Nonlinear Study of Symmetry Breaking in Actin Gels: Implications for Cellular Motility

Karin John,1,* Philippe Peyla,1 Klaus Kassner,2 Jacques Prost,3 and Chaouqi Misbah1,†

1LSP, Universite J. Fourier and CNRS, BP 87, 38402 Grenoble Cedex, France2Institut fur Theoretische Physik, Otto-von-Guericke-Universitat Magdeburg, PF 4120, 39016 Magdeburg, Germany

3Physico-Chimie, UMR168 Institut Curie, 26 rue d’Ulm, 75248 Paris Cedex 05, France(Received 1 July 2007; published 11 February 2008)

Force generation by actin polymerization is an important step in cellular motility and can induce themotion of organelles or bacteria, which move inside their host cells by trailing an actin tail behind.Biomimetic experiments on beads and droplets have identified the biochemical ingredients to induce thismotion, which requires a spontaneous symmetry breaking in the absence of external fields. We find thatthe symmetry breaking can be captured on the basis of elasticity theory and linear flux-force relationships.Furthermore, we develop a phase-field approach to study the fully nonlinear regime and show that actin-comet formation is a robust feature, triggered by growth and mechanical stresses. We discuss theimplications of symmetry breaking for self-propulsion.

DOI: 10.1103/PhysRevLett.100.068101 PACS numbers: 87.17.Jj, 82.35.Pq, 82.39.�k, 87.15.rp

Introduction.—In eukaryotic cells forces can be gener-ated by polymerizing actin monomers into filaments,which form an elastic cross-linked gel. These forces leadto the extension of membrane protrusions in eukaryoticcells, a necessary prerequisite for motility of, e.g., amoebaand cells of the immune system. It is admitted that besidesactin polymerization the motility is assisted by molecularmotors, which bind to actin filaments, and convert ATP(adenosine triphosphate) into mechanical work. However,some bacteria, e.g., Listeria monocytogenes [1], move in-side their host cells by using the actin polymerizationmachinery of the host, without the assistance of molecularmotors. This discovery pointed to the important nontrivialfact that motors are not necessary to induce motion.

Actin polymerization and cross-linking is triggered byan enzyme (ActA or Wasp activates the so called Arp2=3complex which nucleates new actin filaments on preexist-ing ones) on the external side of the bacterial membraneand leads initially to the growth of a symmetric gel aroundthe bacterium. Later, this actin shell undergoes a symme-try breaking and develops into a comet that propels thebacteria forward in their host cell.

Our understanding of the underlying mechanisms hasbeen greatly enhanced by model experiments under con-trolled conditions [1–6]. For example, artificial beadscoated homogeneously with the ActA enzyme and incu-bated in a solution containing actin monomers (and otherspecies, such as ATP, Arp2=3, and so-called capping pro-teins) induce the growth of an actin gel at the bead surface.Initially this gel grows symmetrically until a parity sym-metry breaking is observed and the bead moves forward bytrailing an actin comet behind. Symmetry breaking mayeither occur after the gel has reached a steady state thick-ness or, in the case of unlimited gel growth, it overrides thegrowing gel.

Previous theoretical works [7–12] have identified somekey ingredients, necessary for the symmetry breaking of an

actin gel around a bead. In Brownian ratchet modelssymmetry breaking is governed by stochastic effects inthe filament elongation process [7,8] whereas in elasticmodels symmetry breaking is initiated by mechanicalstresses either through fracture [9,12] or stress dependentdepolymerization [10]. In the latter case the stress distri-bution in the gel and the coupling mechanism betweenelastic stresses and growth is not clear.

Furthermore, a full nonlinear analysis of a basic modelto ascertain the longtime behavior and to relate the non-linear evolution to the force generation, is lacking.

In this Letter we find that a linear elasticity theory issufficient to capture the essence of the symmetry-breakingbifurcation, where the gel layer ahead of the bead ‘‘breaks’’by a process of polymerization or depolymerization, whichdoes not require a fracture mechanism. Furthermore, westudy the nonlinear evolution of the gel envelope aftersymmetry breaking, which leads to the formation of anactin comet, as observed experimentally.

The sharp interface model.—We consider a bead (radiusr1) surrounded by a growing elastic actin gel (radius r2)shown in Fig. 1. For simplicity, and in order to identify thebasic ingredients which are necessary to account for themain phenomenology, the gel is treated within linear, iso-tropic elasticity, where the stresses �ij are related to thestrains uij by Hooke’s law

�ij � 2�uij � �ukk�ij: (1)

� and � denote the Lame coefficients, which are related toYoung’s modulus E and the Poisson ration � by 2� �E=�1� �� and � � E�=�1� ���1� 2��. �ij are thecomponents of the Cauchy stress tensor �. At each growthstate the gel is in mechanical equilibrium, determined by

div ��� � 0: (2)

The shear vanishes at the interfaces (�nt � 0) and the

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hydrostatic pressure at the gel-solution boundary is fixed(p � ��nn � 0). The gel is stressed by a small moleculardisplacement in normal direction ur � L at the bead-gelinterface. This choice is motivated by the microscopicpicture, that for the addition of monomers, enzymes facili-tate a molecular displacement in the gel. The evolution ofthe external gel interface is given by

@tr2 � �M1��1 �M2��2; (3)

where M1 (M2) and ��1 (��2) denote a mobility and thedifference in the chemical potential between a volumeelement in the gel and in solution at the internal (external)interface, respectively. Here we assume that the mobility isassociated with the polymerization/depolymerization ki-netics, which constitutes the prevailing dissipation mecha-nism. Thus the velocity of the gel-solution interfacedepends on a local property ��2 and a nonlocal property��1; i.e., the insertion of mass at the bead-gel interfaceleads to a translocation of the gel-solution interface. Toderive ��i (i � 1, 2) we calculate the change in the Gibbsfree energy upon transporting a small volume element �V0

from the solution into the gel at the interfaces

�G � �F� ��p�V� � ���V0: (4)

�F � �Fe ��Fp � �Fs contains several contributions,the work necessary to increase the elastic strain of thevolume element (�Fe), the change in free energy due tochemical bond formation by polymerization (�Fp ���p�V0), and the work necessary to increase the gelsurface (�Fs � ���V0). ��p is constant (the solutionconstitutes an infinite reservoir for monomers) and inde-pendent of the gel thickness. Indeed, the gel represents aporous structure, where monomers can diffuse freely [13].Furthermore, polymerization (depolymerization) occurs atthe internal (external) interface, i.e., ��p;1 < 0 and��p;2 > 0. In the expression for Fs, �, and � denote thesurface tension and curvature of the gel at the interface,respectively.

The elastic contribution �Ge is given in its differentialform by

dGe � dFe � d�p�V�; (5)

where p denotes the isotropic part of the stress tensor.Using the relations �V � �V0�1� ukk� and p � ��nnand integrating up to finite strains one obtains �Ge ���e�V0, where

��e � �uijuij ��2u2kk � �nn�1� ukk�: (6)

In the following we will restrict ourselves to a gel growingfrom the internal interface, i.e., M2 � 0 with M � M1 and�� � ��1 and neglect the surface tension.

To solve Eqs. (2) and (3) we use spherical coordinateswith the displacement given by ~u � ur ~er � u� ~e� � u ~e.First we consider the completely symmetric situation for aspherical gel with u�0�� � u�0� � 0. The solution to Eq. (2)

has the form u�0�r � ar� b=r2, where a and b are integra-tion constants [14]. For small � L=r1 � 10�3 [9],growth is dominated by ��p and the normal stress ��0�nnat the bead-gel interface and

dr2

dt� �M

���p �

2E�r32 � r

31�

2�1� 2��r31 � �1� ��r

32

�: (7)

dr2=dt is monotonically decreasing with r2 and �� satu-rates at ��p � 2E=�1� �� as r2 ! 1. Thus, this ki-netic equation has either zero or one linearly stable fixedpoint. (i) For 2E=�1� ��>���p elastic forces actsagainst addition of monomers at the internal interface, untila dynamic equilibrium is reached, given by

�r 2 �

�2E� �1� 2����p

2E� �1� ����p

��1=3�

r1: (8)

(ii) For 2E=�1� �� � ���p elasticity fails to actagainst monomer addition, so that the gel grows withoutbound. Both cases have been observed experimentally[13]. In case (ii) the gel growth ultimately stops due to adepletion of monomers.

The linear stability of the homogeneous steady state �r2 isperformed on a series of spherical harmonics with a smallamplitude ". ~u�1� denotes the perturbation proportional tovector spherical harmonics. Then expanding Eq. (3) up toO�"� and using " � "0e!t one finds the dispersion relation! � �M� ~��1�, where ���1� � " �r2Y

ml � ~��1�. For illus-

trating purposes we split the growth rate ! � !�0� �!�1�

into two contributions: the first originating from the linearterm ��1�nn and the second from the quadratic terms inEq. (6).!�0� and !�1� are depicted in Fig. 2. For the growth rate

!�0� we find that �r2 is stable against all perturbationsmodes, except for the mode l � 1, which constitutes aneutral mode (!�0��l�1� � 0). For the contribution from!�1� we find that a homogeneous perturbations as well as

=0σnt

r1

r2

ur

∆µ2∆µ1

bead

gel ν, λ

p=0

solution

FIG. 1 (color online). Schematic view of a bead surrounded byan elastic gel with the Lame coefficients � and �.

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higher order modes are stabilized. In contrast, for 0< l <lc we find a band of unstable modes, with the maximum atl � 1. Since j!�0�j � j!�1�j for l � 1 only the mode l � 1is unstable. Thus we expect that the symmetry breaking isinitiated by the translocation of mass from one side of thegel to the other. The instability is also present at theexternal interface, with the mode l � 1 being the mostunstable mode (data not shown).

Nonlinear evolution.—For the determination of the longtime behavior of the gel shape after symmetry breaking anonlinear analysis is necessary. While the elastic equationsare linear, the problem is highly nonlinear due to the freeboundary character. We study the nonlinear regime byintroducing a two-dimensional phase-field model. Thecrux of the phase-field approach is to introduce a continu-ous (albeit abruptly changing at the interface location)function ��r; ��, where � � 1 corresponds to the geland � � 0 corresponds to the solution. The gel-solutionboundary at r2��� is described by a diffuse interface withthickness �. The center of the bead denotes the origin of apolar coordinate system with radius r and angle �. Thephase field is bounded at (i) r � r1 with the boundaryconditions � � 1, ur � L � 0, and �r� � 0 and at(ii) r � r3 > r2��� (the total computational domain) withthe boundary conditions @n� � 0 and zero hydrostaticpressure; (i) is equivalent to the sharp bead-gel interfaceand (ii) denotes a liquid boundary, far away from thegrowing gel. We write the free energy of the phase field as

F� �ZdA�fb��� �

1

2��2�r��2

�; (9)

where the bulk energy takes the form fb � 2�g��� withg � �2�1���2 and � denotes the surface energy density.The mechanical equilibrium condition is extended to thewhole phase field by writing div��� � 0, where � denotesa stress tensor with the components

�ij � 2h�uij � h�� �1� h��sukk�ij: (10)

�s describes the properties of the shear free (�s � 0),incompressible solution (hydrodynamic flow is disre-garded). The function h is chosen as in [15] h � h��� �

�2�3� 2��. The evolution of the phase field (gel� liquidsolution) encodes the dynamics of the gel-liquid interfaceautomatically and is given by

@t� � �M�

��F��� h0���r1; ��

�; (11)

where the functional (or Frechet) derivative of F is givenby

�F��� ���2

�r2��

2

�2 g0 �

2

����gp��: (12)

� � �r � ~n is the curvature of �, and ~n � r�=jr�j isthe normal. The prime denotes a derivative with respectto �. The last term in Eq. (12) has been added to avoidany spurious surface tension effects inherent in the phase-field formulation [16]. The last term in Eq. (11) accountsfor the nonlocal growth of the gel, where ���r1; �� ���e � ��p with ��e given by Eq. (6) and is calculateddirectly at the bead-gel interface.

The model is efficiently solved by using a finite elementsscheme [17]. Figure 3 shows the result of a phase-fieldcalculation (plane strain). The initial condition corre-sponds to a thin gel with small amplitude perturbations.The gel grows initially with a symmetrical shape until itreaches its equilibrium thickness. Then the gel growthstalls and all initial perturbations decay except for themode l � 1 perturbation, which leads subsequently to anasymmetrical gel evolution. A comet forms that growsindefinitely until it reaches the boundary of the computa-tional domain. We are currently studying dynamics inlarger box sizes with the aim to investigate the possibilityof a larger comet formation.

In experiments [9] the gel thickness decreases dramati-cally at the front of the bead, conveying the strong impres-

FIG. 3 (color online). (a) Symmetry breaking of a circular gel.Shown is the evolution of the gel thickness, starting from ahomogeneous thin gel with random small amplitude pertur-bations. The simulation time is indicated in the right topcorner of each snapshot. (b) Shows the shape of the comet inthe far nonlinear regime with a small surface tension addedto avoid numerical instabilities. Parameters are r1 � 1:0, � �0:04, ��p=E � �0:000 58, � � 0:4, � 0:003, and for(b) E=� � 103. The length scale is r0 � 1 �m and the timescale is r2

0=�M�E�2�.

0 5 10 15 20mode l

-0.03

0

0.03

0.06

0.09

ω

ω(0)

ω(1)/η

FIG. 2 (color online). Scaled growth rates �!�0� and �!�1� de-pending on the perturbation mode. The parameters are �0:003, � � 0:4, and j��pj=E � 8:9� 10�4, giving rise to asteady state �r2=r1 � 1:1. The time scale is given by r1=�ME�.

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sion of the fracture of the gel. Here, however, only poly-merization or depolymerization is sufficient to induce thiseffect, without reference to bond breaking as true fracturewould dictate. Furthermore we find that the gel is undertension along the tangential and under compression in theradial direction. In general, stresses are mainly concen-trated at the bead-gel interface (not shown).

Discussion.—Our results indicate that the growth of asymmetrical gel is mainly counteracted by the normalstress into the radial direction of the gel at the internalinterface. All other contributions from the elastic energyare of higher order. This result holds for the case that themonomers in solution constitute an infinite reservoir andthat growth is not limited by a depletion of actin monomersin the solution.

A linear stability analysis reveals that perturbations withthe mode l � 1 are unstable, which seems to be a generalfeature, also found in more generic models on symme-try breaking in spherical geometries [11]. This parity sym-metry breaking could be at the origin of the formation of anactin comet and the subsequent movement of the bead.

Our work shows that the polymerization at the bead-gelinterface is sufficient to trigger an instability, and that astress dependent depolymerization at the gel-solution in-terface is not mandatory. Conversely, the stress dependentdepolymerization can provide a destabilization mecha-nism, and further work is required to pinpoint the exactorigin of the instability. In addition, in a more biologi-cal context, other factors can contribute to a symme-try breaking, e.g., nonlinear biochemical reactions, diffu-sion, or a heterogeneity in the environment. However, webelieve that the coupling between growth and mechanicalproperties of the actin gel is a crucial step in actin basedmotility.

Our results are in accordance with the experimentalobservation that symmetry breaking occurs only if cappingproteins are present in a sufficient concentration [9], thatblock polymerization/depolymerization from free barbedpolymer ends, thus restricting the polymerization reactionto the interfaces.

From experiments we can estimate time and lengthscales associated with symmetry breaking. Under physi-ological conditions the kinetic rate constant for thepolymerization of actin at the barbed end is kp �11:6 �M�1 s�1 [13] (and references therein). With amonomeric actin concentration of 0:4 �M and an increasein filament length of L � 2:75 nm per added monomer wefind a growth velocity of v � M��p � 12:76 nm s�1 inthe unstressed state. Using the parameters given in Fig. 2with r1 � 1 �m we find a time scale t0 � r1j�j�pj=�vE� � 0:1 s resulting in a characteristic time for thegrowth of the mode l � 1 of � � 300 s, which falls intothe range reported experimentally [9].

The actin-comet formation leads to the evolution of a netnormal force. By introducing a linear relationship betweenthe force and velocity of the bead, the bead could move

forward in a direction opposite to the comet location. Thisrequires identifying the proper dissipation mechanisms.Preliminary estimates indicate that the bead-gel frictionis the main source of dissipation. This question is currentlyunder investigation. Another open question is the influenceof prestresses on the stability problem, whereby the defi-nition of a meaningful prestress will have to include thehistory of the gel growth [18]. Finally, while the assump-tion of an elastic body has proven to capture some essentialfeatures, it must be stressed that a more realistic model ofthe cross-linked actin gel [19] should be investigated in thefuture in order to elucidate more deeply the basic aspects.

We thank D. Caillerie, E. Bonnetier, D. Jamet, andM. Ismail for fruitful discussions. K. J. and C. M. acknowl-edge financial support by the CNES and the Alexandervon Humboldt Foundation.

*kjohn@spectro.ujf-grenoble.fr†cmisbah@spectro.ujf-grenoble.fr

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