Mathematical Analysis of the Escherichia coli Chemotaxis Signalling Pathway

Abstract

We undertake a detailed mathematical analysis of a recent nonlinear ordinary differential equation (ODE) model describing the chemotactic signalling cascade within an 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 do. 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 in 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 paper closes by discussing intercell variability of total protein concentration as a means of ensuring the overall survival of a population as cells are subjected to different environments.

Rhizobial Chemoattractants, the Taste and Preferences of Legume Symbionts

“…The remainder of our knowledge on rhizobial chemotaxis is limited to dated behavioral studies. A complete understanding of rhizobium chemotaxis is an important goal for the following reasons: (1) total knowledge of a rhizobium’s chemotaxis system permits the modeling of its behavior in the environment and the prediction of its performance in host nodulation based on its exudate composition ( Edgington and Tindall, 2018 ); (2) inoculant strains can be optimized to more efficiently nodulate crops, outcompete symbiotically inefficient native strains, and better survive in the soil, which increases the longevity of their application; and (3) chemoreceptors are often conserved among plant and animal pathogens, making information on MCPs translational to numerous fields ( Brewster et al, 2016 ; Compton et al, 2018 ).…”

A Mathematical Description of Bacterial Chemotaxis in Response to Two Stimuli

“…However, the phosphorylation processes and the methylation process occur at different time scales and one can reduce the 4dimensional system into a 3-, 2-, or even a 1-dimensional system. In [8], the authors explained these reductions in detail.…”

“…There are several models for F and G as in [7,8,35,36]. Since our results can be applied to almost any scalar functions F and G, for ease of calculations, we choose a simple model, as discussed in (1) and (2) below.…”

“…Scaleinvariance property of a system means that the system does not distinguish between an input (here, S 1 or S 2 ) and its scaled version (e.g., p 1 S 1 or p 2 S 2 ). For more details see [8] and [45]. Using this assumption, we make the following approximation…”

“…We employ a Fokker-Planck type master equation (also known as balance equation [6]) to describe the bacterial chemotaxis. This (microscopic) model enables us to incorporate the internal dynamics of E. coli representing the chemotaxis signaling pathway [7,8]. However, since this equation is not mathematically tractable, using techniques from [9], we derive a (macroscopic) advection-diffusion equation, which is analogous to the Keller-Segel model [10], in which the temporal rate of the chemical concentrations is zero, as is assumed here.…”