'The Forms of Tissues, or Cell-aggregates': D'Arcy Thompson's influence and its limits

A shape-driven reentrant jamming transition in confluent monolayers of synthetic cell-mimics

Many critical biological processes, like wound healing, require densely packed cell monolayers/tissues to transition from a jammed solid-like to a fluid-like state. Although numerical studies anticipate changes in the cell shape alone can lead to unjamming, experimental support for this prediction is not definitive because, in living systems, fluidization due to density changes cannot be ruled out. Additionally, a cell's ability to modulate its motility only compounds difficulties since even in assemblies of rigid active particles, changing the nature of self-propulsion has non-trivial effects on the dynamics. Here, we design and assemble a monolayer of synthetic cell-mimics and examine their collective behaviour. By systematically increasing the persistence time of self-propulsion, we discovered a cell shape-driven, density-independent, re-entrant jamming transition. Notably, we observed cell shape and shape variability were mutually constrained in the confluent limit and followed the same universal scaling as that observed in confluent epithelia. Dynamical heterogeneities, however, did not conform to this scaling, with the fast cells showing suppressed shape variability, which our simulations revealed is due to a transient confinement effect of these cells by their slower neighbors. Our experiments unequivocally establish a morphodynamic link, demonstrating that geometric constraints alone can dictate epithelial jamming/unjamming.

Poissonian Cellular Potts Models Reveal Nonequilibrium Kinetics of Cell Sorting

Cellular Potts models are broadly applied across developmental biology and cancer research. We overcome limitations of the traditional approach, which reinterprets a modified Metropolis sampling as ad hoc dynamics, by introducing a physical timescale through Poissonian kinetics and by applying principles of stochastic thermodynamics to separate thermal and relaxation effects from athermal noise and nonconservative forces. Our method accurately describes cell-sorting dynamics in mouse-embryo development and identifies the distinct contributions of nonequilibrium processes, e.g., cell growth and active fluctuations.

From soap bubbles to multicellular organisms: Unraveling the role of cell adhesion and physical constraints in tile pattern formation and tissue morphogenesis

Tile patterns, in which numerous cells are arranged in a regular pattern, are found in a variety of multicellular organisms and play important functional roles. Such regular arrangements of cells are regulated by various cell adhesion molecules. On the other hand, cell shape is also known to be regulated by physical constraints similar to those of soap bubbles. In particular, circumference minimization plays an important role, and cell adhesion negatively affects this process, thereby regulating tissue morphogenesis based on physical properties. Here, we focus on the Drosophila compound eye and the mouse auditory epithelium, and summarize the mechanisms of tile pattern formation by cell adhesion molecules such as cadherins, Irre Cell Recognition Modules (IRMs), and nectins. Phenomena that cannot be explained by physical stability based on cortical tension alone have been reported in the tile pattern formation in the compound eye, suggesting that previously unexplored forces such as cellular concentric expansion force may play an important role. We would like to summarize perspectives for future research on the mechanisms of tissue morphogenesis.

Embryo mechanics cartography: inference of 3D force atlases from fluorescence microscopy

Tissue morphogenesis results from a tight interplay between gene expression, biochemical signaling and mechanics. Although sequencing methods allow the generation of cell-resolved spatiotemporal maps of gene expression, creating similar maps of cell mechanics in three-dimensional (3D) developing tissues has remained a real challenge. Exploiting the foam-like arrangement of cells, we propose a robust end-to-end computational method called 'foambryo' to infer spatiotemporal atlases of cellular forces from fluorescence microscopy images of cell membranes. Our method generates precise 3D meshes of cells' geometry and successively predicts relative cell surface tensions and pressures. We validate it with 3D foam simulations, study its noise sensitivity and prove its biological relevance in mouse, ascidian and worm embryos. 3D force inference allows us to recover mechanical features identified previously, but also predicts new ones, unveiling potential new insights on the spatiotemporal regulation of cell mechanics in developing embryos. Our code is freely available and paves the way for unraveling the unknown mechanochemical feedbacks that control embryo and tissue morphogenesis.

Two-point optical manipulation reveals mechanosensitive remodeling of cell-cell contacts in vivo

Biological tissues acquire reproducible shapes during development through dynamic cell behaviors. Most of these behaviors involve the remodeling of cell-cell contacts. During epithelial morphogenesis, contractile actomyosin networks remodel cell-cell contacts by shrinking and extending junctions between lateral cell surfaces. However, actomyosin networks not only generate mechanical stresses but also respond to them, confounding our understanding of how mechanical stresses remodel cell-cell contacts. Here, we develop a two-point optical manipulation method to impose different stress patterns on cell-cell contacts in the early epithelium of the Drosophila embryo. The technique allows us to produce junction extension and shrinkage through different push and pull manipulations at the edges of junctions. We use these observations to expand classical vertex-based models of tissue mechanics, incorporating negative and positive mechanosensitive feedback depending on the type of remodeling. In particular, we show that Myosin-II activity responds to junction strain rate and facilitates full junction shrinkage. Altogether our work provides insight into how stress produces efficient deformation of cell-cell contacts in vivo and identifies unanticipated mechanosensitive features of their remodeling.

On shape forming by contractile filaments in the surface of growing tissues

Growing tissues are highly dynamic, and flow on sufficiently long timescales due to cell proliferation, migration, and tissue remodeling. As a consequence, growing tissues can often be approximated as viscous fluids. This means that the shape of microtissues growing in vitro is governed by their surface stress state, as in fluid droplets. Recent work showed that cells in the near-surface region of fibroblastic or osteoblastic microtissues contract with highly oriented actin filaments, thus making the surface properties highly anisotropic, in contrast to what is expected for an isotropic fluid. Here, we develop a model that includes mechanical anisotropy of the surface generated by contractile fibers and we show that mechanical equilibrium requires contractile filaments to follow geodesic lines on the surface. Constant pressure in the fluid forces these contractile filaments to be along geodesics with a constant normal curvature. We then take this into account to determine equilibrium shapes of rotationally symmetric bodies subjected to anisotropic surface stress states and derive a family of surfaces of revolution. A comparison with recently published shapes of microtissues shows that this theory accurately predicts both the surface shape and the direction of the actin filaments on the surface.

Computational approaches for simulating luminogenesis

Lumens, liquid-filled cavities surrounded by polarized tissue cells, are elementary units involved in the morphogenesis of organs. Theoretical modeling and computations, which can integrate various factors involved in biophysics of morphogenesis of cell assembly and lumens, may play significant roles to elucidate the mechanisms in formation of such complex tissue with lumens. However, up to present, it has not been documented well what computational approaches or frameworks can be applied for this purpose and how we can choose the appropriate approach for each problem. In this review, we report some typical lumen morphologies and basic mechanisms for the development of lumens, focusing on three keywords - mechanics, hydraulics and geometry - while outlining pros and cons of the current main computational strategies. We also describe brief guidance of readouts, i.e., what we should measure in experiments to make the comparison with the model's assumptions and predictions.

Pulsations and flows in tissues as two collective dynamics with simple cellular rules

Collective motions of epithelial cells are essential for morphogenesis. Tissues elongate, contract, flow, and oscillate, thus sculpting embryos. These tissue level dynamics are known, but the physical mechanisms at the cellular level are unclear. Here, we demonstrate that a single epithelial monolayer of MDCK cells can exhibit two types of local tissue kinematics, pulsations and long range coherent flows, characterized by using quantitative live imaging. We report that these motions can be controlled with internal and external cues such as specific inhibitors and substrate friction modulation. We demonstrate the associated mechanisms with a unified vertex model. When cell velocity alignment and random diffusion of cell polarization are comparable, a pulsatile flow emerges whereas tissue undergoes long-range flows when velocity alignment dominates which is consistent with cytoskeletal dynamics measurements. We propose that environmental friction, acto-myosin distributions, and cell polarization kinetics are important in regulating dynamics of tissue morphogenesis.

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Two collective cell motions, pulsations and flows, coexist in MDCK monolayers

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Each collective movement is identified using divergence and velocity correlations

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Motion is controlled by the regulation of substrate friction and cytoskeleton

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A vertex model recapitulates the motion by tuning velocity and polarity alignment

On the origin of universal cell shape variability in confluent epithelial monolayers

Cell shape is fundamental in biology. The average cell shape can influence crucial biological functions, such as cell fate and division orientation. But cell-to-cell shape variability is often regarded as noise. In contrast, recent works reveal that shape variability in diverse epithelial monolayers follows a nearly universal distribution. However, the origin and implications of this universality remain unclear. Here, assuming contractility and adhesion are crucial for cell shape, characterized via aspect ratio (r), we develop a mean-field analytical theory for shape variability. We find that all the system-specific details combine into a single parameter α that governs the probability distribution function (PDF) of r; this leads to a universal relation between the standard deviation and the average of r. The PDF for the scaled r is not strictly but nearly universal. In addition, we obtain the scaled area distribution, described by the parameter μ. Information of α and μ together can distinguish the effects of changing physical conditions, such as maturation, on different system properties. We have verified the theory via simulations of two distinct models of epithelial monolayers and with existing experiments on diverse systems. We demonstrate that in a confluent monolayer, average shape determines both the shape variability and dynamics. Our results imply that cell shape distribution is inevitable, where a single parameter describes both statics and dynamics and provides a framework to analyze and compare diverse epithelial systems. In contrast to existing theories, our work shows that the universal properties are consequences of a mathematical property and should be valid in general, even in the fluid regime.

Dynamic changes in epithelial cell packing during tissue morphogenesis

Cell packing - the spatial arrangement of cells - determines the shapes of organs. Recently, investigations of organ development in a variety of model organisms have uncovered cellular mechanisms that are used by epithelial tissues to change cell packing, and thereby their shapes, to generate functional architectures. Here, we review these cellular mechanisms across a wide variety of developmental processes in vertebrates and invertebrates and identify a set of common motifs in the morphogenesis toolbox that, in combination, appear to allow any change in tissue shape. We focus on tissue elongation, folding and invagination, and branching. We also highlight how these morphogenetic processes are achieved by cell-shape changes, cell rearrangements, and oriented cell division. Finally, we describe approaches that have the potential to engineer three-dimensional tissues for both basic science and translational purposes. This review provides a framework for future analyses of how tissues are shaped by the dynamics of epithelial cell packing.

Active flows and deformable surfaces in development

We review progress in active hydrodynamic descriptions of flowing media on curved and deformable manifolds: the state-of-the-art in continuum descriptions of single-layers of epithelial and/or other tissues during development. First, after a brief overview of activity, flows and hydrodynamic descriptions, we highlight the generic challenge of identifying the dependence on dynamical variables of so-called active kinetic coefficients- active counterparts to dissipative Onsager coefficients. We go on to describe some of the subtleties concerning how curvature and active flows interact, and the issues that arise when surfaces are deformable. We finish with a broad discussion around the utility of such theories in developmental biology. This includes limitations to analytical techniques, challenges associated with numerical integration, fitting-to-data and inference, and potential tools for the future, such as discrete differential geometry.

Synthetic living machines: A new window on life

Increased control of biological growth and form is an essential gateway to transformative medical advances. Repairing of birth defects, restoring lost or damaged organs, normalizing tumors, all depend on understanding how cells cooperate to make specific, functional large-scale structures. Despite advances in molecular genetics, significant gaps remain in our understanding of the meso-scale rules of morphogenesis. An engineering approach to this problem is the creation of novel synthetic living forms, greatly extending available model systems beyond evolved plant and animal lineages. Here, we review recent advances in the emerging field of synthetic morphogenesis, the bioengineering of novel multicellular living bodies. Emphasizing emergent self-organization, tissue-level guided self-assembly, and active functionality, this work is the essential next generation of synthetic biology. Aside from useful living machines for specific functions, the rational design and analysis of new, coherent anatomies will greatly increase our understanding of foundational questions in evolutionary developmental and cell biology.

Inferring cell junction tension and pressure from cell geometry

Recognizing the crucial role of mechanical regulation and forces in tissue development and homeostasis has stirred a demand for in situ measurement of forces and stresses. Among emerging techniques, the use of cell geometry to infer cell junction tensions, cell pressures and tissue stress has gained popularity owing to the development of computational analyses. This approach is non-destructive and fast, and statistically validated based on comparisons with other techniques. However, its qualitative and quantitative limitations, in theory as well as in practice, should be examined with care. In this Primer, we summarize the underlying principles and assumptions behind stress inference, discuss its validity criteria and provide guidance to help beginners make the appropriate choice of its variants. We extend our discussion from two-dimensional stress inference to three dimensional, using the early mouse embryo as an example, and list a few possible extensions. We hope to make stress inference more accessible to the scientific community and trigger a broader interest in using this technique to study mechanics in development.

The plastic cell: mechanical deformation of cells and tissues

Epithelial cells possess the ability to change their shape in response to mechanical stress by remodelling their junctions and their cytoskeleton. This property lies at the heart of tissue morphogenesis in embryos. A key feature of embryonic cell shape changes is that they result from repeated mechanical inputs that make them partially irreversible at each step. Past work on cell rheology has rarely addressed how changes can become irreversible in a complex tissue. Here, we review new and exciting findings dissecting some of the physical principles and molecular mechanisms accounting for irreversible cell shape changes. We discuss concepts of mechanical ratchets and tension thresholds required to induce permanent cell deformations akin to mechanical plasticity. Work in different systems has highlighted the importance of actin remodelling and of E-cadherin endocytosis. We also list some novel experimental approaches to fine-tune mechanical tension, using optogenetics, magnetic beads or stretching of suspended epithelial tissues. Finally, we discuss some mathematical models that have been used to describe the quantitative aspects of accounting for mechanical cell plasticity and offer perspectives on this rapidly evolving field.

From genes to shape during metamorphosis: a history

Metamorphosis (Greek for a state of transcending-form or change-in-shape) refers to a dramatic transformation of an animal's body structure that occurs after development of the embryo or larva in many species. The development of a fly (or butterfly) from a crawling larva (or caterpillar) that forms a pupa (or chrysalis) before eclosing as a flying adult is a classic example of metamorphosis that captures the imagination and has been immortalized in children's books. Powerful genetic experiments in the fruit fly Drosophila melanogaster have revealed how genes can instruct the behaviour of individual cells to control patterns of tissue growth, mechanical force, cell-cell adhesion and cell-matrix adhesion drive morphogenetic change in epithelial tissues. Together, the distribution of mass, force and resistance determines cell shape changes, cell-cell rearrangements, and/or the orientation of cell divisions to generate the final form of the tissue. In organising tissue shape, genes harness the power of self-organisation to determine the collective behaviour of molecules and cells, which can often be reproduced in computer simulations of cell polarity and/or tissue mechanics. This review highlights fundamental discoveries in epithelial morphogenesis made by pioneers who were fascinated by metamorphosis, including D'Arcy Thompson, Conrad Waddington, Dianne Fristrom and Antonio Garcia-Bellido.

Cell Junction Mechanics beyond the Bounds of Adhesion and Tension

Cell-cell junctions, in particular adherens junctions, are major determinants of tissue mechanics during morphogenesis and homeostasis. In attempts to link junctional mechanics to tissue mechanics, many have utilized explicitly or implicitly equilibrium approaches based on adhesion energy, surface energy, and contractility to determine the mechanical equilibrium at junctions. However, it is increasingly clear that they have significant limitations, such as that it remains challenging to link the dynamics of the molecular components to the resulting physical properties of the junction, to its remodeling ability, and to its adhesion strength. In this perspective, we discuss recent attempts to consider the aspect of energy dissipation at junctions to draw contact points with soft matter physics where energy loss plays a critical role in adhesion theories. We set the grounds for a theoretical framework of the junction mechanics that bridges the dynamics at the molecular scale to the mechanics at the tissue scale.

DeepScratch: Single-cell based topological metrics of scratch wound assays

Changes in tissue architecture and multicellular organisation contribute to many diseases, including cancer and cardiovascular diseases. Scratch wound assay is a commonly used tool that assesses cells' migratory ability based on the area of a wound they cover over a certain time. However, analysis of changes in the organisational patterns formed by migrating cells following genetic or pharmacological perturbations are not well explored in these assays, in part because analysing the resulting imaging data is challenging. Here we present DeepScratch, a neural network that accurately detects the cells in scratch assays based on a heterogeneous set of markers. We demonstrate the utility of DeepScratch by analysing images of more than 232,000 lymphatic endothelial cells. In addition, we propose various topological measures of cell connectivity and local cell density (LCD) to characterise tissue remodelling during wound healing. We show that LCD-based metrics allow classification of CDH5 and CDC42 genetic perturbations that are known to affect cell migration through different biological mechanisms. Such differences cannot be captured when considering only the wound area. Taken together, single-cell detection using DeepScratch allows more detailed investigation of the roles of various genetic components in tissue topology and the biological mechanisms underlying their effects on collective cell migration.

From energy to cellular forces in the Cellular Potts Model: An algorithmic approach

Single and collective cell dynamics, cell shape changes, and cell migration can be conveniently represented by the Cellular Potts Model, a computational platform based on minimization of a Hamiltonian. Using the fact that a force field is easily derived from a scalar energy (F = −∇H), we develop a simple algorithm to associate effective forces with cell shapes in the CPM. We predict the traction forces exerted by single cells of various shapes and sizes on a 2D substrate. While CPM forces are specified directly from the Hamiltonian on the cell perimeter, we approximate the force field inside the cell domain using interpolation, and refine the results with smoothing. Predicted forces compare favorably with experimentally measured cellular traction forces. We show that a CPM model with internal signaling (such as Rho-GTPase-related contractility) can be associated with retraction-protrusion forces that accompany cell shape changes and migration. We adapt the computations to multicellular systems, showing, for example, the forces that a pair of swirling cells exert on one another, demonstrating that our algorithm works equally well for interacting cells. Finally, we show forces exerted by cells on one another in classic cell-sorting experiments.

Cells exert forces on their surroundings and on one another. In simulations of cell shape using the Cellular Potts Model (CPM), the dynamics of deforming cell shapes is traditionally represented by an energy-minimization method. We use this CPM energy, the Hamiltonian, to derive and visualize the corresponding forces exerted by the cells. We use the fact that force is the negative gradient of energy to assign forces to the CPM cell edges, and then extend the results to approximate interior forces by interpolation. We show that this method works for single as well as multiple interacting model cells, both static and motile. Finally, we show favorable comparison between predicted forces and real forces measured experimentally.

Cell-Size Pleomorphism Drives Aberrant Clone Dispersal in Proliferating Epithelia

As epithelial tissues develop, groups of cells related by descent tend to associate in clonal populations rather than dispersing within the cell layer. While this is frequently assumed to be a result of differential adhesion, precise mechanisms controlling clonal cohesiveness remain unknown. Here we employ computational simulations to modulate epithelial cell size in silico and show that junctions between small cells frequently collapse, resulting in clone-cell dispersal among larger neighbors. Consistent with similar dynamics in vivo, we further demonstrate that mosaic disruption of Drosophila Tor generates small cells and results in aberrant clone dispersal in developing wing disc epithelia. We propose a geometric basis for this phenomenon, supported in part by the observation that soap-foam cells exhibit similar size-dependent junctional rearrangements. Combined, these results establish a link between cell-size pleomorphism and the control of epithelial cell packing, with potential implications for understanding tumor cell dispersal in human disease.

3D culture models for studying branching morphogenesis in the mammary gland and mammalian lung

The intricate architecture of branched tissues and organs has fascinated scientists and engineers for centuries. Yet-despite their ubiquity-the biophysical and biochemical mechanisms by which tissues and organs undergo branching morphogenesis remain unclear. With the advent of three-dimensional (3D) culture models, an increasingly powerful and diverse set of tools are available for investigating the development and remodeling of branched tissues and organs. In this review, we discuss the application of 3D culture models for studying branching morphogenesis of the mammary gland and the mammalian lung in the context of normal development and disease. While current 3D culture models lack the cellular and molecular complexity observed in vivo, we emphasize how these models can be used to answer targeted questions about branching morphogenesis. We highlight the specific advantages and limitations of using 3D culture models to study the dynamics and mechanisms of branching in the mammary gland and mammalian lung. Finally, we discuss potential directions for future research and propose strategies for engineering the next generation of 3D culture models for studying tissue morphogenesis.

Tension, contraction and tissue morphogenesis

D'Arcy Thompson was a proponent of applying mathematical and physical principles to biological systems, an approach that is becoming increasingly common in developmental biology. Indeed, the recent integration of quantitative experimental data, force measurements and mathematical modeling has changed our understanding of morphogenesis - the shaping of an organism during development. Emerging evidence suggests that the subcellular organization of contractile cytoskeletal networks plays a key role in force generation, while on the tissue level the spatial organization of forces determines the morphogenetic output. Inspired by D'Arcy Thompson's On Growth and Form, we review our current understanding of how biological forms are created and maintained by the generation and organization of contractile forces at the cell and tissue levels. We focus on recent advances in our understanding of how cells actively sculpt tissues and how forces are involved in specific morphogenetic processes.

The old and new faces of morphology: the legacy of D'Arcy Thompson's 'theory of transformations' and 'laws of growth'

In 1917, the publication of On Growth and Form by D'Arcy Wentworth Thompson challenged both mathematicians and naturalists to think about biological shapes and diversity as more than a confusion of chaotic forms generated at random, but rather as geometric shapes that could be described by principles of physics and mathematics. Thompson's work was based on the ideas of Galileo and Goethe on morphology and of Russell on functionalism, but he was first to postulate that physical forces and internal growth parameters regulate biological forms and could be revealed via geometric transformations in morphological space. Such precise mathematical structure suggested a unifying generative process, as reflected in the title of the book. To Thompson it was growth that could explain the generation of any particular biological form, and changes in ontogeny, rather than natural selection, could then explain the diversity of biological shapes. Whereas adaptationism, widely accepted in evolutionary biology, gives primacy to extrinsic factors in producing morphological variation, Thompson's 'laws of growth' provide intrinsic directives and constraints for the generation of individual shapes, helping to explain the 'profusion of forms, colours, and other modifications' observed in the living world.

Morphogenesis one century after On Growth and Form

Summary: This Editorial introduces the special issue – providing a perspective on the influence of D'Arcy Thompson's work and an overview of the articles in this issue.