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

Meredys, a multi-compartment reaction-diffusion simulator using multistate realistic molecular complexes

<p>Abstract</p> <p>Background</p> <p>Most cellular signal transduction mechanisms depend on a few molecular partners whose roles depend on their position and movement in relation to the input signal. This movement can follow various rules and take place in different compartments. Additionally, the molecules can form transient complexes. Complexation and signal transduction depend on the specific states partners and complexes adopt. Several spatial simulator have been developed to date, but none are able to model reaction-diffusion of realistic multi-state transient complexes.</p> <p>Results</p> <p><it>Meredys </it>allows for the simulation of multi-component, multi-feature state molecular species in two and three dimensions. Several compartments can be defined with different diffusion and boundary properties. The software employs a Brownian dynamics engine to simulate reaction-diffusion systems at the reactive particle level, based on compartment properties, complex structure, and hydro-dynamic radii. Zeroth-, first-, and second order reactions are supported. The molecular complexes have realistic geometries. Reactive species can contain user-defined feature states which can modify reaction rates and outcome. Models are defined in a versatile NeuroML input file. The simulation volume can be split in subvolumes to speed up run-time.</p> <p>Conclusions</p> <p><it>Meredys </it>provides a powerful and versatile way to run accurate simulations of molecular and sub-cellular systems, that complement existing multi-agent simulation systems. <it>Meredys </it>is a Free Software and the source code is available at <url>http://meredys.sourceforge.net/</url>.</p

Differential Affinity and Catalytic Activity of CheZ in E. coli Chemotaxis

Push–pull networks, in which two antagonistic enzymes control the activity of a messenger protein, are ubiquitous in signal transduction pathways. A classical example is the chemotaxis system of the bacterium Escherichia coli, in which the kinase CheA and the phosphatase CheZ regulate the phosphorylation level of the messenger protein CheY. Recent experiments suggest that both the kinase and the phosphatase are localized at the receptor cluster, and Vaknin and Berg recently demonstrated that the spatial distribution of the phosphatase can markedly affect the dose–response curves. We argue, using mathematical modeling, that the canonical model of the chemotaxis network cannot explain the experimental observations of Vaknin and Berg. We present a new model, in which a small fraction of the phosphatase is localized at the receptor cluster, while the remainder freely diffuses in the cytoplasm; moreover, the phosphatase at the cluster has a higher binding affinity for the messenger protein and a higher catalytic activity than the phosphatase in the cytoplasm. This model is consistent with a large body of experimental data and can explain many of the experimental observations of Vaknin and Berg. More generally, the combination of differential affinity and catalytic activity provides a generic mechanism for amplifying signals that could be exploited in other two-component signaling systems. If this model is correct, then a number of recent modeling studies, which aim to explain the chemotactic gain in terms of the activity of the receptor cluster, should be reconsidered

Mathematical analysis of the Escherichia coli chemotaxis signalling pathway

We undertake a detailed mathematical analysis of a recent nonlinear ordinary differential equation (ODE) model describing the chemotactic signalling cascade within an {\it Escherichia coli} cell. The model includes a detailed description of the cell signalling cascade and an average approximation of the receptor activity. A steady-state stability analysis reveals the system exhibits one positive real steady-state which is shown to be asymptotically stable. Given the occurrence of a negative feedback between phosphorylated CheB (CheB-P) and the receptor state, we ask under what conditions, the system may exhibit oscillatory type behaviour. A detailed analysis of parameter space reveals that whilst variation in kinetic rate parameters within known biological limits is unlikely to lead to such behaviour, changes in the total concentration of the signalling proteins does. We postulate that experimentally observed overshoot behaviour can actually be described by damped oscillatory dynamics and consider the relationship between overshoot amplitude, total cell protein concentration and the magnitude of the external ligand stimulus. Model reductions of the full ODE model allow us to understand the link between phosphorylation events and the negative feedback between CheB-P and receptor methylation, as well as elucidate why some mathematical models exhibit overshoot and others do not. Our manuscript closes by discussing intercell variability of total protein concentration as means of ensuring the overall survival of a population as cells are subjected to different environments

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

Cell-signalling dynamics in time and space

The specificity of cellular responses to receptor stimulation is encoded by the spatial and temporal dynamics of downstream signalling networks. Computational models provide insights into the intricate relationships between stimuli and responses and reveal mechanisms that enable networks to amplify signals, reduce noise and generate discontinuous bistable dynamics or oscillations. These temporal dynamics are coupled to precipitous spatial gradients of signalling activities, which guide pivotal intracellular processes, but also necessitate mechanisms to facilitate signal propagation across a cell

Amelioration of protein misfolding disease by rapamycin: translation or autophagy?

Rapamycin is an inhibitor of mTOR, a key component of the mTORC1 complex that controls the growth and survival of cells in response to growth factors, nutrients, energy balance and stresses. The downstream targets of mTORC1 include ribosome biogenesis, transcription, translation and macroautophagy. Recently it was proposed that rapamycin and its derivatives enhance the clearance (and/or reduce the accumulation) of mutant intracellular proteins causing proteinopathies such as tau, alpha-synuclein, ataxin-3, and full-length or fragments of huntingtin containing a polyglutamine (polyQ) expansion, by upregulating macroautophagy. We tested this proposal directly using macroautophagy-deficient fibroblasts. We found that rapamycin inhibits the aggregation of a fragment of huntingtin (exon 1) containing 97 polyQs similarly in macroautophagy-proficient (Atg5(+/+)) and macroautophagy-deficient (Atg5(-/-)) cells. These data demonstrate that autophagy is not the only mechanism by which rapamycin can alleviate the accumulation of misfolded proteins. Our data suggest that rapamycin inhibits mutant huntingtin fragment accumulation due to inhibition of protein synthesis. A model illustrates how a modest reduction in polyQ synthesis can lead to a long-lasting reduction in polyQ aggregation. We propose that several mechanisms exist by which rapamycin reduces the accumulation and potential toxicity of misfolded proteins in diseases caused by protein misfolding and aggregation

Accurate Particle-Based Reaction Algorithms for Fixed Timestep Simulators

Particle-based simulators are widely used to study biochemical systems involving spatial transport and chemical reactions on sub-cellular length scales. Fixed time step methods can often offer good performance even when simulating complex many-particle systems. However, current reaction algorithms approximate more detailed molecular dynamics models either inaccurately or slowly. Here, we present new reaction algorithms that better approximate microscopic molecular dynamics models while maintaining good computational efficiency. A “Brownian bridge” algorithm samples reactions using reactant positions both before and after each diffusive step; its simulated dynamics exactly match those of appropriate underlying idealised models. Simpler but less accurate “RDF-matching” algorithms sample reactions by only using reactant positions after diffusive steps; they accurately reproduce the steady-state radial distribution function of the underlying idealised model. These algorithms can accurately approximate both commonly used reaction models and more realistic models that account for intermolecular potentials