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

A Stochastic Modelling Framework for Single Cell Migration: Coupling Contractility and Focal Adhesions

The interaction of the actin cytoskeleton with cell–substrate adhesions is necessary for cell migration. While the trajectories of motile cells have a stochastic character, investigations of cell motility mechanisms rarely elaborate on the origins of the observed randomness. Here, guided by a few fundamental attributes of cell motility, I construct a minimal stochastic cell migration model from ground-up. The resulting model couples a deterministic actomyosin contractility mechanism with stochastic cell–substrate adhesion kinetics, and yields a well-defined piecewise deterministic process. Numerical simulations reproduce several experimentally observed results, including anomalous diffusion, tactic migration and contact guidance. This work provides a basis for the development of cell–cell collision and population migration models.

A hybrid stochastic-deterministic mechanochemical model of cell polarization

Polarization is a crucial component in cell differentiation, development, and motility, but its details are not yet well understood. At the onset of cell locomotion, cells break symmetry to form well-defined cell fronts and rears. This polarity establishment varies across cell types: in Dictyostelium discoideum cells, it is mediated by biochemical signaling pathways and can function in the absence of a cytoskeleton, while in keratocytes, it is tightly connected to cytoskeletal dynamics and mechanics. Theoretical models that have been developed to understand the onset of polarization have explored either signaling or mechanical pathways, yet few have explored mechanochemical mechanisms. However, many motile cells rely on both signaling modules and actin cytoskeleton to break symmetry and achieve a stable polarized state. We propose a general mechanochemical polarization model based on coupling between a stochastic model for the segregation of signaling molecules and a simplified mechanical model for actin cytoskeleton network competition. We find that local linear coupling between minimally nonlinear signaling and cytoskeletal systems, separately not supporting stable polarization, yields a robustly polarized cell state. The model captures the essence of spontaneous polarization of neutrophils, which has been proposed to emerge due to the competition between frontness and backness pathways.

Multi-phasic bi-directional chemotactic responses of the growth cone

The nerve growth cone is bi-directionally attracted and repelled by the same cue molecules depending on the situations, while other non-neural chemotactic cells usually show uni-directional attraction or repulsion toward their specific cue molecules. However, how the growth cone differs from other non-neural cells remains unclear. Toward this question, we developed a theory for describing chemotactic response based on a mathematical model of intracellular signaling of activator and inhibitor. Our theory was first able to clarify the conditions of attraction and repulsion, which are determined by balance between activator and inhibitor, and the conditions of uni- and bi-directional responses, which are determined by dose-response profiles of activator and inhibitor to the guidance cue. With biologically realistic sigmoidal dose-responses, our model predicted tri-phasic turning response depending on intracellular Ca²⁺ level, which was then experimentally confirmed by growth cone turning assays and Ca²⁺ imaging. Furthermore, we took a reverse-engineering analysis to identify balanced regulation between CaMKII (activator) and PP1 (inhibitor) and then the model performance was validated by reproducing turning assays with inhibitions of CaMKII and PP1. Thus, our study implies that the balance between activator and inhibitor underlies the multi-phasic bi-directional turning response of the growth cone.

A Predictive Model for Yeast Cell Polarization in Pheromone Gradients

Budding yeast cells exist in two mating types, a and α, which use peptide pheromones to communicate with each other during mating. Mating depends on the ability of cells to polarize up pheromone gradients, but cells also respond to spatially uniform fields of pheromone by polarizing along a single axis. We used quantitative measurements of the response of a cells to α-factor to produce a predictive model of yeast polarization towards a pheromone gradient. We found that cells make a sharp transition between budding cycles and mating induced polarization and that they detect pheromone gradients accurately only over a narrow range of pheromone concentrations corresponding to this transition. We fit all the parameters of the mathematical model by using quantitative data on spontaneous polarization in uniform pheromone concentration. Once these parameters have been computed, and without any further fit, our model quantitatively predicts the yeast cell response to pheromone gradient providing an important step toward understanding how cells communicate with each other.

Dynamic membrane patterning, signal localization and polarity in living cells

We review the molecular and physical aspects of the dynamic localization of signaling molecules on the plasma membranes of living cells. At the nanoscale, clusters of receptors and signaling proteins play an essential role in the processing of extracellular signals. At the microscale, "soft" and highly dynamic signaling domains control the interaction of individual cells with their environment. At the multicellular scale, individual polarity patterns control the forces that shape multicellular aggregates and tissues.

Moesin and myosin phosphatase confine neutrophil orientation in a chemotactic gradient

Jingsong Xu and colleagues investigate how neutrophils initiate polarized migration toward bacteria or chemoattractants. They find that attractant-induced activation of myosin phosphatase results in the deactivation of moesin at the prospective leading edge and its redistribution to the trailing edge, establishing polarity and directional pseudopod formation.

Neutrophils respond to invading bacteria by adopting a polarized morphology, migrating in the correct direction, and engulfing the bacteria. How neutrophils establish and precisely orient this polarity toward pathogens remains unclear. Here we report that in resting neutrophils, the ERM (ezrin, radixin, and moesin) protein moesin in its active form (phosphorylated and membrane bound) prevented cell polarization by inhibiting the small GTPases Rac, Rho, and Cdc42. Attractant-induced activation of myosin phosphatase deactivated moesin at the prospective leading edge to break symmetry and establish polarity. Subsequent translocation of moesin to the trailing edge confined the formation of a prominent pseudopod directed toward pathogens and prevented secondary pseudopod formation in other directions. Therefore, both moesin-mediated inhibition and its localized deactivation by myosin phosphatase are essential for neutrophil polarization and effective neutrophil tracking of pathogens.

The IQGAP-related protein DGAP1 mediates signaling to the actin cytoskeleton as an effector and a sequestrator of Rac1 GTPases

Proteins are typically categorized into protein families based on their domain organization. Yet, evolutionarily unrelated proteins can also be grouped together according to their common functional roles. Sequestering proteins constitute one such functional class, acting as macromolecular buffers and serving as an intracellular reservoir ready to release large quantities of bound proteins or other molecules upon appropriate stimulation. Another functional protein class comprises effector proteins, which constitute essential components of many intracellular signal transduction pathways. For instance, effectors of small GTP-hydrolases are activated upon binding a GTP-bound GTPase and thereupon participate in downstream interactions. Here we describe a member of the IQGAP family of scaffolding proteins, DGAP1 from Dictyostelium, which unifies the roles of an effector and a sequestrator in regard to the small GTPase Rac1. Unlike classical effectors, which bind their activators transiently leading to short-lived signaling complexes, interaction between DGAP1 and Rac1-GTP is stable and induces formation of a complex with actin-bundling proteins cortexillins at the back end of the cell. An oppositely localized Rac1 effector, the Scar/WAVE complex, promotes actin polymerization at the cell front. Competition between DGAP1 and Scar/WAVE for the common activator Rac1-GTP might provide the basis for the oscillatory re-polarization typically seen in randomly migrating Dictyostelium cells. We discuss the consequences of the dual roles exerted by DGAP1 and Rac1 in the regulation of cell motility and polarity, and propose that similar signaling mechanisms may be of general importance in regulating spatiotemporal dynamics of the actin cytoskeleton by small GTPases.

Identifying network motifs that buffer front-to-back signaling in polarized neutrophils

Neutrophil polarity relies on local, mutual inhibition to segregate incompatible signaling circuits to the leading and trailing edges. Mutual inhibition alone should lead to cells having strong fronts and weak backs or vice versa. However, analysis of cell-to-cell variation in human neutrophils revealed that back polarity remains consistent despite changes in front strength. How is this buffering achieved? Pharmacological perturbations and mathematical modeling revealed a functional role for microtubules in buffering back polarity by mediating positive, long-range crosstalk from front to back; loss of microtubules inhibits buffering and results in anticorrelation between front and back signaling. Furthermore, a systematic, computational search of network topologies found that a long-range, positive front-to-back link is necessary for back buffering. Our studies suggest a design principle that can be employed by polarity networks: short-range mutual inhibition establishes distinct signaling regions, after which directed long-range activation insulates one region from variations in the other.

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.

The tandem PH domain-containing protein 2 (TAPP2) regulates chemokine-induced cytoskeletal reorganization and malignant B cell migration

The intracellular signaling processes controlling malignant B cell migration and tissue localization remain largely undefined. Tandem PH domain-containing proteins TAPP1 and TAPP2 are adaptor proteins that specifically bind to phosphatidylinositol-3,4-bisphosphate, or PI(3,4)P2, a product of phosphoinositide 3-kinases (PI3K). While PI3K enzymes have a number of functions in cell biology, including cell migration, the functions of PI(3,4)P2 and its binding proteins are not well understood. Previously we found that TAPP2 is highly expressed in primary leukemic B cells that have strong migratory capacity. Here we find that SDF-1-dependent migration of human malignant B cells requires both PI3K signaling and TAPP2. Migration in a transwell assay is significantly impaired by pan-PI3K and isoform-selective PI3K inhibitors, or by TAPP2 shRNA knockdown (KD). Strikingly, TAPP2 KD in combination with PI3K inhibitor treatment nearly abolished the migration response, suggesting that TAPP2 may contribute some functions independent of the PI3K pathway. In microfluidic chamber cell tracking assays, TAPP2 KD cells show reduction in percentage of migrating cells, migration velocity and directionality. TAPP2 KD led to alterations in chemokine-induced rearrangement of the actin cytoskeleton and failure to form polarized morphology. TAPP2 co-localized with the stable F-actin-binding protein utrophin, with both molecules reciprocally localizing against F-actin accumulated at the leading edge upon SDF-1 stimulation. In TAPP2 KD cells, Rac was over-activated and localized to multiple membrane protrusions, suggesting that TAPP2 may act in concert with utrophin and stable F-actin to spatially restrict Rac activation and reduce formation of multiple membrane protrusions. TAPP2 function in cell migration is also apparent in the more complex context of B cell migration into stromal cell layers – a process that is only partially dependent on PI3K and SDF-1. In summary, this study identified TAPP2 as a novel regulator of malignant B cell migration and a potential therapeutic intervention target.

Designing synthetic regulatory networks capable of self-organizing cell polarization

How cells form global, self-organized structures using genetically encoded molecular rules remains elusive. Here, we take a synthetic biology approach to investigate the design principles governing cell polarization. First, using a coarse-grained computational model, we searched for all possible simple networks that can achieve polarization. All solutions contained one of three minimal motifs: positive feedback, mutual inhibition, or inhibitor with positive feedback. These minimal motifs alone could achieve polarization under limited conditions; circuits that combined two or more of these motifs were significantly more robust. With these design principles as a blueprint, we experimentally constructed artificial polarization networks in yeast, using a toolkit of chimeric signaling proteins that spatially direct the synthesis and degradation of phosphatidylinositol (3,4,5)-trisphosphate (PIP(3)). Circuits with combinatorial motifs yielded clear foci of synthetic PIP(3) that can persist for nearly an hour. Thus, by harnessing localization-regulated signaling molecules, we can engineer simple molecular circuits that reliably execute spatial self-organized programs.

An investigation of spatial signal transduction in cellular networks

Background

Spatial signal transduction plays a vital role in many intracellular processes such as eukaryotic chemotaxis, polarity generation and cell division. Furthermore it is being increasingly realized that the spatial dimension to signalling may play an important role in other apparently purely temporal signal transduction processes. It is increasingly being recognized that a conceptual basis for studying spatial signal transduction in signalling networks is necessary.

Results

In this work we examine spatial signal transduction in a series of standard motifs/networks. These networks include coherent and incoherent feedforward, positive and negative feedback, cyclic motifs, monostable switches, bistable switches and negative feedback oscillators. In all these cases, the driving signal has spatial variation. For each network we consider two cases, one where all elements are essentially non-diffusible, and the other where one of the network elements may be highly diffusible. A careful analysis of steady state signal transduction provides many insights into the behaviour of all these modules. While in the non-diffusible case for the most part, spatial signalling reflects the temporal signalling behaviour, in the diffusible cases, we see significant differences between spatial and temporal signalling characteristics. Our results demonstrate that the presence of diffusible elements in the networks provides important constraints and capabilities for signalling.

Conclusions

Our results provide a systematic basis for understanding spatial signalling in networks and the role of diffusible elements therein. This provides many insights into the signal transduction capabilities and constraints in such networks and suggests ways in which cellular signalling and information processing is organized to conform to or bypass those constraints. It also provides a framework for starting to understand the organization and regulation of spatial signal transduction in individual processes.

How cells integrate complex stimuli: the effect of feedback from phosphoinositides and cell shape on cell polarization and motility

To regulate shape changes, motility and chemotaxis in eukaryotic cells, signal transduction pathways channel extracellular stimuli to the reorganization of the actin cytoskeleton. The complexity of such networks makes it difficult to understand the roles of individual components, let alone their interactions and multiple feedbacks within a given layer and between layers of signalling. Even more challenging is the question of if and how the shape of the cell affects and is affected by this internal spatiotemporal reorganization. Here we build on our previous 2D cell motility model where signalling from the Rho family GTPases (Cdc42, Rac, and Rho) was shown to organize the cell polarization, actin reorganization, shape change, and motility in simple gradients. We extend this work in two ways: First, we investigate the effects of the feedback between the phosphoinositides (PIs) , and Rho family GTPases. We show how that feedback increases heights and breadths of zones of Cdc42 activity, facilitating global communication between competing cell “fronts”. This hastens the commitment to a single lamellipodium initiated in response to multiple, complex, or rapidly changing stimuli. Second, we show how cell shape feeds back on internal distribution of GTPases. Constraints on chemical isocline curvature imposed by boundary conditions results in the fact that dynamic cell shape leads to faster biochemical redistribution when the cell is repolarized. Cells with frozen cytoskeleton, and static shapes, consequently respond more slowly to reorienting stimuli than cells with dynamic shape changes, the degree of the shape-induced effects being proportional to the extent of cell deformation. We explain these concepts in the context of several in silico experiments using our 2D computational cell model.

Single cells, such as amoeba and white blood cells, change shape and move in response to environmental stimuli. Their behaviour is a consequence of the intracellular properties balanced by external forces. The internal regulation is modulated by several proteins that interact with one another and with membrane lipids. We examine, through in silico experiments using a computational model of a moving cell, the interactions of an important class of such proteins (Rho GTPases) and lipids (phosphoinositides, PIs), their spatial redistribution, and how they affect and are affected by cell shape. Certain GTPases promote the assembly of the actin cytoskeleton. This then leads to the formation of a cell protrusion, the leading edge. The feedback between PIs and GTPases facilitates global communication across the cell, ensuring that multiple, complex, or rapidly changing stimuli can be resolved into a single decision for positioning the leading edge. Interestingly, the cell shape itself affects the intracellular biochemistry, resulting from interactions between the curvature of the chemical fronts and the cell edge. Cells with static shapes consequently respond more slowly to reorienting stimuli than cells with dynamic shape changes. This potential to respond more rapidly to external stimuli depends on the degree of cellular shape deformation.

Noise filtering tradeoffs in spatial gradient sensing and cell polarization response

BACKGROUND

Cells sense chemical spatial gradients and respond by polarizing internal components. This process can be disrupted by gradient noise caused by fluctuations in chemical concentration.

RESULTS

We investigated how external gradient noise affects spatial sensing and response focusing on noise-filtering and the resultant tradeoffs. First, using a coarse-grained mathematical model of gradient-sensing and cell polarity, we characterized three negative consequences of noise: Inhibition of the extent of polarization, degradation of directional accuracy, and production of a noisy output polarization. Next, we explored filtering strategies and discovered that a combination of positive feedback, multiple signaling stages, and time-averaging produced good results. There was an important tradeoff, however, because filtering resulted in slower polarization. Simulations demonstrated that a two-stage filter-amplifier resulted in a balanced outcome. Then, we analyzed the effect of noise on a mechanistic model of yeast cell polarization in response to gradients of mating pheromone. This analysis showed that yeast cells likely also combine the above three filtering mechanisms into a filter-amplifier structure to achieve impressive spatial-noise tolerance, but with the consequence of a slow response time. Further investigation of the amplifier architecture revealed two positive feedback loops, a fast inner and a slow outer, both of which contributed to noise-tolerant polarization. This model also made specific predictions about how orientation performance depended upon the ratio between the gradient slope (signal) and the noise variance. To test these predictions, we performed microfluidics experiments measuring the ability of yeast cells to orient to shallow gradients of mating pheromone. The results of these experiments agreed well with the modeling predictions, demonstrating that yeast cells can sense gradients shallower than 0.1% μm-1, approximately a single receptor-ligand molecule difference between front and back, on par with motile eukaryotic cells.

CONCLUSIONS

Spatial noise impedes the extent, accuracy, and smoothness of cell polarization. A combined filtering strategy implemented by a filter-amplifier architecture with slow dynamics was effective. Modeling and experimental data suggest that yeast cells employ these elaborate mechanisms to filter gradient noise resulting in a slow but relatively accurate polarization response.

Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization

Cell polarization, in which substances previously uniformly distributed become asymmetric due to external or/and internal stimulation, is a fundamental process underlying cell mobility, cell division, and other polarized functions. The yeast cell S. cerevisiae has been a model system to study cell polarization. During mating, yeast cells sense shallow external spatial gradients and respond by creating steeper internal gradients of protein aligned with the external cue. The complex spatial dynamics during yeast mating polarization consists of positive feedback, degradation, global negative feedback control, and cooperative effects in protein synthesis. Understanding such complex regulations and interactions is critical to studying many important characteristics in cell polarization including signal amplification, tracking dynamic signals, and potential trade-off between achieving both objectives in a robust fashion. In this paper, we study some of these questions by analyzing several models with different spatial complexity: two compartments, three compartments, and continuum in space. The step-wise approach allows detailed characterization of properties of the steady state of the system, providing more insights for biological regulations during cell polarization. For cases without membrane diffusion, our study reveals that increasing the number of spatial compartments results in an increase in the number of steady-state solutions, in particular, the number of stable steady-state solutions, with the continuum models possessing infinitely many steady-state solutions. Through both analysis and simulations, we find that stronger positive feedback, reduced diffusion, and a shallower ligand gradient all result in more steady-state solutions, although most of these are not optimally aligned with the gradient. We explore in the different settings the relationship between the number of steady-state solutions and the extent and accuracy of the polarization. Taken together these results furnish a detailed description of the factors that influence the tradeoff between a single correctly aligned but poorly polarized stable steady-state solution versus multiple more highly polarized stable steady-state solutions that may be incorrectly aligned with the external gradient.

PAR-3 oligomerization may provide an actin-independent mechanism to maintain distinct par protein domains in the early Caenorhabditis elegans embryo

Par proteins establish discrete intracellular spatial domains to polarize many different cell types. In the single-cell embryo of the nematode worm Caenorhabditis elegans, the segregation of Par proteins is crucial for proper division and cell fate specification. Actomyosin-based cortical flows drive the initial formation of anterior and posterior Par domains, but cortical actin is not required for the maintenance of these domains. Here we develop a model of interactions between the Par proteins that includes both mutual inhibition and PAR-3 oligomerization. We show that this model gives rise to a bistable switch mechanism, allowing the Par proteins to occupy distinct anterior and posterior domains seen in the early C. elegans embryo, independent of dynamics or asymmetries in the actin cortex. The model predicts a sharp loss of cortical Par protein asymmetries during gradual depletion of the Par protein PAR-6, and we confirm this prediction experimentally. Together, these results suggest both mutual inhibition and PAR-3 oligomerization are sufficient to maintain distinct Par protein domains in the early C. elegans embryo.

A comparison of mathematical models for polarization of single eukaryotic cells in response to guided cues

Polarization, a primary step in the response of an individual eukaryotic cell to a spatial stimulus, has attracted numerous theoretical treatments complementing experimental studies in a variety of cell types. While the phenomenon itself is universal, details differ across cell types, and across classes of models that have been proposed. Most models address how symmetry breaking leads to polarization, some in abstract settings, others based on specific biochemistry. Here, we compare polarization in response to a stimulus (e.g., a chemoattractant) in cells typically used in experiments (yeast, amoebae, leukocytes, keratocytes, fibroblasts, and neurons), and, in parallel, responses of several prototypical models to typical stimulation protocols. We find that the diversity of cell behaviors is reflected by a diversity of models, and that some, but not all models, can account for amplification of stimulus, maintenance of polarity, adaptation, sensitivity to new signals, and robustness.

A model of localised Rac1 activation in endothelial cells due to fluid flow

Endothelial cells respond to fluid flow by elongating in the direction of flow. Cytoskeletal changes and activation of signalling molecules have been extensively studied in this response, including: activation of receptors by mechano-transduction, actin filament alignment in the direction of flow, changes to cell-substratum adhesions, actin-driven lamellipodium extension, and localised activation of Rho GTPases. To study this process we model the force over a single cell and couple this to a model of the Rho GTPases, Rac and Rho, via a Kelvin-body model of mechano-transduction. It is demonstrated that a mechano-transducer can respond to the normal component of the force is likely to be a necessary component of the signalling network in order to establish polarity. Furthermore, the rate-limiting step of Rac1 activation is predicted to be conversion of Rac-GDP to Rac-GTP, rather than activation of upstream components. Modelling illustrates that the aligned endothelial cell morphology could attenuate the signalling network.

ASYMPTOTIC AND BIFURCATION ANALYSIS OF WAVE-PINNING IN A REACTION-DIFFUSION MODEL FOR CELL POLARIZATION

We describe and analyze a bistable reaction-diffusion (RD) model for two interconverting chemical species that exhibits a phenomenon of wave-pinning: a wave of activation of one of the species is initiated at one end of the domain, moves into the domain, decelerates, and eventually stops inside the domain, forming a stationary front. The second ("inactive") species is depleted in this process. This behavior arises in a model for chemical polarization of a cell by Rho GTPases in response to stimulation. The initially spatially homogeneous concentration profile (representative of a resting cell) develops into an asymmetric stationary front profile (typical of a polarized cell). Wave-pinning here is based on three properties: (1) mass conservation in a finite domain, (2) nonlinear reaction kinetics allowing for multiple stable steady states, and (3) a sufficiently large difference in diffusion of the two species. Using matched asymptotic analysis, we explain the mathematical basis of wave-pinning, and predict the speed and pinned position of the wave. An analysis of the bifurcation of the pinned front solution reveals how the wave-pinning regime depends on parameters such as rates of diffusion and total mass of the species. We describe two ways in which the pinned solution can be lost depending on the details of the reaction kinetics: a saddle-node or a pitchfork bifurcation.

From Physics to Pharmacology?

Over the last fifty years there has been an explosion of biological data, leading to the realization that to fully explain biological mechanisms it is necessary to interpret them as complex dynamical systems. The first stage of this interpretation is to determine which components (proteins, genes or metabolites) of the system interact. This is usually represented by a graph, or network. The behavior of this network can then be investigated using mathematical modeling. In vivo these biological networks show several remarkable (and seemingly paradoxical) properties including robustness, plasticity and sensitivity. Erroneous behavior of these networks is often associated with disease. Hence understanding the system-level properties can have important implications for the treatment of disease. Systems biology is an organized approach to quantitatively describe and elucidate the behavior of these complex networks. This review focuses on the progress and future challenges of a systems approach to biology.

An investigation of design principles underlying repulsive and attractive gradient sensing and their switching

Many important cellular processes rely on cellular responses to spatially graded signals. This response may be either attractive, indicating a positive bias, or repulsive indicating a negative bias. In this paper we consider cells which exhibit both repulsive and attractive gradient sensing responses and aim to uncover the underlying design principles and features of how the networks are wired which could allow a cell to exhibit both responses. We use a modular approach to examine different configurations which will allow for a cell to exhibit both responses and analyse how this depends on the basic characteristics of gradient sensing and downstream signal propagation. Overall our analysis provides insights into how gradient responses can be switched and the key factors which affect this switching.

Bayes-optimal chemotaxis

Chemotaxis plays a crucial role in many biological processes, including nervous system development. However, fundamental physical constraints limit the ability of a small sensing device such as a cell or growth cone to detect an external chemical gradient. One of these is the stochastic nature of receptor binding, leading to a constantly fluctuating binding pattern across the cell's array of receptors. This is analogous to the uncertainty in sensory information often encountered by the brain at the systems level. Here we derive analytically the Bayes-optimal strategy for combining information from a spatial array of receptors in both one and two dimensions to determine gradient direction. We also show how information from more than one receptor species can be optimally integrated, derive the gradient shapes that are optimal for guiding cells or growth cones over the longest possible distances, and illustrate that polarized cell behavior might arise as an adaptation to slowly varying environments. Together our results provide closed-form predictions for variations in chemotactic performance over a wide range of gradient conditions.

Rebuilding cytoskeleton roads: active-transport-induced polarization of cells

Many cellular processes require a polarization axis which generally initially emerges as an inhomogeneous distribution of molecular markers in the cell. We present a simple analytical model of a general mechanism of cell polarization taking into account the positive feedback due to the coupled dynamics of molecular markers and cytoskeleton filaments. We find that the geometry of the organization of cytoskeleton filaments, nucleated on the membrane (e.g., cortical actin) or from a center in the cytoplasm (e.g., microtubule asters), dictates whether the system is capable of spontaneous polarization or polarizes only in response to external asymmetric signals. Our model also captures the main features of recent experiments of cell polarization in two considerably different biological systems, namely, mating budding yeast and neuron growth cones.

Calling heads from tails: the role of mathematical modeling in understanding cell polarization

Theorists have long speculated on the mechanisms driving directed and spontaneous cell polarization. Recently, experimentalists have uncovered many of the mechanisms underlying polarization, enabling these models to be directly tested. In the process, they have demonstrated the explanatory and predictive value of these models and, at the same time, uncovered additional complexities not currently explained by them. In this review, we discuss some of main theories regarding cell polarization and highlight how the intersection of mathematical and experimental biology has yielded new insights into these mechanisms in the case of budding yeast and eukaryotic chemotaxis.

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.

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.

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.

Interlinked mutual inhibitory positive feedbacks induce robust cellular memory effects

Mutual inhibitory positive feedback (MIPF), or double-negative feedback, is a key regulatory motif of cellular memory with the capability of maintaining switched states for transient stimuli. Such MIPFs are found in various biological systems where they are interlinked in many cases despite a single MIPF can still realize such a memory effect. An intriguing question then arises about the advantage of interlinking MIPFs instead of exploiting an isolated single MIPF to realize the memory effect. We have investigated the advantages of interlinked MIPF systems through mathematical modeling and computer simulations. Our results revealed that interlinking MIPFs expands the parameter range of achieving the memory effect, or the memory region, thereby making the system more robust to parameter perturbations. Moreover, the minimal duration and amplitude of an external stimulus required for off-to-on state transition are increased and, as a result, external noises can more effectively be filtered out. Hence, interlinked MIPF systems can realize more robust cellular memories with respect to both parameter perturbations and external noises. Our study suggests that interlinked MIPF systems might be an evolutionary consequence acquired for a more reliable memory effect by enhancing robustness against noisy cellular environments.

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.

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.

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.

Receptor-mediated and intrinsic polarization and their interaction in chemotaxing cells

Polarization--the clear and persistent localization of different signaling molecules to opposite ends of the cell-is critical for effective chemotaxis in eukaryotic systems. In many systems, polarization can also occur without an externally imposed chemical gradient. We build a modeling framework to study the relationship between the intrinsic capacity for polarization, and that induced by an external gradient. Working within this framework, we analyze different scenarios for the interaction of these pathways. The models are qualitatively simplified, motivated by known properties of the signaling pathways. We also examine the possible role of nonlinear transitions occurring in the polarization pathways. The modeling framework generates testable predictions regarding the relationship between intrinsic polarization and that induced during chemotaxis, and is the first step toward a systematic analysis of the interaction between these pathways.

Polarization and Movement of Keratocytes: A Multiscale Modelling Approach

Eukariotic cell motility is a complex phenomenon, in which the cytoskeleton and its major constituent, actin, play an essential role. Actin forms polymers of long, stiff filaments that are cross-linked into an anisotropic network inside a thin sheet-like cellular protrusion, the lamellipod. At the leading edge of this structure, polymerization of actin filaments creates the force that pushes out the membrane and leads to translocation of a motile cell. Dynamics of the actin network account for changes in cell shape, crawling motion and turning of the cell in response to external cues. Regulating the dynamics of the cytoskeleton, and playing a central role in signal transduction in the cell, are Cdc42, Rac and Rho (GTPases of the rho family, collectively known as the small G-proteins) and the actin nucleating complex, Arp2/3. In this paper, we use a multiscale modelling approach in a 2D model of a motile cell. We describe the mutual interactions of the small G-proteins, and their effects on capping and side-branching of actin filaments. We incorporate the pushing exerted by oriented actin filament ends on the cell edge, and a Rho-dependent contraction force. Combining these biochemical and mechanical aspects, we investigate the dynamics of a model epidermal fish keratocyte through in silico experiments. Our model gives insight into how, in response to some cue, a cell can polarize, form a leading edge, and move; concomitantly it explains how a keratocyte cell can maintain its shape and polarity, even after removal of the initial stimulus, and how it can change direction quickly in response to changes in its environment. We show that establishment of polarity stems from interactions of Cdc42, Rac and Rho, while maintenance and robustness of polarity is due to the rapid cytosolic diffusion of the inactive (GDI-bound) forms of the small G-proteins. Our model produces a cell shape that closely resembles the keratocytes and correct speeds for biologically reasonable parameter values. Movies of the simulations can be obtained from http://theory.bio.uu.nl/stan/keratocyte.