Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Ludwig A.
Hoffmann
^{a},
Koen
Schakenraad
^{ab},
Roeland M. H.
Merks
^{bc} and
Luca
Giomi
*^{a}
^{a}Instituut-Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands. E-mail: giomi@lorentz.leidenuniv.nl
^{b}Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
^{c}Institute of Biology, Leiden University, P.O. Box 9505, 2300 RA Leiden, The Netherlands

Received
13th September 2019
, Accepted 28th November 2019

First published on 2nd December 2019

Recent experiments on monolayers of spindle-like cells plated on adhesive stripe-shaped domains have provided a convincing demonstration that certain types of collective phenomena in epithelia are well described by active nematic hydrodynamics. While recovering some of the hallmark predictions of this framework, however, these experiments have also revealed a number of unexpected features that could be ascribed to the existence of chirality over length scales larger than the typical size of a cell. In this article we elaborate on the microscopic origin of chiral stresses in nematic cell monolayers and investigate how chirality affects the motion of topological defects, as well as the collective motion in stripe-shaped domains. We find that chirality introduces a characteristic asymmetry in the collective cellular flow, from which the ratio between chiral and non-chiral active stresses can be inferred by particle-image-velocimetry measurements. Furthermore, we find that chirality changes the nature of the spontaneous flow transition under confinement and that, for specific anchoring conditions, the latter has the structure of an imperfect pitchfork bifurcation.

First identified as a broken symmetry in certain types of cell cultures,^{10} and later exploited to decipher their static^{11} and dynamical properties,^{2,4–8} nematic order has surged as one of the central themes in collective cell dynamics. In layers of spindle-like cells, where the local orientation can be unambiguously identified, nematic order is marked by the presence of ±1/2 disclinations,^{4,6}i.e. point-like singularities about which the average cellular orientation rotates by ±π. Consistently with the predictions of active nematic hydrodynamics,^{12,13} these defects self-propel and pairwise annihilate until cell crowding freezes the system into a jammed configuration. Before dynamical arrest, the collective motion of the cells gives rise to a decaying turbulent flow at low Reynolds number, whose statistics, spatial organization and spectral structure are in exceptional agreement with the hydrodynamic picture^{14} (but see ref. 15 for an alternative theoretical picture based on glassy dynamics).

Another remarkable demonstration of active hydrodynamic behavior in eukaryotic cell layers has been recently reported by Duclos et al., upon confining spindle-like RPE1 and C2C12 cells within adhesive stripe-shaped domains.^{8} Depending on the width of the stripe the system was found either in a stationary state, with the cells parallel to the longitudinal direction of the confining region, or in a collectively flowing state characterized by a spontaneous tilt of the cells toward the center of the stripe. The latter picture, often referred to as spontaneous flow transition, had been anticipated for over a decade by Voituriez et al.^{16} and represents one of the hallmarks of active liquid crystals. While confirming this seminal prediction, however, Duclos et al. have also highlighted a number of unexpected features that could be ascribed to the existence of chirality over length scales larger than the typical size of a cell.

The notion of chirality is not new in active matter and has been theoretically explored well before the interest around collective cell dynamics in eukaryotes had started to blossom. Fürthauer et al., for instance, demonstrated that microscopic torque dipoles, such as those arising from rotating molecular motors or flagella, give rise to antisymmetric stresses and angular momentum fluxes, which, in turn, drive rotating flows and other chiral patterns on the large scale.^{17,18} More recently, Banerjee et al. showed that rotational motion at the microscopic scale further enriches the spectrum of hydrodynamical behaviors or chiral active fluids by giving rise to non-dissipative “odd” viscosity,^{19} analogous to that found in quantum Hall fluids.^{20} Whereas undoubtedly interesting and relevant for a broad class of biological and synthetic systems, these mechanisms appear however unsuited to account for the chirality observed in the experiments by Duclos et al., because of the manifest lack of rotational motion at the scale of individual cells.

In this article, we show that macroscopic chirality can arise in nematic cell layers as a consequence of a misalignment between the cell's local orientation and active forces, even in the absence of microscopic rotational motion (Section II). Collectively, this gives rise to a chiral and yet symmetric stress tensor that, in two dimensions, complies with the symmetries of the nematic phase. Next, we explore the effect of such a chiral stress on the active flow generated by ±1/2 disclinations and identify a characteristic signature of chirality from which the ratio between chiral and non-chiral active stresses can be experimentally estimated (Section III). Finally, following Duclos et al.,^{8} we investigate the hydrodynamic stability of a chiral nematic cell monolayer confined on adhesive stripes and subject to various boundary conditions and classify all possible scenarios arising from the interplay between the geometry of the confining region, the extensile/contractile stresses and chirality (Section IV).

σ^{a} = σ_{‖}nn + σ_{⊥}n^{⊥}n^{⊥} + τ(nn^{⊥} + n^{⊥}n). | (1) |

(2) |

(3) |

(4) |

Now, the magnitude of the active stresses P^{a}, α, and τ depend exclusively upon the distribution of the forces exerted by the cells along their contour. The simplest approximation of the force density field f_{c} consists then of a dipole of the form:

f_{c} = F_{c}δ(R_{c} − aν_{c}) − F_{c}δ(R_{c} + aν_{c}), | (5) |

σ^{a} = aρ[2F_{‖}〈ν_{c}ν_{c}〉 + F_{⊥}〈ν_{c}ν^{⊥}_{c} + ν^{⊥}_{c}ν_{c}〉], | (6) |

P^{a} = −aρF_{‖}, α = 2aρSF_{‖}, τ = aρSF_{⊥}, | (7) |

σ^{a}_{ij} = −P^{a}δ_{ij} + α_{0}Q_{ij} − 2τ_{0}ε_{ik}Q_{kj}, | (8) |

Some comments are in order. Although eqn (5) is only a rudimental approximation of the force field generated by an irregularly-shaped cell, considering a more involved force distribution does not change the qualitative picture with respect to the emergence of chiral stresses, as long as this is asymmetric with respect to the cell's longitudinal direction. To illustrate this concept, we discuss in Appendix A the case of a quadrupolar force distribution. Whereas the exact origin of this asymmetry is beyond the scope of the present article, the biophysical literature is not scarce of examples where chirality can be detected at the single-cell level. For instance, various mammalian cells, when plated on micropatterns, can break the left-right symmetry by suitably positioning their internal organelles with respect to the cell body.^{24} Analogously, chirality can emerge at the scale of the entire cell from the self-organization of the actin cytoskeleton.^{25,26} The broken symmetry can furthermore propagate over the mesoscopic scale and bias the cell's collective migratory motion.^{27}

Fig. 2 Pressure (a and b) and velocity (c and d) fields in proximity of ±1/2 defects obtained from the analytical solutions of eqn (9) for an extensile chiral active nematic with α = 4τ < 0. The configuration of the director is indicated by white lines for the case of +1/2 (a) and −1/2 (b) defects. For +1/2 defects, the velocity field at the core, thus the direction of motion of the defect, is tilted by an angle θ_{tilt} = arctan(1/2) ≈ 27° with respect to the defect polarity direction p = . For both chiral and achiral +1/2 defects, the pressure is anisotropic and larger toward the direction of motion of the defects. |

An analytical approximation of the flow driven by the active stresses in the surrounding of a disclination can be obtained by solving the incompressible Stokes equation with a body force resulting from the active stress associated with an isolated ±1/2 defect. Namely:

η∇^{2}v + f^{a}_{±} = ∇P, ∇·v = 0, | (9) |

(10) |

(11) |

∇^{2}P_{±} − ∇·f^{a}_{±} = 0. | (12) |

(13) |

(14) |

To calculate the flow velocity in proximity of a ±1/2 defect, we set, without loss of generality, ψ = 0, thus p = and p^{⊥} = ŷ and we assume the defect at the center of a circular domain of radius R. The exact velocity of the flow at the boundary of such a domain, hence the homogeneous solution v_{0}, is not relevant for the purpose of this discussion. In practice, this will be determined by the chemistry of the substrate and the possible presence of other topological defects in the same region.^{4} Under these assumptions, and carrying out algebraic manipulations as those in ref. 13, the flow velocity caused by a ±1/2 defect can be found from eqn (10), (11) and (13) in the form:

(15a) |

(15b) |

(16a) |

(16b) |

In summary, the presence of a symmetric chiral active stress, such as that embodied by the parameter τ, affects the flow generated by ±1/2 disclinations by stretching and rotating the velocity field in the surrounding of the defects (Fig. 2c and d). Most prominently, this results in a tilt in the direction of motion of +1/2 defects: i.e.v_{self} = v_{+}(r = 0) = R/(4η)(αp + 2τp^{⊥}). Thus +1/2 defects self-propel at an angle θ_{tilt} with respect to their orientation p [see eqn (14)]. Such an angle could in principle be measured in experiments on two-dimensional cell cultures, thus providing a direct measurement of the relative magnitude of the chiral stress. The same behavior has been reported in the case of actively rotating +1/2 defects, in the limit of vanishing angular velocity.^{34}

Fig. 3 Schematic representation of the channel of infinite length in x-direction and width L in y-direction. |

In the following, we extend and generalize the theoretical analysis by Duclos et al.^{8} by considering various experimentally relevant scenario in terms of boundary anchoring and flow. The hydrodynamic equations governing the dynamics of an incompressible (i.e. ∇·v = 0) active nematic liquid crystal are given by:^{16,36,37}

(17a) |

(17b) |

We solve eqn (17) in a rectangular channel which is infinitely long in x-direction and has width L in y-direction (Fig. 3). Because the channel is invariant for translations along the longitudinal direction, we assume for simplicity n and v to be independent of x, both while stationary and spontaneously flowing. Furthermore, incompressibility and mass conservation demand the y-component of the velocity to vanish identically, i.e. v_{y} = 0. Thus taking n = (cosθ,sinθ) and v = (v_{x},0), the eqn (17) reduce to:

(18a) |

(18b) |

(19) |

(20) |

(21) |

θ(0) = θ(L) = θ_{0}. | (22) |

Our analysis is complemented by numerical solutions of eqn (18) with various boundary conditions. For this purpose we rescale time by the viscous time scale τ_{ν} = ρL^{2}/η, length by the channel width L and stress by the viscous stress scale σ_{ν} = ρL^{2}/τ_{ν}^{2}, i.e. t → t/τ_{ν}, y → y/L, σ → σ/σ_{ν}. All the other quantities in Fig. 4 and 5 are rescaled accordingly.

Fig. 4 Bifurcation diagram of the spontaneous flow transition obtained from numerical (dots) and analytical (lines) solutions of eqn (18) for K/γ = L = 1, λ = −0.5 and various τ values. (a) No-slip boundary conditions and parallel anchoring (see Section IV.A.1). The chiral stress τ does not influence the critical α value, but weakly affect the post-transitional configuration of the nematic director. (b) Stress-free boundary conditions and special boundary anchoring θ(0) = θ(L) = −θ_{tilt}/2 (see Section IV.B.3). The chiral active stresses, embodied by the parameter τ, explicitly break the clock-counterclockwise symmetry of the lowest free-energy configuration rendering the pitchfork bifurcation “imperfect”. In this case, only one of the two branches of the bifurcation diagram is connected to the trivial solution, which may then be the only one observed experimentally. |

Fig. 5 Numerical solutions of eqn (18) in the dimensionless quantities defined above and for K/γ = 1, λ = −0.5 as well as different values of τ and α for three sets of boundary conditions. (a) and (b) are numerical solutions for the eqn (18) with parallel anchoring as well as no-slip boundary conditions [panel (a) and analyzed analytically in Section IV.A.1] and stress-free boundary conditions [panel (b) and Section IV.B.1], respectively. (c) Numerical solutions for stress-free walls and the stationary solution θ_{0} = −θ_{tilt}/2 = −arctan(2τ/α)/2 imposed at the boundary, see Section IV.B.3. |

∂_{y}^{2}δθ + q^{2}δθ = 0, | (23) |

(24) |

δθ = Csin(qy), qL = nπ, | (25) |

η∂_{y}v_{x} = δθ(2τsin2θ_{0} − αcos2θ_{0}). | (26) |

(27) |

1. Parallel anchoring.
If the nematic director is parallel to the channel walls θ_{0} = 0, thus:

As in non-chiral active nematics, the instability is triggered by splay deformations (i.e. transverse to the nematic director) and uniquely depends on the non-chiral active stress α. Furthermore, as in non-chiral active nematics,^{13} such a splay instability affects flow-aligning systems (i.e. λ > 1) in the presence of extensile active stresses (i.e. α < 0), and flow-tumbling systems (i.e. λ < 1) in the presence of contractile active stresses (i.e. α > 0). A critical α value is readily found in the form: α_{c} = 8π^{2}ηK/[γL^{2}(1 − λ)].

(28) |

At the onset of the transition, the constant C can be calculated upon expanding eqn (21) up to the third order in δθ, Then, using the solution of the linearized equation yields a cubic equation in C. Solving the latter gives (see Appendix B):

The spontaneous flow instability consists, therefore, of a standard pitchfork bifurcation whose relevant fields, θ and v_{x}, scale like (L − L_{c})^{1/2} at criticality, see Fig. 4a. Despite the fact that the chiral stress τ does not affect the instability of the stationary state, it leaves a clear signature on the post-transitional behavior of the flowing monolayers. This can be seen in Fig. 5a, showing numerical solutions of eqn (18) in the flowing state for various α and τ values. The most prominent effect of chirality, in this case, is evidently to render both the distortion of the nematic director and the associated flow asymmetric with respect to the channel centerline.

2. Homeotropic anchoring.
The case in which the nematic director is perpendicularly anchored to the channel walls, θ_{0} = π/2, yields:

In this case the instability is triggered by bending deformations (i.e. parallel to the nematic director). In contrast with the scenario of Section IV.A.1, flow-aligning systems are unstable in the presence of extensile active stresses, whereas strongly flow-tumbling systems (i.e. λ < −1), are unstable in the presence of contractile active stresses. The critical α value is readily found in the form: α_{c} = −8π^{2}ηK/[γL^{2}(1 + λ)].

(29) |

To conclude this section, we observe that for both parallel and homeotropic anchoring, the spontaneous flow instability crucially relies on the flow-alignment behavior of the system, governed by the phenomenological parameter λ. As for molecular liquid crystals, where λ depends upon the molecules shape and interactions, we expect the flow-alignment parameter to be affected by the cellular shape, which in turn is not fixed, and by the passive and active processes underlying the cell–cell and cell–substrate interactions.

The most striking difference with respect to the case discussed in Section IV.A, as well as the most prominent consequence of the chiral stress τ, is that the stationary and uniformly aligned configuration (i.e. θ = θ_{0} and v_{x} = 0) is not a trivial solution of eqn (21) for non-vanishing α and τ values, unless the chiral and non-chiral active stresses cancel each other identically. Before considering this latter case (see Section IV.B.3), we find approximated expressions for the local orientation θ and the velocity v_{x} in the limits in which the active stresses are either very small or very large.

For very large active stresses, the active terms in eqn (21) overweight the elastic term on the left hand side. As a consequence, the equilibrium configuration of the nematic monolayer consists of a region in the bulk of the channel where the cells have uniform orientation θ = −θ_{tilt}/2 = −arctan(2τ/α)/2 and two boundary layers, whose size is roughly , with σ_{0} = (α/2)sin2θ_{0} + τcos2θ_{0}, where the director interpolates between the bulk and boundary orientation (see Fig. 5b). This phenomenon closely resembles flow-alignment in nematics (see e.g.ref. 40) with −θ_{tilt}/2 playing the role of the Leslie angle θ_{L} = arccos(1/λ)/2. Whereas passive flow-alignment, however, requires λ > 1 (e.g. flow-aligning nematics), such an active flow-alignment occurs at any finite value of α and τ, provided the elastic boundary layer is sufficiently small to have a clear distinction between bulk and boundary alignment.

For small α and τ values, we can postulate that the nematic director will depart only slightly from its orientation at the boundary. Thus, taking again θ(y) = θ_{0} + δθ(y) and linearizing eqn (21) around δθ = 0, we obtain:

∂_{y}^{2}δθ + q^{2}(δθ + δθ_{0}) = 0, | (30) |

(31) |

(32) |

(33) |

In the following, we provide explicit approximated expression for the velocity field in the special cases where θ_{0} = 0 (parallel anchoring) and θ_{0} = π/2. Furthermore, we will investigate the stability of the trivial solution of eqn (21) obtained when θ_{0} is such that the chiral and non-chiral active stresses cancel each other identically.

1. Parallel anchoring.
For θ_{0} = 0 eqn (28) and (31) yield:

The corresponding velocity field is then readily obtained by integrating eqn (33). This gives:

where the wave number q_{‖} is that given in eqn (28). Numerical solutions for this case are displayed in Fig. 5b for various α and τ values.

(34) |

(35) |

We stress that, whereas the flowing configurations resulting from the instability of the stationary state are left-right and clock-counterclockwise symmetric (i.e. the cells are equally likely to flow toward the negative or positive x-direction and, correspondingly, to tilt clock- or counterclockwise, see Section IV.A.1), in this case the direction of the tilt as well as that of the flowing monolayer is set by the signs of the constants α and τ.

2. Homeotropic anchoring.
For θ_{0} = π/2 from eqn (28) and (31) we find that the amplitude δθ_{0} is given, once again, by eqn (34). Thus, the expressions for δθ and v_{x} are formally identical to those given in eqn (32) and (35), but with wave number q_{⊥} as given in eqn (29).

3. Stationary solution.
To conclude this subsection, we consider a special situation where the orientation of the cells at the boundary is fixed, as before, but such that the chiral and non-chiral stresses cancel each other identically. Thus: θ_{0} = −θ_{tilt}/2 = −arctan(2τ/α)/2. In this case the orientation of the nematic director in the bulk of the channel, determined by the balance between the chiral and non-chiral active stress, is equal to that at the boundary. As a consequence, the boundary layer described in Section IV.B disappears and the system can achieve a stationary and uniformly aligned configuration. As those described in Section IV.A, however, the latter is unstable for sufficiently large active stresses or channel width.

Analogously, the velocity is given by eqn (27), but with no constraint on the phase qL, because of the stress-free boundary conditions. As a consequence, the first mode to be excited is n = 1, thus the stationary state becomes unstable when q = q_{c} = π/L or, equivalently, when L = L_{c} = π/q. Some numerical solution of eqn (18), in this regime, are shown in Fig. 5c.

where and . For α > α_{c}, the solution consists of two branches, of which only one is connected with the stationary solution θ(y) = θ_{0}. Furthermore, the gap between the two branches increases monotonically with τ. If the instability is triggered upon applying a small random perturbation to the stationary state, this will always select the closest branch, thus the one connected to the trivial solution. As a consequence, a chiral cellular monolayer driven out of the stationary state by a small perturbation, will systematically tilt and flow in the same direction, which is in turn determined by the sign of the chiral active stress τ. The transition described above is known in bifurcation theory as a perturbed or imperfect pitchfork bifurcation and occurs when a standard pitchfork bifurcation, whose normal form is θ^{3} − μθ = 0, is biased by a small symmetry-breaking perturbation: i.e. θ^{3} − μθ + P_{L} + P_{S}θ^{2} = 0, where μ, P_{L}, and P_{S} are constant parameters. If P_{L} = P_{S} = 0, the equation is invariant under θ → −θ. Thus, for μ > 0, the trivial solution is unstable and the transition is supercritical, while for μ < 0, only the trivial solution is stable, and the bifurcation is subcritical. By contrast, for non-vanishing P_{L} and P_{S}, the equation is no longer invariant under θ → −θ and the bifurcation is no longer symmetric (see ref. 41 and 42 for an overview).

Evidently, this coincides with the normal form of a perturbed bifurcation for any finite τ value. For τ = 0, on the other hand, one recovers the normal form of the symmetric pitchfork bifurcation. Upon increasing τ, the bifurcation is shifted toward smaller α values, until, for τ = π^{2}/ηK(γL^{2}), α_{c} = 0 and the system is never stationary.

Using the same algebraic manipulations adopted in Section IV.A, one can show that the perturbation δθ is again of the form given in eqn (25) with:

(36) |

Notably, the transition from stationary to flowing is, in this case, no longer left-right and clock-counterclockwise symmetric, as in the examples discussed in Section IV.A, for any τ ≠ 0. This is well illustrated by the bifurcation diagram of Fig. 4b, showing the departure in the director orientation from the boundary value at the center of the channel [i.e. θ(1/2) − θ_{0}, with L = 1]. The dots have been obtained from a numerical integration of eqn (18), whereas the solid lines correspond to analytical solutions obtained by solving a third order equation for the constant C in eqn (25), as in Section IV.A.1. We find:

(37) |

In our case, the role of the symmetry-breaking perturbation is played by the chiral stress τ. Thus, in the unperturbed scenario, τ = 0 and the stationary solution is θ = 0, with the critical α value being α_{c} = 2π^{2}ηK/[γL^{2}(1 − λ)]. When τ ≠ 0, on the other hand, expanding eqn (21) around θ = 0 and using ∂_{y}^{2}θ = −q^{2}θ one finds:

(38) |

Further experimental investigations into the influence of chirality would be interesting. In particular, the tilt of the flow around ±1/2 disclinations has, according to our knowledge, not yet been observed. Thus, measurements of the tilt angle and experimental investigations of the flow field are needed to compare the theory with real-life cell monolayers. Additionally, as mentioned, the tilt angle opens a possibility to determine the relative magnitude of the chiral stress directly by particle-image-velocimetry measurements. Furthermore, since the cells used in ref. 8 were only weakly chiral the effects of chirality were not as pronounced. Performing similar experiments with cells with stronger chirality and for different boundary conditions would enable further tests of the presented theory.

f_{c} = F^{(a)}_{c}δ(R_{c} − aν_{c}) − F^{(a)}_{c}δ(R_{c} + aν_{c}) + F^{(b)}_{c}δ(R_{c} − bν^{⊥}_{c}) − F^{(b)}_{c}δ(R_{c} + bν^{⊥}_{c}), | (A1) |

which can be written as

(B1) |

The stress σ_{xy} is determined by the no-slip boundary condition and force balance, i.e.,

(B2) |

(B3) |

(B4) |

(B5) |

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