A continuum model of actin waves in Dictyostelium discoideum

Osteoclasts at Bone Remodeling: Order from Order

Osteoclasts are multinucleated bone-resorbing cells derived from the monocyte/macrophage lineage. The macrophage colony-stimulating factor/receptor activator of nuclear factor κB ligand (M-CSF/RANKL) signaling network governs the differentiation of precursor cells into fusion-competent mononucleated cells. Repetitive fusion of fusion-competent cells produces multinucleated osteoclasts. Osteoclasts are believed to die via apoptosis after bone resorption. However, recent studies have found that osteoclastogenesis in vivo proceeds by replacing the old nucleus of existing osteoclasts with a single newly differentiated mononucleated cell. Thus, the formation of new osteoclasts is minimal. Furthermore, the sizes of osteoclasts can change via cell fusion and fission in response to external conditions. On the other hand, osteoclastogenesis in vitro involves various levels of heterogeneity, including osteoclast precursors, mode of fusion, and properties of the differentiated osteoclasts. To better understand the origin of these heterogeneities and the plasticity of osteoclasts, we examine several processes of osteoclastogenesis in this review. Candidate mechanisms that create heterogeneity involve asymmetric cell division, osteoclast niche, self-organization, and mode of fusion and fission. Elucidation of the plasticity or fluctuation of the M-CSF/RANKL network should be an important topic for future researches.

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

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

Cytoskeletal dysregulation and neurodegenerative disease: Formation, monitoring, and inhibition of cofilin-actin rods

The presence of atypical cytoskeletal dynamics, structures, and associated morphologies is a common theme uniting numerous diseases and developmental disorders. In particular, cytoskeletal dysregulation is a common cellular feature of Alzheimer’s disease, Parkinson’s disease, and Huntington’s disease. While the numerous activators and inhibitors of dysregulation present complexities for characterizing these elements as byproducts or initiators of the disease state, it is increasingly clear that a better understanding of these anomalies is critical for advancing the state of knowledge and plan of therapeutic attack. In this review, we focus on the hallmarks of cytoskeletal dysregulation that are associated with cofilin-linked actin regulation, with a particular emphasis on the formation, monitoring, and inhibition of cofilin-actin rods. We also review actin-associated proteins other than cofilin with links to cytoskeleton-associated neurodegenerative processes, recognizing that cofilin-actin rods comprise one strand of a vast web of interactions that occur as a result of cytoskeletal dysregulation. Our aim is to present a current perspective on cytoskeletal dysregulation, connecting recent developments in our understanding with emerging strategies for biosensing and biomimicry that will help shape future directions of the field.

Versatile Patterns in the Actin Cortex of Motile Cells: Self-Organized Pulses Can Coexist with Macropinocytic Ring-Shaped Waves

Self-organized patterns in the actin cytoskeleton are essential for eukaryotic cellular life. They are the building blocks of many functional structures that often operate simultaneously to facilitate, for example, nutrient uptake and movement of cells. However, identifying how qualitatively distinct actin patterns can coexist remains a challenge. Using bifurcation theory of a mass conserved activator-inhibitor system, we uncover a generic mechanism of how different actin waves-traveling waves and excitable pulses-organize and simultaneously emerge. Live-cell imaging experiments indeed reveal that narrow, planar, and fast-moving excitable pulses may coexist with ring-shaped macropinocytic actin waves in the cortex of motile amoeboid cells.

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.

Three-dimensional stochastic simulation of chemoattractant-mediated excitability in cells

During the last decade, a consensus has emerged that the stochastic triggering of an excitable system drives pseudopod formation and subsequent migration of amoeboid cells. The presence of chemoattractant stimuli alters the threshold for triggering this activity and can bias the direction of migration. Though noise plays an important role in these behaviors, mathematical models have typically ignored its origin and merely introduced it as an external signal into a series of reaction-diffusion equations. Here we consider a more realistic description based on a reaction-diffusion master equation formalism to implement these networks. In this scheme, noise arises naturally from a stochastic description of the various reaction and diffusion terms. Working on a three-dimensional geometry in which separate compartments are divided into a tetrahedral mesh, we implement a modular description of the system, consisting of G-protein coupled receptor signaling (GPCR), a local excitation-global inhibition mechanism (LEGI), and signal transduction excitable network (STEN). Our models implement detailed biochemical descriptions whenever this information is available, such as in the GPCR and G-protein interactions. In contrast, where the biochemical entities are less certain, such as the LEGI mechanism, we consider various possible schemes and highlight the differences between them. Our simulations show that even when the LEGI mechanism displays perfect adaptation in terms of the mean level of proteins, the variance shows a dose-dependence. This differs between the various models considered, suggesting a possible means for determining experimentally among the various potential networks. Overall, our simulations recreate temporal and spatial patterns observed experimentally in both wild-type and perturbed cells, providing further evidence for the excitable system paradigm. Moreover, because of the overall importance and ubiquity of the modules we consider, including GPCR signaling and adaptation, our results will be of interest beyond the field of directed migration.

Though the term noise usually carries negative connotations, it can also contribute positively to the characteristic dynamics of a system. In biological systems, where noise arises from the stochastic interactions between molecules, its study is usually confined to genetic regulatory systems in which copy numbers are small and fluctuations large. However, noise can have important roles when the number of signaling molecules is large. The extension of pseudopods and the subsequent motion of amoeboid cells arises from the noise-induced trigger of an excitable system. Chemoattractant signals bias this triggering thereby directing cell motion. To date, this paradigm has not been tested by mathematical models that account accurately for the noise that arises in the corresponding reactions. In this study, we employ a reaction-diffusion master equation approach to investigate the effects of noise. Using a modular approach and a three-dimensional cell model with specific subdomains attributed to the cell membrane and cortex, we explore the spatiotemporal dynamics of the system. Our simulations recreate many experimentally-observed cell behaviors thereby supporting the biased-excitable network hypothesis.

A phosphoinositide-based model of actin waves in frustrated phagocytosis

Phagocytosis is a complex process by which phagocytes such as lymphocytes or macrophages engulf and destroy foreign bodies called pathogens in a tissue. The process is triggered by the detection of antibodies that trigger signaling mechanisms that control the changes of the cellular cytoskeleton needed for engulfment of the pathogen. A mathematical model of the entire process would be extremely complicated, because the signaling and cytoskeletal changes produce large mechanical deformations of the cell. Recent experiments have used a confinement technique that leads to a process called frustrated phagocytosis, in which the membrane does not deform, but rather, signaling triggers actin waves that propagate along the boundary of the cell. This eliminates the large-scale deformations and facilitates modeling of the wave dynamics. Herein we develop a model of the actin dynamics observed in frustrated phagocytosis and show that it can replicate the experimental observations. We identify the key components that control the actin waves and make a number of experimentally-testable predictions. In particular, we predict that diffusion coefficients of membrane-bound species must be larger behind the wavefront to replicate the internal structure of the waves. Our model is a first step toward a more complete model of phagocytosis, and provides insights into circular dorsal ruffles as well.

Mathematical modeling of chemotaxis guided amoeboid cell swimming

Cells and microorganisms adopt various strategies to migrate in response to different environmental stimuli. To date, many modeling research has focused on the crawling-basedDictyostelium discoideum(Dd) cells migration induced by chemotaxis, yet recent experimental results reveal that even without adhesion or contact to a substrate, Dd cells can still swim to follow chemoattractant signals. In this paper, we develop a modeling framework to investigate the chemotaxis induced amoeboid cell swimming dynamics. A minimal swimming system consists of one deformable Dd amoeboid cell and a dilute suspension of bacteria, and the bacteria produce chemoattractant signals that attract the Dd cell. We use themathematical amoeba modelto generate Dd cell deformation and solve the resulting low Reynolds number flows, and use a moving mesh based finite volume method to solve the reaction-diffusion-convection equation. Using the computational model, we show that chemotaxis guides a swimming Dd cell to follow and catch bacteria, while on the other hand, bacterial rheotaxis may help the bacteria to escape from the predator Dd cell.

Why a Large-Scale Mode Can Be Essential for Understanding Intracellular Actin Waves

During the last decade, intracellular actin waves have attracted much attention due to their essential role in various cellular functions, ranging from motility to cytokinesis. Experimental methods have advanced significantly and can capture the dynamics of actin waves over a large range of spatio-temporal scales. However, the corresponding coarse-grained theory mostly avoids the full complexity of this multi-scale phenomenon. In this perspective, we focus on a minimal continuum model of activator-inhibitor type and highlight the qualitative role of mass conservation, which is typically overlooked. Specifically, our interest is to connect between the mathematical mechanisms of pattern formation in the presence of a large-scale mode, due to mass conservation, and distinct behaviors of actin waves.

The Roles of Signaling in Cytoskeletal Changes, Random Movement, Direction-Sensing and Polarization of Eukaryotic Cells

Movement of cells and tissues is essential at various stages during the lifetime of an organism, including morphogenesis in early development, in the immune response to pathogens, and during wound-healing and tissue regeneration. Individual cells are able to move in a variety of microenvironments (MEs) (A glossary of the acronyms used herein is given at the end) by suitably adapting both their shape and how they transmit force to the ME, but how cells translate environmental signals into the forces that shape them and enable them to move is poorly understood. While many of the networks involved in signal detection, transduction and movement have been characterized, how intracellular signals control re-building of the cyctoskeleton to enable movement is not understood. In this review we discuss recent advances in our understanding of signal transduction networks related to direction-sensing and movement, and some of the problems that remain to be solved.

Thermodynamically consistent treatment of the growth of a biopolymer in the presence of a smooth obstacle interaction potential

We investigate the effect of filament-obstacle interactions on the force-velocity relation of growing biopolymers, via calculations explicitly treating obstacle diffusion and stochastic addition and subtraction of subunits. We first show that the instantaneous subunit on- and off-rates satisfy a rigorous thermodynamic relationship determined by the filament-obstacle interaction potential, which has been violated by several calculations in the literature. The instantaneous rates depend not only on the average force on the obstacle but also on the shape of the potential on the nanometer length scale. Basing obstacle-induced reduction of the on-rate entirely on the force, as previous work has often done, is thermodynamically inconsistent and can overestimate the stall force, sometimes by more than a factor of two. We perform simulations and analytic calculations of the force-velocity relation satisfying the thermodynamic relationship. The force-velocity relation can deviate strongly from the Brownian-Ratchet predictions. For shallow potential wells of depth ∼5k_{B}T, which might correspond to transient filament-membrane attachments, the velocity drops more rapidly than predicted by the Brownian-Ratchet model, in some cases by as much as a factor of 50 at an opposing force of only 1 pN. On the other hand, the zero-force velocity is much less affected than would be expected from naive use of the Boltzmann factor. Furthermore, the growth velocity has a surprisingly strong dependence on the obstacle diffusion coefficient even when the dimensionless diffusion coefficient is large. For deep potential wells, as might result from strong filament-membrane links, both the on- and off-rates are reduced significantly, slowing polymerization. Such potentials can sustain pulling forces while polymerizing but only if the attractive well is relatively flat over a region comparable to or greater than the monomer size. For double-well potentials, which have such a flat region, the slowing of polymerization by external pushing force is almost linear up to the stall force in some parameter ranges.

Cell-Substrate Patterns Driven by Curvature-Sensitive Actin Polymerization: Waves and Podosomes

Cells adhered to an external solid substrate are observed to exhibit rich dynamics of actin structures on the basal membrane, which are distinct from those observed on the dorsal (free) membrane. Here we explore the dynamics of curved membrane proteins, or protein complexes, that recruit actin polymerization when the membrane is confined by the solid substrate. Such curved proteins can induce the spontaneous formation of membrane protrusions on the dorsal side of cells. However, on the basal side of the cells, such protrusions can only extend as far as the solid substrate and this constraint can convert such protrusions into propagating wave-like structures. We also demonstrate that adhesion molecules can stabilize localized protrusions that resemble some features of podosomes. This coupling of curvature and actin forces may underlie the differences in the observed actin-membrane dynamics between the basal and dorsal sides of adhered cells.

A calcium-mediated actin redistribution at egg activation in Drosophila

Egg activation is the essential process in which mature oocytes gain the competency to proceed into embryonic development. Many events of egg activation are conserved, including an initial rise of intracellular calcium. In some species, such as echinoderms and mammals, changes in the actin cytoskeleton occur around the time of fertilization and egg activation. However, the interplay between calcium and actin during egg activation remains unclear. Here, we use imaging, genetics, pharmacological treatment, and physical manipulation to elucidate the relationship between calcium and actin in living Drosophila eggs. We show that, before egg activation, actin is smoothly distributed between ridges in the cortex of the dehydrated mature oocytes. At the onset of egg activation, we observe actin spreading out as the egg swells though the intake of fluid. We show that a relaxed actin cytoskeleton is required for the intracellular rise of calcium to initiate and propagate. Once the swelling is complete and the calcium wave is traversing the egg, it leads to a reorganization of actin in a wavelike manner. After the calcium wave, the actin cytoskeleton has an even distribution of foci at the cortex. Together, our data show that calcium resets the actin cytoskeleton at egg activation, a model that we propose to be likely conserved in other species.

Oscillatory Switches of Dorso-Ventral Polarity in Cells Confined between Two Surfaces

To maneuver in a three-dimensional space, migrating cells need to accommodate to multiple surfaces. In particular, phagocytes have to explore their environment in the search for particles to be ingested. To examine how cells decide between competing surfaces, we exposed single cells of Dictyostelium to a defined three-dimensional space by confining them between two planar surfaces: those of a cover glass and of a wedged microcantilever. These cells form propagating waves of filamentous actin and PIP3 on their ventral substrate-attached surface. The dynamics of wave formation in the confined cells was explored using two-focus fluorescence imaging. When waves formed on one substrate, wave formation on the other substrate was efficiently suppressed. The propensity for wave formation switched between the opposing cell surfaces with periods of 2-5 min by one of two modes: 1) a rolling mode involving the slipping of a wave along the nonattached plasma membrane and 2) de novo initiation of waves on the previously blank cell surface. These data provide evidence for a cell-autonomous oscillator that switches dorso-ventral polarity in a cell simultaneously exposed to multiple substrate surfaces.

The Architecture of Traveling Actin Waves Revealed by Cryo-Electron Tomography

Actin waves are dynamic supramolecular structures involved in cell migration, cytokinesis, adhesion, and neurogenesis. Although wave-like propagation of actin networks is a widespread phenomenon, the actin architecture underlying wave propagation remained unknown. In situ cryo-electron tomography of Dictyostelium cells unveils the wave architecture and provides evidence for wave progression by de novo actin nucleation. Subtomogram averaging reveals the structure of Arp2/3 complex-mediated branch junctions in their native state, and enables quantitative analysis of the 3D organization of branching within the waves. We find an excess of branches directed toward the substrate-attached membrane, and tent-like structures at sites of branch clustering. Fluorescence imaging shows that Arp2/3 clusters follow accumulation of the elongation factor VASP. We propose that filament growth toward the membrane lifts up the actin network as the wave propagates, until depolymerization of oblique filaments at the back causes the collapse of horizontal filaments into a compact layer.

Eukaryotic Cell Dynamics from Crawlers to Swimmers

Movement requires force transmission to the environment, and motile cells are robustly, though not elegantly, designed nanomachines that often can cope with a variety of environmental conditions by altering the mode of force transmission used. As with humans, the available modes range from momentary attachment to a substrate when crawling, to shape deformations when swimming, and at the cellular level this involves sensing the mechanical properties of the environment and altering the mode appropriately. While many types of cells can adapt their mode of movement to their microenvironment (ME), our understanding of how they detect, transduce and process information from the ME to determine the optimal mode is still rudimentary. The shape and integrity of a cell is determined by its cytoskeleton (CSK), and thus the shape changes that may be required to move involve controlled remodeling of the CSK. Motion in vivo is often in response to extracellular signals, which requires the ability to detect such signals and transduce them into the shape changes and force generation needed for movement. Thus the nanomachine is complex, and while much is known about individual components involved in movement, an integrated understanding of motility in even simple cells such as bacteria is not at hand. In this review we discuss recent advances in our understanding of cell motility and some of the problems remaining to be solved.

Parallel signaling pathways regulate excitable dynamics differently for pseudopod formation in eukaryotic chemotaxis

In eukaryotic chemotaxis, parallel signaling pathways regulate the spatiotemporal pseudopod dynamics at the leading edge of a motile cell through characteristic dynamics of an excitable system; however, differences in the excitability and the physiological roles of individual pathways remain to be elucidated. Here we found that two different pathways, soluble guanylyl cyclase (sGC) and phosphatidylinositol 3-kinase (PI3K), exhibited similar all-or-none responses but different refractory periods by simultaneous observations of their excitable properties. Due to the shorter refractory period, sGC signaling responded more frequently to chemoattractants, leading to pseudopod formation with higher frequency. sGC excitability was regulated negatively by its product, cGMP, and cGMP-binding protein C (GbpC) through the suppression of F-actin polymerization, providing the underlying delayed negative feedback mechanism for the cyclical pseudopod formation. These results suggest that parallel pathways respond on different time-scales to environmental cues for chemotactic motility based on their intrinsic excitability. Key words: cGMP signaling, chemotaxis, excitability, pseudopod formation

A master equation approach to actin polymerization applied to endocytosis in yeast

We present a Master Equation approach to calculating polymerization dynamics and force generation by branched actin networks at membranes. The method treats the time evolution of the F-actin distribution in three dimensions, with branching included as a directional spreading term. It is validated by comparison with stochastic simulations of force generation by actin polymerization at obstacles coated with actin “nucleation promoting factors” (NPFs). The method is then used to treat the dynamics of actin polymerization and force generation during endocytosis in yeast, using a model in which NPFs form a ring around the endocytic site, centered by a spot of molecules attaching the actin network strongly to the membrane. We find that a spontaneous actin filament nucleation mechanism is required for adequate forces to drive the process, that partial inhibition of branching and polymerization lead to different characteristic responses, and that a limited range of polymerization-rate values provide effective invagination and obtain correct predictions for the effects of mutations in the active regions of the NPFs.

Endocytosis is a dynamic process by which cells internalize substances from outside the cell. Especially in yeast, endocytosis is mechanically demanding due to the high pressure difference across the cell membrane, or turgor pressure. Polymerization of a branched actin network is the major process providing the mechanical force to overcome the turgor pressure. Understanding the kinetics of the actin network, and the mechanical interaction between the actin network and the cell membrane, is thus crucial for the study of endocytosis. We develop an efficient mathematical framework for actin dynamics that can realistically incorporate these two features, thus providing a practical method for quantitatively modeling actin dynamics during endocytosis. The resulting model mechanistically reveals that spontaneous nucleation at the center of the endocytic site is required for successful endocytosis, distinguishes the roles of branching and polymerization, and predicts several other experimentally testable outcomes. The accuracy and efficiency of the method, in describing both mechanics and chemistry, render it applicable to a broad field of membrane-bending processes.

Fronts and waves of actin polymerization in a bistability-based mechanism of circular dorsal ruffles

During macropinocytosis, cells remodel their morphologies for the uptake of extracellular matter. This endocytotic mechanism relies on the collapse and closure of precursory structures, which are propagating actin-based, ring-shaped vertical undulations at the dorsal (top) cell membrane, a.k.a. circular dorsal ruffles (CDRs). As such, CDRs are essential to a range of vital and pathogenic processes alike. Here we show, based on both experimental data and theoretical analysis, that CDRs are propagating fronts of actin polymerization in a bistable system. The theory relies on a novel mass-conserving reaction–diffusion model, which associates the expansion and contraction of waves to distinct counter-propagating front solutions. Moreover, the model predicts that under a change in parameters (for example, biochemical conditions) CDRs may be pinned and fluctuate near the cell boundary or exhibit complex spiral wave dynamics due to a wave instability. We observe both phenomena also in our experiments indicating the conditions for which macropinocytosis is suppressed.

Circular dorsal ruffles (CDRs) are important for the vesicular uptake of extracellular matter, but the basis of their wave dynamics is not understood. Here, the authors propose and experimentally test a bistable reaction-diffusion system, which they show accounts for the typical CDR expansion and shrinkage and for aberrant formation of pinned waves and spirals.

A plasma membrane template for macropinocytic cups

Macropinocytosis is a fundamental mechanism that allows cells to take up extracellular liquid into large vesicles. It critically depends on the formation of a ring of protrusive actin beneath the plasma membrane, which develops into the macropinocytic cup. We show that macropinocytic cups in Dictyostelium are organised around coincident intense patches of PIP3, active Ras and active Rac. These signalling patches are invariably associated with a ring of active SCAR/WAVE at their periphery, as are all examined structures based on PIP3 patches, including phagocytic cups and basal waves. Patch formation does not depend on the enclosing F-actin ring, and patches become enlarged when the RasGAP NF1 is mutated, showing that Ras plays an instructive role. New macropinocytic cups predominantly form by splitting from existing ones. We propose that cup-shaped plasma membrane structures form from self-organizing patches of active Ras/PIP3, which recruit a ring of actin nucleators to their periphery.

Adhesion-Dependent Wave Generation in Crawling Cells

Dynamic actin networks are excitable. In migrating cells, feedback loops can amplify stochastic fluctuations in actin dynamics, often resulting in traveling waves of protrusion. The precise contributions of various molecular and mechanical interactions to wave generation have been difficult to disentangle, in part due to complex cellular morphodynamics. Here we used a relatively simple cell type-the fish epithelial keratocyte-to define a set of mechanochemical feedback loops underlying actin network excitability and wave generation. Although keratocytes are normally characterized by the persistent protrusion of a broad leading edge, increasing cell-substrate adhesion strength results in waving protrusion of a short leading edge. We show that protrusion waves are due to fluctuations in actin polymerization rates and that overexpression of VASP, an actin anti-capping protein that promotes actin polymerization, switches highly adherent keratocytes from waving to persistent protrusion. Moreover, VASP localizes both to adhesion complexes and to the leading edge. Based on these results, we developed a mathematical model for protrusion waves in which local depletion of VASP from the leading edge by adhesions-along with lateral propagation of protrusion due to the branched architecture of the actin network and negative mechanical feedback from the cell membrane-results in regular protrusion waves. Consistent with our model simulations, we show that VASP localization at the leading edge oscillates, with VASP leading-edge enrichment greatest just prior to protrusion initiation. We propose that the mechanochemical feedbacks underlying wave generation in keratocytes may constitute a general module for establishing excitable actin dynamics in other cellular contexts.

Pattern Formation in Polymerizing Actin Flocks: Spirals, Spots, and Waves without Nonlinear Chemistry

We propose a model solely based on actin treadmilling and polymerization which describes many characteristic states of actin-wave formation: spots, spirals, and traveling waves. In our model, as in experiments on cells recovering motility following actin depolymerization, we choose an isotropic low-density initial condition; polymerization of actin filaments then raises the density towards the Onsager threshold where they align. We show that this alignment, in turn, destabilizes the isotropic phase and generically induces transient actin spots or spirals as part of the dynamical pathway towards a polarized phase which can either be uniform or consist of a series of actin-wave trains (flocks). Our results uncover a universal route to actin-wave formation in the absence of any system-specific nonlinear biochemistry, and it may help to understand the mechanism underlying the observation of actin spots and waves in vivo. They also suggest a minimal setup to design similar patterns in vitro.

Progress and perspectives in signal transduction, actin dynamics, and movement at the cell and tissue level: lessons from Dictyostelium

Movement of cells and tissues is a basic biological process that is used in development, wound repair, the immune response to bacterial invasion, tumour formation and metastasis, and the search for food and mates. While some cell movement is random, directed movement stimulated by extracellular signals is our focus here. This involves a sequence of steps in which cells first detect extracellular chemical and/or mechanical signals via membrane receptors that activate signal transduction cascades and produce intracellular signals. These intracellular signals control the motile machinery of the cell and thereby determine the spatial localization of the sites of force generation needed to produce directed motion. Understanding how force generation within cells and mechanical interactions with their surroundings, including other cells, are controlled in space and time to produce cell-level movement is a major challenge, and involves many issues that are amenable to mathematical modelling.

Modeling large-scale dynamic processes in the cell: polarization, waves, and division

The past decade has witnessed significant developments in molecular biology techniques, fluorescent labeling, and super-resolution microscopy, and together these advances have vastly increased our quantitative understanding of the cell. This detailed knowledge has concomitantly opened the door for biophysical modeling on a cellular scale. There have been comprehensive models produced describing many processes such as motility, transport, gene regulation, and chemotaxis. However, in this review we focus on a specific set of phenomena, namely cell polarization, F-actin waves, and cytokinesis. In each case, we compare and contrast various published models, highlight the relevant aspects of the biology, and provide a sense of the direction in which the field is moving.

Modeling the dynamics of dendritic actin waves in living cells

The actin cytoskeleton in living cells exhibits a high degree of capacity for dynamic self-organization. Recent experiments have observed propagating actin waves in Dictyostelium cells recovering from complete depolymerization of their actin cytoskeleton. The propagation of these waves appear to be dependent on a programmed recruitment of a few proteins that control actin assembly and disassembly. Such waves also arise spontaneously along the plasma membrane of the cell, and it has been suggested that actin waves enable the cell to scan a surface for particles to engulf. Based on known molecular components involved in wave propagation, we present and study a minimal reaction-diffusion model for actin wave production observed in recovering cells.

Actin and PIP3 waves in giant cells reveal the inherent length scale of an excited state

The membrane and actin cortex of a motile cell can autonomously differentiate into two states, one typical of the front, the other of the tail. On the substrate-attached surface of Dictyostelium discoideum cells, dynamic patterns of front-like and tail-like states are generated that are well suited to monitor transitions between these states. To image large-scale pattern dynamics independently of boundary effects, we produced giant cells by electric-pulse-induced cell fusion. In these cells, actin waves are coupled to the front and back of phosphatidylinositol (3,4,5)-trisphosphate (PIP3)-rich bands that have a finite width. These composite waves propagate across the plasma membrane of the giant cells with undiminished velocity. After any disturbance, the bands of PIP3 return to their intrinsic width. Upon collision, the waves locally annihilate each other and change direction; at the cell border they are either extinguished or reflected. Accordingly, expanding areas of progressing PIP3 synthesis become unstable beyond a critical radius, their center switching from a front-like to a tail-like state. Our data suggest that PIP3 patterns in normal-sized cells are segments of the self-organizing patterns that evolve in giant cells.

Modeling self-organized spatio-temporal patterns of PIP₃ and PTEN during spontaneous cell polarization

During spontaneous cell polarization of Dictyostelium discoideum cells, phosphatidylinositol (3,4,5)-triphoshpate (PIP3) and PTEN (phosphatase tensin homolog) have been identified as key signaling molecules which govern the process of polarization in a self-organized manner. Recent experiments have quantified the spatio-temporal dynamics of these signaling components. Surprisingly, it was found that membrane-bound PTEN can be either in a high or low state, that PIP3 waves were initiated in areas lacking PTEN through an excitable mechanism, and that PIP3 was degraded even though the PTEN concentration remained low. Here we develop a reaction-diffusion model that aims to explain these experimental findings. Our model contains bistable dynamics for PTEN, excitable dynamics for PIP3, and postulates the existence of two species of PTEN with different dephosphorylation rates. We show that our model is able to produce results that are in good qualitative agreement with the experiments, suggesting that our reaction-diffusion model underlies the self-organized spatio-temporal patterns observed in experiments.

The association of myosin IB with actin waves in dictyostelium requires both the plasma membrane-binding site and actin-binding region in the myosin tail

F-actin structures and their distribution are important determinants of the dynamic shapes and functions of eukaryotic cells. Actin waves are F-actin formations that move along the ventral cell membrane driven by actin polymerization. Dictyostelium myosin IB is associated with actin waves but its role in the wave is unknown. Myosin IB is a monomeric, non-filamentous myosin with a globular head that binds to F-actin and has motor activity, and a non-helical tail comprising a basic region, a glycine-proline-glutamine-rich region and an SH3-domain. The basic region binds to acidic phospholipids in the plasma membrane through a short basic-hydrophobic site and the Gly-Pro-Gln region binds F-actin. In the current work we found that both the basic-hydrophobic site in the basic region and the Gly-Pro-Gln region of the tail are required for the association of myosin IB with actin waves. This is the first evidence that the Gly-Pro-Gln region is required for localization of myosin IB to a specific actin structure in situ. The head is not required for myosin IB association with actin waves but binding of the head to F-actin strengthens the association of myosin IB with waves and stabilizes waves. Neither the SH3-domain nor motor activity is required for association of myosin IB with actin waves. We conclude that myosin IB contributes to anchoring actin waves to the plasma membranes by binding of the basic-hydrophobic site to acidic phospholipids in the plasma membrane and binding of the Gly-Pro-Gln region to F-actin in the wave.