Periodic migration in a physical model of cells on micropatterns

Deposited footprints let cells switch between confined, oscillatory, and exploratory migration

For eukaryotic cells to heal wounds, respond to immune signals, or metastasize, they must migrate, often by adhering to extracellular matrix (ECM). Cells may also deposit ECM components, leaving behind a footprint that influences their crawling. Recent experiments showed that some epithelial cell lines on micropatterned adhesive stripes move persistently in regions they have previously crawled over, where footprints have been formed, but barely advance into unexplored regions, creating an oscillatory migration of increasing amplitude. Here, we explore through mathematical modeling how footprint deposition and cell responses to footprint combine to allow cells to develop oscillation and other complex migratory motions. We simulate cell crawling with a phase field model coupled to a biochemical model of cell polarity, assuming local contact with the deposited footprint activates Rac1, a protein that establishes the cell's front. Depending on footprint deposition rate and response to the footprint, cells on micropatterned lines can display many types of motility, including confined, oscillatory, and persistent motion. On two-dimensional (2D) substrates, we predict a transition between cells undergoing circular motion and cells developing an exploratory phenotype. Small quantitative changes in a cell's interaction with its footprint can completely alter exploration, allowing cells to tightly regulate their motion, leading to different motility phenotypes (confined vs. exploratory) in different cells when deposition or sensing is variable from cell to cell. Consistent with our computational predictions, we find in earlier experimental data evidence of cells undergoing both circular and exploratory motion.

Deposited footprints let cells switch between confined, oscillatory, and exploratory migration

For eukaryotic cells to heal wounds, respond to immune signals, or metastasize, they must migrate, often by adhering to extracellular matrix. Cells may also deposit extracellular matrix components, leaving behind a footprint that influences their crawling. Recent experiments showed that some epithelial cells on micropatterned adhesive stripes move persistently in regions they have previously crawled over, where footprints have been formed, but barely advance into unexplored regions, creating an oscillatory migration of increasing amplitude. Here, we explore through mathematical modeling how footprint deposition and cell responses to footprint combine to allow cells to develop oscillation and other complex migratory motions. We simulate cell crawling with a phase field model coupled to a biochemical model of cell polarity, assuming local contact with the deposited footprint activates Rac1, a protein that establishes the cell's front. Depending on footprint deposition rate and response to the footprint, cells on micropatterned lines can display many types of motility, including confined, oscillatory, and persistent motion. On two-dimensional substrates, we predict a transition between cells undergoing circular motion and cells developing an exploratory phenotype. Small quantitative changes in a cell's interaction with its footprint can completely alter exploration, allowing cells to tightly regulate their motion, leading to different motility phenotypes (confined vs exploratory) in different cells when deposition or sensing is variable from cell to cell. Consistent with our computational predictions, we find in earlier experimental data evidence of cells undergoing both circular and exploratory motion.

Mesenchymal cell migration on one-dimensional micropatterns

Quantitative studies of mesenchymal cell motion are important to elucidate cytoskeleton function and mechanisms of cell migration. To this end, confinement of cell motion to one dimension (1D) significantly simplifies the problem of cell shape in experimental and theoretical investigations. Here we review 1D migration assays employing micro-fabricated lanes and reflect on the advantages of such platforms. Data are analyzed using biophysical models of cell migration that reproduce the rich scenario of morphodynamic behavior found in 1D. We describe basic model assumptions and model behavior. It appears that mechanical models explain the occurrence of universal relations conserved across different cell lines such as the adhesion-velocity relation and the universal correlation between speed and persistence (UCSP). We highlight the unique opportunity of reproducible and standardized 1D assays to validate theory based on statistical measures from large data of trajectories and discuss the potential of experimental settings embedding controlled perturbations to probe response in migratory behavior.

Three-component contour dynamics model to simulate and analyze amoeboid cell motility in two dimensions

Amoeboid cell motility is relevant in a wide variety of biomedical processes such as wound healing, cancer metastasis, and embryonic morphogenesis. It is characterized by pronounced changes of the cell shape associated with expansions and retractions of the cell membrane, which result in a crawling kind of locomotion. Despite existing computational models of amoeboid motion, the inference of expansion and retraction components of individual cells, the corresponding classification of cells, and the a priori specification of the parameter regime to achieve a specific motility behavior remain challenging open problems. We propose a novel model of the spatio-temporal evolution of two-dimensional cell contours comprising three biophysiologically motivated components: a stochastic term accounting for membrane protrusions and two deterministic terms accounting for membrane retractions by regularizing the shape and area of the contour. Mathematically, these correspond to the intensity of a self-exciting Poisson point process, the area-preserving curve-shortening flow, and an area adjustment flow. The model is used to generate contour data for a variety of qualitatively different, e.g., polarized and non-polarized, cell tracks that visually resemble experimental data very closely. In application to experimental cell tracks, we inferred the protrusion component and examined its correlation to common biomarkers: the F-actin density close to the membrane and its local motion. Due to the low model complexity, parameter estimation is fast, straightforward, and offers a simple way to classify contour dynamics based on two locomotion types: the amoeboid and a so-called fan-shaped type. For both types, we use cell tracks segmented from fluorescence imaging data of the model organism Dictyostelium discoideum. An implementation of the model is provided within the open-source software package AmoePy, a Python-based toolbox for analyzing and simulating amoeboid cell motility.

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.

Coupling cell shape and velocity leads to oscillation and circling in keratocyte galvanotaxis

During wound healing, fish keratocyte cells undergo galvanotaxis where they follow a wound-induced electric field. In addition to their stereotypical persistent motion, keratocytes can develop circular motion without a field or oscillate while crawling in the field direction. We developed a coarse-grained phenomenological model that captures these keratocyte behaviors. We fit this model to experimental data on keratocyte response to an electric field being turned on. A critical element of our model is a tendency for cells to turn toward their long axis, arising from a coupling between cell shape and velocity, which gives rise to oscillatory and circular motion. Galvanotaxis is influenced not only by the field-dependent responses, but also cell speed and cell shape relaxation rate. When the cell reacts to an electric field being turned on, our model predicts that stiff, slow cells react slowly but follow the signal reliably. Cells that polarize and align to the field at a faster rate react more quickly and follow the signal more reliably. When cells are exposed to a field that switches direction rapidly, cells follow the average of field directions, while if the field is switched more slowly, cells follow a "staircase" pattern. Our study indicated that a simple phenomenological model coupling cell speed and shape is sufficient to reproduce a broad variety of different keratocyte behaviors, ranging from circling to oscillation to galvanotactic response, by only varying a few parameters.

Picking winners in cell-cell collisions: Wetting, speed, and contact

Groups of eukaryotic cells can coordinate their crawling motion to follow cues more effectively, stay together, or invade new areas. This collective cell migration depends on cell-cell interactions, which are often studied by colliding pairs of cells together. Can the outcome of these collisions be predicted? Recent experiments on trains of colliding epithelial cells suggest that cells with a smaller contact angle to the surface or larger speeds are more likely to maintain their direction ("win") upon collision. When should we expect shape or speed to correlate with the outcome of a collision? To investigate this question, we build a model for two-cell collisions within the phase field framework, which allows for cell shape changes. We can reproduce the observation that cells with high speed and small contact angles are more likely to win with two different assumptions for how cells interact: (1) velocity aligning, in which we hypothesize that cells sense their own velocity and align to it over a finite timescale, and (2) front-front contact repolarization, where cells polarize away from cell-cell contact, akin to contact inhibition of locomotion. Surprisingly, though we simulate collisions between cells with widely varying properties, in each case, the probability of a cell winning is completely captured by a single summary variable: its relative speed (in the velocity-aligning model) or its relative contact angle (in the contact repolarization model). Both models are currently consistent with reported experimental results, but they can be distinguished by varying cell contact angle and speed through orthogonal perturbations.

Self-organization in amoeboid motility

Amoeboid motility has come to refer to a spectrum of cell migration modes enabling a cell to move in the absence of strong, specific adhesion. To do so, cells have evolved a range of motile surface movements whose physical principles are now coming into view. In response to external cues, many cells—and some single-celled-organisms—have the capacity to turn off their default migration mode. and switch to an amoeboid mode. This implies a restructuring of the migration machinery at the cell scale and suggests a close link between cell polarization and migration mediated by self-organizing mechanisms. Here, I review recent theoretical models with the aim of providing an integrative, physical picture of amoeboid migration.

Sensing the shape of a cell with reaction diffusion and energy minimization

Some dividing cells sense their shape by becoming polarized along their long axis. Cell polarity is controlled in part by polarity proteins, like Rho GTPases, cycling between active membrane-bound forms and inactive cytosolic forms, modeled as a "wave-pinning" reaction-diffusion process. Does shape sensing emerge from wave pinning? We show that wave pinning senses the cell's long axis. Simulating wave pinning on a curved surface, we find that high-activity domains migrate to peaks and troughs of the surface. For smooth surfaces, a simple rule of minimizing the domain perimeter while keeping its area fixed predicts the final position of the domain and its shape. However, when we introduce roughness to our surfaces, shape sensing can be disrupted, and high-activity domains can become localized to locations other than the global peaks and valleys of the surface. On rough surfaces, the domains of the wave-pinning model are more robust in finding the peaks and troughs than the minimization rule, although both can become trapped in steady states away from the peaks and valleys. We can control the robustness of shape sensing by altering the Rho GTPase diffusivity and the domain size. We also find that the shape-sensing properties of cell polarity models can explain how domains localize to curved regions of deformed cells. Our results help to understand the factors that allow cells to sense their shape-and the limits that membrane roughness can place on this process.

Cell motility as an energy minimization process

The dynamics of active matter driven by interacting molecular motors has a nonpotential structure at the local scale. However, we show that there exists a quasipotential effectively describing the collective self-organization of the motors propelling a cell at a continuum active gel level. Such a model allows us to understand cell motility as an active phase transition problem between the static and motile steady-state configurations that minimize the quasipotential. In particular, both configurations can coexist in a metastable fashion and a small stochastic disorder in the gel is sufficient to trigger an intermittent cell dynamics where either static or motile phases are more probable, depending on which state is the global minimum of the quasipotential.

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.

Stick-slip model for actin-driven cell protrusions, cell polarization, and crawling

Cell crawling requires the generation of intracellular forces by the cytoskeleton and their transmission to an extracellular substrate through specific adhesion molecules. Crawling cells show many features of excitable systems, such as spontaneous symmetry breaking and crawling in the absence of external cues, and periodic and propagating waves of activity. Mechanical instabilities in the active cytoskeleton network and feedback loops in the biochemical network of activators and repressors of cytoskeleton dynamics have been invoked to explain these dynamical features. Here, I show that the interplay between the dynamics of cell-substrate adhesion and linear cellular mechanics is sufficient to reproduce many nonlinear dynamical patterns observed in spreading and crawling cells. Using an analytical formalism of the molecular clutch model of cell adhesion, regulated by local mechanical forces, I show that cellular traction forces exhibit stick-slip dynamics resulting in periodic waves of protrusion/retraction and propagating waves along the cell edge. This can explain spontaneous symmetry breaking and polarization of spreading cells, leading to steady crawling or bipedal motion, and bistability, where persistent cell motion requires a sufficiently strong transient external stimulus. The model also highlights the role of membrane tension in providing the long-range mechanical communication across the cell required for symmetry breaking.

Cell position fates and collective fountain flow in bacterial biofilms revealed by light-sheet microscopy

Bacterial biofilms represent a basic form of multicellular organization that confers survival advantages to constituent cells. The sequential stages of cell ordering during biofilm development have been studied in the pathogen and model biofilm-former Vibrio cholerae It is unknown how spatial trajectories of individual cells and the collective motions of many cells drive biofilm expansion. We developed dual-view light-sheet microscopy to investigate the dynamics of biofilm development from a founder cell to a mature three-dimensional community. Tracking of individual cells revealed two distinct fates: one set of biofilm cells expanded ballistically outward, while the other became trapped at the substrate. A collective fountain-like flow transported cells to the biofilm front, bypassing members trapped at the substrate and facilitating lateral biofilm expansion. This collective flow pattern was quantitatively captured by a continuum model of biofilm growth against substrate friction. Coordinated cell movement required the matrix protein RbmA, without which cells expanded erratically. Thus, tracking cell lineages and trajectories in space and time revealed how multicellular structures form from a single founder cell.

Quasi-periodic migration of single cells on short microlanes

Cell migration on microlanes represents a suitable and simple platform for the exploration of the molecular mechanisms underlying cell cytoskeleton dynamics. Here, we report on the quasi-periodic movement of cells confined in stripe-shaped microlanes. We observe persistent polarized cell shapes and directed pole-to-pole motion within the microlanes. Cells depolarize at one end of a given microlane, followed by delayed repolarization towards the opposite end. We analyze cell motility via the spatial velocity distribution, the velocity frequency spectrum and the reversal time as a measure for depolarization and spontaneous repolarization of cells at the microlane ends. The frequent encounters of a boundary in the stripe geometry provides a robust framework for quantitative investigations of the cytoskeleton protrusion and repolarization dynamics. In a first advance to rigorously test physical models of cell migration, we find that the statistics of the cell migration is recapitulated by a Cellular Potts model with a minimal description of cytoskeleton dynamics. Using LifeAct-GFP transfected cells and microlanes with differently shaped ends, we show that the local deformation of the leading cell edge in response to the tip geometry can locally either amplify or quench actin polymerization, while leaving the average reversal times unaffected.

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.

Motility and morphodynamics of confined cells

We introduce a minimal hydrodynamic model of polarization, migration, and deformation of a biological cell confined between two parallel surfaces. In our model, the cell is driven out of equilibrium by an active cytsokeleton force that acts on the membrane. The cell cytoplasm, described as a viscous droplet in the Darcy flow regime, contains a diffusive solute that actively transduces the applied cytoskeleton force. While fairly simple and analytically tractable, this quasi-two-dimensional model predicts a range of compelling dynamic behaviours. A linear stability analysis of the system reveals that solute activity first destabilizes a global polarization-translation mode, prompting cell motility through spontaneous symmetry breaking. At higher activity, the system crosses a series of Hopf bifurcations leading to coupled oscillations of droplet shape and solute concentration profiles. At the nonlinear level, we find traveling-wave solutions associated with unique polarized shapes that resemble experimental observations. Altogether, this model offers an analytical paradigm of active deformable systems in which viscous hydrodynamics are coupled to diffusive force transducers.

Active gel segment behaving as an active particle

We reduce a one-dimensional model of an active segment (AS), which is used, for instance, in the description of contraction-driven cell motility, to a zero-dimensional model of an active particle (AP) characterized by two internal degrees of freedom: position and polarity. Both models give rise to hysteretic force-velocity relations showing that an active agent can support two opposite polarities under the same external force and that it can maintain the same polarity while being dragged by external forces with opposite orientations. This double bistability results in a rich dynamic repertoire which we illustrate by studying static, stalled, motile, and periodically repolarizing regimes displayed by an active agent confined in a viscoelastic environment. We show that the AS and AP models can be calibrated to generate quantitatively similar dynamic responses.

Flower-like patterns in multi-species bacterial colonies

Diverse interactions among species within bacterial colonies lead to intricate spatiotemporal dynamics, which can affect their growth and survival. Here, we describe the emergence of complex structures in a colony grown from mixtures of motile and non-motile bacterial species on a soft agar surface. Time-lapse imaging shows that non-motile bacteria 'hitchhike' on the motile bacteria as the latter migrate outward. The non-motile bacteria accumulate at the boundary of the colony and trigger an instability that leaves behind striking flower-like patterns. The mechanism of the front instability governing this pattern formation is elucidated by a mathematical model for the frictional motion of the colony interface, with friction depending on the local concentration of the non-motile species. A more elaborate two-dimensional phase-field model that explicitly accounts for the interplay between growth, mechanical stress from the motile species, and friction provided by the non-motile species, fully reproduces the observed flower-like patterns.

Dual role of the nucleus in cell migration on planar substrates

Cell migration is essential to sustain life. There have been significant advances in the understanding of the mechanisms that control cell crawling, but the role of the nucleus remains poorly understood. The nucleus exerts a tight control of cell migration in 3D environments, but its influence in 2D migration on planar substrates remains unclear. Here, we study the role of the cell nucleus in 2D cell migration using a computational model of fish keratocytes. Our results indicate that the apparently minor role played by the nucleus emerges from two antagonist effects: While the nucleus modifies the spatial distributions of actin and myosin in a way that reduces cell velocity (e.g., the nucleus displaces myosin to the sides and front of the cell), its mechanical connection with the cytoskeleton alters the intracellular stresses promoting cell migration. Overall, the favorable effect of the nucleus-cytoskeleton connection prevails, which may explain why regular cells usually move faster than enucleated cells.

A minimal computational model for three-dimensional cell migration

During migration, eukaryotic cells can continuously change their three-dimensional morphology, resulting in a highly dynamic and complex process. Further complicating this process is the observation that the same cell type can rapidly switch between different modes of migration. Modelling this complexity necessitates models that are able to track deforming membranes and that can capture the intracellular dynamics responsible for changes in migration modes. Here we develop an efficient three-dimensional computational model for cell migration, which couples cell mechanics to a simple intracellular activator-inhibitor signalling system. We compare the computational results to quantitative experiments using the social amoeba Dictyostelium discoideum. The model can reproduce the observed migration modes generated by varying either mechanical or biochemical model parameters and suggests a coupling between the substrate and the biomechanics of the cell.

Bridging the gap between single-cell migration and collective dynamics

Motivated by the wealth of experimental data recently available, we present a cellular-automaton-based modeling framework focussing on high-level cell functions and their concerted effect on cellular migration patterns. Specifically, we formulate a coarse-grained description of cell polarity through self-regulated actin organization and its response to mechanical cues. Furthermore, we address the impact of cell adhesion on collective migration in cell cohorts. The model faithfully reproduces typical cell shapes and movements down to the level of single cells, yet allows for the efficient simulation of confluent tissues. In confined circular geometries, we find that specific properties of individual cells (polarizability; contractility) influence the emerging collective motion of small cell cohorts. Finally, we study the properties of expanding cellular monolayers (front morphology; stress and velocity distributions) at the level of extended tissues.

Plasticity of cell migration resulting from mechanochemical coupling

Eukaryotic cells can migrate using different modes, ranging from amoeboid-like, during which actin filled protrusions come and go, to keratocyte-like, characterized by a stable morphology and persistent motion. How cells can switch between these modes is not well understood but waves of signaling events are thought to play an important role in these transitions. Here we present a simple two-component biochemical reaction-diffusion model based on relaxation oscillators and couple this to a model for the mechanics of cell deformations. Different migration modes, including amoeboid-like and keratocyte-like, naturally emerge through transitions determined by interactions between biochemical traveling waves, cell mechanics and morphology. The model predictions are explicitly verified by systematically reducing the protrusive force of the actin network in experiments using Dictyostelium discoideum cells. Our results indicate the importance of coupling signaling events to cell mechanics and morphology and may be applicable in a wide variety of cell motility systems.

Cell motility dependence on adhesive wetting

Adhesive cell-substrate interactions are crucial for cell motility and are responsible for the necessary traction that propels cells. These interactions can also change the shape of the cell, analogous to liquid droplet wetting on adhesive substrates. To address how these shape changes affect cell migration and cell speed we model motility using deformable, 2D cross-sections of cells in which adhesion and frictional forces between cell and substrate can be varied separately. Our simulations show that increasing the adhesion results in increased spreading of cells and larger cell speeds. We propose an analytical model which shows that the cell speed is inversely proportional to an effective height of the cell and that increasing this height results in increased internal shear stress. The numerical and analytical results are confirmed in experiments on motile eukaryotic cells.

Microdomain formation is a general property of bacterial membrane proteins and induces heterogeneity of diffusion patterns

Background

Proteins within the cytoplasmic membrane display distinct localization patterns and arrangements. While multiple models exist describing the dynamics of membrane proteins, to date, there have been few systematic studies, particularly in bacteria, to evaluate how protein size, number of transmembrane domains, and temperature affect their diffusion, and if conserved localization patterns exist.

Results

We have used fluorescence microscopy, single-molecule tracking (SMT), and computer-aided visualization methods to obtain a better understanding of the three-dimensional organization of bacterial membrane proteins, using the model bacterium Bacillus subtilis. First, we carried out a systematic study of the localization of over 200 B. subtilis membrane proteins, tagged with monomeric mVenus-YFP at their original gene locus. Their subcellular localization could be discriminated in polar, septal, patchy, and punctate patterns. Almost 20% of membrane proteins specifically localized to the cell poles, and a vast majority of all proteins localized in distinct structures, which we term microdomains. Dynamics were analyzed for selected membrane proteins, using SMT. Diffusion coefficients of the analyzed transmembrane proteins did not correlate with protein molecular weight, but correlated inversely with the number of transmembrane helices, i.e., transmembrane radius. We observed that temperature can strongly influence diffusion on the membrane, in that upon growth temperature upshift, diffusion coefficients of membrane proteins increased and still correlated inversely to the number of transmembrane domains, following the Saffman–Delbrück relation.

Conclusions

The vast majority of membrane proteins localized to distinct multimeric assemblies. Diffusion of membrane proteins can be suitably described by discriminating diffusion coefficients into two protein populations, one mobile and one immobile, the latter likely constituting microdomains. Our results show there is high heterogeneity and yet structural order in the cell membrane, and provide a roadmap for our understanding of membrane organization in prokaryotes.

Electronic supplementary material

The online version of this article (10.1186/s12915-018-0561-0) contains supplementary material, which is available to authorized users.

On the relation between filament density, force generation, and protrusion rate in mesenchymal cell motility

Lamellipodia are flat membrane protrusions formed during mesenchymal motion. Polymerization at the leading edge assembles the actin filament network and generates protrusion force. How this force is supported by the network and how the assembly rate is shared between protrusion and network retrograde flow determines the protrusion rate. We use mathematical modeling to understand experiments changing the F-actin density in lamellipodia of B16-F1 melanoma cells by modulation of Arp2/3 complex activity or knockout of the formins FMNL2 and FMNL3. Cells respond to a reduction of density with a decrease of protrusion velocity, an increase in the ratio of force to filament number, but constant network assembly rate. The relation between protrusion force and tension gradient in the F-actin network and the density dependency of friction, elasticity, and viscosity of the network explain the experimental observations. The formins act as filament nucleators and elongators with differential rates. Modulation of their activity suggests an effect on network assembly rate. Contrary to these expectations, the effect of changes in elongator composition is much weaker than the consequences of the density change. We conclude that the force acting on the leading edge membrane is the force required to drive F-actin network retrograde flow.

Epithelial vertex models with active biochemical regulation of contractility can explain organized collective cell motility

Collective motions of groups of cells are observed in many biological settings such as embryo development, tissue formation, and cancer metastasis. To effectively model collective cell movement, it is important to incorporate cell specific features such as cell size, cell shape, and cell mechanics, as well as active behavior of cells such as protrusion and force generation, contractile forces, and active biochemical signaling mechanisms that regulate cell behavior. In this paper, we develop a comprehensive model of collective cell migration in confluent epithelia based on the vertex modeling approach. We develop a method to compute cell-cell viscous friction based on the vertex model and incorporate RhoGTPase regulation of cortical myosin contraction. Global features of collective cell migration are examined by computing the spatial velocity correlation function. As active cell force parameters are varied, we found rich dynamical behavior. Furthermore, we find that cells exhibit nonlinear phenomena such as contractile waves and vortex formation. Together our work highlights the importance of active behavior of cells in generating collective cell movement. The vertex modeling approach is an efficient and versatile approach to rigorously examine cell motion in the epithelium.

The importance of mechanical constraints for proper polarization and psuedo-cleavage furrow generation in the early Caenorhabditis elegans embryo

Intracellular polarization, where a cell specifies a spatial axis by segregation of specific factors, is a fundamental biological process. In the early embryo of the nematode worm Caenorhabditis elegans (C. elegans), polarization is often accompanied by deformations of the cortex, a highly contractile structure consisting of actin filaments cross-linked by the motor protein myosin (actomyosin). It has been suggested that the eggshell surrounding the early embryo plays a role in polarization although its function is not understood. Here we develop a mathematical model which couples a reaction-diffusion model of actomyosin dynamics with a phase field model of the cell cortex to implicitly track cell shape changes in the early C. elegans embryo. We investigate the potential rigidity effect of the geometric constraint imposed by the presence and size of the eggshell on polarization dynamics. Our model suggests that the geometric constraint of the eggshell is essential for proper polarization and the size of the eggshell also affects the dynamics of polarization. Therefore, we conclude that geometric constraint on a cell might affect the dynamics of a biochemical process.

Polarization, whereby molecules and proteins are asymmetrically distributed throughout the cell, is a vital process for many cellular functions. In the early C. elegans embryo the asymmetric distribution of cell cytoskeleton during the initiation of polarization leads to asymmetric contractions which are higher in the anterior and lower in the posterior of a cell. The C. elegans embryo is surrounded by a rigid body, the eggshell, which functions in numerous cell processes. We investigate the structural support of eggshell during the establishment phase by tracking the moving cell surface. We incorporate protein dynamics involved in polarization into the membrane evolution. We conclude that eggshell might have a role in cell polarization by preventing the distortion of cell surface.

Mechanochemical modeling of neutrophil migration based on four signaling layers, integrin dynamics, and substrate stiffness

Directional neutrophil migration during human immune responses is a highly coordinated process regulated by both biochemical and biomechanical environments. In this paper, we developed an integrative mathematical model of neutrophil migration using a lattice Boltzmann-particle method built in-house to solve the moving boundary problem with spatiotemporal regulation of biochemical components. The mechanical features of the cell cortex are modeled by a series of spring-connected nodes representing discrete cell-substrate adhesive sites. The intracellular signaling cascades responsible for cytoskeletal remodeling [e.g., small GTPases, phosphoinositide-3-kinase (PI3K), and phosphatase and tensin homolog] are built based on our previous four-layered signaling model centered on the bidirectional molecular transport mechanism and implemented as reaction-diffusion equations. Focal adhesion dynamics are determined by force-dependent integrin-ligand binding kinetics and integrin recycling and are thus integrated with cell motion. Using numerical simulations, the model reproduces the major features of cell migration in response to uniform and gradient biochemical stimuli based on the quantitative spatiotemporal regulation of signaling molecules, which agree with experimental observations. The existence of multiple types of integrins with different binding kinetics could act as an adaptation mechanism for substrate stiffness. Moreover, cells can perform reversal, U-turn, or lock-on behaviors depending on the steepness of the reversal biochemical signals received. Finally, this model is also applied to predict the responses of mutants in which PTEN is overexpressed or disrupted.

Cell migration in microengineered tumor environments

Recent advances in microengineered cell migration platforms are discussed critically with a focus on how cell migration is influenced by engineered tumor microenvironments, the medical relevance being to understand how tumor microenvironments may promote or suppress the progression of cancer. We first introduce key findings in cancer cell migration under the influence of the physical environment, which is systematically controlled by microengineering technology, followed by multi-cues of physico-chemical factors, which represent the complexity of the tumor environment. Recognizing that cancer cells constantly communicate not only with each other but also with tumor-associated cells such as vascular, fibroblast, and immune cells, and also with non-cellular components, it follows that cell motility in tumor microenvironments, especially metastasis via the invasion of cancer cells into the extracellular matrix and other tissues, is closely related to the malignancy of cancer-related mortality. Medical relevance of forefront research realized in microfabricated devices, such as single cell sorting based on the analysis of cell migration behavior, may assist personalized theragnostics based on the cell migration phenotype. Furthermore, we urge development of theory and numerical understanding of single or collective cell migration in microengineered platforms to gain new insights in cancer metastasis and in therapeutic strategies.

Complex self-propelled rings: a minimal model for cell motility

Collective behavior of active matter is observed for self-propelled particles, such as vibrated disks and active Brownian particles, as well as for cytoskeletal filaments in motile cells. Here, a system of quasi two-dimensional penetrable self-propelled rods inside rigid rings is used to construct a complex self-propelled particle. The rods interact sterically with each other and with a stationary or mobile ring via a separation-shifted Lennard-Jones potential. They either have a sliding attachment to the inside of the ring at one of their ends, or can move freely within the ring confinement. We study the inner structure and dynamics of the mobile self-propelled rings. We find that these complex particles cannot only be characterized as active Brownian particles, but can also exhibit cell-like motility: random walks, persistent motion, circling, and run-and-circle motion.

Autonomous reciprocating migration of an active material

Periodic to-and-fro migration is a sophisticated mode of locomotion found in many forms of active matter in nature. Providing a general description of periodic migration is challenging, because many details of animal migration remain a mystery. We study periodic migration in a simpler system using a mechanistic model of a photosensitive, active material in which a stimulus-responsive polymer gel is propelled by chemical waves under the regulation of an illumination gradient sensed by the gel, which plays a role analogous to the environment in periodic animal migration. The reciprocating gel migration results from autonomous transitions between retrograde and direct wave locomotion modes arising from the gradient distribution of the illumination intensity. The local dynamics of the chemical waves modulates the asymmetry between push and pull forces to achieve repeated reorientation of the direction of locomotion. Materials that display similar intelligent, self-adaptive locomotion might be tailored for such functions as drug delivery or self-cleaning systems.

Mechanochemical feedback underlies coexistence of qualitatively distinct cell polarity patterns within diverse cell populations

Cell polarization and directional cell migration can display random, persistent, and oscillatory dynamic patterns. However, it is not clear whether these polarity patterns can be explained by the same underlying regulatory mechanism. Here, we show that random, persistent, and oscillatory migration accompanied by polarization can simultaneously occur in populations of melanoma cells derived from tumors with different degrees of aggressiveness. We demonstrate that all of these patterns and the probabilities of their occurrence are quantitatively accounted for by a simple mechanism involving a spatially distributed, mechanochemical feedback coupling the dynamically changing extracellular matrix (ECM)-cell contacts to the activation of signaling downstream of the Rho-family small GTPases. This mechanism is supported by a predictive mathematical model and extensive experimental validation, and can explain previously reported results for diverse cell types. In melanoma, this mechanism also accounts for the effects of genetic and environmental perturbations, including mutations linked to invasive cell spread. The resulting mechanistic understanding of cell polarity quantitatively captures the relationship between population variability and phenotypic plasticity, with the potential to account for a wide variety of cell migration states in diverse pathological and physiological conditions.

A mathematical model coupling polarity signaling to cell adhesion explains diverse cell migration patterns

Protrusion and retraction of lamellipodia are common features of eukaryotic cell motility. As a cell migrates through its extracellular matrix (ECM), lamellipod growth increases cell-ECM contact area and enhances engagement of integrin receptors, locally amplifying ECM input to internal signaling cascades. In contrast, contraction of lamellipodia results in reduced integrin engagement that dampens the level of ECM-induced signaling. These changes in cell shape are both influenced by, and feed back onto ECM signaling. Motivated by experimental observations on melanoma cells lines (1205Lu and SBcl2) migrating on fibronectin (FN) coated topographic substrates (anisotropic post-density arrays), we probe this interplay between intracellular and ECM signaling. Experimentally, cells exhibited one of three lamellipodial dynamics: persistently polarized, random, or oscillatory, with competing lamellipodia oscillating out of phase (Park et al., 2017). Pharmacological treatments, changes in FN density, and substrate topography all affected the fraction of cells exhibiting these behaviours. We use these observations as constraints to test a sequence of hypotheses for how intracellular (GTPase) and ECM signaling jointly regulate lamellipodial dynamics. The models encoding these hypotheses are predicated on mutually antagonistic Rac-Rho signaling, Rac-mediated protrusion (via activation of Arp2/3 actin nucleation) and Rho-mediated contraction (via ROCK phosphorylation of myosin light chain), which are coupled to ECM signaling that is modulated by protrusion/contraction. By testing each model against experimental observations, we identify how the signaling layers interact to generate the diverse range of cell behaviors, and how various molecular perturbations and changes in ECM signaling modulate the fraction of cells exhibiting each. We identify several factors that play distinct but critical roles in generating the observed dynamic: (1) competition between lamellipodia for shared pools of Rac and Rho, (2) activation of RhoA by ECM signaling, and (3) feedback from lamellipodial growth or contraction to cell-ECM contact area and therefore to the ECM signaling level.

Crawling and turning in a minimal reaction-diffusion cell motility model: Coupling cell shape and biochemistry

We study a minimal model of a crawling eukaryotic cell with a chemical polarity controlled by a reaction-diffusion mechanism describing Rho GTPase dynamics. The size, shape, and speed of the cell emerge from the combination of the chemical polarity, which controls the locations where actin polymerization occurs, and the physical properties of the cell, including its membrane tension. We find in our model both highly persistent trajectories, in which the cell crawls in a straight line, and turning trajectories, where the cell transitions from crawling in a line to crawling in a circle. We discuss the controlling variables for this turning instability and argue that turning arises from a coupling between the reaction-diffusion mechanism and the shape of the cell. This emphasizes the surprising features that can arise from simple links between cell mechanics and biochemistry. Our results suggest that similar instabilities may be present in a broad class of biochemical descriptions of cell polarity.

Modeling Contact Inhibition of Locomotion of Colliding Cells Migrating on Micropatterned Substrates

In cancer metastasis, embryonic development, and wound healing, cells can coordinate their motion, leading to collective motility. To characterize these cell-cell interactions, which include contact inhibition of locomotion (CIL), micropatterned substrates are often used to restrict cell migration to linear, quasi-one-dimensional paths. In these assays, collisions between polarized cells occur frequently with only a few possible outcomes, such as cells reversing direction, sticking to one another, or walking past one another. Using a computational phase field model of collective cell motility that includes the mechanics of cell shape and a minimal chemical model for CIL, we are able to reproduce all cases seen in two-cell collisions. A subtle balance between the internal cell polarization, CIL and cell-cell adhesion governs the collision outcome. We identify the parameters that control transitions between the different cases, including cell-cell adhesion, propulsion strength, and the rates of CIL. These parameters suggest hypotheses for why different cell types have different collision behavior and the effect of interventions that modulate collision outcomes. To reproduce the heterogeneity in cell-cell collision outcomes observed experimentally in neural crest cells, we must either carefully tune our parameters or assume that there is significant cell-to-cell variation in key parameters like cell-cell adhesion.

Many cells cooperate with their neighbors to move as a group. However, the mechanisms of these cell-cell interactions are not well understood. One experimental tool to analyze interactions is to allow cells to collide with one another, and see what happens. In order to better understand what features these experiments measure, we develop a computational model of cell-cell collisions, and identify the biochemical and mechanical parameters that lead to different outcomes of collisions. We can recreate all known types of collisions seen in experiments, including cells reversing on contact, sticking, or walking past each other. Our model suggests that what happens in a collision may depend strongly on the mechanical forces between the two cells.

Computational model for amoeboid motion: Coupling membrane and cytosol dynamics

A distinguishing feature of amoeboid motion is that the migrating cell undergoes large deformations, caused by the emergence and retraction of actin-rich protrusions, called pseudopods. Here, we propose a cell motility model that represents pseudopod dynamics, as well as its interaction with membrane signaling molecules. The model accounts for internal and external forces, such as protrusion, contraction, adhesion, surface tension, or those arising from cell-obstacle contacts. By coupling the membrane and cytosol interactions we are able to reproduce a realistic picture of amoeboid motion. The model results are in quantitative agreement with experiments and show how cells may take advantage of the geometry of their microenvironment to migrate more efficiently.

Microtubules stabilize cell polarity by localizing rear signals

Microtubules are known to play an important role in cell polarity; however, the mechanism remains unclear. Using cells migrating persistently on micropatterned strips, we found that depolymerization of microtubules caused cells to change from persistent to oscillatory migration. Mathematical modeling in the context of a local-excitation-global-inhibition control mechanism indicated that this mechanism can account for microtubule-dependent oscillation, assuming that microtubules remove inhibitory signals from the front after a delayed generation. Experiments further supported model predictions that the period of oscillation positively correlates with cell length and that oscillation may be induced by inhibiting retrograde motors. We suggest that microtubules are required not for the generation but for the maintenance of cell polarity, by mediating the global distribution of inhibitory signals. Disassembly of microtubules induces cell oscillation by allowing inhibitory signals to accumulate at the front, which stops frontal protrusion and allows the polarity to reverse.

Polarity mechanisms such as contact inhibition of locomotion regulate persistent rotational motion of mammalian cells on micropatterns

Pairs of endothelial cells on adhesive micropatterns rotate persistently, but pairs of fibroblasts do not; coherent rotation is present in normal mammary acini and kidney cells but absent in cancerous cells. Why? To answer this question, we develop a computational model of pairs of mammalian cells on adhesive micropatterns using a phase field method and study the conditions under which persistent rotational motion (PRM) emerges. Our model couples the shape of the cell, the cell's internal chemical polarity, and interactions between cells such as volume exclusion and adhesion. We show that PRM can emerge from this minimal model and that the cell-cell interface may be influenced by the nucleus. We study the effect of various cell polarity mechanisms on rotational motion, including contact inhibition of locomotion, neighbor alignment, and velocity alignment, where cells align their polarity to their velocity. These polarity mechanisms strongly regulate PRM: Small differences in polarity mechanisms can create significant differences in collective rotation. We argue that the existence or absence of rotation under confinement may lead to insight into the cell's methods for coordinating collective cell motility.

Active elastic dimers: cells moving on rigid tracks

Experiments suggest that the migration of some cells in the three-dimensional extracellular matrix bears strong resemblance to one-dimensional cell migration. Motivated by this observation, we construct and study a minimal one-dimensional model cell made of two beads and an active spring moving along a rigid track. The active spring models the stress fibers with their myosin-driven contractility and α-actinin-driven extendability, while the friction coefficients of the two beads describe the catch and slip-bond behaviors of the integrins in focal adhesions. In the absence of active noise, net motion arises from an interplay between active contractility (and passive extendability) of the stress fibers and an asymmetry between the front and back of the cell due to catch-bond behavior of integrins at the front of the cell and slip-bond behavior of integrins at the back. We obtain reasonable cell speeds with independently estimated parameters. We also study the effects of hysteresis in the active spring, due to catch-bond behavior and the dynamics of cross linking, and the addition of active noise on the motion of the cell. Our model highlights the role of α-actinin in three-dimensional cell motility and does not require Arp2/3 actin filament nucleation for net motion.

Pattern formation by phase-field relaxation of bending energy with fixed surface area and volume

We explore a wide variety of patterns of closed surfaces that minimize the elastic bending energy with fixed surface area and volume. To avoid complicated discretization and numerical instabilities for sharp surfaces, we reformulate the underlying constrained minimization problem by constructing phase-field functionals of bending energy with penalty terms for the constraints and develop stable numerical methods to relax these functionals. We report our extensive computational results with different initial surfaces. These results are discussed in terms of the reduced volume and are compared with the known results obtained using the sharp-interface approach. Finally, we discuss the implications of our numerical findings.

Velocity alignment leads to high persistence in confined cells

Many cell types display random motility on two-dimensional substrates but crawl persistently in a single direction when confined in a microchannel or on an adhesive micropattern. Does this imply that the motility mechanism of confined cells is fundamentally different from that of unconfined cells? We argue that both free- and confined-cell migration may be described by a generic model of cells as "velocity-aligning" active Brownian particles previously proposed to solve a completely separate problem in collective cell migration. Our model can be mapped to a diffusive escape over a barrier and analytically solved to determine the cell's orientation distribution and repolarization rate. In quasi-one-dimensional confinement, velocity-aligning cells maintain their direction for times that can be exponentially larger than their persistence time in the absence of confinement. Our results suggest an important connection between single- and collective-cell migration: high persistence in confined cells corresponds with fast alignment of velocity to cell-cell forces.