Dynamics of Traction Stress Field during Cell Division

Hirokazu Tanimoto* and Masaki Sano

Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan(Received 30 May 2012; published 14 December 2012)

We report a quantitative measurement of traction stress exerted by dividing eukaryotic cells. The stress

field was highly dynamic and sequentially changed as follows: (1) strong and localized as two spots,

(2) weak and broadly distributed, and (3) strong and localized as four spots. At the final stage of

cytokinesis, the dividing cells exerted strong tensile force on the intercellular bridge. The asymmetry of

the traction stress and the orientation of the division axis matched throughout the division process,

suggesting the possible role of the mechanical force as a ‘‘store’’ of the orientational information.

DOI: 10.1103/PhysRevLett.109.248110 PACS numbers: 87.17.Ee, 87.15.La, 87.17.Rt

Active forces generated by dividing adherent cells canbe classified into three types: force working on the cell-solution boundary (surface tension), inside of the cell(intracellular stress), and on the cell-substrate boundary(traction stress). While the first two forces have beenreported by many groups for various cell types [1–4],the last one, traction stress, has not been quantitativelydetermined to date [5]. Quantification of such active forceshas illuminated many aspects of the division process, suchas the inherent unstable nature of the cortex of dividingcells [3] and the molecular mechanism of mitotic cellrounding [4]. There are many unsolved biophysical ques-tions possibly related to the traction stress [6,7]. Forexample, it is suggested that mechanical forces would beresponsible for cleavage furrow thinning and abscission[8,9]. Quantitative measurement of the traction stress ofdividing cells thus must be essential to complete ourbiophysical understanding of the biologically importantphenomenon.

In this Letter, we present a quantitative measurement ofthe traction stress exerted by dividing Dictyostelium cells,which is a simple model organism for the study of eukary-otic cell division [2,10–13]. For the stress measurement,we prepared flexible poly-acrylamide gel (Young modulus800 Pa) embedded with fluorescence beads. We observeddividing cells on the gel with confocal microscopy (TSC-SP5, Leica) and detected the in-plane deformation of thesubstrate by tracking the fluorescence beads adopting theparticle image velocimetry method (see the SupplementalMaterial [14] for the detailed protocol).

We calculated traction stress Tið ~xÞ from the substratedeformation uið ~xÞ in Fourier space (subscripts denote acomponent x or y). By assuming linear elasticity of thesubstrate and neglecting vertical force, the relationship

between Fourier transformed deformation ~uið ~kÞ and stress~Tið ~kÞ can be expressed as ~Tið ~kÞ ¼ ~G�1

ij ð ~kÞ~ujð ~kÞ. We use

Einstein’s summation rule to express the summation with

subscripts, and ~Gijð ~kÞ is the Fourier transformed

Boussinesq Green function [15,16],

~Gð ~kÞ ¼ 2ð1þ �ÞEk3

k2 � �k2x ��kxky

��kxky k2 � �k2y

" #: (1)

Because of the ill-posed nature of the problem, we needsome filtering schemes to avoid the amplification ofhigh-frequency components, which mainly come frommeasurement noise [16]. To this end, we introduced asimple low-pass filtering. Comparing the power spectra

j~uið ~kÞj2 of data (strained) and control (unstrained) sub-strates, we found that the strain signal had significant com-ponents above control only for low wave numbers. Hence

we chose the cutoff frequency to be j ~kj ¼ 0:25 �m�1,where the levels of signal and the background noise becamethe same order [17].Eukaryotic cells entering the mitotic phase stop migration

and round up. Then themorphology changes from circular toelliptic followedby the appearance of the cleavage furrow forcytokinesis, and eventually the twodaughter cells completelyseparate. In the case ofDictyostelium cells, the time spent forcompleting cytokinesis from the circular shape is known tobe about 3 min [10,13], whereas the total mitotic phase is

-10 -5 0 5 10

10

5

0

-5

-10-10

-8

-6

-4

-2

0

Tim

e (min)

-8.7 -7.0 -4.9

-1.5 -0.9 -0.1

Pos

ition

y (µ

m)

Position x (µm)

FIG. 1 (color online). Morphological change of a dividingDictyostelium cell. Left: Snapshots of a dividing cell.Transmission images of a dividing Dictyostelium cell are shown.The cleavage furrow appeared around t ¼ �2 min , and t ¼�0:1 min corresponded with the instant just before cleavagefurrow abscission (t ¼ 0 is the instant when the sister cellsseparate). Number indicates the time in minutes. Right:Sequential cellular outlines. The outlines of a dividing cell(the same cell appeared in the left) are drawn.

PRL 109, 248110 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

14 DECEMBER 2012

0031-9007=12=109(24)=248110(5) 248110-1 � 2012 American Physical Society

around 10 min [18]. The observed time-dependent morpho-logical change of dividing Dictyostelium cells in our setup(Fig. 1) was almost similar to what was observed by formerstudies using agar or glass substrate [2,10,13].

The traction stress of dividing cells was found to behighly dynamic in space and time and the dynamics wasin connection with the morphological change (Fig. 2). Att ¼ �10 min (throughout the Letter, the origin of the timeaxis is set to be the instant when cytokinesis completes),the mother cell already stopped migration but was stillmaking some pseudopodia. The traction stress exerted bythis mother cell was rather strong and localized as twopoint forces at the poles of the cell. This character lastedfor about 4 min (Fig. 2, �11 to �7 min ). Aroundt ¼ �6 min , the mother cell ceased making pseudopodiaand rounded up. On this occasion, the two force spots dis-appeared and the traction stress became smaller and distrib-uted almost uniformly on the cell edge (� 6 to �4 min ).Then the cleavage furrow appeared around t ¼ �3 minand started up thinning (� 3 to �1 min ). The tractionstress got stronger again and finally, after around t ¼�2 min, localized as four point forces at the sides ofboth edges of the cleavage furrow. After completing cyto-kinesis, the traction stress became significantly smaller(0 min). The traction stress was all directed inward, aswe will confirm below. The above features were commonin other observed samples.

Based on the observations, we defined three phases ofthe traction stress dynamics during cell division. Phase 0(� 11 to �7 min , green bar in Fig. 2) is characterized bythe strong traction stress with two point forces and cellmaking pseudopodia. Phase 1 (� 6 to�4 min , blue bar) ischaracterized by the weak and broad stress and the roundedmorphology. Phase 2, the last phase (� 3 to 0 min, red bar)is characterized by the strong stress with four point forcesand the visible cleavage furrow.

We then quantified the spatiotemporal dynamics of thetraction stress field (Fig. 3). The total force exerted bydividing cells was characterized by the integration of theabsolute value of the traction stress (Fig. 3, top section),Z

S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2x þ T2

y

qdS; (2)

where S is the area of interest (typically, we chose an area30 �m� 30 �m to contain the whole cell). The total forcefirst increased from the beginning and took its maximalat �8:4� 2:5 min (N ¼ 5 cells, see the SupplementalMaterial [14] for statistical discussion), which was in themiddle of phase 0. Then the total force decreased and startedincreasing again at �4:9� 1:9 min , at the beginning ofphase 1. The total force kept increasing until the end ofcytokinesis and steeply decreasedwhen the cleavage furrowbroke (the rightmost broken line). The maximum force justbefore the furrow break was 23:4� 1:2 nN, and this wassignificantly larger compared to that of ordinary migratingDictyostelium cells (average 10 nN [17,19]). The sharp dropof the total force at the furrow breaking was around 6:5�1:9 nN, which was suddenly released between the twosuccessive measurement frames (6 sec). We separated thetraction stress into parallel and vertical components withrespect to the division axis (Fig. 3, second section) andfound that this rapid decreasemainly came from the parallelcomponent. The released parallel force was 6:0� 2:4 nN.To investigate the contracting nature of the traction

stress, we detected the center of the traction stress(‘‘sink’’ of the stress vector field). The force center wasdefined using the following integral [20],

N ¼ 1

2�

Ic

~r�ðx; yÞ � d~l; (3)

where �ðx; yÞ is the orientation of the stress vector. Wesearched the point where the above integration had a value

30

Stress (P

a)150

-11 min -10 -9 -8 -7 -6

120

90

60

-5 min -4 -3 -2 -1 0 (divided)

FIG. 2 (color online). Traction stress during cell division. The traction stress exerted by a dividing cell was measured in a timeinterval of 6 sec. The stress vectors (white arrow), their amplitudes (color), and the cellular outline (white line) were plotted for every1 min. Upper color bar indicates three phases of the dividing process (see main text): phase 0 (pro- and metaphase [11], green bar),phase 1 (meta- and anaphase, blue bar), and phase 2 (ana- and telophase, red bar). Bar: 10 �m.

PRL 109, 248110 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

14 DECEMBER 2012

248110-2

of 1 and found that there existed almost only one forcecenter throughout the division process (Fig. 4). Theforce center was located near the cell center, and the forcebalance condition implies that cell feels the counter forceof the traction stress. The result thus indicates that thedividing cells apply strong tensile force to the intercellularbridge in a tug-of-war manner.

We then calculated the force dipole (or the first-ordermoment) of the stress field Mij, which is another measure

of the traction stress including spatial property,

Mij ¼ZSxiTjdS: (4)

Note that since the Nth moment is the Nth derivative of

Fourier components around ~k ¼ ~0, the first-order momentcan be determined only with the lowest Fourier compo-nents and is consequently robust against the details of thefiltering. The matrix of the calculated first-order momentwas diagonalized and its eigenvalues and eigenvectorswere obtained. We will call the eigenvector with the larger(smaller) absolute eigenvalue the major (minor) dipole.

Both the major and minor dipoles were almost alwaysnegative, reflecting the contracting nature of the tractionstress (Fig. 3, third section). The two dipoles behaved inthe same manner as the total force in phases 0 and 1 andboth the major (brown) and the minor (yellow) dipoles hadtheir first peak in the middle of phase 0 and then increasedduring phase 1. The subsequent time evolutions of the two

dipoles were different. While the minor dipole stoppedincreasing at the beginning of phase 2, the major dipolekept increasing until the end of cytokinesis. The twodipoles significantly differed throughout the process.Since the difference between the two dipoles correspondsto the degree of rotational asymmetry of the stress field,this result shows that the traction stress is kept asymmetricfrom the beginning to the end of the dividing process.Finally, we investigated the relationship between the

traction stress asymmetry and the division axis orientation.Themain axis of the force dipole matrix was rather constantthroughout the division process (Fig. 3, bottom section) andwe here found that this orientation agreed well with theorientation of the division axis. We calculated the probabil-ity distribution function (PDF) of the angle formed by thedipole and division axes [Fig. 5(a)]. The resulting PDFhad asharp peak around �� ¼ 0, showing that the two axesagreed well. The orientations of the two axes for differentdividing cells were plotted as a function of the orientation ofthe other axis [Fig. 5(b)], and all data collapsed into a singleline confirming the tight orientational correlation.Our results showed that the integrated absolute value of

the traction stress exerted by dividing cells was around23 nN, which was significantly large compared to cells inthe interphase. The traction stress was directed inwardthroughout the mitotic phase, and there existed almostonly one force center. It is supposed, although havingfew even indirect evidences, that two daughter cells indi-vidually reestablish each contracting traction field beforethey completely separate [21]. Our observation was differ-ent from this accepted scenario and showed that twodaughter cells applied contracting force to the intercellularbridge in a tug-of-war manner. We estimated the tensileforce as the released traction stress parallel to the divisionaxis during furrow abscission. The amplitude of the tensileforce was unexpectedly large, around 6 nN.The mechanism of abscission of the cleavage furrow

occurring at the last step of cytokinesis is largely unknown,and whether cytoskeleton or membrane persists longer iseven unrevealing [8]. Themembrane alone being insufficient

(a)

-10 -8 -6 -4 -2 00

1

2

3

Time (min)

Num

ber

of d

efec

ts

(b)

0

10020Stress (Pa)

Orientation(rad)

π

-π

FIG. 4 (color online). Tug-of-war behavior. (a) Phase of trac-tion stress vectors. The center of the stress field was detectedusing Eq. (3). There was one single defect in this phase space(black circle). Bar: 10 �m. (b) Number of force center. Thenumber was almost kept at one until completing cytokinesis.After separating, each of the two daughter cells has one forcecenter for each; thus, the total number was two.

-120

-80

-40

0

0

10

20

0

10

20

-0.5

0

0.5

Abs

olut

e va

lue

(nN

)

Absolute value (nN

)

Dip

ole

ampl

itude

(fJ

)

Dipole orientation (rad/π )

Time (min)-10 -8 -6 -4 -2 0

-10 -8 -6 -4 -2 0Time (min)

FIG. 3 (color online). Quantification of traction stress dynam-ics. The total absolute force [(blue), colors noted from top tobottom], the absolute forces parallel (red) and perpendicular(green) to the division axis, the amplitudes of the force dipole(brown, major; yellow, minor), and the orientation of the dipoleaxis [(light blue), dotted line corresponds to a continuous curve;see the Supplemental Material [14]] are plotted as a function oftime. Color bar indicates the three dividing phases.

PRL 109, 248110 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

14 DECEMBER 2012

248110-3

to resist the measured nN order tension [22], our observationsuggests that there must exist cytoskeletal structures at thefinal instant of cytokinesis. Recent reports claim that, unlikemammalian and yeast cells, there are no circumferentialoriented actin filaments at the equator of dividingDictyostelium cells and, instead, a meshwork actin networkis observed [2]. The actin meshwork investigated with SEM[2] was composed of around 30 actin filaments parallel to thedivision axis.Assuming that the tensile force is applied on thesides of this filamentous actin (F-actin) network, the tensileforce applied to a single actin filament can be estimated to beabout 100 pN. The breaking force of a single actin filamentin vitro has been reported as 200–400 pN, which is the sameorder as the estimated tensile force [23]. We thus concludethat the applied tensile force may possibly break the actinfilaments at the equator in a purely mechanical way.

We found that the force dipole axis agreed well with thedivision axis and that this coincidence held even beforecell-shape rounding. The duplication of the centrosomeoccurs at the beginning of the mitotic phase. The dupli-cated centrosomes can still rotatory move, and the spindleelongation starts about 5 min before the end of cell division[24,25]. Therefore, the force dipole would be an earlierfactor breaking circular symmetry than the mitotic spindle.

Recent studies insist that the force exerted by dividingcells to their substrate would determine the division axis

through positioning the mitotic spindle [26,27] and, basedon this hypothesis, a theoretical model has been proposed[26]. Inspired by these works, we try to understand ourresult with a simple model. We start from assuming that thepulling force acts on the nearest side of the spindle(reflected by the integral range for torque calculation)and the spindle can only rotate [26].Let � be the orientation of the imaginary spindle

(� �=2<�<�=2) [Fig. 5(c)]. The force acting on thespindle toward the direction ~e� will be

~Fð�;�Þ ¼ ~e�Z R

0j ~Tðs; �; �Þjds; (5)

where s is the polar amplitude from the spindle sides alongthe unit vector ~e�, R is the size of the region of interest

[S in Eqs. (2) and (4)], and ~Tðs;�; �Þ � ~TðL cos�þs cos�; L sin�þ s sin�Þ. Here, we adopt a crude approxi-mation; the traction stress amplitude as a measure ofadhesion strength which is eventually proportional to thepulling force on the spindle by activating membrane cuespositioning astral microtubules [Fig. 5(c)]. The spindle

feels the net torque Nð�Þ ¼ R�þ�=2���=2

~rð�Þ � ~Fð�;�Þjzd�,where ~rð�Þ is the position of the spindle edge. Finally, theeffective free energy Wð�Þ of the system is obtained byintegrating the torque Nð�Þ from ��=2 to �; Wð�Þ ¼�R�

��=2 Nð�0Þd�0.We calculated Wð�Þ of each of the data frames

[Fig. 5(d)]. Most of the calculated potentials have onlyone global minimum and the others have a global and oneother local minimum. We found that the global minimumand the orientation of the real division axis agreed well.This result gives evidence that traction stress would play arole in division axis determination.In summary, we report a quantitative measurement of

the traction stress exerted by dividing eukaryotic cells, toour knowledge, for the first time. Dividing cells were foundto exert much larger force compared to ordinary migratingcells, and a major part of this was the tensile force pullingthe intercellular bridge. The observed orientational rela-tionship between the traction stress asymmetry and thedivision axis was successfully reproduced by the intro-duced simple model. We expect that these results wouldmotivate further studies elucidating biological functions ofmechanical forces in cell division.We are grateful to Dr. Jean-Paul Rieu and his colleagues

for generous sharing of their skills on traction stressmeasurement.

*Present address: Institut Curie, UMR 144 CNRS/IC, 26rue d’Ulm, 75248 Paris Cedex 05, France.thirokazu@daisy.phys.s.u-tokyo.ac.jp

[1] Y. Hiramoto, Exp. Cell Res. 32, 59 (1963).

F(θ,φ)

0 0.50

0.1

0.2

0.3

0.4

0.5

0 0.5

00.10.20.30.40.5

Pro

babi

lity

(a) (b)

(d)(c)

-0.5Angle difference (rad/π) Division axis orientation (rad/π)

For

ce d

ipol

e or

ient

atio

n (r

ad/π

)

-0.1

-0.5-0.4-0.3-0.2

-0.5

θφ

0 0.5

0

50

100

-0.5

-50

-350

-250

-150

-300

-200

-100

Spindle orientation (rad/ π)

Effe

ctiv

e po

tent

ial

-10

-8

-6

-4

-2

0

Tim

e (min)

|T|

FIG. 5 (color online). Orientational relationship between trac-tion stress asymmetry and division axis. (a) PDF of angledifference. The PDF of the angle formed by the dipole and thedivision axes is shown. N ¼ 5 cells. (b) Orientational relation-ship between dipole and division axes. A scatter plot of theorientations of the dipole and the division axes is shown.(c) Model. The traction stress j ~Tj is assumed to apply pullingforce ~F to an imaginary spindle placed at the center of a dividingcell. (d) Effective potential of the system. The calculated effec-tive potential Wð�Þ is plotted as a function of the spindleorientation �. Broken line indicates the real division axis ori-entation. We set the spindle length 2L ¼ 6 �m.

PRL 109, 248110 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

14 DECEMBER 2012

248110-4

[2] E.M. Reichl, Y. Ren, M.K. Morphew, M. Delannoy, J. C.Effler, K.D. Girard, S. Divi, P. A. Iglesias, S. C. Kuo, andD.N. Robinson, Curr. Biol. 18, 471 (2008).

[3] J. Sedzinski, M. Biro, A. Oswald, J.-Y. Tinevez, G.Salbreux, and E. Paluch, Nature (London) 476, 462 (2011).

[4] M. P. Stewart, J. Helenius, Y. Toyoda, S. P. Ramanathan,D. J. Muller, and A.A. Hyman, Nature (London) 469, 226(2011).

[5] K. Burton and D. L. Taylor, Nature (London) 385, 450(1997).

[6] E. Hannezo, J. Prost, and J.-F. Joanny, Phys. Rev. Lett.107, 078104 (2011).

[7] P. Katira, M.H. Zaman, and R. T. Bonnecaze, Phys. Rev.Lett. 108, 028103 (2012).

[8] P. Steigemann and D.W. Gerlich, Trends Cell Biol. 19,606 (2009).

[9] C. C. Poirier, W. P. Ng, D.N. Robinson, and P.A. Iglesias,PLoS Comput. Biol. 8, e1002467 (2012).

[10] J. H. Zang, G. Cavet, J. H. Sabry, P. Wagner, S. L. Moores,and J. A. Spudich, Mol. Biol. Cell 8, 2617 (1997).

[11] S. Yumura and Y. Fukui, J. Cell Sci. 111, 2097 (1998).[12] A. Nagasaki, G. Itoh, S. Yumura, and T. Q. P. Uyeda, Mol.

Biol. Cell 13, 4333 (2002).[13] W. Zhang and D.N. Robinson, Proc. Natl. Acad. Sci.

U.S.A. 102, 7186 (2005).[14] See Supplemental Material at http://link.aps.org/

supplemental/10.1103/PhysRevLett.109.248110 for de-tailed protocol and additional discussion.

[15] J. P. Butler, I.M. Tolic-Norrelykke, B. Fabry, andJ. J. Fredberg, Am. J. Physiol. Cell Physiol. 282, C595(2002).

[16] B. Sabass, M. L. Gardel, C.M. Waterman, and U. S.Schwarz, Biophys. J. 94, 207 (2008).

[17] H. Tanimoto and M. Sano (to be published).[18] Y. Maeda, Int. Rev. Cytol. 244, 287 (2005).[19] H. Delanoe-Ayari, J. P. Rieu, and M. Sano, Phys. Rev. Lett.

105, 248103 (2010).[20] P.M. Chaikin and T. C. Lubensky, Principles of

Condensed Matter Physics (Cambridge University Press,New York, 1995).

[21] M. Thery and M. Bornens, Curr. Opin. Cell Biol. 18, 648(2006).

[22] D. K. Fygenson, J. F. Marko, and A. Libchaber, Phys. Rev.Lett. 79, 4497 (1997).

[23] Y. Tsuda, H. Yasutake, A. Ishijima, and T. Yanagida, Proc.Natl. Acad. Sci. U.S.A. 93, 12 937 (1996).

[24] M. Ueda, M. Schliwa, and U. Euteneuer, Mol. Biol. Cell10, 151 (1999).

[25] A. Rump, T. Scholz, C. Thiel, F. K. Hartmann, P. Uta,M.H. Hinrichs, M.H. Taft, and G. Tsiavaliaris, J. Cell Sci.124, 2521 (2011).

[26] M. Thery, A. Jimenez-Dalmaroni, V. Racine, M. Bornens,and F. Julicher, Nature (London) 447, 493 (2007).

[27] J. Fink, N. Carpi, T. Betz, A. Betard, M. Chebah, A.Azioune, M. Bornens, C. Sykes, L. Fetler, D. Cuvelier,and M. Piel, Nat. Cell Biol. 13, 771 (2011).

PRL 109, 248110 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

14 DECEMBER 2012

248110-5