Vertex-based Model Research Articles

Modelling cellular interactions with the mechanical properties of the environment through a stochastic vertex-based model of single-cell migration

Modelling cellular interactions with the mechanical properties of the environment through a stochastic vertex-based model of single-cell migration

Topological damping in an ultrafast giant cell

Cellular systems are known to exhibit some of the fastest movements in biology, but little is known as to how single cells can dissipate this energy rapidly and adapt to such large accelerations without disrupting internal architecture. To address this, we investigate Spirostomum ambiguum-a giant cell (1-4 mm in length) well-known to exhibit ultrafast contractions (50% of body length) within 5 ms with a peak acceleration of 15[Formula: see text]. Utilizing transmitted electron microscopy and confocal imaging, we identify an association of rough endoplasmic reticulum (RER) and vacuoles throughout the cell-forming a contiguous fenestrated membrane architecture that topologically entangles these two organelles. A nearly uniform interorganelle spacing of 60 nm is observed between RER and vacuoles, closely packing the entire cell. Inspired by the entangled organelle structure, we study the mechanical properties of entangled deformable particles using a vertex-based model, with all simulation parameters matching 10 dimensionless numbers to ensure dynamic similarity. We demonstrate how entangled deformable particles respond to external loads by an increased viscosity against squeezing and help preserve spatial relationships. Because this enhanced damping arises from the entanglement of two networks incurring a strain-induced jamming transition at subcritical volume fractions, which is demonstrated through the spatial correlation of velocity direction, we term this phenomenon "topological damping." Our findings suggest a mechanical role of RER-vacuolar meshwork as a metamaterial capable of damping an ultrafast contraction event.

Moving horizon estimation of Amnioserosa cell dynamics during Drosophila dorsal closure

Dorsal closure is a morphogenetic process similar to wound healing, whereby a gap in the epithelium is closed through the coordination of the elongation of epidermal cells and contraction of Amnioserosa cells (AS). Throughout the early stages of this process, the AS cells exhibit fluctuations of their apical area which provides an ideal system to understand contractile epithelia, both in terms of the cellular mechanisms and how tissue behavior emerges from the activity of individual cells. In this work, we present a vertex-based model to explore the dynamics of Amnioserosa cells and provide insights into the regulation of dorsal closure. Towards this end, moving horizon estimation is used to predict the mechanics of AS cell oscillations and contractions modeled as a quasi-linear parameter-varying system with bounded unknown parameters. We perform estimation through the minimization of the least-squares moving horizon cost with respect to both state variables and unknown parameters simultaneously. For simulation purposes, data from the work published in [13] are used to evaluate the performance of moving-horizon estimation as compared to the extended Kalman filter. The comparative analysis reveals that the moving-horizon approach performs better than those provided by the Kalman filtering and with an increased robustness to measurement and process noises.

Controlled Neighbor Exchanges Drive Glassy Behavior, Intermittency, and Cell Streaming in Epithelial Tissues

Cell neighbor exchanges are integral to tissue rearrangements in biology, including development and repair. Often these processes occur via topological T1 transitions analogous to those observed in foams, grains and colloids. However, in contrast to in non-living materials the T1 transitions in biological tissues are rate-limited and cannot occur instantaneously due to the finite time required to remodel complex structures at cell-cell junctions. Here we study how this rate-limiting process affects the mechanics and collective behavior of cells in a tissue by introducing this important biological constraint in a theoretical vertex-based model as an intrinsic single-cell property. We report in the absence of this time constraint, the tissue undergoes a motility-driven glass transition characterized by a sharp increase in the intermittency of cell-cell rearrangements. Remarkably, this glass transition disappears as T1 transitions are temporally limited. As a unique consequence of limited rearrangements, we also find that the tissue develops spaitally correlated streams of fast and slow cells, in which the fast cells organize into stream-like patterns with leader-follower interactions, and maintain optimally stable cell-cell contacts. The predictions of this work is compared with existing in-vivo experiments in Drosophila pupal development.

Predicting Collective Migration of Heterogeneous Cell Populations

In basic biological processes of disease and development, heterogeneous populations of cells migrate collectively across tissue contexts. We now know that cells attain front-rear polarity, generate forces, and form adhesions to enable such migration. However, it remains unclear how the ability of cells in a population to dynamically repolarize themselves could regulate the collective response. We devise a vertex-based model in which each deformable cell randomly chooses a new polarization direction after every defined time interval, elongates, generates forces, and enables collective migration. Cell types that repolarize at longer time intervals attain more elongated shapes, migrate faster, and roughen the leading edge. By imaging cell monolayers at high temporal resolution, we show that leading-edge cells possess longer repolarization intervals and elongated shapes compared to those within the monolayer. We defined aggressive mutant leader cells as long repolarization interval and minimal cell-cell contact. When these cells were placed at the leading-edge, they generated more aggressive collective cell migration, as opposed to dispersed mutants. Beyond 2D monolayers, in reality, epithelial cells also form 3D clusters and spheroids of heterogeneous populations. To understand how the dynamic coupling of cell-cell contacts, front-rear polarity, and cell-matrix adhesions impact collective invasion in 3D, we built a Potts-based computational model integrated with the mechanotransduction mechanisms developed earlier in the vertex-based model. Our simulations predict that grouped cells with stronger cell-cell junctions are more likely to invade their 3D environment as collective streams as opposed to single cell scattering. Cells with high matrix adhesions leave spheroids singly or in small clusters. Our vertex and Potts-based computational models predict that heterogeneous cell populations of varying inter- and extra-cellular adhesion and repolarization may generate myriad modes of collective cell migration in 2D and 3D environments, which regulate processes from morphogenesis to tumor invasion.

Mechanical Heterogeneity in Tissues Promotes Rigidity and Controls Cellular Invasion

We study the influence of cell-level mechanical heterogeneity in epithelial tissues using a vertex-based model. Heterogeneity is introduced into the cell shape index (p_{0}) that tunes the stiffness at a single-cell level. The addition of heterogeneity can always enhance the mechanical rigidity of the epithelial layer by increasing its shear modulus, hence making it more rigid. There is an excellent scaling collapse of our data as a function of a single scaling variable f_{r}, which accounts for the overall fraction of rigid cells. We identify a universal threshold f_{r}^{*} that demarcates fluid versus solid tissues. Furthermore, this rigidity onset is far below the contact percolation threshold of rigid cells. These results give rise to a separation of rigidity and contact percolation processes that leads to distinct types of solid states. We also investigate the influence of heterogeneity on tumor invasion dynamics. There is an overall impedance of invasion as the tissue becomes more rigid. Invasion can also occur in an intermediate heterogeneous solid state that is characterized by significant spatial-temporal intermittency.

Predicting Collective Migration of Cell Populations Defined by Varying Repolarization Dynamics

Collective migration of heterogeneous cell populations is an essential aspect of fundamental biological processes, including morphogenesis, wound healing, and tumor invasion. Through experiments and modeling, it has been shown that cells attain front-rear polarity, generate forces, and form adhesions to migrate. However, it remains unclear how the ability of individual cells in a population to dynamically repolarize themselves into new directions could regulate the collective response. We present a vertex-based model in which each deformable cell randomly chooses a new polarization direction after every defined time interval, elongates, proportionally generates forces, and causes collective migration. Our simulations predict that cell types that repolarize at longer time intervals attain more elongated shapes, migrate faster, deform the cell sheet, and roughen the leading edge. By imaging collectively migrating epithelial cell monolayers at high temporal resolution, we found longer repolarization intervals and elongated shapes of cells at the leading edge compared to those within the monolayer. Based on these experimental measurements and simulations, we defined aggressive mutant leader cells by long repolarization interval and minimal intercellular contact. The cells with frequent and random repolarization were defined as normal cells. In simulations with uniformly dispersed leader cells in a normal cell population at a 1:10 ratio, the resulting migration and deformation of the heterogeneous cell sheet remained low. However, when the 10% mutant leaders were placed only at the leading edge, we predicted a rise in the migration of an otherwise normal cell sheet. Our model predicts that a repolarization-based definition of leader cells and their placement within a healthy population can generate myriad modes of collective cell migration, which can enhance our understanding of collective cell migration in disease and development.

Dynamics of PAR Proteins Explain the Oscillation and Ratcheting Mechanisms in Dorsal Closure

We present a vertex-based model for Drosophila dorsal closure that predicts the mechanics of cell oscillation and contraction from the dynamics of the PAR proteins. Based on experimental observations of how aPKC, Par-6, and Bazooka translocate from the circumference of the apical surface to the medial domain, and how they interact with each other and ultimately regulate the apicomedial actomyosin, we formulate a system of differential equations that captures the key features of dorsal closure, including distinctive behaviors in its early, slow, and fast phases. The oscillation in cell area in the early phase of dorsal closure results from an intracellular negative feedback loop that involves myosin, an actomyosin regulator, aPKC, and Bazooka. In the slow phase, gradual sequestration of apicomedial aPKC by Bazooka clusters causes incomplete disassembly of the actomyosin network over each cycle of oscillation, thus producing a so-called ratchet. The fast phase of rapid cell and tissue contraction arises when medial myosin, no longer antagonized by aPKC, builds up in time and produces sustained contraction. Thus, a minimal set of rules governing the dynamics of the PAR proteins, extracted from experimental observations, can account for all major mechanical outcomes of dorsal closure, including the transitions between its three distinct phases.

Mechanical characterization of disordered and anisotropic cellular monolayers.

We consider a cellular monolayer, described using a vertex-based model, for which cells form a spatially disordered array of convex polygons that tile the plane. Equilibrium cell configurations are assumed to minimize a global energy defined in terms of cell areas and perimeters; energy is dissipated via dynamic area and length changes, as well as cell neighbor exchanges. The model captures our observations of an epithelium from a Xenopus embryo showing that uniaxial stretching induces spatial ordering, with cells under net tension (compression) tending to align with (against) the direction of stretch, but with the stress remaining heterogeneous at the single-cell level. We use the vertex model to derive the linearized relation between tissue-level stress, strain, and strain rate about a deformed base state, which can be used to characterize the tissue’s anisotropic mechanical properties; expressions for viscoelastic tissue moduli are given as direct sums over cells. When the base state is isotropic, the model predicts that tissue properties can be tuned to a regime with high elastic shear resistance but low resistance to area changes, or vice versa.

Relating cell shape and mechanical stress in a spatially disordered epithelium using a vertex-based model.

Using a popular vertex-based model to describe a spatially disordered planar epithelial monolayer, we examine the relationship between cell shape and mechanical stress at the cell and tissue level. Deriving expressions for stress tensors starting from an energetic formulation of the model, we show that the principal axes of stress for an individual cell align with the principal axes of shape, and we determine the bulk effective tissue pressure when the monolayer is isotropic at the tissue level. Using simulations for a monolayer that is not under peripheral stress, we fit parameters of the model to experimental data for Xenopus embryonic tissue. The model predicts that mechanical interactions can generate mesoscopic patterns within the monolayer that exhibit long-range correlations in cell shape. The model also suggests that the orientation of mechanical and geometric cues for processes such as cell division are likely to be strongly correlated in real epithelia. Some limitations of the model in capturing geometric features of Xenopus epithelial cells are highlighted.

SMPL

We present a learned model of human body shape and pose-dependent shape variation that is more accurate than previous models and is compatible with existing graphics pipelines. Our Skinned Multi-Person Linear model (SMPL) is a skinned vertex-based model that accurately represents a wide variety of body shapes in natural human poses. The parameters of the model are learned from data including the rest pose template, blend weights, pose-dependent blend shapes, identity-dependent blend shapes, and a regressor from vertices to joint locations. Unlike previous models, the pose-dependent blend shapes are a linear function of the elements of the pose rotation matrices. This simple formulation enables training the entire model from a relatively large number of aligned 3D meshes of different people in different poses. We quantitatively evaluate variants of SMPL using linear or dual-quaternion blend skinning and show that both are more accurate than a Blend-SCAPE model trained on the same data. We also extend SMPL to realistically model dynamic soft-tissue deformations. Because it is based on blend skinning, SMPL is compatible with existing rendering engines and we make it available for research purposes.

Multi-cellular rosettes in the mouse visceral endoderm facilitate the ordered migration of anterior visceral endoderm cells.

Author SummaryThe mouse visceral endoderm (VE) is a simple epithelium in the egg cylinder stage mouse embryo. Many functions associated with epithelia require them to undergo extensive remodelling through changes in the shape and relative positions of constituent cells, a process about which we understand relatively little. The anterior visceral endoderm (AVE) is a specialized group of cells in the simple epithelium of the VE, and their stereotypic migratory behaviour is essential for establishing the orientation of the anterior-posterior axis in the early mouse embryo. We show that AVE migration is linked to changes in cell packing in the VE and an increase in “rosettes,” which are striking collections of five or more cells meeting at a central point. To probe the role of rosettes during AVE migration, we have developed a mathematical model of cell movement in the VE. Simulations suggest that rosettes are not essential for AVE migration, but are crucial for the orderliness of this migration. We also explored the role of Planar Cell Polarity (PCP) signalling, which is known to coordinate cell polarization and rearrangement in many different tissues. We find that mutants in which PCP signalling is disrupted have fewer rosettes, altered epithelial packing, and abnormal AVE migration. We suggest that rosettes in the mouse VE are a PCP-dependent arrangement of cells that act to buffer the disturbances in cell packing generated by AVE migration, thereby enabling AVE cells to migrate in an orderly manner.

A curvature-driven vertex model for two-dimensional grain growth

Abstract A new vertex-based model of two-dimensional grain growth is presented. The motion of each vertex is determined by the curvature-driven motions of the adjacent boundaries. Boundary curvatures are approximated by a geometrical construction that is fully specified by the locations of the three neighbouring vertices. Simulations based on this model exhibit parabolic growth kinetics for the grain aggregate, and the Mullins-von Neumann growth law is obtained for individual grains.