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Early-time dynamics of actomyosin polarization in cells of confined shape in elastic matrices† Noam Nisenholz,a Mordechai Bottonb and Assaf Zemel*a The cell shape and the rigidity of the extracellular matrix have been shown to play an important role in the regulation of cytoskeleton structure and force generation. Elastic stresses that develop by actomyosin contraction feedback on myosin activity and govern the anisotropic polarization of stress fibers in the cell. We theoretically study the consequences that the cell shape and matrix rigidity may have on the dynamics and steady state polarization of actomyosin forces in the cell. Actomyosin forces are assumed to polarize in accordance with the stresses that develop in the cytoskeleton. The theory examines this selfpolarization process as a relaxation response determined by two distinct susceptibility factors and two characteristic times. These reveal two canonical polarization responses to local variations in the elastic stress: an isotropic response, in which actomyosin dipolar stress isotropically changes in magnitude, and an orientational response, in which actomyosin forces orient with no net change in magnitude. Actual polarization may show up as a superimposition of the two mechanisms yielding different phases in the polarization response as observed experimentally. The cell shape and elastic moduli of the surroundings

Received 26th September 2013 Accepted 17th January 2014

are shown to govern both the dynamics of the process as well as the steady-state. We predict that in the steady-state, beyond a critical matrix rigidity, spherical cells exert maximal force, and below that rigidity,

DOI: 10.1039/c3sm52524d

elongated or flattened cells exert more force. Similar behaviors are reflected in the rate of the polarization

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process. The theory is also applicable to study the elastic response of whole cell aggregates in a gel.

1

Introduction

The cell shape and the rigidity of the extracellular matrix (ECM) have been recognized as prominent mechanical factors in the determination of cell behavior and fate; they have also been shown to govern the development of cytoskeleton structure, strength of cell adhesion, and the pattern and magnitude of forces exerted by cells into their surroundings.1–3 Elastic stresses that develop in the cell during cell adhesion provide a means of internally transmitting global mechanical and geometrical information from the environment into the cell to alter cell behavior and gene expression regulation. Force-sensitive elements in the cell membrane, adhesion complexes, cytoskeleton and nucleus respond to these stresses – either mechanically and/or by eliciting biochemical signals. The mechanical response of the cytoskeleton to variations of cellular stress governs cell morphology and internal structure.2,4 The actomyosin cytoskeleton is the major force producing network in the cell.3 The forces produced in the cytoskeleton are exerted into the environment via specialized protein complexes a

Institute of Dental Sciences and the Fritz Haber Center for Molecular Dynamics, The Hebrew University-Hadassah Medical Center, Jerusalem, 91120, Israel. E-mail: assaf. [email protected]

b

Racah Institute of Physics, The Hebrew University of Jerusalem, 91904, Israel

† Electronic supplementary 10.1039/c3sm52524d

information

(ESI)

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available.

See

DOI:

called focal adhesions that anchor the cell to the ECM.5 In sufficiently rigid environments, the actomyosin cytoskeleton changes upon cell adhesion from an initially isotropic network, to a more organized and polarized structure with prominent stress ber bundles lling the cell volume.6,7 Stress ber formation occurs on a time scale of tens of minutes and prominent bers appear when sufficient tension develops in the cell.6,7 The matrix rigidity and cell shape govern stress ber and focal adhesion alignment in the cell.7,8 The more rigid the environment, the more prominent and numerous are the stress bers and focal adhesions in the cell;9–11 in so environments, stress bers and focal adhesions oen hardly form. The formation of stress bers in the cell is accompanied by a rise in cellular tension.6,11 These observations, along with a variety of other experimental indications and theoretical investigations, showed that actin bundling and stress ber formation are dependent on the build-up of tension in the cytoskeleton, and that (in the absence of applied elds) they locally develop in parallel to the principal stress direction.2 Because the matrix rigidity and cell shape dictate the magnitude and spatial distribution of elastic stresses in the cell, they were suggested to direct the formation and alignment of stress bers and focal adhesions in the cell in this manner.12–15 A notable manifestation of the coupling between the cell shape and stress ber alignment is the commonly observed development of stress bers in parallel to the long axis of the cell;13,16–19 the more

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elongated the cell is, the stronger is the alignment.13,20 Matrix rigidity modulates the aspect ratio of cells as well as the orientational order of the stress bers. Human mesenchymal stem cells were shown to elongate best and exhibit maximal alignment of stress bers when plated on substrates of intermediate rigidity;13,20 this optimal alignment was found even for cells of equal aspect ratio on the differently rigid substrates. Similar behavior has been reported with cardiomyocytes.18 In contrast, a recent study on broblasts has shown that cell elongation and actomyosin polarization increase monotonically with matrix rigidity.7 How elastic stresses orchestrate the formation dynamics of stress bers during cell adhesion still remains largely unknown. Recent studies combined high resolution imaging techniques and traction force microscopy to simultaneously measure local variations in cytoskeleton organization and force generation, providing exquisite insight into the dynamics of this process.6,7 Computer simulations of interacting laments,14,21,22 spring networks15,23 and nite element calculations12 have shed light on how cell shape and boundary conditions dictate the spatial distribution of cellular stresses and how these may induce the bundling of laments along principal stress directions. We note that a different class of cytoskeletal responses is typically found under the application of high frequency stresses. In these cases stress bers and focal adhesions are seen to destabilize in the direction of stretch and to orient away from the applied eld direction;24,25 the underlying mechanism is believed to be different and a variety of models have been suggested to explain the behavior.2,25–28 Here, we shall focus on the mechanical regulation of stress ber formation during cell adhesion in the absence of applied elds. In this paper we use a continuum approach to theoretically investigate the mechanical coupling between the cell shape and early-time actomyosin polarization in the cell. The cell is modeled as an active elastic spheroidal inclusion, embedded in a large isotropic and homogeneous matrix of different rigidity. Actomyosin forces are modeled by a localized distribution of force dipoles that may dynamically polarize, in magnitude and orientation, in response to local variations in the elastic stress. Based on the coupling between the exerted actomyosin dipolar stress and the local elastic stress we derive two coupled dynamical equations for the spontaneous development of cellular force along the two principal directions of the cell. This enables us to predict the effects of the cell aspect ratio and matrix rigidity on the dynamics of force polarization in the cell. Our approach is complementary to detailed computational studies that simulate actin lament dynamics and network reorganization by myosin forces on the molecular level.14,21,22 Consistent with experiments we demonstrate the tendency of nascent stress bers to polarize in parallel to the long axis of cells; we show that the dependence on matrix rigidity may gradually change from non-monotonic to a monotonic increase, as the asymmetry of adhesion induced stresses increases. In addition, our theory predicts the existence of a critical matrix rigidity, above which spherical cells exert maximal force, and below which elongated or attened cells exert the highest force. The cell shape and rigidity of the matrix are also shown to govern the rate of force polarization and the time to reach the steady state. Consistent with recent experiments, we identify two fundamental time scales

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associated with the change in magnitude and orientation of actomyosin forces in the cell; different mechanisms of force polarization are discussed in terms of our model parameters. The theory is also applicable to study the polarization of whole cell aggregates conned in spheroidal domains in a gel.

2 Theory of actomyosin polarization dynamics We study the response of actomyosin forces to elastic stresses that develop in the cell during cell adhesion to the ECM. These stresses may generally have two contributions arising from (i) myosin contraction and (ii) cell spreading (and stretching) in the 3D gel. We focus here on the response to the former contribution, and for the time being ignore the additional source of stress that may originate due to the spreading of the cell (which is examined below). As the cell adheres to its surrounding matrix, active contractile forces are generated in the cytoskeleton and transmitted via adhesion complexes into the ECM. At these early stages of cell adhesion, the cytoskeleton is still in an isotropic gel state and myosin II motors are homogeneously and isotropically dispersed therein. Myosin contraction produces a tensile stress in the cell and the matrix. The 3D shape of the cell as well as the elastic moduli of the cell and matrix dictate the elastic eld that develops in the system. For simplicity we focus here on cells of spheroidal geometry, dened by two principal axes, a and c, oriented parallel to a Cartesian coordinate system: (x2 + y2)/a2 + z2/ c2 ¼ 1; r ¼ c/a is the aspect ratio of the cell. The dimensions, c and a, refer to the cell in its extended state where it is anchored to the ECM, but before the actomyosin force has been exerted into the matrix, see Fig. 1. Throughout the manuscript we assume that the aspect ratio r is xed. The cell and matrix are further assumed to be homogeneous and isotropic and the matrix to be innitely extended. We denote the elastic moduli tensors of the cell and matrix by Cc and Cm, respectively. Here and everywhere below, we use boldface letters to designate the fourth-rank tensors and a product of the form A gij denotes Aijkl gkl (similarly, ABgij ¼ AijmnBmnkl gkl) where summation over repeated indices is implied. Our choice of spheroidal geometry considerably simplies the problem since under the conditions stated above the elastic eld generated in the cell is predicted to be uniform;29 allowing us to overlook spatial variations in the elastic eld in the cell. The elastic eld produced in the cell is assumed to locally modulate myosin interaction with actin laments and thereby to affect the magnitude and/or the orientation of the forces that myosin motors apply on the laments. A local polarization of actomyosin force may result from strengthening of actomyosin binding, e.g., via a catch bond mechanism,30,31 local bundling of actin laments,6,14 or recruitment of myosin motors.32 This in return alters the elastic stress that builds up in the system and a mechanical feedback loop of actomyosin self-polarization is created. The process we consider pertains to early-time orientation and strengthening of actomyosin cross-bridges in the cytoskeleton which leads to nascent sarcomere formation; this corresponds to the rst minutes to tens of minutes of cell interaction with the extracellular matrix.

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relationship between the mean dipolar stress, pij(t), and the resulting strain in the cell, ucij(t), from general considerations of elastic inclusions in solids:13,40,41

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ucij(t) ¼ ASCm1pij(t),

Fig. 1 Schematic illustration of spontaneous cell deformation and actomyosin polarization. Panel I shows the “free-transformation strain” resulting due to actomyosin contraction in a cell in an infinitely soft environment. The only resistance to myosin contraction is the cell elasticity. Myosin activity in this compressed state of the cell is assumed to be the baseline level of actomyosin contractility. The inset illustrates a microscopic view of the actomyosin force-dipole, showing the myosin forces (red arrows) pulling on two actin filaments (blue lines). Panel II shows the elastic cellular deformation, ucij (t), and the corresponding polarized actomyosin force-dipoles, pij(t), at some time, t, of the deformation, as it occurs in differently rigid substrates. Note that the value of pij is generally higher in the more rigid matrices.

The local, active actomyosin dipolar stress arises from the two equal and opposite forces exerted by myosin motors at two nearby points on actin laments, and is denoted by the force^ is the force dipole tensor filj ¼  ninj;13,33–36 where ~ f ¼ f n exerted at a point situated at a distance ~ l ¼ l^ n from a neighboring point. We denote by pij ¼ rh filii the local spatial average of these dipoles, where r is the dipole density per unit volume.‡ Our idealization of the cell as a homogeneous spheroid, where myosin molecular motors are uniformly dispersed therein, allows us to regard pij as uniform throughout the cell. The local distribution of actomyosin dipoles responds to the evolving stress in the cell, thus their average evolves with time, pij ¼ pij(t). While cells are generally viscoelastic, showing response times on the order of seconds,37,38 we shall consider here active actomyosin responses that occur on longer time scales, typically minutes to tens of minutes.39 We thus deliberately neglect the viscoelastic response of the cell and the matrix, and focus on the inherent time scales associated with the active polarization response. Elastic force balance is assumed to be achieved instantaneously at every given moment in the polarization response. This overlooks the possibility that actomyosin reorganization can include a rapid phase, comparable to the viscus time scale. For an active homogeneous and isotropic ellipsoidal cell with elastic modulus Cc, embedded in an innitely extended 3D matrix with elastic modulus Cm, one can derive the following

‡ We assume that the dipole is in mechanical equilibrium with the cytoskeleton and that viscous forces are negligible compared to the active contractile force,36 hence pij is a symmetric, torqueless tensor.33

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(1)

Here, S is the Eshelby tensor,29,40 a function of the cell aspect ratio, r, and Poisson ratio of the matrix; A ¼ (I + S(Cm1Cc  I))1 is the so-called strain concentration tensor42 and, Iijkl ¼ 1/2(dikdjl + dildjk), is the fourth-rank symmetric unit tensor.40 Eqn (1) characterizes the temporal state of force balance between the cell and the matrix; the dependence on the cell shape and elastic moduli enters via S and A. In the limit that the surroundings are innitely so, Cm / 0 (as is also the case for a oating cell in solution), one nds ASCm1 ¼ Cc1, hence, as expected, the elastic contraction of the cell occurs with the elastic resistance of the cell only. Because the matrix then exerts no force on the cell, we expect it to have no mechanical effect on the cell and therefore to induce no actomyosin polarization. We thus take the active dipolar stress in this case to be the baseline level of actomyosin contraction, and we denote it by, p0ij. From eqn (1) one nds the following relationship, ucij ¼ Cc1p0ij h u0ij. We next consider how the state of force balance in the system inuences the development of pij(t) over time. The baseline level of actomyosin contraction, p0ij ¼ Ccu0ij, dened for a cell in an innitely so matrix, is also likely to be the mean dipolar stress exerted in the initial stages of cell adhesion since it is exerted prior to any feedback from the matrix; we therefore use it as our initial conditions (pij(0) ¼ p0ij) in our equations below. It is these initial forces that trigger the self-polarization response of pij(t), ˜ij(t) ¼ ucij(t)  u0ij, or equivasee below. Any excess cell strain, u ~ij ¼ Cc[ucij(t)  u0ij], arising due to the lently, excess cell stress s elastic resistance of the matrix, is assumed to give rise to a polarization of the mean actomyosin dipolar stress from its baseline (early-time) level. This is based on a variety of experiments indicating that actomyosin forces polarize in response to the local tensile stress.2 We express this mathematically via the following linear response: ðt i   h   ~ t  t0 Cc ucij t0  u0ij dt0 (2) Pij ðtÞ h pij ðtÞ  p0ij ¼  c 0

In this equation, Pij(t) denotes the temporal increase of the mean dipole tensor from its baseline level, p0ij, to its present value, pij(t); we therefore refer to it as the polarization tensor. ~ (t  t0 ) denotes a time-dependent, active susceptibility tensor; c ~ (t  t0 ) couples it is a fourth-rank tensor having units of time1. c c 0 0 0 ~ij(t ) ¼ Cc[uij(t )  uij], at every instant of time, the excess stress, s t0 , to a temporal modulation of the mean dipole tensor. Because the actomyosin response is not instantaneous, an integration is ~ (t  t0 ) carried out to include all accumulating contributions.§ c § We note that eqn (2) is accurate to describe the relatively slow process of actomyosin self-polarization. However, an extension of this equation to treat rapidly oscillating external stresses is not trivial since the cells behave qualitatively different when subjected to high frequency loads.26 For recent models that capture this response see ref. 25–28.

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behaves as a memory kernel and is expected to be a decreasing function of the time interval dt ¼ t  t0 , a property that is oen referred to as a principle of fading memory.43 As a common simplifying approximation,43,44 we take this function to have the following generalized exponential form:{

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~ (dt) ¼ exp[s1 dt]~ c0. c

(3)

~0 ¼ c ~ (dt ¼ 0) denotes the high frequency, or instanHere, c taneous response function, and s is a (forth-rank) tensor that includes the corresponding relaxation times. The bold-face exponent in the above equation is not simply an exponent, but a N X 1 n linear operator dened via: exp½Q ¼ Q , where Q is an n! n¼0 arbitrary fourth-rank tensor; this takes into account the coupling of the relaxation of different dipole components that may occur on different time scales, see below. Similar to other relaxation phenomena,43,44 the characteristic relaxation times associated with actomyosin polarization are assumed to arise from the 3D frictional (weak binding46) interaction of myosin II motors with actin laments as they change their orientation, switch between laments, or slide them apart. Rather than representing these interactions explicitly as in ref. 14 they are ~ (dt). represented here phenomenologically via the decaying of c Combining eqn (2) and (3) allows us to express the temporal evolution of the polarization response in terms of a linear set of differential equations for all distinct elements of the mean polarization tensor, pij(t); substituting eqn (3) in (2) and taking the time derivative from both sides, we nd: h h ii dPij ðtÞ ¼ s1 Pij ðtÞ þ cCc ucij ðtÞ  u0ij (4) dt where we have dened the dimensionless, time independent tensor, c ¼ s~ c0. The tensor c may be regarded as the static actomyosin susceptibility tensor since it relates between the ess steady-state polarization Pss ij and the excess stress, sij , tensors, ss ss as follows, Pij ¼ c seij , as can be seen by taking dPij/dt ¼ 0 on the le hand side of eqn (4). The tensors s and c play an essential role in our theory since they characterize the underlying dynamical mechanism of the polarization response. We shall discuss these considerations in more detail in the next section, however, it is already worthwhile mentioning few generic properties of s and c at this point. Our assumption that the cytoskeleton is isotropic in the early stages of cell adhesion suggests that the polarization response to any ~ij(t) ¼ Cc[ucij(t)  u0ij], symmetric or asymmetric excess stress, s should a priori be independent of any direction. This is true both with regard to the duration of the response, and with regard to the extent of the response. Hence, the most natural rst assumption about s and c, which we adopt in this paper, is that these are isotropic tensors, each given by two independent parameters, one reecting the response to an isotropic (volume) change, the other to a deviatoric (shear) change. In explicit form

~ (dt) can be expanded as a sum of regular decaying exponents { This form of c multiplying an orthonormal basis of the 4th-rank tensors derived from the ~ 0.45 spectral decomposition of s and c

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  1 1 they are therefore given by: cijkl ¼ cv dij dkl þ cs Iijkl  dij dkl 3 3   1 1 13,41 and sijkl ¼ sv dij dkl þ ss Iijkl  dij dkl . Interestingly, the 3 3 relationship between the two (scalar) susceptibility parameters, cs and cv, and the two relaxation times ss and sv, reect different possible dynamical mechanisms of actomyosin polarization; the latter four quantities are the only input parameters in our theory and their consequences will be examined below. In the outset, eqn (4) resembles Debye's relaxation equation for dielectric response;44 Pij is analogous to the electrical polarization, c is analogous to the static electrostatic susceptibility, and s generalizes the characteristic relaxation time. Moreover, the temporal (strain) eld, ucij(t), appearing on the right hand side of eqn (4), is in itself a function of the actomyosin polarization tensor, pij(t), as given by eqn (1); this is similar to the electric eld, ~ E, that depends on the polarization vector, ~ P , in electrostatics. Despite this similarity, there is one essential difference between the two physical processes. Unlike the dielectric response that initiates with application of an external electric eld, here, the actomyosin response is to the eld created by the dipoles themselves, namely to the eld created by the early-time dipolar stress, p0ij. Thus, when combined with eqn (1) one nds a set of self-consistent linear differential equations to solve for the actomyosin polarization dynamics, Pij(t): ~s

dPij þ Pij ¼ Pijss dt

(5)

Taking as initial condition, Pij(0) ¼ 0 (equivalently, pij(0) ¼ p0ij) we nd the following solution: Pij(t) ¼ [I  exp(~t 1t)]Pss ij

(6)

with the (cell shape and rigidity) scaled relaxation time tensor ~t ¼ Zs

(7)

and steady state polarization: 0 Pss ij ¼ [Z(I + c)  I]pij

(8)

˜ m1 Z ¼ CmA1AC

(9)

where

˜ ¼ [I + S(Cm1C ~ c  I)]1 is given in terms of the and (cf. A) A ~ effective cell moduli, Cc ¼ (1 + c)Cc.47 The above equations describe the dynamics of the selfpolarization response. It may readily be seen that this response arises due to the early-time tractions, p0ij, since if p0ij ¼ 0, no response will initiate. The cell shape and elastic modulidependence of the polarization process enters via the tensor, Z, that governs the behavior of both the relaxation times and the steady-state polarization, as seen in eqn (7) and (8). The dependence of Pij(t) on the cell shape is an essential prediction of the present theory. Eqn (8) predicts that even for an initial isotropic mean dipolar stress, p0xx ¼ p0yy ¼ p0zz, the dipole

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elements at any later time can differ from each other, i.e., pxx(t) s pzz(t). Thus, via the asymmetric global shape of the cell, the actomyosin dipoles may locally self-polarize (and orient) in the cytoskeleton. This interesting response occurs due to the breaking of symmetry of the elastic stress in the cell, which depends on the cell shape. Eqn (6) shows that also the time course of the polarization response is shape dependent. We shall discuss these phenomena in more detail in the next section.

3 Results The foregoing formalism allows us to predict the effects of the cell shape and elasticity of the environment on the dynamics of actomyosin polarization in the cell. We begin by examining our predictions for the steady-state, and in the next section analyze the dynamics of the relaxation response.

We take the total magnitude of the adhesion-induced stress to be equal for all cell shapes, Tr[pdij] ¼ pdxx + pdyy + pdzz ¼ pd. Fig. 2 summarizes our major predictions for the steady-state polarization in the cell. It compares the pure self-polarization response, shown in the upper two panels for pdij ¼ 0, to the case shown in the lower two panels, where anisotropic adhesioninduced stresses are taken into account. The magnitude of the mean dipole tensor is given by the trace, p ¼ Tr[pij] ¼ pxx + pyy + pzz, and the anisotropy of the dipole tensor is characterized by 1 the order parameter, S ¼ h3cos2 ðqÞ  1i ¼ ð pzz  pxx Þ=p.13 q is 2 the angle between any actomyosin force-dipole in the cell and the principal axis of the cell along the z-axis. One nds, S ¼ 1, when all actomyosin forces polarize parallel to the z-axis (q ¼ 0),

3.1 Effects of cell shape, matrix rigidity and adhesioninduced stresses on the steady-state polarization of nascent sarcomeres in the cell Our paper is focussed mainly on the self-polarization of actomyosin dipoles, pij(t). We recognize, however, that adhesioninduced stresses that may accompany changes in cell shape during spreading may provide an additional driving force of actomyosin polarization. To study the consequences of these stresses, we extend our denition of the early-time stress exerted by the cell as follows: Ccu0ij ¼ p0ij + pdij. The rst contribution, p0ij, is the early-time actomyosin dipolar stress that later develops in response to the evolving stress in the cell; the second term, pdij, is the adhesion-induced stress. We assume that pdij is acquired with cell adhesion and adds a constant contribution to the overall stress in the cell (the dynamics of cell spreading in the 3D matrix is not included in the present analysis). To incorporate these stresses in our theory, eqn (1) is ~ m1[pij(t) + pdij]; this leads to the generalized to read: ucij(t) ¼ ASC following generalization of eqn (8): ss 0 0 d Pss ij ¼ pij  pij ¼ [Z(I + c)  I](pij + pij)

(10)

where pss ij is the steady-state actomyosin dipolar tensor; other equations in the above theory remain unchanged. Note that this equation predicts that the actomyosin dipoles, pij(t), may polarize in response to the adhesion induced stress, pdij, as well as, self-polarize in response to their own generated stress. In all the results presented below we take the initial dipole 1 distribution to be isotropic, namely, p0ij ¼ p0 dij . In contrast, pdij 3 is assumed to add an anisotropic contribution to the early-time stress in proportion to the aspect ratio of the cell, r. Accordingly, elongated (prolate) cells, r > 1, are assumed to sustain anisotropic adhesion-induced stress along their long axis; oblate cells, r < 1, are assumed to be subjected to (transversely isotropic) adhesion-induced stresses in the x–y-plane; and round cells are assumed to be subjected to an isotropic adhesion-induced stress. This is given by: pdzz ¼ r$pdxx and pdxx ¼ pdyy. This journal is © The Royal Society of Chemistry 2014

Steady-state orientational order parameter (left panels) and magnitude (right panels) of the actomyosin dipolar stress, plotted as a function of the aspect ratio of the cell, for different values of matrix-tocell rigidity ratios, mm/~ mc, where m ~ c ¼ (1 + cs)mc; purple, blue, green, dotted-green, yellow, dotted-red, red, and black correspond to: mm/~ mc ¼ 0, 1/300, 1/30, 1/5, 1, 5, 30, N. Upper panels correspond to a pure self-polarization response (pdij ¼ 0), where early-time isotropic forces trigger the response, and bottom panels demonstrate the effects of the contribution of anisotropic adhesion induced stresses (pdzz/pdxx ¼ r and pd ¼ (5/3)p0). Each curve corresponds to a different rigidity ratio. The inset figures in panels a and c present the orientational order parameter, Sss, as a function of the matrix to cell rigidity ratio, mm/~ mc, for cell aspect ratios of r ¼ 1/5, 1, and 5 (dashed, dotted, and bold lines, respectively). The cell and matrix were assumed to possess equal Poisson ratios (particularly, vc ¼ vm ¼ 0.3), hence the ratio of shear and bulk moduli is equal mm/mc ¼ km/kc. The active susceptibility constants were taken to be cv ¼ cs ¼ 5. We note the interesting role of matrix rigidity in regulating both the orientational order of actomyosin stress fibers in the cell as well as the dependence of the total cell force on the cell shape. The blue rectangle frame indicates a regime of realistic cell aspect ratios. Fig. 2

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S ¼ 0, when the force-dipoles distribute isotropically in the cell, and S ¼ 1/2 when all dipoles orient parallel to the x–y plane (q ¼ p/2) (with zero z-component). Panel a shows that the orientational order parameter, S, increases monotonically with the aspect ratio of the cell; it is positive for prolate spheroids, indicating that the dipoles orient parallel to the long axis of the cell (z-axis), and negative for oblate spheroids indicating their orientation into the x–y plane. These results provide an explanation for the general experimental observation that stress bers spontaneously develop in parallel to the long axis of cells.13,16–19 We nd a non-monotonic dependence of the order parameter on the matrix rigidity (see inset), attaining a maximum for an intermediate ratio of the matrix-to-cell rigidity ratio (mm/mc)*; where mm and mc are the shear moduli of the matrix and cell respectively. This optimal alignment has been conrmed in experiments on stem cells13,20 and has been observed in other cell types as well.18 Furthermore, our theory makes the interesting prediction that this optimal rigidity depends on the cell shape – generally decreasing with the aspect ratio of the cell. That is, more elongated cells polarize the best already in soer environments. Our results show that the matrix better resists internally generated stresses along the long-axis (as compared to the short axis) of the cell. Consequently, more elongated cells can polarize already in soer environments, explaining why they show a lower value of (mm/mc)*. Panel b shows the variations in the magnitude (or trace) of the dipolar tensor, pss, as a function of the aspect ratio of the cell, plotted for different matrix-to-cell rigidity ratios. Strikingly, we nd opposite trends for cells grown in a rigid environment, where mm > m ~ c, and for cells grown in a so environment, where mm < m ~ c and m ~ c ¼ (1 + cs)mc. In a so matrix, round cells (r ¼ 1) exert minimal overall force, while disc-like or rod-like cells exert higher forces. In contrast, in rigid matrices round cells exert maximal force, and more asymmetrically shaped cells exert less force. The critical rigidity where the qualitative behavior changes is mm ¼ m ~ c; for that critical stiffness the trace of the dipole tensor is shape-independent. We emphasize that this special rigidity depends on two intrinsic characteristics of the cell, cs and mc, that enter the effective cellular stiffness, m ~ c ¼ (1 + cs)mc. These intriguing shape effects are most profoundly expressed for some intermediate regime of stiffness of the environment. For very so or very hard environments no effect of the cell shape is observed. For a more in-depth explanation of these behaviors see the ESI.† In the presence of anisotropic adhesion-induced stresses, pdij, our predictions for the qualitative behavior of the order parameter, Sss, and trace, pss, of the dipole tensor are very different. The two major differences are: (1) the order parameter attains a smoother peak at the optimal rigidity, (mm/mc)*, showing signicant polarization for very rigid matrices, as shown in the inset of panel c. (2) The range of matrix stiffnesses, for which round cells exert less force than those with less symmetric structures, extends beyond the critical value, (mm ¼ m ~ c). To explain the rst of these (1), we note that when the matrix becomes rigid enough, shape anisotropy has a lesser impact on the anisotropy of the

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stresses and strains in the cell. Indeed for a passive inclusion in a rigid matrix (Cm / N) the constrained strain is zero, ucij ¼ ASCm1Ccu0ij ¼ 0 (cf. eqn (1)), since the shapedependent tensor, ASCm1Cc / 0; in this limit also, Z / I, in eqn (10). Consequently, for an axial polarization mechanism where cs ¼ cv ¼ c (see below) we nd in this limit 1 0 0 0 d 0 that: pss ij  pij ¼ c(pij + pij). Hence, for, pij ¼ p dij , the order 3 parameter simply reects the anisotropy of the adhesioninduced stress, Sss ¼ (pdzz  pdxx)/pd. The explanation of point (2) above follows from the general tendency of the surroundings to better resist deformations along the long axis of the cell (compared to the short axis of the cell), see ESI.† Adhesion-induced stresses along the long cell axis are thus better supported by the matrix. This effect is signicant even for matrices more rigid than the cell, i.e., for mm $ m ~ c, explaining the behavior exhibited by the orange curve in panel d of Fig. 2. These predictions for how the cell shape might inuence the total cell force are presented here for the rst time and experiments are called for to assess their validity.

3.2 Relaxation dynamics of actomyosin polarization response The dynamical response of nascent actomyosin sarcomeres to local variations in the elastic stress is described here in terms of four independent parameters: two static susceptibility factors, cs and cv, and two corresponding relaxation times, ss and sv. Interestingly, the relationship between the two susceptibility factors reects different mechanisms of actomyosin polarization. We describe three limiting cases of interest: (i) cs ¼ 0, (ii) cv ¼ 0, and (iii) cs ¼ cv, corresponding to isotropic, orientational and axial polarization mechanisms, respectively.41,48 The two characteristic times, sv and ss, govern the time scales associated with isotropic and orientational responses of the actomyosin forces in the cell. Fig. 3 compares our predictions for the three distinct polarization mechanisms using different values of the two relaxation times. Le panels illustrate the simultaneous variations of the two principal components of the dipolar tensor, pzz(t) (solid lines), and pxx(t) ¼ pyy(t) (dashed lines); the middle column panels show the normalized trace (or magnitude) of the dipolar tensor, p(t)/p0; and right panels plot the variations of the order parameter S(t) ¼ [pzz(t)  pxx(t)]/p(t). For simplicity, we focus here on the self-polarization response of actomyosin dipoles and ignore effects due to adhesion induced stresses, i.e., pdij ¼ 0. In addition, we demonstrate the behavior for one single choice of cell aspect ratio and matrix elasticity; the effects of these are examined next. Isotropic polarization (cs ¼ 0). As expected for this case (upper three panels), all three elements of the dipole tensor polarize simultaneously with the same rate, given by sv. Thus p(t) ¼ 3pxx(t) and S(t) ¼ 0. A purely isotropic polarization response may arise in cases where the elastic stress initiates internal molecular signaling that governs the actomyosin polarization response in an isotropic manner; for instance, by uniformly rasing myosin density or activity in the cell. Cells

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Fig. 3 Dynamics of actomyosin polarization response in three limiting mechanisms: panel I: isotropic polarization (cs ¼ 0, cv ¼ 5), panel II: orientational polarization (cv ¼ 0, cs ¼ 5), and panel III: axial polarization (cs ¼ cv ¼ 5), plotted for different ratios of the two characteristic times, ss and sv. In panel I, sv/s0 ¼ 1, 5, 9, where s0 indicated the time units, and ss has no effect on the dynamics. In panel II, ss/s0 ¼ 1, 5, 9 and sv has no effect on the dynamics. The mechanisms shown in Panel III are: (i) fast orientational response, where sv ¼ 9ss ¼ 9s0 (blue line); (ii) fast isotropic response, where ss ¼ 9sv ¼ 9s0 (red line), and (iii) isotropic and orientation rates are equal, sv ¼ ss ¼ 9s0 (black line). Left panels show the normalized elements of the mean dipole tensor, 3pxx(t)/p0 (dashed lines) and 3pzz(t)/p0 (solid lines). Middle column panels show the evolution of the normalized magnitude (trace) of the dipole tensor, p(t)/p0 and right panels show the modulation of the order parameter, S(t). Panel IV plots the temporal variations in the magnitude of the dipole stress, p(t), as a function of the evolving order parameter, S(t), for the three mechanisms of actomyosin polarization response; the isotropic (cs ¼ 0) mechanism shown in orange (vertical line) and the orientational mechanism (cv ¼ 0), in purple (horizontal line). For the axial mechanism three scenarios are possible. (i) Shown in blue is the case where dipole orientation occurs first, and only then an isotropic increase in the dipolar stresses follows; this occurs when ss < sv. (ii) Red curve shows the opposite case where an isotropic increase in the dipolar stresses precedes an orientational response; this occurs when sv < ss. The black curve shows how these responses blend when sv ¼ ss. Other parameters used in this figure are: mm/mc ¼ 1, r ¼ 3, vc ¼ vm ¼ 0.3.

operating via such mechanism would exhibit an isotropic development of stress bers in the cell irrespective of the cell shape or matrix rigidity.

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Orientational polarization (cv ¼ 0). The opposite extreme is a pure orientational response induced by the anisotropy of cell shape; in this case only the orientation but not the magnitude of the actomyosin forces varies in time. This is reected in the evolution of the order parameter, S(t), that occurs on a time scale ss (panel IIc). Panel IIa shows that the two elements of the polarization tensor, Pzz(t) ¼ pzz(t)  p0zz and Pxx(t) ¼ pxx(t)  p0xx, take on opposite signs, and that Pzz(t) ¼ 2Pxx(t). For a prolate spheroid (r > 1), forces along the z axis increase with time, while those in the x–y plane decrease, thus reecting an orientation of the dipoles from the x–y plane to the z direction; the trace of the polarization tensor remains un-changed p(t) ¼ p0 (panel IIb). Axial polarization (cv ¼ cs). When both cv and cs are nonzero, a combination of the two processes is to be observed. In this case, sv and ss dictate which process – orientation or net strengthening of the actomyosin force – occurs rst. Panel IIIa shows time variations of the two elements of the polarization tensor for three different proportions of sv and ss. Blue curves are for ss < sv, red curves are for sv < ss and black curves are for ss ¼ sv. In the former case, ss < sv, an orientation of the forces occurs rst as can be seen in the early-time splitting of the two blue curves: pxx(t)/p0 (dashed blue) decreases with time, while pzz(t)/p0 (solid blue) increases with time. Along with this relatively fast orientation, actomyosin forces strengthen in time eventually giving rise to a gradual increase in both pxx(t)/p0 and pzz(t)/p0. The reverse is observed when sv < ss (red lines in panel III). In this case there is rst an isotropic strengthening of the force and only later a net orientation of the dipoles. We note that this does not necessarily mean that the dipoles actually orient at these longer times, rather, it might mean that the simultaneous variations of the forces in different directions are opposite. Note that for t > sv, there is also a slower second phase in which the dipole magnitude increases on timescale ss; this slower phase goes in parallel to the continued development of S(t). This effect is similar in nature to the effect of pdij on p discussed previously – the additional stability of the long axis allows higher anisotropic stresses to develop. The bottom panel in Fig. 3 summarizes these behaviors in a suitable way for experimental verication. We plot the instantaneous magnitude of the dipole tensor, p(t), as a function of the order parameter, S(t), for different polarization mechanisms. In a purely isotropic response (vertical orange), p(t), changes but the order parameter remains zero, S(t) ¼ 0. In a pure orientational response (horizontal purple), the order parameter gradually increases, but the magnitude of the dipole tensor remains xed at the initial value, p(t) ¼ p0. The three curves in the middle of the plot correspond to the axial polarization case, plotted for three choices of sv and ss. The red curve corresponds to sv < ss hence one sees a rapid isotropic phase (governed by sv) rst, followed by a slower orientational phase (governed by ss). The blue curve shows the opposite, rst an orientational response with ss, and then an isotropic response with sv. The black curve shows how these two processes blend in the case that sv ¼ ss. In the ESI† we provide additional mathematical analysis of these results.

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3.3 Effects of matrix rigidity and cell shape on actomyosin polarization dynamics Eqn (6)–(8) predict that both the steady-state polarization and the dynamics are cell-shape and matrix rigidity dependent. Furthermore, these dependencies enter via the same shape dependent tensor Z. Hence it is expected that the elastic moduli and cell shape would have a similar effect on the steady-state and the dynamics of the relaxation response. Fig. 4 shows the variations of the order parameter, S(t), and the normalized magnitude of the dipole tensor, p(t)/p0, for different values of the aspect ratio, r (panels Ia and IIa), and for different mm/mc ratios (panels Ib and IIb). The two rightmost panels (Ic and IIc) show the corresponding variations of the half-lives, of the order parameter, tS1/2, and of the dipole magnitude, tp1/2, as a function of the cell aspect ratio, and for different matrix-to-cell rigidity ratios. The half-lives are calculated as the time required to reach half the saturation values. Panels Ia and IIa are plotted for mm ¼ mc, hence consistent with the trends shown for the steady-state (panels a and b of Fig. 2), and with experiments,13,18 they show that more elongated cells polarize better both in magnitude and order parameter. Similarly, panels Ib and IIb are consistent with the predictions discussed for the steady-state. The order parameter, S(t), reaches higher levels for intermediate rigidity of the matrix, and the saturation value of p(t) increases monotonically with matrix rigidity.

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The new features examined here are the dependence of the half-lives of these responses on the aspect ratio of the cell and matrix rigidity (panels Ic and IIc). Interestingly, the plots of tS1/2(r) and tp1/2(r) for different mm/mc ratios resemble the behaviors of Sss(r) and pss(r), shown in panels a and b of Fig. 2. We note that both tS1/2 and tp1/2 monotonically increase with matrix rigidity. This predicts that cells would reach the saturation value of Sss and pss more quickly in soer environments, as has recently been observed experimentally.6 Our explanation to this behavior is that the rise in p(t) results in cell contraction that partially relieves the stress that drives the polarization response. This causes the elements of the dipole tensor to reach saturation more quickly; hence both tS1/2 and tp1/2 decrease with the soness of the matrix. Similar to the dependence of pss on the aspect ratio, r, we nd that tp1/2 depends non-monotonically on r, in a manner that depends on the matrix rigidity. In so matrix environments, tp1/2 increases with the shape anisotropy of the cell, but in rigid matrices it decreases. Hence we predict that there exists a regime of so matrices where spherical cells reach saturation the fastest, and a regime of stiffer matrices where spherical cells are the slowest to reach saturation. The explanation of this dependence is similar to the dependence of pss on r. The higher the overall resistance to actomyosin contraction (as dictated by both the shape anisotropy and matrix rigidity) the stronger is the driving force for actomyosin polarization and consequently the slower is the convergence of the polarization response.

4 Concluding remarks

Fig. 4 Effects of the cell shape and matrix rigidity on actomyosin polarization dynamics. The order parameter, S(t) (upper panels), and (normalized) magnitude (bottom panels) p(t)/p0 of the dipole tensor are plotted as a function of time for different cell aspect ratios (panel a), and matrix rigidities (panel b). In panel a the cell aspect ratios are r ¼ 2, 5, 10, shown by the dotted, solid and dashed lines respectively; the matrix-to-cell rigidity ratio is mm/mc ¼ 1. In panel b the matrix-to-cell rigidity ratios are mm/mc ¼ 0.1, 1, and 10, shown in blue, green and red, respectively; aspect ratio is r ¼ 5. Panels c1 and c2 show the respective half-lives in the evolution of the order parameter S(t), and dipole magnitude, p(t)/p0, plotted as a function of the aspect ratio of the cell, for different choices of the matrix-to-cell rigidity ratios; purple, blue, green, red, orange, brown and black lines correspond to mm/mc ¼ 0.01, 0.1, 1, 10, 20, 100, and 1000. Other parameters used in these plots are: cv ¼ cs ¼ 5, and ss ¼ sv ¼ s. Note the similarity in the shape of these curves and those plotted for Sss(r) and pss(r) in Fig. 2.

2460 | Soft Matter, 2014, 10, 2453–2462

The cell shape has been demonstrated to provide global control of stress ber organization in cells.8,49 This coupling has been suggested to arise due to the responsiveness of the actomyosin cytoskeleton to spatial variations in the elastic stress.12,13,50,51 Along these lines we recently presented an elastic explanation for the tendency of stress bers to develop in parallel to the long axis of cells.13,48 Validated by experiments on stem cells we demonstrated that this alignment maximizes when the cell and matrix have comparable rigidities. A related phenomenon is also seen with the optimal elongation of some cell types on substrates of intermediate rigidities.18,20 Here we point out another striking role of matrix rigidity. We predict the existence of a critical rigidity, above which round cells will exert maximal force, and below which round cells will exert minimal force (Fig. 2b). At the critical rigidity, the trace of the dipolar stress is expected to be cell-shape independent. Furthermore, we demonstrate that this critical rigidity strongly depends on the symmetry of adhesion-induced stresses, pdij, that may accompany shape changes during cell spreading (Fig. 2d). In the ESI† we provide an explanation for these predictions based on the generic properties of the deformation of elastic inclusions in solids. To the best of our knowledge there is still no experimental assessment of these predictions. In addition, our formalism reveals two inherent time scales dictating the dynamics of the polarization response, one associated with the isotropic rise in the magnitude of the force, sv, the other with the anisotropic orientation of these forces, ss. Recently Aratyn-Schaus et al.6 described the dynamics of force generation

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and actomyosin bundling in U2OS cells following short treatment with the myosin inhibitor blebbistatin. Aer washing out blebbistatin from their samples, these cells exhibited a fast isotropic contraction phase, lasting 3 min, accompanied by a slower orientational phase in which bundles developed and the local order parameter of F-actin gradually increased over a time scale of 10 min. These results are consistent with a mixed polarization mechanism in which both cv and cs contribute since there is a rise in both the magnitude and orientational order of the forces exerted by the cell. The separation of time scales is generally consistent with the behavior seen by the red curves in Fig. 3III and IV. However, we note that the order parameter in these experiments was evaluated locally and it therefore reects a local organization of laments to form anisotropic bundles, while the process we describe concerns the global alignment of actomyosin forces in parallel to the long axis of the cell. A similar separation of time scales for isotropic and orientational responses has recently been reported for the polarization dynamics of broblasts.7 However, that study showed that focal adhesion redistribution plays a central role in the polarization response. Clearly these additional dynamical mechanisms as well as local and non-linear effects would have to be examined in future developments of the theory. Nevertheless, already the present continuum linear theory presents interesting experimentally testable predictions on the effects of the matrix rigidity and cell shape on the dynamics of force polarization in cells. Simultaneous measurements of p(t) and S(t) for matrices of different rigidity and different cell shapes will provide important insight into the dynamical mechanisms that underlay actomyosin polarization in the cell.

Acknowledgements We thank Florian Rehfeldt and Sarah Koester for fruitful discussions. AZ thanks the Israel Science Foundation and the Niedersachsen, German–Israeli Lower Saxony Cooperation, for their suport.

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