Orientation of chemotactic cells and growth cones: models and mechanisms

Computational modeling of signaling networks for eukaryotic chemosensing

The task of developing and simulating computational models of signaling networks for eukaryotic chemosensing confronts the modeler with several challenges: (1) The stimuli that initiate the cellular responses one wishes to study are provided by extracellular concentration gradients. This means that the computational model must have a spatially resolved representation of extracellular molecular concentrations. (2) The intracellular responses consist of the generation of intracellular accumulations and/or translocations of signaling molecules, requiring spatially resolved computational representations of the simulated cells. (3) The signaling networks responsible for eukaryotic chemosensing comprise a multitude of components acting as receptors, adaptors, (lipid- and protein-) kinases (including GTPases), (lipid- and protein-) phosphatases, and molecule types used by others for membrane attachment. Models of such signaling networks may become quite complicated, unless one wishes to rely on abstract functional modules with certain input-output characteristics as modeling "shortcuts" replacing subnetworks of biological signaling molecules. In this chapter, we describe how modelers can use a modeling tool ("simmune") developed to facilitate the design and simulation of detailed computational models of signaling pathways (for eukaryotic chemosensing here), thereby avoiding the technical difficulties typically associated with building and simulating such quantitative models.

Receptor noise limitations on chemotactic sensing

Chemotactic eukaryotic cells are able to detect chemoattractant gradients that are both shallow and have a low background concentration. Under these conditions, the noise in the number of bound receptors can be significant and needs to be taken into account in determining the directional sensing process. Here, we quantify numerically the number of bound receptors on the membrane of a disk-shaped cell by using a numerical Monte Carlo tool. The obtained time traces of the receptor occupancy can be used as inputs for any directional sensing model. We investigate the response of the local excitation global inhibition model and a recently developed balanced inactivation model. We determine a measure for the motility of the cell for each model, based on the relevant output variable, as a function of experimental parameters, resulting in several experimentally testable predictions. Furthermore, we show that these two models behave in a qualitatively different fashion when the background concentration is varied. Thus, to properly characterize the sensitivity of cells to receptor occupancy, it is not sufficient to examine the input signal. Rather, one needs to take into account the response of the second messenger pathway.

ERM proteins regulate growth cone responses to Sema3A

Axonal growth cones initiate and sustain directed growth in response to cues in their environment. A variety of events such as receptor internalization, kinase activation, and actin rearrangement can be stimulated by guidance cues and are essential for mediating targeted growth cone behavior. Surprisingly little is known about how such disparate actions are coordinated. Our data suggest that ezrin, radixin, and moesin (ERMs), a family of highly homologous, multifunctional proteins may be able to coordinate growth cone responses to the guidance cue Semaphorin 3A (Sema3A). We show that active ERMs concentrate asymmetrically in neocortical growth cones, are rapidly and transiently inactivated by Sema3A, and are required for Sema3A-mediated growth cone collapse and guidance. The FERM domain of active ERMs regulates internalization of the Sema3A receptor, Npn1, and its coreceptor, L1CAM, while the ERM C-terminal domain binds and caps F-actin. Our data support a model in which ERMs can coordinate membrane and actin dynamics in response to Sema3A.

Beyond polymer polarity: how the cytoskeleton builds a polarized cell

Cell polarity relies on the asymmetric organization of cellular components and structures. Actin and microtubules are well suited to provide the structural basis for cell polarization because of their inherent structural polarity along the polymer lattices and intrinsic dynamics that allow them to respond rapidly to polarity cues. In general, the actin cytoskeleton drives the symmetry-breaking process that enables the establishment of a polarized distribution of regulatory molecules, whereas microtubules build on this asymmetry and maintain the stability of the polarized organization. Crosstalk coordinates the functions of the two cytoskeletal systems.

Accuracy of direct gradient sensing by single cells

Many types of cells are able to accurately sense shallow gradients of chemicals across their diameters, allowing the cells to move toward or away from chemical sources. This chemotactic ability relies on the remarkable capacity of cells to infer gradients from particles randomly arriving at cell-surface receptors by diffusion. Whereas the physical limits of concentration sensing by cells have been explored, there is no theory for the physical limits of gradient sensing. Here, we derive such a theory, using as models a perfectly absorbing sphere and a perfectly monitoring sphere, which, respectively, infer gradients from the absorbed surface particle density or the positions of freely diffusing particles inside a spherical volume. We find that the perfectly absorbing sphere is superior to the perfectly monitoring sphere, both for concentration and gradient sensing, because previously observed particles are never remeasured. The superiority of the absorbing sphere helps explain the presence at the surfaces of cells of signal-degrading enzymes, such as PDE for cAMP in Dictyostelium discoideum (Dicty) and BAR1 for mating factor alpha in Saccharomyces cerevisiae (budding yeast). Quantitatively, our theory compares favorably with recent measurements of Dicty moving up a cAMP gradient, suggesting these cells operate near the physical limits of gradient detection.

Modeling robustness tradeoffs in yeast cell polarization induced by spatial gradients

Cells localize (polarize) internal components to specific locations in response to external signals such as spatial gradients. For example, yeast cells form a mating projection toward the source of mating pheromone. There are specific challenges associated with cell polarization including amplification of shallow external gradients of ligand to produce steep internal gradients of protein components (e.g. localized distribution), response over a broad range of ligand concentrations, and tracking of moving signal sources. In this work, we investigated the tradeoffs among these performance objectives using a generic model that captures the basic spatial dynamics of polarization in yeast cells, which are small. We varied the positive feedback, cooperativity, and diffusion coefficients in the model to explore the nature of this tradeoff. Increasing the positive feedback gain resulted in better amplification, but also produced multiple steady-states and hysteresis that prevented the tracking of directional changes of the gradient. Feedforward/feedback coincidence detection in the positive feedback loop and multi-stage amplification both improved tracking with only a modest loss of amplification. Surprisingly, we found that introducing lateral surface diffusion increased the robustness of polarization and collapsed the multiple steady-states to a single steady-state at the cost of a reduction in polarization. Finally, in a more mechanistic model of yeast cell polarization, a surface diffusion coefficient between 0.01 and 0.001 µm(2)/s produced the best polarization performance, and this range is close to the measured value. The model also showed good gradient-sensitivity and dynamic range. This research is significant because it provides an in-depth analysis of the performance tradeoffs that confront biological systems that sense and respond to chemical spatial gradients, proposes strategies for balancing this tradeoff, highlights the critical role of lateral diffusion of proteins in the membrane on the robustness of polarization, and furnishes a framework for future spatial models of yeast cell polarization.

Synthetic activation of endogenous PI3K and Rac identifies an AND-gate switch for cell polarization and migration

Phosphatidylinositol 3-OH kinase (PI3K) has been widely studied as a principal regulator of cell polarization, migration, and chemotaxis [1], [2], [3], [4]. Surprisingly, recent studies showed that mammalian neutrophils and Dictyostelium discoideum cells can polarize and migrate in the absence of PI3K activity [5], [6], [7]. Here we directly probe the roles of PI3K and its downstream effector, Rac, in HL-60 neutrophils by using a chemical biology approach whereby the endogenously present enzymes are synthetically activated in less than one minute [8], [9], [10]. We show that uniform activation of endogenous PI3K is sufficient to polarize previously unpolarized neutrophils and trigger effective cell migration. After a delay following symmetrical phosphatidylinositol (3,4,5)-triphosphate (PIP₃) production, a polarized distribution of PIP₃ was induced by positive feedback requiring actin polymerization. Pharmacological studies argue that this process does not require receptor-coupled trimeric G proteins. Contrary to the current working model, rapid activation of endogenous Rac proteins triggered effective actin polymerization but failed to feed back to PI3K to generate PIP₃ or induce cell polarization. Thus, the increase in PIP₃ concentration at the leading edge is generated by positive feedback with an AND gate logic with a PI3K-Rac-actin polymerization pathway as a first input and a PI3K initiated non-Rac pathway as a second input. This AND-gate control for cell polarization can explain how Rac can be employed for both PI3K-dependent and -independent signaling pathways coexisting in the same cell.

Stochastic control of spontaneous signal generation for gradient sensing in chemotaxis

Chemotaxis is characterized by spontaneous cellular behavior. This spontaneity results, in part, from the stochasticity of intracellular reactions. Spontaneous and random migration of chemotactic cells is regulated by spontaneously generated signals, namely transient local increases in the level of phosphoinositol-3,4,5-triphosphate (PIP3 pulses). In this study, we attempted to elucidate the mechanisms that generate these PIP3 pulses and how the pulses contribute to gradient sensing during chemotaxis. To this end, we constructed a simple biophysical model of intracellular signal transduction consisting of an inositol phospholipid signaling pathway and small GTPases. Our theoretical analysis revealed that an excitable system can emerge from the non-linear dynamics of the model, and that stochastic reactions allow the system to spontaneously become excited, which was corresponded to the PIP3 pulses. Based on these results, we framed a hypothesis of the gradient sensing; a chemical gradient spatially modifies a potential barrier for excitation and then PIP3 pulses are preferentially generated on the side of the cell exposed to the higher chemical concentration. We then validated our hypothesis using stochastic simulations of the signal transduction.

FRET imaging and in silico simulation: analysis of the signaling network of nerve growth factor-induced neuritogenesis

Genetically encoded probes based on Förster resonance energy transfer (FRET) enable us to decipher spatiotemporal information encoded in complex tissues such as the brain. Firstly, this review focuses on FRET probes wherein both the donor and acceptor are fluorescence proteins and are incorporated into a single molecule, i.e. unimolecular probes. Advantages of these probes lie in their easy loading into cells, the simple acquisition of FRET images, and the clear evaluation of data. Next, we introduce our recent study which encompasses FRET imaging and in silico simulation. In nerve growth factor-induced neurite outgrowth in PC12 cells, we found positive and negative signaling feedback loops. We propose that these feedback loops determine neurite-budding sites. We would like to emphasize that it is now time to accelerate crossover research in neuroscience, optics, and computational biology.

Changing directions in the study of chemotaxis

Chemotaxis--the guided movement of cells in chemical gradients--probably first emerged in our single-celled ancestors and even today is recognizably similar in neutrophils and amoebae. Chemotaxis enables immune cells to reach sites of infection, allows wounds to heal and is crucial for forming embryonic patterns. Furthermore, the manipulation of chemotaxis may help to alleviate disease states, including the metastasis of cancer cells. This review discusses recent results concerning how cells orientate in chemotactic gradients and the role of phosphatidylinositol-3,4,5-trisphosphate, what produces the force for projecting pseudopodia and a new role for the endocytic cycle in movement.

Receptor noise and directional sensing in eukaryotic chemotaxis

Chemotacting eukaryotic cells are able to detect very small chemical gradients (approximately 1%) for a large range of background concentrations. For these chemical environments, fluctuations in the number of bound ligands will become important. Here, we investigate the effect of receptor noise in a simplified one-dimensional geometry. The auto- and cross-correlations of the noise sources at the front and the back of the cell are explicitly computed using an effective Monte Carlo simulation tool. The resulting stochastic equations for the investigated directional sensing model can be solved analytically in Fourier space. We determine the chemotactic efficiency, a measure of motility for the cell, as a function of several experimental parameters, leading to explicit experimental predictions.

Mathematics of cell motility: have we got its number?

Mathematical and computational modeling is rapidly becoming an essential research technique complementing traditional experimental biological methods. However, lack of standard modeling methods, difficulties of translating biological phenomena into mathematical language, and differences in biological and mathematical mentalities continue to hinder the scientific progress. Here we focus on one area-cell motility-characterized by an unusually high modeling activity, largely due to a vast amount of quantitative, biophysical data, 'modular' character of motility, and pioneering vision of the area's experimental leaders. In this review, after brief introduction to biology of cell movements, we discuss quantitative models of actin dynamics, protrusion, adhesion, contraction, and cell shape and movement that made an impact on the process of biological discovery. We also comment on modeling approaches and open questions.

Directional sensing during chemotaxis

Cells have the innate ability to sense and move towards a variety of chemoattractants. We investigate the pathways by which cells sense and respond to chemoattractant gradients. We focus on the model system Dictyostelium and compare our understanding of chemotaxis in this system with recent advances made using neutrophils and other mammalian cell types, which share many molecular components and signaling pathways with Dictyostelium. This review also examines models that have been proposed to explain how cells are able to respond to small differences in ligand concentrations between the anterior leading edge and posterior of the cell. In addition, we highlight the overlapping functions of many signaling components in diverse processes beyond chemotaxis, including random cell motility and cell division.

Dynamics of Cdc42 network embodies a Turing-type mechanism of yeast cell polarity

Complex biochemical networks can be understood by identifying their principal regulatory motifs and mode of action. We model the early phase of budding yeast cellular polarization and show that the biochemical processes in the presumptive bud site comprise a Turing-type mechanism. The roles of the prototypical activator and substrate are played by GTPase Cdc42 in its active and inactive states, respectively. We demonstrate that the nucleotide cycling of Cdc42 converts cellular energy into a stable cluster of activated Cdc42. This energy drives a continuous membrane-cytoplasmic exchange of the cluster components to counteract diffusive spread of the cluster. This exchange explains why only one bud forms per cell cycle, because the winner-takes-all competition of candidate sites inevitably selects a single site.

Wave-pinning and cell polarity from a bistable reaction-diffusion system

Motile eukaryotic cells polarize in response to external signals. Numerous mechanisms have been suggested to account for this symmetry breaking and for the ensuing robust polarization. Implicated in this process are various proteins that are recruited to the plasma membrane and segregate at an emergent front or back of the polarizing cell. Among these are PI3K, PTEN, and members of the Rho family GTPases such as Cdc42, Rac, and Rho. Many such proteins, including the Rho GTPases, cycle between active membrane-bound forms and inactive cytosolic forms. In previous work, we have shown that this property, together with appropriate crosstalk, endows a biochemical circuit (Cdc42, Rac, and Rho) with the property of inherent polarizability. Here we show that this property is present in an even simpler system comprised of a single active/inactive protein pair with positive feedback to its own activation. The simplicity of this minimal system also allows us to explain the mechanism using insights from mathematical analysis. The basic idea resides in a well-known property of reaction-diffusion systems with bistable kinetics, namely, propagation of fronts. However, it crucially depends on exchange between active and inactive forms of the chemicals with unequal rates of diffusion, and overall conservation to pin the waves into a stable polar distribution. We refer to these dynamics as wave-pinning and we show that this phenomenon is distinct from Turing-instability-generated pattern formation that occurs in reaction-diffusion systems that appear to be very similar. We explain the mathematical basis of the phenomenon, relate it to spatial segregation of Rho GTPases, and show how it can account for spatial amplification and maintenance of polarity, as well as sensitivity to new stimuli typical in polarization of eukaryotic cells.

Exploring the control circuit of cell migration by mathematical modeling

We have developed a top-down, rule-based mathematical model to explore the basic principles that coordinate mechanochemical events during animal cell migration, particularly the local-stimulation-global-inhibition model suggested originally for chemotaxis. Cells were modeled as a shape machine that protrudes or retracts in response to a combination of local protrusion and global retraction signals. Using an optimization algorithm to identify parameters that generate specific shapes and migration patterns, we show that the mechanism of local stimulation global inhibition can readily account for the behavior of Dictyostelium under a large collection of conditions. Within this collection, some parameters showed strong correlation, indicating that a normal phenotype may be maintained by complementation among functional modules. In addition, comparison of parameters for control and nocodazole-treated Dictyostelium identified the most prominent effect of microtubules as regulating the rates of retraction and protrusion signal decay, and the extent of global inhibition. Other changes in parameters can lead to profound transformations from amoeboid cells into cells mimicking keratocytes, neurons, or fibroblasts. Thus, a simple circuit of local stimulation-global inhibition can account for a wide range of cell behaviors. A similar top-down approach may be applied to other complex problems and combined with molecular manipulations to define specific protein functions.

Modeling of granulocyte cytoskeletal responses following fMLP challenging

Formyl peptides released from Gram-negative bacteria ligate a group of specific mammalian receptors, expressed mainly on granulocytes, monocytes, and macrophages. Receptor ligation activates different transduction cascades, eventually leading to the release of reactive oxygen species and other bactericidal chemical species, and the activation of the actin cytoskeleton with extension of lamellipodia and migration toward the sites of maximal formyl peptide concentration. In vitro, under conditions of nongradient formyl peptide concentrations, lamellipodia form all around the cell contour (chemokinesis). In granulocytes challenged under these conditions with N-formyl-methionyl-leucyl-phenylalanine, (i) the power spectrum of the contour of activated cells shows a peak at a specific periodicity, indicating that the lamellipodial extension is not completely random but stochastically conforms to a deterministic scheme, and (ii) the morphological response (percent of cells exhibiting chemokinesis) tends to reach a maximum at certain drug concentrations, then declining at higher concentrations. Accordingly, the logarithm of the drug concentration-polarizing effect curve is bell-shaped. Herein we illustrate theoretical models for the simulation of these two components of the chemokinetic responses. We show that the main traits of the general morphology and arrangement of lamellipodia may be simulated by an algorithm that starting from a situation of random distribution of active receptors on the cell membrane, encompasses in the successive calculation cycles both a local autocatalytic enhancement of the actin polymerization and a relative inhibition of the actin polymerization at some distance from the more active polymerization foci. In addition, a drug log concentration-polarizing effect bell-shaped curve may be simulated by assuming that the N-formyl-methionyl-leucyl-phenylalanine, while binding with high affinity to the specific receptor, is also able to bind to another lower affinity receptor that may effect depolarizing actions or, more generally, metabolic blocking effects. Under these conditions, at low drug concentrations the polarizing effect brought about by the ligation of the specific receptor is largely predominant. However, as the drug concentration increases and the specific receptors approach saturation, the inhibitory effects become more and more powerful and the net polarizing effect is reduced.

Locally controlled inhibitory mechanisms are involved in eukaryotic GPCR-mediated chemosensing

Gprotein-coupled receptor (GPCR) signaling mediates a balance of excitatory and inhibitory activities that regulate Dictyostelium chemosensing to cAMP. The molecular nature and kinetics of these inhibitors are unknown. We report that transient cAMP stimulations induce PIP3 responses without a refractory period, suggesting that GPCR-mediated inhibition accumulates and decays slowly. Moreover, exposure to cAMP gradients leads to asymmetric distribution of the inhibitory components. The gradients induce a stable accumulation of the PIP3 reporter PHCrac-GFP in the front of cells near the cAMP source. Rapid withdrawal of the gradient led to the reassociation of G protein subunits, and the return of the PIP3 phosphatase PTEN and PHCrac-GFP to their pre-stimulus distribution. Reapplication of cAMP stimulation produces a clear PHCrac-GFP translocation to the back but not to the front, indicating that a stronger inhibition is maintained in the front of a polarized cell. Our study demonstrates a novel spatiotemporal feature of currently unknown inhibitory mechanisms acting locally on the PI3K activation pathway.

Computational modelling of epithelial patterning

The generation of polar cell polarity (PCP) can be regarded as a pattern-forming process. Pattern formation requires local self-enhancement and long-range inhibition that can take place either within a cell or between adjacent cells. A comparison of this general condition with implementations in molecular terms in recent PCP models facilitates an understanding of inherent similarities and differences between them. In addition, it is important to integrate the most interesting and still valid results of classical transplantation experiments that were made some 40 years ago. They remind us that the global polarizing signal is based on graded positional identities carried by the individual cells whose molecular nature is still unknown.

Asymmetric redistribution of GABA receptors during GABA gradient sensing by nerve growth cones analyzed by single quantum dot imaging

During development of the nervous system, the tip of a growing axon, the growth cone (GC), must respond accurately to stimuli that direct its growth. This axonal navigation depends on extracellular concentration gradients of numerous guidance cues, including GABA. GCs can detect even weak directional signals, yet the mechanisms underlying this sensitivity remain unclear. Past studies in other eukaryotic chemotactic systems have pointed to the role of the spatial reorganization of the transduction pathway in their sensitive response. Here we have developed a single-molecule assay to observe individual GABA(A) receptors (GABA(A)Rs) in the plasma membrane of nerve GCs subjected to directional stimuli. We report that in the presence of an external GABA gradient GABA(A)Rs redistribute asymmetrically across the GC toward the gradient source. Single-particle tracking of GABA(A)Rs shows that the redistribution results from transient interactions between the receptors and the microtubules. Moreover, the relocalization is accompanied by an enhancement in the asymmetry of intracellular calcium concentration. Altogether, our results reveal a microtubule-dependent polarized reorganization of chemoreceptors at the cell surface and suggest that this polarization serves as an amplification step in GABA gradient sensing by nerve GCs.

Long-range Ca2+ signaling from growth cone to soma mediates reversal of neuronal migration induced by slit-2

Neuronal migration and growth-cone extension are both guided by extracellular factors in the developing brain, but whether these two forms of guidance are mechanistically linked is unclear. Application of a Slit-2 gradient in front of the leading process of cultured cerebellar granule cells led to the collapse of the growth cone and the reversal of neuronal migration, an event preceded by a propagating Ca(2+) wave from the growth cone to the soma. The Ca(2+) wave was required for the Slit-2 effect and was sufficient by itself to induce the reversal of migration. The Slit-2-induced reversal of migration required active RhoA, which was accumulated at the front of the migrating neuron, and this polarized RhoA distribution was reversed during the migration reversal induced by either the Slit-2 gradient or the Ca(2+) wave. Thus, long-range Ca(2+) signaling coordinates the Slit-2-induced changes in motility at two distant parts of migrating neurons by regulating RhoA distribution.

An essential role for the SHIP2-dependent negative feedback loop in neuritogenesis of nerve growth factor-stimulated PC12 cells

The local accumulation of phosphatidylinositol (3,4,5) trisphosphate (PIP(3)) and resulting activation of Rac1/Cdc42 play a critical role in nerve growth factor (NGF)-induced neurite outgrowth. To further explore the mechanism, we visualized PIP(3), phosphatidylinositol (3,4) bisphosphate, and Rac1/Cdc42 activities by fluorescence resonance energy transfer (FRET) imaging in NGF-stimulated PC12 cells. Based on the obtained FRET images, and with the help of in silico kinetic reaction model, we predicted that PI-5-phosphatase negatively regulates PIP(3) upon NGF stimulation. In agreement with this model, depletion of Src homology 2 domain-containing inositol polyphosphate 5-phosphatase 2 (SHIP2) markedly potentiated NGF-induced Rac1/Cdc42 activation and PIP(3) accumulation and considerably increased the number and the length of neurites in phosphate and tensin homologue-depleted PC12 cells. Further refinement of the computational model predicted Rac1 regulation of PI3-kinase and SHIP2, which was also validated experimentally. We propose that the SHIP2-mediated negative feedback on PIP(3) coordinately works with the PI3-kinase-mediated positive feedback to form an initial protrusive pattern and, later, to punctuate the PIP(3) accumulation to maintain proper neurite outgrowth.

A mass conserved reaction-diffusion system captures properties of cell polarity

Cell polarity is a general cellular process that can be seen in various cell types such as migrating neutrophils and Dictyostelium cells. The Rho small GTP(guanosine 5′-tri phosphate)ases have been shown to regulate cell polarity; however, its mechanism of emergence has yet to be clarified. We first developed a reaction–diffusion model of the Rho GTPases, which exhibits switch-like reversible response to a gradient of extracellular signals, exclusive accumulation of Cdc42 and Rac, or RhoA at the maximal or minimal intensity of the signal, respectively, and tracking of changes of a signal gradient by the polarized peak. The previous cell polarity models proposed by Subramanian and Narang show similar behaviors to our Rho GTPase model, despite the difference in molecular networks. This led us to compare these models, and we found that these models commonly share instability and a mass conservation of components. Based on these common properties, we developed conceptual models of a mass conserved reaction–diffusion system with diffusion–driven instability. These conceptual models retained similar behaviors of cell polarity in the Rho GTPase model. Using these models, we numerically and analytically found that multiple polarized peaks are unstable, resulting in a single stable peak (uniqueness of axis), and that sensitivity toward changes of a signal gradient is specifically restricted at the polarized peak (localized sensitivity). Although molecular networks may differ from one cell type to another, the behaviors of cell polarity in migrating cells seem similar, suggesting that there should be a fundamental principle. Thus, we propose that a mass conserved reaction–diffusion system with diffusion-driven instability is one of such principles of cell polarity.

Eukaryotic cells such as neutrophils and Dictyostelium cells respond to temporal and spatial gradients of extracellular signals with directional movements. In a migrating cell, specific molecular events take place at the front and back edges. The spatially distinctive molecular accumulation inside cells is known as cell polarity. Despite numerous experimental and theoretical studies, its mechanism of emergence has yet to be clarified. We first developed a mathematical model of the Rho small GTP(guanosine 5′-tri phosphate)ases that cooperatively regulate cell polarity, and showed that the model generates specific spatial accumulation of the molecules. Based on our Rho GTPases model and other models, we further established a conceptual model, a mass conserved reaction–diffusion system, and showed that diffusion-driven instability and a mass conservation of molecules that have active and inactive states are sufficient for polarity formation. We numerically and analytically found that molecular accumulations at multiple sites are unstable, resulting in a single stable front–back axis, and that sensitivity toward changes of a signal gradient is specifically restricted at the front of a polarized cell. We propose that a mass conserved reaction–diffusion system is one of the fundamental principles of cell polarity.

Mathematical model for spatial segregation of the Rho-family GTPases based on inhibitory crosstalk

Cdc42, Rac, and Rho are small GTPases known to play a central role in signal transduction to the actin cytoskeleton. These proteins regulate cell motility, by affecting nucleation, uncapping, and depolymerization of actin filaments, and acto-myosin contractility. Studies of crosstalk and mutual feedbacks in these three proteins have led to a number of proposals for their interaction. At the same time, observations of the spatio-temporal dynamics of Rho-family proteins give evidence of spatial polarization and mutual exclusion between Cdc42/Rac and Rho. In this paper, we formulate a mathematical model to account for such observations, based on the known underlying biology of these proteins. We first investigate which of the crosstalk schemes proposed in the literature is consistent with observed dynamics, and then derive a simple model that can correctly describe these dynamics (assuming crosstalk is mediated via Rho GEFs). We show that cooperativity is an essential ingredient in the interactions of the proteins. The co-occurrence of a stable rest state with the possibility of fast spatial segregation can be related to bistability in a set of underlying ODEs in which the inactive forms of these proteins are fixed at a constant level. We show that the fast diffusion of the inactive forms is essential for stabilizing the transition fronts in the PDE formulation of the model, leading to robust spatial polarization, rather than traveling waves.

Central roles of small GTPases in the development of cell polarity in yeast and beyond

SUMMARY

The establishment of cell polarity is critical for the development of many organisms and for the function of many cell types. A large number of studies of diverse organisms from yeast to humans indicate that the conserved, small-molecular-weight GTPases function as key signaling proteins involved in cell polarization. The budding yeast Saccharomyces cerevisiae is a particularly attractive model because it displays pronounced cell polarity in response to intracellular and extracellular cues. Cells of S. cerevisiae undergo polarized growth during various phases of their life cycle, such as during vegetative growth, mating between haploid cells of opposite mating types, and filamentous growth upon deprivation of nutrition such as nitrogen. Substantial progress has been made in deciphering the molecular basis of cell polarity in budding yeast. In particular, it becomes increasingly clear how small GTPases regulate polarized cytoskeletal organization, cell wall assembly, and exocytosis at the molecular level and how these GTPases are regulated. In this review, we discuss the key signaling pathways that regulate cell polarization during the mitotic cell cycle and during mating.

Taxis equations for amoeboid cells

The classical macroscopic chemotaxis equations have previously been derived from an individual-based description of the tactic response of cells that use a "run-and-tumble" strategy in response to environmental cues [17,18]. Here we derive macroscopic equations for the more complex type of behavioral response characteristic of crawling cells, which detect a signal, extract directional information from a scalar concentration field, and change their motile behavior accordingly. We present several models of increasing complexity for which the derivation of population-level equations is possible, and we show how experimentally measured statistics can be obtained from the transport equation formalism. We also show that amoeboid cells that do not adapt to constant signals can still aggregate in steady gradients, but not in response to periodic waves. This is in contrast to the case of cells that use a "run-and-tumble" strategy, where adaptation is essential.

Chemotaxis in shallow gradients is mediated independently of PtdIns 3-kinase by biased choices between random protrusions

Current models of eukaryotic chemotaxis propose that directional sensing causes localized generation of new pseudopods. However, quantitative analysis of pseudopod generation suggests a fundamentally different mechanism for chemotaxis in shallow gradients: first, pseudopods in multiple cell types are usually generated when existing ones bifurcate and are rarely made de novo; second, in Dictyostelium cells in shallow chemoattractant gradients, pseudopods are made at the same rate whether cells are moving up or down gradients. The location and direction of new pseudopods are random within the range allowed by bifurcation and are not oriented by chemoattractants. Thus, pseudopod generation is controlled independently of chemotactic signalling. Third, directional sensing is mediated by maintaining the most accurate existing pseudopod, rather than through the generation of new ones. Finally, the phosphatidylinositol 3-kinase (PI(3)K) inhibitor LY294002 affects the frequency of pseudopod generation, but not the accuracy of selection, suggesting that PI(3)K regulates the underlying mechanism of cell movement, rather than control of direction.

A mathematical model for neutrophil gradient sensing and polarization

Directed cell migration in response to chemical cues, also known as chemotaxis, is an important physiological process involved in wound healing, foraging, and the immune response. Cell migration requires the simultaneous formation of actin polymers at the leading edge and actomyosin complexes at the sides and back of the cell. An unresolved question in eukaryotic chemotaxis is how the same chemoattractant signal determines both the cell's front and back. Recent experimental studies have begun to reveal the biochemical mechanisms necessary for this polarized cellular response. We propose a mathematical model of neutrophil gradient sensing and polarization based on experimentally characterized biochemical mechanisms. The model demonstrates that the known dynamics for Rho GTPase and phosphatidylinositol-3-kinase (PI3K) activation are sufficient for both gradient sensing and polarization. In particular, the model demonstrates that these mechanisms can correctly localize the “front” and “rear” pathways in response to both uniform concentrations and gradients of chemical attractants, including in actin-inhibited cells. Furthermore, the model predictions are robust to the values of many parameters. A key result of the model is the proposed coincidence circuit involving PI3K and Ras that obviates the need for the “global inhibitors” proposed, though never experimentally verified, in many previous mathematical models of eukaryotic chemotaxis. Finally, experiments are proposed to (in)validate this model and further our understanding of neutrophil chemotaxis.

Neutrophils target sites of infection and inflammation by sensing chemical signals produced by damaged tissue and infecting microbes and then move in the direction where their concentration is greatest. An open question is how neutrophils integrate this information to determine the direction of motility. We present a mathematical model for the intracellular signaling network regulating polarization and chemotaxis in neutrophils. We demonstrate how the activation of two antagonizing pathways robustly establishes the front and back of the migrating cell. The model is able to reproduce a number of experimental studies, and new experiments are proposed to test different aspects of the model. A key result is the characterization of a coincidence circuit involving phosphatidylinositol-3-kinase (PI3K) and Ras. We demonstrate that this circuit plays a critical role in selectively localizing F-actin to the front of the cell and actomyosin complexes to the rear. As directed motility in response to chemical cues is critical in a number of processes including wound healing and tumor metastasis, the results and insights gained from the model may be applicable to other cell types and organisms.

Mechanisms of Gradient Detection: A Comparison of Axon Pathfinding with Eukaryotic Cell Migration

The detection of gradients of chemotactic cues is a common task for migrating cells and outgrowing axons. Eukaryotic gradient detection employs a spatial mechanism, meaning that the external gradient has to be translated into an intracellular signaling gradient, which affects cell polarization and directional movement. The sensitivity of gradient detection is governed by signal amplification and adaptation mechanisms. Comparison of the major signal transduction pathways underlying gradient detection in three exemplary chemotaxing cell types, Dictyostelium, neutrophils, and fibroblasts and in neuronal growth cones, reveals conserved mechanisms such as localized PI3 kinase/PIP3 signaling and a common output, the regulation of the cytoskeleton by Rho GTPases. Local protein translation plays a role in directional movement of both fibroblasts and neuronal growth cones. Ca(2+) signaling is prominently involved in growth cone gradient detection. The diversity of signaling between different cell types and its functional implications make sense in the biological context.

Phosphoinositides and Rho proteins spatially regulate actin polymerization to initiate and maintain directed movement in a one-dimensional model of a motile cell

Gradient sensing, polarization, and chemotaxis of motile cells involve the actin cytoskeleton, and regulatory modules, including the phosphoinositides (PIs), their kinases/phosphatases, and small GTPases (Rho proteins). Here we model their individual components (PIP1, PIP2, PIP3; PTEN, PI3K, PI5K; Cdc42, Rac, Rho; Arp2/3, and actin), their interconversions, interactions, and modular functions in the context of a one-dimensional dynamic model for protrusive cell motility, with parameter values derived from in vitro and in vivo studies. In response to a spatially graded stimulus, the model produces stable amplified internal profiles of regulatory components, and initiates persistent motility (consistent with experimental observations). By connecting the modules, we find that Rho GTPases work as a spatial switch, and that the PIs filter noise, and define the front versus back. Relatively fast PI diffusion also leads to selection of a unique pattern of Rho distribution from a collection of possible patterns. We use the model to explore the importance of specific hypothesized interactions, to explore mutant phenotypes, and to study the role of actin polymerization in the maintenance of the PI asymmetry. We also suggest a mechanism to explain the spatial exclusion of Cdc42 and PTEN and the inability of cells lacking active Cdc42 to properly detect chemoattractant gradients.

The region-specific activities of lipid rafts during axon growth and guidance

The generation and control of cell polarity is a fundamental mechanism for directed migration of the cell. In developing neurons, the axonal growth cone recognizes environmental molecular cues and migrates toward its correct target, thereby forming neuronal networks. The spatial information provided by environmental cues directs axon growth and guidance through generating polarity of intracellular signals and cytoskeletal organization in the growth cone. This polarization process is dependent on lipid rafts, specialized microdomains in the cell membrane. Lipid rafts in specific regions of the growth cone are involved in axon growth and guidance. For example, forward migration of the growth cone requires raft membranes in its leading front. Recent experiments have suggested that lipid rafts function as a platform for localized signaling downstream of adhesion molecules and guidance receptors. The rafts assemble into an active membrane domain that captures and reorganizes the cytoskeletal machinery. In this way, the spatial control of signaling through raft membranes plays a critical role in translating extracellular information into polarized motility of the growth cone.

Feedback signaling controls leading-edge formation during chemotaxis

Chemotactic cells translate shallow chemoattractant gradients into a highly polarized intracellular response that includes the localized production of PI(3,4,5)P(3) on the side of the cell facing the highest chemoattractant concentration. Research over the past decade began to uncover the molecular mechanisms involved in this localized signal amplification controlling the leading edge of chemotaxing cells. These mechanisms have been shown to involve multiple positive feedback loops, in which the PI(3,4,5)P(3) signal amplifies itself independently of the original stimulus, as well as inhibitory signals that restrict PI(3,4,5)P(3) to the leading edge, thereby creating a steep intracellular PI(3,4,5)P(3) gradient. Molecules involved in positive feedback signaling at the leading edge include the small G-proteins Rac and Ras, phosphatidylinositol-3 kinase and F-actin, as part of interlinked feedback loops that lead to a robust production of PI(3,4,5)P(3).

Cellular asymmetry and individuality in directional sensing

It is generally assumed that single cells in an isogenic population, when exposed to identical environments, exhibit the same behavior. However, it is becoming increasingly clear that, even in a genetically identical population, cellular behavior can vary significantly among cells. Here we explore this variability in the gradient-sensing response of Dictyostelium cells when exposed to repeated spatiotemporal pulses of chemoattractant. Our experiments show the response of a single cell to be highly reproducible from pulse to pulse. In contrast, a large variability in the response direction and magnitude is observed from cell to cell, even when different cells are exposed to the same pulse. First, these results indicate that the gradient-sensing network has inherent asymmetries that can significantly impact the ability of cells to faithfully sense the direction of extracellular signals (cellular asymmetry). Second, we find that the magnitude of this asymmetry varies greatly among cells. Some cells are able to accurately follow the direction of an extracellular stimulus, whereas, in other cells, the intracellular asymmetry dominates, resulting in a polarization axis that is independent of the direction of the extracellular cue (cellular individuality). We integrate these experimental findings into a model that treats the effective signal a cell detects as the product of the extracellular signal and the asymmetric intracellular signal. With this model we successfully predict the population response. This cellular individuality and asymmetry might fundamentally limit the fidelity of signal detection; in contrast, however, it might be beneficial by diversifying phenotypes in isogenic populations.

To stabilize neutrophil polarity, PIP3 and Cdc42 augment RhoA activity at the back as well as signals at the front

Chemoattractants like f-Met-Leu-Phe (fMLP) induce neutrophils to polarize by triggering divergent signals that promote the formation of protrusive filamentous actin (F-actin; frontness) and RhoA-dependent actomyosin contraction (backness). Frontness locally inhibits backness and vice versa. In neutrophil-like HL60 cells, blocking phosphatidylinositol-3,4,5-tris-phosphate (PIP3) accumulation with selective inhibitors of PIP3 synthesis completely prevents fMLP from activating a PIP3-dependent kinase and Cdc42 but not from stimulating F-actin accumulation. PIP3-deficient cells show reduced fMLP-dependent Rac activity and unstable pseudopods, which is consistent with the established role of PIP3 as a mediator of positive feedback pathways that augment Rac activation at the front. Surprisingly, such cells also show reduced RhoA activation and RhoA-dependent contraction at the trailing edge, leading to the formation of multiple lateral pseudopods. Cdc42 mediates PIP3's positive effect on RhoA activity. Thus, PIP3 and Cdc42 maintain stable polarity with a single front and a single back not only by strengthening pseudopods but also, at longer range, by promoting RhoA-dependent actomyosin contraction at the trailing edge.

Directional sensing in eukaryotic chemotaxis: a balanced inactivation model

Many eukaryotic cells, including Dictyostelium discoideum amoebae, fibroblasts, and neutrophils, are able to respond to chemoattractant gradients with high sensitivity. Recent studies have demonstrated that, after the introduction of a chemoattractant gradient, several chemotaxis pathway components exhibit a subcellular reorganization that cannot be described as a simple amplification of the external gradient. Instead, this reorganization has the characteristics of a switch, leading to a well defined front and back. Here, we propose a directional sensing mechanism in which two second messengers are produced at equal rates. The diffusion of one of them, coupled with an inactivation scheme, ensures a switch-like response to external gradients for a large range of gradient steepness and average concentration. Furthermore, our model is able to reverse the subcellular organization rapidly, and its response to multiple simultaneous chemoattractant sources is in good agreement with recent experimental results. Finally, we propose that the dynamics of a heterotrimeric G protein might allow for a specific biochemical realization of our model.

Is there a pilot in a pseudopod?

The concept of pilot pseudopodia is reconsidered 30 years after its inauguration (Gerisch, G., Hülser, D., Malchow, D., Wick, U., 1975. Cell communication by periodic cyclic-AMP pulses. Phil. Trans. R. Soc. Lond. B 272, 181-192). The original hypothesis stated that protruding pseudopodia serve as dynamic sensory organelles that aid a cell in perceiving variations of chemoattractant concentration and, consequently, in navigation during chemotaxis. This influential idea is reevaluated in the light of recent findings about the mechanisms governing chemotactic cell motility, morphology and dynamics of pseudopodia, and about molecular constituents and regulators of pseudopod extension and retraction. It is proposed that stimulation by a chemoattractant modulates speed of pseudopod protrusion and thereby increases cell elongation. Elongation further enhances chemotactic sensitivity of the cell to shallow chemoattractant gradients, reinforces cell polarization, and finally leads to suppression of lateral pseudopodia and continuation of cell migration in the gradient direction.

Moving toward understanding eukaryotic chemotaxis

The discovery in 1947 of directed cell movement in Dictyostelium discoideum quietly gave a birth to a new line of investigation into the molecular basis of chemotaxis. Some 60 years later, D. discoideum continues to be a key model system for the study of eukaryotic chemotaxis as well as an array of other important biological processes. As one of the most influential scientists, Guenther Gerisch has inspired several generations of researchers with his insightful and rigorous approaches applied to this model system. His studies have greatly contributed to current knowledge of many fundamental processes, such as cell-cell adhesion, phagocytosis, endocytosis, cytokinesis, cell signaling and chemotaxis. In this review, we wish to look back at the journey that has led to our current understanding of chemotaxis of eukaryotic cells.

Spontaneous polarization in eukaryotic gradient sensing: a mathematical model based on mutual inhibition of frontness and backness pathways

A key problem of eukaryotic cell motility is the signaling mechanism of chemoattractant gradient sensing. Recent experiments have revealed the molecular correlate of gradient sensing: Frontness molecules, such as PI3P and Rac, localize at the front end of the cell, and backness molecules, such as Rho and myosin II, accumulate at the back of the cell. Importantly, this frontness-backness polarization occurs spontaneously even if the cells are exposed to uniform chemoattractant profiles. The spontaneous polarization suggests that the gradient sensing machinery undergoes a Turing bifurcation. This has led to several classical activator-inhibitor and activator-substrate models which identify the frontness molecules with the activator. Conspicuously absent from these models is any accounting of the backness molecules. This stands in sharp contrast to experiments which show that the backness pathways inhibit the frontness pathways. Here, we formulate a model based on the mutually inhibitory interaction between the frontness and backness pathways. The model builds upon the mutual inhibition model proposed by Bourne and coworkers [Xu et al., 2003. Divergent signals and cytoskeletal assemblies regulate self-organizing polarity in neutrophils. Cell 114, 201-214.]. We show that mutual inhibition alone, without the help of any positive feedback (autocatalysis), can trigger spontaneous polarization of the frontness and backness pathways. The spatial distribution of the frontness and backness molecules in response to inhibition and activation of the frontness and backness pathways are consistent with those observed in experiments. Furthermore, depending on the parameter values, the model yields spatial distributions corresponding to chemoattraction (frontness pathways in-phase with the external gradient) and chemorepulsion (frontness pathways out-of-phase with the external gradient). Analysis of the model suggests a mechanism for the chemorepulsion-to-chemoattraction transition observed in neurons.

Membrane-bound Turing patterns

Motivated by recent observations in biological cells, we study Turing patterns in bounded regions where the nonlinear chemical reactions occur on the boundary and where reagent transport occurs in the bulk. Within a generic model, we formulate the stability problem and discuss the conditions for the occurrence of a Turing instability. By choosing other model parameters to be unequal, we find that Turing patterns exist even in the case of equal diffusion constants. Finally, a recently introduced computation technique is utilized to follow the nascent pattern into the highly nonlinear regime.

Quantitative elucidation of a distinct spatial gradient-sensing mechanism in fibroblasts

Migration of eukaryotic cells toward a chemoattractant often relies on their ability to distinguish receptor-mediated signaling at different subcellular locations, a phenomenon known as spatial sensing. A prominent example that is seen during wound healing is fibroblast migration in platelet-derived growth factor (PDGF) gradients. As in the well-characterized chemotactic cells Dictyostelium discoideum and neutrophils, signaling to the cytoskeleton via the phosphoinositide 3-kinase pathway in fibroblasts is spatially polarized by a PDGF gradient; however, the sensitivity of this process and how it is regulated are unknown. Through a quantitative analysis of mathematical models and live cell total internal reflection fluorescence microscopy experiments, we demonstrate that PDGF detection is governed by mechanisms that are fundamentally different from those in D. discoideum and neutrophils. Robust PDGF sensing requires steeper gradients and a much narrower range of absolute chemoattractant concentration, which is consistent with a simpler system lacking the feedback loops that yield signal amplification and adaptation in amoeboid cells.

A modelling framework describing the enzyme regulation of membrane lipids underlying gradient perception in Dictyostelium cells II: input-output analysis

Spatial sensing in Dictyostelium involves localization of the phosphoinositide lipids PI(3,4,5)P3 and PI(3,4)P2 at the leading edge of the cell in response to an external gradient. We have previously proposed a modelling framework describing the regulation of these lipids by the enzymes PI3K and PTEN. In this paper we analyse this regulation from an input-output perspective. When the inputs are homogeneous, we obtain explicit analytical expressions for the lipid concentrations as a function of enzyme concentrations and model parameters. We also show that the system can be cast as an open-loop bilinear control system, and employ control engineering tools to show that a local three-dimensional region in the four-dimensional phase space can be accessed by temporally varying either or both enzyme concentrations. For spatially graded enzyme profiles, we show that diffusion limits the extent to which lipid profiles can be manipulated by enzymes. However, we also demonstrate that for certain ranges of network parameters, increasing lipid diffusion can lead to an increase in steady-state leading-edge concentrations of PI(3,4,5)P3 or PI(3,4)P2, even though all lipid diffusion coefficients are equal. Finally, in order to determine the extent to which lipid profiles can be regulated by the enzymes, we formulate and solve inverse problems, where we determine the enzyme profiles required to realize particular lipid profiles at steady state.

A System of Counteracting Feedback Loops Regulates Cdc42p Activity during Spontaneous Cell Polarization

Cellular polarization is often a response to distinct extracellular or intracellular cues, such as nutrient gradients or cortical landmarks. However, in the absence of such cues, some cells can still select a polarization axis at random. Positive feedback loops promoting localized activation of the GTPase Cdc42p are central to this process in budding yeast. Here, we explore spontaneous polarization during bud site selection in mutant yeast cells that lack functional landmarks. We find that these cells do not select a single random polarization axis, but continuously change this axis during the G1 phase of the cell cycle. This is reflected in traveling waves of activated Cdc42p which randomly explore the cell periphery. Our integrated computational and in vivo analyses of these waves reveal a negative feedback loop that competes with the aforementioned positive feedback loops to regulate Cdc42p activity and confer dynamic responsiveness on the robust initiation of cell polarization.

Adaptation is not required to explain the long-term response of axons to molecular gradients

It has been suggested that growth cones navigating through the developing nervous system might display adaptation, so that their response to gradient signals is conserved over wide variations in ligand concentration. Recently however, a new chemotaxis assay that allows the effect of gradient parameters on axonal trajectories to be finely varied has revealed a decline in gradient sensitivity on either side of an optimal concentration. We show that this behavior can be quantitatively reproduced with a computational model of axonal chemotaxis that does not employ explicit adaptation. Two crucial components of this model required to reproduce the observed sensitivity are spatial and temporal averaging. These can be interpreted as corresponding, respectively, to the spatial spread of signaling effects downstream from receptor binding, and to the finite time over which these signaling effects decay. For spatial averaging, the model predicts that an effective range of roughly one-third of the extent of the growth cone is optimal for detecting small gradient signals. For temporal decay, a timescale of about 3 minutes is required for the model to reproduce the experimentally observed sensitivity.

Signaling to cytoskeletal dynamics during chemotaxis

We review insights in signaling pathways controlling cell polarization and cytoskeletal organization during chemotactic movement in Dictyostelium amoebae and neutrophils. We compare and contrast these insights with our current understanding of pathways controlling chemotactic movements in more-complex multicellular developmental contexts.

Deterministic and Stochastic Elements of Axonal Guidance

An enormous literature has been developed on investigations of the growth and guidance of axons during development and after injury. In this review, we provide a guide to this literature as a resource for biomedical investigators. We first review briefly the molecular biology that is known to regulate migration of the growth cone and branching of axonal arbors. We then outline some important fundamental considerations that are important to the modeling of the phenomenology of these guidance effects and of what is known of their underlying internal mechanisms. We conclude by providing some thoughts on the outlook for future biomedical modeling in the field.

Distinguishing modes of eukaryotic gradient sensing

We develop a mathematical model of phosphoinositide-mediated gradient sensing that can be applied to chemotactic behavior in highly motile eukaryotic cells such as Dictyostelium and neutrophils. We generate four variants of our model by adjusting parameters that control the strengths of coupled positive feedbacks and the importance of molecules that translocate from the cytosol to the membrane. Each variant exhibits a qualitatively different mode of gradient sensing. Simulations of characteristic behaviors suggest that differences between the variants are most evident at transitions between efficient gradient detection and failure. Based on these results, we propose criteria to distinguish between possible modes of gradient sensing in real cells, where many biochemical parameters may be unknown. We also identify constraints on parameters required for efficient gradient detection. Finally, our analysis suggests how a cell might transition between responsiveness and nonresponsiveness, and between different modes of gradient sensing, by adjusting its biochemical parameters.