Perfect and Near-Perfect Adaptation in a Model of Bacterial Chemotaxis

Abstract

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.

Relationship between cellular response and behavioral variability in bacterial chemotaxis

“…with the concentration, N, of receptor complexes present in the system. The concentration of active and inactive receptors at steady state, A* tot and A tot , is independent from the ligand concentration (see Materials and Methods), which is consistent with the property of perfect adaptation (7,18). We model the phosphorylation cascade using mechanisms and parameter values similar to those proposed by Sourjik and others (19,20,24): The rate of autophosphorylation of the kinase CheA is proportional to the total kinase activity A* tot and the phosphate transfer from the kinase to CheY and CheB is proportional to the concentration of phosphorylated kinases (Eqs.…”

“…Methylation and demethylation are complex biochemical processes that involve the enhanced recruitment of CheR and CheB to the receptors by a tethering site that is distinct from methylation sites (14,15). As in previous models of chemotaxis (7,(16)(17)(18)(19)(20)(21)(22)(23), we coarse-grain the underlying biochemical details to capture the behavior of the full chemotaxis system obtained from experiments on living cells. In particular, we assume that receptor-kinase complexes can be either active or inactive (23), and we use the Barkai and Leibler activity-dependent feedback in the receptor modification system that yields robust exact adaptation (7,8) (Fig.…”

“…We use Michaelis-Menten kinetics for the methylation-demethylation steps of the complexes. Here, CheR (CheB-P) binds the methylation sites only of inactive (active) receptors (16)(17)(18)(19)(20)(21)(22)(23). But in Sect.…”

“…, mmax indicates the methylation level of the receptor complex. To ensure perfect adaptation, we assume that a0(L) ϭ 0 and am max (L) ϭ 1 (18). The methylation-demethylation steps and the time evolution of the concentrations Xm of free (not bound to enzyme) receptor complexes with m methyl groups (Fig.…”

Quantitative modeling of sensitivity in bacterial chemotaxis: The role of coupling among different chemoreceptor species

“…As shown in previous studies on adaptation (11)(12)(13)(14) one of the key conditions for achieving perfect adaptation is that the methylation͞demethylation rates depend on the activity of the receptor. Assuming that only inactive receptors can be methylated, and only active receptors can be demethylated (12,14), we use the following simplified methylation flux balance equation between any two consecutive methylation levels to determine the steady-state distribution of the receptors in different methylation states:…”