Attractant Gradient Research Articles

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

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

Asymmetry in the clockwise and counterclockwise rotation of the bacterial flagellar motor

Cells of Escherichia coli are able to swim up gradients of chemical attractants by modulating the direction of rotation of their flagellar motors, which spin alternately clockwise (CW) and counterclockwise (CCW). Rotation in either direction has been thought to be symmetric and exhibit the same torques and speeds. The relationship between torque and speed is one of the most important measurable characteristics of the motor, used to distinguish specific mechanisms of motor rotation. Previous measurements of the torque-speed relationship have been made with cells lacking the response regulator CheY that spin their motors exclusively CCW. In this case, the torque declines slightly up to an intermediate speed called the "knee speed" after which it falls rapidly to zero. This result is consistent with a "power-stroke" mechanism for torque generation. Here, we measure the torque-speed relationship for cells that express large amounts of CheY and only spin their motors CW. We find that the torque decreases linearly with speed, a result remarkably different from that for CCW rotation. We obtain similar results for wild-type cells by reexamining data collected in previous work. We speculate that CCW rotation might be optimized for runs, with higher speeds increasing the ability of cells to sense spatial gradients, whereas CW rotation might be optimized for tumbles, where the object is to change cell trajectories. But why a linear torque-speed relationship might be optimum for the latter purpose we do not know.

Theoretical results for chemotactic response and drift of E. coli in a weak attractant gradient

The bacterium Escherichia coli ( E. coli) moves in its natural environment in a series of straight runs, interrupted by tumbles which cause change of direction. It performs chemotaxis towards chemo-attractants by extending the duration of runs in the direction of the source. When there is a spatial gradient in the attractant concentration, this bias produces a drift velocity directed towards its source, whereas in a uniform concentration, E. coli adapts, almost perfectly in case of methyl aspartate. Recently, microfluidic experiments have measured the drift velocity of E. coli in precisely controlled attractant gradients, but no general theoretical expression for the same exists. With this motivation, we study an analytically soluble model here, based on the Barkai–Leibler model, originally introduced to explain the perfect adaptation. Rigorous mathematical expressions are obtained for the chemotactic response function and the drift velocity in the limit of weak gradients and under the assumption of completely random tumbles. The theoretical predictions compare favorably with experimental results, especially at high concentrations. We further show that the signal transduction network weakens the dependence of the drift on concentration, thus enhancing the range of sensitivity.

A Microfluidic Device for Quantifying Bacterial Chemotaxis in Stable Concentration Gradients

Chemotaxis allows bacteria to approach sources of attractant chemicals or to avoid sources of repellent chemicals. Bacteria constantly monitor the concentration of specific chemoeffectors by comparing the current concentration to the concentration detected a few seconds earlier. This comparison determines the net direction of movement. Although multiple, competing gradients often coexist in nature, conventional approaches for investigating bacterial chemotaxis are suboptimal for quantifying migration in response to concentration gradients of attractants and repellents. Here, we describe the development of a microfluidic chemotaxis model for presenting precise and stable concentration gradients of chemoeffectors to bacteria and quantitatively investigating their response to the applied gradient. The device is versatile in that concentration gradients of any desired absolute concentration and gradient strength can be easily generated by diffusive mixing. The device is demonstrated using the response of Escherichia coli RP437 to gradients of amino acids and nickel ions.

Investigation of bacterial chemotaxis in flow-based microfluidic devices

The plug-in-pond and capillary assays are convenient methods for measuring attractant and repellent bacterial chemotaxis. However, these assays do not provide quantitative information on the extent of migration and are not well-suited for investigating repellent taxis. Here, we describe a protocol for a flow-based microfluidic system (microFlow) to quantitatively investigate chemotaxis in response to concentration gradients of attractants and repellents. The microFlow device uses diffusive mixing to generate concentration gradients that are stable throughout the chemotaxis chamber and for the duration of the experiment. The gradients may be of any desired absolute concentration and gradient strength. GFP-expressing bacteria immediately encounter a stable concentration gradient when they enter the chemotaxis chamber, and the migration in response to the gradient is monitored by microscopy. The effects of different parameters that influence the extent of migration in the microFlow device-preparation of the motile bacterial population preparation, strength of the concentration gradient and duration of exposure to the gradient-are discussed in the context of repellent taxis of chemotactically wild-type Escherichia coli cells in a gradient of NiSO(4). Fabrication of the microfluidic device takes 1 d while preparing motile cells and carrying out the chemotaxis experiment takes 4-6 h to complete.

Quantitative Modeling of Escherichia coli Chemotactic Motion in Environments Varying in Space and Time

Escherichia coli chemotactic motion in spatiotemporally varying environments is studied by using a computational model based on a coarse-grained description of the intracellular signaling pathway dynamics. We find that the cell's chemotaxis drift velocity vd is a constant in an exponential attractant concentration gradient [L]∝exp(Gx). vd depends linearly on the exponential gradient G before it saturates when G is larger than a critical value GC. We find that GC is determined by the intracellular adaptation rate kR with a simple scaling law: . The linear dependence of vd on G = d(ln[L])/dx directly demonstrates E. coli's ability in sensing the derivative of the logarithmic attractant concentration. The existence of the limiting gradient GC and its scaling with kR are explained by the underlying intracellular adaptation dynamics and the flagellar motor response characteristics. For individual cells, we find that the overall average run length in an exponential gradient is longer than that in a homogeneous environment, which is caused by the constant kinase activity shift (decrease). The forward runs (up the gradient) are longer than the backward runs, as expected; and depending on the exact gradient, the (shorter) backward runs can be comparable to runs in a spatially homogeneous environment, consistent with previous experiments. In (spatial) ligand gradients that also vary in time, the chemotaxis motion is damped as the frequency ω of the time-varying spatial gradient becomes faster than a critical value ωc, which is controlled by the cell's chemotaxis adaptation rate kR. Finally, our model, with no adjustable parameters, agrees quantitatively with the classical capillary assay experiments where the attractant concentration changes both in space and time. Our model can thus be used to study E. coli chemotaxis behavior in arbitrary spatiotemporally varying environments. Further experiments are suggested to test some of the model predictions.

Nucleation dynamics in two-dimensional cylindrical Ising models and chemotaxis

The aim of our work is to study the effect of geometry variation on nucleation times and to address its role in the context of eukaryotic chemotaxis (i.e., the process which allows cells to identify and follow a gradient of chemical attractant). As a first step in this direction we study the nucleation dynamics of the two-dimensional Ising model defined on a cylindrical lattice whose radius changes as a function of time. Geometry variation is obtained by changing the relative value of the couplings between spins in the compactified (vertical) direction with respect to the horizontal one. This allows us to keep the lattice size unchanged and study in a single simulation the values of the compactification radius which change in time. We show both with theoretical arguments and numerical simulations that squeezing the geometry allows the system to speed up nucleation times even in presence of a very small energy gap between the stable and the metastable states. We then address the implications of our analysis for directional chemotaxis. The initial steps of chemotaxis can be modeled as a nucleation process occurring on the cell membrane as a consequence of the external chemical gradient (which plays the role of energy gap between the stable and metastable phases). In nature most of the cells modify their geometry by extending quasi-one-dimensional protrusions (filopodia) so as to enhance their sensitivity to chemoattractant. Our results show that this geometry variation has indeed the effect of greatly decreasing the time scale of the nucleation process even in presence of very small amounts of chemoattractants.

Responses of Escherichia coli Bacteria to Two Opposing Chemoattractant Gradients Depend on the Chemoreceptor Ratio

Escherichia coli chemotaxis has long served as a simple model of environmental signal processing, and bacterial responses to single chemical gradients are relatively well understood. Less is known about the chemotactic behavior of E. coli in multiple chemical gradients. In their native environment, cells are often exposed to multiple chemical stimuli. Using a recently developed microfluidic chemotaxis device, we exposed E. coli cells to two opposing but equally potent gradients of major attractants, methyl-aspartate and serine. The responses of E. coli cells demonstrated that chemotactic decisions depended on the ratio of the respective receptor number of Tar/Tsr. In addition, the ratio of Tar to Tsr was found to vary with cells' growth conditions, whereby it depended on the culture density but not on the growth duration. These results provide biological insights into the decision-making processes of chemotactic bacteria that are subjected to multiple chemical stimuli and demonstrate the importance of the cellular microenvironment in determining phenotypic behavior.

Second Messengers at the Leading Edge: Calcium Joins PIP3 as an Essential Signal

The leading edge of a chemotaxing cell is the engine that drives the cell up an attractant gradient. In one of the most dramatic spatial redistributions in cell biology, local second messenger signals recruit dozens of regulatory proteins from cytoplasm to the leading edge membrane. In turn these proteins control the actin polymerization and membrane remodeling processes propelling migration. It has long been known that the signaling lipid PIP3 is a crucial second messenger at the leading edge where it recruits PH domain proteins.

Directional persistence and the optimality of run-and-tumble chemotaxis

E. coli does chemotaxis by performing a biased random walk composed of alternating periods of swimming (runs) and reorientations (tumbles). Tumbles are typically modelled as complete directional randomisations but it is known that in wild type E. coli, successive run directions are actually weakly correlated, with a mean directional difference of ∼63°. We recently presented a model of the evolution of chemotactic swimming strategies in bacteria which is able to quantitatively reproduce the emergence of this correlation. The agreement between model and experiments suggests that directional persistence may serve some function, a hypothesis supported by the results of an earlier model. Here we investigate the effect of persistence on chemotactic efficiency, using a spatial Monte Carlo model of bacterial swimming in a gradient, combined with simulations of natural selection based on chemotactic efficiency. A direct search of the parameter space reveals two attractant gradient regimes, (a) a low-gradient regime, in which efficiency is unaffected by directional persistence and (b) a high-gradient regime, in which persistence can improve chemotactic efficiency. The value of the persistence parameter that maximises this effect corresponds very closely with the value observed experimentally. This result is matched by independent simulations of the evolution of directional memory in a population of model bacteria, which also predict the emergence of persistence in high-gradient conditions. The relationship between optimality and persistence in different environments may reflect a universal property of random-walk foraging algorithms, which must strike a compromise between two competing aims: exploration and exploitation. We also present a new graphical way to generally illustrate the evolution of a particular trait in a population, in terms of variations in an evolvable parameter.

Switching of the Bacterial Flagellar Motor Near Zero Load

Flagellated bacteria, such as Escherichia coli, are able to swim up gradients of chemical attractants by modulating the direction of rotation of their flagellar motors, which spin alternately clockwise (CW) and counterclockwise (CCW). Chemotactic behavior has been studied under a variety of conditions, mostly at high loads (at large motor torques). Here, we examine motor switching at low loads. Nano-gold spheres of various sizes were attached to hooks (the flexible coupling at the base of the flagellar filament) of cells lacking flagellar filaments in media containing different concentrations of the viscous agent Ficoll. The speeds and directions of rotation of the spheres were measured. Contrary to the case at high loads, motor switching rates increased appreciably with load. Both the CW → CCW and CCW → CW switching rates increased linearly with motor torque. Evidently, the switch senses stator–rotor interactions as well as the CheY-P concentration.

Two Lipids That Give Direction

Precise and sequential intracellular signaling events involving two phospholipids direct an immune cell toward an attractant molecule gradient.

Whispering gallery mode carousel – a photonic mechanism for enhanced nanoparticle detection in biosensing

Individual nanoparticles in aqueous solution are observed to be attracted to and orbit within the evanescent sensing ring of a Whispering Gallery Mode micro-sensor with only microwatts of driving power. This Carousel trap, caused by attractive optical gradient forces, interfacial interactions, and the circulating momentum flux, considerably enhances the rate of transport to the sensing region, thereby overcoming limitations posed by diffusion on such small area detectors. Resonance frequency fluctuations, caused by the radial Brownian motion of the nanoparticle, reveal the radial trapping potential and the nanoparticle size. Since the attractive forces draw particles to the highest evanescent intensity at the surface, binding steps are found to be uniform.

Optical trapping of coated microspheres

In an optical trap, micron-sized dielectric particles are held by a tightly focused laser beam. The optical force on the particle is composed of an attractive gradient force and a destabilizing scattering force. We hypothesized that using anti-reflection-coated microspheres would reduce scattering and lead to stronger trapping. We found that homogeneous silica and polystyrene microspheres had a sharp maximum trap stiffness at a diameter of around 800 nm--the trapping laser wavelength in water--and that a silica coating on a polystyrene microsphere was a substantial improvement for larger diameters. In addition, we noticed that homogeneous spheres of a correct size demonstrated anti-reflective properties. Our results quantitatively agreed with Mie scattering calculations and serve as a proof of principle. We used a DNA stretching experiment to confirm the large linear range in detection and force of the coated microspheres and performed a high-force motor protein assay. These measurements show that the surfaces of the coated microspheres are compatible with biophysical assays.

MULTISCALE MODELS OF TAXIS-DRIVEN PATTERNING IN BACTERIAL POPULATIONS.

Spatially-distributed populations of various types of bacteria often display intricate spatial patterns that are thought to result from the cellular response to gradients of nutrients or other attractants. In the past decade a great deal has been learned about signal transduction, metabolism and movement in E. coli and other bacteria, but translating the individual-level behavior into population-level dynamics is still a challenging problem. However, this is a necessary step because it is computationally impractical to use a strictly cell-based model to understand patterning in growing populations, since the total number of cells may reach 10(12) - 10(14) in some experiments. In the past phenomenological equations such as the Patlak-Keller-Segel equations have been used in modeling the cell movement that is involved in the formation of such patterns, but the question remains as to how the microscopic behavior can be correctly described by a macroscopic equation. Significant progress has been made for bacterial species that employ a "run-and-tumble" strategy of movement, in that macroscopic equations based on simplified schemes for signal transduction and turning behavior have been derived [14, 15]. Here we extend previous work in a number of directions: (i) we allow for time-dependent signals, which extends the applicability of the equations to natural environments, (ii) we use a more general turning rate function that better describes the biological behavior, and (iii) we incorporate the effect of hydrodynamic forces that arise when cells swim in close proximity to a surface. We also develop a new approach to solving the moment equations derived from the transport equation that does not involve closure assumptions. Numerical examples show that the solution of the lowest-order macroscopic equation agrees well with the solution obtained from a Monte Carlo simulation of cell movement under a variety of temporal protocols for the signal. We also apply the method to derive equations of chemotactic movement that are governed by multiple chemotactic signals.

Optical trapping of coated microspheres

In an optical trap, micron-sized dielectric particles are held by a tightly focused laser beam. The optical force on the particle is composed of an attractive gradient force and a destabilizing scattering force. We hypothesized that using anti-reflection-coated microspheres would reduce scattering and lead to stronger trapping. We found that homogeneous silica and polystyrene microspheres had a sharp maximum trap stiffness at a diameter of around 800 nm--the trapping laser wavelength in water--and that a silica coating on a polystyrene microsphere was a substantial improvement for larger diameters. In addition, we noticed that homogeneous spheres of a correct size demonstrated anti-reflective properties. Our results quantitatively agreed with Mie scattering calculations and serve as a proof of principle. We used a DNA stretching experiment to confirm the large linear range in detection and force of the coated microspheres and performed a high-force motor protein assay. These measurements show that the surfaces of the coated microspheres are compatible with biophysical assays.

Progesterone from the Cumulus Cells Is the Sperm Chemoattractant Secreted by the Rabbit Oocyte Cumulus Complex

Sperm chemotaxis in mammals have been identified towards several female sources as follicular fluid (FF), oviduct fluid, and conditioned medium from the cumulus oophorus (CU) and the oocyte (O). Though several substances were confirmed as sperm chemoattractant, Progesterone (P) seems to be the best chemoattractant candidate, because: 1) spermatozoa express a cell surface P receptor, 2) capacitated spermatozoa are chemotactically attracted in vitro by gradients of low quantities of P; 3) the CU cells produce and secrete P after ovulation; 4) a gradient of P may be kept stable along the CU; and 5) the most probable site for sperm chemotaxis in vivo could be near and/or inside the CU. The aim of this study was to verify whether P is the sperm chemoattractant secreted by the rabbit oocyte-cumulus complex (OCC) in the rabbit, as a mammalian animal model. By means of videomicroscopy and computer image analysis we observed that only the CU are a stable source of sperm attractants. The CU produce and secrete P since the hormone was localized inside these cells by immunocytochemistry and in the conditioned medium by enzyme immunoassay. In addition, rabbit spermatozoa express a cell surface P receptor detected by western blot and localized over the acrosomal region by immunocytochemistry. To confirm that P is the sperm chemoattractant secreted by the CU, the sperm chemotactic response towards the OCC conditioned medium was inhibited by three different approaches: P from the OCC conditioned medium was removed with an anti-P antibody, the attractant gradient of the OCC conditioned medium was disrupted by a P counter gradient, and the sperm P receptor was blocked with a specific antibody. We concluded that only the CU but not the oocyte secretes P, and the latter chemoattract spermatozoa by means of a cell surface receptor. Our findings may be of interest in assisted reproduction procedures in humans, animals of economic importance and endangered species.

Overview of Mathematical Approaches Used to Model Bacterial Chemotaxis I: The Single Cell

Mathematical modeling of bacterial chemotaxis systems has been influential and insightful in helping to understand experimental observations. We provide here a comprehensive overview of the range of mathematical approaches used for modeling, within a single bacterium, chemotactic processes caused by changes to external gradients in its environment. Specific areas of the bacterial system which have been studied and modeled are discussed in detail, including the modeling of adaptation in response to attractant gradients, the intracellular phosphorylation cascade, membrane receptor clustering, and spatial modeling of intracellular protein signal transduction. The importance of producing robust models that address adaptation, gain, and sensitivity are also discussed. This review highlights that while mathematical modeling has aided in understanding bacterial chemotaxis on the individual cell scale and guiding experimental design, no single model succeeds in robustly describing all of the basic elements of the cell. We conclude by discussing the importance of this and the future of modeling in this area.

Relationship between cellular response and behavioral variability in bacterial chemotaxis

Over the last decades, bacterial chemotaxis in Escherichia coli has emerged as a canonical system for the study of signal transduction. A remarkable feature of this system is the coexistence of a robust adaptive behavior observed at the population level with a large fluctuating behavior in single cells [Korobkova E, Emonet T, Vilar JMG, Shimizu TS, Cluzel P (2004) Nature 428:574-578]. Using a unified stochastic model, we demonstrate that this coexistence is not fortuitous but a direct consequence of the architecture of this adaptive system. The methylation and demethylation cycles that regulate the activity of receptor-kinase complexes are ultrasensitive because they operate outside the region of first-order kinetics. As a result, the receptor-kinase that governs cellular behavior exhibits a sigmoidal activation curve. We propose that the steepness of this kinase activation curve simultaneously controls the behavioral variability in nonstimulated individual bacteria and the duration of the adaptive response to small stimuli. We predict that the fluctuating behavior and the chemotactic response of individual cells both peak within the transition region of this sigmoidal curve. Large-scale simulations of digital bacteria suggest that the chemotaxis network is tuned to simultaneously maximize both the random spread of cells in the absence of nutrients and the cellular response to gradients of attractant. This study highlights a fundamental relation from which the behavioral variability of nonstimulated cells is used to infer the timing of the cellular response to small stimuli.

Optical forces on interacting plasmonic nanoparticles in a focused Gaussian beam

We theoretically analyze optical forces on aggregates of metal nanoparticles in a focused Gaussian beam by extending the generalized Mie theory, which includes higher order multipoles and retardation effects. For two interacting metallic particles, an attractive gradient force, mainly caused by multipole plasmon excitation, exists at short interparticle distances, while induced dipolar fields dominate for separations of the order of the particle radius $R$ or larger. The long-range force component can be either attractive or repulsive depending on the phase of the induced dipoles, as determined by the illumination wavelength and the collective dipolar plasmon resonance. In particular, the repulsive force that occurs for illumination near the plasmon resonance wavelength can be so large that it overcomes the optical trapping effect of the Gaussian beam.