Computer simulations of cell sorting due to differential adhesion

A Cellular Potts Model of the interplay of synchronization and aggregation

We investigate the behavior of systems of cells with intracellular molecular oscillators ("clocks") where cell-cell adhesion is mediated by differences in clock phase between neighbors. This is motivated by phenomena in developmental biology and in aggregative multicellularity of unicellular organisms. In such systems, aggregation co-occurs with clock synchronization. To account for the effects of spatially extended cells, we use the Cellular Potts Model (CPM), a lattice agent-based model. We find four distinct possible phases: global synchronization, local synchronization, incoherence, and anti-synchronization (checkerboard patterns). We characterize these phases via order parameters. In the case of global synchrony, the speed of synchronization depends on the adhesive effects of the clocks. Synchronization happens fastest when cells in opposite phases adhere the strongest ("opposites attract"). When cells of the same clock phase adhere the strongest ("like attracts like"), synchronization is slower. Surprisingly, the slowest synchronization happens in the diffusive mixing case, where cell-cell adhesion is independent of clock phase. We briefly discuss potential applications of the model, such as pattern formation in the auditory sensory epithelium.

Cellular segregation in cocultures is driven by differential adhesion and contractility on distinct timescales

Cellular sorting and pattern formation are crucial for many biological processes such as development, tissue regeneration, and cancer progression. Prominent physical driving forces for cellular sorting are differential adhesion and contractility. Here, we studied the segregation of epithelial cocultures containing highly contractile, ZO1/2-depleted MDCKII cells (dKD) and their wild-type (WT) counterparts using multiple quantitative, high-throughput methods to monitor their dynamical and mechanical properties. We observe a time-dependent segregation process governed mainly by differential contractility on short (<5 h) and differential adhesion on long (>5 h) timescales. The overly contractile dKD cells exert strong lateral forces on their WT neighbors, thereby apically depleting their surface area. Concomitantly, the tight junction-depleted, contractile cells exhibit weaker cell-cell adhesion and lower traction force. Drug-induced contractility reduction and partial calcium depletion delay the initial segregation but cease to change the final demixed state, rendering differential adhesion the dominant segregation force at longer timescales. This well-controlled model system shows how cell sorting is accomplished through a complex interplay between differential adhesion and contractility and can be explained largely by generic physical driving forces.

A computational modeling approach for predicting multicell spheroid patterns based on signaling-induced differential adhesion

Physiological and pathological processes including embryogenesis and tumorigenesis rely on the ability of individual cells to work collectively to form multicell patterns. In these heterogeneous multicell systems, cell-cell signaling induces differential adhesion between cells that leads to tissue-level patterning. However, the sensitivity of pattern formation to changes in the strengths of signaling or cell adhesion processes is not well understood. Prior work has explored these issues using synthetically engineered heterogeneous multicell spheroid systems, in which cell subpopulations engage in bidirectional intercellular signaling to regulate the expression of different cadherins. While engineered cell systems provide excellent experimental tools to observe pattern formation in cell populations, computational models of these systems may be leveraged to explore more systematically how specific combinations of signaling and adhesion parameters can drive the emergence of unique patterns. We developed and validated two- and three-dimensional agent-based models (ABMs) of spheroid patterning for previously described cells engineered with a bidirectional signaling circuit that regulates N- and P-cadherin expression. Systematic exploration of model predictions, some of which were experimentally validated, revealed how cell seeding parameters, the order of signaling events, probabilities of induced cadherin expression, and homotypic adhesion strengths affect pattern formation. Unsupervised clustering was also used to map combinations of signaling and adhesion parameters to these unique spheroid patterns predicted by the ABM. Finally, we demonstrated how the model may be deployed to design new synthetic cell signaling circuits based on a desired final multicell pattern.

Adhesion-Based Self-Organization in Tissue Patterning

Since the proposal of the differential adhesion hypothesis, scientists have been fascinated by how cell adhesion mediates cellular self-organization to form spatial patterns during development. The search for molecular tool kits with homophilic binding specificity resulted in a diverse repertoire of adhesion molecules. Recent understanding of the dominant role of cortical tension over adhesion binding redirects the focus of differential adhesion studies to the signaling function of adhesion proteins to regulate actomyosin contractility. The broader framework of differential interfacial tension encompasses both adhesion and nonadhesion molecules, sharing the common function of modulating interfacial tension during cell sorting to generate diverse tissue patterns. Robust adhesion-based patterning requires close coordination between morphogen signaling, cell fate decisions, and changes in adhesion. Current advances in bridging theoretical and experimental approaches present exciting opportunities to understand molecular, cellular, and tissue dynamics during adhesion-based tissue patterning across multiple time and length scales.

Is cell segregation like oil and water: Asymptotic versus transitory regime

Understanding the segregation of cells is crucial to answer questions about tissue formation in embryos or tumor progression. Steinberg proposed that separation of cells can be compared to the separation of two liquids. Such a separation is well described by the Cahn-Hilliard (CH) equations and the segregation indices exhibit an algebraic decay with exponent 1/3 with respect to time. Similar exponents are also observed in cell-based models. However, the scaling behavior in these numerical models is usually only examined in the asymptotic regime and these models have not been directly applied to actual cell segregation data. In contrast, experimental data also reveals other scaling exponents and even slow logarithmic scaling laws. These discrepancies are commonly attributed to the effects of collective motion or velocity-dependent interactions. By calibrating a 2D cellular automaton (CA) model which efficiently implements a dynamic variant of the differential adhesion hypothesis to 2D experimental data from Méhes et al., we reproduce the biological cell segregation experiments with just adhesive forces. The segregation in the cellular automaton model follows a logarithmic scaling initially, which is in contrast to the proposed algebraic scaling with exponent 1/3. However, within the less than two orders of magnitudes in time which are observable in the experiments, a logarithmic scaling may appear as a pseudo-algebraic scaling. In particular, we demonstrate that the cellular automaton model can exhibit a range of exponents ≤1/3 for such a pseudo-algebraic scaling. Moreover, the time span of the experiment falls into the transitory regime of the cellular automaton rather than the asymptotic one. We additionally develop a method for the calibration of the 2D Cahn-Hilliard model and find a match with experimental data within the transitory regime of the Cahn-Hilliard model with exponent 1/4. On the one hand this demonstrates that the transitory behavior is relevant for the experiment rather than the asymptotic one. On the other hand this corroborates the ambiguity of the scaling behavior, when segregation processes can be only observed on short time spans.

Segregation of different cell types is a crucial process for the pattern formation in tissues, in particular during embryogenesis. Since the involved cell interactions are complex and difficult to measure individually in experiments, mathematical modelling plays an increasingly important role to unravel the mechanisms governing segregation. The analysis of these theoretical models focuses mainly on the asymptotic behavior at large times, in a steady regime and for large numbers of cells. Most famously, cell-segregation models based on the minimization of the total surface energy, a mechanism also driving the demixing of immiscible fluids, are known to exhibit asymptotically a particular algebraic scaling behavior. However, it is not clear, whether the asymptotic regime of the numerical models is relevant at the spatio-temporal scales of actual biological processes and in-vitro experiments. By developing a mapping between 2D cell-based models and experimental settings, we are able to directly compare previous experimental data to numerical simulations of cell segregation quantitatively. We demonstrate that the experiments are reproduced by the transitory regime of the models rather than the asymptotic one. Our work puts a new perspective on previous model-driven conclusions on cell segregation mechanisms.

Stem cell-derived synthetic embryos self-assemble by exploiting cadherin codes and cortical tension

Mammalian embryos sequentially differentiate into trophectoderm and an inner cell mass, the latter of which differentiates into primitive endoderm and epiblast. Trophoblast stem (TS), extraembryonic endoderm (XEN) and embryonic stem (ES) cells derived from these three lineages can self-assemble into synthetic embryos, but the mechanisms remain unknown. Here, we show that a stem cell-specific cadherin code drives synthetic embryogenesis. The XEN cell cadherin code enables XEN cell sorting into a layer below ES cells, recapitulating the sorting of epiblast and primitive endoderm before implantation. The TS cell cadherin code enables TS cell sorting above ES cells, resembling extraembryonic ectoderm clustering above epiblast following implantation. Whereas differential cadherin expression drives initial cell sorting, cortical tension consolidates tissue organization. By optimizing cadherin code expression in different stem cell lines, we tripled the frequency of correctly formed synthetic embryos. Thus, by exploiting cadherin codes from different stages of development, lineage-specific stem cells bypass the preimplantation structure to directly assemble a postimplantation embryo.

Mathematical and computational models in spheroid-based biofabrication

Ubiquitous in embryonic development, tissue fusion is of interest to tissue engineers who use tissue spheroids or organoids as building blocks of three-dimensional (3D) multicellular constructs. This review presents mathematical models and computer simulations of the fusion of tissue spheroids. The motivation of this study stems from the need to predict the post-printing evolution of 3D bioprinted constructs. First, we provide a brief overview of differential adhesion, the main morphogenetic mechanism involved in post-printing structure formation. It will be shown that clusters of cohesive cells behave as an incompressible viscous fluid on the time scale of hours. The discussion turns then to mathematical models based on the continuum hydrodynamics of highly viscous liquids and on statistical mechanics. Next, we analyze the validity and practical use of computational models of multicellular self-assembly in live constructs created by tissue spheroid bioprinting. Finally, we discuss the perspectives of the field as machine learning starts to reshape experimental design, and modular robotic workstations tend to alleviate the burden of repetitive tasks in biofabrication. STATEMENT OF SIGNIFICANCE: Bioprinted constructs are living systems, which evolve via morphogenetic mechanisms known from developmental biology. This review presents mathematical and computational tools devised for modeling post-printing structure formation. They help achieving a desirable outcome without expensive optimization experiments. While previous reviews mainly focused on assumptions, technical details, strengths, and limitations of computational models of multicellular self-assembly, this article discusses their validity and practical use in biofabrication. It also presents an overview of mathematical models that proved to be useful in the evaluation of experimental data on tissue spheroid fusion, and in the calibration of computational models. Finally, the perspectives of the field are discussed in the advent of robotic biofabrication platforms and bioprinting process optimization by machine learning.

Matrix adhesion and remodeling diversifies modes of cancer invasion across spatial scales

The metastasis of malignant epithelial tumors begins with the egress of transformed cells from the confines of their basement membrane (BM) to their surrounding collagen-rich stroma. Invasion can be morphologically diverse: when breast cancer cells are separately cultured within BM-like matrix, collagen I (Coll I), or a combination of both, they exhibit collective-, dispersed mesenchymal-, and a mixed collective-dispersed (multimodal)- invasion, respectively. In this paper, we asked how distinct these invasive modes are with respect to the cellular and microenvironmental cues that drive them. A rigorous computational exploration of invasion was performed within an experimentally motivated Cellular Potts-based modeling environment. The model comprised of adhesive interactions between cancer cells, BM- and Coll I-like extracellular matrix (ECM), and reaction-diffusion-based remodeling of ECM. The model outputs were parameters cognate to dispersed- and collective- invasion. A clustering analysis of the output distribution curated through a careful examination of subsumed phenotypes suggested at least four distinct invasive states: dispersed, papillary-collective, bulk-collective, and multimodal, in addition to an indolent/non-invasive state. Mapping input values to specific output clusters suggested that each of these invasive states are specified by distinct input signatures of proliferation, adhesion and ECM remodeling. In addition, specific input perturbations allowed transitions between the clusters and revealed the variation in the robustness between the invasive states. Our systems-level approach proffers quantitative insights into how the diversity in ECM microenvironments may steer invasion into diverse phenotypic modes during early dissemination of breast cancer and contributes to tumor heterogeneity.

A possible billion-year-old holozoan with differentiated multicellularity

Sediments of the Torridonian sequence of the Northwest Scottish Highlands contain a wide array of microfossils, documenting life in a non-marine setting a billion years ago (1 Ga).^1-4 Phosphate nodules from the Diabaig Formation at Loch Torridon preserve microorganisms with cellular-level fidelity,^5,6 allowing for partial reconstruction of the developmental stages of a new organism, Bicellum brasieri gen. et sp. nov. The mature form of Bicellum consists of a solid, spherical ball of tightly packed cells (a stereoblast) of isodiametric cells enclosed in a monolayer of elongated, sausage-shaped cells. However, two populations of naked stereoblasts show mixed cell shapes, which we infer to indicate incipient development of elongated cells that were migrating to the periphery of the cell mass. These simple morphogenetic movements could be explained by differential cell-cell adhesion.^7,8 In fact, the basic morphology of Bicellum is topologically similar to that of experimentally produced cell masses that were shown to spontaneously segregate into two distinct domains based on differential cadherin-based cell adhesion.⁹ The lack of rigid cell walls in the stereoblast renders an algal affinity for Bicellum unlikely: its overall morphology is more consistent with a holozoan origin. Unicellular holozoans are known today to form multicellular stages within complex life cycles,^10-13 so the occurrence of such simple levels of transient multicellularity seen here is consistent with a holozoan affinity. Regardless of precise phylogenetic placement, these fossils demonstrate simple cell differentiation and morphogenic processes that are similar to those seen in some metazoans today.

Continuous Dynamic Modeling of Regulated Cell Adhesion: Sorting, Intercalation, and Involution

Cell-cell adhesion is essential for tissue growth and multicellular pattern formation and crucial for the cellular dynamics during embryogenesis and cancer progression. Understanding the dynamical gene regulation of cell adhesion molecules (CAMs) responsible for the emerging spatial tissue behaviors is a current challenge because of the complexity of these nonlinear interactions and feedback loops at different levels of abstraction-from genetic regulation to whole-organism shape formation. To extend our understanding of cell and tissue behaviors due to the regulation of adhesion molecules, here we present a novel, to our knowledge, model for the spatial dynamics of cellular patterning, growth, and shape formation due to the differential expression of CAMs and their regulation. Capturing the dynamic interplay between genetic regulation, CAM expression, and differential cell adhesion, the proposed continuous model can explain the complex and emergent spatial behaviors of cell populations that change their adhesion properties dynamically because of inter- and intracellular genetic regulation. This approach can demonstrate the mechanisms responsible for classical cell-sorting behaviors, cell intercalation in proliferating populations, and the involution of germ layer cells induced by a diffusing morphogen during gastrulation. The model makes predictions on the physical parameters controlling the amplitude and wavelength of a cellular intercalation interface, as well as the crucial role of N-cadherin regulation for the involution and migration of cells beyond the gradient of the morphogen Nodal during zebrafish gastrulation. Integrating the emergent spatial tissue behaviors with the regulation of genes responsible for essential cellular properties such as adhesion will pave the way toward understanding the genetic regulation of large-scale complex patterns and shapes formation in developmental, regenerative, and cancer biology.

Tuning intercellular adhesion with membrane-anchored oligonucleotides

Adhesive interactions between cells play an integral role in development, differentiation and regeneration. Existing methods for controlling cell-cell cohesion and adhesion by manipulating protein expression are constrained by biological interdependencies, e.g. coupling of cadherins to actomyosin force-feedback mechanisms. We use oligonucleotides conjugated to PEGylated lipid anchors (ssDNAPEGDPPE) to introduce artificial cell-cell adhesion that is largely decoupled from the internal cytoskeleton. We describe cell-cell doublets with a mechanical model based on isotropic, elastic deformation of spheres to estimate the adhesion at the cell-cell interface. Physical manipulation of adhesion by modulating the PEG-lipid to ssDNAPEGDPPE ratio, and conversely treating with actin-depolymerizing cytochalasin D, resulted in decreases and increases in doublet contact area, respectively. Our data are relevant to the ongoing discussion over mechanisms of tissue surface tension and in agreement with models based on opposing cortical and cohesive forces. PEG-lipid modulation of doublet geometries resulted in a well-defined curve indicating continuity, enabling prescriptive calibration for controlling doublet geometry. Our study demonstrates tuning of basic doublet adhesion, laying the foundation for more complex multicellular adhesion control independent of protein expression.

Synthetic development: building mammalian multicellular structures with artificial genetic programs

Synthetic biology efforts began in simple single-cell systems, which were relatively easy to manipulate genetically (Cameron et al., 2014). The field grew exponentially in the last two decades, and one of the latest frontiers are synthetic developmental programs for multicellular mammalian systems (Black et al., 2017; Wieland and Fussenegger, 2012) to genetically control features such as patterning or morphogenesis. These programs rely on engineered cell-cell communications, multicellular gene regulatory networks and effector genes. Here, we contextualize the first of these synthetic developmental programs, examine molecular and computational tools that can be used to generate next generation versions, and present the general logic that underpins these approaches. These advances are exciting as they represent a novel way to address both control and understanding in the field of developmental biology and tissue development (Elowitz and Lim, 2010; Velazquez et al., 2018; White et al., 2018; Morsut, 2017). This field is just at the beginning, and it promises to be of major interest in the upcoming years of biomedical research.

An Interplay Between Reaction-Diffusion and Cell-Matrix Adhesion Regulates Multiscale Invasion in Early Breast Carcinomatosis

The progression of cancer in the breast involves multiple reciprocal interactions between malignantly transformed epithelia, surrounding untransformed but affected stromal cells, and the extracellular matrix (ECM) that is remodeled during the process. A quantitative understanding of the relative contribution of such interactions to phenotypes associated with cancer cells can be arrived at through the construction of increasingly complex experimental and computational models. Herein, we introduce a multiscale three-dimensional (3D) organo- and pathotypic experimental assay that approximates, to an unprecedented extent, the histopathological complexity of a tumor disseminating into its surrounding stromal milieu via both bulk and solitary motility dynamics. End point and time-lapse microscopic observations of this assay allow us to study the earliest steps of cancer invasion as well as the dynamical interactions between the epithelial and stromal compartments. We then simulate our experimental observations using the modeling environment Compucell3D that is based on the Glazier–Graner–Hogeweg model. The computational model, which comprises adhesion between cancer cells and the matrices, cell proliferation and apoptosis, and matrix remodeling through reaction–diffusion–based morphogen dynamics, is first trained to phenocopy controls run with the experimental model, wherein one or the other matrices have been removed. The trained computational model successfully predicts phenotypes of the experimental counterparts that are subjected to pharmacological treatments (inhibition of N-linked glycosylation and matrix metalloproteinase activity) and scaffold modulation (alteration of collagen density). Further parametric exploration-based simulations suggest that specific permissive regimes of cell–cell and cell–matrix adhesions, operating in the context of a reaction–diffusion–regulated ECM dynamics, promote multiscale invasion of breast cancer cells and determine the extent to which the latter migrate through their surrounding stroma.

Rotation of sex combs in Drosophila melanogaster requires precise and coordinated spatio-temporal dynamics from forces generated by epithelial cells

The morphogenesis of sex combs (SCs), a male trait in many species of fruit flies, is an excellent system in which to study the cell biology, genetics and evolution of a trait. In Drosophila melanogaster, where the incipient SC rotates from horizontal to a vertical position, three signal comb properties have been documented: length, final angle and shape (linearity). During SC rotation, in which many cellular processes are occurring both spatially and temporally, it is difficult to distinguish which processes are crucial for which attributes of the comb. We have used a novel approach combining simulations and experiments to uncover the spatio-temporal dynamics underlying SC rotation. Our results indicate that 1) the final SC shape is primarily controlled by the inhomogeneity of initial cell size in cells close to the immature comb, 2) the final angle is primarily controlled by later cell expansion and 3) a temporal sequence of cell expansion mitigates the malformations generally associated with longer rotated SCs. Overall, our work has linked together the morphological diversity of SCs and the cellular dynamics behind such diversity, thus providing important insights on how evolution may affect SC development via the behaviours of surrounding epithelial cells.

The sex comb (SC) is a series of modified bristles on the male forelegs of many species of fruit flies. The size, position and shape of these sex combs vary drastically across different fly species. Therefore, SCs are a model system which illustrates the interaction between evolution and organism development influencing phenotypic features. In this work, we use a combined simulation-experimental approach to study the cellular processes involved in the rotation of developing SCs in common fruit flies (D. melanogaster). Our results indicate that despite the appearance of a complicated set of motions of surrounding cells associated with SC rotation, the final SC attributes only depend on a few selected parameters. We showed that changes in the timing and extent of cell size increase in distal cells altered the extent of SC rotation and breakage. Furthermore, these changes were sufficient to account for the observed variations in SC rotation between different fly species. Thus, our computational model has given us important insights on how evolution may use various cellular processes as a means to manifest the diversity of SCs across different fly species.

Scalable population-level modelling of biological cells incorporating mechanics and kinetics in continuous time

The processes taking place inside the living cell are now understood to the point where predictive computational models can be used to gain detailed understanding of important biological phenomena. A key challenge is to extrapolate this detailed knowledge of the individual cell to be able to explain at the population level how cells interact and respond with each other and their environment. In particular, the goal is to understand how organisms develop, maintain and repair functional tissues and organs. In this paper, we propose a novel computational framework for modelling populations of interacting cells. Our framework incorporates mechanistic, constitutive descriptions of biomechanical properties of the cell population, and uses a coarse-graining approach to derive individual rate laws that enable propagation of the population through time. Thanks to its multiscale nature, the resulting simulation algorithm is extremely scalable and highly efficient. As highlighted in our computational examples, the framework is also very flexible and may straightforwardly be coupled with continuous-time descriptions of biochemical signalling within, and between, individual cells.

A multiscale model for heterogeneous tumor spheroid in vitro

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

Physical Mechanisms Driving Cell Sorting in Hydra

Cell sorting, whereby a heterogeneous cell mixture organizes into distinct tissues, is a fundamental patterning process in development. Hydra is a powerful model system for carrying out studies of cell sorting in three dimensions, because of its unique ability to regenerate after complete dissociation into individual cells. The physicists Alfred Gierer and Hans Meinhardt recognized Hydra's self-organizing properties more than 40 years ago. However, what drives cell sorting during regeneration of Hydra from cell aggregates is still debated. Differential motility and differential adhesion have been proposed as driving mechanisms, but the available experimental data are insufficient to distinguish between these two. Here, we answer this longstanding question by using transgenic Hydra expressing fluorescent proteins and a multiscale experimental and numerical approach. By quantifying the kinematics of single cell and whole aggregate behaviors, we show that no differences in cell motility exist among cell types and that sorting dynamics follow a power law with an exponent of ∼0.5. Additionally, we measure the physical properties of separated tissues and quantify their viscosities and surface tensions. Based on our experimental results and numerical simulations, we conclude that tissue interfacial tensions are sufficient to explain cell sorting in aggregates of Hydra cells. Furthermore, we demonstrate that the aggregate's geometry during sorting is key to understanding the sorting dynamics and explains the exponent of the power law behavior. Our results answer the long standing question of the physical mechanisms driving cell sorting in Hydra cell aggregates. In addition, they demonstrate how powerful this organism is for biophysical studies of self-organization and pattern formation.

iPSC-Derived Vascular Cell Spheroids as Building Blocks for Scaffold-Free Biofabrication

Recently a protocol is established to obtain large quantities of human induced pluripotent stem cells (iPSC)-derived endothelial progenitors, called endothelial colony forming cells (ECFC), and of candidate smooth-muscle forming cells (SMFC). Here, the suitability for assembling in spheroids, and in larger 3D cell constructs is tested. iPSC-derived ECFC and SMFC are labeled with tdTomato and eGFP, respectively. Spheroids are formed in ultra-low adhesive wells, and their dynamic proprieties are studied by time-lapse microscopy, or by confocal microscopy. Spheroids are also tested for fusion ability either in the wells, or assembled on the Regenova 3D bioprinter which laces them in stainless steel micro-needles (the "Kenzan" method). It is found that both ECFC and SMFC formed spheroids in about 24 h. Fluorescence monitoring indicated a continuous compaction of ECFC spheroids, but stabilization in those prepared from SMFC. In mixed spheroids, the cell distribution changed continuously, with ECFC relocating to the core, and showing pre-vascular organization. All spheroids have the ability of in-well fusion, but only those containing SMFC are robust enough to sustain assembling in tubular structures. In these constructs a layered distribution of alpha smooth muscle actin-positive cells and extracellular matrix deposition is found. In conclusion, iPSC-derived vascular cell spheroids represent a promising new cellular material for scaffold-free biofabrication.

Mean-cluster approach indicates cell sorting time scales are determined by collective dynamics

Cell migration is essential to cell segregation, playing a central role in tissue formation, wound healing, and tumor evolution. Considering random mixtures of two cell types, it is still not clear which cell characteristics define clustering time scales. The mass of diffusing clusters merging with one another is expected to grow as t^{d/d+2} when the diffusion constant scales with the inverse of the cluster mass. Cell segregation experiments deviate from that behavior. Explanations for that could arise from specific microscopic mechanisms or from collective effects, typical of active matter. Here we consider a power law connecting diffusion constant and cluster mass to propose an analytic approach to model cell segregation where we explicitly take into account finite-size corrections. The results are compared with active matter model simulations and experiments available in the literature. To investigate the role played by different mechanisms we considered different hypotheses describing cell-cell interaction: differential adhesion hypothesis and different velocities hypothesis. We find that the simulations yield normal diffusion for long time intervals. Analytic and simulation results show that (i) cluster evolution clearly tends to a scaling regime, disrupted only at finite-size limits; (ii) cluster diffusion is greatly enhanced by cell collective behavior, such that for high enough tendency to follow the neighbors, cluster diffusion may become independent of cluster size; (iii) the scaling exponent for cluster growth depends only on the mass-diffusion relation, not on the detailed local segregation mechanism. These results apply for active matter systems in general and, in particular, the mechanisms found underlying the increase in cell sorting speed certainly have deep implications in biological evolution as a selection mechanism.

Cell Sorting and Noise-Induced Cell Plasticity Coordinate to Sharpen Boundaries between Gene Expression Domains

A fundamental question in biology is how sharp boundaries of gene expression form precisely in spite of biological variation/noise. Numerous mechanisms position gene expression domains across fields of cells (e.g. morphogens), but how these domains are refined remains unclear. In some cases, domain boundaries sharpen through differential adhesion-mediated cell sorting. However, boundaries can also sharpen through cellular plasticity, with cell fate changes driven by up- or down-regulation of gene expression. In this context, we have argued that noise in gene expression can help cells transition to the correct fate. Here we investigate the efficacy of cell sorting, gene expression plasticity, and their combination in boundary sharpening using multi-scale, stochastic models. We focus on the formation of hindbrain segments (rhombomeres) in the developing zebrafish as an example, but the mechanisms investigated apply broadly to many tissues. Our results indicate that neither sorting nor plasticity is sufficient on its own to sharpen transition regions between different rhombomeres. Rather the two have complementary strengths and weaknesses, which synergize when combined to sharpen gene expression boundaries.

In many developing systems, chemical gradients control the formation of segmental domains of gene expression, specifying distinct domains that go on to form different tissues and structures, in a concentration-dependent manner. These gradients are noisy however, raising the question of how sharply delineated boundaries between distinct segments form. It is crucial that developing systems be able to cope with stochasticity and generate well-defined boundaries between different segmented domains. Previous work suggests that cell sorting and cellular plasticity help sharpen boundaries between segments. However, it remains unclear how effective each of these mechanisms is and what their role in sharpening may be. Motivated by recent experimental observations, we construct a hybrid stochastic model to investigate these questions. We find that neither mechanism is sufficient on its own to sharpen boundaries between different segments. Rather, results indicate each has its own strengths and weaknesses, and that they work together synergistically to promote the development of precise, well defined segment boundaries. Formation of segmented rhombomeres in the zebrafish hindbrain, which later form different components of the central nervous system, is a motivating case for this study.

Schelling model of cell segregation based only on local information

While biological studies suggest that motility of cells is involved in cell segregation, few computational models have investigated this mechanism. We apply a simple Schelling model, modified to reflect biological conditions, demonstrating how differences in cell motility arising exclusively from differences in the composition of the local environment can be sufficient to drive segregation. The work presented here demonstrates that the segregation behavior observed in the original Schelling model is robust to a relaxation of the requirement for global information and that the Schelling model may yield insight in the context of biological systems. In the model, the time course of cell segregation follows a power law in accord with experimental observations and previous work.

Collective motion of cells: from experiments to models

Swarming or collective motion of living entities is one of the most common and spectacular manifestations of living systems that have been extensively studied in recent years. A number of general principles have been established. The interactions at the level of cells are quite different from those among individual animals, therefore the study of collective motion of cells is likely to reveal some specific important features which we plan to overview in this paper. In addition to presenting the most appealing results from the quickly growing related literature we also deliver a critical discussion of the emerging picture and summarize our present understanding of collective motion at the cellular level. Collective motion of cells plays an essential role in a number of experimental and real-life situations. In most cases the coordinated motion is a helpful aspect of the given phenomenon and results in making a related process more efficient (e.g., embryogenesis or wound healing), while in the case of tumor cell invasion it appears to speed up the progression of the disease. In these mechanisms cells both have to be motile and adhere to one another, the adherence feature being the most specific to this sort of collective behavior. One of the central aims of this review is to present the related experimental observations and treat them in light of a few basic computational models so as to make an interpretation of the phenomena at a quantitative level as well.

Why are enteric ganglia so small? Role of differential adhesion of enteric neurons and enteric neural crest cells

The avian enteric nervous system (ENS) consists of a vast number of unusually small ganglia compared to other peripheral ganglia. Each ENS ganglion at mid-gestation has a core of neurons and a shell of mesenchymal precursor/glia-like enteric neural crest (ENC) cells. To study ENS cell ganglionation we isolated midgut ENS cells by HNK-1 fluorescence-activated cell sorting (FACS) from E5 and E8 quail embryos, and from E9 chick embryos. We performed cell-cell aggregation assays which revealed a developmentally regulated functional increase in ENS cell adhesive function, requiring both Ca ²⁺ -dependent and independent adhesion. This was consistent with N-cadherin and NCAM labelling. Neurons sorted to the core of aggregates, surrounded by outer ENC cells, showing that neurons had higher adhesion than ENC cells. The outer surface of aggregates became relatively non-adhesive, correlating with low levels of NCAM and N-cadherin on this surface of the outer non-neuronal ENC cells. Aggregation assays showed that ENS cells FACS selected for NCAM-high and enriched for enteric neurons formed larger and more coherent aggregates than unsorted ENS cells. In contrast, ENS cells of the NCAM-low FACS fraction formed small, disorganised aggregates.  This suggests a novel mechanism for control of ENS ganglion morphogenesis where i) differential adhesion of ENS neurons and ENC cells controls the core/shell ganglionic structure and ii) the ratio of neurons to ENC cells dictates the equilibrium ganglion size by generation of an outer non-adhesive surface.

Segment-specific adhesion as a driver of convergent extension

Convergent extension, the simultaneous extension and narrowing of tissues, is a crucial event in the formation of the main body axis during embryonic development. It involves processes on multiple scales: the sub-cellular, cellular and tissue level, which interact via explicit or intrinsic feedback mechanisms. Computational modelling studies play an important role in unravelling the multiscale feedbacks underlying convergent extension. Convergent extension usually operates in tissue which has been patterned or is currently being patterned into distinct domains of gene expression. How such tissue patterns are maintained during the large scale tissue movements of convergent extension has thus far not been investigated. Intriguingly, experimental data indicate that in certain cases these tissue patterns may drive convergent extension rather than requiring safeguarding against convergent extension. Here we use a 2D Cellular Potts Model (CPM) of a tissue prepatterned into segments, to show that convergent extension tends to disrupt this pre-existing segmental pattern. However, when cells preferentially adhere to cells of the same segment type, segment integrity is maintained without any reduction in tissue extension. Strikingly, we demonstrate that this segment-specific adhesion is by itself sufficient to drive convergent extension. Convergent extension is enhanced when we endow our in silico cells with persistence of motion, which in vivo would naturally follow from cytoskeletal dynamics. Finally, we extend our model to confirm the generality of our results. We demonstrate a similar effect of differential adhesion on convergent extension in tissues that can only extend in a single direction (as often occurs due to the inertia of the head region of the embryo), and in tissues prepatterned into a sequence of domains resulting in two opposing adhesive gradients, rather than alternating segments.

The process of convergent extension is a major contributor to the formation of the anterior-posterior body axis in the early embryo. Convergent extension refers to the directed movement of cells that leads to the extension of tissue in one direction and narrowing of the tissue in the perpendicular direction. Often, convergent extension occurs in tissue which already contains distinct domains of gene expression such as segments, and it is unclear how these patterns are maintained despite extensive cell movement. Interestingly, experimental evidence suggests that these tissue patterns may drive rather than be compromised by convergent extension. However, existing computational models aimed at unravelling the mechanisms of convergent extension have thus far only studied the process in homogeneous tissues. With our model, we demonstrate that in a segmented tissue, preferential adhesion of cells to other cells within the same segment type is required to maintain the tissue pattern during convergent extension. Furthermore, such segment-specific adhesion is by itself capable of driving convergent extension.

A kinetic mechanism for cell sorting based on local variations in cell motility

Our current understanding of cell sorting relies on physical difference, either in the interfacial properties or motile force, between cell types. But is such asymmetry a prerequisite for cell sorting? We test this using a minimal model in which the two cell populations are identical with respect to their physical properties and differences in motility arise solely from how cells interact with their surroundings. The model resembles the Schelling model used in social sciences to study segregation phenomena at the scale of societies. Our results demonstrate that segregation can emerge solely from cell motility being a dynamic property that changes in response to the local environment of the cell, but that additional mechanisms are necessary to reproduce the envelopment behaviour observed in vitro. The time course of segregation follows a power law, in agreement with the scaling reported from experiment and in other models of motility-driven segregation.

Lattice-based model of ductal carcinoma in situ suggests rules for breast cancer progression to an invasive state

Ductal carcinoma in situ (DCIS) is a heterogeneous group of non-invasive lesions of the breast that result from abnormal proliferation of mammary epithelial cells. Pathologists characterize DCIS by four tissue morphologies (micropapillary, cribriform, solid, and comedo), but the underlying mechanisms that distinguish the development and progression of these morphologies are not well understood. Here we explored the conditions leading to the emergence of the different morphologies of DCIS using a two-dimensional multi-cell lattice-based model that incorporates cell proliferation, apoptosis, necrosis, adhesion, and contractility. We found that the relative rates of cell proliferation and apoptosis governed which of the four morphologies emerged. High proliferation and low apoptosis favored the emergence of solid and comedo morphologies. In contrast, low proliferation and high apoptosis led to the micropapillary morphology, whereas high proliferation and high apoptosis led to the cribriform morphology. The natural progression between morphologies cannot be investigated in vivo since lesions are usually surgically removed upon detection; however, our model suggests probable transitions between these morphologies during breast cancer progression. Importantly, cribriform and comedo appear to be the ultimate morphologies of DCIS. Motivated by previous experimental studies demonstrating that tumor cells behave differently depending on where they are located within the mammary duct in vivo or in engineered tissues, we examined the effects of tissue geometry on the progression of DCIS. In agreement with our previous experimental work, we found that cells are more likely to invade from the end of ducts and that this preferential invasion is regulated by cell adhesion and contractility. This model provides additional insight into tumor cell behavior and allows the exploration of phenotypic transitions not easily monitored in vivo.

Breast cancer is a complex disease that affects women worldwide. One heterogeneous group of lesions, ductal carcinoma in situ (DCIS), often begins as a nonmalignant disease but can readily progress if left untreated. The progression of this disease is not well understood because DCIS is typically removed upon detection. Therefore, computational models might help predict whether DCIS will remain nonmalignant or progress towards invasive ductal carcinoma. Here we used a multi-cell lattice-based model to explore the relative effects of cell proliferation, death, division axis, adhesion and contractility on the development and progression of DCIS. We also examined the emergence and progression of DCIS in physiologically relevant geometries of the mammary duct. Our model suggests several plausible progressions between morphologies of DCIS, and predicts that some regions of a duct are preferential for tumor cell invasion.

Computational modeling reveals that a combination of chemotaxis and differential adhesion leads to robust cell sorting during tissue patterning

Robust tissue patterning is crucial to many processes during development. The "French Flag" model of patterning, whereby naïve cells in a gradient of diffusible morphogen signal adopt different fates due to exposure to different amounts of morphogen concentration, has been the most widely proposed model for tissue patterning. However, recently, using time-lapse experiments, cell sorting has been found to be an alternative model for tissue patterning in the zebrafish neural tube. But it remains unclear what the sorting mechanism is. In this article, we used computational modeling to show that two mechanisms, chemotaxis and differential adhesion, are needed for robust cell sorting. We assessed the performance of each of the two mechanisms by quantifying the fraction of correct sorting, the fraction of stable clusters formed after correct sorting, the time needed to achieve correct sorting, and the size variations of the cells having different fates. We found that chemotaxis and differential adhesion confer different advantages to the sorting process. Chemotaxis leads to high fraction of correct sorting as individual cells will either migrate towards or away from the source depending on its cell type. However after the cells have sorted correctly, there is no interaction among cells of the same type to stabilize the sorted boundaries, leading to cell clusters that are unstable. On the other hand, differential adhesion results in low fraction of correct clusters that are more stable. In the absence of morphogen gradient noise, a combination of both chemotaxis and differential adhesion yields cell sorting that is both accurate and robust. However, in the presence of gradient noise, the simple combination of chemotaxis and differential adhesion is insufficient for cell sorting; instead, chemotaxis coupled with delayed differential adhesion is required to yield optimal sorting.

Differential adhesion between moving particles as a mechanism for the evolution of social groups

The evolutionary stability of cooperative traits, that are beneficial to other individuals but costly to their carrier, is considered possible only through the establishment of a sufficient degree of assortment between cooperators. Chimeric microbial populations, characterized by simple interactions between unrelated individuals, restrain the applicability of standard mechanisms generating such assortment, in particular when cells disperse between successive reproductive events such as happens in Dicyostelids and Myxobacteria. In this paper, we address the evolutionary dynamics of a costly trait that enhances attachment to others as well as group cohesion. By modeling cells as self-propelled particles moving on a plane according to local interaction forces and undergoing cycles of aggregation, reproduction and dispersal, we show that blind differential adhesion provides a basis for assortment in the process of group formation. When reproductive performance depends on the social context of players, evolution by natural selection can lead to the success of the social trait, and to the concomitant emergence of sizeable groups. We point out the conditions on the microscopic properties of motion and interaction that make such evolutionary outcome possible, stressing that the advent of sociality by differential adhesion is restricted to specific ecological contexts. Moreover, we show that the aggregation process naturally implies the existence of non-aggregated particles, and highlight their crucial evolutionary role despite being largely neglected in theoretical models for the evolution of sociality.

Although pervasive in the living world, collective behavior is a puzzle for evolutionary biology. The genetic traits that sustain it are costly for their carriers and make them vulnerable to the exploitation of asocial “free-riders” that benefit from the group without contributing to its cohesion. This paradox has spawned an extensive literature mainly concerned with elaborate cooperative behaviors that might be inoperant for simple biological entities such as microbes. We model successive life cycles of aggregation, reproduction and dispersal in a biological population combining a statistical physics approach to mimic the group formation process and an evolutionary game theory approach to account for the conflict between individual competition and collective success. Our results show a parsimonious way to the advent of sociality based on differential physical adhesion in organisms deprived of complex cognitive abilities. We also stress the key role of ungrouped individuals and specify the conditions on motion properties that make sociality possible. In detailing a mechanism akin to promote social behavior in microbes in the absence of genealogical relatedness, our work might shed light on both the maintenance of facultative multicellular lifestyles and the evolutionary origins of multicellularity.

The effects of cell compressibility, motility and contact inhibition on the growth of tumor cell clusters using the Cellular Potts Model

There are numerous biological examples where genes associated with migratory ability of cells also confer the cells with an increased fitness even though these genes may not have any known effect on the cell mitosis rates. Here, we provide insight into these observations by analyzing the effects of cell migration, compression, and contact inhibition on the growth of tumor cell clusters using the Cellular Potts Model (CPM) in a monolayer geometry. This is a follow-up of a previous study (Thalhauser et al. 2010) in which a Moran-type model was used to study the interaction of cell proliferation, migratory potential and death on the emergence of invasive phenotypes. Here, we extend the study to include the effects of cell size and shape. In particular, we investigate the interplay between cell motility and compressibility within the CPM and find that the CPM predicts that increased cell motility leads to smaller cells. This is an artifact in the CPM. An analysis of the CPM reveals an explicit inverse-relationship between the cell stiffness and motility parameters. We use this relationship to compensate for motility-induced changes in cell size in the CPM so that in the corrected CPM, cell size is independent of the cell motility. We find that subject to comparable levels of compression, clusters of motile cells grow faster than clusters of less motile cells, in qualitative agreement with biological observations and our previous study. Increasing compression tends to reduce growth rates. Contact inhibition penalizes clumped cells by halting their growth and gives motile cells an even greater advantage. Finally, our model predicts cell size distributions that are consistent with those observed in clusters of neuroblastoma cells cultured in low and high density conditions.

Segregation mechanisms of tissue cells: from experimental data to models

Considerable advance has been made in recent years in the research field of pattern formation by segregation of tissue cells. Research has become more quantitative partly due to more in-depth analysis of experimental data and the emergence modeling approaches. In this review we present experimental observations, including some of our new results, on various aspects of two and three dimensional segregation events and then summarize the computational modeling approaches.

Multi-scale modeling in morphogenesis: a critical analysis of the cellular Potts model

Cellular Potts models (CPMs) are used as a modeling framework to elucidate mechanisms of biological development. They allow a spatial resolution below the cellular scale and are applied particularly when problems are studied where multiple spatial and temporal scales are involved. Despite the increasing usage of CPMs in theoretical biology, this model class has received little attention from mathematical theory. To narrow this gap, the CPMs are subjected to a theoretical study here. It is asked to which extent the updating rules establish an appropriate dynamical model of intercellular interactions and what the principal behavior at different time scales characterizes. It is shown that the longtime behavior of a CPM is degenerate in the sense that the cells consecutively die out, independent of the specific interdependence structure that characterizes the model. While CPMs are naturally defined on finite, spatially bounded lattices, possible extensions to spatially unbounded systems are explored to assess to which extent spatio-temporal limit procedures can be applied to describe the emergent behavior at the tissue scale. To elucidate the mechanistic structure of CPMs, the model class is integrated into a general multiscale framework. It is shown that the central role of the surface fluctuations, which subsume several cellular and intercellular factors, entails substantial limitations for a CPM's exploitation both as a mechanistic and as a phenomenological model.

Multi-scale modeling of tissues using CompuCell3D

The study of how cells interact to produce tissue development, homeostasis, or diseases was, until recently, almost purely experimental. Now, multi-cell computer simulation methods, ranging from relatively simple cellular automata to complex immersed-boundary and finite-element mechanistic models, allow in silico study of multi-cell phenomena at the tissue scale based on biologically observed cell behaviors and interactions such as movement, adhesion, growth, death, mitosis, secretion of chemicals, chemotaxis, etc. This tutorial introduces the lattice-based Glazier-Graner-Hogeweg (GGH) Monte Carlo multi-cell modeling and the open-source GGH-based CompuCell3D simulation environment that allows rapid and intuitive modeling and simulation of cellular and multi-cellular behaviors in the context of tissue formation and subsequent dynamics. We also present a walkthrough of four biological models and their associated simulations that demonstrate the capabilities of the GGH and CompuCell3D.