On shape forming by contractile filaments in the surface of growing tissues

Abstract

Growing tissues are highly dynamic, and flow on sufficiently long timescales due to cell proliferation, migration, and tissue remodeling. As a consequence, growing tissues can often be approximated as viscous fluids. This means that the shape of microtissues growing in vitro is governed by their surface stress state, as in fluid droplets. Recent work showed that cells in the near-surface region of fibroblastic or osteoblastic microtissues contract with highly oriented actin filaments, thus making the surface properties highly anisotropic, in contrast to what is expected for an isotropic fluid. Here, we develop a model that includes mechanical anisotropy of the surface generated by contractile fibers and we show that mechanical equilibrium requires contractile filaments to follow geodesic lines on the surface. Constant pressure in the fluid forces these contractile filaments to be along geodesics with a constant normal curvature. We then take this into account to determine equilibrium shapes of rotationally symmetric bodies subjected to anisotropic surface stress states and derive a family of surfaces of revolution. A comparison with recently published shapes of microtissues shows that this theory accurately predicts both the surface shape and the direction of the actin filaments on the surface.

Keywords: mechanobiology; surface stress; tissue growth.

Fig. 1.

Experimental observations and definition of surface coordinates. (A) Image of a catenoid, i.e., the surface of revolution of a catenary, which satisfies Young–Laplace equation for zero pressure difference over the membrane. (B) Projection of bone-like tissue grown on a polymeric surface of revolution (capillary bridge) fixed on a central pin (dark line). The light arrow indicates the boundary of the polymeric surface, and the dark arrow indicates the position of the tissue after 30 days growth. and are the radius and separation of two circular disks corresponding to the upper and the lower boundary of the tissue. (C) Projection of a 3D light sheet fluorescence microscopy image of tissue stained for actin (green fibers). Note the strong orientation of the actin stress fibers. (D) The Young–Laplace equation (Eq. 1) can be understood by the tension balance over a surface patch, with two principal curvatures and and membrane forces and Eq. 2. (E) The pressure of the fluid within the volume generates a force directed along the normal to the surface . For isotropic mechanical surface properties, the resultant membrane force is also perpendicular to the surface but pointing in opposite direction . There is equilibrium if and have the same magnitude. (F) If the local mechanical response of the membrane is only generated by a fiber on the surface with tension , then the resultant local force lies along , within the osculating plane of the fiber, and is not necessarily colinear with (i.e., 𝛼 ≠ 0). Therefore, the conditions for equilibrium of the surface are (i) that the direction of (i.e., ) lies within the osculating plane of the curve describing the fiber (i.e., the angle 𝛼 = 0), and (ii) that the magnitudes of and are the same, which correspond to Eqs. 4a and 3, respectively. The fibril angle is measured between the fiber direction and . The images in panels B and C are reproduced from ref. [14], under the CC BY-NC licence.

Fig. 2.

Surfaces of revolution satisfying Eqs. 11a and b and the boundary conditions 14 as indicated in the inset top left, characterized by the (constant) curvature of the fibers which stabilize the shape and the neck radius . Fibers follow spiraling paths, and the fibril angle at the equator is indicated for each set of curvature and neck radius. The inset (top-left) shows one such surface of revolution consisting of fibers of constant curvature that cross the equator with a “microfibril” angle, . The dashed line on the inset indicates the path of one such fiber. Grey lines in the main diagram give the relationship between and for a fixed fibril angle at the equator. The dashed red lines indicate the limits below or above which no solutions can be found. The solid red line shows the range of solutions satisfied by hyperboloids of one sheet, the dashed black line shows the range of cylindrical solutions. The full black circle indicates the solution given by one spherical segment. All solutions with microfibril angles less than 58° pass through this point. The dotted blue line indicates the relationship between and or a stack of two spherical segments, and the solid blue line for a stack of three spherical segments. Note that these solutions are not differentiable at the joint between the spherical segments and can be considered as limit cases. The lower part of the graph ( < 1) corresponds to necked structures as shown in the inset. The upper part of the graph describes bulged structures ( > 1), akin to the single spherical segment. The light grey region between the dotted blue line and the solid blue line is shown in more detail in the supplementary information (Fig. S2).

Fig. 3.

Alternative representation of Fig. 2, showing the neck radius, , versus the mean curvature at the neck, , for different “mifrofibril” angle, , at the equator. Grey lines give solutions in which the microfibril angle at the neck is fixed. The dashed red lines indicate the limits beyond, which no solutions can be found. The solid red line shows the range of solutions satisfied by hyperboloids of one sheet, the solid blue line indicates solutions for stacks of three spherical segments. The black dot indicates cylindrical solutions, the blue dot shows the solution for one segmented sphere. The dashed black line shows the relationship between neck radius and mean curvature for Delaunay surfaces that satisfy the boundary conditions. Note that this curve is close to the solutions for = 35° for the neck ranging from about = 0.4 to = 0.7.

Fig. 4.

Comparison of the model based on anisotropic surface contraction to the experiments of ref. [14]. Panel A is an enlargement of a portion of Fig. 3 with dots indicating all the tissue growth experiments performed in ref. [14]. It is quite remarkable that these dots are simultaneously close to the line for Delaunay surfaces (dotted line) and to fiber-stabilized volumes with a microfibril angle at the neck between 30° and 35°. It is also important to realize that the size of these dots is not representative for the experimental uncertainty in measuring curvatures. To account for this, they should be much larger and make the figure unreadable. Note that the induced tissue pressure is positive for all configurations in the blue area above the red line that corresponds to hyperboloids. Below this line, the tissue pressure induced by contraction of the fibers is negative and favors tissue growth. For isotropic mechanical surface properties, the surface stress induces a positive tissue pressure on the right side of the vertical line with zero mean curvature (sepia area in panel A). Panel B shows a typical microtissue grown in vitro under the boundary conditions used in our model calculations. The green coloration is due to actin staining and the microfibril angle in the neck is around 30° to 35° for most of the experiments. Panel C shows a solution to Eq. (11) with predicted fiber paths shown in green and a microfibril angle at the neck of 30°.

Fig. 5.

Comparison of Delaunay shapes (red) for different values of neck radii (0.25, 0.5, and 0.75) with solutions of Eq. 11 (grey lines) using values of the fiber angle at the neck, , as indicated. While the surfaces are mathematically distinct, they are essentially indistinguishable in a typical tissue culture experiment. A more detailed comparison is shown in Fig. S4.