A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices

The mechanical behaviour of solid biological tissues has long been described using models based on classical continuum mechanics. However, the classical continuum theories of elasticity and viscoelasticity cannot easily capture the continual remodelling and associated structural changes in biological tissues. Furthermore, models drawn from plasticity theory are difficult to apply and interpret in this context, where there is no equivalent of a yield stress or flow rule. In this work, we describe a novel one-dimensional mathematical model of tissue remodelling based on the multiplicative decomposition of the deformation gradient. We express the mechanical effects of remodelling as an evolution equation for the effective strain, a measure of the difference between the current state and a hypothetical mechanically relaxed state of the tissue. This morphoelastic model combines the simplicity and interpretability of classical viscoelastic models with the versatility of plasticity theory. A novel feature of our model is that while most models describe growth as a continuous quantity, here we begin with discrete cells and develop a continuum representation of lattice remodelling based on an appropriate limit of the behaviour of discrete cells. To demonstrate the utility of our approach, we use this framework to capture qualitative aspects of the continual remodelling observed in fibroblast-populated collagen lattices, in particular its contraction and its subsequent sudden re-expansion when remodelling is interrupted.     

Getting in shape and swimming: the role of cortical forces and membrane heterogeneity in eukaryotic cells

Recent research has shown that motile cells can adapt their mode of propulsion to the mechanical properties of the environment in which they find themselves—crawling in some environments while swimming in others. The latter can involve movement by blebbing or other cyclic shape changes, and both highly-simplified and more realistic models of these modes have been studied previously. Herein we study swimming that is driven by membrane tension gradients that arise from flows in the actin cortex underlying the membrane, and does not involve imposed cyclic shape changes. Such gradients can lead to a number of different characteristic cell shapes, and our first objective is to understand how different distributions of membrane tension influence the shape of cells in an inviscid quiescent fluid. We then analyze the effects of spatial variation in other membrane properties, and how they interact with tension gradients to determine the shape. We also study the effect of fluid–cell interactions and show how tension leads to cell movement, how the balance between tension gradients and a variable bending modulus determine the shape and direction of movement, and how the efficiency of movement depends on the properties of the fluid and the distribution of tension and bending modulus in the membrane.

     

Tensional Homeostasis in Single Fibroblasts

Adherent cells generate forces through acto-myosin contraction to move, change shape, and sense the mechanical properties of their environment. They are thought to maintain defined levels of tension with their surroundings despite mechanical perturbations that could change tension, a concept known as tensional homeostasis. Misregulation of tensional homeostasis has been proposed to drive disorganization of tissues and promote progression of diseases such as cancer. However, whether tensional homeostasis operates at the single cell level is unclear. Here, we directly test the ability of single fibroblast cells to regulate tension when subjected to mechanical displacements in the absence of changes to spread area or substrate elasticity. We use a feedback-controlled atomic force microscope to measure and modulate forces and displacements of individual contracting cells as they spread on a fibronectin-patterned atomic-force microscope cantilever and coverslip. We find that the cells reach a steady-state contraction force and height that is insensitive to stiffness changes as they fill the micropatterned areas. Rather than maintaining a constant tension, the fibroblasts altered their contraction force in response to mechanical displacement in a strain-rate-dependent manner, leading to a new and stable steady-state force and height. This response is influenced by overexpression of the actin crosslinker α-actinin, and rheology measurements reveal that changes in cell elasticity are also strain- rate-dependent. Our finding of tensional buffering, rather than homeostasis, allows cells to transition between different tensional states depending on how they are displaced, permitting distinct responses to slow deformations during tissue growth and rapid deformations associated with injury.     

Tensional Homeostasis in Single Fibroblasts

Adherent cells generate forces through acto-myosin contraction to move, change shape, and sense the mechanical properties of their environment. They are thought to maintain defined levels of tension with their surroundings despite mechanical perturbations that could change tension, a concept known as tensional homeostasis. Misregulation of tensional homeostasis has been proposed to drive disorganization of tissues and promote progression of diseases such as cancer. However, whether tensional homeostasis operates at the single cell level is unclear. Here, we directly test the ability of single fibroblast cells to regulate tension when subjected to mechanical displacements in the absence of changes to spread area or substrate elasticity. We use a feedback-controlled atomic force microscope to measure and modulate forces and displacements of individual contracting cells as they spread on a fibronectin-patterned atomic-force microscope cantilever and coverslip. We find that the cells reach a steady-state contraction force and height that is insensitive to stiffness changes as they fill the micropatterned areas. Rather than maintaining a constant tension, the fibroblasts altered their contraction force in response to mechanical displacement in a strain-rate-dependent manner, leading to a new and stable steady-state force and height. This response is influenced by overexpression of the actin crosslinker α-actinin, and rheology measurements reveal that changes in cell elasticity are also strain- rate-dependent. Our finding of tensional buffering, rather than homeostasis, allows cells to transition between different tensional states depending on how they are displaced, permitting distinct responses to slow deformations during tissue growth and rapid deformations associated with injury.     

Patterned cell adhesion associated with tissue deformations during dorsal closure in Drosophila

Cell shape changes within epithelia require the regulation of adhesive molecules that maintain tissue integrity. How remodelling of cell contacts is achieved while tissue integrity is maintained remains a fundamental question in morphogenesis. Dorsal Closure is a good system to study the dynamics of DE-Cadherin during morphogenesis. It relies on concerted cell shape changes of two epithelial sheets: amnioserosa cell contraction and epidermal cell elongation. To investigate the modulation of DE-Cadherin we performed antibody uptake experiments in live embryos during Dorsal Closure. We found that some antibodies access certain epitopes of the extracellular domain of native DE-Cadherin only in the amnioserosa and epidermal cells attached to the amnioserosa, which has never been observed in fixed DE-Cadherin in Drosophila embryos. These differences correlate with the different cell behaviour of these regions and therefore we suggest that DE-Cadherin exists in different forms that confer different adhesive strengths. We propose this to be a widespread mechanism for the differential modulation of adhesion during morphogenesis.     

A theoretical model of inflammation- and mechanotransduction-driven asthmatic airway remodelling

Inflammation, airway hyper-responsiveness and airway remodelling are well-established hallmarks of asthma, but their inter-relationships remain elusive. In order to obtain a better understanding of their inter-dependence, we develop a mechanochemical morphoelastic model of the airway wall accounting for local volume changes in airway smooth muscle (ASM) and extracellular matrix in response to transient inflammatory or contractile agonist challenges. We use constrained mixture theory, together with a multiplicative decomposition of growth from the elastic deformation, to model the airway wall as a nonlinear fibre-reinforced elastic cylinder. Local contractile agonist drives ASM cell contraction, generating mechanical stresses in the tissue that drive further release of mitogenic mediators and contractile agonists via underlying mechanotransductive signalling pathways. Our model predictions are consistent with previously described inflammation-induced remodelling within an axisymmetric airway geometry. Additionally, our simulations reveal novel mechanotransductive feedback by which hyper-responsive airways exhibit increased remodelling, for example, via stress-induced release of pro-mitogenic and pro-contractile cytokines. Simulation results also reveal emergence of a persistent contractile tone observed in asthmatics, via either a pathological mechanotransductive feedback loop, a failure to clear agonists from the tissue, or a combination of both. Furthermore, we identify various parameter combinations that may contribute to the existence of different asthma phenotypes, and we illustrate a combination of factors which may predispose severe asthmatics to fatal bronchospasms.     

The emergence of ECM mechanics and cytoskeletal tension as important regulators of cell function

The ability to harvest and maintain viable cells from mammalian tissues represented a critical advance in biomedical research, enabling individual cells to be cultured and studied in molecular detail. However, in these traditional cultures, cells are grown on rigid glass or polystyrene substrates, the mechanical properties of which often do not match those of the in vivo tissue from which the cells were originally derived. This mechanical mismatch likely contributes to abrupt changes in cellular phenotype. In fact, it has been proposed that mechanical changes in the cellular microenvironment may alone be responsible for driving specific cellular behaviors. Recent multidisciplinary efforts from basic scientists and engineers have begun to address this hypothesis more explicitly by probing the effects of ECM mechanics on cell and tissue function. Understanding the consequences of such mechanical changes is physiologically relevant in the context of a number of tissues in which altered mechanics may either correlate with or play an important role in the onset of pathology. Examples include changes in the compliance of blood vessels associated with atherosclerosis and intimal hyperplasia, as well as changes in the mechanical properties of developing tumors. Compelling evidence from 2-D in vitro model systems has shown that substrate mechanical properties induce changes in cell shape, migration, proliferation, and differentiation, but it remains to be seen whether or not these same effects translate to 3-D systems or in vivo. Furthermore, the molecular “mechanotransduction” mechanisms by which cells respond to changes in ECM mechanics remain unclear. Here, we provide some historical context for this emerging area of research, and discuss recent evidence that regulation of cytoskeletal tension by changes in ECM mechanics (either directly or indirectly) may provide a critical switch that controls cell function.     

Pulsation and stabilization: Contractile forces that underlie morphogenesis

Embryonic development involves global changes in tissue shape and architecture that are driven by cell shape changes and rearrangements within cohesive cell sheets. Morphogenetic changes at the cell and tissue level require that cells generate forces and that these forces are transmitted between the cells of a coherent tissue. Contractile forces generated by the actin-myosin cytoskeleton are critical for morphogenesis, but the cellular and molecular mechanisms of contraction have been elusive for many cell shape changes and movements. Recent studies that have combined live imaging with computational and biophysical approaches have provided new insights into how contractile forces are generated and coordinated between cells and tissues. In this review, we discuss our current understanding of the mechanical forces that shape cells, tissues, and embryos, emphasizing the different modes of actomyosin contraction that generate various temporal and spatial patterns of force generation.     

Mechanics of the cellular actin cortex: From signalling to shape change

The actin cortex is a thin layer of actin, myosin and actin-binding proteins that underlies the membrane of most animal cells. It is highly dynamic and can undergo remodelling on timescales of tens of seconds, thanks to protein turnover and myosin-mediated contractions. The cortex enables cells to resist external mechanical stresses, controls cell shape and allows cells to exert forces on their neighbours. Thus, its mechanical properties are the key to its physiological function. Here, we give an overview of how cortex composition, structure and dynamics control cortex mechanics and cell shape. We use mitosis as an example to illustrate how global and local regulation of cortex mechanics gives rise to a complex series of cell shape changes.     

Cytoskeletal Reorganization Drives Mesenchymal Condensation and Regulates Downstream Molecular Signaling

Skeletal condensation occurs when specified mesenchyme cells self-organize over several days to form a distinctive cartilage template. Here, we determine how and when specified mesenchyme cells integrate mechanical and molecular information from their environment, forming cartilage condensations in the pharyngeal arches of chick embryos. By disrupting cytoskeletal reorganization, we demonstrate that dynamic cell shape changes drive condensation and modulate the response of the condensing cells to Fibroblast Growth Factor (FGF), Bone Morphogenetic Protein (BMP) and Transforming Growth Factor beta (TGF-β) signaling pathways. Rho Kinase (ROCK)-driven actomyosin contractions and Myosin II-generated differential cell cortex tension regulate these cell shape changes. Disruption of the condensation process inhibits the differentiation of the mesenchyme cells into chondrocytes, demonstrating that condensation regulates the fate of the mesenchyme cells. We also find that dorsal and ventral condensations undergo distinct cell shape changes. BMP signaling is instructive for dorsal condensation-specific cell shape changes. Moreover, condensations exhibit ventral characteristics in the absence of BMP signaling, suggesting that in the pharyngeal arches ventral morphology is the ground pattern. Overall, this study characterizes the interplay between cytoskeletal dynamics and molecular signaling in a self-organizing system during tissue morphogenesis.     

From the regulation of peptidoglycan synthesis to bacterial growth and morphology

How bacteria grow and divide while retaining a defined shape is a fundamental question in microbiology, but technological advances are now driving a new understanding of how the shape-maintaining bacterial peptidoglycan sacculus grows. In this Review, we highlight the relationship between peptidoglycan synthesis complexes and cytoskeletal elements, as well as recent evidence that peptidoglycan growth is regulated from outside the sacculus in Gram-negative bacteria. We also discuss how growth of the sacculus is sensitive to mechanical force and nutritional status, and describe the roles of peptidoglycan hydrolases in generating cell shape and of D-amino acids in sacculus remodelling.     

An epigenetic model for pigment patterning based on mechanical and cellular interactions

Pigment patterning in animals generally occurs during early developmental stages and has ecological, physiological, ethological, and evolutionary significance. Despite the relative simplicity of color patterns, their emergence depends upon multilevel complex processes. Thus, theoretical models have become necessary tools to further understand how such patterns emerge. Recent studies have reevaluated the importance of epigenetic, as well as genetic factors in developmental pattern formation. Yet epigenetic phenomena, specially those related to physical constraints that might be involved in the emergence of color patterns, have not been fully studied. In this article, we propose a model of color patterning in which epigenetic aspects such as cell migration, cell-tissue interactions, and physical and mechanical phenomena are central. This model considers that motile cells embedded in a fibrous, viscoelastic matrix-mesenchyme-can deform it in such a way that tension tracks are formed. We postulate that these tracks act, in turn, as guides for subsequent cell migration and establishment, generating long-range phenomenological interactions. We aim to describe some general aspects of this developmental phenomenon with a rather simple mathematical model. Then we discuss our model in the context of available experimental and morphological evidence for reptiles, amphibians, and fishes, and compare it with other patterning models. We also put forward novel testable predictions derived from our model, regarding, for instance, the localization of the postulated tension tracks, and we propose new experiments. Finally, we discuss how the proposed mechanism could constitute a dynamic patterning module accounting for pattern formation in many animal lineages.     

Apoptotic force: active mechanical function of cell death during morphogenesis

Apoptosis, or programmed cell death, is an essential process for the elimination of unnecessary cells during embryonic development, tissue homeostasis, and certain pathological conditions. Recently, an active mechanical function of apoptosis called apoptotic force has been demonstrated during a tissue fusion process of Drosophila embryogenesis. The mechanical force produced during apoptosis is used not only to force dying cells out from tissues in order to keep tissue integrity, but also to change the morphology of neighboring cells to fill the space originally occupied by the dying cell. Furthermore, the occurrence of apoptosis correlates with tissue movement and tension of the tissue. This finding suggests that apoptotic forces might be harnessed throughout cell death-related morphogenesis; however, this concept remains to be fully investigated. While the investigation of this active mechanical function of apoptosis has just begun, here we summarize the current understandings of this novel function of apoptosis, and discuss some possible developmental processes in which apoptosis may play a mechanical role. The concept of apoptotic force prompts a necessity to rethink the role of programmed cell death during morphogenesis.     

Bronchoconstriction: a potential missing link in airway remodelling

In asthma, progressive structural changes of the airway wall are collectively termed airway remodelling. Despite its deleterious effect on lung function, airway remodelling is incompletely understood. As one of the important causes leading to airway remodelling, here we discuss the significance of mechanical forces that are produced in the narrowed airway during asthma exacerbation, as a driving force of airway remodelling. We cover in vitro, ex vivo and in vivo work in this field, and discuss up-to-date literature supporting the idea that bronchoconstriction may be the missing link in a comprehensive understanding of airway remodelling in asthma.     

Mechanical control of tissue morphogenesis during embryological development

Twenty years ago, we proposed a model of developmental control based on tensegrity architecture, in which tissue pattern formation in the embryo is controlled through mechanical interactions between cells and extracellular matrix (ECM) which place the tissue in a state of isometric tension (prestress). The model proposed that local changes in the mechanical compliance of the ECM, for example, due to regional variations in basement membrane degradation beneath growing epithelium, may result in local stretching of the ECM and associated adherent cells, much like a "run-in-a-stocking". Cell growth and function would be controlled locally though physical distortion of the associated cells, or changes in cytoskeletal tension. Importantly, experimental studies have demonstrated that cultured cells can be switched between different fates, including growth, differentiation, apoptosis, directional motility and different stem cell lineages, by modulating cell shape. Experiments in whole embryonic organ rudiments also have confirmed the tight correlation between basement membrane thinning, cell tension generation and new bud and branch formation during tissue morphogenesis and that this process can be inhibited or accelerated by dissipating or enhancing cytoskeletal tension, respectively. Taken together, this work confirms that mechanical forces generated in the cytoskeleton of individual cells and exerted on ECM scaffolds, play a critical role in the sculpting of the embryo.     

The mechanical basis of cell rearrangement. I. Epithelial morphogenesis during Fundulus epiboly

Many morphogenetic processes are accomplished by coordinated cell rearrangements. These rearrangements are accompanied by substantial shifts in the neighbor relationships between cells. Here we propose a model for studying morphogenesis in epithelial sheets by directed cell neighbor change. Our model describes cell rearrangements by accounting for the balance of forces between neighboring cells within an epithelium. Cell rearrangement and cell shape changes occur when these forces are not in mechanical equilibrium. We will show that cell rearrangement within the epidermal enveloping layer (EVL) of the teleost fish Fundulus during epiboly can be explained solely in terms of the balance of forces generated among constituent epithelial cells. Within a cell, we account for circumferential elastic forces and the force generated by hydrostatic and osmotic pressure. The model treats epithelial cells as two-dimensional polygons where the mechanical forces are applied to the polygonal nodes. A cell node protrudes or contracts when the nodal forces are not in mechanical equilibrium. In an epithelial sheet, adjacent cells share common boundary nodes; in this way, mechanical force is transmitted from cell to cell, mimicking junctional coupling. These junctional nodes can slide, and nodes may appear or disappear, so that the number of polygonal sides is variable. Computer graphics allows us to compare numerical simulations of the model with time-lapse cinemicroscopy of cell rearrangements in the living embryo, and data obtained from fixed and silver stained embryos. By manipulating the mechanical properties of the model cells we can study the conditions necessary to reproduce normal cell behavior during Fundulus epiboly. We find that simple stress relaxation is sufficient to account for cell rearrangements among interior cells of the EVL when they are isotropically contractile. Experimental observations show that the number of EVL marginal cells continuously decreases throughout epiboly. In order for the simulation to reproduce this behavior, cells at the EVL boundary must generate protrusive forces rather than contractile tension forces. Therefore, the simulation results suggest that the mechanical properties of EVL marginal cells at their leading edge must be quite different from EVL interior cells.     

Mechanosensitivity and compositional dynamics of cell–matrix adhesions

Cells perceive information about the biochemical and biophysical properties of their tissue microenvironment through integrin‐mediated cell–matrix adhesions, which connect the cytoskeleton with the extracellular matrix and thereby allow cohesion and long‐range mechanical connections within tissues. The formation of cell–matrix adhesions and integrin signalling involves the dynamic recruitment and assembly of an inventory of proteins, collectively termed the ‘adhesome’, at the adhesive site. The recruitment of some adhesome proteins, most notably the Lin11‐, Isl1‐ and Mec3‐domain‐containing proteins, depends on mechanical tension generated by myosin II‐mediated contractile forces exerted on cell–matrix adhesions. When exposed to force, mechanosensitive adhesome proteins can change their conformation or expose cryptic‐binding sites leading to the recruitment of proteins, rearrangement of the cytoskeleton, reinforcement of the adhesive site and signal transduction. Biophysical methods and proteomics revealed force ranges within the adhesome and cytoskeleton, and also force‐dependent changes in adhesome composition. In this review, we provide an overview of the compositional dynamics of cell–matrix adhesions, discuss the most prevalent functional domains in adhesome proteins and review literature and concepts about mechanosensing mechanisms that operate at the adhesion site.     

Biophysics of substrate interaction: influence on neural motility, differentiation, and repair

The identity and behavior of a cell is shaped by the molecular and mechanical composition of its surroundings. Molecular cues have firmly established roles in guiding both neuronal fate decisions and the migration of cells and axons. However, there is growing evidence that topographical and rigidity cues in the extracellular environment act synergistically with these molecular cues. Like chemical cues, physical factors do not elicit a fixed response, but rather one that depends on the sensory makeup of the cell. Moreover, from developmental studies and the plasticity of neural tissue, it is evident that there is dynamic feedback between physical and chemical factors to produce the final morphology. Here, we focus on our current understanding of how these physical cues shape cellular differentiation and migration, and discuss their relevance to repairing the injured nervous system.