Dendritic actin filament nucleation causes traveling waves and patches

Supramolecular fibrillation in coacervates and other confined systems towards biomimetic function

As in natural cytoskeletons, the cooperative assembly of fibrillar networks can be hosted inside compartments to engineer biomimetic functions, such as mechanical actuation, transport, and reaction templating. Coacervates impose an optimal liquid-liquid phase separation within the aqueous continuum, functioning as membrane-less compartments that can organise such self-assembling processes as well as the exchange of information with their environment. Furthermore, biological fibrillation can often be controlled or assisted by intracellular compartments. Thus, the reconstitution of analogues of natural filaments in simplified artificial compartments, such as coacervates, offer a suitable model to unravel, mimic, and potentially exploit cellular functions. This perspective summarises the latest developments towards assembling fibrillar networks under confinement inside coacervates and related compartments, including a selection of examples ranging from biological to fully synthetic monomers. Comparative analysis between coacervates, lipid vesicles, and droplet emulsions showcases the interplay between supramolecular fibres and the boundaries of the corresponding compartment. Combining inspiration from natural systems and the custom properties of tailored synthetic fibrillators, rational monomer and compartment design will contribute towards engineering increasingly complex and more realistic artificial protocells.

Modeling the dynamic growth and branching of actin filaments

As an essential component of the cytoskeleton, actin filaments play a key role in a variety of cellular physiological activities. To better understand the function of actin filaments, which are a special kind of polymer chain, researchers have started to focus on the Brownian dynamics of polymers. Currently, to study the dynamics of polymers, classical explicit bead-spring models and finite-element methods (FEMs) have both been broadly used. However, compared to bead-spring models, FEMs can address the mechanical properties of actin filaments and actin networks with more detail and better accuracy. However, current FEMs do not consider the dynamic assembly of actin into an actin filament network. Here, we extend the traditional FEM and present a new framework of the FEM based on the co-rotational grid method, which allows us to simulate the dynamic growth and branching of actin filaments. Several examples are studied. The proposed numerical model is capable of capturing the dynamic assembly of actin filaments.

Supramolecular fibrillation of peptide amphiphiles induces environmental responses in aqueous droplets

One-dimensional (1D) supramolecular polymers are commonly found in natural and synthetic systems to prompt functional responses that capitalise on hierarchical molecular ordering. Despite amphiphilic self-assembly being significantly studied in the context of aqueous encapsulation and autopoiesis, very little is currently known about the physico-chemical consequences and functional role of 1D supramolecular polymerisation confined in aqueous compartments. Here, we describe the different phenomena that resulted from the chemically triggered supramolecular fibrillation of synthetic peptide amphiphiles inside water microdroplets. The confined connection of suitable dormant precursors triggered a physically autocatalysed chemical reaction that resulted in functional environmental responses such as molecular uptake, fusion and chemical exchange. These results demonstrate the potential of minimalistic 1D supramolecular polymerisation to modulate the behaviour of individual aqueous entities with their environment and within communities.

Reproducing asymmetrical spine shape fluctuations in a model of actin dynamics predicts self-organized criticality

Dendritic spines change their size and shape spontaneously, but the function of this remains unclear. Here, we address this in a biophysical model of spine fluctuations, which reproduces experimentally measured spine fluctuations. For this, we characterize size- and shape fluctuations from confocal microscopy image sequences using autoregressive models and a new set of shape descriptors derived from circular statistics. Using the biophysical model, we extrapolate into longer temporal intervals and find the presence of 1/f noise. When investigating its origins, the model predicts that the actin dynamics underlying shape fluctuations self-organizes into a critical state, which creates a fine balance between static actin filaments and free monomers. In a comparison against a non-critical model, we show that this state facilitates spine enlargement, which happens after LTP induction. Thus, ongoing spine shape fluctuations might be necessary to react quickly to plasticity events.

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.

Quasi-periodic migration of single cells on short microlanes

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

Correlated random walks inside a cell: actin branching and microtubule dynamics

Correlated random walks (CRW) have been explored in many settings, most notably in the motion of individuals in a swarm or flock. But some subcellular systems such as growth or disassembly of bio-polymers can also be described with similar models and understood using related mathematical methods. Here we consider two examples of growing cytoskeletal elements, actin and microtubules. We use CRW or generalized CRW-like PDEs to model their spatial distributions. In each case, the linear models can be reduced to a Telegrapher's equation. A combination of explicit solutions (in one case) and numerical solutions (in the other) demonstrates that the approach to steady state can be accompanied by (decaying) waves.

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.

The Sealing Zone in Osteoclasts: A Self-Organized Structure on the Bone

Osteoclasts form a specialized cell-matrix adhesion structure, known as the "sealing zone", during bone resorption. The sealing zone is a dynamic actin-rich structure that defines the resorption area of the bone. The detailed dynamics and fine structure of the sealing zone have been elusive. Osteoclasts plated on glass do not form a sealing zone, but generate a separate supra-molecular structure called the "podosome belt". Podosomes are integrin-based adhesion complexes involved in matrix adhesion, cell migration, matrix degradation, and mechanosensing. Invadopodia, podosome-like protrusions in cancer cells, are involved in cell invasion into other tissues by promoting matrix degradation. Both podosomes and invadopodia exhibit actin pattern transitions during maturation. We previously found that Arp2/3-dependent actin flow occurs in all observed assembly patterns of podosomes in osteoclasts on glass. It is known that the actin wave in Dictyostelium cells exhibits a similar pattern transition in its evolution. Because of significant advances in our understanding regarding the mechanism of podosomes/invadopodia formation over the last decade, we revisited the structure and function of the sealing zone in this review, highlighting the possible involvement of self-organized actin waves in the organogenesis of the sealing zone.

Cell protrusion and retraction driven by fluctuations in actin polymerization: A two-dimensional model

Animal cells that spread onto a surface often rely on actin-rich lamellipodial extensions to execute protrusion. Many cell types recently adhered on a two-dimensional substrate exhibit protrusion and retraction of their lamellipodia, even though the cell is not translating. Travelling waves of protrusion have also been observed, similar to those observed in crawling cells. These regular patterns of protrusion and retraction allow quantitative analysis for comparison to mathematical models. The periodic fluctuations in leading edge position of XTC cells have been linked to excitable actin dynamics using a one-dimensional model of actin dynamics, as a function of arc-length along the cell. In this work we extend this earlier model of actin dynamics into two dimensions (along the arc-length and radial directions of the cell) and include a model membrane that protrudes and retracts in response to the changing number of free barbed ends of actin filaments near the membrane. We show that if the polymerization rate at the barbed ends changes in response to changes in their local concentration at the leading edge and/or the opposing force from the cell membrane, the model can reproduce the patterns of membrane protrusion and retraction seen in experiment. We investigate both Brownian ratchet and switch-like force-velocity relationships between the membrane load forces and actin polymerization rate. The switch-like polymerization dynamics recover the observed patterns of protrusion and retraction as well as the fluctuations in F-actin concentration profiles. The model generates predictions for the behavior of cells after local membrane tension perturbations.

Symmetrical retrograde actin flow in the actin fusion structure is involved in osteoclast fusion

The aim of this study was to elucidate the role of the zipper-like structure (ZLS), a podosome-related structure that transiently appears at the cell contact zone, in osteoclast fusion. Live-cell imaging of osteoclasts derived from RAW264.7 cells transfected with EGFP-actin revealed consistent symmetrical retrograde actin flow in the ZLS, but not in the podosome cluster, the podosome ring or the podosome belt. Confocal imaging showed that the distributions of F-actin, vinculin, paxillin and zyxin in the ZLS were different from those in the podosome belt. Thick actin filament bundles running outside the ZLS appeared to recruit non-muscle myosin IIA. The F-actin-rich domain of the ZLS contained actin-related protein 2/3 complex (Arp2/3). Inhibition of Arp2/3 activity disorganized the ZLS, disrupted actin flow, deteriorated cell-cell adhesion and inhibited osteoclast hypermultinucleation. In contrast, ML-7, an inhibitor of myosin light chain kinase, had little effect on the structure of ZLS and promoted osteoclast hypermultinucleation. These results reveal a link between actin flow in the ZLS and osteoclast fusion. Osteoclast fusion was promoted by branched actin elongation and negatively regulated by actomyosin contraction.

Summary: Multinucleated osteoclasts form a podosome-derived fusion structure during cell fusion. Juxtaposition of fusion partner cells is probably maintained via force generated by symmetrical retrograde actin flow in the fusion structure.

Diversity of actin architecture in human osteoclasts: network of curved and branched actin supporting cell shape and intercellular micrometer-level tubes

Osteoclasts are multinucleated bone-resorbing cells with a dynamic actin cytoskeleton. Osteoclasts are derived from circulating mononuclear precursors. Confocal and stimulated emission depletion (STED) super-resolution microscopy was used to investigate peripheral blood-derived human osteoclasts cultured on glass surfaces. STED and confocal microscopy demonstrated that the actin was curved and branched, for instance, in the vicinity of membrane ruffles. The overall architecture of the curved actin network extended from the podosomes to the top of the cell. The other novel finding was that a micrometer-level tube containing actin bridged the osteoclasts well above the level of the culture glass. The actin filaments of the tubes originated from the network of curved actin often surrounding a group of nuclei. Furthermore, nuclei were occasionally located inside the tubes. Our findings demonstrated the accumulation of c-Src, cortactin, cofilin, and actin around nuclei suggesting their role in nuclear processes such as the locomotion of nuclei. ARP2/3 labeling was abundant at the substratum level of osteoclasts and in the branched actin network, where it localized to the branching points. We speculate that the actin-containing tubes of osteoclasts may provide a means of transportation of nuclei, e.g., during the fusion of osteoclasts. These novel findings can pave the way for future studies aiming at the elucidation of the differentiation of multinucleated osteoclasts.

Spatiotemporal Patterns of Noise-Driven Confined Actin Waves in Living Cells

Cells utilize waves of polymerizing actin to reshape their morphologies, which is central to physiological and pathological processes alike. Here, we force dorsal actin waves to propagate on one-dimensional domains with periodic boundary conditions, which results in striking spatiotemporal patterns with a clear signature of noise-driven dynamics. We show that these patterns can be very closely reproduced with a noise-driven active medium at coherence resonance.

Actin-Regulator Feedback Interactions during Endocytosis

Endocytosis mediated by clathrin, a cellular process by which cells internalize membrane receptors and their extracellular ligands, is an important component of cell signaling regulation. Actin polymerization is involved in endocytosis in varying degrees depending on the cellular context. In yeast, clathrin-mediated endocytosis requires a pulse of polymerized actin and its regulators, which recruit and activate the Arp2/3 complex. In this article, we seek to identify the main protein-protein interactions that 1) cause actin and its regulators to appear in pulses, and 2) determine the effects of key mutations and drug treatments on actin and regulator assembly. We perform a joint modeling/experimental study of actin and regulator dynamics during endocytosis in the budding yeast Saccharomyces cerevisiae. We treat both a stochastic model that grows an explicit three-dimensional actin network, and a simpler two-variable Fitzhugh-Nagumo type model. The models include a negative-feedback interaction of F-actin onto the Arp2/3 regulators. Both models explain the pulse time courses and the effects of interventions on actin polymerization: the surprising increase in the peak F-actin count caused by reduced regulator branching activity, the increase in F-actin resulting from slowing of actin disassembly, and the increased Arp2/3 regulator lifetime resulting from latrunculin treatment. In addition, they predict that decreases in the regulator branching activity lead to increases in accumulation of regulators, and we confirmed this prediction with experiments on yeast harboring mutations in the Arp2/3 regulators, using quantitative fluorescence microscopy. Our experimental measurements suggest that the regulators act quasi-independently, in the sense that accumulation of a particular regulator is most strongly affected by mutations of that regulator, as opposed to the others.

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.

Competition for actin between two distinct F-actin networks defines a bistable switch for cell polarization

Symmetry-breaking polarization enables functional plasticity of cells and tissues and is yet not well understood. Here we show that epithelial cells, hard-wired to maintain a static morphology and to preserve tissue organization, can spontaneously switch to a migratory polarized phenotype upon relaxation of the actomyosin cytoskeleton. We find that myosin-II engages actin in the formation of cortical actomyosin bundles and thus makes it unavailable for deployment in the process of dendritic growth normally driving cell motility. At low contractility regimes epithelial cells polarize in a front-back manner due to emergence of actin retrograde flows powered by dendritic polymerization of actin. Coupled to cell movement, the flows transport myosin-II from the front to the back of the cell, where the motor locally “locks” actin in contractile bundles. This polarization mechanism could be employed by embryonic and cancer epithelial cells in microenvironments where high contractility-driven cell motion is inefficient.

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.

Evolutionarily conserved coupling of adaptive and excitable networks mediates eukaryotic chemotaxis

Numerous models explain how cells sense and migrate towards shallow chemoattractant gradients. Studies show that an excitable signal transduction network acts as a pacemaker that controls the cytoskeleton to drive motility. Here we show that this network is required to link stimuli to actin polymerization and chemotactic motility and we distinguish the various models of chemotaxis. First, signalling activity is suppressed towards the low side in a gradient or following removal of uniform chemoattractant. Second, signalling activities display a rapid shut off and a slower adaptation during which responsiveness to subsequent test stimuli decline. Simulations of various models indicate that these properties require coupled adaptive and excitable networks. Adaptation involves a G-protein-independent inhibitor, as stimulation of cells lacking G-protein function suppresses basal activities. The salient features of the coupled networks were observed for different chemoattractants in Dictyostelium and in human neutrophils, suggesting an evolutionarily conserved mechanism for eukaryotic chemotaxis.

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.

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.

Continuum descriptions of cytoskeletal dynamics

This tutorial presents an introduction into continuum descriptions of cytoskeletal dynamics. In contrast to discrete models in which each molecule keeps its identity, such descriptions are given in terms of averaged quantities per unit volume like the number density of a certain molecule. Starting with a discrete description for the assembly dynamics of cytoskeletal filaments, we derive the continuity equation, which serves as the basis of many continuum theories. We illustrate the use of this approach with an investigation of spontaneous cytoskeletal polymerization waves. Such waves have by now been observed in various cell types and might help to orchestrate cytoskeletal dynamics during cell spreading and locomotion. Our analysis shows how processes at the scale of single molecules, namely, the nucleation of new filaments and filament treadmilling, can lead to the spontaneous appearance of coherent traveling waves on scales spanning many filament lengths. For readers less familiar with calculus, we include an informal introduction to the Taylor expansion.

Formation of transient lamellipodia

Cell motility driven by actin polymerization is pivotal to the development and survival of organisms and individual cells. Motile cells plated on flat substrates form membrane protrusions called lamellipodia. The protrusions repeatedly appear and retract in all directions. If a lamellipodium is stabilized and lasts for some time, it can take over the lead and determine the direction of cell motion. Protrusions traveling along the cell perimeter have also been observed. Their initiation is in some situations the effect of the dynamics of the pathway linking plasma membrane receptors to actin filament nucleation, e.g. in chemotaxis. However, lamellipodia are also formed in many cells incessantly during motion with a constant state of the signaling pathways upstream from nucleation promoting factors (NPFs), or spontaneously in resting cells. These observations strongly suggest protrusion formation can also be a consequence of the dynamics downstream from NPFs, with signaling setting the dynamic regime but not initiating the formation of individual protrusions. A quantitative mechanism for this kind of lamellipodium dynamics has not been suggested yet. Here, we present a model exhibiting excitable actin network dynamics. Individual lamellipodia form due to random supercritical filament nucleation events amplified by autocatalytic branching. They last for about 30 seconds to many minutes and are terminated by filament bundling, severing and capping. We show the relevance of the model mechanism for experimentally observed protrusion dynamics by reproducing in very good approximation the repetitive protrusion formation measured by Burnette et al. with respect to the velocities of leading edge protrusion and retrograde flow, oscillation amplitudes, periods and shape, as well as the phase relation between protrusion and retrograde flow. Our modeling results agree with the mechanism of actin bundle formation during lamellipodium retraction suggested by Burnette et al. and Koestler et al.

Spiral and never-settling patterns in active systems

We present a combined numerical and analytical study of pattern formation in an active system where particles align, possess a density-dependent motility, and are subject to a logistic reaction. The model can describe suspensions of reproducing bacteria, as well as polymerizing actomyosin gels in vitro or in vivo. In the disordered phase, we find that motility suppression and growth compete to yield stable or blinking patterns, which, when dense enough, acquire internal orientational ordering to give asters or spirals. We predict these may be observed within chemotactic aggregates in bacterial fluids. In the ordered phase, the reaction term leads to previously unobserved never-settling patterns which can provide a simple framework to understand the formation of motile and spiral patterns in intracellular actin systems.

A model for intracellular actin waves explored by nonlinear local perturbation analysis

Waves and dynamic patterns in chemical and physical systems have long interested experimentalists and theoreticians alike. Here we investigate a recent example within the context of cell biology, where waves of actin (a major component of the cytoskeleton) and its regulators (nucleation promoting factors, NPFs) are observed experimentally. We describe and analyze a minimal reaction diffusion model depicting the feedback between signalling proteins and filamentous actin (F-actin). Using numerical simulation, we show that this model displays a rich variety of patterning regimes. A relatively recent nonlinear stability method, the Local Perturbation Analysis (LPA), is used to map the parameter space of this model and explain the genesis of patterns in various linear and nonlinear patterning regimes. We compare our model for actin waves to others in the literature, and focus on transitions between static polarization, transient waves, periodic wave trains, and reflecting waves. We show, using LPA, that the spatially distributed model gives rise to dynamics that are absent in the kinetics alone. Finally, we show that the width and speed of the waves depend counter-intuitively on parameters such as rates of NPF activation, negative feedback, and the F-actin time scale.

A continuum model of actin waves in Dictyostelium discoideum

Actin waves are complex dynamical patterns of the dendritic network of filamentous actin in eukaryotes. We developed a model of actin waves in PTEN-deficient Dictyostelium discoideum by deriving an approximation of the dynamics of discrete actin filaments and combining it with a signaling pathway that controls filament branching. This signaling pathway, together with the actin network, contains a positive feedback loop that drives the actin waves. Our model predicts the structure, composition, and dynamics of waves that are consistent with existing experimental evidence, as well as the biochemical dependence on various protein partners. Simulation suggests that actin waves are initiated when local actin network activity, caused by an independent process, exceeds a certain threshold. Moreover, diffusion of proteins that form a positive feedback loop with the actin network alone is sufficient for propagation of actin waves at the observed speed of . Decay of the wave back can be caused by scarcity of network components, and the shape of actin waves is highly dependent on the filament disassembly rate. The model allows retraction of actin waves and captures formation of new wave fronts in broken waves. Our results demonstrate that a delicate balance between a positive feedback, filament disassembly, and local availability of network components is essential for the complex dynamics of actin waves.

Cell polarity: mechanochemical patterning

Nearly every cell type exhibits some form of polarity, yet the molecular mechanisms vary widely. Here we examine what we term 'chemical systems' where cell polarization arises through biochemical interactions in signaling pathways, 'mechanical systems' where cells polarize due to forces, stresses and transport, and 'mechanochemical systems' where polarization results from interplay between mechanics and chemical signaling. To reveal potentially unifying principles, we discuss mathematical conceptualizations of several prototypical examples. We suggest that the concept of local activation and global inhibition - originally developed to explain spatial patterning in reaction-diffusion systems - provides a framework for understanding many cases of cell polarity. Importantly, we find that the core ingredients in this framework - symmetry breaking, self-amplifying feedback, and long-range inhibition - involve processes that can be chemical, mechanical, or even mechanochemical in nature.

Traveling waves in actin dynamics and cell motility

Much of current understanding of cell motility arose from studying steady treadmilling of actin arrays. Recently, there have been a growing number of observations of a more complex, non-steady, actin behavior, including self-organized waves. It is becoming clear that these waves result from activation and inhibition feedbacks in actin dynamics acting on different scales, but the exact molecular nature of these feedbacks and the respective roles of biomechanics and biochemistry are still unclear. Here, we review recent advances achieved in experimental and theoretical studies of actin waves and discuss mechanisms and physiological significance of wavy protrusions.

Regimes of wave type patterning driven by refractory actin feedback: transition from static polarization to dynamic wave behaviour

Patterns of waves, patches, and peaks of actin are observed experimentally in many living cells. Models of this phenomenon have been based on the interplay between filamentous actin (F-actin) and its nucleation promoting factors (NPFs) that activate the Arp2/3 complex. Here we present an alternative biologically-motivated model for F-actin-NPF interaction based on properties of GTPases acting as NPFs. GTPases (such as Cdc42, Rac) are known to promote actin nucleation, and to have active membrane-bound and inactive cytosolic forms. The model is a natural extension of a previous mathematical mini-model of small GTPases that generates static cell polarization. Like other modellers, we assume that F-actin negative feedback shapes the observed patterns by suppressing the trailing edge of NPF-generated wave-fronts, hence localizing the activity spatially. We find that our NPF-actin model generates a rich set of behaviours, spanning a transition from static polarization to single pulses, reflecting waves, wave trains, and oscillations localized at the cell edge. The model is developed with simplicity in mind to investigate the interaction between nucleation promoting factor kinetics and negative feedback. It explains distinct types of pattern initiation mechanisms, and identifies parameter regimes corresponding to distinct behaviours. We show that weak actin feedback yields static patterning, moderate feedback yields dynamical behaviour such as travelling waves, and strong feedback can lead to wave trains or total suppression of patterning. We use a recently introduced nonlinear bifurcation analysis to explore the parameter space of this model and predict its behaviour with simulations validating those results.

Confinement induces actin flow in a meiotic cytoplasm

In vivo, F-actin flows are observed at different cell life stages and participate in various developmental processes during asymmetric divisions in vertebrate oocytes, cell migration, or wound healing. Here, we show that confinement has a dramatic effect on F-actin spatiotemporal organization. We reconstitute in vitro the spontaneous generation of F-actin flow using Xenopus meiotic extracts artificially confined within a geometry mimicking the cell boundary. Perturbations of actin polymerization kinetics or F-actin nucleation sites strongly modify the network flow dynamics. A combination of quantitative image analysis and biochemical perturbations shows that both spatial localization of F-actin nucleators and actin turnover play a decisive role in generating flow. Interestingly, our in vitro assay recapitulates several symmetry-breaking processes observed in oocytes and early embryonic cells.

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.

Excitable actin dynamics in lamellipodial protrusion and retraction

Many animal cells initiate crawling by protruding lamellipodia, consisting of a dense network of actin filaments, at their leading edge. We imaged XTC cells that exhibit flat lamellipodia on poly-L-lysine-coated coverslips. Using active contours, we tracked the leading edge and measured the total amount of F-actin by summing the pixel intensities within a 5-μm band. We observed protrusion and retraction with period 130-200 s and local wavelike features. Positive (negative) velocities correlated with minimum (maximum) integrated actin concentration. Approximately constant retrograde flow indicated that protrusions and retractions were driven by fluctuations of the actin polymerization rate. We present a model of these actin dynamics as an excitable system in which a diffusive, autocatalytic activator causes actin polymerization; F-actin accumulation in turn inhibits further activator accumulation. Simulations of the model reproduced the pattern of actin polymerization seen in experiments. To explore the model's assumption of an autocatalytic activation mechanism, we imaged cells expressing markers for both F-actin and the p21 subunit of the Arp2/3 complex. We found that integrated Arp2/3-complex concentrations spike several seconds before spikes of F-actin concentration. This suggests that the Arp2/3 complex participates in an activation mechanism that includes additional diffuse components. Response of cells to stimulation by fetal calf serum could be reproduced by the model, further supporting the proposed dynamical picture.

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.

Self-feedback in actin polymerization

Polymerization of actin, which is crucial for functions such as cell migration, membrane ruffling, cytokinesis, and endocytosis, must be tightly regulated in order to preserve an adequate supply of free actin monomers to respond to changing external conditions. The paper will describe mechanisms by which F-actin feeds back on its own assembly, thus regulating itself. I will present the experimental evidence for such feedback terms, discuss their use in current models of actin dynamics in cells, and present preliminary calculations for the role of feedback in transient endocytic actin patches. These calculations suggest a partial homeostasis of F-actin, in which the F-actin peak height depends only weakly on the actin filament nucleation rate.

Modeling morphodynamic phenotypes and dynamic regimes of cell motion

Many cellular processes and signaling pathways converge onto cell morphology and cell motion, which share important components. The mechanisms used for propulsion could also be responsible for shape changes, if they are capable of generating the rich observed variety of dynamic regimes. Additionally, the analysis of cell shape changes in space and time promises insight into the state of the cytoskeleton and signaling pathways controlling it. While this has been obvious for some time by now, little effort has been made to systematically and quantitatively explore this source of information. First pioneering experimental work revealed morphodynamic phenotypes which can be associated with dynamic regimes like oscillations and excitability. Here, we review the current state of modeling of morphodynamic phenotypes, the experimental results and discuss the ideas on the mechanisms driving shape changes which are suggested by modeling.

Cell motility resulting from spontaneous polymerization waves

The crawling of cells on a substrate is in many cases driven by the actin cytoskeleton. How actin filaments and associated proteins are organized to generate directed motion is still poorly understood. Recent experimental observations suggest that spontaneous cytoskeletal waves might orchestrate the actin-filament network to produce directed motion. We investigate this possibility by studying a mean-field description of treadmilling filaments interacting with nucleating proteins, a system that is known to self-organize into waves. Confining the system by a boundary that shares essential features of membranes, we find that spontaneous waves can generate directional motion. We also find that it can produce lateral waves along the confining membrane as are observed in spreading cells.

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.

Adhesive F-actin waves: a novel integrin-mediated adhesion complex coupled to ventral actin polymerization

At the leading lamellipodium of migrating cells, protrusion of an Arp2/3-nucleated actin network is coupled to formation of integrin-based adhesions, suggesting that Arp2/3-mediated actin polymerization and integrin-dependent adhesion may be mechanistically linked. Arp2/3 also mediates actin polymerization in structures distinct from the lamellipodium, in “ventral F-actin waves” that propagate as spots and wavefronts along the ventral plasma membrane. Here we show that integrins engage the extracellular matrix downstream of ventral F-actin waves in several mammalian cell lines as well as in primary mouse embryonic fibroblasts. These “adhesive F-actin waves” require a cycle of integrin engagement and disengagement to the extracellular matrix for their formation and propagation, and exhibit morphometry and a hierarchical assembly and disassembly mechanism distinct from other integrin-containing structures. After Arp2/3-mediated actin polymerization, zyxin and VASP are co-recruited to adhesive F-actin waves, followed by paxillin and vinculin, and finally talin and integrin. Adhesive F-actin waves thus represent a previously uncharacterized integrin-based adhesion complex associated with Arp2/3-mediated actin polymerization.

Protein self-organization: lessons from the min system

One of the most fundamental features of biological systems is probably their ability to self-organize in space and time on different scales. Despite many elaborate theoretical models of how molecular self-organization can come about, only a few experimental systems of biological origin have so far been rigorously described, due mostly to their inherent complexity. The most promising strategy of modern biophysics is thus to identify minimal biological systems showing self-organized emergent behavior. One of the best-understood examples of protein self-organization, which has recently been successfully reconstituted in vitro, is represented by the oscillations of the Min proteins in Escherichia coli. In this review, we summarize the current understanding of the mechanism of Min protein self-organization in vivo and in vitro. We discuss the potential of the Min oscillations to sense the geometry of the cell and suggest that spontaneous protein waves could be a general means of intracellular organization. We hypothesize that cooperative membrane binding and unbinding, e.g., as an energy-dependent switch, may act as an important regulatory mechanism for protein oscillations and pattern formation in the cell.

Propagating cell-membrane waves driven by curved activators of actin polymerization

Cells exhibit propagating membrane waves which involve the actin cytoskeleton. One type of such membranal waves are Circular Dorsal Ruffles (CDR) which are related to endocytosis and receptor internalization. Experimentally, CDRs have been associated with membrane bound activators of actin polymerization of concave shape. We present experimental evidence for the localization of convex membrane proteins in these structures, and their insensitivity to inhibition of myosin II contractility in immortalized mouse embryo fibroblasts cell cultures. These observations lead us to propose a theoretical model which explains the formation of these waves due to the interplay between complexes that contain activators of actin polymerization and membrane-bound curved proteins of both types of curvature (concave and convex). Our model predicts that the activity of both types of curved proteins is essential for sustaining propagating waves, which are abolished when one type of curved activator is removed. Within this model waves are initiated when the level of actin polymerization induced by the curved activators is higher than some threshold value, which allows the cell to control CDR formation. We demonstrate that the model can explain many features of CDRs, and give several testable predictions. This work demonstrates the importance of curved membrane proteins in organizing the actin cytoskeleton and cell shape.