Leah Edelstein-Keshet

Neutrophils move through narrow pores, convoluted channels, and tight spaces in tissue to find infection sites. Their ability to sense weak chemical gradients, undergo directed motion, and solve such path-finding problems rests on internal GTPase signaling circuits that control the front protrusion and rear retraction of the cell. Here we explore several variants of known core polarity circuits, with local and long-ranged negative feedback, including inhibitor downstream of Rac, Rac-Rho antagonism, and effects of membrane tension. The resulting reaction-diffusion (RD) equations for Rac and Rho are then used to simulate protrusion-retractions along the edge of a simulated motile cell. We visualize how cells navigate through narrow tracks with sharp corners and weak chemical gradients in 2D. Our metrics for cell performance include polarity initiation, wall-collision intensity, and track completion. In this way, we expose how Rac and Rho, together with their immediate down and upstream components can fine-tune neutrophil motility through complex environments.

Capsular contracture is a pathological response to implant-based reconstructive breast surgery, where the “capsule” (tissue surrounding an implant) painfully thickens, contracts and deforms. It is known to affect breast-cancer survivors at higher rates than healthy women opting for cosmetic breast augmentation with implants. We model the early stages of capsular contracture based on stress-dependent recruitment of contractile and mechanosensitive cells to the implant site. We derive a one-dimensional continuum spatial model for the spatio-temporal evolution of cells and collagen densities away from the implant surface. Various mechanistic assumptions are investigated for linear versus saturating mechanical cell responses and cell traction forces. Our results point to specific risk factors for capsular contracture, and indicate how physiological parameters, as well as initial states (such as inflammation after surgery) contribute to patient susceptibility.

For polarization and directed migration, cells use a combination of local positive feedback and long-range inhibition. We have previously used mathematical models to show the ability of this core circuit to regulate directed cell movement. However, this wave pinning model lacks important additional feedback circuits, including the recently demonstrated local negative feedback from Town and Weiner. Here we extend our models to investigate the consequences of this additional link on cell physiology. We model responses of neutrophil-like HL-60 cells to spatially-controlled optogenetic stimulation of PI3K, leading (via PIP3) to Rac activity. We sequentially build up and investigate partial differential equation (PDE) models of the key Rac, Rac-Inhibitor, and PIP3-Rac-Inhibitor circuits. We fit model parameters to temporal and spatial (cell trajectory) data. Cell shapes, motility, and responses to stimuli are modeled in 2D cell-based simulations, with PDEs for Rac and the other regulatory components solved along the cell edge. We demonstrate that the ability of modeled cells to respond to temporal as well as spatial features of guidance cues depends on the addition of the local negative feedback circuit. Furthermore, the local Rac inhibitor improves the ability of modeled cells to respond to noisy or dynamic extracellular gradients. Our work demonstrates how local negative feedback enhances dynamic polarity and gradient sensing in migratory cells.

Motile eukaryotic cells display distinct modes of migration that often occur within the same cell type. It remains unclear, however, whether transitions between the migratory modes require changes in external conditions, or whether the different modes are coexisting states that emerge from the underlying signaling network. Using a simplified mass-conserved reaction–diffusion model of small GTPase signaling with F-actin mediated feedback, we uncover a distinct bistable mechanism (involving gradient-like phase-separation and traveling waves) and a regime where a polarized mode of migration coexists with spatiotemporal oscillations; the latter, in larger domains, including in three-dimensional surface geometry, result in disordered patterns even in the absence of noise or shape deformations. Indeed, experimental observations of Dictyostelium discoideum show that, upon collision with a rigid boundary, cells may switch from polarized to disordered motion.

We survey a combination of classical and recent experimental and modeling developments in eukaryotic cell migration, from the subcellular processes and regulation, to single and collective cell dynamics. We showcase several examples that demonstrate simulations at several hierarchies: subcellular actin waves, corresponding migratory cell behavior, and collective behavior of several multicellular systems. We argue that the use of shared open-source software packages (such as Morpheus, in our case) to simulate multiscale models would be a boon to the community, allowing us to recreate, investigate, and build on existing work. A brief summary of currently available software is provided.

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.

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.

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.

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.