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PRL 113, 148102 (2014)

week ending 3 OCTOBER 2014

PHYSICAL REVIEW LETTERS

Spontaneous Oscillations of Elastic Contractile Materials with Turnover Kai Dierkes,1,2 Angughali Sumi,1,2 Jérôme Solon,1,2 and Guillaume Salbreux3 1

Centre for Genomic Regulation (CRG), Dr. Aiguader 88, 08003 Barcelona, Spain 2 Universitat Pompeu Fabra (UPF), 08003 Barcelona, Spain 3 Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany (Received 9 April 2014; revised manuscript received 4 July 2014; published 1 October 2014) Single and collective cellular oscillations driven by the actomyosin cytoskeleton have been observed in numerous biological systems. Here, we propose that these oscillations can be accounted for by a generic oscillator model of a material turning over and contracting against an elastic element. As an example, we show that during dorsal closure of the Drosophila embryo, experimentally observed changes in actomyosin concentration and oscillatory cell shape changes can, indeed, be captured by the dynamic equations studied here. We also investigate the collective dynamics of an ensemble of such contractile elements and show that the relative contribution of viscous and friction losses yields different regimes of collective oscillations. Taking into account the diffusion of force-producing molecules between contractile elements, our theoretical framework predicts the appearance of traveling waves, resembling the propagation of actomyosin waves observed during morphogenesis. DOI: 10.1103/PhysRevLett.113.148102

PACS numbers: 87.10.-e, 05.45.Xt, 87.16.Ln, 87.17.Pq

The ability of cells and tissues to dynamically adjust their shapes and morphological organization on time scales ranging from a few minutes to several hours is crucial for the proper formation of organs during development [1,2]. A striking example of such deformations in morphogenesis is cell area oscillations with periods of several minutes observed during Drosophila, C. elegans, and Xenopus development [3–7]. A key player in determining cell shape is the actomyosin cortex, a thin layer of cross-linked actin filaments located beneath the plasma membrane of cells and concentrated on the apical side of tissues [8,9]. Within the cortex, myosin molecular motors generate stresses, which can result in flows and deformations occurring at the cellular and tissue level [8,10,11]. However, while it has been recognized that periodic shape changes in cells and tissues are, indeed, accompanied by nearly antiphasic cycles of cortical actomyosin density [see Figs. 1(b) and 1(c)] [3,4,12–14], the biophysical mechanism underlying the observed oscillatory instabilities is still elusive. Although also involving actomyosin dynamics, these oscillations are much slower than oscillations of sarcomeric units (of the order of 1–10 s) for which physical descriptions based on the collective dynamics of molecular motors have been proposed [15–18]. A fundamental property of the actomyosin cortex is the continuous renewal of its constituents on the time scale of minutes [Fig. 1(a)]. A recent study highlighted the importance of this time scale inherent to cortical dynamics for shape oscillations observed during cytokinesis of individual cells [12]. Here, we study a minimal, generic description of cell shape changes driven by a continuously renewing actomyosin cortex, predicting spontaneous oscillations, different patterns of collective oscillations, and 0031-9007=14=113(14)=148102(5)

wave propagation. Oscillations arise from the inherent coupling between the mechanical and chemical degrees of freedom and do not rely on the presence of a chemical network [19]. Note that pulsatory patterns can arise in a fluid when the active stress is regulated by two species [20]. The oscillatory mechanism proposed here is linked to the elastic properties of the material. We argue that this

(a)

(c)

(b)

(d)

FIG. 1 (color online). (a) Schematic of the apical cortex of an epithelial cell. Myosin molecular motors accumulate when the network contracts and exchange with a reservoir with rates kon and koff . (b) Snapshots of an oscillating amnioserosa cell during dorsal closure in the Drosophila embryo expressing Tomato E cadherin (red) [21] and myosin, GFP-sqh (green) [22]. Scale bar, 10 μm. (c) Time series of the normalized area and average myosin concentration c at the cell apical surface, for a representative cell in the amnioserosa. (d) Plot of the rate ð1=Ac0 Þ½dðcAÞ=dt vs normalized concentration c=c0 for the data shown in (c). The points align on a line, in accordance with Eq. (1). The slope of the line is related to the turnover time scale τ.

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description captures the main features of cell shape oscillations during Drosophila dorsal closure and constitutes a general framework for understanding periodic shape changes in developmental contexts. Minimal contractile unit.—We consider a system consisting of (1) a spatially homogeneous contractile element whose constituents are turning over, (2) an elastic element, and (3) a viscous damper, i.e., a dissipative element [Fig. 2(a)]. The contractile element exchanges forceproducing molecules (e.g., myosins) of concentration c with a reservoir, such that c follows the dynamic equation dc 1 c dl ¼ − ðc − c0 Þ − ; dt τ l dt

(a)

(c)

[Figs. 1(b), 1(c), and Supplemental Material Fig. S2 [23]]. Amnioserosa cells have been shown to exhibit oscillations of their apical surface with a period of ∼230 s [4]. As predicted by Eq. (1) with l substituted by A, during oscillations of the apical surface area of the amnioserosa cell the rate ð1=AÞ½dðcAÞ=dt is proportional to the myosin concentration c [see Fig. 1(d) and Supplemental Material Fig. S2 [23]]. The mechanics of the contractile unit is described by a dynamic equation for its length l, μ

ð1Þ

where l is the length of the contractile element. Here, the dynamics of c arises from two effects: (i) exchange of force-producing molecules with the reservoir occurring with binding rate kon ¼ c0 =τ and unbinding rate koff c ¼ c=τ, and (ii) matter conservation, imposing that in the absence of turnover the product lc is constant, giving rise to the second term on the right-hand side of Eq. (1). To verify whether Eq. (1) can account for the dynamics of the actomyosin cytoskeleton density in a tissue, we have measured the myosin intensity and cell surface area A in cells of the amnioserosa during Drosophila dorsal closure (b)

(d)

FIG. 2 (color online). (a) Schematic of a minimal model of a mechanochemical oscillator. A spring and a dashpot are in parallel with a contractile element whose force-producing molecules are turning over. The contractile material exerts a tension TðcÞ, depending on the concentration c. (b) Phase diagram for the behavior of the unit shown in (a), for k3 l20 =k1 ¼ 15. The white line corresponds to a Hopf bifurcation to spontaneous oscillations. Above the dashed upper line, the unit collapses (l → 0). The color code corresponds to the variance of the length lðtÞ=l0 in the oscillatory steady state, i.e., to hlðtÞ2 =l20 i − hlðtÞ=l0 i2 , where h…i denotes a time average after transients have decayed. [(c), (d)] Example trajectories of the unit in the three regions of the phase diagram, in a phase plot (c) and in a time series (d).


dl ¼ T e − TðcÞ − KðlÞ; dt

ð2Þ

where the tension generated by the contractile unit TðcÞ is a function of the concentration of force-producing molecules [24,25], while KðlÞ is the elastic restoring force of the spring element. Both TðcÞ and KðlÞ are assumed to be monotonic functions of their arguments; μ is a damping coefficient, and T e is an external tension opposing deformation of the unit [Fig. 2(a)]. At steady state, the external tension balances the internal tension such that T e ¼ Tðc0 Þ þ Kðl0 Þ. Expanding around the steady state, we write TðcÞ ¼ Tðc0 Þ þ t1 ðc − c0 Þ and KðlÞ ¼ Kðl0 Þþ k1 ðl − l0 Þ þ k3 ðl − l0 Þ3 . For simplicity, we have assumed here a symmetric spring response k2 ¼ 0, leaving three nondimensional parameters determining the dynamics of the system: ðt1 c0 Þ=ðk1 l0 Þ, μ=ðk1 τÞ, and k3 =ðk1 l20 Þ (Supplemental Material [23]). A linear stability analysis shows that the system undergoes a Hopf bifurcation [26–28] for ðt1 c0 Þ=ðk1 l0 Þ ¼ 1 þ μ=ðk1 τÞ (Supplemental Material [23]). The instability occurs as contraction of the material leads to an increase of its density, leading to an even larger contractile force [29–31]. Turnover balances out this effect by attempting to restore the reference concentration c0 . Above the bifurcation, a limit cycle appears and the system undergoes spontaneousposcillations. ffiffiffiffiffiffiffiffiffiffiffiffi We find that the period at bifurcation is τ0 ¼ τμ=k1 , and the transition is supercritical for large enough nonlinear elasticity, 3½ðk3 l20 Þ=k1 ½μ=ðk1 τÞ þ 1 > 2½μ=ðk1 τÞ2 þ 6½μ=ðk1 τÞ þ 4. Further away from the bifurcation, we turn to numerical simulations to investigate the dynamics of the contractile unit. k3 is chosen large enough to ensure that the transition is supercritical. The complete phase diagram is plotted in Fig. 2(b) as a function of the reduced parameters ðt1 c0 Þ=ðk1 l0 Þ and μ=ðk1 τÞ. On an increase of the tension t1 away from the Hopf bifurcation, a region appears where the system collapses to l ¼ 0 within a finite time [see Figs. 2(c) and 2(d), green lines]. The exact position of the transition separating stable oscillations from collapse depends on the nonlinearities in the functions TðcÞ and KðlÞ and is numerically evaluated for k3 l20 =k1 ¼ 15 in Fig. 2(b). This collapsing behavior could possibly be related to the delamination of cells whose apical area vanishes [32,33].

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To compare these results with experimental observations, we note that the slope of the curve in Fig. 1(d) allows us to estimate the turnover time scale τ ≃ 108 66 s (n ¼ 73). The time scale μ=k1 is of the order 90 50 s [4]. With these estimates, the period at threshold would be τ0 ¼ 425 150 s, comparable to the measured period of cell oscillations, τ0 ¼ 230 76 s [4]. Collective oscillations.—In a tissue, adjacent oscillating cells are mechanically coupled via adhesion proteins. Therefore, we now investigate the collective dynamics of a one-dimensional chain of N mechanically coupled oscillating units [Fig. 3(a)]. We denote by xn the position of the nth boundary between two contractile units and by ln ¼ xnþ1 − xn the length of one unit. An additional friction force with friction coefficient λ opposes movement of the boundaries between the units relative to a fixed external substrate. The mechanical equation for the position of one lattice point xn then reads λ

dxn ¼ f n − f n−1 ; dt

f n ¼ Tðcn Þ þ Kðln Þ þ μ

dln ; ð3Þ dt

where f n is the total force exerted by one contractile unit. The concentration of the nth contractile unit cn still follows Eq. (1). We start by considering the case of vanishing external friction λ → 0. In that limit, the total force in each contractile unit is constant, f n ¼ f. The chain is assumed to be in contact with a spring of elasticity kext at its boundaries, such that the total force is given by f ¼ ðkext =2ÞðNl0 − LÞ, with L ¼ xN − x0 the total length of the chain. In the limit of a free chain, i.e., kext → 0, the oscillators are completely uncoupled and oscillate independently. Taking kext ≠ 0 introduces a global coupling between the oscillators, as the contraction of one oscillator induces a reduction of the total length and, thus, an increase in the total force along the chain that acts equally on every oscillator [Fig. 3(b)]. To investigate the properties of the globally coupled oscillators, we perform a linear stability analysis of the regular lattice distribution by decomposing the motion of the lattice points in phonons x~ k ¼ P 2πink . At linear order, all modes k ≠ 0 n ðxn − nl0 Þe behave as a single unit with elastic modulus k1 and undergo a Hopf bifurcation at the same threshold value of t1 c0 =k1 l0 (see Supplemental Material [23]). Although all collective modes are equally unstable at linear order, nonlinearities can give rise to patterns of collective oscillations. To investigate those, we have simulated a system of N oscillators [Fig. 3(c)] and numerically performed a phase reduction of the dynamical system, associating a phase 0 < ϕ < 2π such that ϕ_ ≡ const on the limit cycle. By analyzing the distribution of phases [Fig. 3(e)], we find that the chain of oscillators transitions from an uncoupled state with random phases for small external coupling kext to clustered states [Fig. 3(d)]. In a clustered state, the oscillators segregate into subpopulations

FIG. 3 (color online). (a) Schematic of a chain of oscillators, experiencing a friction relative to an external substrate. (b) Without friction, contraction of one unit (black) drives expansion of all other units (gray). With friction, a local contraction first triggers expansion of the nearest neighbors. [(c),(f)] Kymographs showing the time evolution of N ¼ 60 oscillators, with gray level coding for the concentration in each oscillator, without (c) or with (f) external friction. (d) Phase diagram of the collective oscillation modes for the chain shown in (a) (k3 =k1 ¼ 6, t1 c0 =k1 l0 ¼ 1.4, N ¼ 60). Regions correspond to clustering into two or more groups of synchronized oscillators (PO ¼ partially ordered; UO ¼ unordered). (e) Histograms of phase distributions in different regions of the diagram in (d). (g) Real parts of the eigenvalues of the linearized dynamics around the homogeneous state sðkÞ, as a function of the wave vector k, for ðτk1 Þ=μ ¼ 1 and μ=λ ¼ 0. k ¼ 12 corresponds to a mode with nearest neighbors oscillating in antiphase.

that are fully synchronized, giving rise to several peaks in phase histograms [Fig. 3(d)]. We quantify clustering by an order parameter defined as Z κ ¼ max κn ; κ n ¼ n

0

2π

−inϕ

pðϕÞe

dϕ;

ð4Þ

where pðϕÞ is the density of phases at a given time. We define a system to be clustered when κ > 0.93 and partially

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ordered when κ > 0.3. The number of clusters appears to depend on the oscillator parameters as well as on the external spring modulus [Fig. 3(d)]. We never find fully synchronized states. Because the system is invariant under permutation of the oscillators, there is no spatial ordering of the clusters [Fig. 3(b)]. Because contraction of one unit leads to expansion of other units, the coupling between unit oscillators is effectively repulsive. The frictionless chain studied here is, therefore, in the class of globally coupled oscillators with a repulsive coupling [18,34,35]. We now consider the case of nonvanishing friction λ ≠ 0 for a periodic chain of oscillators of constant length L. Linear stability analysis around the homogeneous state indicates that the nearest-neighbor antiphase oscillations mode k ¼ 12 is the first mode to become unstable as the tension t1 is increased, at the threshold ðt1 c0 Þ=ðk1 l0 Þ ¼ 1 þ μ=ðτk1 Þ þ λ=ð4τk1 Þ [Fig. 3(g)]. Above the threshold, the units indeed oscillate in phase opposition in simulations [Fig. 3(f)]. The difference in oscillation patterns with and without friction can be understood as follows: without friction, the contraction of one unit results in a force acting equally on all other units, as the total force f is conserved along the chain. With friction, contraction of one unit initially only drives expansion of the nearest neighbors [Fig. 3(b)]. Diffusion of force-producing molecules.—We have considered so far contractile elements with independent concentrations c, describing mechanically connected cells that do not exchange material. However, within individual amnioserosa cells, the actomyosin cortex also exhibits pulsing dynamics, with propagating actomyosin contractile waves appearing at the cell surface [14,36] [Fig. 4(a) and Supplemental Material Fig. S3 [23]]. We measured the velocity of waves in a subset of cells exhibiting unidirectional propagation [N ¼ 15 cells, Fig. 4(b), Supplemental Material Fig. S3 [23]] and found that one pulse was propagating per cell with an average velocity of 13.4 4.6 μm · min−1 . To explain these observations, we now turn to the case where the concentration in each oscillating unit diffuses between nearest neighbors. The chain of contractile units can then be replaced by a continuum description in the limit l0 → 0, N → ∞. Such a description is appropriate to describe the actomyosin network within cells [Fig. 4(a)], where force producing elements are free to diffuse. The equations in the continuum limit read 1 c ∂ t c ¼ ðc0 − cÞ − ∂ v þ D∂ 2x c; τ 1 þ ∂ x ux x x

ð5Þ

f x ¼ E∂ x ux þ E3 ð∂ x ux Þ3 þ η∂ x vx þ TðcÞ;

ð6Þ

∂ x f x ¼ αvx ;

ð7Þ

where E ¼ k1 l0 , E3 ¼ k3 l30 are two elastic moduli, η ¼ μl0 is the material viscosity, α ¼ λ=l0 is a friction coefficient,

(a)

(b)

(d)

(c)

(e)

FIG. 4 (color online). (a) Snapshots of a traveling actomyosin pulse (white arrows) in an amnioserosa cell [labeling as in Fig. 1(b)]. (b) Kymograph of the myosin intensity along the dashed line in (a). (c) Schematic for a chain of oscillators with diffusion of the concentration between different contractile units. (d) Snapshot of a result of a simulation of Eqs. (5)–(7) with periodic boundary conditions [ðDαÞ=E ¼ 0.1, ðc0 t1 Þ=E ¼ 3.1, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η=ðτEÞ ¼ 1, E3 =E ¼ 15, η=ðαL2 Þ ¼ 0.04]. The direction of wave propagation depends on initial conditions (black dashed arrow). Inset: kymograph with gray level coding for different levels of the concentration c. (e) Wave speed as a function of ðc0 t1 Þ=E [parameters as in D, except ðDαÞ=E ¼ 0.05]. Away from threshold, solutions exist for different number of peaks N. Dashed lines: instability threshold (vertical), wave speed at threshold (horizontal).

x ¼ nl0 is a coordinate labeling units in the initial resting frame, ux ðxÞ is a displacement field such that the position of the nth oscillator is xn ¼ nl0 þ ux ðnl0 Þ, and vx ¼ dux =dt. Performing a spatial Fourier transform of Eqs. (5)–(7), we find that the homogeneous state becomes unstable to an oscillatory periodic instability above the threshold tension pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t1 c0 ¼ Eþη=τ þDαþ2 ðDαηÞ=τ (Supplemental Material [23]). For nonzero friction α ≠ 0 and at threshold, the system admits left or right traveling wave solutions with velocity v ¼ ½ðDE2 Þ=ðαητÞ1=4 and wavelength λ ¼ 2π½Dðη=αÞτ1=4 . Simulations of Eqs. (5)–(7) in a box of size 2L with a spectral method, indeed, yield traveling wave solutions whose velocity increases with increasing t1 pffiffiffiffiffiffiffi ffi [Figs. 4(d)–4(e)]. Taking η=α ≃ 14 μm [10], η=E ¼ μ=k1 ≃ 90 s, τ ≃ 108 s, and D ¼ 0.01 μm2 · s−1 , we find a velocity at threshold v ≃ 2.4 μm · min−1 and wavelength λ ≃ 24 μm, comparable to the pulse velocity and size of an amnioserosa cell. Summary.—Motivated by actomyosin-driven cell shape changes, we have studied the individual and collective dynamics exhibited by a generic contractile unit with

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turnover. Our description exhibits several dynamic regimes and captures essential features of cells pulsating in vivo during dorsal closure of the Drosophila embryo. On the tissue scale, we predict patterns of phase distributions, the experimental analysis of which could glean insights into the relative importance of viscous vs friction losses in morphogenetic contexts. We are grateful to the Advanced Light Microscopy Unit (ALMU) at the CRG. The research leading to these results has received funding from the Spanish Ministry of Economy and Competitiveness (Ministerio de Economia y Competividad; Plan Nacional BFU2010-16546) and from the Max Planck Society.

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