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

Mating yeast cells use an intrinsic polarity site to assemble a pheromone-gradient tracking machine

The mating of budding yeast depends on chemotropism, a fundamental cellular process. The two yeast mating types secrete peptide pheromones that bind to GPCRs on cells of the opposite type. Cells find and contact a partner by determining the direction of the pheromone source and polarizing their growth toward it. Actin-directed secretion to the chemotropic growth site (CS) generates a mating projection. When pheromone-stimulated cells are unable to sense a gradient, they form mating projections where they would have budded in the next cell cycle, at a position called the default polarity site (DS). Numerous models have been proposed to explain yeast gradient sensing, but none address how cells reliably switch from the intrinsically determined DS to the gradient-aligned CS, despite a weak spatial signal. Here we demonstrate that, in mating cells, the initially uniform receptor and G protein first polarize to the DS, then redistribute along the plasma membrane until they reach the CS. Our data indicate that signaling, polarity, and trafficking proteins localize to the DS during assembly of what we call the gradient tracking machine (GTM). Differential activation of the receptor triggers feedback mechanisms that bias exocytosis upgradient and endocytosis downgradient, thus enabling redistribution of the GTM toward the pheromone source. The GTM stabilizes when the receptor peak centers at the CS and the endocytic machinery surrounds it. A computational model simulates GTM tracking and stabilization and correctly predicts that its assembly at a single site contributes to mating fidelity.

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.

Interaction of motility, directional sensing, and polarity modules recreates the behaviors of chemotaxing cells

Chemotaxis involves the coordinated action of separable but interrelated processes: motility, gradient sensing, and polarization. We have hypothesized that these are mediated by separate modules that account for these processes individually and that, when combined, recreate most of the behaviors of chemotactic cells. Here, we describe a mathematical model where the modules are implemented in terms of reaction-diffusion equations. Migration and the accompanying changes in cellular morphology are demonstrated in simulations using a mechanical model of the cell cortex implemented in the level set framework. The central module is an excitable network that accounts for random migration. The response to combinations of uniform stimuli and gradients is mediated by a local excitation, global inhibition module that biases the direction in which excitability is directed. A polarization module linked to the excitable network through the cytoskeleton allows unstimulated cells to move persistently and, for cells in gradients, to gradually acquire distinct sensitivity between front and back. Finally, by varying the strengths of various feedback loops in the model we obtain cellular behaviors that mirror those of genetically altered cell lines.

Chemotaxis is the movement of cells in response to spatial gradients of chemical cues. While single-celled organisms rely on sensing and responding to chemical gradients to search for nutrients, chemotaxis is also an essential component of the mammalian immune system. However, chemotaxis can also be deleterious, since chemotactic tumor cells can lead to metastasis. Due to its importance, understanding the process by which cells sense and respond to chemical gradients has attracted considerable interest. Moreover, because of the complexity of chemotactic signaling, which includes multiple feedback loops and redundant pathways, this has been a research area in which computational models have had a significant impact in understanding experimental findings. Here, we propose a modular description of the signaling network that regulates chemotaxis. The modules describe different processes that are observed in chemotactic cells. In addition to accounting for these behaviors individually, we show that the overall system recreates many features of the directed motion of migrating cells. The signaling described by our modules is implemented as a series of equations, whereas movement and the accompanying cellular deformations are simulated using a mechanical model of the cell and implemented using level set methods, a method that allows simulations of cells as they change morphology.

Excitable behavior in amoeboid chemotaxis

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

A computational model of cell polarization and motility coupling mechanics and biochemistry

The motion of a eukaryotic cell presents a variety of interesting and challenging problems from both a modeling and a computational perspective. The processes span many spatial scales (from molecular to tissue) as well as disparate time scales, with reaction kinetics on the order of seconds, and the deformation and motion of the cell occurring on the order of minutes. The computational difficulty, even in 2D, resides in the fact that the problem is inherently one of deforming, non-stationary domains, bounded by an elastic perimeter, inside of which there is redistribution of biochemical signaling substances. Here we report the results of a computational scheme using the immersed boundary method to address this problem. We adopt a simple reaction-diffusion system that represents an internal regulatory mechanism controlling the polarization of a cell, and determining the strength of protrusion forces at the front of its elastic perimeter. Using this computational scheme we are able to study the effect of protrusive and elastic forces on cell shapes on their own, the distribution of the reaction-diffusion system in irregular domains on its own, and the coupled mechanical-chemical system. We find that this representation of cell crawling can recover important aspects of the spontaneous polarization and motion of certain types of crawling cells.

Signaling pathways that control cell migration: models and analysis

Dissecting the intracellular signaling mechanisms that govern the movement of eukaryotic cells presents a major challenge, not only because of the large number of molecular players involved, but even more so because of the dynamic nature of their regulation by both biochemical and mechanical interactions. Computational modeling and analysis have emerged as useful tools for understanding how the physical properties of cells and their microenvironment are coupled with certain biochemical pathways to actuate and control cell motility. In this focused review, we highlight some of the more recent applications of quantitative modeling and analysis in the field of cell migration. Both in modeling and experiment, it has been prudent to follow a reductionist approach in order to characterize what are arguably the principal modules: spatial polarization of signaling pathways, regulation of the actin cytoskeleton, and dynamics of focal adhesions. While it is important that we 'cut our teeth' on these subsystems, focusing on the details of certain aspects while ignoring or coarse-graining others, it is clear that the challenge ahead will be to characterize the couplings between them in an integrated framework.

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.

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.

A mathematical modelling framework for understanding chemorepulsive signal transduction in Dictyostelium

Chemorepulsion is the process by which an organism or a cell moves in the direction of decreasing chemical concentration. While a few experimental studies have been performed, no mathematical models exist for this process. In this paper we have modelled gradient sensing, the first subprocess of chemorepulsion, in Dictyostelium discoideum-a well characterized model eukaryotic system. We take the first steps towards achieving a comprehensive mechanistic understanding of chemorepulsion in this system. We have used, as a basis, the biochemical network of the Keizer-Gunnink et al. (2007) to develop the mathematical modelling framework. This network describes the underlying pathways of chemorepellent gradient sensing in D. discoideum. Working within this modelling framework we address whether the postulated interactions of the pathways and species in this network can lead to a chemorepulsive response. We also analyse the possible role of additional regulatory effects (such as additional receptor regulation of enzymes in this network) and if this is necessary to achieve this behaviour. Thus we have investigated the receptor regulation of important enzymes and feedback effects in the network. This modelling framework generates important insights into and testable predictions regarding the role of key components and feedback loops in regulating chemorepulsive gradient sensing, and what factors might be important for generating a chemorepulsive response; it serves as a first step towards a comprehensive mechanistic understanding of this process.

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.

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.

On the spontaneous emergence of cell polarity

Diverse cell polarity networks require positive feedback for locally amplifying distributions of signalling molecules at the plasma membrane. Additional mechanisms, such as directed transport or coupled inhibitors, have been proposed to be required for reinforcing a unique axis of polarity. Here we analyse a simple model of positive feedback, with strong analogy to the 'stepping stone' model of population genetics, in which a single species of diffusible, membrane-bound signalling molecules can self-recruit from a cytoplasmic pool. We identify an intrinsic stochastic mechanism through which positive feedback alone is sufficient to account for the spontaneous establishment of a single site of polarity. We find that the polarization frequency has an inverse dependence on the number of signalling molecules: the frequency of polarization decreases as the number of molecules becomes large. Experimental observation of polarizing Cdc42 in budding yeast is consistent with this prediction. Our work suggests that positive feedback can work alone or with additional mechanisms to create robust cell polarity.

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.