Cellular Potts modeling of complex multicellular behaviors in tissue morphogenesis

Poissonian Cellular Potts Models Reveal Nonequilibrium Kinetics of Cell Sorting

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

Neuraminidase inhibition promotes the collective migration of neurons and recovery of brain function

In the injured brain, new neurons produced from endogenous neural stem cells form chains and migrate to injured areas and contribute to the regeneration of lost neurons. However, this endogenous regenerative capacity of the brain has not yet been leveraged for the treatment of brain injury. Here, we show that in healthy brain chains of migrating new neurons maintain unexpectedly large non-adherent areas between neighboring cells, allowing for efficient migration. In instances of brain injury, neuraminidase reduces polysialic acid levels, which negatively regulates adhesion, leading to increased cell-cell adhesion and reduced migration efficiency. The administration of zanamivir, a neuraminidase inhibitor used for influenza treatment, promotes neuronal migration toward damaged regions, fosters neuronal regeneration, and facilitates functional recovery. Together, these findings shed light on a new mechanism governing efficient neuronal migration in the adult brain under physiological conditions, pinpoint the disruption of this mechanism during brain injury, and propose a promising therapeutic avenue for brain injury through drug repositioning.

Toward a predictive understanding of epithelial cell death

Epithelial cell death is highly prevalent during development and tissue homeostasis. While we have a rather good understanding of the molecular regulators of programmed cell death, especially for apoptosis, we still fail to predict when, where, how many and which specific cells will die in a tissue. This likely relies on the much more complex picture of apoptosis regulation in a tissular and epithelial context, which entails cell autonomous but also non-cell autonomous factors, diverse feedback and multiple layers of regulation of the commitment to apoptosis. In this review, we illustrate this complexity of epithelial apoptosis regulation by describing these different layers of control, all demonstrating that local cell death probability is a complex emerging feature. We first focus on non-cell autonomous factors that can locally modulate the rate of cell death, including cell competition, mechanical input and geometry as well as systemic effects. We then describe the multiple feedback mechanisms generated by cell death itself. We also outline the multiple layers of regulation of epithelial cell death, including the coordination of extrusion and regulation occurring downstream of effector caspases. Eventually, we propose a roadmap to reach a more predictive understanding of cell death regulation in an epithelial context.

The free energy landscape of small-world networks of cells

The properties of organs, tissues, organoids, and other systems of cells, are influenced by the spatial localization and distribution of their elements. Here, we used networks to describe distributions of cells on a surface where the small-world coefficient (SW) of the networks was varied between SW~1 (random uniform distributions) and SW~10 (clustered distributions). The small-world coefficient is a topological measure of graphs: networks with SW>1 are topologically biased to transmit information. For each system configuration, we then determined the total energy U as the sum of the energies that describe cell-cell interactions - approximated by a harmonic potential. The graph of energy (U) across the configuration space of the networks (SW) is the energy landscape: it indicates which configuration a system of cells will likely assume over time. We found that, depending on the model parameters, the energy landscapes of 2D distributions of cells may be of different types: from type I to type IV. Type I and type II systems have high probability to evolve into random distributions. Type III and type IV systems have a higher probability to form clustered architectures. A great many of simulations indicated that cultures of cells with high initial density and limited sensing range could evolve into clustered configurations with enhanced topological characteristics. Moreover, the strongest the binding between cells, the greater the likelihood that they will assume configurations characterized by finite values of SW. Results of the work are relevant for those working the field of tissue engineering, regenerative medicine, the formation of in-vitro-models, the analysis of neuro-degenerative diseases.

Assessing mechanical agency during apical apoptotic cell extrusion

Epithelial tissues maintain homeostasis through the continual addition and removal of cells. Homeostasis is necessary for epithelia to maintain barrier function and prevent the accumulation of defective cells. Unfit, excess, and dying cells can be removed from epithelia by the process of extrusion. Controlled cell death and extrusion in the epithelium of the larval zebrafish tail fin coincides with oscillation of cell area, both in the extruding cell and its neighbors. Both cell-autonomous and non-autonomous factors have been proposed to contribute to extrusion but have been challenging to test by experimental approaches. Here we develop a dynamic cell-based biophysical model that recapitulates the process of oscillatory cell extrusion to test and compare the relative contributions of these factors. Our model incorporates the mechanical properties of individual epithelial cells in a two-dimensional simulation as repelling active particles. The area of cells destined to extrude oscillates with varying durations or amplitudes, decreasing their mechanical contribution to the epithelium and surrendering their space to surrounding cells. Quantitative variations in cell shape and size during extrusion are visualized by a hybrid weighted Voronoi tessellation technique that renders individual cell mechanical properties directly into an epithelial sheet. To explore the role of autonomous and non-autonomous mechanics, we vary the biophysical properties and behaviors of extruding cells and neighbors such as the period and amplitude of repulsive forces, cell density, and tissue viscosity. Our data suggest that cell autonomous processes are major contributors to the dynamics of extrusion, with the mechanical microenvironment providing a less pronounced contribution. Our computational model based on in vivo data serves as a tool to provide insights into the cellular dynamics and localized changes in mechanics that promote elimination of unwanted cells from epithelia during homeostatic tissue maintenance.

Cyclic stretching combined with cell-cell adhesion is sufficient for inducing cell intercalation

Although the important role of cell intercalation within a collective has long been recognized particularly for morphogenesis, the underlying mechanism remains poorly understood. Here we investigate the possibility that cellular responses to cyclic stretching play a major role in this process. By applying synchronized imaging and cyclic stretching to epithelial cells cultured on micropatterned polyacrylamide (PAA) substrates, we discovered that uniaxial cyclic stretching induces cell intercalation along with cell shape change and cell-cell interfacial remodeling. The process involved intermediate steps as previously reported for cell intercalation during embryonic morphogenesis, including the appearance of cell vertices, anisotropic vertex resolution, and directional expansion of cell-cell interface. Using mathematical modeling, we further found that cell shape change in conjunction with dynamic cell-cell adhesions was sufficient to account for the observations. Further investigation with small-molecule inhibitors indicated that disruption of myosin II activities suppressed cyclic stretching-induced intercalation while inhibiting the appearance of oriented vertices. Inhibition of Wnt signaling did not suppress stretch-induced cell shape change but disrupted cell intercalation and vertex resolution. Our results suggest that cyclic stretching, by inducing cell shape change and reorientation in the presence of dynamic cell-cell adhesions, can induce at least some aspects of cell intercalation and that this process is dependent in distinct ways on myosin II activities and Wnt signaling.

Modelling contractile ring formation and division to daughter cells for simulating proliferative multicellular dynamics

Cell proliferation is a fundamental process underlying embryogenesis, homeostasis, wound healing, and cancer. The process involves multiple events during each cell cycle, such as cell growth, contractile ring formation, and division to daughter cells, which affect the surrounding cell population geometrically and mechanically. However, existing methods do not comprehensively describe the dynamics of multicellular structures involving cell proliferation at a subcellular resolution. In this study, we present a novel model for proliferative multicellular dynamics at the subcellular level by building upon the nonconservative fluid membrane (NCF) model that we developed in earlier research. The NCF model utilizes a dynamically-rearranging closed triangular mesh to depict the shape of each cell, enabling us to analyze cell dynamics over extended periods beyond each cell cycle, during which cell surface components undergo dynamic turnover. The proposed model represents the process of cell proliferation by incorporating cell volume growth and contractile ring formation through an energy function and topologically dividing each cell at the cleavage furrow formed by the ring. Numerical simulations demonstrated that the model recapitulated the process of cell proliferation at subcellular resolution, including cell volume growth, cleavage furrow formation, and division to daughter cells. Further analyses suggested that the orientation of actomyosin stress in the contractile ring plays a crucial role in the cleavage furrow formation, i.e., circumferential orientation can form a cleavage furrow but isotropic orientation cannot. Furthermore, the model replicated tissue-scale multicellular dynamics, where the successive proliferation of adhesive cells led to the formation of a cell sheet and stratification on the substrate. Overall, the proposed model provides a basis for analyzing proliferative multicellular dynamics at subcellular resolution.

Hybrid cellular Potts and bead-spring modeling of cells in fibrous extracellular matrix

The mechanical interaction between cells and the extracellular matrix (ECM) is fundamental to coordinate collective cell behavior in tissues. Relating individual cell-level mechanics to tissue-scale collective behavior is a challenge that cell-based models such as the cellular Potts model (CPM) are well-positioned to address. These models generally represent the ECM with mean-field approaches, which assume substrate homogeneity. This assumption breaks down with fibrous ECM, which has nontrivial structure and mechanics. Here, we extend the CPM with a bead-spring model of ECM fiber networks modeled using molecular dynamics. We model a contractile cell pulling with discrete focal adhesion-like sites on the fiber network and demonstrate agreement with experimental spatiotemporal fiber densification and displacement. We show that at high network cross-linking, contractile cell forces propagate over at least eight cell diameters, decaying with distance with power law exponent n= 0.35 - 0.65 typical of viscoelastic ECMs. Further, we use in silico atomic force microscopy to measure local cell-induced network stiffening consistent with experiments. Our model lays the foundation for investigating how local and long-ranged cell-ECM mechanobiology contributes to multicellular morphogenesis.

Apoptotic extracellular vesicle formation via local phosphatidylserine exposure drives efficient cell extrusion

Cell extrusion is a universal mode of cell removal from tissues, and it plays an important role in regulating cell numbers and eliminating unwanted cells. However, the underlying mechanisms of cell delamination from the cell layer are unclear. Here, we report a conserved execution mechanism of apoptotic cell extrusion. We found extracellular vesicle (EV) formation in extruding mammalian and Drosophila cells at a site opposite to the extrusion direction. Lipid-scramblase-mediated local exposure of phosphatidylserine is responsible for EV formation and is crucial for executing cell extrusion. Inhibition of this process disrupts prompt cell delamination and tissue homeostasis. Although the EV has hallmarks of an apoptotic body, its formation is governed by the mechanism of microvesicle formation. Experimental and mathematical modeling analysis illustrated that EV formation promotes neighboring cells' invasion. This study showed that membrane dynamics play a crucial role in cell exit by connecting the actions of the extruding cell and neighboring cells.

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.

Long-term adherent cell dynamics emerging from energetic and frictional interactions at the interface

Cell adhesion plays an important role in a wide range of biological situations, including embryonic development, cancer invasion, and wound healing. Although several computational models describing adhesion dynamics have been proposed, models applicable to long-term, large-length-scale cell dynamics are lacking. In this study we investigated possible states of long-term adherent cell dynamics in three-dimensional space by constructing a continuum model of interfacial interactions between adhesive surfaces. In this model a pseudointerface is supposed between each pair of triangular elements that discretize cell surfaces. By introducing a distance between each pair of elements, the physical properties of the interface are given by interfacial energy and friction. The proposed model was implemented into the model of a nonconservative fluid cell membrane where the cell membrane dynamically flows with turnover. Using the implemented model, numerical simulations of adherent cell dynamics on a substrate under flow were performed. The simulations not only reproduced the previously reported dynamics of adherent cells, such as detachment, rolling, and fixation on the substrate, but also discovered other dynamic states, including cell slipping and membrane flow patterns, corresponding to behaviors that occur on much longer timescales than the dissociation of adhesion molecules. These results illustrate the variety of long-term adherent cell dynamics, which are more diverse than the short-term ones. The proposed model can be extended to arbitrarily shaped membranes, thus being useful for the mechanical analysis of a wide range of long-term cell dynamics where adhesion is essential.

Modelling the evolution of novelty: a review

Evolution has been an inventive process since its inception, about 4 billion years ago. It has generated an astounding diversity of novel mechanisms and structures for adaptation to the environment, for competition and cooperation, and for organisation of the internal and external dynamics of the organism. How does this novelty come about? Evolution builds with the tools available, and on top of what it has already built - therefore, much novelty consists in repurposing old functions in a different context. In the process, the tools themselves evolve, allowing yet more novelty to arise. Despite evolutionary novelty being the most striking observable of evolution, it is not accounted for in classical evolutionary theory. Nevertheless, mathematical and computational models that illustrate mechanisms of evolutionary innovation have been developed. In the present review, we present and compare several examples of computational evo-devo models that capture two aspects of novelty: 'between-level novelty' and 'constructive novelty.' Novelty can evolve between predefined levels of organisation to dynamically transcode biological information across these levels - as occurs during development. Constructive novelty instead generates a level of biological organisation by exploiting the lower level as an informational scaffold to open a new space of possibilities - an example being the evolution of multicellularity. We propose that the field of computational evo-devo is well-poised to reveal many more exciting mechanisms for the evolution of novelty. A broader theory of evolutionary novelty may well be attainable in the near future.

Listeria motility increases the efficiency of epithelial invasion during intestinal infection

Listeria monocytogenes (Lm) is a food-borne pathogen that causes severe bacterial gastroenteritis, with high rates of hospitalization and mortality. Lm is ubiquitous in soil, water and livestock, and can survive and proliferate at low temperatures. Following oral ingestion of contaminated food, Lm crosses the epithelium through intestinal goblet cells in a mechanism mediated by Lm InlA binding host E-cadherin. Importantly, human infections typically occur with Lm growing at or below room temperature, which is flagellated and motile. Even though many important human bacterial pathogens are flagellated, little is known regarding the effect of Lm motility on invasion and immune evasion. Here, we used complementary imaging and computer modeling approaches to test the hypothesis that bacterial motility helps Lm locate and engage target cells permissive for invasion. Imaging explanted mouse and human intestine, we showed that Lm grown at room temperature uses motility to scan the epithelial surface and preferentially attach to target cells. Furthermore, we integrated quantitative parameters from our imaging experiments to construct a versatile "layered" cellular Potts model (L-CPM) that simulates host-pathogen dynamics. Simulated data are consistent with the hypothesis that bacterial motility enhances invasion by allowing bacteria to search the epithelial surface for their preferred invasion targets. Indeed, our model consistently predicts that motile bacteria invade twice as efficiently over the first hour of infection. We also examined how bacterial motility affected interactions with host cellular immunity. In a mouse model of persistent infection, we found that neutrophils migrated to the apical surface of the epithelium 5 hours post infection and interacted with Lm. Yet in contrast to the view that neutrophils "hunt" for bacteria, we found that these interactions were driven by motility of Lm-which moved at least ~50x faster than neutrophils. Furthermore, our L-CPM predicts that motile bacteria maintain their invasion advantage even in the presence of host phagocytes, with the balance between invasion and phagocytosis governed almost entirely by bacterial motility. In conclusion, our simulations provide insight into host pathogen interaction dynamics at the intestinal epithelial barrier early during infection.

Computational approaches for simulating luminogenesis

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

Appropriate tension sensitivity of α-catenin ensures rounding morphogenesis of epithelial spheroids

The adherens junction (AJ) is an actin filament-anchoring junction. It plays a central role in epithelial morphogenesis through cadherin-based recognition and adhesion among cells. The stability and plasticity of AJs are required for the morphogenesis. An actin-binding α-catenin is an essential component of the cadherin-catenin complex and functions as a tension transducer that changes its conformation and induces AJ development in response to tension. Despite much progress in understanding molecular mechanisms of tension sensitivity of α-catenin, its significance on epithelial morphogenesis is still unknown. Here we show that the tension sensitivity of α-catenin is essential for epithelial cells to form round spheroids through proper multicellular rearrangement. Using a novel in vitro suspension culture model, we found that epithelial cells form round spheroids even from rectangular-shaped cell masses with high aspect ratios without using high tension and that increased tension sensitivity of α-catenin affected this morphogenesis. Analyses of AJ formation and cellular tracking during rounding morphogenesis showed cellular rearrangement, probably through AJ remodeling. The rearrangement occurs at the cell mass level, but not single-cell level. Hypersensitive α-catenin mutant-expressing cells did not show cellular rearrangement at the cell mass level, suggesting that the appropriate tension sensitivity of α-catenin is crucial for the coordinated round morphogenesis.Key words: α-catenin, vinculin, adherens junction, morphogenesis, mechanotransduction.

A Review of Mathematical and Computational Methods in Cancer Dynamics

Cancers are complex adaptive diseases regulated by the nonlinear feedback systems between genetic instabilities, environmental signals, cellular protein flows, and gene regulatory networks. Understanding the cybernetics of cancer requires the integration of information dynamics across multidimensional spatiotemporal scales, including genetic, transcriptional, metabolic, proteomic, epigenetic, and multi-cellular networks. However, the time-series analysis of these complex networks remains vastly absent in cancer research. With longitudinal screening and time-series analysis of cellular dynamics, universally observed causal patterns pertaining to dynamical systems, may self-organize in the signaling or gene expression state-space of cancer triggering processes. A class of these patterns, strange attractors, may be mathematical biomarkers of cancer progression. The emergence of intracellular chaos and chaotic cell population dynamics remains a new paradigm in systems medicine. As such, chaotic and complex dynamics are discussed as mathematical hallmarks of cancer cell fate dynamics herein. Given the assumption that time-resolved single-cell datasets are made available, a survey of interdisciplinary tools and algorithms from complexity theory, are hereby reviewed to investigate critical phenomena and chaotic dynamics in cancer ecosystems. To conclude, the perspective cultivates an intuition for computational systems oncology in terms of nonlinear dynamics, information theory, inverse problems, and complexity. We highlight the limitations we see in the area of statistical machine learning but the opportunity at combining it with the symbolic computational power offered by the mathematical tools explored.

Integrative lymph node-mimicking models created with biomaterials and computational tools to study the immune system

The lymph node (LN) is a vital organ of the lymphatic and immune system that enables timely detection, response, and clearance of harmful substances from the body. Each LN comprises of distinct substructures, which host a plethora of immune cell types working in tandem to coordinate complex innate and adaptive immune responses. An improved understanding of LN biology could facilitate treatment in LN-associated pathologies and immunotherapeutic interventions, yet at present, animal models, which often have poor physiological relevance, are the most popular experimental platforms. Emerging biomaterial engineering offers powerful alternatives, with the potential to circumvent limitations of animal models, for in-depth characterization and engineering of the lymphatic and adaptive immune system. In addition, mathematical and computational approaches, particularly in the current age of big data research, are reliable tools to verify and complement biomaterial works. In this review, we first discuss the importance of lymph node in immunity protection followed by recent advances using biomaterials to create in vitro/vivo LN-mimicking models to recreate the lymphoid tissue microstructure and microenvironment, as well as to describe the related immuno-functionality for biological investigation. We also explore the great potential of mathematical and computational models to serve as in silico supports. Furthermore, we suggest how both in vitro/vivo and in silico approaches can be integrated to strengthen basic patho-biological research, translational drug screening and clinical personalized therapies. We hope that this review will promote synergistic collaborations to accelerate progress of LN-mimicking systems to enhance understanding of immuno-complexity.

Implementing computational modeling in tissue engineering: where disciplines meet

In recent years, the mathematical and computational sciences have developed novel methodologies and insights that can aid in designing advanced bioreactors, microfluidic set-ups or organ-on-chip devices, in optimizing culture conditions, or predicting long-term behavior of engineered tissues in vivo. In this review, we introduce the concept of computational models and how they can be integrated in an interdisciplinary workflow for Tissue Engineering and Regenerative Medicine (TERM). We specifically aim this review of general concepts and examples at experimental scientists with little or no computational modeling experience. We also describe the contribution of computational models in understanding TERM processes and in advancing the TERM field by providing novel insights.

Quantitative modeling identifies critical cell mechanics driving bile duct lumen formation

Biliary ducts collect bile from liver lobules, the smallest functional and anatomical units of liver, and carry it to the gallbladder. Disruptions in this process caused by defective embryonic development, or through ductal reaction in liver disease have a major impact on life quality and survival of patients. A deep understanding of the processes underlying bile duct lumen formation is crucial to identify intervention points to avoid or treat the appearance of defective bile ducts. Several hypotheses have been proposed to characterize the biophysical mechanisms driving initial bile duct lumen formation during embryogenesis. Here, guided by the quantification of morphological features and expression of genes in bile ducts from embryonic mouse liver, we sharpened these hypotheses and collected data to develop a high resolution individual cell-based computational model that enables to test alternative hypotheses in silico. This model permits realistic simulations of tissue and cell mechanics at sub-cellular scale. Our simulations suggest that successful bile duct lumen formation requires a simultaneous contribution of directed cell division of cholangiocytes, local osmotic effects generated by salt excretion in the lumen, and temporally-controlled differentiation of hepatoblasts to cholangiocytes, with apical constriction of cholangiocytes only moderately affecting luminal size.

The initial step in bile duct development is the formation of a biliary lumen, a process which involves several cellular mechanisms, such as cell division and polarization, and secretion of fluid. However, how these mechanisms are orchestrated in time and space is difficult to understand. Here, we built a computational model of biliary lumen formation which represents every cell and its function in detail. With the model we can simulate the effect of biophysical aspects that affect duct formation. We have tested the individual and combined effects of directed cell division, apical constriction, and osmotic effects on lumen expansion by varying the parameters that control their relative strength. Our simulations suggest that successful bile duct lumen formation requires the simultaneous contribution of directed cell division of cholangiocytes, local osmotic effects generated by salt excretion in the lumen, and temporally-controlled differentiation of hepatoblasts to cholangiocytes, with apical constriction of cholangiocytes only moderately affecting luminal size.

Generation, Transmission, and Regulation of Mechanical Forces in Embryonic Morphogenesis

Embryonic morphogenesis is a biological process which depicts shape forming of tissues and organs during development. Unveiling the roles of mechanical forces generated, transmitted, and regulated in cells and tissues through these processes is key to understanding the biophysical mechanisms governing morphogenesis. To this end, it is imperative to measure, simulate, and predict the regulation and control of these mechanical forces during morphogenesis. This article aims to provide a comprehensive review of the recent advances on mechanical properties of cells and tissues, generation of mechanical forces in cells and tissues, the transmission processes of these generated forces during cells and tissues, the tools and methods used to measure and predict these mechanical forces in vivo, in vitro, or in silico, and to better understand the corresponding regulation and control of generated forces. Understanding the biomechanics and mechanobiology of morphogenesis will not only shed light on the fundamental physical mechanisms underlying these concerted biological processes during normal development, but also uncover new information that will benefit biomedical research in preventing and treating congenital defects or tissue engineering and regeneration.

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

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

Quasistatic rheology of soft cellular systems using the cellular Potts model

Soft cellular systems, such as foams or biological tissues, exhibit highly complex rheological properties, even in the quasistatic regime, that numerical modeling can help to apprehend. We present a numerical implementation of quasistatic strain within the widely used cellular Potts model (CPM). The accuracy of the method is tested by simulating the quasistatic strain of two-dimensional dry foams, both ordered and disordered. The implementation of quasistatic strain in CPM allows the investigation of sophisticated interplays between stress-strain relationship and structural changes that take place in cellular systems.

Computational modelling of nephron progenitor cell movement and aggregation during kidney organogenesis

During early kidney organogenesis, nephron progenitor (NP) cells move from the tip to the corner region of the ureteric bud (UB) branches in order to form the pretubular aggregate, the early structure giving rise to nephron formation. NP cells derive from metanephric mesenchymal cells and physically interact with them during the movement. Chemotaxis and cell-cell adhesion differences are believed to drive the cell patterning during this critical period of organogenesis. However, the effect of these forces to the cell patterns and their respective movements are known in limited details. We applied a Cellular Potts Model to explore how these forces and organizations contribute to directed cell movement and aggregation. Model parameters were estimated based on fitting to experimental data obtained in ex vivo kidney explant and dissociation-reaggregation organoid culture studies. Our simulations indicated that optimal enrichment and aggregation of NP cells in the UB corner niche requires chemoattractant secretion from both the UB epithelial cells and the NP cells themselves, as well as differences in cell-cell adhesion energies. Furthermore, NP cells were observed, both experimentally and by modelling, to move at higher speed in the UB corner as compared to the tip region where they originated. The existence of different cell speed domains along the UB was confirmed using self-organizing map analysis. In summary, we saw faster NP cell movements near aggregation. The applicability of Cellular Potts Model approach to simulate cell movement and patterning was found to be good during for this early nephrogenesis process. Further refinement of the model should allow us to recapitulate the effects of developmental changes of cell phenotypes and molecular crosstalk during further organ development.

A mesoscale mechanical model of cellular interactions

Computational models of cell mechanics allow the precise interrogation of cell shape change. These morphological changes are required for cells to survive in diverse tissue environments. Here, we present a mesoscale mechanical model of cell-substrate interactions using the level set method based on experimentally measured parameters. By implementing a viscoelastic mechanical equivalent circuit, we accurately model whole-cell deformations that are important for a variety of cellular processes. To effectively model shape changes as a cell interacts with a substrate, we have included receptor-mediated adhesion, which is governed by catch-slip bond behavior. The effect of adhesion was explored by subjecting cells to a variety of different substrates including flat, curved, and deformable surfaces. Finally, we increased the accuracy of our simulations by including a deformable nucleus in our cells. This model sets the foundation for further exploration into computational analyses of multicellular interactions.

Large-scale simulations of biological cell sorting driven by differential adhesion follow diffusion-limited domain coalescence regime

Cell sorting, whereby a heterogeneous cell mixture segregates and forms distinct homogeneous tissues, is one of the main collective cell behaviors at work during development. Although differences in interfacial energies are recognized to be a possible driving source for cell sorting, no clear consensus has emerged on the kinetic law of cell sorting driven by differential adhesion. Using a modified Cellular Potts Model algorithm that allows for efficient simulations while preserving the connectivity of cells, we numerically explore cell-sorting dynamics over very large scales in space and time. For a binary mixture of cells surrounded by a medium, increase of domain size follows a power-law with exponent n = 1/4 independently of the mixture ratio, revealing that the kinetics is dominated by the diffusion and coalescence of rounded domains. We compare these results with recent numerical studies on cell sorting, and discuss the importance of algorithmic differences as well as boundary conditions on the observed scaling.

Theory and simulation for equilibrium glassy dynamics in cellular Potts model of confluent biological tissue

Glassy dynamics in a confluent monolayer is indispensable in morphogenesis, wound healing, bronchial asthma, and many others; a detailed theoretical framework for such a system is, therefore, important. Vertex-model (VM) simulations have provided crucial insights into the dynamics of such systems, but their nonequilibrium nature makes theoretical development difficult. The cellular Potts model (CPM) of confluent monolayers provides an alternative model for such systems with a well-defined equilibrium limit. We combine numerical simulations of the CPM and an analytical study based on one of the most successful theories of equilibrium glass, the random first-order transition theory, and develop a comprehensive theoretical framework for a confluent glassy system. We find that the glassy dynamics within the CPM is qualitatively similar to that in the VM. Our study elucidates the crucial role of geometric constraints in bringing about two distinct regimes in the dynamics, as the target perimeter P_{0} is varied. The unusual sub-Arrhenius relaxation results from the distinctive interaction potential arising from the perimeter constraint in such systems. The fragility of the system decreases with increasing P_{0} in the low-P_{0} regime, whereas the dynamics is independent of P_{0} in the other regime. The rigidity transition, found in the VM, is absent within the CPM; this difference seems to come from the nonequilibrium nature of the former. We show that the CPM captures the basic phenomenology of glassy dynamics in a confluent biological system via comparison of our numerical results with existing experiments on different systems.

Artistoo, a library to build, share, and explore simulations of cells and tissues in the web browser

The cellular Potts model (CPM) is a powerful in silico method for simulating biological processes at tissue scale. Their inherently graphical nature makes CPMs very accessible in theory, but in practice, they are mostly implemented in specialised frameworks users need to master before they can run simulations. We here present Artistoo (Artificial Tissue Toolbox), a JavaScript library for building ‘explorable’ CPM simulations where viewers can change parameters interactively, exploring their effects in real time. Simulations run directly in the web browser and do not require third-party software, plugins, or back-end servers. The JavaScript implementation imposes no major performance loss compared to frameworks written in C++; Artistoo remains sufficiently fast for interactive, real-time simulations. Artistoo provides an opportunity to unlock CPM models for a broader audience: interactive simulations can be shared via a URL in a zero-install setting. We discuss applications in CPM research, science dissemination, open science, and education.

Multi-scale simulation of early kidney branching morphogenesis

An important feature of the branch morphogenesis during kidney development is the termination of the tips on the outer surface of a kidney. This feature requires the avoidance of the intersection between the tips and existing ducts inside the kidney. Here, we started from a continuous model and implemented the coarse grained rules into a fast and discrete simulations. The ligand-receptor-based Turing mechanism suggests a repulsion that decreases exponentially with distance between interacting branches, preventing the intersection between neighboring branches. We considered this repulsive effect in numerical simulations and successfully reproduce the key features of the experimentally observed branch morphology for an E15.5 kidney. We examine the similarity of several geometrical parameters between the simulation results and experimental observations. The good agreement between the simulations and experiments suggests that the concentration decay caused by the absorption of glial cell line derived neurotrophic factor might be the key factor to affect the geometry in early kidney development.

Thermal shape fluctuations of a two-dimensional compressible droplet

Analysis of thermal capillary waves on the surface of a liquid usually assumes incompressibility of the bulk fluid. However, for droplets or bubbles with submicronic size, or for epithelial cells whose out-of-plane elongation can be modeled by an effective 2D bulk modulus, compressibility of the internal fluid must be taken into account for the characterization of their shape fluctuations. We present a theoretical analysis of the fluctuations of a two-dimensional compressible droplet. Analytical expressions for area, perimeter and energy fluctuations are derived and compared with Cellular Potts Model (CPM) simulations. This comparison shows a very good agreement between theory and simulations, and offers a precise calibration method for CPM simulations.

Self-organized multicellular structures from simple cell signaling: a computational model

Recent synthetic biology experiments reveal that signaling modules designed to target cell-cell adhesion enable self-organization of multicellular structures Toda et al (2018 Science 361 156-162). Changes in homotypic adhesion that arise through contact-dependent signaling networks result in sorting of an aggregate into two- or three-layered structures. Here we investigate the formation, maintenance, and robustness of such self-organization in the context of a computational model. To do so, we use an established model for Notch/ligand signaling within cells to set up differential E-cadherin expression. This signaling model is integrated with the cellular Potts model to track state changes, adhesion, and cell sorting in a group of cells. The resulting multicellular structures are in accordance with those observed in the experimental reference. In addition to reproducing these experimental results, we track the dynamics of the evolving structures and cell states to understand how such morphologies are dynamically maintained. This appears to be an important developmental principle that was not emphasized in previous models. Our computational model facilitates more detailed understanding of the link between intra- and intercellular signaling, spatio-temporal rearrangement, and emergent behavior at the scale of hundred(s) of cells.

Agent-Based Modeling and Simulation in Mathematics and Biology Education

With advances in computing, agent-based models (ABMs) have become a feasible and appealing tool to study biological systems. ABMs are seeing increased incorporation into both the biology and mathematics classrooms as powerful modeling tools to study processes involving substantial amounts of stochasticity, nonlinear interactions, and/or heterogeneous spatial structures. Here we present a brief synopsis of the agent-based modeling approach with an emphasis on its use to simulate biological systems, and provide a discussion of its role and limitations in both the biology and mathematics classrooms.

Adhesion-stabilizing long-distance transport of cells on tissue surface

The stable transport of migrating eukaryotic cells is essential in organ development and repair processes. However, the mechanism that preserves transport stability over long distances in organs is not fully understood. As the driving mechanism of cell migration, the expressions of heterophilic cell-cell adhesion between moving cells and scaffolding tissue have been observed in such transport. In this paper, we theoretically investigate this heterophilic adhesion, which is persistently polarized in the migrating cell, as a possible transport stabilization mechanism. The adhesion was examined on the basis of the cellular Potts model, and our results confirm the stabilization of the transport to be an effect of the persistence.

ERK-Mediated Mechanochemical Waves Direct Collective Cell Polarization

During collective migration of epithelial cells, the migration direction is aligned over a tissue-scale expanse. Although the collective cell migration is known to be directed by mechanical forces transmitted via cell-cell junctions, it remains elusive how the intercellular force transmission is coordinated with intracellular biochemical signaling to achieve collective movements. Here, we show that intercellular coupling of extracellular signal-regulated kinase (ERK)-mediated mechanochemical feedback yields long-distance transmission of guidance cues. Mechanical stretch activates ERK through epidermal growth factor receptor (EGFR) activation, and ERK activation triggers cell contraction. The contraction of the activated cell pulls neighboring cells, evoking another round of ERK activation and contraction in the neighbors. Furthermore, anisotropic contraction based on front-rear polarization guarantees unidirectional propagation of ERK activation, and in turn, the ERK activation waves direct multicellular alignment of the polarity, leading to long-range ordered migration. Our findings reveal that mechanical forces mediate intercellular signaling underlying sustained transmission of guidance cues for collective cell migration.

Thermally Driven Order-Disorder Transition in Two-Dimensional Soft Cellular Systems

Many systems, including biological tissues and foams, are made of highly packed units having high deformability but low compressibility. At two dimensions, these systems offer natural tesselations of a plane with fixed density, in which transitions from ordered to disordered patterns are often observed, in both directions. Using a modified cellular Potts model algorithm that allows rapid thermalization of extensive systems, we numerically explore the order-disorder transition of monodisperse, two-dimensional cellular systems driven by thermal agitation. We show that the transition follows most of the predictions of Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory developed for melting of 2D solids, extending the validity of this theory to systems with many-body interactions. In particular, we show the existence of an intermediate hexatic phase, which preserves the orientational order of the regular hexagonal tiling but loses its positional order. In addition to shedding light on the structural changes observed in experimental systems, our study shows that soft cellular systems offer macroscopic systems in which the KTHNY melting scenario can be explored, in the continuation of Bragg's experiments on bubble rafts.

A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation

We discuss several continuum cell-cell adhesion models based on the underlying microscopic assumptions. We propose an improvement on these models leading to sharp fronts and intermingling invasion fronts between different cell type populations. The model is based on basic principles of localized repulsion and nonlocal attraction due to adhesion forces at the microscopic level. The new model is able to capture both qualitatively and quantitatively experiments by Katsunuma et al. (2016). We also review some of the applications of these models in other areas of tissue growth in developmental biology. We finally explore the resulting qualitative behavior due to cell-cell repulsion.

Mathematical Approaches of Branching Morphogenesis

Many organs require a high surface to volume ratio to properly function. Lungs and kidneys, for example, achieve this by creating highly branched tubular structures during a developmental process called branching morphogenesis. The genes that control lung and kidney branching share a similar network structure that is based on ligand-receptor reciprocal signalling interactions between the epithelium and the surrounding mesenchyme. Nevertheless, the temporal and spatial development of the branched epithelial trees differs, resulting in organs of distinct shape and size. In the embryonic lung, branching morphogenesis highly depends on FGF10 signalling, whereas GDNF is the driving morphogen in the kidney. Knockout of Fgf10 and Gdnf leads to lung and kidney agenesis, respectively. However, FGF10 plays a significant role during kidney branching and both the FGF10 and GDNF pathway converge on the transcription factors ETV4/5. Although the involved signalling proteins have been defined, the underlying mechanism that controls lung and kidney branching morphogenesis is still elusive. A wide range of modelling approaches exists that differ not only in the mathematical framework (e.g., stochastic or deterministic) but also in the spatial scale (e.g., cell or tissue level). Due to advancing imaging techniques, image-based modelling approaches have proven to be a valuable method for investigating the control of branching events with respect to organ-specific properties. Here, we review several mathematical models on lung and kidney branching morphogenesis and suggest that a ligand-receptor-based Turing model represents a potential candidate for a general but also adaptive mechanism to control branching morphogenesis during development.

Perspective: The promise of multi-cellular engineered living systems

Recent technological breakthroughs in our ability to derive and differentiate induced pluripotent stem cells, organoid biology, organ-on-chip assays, and 3-D bioprinting have all contributed to a heightened interest in the design, assembly, and manufacture of living systems with a broad range of potential uses. This white paper summarizes the state of the emerging field of "multi-cellular engineered living systems," which are composed of interacting cell populations. Recent accomplishments are described, focusing on current and potential applications, as well as barriers to future advances, and the outlook for longer term benefits and potential ethical issues that need to be considered.

Fibroblast state switching orchestrates dermal maturation and wound healing

Murine dermis contains functionally and spatially distinct fibroblast lineages that cease to proliferate in early postnatal life. Here, we propose a model in which a negative feedback loop between extracellular matrix (ECM) deposition and fibroblast proliferation determines dermal architecture. Virtual-tissue simulations of our model faithfully recapitulate dermal maturation, predicting a loss of spatial segregation of fibroblast lineages and dictating that fibroblast migration is only required for wound healing. To test this, we performed in vivo live imaging of dermal fibroblasts, which revealed that homeostatic tissue architecture is achieved without active cell migration. In contrast, both fibroblast proliferation and migration are key determinants of tissue repair following wounding. The results show that tissue-scale coordination is driven by the interdependence of cell proliferation and ECM deposition, paving the way for identifying new therapeutic strategies to enhance skin regeneration.