Active morphogenesis of patterned epithelial shells

  1. Diana Khoromskaia
  2. Guillaume Salbreux  Is a corresponding author
  1. The Francis Crick Institute, United Kingdom
  2. University of Geneva, Switzerland
13 figures and 1 additional file

Figures

Figure 1 with 8 supplements
A two-dimensional surface with nematic order represents an epithelial sheet undergoing active deformations.

(a) Schematic of an epithelial tissue with a cellular state pattern. (b) Parametrisation of the axially symmetric shell and its deformation with the flow v, and components of the tension and torque tensors. We note that mϕs=m¯ϕϕx and msϕ=-m¯ss/x. (c) Stresses integrated across the thickness of the sheet result in tensions tij and bending moments mij acting on the midsurface. Anisotropic and possibly different tensions (dark-blue arrow crosses) on the apical and basal sides of the epithelium result in anisotropies in tij and m¯ij, which can be captured by a nematic order parameter Qij (e.g. blue rods on the top surface).

Figure 1—video 1
Deformation of an epithelial shell with free volume, la/L0=0.85, δζc=15κ/R0, with active torque colour coded.
Figure 1—video 2
Deformation of an epithelial shell with free volume, la/L0=0.85, δζc=15κ/R0, with tangential velocity shown as white arrows and colour coded.
Figure 1—video 3
Deformation of an epithelial shell with free volume, la/L0=0.35, δζc=12.5κ/R0, with active torque colour coded.
Figure 1—video 4
Deformation of an epithelial shell with free volume, la/L0=0.35, δζc=12.5κ/R0, with tangential velocity shown as white arrows and colour coded.
Figure 1—video 5
Deformation of an epithelial shell with conserved volume, la/L0=0.1, δζc=40κ/R0, with active torque colour coded.
Figure 1—video 6
Deformation of an epithelial shell with conserved volume, la/L0=0.1, δζc=40κ/R0, with tangential velocity shown as white arrows and colour coded.
Figure 1—video 7
Deformation of an epithelial shell with conserved volume, la/L0=0.7, δζc=110κ/R0, with active torque colour coded.
Figure 1—video 8
Deformation of an epithelial shell with conserved volume, la/L0=0.7, δζc=110κ/R0, with tangential velocity shown as white arrows and colour coded.
Figure 2 with 10 supplements
Deformations of epithelial shells due to active bending moments, with free (a–d) and conserved (e–i) volume.

(a, e) Shape diagram. (b, f) Details of shape diagram illustrating different behaviours of solution branches. The ideal neck line (green) represents the bending moment difference required to create budded shapes consisting of two spheres with u=0, as given by Equation 24. (c) Examples of solution branches in the (δζc,V)-plane corresponding to four different regions in (b). (g) Examples of solution branches in the (δζc,P)-plane chosen from three different regions in (e). (d, h) Dynamic simulations of shape changes, for parameter values indicated in the shape diagrams (a, e). (i) Neck radius and curvatures at the neck as functions of δζc for the example la/L0=0.04 in (g). Other parameters: K~=103,η~cb=10-2, η~V=10-4.

Figure 2—figure supplement 1
Details of the steady-state solutions with nearly closed necks formed by isotropic bending moments for free volume (a) and conserved volume (b), and la/L0=0.9.

The location of the neck, taken as the point where Cϕϕ is maximal, is marked by a grey line in the plots. (a) The shape is characterised by tss=tns=u=0 and constant.m¯ss. (b) Here, tss changes sign and m¯ss is continuous across the neck.

Figure 2—figure supplement 2
Maximal relative surface area of the steady-state shapes measured along a solution branch for each la/L0 in the case of conserved volume, corresponding to shapes shown in Figure 2e–g.
Figure 2—video 1
Deformation of an epithelial shell with free volume, la/L0=1, ζn=1.5κ/R02, with nematic director shown with black lines and active nematic tension colour coded.
Figure 2—video 2
Deformation of an epithelial shell with free volume, la/L0=1, ζn=1.5κ/R02, with tangential velocity shown as white arrows and colour coded.
Figure 2—video 3
Deformation of an epithelial shell with free volume, la/L0=1, ζn=3κ/R02, with nematic director shown with black lines and active nematic tension colour coded.
Figure 2—video 4
Deformation of an epithelial shell with free volume, la/L0=1, ζn=3κ/R02, with tangential velocity shown as white arrows and colour coded.
Figure 2—video 5
Deformation of an epithelial shell with conserved volume, la/L0=0.3, δζn=40κ/R02, with nematic director shown with black lines (where δζn0) and active nematic tension colour coded.
Figure 2—video 6
Deformation of an epithelial shell with conserved volume, la/L0=0.3, ζn=40κ/R02, with tangential velocity shown as white arrows and colour coded.
Figure 2—video 7
Deformation of an epithelial shell with conserved volume, la/L0=0.7, δζn=60κ/R02, with nematic director shown with black lines (where δζn0) and active nematic tension colour coded.
Figure 2—video 8
Deformation of an epithelial shell with conserved volume, la/L0=0.7, ζn=60κ/R02, with tangential velocity shown as white arrows and colour coded.
Nematic order on a sphere.

(a) Two possible configurations for the nematic order parameter Qij on a sphere with a + 1 topological defect at each pole: meridional (left) or circumferential (right) alignment. The order parameter minimises an effective energy (Equation 9 with lc=0.1R0). (b) Order parameter q(s)=Qϕϕ(s) as a solution of the Euler–Lagrange Equation 16 on a sphere with R0=1 and lc=0.1R0;q=1 at the equator and q=0 at the locations of the defects (poles). For uniform ζn, ζniQis is the active nematic contribution to the tangential force balance (Equation 63) and, close to the equator, results in the elongation of the surface along the axis of symmetry for ζn>0, and its contraction for ζn<0.

Figure 4 with 13 supplements
Deformations of epithelial shells due to nematic tensions, with free (a–e) and conserved (f–i) volume.

(a, e) Shape diagrams. (b, g) Details of shape diagram illustrating the behaviour of solution branches. (d) Curvature at the south pole for extensile stress. (c, e, h, i) Dynamic simulations of shell shape changes, for parameter values indicated in the phase diagrams (a, f). Other parameters: K~=103,η~cb=10-2, η~V=10-4, l~c=0.1.

Figure 4—figure supplement 1
Details of dynamics simulations for shells with (a, b) homogeneous and (c) patterned nematic tension, which result in one or two constricting necks.

Surface quantities are shown for the last plotted time step in Figure 4. The location of the neck, taken as the point where Cϕϕ is maximal, is marked by a grey line in the plots. In (a, b) the radius at the equator, the volume, and the pole–pole distance are shown as functions of time. In (c) the smooth sigmoidal pattern of ζn(s) is visualised as two discrete regions (colour coded as red and blue) for simplicity. The parameter values correspond to the examples in Figure 4c, e and h in the main text.

Figure 4—video 1
Deformation of an epithelial shell with free volume, la/L0=1, ζcn=-5κ/R0, with nematic director (black lines, as described in figure caption) and active nematic torque colour coded.
Figure 4—video 2
Deformation of an epithelial shell with free volume, la/L0=1, ζcn=-5κ/R0, with tangential velocity shown as white arrows and colour coded.
Figure 4—video 3
Deformation of an epithelial shell with free volume, la/L0=0.3, δζcn=-20κ/R0, with nematic director (black lines, as described in figure caption) and active nematic torque colour coded.
Figure 4—video 4
Deformation of an epithelial shell with free volume, la/L0=0.3, δζcn=-20κ/R0, with tangential velocity shown as white arrows and colour coded.
Figure 4—video 5
Deformation of an epithelial shell with free volume, la/L0=1, ζcn=4κ/R0, with nematic director (black lines, as described in figure caption) and active nematic torque colour coded.
Figure 4—video 6
Deformation of an epithelial shell with free volume, la/L0=1, ζcn=4κ/R0, with tangential velocity shown as white arrows and colour coded.
Figure 4—video 7
Deformation of an epithelial shell with free volume, la/L0=0.3, δζcn=15κ/R0, with nematic director (black lines, as described in figure caption) and active nematic torque colour coded.
Figure 4—video 8
Deformation of an epithelial shell with free volume, la/L0=0.3, δζcn=15κ/R0, with tangential velocity shown as white arrows and colour coded.
Figure 4—video 9
Deformation of an epithelial shell with conserved volume, la/L0=0.3, δζcn=-150κ/R0, with nematic director (black lines, as described in figure caption) and active nematic torque colour coded.
Figure 4—video 10
Deformation of an epithelial shell with conserved volume, la/L0=0.3, δζcn=-150κ/R0, with tangential velocity shown as white arrows and colour coded.
Figure 4—video 11
Deformation of an epithelial shell with conserved volume, la/L0=0.5, δζcn=50κ/R0, with nematic director (black lines, as described in figure caption) and active nematic torque colour coded.
Figure 4—video 12
Deformation of an epithelial shell with conserved volume, la/L0=0.5, δζcn=50κ/R0, with tangential velocity shown as white arrows and colour coded.
Figure 5 with 1 supplement
Deformations of epithelial shells due to nematic bending moments, with free (a–c) and conserved (d, e) volume.

(a, d) Shape diagrams. (b, e) Details of shape diagram illustrating the behaviour of solution branches. (c, f) Dynamic simulations of shell shape changes, for parameter values indicated in the phase diagrams (a, d). In both cases in (f) the dynamics results in self-intersection. (g) Comparison of curvature and length of the cylindrical tubes for la/L0=1,0.7,0.3, δζcn<0 with analytical predictions. The tube length is measured on the steady-state shape as the arc length of the deformed active region, stube=s(s0=la), and the tube curvature as Cϕϕ(stube/2). Other parameters: K~=1000,η~cb=10-2, η~V=10-4, l~c=0.1. In (c), (f), for δζcn,ζcn<0 the orientation of the director field drawn on the surface (black lines) is set by -Qij.

Figure 5—figure supplement 1
Details of steady-state shapes resulting from nematic bending moments with ζcn<0 and free volume.

(a) Closed cylinder; (b) shape with cylindrical appendage. Such solutions are characterised by tss=tns=u=0 everywhere, and a cylindrical part where Css=0 and m¯ss is constant.

Summary of shape changes obtained through patterning of isotropic and anisotropic active tensions and bending moments.

Active tensions and bending moments are present only in the red region of the surface. For ζcn<0 the director field orientation (black lines) is set by -Qij.

Appendix 1—figure 1
Schematic of the surface S1 used to derive the integral of the normal force balance.
Appendix 4—figure 1
Schematic of notations used for the asymptotic analysis of a bilayered disc.
Appendix 4—figure 2
Details of shape and nematic profiles for flattened steady-state shapes resulting from a homogeneous nematic tension.

(a) Profile of nematic order parameter q, which decreases for increasing |ζn|. (b) Distance between the poles of the steady-state solution for different values of |ζn| and K~, and corresponding prediction of Equation 134 (dotted lines). (c) Profile of ψ(s) for different values of ζn. The profile is invariant with respect to ζn, for large values of |ζn|.

Appendix 7—figure 1
Convergence analysis of a dynamics simulation to a steady state obtained from direct calculation, for the example shape shown in the inset of (a).

For different tolt (time step) the error in the shape in (a) and error in the external force integral in (b) are shown.

Appendix 8—figure 1
Rescaling of the size of the active region with active tension difference.

(a) Plot of la/L as given by Equations 213; 214. This illustrates the rescaling of the active region size as a function of the tension difference δζ, for initial values la/L0=0.1,0.3,0.5,0.7,0.9 (blue to purple). (b) Shapes corresponding to two points on the la/L0=0.5 curve.

Author response image 1
Shapes obtained with two contiguous domains of extensile and contractile nematic active tension (left) or nematic active bending moments (right).

Parameter values with + and – subscripts indicate values in the contractile and extensile regions. The contractile region is at the bottom of the shape.

Author response image 2
Results of dynamics simulations that converge to a steady state (light blue circles) overlaid with solution branches from the main text.

Since in the dynamics a smooth profile is used, rather than a step function as in the direct calculation, there is a small deviation between the two in some cases. We checked on two cases that if the smooth profile is used in the steady-state equation the match becomes excellent (b, blue dotted lines).

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  1. Diana Khoromskaia
  2. Guillaume Salbreux
(2023)
Active morphogenesis of patterned epithelial shells
eLife 12:e75878.
https://doi.org/10.7554/eLife.75878