Investigating circular dorsal ruffles through varying substrate stiffness and mathematical modeling

Circular dorsal ruffles (CDRs) are transient actin-rich ringlike structures that form on the dorsal surface of growth-factor stimulated cells. However, the dynamics and mechanism of formation of CDRs are still unknown. It has been observed that CDR formation leads to stress fibers disappearing near the CDRs. Because stress fiber formation can be modified by substrate stiffness, we examined the effect of substrate stiffness on CDR formation by seeding NIH 3T3 fibroblasts on glass and polydimethylsiloxane substrates of varying stiffnesses from 20 kPa to 1800 kPa. We found that increasing substrate stiffness increased the lifetime of the CDRs. We developed a mathematical model of the signaling pathways involved in CDR formation to provide insight into this lifetime and size dependence that is linked to substrate stiffness via Rac-Rho antagonism. From the model, increasing stiffness raised mDia1-nucleated stress fiber formation due to Rho activation. The increased stress fibers present increased replenishment of the G-actin pool, therefore prolonging Arp2/3-nucleated CDR formation due to Rac activation. Negative feedback by WAVE-related RacGAP on Rac explained how CDR actin propagates as an excitable wave, much like wave propagation in other excitable medium, e.g., nerve signal transmission.     

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.     

A Comparative Study of Early Afterdepolarization-Mediated Fibrillation in Two Mathematical Models for Human Ventricular Cells

Early afterdepolarizations (EADs), which are abnormal oscillations of the membrane potential at the plateau phase of an action potential, are implicated in the development of cardiac arrhythmias like Torsade de Pointes. We carry out extensive numerical simulations of the TP06 and ORd mathematical models for human ventricular cells with EADs. We investigate the different regimes in both these models, namely, the parameter regimes where they exhibit (1) a normal action potential (AP) with no EADs, (2) an AP with EADs, and (3) an AP with EADs that does not go back to the resting potential. We also study the dependence of EADs on the rate of at which we pace a cell, with the specific goal of elucidating EADs that are induced by slow or fast rate pacing. In our simulations in two- and three-dimensional domains, in the presence of EADs, we find the following wave types: (A) waves driven by the fast sodium current and the L-type calcium current (Na-Ca-mediated waves); (B) waves driven only by the L-type calcium current (Ca-mediated waves); (C) phase waves, which are pseudo-travelling waves. Furthermore, we compare the wave patterns of the various wave-types (Na-Ca-mediated, Ca-mediated, and phase waves) in both these models. We find that the two models produce qualitatively similar results in terms of exhibiting Na-Ca-mediated wave patterns that are more chaotic than those for the Ca-mediated and phase waves. However, there are quantitative differences in the wave patterns of each wave type. The Na-Ca-mediated waves in the ORd model show short-lived spirals but the TP06 model does not. The TP06 model supports more Ca-mediated spirals than those in the ORd model, and the TP06 model exhibits more phase-wave patterns than does the ORd model.     

Dynamic actin patterns and Arp2/3 assembly at the substrate-attached surface of motile cells

BACKGROUND: In the cortical region of motile cells, the actin network rapidly reorganizes as required for movement in various directions and for cell-to-substrate adhesion. The analysis of actin network dynamics requires the combination of high-resolution imaging with a specific fluorescent probe that highlights the filamentous actin structures in live cells. RESULTS: Combining total internal reflection fluorescence (TIRF) microscopy with a method for labeling actin filaments, we analyze the dynamics of actin patterns in the highly motile cells of Dictyostelium. A rapidly restructured network of single or bundled actin filaments provides a scaffold for the assembly of differentiated actin complexes. Recruitment of the Arp2/3 complex characterizes stationary foci with a lifetime of 7-10 s and traveling waves. These structures are also formed in the absence of myosin-II. Arp2/3-actin assemblies similar to those driving the protrusion of a leading edge form freely at the inner face of the plasma membrane. CONCLUSIONS: The actin system of highly motile cells runs far from equilibrium and generates a multitude of patterns within a dynamic filamentous network. Traveling waves are the most complicated patterns based on recruitment of the Arp2/3 complex. They are governed by the propagated induction of actin polymerization. We hypothesize that the actin system autonomously generates primordia of specialized structures such as phagocytic cups or lamellipodia. These primordia would represent an activated state of the actin system and enable cells to respond within seconds to local stimuli by chemotaxis or phagocytic-cup formation.     

The Three-Dimensional Dynamics of Actin Waves, a Model of Cytoskeletal Self-Organization

Actin polymerization is typically initiated at specific sites in a cell by membrane-bound protein complexes, and the resulting structures are involved in specialized cellular functions, such as migration, particle uptake, or mitotic division. Here we analyze the potential of the actin system to self-organize into waves that propagate on the planar, substrate-attached membrane of a cell. We show that self-assembly involves the ordered recruitment of proteins from the cytoplasmic pool and relate the organization of actin waves to their capacity for applying force. Three proteins are shown to form distinct three-dimensional patterns in the actin waves. Myosin-IB is enriched at the wave front and close to the plasma membrane, the Arp2/3 complex is distributed throughout the waves, and coronin forms a sloping layer on top of them. CARMIL, a protein that links myosin-IB to the Arp2/3 complex, is also recruited to the waves. Wave formation does not depend on signals transmitted by heterotrimeric G-proteins, nor does their propagation require SCAR, a regulator upstream of the Arp2/3 complex. Propagation of the waves is based on an actin treadmilling mechanism, indicating a program that couples actin assembly to disassembly in a three-dimensional pattern. When waves impinge on the cell perimeter, they push the edge forward; when they reverse direction, the cell border is paralyzed. These data show that force-generating, highly organized supramolecular networks are autonomously formed in live cells from molecular motors and proteins controlling actin polymerization and depolymerization.     

Direction of regeneration waves in grid-based models for forest dynamics

Progressing waves of regeneration are observed in forest ecosystems such as Shimagare fir forests. The patterns generated by lattice models for forest dynamics often show similar waves of disturbance and recovery. This paper introduces a method to detect and quantify the directional movement of these waves. The method is based only on the disturbance times of the sites and allows to distinguish three types of wave patterns: patterns with global direction, patterns with local direction, and patterns without direction. We apply this to several grid-based models for forest dynamics which are evaluated analytically or by simulation. The results reveal a clear distinction of the models which earlier studies were not able to detect.     

Refinement and Pattern Formation in Neural Circuits by the Interaction of Traveling Waves with Spike-Timing Dependent Plasticity

Traveling waves in the developing brain are a prominent source of highly correlated spiking activity that may instruct the refinement of neural circuits. A candidate mechanism for mediating such refinement is spike-timing dependent plasticity (STDP), which translates correlated activity patterns into changes in synaptic strength. To assess the potential of these phenomena to build useful structure in developing neural circuits, we examined the interaction of wave activity with STDP rules in simple, biologically plausible models of spiking neurons. We derive an expression for the synaptic strength dynamics showing that, by mapping the time dependence of STDP into spatial interactions, traveling waves can build periodic synaptic connectivity patterns into feedforward circuits with a broad class of experimentally observed STDP rules. The spatial scale of the connectivity patterns increases with wave speed and STDP time constants. We verify these results with simulations and demonstrate their robustness to likely sources of noise. We show how this pattern formation ability, which is analogous to solutions of reaction-diffusion systems that have been widely applied to biological pattern formation, can be harnessed to instruct the refinement of postsynaptic receptive fields. Our results hold for rich, complex wave patterns in two dimensions and over several orders of magnitude in wave speeds and STDP time constants, and they provide predictions that can be tested under existing experimental paradigms. Our model generalizes across brain areas and STDP rules, allowing broad application to the ubiquitous occurrence of traveling waves and to wave-like activity patterns induced by moving stimuli.     

Reaction-diffusion waves of actin filament polymerization/depolymerization in Dictyostelium pseudopodium extension and cell locomotion

Cell surface movements and the intracellular spatial patterns and dynamics of actin filament (F-actin) were investigated in living and formalin-fixed cells of Dictyostelium discoideum by confocal microscopy. Excitation waves of F-actin assembly developed and propagated several micrometers at up to 26 microm/min in cells which had been intracellularly loaded with fluorescently labeled actin monomer. Wave propagation and extinction corresponded with the initiation and attenuation of pseudopodium extension and cell advance, respectively. The identification of chemical waves was supported by the ring, sphere, spiral and scroll wave patterns, which were observed in the extensions of fixed cells stained with phalloidin-rhodamine, and by the similar, asymmetrical [F-actin] distribution in wavefronts in living and fixed cells. These F-actin patterns and dynamics in Dictyostelium provide evidence for a new supramolecular state of actin, which propagates as a self-organized, reaction-diffusion wave of reversible F-actin assembly and affects pseudopodium extension. Actin's properties of oscillation and self-organization might also fundamentally determine the nature of the eukaryotic cell's reactions of adaptation, timing and signal response.     

Coupling traction force patterns and actomyosin wave dynamics reveals mechanics of cell motion

Motile cells can use and switch between different modes of migration. Here, we use traction force microscopy and fluorescent labeling of actin and myosin to quantify and correlate traction force patterns and cytoskeletal distributions in Dictyostelium discoideum cells that move and switch between keratocyte‐like fan‐shaped, oscillatory, and amoeboid modes. We find that the wave dynamics of the cytoskeletal components critically determine the traction force pattern, cell morphology, and migration mode. Furthermore, we find that fan‐shaped cells can exhibit two different propulsion mechanisms, each with a distinct traction force pattern. Finally, the traction force patterns can be recapitulated using a computational model, which uses the experimentally determined spatiotemporal distributions of actin and myosin forces and a viscous cytoskeletal network. Our results suggest that cell motion can be generated by friction between the flow of this network and the substrate.     

Self-organizing actin waves as planar phagocytic cup structures

Actin waves that travel on the planar membrane of a substrate-attached cell underscore the capability of the actin system to assemble into dynamic structures by the recruitment of proteins from the cytoplasm. The waves have no fixed shape, can reverse their direction of propagation and can fuse or divide. Actin waves separate two phases of the plasma membrane that are distinguished by their lipid composition. The area circumscribed by a wave resembles in its phosphoinositide content the interior of a phagocytic cup, leading us to explore the possibility that actin waves are in-plane phagocytic structures generated without the localized stimulus of an attached particle. Consistent with this view, wave-forming cells were found to exhibit a high propensity for taking up particles. Cells fed rod-shaped particles produced elongated phagocytic cups that displayed a zonal pattern that reflected in detail the actin and lipid pattern of free-running actin waves. Neutrophils and macrophages are known to spread on surfaces decorated with immune complexes, a process that has been interpreted as “frustrated” phagocytosis. We suggest that actin waves enable a phagocyte to scan a surface for particles that might be engulfed.Key words: actin waves, Dictyostelium, membrane tension, pattern formation, phagocytosis, PI3-kinase, PI(3,4,5)P3, self-organization     

An individual based model of rippling movement in a myxobacteria population

Migrating cells of Myxococcus xanthus (MX) in the early stages of starvation-induced development exhibit elaborate patterns of propagating waves. These so-called rippling patterns are formed by two sets of waves travelling in opposite directions. It has been experimentally shown that formation of these waves is mediated by cell-cell contact signalling (C-signalling). Here, we develop an individual-based model to study the formation of rippling patterns in MX populations. Following the work of Igoshin et al. (Proc. Natl. Acad. Sci. 98 (2001) 14913) we consider each moving cell to have an internal clock which controls its turning behaviour and sensitivity to C-signal. Specifically, we examine the effects of changing: C-signal strength, sensitivity/refractoriness, cell density, and noise upon the formation and structure of the rippling patterns. We also consider three modified models that have no explicit refractory period and examine their ability to produce rippling patterns.     

On the importance of dimensionality of space in models of space-mediated population persistence

Spatially explicit models have become widely used in today's mathematical ecology to study persistence of populations. For the sake of simplicity, population dynamics is often analyzed with 1-D models. An important question is: how adequate is such 1-D simplification of 2-D (or 3-D) dynamics for predicting species persistence. Here we show that dimensionality of the environment can play a critical role in the persistence of predator-prey interactions. We consider 1-D and 2-D dynamics of a predator-prey model with the prey growth damped by the Allee effect. We show that adding a second space coordinate into the 1-D model results in a pronounced increase of size of the domain in the parametric space where predator-prey coexistence becomes possible. This result is due to the possibility of formation of a number of 2-D patterns, which is impossible in the 1-D model. The 1-D and the 2-D models exhibit different qualitative responses to variations of system parameters. We show that in ecosystems having a narrow width (e.g. mountain valleys, vegetation patterns along canals in dry areas, etc.), extinction of species is more probable compared to ecosystems having a pronounced second dimension. In particular, the width of a long narrow natural reserve should be large enough to guarantee nonextinction of species via interaction of 2-D population patches.     

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.     

Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications

A mathematical model of the spatio-temporal dynamics of a two host, two parasitoid system is presented. There is a coupling of the four species through parasitism of both hosts by one of the parasitoids. The model comprises a system of four reaction-diffusion equations. The underlying system of ordinary differential equations, modelling the host-parasitoid population dynamics, has a unique positive steady state and is shown to be capable of undergoing Hopf bifurcations, leading to limit cycle kinetics which give rise to oscillatory temporal dynamics. The stability of the positive steady state has a fundamental impact on the spatio-temporal dynamics: stable travelling waves of parasitoid invasion exhibit increasingly irregular periodic travelling wave behaviour when key parameter values are increased beyond their Hopf bifurcation point. These irregular periodic travelling waves give rise to heterogeneous spatio-temporal patterns of host and parasitoid abundance. The generation of heterogeneous patterns has ecological implications and the concepts of temporary host refuge and niche formation are considered.     

An actin-based wave generator organizes cell motility

Although many of the regulators of actin assembly are known, we do not understand how these components act together to organize cell shape and movement. To address this question, we analyzed the spatial dynamics of a key actin regulator—the Scar/WAVE complex—which plays an important role in regulating cell shape in both metazoans and plants. We have recently discovered that the Hem-1/Nap1 component of the Scar/WAVE complex localizes to propagating waves that appear to organize the leading edge of a motile immune cell, the human neutrophil. Actin is both an output and input to the Scar/WAVE complex: the complex stimulates actin assembly, and actin polymer is also required to remove the complex from the membrane. These reciprocal interactions appear to generate propagated waves of actin nucleation that exhibit many of the properties of morphogenesis in motile cells, such as the ability of cells to flow around barriers and the intricate spatial organization of protrusion at the leading edge. We propose that cell motility results from the collective behavior of multiple self-organizing waves.     

Actomyosin dynamics modulate microtubule deformation and growth during T-cell activation

Activation of T-cells leads to the formation of immune synapses (ISs) with antigen-presenting cells. This requires T-cell polarization and coordination between the actomyosin and microtubule cytoskeletons. The interactions between these two cytoskeletal components during T-cell activation are not well understood. Here, we elucidate the interactions between microtubules and actin at the IS with high-resolution fluorescence microscopy. We show that microtubule growth dynamics in the peripheral actin-rich region is distinct from that in the central actin-free region. We further demonstrate that these differences arise from differential involvement of Arp2/3- and formin-nucleated actin structures. Formin inhibition results in a moderate decrease in microtubule growth rates, which is amplified in the presence of integrin engagement. In contrast, Arp2/3 inhibition leads to an increase in microtubule growth rates. We find that microtubule filaments are more deformed and exhibit greater shape fluctuations in the periphery of the IS than at the center. Using small molecule inhibitors, we show that actin dynamics and actomyosin contractility play key roles in defining microtubule deformations and shape fluctuations. Our results indicate a mechanical coupling between the actomyosin and microtubule systems during T-cell activation, whereby different actin structures influence microtubule dynamics in distinct ways.     

Surface waves of Min-proteins

In the bacterium Escherichia coli, the Min-proteins show pronounced pole-to-pole oscillations. They are functional for suppressing cell division at the cell ends, leaving the center as the only possible site for division. Analyzing different models of Min-protein dynamics in a bacterial geometry, we find waves on the cytoplasmic membrane. Interestingly, the surface wave solutions of different models belong to different symmetry classes. We suggest that experiments on Min-protein surface waves in vitro are helpful in distinguishing between different classes of models of Min-protein dynamics.     

A Study of Early Afterdepolarizations in a Model for Human Ventricular Tissue

Sudden cardiac death is often caused by cardiac arrhythmias. Recently, special attention has been given to a certain arrhythmogenic condition, the long-QT syndrome, which occurs as a result of genetic mutations or drug toxicity. The underlying mechanisms of arrhythmias, caused by the long-QT syndrome, are not fully understood. However, arrhythmias are often connected to special excitations of cardiac cells, called early afterdepolarizations (EADs), which are depolarizations during the repolarizing phase of the action potential. So far, EADs have been studied mainly in isolated cardiac cells. However, the question on how EADs at the single-cell level can result in fibrillation at the tissue level, especially in human cell models, has not been widely studied yet. In this paper, we study wave patterns that result from single-cell EAD dynamics in a mathematical model for human ventricular cardiac tissue. We induce EADs by modeling experimental conditions which have been shown to evoke EADs at a single-cell level: by an increase of L-type Ca currents and a decrease of the delayed rectifier potassium currents. We show that, at the tissue level and depending on these parameters, three types of abnormal wave patterns emerge. We classify them into two types of spiral fibrillation and one type of oscillatory dynamics. Moreover, we find that the emergent wave patterns can be driven by calcium or sodium currents and we find phase waves in the oscillatory excitation regime. From our simulations we predict that arrhythmias caused by EADs can occur during normal wave propagation and do not require tissue heterogeneities. Experimental verification of our results is possible for experiments at the cell-culture level, where EADs can be induced by an increase of the L-type calcium conductance and by the application of I blockers, and the properties of the emergent patterns can be studied by optical mapping of the voltage and calcium.     

Pulsatile contractions and pattern formation in excitable actomyosin cortex

The actin cortex is an active adaptive material, embedded with complex regulatory networks that can sense, generate, and transmit mechanical forces. The cortex exhibits a wide range of dynamic behaviours, from generating pulsatory contractions and travelling waves to forming organised structures. Despite the progress in characterising the biochemical and mechanical components of the actin cortex, the emergent dynamics of this mechanochemical system is poorly understood. Here we develop a reaction-diffusion model for the RhoA signalling network, the upstream regulator for actomyosin assembly and contractility, coupled to an active actomyosin gel, to investigate how the interplay between chemical signalling and mechanical forces regulates stresses and patterns in the cortex. We demonstrate that mechanochemical feedback in the cortex acts to destabilise homogeneous states and robustly generate pulsatile contractions. By tuning active stress in the system, we show that the cortex can generate propagating contraction pulses, form network structures, or exhibit topological turbulence.     

Self-organizing actin waves as planar phagocytic cup structures

Actin waves that travel on the planar membrane of a substrate-attached cell underscore the capability of the actin system to assemble into dynamic structures by the recruitment of proteins from the cytoplasm. The waves have no fixed shape, can reverse their direction of propagation, and can fuse or divide. Actin waves separate two phases of the plasma membrane that are distinguished by their lipid composition. The area circumscribed by a wave resembles in its phosphoinositide content the interior of a phagocytic cup, leading us to explore the possibility that actin waves are in-plane phagocytic structures generated without the localized stimulus of an attached particle. Consistent with this view, wave-forming cells were found to exhibit a high propensity for taking up particles. Cells fed rod-shaped particles produced elongated phagocytic cups that displayed a zonal pattern that reflected in detail the actin and lipid pattern of free-running actin waves. Neutrophils and macrophages are known to spread on surfaces decorated with immune complexes, a process that has been interpreted as “frustrated” phagocytosis. We suggest that actin waves enable a phagocyte to scan a surface for particles that might be engulfed.