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Mechanical Control of Growth: Ideas, Facts and Challenges Kenneth D © 2017. Published by The Company of Biologists Ltd | Development (2017) 144, 4238-4248 doi:10.1242/dev.151902 REVIEW Mechanical control of growth: ideas, facts and challenges Kenneth D. Irvine1,* and Boris I. Shraiman2,* ABSTRACT interpretation of Thompson’s dictum ‘…the form of an object is a “ ”’ In his classic book On Growth and Form,D’Arcy Thompson diagram of forces , to describe a specific mathematical framework discussed the necessity of a physical and mathematical approach relating geometric form to the history of the growth process. This to understanding the relationship between growth and form. The past relation will immediately require assumptions concerning tissue century has seen extraordinary advances in our understanding of mechanics and will illustrate that some spatiotemporal patterns of ‘ ’ biological components and processes contributing to organismal growth might be in certain ways mechanically better than others. morphogenesis, but the mathematical and physical principles This will in turn bring us to the question of controlling growth as a involved have not received comparable attention. The most obvious function of spatial position and time in order to achieve correct entry of physics into morphogenesis is via tissue mechanics. In this developmental outcome: a correctly shaped and sized organ or limb. Review, we discuss the fundamental role of mechanical interactions We discuss this in the context of a biologically well-studied model between cells induced by growth in shaping a tissue. Non-uniform of organ growth, the Drosophila wing, and with particular reference – – growth can lead to accumulation of mechanical stress, which in the to a signaling network the Hippo pathway that integrates context of two-dimensional sheets of tissue can specify the shape it multiple molecular signals with mechanical input. We also discuss assumes in three dimensions. A special class of growth patterns – the issues that one encounters in connecting cellular scale conformal growth – does not lead to the accumulation of stress and cytoskeletal mechanics, which provides a mechanosensor linked can generate a rich variety of planar tissue shapes. Conversely, into growth control, with the mechanical behavior on the scale of the mechanical stress can provide a regulatory feedback signal into the tissue. Our Review is not intended as a general overview of growth growth control circuit. Both theory and experiment support a key role control mechanisms, but rather an argument for, and an exposition for mechanical interactions in shaping tissues and, via mechanical of, a fundamental role for tissue mechanics in growth control. feedback, controlling epithelial growth. A mathematical connection between ‘growth’ and ‘form’ KEY WORDS: Stress, Growth, Mechanics, Hippo If we consider a change in form that accompanies growth, such as a ball of cells in a limb bud growing into an elongated appendage, we Introduction can anticipate that there may be many different spatiotemporal In On Growth and Form, first published 100 years ago, D’Arcy patterns of growth that arrive at the same geometric shape. Can we Thompson famously considered the study of growth and form of begin to classify different growth strategies? How can these strategies plants and animals to be in the domain of physics and mathematics, be plausibly ‘encoded’ in cell behavior during the developmental stating: ‘the morphologist is, ipso facto, a student of physical science’ process to yield a reproducible shape? For simplicity, we proceed by (p. 8, Thompson, 1917). Yet, our present understanding of animal and considering an abstract two-dimensional (2D) problem, which will plant development owes much more to genetics, cell biology and illustrate the main ideas that can be applied in specific biological biochemistry than to physics and mathematics. The immense progress contexts, and which generalizes naturally to 3D. of modern developmental biology has provided us with a great deal of Consider a growing 2D planar ‘body’ defined by the shape of its insight into the genetic and molecular basis of developmental boundary. Suppose the initial body has a shape of a disc: what processes and the biochemical signals that control them. However, a growth process could create a given final shape? Two types century later, the central question of how controlled growth defines of growth processes can immediately be imagined: (1) ‘boundary biological form, sublimated from D’Arcy Thompson’s writings, growth’ corresponding to proliferation of cells only at the edge of remains unanswered. Hence, it seems timely to revisit D’Arcy the body, and (2) ‘bulk growth’ that occurs throughout the body. In Thompson’s agenda of understanding physical and geometric aspects either case, to understand the overall shape of the body we have to of morphogenesis in the context of our current knowledge of the understand the motion of the boundary, induced by growth. Both cellular and molecular genetic aspects of developmental biology. types of growth exist in nature, with boundary growth typically Starting on this path, one immediately arrives at the problem of observed for hard tissues such as bone, antlers and shells, and bulk mechanics in development, which has attracted considerable recent growth typically observed for soft tissues such as heart or intestine attention (Coen et al., 2004; Heer and Martin, 2017; Heisenberg and (Cowin, 2004). Here, we focus our discussion on the latter, bulk Bellaïche, 2013; Lecuit and Yap, 2015; Munjal and Lecuit, 2014; growth scenario, because it is commonly observed and it illustrates Tallinen et al., 2016). In this Review, we begin with an almost literal the role of mechanics in relating the process of growth to the resulting shape. In the mathematical discussion that follows, terms 1Waksman Institute and Department of Molecular Biology and Biochemistry, used in the equations are defined in Box 1, and the supplementary Rutgers University, Piscataway NJ 08854, USA. 2Department of Physics, Kavli information provides further information to the interested reader. Institute for Theoretical Physics, University of California, Santa Barbara, CA 93101, – USA. In the simplest case, growth is locally isotropic which is to say cell division axes are randomly distributed – and can be described *Authors for correspondence ([email protected]; simply by the rate of cell division gð~r; tÞ, which can vary depending [email protected]) upon position, ~r, within the body and time, t. As cells grow and K.D.I., 0000-0002-0515-3562; B.I.S., 0000-0003-0886-8990 divide, the total volume (or area in our 2D case) of the body will DEVELOPMENT 4238 REVIEW Development (2017) 144, 4238-4248 doi:10.1242/dev.151902 Box 1. Glossary of mathematical symbols Box 2. Glossary of physical and mathematical terms γ growth rate Cellular flow. Change in position of a cell over time. r position Conformal map. A transformation of a planar object that preserves local t time angles. ρ label identifying a particular cell Einstein’s convention. A mathematical notation that abbreviates V velocity vector formulas by implicit summation of repeated indices. u displacement vector Elastic. An object or material that resists being compressed or stretched s strain tensor and returns to its normal shape after external force is lifted. σ stress tensor Elastic modulus. The ratio of stress to strain, which describes how μ elastic modulus deformable a material is in response to an applied force. δ Kroneker delta Harmonic function. A function for which second order partial À1 derivatives sum to zero (i.e. satisfy the Laplace equation). ts stress relaxation rate Harmonic/stressless growth. Growth in a harmonic spatial pattern, mab myosin activity M growth factor concentration which generates no change in stress. Strain. Local deformation of a material arising from spatially varying displacement of its elements. Strain tensor. A mathematical description of local deformation based on increase, but what will be the shape? The answer depends on the the spatial variation of vectorial displacements of material points. In two pattern of growth, but also on the details of tissue mechanics. In dimensions, spatial components of a tensor form a 2×2 matrix. particular, it matters how deformable cells are, how much they stick Stress. Force per unit area that components of a material exert on each ‘ ’ other. to each other, and how free they are to rearrange and slide by each Stress tensor. A mathematical description of a force vector acting on the other. Independent of the details, however, growth results in a slow small element of the surface with a given spatial orientation. displacement of cells that can be described by a velocity vector field Velocity vector field. An area (in 2D) or volume (in 3D) in which each ~ (see Glossary, Box 2) Vð~r; tÞ¼½Vxð~r; tÞ; Vyð~r; tÞ as a function of point is assigned a vector indicating its speed and direction. position within the body (we will often use an alternative notation Vað~r; tÞ using index a as a stand-in for either x or y components of a vector). Given this cell displacement velocity field as a function of The strain is described mathematically by a tensor ð~Þ¼@ ð~Þþ@ ð~Þ time, one can determine the positions of all cells as they are carried sab r aub r bua r , where indices a and b stand in for (x ð~Þ along within the mass of proliferating cells. The positions of cells in or y) vector and tensor component labels, so that vector ua r the body ~rðr; tÞ are governed by: denotes the displacement of a material point (of the elastic sheet) away from its unperturbed position and ∂a denotes a partial d derivative with respect to ra. Thus, different components of sab are raðr; tÞ¼Vað~rðr; tÞ; tÞ; ð1Þ dt defined by how components of the displacement vector ua change where ρ identifies a particular cell within the body and the change in with position (Fig.
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