Growth-induced buckling of an epithelial layer

Mechanobiological tissue instability induced by stress-modulated growth

The growth of biological tissues, which is regulated by a variety of factors, can induce stresses that may, in turn, destabilize the tissues into diverse patterns. In most previous studies, however, tissue growth was usually assumed as a prescribed parameter independent of stresses, limiting our understanding of the mechanobiological morphogenesis of real tissues. In this paper, we propose a theoretical model to investigate the mechanobiological response of soft tissues undergoing stress-modulated growth. Linear stability analysis is first performed to elucidate the surface instability mechanism induced by stress-modulated volumetric growth. We further conduct finite element simulations to validate the theoretical prediction and, particularly, to capture the post-buckling pattern evolution. Our results show that the non-uniform stresses, which evolve with the tissue growth and morphogenesis, exert mechanical feedback on the growth itself, producing up-down asymmetric surface morphologies as observed in, for example, the gyrification of human brains and brain organoids. It is also revealed that large residual stresses are unnecessary to cause mechanobiological instability and subsequent asymmetric patterning, which has long been believed to be driven by sufficiently high stresses. The present work could help us to understand the morphological changes of biological tissues under physiological and pathological conditions.

Modelling realistic 3D deformations of simple epithelia in dynamic homeostasis

The maintenance of tissue and organ structures during dynamic homeostasis is often not well understood. In order for a system to be stable, cell renewal, cell migration and cell death must be finely balanced. Moreover, a tissue's shape must remain relatively unchanged. Simple epithelial tissues occur in various structures throughout the body, such as the endothelium, mesothelium, linings of the lungs, saliva and thyroid glands, and gastrointestinal tract. Despite the prevalence of simple epithelial tissues, there are few models which accurately describe how these tissues maintain a stable structure. Here, we present a novel, 3D, deformable, multilayer, cell-centre model of a simple epithelium. Cell movement is governed by the minimisation of a bending potential across the epithelium, cell-cell adhesion, and viscous effects. We show that the model is capable of maintaining a consistent tissue structure while undergoing self renewal. We also demonstrate the model's robustness under tissue renewal, cell migration and cell removal. The model presented here is a valuable advancement towards the modelling of tissues and organs with complex and generalised structures.

Mechanical forces directing intestinal form and function

The vertebrate intestine experiences a range of intrinsically generated and external forces during both development and adult homeostasis. It is increasingly understood how the coordination of these forces shapes the intestine through organ-scale folding and epithelial organization into crypt-villus compartments. Moreover, accumulating evidence shows that several cell types in the adult intestine can sense and respond to forces to regulate key cellular processes underlying adult intestinal functions and self-renewal. In this way, transduction of forces may direct both intestinal homeostasis as well as adaptation to external stimuli, such as food ingestion or injury. In this review, we will discuss recent insights from complementary model systems into the force-dependent mechanisms that establish and maintain the unique architecture of the intestine, as well as its homeostatic regulation and function throughout adult life.

Mechanobiology of Colorectal Cancer

Simple Summary

It is well documented that colorectal cancer (CRC) is the third most common cancer type, responsible for high mortality in developed countries, resulting in a high socio-economic impact. Several biochemical and gene expression pathways explaining the manifestation of this cancer in humans have already been identified. However, explanations for some of the related biophysical mechanisms and their influence on CRC remain elusive. In CRC, biophysics and medical research have already revealed the importance of studying the effects of the stiffness and viscoelasticity of the substrate on cells, as well as the effect of the shear stress of blood and lymphatic vessels on the behavior of cells and tissues. A deeper understanding of the relationship between the biophysical cues and biochemical signals could be advantageous to develop new diagnostic techniques and therapeutic strategies. Being a disease with a high mortality rate, it becomes crucial to dedicate efforts to finding effective, alternative therapeutic strategies.

Abstract

In this review, the mechanobiology of colorectal cancer (CRC) are discussed. Mechanotransduction of CRC is addressed considering the relationship of several biophysical cues and biochemical pathways. Mechanobiology is focused on considering how it may influence epithelial cells in terms of motility, morphometric changes, intravasation, circulation, extravasation, and metastization in CRC development. The roles of the tumor microenvironment, ECM, and stroma are also discussed, taking into account the influence of alterations and surface modifications on mechanical properties and their impact on epithelial cells and CRC progression. The role of cancer-associated fibroblasts and the impact of flow shear stress is addressed in terms of how it affects CRC metastization. Finally, some insights concerning how the knowledge of biophysical mechanisms may contribute to the development of new therapeutic strategies and targeting molecules and how mechanical changes of the microenvironment play a role in CRC disease are presented.

Decellularized Colorectal Cancer Matrices as Bioactive Scaffolds for Studying Tumor-Stroma Interactions

More than a physical structure providing support to tissues, the extracellular matrix (ECM) is a complex and dynamic network of macromolecules that modulates the behavior of both cancer cells and associated stromal cells of the tumor microenvironment (TME). Over the last few years, several efforts have been made to develop new models that accurately mimic the interconnections within the TME and specifically the biomechanical and biomolecular complexity of the tumor ECM. Particularly in colorectal cancer, the ECM is highly remodeled and disorganized and constitutes a key component that affects cancer hallmarks, such as cell differentiation, proliferation, angiogenesis, invasion and metastasis. Therefore, several scaffolds produced from natural and/or synthetic polymers and ceramics have been used in 3D biomimetic strategies for colorectal cancer research. Nevertheless, decellularized ECM from colorectal tumors is a unique model that offers the maintenance of native ECM architecture and molecular composition. This review will focus on innovative and advanced 3D-based models of decellularized ECM as high-throughput strategies in colorectal cancer research that potentially fill some of the gaps between in vitro 2D and in vivo models. Our aim is to highlight the need for strategies that accurately mimic the TME for precision medicine and for studying the pathophysiology of the disease.

Axial elongation of caudalized human organoids mimics aspects of neural tube development

Axial elongation of the neural tube is crucial during mammalian embryogenesis for anterior-posterior body axis establishment and subsequent spinal cord development, but these processes cannot be interrogated directly in humans as they occur post-implantation. Here, we report an organoid model of neural tube extension derived from human pluripotent stem cell (hPSC) aggregates that have been caudalized with Wnt agonism, enabling them to recapitulate aspects of the morphological and temporal gene expression patterns of neural tube development. Elongating organoids consist largely of neuroepithelial compartments and contain TBXT+SOX2+ neuro-mesodermal progenitors in addition to PAX6+NES+ neural progenitors. A critical threshold of Wnt agonism stimulated singular axial extensions while maintaining multiple cell lineages, such that organoids displayed regionalized anterior-to-posterior HOX gene expression with hindbrain (HOXB1) regions spatially distinct from brachial (HOXC6) and thoracic (HOXB9) regions. CRISPR interference-mediated silencing of TBXT, a Wnt pathway target, increased neuroepithelial compartmentalization, abrogated HOX expression and disrupted uniaxial elongation. Together, these results demonstrate the potent capacity of caudalized hPSC organoids to undergo axial elongation in a manner that can be used to dissect the cellular organization and patterning decisions that dictate early human nervous system development.

Organoid-based modeling of intestinal development, regeneration, and repair

The intestinal epithelium harbors a remarkable adaptability to undergo injury-induced repair. A key part of the regenerative response is the transient reprogramming of epithelial cells into a fetal-like state, which drives uniform proliferation, tissue remodeling, and subsequent restoration of the homeostatic state. In this review, we discuss how Wnt and YAP signaling pathways control the intestinal repair response and the transitioning of cell states, in comparison with the process of intestinal development. Furthermore, we highlight how organoid-based applications have contributed to the characterization of the mechanistic principles and key players that guide these developmental and regenerative events.

The role of mechanics in the growth and homeostasis of the intestinal crypt

We present a mechanical model of tissue homeostasis that is specialised to the intestinal crypt. Growth and deformation of the crypt, idealised as a line of cells on a substrate, are modelled using morphoelastic rod theory. Alternating between Lagrangian and Eulerian mechanical descriptions enables us to precisely characterise the dynamic nature of tissue homeostasis, whereby the proliferative structure and morphology are static in the Eulerian frame, but there is active migration of Lagrangian material points out of the crypt. Assuming mechanochemical growth, we identify the necessary conditions for homeostasis, reducing the full, time-dependent system to a static boundary value problem characterising a spatially heterogeneous "treadmilling" state. We extract essential features of crypt homeostasis, such as the morphology, the proliferative structure, the migration velocity, and the sloughing rate. We also derive closed-form solutions for growth and sloughing dynamics in homeostasis, and show that mechanochemical growth is sufficient to generate the observed proliferative structure of the crypt. Key to this is the concept of threshold-dependent mechanical feedback, that regulates an established Wnt signal for biochemical growth. Numerical solutions demonstrate the importance of crypt morphology on homeostatic growth, migration, and sloughing, and highlight the value of this framework as a foundation for studying the role of mechanics in homeostasis.

Matrix Stiffness and Colorectal Cancer

In recent years, a growing consensus is emerging that the mechanical microenvironment of tumors is far more critical in the onset of tumor, tumor progression, invasion, and metastasis. Matrix stiffness, one of the sources of mechanical stimulation, affects tumor cells as well as non-tumor cells in multiple different molecular signaling pathways in solid tumors such as colorectal tumors, which lead to tumor invasion and metastasis, immune evasion and drug resistance. This review will illustrate the relationship between matrix stiffness and colorectal cancer from the following aspects. First, briefly introduce the mechanical microenvironment and colorectal cancer, then explain the origin of colorectal cancer extracellular matrix stiffness, and then synthesize the study of matrix stiffness of colorectal cancer in recent years to elaborate the effects of extracellular matrix stiffness in colorectal cancer’s biological behavior and signaling pathways, and finally we will discuss the transformation treatment for the matrix stiffness of colorectal cancer. An in-depth understanding of matrix stiffness and colorectal cancer can help researchers conduct further experiments to find new targets for the treatment of colorectal cancer.

Tunable corrugated patterns in an active nematic sheet

Active matter locally converts chemical energy into mechanical work and, for this reason, it provides new mechanisms of pattern formation. In particular, active nematic fluids made of protein motors and filaments are far-from-equilibrium systems that may exhibit spontaneous motion, leading to actively driven spatiotemporally chaotic states in 2 and 3 dimensions and coherent flows in 3 dimensions (3D). Although these dynamic flows reveal a characteristic length scale resulting from the interplay between active forcing and passive restoring forces, the observation of static and large-scale spatial patterns in active nematic fluids has remained elusive. In this work, we demonstrate that a 3D solution of kinesin motors and microtubule filaments spontaneously forms a 2D free-standing nematic active sheet that actively buckles out of plane into a centimeter-sized periodic corrugated sheet that is stable for several days at low activity. Importantly, the nematic orientational field does not display topological defects in the corrugated state and the wavelength and stability of the corrugations are controlled by the motor concentration, in agreement with a hydrodynamic theory. At higher activities these patterns are transient and chaotic flows are observed at longer times. Our results underline the importance of both passive and active forces in shaping active matter and demonstrate that a spontaneously flowing active fluid can be sculpted into a static material through an active mechanism.

Apical constriction is necessary for crypt formation in small intestinal organoids

Small intestinal organoids have become an important tool to study crypt homeostasis, cell fate dynamics and tissue biomechanics. Yet, the mechanisms that drive the budding of crypts from the smooth organoid epithelium remain incompletely understood. Locally enhanced proliferation has been suggested to induce tissue buckling and crypt initiation. Here we report that changes in cell morphology play a crucial role in crypt formation. Crypt formation is preceded by local epithelial thickening, apicobasal elongation, and apical narrowing, resulting in a wedge-like cell-shape, followed by apical evagination and crypt outgrowth. Myosin II activity is found to coincide with apical constriction of cells, while inhibition of myosin suppresses apical constriction and bud formation. The data suggest that myosin-driven apical constriction is a key driving force of bud initiation in small intestinal organoids.

Relating cell shape and mechanical stress in a spatially disordered epithelium using a vertex-based model

Using a popular vertex-based model to describe a spatially disordered planar epithelial monolayer, we examine the relationship between cell shape and mechanical stress at the cell and tissue level. Deriving expressions for stress tensors starting from an energetic formulation of the model, we show that the principal axes of stress for an individual cell align with the principal axes of shape, and we determine the bulk effective tissue pressure when the monolayer is isotropic at the tissue level. Using simulations for a monolayer that is not under peripheral stress, we fit parameters of the model to experimental data for Xenopus embryonic tissue. The model predicts that mechanical interactions can generate mesoscopic patterns within the monolayer that exhibit long-range correlations in cell shape. The model also suggests that the orientation of mechanical and geometric cues for processes such as cell division are likely to be strongly correlated in real epithelia. Some limitations of the model in capturing geometric features of Xenopus epithelial cells are highlighted.

Post-buckling behaviour of a growing elastic rod

We consider mechanically-induced pattern formation within the framework of a growing, planar, elastic rod attached to an elastic foundation. Through a combination of weakly nonlinear analysis and numerical methods, we identify how the shape and type of buckling (super- or subcritical) depend on material parameters, and a complex phase-space of transition from super- to subcritical is uncovered. We then examine the effect of heterogeneity on buckling and post-buckling behaviour, in the context of a heterogeneous substrate adhesion, elastic stiffness, or growth. We show how the same functional form of heterogeneity in different properties is manifest in a vastly differing post-buckled shape. Finally, a fourth form of heterogeneity, an imperfect foundation, is incorporated and shown to have a more dramatic impact on the buckling instability, a difference that can be qualitatively understood via the weakly nonlinear analysis.

Changes in cellular mechanical properties during onset or progression of colorectal cancer

Colorectal cancer (CRC) development represents a multistep process starting with specific mutations that affect proto-oncogenes and tumour suppressor genes. These mutations confer a selective growth advantage to colonic epithelial cells that form first dysplastic crypts, and then malignant tumours and metastases. All these steps are accompanied by deep mechanical changes at the cellular and the tissue level. A growing consensus is emerging that such modifications are not merely a by-product of the malignant progression, but they could play a relevant role in the cancer onset and accelerate its progression. In this review, we focus on recent studies investigating the role of the biomechanical signals in the initiation and the development of CRC. We show that mechanical cues might contribute to early phases of the tumour initiation by controlling the Wnt pathway, one of most important regulators of cell proliferation in various systems. We highlight how physical stimuli may be involved in the differentiation of non-invasive cells into metastatic variants and how metastatic cells modify their mechanical properties, both stiffness and adhesion, to survive the mechanical stress associated with intravasation, circulation and extravasation. A deep comprehension of these mechanical modifications may help scientist to define novel molecular targets for the cure of CRC.

Feedback, Lineages and Self-Organizing Morphogenesis

Feedback regulation of cell lineage progression plays an important role in tissue size homeostasis, but whether such feedback also plays an important role in tissue morphogenesis has yet to be explored. Here we use mathematical modeling to show that a particular feedback architecture in which both positive and negative diffusible signals act on stem and/or progenitor cells leads to the appearance of bistable or bi-modal growth behaviors, ultrasensitivity to external growth cues, local growth-driven budding, self-sustaining elongation, and the triggering of self-organization in the form of lamellar fingers. Such behaviors arise not through regulation of cell cycle speeds, but through the control of stem or progenitor self-renewal. Even though the spatial patterns that arise in this setting are the result of interactions between diffusible factors with antagonistic effects, morphogenesis is not the consequence of Turing-type instabilities.

An individual based computational model of intestinal crypt fission and its application to predicting unrestrictive growth of the intestinal epithelium

Intestinal crypt fission is a homeostatic phenomenon, observable in healthy adult mucosa, but which also plays a pathological role as the main mode of growth of some intestinal polyps. Building on our previous individual based model for the small intestinal crypt and on in vitro cultured intestinal organoids, we here model crypt fission as a budding process based on fluid mechanics at the individual cell level and extrapolated predictions for growth of the intestinal epithelium. Budding was always observed in regions of organoids with abundant Paneth cells. Our data support a model in which buds are biomechanically initiated by single stem cells surrounded by Paneth cells which exhibit greater resistance to viscoelastic deformation, a hypothesis supported by atomic force measurements of single cells. Time intervals between consecutive budding events, as simulated by the model and observed in vitro, were 2.84 and 2.62 days, respectively. Predicted cell dynamics was unaffected within the original crypt which retained its full capability of providing cells to the epithelium throughout fission. Mitotic pressure in simulated primary crypts forced upward migration of buds, which simultaneously grew into new protruding crypts at a rate equal to 1.03 days(-1) in simulations and 0.99 days(-1) in cultured organoids. Simulated crypts reached their final size in 4.6 days, and required 6.2 days to migrate to the top of the primary crypt. The growth of the secondary crypt is independent of its migration along the original crypt. Assuming unrestricted crypt fission and multiple budding events, a maximal growth rate of the intestinal epithelium of 0.10 days(-1) is predicted and thus approximately 22 days are required for a 10-fold increase of polyp size. These predictions are in agreement with the time reported to develop macroscopic adenomas in mice after loss of Apc in intestinal stem cells.

Discrete Element Framework for Modelling Extracellular Matrix, Deformable Cells and Subcellular Components

This paper presents a framework for modelling biological tissues based on discrete particles. Cell components (e.g. cell membranes, cell cytoskeleton, cell nucleus) and extracellular matrix (e.g. collagen) are represented using collections of particles. Simple particle to particle interaction laws are used to simulate and control complex physical interaction types (e.g. cell-cell adhesion via cadherins, integrin basement membrane attachment, cytoskeletal mechanical properties). Particles may be given the capacity to change their properties and behaviours in response to changes in the cellular microenvironment (e.g., in response to cell-cell signalling or mechanical loadings). Each particle is in effect an ‘agent’, meaning that the agent can sense local environmental information and respond according to pre-determined or stochastic events. The behaviour of the proposed framework is exemplified through several biological problems of ongoing interest. These examples illustrate how the modelling framework allows enormous flexibility for representing the mechanical behaviour of different tissues, and we argue this is a more intuitive approach than perhaps offered by traditional continuum methods. Because of this flexibility, we believe the discrete modelling framework provides an avenue for biologists and bioengineers to explore the behaviour of tissue systems in a computational laboratory.

Modelling is an important tool in understanding the behaviour of biological tissues. In this paper we advocate a new modelling framework in which cells and tissues are represented by a collection of particles with associated properties. The particles interact with each other and can change their behaviour in response to changes in their environment. We demonstrate how the propose framework can be used to represent the mechanical behaviour of different tissues with much greater flexibility as compared to traditional continuum based methods.

Secondary instabilities modulate cortical complexity in the mammalian brain

Disclosing the origin of convolutions in the mammalian brain remains a scientific challenge. Primary folds form before we are born: they are static, well defined, and highly preserved across individuals. Secondary folds occur and disappear throughout our entire life time: they are dynamic, irregular, and highly variable among individuals. While extensive research has improved our understanding of primary folding in the mammalian brain, secondary folding remains understudied and poorly understood. Here, we show that secondary instabilities can explain the increasing complexity of our brain surface as we age. Using the nonlinear field theories of mechanics supplemented by the theory of finite growth, we explore the critical conditions for secondary instabilities. We show that with continuing growth, our brain surface continues to bifurcate into increasingly complex morphologies. Our results suggest that even small geometric variations can have a significant impact on surface morphogenesis. Secondary bifurcations, and with them morphological changes during childhood and adolescence, are closely associated with the formation and loss of neuronal connections. Understanding the correlation between neuronal connectivity, cortical thickness, surface morphology, and ultimately behavior, could have important implications on the diagnostics, classification, and treatment of neurological disorders.

Growing matter: a review of growth in living systems

Living systems can grow, develop, adapt, and evolve. These phenomena are non-intuitive to traditional engineers and often difficult to understand. Yet, classical engineering tools can provide valuable insight into the mechanisms of growth in health and disease. Within the past decade, the concept of incompatible configurations has evolved as a powerful tool to model growing systems within the framework of nonlinear continuum mechanics. However, there is still a substantial disconnect between the individual disciplines, which explore the phenomenon of growth from different angles. Here we show that the nonlinear field theories of mechanics provide a unified concept to model finite growth by means of a single tensorial internal variable, the second order growth tensor. We review the literature and categorize existing growth models by means of two criteria: the microstructural appearance of growth, either isotropic or anisotropic; and the microenvironmental cues that drive the growth process, either chemical or mechanical. We demonstrate that this generic concept is applicable to a broad range of phenomena such as growing arteries, growing tumors, growing skin, growing airway walls, growing heart valve leaflets, growing skeletal muscle, growing plant stems, growing heart valve annuli, and growing cardiac muscle. The proposed approach has important biological and clinical applications in atherosclerosis, in-stent restenosis, tumor invasion, tissue expansion, chronic bronchitis, mitral regurgitation, limb lengthening, tendon tear, plant physiology, dilated and hypertrophic cardiomyopathy, and heart failure. Understanding the mechanisms of growth in these chronic conditions may open new avenues in medical device design and personalized medicine to surgically or pharmacologically manipulate development and alter, control, or revert disease progression.

Buckling of a growing tissue and the emergence of two-dimensional patterns

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We model the growth of gut epithelial cells cultured upon a deformable substrate.

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Growth generates buckling instabilities, contributing to crypt formation in vivo.

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Variations in mechanical properties have little effect on resulting configurations.

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Configurations are controlled by growth patterns & interactions with strata below.

The process of biological growth and the associated generation of residual stress has previously been considered as a driving mechanism for tissue buckling and pattern selection in numerous areas of biology. Here, we develop a two-dimensional thin plate theory to simulate the growth of cultured intestinal epithelial cells on a deformable substrate, with the goal of elucidating how a tissue engineer might best recreate the regular array of invaginations (crypts of Lieberkühn) found in the wall of the mammalian intestine. We extend the standard von Kármán equations to incorporate inhomogeneity in the plate’s mechanical properties and surface stresses applied to the substrate by cell proliferation. We determine numerically the configurations of a homogeneous plate under uniform cell growth, and show how tethering to an underlying elastic foundation can be used to promote higher-order buckled configurations. We then examine the independent effects of localised softening of the substrate and spatial patterning of cellular growth, demonstrating that (within a two-dimensional framework, and contrary to the predictions of one-dimensional models) growth patterning constitutes a more viable mechanism for control of crypt distribution than does material inhomogeneity.

On the mechanics of thin films and growing surfaces

Many living structures are coated by thin films, which have distinct mechanical properties from the bulk. In particular, these thin layers may grow faster or slower than the inner core. Differential growth creates a balanced interplay between tension and compression and plays a critical role in enhancing structural rigidity. Typical examples with a compressive outer surface and a tensile inner core are the petioles of celery, caladium, or rhubarb. While plant physiologists have studied the impact of tissue tension on plant rigidity for more than a century, the fundamental theory of growing surfaces remains poorly understood. Here, we establish a theoretical and computational framework for continua with growing surfaces and demonstrate its application to classical phenomena in plant growth. To allow the surface to grow independently of the bulk, we equip it with its own potential energy and its own surface stress. We derive the governing equations for growing surfaces of zero thickness and obtain their spatial discretization using the finite-element method. To illustrate the features of our new surface growth model we simulate the effects of growth-induced longitudinal tissue tension in a stalk of rhubarb. Our results demonstrate that different growth rates create a mechanical environment of axial tissue tension and residual stress, which can be released by peeling off the outer layer. Our novel framework for continua with growing surfaces has immediate biomedical applications beyond these classical model problems in botany: it can be easily extended to model and predict surface growth in asthma, gastritis, obstructive sleep apnoea, brain development, and tumor invasion. Beyond biology and medicine, surface growth models are valuable tools for material scientists when designing functionalized surfaces with distinct user-defined properties.

Toward a quantitative understanding of the Wnt/β-catenin pathway through simulation and experiment

Wnt signaling regulates cell survival, proliferation, and differentiation throughout development and is aberrantly regulated in cancer. The pathway is activated when Wnt ligands bind to specific receptors on the cell surface, resulting in the stabilization and nuclear accumulation of the transcriptional co-activator β-catenin. Mathematical and computational models have been used to study the spatial and temporal regulation of the Wnt/β-catenin pathway and to investigate the functional impact of mutations in key components. Such models range in complexity, from time-dependent, ordinary differential equations that describe the biochemical interactions between key pathway components within a single cell, to complex, multiscale models that incorporate the role of the Wnt/β-catenin pathway target genes in tissue homeostasis and carcinogenesis. This review aims to summarize recent progress in mathematical modeling of the Wnt pathway and to highlight new biological results that could form the basis for future theoretical investigations designed to increase the utility of theoretical models of Wnt signaling in the biomedical arena.

On the mechanics of continua with boundary energies and growing surfaces

Many biological systems are coated by thin films for protection, selective absorption, or transmembrane transport. A typical example is the mucous membrane covering the airways, the esophagus, and the intestine. Biological surfaces typically display a distinct mechanical behavior from the bulk; in particular, they may grow at different rates. Growth, morphological instabilities, and buckling of biological surfaces have been studied intensely by approximating the surface as a layer of finite thickness; however, growth has never been attributed to the surface itself. Here, we establish a theory of continua with boundary energies and growing surfaces of zero thickness in which the surface is equipped with its own potential energy and is allowed to grow independently of the bulk. In complete analogy to the kinematic equations, the balance equations, and the constitutive equations of a growing solid body, we derive the governing equations for a growing surface. We illustrate their spatial discretization using the finite element method, and discuss their consistent algorithmic linearization. To demonstrate the conceptual differences between volume and surface growth, we simulate the constrained growth of the inner layer of a cylindrical tube. Our novel approach towards continua with growing surfaces is capable of predicting extreme growth of the inner cylindrical surface, which more than doubles its initial area. The underlying algorithmic framework is robust and stable; it allows to predict morphological changes due to surface growth during the onset of buckling and beyond. The modeling of surface growth has immediate biomedical applications in the diagnosis and treatment of asthma, gastritis, obstructive sleep apnoea, and tumor invasion. Beyond biomedical applications, the scientific understanding of growth-induced morphological instabilities and surface wrinkling has important implications in material sciences, manufacturing, and microfabrication, with applications in soft lithography, metrology, and flexible electronics.

A biomechanical model of anther opening reveals the roles of dehydration and secondary thickening

Understanding the processes that underlie pollen release is a prime target for controlling fertility to enable selective breeding and the efficient production of hybrid crops. Pollen release requires anther opening, which involves changes in the biomechanical properties of the anther wall. In this research, we develop and use a mathematical model to understand how these biomechanical processes lead to anther opening. Our mathematical model describing the biomechanics of anther opening incorporates the bilayer structure of the mature anther wall, which comprises the outer epidermal cell layer, whose turgor pressure is related to its hydration, and the endothecial layer, whose walls contain helical secondary thickening, which resists stretching and bending. The model describes how epidermal dehydration, in association with the thickened endothecial layer, creates forces within the anther wall causing it to bend outwards, resulting in anther opening and pollen release. The model demonstrates that epidermal dehydration can drive anther opening, and suggests why endothecial secondary thickening is essential for this process (explaining the phenotypes presented in the myb26 and nst1nst2 mutants). The research hypothesizes and demonstrates a biomechanical mechanism for anther opening, which appears to be conserved in many other biological situations where tissue movement occurs.

A two-dimensional model of the colonic crypt accounting for the role of the basement membrane and pericryptal fibroblast sheath

The role of the basement membrane is vital in maintaining the integrity and structure of an epithelial layer, acting as both a mechanical support and forming the physical interface between epithelial cells and the surrounding connective tissue. The function of this membrane is explored here in the context of the epithelial monolayer that lines the colonic crypt, test-tube shaped invaginations that punctuate the lining of the intestine and coordinate a regular turnover of cells to replenish the epithelial layer every few days. To investigate the consequence of genetic mutations that perturb the system dynamics and can lead to colorectal cancer, it must be possible to track the emerging tissue level changes that arise in the crypt. To that end, a theoretical crypt model with a realistic, deformable geometry is required. A new discrete crypt model is presented, which focuses on the interaction between cell- and tissue-level behaviour, while incorporating key subcellular components. The model contains a novel description of the role of the surrounding tissue and musculature, based upon experimental observations of the tissue structure of the crypt, which are also reported. A two-dimensional (2D) cross-sectional geometry is considered, and the shape of the crypt is allowed to evolve and deform. Simulation results reveal how the shape of the crypt may contribute mechanically to the asymmetric division events typically associated with the stem cells at the base. The model predicts that epithelial cell migration may arise due to feedback between cell loss at the crypt collar and density-dependent cell division, an hypothesis which can be investigated in a wet lab. This work forms the basis for investigation of the deformation of the crypt structure that can occur due to proliferation of cells exhibiting mutant phenotypes, experiments that would not be possible in vivo or in vitro.