oSci
Un
Cell migration
Centrifuge microscope
Turgor pressure
a c
qu
e un
repo
l gr
oe
he
against the applied gravity field. The high g-load bend the pseudopodium thereby preventing its
attachment to the target point directly ahead of the cell. In this paper we further refine this idea by
at dist
nd ani
single
pivotal
e who
espons
The measurement of the actual forces exerted by crawling cells
been employed in order to perform such measurements (Oliver
ising
ling
was
ually
buoyant in it, it nevertheless stalled, even though, in this case, the
et al. observed that what appeared to limit the crawling ability of
Contents lists available at ScienceDirect
.e
Journal of Theor
Journal of Theoretical Biology 271 (2011) 202–204directly ahead of the amoeba (see Fig. 1). In the present study, wey-fukui@northwestern.edu (Y. Fukui).the amoebawas not the net force on the cell but rather the inability
of the cell to extend a pseudopodiumagainst the large gravitational
field generated by the centrifuge; the pseudopodiumwas observed
to bend and therefore not able to attach to a point on the substrate
0022-5193/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2010.11.036
 Corresponding author.
E-mail addresses: s-ghosal@northwestern.edu (S. Ghosal),on substrates is of obvious importance and various techniques have gravity field was pushing the cell in the direction of motion. FukuiThe mechanics of crawling (Fletcher and Theriot, 2004; Stossel,
1993, 1994; Fukui, 1993; Keller and Falkovitz, 1983) is a funda-
mental problem in biomechanics that has not yet been fully
understood.
direction opposite to the gravity field. However, a rather surpr
finding was that the net force alone did not limit the craw
ability of the amoeba. Indeed, when the culture medium
replacedby one of a higher density, so that the amoebawas actof form and structure in embryogenesis. The movement of cells is
crucial in the process ofwoundhealing and in the functioningof the
immune system and unfortunately in metastasis, where cancer
cells spread from the primary tumor to invade other organs of the
body (Li et al., 2005).When the environment of the cell is a fluid, the
cell swims, but on a substrate or in a fibrous network, cells crawl.
doesnot require anything to be attached to the cell and it is the least
likely to affect the natural behavior of the cell. Fukui et al. (2000)
reported on an experiment in which amoebae of Dictyostelium
discoideum were allowed to crawl against an artificial gravity field
created by a centrifuge. They determined the maximum g-force at
which the amoeba ‘‘stalled’’—that is, was unable to crawl in aIntroduction
Motility is a fundamental trait th
The macroscopic motion of plants a
reduced to motion on the level of
movement of single cells also play a
directly related to movement of th
morphogenetic migration of cells is ridentifying the bending of the pseudopodiumwith the onset of elastic instability of a beam under its own
weight. It is shown that the principal features of the experiment may be understood through this model
and an estimate for the limiting g-load in reasonable accord with the experimental measurements is
recovered.
& 2010 Elsevier Ltd. All rights reserved.
inguishes living things.
mals can ultimately be
cells (Bray, 1992). The
role in phenomena not
le organism. Thus, the
ible for the appearance
et al., 1994). One method is to restrain the cell in some way, for
example by holding a micro-needle in its path or by applying a
suction force with a micro-pippette. An ingenious non-invasive
technique involves imaging the wrinkles on an elastic substrate
from which the applied force may be inferred (Dembo and Wang,
1999). Restraining forces can be applied to crawling cells by
attaching magnetic beads to them and pulling with magnetic
fields. Gravity is an excellent candidate for an external force as itDictyostelium discoideum
Euler–Bernoulli theoryDoes buckling instability of the pseudop
can climb?
Sandip Ghosal a,, Yoshio Fukui b
a Department of Mechanical Engineering, McCormick School of Engineering and Applied
b Department of Cell and Molecular Biology, Feinberg School of Medicine, Northwestern
a r t i c l e i n f o
Article history:
Received 18 August 2010
Received in revised form
22 November 2010
Accepted 24 November 2010
Available online 2 December 2010
Keywords:
a b s t r a c t
The maximum force that
biomechanics. One way of
adjustable restraining forc
force. Fukui et al. (2000)
crawled against an artificia
net applied force on the am
Instead, it appeared that t
journal homepage: wwwdium limit how well an amoeba
ence, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, United States
iversity, 303 East Chicago Avenue, Chicago, IL 60611, United States
rawling cell can exert on a substrate is a quantity of interest in cell
antifying this force is to allow the cell to crawl against a measurable and
til the cell is no longer able to move in a direction opposite to the applied
rted on an experiment where amoeboid cells were imaged while they
avity field created by a centrifuge. An unexpected observationwas that the
ba did not seem to be the primary factor that limited its ability to climb.
amoeba stalled when it was no longer able to support a pseudopodium
lsevier.com/locate/yjtbi
etical Biology

turgor pressure in the pseudopodium by virtue of the contractile
(1881). Eq. (2) was found to hold except that a 1:99. For tapered
beams Eq. (2) may still be applied if I is regarded as the area
moment of inertia of the base. Greenhill showed that for a right
circular cone a 2:54 and for a paraboloid of revolution a 2:47.
Keller and Niordson (1966) calculated the greatest height L that a
beam of fixed weight W can have if it is allowed to taper in an
S. Ghosal, Y. Fukui / Journal of Theoretical Biology 271 (2011) 202–204 203forces in the posterior part of the cell cortex (Fukui, 1993).
Theory
If one assumes, based on these observations, that the structural
stability of the actin-rich pseudopodium against the gravity field
limits the crawling ability of the cell, then it should be possible to
use classical results on the stability of elastic structures undershow that this proposal of Fukui et al. that the buckling instability
of the pseudopodium is themajor factor limiting the ability of cells
to crawl against high gravitational forces, is supported by a
quantitative analysis based on mechanics. We further suggest that
the ability to crawl against high gravitational forces is significantly
improved in cells that are able to actively generate an internal
Fig. 1. The amoeba stalls (is unable tomove ‘‘up’’ against the artificial gravity field g
of the centrifuge) when the pseudopodium advancing in the direction of intended
motion is bent by gravitational forces. Thus, the pseudopodium fails to attach at the
target location A on the substrate directly above the cell body. Downward or lateral
crawling of the cell is not prevented.gravity to estimate the stalling acceleration (gc). The relevant result
is the classical Euler–Bernoulli theorywhichpredicts that an elastic
beam of uniform cross-section buckles under a compressive force
(F) if this force exceeds a critical value given by
F ¼
aEI
L2
, ð1Þ
where E is the Young’s modulus of the material, I is the area
moment of inertia of the cross-section, L is its length and the value
of the numerical constant a depends on the conditions at the ends
of the beam; for a beam clamped at one end and free at the other,
a¼ p2=4. The Euler–Bernoulli theory proceeds from the assump-
tions ofmechanical equilibrium and small deformations that result
in a linear boundary value problem for the beam centerline. The
requirement that this equation admit nonzero solutions results in
an eigenvalue problem, and Eq. (1), corresponds to the lowest
eigenvalue (Timoshenko and Gere, 2009). If the compressive force
is the weight of the beam (W), a simple estimate for the maximum
weight up to which the vertical configuration could be stable may
be obtained by assuming all of theweight to be concentrated at the
center of mass, so that
W ¼
4aEI
L2
: ð2Þ
A more careful analysis where the weight is assumed to be
uniformly distributed along the beam was provided by Greenhillarbitrarywaywhile preserving the shape of the cross-section. Once
again, the result can be expressed in the form of Eq. (2) with
a 8:23. Thus, Eq. (2), where a is a numerical constant that has a
value roughly between one and ten, may be used to estimate the
critical acceleration gc of the centrifuge. For this purpose, one can
rewrite Eq. (2) as
gc ¼
4aEI
ðDrÞVL2
, ð3Þ
using the apparent weight (Kokkinis and Bernitsas, 1987) (in place
of W) of the pseudopodium which is the difference in density
between the pseudopodium and the culture medium (Dr) multi-
plied by the volume (V) of the pseudopodium.1
If the pseudopodium is regarded as a circular cylinder of
diameter 2 mm and length 5 mm, then I  0:8  1024 m4 and
V  1:6 1017 m3. Fukui et al. (2000) estimate that the density
of the actin-rich pseudopodium must be at least r¼
1:124 gm=cm3. If we accept this value, then, since the medium
density (at 0% Percoll) is r0 ¼ 1:005 gm=cm3, Dr¼ rr0 ¼
0:119 gm=cm3. The greatest uncertainty arises in estimating the
Young’s modulus E. If the pseudopodium is presumed to be
supported predominantly by the mechanical strength of the actin
network one could use in vitro measurements of the strength of
actin gels. Such measurements are usually expressed in terms of
the shear modulus G which is related to E and the bulk modulus KV
as E¼ 9GKV=ð3KV þGÞ  3G (since G/KV is smaller than 107). In
vitro measurements show that G for actin networks vary from
about 300 Pa in the ‘‘gel’’ state to a value three orders of magnitude
lower (Janmey et al., 1994; Gardel et al., 2004) in the fluid state. The
measured value depends primarily on the density and length of
actin filaments and the density of cross-links created by various
actin binding proteins (ABPs). If we take G 300 Pa and a 2:5
(corresponding to a structure of paraboloid shape) Eq. (3) yields a
numerical estimate gc  5  104 m=s2.
In the experiment Fukui et al. (2000) found that the myosin II
knockout mutant (HS1) of the Dictyostelium amoeba stalled at
gc  3:9  104 m=s2 (in the buffer without Percoll) which is in
reasonable accord with the above estimate. The wild type strain
(NC4) containing myosin II did not stall even at the highest
accelerations tested (about 11:2  104 m=s2). This is probably
because the wild type cells are able to create considerable turgor
pressure due to the myosin II-dependent contractile forces in the
actin cortex thereby stiffening the pseudopodium. This case is
discussed next.
In order to understand the effect of turgor pressure one needs to
recognize that the pseudopodium is a poroelastic solid. When a
beam made of such a material bends, the compression of the pores
on one side of the neutral line results in a pressure rise in the
interstitial fluid which then drives a flow across the beam. On
account of viscous resistance, the fluid responds to the bend with a
time lag so that the system behaves much like a coupled mass-
spring-damper system (Skotheim and Mahadevan, 2004).
1 This correction in the formula for the stability threshold is necessary because
the equation for the deformation y(x) of the beam is obtained by minimizing the
energy functional
R L
0 dx½EIðy
00Þ2=2þðrr0ÞgdX where dX is the vertical displace-
ment of a point on the beam at location ‘x’ due to beam curvature and r0 is the
density of the external medium. This differs from the corresponding expression in
the absence of the external medium in that the density difference Dr¼ rr0
replaces the density of the material of the beam, r.

against a strong gravity field limits its ability to crawl against such a
field appear to be supported by the present analysis based on the
mechanics of the buckling of elastic structures. Similar ideas have
long been used in the field of plant biomechanics (Vogel, 2006), but
the centrifuge experiments of Fukui et al. present an opportunity
S. Ghosal, Y. Fukui / Journal of Theoretical Biology 271 (2011) 202–204204However, thismechanismdoes not alter the stability limit since the
bifurcation at the onset of elastic instability takes place at zero
frequency. Internal hydrostatic pressure stiffens the structure by
one of twomechanisms (a) the swelling and consequent stretching
of elastic elements may put it in a regime where the stress strain
relation is no longer linear (b) the swelling may alter geometrical
parameters (specifically, the parameter I in Eq. (3)). The first of
these effects is less likely, though a careful estimate is difficult as
not much information is available on the nonlinear elasticity of
actin gels. However, the second of these effects is readily estimated.
The situation is depicted in Fig. 2. Since an element of the cell
membrane is in equilibrium due to the balance of an outward
pressure ðDpÞ and an inward elastic stress EðdR=RÞ where dR is the
increase in the radius R, we have dR=R¼ ðDpÞ=E. Thus,
1þ dgc
gc
¼ 1þ dR
R
 4
¼ 1þ Dp
E
 4
ð4Þ
since the area moment of inertia of a cylinder is I ¼ pR4=4. In the
experiment (Fukui et al., 2000) the wild type strain (NC4) containing
myosin II did not stall at the maximum acceleration of gc ¼
11:2  104 m=s2, suggesting that dgc=gc41:9. Thus, Dp40:305
Fig. 2. Diagram illustrating the radial expansion of the pseudopodium as a
consequence of the rise in internal turgor pressure, resulting in a ‘‘stiffening’’ of
the structure against elastic buckling.E¼ 275 Pa,using thevalueE¼3G¼900 Pacitedearlier. Pasternaketal.
(1989) report a difference DT  0:13 mdyn per micron in the cortical
tension between the strains AX4 and themyosin lackingmhcAstrain
of Dictyosteliumboth in the resting phase. If one converts this number
to a pressure using the Laplace formula for surface tension, one obtains
the estimate for themyosin generatedpressure:Dp¼ 2DT=R 520 Pa
where R 5 mm is taken as a characteristic radius of the cell. Thus, our
rough estimate Dp4275 Pa is not inconsistent with reported values
for myosin II-dependent pressures in the cell cortex that may be
inferred on the basis of existing experimental data.
Conclusion
In conclusion, the hypothesis advanced by Fukui et al. that the
ability of the Dictyostelium amoeba to support a pseudopodium
Li, S., Guan, J.L., Chien, S., 2005. Biochemistry and biomechanics of cell motility.
Annual Review of Biomedical Engineering 7, 105–150.Oliver, T., Lee, J., Jacobson, K., 1994. Forces exerted by locomoting cells. Seminars in
Cell Biology 5, 139–147 PMID: 7919227.
Pasternak, C., Spudich, J.A., Elson, E.L., 1989. Capping of surface receptors and
concomitant cortical tension are generated by conventionalmyosin.Nature 341,
549–551.
Skotheim, J.M.,Mahadevan, L., 2004. Dynamics of poroelastic filaments. Proceedings
of the Royal Society of London (A) 460, 1995–2020.
Stossel, T.P., 1993. On the crawling of animal cells. Science 260, 1086–1094.
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Experimental Cell Research 8, 260–281.crawling. In a broader context, we illustrate that the interpretation
of experiments inwhich one attempts to quantify the force applied
by a crawling cell on a substrate by measuring an applied
restraining force (Oliver et al., 1994) may be subtle, because, the
limiting factor may not be the ability of the cell to pull against
the applied force but rather a failure in some other aspect of the
motilitymechanismof the cell (e.g. contact inhibitionWeiss, 1961).
Wewould like to thank Howard A. Stone and Joseph B. Keller for
reading and commenting on a draft of the manuscript and
L. Mahadevan for helpful discussions relating to the bending of
poroelastic beams.
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