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.     

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.     

Corollary discharge enables proprioception from lateral line sensory feedback

Animals modulate sensory processing in concert with motor actions. Parallel copies of motor signals, called corollary discharge (CD), prepare the nervous system to process the mixture of externally and self-generated (reafferent) feedback that arises during locomotion. Commonly, CD in the peripheral nervous system cancels reafference to protect sensors and the central nervous system from being fatigued and overwhelmed by self-generated feedback. However, cancellation also limits the feedback that contributes to an animal’s awareness of its body position and motion within the environment, the sense of proprioception. We propose that, rather than cancellation, CD to the fish lateral line organ restructures reafference to maximize proprioceptive information content. Fishes’ undulatory body motions induce reafferent feedback that can encode the body’s instantaneous configuration with respect to fluid flows. We combined experimental and computational analyses of swimming biomechanics and hair cell physiology to develop a neuromechanical model of how fish can track peak body curvature, a key signature of axial undulatory locomotion. Without CD, this computation would be challenged by sensory adaptation, typified by decaying sensitivity and phase distortions with respect to an input stimulus. We find that CD interacts synergistically with sensor polarization to sharpen sensitivity along sensors’ preferred axes. The sharpening of sensitivity regulates spiking to a narrow interval coinciding with peak reafferent stimulation, which prevents adaptation and homogenizes the otherwise variable sensor output. Our integrative model reveals a vital role of CD for ensuring precise proprioceptive feedback during undulatory locomotion, which we term external proprioception.

Animals modulate sensory processing in concert with motor actions. A study of the corollary discharge in zebrafish reveals that it modulates the sensitivity of the lateral line during swimming to prevent sensor adaptation and maintain the high-quality feedback necessary for kinematic control.     

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.     

Spatiotemporal Spike Coding of Behavioral Adaptation in the Dorsal Anterior Cingulate Cortex

The frontal cortex controls behavioral adaptation in environments governed by complex rules. Many studies have established the relevance of firing rate modulation after informative events signaling whether and how to update the behavioral policy. However, whether the spatiotemporal features of these neuronal activities contribute to encoding imminent behavioral updates remains unclear. We investigated this issue in the dorsal anterior cingulate cortex (dACC) of monkeys while they adapted their behavior based on their memory of feedback from past choices. We analyzed spike trains of both single units and pairs of simultaneously recorded neurons using an algorithm that emulates different biologically plausible decoding circuits. This method permits the assessment of the performance of both spike-count and spike-timing sensitive decoders. In response to the feedback, single neurons emitted stereotypical spike trains whose temporal structure identified informative events with higher accuracy than mere spike count. The optimal decoding time scale was in the range of 70–200 ms, which is significantly shorter than the memory time scale required by the behavioral task. Importantly, the temporal spiking patterns of single units were predictive of the monkeys’ behavioral response time. Furthermore, some features of these spiking patterns often varied between jointly recorded neurons. All together, our results suggest that dACC drives behavioral adaptation through complex spatiotemporal spike coding. They also indicate that downstream networks, which decode dACC feedback signals, are unlikely to act as mere neural integrators.     

The Role of Adaptation in Bacterial Speed Races

Evolution of biological sensory systems is driven by the need for efficient responses to environmental stimuli. A paradigm among prokaryotes is the chemotaxis system, which allows bacteria to navigate gradients of chemoattractants by biasing their run-and-tumble motion. A notable feature of chemotaxis is adaptation: after the application of a step stimulus, the bacterial running time relaxes to its pre-stimulus level. The response to the amino acid aspartate is precisely adapted whilst the response to serine is not, in spite of the same pathway processing the signals preferentially sensed by the two receptors Tar and Tsr, respectively. While the chemotaxis pathway in E. coli is well characterized, the role of adaptation, its functional significance and the ecological conditions where chemotaxis is selected, are largely unknown. Here, we investigate the role of adaptation in the climbing of gradients by E. coli. We first present theoretical arguments that highlight the mechanisms that control the efficiency of the chemotactic up-gradient motion. We discuss then the limitations of linear response theory, which motivate our subsequent experimental investigation of E. coli speed races in gradients of aspartate, serine and combinations thereof. By using microfluidic techniques, we engineer controlled gradients and demonstrate that bacterial fronts progress faster in equal-magnitude gradients of serine than aspartate. The effect is observed over an extended range of concentrations and is not due to differences in swimming velocities. We then show that adding a constant background of serine to gradients of aspartate breaks the adaptation to aspartate, which results in a sped-up progression of the fronts and directly illustrate the role of adaptation in chemotactic gradient-climbing.     

An Adapting Auditory-motor Feedback Loop Can Contribute to Generating Vocal Repetition

Consecutive repetition of actions is common in behavioral sequences. Although integration of sensory feedback with internal motor programs is important for sequence generation, if and how feedback contributes to repetitive actions is poorly understood. Here we study how auditory feedback contributes to generating repetitive syllable sequences in songbirds. We propose that auditory signals provide positive feedback to ongoing motor commands, but this influence decays as feedback weakens from response adaptation during syllable repetitions. Computational models show that this mechanism explains repeat distributions observed in Bengalese finch song. We experimentally confirmed two predictions of this mechanism in Bengalese finches: removal of auditory feedback by deafening reduces syllable repetitions; and neural responses to auditory playback of repeated syllable sequences gradually adapt in sensory-motor nucleus HVC. Together, our results implicate a positive auditory-feedback loop with adaptation in generating repetitive vocalizations, and suggest sensory adaptation is important for feedback control of motor sequences.     

An inexpensive photosynthetic irradiance sensor for ecological field studies

The design and characteristics of inexpensive and simply constructed equal-energy response photosynthetic irradiance sensors is described for use particularly where several cells are required in comparative ecological studies either above or below water. The dimensions of the sensors can be changed proportionally to suit different applications or components. The response of the sensor to irradiance at varying angles corresponds very closely to that required by the cosine law. The sensor is comparatively insensitive to other environmental variables in field use and gave a stable output; the long term drift was proportional to electrical output but in continuous use, drift is regular and could reach -0.08 year^-1 of the total. The spectral range and cosine response is discussed in comparison to other more expensive (x 5–10) commercially available, sensors and to local standards.     

Models of Eukaryotic Gradient Sensing: Application to Chemotaxis of Amoebae and Neutrophils

Eukaryotic cells can detect shallow gradients of chemoattractants with exquisite precision and respond quickly to changes in the gradient steepness and direction. Here, we describe a set of models explaining both adaptation to uniform increases in chemoattractant and persistent signaling in response to gradients. We demonstrate that one of these models can be mapped directly onto the biochemical signal-transduction pathways underlying gradient sensing in amoebae and neutrophils. According to this scheme, a locally acting activator (PI3-kinase) and a globally acting inactivator (PTEN or a similar phosphatase) are coordinately controlled by the G-protein activation. This signaling system adapts perfectly to spatially homogeneous changes in the chemoattractant. In chemoattractant gradients, an imbalance between the action of the activator and the inactivator results in a spatially oriented persistent signaling, amplified by a substrate supply-based positive feedback acting through small G-proteins. The amplification is activated only in a continuous presence of the external signal gradient, thus providing the mechanism for sensitivity to gradient alterations. Finally, based on this mapping, we make predictions concerning the dynamics of signaling. We propose that the underlying principles of perfect adaptation and substrate supply-based positive feedback will be found in the sensory systems of other chemotactic cell types.     

Logarithmic Sensing in Escherichia coli Bacterial Chemotaxis

We studied the response of swimming Escherichia coli (E. coli) bacteria in a comprehensive set of well-controlled chemical concentration gradients using a newly developed microfluidic device and cell tracking imaging technique. In parallel, we carried out a multi-scale theoretical modeling of bacterial chemotaxis taking into account the relevant internal signaling pathway dynamics, and predicted bacterial chemotactic responses at the cellular level. By measuring the E. coli cell density profiles across the microfluidic channel at various spatial gradients of ligand concentration grad[L] and the average ligand concentration near the peak chemotactic response region, we demonstrated unambiguously in both experiments and model simulation that the mean chemotactic drift velocity of E. coli cells increased monotonically with grad [L]/ or ∼grad(log[L])—that is E. coli cells sense the spatial gradient of the logarithmic ligand concentration. The exact range of the log-sensing regime was determined. The agreements between the experiments and the multi-scale model simulation verify the validity of the theoretical model, and revealed that the key microscopic mechanism for logarithmic sensing in bacterial chemotaxis is the adaptation kinetics, in contrast to explanations based directly on ligand occupancy.     

Principles of goal-directed spatial robot navigation in biomimetic models

Mobile robots and animals alike must effectively navigate their environments in order to achieve their goals. For animals goal-directed navigation facilitates finding food, seeking shelter or migration; similarly robots perform goal-directed navigation to find a charging station, get out of the rain or guide a person to a destination. This similarity in tasks extends to the environment as well; increasingly, mobile robots are operating in the same underwater, ground and aerial environments that animals do. Yet despite these similarities, goal-directed navigation research in robotics and biology has proceeded largely in parallel, linked only by a small amount of interdisciplinary research spanning both areas. Most state-of-the-art robotic navigation systems employ a range of sensors, world representations and navigation algorithms that seem far removed from what we know of how animals navigate; their navigation systems are shaped by key principles of navigation in ‘real-world’ environments including dealing with uncertainty in sensing, landmark observation and world modelling. By contrast, biomimetic animal navigation models produce plausible animal navigation behaviour in a range of laboratory experimental navigation paradigms, typically without addressing many of these robotic navigation principles. In this paper, we attempt to link robotics and biology by reviewing the current state of the art in conventional and biomimetic goal-directed navigation models, focusing on the key principles of goal-oriented robotic navigation and the extent to which these principles have been adapted by biomimetic navigation models and why.     

Bias in the gradient-sensing response of chemotactic cells

We apply linear stability theory and perform perturbation studies to better characterize, and to generate new experimental predictions from, a model of chemotactic gradient sensing in eukaryotic cells. The model uses reaction-diffusion equations to describe 3(') phosphoinositide signaling and its regulation at the plasma membrane. It demonstrates a range of possible gradient-sensing mechanisms and captures such characteristic behaviors as strong polarization in response to static gradients, adaptation to differing mean levels of stimulus, and plasticity in response to changing gradients. An analysis of the stability of polarized steady-state solutions indicates that the model is most sensitive to off-axis perturbations. This biased sensitivity is also reflected in responses to localized external stimuli, and leads to a clear experimental prediction, namely, that a cell which is polarized in a background gradient will be most sensitive to transient point-source stimuli lying within a range of angles that are oblique with respect to the polarization axis. Stimuli at angles below this range will elicit responses whose directions overshoot the stimulus angle, while responses to stimuli applied at larger angles will undershoot the stimulus angle. We argue that such a bias is likely to be a general feature of gradient sensing in highly motile cells, particularly if they are optimized to respond to small gradients. Finally, an angular bias in gradient sensing might lead to preferred turn angles and zigzag movements of cells moving up chemotactic gradients, as has been noted under certain experimental conditions.     

Alternative splicing and interaction with di- and polyvalent cations control the dynamic range of acid-sensing ion channel 1 (ASIC1)

Homomeric acid-sensing ion channel 1 (ASIC1) can be activated by extracellular H(+) in the physiological pH range and may, therefore, contribute to neurotransmission and peripheral pain perception. ASIC1a and ASIC1b are alternative splice products of the ASIC1 gene. Here we show that both splice variants show steady-state inactivation when exposed to slightly decreased pH, limiting their operational range. Compared with ASIC1a, steady-state inactivation and pH activation of ASIC1b are shifted to more acidic values by 0.25 and 0.7 pH units, respectively, extending the dynamic range of ASIC1. Shifts of inactivation and activation are intimately linked; only two amino acids in the ectodomain, which are exchanged by alternative splicing, control both properties. Moreover, we show that extracellular, divalent cations like Ca(2+) and Mg(2+) as well as the polyvalent cation spermine shift the steady-state inactivation of ASIC1a and ASIC1b to more acidic values. This leads to a potentiation of the channel response and is due to a stabilization of the resting state. Our results indicate that ASIC1b is an effective sensor of transient H(+) signals during slight acidosis and that, in addition to alternative splicing, interaction with di- and polyvalent cations extends the dynamic range of ASIC H(+) sensors.     

Dictyostelium discoideum chemotaxis: threshold for directed motion

The chemotactic response of Dictyostelium discoideum cells to stationary, linear gradients of cyclic adenosine 3',5'-monophosphate (cAMP) was studied using microfluidic devices. In shallow gradients of less than 10(-3) nM/microm, the cells showed no directional response and exhibited a constant basal motility. In steeper gradients, cells moved up the gradient on average. The chemotactic speed and the motility increased with increasing steepness up to a plateau at around 10(-1) nM/microm. In very steep gradients, above 10 nM/microm, the cells lost directionality and the motility returned to the sub-threshold level. In the regime of optimal response the difference in receptor occupancy at the front and back of the cell is estimated to be only about 100 molecules.     

Quantitative elucidation of a distinct spatial gradient-sensing mechanism in fibroblasts

Migration of eukaryotic cells toward a chemoattractant often relies on their ability to distinguish receptor-mediated signaling at different subcellular locations, a phenomenon known as spatial sensing. A prominent example that is seen during wound healing is fibroblast migration in platelet-derived growth factor (PDGF) gradients. As in the well-characterized chemotactic cells Dictyostelium discoideum and neutrophils, signaling to the cytoskeleton via the phosphoinositide 3-kinase pathway in fibroblasts is spatially polarized by a PDGF gradient; however, the sensitivity of this process and how it is regulated are unknown. Through a quantitative analysis of mathematical models and live cell total internal reflection fluorescence microscopy experiments, we demonstrate that PDGF detection is governed by mechanisms that are fundamentally different from those in D. discoideum and neutrophils. Robust PDGF sensing requires steeper gradients and a much narrower range of absolute chemoattractant concentration, which is consistent with a simpler system lacking the feedback loops that yield signal amplification and adaptation in amoeboid cells.     

Time response and stability of porous silicon capacitive immunosensors

The time response of affinity sensors made with nanostructured materials is a topic of considerable interest, since affinity sensors made with nanostructured materials provide greater sensitivities than corresponding planar crystalline devices but at the cost of stability and drift. We present a study of the time response of capacitive immunosensors made using porous silicon and ultrathin room temperature anodic oxide. It was found that sensor drift can be substantial but can be reduced by subjecting the capacitive immunosensor in buffer to an anodic bias that is larger than the bias at which sensor capacitance is measured. By measuring sensor response before the addition of the analyte and using it for baseline correction after addition of the analyte, the effect of nonspecific sensor drift can be further reduced. We observed that after the addition of the analyte to the porous silicon immunocapacitor, there is a fast decrease in capacitance (order of tens of seconds) followed by a slow increase (order of tens of minutes), which models well as a sum of exponents with a fast exponential decay followed by a slow exponential rise. Possible processes that can give rise to such a response are perturbations of the double layer for the fast decay and column resistance switching for the slow rise.     

Stochastic model of chemoattractant receptor dynamics in leukocyte chemosensory movement

Mammalian white blood cells are known to bias the direction of their movement along concentration gradients of specific chemical stimuli, a phenomenon called chemotaxis. Chemotaxis of leukocyte cells is central to the acute inflammatory response in living organisms and other critical physiological functions. On a molecular level, these cells sense the stimuli termed chemotactic factor (CF) through specific cell surface receptors that bind CF molecules. This triggers a complex signal transduction process involving intracellular biochemical pathways and biophysical events, eventually leading to the observable chemotactic response. Several investigators have shown theoretically that statistical fluctuations in receptor binding lead to “noisy” intracellular signals, which may explain the observed imperfect chemotactic response to a CF gradient. The most recent dynamic model (Tranquillo and Lauffenburger,J. Math. Biol. 25, 229–262. 1987) couples a scheme for intracellular signal transduction and cell motility response with fluctuations in receptor binding. However, this model employs several assumptions regarding receptor dynamics that are now known to be oversimplifications. We extend the earlier model by accounting for several known and speculated chemotactic receptor dynamics, namely, transient G-protein signaling, cytoskeletal association, and receptor internalization and recycling, including statistical fluctuations in the numbers of receptors among the various states. Published studies are used to estimate associated constants and ensure the predicted receptor distribution is accurate. Model analysis indicates that directional persistence in uniform CF concentrations is enhanced by increasing rate constants for receptor cytoskeletal inactivation, ternary complex dissociation, and binary complex dissociation, and by decreasing rate constants for receptor internalization and recycling. For most rate constants, we have detected an optimal range that maximizes orientation bias in CF gradients. We have also examined different desensitization and receptor recycling mechanisms that yield experimentally documented orientation behavior. These yield novel insights into the relationship between receptor dynamics and leukocyte chemosensory movement behavior.     

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.     

Species range expansion by beneficial mutations

A species' range can be limited when there is no genetic variation for a trait that allows for adaptation to more extreme environments. We study how range expansion occurs by the establishment of a new mutation that affects a quantitative trait in a spatially continuous population. The optimal phenotype for the trait varies linearly in space. The survival probabilities of new mutations affecting the trait are found by simulation. Shallow environmental gradients favour mutations that arise nearer to the range margin and that have smaller phenotypic effects than do steep gradients. Mutations that become established in shallow environmental gradients typically result in proportionally larger range expansions than those that establish in steep gradients. Mutations that become established in populations with high maximum growth rates tend to originate nearer to the range edge and to cause relatively smaller range expansion than mutations that establish in populations with low maximum growth rates. Under plausible parameter values, mutations that allow for range expansion tend to have large phenotypic effects (more than one phenotypic standard deviation) and cause substantial range expansions (15% or more). Sexual reproduction allows for larger range expansions and adaptation to more extreme environments than asexual reproduction.