Vertex Models of Epithelial Morphogenesis

The dynamic behavior of epithelial cell sheets plays a central role during numerous developmental processes. Genetic and imaging studies of epithelial morphogenesis in a wide range of organisms have led to increasingly detailed mechanisms of cell sheet dynamics. Computational models offer a useful means by which to investigate and test these mechanisms, and have played a key role in the study of cell-cell interactions. A variety of modeling approaches can be used to simulate the balance of forces within an epithelial sheet. Vertex models are a class of such models that consider cells as individual objects, approximated by two-dimensional polygons representing cellular interfaces, in which each vertex moves in response to forces due to growth, interfacial tension, and pressure within each cell. Vertex models are used to study cellular processes within epithelia, including cell motility, adhesion, mitosis, and delamination. This review summarizes how vertex models have been used to provide insight into developmental processes and highlights current challenges in this area, including progressing these models from two to three dimensions and developing new tools for model validation.     

Vertex Models of Epithelial Morphogenesis

The dynamic behavior of epithelial cell sheets plays a central role during numerous developmental processes. Genetic and imaging studies of epithelial morphogenesis in a wide range of organisms have led to increasingly detailed mechanisms of cell sheet dynamics. Computational models offer a useful means by which to investigate and test these mechanisms, and have played a key role in the study of cell-cell interactions. A variety of modeling approaches can be used to simulate the balance of forces within an epithelial sheet. Vertex models are a class of such models that consider cells as individual objects, approximated by two-dimensional polygons representing cellular interfaces, in which each vertex moves in response to forces due to growth, interfacial tension, and pressure within each cell. Vertex models are used to study cellular processes within epithelia, including cell motility, adhesion, mitosis, and delamination. This review summarizes how vertex models have been used to provide insight into developmental processes and highlights current challenges in this area, including progressing these models from two to three dimensions and developing new tools for model validation.     

Transmission of Mechanical Stresses within the Cytoskeleton of Adherent Cells: A Theoretical Analysis Based on a Multi-Component Cell Model

How environmental mechanical forces affect cellular functions is a central problem in cell biology. Theoretical models of cellular biomechanics provide relevant tools for understanding how the contributions of deformable intracellular components and specific adhesion conditions at the cell interface are integrated for determining the overall balance of mechanical forces within the cell. We investigate here the spatial distributions of intracellular stresses when adherent cells are probed by magnetic twisting cytometry. The influence of the cell nucleus stiffness on the simulated nonlinear torque-bead rotation response is analyzed by considering a finite element multi-component cell model in which the cell and its nucleus are considered as different hyperelastic materials. We additionally take into account the mechanical properties of the basal cell cortex, which can be affected by the interaction of the basal cell membrane with the extracellular substrate. In agreement with data obtained on epithelial cells, the simulated behaviour of the cell model relates the hyperelastic response observed at the entire cell scale to the distribution of stresses and strains within the nucleus and the cytoskeleton, up to cell adhesion areas. These results, which indicate how mechanical forces are transmitted at distant points through the cytoskeleton, are compared to recent data imaging the highly localized distribution of intracellular stresses.     

Embryonic wound healing by apical contraction and ingression in Xenopus laevis

We have characterized excisional wounds in the animal cap of early embryos of the frog Xenopus laevis and found that these wounds close accompanied by three distinct processes: (1) the assembly of an actin purse-string in the epithelial cells at the wound margin, (2) contraction and ingression of exposed deep cells, and (3) protrusive activity of epithelial cells at the margin. Microsurgical manipulation allowing fine control over the area and depth of the wound combined with videomicroscopy and confocal analysis enabled us to describe the kinematics and challenge the mechanics of the closing wound. Full closure typically occurs only when the deep, mesenchymal cell-layer of the ectoderm is left intact; in contrast, when deep cells are removed along with the superficial, epithelial cell-layer of the ectoderm, wounds do not close. Actin localizes to the superficial epithelial cell-layer at the wound margin immediately after wounding and forms a contiguous "purse-string" in those cells within 15 min. However, manipulation and closure kinematics of shaped wounds and microsurgical cuts made through the purse-string rule out a major force-generating role for the purse-string. Further analysis of the cell behaviors within the wound show that deep, mesenchymal cells contract their apical surfaces and ingress from the exposed surface. High resolution time-lapse sequences of cells at the leading edge of the wound show that these cells undergo protrusive activity only during the final phases of wound closure as the ectoderm reseals. We propose that assembly of the actin purse-string works to organize and maintain the epithelial sheet at the wound margin, that contraction and ingression of deep cells pulls the wound margins together, and that protrusive activity of epithelial cells at the wound margin reseals the ectoderm and re-establishes tissue integrity during wound healing in the Xenopus embryonic ectoderm.     

Heterogeneity Profoundly Alters Emergent Stress Fields in Constrained Multicellular Systems

Stress fields emerging from the transfer of forces between cells within multicellular systems are increasingly being recognized as major determinants of cell fate. Current analytical and numerical models used for the calculation of stresses within cell monolayers assume homogeneous contractile and mechanical cellular properties; however, cell behavior varies by region within constrained tissues. Here, we show the impact of heterogeneous cell properties on resulting stress fields that guide cell phenotype and apoptosis. Using circular micropatterns, we measured biophysical metrics associated with cell mechanical stresses. We then computed cell-layer stress distributions using finite element contraction models and monolayer stress microscopy. In agreement with previous studies, cell spread area, alignment, and traction forces increase, whereas apoptotic activity decreases, from the center of cell layers to the edge. The distribution of these metrics clearly indicates low cell stress in central regions and high cell stress at the periphery of the patterns. However, the opposite trend is predicted by computational models when homogeneous contractile and mechanical properties are assumed. In our model, utilizing heterogeneous cell-layer contractility and elastic moduli values based on experimentally measured biophysical parameters, we calculate low cell stress in central areas and high anisotropic stresses in peripheral regions, consistent with the biometrics. These results clearly demonstrate that common assumptions of uniformity in cell contractility and stiffness break down in postconfluence confined multicellular systems. This work highlights the importance of incorporating regional variations in cell mechanical properties when estimating emergent stress fields from collective cell behavior.     

Fine structure, functional organization and supportive role of neuroglia in Nereis

Fluorescence and electron microscopy reveal a dynamic architectural pattern in the organization of neuroglia in the central nervous system of nereid polychaetes. Fibrous glial processes intertwine among neuronal elements, binding them together, and anchor the nerve cord to the epidermis. Rigidity and resilience are provided to this supportive framework by glial filaments and desmosomes. Tensive stresses, arising from contraction of locomotory muscles attached to the nerve cord, may be dissipated over the web-like meshwork, thus reducing the potential influence of deformative forces on neuronal elements within the neuropile. The voluminous glial processes forming the peripheral zone of the nerve cord may protect neurons from compressive forces.     

Mapping of mechanical strains and stresses around quiescent engineered three-dimensional epithelial tissues

Understanding how physical signals guide biological processes requires qualitative and quantitative knowledge of the mechanical forces generated and sensed by cells in a physiologically realistic three-dimensional (3D) context. Here, we used computational modeling and engineered epithelial tissues of precise geometry to define the experimental parameters that are required to measure directly the mechanical stress profile of 3D tissues embedded within native type I collagen. We found that to calculate the stresses accurately in these settings, we had to account for mechanical heterogeneities within the matrix, which we visualized and quantified using confocal reflectance and atomic force microscopy. Using this technique, we were able to obtain traction forces at the epithelium-matrix interface, and to resolve and quantify patterns of mechanical stress throughout the surrounding matrix. We discovered that whereas single cells generate tension by contracting and pulling on the matrix, the contraction of multicellular tissues can also push against the matrix, causing emergent compression. Furthermore, tissue geometry defines the spatial distribution of mechanical stress across the epithelium, which communicates mechanically over distances spanning hundreds of micrometers. Spatially resolved mechanical maps can provide insight into the types and magnitudes of physical parameters that are sensed and interpreted by multicellular tissues during normal and pathological processes.     

The effects of mechanical forces on intestinal physiology and pathology

The epithelial and non-epithelial cells of the intestinal wall experience a myriad of physical forces including strain, shear, and villous motility during normal gut function. Pathologic conditions alter these forces, leading to changes in the biology of these cells. The responses of intestinal epithelial cells to forces vary with both the applied force and the extracellular matrix proteins with which the cells interact, with differing effects on proliferation, differentiation, and motility, and the regulation of these effects involves similar but distinctly different signal transduction mechanisms. Although normal epithelial cells respond to mechanical forces, malignant gastrointestinal epithelial cells also respond to forces, most notably by increased cell adhesion, a critical step in tumor metastasis. This review will focus on the phenomenon of mechanical forces influencing cell biology and the mechanisms by which the gut responds these forces in both the normal as well as pathophysiologic states when forces are altered. Although more is known about epithelial responses to force, information regarding mechanosensitivity of vascular, neural, and endocrine cells within the gut wall will also be discussed, as will, the mechanism by which forces can regulate epithelial tumor cell adhesion.     

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.     

A multi-modular tensegrity model of an actin stress fiber

Stress fibers are contractile bundles in the cytoskeleton that stabilize cell structure by exerting traction forces on the extracellular matrix. Individual stress fibers are molecular bundles composed of parallel actin and myosin filaments linked by various actin-binding proteins, which are organized end-on-end in a sarcomere-like pattern within an elongated three-dimensional network. While measurements of single stress fibers in living cells show that they behave like tensed viscoelastic fibers, precisely how this mechanical behavior arises from this complex supramolecular arrangement of protein components remains unclear. Here we show that computationally modeling a stress fiber as a multi-modular tensegrity network can predict several key behaviors of stress fibers measured in living cells, including viscoelastic retraction, fiber splaying after severing, non-uniform contraction, and elliptical strain of a puncture wound within the fiber. The tensegrity model can also explain how they simultaneously experience passive tension and generate active contraction forces; in contrast, a tensed cable net model predicts some, but not all, of these properties. Thus, tensegrity models may provide a useful link between molecular and cellular scale mechanical behaviors and represent a new handle on multi-scale modeling of living materials.     

Thrombin-induced contraction in alveolar epithelial cells probed by traction microscopy.

Contractile tension of alveolar epithelial cells plays a major role in the force balance that regulates the structural integrity of the alveolar barrier. The aim of this work was to study thrombin-induced contractile forces of alveolar epithelial cells. A549 alveolar epithelial cells were challenged with thrombin, and time course of contractile forces was measured by traction microscopy. The cells exhibited basal contraction with total force magnitude 55.0 +/- 12.0 nN (mean +/- SE, n = 12). Traction forces were exerted predominantly at the cell periphery and pointed to the cell center. Thrombin (1 U/ml) induced a fast and sustained 2.5-fold increase in traction forces, which maintained peripheral and centripetal distribution. Actin fluorescent staining revealed F-actin polymerization and enhancement of peripheral actin rim. Disruption of actin cytoskeleton with cytochalasin D (5 microM, 30 min) and inhibition of myosin light chain kinase with ML-7 (10 microM, 30 min) and Rho kinase with Y-27632 (10 microM, 30 min) markedly depressed basal contractile tone and abolished thrombin-induced cell contraction. Therefore, the contractile response of alveolar epithelial cells to the inflammatory agonist thrombin was mediated by actin cytoskeleton remodeling and actomyosin activation through myosin light chain kinase and Rho kinase signaling pathways. Thrombin-induced contractile tension might further impair alveolar epithelial barrier integrity in the injured lung.     

Tracheole migration in an insect wing

Summary Insect tissues are supplied with oxygen by a system of long and highly branched cuticular tubes known as tracheae and tracheoles. During the growth of with imaginal discs in moths and butterflies, tracheole cells migrate distally from the base of the disc. Tracheoles radiate in a distal direction through the extracellular space sandwiched between the upper and lower epithelial surfaces of the wing.Migration of most cells is assumed to be governed by forces intrinsic to the cell. However, the movement of tracheoles is apparently a passive process whose motive force resides in adjacent epithelial cells. After epithelial cells are exposed to ecdysteroid hormones, these cells extend basal processes that are attracted to oxygen-rich tracheoles. By applying traction to the tracheoles with which they establish intimate contact, epithelial cells may control the pattern of their distribution within wing tissue.     

Mechanical stress inference for two dimensional cell arrays

Many morphogenetic processes involve mechanical rearrangements of epithelial tissues that are driven by precisely regulated cytoskeletal forces and cell adhesion. The mechanical state of the cell and intercellular adhesion are not only the targets of regulation, but are themselves the likely signals that coordinate developmental process. Yet, because it is difficult to directly measure mechanical stress in vivo on sub-cellular scale, little is understood about the role of mechanics in development. Here we present an alternative approach which takes advantage of the recent progress in live imaging of morphogenetic processes and uses computational analysis of high resolution images of epithelial tissues to infer relative magnitude of forces acting within and between cells. We model intracellular stress in terms of bulk pressure and interfacial tension, allowing these parameters to vary from cell to cell and from interface to interface. Assuming that epithelial cell layers are close to mechanical equilibrium, we use the observed geometry of the two dimensional cell array to infer interfacial tensions and intracellular pressures. Here we present the mathematical formulation of the proposed Mechanical Inverse method and apply it to the analysis of epithelial cell layers observed at the onset of ventral furrow formation in the Drosophila embryo and in the process of hair-cell determination in the avian cochlea. The analysis reveals mechanical anisotropy in the former process and mechanical heterogeneity, correlated with cell differentiation, in the latter process. The proposed method opens a way for quantitative and detailed experimental tests of models of cell and tissue mechanics.     

The interaction of epidermal keratinocytes and fibroblasts during the formation of a live skin equivalent]

The effect of human fetal fibroblasts and adult keratinocytes on collagen contraction was studied. Keratinocytes embedded in collagen lattices did not spread and produced only a slight contraction. When keratinocytes were seeded on the surface of tht gel, the contraction began within 24 h and correlated with the formation of epithelial colonies. Transplantation of multilayered epithelial sheets on the gel significantly accelerated the onset of contraction. Keratinocytes seeded on and fibroblasts grown in collagen lattices cooperatively contracted the gel, and keratinocytes were able to stimulate gel contraction even when they had no contact with the collagen roughly populated with fibroblasts. Swiss 3T3 cells remained spherical in collagen lattices and did not contract the gel but when cultivated with keratinocytes they stimulated gel contraction. In their turn, keratinocytes influenced the behaviour of Swiss 3T3 cells which elongated and produced processes. We suggest that both keratinocytes and mesenchymal cells can affect gel contraction 1) by a direct contact with collagen lattices, and 2) through potentiation of the ability of another cell type to contract the gel.     

Organization of the nematocyst battery in the tentacle of hydra: Arrangement of the complex anchoring junctions between nematocytes,epithelial cells,and basement membrane

Summary Nematocytes (stinging cells) of hydra tentacles are anchored to the basement membrane by peculiar complex junctions in which a flattened tongue of an epithelial cell is interposed between the nematocyte and the basement membrane. In this paper we describe the arrangement of these junctions with emphasis on how they are related to the architecture of the epithelial cell. Each epithelial cell, called a battery cell, harbors 10–20 nematocytes and bears muscle processes that extend along the basement membrane. The epithelial cell component of the complex junction is usually a lateral extension of a muscle process. All nematocytes within a battery cell make junctions with muscle processes of the same (resident) epithelial battery cell despite the presence of numerous muscle processes from adjacent (foreign) cells. Some nematocytes make junctions with several resident processes, spanning the foreign processes to do so. Most junctions reside near the proximal ends of the muscle processes. New findings are reported on the substructure of the junctions. They are composed of aggregates of smaller elements, and the cytoskeleton within the complexes has a pronounced longitudinal organization. These observations are consistent with a suggestion that the complex junctions develop by aggregation of smaller junctional units originating elsewhere on the cells.     

On the dynamics of cell cleavage

A completely fluid model of cleavage dynamics is studied in which the forces exerted within the boundary structure of a cell are approximated by an effective surface tension. The hypothesis that surface tension depends in part on the concentration of tension elements implies a contraction of the surface towards the equator resulting from a dynamical instability that once triggered develops spontaneously into cleavage. The circulation of the cytoplasm induced by surface stresses collects and aligns the surface-bound tension elements into an equatorial belt. This flow may be a means of assembling a contractile ring.     

Mechanical modelling of cell/ECM and cell/cell interactions during the contraction of a fibroblast-populated collagen microsphere: theory and model simulation

The cell-derived forces generated during wound healing may be beneficial in reducing the wound size by contraction, but are also detrimental because of the high mechanical stresses in and around the scar that can cause pain, disfigurement and loss of tissue function. The fibroblasts seeded collagen matrix is regarded as an in vitro model for this process and as a suitable way to study these mechanical aspects which are poorly understood. It is proposed here, to improve the continuum theory of Murray-Oster by assuming that more than one control system may be operative in wound contraction regulation. In particular, it is suggested that the wound contraction mechanism is not exclusively due to cell/ECM interaction forces but rather that both ECM/cell and the cell/cell interactions operate together in such process.     

Cells Actively Stiffen Fibrin Networks by Generating Contractile Stress

During wound healing and angiogenesis, fibrin serves as a provisional extracellular matrix. We use a model system of fibroblasts embedded in fibrin gels to study how cell-mediated contraction may influence the macroscopic mechanical properties of their extracellular matrix during such processes. We demonstrate by macroscopic shear rheology that the cells increase the elastic modulus of the fibrin gels. Microscopy observations show that this stiffening sets in when the cells spread and apply traction forces on the fibrin fibers. We further show that the stiffening response mimics the effect of an external stress applied by mechanical shear. We propose that stiffening is a consequence of active myosin-driven cell contraction, which provokes a nonlinear elastic response of the fibrin matrix. Cell-induced stiffening is limited to a factor 3 even though fibrin gels can in principle stiffen much more before breaking. We discuss this observation in light of recent models of fibrin gel elasticity, and conclude that the fibroblasts pull out floppy modes, such as thermal bending undulations, from the fibrin network, but do not axially stretch the fibers. Our findings are relevant for understanding the role of matrix contraction by cells during wound healing and cancer development, and may provide design parameters for materials to guide morphogenesis in tissue engineering.     

Cells Actively Stiffen Fibrin Networks by Generating Contractile Stress

During wound healing and angiogenesis, fibrin serves as a provisional extracellular matrix. We use a model system of fibroblasts embedded in fibrin gels to study how cell-mediated contraction may influence the macroscopic mechanical properties of their extracellular matrix during such processes. We demonstrate by macroscopic shear rheology that the cells increase the elastic modulus of the fibrin gels. Microscopy observations show that this stiffening sets in when the cells spread and apply traction forces on the fibrin fibers. We further show that the stiffening response mimics the effect of an external stress applied by mechanical shear. We propose that stiffening is a consequence of active myosin-driven cell contraction, which provokes a nonlinear elastic response of the fibrin matrix. Cell-induced stiffening is limited to a factor 3 even though fibrin gels can in principle stiffen much more before breaking. We discuss this observation in light of recent models of fibrin gel elasticity, and conclude that the fibroblasts pull out floppy modes, such as thermal bending undulations, from the fibrin network, but do not axially stretch the fibers. Our findings are relevant for understanding the role of matrix contraction by cells during wound healing and cancer development, and may provide design parameters for materials to guide morphogenesis in tissue engineering.