractile, and adhesion forces. We

present experimental evidence to support a physical model

for oscillation in which periodic retraction of the trailing

edge is the result of elastic coupling with the leading edge.

METHODS

Cell culture and sample preparation

Keratocytes were cultured from the scales of the Central American cichlid

Hypsophrys nicaraguensis as previously described (13). Briefly, scales

were sandwiched between two acid-washed 25-mm glass coverslips and

cultured at room temperature for 16–20 h using Leibovitz’s L-15 medium

(Gibco BRL, Carlsbad, CA) supplemented with 14.2 mM HEPES,

pH 7.4, 10% FBS, and 1% antibiotic-antimycotic (Gibco BRL). Individual

cells were obtained by disaggregating sheets of keratocytes with 2.5 mM

EGTA in 85% PBS, pH 7.4, for 5–10 min. To facilitate tracking of the

cell body, cells were loaded with the fluorescent volume marker CMFDA

(Molecular Probes, Eugene, OR). Latrunculin A (Molecular Probes) or

GRGSS peptides (Stanford PAN Facility, Stanford, CA) were added to cells

in full media at a final concentration of 10 nM or 100 mg/ml, respectively.

Cells were imaged 10–30 min after treatment.

Measurement of cell body and cell shape

oscillations

Cells were imaged at 2- to 5-s intervals over 5–20 min, and the cell body

centroid was determined by the weighted pixel intensity of the cell body.

The path of the cell body was fit to a smooth curve generated by calculating

the weighted moving average for each x and y coordinate, and the distance

between the cell body centroid and the fit line was calculated at each time

point. To measure edge velocities, cell outlines were extracted from phase

images using Celltool, an open source collection of tools for quantifying

cell shape, as described (13,23), and the velocity of the cell perimeter was

calculated at evenly spaced points along the perimeter. The length of the

cell on either side of the cell body from leading edge to trailing edge was

measured in ImageJ. The frequencies of cell-body, cell-length, and edge-

velocity oscillations were determined by fast Fourier transform. The signif-

icance of the measured frequency was assessed by determining the proba-

bility of gettingthe samepeakin the powerspectrumfor randomlygenerated

signals. Data analysis was performed using custom-written code in

MATLAB 7 (The MathWorks, Natick, MA) and Python.

Model parameter choices

We estimated many of the parameter values for our model for keratocyte

oscillations from measurements of keratocyte shape and speed. We chose

n0¼ 0.2 mm/s, the average cell speed for a population of oscillating kerato-

cytes, and W ¼ 20 mm, the average cell width. Average cell length is

~15 mm, so we chose L0¼ 10 mm and L0¼ 5 mm. To estimaten1, n2, a,

and b, we measured the length of the cell, from leading edge to trailing

edge, on either side of the cell body for four oscillating keratocytes that

rangedin speed from 0.14mm/sto 0.3 mm/s. The lengthof each cell on either

side of the cell body oscillated in an anticorrelated, periodic fashion, as

shown for one cell in Fig. 1 D, with an average Dd ¼ 2.5mm. We resampled

and averaged each cycle of retraction for each of the four cells and estimated

the speed of the trailing edge as vzv0?_L for an average cycle of retraction.

Based on this, we estimated n1~ 0.08 mm/s, n2~ 1 mm/s, a ~ 0.5 s, and

b ~ 25 s. Motile keratocytes form adhesions to the substrate under the lamel-

lipodium, but these adhesions are smaller than those formed at the trailing

edge (19). Therefore, we assumed that g, the friction coefficient at position

x, was greater than a but less than b, and chose g ¼ 4 s. The elastic modulus

of the lamellipodia of moving keratocytes, measured by atomic force

microscopy (AFM), ranges between 10 kPa at the base of the lamellipodium,

near the cell body, and 55 kPa at the leading edge (17). For a keratocyte with

typical dimensions (20 mm wide, 15 mm long, and 200 nm high), this corre-

spondsto astiffness of~3–15 nN/mm. However, thelamellipodiumisso thin

that AFM likely overestimates the elastic modulus due to the influence of the

underlying, stiff glass substrate, and so we chose K ¼ 1 nN/mm for the stiff-

ness of the base of the lamellipodium and explored values for K, the stiffness

of the lamellipodium close to the leading edge, between 1 and 10 nN/mm.

ForKw,the stiffnessof the springconnecting X21and X2r,weexploredvalues

between 0.1 and 100 nN/mm. Finally, we treated g, the memory term, as

a free parameter and explored values between 0.01 and 10 s2nN/mm.

Model simulations

The dynamic equations for keratocyte motion (Eqs. 4–7) in the main text)

were evaluated numerically in Mathematica with the parameter values dis-

cussed above. To determine the relative phase lag, f, between the length

oscillations for the left and right sides of the cell, we first calculated the

period of oscillation, T, for both sides by fast Fourier transform. We then

calculated the correlation coefficient, C, between the left side at time t and

the right side at time t þ P for P ¼ 0 to P ¼ T. f was given by 1 - (Pmax/T),

where Pmaxis the value for P with the highest correlation coefficient. Simu-

lated oscillations were classified as either in-phase (0 < f < 0.1) or anti-

phase (0.4 < f < 0.5), and either stable (C > 0.8) or irregular (C < 0.8).

The correlation coefficient was defined as Cl;r¼Pðli? lÞðri? rÞ=

deviations for the left and right sides of the cell, respectively. The sum was

evaluated from t ¼ i ? n, where n is the number of time points.

ðn ? 1Þslsr, where l and r are the mean lengths and sland srare the standard

RESULTS

By imaging moving keratocytes at high spatial and temporal

resolution, we observed a common mode of migration for

these cells in which retraction of the trailing edge on one

side of the cell was out of phase with retraction on the other

side (Fig.1, Aand F–H,and Movie S1,MovieS2, andMovie

S3 in the Supporting Material). Antiphase retraction of the

trailing edge resulted in periodic lateral oscillation of the

cell body (Fig. 1, A–C). In addition, the length of the cell

from leading edge to trailing edge on either side of the cell

body oscillated, with the cell length on one side increasing

and decreasing out of phase with the other side, with the

same frequency as the lateral movement of the cell body

(Fig. 1, D and E). Oscillations were confined to the rear of

the cell; the leading edge moved forward with constant

velocity(Fig.1,F–H).Wedeterminedtheprevalenceofthese

oscillations by measuring the frequency of cell-body oscilla-

tion in a population of randomly selected cells (n ¼ 50).

Representative examples of oscillating and nonoscillating

cells are shown in Fig. 2. Seventy-four percent of keratocytes

oscillated with significant power, with periods ranging from

25 to 130 s and amplitudes between 0.5 and 3 mm for the

lateral movement of the cell body.

Biophysical Journal 98(6) 933–942

934Barnhart et al.

Page 3

Keratocytes range from coherent, fan-shaped, fast-moving

cells with smooth leading edges, to decoherent, round, slow-

moving cells with rough leading edges (13,14). To elucidate

the difference between oscillating and nonoscillating cells,

we measured cell speed, area, aspect ratio (the width of the

cell perpendicular to the direction of motion divided by the

length of the cell from leading edge to trailing edge), and

leading-edgemorphology(Fig.2,D–G).Oscillatingandnon-

oscillating cells were on average the same size (average cell

area ¼ 412 mm2and 430 mm2, respectively (Fig. 2 E)), but

oscillating cells were faster and more fan-shaped (average

cellspeed,0.19mm/s;averageaspectratio,1.8)thannonoscil-

lating cells (average cell speed, 0.09 mm/s, p < 0.001,

Student’s t-test; average aspect ratio, 1.6, p < 0.05, Fig. 2,

D and F). To quantify leading-edge roughness, we measured

localleading-edgecurvature(13,14);cellswithroughleading

edges have high local curvature values and cells with smooth

leading edges have low local curvature values. On average,

oscillating cells had smoother leading edges (average local

leading-edge curvature, 4.4) than nonoscillating cells

(averagelocalcurvature,8.0,p<0.005,Fig.2G).Oscillating

cells,then,weremorecoherentthennonoscillatingcells,indi-

cating that cell-body oscillation correlates with rapid cell

movement and efficient front-to-back coordination.

We found that two possible candidates for contributing to

retraction of the trailing edge—myosin and calcium

points

0

50

100

150

200

0 0.5-0.5

edge velocity

(µm/s)

0

seconds

2550 75100

cell body

position

fit line

020 4060 80100

µm

0

20

µm

0

1

2

3

power

00.020.04

frequency (s-1)

0 100 200 300

seconds

cell body lateral

displacement (µm)

1.5

0

-1.5

16

14

12

cell length (µm)

0 100 200 300

seconds

left right

0 100 200 300

seconds

0

-0.3

-0.6

edge velocity,

right rear (µm/s)

0100 200 300

seconds

0.6

0.3

0

edge velocity,

front (µm/s)

0100 200 300

seconds

0

-0.3

-0.6

edge velocity,

left rear (µm/s)

00.02 0.04

frequency (s-1)

0

2

4

power

00.020.04

frequency (s-1)

0

2

4

power

00.020.04

frequency (s-1)

0

2

4

power

edge velocity

map

seconds

0300150

A

CDE

FGH

0

50

100

150

B

10 µm

FIGURE 1

A locomoting keratocyte was imaged at 5-s intervals for

6 min, and the frequency of oscillation of cell body, cell

length, and edge velocity were measured by fast Fourier

transform. (A) Individual frames from the movie are shown

at 25-s intervals; the outline of the cell from the preceding

image is superimposed on each frame. Scale bar, 10 mm.

(B) The position of the cell body centroid plotted in two

dimensions (red line), fit to a smooth curve (black line).

(C) Lateral displacement of the cell body. The distance

between the cell body centroid and the fit line shown in B

is plotted over time. (D) Cell length on the left (blue line)

and right (green line) sides of the cell body over time. (E)

Power spectra for cell body (red line) and cell length

(blueand green lines) oscillations.The power of each trans-

form is plotted versus frequency. (F) Edge velocity map.

Cell outlines were extracted from each image and re-

sampled such that point 0 is the cell rear and point 100 is

the cell front. The velocity at each point on the cell perim-

eter is plotted over time. Negative velocities (blue) are

retractions and positive values (red) are protrusions. (G)

Velocities of the trailing edge to the left of the cell body

(point 175; upper graph), the center of the leading edge

(point 100; middle graph), and the trailing edge to the right

of the cell body (point 25; lower graph) are plotted over

time. (H) Power spectra for the velocities of the leading

edge and trailing edge on either side of the cell body. The

frequency of oscillation of the cell body, cell length, and

trailing edge velocity for this cell is ~42 s.

Periodic shape and movement oscillations.

Biophysical Journal 98(6) 933–942

Bipedal Locomotion in Crawling Cells 935

Page 4

transients—were not required for keratocyte oscillation.

Although Ca2þtransients have been shown to correlate

with rear retraction in slow-moving, fibroblastlike kerato-

cytes (24), Ca2þtransients observed in oscillating cells did

not correlate with cell-body oscillation (Fig. S1, A and B),

and inhibition of calcium transients did not abolish oscilla-

tion (Fig. S1 C). Inhibition of myosin II contraction with

the small-molecule inhibitor blebbistatin (25) has been

shown to abolish inward-directed actin network flow in the

keratocyte rear (21) and in this study caused focal adhesions

to shrink (Fig. S2) but did not inhibit oscillation or affect its

frequency (Fig. S3). In keratocytes, myosin contraction

exerts force on adhesion bonds perpendicular, rather than

parallel, to the direction of migration (10), suggesting that

adhesion strength and myosin contraction are normally

balanced perpendicular to the direction of cell movement,

and that elastic force parallel to the direction of motion is

sufficient for oscillation.

We propose a model for keratocyte oscillation in which

retraction of the trailing edge is the result of elastic coupling

with the leading edge. It is known that cells are viscoelastic

materials that display predominantly elastic behaviors over

timescales up to tens of seconds (26,27), which is the rele-

vant timescale for keratocyte migration. We have previously

observed that keratocytes maintain the same shape over long

time periods while migrating for tens or hundreds of cell

diameters (14). This suggests that keratocytes can be treated

as elastic bodies with preferred shapes such that any defor-

mation from the default shape results in a restoring force.

This general phenomenological description does not depend

on the detailed nature of the elastic restoring force; multiple

mechanical components of the cells may contribute,

including the cytoskeleton and the rounded cell body. In

an oscillating keratocyte, actin polymerization drives protru-

sion of the leading edge (4) and adhesions in the rear oppose

retraction of the trailing edge (19). The front of the cell

advances while the rear remains behind, cell length

increases, and an elastic restoring force builds up. We

assume that adhesions at the leading edge are much stronger

than adhesions in the rear (19), and so when the elastic force

generated by protrusion exceeds a critical force, adhesion

bonds in the rear rupture, the trailing edge jumps forward,

and the length of the cell decreases. As the trailing edge

slows down again, adhesions reform, the cell stretches, and

elastic forces increase until they again exceed the critical

force where adhesions break, resulting in oscillations in

cell length, as observed experimentally (Fig. 1 D).

We developed a quantitative model for keratocyte length

oscillations, first in one dimension. In this simplified model,

a moving cell has its front at position x1and its rear at posi-

tion x2, and the cell length is described by L ¼ x1-x2. The

front of the cell moves forward with constant velocity,

A

B

CG

D

E

F

FIGURE 2

and more coherent than nonoscillating

cells. A population of 50 randomly

selected cells were imaged at 5-s inter-

vals for 10 min, and the frequency of

lateralcellbody

measured by fast Fourier transform. Of

the 50 cells, 37 oscillated with signifi-

cant power (see Methods). (A–C) Re-

presentative examples of an oscillating

cell (right) and a nonoscillating cell

(left) are shown. (A) Phase images. (B)

Lateral displacement of the cell body.

The cell body position for each cell

was fit to a smooth curve, and the

distance between the fit line and the

cell body position is plotted over time.

(C) Power spectra for the cell body

oscillations. (D–G) Box and whisker

plots for cell speed (D), area (E), aspect

ratio (F), and front roughness (G) for

populations of oscillating and nonoscil-

lating cells. The plots indicate the 25th

percentile (lower bound), median (red

line), 75th percentile (upper bound),

andobservations within1.5 times thein-

terquartile range (whiskers). Asterisks

indicate significant differences between

the oscillating and nonoscillating popu-

lations (*p < 0.05; **p < 0.005).

Oscillating cells are faster

oscillationwas

Biophysical Journal 98(6) 933–942

936Barnhart et al.

Page 5

_ x1¼ v0(Fig. 3 A), driven by actin polymerization (4). The

front of the cell is connected to the trailing edge by an elastic

element with spring constant K and rest length L0. The elastic

force acting on the back from the front is Fe¼ Kd, where d is

the displacement of the elastic element from its rest length:

d ¼ x1? x2? L0. The rear of the cell adheres to the substrate,

and as it slides along the substrate with velocity _ x2¼ v, it

exerts force on the substrate (10,11). For constant velocity,

v, of the trailing edge, the adhesion force is a nonlinear func-

tion of v and thus an effective friction force: Fa¼ F(v) ¼ lv,

where l is a friction coefficient. The friction coefficient l

comes from the binding and detachment of adhesion bonds

(28) (see Appendix) and cannot be easily measured. There-

fore, we relate the friction coefficient l in the system to the

relaxation time t ¼ l/K of the elastic element, which we

can estimate from measured changes in L (see Methods).

At smalln < n1, the rear of the cell forms adhesions to the

substrate, and we approximate F(n) z K bn, where b is

the relaxation time associated with cell adhesion. At larger

velocity, n > n2, the adhesions are broken, and we write

F(n) z K an, where a is the relaxation time for nonspecific

friction, which obeys a ? b. For n1< n < n2, we estimate the

unstable branch of the friction force as F(n) ¼ ? K bn ? an2/

n2? n1(n ? n1) þ Kbn1(Fig. 3 B). If velocity v changes in

time, the friction force gradually changes via the kinetics

of binding and detachment of adhesion molecules. This

relaxation is captured by a coefficient, g, in the adhesion

force: Fa¼ FðvÞ þ g_ v (see Appendix). The dynamic equa-

tion for the position of x2results from the balance of elastic

and adhesive forces, Fe¼ Fa, and reads

g€ x2 ¼ Kðx1? x2? L0Þ ? F?_ x2

In the limit where adhesion forces change instantaneously,

g ¼ 0, the period, T, of oscillation is given by

?

where Dd ¼ dmax ? dmin, and dmax and dmin are the

maximum and minimum displacements, respectively, of

the elastic element. The first term in this expression for the

period T represents the rising phase, where the rear of the

cell adheres to the substrate and cell length increases, and

the second term represents the falling phase of the oscillation

cycle, where adhesions in the rear are broken and cell length

decreases. For simplicity, we assume that the rising phase is

much longer than the falling phase, and approximate T as

?:

(1)

T ¼ b ln

1 þ

Dd

bðv0? v1Þ

?

þ a ln

?

1 þ

Dd

aðv2? v0Þ

?

; (2)

Tz

Dd

ðv0? v1Þ:

(3)

We estimated parameter values for this model from measure-

ments of keratocyte shape and speed (see Methods). Values

for T calculated with these parameter values using either Eq.

2 or Eq. 3 were within 15% of each other, which is compa-

rable to the variability in the experimental data. Thus, Eq. 3

is a reasonable approximation of Eq. 2 for the keratocyte

system.

This simple one-dimensional model makes a number of

specific quantitative predictions about the behavior of oscil-

lating keratocytes. First, if elastic force generated by protru-

sion of the leading edge causes retraction of the trailing edge,

then the faster a cell moves, the faster the elastic force

reaches the critical force, and the higher the frequency of

oscillation (see Eq. 3). The frequency of oscillation should

therefore correlate with n0. To test this, we measured cell-

speed and cell-body oscillation in a large population of cells

(n ¼ 98), and the frequency of oscillation did in fact correlate

with the speed of the cell (Fig. 4 A). Moreover, by rewriting

Eq. 3 as n0¼ fDd þ n1, where the frequency of oscillation is

f ¼ 1/T, we can see that the slope of the fit line in Fig. 4 A is

approximately equal to Dd, the change in cell length in one

oscillation cycle, and the y intercept is equal to v1, the critical

velocity where adhesions in the rear rupture. The Dd

dmax

dmin

vmin

velocity of the trailing edge

vmax

v1

v2

d=F(v)

K

a

b

B

A

x1

v0

x2

L0

displacement of

cell length

FIGURE 3

leading edge (position x1) moves forward with constant velocity v0. The

trailing edge (position x2) adheres to the substrate. The leading and trailing

edges are connected by an elastic element with stiffness K and rest length L0.

(B) Schematic plot of d, the displacement of the cell length from rest, as

a function of v, the velocity of the trailing edge. At small velocity, n < n1,

the rear of the cell forms adhesions to the substrate and moves more slowly

than the front, and the cell stretches, with displacement d ¼ F(n)/K z bn.

When the speed of the trailing edge reaches n1, the adhesions between the

rear of the cell and substrate rupture, and v quickly increases to nmax. As

the rear of the cell then begins to slow down, at large n > n2the length of

the cell decreases with d z an. When n < n2, adhesions reform between

the cell and the substrate, the velocity of the rear slows to nmin, and the length

of the cell again increases with d z bn.

One-dimensional model for keratocyte oscillation. (A) The

Biophysical Journal 98(6) 933–942

Bipedal Locomotion in Crawling Cells937