Mechanical control of growth: ideas, facts and challenges

Abstract

In his classic book On Growth and Form, D'Arcy Thompson discussed the necessity of a physical and mathematical approach to understanding the relationship between growth and form. The past century has seen extraordinary advances in our understanding of biological components and processes contributing to organismal morphogenesis, but the mathematical and physical principles involved have not received comparable attention. The most obvious entry of physics into morphogenesis is via tissue mechanics. In this Review, we discuss the fundamental role of mechanical interactions between cells induced by growth in shaping a tissue. Non-uniform growth can lead to accumulation of mechanical stress, which in the context of two-dimensional sheets of tissue can specify the shape it assumes in three dimensions. A special class of growth patterns - conformal growth - does not lead to the accumulation of stress and can generate a rich variety of planar tissue shapes. Conversely, mechanical stress can provide a regulatory feedback signal into the growth control circuit. Both theory and experiment support a key role for mechanical interactions in shaping tissues and, via mechanical feedback, controlling epithelial growth.

Keywords: Growth; Hippo; Mechanics; Stress.

Fig. 1.

A toy example of conformal growth. (A) Individual cells (illustrated by blue or red hexagons) move as the tissue changes shape as a result of growth: a circular body (gray) (left) undergoing an imprinted growth profile (shown by the gray shading) leads to a more complex shape (right). Dashed outlines indicate the initial locations of the body and cells. ‘Cellular flow’ corresponds to the continuous displacement of cells as a function of time. (B) A harmonic growth pattern, γ(x, y)=2+x³−3xy², which is the simplest 3-fold symmetric harmonic function, was ‘imprinted’ in the circular body (left), defining the alternating sectors of faster (yellow) and slower (blue) growth. It was assumed that growth rate remains constant along the trajectory of each point. Resulting growth greatly expands the domain of faster growth compared with the slow-growing regions, changing the shape of the 2D body (right). (C) The conformal mapping of initially polar coordinates onto the final shape. The conformal map is given by , where z=x+iy is a complex number constructed from spatial coordinates (x, y) generated by the growth profile γ specified above (see supplementary information part D). To relate this continuum analysis with growth of cellular tissue, one would assume that growth rate is constant along cell lineage, with newborn cells growing at the rate that interpolates the growth rate of their neighbors.

Fig. 2.

The effects of displacement and strain. (A) Faster growth of the inner (red) region of an elastic tissue layer results in compression of the faster-growing region and causes radial compression and azimuthal stretching of the surrounding (blue) region. (B) A cellular perspective on strain. Deformation of cells from the initial shape (black) to a later shape (red) is associated displacement of their centroids a,b,c. (C) Continuum elasticity describes tissue deformation on supracellular scale. These deformations can be described mathematically by strain tensors. To illustrate this, we show two modes of deformations that do not change the area of an arbitrary region, and how this is related to components of the strain tensor (s).

Fig. 3.

Patterns of Dpp activity and cell proliferation in the wing disc. Schematics of part of the Drosophila wing imaginal disc. (A) The morphogen growth factor Dpp is produced from cells at a localized source along the center of the disc, and spreads out forming a concentration gradient. (B) Cell proliferation (shown by the dots) is essentially evenly distributed throughout the wing disc. Lines mark the compartment boundaries. (C) When Jub-mediated mechanical feedback is blocked (Pan et al., 2016) proliferation becomes unevenly distributed, with higher levels where Dpp signaling is higher.

Fig. 4.

Hippo signaling and mechanical feedback. (A) Simplified depiction of regulatory connections between some key components of the Hippo pathway, with Drosophila names above and vertebrate names of homologous proteins in parentheses below. (B) Schematics illustrating the consequences of differential growth rates in the wing disc epithelium on myosin levels (indicative of tension), junctional Jub levels, and Yki activity, with green indicating wild-type levels, red indicating higher levels, and blue indicating lower levels. (i) Marked clones of cells (outlined by thin black lines) growing at the same rate as surrounding tissue do not affect Myosin, Jub or Yki activity. (ii) Clones of cells growing at an abnormally fast rate and surrounded by tissue growing at a normal rate become compressed, leading to lower levels of Myosin, Jub and Yki activity. (iii) Clones of cells growing at a normal rate and surrounded by tissue growing at an abnormally slow rate also become compressed, leading to lower levels of Myosin, Jub and Yki activity. Adapted from Pan et al. (2016).