Exploring the effects of mechanical feedback on epithelial topology

Competing mechanisms for the buckling of an epithelial monolayer identified using multicellular simulation

A model using the rigid body multi-cellular framework (RBMCF) is implemented to investigate the mechanisms of buckling of an epithelial monolayer. Specifically, the deformation of a monolayer of epithelial cells which are attached to a basement membrane and the surrounding stromal tissue. The epithelial monolayer, supporting basement membrane and stromal tissue are modelled using two separate vertex dynamics models (one for the epithelial monolayer layer and one for the basement membrane and stromal tissue combined) and interactions between the two are considered using the RBMCF to ensure biologically realistic interactions. Model simulations are used to investigate the effects of cell-stromal attachment and membrane rigidity on buckling behaviour. We demonstrate that there are two competing modes of buckling, stromal deformation and stromal separation.

Active shape programming drives Drosophila wing disc eversion

How complex 3D tissue shape emerges during animal development remains an important open question in biology and biophysics. Here, we discover a mechanism for 3D epithelial shape change based on active, in-plane cellular events that is analogous to inanimate "shape programmable" materials, which undergo blueprinted 3D shape transformations from in-plane gradients of spontaneous strains. We study eversion of the Drosophila wing disc pouch, when the epithelium transforms from a dome into a curved fold, quantifying 3D tissue shape changes and mapping spatial patterns of cellular behaviors on the evolving geometry using cellular topology. Using a physical model inspired by shape programming, we find that active cell rearrangements are the major contributor to pouch eversion and validate this conclusion using a knockdown of MyoVI, which reduces rearrangements and disrupts morphogenesis. This work shows that shape programming is a mechanism for animal tissue morphogenesis and suggests that patterns in nature could present design strategies for shape-programmable materials.

Choice of friction coefficient deeply affects tissue behaviour in stochastic epithelial vertex models

To understand the mechanisms that coordinate the formation of biological tissues, the use of numerical implementations is necessary. The complexity of such models involves many assumptions and parameter choices that result in unpredictable consequences, obstructing the comparison with experimental data. Here, we focus on vertex models, a family of spatial models used extensively to simulate the dynamics of epithelial tissues. Usually, in the literature, the choice of the friction coefficient is not addressed using quasi-static deformation arguments that generally do not apply to realistic scenarios. In this manuscript, we discuss the role that the choice of friction coefficient has on the relaxation times and consequently in the conditions of cell cycle progression and division. We explore the effects that these changes have on the morphology, growth rate and topological transitions of the tissue dynamics. These results provide a deeper understanding of the role that an accurate mechanical description plays in the use of vertex models as inference tools. This article is part of a discussion meeting issue 'Causes and consequences of stochastic processes in development and disease'.

Topological balance of cell distributions in plane monolayers

Most of normal proliferative epithelia of plants and metazoans are topologically invariant and characterized by similar cell distributions according to the number of cell neighbors (DCNs). Here we study peculiarities of these distributions and explain why the DCN obtained from the location of intercellular boundaries and that based on the Voronoi tessellation with nodes located on cell nuclei may differ from each other. As we demonstrate, special microdomains where four or more intercellular boundaries converge are topologically charged. Using this fact, we deduce a new equation describing the topological balance of the DCNs. The developed theory is applied for a series of microphotographs of non-tumoral epithelial cells of the human cervix (HCerEpiC) to improve the image processing near the edges of microphotographs and reveal the topological invariance of the examined monolayers. Special contact microdomains may be present in epithelia of various natures, however, considering the well-known vertex model of epithelium, we show that such contacts are absent in the usual solid-like state of the model and appear only in the liquid-like cancer state. Also, we discuss a possible biological role of special contacts in context of proliferative epithelium dynamics and tissue morphogenesis.

Adhesion and Polarity protein distribution-regulates hexagon dominated plasma membrane organization in Drosophila blastoderm embryos

Epithelial cells contain polarity complexes on the lateral membrane and are organized in a hexagon-dominated polygonal array. The mechanisms regulating the organization of polygonal architecture in metazoan embryogenesis are not completely understood. Drosophila embryogenesis enables mechanistic analysis of epithelial polarity formation and its impact on polygonal organization. The plasma membrane (PM) of syncytial Drosophila blastoderm embryos is organized as a polygonal array with pseudocleavage furrow formation during the almost synchronous cortical division cycles. We find that polygonal (PM) organization arises in the metaphase (MP) of division cycle 11, and hexagon dominance occurs with an increase in furrow length in the metaphase of cycle 12. There is a decrease in cell shape index in metaphase from cycles 11 to 13. This coincides with Drosophila E-cad (DE-cadherin) and Bazooka enrichment at the edges and the septin, Peanut at the vertices of the furrow. We further assess the role of polarity and adhesion proteins in pseudocleavage furrow formation and its organization as a polygonal array. We find that DE-cadherin depletion leads to decreased furrow length, loss of hexagon dominance, and increased cell shape index. Bazooka and Peanut depletion lead to decreased furrow length, delay in onset of hexagon dominance from cycle 12 to 13, and increased cell shape index. Hexagon dominance occurs with an increase in furrow length in cycle 13 and increased DE-cadherin, possibly due to the inhibition of endocytosis. We conclude that polarity protein recruitment and regulation of endocytic pathways enable pseudocleavage furrow stability and the formation of a hexagon-dominated polygon array.

Is Drosophila Dpp/BMP morphogen spreading required for wing patterning and growth?

Secreted signaling molecules act as morphogens to control patterning and growth in many developing tissues. Since locally produced morphogens spread to form a concentration gradient in the surrounding tissue, spreading is generally thought to be the key step in the non-autonomous actions. Here, we review recent advances in tool development to investigate morphogen function using the role of decapentaplegic (Dpp)/bone morphogenetic protein (BMP)-type ligand in the Drosophila wing disc as an example. By applying protein binder tools to distinguish between the roles of Dpp spreading and local Dpp signaling, we found that Dpp signaling in the source cells is important for wing patterning and growth but Dpp spreading from this source cells is not as strictly required as previously thought. Given recent studies showing unexpected requirements of long-range action of different morphogens, manipulating endogenous morphogen gradients by synthetic protein binder tools could shed more light on how morphogens act in developing tissues.

Topological properties and shape of proliferative and nonproliferative cell monolayers

During embryonic development, structures with complex geometry can emerge from planar epithelial monolayers; studying these shape transitions is of key importance for revealing the biophysical laws involved in the morphogenesis of biological systems. Here, using the example of normal proliferative monkey kidney (COS) cell monolayers, we investigate global and local topological characteristics of this model system in dependence on its shape. The obtained distributions of cells by their valence demonstrate a difference between the spherical and planar monolayers. In addition, in both spherical and planar monolayers, the probability of observing a pair of neighboring cells with certain valences depends on the topological charge of the pair. The zero topological charge of the cell pair can increase the probability for the cells to be the nearest neighbors. We then test and confirm that analogous relationships take place in a more ordered spherical system with a larger fraction of 6-valent cells, namely, in the nonproliferative epithelium (follicular system) of ascidian species oocytes. However, unlike spherical COS cell monolayers, ascidian monolayers are prone to nonrandom agglomeration of 6-valent cells and have linear topological defects called scars and pleats. The reasons for this difference in morphology are discussed. The morphological peculiarities found are compared with predictions of the widely used vertex model of epithelium.

Game of clones: Battles in the field of carcinogenesis

Recent advances in bulk sequencing approaches as well as genomic decoding at the single-cell level have revealed surprisingly high somatic mutational burdens in normal tissues, as well as increased our understanding of the landscape of "field cancerization", that is, molecular and immune alterations in mutagen-exposed normal-appearing tissues that recapitulated those present in tumors. Charting the somatic mutational landscapes in normal tissues can have strong implications on our understanding of how tumors arise from mutagenized epithelium. Making sense of those mutations to understand the progression along the pathologic continuum of normal epithelia, preneoplasias, up to malignant tissues will help pave way for identification of ideal targets that can guide new strategies for preventing or eliminating cancers at their earliest stages of development. In this review, we will provide a brief history of field cancerization and its implications on understanding early stages of cancer pathogenesis and deviation from the pathologically "normal" state. The review will provide an overview of how mutations accumulating in normal tissues can lead to a patchwork of mutated cell clones that compete while maintaining an overall state of functional homeostasis. The review also explores the role of clonal competition in directing the fate of normal tissues and summarizes multiple mechanisms elicited in this phenomenon and which have been linked to cancer development. Finally, we highlight the importance of understanding mutations in normal tissues, as well as clonal competition dynamics (in both the epithelium and the microenvironment) and their significance in exploring new approaches to combatting cancer.

Avalanches during epithelial tissue growth; Uniform Growth and a drosophila eye disc model

Epithelial tissues constitute an exotic type of active matter with non-linear properties reminiscent of amorphous materials. In the context of a proliferating epithelium, modeled by the quasistatic vertex model, we identify novel discrete tissue scale rearrangements, i.e. cellular rearrangement avalanches, which are a form of collective cell movement. During the avalanches, the vast majority of cells retain their neighbors, and the resulting cellular trajectories are radial in the periphery, a vortex in the core. After the onset of these avalanches, the epithelial area grows discontinuously. The avalanches are found to be stochastic, and their strength is correlated with the density of cells in the tissue. Overall, avalanches redistribute accumulated local spatial pressure along the tissue. Furthermore, the distribution of avalanche magnitudes is found to obey a power law, with an exponent consistent with sheer induced avalanches in amorphous materials. To understand the role of avalanches in organ development, we simulate epithelial growth of the Drosophila eye disc during the third instar using a computational model, which includes both chemical and mechanistic signaling. During the third instar, the morphogenetic furrow (MF), a ~10 cell wide wave of apical area constriction propagates through the epithelium. These simulations are used to understand the details of the growth process, the effect of the MF on the growth dynamics on the tissue scale, and to make predictions for experimental observations. The avalanches are found to depend on the strength of the apical constriction of cells in the MF, with a stronger apical constriction leading to less frequent and more pronounced avalanches. The results herein highlight the dependence of simulated tissue growth dynamics on relaxation timescales, and serve as a guide for in vitro experiments.

The role of cell geometry and cell-cell communication in gradient sensing

Cells can measure shallow gradients of external signals to initiate and accomplish a migration or a morphogenetic process. Recently, starting from mathematical models like the local-excitation global-inhibition (LEGI) model and with the support of empirical evidence, it has been proposed that cellular communication improves the measurement of an external gradient. However, the mathematical models that have been used have over-simplified geometries (e.g., they are uni-dimensional) or assumptions about cellular communication, which limit the possibility to analyze the gradient sensing ability of more complex cellular systems. Here, we generalize the existing models to study the effects on gradient sensing of cell number, geometry and of long- versus short-range cellular communication in 2D systems representing epithelial tissues. We find that increasing the cell number can be detrimental for gradient sensing when the communication is weak and limited to nearest neighbour cells, while it is beneficial when there is long-range communication. We also find that, with long-range communication, the gradient sensing ability improves for tissues with more disordered geometries; on the other hand, an ordered structure with mostly hexagonal cells is advantageous with nearest neighbour communication. Our results considerably extend the current models of gradient sensing by epithelial tissues, making a step further toward predicting the mechanism of communication and its putative mediator in many biological processes.

Asymmetric requirement of Dpp/BMP morphogen dispersal in the Drosophila wing disc

How morphogen gradients control patterning and growth in developing tissues remains largely unknown due to lack of tools manipulating morphogen gradients. Here, we generate two membrane-tethered protein binders that manipulate different aspects of Decapentaplegic (Dpp), a morphogen required for overall patterning and growth of the Drosophila wing. One is "HA trap" based on a single-chain variable fragment (scFv) against the HA tag that traps HA-Dpp to mainly block its dispersal, the other is "Dpp trap" based on a Designed Ankyrin Repeat Protein (DARPin) against Dpp that traps Dpp to block both its dispersal and signaling. Using these tools, we found that, while posterior patterning and growth require Dpp dispersal, anterior patterning and growth largely proceed without Dpp dispersal. We show that dpp transcriptional refinement from an initially uniform to a localized expression and persistent signaling in transient dpp source cells render the anterior compartment robust against the absence of Dpp dispersal. Furthermore, despite a critical requirement of dpp for the overall wing growth, neither Dpp dispersal nor direct signaling is critical for lateral wing growth after wing pouch specification. These results challenge the long-standing dogma that Dpp dispersal is strictly required to control and coordinate overall wing patterning and growth.

Dissecting Organismal Morphogenesis by Bridging Genetics and Biophysics

Multicellular organisms develop complex shapes from much simpler, single-celled zygotes through a process commonly called morphogenesis. Morphogenesis involves an interplay between several factors, ranging from the gene regulatory networks determining cell fate and differentiation to the mechanical processes underlying cell and tissue shape changes. Thus, the study of morphogenesis has historically been based on multidisciplinary approaches at the interface of biology with physics and mathematics. Recent technological advances have further improved our ability to study morphogenesis by bridging the gap between the genetic and biophysical factors through the development of new tools for visualizing, analyzing, and perturbing these factors and their biochemical intermediaries. Here, we review how a combination of genetic, microscopic, biophysical, and biochemical approaches has aided our attempts to understand morphogenesis and discuss potential approaches that may be beneficial to such an inquiry in the future. Expected final online publication date for the Annual Review of Genetics, Volume 55 is November 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Self-organized patterning of cell morphology via mechanosensitive feedback

Tissue organization is often characterized by specific patterns of cell morphology. How such patterns emerge in developing tissues is a fundamental open question. Here, we investigate the emergence of tissue-scale patterns of cell shape and mechanical tissue stress in the Drosophila wing imaginal disc during larval development. Using quantitative analysis of the cellular dynamics, we reveal a pattern of radially oriented cell rearrangements that is coupled to the buildup of tangential cell elongation. Developing a laser ablation method, we map tissue stresses and extract key parameters of tissue mechanics. We present a continuum theory showing that this pattern of cell morphology and tissue stress can arise via self-organization of a mechanical feedback that couples cell polarity to active cell rearrangements. The predictions of this model are supported by knockdown of MyoVI, a component of mechanosensitive feedback. Our work reveals a mechanism for the emergence of cellular patterns in morphogenesis.

FACEts of mechanical regulation in the morphogenesis of craniofacial structures

During embryonic development, organs undergo distinct and programmed morphological changes as they develop into their functional forms. While genetics and biochemical signals are well recognized regulators of morphogenesis, mechanical forces and the physical properties of tissues are now emerging as integral parts of this process as well. These physical factors drive coordinated cell movements and reorganizations, shape and size changes, proliferation and differentiation, as well as gene expression changes, and ultimately sculpt any developing structure by guiding correct cellular architectures and compositions. In this review we focus on several craniofacial structures, including the tooth, the mandible, the palate, and the cranium. We discuss the spatiotemporal regulation of different mechanical cues at both the cellular and tissue scales during craniofacial development and examine how tissue mechanics control various aspects of cell biology and signaling to shape a developing craniofacial organ.

Emerging mechanisms of cell competition

The growth and survival of cells within tissues can be affected by 'cell competition' between different cell clones. This phenomenon was initially recognized between wild-type cells and cells with mutations in ribosomal protein (Rp) genes in Drosophila melanogaster. However, competition also affects D. melanogaster cells with mutations in epithelial polarity genes, and wild-type cells exposed to 'super-competitor' cells with mutation in the Salvador-Warts-Hippo tumour suppressor pathway or expressing elevated levels of Myc. More recently, cell competition and super-competition were recognized in mammalian development, organ homeostasis and cancer. Genetic and cell biological studies have revealed that mechanisms underlying cell competition include the molecular recognition of 'different' cells, signalling imbalances between distinct cell populations and the mechanical consequences of differential growth rates; these mechanisms may also involve innate immune proteins, p53 and changes in translation.

Neuronal differentiation influences progenitor arrangement in the vertebrate neuroepithelium

Cell division, movement and differentiation contribute to pattern formation in developing tissues. This is the case in the vertebrate neural tube, in which neurons differentiate in a characteristic pattern from a highly dynamic proliferating pseudostratified epithelium. To investigate how progenitor proliferation and differentiation affect cell arrangement and growth of the neural tube, we used experimental measurements to develop a mechanical model of the apical surface of the neuroepithelium that incorporates the effect of interkinetic nuclear movement and spatially varying rates of neuronal differentiation. Simulations predict that tissue growth and the shape of lineage-related clones of cells differ with the rate of differentiation. Growth is isotropic in regions of high differentiation, but dorsoventrally biased in regions of low differentiation. This is consistent with experimental observations. The absence of directional signalling in the simulations indicates that global mechanical constraints are sufficient to explain the observed differences in anisotropy. This provides insight into how the tissue growth rate affects cell dynamics and growth anisotropy and opens up possibilities to study the coupling between mechanics, pattern formation and growth in the neural tube.

Summary: A mechanical model of the vertebrate neuroepithelium, based on experimental observations, suggests that the rate of neuronal differentiation influences tissue growth and the shape of lineage-related clones.

The Hippo pathway integrates PI3K-Akt signals with mechanical and polarity cues to control tissue growth

The Hippo signalling pathway restricts cell proliferation in animal tissues by inhibiting Yes-associated protein (YAP or YAP1) and Transcriptional Activator with a PDZ domain (TAZ or WW-domain-containing transcriptional activator [WWTR1]), coactivators of the Scalloped (Sd or TEAD) DNA-binding transcription factor. Drosophila has a single YAP/TAZ homolog named Yorkie (Yki) that is regulated by Hippo pathway signalling in response to epithelial polarity and tissue mechanics during development. Here, we show that Yki translocates to the nucleus to drive Sd-mediated cell proliferation in the ovarian follicle cell epithelium in response to mechanical stretching caused by the growth of the germline. Importantly, mechanically induced Yki nuclear localisation also requires nutritionally induced insulin/insulin-like growth factor 1 (IGF-1) signalling (IIS) via phosphatidyl inositol-3-kinase (PI3K), phosphoinositide-dependent kinase 1 (PDK1 or PDPK1), and protein kinase B (Akt or PKB) in the follicular epithelium. We find similar results in the developing Drosophila wing, where Yki becomes nuclear in the mechanically stretched cells of the wing pouch during larval feeding, which induces IIS, but translocates to the cytoplasm upon cessation of feeding in the third instar stage. Inactivating Akt prevents nuclear Yki localisation in the wing disc, while ectopic activation of the insulin receptor, PI3K, or Akt/PKB is sufficient to maintain nuclear Yki in mechanically stimulated cells of the wing pouch even after feeding ceases. Finally, IIS also promotes YAP nuclear localisation in response to mechanical cues in mammalian skin epithelia. Thus, the Hippo pathway has a physiological function as an integrator of epithelial cell polarity, tissue mechanics, and nutritional cues to control cell proliferation and tissue growth in both Drosophila and mammals.

Topology of dividing planar tilings: Mitosis and order in epithelial tissues

We investigate a range of rule-based models of the in-plane structure of growing single-cell-thick epithelia represented by the distribution of frequencies of polygon classes. Within the Markovian framework introduced by Gibson et al. [Nature (London) 442, 1038 (2006)10.1038/nature05014], we discuss various topologically allowed cell division schemes assumed to control the structure of the tissue as well as a phenomenological Gaussian scheme, and we compute the stationary distributions for all of them. Some of the distributions reproduce those seen in tissues characterized by unbiased mitotic events but also in certain tissues with a preferred orientation of the mitotic plane or a cell-rearrangement process such as neighbor exchange. In addition, we propose the asynchronous-division variant of the model, which builds on the Lewis law and on the Aboav-Weaire law as well as on the fact that the dividing cells are larger than the resting cells. This generalization a posteriori validates the original model.

Cell-Cell Adhesion and Cortical Actin Bending Govern Cell Elongation on Negatively Curved Substrates

Physiologically, cells experience and respond to a variety of mechanical stimuli such as rigidity and topography of the extracellular matrix. However, little is known about the effects of substrate curvature on cell behavior. We developed a novel, to our knowledge, method to fabricate cell culture substrates with semicylindrical grooves of negative curvatures (radius of curvature, R_c = 20-100 μm). We found that negative substrate curvatures induced elongation of mesenchymal and epithelial cells along the cylinder axis. As R_c decreases, mesenchymal National Institutes of Health 3T3 fibroblasts increasingly elongate along the long axis of the grooves, whereas elongation of epithelial Madin-Darby Canine Kidney (MDCK) cells is biphasic with maximal cell elongation when R_c = 40 μm. Addition of blebbistatin to MDCK cells to reduce cortical actin rigidity resulted in a decrease in cell elongation across all curvatures while preserving the biphasic trend. However, addition of calyculin A or ethylene glycol-bis(2-aminoethylether)-N,N,N',N'-tetraacetic acid, to increase cortical rigidity or reduce intercellular adhesion, respectively, resulted in a monotonic increase in MDCK cell elongation with decreasing R_c. Using an energy minimization model, we showed that cell elongation in epithelial cell sheet is governed by the competition between two energies as R_c decreases: curvature-dependent intercellular adhesion that prevents elongation; and intracellular cortical actin bending that enhances elongation. Therefore, our results of cellular elongation induced by negatively curved substrates offer insights into how tubule elongation or growth of tubular structures such as kidney tubules can be controlled by the substrate curvature in vivo.

Statistics of noisy growth with mechanical feedback in elastic tissues

Tissue growth is a fundamental aspect of development and is intrinsically noisy. Stochasticity has important implications for morphogenesis, precise control of organ size, and regulation of tissue composition and heterogeneity. However, the basic statistical properties of growing tissues, particularly when growth induces mechanical stresses that can in turn affect growth rates, have received little attention. Here, we study the noisy growth of elastic sheets subject to mechanical feedback. Considering both isotropic and anisotropic growth, we find that the density-density correlation function shows power law scaling. We also consider the dynamics of marked, neutral clones of cells. We find that the areas (but not the shapes) of two clones are always statistically independent, even when they are adjacent. For anisotropic growth, we show that clone size variance scales like the average area squared and that the mode amplitudes characterizing clone shape show a slow [Formula: see text] decay, where n is the mode index. This is in stark contrast to the isotropic case, where relative variations in clone size and shape vanish at long times. The high variability in clone statistics observed in anisotropic growth is due to the presence of two soft modes-growth modes that generate no stress. Our results lay the groundwork for more in-depth explorations of the properties of noisy tissue growth in specific biological contexts.

Oriented Cell Divisions Are Not Required for Drosophila Wing Shape

Formation of correctly shaped organs is vital for normal function. The Drosophila wing has an elongated shape, which has been attributed in part to a preferential orientation of mitotic spindles along the proximal-distal axis [1, 2]. Orientation of mitotic spindles is believed to be a fundamental morphogenetic mechanism in multicellular organisms [3-6]. A contribution of spindle orientation to wing shape was inferred from observations that mutation of Dachsous-Fat pathway genes results in both rounder wings and loss of the normal proximal-distal bias in spindle orientation [1, 2, 7]. To directly evaluate the potential contribution of spindle orientation to wing morphogenesis, we assessed the consequences of loss of the Drosophila NuMA homolog Mud, which interacts with the dynein complex and has a conserved role in spindle orientation [8, 9]. Loss of Mud randomizes spindle orientation but does not alter wing shape. Analysis of growth and cell dynamics in developing discs and in ex vivo culture suggests that the absence of oriented cell divisions is compensated for by an increased contribution of cell rearrangements to wing shape. Our results indicate that oriented cell divisions are not required for wing morphogenesis, nor are they required for the morphogenesis of other Drosophila appendages. Moreover, our results suggest that normal organ shape is not achieved through locally specifying and then summing up individual cell behaviors, like oriented cell division. Instead, wing shape might be specified through tissue-wide stresses that dictate an overall arrangement of cells without specifying the individual cell behaviors needed to achieve it.

Development of epithelial tissues: How are cleavage planes chosen?

The cross-section of a cell in a monolayer epithelial tissue can be modeled mathematically as a k-sided polygon. Empirically studied distributions of the proportions of k-sided cells in epithelia show remarkable similarities in a wide range of evolutionarily distant organisms. A variety of mathematical models have been proposed for explaining this phenomenon. The highly parsimonious simulation model of (Patel et al., PLoS Comput. Biol., 2009) that takes into account only the number of sides of a given cell and cell division already achieves a remarkably good fit with empirical distributions from Drosophila, Hydra, Xenopus, Cucumber, and Anagallis. Within the same modeling framework as in that paper, we introduce additional options for the choice of the endpoints of the cleavage plane that appear to be biologically more realistic. By taking the same data sets as our benchmarks, we found that combinations of some of our new options consistently gave better fits with each of these data sets than previously studied ones. Both our algorithm and simulation data are made available as research tools for future investigations.

Approximate Bayesian computation reveals the importance of repeated measurements for parameterising cell-based models of growing tissues

The growth and dynamics of epithelial tissues govern many morphogenetic processes in embryonic development. A recent quantitative transition in data acquisition, facilitated by advances in genetic and live-imaging techniques, is paving the way for new insights to these processes. Computational models can help us understand and interpret observations, and then make predictions for future experiments that can distinguish between hypothesised mechanisms. Increasingly, cell-based modelling approaches such as vertex models are being used to help understand the mechanics underlying epithelial morphogenesis. These models typically seek to reproduce qualitative phenomena, such as cell sorting or tissue buckling. However, it remains unclear to what extent quantitative data can be used to constrain these models so that they can then be used to make quantitative, experimentally testable predictions. To address this issue, we perform an in silico study to investigate whether vertex model parameters can be inferred from imaging data, and explore methods to quantify the uncertainty of such estimates. Our approach requires the use of summary statistics to estimate parameters. Here, we focus on summary statistics of cellular packing and of laser ablation experiments, as are commonly reported from imaging studies. We find that including data from repeated experiments is necessary to generate reliable parameter estimates that can facilitate quantitative model predictions.

Geometric constraints alter cell arrangements within curved epithelial tissues

Organ and tissue formation are complex three-dimensional processes involving cell division, growth, migration, and rearrangement, all of which occur within physically constrained regions. However, analyzing such processes in three dimensions in vivo is challenging. Here, we focus on the process of cellularization in the anterior pole of the early Drosophila embryo to explore how cells compete for space under geometric constraints. Using microfluidics combined with fluorescence microscopy, we extract quantitative information on the three-dimensional epithelial cell morphology. We observed a cellular membrane rearrangement in which cells exchange neighbors along the apical-basal axis. Such apical-to-basal neighbor exchanges were observed more frequently in the anterior pole than in the embryo trunk. Furthermore, cells within the anterior pole skewed toward the trunk along their long axis relative to the embryo surface, with maximum skew on the ventral side. We constructed a vertex model for cells in a curved environment. We could reproduce the observed cellular skew in both wild-type embryos and embryos with distorted morphology. Further, such modeling showed that cell rearrangements were more likely in ellipsoidal, compared with cylindrical, geometry. Overall, we demonstrate that geometric constraints can influence three-dimensional cell morphology and packing within epithelial tissues.

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'.

A hybrid computational model to explore the topological characteristics of epithelial tissues

Epithelial tissues show a particular topology where cells resemble a polygon-like shape, but some biological processes can alter this tissue topology. During cell proliferation, mitotic cell dilation deforms the tissue and modifies the tissue topology. Additionally, cells are reorganized in the epithelial layer and these rearrangements also alter the polygon distribution. We present here a computer-based hybrid framework focused on the simulation of epithelial layer dynamics that combines discrete and continuum numerical models. In this framework, we consider topological and mechanical aspects of the epithelial tissue. Individual cells in the tissue are simulated by an off-lattice agent-based model, which keeps the information of each cell. In addition, we model the cell-cell interaction forces and the cell cycle. Otherwise, we simulate the passive mechanical behaviour of the cell monolayer using a material that approximates the mechanical properties of the cell. This continuum approach is solved by the finite element method, which uses a dynamic mesh generated by the triangulation of cell polygons. Forces generated by cell-cell interaction in the agent-based model are also applied on the finite element mesh. Cell movement in the agent-based model is driven by the displacements obtained from the deformed finite element mesh of the continuum mechanical approach. We successfully compare the results of our simulations with some experiments about the topology of proliferating epithelial tissues in Drosophila. Our framework is able to model the emergent behaviour of the cell monolayer that is due to local cell-cell interactions, which have a direct influence on the dynamics of the epithelial tissue.

Inference of Cell Mechanics in Heterogeneous Epithelial Tissue Based on Multivariate Clone Shape Quantification

Cell populations in multicellular organisms show genetic and non-genetic heterogeneity, even in undifferentiated tissues of multipotent cells during development and tumorigenesis. The heterogeneity causes difference of mechanical properties, such as, cell bond tension or adhesion, at the cell–cell interface, which determine the shape of clonal population boundaries via cell sorting or mixing. The boundary shape could alter the degree of cell–cell contacts and thus influence the physiological consequences of sorting or mixing at the boundary (e.g., tumor suppression or progression), suggesting that the cell mechanics could help clarify the physiology of heterogeneous tissues. While precise inference of mechanical tension loaded at each cell–cell contacts has been extensively developed, there has been little progress on how to distinguish the population-boundary geometry and identify the cause of geometry in heterogeneous tissues. We developed a pipeline by combining multivariate analysis of clone shape with tissue mechanical simulations. We examined clones with four different genotypes within Drosophila wing imaginal discs: wild-type, tartan (trn) overexpression, hibris (hbs) overexpression, and Eph RNAi. Although the clones were previously known to exhibit smoothed or convoluted morphologies, their mechanical properties were unknown. By applying a multivariate analysis to multiple criteria used to quantify the clone shapes based on individual cell shapes, we found the optimal criteria to distinguish not only among the four genotypes, but also non-genetic heterogeneity from genetic one. The efficient segregation of clone shape enabled us to quantitatively compare experimental data with tissue mechanical simulations. As a result, we identified the mechanical basis contributed to clone shape of distinct genotypes. The present pipeline will promote the understanding of the functions of mechanical interactions in heterogeneous tissue in a non-invasive manner.

Possible roles of mechanical cell elimination intrinsic to growing tissues from the perspective of tissue growth efficiency and homeostasis

Cell competition is a phenomenon originally described as the competition between cell populations with different genetic backgrounds; losing cells with lower fitness are eliminated. With the progress in identification of related molecules, some reports described the relevance of cell mechanics during elimination. Furthermore, recent live imaging studies have shown that even in tissues composed of genetically identical cells, a non-negligible number of cells are eliminated during growth. Thus, mechanical cell elimination (MCE) as a consequence of mechanical cellular interactions is an unavoidable event in growing tissues and a commonly observed phenomenon. Here, we studied MCE in a genetically-homogeneous tissue from the perspective of tissue growth efficiency and homeostasis. First, we propose two quantitative measures, cell and tissue fitness, to evaluate cellular competitiveness and tissue growth efficiency, respectively. By mechanical tissue simulation in a pure population where all cells have the same mechanical traits, we clarified the dependence of cell elimination rate or cell fitness on different mechanical/growth parameters. In particular, we found that geometrical (specifically, cell size) and mechanical (stress magnitude) heterogeneities are common determinants of the elimination rate. Based on these results, we propose possible mechanical feedback mechanisms that could improve tissue growth efficiency and density/stress homeostasis. Moreover, when cells with different mechanical traits are mixed (e.g., in the presence of phenotypic variation), we show that MCE could drive a drastic shift in cell trait distribution, thereby improving tissue growth efficiency through the selection of cellular traits, i.e. intra-tissue "evolution". Along with the improvement of growth efficiency, cell density, stress state, and phenotype (mechanical traits) were also shown to be homogenized through growth. More theoretically, we propose a mathematical model that approximates cell competition dynamics, by which the time evolution of tissue fitness and cellular trait distribution can be predicted without directly simulating a cell-based mechanical model.

Active Vertex Model for cell-resolution description of epithelial tissue mechanics

We introduce an Active Vertex Model (AVM) for cell-resolution studies of the mechanics of confluent epithelial tissues consisting of tens of thousands of cells, with a level of detail inaccessible to similar methods. The AVM combines the Vertex Model for confluent epithelial tissues with active matter dynamics. This introduces a natural description of the cell motion and accounts for motion patterns observed on multiple scales. Furthermore, cell contacts are generated dynamically from positions of cell centres. This not only enables efficient numerical implementation, but provides a natural description of the T1 transition events responsible for local tissue rearrangements. The AVM also includes cell alignment, cell-specific mechanical properties, cell growth, division and apoptosis. In addition, the AVM introduces a flexible, dynamically changing boundary of the epithelial sheet allowing for studies of phenomena such as the fingering instability or wound healing. We illustrate these capabilities with a number of case studies.

A new hypothesis for foregut and heart tube formation based on differential growth and actomyosin contraction

For decades, it was commonly thought that the bilateral heart fields in the early embryo fold directly towards the midline, where they meet and fuse to create the primitive heart tube. Recent studies have challenged this view, however, suggesting that the heart fields fold diagonally. As early foregut and heart tube morphogenesis are intimately related, this finding also raises questions concerning the traditional view of foregut formation. Here, we combine experiments on chick embryos with computational modeling to explore a new hypothesis for the physical mechanisms of heart tube and foregut formation. According to our hypothesis, differential anisotropic growth between mesoderm and endoderm drives diagonal folding. Then, active contraction along the anterior intestinal portal generates tension to elongate the foregut and heart tube. We test this hypothesis using biochemical perturbations of cell proliferation and contractility, as well as computational modeling based on nonlinear elasticity theory including growth and contraction. The present results generally support the view that differential growth and actomyosin contraction drive formation of the foregut and heart tube in the early chick embryo.

Vertex models: from cell mechanics to tissue morphogenesis

Tissue morphogenesis requires the collective, coordinated motion and deformation of a large number of cells. Vertex model simulations for tissue mechanics have been developed to bridge the scales between force generation at the cellular level and tissue deformation and flows. We review here various formulations of vertex models that have been proposed for describing tissues in two and three dimensions. We discuss a generic formulation using a virtual work differential, and we review applications of vertex models to biological morphogenetic processes. We also highlight recent efforts to obtain continuum theories of tissue mechanics, which are effective, coarse-grained descriptions of vertex models.This article is part of the themed issue 'Systems morphodynamics: understanding the development of tissue hardware'.

Modelling apical columnar epithelium mechanics from circumferential contractile fibres

Simple columnar epithelia are formed by individual epithelial cells connecting together to form single cell high sheets. They are a main component of many important body tissues and are heavily involved in both normal and cancerous cell activities. Prior experimental observations have identified a series of contractile fibres around the circumference of a cross section located in the upper (apical) region of each cell. While other potential mechanisms have been identified in both the experimental and theoretical literature, these circumferential fibres are considered to be the most likely mechanism controlling movement of this cross section. Here, we investigated the impact of circumferential contractile fibres on movement of the cross section by creating an alternate model where movement is driven from circumferential contractile fibres, without any other potential mechanisms. In this model, we utilised a circumferential contractile fibre representation based on investigations into the movement of contractile fibres as an individual system, treated circumferential fibres as a series of units, and matched our model simulation to experimental geometries. By testing against laser ablation datasets sourced from existing literature, we found that circumferential fibres can reproduce the majority of cross-sectional movements. We also investigated model predictions related to various aspects of cross-sectional movement, providing insights into epithelium mechanics and demonstrating the usefulness of our modelling approach.

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.

Local Cell Death Changes the Orientation of Cell Division in the Developing Drosophila Wing Imaginal Disc Without Using Fat or Dachsous as Orienting Signals

Drosophila imaginal disc cells exhibit preferred cell division orientations according to location within the disc. These orientations are altered if cell death occurs within the epithelium, such as is caused by cell competition or by genotypes affecting cell survival. Both normal cell division orientations, and their orientations after cell death, depend on the Fat-Dachsous pathway of planar cell polarity (PCP). The hypothesis that cell death initiates a planar polarity signal was investigated. When clones homozygous for the pineapple eye (pie) mutation were made to initiate cell death, neither Dachsous nor Fat was required in pie cells for the re-orientation of nearby cells, indicating a distinct signal for this PCP pathway. Dpp and Wg were also not needed for pie clones to re-orient cell division. Cell shapes were evaluated in wild type and mosaic wing discs to assess mechanical consequences of cell loss. Although proximal wing disc cells and cells close to the dorso-ventral boundary were elongated in their preferred cell division axes in wild type discs, cell shapes in much of the wing pouch were symmetrical on average and did not predict their preferred division axis. Cells in pie mutant clones were slightly larger than their normal counterparts, consistent with mechanical stretching following cell loss, but no bias in cell shape was detected in the surrounding cells. These findings indicate that an unidentified signal influences PCP-dependent cell division orientation in imaginal discs.

Robust cell tracking in epithelial tissues through identification of maximum common subgraphs

Tracking of cells in live-imaging microscopy videos of epithelial sheets is a powerful tool for investigating fundamental processes in embryonic development. Characterizing cell growth, proliferation, intercalation and apoptosis in epithelia helps us to understand how morphogenetic processes such as tissue invagination and extension are locally regulated and controlled. Accurate cell tracking requires correctly resolving cells entering or leaving the field of view between frames, cell neighbour exchanges, cell removals and cell divisions. However, current tracking methods for epithelial sheets are not robust to large morphogenetic deformations and require significant manual interventions. Here, we present a novel algorithm for epithelial cell tracking, exploiting the graph-theoretic concept of a 'maximum common subgraph' to track cells between frames of a video. Our algorithm does not require the adjustment of tissue-specific parameters, and scales in sub-quadratic time with tissue size. It does not rely on precise positional information, permitting large cell movements between frames and enabling tracking in datasets acquired at low temporal resolution due to experimental constraints such as phototoxicity. To demonstrate the method, we perform tracking on the Drosophila embryonic epidermis and compare cell-cell rearrangements to previous studies in other tissues. Our implementation is open source and generally applicable to epithelial tissues.

A Mechanistic Collective Cell Model for Epithelial Colony Growth and Contact Inhibition

We present a mechanistic hybrid continuum-discrete model to simulate the dynamics of epithelial cell colonies. Collective cell dynamics are modeled using continuum equations that capture plastic, viscoelastic, and elastic deformations in the clusters while providing single-cell resolution. The continuum equations can be viewed as a coarse-grained version of previously developed discrete models that treat epithelial clusters as a two-dimensional network of vertices or stochastic interacting particles and follow the framework of dynamic density functional theory appropriately modified to account for cell size and shape variability. The discrete component of the model implements cell division and thus influences cell size and shape that couple to the continuum component. The model is validated against recent in vitro studies of epithelial cell colonies using Madin-Darby canine kidney cells. In good agreement with experiments, we find that mechanical interactions and constraints on the local expansion of cell size cause inhibition of cell motion and reductive cell division. This leads to successively smaller cells and a transition from exponential to quadratic growth of the colony that is associated with a constant-thickness rim of growing cells at the cluster edge, as well as the emergence of short-range ordering and solid-like behavior. A detailed analysis of the model reveals a scale invariance of the growth and provides insight into the generation of stresses and their influence on the dynamics of the colonies. Compared to previous models, our approach has several advantages: it is independent of dimension, it can be parameterized using classical elastic properties (Poisson's ratio and Young's modulus), and it can easily be extended to incorporate multiple cell types and general substrate geometries.

Looking Beyond the Genes: The Interplay Between Signaling Pathways and Mechanics in the Shaping and Diversification of Epithelial Tissues

The core of Evo-Devo lies in the intuition that the way tissues grow during embryonic development, the way they sustain their structure and function throughout lifetime, and the way they evolve are closely linked. Epithelial tissues are ubiquitous in metazoans, covering the gut and internal branched organs, as well as the skin and its derivatives (ie, teeth). Here, we discuss in vitro, in vivo, and in silico studies on epithelial tissues to illustrate the conserved, dynamical, and complex aspects of their development. We then explore the implications of the dynamical and nonlinear nature of development on the evolution of their size and shape at the phenotypic and genetic levels. In rare cases, when the interplay between signaling and mechanics is well understood at the cell level, it is becoming clear that the structure of development leads to covariation of characters, an integration which in turn provides some predictable structure to evolutionary changes. We suggest that such nonlinear systems are prone to genetic drift, cryptic genetic variation, and context-dependent mutational effects. We argue that experimental and theoretical studies at the cell level are critical to our understanding of the phenotypic and genetic evolution of epithelial tissues, including carcinomas.

Fundamental physical cellular constraints drive self-organization of tissues

Morphogenesis is driven by small cell shape changes that modulate tissue organization. Apical surfaces of proliferating epithelial sheets have been particularly well studied. Currently, it is accepted that a stereotyped distribution of cellular polygons is conserved in proliferating tissues among metazoans. In this work, we challenge these previous findings showing that diverse natural packed tissues have very different polygon distributions. We use Voronoi tessellations as a mathematical framework that predicts this diversity. We demonstrate that Voronoi tessellations and the very different tissues analysed share an overriding restriction: the frequency of polygon types correlates with the distribution of cell areas. By altering the balance of tensions and pressures within the packed tissues using disease, genetic or computer model perturbations, we show that as long as packed cells present a balance of forces within tissue, they will be under a physical constraint that limits its organization. Our discoveries establish a new framework to understand tissue architecture in development and disease.

Capabilities and Limitations of Tissue Size Control through Passive Mechanical Forces

Embryogenesis is an extraordinarily robust process, exhibiting the ability to control tissue size and repair patterning defects in the face of environmental and genetic perturbations. The size and shape of a developing tissue is a function of the number and size of its constituent cells as well as their geometric packing. How these cellular properties are coordinated at the tissue level to ensure developmental robustness remains a mystery; understanding this process requires studying multiple concurrent processes that make up morphogenesis, including the spatial patterning of cell fates and apoptosis, as well as cell intercalations. In this work, we develop a computational model that aims to understand aspects of the robust pattern repair mechanisms of the Drosophila embryonic epidermal tissues. Size control in this system has previously been shown to rely on the regulation of apoptosis rather than proliferation; however, to date little work has been done to understand the role of cellular mechanics in this process. We employ a vertex model of an embryonic segment to test hypotheses about the emergence of this size control. Comparing the model to previously published data across wild type and genetic perturbations, we show that passive mechanical forces suffice to explain the observed size control in the posterior (P) compartment of a segment. However, observed asymmetries in cell death frequencies across the segment are demonstrated to require patterning of cellular properties in the model. Finally, we show that distinct forms of mechanical regulation in the model may be distinguished by differences in cell shapes in the P compartment, as quantified through experimentally accessible summary statistics, as well as by the tissue recoil after laser ablation experiments.

Cell shapes and patterns as quantitative indicators of tissue stress in the plant epidermis

In a confluent, single-cell tissue layer, we show that cell shapes and statistics correlate directly with the tissue's mechanical properties, described by an energy functional with generic interfacial terms only. Upon increasing the cohesive component of the model, we observe a clear transition from a tense state with isotropic cells to a relaxed state with anisotropic cells. Signatures of the transition are present in the interfacial mechanics, the domain geometry, and the domain statistics, thus linking all three fields of study. This transition persists for all cell size distributions, but its exact position is crucially dependent on fluctuations in the parameter values of the functional (quenched disorder). The magnitude of fluctuations can be matched to the observed shape distribution of cells, so that visual observation of cell shapes and statistics provides information about the mechanical state of the tissue. Comparing with experimental data from the Cucumis epidermis, we find that the system is located right at the transition, allowing the tissue to relieve most of the local stress while maintaining integrity.

Patterning of wound-induced intercellular Ca(2+) flashes in a developing epithelium

Differential mechanical force distributions are increasingly recognized to provide important feedback into the control of an organ's final size and shape. As a second messenger that integrates and relays mechanical information to the cell, calcium ions (Ca(2+)) are a prime candidate for providing important information on both the overall mechanical state of the tissue and resulting behavior at the individual-cell level during development. Still, how the spatiotemporal properties of Ca(2+) transients reflect the underlying mechanical characteristics of tissues is still poorly understood. Here we use an established model system of an epithelial tissue, the Drosophila wing imaginal disc, to investigate how tissue properties impact the propagation of Ca(2+) transients induced by laser ablation. The resulting intercellular Ca(2+) flash is found to be mediated by inositol 1,4,5-trisphosphate and depends on gap junction communication. Further, we find that intercellular Ca(2+) transients show spatially non-uniform characteristics across the proximal-distal axis of the larval wing imaginal disc, which exhibit a gradient in cell size and anisotropy. A computational model of Ca(2+) transients is employed to identify the principle factors explaining the spatiotemporal patterning dynamics of intercellular Ca(2+) flashes. The relative Ca(2+) flash anisotropy is principally explained by local cell shape anisotropy. Further, Ca(2+) velocities are relatively uniform throughout the wing disc, irrespective of cell size or anisotropy. This can be explained by the opposing effects of cell diameter and cell elongation on intercellular Ca(2+) propagation. Thus, intercellular Ca(2+) transients follow lines of mechanical tension at velocities that are largely independent of tissue heterogeneity and reflect the mechanical state of the underlying tissue.

Large Scale Tissue Morphogenesis Simulation on Heterogenous Systems Based on a Flexible Biomechanical Cell Model

The complexity of biological tissue morphogenesis makes in silico simulations of such system very interesting in order to gain a better understanding of the underlying mechanisms ruling the development of multicellular tissues. This complexity is mainly due to two elements: firstly, biological tissues comprise a large amount of cells; secondly, these cells exhibit complex interactions and behaviors. To address these two issues, we propose two tools: the first one is a virtual cell model that comprise two main elements: firstly, a mechanical structure (membrane, cytoskeleton, and cortex) and secondly, the main behaviors exhibited by biological cells, i.e., mitosis, growth, differentiation, molecule consumption, and production as well as the consideration of the physical constraints issued from the environment. An artificial chemistry is also included in the model. This virtual cell model is coupled to an agent-based formalism. The second tool is a simulator that relies on the OpenCL framework. It allows efficient parallel simulations on heterogenous devices such as micro-processors or graphics processors. We present two case studies validating the implementation of our model in our simulator: cellular proliferation controlled by cell signalling and limb growth in a virtual organism.

Mechanical model of geometric cell and topological algorithm for cell dynamics from single-cell to formation of monolayered tissues with pattern

Geometric and mechanical properties of individual cells and interactions among neighboring cells are the basis of formation of tissue patterns. Understanding the complex interplay of cells is essential for gaining insight into embryogenesis, tissue development, and other emerging behavior. Here we describe a cell model and an efficient geometric algorithm for studying the dynamic process of tissue formation in 2D (e.g. epithelial tissues). Our approach improves upon previous methods by incorporating properties of individual cells as well as detailed description of the dynamic growth process, with all topological changes accounted for. Cell size, shape, and division plane orientation are modeled realistically. In addition, cell birth, cell growth, cell shrinkage, cell death, cell division, cell collision, and cell rearrangements are now fully accounted for. Different models of cell-cell interactions, such as lateral inhibition during the process of growth, can be studied in detail. Cellular pattern formation for monolayered tissues from arbitrary initial conditions, including that of a single cell, can also be studied in detail. Computational efficiency is achieved through the employment of a special data structure that ensures access to neighboring cells in constant time, without additional space requirement. We have successfully generated tissues consisting of more than 20,000 cells starting from 2 cells within 1 hour. We show that our model can be used to study embryogenesis, tissue fusion, and cell apoptosis. We give detailed study of the classical developmental process of bristle formation on the epidermis of D. melanogaster and the fundamental problem of homeostatic size control in epithelial tissues. Simulation results reveal significant roles of solubility of secreted factors in both the bristle formation and the homeostatic control of tissue size. Our method can be used to study broad problems in monolayered tissue formation. Our software is publicly available.

The role of mathematical models in understanding pattern formation in developmental biology

In a Wall Street Journal article published on April 5, 2013, E. O. Wilson attempted to make the case that biologists do not really need to learn any mathematics-whenever they run into difficulty with numerical issues, they can find a technician (aka mathematician) to help them out of their difficulty. He formalizes this in Wilsons Principle No. 1: "It is far easier for scientists to acquire needed collaboration from mathematicians and statisticians than it is for mathematicians and statisticians to find scientists able to make use of their equations." This reflects a complete misunderstanding of the role of mathematics in all sciences throughout history. To Wilson, mathematics is mere number crunching, but as Galileo said long ago, "The laws of Nature are written in the language of mathematics[Formula: see text] the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word." Mathematics has moved beyond the geometry-based model of Galileo's time, and in a rebuttal to Wilson, E. Frenkel has pointed out the role of mathematics in synthesizing the general principles in science (Both point and counter-point are available in Wilson and Frenkel in Notices Am Math Soc 60(7):837-838, 2013). We will take this a step further and show how mathematics has been used to make new and experimentally verified discoveries in developmental biology and how mathematics is essential for understanding a problem that has puzzled experimentalists for decades-that of how organisms can scale in size. Mathematical analysis alone cannot "solve" these problems since the validation lies at the molecular level, but conversely, a growing number of questions in biology cannot be solved without mathematical analysis and modeling. Herein, we discuss a few examples of the productive intercourse between mathematics and biology.

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.

Cell division and the maintenance of epithelial order

Epithelia are polarized layers of adherent cells that are the building blocks for organ and appendage structures throughout animals. To preserve tissue architecture and barrier function during both homeostasis and rapid growth, individual epithelial cells divide in a highly constrained manner. Building on decades of research focused on single cells, recent work is probing the mechanisms by which the dynamic process of mitosis is reconciled with the global maintenance of epithelial order during development. These studies reveal how symmetrically dividing cells both exploit and conform to tissue organization to orient their mitotic spindles during division and establish new adhesive junctions during cytokinesis.

Multiscale Modeling of Cellular Epigenetic States: Stochasticity in Molecular Networks, Chromatin Folding in Cell Nuclei, and Tissue Pattern Formation of Cells

Genome sequences provide the overall genetic blueprint of cells, but cells possessing the same genome can exhibit diverse phenotypes. There is a multitude of mechanisms controlling cellular epigenetic states and that dictate the behavior of cells. Among these, networks of interacting molecules, often under stochastic control, depending on the specific wirings of molecular components and the physiological conditions, can have a different landscape of cellular states. In addition, chromosome folding in three-dimensional space provides another important control mechanism for selective activation and repression of gene expression. Fully differentiated cells with different properties grow, divide, and interact through mechanical forces and communicate through signal transduction, resulting in the formation of complex tissue patterns. Developing quantitative models to study these multi-scale phenomena and to identify opportunities for improving human health requires development of theoretical models, algorithms, and computational tools. Here we review recent progress made in these important directions.