Modeling cellular deformations using the level set formalism

Ras suppression potentiates rear actomyosin contractility-driven cell polarization and migration

Ras has been extensively studied as a promoter of cell proliferation, whereas few studies have explored its role in migration. To investigate the direct and immediate effects of Ras activity on cell motility or polarity, we focused on RasGAPs, C2GAPB in Dictyostelium amoebae and RASAL3 in HL-60 neutrophils and macrophages. In both cellular systems, optically recruiting the respective RasGAP to the cell front extinguished pre-existing protrusions and changed migration direction. However, when these respective RasGAPs were recruited uniformly to the membrane, cells polarized and moved more rapidly, whereas targeting to the back exaggerated these effects. These unexpected outcomes of attenuating Ras activity naturally had strong, context-dependent consequences for chemotaxis. The RasGAP-mediated polarization depended critically on myosin II activity and commenced with contraction at the cell rear, followed by sustained mTORC2-dependent actin polymerization at the front. These experimental results were captured by computational simulations in which Ras levels control front- and back-promoting feedback loops. The discovery that inhibiting Ras activity can produce counterintuitive effects on cell migration has important implications for future drug-design strategies targeting oncogenic Ras.

Self-organization of PIP3 waves is controlled by the topology and curvature of cell membranes

Phosphatidylinositol (3,4,5)-trisphosphate (PIP3) is a signaling lipid on the plasma membrane that plays a fundamental role in cell signaling with a strong impact on cell physiology and diseases. It is responsible for the protruding edge formation, cell polarization, macropinocytosis, and other membrane remodeling dynamics in cells. It has been shown that the membrane confinement and curvature affects the wave formation of PIP3 and F-actin. But, even in the absence of F-actin, a complex self-organization of the spatiotemporal PIP3 waves is observed. In recent findings, we have shown that these waves can be guided and pinned on strongly bended Dictyostelium membranes caused by molecular crowding and curvature-limited diffusion. Based on these experimental findings, we investigate the spatiotemporal PIP3 wave dynamics on realistic three-dimensional cell-like membranes to explore the effect of curvature-limited diffusion, as observed experimentally. We use an established stochastic reaction-diffusion model with enzymatic Michaelis-Menten-type reactions that mimics the dynamics of Dictyostelium cells. As these cells mimic the three-dimensional shape and size observed experimentally, we found that the PIP3 wave directionality can be explained by a Hopf-like and a reverse periodic-doubling bifurcation for uniform diffusion and curvature-limited diffusion properties. Finally, we compare the results with recent experimental findings and discuss the discrepancy between the biological and numerical results.

Complementary Cytoskeletal Feedback Loops Control Signal Transduction Excitability and Cell Polarity

To move through complex environments, cells must constantly integrate chemical and mechanical cues. Signaling networks, such as those comprising Ras and PI3K, transmit chemical cues to the cytoskeleton, but the cytoskeleton must also relay mechanical information back to those signaling systems. Using novel synthetic tools to acutely control specific elements of the cytoskeleton in Dictyostelium and neutrophils, we delineate feedback mechanisms that alter the signaling network and promote front- or back-states of the cell membrane and cortex. First, increasing branched actin assembly increases Ras/PI3K activation while reducing polymeric actin levels overall decreases activation. Second, reducing myosin II assembly immediately increases Ras/PI3K activation and sensitivity to chemotactic stimuli. Third, inhibiting branched actin alone increases cortical actin assembly and strongly blocks Ras/PI3K activation. This effect is mitigated by reducing filamentous actin levels and in cells lacking myosin II. Finally, increasing actin crosslinking with a controllable activator of cytoskeletal regulator RacE leads to a large decrease in Ras activation both globally and locally. Curiously, RacE activation can trigger cell spreading and protrusion with no detectable activation of branched actin nucleators. Taken together with legacy data that Ras/PI3K promotes branched actin assembly and myosin II disassembly, our results define front- and back-promoting positive feedback loops. We propose that these loops play a crucial role in establishing cell polarity and mediating signal integration by controlling the excitable state of the signal transduction networks in respective regions of the membrane and cortex. This interplay enables cells to navigate intricate topologies like tissues containing other cells, the extracellular matrix, and fluids.

Biophysical models of early mammalian embryogenesis

Embryo development is a critical and fascinating stage in the life cycle of many organisms. Despite decades of research, the earliest stages of mammalian embryogenesis are still poorly understood, caused by a scarcity of high-resolution spatial and temporal data, the use of only a few model organisms, and a paucity of truly multidisciplinary approaches that combine biological research with biophysical modeling and computational simulation. Here, we explain the theoretical frameworks and biophysical processes that are best suited to modeling the early mammalian embryo, review a comprehensive list of previous models, and discuss the most promising avenues for future work.

Cells use molecular working memory to navigate inchanging chemoattractant fields

In order to migrate over large distances, cells within tissues and organisms rely on sensing local gradient cues which are irregular, conflicting, and changing over time and space. The mechanism how they generate persistent directional migration when signals are disrupted, while still remaining adaptive to signal's localization changes remain unknown. Here we find that single cells utilize a molecular mechanism akin to a working memory to satisfy these two opposing demands. We derive theoretically that this is characteristic for receptor networks maintained away from steady states. Time-resolved live-cell imaging of Epidermal growth factor receptor (EGFR) phosphorylation dynamics shows that cells transiently memorize position of encountered signals via slow-escaping remnant of the polarized signaling state, a dynamical 'ghost', driving memory-guided persistent directional migration. The metastability of this state further enables migrational adaptation when encountering new signals. We thus identify basic mechanism of real-time computations underlying cellular navigation in changing chemoattractant fields.

Two Complementary Signaling Pathways Depict Eukaryotic Chemotaxis: A Mechanochemical Coupling Model

Many eukaryotic cells, including neutrophils and Dictyostelium cells, are able to undergo correlated random migration in the absence of directional cues while reacting to shallow gradients of chemoattractants with exquisite precision. Although progress has been made with regard to molecular identities, it remains elusive how molecular mechanics are integrated with cell mechanics to initiate and manipulate cell motility. Here, we propose a two dimensional (2D) cell migration model wherein a multilayered dynamic seesaw mechanism is accompanied by a mechanical strain-based inhibition mechanism. In biology, these two mechanisms can be mapped onto the biochemical feedback between phosphoinositides (PIs) and Rho GTPase and the mechanical interplay between filamin A (FLNa) and FilGAP. Cell migration and the accompanying morphological changes are demonstrated in numerical simulations using a particle-spring model, and the diffusion in the cell membrane are simulations using a one dimensional (1D) finite differences method (FDM). The fine balance established between endogenous signaling and a mechanically governed inactivation scheme ensures the endogenous cycle of self-organizing pseudopods, accounting for the correlated random migration. Furthermore, this model cell manifests directional and adaptable responses to shallow graded signaling, depending on the overwhelming effect of the graded stimuli guidance on strain-based inhibition. Finally, the model cell becomes trapped within an obstacle-ridden spatial region, manifesting a shuttle run for local explorations and can chemotactically “escape”, illustrating again the balance required in the complementary signaling pathways.

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.

Computational models of cancer cell transport through the microcirculation

The transport of cancerous cells through the microcirculation during metastatic spread encompasses several interdependent steps that are not fully understood. Computational models which resolve the cellular-scale dynamics of complex microcirculatory flows offer considerable potential to yield needed insights into the spread of cancer as a result of the level of detail that can be captured. In recent years, in silico methods have been developed that can accurately and efficiently model the circulatory flows of cancer and other biological cells. These computational methods are capable of resolving detailed fluid flow fields which transport cells through tortuous physiological geometries, as well as the deformation and interactions between cells, cell-to-endothelium interactions, and tumor cell aggregates, all of which play important roles in metastatic spread. Such models can provide a powerful complement to experimental works, and a promising approach to recapitulating the endogenous setting while maintaining control over parameters such as shear rate, cell deformability, and the strength of adhesive binding to better understand tumor cell transport. In this review, we present an overview of computational models that have been developed for modeling cancer cells in the microcirculation, including insights they have provided into cell transport phenomena.

Mathematical modelling in cell migration: tackling biochemistry in changing geometries

Directed cell migration poses a rich set of theoretical challenges. Broadly, these are concerned with (1) how cells sense external signal gradients and adapt; (2) how actin polymerisation is localised to drive the leading cell edge and Myosin-II molecular motors retract the cell rear; and (3) how the combined action of cellular forces and cell adhesion results in cell shape changes and net migration. Reaction-diffusion models for biological pattern formation going back to Turing have long been used to explain generic principles of gradient sensing and cell polarisation in simple, static geometries like a circle. In this minireview, we focus on recent research which aims at coupling the biochemistry with cellular mechanics and modelling cell shape changes. In particular, we want to contrast two principal modelling approaches: (1) interface tracking where the cell membrane, interfacing cell interior and exterior, is explicitly represented by a set of moving points in 2D or 3D space and (2) interface capturing. In interface capturing, the membrane is implicitly modelled analogously to a level line in a hilly landscape whose topology changes according to forces acting on the membrane. With the increased availability of high-quality 3D microscopy data of complex cell shapes, such methods will become increasingly important in data-driven, image-based modelling to better understand the mechanochemistry underpinning cell motion.

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.

Model for deformation of cells from external electric fields at or near resonant frequencies

This paper presents a numerical model to investigate the deformation of biological cells by applying external electric fields operating at or near cell resonant frequencies. Cells are represented as pseudo solids with high viscosity suspended in liquid media. The electric field source is an atmospheric plasma jet developed inhouse, for which the emitted energy distribution has been measured. Viscoelastic response is resolved in the entire cell structure by solving a deformation matrix assuming an isotropic material with a prescribed modulus of elasticity. To investigate cell deformation at resonant frequencies, one mode of natural cell oscillation is considered in which the cell membrane is made to radially move about its eigenfrequency. An electromagnetic wave source interacts with the cell and induces oscillation and viscoelastic response. The source carries energy in the form of a distribution function which couples a range of oscillating frequencies with electric field amplitudes. Results show that cell response may be increased by the external electric field operating at or near resonance. In the elastic regime, response increases until a steady threshold value, and the structure moves as a damped oscillator. Generally, this response is a function of both frequency and magnitude of the source, with a maximum effect found at resonance. To understand the full effect of the source energy spectrum, the system is solved by considering five frequency-amplitude couplings. Results show that the total solution is a nonlinear combination of the individual solutions. Additionally, sources with different signal phases are simulated to determine the effect of initial conditions on the evolution of the system, and the result suggests that there may be multiple solutions within the same order of magnitude for elastic response and velocity. Cell rupture from electric stress may occur during application given a high energy source.

Traveling and standing waves mediate pattern formation in cellular protrusions

The mechanisms regulating protrusions during amoeboid migration exhibit excitability. Theoretical studies have suggested the possible coexistence of traveling and standing waves in excitable systems. Here, we demonstrate the direct transformation of a traveling into a standing wave and establish conditions for the stability of this conversion. This theory combines excitable wave stopping and the emergence of a family of standing waves at zero velocity, without altering diffusion parameters. Experimentally, we show the existence of this phenomenon on the cell cortex of some Dictyostelium and mammalian mutant strains. We further predict a template that encompasses a spectrum of protrusive phenotypes, including pseudopodia and filopodia, through transitions between traveling and standing waves, allowing the cell to switch between excitability and bistability. Overall, this suggests that a previously-unidentified method of pattern formation, in which traveling waves spread, stop, and turn into standing waves that rearrange to form stable patterns, governs cell motility.

Wave patterns organize cellular protrusions and control cortical dynamics

Cellular protrusions are typically considered as distinct structures associated with specific regulators. However, we found that these regulators coordinately localize as propagating cortical waves, suggesting a common underlying mechanism. These molecular events fell into two excitable networks, the signal transduction network STEN and the cytoskeletal network CEN with different wave substructures. Computational studies using a coupled-network model reproduced these features and showed that the morphology and kinetics of the waves depended on strengths of feedback loops. Chemically induced dimerization at multiple nodes produced distinct, coordinated alterations in patterns of other network components. Taken together, these studies indicate: STEN positive feedback is mediated by mutual inhibition between Ras/Rap and PIP2, while negative feedback depends on delayed PKB activation; PKBs link STEN to CEN; CEN includes positive feedback between Rac and F-actin, and exerts fast positive and slow negative feedbacks to STEN The alterations produced protrusions resembling filopodia, ruffles, pseudopodia, or lamellipodia, suggesting that these structures arise from a common regulatory mechanism and that the overall state of the STEN-CEN system determines cellular morphology.

Unite to divide - how models and biological experimentation have come together to reveal mechanisms of cytokinesis

Cytokinesis is the fundamental and ancient cellular process by which one cell physically divides into two. Cytokinesis in animal and fungal cells is achieved by contraction of an actomyosin cytoskeletal ring assembled in the cell cortex, typically at the cell equator. Cytokinesis is essential for the development of fertilized eggs into multicellular organisms and for homeostatic replenishment of cells. Correct execution of cytokinesis is also necessary for genome stability and the evasion of diseases including cancer. Cytokinesis has fascinated scientists for well over a century, but its speed and dynamics make experiments challenging to perform and interpret. The presence of redundant mechanisms is also a challenge to understand cytokinesis, leaving many fundamental questions unresolved. For example, how does a disordered cytoskeletal network transform into a coherent ring? What are the long-distance effects of localized contractility? Here, we provide a general introduction to 'modeling for biologists', and review how agent-based modeling and continuum mechanics modeling have helped to address these questions.

Frequency-induced morphology alterations in microconfined biological cells

Low-intensity therapeutic ultrasound has demonstrated an impetus in bone signaling and tissue healing for decades now. Though this technology is clinically well proven, still there are breaches in studies to understand the fundamental principle of how osteoblast tissue regenerates physiologically at the cellular level with ultrasound interaction as a form of acoustic wave stimuli. Through this article, we illustrate an analysis for cytomechanical changes of cell membrane periphery as a basic first physical principle for facilitating late downstream biochemical pathways. With the help of in situ single-cell direct analysis in a microfluidic confinement, we demonstrate that alteration of low-intensity pulse ultrasound (LIPUS) frequency would physically perturb cell membrane and establish inherent cell oscillation. We experimentally demonstrate here that, at LIPUS resonance near 1.7 MHz (during 1-3 MHz alteration), cell membrane area would expand to 6.85 ± 0.7% during ultrasound exposure while it contracts 44.68 ± 0.8% in post actuation. Conversely, cell cross-sectional area change (%) from its previous morphology during and after switching off LIPUS was reversibly different before and after resonance. For instance, at 1.5 MHz, LIPUS exposure produced 1.44 ± 0.5% expansion while in contrast 2 MHz instigates 1.6 ± 0.3% contraction. We conclude that alteration of LIPUS frequency from 1-3 MHz keeping other ultrasound parameters like exposure time, pulse repetition frequency (PRF), etc., constant, if applied to a microconfined biological single living cell, would perturb physical structure reversibly based on the system resonance during and post exposure ultrasound pulsing. We envision, in the near future, our results would constitute the foundation of mechanistic effects of low-intensity therapeutic ultrasound and its allied potential in medical applications. Graphical Abstract Frequency Dependent Characterization of Area Strain in Cell Membrane by Microfluidic Based Single Cell Analysis.

The threshold of an excitable system serves as a control mechanism for noise filtering during chemotaxis

Chemotaxis, the migration of cells in the direction of a chemical gradient, is of utmost importance in various biological processes. In recent years, research has demonstrated that the underlying mechanism that controls cell migration is an excitable network. One of the properties that characterizes excitability is the presence of a threshold for activation. Here, we show that excitable systems possess noise filtering capabilities that enable faster and more efficient directed migration compared to other systems that also include a threshold, such as ultrasensitive switches. We demonstrate that this filtering ability is a consequence of the varying responses of excitable systems to step and pulse stimuli. Whereas the response to step inputs is determined solely by the magnitude of the stimulus, for pulse stimuli, the response depends on both the magnitude and duration of the stimulus. We then show that these two forms of threshold behavior can be decoupled from one another, allowing finer control in chemotaxis. Finally, we use a simple model of chemotaxis to demonstrate that cells that rely on an excitable system display faster and more effective directed migration that a hypothetical cell guided by an ultra-sensitive switch.

Micropipette Aspiration of Oocytes to Assess Cortical Tension

Just as it is important to understand the cell biology of signaling pathways, it is valuable also to understand mechanical forces in cells. The field of mechanobiology has a rich history, including study of cellular mechanics during mitosis and meiosis in echinoderm oocytes and zygotes dating back to the 1930s. This chapter addresses the use of micropipette aspiration (MPA) to assess cellular mechanics, specifically cortical tension, in mammalian oocytes.

Mechanochemical Signaling Directs Cell-Shape Change

For specialized cell function, as well as active cell behaviors such as division, migration, and tissue development, cells must undergo dynamic changes in shape. To complete these processes, cells integrate chemical and mechanical signals to direct force production. This mechanochemical integration allows for the rapid production and adaptation of leading-edge machinery in migrating cells, the invasion of one cell into another during cell-cell fusion, and the force-feedback loops that ensure robust cytokinesis. A quantitative understanding of cell mechanics coupled with protein dynamics has allowed us to account for furrow ingression during cytokinesis, a model cell-shape-change process. At the core of cell-shape changes is the ability of the cell's machinery to sense mechanical forces and tune the force-generating machinery as needed. Force-sensitive cytoskeletal proteins, including myosin II motors and actin cross-linkers such as α-actinin and filamin, accumulate in response to internally generated and externally imposed mechanical stresses, endowing the cell with the ability to discern and respond to mechanical cues. The physical theory behind how these proteins display mechanosensitive accumulation has allowed us to predict paralog-specific behaviors of different cross-linking proteins and identify a zone of optimal actin-binding affinity that allows for mechanical stress-induced protein accumulation. These molecular mechanisms coupled with the mechanical feedback systems ensure robust shape changes, but if they go awry, they are poised to promote disease states such as cancer cell metastasis and loss of tissue integrity.

Altering the threshold of an excitable signal transduction network changes cell migratory modes

The diverse migratory modes displayed by different cell types are generally believed to be idiosyncratic. Here we show that the migratory behavior of Dictyostelium was switched from amoeboid to keratocyte-like and oscillatory modes by synthetically decreasing PIP2 levels or increasing Ras/Rap-related activities. The perturbations at these key nodes of an excitable signal transduction network initiated a causal chain of events: The threshold for network activation was lowered, the speed and range of propagating waves of signal transduction activity increased, actin driven cellular protrusions expanded and, consequently, the cell migratory mode transitions ensued. Conversely, innately keratocyte-like and oscillatory cells were promptly converted to amoeboid by inhibition of Ras effectors with restoration of directed migration. We use computational analysis to explain how thresholds control cell migration and discuss the architecture of the signal transduction network that gives rise to excitability.

Unified model of brain tissue microstructure dynamically binds diffusion and osmosis with extracellular space geometry

We present a universal model of brain tissue microstructure that dynamically links osmosis and diffusion with geometrical parameters of brain extracellular space (ECS). Our model robustly describes and predicts the nonlinear time dependency of tortuosity (λ=sqrt[D/D^{*}]) changes with very high precision in various media with uniform and nonuniform osmolarity distribution, as demonstrated by previously published experimental data (D = free diffusion coefficient, D^{*} = effective diffusion coefficient). To construct this model, we first developed a multiscale technique for computationally effective modeling of osmolarity in the brain tissue. Osmolarity differences across cell membranes lead to changes in the ECS dynamics. The evolution of the underlying dynamics is then captured by a level set method. Subsequently, using a homogenization technique, we derived a coarse-grained model with parameters that are explicitly related to the geometry of cells and their associated ECS. Our modeling results in very accurate analytical approximation of tortuosity based on time, space, osmolarity differences across cell membranes, and water permeability of cell membranes. Our model provides a unique platform for studying ECS dynamics not only in physiologic conditions such as sleep-wake cycles and aging but also in pathologic conditions such as stroke, seizure, and neoplasia, as well as in predictive pharmacokinetic modeling such as predicting medication biodistribution and efficacy and novel biomolecule development and testing.

Immersed Boundary Models for Quantifying Flow-Induced Mechanical Stimuli on Stem Cells Seeded on 3D Scaffolds in Perfusion Bioreactors

Perfusion bioreactors regulate flow conditions in order to provide cells with oxygen, nutrients and flow-associated mechanical stimuli. Locally, these flow conditions can vary depending on the scaffold geometry, cellular confluency and amount of extra cellular matrix deposition. In this study, a novel application of the immersed boundary method was introduced in order to represent a detailed deformable cell attached to a 3D scaffold inside a perfusion bioreactor and exposed to microscopic flow. The immersed boundary model permits the prediction of mechanical effects of the local flow conditions on the cell. Incorporating stiffness values measured with atomic force microscopy and micro-flow boundary conditions obtained from computational fluid dynamics simulations on the entire scaffold, we compared cell deformation, cortical tension, normal and shear pressure between different cell shapes and locations. We observed a large effect of the precise cell location on the local shear stress and we predicted flow-induced cortical tensions in the order of 5 pN/μm, at the lower end of the range reported in literature. The proposed method provides an interesting tool to study perfusion bioreactors processes down to the level of the individual cell's micro-environment, which can further aid in the achievement of robust bioprocess control for regenerative medicine applications.

Modelling of Yeast Mating Reveals Robustness Strategies for Cell-Cell Interactions

Mating of budding yeast cells is a model system for studying cell-cell interactions. Haploid yeast cells secrete mating pheromones that are sensed by the partner which responds by growing a mating projection toward the source. The two projections meet and fuse to form the diploid. Successful mating relies on precise coordination of dynamic extracellular signals, signaling pathways, and cell shape changes in a noisy background. It remains elusive how cells mate accurately and efficiently in a natural multi-cell environment. Here we present the first stochastic model of multiple mating cells whose morphologies are driven by pheromone gradients and intracellular signals. Our novel computational framework encompassed a moving boundary method for modeling both a-cells and α-cells and their cell shape changes, the extracellular diffusion of mating pheromones dynamically coupled with cell polarization, and both external and internal noise. Quantification of mating efficiency was developed and tested for different model parameters. Computer simulations revealed important robustness strategies for mating in the presence of noise. These strategies included the polarized secretion of pheromone, the presence of the α-factor protease Bar1, and the regulation of sensing sensitivity; all were consistent with data in the literature. In addition, we investigated mating discrimination, the ability of an a-cell to distinguish between α-cells either making or not making α-factor, and mating competition, in which multiple a-cells compete to mate with one α-cell. Our simulations were consistent with previous experimental results. Moreover, we performed a combination of simulations and experiments to estimate the diffusion rate of the pheromone a-factor. In summary, we constructed a framework for simulating yeast mating with multiple cells in a noisy environment, and used this framework to reproduce mating behaviors and to identify strategies for robust cell-cell interactions.

One of the riddles of Nature is how cells interact with one another to create complex cellular networks such as the neural networks in the brain. Forming precise connections between irregularly shaped cells is a challenge for biology. We developed computational methods for simulating these complex cell-cell interactions. We applied these methods to investigate yeast mating in which two yeast cells grow projections that meet and fuse guided by pheromone attractants. The simulations described molecules both inside and outside of the cell, and represented the continually changing shapes of the cells. We found that positioning the secretion and sensing of pheromones at the same location on the cell surface was important. Other key factors for robust mating included secreting a protein that removed excess pheromone from outside of the cell so that the signal would not be too strong. An important advance was being able to simulate as many as five cells in complex mating arrangements. Taken together we used our novel computational methods to describe in greater detail the yeast mating process, and more generally, interactions among cells changing their shapes in response to their neighbors.

Cytokinesis from nanometers to micrometers and microseconds to minutes

Cytokinesis, a model cell shape change event, is controlled by an integrated system that coordinates the mitotic spindle signals with a mechanoresponsive cytoskeletal network that drives contractility and furrow ingression. Quantitative methods that measure cell mechanics, mechanoresponse (mechanical stress-induced protein accumulation), protein dynamics, and molecular interactions are necessary to provide insight into both the mechanical and biochemical components involved in cytokinesis and cell shape regulation. Micropipette aspiration, fluorescence correlation and cross-correlation spectroscopy, and fluorescence recovery after photobleaching are valuable methods for measuring cell mechanics and protein dynamics in vivo that occur on nanometer to micron length-scales, and microsecond to minute timescales. Collectively, these methods provide the ability to quantify the molecular interactions that control the cell's ability to change shape and undergo cytokinesis.

Cell shape regulation through mechanosensory feedback control

Cells undergo controlled changes in morphology in response to intracellular and extracellular signals. These changes require a means for sensing and interpreting the signalling cues, for generating the forces that act on the cell's physical material, and a control system to regulate this process. Experiments on Dictyostelium amoebae have shown that force-generating proteins can localize in response to external mechanical perturbations. This mechanosensing, and the ensuing mechanical feedback, plays an important role in minimizing the effect of mechanical disturbances in the course of changes in cell shape, especially during cell division, and likely in other contexts, such as during three-dimensional migration. Owing to the complexity of the feedback system, which couples mechanical and biochemical signals involved in shape regulation, theoretical approaches can guide further investigation by providing insights that are difficult to decipher experimentally. Here, we present a computational model that explains the different mechanosensory and mechanoresponsive behaviours observed in Dictyostelium cells. The model features a multiscale description of myosin II bipolar thick filament assembly that includes cooperative and force-dependent myosin-actin binding, and identifies the feedback mechanisms hidden in the observed mechanoresponsive behaviours of Dictyostelium cells during micropipette aspiration experiments. These feedbacks provide a mechanistic explanation of cellular retraction and hence cell shape regulation.

Modeling Excitable Dynamics of Chemotactic Networks

The study of chemotaxis has benefited greatly from computational models that describe the response of cells to chemoattractant stimuli. These models must keep track of spatially and temporally varying distributions of numerous intracellular species. Moreover, recent evidence suggests that these are not deterministic interactions, but also include the effect of stochastic variations that trigger an excitable network. In this chapter we illustrate how to create simulations of excitable networks using the Virtual Cell modeling environment.

Crawling and Gliding: A Computational Model for Shape-Driven Cell Migration

Cell migration is a complex process involving many intracellular and extracellular factors, with different cell types adopting sometimes strikingly different morphologies. Modeling realistically behaving cells in tissues is computationally challenging because it implies dealing with multiple levels of complexity. We extend the Cellular Potts Model with an actin-inspired feedback mechanism that allows small stochastic cell rufflings to expand to cell protrusions. This simple phenomenological model produces realistically crawling and deforming amoeboid cells, and gliding half-moon shaped keratocyte-like cells. Both cell types can migrate randomly or follow directional cues. They can squeeze in between other cells in densely populated environments or migrate collectively. The model is computationally light, which allows the study of large, dense and heterogeneous tissues containing cells with realistic shapes and migratory properties.

Cell migration is involved in vital processes like morphogenesis, regeneration and immune system responses, but can also play a central role in pathological processes like metastasization. Computational models have been successfully employed to explain how single cells migrate, and to study how diverse cell-cell interactions contribute to tissue level behavior. However, there are few models that implement realistic cell shapes in multicellular simulations. The method we present here is able to reproduce two different types of motile cells—amoeboid and keratocyte-like cells. Amoeboid cells are highly motile and deform frequently; many cells can act amoeboid in certain circumstances e.g., immune system cells, epithelial cells, individually migrating cancer cells. Keratocytes are (fish) epithelial cells which are famous for their ability to preserve their shape and direction when migrating individually; during wound healing, keratocytes migrate collectively, in sheets, to the site needing reepithelialization. Our method is computationally simple, improves the realism of multicellular simulations and can help assess the tissue level impact of specific cell shapes. For example, it can be employed to study the tissue scanning strategies of leukocytes, the circumstances in which cancer cells adopt amoeboid migration strategies, or the collective migration of keratocytes.

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.

An excitable signal integrator couples to an idling cytoskeletal oscillator to drive cell migration

It is generally believed that cytoskeletal activities drive random cell migration while signal transduction events initiated by receptors regulate the cytoskeleton to guide cells. However, we find that the cytoskeletal network, involving Scar/Wave, Arp 2/3, and actin binding proteins, is only capable of generating rapid oscillations and undulations of the cell boundary. The signal transduction network, comprising multiple pathways that include Ras GTPases, PI3K, and Rac GTPases, is required to generate the sustained protrusions of migrating cells. The signal transduction network is excitable, displaying wave propagation, refractoriness, and maximal response to suprathreshold stimuli, even in the absence of the cytoskeleton. We suggest that cell motility results from coupling of “pacemaker” signal transduction and “idling motor” cytoskeletal networks, and various guidance cues that modulate the threshold for triggering signal transduction events are integrated to control the mode and direction of migration.

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.

Prophase I mouse oocytes are deficient in the ability to respond to fertilization by decreasing membrane receptivity to sperm and establishing a membrane block to polyspermy

Changes occurring as the prophase I oocyte matures to metaphase II are critical for the acquisition of competence for normal egg activation and early embryogenesis. A prophase I oocyte cannot respond to a fertilizing sperm as a metaphase II egg does, including the ability to prevent polyspermic fertilization. Studies here demonstrate that the competence for the membrane block to polyspermy is deficient in prophase I mouse oocytes. In vitro fertilization experiments using identical insemination conditions result in monospermy in 87% of zona pellucida (ZP)-free metaphase II eggs, while 92% of ZP-free prophase I oocytes have four or more fused sperm. The membrane block is associated with a postfertilization reduction in the capacity to support sperm binding, but this reduction in sperm-binding capacity is both less robust and slower to develop in fertilized prophase I oocytes. Fertilization of oocytes is dependent on the tetraspanin CD9, but little to no release of CD9 from the oocyte membrane is detected, suggesting that release of CD9-containing vesicles is not essential for fertilization. The deficiency in membrane block establishment in prophase I oocytes correlates with abnormalities in two postfertilization cytoskeletal changes: sperm-induced cortical remodeling that results in fertilization cone formation and a postfertilization increase in effective cortical tension. These data indicate that cortical maturation is a component of cytoplasmic maturation during the oocyte-to-egg transition and that the egg cortex has to be appropriately primed and tuned to be responsive to a fertilizing sperm.

A mechanism for effective cell-seeding in rigid, microporous substrates

Seeding cells into porous ceramic substrates has been shown to improve outcomes in surgical repair of large bone defects, but the physics underlying cellular ingress into such scaffolds remains elusive. This paper demonstrates capillary forces as a novel, yet simple, self-loading or self-seeding mechanism for rigid, microporous substrates. Capillary forces were found to draw cells through a microporous network with interconnections smaller than the diameter of the cells in suspension. Work here emphasizes CaP-based bone scaffolds containing both macroporosity (>100μm) and microporosity (5-50μm); these have been shown to improve bone formation in vivo as compared to their macroporous counterparts and also performed better than microporous scaffolds containing BMP-2 by some measures of bone regeneration. We hypothesize that capillary force driven self-seeding in both macro- and micropores may underlie this improvement, and present a mathematical model and experiments that support this hypothesis. The cell localization and penetration depth within these two-dimensional substrates in vitro depends upon both the cell type (size and stiffness) and the capillary forces generated by the microstructure. Additional experiments showing that cell penetration depth in vitro depends on cell size and stiffness suggest that microporosity could be tailored to optimize cell infiltration in a cell-specific way. Endogenous cells are also drawn into the microporous network in vivo. Results have important implications for design of scaffolds for the healing of large bone defects, and for controlled release of drugs in vivo.

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.

Micropipette aspiration for studying cellular mechanosensory responses and mechanics

Micropipette aspiration (MPA) is a widely applied method for studying cortical tension and deformability. Based on simple hydrostatic principles, this assay allows the application of a specific magnitude of mechanical stress on cells. This powerful method has revealed insights about cell mechanics and mechanosensing, not only in Dictyostelium discoideum but also in other cell types. In this chapter, we present how to set up a micropipette aspiration system and the experimental procedures for determining cortical tension and mechanosensory responses.

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.

Cytokinesis mechanics and mechanosensing

Cytokinesis shape change occurs through the interfacing of three modules, cell mechanics, myosin II-mediated contractile stress generation and sensing, and a control system of regulatory proteins, which together ensure flexibility and robustness. This integrated system then defines the stereotypical shape changes of successful cytokinesis, which occurs under a diversity of mechanical contexts and environmental conditions.

Modelling cell motility and chemotaxis with evolving surface finite elements

We present a mathematical and a computational framework for the modelling of cell motility. The cell membrane is represented by an evolving surface, with the movement of the cell determined by the interaction of various forces that act normal to the surface. We consider external forces such as those that may arise owing to inhomogeneities in the medium and a pressure that constrains the enclosed volume, as well as internal forces that arise from the reaction of the cells' surface to stretching and bending. We also consider a protrusive force associated with a reaction-diffusion system (RDS) posed on the cell membrane, with cell polarization modelled by this surface RDS. The computational method is based on an evolving surface finite-element method. The general method can account for the large deformations that arise in cell motility and allows the simulation of cell migration in three dimensions. We illustrate applications of the proposed modelling framework and numerical method by reporting on numerical simulations of a model for eukaryotic chemotaxis and a model for the persistent movement of keratocytes in two and three space dimensions. Movies of the simulated cells can be obtained from http://homepages.warwick.ac.uk/∼maskae/CV_Warwick/Chemotaxis.html.

Deconvolution of the cellular force-generating subsystems that govern cytokinesis furrow ingression

Cytokinesis occurs through the coordinated action of several biochemically-mediated stresses acting on the cytoskeleton. Here, we develop a computational model of cellular mechanics, and using a large number of experimentally measured biophysical parameters, we simulate cell division under a number of different scenarios. We demonstrate that traction-mediated protrusive forces or contractile forces due to myosin II are sufficient to initiate furrow ingression. Furthermore, we show that passive forces due to the cell's cortical tension and surface curvature allow the furrow to complete ingression. We compare quantitatively the furrow thinning trajectories obtained from simulation with those observed experimentally in both wild-type and myosin II null Dictyostelium cells. Our simulations highlight the relative contributions of different biomechanical subsystems to cell shape progression during cell division.

Cytokinesis, the physical separation of a mother cell into two daughter cells, requires force to deform the cell. Though there is ample evidence in many systems that myosin II provides some of this force, it is also well known that some cell types can divide in the absence of myosin II. To elucidate the mechanisms by which cells control furrow ingression, we developed a computational model of cellular dynamics during cytokinesis in the social amoeba, Dictyostelium discoideum. We took advantage of a large number of experimentally measured parameters and well-characterized furrow ingression dynamics for a number of different strains. Our simulations demonstrate that there are distinct phases of cytokinesis. Myosin II plays a role providing the stress that initiates furrow ingression. In its absence, however, this force can be supplied by a combination of adhesion and protrusion-mediated stresses. Thereafter, Laplace-like pressures take over and provide stresses that enable the cell to divide. Overall, we show how various mechanical parameters quantitatively impact furrow ingression kinetics, accounting for the cytokinesis dynamics of wild type and mutant cell-lines.

Coupling actin flow, adhesion, and morphology in a computational cell motility model

Cell migration is a pervasive process in many biology systems and involves protrusive forces generated by actin polymerization, myosin dependent contractile forces, and force transmission between the cell and the substrate through adhesion sites. Here we develop a computational model for cell motion that uses the phase-field method to solve for the moving boundary with physical membrane properties. It includes a reaction-diffusion model for the actin-myosin machinery and discrete adhesion sites which can be in a "gripping" or "slipping" mode and integrates the adhesion dynamics with the dynamics of the actin filaments, modeled as a viscous network. To test this model, we apply it to fish keratocytes, fast moving cells that maintain their morphology, and show that we are able to reproduce recent experimental results on actin flow and stress patterns. Furthermore, we explore the phase diagram of cell motility by varying myosin II activity and adhesion strength. Our model suggests that the pattern of the actin flow inside the cell, the cell velocity, and the cell morphology are determined by the integration of actin polymerization, myosin contraction, adhesion forces, and membrane forces.

Separation anxiety: stress, tension and cytokinesis

Cytokinesis, the physical separation of a mother cell into two daughter cells, progresses through a series of well-defined changes in morphology. These changes involve distinct biochemical and mechanical processes. Here, we review the mechanical features of cells during cytokinesis, discussing both the material properties as well as sources of stresses, both active and passive, which lead to the observed changes in morphology. We also describe a mechanosensory feedback control system that regulates protein localization and shape progression during cytokinesis.

A review of models of fluctuating protrusion and retraction patterns at the leading edge of motile cells

A characteristic feature of motile cells as they undergo a change in motile behavior is the development of fluctuating exploratory motions of the leading edge, driven by actin polymerization. We review quantitative models of these protrusion and retraction phenomena. Theoretical studies have been motivated by advances in experimental and computational methods that allow controlled perturbations, single molecule imaging, and analysis of spatiotemporal correlations in microscopic images. To explain oscillations and waves of the leading edge, most theoretical models propose nonlinear interactions and feedback mechanisms among different components of the actin cytoskeleton system. These mechanisms include curvature-sensing membrane proteins, myosin contraction, and autocatalytic biochemical reaction kinetics. We discuss how the combination of experimental studies with modeling promises to quantify the relative importance of these biochemical and biophysical processes at the leading edge and to evaluate their generality across cell types and extracellular environments.

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.

Biased excitable networks: how cells direct motion in response to gradients

The actin cytoskeleton in motile cells has many of the hallmarks of an excitable medium, including the presence of propagating waves. This excitable behavior can account for the spontaneous migration of cells. A number of reports have suggested that the chemoattractant-mediated signaling can bias excitability, thus providing a means by which cell motility can be directed. In this review, we discuss some of these observations and theories proposed to explain them. We also suggest a mechanism for cell polarity that can be incorporated into the existing framework.

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.

Activated membrane patches guide chemotactic cell motility

Many eukaryotic cells are able to crawl on surfaces and guide their motility based on environmental cues. These cues are interpreted by signaling systems which couple to cell mechanics; indeed membrane protrusions in crawling cells are often accompanied by activated membrane patches, which are localized areas of increased concentration of one or more signaling components. To determine how these patches are related to cell motion, we examine the spatial localization of RasGTP in chemotaxing Dictyostelium discoideum cells under conditions where the vertical extent of the cell was restricted. Quantitative analyses of the data reveal a high degree of spatial correlation between patches of activated Ras and membrane protrusions. Based on these findings, we formulate a model for amoeboid cell motion that consists of two coupled modules. The first module utilizes a recently developed two-component reaction diffusion model that generates transient and localized areas of elevated concentration of one of the components along the membrane. The activated patches determine the location of membrane protrusions (and overall cell motion) that are computed in the second module, which also takes into account the cortical tension and the availability of protrusion resources. We show that our model is able to produce realistic amoeboid-like motion and that our numerical results are consistent with experimentally observed pseudopod dynamics. Specifically, we show that the commonly observed splitting of pseudopods can result directly from the dynamics of the signaling patches.

Different types of cells are able to directionally migrate, responding to spatially-varying environmental cues. To do so, the cell needs to sense its environment, decide on the correct direction, and finally implement the needed mechanical changes in order to actually move. In this work we study the relation between the sensing-signaling system and the mechanical motion. We first show that membrane protrusions which drive the overall translocation occur exactly at the same locations at which membrane-bound signal-transduction effectors accumulate. These high concentration areas, also termed “patches”, exhibit interesting dynamics of disappearing and reappearing. Based on these findings, we develop a mathematical-computational model, in which membrane protrusions are driven by these membrane “patches”. These protrusions are then coupled to other cellular forces and the overall model predicts motion and its relationship to shape changes. Using our approach, we show that several observed features of cellular motility, for example the splitting of the cell tip, can be explained by the upstream signaling dynamics.

A multiphysical model of cell migration integrating reaction-diffusion, membrane and cytoskeleton

Cellular motility is a complicated phenomenon that involves multiphysics, including the cytoskeleton, the plasma membrane and intracellular signal transduction. In this study, a hybrid computational model was developed for the simulation of whole-cell migration behaviors. The model integrates sub-models of reaction-diffusion, actin filaments (F-actin) and the plasma membrane. Reaction-diffusion was calculated as if enclosed by a moving membrane. Individual F-actins were reorganized on the basis of stochastic kinetic events, such as polymerization, capping, branching and severing. Membrane dynamics were modeled using an optimization of energy function that depends on cell volume, surface area, smoothness and the elasticity of F-actin against the membrane. Simulations of this model demonstrated self-organization of F-actin networks, as in lamellipodia, and chemotactic migration. Furthermore, this method was extended to address external obstacles to simulate the dynamic cellular morphological changes seen during invasive migration.

Models at the single cell level

Many cellular behaviors cannot be completely captured or appropriately described at the cell population level. Noise induced by stochastic chemical reactions, spatially polarized signaling networks, and heterogeneous cell-cell communication are among the many phenomena that require fine-grained analysis. Accordingly, the mathematical models used to describe such systems must be capable of single cell or subcellular resolution. Here, we review techniques for modeling single cells, including models of stochastic chemical kinetics, spatially heterogeneous intracellular signaling, and spatial stochastic systems. We also briefly discuss applications of each type of model.

Use of the parameterised finite element method to robustly and efficiently evolve the edge of a moving cell

In this paper we present a computational tool that enables the simulation of mathematical models of cell migration and chemotaxis on an evolving cell membrane. Recent models require the numerical solution of systems of reaction-diffusion equations on the evolving cell membrane and then the solution state is used to drive the evolution of the cell edge. Previous work involved moving the cell edge using a level set method (LSM). However, the LSM is computationally very expensive, which severely limits the practical usefulness of the algorithm. To address this issue, we have employed the parameterised finite element method (PFEM) as an alternative method for evolving a cell boundary. We show that the PFEM is far more efficient and robust than the LSM. We therefore suggest that the PFEM potentially has an essential role to play in computational modelling efforts towards the understanding of many of the complex issues related to chemotaxis.