Perfect and near-perfect adaptation in a model of bacterial chemotaxis

The signaling apparatus mediating bacterial chemotaxis can adapt to a wide range of persistent external stimuli. In many cases, the bacterial activity returns to its prestimulus level exactly, and this perfect adaptability is robust against variations in various chemotaxis protein concentrations. We model the bacterial chemotaxis signaling pathway, from ligand binding to CheY phosphorylation. By solving the steady-state equations of the model analytically, we derive a full set of conditions for the system to achieve perfect adaptation. The conditions related to the phosphorylation part of the pathway are discovered for the first time, while other conditions are generalizations of the ones found in previous works. Sensitivity of the perfect adaptation is evaluated by perturbing these conditions. We find that, even in the absence of some of the perfect adaptation conditions, adaptation can be achieved with near-perfect precision as a result of the separation of scales in both chemotaxis protein concentrations and reaction rates, or specific properties of the receptor distribution in different methylation states. Since near-perfect adaptation can be found in much larger regions of the parameter space than that defined by the perfect adaptation conditions, their existence is essential to understand robustness in bacterial chemotaxis.     

Precision and kinetics of adaptation in bacterial chemotaxis

The chemotaxis network of the bacterium Escherichia coli is perhaps the most studied model for adaptation of a signaling system to persistent stimuli. Although adaptation in this system is generally considered to be precise, there has been little effort to quantify this precision, or to understand how and when precision fails. Using a Förster resonance energy transfer-based reporter of signaling activity, we undertook a systematic study of adaptation kinetics and precision in E. coli cells expressing a single type of chemoreceptor (Tar). Quantifiable loss of precision of adaptation was observed at levels of the attractant MeAsp as low 10 μM, with pronounced differences in both kinetics and precision of adaptation between addition and removal of attractant. Quantitative modeling of the kinetic data suggests that loss of precise adaptation is due to a slowing of receptor methylation as available modification sites become scarce. Moreover, the observed kinetics of adaptation imply large cell-to-cell variation in adaptation rates—potentially providing genetically identical cells with the ability to “hedge their bets” by pursuing distinct chemotactic strategies.     

Studies of bacterial chemotaxis in defined concentration gradients. A model for chemotaxis toward L-serine

The details of the chemotactic response of Salmonella typhimurium to gradients of L-serine have been examined in some detail. Two relatively macroscopic techniques have been employed to measure the bacterial response. These include measurements of the average velocity as the bacterial population moves toward attractants, and measurement of the upward-to-downward flux ratio, R, in the stable preformed attractant gradients. The dependence of the average velocity on gradient appears to be hyperbolic in nature, while the flux ratio depends linearly on the gradient. These data suggest a microscopic model for the dependence of bacterial behavior on the serine gradient. The model involves a linear dependence of the mean lifetime of a bacterial trajectory on the gradient for those bacteria moving toward higher attractant concentration. Those moving toward low concentrations of attractant do not change the mean duration of their trajectories, or the speed at which a given bacterium swims through the solution. This model generates the observed dependences of the average velocity and flux ratio on gradient. Interpretation of the experimental data suggests that a gradient which increases serine concentration by a factor of 2 in 10 mm is sufficient to double the average duration of a trajectory for a bacterium moving directly up the gradient. The concentration dependence of the chemotactic response to serine is more complicated. It suggests that more than one receptor of serine may be involved in determining chemotactic behavior to this attractant.     

Lattice-Boltzmann model for bacterial chemotaxis

We present a new numerical approach for modeling bacterial chemotaxis and the fate and transport of a chemoattractant in bulk liquids. This Lattice-Boltzmann method represents the microorganisms and the chemoattractant by quasi-particles that move, collide, and react with each other on a two-dimensional numerical lattice. We use the model to simulate traveling bands of bacteria along self-generated gradients in substrate concentration in bulk liquids. Particularly, we simulate Pseudomonas putida that respond chemotactically to naphthalene dissolved in water. We find that only a fraction of a bacterial slug injected into a domain containing the chemoattractant at constant concentration forms a traveling band as the slug length exceeds a critical value. An expanding bacterial ring forms as one injects a droplet of bacteria into a two-dimensional domain.     

Simple molecular model for sensing and adaptation based on receptor modification with application to bacterial chemotaxis

A series of simple models to explain adaptation in a sensory system based on reversible covalent modification is developed. The models are applied to the reversible methylation of chemoreceptors in bacteria and by analogy to other sensory transduction systems. The receptor modification system exhibits sensing and adaptation, i.e. raising the stimulus to a new level generates a transient response followed by a return to prestimulus behavior. By means of an analytical solution of the kinetic equation that governs the evolution of the receptor system. an exact expression is obtained for the time required for adaptation. The results account for the most conspicuous properties of the bacterial sensory system; namely, the response times in relation to stimulus changes, the proportionality of receptor modification to receptor occupancy, and the additivity of response times. The analysis indicates how these properties depend upon the parameters of the system, e.g. the rates of covalent modification and demodification, the accuracy of the detector, and the molecular nature of the response regulator. The theory developed for analysis of the bacterial system revealed properties that will be applicable to any system processing sensory information.     

Adaptation kinetics in bacterial chemotaxis.

Cells of Escherichia coli, tethered to glass by a single flagellum, were subjected to constant flow of a medium containing the attractant alpha-methyl-DL-aspartate. The concentration of this chemical was varied with a programmable mixing apparatus over a range spanning the dissociation constant of the chemoreceptor at rates comparable to those experienced by cells swimming in spatial gradients. When an exponentially increasing ramp was turned on (a ramp that increases the chemoreceptor occupancy linearly), the rotational bias of the cells (the fraction of time spent spinning counterclockwise) changed rapidly to a higher stable level, which persisted for the duration of the ramp. The change in bias increased with ramp rate, i.e., with the time rate of change of chemoreceptor occupancy. This behavior can be accounted for by a model for adaptation involving proportional control, in which the flagellar motors respond to an error signal proportional to the difference between the current occupancy and the occupancy averaged over the recent past. Distributions of clockwise and counterclockwise rotation intervals were found to be exponential. This result cannot be explained by a response regular model in which transitions between rotational states are generated by threshold crossings of a regular subject to statistical fluctuation; this mechanism generates distributions with far too many long events. However, the data can be fit by a model in which transitions between rotational states are governed by first-order rate constants. The error signal acts as a bias regulator, controlling the values of these constants.     

A spatially extended stochastic model of the bacterial chemotaxis signalling pathway

We have combined two distinct but related stochastic approaches to model the Escherichia coli chemotaxis pathway. Reactions involving cytosolic components of the pathway were assumed to obey the laws of conventional stochastic chemical kinetics, while the clustered membrane receptors were represented in two-dimensional arrays similar to the Ising model. Receptors were assumed to flip between an active and an inactive state with probabilities dependent upon three energy inputs: ligand binding, methylation level due to adaptation, and the activity of neighbouring receptors. Examination of models with different lattice size and geometry showed that the sensitivity to stimuli increases with lattice size and the nearest-neighbour coupling strength up to a critical point, but this amplification was also accompanied by a proportional increase in steady-state noise. Multiple methylation of receptors resulted in diminished signal-to-noise ratio, but showed improved stability to variation in the coupling strength and increased gain. Under the best conditions the simulated output of a coupled lattice of receptors closely matched the time-course and amplitude found experimentally in living bacteria. The model also has some of the properties of a cellular automaton and shows an unexpected emergence of spatial patterns of methylation within the receptor lattice.     

Reconstitution of signaling in bacterial chemotaxis.

Strains missing several genes required for chemotaxis toward amino acids, peptides, and certain sugars were tethered and their rotational behavior was analyzed. Null strains (called gutted) were deleted for genes that code for the transducers Tsr, Tar, Tap, and Trg and for the cytoplasmic proteins CheA, CheW, CheR, CheB, CheY, and CheZ. Motor switch components were wild type, flaAII(cheC), or flaBII(cheV). Gutted cells with wild-type motors spun exclusively counterclockwise, while those with mutant motors changed their directions of rotation. CheY reduced the bias (the fraction of time that cells spun counterclockwise) in either case. CheZ offset the effect of CheY to an extent that varied with switch allele but did not change the bias when tested alone. Transducers also increased the bias in the presence of CheY but not when tested alone. However, cells containing transducers and CheY failed to respond to attractants or repellents normally detected in the periplasm. This sensitivity was restored by addition of CheA and CheW. Thus, CheY both enhances clockwise rotation and couples the transducers to the flagella. CheZ acts, at the level of the motor, as a CheY antagonist. CheA or CheW or both are required to complete the signal pathway. A model is presented that explains these results and is consistent with other data found in the literature.     

An allosteric model for transmembrane signaling in bacterial chemotaxis

Bacteria are able to sense chemical gradients over a wide range of concentrations. However, calculations based on the known number of receptors do not predict such a range unless receptors interact with one another in a cooperative manner. A number of recent experiments support the notion that this remarkable sensitivity in chemotaxis is mediated by localized interactions or crosstalk between neighboring receptors. A number of simple, elegant models have proposed mechanisms for signal integration within receptor clusters. What is a lacking is a model, based on known molecular mechanisms and our accumulated knowledge of chemotaxis, that integrates data from multiple, heterogeneous sources. To address this question, we propose an allosteric mechanism for transmembrane signaling in bacterial chemotaxis based on the "trimer of dimers" model, where three receptor dimers form a stable complex with CheW and CheA. The mechanism is used to integrate a diverse set of experimental data in a consistent framework. The main predictions are: (1) trimers of receptor dimers form the building blocks for the signaling complexes; (2) receptor methylation increases the stability of the active state and retards the inhibition arising from ligand-bound receptors within the signaling complex; (3) trimer of dimer receptor complexes aggregate into clusters through their mutual interactions with CheA and CheW; (4) cooperativity arises from neighboring interaction within these clusters; and (5) cluster size is determined by the concentration of receptors, CheA, and CheW. The model is able to explain a number of seemingly contradictory experiments in a consistent manner and, in the process, explain how bacteria are able to sense chemical gradients over a wide range of concentrations by demonstrating how signals are integrated within the signaling complex.     

Response regulator output in bacterial chemotaxis.

Chemotaxis responses in Escherichia coli are mediated by the phosphorylated response-regulator protein P-CheY. Biochemical and genetic studies have established the mechanisms by which the various components of the chemotaxis system, the membrane receptors and Che proteins function to modulate levels of CheY phosphorylation. Detailed models have been formulated to explain chemotaxis sensing in quantitative terms; however, the models cannot be adequately tested without knowledge of the quantitative relationship between P-CheY and bacterial swimming behavior. A computerized image analysis system was developed to collect extensive statistics on freeswimming and individual tethered cells. P-CheY levels were systematically varied by controlled expression of CheY in an E.coli strain lacking the CheY phosphatase, CheZ, and the receptor demethylating enzyme CheB. Tumbling frequency was found to vary with P-CheY concentration in a weakly sigmoidal fashion (apparent Hill coefficient approximately 2.5). This indicates that the high sensitivity of the chemotaxis system is not derived from highly cooperative interactions between P-CheY and the flagellar motor, but rather depends on nonlinear effects within the chemotaxis signal transduction network. The complex relationship between single flagella rotation and free-swimming behavior was examined; our results indicate that there is an additional level of information processing associated with interactions between the individual flagella. An allosteric model of the motor switching process is proposed which gives a good fit to the observed switching induced by P-CheY. Thus the level of intracellular P-CheY can be estimated from behavior determinations: approximately 30% of the intracellular pool of CheY appears to be phosphorylated in fully adapted wild-type cells.     

A biochemical mechanism for bacterial chemotaxis

Peritrichous bacteria alternately swim and tumble (thrash about with little forward progress). By selective modulation of tumbling frequency, these bacteria carry out chemotaxis, which is migration to higher concentrations of attractant or lower concentrations of repellent. A model for chemotaxis is presented in which tumbling frequency is regulated by concentration of Ca²⁺ ion at the switch that controls tumbling and swimming. Attractants cause decreased levels of free cytoplasmic Ca²⁺ ion due to binding of Ca²⁺ ion by specific proteins. This Ca²⁺ ion is released when these proteins become methylated. An alternative model. involving a cytoplasmic metabolite “compound X”, is discussed.     

Ribose and glucose-galactose receptors. Competitors in bacterial chemotaxis.

The periplasmic ribose and glucose-galactose receptors (binding proteins) of Gram-negative bacteria compete for a common inner membrane receptor in bacterial chemotaxis, as well as being the essential primary receptors for their respective membrane transport systems. The high-resolution structures of the periplasmic receptors for ribose (from Escherichia coli) and glucose or galactose (from both Salmonella typhimurium and E. coli) are compared here to outline some features that may be important in their dual functions. The overall structure of each protein consists of two similar domains, both of which are made up of two non-contiguous segments of amino acid chain. Each domain is composed of a core of beta-sheet flanked on both sides with alpha-helices. The two domains are related to each other by an almost perfect intramolecular axis of symmetry. The ribose receptor is smaller as a result of a number of deletions in its sequence relative to the glucose-galactose receptor, mostly occurring in the loop regions; as a result, this protein is also more symmetrical. Many structural features, including some hydrophobic core interactions, a buried aspartate residue and several unusual turns, are conserved between the two proteins. The binding sites for ligand are in similar locations, and built along similar principles, although none of the specific interactions with the sugars is conserved. A comparison shows further that slightly different rotations relate the domains to each other in the three proteins, with the ribose receptor being the most closed, and the Salmonella glucose-galactose receptor the most open. The primary axis of relative rotation is almost perpendicular to that which describes the intramolecular symmetry in each case. These relative rotations of the domains are accompanied by the sliding of some helices as the structures adjust themselves to relieve strain. The hinges which are responsible for most of these relative domain rotations are very similar in the three proteins, consisting of a symmetrical arrangement of beta-strands and alpha-helices and two conserved water molecules that are critical to the hydrogen bonding in the important interdomain region. A region of high sequence and structural similarity between the ribose and glucose-galactose receptors is also located around the intramolecular symmetry axis, on the opposite side of the proteins from the hinge region. This region is that which is altered most by the relative rotations, and is the location of most of the known mutations which affect chemotaxis and transport in the ribose receptor.     

Oxygen as attractant and repellent in bacterial chemotaxis.

Studies of bacterial chemotaxis to oxygen (aerotaxis) over a broad range of oxygen concentrations showed that at high concentrations, oxygen was a repellent of Salmonella typhimurium, Escherichia coli, and some bacilli, whereas it is known that at lower concentrations (less than or equal to 0.25 mM dissolved oxygen), oxygen is an attractant. In a temporal assay of aerotaxis, S. typhimurium in medium equilibrated with air (0.25 mM dissolved oxygen) and then exposed to pure oxygen (1.2 mM) tumbled continuously for approximately 20 s. The oxygen concentration that elicited a half-maximal negative (repellent) response was 1.0 mM for both S. typhimurium and E. coli. The receptor for the negative chemoresponse to high concentrations of oxygen is apparently different from the receptor for the positive chemoresponse to low concentrations of oxygen, since the oxygen concentration that elicits a half-maximal positive (attractant) response in S. typhimurium and E. coli is reported to be 0.7 microM. Adaptation to high concentrations of oxygen, like adaptation to low concentrations of oxygen, was independent of methylation of a transducer protein. Only the response to low oxygen concentrations, however, was altered by interaction with the amidated Tsr transducer in cheB mutants.     

Phosphorylating and dephosphorylating protein complexes in bacterial chemotaxis.

During optimal motility conditions, a 1:1 stoichiometry of CheA(L) (654 amino acids) to CheA(S) (557 amino acids) was determined. It was also found that CheZ binding to CheA(S) was inhibited by CheA(L)-CheA(S)-CheW interaction. This suggests that CheA(S) has different functions in the phosphorylating complex (CheA(L)-CheA(S)-CheW) and in the dephosphorylating complex (CheA(S)-CheZ).     

Sensory adaptation during negative chemotaxis in Myxococcus xanthus.

Myxococcus xanthus exhibits many tactic movements that require the frz signal transduction system, such as colony swarming and cellular aggregation during fruiting body formation. Previously we demonstrated that the Frz proteins control the chemotactic movements of M. xanthus (W. Shi, T. Köhler, and D. R. Zusman, Mol. Microbiol. 9:601-611, 1993). However it was unclear from that study how chemotaxis might be achieved at the cellular level. In this study, we showed that M. xanthus cells not only modulate the reversal frequency of cell movement in response to repellent stimuli but also exhibit sensory adaptation in response to the continuous presence of nonsaturating repellent stimuli. The sensory adaptation behavior requires FrzF (a putative methyltransferase) and is correlated with the methylation-demethylation of FrzCD, a methyl-accepting chemotaxis protein. These results indicate that negative chemotaxis in M. xanthus is achieved by chemokinesis plus sensory adaptation in a manner analogous to that of the free-swimming enteric bacteria.     

A hemerythrin-like domain in a bacterial chemotaxis protein

Hemerythrin (Hr) is an O(2)-carrying protein found in some marine invertebrates. A conserved sequence motif in all Hrs provides five histidine and two carboxylate ligands to an oxo-/hydroxo-bridged diiron active site, as well as a hydrophobic O(2) binding pocket. Database searches located a previously unrecognized Hr-like sequence motif at the 3' end of the gene, dcrH, from the anaerobic sulfate-reducing bacterium, Desulfovibrio (D.) vulgaris (Hildenborough). This gene encodes a putative methyl-accepting chemotaxis protein, DcrH. We have established by immunoblotting that a full-length DcrH, including the Hr-like domain, is expressed in D. vulgaris (Hildenborough). The C-terminal domain of DcrH, when expressed separately in recombinant form in Escherichia coli, was found to fold into a stable protein, DcrH-Hr. The UV-vis absorption and resonance Raman spectra of DcrH-Hr, and of its azide adduct, provide clear evidence for an oxo-bridged diiron(III) site very similar to that found in Hr. Based on UV-vis absorption spectra, exposure of the reduced (colorless, presumably diferrous) DcrH-Hr to air resulted in formation of an O(2) adduct also very similar to that of Hr. Unlike that of Hr, the O(2) adduct of DcrH-Hr autoxidized within a few minutes at room temperature. The O(2) binding pocket of DcrH-Hr appears to be larger than that of Hr. Given the air-sensitive nature of D. vulgaris and the putative chemotactic function of DcrH, one possible role for the Hr-like domain of DcrH is O(2)-sensing. DcrH-Hr is the first characterized example of a Hr-like protein from any microorganism.     

A model of chemotaxis and associative learning in C. elegans

The nematode C. elegans has attracted a great deal of interest from the neuroscience community due to the simplicity of its nervous system, which in the hermaphrodite is composed of just 302 neurons. C. elegans is known to engage in a number of sophisticated behaviours such as chemo- and thermotaxis. Experimental work has shown that these behaviours can be modified by experience and that C. elegans is capable of associative learning. In this paper, we focus on the chemotactic response of C. elegans to sodium chloride mediated by the ASE sensory neurons. We construct a biophysical model of the ASEL and ASER neurons that captures the time course of the ASE responses in response to up- and down-steps in NaCl concentration. We use this model to show that the time course of the ASE responses provide sufficient temporal resolution to successfully drive chemotaxis in C. elegans via steering, pirouettes and control of final turn angle. We show that these different locomotion strategies are individually capable of driving chemotaxis and that by working together they produce the best chemotactic response. We find that there is a separation into upward and downward drives mediated by the left and right ASE neurons. We show that the connectivity from ASEL and ASER must be of opposite polarity and that ASER, and the concomitant ability to sense when the worm is moving down the gradient, is more important for chemotaxis than ASEL, findings that are consistent with existing modelling studies in the literature. Finally, we examine associative learning in the network and show that experimental data can be explained by changes that occur at either the synaptic or sensory neuron level, the choice of which has distinct consequences for network function.