Stochastic signal inputs for chemotactic response in Dictyostelium cells revealed by single molecule imaging techniques

Chemotactic cells can exhibit extreme sensitivity to chemical gradients. Theoretical estimations of the signal inputs required for chemotaxis suggest that the response can be achieved under the strong influence of stochastic input noise generated by the receptors during the transmembrane signaling. This arises a fundamental question regarding the mechanisms for directional sensing: how do cells obtain reliable information regarding gradient direction by using stochastically operating receptors and the downstream molecules? To address this question, we have developed single molecule imaging techniques to visualize signaling molecules responsible for chemotaxis in living Dictyostelium cells, allowing us to monitor the stochastic signaling processes directly. Single molecule imaging of a chemoattractant bound to a receptor demonstrates that signal inputs fluctuate with time and space. Downstream signaling molecules, such as PTEN and a PH domain-containing protein that are constituent parts of chemotactic signaling system, can also be followed at single molecule level in living cells, illuminating the stochastic nature of chemotactic signaling processes. In this report, we start with a brief introduction of chemotactic response of the eukaryotic cells, followed by an explanation for single molecule imaging techniques, and finally discuss these applications to chemotactic signaling system of Dictyostelium cells.     

Stochastic signal processing and transduction in chemotactic response of eukaryotic cells

Single-molecule imaging analysis of chemotactic response in eukaryotic cells has revealed a stochastic nature in the input signals and the signal transduction processes. This leads to a fundamental question about the signaling processes: how does the signaling system operate under stochastic fluctuations or noise? Here, we report a stochastic model of chemotactic signaling in which noise and signal propagation along the transmembrane signaling pathway by chemoattractant receptors can be analyzed quantitatively. The results obtained from this analysis reveal that the second-messenger-production reactions by the receptors generate noisy signals that contain intrinsic noise inherently generated at this reaction and extrinsic noise propagated from the ligand-receptor binding. Such intrinsic and extrinsic noise limits the directional sensing ability of chemotactic cells, which may explain the dependence of chemotactic accuracy on chemical gradients that has been observed experimentally. Our analysis also reveals regulatory mechanisms for signal improvement in the stochastically operating signaling system by analyzing how the SNR of chemotactic signals can be improved on or deteriorated by the stochastic properties of receptors and second-messenger molecules. Theoretical consideration of noisy signal transduction by chemotactic signaling systems can further be applied to signaling systems in general.     

Traction forces of neutrophils migrating on compliant substrates

Proper functioning of the innate immune response depends on migration of circulating neutrophils into tissues at sites of infection and inflammation. Migration of highly motile, amoeboid cells such as neutrophils has significant physiological relevance, yet the traction forces that drive neutrophil motion in response to chemical cues are not well characterized. To better understand the relationship between chemotactic signals and the organization of forces in motile neutrophils, force measurements were made on hydrogel surfaces under well-defined chemotactic gradients created with a microfluidic device. Two parameters, the mean chemoattractant concentration (C_M) and the gradient magnitude (Δc/Δx) were varied. Cells experiencing a large gradient with C_M near the chemotactic receptor K_D displayed strong punctate centers of uropodial contractile force and strong directional motion on stiff (12 kPa) surfaces. Under conditions of ideal chemotaxis—cells in strong gradients with mean chemoattractant near the receptor K_D and on stiffer substrates—there is a correlation between the magnitude of force generation and directional motion as measured by the chemotactic index. However, on soft materials or under weaker chemotactic conditions, directional motion is uncorrelated with the magnitude of traction force. Inhibition of either β₂ integrins or Rho-associated kinase, a kinase downstream from RhoA, greatly reduced rearward traction forces and directional motion, although some vestigial lamellipodium-driven motility remained. In summary, neutrophils display a diverse repertoire of methods for organizing their internal machinery to generate directional motion.     

Stochastic control of spontaneous signal generation for gradient sensing in chemotaxis

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

3'-phosphoinositides regulate the coordination of speed and accuracy during chemotaxis

The PI3K/PTEN pathway, as the regulator of 3′-phosphoinositide (3′-PI) dynamics, has emerged as a key regulator of chemoattractant gradient sensing during chemotaxis in Dictyostelium and other cell types. Previous results have shown 3′-PIs to be important for regulating basal cell motility and sensing the direction and strength of the chemoattractant gradient. We examined the chemotaxis of wild-type cells and cells lacking PTEN or PI3K1 and 2 using analytical methods that allowed us to quantitatively discern differences between the genotype's ability to sense and efficiently respond to changes in gradient steepness during chemotaxis. We found that cells are capable of increasing their chemotactic accuracy and speed as they approach a micropipette in a manner that is dependent on the increasing strength of the concentration gradient and 3′-PI signaling. Further, our data show that 3′-PI signaling affects a cell's ability to coordinate speed and direction to increase chemotactic efficiency. Using to our knowledge a new measurement of chemotactic efficiency that reveals the degree of coordination between speed and accuracy, we found that cells also have the capacity to increase their chemotactic efficiency as they approach the micropipette. Like directional accuracy and speed, the increase in chemotactic efficiency of cells with increased gradient strength is sensitive to 3′-PI dysregulation. Our evidence suggests that receptor-driven 3′-PI signaling regulates the ability of a cell to capitalize on stronger directional inputs and minimize the effects of inaccurate turns to increase chemotactic efficiency.     

Deterministic Versus Stochastic Cell Polarisation Through Wave-Pinning

Cell polarization is an important part of the response of eukaryotic cells to stimuli, and forms a primary step in cell motility, differentiation, and many cellular functions. Among the important biochemical players implicated in the onset of intracellular asymmetries that constitute the early phases of polarization are the Rho GTPases, such as Cdc42, Rac, and Rho, which present high active concentration levels in a spatially localized manner. Rho GTPases exhibit positive feedback-driven interconversion between distinct active and inactive forms, the former residing on the cell membrane, and the latter predominantly in the cytosol. A?deterministic model of the dynamics of a single Rho GTPase described earlier by Mori et al.?exhibits sustained polarization by a wave-pinning mechanism. It remained, however, unclear how such polarization behaves at typically low cellular concentrations, as stochasticity could significantly affect the dynamics. We therefore study the low copy number dynamics of this model, using a stochastic kinetics framework based on the Gillespie algorithm, and propose statistical and analytic techniques which help us analyse the equilibrium behaviour of our stochastic system. We use local perturbation analysis to predict parameter regimes for initiation of polarity and wave-pinning in our deterministic system, and compare these predictions with deterministic and stochastic spatial simulations. Comparing the behaviour of the stochastic with the deterministic system, we determine the threshold number of molecules required for robust polarization in a given effective reaction volume. We show that when the molecule number is lowered wave-pinning behaviour is lost due to an increasingly large transition zone as well as increasing fluctuations in the pinning position, due to which a broadness can be reached that is unsustainable, causing the collapse of the wave, while the variations in the high and low equilibrium levels are much less affected.     

Chemotropism indices for polymorphonuclear leukocytes.

Trajectories of polymorphonuclear leukocytes which are responding to a chemical gradient are analyzed in order to deduce probability distributions of the angles between successive path segments. The turn angle probability distributions thus obtained are seen to be strongly dependent on the direction of locomotion prior to a turn, in that cells usually turn to maintain alignment along an axis directed towards the chemoattractant source. A mathematical model based on these observations is developed in order to understand the relationship between net chemotactic response and parameters characterizing stochastic movements of individual cells. In particular, the manner in which the chemotropism index depends on details of the turn-angle distributions is examined. When bias in the direction of turn is induced by a chemotactic field, transition from random motion to directed response occurs most abruptly if the turn-angle distribution is narrow. "Accommodation," viz., a dependence of the mean angle of turn upon prior orientation, is found to have relatively little effect on the magnitude of the response.     

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.     

Three-dimensional simulation of obstacle-mediated chemotaxis

Amoeboid cells exhibit a highly dynamic motion that can be directed by external chemical signals, through the process of chemotaxis. Here, we propose a three-dimensional model for chemotactic motion of amoeboid cells. We account for the interactions between the extracellular substances, the membrane-bound proteins, and the cytosolic components involved in the signaling pathway that originates cell motility. We show two- and three-dimensional simulations of cell migration on planar substrates, flat surfaces with obstacles, and fibrous networks. The results show that our model reproduces the main features of chemotactic amoeboid motion. Our simulations unveil a complicated interplay between the geometry of the cell’s environment and the chemoattractant dynamics that tightly regulates cell motion. The model opens new opportunities to simulate the interactions between extra- and intra-cellular compounds mediated by the matrix geometry.     

Noise generation, amplification and propagation in chemotactic signaling systems of living cells

Theoretical considerations of stochastic signal transduction in living cells have revealed the gain-fluctuation relation, which provides a theoretical framework to describe quantitatively how noise is generated, amplified and propagated along a signaling cascade in living cells. We chose the chemotactic signaling of bacteria and eukaryotic cells as a typical example of noisy signal transduction and applied the gain-fluctuation relation to these signaling systems in order to analyze the effects of noise on signal transduction. Comparing our theoretical analysis with the experimental results of chemotaxis in bacteria Escherichia coli and eukaryote Dictyostelium discoideum revealed that noise in signal transduction systems limits the cells' chemotactic ability and contributes to their behavioral variability. Based on the kinetic properties of signaling molecules in living cells, the gain-fluctuation relation can quantitatively explain stochastic cellular behaviors.     

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.     

Global parameter estimation methods for stochastic biochemical systems

Background  

The importance of stochasticity in cellular processes having low number of molecules has resulted in the development of stochastic models such as chemical master equation. As in other modelling frameworks, the accompanying rate constants are important for the end-applications like analyzing system properties (e.g. robustness) or predicting the effects of genetic perturbations. Prior knowledge of kinetic constants is usually limited and the model identification routine typically includes parameter estimation from experimental data. Although the subject of parameter estimation is well-established for deterministic models, it is not yet routine for the chemical master equation. In addition, recent advances in measurement technology have made the quantification of genetic substrates possible to single molecular levels. Thus, the purpose of this work is to develop practical and effective methods for estimating kinetic model parameters in the chemical master equation and other stochastic models from single cell and cell population experimental data.     

Testing human sperm chemotaxis: how to detect biased motion in population assays

Biased motion of motile cells in a concentration gradient of a chemoattractant is frequently studied on the population level. This approach has been particularly employed in human sperm chemotactic assays, where the fraction of responsive cells is low and detection of biased motion depends on subtle differences. In these assays, statistical measures such as population odds ratios of swimming directions can be employed to infer chemotactic performance. Here, we report on an improved method to assess statistical significance of experimentally determined odds ratios and discuss the strong impact of data correlations that arise from the directional persistence of sperm swimming.     

A Highlights from MBoC Selection: Polarization of the Yeast Pheromone Receptor Requires Its Internalization but Not Actin-dependent Secretion

In the best understood models of eukaryotic directional sensing, chemotactic cells maintain a uniform distribution of surface receptors even when responding to chemical gradients. The yeast pheromone receptor is also uniformly distributed on the plasma membrane of vegetative cells, but pheromone induces its polarization into “crescents” that cap the future mating projection. Here, we find that in pheromone-treated cells, receptor crescents are visible before detectable polarization of actin cables and that the receptor can polarize in the absence of actin-dependent directed secretion. Receptor internalization, in contrast, seems to be essential for the generation of receptor polarity, and mutations that deregulate this process confer dramatic defects in directional sensing. We also show that pheromone induces the internalization and subsequent polarization of the mating-specific Gα and Gβ proteins and that the changes in G protein localization depend on receptor internalization and receptor–Gα coupling. Our data suggest that the polarization of the receptor and its G protein precedes actin polarization and is important for gradient sensing. We propose that the establishment of receptor/G protein polarity depends on a novel mechanism involving differential internalization and that this serves to amplify the shallow gradient of activated receptor across the cell.     

Stochastic Petri Net extension of a yeast cell cycle model

This paper presents the definition, solution and validation of a stochastic model of the budding yeast cell cycle, based on Stochastic Petri Nets (SPN). A specific family of SPNs is selected for building a stochastic version of a well-established deterministic model. We describe the procedure followed in defining the SPN model from the deterministic ODE model, a procedure that can be largely automated. The validation of the SPN model is conducted with respect to both the results provided by the deterministic one and the experimental results available from literature. The SPN model catches the behavior of the wild type budding yeast cells and a variety of mutants. We show that the stochastic model matches some characteristics of budding yeast cells that cannot be found with the deterministic model. The SPN model fine-tunes the simulation results, enriching the breadth and the quality of its outcome.     

Stochastic exit from mitosis in budding yeast

Unlike many mutants that are completely viable or inviable, the CLB2-dbΔ clb5Δ mutant of Saccharomyces cerevisiae is inviable in glucose but partially viable on slower growth media such as raffinose. On raffinose, the mutant cells can bud and divide but in each cycle there is a chance that a cell will fail to divide (telophase arrest), causing it to exit the cell cycle. This effect gives rise to a stochastic phenotype that cannot be explained by a deterministic model. We measure the inter-bud times of wild type and mutant cells growing on raffinose and compute statistics and distributions to characterize the mutant’s behavior. We convert a detailed deterministic model of the budding yeast cell cycle to a stochastic model and determine the extent to which it captures the stochastic phenotype of the mutant strain. Predictions of the mathematical model are in reasonable agreement with our experimental data and suggest directions for improving the model. Ultimately, the ability to accurately model stochastic phenotypes may prove critical to understanding disease and therapeutic interventions in higher eukaryotes.     

A Novel Chemotaxis Assay in 3-D Collagen Gels by Time-Lapse Microscopy

The directional cell response to chemical gradients, referred to as chemotaxis, plays an important role in physiological and pathological processes including development, immune response and tumor cell invasion. Despite such implications, chemotaxis remains a challenging process to study under physiologically-relevant conditions in-vitro, mainly due to difficulties in generating a well characterized and sustained gradient in substrata mimicking the in-vivo environment while allowing dynamic cell imaging. Here, we describe a novel chemotaxis assay in 3D collagen gels, based on a reusable direct-viewing chamber in which a chemoattractant gradient is generated by diffusion through a porous membrane. The diffusion process has been analysed by monitoring the concentration of FITC-labelled dextran through epifluorescence microscopy and by comparing experimental data with theoretical and numerical predictions based on Fick''s law. Cell migration towards chemoattractant gradients has been followed by time-lapse microscopy and quantified by cell tracking based on image analysis techniques. The results are expressed in terms of chemotactic index (I) and average cell velocity. The assay has been tested by comparing the migration of human neutrophils in isotropic conditions and in the presence of an Interleukin-8 (IL-8) gradient. In the absence of IL-8 stimulation, 80% of the cells showed a velocity ranging from 0 to 1 µm/min. However, in the presence of an IL-8 gradient, 60% of the cells showed an increase in velocity reaching values between 2 and 7 µm/min. Furthermore, after IL-8 addition, I increased from 0 to 0.25 and 0.25 to 0.5, respectively, for the two donors examined. These data indicate a pronounced directional migration of neutrophils towards the IL-8 gradient in 3D collagen matrix. The chemotaxis assay described here can be adapted to other cell types and may serve as a physiologically relevant method to study the directed locomotion of cells in a 3D environment in response to different chemoattractants.     

Stochastic exit from mitosis in budding yeast: Model predictions and experimental observations

Unlike many mutants that are completely viable or inviable, the CLB2-dbΔ clb5Δ mutant of Saccharomyces cerevisiae is inviable in glucose but partially viable on slower growth media such as raffinose. On raffinose, the mutant cells can bud and divide but in each cycle there is a chance that a cell will fail to divide (telophase arrest), causing it to exit the cell cycle. This effect gives rise to a stochastic phenotype that cannot be explained by a deterministic model. We measure the interbud times of wild-type and mutant cells growing on raffinose and compute statistics and distributions to characterize the mutant''s behavior. We convert a detailed deterministic model of the budding yeast cell cycle to a stochastic model and determine the extent to which it captures the stochastic phenotype of the mutant strain. Predictions of the mathematical model are in reasonable agreement with our experimental data and suggest directions for improving the model. Ultimately, the ability to accurately model stochastic phenotypes may prove critical to understanding disease and therapeutic interventions in higher eukaryotes.Key words: stochastic phenotype, mitotic exit, non-genetic variability, cell cycle modeling, computational biology, stochastic modeling, deterministic modeling