Modeling the cell's guidance system

Three-dimensional stochastic simulation of chemoattractant-mediated excitability in cells

During the last decade, a consensus has emerged that the stochastic triggering of an excitable system drives pseudopod formation and subsequent migration of amoeboid cells. The presence of chemoattractant stimuli alters the threshold for triggering this activity and can bias the direction of migration. Though noise plays an important role in these behaviors, mathematical models have typically ignored its origin and merely introduced it as an external signal into a series of reaction-diffusion equations. Here we consider a more realistic description based on a reaction-diffusion master equation formalism to implement these networks. In this scheme, noise arises naturally from a stochastic description of the various reaction and diffusion terms. Working on a three-dimensional geometry in which separate compartments are divided into a tetrahedral mesh, we implement a modular description of the system, consisting of G-protein coupled receptor signaling (GPCR), a local excitation-global inhibition mechanism (LEGI), and signal transduction excitable network (STEN). Our models implement detailed biochemical descriptions whenever this information is available, such as in the GPCR and G-protein interactions. In contrast, where the biochemical entities are less certain, such as the LEGI mechanism, we consider various possible schemes and highlight the differences between them. Our simulations show that even when the LEGI mechanism displays perfect adaptation in terms of the mean level of proteins, the variance shows a dose-dependence. This differs between the various models considered, suggesting a possible means for determining experimentally among the various potential networks. Overall, our simulations recreate temporal and spatial patterns observed experimentally in both wild-type and perturbed cells, providing further evidence for the excitable system paradigm. Moreover, because of the overall importance and ubiquity of the modules we consider, including GPCR signaling and adaptation, our results will be of interest beyond the field of directed migration.

Though the term noise usually carries negative connotations, it can also contribute positively to the characteristic dynamics of a system. In biological systems, where noise arises from the stochastic interactions between molecules, its study is usually confined to genetic regulatory systems in which copy numbers are small and fluctuations large. However, noise can have important roles when the number of signaling molecules is large. The extension of pseudopods and the subsequent motion of amoeboid cells arises from the noise-induced trigger of an excitable system. Chemoattractant signals bias this triggering thereby directing cell motion. To date, this paradigm has not been tested by mathematical models that account accurately for the noise that arises in the corresponding reactions. In this study, we employ a reaction-diffusion master equation approach to investigate the effects of noise. Using a modular approach and a three-dimensional cell model with specific subdomains attributed to the cell membrane and cortex, we explore the spatiotemporal dynamics of the system. Our simulations recreate many experimentally-observed cell behaviors thereby supporting the biased-excitable network hypothesis.

Mechanistic models of PLC/PKC signaling implicate phosphatidic acid as a key amplifier of chemotactic gradient sensing

Chemotaxis of fibroblasts and other mesenchymal cells is critical for embryonic development and wound healing. Fibroblast chemotaxis directed by a gradient of platelet-derived growth factor (PDGF) requires signaling through the phospholipase C (PLC)/protein kinase C (PKC) pathway. Diacylglycerol (DAG), the lipid product of PLC that activates conventional PKCs, is focally enriched at the up-gradient leading edge of fibroblasts responding to a shallow gradient of PDGF, signifying polarization. To explain the underlying mechanisms, we formulated reaction-diffusion models including as many as three putative feedback loops based on known biochemistry. These include the previously analyzed mechanism of substrate-buffering by myristoylated alanine-rich C kinase substrate (MARCKS) and two newly considered feedback loops involving the lipid, phosphatidic acid (PA). DAG kinases and phospholipase D, the enzymes that produce PA, are identified as key regulators in the models. Paradoxically, increasing DAG kinase activity can enhance the robustness of DAG/active PKC polarization with respect to chemoattractant concentration while decreasing their whole-cell levels. Finally, in simulations of wound invasion, efficient collective migration is achieved with thresholds for chemotaxis matching those of polarization in the reaction-diffusion models. This multi-scale modeling framework offers testable predictions to guide further study of signal transduction and cell behavior that affect mesenchymal chemotaxis.

Cell movement directed by external gradients of chemical composition is critical for immune responses, wound healing, and development. Although theoretical concepts explaining how shallow external gradients might definitively polarize a cell’s motility have been offered over the past two decades, mathematical models cast in terms of defined molecules and mechanisms are uncommon in this context. Based on both recent and older insights from the literature, we offer mechanistic models that are able to explain experimentally observed polarization of signal transduction elicited by shallow attractant gradients. A novel insight of our models is the implicated role of phosphatidic acid, a membrane lipid produced by at least two enzymatic pathways, in two positive feedback loops that amplify signal transduction locally. In separate simulations, we explored the implications of polarization for efficient cell invasion during wound healing. We expected that the ability to polarize in response to shallow gradients would enhance the speed of wound invasion, but an unexpected finding is that this property can promote intermittent polarization throughout the wound.

The engineering principles of combining a transcriptional incoherent feedforward loop with negative feedback

Background

Regulation of gene expression is of paramount importance in all living systems. In the past two decades, it has been discovered that certain motifs, such as the feedforward motif, are overrepresented in gene regulatory circuits. Feedforward loops are also ubiquitous in process control engineering, and are nearly always structured so that one branch has the opposite effect of the other, which is a structure known as an “incoherent” feedforward loop in biology. In engineered systems, feedforward control loops are subject to several engineering constraints, including that (1) they are finely-tuned so that the system returns to the original steady state after a disturbance occurs (perfect adaptation), (2) they are typically only implemented in the combination with negative feedback, and (3) they can greatly improve the stability and dynamical characteristics of the conjoined negative feedback loop. On the other hand, in biology, incoherent feedforward loops can serve many purposes, one of which may be perfect adaptation. It is an open question as to whether those that achieve perfect adaptation are subject to the above engineering principles.

Results

We analyzed an incoherent feedforward gene regulatory motif from the standpoint of the above engineering principles. In particular, we showed that an incoherent feedforward loop Type 1 (I1-FFL), from within a gene regulatory circuit, can be finely-tuned for perfect adaptation after a stimulus, and that the robustness of this behavior is increased by the presence of moderate negative feedback. In addition, we analyzed the advantages of adding a feedforward loop to a system that already operated under negative feedback, and found that the dynamical properties of the combined feedforward/feedback system were superior.

Conclusions

Our analysis shows that many of the engineering principles used in engineering design of feedforward control are also applicable to feedforward loops in biological systems. We speculate that principles found in other domains of engineering may also be applicable to analogous structures in biology.

Electronic supplementary material

The online version of this article (10.1186/s13036-019-0190-3) contains supplementary material, which is available to authorized users.

The threshold of an excitable system serves as a control mechanism for noise filtering during chemotaxis

Chemotaxis, the migration of cells in the direction of a chemical gradient, is of utmost importance in various biological processes. In recent years, research has demonstrated that the underlying mechanism that controls cell migration is an excitable network. One of the properties that characterizes excitability is the presence of a threshold for activation. Here, we show that excitable systems possess noise filtering capabilities that enable faster and more efficient directed migration compared to other systems that also include a threshold, such as ultrasensitive switches. We demonstrate that this filtering ability is a consequence of the varying responses of excitable systems to step and pulse stimuli. Whereas the response to step inputs is determined solely by the magnitude of the stimulus, for pulse stimuli, the response depends on both the magnitude and duration of the stimulus. We then show that these two forms of threshold behavior can be decoupled from one another, allowing finer control in chemotaxis. Finally, we use a simple model of chemotaxis to demonstrate that cells that rely on an excitable system display faster and more effective directed migration that a hypothetical cell guided by an ultra-sensitive switch.

A model for the integration of conflicting exogenous and endogenous signals by dendritic cells

Cells of the immune system are confronted with opposing pro- and anti-inflammatory signals. Dendritic cells (DC) integrate these cues to make informed decisions whether to initiate an immune response. Confronted with exogenous microbial stimuli, DC endogenously produce both anti- (IL-10) and pro-inflammatory (TNFα) cues whose joint integration controls the cell's final decision. Backed by experimental measurements we present a theoretical model to quantitatively describe the integration mode of these opposing signals. We propose a two step integration model that modulates the effect of the two types of signals: an initial bottleneck integrates both signals (IL-10 and TNFα), the output of which is later modulated by the anti-inflammatory signal. We show that the anti-inflammatory IL-10 signaling is long ranged, as opposed to the short-ranged pro-inflammatory TNFα signaling. The model suggests that the population averaging and modulation of the pro-inflammatory response by the anti-inflammatory signal is a safety guard against excessive immune responses.

A Model for Direction Sensing in Dictyostelium discoideum: Ras Activity and Symmetry Breaking Driven by a Gβγ-Mediated, Gα2-Ric8 -- Dependent Signal Transduction Network

Chemotaxis is a dynamic cellular process, comprised of direction sensing, polarization and locomotion, that leads to the directed movement of eukaryotic cells along extracellular gradients. As a primary step in the response of an individual cell to a spatial stimulus, direction sensing has attracted numerous theoretical treatments aimed at explaining experimental observations in a variety of cell types. Here we propose a new model of direction sensing based on experiments using Dictyostelium discoideum (Dicty). The model is built around a reaction-diffusion-translocation system that involves three main component processes: a signal detection step based on G-protein-coupled receptors (GPCR) for cyclic AMP (cAMP), a transduction step based on a heterotrimetic G protein Gα2βγ, and an activation step of a monomeric G-protein Ras. The model can predict the experimentally-observed response of cells treated with latrunculin A, which removes feedback from downstream processes, under a variety of stimulus protocols. We show that [Formula: see text] cycling modulated by Ric8, a nonreceptor guanine exchange factor for [Formula: see text] in Dicty, drives multiple phases of Ras activation and leads to direction sensing and signal amplification in cAMP gradients. The model predicts that both [Formula: see text] and Gβγ are essential for direction sensing, in that membrane-localized [Formula: see text], the activated GTP-bearing form of [Formula: see text], leads to asymmetrical recruitment of RasGEF and Ric8, while globally-diffusing Gβγ mediates their activation. We show that the predicted response at the level of Ras activation encodes sufficient 'memory' to eliminate the 'back-of-the wave' problem, and the effects of diffusion and cell shape on direction sensing are also investigated. In contrast with existing LEGI models of chemotaxis, the results do not require a disparity between the diffusion coefficients of the Ras activator GEF and the Ras inhibitor GAP. Since the signal pathways we study are highly conserved between Dicty and mammalian leukocytes, the model can serve as a generic one for direction sensing.

Cell-cell communication enhances the capacity of cell ensembles to sense shallow gradients during morphogenesis

Collective cell responses to exogenous cues depend on cell-cell interactions. In principle, these can result in enhanced sensitivity to weak and noisy stimuli. However, this has not yet been shown experimentally, and little is known about how multicellular signal processing modulates single-cell sensitivity to extracellular signaling inputs, including those guiding complex changes in the tissue form and function. Here we explored whether cell-cell communication can enhance the ability of cell ensembles to sense and respond to weak gradients of chemotactic cues. Using a combination of experiments with mammary epithelial cells and mathematical modeling, we find that multicellular sensing enables detection of and response to shallow epidermal growth factor (EGF) gradients that are undetectable by single cells. However, the advantage of this type of gradient sensing is limited by the noisiness of the signaling relay, necessary to integrate spatially distributed ligand concentration information. We calculate the fundamental sensory limits imposed by this communication noise and combine them with the experimental data to estimate the effective size of multicellular sensory groups involved in gradient sensing. Functional experiments strongly implicated intercellular communication through gap junctions and calcium release from intracellular stores as mediators of collective gradient sensing. The resulting integrative analysis provides a framework for understanding the advantages and limitations of sensory information processing by relays of chemically coupled cells.

Signaling networks that regulate cell migration

SUMMARY

Stimuli that promote cell migration, such as chemokines, cytokines, and growth factors in metazoans and cyclic AMP in Dictyostelium, activate signaling pathways that control organization of the actin cytoskeleton and adhesion complexes. The Rho-family GTPases are a key convergence point of these pathways. Their effectors include actin regulators such as formins, members of the WASP/WAVE family and the Arp2/3 complex, and the myosin II motor protein. Pathways that link to the Rho GTPases include Ras GTPases, TorC2, and PI3K. Many of the molecules involved form gradients within cells, which define the front and rear of migrating cells, and are also established in related cellular behaviors such as neuronal growth cone extension and cytokinesis. The signaling molecules that regulate migration can be integrated to provide a model of network function. The network displays biochemical excitability seen as spontaneous waves of activation that propagate along the cell cortex. These events coordinate cell movement and can be biased by external cues to bring about directed migration.

Cellular memory in eukaryotic chemotaxis

Natural chemical gradients to which cells respond chemotactically are often dynamic, with both spatial and temporal components. A primary example is the social amoeba Dictyostelium, which migrates to the source of traveling waves of chemoattractant as part of a self-organized aggregation process. Despite its physiological importance, little is known about how cells migrate directionally in response to traveling waves. The classic back-of-the-wave problem is how cells chemotax toward the wave source, even though the spatial gradient reverses direction in the back of the wave. Here, we address this problem by using microfluidics to expose cells to traveling waves of chemoattractant with varying periods. We find that cells exhibit memory and maintain directed motion toward the wave source in the back of the wave for the natural period of 6 min, but increasingly reverse direction for longer wave periods. Further insights into cellular memory are provided by experiments quantifying cell motion and localization of a directional-sensing marker after rapid gradient switches. The results can be explained by a model that couples adaptive directional sensing to bistable cellular memory. Our study shows how spatiotemporal cues can guide cell migration over large distances.

Adaptive molecular networks controlling chemotactic migration: dynamic inputs and selection of the network architecture

Eukaryotic signalling networks underlying the cell's ability to sense the gradient of chemotactic cues frequently have the dual property of perfect adaptation to spatially homogeneous inputs, and persistent activation by inputs that are spatially graded. This property is also shared by bacterial chemotaxis networks, raising the question of whether these two types of chemotactic processes also have similar organization of the underlying biomolecular processes. Interestingly, perfect adaptation can only be achieved robustly by a handful of mechanisms, and while eukaryotic chemotactic networks appear to rely on one of these-the incoherent feed-forward loop, bacterial chemotaxis depends on another-the negative feedback loop. In this review, we discuss how this conclusion can be reached even if the details of the molecular networks are incompletely understood. Furthermore, we argue that the use of distinct network architectures is not accidental and may be a consequence of the nature of the signalling inputs and the limitations of the sensory properties of different cell types.

Spatial stochastic dynamics enable robust cell polarization

Although cell polarity is an essential feature of living cells, it is far from being well-understood. Using a combination of computational modeling and biological experiments we closely examine an important prototype of cell polarity: the pheromone-induced formation of the yeast polarisome. Focusing on the role of noise and spatial heterogeneity, we develop and investigate two mechanistic spatial models of polarisome formation, one deterministic and the other stochastic, and compare the contrasting predictions of these two models against experimental phenotypes of wild-type and mutant cells. We find that the stochastic model can more robustly reproduce two fundamental characteristics observed in wild-type cells: a highly polarized phenotype via a mechanism that we refer to as spatial stochastic amplification, and the ability of the polarisome to track a moving pheromone input. Moreover, we find that only the stochastic model can simultaneously reproduce these characteristics of the wild-type phenotype and the multi-polarisome phenotype of a deletion mutant of the scaffolding protein Spa2. Significantly, our analysis also demonstrates that higher levels of stochastic noise results in increased robustness of polarization to parameter variation. Furthermore, our work suggests a novel role for a polarisome protein in the stabilization of actin cables. These findings elucidate the intricate role of spatial stochastic effects in cell polarity, giving support to a cellular model where noise and spatial heterogeneity combine to achieve robust biological function.

Cell polarity is the fundamental process of breaking symmetry to create asymmetric cellular structures. It is an open question how randomness (stochasticity) in the cell hinders or helps cell polarity. In this work, we focus on the ability of yeast cells to sense a spatial gradient of mating pheromone and respond by forming a projection in the direction of the mating partner. A key element is the polarisome, which is at the tip of the mating projection. We introduce the first model of polarisome formation in yeast. The model is well-supported by experimental data. We perform modeling to explore the role of noise in the formation of the polarisome. By running simulations with and without noise, we arrive at the surprising conclusion, that gradient-dependent polarization is enhanced by stochasticity. Both the tight localization (amplification) and the ability to respond to directional change of the input (tracking) are enhanced by stochastic dynamics, resulting in a more robust behavior. Mutants in which key polarisome proteins have been deleted exhibit broader, noisier polarisome than the wild type. The mutant phenotype is accurately captured by our stochastic simulations. These results demonstrate the importance of stochasticity in the study of cell polarity.

Gradient sensing during chemotaxis

Eukaryotic cells have the ability to sense chemoattractant gradients and to migrate toward the sources of attractants. The chemical gradient-guided cell movement is referred to as chemotaxis. Chemoattractants are detected by members of G-protein-coupled receptors (GPCRs) that link to heterotrimeric G-proteins. The GPCR/G-protein sensing machinery is able to translate external chemoattractants fields into intercellular cues, which direct reorganization of the actin cytoskeleton that drives cell movement. Here, I review our current understanding of the formation of chemoattractant gradients in vivo, the GPCR-mediated gradient sensing, and the sophisticated signaling network that guides the function of the actin cytoskeleton.

Chemoattractant signaling in dictyostelium: adaptation and amplification

The social amoebae Dictyostelium discoideum has long proved a powerful model organism for studying how cells sense and interpret chemoattractant gradients. Because of the rich behavior observed in its response to chemoattractants, as well as the complex nature of the signaling pathways involved, this research has attracted and benefited from the use of theoretical models. Recent quantitative experiments provide support for a popular model: the local excitation, global inhibition mechanism of gradient sensing. Here, I discuss these findings and suggest some important open problems.

Diverse sensitivity thresholds in dynamic signaling responses by social amoebae

The complex transition from a single-cell to a multicellular life form during the formation of a fruiting body by the amoeba Dictyostelium discoideum is accompanied by a pulsatile collective signaling process that instigates chemotaxis of the constituent cells. Although the cells used for the analysis of this phenomenon are normally genetically identical (isogenic), it is not clear whether they are equally responsive to the waves of the signaling stimulus, nor is it clear how responses across the population influence collective cell behavior. Here, we found that isogenic Dictyostelium cells displayed differing sensitivities to the chemoattractant cyclic adenosine monophosphate (cAMP). Furthermore, the resulting signaling responses could be explained by a model in which cells are refractory to further stimulation for 5 to 6 min after the initial input and the signaling output is amplified, with the amplification threshold varying across the cells in the population. This pathway structure could explain intracellular amplification of the chemoattractant gradient during cell migration. The new model predicts that diverse cell responsiveness can facilitate collective cell behavior, specifically due to the presence of a small number of cells in the population with increased responsiveness that aid in propagating the initial cAMP signaling wave across the cell population.

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.

Modeling cell gradient sensing and migration in competing chemoattractant fields

Directed cell migration mediates physiological and pathological processes. In particular, immune cell trafficking in tissues is crucial for inducing immune responses and is coordinated by multiple environmental cues such as chemoattractant gradients. Although the chemotaxis mechanism has been extensively studied, how cells integrate multiple chemotactic signals for effective trafficking and positioning in tissues is not clearly defined. Results from previous neutrophil chemotaxis experiments and modeling studies suggested that ligand-induced homologous receptor desensitization may provide an important mechanism for cell migration in competing chemoattractant gradients. However, the previous mathematical model is oversimplified to cell gradient sensing in one-dimensional (1-D) environment. To better understand the receptor desensitization mechanism for chemotactic navigation, we further developed the model to test the role of homologous receptor desensitization in regulating both cell gradient sensing and migration in different configurations of chemoattractant fields in two-dimension (2-D). Our results show that cells expressing normal desensitizable receptors preferentially orient and migrate toward the distant gradient in the presence of a second local competing gradient, which are consistent with the experimentally observed preferential migration of cells toward the distant attractant source and confirm the requirement of receptor desensitization for such migratory behaviors. Furthermore, our results are in qualitative agreement with the experimentally observed cell migration patterns in different configurations of competing chemoattractant fields.

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.

GPCR-controlled chemotaxis in Dictyostelium discoideum

Dictyostelium discoideum has been chosen as the key model organism for the study of eukaryotic chemotaxis. Studies in this lower eukaryotic organism have allowed us to discover eukaryotic chemotaxis behavior and to gradually understand the mechanism of chemotaxis. Investigations in this simple organism often guide the direction of chemotaxis studies in areas such as forming concepts, discovering molecular components, revealing pathways and networks. The cooperation between experimental approaches and computational modeling has helped us to comprehend the signaling network as a system. To further reveal the relationships among the molecular mechanisms of individual signaling steps, a continuous interplay between model development and refinement and experimental testing and verification will be useful. This article focuses on a chemoattractant G-protein-coupled receptor (GPCR)/G-protein gradient sensing machinery, which is monitored by PIP(3) responses and investigated by the interplay between live cell imaging experiments and computational modeling. We believe that such an approach will lead to a much better understanding of GPCR-controlled chemotaxis of all eukaryotic cells.

Models at the single cell level

Many cellular behaviors cannot be completely captured or appropriately described at the cell population level. Noise induced by stochastic chemical reactions, spatially polarized signaling networks, and heterogeneous cell-cell communication are among the many phenomena that require fine-grained analysis. Accordingly, the mathematical models used to describe such systems must be capable of single cell or subcellular resolution. Here, we review techniques for modeling single cells, including models of stochastic chemical kinetics, spatially heterogeneous intracellular signaling, and spatial stochastic systems. We also briefly discuss applications of each type of model.

Tools for analyzing cell shape changes during chemotaxis

Chemotaxis refers to the ability of cells to sense the direction of external chemical gradients and respond by migrating towards the source. A thorough understanding of the chemotactic response of amoebae and neutrophils requires careful quantification of the cell shape changes observed during cell movement. The stochastic nature of this response calls for a statistical characterization of cellular morphology and this requires the processing of large data sets. For this reason, automatic image analysis algorithms are highly desirable and are becoming increasingly available. These usually include a combination of techniques from image segmentation, morphological transformations, as well as the incorporation of numerical algorithms and physical models. Here we review recent developments in the tracking and understanding of motile chemotaxing cells, with a particular emphasis on the description of pseudopodial activity in chemotactic Dictyostelium cells.

Coupling mechanism of a GPCR and a heterotrimeric G protein during chemoattractant gradient sensing in Dictyostelium

The coupling of heterotrimeric guanine nucleotide-binding protein (G protein)-coupled receptors (GPCRs) with G proteins is fundamental for GPCR signaling; however, the mechanism of coupling is still debated. Moreover, how the proposed mechanisms affect the dynamics of downstream signaling remains unclear. Here, through experiments involving fluorescence recovery after photobleaching and single-molecule imaging, we directly measured the mobilities of cyclic adenosine monophosphate (cAMP) receptor 1 (cAR1), a chemoattractant receptor, and a G protein βγ subunit in live cells. We found that cAR1 diffused more slowly in the plasma membrane than did Gβγ. Upon binding of ligand to the receptor, the mobility of cAR1 was unchanged, whereas the speed of a fraction of the faster-moving Gβγ subunits decreased. Our measurements showed that cAR1 was relatively immobile and Gβγ diffused freely, suggesting that chemoattractant-bound cAR1 transiently interacted with G proteins. Using models of possible coupling mechanisms, we computed the temporal kinetics of G protein activation. Our fluorescence resonance energy transfer imaging data showed that fully activated cAR1 induced the sustained dissociation of G protein α and βγ subunits, which indicated that ligand-bound cAR1 activated G proteins continuously. Finally, simulations indicated that a high-affinity coupling of ligand-bound receptors and G proteins was essential for cAR1 to translate extracellular gradient signals into directional cellular responses. We suggest that chemoattractant receptors use a ligand-induced coupling rather than a precoupled mechanism to control the activation of G proteins during chemotaxis.

The role of Hopf bifurcation dynamics in sensory processes

We emphasize here the role of the Hopf bifurcation in detection of stimuli in sensory processes--we discuss in particular chemosensors. It is shown that the essential nonlinearities inherent in the signal transduction mechanism can take advantage of the noise from the environment the system is subject to, to display a highly amplified response to stimuli in a frequency-selective manner. It is shown that in the absence of any externally applied stimulus, the feedback mechanisms playing a regulatory role in the transduction mechanism can give rise, in the presence of noise, to peaks in the spectral power density, suggesting enhanced spontaneous activity in sensory cells. The power law in this spectrum is determined.

Functional motifs in biochemical reaction networks

The signal-response characteristics of a living cell are determined by complex networks of interacting genes, proteins, and metabolites. Understanding how cells respond to specific challenges, how these responses are contravened in diseased cells, and how to intervene pharmacologically in the decision-making processes of cells requires an accurate theory of the information-processing capabilities of macromolecular regulatory networks. Adopting an engineer's approach to control systems, we ask whether realistic cellular control networks can be decomposed into simple regulatory motifs that carry out specific functions in a cell. We show that such functional motifs exist and review the experimental evidence that they control cellular responses as expected.

Cell stimulation with optically manipulated microsources

Molecular gradients are important for various biological processes including the polarization of tissues and cells during embryogenesis and chemotaxis. Investigations of these phenomena require control over the chemical microenvironment of cells. We present a technique to set up molecular concentration patterns that are chemically, spatially and temporally flexible. Our strategy uses optically manipulated microsources, which steadily release molecules. Our technique enables the control of molecular concentrations over length scales down to about 1 microm and timescales from fractions of a second to an hour. We demonstrate this technique by manipulating the motility of single human neutrophils. We induced directed cell polarization and migration with microsources loaded with the chemoattractant formyl-methionine-leucine-phenylalanine. Furthermore, we triggered highly localized retraction of lamellipodia and redirection of polarization and migration with microsources releasing cytochalasin D, an inhibitor of actin polymerization.

Modeling the role of homologous receptor desensitization in cell gradient sensing

G-protein-coupled chemoattractant receptors signal transiently upon ligand binding to effect cell orientation and motility but then are rapidly desensitized. The importance of desensitization has been unclear, because mutated nondesensitizable receptors mediate efficient chemotaxis. We hypothesized that homologous receptor desensitization is required for cellular navigation in fields of competing attractants. Modeling of receptor-mediated orientation shows that desensitization allows integration of attractant signals. Cells expressing normal receptors are predicted to 1) orient preferentially to distant gradients; 2) seek an intermediate position between balanced agonist sources; 3) and can be repositioned between chemoattractant-defined microenvironmental domains by modest changes in receptor number. In contrast, in the absence of desensitization, orientation is dominated by local agonist sources, precluding continued navigation. Furthermore, cell orientation in competing ligand gradients depends on the relative kinetic rates of receptor desensitization and recycling. We propose that homologous receptor desensitization is critical for cellular navigation in complex chemoattractant fields.

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.

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.

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.

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.

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.

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.

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.

Key role of local regulation in chemosensing revealed by a new molecular interaction-based modeling method

The signaling network underlying eukaryotic chemosensing is a complex combination of receptor-mediated transmembrane signals, lipid modifications, protein translocations, and differential activation/deactivation of membrane-bound and cytosolic components. As such, it provides particularly interesting challenges for a combined computational and experimental analysis. We developed a novel detailed molecular signaling model that, when used to simulate the response to the attractant cyclic adenosine monophosphate (cAMP), made nontrivial predictions about Dictyostelium chemosensing. These predictions, including the unexpected existence of spatially asymmetrical, multiphasic, cyclic adenosine monophosphate–induced PTEN translocation and phosphatidylinositol-(3,4,5)P₃ generation, were experimentally verified by quantitative single-cell microscopy leading us to propose significant modifications to the current standard model for chemoattractant-induced biochemical polarization in this organism. Key to this successful modeling effort was the use of “Simmune,” a new software package that supports the facile development and testing of detailed computational representations of cellular behavior. An intuitive interface allows user definition of complex signaling networks based on the definition of specific molecular binding site interactions and the subcellular localization of molecules. It automatically translates such inputs into spatially resolved simulations and dynamic graphical representations of the resulting signaling network that can be explored in a manner that closely parallels wet lab experimental procedures. These features of Simmune were critical to the model development and analysis presented here and are likely to be useful in the computational investigation of many aspects of cell biology.

Cells can orient their migration in response to small local differences in the concentration of extracellular chemicals (chemoattractants). Understanding this process (chemosensing) requires analyzing the time and position-dependent behavior of the signaling molecules within the responding cell, making it an especially interesting challenge for both experimental and computational investigation. Here, the authors report the development and testing of a new detailed molecular model of the chemosensing apparatus of the amoeba Dictyostelium discoidium reacting to the chemoattractant cyclic adenosine monophosphate. Computer simulations performed using this model predicted unexpected and previously unreported patterns of changes in the concentration and location of two important intracellular signaling molecules. These predictions were experimentally verified using microscopy, suggesting the need for modifications to the current “standard” model of eukaryotic chemosensing. The high degree of detail in their model was made possible by a new software suite called “Simmune,” which allows biologists to enter information about molecular interactions using a graphical interface. Without requiring the user to write any equations, the software automatically constructs the overall reaction network, simulates the model, and provides several ways to view the biochemistry of simulated cells. This new tool should help biologists to translate qualitative representations of cell biological processes into quantitative, predictive models.

The Multiple Personalities of the Chemokine Receptor CCR7 in Dendritic Cells

CCR7 was described initially as a potent leukocyte chemotactic receptor that was later shown to be responsible of directing the migration of dendritic cells (DCs) to the lymph nodes where these cells play an important role in the initiation of the immune response. Recently, a variety of reports have indicated that, apart from chemotaxis, CCR7 controls the cytoarchitecture, the rate of endocytosis, the survival, the migratory speed, and the maturation of the DCs. Some of these functions of CCR7 and additional ones also have been described in other cell types. Herein we discuss how this receptor may contribute to modulate the immune response by regulating different functions in DCs. Finally, we also suggest a possible mechanism whereby CCR7 may control its multiple tasks in these cells.

Positive feedback may cause the biphasic response observed in the chemoattractant-induced response of Dictyostelium cells

After stimulation by chemoattractant, Dictyostelium cells exhibit a rapid response. The concentrations of several intracellular proteins rise rapidly reaching their maximum levels approximately 5-10 seconds, after which they return to prestimulus levels. This response, which is found in many other chemotaxing cells, is an example of a step disturbance rejection, a process known to biologists as perfect adaptation. Unlike other cells, however, the initial first peak observed in the chemoattractant-induced response of Dictyostelium cells is then followed by a slower, smaller phase peaking approximately one to two minutes after the stimulus. Until recently, the nature of this biphasic response has been poorly understood. Moreover, the origin for the second phase is unknown. In this paper we conjecture the existence of a feedback path between the response and stimulus. Using a mathematical model of the chemoattractant-induced response in cells, and standard tools from control engineering, we show that positive feedback may elicit this second peak.

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.

A hybrid systems framework for cellular processes

With the availability of technologies that allow us to obtain stimulus-response time series data for modeling and system identification, there is going to be an increasing need for conceptual frameworks in which to formulate and test hypotheses about intra- and inter-cellular dynamics, in general and not just dependent on a particular cell line, cell type, organism, or technology. While the semantics can be quite different, biologists and systems scientists use in many cases a similar language (notion of feedback, regulation, etc.). A more abstract system-theoretic framework for signals, systems, and control could provide the biologist with an interface between the domains. Apart from recent examples to identify functional elements and describing them in engineering terms, there have been various more abstract developments to describe dynamics at the cell level in the past. This includes Rosen's (M,R)-systems. This paper presents an abstract and general compact mathematical framework of intracellular dynamics, regulation and regime switching inspired by (M,R)-theory and based on hybrid automata.

Quantitative imaging of single live cells reveals spatiotemporal dynamics of multistep signaling events of chemoattractant gradient sensing in Dictyostelium

Activation of G-protein-coupled chemoattractant receptors triggers dissociation of Galpha and Gbetagamma subunits. These subunits induce intracellular responses that can be highly polarized when a cell experiences a gradient of chemoattractant. Exactly how a cell achieves this amplified signal polarization is still not well understood. Here, we quantitatively measure temporal and spatial changes of receptor occupancy, G-protein activation by FRET imaging, and PIP3 levels by monitoring the dynamics of PH(Crac)-GFP translocation in single living cells in response to different chemoattractant fields. Our results provided the first direct evidence that G-proteins are activated to different extents on the cell surface in response to asymmetrical stimulations. A stronger, uniformly applied stimulation triggers not only a stronger G-protein activation but also a faster adaptation of downstream responses. When naive cells (which have not experienced chemoattractant) were abruptly exposed to stable cAMP gradients, G-proteins were persistently activated throughout the entire cell surface, whereas the response of PH(Crac)-GFP translocation surprisingly consisted of two phases, an initial transient and asymmetrical translocation around the cell membrane, followed by a second phase producing a highly polarized distribution of PH(Crac)-GFP. We propose a revised model of gradient sensing, suggesting an important role for locally controlled components that inhibit PI3Kinase activity.

Two complementary, local excitation, global inhibition mechanisms acting in parallel can explain the chemoattractant-induced regulation of PI(3,4,5)P3 response in dictyostelium cells

Chemotaxing cells, such as Dictyostelium and mammalian neutrophils, sense shallow chemoattractant gradients and respond with highly polarized changes in cell morphology and motility. Uniform chemoattractant stimulation induces the transient translocations of several downstream signaling components, including phosphoinositide 3-kinase (PI3K), tensin homology protein (PTEN), and phosphatidylinositol 3,4,5-trisphosphate (PI(3,4,5)P3). In contrast, static spatial chemoattractant gradients elicit the persistent, amplified localization of these molecules. We have proposed a model in which the response to chemoattractant is regulated by a balance of a local excitation and a global inhibition, both of which are controlled by receptor occupancy. This model can account for both the transient and spatial responses to chemoattractants, but alone does not amplify the external gradient. In this article, we develop a model in which parallel local excitation, global inhibition mechanisms control the membrane binding of PI3K and PTEN. Together, the action of these enzymes induces an amplified PI(3,4,5)P3 response that agrees quantitatively with experimentally obtained plekstrin homology-green fluorescent protein distributions in latrunculin-treated cells. We compare the model's performance with that of several mutants in which one or both of the enzymes are disrupted. The model accounts for the observed response to multiple, simultaneous chemoattractant cues and can recreate the cellular response to combinations of temporal and spatial stimuli. Finally, we use the model to predict the response of a cell where only a fraction is stimulated by a saturating dose of chemoattractant.

Chemoattractant-induced phosphatidylinositol 3,4,5-trisphosphate accumulation is spatially amplified and adapts, independent of the actin cytoskeleton

Experiments in amoebae and neutrophils have shown that local accumulations of phosphatidylinositol 3,4,5-trisphosphate [PI(3,4,5)P(3)] mediate the ability of cells to migrate during gradient sensing. To define the nature of this response, we subjected Dictyostelium discoideum cells to measurable temporal and spatial chemotactic inputs and analyzed the accumulation of PI(3,4,5)P(3) on the membrane, as well as the recruitment of the enzymes phosphoinositide 3-kinase and PTEN. In latrunculin-treated cells, spatial gradients elicited a PI(3,4,5)P(3) response only on the front portion of the cell where the response increased more steeply than the gradient and did not depend on its absolute concentration. Phosphoinositide 3-kinase bound to the membrane only at the front, although it was less sharply localized than PI(3,4,5)P(3). Membrane-bound PTEN was highest at the rear and varied inversely with receptor occupancy. The localization of PI(3,4,5)P(3) was enhanced further in untreated polarized cells containing an intact cytoskeleton. Interestingly, the treated cells could respond to two independent gradients simultaneously, demonstrating that a response at the front does not necessarily inhibit the back. Combinations of temporal and spatial stimuli provided evidence of an inhibitory process and showed that a gradient generates a persistent steady-state response independent of a previous history of exposure to chemoattractant. These results support a local excitation/global inhibition model and argue against other schemes proposed to explain directional sensing.

Do phosphoinositide 3-kinases direct lymphocyte navigation?

Phosphoinositide 3-kinase (PI3K)-dependent signalling pathways have been suggested to have pivotal roles in determining the polarity of leukocytes moving toward a chemotactic gradient, a process termed chemotaxis. Current perceptions concerning the role of PI3K in leukocyte migration are based predominantly around evidence derived from single-cell organisms and analysis of neutrophil migration from mice deficient in the gamma-isoform of the p110 catalytic subunit. With regard to directed T-lymphocyte migration, there is convincing evidence for the activation of PI3K isoforms in T lymphocytes by several chemokines. However, there is a growing body of evidence, which now indicates that in T lymphocytes at least, PI3K activation can be a dispensable signal for directed cell migration in certain settings. In fact, evidence is emerging that during directed cell migration, T lymphocytes use biochemical pathways distinct from those adopted by monocytes. The non-universal role of PI3K in directional cell migration and the existence of cell-specific signalling pathways for chemotactic responses has important implications for the validation of effective new targets for inflammation, where one aim is to block migration of leukocytes to the site of inflammatory lesion.

Visualizing signals moving in cells

Cells display a highly complex spatiotemporal organization, required to exert a wide variety of different functions, for example, detection, processing, and propagation of nerve impulses by neurons; contraction and relaxation by muscle cells; movement by leukocytes; and adsorption and secretion of nutrients and metabolites by epithelial cells lining the gut. Successful execution of these complex processes requires highly dynamic information transfer between different regions and compartments within cells. Through the development of fluorescent sensors for intracellular signaling molecules coupled with improved microscopic imaging techniques, it has now become possible to investigate signal propagation in cells with high spatial and temporal resolution.

Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell

The physiological responses of cells to external and internal stimuli are governed by genes and proteins interacting in complex networks whose dynamical properties are impossible to understand by intuitive reasoning alone. Recent advances by theoretical biologists have demonstrated that molecular regulatory networks can be accurately modeled in mathematical terms. These models shed light on the design principles of biological control systems and make predictions that have been verified experimentally.