Modeling the self-organized phosphatidylinositol lipid signaling system in chemotactic cells using quantitative image analysis

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.

Control of phosphatidylinositol-3-kinase signaling by nanoscale membrane compartmentalization

Phosphatidylinositol-3-kinases (PI3Ks) are lipid kinases that produce 3-phosphorylated derivatives of phosphatidylinositol upon activation by various cues. These 3-phosphorylated lipids bind to various protein effectors to control many cellular functions. Lipid phosphatases such as phosphatase and tensin homolog (PTEN) terminate PI3K-derived signals and are critical to ensure appropriate signaling outcomes. Many lines of evidence indicate that PI3Ks and PTEN, as well as some specific lipid effectors are highly compartmentalized, either in plasma membrane nanodomains or in endosomal compartments. We examine the evidence for specific recruitment of PI3Ks, PTEN, and other related enzymes to membrane nanodomains and endocytic compartments. We then examine the hypothesis that scaffolding of the sources (PI3Ks), terminators (PTEN), and effectors of these lipid signals with a common plasma membrane nanodomain may achieve highly localized lipid signaling and ensure selective activation of specific effectors. This highlights the importance of spatial regulation of PI3K signaling in various physiological and disease contexts.

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.

Local Membrane Curvature Pins and Guides Excitable Membrane Waves in Chemotactic and Macropinocytic Cells - Biomedical Insights From an Innovative Simple Model

PIP3 dynamics observed in membranes are responsible for the protruding edge formation in cancer and amoeboid cells. The mechanisms that maintain those PIP3 domains in three-dimensional space remain elusive, due to limitations in observation and analysis techniques. Recently, a strong relation between the cell geometry, the spatial confinement of the membrane, and the excitable signal transduction system has been revealed by Hörning and Shibata (2019) using a novel 3D spatiotemporal analysis methodology that enables the study of membrane signaling on the entire membrane (Hörning and Shibata, 2019). Here, using 3D spatial fluctuation and phase map analysis on actin polymerization inhibited Dictyostelium cells, we reveal a spatial asymmetry of PIP3 signaling on the membrane that is mediated by the contact perimeter of the plasma membrane — the spatial boundary around the cell-substrate adhered area on the plasma membrane. We show that the contact perimeter guides PIP3 waves and acts as a pinning site of PIP3 phase singularities, that is, the center point of spiral waves. The contact perimeter serves as a diffusion influencing boundary that is regulated by a cell size- and shape-dependent curvature. Our findings suggest an underlying mechanism that explains how local curvature can favor actin polymerization when PIP3 domains get pinned at the curved protrusive membrane edges in amoeboid cells.

Using Live-Cell Imaging and Synthetic Biology to Probe Directed Migration in Dictyostelium

For decades, the social amoeba Dictyostelium discoideum has been an invaluable tool for dissecting the biology of eukaryotic cells. Its short growth cycle and genetic tractability make it ideal for a variety of biochemical, cell biological, and biophysical assays. Dictyostelium have been widely used as a model of eukaryotic cell motility because the signaling and mechanical networks which they use to steer and produce forward motion are highly conserved. Because these migration networks consist of hundreds of interconnected proteins, perturbing individual molecules can have subtle effects or alter cell morphology and signaling in major unpredictable ways. Therefore, to fully understand this network, we must be able to quantitatively assess the consequences of abrupt modifications. This ability will allow us better control cell migration, which is critical for development and disease, in vivo. Here, we review recent advances in imaging, synthetic biology, and computational analysis which enable researchers to tune the activity of individual molecules in single living cells and precisely measure the effects on cellular motility and signaling. We also provide practical advice and resources to assist in applying these approaches in Dictyostelium.

Mechanobiology in cortical waves and oscillations

Cortical actin waves have emerged as a widely prevalent phenomena and brought pattern formation to many fields of cell biology. Cortical excitabilities, reminiscent of the electric excitability in neurons, are likely fundamental property of the cell cortex. Although they have been mostly considered to be biochemical in nature, accumulating evidence support the role of mechanics in the pattern formation process. Both pattern formation and mechanobiology approach biological phenomena at the collective level, either by looking at the mesoscale dynamical behavior of molecular networks or by using collective physical properties to characterize biological systems. As such they are very different from the traditional reductionist, bottom-up view of biology, which brings new challenges and potential opportunities. In this essay, we aim to provide our perspectives on what the proposed mechanochemical feedbacks are and open questions regarding their role in cortical excitable and oscillatory dynamics.

Cellular self-organization: generating order from the abyss

While the organization of inanimate systems such as gases or liquids is predominantly thermodynamically driven—a mixture of two gases will tend to mix until they reach equilibrium—biological systems frequently exhibit organization that is far from a well-mixed equilibrium. The anisotropies displayed by cells are evident in some of the dynamic processes that constitute life including cell development, movement, and division. These anisotropies operate at different length-scales, from the meso- to the nanoscale, and are proposed to reflect self-organization, a characteristic of living systems that is becoming accessible to reconstitution from purified components, and thus a more thorough understanding. Here, some examples of self-organization underlying cellular anisotropies at the cellular level are reviewed, with an emphasis on Rho-family GTPases operating at the plasma membrane. Given the technical challenges of studying these dynamic proteins, some of the successful approaches that are being employed to study their self-organization will also be considered.

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.

Single-molecule imaging of PI(4,5)P2 and PTEN in vitro reveals a positive feedback mechanism for PTEN membrane binding

PTEN, a 3-phosphatase of phosphoinositide, regulates asymmetric PI(3,4,5)P3 signaling for the anterior-posterior polarization and migration of motile cells. PTEN acts through posterior localization on the plasma membrane, but the mechanism for this accumulation is poorly understood. Here we developed an in vitro single-molecule imaging assay with various lipid compositions and use it to demonstrate that the enzymatic product, PI(4,5)P2, stabilizes PTEN's membrane-binding. The dissociation kinetics and lateral mobility of PTEN depended on the PI(4,5)P2 density on artificial lipid bilayers. The basic residues of PTEN were responsible for electrostatic interactions with anionic PI(4,5)P2 and thus the PI(4,5)P2-dependent stabilization. Single-molecule imaging in living Dictyostelium cells revealed that these interactions were indispensable for the stabilization in vivo, which enabled efficient cell migration by accumulating PTEN posteriorly to restrict PI(3,4,5)P3 distribution to the anterior. These results suggest that PI(4,5)P2-mediated positive feedback and PTEN-induced PI(4,5)P2 clustering may be important for anterior-posterior polarization.

Rhythmicity and waves in the cortex of single cells

Emergence of dynamic patterns in the form of oscillations and waves on the cortex of single cells is a fascinating and enigmatic phenomenon. Here we outline various theoretical frameworks used to model pattern formation with the goal of reducing complex, heterogeneous patterns into key parameters that are biologically tractable. We also review progress made in recent years on the quantitative and molecular definitions of these terms, which we believe have begun to transform single-cell dynamic patterns from a purely observational and descriptive subject to more mechanistic studies. Specifically, we focus on the nature of local excitable and oscillation events, their spatial couplings leading to propagating waves and the role of active membrane. Instead of arguing for their functional importance, we prefer to consider such patterns as basic properties of dynamic systems. We discuss how knowledge of these patterns could be used to dissect the structure of cellular organization and how the network-centric view could help define cellular functions as transitions between different dynamical states. Last, we speculate on how these patterns could encode temporal and spatial information.

This article is part of the theme issue ‘Self-organization in cell biology’.

Excitable dynamics of Ras triggers spontaneous symmetry breaking of PIP3 signaling in motile cells

Spontaneous cell movement is underpinned by an asymmetric distribution of signaling molecules including small G proteins and phosphoinositides on the cell membrane. However, the molecular network necessary for spontaneous symmetry breaking has not been fully elucidated. Here, we report that, in Dictyostelium discoideum, the spatiotemporal dynamics of GTP bound Ras (Ras-GTP) breaks the symmetry due its intrinsic excitability even in the absence of extracellular spatial cues and downstream signaling activities. A stochastic excitation of local and transient Ras activation induced phosphatidylinositol (3,4,5)-trisphosphate (PIP3) accumulation via direct interaction with Phosphoinositide 3-kinase (PI3K), causing tightly coupled traveling waves that propagated along the membrane. Comprehensive phase analysis of the waves of Ras-GTP and PIP3 metabolism-related molecules revealed the network structure of the excitable system including positive-feedback regulation of Ras-GTP by the downstream PIP3. A mathematical model reconstituted a series of the observed symmetry-breaking phenomena, illustrating the essential involvement of Ras excitability in the cellular decision-making process.

Three-Dimensional Cell Geometry Controls Excitable Membrane Signaling in Dictyostelium Cells

Phosphatidylinositol (3-5)-trisphosphate (PtdInsP3) is known to propagate as waves on the plasma membrane and is related to the membrane-protrusive activities in Dictyostelium and mammalian cells. Although there have been a few attempts to study the three-dimensional (3D) dynamics of these processes, most studies have focused on the dynamics extracted from single focal planes. However, the relation between the dynamics and 3D cell shape remains elusive because of the lack of signaling information about the unobserved part of the membrane. Here, we show that PtdInsP3 wave dynamics are directly regulated by the 3D geometry (i.e., size and shape) of the plasma membrane. By introducing an analysis method that extracts the 3D spatiotemporal activities on the entire cell membrane, we show that PtdInsP3 waves self-regulate their dynamics within the confined membrane area. This leads to changes in speed, orientation, and pattern evolution, following the underlying excitability of the signal transduction system. Our findings emphasize the role of the plasma membrane topology in reaction-diffusion-driven biological systems and indicate its importance in other mammalian systems.

Mutual inhibition between PTEN and PIP3 generates bistability for polarity in motile cells

Phosphatidylinositol 3,4,5-trisphosphate (PIP3) and PIP3 phosphatase (PTEN) are enriched mutually exclusively on the anterior and posterior membranes of eukaryotic motile cells. However, the mechanism that causes this spatial separation between the two molecules is unknown. Here we develop a method to manipulate PIP3 levels in living cells and used it to show PIP3 suppresses the membrane localization of PTEN. Single-molecule measurements of membrane-association and -dissociation kinetics and of lateral diffusion reveal that PIP3 suppresses the PTEN binding site required for stable PTEN membrane binding. Mutual inhibition between PIP3 and PTEN provides a mechanistic basis for bistability that creates a PIP3-enriched/PTEN-excluded state and a PTEN-enriched/PIP3-excluded state underlying the strict spatial separation between PIP3 and PTEN. The PTEN binding site also mediates the suppression of PTEN membrane localization in chemotactic signaling. These results illustrate that the PIP3-PTEN bistable system underlies a cell's decision-making for directional movement irrespective of the environment.

Modeling random crawling, membrane deformation and intracellular polarity of motile amoeboid cells

Amoeboid movement is one of the most widespread forms of cell motility that plays a key role in numerous biological contexts. While many aspects of this process are well investigated, the large cell-to-cell variability in the motile characteristics of an otherwise uniform population remains an open question that was largely ignored by previous models. In this article, we present a mathematical model of amoeboid motility that combines noisy bistable kinetics with a dynamic phase field for the cell shape. To capture cell-to-cell variability, we introduce a single parameter for tuning the balance between polarity formation and intracellular noise. We compare numerical simulations of our model to experiments with the social amoeba Dictyostelium discoideum. Despite the simple structure of our model, we found close agreement with the experimental results for the center-of-mass motion as well as for the evolution of the cell shape and the overall intracellular patterns. We thus conjecture that the building blocks of our model capture essential features of amoeboid motility and may serve as a starting point for more detailed descriptions of cell motion in chemical gradients and confined environments.

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.

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.

Wave Patterns in Cell Membrane and Actin Cortex Uncoupled from Chemotactic Signals

When cells of Dictyostelium discoideum orientate in a gradient of chemoattractant, they are polarized into a protruding front pointing toward the source of attractant, and into a retracting tail. Under the control of chemotactic signal inputs, Ras is activated and PIP3 is synthesized at the front, while the PIP3-degrading phosphatase PTEN decorates the tail region. As a result of signal transduction, actin filaments assemble at the front into dendritic structures associated with the Arp2/3 complex, in contrast to the tail region where a loose actin meshwork is associated with myosin-II and cortexillin, an antiparallel actin-bundling protein. In axenically growing strains of D. discoideum, wave patterns built by the same components evolve in the absence of any external signal input. Since these autonomously generated patterns are constrained to the plane of the substrate-attached cell surface, they are optimally suited to the optical analysis of state transitions between front-like and tail-like states of the membrane and the actin cortex. Here, we describe imaging techniques using fluorescent proteins to probe for the state of the membrane, the reorganization of the actin network, and the dynamics of wave patterns.

Spontaneous Behaviors and Wall-Curvature Lead to Apparent Wall Preference in Planarian

The planarian Dugesia japonica tends to stay near the walls of its breeding containers and experimental dishes in the laboratory, a phenomenon called “wall preference”. This behavior is thought to be important for environmental adaptation, such as hiding by planarians in nature. However, the mechanisms regulating wall-preference behavior are not well understood, since this behavior occurs in the absence of any particular stimulation. Here we show the mechanisms of wall-preference behavior. Surprisingly, planarian wall-preference behavior was also shown even by the head alone and by headless planarians. These results indicate that planarian “wall-preference” behavior only appears to be a “preference” behavior, and is actually an outcome of spontaneous behaviors, rather than of brain function. We found that in the absence of environmental cues planarians moved basically straight ahead until they reached a wall, and that after reaching a wall, they changed their direction of movement to one tangential to the wall, suggesting that this spontaneous behavior may play a critical role in the wall preference. When we tested another spontaneous behavior, the wigwag movement of the planarian head, using computer simulation with various wigwag angles and wigwag intervals, large wigwag angle and short wigwag interval reduced wall-preference behavior. This indicated that wigwag movement may determine the probability of staying near the wall or leaving the wall. Furthermore, in accord with this simulation, when we tested planarian wall-preference behavior using several assay fields with different curvature of the wall, we found that concavity and sharp curvature of walls negatively impacted wall preference by affecting the permissible angle of the wigwag movement. Together, these results indicate that planarian wall preference may be involuntarily caused by the combination of two spontaneous planarian behaviors: moving straight ahead until reaching a wall and then moving along it in the absence of environmental cues, and wigwag movements of the head.

Activator-inhibitor coupling between Rho signalling and actin assembly makes the cell cortex an excitable medium

Animal cell cytokinesis results from patterned activation of the small GTPase Rho, which directs assembly of actomyosin in the equatorial cortex. Cytokinesis is restricted to a portion of the cell cycle following anaphase onset in which the cortex is responsive to signals from the spindle. We show that shortly after anaphase onset oocytes and embryonic cells of frogs and echinoderms exhibit cortical waves of Rho activity and F-actin polymerization. The waves are modulated by cyclin-dependent kinase 1 (Cdk1) activity and require the Rho GEF (guanine nucleotide exchange factor), Ect2. Surprisingly, during wave propagation, while Rho activity elicits F-actin assembly, F-actin subsequently inactivates Rho. Experimental and modeling results show that waves represent excitable dynamics of a reaction diffusion system with Rho as the activator and F-actin the inhibitor. We propose that cortical excitability explains fundamental features of cytokinesis including its cell cycle regulation.

Excitable signal transduction induces both spontaneous and directional cell asymmetries in the phosphatidylinositol lipid signaling system for eukaryotic chemotaxis

Intracellular asymmetry in the signaling network works as a compass to navigate eukaryotic chemotaxis in response to guidance cues. Although the compass variable can be derived from a self-organization dynamics, such as excitability, the responsible mechanism remains to be clarified. Here, we analyzed the spatiotemporal dynamics of the phosphatidylinositol 3,4,5-trisphosphate (PtdInsP3) pathway, which is crucial for chemotaxis. We show that spontaneous activation of PtdInsP3-enriched domains is generated by an intrinsic excitable system. Formation of the same signal domain could be triggered by various perturbations, such as short impulse perturbations that triggered the activation of intrinsic dynamics to form signal domains. We also observed the refractory behavior exhibited in typical excitable systems. We show that the chemotactic response of PtdInsP3 involves biasing the spontaneous excitation to orient the activation site toward the chemoattractant. Thus, this biased excitability embodies the compass variable that is responsible for both random cell migration and biased random walk. Our finding may explain how cells achieve high sensitivity to and robust coordination of the downstream activation that allows chemotactic behavior in the noisy environment outside and inside the 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.

Intracellular encoding of spatiotemporal guidance cues in a self-organizing signaling system for chemotaxis in Dictyostelium cells

Even in the absence of guidance cues, chemotactic cells are often spontaneously motile, which should accompany a spontaneous symmetry breaking inside the cells. A shallow chemoattractant gradient can induce these cells to move directionally without much change in cell morphology. As the gradient becomes steeper, the accuracy of chemotaxis increases. It is not clear how the steepness is expressed or encoded internally in the signaling network, which in turn coordinately activates the motile apparatus for chemotaxis. In Dictyostelium cells, self-organizing polarization activities in the signaling network have been reported. In this paper, we conducted a theoretical study of the response of this self-organizing system to guidance cues. Our analyses indicate that self-organizing systems respond sharply to a shallow external gradient by increasing the precision of polarity direction and modulating the frequency of self-polarization. We also show how the precision increase and frequency modulation are achieved. Our results indicate that self-organizing activity, independent of external cues, is the basis for the sensitive and robust response to shallow gradients. Finally, we show that the system can sense the direction of space-time waves of a stimulus, for which Dictyostelium cells exhibit chemotaxis in the developmental process.

Excitable behavior in amoeboid chemotaxis

Chemotaxis, the directed motion of cells in response to chemical gradients, is a fundamental process. Eukaryotic cells detect spatial differences in chemoattractant receptor occupancy with high precision and use these differences to bias the location of actin-rich protrusions to guide their movement. Research into chemotaxis has benefitted greatly from a systems biology approach that combines novel experimental and computational tools to pose and test hypotheses. Recently, one such hypothesis has been postulated proposing that chemotaxis in eukaryotic cells is mediated by locally biasing the activity of an underlying excitable system. The excitable system hypothesis can account for a number of cellular behaviors related to chemotaxis, including the stochastic nature of the movement of unstimulated cells, the directional bias imposed by chemoattractant gradients, and the observed spatial and temporal distribution of signaling and cytoskeleton proteins.

Envisioning migration: mathematics in both experimental analysis and modeling of cell behavior

The complex nature of cell migration highlights the power and challenges of applying mathematics to biological studies. Mathematics may be used to create model equations that recapitulate migration, which can predict phenomena not easily uncovered by experiments or intuition alone. Alternatively, mathematics may be applied to interpreting complex data sets with better resolution--potentially empowering scientists to discern subtle patterns amid the noise and heterogeneity typical of migrating cells. Iteration between these two methods is necessary in order to reveal connections within the cell migration signaling network, as well as to understand the behavior that arises from those connections. Here, we review recent quantitative analysis and mathematical modeling approaches to the cell migration problem.

Theoretical model for cell migration with gradient sensing and shape deformation

Amoeboid cells take various shapes during migration, depending on the cell type and its environment. Deformability of the cell shape can then affect the migrating behavior. In this article, we introduce a theoretical model of chemotactic cell migration with elliptical shape deformation. Based on the model, we calculate the stationary distributions of the migration directions analytically. As a result, we find that the distributions show different characteristics depending on the difference in the interdependence of the internal polarity, cell morphology and gradient sensing.

A dual role model for active Rac1 in cell migration

Over time we have come to appreciate that the complex regulation of Rho GTPases involves additional mechanisms beyond the activating role of RhoGEFs, the inactivating function of RhoGAPs and the sequestering activity of RhoGDIs. One class of regulatory mechanisms includes direct modifications of Rho proteins such as isoprenylation, phosphorylation and SUMOylation. Rho GTPases can also regulate each other by means of crosstalk signaling, which is again mostly mediated by GEFs, GAPs and GDIs. More complex mutual regulation ensues when and where two or more Rho proteins activate a common molecular target, i.e., share a common effector. We have recently unraveled a reciprocal mechanism wherein spatiotemporal dynamics of Rac1 activity during migration of Dictyostelium cells is apparently regulated by antagonizing interactions of Rac1-GTP with two distinct effectors. By monitoring specific fluorescent probes, activated Rac1 is simultaneously present at the leading edge, where it participates in Scar/WAVE-mediated actin polymerization, and at the trailing edge, where it induces formation of a DGAP1/cortexillin actin-bundling complex. Strikingly, in addition to their opposed localization, the two populations of activated Rac1 also display opposite kinetics of recruitment to the plasma membrane upon stimulation by chemoattractants. These findings with respect to Rac1 in Dictyostelium suggest a novel principle for regulation of Rho GTPase activity that might also play a role in other cell types and for other Rho family members.

Phase geometries of two-dimensional excitable waves govern self-organized morphodynamics of amoeboid cells

In both randomly moving Dictyostelium and mammalian cells, phosphatidylinositol (3,4,5)-trisphosphate and F-actin are known to propagate as waves at the membrane and act to push out the protruding edge. To date, however, the relationship between the wave geometry and the patterns of amoeboid shape change remains elusive. Here, by using phase map analysis, we show that morphology dynamics of randomly moving Dictyostelium discoideum cells can be characterized by the number, topology, and position of spatial phase singularities, i.e., points that represent organizing centers of rotating waves. A single isolated singularity near the cellular edge induced a rotational protrusion, whereas a pair of singularities supported a symmetric extension. These singularities appeared by strong phase resetting due to de novo nucleation at the back of preexisting waves. Analysis of a theoretical model indicated excitability of the system that is governed by positive feedback from phosphatidylinositol (3,4,5)-trisphosphate to PI3-kinase activation, and we showed experimentally that this requires F-actin. Furthermore, by incorporating membrane deformation into the model, we demonstrated that geometries of competing waves explain most of the observed semiperiodic changes in amoeboid morphology.

Asymmetric PTEN distribution regulated by spatial heterogeneity in membrane-binding state transitions

The molecular mechanisms that underlie asymmetric PTEN distribution at the posterior of polarized motile cells and regulate anterior pseudopod formation were addressed by novel single-molecule tracking analysis. Heterogeneity in the lateral mobility of PTEN on a membrane indicated the existence of three membrane-binding states with different diffusion coefficients and membrane-binding lifetimes. The stochastic state transition kinetics of PTEN among these three states were suggested to be regulated spatially along the cell polarity such that only the stable binding state is selectively suppressed at the anterior membrane to cause local PTEN depletion. By incorporating experimentally observed kinetic parameters into a simple mathematical model, the asymmetric PTEN distribution can be explained quantitatively to illustrate the regulatory mechanisms for cellular asymmetry based on an essential causal link between individual stochastic reactions and stable localizations of the ensemble.