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.

Overview of mathematical approaches used to model bacterial chemotaxis II: bacterial populations

We review the application of mathematical modeling to understanding the behavior of populations of chemotactic bacteria. The application of continuum mathematical models, in particular generalized Keller-Segel models, is discussed along with attempts to incorporate the microscale (individual) behavior on the macroscale, modeling the interaction between different species of bacteria, the interaction of bacteria with their environment, and methods used to obtain experimentally verified parameter values. We allude briefly to the role of modeling pattern formation in understanding collective behavior within bacterial populations. Various aspects of each model are discussed and areas for possible future research are postulated.

Theoretical insights into bacterial chemotaxis

Research into understanding bacterial chemotactic systems has become a paradigm for Systems Biology. Experimental and theoretical researchers have worked hand-in-hand for over 40 years to understand the intricate behavior driving bacterial species, in particular how such small creatures, usually not more than 5 µm in length, detect and respond to small changes in their extracellular environment. In this review we highlight the importance that theoretical modeling has played in providing new insight and understanding into bacterial chemotaxis. We begin with an overview of the bacterial chemotaxis sensory response, before reviewing the role of theoretical modeling in understanding elements of the system on the single cell scale and features underpinning multiscale extensions to population models. WIREs Syst Biol Med 2012 doi: 10.1002/wsbm.1168 For further resources related to this article, please visit the WIREs website.

Two binding modes reveal flexibility in kinase/response regulator interactions in the bacterial chemotaxis pathway

The crystal structure at 2.0-Å resolution of the complex of the Escherichia coli chemotaxis response regulator CheY and the phosphoacceptor-binding domain (P2) of the kinase CheA is presented. The binding interface involves the fourth and fifth helices and fifth β-strand of CheY and both helices of P2. Surprisingly, the two heterodimers in the asymmetric unit have two different binding modes involving the same interface, suggesting some flexibility in the binding regions. Significant conformational changes have occurred in CheY compared with previously determined unbound structures. The active site of CheY is exposed by the binding of the kinase domain, possibly to enhance phosphotransfer from CheA to CheY. The conformational changes upon complex formation as well as the observation that there are two different binding modes suggest that the plasticity of CheY is an essential feature of response regulator function.

A model of excitation and adaptation in bacterial chemotaxis

Bacterial chemotaxis is widely studied because of its accessibility and because it incorporates processes that are important in a number of sensory systems: signal transduction, excitation, adaptation, and a change in behavior, all in response to stimuli. Quantitative data on the change in behavior are available for this system, and the major biochemical steps in the signal transduction/processing pathway have been identified. We have incorporated recent biochemical data into a mathematical model that can reproduce many of the major features of the intracellular response, including the change in the level of chemotactic proteins to step and ramp stimuli such as those used in experimental protocols. The interaction of the chemotactic proteins with the motor is not modeled, but we can estimate the degree of cooperativity needed to produce the observed gain under the assumption that the chemotactic proteins interact directly with the motor proteins.

Signal termination in bacterial chemotaxis: CheZ mediates dephosphorylation of free rather than switch-bound CheY.

Chemotaxis in bacteria is controlled by regulating the direction of flagellar rotation. The regulation is carried out by the chemotaxis protein CheY. When phosphorylated, CheY binds to FliM, which is one of the proteins that constitute the "gear box" (or "switch") of the flagellar motor. Consequently, the motor shifts from the default direction of rotation, counterclockwise, to clockwise rotation. This biased rotation is terminated when CheY is dephosphorylated either spontaneously or, faster, by a specific phosphatase, CheZ. Logically, one might expect CheZ to act directly on FliM-bound CheY. However, here we provide direct biochemical evidence that, in contrast to this expectation, phosphorylated CheY (CheY approximately P), bound to FliM, is protected from dephosphorylation by CheZ. The complex between CheY approximately P and FliM was trapped by cross-linking with dimethylsuberimidate, and its susceptibility to CheZ was measured. CheY approximately P complexed with FliM, unlike free CheY approximately P, was not dephosphorylated by CheZ. However, it did undergo spontaneous dephosphorylation. Nonspecific cross-linked CheY dimers, measured as a control, were dephosphorylated by CheZ. No significant binding between CheZ and any of the switch proteins was detected. It is concluded that, in the termination mechanism of signal transduction in bacterial chemotaxis, CheZ acts only on free CheY approximately P. We suggest that CheZ affects switch-bound CheY approximately P by shifting the equilibrium between bound and free CheY approximately P.

Modeling chemotaxis reveals the role of reversed phosphotransfer and a bi-functional kinase-phosphatase

Understanding how multiple signals are integrated in living cells to produce a balanced response is a major challenge in biology. Two-component signal transduction pathways, such as bacterial chemotaxis, comprise histidine protein kinases (HPKs) and response regulators (RRs). These are used to sense and respond to changes in the environment. Rhodobacter sphaeroides has a complex chemosensory network with two signaling clusters, each containing a HPK, CheA. Here we demonstrate, using a mathematical model, how the outputs of the two signaling clusters may be integrated. We use our mathematical model supported by experimental data to predict that: (1) the main RR controlling flagellar rotation, CheY6, aided by its specific phosphatase, the bifunctional kinase CheA3, acts as a phosphate sink for the other RRs; and (2) a phosphorelay pathway involving CheB2 connects the cytoplasmic cluster kinase CheA3 with the polar localised kinase CheA2, and allows CheA3-P to phosphorylate non-cognate chemotaxis RRs. These two mechanisms enable the bifunctional kinase/phosphatase activity of CheA3 to integrate and tune the sensory output of each signaling cluster to produce a balanced response. The signal integration mechanisms identified here may be widely used by other bacteria, since like R. sphaeroides, over 50% of chemotactic bacteria have multiple cheA homologues and need to integrate signals from different sources.

Response kinetics in the complex chemotaxis signalling pathway of Rhodobacter sphaeroides

Chemotaxis is one of the best characterised signalling systems in biology. It is the mechanism by which bacteria move towards optimal environments and is implicated in biofilm formation, pathogenesis and symbiosis. The properties of the bacterial chemosensory response have been described in detail for the single chemosensory pathway of Escherichia coli. We have characterised the properties of the chemosensory response of Rhodobacter sphaeroides, an -proteobacterium with multiple chemotaxis pathways, under two growth conditions allowing the effects of protein expression levels and cell architecture to be investigated. Using tethered cell assays we measured the responses of the system to step changes in concentration of the attractant propionate and show that, independently of the growth conditions, R. sphaeroides is chemotactic over at least five orders of magnitude and has a sensing profile following Weber’s law. Mathematical modelling also shows that, like E. coli, R. sphaeroides is capable of showing Fold-Change Detection (FCD). Our results indicate that general features of bacterial chemotaxis such as the range and sensitivity of detection, adaptation times, adherence to Weber’s law and the presence of FCD may be integral features of chemotaxis systems in general, regardless of network complexity, protein expression levels and cellular architecture across different species.

Conformational coupling in the chemotaxis response regulator CheY

CheY, a response regulator protein in bacterial chemotaxis, serves as a prototype for the analysis of response regulator function in two-component signal transduction. Phosphorylation of a conserved aspartate at the active site mediates a conformational change at a distal signaling surface that modulates interactions with the flagellar motor component FliM, the sensor kinase CheA, and the phosphatase CheZ. The objective of this study was to probe the conformational coupling between the phosphorylation site and the signaling surface of CheY in the reverse direction by quantifying phosphorylation activity in the presence and absence of peptides of CheA, CheZ, and FliM that specifically interact with CheY. Binding of these peptides dramatically impacted autophosphorylation of CheY by small molecule phosphodonors, which is indicative of reverse signal propagation in CheY. Autodephosphorylation and substrate affinity, however, were not significantly affected. Kinetic characterization of several CheY mutants suggested that conserved residues Thr-87, Tyr-106, and Lys-109, implicated in the activation mechanism, are not essential for conformational coupling. These findings provide structural and conceptual insights into the mechanism of CheY activation. Our results are consistent with a multistate thermodynamic model of response regulator activation.

A geometric construction of travelling wave solutions to the Keller–Segel model

We study a version of the Keller–Segel model for bacterial chemotaxis, for which exact travelling wave solutions are explicitly known in the zero attractant diffusion limit. Using geometric singular perturbation theory, we construct travelling wave solutions in the small diffusion case that converge to these exact solutions in the singular limit.

Numerical computation of an Evans function for travelling waves

We demonstrate a geometrically inspired technique for computing Evans functions for the linearised operators about travelling waves. Using the examples of the F-KPP equation and a Keller–Segel model of bacterial chemotaxis, we produce an Evans function which is computable through several orders of magnitude in the spectral parameter and show how such a function can naturally be extended into the continuous spectrum. In both examples, we use this function to numerically verify the absence of eigenvalues in a large region of the right half of the spectral plane. We also include a new proof of spectral stability in the appropriate weighted space of travelling waves of speed c≥sqrt(2δ) in the F-KPP equation.

A chemotactic-based model for spatial activity recognition

Spatial activity recognition in everyday environments is particularly challenging due to noise incorporated during video-tracking. We address the noise issue of spatial recognition with a biologically inspired chemotactic model that is capable of handling noisy data. The model is based on bacterial chemotaxis, a process that allows bacteria to survive by changing motile behaviour in relation to environmental dynamics. Using chemotactic principles, we propose the chemotactic model and evaluate its classification performance in a smart house environment. The model exhibits high classification accuracy (99%) with a diverse 10 class activity dataset and outperforms the discrete hidden Markov model (HMM). High accuracy (>89%) is also maintained across small training sets and through incorporation of varying degrees of artificial noise into testing sequences. Importantly, unlike other bottom–up spatial activity recognition models, we show that the chemotactic model is capable of recognizing simple interwoven activities.

Spatial activity recognition in a smart home environment using a chemotactic model

Spatial activity recognition is challenging due to the amount of noise incorporated during video tracking in everyday environments. We address the spatial recognition problem with a biologically-inspired chemotactic model that is capable of handling noisy data. The model is based on bacterial chemotaxis, a process that allows bacteria to change motile behaviour in relation to environmental gradients. Through adoption of chemotactic principles, we propose the chemotactic model and evaluate its performance in a smart house environment. The model exhibits greater than 99% recognition performance with a diverse six class dataset and outperforms the Hidden Markov Model (HMM). The approach also maintains high accuracy (90-99%) with small training sets of one to five sequences. Importantly, unlike other low-level spatial activity recognition models, we show that the chemotactic model is capable of recognising simple interwoven activities.