Minimizing the Elastic Energy of Growing Leaves by Conformal Mapping.

During morphogenesis, the shape of living species results from growth, stress relaxation, and remodeling. When the growth does not generate any stress, the body shape only reflects the growth density. In two dimensions, we show that stress free configurations are simply determined by the time evolution of a conformal mapping which concerns not only the boundary but also the displacement field during an arbitrary period of time inside the sample. Fresh planar leaves are good examples for our study: they have no elastic stress, almost no weight, and their shape can be easily represented by holomorphic functions. The growth factor, isotropic or anisotropic, is related to the metrics between the initial and current conformal maps. By adjusting the mathematical shape function, main characteristics such as tips (convex or concave or sharp-pointed), undulating borders, and veins can be mathematically recovered, which are in good agreement with observations. It is worth mentioning that this flexible method allows us to study complex morphologies of growing leaves such as the fenestration process in Monstera deliciosa, and can also shed light on many other 2D biological patterns.

Physics of growing biological tissues: the complex cross-talk between cell activity, growth and resistance.

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'.

Assessing the Contribution of Active and Passive Stresses in C. elegans Elongation.

The role of the actomyosin network is investigated in the elongation of C. elegans during embryonic morphogenesis. We present a model of active elongating matter that combines prestress and passive stress in nonlinear elasticity. Using this model we revisit recently published data from laser ablation experiments to account for why cells under contraction can lead to an opening fracture. By taking into account the specific embryo geometry, we obtain quantitative predictions for the contractile forces exerted by the molecular motors myosin II for an elongation up to 70% of the initial length. This study demonstrates the importance of active processes in embryonic morphogenesis and the interplay between geometry and nonlinear mechanics during morphological events. In particular, it outlines the role of each connected layer of the epidermis compressed by an apical extracellular matrix that distributes the stresses during elongation.

Anisotropic growth shapes intestinal tissues during embryogenesis.

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.

Cyclic muscle contractions reinforce the actomyosin motors and mediate the full elongation of C. elegans embryo.

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.

Chemotaxis migration and morphogenesis of living colonies.

Development of forms in living organisms is complex and fascinating. Morphogenetic theories that investigate these shapes range from discrete to continuous models, from the variational elasticity to time-dependent fluid approach. Here a mixture model is chosen to describe the mass transport in a morphogenetic gradient: it gives a mathematical description of a mixture involving several constituents in mechanical interactions. This model, which is highly flexible can incorporate many biological processes but also complex interactions between cells as well as between cells and their environment. We use this model to derive a free-boundary problem easier to handle analytically. We solve it in the simplest geometry: an infinite linear front advancing with a constant velocity. In all the cases investigated here as the 3 D diffusion, the increase of mitotic activity at the border, nonlinear laws for the uptake of morphogens or for the mobility coefficient, a planar front exists above a critical threshold for the mobility coefficient but it becomes unstable just above the threshold at long wavelengths due to the existence of a Goldstone mode. This explains why sparsely bacteria exhibit dendritic patterns experimentally in opposition to other colonies such as biofilms and epithelia which are more compact. In the most unstable situation, where all the laws: diffusion, chemotaxis driving and chemoattractant uptake are linear, we show also that the system can recover a dynamic stability. A second threshold for the mobility exists which has a lower value as the ratio between diffusion coefficients decreases. Within the framework of this model where the biomass is treated mainly as a viscous and diffusive fluid, we show that the multiplicity of independent parameters in real biologic experimental set-up may explain varieties of observed patterns.

Stochasticity and Drug Effects in Dynamical Model for Cancer Stem Cells.

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.

Curvature in Biological Systems: Its Quantification, Emergence, and Implications across the Scales.

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.

Shape transition in artificial tumors: from smooth buckles to singular creases.

Using swelling hydrogels, we study the evolution of a thin circular artificial tumor whose growth is confined at the periphery. When the volume of the outer proliferative ring increases, the tumor loses its initial symmetry and bifurcates towards an oscillatory shape. Depending on the geometrical and elastic parameters, we observe either a smooth large-wavelength undulation of the swelling layer or the formation of sharp creases at the free boundary. Our experimental results as well as previous observations from other studies are in very good agreement with a nonlinear poroelastic model.

Morphological changes in early melanoma development: influence of nutrients, growth inhibitors and cell-adhesion mechanisms.

Current diagnostic methods for skin cancers are based on some morphological characteristics of the pigmented skin lesions, including the geometry of their contour. The aim of this article is to model the early growth of melanoma accounting for the biomechanical characteristics of the tumor micro-environment, and evaluating their influence on the tumor morphology and its evolution. The spatial distribution of tumor cells and diffusing molecules are explicitly described in a three-dimensional multiphase model, which incorporates general cell-to-cell mechanical interactions, a dependence of cell proliferation on contact inhibition, as well as a local diffusion of nutrients and inhibiting molecules. A two-dimensional model is derived in a lubrication limit accounting for the thin geometry of the epidermis. First, the dynamical and spatial properties of planar and circular tumor fronts are studied, with both numerical and analytical techniques. A WKB method is then developed in order to analyze the solution of the governing partial differential equations and to derive the threshold conditions for a contour instability of the growing tumor. A control parameter and a critical wavelength are identified, showing that high cell proliferation, high cell adhesion, large tumor radius and slow tumor growth correlate with the occurrence of a contour instability. Finally, comparing the theoretical results with a large amount of clinical data we show that our predictions describe accurately both the morphology of melanoma observed in vivo and its variations with the tumor growth rate. This study represents a fundamental step to understand more complex microstructural patterns observed during skin tumor growth. Its results have important implications for the improvement of the diagnostic methods for melanoma, possibly driving progress towards a personalized screening.

Self-contact and instabilities in the anisotropic growth of elastic membranes.

We investigate the morphology of thin discs and rings growing in the circumferential direction. Recent analytical results suggest that this growth produces symmetric excess cones (e cones). We study the stability of such solutions considering self-contact and bending stress. We show that, contrary to what was assumed in previous analytical solutions, beyond a critical growth factor, no symmetric e cone solution is energetically minimal any more. Instead, we obtain skewed e cone solutions having lower energy, characterized by a skewness angle and repetitive spiral winding with increasing growth. These results are generalized to discs with varying thickness and rings with holes of different radii.

Local membrane mechanics of pore-spanning bilayers.

The mechanical behavior of lipid bilayers spanning the pores of highly ordered porous silicon substrates was scrutinized by local indentation experiments as a function of surface functionalization, lipid composition, solvent content, indentation velocity, and pore radius. Solvent-containing nano black lipid membranes (nano-BLMs) as well as solvent-free pore-spanning bilayers were imaged by fluorescence and atomic force microscopy prior to force curve acquisition, which allows distinguishing between membrane-covered and uncovered pores. Force indentation curves on pore-spanning bilayers attached to functionalized hydrophobic porous silicon substrates reveal a predominately linear response that is mainly attributed to prestress in the membranes. This is in agreement with the observation that indentation leads to membrane lysis well below 5% area dilatation. However, membrane bending and lateral tension dominate over prestress and stretching if solvent-free supported membranes obtained from spreading giant liposomes on hydrophilic porous silicon are indented. An elastic regime diagram is presented that readily allows determining the dominant contribution to the mechanical response upon indentation as a function of load and pore radius.

Clonal pattern dynamics in tumor: the concept of cancer stem cells.

We present a multiphase model for solid tumor initiation and progression focusing on the properties of cancer stem cells (CSC). CSCs are a small and singular cell sub-population having outstanding capacities: high proliferation rate, self-renewal and extreme therapy resistance. Our model takes all these factors into account under a recent perspective: the possibility of phenotype switching of differentiated cancer cells (DC) to the stem cell state, mediated by chemical activators. This plasticity of cancerous cells complicates the complete eradication of CSCs and the tumor suppression. The model in itself requires a sophisticated treatment of population dynamics driven by chemical factors. We analytically demonstrate that the rather important number of parameters, inherent to any biological complexity, is reduced to three pivotal quantities.Three fixed points guide the dynamics, and two of them may lead to an optimistic issue, predicting either a control of the cancerous cell population or a complete eradication. The space environment, critical for the tumor outcome, is introduced via a density formalism. Disordered patterns are obtained inside a stable growing contour driven by the CSC. Somewhat surprisingly, despite the patterning instability, the contour maintains its circular shape but ceases to grow for a typical size independently of segregation patterns or obstacles located inside.

Growth and remodelling of living tissues: perspectives, challenges and opportunities.

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.

On the definition and modeling of incremental, cumulative, and continuous growth laws in morphoelasticity.

In the theory of elastic growth, a growth process is modeled by a sequence of growth itself followed by an elastic relaxation ensuring integrity and compatibility of the body. The description of this process is local in time and only corresponds to an incremental step in the total growth process. As time evolves, these incremental growth steps are compounded and a natural question is the description of the overall cumulative growth and whether a continuous description of this process is possible. These ideas are discussed and further studied in the case of incompressible shells.

Morphoelasticity in the development of brown alga Ectocarpus siliculosus: from cell rounding to branching.

A biomechanical model is proposed for the growth of the brown alga Ectocarpus siliculosus Featuring ramified uniseriate filaments, this alga has two modes of growth: apical growth and intercalary growth with branching. Apical growth occurs upon the mitosis of a young cell at one extremity and leads to a new tip cell followed by a cylindrical cell, whereas branching mainly occurs when a cylindrical cell becomes rounded and swells, forming a spherical cell. Given the continuous interplay between cell growth and swelling, a poroelastic model combining osmotic pressure and volumetric growth is considered for the whole cell, cytoplasm and cell wall. The model recovers the morphogenetic transformations of mature cells: transformation of a cylindrical shape into spherical shape with a volumetric increase, and then lateral branching. Our simulations show that the poro-elastic model, including the Mooney-Rivlin approach for hyper-elastic materials, can correctly reproduce the observations. In particular, branching appears to be a plasticity effect due to the high level of tension created after the increase in volume of mature cells.

The interplay of stiffness and force anisotropies drives embryo elongation.

The morphogenesis of tissues, like the deformation of an object, results from the interplay between their material properties and the mechanical forces exerted on them. The importance of mechanical forces in influencing cell behaviour is widely recognized, whereas the importance of tissue material properties, in particular stiffness, has received much less attention. Using Caenorhabditis elegans as a model, we examine how both aspects contribute to embryonic elongation. Measuring the opening shape of the epidermal actin cortex after laser nano-ablation, we assess the spatiotemporal changes of actomyosin-dependent force and stiffness along the antero-posterior and dorso-ventral axis. Experimental data and analytical modelling show that myosin-II-dependent force anisotropy within the lateral epidermis, and stiffness anisotropy within the fiber-reinforced dorso-ventral epidermis are critical in driving embryonic elongation. Together, our results establish a quantitative link between cortical tension, material properties and morphogenesis of an entire embryo.

Towards a unified approach in the modeling of fibrosis: A review with research perspectives.

Pathological fibrosis is the result of a failure in the wound healing process. The comprehension and the related modeling of the different mechanisms that trigger fibrosis are a challenge of many researchers that work in the field of medicine and biology. The modern scientific analysis of a phenomenon generally consists of three major approaches: theoretical, experimental, and computational. Different theoretical tools coming from mathematics and physics have been proposed for the modeling of the physiological and pathological fibrosis. However a complete framework is missing and the development of a general theory is required. This review aims at finding a unified approach in the modeling of fibrosis diseases that takes into account the different phenomena occurring at each level: molecular, cellular and tissue. Specifically by means of a critical analysis of the different models that have been proposed in the mathematical, computational and physical biology, from molecular to tissue scales, a multiscale approach is proposed, an approach that has been strongly recommended by top level biologists in the past decades.

Onset of nonlinearity in a stochastic model for auto-chemotactic advancing epithelia.

We investigate the role of auto-chemotaxis in the growth and motility of an epithelium advancing on a solid substrate. In this process, cells create their own chemoattractant allowing communications among neighbours, thus leading to a signaling pathway. As known, chemotaxis provokes the onset of cellular density gradients and spatial inhomogeneities mostly at the front, a phenomenon able to predict some features revealed in in vitro experiments. A continuous model is proposed where the coupling between the cellular proliferation, the friction on the substrate and chemotaxis is investigated. According to our results, the friction and proliferation stabilize the front whereas auto-chemotaxis is a factor of destabilization. This antagonist role induces a fingering pattern with a selected wavenumber k0. However, in the planar front case, the translational invariance of the experimental set-up gives also a mode at k = 0 and the coupling between these two modes in the nonlinear regime is responsible for the onset of a Hopf-bifurcation. The time-dependent oscillations of patterns observed experimentally can be predicted simply in this continuous non-linear approach. Finally the effects of noise are also investigated below the instability threshold.

Growth and remodelling for profound circular wounds in skin.

Wound healing studies both in vitro and in vivo have received a lot of attention recently. In vivo wound healing is a multi-step process involving physiological factors such as fibrinogen forming the clot, the infiltrated inflammatory cells, the recruited fibroblasts and the differentiated myofibroblasts as well as deposited collagens. All these actors play their roles at different times, aided by a cascade of morphogenetic agents and the result for the repair is approximatively successful but the imperfection is remained for large scars with fibrosis. Here, we want to study wound healing from the viewpoint of skin biomechanics, integrating the particular layered geometry of the skin, and the role of the neighbouring wound epidermis. After 2 days post-injury, it migrates towards the wound centre to cover the hole, the migration being coupled to proliferation at the wound border. Such a process is dominated by the skin properties which varies with ages, locations, pathologies, radiations, etc. It is also controlled by passive (actin, collagen) and active (myo-fibroblasts) fibres. We explore a growth model in finite elasticity of a bilayer surrounding a circular wound, only the interior one being proliferative and contractile. We discuss the occurrence of an irregular wound geometry generated by stresses and show quantitatively that it results from the combined effects of the stiffness, the size of the wound, eventually weakened by actin cables. Comparison of our findings is made with known observations or experiments in vivo.