Hydrodynamics and multiscale order in confluent epithelia

  1. Josep-Maria Armengol-Collado
  2. Livio Nicola Carenza
  3. Luca Giomi  Is a corresponding author
  1. Instituut-Lorentz, Leiden University, Netherlands
6 figures


Epithelia and hexanematic order.

(a) Example of multiscale hexanematic order in an in vitro layer of Madin–Darby canine kidney (MDCK) cells (b) and its computer-constructed segmentation. Both panels are adapted from Figure 3 of Armengol-Collado et al., 2023. The six-legged stars in the shaded region denote the sixfold orientation of the cells obtained using the approach summarized in Methods. The colored stripes mark the configuration of the nematic director at the length scale of the light-blue disk. (c) Schematic representation of the sixfold symmetric force complexion exerted by cells. The red arrows indicate the structure of the contractile forces acting within the cellular junctions.

Structure factor S(|q|) obtained from the analytical solutions of the linearized hydrodynamic equations in the presence of two different noise fields: purely rotational (blue) and rototranslational (red).

The full analytical expression of S(|q|) is given in Methods, together with a derivation of the exact asymptotic expansions of Equation (8). (a) As long as viscous dissipation takes place (i.e. ‘wet’ regime), S(|q|)|q|4 in the limit |q|, irrespective of the type of noise. (b) On the other hand, when friction is the sole momentum dissipation mechanism at play (‘dry’ regime), S(|q|)|q|6 in case of rotational noise and S(|q|)|q|10 when noise is both rotational and translational. In both panels, the wave number |q| is rescaled by q¯=2π/¯, with ¯=(2+6)/2 and 2 and 6 as defined in Equation (7).

Density fluctuations in model epithelia.

(a) Structure factor of model-epithelia calculated from a numerical integration of Equation 2a (black line) and from simulations of two different cell-resolved models: i.e. the self-propelled Voronoi model (SPV, red) and the multiphase field (MPF) model (blue), for a particular choice of parameters. The dashed diagonal lines mark the scaling regimes obtained analytically at the linear order, Equation (8), and the wave number |q| is rescaled by qcell=2π/ΔxLB, where ΔxLB is the grid size used by the Lattice Boltzmann integrator (see Methods for details). (b) The exponent β, as defined in Equation (8), versus the Péclet number Pe, reflecting the persistence of directed cellular motion in front of diffusion. Error bars calculated as standard error over $n=250$ configurations for both the SPV and the MPF models. Insets: typical configurations of the SPV (bottom left) and MPF (top right) models.

Cross-correlation function C26(r), as defined in Equation 9, obtained from a numerical integration of Equation 2a augmented with rotational noise.

(a) In the passive case, when α2=0 and α6=0, the correlation function decays with |r| at a rate i.e. lower at short distances, where the dynamics of the hexatic and nematic orientations is dominated by fluctuations, and larger at long distances, where the orientations are ‘locked’ in a parallel configuration, or tilted by π/6 with respect to each other. (b) In the active case, conversely, the cross-correlation function has a damped oscillatory behavior. Consistently with Equation (7) and the related discussion, the range of the oscillations, corresponding to the distance at which these are fully damped, increases with the hexatic activity α6, indicating an enhancement of hexatic order at larger length scales. Shaded region corresponding to the standard deviation of n=500 configurations. Distance is expressed in terms of the grid size ΔxLB used by the Lattice Boltzmann integrator (see Methods for details).

Nematic and hexatic shape function.

(a) Irregular polygonal cell with a red cross marking its center of mass and rv and ϕv the radial vector and the angle to one of the six vertices, respectively. (b) and (c) show the same tessellation of the plane with cells of different shapes and the shape analysis using the function in Equation 10 for the nematic (p=2) and hexatic (p=6) case. Rods and stars are oriented according to the phase of γp and the color corresponds to its magnitude.

Hexanematic symmetry of Voronoi model.

(a) Coarse-grained nematic orientation θ obtained from averaging the local shape of cells over domains of size 30cell, with cell the average size of individual cells. Regions with the same color represent domains of coherent nematic orientation. (b) Part of the system where we use Γ2 to characterize the nematic phase. Solid lines represent the nematic director and the color inditicates the magnitude of the nematic shape function. (c) Voronoi cell structure of a region where the nematic field is uniform. Polygons are colored according to |γ6| and the stars are oriented according to Arg(γ6)/6.

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  1. Josep-Maria Armengol-Collado
  2. Livio Nicola Carenza
  3. Luca Giomi
Hydrodynamics and multiscale order in confluent epithelia
eLife 13:e86400.