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

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

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

Apical Junctional Fluctuations Lead to Cell Flow while Maintaining Epithelial Integrity

Epithelial sheet integrity is robustly maintained during morphogenesis, which is essential to shape organs and embryos. While maintaining the planar monolayer in three-dimensional space, cells dynamically flow via rearranging their connections between each other. However, little is known about how cells maintain the plane sheet integrity in three-dimensional space and provide cell flow in the in-plane sheet. In this study, using a three-dimensional vertex model, we demonstrate that apical junctional fluctuations allow stable cell rearrangements while ensuring monolayer integrity. In addition to the fluctuations, direction-dependent contraction on the apical cell boundaries, which corresponds to forces from adherens junctions, induces cell flow in a definite direction. We compared the kinematic behaviors of this apical-force-driven cell flow with those of typical cell flow that is driven by forces generated on basal regions and revealed the characteristic differences between them. These differences can be used to distinguish the mechanism of epithelial cell flow observed in experiments, i.e., whether it is apical- or basal-force-driven. Our numerical simulations suggest that cells actively generate fluctuations and use them to regulate both epithelial integrity and plasticity during morphogenesis.

Strain-triggered mechanical feedback in self-organizing optic-cup morphogenesis

Organogenesis is a self-organizing process of multiple cells in three-dimensional (3D) space, where macroscopic tissue deformations are robustly regulated by multicellular autonomy. It is clear that this robust regulation requires cells to sense and modulate 3D tissue formation across different scales, but its underlying mechanisms are still unclear. To address this question, we developed a versatile computational model of 3D multicellular dynamics at single-cell resolution and combined it with the 3D culture system of pluripotent stem cell-derived optic-cup organoid. The complementary approach enabled quantitative prediction of morphogenesis and its corresponding verification and elucidated that the macroscopic 3D tissue deformation is fed back to individual cellular force generations via mechanosensing. We hereby conclude that mechanical force plays a key role as a feedback regulator to establish the robustness of organogenesis.

Cancer cell mechanics with altered cytoskeletal behavior and substrate effects: A 3D finite element modeling study

A robust computational model of a cancer cell is presented using finite element modeling. The model accurately captures nuances of the various components of the cellular substructure. The role of degradation of cytoskeleton on overall elastic properties of the cancer cell is reported. The motivation for degraded cancer cellular substructure, the cytoskeleton is the observation that the innate mechanics of cytoskeleton is disrupted by various anti-cancer drugs as therapeutic treatments for the destruction of the cancer tumors. We report a significant influence on the degradation of the cytoskeleton on the mechanics of cancer cell. Further, a simulations based study is reported where we evaluate mechanical properties of the cancer cell attached to a variety of substrates. The loading of the cancer cell is less influenced by nature of the substrate, but low modulus substrates such as osteoblasts and hydrogels indicate a significant change in unloading behavior and also the plastic deformation. Overall, softer substrates such as osteoblasts and other bone cells result in a much altered unloading response as well as significant plastic deformation. These substrates are relevant to metastasis wherein certain type of cancers such as prostate and breast cancer cells migrate to the bone and colonize through mesenchymal to epithelial transition. The modeling study presented here is an important first step in the development of strong predictive methodologies for cancer progression.

Three-dimensional vertex model for simulating multicellular morphogenesis

During morphogenesis, various cellular activities are spatiotemporally coordinated on the protein regulatory background to construct the complicated, three-dimensional (3D) structures of organs. Computational simulations using 3D vertex models have been the focus of efforts to approach the mechanisms underlying 3D multicellular constructions, such as dynamics of the 3D monolayer or multilayer cell sheet like epithelia as well as the 3D compacted cell aggregate, including dynamic changes in layer structures. 3D vertex models enable the quantitative simulation of multicellular morphogenesis on the basis of single-cell mechanics, with complete control of various cellular activities such as cell contraction, growth, rearrangement, division, and death. This review describes the general use of the 3D vertex model, along with its applications to several simplified problems of developmental phenomena.

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

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

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

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

Vertex models of epithelial morphogenesis

The dynamic behavior of epithelial cell sheets plays a central role during numerous developmental processes. Genetic and imaging studies of epithelial morphogenesis in a wide range of organisms have led to increasingly detailed mechanisms of cell sheet dynamics. Computational models offer a useful means by which to investigate and test these mechanisms, and have played a key role in the study of cell-cell interactions. A variety of modeling approaches can be used to simulate the balance of forces within an epithelial sheet. Vertex models are a class of such models that consider cells as individual objects, approximated by two-dimensional polygons representing cellular interfaces, in which each vertex moves in response to forces due to growth, interfacial tension, and pressure within each cell. Vertex models are used to study cellular processes within epithelia, including cell motility, adhesion, mitosis, and delamination. This review summarizes how vertex models have been used to provide insight into developmental processes and highlights current challenges in this area, including progressing these models from two to three dimensions and developing new tools for model validation.

Light sheet-based imaging and analysis of early embryogenesis in the fruit fly

The fruit fly is an excellent model system for investigating the sequence of epithelial tissue invaginations constituting the process of gastrulation. By combining recent advancements in light sheet fluorescence microscopy (LSFM) and image processing, the three-dimensional fly embryo morphology and relevant gene expression patterns can be accurately recorded throughout the entire process of embryogenesis. LSFM provides exceptionally high imaging speed, high signal-to-noise ratio, low level of photoinduced damage, and good optical penetration depth. This powerful combination of capabilities makes LSFM particularly suitable for live imaging of the fly embryo.The resulting high-information-content image data are subsequently processed to obtain the outlines of cells and cell nuclei, as well as the geometry of the whole embryo tissue by image segmentation. Furthermore, morphodynamics information is extracted by computationally tracking objects in the image. Towards that goal we describe the successful implementation of a fast fitting strategy of Gaussian mixture models.The data obtained by image processing is well-suited for hypothesis testing of the detailed biomechanics of the gastrulating embryo. Typically this involves constructing computational mechanics models that consist of an objective function providing an estimate of strain energy for a given morphological configuration of the tissue, and a numerical minimization mechanism of this energy, achieved by varying morphological parameters.In this chapter, we provide an overview of in vivo imaging of fruit fly embryos using LSFM, computational tools suitable for processing the resulting images, and examples of computational biomechanical simulations of fly embryo gastrulation.

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

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

A theoretical model of dentinogenesis: dentin and dentinal tubule formation

INTRODUCTION

Dentinogenesis, odontoblast dentin formation, includes dentinal growth, mineralization and dentinal tubule formation. Odontoblasts synthesize collagen resulting in collagen apposition contributing to dentinogenesis. Furthermore, within the tubule, they express non-collagenous proteins, such as dentin phosphoprotein (DPP), associated with hydroxyapatite crystal formation and growth. The aim of this work was to determine patterns of growth and dentin formation and quantification of its mineralization. Findings from our work are relevant to endodontics for future regenerative treatment.

METHODS

We formulated a 3D domain mathematical model, which recreates the events that lead to dentinal tubule mineralization. As reference we used collagen apposition and DPP activity.

RESULTS

We obtained a model depicting predentin's mineralization distribution during dentin development. Furthermore, we verified different DPP diffusion coefficients to test the model's sensitivity.

CONCLUSIONS

We present a model to shed light on the process of dentin and dentinal tubule formation, and its relation to diffusion and mineralization processes.

A multi-structural single cell model of force-induced interactions of cytoskeletal components

Several computational models based on experimental techniques and theories have been proposed to describe cytoskeleton (CSK) mechanics. Tensegrity is a prominent model for force generation, but it cannot predict mechanics of individual CSK components, nor explain the discrepancies from the different single cell stimulating techniques studies combined with cytoskeleton-disruptors. A new numerical concept that defines a multi-structural 3D finite element (FE) model of a single-adherent cell is proposed to investigate the biophysical and biochemical differences of the mechanical role of each cytoskeleton component under loading. The model includes prestressed actin bundles and microtubule within cytoplasm and nucleus surrounded by the actin cortex. We performed numerical simulations of atomic force microscopy (AFM) experiments by subjecting the cell model to compressive loads. The numerical role of the CSK components was corroborated with AFM force measurements on U2OS-osteosarcoma cells and NIH-3T3 fibroblasts exposed to different cytoskeleton-disrupting drugs. Computational simulation showed that actin cortex and microtubules are the major components targeted in resisting compression. This is a new numerical tool that explains the specific role of the cortex and overcomes the difficulty of isolating this component from other networks in vitro. This illustrates that a combination of cytoskeletal structures with their own properties is necessary for a complete description of cellular mechanics.

A multiscale road map of cancer spheroids – incorporating experimental and mathematical modelling to understand cancer progression

Computational models represent a highly suitable framework, not only for testing biological hypotheses and generating new ones but also for optimising experimental strategies. As one surveys the literature devoted to cancer modelling, it is obvious that immense progress has been made in applying simulation techniques to the study of cancer biology, although the full impact has yet to be realised. For example, there are excellent models to describe cancer incidence rates or factors for early disease detection, but these predictions are unable to explain the functional and molecular changes that are associated with tumour progression. In addition, it is crucial that interactions between mechanical effects, and intracellular and intercellular signalling are incorporated in order to understand cancer growth, its interaction with the extracellular microenvironment and invasion of secondary sites. There is a compelling need to tailor new, physiologically relevant in silico models that are specialised for particular types of cancer, such as ovarian cancer owing to its unique route of metastasis, which are capable of investigating anti-cancer therapies, and generating both qualitative and quantitative predictions. This Commentary will focus on how computational simulation approaches can advance our understanding of ovarian cancer progression and treatment, in particular, with the help of multicellular cancer spheroids, and thus, can inform biological hypothesis and experimental design.

The mechanics of metastasis: insights from a computational model

Although it may seem obvious that mechanical forces are required to drive metastatic cell movements, understanding of the mechanical aspects of metastasis has lagged far behind genetic and biochemical knowledge. The goal of this study is to learn about the mechanics of metastasis using a cell-based finite element model that proved useful for advancing knowledge about the forces that drive embryonic cell and tissue movements. Metastasis, the predominant cause of cancer-related deaths, involves a series of mechanical events in which one or more cells dissociate from a primary tumour, migrate through normal tissue, traverse in and out of a multi-layer circulatory system vessel and resettle. The present work focuses on the dissemination steps, from dissociation to circulation. The model shows that certain surface tension relationships must be satisfied for cancerous cells to dissociate from a primary tumour and that these equations are analogous to those that govern dissociation of embryonic cells. For a dissociated cell to then migrate by invadopodium extension and contraction and exhibit the shapes seen in experiments, the invadopodium must generate a contraction equal to approximately twice that produced by the interfacial tension associated with surrounding cells. Intravasation through the wall of a vessel is governed by relationships akin to those in the previous two steps, while release from the vessel wall is governed by equations that involve surface and interfacial tensions. The model raises a number of potential research questions. It also identifies how specific mechanical properties and the sub-cellular structural components that give rise to them might be changed so as to thwart particular metastatic steps and thereby block the spread of cancer.

Cellular interfacial and surface tensions determined from aggregate compression tests using a finite element model

Although previous studies suggested that the interfacial tension gamma(cc) acting along cell-cell boundaries and the effective viscosity mu of the cell cytoplasm could be measured by compressing a spherical aggregate of cells between parallel plates, the mechanical understanding necessary to extract this information from these tests-tests that have provided the surface tension sigma(cm) acting along cell-medium interfaces-has been lacking. These tensions can produce net forces at the subcellular level and give rise to cell motions and tissue reorganization, the rates of which are regulated by mu. Here, a three-dimensional (3D) cell-based finite element model provides insight into the mechanics of the compression test, where these same forces are at work, and leads to quantitative relationships from which the effective viscosity mu of the cell cytoplasm, the tension gamma(cc) that acts along internal cell-cell interfaces and the surface tension sigma(cp) along the cell-platen boundaries can be determined from force-time curves and aggregate profiles. Tests on 5-day embryonic chick mesencephalon, neural retina, liver, and heart aggregates show that all of these properties vary significantly with cell type, except gamma(cc), which is remarkably constant. These properties are crucial for understanding cell rearrangement and tissue self-organization in contexts that include embryogenesis, cancer metastases, and tissue engineering.

Mechanical determinants of epithelium thickness in early-stage embryos

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

Cell sorting in three dimensions: topology, fluctuations, and fluidlike instabilities

Previous 2D and 3D models concluded that cell sorting requires cytoskeletal fluctuations and is stalled by high tension at heterotypic interfaces. New deterministic and stochastic models show that this is not true in 3D. Sorting in 3D involves both topological untangling and domain coalescence. Coalescence requires fluctuations and low tension, but untangling does not. It occurs by a Plateau-Rayleigh instability of cell threads-deterministically driven by high tension. At high minority-cell fractions, untangling dominates and significant partial sorting can occur without fluctuations.

Modeling and simulation of chemomechanics at the cell-matrix interface

Chemomechanical characteristics of the extracellular materials with which cells interact can have a profound impact on cell adhesion and migration. To understand and modulate such complex multiscale processes, a detailed understanding of the feedback between a cell and the adjacent microenvironment is crucial. Here, we use computational modeling and simulation to examine the cell-matrix interaction at both the molecular and continuum lengthscales. Using steered molecular dynamics, we consider how extracellular matrix (ECM) stiffness and extracellular pH influence the interaction between cell surface adhesion receptors and extracellular matrix ligands, and we predict potential consequences for focal adhesion formation and dissolution. Using continuum level finite element simulations and analytical methods to model cell-induced ECM deformation as a function of ECM stiffness and thickness, we consider the implications toward design of synthetic substrata for cell biology experiments that intend to decouple chemical and mechanical cues.

"Developmental mechanics": cellular patterns controlled by adhesion, cortical tension and cell division

How embryos are shaped during development has inspired the work of many, embryologists, geneticists, but also mathematicians such as Turing, and physicists. Despite the inherent complexity of the problems it tackles, developmental biology has produced one of the most spectacular conceptual achievements, demonstrating that embryos are built with conserved molecules that orchestrate pattern and morphogenesis. As the logic of development now emerges, new challenges arise, such as how tissue mechanics is controlled. Quantitative approaches and computational models are essential to predict tissue organization and cell shapes. I review briefly how physical concepts have fueled this research in the past decades.