How Cells Stay Together: A Mechanism for Maintenance of a Robust Cluster Explored by Local and Non-local Continuum Models.

Formation of organs and specialized tissues in embryonic development requires migration of cells to specific targets. In some instances, such cells migrate as a robust cluster. We here explore a recent local approximation of non-local continuum models by Falcó et al. (SIAM J Appl Math 84:17-42, 2023). We apply their theoretical results by specifying biologically-based cell-cell interactions, showing how such cell communication results in an effective attraction-repulsion Morse potential. We then explore the clustering instability, the existence and size of the cluster, and its stability. For attractant-repellent chemotaxis, we derive an explicit condition on cell and chemical properties that guarantee the existence of robust clusters. We also extend their work by investigating the accuracy of the local approximation relative to the full non-local model.

Modeling the Growth and Size Distribution of Human Pluripotent Stem Cell Clusters in Culture.

Human pluripotent stem cells (hPSCs) hold promise for regenerative medicine to replace essential cells that die or become dysfunctional. In some cases, these cells can be used to form clusters whose size distribution affects the growth dynamics. We develop models to predict cluster size distributions of hPSCs based on several plausible hypotheses, including (0) exponential growth, (1) surface growth, (2) Logistic growth, and (3) Gompertz growth. We use experimental data to investigate these models. A partial differential equation for the dynamics of the cluster size distribution is used to fit parameters (rates of growth, mortality, etc.). A comparison of the models using their mean squared error and the Akaike Information criterion suggests that Models 1 (surface growth) or 2 (Logistic growth) best describe the data.

Implant Texture and Capsular Contracture: A Review of Cellular and Molecular Pathways.

Background: Capsular contracture (CC) is a leading cause of morbidity in implant-based breast surgery. Implant surface texture has been implicated in CC development, yet its etiopathogenesis remains unclear. We conducted a systematic review to determine the influence of implant surface texture on cellular and molecular mechanisms involved in the etiopathogenesis of CC. Methods: A systematic review of the MEDLINE, Embase, Web of Science, and Scopus databases was completed to examine the influence of implant texture on cellular and molecular pathways leading to CC. Excluded articles were reviews and those examining solely the clinical presentation of CC. Results: Development of CC includes prolonged inflammation, increased myofibroblast density, parallel arrangement of collagen fibers, and biofilm formation. When compared with textured implants, smooth implants are associated with reduction in parallel collagen, capsule thickness, and sheer frictional force. Microtextured implants trigger a reduced macrophage response and decreased fibroblast activation as compared with smooth and macrotextured surfaces. Bacterial counts on microtextured and smooth surfaces are significantly lower than that of macrotextured surfaces. Both micro- and macrotextured implants have increased matrix metalloproteinases and activation of tumor necrosis factor α pathway, with increased activation of the transforming growth factor ß1 pathway relative to smooth implants. Conclusions: Implant surface texture alters the cellular and molecular mechanisms in the chronic inflammatory process leading to CC. Given the complex biological system of cellular and molecular events in CC, a mathematical model integrating these influences may be optimal to deduce the etiopathogenesis.

Models for Implant-Induced Capsular Contracture Post Breast Cancer Surgery.

Capsular contracture is a painful deformation of scar-tissue that may form around an implant in post-breast cancer reconstruction or cosmetic surgery. Inflammation due to surgical trauma or contamination in the tissue around the implant could account for recruitment of immune cells, and transdifferentiation of resident fibroblasts into cells that deposit abnormally thick collagen. Here we examine this hypothesis using a mathematical model for interacting macrophages, fibroblasts, myofibroblasts, and collagen. Our model demonstrates that cellular response can, together with inflammatory cell recruitment, account for prognoses.

From actin waves to mechanism and back: How theory aids biological understanding.

Actin dynamics in cell motility, division, and phagocytosis is regulated by complex factors with multiple feedback loops, often leading to emergent dynamic patterns in the form of propagating waves of actin polymerization activity that are poorly understood. Many in the actin wave community have attempted to discern the underlying mechanisms using experiments and/or mathematical models and theory. Here, we survey methods and hypotheses for actin waves based on signaling networks, mechano-chemical effects, and transport characteristics, with examples drawn from Dictyostelium discoideum, human neutrophils, Caenorhabditis elegans, and Xenopus laevis oocytes. While experimentalists focus on the details of molecular components, theorists pose a central question of universality: Are there generic, model-independent, underlying principles, or just boundless cell-specific details? We argue that mathematical methods are equally important for understanding the emergence, evolution, and persistence of actin waves and conclude with a few challenges for future studies.

Predicting the number of lifetime divisions for hematopoietic stem cells from telomere length measurements.

How many times does a typical hematopoietic stem cell (HSC) divide to maintain a daily production of over 1011 blood cells over a human lifetime? It has been predicted that relatively few, slowly dividing HSCs occupy the top of the hematopoietic hierarchy. However, tracking HSCs directly is extremely challenging due to their rarity. Here, we utilize previously published data documenting the loss of telomeric DNA repeats in granulocytes, to draw inferences about HSC division rates, the timing of major changes in those rates, as well as lifetime division totals. Our method uses segmented regression to identify the best candidate representations of the telomere length data. Our method predicts that, on average, an HSC divides 56 times over an 85-year lifespan (with lower and upper bounds of 36 and 120, respectively), with half of these divisions during the first 24 years of life.

Cell Repolarization: A Bifurcation Study of Spatio-Temporal Perturbations of Polar Cells.

The intrinsic polarity of migrating cells is regulated by spatial distributions of protein activity. Those proteins (Rho-family GTPases, such as Rac and Rho) redistribute in response to stimuli, determining the cell front and back. Reaction-diffusion equations with mass conservation and positive feedback have been used to explain initial polarization of a cell. However, the sensitivity of a polar cell to a reversal stimulus has not yet been fully understood. We carry out a PDE bifurcation analysis of two polarity models to investigate routes to repolarization: (1) a single-GTPase ("wave-pinning") model and (2) a mutually antagonistic Rac-Rho model. We find distinct routes to reversal in (1) vs. (2). We show numerical simulations of full PDE solutions for the RD equations, demonstrating agreement with predictions of the bifurcation results. Finally, we show that simulations of the polarity models in deforming 1D model cells are consistent with biological experiments.

A multiscale computational model of YAP signaling in epithelial fingering behavior.

In epithelial-mesenchymal transition (EMT), cells organized into sheets break away and become motile mesenchymal cells. EMT plays a crucial role in wound healing, embryonic development, and cancer metastasis. Intracellular signaling in response to mechanical, topographic, or chemical stimuli can promote EMT. We present a multiscale model for EMT downstream of the protein YAP, which suppresses the cell-cell adhesion protein E-cadherin and activates the GTPase Rac1 that enhances cell migration. We first propose an ordinary differential equation (ODE) model for intracellular YAP/Rac1/E-cadherin interactions. The ODE model dynamics are bistable, accounting for both motile loose cells and adherent slower cells. We incorporate this model into a cellular Potts model simulation of two-dimensional wound healing using the open-source platform Morpheus. We show that, under suitable stimuli representing topographic cues, the sheet exhibits finger-like projections and EMT. Morphological differences and quantitative differences in YAP levels as well as variations in cell speed across the sheet are consistent with previous experimental observations of epithelial sheets grown on topographic features in vitro. The simulation is also consistent with experiments that knock down or overexpress YAP, inhibit Rac1, or block E-cadherin.

Cellular Tango: how extracellular matrix adhesion choreographs Rac-Rho signaling and cell movement.

The small GTPases Rac and Rho are known to regulate eukaryotic cell shape, promoting front protrusion (Rac) or rear retraction (Rho) of the cell edge. Such cell deformation changes the contact and adhesion of cell to the extracellular matrix (ECM), while ECM signaling through integrin receptors also affects GTPase activity. We develop and investigate a model for this three-way feedback loop in 1D and 2D spatial domains, as well as in a fully deforming 2D cell shapes with detailed adhesion-bond biophysics. The model consists of reaction-diffusion equations solved numerically with open-source software, Morpheus, and with custom-built cellular Potts model simulations. We find a variety of patterns and cell behaviors, including persistent polarity, flipped front-back cell polarity oscillations, spiral waves, and random protrusion-retraction. We show that the observed spatial patterns depend on the cell shape, and vice versa.

Cross-talk-dependent cortical patterning of Rho GTPases during cell repair.

Rho GTPases such as Rho, Rac, and Cdc42 are important regulators of the cortical cytoskeleton in processes including cell division, locomotion, and repair. In these processes, Rho GTPases assume characteristic patterns wherein the active GTPases occupy mutually exclusive "zones" in the cell cortex. During cell wound repair, for example, a Rho zone encircles the wound edge and is in turn encircled by a Cdc42 zone. Here we evaluated the contributions of cross-talk between Rho and Cdc42 to the patterning of their respective zones in wounded Xenopus oocytes using experimental manipulations in combination with mathematical modeling. The results show that the position of the Cdc42 zone relative to the Rho zone and relative to the wound edge is controlled by the level of Rho activity. In contrast, the outer boundary of the Rho zone is limited by the level of Cdc42 activity. Models based on positive feedback within zones and negative feedback from Rho to the GEF-GAP Abr to Cdc42 capture some, but not all, of the observed behaviors. We conclude that GTPase zone positioning is controlled at the level of Rho activity and we speculate that the Cdc42 zone or something associated with it limits the spread of Rho activity.

Spots, stripes, and spiral waves in models for static and motile cells : GTPase patterns in cells.

The polarization and motility of eukaryotic cells depends on assembly and contraction of the actin cytoskeleton and its regulation by proteins called GTPases. The activity of GTPases causes assembly of filamentous actin (by GTPases Cdc42, Rac), resulting in protrusion of the cell edge. Mathematical models for GTPase dynamics address the spontaneous formation of patterns and nonuniform spatial distributions of such proteins in the cell. Here we revisit the wave-pinning model for GTPase-induced cell polarization, together with a number of extensions proposed in the literature. These include introduction of sources and sinks of active and inactive GTPase (by the group of A. Champneys), and negative feedback from F-actin to GTPase activity. We discuss these extensions singly and in combination, in 1D, and 2D static domains. We then show how the patterns that form (spots, waves, and spirals) interact with cell boundaries to create a variety of interesting and dynamic cell shapes and motion.

Symmetry and fluctuation of cell movements in neural crest-derived facial mesenchyme.

In the face, symmetry is established when bilateral streams of neural crest cells leave the neural tube at the same time, follow identical migration routes and then give rise to the facial prominences. However, developmental instability exists, particularly surrounding the steps of lip fusion. The causes of instability are unknown but inability to cope with developmental fluctuations are a likely cause of congenital malformations, such as non-syndromic orofacial clefts. Here, we tracked cell movements over time in the frontonasal mass, which forms the facial midline and participates in lip fusion, using live-cell imaging of chick embryos. Our mathematical examination of cell velocity vectors uncovered temporal fluctuations in several parameters, including order/disorder, symmetry/asymmetry and divergence/convergence. We found that treatment with a Rho GTPase inhibitor completely disrupted the temporal fluctuations in all measures and blocked morphogenesis. Thus, we discovered that genetic control of symmetry extends to mesenchymal cell movements and that these movements are of the type that could be perturbed in asymmetrical malformations, such as non-syndromic cleft lip. This article has an associated 'The people behind the papers' interview.

Bridging from single to collective cell migration: A review of models and links to experiments.

Mathematical and computational models can assist in gaining an understanding of cell behavior at many levels of organization. Here, we review models in the literature that focus on eukaryotic cell motility at 3 size scales: intracellular signaling that regulates cell shape and movement, single cell motility, and collective cell behavior from a few cells to tissues. We survey recent literature to summarize distinct computational methods (phase-field, polygonal, Cellular Potts, and spherical cells). We discuss models that bridge between levels of organization, and describe levels of detail, both biochemical and geometric, included in the models. We also highlight links between models and experiments. We find that models that span the 3 levels are still in the minority.

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.

Cell Size, Mechanical Tension, and GTPase Signaling in the Single Cell.

Cell polarization requires redistribution of specific proteins to the nascent front and back of a eukaryotic cell. Among these proteins are Rac and Rho, members of the small GTPase family that regulate the actin cytoskeleton. Rac promotes actin assembly and protrusion of the front edge, whereas Rho activates myosin-driven contraction at the back. Mathematical models of cell polarization at many levels of detail have appeared. One of the simplest based on "wave-pinning" consists of a pair of reaction-diffusion equations for a single GTPase. Mathematical analysis of wave-pinning so far is largely restricted to static domains in one spatial dimension. Here, we extend the analysis to cells that change in size, showing that both shrinking and growing cells can lose polarity. We further consider the feedback between mechanical tension, GTPase activation, and cell deformation in both static, growing, shrinking, and moving cells. Special cases (spatially uniform cell chemistry, the absence or presence of mechanical feedback) are analyzed, and the full model is explored by simulations in 1D. We find a variety of novel behaviors, including "dilution-induced" oscillations of Rac activity and cell size, as well as gain or loss of polarization and motility in the model cell.

From energy to cellular forces in the Cellular Potts Model: An algorithmic approach.

Single and collective cell dynamics, cell shape changes, and cell migration can be conveniently represented by the Cellular Potts Model, a computational platform based on minimization of a Hamiltonian. Using the fact that a force field is easily derived from a scalar energy (F = -∇H), we develop a simple algorithm to associate effective forces with cell shapes in the CPM. We predict the traction forces exerted by single cells of various shapes and sizes on a 2D substrate. While CPM forces are specified directly from the Hamiltonian on the cell perimeter, we approximate the force field inside the cell domain using interpolation, and refine the results with smoothing. Predicted forces compare favorably with experimentally measured cellular traction forces. We show that a CPM model with internal signaling (such as Rho-GTPase-related contractility) can be associated with retraction-protrusion forces that accompany cell shape changes and migration. We adapt the computations to multicellular systems, showing, for example, the forces that a pair of swirling cells exert on one another, demonstrating that our algorithm works equally well for interacting cells. Finally, we show forces exerted by cells on one another in classic cell-sorting experiments.

Correlated random walks inside a cell: actin branching and microtubule dynamics.

Correlated random walks (CRW) have been explored in many settings, most notably in the motion of individuals in a swarm or flock. But some subcellular systems such as growth or disassembly of bio-polymers can also be described with similar models and understood using related mathematical methods. Here we consider two examples of growing cytoskeletal elements, actin and microtubules. We use CRW or generalized CRW-like PDEs to model their spatial distributions. In each case, the linear models can be reduced to a Telegrapher's equation. A combination of explicit solutions (in one case) and numerical solutions (in the other) demonstrates that the approach to steady state can be accompanied by (decaying) waves.

Coupling mechanical tension and GTPase signaling to generate cell and tissue dynamics.

Regulators of the actin cytoskeleton such Rho GTPases can modulate forces developed in cells by promoting actomyosin contraction. At the same time, through mechanosensing, tension is known to affect the activity of Rho GTPases. What happens when these effects act in concert? Using a minimal model (1 GTPase coupled to a Kelvin-Voigt element), we show that two-way feedback between signaling ('RhoA') and mechanical tension (stretching) leads to a spectrum of cell behaviors, including contracted or relaxed cells, and cells that oscillate between these extremes. When such 'model cells' are connected to one another in a row or in a 2D sheet ('epithelium'), we observe waves of contraction/relaxation and GTPase activity sweeping through the tissue. The minimal model lends itself to full bifurcation analysis, and suggests a mechanism that explains behavior observed in the context of development and collective cell behavior.

Correction To: Application of Quasi-Steady-State Methods to Nonlinear Models of Intracellular Transport by Molecular Motors.

We apologize for the error in the references.

A Rho-GTPase based model explains spontaneous collective migration of neural crest cell clusters.

We propose a model to explain the spontaneous collective migration of neural crest cells in the absence of an external gradient of chemoattractants. The model is based on the dynamical interaction between Rac1 and RhoA that is known to regulate the polarization, contact inhibition and co-attraction of neural crest cells. Coupling the reaction-diffusion equations for active and inactive Rac1 and RhoA on the cell membrane with a mechanical model for the overdamped motion of membrane vertices, we show that co-attraction and contact inhibition cooperate to produce persistence of polarity in a cluster of neural crest cells by suppressing the random onset of Rac1 hotspots that may mature into new protrusion fronts. This produces persistent directional migration of cell clusters in corridors. Our model confirms a prior hypothesis that co-attraction and contact inhibition are key to spontaneous collective migration, and provides an explanation of their cooperative working mechanism in terms of Rho GTPase signaling. The model shows that the spontaneous migration is more robust for larger clusters, and is most efficient in a corridor of optimal confinement.