The Moving Boundary Node Method: A level set-based, finite volume algorithm with applications to cell motility

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.

Pseudopod Tracking and Statistics During Cell Movement in Buffer and Chemotaxis

Amoeboid cells such as the protist Dictyostelium, human neutrophils, and the fungus B.d. chytrid move by extending pseudopods. The trajectories of cell movement depend on the size, rhythm, and direction of long series of pseudopods. These pseudopod properties are regulated by internal factors such as memory of previous directions and by external factors such as gradients of chemoattractants or electric currents. Here a simple method is described that defines the X, Y time coordinates of a pseudopod at the start and the end of the extension phase. The connection between the start and end of an extending pseudopod defines a vector, which is the input of different levels of analysis that defines cell movement. The primary information of the vector is its spatial length (pseudopod size), temporal length (extension time), extension rate (size divided by time), and direction. The second layer of information describes the sequence of two (or more) pseudopods: the direction of the second pseudopod relative to the direction of the first pseudopod, the start of the second pseudopod relative to the extension phase of the first pseudopod (the second starts while the first is still extending or after the first has stopped), and the alternating right/left extension of pseudopods. The third layer of information is provided by specific and detailed statistical analysis of these data and addresses question such as: is pseudopod extension in buffer in random direction or has the system internal directional memory, and how do shallow external electrical or chemical gradients bias the intrinsic pseudopod extension. The method is described for Dictyostelium, but has been used successfully for fast-moving neutrophils, slow-moving stem cells, and the fungus B.d. chytrid.

A Hands-on Guide to AmoePy - a Python-Based Software Package to Analyze Cell Migration Data

Amoeboid cell motility is fundamental for a multitude of biological processes such as embryogenesis, immune responses, wound healing, and cancer metastasis. It is characterized by specific cell shape changes: the extension and retraction of membrane protrusions, known as pseudopodia. A common approach to investigate the mechanisms underlying this type of cell motility is to study phenotypic differences in the locomotion of mutant cell lines. To characterize such differences, methods are required to quantify the contour dynamics of migrating cells. AmoePy is a Python-based software package that provides tools for cell segmentation, contour detection as well as analyzing and simulating contour dynamics. First, a digital representation of the cell contour as a chain of nodes is extracted from each frame of a time-lapse microscopy recording of a moving cell. Then, the dynamics of these nodes-referred to as virtual markers-are tracked as the cell contour evolves over time. From these data, various quantities can be calculated that characterize the contour dynamics, such as the displacement of the virtual markers or the local stretching rate of the marker chain. Their dynamics is typically visualized in space-time plots, the so-called kymographs, where the temporal evolution is displayed for the different locations along the cell contour. Using AmoePy, you can straightforwardly create kymograph plots and videos from stacks of experimental bright-field or fluorescent images of motile cells. A hands-on guide on how to install and use AmoePy is provided in this chapter.

Biomechanics of cells and subcellular components: A comprehensive review of computational models and applications

Cells are a fundamental structural, functional and biological unit for all living organisms. Up till now, considerable efforts have been made to study the responses of single cells and subcellular components to an external load, and understand the biophysics underlying cell rheology, mechanotransduction and cell functions using experimental and in silico approaches. In the last decade, computational simulation has become increasingly attractive due to its critical role in interpreting experimental data, analysing complex cellular/subcellular structures, facilitating diagnostic designs and therapeutic techniques, and developing biomimetic materials. Despite the significant progress, developing comprehensive and accurate models of living cells remains a grand challenge in the 21st century. To understand current state of the art, this review summarises and classifies the vast array of computational biomechanical models for cells. The article covers the cellular components at multi-spatial levels, that is, protein polymers, subcellular components, whole cells and the systems with scale beyond a cell. In addition to the comprehensive review of the topic, this article also provides new insights into the future prospects of developing integrated, active and high-fidelity cell models that are multiscale, multi-physics and multi-disciplinary in nature. This review will be beneficial for the researchers in modelling the biomechanics of subcellular components, cells and multiple cell systems and understanding the cell functions and biological processes from the perspective of cell mechanics.

A mesoscale mechanical model of cellular interactions

Computational models of cell mechanics allow the precise interrogation of cell shape change. These morphological changes are required for cells to survive in diverse tissue environments. Here, we present a mesoscale mechanical model of cell-substrate interactions using the level set method based on experimentally measured parameters. By implementing a viscoelastic mechanical equivalent circuit, we accurately model whole-cell deformations that are important for a variety of cellular processes. To effectively model shape changes as a cell interacts with a substrate, we have included receptor-mediated adhesion, which is governed by catch-slip bond behavior. The effect of adhesion was explored by subjecting cells to a variety of different substrates including flat, curved, and deformable surfaces. Finally, we increased the accuracy of our simulations by including a deformable nucleus in our cells. This model sets the foundation for further exploration into computational analyses of multicellular interactions.

Analysis of protrusion dynamics in amoeboid cell motility by means of regularized contour flows

Amoeboid cell motility is essential for a wide range of biological processes including wound healing, embryonic morphogenesis, and cancer metastasis. It relies on complex dynamical patterns of cell shape changes that pose long-standing challenges to mathematical modeling and raise a need for automated and reproducible approaches to extract quantitative morphological features from image sequences. Here, we introduce a theoretical framework and a computational method for obtaining smooth representations of the spatiotemporal contour dynamics from stacks of segmented microscopy images. Based on a Gaussian process regression we propose a one-parameter family of regularized contour flows that allows us to continuously track reference points (virtual markers) between successive cell contours. We use this approach to define a coordinate system on the moving cell boundary and to represent different local geometric quantities in this frame of reference. In particular, we introduce the local marker dispersion as a measure to identify localized membrane expansions and provide a fully automated way to extract the properties of such expansions, including their area and growth time. The methods are available as an open-source software package called AmoePy, a Python-based toolbox for analyzing amoeboid cell motility (based on time-lapse microscopy data), including a graphical user interface and detailed documentation. Due to the mathematical rigor of our framework, we envision it to be of use for the development of novel cell motility models. We mainly use experimental data of the social amoeba Dictyostelium discoideum to illustrate and validate our approach.

Amoeboid motion is a crawling-like cell migration that plays an important key role in multiple biological processes such as wound healing and cancer metastasis. This type of cell motility results from expanding and simultaneously contracting parts of the cell membrane. From fluorescence images, we obtain a sequence of points, representing the cell membrane, for each time step. By using regression analysis on these sequences, we derive smooth representations, so-called contours, of the membrane. Since the number of measurements is discrete and often limited, the question is raised of how to link consecutive contours with each other. In this work, we present a novel mathematical framework in which these links are described by regularized flows allowing a certain degree of concentration or stretching of neighboring reference points on the same contour. This stretching rate, the so-called local dispersion, is used to identify expansions and contractions of the cell membrane providing a fully automated way of extracting properties of these cell shape changes. We applied our methods to time-lapse microscopy data of the social amoeba Dictyostelium discoideum.

Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry

Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.

Tools for computational analysis of moving boundary problems in cellular mechanobiology

A cell's ability to change shape is one of the most fundamental biological processes and is essential for maintaining healthy organisms. When the ability to control shape goes awry, it often results in a diseased system. As such, it is important to understand the mechanisms that allow a cell to sense and respond to its environment so as to maintain cellular shape homeostasis. Because of the inherent complexity of the system, computational models that are based on sound theoretical understanding of the biochemistry and biomechanics and that use experimentally measured parameters are an essential tool. These models involve an inherent feedback, whereby shape is determined by the action of regulatory signals whose spatial distribution depends on the shape. To carry out computational simulations of these moving boundary problems requires special computational techniques. A variety of alternative approaches, depending on the type and scale of question being asked, have been used to simulate various biological processes, including cell motility, division, mechanosensation, and cell engulfment. In general, these models consider the forces that act on the system (both internally generated, or externally imposed) and the mechanical properties of the cell that resist these forces. Moving forward, making these techniques more accessible to the non-expert will help improve interdisciplinary research thereby providing new insight into important biological processes that affect human health. This article is categorized under: Cancer > Cancer>Computational Models Cancer > Cancer>Molecular and Cellular Physiology.

Membrane Tension Can Enhance Adaptation to Maintain Polarity of Migrating Cells

Migratory cells are known to adapt to environments that contain wide-ranging levels of chemoattractant. Although biochemical models of adaptation have been previously proposed, here, we discuss a different mechanism based on mechanosensing, in which the interaction between biochemical signaling and cell tension facilitates adaptation. We describe and analyze a model of mechanochemical-based adaptation coupling a mechanics-based physical model of cell tension coupled with the wave-pinning reaction-diffusion model for Rac GTPase activity. The mathematical analysis of this model, simulations of a simplified one-dimensional cell geometry, and two-dimensional finite element simulations of deforming cells reveal that as a cell protrudes under the influence of high stimulation levels, tension-mediated inhibition of Rac signaling causes the cell to polarize even when initially overstimulated. Specifically, tension-mediated inhibition of Rac activation, which has been experimentally observed in recent years, facilitates this adaptation by countering the high levels of environmental stimulation. These results demonstrate how tension-related mechanosensing may provide an alternative (and potentially complementary) mechanism for cell adaptation.

Active random forces can drive differential cellular positioning and enhance motor-driven transport

Cells are remarkable machines capable of performing an exquisite range of functions, many of which depend crucially on the activity of molecular motors that generate forces. Recent experiments have shown that intracellular random movements are not solely thermal in nature but also arise from stochasticity in the forces from these molecular motors. Here we consider the effects of these nonthermal random forces. We show that stochastic motor force not only enhances diffusion but also leads to size-dependent transport of objects that depends on the local density of the cytoskeletal filaments on which motors operate. As a consequence, we find that objects that are larger than the mesh size of the cytoskeleton should be attracted to regions of high cytoskeletal density, while objects that are smaller than the mesh size will preferentially avoid these regions. These results suggest a mechanism for size-based organelle positioning and also suggest that motor-driven random forces can additionally enhance motor-driven transport.

A minimal mechanosensing model predicts keratocyte evolution on flexible substrates

A mathematical model is proposed for shape evolution and locomotion of fish epidermal keratocytes on elastic substrates. The model is based on mechanosensing concepts: cells apply contractile forces onto the elastic substrate, while cell shape evolution depends locally on the substrate stress generated by themselves or external mechanical stimuli acting on the substrate. We use the level set method to study the behaviour of the model numerically, and predict a number of distinct phenomena observed in experiments, such as (i) symmetry breaking from the stationary centrosymmetric to the well-known steadily propagating crescent shape, (ii) asymmetric bipedal oscillations and travelling waves in the lamellipodium leading edge, (iii) response to remote mechanical stress externally applied to the substrate (tensotaxis) and (iv) changing direction of motion towards an interface with a rigid substrate (durotaxis).

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.

Swimming bacteria power microspin cycles

Dense suspensions of swimming bacteria are living fluids, an archetype of active matter. For example, Bacillus subtilis confined within a disc-shaped region forms a persistent stable vortex that counterrotates at the periphery. Here, we examined Escherichia coli under similar confinement and found that these bacteria, instead, form microspin cycles: a single vortex that periodically reverses direction on time scales of seconds. Using experimental perturbations of the confinement geometry, medium viscosity, bacterial length, density, and chemotaxis pathway, we show that morphological alterations of the bacteria transition a stable vortex into a periodically reversing one. We develop a mathematical model based on single-cell biophysics that quantitatively recreates the dynamics of these vortices and predicts that density gradients power the reversals. Our results define how microbial physics drives the active behavior of dense bacterial suspensions and may allow one to engineer novel micromixers for biomedical and other microfluidic applications.

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.

Going with the Flow: Water Flux and Cell Shape during Cytokinesis

Cell shape changes during cytokinesis in eukaryotic cells have been attributed to contractile forces from the actomyosin ring and the actomyosin cortex. Here we propose an additional mechanism where active pumping of ions and water at the cell poles and the division furrow can also achieve the same type of shape change during cytokinesis without myosin contraction. We develop a general mathematical model to examine shape changes in a permeable object subject to boundary fluxes. We find that hydrodynamic flows in the cytoplasm and the relative drag between the cytoskeleton network phase and the water phase also play a role in determining the cell shape during cytokinesis. Forces from the actomyosin contractile ring and cortex do contribute to the cell shape, and can work together with water permeation to facilitate cytokinesis. To influence water flow, we osmotically shock the cell during cell division, and find that the cell can actively adapt to osmotic changes and complete division. Depolymerizing the actin cytoskeleton during cytokinesis also does not affect the contraction speed. We also explore the role of membrane ion channels and pumps in setting up the spatially varying water flux.

New Continuum Approaches for Determining Protein-Induced Membrane Deformations

The influence of the membrane on transmembrane proteins is central to a number of biological phenomena, notably the gating of stretch activated ion channels. Conversely, membrane proteins can influence the bilayer, leading to the stabilization of particular membrane shapes, topological changes that occur during vesicle fission and fusion, and shape-dependent protein aggregation. Continuum elastic models of the membrane have been widely used to study protein-membrane interactions. These mathematical approaches produce physically interpretable membrane shapes, energy estimates for the cost of deformation, and a snapshot of the equilibrium configuration. Moreover, elastic models are much less computationally demanding than fully atomistic and coarse-grained simulation methodologies; however, it has been argued that continuum models cannot reproduce the distortions observed in fully atomistic molecular dynamics simulations. We suggest that this failure can be overcome by using chemically and geometrically accurate representations of the protein. Here, we present a fast and reliable hybrid continuum-atomistic model that couples the protein to the membrane. We show that the model is in excellent agreement with fully atomistic simulations of the ion channel gramicidin embedded in a POPC membrane. Our continuum calculations not only reproduce the membrane distortions produced by the channel but also accurately determine the channel's orientation. Finally, we use our method to investigate the role of membrane bending around the charged voltage sensors of the transient receptor potential cation channel TRPV1. We find that membrane deformation significantly stabilizes the energy of insertion of TRPV1 by exposing charged residues on the S4 segment to solution.

Physical Mechanisms of Cancer in the Transition to Metastasis

Whether a tumor is metastatic is one of the most significant factors that influence the prognosis for a cancer patient. The transition from a nonmetastatic tumor to a metastatic one is accompanied by a number of genetic and proteomic changes within the tumor cells. These protein-level changes conspire to produce behavioral changes in the cells: cells that had been relatively stationary begin to move, often as a group. In this study we ask the question of what cell-level biophysical changes are sufficient to initiate evasion away from an otherwise static tumor. We use a mathematical model developed to describe the biophysics of epithelial tissue to explore this problem. The model is first validated against in vitro wound healing experiments with cancer cell lines. Then we simulate the behavior of a group of mutated cells within a sea of healthy tissue. We find that moderate increases in adhesion between the cell and extracellular matrix (ECM) accompanied by a decrease in cell-cell adhesion and/or Rho family of small GTPase activation can cause a group of cells to break free from a tumor and spontaneously migrate. This result may explain why some metastatic cells have been observed to upregulate integrin, downregulate cadherin, and activate Rho family signaling.

Mechanisms of Cell Polarization

Cell polarization is a key step in the migration, development, and organization of eukaryotic cells, both at the single cell and multicellular level. Research on the mechanisms that give rise to polarization of a given cell, and organization of polarity within a tissue has led to new understanding across cellular and developmental biology. In this review, we describe some of the history of theoretical and experimental aspects of the field, as well as some interesting questions and challenges for the future.

Continuum descriptions of membranes and their interaction with proteins: Towards chemically accurate models

Biological membranes deform in response to resident proteins leading to a coupling between membrane shape and protein localization. Additionally, the membrane influences the function of membrane proteins. Here we review contributions to this field from continuum elastic membrane models focusing on the class of models that couple the protein to the membrane. While it has been argued that continuum models cannot reproduce the distortions observed in fully-atomistic molecular dynamics simulations, we suggest that this failure can be overcome by using chemically accurate representations of the protein. We outline our recent advances along these lines with our hybrid continuum-atomistic model, and we show the model is in excellent agreement with fully-atomistic simulations of the nhTMEM16 lipid scramblase. We believe that the speed and accuracy of continuum-atomistic methodologies will make it possible to simulate large scale, slow biological processes, such as membrane morphological changes, that are currently beyond the scope of other computational approaches. This article is part of a Special Issue entitled: Membrane Proteins edited by J.C. Gumbart and Sergei Noskov.

A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis

In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk-surface reaction-diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk-surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane.

The chaser and the chased: a phase-field model of an immune response

In this paper we present a model for an immune response to an invading pathogen. Particularly, we follow the motion of a neutrophil as it migrates to the site of infection guided by chemical cues, a mechanism termed chemotaxis, with the ability to reorient itself as the pathogen changes its position. In the process, the cell undergoes morphological alterations, in addition to the structural changes observed at its leading edge. Also, we derive a condition for a successful immune reaction by relating the speed of the neutrophil to that of the pathogen and to the diffusion coefficient of the chemical attractant.

A conservative algorithm for parabolic problems in domains with moving boundaries

We describe a novel conservative algorithm for parabolic problems in domains with moving boundaries developed for modeling in cell biology. The spatial discretization is accomplished by applying Voronoi decomposition to a fixed rectangular grid. In the vicinity of the boundary, the procedure generates irregular Voronoi cells that conform to the domain shape and merge seamlessly with regular control volumes in the domain interior. Consequently, our algorithm is free of the CFL stability issue due to moving interfaces and does not involve cell-merging or mass redistribution. Local mass conservation is ensured by finite-volume discretization and natural-neighbor interpolation. Numerical experiments with two-dimensional geometries demonstrate exact mass conservation and indicate an order of convergence in space between one and two. The use of standard meshing techniques makes extension of the method to three dimensions conceptually straightforward.

From simple to detailed models for cell polarization

Many mathematical models have been proposed for the process of cell polarization. Some of these are 'functional models' that capture a class of dynamical behaviour, whereas others are derived from features of signalling molecules. Some mechanistic models are detailed, and therefore complex, whereas others are simplified. Each type contributes to our understanding of cell polarization. However, the huge variety at different levels of detail makes comparisons challenging. Here, we provide examples of both elementary and more detailed models for polarization. We also display how a recent mathematical method, local perturbation analysis, can provide an appropriate tool for such comparisons. This technique simplifies and speeds up the model development process by revealing the effect of model extensions, parameter variations and in silico manipulations such as knock-out or over-expression of key molecules. Finally, simulations in both one dimension and two dimensions, and particularly in deforming two-dimensional 'cells', can highlight behaviour not captured by traditional simulation methods.

Phase-field approach to chemotactic driving of neutrophil morphodynamics

To simulate the motion of neutrophils and their morphodynamics in response to chemical cues, we construct a model based on the phase-field method utilizing a description with a free-energy functional and associated dynamics which captures the basic features of the phenomenon. We additionally incorporate spatial sensing by introducing an auxiliary field which depicts the polymerization of the region of the cell facing the highest concentration of the chemical attractant.

Interaction of motility, directional sensing, and polarity modules recreates the behaviors of chemotaxing cells

Chemotaxis involves the coordinated action of separable but interrelated processes: motility, gradient sensing, and polarization. We have hypothesized that these are mediated by separate modules that account for these processes individually and that, when combined, recreate most of the behaviors of chemotactic cells. Here, we describe a mathematical model where the modules are implemented in terms of reaction-diffusion equations. Migration and the accompanying changes in cellular morphology are demonstrated in simulations using a mechanical model of the cell cortex implemented in the level set framework. The central module is an excitable network that accounts for random migration. The response to combinations of uniform stimuli and gradients is mediated by a local excitation, global inhibition module that biases the direction in which excitability is directed. A polarization module linked to the excitable network through the cytoskeleton allows unstimulated cells to move persistently and, for cells in gradients, to gradually acquire distinct sensitivity between front and back. Finally, by varying the strengths of various feedback loops in the model we obtain cellular behaviors that mirror those of genetically altered cell lines.

Chemotaxis is the movement of cells in response to spatial gradients of chemical cues. While single-celled organisms rely on sensing and responding to chemical gradients to search for nutrients, chemotaxis is also an essential component of the mammalian immune system. However, chemotaxis can also be deleterious, since chemotactic tumor cells can lead to metastasis. Due to its importance, understanding the process by which cells sense and respond to chemical gradients has attracted considerable interest. Moreover, because of the complexity of chemotactic signaling, which includes multiple feedback loops and redundant pathways, this has been a research area in which computational models have had a significant impact in understanding experimental findings. Here, we propose a modular description of the signaling network that regulates chemotaxis. The modules describe different processes that are observed in chemotactic cells. In addition to accounting for these behaviors individually, we show that the overall system recreates many features of the directed motion of migrating cells. The signaling described by our modules is implemented as a series of equations, whereas movement and the accompanying cellular deformations are simulated using a mechanical model of the cell and implemented using level set methods, a method that allows simulations of cells as they change morphology.

A comparison of computational models for eukaryotic cell shape and motility

Eukaryotic cell motility involves complex interactions of signalling molecules, cytoskeleton, cell membrane, and mechanics interacting in space and time. Collectively, these components are used by the cell to interpret and respond to external stimuli, leading to polarization, protrusion, adhesion formation, and myosin-facilitated retraction. When these processes are choreographed correctly, shape change and motility results. A wealth of experimental data have identified numerous molecular constituents involved in these processes, but the complexity of their interactions and spatial organization make this a challenging problem to understand. This has motivated theoretical and computational approaches with simplified caricatures of cell structure and behaviour, each aiming to gain better understanding of certain kinds of cells and/or repertoire of behaviour. Reaction–diffusion (RD) equations as well as equations of viscoelastic flows have been used to describe the motility machinery. In this review, we describe some of the recent computational models for cell motility, concentrating on simulations of cell shape changes (mainly in two but also three dimensions). The problem is challenging not only due to the difficulty of abstracting and simplifying biological complexity but also because computing RD or fluid flow equations in deforming regions, known as a “free-boundary” problem, is an extremely challenging problem in applied mathematics. Here we describe the distinct approaches, comparing their strengths and weaknesses, and the kinds of biological questions that they have been able to address.

Signaling regulated endocytosis and exocytosis lead to mating pheromone concentration dependent morphologies in yeast

Polarized cell morphogenesis requires actin cytoskeleton rearrangement for polarized transport of proteins, organelles and secretory vesicles, which fundamentally underlies cell differentiation and behavior. During yeast mating, Saccharomyces cerevisiae responds to extracellular pheromone gradients by extending polarized projections, which are likely maintained through vesicle transport to (exocytosis) and from (endocytosis) the membrane. We experimentally demonstrate that the projection morphology is pheromone concentration-dependent, and propose the underlying mechanism through mathematical modeling. The inclusion of membrane flux and dynamically evolving cell boundary into our yeast mating signaling model shows good agreement with experimental measurements, and provides a plausible explanation for pheromone-induced cell morphology.

Tracking monotonically advancing boundaries in image sequences using graph cuts and recursive kernel shape priors

We introduce a probabilistic computer vision technique to track monotonically advancing boundaries of objects within image sequences. Our method incorporates a novel technique for including statistical prior shape information into graph-cut based segmentation, with the aid of a majorization-minimization algorithm. Extension of segmentation from single images to image sequences then follows naturally using sequential Bayesian estimation. Our methodology is applied to two unrelated sets of real biomedical imaging data, and a set of synthetic images. Our results are shown to be superior to manual segmentation.

How cells integrate complex stimuli: the effect of feedback from phosphoinositides and cell shape on cell polarization and motility

To regulate shape changes, motility and chemotaxis in eukaryotic cells, signal transduction pathways channel extracellular stimuli to the reorganization of the actin cytoskeleton. The complexity of such networks makes it difficult to understand the roles of individual components, let alone their interactions and multiple feedbacks within a given layer and between layers of signalling. Even more challenging is the question of if and how the shape of the cell affects and is affected by this internal spatiotemporal reorganization. Here we build on our previous 2D cell motility model where signalling from the Rho family GTPases (Cdc42, Rac, and Rho) was shown to organize the cell polarization, actin reorganization, shape change, and motility in simple gradients. We extend this work in two ways: First, we investigate the effects of the feedback between the phosphoinositides (PIs) , and Rho family GTPases. We show how that feedback increases heights and breadths of zones of Cdc42 activity, facilitating global communication between competing cell “fronts”. This hastens the commitment to a single lamellipodium initiated in response to multiple, complex, or rapidly changing stimuli. Second, we show how cell shape feeds back on internal distribution of GTPases. Constraints on chemical isocline curvature imposed by boundary conditions results in the fact that dynamic cell shape leads to faster biochemical redistribution when the cell is repolarized. Cells with frozen cytoskeleton, and static shapes, consequently respond more slowly to reorienting stimuli than cells with dynamic shape changes, the degree of the shape-induced effects being proportional to the extent of cell deformation. We explain these concepts in the context of several in silico experiments using our 2D computational cell model.

Single cells, such as amoeba and white blood cells, change shape and move in response to environmental stimuli. Their behaviour is a consequence of the intracellular properties balanced by external forces. The internal regulation is modulated by several proteins that interact with one another and with membrane lipids. We examine, through in silico experiments using a computational model of a moving cell, the interactions of an important class of such proteins (Rho GTPases) and lipids (phosphoinositides, PIs), their spatial redistribution, and how they affect and are affected by cell shape. Certain GTPases promote the assembly of the actin cytoskeleton. This then leads to the formation of a cell protrusion, the leading edge. The feedback between PIs and GTPases facilitates global communication across the cell, ensuring that multiple, complex, or rapidly changing stimuli can be resolved into a single decision for positioning the leading edge. Interestingly, the cell shape itself affects the intracellular biochemistry, resulting from interactions between the curvature of the chemical fronts and the cell edge. Cells with static shapes consequently respond more slowly to reorienting stimuli than cells with dynamic shape changes. This potential to respond more rapidly to external stimuli depends on the degree of cellular shape deformation.

Redundant mechanisms for stable cell locomotion revealed by minimal models

Crawling of eukaryotic cells on flat surfaces is underlain by the protrusion of the actin network, the contractile activity of myosin II motors, and graded adhesion to the substrate regulated by complex biochemical networks. Some crawling cells, such as fish keratocytes, maintain a roughly constant shape and velocity. Here we use moving-boundary simulations to explore four different minimal mechanisms for cell locomotion: 1), a biophysical model for myosin contraction-driven motility; 2), a G-actin transport-limited motility model; 3), a simple model for Rac/Rho-regulated motility; and 4), a model that assumes that microtubule-based transport of vesicles to the leading edge limits the rate of protrusion. We show that all of these models, alone or in combination, are sufficient to produce half-moon steady shapes and movements that are characteristic of keratocytes, suggesting that these mechanisms may serve redundant and complementary roles in driving cell motility. Moving-boundary simulations demonstrate local and global stability of the motile cell shapes and make testable predictions regarding the dependence of shape and speed on mechanical and biochemical parameters. The models shed light on the roles of membrane-mediated area conservation and the coupling of mechanical and biochemical mechanisms in stabilizing motile cells.

A computational model of cell polarization and motility coupling mechanics and biochemistry

The motion of a eukaryotic cell presents a variety of interesting and challenging problems from both a modeling and a computational perspective. The processes span many spatial scales (from molecular to tissue) as well as disparate time scales, with reaction kinetics on the order of seconds, and the deformation and motion of the cell occurring on the order of minutes. The computational difficulty, even in 2D, resides in the fact that the problem is inherently one of deforming, non-stationary domains, bounded by an elastic perimeter, inside of which there is redistribution of biochemical signaling substances. Here we report the results of a computational scheme using the immersed boundary method to address this problem. We adopt a simple reaction-diffusion system that represents an internal regulatory mechanism controlling the polarization of a cell, and determining the strength of protrusion forces at the front of its elastic perimeter. Using this computational scheme we are able to study the effect of protrusive and elastic forces on cell shapes on their own, the distribution of the reaction-diffusion system in irregular domains on its own, and the coupled mechanical-chemical system. We find that this representation of cell crawling can recover important aspects of the spontaneous polarization and motion of certain types of crawling cells.

Advent of complex flows in epithelial tissues

The collective migration of cells in tissue pervades many important biological processes, such as wound healing, organism development, and cancer metastasis. Recent experiments on wound healing show that the collective migratory behavior of cells can be quite complex, including transient vortices and long-range correlations. Here, we explore cellular flows in epithelial tissues using a model that considers the force distribution and polarity of a single cell along with cell-cell adhesion. We show that the dipole nature of a crawling cell's force distribution destabilizes steady cellular motion. We determine the values of the physical parameters that are necessary to produce these complex motions and use numerical simulation to verify the linear analysis and to demonstrate the complex flows. We find that the tendency for cells to align is the dominant physical parameter that determines the stability of steady flows in the epithelium.

Crawling cells can close wounds without purse strings or signaling

When a gash or gouge is made in a confluent layer of epithelial cells, the cells move to fill in the “wound.” In some cases, such as in wounded embryonic chick wing buds, the movement of the cells is driven by cortical actin contraction (i.e., a purse string mechanism). In adult tissue, though, cells apparently crawl to close wounds. At the single cell level, this crawling is driven by the dynamics of the cell's actin cytoskeleton, which is regulated by a complex biochemical network, and cell signaling has been proposed to play a significant role in directing cells to move into the denuded area. However, wounds made in monolayers of Madin-Darby canine kidney (MDCK) cells still close even when a row of cells is deactivated at the periphery of the wound, and recent experiments show complex, highly-correlated cellular motions that extend tens of cell lengths away from the boundary. These experiments suggest a dominant role for mechanics in wound healing. Here we present a biophysical description of the collective migration of epithelial cells during wound healing based on the basic motility of single cells and cell-cell interactions. This model quantitatively captures the dynamics of wound closure and reproduces the complex cellular flows that are observed. These results suggest that wound healing is predominantly a mechanical process that is modified, but not produced, by cell-cell signaling.

Wound healing is driven by the collective migration of groups of epithelial cells. Experiments have shown that the motions of cells during wound healing are not as simple as had once been thought. Indeed, cells do not just move out to fill in the wounded area but rather undergo a number of complex but coordinated motions. Furthermore, wound healing is not just a response to chemical cues and can be driven by cells that are not immediately at the edge of the wound. In this paper, we develop a mathematical model based on the mechanical behavior of single crawling cells and also includes cell-cell adhesion. We show that this model is capable of explaining quantitatively the dynamics that occur during wound healing assays. This suggests that wound healing is largely a mechanical process where chemical signaling merely acts to augment the overall behavior.