Activity induces traveling waves, vortices and spatiotemporal chaos in a model actomyosin layer

Inspired by the actomyosin cortex in biological cells, we investigate the spatiotemporal dynamics of a model describing a contractile active polar fluid sandwiched between two external media. The external media impose frictional forces at the interface with the active fluid. The fluid is driven by a spatially-homogeneous activity measuring the strength of the active stress that is generated by processes consuming a chemical fuel. We observe that as the activity is increased over two orders of magnitude the active polar fluid first shows spontaneous flow transition followed by transition to oscillatory dynamics with traveling waves and traveling vortices in the flow field. In the flow-tumbling regime, the active polar fluid also shows transition to spatiotemporal chaos at sufficiently large activities. These results demonstrate that level of activity alone can be used to tune the operating point of actomyosin layers with qualitatively different spatiotemporal dynamics.


Model
We consider a two-dimensional active polar fluid in the x-y plane described by a continuum hydrodynamic theory (see Hydrodynamic equations of active polar fluids in Sec. Methods). This corresponds to the case of a three-dimensional system with translational invariance and zero polarity component in the z-direction. The x and y components of the polarity field p at each point is denoted by p x and p y , such that = p 1. The components of the velocity field υ are denoted by υ x and υ y . The fluid has a thickness L y in the y-direction, and length L x in the x-direction. We impose a friction boundary condition for the flow along = y 0 and = y L y so that the shear stress σ µ υ ( , = ) = ( , = ) x y x y 0 0 xy b x and σ µ υ ( , = ) = − ( , = ) x y L xy L xy y t x y , where µ b and µ t denote the friction coefficients at the bottom and top surfaces respectively. This flow boundary condition is a generalized slip boundary condition that models the effect of (different) frictions due to the cytosol on one side of the actomyosin gel and the membrane on the other. The normal component of the velocity υ y at = y 0 and = y L y vanishes. The polarity along the surface = y 0 and = y L y are anchored parallel to the surface (see Fig. 1 for an illustration of the model).
The hydrodynamic model is parametrized by the following material constants: viscosity of the fluid η, orientational friction of the polar filaments γ, the elastic constants of the polarity field K (considering , and a coefficient ν coupling the rate of change of polarity with the strain rate (see Hydrodynamic equations of active polar fluids in Sec. Methods for more details). The fluid is subjected to activity α that is spatially-homogeneous. We choose these parameters by constraining our active polar fluid model to be contractile 11,14 and spontaneously flowing like an actomyosin cortex in biological cells 7,8 . First, in order to ensure that the active stresses are contractile, we enforce that α > 0 (see Eq. 1). In addition, spontaneous flow of the fluid requires that the coupling coefficient ν > −1 (see Sections Critical activity α c for spontaneous flow transition and Model parameters in Sec. Methods for details). If ν < 1, the active fluid model is known to be in a flow-tumbling regime and in the flow-aligning regime otherwise 17,22 . Individual actin filaments have been observed to be flow-tumbling 30 supported by theoretical predictions and experimental observations of similar rod-like polymeric liquid crystals 31 . Response of individual actin filaments that are chemically interacting with other biomolecules in an in vivo actomyosin cortex is, however, unclear. We therefore study the flow patterns in the flow-tumbling as well in the flow-aligning regimes. See Model Parameters in Sec. Methods for the values of all parameters for the two regimes used in our study.
With the given parametrization of the model and periodic boundary condition in the x-direction, we numerically solve the equations governing the hydrodynamics using a recently developed general hybrid particle-mesh method for incompressible active polar viscous gels 24 . The method imposes the unit vector polarity and incompressibility constraints exactly, and has shown to be consistent, stable and therefore convergent 24 . The initial condition and settings used for the numerical simulations are presented in Sec. Methods (see Settings for numerical simulation).

Results
We present the numerical solution to the hydrodynamic equation governing the model system 24 . We study the polarity and flow dynamics as activity α induces larger contractile stresses in the active polar fluid model. Specifically, we characterize the polarity and flow dynamics as α α / c is increased over two orders of magnitude where α c is the critical activity (lower bound of Eq. 9) beyond which spontaneous flows occur. Over this range of activities, nonlinearities in the hydrodynamics become significant and cannot be ignored. As the activity is increased, we observe that the contractile active polar fluid in the flow-tumbling regime undergoes 3 transitions: spontaneous flow transition, transition to oscillatory dynamics, and finally a transition to spatiotemporal chaos. In contrast, the flow-aligning regime only shows 2 transitions: the spontaneous flow transition and the transition to oscillatory dynamics. The oscillatory dynamics and the chaotic dynamics are not predicted by the linearized equations. These transitions are therefore due to the nonlinearities in the hydrodynamic equations that are neglected in the linearized regime. In the following sections, we present the results in the flow-tumbling regime before summarizing the differences for the flow-aligning regime towards the end of the section.

Spontaneous flow.
For activities such that 0 < (α/α c ) < 1, the steady-state polarity field is spatial homogeneous and the velocity is zero. When α α ( / ) c is increased beyond 1, the active polar fluid undergoes a spontaneous flow transition as predicted by the linear perturbation analysis (see Critical activity α c for spontaneous flow transition in Sec. Methods). Figure 2 shows the steady-state polarity and velocity fields when (α/α c ) = 3 for the flow-tumbling regime. We observe that the steady-state polarity and velocity fields are translationally invariant along the x-direction with a finite velocity in the x-direction. The translational invariance in the x-direction along with the incompressibility constraint render velocity υ y in the y-direction zero. The velocity υ x along the x-direction, however, is finite owing to non-zero gradients in the polarity field in the y-direction.
In summary, the numerical solution confirms the theoretical prediction of spontaneous flow transition beyond critical activity. Previous numerical studies have also confirmed such spontaneous flow transition in active polar fluids albeit with no slip boundaries 22,27 instead of the friction boundary conditions used in the current study. Traveling waves and traveling vortices. As the activity α is further increased, we observe that the translational symmetry in the x-direction is spontaneously broken beyond α α ( / ) ≈ 12 c . Figure 3(A) shows a snapshot of the polarity and velocity fields after a long time for α α ( / ) = 15 c in the flow-tumbling regime. The polarity and velocity fields in Fig. 3(A) show no translational invariance in the x-direction, rendering υ y non-zero. In addition, closed streamlines of the velocity field show the presence of vortices in the flow fields.
The spatial polarity and velocity pattern observed in Fig. 3(A) travel in the x-direction with time. In order to demonstrate the traveling wave pattern, we compute the spatiotemporal correlation function δ δ δ ( , , ) C x y t of the polarity field (see Spatiotemporal correlation in Sec. Methods). Figure 3(B) shows δ δ δ ( , , ) C x y t for three values of δ = , t 0 2 and 4. We observe that δ δ δ ( , , ) C x y t is merely translated in the x-direction as δt is increased beyond 0. This observation shows that the polarity field and the velocity field travel in the x-direction with time.
We next analyze the spatiotemporal frequency spectrum of the polarity field to characterize the nature of the traveling wave pattern. The spatiotemporal Fourier spectrum θ ω ( , , ) k k x y is computed as a function of angular wavenumber k x along the x-direction, angular wavenumber k y along the y-direction and angular temporal frequency ω. Subsequently, the power spectrum θ ω | ( , , )| k k x y 2 is computed normalized by the total power. after integrating out the dependence on k y . We observe that a significant fraction of the power is concentrated around ω = −c k gx x with ≈ . c 1 3352 gx . A constant c gx indicates that the traveling wave is non-dispersive with a constant group velocity c gx in the x-direction. In summary, as the activity increases the contractile active fluid in the flow-tumbling regime shows oscillating spatiotemporal patterns. These spatiotemporal patterns consist of non-dispersive traveling waves accompanied by traveling vortices in the flow field. Such oscillating spatiotemporal patterns are not predicted by the linear perturbation analysis (see Critical activity α c for spontaneous flow transition in Sec. Methods), and is therefore an effect mediated by nonlinearities in the model.

Spatiotemporal chaos. As the activity is further increased beyond α α
( / ) ≈ 90 c , we observe that the traveling waves in the polarity and velocity fields disappear. Figure 4(A) shows the polarity and the velocity fields at two close time-points for α α ( / ) = 100 c in the flow-tumbling regime. Visual inspection of the polarity and the velocity fields indicates irregular spatiotemporal dynamics. In addition, the velocity fields in Fig. 4(A) shows several vortices characteristic of a turbulent flow pattern.
In order to investigate whether the dynamics is chaotic, we compute the maximum Lyapunov exponent 32 (see Maximum Lyapunov exponant in Sec. Methods). The maximum Lyapunov exponent λ is a measure of sensitivity to small perturbations, with a positive value signifying chaos 32,33 . We find that λ ≈ . ± . 0 14 0 025. A positive maximum Lyapunov exponent indicates that a small perturbation in the polarity field gets amplified over time making the dynamics temporally decorrelated. Since the velocity field is coupled to the polarity field, we conclude that both the polarity and velocity fields shown in Fig. 4(A) are therefore temporally chaotic.
To investigate if the spatiotemporal dynamics are irregular, we compute the spatiotemporal correlation function 33 δ δ δ ( , , ) C x y t of the polarity field (see Spatiotemporal Correlation in Sec. Methods). Figure 4(B) shows δ δ δ ( , , ) C x y t for increasing δt. We observe that spatial correlations progressively disappear as δt is increased. This implies that the polarity field gets spatially decorrelated in time 33 . Taken together with the positive maximum Lyapunov exponent, this shows that both the polarity and velocity fields are spatially and temporally irregular revealing characteristics of spatiotemporal chaos.
In summary, for large activities the contractile active fluid in the flow-tumbling regime exhibits spatiotemporal chaos characterized by irregular spatiotemporal patterns. Like the oscillatory dynamics, the linear perturbation analysis (see Critical activity α c for spontaneous flow transition in Sec. Methods) does not predict spatiotemporal chaos, and is an effect of nonlinearities in the model equations.
Comparison and summary of dynamics for flow-tumbling and flow-aligning regimes. , the steady state polarity field has non-zero gradients in y-direction and is translationally invariant along the x-direction. As a consequence the velocity along the x-direction is non-zero (see Sec. Spontaneous flow). For   α α ( / ) 12 9 0 c , λ is 0 and the spatiotemporal dynamics of the active polar fluid is oscillatory. The oscillatory dynamics is characterized by traveling waves accompanied by traveling vortices in the flow field (see Sec. Traveling waves and traveling vortices). For  α α ( / ) 90 c , λ is greater than 0 indicating that the dynamics is temporally chaotic. In addition, spatiotemporal correlations disappear over a short time interval and the dynamics is therefore an instance of spatiotemporal chaos (see Sec. Spatiotemporal chaos). Thus, the contractile active polar fluid in the flow-tumbling regime undergoes 3 transitions: spontaneous flow transition, transition to oscillatory dynamics, and a transition to spatiotemporal chaos.
In contrast, in the flow-aligning regime, the contractile active polar fluid does not exhibit spatiotemporal chaos as α α ( / ) c is varied over two orders of magnitude (from . 0 7 to 240). Over this range of activity the contractile active polar fluid in the flow-aligning regime only shows two transitions, namely the spontaneous flow transition and transition to oscillatory dynamics. Figure 5 Next, we investigate the effect of friction at the boundaries on the activity required for the transition from a steady-state flow to traveling waves for both the flow tumbling and flow aligning regimes. We also study the activity required for transition from traveling waves to spatiotemporal chaos for the flow tumbling regime. For simplicity, we choose the friction coefficients at both boundaries to be equal, that is µ µ µ = = t b . A friction coefficient μ = 0 corresponds a stress-free boundary while the limit of very large μ corresponds to a no slip boundary. Figure 5(C,D) show the activity thresholds for transition in the flow-tumbling and flow-aligning cases respectively, as μ is increased within a range from 10 −4 to 10 4 . We observe that for the flow tumbling case (Fig. 5(C)), the activities required for the transition from spontaneous flow to traveling waves is independent of μ within the numerical uncertainty of the threshold. For the flow aligning regime (Fig. 5(D)), this threshold, however, increases before saturating at large μ. The activities required for the transition from traveling waves to spatiotemporal chaos that is observed only in the flow tumbling regime also increase before saturating at large μ Scientific RepoRts | 6:20838 | DOI: 10.1038/srep20838 ( Fig. 5(C)). Nevertheless, we find that the required activities for all of the observed transitions increase only by at most 50% as μ is increased over 8 orders of magnitude.

Discussion
We have studied the dynamics of active polar fluids at low Reynolds numbers as a function of a spatially-homogeneous activity measuring the strength of the active stress in the system. We consider a two-dimensional contractile, active polar fluid sandwiched between two surfaces. The surfaces impose frictional forces at the interface modeling the effect of membrane on one side and the cytosol on the other side of an actomyosin cortex in biological cells. The spatiotemporal dynamics of such an active fluid is described by a nonlinear continuum hydrodynamic description 1-3 . We numerically solve the hydrodynamic equations using a hybrid particle-mesh method 24 when the fluid is subjected to activities over two orders of magnitude. The active polar fluid not only shows transition to spontaneous flow as predicted by linear perturbation analysis, but also transitions to oscillatory spatiotemporal patterns, and even spatiotemporal chaos as the activity is increased. In the flow-aligning regime, where the polarity field tends to align with local shear, the model exhibits spontaneous flow transitions and transitions to oscillatory spatiotemporal patterns. The oscillatory spatiotemporal dynamics is accompanied by traveling waves and traveling vortices in the flow field. In the flow-tumbling regime, where the polarity field tends to tumble in local shear flow, the model also exhibits a transition to spatiotemporal chaos as the activity is increased, resulting in irregular spatiotemporal dynamics. The chaotic regime is characterized by the maximum Lyapunov exponent of the spatiotemporal dynamics which we determine numerically. The transitions are mediated by nonlinearities in the hydrodynamic description. This is the first time such transitions have been shown in active polar fluids subjected to spatially homogeneous activity. The results therefore suggest that the level of activity alone can tune the operating point of an actomyosin layer characterized by qualitatively different spatiotemporal dynamics.
These results suggest possible mechanisms for some observed biological phenomena and experimentally testable predictions. For example, oscillations and traveling waves in actomyosin cortex have been observed in amnioserosa cells during dorsal closure in Drosophila embryos 13 and in periodically protruding cells 34 . Even though the presented model may be too simple for a direct comparison to these experiments, our results show that a homogeneous level of activity alone is sufficient for generating such behavior. In addition, our model predicts onset of irregular spatiotemporal dynamics as the activity is increased.
In our study, we have used several simplifications. The model is two-dimensional with the third dimension considered transitionally invariant. The finite magnitude of small perturbations to the initial condition limit the precision of numerically determining the transition thresholds. Further, we consider the case where the activity of the system is spatially homogeneous and are not regulated by other components in the system. Such regulatory mechanisms might either increase or decrease the activity needed for spatiotemporal chaos in in vivo actomyosin layers. We have also not considered the effect of actin filament turnover, actin polymerization and . multicomponent nature of in vivo actomyosin layers. Intrinsic fluctuations in actomyosin layers that might be play an important role in mediating its dynamics have also not been considered [35][36][37] . Additionally, due to the constant magnitude constraint of the polarity field and the polarity boundary conditions, our model does not show topological defects in the polarity field. The effect of defects on the spatiotemporal dynamics of our model active fluid remains to be investigated. This investigation requires removing the constant magnitude constraint of the polarity field. Future work will focus on relaxing some of these simplifications to gain more insight into active polar fluids for modeling the actomyosin cortex. We envision that a systematic investigation of such model actomyosin layers together with in vitro and in vivo experiments will help improve our understanding on the role of activity in actomyosin cortex mediating crucial morphogenetic phenomena in developing organisms.

Methods
Hydrodynamic equations of active polar fluids. Denoting the polarity and velocity at position r at time t by ( , ) r p t i and υ ( , ) r t i respectively (i ∈ {x, y}), the hydrodynamic description of incompressible active polar fluids in two-dimensions assuming negligible inertial forces is made up of a constitutive relation, the Onsager relation for the polarity field, force balance condition and the incompressibility constraint [1][2][3]16,20,24 .
The constitutive relation of active polar fluid reads where σ ( ) s is the symmetric part of the deviatoric stress tensor with components σ ( ) ij s , h is the molecular field vector with components h i , δ ij are the components of the Kronecker-delta tensor such that δ ij = 1 if = i j and 0 otherwise, and k k ij are the components of the symmetric, traceless part of the velocity-gradient tensor. In Eq. 1, the parameter η is the viscosity of the fluid, ν is the coefficient coupling mechanical stress to polarity field, and α is the activity measuring the active stresses induced by consumption of energy. If α < 0, the active stress is extensile, and if α > 0 the active stresses are contractile.
The equation of motion for the polarity field is given by the Onsager relation: i j are the components of the vorticity tensor (the anti-symmetric part of the velocity-gradient tensor). In Eq. 2, the parameter γ is the orientational friction, and the same coefficient ν in Eq. 1 acts a coefficient between the rate of change of polarity and strain-rate u ij . The coefficient ν in Eq. 2 describes alignment or tumbling of the polarity field by local shear flow 2,17 . If ν < 1, the polarity field tumbles in the local shear flow whereas for ν ≥ 1 the polarity field tends to align with the local shear flow.
The force balance condition and the incompressibility constraint are given by respectively. Here, Π is the pressure and σ ij is one of the four components of the deviatoric stress tensor σ. The deviatoric stress σ is a sum of a symmetric stress σ ( ) s (Eq. 1), an antisymmetric stress σ ( ) a : i j j i a and the Ericksen stress σ ( ) e which is an equilibrium stress generalizing the hydrostatic pressure to anisotropic fluids 38,39 .
The components σ ( ) ij e of the Ericksen stress tensor and the components h i of the molecular field vector h are defined as a function of a distortion free-energy density f: The distortion or the Franck free-energy density defines the increase in the energy density due to distortions in the polar nematic liquid crystals from its uniformly aligned configuration and is defined as where  ij are the components of the permutation (Levi-Civita) tensor. The free-energy density f is parametrised by K s , the splay elastic constant and K b , the bend elastic constant. We neglect the twist elastic constant since it is irrelevant in two dimensions. The free-energy density also includes a contribution from a component h 0 of the molecular field assuming that fluctuations in polarity orientation dominate the fluctuations in polarity amplitudes. This assumption implies that the amplitude p p k k is a constant and can be assumed to be 1 without loss of generality 2  Equations 1, 2 and 3, along with boundary conditions and initial polarity field fully describe the hydrodynamics of incompressible active polar fluids. We numerically solve these equations using a recently developed hybrid particle-mesh method for incompressible active polar fluids. For details on the computational method, refer to Ramaswamy et al. (2015) 24 .
Critical activity α c for spontaneous flow transition. We consider an active polar fluid that is translationally invariant along the x-direction and has a thickness L y in the y-direction. The surface of the fluid at = y 0 and = y L y is impenetrable (υ ( , , ) =  A trivial solution to these equations (satisfying the boundary conditions) at steady state is θ = 0 ( = p 1 x and = p 0 y ) and υ = 0 x . Expanding Eq. 8 around a small perturbation ε to the steady state configuration of polarity, and using the polarity and friction boundary conditions at = y 0 and = y L y , we find that ε is finite and non-zero for activities α α / > 1 c where the critical activity α c is within an interval given by π η γ ν γ ν α π η γ ν γ ν For activities α α ( / ) > 1 c , the polarity θ is finite leading to a spontaneous flow transition since υ > 0 x due to gradients in polarity along the y-direction 2,18,22,24 . This transition is similar to the classical Fréedericksz transition of nematic liquid crystals in which the transition is brought about by an external magnetic field and not by internal active stresses 2,18,38 . Model parameters. We define the parameters in our model scaled by units for length l, time τ and stress σ.
where the elastic constants in Eq. 5. The unit for stress σ = / K l 2 . The unit for length l is chosen to be 1. In these units, we choose viscosity η = 1. The dimensionless coupling coefficient ν is chosen in order to reflect properties of contractile 11,14 and spontaneously flowing 7-9 active systems like actomyosin gels. In our hydrodynamic description (Sec. Hydrodynamic equations of active polar fluids), contractile active stress in Eq. 1 requires that activity α contributes to positive stress in the direction of polarity 2,18 . Positive active stress parallel to polar direction is ensured when α > 0. In addition, for contractile active fluids to exhibit spontaneous flow for α α ( / ) > 1 c , the critical activity α c (Eq. 9) needs to be positive. Positive critical activity is ensured if ν > − 1 in Eq. 9. Therefore, contractile active polar fluids capable of exhibiting spontaneous flow can be realized in a flow-tumbling regime where ν − < < 1 1 or in a flow-aligning regime where ν ≥ 1 2,17,18,22 . For the flow-tumbling regime, we choose ν = − / 2 5 and for the flow-aligning regime, we choose ν = / 6 5.
Friction at the boundaries. We impose a friction boundary condition along = y 0 and = y L y such that the shear stress σ µ υ ( , = ) = ( , = ) x y x y 0 0 xy b x and σ µ υ ( , = ) = − ( , = ) x y L xy L xy y t x y . We set µ = 1 t . Assuming that the friction coefficients are different at the two surfaces in general, we set µ µ µ ( / ) = t b ratio . Without loss of generality, we assume that the friction coefficient µ t at = y L y is greater than or equal to the friction coefficient µ b at = y 0. Therefore, µ ≥ 1 ratio . For all simulations presented in the paper, we set µ = 4 ratio . We, however, verified that the results are qualitatively unaltered as µ ratio is varied between 1 to 10.
Settings for numerical simulation. For the numerical simulation using the hybrid particle-mesh method 24 , we use = = L L 10 x y , and discretize the computational domain into 65 mesh nodes in each direction. The time integration is performed using a time-step length of 0.0004 using fourth-order Runge-Kutta time integration scheme 24 .
The initial condition for the polarity field p is a trivial spatially homogeneous steady state of the model active polar viscous layer except for a small perturbation to the polarity field at the centre of the computational domain.