The mechanics of cell sorting and envelopment

Versatile system cores as a conceptual basis for generality in cell and developmental biology

The discovery of general principles underlying the complexity and diversity of cellular and developmental systems is a central and long-standing aim of biology. While new technologies collect data at an ever-accelerating rate, there is growing concern that conceptual progress is not keeping pace. We contend that this is due to a paucity of conceptual frameworks that support meaningful generalizations. This led us to develop the core and periphery (C&P) hypothesis, which posits that many biological systems can be decomposed into a highly versatile core with a large behavioral repertoire and a specific periphery that configures said core to perform one particular function. Versatile cores tend to be widely reused across biology, which confers generality to theories describing them. Here, we introduce this concept and describe examples at multiple scales, including Turing patterning, actomyosin dynamics, multi-cellular morphogenesis, and vertebrate gastrulation. We also sketch its evolutionary basis and discuss key implications and open questions. We propose that the C&P hypothesis could unlock new avenues of conceptual progress in mesoscale biology.

Self-organized collective cell behaviors as design principles for synthetic developmental biology

Over the past two decades, molecular cell biology has graduated from a mostly analytic science to one with substantial synthetic capability. This success is built on a deep understanding of the structure and function of biomolecules and molecular mechanisms. For synthetic biology to achieve similar success at the scale of tissues and organs, an equally deep understanding of the principles of development is required. Here, we review some of the central concepts and recent progress in tissue patterning, morphogenesis and collective cell migration and discuss their value for synthetic developmental biology, emphasizing in particular the power of (guided) self-organization and the role of theoretical advances in making developmental insights applicable in synthesis.

Biophysical models of early mammalian embryogenesis

Embryo development is a critical and fascinating stage in the life cycle of many organisms. Despite decades of research, the earliest stages of mammalian embryogenesis are still poorly understood, caused by a scarcity of high-resolution spatial and temporal data, the use of only a few model organisms, and a paucity of truly multidisciplinary approaches that combine biological research with biophysical modeling and computational simulation. Here, we explain the theoretical frameworks and biophysical processes that are best suited to modeling the early mammalian embryo, review a comprehensive list of previous models, and discuss the most promising avenues for future work.

Cell mechanics and cell-cell recognition controls by Toll-like receptors in tissue morphogenesis and homeostasis

Signal transduction by the Toll-like receptors (TLRs) is conserved and essential for innate immunity in metazoans. The founding member of the TLR family, Drosophila Toll-1, was initially identified for its role in dorsoventral axis formation in early embryogenesis. The Drosophila genome encodes nine TLRs that display dynamic expression patterns during development, suggesting their involvement in tissue morphogenesis and homeostasis. Recent progress on the developmental functions of TLRs beyond dorsoventral patterning has revealed not only their diverse functions in various biological processes, but also unprecedented molecular mechanisms in directly regulating cell mechanics and cell-cell recognition independent of the canonical signal transduction pathway involving transcriptional regulation of target genes. In this review, I feature and discuss the non-immune functions of TLRs in the control of epithelial tissue homeostasis, tissue morphogenesis, and cell-cell recognition between cell populations with different cell identities.

EPH/EPHRIN regulates cellular organization by actomyosin contractility effects on cell contacts

EPH/EPHRIN signaling is essential to many aspects of tissue self-organization and morphogenesis, but little is known about how EPH/EPHRIN signaling regulates cell mechanics during these processes. Here, we use a series of approaches to examine how EPH/EPHRIN signaling drives cellular self-organization. Contact angle measurements reveal that EPH/EPHRIN signaling decreases the stability of heterotypic cell:cell contacts through increased cortical actomyosin contractility. We find that EPH/EPHRIN-driven cell segregation depends on actomyosin contractility but occurs independently of directed cell migration and without changes in cell adhesion. Atomic force microscopy and live cell imaging of myosin localization support that EPH/EPHRIN signaling results in increased cortical tension. Interestingly, actomyosin contractility also nonautonomously drives increased EPHB2:EPHB2 homotypic contacts. Finally, we demonstrate that changes in tissue organization are driven by minimization of heterotypic contacts through actomyosin contractility in cell aggregates and by mouse genetics experiments. These data elucidate the biomechanical mechanisms driving EPH/EPHRIN-based cell segregation wherein differences in interfacial tension, regulated by actomyosin contractility, govern cellular self-organization.

Differential cell adhesion implemented by Drosophila Toll corrects local distortions of the anterior-posterior compartment boundary

Maintaining lineage restriction boundaries in proliferating tissues is vital to animal development. A long-standing thermodynamics theory, the differential adhesion hypothesis, attributes cell sorting phenomena to differentially expressed adhesion molecules. However, the contribution of the differential adhesion system during tissue morphogenesis has been unsubstantiated despite substantial theoretical support. Here, we report that Toll-1, a transmembrane receptor protein, acts as a differentially expressed adhesion molecule that straightens the fluctuating anteroposterior compartment boundary in the abdominal epidermal epithelium of the Drosophila pupa. Toll-1 is expressed across the entire posterior compartment under the control of the selector gene engrailed and displays a sharp expression boundary that coincides with the compartment boundary. Toll-1 corrects local distortions of the boundary in the absence of cable-like Myosin II enrichment along the boundary. The reinforced adhesion of homotypic cell contacts, together with pulsed cell contraction, achieves a biased vertex sliding action by resisting the separation of homotypic cell contacts in boundary cells. This work reveals a self-organizing system that integrates a differential adhesion system with pulsed contraction of cells to maintain lineage restriction boundaries.

Cell sorting and morphogenesis in early Drosophila embryos

The regionalisation of growing tissues into compartments that do not mix is thought to be a common motif of animal development. Compartments and compartmental boundaries were discovered by lineage studies in the model organism Drosophila. Since then, many compartment boundaries have been identified in developing tissues, from insects to vertebrates. These are important for animal development, because boundaries localize signalling centres that control tissue morphogenesis. Compartment boundaries are boundaries of lineage restriction, where specific mechanisms keep boundaries straight and cells segregated. Here, we review the mechanisms of cell sorting at boundaries found in early Drosophila embryos. The parasegmental boundaries, separating anterior from posterior compartments in the embryo, keep cells segregated by increasing actomyosin contractility at boundary cell-cell interfaces. Differential actomyosin contractility in turn promotes fold formation and orients cell division. Earlier in development, actomyosin differentials are also important for cell sorting during axis extension. Specific cell surface asymmetries and signalling pathways are required to initiate and maintain these actomyosin differentials.

Dynamic cell–cell adhesion mediated by pericellular matrix interaction – a hypothesis

Cell–cell adhesion strength, measured as tissue surface tension, spans an enormous 1000-fold range when different cell types are compared. However, the examination of basic mechanical principles of cell adhesion indicates that cadherin-based and related mechanisms are not able to promote the high-strength adhesion experimentally observed in many late embryonic or malignant tissues. Therefore, the hypothesis is explored that the interaction of the pericellular matrices of cells generates strong adhesion by a mechanism akin to the self-adhesion/self-healing of dynamically cross-linked hydrogels. Quantitative data from biofilm matrices support this model. The mechanism links tissue surface tension to pericellular matrix stiffness. Moreover, it explains the wide, matrix-filled spaces around cells in liquid-like, yet highly cohesive, tissues, and it rehabilitates aspects of the original interpretation of classical cell sorting experiments, as expressed in Steinberg's differential adhesion hypothesis: that quantitative differences in adhesion energies between cells are sufficient to drive sorting.

Role of forward and reverse signaling in Eph receptor and ephrin mediated cell segregation

Eph receptor and ephrin signaling has a major role in segregating distinct cell populations to form sharp borders. Expression of interacting Ephs and ephrins typically occurs in complementary regions, such that polarised activation of both components occurs at the interface. Forward signaling through Eph receptors can drive cell segregation, but it is unclear whether reverse signaling through ephrins can also contribute. We have tested the role of reverse signaling, and of polarised versus non-polarised activation, in assays in which contact repulsion drives cell segregation and border sharpening. We find that polarised forward signaling drives stronger segregation than polarised reverse signaling. Nevertheless, reverse signaling contributes since bidirectional Eph and ephrin activation drives stronger segregation than unidirectional forward signaling alone. In contrast, non-polarised Eph activation drives little segregation. We propose that although polarised forward signaling is the principal driver of segregation, reverse signaling enables bidirectional repulsion which prevents mingling of each population into the other.

Elongated Cells Drive Morphogenesis in a Surface-Wrapped Finite-Element Model of Germband Retraction

During Drosophila embryogenesis, the germband first extends to curl around the posterior end of the embryo and then retracts back; however, retraction is not simply the reversal of extension. At a tissue level, extension is coincident with ventral furrow formation, and at a cellular level, extension occurs via convergent cell neighbor exchanges in the germband, whereas retraction involves only changes in cell shape. To understand how cell shapes, tissue organization, and cellular forces drive germband retraction, we investigate this process using a whole-embryo, surface-wrapped cellular finite-element model. This model represents two key epithelial tissues-amnioserosa and germband-as adjacent sheets of two-dimensional cellular finite elements that are wrapped around an ellipsoidal three-dimensional approximation of an embryo. The model reproduces the detailed kinematics of in vivo retraction by fitting just one free model parameter, the tension along germband cell interfaces; all other cellular forces are constrained to follow ratios inferred from experimental observations. With no additional parameter adjustments, the model also reproduces quantitative assessments of mechanical stress using laser dissection and failures of retraction when amnioserosa cells are removed via mutations or microsurgery. Surprisingly, retraction in the model is robust to changes in cellular force values but is critically dependent on starting from a configuration with highly elongated amnioserosa cells. Their extreme cellular elongation is established during the prior process of germband extension and is then used to drive retraction. The amnioserosa is the one tissue whose cellular morphogenesis is reversed from germband extension to retraction, and this reversal coordinates the forces needed to retract the germband back to its pre-extension position and shape. In this case, cellular force strengths are less important than the carefully established cell shapes that direct them. VIDEO ABSTRACT.

A multiscale model for heterogeneous tumor spheroid in vitro

In this paper, a novel multiscale method is proposed for the study of heterogeneous tumor spheroid growth in vitro. The entire tumor spheroid is described by an ellipsoid-based model while nutrient and other environmental factors are treated as continua. The ellipsoid-based discrete component is capable of incorporating mechanical effects and deformability, while keeping a minimum set of free variables to describe complex shape variations. Moreover, our purely cell-based description of tumor avoids the complex mutual conversion between a cell-based model and continuum model within a tumor, such as force and mass transformation. This advantage makes it highly suitable for the study of tumor spheroids in vitro whose size are normally less than 800 μm in diameter. In addition, our numerical scheme provides two computational options depending on tumor size. For a small or medium tumor spheroid, a three-dimensional (3D) numerical model can be directly applied. For a large spheroid, we suggest the use of a 3D-adapted 2D cross section configuration, which has not yet been explored in the literature, as an alternative for the theoretical investigation to bridge the gap between the 2D and 3D models. Our model and its implementations have been validated and applied to various studies given in the paper. The simulation results fit corresponding in vitro experimental observations very well.

Inferring cellular forces from image stacks

Although the importance of cellular forces to a wide range of embryogenesis and disease processes is widely recognized, measuring these forces is challenging, especially in three dimensions. Here, we introduce CellFIT-3D, a force inference technique that allows tension maps for three-dimensional cellular systems to be estimated from image stacks. Like its predecessors, video force microscopy and CellFIT, this cell mechanics technique assumes boundary-specific interfacial tensions to be the primary drivers, and it constructs force-balance equations based on triple junction (TJ) dihedral angles. The technique involves image processing, segmenting of cells, grouping of cell outlines, calculation of dihedral planes, averaging along three-dimensional TJs, and matrix equation assembly and solution. The equations tend to be strongly overdetermined, allowing indistinct TJs to be ignored and solution error estimates to be determined. Application to clean and noisy synthetic data generated using Surface Evolver gave tension errors of 1.6-7%, and analyses of eight-cell murine embryos gave estimated errors smaller than the 10% uncertainty of companion aspiration experiments. Other possible areas of application include morphogenesis, cancer metastasis and tissue engineering.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

Mechanocellular models of epithelial morphogenesis

Embryonic epithelia achieve complex morphogenetic movements, including in-plane reshaping, bending and folding, through the coordinated action and rearrangement of individual cells. Technical advances in molecular and live-imaging studies of epithelial dynamics provide a very real opportunity to understand how cell-level processes facilitate these large-scale tissue rearrangements. However, the large datasets that we are now able to generate require careful interpretation. In combination with experimental approaches, computational modelling allows us to challenge and refine our current understanding of epithelial morphogenesis and to explore experimentally intractable questions. To this end, a variety of cell-based modelling approaches have been developed to describe cell-cell mechanical interactions, ranging from vertex and 'finite-element' models that approximate each cell geometrically by a polygon representing the cell's membrane, to immersed boundary and subcellular element models that allow for more arbitrary cell shapes. Here, we review how these models have been used to provide insights into epithelial morphogenesis and describe how such models could help future efforts to decipher the forces and mechanical and biochemical feedbacks that guide cell and tissue-level behaviour. In addition, we discuss current challenges associated with using computational models of morphogenetic processes in a quantitative and predictive way.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

Interstitial fluid osmolarity modulates the action of differential tissue surface tension in progenitor cell segregation during gastrulation

The segregation of different cell types into distinct tissues is a fundamental process in metazoan development. Differences in cell adhesion and cortex tension are commonly thought to drive cell sorting by regulating tissue surface tension (TST). However, the role that differential TST plays in cell segregation within the developing embryo is as yet unclear. Here, we have analyzed the role of differential TST for germ layer progenitor cell segregation during zebrafish gastrulation. Contrary to previous observations that differential TST drives germ layer progenitor cell segregation in vitro, we show that germ layers display indistinguishable TST within the gastrulating embryo, arguing against differential TST driving germ layer progenitor cell segregation in vivo. We further show that the osmolarity of the interstitial fluid (IF) is an important factor that influences germ layer TST in vivo, and that lower osmolarity of the IF compared with standard cell culture medium can explain why germ layers display differential TST in culture but not in vivo. Finally, we show that directed migration of mesendoderm progenitors is required for germ layer progenitor cell segregation and germ layer formation.

Highlighted Article: Segregation of the germ layer progenitors in the zebrafish gastrula is driven by directed cell migration, rather than differential tissue surface tension.

A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis

Cellular adhesion provides one of the fundamental forms of biological interaction between cells and their surroundings, yet the continuum modelling of cellular adhesion has remained mathematically challenging. In 2006, Armstrong et al. proposed a mathematical model in the form of an integro-partial differential equation. Although successful in applications, a derivation from an underlying stochastic random walk has remained elusive. In this work we develop a framework by which non-local models can be derived from a space-jump process. We show how the notions of motility and a cell polarization vector can be naturally included. With this derivation we are able to include microscopic biological properties into the model. We show that particular choices yield the original Armstrong model, while others lead to more general models, including a doubly non-local adhesion model and non-local chemotaxis models. Finally, we use random walk simulations to confirm that the corresponding continuum model represents the mean field behaviour of the stochastic random walk.

Comparing individual-based approaches to modelling the self-organization of multicellular tissues

The coordinated behaviour of populations of cells plays a central role in tissue growth and renewal. Cells react to their microenvironment by modulating processes such as movement, growth and proliferation, and signalling. Alongside experimental studies, computational models offer a useful means by which to investigate these processes. To this end a variety of cell-based modelling approaches have been developed, ranging from lattice-based cellular automata to lattice-free models that treat cells as point-like particles or extended shapes. However, it remains unclear how these approaches compare when applied to the same biological problem, and what differences in behaviour are due to different model assumptions and abstractions. Here, we exploit the availability of an implementation of five popular cell-based modelling approaches within a consistent computational framework, Chaste (http://www.cs.ox.ac.uk/chaste). This framework allows one to easily change constitutive assumptions within these models. In each case we provide full details of all technical aspects of our model implementations. We compare model implementations using four case studies, chosen to reflect the key cellular processes of proliferation, adhesion, and short- and long-range signalling. These case studies demonstrate the applicability of each model and provide a guide for model usage.

Cell adhesion strength from cortical tension - an integration of concepts

Morphogenetic mechanisms such as cell movement or tissue separation depend on cell attachment and detachment processes, which involve adhesion receptors as well as the cortical cytoskeleton. The interplay between the two components is of stunning complexity. Most strikingly, the binding energy of adhesion molecules is usually too small for substantial cell-cell attachment, pointing to a main deficit in our present understanding of adhesion. In this Opinion article, I integrate recent findings and conceptual advances in the field into a coherent framework for cell adhesion. I argue that active cortical tension is best viewed as an integral part of adhesion, and propose on this basis a non-arbitrary measure of adhesion strength - the tissue surface tension of cell aggregates. This concept of adhesion integrates heterogeneous molecular inputs into a single mechanical property and simplifies the analysis of attachment-detachment processes. It draws attention to the enormous variation of adhesion strengths among tissues, whose origin and function is little understood.

Mouse Embryo Compaction

Compaction is a critical first morphological event in the preimplantation development of the mammalian embryo. Characterized by the transformation of the embryo from a loose cluster of spherical cells into a tightly packed mass, compaction is a key step in the establishment of the first tissue-like structures of the embryo. Although early investigation of the mechanisms driving compaction implicated changes in cell-cell adhesion, recent work has identified essential roles for cortical tension and a compaction-specific class of filopodia. During the transition from 8 to 16 cells, as the embryo is compacting, it must also make fundamental decisions regarding cell position, polarity, and fate. Understanding how these and other processes are integrated with compaction requires further investigation. Emerging imaging-based techniques that enable quantitative analysis from the level of cell-cell interactions down to the level of individual regulatory molecules will provide a greater understanding of how compaction shapes the early mammalian embryo.

A Sawtooth Pattern of Cadherin 2 Stability Mechanically Regulates Somite Morphogenesis

Differential cadherin (Cdh) expression is a classical mechanism for in vitro cell sorting. Studies have explored the roles of differential Cdh levels in cell aggregates and during vertebrate gastrulation, but the role of differential Cdh activity in forming in vivo tissue boundaries and boundary extracellular matrix (ECM) is unclear. Here, we examine the interactions between cell-cell and cell-ECM adhesion during somitogenesis, the formation of the segmented embryonic precursors of the vertebral column and musculature. We identify a sawtooth pattern of stable Cdh2 adhesions in which there is a posterior-to-anterior gradient of stable Cdh2 within each somite, while there is a step-like drop in stable Cdh2 along the somite boundary. Moreover, we find that the posterior somite boundary cells with high levels of stable Cdh2 have the most columnar morphology. Cdh2 is required for maximal cell aspect ratio and thus full epithelialization of the posterior somite. Loss-of-function analysis also indicates that Cdh2 acts with the fibronectin (FN) receptor integrin α5 (Itgα5) to promote somite boundary formation. Using genetic mosaics, we demonstrate that differential Cdh2 levels are sufficient to induce boundary formation, Itgα5 activation, and FN matrix assembly in the paraxial mesoderm. Elevated cytoskeletal contractility is sufficient to replace differential Cdh2 levels in genetic mosaics, suggesting that Cdh2 promotes ECM assembly by increasing cytoskeletal and tissue stiffness along the posterior somite boundary. Throughout somitogenesis, Cdh2 promotes ECM assembly along tissue boundaries and inhibits ECM assembly in the tissue mesenchyme.

Differential interaction forces govern bacterial sorting in early biofilms

Bacterial biofilms can generate micro-heterogeneity in terms of surface structures. However, little is known about the associated changes in the physics of cell–cell interaction and its impact on the architecture of biofilms. In this study, we used the type IV pilus of Neisseria gonorrhoeae to test whether variation of surface structures induces cell-sorting. We show that the rupture forces between pili are fine-tuned by post-translational modification. Bacterial sorting was dependent on pilus post-translational modification and pilus density. Active force generation was necessary for defined morphologies of mixed microcolonies. The observed morphotypes were in remarkable agreement with the differential strength of adhesion hypothesis proposing that a tug-of-war among surface structures of different cells governs cell sorting. We conclude that in early biofilms the density and rupture force of bacterial surface structures can trigger cell sorting based on similar physical principles as in developing embryos.

DOI:http://dx.doi.org/10.7554/eLife.10811.001

Communities of bacterial cells can live together embedded within a slime-like molecular matrix as a biofilm. This allows the bacteria to hide from external stresses. A single bacterium can replicate itself and develop into a biofilm, and over time the bacterial cells in specific regions of the biofilm will start to interact with their neighbors in different ways. These interactions occur via structures on the surface of the bacterial cells, and the differences in these interactions resemble those that occur as cells specialize during the development of animal embryos. Previous research into embryonic development has shown how differences in the physical interactions between embryonic cells are essential for sorting the cells into their correct locations and shaping the embryo. However, little is known about which processes govern the development of biofilms.

Now, Oldewurtel et al. have asked whether differences in the physical interactions between bacteria trigger cell sorting during the early stages of biofilm development. The experiments involved measuring the force required to break the cell–cell connections (called the ‘rupture force’) in biofilms of a bacterium called Neisseria gonorrhoeae. Oldewurtel et al. found that, in agreement with previous predictions, physical interactions were important for sorting bacterial cells into clusters based on the structures on their surfaces. Bacterial cells actively pull on the surface structures of their neighbors, which allows the cells to sort themselves in a tug-of-war fashion. This means that a cell will move in the direction where it can pull the strongest (i.e., in the direction where the rupture force is highest).

While bacteria and embryos use different molecules to generate these pulling forces, these findings indicate that the basic physical principles are similar in both systems. One of the next challenges will be to evaluate how biofilms might benefit from the structures that develop due to cell sorting.

DOI:http://dx.doi.org/10.7554/eLife.10811.002

Integration of cell-cell and cell-ECM adhesion in vertebrate morphogenesis

In this review, we highlight recent re-evaluations of the classical cell sorting models and their application to understanding embryonic morphogenesis. Modern genetic and biophysical techniques reveal that tissue self-assembly is not solely a result of differential adhesion, but rather incorporates dynamic cytoskeletal tension and extracellular matrix assembly. There is growing evidence that these biomechanical modules cooperate to organize developing tissues. We describe the contributions of Cadherins and Integrins to tissue assembly and propose a model in which these very different adhesive regimes affect the same outcome through separate but convergent mechanisms.

Non-straight cell edges are important to invasion and engulfment as demonstrated by cell mechanics model

Computational models of cell–cell mechanical interactions typically simulate sorting and certain other motions well, but as demands on these models continue to grow, discrepancies between the cell shapes, contact angles and behaviours they predict and those that occur in real cells have come under increased scrutiny. To investigate whether these discrepancies are a direct result of the straight cell–cell edges generally assumed in these models, we developed a finite element model that approximates cell boundaries using polylines with an arbitrary number of segments. We then compared the predictions of otherwise identical polyline and monoline (straight-edge) models in a variety of scenarios, including annealing, single- and multi-cell engulfment, sorting, and two forms of mixing—invasion and checkerboard pattern formation. Keeping cell–cell edges straight influences cell motion, cell shape, contact angle, and boundary length, especially in cases where one cell type is pulled between or around cells of a different type, as in engulfment or invasion. These differences arise because monoline cells have restricted deformation modes. Polyline cells do not face these restrictions, and with as few as three segments per edge yielded realistic edge shapes and contact angle errors one-tenth of those produced by monoline models, making them considerably more suitable for situations where angles and shapes matter, such as validation of cellular force–inference techniques. The findings suggest that non-straight cell edges are important both in modelling and in nature.

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.

Computational models for mechanics of morphogenesis

In the developing embryo, tissues differentiate, deform, and move in an orchestrated manner to generate various biological shapes driven by the complex interplay between genetic, epigenetic, and environmental factors. Mechanics plays a key role in regulating and controlling morphogenesis, and quantitative models help us understand how various mechanical forces combine to shape the embryo. Models allow for the quantitative, unbiased testing of physical mechanisms, and when used appropriately, can motivate new experimentaldirections. This knowledge benefits biomedical researchers who aim to prevent and treat congenital malformations, as well as engineers working to create replacement tissues in the laboratory. In this review, we first give an overview of fundamental mechanical theories for morphogenesis, and then focus on models for specific processes, including pattern formation, gastrulation, neurulation, organogenesis, and wound healing. The role of mechanical feedback in development is also discussed. Finally, some perspectives aregiven on the emerging challenges in morphomechanics and mechanobiology.

Computational modeling of epithelial tissues

There is an extensive literature on the computational modeling of epithelial tissues at all levels from subcellular to whole tissue. This review concentrates on behavior at the individual cell to whole tissue level, and particularly on organizational aspects, and provides an indication of where information from other areas, such as the modeling of angiogenesis, is relevant. The skin, and the lining of all of the body cavities (lung, gut, cervix, bladder etc) are epithelial tissues, which in a topological sense are the boundary between inside and outside the body. They are thin sheets of cells (usually of the order of 0.5 mm thick) without extracellular matrix, have a relatively simple structure, and contain few types of cells. They have important barrier, secretory and transport functions, which are essential for the maintenance of life, so homeostasis and wound healing are important aspects of the behavior of epithelial tissues. Carcinomas originate in epithelial tissues.There are essentially two approaches to modeling tissues--to start at the level of the tissue (i.e., a length scale of the order of 1 mm) and develop generalized equations for behavior (a continuum approach); or to start at the level of the cell (i.e., a length scale of the order of 10 µm) and develop tissue behavior as an emergent property of cellular behavior (an individual-based approach). As will be seen, these are not mutually exclusive approaches, and they come in a variety of flavors.

Cell sorting in development

During the development of multicellular organisms, cell fate specification is followed by the sorting of different cell types into distinct domains from where the different tissues and organs are formed. Cell sorting involves both the segregation of a mixed population of cells with different fates and properties into distinct domains, and the active maintenance of their segregated state. Because of its biological importance and apparent resemblance to fluid segregation in physics, cell sorting was extensively studied by both biologists and physicists over the last decades. Different theories were developed that try to explain cell sorting on the basis of the physical properties of the constituent cells. However, only recently the molecular and cellular mechanisms that control the physical properties driving cell sorting, have begun to be unraveled. In this review, we will provide an overview of different cell-sorting processes in development and discuss how these processes can be explained by the different sorting theories, and how these theories in turn can be connected to the molecular and cellular mechanisms driving these processes.

Identifying same-cell contours in image stacks: a key step in making 3D reconstructions

Identification of contours belonging to the same cell is a crucial step in the analysis of confocal stacks and other image sets in which cell outlines are visible, and it is central to the making of 3D cell reconstructions. When the cells are close packed, the contour grouping problem is more complex than that found in medical imaging, for example, because there are multiple regions of interest, the regions are not separable from each other by an identifiable background and regions cannot be distinguished by intensity differences. Here, we present an algorithm that uses three primary metrics-overlap of contour areas in adjacent images, co-linearity of the centroids of these areas across three images in a stack, and cell taper-to assign cells to groups. Decreasing thresholds are used to successively assign contours whose membership is less obvious. In a final step, remaining contours are assigned to existing groups by setting all thresholds to zero and groups having strong hour-glass shapes are partitioned. When applied to synthetic data from isotropic model aggregates, a curved model epithelium in which the long axes of the cells lie at all possible angles to the transection plane, and a confocal image stack, algorithm assignments were between 97 and 100% accurate in sets having at least four contours per cell. The algorithm is not particularly sensitive to the thresholds used, and a single set of parameters was used for all of the tests. The algorithm, which could be extended to time-lapse data, solves a key problem in the translation of image data into cell information.

Geometric control of human stem cell morphology and differentiation

During tissue morphogenesis, stem cells and progenitor cells migrate, proliferate, and differentiate, with striking changes in cell shape, size, and acting mechanical stresses. The local cellular function depends on the spatial distribution of cytokines as well as local mechanical microenvironments in which the cells reside. In this study, we controlled the organization of human adipose derived stem cells using micro-patterning technologies, to investigate the influence of multi-cellular form on spatial distribution of cellular function at an early stage of cell differentiation. The underlying role of cytoskeletal tension was probed through drug treatment. Our results show that the cultivation of stem cells on geometric patterns resulted in pattern- and position-specific cell morphology, proliferation and differentiation. The highest cell proliferation occurred in the regions with large, spreading cells (such as the outer edge of a ring and the short edges of rectangles). In contrast, stem cell differentiation co-localized with the regions containing small, elongated cells (such as the inner edge of a ring and the regions next to the short edges of rectangles). The application of drugs that inhibit the formation of actomyosin resulted in the lack of geometrically specific differentiation patterns. This study confirms the role of substrate geometry on stem cell differentiation, through associated physical forces, and provides a simple and controllable system for studying biophysical regulation of cell function.

Biology and physics of cell shape changes in development

Together with cell growth, division and death, changes in cell shape are of central importance for tissue morphogenesis during development. Cell shape is the product of a cell's material and active properties balanced by external forces. Control of cell shape, therefore, relies on both tight regulation of intracellular mechanics and the cell's physical interaction with its environment. In this review, we first discuss the biological and physical mechanisms of cell shape control. We next examine a number of developmental processes in which cell shape change - either individually or in a coordinated manner - drives embryonic morphogenesis and discuss how cell shape is controlled in these processes. Finally, we emphasize that cell shape control during tissue morphogenesis can only be fully understood by using a combination of cellular, molecular, developmental and biophysical approaches.

Biomechanics: cell research and applications for the next decade

With the recent revolution in Molecular Biology and the deciphering of the Human Genome, understanding of the building blocks that comprise living systems has advanced rapidly. We have yet to understand, however, how the physical forces that animate life affect the synthesis, folding, assembly, and function of these molecular building blocks. We are equally uncertain as to how these building blocks interact dynamically to create coupled regulatory networks from which integrative biological behaviors emerge. Here we review recent advances in the field of biomechanics at the cellular and molecular levels, and set forth challenges confronting the field. Living systems work and move as multi-molecular collectives, and in order to understand key aspects of health and disease we must first be able to explain how physical forces and mechanical structures contribute to the active material properties of living cells and tissues, as well as how these forces impact information processing and cellular decision making. Such insights will no doubt inform basic biology and rational engineering of effective new approaches to clinical therapy.

Mechanical determinants of epithelium thickness in early-stage embryos

To address the fundamental question, "How is the thickness of an embryonic epithelium determined?" equations of force equilibrium are derived and used to construct a theory applicable to embryonic epithelia and other labile monolayers, including confluent cultured cells. Under typical physiological conditions, the underlying mathematical relationships can be reworked into one simple algebraic equation. The theory shows that sheet thickness is determined by mechanical interactions between microfilaments, microtubules, cell adhesion systems and in-plane loading. It also explains the perplexing negative in-plane Poisson's ratio observed in some uniaxial tensile tests of embryonic epithelia. Thickness anomalies can affect phenotype, and the equations make it possible to calculate the degree to which gene expression, signalling pathways or other factors must alter the forces at work in order to produce a specified change in sheet thicknesses. Key findings are confirmed using a cell-based finite element model.

Growth based morphogenesis of vertebrate limb bud

Many genes and their regulatory relationships are involved in developmental phenomena. However, by chemical information alone, we cannot fully understand changing organ morphologies through tissue growth because deformation and growth of the organ are essentially mechanical processes. Here, we develop a mathematical model to describe the change of organ morphologies through cell proliferation. Our basic idea is that the proper specification of localized volume source (e.g., cell proliferation) is able to guide organ morphogenesis, and that the specification is given by chemical gradients. We call this idea "growth-based morphogenesis." We find that this morphogenetic mechanism works if the tissue is elastic for small deformation and plastic for large deformation. To illustrate our concept, we study the development of vertebrate limb buds, in which a limb bud protrudes from a flat lateral plate and extends distally in a self-organized manner. We show how the proportion of limb bud shape depends on different parameters and also show the conditions needed for normal morphogenesis, which can explain abnormal morphology of some mutants. We believe that the ideas shown in the present paper are useful for the morphogenesis of other organs.

A three-dimensional finite element model for the mechanics of cell-cell interactions

Technical challenges, including significant ones associated with cell rearrangement, have hampered the development of three-dimensional finite element models for the mechanics of embryonic cells. These challenges have been overcome by a new formulation in which the contents of each cell, assumed to have a viscosity mu, are modeled using a system of orthogonal dashpots. This approach overcomes a stiffening artifact that affects more traditional models, in which space-filling viscous elements are used to model the cytoplasm. Cells are assumed to be polyhedral in geometry, and each n-sided polygonal face is subdivided into n triangles with a common node at the face center so that it needs not remain flat. A constant tension gamma is assumed to act along each cell-cell interface, and cell rearrangements occur through one of two complementary topological transformations. The formulation predicts mechanical interactions between pairs of similar or dissimilar cells that are consistent with experiments, two-dimensional simulations, contact angle theory, and intracellular pressure calculations. Simulations of the partial engulfment of one tissue type by another show that the formulation is able to model aggregates of several hundred cells without difficulty. Simulations carried out using this formulation suggest new experimental approaches for measuring cell surface tensions and interfacial tensions. The formulation holds promise as a tool for gaining insight into the mechanics of isolated or aggregated embryonic cells and for the design and interpretation of experiments that involve them.

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A three-dimensional vertex dynamics cell model of space-filling polyhedra simulating cell behavior in a cell aggregate

We developed a three-dimensional (3D) cell model of a multicellular aggregate consisting of several polyhedral cells to investigate the deformation and rearrangement of cells under the influence of external forces. The polyhedral cells fill the space in the aggregate without gaps or overlaps, consist of contracting interfaces and maintain their volumes. The interfaces and volumes were expressed by 3D vertex coordinates. Vertex movements obey equations of motion that rearrange the cells to minimize total free energy, and undergo an elementary process that exchanges vertex pair connections when vertices approach each other. The total free energy includes the interface energy of cells and the compression or expansion energy of cells. Computer simulations provided the following results: An aggregate of cells becomes spherical to minimize individual cell surface areas; Polygonal interfaces of cells remain flat; Cells within the 3D cell aggregate can move and rearrange despite the absence of free space. We examined cell rearrangement to elucidate the viscoelastic properties of the aggregate, e.g. when an external force flattens a cell aggregate (e.g. under centrifugation) its component cells quickly flatten. Under a continuous external force, the cells slowly rearrange to recover their original shape although the cell aggregate remains flat. The deformation and rearrangement of individual cells is a two-step process with a time lag. Our results showed that morphological and viscoelastic properties of the cell aggregate with long relaxation time are based on component cells where minimization of interfacial energy of cells provides a motive force for cell movement.

Computer simulations of mitosis and interdependencies between mitosis orientation, cell shape and epithelia reshaping

Finite element-based computer simulations are used to investigate mitosis and how mitosis, cell shape, and epithelium reshaping depend on each other. Frame- and cell-oriented patterns of mitosis with growing and non-growing daughter cells are considered. Previous simulations have shown that applied stresses or strains can reshape cells so that their long axes are aligned in the principal stretch direction. The simulations reported here show that this can produce global alignment of the mitosis cleavage planes. Other simulations reported here show that mitoses with suitably aligned cleavage planes can drive epithelium reshaping. Formulas that quantify these and other dependencies are derived. These formulas provide quantitative relationships against which current hypotheses regarding epithelia reshaping in real biological systems can be evaluated.

The Differential Interfacial Tension Hypothesis (DITH): a comprehensive theory for the self-rearrangement of embryonic cells and tissues

A comprehensive theory, herein named the Differential Interfacial Tension Hypothesis for the self-rearrangement of embryonic cells and tissues is presented. These rearrangements include sorting, mixing and formation of checkerboard patterns in heterotypic aggregates of embryonic cells, and total or partial engulfment, separation and dissociation of tissues. This broadly-based theory accounts for the action of all currently known cytoskeletal components and cell adhesion mechanisms. The theory is used to derive conditions for the cell and tissue rearrangements named above. Finite element-based computer simulations involving two or more cell types confirm these conditions.

Formation and maintenance of distinctive cell patterns by coexpression of membrane-bound ligands and their receptors

We show that graded or checkerboard-like cell patterns, and segmental domains along a body axis, can be generated by cell behaviors involving differences in intercellular repulsion. A membrane-bound signal transduction system mediating contact-dependent cell interactions includes membrane-bound ligands (ephrins) and their receptors with tyrosine-kinase activity (Eph proteins). These molecules mediate both repulsive and attractive interactions under bilateral threshold control, i.e., cells expressing the receptors adhere to a surface bearing a critical density of ligand reciprocal to the density of receptor but are repelled by a surface with other densities of ligand (Honda [1998] J. Theor. Biol. 192:235-246). We extend this model. General membrane-bound ligands (not always ephrins) and their receptors are presumably coexpressed in a single cell under bilateral threshold control. Computer simulations of cell pattern formation showed that when coexpression of the ligand and receptor is reciprocal, the cells self-organize into a pattern of segmental domains or a graded cell arrangement along the body axis. The latter process interprets positional information in terms of protein molecules. When coexpression of the two species of molecules is not always reciprocal, the cells generate various patterns including checkerboard and kagome (star) patterns. The case of separate expression of ligands and receptors in different cells is also examined. The mechanism of differences in cell repulsion is compared with the differential cell adhesion hypothesis, which has been used to explain cell sorting.

The mechanics of heterotypic cell aggregates: insights from computer simulations

Finite element-based computer simulations are used to investigate a number of phenomena, including tissue engulfment, cell sorting, and checkerboard-pattern formation, exhibited by heterotypic cell aggregates. The simulations show that these phenomena can be driven by a single equivalent force, namely a surface (or interfacial) tension, that results from cytoskeletal components and cell-cell adhesions. They also reveal that tissue engulfment, cell sorting, and checkerboard-pattern formation involve several discernible mechanical features or stages. With the aid of analytical arguments, we identify the conditions necessary for each of these phenomena. These findings are consistent with previous experimental investigations and computer simulations, but pose significant challenges to current theories of cell sorting and tissue engulfment.