Embryogenesis offers a real laboratory for pattern formation, buckling, and postbuckling induced by growth of soft tissues. Each part of our body is structured in multiple adjacent layers: the skin, ...the brain, and the interior of organs. Each layer has a complex biological composition presenting different elasticity. Generated during fetal life, these layers will experience growth and remodeling in the early postfertilization stages. Here, we focus on a herringbone pattern occurring in fetal intestinal tissues. Common to many mammalians, this instability is a precursor of the villi, finger-like projections into the lumen. For avians (chicks’ and turkeys’ embryos), it has been shown that, a few days after fertilization, the mucosal epithelium of the duodenum is smooth, and then folds emerge, which present 2 d later a pronounced zigzag instability. Many debates and biological studies are devoted to this specific morphology, which regulates the cell renewal in the intestine. After reviewing experimental results about duodenum morphogenesis, we show that a model based on simplified hypothesis for the growth of the mesenchyme can explain buckling and postbuckling instabilities. Being completely analytical, it is based on biaxial compressive stresses due to differential growth between layers and it predicts quantitatively the morphological changes. The growth anisotropy increasing with time, the competition between folds and zigzags, is proved to occur as a secondary instability. The model is compared with available experimental data on chick’s duodenum and can be applied to other intestinal tissues, the zigzag being a common and spectacular microstructural pattern of intestine embryogenesis.
The paramount importance of mechanical forces in morphogenesis and embryogenesis is widely recognized, but understanding the mechanism at the cellular and molecular level remains challenging. Because ...of its simple internal organization, Caenorhabditis elegans is a rewarding system of study. As demonstrated experimentally, after an initial period of steady elongation driven by the actomyosin network, muscle contractions operate a quasi-periodic sequence of bending, rotation, and torsion, that leads to the final fourfold size of the embryos before hatching. How actomyosin and muscles contribute to embryonic elongation is investigated here theoretically. A filamentary elastic model that converts stimuli generated by biochemical signals in the tissue into driving forces, explains embryonic deformation under actin bundles and muscle activity, and dictates mechanisms of late elongation based on the effects of energy conversion and dissipation. We quantify this dynamic transformation by stretches applied to a cylindrical structure that mimics the body shape in finite elasticity, obtaining good agreement and understanding of both wild-type and mutant embryos at all stages.
We revisit the classical theory of indentation for very soft materials. Many experiments consist in extracting the stiffness of a membrane from the cubic answer of the force versus indentation depth. ...However, this law is restricted to a perfect membrane under a sharp point loading. In biophysical experiments, where this technique recovers some success at low scales thanks to AFM, the thin samples are highly deformable, hyper-elastic, pre-stretched and often attached to a substrate which cannot be neglected. In addition microscopic tiny pores may exist or may be created by the indenter. This diversity requires specific studies with the correct elasticity: here we choose the neo-Hookean elastic model at large membrane deformations. We show that, the weak loading regime is extremely sensitive to any physical properties of the thin layer but also of the indenter geometry. In addition, when a hole exists, at finite forcing or finite indentation depth, we discover a topological bifurcation with abrupt dynamical jump variation of typical quantities such as the hole size. This bifurcation is similar to the famous catenoid instabilities which are also due to a topological bifurcation, when the two rings of support are pulled apart. This bifurcation is robust in the sense that it always exists whatever the physical properties of the sample.
One of the most remarkable differences between classical engineering materials and living matter is the ability of the latter to grow and remodel in response to diverse stimuli. The mechanical ...behaviour of living matter is governed not only by an elastic or viscoelastic response to loading on short time scales up to several minutes, but also by often crucial growth and remodelling responses on time scales from hours to months. Phenomena of growth and remodelling play important roles, for example during morphogenesis in early life as well as in homeostasis and pathogenesis in adult tissues, which often adapt to changes in their chemo-mechanical environment as a result of ageing, diseases, injury or surgical intervention. Mechano-regulated growth and remodelling are observed in various soft tissues, ranging from tendons and arteries to the eye and brain, but also in bone, lower organisms and plants. Understanding and predicting growth and remodelling of living systems is one of the most important challenges in biomechanics and mechanobiology. This article reviews the current state of growth and remodelling as it applies primarily to soft tissues, and provides a perspective on critical challenges and future directions.
The Cancer Stem Model allows for a dynamical description of cancer colonies which accounts for the existence of different families of cells, namely stem cells, highly proliferating and ...quasi-immortal, and differentiated cells, both undergoing cellular processes under numerous activated pathways. In the present work, we investigate a dynamical model numerically, as a system of coupled differential equations, and include a plasticity mechanism, of differentiated cells turning into a stem state if the stem concentration drops low. We are particularly interested in the stability of the model once we introduce stochastically evolving parameters, associated with environmental and cellular intrinsic variabilities, as well as the response of the model after introducing a drug therapy. As long as we stay within the characteristic time scale of the system, defined on the base of the needed time for the trajectories to converge on stable states, we observe that the system remains stable for the main parameters evolving stochastically according to white noise. As for the drug treatments, we discuss a model both for the kinetics and the dynamics of the substance in the organism, and then consider the impact of different types of therapies in a few particular examples, outlining some interesting mechanisms, such as the tumor growth paradox, that possibly impact the outcome of therapy significantly.
Nowadays, new experimental set-ups allow to determine the strains which appear in biological tissues under small controlled solicitations as stretch (uniaxial and bi-axial) or shear. Automatically ...recorded by camera, these data exhibit hysteresis even for very slow and weak forcing. The aim of this work is to revisit the modeling of hyper-elasticity and visco-elasticity for tissues with a rather disordered structure. Despite the weakness of the imposed stretch values, strong nonlinearities are observed for these tissues that polymeric gels do not show under the same conditions. Various classical models have been tested on surgical mammary tissues and it turns out that the simplest Neo-Hookean model with its viscous counter-part explain perfectly the data. Viscosity and elasticity compete quite equivalently showing that, in the selected range of forcing values, visco-elasticity cannot be discarded.
Surface curvature both emerges from, and influences the behavior of, living objects at length scales ranging from cell membranes to single cells to tissues and organs. The relevance of surface ...curvature in biology is supported by numerous experimental and theoretical investigations in recent years. In this review, first, a brief introduction to the key ideas of surface curvature in the context of biological systems is given and the challenges that arise when measuring surface curvature are discussed. Giving an overview of the emergence of curvature in biological systems, its significance at different length scales becomes apparent. On the other hand, summarizing current findings also shows that both single cells and entire cell sheets, tissues or organisms respond to curvature by modulating their shape and their migration behavior. Finally, the interplay between the distribution of morphogens or micro‐organisms and the emergence of curvature across length scales is addressed with examples demonstrating these key mechanistic principles of morphogenesis. Overall, this review highlights that curved interfaces are not merely a passive by‐product of the chemical, biological, and mechanical processes but that curvature acts also as a signal that co‐determines these processes.
Curvature as a local descriptor for shape has been revealed to play a fundamental role in the development of biological systems. Advanced 3D characterization methods allow its quantification across time and length scales indicating that cells and tissue growth can cause emergence of curved surfaces but in turn curvature also acts as a trigger for specific biological processes.
For many organisms, shapes emerge from growth, which generates stresses, which in turn can feedback on growth. In this review, theoretical methods to analyse various aspects of morphogenesis are ...discussed with the aim to determine the most adapted method for tissue mechanics. We discuss the need to work at scales intermediate between cells and tissues and emphasize the use of finite elasticity for this. We detail the application of these ideas to four systems: active cells embedded in tissues, brain cortical convolutions, the cortex of Caenorhabditis elegans during elongation and finally the proliferation of epithelia on extracellular matrix. Numerical models well adapted to inhomogeneities are also presented. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.
The purpose of this work is to provide a theoretical analysis of the mechanical behavior of the growth of soft materials under geometrical constraints. In particular, we focus on the swelling of a ...gel layer clamped to a substrate, which is still the subject of many experimental tests. Because the constrained swelling process induces compressive stresses, all these experiments exhibit surface instabilities, which ultimately lead to cusp formation. Our model is based on fixing a neo-Hookean constitutive energy together with the incompressibility requirement for a volumetric, homogeneous mass addition. Our approach is developed mostly, but not uniquely, in the plane strain configuration. We show how the standard equilibrium equations from continuum mechanics have a similarity with the two-dimensional Stokes flows, and we use a nonlinear stream function for the exact treatment of the incompressibility constraint. A free energy approach allows the extension both to arbitrary hyperelastic strain energies and to additional interactions, such as surface energies. We find that, at constant volumetric growth, the threshold for a wavy instability is completely governed by the amount of growth. Nevertheless, the determination of the wavelength at threshold, which scales with the initial thickness of the gel layer, requires the coupling with a surface effect. Our findings, which are valid in proximity of the threshold, are compared to experimental results. The proposed treatment can be extended to weakly nonlinearities within the aim of the theory of bifurcations.
The shape of plants and other living organisms is a crucial element of their biological functioning. Morphogenesis is the result of complex growth processes involving biological, chemical and ...physical factors at different temporal and spatial scales.
This study aims at describing stresses and strains induced by the production and reorganization of the material. The mechanical properties of soft tissues are modeled within the framework of continuum mechanics in finite elasticity. The kinematical description is based on the multiplicative decomposition of the deformation gradient tensor into an elastic and a growth term. Using this formalism, the authors have studied the growth of thin hyperelastic samples. Under appropriate assumptions, the dimensionality of the problem can be reduced, and the behavior of the plate is described by a two-dimensional surface.
The results of this theory demonstrate that the corresponding equilibrium equations are of the Föppl–von Kármán type where growth acts as a source of mean and Gaussian curvatures. Finally, the cockling of paper and the rippling of a grass blade are considered as two examples of growth-induced pattern formation.
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