Deterministic model of dermal wound invasion incorporating receptor-mediated signal transduction and spatial gradient sensing

During dermal wound healing, platelet-derived growth factor (PDGF) serves as both a chemoattractant and mitogen for fibroblasts, potently stimulating their invasion of the fibrin clot over a period of several days. A mathematical model of this process is presented, which accurately accounts for the sensitivity of PDGF gradient sensing through PDGF receptor/phosphoinositide 3-kinase-mediated signal transduction. Analysis of the model suggests that PDGF receptor-mediated endocytosis and degradation of PDGF allows a constant PDGF concentration profile to be maintained at the leading front of the fibroblast density profile as it propagates, at a constant rate, into the clot. Thus, the constant PDGF gradient can span the optimal concentration range for asymmetric phosphoinositide 3-kinase signaling and fibroblast chemotaxis, with near-maximal invasion rates elicited over a relatively broad range of PDGF secretion rates. A somewhat surprising finding was that extremely sharp PDGF gradients do not necessarily stimulate faster progression through the clot, because maintaining such a gradient through PDGF consumption is a potentially rate-limiting process.     

A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis. II. The effects of non-equilibrated ligand binding

Cells generally chemotax along a direction in which their receptor occupancy gradient--whether spatial or temporal--is maximum. Occupancy differentials are, however, often so small as to be masked by thermal noise; i.e., by fluctuations inherent in the stochastic nature of ligand binding. Such fluctuations therefore impose a fundamental limit on the sensitivity of a cell's ability to detect a chemoattractant gradient. In order to pursue the implications of this limit, fluctuation theories have been developed. The theories assume that the signal is some function of the receptor occupancy gradient, allow an estimate of the standard deviation about the mean signal, and permit an evaluation of, among other things, the extent to which a receptor defect can impair an effective response. Previous theories have assumed an equilibrated ligand-receptor interaction. In this paper we introduce a generalized definition of a signal caused by a receptor occupancy gradient that allows us to develop a non-equilibrium theory of thermal noise. We show that previous formulations are a special case of the current development. More specifically, we find the following. Swimming cells subject to Brownian tumbling must generally average their signals over a very long time period to achieve a signal-to-noise ratio less than or equal to 1. Spatial gradient detection is possible with ligand-receptor equilibrium constants less than 10(3)M-1, but since such ligands are rare, theory predicts that tumbling cells will generally not detect gradients by measuring spatial occupancy differentials. These conclusions hold irrespective of whether chemical equilibrium is achieved. For crawling cells not subject to Brownian tumbling, a range of affinities exists in which spatial or temporal gradient detection is possible. In general a spatial mechanism is more efficient for low affinity ligands (dissociation times less than 0.3 s), whereas a temporal mechanism is more efficient for higher K. In this case the detection of gradients in slowly dissociating ligand will be facilitated if signal processing begins prior to chemical equilibration. An important new parameter is indicated by the theory. The definitions of a temporal gradient signal is based on estimating and comparing average occupancy over two time intervals displaced by a time t1. The theory predicts an optimal t1, of order milliseconds, that leads to the shortest minimum averaging time. For t1 values at and longer than the optimum, and for all averaging times exceeding some minimum, the cell will detect a temporal signal.(ABSTRACT TRUNCATED AT 400 WORDS)     

Phosphoinositide 3-kinase activity controls the chemoattractant-mediated activation and adaptation of adenylyl cyclase

The binding of chemoattractants to cognate G protein-coupled receptors activates a variety of signaling cascades that provide spatial and temporal cues required for chemotaxis. When subjected to uniform stimulation, these responses are transient, showing an initial peak of activation followed by a period of adaptation, in which activity subsides even in the presence of stimulus. A tightly regulated balance between receptor-mediated stimulatory and inhibitory pathways controls the kinetics of activation and subsequent adaptation. In Dictyostelium, the adenylyl cyclase expressed during aggregation (ACA), which synthesizes the chemoattractant cAMP, is essential to relay the signal to neighboring cells. Here, we report that cells lacking phosphoinositide 3-kinase (PI3K) activity are deficient in signal relay. In LY294002-treated cells, this defect is because of a loss of ACA activation. In contrast, in cells lacking PI3K1 and PI3K2, the signal relay defect is because of a loss of ACA adaptation. We propose that the residual low level of 3-phosphoinositides in pi3k(1-/2-) cells is sufficient to generate the initial peak of ACA activity, yet is insufficient to sustain the inhibitory phase required for its adaptation. Thus, PI3K activity is poised to regulate both ACA activation and adaptation, thereby providing a link to ensure the proper balance of counteracting signals required to maintain optimal chemoresponsiveness.     

A Mechanism for the Polarity Formation of Chemoreceptors at the Growth Cone Membrane for Gradient Amplification during Directional Sensing

Accurate response to external directional signals is essential for many physiological functions such as chemotaxis or axonal guidance. It relies on the detection and amplification of gradients of chemical cues, which, in eukaryotic cells, involves the asymmetric relocalization of signaling molecules. How molecular events coordinate to induce a polarity at the cell level remains however poorly understood, particularly for nerve chemotaxis. Here, we propose a model, inspired by single-molecule experiments, for the membrane dynamics of GABA chemoreceptors in nerve growth cones (GCs) during directional sensing. In our model, transient interactions between the receptors and the microtubules, coupled to GABA-induced signaling, provide a positive-feedback loop that leads to redistribution of the receptors towards the gradient source. Using numerical simulations with parameters derived from experiments, we find that the kinetics of polarization and the steady-state polarized distribution of GABA receptors are in remarkable agreement with experimental observations. Furthermore, we make predictions on the properties of the GC seen as a sensing, amplification and filtering module. In particular, the growth cone acts as a low-pass filter with a time constant ∼10 minutes determined by the Brownian diffusion of chemoreceptors in the membrane. This filtering makes the gradient amplification resistent to rapid fluctuations of the external signals, a beneficial feature to enhance the accuracy of neuronal wiring. Since the model is based on minimal assumptions on the receptor/cytoskeleton interactions, its validity extends to polarity formation beyond the case of GABA gradient sensing. Altogether, it constitutes an original positive-feedback mechanism by which cells can dynamically adapt their internal organization to external signals.     

Responding to chemical gradients: bacterial chemotaxis

Chemotaxis allows bacteria to follow gradients of nutrients and other environmental stimuli. The bacterium Escherichia coli performs chemotaxis via a run-and-tumble strategy in which sensitive temporal comparisons lead to a biased random walk, with longer runs in the preferred gradient direction. The chemotaxis network of E. coli has developed over the years into one of the most thoroughly studied model systems for signal transduction and behavior, yielding general insights into such properties of cellular networks as signal amplification, signal integration, and robustness. Despite its relative simplicity, the operation of the E. coli chemotaxis network is highly refined and evolutionarily optimized at many levels. For example, recent studies revealed that the network adjusts its signaling properties dependent on the extracellular environment, apparently to optimize chemotaxis under particular conditions. The network can even utilize potentially detrimental stochastic fluctuations in protein levels and reaction rates to maximize the chemotactic performance of the population.     

Chemoattraction and chemotaxis in Dictyostelium discoideum: myxamoeba cannot read spatial gradients of cyclic adenosine monophosphate

Myxamoebae of the morphogenetic cellular slime mold Dictyostelium discoideum are thought to be able to accurately read and respond to directional information in spatial gradients of cyclic AMP. We examined the spatial and temporal mechanisms proposed for chemotaxis by comparing the behavior of spreading or evenly distributed cell populations after exposure to well-defined spatial gradients. The effects of gradient generation on cells were avoided by using predeveloped gradients. Qualitatively different responses were obtained using (a) isotropic, (b) static spatial, or (c) temporal (impulse) gradients in a simple chamber of penetrable micropore filters. We simulated models of chemotaxis and chemokinesis to aid our interpretations. The attractive and locomotory responses of populations were maximally stimulated by 0.05 microM cyclic AMP, provided that cellular phosphodiesterase was inhibited. But a single impulse of cyclic AMP during gradient development caused a greater and qualitatively different attraction. Attraction in spatial gradients was only transient, in that populations eventually developed a random distribution when confined to a narrow territory. Populations never accumulated nor lost their random distribution even in extremely steep spatial gradients. Attraction in spatial gradients was inducible only in spreading populations, not randomly distributed ones. Thus, spatial gradients effect biased-random locomotion: i.e., chemokinesis without adaptation. Cells cannot read gradients; the reaction of the cells is stochastic. Spatial gradients do not cause chemotaxis, which probably requires a sharp stimulant concentration increase (a temporal gradient) as a pulse or impulse. The results also bear on concepts of how embryonic cells might be able to decipher the positional information in a morphogen spatial gradient during development.     

An information-theoretic characterization of the optimal gradient sensing response of cells

Many cellular systems rely on the ability to interpret spatial heterogeneities in chemoattractant concentration to direct cell migration. The accuracy of this process is limited by stochastic fluctuations in the concentration of the external signal and in the internal signaling components. Here we use information theory to determine the optimal scheme to detect the location of an external chemoattractant source in the presence of noise. We compute the minimum amount of mutual information needed between the chemoattractant gradient and the internal signal to achieve a prespecified chemotactic accuracy. We show that more accurate chemotaxis requires greater mutual information. We also demonstrate that a priori information can improve chemotaxis efficiency. We compare the optimal signaling schemes with existing experimental measurements and models of eukaryotic gradient sensing. Remarkably, there is good quantitative agreement between the optimal response when no a priori assumption is made about the location of the existing source, and the observed experimental response of unpolarized Dictyostelium discoideum cells. In contrast, the measured response of polarized D. discoideum cells matches closely the optimal scheme, assuming prior knowledge of the external gradient-for example, through prolonged chemotaxis in a given direction. Our results demonstrate that different observed classes of responses in cells (polarized and unpolarized) are optimal under varying information assumptions.     

A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis

In order that cells respond to environmental cues, they must be able to measure ambient ligand concentration. Concentrations fluctuate, however, because of thermal noise, and one can readily show that estimates based on concentration values at a particular moment will be subject to substantial error. Cells are therefore expected to average their estimates over some limited time period. In this paper we assume that a cell uses fractional receptor occupancy as a measure of ambient ligand concentration and develop general expressions for the error a cell makes because the length of the averaging period is necessarily limited. Our analysis is general, relieving many of the assumptions underlying the seminal work of Berg and Purcell. The most important formal difference is our inclusion of occupancy-dependent dissociation--a phenomenon that has been well-documented for many systems. In addition, our formulation permits signal averaging to begin before chemical equilibrium has been established and it allows binding kinetics to be nonlinear (i.e., biomolecular rather than pseudo-first-order). The results are applied to spatial and temporal concentration gradients. In particular we estimate the minimum averaging times required for cells to detect such gradients under typical in vitro conditions. These estimates involve assigning numerical values to receptor ligand rate constants. If the rate constants are at their maximum possible values (limited only by center of mass diffusion), then either temporal or spatial gradients can be detected in minutes or less. If, however, as suggested by experiments, the rate constants are several orders of magnitude below their diffusion-limited values, then under typical constant gradient conditions the time required to detect a spatial gradient is prohibitively long, whereas temporal gradients can still be detected in reasonable lengths of time. This result was obtained for large cells such as lymphocytes, as well as for the smaller, bacterial cells. The ratio of averaging times for the two mechanisms--amounting to several orders of magnitude--is well beyond what could be reconciled by limitations of the calculation, and strongly suggests heavy reliance on temporal sensing mechanisms under typical in vitro conditions with constant spatial gradients.     

Adaptive-Control Model for Neutrophil Orientation in the Direction of Chemical Gradients

Neutrophils have a remarkable ability to detect the direction of chemoattractant gradients and move directionally in response to bacterial infections and tissue injuries. For their role in health and disease, neutrophils have been extensively studied, and many of the molecules involved in the signaling mechanisms of gradient detection and chemotaxis have been identified. However, the cellular-scale mechanisms of gradient sensing and directional neutrophil migration have been more elusive, and existent models provide only limited insight into these processes. Here, we propose a what we believe is a novel adaptive-control model for the initiation of cell polarization in response to gradients. In this model, the neutrophils first sample the environment by extending protrusions in random directions and subsequently adapt their sensitivity depending on localized, temporal changes in stimulation levels. Our results suggest that microtubules may play a critical role in integrating all the sensing events from the cellular periphery through their redistribution inside the neutrophils, and may also be involved in modulating local signaling. An unexpected finding was that model neutrophils exhibit significant randomness in timing and directionality of activation, comparable to our experimental observations in microfluidic devices. Moreover, their responses are robust against alterations of the rate and amplitude of the signaling reactions, and for a broad range in chemoattractant concentrations and spatial gradients.     

The Relation of Signal Transduction to the Sensitivity and Dynamic Range of Bacterial Chemotaxis

Complex networks of interacting molecular components of living cells are responsible for many important processes, such as signal processing and transduction. An important challenge is to understand how the individual properties of these molecular interactions and biochemical transformations determine the system-level properties of biological functions. Here, we address the issue of the accuracy of signal transduction performed by a bacterial chemotaxis system. The chemotaxis sensitivity of bacteria to a chemoattractant gradient has been measured experimentally from bacterial aggregation in a chemoattractant-containing capillary. The observed precision of the chemotaxis depended on environmental conditions such as the concentration and molecular makeup of the chemoattractant. In a quantitative model, we derived the chemotactic response function, which is essential to describing the signal transduction process involved in bacterial chemotaxis. In the presence of a gradient, an analytical solution is derived that reveals connections between the chemotaxis sensitivity and the characteristics of the signaling system, such as reaction rates. These biochemical parameters are integrated into two system-level parameters: one characterizes the efficiency of gradient sensing, and the other is related to the dynamic range of chemotaxis. Thus, our approach explains how a particular signal transduction property affects the system-level performance of bacterial chemotaxis. We further show that the two parameters can be derived from published experimental data from a capillary assay, which successfully characterizes the performance of bacterial chemotaxis.     

Mechanisms of Gradient Sensing and Chemotaxis: Conserved Pathways,Diverse Regulation

Directed cell migration is critical for normal development, immune responses, and wound healing and plays a prominent role in tumor metastasis. In eukaryotes, cell orientation is biased by an external chemoattractant gradient through a spatial contrast in chemoattractant receptor-mediated signal transduction processes that differentially affect cytoskeletal dynamics at the cell front and rear. Mechanisms of spatial gradient sensing and chemotaxis have been studied extensively in the social amoeba Dictyostelium discoideum and mammalian leukocytes (neutrophils), which are similar in their remarkable sensitivity to shallow gradients and robustness of response over a broad range of chemoattractant concentration. Recently, we have quantitatively characterized a different gradient sensing system, that of platelet-derived growth factor-stimulated fibroblasts, an important component of dermal wound healing. The marked differences between this system and the others have led us to speculate on the diversity of gradient sensing mechanisms and their biological implications.     

Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms

We study kinetic models for chemotaxis, incorporating the ability of cells to assess temporal changes of the chemoattractant concentration as well as its spatial variations. For prescribed smooth chemoattractant density, the macroscopic limit is carried out rigorously. It leads to a drift equation with a chemotactic sensitivity depending on the time derivative of the chemoattractant density. As an application it is shown by numerical experiments that the new model can resolve the chemotactic wave paradox. For this purpose, the macroscopic equation is coupled to a simple activation-inhibition model for the chemoattractant which produces the chemoattractant waves typical for the slime mold Dictyostelium discoideum.     

Asymmetric localization of calpain 2 during neutrophil chemotaxis

Chemoattractants induce neutrophil polarization through localized polymerization of F-actin at the leading edge. The suppression of rear and lateral protrusions is required for efficient chemotaxis and involves the temporal and spatial segregation of signaling molecules. We have previously shown that the intracellular calcium-dependent protease calpain is required for cell migration and is involved in regulating neutrophil chemotaxis. Here, we show that primary neutrophils and neutrophil-like HL-60 cells express both calpain 1 and calpain 2 and that chemoattractants induce the asymmetric recruitment of calpain 2, but not calpain 1, to the leading edge of polarized neutrophils and differentiated HL-60 cells. Using time-lapse microscopy, we show that enrichment of calpain 2 at the leading edge occurs during early pseudopod formation and that its localization is sensitive to changes in the chemotactic gradient. We demonstrate that calpain 2 is recruited to lipid rafts and that cholesterol depletion perturbs calpain 2 localization, suggesting that its enrichment at the front requires proper membrane organization. Finally, we show that catalytic activity of calpain is required to limit pseudopod formation in the direction of chemoattractant and for efficient chemotaxis. Together, our findings identify calpain 2 as a novel component of the frontness signal that promotes polarization during chemotaxis.     

Imprecision of Adaptation in Escherichia coli Chemotaxis

Adaptability is an essential property of many sensory systems, enabling maintenance of a sensitive response over a range of background stimulus levels. In bacterial chemotaxis, adaptation to the preset level of pathway activity is achieved through an integral feedback mechanism based on activity-dependent methylation of chemoreceptors. It has been argued that this architecture ensures precise and robust adaptation regardless of the ambient ligand concentration, making perfect adaptation a celebrated property of the chemotaxis system. However, possible deviations from such ideal adaptive behavior and its consequences for chemotaxis have not been explored in detail. Here we show that the chemotaxis pathway in Escherichia coli shows increasingly imprecise adaptation to higher concentrations of attractants, with a clear correlation between the time of adaptation to a step-like stimulus and the extent of imprecision. Our analysis suggests that this imprecision results from a gradual saturation of receptor methylation sites at high levels of stimulation, which prevents full recovery of the pathway activity by violating the conditions required for precise adaptation. We further use computer simulations to show that limited imprecision of adaptation has little effect on the rate of chemotactic drift of a bacterial population in gradients, but hinders precise accumulation at the peak of the gradient. Finally, we show that for two major chemoeffectors, serine and cysteine, failure of adaptation at concentrations above 1 mM might prevent bacteria from accumulating at toxic concentrations of these amino acids.     

Dependence of Bacterial Chemotaxis on Gradient Shape and Adaptation Rate

Simulation of cellular behavior on multiple scales requires models that are sufficiently detailed to capture central intracellular processes but at the same time enable the simulation of entire cell populations in a computationally cheap way. In this paper we present RapidCell, a hybrid model of chemotactic Escherichia coli that combines the Monod-Wyman-Changeux signal processing by mixed chemoreceptor clusters, the adaptation dynamics described by ordinary differential equations, and a detailed model of cell tumbling. Our model dramatically reduces computational costs and allows the highly efficient simulation of E. coli chemotaxis. We use the model to investigate chemotaxis in different gradients, and suggest a new, constant-activity type of gradient to systematically study chemotactic behavior of virtual bacteria. Using the unique properties of this gradient, we show that optimal chemotaxis is observed in a narrow range of CheA kinase activity, where concentration of the response regulator CheY-P falls into the operating range of flagellar motors. Our simulations also confirm that the CheB phosphorylation feedback improves chemotactic efficiency by shifting the average CheY-P concentration to fit the motor operating range. Our results suggest that in liquid media the variability in adaptation times among cells may be evolutionary favorable to ensure coexistence of subpopulations that will be optimally tactic in different gradients. However, in a porous medium (agar) such variability appears to be less important, because agar structure poses mainly negative selection against subpopulations with low levels of adaptation enzymes. RapidCell is available from the authors upon request.     

Theoretical analysis of gradient detection by growth cones.

Gradients of diffusible and substrate-bound molecules play an important role in guiding axons to appropriate targets in the developing nervous system. Although some of the molecules involved have recently been identified, little is known about the physical mechanisms by which growth cones sense gradients. This article applies the seminal Berg and Purcell (1977) model of gradient sensing to this problem. The model provides estimates for the statistical fluctuations in the measurement of concentration by a small sensing device. By assuming that gradient detection consists of the comparison of concentrations at two spatially or temporally separated points, the model therefore provides an estimate for the steepness of gradient that can be detected as a function of physiological parameters. The model makes the following specific predictions. (a) It is more likely that growth cones use a spatial rather than temporal sensing strategy. (b) Growth cone sensitivity increases with the concentration of ligand, the speed of ligand diffusion, the size of the growth cone, and the time over which it averages the gradient signal. (c) The minimum detectable gradient steepness for growth cones is roughly in the range 1-10%. (d) This value varies depending on whether a bound or freely diffusing ligand is being sensed, and on whether the sensing occurs in three or two dimensions. The model also makes predictions concerning the role of filopodia in gradient detection.     

Variable sizes of Escherichia coli chemoreceptor signaling teams

Like many sensory receptors, bacterial chemotaxis receptors form clusters. In bacteria, large‐scale clusters are subdivided into signaling teams that act as ‘antennas’ allowing detection of ligands with remarkable sensitivity. The range of sensitivity is greatly extended by adaptation of receptors to changes in concentrations through covalent modification. However, surprisingly little is known about the sizes of receptor signaling teams. Here, we combine measurements of the signaling response, obtained from in vivo fluorescence resonance energy transfer, with the statistical method of principal component analysis, to quantify the size of signaling teams within the framework of the previously successful Monod–Wyman–Changeux model. We find that size of signaling teams increases 2‐ to 3‐fold with receptor modification, indicating an additional, previously unrecognized level of adaptation of the chemotaxis network. This variation of signaling‐team size shows that receptor cooperativity is dynamic and likely optimized for sensing noisy ligand concentrations.     

Dynamic coordination of innate immune signaling and insulin signaling regulates systemic responses to localized DNA damage

Metazoans adapt to changing environmental conditions and to harmful challenges by attenuating growth and metabolic activities systemically. Recent studies in mice and flies indicate that endocrine signaling interactions between insulin/IGF signaling (IIS) and innate immune signaling pathways are critical for this adaptation, yet the temporal and spatial hierarchy of these signaling events remains elusive. Here, we identify and characterize a program of signaling interactions that regulates the systemic response of the Drosophila larva to localized DNA damage. We provide evidence that epidermal DNA damage induces an innate immune response that is kept in check by systemic repression of IIS activity. IIS repression induces NFκB/Relish signaling in the fat body, which is required for recovery of IIS activity in a second phase of the systemic response to DNA damage. This systemic response to localized DNA damage thus coordinates growth and metabolic activities across tissues, ensuring growth homeostasis and survival of the animal.